Copyright © 2002 IFAC 15th Triennial World Congress, Barcelona, Spain SLIDING MODE SIMPLEX CONTROL METHODS FOR MECHANICAL SYSTEMS Giorgio Bartolini ∗ Elisabetta Punta ∗∗ Tullio Zolezzi ∗∗∗ ∗ University of Cagliari, DIEE, Cagliari, Italy e-mail: [email protected] ∗∗ University of Genova, DIST, Genova, Italy e-mail: [email protected] ∗∗∗ University of Genova, DIMA, Genova, Italy e-mail: [email protected] Abstract: We consider the control of mechanical systems based on sliding mode control techniques. Recently developed simplex control methods are shown to converge in a finite time when applied to nonlinear systems under bounded deterministic uncertainty. Applications are considered to the control of mechanical systems in which the control action is provided by monodirectional devices. Keywords: Control, sliding mode control, control of nonlinear uncertain systems, control of mechanical systems. 1. INTRODUCTION AND PROBLEM u ∈ U. (2) STATEMENT The system is subject to unknown additive uncer- In this note a class of nonlinear uncertain con- tain perturbations of deterministic nature. Thus trolled objects, sufficiently wide to include the every uncertain system’s state y evolves according representation of most of the robotic struc- to tures, is considered. The control problem con- ≥ sists in forcing this class of objects to fulfill the y˙ = f(t, y, u)+ϕ(t, y, u),t 0 (3) control aim with the constraint that the con- where ϕ represents some uncertainty acting on trol vector u belongs to a finite set of vectors K the nominal system (1). Here the control vector ui ∈ R : i =1,...,p . The control design con- u ∈ RK and the state variables x, y ∈ RN . sists in the choice of the vectors ui, the integer value p, the set of events together with the suitable A given sliding manifold commutation criterion in order to guarantee the achievement of a prespecified control objective s(t, x) = 0 (4) through a suitable sliding condition. The formula- tion of the above control problem, in many cases, is fixed in order to fulfill prescribed control aims, ∈ M implies a representation of the controlled system where s(t, x) R . The sliding manifold is in terms of discontinuous differential equations. designed in such a way that, when the trajectories of system (3) belong to it, then the desired control The basic problem is the following. We consider a objectives, e.g. stabilization or model tracking, are fixed known nominal control system satisfied. x˙ = f(t, x, u),t≥ 0 (1) The control aim is to select an admissible feedback control law u = u∗(t, x) ∈ U such that the corre- with control constraint sponding state y through (3), issued from a given initial position at time t = 0, reaches in a finite taking values in U, such that the vectors time t∗ some point fulfilling (4) (attainability)and remains on the sliding manifold (4) for all t ≥ t∗ gi = gi(t, x)=Ds(x)f[t, x, ui(t, x)] (sliding). fulfill A known constant L>0 is fixed. We want to T 2 control those uncertain state variables y = y(t)of 0 <a≤|gi| and gi gh ≤−c |gi||gh| (7) the control system (3) which fulfill the condition if i = h |y(t)|≤L, t ≥ 0 (5) for every (t, x) and some constants a, c =0. in order to guarantee the sliding property If (7) holds, then (for any given (t, x)) the vectors M s [t, y(t)] = 0 (6) gi, i =1,...,M+ 1 define a simplex in R in the following sense. RM is partitioned in M + 1 cones for every t sufficiently large. We assume that s [t, y(t)] is available to the controller for feedback Qh =cone(gi : i = h)= ≥ purposes for each t 0 and every uncertain state y. ≥ = αigi : αi 0ifi = h , (8) We consider plants (3) affected by noise in such i=h a way that the statistical informations needed to h =1,...,M +1 employ stochastic control techniques are unavail- able to the controller. Instead we rely only on the nominal system (1), on available bounds about with pairwise disjoint interiors. Thus for every ≥ uncertainties and some qualitative features of the t 0andx with s(t, x) = 0 there exist coefficients ≥ dynamics. αi = αi(t, x) 0 such that At least three methods are known to control a s(t, x)= αigi(t, x) (9) dynamical system by sliding mode techniques, i.e. i=h in order to fulfill (4). The first two, namely component-wise sliding con- with the smallest possible index h = h(t, x). Then trol and unit control, have been widely investi- the moving simplex control law is defined by ∗ gated, see (Utkin, 1992) for a survey. u (t, x)=uh(t, x). (10) In the following we focus on the less known sim- M ∗ plex methods (originated by (Bajda and Izosi- The cones Qh given by (8) cover R , hence u mov, 1985)) which we now describe in a more given by (10) is well defined. general and new setting. ∗ By (10), the control law u undergoes discontinu- ities as a function of x. By injecting u∗ into (3) 2. MOVING SIMPLEX CONTROL UNDER we consider the system UNCERTAINTY y˙(t) = (11) Let Ω be an open set of RN containing the closed = f (t, y(t),u∗ [t, y(t)]) + ϕ (t, y(t),u∗ [t, y(t)]) ball of center 0 and radius L. We assume that K N ≥ M, U is a closed set in R ,and where the dynamics f [t, x, u∗(t, x)], ϕ [t, x, u∗(t, x)] s :[0, +∞) × Ω → RM are now discontinuous functions of x. Based on the Filippov notion of solution, it is possible to build a rigorous theory, see (Filippov, 1988) and is a smooth mapping; (Utkin, 1992), in good agreement with the ob- f,ϕ :[0, +∞) × Ω × U → RN served behaviour of some real control systems, see (Utkin, 1978). are Carath´eodory maps, for every uncertain dy- In the following, states of (3) corresponding to u∗ × namics ϕ. Denote by Ds(t, x)theM N jacobian will be understood as Filippov solutions of (11). ∂si(t,x) matrix ∂xh , i =1,...,M, h =1,...,N and assume that Ds(t, x) has maximum rank every- We emphasize that no information beyond those where. available about the nominal system (1), (2), (4) are needed in order to obtain the moving simplex The basic assumption is the following. There exist control law u∗ in (10). Carath´eodory functions Under explicit conditions involving the known u1(t, x),...,uM+1(t, x) nominal system, the geometry of the simplex with obtuse angles made by g1,...,gM+1,andan Theorem 1 generalizes the convergence result of estimate of the maximal amount of uncertainty, (Bartolini et al., 1999) to control systems subject the feedback u∗ guarantees the sliding condition to uncertainty and time-dependent sliding mani- (6) for every uncertain state fulfilling (5). fold. Suppose that there exist constants A∗, H such that 3. FIXED SIMPLEX CONTROL ∗ |f[t, x, ui(t, x)]|≤A and A different but related control method deals with |ϕ[t, x, ui(t, x)]|≤H (12) a fixed simplex, i.e. the edges do not depend of t, x and the dynamics, as follows. Suppose that, as in for all t, x, i =1,...,M +1,H explicitly known most mechanical systems, the nominal dynamics to the controller. Finally assume that for some (1) are affine in the control variables. Moreover let constants W0, W we have the nominal system be autonomous, namely ∂s(t, x) x˙ = A(x)+B(x)u (16) ≤ W0 and |Ds(t, x)|≤W (13) ∂t with control constraint everywhere. |u|≤ρ (17) Theorem 1 K = M, and sliding manifold s(t, x)=Cx + d(t)=0. (18) Suppose that (7), (12), (13) hold and 2 ac > (W0 + WH) E (14) Let the control system be described by Lagrangian coordinates q,sox =(qT , q˙T )T . Assume that where B = B(q) only and that deterministic uncertainty acts on the nominal system (16) as | | i∈I αi gi E = max : αi ≥ 0, |α| =1 , y˙ = A +∆A +(B +∆B)u. i∈I αigi Constants A0, B0 are known such that the un- I of M elements. known dynamics fulfill Then every uncertain state fulfilling (5) verifies | |≤ | |≤ the sliding condition (6) for every t sufficiently ∆A(t, x, u) A0 and ∆B(t, x, u) B0 large. M for all t, x, u. Fix points u1,...,uM+1 ∈ R such Given the maximal amount H of uncertainty, that condition (14) requires sufficient control authority about the nominal system in order to fix a>0 |ui| = ρ, i =1,...,M +1and sufficiently large, because of (7). T 2 ui uh ≤−c |ui||uh| ,i= h We sketch the proof of Theorem 1 in the (very particular) case when s = s(x) only, and no for some constant c = 0. Then for every t ≥ 0, uncertainty acts on the system (hence W0 =0= |x|≤L with s(t, x) =0wehave H). s(t, x) ∈ cone(ui : i = h) Let s(y)=s[y(t)] fulfill (6) in a given time inter- val. If y corresponds to u∗ we formally compute with the least possible h = h(t, x). Then define T T T the fixed simplex control law as s (y)˙s(y)=s (y)Ds(y)˙y = αigi gh (15) i=h u∗(x)=uh. 2 2 ≤−c αi |gi||gh|≤−ac |s(y)| i=h Accordingly, the fixed simplex control law u∗ is constant in every region of [0, +∞) × RN where by (7).