Robust output tracking control of nonlinear MIMO systems via sliding mode technique

Automatica, Vol. 28. No. 1, pp. 145-151, 1992 Printed in Great Britain.

0005-1098/92 $5.00 + 0.00 Pergamon Press p|c ~) 1991 International Federation of Automatic Control

Brief Paper

Robust Output Tracking Control of Nonlinear MIMO Systems via Sliding Mode Technique* HAKAN ELMALI~" and NEJAT OLGAC~':~ Key Words--Automatic control; variable structure systems; nonlinear control systems; nonlinear systems; robust control; stability; smoothing; linearization techniques.

venue, a powerful algorithm following Chang (1990) is introduced: second order sliding mode control. I/O linearization of a nonlinear system relies on the exact cancellation of the nonlinear terms. In the presence of modeling uncertainties and disturbances, however, this linearization may not be possible. But, for a certain class of disturbances which obey the so called "matching conditions", I/O linearization is guaranteed. Consequently the robust control strategy (SMC) can be performed effectively. It is important to note that there may be a remainder of the state dynamics which does not appear in the I/O linearized structure. This portion of the dynamics (also named zero dynamics) should behave bounded if not asymptotically stable. This point is discussed in Section 2. A characteristic component of SMC applications is s(x,t) =0, which represents the sliding hyperplanes. The second order SMC was originally conceived to introduce a low-pass filter for the s dynamics (i.e. the time variation of s) and a high-pass (or band-pass) filter for the output tracking error dynamics in series. This form of the second-order s dynamics rejects unwanted high-frequency signals resulting from the disturbances and uncertainties. The bandwidth of the s dynamics should be designed low to minimize the high frequency excitations to error dynamics. Therefore, the selection of SMC parameters for the s dynamics and the error dynamics are crucial to obtain the desired tracking performance. The difficulties associated with this philosophy are elaborated upon in Section 3. The novelty in this paper is at the junction of the two methodologies; I/O linearization and SMC, for nonlinear MIMO systems. Parallelism between these two operations is pointed out in Section 4. Computer simulations have proven very successful for the attitude control of a spacecraft on a circular orbit. In Section 5, several cases of parameter selection are presented.

Abstract--The robust output tracking control problem of general nonlinear multi-input multi-output (MIMO) systems is discussed. The robustness against parameter uncertainties and unknown disturbances is considered. A second order sliding mode control (SMC) technique is used to establish the desired tracking. Input/output (I/O) linearization, relative degree, minimum phase and matching condition concepts are reviewed. Some earlier SMC strategies which are restricted to the systems in canonical form are extended to a much broader class of nonlinear dynamics. It is also shown that for unperturbed dynamics, the sliding phase of the SMC applications have a direct correspondence to the I/O iinearization operations. Interesting parametric flexibilities emanate within the formation of the second order SMC, designating the "s dynamics" and the "error dynamics" segments as frequency domain filters. However, a critical impasse is posed in the off-line selections of the design parameters. A set of example cases is presented for a spacecraft attitude control problem. These examples manifest that the proposed control strategy is tunable to a desired response despite the disturbances and uncertainties.

1. Introduction GENERALCLASSo f nonlinear control systems is considered in this paper. Towards a robust output tracking control of this system, an I/O linearization is performed, first. Then a second order SMC is applied on the I/O iinearized system. The relation between I/O linearization and SMC technique is discussed. It is shown that much broader class of nonlinear dynamics can be treated with this strategy, compared to the earlier applications of other investigators on the systems in canonical forms. The recent developments on the I]O linearization methods Isidori (1985), Sastry and Isidori (1989) and Byrnes and Isidori (1987) have brought a profound insight to the subtleties in the above mentioned procedures. We base this study upon these clarifications in I/O linearization. After the nonlinear transformation which converts any nonlinear system into a linear format, we come to the second level of endeavor: guaranteeing the robustness of the control strategy against modeling uncertainties and disturbances. At this m

2. I / 0 linearization of MlM O nonlinear systems with uncertainties A MIMO nonlinear control system is taken as: m

i = t(x) + At(x, t) + ~ [gi(x) +

Agi(x)lui

i~l

Yi=hi(x),

i=l .....

m

(la)

where

x(.):~+--~fi~_~n is the plant state vector, are system input and output vectors, respectively, f(-), g~(.) : ~ n - ~ ~ ' , i = 1. . . . . m, are smooth vector fields and h i ( . ) : ~ n - - ~ , i = 1. . . . . m, are smooth functions. At(x, t) and Ag~(x), i = 1. . . . . m, represent the disturbances and modeling uncertainties. For convenience the above equation will be rewritten in a condensed form:

*Received 7 November 1990; revised 16 May 1991; received in final form 1 June 1991. The original version of this paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor V. I. Utkin under the direction of Editor H. Kwakernaak. ~-University of Connecticut, Mechanical Engineering Department, Storrs, CT 06269, U.S.A. ~Author to whom all correspondence should be addressed.

u(.),y(.):~+--~'~

i = t(x) + At(x, t) + [G(x) + At;(x)]u y = h(x) 145

(lb)

146

Brief Paper a0((~ , q) = L ~ L ~ ' - ' h , ( T

where

Um) Ym)

U = c o l (U 1 . . . . .

y : c o l (Yl . . . . .

G(x) = [g,(x) . . . . .

Aaq(~, q) = L,x~L~f ~ lhi(T-'(~, q))

Abi([, q)

Ag~(x)].

Throughout this paper, the boldface lower-case letters indicate vectors and the upper-case, matrices. The goal of the control problem is to guide the output y(x) along a desired trajectory ya(t) for the given system in equation (1). The control strategy utilized should be robust enough to handle the modeling uncertainties and unknown disturbances (Af and AG). The upper bounds of these variations are: IA~(x, t)l -< a'i(x, t) IAgi~(x)l