A Generalized Approach for Analysis and Control ... - Semantic Scholar
A Generalized Approach for Analysis and Control of Discrete-Time Piecewise Affine and Hybrid Systems Francesco Alessandro Cuzzola? and Manfred Morari Institut f¨ ur Automatik, ETH - Swiss Federal Institute of Technology, ETHZ - ETL, CH 8092 Zurich, Switzerland {cuzzola, morari}@aut.ee.ethz.ch
Abstract. In this paper we investigate some analysis and control problems for discrete-time hybrid systems in the piece-wise affine form. By using arguments from the dissipativity theory for nonlinear systems, we show that H∞ analysis and synthesis problems can be formulated and solved via Linear Matrix Inequalities by taking into account the switching structure of the considered system. In this paper we address the generalized problem of controlling hybrid systems whose switching structure does not depend only on the state but also on the control input.
1
Introduction
Piece-Wise Affine (PWA) systems have been receiving increasing attention by the control community because they provide a useful modeling framework for hybrid systems. In fact, discrete-time PWA systems are equivalent to interconnections of linear systems and finite automata [17], to complementarity systems [9] and also hybrid systems in the Mixed Logic Dynamical (MLD) form [1]. In particular, the MLD form is capable to model a large class of hybrid systems including linear hybrid dynamical systems, hybrid automata, some classes of discrete-event systems, and systems with qualitative inputs/outputs [1,3]. The algorithm to obtain the discrete-time PWA representation of an MLD system and vice-versa is reported in [3]. In order to stress the importance of PWA systems it is worth recalling that in [2] the explicit form of Model Predictive Control (MPC) for linear constrained systems was derived and, besides providing an algorithm for its computation, it was shown that the closed-loop system has a PWA structure. Also in this case the closed-loop system turns out to be a PWA model. An important feature of a PWA model is that the state-update map can be discontinuous along the boundary of the regions. For instance, when considering PWA systems stemming from hybrid systems in the MLD form, discontinuities can arise from the representation of logic conditions. The control synthesis problem for MLD systems and consequently PWA systems ?
This research has been supported by the Swiss National Science Foundation.
M.D. Di Benedetto, A. Sangiovanni-Vincentelli (Eds.): HSCC 2001, LNCS 2034, pp. 189–203, 2001. c Springer-Verlag Berlin Heidelberg 2001
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is computationally difficult: in [1] a Mixed Integer Quadratic Programming (MIQP) approach is proposed in order to solve the control problem of MLD systems by means of MPC techniques. Needless to say, the computational complexity of this approach may increase exponentially with the prediction horizon considered. The use of Linear Matrix Inequalities (LMI) techniques, for which computationally advantageous and numerically reliable algorithms as well as toolboxes are available (see [8]) would seem to be a promising alternative. Concerning the stability analysis of PWA systems, the authors presented various algorithms with different degrees of conservativeness in [15]. Similarly to [12, 13], where a particular class of continuous-time PWA systems was considered, such procedures exploit Piece-Wise Quadratic (PWQ) Lyapunov functions that can be computed as the solution of a set of LMIs. For the sake of completeness, the main stability test of [15] is reported in Section 2 in a suitable form. In this work, we consider both analysis and synthesis problems for the general class of PWA models whose switching sequence depends on both state and input trajectories. As pointed out in [3] the dependence of the switching sequence on the input can be met by translating an MLD system into a PWA form. Moreover, the dependence of the switching sequence on the input signal is common in real systems: for example, it could be caused by saturation effects or limitations on the control signal. It is worthwhile emphasizing that this type of PWA models is more general than that considered in [12,13] and [15]: indeed, in these works the switching structure depended on the state only. Furthermore, we generalize the results of [15] by considering analysis and synthesis problems with performance for PWA systems. We focus on the H∞ norm showing that the H∞ -analysis and the H∞ -synthesis of a piecewise linear state-feedback can be addressed by resorting to LMI-based algorithms. The rationale of our derivation hinges on the use of passivity theory for nonlinear systems [14]. We point out that a significant application of the H∞ analysis test is the possibility of checking a posteriori the performance of MPC for both linear and MLD systems. As mentioned before this can be done by exploiting the explicit PWA form of the closed-loop system. The results are presented in Sections 3 and 4. An illustrative example is provided in Section 5.
Notation: The symbol ∗ will be used in some matrix expressions to induce a symmetric structure. For example, if L and R are symmetric matrices, then L+M +∗ ∗ L + M + MT NT . (1) := N R N R Moreover, we define kxk kl2 (0,N ) :=
N X
k=0
1
{xTk xk } 2 .
(2)
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Stability and State-Feedback Stabilization of PWA Systems
A linear discrete-time piecewise affine system is defined by the state-space equation xk (3) ∈ χi xk+1 = Ai xk + Bi uk + ai , for uk where xk ∈ IRn is the state and uk ∈ IRm is the control input. The set X ⊆ IRn+m T is either IRn+m or a polyhedron containing the of every possible vector xTk uTk origin, {χi }si=1 is a polyhedral partition 1 of X and ai ∈ IRn , are constant vectors. We refer to each χi as a cell. Moreover, in order to simplify the exposition, we assume that our cells are polyhedra defined by matrices Fix , Fiu , fix and fiu as follows o n x x u u T T T x u (4) χi := such that Fi x ≥ fi and Fi u ≥ fi . The results presented in this paper can be extended to systems whose cells χi have a more complicate structure. Moreover, it is worth introducing the following notation: χ ¯i := {x such that Fix x ≥ fix }
(5)
and n
T
¯j , x u Sj := i such that ∃x, u with x ∈ χ
T
T
o
∈ χi .
(6)
In a nutshell, Sj is the set of all indices i such that χi is a cell containing a T vector xT uT for which the condition x ∈ χ ¯j is satisfied. We denote with I = {1, . . . , s} the set of indices of the cells χi whereas the symbol J = {1, . . . , t} will be used to denote the set of indices of the cells χ ¯j . It is important to observe that: t [
Sj = I.
(7)
j=1
Furthermore, if cells χi have the structure pointed out in eq. (4) then the sets Sj are disjoint whereas if cells χi have a more complicate structure (for instance when mixed state-input constraints are used to define each cell χi ) then the sets Sj could be overlapping. In the latter case the results we are going to present could become more conservative. When we focus on the stability of the origin, we consider autonomous 1
Each Ss set χi is a (not necessarily closed) convex polyhedron s.t. χi i=1 χi = X.
T
χj = ∅, ∀i 6= j,
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PWA systems and we assume that x = 0 is an equilibrium point. To begin with it is necessary to observe that an autonomous system can be obtained from system (3) by applying a suitable control law. In the following we consider a piecewise linear state feedback with the structure xk (8) ∈ χi . uk = Ki xk , for uk By applying the controller (8) to the system (3) we achieve the following closedloop dynamic system xk ∈ χi (9) xk+1 = Ai xk + ai , for uk where Ai = Ai + Bi Ki and uk = Ki xk . We note that the the evolution of closed-loop system (9) depends on the “hidden” variable uk since it influences the index i of the current cell χi . As customary for constrained systems, we assume that the state trajectories T T T T x k uk ∈ X, ∀k ∈ IN. generated by the control law (8) satisfy xTk uTk In [15] the stability of the origin of PWA system was characterized by using Piece-Wise Quadratic (PWQ) Lyapunov functions. In the following theorem we report the main result of [15] valid for the case ai = 0, ∀i ∈ I and adapted to the closed loop system (9). Theorem 1. Consider the system (9). If there exist matrices Pi = PiT > 0, T ∈ χi ∀i ∈ I such that the positive-definite function V (x, u) = xT Pi x, ∀ xTk uTk satisfies V (xk+1 , uk+1 ) − V (xk , uk ) < 0, then the origin of the PWA systems (9) is exponentially stable and limk→+∞ kxk k = 0 for all system trajectories fulfilling T T T x k uk ∈ X, ∀k ∈ IN. t u The Lyapunov function appearing in Theorem 1 can be computed by solving the LMIs ATj Pi Aj − Pj < 0, Pi = PiT > 0,
∀(i, j) ∈ S
(10)
∀i ∈ I
(11)
where
xk xk+1 , ∈X S := (i, j) : i, j ∈ I and ∃k ∈ N0 , ∃ uk uk+1 xk+1 xk ∈ χj and ∈ χi . such that uk uk+1
(12)
In other words, the set S contains all the ordered pairs of indices denoting the possible switches from cell j to cell i and it can be computed via reachability analysis for MLD systems [4]. Then, the inequalities (10) take into account all the
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admissible switches between different regions and guarantee that the Lyapunov function is decreasing along all possible state trajectories. When there exist matrices Pi such that the LMIs (10) and (11) are satisfied, the PWA system and the corresponding controller (eq. (8)) are termed PWQ-stable and PWQstabilizing respectively. We refer the interested reader to [15] for further details. Remark 1. Conservativeness. The conservativeness of the LMIs conditions for stability analysis can be reduced by exploiting the so-called S-Procedure [20] in order to avoid imposing xT Pi x > 0 T for xT uT ∈ χj , j 6= i [15]. This modification was proposed in [12] for continuous-time PWA systems and can be easily generalized to the discretetime case. We point out that similar modifications can be applied to all the analysis LMIs we derive in the following. It is important to highlight that with respect to the continuous-time case (see [12]) in the discrete-time case there is no need to guarantee the continuity of the Lyapunov function over the whole state-space. This fact can determine a reduced degree of conservativeness of the results that we are going to present with respect to those presented in [12]. Finally, following the lead given in [11], the authors proposed in [7] discrete-time performance analysis results with a notably reduced degree of conservativeness. t u Remark 2. Extension of Theorem 1. Theorem 1 can be extended to the case ai 6= 0 as done in [12,13] by introducing the extended state x ¯k = [ xTk 1 ]T and rewriting the system (3) as follows: ¯i uk for ¯k + B x ¯k+1 = A¯i x
xk uk
∈ χi
(13)
where Ai ai ¯ Bi ¯ Bi = . Ai = 0 1 0
(14) t u
When designing the controller i.e. when the controller gains Ki appearing in the inequalities (10) are unknown, the set of all possible switches is generally not known in advance, and it could be necessary to consider all the pairs of indices in Sall := I × I instead of S. Furthermore, we note that the design of a controller of type (8) could be a very hard task because, at each time instant, the vector uk has to be
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calculated by means of a control gain K¯i whose index ¯i is found on the basis of the admissibility condition xk (15) ∈ χ¯i . uk This implies that in general it is not possible to calculate uk since the index ¯i for which the condition (15) is satisfied, is difficult to know in advance. Therefore, we turn our problem into one of designing a controller with the following structure u k = Kj x k ,
xk ∈ χ ¯j .
(16)
Thus we consider a different control gain not for all the cells χi with i ∈ I but for all cells χ ¯j with j ∈ J . Despite this restricted controller structure, in order to design a control law of type (16) one must exploit a different Lyapunov matrix Pi for each cell χi with i ∈ I (see the corresponding analysis result of Theorem 1) to reduce the conservativeness.
3
Synthesis of a Stabilizing State Feedback
In this section we consider the problem of finding a state feedback control law of type (16) for the system (3). For this purpose we start from the analysis condition (10) rewritten for the closed-loop system: xk ¯j (17) ∈ χi , x k ∈ χ xk+1 = Aij xk , for uk where Aij = Ai + Bi Kj and uk = Kj xk . More precisely, eq. (10) rewritten for the closed loop system (17) assumes the following form ATij Pl Aij − Pi < 0 Pi = PiT > 0,
∀j ∈ J , ∀i ∈ Sj , ∀l such that (l, i) ∈ Sall , ∀i ∈ I.
(18) (19)
Inequalities (18)-(19) represent a closed-loop stability condition. For each cell χ ¯j (with j ∈ J ) we want to calculate a state feedback control law represented by the gain matrix Kj . The control gain Kj is used when xTk uTk belongs ¯j . Furthermore, this to any cell χi such that i ∈ Sj or, equivalently, if xk ∈ χ T controller is applied independently of the subcell χl in which xTk+1 uTk+1 is contained (obviously, the pair (l, i) has to belong to the set of all possible switches i.e. Sall ). Clearly, in view of eq. (7) these inequalities are exhaustive stability conditions since they cover all possible transitions of the set Sall . Because each matrix Pi is positive definite we can rewrite (18) by resorting to the Schur lemma as follows: −Qi Qi ATij < 0, ∀j ∈ J , ∀i ∈ Sj , ∀l such that (l, i) ∈ Sall (20) Aij Qi −Ql
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where Qi := Pi−1 . We will show that (20) is guaranteed if there exist matrices Gj with j ∈ J of suitable dimensions such that the following alternative inequalities are satisfied Qi − Gj − GTj GTj ATij < 0, ∀j ∈ J , ∀i ∈ Sj , ∀l such that (l, i) ∈ Sall (21) Aij Gj −Ql where Gj , j ∈ J are matrices of suitable dimensions. In order to demonstrate that (21) implies inequalities (20) we first observe that matrices Gj are nonsingular since we have assumed Qi > 0 ∀i ∈ I whereas the element {1, 1} of (21) implies that Gj +GTj > Qi . Secondly, if Qi > 0 the matrix (GTj −Qi )Q−1 i (Gj −Qi ) is nonnegative definite and consequently: 0 < Gj + GTj − Qi ≤ GTj Q−1 i Gj . Moreover, because of (22) inequalities (21) imply T T A G G −GTj Q−1 j ij < 0, j i ∀j ∈ J , ∀i ∈ Sj , ∀l such that (l, i) ∈ Sall . Aij Gj −Ql
(22)
(23)
Finally, recalling that the matrices Gj are nonsingular we can obtain (20) from −T (23) by multiplying (23) from the right by diag Qi Gj , I and from the left Q , I . by diag G−1 i j These considerations lead to the following algorithm to calculate a stabilizing state-feedback control law. Indeed, in the following theorem we propose calculating a state-feedback controller of type (16) by exploiting a Piece-Wise Quadratic (PWQ) Lyapunov function defined by s matrices Pi with i ∈ I: Theorem 2. Consider the PWA system (3). There exists a state feedback control law of type (16) guaranteeing PWQ stability if there exist matrices Qi = QTi > 0 with i ∈ I and matrices Gj , Yj with j ∈ J , such that ∀j ∈ J , ∀i ∈ Sj and ∀l with (l, i) ∈ Sall Qi − Gj − GTj GTj ATi + YjT BiT < 0. (24) Ai Gj + Bi Yj −Ql The feedback gains Kj are given by: Kj := Yj G−1 j ,j ∈ J.
(25) t u
4
H∞ Performance of Piecewise Affine Systems
Consider the PWA system xk+1 = Ai xk + Bi uk + Biw wk + ai zk = Ci xk + Di uk + Diw wk ,
xk uk
¯j ∈ χi , xk ∈ χ
(26)
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where wk ∈ IRr is a disturbance signal and zk ∈ IRs is a performance output that can model, for instance, tracking errors or the cost of the input. First, to simplify the exposition we consider the case ai = 0, ∀i ∈ I (Subsection 4.1). Then, we extend our results to the case ai 6= 0 (Subsection 4.2). In any case, we assume that the system (26) admits x = 0 as an equilibrium point. As customary in control of nonlinear systems [14] we consider performance indices defined over a finite time horizon. In this section we focus on the disturbance attenuation problem in an H∞ framework: given a real number γ > 0, the exogenous signal w is attenuated by γ if, assuming x0 = 0, for each integer N ≥ 0 and for every w ∈ l2 ([0, N ] , IRr ) N X
k=0
2
kzk k < γ
2
N X
kwk k2 .
(27)
k=0
The control problem of discrete-time nonlinear systems can be very difficult due to the lack of geometric properties [14]. We will show that for PWA systems this task turns out to be less impervious provided the use of some fundamental LMI techniques [16,6]. To begin with, we present some analysis results for the following closedloop system obtained by applying a feedback control law of type (16) to system (26): xk+1 = Aij xk + Biw wk xk ¯j (28) ∈ χi , x k ∈ χ zk = Cij xk + Diw wk , uk where Aij = Ai + Bi Kj , Cij = Ci + Di Kj and uk = Kj xk . We observe again that the evolution of the closed-loop system (28) depends on the “hidden” variable uk since it influences the index i of the cell χi . A discrete-time nonlinear system (as the PWA system (28)) is strictly dissipative with supply rate W : IRs × IRr → IR if there exists a non-negative function V : IRn × IRm → IR termed storage function such that ∀w ∈ IRr , ∀k ≥ 0,
V (xk+1 , uk+1 ) − V (xk , uk ) < W (zk , wk )
(29)
and V (0, u) = 0, ∀u [5]. Condition (29), is the so-called dissipation inequality that can be equivalently represented through the condition [14,19] ∀wk , ∀N ≥ 0, ∀x0 ,
V (xN +1 , uN +1 ) − V (x0 , u0 )
0.
(31)
As will be shown, the supply rate W∞ (z, w) is related to the H∞ performance of the PWA system.
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H∞ Analysis and Synthesis for PWA Systems without Displacement Terms
The rationale presented in this section hinges on the assumption that the pair (Ai , Bi ), i ∈ I is PWQ stabilizable: that is, we assume that there exists Kj , j ∈ J and Pi = PiT > 0 with i ∈ I such that ∀j ∈ J , ∀i ∈ Sj , ∀l with (l, i) ∈ Sall ATij Pl Aij − Pi < 0
(32)
where Aij = Ai + Bi Kj if i ∈ Sj and Sall is the set of all possible switches. The next Lemma, which is a generalization of the classical Bounded Real Lemma [18,14] to PWA systems, allows to analyze the H∞ performance. Lemma 1. Consider the system (28) with zero initial condition x0 = 0. If there T exists a function V (x, u) = xT Pi x for xT uT ∈ χi with Pi = PiT > 0 satisfying the dissipativity inequality (29) with supply rate (31), i.e. ∀k, V (xk+1 , uk+1 ) − V (xk , uk ) < (γ 2 kwk k2 − kzk k2 ),
(33)
then, the H∞ performance condition (27) is satisfied. Furthermore, condition (33) is fulfilled if the following matrix inequalities are satisfied ∀j ∈ J , ∀i ∈ Sj , ∀l with (l, i) ∈ S, Ml,ij < 0.
(34)
where Ml,ij
T ATij Pl Aij − Pi + Cij Cij ∗ := . DiT Cij + BiT Pl Aij BiT Pl Bi + DiT Di − γ 2 I
(35)
In this last case the system (28) is PWQ stable. Proof. By recalling that x0 = 0, from (33) it follows that, ∀N ≥ 0 V (xN +1 , uN +1 )
0 with i ∈ I and matrices Gj , Yj with j ∈ J , such that ∀j ∈ J , ∀i ∈ Sj , ∀l with (l, i) ∈ Sall ∗ ∗ Qi − Gj − GTj ∗ Ai Gj + Bi Yj −Ql ∗ ∗ < 0. Ci Gj + Di Yj 0 −I ∗ 0 Biw T Diw T −γ 2 I
(39)
The feedback gains Kj with j ∈ J are given by: Kj := Yj G−1 j .
(40) t u
The proof of this theorem can be achieved form the results reported in Lemma 1 by applying the same line of reasoning used to demonstrate Theorem 2. 4.2
Extension to PWA Systems with Displacement Terms
Some analysis results have been extended to the case ai 6= 0 by considering an extended state space (see eq. (13)) [12,13]. Unfortunately, this approach is very restrictive for synthesis problems because the extended dynamic matrix A¯i is never a stability matrix (A¯i contains an unreachable eigenvalue at 1) and consequently it is never possible to find P = P T > 0 satisfying the Lyapunov stability condition A¯Ti P A¯i − P < 0.
(41)
On the other hand, the set Sall of all possible transitions contains also the transitions of type (i, i) i.e. from region i to the same region. This implies that the synthesis approach proposed in the previous part of this section can never be applied to a system obtained by extending the state vector as proposed in [12,13]. Therefore we consider a different approach based on the extension of the input signal wk as follows: wk . (42) w ˜k := ai
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Thus, the system (26) can be rewritten as: ˜ww xk+1 = Ai xk + Bi uk + B i ˜k w ˜ ˜k , zk = C i x k + D i u k + D i w
xk uk
¯j ∈ χi , x k ∈ χ
(43)
where w w ˜ = D 0 . ˜ w = Bw I D B i i i i
(44)
The H∞ framework considered here, is based on a finite horizon definition of the l2 gain and, consequently, the proposed extension of the disturbance input is sensible. Clearly, it is possible to apply the control approach proposed in Theorem 3 directly to the extended system (43). This can be conservative because ai is not an unknown disturbance but a known term. Unfortunately, in general, ai is known only when the control signal uk has already been calculated. Notwithstanding this, under the standard assumption ¯j , ∀i ∈ Sj , ∀j ∈ J , ai = a
(45)
an alternative control strategy can be proposed. More precisely, the control is assumed to have the following structure: 1 2 xk ¯j . (46) , xk ∈ χ u k = Kj K j a ¯j In this way the controller can take into account also the displacement term ˜k where ai = D w D := 0 I . By applying the control law (46) to the PWA system (43) we obtain the closedloop PWA system: w w ˜k xk+1 = Aij xk + B˜ij w ˜ w zk = Cij xk + D ij ˜k ,
xk uk
∈ χi , x k ∈ χ ¯j
(47)
where w ˜ w + Bi K 2 D =B Aij = Ai + Bi Kj1 B˜ij j i ˜ w + Di K 2 D. ˜w = D Cij = Ci + Di Kj1 D j i ij
(48)
Now, we can apply the H∞ result of Lemma 1 to the closed-loop PWA system (47) to arrive at the synthesis procedure summarized in the subsequent theorem. In this case, the controller gain is composed of two different parts, Kj1 and Kj2 , that constitute two unknowns of a suitable LMI problem:
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Theorem 4. Consider the PWA system (26). There exists a state feedback control law of type (46) guaranteeing PWQ Lyapunov stability and fulfilling the dissipativity constraint (29) with supply rate ˜ ∞ (zk , wk ) = (γ 2 k wT aT T k2 − kzk k2 ) = W k i x 2 2 2 2 = (γ kwk k + kai k − kzk k ), γ > 0, k ∈ χi uk
(49) (50)
if there exist matrices Qi = QTi > 0 with i ∈ I and matrices Gj , Yj , Kj2 with j ∈ J , such that ∀j ∈ J , ∀i ∈ Sj , ∀l with (l, i) ∈ Sall Qi − Gj − GTj ∗ ∗ ∗ A i Gj + B i Y j −Ql ∗ ∗ (51) < 0. C i Gj + D i Y j 0 −I ∗ T T ˜ w + Di K 2 D ˜ w + Bi K 2 D D −γ 2 I 0 B i
j
i
j
The feedback gain matrices Kj1 with j ∈ J are given by: Kj1 := Yj G−1 j .
(52) t u
5
Numerical Example: The Tank Case
The example we consider here is inspired by the three-tank benchmark described in [10] that will be the subject of future investigation. It consists of a single tank with cross section section A. It is filled by means of a pump whose mass flow
Pump
u K2 x
Tank x2
x
K1x
x1
f
Fig. 1. Tank configuration
rate is given by the control variable u (see Figure 1). Obviously, we suppose that ¯1 and x ¯2 we 0 ≤ u ≤ umax . The tank level is denoted by x. At the heights x assume there are two pipes through which we have the output mass flow rates
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K1 x and K2 x respectively. Finally, at the bottom we have a constant output mass flow rate f . In our case we have chosen the following numerical values with suitable dimensions: A = 1, K1 = 0.2, K2 = 0.1, ¯2 = 0.6, f = 0.01, umax = 0.019. x ¯1 = 0.3, x
(53)
In order to introduce the tank model we adopt the following notation. Let b be a boolean expression, then we denote with | · | the function 1 if b = T RU E |b| = (54) 0 if b = F ALSE Then, a possible continuous-time model for the tank of Figure 1 is given as follows: Ax˙ =u|(u ≥ 0) ∧ (u ≤ umax )| + umax |u > umax | + ¯1 | − K2 x|x ≥ x ¯2 |. − f − K1 x|x ≥ x
(55)
In this model we have neglected the obvious physical condition x ≥ 0. Moreover, it is very simple to obtain from (55) the following PWA continuous-time model: −f u−f umax − f −f − K1 x u − f − K1 x Ax˙ = u − f − K1 x max −f − (K 1 + K2 )x u − f − (K 1 + K2 )x umax − f − (K1 + K2 )x
if u < 0, x<x ¯1 if 0 ≤ u ≤ umax , x<x ¯1 if u > umax , x<x ¯1 if u < 0, x ¯1 ≤ x < x ¯2 if 0 ≤ u ≤ umax , x ¯1 ≤ x < x ¯2 if u > umax , x ¯1 ≤ x < x ¯2 if u < 0, x≥x ¯2 if 0 ≤ u ≤ umax , x≥x ¯2 if u > umax , x≥x ¯2 .
(56)
This model can be reduced to a discrete-time PWA system of type (26) by discretization (employing the implicit Euler’s rule with a discretization time equal to 0.5 sec.). Finally, the PWA discrete-time model has 9 cells χi and 3 cells χ ¯j . Furthermore, we do not consider any disturbance inputs of type w and we consider the problem of regulating the level z := x around 0.1. For this
9
0.6 8
0.5 0.4
(a)
0.3
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Switching history
Tank Level − x
0.7
0.1 0
0
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2
2.5
3
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4
Time Control Variable − u
0.02
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(c)
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(b) 2
0.005
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2.5
3
3.5
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Fig. 2. Closed Loop simulation - (a) State, (b) Control Input, (c) Switching history
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purpose we have applied to the discretized model an H∞ regulator obtained by means of the synthesis procedure of Theorems 3 and 4. In Figures 2.(a)-2.(b) we report the time-histories of the state variable and of the control input (the initial state considered is x0 = 0.7). Finally, in Figure 2.(c) we show the corresponding switching history (we recall that we have 9 cells of type χi and in this picture T we report the index of the cell χi in which the vector xTk uTk is contained).
6
Conclusions
In this paper we derived LMIs-based procedures to solve H∞ analysis and synthesis problems for PWA systems whose switching sequence depends on the state and on the control input. These PWA systems can be found by translating an MLD system into PWA form. The analysis tests can be applied to assess the performance of MPC control schemes applied both to linear and hybrid systems. Moreover, the state-feedback design methodologies provide an alternative way to synthesize controllers with a prescribed degree of performance. All the proposed synthesis procedures are clearly only sufficient i.e. nothing can be said if the LMIs are infeasible. A thorough analysis of their conservativeness will be subject of further investigations.
References 1. Bemporad, A., Morari, M.: Control of Systems Integrating Logic, Dynamics, and Constraints. Automatica, 35(3), (1999), 407-427. 2. Bemporad, A., Morari, M., Dua, V., Pistikopoulos, E. N.: The Explicit Linear Quadratic Regulator for Constrained Systems. American Control Conference, Chicago, IL, (2000). 3. Bemporad, A., Ferrari-Trecate, G., Morari, M.: Observability and Controllability of Piecewise Affine and Hybrid Systems. IEEE Transactions on Automatic Control, 45(10), (2000), 1864–1876. 4. Bemporad, A., Torrisi, F.D., Morari, M.: Optimization-Based Verification and Stability Characterization of Piecewise Affine and Hybrid Systems. Proceedings 3rd International Workshop on Hybrid Systems, Lecture Notes in Computer Science, Springer-Verlag, Pittsburgh, USA (2000). 5. Byrnes, C. I., Lin, W.: Passivity and absolute stabilization of a class of discretetime nonlinear systems. Automatica, 31(2), (1995), 263–268. 6. de Oliveira, M. C., Bernussou, J., Geromel, J. C.: A new discrete-time robust stability condition. System & Control Letters, 37, (1999), 261–265. 7. Ferrari-Trecate, G., Cuzzola, F. A., Mignone, D., Morari, M.: Analysis of DiscreteTime Piecewise Affine and Hybrid Systems. Submitted for publication. 8. Gahinet, P., Nemirowski, A., Laub, A. J., Chilali, M.: LMI Control Toolbox, The MathWorks Inc., (1994). 9. Heemels, W.P.M.H., De Schutter, B.: On the Equivalence of Classes of Hybrid Systems: Mixed Logical Dynamical and Complementarity Systems. T.R. 00 I/04, Technische Universiteit Eindhoven, (2000).
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10. Heiming, B., Lunze, J.: Definition of the Three-Tank Benchmark Problem for Controller Reconfiguration, European Control Conference, Karlshrue, Germany, (1999). 11. Johansen, T. A.: Computation of Lyapunov functions for smooth nonlinear systems using convex optimisation. Automatica, 36, (2000), 1617–1626. 12. Johansson, M., Rantzer, A.: Computation of Piecewise Quadratic Lyapunov Functions for Hybrid Systems. IEEE Transactions on Automatic Control, 43(4), (2000), 555–559. 13. Johansson, M., Rantzer, A.: Piecewise Linear Quadratic Optimal Control. IEEE Transactions on Automatic Control, 43(4), (2000), 629–637. 14. Lin, W., Byrnes, C. I.: H∞ Control of Discrete-Time Nonlinear Systems. IEEE Transactions on Automatic Control, 41(4), (1996), 494–510. 15. Mignone, D., Ferrari-Trecate, G., Morari, M.: Stability and Stabilization of Piecewise Affine and Hybrid Systems: An LMI Approach, IEEE Conference on Decision and Control, Sydney, Australia, (2000). 16. Scherer, C. W., Gahinet, P., Chilali, M.: Multi-Objective Output-Feedback Control via LMI Optimization. IEEE Transactions on Automatic Control, 42(7), (1997), 896–911. 17. Sontag, E.D.: Interconnected automata and linear systems: A theoretical framework in discrete-time. Hybrid Systems III - Verification and Control, R. Alur, T.A. Henzinger and E.D. Sontag eds., 1066, Lecture Notes in Computer Science. Springer-Verlag, Pittsburgh, USA, (1996), 436–448. 18. Van der Schaft, A. J.: L2 -gain analysis of nonlinear systems and nonlinear H∞ control, IEEE Transactions on Automatic Control, 37, (1992), 770–784. 19. Willems, J. C.: Dissipative dynamic systems, Arch. Rational Mechanics Analysis, 45, (1972), 321–393. 20. Yakubovich, V. A.: S-Procedure in nonlinear control theory. Vestnik Leninggradskogo Universiteta, Ser. Matematika, (1971), 62–77.
Abstract. In this paper we investigate some analysis and control problems for discrete-time hybrid systems in the piece-wise affine form. By using arguments from the dissipativity theory for nonlinear systems, we show that H∞ analysis and synthesis problems can be formulated and solved via Linear Matrix Inequalities by taking into account the switching structure of the considered system. In this paper we address the generalized problem of controlling hybrid systems whose switching structure does not depend only on the state but also on the control input.
1
Introduction
Piece-Wise Affine (PWA) systems have been receiving increasing attention by the control community because they provide a useful modeling framework for hybrid systems. In fact, discrete-time PWA systems are equivalent to interconnections of linear systems and finite automata [17], to complementarity systems [9] and also hybrid systems in the Mixed Logic Dynamical (MLD) form [1]. In particular, the MLD form is capable to model a large class of hybrid systems including linear hybrid dynamical systems, hybrid automata, some classes of discrete-event systems, and systems with qualitative inputs/outputs [1,3]. The algorithm to obtain the discrete-time PWA representation of an MLD system and vice-versa is reported in [3]. In order to stress the importance of PWA systems it is worth recalling that in [2] the explicit form of Model Predictive Control (MPC) for linear constrained systems was derived and, besides providing an algorithm for its computation, it was shown that the closed-loop system has a PWA structure. Also in this case the closed-loop system turns out to be a PWA model. An important feature of a PWA model is that the state-update map can be discontinuous along the boundary of the regions. For instance, when considering PWA systems stemming from hybrid systems in the MLD form, discontinuities can arise from the representation of logic conditions. The control synthesis problem for MLD systems and consequently PWA systems ?
This research has been supported by the Swiss National Science Foundation.
M.D. Di Benedetto, A. Sangiovanni-Vincentelli (Eds.): HSCC 2001, LNCS 2034, pp. 189–203, 2001. c Springer-Verlag Berlin Heidelberg 2001
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is computationally difficult: in [1] a Mixed Integer Quadratic Programming (MIQP) approach is proposed in order to solve the control problem of MLD systems by means of MPC techniques. Needless to say, the computational complexity of this approach may increase exponentially with the prediction horizon considered. The use of Linear Matrix Inequalities (LMI) techniques, for which computationally advantageous and numerically reliable algorithms as well as toolboxes are available (see [8]) would seem to be a promising alternative. Concerning the stability analysis of PWA systems, the authors presented various algorithms with different degrees of conservativeness in [15]. Similarly to [12, 13], where a particular class of continuous-time PWA systems was considered, such procedures exploit Piece-Wise Quadratic (PWQ) Lyapunov functions that can be computed as the solution of a set of LMIs. For the sake of completeness, the main stability test of [15] is reported in Section 2 in a suitable form. In this work, we consider both analysis and synthesis problems for the general class of PWA models whose switching sequence depends on both state and input trajectories. As pointed out in [3] the dependence of the switching sequence on the input can be met by translating an MLD system into a PWA form. Moreover, the dependence of the switching sequence on the input signal is common in real systems: for example, it could be caused by saturation effects or limitations on the control signal. It is worthwhile emphasizing that this type of PWA models is more general than that considered in [12,13] and [15]: indeed, in these works the switching structure depended on the state only. Furthermore, we generalize the results of [15] by considering analysis and synthesis problems with performance for PWA systems. We focus on the H∞ norm showing that the H∞ -analysis and the H∞ -synthesis of a piecewise linear state-feedback can be addressed by resorting to LMI-based algorithms. The rationale of our derivation hinges on the use of passivity theory for nonlinear systems [14]. We point out that a significant application of the H∞ analysis test is the possibility of checking a posteriori the performance of MPC for both linear and MLD systems. As mentioned before this can be done by exploiting the explicit PWA form of the closed-loop system. The results are presented in Sections 3 and 4. An illustrative example is provided in Section 5.
Notation: The symbol ∗ will be used in some matrix expressions to induce a symmetric structure. For example, if L and R are symmetric matrices, then L+M +∗ ∗ L + M + MT NT . (1) := N R N R Moreover, we define kxk kl2 (0,N ) :=
N X
k=0
1
{xTk xk } 2 .
(2)
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Stability and State-Feedback Stabilization of PWA Systems
A linear discrete-time piecewise affine system is defined by the state-space equation xk (3) ∈ χi xk+1 = Ai xk + Bi uk + ai , for uk where xk ∈ IRn is the state and uk ∈ IRm is the control input. The set X ⊆ IRn+m T is either IRn+m or a polyhedron containing the of every possible vector xTk uTk origin, {χi }si=1 is a polyhedral partition 1 of X and ai ∈ IRn , are constant vectors. We refer to each χi as a cell. Moreover, in order to simplify the exposition, we assume that our cells are polyhedra defined by matrices Fix , Fiu , fix and fiu as follows o n x x u u T T T x u (4) χi := such that Fi x ≥ fi and Fi u ≥ fi . The results presented in this paper can be extended to systems whose cells χi have a more complicate structure. Moreover, it is worth introducing the following notation: χ ¯i := {x such that Fix x ≥ fix }
(5)
and n
T
¯j , x u Sj := i such that ∃x, u with x ∈ χ
T
T
o
∈ χi .
(6)
In a nutshell, Sj is the set of all indices i such that χi is a cell containing a T vector xT uT for which the condition x ∈ χ ¯j is satisfied. We denote with I = {1, . . . , s} the set of indices of the cells χi whereas the symbol J = {1, . . . , t} will be used to denote the set of indices of the cells χ ¯j . It is important to observe that: t [
Sj = I.
(7)
j=1
Furthermore, if cells χi have the structure pointed out in eq. (4) then the sets Sj are disjoint whereas if cells χi have a more complicate structure (for instance when mixed state-input constraints are used to define each cell χi ) then the sets Sj could be overlapping. In the latter case the results we are going to present could become more conservative. When we focus on the stability of the origin, we consider autonomous 1
Each Ss set χi is a (not necessarily closed) convex polyhedron s.t. χi i=1 χi = X.
T
χj = ∅, ∀i 6= j,
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PWA systems and we assume that x = 0 is an equilibrium point. To begin with it is necessary to observe that an autonomous system can be obtained from system (3) by applying a suitable control law. In the following we consider a piecewise linear state feedback with the structure xk (8) ∈ χi . uk = Ki xk , for uk By applying the controller (8) to the system (3) we achieve the following closedloop dynamic system xk ∈ χi (9) xk+1 = Ai xk + ai , for uk where Ai = Ai + Bi Ki and uk = Ki xk . We note that the the evolution of closed-loop system (9) depends on the “hidden” variable uk since it influences the index i of the current cell χi . As customary for constrained systems, we assume that the state trajectories T T T T x k uk ∈ X, ∀k ∈ IN. generated by the control law (8) satisfy xTk uTk In [15] the stability of the origin of PWA system was characterized by using Piece-Wise Quadratic (PWQ) Lyapunov functions. In the following theorem we report the main result of [15] valid for the case ai = 0, ∀i ∈ I and adapted to the closed loop system (9). Theorem 1. Consider the system (9). If there exist matrices Pi = PiT > 0, T ∈ χi ∀i ∈ I such that the positive-definite function V (x, u) = xT Pi x, ∀ xTk uTk satisfies V (xk+1 , uk+1 ) − V (xk , uk ) < 0, then the origin of the PWA systems (9) is exponentially stable and limk→+∞ kxk k = 0 for all system trajectories fulfilling T T T x k uk ∈ X, ∀k ∈ IN. t u The Lyapunov function appearing in Theorem 1 can be computed by solving the LMIs ATj Pi Aj − Pj < 0, Pi = PiT > 0,
∀(i, j) ∈ S
(10)
∀i ∈ I
(11)
where
xk xk+1 , ∈X S := (i, j) : i, j ∈ I and ∃k ∈ N0 , ∃ uk uk+1 xk+1 xk ∈ χj and ∈ χi . such that uk uk+1
(12)
In other words, the set S contains all the ordered pairs of indices denoting the possible switches from cell j to cell i and it can be computed via reachability analysis for MLD systems [4]. Then, the inequalities (10) take into account all the
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admissible switches between different regions and guarantee that the Lyapunov function is decreasing along all possible state trajectories. When there exist matrices Pi such that the LMIs (10) and (11) are satisfied, the PWA system and the corresponding controller (eq. (8)) are termed PWQ-stable and PWQstabilizing respectively. We refer the interested reader to [15] for further details. Remark 1. Conservativeness. The conservativeness of the LMIs conditions for stability analysis can be reduced by exploiting the so-called S-Procedure [20] in order to avoid imposing xT Pi x > 0 T for xT uT ∈ χj , j 6= i [15]. This modification was proposed in [12] for continuous-time PWA systems and can be easily generalized to the discretetime case. We point out that similar modifications can be applied to all the analysis LMIs we derive in the following. It is important to highlight that with respect to the continuous-time case (see [12]) in the discrete-time case there is no need to guarantee the continuity of the Lyapunov function over the whole state-space. This fact can determine a reduced degree of conservativeness of the results that we are going to present with respect to those presented in [12]. Finally, following the lead given in [11], the authors proposed in [7] discrete-time performance analysis results with a notably reduced degree of conservativeness. t u Remark 2. Extension of Theorem 1. Theorem 1 can be extended to the case ai 6= 0 as done in [12,13] by introducing the extended state x ¯k = [ xTk 1 ]T and rewriting the system (3) as follows: ¯i uk for ¯k + B x ¯k+1 = A¯i x
xk uk
∈ χi
(13)
where Ai ai ¯ Bi ¯ Bi = . Ai = 0 1 0
(14) t u
When designing the controller i.e. when the controller gains Ki appearing in the inequalities (10) are unknown, the set of all possible switches is generally not known in advance, and it could be necessary to consider all the pairs of indices in Sall := I × I instead of S. Furthermore, we note that the design of a controller of type (8) could be a very hard task because, at each time instant, the vector uk has to be
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calculated by means of a control gain K¯i whose index ¯i is found on the basis of the admissibility condition xk (15) ∈ χ¯i . uk This implies that in general it is not possible to calculate uk since the index ¯i for which the condition (15) is satisfied, is difficult to know in advance. Therefore, we turn our problem into one of designing a controller with the following structure u k = Kj x k ,
xk ∈ χ ¯j .
(16)
Thus we consider a different control gain not for all the cells χi with i ∈ I but for all cells χ ¯j with j ∈ J . Despite this restricted controller structure, in order to design a control law of type (16) one must exploit a different Lyapunov matrix Pi for each cell χi with i ∈ I (see the corresponding analysis result of Theorem 1) to reduce the conservativeness.
3
Synthesis of a Stabilizing State Feedback
In this section we consider the problem of finding a state feedback control law of type (16) for the system (3). For this purpose we start from the analysis condition (10) rewritten for the closed-loop system: xk ¯j (17) ∈ χi , x k ∈ χ xk+1 = Aij xk , for uk where Aij = Ai + Bi Kj and uk = Kj xk . More precisely, eq. (10) rewritten for the closed loop system (17) assumes the following form ATij Pl Aij − Pi < 0 Pi = PiT > 0,
∀j ∈ J , ∀i ∈ Sj , ∀l such that (l, i) ∈ Sall , ∀i ∈ I.
(18) (19)
Inequalities (18)-(19) represent a closed-loop stability condition. For each cell χ ¯j (with j ∈ J ) we want to calculate a state feedback control law represented by the gain matrix Kj . The control gain Kj is used when xTk uTk belongs ¯j . Furthermore, this to any cell χi such that i ∈ Sj or, equivalently, if xk ∈ χ T controller is applied independently of the subcell χl in which xTk+1 uTk+1 is contained (obviously, the pair (l, i) has to belong to the set of all possible switches i.e. Sall ). Clearly, in view of eq. (7) these inequalities are exhaustive stability conditions since they cover all possible transitions of the set Sall . Because each matrix Pi is positive definite we can rewrite (18) by resorting to the Schur lemma as follows: −Qi Qi ATij < 0, ∀j ∈ J , ∀i ∈ Sj , ∀l such that (l, i) ∈ Sall (20) Aij Qi −Ql
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where Qi := Pi−1 . We will show that (20) is guaranteed if there exist matrices Gj with j ∈ J of suitable dimensions such that the following alternative inequalities are satisfied Qi − Gj − GTj GTj ATij < 0, ∀j ∈ J , ∀i ∈ Sj , ∀l such that (l, i) ∈ Sall (21) Aij Gj −Ql where Gj , j ∈ J are matrices of suitable dimensions. In order to demonstrate that (21) implies inequalities (20) we first observe that matrices Gj are nonsingular since we have assumed Qi > 0 ∀i ∈ I whereas the element {1, 1} of (21) implies that Gj +GTj > Qi . Secondly, if Qi > 0 the matrix (GTj −Qi )Q−1 i (Gj −Qi ) is nonnegative definite and consequently: 0 < Gj + GTj − Qi ≤ GTj Q−1 i Gj . Moreover, because of (22) inequalities (21) imply T T A G G −GTj Q−1 j ij < 0, j i ∀j ∈ J , ∀i ∈ Sj , ∀l such that (l, i) ∈ Sall . Aij Gj −Ql
(22)
(23)
Finally, recalling that the matrices Gj are nonsingular we can obtain (20) from −T (23) by multiplying (23) from the right by diag Qi Gj , I and from the left Q , I . by diag G−1 i j These considerations lead to the following algorithm to calculate a stabilizing state-feedback control law. Indeed, in the following theorem we propose calculating a state-feedback controller of type (16) by exploiting a Piece-Wise Quadratic (PWQ) Lyapunov function defined by s matrices Pi with i ∈ I: Theorem 2. Consider the PWA system (3). There exists a state feedback control law of type (16) guaranteeing PWQ stability if there exist matrices Qi = QTi > 0 with i ∈ I and matrices Gj , Yj with j ∈ J , such that ∀j ∈ J , ∀i ∈ Sj and ∀l with (l, i) ∈ Sall Qi − Gj − GTj GTj ATi + YjT BiT < 0. (24) Ai Gj + Bi Yj −Ql The feedback gains Kj are given by: Kj := Yj G−1 j ,j ∈ J.
(25) t u
4
H∞ Performance of Piecewise Affine Systems
Consider the PWA system xk+1 = Ai xk + Bi uk + Biw wk + ai zk = Ci xk + Di uk + Diw wk ,
xk uk
¯j ∈ χi , xk ∈ χ
(26)
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where wk ∈ IRr is a disturbance signal and zk ∈ IRs is a performance output that can model, for instance, tracking errors or the cost of the input. First, to simplify the exposition we consider the case ai = 0, ∀i ∈ I (Subsection 4.1). Then, we extend our results to the case ai 6= 0 (Subsection 4.2). In any case, we assume that the system (26) admits x = 0 as an equilibrium point. As customary in control of nonlinear systems [14] we consider performance indices defined over a finite time horizon. In this section we focus on the disturbance attenuation problem in an H∞ framework: given a real number γ > 0, the exogenous signal w is attenuated by γ if, assuming x0 = 0, for each integer N ≥ 0 and for every w ∈ l2 ([0, N ] , IRr ) N X
k=0
2
kzk k < γ
2
N X
kwk k2 .
(27)
k=0
The control problem of discrete-time nonlinear systems can be very difficult due to the lack of geometric properties [14]. We will show that for PWA systems this task turns out to be less impervious provided the use of some fundamental LMI techniques [16,6]. To begin with, we present some analysis results for the following closedloop system obtained by applying a feedback control law of type (16) to system (26): xk+1 = Aij xk + Biw wk xk ¯j (28) ∈ χi , x k ∈ χ zk = Cij xk + Diw wk , uk where Aij = Ai + Bi Kj , Cij = Ci + Di Kj and uk = Kj xk . We observe again that the evolution of the closed-loop system (28) depends on the “hidden” variable uk since it influences the index i of the cell χi . A discrete-time nonlinear system (as the PWA system (28)) is strictly dissipative with supply rate W : IRs × IRr → IR if there exists a non-negative function V : IRn × IRm → IR termed storage function such that ∀w ∈ IRr , ∀k ≥ 0,
V (xk+1 , uk+1 ) − V (xk , uk ) < W (zk , wk )
(29)
and V (0, u) = 0, ∀u [5]. Condition (29), is the so-called dissipation inequality that can be equivalently represented through the condition [14,19] ∀wk , ∀N ≥ 0, ∀x0 ,
V (xN +1 , uN +1 ) − V (x0 , u0 )
0.
(31)
As will be shown, the supply rate W∞ (z, w) is related to the H∞ performance of the PWA system.
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H∞ Analysis and Synthesis for PWA Systems without Displacement Terms
The rationale presented in this section hinges on the assumption that the pair (Ai , Bi ), i ∈ I is PWQ stabilizable: that is, we assume that there exists Kj , j ∈ J and Pi = PiT > 0 with i ∈ I such that ∀j ∈ J , ∀i ∈ Sj , ∀l with (l, i) ∈ Sall ATij Pl Aij − Pi < 0
(32)
where Aij = Ai + Bi Kj if i ∈ Sj and Sall is the set of all possible switches. The next Lemma, which is a generalization of the classical Bounded Real Lemma [18,14] to PWA systems, allows to analyze the H∞ performance. Lemma 1. Consider the system (28) with zero initial condition x0 = 0. If there T exists a function V (x, u) = xT Pi x for xT uT ∈ χi with Pi = PiT > 0 satisfying the dissipativity inequality (29) with supply rate (31), i.e. ∀k, V (xk+1 , uk+1 ) − V (xk , uk ) < (γ 2 kwk k2 − kzk k2 ),
(33)
then, the H∞ performance condition (27) is satisfied. Furthermore, condition (33) is fulfilled if the following matrix inequalities are satisfied ∀j ∈ J , ∀i ∈ Sj , ∀l with (l, i) ∈ S, Ml,ij < 0.
(34)
where Ml,ij
T ATij Pl Aij − Pi + Cij Cij ∗ := . DiT Cij + BiT Pl Aij BiT Pl Bi + DiT Di − γ 2 I
(35)
In this last case the system (28) is PWQ stable. Proof. By recalling that x0 = 0, from (33) it follows that, ∀N ≥ 0 V (xN +1 , uN +1 )
0 with i ∈ I and matrices Gj , Yj with j ∈ J , such that ∀j ∈ J , ∀i ∈ Sj , ∀l with (l, i) ∈ Sall ∗ ∗ Qi − Gj − GTj ∗ Ai Gj + Bi Yj −Ql ∗ ∗ < 0. Ci Gj + Di Yj 0 −I ∗ 0 Biw T Diw T −γ 2 I
(39)
The feedback gains Kj with j ∈ J are given by: Kj := Yj G−1 j .
(40) t u
The proof of this theorem can be achieved form the results reported in Lemma 1 by applying the same line of reasoning used to demonstrate Theorem 2. 4.2
Extension to PWA Systems with Displacement Terms
Some analysis results have been extended to the case ai 6= 0 by considering an extended state space (see eq. (13)) [12,13]. Unfortunately, this approach is very restrictive for synthesis problems because the extended dynamic matrix A¯i is never a stability matrix (A¯i contains an unreachable eigenvalue at 1) and consequently it is never possible to find P = P T > 0 satisfying the Lyapunov stability condition A¯Ti P A¯i − P < 0.
(41)
On the other hand, the set Sall of all possible transitions contains also the transitions of type (i, i) i.e. from region i to the same region. This implies that the synthesis approach proposed in the previous part of this section can never be applied to a system obtained by extending the state vector as proposed in [12,13]. Therefore we consider a different approach based on the extension of the input signal wk as follows: wk . (42) w ˜k := ai
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Thus, the system (26) can be rewritten as: ˜ww xk+1 = Ai xk + Bi uk + B i ˜k w ˜ ˜k , zk = C i x k + D i u k + D i w
xk uk
¯j ∈ χi , x k ∈ χ
(43)
where w w ˜ = D 0 . ˜ w = Bw I D B i i i i
(44)
The H∞ framework considered here, is based on a finite horizon definition of the l2 gain and, consequently, the proposed extension of the disturbance input is sensible. Clearly, it is possible to apply the control approach proposed in Theorem 3 directly to the extended system (43). This can be conservative because ai is not an unknown disturbance but a known term. Unfortunately, in general, ai is known only when the control signal uk has already been calculated. Notwithstanding this, under the standard assumption ¯j , ∀i ∈ Sj , ∀j ∈ J , ai = a
(45)
an alternative control strategy can be proposed. More precisely, the control is assumed to have the following structure: 1 2 xk ¯j . (46) , xk ∈ χ u k = Kj K j a ¯j In this way the controller can take into account also the displacement term ˜k where ai = D w D := 0 I . By applying the control law (46) to the PWA system (43) we obtain the closedloop PWA system: w w ˜k xk+1 = Aij xk + B˜ij w ˜ w zk = Cij xk + D ij ˜k ,
xk uk
∈ χi , x k ∈ χ ¯j
(47)
where w ˜ w + Bi K 2 D =B Aij = Ai + Bi Kj1 B˜ij j i ˜ w + Di K 2 D. ˜w = D Cij = Ci + Di Kj1 D j i ij
(48)
Now, we can apply the H∞ result of Lemma 1 to the closed-loop PWA system (47) to arrive at the synthesis procedure summarized in the subsequent theorem. In this case, the controller gain is composed of two different parts, Kj1 and Kj2 , that constitute two unknowns of a suitable LMI problem:
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Theorem 4. Consider the PWA system (26). There exists a state feedback control law of type (46) guaranteeing PWQ Lyapunov stability and fulfilling the dissipativity constraint (29) with supply rate ˜ ∞ (zk , wk ) = (γ 2 k wT aT T k2 − kzk k2 ) = W k i x 2 2 2 2 = (γ kwk k + kai k − kzk k ), γ > 0, k ∈ χi uk
(49) (50)
if there exist matrices Qi = QTi > 0 with i ∈ I and matrices Gj , Yj , Kj2 with j ∈ J , such that ∀j ∈ J , ∀i ∈ Sj , ∀l with (l, i) ∈ Sall Qi − Gj − GTj ∗ ∗ ∗ A i Gj + B i Y j −Ql ∗ ∗ (51) < 0. C i Gj + D i Y j 0 −I ∗ T T ˜ w + Di K 2 D ˜ w + Bi K 2 D D −γ 2 I 0 B i
j
i
j
The feedback gain matrices Kj1 with j ∈ J are given by: Kj1 := Yj G−1 j .
(52) t u
5
Numerical Example: The Tank Case
The example we consider here is inspired by the three-tank benchmark described in [10] that will be the subject of future investigation. It consists of a single tank with cross section section A. It is filled by means of a pump whose mass flow
Pump
u K2 x
Tank x2
x
K1x
x1
f
Fig. 1. Tank configuration
rate is given by the control variable u (see Figure 1). Obviously, we suppose that ¯1 and x ¯2 we 0 ≤ u ≤ umax . The tank level is denoted by x. At the heights x assume there are two pipes through which we have the output mass flow rates
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K1 x and K2 x respectively. Finally, at the bottom we have a constant output mass flow rate f . In our case we have chosen the following numerical values with suitable dimensions: A = 1, K1 = 0.2, K2 = 0.1, ¯2 = 0.6, f = 0.01, umax = 0.019. x ¯1 = 0.3, x
(53)
In order to introduce the tank model we adopt the following notation. Let b be a boolean expression, then we denote with | · | the function 1 if b = T RU E |b| = (54) 0 if b = F ALSE Then, a possible continuous-time model for the tank of Figure 1 is given as follows: Ax˙ =u|(u ≥ 0) ∧ (u ≤ umax )| + umax |u > umax | + ¯1 | − K2 x|x ≥ x ¯2 |. − f − K1 x|x ≥ x
(55)
In this model we have neglected the obvious physical condition x ≥ 0. Moreover, it is very simple to obtain from (55) the following PWA continuous-time model: −f u−f umax − f −f − K1 x u − f − K1 x Ax˙ = u − f − K1 x max −f − (K 1 + K2 )x u − f − (K 1 + K2 )x umax − f − (K1 + K2 )x
if u < 0, x<x ¯1 if 0 ≤ u ≤ umax , x<x ¯1 if u > umax , x<x ¯1 if u < 0, x ¯1 ≤ x < x ¯2 if 0 ≤ u ≤ umax , x ¯1 ≤ x < x ¯2 if u > umax , x ¯1 ≤ x < x ¯2 if u < 0, x≥x ¯2 if 0 ≤ u ≤ umax , x≥x ¯2 if u > umax , x≥x ¯2 .
(56)
This model can be reduced to a discrete-time PWA system of type (26) by discretization (employing the implicit Euler’s rule with a discretization time equal to 0.5 sec.). Finally, the PWA discrete-time model has 9 cells χi and 3 cells χ ¯j . Furthermore, we do not consider any disturbance inputs of type w and we consider the problem of regulating the level z := x around 0.1. For this
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Fig. 2. Closed Loop simulation - (a) State, (b) Control Input, (c) Switching history
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purpose we have applied to the discretized model an H∞ regulator obtained by means of the synthesis procedure of Theorems 3 and 4. In Figures 2.(a)-2.(b) we report the time-histories of the state variable and of the control input (the initial state considered is x0 = 0.7). Finally, in Figure 2.(c) we show the corresponding switching history (we recall that we have 9 cells of type χi and in this picture T we report the index of the cell χi in which the vector xTk uTk is contained).
6
Conclusions
In this paper we derived LMIs-based procedures to solve H∞ analysis and synthesis problems for PWA systems whose switching sequence depends on the state and on the control input. These PWA systems can be found by translating an MLD system into PWA form. The analysis tests can be applied to assess the performance of MPC control schemes applied both to linear and hybrid systems. Moreover, the state-feedback design methodologies provide an alternative way to synthesize controllers with a prescribed degree of performance. All the proposed synthesis procedures are clearly only sufficient i.e. nothing can be said if the LMIs are infeasible. A thorough analysis of their conservativeness will be subject of further investigations.
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