LMI robust control design for boost PWM converters - Semantic Scholar

www.ietdl.org Published in IET Power Electronics Received on 13th August 2008 Revised on 11th November 2008 doi: 10.1049/iet-pel.2008.0271

ISSN 1755-4535

LMI robust control design for boost PWM converters C. Olalla1 R. Leyva1 A. El Aroudi1 P. Garce´s1 I. Queinnec2 1

Departament d’Enginyeria Electro`nica, Ele`ctrica i Automa`tica, Escola Te`cnica Superior d’Enginyeria, Universitat Rovira i Virgili, Campus Sescelades 43007, Tarragona, Spain 2 CNRS, LAAS, University of Toulouse, 7 Avenue du Colonel Roche, Toulouse F-31077, France E-mail: [email protected]

Abstract: This work presents an analytical study and an experimental verification of a robust control design based on a linear matrix inequalities (LMI) framework for boost regulators. With the proposed LMI method, nonlinearities and uncertainties are modelled as a convex polytope. Thus, the LMI constraints permit to robustly guarantee a certain perturbation rejection level and a region of pole location. With this approach, the multiobjective robust controller is computed automatically by a standard optimisation algorithm. The proposed method results in a state-feedback law efficiently implementable by operational amplifiers. PSIM simulations and experimental results obtained from a prototype are used to validate this approach. The results obtained are compared with a conventional PID controller.

1

Introduction

Dc – dc switched-mode converters are power efficient devices used to match the voltage level of an energy source to the specifications of the load. The dynamics of such power converters is described by non-linear models. Despite the non-linearities, dc – dc converters are usually driven by means of linear (state or output) feedback controllers, thus reducing the complexity and cost of the control system. The control objective of such devices is (i) to maintain regulation of the output voltage at the desired value, (ii) to maximise the bandwidth of the closed-loop system in order to reject disturbances and (iii) to satisfy certain transient characteristics (as e.g. to minimise output overshoot). Such linear controllers are usually designed considering a linearisation of the model at a certain operation point. In this case, large-signal transients may deteriorate the output signal or even make the system diverge from the desired operation point. Owing to the intrinsic non-linear nature of the switched regulators, several authors have proposed non-linear controllers in order to maintain stability over a certain range of operating conditions. Some of the first works on nonlinear control for power converters can be found in [1, 2], IET Power Electron., 2010, Vol. 3, Iss. 1, pp. 75– 85 doi: 10.1049/iet-pel.2008.0271

where the authors propose non-linear strategies based on quadratic Lyapunov functions. More recently, Cortes et al. [3], He and Luo [4] and Leyva et al. [5], derive robust nonlinear controllers for power converters. These last works seek large-signal stability of the regulator dynamics when a complex control law is applied. The main disadvantages of the previous non-linear controllers are the difficulty to predict transient performances and the complexity of the implementation. Besides of non-linear control, other authors have adapted linear robust control techniques to power converters with the aim to assure stability under different operating conditions. Linear control laws are, a priori, easily implementable as, for example, PID controllers [6 –8]. Furthermore, the robust linear control techniques, unlike conventional and nonlinear control, allow to take into account parametric uncertainty. A correct treatment of uncertainties is of major importance in power converters, since some of the regulator parameters such as the storage elements or the load are usually time dependent or partially unknown. Some of the robust methods successfully adapted to power electronics are H1 [9– 11], m-synthesis [12], quantitative feedback theory (QFT) [13, 14] and approaches based on linear matrix inequalities (LMI) [15 – 18]. It is worth to point out some of

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www.ietdl.org their characteristics in order to compare such techniques: H1 and m-synthesis methods [9– 12] require that the designer arbitrarily choose weighting functions to specify the required performances and to approximate the transient properties. On the contrary, LMI-based control does not necessarily require weighting functions and it can also deal with transient requirements with pole placement constraints. Also, it can be pointed out that the controller expression is optimised manually in the QFT technique [13, 14], whereas LMI control permits to optimise the controller parameters automatically. Since LMI provides the above advantages, the goal of this work is to adapt LMI robust control concepts to dc/dc regulator design. The fact that LMIs can be solved automatically by efficient standard numerical algorithms [19 – 21] has prompted a great number of researchers to describe different control problems in terms of LMIs. Therefore this technique can be considered as mature in the field of Control Theory. The controller design shown in this work consists of describing in terms of LMIs the following restrictions: first, the converter stability (in Lyapunov’s sense); second, a minimum level of perturbation rejection; and third, several constraints in pole location. In all the cases, the design takes into account uncertainty in the operation point and load value. Finally, the design is completed when the restrictions are numerically solved. The result is an efficient state feedback controller. The main contribution of this work is the experimental implementation of a prototype which validates the feasibility of an LMI approach, successfully applied in many engineering domains, in the area of power electronics. It is worth to mention that LMI control has been previously reported in [15–17] to design a state-feedback controller for a dc/dc converter. The approach shown here differs from the previous works since we consider a step-up converter, whereas the previous references consider a buck converter. Therefore our work takes into account a non-linear system, whose linearisation yields a non-minimum phase plant, whereas [15–17] deal with a minimum phase linear case. In addition,

[15–17] do not describe any experimental prototype, whereas we present accurate experimental measures. In order to evaluate the performance achieved by the LMI controller, we have compared the results with a conventional output-feedback PID law. Both controls can be easily implemented by operational amplifiers, a PWM circuit and standard analogue elements. The remaining sections are organised as follows. Section 2 shows the uncertain model of the non-minimum phase converter. Section 3 introduces the LMIs and the controller constraints taken into account. The validity of the design is verified with experimental results from a boost prototype in Section 4, where we compare the performance of the LMI controller with the conventional PID case numerically using PSIM package. Section 5 summarises the key aspects of the design method and presents some conclusions.

2 Boost converter uncertainty model Fig. 1 shows the schematic circuit diagram of a dc – dc step-up (boost) converter and the relevant control signals. In Fig. 1, vo is the output voltage, vg the line voltage and iload the output disturbance. The output voltage must be kept at a given constant value Vref . R models the converter load, while C and L represent, respectively, capacitor and inductor values. The binary signal (ub) that turns on and off the switches is controlled by means of a fixed-frequency pulse width modulation (PWM) circuit (see Fig. 1b). The constant switching frequency is 1=Ts , where Ts is the switching period equal to the sum of Ton (when ub ¼ 1) and Toff (when ub ¼ 0) where the ratio Ton =(Ton þ Toff ) is the duty cycle dd . The duty cycle is compared with a sawtooth signal vs of amplitude VM . We assume that the converter operates in continuous conduction mode (CCM) and that the inductor current is not saturated. The following expressions show the state-space averaged and linearised model of the boost converter. Averaged

Figure 1 Schematic diagram of a boost converter and its control circuit a Electric circuit b PWM waveforms

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IET Power Electron., 2010, Vol. 3, Iss. 1, pp. 75– 85 doi: 10.1049/iet-pel.2008.0271

www.ietdl.org models of dc – dc converters neglect the high-frequency dynamics because of the switching action, since we consider Ts much smaller than the converter time constants. The model is then valid for a frequency range up to half the switching frequency [22] in the vicinity of the operation point, since it has been linearised. The state-space representation is written as

x_ (t) ¼ Ax(t) þ Bw w(t) þ Bu u(t) þ Bref Vref z(t) ¼ C z x(t) þ Dzw w(t) þ Dzu u(t)

(1)

where A [ Rnn , Bw [ Rnr , Bu [ Rnm , C z [ Rpn , pr Dzw [ R , Dzu [ Rpm and 2

the changes in parameters Dd0 and R, we express the system matrices (1) as function of these parameters

x_ (t) ¼ A(p)x(t) þ Bw w(t) þ Bu (p)u(t) þ Bref Vref z(t) ¼ C z x(t) þ Dzw w(t) þ Dzu u(t)

Only matrices A and Bu depend on the uncertain terms, which have been grouped in a vector p. In a general case, the vector p consists of N uncertain parameters p ¼ (p1 , . . . , pN ), where each uncertain parameter pi is bounded between a minimum and a maximum value pi and pi h i pi [ pi , pi

3

i L (t) 6 7 x(t) ¼ 4 vo (t) 5, w(t) ¼ i load (t) , x3 (t) z(t) ¼ vo (t)

u(t) ¼ dd (t) ,

The state variable x3 is the integral of the error signal obtained from the difference between the reference Vref and the output voltage vo . At the equilibrium state, the voltage error is zero. Therefore x3 is constant. The line voltage is considered a dc value vg ¼ Vg . The disturbance vector w has been defined as an output current source in order to characterise the output impedance of the converter. Such output impedance describes the output voltage vo behaviour in presence of changes in the output current iload . The output z is the output voltage vo and represents the controlled output whose response has to fulfil the control requirements. The matrices of the state-space representation are as follows 2

D0 d L 1 RC 1

3

2

3

07 0 7 17 7, B w ¼ 6 4 5, 7 C 05 0 0 3 0 Vg 2 3 6 D0 L 7 0 6 7 d 6 7 6 7 Bu ¼ 6 Vg 7, Bref ¼ 4 0 5 6 7 4 (D02 R)C 5 1 d 0

6 0 6 0 A¼6 6 Dd 4C 0 2

(2)

1 0 ,

Dzw ¼ [0],

Dzu ¼ [0]

[A(p), Bu (p)] [ Co{G1 , . . . , GL } ( ) L L X X li Gi , li 0, li ¼ 1 :¼ i¼1


(6)

i¼1

For an in-deep explanation of polytopic models of uncertainty see [19 (Ch. 2), 23, 24 (Ch. 4)]. Since the boost converter matrices A and Bu do not depend linearly on the uncertain parameters D d0 and R, we define two new uncertain variables d ¼ 1=Dd0 and b ¼ 1=(Dd02 R) in order to meet with a linear dependence. Thus, the parameter vector is defined as p ¼ R, Dd0 , d, b . By using this parameter vector we can bound the uncertainty inside a convex polytope. Based on the uncertainty model described above, the synthesis objective is to find a state-feedback gain K (u ¼ Kx), where uncertainty is restricted inside the following intervals R [ [Rmin , Rmax ]

0 0 , Ddmax ] Dd0 [ [Ddmin

0 0 d [ [1=Ddmax , 1=Ddmin ]

(7)

02 02 b [ [1=(Ddmax Rmax ), 1=(Ddmin Rmin )]

(3)

We consider that the load R and the duty-cycle Dd0 at the operating point are uncertain or time-varying parameters. We also consider that all other parameters are well known. It is worth to point out that the same procedure can be used to take into account more uncertain terms. Nevertheless, the more uncertainty is considered in the converter, the lower performance level can be assured. Thus, in order to deal with

(5)

The admissible values of vector p are constrained in an hyperrectangle in the parameter space RN with L ¼ 2N vertices {v1 , . . . , vL }. The images of the matrix [A(p), Bu (p)] for each vertex vi corresponds to a set {G1 , . . . , GL }. The components of the set {G1 , . . . , GL } are the extrema of a convex polytope, noted Co{G1 , . . . , GL }, which contains the images for all admissible values of p if the matrix [A(p), Bu (p)] depends linearly on p, that is

where Dd is the operating point duty cycle and its complementary corresponds to Dd0 ¼ 1 Dd . The remaining state-space matrices are Cz ¼ 0

(4)

Note that the uncertain model is inside a polytopic domain formed by L ¼ 24 vertices. The introduction of these new parameters d and b implies a relaxation in the uncertainty restrictions, since we assume the independence between uncertain parameters in order to have linear relations. Consequently, the new relaxed model

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www.ietdl.org considers dynamic responses that do not correspond to any real case. Therefore we will obtain a potentially conservative solution. The objectives of the design are to guarantee the stability, to assure a minimum level of perturbation rejection and to constraint the transient performance, for all the possible cases [A( p), Bu( p)]. In next section, we will establish the LMI conditions which satisfy these objectives.

3.2 Quadratic stability LMIs The following theorem [23] adapts the quadratic stability inequality (11) for a closed-loop system with a state feedback u ¼ Kx.

Theorem 3.1: System (1) is stabilisable by state feedback

u ¼ K x if and only if there exists a symmetric matrix W [ Rnn and a matrix Y [ Rmn such that

3

LMI design constraints

This section introduces the concept of LMI and presents the constraints used in the controller synthesis problem.

3.1 Lyapunov-based stability The use of matrix inequalities to demonstrate certain properties of dynamical systems can already be found in about 1890, when Lyapunov establishes his well-known stability method, whose linear case has been reproduced here for completeness [24]. Given a linear time-invariant (LTI) system x_ ¼ Ax

8x = 0

8x = 0

s.t. A 0 P þ PA , 0

(9)

(10)

(11)

In this case, P is the matrix variable to be found to prove the stability. Ref. [25] showed that these inequalities that presented linear dependence on the variables, afterwards called LMIs, can be solved by convex optimisation methods. We will take advantage of such convex optimisation methods, that have been implemented in computer algorithms [19 – 21], in order to solve the LMIs that arise in the control of the boost converter. In the following subsections we introduce the LMIs to ensure the quadratic stability, to deal appropriately with disturbances and to constraint the pole placement of the closed-loop system. 78

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The decomposition of K in matrix variables Y and W allows to satisfy the linearity condition of these inequalities. A detailed proof can be obtained in [23]. The case with polytopic uncertainty in matrices A and Bu directly extends by computing (12) at all the vertices {G1 , . . . , GL } of the convex polytope Co{G1 , . . . , GL } [24, Ch. 5]. This extension allows us to assure the quadratic stability of a uncertain plant.

The H1 norm of a stable scalar transfer function f (s) is the peak value of jf (j v)j as a function of frequency [26]. It is used as a measure of the performance of a system, for example, to evaluate the minimum attenuation level of an external disturbance. Considering the transfer function H(s) from disturbances w to outputs z, the H1 norm of such system is equal to

Thus, the system is stable if and only if there exists a symmetric matrix P that is positive definite (in the following, the notation P . 0 means that the matrix P is positive definite) for which V_ (x) , 0. Inequality (10) is satisfied if and only if the term (A 0 P þ PA) is negative definite, that is 9P . 0

a controller for such state feedback is given by K ¼ YW 1 .

3.3 H1 control LMIs

that satisfies V_ (x) , 0 is a necessary and sufficient condition to assure that the system is stable (i.e. all trajectories converge to zero) [24]. Since V(x) has quadratic form, this condition is referred as quadratic stability. In this case, the condition V_ (x) , 0 can be rewritten as follows V_ (x) ¼ x0 (A0 P þ PA)x , 0,

(12)

(8)

the existence of a quadratic function of the form V (x) ¼ x0 Px . 0,

W .0 AW þ WA 0 þ Bu Y þ Y 0 B0u , 0

kzk2 kwk 2 v(t)=0

kH (s)k1 W sup

(13)

where k k1 and k k2 stand for the infinity and the Euclidian norms, respectively. The following theorem, adapted from [27], guarantees a maximum H1 norm g (i.e. a minimum level of disturbance attenuation).

Theorem 3.2: System (1) is stabilisable by state-feedback

u ¼ K x and kzk2 =kwk2 , g if and only if there exists a symmetric definite positive matrix W [ Rnn and a matrix Y [ Rmn such that the following inequality holds 2 6 4

AW þ WA 0 þ Bu Y þ Y 0 B0u B0w C y W þ Dyu Y

Bw

W Cy0 þ Y 0 D0yu

g1 0

0 g 1

3 7 5,0 (14)

a controller for such state feedback is given by K ¼ YW21. IET Power Electron., 2010, Vol. 3, Iss. 1, pp. 75– 85 doi: 10.1049/iet-pel.2008.0271

www.ietdl.org A proof is given in [27]. Note that satisfaction of inequality (14) implies satisfaction of (12) and therefore this theorem also ensures quadratic stability. The polytopic case is directly applicable by satisfying (14) for all the vertices G1 , . . . , GL of the polytopic domain.

3.4 Pole placement LMIs It is a desirable property of the closed-loop system that its poles are located in a certain region of the complex plane, in order to assure some dynamical properties like overshoot and settling time. In [28], a region of the complex plane S(a, r, u), depicted in Fig. 2, is defined such that the (complex) poles of the system in the form x + jy satisfy x , a , 0,

jx + jyj , r,

y , cot (u)x

(15)

In such a case the poles of the system x + jy ¼ zvn + j vd ensure a certain damping at the desired rate (see [29]). The presented region is equivalent to a minimum decay rate a, a minimum damping ratio z . sin u and a maximum vd , r cos u, where damped pffiffiffiffiffiffiffiffiffiffiffiffi natural frequency ffi v d ¼ v n 1 z2 . The following theorem allows to constraint the location of the closed-loop poles in the region S(a, r, u).

Theorem 3.3: The closed-loop poles of the system (1) with

a state-feedback u ¼ Kx are inside the region S(a, r, u) if there exists a symmetric definite positive matrix W and a matrix Y such that AW þ WA 0 þ Bu Y þ Y 0 B0u þ 2aW , 0

WA 0 þ Y 0 B0u rW ,0 AW þ Bu Y rW

cos u(AW þ WA 0 þ Bu Y þ Y 0 B0u ) sin u(AW þ WA 0 Bu Y þ Y 0 B0u ) sin u(AW WA 0 þ Bu Y Y 0 B0u ) ,0 cos u(AW þ WA 0 þ Bu Y þ Y 0 B0u ) and K ¼ YW 21 is the state feedback gain.

Figure 2 LMI region S(a, r, u) IET Power Electron., 2010, Vol. 3, Iss. 1, pp. 75– 85 doi: 10.1049/iet-pel.2008.0271

(16)

A detailed proof is reported in [28], where the pole placement in generic regions of the complex plane is explored. A different point of view on robust pole placement is given in [30]. Once again, the polytopic case directly extends by satisfying (16) – (18) for all G1 , . . . , GL . Therefore by taking into account these conditions in each vertex of the convex polytope Co G1 , . . . , GL , we will ensure a proper performance despite uncertainty.

3.5 Remarks We summarise the LMI synthesis method as follows. The design of a robust control for a boost converter consists of solving inequalities (14) and (16)–(18) to find the matrix variables Y and W that minimise the H1 norm g and that satisfy the pole placement constraints of the region S(a, r, u) for all the extrema of the polytopic model G1 , . . . , GL min g Y ,W

under conditions (14), (16), (17) and (18) 8 Gi , i ¼ 1, . . . , L

(19)

It is necessary to remark that the stability and the H1 bound of the closed-loop system is guaranteed for arbitrarily fast changes in Dd0 and R. However, the pole placement constraint is only satisfied if the time-dependent parameters changes are slowly enough to recover the steady state of the system. For a survey on how the rate change of parameters can be taken into account, see [17]. In the next section, we develop in detail the given procedure for a specified boost converter.

(17)

4 Control implementation and experimental results (18)

In this section, we present the results of the LMI design method proposed above. First, we identify the parameter values of the boost converter. Next, the state-feedback gain K is numerically found. The resulting controller has been simulated with a switched model of the converter by using PSIM software. We have compared the simulated response of the proposed controller with a conventional PID controller, in order to evaluate the robustness and performance achieved with our approach. Finally, the simulation results have been verified with an experimental set-up that is explained in detail at the end of this section. The parameters of the dc – dc converter, that have been taken from [13], can be found in Table 1. The nominal values of the uncertain converter parameters are R ¼ 50 V and D d0 ¼ 0.5. Note that since the output voltage vo is considered constant, an uncertain operation point Dd0 in the interval [Dd0 min , Dd0 max ] introduces a range of possible line 0 0 voltages Vg [ [vo Ddmin ,vo Ddmax ] that are included in the polytope Co G1 , . . . , GL defined in (6).

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www.ietdl.org Table 1 Boost converter parameters Parameter

Value

R

[10, 50] V

D d0

[0.3, 1]

vo , Vref

24 V

C

600 mF

L

310 mH

Ts

5 ms

Given the parameters of Table 1, the uncertain variables d and b, defined in (7), belong to the intervals [1, 3.33] and [0.02, 1.11], respectively.

4.1 Controller synthesis As it has been stated before the synthesis objective is to minimise g while a pole placement constraint S(a, r, u) is satisfied. The values of (a, r, u) can be found in Table 2. In order to have the poles of the system inside the valid frequency range of the model, r has been set to 1/10 of the switching frequency. For a minimum damping ratio of Table 2 Pole placement parameters Parameter Value

a r

u

130 2p 10Ts 258

0.4, u has been set to 258. Finally, for a fast decay rate, we have tested several different values of a. For the present case, the parameter a has been set to 130; a higher value of a results in an unfeasible problem for this parameter set. Solving problem (19) using Matlab’s standard LMI toolbox [19], which yields a state-feedback controller K K ¼ [1:95

2:00

725:22]

(20)

and a guaranteed H1 bound from disturbances to outputs of g ¼ 2.89, which is equivalent to 9.21 dB. The corresponding control law that yields the duty cycle is dd (t) ¼ 1:95iL (t) 2:00vo (t) þ 725:22x3 (t)

(21)

4.2 Simulations The switched model of Fig. 1a with the controller K has been implemented using PSIM simulator [31], taking the nominal values of the converter. In order to show the robustness of the proposed controller, we have simulated the transient behaviour in presence of an abrupt load change under nominal conditions and out of nominal conditions. Fig. 3a depicts the output voltage when the converter operates at the nominal equilibrium point and reacts in front of a load step change of 0.5 A. It can be noted that the output voltage response presents a time constant of approximately 10 ms, that corresponds to a decay rate of 400, which as expected is larger than the minimum guaranteed decay rate (a ¼ 130). Furthermore, the output voltage exhibits a small overshoot that agrees with the damping ratio restriction. Fig. 3b shows the simulation out of the nominal condition, without any change in the controller parameters. The duty cycle at the new equilibrium point is D d0 ¼ 0.3, which corresponds to a Vg equal to 7.2 V. Again, we simulate the regulator

Figure 3 PSIM simulated response for a loading step of 0.5 A with the nominal load R ¼ 50 V

a Under nominal duty cycle D d0 ¼ 0.5 b Out of nominal duty cycle D d0 ¼ 0.3

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Figure 4 PSIM simulated response for a loading step of 0.5 A with the nonnominal load R ¼ 10 V

a Under nominal duty cycle D d0 ¼ 0.5 b Out of nominal duty cycle D d0 ¼ 0.3

response in front of a step load perturbation of 0.5 A. It can be observed that despite the new operation point, the proposed controller maintains a small overshoot of the output voltage. Although this overshoot is larger than in the nominal case, it is yet inside of the specified damping ratio restriction. Besides, the decay rate is approximately 200 which is smaller than in the nominal case but larger than the specified a. Fig. 4 shows the same simulations of Fig. 3 with a different non-nominal load of R ¼ 10 V. The simulations show that the LMI controller satisfies the damping ratio specification and also the decay rate is better than the specified a. In order to contrast the performance and robustness of the proposed regulator, we have also simulated the boost converter with a conventional PID controller, whose control law at low frequencies is given in (22). This comparison has been made with the aim to show the benefits of using a robust control method compared with a conventional approach. Obviously, this comparison could have been done with another state-feedback or current-mode controller. Nevertheless, the robust method would keep, to a great extent, its advantages over a design disregarding uncertainty.

130, which are similar to those of the LMI controller. To select such PID controller we have used sisotool of Matlab. Vg (R (L=Dd02 )s) vo (s) ¼ u(s) CLRs2 þ Ls þ Dd02 R PID(s) ¼

0:0000447531s2 þ 0:0209432186s þ 31:25 s((s=1 105 ) þ 1)((s=1 105 ) þ 1)

(23) (24)

Fig. 3a depicts the transient simulation of the converter with the PID controller for a load step change of 0.5 A in nominal conditions. When compared with the LMI simulation, it can be observed that the response of the PID controllers is better in terms of decay rate and perturbation rejection, but it is slightly worse in terms of damping ratio, since they have been built using the concepts of phase and gain margin. However it is important to remark that all the controllers present adequate damping properties at the nominal operation point.

ð dd (t) ¼ Ki (Vref vo (t)) þ Kp (Vref vo (t)) þ Kd

d(Vref vo (t)) dt

(22)

Equation (23) shows the control-to-output transfer function of the boost converter, whereas (24) is the transfer function of the PID controller, which corresponds with a PID controller with Ki ¼ 31.25, Kp ¼ 0.0209432186 and Kd ¼ 0.0000447531. This PID controller has been designed at the nominal model of the converter. We impose as PID specifications a phase margin greater than 608 and a gain margin greater than 20 dB, which are common specifications in power electronics control. We also impose that the closed-loop poles present a damping ratio greater than 0.4 and a decay rate greater than IET Power Electron., 2010, Vol. 3, Iss. 1, pp. 75– 85 doi: 10.1049/iet-pel.2008.0271

Figure 5 PSIM simulated output impedance of the converter with the LMI controller. The simulation was made with a current sink of 200 mA at nominal duty cycle D d0 ¼ 0.5

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www.ietdl.org Out of these nominal conditions, in Fig. 3b, it is shown the time-response simulation in presence of the same perturbation of 0.5 A when the converter operates at a duty cycle equal to Dd0 ¼ 0.3. Again, we have maintained the previous controller parameters. Since the PID controller was not designed taking into account the uncertainty of the converter, the regulator with PID controller loses its damping

properties and exhibits poor response when the duty cycle Dd0 decreases far from the nominal value. The simulations with the non-nominal load R ¼ 10 V with the PID controller are shown in Fig. 4. While in the nominal duty cycle Dd0 ¼ 0:5 the behaviour is close to the case with the nominal load R ¼ 50 V, when the duty cycle is out of its nominal value Dd0 ¼ 0:3, the output voltage behaviour is worse than the

Figure 6 Implementation diagram of a boost converter with the proposed LMI state-feedback regulation

Figure 7 Detail of the circuit implementation of the LMI state-feedback controller and the PWM regulator 82

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www.ietdl.org LMI controller in terms of perturbation rejection, decay rate and damping ratio. Besides the transient response characteristics, we have simulated the steady-state output impedance of the regulator with the LMI controller. The frequency response for a perturbation iload ¼ 200 mA around the nominal duty cycle D d0 ¼ 0.5 has been depicted in Fig. 5 for the cases of the nominal load R ¼ 50 V and a non-nominal load R ¼ 10 V. The output impedance gain peak with the LMI controller is approximately 7 dB, which agrees with the H1 norm specification bound g ¼ 9.21 dB found in the LMI synthesis.

4.3 Experimental verification To demonstrate the advantages of the proposed control scheme, an experimental prototype of a boost converter has been implemented. The structure of the converter with the state-feedback controller is shown in Fig. 6. The statefeedback controller, whose electric diagram is also given in Fig. 7, only requires a small number of operational amplifiers and discrete components.

Figure 9 Experimental response for a loading step of 0.5 A out of nominal duty cycle D d0 ¼ 0.3. Upper traces are output voltage. Lower traces are output current

Fig. 8 shows the prototype transient response under a loading step of 0.5 A, as it was carried out in the simulations. The slew rate of the current step is greater than 150 A/ms. For this figure, the converter was working at the nominal operation point. The transient closely resembles the simulation in Fig. 3. The robustness of the controller is verified in Fig. 9, where the loading step is applied out of the nominal conditions (D d0 ¼ 0.3). Once again, the experimental waveform agrees with the simulation in Fig. 4. Besides the transient experiments, the output impedance of the regulator has been measured with the help of a frequency

Figure 10 Experimental output impedance of the converter with the LMI controller. The measurement was made with a current sink of 200 mA at nominal duty cycle D d0 ¼ 0.5 analyser connected to a voltage-controlled current sink of 200 mA. The measurements have been carried out for R ¼ 10 and 50 V, at the nominal duty cycle Dd0 ¼ 0.5. The measurements, depicted in Fig. 10, show a perfect agreement with the simulation results and demonstrate that despite of uncertainty, the LMI controller fulfils the objectives marked in the synthesis procedure.

Figure 8 Experimental response for a loading step of 0.5 A under nominal duty cycle D d0 ¼ 0.5. Upper traces are output voltage. Lower traces are output current IET Power Electron., 2010, Vol. 3, Iss. 1, pp. 75– 85 doi: 10.1049/iet-pel.2008.0271

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Conclusions

This work has presented a robust controller design framework based on LMIs for switched-mode dc – dc converters. The

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www.ietdl.org LMI design method has been applied to a boost converter and the results have been compared with a conventional PID controller by means of both numerical simulations. Also the experimental measurements from a laboratory prototype of a boost converter have verified the results of the LMI controller. The LMI control circuit implementation consists of a standard current sensor, some OPAMPs and a PWM device. We have shown that the state-feedback LMI approach is a valid synthesis method for non-minimum phase converters. With the proposed method, transient and steady-state performances can be taken into account and the resulting design has, in this case, good performance characteristics, despite the conservatism in the uncertainty model. The main advantage of this approach is that the state-feedback controller can be synthesised automatically, differently from other robust control methods, for which the controller must be synthesised manually or using CAD tools. Since the synthesis is carried out by means of a Matlab’s LMI toolbox, the method can be readily extensible in order to consider parasitic resistances in the inductor and capacitor by modifying the state-space matrices of the model. Finally, experimental verifications show a perfect agreement with the design constraints despite uncertainty. Future work will deal with the application of the method to more complex power converters such as high-order circuits, multiphase and multilevel converters.

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Acknowledgment

This work was partially supported by the Spanish Ministerio de Educacioń y Ciencia under grants TEC2004-05608C02-02 and TEC2007-67988-C02-02.

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References

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