H∞ and Generalized H2 Estimation of Continuous-time Piecewise Linear Systems Jun Xu ∗ and Lihua Xie∗† and Yeng Chai Soh∗ ∗ BLK S2, School of Electrical and Electronic Engineering Nanyang Technological University, Nanyang Avenue, Singapore 639798 † Email:
[email protected] Abstract This paper is concerned with the H∞ and generalized H2 estimation problems for a class of continuoustime piecewise linear systems. We present a linear matrix inequality (LMI) based approach to the design of generalized H2 and H∞ estimators. Our approach employs S-procedure and partition-dependent slack variables to reduce design conservatism. Examples demonstrate the advantages and good estimation performance of our design.
1 Introduction In this paper, we investigate the estimation problems of continuous-time piecewise linear (PWL) systems. The PWL systems have the characteristic feature that they switch between different state evolution maps, depending on whether the state belongs to a specified subspace. One typical switching engineering system is an electrical circuit with many relay components [4]. Other examples may exist in power systems, air conditioners and communication networks [11, 10]. More generally, PWL systems can be used to approximate general nonlinear systems which lack systematic methods for analysis and design. PWL systems have attracted a lot of attention recently; see e.g. [13, 7, 12, 5]. In [5], Feng proposed a novel synthesis method for an observer-based output feedback controller design. Rodrigues [13] addressed the problem of how to avoid sliding modes in control design for PWL systems. These works are based on piecewise quadratic Lyapunov functions and LMIs [3]. The observer or estimator design problem for linear systems has a long history and is now a well established area. However, there have been few existing works for linear PWL systems. In [1], Alessadri presented a Lur’e observer for both discrete-time and continuous-time systems and a design based on a projection method to minimize the estimation error was given in [2]. In [9, 8], Juloski et al. introduced a design procedure to the Lur’e observer, which does not require information on the currently active dynamics of the PWL system. Also partition information is included there to alleviate the design conservatism similar to the works of [7, 12] where the S-procedure is introduced for PWL systems. In this paper, we investigate the H∞ and H2 estimations for PWL systems. For the sake of simplicity of presentation, we only consider a simple case where there are only two modes or subspaces although the proposed approach can be extended to more general cases. Our solutions to both estimation problems are
given in terms of LMIs. Hence, the H∞ and the generalized H2 estimators can be obtained via convex optimization. In our design, we apply the S-procedure and introduce partition-dependent slack variables which incorporate partition information of the system for alleviating the design conservatism. The rest of the paper is organized as follows. In Section 2, we describe the systems and state the problems under investigation. In Section 3, we present an H∞ analysis result and apply it to the H∞ estimator design. Section 4 extends the work of H∞ estimation to the generalized H2 estimation problem. Examples are given to illustrate the proposed design in Section 6. Some conclusions are drawn in Section 7. For convenience, we introduce the following notations: A > 0 (A < 0) means that A is positive definite (negative definite). A 0 implies that A is copositive, i.e. every element of the matrix is non-negative. Also, ∞ e 2 = eT e and e 22 = 0 eT e dt.
2 Problem Statement Consider the following continuous-time PWL system: A1 x + B1 w if H T x ≤ 0 x˙ = (1) A2 x + B2 w if H T x > 0 y z
= Cx + Dw = Ex
(2) (3)
where x ∈ Rn is the state vector with initial state x0 , y ∈ Rm is the measurement, w ∈ Rr is the noise, and z ∈ Rp is the signal to be estimated. We assume that all matrices mentioned in this paper are appropriately dimensioned. We consider the following observer-based estimator for the above PWL system: A1 x ˆ + L1 (y − C x ˆ) if H T xˆ ≤ 0 ˙xˆ = (4) A2 x ˆ + L2 (y − C x ˆ) if H T xˆ > 0 zˆ = E x ˆ n
(5) p
where x ˆ ∈ R is the estimator state and zˆ ∈ R is an estimate of z, and L1 and L2 are estimator gains to be determined. The initial state of the estimator is denoted by x ˆ0 . Let I and J be the index sets of the partitions in the system (1) and the estimator (4), respectively. It is clear that I = {1, 2} and J = {1, 2} and we have the
following combinations for the combined dynamics of the system and the estimator: ⎧ Case 1 : H T x ≤ 0 ∩ H T xˆ ≤ 0 ⎪ ⎨ Case 2 : H T x ≤ 0 ∩ H T xˆ > 0 (6) T T ⎪ ⎩ Case 3 : H T x > 0 ∩ H T xˆ ≤ 0 Case 4 : H x > 0 ∩ H xˆ > 0
Remark 2 Note that an observer design problem for system (1)-(2) has been considered in [8, 9]. In [8], an observer that guarantees an asymptotic stability of error dynamics has been derived when the system has continuous dynamics over the switching plane. The case of discontinuous dynamics has also been investigated. A particle filtering approach is studied in [9].
Denote the above four combinations by the set K = {1, 2, 3, 4}. Then, we have the set of all possible (k, i, j) as
3 H∞ Estimation
∆
S = {(1, 1, 1), (2, 1, 2), (3, 2, 1), (4, 2, 2)} ⊆ K × I × J (7) which means that, for a given k ∈ K, there is a corresponding pair (i, j) ∈ I ×J . For example, when k = 1, (i, j) = (1, 1). ∆
∆
Define e = x − x ˆ and ξ = [xT eT ]T . Thus, it follows from (1)-(6) that the estimation error system can be given by ˜k w, f or (k, i, j) ∈ S (Σ) : ξ˙ = A˜k ξ + B
(8)
˜ z − zˆ = Eξ
(9)
where: A˜k =
0 Ai Bi ˜ ∆Aij Aj − Lj C , Bk = Bi − Lj D , ˜ = [0 E] ∆Aij = Ai − Aj , E (10)
Remark 1 We note that for Case k (k = 1, 2, 3, 4), the switching plane of the system (8) is given by ˜ kT ξ ≤ 0, k ∈ K H where: ˜ 1T = H ˜ 3T = H
T
H HT −H T HT
0 −H T 0 −H T
(11)
T
H 0 , −H T H T
−H T 0 T ˜ , H4 = −H T H T ˜ 2T = ,H
The problems under consideration are stated as follows: H∞ estimation problem: Consider the linear PWL system (1)-(3). Given a scalar γ > 0, design an estimator of the form (4)-(5) such that the error system (8)-(9) is stable and ˆ0 ) z − zˆ 22 < γ 2 w 22 +v(x0 , x
(12)
for any non-zero w and some function v(·, ·) ≥ 0. Generalized H2 estimation problem: Assume ˆ0 = 0. Given a positive scalar τ > 0, that x0 = 0 and x our objective is to design an estimator of the form (4)(5) such that the error system (8)-(9) is stable and the generalized H2 norm of the system defined as: tf ∆ Σ2 = sup z(tf ) − zˆ(tf ) 2 | wT wdt ≤ 1 0
is less than τ for any tf > 0.
(13)
In this section, we consider the H∞ estimation problem for the system (1)-(3). We shall develop an H∞ estimator design method in terms of a linear matrix inequality. We shall first give an analysis result. Theorem 1 Consider the PWL system defined by (1)(3). Given a scalar γ > 0, the estimator (4)-(5) solves the H∞ estimation problem if there exists a solution (P > 0, Uk 0, k ∈ K) such that the following inequalities are satisfied:
˜k + A ˜Tk P + E ˜T E ˜ +H ˜ k Uk H ˜ kT PA ˜kT P B
˜k PB −γ 2 I
< 0, ∀k ∈ K (14)
Proof: We define a Lyapunov function candidate as follows: V (ξ) = ξ T P ξ (15) Then, along the state trajectory of the system (8),
˜k P A˜k + A˜Tk P P B V˙ = ζ T ζ, k ∈ K (16) ˜T P B 0 k where ζ = [ξ T wT ]T . The existence of (P > 0, Uk 0, k ∈ K) to (14) implies that ˜ +H ˜ k Uk H ˜ kT < 0 ˜T E P A˜k + A˜Tk P + E which implies the stability of the system (8) [7]. Next, (14) can be rewritten as:
˜k ˜T ˜ P A˜k + A˜Tk P P B + E E T ˜ 0 B P 0 k
˜T ˜ + Hk U k H k 0 < 0 0 0
0 −γ 2 I
(17)
Multiplying (17) by ζ T and ζ from the left and the right, respectively, we have:
˜ k Uk H ˜Tξ < 0 V˙ + z − zˆ 2 −γ 2 w 2 ≤ −ξ T H k (18) By integrating (18) from 0 to tf , we obtain tf V (ξ(tf )) − V (ξ(0)) + ( z − zˆ 2 −γ 2 w 2 )dt < 0 0
(19) for any tf > 0. Since V (ξ(tf )) ≥ 0, this implies that z − zˆ 22 < γ 2 w 22 +V (x0 , x ˆ0 )
(20)
Remark 3 In order to reduce the design conservatism, in [9] S-procedure is applied to the partition (11) for the design of observer by considering:
Remark 4 Note that Corollary 2 requires that the system (1)-(2) be stable with a fixed Lyapunov matrix P1 satisfying ([7])
¯ T ξ ≥ 0, k ∈ K ξT H k
˜ H T < 0, ∀i ∈ I ATi P1 + P1 Ai + H U k
(21)
˜ (1) 0. where U k
where ¯ 1T = H ¯ 4T = H
¯T = H ¯T = H 2 3
HH T − 12 HH T
−HH T 1 T 2 HH
− 21 HH T 0 1 T 2 HH
,
The above result is in general conservative due to the requirements that P be block-diagonal and fixed for all the four partitions in K. In the following, we shall focus on how this conservatism can be alleviated. To this end, we first present a technical lemma.
0
In Theorem 1, we incorporated the partition informa˜ k Uk H ˜T tion of the system (8) by introducing the term H k ¯ k of (21) in (14). It can be easily known that H ˜ T . In fact, by letting ˜ k Uk H is a special case of H k (1)
Uk =
H 0
Uk (2)T Uk 0 ˜k U H
(2)
Uk ˜T = ˜ k Uk H 0, we have H k (3) Uk H 0 T ≥ 0, where for (k, i, j) ∈ S 0 H
˜ (2) ˜ (1) U U k k ˜ Uk = = T ˜ (2) ˜ (3) U U k k (1) (2) (2)T (3) Uk + (−1)i+j (Uk + Uk ) + Uk T (2) (3) −(−1)i+j Uk − Uk
∗
(3)
Uk
(22)
Hence, the result of [9] can be obtained by setting γ → (1) (3) (2) ∞, Uk = Uk = 0 and Uk = 12 λi I, k, i ∈ S. A simple result for the H∞ estimator design can be a structurally constrained P , i.e. P = obtained by P1 0 0 P . The result is given in the corollary below. 2
Corollary 2 Consider the system defined by (1)-(3). Given a scalar γ > 0, the H∞ estimation problem is solvable if there exits a solution (P1 > 0, P2 > 0, Uk 0, Wj , k ∈ K, j ∈ J ) to the following LMIs for ∀(k, i, j) ∈ S:
˜ (1) H T AT P1 + P1 Ai + H U i k ˜ (2)T H T + P ∆A HU 2 k k B T P1 i
∗
∗
Θk B T P2 − DT Wj i
−γ 2 I
∗
(1)
< 0
(23)
where Θk = ATj P2 + P2 Aj − C T Wj − WjT C + E T E + (3)
HUk H T . In this situation, the estimator gains are given by Lj = P2−T WjT , j ∈ J Proof: The result can be easily established by substituting P = diag{P1 , P2 } into (14) and letting Wj = LTj P2 , j = 1, 2. Clearly, (23) is linear in P, Uk , Wj . Hence, they can be solved by a convex optimization.
Lemma 1 There exists a matrix P > 0 to (14) if and only if for some positive scalar ε, there exists a solution (Gk , k ∈ K, P = P T > 0) to the following matrix inequality T ˜ ˜T ˜ ˜T ˜T ˜ G Ak + A Gk + E E + Hk Uk H k k k ˜T G B k k ˜ Gk − P − εGT A k k
∗
∗
−γ 2 I ˜ −εGT B k k
−ε
∗
Gk + GT k
< 0 (24)
where no constraint on the matrices Gk has been imposed. Proof: ” =⇒ ”: By setting Gk = GTk = P and letting ε → 0, (24) follows from (14). ” ⇐= ”: Note that Gk is invertible since Gk + GTk > 0. I 0 0 ∆ 0 I 0 . Multiplying (24) by Define Γ = ˜k I −A˜k −B ΓT from the left and Γ from the right, respectively, we have ˜ ˜T ˜T ˜ ˜ ˜T P Ak + A P + E E + Hk Uk H k k ˜T P B k ˜ Gk − P + εGk Ak
∗
∗
−γ 2 I ˜ εGk B k
−ε
∗
Gk + GT k
< 0 (25)
which implies that (14) holds. The above result shows the equivalence between (14) and (24). This indicates that (24) has no advantage over (14) in terms of H∞ performance analysis of a given estimator. It should, however, be noted that (24) will provide a less conservative design of an estimator due to: 1) no structural constraint may be required on the Lyapunov matrix P as in Corollary 2 when applying (24) to the estimator design; 2) the introduction of the partition-dependent slack variables Gk , k ∈ K. In order to apply Lemma 1 to the design of estimator, in (24) we let
(1) (2) P1 P2 Gk Gk P = , k ∈ K, , Gk = (3) P2T P3 0 Gk (26) (3) (3) (3) (3) where G1 = G3 , G2 = G4 . Since Gk is invertible, (3) so is Gk . By substituting (26) into (24) and invoking (10), we have the following result. Theorem 3 Consider the system defined by (1)-(3). Given a scalar γ > 0, there exists an estimator (4)-(5) that solves the H∞ estimation problem if for some ε >
0, there exists a solution (P > 0, Gk , Uk 0, Wj , k ∈ K) to the following LMIs: ⎡ ⎢ ⎢ ⎣
(1) (1)T ˜ (1) H T AT G +G Ai + H U i k k k ˜ (2)T H T + G(2)T A + G(3)T ∆A HU i ij k k k (1) BT G i k (1) (1)T G − P1 − εG Ai k k (2)T (3)T T −P − εG Ai − εG ∆Aij 2 k k
∗ Θk (2) (3) BT G + BT G − DT Wj i i k k (2)T G − P2 k (3) (3)T G − P3 − ε(G Aj − W T C) j k k
(1) (1)T −ε(G +G ) k k (2)T −εG k
G Ak + A Gk + Hk Uk H k k k ˜T G B k k ˜ Gk − P − εGT A k k
∗
⎤ ⎦ 0, the estimator (4)-(5) solves the generalized H2 estimation problem if there exits a solution (P > 0, Uk 0, k ∈ K) to the following matrix inequality 1 ˜≥0 (28) P − E˜ T E τ
˜T PB ˜ k Uk H ˜k P A˜k + A˜Tk P + H k < 0, ∀k ∈ K ˜T P B −I k (29) Similar to Lemma 1, we have:
−ε
∗
Gk + GT k
< 0
τ
(31)
subject to (28) and the following constraint: ⎡ (1)T T (1) ˜ (1) T
Aj − C T Wj − WjT C + E T E + HUk H T .
Lj = Gj
˜ −εGT B k k
Theorem 5 Consider the system defined by (1)-(3). The optimal generalized H2 estimator can be obtained by the following optimization
(27)
for all k ∈ K, where ∆Aij = Ai − Aj , Θk = ATj Gk + Gk
∗
We choose Gk with the same structure as in (26). Then, the following result follows.
(1)T −εG Bi k (2)T (3)T −εG Bi − ε(G Bi − W T D) j k k ∗ ∗ ∗
∗
−I
(30)
∗ ∗ −γ 2 I
∗ ∗ ∗
Lemma 2 There exists a matrix P = P T to (29) if and only if for some positive scalar ε, there exists a solution (Gk , P = P T > 0, k ∈ K) to (30) for ∀k ∈ K: T ˜ ˜T ˜ ˜T
A G +G Ai + H U H i k k k ˜ (2)T H T + G(2)T A + G(3)T ∆A HU i ij k k k (1) BT G i k (1) (1)T G − P1 − εG Ai k k (2)T (3)T −P T − εG Ai − εG ∆Aij 2 k k
∗ Θk (2) (3) BT G + BT G − DT Wj i i k k (2)T G − P2 k (3) (3)T G − P3 − ε(G Aj − W T C) j k k
∗ ∗ −I (1)T −εG Bi k (2)T (3)T −εG Bi − ε(G Bi − W T D) j k k
∗ ∗ ∗
(1) (1)T −ε(G +G ) k k (2)T −εG k
∗ ∗ ∗ ∗
⎤ ⎦ 0 and Uk 0, Θk = ATj Gk +
(3)T Gk Aj
− C T Wj − WjT C +
(3) HUk H T
k ∈ K.
In this situation, suitable estimator gains can be chosen as (3)−T WjT , j ∈ J Lj = Gj Remark 7 Theorem 5 presents a solution to the generalized H2 estimation for switching systems. Similar to the H∞ case, the optimal solution involves a onedimensional search over the scaling parameter ε which can be carried out by employing the fminsearch algorithm from the Matlab Optimization Toolbox.
5 Sliding Mode Analysis Sliding mode may occur in the bi-modal system (1)-(2) and/or the observer (4) along their switching planes. For the system (1)-(2) (respectively, the observer (4)), ˆ= the switching plane is H T x = 0 (respectively, H T x 0). Following the discussions in [9, 6], we shall show that the H∞ or generalized H2 performance is guaranteed even when the sliding mode occurs. There are three cases of sliding mode. The first is that the sliding mode occurs in the system (1)-(2) itself. In this situation, consider the convex combination of the constituting linear dynamic: x˙ = µ(A1 x + B1 w) + (1 − µ)(A2 x + B2 w), 1 ≥ µ ≥ 0 y = Cx + Dw (33)
If the observed state x ˆ is in the mode H T x ˆ ≤ 0, then the error dynamic is given by:
6 Examples
e˙ = x˙ − x ˆ˙ = µ[(A1 − L1 C)e + (B1 − L1 D)w]+ (1 − µ)[(A2 − A1 )x + (A1 − L1 C)e + (B2 − L1 D)w] (34) Thus for ξ = [xT eT ]T we have
Example 1 Consider the system (1)-(3) with the following parameters
˜1 w) + (1 − µ)(A˜3 ξ + B ˜3 w) ξ˙ = µ(A˜1 ξ + B
(35)
which is the convex combination of the Case 1 and Case 3 in (6). On the other hand, if the observed state x ˆ is in the mode H T x ˆ > 0, we have: ˜2 w) + (1 − µ)(A˜4 ξ + B ˜4 w) ξ˙ = µ(A˜2 ξ + B
(36)
which is the convex combination of Case 2 and Case 4 of (6). Note that (35) and (36) can be represented by ˜1 + (1 − µ)A ˜3 ]ξ + [µB ˜1 + (1 − µ)B ˜3 ]w, H T x [µA ˆ≤0 ξ˙ =
˜4 ]ξ + [µB ˜2 + (1 − µ)B ˜4 ]w, H T x ˜2 + (1 − µ)A ˆ>0 [µA (37)
Denote: Ξk =
˜k + A ˜Tk P + E ˜T E ˜ +H ˜ k Uk H ˜ kT PA ˜kT P B
˜k PB −γ 2 I
, k∈K (38)
If Theorem 1 holds, we have: µΞ1 + (1 − µ)Ξ3 < 0, H T x ˆ≤0 ˆ>0 µΞ2 + (1 − µ)Ξ4 < 0, H T x
(39)
−1 −0.02 0.02 −1
, A2 =
−1 0.02 −0.02 −0.3
,
10 0 1 0 1 0 C= 0 1 , D= 0 , H= 1 , E=I B1 =
0 10
, B2 =
Using Corollary 2, we obtain the optimal H∞ performance γ = 0.01848. Using Theorem 3, by employing the fminsearch, we obtain the optimal H∞ performance γ = 0.01785 when ε = 1.10e − 4. The optimal estimator gains are L1 =
8.192 1.621
−4.935e4 4.973e4
, L2 =
9.780 0.213
−1.481e4 1.483e4
To simulate the tracking performance, we let the input be a white noise with unit variance. The simulation results with initial state [−5 − 1]T for the actual system state and its estimated state are shown in Figures 1 and 2 where the estimated and the actual states overlap with each other. system and observer response for x1 15
system state estimation state
10
which guarantee the H∞ performance γ of the system (35) or (36). Therefore, if Corollary 2 or Theorem 3 holds, the designed estimator also guarantees the corresponding performance in the case when there exists a sliding mode in the system (1)-(2). A similar argument follows for the generalized H2 performance. The second situation is that the sliding mode occurs in the observer (4). The dynamic of the observer is given by : ˆ)] + (1 − λ)[A2 x + L2 (y − C x ˆ)] x ˆ˙ = λ[A1 x + L1 (y − C x 0≤λ≤1
A1 =
5
0
−5
−10
−15
−20
0
5
10
15
20
25
30
Figure 1: Actual state x1 and its estimate xˆ1 via the H∞ estimator.
(40)
Similar to the discussion in the first situation, we can also conclude that the H∞ or generalized H2 performance is ensured in this situation when the conditions in Theorem 3 or Theorem 5 are satisfied. The third situation is that the sliding mode occurs in both the system and the observer. The corresponding dynamic is given by:
system and observer response for x2 2
system state estimation state
0
−2
−4
−6
e˙ = (µ − λ)[(A1 − A2 )x + (B1 − B2 )w] + λ[(A1 − L1 C)e +(B1 − L1 D)w] + (1 − λ)[(A1 − L1 C)e + (B2 − L1 D)w] (41)
If λ − µ > 0, from (41), we have:
˜1 w)+(λ−µ)(A˜3 ξ+ B ˜ 3 w)+(1−λ)(A˜4 ξ+ B ˜4 w) ˜1 ξ+ B ξ˙ = µ(A (42)
If λ − µ ≤ 0, we have:
˜ 1 w)+(µ−λ)(A˜2 ξ+ B ˜2 w)+(1−µ)(A˜4 ξ+ B ˜4 w) ˜1 ξ+ B ξ˙ = λ(A (43)
It is obvious that the above two equations are also convex combinations of (8). Thus, the results of Theorem 3 or Theorem 5 guarantee the corresponding performances.
−8
−10
−12
0
5
10
15
20
25
30
Figure 2: Actual state x2 and its estimate xˆ2 via the H∞ estimator.
The suboptimal generalized H2 performance of the estimator is τ = 1.7668e − 3 with ε = 0.9875. Under the white noise with unit variance, the simulation results are shown in Figures 3 and 4.
system and observer response for x1 15
type of estimator has been proposed to achieve the H∞ and generalized H2 performances, respectively. A less conservative design is achieved by applying the Sprocedure and partition-dependent slack variables in optimization. Our results are given in terms of LMIs, which can be easily solved using convex optimization.
system state estimation state
10
5
0
−5
−10
References
−15
−20
0
5
10
15
20
25
30
Figure 3: Actual state x1 and its estimate xˆ1 via the generalized H2 estimator. system and observer response for x2 5
system state estimation state
0
−5
−10
−15
−20
0
5
10
15
20
25
30
Figure 4: Actual state x2 and its estimate xˆ2 via the generalized H2 estimator.
Example 2 Consider the system (1)-(3) with the following parameters −1 −0.02 −1 0.02 , A = A1 = 0.02 2 −1 −0.02 −0.3 ,
0 0.5 1 ], D = [ 1 ], H = 0 1 , E=I
B1 = C=[ 1
0.5 1
, B2 =
By applying Theorem 3, we obtain the optimal H∞ performance of γ = 0.212 with ε = 0.650. The corresponding estimator gains are 0.247 0.249 L1 = 0.720 , L2 = 0.756 Note that we can get the optimal performance γ = 0.236 using Corollary 2. Clearly, Theorem 3 gives a less conservative design. As for the generalized H2 esimation, with ε = 0.0012008, we get the optimal performance τ = 0.0635. The corresponding estimator gains are 0.245 0.250 L1 = 0.736 , L2 = 0.753
7 Conclusion In this paper, we have considered the H∞ and generalized H2 state estimation problems for a class of continuous-time piecewise linear systems. A Lur’e
[1] A. Alessandri and P. Colleta, “Design of luenberger observers for a class of hybrid linear systems,” in Proc. of Hybrid Systems: Computation and Control, Rome, Italy, 6 2001. [2] ——, “Switching observers for continuous-time and discrete-time linear systems,” in Proc. of the American Control Conference, Arlington, VA, 6 2001, pp. 2516–2521. [3] S. Boyd, L. E. Ghaoui, E. Feron, and V. Balakrishna, Linear Matrix Inequalities in System and Control Theory. SIAM, 1994. [4] L. O. Chua, C. A. Desoer, and E. S. Kuh, Linear and Nonlinear Circuits. McGraw-Hill, New-York, 1987. [5] G. Feng, “Output feedback controller design of piecewise discrete time linear systems,” IEEE trans. on Circuit and Systems I, vol. 50, no. 3, 2003. [6] A. F. Filippov, “Differential equations with discontinuous righthand sides,” Mathematics and its Applications, 1988, kluwer, Dordrecht, Netherlands. [7] M. Johansson, “Piecewise linear control systems,” PhD thesis, Lund Institute of Technology, Sweden, 1999. [8] A. L. Juloski, W. P. M. H. Heemels, Y. Boers, and F. Verschure, “Two approaches to state estimation for a class of piecewise affine systems,” in Proc. 42st IEEE Conference on Decision and control, 2003. [9] A. L. Juloski, W. P. M. H. Heemels, and S. Weiland, “Observer design for a class of piece-wise affine system,” in Proc. 41st IEEE Conference on Decision and control, Las Vegas, USA, 12 2002, pp. 2602– 2611. [10] D. Liberzon and A. S. Morse, “Basic problems in stability and design of switched systems,” IEEE Control Magazine, no. 5, pp. 59–70, Oct 1999. [11] A. S. Morse, Ed., Control Using logic-based Switching. London:Springer-verlag, 1997. [12] A. Rantzer and M. Johansson, “Piecewise linear quadratic optimal control,” IEEE trans. on Automatic Control, vol. 45, pp. 629–637, April 2000. [13] L. Rodrigues, “Dynamic output feedback controller synthesis for piecewise-affine systems,” PhD thesis, Stanford University, 2002.
