**3. Equivalent Control of Modified Cuk Converter ´**

A cascaded PI+SMC controller structure could be used for ease of implementation to modified Cuk converter as depicted in Figure ´ 5. A simple external voltage controller can generate input current reference, while equivalent controller controls the input current [29,30].

**Figure 5.** Cascaded control of modified Cuk converter. PI: proportional integral; SMC: sliding ´ mode controller.

Although the modified Cuk converter is a third-order nonlinear model, only an input current ´ equation is required to construct an equivalent controller. This is the main advantage of SMC-based equivalent controllers, since the performance is independent of all system dynamics and parameter variations. Input current of the modified Cuk converter in terms of Kirchhoff's voltage law can be ´ written in the following form:

$$\frac{di\_{L1}}{dt} = \frac{1}{L\_1}(V\_i - (1 - \mu)V\_{\mathbb{C}1}) \tag{12}$$

The terms *Vi* and *Vc*<sup>1</sup> are input *C*<sup>1</sup> voltages, and *u* is the switching signal of semiconductor switch as explained below.

$$u(t)\begin{bmatrix} 1 & Switch \rightarrow ON & D\_o \rightarrow OFF \\ 0 & Switch \rightarrow OFF & D\_o \rightarrow ON \end{bmatrix} \tag{13}$$

The external PI controller aims to achieve the reference voltage target, and the output of the PI controller acts as reference current (*i \** ). The internal SMC-based equivalent current controller aims to track current trajectory, and the analytical design procedure is detailed below. The switching surface can be given as [29,30]:

$$
\sigma = i\_{L1} - i^\* \tag{14}
$$

The time derivative of the switching surface is:

$$
\dot{\sigma} = \dot{i}\_{L1}^{\cdot} - \dot{i}^{\ast} \tag{15}
$$

If the input current equation of the modified Cuk converter in (12) is written to derivative of the ´ switching surface in Equation (15), the following equation can be obtained:

$$
\dot{\sigma} = \frac{V\_i}{L\_1} - \frac{1}{L\_1}(1 - \mu)\dot{V}\_{c1} - \dot{i}^\* \tag{16}
$$

If . *σ* is assumed to be zero at steady-state, the equivalent control signal can be generated as given below.

$$
\mu\_{c\eta} = 1 + \frac{L\_1}{V\_{c1}} i^\* - \frac{V\_i}{V\_{c1}} \tag{17}
$$

The switching surface can be simplified as given below, considering . *σ* is zero at steady-state. The continuous function *ueq* will be converted into discontinuous form as follows:

$$
\dot{\sigma} = \mu - u\_{c\eta} \tag{18}
$$

Closed loop control signal from switching surface can be given as:

$$
\hbar \mu = \hbar\_{eq} - \text{K}\sigma \tag{19}
$$

The term K is positive definite control gain, and if (19) is inserted in (18), the switching surface can be written as follows: .

$$
\dot{\sigma} = \dot{\mathfrak{u}}\_{cq} - \mathrm{K}\sigma - \mathfrak{u}\_{cq} \tag{20}
$$

where *u*ˆ*eq* is the estimated equivalent control input. It can be written in steady-state that *u*ˆ*eq* = *ueq*.

$$
\dot{\sigma} = -\mathcal{K}\sigma\tag{21}
$$

Finally, stability and existing conditions for sliding mode control must be clarified [18]. The definition . *σσ* < 0 must be satisfied, and it can be derived from (21) that;

$$\begin{cases} \sigma > 0 & \dot{\sigma} < 0 \\ \sigma < 0 & \dot{\sigma} > 0 \end{cases} \tag{22}$$

Thus, the stability of the sliding surface is satisfied. Controller structure can be constructed by estimating *u*ˆ*eq*. Estimation of the equivalent control can be formed as:

$$\nu = \text{fl}\_{cq} + l\sigma \tag{23}$$

Where the term l is the filter gain of the estimator. It is assumed that *u*ˆ*eq* is constant, and the time derivative of (23) can be written as given below:

$$
\dot{v} = \text{l}\dot{\sigma}\tag{24}
$$

The time derivative of . *σ* in (18) can be inserted into (23) as given below:

$$
\dot{\upsilon} = l \left( \mu - \mu\_{eq} \right) \tag{25}
$$

It can be stated that *u*ˆ*eq* = *ueq* in steady state:

$$
\dot{\boldsymbol{w}} = \boldsymbol{l} \left( \boldsymbol{u} - \boldsymbol{\mathfrak{n}}\_{eq} \right) \tag{26}
$$

If (26) is written in the form of *u*ˆ*eq* = *v* − *lσ*, the following equation can be obtained [29,30]:

$$
\dot{\upsilon} = l(\mu - \upsilon + l\sigma) \tag{27}
$$

Finally, the simple equivalent controller structure in Figure 6 can be obtained from the definitions written above.

**Figure 6.** SMC-based equivalent controller.

As detailed in [25,29,30], Figure 6 shows that a dynamic system can be formed as a series of integrators, and it can be assumed that the output of this system can be estimated by an upper bound of the integral. Finally, a dynamic relay with hysteresis function as given in Figure 7 can be applied to control a signal to generate a sliding surface which oscillates with the magnitude of M.

**Figure 7.** Dynamic relay with hysteresis function.

The main disadvantage of the SMC-based equivalent controller is variable switching frequency (SF), because the magnitude of oscillations in sliding surface is highly dependent on circuit parameters and operating conditions due to the nonlinear behavior of the converter. One of the methods that can constrain the sliding surface to constant SF operation can be a dynamic hysteresis controller [35] which dynamically changes the magnitude of sliding surface *σ* according to the desired switching frequency value. An additional PI controller which intermittently operates to bring back the SF to the desired value can be a simple and practical solution. The output of the PI controller dynamically changes the hysteresis of dynamic relay (M). Thus, SF can settle to a desired interval accordingly.

Another drawback of the method is the requirement of the SF measurement. A SF measurement algorithm could be implemented by counting the rising edge of the gate signals at certain instants. If the number of rising signals is divided into a predefined time interval, SF can be easily calculated. As a result, an intermittent PI controller structure can keep the SF constant at a specified interval as shown in Figure 8. The output of the intermittent PI controller is the resultant M value of the dynamic relay with hysteresis function.

**Figure 8.** Switching frequency (SF) measurement and intermittent PI controller.

#### **4. Simulation Results**

Four different scenarios are implemented in a single simulation in MATLAB/Simulink SimPowerSystem platform at different time instants. Variable Step Ode23tb (stiff/TR-BDF2) solver is used in simulations. Circuit parameters given in Table 1 are used in simulation. Stable gains for external PI controller and equivalent controller are given in Table 2.

**Table 2.** Controller parameters of modified Cuk converter. ´


Standard Routh–Hurwitz criterion for determination of PI gain values is omitted in this paper, and all gain values are determined using trial–error methods. For further details of Routh–Hurwitz criterion for external PI controllers, one can refer to [24]. The target SF value was selected as 150 kHz, and intermittent PI hysteresis controller was only enabled when the absolute value of error exceeded 10, as shown in Figure 8. It is not possible to realize a precise controller for SF control in all perturbations due to unexpected oscillations in M value.

Applied step and disturbance instants are given below.

**0.03–0.06 s:** Output voltage reference is changed from −150 V to −50 V. **0.08–0.11 s:** Load resistance is decreased from 100 Ω to 50 Ω (100% load increase) **0.13–0.16 s:** Input voltage is decreased from 15 V to 12 V (20% input voltage dip) **0.18–0.21 s:** Input inductor (L1) is decreased from 100 μH to 85 μH (15% *L*<sup>1</sup> reduction)

Figure 9 shows the output voltage and input current response at different perturbations. Required trajectories are successfully tracked at all disturbances. Figure 9a shows the output voltage trajectory at all peak values at the instants of perturbations. The controller successfully passed all perturbations. Figure 9b,c shows the input and output currents of the converter. All ripple values are zoomed, and matches the design circuit results of Figure 3.

**Figure 9.** Performance of proposed equivalent controller. (**a**) Output Voltage (V); (**b**) Input Current (A); (**c**) Output Current (A).

Figure 10 shows the performance of the intermittent hysteresis controller. Figure 10a shows the load resistance variation to validate the applied load variation. Figure 10b shows the SF variations at all perturbations. Target SF is achieved at all different perturbations by changing M value as shown in Figure 10c. SF exceeds the target value at transient conditions, but settles to target interval at all steady-state instants of perturbations. Figure 11 shows the output voltage and input current ripples at the instants of semiconductor ON and OFF states in simulation. Figure 11a shows the instants of the gate signal, and Figure 11b shows the control signal u generated by the SMC-based equivalent controller. Figure 11c,d show the ripple contents of output voltage and input current. The ripple values verify the ripple values at design circuit simulation in Figure 3.

**Figure 10.** Performance of the proposed equivalent controller. (**a**) *R*L; (**b**) SF; (**c**) M.

**Figure 11.** Output voltage and current ripple transients. (**a**) Gate Signal; (**b**) Control Signal (u); (**c**) Input current; (**d**) Output voltage (*Vo*).

A comparison between classical linear control methods and the proposed equivalent controller was also attempted to show the effectiveness of proposed method. A cascaded controller structure which consists of voltage and current PI controllers is depicted in Figure 12. In particular, some studies use single voltage PI controllers to control a DC-DC converter. However, this type of controller has a very limited operational range, and comparison of a single PI controller would be inconsistent due to the cascaded controller structure of the proposed equivalent controller. An additional current controller introduces additional left half plane zeros to the controller and increases the performance of the controller structure. It is a difficult task to design a linear controller for a third-order modified Cuk ´ converter, and all states must be observed or measured. Perturbations applied to the controller are given below.

**0.03–0.06 s:** Output voltage reference is changed from −145 V to −45 V. **0.08–0.11 s:** Load resistance is decreased from 100 Ω to 85 Ω (15% load increase) **0.13–0.16 s:** Input voltage is decreased from 15 V to 12 V (20% input voltage dip) **0.18–0.21 s:** Input inductor (*L1*) is decreased from 100 μH to 85 μH (15% *L*<sup>1</sup> reduction)

**Figure 12.** Comparison controller structure.

Higher perturbations as applied to proposed SMC equivalent controller could not be accomplished due to stability problems. Maximum allowable *δ* is 0.9, and higher *δ* values could not be achieved. Therefore, −150 V output voltage reference could not be accomplished.

PI controller gain values are optimized with trial and error methods, which are given in Table 3. Controllers are tuned at maximum allowable proportional and integral coefficients to achieve the highest dynamic performance. Higher proportional and integral gains could not be achieved due to higher oscillations in voltage output.

**Table 3.** Controller parameters of comparison PI controller.


Figure 12 shows the performance comparison of PI controller. Figure 12a shows that output voltage performance could not be achieved for all perturbations. Dynamic response is more sluggish compared to proposed SMC based equivalent controller, and load resistance change at 0.11th second causes steady state error and oscillations. Input voltage perturbation cannot be responded due to unavailability of higher *δ* than 0.9. Steady state error exists at the instant of input voltage perturbation due to sluggish dynamic performance and unavailability of higher *δ*. Figure 12b shows the responded input currents, and Figure 12c depicts the resultant *δ*.

Finally, performance indices of the proposed controller structure at PI controller. Figure 13a shows that output voltage performance could not be achieved for all perturbations. Dynamic response is more sluggish compared to the proposed SMC-based equivalent controller, and load resistance change at the 0.11th second causes steady-state error and oscillations. Input voltage perturbation cannot be responded due to the unavailability of higher *δ* than 0.9. Steady-state error exists at the instant of input voltage perturbation due to sluggish dynamic performance and unavailability of higher *δ*. Figure 13b shows the responded input currents, and Figure 13c depicts the resultant *δ*. All perturbations are summarized in Table 4. Maximum peak overshoots are outlined and it is shown that the controller passed high load impact values and other disturbances. The value of M is dynamically changed according to SF requirements at different conditions. Input current and output voltage ripples changed according to varying M value.

**Figure 13.** Comparison controller performance. (**a**) Output voltage; (**b**) Input current (u); (**c**) *δ*.


**Table 4.** Performance indices of modified Cuk converter. ´
