Kugi, A.; Schlacher, K.

Nonlinear Controller Design for PWM–Controlled Converters

This contribution is concerned with the nonlinear controller design for a certain class of PWM(pulse widthmodulation)–controlled converter systems. It will be shown that under certain assumptions the SSA(state spaceaveraging)–model of the PWM–controlled converter with the duty ratio as the plant input has a very special mathe-matical structure. Based on this mathematical model a modified version of the nonlinear H2–design where an integralterm is systematically included in the nonlinear controller will be presented.

1. SSA–Model of a PWM–Converter System

Throughout this contribution we deal with PWM–controlled converter systems with N switches {S1, . . . , SN} andeach switch Sj , j = 1, . . . , N has two positions: an on–position denoted by A and an off–position denoted by B. It isassumed that the N–switch tuple {S1, . . . , SN} is turned on and off simultaneously in a CCM(continuous conductionmode)–operation, see, e.g., [1]. Suppose that the associated digraph of the PWM–controlled converter is connectedfor both switch positions, A and B. Let us further assume that all components (resistors, inductors and capacitors)are linear and only uncontrolled voltage and current sources are allowed. If in addition the coordinate chart ofthe network remains the same in both switch positions, then the considered PWM–controlled converters can bedescribed by two systems of differential equations of the form

ddt

x = fA (x) = FAx + GAs0 t ∈ (iT, (i + dA)T ] {S1 , . . . , SN} in A

ddt

x = fB (x) = FBx + GBs0 t ∈ ((i + dA)T, (i + 1)T ] {S1, . . . , SN} in B

(1)

for i = 0, 1, . . . with the state x (independent inductor currents and capacitor voltages), the vector s0 combining theconstant voltages and currents of the uncontrolled voltage and current sources and the matrices FA, FB , GA andGB with suitable dimensions. Here, dA, 0 ≤ dA ≤ 1, denotes the so–called duty ratio, which specifies the ratio ofthe duration of the N–switch tuple {S1, . . . , SN} in position A to the fixed modulation period T .

Let ϕfA

t (x) denote the flow of the electric circuit for the N–switch tuple {S1, . . . , SN} in position A and ϕfB

t (x)in position B. Then a solution γ (t) of the PWM–controlled converter (1) for t = iT , i = 0, 1, . . . meets the relation

γ ((i + 1)T ) = ϕfA

dAT ◦ ϕfB

(1−dA)T (γ (iT )) . (2)

Under the assumption that the modulation frequency is much higher than the natural frequencies of the convertersystem and the switches Sj are realized with common power semiconductor devices, we can derive the so–calledSSA(state space averaging)–model for the PWM–controlled converter (1), see, e.g., [1], [2]. The SSA–model of (1)

ddt

xa = fAdA + fB (1 − dA) = FBxa + GBs0 + ((FA − FB)xa + (GA − GB) s0) dA (3)

with the average state xa is the result of the first order approximation of γ (t) with respect to T by γa (t)

ddt

γa (iT ) = limT→0

∂T ϕfA

dAT ◦ ϕfB

(1−dA)T (γa (iT )) for t = iT and i = 0, 1, . . . . (4)

By means of a simple change of coordinates x = xa − xa and u = dA − dA the operating point(dA, xa

)of (3) can

be shifted to the origin and we obtain a nonlinear mathematical model of the form

ddt

x = Fx + g (x)u withF = FB + (FA − FB) dA

g (x) = (GA − GB) s0 + (FA − FB) (x + xa) .(5)

At this point it is worth mentioning that the SSA–model can also be derived from the averaged generalized potentialsin an energy–based description due to the famous theorem of Brayton–Moser, see, e.g., [2].

For a certain class of PWM–controlled DC–DC–converters with the output voltage taken as the plant output,

PAMM · Proc. Appl. Math. Mech. 2, 96–97 (2003) / DOI 10.1002/pamm.200310034

we are confronted with an unstable zero–dynamics. Thus, the direct application of the nonlinear control designtechnique of exact input/output linearization fails. Furthermore, a nonlinear state controller in general cannotcompensate for changes of the operating point due to load variations or parameter fluctuations. In order to overcomethese deficiencies we present a nonlinear controller design in the next section, where an integral term in the plantoutput is systematically included in the controller.

2. Nonlinear H2–Design with Integral Term

In the following we will restrict ourselves to the nonlinear SISO-system (5) with the state x, the plant input u and aplant output y of the form y = hT x. The goal of the nonlinear H2–design with integral term is to find a control law

ddt

xI = y = hT x and u = u (x, xI) with u (0, 0) = 0 (6)

such that the origin of the closed–loop is rendered asymptotically stable and the objective function

J2 = infu∈L2[0,∞)

∫ ∞

0

12

(βy2 + u2

)dt , β > 0 (7)

is minimized with respect to u. Under certain observability conditions it can be shown, see, e.g., [2], [3], that thestate feedback law

u∗ (x, xI) = − ∂

∂xV (x, xI) g (x) (8)

solves the nonlinear H2–design problem, if we succeed in finding a positive definite solution V (x, xI) of the associatedHamilton–Jacobi–Bellman equation (HJBe)

∂

∂xV (x, xI) Fx +

∂

∂xIV (x, xI)hT x +

12

(βxT hhT x− u∗ (x, xI)

2)

= 0. (9)

In general this is a very difficult task, which is why we often have to be content with a positive definite solution ofthe Hamilton–Jacobi–Bellman inequality (HJBi), where in (9) the “=” is replaced by “≤”. In this case, we say thatthe nonlinear H2–design problem is only solved suboptimally. However, for the PWM–controlled converters (5) withthe plant output y = hT x we may formulate the following proposition (see [2] for the proof):

P ropo s i t i o n Given the system (5) with the plant output y = hT x and suppose the matrix F is Hurwitz,the pair

(hT , F

)is observable and the condition hT F−1g (0) �= 0 is satisfied. Then the nonlinear state feedback

controller with integral part

ddt

xI = hT x

u = −xT(P11 +

(F−1

)ThhT F−1p22

)g (x) + p22h

T F−1g (x)xI ,(10)

with P11 as the unique positive definite solution of the Lyapunov equation

P11F + F T P11 + βhhT = 0 (11)

solves the suboptimal nonlinear H2–design problem for the controller parameters β, p22 > 0. Furthermore, theparameters β and p22 can be used to adjust the closed–loop performance.

3. References

1 Kassakian, J.G.; Schlecht M.F.; Verghese G.C.: Principles of Power Electronics, Addison Wesley, New York, (1992).2 Kugi, A.: Non–linear Control Based on Physical Models: Electrical, Mechanical and Hydraulic Systems, Lecture Notes in

Control and Information Sciences 260, Springer, London, (2001).3 van der Schaft, A.J.: L2–Gain and Passivity Techniques in Nonlinear Control, Springer, London, (2000).

Prof. Dr.techn. Andreas Kugi, Chair of System Theory and Automatic Control, Saarland Uni-versity, Tel.: 0049 (0)681 302-64720, E-mail: andreas.kugi@lsr.uni-saarland.de, Gebaude 13, 66123Saarbrucken, Germany and Prof. Dr.techn. Kurt Schlacher, Department of Automatic Controland Control Systems Technology, Johannes Kepler University of Linz, Altenbergerstr. 69, 4040Linz, Austria.

Section 2: Stability and Control Theory 97