35.3 Sliding-Mode Control of Switching Power Converters

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Example 35.9

Output voltage control in three-phase voltage-source inverters using sinusoidal wave PWM (SPWM) and space-vector modulation (SVM)

Sinusoidal wave PWM

Voltage-source three-phase inverters (Fig. 35.21) are often used to drive squirrel cage induction motors (IM) in variable speed applications.

f35-21-9780128114070 — Fig. 35.21 IGBT-based voltage-sourced three-phase inverter with induction motor.

Considering almost ideal power semiconductors, the output voltage u_bk(k∈{1, 2, 3}) dynamics of the inverter is negligible as the output voltage can hardly be considered a state variable in the time scale describing the motor behavior. Therefore, the best known method to create sinusoidal output voltages uses an open-loop modulator with low-frequency sinusoidal waveforms sin(ωt), with the amplitude defined by the modulation index m_i (m_i∈[0, 1]), modulating high-frequency triangular waveforms r(t) (carriers) (Fig. 35.22), a process similar to the one described in Section 35.2.4. The frequency and phase of the triangular carriers should guarantee half-wave and quarter-wave symmetry for minimum harmonic distortion.

f35-22-9780128114070 — Fig. 35.22 (A) SPWM modulator schematic and (B) main SPWM signals.

This sinusoidal wave PWM (SPWM) modulator generates the variable γ_k, represented in Fig. 35.22 by the rectangular waveform, which describes the inverter k leg state:

$\begin{array}{l} γ_{k} & = \{\begin{array}{c} 1 \to when m_{i} sin (ω t) > r (t) \\ 0 \to when m_{i} sin (ω t) < r (t) \end{array} \end{array}$

si202_e (35.70)

The turn-on and turn-off signals for the k leg inverter switches are related with the variable γ_k as follows:

$\begin{array}{l} γ_{k} & = \{\begin{array}{c} 1 \to when S u_{k} is on and S l_{k} is off \\ 0 \to when S u_{k} is off and S l_{k} is on \end{array} \end{array}$

si203_e (35.71)

This applies constant-frequency sinusoidally weighted PWM signals to the gates of each insulated-gate bipolar transistor (IGBT). The PWM signals for all the upper IGBTs (Su_k, k∈{1, 2, 3}) must be 120° out of phase, and the PWM signal for the lower IGBT Sl_k must be the complement of the Su_k signal. Since transistor turn-on times are usually shorter than turn-off times, some dead time must be included between the Su_k and Sl_k pulses to prevent internal short circuits.

Sinusoidal PWM can be easily implemented using a microprocessor or two digital counters/timers generating the addresses for two lookup tables (one for the triangular function and another for supplying the per unit basis of the sine, whose frequency can vary). Tables can be stored in read-only memories, ROM, or erasable programmable ROM, EPROM. One multiplier for the modulation index (perhaps into the digital-to-analog (D/A) converter for the sine ROM output) and one hysteresis comparator must also be included.

With SPWM, the first harmonic maximum amplitude of the obtained line-to-line voltage is only about 86% of the inverter dc supply voltage V_a. Since it is expectable that this amplitude should be closer to V_a, different modulating voltages (e.g., adding a third-order harmonic with one-fourth of the fundamental sine amplitude) can be used as long as the fundamental harmonic of the line-to-line voltage is kept sinusoidal. Another way is to leave SPWM and consider the eight possible inverter output voltages trying to directly use then. This will lead to space-vector modulation.

Space vector modulation

Space vector modulation (SVM) is based on the polar representation (Fig. 35.23) of the eight possible base output voltages of the three-phase inverter (Table 35.1, where v_α and v_β are the vector components of vector ${\vec{V}}_{g}, g \in \{0, 1, 2, 3, 4, 5, 6, 7\}$ , obtained with Eq. (35.72)). Therefore, as all the available voltages can be used, SVM does not present the voltage limitation of SPWM.

f35-23-9780128114070 — Fig. 35.23 α and β space-vector representation of the three-phase bridge inverter leg base vectors.

Table 35.1

The three-phase inverter with eight possible γk combinations, vector numbers, and respective α and β components

γ₁	γ₂	γ₃	u_bk	u_bk−u_bk+1	v_α	v_β	Vector
0	0	0	0	0	0	0	${\vec{V}}_{0}$
1	0	0	γ_k V_a	(γ_k−γ_k+1)V_a	$\sqrt{2 / 3} V_{a}$	0	${\vec{V}}_{1}$
1	1	0	γ_k V_a	(γ_k−γ_k+1)V_a	$V_{a} / \sqrt{6}$	$V_{a} / \sqrt{2}$	${\vec{V}}_{2}$
0	1	0	γ_k V_a	(γ_k−γ_k+1)V_a	$- V_{a} / \sqrt{6}$	$V_{a} / \sqrt{2}$	${\vec{V}}_{3}$
0	1	1	γ_k V_a	(γ_k−γ_k+1)V_a	$- \sqrt{2 / 3} V_{a}$		${\vec{V}}_{4}$
1	1	1	V_a	0	0	0	${\vec{V}}_{7}$
1	0	1	γ_k V_a	(γ_k–γ_k+1)V_a	$V_{a} / \sqrt{6}$	$- V_{a} / \sqrt{2}$	${\vec{V}}_{6}$
0	0	1	γ_k V_a	(γ_k–γ_k+1)V_a	$- V_{a} / \sqrt{6}$	$- V_{a} / \sqrt{2}$	${\vec{V}}_{5}$

t0010

Furthermore, being a vector technique, SVM fits nicely with the vector control methods often used in IM drives:

$[\begin{array}{l} v_{α} \\ v_{β} \end{array}] = \sqrt{\frac{2}{3}} [\begin{array}{c} 1 & - 1 / 2 & - 1 / 2 \\ 0 & \frac{\sqrt{3}}{2} & - \frac{\sqrt{3}}{2} \end{array}] [\begin{array}{l} γ_{1} \\ γ_{2} \\ γ_{3} \end{array}] V_{a}$

si205_e (35.72)

Consider that the vector ${\vec{V}}_{s}$ (magnitude V_s and angle [Fcy]) must be applied to the IM. Since there is no such vector available directly, SVM uses an averaging technique to apply the two vectors, ${\vec{V}}_{1}$ and ${\vec{V}}_{2}$ , closest to ${\vec{V}}_{s}$ . The vector ${\vec{V}}_{1}$ will be applied during δ_AT_s, while vector ${\vec{V}}_{2}$ will last δ_BT_s (where 1/T_s is the inverter switching frequency and δ_A and δ_B are duty cycles, δ_A and δ_B∈[0, 1]). If there is any leftover time in the PWM period T_s, then the zero vector is applied during time δ₀T_s=T_s−δ_AT_s−δ_BT_s. Since there are two zero vectors ( ${\vec{V}}_{0}$ and ${\vec{V}}_{7}$ ), a symmetrical PWM can be devised, which uses both ${\vec{V}}_{0}$ and ${\vec{V}}_{7}$ , as shown in Fig. 35.24. Such a PWM arrangement minimizes the power semiconductor switching frequency and IM torque ripples

f35-24-9780128114070 — Fig. 35.24 Symmetrical SVM.

The input to the SVM algorithm is the space vector ${\vec{V}}_{s}$ , into the sector s_n, with magnitude V_s and angle Φ_s. This vector can be rotated to fit into sector 0 (Fig. 35.23) reducing Φ_s to the first sector, Φ=Φ_s−s_nπ/3. For any ${\vec{V}}_{s}$ that is not exactly along one of the six nonnull inverter base vectors (Fig. 35.23), SVM must generate an approximation by applying the two adjacent vectors during an appropriate amount of time. The algorithm can be devised considering that the projections of ${\vec{V}}_{s}$ , onto the two closest base vectors, are values proportional to δ_A and δ_B duty cycles. Using simple trigonometric relations in sector 0 (0<Φ<π/3) (Fig. 35.23) and considering K_T the proportional ratio, δ_A and δ_B are, respectively, $δ_{A} = K_{T} \bar{O A}$ and $δ_{B} = K_{T} \bar{O A}$ , yielding

$\begin{array}{l} δ_{A} = K_{T} \frac{2 V_{s}}{\sqrt{3}} sin (\frac{π}{3} - Φ) \\ δ_{B} = K_{T} \frac{2 V_{s}}{\sqrt{3}} sin Φ \end{array}$

si221_e (35.73)

The K_T value can be found if we notice that when ${\vec{V}}_{s} = {\vec{V}}_{1}$ , δ_A=1, and δ_B=0 (or when ${\vec{V}}_{s} = {\vec{V}}_{2}$ , δ_A=0, and δ_B=1). Therefore, since when ${\vec{V}}_{s} = {\vec{V}}_{1}$ , $V_{s} = \sqrt{v_{α}^{2} + v_{β}^{2}} = \sqrt{2 / 3} V_{a}$ , Φ=0, or when ${\vec{V}}_{s} = {\vec{V}}_{2}$ , $V_{s} = \sqrt{2 / 3} V_{a}$ , Φ=π/3, the K_T constant is $K_{T} = \sqrt{3} / (\sqrt{2} V_{a})$ . Hence,

$\begin{array}{l} δ_{A} = \frac{\sqrt{2} V_{s}}{V_{a}} sin (\frac{π}{3} - Φ) \\ δ_{B} = \frac{\sqrt{2} V_{s}}{V_{a}} sin Φ \\ δ_{0} = 1 - δ_{A} - δ_{B} \end{array}$

si229_e (35.74)

The obtained resulting vector ${\vec{V}}_{s}$ cannot extend beyond the hexagon of Fig. 35.23. This can be understood if the maximum magnitude V_sm of a vector with Φ=π/6 is calculated. For Φ=π/6, the maximum duty cycles are δ_A=1/2 and δ_B=1/2. Then, from Eq. (35.74,) $V_{s m} = V_{a} / \sqrt{2}$ is obtained. This magnitude is lower than that of the vector ${\vec{V}}_{1}$ since the ratio between these magnitudes is $\sqrt{3 / 2}$ . To generate sinusoidal voltages, the vector ${\vec{V}}_{s}$ must be inside the inner circle of Fig. 35.23, so that it can be rotated without crossing the hexagon boundary. Vectors with tips between this circle and the hexagon are reachable but produce nonsinusoidal line-to-line voltages.

For sector 0 (Fig. 35.23), SVM symmetrical PWM switching variables (γ₁, γ₂, and γ₃) and intervals (Fig. 35.24) can be obtained by comparing a triangular wave with amplitude u_cmax (Fig. 35.24, where r(t)=2u_cmaxt/T_s and t∈[0, T_s/2]), with the following values:

$\begin{array}{l} C_{0} = \frac{u_{c max}}{2} δ_{0} = \frac{u_{c max}}{2} (1 - δ_{A} - δ_{B}) \\ C_{A} = \frac{u_{c max}}{2} (\frac{δ_{0}}{2} + δ_{A}) = \frac{{u_{c}}_{max}}{2} (1 + δ_{A} - δ_{B}) \\ C_{B} = \frac{{u_{c}}_{max}}{2} (\frac{δ_{0}}{2} + δ_{A} + δ_{B}) = \frac{{u_{c}}_{max}}{2} (1 + δ_{A} + δ_{B}) \end{array}$

si235_e (35.75)

Extension of Eq. (35.75) to all six sectors can be done if the sector number s_n is considered, together with the auxiliary matrix Ξ:

$Ξ^{T} = [\begin{array}{c} - 1 & - 1 & 1 & 1 & 1 & - 1 \\ - 1 & 1 & 1 & 1 & - 1 & - 1 \end{array}]$

si236_e (35.76)

Generalization of the values C₀, C_A, and C_B, denoted C_0sn, C_Asn, and C_Bsn, is written in Eq. (35.77), knowing that, for example, $Ξ_{({(s n + 4)}_{mod 6} + 1)}$ is the Ξ matrix row with number (s_n+4)_{mod 6}+1:

$\begin{array}{l} C_{0 s n} = \frac{{u_{c}}_{max}}{2} (1 + Ξ_{({(S n)}_{\mod 6} + 1)} [\begin{array}{c} δ_{A} \\ δ_{B} \end{array}]) \\ C_{A s n} = \frac{{u_{c}}_{max}}{2} (1 + Ξ_{({(S n + 4)}_{\mod 6} + 1)} [\begin{array}{c} δ_{A} \\ δ_{B} \end{array}]) \\ C_{B s n} = \frac{{u_{c}}_{max}}{2} (1 + Ξ_{({(S n + 2)}_{\mod 6} + 1)} [\begin{array}{c} δ_{A} \\ δ_{B} \end{array}]) \end{array}$

si238_e (35.77)

Therefore, γ₁, γ₂, and γ₃ are

$\begin{array}{l} γ_{1} = \{\begin{array}{c} 0 \to when r (t) < C_{0 s n} \\ 1 \to when r (t) > C_{0 s n} \end{array} \\ γ_{2} = \{\begin{array}{c} 0 \to when r (t) < C_{A s n} \\ 1 \to when r (t) > C_{A s n} \end{array} \\ γ_{3} = \{\begin{array}{c} 0 \to when r (t) < C_{B s n} \\ 1 \to when r (t) > C_{B s n} \end{array} \end{array}$

si239_e (35.78)

Supposing that the space vector ${\vec{V}}_{s}$ is now specified in the orthogonal coordinates $α, β ({\vec{V}}_{α}, {\vec{V}}_{β})$ , instead of magnitude V_s and angle Φ_s, the duty cycles δ_A and δ_B can be easily calculated knowing that v_α=V_s cos Φ and v_β=V_s sin Φ and using Eq. (35.74):

$\begin{array}{l} δ_{A} = \frac{\sqrt{2}}{2 V_{a}} (\sqrt{3} v_{α} - v_{β}) \\ δ_{B} = \frac{\sqrt{2}}{V_{a}} v_{β} \end{array}$

si242_e (35.79)

This equation enables the use of Eqs. (35.77) and (35.78) to obtain SVM in orthogonal coordinates.

Using SVM or SPWM, the closed-loop control of the inverter output currents (induction motor stator currents) can be performed using an approach similar to that outlined in Example 35.6 and decoupling the currents expressed in a d and q rotating frame.

35.3 Sliding-Mode Control of Switching Power Converters

35.3.1 Introduction

All the designed controllers for switching power converters are in fact variable structure controllers, in the sense that the control action changes rapidly from one to another of, usually, two possible δ(t) values, cyclically changing the converter topology. This is accomplished by the modulator (Fig. 35.6), which creates the switching variable δ(t) imposing δ(t)=1 or δ(t)=0, to turn on or off the power semiconductors. As a consequence of this discontinuous control action, indispensable for efficiency reasons, state trajectories move back and forth around a certain average surface in the state-space, and variables present some ripple. To avoid the effects of this ripple in the modeling and to apply linear control methodologies to time-variant systems, average values of state variables and state-space averaged models or circuits were presented (Section 35.2). However, a nonlinear approach to the modeling and control problem, taking advantage of the inherent ripple and variable structure behavior of switching power converters, instead of just trying to live with them, would be desirable, especially if enhanced performances could be attained.

In this approach, switching power converter topologies, as discrete nonlinear time-variant systems, are controlled to switch from one dynamics to another when just needed. If this switching occurs at a very high frequency (theoretically infinite), the state dynamics described as in Eq. (35.4) can be enforced to slide along a certain prescribed state-space trajectory. The converter is said to be in sliding mode, the allowed deviations from the trajectory (the ripple) imposing the practical switching frequency.

Sliding-mode control of variable structure systems, such as switching power converters, is particularly interesting because of the inherent robustness [10,11], capability of system order reduction, and appropriateness to the on/off switching of power semiconductors. The control action, being the control equivalent of the management paradigm “just in time” (JIT), provides timely and precise control actions, determined by the control law and the allowed ripple. Therefore, the switching frequency is not constant over all operating regions of the converter.

This section treats the derivation of the control (sliding surface) and switching laws, robustness, stability, constant-frequency operation, and steady-state error elimination necessary for sliding-mode control of switching power converters, also giving some examples.

35.3.2 Principles of Sliding-mode Control

Consider the state-space switched model Eq. (35.4) of a switching converter subsystem and input-output linearization or another technique, to obtain, from state-space equations, one Eq. (35.80), for each controllable subsystem output y=x. In the controllability canonical form [12] (also known as input-output decoupled or companion form), Eq. (35.80) is

$\begin{array}{l} \frac{d}{d t} {[x_{h}, \dots x_{j - 1}, x_{j}]}^{T} = [x_{h + 1}, \dots, x_{j}, - f_{h} (x) - p_{h} (t) \\ + b_{h} (x) u_{h} (t)]^{T} \end{array}$

si243_e (35.80)

where x=[x_h, …, x_j−1, x_j]^T is the subsystem state vector, f_h(x) and b_h(x) are functions of x, p_h(t) represents the external disturbances, and u_h(t) is the control input. In this special form of state-space modeling, the state variables are chosen so that the x_i+1 variable (i∈{h, …, j−1}) is the time derivative of x_i, that is, $x = {[x_{h} {\dot{x}}_{h} x_{h} \dots {\overset{m}{x}}_{h}]}^{T}$ , where m=j−h [13].

35.3.2.1 Control Law (Sliding Surface)

The required closed-loop dynamics for the subsystem output vector y=x can be chosen to verify Eq. (35.81) with selected k_i values. This is a model reference adaptive control approach to impose a state trajectory that advantageously reduces the system order (j−h+1):

$\frac{d x_{j}}{d t} = - \sum_{i = h}^{j - 1} \frac{k_{i}}{k_{j}} x_{i + 1}$

si245_e (35.81)

Effectively, in a single-input single-output (SISO) subsystem, the order is reduced by unity, applying the restriction Eq. (35.81). In a multiple-input multiple-output (MIMO) system, in which ν-independent restrictions could be imposed (usually with ν degrees of freedom), the order could often be reduced in ν units. Indeed, from Eq. (35.81), the dynamics of the jth term of x is linearly dependent from the j−h first terms:

$\frac{d x_{j}}{d t} = - \sum_{i = h}^{j - 1} \frac{k_{i}}{k_{j}} x_{i + 1} = - \sum_{i = h}^{j - 1} \frac{k_{i}}{k_{j}} \frac{d x_{i}}{d t}$

si246_e (35.82)

The controllability canonical model allows the direct calculation of the needed control input to achieve the desired dynamics Eq. (35.81). In fact, as the control action should enforce the state vector x, to follow the reference vector $x_{r} = {[x_{h_{r}} {\dot{x}}_{h_{r}} {\ddot{x}}_{h_{r}} \dots {\overset{m}{x}}_{h_{r}}]}^{T}$ , the tracking error vector will be $e = {[x_{h_{r}} - x_{h}, \dots, x_{j - 1 r} - x_{j - 1}, x_{j r} - x_{j}]}^{T}$ or $e = {[e_{x_{h}} \dots e_{x j - 1} e_{x_{j}}]}^{T}$ . Thus, equating the subexpressions for dx_j/dt of Eqs. (35.80) and (35.81), the necessary control input u_h(t) is

$\begin{array}{l} u_{h} (t) = \frac{p_{h} (t) + f_{h} (x) + \frac{d x_{j}}{d t}}{b_{h} (x)} \\ = \frac{p_{h} (t) + f_{h} (x) - \sum_{i = h}^{j - 1} \frac{k_{i}}{k_{j}} x_{i + 1_{r}} + \sum_{i = h}^{j - 1} \frac{k_{i}}{k_{j}} {e_{x}}_{_{i + 1}}}{b_{h} (x)} \end{array}$

si250_e (35.83)

This expression is the required closed-loop control law, but unfortunately, it depends on the system parameters and on external perturbations and is difficult to compute. Moreover, for some output requirements, Eq. (35.83) would give extremely high values for the control input u_h(t), which would be impractical or almost impossible.

In most switching power converters, u_h(t) is discontinuous. Yet, if we assume one or more discontinuity borders dividing the state-space into subspaces, the existence and uniqueness of the solution are guaranteed out of the discontinuity borders, since in each subspace the input is continuous. The discontinuity borders are subspace switching hypersurfaces, whose order is the space order minus one, along which the subsystem state slides, since its intersections with the auxiliary equations defining the discontinuity surfaces can give the needed control input.

Within the sliding-mode control (SMC) theory, assuming a certain dynamic error driven to zero, one auxiliary equation (sliding surface) and the equivalent control input u_h(t) can be obtained, integrating both sides of Eq. (35.82) with null initial conditions:

$k_{i} x_{j} + \sum_{i = h}^{j - 1} k_{i} x_{i} = \sum_{i = h}^{j} k_{i} x_{i} = 0$

si251_e (35.84)

This equation represents the discontinuity surface (hyperplane) and just defines the necessary sliding surface S(x_i, t) to obtain the prescribed dynamics of Eq. (35.81):

$S (x_{i}, t) = \sum_{i = h}^{j} k_{i} x_{i} = 0$

si252_e (35.85)

In fact, by taking the first time derivative of S(x_i, t), $\dot{S} (x_{i}, t) = 0$ ; solving it for dx_j/dt; and substituting the result in Eq. (35.83), the dynamics specified by Eq. (35.81) is obtained. This means that the control problem is reduced to a first-order problem, since it is only necessary to calculate the time derivative of Eq. (35.85) to obtain the dynamics (35.81) and the needed control input u_h(t).

The sliding surface Eq. (35.85), as the dynamics of the converter subsystem, must be a Routh-Hurwitz polynomial and verify the sliding manifold invariance conditions, S(x_i, t)=0 and $\dot{S} (x_{i}, t) = 0$ . Consequently, the closed-loop controlled system behaves as a stable system of order j−h, whose dynamics is imposed by the coefficients k_i, which can be chosen by pole placement of the poles of the order m=j−h polynomial. Alternatively, certain kinds of polynomials can be advantageously used [14]: Butterworth, Bessel, Chebyshev, elliptic (or Cauer), binomial, and minimum integral of time absolute error product (ITAE). Most useful are Bessel polynomials B_E(s) (Eq. (35.88)), which minimize the system response time t_r, providing no overshoot; the polynomials I_TAE(s) (Eq. (35.87)) that minimize the ITAE criterion for a system with desired natural oscillating frequency ω_o; and binomial polynomials B_I(s) (Eq. (35.86)). For m>1, ITAE polynomials give faster responses than binomial polynomials:

$\begin{array}{l} B_{I} {(s)}_{m} = {(s + ω_{o})}^{m} \\ = \{\begin{cases} m = 0 \Rightarrow B_{I} (s) = 1 \\ m = 1 \Rightarrow B_{I} (s) = s + ω_{o} \\ m = 2 \Rightarrow B_{I} (s) = s^{2} + 2 ω_{o} s + ω_{0}^{2} \\ m = 3 \Rightarrow B_{I} (s) = s^{3} + 3 ω_{o} s^{2} + 3 ω_{0}^{2} s + ω_{0}^{3} \\ m = 4 \Rightarrow B_{I} (s) = s^{4} + 4 ω_{o} s^{3} + 6 ω_{0}^{2} s^{2} + 4 ω_{0}^{3} s + ω_{0}^{4} \\ \dots \end{cases} \end{array}$

si255_e (35.86)

$\begin{array}{l} I_{T A E} {(s)}_{m} = \{\begin{cases} m = 0 \Rightarrow I_{T A E} (s) = 1 \\ m = 1 \Rightarrow I_{T A E} (s) = s + ω_{o} \\ m = 2 \Rightarrow I_{T A E} (s) = s^{2} + 1.4 ω_{o} s + ω_{0}^{2} \\ m = 3 \Rightarrow I_{T A E} (s) = s^{3} + 1.75 ω_{o} s^{2} + 2.15 ω_{0}^{2} s + ω_{0}^{3} \\ \begin{array}{l} m = 4 \Rightarrow I_{T A E} (s) = s^{4} + 2.1 ω_{o} s^{3} + 3.4 ω_{0}^{2} s^{2} \\ + 2.7 ω_{0}^{3} s + ω_{0}^{4} \end{array} \\ \dots \end{cases} \end{array}$

si256_e (35.87)

$\begin{array}{l} B_{E} {(s)}_{m} = \{\begin{cases} \begin{array}{l} m = 0 \Rightarrow B_{E} (s) & = 1 \\ m = 1 \Rightarrow B_{E} (s) & = s t_{r} + 1 \\ m = 2 \Rightarrow B_{E} (s) & = \frac{{(s t_{r})}^{2} + 3 s t_{r} + 3}{3} \\ m = 3 \Rightarrow B_{E} (s) & = \frac{({(s t_{r})}^{2} + 3.678 s t_{r} + 6.459) (s t_{r} + 2.322)}{15} \\ = \frac{{(s t_{t})}^{3} + 6 {(s t_{r})}^{2} + 15 s t_{r} + 15}{15} \\ m = 4 \Rightarrow B_{E} (s) & = \frac{{(s t_{r})}^{4} + 10 {(s t_{r})}^{3} + 45 {(s t_{r})}^{2} + 105 (s t_{r}) + 105}{105} \end{array} \\ \dots \end{cases} \end{array}$

si257_e (35.88)

These polynomials can be the reference model for this model reference adaptive control method.

35.3.2.2 Closed-Loop Control Input–Output Decoupled Form

For closed-loop control applications, instead of the state variables x_i, it is worthy to consider, as new state variables, the errors $e_{x_{i}}$ , components of the error vector $e = {[e_{x_{h}} {\dot{e}}_{x_{h}} e_{x_{h}} \dots {\overset{m}{x}}_{x_{h}}]}^{T}$ of the state-space variables x_i, relative to a given reference $x_{i_{r}}$ Eq. (35.90). The new controllability canonical model of the system is

$\frac{d}{d t} {[e_{x_{h}} \dots e_{x_{j - 1}} e_{x_{j}}]}^{T} = {[e_{x_{h + 1}}, \dots, e_{x_{j}}, - f_{e} (e) + p_{e} (t) - b_{e} (e) u_{h} (t)]}^{T}$

(35.89)

where f_e(e), p_e(t), and b_e(e) are functions of the error vector e.

As the transformation of variables

$\begin{array}{l} e_{x_{i}} = x_{i_{r}} - x_{i} & with & i = h, \dots, j \end{array}$

(35.90)

is linear, the Routh-Hurwitz polynomial for the new sliding surface $S (e_{x_{i}}, t)$ is

$S (e_{x_{i}}, t) = \sum_{i = h}^{j} k_{i} e_{x_{i}} = 0$

si264_e (35.91)

Since $e_{x_{i + 1}} (s) = s e_{x_{i}} (s)$ , this control law, from Eqs. (35.86) to (35.88), can be written as $S (e, s) = e_{x_{i}} {(s + ω_{o})}^{m}$ and is robust as it does not depend on circuit parameters, disturbances, or operating conditions, but only on the imposed k_i parameters and on the state variable errors $e_{x_{i}}$ , which can usually be measured or estimated. The control law Eq. (35.91) enables the desired dynamics of the output variable(s), if the semiconductor switching strategy is designed to guarantee the system stability. In practice, the finite switching frequency of the semiconductors will impose a certain dynamic error ɛ steered to zero. The control law Eq. (35.91) is the required controller for the closed-loop SISO subsystem with output y.

35.3.2.3 Stability

Existence condition. The existence of the operation in sliding mode implies $S (e_{x_{i}}, t) = 0$ . Also, to stay in this regime, the control system should guarantee $\dot{S} (e_{x_{i}}, t) = 0$ . Therefore, the semiconductor switching law must ensure the stability condition for the system in sliding mode, written as

$S (e_{x_{i}}, t) \dot{S} (e_{x_{i}}, t) < 0$

(35.92)

The fulfillment of this inequality ensures the convergence of the system state trajectories to the sliding surface $S (e_{x_{i}}, t) = 0$ , since

– if $S (e_{x_{i}}, t) > 0$ and $\dot{S} (e_{x_{i}}, t) < 0$ , then $S (e_{x_{i}}, t)$ will decrease to zero,

– if $S (e_{x_{i}}, t) < 0$ and $\dot{S} (e_{x_{i}}, t) > 0$ , then $S (e_{x_{i}}, t)$ will increase toward zero.

Hence, if Eq. (35.92) is verified, then $S (e_{x_{i}}, t)$ will converge to zero. The condition (35.92) is the manifold $S (e_{x_{i}}, t)$ invariance condition or the sliding-mode existence condition.

Given the state-space model Eq. (35.89) as a function of the error vector e and, from $\dot{S} (e_{x_{i}}, t) = 0$ , the equivalent average control input U_eq(t) that must be applied to the system in order that the system state slides along the surface, Eq. (35.91) is given by

$U_{eq} (t) = \frac{k_{h} \frac{d e_{x_{h}}}{d t} + k_{h + 1} \frac{d e_{x_{h + 1}}}{d t} + \dots + k_{j - 1} + \frac{d e_{x_{j - 1}}}{d t} + k_{j} (- f_{e} (e) + p_{e} (t))}{k_{j} b_{e} (e)}$

si281_e (35.93)

This control input U_eq(t) ensures the converter subsystem operation in the sliding mode.

Reaching condition. The fulfillment of $S ({e_{x}}_{_{i}}, t) \dot{S} ({e_{x}}_{_{i}}, t) < 0$ as $S ({e_{x}}_{_{i}}, t) \dot{S} ({e_{x}}_{_{i}}, t) = (1 / 2) {\dot{S}}^{2} ({e_{x}}_{_{i}}, t)$ implies that the distance between the system state and the sliding surface will tend to zero, since $S^{2} ({e_{x}}_{_{i}}, t)$ can be considered as a measure for this distance. This means that the system will reach sliding mode. Additionally, from Eq. (35.89), it can be written as

$\frac{d e_{x_{j}}}{d t} = - f_{e} (e) + p_{e} (t) - b_{e} (e) u_{h} (t)$

si285_e (35.94)

From Eq. (35.91), Eq. (35.95) is obtained:

$\begin{array}{l} S ({e_{x}}_{_{i}}, t) = \sum_{i = h}^{j} k_{i} e_{x_{i}} = k_{h} e_{x_{h}} + k_{h + 1} \frac{d e_{x_{h}}}{d t} + k_{h + 2} \frac{d^{2} e_{x_{h}}}{d t^{2}} \\ + \dots + k_{j} \frac{d^{m} e_{x_{h}}}{d t^{m}} \end{array}$

si286_e (35.95)

If $S (e_{x_{i}}, t) > 0$ , from the Routh-Hurwitz property of Eq. (35.91), then $e_{x_{j}} > 0$ . In this case, to reach $S (e_{x_{i}}, t) = 0$ , it is necessary to impose−b_e(e)u_h(t)=−U in Eq. (35.94), with U chosen to guarantee $d e_{x_{j}} / d t < 0$ . After a certain time, $e_{x_{j}}$ will be $e_{x_{j}} = d^{m} e_{x_{h}} / d t^{m} < 0$ , implying along with Eq. (35.95) that $\dot{S} (e_{x_{i}}, t) < 0$ , thus verifying Eq. (35.92). Therefore, every term of $S (e_{x_{i}}, t)$ will be negative, which implies, after a certain time, an error $e_{x_{h}} < 0$ and $S (e_{x_{i}}, t) < 0$ . Hence, the system will reach sliding mode, staying there if U=U_eq(t). This same reasoning can be made for $S (e_{x_{i}}, t) < 0$ ; it is now being necessary to impose−b_e(e)u_h(t)=+U, with U high enough to guarantee $d e_{x_{j}} / d t > 0$ .

To ensure that the system always reaches sliding-mode operation, it is necessary to calculate the maximum value of U_eq(t) and U_eqmax and also impose the reaching condition:

$U > U_{eqmax}$

(35.96)

This means that the power supply voltage values U should be chosen high enough to additionally account for the maximum effects of the perturbations. With step inputs, even with U>U_eqmax, the converter usually loses sliding mode, but it will reach it again, even if the U_eqmax is calculated considering only the maximum steady-state values for the perturbations.

35.3.2.4 Switching Law

From the foregoing considerations, supposing a system with two possible structures, the semiconductor switching strategy must ensure $S ({e_{x}}_{_{i}}, t) \dot{S} ({e_{x}}_{_{i}}, t) < 0$ . Therefore, if $S (e_{x_{i}}, t) > 0$ , then $\dot{S} (e_{x_{i}}, t) < 0$ , which implies, as seen, −b_e(e)u_h(t)=−U (the sign of b_e(e) must be known). Also, if $S (e_{x_{i}}, t) < 0$ , then $\dot{S} (e_{x_{i}}, t) > 0$ , which implies−b_e(e)u_h(t)=+U. This imposes the switching between two structures at infinite frequency. Since power semiconductors can only switch at finite frequency, in practice, a small enough error for $S (e_{x_{i}}, t)$ must be allowed $(- ɛ < S (e_{x_{i}}, t) < + ɛ)$ . Hence, the switching law between the two possible system structures might be

$u_{h} (t) = \{\begin{array}{c} U / b_{e} (e) f o r S (e_{x_{i}}, t) > + ε \\ - U / b_{e} (e) f o r S (e_{x_{i}}, t) < - ε \end{array}$

si307_e (35.97)

The condition Eq. (35.97) determines the control input to be applied and therefore represents the semiconductor switching strategy or switching function. This law determines a two-level pulse-width modulator with JIT switching (variable frequency).

35.3.2.5 Robustness

The dynamics of a system, with closed-loop control using the control law Eq. (35.91) and the switching law Eq. (35.97), does not depend on the system operating point, load, circuit parameters, power supply, or bounded disturbances, as long as the control input u_h(t) is large enough to maintain the converter subsystem in sliding mode. Therefore, it is said that the switching power converter dynamics, operating in sliding mode, is robust against changing operating conditions, variations of circuit parameters, and external disturbances. The desired dynamics for the output variable(s) is determined only by the k_i coefficients of the control law Eq. (35.91), as long as the switching law (35.97) maintains the converter in sliding mode.

35.3.3 Constant-Frequency Operation

Prefixed switching frequency can be achieved, even with the sliding-mode controllers, at the cost of losing the JIT action. As the sliding-mode controller changes the control input when needed and not at a certain prefixed rhythm, applications needing constant switching frequency (such as thyristor rectifiers or resonant converters) must compare S(e_xi, t) (hysteresis width 2ι much narrower than 2ɛ) with auxiliary triangular waveforms (Fig. 35.25A), auxiliary sawtooth functions (Fig. 35.25B), three-level clocks (Fig. 35.25C), or phase-locked loop control of the comparator hysteresis variable width 2ɛ [15]. However, as illustrated in Fig. 35.25D, steady-state errors do appear. Often, they should be eliminated as described in Section 35.3.4.

f35-25-9780128114070 — Fig. 35.25 Auxiliary functions and methods to obtain constant switching frequency with sliding-mode controllers. (A) Using a triangular carrier; (B) using a sawtooth carrier; (C) using a three-level clock; (D) generation of steady-state error.

35.3.4 Steady-State Error Elimination in Converters with Continuous Control Inputs

In the ideal sliding mode, state trajectories are directed toward the sliding surface (35.91) and move exactly along the discontinuity surface, switching between the possible system structures at infinite frequency. Practical sliding modes cannot switch at infinite frequency and therefore exhibit phase-plane trajectory oscillations inside a hysteresis band of width 2ɛ, centered in the discontinuity surface.

The switching law Eq. (35.91) permits no steady-state errors as long as $S (e_{x_{i}}, t)$ tends to zero, which implies no restrictions on the commutation frequency. Control circuits operating at constant frequency or needed continuous inputs or particular limitations of the power semiconductors, such as minimum on or off times, can originate $S (e_{x_{i}}, t) = ɛ_{1} \neq 0$ . The steady-state error $(e_{x_{h}})$ of the x_h variable $x_{h_{r}} - x_{h} = ɛ_{1} / k_{h}$ can be eliminated, increasing the system order by 1. The new state-space controllability canonical form, considering the error $e_{x_{j}}$ , between the variables and their references, as the state vector, is

$\begin{array}{l} \frac{d}{d t} {[\int e_{x_{h}} d t, e_{x_{h}}, \dots, e_{x_{j - 1}}, e_{x_{j}}]}^{T} \\ = {[e_{x_{h}}, e_{x_{h + 1}}, \dots, e_{x_{j}}, - f_{e} (e) - p_{e} (t) - b_{e} (e) u_{h} (t)]}^{T} \end{array}$

si313_e (35.98)

The new sliding surface $S (e_{x_{i}}, t)$ written from Eq. (35.91), considering the new system in Eq. (35.98), is

$S (e_{x_{i}}, t) = k_{0} \int e_{x_{h}} d t + \sum_{i = h}^{j} k_{i} e_{x_{i}} = 0$

si315_e (35.99)

This sliding surface offers zero-state error, even if $S (e_{x_{i}}, t) = ɛ_{1}$ due to the hardware errors or fixed (or limited) frequency switching. Indeed, at the steady state, the only nonzero term is $k_{0} \int e_{x_{h}} d t = ɛ_{1}$ . Also, like Eq. (35.91), this closed-loop control law does not depend on system parameters or perturbations to ensure a prescribed closed-loop dynamics similar to Eq. (35.81) with an error approaching zero.

The approach outlined herein precisely defines the control law (sliding surface (35.91) or (35.99)) needed to obtain the selected dynamics and the switching law Eq. (35.97). As the control law allows the implementation of the system controller and the switching law gives the PWM modulator, there is no need to design linear or nonlinear controllers, based on linear converter models, or devise offline PWM modulators. Therefore, sliding-mode control theory, applied to switching power converters, provides a systematic method to generate both the controller(s) (usually nonlinear) and the modulator(s) that will ensure a model reference robust dynamics, solving the control problem of switching power converters.

In the next examples, it is shown that the sliding-mode controllers use (nonlinear) state feedback, therefore, needing to measure the state variables and often other variables, since they use more system information. This is a disadvantage since more sensors are needed. However, the straightforward control design and obtained performances are much better than those obtained with the averaged models, the use of more sensors being really valued. Alternatively to the extra sensors, state observers can be used [12,13].

35.3.5 Examples: Buck–Boost DC/DC Converter, Half-bridge Inverter, 12-Pulse Parallel Rectifiers, Audio Power Amplifiers, Near Unity Power Factor Rectifiers, Multilevel Inverters, Matrix Converters

Example 35.10

Sliding-mode control of the buck–boost dc/dc converter

Consider again the buck-boost converter of Fig. 35.1 and assume the converter output voltage v_o to be the controlled output. From Section 35.2, using the switched state-space model of Eq. (35.11), making dv_o/dt=θ, and calculating the first time derivative of θ, the controllability canonical model (35.100), where i_o=v_o/R_o, is obtained:

$\begin{array}{l} \frac{d v_{o}}{d t} = θ = \frac{1 - δ (t)}{C_{o}} i_{L} - \frac{i_{o}}{C_{o}} \\ \frac{d θ}{d t} = - \frac{{(1 - δ (t))}^{2}}{L_{i} C_{o}} v_{o} - \frac{C_{o} θ + i_{o}}{C_{o} (1 - δ (t))} \frac{d δ (t)}{d t} \\ - \frac{1}{C_{o}} \frac{d i_{o}}{d t} + \frac{δ (t) (1 - δ (t))}{C_{o} L_{i}} V_{D C} \end{array}$

si318_e (35.100)

This model, written in the form of Eq. (35.80), contains two state variables, v_o and θ. Therefore, from Eq. (35.91) and considering $e_{v_{o}} = v_{o_{r}} - v_{o}, e_{θ} = θ_{r} - θ$ , the control law (sliding surface) is

$\begin{array}{l} S (e_{x_{i}}, t) = \sum_{i = h}^{2} k_{i} e_{x_{i}} = k_{1} (v_{o_{r}} - v_{o}) + k_{2} \frac{d v_{o_{r}}}{d t} - k_{2} \frac{d v_{o}}{d t} \\ = k_{1} (v_{o_{r}} - v_{o}) + k_{2} \frac{d v_{or}}{d t} - \frac{k_{2}}{C_{o}} (1 - δ (t)) i_{L} \\ + \frac{k_{2}}{C_{o}} i_{o} = 0 \end{array}$

si320_e (35.101)

This sliding surface depends on the variable δ(t), which should be precisely the result of the application, in Eq. (35.101), of a switching law similar to Eq. (35.97). Assuming an ideal up-down converter and slow variations, from Eq. (35.31), the variable δ(t) can be averaged to δ₁=v_o/(v_o+V_DC). Substituting this relation in Eq. (35.101) and rearranging, Eq. (35.102) is derived:

$\begin{array}{l} S (e_{x_{i}}, t) = \frac{C_{o} k_{1}}{k_{2}} (\frac{v_{o} + V_{D C}}{v_{o}}) \\ \times (({v_{o}}_{r} - v_{0}) + \frac{k_{2}}{k_{1}} \frac{d v_{o_{r}}}{d t} + \frac{k_{2}}{k_{1}} \frac{1}{C_{o}} i_{o}) - i_{L} = 0 \end{array}$

si321_e (35.102)

This control law shows that the power supply voltage V_DC must be measured, as well as the output voltage v_o and the currents i_o and i_L.

To obtain the switching law from stability considerations (35.92), the time derivative of $S (e_{x_{i}}, t)$ , supposing (v_o+V_DC)/v_o almost constant, is

$\begin{array}{l} \dot{S} (e_{x_{i}}, t) = \frac{C_{o} k_{1}}{k_{2}} (\frac{v_{o} + V_{D C}}{v_{o}}) \\ \times (\frac{d e_{v_{o}}}{d t} + \frac{k_{2}}{k_{1}} \frac{d^{2} v_{o_{r}}}{d t^{2}} + \frac{k_{2}}{k_{1} C_{o}} \frac{d i_{o}}{d t}) - \frac{d i_{L}}{d t} \end{array}$

si323_e (35.103)

If $S (e_{x_{i}}, t) > 0$ , then, from Eq. (35.92), $\dot{S} (e_{x_{i}}, t) < 0$ must hold. Analyzing Eq. (35.103), we can conclude that, if $S (e_{x_{i}}, t) > 0$ , $\dot{S} (e_{x_{i}}, t)$ is negative if, and only if, di_L/dt>0. Therefore, for positive errors $e_{v_{o}} > 0$ , the current i_L must be increased, which implies δ(t)=1. Similarly, for $S (e_{x_{i}}, t) < 0$ , di_L/dt<0, and δ(t)=0. Thus, a switching law similar to Eq. (35.97) is obtained:

$δ (t) = \{\begin{array}{c} 1 & f o r S (e_{x_{i}}, t) > + e \\ 0 & f o r S (e_{x_{i}}, t) < - e \end{array}$

si330_e (35.104)

The same switching law could be obtained from knowing the dynamic behavior of this nonminimum-phase up-down converter: to increase (decrease) the output voltage, a previous increase (decrease) of the i_L current is mandatory.

Eq. (35.101) shows that, if the buck-boost converter is into the sliding mode $(S (e_{x_{i}}, t) = 0)$ , the dynamics of the output voltage error tends exponentially to zero with time constant k₂/k₁ (k₂/k₁>0). Since during step transients, the converter is in the reaching mode, the time constant k₂/k₁ cannot be designed to originate error variations larger than the one allowed by the self-dynamics of the converter excited by a certain maximum permissible i_L current. Given the polynomials (35.86–35.88) with m=1, k₁/k₂=ω_o should be much lower than the finite switching frequency (1/T) of the converter. Therefore, the time constant must obey k₂/k₁≫T. Then, knowing that k₂ and k₁ are both imposed, the control designer can tailor the time constant as needed, provided that the above restrictions are observed.

Short-circuit-proof operation for the sliding-mode-controlled buck-boost converter can be derived from Eq. (35.102), noting that all the terms to the left of i_L represent the set point for this current. Therefore, limiting these terms (Fig. 35.26, saturation block, with i_Lmax=40 A), the switching law (35.104) ensures that the output current will not rise above the maximum imposed limit. Given the converter nonminimum-phase behavior, this i_L current limit is fundamental to reach the sliding mode of operation with step disturbances.

f35-26-9780128114070 — Fig. 35.26 (A) Block diagram of the sliding-mode nonlinear controller for the buck-boost converter and (B) transient responses of the sliding-mode-controlled buck-boost converter. At t=0.005 s, v_oref step from 23 to 26 V. At t=0.02 s, V_DC step from 26 to 23 V. Top graph: step reference v_oref and output voltage v_o. Bottom graph: trace starting at 20 is i_L current; trace starting at zero is 10×(v_oref−v_o).

The block diagram (Fig. 35.26A) of the implemented control law Eq. (35.102) (with C_ok₁/k₂=4) and switching law (35.103) (with ɛ=0.3) does not include the time derivative of the reference $(d v_{o_{r}} / d t)$ since, in a dc/dc converter, its value is considered zero. The controller hardware (or software), derived using just the sliding-mode approach, operates only in a closed-loop.

The resulting performance (Fig. 35.26B) is much better than that obtained with the PID notch filter (compare with Example 35.4 and Fig. 35.9B), with a higher response speed and robustness against power supply variations.

35.3.5.1 Example 35.11: Sliding Mode Control of the Single-Phase Half-Bridge Converter

Consider the half-bridge four-quadrant converter of Fig. 35.27 with the output filter and the inductive load (V_DCmax=300 V, V_DCmin=230 V, R_i=0.1 Ω, L_o=4 mH, C_o=470 μF, inductive load with nominal values R_o=7 Ω, and L_o=1 mH).

f35-27-9780128114070 — Fig. 35.27 Half-bridge power inverter with insulated-gate bipolar transistors, output filter, and load.

Assuming that power switches, output filter capacitor, and power supply are all ideal and a generic load with allowed slow variations, the switched state-space model of the converter, with state variables v_o and i_L, is

$\begin{array}{l} \frac{d}{d t} [\begin{array}{c} v_{o} \\ i_{L} \end{array}] = [\begin{array}{c} 0 & 1 / C_{o} \\ - 1 / L_{o} & - R_{i} / L_{o} \end{array}] [\begin{array}{c} v_{o} \\ i_{L} \end{array}] \\ + [\begin{array}{c} - 1 / C_{o} & 0 \\ 0 & 1 / L_{o} \end{array}] [\begin{array}{c} i_{o} \\ δ (t) V_{D C} \end{array}] \end{array}$

si333_e (35.105)

where i_o is the generic load current and v_PWM=δ(t)V_DC is the extended PWM output voltage (δ(t)=+1 when one of the upper main semiconductors of Fig. 35.27 is conducting and δ(t)=−1 when one of the lower semiconductors is on).

Output current control (current-mode control)

To perform as a $v_{i_{L}}$ voltage-controlled i_L current source (or sink) with transconductance g_m $(g_{m} = i_{L} / v_{i_{L}})$ , this converter must supply a current i_L to the output inductor, obeying $i_{L} = g_{m} v_{i_{L}}$ . Using a bounded $v_{i_{L}}$ voltage to provide output short-circuit protection, the reference current for a sliding-mode controller must be ${i_{L}}_{_{r}} = g_{m} v_{i L}$ . Therefore, the controlled output is the i_L current, and the controllability canonical model (35.106) is obtained from the second equation of (35.105), since the dynamics of this subsystem, being governed by δ(t)V_DC, is already in the controllability canonical form for this chosen output:

$\frac{d i_{L}}{d t} = - \frac{R_{i}}{L_{o}} i_{L} - \frac{1}{L_{o}} v_{o} + \frac{δ (t) V_{D C}}{L_{o}}$

si339_e (35.106)

A suitable sliding surface (35.107) is obtained from Eq. (35.91), making $e_{i_{L}} = i_{L_{r}} - i_{L}$ :

$S (e_{i_{L}}, t) = k_{p} e_{i_{L}} = k_{p} (i_{L_{r}} - i_{L}) = k_{p} (g_{m} v_{i_{L}} - i_{L}) = 0$

(35.107)

The switching law Eq. (35.108) can be devised calculating the time derivative of Eq. (35.107) $\dot{S} (e_{i_{L}}, t)$ and applying Eq. (35.92). If $S (e_{i_{L}}, t) > 0$ , then di_L/dt>0 must hold to obtain $S (e_{i_{L}}, t) < 0$ , implying δ(t)=1:

$δ (t) = \{\begin{array}{c} 1 & f o r S (e_{i_{L}}, t) > + ε \\ - 1 & f o r S (e_{i_{L}}, t) < - ε \end{array}$

si345_e (35.108)

The k_p value and the allowed ripple ɛ define the instantaneous value of the variable switching frequency. The sliding-mode controller is represented in Fig. 35.28A. Step response (Fig. 35.29A) shows the variable-frequency operation, a very short rise time (limited only by the available power supply), and confirms the expected robustness against supply variations.

f35-28-9780128114070 — Fig. 35.28 (A) Implementation of short-circuit-proof sliding-mode current controller (variable frequency) and (B) implementation of fixed-frequency, short-circuit-proof sliding-mode current controller using a triangular waveform.

f35-29-9780128114070 — Fig. 35.29 Performance of the transconductance amplifier; response to an i_Lr step from −20 to 20 A at t=0.001 s and to a V_DC step from 300 to 230 V at t=0.015 s: (A) variable-frequency sliding-mode controller and (B) fixed-frequency sliding-mode controller.

For systems where fixed-frequency operation is needed, a triangular wave, with frequency (10 kHz) slightly greater than the maximum variable frequency, can be added (Fig. 35.28B) to the sliding-mode controller, as explained in Section 35.3.3. Performances (Fig. 35.29B) are comparable with those of the variable-frequency sliding-mode controller (Fig. 35.29A). Fig. 35.29B shows not only the constant switching frequency but also a steady-state error dependent on the operating point.

To eliminate this error, a new sliding surface Eq. (35.109), based on Eq. (35.99), should be used. The constants k_p and k₀ can be calculated, as discussed in Example 35.10:

$S (e_{i_{L}}, t) = k_{0} \int e_{i_{L}} d t + k_{p} {e_{i}}_{_{L}} = 0$

si346_e (35.109)

The new constant-frequency sliding-mode current controller (Fig. 35.30A), with added antiwindup techniques (Example 35.6), since a saturation (errMax) is needed to keep the frequency constant, now presents no steady-state error (Fig. 35.30B). Performances are comparable with those of the variable-frequency controller, and no robustness loss is visible. The applied sliding-mode approach led to the derivation of the known average current-mode controller.

f35-30-9780128114070 — Fig. 35.30 (A) Block diagram of the average current-mode controller (sliding mode) and (B) performance of the fixed-frequency sliding-mode controller with removed steady-state error: response to an ${i_{L}}_{_{r}}$ step from −20 to 20 A at t=0.001 s and to a V_DC step from 300 to 230 V at t=0.015 s.

Output voltage control

To obtain a power operational amplifier suitable for building uninterruptible power supplies, power filters, power gyrators, inductance simulators, or power factor active compensators, v_o must be the controlled converter output. Therefore, using the input-output linearization technique, it is seen that the first time derivative of the output (dv_o/dt)=(i_L−i_o)/C_o=θ does not explicitly contain the control input δ(t)V_DC. Then, the second derivative must be calculated. Taking into account Eq. (35.105), as θ=(i_L−i_o)/C_o, Eq. (35.110) is derived

$\begin{array}{l} \frac{d^{2} v_{o}}{d t^{2}} = \frac{d}{d t} θ = \frac{d}{d t} (\frac{i_{L} - i_{o}}{C_{o}}) \\ = - \frac{R_{i}}{L_{o}} θ - \frac{1}{L_{o} C_{o}} v_{o} - \frac{R_{i}}{L_{o} C_{o}} i_{o} - \frac{1}{C_{o}} \frac{d i_{o}}{d t} + \frac{1}{L_{o} C_{o}} δ (t) V_{D C} \end{array}$

si347_e (35.110)

This expression shows that the second derivative of the output depends on the control input δ(t)V_DC. No further time derivative is needed, and the state-space equations of the equivalent circuit, written in the phase canonical form, are

$\frac{d}{d t} [\begin{array}{l} v_{o} \\ θ \end{array}] = [\begin{array}{c} θ \\ - \frac{R_{i}}{L_{o}} θ - \frac{1}{L_{o} C_{o}} v_{o} - \frac{R_{i}}{L_{o} C_{o}} i_{o} - \frac{1}{C_{o}} \frac{d i_{o}}{d t} + \frac{1}{L_{o} C_{o}} δ (t) V_{D C} \end{array}]$

si348_e (35.111)

According to Eqs. (35.91), (35.111), and (35.105), considering that $e_{v_{o}}$ is the feedback error $e_{v_{o}} = v_{o_{r}} - v_{o}$ , a sliding surface $S (e_{v_{o}}, t)$ can be chosen:

$\begin{array}{l} S (e_{v_{o}}, t) = k_{1} e_{v_{o}} + k_{2} \frac{d e_{v_{o}}}{d t} = e_{v_{o}} + \frac{k_{2}}{k_{1}} \frac{d e_{v_{o}}}{d t} \\ = e_{v_{o}} + β \frac{d {e_{v}}_{_{o}}}{d t} = \frac{C_{o}}{β} (v_{o_{r}} - v_{o}) + C_{o} \frac{d v_{o_{r}}}{d t} + i_{o} - i_{L} = 0 \end{array}$

si352_e (35.112)

where β is the time constant of the desired first-order response of output voltage (β≫T>0), as the strong relative degree [13] of this system is 2, and the sliding-mode operation reduces by one, the order of this system (the strong relative degree of a system is the least positive integer r for which the rth derivative of the output d^rv_o/dt^r is an explicit function of the control input u, being dⁱv_o/du=0 for 0≤i≤r−1 and d^rv_o/du≠0).

Calculating $\dot{S} (e_{v_{o}}, t)$ , the control strategy (switching law) Eq. (35.113) can be devised since, if $S (e_{v_{o}}, t) > 0$ , then di_L/dt must be positive to obtain $\dot{S} (e_{i_{L}}, t) < 0$ , implying δ(t)=1. Otherwise, δ(t)=−1:

$δ (t) = \{\begin{array}{c} 1 & f o r S (e_{v_{o}}, t) > 0 (v_{P W M} = + V_{D C}) \\ - 1 & f o r S (e_{v_{o}}, t) < 0 (v_{P W M} = - V_{D C}) \end{array}$

si356_e (35.113)

In the ideal sliding-mode dynamics, the filter input voltage v_PWM switches between V_DC and −V_DC with the infinite frequency. This switching generates the equivalent control voltage V_eq that must satisfy the sliding manifold invariance conditions, $S (e_{v_{o}}, t) = 0$ and $\dot{S} (e_{v_{o}}, t) = 0$ . Therefore, from $\dot{S} (e_{v_{o}}, t) = 0$ , using Eqs. (35.112) and (35.105) (or from Eq. (35.110)), V_eq is

$V_{eq} = L_{o} C_{o} [\frac{d^{2} v_{o_{r}}}{d t^{2}} + \frac{1}{β} \frac{d v_{o_{r}}}{d t} + \frac{v_{o}}{L_{o} C_{o}} + \frac{(β R_{i} - L_{o}) i_{L}}{β L_{o} C_{o}} \frac{i_{o}}{β C_{o}} + \frac{1}{C_{o}} \frac{d i_{o}}{d t}]$

si360_e (35.114)

This equation shows that only smooth input ${v_{o}}_{_{r}}$ signals (“smooth” functions) can be accurately reproduced at the inverter output, as it contains derivatives of the ${v_{o}}_{_{r}}$ signal. This fact is a consequence of the stored electromagnetic energy. The existence of the sliding-mode operation implies the following necessary and sufficient condition:

$- V_{D C} < V_{eq} < V_{D C}$

(35.115)

Eq. (35.115) enables the determination of the minimum input voltage V_DC needed to enforce the sliding-mode operation. Moreover, even in the case of |V_eq|>|V_DC|, the system experiences only a saturation transient and eventually reaches the region of sliding-mode operation, except if the operating point and disturbances enforce |V_eq|>|V_DC| in steady state.

In the ideal sliding mode, at infinite switching frequency, state trajectories are directed toward the sliding surface and move exactly along the discontinuity surface. Practical switching power converters cannot switch at infinite frequency, so a typical implementation of Eq. (35.112) (Fig. 35.31A) with neglected ${\dot{v}}_{o_{r}}$ features a comparator with hysteresis 2ɛ, switching occurring at $|S (e_{v_{o}}, t)| > ɛ$ with frequency depending on the slopes of i_L. This hysteresis causes phase-plane trajectory oscillations of width 2ɛ around the discontinuity surface $S (e_{v_{o}}, t) = 0$ , but the V_eq voltage is still correctly generated, since the resulting duty cycle is a continuous variable (except for error limitations in the hardware or software, which can be corrected using the approach pointed out by Eq. (35.98)).

f35-31-9780128114070 — Fig. 35.31 (A) Implementation of short-circuit-proof, sliding-mode output voltage controller (variable frequency) and (B) implementation of antiwindup PI current-mode (fixed frequency) controller.

The design of the compensator and the modulator is integrated with the same theoretical approach, since the signal $S (e_{v_{o}}, t)$ applied to a comparator generates the pulses for the power semiconductors drives. If the short-circuit-proof operation is built into the power semiconductor drives, there is the possibility to measure only the capacitor current (i_L−i_o).

Short-circuit protection and fixed-frequency operation of the power operational amplifier

If we note that all the terms to the left of i_L in Eq. (35.112) represent the value of $i_{L_{r}}$ , a simple way to provide short-circuit protection is to bound the sum of all these terms (Fig. 35.31A with $i_{L_{rmax}} = 100 A$ ). Alternatively, the output current controllers of Fig. 35.28 can be used, comparing Eq. (35.107) with Eq. (35.112), to obtain $i_{L_{r}} = S (e_{v_{o}}, t) / k_{p} + i_{L}$ . Therefore, the block diagram of Fig. 35.31A provides the $i_{L_{r}}$ output (for k_p=1) to be the input of the current controllers (Figs. 35.28 and 35.31A). As seen, the controllers of Figs. 35.28B and 35.30A also ensure fixed-frequency operation.

For comparison purposes, a proportional-integral (PI) controller, with antiwindup (Fig. 35.31B) for output voltage control, was designed, supposing the current-mode control of the half bridge $(i_{L_{r}} = g_{m} v_{i_{L}} / (1 + s T_{d})$ considering a small delay T_d) and a pure resistive load R_o and using the approach outlined in Examples 35.6 and 35.8 (k_v=1, g_m=1, ζ²=0.5, and T_d=600 μs). The obtained PI (35.50) parameters are

$\begin{array}{l} T_{z} = R_{o} C_{o} \\ T_{p} = 4 ζ^{2} g_{m} k_{v} R_{o} T_{d} \end{array}$

si373_e (35.116)

Both variable frequency (Fig. 35.32) and constant frequency (Fig. 35.33) sliding-mode output voltage controllers present excellent performance and robustness with nominal loads. With loads much higher than the nominal value (Figs. 35.32B and 35.33B), the performance and robustness are also excellent. The sliding-mode constant-frequency PWM controller presents the additional advantage of injecting lower ripple in the load.

f35-32-9780128114070 — Fig. 35.32 Performance of the power operational amplifier; response to a ${v_{o}}_{_{r}}$ step from −200 to 200 V at t=0.001 s and to a V_DC step from 300 to 230 V at t=0.015 s: (A) variable-frequency sliding mode (nominal load) and (B) variable-frequency sliding mode (R_o×20).

f35-33-9780128114070 — Fig. 35.33 Performance of the power operational amplifier; response to a ${v_{o}}_{_{r}}$ step from −200 to 200 V at t=0.001 s and to a V_DC step from 300 to 230 V at t=0.015 s: (A) fixed-frequency sliding mode (nominal load) and (B) fixed-frequency sliding mode (R_o×20).

As expected, the PI regulator presents lower performance (Fig. 35.34). The response speed is lower, and the insensitivity to power supply and load variations (Fig. 35.34B) is not as high as with the sliding mode. Nevertheless, the PI performances are acceptable, since its design was carried considering a slow and fast manifold sliding-mode approach: the fixed-frequency sliding-mode current controller (35.109) for the fast manifold (the i_L current dynamics) and the antiwindup PI for the slow manifold (the v_o voltage dynamics, usually much slower than the current dynamics).

f35-34-9780128114070 — Fig. 35.34 Performance of the PI-controlled power operational amplifier; response to a ${v_{o}}_{_{r}}$ step from −200 to 200 V at t=0.001 s and to a V_DC step from 300 to 230 V at t=0.015 s: (A) PI current-mode controller (nominal load) and (B) PI current-mode controller (R_o×20).

35.3.5.2 Example 35.12: Constant-Frequency Sliding-Mode Control of p Pulse Parallel Rectifiers

This example presents a new paradigm to the control of thyristor rectifiers. Since p pulse rectifiers are variable-structure systems, sliding-mode control is applied here to 12-pulse rectifiers, still useful for very high-power applications [3]. The design determines the variables to be measured, and the controlled rectifier presents robustness and much shorter response times, even with the parameter uncertainty, perturbations, noise, and nonmodeled dynamics. These performances are not feasible using linear controllers, obtained here for comparison purposes.

Modeling the 12-pulse parallel rectifier

The 12-pulse rectifier (Fig. 35.35A) is built with four three-phase half-wave rectifiers, connected in parallel with current-sharing inductances l and l′ merged with capacitors C′ and C₂, to obtain a second-order LC filter. This allows low-ripple output voltage and continuous mode of operation (laboratory model with l=44 mH, l′=13 mH, C′=C₂=10 mF, star-delta-connected ac sources with E_RMS≈65 V, and power rating 2.2 kW, load approximately resistive R_o≈3–5 Ω).

f35-35-9780128114070 — Fig. 35.35 (A) Twelve-pulse rectifier with interphase reactors and intermediate capacitors, (B) rectifier model neglecting the half-wave rectifier dynamics, and (C) low-order averaged equivalent circuit for the 12-pulse rectifier with the resulting output double LC filter.

To control the output voltage v_c2, given the complexity of the whole system, the best approach is to derive a low-order model. By averaging the four half-wave rectifiers, neglecting the rectifier dynamics and mutual couplings, the equivalent circuit of Fig. 35.35B is obtained (l₁=l₂=l₃=l₄=l, l₅=l₆=l′, and C₁₁=C₁₂=C′). Since the rectifiers are identical, the equivalent 12-pulse rectifier model of Fig. 35.35C is derived, simplifying the resulting parallel associations (L₁=l/4, L₂=l/2, and C₁=2C′).

Considering the load current i_o as an external perturbation and v_i the control input, the state-space model of the equivalent circuit of Fig. 35.35C is

$\begin{array}{l} \frac{d}{d t} [\begin{array}{l} i_{L_{1}} \\ i_{L 2} \\ v_{c_{1}} \\ v_{c_{2}} \end{array}] = [\begin{array}{c} 0 & 0 & - 1 / L_{1} & 0 \\ 0 & 0 & 1 / L_{2} & - 1 / L_{2} \\ 1 / C_{1} & - 1 / C_{1} & 0 & 0 \\ 0 & 1 / C_{2} & 0 & 0 \end{array}] [\begin{array}{l} i_{L_{1}} \\ i_{L 2} \\ v_{c_{1}} \\ v_{c_{2}} \end{array}] \\ + [\begin{array}{c} 1 / L_{1} & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & - 1 / C_{2} \end{array}] [\begin{array}{c} v_{i} \\ i_{o} \end{array}] \end{array}$

si374_e (35.117)

Sliding-mode control of the 12-pulse parallel rectifier

Since the output voltage $v_{c_{2}}$ of the system must follow the reference $v_{c_{2 r}}$ , the system equations in the phase canonical (or controllability) form must be written, using the error $e_{v_{2}} = v_{c_{2 r}} - v_{c_{2}}$ and its time derivatives as new state error variables, as done in Example 35.11:

$\frac{d}{d t} [\begin{array}{c} e_{v_{c_{2}}} \\ e_{θ} \\ e γ \\ e_{β} \end{array}] = [\begin{array}{c} e_{θ} \\ e_{γ} \\ e_{β} \\ - (\frac{1}{C_{1} L_{1}} + \frac{1}{C_{1} L_{2}} + \frac{1}{C_{2} L_{2}}) e_{γ} - \frac{e_{v_{c_{2}}}}{C_{1} L_{1} C_{2} L_{2}} - (\frac{1}{C_{1} L_{1} C_{2}} \\ + \frac{1}{C_{1} C_{2} L_{2}}) \frac{d i_{o}}{d t} - \frac{1}{C_{2}} \frac{d^{3} i_{o}}{d t^{3}} - \frac{v_{i}}{C_{1} L_{1} C_{2} L_{2}} \end{array}]$

si378_e (35.118)

The sliding surface $S (e_{x_{i}}, t)$ , designed to reduce the system order, is a linear combination of all the phase canonical state variables. Considering Eqs. (35.118) and (35.117) and the errors $e_{v_{c_{2}}}$ , e_θ, e_γ, and e_β, the sliding surface can be expressed as a combination of the rectifier currents, voltages, and their time derivatives:

$\begin{array}{l} S (e_{x_{i}}, t) = e_{v_{c_{2}}} + k_{θ} e_{θ} + k_{γ} e_{γ} + k_{β} e_{β} \\ = v_{c_{2 r}} + k_{θ} θ_{r} + k_{γ} γ_{r} + k_{β} β_{r} - (1 - \frac{k_{γ}}{C_{2} L_{2}}) v_{c_{2}} \\ - \frac{k_{γ}}{C_{2} L_{2}} v_{c_{1}} + (\frac{k_{θ}}{C_{2}} - \frac{k_{β}}{C_{2}^{2} L_{2}}) i_{o} + \frac{k_{γ}}{C_{2}} \frac{d i_{o}}{d t} + \frac{k_{β}}{C_{2}} \frac{d^{2} i_{o}}{d t^{2}} \\ - \frac{k_{β}}{C_{1} C_{2} L_{2}} i_{L_{1}} - (\frac{k_{θ}}{C_{2}} - \frac{k_{β}}{C_{1} C_{2} L_{2}} - \frac{k_{β}}{C_{2}^{2} L_{2}}) i_{L_{2}} = 0 \end{array}$

si381_e (35.119)

Eq. (35.119) shows the variables to be measured $(v_{c_{2}}, v_{c_{1}}, i_{o}, i_{L_{1}}, and i_{L_{2}})$ . Therefore, it can be concluded that the output current of each three-phase half-wave rectifier must be measured.

The existence of the sliding mode implies $S (e_{x_{i}}, t) = 0$ and $\dot{S} (e_{x_{i}}, t) = 0$ . Given the state models (35.117) and (35.118) and from $\dot{S} (e_{x_{i}}, t) = 0$ , the available voltage of the power supply v_i must exceed the equivalent average dc input voltage V_eq (35.120), which should be applied at the filter input, in order that the system state slides along the sliding surface (35.119):

$\begin{array}{l} V_{eq} & = \frac{C_{1} L_{1} C_{2} L_{2}}{k_{β}} (θ_{r} + k_{θ} γ_{r} + k_{γ} β_{r} + k_{β} {\dot{β}}_{r}) + v_{c_{2}} - \frac{C_{1} L_{1} C_{2} L_{2}}{k_{β}} \\ \times (θ + k_{γ} β) + (C_{2} L_{2} + C_{2} L_{1} + C_{1} L_{1} - C_{1} L_{1} C_{2} L_{2} \frac{k_{θ}}{k_{β}}) γ \\ + (L_{1} + L_{2}) \frac{d i_{o}}{d t} + C_{1} L_{1} L_{2} \frac{d^{3} i_{o}}{d t^{3}} \end{array}$

si386_e (35.120)

This means that the power supply root mean square (RMS) voltage values should be chosen high enough to account for the maximum effects of the perturbations. This is almost the same criterion adopted when calculating the RMS voltage values needed with linear controllers. However, as the V_eq voltage contains the derivatives of the reference voltage, the system will not be able to stay in sliding mode with a step as the reference.

The switching law would be derived, considering that, from Eq. (35.97), b_e(e)>0. Therefore, from Eq. (35.119), if $S (e_{x_{i}}, t) > + ɛ$ , then $v_{i} (t) = V_{e q_{\max}}$ , else if $S (e_{x_{i}}, t) < - ɛ$ , then $v_{i} (t) = - V_{e q_{\max}}$ . However, because of the lack of gate turn-off capability of the rectifier thyristors, power rectifiers cannot generate the high-frequency switching voltage v_i(t), since the statistical mean delay time is T/2p (T=20 ms) and reaches T/2 when switching from $+ V_{e q_{\max}}$ to $- V_{e q_{\max}}$ . To control mains switched rectifiers, the described constant-frequency sliding-mode operation method is used, in which the sliding surface $S (e_{x_{i}}, t)$ instead of being compared with zero is compared with an auxiliary constant-frequency function r(t) (Fig. 35.6B) synchronized with the mains frequency. The new switching law is

$\begin{array}{l} if k_{p} S (e_{x_{i}}, t) > r (t) + ι \Rightarrow Trigger the next thyristor \\ if k_{p} S (e_{x_{i}}, t) < r (t) - ι \Rightarrow Do not trigger any thyristor \end{array}\} \Rightarrow v_{i} (t)$

si394_e (35.121)

Since now $S (e_{x_{i}}, t)$ is not near zero, but around some value of r(t), a steady-state error $e_{v_{c_{2 a v}}}$ appears $(min [r (t)] / k_{p} < e_{v_{c_{2 a v}}} < max [r (t)] / k_{p})$ , as seen in Example 35.11. Increasing the value of k_p (toward the ideal saturation control) does not overcome this drawback, since oscillations would appear even for moderate k_p gains, because of the rectifier dynamics. Instead, the sliding surface (35.122), based on Eq. (35.99), should be used. It contains an integral term, which, given the canonical controllability form and the Routh-Hurwitz property, is the only nonzero term at steady state, enabling the complete elimination of the steady-state error:

$S_{i} (e_{x_{i}}, t) = \int e_{v_{c_{2}}} d t + k_{1 v} e_{v_{c_{2}}} + k_{1 θ} e_{θ} + k_{1 γ} e_{γ} + {k_{1}}_{β} e_{β}$

si398_e (35.122)

To determine the k constants of Eq. (35.122), a pole-placement technique is selected, according to a fourth-order Bessel polynomial B_E(s)_m, m=4, from Eq. (35.88), in order to obtain the smallest possible response time with almost no overshoot. For a delay characteristic as flat as possible, the delay t_r is taken inversely proportional to a frequency f_ci just below the lowest cutoff frequency (f_ci<8.44 Hz) of the double LC filter. For this fourth-order filter, the delay is t_r=2.8/(2πf_ci). By choosing f_ci=7 Hz (t_r≈64 ms) and dividing all the Bessel polynomial terms by st_r, the characteristic polynomial (35.123) is obtained:

$S_{i} (e_{x_{i}}, s) = \frac{1}{s t_{r}} + 1 + \frac{45}{105} s t_{r} + \frac{10}{105} s^{2} t_{r}^{2} + \frac{1}{105} s^{3} t_{r}^{3}$

si399_e (35.123)

This polynomial must be applied to Eq. (35.122) to obtain the four sliding functions needed to derive the thyristor trigger pulses of the four three-phase half-wave rectifiers. These sliding functions will enable the control of the output current $(i_{l_{1}}, i_{l_{2}}, i_{l_{3}} and i_{l_{4}})$ of each half-wave rectifier, improving the current sharing among them (Fig. 35.35B). Supposing equal current share, the relation between the $i_{L_{1}}$ current and the output currents of each three-phase rectifier is $i_{L_{1}} = 4 i_{l_{1}} = 4 i_{l_{2}} = 4 i_{l_{3}} = 4 i_{l_{4}}$ . Therefore, for the nth half-wave three-phase rectifier, since, for n=1 and n=2, $v_{c_{1}} = v_{c_{11}}$ and $i_{L_{2}} = 2 i_{l_{5}}$ and, for n=3 and n=4, $v_{c_{1}} = v_{c_{12}}$ and $i_{L_{2}} = 2 i_{l_{6}}$ , the four sliding surfaces are (k_1v=1)

$\begin{array}{l} S_{i} {(e_{x_{i}}, t)}_{n} & = [k_{1 v} v_{c_{2 r}} + \frac{45 t_{r}}{105} θ_{r} + \frac{10 t_{r}^{2}}{105} γ_{r} + \frac{t_{r}^{3}}{105} β_{r} \\ + \frac{1}{t_{r}} \int (v_{c_{2 r}} - v_{c_{2}}) d t - (\frac{{k_{1}}_{v}}{C_{2} L_{2}} - \frac{10 t_{r}^{2}}{105 C_{2} L_{2}}) v_{c_{2}} \\ - \frac{10 t_{r}^{2}}{105 C_{2} L_{2}} v_{c_{1, 2}} + (\frac{45 t_{r}}{105 C_{2}} - \frac{t_{r}^{3}}{105 C_{2}^{2} L_{2}}) i_{o} \\ + (\frac{10 t_{r}^{2}}{105 C_{2}}) \frac{d i_{o}}{d t} + (\frac{t_{r}^{3}}{105 C_{2}}) \frac{d^{2} i_{o}}{d t^{2}}] / 4 \\ - [(\frac{45 t_{r}}{105 C_{2}} - \frac{t_{r}^{3}}{105 C_{^{2}}^{2} L_{2}} - \frac{t_{r}^{3}}{105 C_{1} L_{2} C_{2}}) i_{l_{5, 6}}] / 2 \\ - (\frac{t_{r}^{3}}{105 C_{1} L_{2} C_{2}}) i_{l_{n}} \end{array}$

si407_e (35.124)

If an inexpensive analog controller is desired, the successive time derivatives of the reference voltage and the output current of Eq. (35.124) can be neglected (furthermore, their calculation is noise prone). Nonzero errors on the first-, second-, and third-order derivatives of the controlled variable will appear, worsening the response speed. However, the steady-state error is not affected.

To implement the four equations (35.124), the variables $v_{c_{2}}, v_{c 11}, v_{c 12}, i_{o}, i_{l_{5}}, i_{l_{6}}, i_{l_{1}}, i_{l_{2}}, i_{l_{3}} and i_{l_{4}}$ must be measured. Although this could be done easily, it is very convenient to further simplify the practical controller, keeping its complexity and cost at the level of linear controllers, while maintaining the advantages of sliding mode. Therefore, the voltages $v_{c_{11}} and v_{c_{12}}$ are assumed almost constant over one period of the filter input current, and $v_{c_{11}} = v_{c_{12}} = v_{c_{2}}$ , meaning that $i_{l_{5}} = i_{l_{6}} = i_{o} / 2$ . With these assumptions, valid as the values of C′ and C₂ are designed to provide an output voltage with very low ripple, the new sliding-mode functions are

$\begin{array}{l} S_{i} {(e_{x_{i}}, t)}_{n} \approx \frac{\frac{1}{t_{r}} \int ({v_{c}}_{2 r} - {v_{c}}_{2}) d t + {k_{1}}_{v} (v_{c_{2 r}} - v_{c_{2}}) + \frac{t_{r}^{3}}{105} \frac{1}{C_{1} C_{2} L_{2}} i_{o}}{4} \\ - (\frac{t_{r}^{3}}{105 C_{1} L_{2} C_{2}}) i_{\ln} \end{array}$

si412_e (35.125)

These approximations disregard only the high-frequency content of $v_{c 11}, v_{c 12}, i_{l_{5}}, and i_{l_{6}}$ and do not affect the rectifier steady-state response, but the step response will be a little slower, although still much faster (150 ms, Fig. 35.39) than that obtained with linear controllers (280 ms, Fig. 35.38). Regardless of all the approximations, the low switching frequency of the rectifier would not allow the elimination of the dynamic errors. As a benefit of these approximations, the sliding-mode controller (Fig. 35.36A) will need only an extra current sensor (or a current observer) and an extra operational amplifier in comparison with linear controllers derived hereafter (which need four current sensors and six operational amplifiers). Compared with the total cost of the 12-pulse rectifier plus output filter, the control hardware cost is negligible in both cases, even for medium-power applications.

f35-36-9780128114070 — Fig. 35.36 (A) Sliding-mode controller block diagram and (B) linear control hierarchy for the 12-pulse rectifier.

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Table of Contents for 35.3 Sliding-Mode Control of Switching Power Converters

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35.3 Sliding-Mode Control of Switching Power Converters

35.3.1 Introduction

35.3.2 Principles of Sliding-mode Control

35.3.2.1 Control Law (Sliding Surface)

35.3.2.2 Closed-Loop Control Input–Output Decoupled Form

35.3.2.3 Stability

35.3.2.4 Switching Law

35.3.2.5 Robustness

35.3.3 Constant-Frequency Operation

35.3.4 Steady-State Error Elimination in Converters with Continuous Control Inputs

35.3.5 Examples: Buck–Boost DC/DC Converter, Half-bridge Inverter, 12-Pulse Parallel Rectifiers, Audio Power Amplifiers, Near Unity Power Factor Rectifiers, Multilevel Inverters, Matrix Converters

35.3.5.1 Example 35.11: Sliding Mode Control of the Single-Phase Half-Bridge Converter

Output current control (current-mode control)

Output voltage control

Short-circuit protection and fixed-frequency operation of the power operational amplifier

35.3.5.2 Example 35.12: Constant-Frequency Sliding-Mode Control of p Pulse Parallel Rectifiers

Modeling the 12-pulse parallel rectifier

Sliding-mode control of the 12-pulse parallel rectifier

Table of Contents for
35.3 Sliding-Mode Control of Switching Power Converters