35.2.2.1 Switched State-Space Model

Given the two binary values of the switching variable δ(t), Eqs. (35.2) and (35.3) can be combined to obtain the nonlinear and time-variant switched state-space model of the switching converter circuit, Eq. (35.4) or (35.5):

$\begin{array}{l}\stackrel{.}{\mathbf{x}}=\left[{\mathbf{A}}_1\delta \left(t\right)+{\mathbf{A}}_2\left(1-\delta \left(t\right)\right)\right]\mathbf{x}+\left[{\mathbf{B}}_1\delta \left(t\right)+{\mathbf{B}}_2\left(1-\delta \left(t\right)\right)\right]\mathbf{u}\\ \mathbf{y}=\left[{\mathbf{C}}_1\delta \left(t\right)+{\mathbf{C}}_2\left(1-\delta \left(t\right)\right)\right]\mathbf{x}+\left[{\mathbf{D}}_1\delta \left(t\right)+{\mathbf{D}}_2\left(1-\delta \left(t\right)\right)\right]\mathbf{u}\end{array}$

$\begin{array}{c}\hfill \stackrel{.}{\mathbf{x}}={\mathbf{A}}_S\mathbf{x}+{\mathbf{B}}_S\mathbf{u}\hfill \\ \hfill \mathbf{y}={\mathbf{C}}_S\mathbf{x}+{\mathbf{D}}_S\mathbf{u}\hfill \end{array}$

where A_S=[A₁δ(t)+A₂(1−δ(t))], B_S=[B₁δ (t)+B₂ (1−δ (t))], C_S=[C₁δ (t)+C₂ (1−δ (t))], and D_S=[D₁δ (t)+D₂ (1−δ (t))].

35.2.2.2 State-Space Averaged Model

Since the state variables of the x vector are continuous, using Eq. (35.4), with the initial conditions x₁(0)=x₂(T) and x₂(δ₁ T)=x₁(δ₁ T), and considering the duty cycle δ₁ as the average value of δ(t), the time evolution of the converter state variables can be obtained, integrating Eq. (35.4) over the intervals 0≤t≤δ₁ T and δ₁ T≤t≤T, although it often requires excessive calculation effort. However, a convenient approximation can be devised, considering λ_max, the maximum of the absolute values of all eigenvalues of A (usually, λ_max is related to the cutoff frequency f_c of an equivalent low-pass filter with f_c≪1/T). For λ_max T≪1, the exponential matrix (or state transition matrix) e^At=I+At+A²t ²/2+…+Aⁿt ⁿ/n!, where I is the identity or unity matrix, can be approximated by e^At≈I+At. Therefore, e^{A₁δ₁ t}·e^{A₂(1−δ₁)t}≈I + [A₁δ₁+A₂(1−δ₁)]t. Hence, the solution over the period T, for the system represented by Eq. (35.4), is found to be

$\begin{array}{c}\mathbf{x}\left(T\right)\cong e^{\left[{\mathbf{A}}_1{\delta }_1+{\mathbf{A}}_2\left(1-{\delta }_1\right)\right]T}{\mathbf{x}}_1\left(0\right)\\ +{\int }_0^T{e}^{\left[{\mathbf{A}}_1{\delta }_1+{\mathbf{A}}_2\left(1-{\delta }_1\right)\right]\left(T-\tau \right)}\left[{\mathbf{B}}_1{\delta }_1+{\mathbf{B}}_2\left(1-{\delta }_1\right)\right]\mathbf{u}\mathrm{d}\tau \end{array}$

This approximate response of Eq. (35.4) is identical to the exact response obtained from the nonlinear continuous time-invariant state-space model (35.7), supposing that the average values of x, denoted $\overline{\mathrm{x}}$, are the new state variables and considering δ₂=1−δ₁. Moreover, if A₁ A₂=A₂ A₁, the approximation is exact:

$\begin{array}{c}\hfill \stackrel{.}{\overline{\mathbf{x}}}=\left[{\mathbf{A}}_1{\delta }_1+{\mathbf{A}}_2{\delta }_2\right]\overline{\mathbf{x}}+\left[{\mathbf{B}}_1{\delta }_1+{\mathbf{B}}_2{\delta }_2\right]\overline{\mathbf{u}}\hfill \\ \hfill \overline{\mathbf{y}}=\left[{\mathbf{C}}_1{\delta }_1+{\mathbf{C}}_2{\delta }_2\right]\overline{\mathbf{x}}+\left[{\mathbf{D}}_1{\delta }_1+{\mathbf{D}}_2{\delta }_2\right]\overline{\mathbf{u}}\hfill \end{array}$

For λ_maxT≪1, the model (35.7), often referred to as the state-space averaged model, is also said to be obtained by “averaging” Eq. (35.4) over one period, under small ripple and slow variations, as the average of products is approximated by products of the averages. Comparing Eq. (35.7) with Eq. (35.1), the relations (35.8), defining the state-space averaged model, are obtained:

$\begin{array}{cc}\hfill \mathbf{A}=\left[{\mathbf{A}}_1{\delta }_1+{\mathbf{A}}_2{\delta }_2\right];\hfill & \hfill \mathbf{B}=\left[{\mathbf{B}}_1{\delta }_1+{\mathbf{B}}_2{\delta }_2\right];\hfill \\ \hfill \mathbf{C}=\left[{\mathbf{C}}_1{\delta }_1+{\mathbf{C}}_2{\delta }_2\right];\hfill & \hfill \mathbf{D}=\left[{\mathbf{D}}_1{\delta }_1+{\mathbf{D}}_2{\delta }_2\right]\hfill \end{array}$

Example 35.1

State-space models for the buck–boost dc/dc converter

Consider the simplified circuitry of the buck-boost converter of Fig. 35.1 switching at f_s=20 kHz (T=50 μs) with V_DCmax=28 V, V_DCmin=22 V, V_o=24 V, L_i=400 μH, C_o=2700 μF, and R_o=2 Ω.

The differential equations governing the dynamics of the state vector x=[i_L, v_o]^T (^T denotes the transpose of vectors or matrices) are

$\begin{array}{c}{L}_i\frac{\mathrm{d}{i}_L}{\mathrm{d}t}=V_{\mathrm{D}\mathrm{C}}\\ C_o\frac{\mathrm{d}{v}_o}{\mathrm{d}t}=-\frac{v_o}{R_o}\end{array}\begin{array}{c}\mathrm{for}0\le t\le {\delta }_{1}T\left(\delta \left(t\right)=1,{\mathrm{Q}}_1\mathrm{is}\mathrm{on}\mathrm{and}{\mathrm{D}}_1\mathrm{is}\mathrm{off}\right)\\ \end{array}$

  (35.9)

$\begin{array}{l}{L}_i\frac{\mathrm{d}{i}_L}{\mathrm{d}t}=-v_o\\ C_o\frac{\mathrm{d}{v}_o}{\mathrm{d}t}=i_L-\frac{v_o}{R_o}\end{array}\begin{array}{c}\mathrm{for}{\delta }_{1}T\le t\le T\left(\delta \left(t\right)=0;{\mathrm{Q}}_1\mathrm{is}\mathrm{off}\mathrm{and}{\mathrm{D}}_1\mathrm{is}\mathrm{on}\right)\\ \end{array}$

  (35.10)

Comparing Eqs. (35.9) and (35.10) with Eqs. (35.2) and (35.3) and considering y=[v_o, i_L]^T, the following matrices can be identified:

$\begin{array}{cc}\hfill {\mathbf{A}}_1=\left[\begin{array}{cc}\hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill -1/\left(R_o{C}_o\right)\hfill \end{array}\right];\hfill & \hfill {\mathbf{A}}_2=\left[\begin{array}{cc}\hfill 0\hfill & \hfill -1/L_i\hfill \\ \hfill -1/C_o\hfill & \hfill -1/\left(R_o{C}_o\right)\hfill \end{array}\right]\text{;}\hfill \end{array}$

$\begin{array}{ccc}\hfill {\mathbf{B}}_1={\left[1/L_i,0\right]}^T;\hfill & \hfill {\mathbf{B}}_2={\left[0,0\right]}^T;\hfill & \hfill \mathbf{u}\hfill \end{array}=\left[V_{\mathrm{D}\mathrm{C}}\right]\text{;}$

$\begin{array}{cc}\hfill {\mathbf{C}}_1=\left[\begin{array}{cc}\hfill 0\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 0\hfill \end{array}\right];\hfill & \hfill {\mathbf{C}}_2=\left[\begin{array}{cc}\hfill 0\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 0\hfill \end{array}\right]\text{;}\hfill \end{array}$

$\begin{array}{cc}\hfill {\mathbf{D}}_1={\left[0,0\right]}^T;\hfill & \hfill {\mathbf{D}}_2={\left[0,0\right]}^T\hfill \end{array}$

From Eqs. (35.4) and (35.5), the switched state-space model of this switching converter is

$\begin{array}{l}\begin{array}{l}\left[\begin{array}{c}{\stackrel{.}{i}}_L\\ {\stackrel{.}{v}}_o\end{array}\right]=\left[\begin{array}{cc}0& -\left(1-\delta \left(t\right)\right)/L_i\\ \left(1-\delta \left(t\right)\right)/C_o& -1/\left(R_o{C}_o\right)\end{array}\right]\left[\begin{array}{c}{i}_L\\ v_o\end{array}\right]+\left[\begin{array}{c}\delta \left(t\right)/L_i\\ 0\end{array}\right]V_{DC}\\ \end{array}\\ \left[\begin{array}{c}{v}_o\\ i_L\end{array}\right]=\left[\begin{array}{ll}0& 1\\ 1& 0\end{array}\right]\left[\begin{array}{c}{i}_L\\ v_o\end{array}\right]+\left[\begin{array}{c}0\\ 0\end{array}\right]\left[V_{DC}\right]\end{array}$

  (35.11)

Now, applying Eq. (35.7), Eqs. (35.12) and (35.13) can be obtained:

$\begin{array}{l}\left[\frac{{\stackrel{.}{\overline{i}}}_L}{{\stackrel{.}{\overline{v}}}_o}\right]=\left[\left[\begin{array}{cc}0& 0\\ 0& -1/\left(R_o{C}_o\right)\end{array}\right]{\delta }_1+\left[\begin{array}{ll}0& -1/L_i\\ 1/C_o& -1/R_o{C}_o\end{array}\right]{\delta }_2\right]\\ \times \left[\begin{array}{c}{\overline{i}}_L\\ {\overline{v}}_o\end{array}\right]+\left[\left[\begin{array}{c}1/L_i\\ 0\end{array}\right]{\delta }_1+\left[\begin{array}{c}0\\ 0\end{array}\right]{\delta }_2\right]\left[{\overline{V}}_{DC}\right]\end{array}$

  (35.12)

$\begin{array}{l}\left[\begin{array}{c}\hfill {\overline{v}}_o\hfill \\ \hfill {\overline{i}}_L\hfill \end{array}\right]=\left[\left[\begin{array}{ll}0\hfill & 1\hfill \\ 1\hfill & 0\hfill \end{array}\right]{\delta }_1+\left[\begin{array}{ll}0\hfill & 1\hfill \\ 1\hfill & 0\hfill \end{array}\right]{\delta }_2\right]\left[\begin{array}{l}{\overline{i}}_L\hfill \\ {\overline{v}}_o\hfill \end{array}\right]\\ +\left[\left[\begin{array}{l}0\hfill \\ 0\hfill \end{array}\right]{\delta }_1+\left[\begin{array}{l}0\hfill \\ 0\hfill \end{array}\right]{\delta }_2\right]\left[{\overline{V}}_{DC}\right]\end{array}$

  (35.13)

From Eqs. (35.12) and (35.13), the state-space averaged model, written as a function of δ₁, is

$\left[\frac{{\stackrel{.}{\overline{i}}}_L}{{\stackrel{.}{\overline{v}}}_o}\right]=\left[\begin{array}{cc}0& -\left(1-{\delta }_1\right)/L_i\\ \left(1-{\delta }_1\right)/C_o& -1/\left(R_o{C}_o\right)\end{array}\right]\left[\begin{array}{c}{\overline{i}}_L\\ {\overline{v}}_o\end{array}\right]+\left[\begin{array}{c}{\delta }_1/L_i\\ 0\end{array}\right]\left[{\overline{V}}_{DC}\right]$

  (35.14)

$\left[\begin{array}{l}{\overline{v}}_o\\ {\overline{i}}_L\end{array}\right]=\left[\begin{array}{cc}\hfill 0\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 0\hfill \end{array}\right]\left[\begin{array}{c}\hfill {\overline{i}}_L\hfill \\ \hfill {\overline{v}}_o\hfill \end{array}\right]+\left[\begin{array}{c}\hfill 0\hfill \\ \hfill 0\hfill \end{array}\right]\left[{\overline{V}}_{DC}\right]$

  (35.15)

The eigenvalues $s_{b{b}_{1,2}}$ or characteristic roots of A are the roots of |sI−A|. Therefore,

$s_{b{b}_{1,2}}=\frac{-1}{2R_o{C}_o}\pm \sqrt{\frac{1}{4{\left(R_o{C}_o\right)}^2}-\frac{{\left(1-{\delta }_1\right)}^2}{L_i{C}_o}}$

  (35.16)

Since λ_max is the maximum of the absolute values of all the eigenvalues of A, the model (35.14) and (35.15) is valid for switching frequencies f_s (f_s=1/T) that verify λ_maxT≪1. Therefore, as T≪1/λ_max, the values of T that approximately verify this restriction are T≪1/max(|s_bb1,2|). Given these buck-boost converter data, T≪2 ms is obtained. Therefore, the converter switching frequency must obey f_s≫max(|s_bb1,2|), implying switching frequencies above, say, 5 kHz. Consequently, the buck-boost switching frequency, the inductor value, and the capacitor value were chosen accordingly.

This restriction can be further used to discuss the maximum frequency ω_max for which the state-space averaged model is still valid, given a certain switching frequency. As λ_max can be regarded as a frequency, the preceding constraint brings ω_max≪2πf_s, say ω_max<2πf_s/10, which means that the state-space averaged model is a good approximation at frequencies under one-tenth of the power converter switching frequency.

The state-space averaged model (35.14) and (35.15) is also the state-space model of the circuit represented in Fig. 35.2. Hence, this circuit is often named “the averaged equivalent circuit” of the buck-boost converter and allows the determination, under small ripple and slow variations, of the average equivalent circuit of the converter switching cell (power transistor plus diode).

The average equivalent circuit of the switching cell (Fig. 35.3A) is represented in Fig. 35.3B and emerges directly from the state-space averaged model (35.14) and (35.15). This equivalent circuit can be viewed as the model of an “ideal transformer” (Fig. 35.3C), whose primary to secondary ratio (v₁/v₂) can be calculated applying Kirchhoff's voltage law to obtain−v₁+v_s−v₂=0. As v₂=δ₁v_s, it follows that v₁=v_s(1−δ₁), giving (v₁/v₂)=(1−δ₁)/δ₁. The same ratio could be obtained beginning with i_L=i₁+i₂ and i₁=δ₁i_L (Fig. 35.3B), which gives i₂=i_L(1−δ₁) and (i₂/i₁)=δ₂/δ₁.

The average equivalent circuit concept, obtained from Eq. (35.7) or Eqs. (35.14) and (35.15), can be applied to other switching converters, with or without a similar switching cell, to obtain transfer functions or to simulate the converter average behavior. The average equivalent circuit of the switching cell can be applied to converters with the same switching cell operating in the continuous conduction mode. However, note that the state variables of Eq. (35.7) or Eqs. (35.14) and (35.15) are the mean values of the converter instantaneous variables and, therefore, do not represent their ripple components. The inputs of the state-space averaged model are the mean values of the converter inputs over one switching period.

35.2.2.3 Linearized State-Space Averaged Model

The converter outputs $\overline{\mathbf{y}}$ must be regulated actuating on the duty cycle δ(t), and the converter inputs ū usually present perturbations due to the load and power supply variations. State variables are decomposed in small ac perturbations (denoted by “~”) and dc steady-state quantities (represented by uppercase letters). Therefore,

$\begin{array}{l}\overline{\mathbf{x}}=\mathbf{X}+\tilde{\mathbf{x}}\hfill \\ \overline{\mathbf{y}}=\mathbf{Y}+\tilde{\mathbf{y}}\hfill \\ \overline{\mathbf{u}}=\mathbf{U}+\tilde{\mathbf{u}}\hfill \\ {\delta }_1={\mathrm{\Delta }}_1+\tilde{\delta }\hfill \\ {\delta }_2={\mathrm{\Delta }}_2-\tilde{\delta }\hfill \end{array}$

Using Eq. (35.17) in Eq. (35.7) and rearranging terms, it is obtained:

$\begin{array}{l}\stackrel{.}{\tilde{\mathbf{x}}}=\left[{\mathbf{A}}_1{\mathrm{\Delta }}_1+{\mathbf{A}}_2{\mathrm{\Delta }}_2\right]\mathbf{X}+\left[{\mathbf{B}}_1{\mathrm{\Delta }}_1+{\mathbf{B}}_2{\mathrm{\Delta }}_2\right]\mathbf{U}\\ +\left[{\mathbf{A}}_1{\mathrm{\Delta }}_1+{\mathbf{A}}_2{\mathrm{\Delta }}_2\right]\tilde{\mathbf{x}}+\left[\left({\mathbf{A}}_1-{\mathbf{A}}_2\right)\mathbf{X}+\left({\mathbf{B}}_1-{\mathbf{B}}_2\right)\mathbf{U}\right]\tilde{\delta }\\ +\left[{\mathbf{B}}_1{\mathrm{\Delta }}_1+{\mathbf{B}}_2{\mathrm{\Delta }}_2\right]\tilde{\mathbf{u}}+\left[\left({\mathbf{A}}_1-{\mathbf{A}}_2\right)\right]\tilde{\mathbf{x}}+\left({\mathbf{B}}_1-{\mathbf{B}}_2\right)\tilde{\mathbf{u}}]\tilde{\delta }\end{array}$

$\begin{array}{l}\mathbf{Y}+\tilde{\mathbf{y}}=\left[{\mathbf{C}}_1{\mathrm{\Delta }}_1+{\mathbf{C}}_2{\mathrm{\Delta }}_2\right]\mathbf{X}+\left[{\mathbf{D}}_1{\mathrm{\Delta }}_1+{\mathbf{D}}_2{\mathrm{\Delta }}_2\right]\mathbf{U}\\ +\left[{\mathbf{C}}_1{\mathrm{\Delta }}_1+{\mathbf{C}}_2{\mathrm{\Delta }}_2\right]\tilde{\mathbf{x}}+\left[\left({\mathbf{C}}_1-{\mathbf{C}}_2\right)\mathbf{X}+\left({\mathbf{D}}_1-{\mathbf{D}}_2\right)\mathbf{U}\right]\tilde{\delta }\\ +\left[{\mathbf{D}}_1{\mathrm{\Delta }}_1+{\mathbf{D}}_2{\mathrm{\Delta }}_2\right]\tilde{\mathbf{u}}+\left[\left({\mathbf{C}}_1-{\mathbf{C}}_2\right)\right]\tilde{\mathbf{x}}+\left({\mathbf{D}}_1-{\mathbf{D}}_2\right)\tilde{\mathbf{u}}]\tilde{\delta }\end{array}$

The terms [A₁Δ₁+A₂ Δ₂] X + [B₁ Δ₁+B₂ Δ₂] U and [C₁ Δ₁+C₂ Δ₂] X + [D₁ Δ₁+D₂ Δ₂] U, respectively, from Eqs. (35.18) and (35.19) represent the steady-state behavior of the system. As in steady-state $\stackrel{.}{\mathrm{X}}=0$, the following relationships hold:

$\mathbf{0}=\left[{\mathbf{A}}_1,{\mathrm{\Delta }}_1,+,{\mathbf{A}}_2,{\mathrm{\Delta }}_2\right]\mathbf{X}+\left[{\mathbf{B}}_1,{\mathrm{\Delta }}_1,+,{\mathbf{B}}_2,{\mathrm{\Delta }}_2\right]\mathbf{U}$

$\mathbf{Y}=\left[{\mathrm{C}}_1,{\mathrm{\Delta }}_1,+,{\mathbf{C}}_2,{\mathrm{\Delta }}_2\right]\mathbf{X}+\left[{\mathbf{D}}_1,{\mathrm{\Delta }}_1,+,{\mathbf{D}}_2,{\mathrm{\Delta }}_2\right]\mathbf{U}$

Neglecting higher-order terms $(\left[\left({\mathbf{A}}_1-{\mathbf{A}}_2\right)\tilde{\mathbf{x}}+\left({\mathbf{B}}_1-{\mathbf{B}}_2\right)\tilde{\mathbf{u}})\right]\tilde{\delta }\approx 0$ of Eqs. (35.18) and (35.19), the linearized small-signal state-space averaged model is

$\begin{array}{l}\stackrel{.}{\tilde{\mathbf{x}}}=\left[{\mathbf{A}}_1{\mathrm{\Delta }}_1+{\mathbf{A}}_2{\mathrm{\Delta }}_2\right]\tilde{\mathbf{x}}+\left[\left({\mathbf{A}}_1-{\mathbf{A}}_2\right)\mathbf{X}+\left({\mathbf{B}}_1-{\mathbf{B}}_2\right)\mathbf{U}\right]\tilde{\delta }\\ +\left[{\mathbf{B}}_1{\mathrm{\Delta }}_1+{\mathbf{B}}_2{\mathrm{\Delta }}_2\right]\tilde{\mathbf{u}}\\ \tilde{\mathbf{y}}=\left[{\mathbf{C}}_1{\mathrm{\Delta }}_1+{\mathbf{C}}_2{\mathrm{\Delta }}_2\right]\tilde{\mathbf{x}}+\left[\left({\mathbf{C}}_1-{\mathbf{C}}_2\right)\mathbf{X}+\left({\mathbf{D}}_1-{\mathbf{D}}_2\right)\mathbf{U}\right]\tilde{\delta }\\ +\left[{\mathbf{D}}_1{\mathrm{\Delta }}_1+{\mathbf{D}}_2{\mathrm{\Delta }}_2\right]\tilde{\mathbf{u}}\end{array}$

or

$\begin{array}{l}\stackrel{.}{\tilde{\mathbf{x}}}={\mathbf{A}}_{av}\tilde{\mathbf{x}}+{\mathbf{B}}_{av}\tilde{\mathbf{u}}+\left[\left({\mathbf{A}}_1-{\mathbf{A}}_2\right)\mathbf{X}+\left({\mathbf{B}}_1-\mathbf{B}{}_2\right)\mathbf{U}\right]\tilde{\delta }\hfill \\ \tilde{\mathbf{y}}={\mathbf{C}}_{av}\tilde{\mathbf{x}}+{\mathbf{D}}_{av}\tilde{\mathbf{u}}+\left[\left({\mathbf{C}}_1-{\mathbf{C}}_2\right)\mathbf{X}+\left({\mathbf{D}}_1-\mathbf{D}{}_2\right)\mathbf{U}\right]\tilde{\delta }\hfill \end{array}$

with

$\begin{array}{l}{\mathbf{A}}_{av}=\left[{\mathbf{A}}_1{\mathrm{\Delta }}_1+{\mathbf{A}}_2{\mathrm{\Delta }}_2\right]\hfill \\ {\mathbf{B}}_{av}=\left[{\mathbf{B}}_1{\mathrm{\Delta }}_1+{\mathbf{B}}_2{\mathrm{\Delta }}_2\right]\hfill \\ {\mathbf{C}}_{av}=\left[{\mathbf{C}}_1{\mathrm{\Delta }}_1+{\mathbf{C}}_2{\mathrm{\Delta }}_2\right]\hfill \\ {\mathbf{D}}_{av}=\left[{\mathbf{D}}_1{\mathrm{\Delta }}_1+{\mathbf{D}}_2{\mathrm{\Delta }}_2\right]\hfill \end{array}$

35.2.5.2 Compensator Selection and Design

The procedure to select the compensator and to design its parameters can be outlined as follows:

1. Compensator selection: In general, since V_DC perturbations exist, null steady-state error guarantee is needed. High-frequency poles are usually necessary, if the transfer function shows a −6 dB/octave roll-off due to high-frequency left plane zeros. Therefore, in general, two types of compensation schemes with integral action (35.51) or (35.50) and (35.52) or (35.53) can be tried. Compensator (35.52) is usually convenient for systems with lightly damped complex poles.

2. Unity gain frequency ω_0dB choice:

• If the selected compensator has no complex zeros, it is better to be conservative, choosing ω_0dB sufficiently below the frequency of the lightly damped poles of the converter (or the frequency of the right half-plane zeros if lower). However, because of the resonant peak of most converter transfer functions, the phase margin can be obtained at a frequency near the resonance. If the phase margin is not enough, the compensator gain must be lowered.

• If the selected compensator has complex zeros, ω_0dB can be chosen slightly above the frequency of the lightly damped poles.

3. Desired phase margin (φ_M) specification φ_M≥30° (preferably between 45 and 70°).

4. Compensator zero-pole placement to achieve the desired phase margin:

• With the integral plus zero-pole compensation type (35.51), the compensator phase φ_cp, at the maximum frequency of unity gain (often ω_0dB), equals the phase margin (φ_M) minus 180° and minus the converter phase φ_cv,(φ_cp=φ_M−180°−φ_cv). The zero-pole position can be obtained calculating the factor f_ct=tg (π/2+ φ_cp/2) being ω_z=ω_0dB/f_ct and ω_M=ω_0dBf_ct.

• With the PID notch filter type (35.52) controller, the two complex zeros are placed to have a damping factor ξ_p roughly equal to two times the damping ξ_z of the converter complex poles and oscillating frequency ω_0cp 30% smaller than the complex poles frequency ω_p. The high-frequency pole ω_p1 is placed to achieve the needed phase margin (ω_p1≈(ω_0cp·ω_0dB)^1/2f_ct² with f_ct=tg (π/2+φ_cp/2) and φ_cp=φ_M−180°−φ_cv [5]). Alternatively, since for stability ξ_z ω_0cp>ξ_p ω_p and ${\omega }_p\sqrt{1-{\xi }_p^2}>{\omega }_{0cp}\sqrt{1-{\xi }_z^2}$, it is obtained 1>ξ_z>ξ_p and using the geometric mean, ${\omega }_{0cp}={\omega }_p\sqrt{\frac{{\xi }_p}{{\xi }_z}\sqrt{\frac{1-{\xi }_p^2}{1-{\xi }_z^2}}}$.

5. Compensator gain calculation (the product of the converter and compensator gains at the ω_0dB frequency must be one).

6. Stability margin verification using Bode plots and root locus.

7. Result evaluation. Restarting the compensator selection and design, if the attained results are still not good enough.

Note that several modern computer programs, such as MATLAB, provide very easy and efficient tools (such as SISOTOOL) to design these controllers, even considering that switching power converters are discrete systems. Optimum controllers such as linear-quadratic-Gaussian (LQG, LQR) controllers, whose performances might be much better than linear controllers, can also be designed using the same tools. However, in evaluating optimum controller results, care must be taken in order to guarantee that the controlled converter frequency bandwidth and gain are not beyond the validity limits of the used models.