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Robust pole placement controller design in LMI region for uncertain and disturbed switched systems
Nonlinear Analysis: Hybrid Systems 2 (2008) 1136–1143
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Nonlinear Analysis: Hybrid Systems journal homepage: www.elsevier.com/locate/nahs
Robust pole placement controller design in LMI region for uncertain and disturbed switched systems B. Mansouri a , N. Manamanni a,∗ , K. Guelton a , M. Djemai b a
CReSTIC, EA3804, University of Reims Champagne Ardenne, Moulin de la House BP1039, 51687 Reims Cedex 2, France
b
LAMIH, UMR CNRS 8530, University of Valenciennes, 59313 Valenciennes, Cedex 9, France
article
info
Article history: Received 6 December 2007 Accepted 4 September 2008 Keywords: Hybrid dynamical systems Linear switched systems Switched control Quadratic Lyapunov function LMI (Linear Matrix Inequalities) H∞ control
a b s t r a c t This paper concerns the state feedback control for continuous-time, disturbed and uncertain linear switched systems with arbitrary switching rules. The main result of this work consists in getting a LMI (Linear Matrix Inequalities) condition guaranteeing a robust pole placement according to some desired specifications. Then, external disturbance attenuation with a fixed rate according to the H∞ criterion is ensured. This is obtained thanks to the existence of a common quadratic Lyapunov function for all sub-systems. Finally, an academic example illustrates the efficiency of the developed approach. © 2008 Elsevier Ltd. All rights reserved.
1. Introduction Hybrid dynamic systems are defined as a set of continuous and discrete sub-systems interacting together. Thus, they associate continuous and discrete dynamic, as well as continuous and discrete control [2,15,17]. The hybrid dynamic systems (HDS) have many varied applications; one finds them in the control of mechanical systems, motorized industry, aeronautics, power electric converters and robotics. The main difficulty in the definition of HDS is that the hybrid term is not restrictive; the interpretation of this term can be extended to any dynamic system. A standard and reasonable definition of a HDS would be to consider the structure only to indicate the objective to be reached and the fixed terminology. So, in this paper, the work will restricted to a particular class of HDS, namely the switched systems, which constitute a set of continuous sub-systems and a switching rule that orchestrates the commutation between them [5,7,11,16,20]. The various studies dedicated to switched systems consider mainly the stability problem. Indeed, three basic problems involved in the stability of the switched system were raised by Liberzon [7] and then developed in the literature [14,17]. One of these problems is to find the conditions that guarantee the system’s asymptotic stability under any switching rule. i.e. arbitrary switching sequence. A necessary condition for asymptotic stability under arbitrary switching sequence is that each sub-system must be asymptotically stable. The authors showed in [17] that this condition is not sufficient, since, it is possible to lead to instability for some classes of switching signals. Thus, to solve this problem, it has been shown in [8] that the existence of a common Lyapunov function is a necessary and sufficient condition. This approach is also reported in several works, see [11–13,18,19,1]. This paper deals primarily with the last problem described above. It concerns the class of continuous-time uncertain, linear switched systems without jump at the switching instant. It is also assumed that the number of switching is finite in
∗
Corresponding author. Tel.: +33 3 26 91 83 86; fax: +33 3 26 91 31 06. E-mail addresses: [email protected] (B. Mansouri), [email protected] (N. Manamanni), [email protected] (K. Guelton), [email protected] (M. Djemai). 1751-570X/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.nahs.2008.09.010
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a finite time (non-zenon). In [10], a quasi-LMI condition ensuring the design of a control law for a disturbed linear switched system is proposed. In this paper, the robustness of the approach has been improved by considering an uncertain system. In fact, in this case it is not obvious to get a strict LMI condition for the asymptotic stability. Thus, the goal of this paper is to provide a strict LMI condition for the synthesis of a switched state feedback control law that guarantees asymptotic stability of the closed-loop system for any arbitrary switching rule and ensures three different performances at the same time. The first is the asymptotic stability with some guaranteed specifications, namely; the transient response and the damping factor [6] by the pole location of each closed-loop linear sub-system. The second performance is to attenuate the disturbance according to the H∞ criterion. Finally, the main contribution of this paper is to obtain state feedback gains thanks to the LMI formulation by considering an uncertain structure of the switched system. A numerical example will illustrate the proposed approach. 2. Uncertain switched systems Let us consider the following uncertain switched system:
x˙ (t ) = Aσ (t ) (t ) x (t ) + B1σ (t ) (t ) ϕ (t ) + B2σ (t ) (t ) u (t ) y (t ) = C σ (t ) (t ) x (t ) + D1σ (t ) (t ) ϕ (t ) + D2σ (t ) (t ) u (t )
(1)
with Aσ (t ) (t ) = Aσ (t ) + ∆Aσ (t ) (t ), B1σ (t ) (t ) = B1σ (t ) + ∆B1σ (t ) (t ), B2σ (t ) (t ) = B2σ (t ) + ∆B2σ (t ) (t ), C σ (t ) (t ) = Cσ (t ) + ∆Cσ (t ) (t ), D1σ (t ) (t ) = D1σ (t ) + ∆D1σ (t ) (t ), D2σ (t ) (t ) = D2σ (t ) + ∆D2σ (t ) (t ) and where σ (t ) is the switching rule defined as follows: Let I = {1, 2, . . . , N } an index’s compact set of the sub-systems, σ (t ) is defined by the mapping: σ (t ) : R+ → I, i.e. the linear mode (A, B1 , B2 , C , D1 , D2 )l , is active if σ (t ) = l with l ∈ I. x (t ) ∈ Rn is the state vector, u (t ) ∈ Rm is the control input vector, ϕ (t ) ∈ Rr is an exogenous input, Aσ (t ) ∈ Rn×n , B1σ (t ) ∈ Rn×r , B2σ (t ) ∈ Rn×m , D1σ (t ) ∈ Rp×r , D2σ (t ) ∈ Rp×m and Cσ (t ) ∈ Rp×n are the sub-system matrices. ∆Aσ (t ) (t ) , ∆B1σ (t ) (t ), ∆B2σ (t ) (t ), ∆D1σ (t ) (t ), ∆D2σ (t ) (t ) and ∆Cσ (t ) (t ) contain all the modelling uncertainties, which can be represented by:
∆Aσ (t ) (t ) = Haσ (t ) ∆aσ (t ) (t ) Naσ (t ) , ∆B1σ (t ) (t ) = H1bσ (t ) ∆1bσ (t ) (t ) N1bσ (t ) , ∆B2σ (t ) (t ) = H2bσ (t ) ∆2bσ (t )σ (t ) (t ) N2bσ (t ) , ∆D1σ (t ) (t ) = H1dσ (t ) ∆1dσ (t ) (t ) N1dσ (t ) , ∆D2σ (t ) (t ) = H2dσ (t ) ∆2dσ (t ) (t ) N2dσ (t ) and ∆Cσ (t ) (t ) = Hc σ (t ) ∆c σ (t ) (t ) Nc σ (t ) where Haσ , H1bσ , H2bσ , H1dσ , H2dσ , Hc σ , Naσ , N1bσ , N2bσ , N1dσ , N2dσ and Nc σ are constant matrices and, ∆aσ (t ) (t ), ∆1bσ (t ) (t ), ∆2bσ (t ) (t ), ∆1dσ (t ) (t ), ∆2dσ (t ) (t ) and ∆c σ (t ) (t ) are the normalized uncertainty matrices, verifying the following conditions:
∆Taσ (t ) (t ) ∆aσ (t ) (t ) ≤ I , ∆T1di (t ) ∆1di (t ) ≤ I ,
∆T1bσ (t ) (t ) ∆1bσ (t ) (t ) ≤ I ,
∆T2bσ (t ) (t ) ∆2bσ (t ) (t ) ≤ I ,
∆T2dσ (t ) (t ) ∆2dσ (t ) (t ) ≤ I and ∆Tc σ (t ) (t ) ∆c σ (t ) (t ) ≤ I .
In this study, we assume that the state vector is available to perform a state feedback and that the discrete control law σ (t ) is not a priori known but is available in real time. Now, we consider the following state feedback control law: u (t ) = Kσ (t ) x (t ) .
(2)
From (1) and (2) the closed-loop switched system can be written as follows:
x˙ (t ) = A˜ σ (t ) x (t ) + B1σ (t ) ϕ (t ) y (t ) = C˜ σ (t ) x (t ) + D1σ (t ) ϕ (t )
(3)
where: A˜ σ (t ) = Aσ (t ) + B2σ (t ) Kσ C˜ σ (t ) = C σ (t ) + D2σ (t ) Kσ .
(4)
The objective is then to determine the state feedback gains Kσ ensuring the stability of the closed-loop switched system (3). 3. Robust pole placement and H∞ performances In the first step, one determines the state feedback gains with pole placement of each closed-loop uncertain linear sub-system, such that the eigenvalues of A˜ σ (t ) are located inside a disk-pole with centre at (− (rσ + dσ ) , 0) with the radius rσ and the distance dσ from the imaginary axis as set in Fig. 1. Using the parameter dσ , it is possible to determine an upper bound for the settling time of the transient response given by 3τσ or 5τσ for each sub-system σ , with τσ = 1/dσ . The value rσ gives the upper bound on the natural frequency of oscillation
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Fig. 1. Pole placement for the sub-system σ .
for the transient response. Notice also that a lower bound on the dumping factor ξσ , which determines the overshoot, can be computed as follows [10]:
p (rσ + dσ )2 − rσ2 ξσ = . rσ + dσ
(5)
Using the well known common quadratic Lyapunov function: V (x) = xT Px,
P = PT > 0
(6)
a necessary and sufficient condition such that all the eigenvalues of A˜ σ (t ) of system (3) lie inside the circular region ς (dσ , rσ ) as depicted in Fig. 1, is given by the existence of a positive definite symmetric matrix W = P −1 such that the following inequality holds [3]:
A˜ Tσ (t ) + dσ I W + W A˜ σ (t ) + dσ I
T
+
T 1 A˜ σ (t ) + dσ I W A˜ σ (t ) + dσ I < 0, rσ
σ = 1, 2, . . . , N .
(7)
On the other hand, in the second step, one needs to attenuate the disturbances ϕ (t ) with a minimum rate η, according to the H∞ criterion:
ky (t )k2 ≤ η kϕ (t )k2 .
(8)
A sufficient condition ensuring the H∞ criterion (8) is given by the following matrix inequality [4]: A˜ σ (t ) W + W A˜ Tσ (t ) BT1σ (t ) ˜Cσ (t ) W
(∗) −I D1σ (t )
(∗) (∗) < 0, −η2 I
σ = 1, 2, . . . , N .
(9)
Finally, to achieve both the performances at the same time; i.e. disturbance’s attenuation according to the H∞ criterion and pole placements (inside the disk-pole ς (dσ , rσ )) leading to a condition that verifies both (7) and (9), we use the following lemma [10]: Lemma 1. If there exists a symmetric positive definite matrix W such that: A˜ σ (t ) W + W A˜ Tσ (t ) + 2dσ W W A˜ Tσ (t ) + dσ W BT1σ (t ) ˜Cσ (t ) W
(∗) −r σ W 0 0
(∗)
(∗)
0 −I
0
D1σ (t )
< 0, (∗) −η2 I
σ = 1, 2, . . . , N
(10)
then, the stability of the closed-loop switched system (3) is ensured under arbitrary switching rule with guaranteed disturbance attenuation level η, defined in (8), and the pole location of each linear sub-system inside the disk-pole ς (dσ , rσ ) depicted in Fig. 1. Proof. Let us consider the following inequality:
A˜ σ (t ) W + W A˜ Tσ (t ) + 2dσ W 1 + A˜ σ (t ) + dσ I W A˜ σ (t ) + dσ I T rσ B˜ T1σ (t ) ˜Cσ (t ) W
(∗) −I D1σ (t )
(∗) < 0, (∗) −η2 I
σ = 1, 2, . . . , N .
(11)
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A necessary condition such that inequality (11) holds is that the first diagonal block of (11) must be negative definite, i.e. A˜ σ (t ) W + W A˜ Tσ (t ) + 2dσ W +
1 rσ
A˜ σ (t ) + dσ I W A˜ σ (t ) + dσ I
T
< 0.
(12)
That is equivalent to: 1
A˜ σ (t ) W + W A˜ Tσ (t ) < −2dσ W −
rσ
A˜ σ (t ) + dσ I W A˜ σ (t ) + dσ I
T
<0
leading to inequality (9) of the H∞ . Then, by applying the Shur complement in the block (1, 1) of inequality (11), inequality (10) is obtained. This ends the proof. Note that, inequality (11) given by Lemma 1 is not a LMI condition since, in our case, it contains varying time uncertainties which can be considered unknown with known upper bounds. Thus, in the following section, the goal is to get a LMI constraint by considering the uncertainty structure mentioned in the first section. 4. LMI formulation Due to the uncertainties, checking a LMI condition for the switched system, with performances stipulated above, is not trivial [9]. In order to obtain a LMI condition to synthesize the state feedback gains, one needs to use the following corollaries: Corollary 1 ([21]). For real matrices A, B and S = S T > 0 with appropriate dimensions and a positive constant τ , we have: AT B + BT A ≤ γ AT A + γ −1 BT B
(13)
X T Y + Y T X ≤ X T S −1 X + Y T SY .
(14)
and
Corollary 2. For real matrices A, B, W , Y , Z and a regular matrix Q with appropriate dimensions we have:
W T + BT AT Z
Y W + AB
<0⇔
Y + BT Q −1 B W
WT Z + AQAT
< 0.
(15)
Proof of corollary i2. For real matrices A, B, matrix Q with appropriate dimensions, the matrix h h W , YT,i Z and h a regular i Y W T + BT AT Y W 0 BT AT < 0 can be rewritten as + < 0. W + AB Z W Z AB 0 From inequality (13) we have:
0 B A
T
0 +
B 0
0
AT
0 M 0 A
≤
that leads to (15) and ends the proof.
AT +
T B 0
M −1 B
0
(16)
Now, let us make a bijective change of variable on the state feedback gains; Zσ = Kσ W . Then, the main result can be summarized by the following theorem: Theorem 1. If there exist a symmetric positive definite matrix W , matrices Zσ and positive constants τ1 , τ2 , τ3 , τ4 , τ5 , τ6 and τ7 such that:
Ξ Σ
(∗) Ψ
<0
with:
T Ξ = Aσ W + B2σ Zσ + WATσ + ZσT BT2σ + 2dσ W + τ1−1 + τ3−3 Haσ HaTσ + τ2−1 + τ4−1 H2bσ H2b σ, N1bσ N2bσ Zσ Nc σ W N2dσ Zσ Naσ W T T T Σ = WAσ + Zσ B2σ + dσ W 0 0 T B 1σ Cσ W + D2σ Zσ 0
(17)
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and
Ψ 1 ,1 Ψ =0
0
−r σ W Naσ W N2bσ Zσ 0 0 0
and where Ψ1,1 = diag −τ5
(∗)
(∗) −τ3−1
0
0 0 0 0
−τ4−1
0 0 0
0 0 0
D1σ 0
−τ2−1
−τ6
0 0 0
(∗) Ψ6,6
Ψ5 ,5
−τ7
−1 T T Ψ5,5 = −I + τ5 H1bσ H1b σ + τ8 N1dσ N1dσ ,
0 0 0 0
,
(∗) −τ8−1
T H1d σ
−τ1−1 , T Ψ6,6 = −η2 I + τ6 Hc σ HcTσ + τ7 H2dσ H2d σ
for σ = 1, 2, . . . , N. Then, the quadratic stability of the closed-loop switched system (3) is ensured under an arbitrary switching rule, with guaranteed disturbance attenuation level η and with the pole location of each linear sub-system inside the disk-pole ς (dσ , rσ ) depicted in Fig. 1. Proof. Starting from inequality (10) and separating the uncertainties, (10) becomes:
Π + ∆Π < 0
(18)
with:
(Aσ + B2σ Kσ ) W + (∗) + 2dσ W W (Aσ + B2σ Kσ )T + dσ W Π = BT1σ (Cσ + D2σ Kσ ) W
(∗) −r σ W
(∗)
0 0
−I
(∗)
0
0 (∗) −η2 I
D1σ
and
(∆Aσ (t ) + ∆B2σ (t ) Kσ ) W + (∗) W (∆Aσ (t ) + ∆B2σ (t ) Kσ )T ∆Π = ∆BT1σ (t ) (∆Cσ (t ) + ∆D2σ (t ) Kσ ) W
(∗)
(∗)
(∗)
0 0 0
0 0
0
(∗)
.
∆D1σ (t )
0
After the bijective change of variable Zσ = Kσ W , one obtains: Aσ W + (∗) + B2σ Zσ + (∗) + 2dσ W WATσ + ZσT BT2σ + dσ W Π = BT1σ Cσ W + D2σ Zσ
(∗) −rσ W
(∗)
0 0
−I
(∗)
0
0 (∗) −η2 I
D1σ
and
∆Aσ (t ) W + (∗) + ∆B2σ (t ) Zσ + (∗) W ∆ATσ (t ) + ZσT ∆BT2σ (t ) ∆Π = ∆BT1σ (t ) ∆Cσ (t ) W + ∆D2σ (t ) Zσ
(∗)
(∗)
(∗)
0 0 0
0 0
0
(∗)
.
∆D1σ (t )
0
Holding the uncertainty structure shown in Section 1, thus ∆Π is written as:
∆Π1,1 ∆Π2,1 ∆Π = ∆Π3,1 ∆Π4,1 ∆Π1,1 = Haσ ∆aσ
(∗)
(∗)
0 0 0
0 0
(∗)
0 , (∗)
H1d ∆1dσ (t ) N1dσ 0 (t ) Naσ W + (∗) + H2b ∆2bσ (t ) N2b Zσ + (∗) ,
T T T ∆Π2,1 = WNaTσ ∆Taσ (t ) HaTσ + ZσT N2b σ ∆2bσ (t ) H2bσ , ∆Π3,1 = H1bσ ∆1bσ (t ) N1bσ
and
∆Π4,1 = Hc σ ∆c σ (t ) Nc σ W + H2dσ ∆2dσ (t ) N2dσ Zσ .
B. Mansouri et al. / Nonlinear Analysis: Hybrid Systems 2 (2008) 1136–1143
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Fig. 2. State trajectories x1 and x2 .
Now, by the means of Corollaries 1 and 2, the matrix ∆Π containing the uncertainties can be bounded in order to find some scalar constants as the following:
∆Π < diag [Γ ] where: −1 −3 −1 T T T T Γ1,1 = τ1 WNaTσ Naσ W + τ1−1 Haσ HaTσ + τ2 ZσT N2b σ N2bσ Zσ + τ2 H2bσ H2bσ + τ3 Haσ Haσ + τ4 H2bσ H2bσ −1 −1 T T T T + τ5−1 N1b σ N1bσ + τ6 WNc σ Nc σ W + τ7 Zσ N2dσ N2dσ Zσ , T Γ2,2 = τ3 WNaTσ Naσ W + τ4 ZσT N2b N2b Zσ ,
−1 T T Γ3,3 = τ5 H1bσ H1b σ + τ8 N1dσ N1dσ
and T T Γ4,4 = τ6 Hc σ HcTσ + τ7 H2dσ H2d σ + τ8 H1dσ H1dσ .
Finally, by adding matrices Π and ∆Π , and using the Shur complement on the diagonal block terms, one finds easily LMI conditions of Theorem 1. 5. Example and simulation To illustrate the results of Theorem 1, we consider a numerical example that is composed of the following matrices:
0 A1 = −3
1 , −0.2
A2 =
−1 2
1 , 0
0 = sin (t ) 0.01 0 = Ha1 ∆a1 (t ) Na1 , 0.05 0.001 cos (t ) 0 0.1 ∆A2 (t ) = = cos (t ) 0.01 0 = Ha2 ∆a2 (t ) Na2 , 0 0 0 0 1 B11 = B12 = I2×2 , ∆B11 (t ) = ∆B12 (t ) = 02×2 , B21 = , B22 = , 1 0 0 0 ∆B21 (t ) = = cos (t ) = H2b1 ∆2b1 (t ) N2b1 , 0.01 cos (t ) 0.01 0.01 0.01 sin (t ) ∆B22 (t ) = = sin (t ) = H2b2 ∆2b2 (t ) N2b2 , 0 0 C1 = C2 = 1 0 , ∆C1 (t ) = ∆C2 (t ) = 01×2 , D11 = D12 = 02×1 , ∆D11 (t ) = ∆D12 (t ) = 02×1 , D21 = D22 = 1 and ∆D21 (t ) = ∆D22 (t ) = 0. The external disturbance of the switched system is given by the vector: ϕ (t ) = 0.001 sin (2t ) 2 cos (2t ) , this one is attenuated at the level η = 0.45. By choosing the following specifications: r1 = 1, r2 = 1.5, d1 = 0.9, d2 = 1.2, ∆A1 (t ) =
0 0.0005 sin (t )
0 0
corresponding to the radius and the distance of the disk-pole from the origin of complex plan for the pole of each sub-system as depicted in Fig. 1.
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Fig. 3. Phase plan.
Fig. 4. Control input signal.
Fig. 5. Switching signal.
After solving the LMI (17) of Theorem 1, one obtains the following state feedback gains: K1 = 0.8516
−3.0497
−2.8451 and the matrix P = W −1 =
0.1812 0.1240
h
0.1240 0.0958
i
.
−2.6653 , K2 =
B. Mansouri et al. / Nonlinear Analysis: Hybrid Systems 2 (2008) 1136–1143
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The switched system is subjected to a known commutation law, whose rule consists in fixing a dwell time for each one. We choose to switch regularly at every 0.25 s from one mode of operation to another by starting from sub-system 2, as illustrated in Fig. 5. The initial state conditions are fixed at x1 (0) = 1, x2 (0) = 0 and represented by * in the phase diagram, Fig. 3. Thus, this shows the convergence of the system to the equilibrium point. The system is stabilised with a settling time less than four seconds as depicted in Fig. 2. Finally, the control signal is illustrated by Fig. 4. Fig. 5 shows the switching signal. 6. Conclusion This work deals with robust control synthesis for a class of hybrid dynamical systems. This one concerns a set of uncertain linear and switched systems with arbitrary switching rules. In fact, based on a state feedback control, we have developed sufficient conditions in terms of LMI that ensures the attenuation of the external disturbances according to the H∞ criterion. Then, robust pole placements according to the desired specifications, namely, the damping factor and time response of each sub-system, were considered. Our future work intends considering the fact that the states are not available for measurements to get LMI conditions in the case of hybrid observer synthesis. Acknowledgements This work was supported by the GIS 3SGS within the framework of the project COSMOS. The authors would like to thanks Mr Elie Pacey for his valuable comments. References [1] D. Angeli, A note on stability of arbitrarily switched homogeneous systems, Systems and Control Letters, Nonlinear Control Abstracts (2000). [2] P.J. Antsaklis, Hybrid systems: Theory & applications, Proceedings of the IEEE 88 (2000) (special issue). [3] J. Bernussou, G. Garcia, Disk pole assignment for uncertain systems with norm bounded uncertainty, in: IFAC International Workshop on Robust Control, airns Queensland Australia, 1993. [4] S. Boyd, L. El Ghaoui, E. Feron, V. Balakrishnan, Linear matrix inequalities in system and control theory, in: SIAM Studies in Applied Mathematics, Philadelphia, PA, 1994. [5] M.S. Branicky, Multiple lyapunov functions and other analysis tools for switched and hybrid systems, IEEE Transactions on Automatic Control 43 (4) (1998) 475–482. [6] W.M. Haddad, D.S. Bernstein, Controller design with regional pole constraints, IEEE Transactions on Automatic Control 37 (1) (1992) 54–69. [7] D. Liberzon, A.S. Morse, Basic problems in stability and design of switched systems, IEEE Control Systems 19 (5) (1999) 59–70. [8] Y. Lin, E.D. Sontag, Y. Wang, A Smooth converse Lyapunov theorem for robust stability, SIAM Journal on Control and Optimization 34 (1996) 124–160. [9] B. Mansouri, N. Manamanni, K. Guelton, A. Kruszewski, T.M. Guerra, Output feedback LMI tracking control conditions with H∞ criterion for uncertain and disturbed TS models, Information Sciences, in press (doi:10.1016/j.ins.2008.10.007). [10] V.F. Montagner, V.J.S. Leite, R.C.L.F. Oliveira, P.L.D Peres, State feedback control of switched linear systems: An LMI approach, Journal of Computational and Applied Mathematics 194 (2006) 192–206. [11] Y. Mori, T. Mori, Y. Kuroe, On a class of linear constant systems which have a common quadratic Lyapunov function, in: 37th IEEE Conference on Decision and Control, 1998, pp. 2808–2809. [12] K.S. Narendra, J. Balakrishnan, A common lyapunov function for stable lti systems with commuting a-matrices, IEEE Transactions on Automatic Control 39 (1994) 2469–2471. [13] T. Ooba, Y. Funahashi, Two conditions concerning common quadratic lyapunov functions for linear systems, IEEE Transactions on Automatic Control 42 (1997) 719–721. [14] P. Peleties, R. DeCarlo, Asymptotic stability of m-switched systems using Lyapunov-like functions, American Control Conference (1991) 1679–1684. [15] S. Pettersson, Analysis and design of hybrid systems, Ph.D. Thesis, Chalmers University of Technology, Sweden, 1999. [16] H. Saadaoui, N. Manamanni, M. Djemai, J.P. Barbot, T. Floquet, Exact differentiation and sliding mode observer for switched mechanical systems, in: Hybrid Systems and Applications, Nonlinear Analysis 65 (5) (2006) 1050–1069 (special issue (4)). [17] A.V. Schaft, H. Shumacher, An Introduction to Hybrid Dynamical Systems, in: Lecture Notes in Control and Information Sciences, vol. 251, Springer, 2000. [18] H. Shim, D.J. Noh, J.H. Seo, Common Lyapunov function for exponentially stable nonlinear systems, in: 14th SIAM Conf. On Control and its Applications, Jaksonville, FL, 1998. [19] R.N. Shorten, K.S. Narenda, A sufficient condition for the existence of a common Lyapunov function for two second-order linear. [20] X. Xu, Analysis and design of switched systems, Ph.D. Thesis, University of Notre Dame, Indiana, 2001. [21] K. Zhou, P.P. Khargonedkar, Robust stabilization of linear systems with norm-bounded time-varying uncertainty, Systems and Control Letters 10 (1988) 17–20.
Contents lists available at ScienceDirect
Nonlinear Analysis: Hybrid Systems journal homepage: www.elsevier.com/locate/nahs
Robust pole placement controller design in LMI region for uncertain and disturbed switched systems B. Mansouri a , N. Manamanni a,∗ , K. Guelton a , M. Djemai b a
CReSTIC, EA3804, University of Reims Champagne Ardenne, Moulin de la House BP1039, 51687 Reims Cedex 2, France
b
LAMIH, UMR CNRS 8530, University of Valenciennes, 59313 Valenciennes, Cedex 9, France
article
info
Article history: Received 6 December 2007 Accepted 4 September 2008 Keywords: Hybrid dynamical systems Linear switched systems Switched control Quadratic Lyapunov function LMI (Linear Matrix Inequalities) H∞ control
a b s t r a c t This paper concerns the state feedback control for continuous-time, disturbed and uncertain linear switched systems with arbitrary switching rules. The main result of this work consists in getting a LMI (Linear Matrix Inequalities) condition guaranteeing a robust pole placement according to some desired specifications. Then, external disturbance attenuation with a fixed rate according to the H∞ criterion is ensured. This is obtained thanks to the existence of a common quadratic Lyapunov function for all sub-systems. Finally, an academic example illustrates the efficiency of the developed approach. © 2008 Elsevier Ltd. All rights reserved.
1. Introduction Hybrid dynamic systems are defined as a set of continuous and discrete sub-systems interacting together. Thus, they associate continuous and discrete dynamic, as well as continuous and discrete control [2,15,17]. The hybrid dynamic systems (HDS) have many varied applications; one finds them in the control of mechanical systems, motorized industry, aeronautics, power electric converters and robotics. The main difficulty in the definition of HDS is that the hybrid term is not restrictive; the interpretation of this term can be extended to any dynamic system. A standard and reasonable definition of a HDS would be to consider the structure only to indicate the objective to be reached and the fixed terminology. So, in this paper, the work will restricted to a particular class of HDS, namely the switched systems, which constitute a set of continuous sub-systems and a switching rule that orchestrates the commutation between them [5,7,11,16,20]. The various studies dedicated to switched systems consider mainly the stability problem. Indeed, three basic problems involved in the stability of the switched system were raised by Liberzon [7] and then developed in the literature [14,17]. One of these problems is to find the conditions that guarantee the system’s asymptotic stability under any switching rule. i.e. arbitrary switching sequence. A necessary condition for asymptotic stability under arbitrary switching sequence is that each sub-system must be asymptotically stable. The authors showed in [17] that this condition is not sufficient, since, it is possible to lead to instability for some classes of switching signals. Thus, to solve this problem, it has been shown in [8] that the existence of a common Lyapunov function is a necessary and sufficient condition. This approach is also reported in several works, see [11–13,18,19,1]. This paper deals primarily with the last problem described above. It concerns the class of continuous-time uncertain, linear switched systems without jump at the switching instant. It is also assumed that the number of switching is finite in
∗
Corresponding author. Tel.: +33 3 26 91 83 86; fax: +33 3 26 91 31 06. E-mail addresses: [email protected] (B. Mansouri), [email protected] (N. Manamanni), [email protected] (K. Guelton), [email protected] (M. Djemai). 1751-570X/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.nahs.2008.09.010
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a finite time (non-zenon). In [10], a quasi-LMI condition ensuring the design of a control law for a disturbed linear switched system is proposed. In this paper, the robustness of the approach has been improved by considering an uncertain system. In fact, in this case it is not obvious to get a strict LMI condition for the asymptotic stability. Thus, the goal of this paper is to provide a strict LMI condition for the synthesis of a switched state feedback control law that guarantees asymptotic stability of the closed-loop system for any arbitrary switching rule and ensures three different performances at the same time. The first is the asymptotic stability with some guaranteed specifications, namely; the transient response and the damping factor [6] by the pole location of each closed-loop linear sub-system. The second performance is to attenuate the disturbance according to the H∞ criterion. Finally, the main contribution of this paper is to obtain state feedback gains thanks to the LMI formulation by considering an uncertain structure of the switched system. A numerical example will illustrate the proposed approach. 2. Uncertain switched systems Let us consider the following uncertain switched system:
x˙ (t ) = Aσ (t ) (t ) x (t ) + B1σ (t ) (t ) ϕ (t ) + B2σ (t ) (t ) u (t ) y (t ) = C σ (t ) (t ) x (t ) + D1σ (t ) (t ) ϕ (t ) + D2σ (t ) (t ) u (t )
(1)
with Aσ (t ) (t ) = Aσ (t ) + ∆Aσ (t ) (t ), B1σ (t ) (t ) = B1σ (t ) + ∆B1σ (t ) (t ), B2σ (t ) (t ) = B2σ (t ) + ∆B2σ (t ) (t ), C σ (t ) (t ) = Cσ (t ) + ∆Cσ (t ) (t ), D1σ (t ) (t ) = D1σ (t ) + ∆D1σ (t ) (t ), D2σ (t ) (t ) = D2σ (t ) + ∆D2σ (t ) (t ) and where σ (t ) is the switching rule defined as follows: Let I = {1, 2, . . . , N } an index’s compact set of the sub-systems, σ (t ) is defined by the mapping: σ (t ) : R+ → I, i.e. the linear mode (A, B1 , B2 , C , D1 , D2 )l , is active if σ (t ) = l with l ∈ I. x (t ) ∈ Rn is the state vector, u (t ) ∈ Rm is the control input vector, ϕ (t ) ∈ Rr is an exogenous input, Aσ (t ) ∈ Rn×n , B1σ (t ) ∈ Rn×r , B2σ (t ) ∈ Rn×m , D1σ (t ) ∈ Rp×r , D2σ (t ) ∈ Rp×m and Cσ (t ) ∈ Rp×n are the sub-system matrices. ∆Aσ (t ) (t ) , ∆B1σ (t ) (t ), ∆B2σ (t ) (t ), ∆D1σ (t ) (t ), ∆D2σ (t ) (t ) and ∆Cσ (t ) (t ) contain all the modelling uncertainties, which can be represented by:
∆Aσ (t ) (t ) = Haσ (t ) ∆aσ (t ) (t ) Naσ (t ) , ∆B1σ (t ) (t ) = H1bσ (t ) ∆1bσ (t ) (t ) N1bσ (t ) , ∆B2σ (t ) (t ) = H2bσ (t ) ∆2bσ (t )σ (t ) (t ) N2bσ (t ) , ∆D1σ (t ) (t ) = H1dσ (t ) ∆1dσ (t ) (t ) N1dσ (t ) , ∆D2σ (t ) (t ) = H2dσ (t ) ∆2dσ (t ) (t ) N2dσ (t ) and ∆Cσ (t ) (t ) = Hc σ (t ) ∆c σ (t ) (t ) Nc σ (t ) where Haσ , H1bσ , H2bσ , H1dσ , H2dσ , Hc σ , Naσ , N1bσ , N2bσ , N1dσ , N2dσ and Nc σ are constant matrices and, ∆aσ (t ) (t ), ∆1bσ (t ) (t ), ∆2bσ (t ) (t ), ∆1dσ (t ) (t ), ∆2dσ (t ) (t ) and ∆c σ (t ) (t ) are the normalized uncertainty matrices, verifying the following conditions:
∆Taσ (t ) (t ) ∆aσ (t ) (t ) ≤ I , ∆T1di (t ) ∆1di (t ) ≤ I ,
∆T1bσ (t ) (t ) ∆1bσ (t ) (t ) ≤ I ,
∆T2bσ (t ) (t ) ∆2bσ (t ) (t ) ≤ I ,
∆T2dσ (t ) (t ) ∆2dσ (t ) (t ) ≤ I and ∆Tc σ (t ) (t ) ∆c σ (t ) (t ) ≤ I .
In this study, we assume that the state vector is available to perform a state feedback and that the discrete control law σ (t ) is not a priori known but is available in real time. Now, we consider the following state feedback control law: u (t ) = Kσ (t ) x (t ) .
(2)
From (1) and (2) the closed-loop switched system can be written as follows:
x˙ (t ) = A˜ σ (t ) x (t ) + B1σ (t ) ϕ (t ) y (t ) = C˜ σ (t ) x (t ) + D1σ (t ) ϕ (t )
(3)
where: A˜ σ (t ) = Aσ (t ) + B2σ (t ) Kσ C˜ σ (t ) = C σ (t ) + D2σ (t ) Kσ .
(4)
The objective is then to determine the state feedback gains Kσ ensuring the stability of the closed-loop switched system (3). 3. Robust pole placement and H∞ performances In the first step, one determines the state feedback gains with pole placement of each closed-loop uncertain linear sub-system, such that the eigenvalues of A˜ σ (t ) are located inside a disk-pole with centre at (− (rσ + dσ ) , 0) with the radius rσ and the distance dσ from the imaginary axis as set in Fig. 1. Using the parameter dσ , it is possible to determine an upper bound for the settling time of the transient response given by 3τσ or 5τσ for each sub-system σ , with τσ = 1/dσ . The value rσ gives the upper bound on the natural frequency of oscillation
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Fig. 1. Pole placement for the sub-system σ .
for the transient response. Notice also that a lower bound on the dumping factor ξσ , which determines the overshoot, can be computed as follows [10]:
p (rσ + dσ )2 − rσ2 ξσ = . rσ + dσ
(5)
Using the well known common quadratic Lyapunov function: V (x) = xT Px,
P = PT > 0
(6)
a necessary and sufficient condition such that all the eigenvalues of A˜ σ (t ) of system (3) lie inside the circular region ς (dσ , rσ ) as depicted in Fig. 1, is given by the existence of a positive definite symmetric matrix W = P −1 such that the following inequality holds [3]:
A˜ Tσ (t ) + dσ I W + W A˜ σ (t ) + dσ I
T
+
T 1 A˜ σ (t ) + dσ I W A˜ σ (t ) + dσ I < 0, rσ
σ = 1, 2, . . . , N .
(7)
On the other hand, in the second step, one needs to attenuate the disturbances ϕ (t ) with a minimum rate η, according to the H∞ criterion:
ky (t )k2 ≤ η kϕ (t )k2 .
(8)
A sufficient condition ensuring the H∞ criterion (8) is given by the following matrix inequality [4]: A˜ σ (t ) W + W A˜ Tσ (t ) BT1σ (t ) ˜Cσ (t ) W
(∗) −I D1σ (t )
(∗) (∗) < 0, −η2 I
σ = 1, 2, . . . , N .
(9)
Finally, to achieve both the performances at the same time; i.e. disturbance’s attenuation according to the H∞ criterion and pole placements (inside the disk-pole ς (dσ , rσ )) leading to a condition that verifies both (7) and (9), we use the following lemma [10]: Lemma 1. If there exists a symmetric positive definite matrix W such that: A˜ σ (t ) W + W A˜ Tσ (t ) + 2dσ W W A˜ Tσ (t ) + dσ W BT1σ (t ) ˜Cσ (t ) W
(∗) −r σ W 0 0
(∗)
(∗)
0 −I
0
D1σ (t )
< 0, (∗) −η2 I
σ = 1, 2, . . . , N
(10)
then, the stability of the closed-loop switched system (3) is ensured under arbitrary switching rule with guaranteed disturbance attenuation level η, defined in (8), and the pole location of each linear sub-system inside the disk-pole ς (dσ , rσ ) depicted in Fig. 1. Proof. Let us consider the following inequality:
A˜ σ (t ) W + W A˜ Tσ (t ) + 2dσ W 1 + A˜ σ (t ) + dσ I W A˜ σ (t ) + dσ I T rσ B˜ T1σ (t ) ˜Cσ (t ) W
(∗) −I D1σ (t )
(∗) < 0, (∗) −η2 I
σ = 1, 2, . . . , N .
(11)
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A necessary condition such that inequality (11) holds is that the first diagonal block of (11) must be negative definite, i.e. A˜ σ (t ) W + W A˜ Tσ (t ) + 2dσ W +
1 rσ
A˜ σ (t ) + dσ I W A˜ σ (t ) + dσ I
T
< 0.
(12)
That is equivalent to: 1
A˜ σ (t ) W + W A˜ Tσ (t ) < −2dσ W −
rσ
A˜ σ (t ) + dσ I W A˜ σ (t ) + dσ I
T
<0
leading to inequality (9) of the H∞ . Then, by applying the Shur complement in the block (1, 1) of inequality (11), inequality (10) is obtained. This ends the proof. Note that, inequality (11) given by Lemma 1 is not a LMI condition since, in our case, it contains varying time uncertainties which can be considered unknown with known upper bounds. Thus, in the following section, the goal is to get a LMI constraint by considering the uncertainty structure mentioned in the first section. 4. LMI formulation Due to the uncertainties, checking a LMI condition for the switched system, with performances stipulated above, is not trivial [9]. In order to obtain a LMI condition to synthesize the state feedback gains, one needs to use the following corollaries: Corollary 1 ([21]). For real matrices A, B and S = S T > 0 with appropriate dimensions and a positive constant τ , we have: AT B + BT A ≤ γ AT A + γ −1 BT B
(13)
X T Y + Y T X ≤ X T S −1 X + Y T SY .
(14)
and
Corollary 2. For real matrices A, B, W , Y , Z and a regular matrix Q with appropriate dimensions we have:
W T + BT AT Z
Y W + AB
<0⇔
Y + BT Q −1 B W
WT Z + AQAT
< 0.
(15)
Proof of corollary i2. For real matrices A, B, matrix Q with appropriate dimensions, the matrix h h W , YT,i Z and h a regular i Y W T + BT AT Y W 0 BT AT < 0 can be rewritten as + < 0. W + AB Z W Z AB 0 From inequality (13) we have:
0 B A
T
0 +
B 0
0
AT
0 M 0 A
≤
that leads to (15) and ends the proof.
AT +
T B 0
M −1 B
0
(16)
Now, let us make a bijective change of variable on the state feedback gains; Zσ = Kσ W . Then, the main result can be summarized by the following theorem: Theorem 1. If there exist a symmetric positive definite matrix W , matrices Zσ and positive constants τ1 , τ2 , τ3 , τ4 , τ5 , τ6 and τ7 such that:
Ξ Σ
(∗) Ψ
<0
with:
T Ξ = Aσ W + B2σ Zσ + WATσ + ZσT BT2σ + 2dσ W + τ1−1 + τ3−3 Haσ HaTσ + τ2−1 + τ4−1 H2bσ H2b σ, N1bσ N2bσ Zσ Nc σ W N2dσ Zσ Naσ W T T T Σ = WAσ + Zσ B2σ + dσ W 0 0 T B 1σ Cσ W + D2σ Zσ 0
(17)
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and
Ψ 1 ,1 Ψ =0
0
−r σ W Naσ W N2bσ Zσ 0 0 0
and where Ψ1,1 = diag −τ5
(∗)
(∗) −τ3−1
0
0 0 0 0
−τ4−1
0 0 0
0 0 0
D1σ 0
−τ2−1
−τ6
0 0 0
(∗) Ψ6,6
Ψ5 ,5
−τ7
−1 T T Ψ5,5 = −I + τ5 H1bσ H1b σ + τ8 N1dσ N1dσ ,
0 0 0 0
,
(∗) −τ8−1
T H1d σ
−τ1−1 , T Ψ6,6 = −η2 I + τ6 Hc σ HcTσ + τ7 H2dσ H2d σ
for σ = 1, 2, . . . , N. Then, the quadratic stability of the closed-loop switched system (3) is ensured under an arbitrary switching rule, with guaranteed disturbance attenuation level η and with the pole location of each linear sub-system inside the disk-pole ς (dσ , rσ ) depicted in Fig. 1. Proof. Starting from inequality (10) and separating the uncertainties, (10) becomes:
Π + ∆Π < 0
(18)
with:
(Aσ + B2σ Kσ ) W + (∗) + 2dσ W W (Aσ + B2σ Kσ )T + dσ W Π = BT1σ (Cσ + D2σ Kσ ) W
(∗) −r σ W
(∗)
0 0
−I
(∗)
0
0 (∗) −η2 I
D1σ
and
(∆Aσ (t ) + ∆B2σ (t ) Kσ ) W + (∗) W (∆Aσ (t ) + ∆B2σ (t ) Kσ )T ∆Π = ∆BT1σ (t ) (∆Cσ (t ) + ∆D2σ (t ) Kσ ) W
(∗)
(∗)
(∗)
0 0 0
0 0
0
(∗)
.
∆D1σ (t )
0
After the bijective change of variable Zσ = Kσ W , one obtains: Aσ W + (∗) + B2σ Zσ + (∗) + 2dσ W WATσ + ZσT BT2σ + dσ W Π = BT1σ Cσ W + D2σ Zσ
(∗) −rσ W
(∗)
0 0
−I
(∗)
0
0 (∗) −η2 I
D1σ
and
∆Aσ (t ) W + (∗) + ∆B2σ (t ) Zσ + (∗) W ∆ATσ (t ) + ZσT ∆BT2σ (t ) ∆Π = ∆BT1σ (t ) ∆Cσ (t ) W + ∆D2σ (t ) Zσ
(∗)
(∗)
(∗)
0 0 0
0 0
0
(∗)
.
∆D1σ (t )
0
Holding the uncertainty structure shown in Section 1, thus ∆Π is written as:
∆Π1,1 ∆Π2,1 ∆Π = ∆Π3,1 ∆Π4,1 ∆Π1,1 = Haσ ∆aσ
(∗)
(∗)
0 0 0
0 0
(∗)
0 , (∗)
H1d ∆1dσ (t ) N1dσ 0 (t ) Naσ W + (∗) + H2b ∆2bσ (t ) N2b Zσ + (∗) ,
T T T ∆Π2,1 = WNaTσ ∆Taσ (t ) HaTσ + ZσT N2b σ ∆2bσ (t ) H2bσ , ∆Π3,1 = H1bσ ∆1bσ (t ) N1bσ
and
∆Π4,1 = Hc σ ∆c σ (t ) Nc σ W + H2dσ ∆2dσ (t ) N2dσ Zσ .
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Fig. 2. State trajectories x1 and x2 .
Now, by the means of Corollaries 1 and 2, the matrix ∆Π containing the uncertainties can be bounded in order to find some scalar constants as the following:
∆Π < diag [Γ ] where: −1 −3 −1 T T T T Γ1,1 = τ1 WNaTσ Naσ W + τ1−1 Haσ HaTσ + τ2 ZσT N2b σ N2bσ Zσ + τ2 H2bσ H2bσ + τ3 Haσ Haσ + τ4 H2bσ H2bσ −1 −1 T T T T + τ5−1 N1b σ N1bσ + τ6 WNc σ Nc σ W + τ7 Zσ N2dσ N2dσ Zσ , T Γ2,2 = τ3 WNaTσ Naσ W + τ4 ZσT N2b N2b Zσ ,
−1 T T Γ3,3 = τ5 H1bσ H1b σ + τ8 N1dσ N1dσ
and T T Γ4,4 = τ6 Hc σ HcTσ + τ7 H2dσ H2d σ + τ8 H1dσ H1dσ .
Finally, by adding matrices Π and ∆Π , and using the Shur complement on the diagonal block terms, one finds easily LMI conditions of Theorem 1. 5. Example and simulation To illustrate the results of Theorem 1, we consider a numerical example that is composed of the following matrices:
0 A1 = −3
1 , −0.2
A2 =
−1 2
1 , 0
0 = sin (t ) 0.01 0 = Ha1 ∆a1 (t ) Na1 , 0.05 0.001 cos (t ) 0 0.1 ∆A2 (t ) = = cos (t ) 0.01 0 = Ha2 ∆a2 (t ) Na2 , 0 0 0 0 1 B11 = B12 = I2×2 , ∆B11 (t ) = ∆B12 (t ) = 02×2 , B21 = , B22 = , 1 0 0 0 ∆B21 (t ) = = cos (t ) = H2b1 ∆2b1 (t ) N2b1 , 0.01 cos (t ) 0.01 0.01 0.01 sin (t ) ∆B22 (t ) = = sin (t ) = H2b2 ∆2b2 (t ) N2b2 , 0 0 C1 = C2 = 1 0 , ∆C1 (t ) = ∆C2 (t ) = 01×2 , D11 = D12 = 02×1 , ∆D11 (t ) = ∆D12 (t ) = 02×1 , D21 = D22 = 1 and ∆D21 (t ) = ∆D22 (t ) = 0. The external disturbance of the switched system is given by the vector: ϕ (t ) = 0.001 sin (2t ) 2 cos (2t ) , this one is attenuated at the level η = 0.45. By choosing the following specifications: r1 = 1, r2 = 1.5, d1 = 0.9, d2 = 1.2, ∆A1 (t ) =
0 0.0005 sin (t )
0 0
corresponding to the radius and the distance of the disk-pole from the origin of complex plan for the pole of each sub-system as depicted in Fig. 1.
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Fig. 3. Phase plan.
Fig. 4. Control input signal.
Fig. 5. Switching signal.
After solving the LMI (17) of Theorem 1, one obtains the following state feedback gains: K1 = 0.8516
−3.0497
−2.8451 and the matrix P = W −1 =
0.1812 0.1240
h
0.1240 0.0958
i
.
−2.6653 , K2 =
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The switched system is subjected to a known commutation law, whose rule consists in fixing a dwell time for each one. We choose to switch regularly at every 0.25 s from one mode of operation to another by starting from sub-system 2, as illustrated in Fig. 5. The initial state conditions are fixed at x1 (0) = 1, x2 (0) = 0 and represented by * in the phase diagram, Fig. 3. Thus, this shows the convergence of the system to the equilibrium point. The system is stabilised with a settling time less than four seconds as depicted in Fig. 2. Finally, the control signal is illustrated by Fig. 4. Fig. 5 shows the switching signal. 6. Conclusion This work deals with robust control synthesis for a class of hybrid dynamical systems. This one concerns a set of uncertain linear and switched systems with arbitrary switching rules. In fact, based on a state feedback control, we have developed sufficient conditions in terms of LMI that ensures the attenuation of the external disturbances according to the H∞ criterion. Then, robust pole placements according to the desired specifications, namely, the damping factor and time response of each sub-system, were considered. Our future work intends considering the fact that the states are not available for measurements to get LMI conditions in the case of hybrid observer synthesis. Acknowledgements This work was supported by the GIS 3SGS within the framework of the project COSMOS. The authors would like to thanks Mr Elie Pacey for his valuable comments. References [1] D. Angeli, A note on stability of arbitrarily switched homogeneous systems, Systems and Control Letters, Nonlinear Control Abstracts (2000). [2] P.J. Antsaklis, Hybrid systems: Theory & applications, Proceedings of the IEEE 88 (2000) (special issue). [3] J. Bernussou, G. Garcia, Disk pole assignment for uncertain systems with norm bounded uncertainty, in: IFAC International Workshop on Robust Control, airns Queensland Australia, 1993. [4] S. Boyd, L. El Ghaoui, E. Feron, V. Balakrishnan, Linear matrix inequalities in system and control theory, in: SIAM Studies in Applied Mathematics, Philadelphia, PA, 1994. [5] M.S. Branicky, Multiple lyapunov functions and other analysis tools for switched and hybrid systems, IEEE Transactions on Automatic Control 43 (4) (1998) 475–482. [6] W.M. Haddad, D.S. Bernstein, Controller design with regional pole constraints, IEEE Transactions on Automatic Control 37 (1) (1992) 54–69. [7] D. Liberzon, A.S. Morse, Basic problems in stability and design of switched systems, IEEE Control Systems 19 (5) (1999) 59–70. [8] Y. Lin, E.D. Sontag, Y. Wang, A Smooth converse Lyapunov theorem for robust stability, SIAM Journal on Control and Optimization 34 (1996) 124–160. [9] B. Mansouri, N. Manamanni, K. Guelton, A. Kruszewski, T.M. Guerra, Output feedback LMI tracking control conditions with H∞ criterion for uncertain and disturbed TS models, Information Sciences, in press (doi:10.1016/j.ins.2008.10.007). [10] V.F. Montagner, V.J.S. Leite, R.C.L.F. Oliveira, P.L.D Peres, State feedback control of switched linear systems: An LMI approach, Journal of Computational and Applied Mathematics 194 (2006) 192–206. [11] Y. Mori, T. Mori, Y. Kuroe, On a class of linear constant systems which have a common quadratic Lyapunov function, in: 37th IEEE Conference on Decision and Control, 1998, pp. 2808–2809. [12] K.S. Narendra, J. Balakrishnan, A common lyapunov function for stable lti systems with commuting a-matrices, IEEE Transactions on Automatic Control 39 (1994) 2469–2471. [13] T. Ooba, Y. Funahashi, Two conditions concerning common quadratic lyapunov functions for linear systems, IEEE Transactions on Automatic Control 42 (1997) 719–721. [14] P. Peleties, R. DeCarlo, Asymptotic stability of m-switched systems using Lyapunov-like functions, American Control Conference (1991) 1679–1684. [15] S. Pettersson, Analysis and design of hybrid systems, Ph.D. Thesis, Chalmers University of Technology, Sweden, 1999. [16] H. Saadaoui, N. Manamanni, M. Djemai, J.P. Barbot, T. Floquet, Exact differentiation and sliding mode observer for switched mechanical systems, in: Hybrid Systems and Applications, Nonlinear Analysis 65 (5) (2006) 1050–1069 (special issue (4)). [17] A.V. Schaft, H. Shumacher, An Introduction to Hybrid Dynamical Systems, in: Lecture Notes in Control and Information Sciences, vol. 251, Springer, 2000. [18] H. Shim, D.J. Noh, J.H. Seo, Common Lyapunov function for exponentially stable nonlinear systems, in: 14th SIAM Conf. On Control and its Applications, Jaksonville, FL, 1998. [19] R.N. Shorten, K.S. Narenda, A sufficient condition for the existence of a common Lyapunov function for two second-order linear. [20] X. Xu, Analysis and design of switched systems, Ph.D. Thesis, University of Notre Dame, Indiana, 2001. [21] K. Zhou, P.P. Khargonedkar, Robust stabilization of linear systems with norm-bounded time-varying uncertainty, Systems and Control Letters 10 (1988) 17–20.