*Article* **Implementation of an FPGA-Based Current Control and SVPWM ASIC with Asymmetric Five-Segment Switching Scheme for AC Motor Drives**

**Ming-Fa Tsai \*, Chung-Shi Tseng and Po-Jen Cheng**

Department of Electrical Engineering, Minghsin University of Science and Technology, No. 1, Xinxing Rd., Xinfeng, Hsinchu 30401, Taiwan; cstseng@must.edu.tw (C.-S.T.); pjcheng3255@gmail.com (P.-J.C.) **\*** Correspondence: mftsai@must.edu.tw; Tel.: +886-3-5593142 (ext. 3070); Fax: +886-3-5573895

**Abstract:** This paper presents the design and implementation of an application-specific integrated circuit (ASIC) for a discrete-time current control and space-vector pulse-width modulation (SVPWM) with asymmetric five-segment switching scheme for AC motor drives. As compared to a conventional three-phase symmetric seven-segment switching SVPWM scheme, the proposed method involves five-segment two-phase switching in each switching period, so the inverter switching times and power loss can be reduced by 33%. In addition, the produced PWM signal is asymmetric with respect to the center-symmetric triangular carrier wave, and the voltage command signal from the discretetime current control output can be given in each half period of the PWM switching time interval, hence increasing the system bandwidth and allowing the motor drive system with better dynamic response. For the verification of the proposed SVPWM modulation scheme, the current control function in the stationary reference frame is also included in the design of the ASIC. The design is firstly verified by using PSIM simulation tool. Then, a DE0-nano field programmable gate array (FPGA) control board is employed to drive a 300W permanent-magnet synchronous motor (PMSM) for the experimental verification of the ASIC.

**Keywords:** SVPWM ASIC; asymmetric five-segment switching; AC motor drives; current control; FPGA control

## **1. Introduction**

The AC motor drives, including induction motors (IM), permanent-magnet synchronous motor (PMSM) drives, synchronous reluctance motors (SynRM), and others, have been very popular, being applied to electric vehicles, railway traction engines, and industrial applications such as CNC tools and robots, because they have higher power density and efficiency as compared to DC motor drives [1–4]. Since the AC motor is a time-varying, multi-variable, and nonlinear control system with very complicated dynamic characteristics, it can be reduced to a simpler linear control system by using the field-oriented vector control method [5–11]. In vector-controlled AC motor drives, various pulse-width modulation (PWM) techniques exist to determine the inverter switch-on and switch-off instants from the control output modulating signals. Popular examples include sinusoidal PWM (SPWM), hysteresis PWM, and space-vector PWM (SVPWM). Among them, the SPWM method generates the inverter switching signals by comparing the modulating signals with a common triangular carrier wave, in which the intersection points determine the switching points of the inverter power devices [12,13]. The hysteresis PWM method controls the inverter output currents to track the reference inputs within the hysteresis band, but with various switching frequencies [14,15]. The SVPWM method is based on the concept of voltage vector space, which can be divided into six sectors, and calculates the switch dwelling or firing time in each sector to generate the PWM signals [16–40]. The relationship between SVPWM and SPWM was presented in [18], which indicated that the SVPWM can

 

**Citation:** Tsai, M.-F.; Tseng, C.-S.; Cheng, P.-J. Implementation of an FPGA-Based Current Control and SVPWM ASIC with Asymmetric Five-Segment Switching Scheme for AC Motor Drives. *Energies* **2021**, *14*, 1462. https://doi.org/10.3390/ en14051462

Academic Editor: Mario Marchesoni

Received: 1 February 2021 Accepted: 27 February 2021 Published: 7 March 2021

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**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

be viewed as a particular form of asymmetric regular-sampled PWM. In [24], the relationship between SVPWM and three-phase SPWM was also analyzed and the implementation of them in a closed-loop feedback converter was discussed as well. The SVPWM technique is the preferred approach in most applications due to the higher voltage utilization and lower harmonic distortion (THD) than the other two methods, and hence it can extend the linear range of the vector-controlled AC motor drives. Furthermore, sensorless and direct torque control (DTC) cooperating with an SVPWM scheme for a flux-modulated permanent-magnet wheel motor and induction motor has been described owing to several advantages, such as low torque/flux ripples in motor drive and reduced direct axis current, over the conventional hysteresis direct torque control method when the motor is operated at a light or a sudden increased load [34,37–40]. The SVPWM scheme has also been extensively applied to the control of a five-phase permanent-magnet motor and multilevel inverters [35,36].

Different SVPWM techniques can be used for three-phase inverters according to the choice of the null vector on the voltage space and the number of samplings during a PWM switching period. They can be divided into seven-segment, five-segment, and threesegment techniques for the switching sequences and the choice of null vector, and classified into symmetric and asymmetric techniques according to the number of samplings during a PWM switching period as well. For the symmetric technique, the sampling period is equal to the switching period, while the sampling period is one half of the switching period for the asymmetric technique. So, the asymmetric methods have only 50% switching action in each switching period as compared with symmetric methods, and hence the sampling frequency of the current control can be doubled at a certain switching frequency. In [28], three SVPWM strategies, including symmetric seven-segment technique, symmetric fivesegment technique, and asymmetric three-segment technique, were compared through simulation analysis. The switching loss of the three-segment scheme is the lowest and the seven-segment scheme performs better in terms of the THD of the output line voltage.

Figure 1 shows the rotor flux-oriented control (RFOC) structure of a PMSM motor drive with a SVPWM modulation scheme, in which an asymmetric five-segment switching is proposed rather than the conventional symmetric seven-segment switching so as to double the sampling frequency and reduce the switching loss. This control structure, from the inner to the outer loops, includes the SVPWM modulation, current control loop, torque and flux control loops, and speed control loop. Conventionally, the execution of all the control tasks is performed by using a high-performance microprocessor or dual digital signal processor (DSP) [21,41,42]. It may take a lot of computation time for the microprocessor or DSP because all the computation of the tasks is very complicated. If the computation of the current control and the SVPWM tasks can be executed by a field programmable gate array (FPGA) device, the computation load can be greatly reduced [16,17,22,25,30]. In such cases, the microprocessor or DSP can have enough time to process higher level tasks, such as position control, motion control, adaptive, fuzzy, or neural-network learning and intelligent control [43]. In [17], an eight-bit SVPWM control IC was realized with a symmetrical five-segment switching scheme, but without current control function included in the chip. In [22], a microcoded machine with a 16-bit computational ALU unit and control sequencer was designed in an FPGA for the execution of the current control and SVPWM modulation of an induction motor with symmetrical seven-segment switching scheme. However, the microcoded machine structure is very complicated.

**Figure 1.** The rotor flux-oriented control (RFOC) structure of a permanent-magnet synchronous motor (PMSM) AC motor drive.

This paper describes the design and implementation of an application-specific integrated circuit (ASIC) for a discrete-time current control and space-vector pulse-width modulation (SVPWM) with asymmetric five-segment switching scheme AC motor drives. It can not only reduce the computation load of a microprocessor or DSP, but also reduce the power transistor switching loss of the inverter. Traditionally, the current control loop is performed in the synchronously *d-q* rotating reference frame [3–7]. However, as indicated in Figure 1, the presented current control loop is performed in the stationary reference frame rather than the synchronously rotating reference frame in order to simplify the computation complexity in the FPGA device. Furthermore, because the current reference commands can be updated two times during one switching period, the current control system can also increase the sampling frequency two times so as to increase the bandwidth.

The algorithm of the proposed current control and asymmetric five-segment switching SVPWM scheme was firstly verified by using PSIM simulation tool as applied to the current control of a PMSM AC motor. The proposed SVPWM scheme and the current controller function were implemented on a DE0-nano control board with Altera Cyclone IV E FPGA device for the experimental verification.

This paper is organized as follows. The principle of the asymmetric five-segment switching SVPWM modulation scheme is described in Section 2. The simulation verification in PSIM simulation tool is analyzed in Section 3 and the FPGA implementation and the experimental results are presented in Section 4. A discussion is then given in Section 5. Finally, the conclusion is given in Section 6.

#### **2. The Principle of the Asymmetrical Five-Segment Switching SVPWM Modulation**

Firstly, as shown in Figure 2, for a three-phase PWM inverter of Y-connected AC motor with neutral point *n*, the three-phase stator voltages with respect to the inverter ground can be written as

$$
v\_{a0} = v\_{an} + v\_{n0} \tag{1}$$

$$
v\_{b0} = v\_{bn} + v\_{n0} \tag{2}$$

$$
v\_{c0} = v\_{cn} + v\_{n0} \tag{3}$$

where *van*, *vbn*, and *vcn* are the three-phase voltages with respect to the neutral point of the motor, and *vn*<sup>0</sup> is the neutral-point voltage with respect to the inverter ground.

**Figure 2.** The pulse-width modulation (PWM) inverter circuit for a Y-connected AC motor.

For balanced three-phase supply, it can be written as

$$
\upsilon\_{an} + \upsilon\_{bn} + \upsilon\_{cn} = 0 \tag{4}
$$

Thus, by adding (1)–(3) with the condition of (4), the neutral-point voltage with respect to the inverter ground, *vn*0, can be written as

$$v\_{n0} = \frac{1}{3}(v\_{d0} + v\_{b0} + v\_{c0})\tag{5}$$

Substituting (5) into (1)–(3) yields

$$
\begin{bmatrix} v\_{an} \\ v\_{bn} \\ v\_{cn} \end{bmatrix} = \begin{bmatrix} \frac{2}{5} - \frac{1}{5} - \frac{1}{5} \\ -\frac{1}{5} & \frac{2}{5} - \frac{1}{5} \\ -\frac{1}{5} - \frac{1}{5} & \frac{2}{5} \end{bmatrix} \begin{bmatrix} v\_{a0} \\ v\_{b0} \\ v\_{c0} \end{bmatrix} \tag{6}
$$

Secondly, because the three-phase stator voltages are dependent upon (4), the inputs of the SVPWM circuit can be reduced from the three-phase variables into *α*-*β* two-axes components with the Clarke transformation given by (7).

$$
\begin{bmatrix} v\_{\alpha} \\ v\_{\beta} \end{bmatrix} = \begin{bmatrix} 1 \ 0 \ 0 \\ 0 \ \frac{1}{\sqrt{3}} \frac{-1}{\sqrt{3}} \end{bmatrix} \begin{bmatrix} v\_{\alpha n} \\ v\_{b n} \\ v\_{c n} \end{bmatrix} \tag{7}
$$

where *v<sup>α</sup>* and *v<sup>β</sup>* are the *α*-*β* axes component voltages in the stationary reference frame. Substituting (6) into (7) yields

$$
\begin{bmatrix} v\_{\alpha} \\ v\_{\beta} \end{bmatrix} = \begin{bmatrix} \frac{2}{3} - \frac{1}{3} - \frac{1}{3} \\ 0 \frac{1}{\sqrt{3}} - \frac{1}{\sqrt{3}} \end{bmatrix} \begin{bmatrix} v\_{d0} \\ v\_{b0} \\ v\_{c0} \end{bmatrix} \tag{8}
$$

Thus, using (8), one can get the eight output voltage vectors, *V*<sup>0</sup> − *V*7, corresponding to the eight switching states of the inverter, as shown in Table 1. The resulted space vector diagram is shown in Figure 3, which is divided into six sectors from Sector I to Sector VI. As can be seen, among the eight voltage vectors, *V*<sup>0</sup> and *V*<sup>7</sup> are the null vectors and *V*<sup>1</sup> − *V*<sup>6</sup> are located on the six corners of the hexagonal diagram.



**Figure 3.** Inverter's eight output voltage space vectors.

The switching waveform generations of the inverter for driving an AC motor can be accomplished by rotating a reference voltage vector around the vector space. Any instance of the reference voltage vector can be produced by the two nearest adjacent vectors and a null vector in an arbitrary sector. For example, as shown in Figure 4, for providing the reference voltage vector *Vs* in Sector I to the motor, the voltage-second balance equation in this sector determines the time length of the two adjacent active inverter states in the following:

$$V\_s T = V\_A T\_1 + V\_B T\_2 \tag{9}$$

where *T*<sup>1</sup> and *T*<sup>2</sup> are the dwelling time length for *V*<sup>1</sup> and *V*2, respectively, and *T* is the sampling period, which is one half of the inverter switching period. In order to get the solution of *T*<sup>1</sup> and *T*<sup>2</sup> in (9), it follows that

$$
\begin{bmatrix} v\_{\alpha} \\ v\_{\beta} \end{bmatrix} T = \frac{2}{3} V\_{dc} \left( T\_1 \begin{bmatrix} 1 \\ 0 \end{bmatrix} + T\_2 \begin{bmatrix} \frac{1}{2} \\ \frac{\sqrt{3}}{2} \end{bmatrix} \right) \tag{10}
$$

Thus, *T*<sup>1</sup> and *T*<sup>2</sup> can be, respectively, solved as

$$T\_1 = \frac{\sqrt{3}T}{V\_{dc}}(\frac{\sqrt{3}}{2}v\_\alpha - \frac{1}{2}v\_\beta) \tag{11}$$

and

$$T\_2 = \frac{\sqrt{3}T}{V\_{dc}}v\_{\beta} \tag{12}$$

Let

 *v<sup>α</sup> vβ* = *Vs* cos *γ* sin *γ* (13)

then (11) and (12) can be rewritten as the following equations [18]:

$$T\_1 = Ta \frac{2}{\sqrt{3}} \sin(\frac{\pi}{3} - \gamma) \tag{14}$$

$$T\_2 = \text{Ta} \frac{2}{\sqrt{3}} \sin(\gamma) \tag{15}$$

where

$$a = \frac{V\_s}{\frac{2}{3}V\_{dc}}\tag{16}$$

Similarly, the switching time interval in the other sectors can be derived. Table 2 summaries the results, in which the condition for sector number selection is also illustrated.

**Table 2.** Dwelling-time interval calculation in each sector.


For minimizing the number of switchings in a switching period, the asymmetrically alternating-reversing pulse sequence with five-segment (*V*<sup>0</sup> − *V*<sup>1</sup> − *V*<sup>2</sup> − *V*<sup>1</sup> − *V*0) switching technique is employed without using the null vector *V*7, as shown in Figure 5a. The pulse patterns for two consecutive sampling intervals can be configured by beginning with the null vector *V*<sup>0</sup> on the *kT* sampling time and also by ending with the same null vector *V*<sup>0</sup> on the (*k +* 1)*T* sampling time, where *k* = 1, 3, 5, ··· . The time length *T*<sup>0</sup> on the *kT* and (*k +* 1)*T* sampling time intervals can be different, either for *T*<sup>1</sup> or *T*2. Therefore, the PWM pulses are asymmetric with respect to the center point of the switching period. The technique benefits from one of the inverter legs not switching during a full switching period, only one inverter leg switching at a time, and only two commutations per sampling period. Thus, the number of switchings in a switching period is four and is less than the conventional symmetric seven-segment SVPWM method, in which the number of switchings in a switching period is six.

**Figure 5.** *Cont.*

**Figure 5.** The asymmetric five-segment switching space-vector pulse-width modulation (SVPWM) pulse waveform in each sector: (**a**) Sector I; (**b**) Sector II; (**c**) Sector III; (**d**) Sector IV; (**e**) Sector V; (**f**) Sector VI.

Furthermore, as can be seen from Table 2, the switch dwelling time length *T*<sup>1</sup> and *T*<sup>2</sup> are a function of the component voltages, *v<sup>α</sup>* and *vβ*, the sampling period, *T*, and the DC bus voltage, *Vdc*. The calculations of the functions are very simple, and hence can be easily carried out by the FPGA-based digital hardware. The pulse waveforms of the proposed asymmetric five-segment switching SVPWM scheme in other sectors are also shown in Figure 5. However, there may be the cases of the reference voltage vector travelling across the sector on the trailing sampling interval. In this case, the sector pulse pattern in the leading sampling interval must change to the next sector pulse waveforms on the trailing sampling interval. Figure 6 illustrates a pulse waveform crossing from Sector I to the other five sectors and from Sector II to I on the trailing sampling interval.

**Figure 6.** *Cont.*

**Figure 6.** The pulse waveforms of sector crossing on the trailing sampling time: (**a**) Sector I to II, (**b**) Sector I to III, (**c**) Sector I to IV, (**d**) Sector I to V, (**e**) Sector I to VI, (**f**) Sector II to I.

The scaled reference commands can be routed to the circuit block for the calculation of the dwelling time duration *T*<sup>1</sup> and *T*<sup>2</sup> in each sector for generating the PWM signals [17]. However, for reducing the computation complexity, an alternative method is used by calculating the firing time, which is defined as the time interval from the start to the leading edge of the PWM pulse. For example, in Sector I, the PWM firing times for the switches *Su*, *Sv* and *Sw* in the *kT* sampling period can be obtained as follows:

$$\begin{array}{c} f\_{\iota} = T\_0 \\ = T - \frac{\sqrt{3}T}{V\_{\rm dc}} (\frac{\sqrt{3}}{2} v\_{\alpha} + \frac{1}{2} v\_{\beta}) \end{array} \tag{17}$$

$$\begin{split} f\_{\upsilon} &= T\_0 + T\_1 = T\_s - T\_2 \\ &= T - \frac{\sqrt{3}T}{V\_{dc}} v\_{\beta} \end{split} \tag{18}$$

$$f\_w = T \tag{19}$$

where *T*<sup>0</sup> = *T* − *T*<sup>1</sup> − *T*2. The resulting equations for the PWM firing time in another sector can be obtained in the same way. Table 3 summarizes the results. As can be seen, the firing-time equations in Sector I are the same as in Sector II. They are the same in Sector III and IV and in Sector V and VI as well. Thus the circuit realization for the sectors with the same firing-time equation can share the same circuit as each other. This can simplify the circuit complexity for the implementation of the ASIC. As shown in Figure 7, the PWM pulse pattern on each phase can be generated by comparing the firing-time signal with a common center-symmetrical triangular wave with amplitude of *T* and period of 2*T*, which can be implemented by an up-down counter. The pulse is low when the firing-time signal is larger than the magnitude of the triangular wave and is high otherwise.


**Table 3.** PWM firing time calculation in each sector.

**Figure 7.** The SVPWM signal generation on one phase.

## **3. Simulation Verification Using PSIM**

Figure 8 shows the simulation verification of the proposed asymmetric five-segment switching SVPWM modulation scheme to drive a PMSM motor by using the PSIM simulation tool, in which the PMSM motor model (Sinano 7CB30-2DE6FKS) was constructed, as shown in Figure 8b. In this simulation, the sampling period is *T* = 40*μs*, the DC bus voltage is *Vdc* = 40V,*v<sup>α</sup>* = 23 cos(40*πt*), and *v<sup>β</sup>* = 23 sin(40*πt*). The SVPWM modulation scheme algorithm was written in C language inside the C block function of PSIM with the flowchart shown in Figure 9. Figure 10 shows the simulation results of current and speed responses, the three-phase firing-time signals, and the firing-time signal vector trajectory with the 20 Hz voltage reference inputs given above and 0.3 Nm load torque at 0.1 s. As can be seen, the speed is 31.4 rad/s in the steady-state, which can verify the correction of the algorithm of the proposed SVPWM modulation scheme.

**Figure 8.** Simulation verification of the SVPWM scheme: (**a**) simulation model using C block, (**b**) PMSM motor model with pole number equal to 8.

**Figure 9.** Flowchart of the proposed asymmetric five-segment switching SVPWM scheme algorithm.

**Figure 10.** Simulation results with 0.3 Nm load torque at 0.1 s: (**a**) the reference voltages, current and speed responses, (**b**) the three-phase firing-time signals, (**c**) the firing-time signal vector trajectory.

Figure 11 shows the simulation verification of the discrete-time current control of the PMSM motor using the proposed SVPWM scheme. A discrete-time proportional-integral (PI) controller is designed with the pulse transfer function in (20) and the zero-order holder with sampling frequency of 25 kHz, which is two times of the inverter switching frequency. As can be seen, the current responses can track the current reference with the amplitude of 3A and frequency of 20 Hz.

$$H(z) = \frac{k\_p + k\_l T - k\_p z^{-1}}{1 - z^{-1}} \tag{20}$$

**Figure 11.** Simulation verification of the current control with 0.3 Nm load torque at 0.4 s: (**a**) simulation model, (**b**) the reference currents, current, and speed responses.

## **4. FPGA Implementation**

A DE0-Nano FPGA control board by Terasic Inc. with Altera Cyclone IV-E device (EP4CE22F17C6) was employed for the implementation and experiment verification of the proposed current control and SVPWM ASIC for driving a PMSM servo motor (Sinano) with rated power of 300 W. Figure 12 shows the designed SVPWM ASIC structure with optional current control function included. The DE0-Nano board includes a 50 MHz oscillator, which can be used as a source clock to the designed multi-clock generation circuit to give various frequency clock signals, such as 2.5 MHz, 200 kHz, 100 kHz, and 10 kHz, for the design of the ASIC. A cosine and a sine look-up table, each with 500 words and 10 bits/word, were created by using LPM\_ROM function to generate the voltage references, *v<sup>α</sup>* and *vβ*, in which the frequency can be determined through the modulus-500 counter as a frequency divider. The voltage references then can be used as the inputs of the sector selection circuit according to Table 2 and the inputs of the firing time calculation circuit in each sector according to Table 3. Because some firing time equations are the same, they can be divided into three groups. The 3:1 multiplexers are used in order to select the three firing-time signals *fu*, *fv*, and *fw* in each sector. The 50 MHz clock is used as the input of an up-and-down counter, which counts up from zero to 2000 and then counts down to zero, for the generation of a triangular carrier signal with an 80 *μs* period. The three firing time signals are then compared with the common triangular carrier counter value, respectively, to generate the three-phase SVPWM signals. A dead-time interval of 2 *μs* in each phase is also inserted to generate the six gate signals to drive the inverter. A SPI-DAC interface to a DAC7513 converter was also designed to illustrate the three-phase firing time signals on the oscilloscope.

**Figure 12.** The SVPWM application-specific integrated circuit (ASIC) structure with optional current control function.

The SVPWM ASIC in the FPGA chip was designed, as shown in Figure 13, by using Altera Quartus II software development system of version 13.1. The experimental results of the SVPWM three-phase switching signals in each sector are shown in Figure 14. They are consistent with the expected pulse waveforms in Figure 5. The open-loop experiment result of the firing-time signal vector trajectory in a steady state is shown in Figure 15, which is also consistent with the simulation result in Figure 10c. Figure 16 shows the experimental result of the open-loop current vector trajectory in steady state with the SVPWM scheme. Two of the three-phase currents are sensed by using two LEM-55P current sensors and filtered through low-pass filters. The filtered currents are shifted up 1.65 V, being the inputs to an ADC128S022 A/D converter, in which the input voltage range is between zero and 3.3 V, for the feedback of the PI current control function with 25 kHz sampling frequency.

**Figure 13.** FPGA implementation of the proposed SVPWM scheme by using Quartus II version 13.1 tool.

**Figure 14.** Experiment verification of the three-phase switching signals in each sector of the proposed SVPWM scheme: (**a**) Sector I; (**b**) Sector II; (**c**) Sector III; (**d**) Sector IV; (**e**) Sector V; (**f**) Sector VI.

**Figure 15.** The open-loop experimental result of the firing-time signal vector trajectory in steady state.

**Figure 16.** The open-loop current vector trajectory in steady state with the SVPWM scheme (7.5 A/V).

For saving the hardware resources, the computation architecture of the PI controller in the ASIC is shown in Figure 17, which has a control unit and a data path, which contains a 12-bit adder/subtracter, a 12-bit multiplier, and a limiter. The control unit is a finite state machine (FSM) which generates the control signals to the data path to control the computation procedures of the PI controller. The computation procedures of the PI controller according to the pulse transfer function in (20) are shown in Figure 18. There are five steps (s1–s5) for the computation of the PI control function. For the 25 kHz sampling frequency, the input clock frequency for the PI control function is 200 kHz. Therefore, there are eight clocks, in which three clocks are used for waiting state, needed to accomplish the computation of the PI controller. Figure 19 shows the experimental result for the closed-loop current vector trajectory in steady state with different DC bus voltage values. As can be seen, it can be a circular in the range from 55 to 63 V.

**Figure 17.** The computation architecture of the ASIC.

**Figure 18.** The computation procedures of the proportional-integral (PI) controller.

**Figure 19.** The closed-loop current vector trajectory in steady state with different DC bus voltage (7.5 A/V): (**a**) *Vdc* = 55 V, (**b**) *Vdc* = 63 V, (**c**) *Vdc* = 45 V, (**d**) *Vdc* = 50 V.

#### **5. Discussion**

Several practical aspects for the implementation of the proposed current control and SVPWM ASIC are discussed as follows:

Firstly, as shown in Figure 7, the firing-time signals are the modulating signals and used to compare with a common center-symmetrical triangular carrier wave for generating the PWM gating signals of the inverter in the proposed SVPWM scheme. This is different from the conventional SPWM implementation method, in which the sinusoidal control outputs are the modulating signals. For convenience, the firing time equation of *fv* in Table 3 is rewritten as follows:

$$f\_{\upsilon} = T - \frac{\sqrt{3}T}{V\_{dc}}v\_{\beta} = T(1 - \frac{\sqrt{3}}{V\_{dc}}v\_{\beta}) \tag{21}$$

The second term on the right-hand side of the equals sign has a relationship with the duty ratio of the gating signal expressed as

$$d\_{\upsilon} = \frac{\sqrt{3}}{V\_{dc}} v\_{\beta} \tag{22}$$

Because the duty ratio is 0 ≤ *d* ≤ 1, it follows that

$$
v\_{\beta} \le \frac{V\_{dc}}{\sqrt{3}}\tag{23}$$

So, the maximum value of the control signals *v<sup>β</sup>* or *v<sup>α</sup>* is *Vdc*/ <sup>√</sup><sup>3</sup> and is larger than *Vdc*/2, which is the maximum value of the modulating signals in SPWM method. Thus, as is well known, the SVPWM method has higher voltage utilization by about 115% than the SPWM modulation method.

Secondly, the term <sup>√</sup>3/*Vdc* in (22) can be seen as the gain from the control signal *<sup>v</sup><sup>β</sup>* to the duty ratio to generate the gating signal in the SVPWM modulation scheme. Thus, this gain can be expressed as

$$K\_{svpwm} = \frac{d\_v}{v\_\beta} = \frac{\sqrt{3}}{V\_{dc}}\tag{24}$$

It was found that this SVPWM gain can be included in the PI current controller parameters. So, although the firing time equations are relative to the dc bus voltage, the multiplication of the SVPWM gain to the control output signal *v<sup>β</sup>* is not necessary for the implementation of the firing-time signal. That means the small perturbation of the DC bus voltage will not affect the performance of the current control and SVPWM ASIC. As can be seen from the experimental results in Figure 19, the closed-loop current vector trajectory in steady state can be a circular for the dc bus voltage of 55 and 63 V, respectively. This finding makes the implementation circuit simpler.

Thirdly, although it is the digital hardware circuit for the computation of the current control and SVPWM scheme, the timing sequence during the sampling interval from the sampling of the feedback currents to the current control and firing-time signal output must be considered for the synchronization, as shown in Figure 20. The computation of the timing signals must be completed before the next sampling time. The resulted firing-time signal values are then loaded into the PWM signal generation circuit for the comparison with an up-down counter at the next sampling time in which the counter starts to count up from zero or count down from 2000.

**Figure 20.** The timing sequence of sampling and loading for the proposed current control and SVPWM scheme.

Fourthly, the conventional SVPWM scheme including the symmetric seven-segment technique, symmetric five-segment technique, and asymmetric three-segment technique are shown in Figure 21. As can be seen, the symmetric five-segment technique has four switching times and is the lowest during a switching period, so the switching loss of the scheme is significantly reduced in comparison with the others. However, in the asymmetric three-segment technique, the sampling frequency can be doubled at a certain switching frequency. The proposed asymmetric five-segment technique can not only have the advantages of minimum switching times, but also can double the sampling frequency in the current control loop so as to improve the control performance.

**Figure 21.** SVPWM switching sequences: (**a**) symmetric 7-segment, (**b**) symmetric 5-segment, (**c**) asymmetric 3-segment.

## **6. Conclusions**

In this work, the design and implementation of an FPGA-based SVPWM ASIC with an asymmetric five-segment switching scheme for AC motor drives have been performed. The inverter switch dwelling and firing times on each sector have been derived. It was found that the firing-time equations in Sector I and II are the same. They are the same in Sector III and IV and in Sector V and VI as well. These finding allow us to simplify the circuit complexity for the implementation of the ASIC. Compared with the conventionally symmetric seven-segment three-phase switching scheme, the inverter switching times and power loss of this proposed scheme can not only be reduced by 33%, but also the asymmetric characteristics mean that the reference voltage command signal can be given in the half period of the PWM switching time interval. Therefore, one can design the closed-loop current control while doubling the sampling frequency, hence increasing the bandwidth, and allowing the motor drive system with better dynamic response. For the verification of the proposed SVPWM modulation scheme, the closed-loop current control function in the stationary reference frame has been also included in the design of the ASIC. The ASIC function is firstly verified by using the PSIM simulation tool. Then, a DE0 nano FPGA control board has been employed to drive a 300 W PMSM AC motor for the experimental verification. The simulation and experimental results show the performance of the proposed SVPWM ASIC both in the open-loop pulse-width modulation and in the current control loop. The proposed current control and SVPWM ASIC can not only be used in PMSM motor drives, but can also be applied in other AC motor drives.

**Author Contributions:** Conceptualization, M.-F.T. and C.-S.T.; methodology, M.-F.T.; software, P.-J.C.; validation, M.-F.T., C.-S.T. and P.-J.C.; formal analysis, M.-F.T.; investigation, M.-F.T.; resources, M.-F.T. and P.-J.C.; data curation, M.-F.T.; writing—original draft preparation, M.-F.T.; writing—review and editing, M.-F.T. and C.-S.T.; visualization, M.-F.T.; supervision, C.-S.T.; project administration, C.-S.T.; funding acquisition, M.-F.T. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by Project number: MUST-110-mission-3, from Ministry of Education (MOE), Taiwan.

**Conflicts of Interest:** The authors declare no conflict of interest.

## **References**


## *Article* **Development of Three-Phase Permanent-Magnet Synchronous Motor Drive with Strategy to Suppress Harmonic Current**

**Wei-Tse Kao , Jonq-Chin Hwang \* and Jia-En Liu**

Department of Electrical Engineering, National Taiwan University of Science and Technology, Taipei 106, Taiwan; D10107202@ntust.edu.tw (W.-T.K.); M10307204@ntust.edu.tw (J.-E.L.)

**\*** Correspondence: jchwang@ntust.edu.tw

**Abstract:** This study aimed to develop a three-phase permanent-magnet synchronous motor drive system with improvement in current harmonics. Considering the harmonic components in the induced electromotive force of a permanent-magnet synchronous motor, the offline response of the induced electromotive force (EMF) was measured for fast Fourier analysis, the main harmonic components were obtained, and the voltage required to reduce the current harmonic components in the corresponding direct (d-axis) and quadrature (q-axis) axes was calculated. In the closed-loop control of the direct axis and quadrature axis current in the rotor reference frame, the compensation amount of the induced EMF with harmonic components was added. Compared with the online adjustment of current harmonic injection, this simplifies the control strategy. The drive system used a 32-bit digital signal processor (DSP) TMS320F28069 as the control core, the control strategies were implemented in software, and a resolver with a resolver-to-digital converter (RDC) was used for the feedback of angular position and speed. The actual measurement results of the current harmonic improvement control show that the total harmonic distortion of the three-phase current was reduced from 5.30% to 2.31%, and the electromagnetic torque ripple was reduced from 15.28% to 5.98%. The actual measurement results verify the feasibility of this method.

**Keywords:** motor drive; current harmonic reduction; torque ripple reduction

## **1. Introduction**

The induced electromotive force of a permanent-magnet motor may contain harmonic components because of the design, which produces electromagnetic torque ripples after multiplying with the phase current of the motor (which also contains harmonic components) [1,2]. Therefore, reducing the harmonic components of the motor current provides a reduction in torque ripples. By performing the harmonic analysis of the current to obtain the harmonic components contained and then adding them to the current command, the influence of the current on the torque can be reduced [3,4]. To reduce the current ripple due to the induced EMF, the harmonic components of the induced EMF can be analyzed and injected into the voltage command, as shown in [5]. Most methods add the harmonic components of the induced EMF to the three-phase voltage command. The compensation parameters of current or induced EMF are based on the analysis of its harmonic components. It is known that the three-phase signals containing harmonics can be transformed into the rotating rotor reference frame and projected on the 0-axis, direct axis (d-axis), and quadrature axis (q-axis) [6]. With open-end windings and driven each phase current independently, the zero-sequence harmonics of the induced EMF are added to the 0-axis voltage command directly, in conjunction with field-oriented control (FOC) can reduce current harmonics [7]. Torque ripple can be estimated from the calculated energy and co-energy through the voltage and current feedback, which is used as the feed-forward compensation amount to reduce the torque ripple and current ripple component [8]. Recently, different control methods have been applied to reduce current harmonics or torque ripple. By using the Kalman filter, the stator current and permanent-magnet (PM) rotor

**Citation:** Kao, W.-T.; Hwang, J.-C.; Liu, J.-E. Development of Three-Phase Permanent-Magnet Synchronous Motor Drive with Strategy to Suppress Harmonic Current. *Energies* **2021**, *14*, 1583. https://doi.org/10.3390/en14061583

Academic Editor: Mario Marchesoni

Received: 30 January 2021 Accepted: 9 March 2021 Published: 12 March 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

flux are used to track the flux linkage and compensate for the torque ripple caused by the demagnetization [9]. Artificial neural networks are used to reduce the torque ripple of the permanent-magnet synchronous motor (PMSM) with non-sinusoidal induced EMF and cogging torque [10]. Predictive torque control is also used to reduce torque ripple and improve its control accuracy by compensating for the current prediction errors [11]. Selective current harmonic suppression method is proposed to reduce current harmonics in case of high-speed operation [12].

In this study, the methods presented in [13,14] were used to conduct fast Fourier transform (FFT) analysis of induced EMF. The induced EMF of the fifth and seventh harmonic components was obtained, the contents and offset angles of which were further analyzed. Moreover, the rotor's rotating coordinate system was used to obtain the sixth harmonic component in direct axis and quadrature axis. Compensation was applied to the q-axis and the d-axis to reduce current harmonic components and torque ripple. Compared with the techniques mentioned above, the method proposed in this paper requires less computational burden and does not need to change the control scheme; however, it is necessary to measure the induced EMF of the motor and analyze its harmonic components as the reference of compensation.

In this paper, Section 2 introduces the mathematical model of the permanent-magnet synchronous motor considering the fifth and seventh harmonics; Section 3 proposes a control strategy to reduce torque ripple with two methods of compensation for abc or qd coordinate systems; Section 4 first measures the induced EMF of the permanent-magnet synchronous motor and determines the compensation amount by spectrum analysis, and compares the current spectrum and torque ripple components with different control strategies.

### **2. Current and Torque Ripples of Permanent-Magnet Synchronous Motors**

If the salience effect is neglected and the d-axis and q-axis inductances are the same, the voltages of a three-phase permanent-magnet synchronous motor can be expressed as follows:

$$\mathbf{v}\_a = \mathbf{R}\_s \mathbf{i}\_a + \mathbf{L}\_s \frac{d}{dt} \mathbf{i}\_a + \mathbf{e}\_{af} \tag{1}$$

$$\nu\_b = \mathcal{R}\_\text{s} i\_b + L\_\text{s} \frac{d}{dt} i\_b + e\_{bf} \tag{2}$$

$$\nu\_c = R\_s i\_c + L\_s \frac{d}{dt} i\_c + e\_{cf} \tag{3}$$

where *va*, *vb*, and *vc* are the input phase voltages of the motor; *ia*, *ib*, and *ic* are the threephase stator currents; *Rs* is the equivalent resistance and *Ls* is the equivalent inductance of the stator; *ea f* , *eb f* , and *ec f* are the three-phase induced EMF of the motor. Due to the symmetry of the rotor, the induced EMF contains no even harmonic components. For the motor with the Y connection, the zero-sequence harmonics can be neglected as they are coupled to both ends of each phase [15]. Therefore, only the odd harmonics of the positive and negative sequences need to be considered.

The matrix for transforming the three-phase system to the rotor reference frame and its inverse transformation is as follows:

$$T\_{qd}^r = \frac{2}{3} \begin{bmatrix} \cos \hat{\theta}\_r & \cos(\hat{\theta}\_r - 120^\circ) & \cos(\hat{\theta}\_r + 120^\circ) \\ \sin \hat{\theta}\_r & \sin(\hat{\theta}\_r - 120^\circ) & \sin(\hat{\theta}\_r + 120^\circ) \end{bmatrix} \tag{4}$$

$$T\_{qd}^{r^{-1}} = \begin{bmatrix} \cos \theta\_r & \sin \theta\_r \\ \cos(\theta\_r - 120^\circ) & \sin(\theta\_r - 120^\circ) \\ \cos(\theta\_r + 120^\circ) & \sin(\theta\_r + 120^\circ) \end{bmatrix} \tag{5}$$

Under balanced conditions, the three-phase motor uses the transformation matrix *T<sup>r</sup> qd* of the rotor reference frame, and the voltages of the q-axis and the d-axis are:

$$v\_q^r = R\_s i\_q^r + L\_q \frac{d}{dt} i\_q^r + \omega\_r (\lambda\_m^\prime + L\_d i\_d^r)\_\prime \tag{6}$$

$$v\_d^r = \mathcal{R}\_s \dot{i}\_d^r + L\_d \frac{d}{dt} \dot{i}\_d^r - \omega\_r L\_q \dot{i}\_{q'}^r \tag{7}$$

where *v<sup>r</sup> <sup>q</sup>* and *v<sup>r</sup> <sup>d</sup>* are the input voltages; *i r <sup>q</sup>* and *i r <sup>d</sup>* are the currents of the q-axis and d-axis, respectively.

The three-phase permanent-magnet synchronous motor used in this study mostly contains the fifth and seventh harmonic components, and the other harmonics barely exist. Therefore, the fifth and seventh harmonic components were added to the estimated values in this section. The resultant induced EMF estimates *e*ˆ*a*, *e*ˆ*b*, and *e*ˆ*<sup>c</sup>* are:

$$\pounds\_{a} = \hat{\omega}\_{r} \lambda\_{m}^{\prime} \cos \hat{\theta}\_{r} + h\_{\hat{\tau}} \hat{\omega}\_{r} \lambda\_{m}^{\prime} \cos[5(\hat{\theta}\_{r} + \frac{\delta\_{5}}{5})] + h\_{\hat{\tau}} \hat{\omega}\_{r} \lambda\_{m}^{\prime} \cos[7(\hat{\theta}\_{r} + \frac{\delta\_{7}}{7})],\tag{8}$$

$$\mathcal{E}\_b = \hat{\omega}\_r \lambda\_m' \cos(\theta\_r - 120^\circ) + h\_5 \hat{\omega}\_r \lambda\_m' \cos[5(\hat{\theta}\_r + \frac{\delta\_5}{5} - 120^\circ)] + h\_7 \hat{\omega}\_r \lambda\_m' \cos[7(\hat{\theta}\_r + \frac{\delta\_7}{7} - 120^\circ)],\tag{9}$$

$$\hat{\alpha}\_{\varepsilon} = \hat{\omega}\_{\prime} \lambda\_{m}^{\prime} \cos(\hat{\theta}\_{\prime} + 120^{\circ}) + h\_{5} \hat{\omega}\_{\prime} \lambda\_{m}^{\prime} \cos[5(\hat{\theta}\_{\prime} + \frac{\delta\_{5}}{5} + 120^{\circ})] + h\_{7} \hat{\omega}\_{\prime} \lambda\_{m}^{\prime} \cos[7(\hat{\theta}\_{\prime} + \frac{\delta\_{7}}{7} + 120^{\circ})],\tag{10}$$

where *E*ˆ*<sup>m</sup>* = *ω*ˆ *<sup>r</sup>λ <sup>m</sup>* is the estimated value of the peak induced EMF; *h*<sup>5</sup> and *h*<sup>7</sup> are the percentages of the fifth and seventh harmonic components of the induced EMF; *δ*<sup>5</sup> and *δ*<sup>7</sup> are the phase differences between the fifth and seventh harmonics and the fundamental wave of the induced EMF, respectively.

Using the transformation matrix *T<sup>r</sup> qd* of the rotor's synchronous rotating coordinate system, the induced EMF containing the fifth and seventh harmonic components is transformed into components along the q-axis and d-axis, *e*ˆ *r <sup>q</sup>* and *e*ˆ *r <sup>d</sup>*, respectively, expressed as:

$$\begin{aligned} \mathcal{\ell}\_{q}^{r} &= \hat{\omega}\_{r} \lambda\_{m}^{\prime} + h\_{5} \hat{\omega}\_{r} \lambda\_{m}^{\prime} \cos(6\hat{\theta}\_{r} + \delta\_{5}) + h\_{7} \hat{\omega}\_{r} \lambda\_{m}^{\prime} \cos(6\hat{\theta}\_{r} + \delta\_{7}) \\ &= \hat{\omega}\_{r} \lambda\_{m}^{\prime} + h\_{6q} \hat{\omega}\_{r} \lambda\_{m}^{\prime} \cos(6\hat{\theta}\_{r} + \delta\_{6q}) \end{aligned} \tag{11}$$

$$\begin{array}{rcl} \mathcal{E}\_{d}^{r} &= h\_{5} \hat{\omega}\_{r} \lambda\_{m}^{\prime} \sin(6\hat{\theta}\_{r} + \delta\_{5}) - h\_{7} \hat{\omega}\_{r} \lambda\_{m}^{\prime} \sin(6\hat{\theta}\_{r} + \delta\_{7}) \\ &= h\_{6d} \hat{\omega}\_{r} \lambda\_{m}^{\prime} \sin(6\hat{\theta}\_{r} + \delta\_{6d}) \end{array} \tag{12}$$

where the harmonics of the induced EMF along the q-axis and d-axis, *e*ˆ *r <sup>q</sup>*−*<sup>h</sup>* and *<sup>e</sup>*<sup>ˆ</sup> *r <sup>d</sup>*−*h*, respectively, are expressed as:

$$\hat{\epsilon}\_{q\to h}^{r} = h\_{6q} \hat{\omega}\_r \lambda\_m' \cos(\theta \hat{\theta}\_r + \delta\_{6q}),\tag{13}$$

$$
\mathscr{E}\_{d-h}^{r} = h\_{6d} \hat{\omega}\_r \lambda\_m' \sin(6\hat{\theta}\_r + \delta\_{6d}).\tag{14}
$$

It can be seen from Equations (11) and (12) that, when the fifth and seventh harmonic components of the induced EMF are transformed into the rotor reference frame and further simplified by the polar coordinate system, they will contain the sixth harmonic components. Therefore, the sixth harmonic components can be used for compensation in the rotor reference frame. In the equations above: *h*6*<sup>q</sup>* is the coefficient indicating the percentage of the sixth harmonic component along the q-axis; *δ*6*<sup>q</sup>* is the phase difference between the sixth harmonic and the fundamental wave along the q-axis; *h*6*<sup>d</sup>* is the coefficient indicating the percentage of the sixth harmonic component along the d-axis; *δ*6*<sup>d</sup>* is the phase difference between the sixth harmonic component and the fundamental wave along the d-axis.

## **3. Current Harmonic Improvement Strategy for Closed-Loop Control of Current along q-Axis and d-Axis**

## *3.1. Closed-Loop Control Strategy for q-Axis and d-Axis Currents with Compensation for Fundamental Wave of Induced EMF*

The closed-loop control of the speed and the q-axis and d-axis currents of the threephase permanent-magnet synchronous motor in this study are shown in Figure 1. The speed feedback *ω*ˆ *<sup>m</sup>* and position feedback ˆ *θ<sup>r</sup>* in Figure 1 are obtained from the the resolver and RDC. The three-phase feedback currents ˆ*ia*, ˆ*ib*, and ˆ*ic* of the motor are transformed to the rotor reference frame to obtain the q-axis current ˆ*i r <sup>q</sup>* and the d-axis current ˆ*i r <sup>d</sup>*. Moreover, the q-axis and d-axis current control strategy is used to obtain the q-axis and d-axis voltage commands *v<sup>r</sup> q* <sup>∗</sup> and *v<sup>r</sup> d* ∗. Finally, the inverse transformation matrix of the rotor reference frame is used to obtain the three-phase voltage commands *v*∗ *<sup>a</sup>* , *v*<sup>∗</sup> *<sup>b</sup>* , and *v*<sup>∗</sup> *<sup>c</sup>* , and then feed to the space vector pulse width modulation (SVPWM) module to produce the switching signals.

**Figure 1.** Three-phase permanent-magnet synchronous motor speed and q-axis/d-axis current closed-loop control block.

The closed-loop control block of the q-axis and d-axis currents is shown in Figure 1. The three-phase sinusoidal signal without the harmonic components is transformed into two direct current (DC) signals, direct axis and quadrature axis signals respectively in the rotor reference frame. The transformation of the rotor reference frame can reduce the influence of external disturbance and improve the robustness of the control strategy. The q-axis and d-axis current regulator of the synchronous motor can be obtained according to voltage Equations (6) and (7). By reducing the coupling terms *ωr*(*Ldi r <sup>d</sup>* + *λ m*) and −*ωrLqi r <sup>q</sup>* in voltage Equations (6) and (7), the linearized voltages *u<sup>r</sup> <sup>q</sup>* and *u<sup>r</sup> <sup>d</sup>* along the q-axis and d-axis are defined as, respectively:

$$\begin{array}{ll} u\_q^r &= R\_s \dot{i}\_q^r + L\_q \frac{d}{dt} \dot{i}\_q^r \\ &= v\_q^r - \omega\_r (L\_d \dot{i}\_d^r + \lambda\_m') \ ' \end{array} \tag{15}$$

$$\begin{array}{rcl} \mu\_d^r &= R\_s \dot{\imath}\_d^r + L\_d \frac{d}{dt} \dot{\imath}\_d^r \\ &= \upsilon\_d^r + \omega\_r L\_q \dot{\imath}\_q^r \end{array} \tag{16}$$

Decoupling *u<sup>r</sup> <sup>q</sup>* and *i r <sup>q</sup>* in Equation (15), and *u<sup>r</sup> <sup>d</sup>* and *i r <sup>d</sup>* in Equation (16), independent linear systems can be obtained, and the q-axis and d-axis current regulator outputs are, respectively:

$$
\mu\_q^{r\*} = \mathbf{G}\_q \circ (\dot{\mathbf{i}}\_q^{r\*} - \dot{\mathbf{i}}\_q^r),
\tag{17}
$$

$$\mathfrak{u}\_d^{r,\*} = \mathcal{G}\_d \circ (\mathfrak{i}\_d^{r,\*} - \mathfrak{i}\_d^r),\tag{18}$$

where *i r q* ∗ and *i r d* ∗ are the q-axis and d-axis current commands. The current regulators *Gq* and *Gd* are both proportional-integral controllers with "◦" as the symbol of proportionalintegral operation. The transfer function of the current regulator is expressed as

$$G\_q = k\_{pq} + \frac{k\_{iq}}{s},\tag{19}$$

$$G\_d = k\_{pd} + \frac{k\_{id}}{s},\tag{20}$$

where *kpq* and *kpd* are the proportional gain; *kiq* and *kid* are the integral gain. The q-axis and d-axis voltage commands *v<sup>r</sup> q* <sup>∗</sup> and *v<sup>r</sup> d* ∗ can be expressed as

$$
\omega\_q^{r \ast} = \mathfrak{u}\_q^{r \ast} + \mathring{\mathcal{R}}\_s \mathring{\mathfrak{i}}\_q^r + \omega\_r (\mathring{\mathcal{L}}\_d \mathring{\mathfrak{i}}\_d^r + \mathring{\lambda}\_m') \,. \tag{21}
$$

$$
\sigma\_d^{r\*} = \mathfrak{u}\_d^{r\*} + \mathring{\mathcal{R}}\_s \mathring{\mathfrak{i}}\_d^r - \omega\_r \mathring{\mathcal{L}}\_q \mathring{\mathfrak{i}}\_{q'}^r \tag{22}
$$

The pre-computed feed-forward compensation *Rsi r <sup>q</sup>* and *Rsi r <sup>d</sup>* can be added into the q-axis and d-axis voltage commands, respectively, to increase system response speed.

## *3.2. Compensation Method of Induced EMF Harmonics along Phase-a, Phase-b, and Phase-c*

The compensation strategy along the a-axis, b-axis, and c-axis is based on the speed and the q-axis and d-axis current closed-loop control strategy shown in Figure 1, with additional compensation for the fifth and seventh harmonic components of the three-phase induced EMF, as shown in Figure 2. As the induced EMF compensation in Figure 1 contains only that for the fundamental wave, the fifth and seventh harmonic components of the induced EMF are not compensated. After transforming the q-axis and d-axis voltage commands *v<sup>r</sup> q* <sup>∗</sup> and *v<sup>r</sup> d* ∗ with the rotor's synchronous reference system, compensation for the fifth and seventh harmonic components of the induced EMF along the a-axis, b-axis, and c-axis are added, as shown in Figure 2.

## *3.3. Compensation Method of Induced EMF Harmonics along q-Axis and d-Axis*

The fifth and seventh harmonic components of the induced EMF are mapped to the sixth harmonic component after transformed to the rotor reference frame, as shown in Figure 3. By compensating on the qd-axis, the calculation of the control strategy can be simplified. The compensation strategy for harmonics in the induced EMF along the q-axis and d-axis adds the sixth harmonic component in Figure 1, as shown in Equations (13) and (14). By adding Equations (13) and (14) into Figure 1 as the compensation, the resulting control block is as shown in Figure 3. The software was developed based on Figure 3.

**Figure 2.** Current harmonics improvement control block of compensation for the induced electromotive force (EMF) harmonics along the a-axis, b-axis, and c-axis in the q-axis/d-axis current closed-loop control.

**Figure 3.** Current harmonics improvement control block of compensation for the induced EMF harmonics along the q-axis and d-axis in the q-axis/d-axis current closed-loop control.

## **4. Measurement Results**

A laboratory prototype based on DSP TMS320F28069 is established to verify the proposed method. The photograph of the motor driver and testbed is shown in Figure 4. The switching frequency of the three-phase inverter is 10 kHz, the sampling time of the closed-loop current control is 100 μs, and the resolution of the A/D converter is 16-bit, corresponding to −5~5 V. The resolution of the D/A converter is 12-bit, corresponding to 0~3.3 V, and the measurement bandwidth of the current probe is 100 kHz. The parameters of the driven motor are shown in Appendix A.

**Figure 4.** Laboratory prototype photos: (**a**) motor driver; (**b**) test bed.

## *4.1. Measurement of Induced EMF*

The prime mover is used to drive the three-phase permanent-magnet synchronous motor at a speed of 1500 rpm. The parameters are *ω*ˆ *<sup>r</sup>* = 2*π*( <sup>1500</sup> <sup>60</sup> )( *Np* <sup>2</sup> ) (rad/s), *λ <sup>m</sup>* = 0.0153 (V/rad/s), *h*<sup>5</sup> = 3.30%, *h*<sup>7</sup> = 1.55%, *δ*<sup>5</sup> = 31.51◦ , and *δ*<sup>7</sup> = 77.35◦ . After the simplification by the polar coordinate system, *h*6*<sup>q</sup>* = 4.52%, *h*6*<sup>d</sup>* = 2.48%, *δ*6*<sup>q</sup>* = 45.76◦ , *δ*6*<sup>d</sup>* = 4.91◦ are obtained, and the estimated results are shown in Figure 5. Figure 5a shows the estimated induced EMF *e*ˆ*a*, which is very close to the measured *ean* shown in Figure 5e, verifying the correctness of the estimation in this study. In addition, the estimated values of the fundamental wave, fifth harmonic component, and seventh harmonic component are shown in Figure 5b–d, respectively. By using the synchronously rotating reference system to transform from abc to qd reference systems in Equation (4), *e*ˆ *r <sup>q</sup>* and *e*ˆ *r <sup>d</sup>* can be obtained. In addition, *e*ˆ *r <sup>q</sup>*−*<sup>h</sup>* and

*e*ˆ *r <sup>d</sup>*−*<sup>h</sup>* can be obtained using Equations (13) and (14) derived in this study, as shown in Figure 6. Figure 6 shows *e*ˆ *r <sup>q</sup>* and *e*ˆ *r <sup>d</sup>*. As *e*ˆ *r <sup>q</sup>* contains the fundamental value *ω*ˆ *<sup>r</sup>λ <sup>m</sup>* and the sixth harmonic component, whereas *e*ˆ *r <sup>d</sup>* only contains the sixth harmonic component, their respective responses are the same as shown in Equations (11) and (12).

**Figure 5.** Measured and estimated results of induced EMF of three-phase permanent-magnet synchronous motor driven by the prime mover at a speed of 1500 rpm: (**a**) estimated *e*ˆ*a* of a-phase induced EMF; (**b**) *E*ˆ*<sup>m</sup>* cos ˆ *θr*; (**c**) *h*5*E*ˆ*<sup>m</sup>* cos(5 ˆ *θ<sup>r</sup>* + *δ*5); (**d**) *h*7*E*ˆ*<sup>m</sup>* cos(7 ˆ *θ<sup>r</sup>* + *δ*7); (**e**) *ean* .

**Figure 6.** Estimated results of the q-axis and d-axis of the three-phase permanent-magnet synchronous motor driven by the prime mover at a speed of 1500 rpm: (**a**) *e*ˆ *r <sup>q</sup>*; (**b**) *e*ˆ *r d*.

## *4.2. Measured Results of Induced EMF without Compensation for Harmonic Components*

The current harmonic improvement control strategy is not added to the control block of Figure 1. With the effective value of the rated phase current of 23.3 A, the rated rotational speed of 1500 rpm, and the rated electromagnetic torque of 3.5 N·m, the measured results are as shown in Figure 7. It can be seen from Figure 7 that the phase current and the fundamental wave of induced EMF are in phase, and the peak value of the phase current is 33.04 A. The total harmonic distortion rate of the current is 5.30%, with 3.30% for the fifth harmonic component and 2.97% for the seventh harmonic component of the phase current. The current response transformed onto the q-axis and d-axis is shown in Figure 8. It can be seen from Figure 8 that the average value of the q-axis current is 32.75 A, and the average value of the d-axis current is not 0 A. The measurement results for speed response and electromagnetic torque are shown in Figure 9. In this study, electromagnetic torque is taken as equivalent to *kT* ˆ*i r <sup>q</sup>*, which is then measured by the D/A converter. It can be seen from Figure 9 that the average value of the electromagnetic torque is 3.01 N·m, and the peak-to-peak value of the electromagnetic torque is 0.46 N·m. As torque ripple is the peak-to-peak value of electromagnetic torque divided by the average value of electromagnetic torque, the torque ripple is 15.28%.

**Figure 7.** Measured phase currents of a three-phase permanent-magnet synchronous motor driven at 1500 rpm without compensation for harmonic components in induced EMF: (**a**–**d**) measured phase currents ˆ*ia*, ˆ*ib* and ˆ*ic*; (**e**) spectrum of phase current ˆ*ia*.

**Figure 8.** Measured q-axis and d-axis currents of a three-phase permanent-magnet synchronous motor driven at 1500 rpm without compensation for harmonic components in induced EMF: (**a**) measured ˆ*i r <sup>q</sup>*; (**b**) measured <sup>ˆ</sup>*<sup>i</sup> r d*.

**Figure 9.** Measured rotational speed and electromagnetic torque of a three-phase permanent-magnet synchronous motor driven at 1500 rpm without compensation for harmonic components in induced EMF: (**a**) measured rotational speed command *ω*∗ *<sup>m</sup>* and rotational speed feedback *ω*ˆ *<sup>m</sup>*; (**b**) measured electromagnetic torque *T*ˆ *e*.

## *4.3. Measured Results of Induced EMF along Phase-a, Phase-b, and Phase-c with Compensation for Harmonic Components*

The control block in Figure 2 shows the control strategy for induced EMF along the a-axis, b-axis, and c-axis with compensation for harmonics. The measured results are as shown in Figure 10. It can be seen from Figure 10 that the phase current and the fundamental wave of induced EMF are in phase, and the peak value of the phase current is 33.55 A. The total harmonic distortion rate of the current is 2.57%, with 0.43% for the fifth harmonic component and 0.72% for the seventh harmonic component of the phase current. The current response transformed onto the q-axis and d-axis is as shown in Figure 11. It can be seen from Figure 11 that the average value of the q-axis current is 33.25 A, and the average value of the d-axis current is approximately 0 A. The measurement results of the speed response and electromagnetic torque are shown in Figure 12. It can be seen from Figure 12 that the average value of the electromagnetic torque is 3.05 N·m, the peak-to-peak value of the electromagnetic torque is 0.19 N·m, and the torque ripple is 6.23%.

#### *4.4. Measured Results of Induced EMF along q-Axis and d-Axis with Compensation Strategy for Harmonic Components*

The control block in Figure 3 shows the control strategy for induced EMF along the q-axis and d-axis with compensation for harmonics. The measured results are as shown in Figure 13. It can be seen from Figure 13 that the phase current and the fundamental wave of induced EMF are in phase, and the peak value of the phase current is 33.70 A. The total harmonic distortion rate of the current is 2.31%, with 0.61% for the fifth harmonic component and 0.35% for the seventh harmonic component of the phase current. Compared to the result without compensation, the fifth harmonic component is reduced from 3.30% to 0.61%, and the seventh harmonic component is reduced from 2.97% to 0.35%. With reference to the experimental results of the selective current harmonic suppression method [12], the fifth harmonic content is reduced from 19.24% to 5.62%, and the seventh harmonic content is reduced from 12.87% to 4.63%. In both methods, the fifth and seventh harmonic components are reduced.

**Figure 10.** Measured phase currents of a three-phase permanent-magnet synchronous motor driven at 1500 rpm with compensation strategy for harmonic components in induced EMF along the a-axis, b-axis, and c-axis: (**a**–**d**) measured phase current ˆ*ia*, ˆ*ib* and ˆ*ic*; (**e**) spectrum of phase current ˆ*ia*.

**Figure 11.** Measured q-axis and d-axis currents of a three-phase permanent-magnet synchronous motor driven at 1500 rpm with compensation strategy for harmonic components in induced EMF along the a-axis, b-axis, and c-axis: (**a**) measured ˆ*i r <sup>q</sup>*; (**b**) measured <sup>ˆ</sup>*<sup>i</sup> r d*.

**Figure 12.** Measured rotational speed and electromagnetic torque of a three-phase permanent-magnet synchronous motor driven at 1500 rpm with a compensation strategy for harmonic components in induced EMF along the a-axis, b-axis, and c-axis: (**a**) measured rotational speed *ω*∗ *<sup>m</sup>* and rotational speed feedback *ω*ˆ *<sup>m</sup>*; (**b**) measured electromagnetic torque *T*ˆ *e*.

**Figure 13.** Measured phase currents of a three-phase permanent-magnet synchronous motor driven at 1500 rpm with a compensation strategy for harmonic components in induced EMF along the q-axis and d-axis: (**a**–**d**) measured phase currents ˆ*ia*, ˆ*ib* and ˆ*ic*; (**e**) spectrum of phase current ˆ*ia*.

The current response transformed onto the q-axis and d-axis is as shown in Figure 14. It can be seen from Figure 14 that the average value of the q-axis current is 32.74 A, and the average value of the d-axis current is approximately 0 A. The measurement results of speed response and electromagnetic torque are shown in Figure 15. It can be seen from Figure 15 that the average value of the electromagnetic torque is 3.01 N·m, the peak-to-peak value of the electromagnetic torque is 0.18 N·m, and the torque ripple is 5.98%.

**Figure 14.** Measured q-axis and d-axis currents of a three-phase permanent-magnet synchronous motor driven at 1500 rpm with a compensation strategy for harmonic components in induced EMF along the q-axis and d-axis: (**a**) measured ˆ*i r <sup>q</sup>*; (**b**) measured <sup>ˆ</sup>*<sup>i</sup> r d*.

**Figure 15.** Measured rotational speed and electromagnetic torque of a three-phase permanent-magnet synchronous motor driven at 1500 rpm with a compensation strategy for harmonic components in induced EMF along the q-axis and d-axis: (**a**) measured rotational speed command *ω*∗ *<sup>m</sup>* and rotational speed feedback *ω*ˆ *<sup>m</sup>*; (**b**) measured electromagnetic torque *T*ˆ *e*.

## **5. Conclusions**

This study explains the closed-loop control strategy for rotational speed and q-axis and d-axis currents, as well as the improvement strategy for current harmonics, including the compensation strategy for induced EMF along an abc reference frame and a qd reference frame. With the compensation for harmonics in the induced EMF along the q-axis and d-axis, the total harmonic distortion rate is reduced from 5.30% to 2.31%, in which the value is reduced from 3.30% to 0.61% for the fifth harmonic component and from 2.97% to 0.35% for the seventh harmonic component. The peak-to-peak electromagnetic torque is reduced from 0.46 to 0.18 N·m, whereas the electromagnetic torque ripple is reduced from 15.28% to 5.98%. The parameters are summarized in Table 1. It can be seen that the compensation method for harmonics is considerably effective. The method can be used to improve the jitters and ripples of the electromagnetic torque to enhance the performance of the servo drive.


**Table 1.** Comparison of the closed-loop control strategy for three-phase permanent-magnet synchronous motor speed and q-axis/d-axis currents.

> **Author Contributions:** Conceptualization, J.-C.H. and W.-T.K.; methodology, W.-T.K.; software, W.- T.K. and J.-E.L.; validation, J.-C.H., W.-T.K., and J.-E.L.; formal analysis, W.-T.K.; investigation, J.-C.H. and W.-T.K.; resources, J.-C.H. and W.-T.K.; data curation, J.-E.L.; writing—original draft preparation, W.-T.K.; writing—review and editing, J.-C.H. and W.-T.K.; visualization, J.-E.L.; supervision, J.-C.H.; project administration, J.-C.H.; funding acquisition, J.-C.H. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Acknowledgments:** The authors would like to acknowledge the financial support of the Ministry of Science and Technology of Taiwan under grant MOST 107-2221-E-011-109-MY3.

**Conflicts of Interest:** The authors declare no conflict of interest.

## **Appendix A**

**Table A1.** Measured results of three-phase permanent-magnet synchronous motor parameters.


## **Appendix B**


## **References**


## *Article* **Inverse Optimal Control in State Derivative Space System with Applications in Motor Control**

**Feng-Chi Lee 1, Yuan-Wei Tseng 2,\*, Rong-Ching Wu 2, Wen-Chuan Chen <sup>1</sup> and Chin-Sheng Chen <sup>1</sup>**


**Abstract:** This paper mathematically explains how state derivative space (SDS) system form with state derivative related feedback can supplement standard state space system with state related feedback in control designs. Practically, inverse optimal control is attractive because it can construct a stable closed-loop system while optimal control may not have exact solution. Unlike the previous algorithms which mainly applied state feedback, in this paper inverse optimal control are carried out utilizing state derivative alone in SDS system. The effectiveness of proposed algorithms are verified by design examples of DC motor tracking control without tachometer and very challenging control problem of singular system with impulse mode. Feedback of direct measurement of state derivatives without integrations can simplify implementation and reduce cost. In addition, the proposed design methods in SDS system with state derivative feedback are analogous to those in state space system with state feedback. Furthermore, with state derivative feedback control in SDS system, wider range of problems such as singular system control can be handled effectively. These are main advantages of carrying out control designs in SDS system.

**Keywords:** inverse optimal control; state derivative space (SDS) system; state derivative feedback; DC motor control; singular system; nonlinear control

## **1. Introduction**

In modern control, state space system is used to carry out state related feedback control design. In state space system, state derivative vector . *x*(*t*) is a dependent function of both control input vector *u*(*t*) and state vector *x*(*t*) as follows.

$$\dot{\mathbf{x}}(t) = f(\mathbf{x}(t), \boldsymbol{\mu}(t)) \tag{1}$$

Previously, in most researches, state related feedback control algorithms *u*(*t*) = *φ*(*x*(*t*)) were developed in state space system form so that the following is a stable closed loop system.

$$
\dot{\mathbf{x}}(t) = f(\mathbf{x}(t), \boldsymbol{\phi}(\mathbf{x}(t))) \tag{2}
$$

However, in reality the control design approach of carrying out state related feedback in state space system has some limitations. For instance, not every system can have its state space system form. Singular systems [1] with pole at infinity are such cases. For example, electrical circuits [2], aerospace vehicles [3], piezoelectric smart structures [4], and chemical systems [5] are actually singular systems. Control design of singular systems were mainly developed in the following generalized state space system or descriptor system form [6,7] where *E* is a singular matrix.

$$E\dot{\mathbf{x}}(t) = F(\mathbf{x}(t), \boldsymbol{\mu}(t)). \tag{3}$$

Control designs for such systems are carried out in large augmented systems and usually require feedbacks of both state and state derivative variables [7–11].

**Citation:** Lee, F.-C.; Tseng, Y.-W.; Wu, R.-C.; Chen, W.-C.; Chen, C.-S. Inverse Optimal Control in State Derivative Space System with Applications in Motor Control. *Energies* **2021**, *14*, 1775. https:// doi.org/10.3390/en14061775

Academic Editor: Frede Blaabjerg

Received: 31 January 2021 Accepted: 18 March 2021 Published: 23 March 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

Therefore, comparing with the design processes for standard state space system, control design processes for singular systems are much more complicate. In the analysis, singular systems are further classified into impulse-free mode and impulse mode [7,12]. When a singular system has impulse mode, designers have to further investigate if the system is impulse controllable and if the impulse mode can be eliminated [7]. In the best case, applying state feedback control only can stabilize singular systems with impulse mode. Therefore, state feedback control design for a singular systems with impulse mode is usually considered as very challenging task.

Moreover, in many systems, the direct measurements by sensor are not state signals but state derivative signals. For example, accelerations sensed by accelerometers [13] and voltages or more precisely speaking current derivatives sensed by inductors are directly measured state derivative related signals in many applications. Especially, velocities and accelerations which can be modelled as state derivative vector are easily available from measurements in vehicle dynamic systems [14–18] and piezoelectric smart structure systems [4]. For those applications, we should not insist to apply state related feedback in control designs because additional numerical integrations or integrators are needed in implementation that result in complex and expensive controllers. Instead, state derivative related feedback should be applied. However, it is not convenient to develop state derivative related feedback algorithms under standard state space system form. Another system form for people to conveniently develop state derivative feedback is needed.

Inspired by the above analysis, the correspondence author of this paper proposed the following state derivative space system form, abbreviated as SDS systems and dedicated for state derivative related feedback control designs.

$$\dot{\mathbf{x}}(t) = F(\dot{\mathbf{x}}(t), \boldsymbol{\mu}(t)). \tag{4}$$

In SDS systems, state vector *<sup>x</sup>*(*t*) is an explicit function vector of state derivative vector . *x*(*t*) and control vector *u*(*t*).

When state derivative related feedback control law *u*(*t*) = *φ* . *x*(*t*) is properly designed and applied, one can obtain a stable closed loop system as

$$\dot{\mathbf{x}}(t) = F(\dot{\mathbf{x}}(t), \boldsymbol{\phi}(\dot{\mathbf{x}}(t))). \tag{5}$$

The linear time invariant system of SDS system, namely, Reciprocal State Space (RSS) system can be described as

$$\mathbf{x} = f\dot{\mathbf{x}} + \mathbf{g}u \tag{6}$$

where *<sup>f</sup>* and *<sup>g</sup>* are constant matrices. When *<sup>u</sup>* <sup>=</sup> <sup>−</sup>*<sup>K</sup>* . *x* is properly designed and applied, the following closed loop system is stable.

$$\mathbf{x} = (f - \mathbf{g}\mathbf{K})\dot{\mathbf{x}} = f\_{\mathbf{c}}\dot{\mathbf{x}}\tag{7}$$

It is well known that the eigenvalues of an invertible matrix and the eigenvalues of its inverse matrix are actually reciprocals to each other and that was why the name of Reciprocal State Space system was given. Therefore, closed loop system poles are the reciprocals of the eigenvalues of matrix *fc* in (7). To construct a stable closed loop RSS system, all eigenvalues of matrix *fc* in (7) must have negative real parts by design of feedback gain *K*.

Both SDS system and RSS system forms were proposed by the correspondence author of this paper. State derivative related feedback control algorithms such as sliding mode control [19,20], H infinity control [21,22], optimal, and LQR control [23] have been developed in SDS system or RSS system form. Even the complicated singular system with impulse mode were successfully controlled in SDS system with state derivative related feedback control laws [22,23].

When systems' global operations are accurately modeled, they are mostly nonlinear systems. However, control design of nonlinear systems are more difficult than of linear systems. Optimal control is among the handful design approaches that can systematically handle nonlinear systems.

Mathematically speaking, problems of nonlinear optimal control can be solved based on a Hamilton–Jacobi–Bellman (HJB) equation to obtain a Lyapunov function of closed-loop system (or control Lyapunov function) and correlated optimal control law that minimize a given performance functional. However, it is not easy to solve this equation. In general cases, exact solution may not even exist [24,25]. For unstable nonlinear systems, the fundamental requirement is to find control laws to stabilize them but this requirement may not be achieved with optimal control. In 1964, Kalman proposed inverse optimal control (IOC) as the alternative for finding control laws that can stabilize nonlinear systems. In design approach of inverse optimal control, a control Lyapunov function is selected at the beginning. Therefore, solving a HJB equation is circumvented. Followed by design steps according to the Lyapunov stability theorem and the coupling in HJB equation, one can find an optimal controller related to a meaningful performance integrand [24,26]. More precisely speaking, the performance integrand to be constructed is related to the control Lyapunov function, system dynamic and feedback control law because they are coupled in the HJB equation. Therefore, Inverse optimal control has great design flexibility by varying parameters in both the performance integrand and the control Lyapunov function to characterize globally stabilizing controller to meet response constraints of closed loop system [27]. Hence, for unstable nonlinear systems, inverse optimal control is usually considered as the last resort to stabilize them.

Inverse optimal control has been widely applied in robotic control [28,29], biological systems [30,31], aerospace vehicles [24,32–34] and power systems [35–37]. In this paper, inverse optimal control in SDS system with state derivative related feedback is presented. To authors' best knowledge, this type of research have not been reported before.

To verify the proposed algorithm, a non-traditional speed tracking controller and torque tracking controller of a DC motor without tachometer by feeding back the voltage of a small inductor externally connected in series with armature circuit of a DC motor are provided as one application example. The small inductor serves as sensor in the DC motor tracking control. Unlike the traditional DC motor controls which apply state related feedback of speed or current, the inductor's voltage is state derivative related measurement feedback of current which is well suitable to apply the proposed IOC algorithm based on state derivative feedback. The advantages of the proposed controllers with inductor's voltage include 1. Inductor's average power is zero so it does not damage the armature circuit. 2. No tachometer is needed so it can save the implementation cost. Another example of a challenging singular system with impulse mode and bounded disturbance is also provided.

The organization of paper is described as follows. In Section 2, we introduce the inverse optimal control design algorithms for SDS systems with state derivative related feedback. In Section 3, we present illustrative examples and simulation results. Finally, we discuss the results and potential of constructing compact and cheap controller for system with direct state derivative measurement in Section 4 and conclusions in Section 5.

## **2. Inverse Optimal Control in State Derivative Space (SDS) System with State Derivative Related Feedback**

This section first introduces stability analysis of SDS system, followed by the algorithms of carrying out inverse optimal control in SDS system with state derivative related feedback building on the inspirations of inverse optimal control deign in state space system with state feedback in [26,27,31].

#### *2.1. Stability Analysis of SDS Systems*

Consider the following SDS system with proper dimensions.

$$\mathbf{x} = f\left(\dot{\mathbf{x}}(t)\right), \mathbf{x}(0) = \mathbf{x}\_0, \ t \ge 0 \tag{8}$$

A Lyapunov function *V*(*x*) should be continuously differentiable and meet the following requirements.

$$V(\mathbf{x}) > 0, \text{ if and only if } \mathbf{x} \neq \mathbf{0} \text{ and } V(\mathbf{0}) = \mathbf{0}. \tag{9}$$

For *x* = 0, taking derivative of *V*(*x*) with respect to time t and substituting system Equation in (8), if the result is negative, the SDS system is stable.

$$\dot{V}(\mathbf{x}) = \frac{dV(\mathbf{x})}{dt} = \frac{dV(\mathbf{x})}{d\mathbf{x}} \frac{d\mathbf{x}}{dt} = V'(\mathbf{x})\dot{\mathbf{x}} = \dot{\mathbf{x}}^T V'^T \left(f(\dot{\mathbf{x}}(t))\right) < 0 \tag{10}$$

For a stable system, when *t* → ∞, *V*(*x*(∞)) → 0 as *x*(∞) → 0.

For simplicity of presentation and for people to better understand that in formula derivation of SDS system control designs, state x should be substituted by its SDS system equation. In this paper, a popular quadratic Lyapunov function *V*(*x*) that meets the requirements in (9) is selected for formula derivation as follows.

$$V(\mathbf{x}) = \frac{1}{2}\mathbf{x}^T \mathbf{P} \mathbf{x} > \mathbf{0},\tag{11}$$

where *P* is a positive definite and symmetric matrix.

Consequently, using SDS system Equation in (8), if

$$\dot{V}(\mathbf{x}) = \frac{dV(\mathbf{x})}{dt} = \dot{\mathbf{x}}^T P \dot{\mathbf{x}} = \dot{\mathbf{x}}^T P \mathbf{x} = \dot{\mathbf{x}}^T P f(\dot{\mathbf{x}}) < 0 \tag{12}$$

the SDS system is stable.

Hence, for a stable SDS system, let the performance integrand as

$$L\left(\dot{\mathbf{x}}(t)\right) = -\dot{\mathbf{x}}^T P f\left(\dot{\mathbf{x}}\right) = -\dot{V}\left(\mathbf{x}(t)\right) > 0\tag{13}$$

We have the following positive performance functional

$$J(\mathbf{x}\_0) = \int\_0^\infty L(\dot{\mathbf{x}}(t))dt = -V(\mathbf{x}(\infty)) + V(\mathbf{x}(0)) = V(\mathbf{x}\_0) = \frac{1}{2}\mathbf{x}\_0^T \mathbf{P} \mathbf{x}\_0 > 0\tag{14}$$

The value of performance functional is bounded, greater than zero and related to the initial condition *x*(0) = *x*0.

## *2.2. Inverse Optimal Control for SDS Systems with State Derivative Related Feedback*

In this section, we explain the inverse optimal nonlinear control design process for SDS systems with state derivative feedback.

Consider the following nonlinear controlled dynamic SDS system with proper dimensions and initial condition.

$$\mathbf{x} = F(\dot{\mathbf{x}}(t), \boldsymbol{\mu}(t)), \mathbf{x}(0) = \mathbf{x}\_0, \ t \ge 0,\tag{15}$$

with performance functional as

$$J(\mathbf{x}\_{0\prime}u(\cdot)) = \int\_0^\infty L\left(\dot{\mathbf{x}}(t), u(t)\right)dt\tag{16}$$

where *u*(·) is an admissible control.

The following process is to construct an inverse optimal globally stabilizing control law.

$$u(t) = \phi(\dot{\mathbf{x}}(t))\tag{17}$$

First, a symmetric and positive definite matrix *P* serving as the design parameter should be selected for Lyapunov function in (11).

Therefore, when the control law *u*(*t*) is properly designed and substitute SDS system Equation in (15), we should have

$$\dot{V}(\mathbf{x}) = \frac{dV(\mathbf{x})}{dt} = \mathbf{x}^T P \dot{\mathbf{x}} = \dot{\mathbf{x}}^T P \mathbf{x} = \dot{\mathbf{x}}^T P F(\dot{\mathbf{x}}(t), u(t)) < 0 \tag{18}$$

For simplicity of presentation, we omit (*t*) in the following formula derivation.

Second, select another design parameter, namely the performance integrand *L* . *x*, *u* and applying (15) so that we have the following Hamiltonian for the SDS system in (15) with the performance functional in (16).

$$H(\dot{\mathbf{x}}, u) = L(\dot{\mathbf{x}}, u) + \dot{V}(\mathbf{x}) = L(\dot{\mathbf{x}}, u) + \dot{\mathbf{x}}^T P \mathbf{x} = L(\dot{\mathbf{x}}, u) + \dot{\mathbf{x}}^T P F(\dot{\mathbf{x}}, u) \ge 0 \tag{19}$$

Third, one can have the inverse optimal feedback control law in (17) by setting

$$\frac{\partial H(\dot{\mathbf{x}}, \mu)}{\partial \mu} = 0 \tag{20}$$

Fourth, applying the obtained inverse optimal feedback control law in the third step, if

$$\dot{V}(\mathbf{x}) = \frac{dV(\mathbf{x})}{dt} = \mathbf{x}^T P \dot{\mathbf{x}} = \dot{\mathbf{x}}^T P \mathbf{x} = \dot{\mathbf{x}}^T P \mathbf{F} \left( \dot{\mathbf{x}}, \phi(\dot{\mathbf{x}}) \right) < 0 \tag{21}$$

and the steady-state Hamilton–Jacobi–Bellman Equation is zero as follows.

$$H(\dot{\mathbf{x}}, \phi(\dot{\mathbf{x}})) = 0 \tag{22}$$

Then, the following closed-loop *SDS* system is stable.

$$\propto = F\left(\dot{\mathbf{x}}(t), \boldsymbol{\phi}\left(\dot{\mathbf{x}}(t)\right)\right) \tag{23}$$

Therefore, the selection of design parameter *L* . *x*, *u* should meet the requirement of Hamiltonian in (19). Consequently, the inverse optimal feedback control law in (17) obtained from solving (20) should satisfy both (21) and (22) to guarantee the global asymptotic stability of the closed-loop SDS system in (23).

Furthermore, from (19), we have the following performance integrand.

$$L(\dot{\mathbf{x}}, u(t)) = -\dot{V}(\mathbf{x}) + H(\dot{\mathbf{x}}, u(t)). \tag{24}$$

Taking integrals of both sides of (24) and using (19) and (22), it follows that

$$\begin{split} J(\mathbf{x}\_{0},\boldsymbol{\mu}(\cdot)) &= \int\_{0}^{\infty} L(\dot{\mathbf{x}}(t),\boldsymbol{\mu}(t))dt = \int\_{0}^{\infty} \Big[ -\dot{V}(\mathbf{x}) + H(\dot{\mathbf{x}}(t),\boldsymbol{\mu}(t)) \Big] dt \\ &= -\lim\_{\begin{subarray}{c} t \to \infty \\ \end{subarray}} V(\mathbf{x}(t)) + V(\mathbf{x}\_{0}) + \int\_{0}^{\infty} H(\dot{\mathbf{x}}(t),\boldsymbol{\mu}(t))dt \\ &= V(\mathbf{x}\_{0}) + \int\_{0}^{\infty} H(\dot{\mathbf{x}}(t),\boldsymbol{\mu}(t))dt \ge V(\mathbf{x}\_{0}) + \int\_{0}^{\infty} H(\dot{\mathbf{x}}(t),\boldsymbol{\mu}(\dot{\mathbf{x}}(t)))dt \\ &\ge V(\mathbf{x}\_{0}) = J(\mathbf{x}\_{0},\boldsymbol{\mu}(\dot{\mathbf{x}}(\cdot))), \end{split} \tag{25}$$

Hence, when we apply inverse optimal control law in (17), we have (22). Consequently, performance functional is the minimum as follows.

$$J(\mathbf{x}\_0, \boldsymbol{\phi}(\dot{\mathbf{x}})) = \min l(\mathbf{x}\_0, \boldsymbol{\mu}(\cdot)) = V(\mathbf{x}\_0) = \frac{1}{2} \mathbf{x}\_0^T P \mathbf{x}\_0 > 0 \tag{26}$$

## *2.3. Inverse Optimal Control for Affine SDS Systems with State Derivative Related Feedback*

Affine systems are nonlinear systems that are linear in the input. Consider the nonlinear affine *SDS* system with dimension notations given by

$$\mathbf{x}\_{n \times 1} = f\_{n \times 1} \left( \dot{\mathbf{x}}(t) \right) + g\_{n \times m} \left( \dot{\mathbf{x}}(t) \right) \\ u\_{m \times 1}(t) \mathbf{x}(0) = \mathbf{x}\_0 \\ t \ge 0,\tag{27}$$

with performance functional as

$$J(\mathbf{x}\_0, \boldsymbol{\mu}(\cdot)) = \int\_0^\infty L\left(\dot{\boldsymbol{x}}(t), \boldsymbol{\mu}(t)\right) dt. \tag{28}$$

The following process is to construct an inverse optimal globally stabilizing control law. First, a symmetric and positive definite matrix *P* serving as the first design parameter should be selected for Lyapunov function *V*(*x*) in (11). Therefore, when the control law *u*(*t*) is properly designed and substitute SDS system Equation in (27), we should have

$$\dot{V}(\mathbf{x}) = \frac{dV(\mathbf{x})}{dt} = \mathbf{x}^T P \dot{\mathbf{x}} = \dot{\mathbf{x}}^T P \mathbf{x} = \dot{\mathbf{x}}^T P \left[ f(\dot{\mathbf{x}}(t)) + g(\dot{\mathbf{x}}(t)) u(t) \right] < 0 \tag{29}$$

Second, we consider the performance integrand *L* . *x*(*t*), *u*(*t*) which is also a design parameter of the form

$$L\_{1 \times 1} \left( \dot{\mathbf{x}}(t), u(t) \right) = L\_{1\_{1 \times 1}} \left( \dot{\mathbf{x}}(t) \right) + L\_{2\_{1 \times m}} \left( \dot{\mathbf{x}}(t) \right) u\_{m \times 1} + u\_{1 \times m} \, ^T(t) R\_{m \times m} \left( \dot{\mathbf{x}} \right) u\_{m \times 1}(t). \tag{30}$$

Therefore, *L* . *x*(*t*), *u*(*t*) is decomposed into three design parameters, namely, *L*11×<sup>1</sup> . *x*(*t*) ,*L*21×*<sup>m</sup>* . *x*(*t*) and *Rm*×*<sup>m</sup>* . *x* . (31)

For simplicity of presentation, we omit (*t*) and dimension notations in the following formula derivation.

Third, use (29) and define following Hamiltonian for the SDS system in (27) with the performance functional specified in (28).

$$H(\dot{\mathbf{x}}, u) = L(\dot{\mathbf{x}}, u) + \dot{V}(\mathbf{x}) = L\_1(\dot{\mathbf{x}}) + L\_2(\dot{\mathbf{x}})u + u^T R(\dot{\mathbf{x}})u + \dot{\mathbf{x}}^T P[f(\dot{\mathbf{x}}) + g(\dot{\mathbf{x}})u] \ge 0 \tag{32}$$

We should first select a positive definite *R* . *x* so that *uTR* . *x u* > 0 in (32). Setting the partial derivative of the Hamiltonian with respect to *u* to zero,

$$\frac{\partial H(\dot{\mathbf{x}}, \mu)}{\partial \mu} = L\_2^T(\dot{\mathbf{x}}) + 2R(\dot{\mathbf{x}})u + \mathbf{g}^T(\dot{\mathbf{x}})P\dot{\mathbf{x}} = \mathbf{0},\tag{33}$$

the inverse optimal state derivative related feedback control law is obtained as follows.

$$\mu = \phi(\dot{\mathbf{x}}) = \frac{-1}{2} R^{-1}(\dot{\mathbf{x}}) \left[ L\_2^T(\dot{\mathbf{x}}) + \mathbf{g}^T(\dot{\mathbf{x}}) P \dot{\mathbf{x}} \right]. \tag{34}$$

From (34), we have

$$\left[L\_2(\dot{\mathbf{x}}) + \dot{\mathbf{x}}^T \mathcal{P} \mathbf{g}(\dot{\mathbf{x}})\right] = -2\boldsymbol{\phi}^T(\dot{\mathbf{x}})\mathcal{R}\left(\dot{\mathbf{x}}\right). \tag{35}$$

Fourth, substituting (34) into (29), we should have

$$\begin{split} \dot{V}(\mathbf{x}) &= \frac{dV(\mathbf{x})}{dt} = \dot{\mathbf{x}}^T P \dot{\mathbf{x}} = \dot{\mathbf{x}}^T P \mathbf{x} = \dot{\mathbf{x}}^T P \left[ f(\dot{\mathbf{x}}) + \mathbf{g}(\dot{\mathbf{x}}) \boldsymbol{\phi}(\dot{\mathbf{x}}) \right] \\ &= \dot{\mathbf{x}}^T P \left[ f(\dot{\mathbf{x}}) - \frac{1}{2} \mathbf{g}(\dot{\mathbf{x}}) R^{-1}(\dot{\mathbf{x}}) L\_2^T(\dot{\mathbf{x}}) - \frac{1}{2} \mathbf{g}(\dot{\mathbf{x}}) R^{-1}(\dot{\mathbf{x}}) \mathbf{g}^T(\dot{\mathbf{x}}) P \dot{\mathbf{x}} \right]. \end{split} \tag{36}$$

Therefore, to ensure (34) is a stabilizing control law, *L*<sup>2</sup> . *x* should be selected such that

$$\dot{\mathbf{x}}^T P \left[ f(\dot{\mathbf{x}}) - \frac{1}{2} g(\dot{\mathbf{x}}) R^{-1}(\dot{\mathbf{x}}) L\_2^T(\dot{\mathbf{x}}) - \frac{1}{2} g(\dot{\mathbf{x}}) R^{-1}(\dot{\mathbf{x}}) g^T(\dot{\mathbf{x}}) P \dot{\mathbf{x}} \right] < 0. \tag{37}$$

Fifth, using (32) and (35), *L*<sup>1</sup> . *x* should be selected as

$$L\_1(\dot{\mathbf{x}}) = \phi^T(\dot{\mathbf{x}})R(\dot{\mathbf{x}})\phi(\dot{\mathbf{x}}) - \dot{\mathbf{x}}^T P f(\dot{\mathbf{x}}) \tag{38}$$

The following is the proof.

Substituting (35) and (38) into (32), it can be shown that

$$\begin{cases} H(\dot{\mathbf{x}}, u) = L\_1(\dot{\mathbf{x}}) + L\_2(\dot{\mathbf{x}})u + u^T R(\dot{\mathbf{x}})u + \dot{\mathbf{x}}^T P \left[ f(\dot{\mathbf{x}}) + g(\dot{\mathbf{x}})u \right] \\ = \boldsymbol{\Phi}^T(\dot{\mathbf{x}}) R(\dot{\mathbf{x}}) \boldsymbol{\phi}(\dot{\mathbf{x}}) - \dot{\mathbf{x}}^T P f(\dot{\mathbf{x}}) + L\_2(\dot{\mathbf{x}})u + u^T R(\dot{\mathbf{x}})u + \dot{\mathbf{x}}^T P \left[ f(\dot{\mathbf{x}}) + g(\dot{\mathbf{x}})u \right] \\ = \boldsymbol{\Phi}^T(\dot{\mathbf{x}}) R(\dot{\mathbf{x}}) \boldsymbol{\phi}(\dot{\mathbf{x}}) + \left[ L\_2(\dot{\mathbf{x}}) + \dot{\mathbf{x}}^T P g(\dot{\mathbf{x}}) \right] u + u^T R(\dot{\mathbf{x}}) u \\ = \boldsymbol{\Phi}^T(\dot{\mathbf{x}}) R(\dot{\mathbf{x}}) \boldsymbol{\phi}(\dot{\mathbf{x}}) - 2\boldsymbol{\Phi}^T(\dot{\mathbf{x}}) R(\dot{\mathbf{x}}) + u^T R(\dot{\mathbf{x}}) u \\ = \left[ u - \boldsymbol{\phi}(\dot{\mathbf{x}}) \right]^T R(\dot{\mathbf{x}}) \left[ u - \boldsymbol{\phi}(\dot{\mathbf{x}}) \right] \ge 0 \end{cases} \tag{39}$$

Based on (39), applying the inverse optimal control law in (34), the steady-state Hamilton–Jacobi–Bellman equation is zero as follows.

$$H(\dot{\mathbf{x}}, \phi(\dot{\mathbf{x}})) = 0 \tag{40}$$

Consequently, the performance integrand in (30) is obtained as

$$\begin{cases} L(\dot{\mathbf{x}}, u) = \boldsymbol{\phi}^T(\dot{\mathbf{x}}) \mathcal{R}(\dot{\mathbf{x}}) \boldsymbol{\phi}(\dot{\mathbf{x}}) - \dot{\mathbf{x}}^T P \boldsymbol{f}(\dot{\mathbf{x}}) + L\_2(\dot{\mathbf{x}}) u + u^T \mathcal{R}(\dot{\mathbf{x}}) u \\ = \boldsymbol{\phi}^T(\dot{\mathbf{x}}) \mathcal{R}(\dot{\mathbf{x}}) \boldsymbol{\phi}(\dot{\mathbf{x}}) + L\_2(\dot{\mathbf{x}}) u + u^T \mathcal{R}(\dot{\mathbf{x}}) u - \dot{\mathbf{x}}^T P \left[ f(\dot{\mathbf{x}}) + g(\dot{\mathbf{x}}) \boldsymbol{\phi}(\dot{\mathbf{x}}) \right] + \dot{\mathbf{x}}^T P g(\dot{\mathbf{x}}) \boldsymbol{\phi}(\dot{\mathbf{x}}) \end{cases} \tag{41}$$

From (35), we have

$$L\_2(\dot{\mathbf{x}}) = -2\phi^T(\dot{\mathbf{x}})\mathcal{R}(\dot{\mathbf{x}}) - \dot{\mathbf{x}}^T \mathcal{P}\mathbf{g}(\dot{\mathbf{x}}).\tag{42}$$

Substituting (34) and (42) into (41) and using (29) yields

$$\begin{split} L(\dot{\mathbf{x}}, \boldsymbol{\phi}(\dot{\mathbf{x}})) &= \boldsymbol{\phi}^{T}(\dot{\mathbf{x}}) \boldsymbol{R}(\dot{\mathbf{x}}) \boldsymbol{\phi}(\dot{\mathbf{x}}) + \left(-2\boldsymbol{\phi}^{T}(\dot{\mathbf{x}}) \boldsymbol{R}(\dot{\mathbf{x}}) - \dot{\mathbf{x}}^{T} \boldsymbol{P} \boldsymbol{g}(\dot{\mathbf{x}})\right) \boldsymbol{\phi}(\dot{\mathbf{x}}) + \boldsymbol{\phi}^{T}(\dot{\mathbf{x}}) \boldsymbol{R}(\dot{\mathbf{x}}) \boldsymbol{\phi}(\dot{\mathbf{x}}) \\ &- \dot{\mathbf{x}}^{T} \boldsymbol{P} \left[ f(\dot{\mathbf{x}}) + \mathbf{g}(\dot{\mathbf{x}}) \boldsymbol{\phi}(\dot{\mathbf{x}}) \right] + \dot{\mathbf{x}}^{T} \boldsymbol{P} \boldsymbol{g}(\dot{\mathbf{x}}) \boldsymbol{\phi}(\dot{\mathbf{x}}) \\ &= -\dot{\mathbf{x}}^{T} \boldsymbol{P} \left[ f(\dot{\mathbf{x}}) + \mathbf{g}(\dot{\mathbf{x}}) \boldsymbol{\phi}(\dot{\mathbf{x}}) \right] = -\dot{\mathbf{V}}(\mathbf{x}) \end{split} \tag{43}$$

Therefore, based on (43) when inverse optimal law in (34) is applied, the closed loop SDS system is stable, performance functional in (28) becomes

$$J(\mathbf{x}\_0, \boldsymbol{\phi}(\dot{\mathbf{x}})) = \int\_0^\infty L(\dot{\mathbf{x}}, \boldsymbol{\phi}(\dot{\mathbf{x}})) dt = -\int\_0^\infty \dot{V}(\mathbf{x}) dt = -\lim\_{t \to \infty} V(\mathbf{x}(t)) + V(\mathbf{x}\_0) = \frac{1}{2} \mathbf{x}\_0^T P \mathbf{x}\_0 > 0 \tag{44}$$

Hence, to have a small value of performance functional, one may consider to select a diagonal *P* matrix with positive but small diagonal elements.

## *2.4. Inverse Optimal Control for Affine SDS Systems with L*<sup>2</sup> *Disturbance*

Consider the nonlinear affine SDS system with bounded *L*<sup>2</sup> input disturbance *ω*(*t*) [27] in the following form.

$$\dot{\omega}(t) = f\left(\dot{\mathbf{x}}(t)\right) + g\left(\dot{\mathbf{x}}(t)\right)u + f\_1\left(\dot{\mathbf{x}}(t)\right)\omega(t), \\ \mathbf{x}(0) = \mathbf{x}\_0, \omega(\cdot) \in L\_2, t \ge 0 \tag{45}$$

with the following performance variables.

$$z = h(\dot{\mathbf{x}}(t)) + f(\dot{\mathbf{x}}(t))u(t),\tag{46}$$

We consider the non-expansivity case [27] so that the supply rate is given by

$$r(z, w) = \gamma^2 \omega^T \omega - z^T z \tag{47}$$

where *γ* > 0.

An inverse optimal globally stabilizing control law should be designed so that the closed loop system satisfies the non-expansivity constraint [27].

$$\int\_{0}^{T} z^{T} z dt \le \int\_{0}^{T} \gamma^{2} \omega^{T} \omega dt + V(\mathfrak{x}\_{0}), \ T \ge 0, \ \omega(\cdot) \in L\_{2} \tag{48}$$

The performance integrand is considered as

$$L(\dot{\mathbf{x}}, \boldsymbol{\mu}) = L\_1(\dot{\mathbf{x}}) + L\_2(\dot{\mathbf{x}})\boldsymbol{\mu} + \boldsymbol{\mu}^T \mathcal{R}(\dot{\mathbf{x}})\boldsymbol{\mu} \tag{49}$$

Therefore, the performance functional becomes

$$J(\mathbf{x}\_{0}, u(\cdot)) = \int\_{0}^{\infty} \left[ L\_{1}(\dot{\mathbf{x}}) + L\_{2}(\dot{\mathbf{x}})u + u^{T}R(\dot{\mathbf{x}})u \right] dt \tag{50}$$

The following process is to construct an inverse optimal globally stabilizing control law *φ* . *x* with state derivative feedback.

First, a symmetric and positive definite matrix *P* serving as the first design parameter should be selected for Lyapunov function in (11). Consequently, substituting SDS system Equation in (45), we have

$$\dot{V}(\mathbf{x}) = \frac{dV(\mathbf{x})}{dt} = \mathbf{x}^T P \dot{\mathbf{x}} = \dot{\mathbf{x}}^T P \mathbf{x} = \dot{\mathbf{x}}^T P \left[ f(\dot{\mathbf{x}}(t)) + \mathbf{g}(\dot{\mathbf{x}}(t))u + f\_1(\dot{\mathbf{x}}(t))\omega(t) \right] \tag{51}$$

In [22], *H*<sup>∞</sup> control has been carried out for the same SDS system in (45), and the *ωTω* is maximum when disturbance is

$$
\omega = \omega^\* = \frac{1}{2\gamma^2} I\_1^T(\dot{\mathbf{x}}) P \dot{\mathbf{x}} \text{ and } \omega^{\*T} \omega^\* = \frac{1}{4\gamma^2} \dot{\mathbf{x}}^T P f\_1(\dot{\mathbf{x}}) I\_1^T(\dot{\mathbf{x}}) P \dot{\mathbf{x}} \tag{52}
$$

Considering (46) and (52), when an inverse optimal globally stabilizing control law *φ* . *x* is obtained, we should have the following conditions in (53)–(55).

$$\Gamma\left(\dot{\mathbf{x}}, \boldsymbol{\phi}(\dot{\mathbf{x}})\right) \ge 0 \\ \text{with } \Gamma\left(\dot{\mathbf{x}}, u\right) = \frac{1}{4\gamma^2} \dot{\mathbf{x}}^T P \mathbf{J}\_1(\dot{\mathbf{x}}) \mathbf{J}\_1^T(\dot{\mathbf{x}}) P \dot{\mathbf{x}} + \left[h(\dot{\mathbf{x}}) + \mathbf{J}(\dot{\mathbf{x}})u\right]^T \left[h(\dot{\mathbf{x}}) + \mathbf{J}(\dot{\mathbf{x}})u\right] \\ \tag{53}$$
 
$$\therefore ^T \mathbf{n} \left[\boldsymbol{\epsilon}(\dot{\boldsymbol{\omega}}(\mathbf{u})) \quad \boldsymbol{\iota}\_a(\dot{\boldsymbol{\omega}}(\mathbf{u})) \boldsymbol{\iota}(\dot{\mathbf{x}})\right] < 0 \\ \tag{54}$$

$$\dot{\mathbf{x}}^T P \left[ f\left(\dot{\mathbf{x}}(t)\right) + g\left(\dot{\mathbf{x}}(t)\right) \phi\left(\dot{\mathbf{x}}\right) \right] < 0 \tag{54}$$

$$\dot{\mathbf{x}}^T P \mathbf{J}\_1(\dot{\mathbf{x}}) \omega \le \gamma^2 \omega^T \omega - \mathbf{z}^T \mathbf{z} + \mathcal{L}(\dot{\mathbf{x}}, \phi(\dot{\mathbf{x}})) + \Gamma(\dot{\mathbf{x}}, \phi(\dot{\mathbf{x}})) \tag{55}$$

Therefore, applying (55) yields

.

$$\begin{split} \dot{V}(\mathbf{x}) &= \frac{dV(\mathbf{x})}{dt} = \dot{\mathbf{x}}^T P \left[ f(\dot{\mathbf{x}}) + g(\dot{\mathbf{x}}) \boldsymbol{\phi}(\dot{\mathbf{x}}) + f\_1(\dot{\mathbf{x}}) \boldsymbol{\omega}(t) \right] \\ &\leq \dot{\mathbf{x}}^T P \left[ f(\dot{\mathbf{x}}) + g(\dot{\mathbf{x}}) \boldsymbol{\phi}(\dot{\mathbf{x}}) \right] + \gamma^2 \boldsymbol{\omega}^T \boldsymbol{\omega} - \mathbf{z}^T \boldsymbol{z} + L \left( \dot{\mathbf{x}}, \boldsymbol{\phi}(\dot{\mathbf{x}}) \right) + \Gamma \left( \dot{\mathbf{x}}, \boldsymbol{\phi}(\dot{\mathbf{x}}) \right) \end{split} \tag{56}$$

Second, an auxiliary cost functional is specified as

$$\mathbb{S}(\mathbf{x}\_{0\prime}u(\cdot)) = \int\_{0}^{\infty} \left[ L(\dot{\mathbf{x}}, u) + \Gamma(\dot{\mathbf{x}}, u) \right] dt \tag{57}$$

From (50), (53), and (57) yields

$$J(\mathbf{x}\_{0\prime}\boldsymbol{\phi}(\dot{\boldsymbol{x}})) \le \mathbb{S}(\mathbf{x}\_{0\prime}\boldsymbol{\phi}(\dot{\boldsymbol{x}})) = \int\_{0}^{\infty} \left[ L(\dot{\boldsymbol{x}}, \boldsymbol{\phi}(\dot{\boldsymbol{x}})) + \Gamma(\dot{\boldsymbol{x}}, \boldsymbol{\phi}(\dot{\boldsymbol{x}})) \right] dt \tag{58}$$

Third, with (45), (49), and (53), and the Hamiltonian has the form

$$\begin{cases} H(\dot{\mathbf{x}}, u) = L\_1(\dot{\mathbf{x}}) + L\_2(\dot{\mathbf{x}})u + u^T R(\dot{\mathbf{x}})u + \dot{\mathbf{x}}^T P \left[ f(\dot{\mathbf{x}}) + g(\dot{\mathbf{x}})u \right] \\ + \frac{1}{4\gamma^2} \dot{\mathbf{x}}^T P f\_1(\dot{\mathbf{x}}) f\_1^T(\dot{\mathbf{x}}) P \dot{\mathbf{x}} + \left[ h(\dot{\mathbf{x}}) + f(\dot{\mathbf{x}}) u \right]^T \left[ h(\dot{\mathbf{x}}) + f(\dot{\mathbf{x}}) u \right] \ge 0 \end{cases} \tag{59}$$

Then, with the feedback control law *φ* . *x* , there exists a neighborhood of the origin such that if *x*<sup>0</sup> within this neighborhood and when SDS system in (45) is undisturbed (*ω*(*t*) ≡ 0), the zero solution *x*(*t*) ≡ 0 of the closed loop system is locally asymptotically stable.

We should select a positive definite *R* . *x* so that *uTR* . *x u* > 0 in (49), followed by setting

$$\frac{\partial H(\dot{\mathbf{x}},u)}{\partial u} = L\_2^T(\dot{\mathbf{x}}) + 2R(\dot{\mathbf{x}})u + g^T(\dot{\mathbf{x}})P\dot{\mathbf{x}} + 2f^T(\dot{\mathbf{x}})f(\dot{\mathbf{x}})u + 2f(\dot{\mathbf{x}})h(\dot{\mathbf{x}}) = 0,\tag{60}$$

and define

$$R\_a(\dot{\mathbf{x}}) = R(\dot{\mathbf{x}}) + f^T(\dot{\mathbf{x}})f(\dot{\mathbf{x}}) \tag{61}$$

the inverse optimal state derivative related feedback control law is obtained as follows.

$$u = \phi\left(\dot{\mathbf{x}}\right) = \frac{-1}{2} R\_a^{-1}\left(\dot{\mathbf{x}}\right) \left[L\_2^T\left(\dot{\mathbf{x}}\right) + \mathbf{g}^T\left(\dot{\mathbf{x}}\right)P\dot{\mathbf{x}} + 2f^T\left(\dot{\mathbf{x}}\right)h\left(\dot{\mathbf{x}}\right)\right] \tag{62}$$

Consequently, from (62) yields

$$\left[L\_2(\dot{\mathbf{x}}) + \dot{\mathbf{x}}^T P \mathbf{g}(\dot{\mathbf{x}}) + 2h^T(\dot{\mathbf{x}})f(\dot{\mathbf{x}})\right] = -2\phi(\dot{\mathbf{x}})\mathcal{R}\_d(\dot{\mathbf{x}})\tag{63}$$

*L*2 . *x* should be selected such that

$$\times^T \boldsymbol{P} \left[ \boldsymbol{f} \left( \dot{\boldsymbol{x}} \right) - \frac{1}{2} \boldsymbol{g} \left( \dot{\boldsymbol{x}} \right) \boldsymbol{R}\_a^{-1} \left( \boldsymbol{L}\_2^T \left( \dot{\boldsymbol{x}} \right) + \boldsymbol{g}^T \left( \dot{\boldsymbol{x}} \right) \boldsymbol{P} \dot{\boldsymbol{x}} + 2 \boldsymbol{J}^T \left( \dot{\boldsymbol{x}} \right) \boldsymbol{h} \left( \dot{\boldsymbol{x}} \right) \right) \right] + \Gamma \left( \dot{\boldsymbol{x}}, \boldsymbol{\phi} \left( \dot{\boldsymbol{x}} \right) \right) < 0 \tag{64}$$

According to (53), (64) implies (54).

In addition, the auxiliary cost functional in (57), with

$$L\_1(\dot{\mathbf{x}}) = \boldsymbol{\phi}^T(\dot{\mathbf{x}}) \mathcal{R}\_d(\dot{\mathbf{x}}) \boldsymbol{\phi}(\dot{\mathbf{x}}) - \dot{\mathbf{x}}^T P f(\dot{\mathbf{x}}) - h^T(\dot{\mathbf{x}}) h(\dot{\mathbf{x}}) - \frac{1}{4\gamma^2} \dot{\mathbf{x}}^T P f\_1(\dot{\mathbf{x}}) f\_1^T(\dot{\mathbf{x}}) P \dot{\mathbf{x}} \tag{65}$$

in the sense that

$$\odot(\mathfrak{x}\_{0\prime}\phi(\dot{\mathfrak{x}})) = \min \odot(\mathfrak{x}\_{0\prime}u(\cdot))\tag{66}$$

Applying (53), (59), (63), and (65), we have

$$\begin{aligned} \boldsymbol{H}(\dot{\mathbf{x}},u) &= L\_1(\dot{\mathbf{x}}) + L\_2(\dot{\mathbf{x}})u + u^T \boldsymbol{R}(\dot{\mathbf{x}})u + \Gamma(\dot{\mathbf{x}},u) + \dot{\mathbf{x}}^T \boldsymbol{P} \left[ f(\dot{\mathbf{x}}) + \mathbf{g}(\dot{\mathbf{x}})u \right] \\ &= \boldsymbol{\phi}^T(\dot{\mathbf{x}}) \boldsymbol{R}\_a(\dot{\mathbf{x}}) \boldsymbol{\phi}(\dot{\mathbf{x}}) + \left[ L\_2(\dot{\mathbf{x}}) + \dot{\mathbf{x}}^T \boldsymbol{P} \boldsymbol{g}(\dot{\mathbf{x}}) + 2h^T(\dot{\mathbf{x}})f(\dot{\mathbf{x}}) \right] u + u^T \left[ \boldsymbol{R}(\dot{\mathbf{x}}) + f\_1^T(\dot{\mathbf{x}})f\_1(\dot{\mathbf{x}}) \right] u \\ &= \boldsymbol{\phi}^T(\dot{\mathbf{x}}) \boldsymbol{R}\_a(\dot{\mathbf{x}}) \boldsymbol{\phi}(\dot{\mathbf{x}}) - 2\boldsymbol{\phi}(\dot{\mathbf{x}}) \boldsymbol{R}\_a(\dot{\mathbf{x}}) u + u^T \boldsymbol{R}\_a(\dot{\mathbf{x}}) u \\ &= \left[ u - \boldsymbol{\phi}(\dot{\mathbf{x}}) \right]^T \boldsymbol{R}\_a(\dot{\mathbf{x}}) \left[ u - \boldsymbol{\phi}(\dot{\mathbf{x}}) \right] \ge 0 \end{aligned} \tag{67}$$

Hence,

$$H(\dot{\mathbf{x}}, \phi(\dot{\mathbf{x}})) = 0 \tag{68}$$

Furthermore, (67) and (68) imply that

$$\begin{cases} L\left(\dot{\mathbf{x}}, \boldsymbol{\phi}(\dot{\mathbf{x}})\right) + \Gamma\left(\dot{\mathbf{x}}, \boldsymbol{\phi}(\dot{\mathbf{x}})\right) = L\_1\left(\dot{\mathbf{x}}\right) + L\_2\left(\dot{\mathbf{x}}\right)\boldsymbol{\phi}(\dot{\mathbf{x}}) + \boldsymbol{\phi}(\dot{\mathbf{x}})R\left(\dot{\mathbf{x}}\right)\boldsymbol{\phi}(\dot{\mathbf{x}}) + \Gamma\left(\dot{\mathbf{x}}, \boldsymbol{\phi}(\dot{\mathbf{x}})\right) \\ = -\dot{\mathbf{x}}^T P\left[f\left(\dot{\mathbf{x}}\right) + g\left(\dot{\mathbf{x}}\right)\boldsymbol{\phi}\left(\dot{\mathbf{x}}\right)\right] > 0 \end{cases} \tag{69}$$

Substituting (69) into (56) yields

$$\dot{V}(\mathbf{x}) = \frac{dV(\mathbf{x})}{dt} = \dot{\mathbf{x}}^T P[f(\dot{\mathbf{x}}) + \mathbf{g}(\dot{\mathbf{x}})\boldsymbol{\phi}(\dot{\mathbf{x}}) + f\_1(\dot{\mathbf{x}})\boldsymbol{\omega}(t)] \le \gamma^2 \omega^T \omega - \mathbf{z}^T \mathbf{z} \tag{70}$$

Integrating over 0, *T*

$$V(\mathbf{x}(T)) - V(\mathbf{x}\_0) \le \int\_0^T \left(\gamma^2 \omega^T \omega - z^T z\right) dt\tag{71}$$

$$V(\mathbf{x}(T)) + \int\_0^T z^T z dt \le \int\_0^T \gamma^2 \omega^T \omega dt + V(\mathbf{x}\_0) \tag{72}$$

$$\mathbf{r} \colon V(\mathfrak{x}(T)) \ge \mathbf{0} \tag{73}$$

$$\int\_{0}^{T} z^{T} z dt \le \int\_{0}^{T} \gamma^{2} \omega^{T} \omega dt + V(\mathbf{x}\_{0}) \tag{74}$$

Therefore, applying the inverse optimal control law in (62), the closed loop system satisfies the non-expansivity constraint in (74).

#### *2.5. Brief Mathematical Review of Singular System with Impulse Mode*

As mentioned in the introduction, singular systems with impulse mode are difficult in control designs with state related feedback alone. However, some of singular systems with impulse mode can be expressed in RSS system form and can be fully controlled with state derivative feedback. An example of such system will be provided in next section to verify the proposed design process, but before that, the limitation of applying state feedback alone to control singular system with impulse mode is reviewed in this subsection.

The researches of linear singular system control mainly focus on the impulse-free mode. For the following linear and time invariant singular system

$$E\dot{\mathbf{x}} = F\mathbf{x} + \mathbf{N}u \tag{75}$$

When matrix *E* in (75) has zero eigenvalues, it cannot be expressed in state space system form. For people to better understand the nature of this kind of system, singular value decomposition (SVD) can be applied to convert the system to new coordinates as follows.

$$
\begin{bmatrix} I & 0 \\ 0 & 0 \end{bmatrix} \begin{bmatrix} \dot{q}\_1 \\ \dot{q}\_2 \end{bmatrix} = \begin{bmatrix} F\_{11} & F\_{12} \\ F\_{21} & F\_{22} \end{bmatrix} \begin{bmatrix} q\_1 \\ q\_2 \end{bmatrix} + \begin{bmatrix} N\_1 \\ N\_2 \end{bmatrix} u \tag{76}
$$

The singular system is *impulse-free* when matrix *F*<sup>22</sup> is invertible. Consequently, *q*<sup>2</sup> and *q*<sup>1</sup> are coupled by the following equation.

$$q\_2 = -F\_{22}^{-1}F\_{21}q\_1 - F\_{22}^{-1}N\_2\mu\tag{77}$$

Substituting (77) into the first equation in (76) gives the following subsystem in state space system form with state vector of *q*1.

$$\dot{q}\_1 = (F\_{11} - F\_{12}F\_{22}^{-1}F\_{21})q\_1 + (N\_1 - F\_{12}F\_{22}^{-1}N\_2)u \tag{78}$$

Therefore, the state vector *q*<sup>1</sup> can be fully controlled with state feedback design if the subsystem in (78) is controllable. However, through the coupling in (77), state vector *q*<sup>2</sup> is only stabilized but not fully controlled.

When matrix *F*<sup>22</sup> is noninvertible, the singular system has impulse mode. This kind of system is usually very difficult to be controlled with state feedback alone. If it is impulse controllable, applying proper state feedback control may obtain a stabilizing closed loop systems as follows.

$$
\begin{bmatrix} I & \mathbf{0} \\ \mathbf{0} & \mathbf{0} \end{bmatrix} \begin{bmatrix} \dot{q}\_1 \\ \dot{q}\_2 \end{bmatrix} = \begin{bmatrix} F\_{c11} & F\_{c12} \\ F\_{c21} & F\_{c22} \end{bmatrix} \begin{bmatrix} q\_1 \\ q\_2 \end{bmatrix} \tag{79}
$$

In such case, matrix *F*−<sup>1</sup> *<sup>c</sup>*<sup>22</sup> must exist. Similarly, state vector *q*<sup>2</sup> still can only be stabilized through its coupling with *q*1. If the system is impossible to apply state feedback to obtain an invertible matrix *Fc*22, it is called impulse uncontrollable in the research literature. Consequently, the system is not stabilizable by applying state feedback alone. In short, applying state feedback alone cannot control the entire singular system. Singular systems can be stabilized with state feedback control only if it is impulse free or impulse controllable.

However, if matrix *F* of the singular system with impulse mode in (75) is invertible, we may express it in the following SDS form to fully control it with state derivative feedback alone.

$$\mathbf{x} = F^{-1}E\dot{\mathbf{x}} - F^{-1}\mathbf{N}u = f\dot{\mathbf{x}} + \mathbf{g}u. \tag{80}$$

In next section, we will provide an example that verify the proposed design process in this section.

## **3. Examples and Results**

In this section, we provide four examples to verify the effectiveness of the proposed design method. In addition, from both the implementation and mathematical point of views, the advantages of using direct state derivative measurement in control design of SDS system are explained.

**Example 1.** *There are part (a) and part (b) in this example for different purposes.*

*(a) To illustrate the utility of the proposed design process for SDS systems and to emphasize that some SDS systems have no equivalent state space form, we consider*

$$\mathbf{x}(t) = \begin{bmatrix} \mathbf{x}\_1(t) \\ \mathbf{x}\_2(t) \end{bmatrix} = \begin{bmatrix} -\dot{\mathbf{x}}\_1^5(t) + \dot{\mathbf{x}}\_2^2(t) \\ \dot{\mathbf{x}}\_1^2(t) \end{bmatrix} + \begin{bmatrix} 0 \\ 1 \end{bmatrix} u(t) = f\left(\dot{\mathbf{x}}(t)\right) + g\left(\dot{\mathbf{x}}(t)\right)u(t) \tag{81}$$

*Please note that we cannot convert the SDS form in (81) to state space form because the characteristics of original SDS system will be lost after using square root or power of even number order operation.*

$$\begin{aligned} \text{First, we select } \mathbf{R} \begin{pmatrix} \dot{\mathbf{x}} \end{pmatrix} &= 1 \text{ and } V(\mathbf{x}) = \frac{1}{2} \mathbf{x}^T \mathbf{P} \mathbf{x} = \frac{1}{2} \begin{bmatrix} \mathbf{x}\_1 & \mathbf{x}\_2 \end{bmatrix} \begin{bmatrix} 2 & 0 \\ 0 & 2 \end{bmatrix} \begin{bmatrix} \mathbf{x}\_1 \\ \mathbf{x}\_2 \end{bmatrix} = \mathbf{x}\_1^2 + \mathbf{x}\_2^2. \\ \text{Based on (36) and (37), L}\_{\text{(c)}} \text{ should be selected such that} \end{aligned}$$

*Based on (36) and (37), L*<sup>2</sup> *x should be selected such that*

$$\dot{V}(\mathbf{x}) = \begin{bmatrix} 2\dot{\mathbf{x}}\_1 & 2\dot{\mathbf{x}}\_2 \end{bmatrix} \begin{bmatrix} -\dot{\mathbf{x}}\_1^5 + \dot{\mathbf{x}}\_2^2\\ \dot{\mathbf{x}}\_1^2 - \frac{1}{2}L\_2^T(\dot{\mathbf{x}}) - \frac{1}{2}(2\dot{\mathbf{x}}\_2) \end{bmatrix} = -2\dot{\mathbf{x}}\_1^6 + 2\dot{\mathbf{x}}\_1\dot{\mathbf{x}}\_2^2 + 2\dot{\mathbf{x}}\_1^2\dot{\mathbf{x}}\_2 - L\_2^T(\dot{\mathbf{x}})\dot{\mathbf{x}}\_2 - 2\dot{\mathbf{x}}\_2^2 < 0$$
 
$$\text{Select } L\_2(\dot{\mathbf{x}}) = 2\left(\dot{\mathbf{x}}\_1\dot{\mathbf{x}}\_2 + \dot{\mathbf{x}}\_1^2\right), \text{ we have}$$

$$\dot{V}(\mathbf{x}) = \begin{bmatrix} 2\dot{\mathbf{x}}\_1 & 2\dot{\mathbf{x}}\_2 \end{bmatrix} \begin{bmatrix} -\dot{\mathbf{x}}\_1^5 + \dot{\mathbf{x}}\_2^2\\ \dot{\mathbf{x}}\_1^2 - \frac{1}{2}L\_2^T(\dot{\mathbf{x}}) - \frac{1}{2}(2\dot{\mathbf{x}}\_2) \end{bmatrix} = -2\dot{\mathbf{x}}\_1^6 + 2\dot{\mathbf{x}}\_1\dot{\mathbf{x}}\_2^2 + 2\dot{\mathbf{x}}\_1^2\dot{\mathbf{x}}\_2 - L\_2^T(\dot{\mathbf{x}})\dot{\mathbf{x}}\_2 - 2\dot{\mathbf{x}}\_2^2 < 0\\ \dot{V}(\mathbf{x}) = -2\dot{\mathbf{x}}\_1^6 - 2\dot{\mathbf{x}}\_2^2 < 0$$

*Therefore, the closed loop system is stable and the corresponding inverse optimal control law φ* . *x is obtained using (34) as*

$$\phi(\dot{\mathbf{x}}) = \frac{-1}{2} \boldsymbol{R}^{-1}(\dot{\mathbf{x}}) \left[ L\_2^T(\dot{\mathbf{x}}) + \boldsymbol{g}^T(\dot{\mathbf{x}}) \boldsymbol{P} \dot{\mathbf{x}} \right] = -\dot{\boldsymbol{x}}\_1 \dot{\mathbf{x}}\_2 - \dot{\boldsymbol{x}}\_1^2 - \dot{\mathbf{x}}\_2.$$

*Next, using (38) obtains*

$$L\_1(\dot{\mathbf{x}}) = \phi^T(\dot{\mathbf{x}}) \mathbb{R}(\dot{\mathbf{x}}) \phi(\dot{\mathbf{x}}) - \dot{\mathbf{x}}^T P f(\dot{\mathbf{x}}) = \left(\dot{\mathbf{x}}\_1 \dot{\mathbf{x}}\_2 + \dot{\mathbf{x}}\_1^2 + \dot{\mathbf{x}}\_2\right)^2 - 2\dot{\mathbf{x}}\_1 \left(-\dot{\mathbf{x}}\_1^5 + \dot{\mathbf{x}}\_2^2\right) - 2\dot{\mathbf{x}}\_2 \dot{\mathbf{x}}\_1^2.$$

*Consequently, using (30) obtains the performance integrand in (28) as*

$$\begin{aligned} \mathbf{L}\left(\dot{\mathbf{x}},\mu\right) &= L\_1\left(\dot{\mathbf{x}}\right) + L\_2\left(\dot{\mathbf{x}}\right)\mu + \mathbf{u}^T \mathbf{R}\left(\dot{\mathbf{x}}\right)\mu \\ &= \left(\dot{\mathbf{x}}\_1 \dot{\mathbf{x}}\_2 + \dot{\mathbf{x}}\_1^2 + \dot{\mathbf{x}}\_2\right)^2 + 2\dot{\mathbf{x}}\_1^6 - 2\dot{\mathbf{x}}\_1 \dot{\mathbf{x}}\_2^2 - 2\dot{\mathbf{x}}\_2 \dot{\mathbf{x}}\_1^2 + 2\left(\dot{\mathbf{x}}\_1 \dot{\mathbf{x}}\_2 + \dot{\mathbf{x}}\_1^2\right)\mu + \mathbf{u}^T \mathbf{u}. \end{aligned}$$

*Furthermore, it can be shown that*

$$H(\dot{\mathbf{x}}, \boldsymbol{\phi}(\dot{\mathbf{x}})) = L(\dot{\mathbf{x}}, \boldsymbol{\phi}(\dot{\mathbf{x}})) + \dot{V}(\mathbf{x}) = \mathbf{0}.$$

*(b) To emphasize the possibility of constructing a stable closed loop system in SDS form with state feedback control, we consider*

$$\mathbf{x} = \begin{bmatrix} \mathbf{x}\_1 \\ \mathbf{x}\_2 \end{bmatrix} = \begin{bmatrix} -\dot{\mathbf{x}}\_1 + \dot{\mathbf{x}}\_2 \\ \dot{\mathbf{x}}\_1 + \dot{\mathbf{x}}\_2^3 \end{bmatrix} + \begin{bmatrix} 0 \\ 1 \end{bmatrix} \boldsymbol{u} = f(\dot{\mathbf{x}}) + \mathbf{g}(\dot{\mathbf{x}}) \mathbf{u} \tag{82}$$

*x*2

*Please note that the spirit of handling nonlinear control in SDS system is to construct a stable closed loop SDS system with feedback control regardless of using state derivative feedback or state feedback. In this example, if we use state feedback control u* = 2*x*2*, the closed loop SDS system becomes* <sup>−</sup> . *<sup>x</sup>*<sup>1</sup> <sup>+</sup> .

− . *<sup>x</sup>*1<sup>−</sup> . *x* 3 2

,

*If*

*P* = 2 0 0 2 

*x* =

 *x*<sup>1</sup> *x*2 =

*we have*

$$\dot{V}(\mathbf{x}) = \dot{\mathbf{x}}^T P \mathbf{x} = \begin{bmatrix} 2\dot{\mathbf{x}}\_1 & 2\dot{\mathbf{x}}\_2 \end{bmatrix} \begin{bmatrix} -\dot{\mathbf{x}}\_1 + \dot{\mathbf{x}}\_2 \\ -\dot{\mathbf{x}}\_1 - \dot{\mathbf{x}}\_2^3 \end{bmatrix} = -2\dot{\mathbf{x}}\_1^2 - 2\dot{\mathbf{x}}\_2^4 < 0.$$

*Therefore, closed loop SDS system is stable.*

*Similarly, it is also possible to apply state derivative feedback to stabilize a nonlinear system in state space form. For example, for the following nonlinear state space system*

$$
\dot{\mathbf{x}} = \begin{bmatrix} \dot{\mathbf{x}}\_1 \\ \dot{\mathbf{x}}\_2 \end{bmatrix} = \begin{bmatrix} -\mathbf{x}\_1 + \mathbf{x}\_2 \\ \mathbf{x}\_1 + \mathbf{x}\_2^3 \end{bmatrix} + \begin{bmatrix} 0 \\ 1 \end{bmatrix} \boldsymbol{\mu}
$$

*If we apply the following state derivative feedback control u* = 2 . *x*2*, the closed loop SDS system becomes*

$$
\dot{\boldsymbol{x}} = \begin{bmatrix}
\dot{\boldsymbol{\pi}}\_1 \\
\dot{\boldsymbol{\pi}}\_2
\end{bmatrix} = \begin{bmatrix}
\end{bmatrix}.
$$

*and also select the same V*(*x*)*, it follows*

$$\dot{V}(\mathbf{x}) = \mathbf{x}^T P \dot{\mathbf{x}} = \begin{bmatrix} 2\mathbf{x}\_1 & 2\mathbf{x}\_2 \end{bmatrix} \begin{bmatrix} -\mathbf{x}\_1 + \mathbf{x}\_2 \\ -\mathbf{x}\_1 - \mathbf{x}\_2^3 \end{bmatrix} = -2\mathbf{x}\_1^2 - 2\mathbf{x}\_2^4 < 0$$

*Therefore, closed loop state space system is stable.*

**Example 2.** *A popular actuator widely used in control systems is DC motor. The DC motor model in [38] is modified and adapted for this example. Figure 1 shows the free-body diagram of the rotor and the equivalent circuit of the armature. As seen in the figure, a small inductor L1 externally connected in series with armature circuit of a DC motor serves as the only sensor of the control system and we assume that tachometer is not installed to measure the rotational speed* . *θ of the shaft. The voltage of L1 is measured and used in feedback controller design. Unlike the traditional DC motor controls which apply state related feedback of angular velocity or current, the inductor's voltage* (L1 *<sup>d</sup>*<sup>i</sup> *dt*) *is state derivative related measurement feedback of current derivative which is well suitable to apply the proposed IOC algorithm based on state derivative feedback. Furthermore, inductor's average power is zero so it does not damage the armature circuit. Therefore, the controller can save implementation cost and avoid power lose.*

**Figure 1.** The rotor free-body and the electric equivalent armature circuit diagram.

In this example, torque tracking controller and rotational speed tracking controller are constructed based on the design algorithm of inverse optimal control for affine SDS Systems with state derivative related feedback. This example is used to illustrate the design method in Section 2.3 for linear time invariant SDS systems, namely RSS systems.

From the above figure, according to Newton's second law and Kirchhoff's voltage law, we can get the following governing Equations.

$$\mathbf{J}\ddot{\boldsymbol{\theta}} + b\dot{\boldsymbol{\theta}} = K\mathbf{i}$$

$$(\mathbf{L} + \mathbf{L}1)\frac{d\mathbf{i}}{dt} + \mathbf{i}R = \mathbf{v} - K\dot{\boldsymbol{\theta}}$$

where i is armature circuit's current, *θ* is rotor's rotational angle, R is electric resistance, L is electric inductance, J is rotor's moment of inertia, *b* is motor viscous friction constant, L1 is the external inductor sensor connected in series with armature circuit, and *K* represents both electromotive force constant and motor torque constant in SI unit.

For simulation purpose, the DC motor's physical parameters in this example are given as R:1 Ω, L: 0.49 H, J: 0.01 *kg*.*m*2, L1: 0.01 H and *K* : 0.01 *V*/*rad*/*sec* for electromotive force constant and 0.01 *N*.*m*/*Amp* for motor torque constant.

Defining state vector as *x* = . *θ* i , using the governing equations, one can obtain the following SDS system.

$$\mathbf{x} = \begin{bmatrix} \dot{\theta} \\ \mathbf{i} \end{bmatrix} = \frac{1}{\mathbf{R}b + \mathbf{K}^2} \begin{bmatrix} -\mathbf{R} \mathbf{J} & -\mathbf{K}(\mathbf{L} + \mathbf{L}\mathbf{1}) \\ \mathbf{K} \mathbf{J} & -(\mathbf{L} + \mathbf{L}\mathbf{1}) \end{bmatrix} \begin{bmatrix} \ddot{\theta} \\ \frac{d\mathbf{i}}{d\mathbf{l}} \end{bmatrix} + \frac{1}{\mathbf{R}b + \mathbf{K}^2} \begin{bmatrix} \mathbf{K} \\ b \end{bmatrix} \mathbf{v} \tag{83}$$

Substituting the physical parameters of the DC motor into above SDS system, one obtains

$$\mathbf{x} = \begin{bmatrix} \dot{\theta} \\ \dot{\mathbf{i}} \end{bmatrix} = \begin{bmatrix} -0.0999 & -0.05 \\ 0.001 & -4.995 \end{bmatrix} \begin{bmatrix} \ddot{\theta} \\ \frac{d\dot{\mathbf{i}}}{dt} \end{bmatrix} + \begin{bmatrix} 0.0999 \\ 0.9990 \end{bmatrix} \mathbf{v} = f\dot{\mathbf{x}} + g\mathbf{u} \tag{84}$$

Since the L1 inductor's voltage is state derivative related measurement, the measurement of the system is given as

$$\mathbf{y} = \begin{bmatrix} \ 0 & \mathbf{L}1 \ \end{bmatrix} \begin{bmatrix} \ \ddot{\theta} \\ \ \frac{d\dot{\mathbf{i}}}{dt} \end{bmatrix} = \begin{bmatrix} \ 0 & \mathbf{0}.01 \ \end{bmatrix} \begin{bmatrix} \ \ddot{\theta} \\ \ \frac{d\dot{\mathbf{i}}}{dt} \end{bmatrix} = \mathbf{C}\dot{\mathbf{x}} \tag{85}$$

The measurement feedback control law is given as

$$u = -ky = -kC\dot{x}$$

where *k* is the measurement feedback gain.

If the rotational speed of the shaft . *θ* needs to track a reference command *r*0, the performance output equation is given as follows.

$$z = \begin{bmatrix} 1 & 0 \ \end{bmatrix} \begin{bmatrix} \dot{\theta} \\ \ \dot{\mathbf{i}} \end{bmatrix} = H\_s \mathbf{x} \tag{86}$$

If the torque *T* = *K*i needs to track a reference command *r*0, the performance output equation is given as follows.

$$z = \begin{bmatrix} \ 0 & K \ \end{bmatrix} \begin{bmatrix} \dot{\theta} \\ \ \dot{\mathbf{i}} \end{bmatrix} = \begin{bmatrix} \ 0 & 0.01 \ \end{bmatrix} \begin{bmatrix} \dot{\theta} \\ \ \dot{\mathbf{i}} \end{bmatrix} = H\_l \mathbf{x} \tag{87}$$

From (83) to (84), the DC motor model is linear and time invariant. Therefore, the SDS system is also a RSS system as mentioned in introduction section. Therefore, the open loop system poles: −9.9975 and −2.0025 are the reciprocals of the eigenvalues of matrix *f* in the system. Although the open loop RSS system is stable, its tracking performance can be further improved.

To carry out tracking control, the closed loop system should be stable.

First, a symmetric and positive definite matrix *P* serving as the design parameter for Lyapunov function *V*(*x*) in (11) is selected as follows.

$$P = \left[\begin{array}{cc} 1 & 0\\ 0 & 0.5 \end{array}\right].$$

Followed by selecting another design parameter *R* . *x* as

$$R
\left(\dot{x}\right) = 0.0004.1$$

To ensure the closed loop system is stable, based on (37), the *L*<sup>2</sup> . *x* is selected as

$$L\_2(\dot{\mathbf{x}}) = \left[ \begin{array}{cc} \ddot{\boldsymbol{\theta}} & \frac{d\mathbf{i}}{d\boldsymbol{\theta}} \end{array} \right] \begin{bmatrix} -0.0999 \\ -0.4999 \end{bmatrix}.$$

Using (34), the inverse optimal law *φ* . *x* is obtained as

$$\phi(\dot{\mathbf{x}}) = 0.497976 \frac{d\mathbf{i}}{dt} = -k\mathbf{C}\dot{\mathbf{x}} = -k \begin{bmatrix} 0 & 0.01 \ \end{bmatrix} \begin{bmatrix} \ddot{\theta} \\ \frac{d\dot{\mathbf{i}}}{dt} \end{bmatrix} = -0.01k\frac{d\dot{\mathbf{i}}}{dt}.$$

Therefore, we obtain the measurement feedback gain *k* in (85) as

$$k = -49.7976...$$

Consequently, use (38) to obtain *L*<sup>1</sup> . *x* as

$$L\_1(\dot{\mathbf{x}}) = \begin{bmatrix} \ddot{\theta} & \frac{d\dot{\mathbf{u}}}{dt} \end{bmatrix} \begin{bmatrix} 0.0999 & 0.0500 \\ -0.0005 & 0.2498 \end{bmatrix} \begin{bmatrix} \ddot{\theta} \\ \frac{d\dot{\mathbf{u}}}{dt} \end{bmatrix}$$

and use (41) to obtain the performance integrand *L*( . *x*, *u*).

Applying the obtained control law with feedback gain, the close loop system poles have been moved to better locations at −10.0102 and −493.9479. We use the following tracking controller to illustrate the improvement of closed loop system.

$$u = -ky + Nr\_0 = -kC\dot{x} + Nr\_0$$

where *r*<sup>0</sup> is given reference command vector to be tracked by performance output and *N* is a feedforward gain to be designed. .

The steady state derivative is zero ( *x*(∞) = 0) when the RSS closed loop system is stable. In that case, the steady state is

$$\overline{\mathbf{x}} = (f - \mathbf{g}k\mathbf{C})\dot{\overline{\mathbf{x}}} + \mathbf{g}Nr\_0 = \mathbf{g}Nr\_0.$$

If the performance output is *z* = *Hx*, to have zero tracking error of steady state, let

$$\overline{\varepsilon}(\infty) = r\_0 - H\overline{\infty} = r\_0 - H\text{g}\\Nr\_0 = (I - H\text{g}N)r\_0 = 0.1$$

Consequently, the feedforward gain *N* is obtained as

$$N = (H\mathfrak{g})\_{right}{}^{-1}\tag{88}$$

where (*Hg*) <sup>−</sup><sup>1</sup> *right* is the right inverse of matrix *Hg*.

If *Hg* is a full rank matrix with the size of *m* × *n* and *m* ≤ *n*, we have

$$\left(\left(\text{Hg}\right)\_{right}^{-1} = \left(\text{Hg}\right)^{T}\left(\text{Hg}\left(\text{Hg}\right)^{T}\right)^{-1}$$

The feedforward gain *N* and feedback gain *k* can be designed separately because they are independent of each other.

Therefore, according to (86), for rotational speed tracking, we have feedforward gain *Ns* as

$$N\_s = (H\_{\text{s}}\text{g})\_{right}{}^{-1} = 10.01$$

Similarly, for motor torque tracking, from (87) we have feedforward gain *Nt* as

$$N\_{\rm f} = (H\_{\rm f}\mathcal{g})\_{right}{}^{-1} = 100.1$$

Since the open loop system is stable, we can give the reference command for system to track. As seen in Figure 2, there are obvious attenuation and phase lag for tracking a sin *<sup>t</sup>* reference command of rotational speed . *θ*.

Furthermore, as seen in Figure 3, both attenuation and phase lag are large for tracking a sin 50*t* waveform of reference command of torque *K*i.

**Figure 3.** Open loop tracking result of torque *K*i with reference command sin 50*t*.

Therefore, tracking performance should be improved. Applying the obtained inverse optimal law, as seen in Figure 4, rotational speed . *θ* can better track sin *t* reference command. There is no attenuation and phase lag is only 5.72◦ .

**Figure 4.** Closed loop tracking result of rotational speed . *θ* with reference command sin *t*.

Applying the obtained inverse optimal law, as seen in Figure 5, torque *K*i can track sin 50*t* reference command much better. There is no attenuation and phase lag is only 5.72◦ .

**Figure 5.** Closed loop tracking result of torque *K*i with reference command sin 50*t*.

Therefore, the control law works well to improve the tracking performance.

**Example 3.** *The following circuit in Figure 6 is a typical singular system with impulse mode from [39] with C = 1. It is unstable and has pole at infinity. This example is used to illustrate the design approach of Inverse Optimal Control for Affine SDS Systems with L*<sup>2</sup> *Disturbance in Section 2.4*.

**Figure 6.** Singular circuit with impulse mode.

$$E\dot{\mathbf{x}} = \begin{bmatrix} 1 & 0\\ 0 & 0 \end{bmatrix} \dot{\mathbf{x}} = \begin{bmatrix} 0 & 1\\ 1 & 0 \end{bmatrix} \mathbf{x} + \begin{bmatrix} 0\\ 1 \end{bmatrix} u = F\mathbf{x} + N\mathbf{u}, \ \mathbf{x} = \begin{bmatrix} v\_c\\ i\_E \end{bmatrix} = \begin{bmatrix} \boldsymbol{\chi}\_1\\ \boldsymbol{\chi}\_2 \end{bmatrix}$$

Since matrix *F* is invertible, the singular system with impulse mode can be expressed in the following SDS system form.

$$\mathbf{x} = F^{-1}E\dot{\mathbf{x}} - F^{-1}\mathbf{N}u = f\dot{\mathbf{x}} + \mathbf{g}u \tag{89}$$

For verifying the proposed algorithm, external disturbance *ω* is added to the system as follows.

$$\begin{aligned} \mathbf{x} &= \begin{bmatrix} 0 & 0 \\ 2 & 0 \end{bmatrix} \dot{\mathbf{x}} + \begin{bmatrix} -1 \\ 0 \end{bmatrix} \boldsymbol{\mu} + \begin{bmatrix} 0.1 \\ -0.1 \end{bmatrix} \boldsymbol{\omega} = f \dot{\mathbf{x}} + g \boldsymbol{\mu} + f\_1 \boldsymbol{\omega} \\\ \mathbf{z} &= \begin{bmatrix} 2 & 1 \end{bmatrix} \dot{\mathbf{x}} + 0.8 \boldsymbol{\mu} = \dot{h} \dot{\mathbf{x}} + f \boldsymbol{\mu} \\\ \boldsymbol{\mu} &= -\mathbf{K}\_{\infty} \dot{\mathbf{x}}\_{\prime} \ \boldsymbol{\gamma} = 0.6 \end{aligned}$$

First, a symmetric and positive definite matrix *P* serving as the design parameter for Lyapunov function *V*(*x*) in (11) is selected as follows.

$$P = \left[\begin{array}{cc} 1 & 0\\ 0 & 1 \end{array}\right].$$

Followed by selecting another design parameter *R* . *x* as

$$\mathcal{R}(\dot{\mathfrak{x}}) = 1.$$

Consequently,

$$
\Gamma(\dot{\mathbf{x}},u) = \dot{\mathbf{x}}^T \begin{bmatrix} 4.0069 & 1.9931 \\ 1.9931 & 1.0069 \end{bmatrix} \dot{\mathbf{x}} + \dot{\mathbf{x}}^T \begin{bmatrix} 3.2 \\ 1.6 \end{bmatrix} u + 0.64u^T u
$$

To ensure the closed loop system is stable, based on (64), the *L*<sup>2</sup> . *x* is selected as

$$L\_2(\dot{\mathbf{x}}) = \begin{bmatrix} \dot{\mathbf{x}}\_1 & \dot{\mathbf{x}}\_2 \end{bmatrix} \begin{bmatrix} -9.9759 \\ -4.9702 \end{bmatrix} \tag{90}$$

Then using (62) yields

$$
\phi(\dot{x}) = \begin{bmatrix} 2.3707 & 1.0275 \end{bmatrix} \dot{x} = -K\_{\infty}\dot{x} \tag{91}
$$

Therefore, the obtained full state derivative feedback gain is

$$K\_{\infty} = \begin{bmatrix} \ -2.3707 & -1.0275 \ \end{bmatrix}.$$

Applying the control law, the closed loop system poles locate at −0.5768 ± 0.3923i. Since their real parts are all negative, the closed loop system are stable.

Applying (65), we have

$$L\_1(\dot{\mathbf{x}}) = \dot{\mathbf{x}}^T \begin{bmatrix} 2.2102 & 2.0018\\ 0.0018 & 0.7245 \end{bmatrix} \dot{\mathbf{x}}$$

Furthermore, applying (91) obtains

$$
\Gamma \left( \dot{\mathbf{x}}, \phi \left( \dot{\mathbf{x}} \right) \right) = \dot{\mathbf{x}}^T \begin{bmatrix} 15.1901 & 7.0926 \\ 7.0926 & 3.3266 \end{bmatrix} \dot{\mathbf{x}} \tag{92}
$$

Substituting (90) and (92) into (64) yields

$$-12.5019x^T x < 0$$

It proved that *L*<sup>2</sup> . *x* is properly selected.

In addition, this SDS system is actually controllable with state derivative feedback control. For example, if we want to assign the closed loop poles at −2 and −4 using full state derivative feedback control law *<sup>u</sup>* <sup>=</sup> <sup>−</sup>*<sup>K</sup>* . *x*, the closed loop SDS system becomes

$$\mathbf{x} = f\dot{\mathbf{x}} + \mathbf{g}u = (f - \mathbf{g}K)\dot{\mathbf{x}}.$$

The gain *K* should be designed such that matrix (*f* − *gK*) has eigenvalues at −0.5 and −0.25 because they are the reciprocals of −2 and −4, respectively. In this case, using Matlab command *place*, one can easily find *K* = −0.7500 −0.0625 . Therefore, using state derivative feedback control, it is possible to assign all closed loop poles for some singular systems with impulse mode if they can be expressed in a controllable SDS system form. However, using state feedback control, it is impossible to assign all closed loop poles for any singular systems. It is only possible to stabilize some of the singular systems with state feedback control.

**Example 4.** *It is interesting to compare the proposed inverse optimal control (IOC) in Section 2.4 with sliding mode control (SMC) design for SDS system with matched disturbance because they are both developed based on Lyapunov stability theorem. Consider the following unstable SDS system with matched disturbance given in [20].*

$$\dot{x}(t) = f\dot{x}(t) + gu(t) + f\_1\omega(t) = f\dot{x} + g(u + \omega(t)), \; t \ge 0 \tag{93}$$

where *f* = ⎡ ⎣ 1 −0.5 0.25 0 0.5 −0.25 0 0 0.5 ⎤ <sup>⎦</sup>, *<sup>g</sup>* <sup>=</sup> *<sup>J</sup>*<sup>1</sup> <sup>=</sup> ⎡ ⎣ −0.25 0.25 −0.5 ⎤ <sup>⎦</sup> (for matched disturbance), *x*(*t*) = ⎡ ⎣ x1 x2 x3 ⎤ <sup>⎦</sup>, and *<sup>ω</sup>*(*t*) = 0.2 sin 0.3333*t*, with the following full state derivative perfor-

mance variables.

$$z = h\dot{x}(t) + f\mu(t) = \dot{x}(t)$$

where *h* = *I*3×<sup>3</sup> (identity matrix) and *J* = [0 00] T.

(a) For sliding mode control (SMC) [20], the sliding surface is selected as

$$s = \begin{bmatrix} \ -84 & -180 & -50 \ \end{bmatrix} \\ x = \mathbf{C}x$$

Consequently, we have *Cg* = 1. The ideal controller is given as

$$u(t) := -\left(\mathbb{C}g\right)^{-1}\mathbb{C}f\dot{\mathbf{x}}(t) - \left(\mathbb{C}g\right)^{-1}(\gamma + a)\operatorname{sign}\left(\dot{s}(t)\right) \tag{94}$$

where *γ* > *Cgω*(*t*) = 0.2 in this example for countermeasure the matched disturbance and *α* > 0 should be selected to ensure that the approaching condition can happen.

In [20], it has been proven that applying the ideal controller in (94), the following approaching condition happens.

$$|s^T(t) \cdot \dot{s}(t) < -\alpha \cdot ||\dot{s}(t)|| < 0$$

To avoid or reduce "chattering phenomenon" due to *sign* . *s*(*t*) switching function in (94), the following modified controller is used.

$$\mu(t) := -(\mathbb{C}\mathfrak{g})^{-1}\mathbb{C}f\dot{\mathfrak{x}}(t) - (\mathbb{C}\mathfrak{g})^{-1}(\gamma + a)\text{sat}\left(\dot{\mathfrak{s}}(t), \varepsilon\right) \tag{95}$$

where *"sat"* is a saturation function to smoothly handle the switching as follows.

$$sat(\dot{s}, \varepsilon) = \begin{cases} 1 & \dot{s} > \varepsilon \\ \frac{\dot{s}}{\varepsilon} & \left| \dot{s} \right| \le \varepsilon \\ -1 & \dot{s} < -\varepsilon \end{cases} = \begin{cases} sign\left(\dot{s}\right) & \left| \dot{s}\right| > \varepsilon \\ \frac{\dot{s}}{\varepsilon} & \left| \dot{s} \right| \le \varepsilon \end{cases}$$

Here *<sup>ε</sup>* is a small positive value as the bound of the differential sliding surface . *s* such that

$$\left|\dot{s}\right| \le \varepsilon$$

In this example *ε* = 0.5,*γ* = 0.4 and *α* = 0.05 are used in the simulation. When sliding surface is selected, we need to tune design parameters of *ε*, *γ* and *α* to have a SMC controller with good enough performance.

(b) For applying the inverse optimal control (IOC) design method in Section 2.4, we can follow the same design steps and use the same notations in example 3. In this example, we have

 $h = I\_{\ $ \times \$ } \text{ (identity matrix) and } J = [0 \ 0 \ 0]^{\top}$ 

First, we select the following design parameters as

 $\gamma = 0.6$ ,  $P = I\_{3 \times 3}$  (identity matrix) and  $R\left(\dot{x}\right) = 1$ .

To ensure the closed loop system is stable, based on (64), the *L*<sup>2</sup> . *x* is then selected as

$$L\_2(\dot{\mathbf{x}}) = \begin{bmatrix} \dot{\mathbf{x}}1 & \dot{\mathbf{x}}2 & \dot{\mathbf{x}}3 \end{bmatrix} \begin{bmatrix} -79.75 \\ -76.25 \\ -17.50 \end{bmatrix}$$

Consequently, using (62) yields

$$
\phi(\dot{\mathbf{x}}) = \begin{bmatrix} 40.00 & 38.00 & 9.00 \end{bmatrix} \dot{\mathbf{x}} = -K\_{\infty} \dot{\mathbf{x}}
$$

Therefore, the obtained full state derivative feedback gain is

$$K\_{\infty} = \begin{bmatrix} \ -40.00 & -38.00 & \ -9.00 & \ \end{bmatrix}.$$

Applying this control law, the closed loop system poles locate at −0.5 ± 0.5i and −1. Since their real parts are all negative, the closed loop system are stable.

The state responses and control effort of both SMC and IOC are plotted in the following figures for comparisons.

The unstable system in this example can be properly controlled by both sliding mode control (SMC) and inverse optimal control (IOC) to have bounded closed loop state responses. As shown in Figures 7–10, for matched disturbance case in this example, when SMC is properly designed, its performance could be better than that of IOC because SMC can have smaller state responses in Figures 7–9 by applying smaller control effort in Figure 10.

**Figure 7.** State ×1 responses.

**Figure 8.** State ×2 responses.

**Figure 9.** State ×3 responses.

**Figure 10.** Control u responses.

#### **4. Discussion**

The proposed design methods are inspired by other previous work [26,27,31] in inverse optimal control in state space system with state feedback design approach. The results show that the use of state derivative feedback for control design in an SDS system is as simple as the use of state feedback for control design in a state space system.

Therefore, with the understanding of SDS systems, many design tools developed in state space system with state feedback can be modified and adapted for control designs in SDS system with state derivative feedback. Since the Hamilton–Jacobi–Bellman equation is not always solvable to obtain the control Lyapunov function, the existence of optimal control solution is not always guaranteed. On the other hand, if we can solve for a control Lyapunov function from HJB equation, we can find from it a control law that achieves the minimum of performance functional and the resulting closed loop system has a unique solution forward in time. Please refer to Chapter 6 in [27] for details, the descriptions and formula derivations are analogous to those for SDS system case. Since the control Lyapunov function is predefined in inverse optimal control design process and no need to solve HJB equation, it is very suitable to find stabilizing control laws for unstable nonlinear systems.

Regarding the examples to verify the proposed methods, Example 1 demonstrates the design steps for the design approach of inverse optimal control for affine SDS systems with state derivative related feedback. In the same example, it also suggests that people should free their mindset in control designs. No matter the system in state space form or SDS form, the possibilities of applying state feedback or state derivative feedback should be both checked so that the controller can be simple while perform well. The DC motor tracking control without tachometer in Example 2 discusses the possibility of using alternative measurement of inductor voltage in DC motor control based on the obtained state derivative feedback algorithms. This idea seems promising, because the method we propose can construct a cheap and compact controller without the need for an expensive tachometer. Furthermore, unlike resistor sensor, the average power of inductor sensor is zero, it will not damage the armature circuit or cause power loss. Example 3 is a singular system with impulse mode. This is a very challenging design problem in previous researches using state feedback in control design of the generalized state space system. Since it can be expressed in SDS system, the design approach is straightforward. Therefore, some systems that are difficult to control through state feedback can be controlled through state derivative feedback in SDS system form.

Since both sliding mode control (SMC) [19,20] and the inverse optimal control in SDS system form are developed based on Lyapunov stability theorem and their performances are dependent on tuning their design parameters, we compare them in terms of conditions to use and limitations. SMC methods in [19,20] have low sensitivity to parameter uncertainties, can work with matched uncertainties as well as matched disturbance that enter into control inputs and apply discontinuous switching control law to ensure the finite time convergence. Those are their advantages. However, SMC methods in [19,20] may suffer from chattering phenomenon and when uncertainties and disturbance are not matched ones, the performance could be downgraded. On the contrary, the inverse optimal control method in Section 2.4 can handle bounded disturbance which are not from control inputs. As shown in Example 4, for system with matched disturbance, the SMC controller could use smaller control effort to obtain smaller state responses than IOC controller. However, from the implementation point of view, the structure of IOC controller is simpler than that of SMC controller and consequently the implementation cost of IOC controller could be cheaper. The inverse optimal control methods in this paper are suitable for controlling the system with precise parameters, such as the DC motor used for tracking control in Example 2. Therefore, designers can make tradeoff between IOC and SMC in terms of performance and cost.

Regarding the future works, for implementation of the DC motor application in example 2, as seen in Figure 1, there is large electric inductance L in armature circuit in the model of DC motor. No matter what kind of sensor we use, one potential problem that is common in DC motor control is inductive kick (kickback) phenomenon or so called Ldi/dt voltages [39]. Since the windings of the DC motor will produce current conversion during commutation, the current conversion will cause the inductive kickback that disturbs both the voltage and current of armature circuit as shown in Figure 11. Therefore the measured voltage of L1 sensor in Figure 1 will also be disturbed.

**Figure 11.** Disturbances due to inductive kickback.

Conventionally, in implementation, various absorption circuits of inductive load kickback [40] can be used as the countermeasure for disturbance caused by inductive kick. Other than applying absorption circuits, we are considering another solution of eliminating disturbance of L1 sensor voltage due to inductive kick in our future research. The basic idea is as follows. Since the inductive kick is formed by periodic commutation, it has an average of 0 characteristics, if the T is the time of every commutator segment passes through a brush, selecting the signal window of L1 sensor voltage with a period as a multiple of T and calculating the average voltage value of window, the L1 voltage disturbance caused by the inductive kick of commutators could be considerably decreased. The control voltage is then generated by feedback of L1 voltage with reduced disturbance. In addition, since this disturbance of L1 sensor voltage is through control input channel, it can be considered as matched disturbance. Hence, we will also consider to apply sliding mode control with state derivative measurement feedback to control it in future.

In this paper, we have proven that the inverse optimal control can be carried out in SDS system form with state derivative feedback. In future, more challenging problems such as stochastic systems can be explored. If a system has randomness associated with it, it is called a stochastic system and does not always produce the same output for a given input. Stochastic systems exist in many applications such as communication systems, markets, social systems, and epidemiology. Optimal control [41,42] and inverse optimal control [42,43] for stochastic systems in state space system form have been solved with stochastic Hamilton–Jacobi–Bellman equation to obtain state related feedback control laws. In future, for people who want to develop inverse optimal control for stochastic systems with state derivative feedback, it is highly recommended to first study [42] because the design approaches in [42] and this paper are both built on [27].

#### **5. Conclusions**

In this paper we have discussed about how SDS system with state derivative feedback can be supplement of state space system with state feedback in control designs. Followed by developing inverse optimal control methods in SDS systems with solely state derivative feedback. As far as the authors know, no similar results have been reported. Inverse optimal control can construct a stable closed-loop system while nonlinear optimal control may not have exact solution. Hence, inverse optimal control should be collaboratively used together with optimal control for designs. The proposed methods are very suitable to find stabilizing control laws for unstable nonlinear systems. The correctness of proposed methods has been properly verified by numerical examples and simulations. Especially, in the third example, a classic difficult problem in control, namely singular system with impulse mode is fully controllable by state derivative feedback in SDS system form and satisfy the non-expansivity constraint when the system is subjected to disturbance. On the contrary, the same system can only be stabilized by state feedback control. The above is the summary of academic contributions of this paper.

From application points of view, in vibration systems of vehicle dynamics and smart structure, accelerations and velocities are available measurements of state derivative vector. In addition, the inductor voltages in electrical systems are also state derivative related measurement. For those systems, using state derivative feedback design in SDS system form are very likely to have more simple, cheap and compact controllers because integrators or numerical integrations are not needed. Therefore, the idea of connecting a small inductor in series with an armature circuit as the only sensor of a DC motor control system in Example 3 is very promising because average power loss of inductor is zero and no tachometer is needed. DC motors are widely used in many industries and facilities for daily life. For example, in automotive body electronics, the estimated demand for automotive DC motors in body domain was 2 billion units in 2020 [44]. So the proposed design approach of this paper can have a wide range of practical applications.

With understandings and awareness of SDS system form, state derivative feedback and inverse optimal control, designers can solve more control problems and develop more new applications based on their previous knowledge and experience in state feedback designs in state space system without applying too much of advanced mathematics.

## **6. Patents**

Authors, Yuan-Wei Tseng and Rong-Ching Wu hold the following patents related to state derivative feedback control in SDS systems.


**Author Contributions:** Conceptualization, Y.-W.T. and F.-C.L.; methodology, F.-C.L.; software, F.-C.L.; validation, R.-C.W., F.-C.L., and W.-C.C.; formal analysis, Y.-W.T.; investigation, F.-C.L. and C.-S.C.; resources, W.-C.C.; data curation, F.-C.L.; writing—original draft preparation, Y.-W.T.; writing—review and editing, C.-S.C. and R.-C.W.; visualization, F.-C.L.; supervision, Y.-W.T.; project administration, W.-C.C. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


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