CHAPTER 11

Linearized Model for Dynamic Analysis

The automatic control is based on the closed-loop concept, which adopts sensing, computation, and correction. Figure 11.1a illustrates a typical feedback control system that regulates power converters. Depending on the system requirement, the control loop is commonly specified and designed to achieve one of the following functions:

FIGURE 11.1 Feedback control diagram: (a) application; (b) modeled.

•   Voltage regulation of either input or output.

•   Current regulation of either input or output.

•   Power regulation of either input or output.

Dynamic modeling is an important step to reveal the important feature of a dynamic system in terms of response speed, damping, steady state, stability, and robustness. It is considered the first step but critical for controller development. After modeling, a dynamic power system can be represented by the transfer function in the frequency domain, as illustrated in Figure 11.1b. The objective of dynamic modeling aims to find the mathematical model of G(s) to represent the key dynamics. The on/off switching of power converters is discontinuous, which cannot be directly represented by a linear model.

The dynamic models of buck converters have been completely developed using averaging techniques when only the continuous conduction mode (CCM) is considered. The modeling has been described in Sec. 10.2.1 and expressed by the transfer functions in (10.8) and (10.9). The models are ready to be utilized for the dynamic analysis and feedback controller design. However, the averaging process of other topologies, e.g., boost and buck-boost, leads to the nonlinearity of differential equations, which cannot be directly applied for the linear control analysis and design. The switching between the CCM and DCM increases the modeling complexity and leads to nonlinearity. Linearization is required to derive small-signal models that can be analyzed and evaluated by the well-developed linear control theory.

11.1  General Linearization

A function is represented by y = f(x), linear or nonlinear. If f(x) is infinitely differentiable, the Taylor series expansion regarding an equilibrium point (x₀, y₀) can be expressed by

where n! denotes the factorial of n, and f⁽ⁿ⁾(x₀) denotes the nth derivative evaluated at the point x₀. The higher value of n indicates a better approximation of the function of f(x) by the Taylor series. Neglecting the higher-order terms, the first-order approximation is the simplest and expressed by (11.2) when n = 1.

where C₁ becomes a constant parameter showing the direction and speed. The function f(x) = f(x_o) + C₁(x − x₀) is the approximation of the nonlinear function of f(x) near the equilibrium point (x = x₀). The approximation is only valid when the deviation is not far from the equilibrium point (x₀, y₀). When a differential equation, , is needed, the first-order approximation of Taylor series can be applied to f(x), which is expressed by

A small perturbation or deviation is defined as . The small-signal model can be developed by (11.3) into the linear representation:

The above operation shows the concept of linearization and represents a nonlinear system into the linearized or small-signal model. The linearized model is only valid or representative near the equilibrium point (x₀, y₀). The same approximation can be used for the nonlinear equations, which show multiple state variables and inputs. A simple system with two state variables (x₁, x₂) and one control input (u) is expressed by

where f(x₁, x₂, u) and g(x₁, x₂, u) are nonlinear. The nonlinear model is representative and can be used to represent the converter dynamics. For example, the x₁ can refer to the inductor current, x₂ symbolizes the output voltage, and u is the duty ratio of the PWM. The values of X₁, X₂, and U are the state variables and control input at the equilibrium point or steady state, expressed by

Applying partial differentiation, the linearized dynamics of the two state variables can be expressed by

where the perturbation or small variant is indicated by “tilde” and defined by

Following the linearization, the variables of represent the small signals; meanwhile, the values of X₁, X₂, and U are the steady-state equilibrium and become constant parameters. Thus, the state-space format can be formed by a group of the first-order differential equations

where a₁₁, a₁₂, a₂₁, and a₂₂ are the constant parameters to form the dynamic matrix, while the constants of b₁ and b₂ construct the control matrix in the state-space representation. The linearized model is only valid near the equilibrium point of the steady state, X₁, X₂, and U. The state-space model can be used directly for controller design or converted into a single-input-single-output (SISO) transfer function in s-domain. For SISO, either or is selected as the controlled output, y, according to the system specification, to form a transfer function, .

11.2  Linearization of Dual Active Bridge

The dual active bridge (DAB) is an isolated bidirectional DC/DC converter, which was introduced in Sec. 9.2. The topology allows its averaged power flow to be controlled by the predefined equation in (9.22). The converter is controlled by the phase shift, represented by the angle value, ϕ. The inductor current in a DAB is studied for the steady-state analysis and power conversion evaluation. In general, the interlink inductor shows low value in inductance to cooperate the high switching operation of DAB, which shows very fast dynamics in terms of power variation in response to the phase shift. When the voltage dynamics at the right terminal is concerned, the equivalent circuit can be plotted in Fig. 11.2. For dynamic modeling, the DAB is divided into two regions, namely the nonlinear and linear. The circuit indicates the forward power flow from the primary side (V_DCP) to the right side, which is formed by the capacitor, C_DCS, and the equivalent load resistance, R_EQ.

FIGURE 11.2 Circuit analysis of dual active bridge for dynamic modeling.

The interlinking inductance L forms an impedance effect to limit the level of power flow. The inductor current, i_L, shows AC waveform without significant energy storage effect. Therefore, the high-frequency dynamics of the inductor are classified into the nonlinear region because the dominant frequency results from the low-frequency components, which comes from the linear region. The forward power flow has been introduced in Sec. 9.2.1. The averaged value is expressed in the nonlinear equation of (11.8) referring to the circuit in Fig. 11.2. An equivalent load resistance, R_EQ, is included in the circuit for dynamic analysis. The condition at the terminal of v_dcs becomes the targeted variable, which can be controlled by the phase shift, ϕ.

The linear region in the RC circuit is represented by C_DCS and R_EQ. The dynamics can be represented by (11.9) and derived into (11.10). Following (11.8) and (11.10), the differential equation is derived that shows the nonlinear dynamics in (11.11).

A nonlinear equation is present as f(v_dcs, ϕ) including the model output of v_dcs and input of ϕ. The linearization process is expressed in (11.12), which results in the small-signal model in (11.13).

where Φ represents the value of the phase shift at the equilibrium point. The small-signal deviations are symbolized as and representing the output and input. The differential equation in (11.12) can be transformed into the s-domain transfer function as

where and τ₀ = R_EQC_DCS. The output voltage shows the first-order dynamics in response to a small perturbation by the phase shift.

The small-signal model can be verified by comparing the simulation results with the high-resolution switching model, which was developed in Sec. 9.2.6 and demonstrated in Fig. 9.24. The comparison is illustrated in Fig. 11.3, showing the output of the linearized model and the simulation result by the switching model. The case study is based on the same parameters presented in Table 9.6, where the nominal voltage and load condition are: V_DCS = 380 V, C_DCS = 1 μF, R_EQ = 193 Ω. The equilibrium point is based on Φ = 45° and p_avg = 750 W. A periodical small perturbation () is intentionally added to the steady-state phase shift angle (Φ), which is rated as ±2.5 mrad. For correct comparison, the offset of the equilibrium point should be added to match the waveform out of the large-signal model. The output of the small-signal model shows no information on the switching ripples but captures the critical dynamics in response to the step variation of in the small perturbation level. The first-order dynamic response is shown in Fig. 11.3, as predicted by the mathematical model in (11.14).