LMI-based robust control of uncertain discrete-time ... - Springer Link

J Control Theory Appl 2010 8 (4) 496–502 DOI 10.1007/s11768-010-8094-2

LMI-based robust control of uncertain discrete-time piecewise affine systems Zhiyuan LIU 1 , Yahui GAO 1 , Hong CHEN 2 (1.Department of Control Science and Engineering, Harbin Institute of Technology, Harbin Heilongjiang 150001, China; 2.Department of Control Science and Engineering, Jilin University, Changchun Jilin 130025, China)

Abstract: The main contribution of this paper is to present stability synthesis results for discrete-time piecewise affine (PWA) systems with polytopic time-varying uncertainties and for discrete-time PWA systems with norm-bounded uncertainties respectively. The basic idea of the proposed approaches is to construct piecewise-quadratic (PWQ) Lyapunov functions to guarantee the stability of the closed-loop systems. The partition information of the PWA systems is taken into account and each polytopic operating region is outer approximated by an ellipsoid, then sufficient conditions for the robust stabilization are derived and expressed as a set of linear matrix inequalities (LMIs). Two examples are given to illustrate the proposed theoretical results. Keywords: Discrete-time systems; Linear matrix inequality; Piecewise affine systems; Piecewise Lyapunov function; Robust control

1

Introduction

PWA systems have been receiving increasing attention from the control community in recent years because of their wide scopes of applications. In fact, many chaotic systems, including Chua’s circuit, the Lozi map and the so-called L system, can be written in the form of PWA models [1∼3]. In addition, many other classes of nonlinear systems can also be approximated by PWA systems [4, 5]. Furthermore, several hybrid systems, such as mixed logical dynamical (MLD) systems, linear hybrid automata (LHA), etc., are equivalent to PWA systems [6, 7]. During the last decade, many results on controller synthesis methods have been developed for PWA systems. In the continuous-time case, Rodrigues [8] discussed state and output feedback controller designs for PWA systems which may involve multiple equilibria, but the controller design results can only be cast as bilinear matrix inequalities (BMIs) that are non-convex and hard to solve [9]. In [10], each polytopic operating region was outer approximated by a union of ellipsoids, then based on a global quadratic Lyapunov function, sufficient conditions for stabilization were cast as LMIs [11] that are numerically feasible with commercially available software. Inspired by [10], PWA stability controller was designed in [12], and the controller design problem was formulated as a set of LMIs parameterized by a vector. Then, this formulation was extended to the H∞ controller design in [13]. In the discrete-time case, an approach was presented in [14] to stabilization based on a global-quadratic Lyapunov function. In [15,16], several stability and H∞ control schemes were given based on a less conservative PWQ Lyapunov function. On the other hand, many practical systems are always subject to various kinds of uncertainties and the PWA systems are no exception. Many results have also been obtained on robust controller design for uncertain continuous-time

PWA systems. Feng [17] developed two constructive controller design methods, and the synthesis problem can be formulated as LMIs provided that each subsystem is stable. Recently, the state feedback and output feedback controller synthesis approaches, which remove the restriction appeared in [17], were proposed in [18,19] respectively, and the design procedures are cast as BMIs. Unfortunately, to the best of our knowledge, there are few results on controller design for uncertain discrete-time PWA systems in the open literature. What is more, it seems not easy to extend the results in [17∼19] to the discrete-time case, because in this case the difference of Lyapunov function is hard to handle due to the use of the S-procedure. Motivated by this situation, this paper suggests two novel controller design methods for uncertain discrete-time PWA systems whose operating regions can be described by ellipsoids. The polytopic time-varying uncertainties and normbounded uncertainties, which are more popular than the one considered in [17, 18], are considered in this paper. The stability restriction of subsystems appeared in [17] is also removed here. In addition, the PWQ Lyapunov function technique is used, and the sufficient conditions for the robust stabilization are finally formulated in the form of LMIs. The rest of this paper is organized as follows. Section 2 provides the problem statements. Robust controller is designed for discrete-time PWA systems with polytopic timevarying uncertainties in Section 3, and for discrete-time PWA systems with norm-bounded uncertainties in Section 4. Then two examples are given in Section 5, which is followed by conclusion in Section 6. For convenience, the following notations are introduced. Rn denotes the n dimensional Euclidean space. The superscript T represents the transpose. ∗ is used as an ellipsis for terms that are induced by symmetry. I denotes an identity matrix of appropriate dimension.

Received 23 May 2008; revised 13 January 2009. This work was supported by the National Science Fund of China for Distinguished Young Scholars (No.60725311). c South China University of Technology and Academy of Mathematics and Systems Science, CAS and Springer-Verlag Berlin Heidelberg 2010

497

Z. LIU et al. / J Control Theory Appl 2010 8 (4) 496–502

2

Problem statement

method mentioned in [12].

Consider two types of uncertainties for robust control. The first one is “polytopic time-varying uncertainties” and the second one is the more popular “norm-bounded uncertainties”. The discrete-time PWA systems with these uncertainties can be written as (1) x(k + 1) = Ai (k)x(k) + Bi (k)u(k) + bi (k) for x(k) ∈ Si , i ∈ ℘, where Si = {x|Hi x + hi 0}i∈℘ ∈ Rn denotes a partition of the state space into a number of closed polyhedral subspaces, ℘ is the index set of these subspaces, x(k) ∈ Rn is the system state variable and u(k) ∈ Rm is the system input variable. All the matrices mentioned in this paper are appropriately dimensioned. For systems with polytopic time-varying uncertainties, the structure of the ith local model is assumed to be of the form [ Ai (k) Bi (k) bi (k) ] = ξili (k) 0,

Li li =1

Li li =1

ξili (k)[ Aili Bili bili ],

3 Controller design of PWA systems with polytopic time-varying uncertainties If the uncertainties of system (1) are defined by (2), the following parameter dependent piecewise linear (PDPWL) control law is considered li =1

while for systems with norm-bounded uncertainties, the structure of the ith local model is assumed to be of the form Ai (k) = Ai + Eai Δai Fai , (3) Bi (k) = Bi + Ebi Δbi Fbi , bi (k) = bi + Eoi Δoi Foi , where Δg represent unknown time-varying matrices with ¯ (Δg ) 1, g = ai, bi, oi. Δg 2 ∼ =σ ℘ is partitioned as ℘ = ℘0 ℘1 , where ℘0 is the index set of the subspaces that contain the origin, i.e. bi (k) = 0, and ℘1 is the index set of the subspaces otherwise. For future use, let Ω represent index pairs denoting the possible switch from one region to itself or another region, that is, Ω∼ = {i, j | x(k) ∈ Si , x(k + 1) ∈ Sj , i, j ∈ ℘} . (4) It is assumed that given any initial condition x(0) = x0 , the difference (1) has a unique solution. For the definition of solution to the PWA system (1), please refer to [4, 20] for details. It is also assumed that when the state of the system transits from the region Si to Sj at the time k, the dynamics of the system is governed by the dynamics of the local model of Si at that time. It is further assumed that matrix Ei and scalar ei exist such that Si ⊆ εi where εi = x(k) Ei x(k) + ei 1 . (5) There are many methods ( [11, 21], etc.) to compute this ellipsoidal outer approximation. The approximation is especially useful when Si is a slab, because in this case an ellipsoid can be found to cover Si exactly. In other words, 2cT i and if Si = x(k) | d1 cT i x(k) d2 , Ei = d2 − d1 d 2 + d1 ei = − can be taken with the result that Si ⊆ εi d2 − d1 and εi ⊆ Si [10, 12]. The goal of this paper is to stabilize system (1) to the origin. Remark 1 If anyone wants to stabilize the system to a given point that is not the origin, he can transform the coordinate of the given point to the origin according to the

ξili (k)Kili x(k).

(6)

By (1) and (6) defined as above, the closed-loop system admits the realization (7) x(k + 1) = Ai (k)x(k) + bi (k), where Ai (k) = Ai (k) + Bi (k)

Li li =1

ξili (k)Kili . Consider

a parameter dependent piecewise-quadratic (PDPWQ) Lyapunov function of the form V (x(k)) = x(k)T Pi (k)x(k)

(2)

ξili (k) = 1,

Li

u(k) =

with Pi (k) =

Li li =1

(8)

ξili (k)Pili and Pili = PilTi > 0.

Definition 1 The closed-loop system (7) is said to be poly-PWQ stable if there exists a PDPWQ Lyapunov function (8) whose difference is negative definite decrescent. The difference of (8) along the solution of (7) is given by V (x(k + 1)) − V (x(k)) = x(k+1)T Pj (k+1)x(k+1)−x(k)T Pi (k)x(k), (9) Lj

where Pj (k+1) =

lj =1

ξjlj (k + 1)Pjlj . Therefore, the Lya-

punov difference is negative if x(k) 1

T

Θ1ij ∗ Θ2ij Θ3ij

x(k) 0, 1

(10)

where Θ1ij = Ai (k)T Pj (k + 1)Ai (k) − Pi (k), Θ2ij = bi (k)T Pj (k + 1)Ai (k), and Θ3ij = bi (k)T Pj (k + 1)bi (k). Then the partition information (5) is taken into account. According to applying the S-procedure [11], (10) is satisfied if ∃λi > 0 such that x(k) 1 −λi

T

x(k) 1

Θ1ij ∗ Θ2ij Θ3ij T

x(k) 1

EiT Ei ∗ fiT Ei fiT fi − 1

x(k) 0. (11) 1

One sufficient condition of (11) is Θ1ij − λi EiT Ei ∗ Θ2ij − λi fiT Ei Θ3ij −λi fiT fi −1 which is equivalent to

< 0,

(12)

Pi (k) + λi EiT Ei ∗ λi fiT fi − 1 λi fiT Ei T

A (k)T P (k+1) Ai (k)T Pj (k+1) − i T j Pj (k+1)−1 bi (k) Pj (k+1) bi (k)T Pj (k+1) > 0. (13) By applying the Schur complement [11] and then per-

498


forming a congruence transformation via diag{I,

0 ∗ }, I 0

(13) can be written as Ψij (k) Pi (k) + λi EiT Ei ∗ ∗ = Pj (k + 1)Ai (k) Pj (k + 1) ∗ bi (k)T Pj (k + 1) λi fiT fi − 1 λi fiT Ei > 0. (14) Notice that Lj

Ψij (k) =

lj =1

where Ψij = with A¯i = Aili satisfied if

ξjlj (k + 1)

Li li =1

ξili (k)

Li li =1

ξili (k)Ψij ,

Pili + λi EiT Ei ∗ ∗ Pjlj ∗ Pjlj A¯i T bT P λ f λi fiT Ei i fi − 1 ili jlj i + Bili Kili . So, according to (2), (14) is Ψij > 0.

(15)

With the leading of Daafouz [22], the following result can be obtained. and Lemma 1 With the substitution of Qili = Pil−1 i −1 Qjlj = Pjlj , condition (15) is satisfied if and only if there exists a matrix Gili such that ϕili jlj Gili +GT ili −Qili

∗ ∗ −1 T = Qjlj +λi bili bili ∗ −1 T λ−1 f b λ f fiT −I i i i i ili > 0. (16) −1 T Proof (16) ⇒ (15): Due to Gili Qili Gili Gili + GT ili − Qili [22], the satisfaction of (16) guarantees A¯i Gili Ei Gili

−1 GT ∗ ∗ ili Qili Gili −1 T ¯ Ai Gili Qjlj + λi bili bili ∗ −1 T λ−1 f b λ f fiT − I Ei Gili i i i i ili > 0. (17)

Multiply (17) from the left by diag(G−T ili , I, I) and from −1 the right by diag(Gili , I, I), then the following LMI is achieved. Q−1 ∗ ∗ ili T > 0. (18) A¯i Qjlj + λ−1 b b ∗ il i ili i −1 T T λ−1 f b λ f f − I Ei i ili i i i i Applying the Schur complement to (18) leads to Q−1 ∗ ili T A¯i Qjlj + λ−1 i bili bili +

EiT −1 λi bili fiT

λi I − fi fiT

−1

(19) −1

The matrix inversion lemma (A + BCD) = A A−1 B(C −1 + DA−1 B)−1 DA−1 [23] states that −1

−1

−

−1

I − fiT fi = I + fiT I − fi fiT fi , −1 T T T −1 T fi I − fi fi = I − fi fi fi .

T Q−1 ili + λi Ei Ei ∗ A¯i Qjl

j

−1 λ ETf + i i i λ−1 1 − fiT fi λi fiT Ei bT i ili bili > 0. (21) Then (15) can be achieved by applying the Schur complement and performing a congruence transformation via diag{I, Pjlj , I}. (15) ⇒ (16): From the proof of (16) ⇒ (15), it is seen that (15) implies (18). We apply the Schur complement to (18) to get

Tili lj =

T Qjlj + λ−1 ∗ i bili bili −1 T T λ−1 f λi fi bili i fi − I i A¯i T . − Qili A¯T i Ei Ei

(20)

(22)

Let Gili = Qili + gili Ii with gili a positive scalar. There exists a sufficiently small gili such that A¯i T T −1 gil−2 (Qili + 2gili Ii ) > A¯T , (23) i Ei ili lj i Ei which is equivalent, by applying the Schur complement, to Qili + 2gili Ii −A¯i gil i

−Ei gili

∗ Tili lj

> 0.

(24)

Notice that (24) can be written as T I 0 0 I −A¯T i −Ei > 0, (25) 0 −A¯i I 0 ϕili jlj 0 I −Ei 0 I 0 0 I which guarantees the satisfaction of (16). This completes the proof. Remark 2 When fi fiT − I < 0, (16) is no longer feasible. fi fiT − I < 0 means that the origin lies inside the ellipsoid εi [10]. In this case, the Lyapunov difference (9) is negative if Ai (k)T Pj (k + 1)Ai (k) − Pi (k) < 0, which is equivalent to Gili + GT ili − Qili ∗ Qjlj A¯i Gili

> 0.

(26)

With the substitution of αi = λ−1 i , the above discussion can be summarized as follows. Theorem 1 Discrete-time PWA system (1) with polytopic time-varying uncertainties (2) is poly-PWQ stabilizable by a PDPWL control law (6) if there exist symmetric positive definite matrices Qili , Qjlj , matrices Gili , Rili and positive scalars αi such that Gili + GT ∗ ili − Qili Aili Gili + Bili Rili Qjlj

T Ei λ−1 i fi bili

> 0.

Combining (19) with (20) results in

> 0, i ∈ ℘0 ,

(27)

Gili + GT ∗ ∗ ili − Qili ∗ Aili Gili + Bili Rili Qjlj + αi bili bT ili T αi fi bili αi fi fiT − I Ei Gili > 0, i ∈ ℘1 (28) for all i, j ∈ Ω, li = 1, · · · , Li , and lj = 1, · · · , Lj . The

499


controller gains of (6) are computed by Kili = Rili G−1 ili . Proof With the change of variable Rili = Kili Gili , (16) and (26) follow directly from (28) and (27). Moreover, (16) and (26) imply that the Lyapunov difference (9) is negative, so the closed-loop control system is poly-PWQ stable. Remark 3 The precondition of using PDPWL control law (6) is that the parameter vector ξili (k) for li = 1, · · · , Li is real-time measurable. If the precondition is unsatisfied, we can only use piecewise linear (PWL) control law u(k) = Ki x(k) with Ki = Ri G−1 i . Ri , Gi can be solved from (27) and (28) by imposing Gili = Gi , Rili = Ri for all li = 1, · · · , Li . PWL control law is a little more conservative than PDPWL control law.

4

Controller design of PWA systems with norm-bounded uncertainties

If the uncertainties of system (1) are defined by (3), the following PWL control law is considered (29) u(k) = Ki x(k).

T T ]T , Fi1 = [ Fai 0 0 ]T , Fi2 = [ 0 FbiT 0 ]T , Fi3 = [ 0 0 Foi

Δi = diag[ Δai Δbi Δoi ]. Then the closed-loop system admits the realization x(k + 1) = A¯i x(k) + bi + Epi p(k), q(k) = F¯i x(k) + Fi3 ,

(30)

p(k) = Δi q(k), where A¯i = Ai + Bi Ki and F¯i = Fi1 + Fi2 Ki . Consider a PWQ Lyapunov function V (x(k)) = x(k)T Pi x(k) with Pi = PiT > 0. The closed-loop system (30) is PWQ stable if V (x(k + 1)) − V (x(k)) = x(k + 1)T Pj x(k + 1) − x(k)T Pi x(k) < 0. (31) Substituting the state space (30) in (31) leads to T ¯ A¯T ∗ ∗ x(k) x(k) i Pj Ai − Pi T T Pj Epi ∗ Epi Pj A¯i Epi p(k) p(k) T T T ¯ 1 1 bi Pj Epi bi Pj bi bi Pj Ai 0 (32) with T ¯ ps (k)T ps (k) F¯i,s x(k) + Fi3,s Fi,s x(k) + Fi3,s , (33) where s = a, b, o. With the use of the S-procedure to incorporate (33) to (32), condition (31) can be guaranteed by

Ξij =

T

λia Inai ∗ ∗ ∗ 0 λib Inbi 0 0 λio Inoi

Λi =

> 0,

¯ ¯T ¯ A¯T i Pj Ai − Pi + Fi Λi Fi T Epi Pj A¯i bT Pj A¯i + F T Λi F¯i i

Ξij − λi

x(k) p(k) 1

T

EiT Ei ∗ ∗ 0 0 ∗ T T fi Ei 0 fi fi − 1

∗ ∗ T Pj Epi − Λi ∗ Epi T bT bT i Pj Epi i Pj bi + Fi3 Λi Fi3 0

x(k) p(k) 1 (34)

x(k) p(k) 1

0.

(36)

One sufficient condition of (36) is T ¯ ¯T ¯ A¯T ∗ i Pj Ai − Pi + Fi Λi Fi − λi Ei Ei T T Epi Pj Epi − Λi Epi Pj A¯i T T ¯ ¯ bT bT i Pj Ai + Fi3 Λi Fi − λi fi Ei i Pj Epi

< 0.

(37)

T T bT i Pj bi + Fi3 Λi Fi3 − λi fi fi − 1

and Pj = Q−1 By substituting Pi = Q−1 i j , using the Schur complement and after some straightforward manipulations, (37) can be written as T Q−1 ∗ ∗ i + λ i Ei Ei −1 T ¯ Ai Qj − Epi Λi Epi ∗ 0 Λ−1 F¯i i

λi EiT fi + bi Fi3

−1

λ−1 1−fiT fi i

λi EiT fi bi Fi3

T

> 0. (38)

Combining (38) with (20) results in Q−1 ∗ ∗ i −1 T T A¯i Qj −Epi Λ−1 E +λ b b ∗ i pi i i i −1 −1 T T F¯i λ−1 F b Λ +λ i3 i i i i Fi3 Fi3 EiT −1 + λi bi fiT T λ−1 i Fi3 fi

λi

−1 I −fi fiT

EiT −1 λi bi fiT T λ−1 i Fi3 fi

T

> 0. (39)

Applying the Schur complement to (39) leads to Q−1 ∗ i −1 T T A¯i Qj − Epi Λ−1 i Epi + λi bi bi −1 T F¯i λ Fi3 b i

Ei

i

T λ−1 i fi bi

∗ ∗ ∗ ∗ −1 T + λ F F ∗ Λ−1 i3 i3 i i −1 T T f F λ f λ−1 i i3 i fi − I i i

i3

(35)

where λia , λib , λio > 0. Then the partition information (5) is taken into account. According to applying the Sprocedure, (34) is satisfied if ∃λi > 0 such that

∗ ∗

Define Epi = [ Eai Ebi Eoi ],

x(k) p(k) 1

with

> 0. (40)

Remark 4 Similar to the derivation of (16) ⇔ (18) in the proof of Lemma 1, it can be also derived out that condition (40) is satisfied if and only if there exists a matrix Gi

500


such that Gi + GT ∗ i − Qi −1 T T A¯i Gi Qj − Epi Λ−1 i Epi + λi bi bi −1 T ¯ Fi Gi λi Fi3 bi T Ei G i λ−1 i fi bi ∗ ∗ ∗ ∗ > 0. (41) −1 −1 T ∗ Λi + λi Fi3 Fi3 T λ−1 fi fiT − I λ−1 i fi Fi3 i Remark 5 In fact, the motivation of introducing the extra slack matrix variable Gi is to reduce the conservativeness of the stability condition [22]. Remark 6 According to Remark 2, it is seen that when the origin lies inside the ellipsoid εi , (41) is no longer feasible. In this case, condition (31) can be guaranteed by Gi + GT ∗ ∗ i − Qi T > 0 (42) Qj − E0pi Λ−1 E ∗ A¯i Gi 0pi i −1 ¯ F0i Gi 0 Λ0i with λia Inai ∗ Λ0i = > 0, (43) 0 λib Inbi where E0pi = [ Eai Ebi ], F¯0i = F0i1 + F0i2 Ki , F0i1 =

T [ Fai 0 ]T , and F0i2 = [ 0 FbiT ]T .

−1 With the substitution of Θi = Λ−1 i , Θ0i = Λ0i , δia = −1 −1 −1 δib = λib , δio = λio , αi = λi , the above discussion can be summarized as follows. Theorem 2 Discrete-time PWA system (1) with normbounded uncertainties (3) is PWQ stabilizable by a PWL control law (29) if there exist symmetric positive definite matrices Qi , Qj , matrices Gi , Ri and positive scalars δia , δib , δio , αi such that Gi + GT ∗ ∗ i − Qi T Ai Gi + Bi Ri Qj −E0pi Θ0i E0pi ∗ > 0, i ∈ ℘0 , 0 Θ0i F0i1 Gi +F0i2 Ri (44) T Gi + Gi − Qi ∗ T + αi bi bT Ai Gi + Bi Ri Qj − Epi Θi Epi i T Fi1 Gi + Fi2 Ri αi Fi3 bi Ei G i αi fi bT i

λ−1 ia ,

∗ ∗ ∗ ∗ T ∗ Θi + αi Fi3 Fi3 T αi fi fiT − I αi fi Fi3

and (42) follow directly from (45) and (44). Moreover, (41) and (42) imply (31), so the closed-loop control system is PWQ stable.

5 Two examples Example 1 The objective of this example is to design a controller that forces a cart on the x-y plane to follow the straight line y = 0 with a constant velocity u0 = 10 m/s. It is assumed that a controller has already been designed to maintain a constant forward velocity. The cart’s path is then controlled by the torque T about the z-axis according to the following dynamics: ψ˙ r˙ y˙

δia Inai ∗ , 0 δib Inbi δia Inai ∗ ∗ Θi = 0 δib Inbi ∗ 0 0 δio Inoi for all i, j ∈ Ω. The controller gains of (29) are computed by Ki = Ri G−1 i . Proof With the change of variable Ri = Ki Gi , (41)

ψ r y

+

0 0 u0 sin ψ

+

0 1 I 0

T,

(46) where ψ is the heading angle with time derivative r, I = 1 (kg·m2 ) is the moment of inertia of the cart with respect to the center of mass, k = 0.01 (kg·m2 /s) is the damping coefficient, and T is the control torque. The state of the system is (x1 , x2 , x3 ) = (ψ, r, y). Assume the trajectories can start from any possible initial angle in the range 3π 3π ) and any initial distance from the line [12]. ψ0 ∈ (− , 5 5 The function sin ψ is approximated by a PWA function 3π π π π with the following regions: x1 ∈ (− , − ), (− , − ), 5 5 5 15 π π π 3π π π ) which are denoted by S1 , (− , ), ( , ), ( , 15 15 15 5 5 5 S2 , S3 , S4 and S5 respectively. The partition information for the regions is described in the form of (5) with 5 15 E1 = E5 = [ 0 0 ], E2 = E3 = E4 = [ 0 0 ], π π f1 = f2 = −f4 = −f5 = 2 and f3 = 0. The approximation method used here is reported in [5] and the approximation effect is shown in Fig.1. It is found that the PWA system obtained is subject to uncertainties due to approximation error, especially in regions S1 and S5 . Describing the uncertainties as polytopic time-varying type and discretizing the resulting uncertain PWA system with scheme x(k + 1) − x(k) (Ts = 0.001 s), we get x˙ = Ts 2

Ai (k) Bi (k) bi (k) = ξili (k) 0,

> 0, i ∈ ℘1 (45)

with

=

0 1 0 k 0− 0 I 0 0 0

2 li =1

li =1

ξili (k) Ai B bili ,

ξili (k) = 1, i = 1, · · · , 5

with A1 = A5 =

1 0.001 0 0 0.99999 0 , 0.002891 0 1

A2 = A4 =

1 0.001 0 0 0.99999 0 , 0.009069 0 1

Θ0i =

A3 =

1 0.001 0 0 0.99999 0 , 0.009926 0 1

501


B = 0 0.001 0

T

, T

b11 = −b51 = 0 0 −0.004061 b12 = −b52 = 0 0 −0.0059 b21 = −b41 = 0 0 −0.00018 b22 = −b42 = 0 0 −0.0003 b31 = b32 = 0 0 0

T

T

,

T T

,

,

,

.

For this system, based on the result of Theorem 1, a PDPWL controller is designed. A number of simulations have been carried out and it is found that the cart can be forced to follow the desired straight line from ψ0 ∈ 3π 3π ). A typical result for this controller is shown in (− , 5 5 Fig.2.

and u is the force applied to the cart. In this simulation, the pendulum parameters are chosen as m = 2 kg, M = 8 kg, and l = 0.5 m [18]. π π Consider the working range of state x1 ∈ (− , ) and 2 2 π π π π π π the following regions: x1 ∈ (− , − ), (− , ), ( , ) 2 4 4 4 4 2 which are denoted by S3 , S1 and S2 respectively. The partition information for the regions is described in the form of 4 8 (5) with E1 = [ 0 ], E2 = E3 = [ 0 ], f1 = 0 and π π f2 = −f3 = −3. Linearise the plant around the origin and x = (±88◦ , 0) and consider the differences between the linearised local model and the original nonlinear model as norm-bounded uncertainties. Discretizing the resulting uncertain PWA system with the same scheme as the one used in Example 1, we get x(k + 1) = (Ai + Eai Δai Fai ) x(k) + (Bi + Ebi Δbi Fbi ) u(k) +bi + Eoi Δoi Foi , i = 1, 2, 3 with 1 0.001 , A1 = 0.0172941 1 1 0.001 , 0.0002395 1

A2 = A3 =

B1 = 0 −1.765

T

× 10−4 ,

B2 = B3 = 0 −5.2 Fig. 1 PWA approximation of sin ψ.

b1 = 0 0

T

T

× 10−6 ,

, b2 = −b3 = 0 −3.679

Eai = Ebi = Eoi =

1 0 , 0 1

Fa1 = Fa2 = Fa3 =

0 7 × 10−4 , 5 0

Fb1 = Fb2 = Fb3 = 0 1 Fo1 = 0 0

Fig. 2 The x − y trajectory for ψ0 = − y0 = 350 m.

3π , r0 = 0 rad/s, and 5

T

T

T

× 10−6 ,

, Fo2 = Fo3 = 0 2

T

× 10−4 .

For this system, based on the result of Theorem 2, a PWL controller is designed. A number of simulations have been carried out and it is found that the pendulum can be balanced from x1 ∈ (−89◦ , 89◦ ). A typical result for this controller is shown in Fig.3.

Example 2 The objective of this example is to balance an inverted pendulum on a cart. The equations of motion of the pendulum are x˙ 1 = x2 , x˙ 2 =

g sin x1 −

amlx22 sin(2x1 ) − a(cos x1 )u (47) 2 , 4l 2 3 − aml cos x1

where x1 denotes the angle of the pendulum from the vertical, and x2 is the angular velocity. g = 9.8 m/s2 is the grav1 ity constant, m is the mass of the pendulum, a = , m+M M is the mass of the cart, 2l is the length of the pendulum,

× 10−4 ,

Fig. 3 Time response for x(0) = (−89◦ , 0).

502

6


Conclusions

A PDPWL controller is firstly designed for the discretetime PWA systems with polytopic time-varying uncertainties, and the resulting closed-loop control systems are polyPWQ stable. Then a PWL controller is designed for the discrete-time PWA systems with norm-bounded uncertainties, and the resulting closed-loop control systems are PWQ stable. To approximate each operating region by an ellipsoid, both robust control schemes are cast as solving a set of LMIs. At last, two examples are presented to demonstrate the controller performance. References [1] G. Feng, T. Zhang. Output regulation of discrete-time piecewiselinear systems with application to controlling chaos[J]. IEEE Transactions on Circuits and Systems-Part II: Express Briefs, 2006, 53(4): 249 – 253. [2] R. Lozi. Un attracteur etrange du type attracteur de Henon[J]. Journal of Physics (Paris), 1978, 39(C5): 9 – 10. [3] S. J. Linz, J. C. Sprott. Elementary chaotic flow[J]. Physics Letters A, 1999, 259(3/4): 240 – 245. [4] M. Johansson. Piecewise Linear Control Systems[M]. New York: Springer-Verlag, 2003. [5] L. Rodrigues, J. P. How. Automated control design for a piecewiseaffine approximation of a class of nonlinear systems[C]//Proceeding of American Control Conference. New York: IEEE, 2001: 3189 – 3194. [6] W. P. M. H. Heemels, B. D. Schutter, A. Bemporad. Equivalence of hybrid dynamical models[J]. Automatica, 2001, 37(7): 1085 – 1091. [7] S. D. Cairano, A. Bemporad. An equivalence result between linear hybrid automata and piecewise affine systems[C]//Proceeding of the 45th Conference on Decision and Control. Piscataway: IEEE, 2006: 2631 – 2636. [8] L. Rodrigues, J. P. How. Observer-based control of piecewise-affine systems[J]. International Journal of Control, 2003, 76(5): 459 – 477. [9] K. C. Goh, L. Turan, M. G. Safonov, et al. Biaffine matrix inequality properties and computational methods[C]//Proceeding of American Control Conference. New York: IEEE, 1994: 850 – 855. [10] A. Hassibi, S. Boyd. Quadratic stabilization and control of piecewiselinear systems[C]//Proceeding of American Control Conference. New York: IEEE, 1998: 3659 – 3664. [11] S. Boyd, L. Ghaoui, E. Feron, et al. Linear Matrix Inequalities in System and Control Theory[M]. Philadephia: SIAM, 1994. [12] L. Rodrigues, S. Boyd. Piecewise-affine state feedback for piecewiseaffine slab systems using convex optimization[J]. Systems & Control Letters, 2005, 54(9): 835 – 853. [13] L. Rodrigues, E. K. Boukas. Piecewise-linear H∞ controller synthesis with applications to inventory control of switched production systems[J]. Automatica, 2006, 42(8): 1245 – 1254. [14] O. Slupphaug, B. A. Foss. Constrained quadratic stabilization of discrete-time uncertain nonlinear multi-model systems using piecewise affine state feedback[J]. International Journal of Control, 1999, 72(7): 686 – 701. [15] J. Xu, L. Xie, G. Feng. Feedback control design for discrete-time piecewise affine systems[C]//International Conference on Control and Automation. Piscataway: IEEE, 2005: 425 – 430.

[16] J. Xu, L. Xie. H∞ state feedback control of discrete-time piecewise affine systems[C]//Proceeding of the 16th IFAC World Congress. Prague, Czech, 2005: Mo-A02-TP/5. [17] G. Feng. Controller design and analysis of uncertain piecewiselinear systems[J]. IEEE Transactions on Circuits and Systems-Part I: Fundamental Theory and Applications, 2002, 49(2): 224 – 232. [18] Y. Zhu, D. Li, G. Feng. H∞ controller synthesis of uncertain piecewise continuous-time linear systems[J]. IEE ProceedingsControl Theory and Applications, 2005, 152(5): 513 – 519. [19] J. Zhang, W. Tang. Output feedback H∞ control for uncertain piecewise linear systems[J]. Journal of Dynamical and Control Systems, 2008, 14(1): 121 – 144. [20] M. Johansson, A. Rantzer. Computation of piecewise quadratic Lyapunov functions for hybrid systems[J]. IEEE Transactions on Automatic Control, 1998, 43(4): 555 – 559. [21] L. Vandenberghe, S. Boyd, S. P. Wu. Determinant maximization with linear matrix inequality constraints[J]. SIAM Journal on Matrix Analysis and Applications, 1998, 19(2): 499 – 533. [22] J. Daafouz, J. Bernusou. Parameter dependent Lyapunov functions for discrete time systems with time varying parametric uncertainties[J]. Systems & Control Letters, 2001, 43(5): 355 – 359. [23] T. Kailath. Linear Systems[M]. Englewood Cliffs: Prentice-Hall, 1989. Zhiyuan LIU was born in 1957. He received his Ph.D. degree from Harbin Institute of Technology in 1992. Now he is a professor at the Harbin Institute of Technology. His research interests cover automotive elective control, robotics, robust control, and model predictive control. E-mail: liuzy [email protected].

Yahui GAO was born in 1981. He is a Ph.D. candidate in Control Theory and Control Engineering at Harbin Institute of Technology. He received his B.S. and M.S. degrees from Harbin Institute of Technology in 2003 and 2005, respectively. His research interests cover hybrid system control, robust control, and model predictive control. E-mail: [email protected]. Hong CHEN was born in 1963. She received her B.S. and M.S. degrees in Process Control from Zhejiang University in 1983 and 1986, respectively, and Ph.D. degree from University of Stuttgart, Germany, in 1997. In 1986, she joined Jilin University of Technology. From 1993 to 1997, she was a “Wissenschaftlicher Mitarbeiter” at Institut fuer Systemdynamik und Regelungstechnik, University of Stuttgart. Since 1999, she has been a professor at Jilin University. Her research interests cover model predictive control, optimal and robust control, and applications in process engineering and mechatronic systems. E-mail: [email protected].