Implicit Euler Numerical Simulation of Sliding Mode Systems

Transcription

1 Implicit Euler Numerical Simulation of Sliding Mode Systems Vincent Acary and Bernard Brogliato, INRIA Grenoble Rhône-Alpes Conference in honour of E. Hairer s 6th birthday Genève, 17-2 June 29

2 Objectives Study the implicit Euler discretization of a class of differential inclusions with sliding surfaces ( Filippov s systems) Show that this numerical method permits a smooth stabilization on the sliding surface, in a finite number of steps Show how this may be used in real-time implementations of sliding mode control

3 To start with we consider the simplest case: 1 if x(t) < ẋ(t) sgn(x(t)) = 1 if x(t) > [,1] if x(t) =, x() = x (1) with x(t) IR. This system possesses a unique Lipschitz continuous solution for any x. The backward Euler discretization of (1) reads as: x k+1 x k = hs k+1 (2) s k+1 sgn(x k+1 )

4 As is known the explicit Euler discretization of such discontinuous systems yields spurious oscillations around the switching surface [Galias et al, IEEE TAC and CAS 26, 27, 28]. this means that the derivative of the switching function while sliding occurs, is very badly estimated. Both the explicit and the implicit methods converge (the approximated solution x N ( ) tends to the Filippov s solution as h ). However or the backward Euler method the following holds: Lemma For all h > and x IR, there exists k such that x k +n = and x k +n+1 x k +n = for all n 1. h

5 On this simple case this has the following graphical interpretation, as the intersection of two graphs: s i h x k 1 x k x k+1 x k+2 x i h Figure: Iterations of the backward Euler method.

6 An interesting property is that the smooth stabilization and the finite-time convergence on the switching surface, hold (more or less) independently of the step h > : 1 x(t) s(t) 1 x(t) s(t) 1 x(t) s(t) t (a) h = t (b) h = t (c) h =.1 Figure: A simple example for x = 1.1 at t =.

7 EXTENSIONS We shall focus on inclusions of the form: ẋ(t) f (t,x(t)) B Sgn(Cx(t) + D), a.e. on (,T) x() = x (3) with B IR n m Sgn(Cx(t) + D) = (sgn(c 1 x + D 1 ),...,sgn(c m x + D m )) T IR m, where sgn( ) is multivalued at.

8 Well-posedness of the differential inclusions (3) Proposition Consider the differential inclusion in (3). Suppose that There exists L such that for all t [, T], for all x 1, x 2 IR n, one has f (t, x 1 ) f (t, x 2 ) L x 1 x 2. There exists a function Φ( ) such that for all R : { Φ(R) = sup f } t (, v) L 2 ((,T);IR n ) v L 2 ((,T);IR n ) R < +. If there exists an n n matrix P = P T > such that PB i = C T i (4) for all 1 i m, then for any initial data the differential inclusion (3) has a unique solution x : (,T) IR n that is Lipschitz continuous.

9 Proof of Proposition 1: The proof uses the change of state variables z = Rx where R = R T > and R 2 = P. After some manipulations the system is rewritten as ż(t) Rf (t,r 1 z(t)) m f i (z(t)) (5) where f i (z) = C i R 1 z + D i. The multivalued mapping z m i=1 f i(z(t)) is maximal monotone. We then use a result in [Bastien-Schatzman ESAIM M2AN 22] to conclude. i=1

10 The existence of a positive definite P such that PB = C T is satisfied in many instances of sliding-mode control: observer-based sliding-mode control, Lyapunov-based discontinuous robust control. This is an input-output constraint on the system, constraining the relative degree of the triple (A,B,C). It is satisfied when (A,B,C) is positive real (dissipative).

11 Time-discretization of (3) The differential inclusion in (3) is therefore discretized as follows: { xk+1 x k f (t k,x k ) BSgn(Cx k+1 + D), a.e. on (,T) h x() = x (6) From [Bastien-Schatzman ESAIM M2AN 22] we have that: Proposition Under Proposition 1 conditions, there exists η such that for all h > one has For all t [,T], x(t) x N (t) η h (7) Moreover lim h + max t [,T] x(t) x N (t) 2 + t x(s) xn (s) 2 ds =.

12 However we have more: the discrete state reaches the sliding surface (when it exists) in a finite number of steps, and stabilizes on it in a smooth way. Let y(t) = Cx(t) + D. Lemma Let us assume that a sliding mode occurs for the index α {1... m}, that is y α (t) =,t > t. Let C and B be such that (4) holds and C α B α >. Then there exists h c > such that h < h c, there exists k IN such that y k +n = Cx k +n+1 + D = for all integers n 1. Such algorithms are similar to proximal algorithms which possess finite-time stabilization properties [Baji and Cabot, Set-Valued Analysis 26].

13 Remarks Contrarily to other methods that reduce (not suppress...) chattering, the discrete-time sliding surface is equal to the continuous-time sliding surface. At each step one has to solve a generalized equation with unknown x k+1 that takes the form of a mixed linear complementarity system (MLCP). Specific MLCP solvers are needed to implement the method.

14 NUMERICAL EXPERIMENTS Let us consider the following two examples: [ ] [ ] 1 ẋ = x sgn( [ c c 1 α 1 1 ] x). (8) (codimension one sliding surface) B = [ ] [ 1 2, C = 2 1 (codimension two sliding surface) ], D =, f (x(t),t) = (9)

15 x2(t) x2(t) x2(t) x2(t) Explicit Euler Explicit Euler x1(t) x1(t) x1(t) x1(t) (a) h =.3. (b) h =.1. (c) h = 1. Im- (d) h =.3. plicit Euler Implicit Euler x2(t) x2(t) x1(t) x1(t) (e) h =.1. (f) h =.5. Implicit Euler Implicit Euler Figure: Equivalent control based SMC, c 1 = 1, α = 1 and x = [, 2.21] T. State x 1 (t) versus x 2 (t).

16 1 x1(t) x2(t) 1 s1(t) s2(t) -.2 state x1(t) and x2(t) x2(t) s values time t (a) state x 1(t) and x 2(t) versus time x1(t) (b) phase portrait x 2(t) (c) sgn function s 1(t) and versus x 1(t) s 2(t) time t Figure: Multiple Sliding surface. h =.2, x() = [1., 1.] T The system reaches firstly the sliding surface 2x 2 + x 1 = without any chattering, The system then slides on the surface up to reaching the second sliding surface 2x 1 x 2 = and comes to rest at the origin.

17 The Filippov s example with switches accumulation B = [ ] [ 1, C = 1 ], D =, f (x(t),t) =. (1) The trajectories may slide on the codimension 2 surface given by Cx =. The origin is attained after an infinite number of switches in finite time.

18 time t x1(t) x2(t) x1(t) time t s1(t) s2(t) state x2(t) s values (a) state x 1 (t) and (b) phase portrait (c) sgn function s 1 (t) x 2 (t) versus time x 2 (t) versus x 1 (t) and s 2 (t) Figure: Multiple Sliding surface. Filippov Example. h =.2, x() = [1., 1.] T The results show that the system reaches the origin without any chattering.

19 The case of Zero Holding discretization (ZOH) for control implementation The ZOH discretization of linear time invariant systems ẋ(t) = Fx(t) + Gu(t) with an ECB-SMC controller, u(x) = (CG) 1 (CFx + αsgn(cx)),α > results in a discrete-time system of the form: x k+1 = Φx k Γs k for all t [kh,(k + 1)h) (11) where h > is the sampling period, and Φ = exp(fh) Γ = h h exp(fτ)dτg(cg) 1 CF (12) exp(fτ)g(cg) 1 dτ (13) with G IR n m, C IR m n, when an explicit Euler implementation of the control is performed.

20 Similarly to the explicit Euler method, the explicit ZOH discretization may yield spurious oscillations as shown in [Galias-Yu, IEEE CAS 28]. For an implicit Euler implementation, let us set { uk = (CG) 1 (CFx k + s k+1 ) s k+1 = Sgn(Cx k+1 ), (14) which corresponds to the implicit discrete time version of the ECB-SMC controller. We therefore get on each sampling period: x k+1 = Φx k Γs k+1 for all t [kh,(k + 1)h) (15)

21 At each time step, one has to solve x k+1 = Φx k Γs k+1 y k+1 = Cx k+1 + D s k+1 Sgn(y k+1 ). (16) Inserting the first line of (16) into the second line we obtain the following one step system (MLCP) { yk+1 = CΦx k + D CΓs k+1. (17) s k+1 Sgn(y k+1 )

22 The control scheme looks like: u k x k CFx k (CG) 1 Discrete-time Plant s k+1 MLCP solver Figure: Control system scheme with implicit Euler implementation.

23 A numerical example The LTI system with an ECB-SMC controller is defined by the following data, [ ] [ ] 1 F =, G =, C = [ c a 1 a ]. (18) Starting from the initial data, x = [.55,,55] T, [Galias-Yu, IEEE CAS 28] have shown that the Explicit ZOH discretization of the system with a 1 = 2, a 2 = 2, c 1 = 1 and h =.3 exhibits a period 2 orbit.

24 x2(t) -.2 x2(t) x1(t) (a) h =.3. Explicit ZOH x1(t) (b) h =.3. Implicit ZOH Figure: Equivalent control based SMC, a 1 = 2, a 2 = 2, c 1 = 1 and h =.3. x = [.55,, 55] T State x 1 (t) versus x 2 (t). the chattering on the switching surface is suppressed.

25 CONCLUSIONS The implicit Euler method allows one to nicely simulate the main features of sliding-mode systems: Finite-time stabilization on the switching surface (of codimension 1) Smooth stabilization on the switching surface It extends to the discrete-time implementation with ZOH discretization: looks like a promising solution for discrete-time sliding modes.