*2.2. Model-Based Control Synthesis*

Youla parameterization is a robust control method that leverages closed-loop frequency shaping to attain the desired closed-loop behavior. These closed-loop transfer functions consist of (*Ty*), known as a complementary sensitivity transfer function, sensitivity transfer function *Sy*, and Youla *Y* transfer function (explained below). This method shapes closedloop transfer functions while ensuring internal stability along with disturbance rejection at low frequencies and sensor noise and unmodeled disturbance rejections at high frequencies. This method was selected because of its ease of control design using the model-based approach for designing appropriate low-level controllers based on the developed bond graph models [36].

The central notion in this method is to form a closed-loop transfer function (*Ty*) with a transfer function named Youla (*Y*(*s*)). Multiply Youla by the plant transfer function (*Gp*) to create the desired closed-loop transfer function (Equation (11)). For good tracking performance in steady-state, the magnitude of *Ty*(*s*) should be set to one at low frequencies. To ensure high-frequency noise rejection, *Ty*(*s*) should be small at high frequencies:

$$T\_{\mathcal{Y}}(\mathbf{s}) = \mathcal{Y}(\mathbf{s}) \times G\_{\mathcal{P}}(\mathbf{s}) \tag{11}$$

As a result, we can shape the closed-loop transfer function using Equation (11) (given we meet all the interpolation conditions for ensuring internal stability mentioned below). It should be noted that the Youla transfer function maps the desired reference signal to the actuator effort. With good target following, such as |*Ty*(*s*)| = 1 at low frequency, the Youla transfer function is approximately equal to the inverse of the plant transfer function at low frequency and equal to the controller *Gc*(*s*) transfer function at high frequency. Thus, keeping Youla's magnitude small at high frequencies would reduce actuator effort and minimize the impact of sensor noise on the actuator.

The closed-loop transfer function (*Ty*(*s*)) and the sensitivity transfer function (*Sy*(*s*)) are complementary to each other, as shown by (Equation (12)). Due to this algebraic constraint, the sensitivity transfer function should be small at low frequencies (to reject low-frequency disturbances) and equal to one in magnitude at high frequencies:

$$S\_y(\mathbf{s}) = 1 - T\_y(\mathbf{s}) \tag{12}$$

If *Gp* is stable, the feedback loop would be internally stable if and only if *Y*(*s*) is selected to be a stable transfer function. In this regard, *Yy*(*s*), *Sy*(*s*), *Ty*(*s*), and *Gp* × *Sy* should all be stable to make the feedback loop internally stable. Consequently, to meet these conditions in case of an unstable pole (*αp*) which is repeated n-times in the plant (*Gp*), Equations (13) and (14) define rational interpolation conditions, which must be met to enforce internal stability. If it is a single unstable pole (not repeated), Equation (13) is the only interpolation condition that needs to be satisfied:

$$T\_y(\alpha\_p) = 1,\\ S\_y(\alpha\_p) = 0 \tag{13}$$

$$\frac{d^k T\_y}{ds^k}(\alpha\_p) = 0, \frac{d^k S\_y}{ds^k}(\alpha\_p) = 0, \quad \forall k \in \{1, n\} \tag{14}$$

If there is a repeated non-minimum phase zero (*αz*), zeros in the RHP (Right Half Plane), the interpolation conditions are met by Equations (15) and (16). If the unstable zero is only repeated once, Equation (15) is the only interpolation condition that must be satisfied [36]:

$$S(a\_z) = 1,\\
T(a\_z) = 0\tag{15}$$

$$\frac{d^k S\_y}{ds^k}(\alpha\_z) = 0, \frac{d^k T\_y}{ds^k}(\alpha\_z) = 0, \quad \forall k \in \{1, n\} \tag{16}$$

Once we ensure that the conditions in the Equations (13)–(16) are met, we can acquire the controller using Equation (17):

$$G\_{\mathcal{E}}(\mathbf{s}) = \mathcal{Y}(\mathbf{s}) \times S\_{\mathcal{Y}}(\mathbf{s})^{-1} \tag{17}$$

#### Cascaded Control

Since the brake-by-wire smart actuators are Single Input Multiple Output (SIMO) problems, we consider the cascaded control scheme. Cascaded control enables systems with relatively more nonlinearities to perform better and be more robust. Therefore, the controllers were designed using cascaded control to mitigate different nonlinearities in the brake actuators (e.g., mechanical friction, pressure nonlinearities). Each inner closedloop is an open-loop for their outer loop controller design. The controller design of each

plant is conducted through the Youla parameterization approach, as discussed in the previous section. Figure 3 shows this cascaded control design for EMB and EWB actuators. In the EMB and EWB, for the first loop, motor voltage is input, and motor current is the output. In the second loop, the motor's desired current is input, and the motor's angular velocity is output. Finally, for the outermost loop, the input is the desired motor angular velocity, and the clamping force is the output (the normal force of the brake pad on the wheel). The shaft's current and angular velocity can be measured directly and is readily available, but the clamping force must be estimated or measured with a force sensor.

One important part of the EMB/EWB plant, which is not shown in Figure 3, is the current and voltage saturation. Current is saturated at ±25 A, and voltage is set to saturate at ±42 Volts. Because of these, the controllers might saturate and, therefore, make the plant unstable. A simple gain anti-windup was used to address the current saturation and to mitigate this issue (Figure 4). There could also be another anti-windup for the voltage saturation; however, normally, the voltage does not reach saturation levels if the current saturation has been addressed. Furthermore, adding an extra anti-windup may result in limiting the bandwidth of the closed-loop system. In addition, other anti-windup strategies such as the one in [37] or [38], which is specifically for cascaded controllers, could have been utilized.

In the EHB control design, a SISO controller was designed based on the linearized equations mentioned in Equations (5)–(7). The operating points taken for this linearization are *u*<sup>0</sup> = 0.3 and *qc*<sup>0</sup> = 0.3 × *q*0. *q*<sup>0</sup> is the steady-state value of *qcyl*. In addition, as mentioned before, it is assumed that *ud* = 1 − *ub*. This continuous control law works well when building and dumping the pressure in the cylinder chamber. However, in the case of keeping constant pressure during the steady-state, we might run into the issue of having both build and dump valves open partially at the same time and therefore losing some of the master cylinder's pressure which wastes energy. For example, *ud* = 0.7 and *ub* = 0.3 would hold the constant cylinder pressure, but this is not energy efficient as the pumps keep running. For this reason, a switching logic was added to the continuous Youla controller. This switching statement changes the values of *ub* and *ud* to zero once the clamping force error is within the desired threshold. Otherwise, it passes the same values from the controller to the plant as shown in Figure 5.

**Figure 4.** Anti-windup gain used to compensate for the current saturation.

**Figure 5.** Control scheme for the EHB.

#### *2.3. Optimization*

#### 2.3.1. Linear Optimization: Using Transfer Functions

In this section, a multi-objective optimization scheme for brake-by-wire actuators based on their transfer functions is considered. This optimization considers dynamic responsiveness and the actuator's effort as objective metrics.

After linearizing the plants, we can obtain their transfer functions. We then synthesize the controllers using the Youla parameterization technique discussed in Section 2.2. There are a few assumptions made when designing the controllers during the optimization process. Controllers are designed to create closed-loop transfer functions to be in a certain form. Equations (18), (19), (20), (21a), (21b), (22), (23a) and (23b) describe the form of Plant transfer function (Gp), Youla transfer function (Y), and closed-loop transfer function (T) for the first, second, and the third loop of EMB (EWB follows a similar control design pattern, and EHB follows the same pattern, but it only has one control loop, and therefore, the design choice is similar to the one in Equation (19)). The goal here is to design a closedloop transfer function with the frequency shape of a second-order Butterworth filter and add extra first-order filters whenever necessary. For example, in Equation (21a), the second open-loop transfer function has an integrator that should not be canceled by the Youla transfer function. Hence, a high pass filter was added to the Youla transfer function ( <sup>1</sup> *<sup>s</sup>*+*W*<sup>1</sup> ). *W*<sup>1</sup> is the pole for the filter that is added to the Youla transfer function. This can be chosen in such a way that it does not affect the bandwidth of the closed-loop system.. In this equation, *G <sup>p</sup>*<sup>2</sup> represents the plant transfer function of the second loop without the s in the numerator. Moreover, in the case of Equation (23a), a repeated first-order transfer function was added to the Youla transfer function to make this transfer function proper (*Gp*<sup>3</sup> is a fourth-order transfer function; for simplicity, we choose to use first-order poles. N-th order Butterworth filter could also be used in this case). More details on the design of these controllers are provided in [31]. It should be noted that *ωn*<sup>1</sup> , *ωn*<sup>2</sup> , and *ωn*<sup>3</sup> are chosen for each loop to have a specific bandwidth (*ωn*<sup>1</sup> , *ωn*<sup>2</sup> , and *ωn*<sup>3</sup> are the Butterworth filter's cut-off frequencies for the different added Butterworth filter to Youla transfer functions). In the case of EMB, this is 200 Hz, 10 Hz, and 2 Hz for the first, second, and last loop, respectively. For the EWB, they are chosen to be 500 Hz, 400 Hz, and 2 Hz. Finally, for EHB, it is chosen to be 2 Hz. Therefore, all of the brake-by-wire actuators have the same final closed-loop of 2 Hz for the clamp force loop. This is a deliberate choice to make sure all the brake-by-wire actuators have the same bandwidth (for the clamping force) for the final comparison in terms of energy and responsiveness metrics. The chosen control parameters mentioned here (such as *ω<sup>n</sup>* and *ξ*) will remain the same over the course of all optimizations. This is performed to have fixed control design for the optimization procedure, and the only change will be the physical parameters of the system. Figure 6 shows the Bode magnitude plots of T, S, and Y for this type of control design.

$$G\_{p\_l} = \frac{f\_m s^2 + D\_m s + (N\_s N\_p)^2 K\_{\rm cal}}{(L\_m l\_m) \times s^3 + (R\_m l\_m + L\_m D\_m) \times s^2 + (R\_m D\_m + L\_m (N\_s N\_p)^2 K\_{\rm cal} + K\_t^2) \times s + R\_m (N\_s N\_p)^2 K\_{\rm cal}} \tag{18}$$

$$Y\_1 = \frac{1}{G\_{p\_1}} \times \frac{\omega\_{n\_1}^2}{s^2 + 2 \times \frac{\pi}{5} \times \omega\_{n\_1} \times s + \omega\_{n\_1}^2},\\ T\_1 = \frac{\omega\_{n\_1}^2}{s^2 + 2 \times \frac{\pi}{5} \times \omega\_n \times s + \omega\_{n\_1}^2} \tag{19}$$

$$\mathcal{G}\_{p\_2} = T\_1 \times \mathcal{G}\_{p\_\omega} = \frac{\omega\_{n\_1}^2}{s^2 + 2 \times \frac{\pi}{8} \times \omega\_{\text{n}} \times s + \omega\_{n\_1}^2} \times \frac{\mathcal{K}\_l \times s}{J\_{\text{m}}s^2 + D\_{\text{m}}s + (\mathcal{N}\_s \mathcal{N}\_p)^2 \mathcal{K}\_{\text{cal}}} \tag{20}$$

$$Y\_2 = \frac{1}{G\_{p\_2}'} \times \frac{\omega\_{n\_2}^2}{s^2 + 2 \times \frac{\pi}{s} \times \omega\_{n\_2} \times s + \omega\_{n\_2}^2} \times \frac{1}{s + W\_1} \times (\frac{W\_2}{s + W\_2})^2 \tag{21a}$$

$$T\_2 = \frac{\omega\_{m\_2}^2}{s^2 + 2 \times \xi \times \omega\_{m\_2} \times s + \omega\_{m\_2}^2} \times \frac{s}{s + W\_1} \times (\frac{W\_2}{s + W\_2})^2 \tag{21b}$$

$$G\_{p\_3} = T\_2 \times G\_{p\_F} = \frac{\omega\_{n\_2}^2}{s^2 + 2 \times \frac{\pi}{s} \times \omega\_{n\_2} \times s + \omega\_{n\_2}^2} \times \frac{s}{s + W\_1} \times (\frac{W\_2}{s + W\_2})^2 \times \frac{K\_{\text{cal}} N\_s N\_p}{s} \tag{22}$$

$$Y\_3 = \frac{1}{G\_{p\_3}} \times \frac{\omega\_{n\_3}^2}{s^2 + 2 \times \xi \times \omega\_{n\_3} \times s + \omega\_{n\_3}^2} \times (\frac{W\_3}{s + W\_3})^4 \tag{23a}$$

$$T\_3 = \frac{\omega\_{\mathfrak{n}\_3}^2}{s^2 + 2 \times \xi \times \omega\_{\mathfrak{n}\_3} \times s + \omega\_{\mathfrak{n}\_3}^2} \times (\frac{W\_3}{s + W\_3})^4 \tag{23b}$$

Frequency (Hz) **Figure 6.** An example design of cascaded controllers for the EMB/EWB.

After designing the controllers, we then utilized the aforementioned transfer functions to optimize actuator response and actuator usage. The process of optimization starts with designing the controllers based on the physical parameters of the system and deriving these transfer functions as a function of the physical system parameters (since the parameters change, the *Gp*s change, and therefore, these transfer functions will be different for each set of physical parameters).

The bandwidth of a plant is related to its dynamic response, and therefore, increasing the bandwidth would result in faster system response. The bandwidth of a system/plant is defined as the frequency range where the magnitude of the system gain does not drop below −3 dB. For EMB/EWB, the plant transfer function, which is chosen (denoted as *Gp* here) for calculating the system bandwidth, is from voltage input to the clamping force output. For EHB, the plant transfer function maps *ub* to the clamping force. The bandwidth of the plant is denoted as *Bandwidth*{*Gp*}.

Another factor to consider is the actuator's power usage which is related to the actuator's effort, and hence, it is related to the Youla transfer function, denoted as *Y*(*s*). The Youla transfer function for the overall system in case of cascaded control design would become *Ysys* = *Y*<sup>1</sup> × *Y*<sup>2</sup> × *Y*3. In here, *Ysys* stands for the Youla transfer function from the clamping force reference to the voltage input (or to the *ub* for the EHB), which is the Youla transfer function of the overall control system. It can be shown if the magnitude of *Ty*(*s*) or the gain of closed-loop transfer function is one at low frequencies, the Youla transfer function at low frequencies is inversely related to the plant transfer function *Gp*(*s*) (From Equation (11), if *Ty* = 1, then *Y* = <sup>1</sup> *Gp* ). By increasing the plant gain at low frequency (approximately its DC gain, denoted as *DC*{*Gp*} in this section), *Y*(*s*) will decrease at low frequencies.

Furthermore, we need to lower the overall values of the Youla transfer function magnitude at other important frequencies, especially around the plant bandwidth. This will ensure the reduction of the actuator effort in all possible frequencies. To this end, we can use an H2 norm of this transfer function. H2 norm is related to the output signal energy when the system input is an impulse [36]. Since we are interested in a specific frequency region of the Youla transfer function, a band-pass filter, see Figure 7, is used to emphasize the frequency region of interest. *ω<sup>L</sup>* and *ω<sup>H</sup>* are chosen to be 0.1 × *Bandwidth*{*Gp*} and 1*e*4 × *Bandwidth*{*Gp*}. This will ensure that the Youla transfer function magnitudes at low and mid-range frequencies stay low. The optimization problem is then formulated by combining all these costs as given in Equation (24):

$$\begin{aligned} \text{minimize} \quad & \& f(\mathbf{x}) = \mathbf{a}\_1 \times ||\boldsymbol{\chi}\_{\text{sys}} \times \mathbf{W}\_Y||\_2 + \mathbf{a}\_2 \times \frac{1}{D \mathbb{C} \{\mathbf{G}\_p\}} + \mathbf{a}\_3 \times \frac{1}{Bandwidth \{\mathbf{G}\_p\}}\\ & \text{subject to} \quad & \& \mathbf{c} \in [\mathbf{x}\_{\text{min}}, \mathbf{x}\_{\text{max}}]\_\prime \end{aligned} \tag{24}$$

where *α*1, *α*2, and *α*<sup>3</sup> are tuning parameters, and *x* is the vector of physical parameters of the system that can be changed during the design process (e.g., gear ratios, moments of inertia, and motor's inductance). *xmin* and *xmax* denote the minimum and maximum of the parameter set, respectively. *WY* is the frequency weighting function for *H*<sup>2</sup> norm optimization. It should be noted that each cost in Equation (24) is normalized by its nominal value to ensure the minimization of the three costs is done without bias. A choice of physical parameters for each actuator, their initial and optimized value are given in Table 1.

**Figure 7.** Band-pass filter, *WY*, is used to emphasize specific frequency region of Youla transfer function in the *H*<sup>2</sup> norm optimization.


**Table 1.** Initial and optimized physical parameter values of EMB, EWB, and EHB and the range of the parameters.
