The controller design uses a sliding mode technique and is divided in two phases: slow feedback control and fast feedback control so that a final composite con-.
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Dynamics and Control. 1 1, 2 5 4 6 , 2001 B 200001 Kluwer A c d m ~ i cP~kdishen.Manufactured in The Netherlands.
Sliding Mode Control and State Estimation for a Class of Nonlinear Singularly Perturbed Systems R. CASTRO-LINARES Deportment of Electricol Engineering, CINVESTAV-IPN, Apdo. Posml 14-740. 07300 Mexico, D.E, Mexico JA. ALVAREZ-GALLEGOS Deportment of Electrical Engineering, CINVESTAV-IPN, Apdo. Pusrcrl 14-740. 07300 Mexico, D.E, Mexico v. VASQUEZ-LOPEZ Departnrenr of Electricol Engineering, CINVESTAV-IPN. Apdo. Postid 1-1-740, 07300 Mexico, D . E , Mexico
Editor: H. Sira-Ramirez Received December 10. 1999; Revised Jcrnuary 26, 2001: Accepted Morch 16, 2001 Abstract. This paper is concerned with the design of a controller-observer scheme for the exponential stabilization of a class of singularly perturbed nonlinear systems. The controller design uses a sliding mode technique and is divided in two phases: slow feedback control and fast feedback control so that a final composite control is obtained. Assuming that only the fast state is available and the system's output is a function of the slow state, an observer design is presented. A stability analysis is also made to provide sufficient conditions for the ultimate boundedness of the full order closed-loop system when the slow state is estimated by means of the observer. An application to the model of a permanent magnet stepper motor is given to show the controller-obsewer methodology and stability analysis. Keywords: nonlinear systems, singular perturbations, state estimation, sliding-mode, stabilization
1. Introduction Frequently, the dynamic models obtained from theoretical considerations are so large that they may be impractical for system analysis and control design purposes. Thus, several methods have been suggested in the literature for deriving reduced order dynamic models from high order models. However, such methods cannot be applied to nonlinear systems. A method used for model reduction of large-scale systems is the singular perturbation method (see, for example, [22,26], and the references therein). Some of the advantages of the singular perturbation method are its applicability to nonlinear systems as well as its simplicity and good performance in many practical control situations [24,31]. When the singular perturbation method is applied, the original system is decomposed into two subsystems of lower dimension, both described in different time scales. From this decomposition, a state feedback may be designed for each lower order subsystem combining them in a so-called composite feedback that is applied to the original system (see [17,19]). On the other hand, the sliding mode control techniques have been extensively used when a robust control scheme is required [7,32]. Thus, the idea of combining the singular perturbation method and the sliding mode control techniques represents a good pos-
26
CASTRO-LINARES, ALVAREZ-GALLEGOS A N D V A S Q U E Z - L ~ P E Z
sibility to achieve classical control objectives for nonlinear systems having unmodeled or parasitic dynamics and parametric uncertainties. In some papers, this idea has been developed by designing a discontinuous control for the fast dynamics thus leading to an improved slow reduced order model [27,29]. More recently, sliding mode controllers have been designed in [I] for a class of nonlinear singularly perturbed systems with stable fast dynamics. Also, there are some works that use a Lyapunov-based approach to design stabilizing controllers for singularly perturbed systems with uncertainties satisfying strict andlor extended matching conditions [6,10,15] as well as mismatching conditions [21]. The controller resulting from the use of the singular perturbation and the sliding mode techniques needs. however, to have information about the state vector of the plant. Then, it is necessary to estimate the states of the system by using state observers. At the present time, several methods have been suggested for the design of the observers. In [4] an excellent survey is made of different approaches proposed for such designs. In this paper, one considers a class of nonlinear singularly perturbed systems described form by the so-called standard singularly pert~rrbed
where to 0, x E & c Rnis the slow state, z E Bz c Rm is the fast state, ~r E Rr is the control input and E E (0, I) is a small positiveperturbation parameter. f i , f2, the columns of the matrices F I , F2 and g2 are assumed to be bounded with their components being meromorphic functions of x . & and B: denote closed and bounded subsets centered at the origin. It is also supposed that f i (0) = f2(0) = 0 and, for u = 0, the origin ( x , z) = (0,O) is an isolated equilibrium state, and that F2(x) is nonsingular for all x E Bx. The goals of the paper are to review the design of a stabilizing and regulation control law based on sliding mode techniques for the class of nonlinear singularly perturbed systems (I), ( 2 ) . recently reported in [I], and to design an state estimator for the slow state when the system's output is a function of this state only. Also, a study of the stability properties of the resultant closed-loop system is presented when the slow state is replaced by its estimate. The observer design is based on a new approach to exact linearization of nonlinear systems by state transformation and input-output injection [I 1,251, as well as sliding mode techniques developed for observer design [28,36]. For the stability analysis of the controller-observer scheme, sufficient conditions are given to assure the ultimate boundedness of the resultant cIosed-loop system. The class of singularly perturbed systems considered in the present paper is a subclass of the one considered in [I], but it is a significant one since many models of processes can be put in the form (I), (2) (e.g., electrical machines and robot manipulators). The paper is organized as follows. In Section 2, the two time scale sliding mode control design is reviewed. In Section 3, the observer design is presented together with an analysis of its convergence properties. The stability properties of the resultant closed-loop system, when the slow state variable is substituted by its estimate, are discussed in Section 4. In Section 5, the controller-observer scheme proposed is illustrated through an example of
SLIDING MODE CONTROL AND STATE ESTIMATION
angular position regulation in the model of a permanent magnet stepper motor. Finally, some conclusions are given in Section 6.
2. Reduced Order Sliding Mode Control Let us consider the class of nonlinear singularly perturbed system described by (I), (2). The slow reduced subsystem is found by making s = 0 in (2), obtaining the nth order slow system
where x,, zs and us denote the slow components of the original variables x, z and u, respectively, and
In the same manner, L~,(X,)in (3) denotes the slow state feedback which only depends on x,. Then, the slow invariant manifold associated to (1), (2) is defined by
where the functions @,
* and u,,
satisfy the so-called manifold condition
for all x, E B, and s sufficiently small, where the subscript "e" stands for exact solutions. The fast dynamics, (boundary layer system), is obtained by transforming the (slow) time scale t to the (fast) time scale s := (t - to)/&and introducing the deviation of z from M,, i.e., q := z - h,(x, s). The original system (1), (2) then becomes
+
ah, di F Z ( ~ ) [+he(:, V E)] g?(i)u - -d7 ax d s where ~ ( 0 = ) zo - he(xo, E ), Z(s) := ~ ( s +to), s with Z(0) = ZO,and i ( s ) := x ( s s +to) with i ( 0 ) = xo. The so-called composite control for the original system (1), (2) is defined by dV - = f2(?)
+
where u,, and u,f denotes the slow and fast components of the control, respectively. The component lief is used to make M , attractive and vanishes there, i.e., uef (x, 0, E) = 0. If
CASTRO-LINARES, ALVAREZ-GALLEGOS AND V A S Q U E Z - L ~ P E Z
LI,,(.?, E) and ah,/a.i are bounded and 2 remains relatively constant with respect to s , then the term [ah,/a.i][ai/as] can be neglected for E sufficiently small. Since Equation (10) defines the fasf reduced subsystem, an O(E) approximation can be obtained for this subsystem using Equation (4) and setting E = 0 in (9), (lo), this is (12)
ds
where fj, h,(2, 0) = h(2) and uf are O(E) approximations for 17, he(;, E) and u,f during the initial boundary layer, and {(O) = zo - h(xo, 0).
2.1. Sliding-Mode Control Design The sliding mode control for the system (I), (2) is designed in two stages. First, the slow corlfrol is designed for the slow subsystem (3). Let us first consider an (n - r)-dimensional slow nonlinear switching surface defined by
where each function a,, : BX -+ 'R is a C ' function such that asi(0) = 0. The equivalent control method [32] is used to determine the slow reduced subsystem motion restricted to the slow switching surface as(x,) = 0, obtaining the slow equivalent confrol
[2
us,(xs) = - - g(x,)
I-' ;:
- f (x,)
where the matrix [aa,/ax,]g(x,) is assumed to be nonsingular for all x, E & . Substitution of (14) in (3) yields the slow sliding-mode equation
where In denotes the n x n identity matrix. In order to complete the slow control design one sets [7]
where us, is the slow equivalent control (14) which acts when the slow reduced systems is restricted to a(x,) = 0, while us, acts when a(x,) # 0. In this work the control us, is selected as
where L,(x,) is an r x r positive-definite matrix whose components are C0 bounded nonlinear real functions of x,, such that IILs(xs)ll 6 ps for all x, E Ox with a constant Ps > 0.
29
SLIDING MODE CONTROL AND STATE ESTIMATION
The equation that describes the projection of the slow subsystem motion outside u(x,) = 0 is given by
The stability properties of ~ ( x , )5 0 in (18) can be studied by means of the Lyapunov function candidate V(x,) = (1/2)a~(xs)us(xs)whose time derivative along (18) satisfies v(x,) = - ~ ~ ( X ~ ) L , ~ ( ~ ~< )0Ufor ~ (allXx, ,~E) 4,. since L,(.r,) is positive definite in B(x). Thus, in accordance to [13], the existence of a slow sliding-mode can be concluded. The system (3) with the control (16) yields the slow reduced closed-loop system
If the slow system (3) is stabilizable in 13,,one chooses us(xs) in such a way that the stability of the system can be guaranteed on the switching surface us(xs) = 0. More precisely, us(x,) is selected such that the equilibrium point x, = 0 of the dynamics (19) is locally exponentially stable. Thus, by a converse theorem of Lyapunov [17], there exists a Lyapunov function Vs(xs) which satisfies
for some positive constants ci, c2, cj and cq. The fast control design for the fast subsystem (12) can be obtained in a similar way to the one used for the slow control. This is, one considers an (m - r)-dimensional fast switching surface defined by / (6) = coqa f, (6), . . . ,
f,
(6))
(21)
=0
where each function ufi (6) : BZ 4 R is also a C' function such that ufi (0) = 0. The complete fast control takes the form uf = ufc + uf, where ufc is the fast equivalent control given by
and
In (22), (23), the matrix [auf/aij]g2(f) is assumed to be nonsingular for all (2,2) E Bx x 23:. and L (6) is a positive-definite matrix of dimension r x r, whose components are Co bounded nonlinear real functions of 6 such that 11 L (ij) 11 pf for all ( i , 2) E Br x B,
0. Thus, a sufficient condition for the existence of a sliding mode is that llell
- py < a MCP,-lcT (C)aM(A - HC)
'
The following results states the exponential convergence of the estimation error to zero. T HEOREM 1 Consider the transformed observable system (34), (35). Let H E R"~' be a constant vector such that ( A - H C ) is Hunr~itt,a discontinuous external signal = sgn(e,) and a matrix A = P;'cT. Then, the obsen*er(36), (37) guarantees the exponential convergence of the estimation error e ( t ) to the origin.
Proof: Choose the Lyapunov function candidate V, = e T Pee. The time derivative of this function along the trajectories of the dynamics ( 38) becomes V, = e T [ p e (-~ H C ) + ( A - H C ) ~ P , ] ~2 e T p e ~ v . From (40)and the selection of v and A , one obtains T
V e = -e Qee - 21Cel
Then limt,,
< -eT Q,e
< 0.
e ( t ) = 0 . Moreover, from the fact that V, is positive definite one has
Set now p, = min{eT Qee: eT Pe = 11, thus Ve
-
ve
< -p,
and
