CHAPTER 12

Control and Regulation

Control engineering is an important subject within power electronics, as discussed earlier in Sec. 1.2. Power conversion cannot be properly operated without control and regulation of the level of voltage, current, or power. A typical two-degreeof-freedom (2DOF) control diagram for power electronics is demonstrated in Fig. 12.1. The desired output is indicated as r, which is also called the reference or set point. The output variable is symbolized as y, which is expected to follow the value of r. In power electronics, r and y commonly refer to the values of voltage, current, or power. The difference between the desired and the actual value is called the error, calculated by e = r − y. The control action is represented as u, which usually refers to either switching duty ratio or phase shift angle. When u is properly applied, the control objective of y = r can be achieved in steady states.

FIGURE 12.1 Typical control configuration for power converters.

The feedforward controller (FFC), shown in Fig. 12.1, is an effective way to improve the performance in terms of reference following and disturbance rejection. The direct format of FFC is simple for stability analysis. The concept can be easily understood by an example. In the CCM, the voltage conversion ratio of a buck converter is theoretically proportional to the applied duty ratio of pulse width modulation (PWM). When the voltage levels of the input and the expected output are known or predictable, the control variable of the duty ratio can be directly determined and used as the straightforward form of the FFC. The configuration can save the significant correction effort through the feedback control loop and shorten the settling time to reach the steady state. The FFC is typically designed by the well-known system information between the control action and system response.

The feedback controller (FBC) has been widely utilized and forms the closed control loop, which includes sensing, feedback, and correction, as shown in Fig. 12.1. The real-time correction is capable of maintaining the desired output and mitigating non-ideal factors regarding disturbance and model uncertainty. Thus, the FBC is considered as the mainstream in controlling power conversion to reach the desired performance. The FFC becomes an additional function to coordinate more variables into the control system, and therefore improves the performance. In this chapter, the linear control theory is studied and applied to design and operates power converters to meet the system requirement.

12.1  Stability and Performance

A control system should be evaluated and based on the following specifications:

•   Absolute stability between input and output is the fundamental requirement for control engineering. It is defined that the system output should always be bounded if the input is bounded. It is referred to the general stability definition in terms of the bounded input and bounded output (BIBO).

•   Internal stability should be guaranteed that all signals inside the control system are always bounded.

•   System robustness should be guaranteed, which refers to the relative stability. It measures the distance from the boundary of instability. Highly robust systems can effectively prevent from any instability caused by unpredictable factors, such as model uncertainty, noise, disturbance, time variance, and temperature effect.

•   Ideal control performance indicates the zero value of steady-state error.

•   Fast response is generally required to demonstrate control performance in response to the setpoint change or against disturbance. The time from one steady state to another should be short. Meanwhile, the transient response is expected to be smooth, without significant deviation or oscillation.

When a closed-loop control system is designed, it is important to follow the procedure to evaluate its stability, robustness, and performance, as recommended in Fig. 12.2. The absolute stability is critically required for all expected operating conditions. The control performance can only be improved with guaranteed sufficient relative stability or robustness. It is mostly a trade-off between the system’s robustness and the control performance. When the absolute instability is guaranteed, the controller can be iteratively tuned to reach the optimal balance between the system robustness and performance.

FIGURE 12.2 Evaluation procedure of closed-loop control systems.

12.2  On/Off Control

The on/off switching is widely used in power electronics to modulate and/or regulate power flow in terms of voltage and current. The switching concept is simple but can be directly utilized for control and regulation. It follows the feedback control mechanism to achieve the function of error detection and correction, and forces the output, y, to follow the setpoint, r. The mechanism follows the simple idea of “true or false” or the boolean logic of “1 or 0.” A simple on/off control scheme is shown in Fig. 12.3a, which can be mathematically described by

FIGURE 12.3 Demonstration of on/off control: (a) ideal concept; (b) hysteresis.

where U_MAX leads to the increase in y to be close to r if r > y; meanwhile, U_MIN results in the decrease in y if y > r. The on/off operation eventually keeps the output, y, close to the setpoint and e ≈ 0.

In power converters, the U_MAX can be referred to as turning on an active switch, which leads to an increase in inductor current. On the contrary, the action of U_MIN is the extreme action of turning off the active switch and causing a decrease in the current. The on/off controller directly produces switching signals to drive active switches, which can neglect the dedicated PWM mechanism. The operation is demonstrated in Fig. 12.3a, which aims to correct the difference between y and r and forces e ≈ 0. However, the control implementation is sensitive to noise, which can make u switch randomly between U_MAX and U_MIN at an uncontrollably high frequency. Therefore, the on/off control introduces a hysteresis band or tolerance to improve its robustness against noise.

12.2.1  Hysteresis Control

The hysteresis controller is often called the bang-bang controller, which is expressed in (12.2) and illustrated in Fig. 12.3b. The approach adds tolerance to avoid extremely frequent transitions between U_MAX and U_MIN but sacrifices the steady-state performance. When the error, e, is within the range between −ε and +ε, the controller maintains its current output, unchanged, until the condition for the alternative state is satisfied. Thus, the output, y, is controlled to oscillate around the reference signal within the hysteresis band of ±ε.

The assigned value of ε shows the trade-off between the steady-state error and the on/off switching frequency. It is known that all physical components show switching speed limit; meanwhile, the high-frequency switching commonly results in power loss, as discussed previously. The hysteresis controller follows a nonlinear approach to form the feedback control loop. The design is simple, because mathematical modeling is unnecessary. It has been widely used in power converters to regulate inductor current.

12.2.2  Case Study and Simulation

The case study is based on the boost converter, the parameters of which have been specified in Sec. 3.4.5. The steady-state analysis shows that the inductor current increases when the active switch is on. Otherwise, the inductor current of the boost converter decreases. A simulation model is constructed to demonstrate the control operation to regulate the inductor current, as shown in Fig. 12.4. A hysteresis controller is constructed by Simulink using the “Relay” block. The output for the “Relay” block switches between two specified values, “0” and “1,” which can control power switching. The hysteresis band (±ε) can be programmed inside the block. The inductor current is fed back for the hysteresis controller for regulation. The inductor current is expected to follow the reference signal indicated in the Simulink diagram. The assigned value of ε is 0.3 A, which is specified in the “Relay” block for the hysteresis control operation. The simulation result is illustrated in Fig. 12.5.

FIGURE 12.4 Simulation model for hysteresis control to regulate inductor current of a boost converter.

FIGURE 12.5 Simulation result for hysteresis control to regulate inductor current of a boost converter.

The converter is controlled for the inductor current to be 3 A at the start, which represents the nominal condition. At the moment of 2.5 ms, the current reference is changed to 2 A. At the moment of 5 ms, a disturbance is introduced by a step change of the input voltage (V_in) from 12 to 14 V, as indicated in Fig. 12.5. The hysteresis controller proves its effectiveness and maintains the inductor current, i_L, to follow the command signal, i_ref, changing from 3 to 2 A. The peak-to-peak ripple of i_L is 0.6 A, which agrees with the converter specifications presented in Table 3.3. The switching frequency can be measured as 50 kHz to follow the current ripple rating at the nominal condition. The inductor current variation leads to the change in the output voltage, v_o, since the load resistance is constant. When the output voltage is treated as the controlled variant, the inductor current regulation becomes the inner feedback loop. The cascade approach can be implemented to reach the voltage regulation, which will be discussed in Sec. 12.5.

12.3  Affine Parameterization

Affine parameterization is a controller design technique, which is based on mathematical modeling and dynamic analysis. The method makes controller design straightforward and guarantees the internal stability and control performance when the system modeling is properly performed. It is sometimes called the Youla parameterization and Q design. For a SISO system, the design procedure is as demonstrated in Fig. 12.6, where R is the reference and Y symbolizes the controlled variable.

FIGURE 12.6 Concept of affine parameterization for design and implementation.

During the design stage, the transfer function of Q(s) is introduced for the open-loop system analysis. For the series formation, as expressed in (12.3), the absolute and internal stability can be guaranteed when the transfer functions of Q(s) and G(s) show the feature of BIBO. The series form of Q(s)G(s) is straightforward for stability analysis, but shows no correction mechanism.

For practical implementation, the control performance cannot be guaranteed if Y is deviated from R due to unpredictable disturbance or non-ideal factors. Thus, the mainstream of control configuration is based on the negative feedback mechanism. The same formation of Q(s)G(s) can be equivalently transferred to the closed-loop format that includes the correction function and the feedback controller, C(s), as illustrated in Fig. 12.6. The closed-loop implementation indicates the correction mechanism that any error between the reference, R, and the plant output, Y, can be detected and corrected by the controller. The transfer function of the closed-loop system demonstrates a rational form.

The individual stability of the transfer functions of C(s) and G(s) cannot indicate the internal stability of the closed-loop showing the relation between Y and R. Therefore, the design stage starts from the synthesis of Q(s)G(s) to guarantee system stability. The equivalence between the transfer functions in (12.3) and (12.4) leads to the feedback controller that can be synthesized by (12.5). Following affine parameterization, the internal stability of the closed-loop system can be guaranteed. The controller synthesis simply takes advantage of both the open-loop stability analysis and closed-loop control implementation.

12.3.1  Design Procedure

Through the modeling process introduced in Chaps. 10 and 11, the mathematical model, G₀(s), should be available to represent the dynamics of power converters in either the large-scale or small signals. The transfer function, G₀(s), should be verified by either simulation or experiment before the controller synthesis is proceeded. Figure 12.7 illustrates the recommended design procedure using affine parameterization. The desired closed-loop transfer function, , should be first defined according to the system requirement and the dynamic analysis of G₀(s). Since the closed-loop transfer function is defined by (12.3), the intermediate function Q(s) is then derived by (12.6) for stability analysis. The transfer function of Q(s) should be stable and proper to indicate the internal stability of F_Q(s) = Q(s)G₀(s).

FIGURE 12.7 Design procedure of affine parameterization.

The transfer function of the feedback controller is then derived by (12.5). The relative stability or robustness should be evaluated by analyzing the transfer function of C(s)G₀(s) regarding the phase margin, gain margin, and sensitivity peak. When the robustness is satisfied, the feedback controller is ready for the closed-loop implementation.

12.3.2  Desired Closed Loop

When a converter model is derived and verified, the relative degree of G₀(s) can be determined, which shows the difference between the number of poles (N_pole) and the number of minimal-phase (MP) zeros, N_zero. The relative degree is computed by N_pole − N_zero, which is generally zero for a physical system. Based on the dynamic modeling in Chaps. 10 and 11, the converter models can be summarized into four types, which are expressed from (12.7) to (12.10). G₀(s) is generally symbolized to represent a group of converter models for the following synthesis regardless of the difference in types and parameters. The relative degree of (12.7) and (12.9) is 1, where β_i > 0. The transfer function in (12.8) shows the relative degree of 2. The model in (12.10) shows a non-minimalphase (NMP) zero since β_v > 0.

Following G₀(s), the next step is to specify the desired closed-loop transfer function, F_Q(s), according to the design procedure, as shown in Fig. 12.7. The step is critical with the consideration of the trade-off between the closed-loop performance and stability. The first-order transfer function, as expressed in (12.7), has been developed to represent the dynamics for the dual active bridge (DAB). Meanwhile, the converters of buck, boost, and buck-boost also show the first-order dynamics when the DCM is considered, which are discussed in Sec. 11.4. The F_Q(s) should be defined as a first-order transfer function expressed in (12.11) to match the relative degree.

The transfer function in (12.9) represents another group when the inductor current is the controlled target, which includes the topologies of buck, boost, and buck-boost. Such transfer functions are shown in (10.9), (11.29), and (11.42), which also indicate the relative degree of 1. The F_Q(s) should be defined the same way as in (12.11) to match the relative degree. The DC gain of F_Q(s) is 1, indicating the zero steady-state error. The speed of dynamic response is specified by the value of α_cl in (12.11). The settling time of the step response can be estimated to be 4α_cl, which is used to determine the value of α_cl.

A converter’s model can show the relative degree of 2, which is represented by (12.8). This model has been developed in Sec. 10.2.1 for the buck converter in the continuous conduction mode (CCM). The desired closed-loop function for such systems should be specified by (12.12), which can also be converted into (12.13) where and α₁ = . The parameters in F_Q(s) should be specified, which include the damping ratio, ξ_cl, and the undamped natural frequency, ω_cl. According to (12.12), the DC gain is assigned to one for the output to follow the reference command in steady states. The damping ratio is typically assigned to be from 0.7 to 2 to balance the response speed and overshoot level. The damping factor, ξ_cl, in F_Q(s) is considered as the measurement of the oscillation scale in terms of the percentage of overshoot (PO) at the step response. The specification of ξ_cl can refer to Table 10.1, indicating the scale of overshoot. The ω_cl is commonly chosen to be equal or higher than ω_n for faster response. The settling time of the step response in the closed-loop system can be roughly approximated as , which is used to expect the closed-loop performance.

The NMP system has been modeled in (11.27) and (11.40) regarding the dynamics of boost and buck-boost converters. The general form is shown in (12.10) for the following analysis. Figure 12.7 shows that the development of Q(s) requires the inverse transfer function, G₀(s)⁻¹. However, the inverse transfer function of the NMP G₀(s) results in a right-hand-plane (RHP) poles that become unstable. To avoid any harmful zero-pole cancellation, the NMP zero should remain in F_Q(s) according to affine parameterization. Thus, the transfer function for the closed-loop system should be specified as (12.14) or (12.15) to admit the existence of the NMP zero. The value of β_v follows the same in (12.10). Due to the negative effect of the NMP zero, the closed-loop specification should be conservatively designed, which is required to maintain sufficient stability margin and robustness.

12.3.3  Derivation of Q(s) and C(s)

For the first-order system shown in the expression of (12.7), the desired closed-loop transfer function in (12.11) is specified. The transfer function of Q(s) can be derived and expressed as in (12.16), which is considered to be stable and proper. Following (12.5), the function of Q(s) and G₀(s) results in the transfer function of the feedback controller, C(s), as expressed in (12.17). The controller can be transferred to the standard format of a proportional-integral (PI) controller showing the proportional gain of and integral gain of

Following the converter transfer functions in the format of (12.9), the desired closed-loop transfer function is also specified by (12.11) since the relative degree of G₀(s) is 1. Following (12.6) and the converter transfer function, the Q(s) can be derived and expressed in (12.18). When the function Q(s) is verified for stability, the design process leads to the feedback controller transfer function, as expressed in (12.19). The transfer function can be converted into a proportional-integral-derivative (PID) controller, which will be discussed further in Sec. 12.4.

When the relative degree of 2 is concerned, such as the converter transfer function for the output voltage of buck converters, the desired closed-loop system should be specified by (12.13). The transfer function of Q(s) can be derived into (12.20) according to the converter transfer function. When Q(s) is verified to be stable and proper, the design process can be continued by (12.5) into the feedback controller transfer function in (12.21). The transfer function of C(s) can also be converted into a PID format.

When a NMP system is considered, the desired closed-loop system is specified by (12.15) to avoid any harmful pole-zero cancellation. The transfer function of Q(s) can be derived into (12.22) according to (12.6) and (12.10). After the Q(s) is verified to be stable and proper, the design process leads to the feedback controller transfer function that is the same as (12.21). In general, the Q(s) is an interlinking function that leads to the design of a feedback controller for the closed-loop implementation.

12.3.4  Relative Stability and Robustness

Affine parameterization leads to the design of a closed-loop system with the absolute and internal stability. However, a real-world control system is coupled with non-ideal factors, such as disturbance and noise, as illustrated in Fig. 12.8. The indicators of D_i, D_o, and D_m represent the input disturbance, output disturbance, and measurement noise, respectively. For power converters, the inaccuracy of duty ratio or contaminated PWM signal results in the input disturbance. Any sudden or unpredictable load variation leads to the output disturbance, D_o. It is known that all real-world signals cannot avoid noise coupling. Further, the power switching operation is nonlinear in nature. The mathematical models are based on the approximation techniques of averaging, linearization, and various levels of assumption. Model uncertainty is another concern. Therefore, the robustness in control system design should be of concern and analyzed.

FIGURE 12.8 Illustration of real-world feedback control systems.

When a feedback controller, C(s), is synthesized, one important measure is the relative stability of the closed-loop system. The relative stability indicates the system robustness and measures how far a nominally stable system enters endless oscillation or instability with the consideration of unpredictable disturbance, non-ideal environment, modeling inaccuracy, and model uncertainty. The common index for the measure includes the phase margin, gain margin, and sensitivity peak. The sensitivity function is defined by the linear control theory and expressed by (12.23). The function represents the transfer function between the output disturbance (D_o) and the closed-loop output, Y, as indicated in Fig. 12.8. The S_o(s) also indicates how sensitive a closed-loop stability is affected by the model uncertainty. The sensitivity peak is the highest amplitude that happens at a certain frequency. It becomes an important measure of the system robustness, which is expected to be low to achieve high system robustness.

Based on the derived transfer function, C(s)G₀(s), the stability margins and sensitivity peak can be detected and evaluated. Either Bode diagrams or Nyquist plots can be applied for analyzing the relative stability. In MATLAB, the functions of “margin” and “nyquist” are commonly used for the Bode and Nyquist analyses, respectively. For demonstration only, a plant transfer function is shown as . A feedback controller is designed to be the PI format and expressed as . Using the “margin” function for C(s)G₀(s), the phase margin is measured as Φ_m = 32.5° at the frequency of 0.76 rad/s. The phase margin is one indicator of the distance from the endless system oscillation. The gain margin is detected to be 2.52 in the absolute value or 8.02 in decibels (dB) at the crossover frequency of 1.29 rad/s. These stability margins can be shown by the Bode diagram of G₀(s)C(s), as in Fig.