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On uniform controller design for linear switched systems
Nonlinear Analysis: Hybrid Systems 4 (2010) 189–198
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Nonlinear Analysis: Hybrid Systems journal homepage: www.elsevier.com/locate/nahs
On uniform controller design for linear switched systems G. Zheng a,∗ , D. Boutat b , J.P. Barbot a,c a
INRIA Lille-Nord Europe, 40 avenue Halley, 59650 Villeneuve d’Ascq, France
b
LVR/ENSI, 10 Boulevard de Lahitolle, 18020 Bourges, France
c
ECS/ENSEA, 6 Avenue du Ponceau, 95014, Cergy-Pontoise, France
article
info
Article history: Received 3 April 2009 Accepted 11 September 2009 Keywords: Switched systems Feedback control Stability
abstract Traditional method to design a controller for each subsystem of switched systems will increase the complexity of the controller’s realization. Hence this paper gives sufficient conditions for designing a uniform output feedback controller for linear switched systems, and this common controller can be used for all subsystems of the switched systems. Then the output stabilization problem for a particular class of linear switched systems under this uniform output feedback controller has been studied. An illustrative example is given in order to highlight the proposed method. © 2009 Elsevier Ltd. All rights reserved.
1. Introduction The study of switched systems comes partly from the fact that switched systems have numerous applications in control of mechanical systems, process control, automotive industry, power systems, aircraft, traffic control, biology, network and many other fields (see for example [1–6]). The stabilization of switched systems is nowadays a widely investigated problem and many significant contributions were given with different techniques: A multiple Lyapunov approach [7,8], LaSalle invariant principle (linear [5] or nonlinear [9,10]), Lie algebra approach [11,12], smooth Lyapunov functions [13], small-gain theorem [14], generalized piecewise linear feedback [15] and switched systems with unknown time-varying delays [16]. For designing an output feedback controller, traditional method is to design different output feedback controllers associated to different subsystems of switched systems, and it is can be seen as designing a hybrid output feedback controller. Obviously, it will increase the complexity of the controller’s realization. In this paper, we just investigate the output stabilization problem of linear switched systems by common output for all subsystems, called uniform controller, with the concept of flat system [17]. The common output is used to overcome the difficulty to measure or at least observe [18] the discrete state, and this greatly simplifies the controller implementation. Hence sufficient conditions which guarantee the existence of a uniform output feedback controller should be studied. This paper is organized as follows. Motivations and problem statements are given in Section 2. In Section 3, we deduce sufficient conditions which guarantee the existence of a uniform output feedback controller for linear switched systems. Section 4 is devoted to studying the output stabilization problem for a particular class of linear switched systems via the uniform output feedback controller. This paper ends with an illustrative example for the purpose of highlighting the feasibility of the proposed results.
∗
Corresponding author. E-mail address: [email protected] (G. Zheng).
1751-570X/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.nahs.2009.09.008
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2. Problem statements Let us consider a class of switched systems as follows: x˙ = fσ (t ) (x) + gσ (t ) (x) u y = cσ (t ) (x)
(1)
where x ∈ Rn , y ∈ R, u ∈ R, fσ (t ) : Rn → Rn , gσ (t ) : Rn → Rn , cσ (t ) : Rn → R are sufficiently smooth where σ (t ) : R+ → Σ is a piecewise constant function representing the switching signal with Σ = {1, 2, . . . , m}. In order to design an output feedback controller, basic method is to design different output feedback controllers associated to different subsystems of (1), such as u = ϕi y, y˙ , . . . , y(n)
for i ∈ Σ .
(2)
By detecting the active mode of (1) from its output [19–22], we can then apply the pre-designed output feedback controller to achieve the control goal, such as stabilization of output. In (2), normally we have
ϕi 6= ϕj for i ∈ Σ and j ∈ Σ
(3)
which implies a hybrid controller should be synthesized, even if the desired output trajectory is the same for each subsystem. And this will absolutely increase the complexity of the controller’s structure. Motivated by this point, this paper considers the following problem: Is it possible to design an output feedback uniform controller for (1) to achieve the control goal? More precisely, we are looking for a controller for system (1) in the following form: u = ϕ y, y˙ , . . . , y(n)
∀i ∈ Σ
(4)
which means the functions ϕi and ϕj in (3) are equal to ϕ in (4). The interest of designing such a uniform output feedback controller can evidently simplify the structure of controller, since we do not need to design a sub-controller for each subsystem of (1). The structure of the controller (4) will keep unchanged no matter which subsystem of (1) is active. Moreover, unlike hybrid controller (2) which needs to use firstly an observer or estimator to detect which subsystem of (1) is active and then apply the associated controller, the uniform controller (4) does not need such kind of instrument. In fact, our main result is based on the following claim. Claim 2.1. For each subsystem of (1), if the following conditions are satisfied: 1. System (1) is 0-flat 1 for all i ∈ Σ ; 2. System (1) has a unique input–output relation for all i ∈ Σ ; then there exists a uniform output feedback controller for system (1).2 Indeed, the first condition of Claim 2.1 implies that, for each subsystem of (1), we can always obtain an input–output relation, such as u = Fi y, y˙ , . . . , y(n)
for all i ∈ Σ . The second condition implies that for all i ∈ Σ , we have Fi = F , which yields u = F y, y˙ , . . . , y(n) .
(5)
Hence for all i ∈ Σ , by well choosing a common function of the output and its successive derivatives as the output feedback control law, noted as u = G y, y˙ , . . . , y(n) , then (5) becomes G y, y˙ , . . . , y(n) = F y, y˙ , . . . , y(n)
which enables us to stabilize the output of system (1). In such a way, we can freely design a uniform output feedback controller for system (1) and it can be described in Fig. 1. This paper will treat the problems of unique input–output relation and stabilization of the switched systems by uniform output feedback controller only for linear switched systems. Based on Claim 2.1, this problem will be studied in two steps: the first step is to give sufficient conditions which guarantee the existence of a uniform output feedback controller for linear switched systems, and the second one is to study the output stabilization problem through the uniform controller.
1 0-flat means that the flat output depends only on the states [23,24]. 2 This claim ensures the same input–output behavior for each subsystem, but does not guarantee any stabilization property.
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Subsystem 1 Subsystem 2
Subsystem m
uniform controller Fig. 1. Illustrative diagram of uniform feedback controller design for switched systems.
3. Existence of a uniform output feedback controller for linear switched systems In this section, based on Claim 2.1 we will give sufficient conditions which guarantee the existence of a uniform output feedback controller for a class of linear switched systems. More precisely, let us consider the following dynamical system:
x˙ = Aσ (t ) x + Bσ (t ) u y = Cσ ( t ) x
(6)
where Aσ (t ) is n × n matrix, Bσ (t ) is n × 1 matrix and Cσ (t ) is 1 × n matrix for all σ (t ) ∈ Σ = {1, 2, . . . m} and our goal is to design a uniform output feedback controller in order to stabilize the output of system (6). It is obvious that, in order to guarantee the existence of a uniform output feedback controller for (6), one sufficient condition is the existence of a unique input–output relation for all i ∈ Σ . For this, let us note
C i Ci A i Ti = ...
.
Ci Ani −1
Theorem 3.1. If the following conditions are fulfilled: 1. Ci Aki Bi = 0 for all i ∈ Σ and 0 ≤ k ≤ n − 2; 2. Ci Ain−1 Bi = Cj Anj −1 Bj = γ 6= 0 for all i ∈ Σ and j ∈ Σ ; 3. C1 An1 , . . . , Cm Anm ∈ Im (T1 , . . . , Tm )
then we can deduce the following unique input–output relation: u=
1
γ
(n)
y
−
n −1 X
! lj y
(j)
(7)
j=0
where lj ∈ R, which implies there exists a uniform output feedback controller for (6). Proof. Let us consider system (6) for all i ∈ Σ , if conditions (1) and (2) of Theorem 3.1 are satisfied, then we have rank (Ti ) = n and the successive derivation of y gives y(j) = Ci Ai x j
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for 0 ≤ j ≤ n − 1 and y(n) = Ci Ani x + Ci Ani −1 Bi u
= Ci Ani x + γ u.
(8)
Moreover, if condition (3) of Theorem 3.1 is fulfilled, then there exists lj ∈ R such that C1 An1 , . . . , Cm Anm =
n −1 h X
j
lj C1 A1 , . . . , Cm Ajm
i
j =0
which implies, for all i ∈ Σ , we have Ci Ani x =
n −1 X
j
lj Ci Ai x =
j =0
n −1 X
lj y(j) .
j =0
Finally, (8) can be written as follows: u =
=
1
γ 1
γ
y(n) − Ci Ani x y(n) −
!
n −1 X
lj y(j)
j=0
which is a unique input–output relation independent of i ∈ Σ . Hence there exists a uniform output feedback controller for system (6). Remark 3.1. Conditions (1) and (2) are equivalent to the fact that each Ci x for i ∈ Σ is a flat output for the considered subsystem. Condition (3) is very restrictive and implies that the controller for each subsystem can be with the same structure. Remark 3.2. Theorem 3.1 gives only sufficient conditions to guarantee the existence of a uniform output feedback controller for system (6), and the deduced unique input–output relation is in the form of (7). In the next section, we will study the output stabilization problem for linear switched systems under the uniform output feedback control law. 4. Stability by the uniform output feedback control Let us reconsider system (6), for all σ (t ) ∈ Σ we note
eσ (t ),1 = y = Cσ (t ) x eσ (t ),j = e˙ σ (t ),j−1 = y(j−1)
for 2 ≤ j ≤ n and eσ (t ) = eσ (t ),1 , eσ (t ),2 , . . . , eσ (t ),n
T
.
Since Theorem 3.1 is satisfied, then the successive derivation of y gives y(n) = Cσ (t ) Anσ (t ) x + Cσ (t ) Anσ−(t1) Bσ (t ) u and finally we have 0
..
. e˙ σ (t ) = 0 0
1
.. .
0 0
··· .. . ··· ···
0
.. . eσ (t ) + 1 0
0
.. .
0 Cσ (t ) Anσ (t ) x + Cσ (t ) Aσn−(t1) Bσ (t ) u
(9)
for σ (t ) ∈ Σ . Remark 4.1. It should be noted that the output y is continuous for each subsystem, however discontinuous at the switching instant. Hence in this paper, we consider the Dini derivative [25] for y just after the switching instant tk , defined as follows: D+ y(tk ) = lim sup h→0+
y(tk + h) − y(tk ) h
and D+ y(tk ) = lim inf h→0+
y(tk + h) − y(tk ) h
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while in our case we have D+ y(tk ) = D+ y(tk ) = lim
y(tk + h) − y(tk ) h
h→0+
.
And for the control input, we keep u(tk ) = u(tk− ) at the switching instant. Let us assume that Theorem 3.1 is satisfied, then there exists a uniform feedback controller for system (6). By choosing the following control law: u=
1
γ
−
n −1 X
(j)
lj y
−
n X
j =0
! (j)
kj y
(10)
j =1
where kj ∈ R, which is chosen such that the polynomial k1 + k2 s + · · · + kn sn−1 is Hurwitz, (9) becomes: e˙ σ (t ) = Aeσ (t )
(11)
where
0
1
.
.. A= 0 −k1
.. .
0 −k2
··· .. . ··· ···
0
.. . .
(12)
1 −kn
Since eσ (t ) = Tσ (t ) x, hence the stability problem of (11) can be converted to study that of the following system: x˙ = Tσ−(1t ) ATσ (t ) x.
(13)
For system (13), some techniques are already developed to study the stability problem based on Lyapunov approaches. For example, under the asymptotic stability of each subsystem, a common Lyapunov function exists when the subsystems matrices are pairwise commutative [26,27]. In [11,12], the authors proposed a generalization of the commutativity notion, based on the solvability of the Lie algebra generated by the subsystems state matrices, i.e., state matrices are uppertriangularizable in a same reference frame. Based on the existing results, we note A¯ i = Ti−1 ATi for all i ∈ Σ . Let us remark that A defined in (12) can make the corresponding system exponentially stable with well chosen kj > 0 for 1 ≤ j ≤ n, hence A¯ i can also make (13) exponentially stable. If we note G the Lie algebra generated by A¯ i , the following proposition is obvious due to Theorem 3.1 of this paper and the results with respect to G stated in [11] and [12]. Proposition 4.1. Suppose that assumptions of Theorem 3.1 are satisfied. If there exists a common output feedback controller u in (10) such that all A¯ i for 1 ≤ i ≤ m are Hurwitz, and G the Lie algebra generated by A¯ i is solvable, then (13) is stabilizable via the uniform output feedback controller. A particular case of solvable Lie algebra is when the following commutative condition is satisfied:
[A¯ i , A¯ j ] = A¯ i A¯ j − A¯ j A¯ i = 0. It is obvious that it is not a difficult task to find a uniform output feedback controller in the form of (10) which makes all A¯ i for 1 ≤ i ≤ m exponentially stable, however it is generally difficult to guarantee all A¯ i satisfy the commutative condition. Hence, in the following, we will focus on a particular class of linear switched systems whose switching signal σ (t ) depends on its states as follows:
x˙ = A1 x + B1 u x˙ = A2 x + B2 u
and y = C1 x and y = C2 x
if Hx < 0 if Hx ≥ 0
(14)
where Ai is n × n matrix, Bi is n × 1 matrix, Ci and H are 1 × n matrices for all i ∈ Σ = {1, 2}. 4.1. Continuity of Lyapunov function on the commutative surface As discussed in the last section, after applying the uniform controller (10), we can deduce the following observation error: e˙ σ (t ) = Aeσ (t ) . Then let us consider the following manifold: S = x | Hx = 0, x ∈ Rn
which represents the commutation surface between the first and the second subsystem of (14). Then we can denote the projection matrix Γ to represent the projection of x on S.
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Theorem 4.1. Suppose that Theorem 3.1 is satisfied. If there exist two positive definitive symmetric matrices P, Q such that AT P + PA = −Q and if the following condition holds: T1 Γ = T2 Γ
where Γ is the projection matrix of x on S, Ti = CiT , (Ci Ai )T , . . . , Ci Ain−1
T T
for all i ∈ Σ , then the uniform controller (10)3
exponentially stabilizes the output of the linear switched systems (14). Proof. Without loss of generality, let us consider the following time sequence: t0 < t1 < · · · < ts , and suppose that when t ∈ [ti−1 , ti [ the first subsystem of (14) is active and when t ∈ [ti , ti+1 [ the second subsystem of (14) becomes active where ti is the switching instant from the first subsystem to the second one through the commutation surface S. Let us consider the following Lyapunov candidate function: Vσ (t ) (e) = eTσ (t ) Peσ (t ) with P a positive definitive symmetric matrix. Then when t ∈ [ti−1 , ti [, we have V˙ σ (t ) (e) = eTσ (t ) AT P + PA eσ (t )
= −eTσ (t ) Qeσ (t ) ≤ −α Vσ (t ) (e) with α = λmin { Q } /λmax {P } where λmax {Λ} (or λmin {Λ}) represents the maximal (or minimal) eigenvalue of Λ. And this leads V − (e) ≤ e−α (ti −ti−1 ) V + (e) σ t σ t i−1
i
+
−
where t (or t ) represents the instant just after (or before) the switching instant. Analogously, for t ∈ [ti , ti+1 [, with the same Lyapunov function, we obtain V˙ σ (t ) (e) ≤ −α Vσ (t ) (e) which yields V
σ ti− +1
(e) ≤ e−α(ti+1 −ti ) Vσ t + (e) . i
At the moment of switching instant t = ti , considering the projection of x on S, the Lyapunov function Vσ (t ) (e) becomes
T
P T1 Γ x ti−
V − (e) = T2 Γ x ti− σ t
T
P T2 Γ x ti−
= T2 Γ x ti+ = Vσ t + (e)
T
P T2 Γ x ti+
V − (e) = T1 Γ x ti− σ t i
.
If T1 Γ = T2 Γ , then i
i
which implies the continuity of the Lyapunov function V − and V + on S. Hence we have σ ti
V
σ ti+ +1
(e) = Vσ
− ti+1
σ ti
(e)
≤ e−α(ti+1 −ti ) Vσ t + (e) i
≤e
−α (ti+1 −ti−1 )
V
σ ti+ −1
(e) .
By induction, for t ∈ [tk , tk+1 [ with k tends to infinity, we have Vσ (t ) (e) ≤ e−α(t −t1 ) V + (e), which implies exponential σ t 0
convergence of y to zero. Remark 4.2. The equality T1 Γ = T2 Γ in Theorem 4.1 implies the output trajectory is continuous on the commutation surface, when switching from one active subsystem to another one. This condition is a little restrictive and it will be relaxed in the following.
3 Obviously if P and Q exist, this implies that k + k s + · · · + k sn−1 is a Hurwitz polynomial where k is given in (12). 1 2 n i
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4.2. Minimal dwell time between the commutations Similar to the above section, let us suppose that there exist two positive definitive symmetric matrices P, Q , such that AT P + PA = −Q
(15)
hence we can define α = λmin { Q } /λmax {P }. Then let us chose for each subsystem of (14) a Lyapunov function as follows: Vσ (t ) (e) = eTσ (t ) Peσ (t ) .
(16)
Denote βi = λmax TiT PTi , βi = λmin
n o TiT PTi for 1 ≤ i ≤ 2 and set β = max1≤i≤2 βi , β = max1≤i≤2 βi and µ = β/β .
Lemma 4.1. For each subsystem of (14), the chosen Lyapunov function (16) satisfies the following inequality: V + (e) ≤ µV − (e) σ ti σ ti
(17)
where ti ∈ R+ represents the switching instant. Proof. Consider the switching instant t = ti without generality we suppose that σ ti_ = 1 and σ ti+ = 2, then on the commutation surface S with the projection matrix Γ , we have
V + (e) = eT + Pe + σ ti
σ ti
σ ti
= xT ti Γ T T2T PT2 Γ x ti+ . Since β > β2 = λmax T2T PT2 and β < β1 = λmin T1T PT1 , we have V + (e) ≤ β2 xT ti+ Γ T Γ x ti+ σ ti ≤ β xT ti+ Γ T Γ x ti+ +
≤
β T − T T x ti Γ T1 PT1 Γ x ti− βi
≤
β T e Pe − = µV − (e) σ ti β σ ti− σ ti
and this ends the proof.
The inequality (17) means that the change of Lyapunov function (16) before and after the switching time is bounded, and Eq. (15) implies the stability of each subsystem of (14) via the uniform controller. Many methods are proposed to study the global stability of such type of linear switched systems. Roughly speaking, if we know the minimal dwell time for each subsystem of (14), then the well-known theorem based on dwell time [1,28,29] can be stated as follows: Theorem 4.2. If there exists a minimal dwell time τmin between the commutations, such that
τmin >
log µ
α
then there exists a uniform controller 4 of the form (10) which asymptotically stabilizes the output of the linear switched systems (14). Remark 4.3. Since system (14) is state dependent, alternative conditions on dwell time and poles placement for such type of system are also largely studied in the literature (see [1,30,31] for details). 5. Illustrative example Let us consider the following linear switched system:
x˙ =
−1
0 1
−β
y = 0
1 0
0 0 1
1
γ
βγ
!
1 0 u 0
!
x+
−γ x
4 which ensures the existence of P and Q .
if x3 < 0 (18)
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G. Zheng et al. / Nonlinear Analysis: Hybrid Systems 4 (2010) 189–198
and
x˙ =
−1
0 1
−1
0 0 1
β
y = 0
1 0 u 0
!
0
!
x+
−γ −1 x. βγ
γ
if x3 ≥ 0 (19)
For systems (18) and (19) we have
0
C1 T1 = C1 A 1 = 0 2 C1 A 1 1
1
1
γ
βγ 1 − β γ β
0 0
and
0
C2 T2 = C2 A 2 = 0 C2 A22 1
1
−
γ
1
βγ 1 . β γ − β
0 0
Moreover it is easy to check that
C1 B1 = C2 B2 = 0 C1 A1 B1 = C2 A2 B2 = 0 C A2 B = C A2 B = 1 1 1 1 2 2 2 and
C1 A31 = −γ C2 A32 = −γ
−γ T1 −γ T2
−β − 1 −β − 1
which implies that C1 A31 , C2 A32 ∈ Im (T1 , T2 ) .
Hence Theorem 3.1 is satisfied and we can design a uniform controller for systems (18) and (19). According to Theorem 4.1, we design the following uniform output feedback controller: u = −γ y − (β + 1) y˙ − γ y¨ −
3 X
kj y(j)
(20)
j =1
where kj ∈ R for 1 ≤ j ≤ 3 is chosen such that the polynomial k1 + k2 s + k3 s2 is Hurwitz. For this example, we take β = γ = 1, k1 = 1, k2 = 0.1 and k3 = 1. Now let us check whether this uniform controller can stabilize the output or not. The straightforward calculation shows that the commutative condition of Proposition 4.1 is not satisfied, i.e.
[A¯ 1 , A¯ 2 ] 6= 0 where A¯ i = Ti−1 ATi with A =
0 0 −k1
1 0 −k2
0 1 −k3
. Hence Proposition 4.1 cannot be applied.
However, with the chosen kj for 1 ≤ j ≤ 3, based on the definition of A in (12), it is easy to find two positive definitive symmetric matrices P and Q , such that AT P + PA = −Q . For (18) and (19), it is obvious that the commutation surface between these two subsystems is defined as follows: S1 = x | x3 = 0, x ∈ Rn
which gives the projection matrix Γ of x on S1 as follows:
Γ =
1 0 0
0 1 0
0 0 0
!
G. Zheng et al. / Nonlinear Analysis: Hybrid Systems 4 (2010) 189–198
197
3 2 1 0 –1 –2 x1 x2 x3
–3 –4
0
5
10
15 20 time (s)
25
30
35
Fig. 2. The trajectory of x under the uniform output feedback controller (20).
1 0.5 0 –0.5 –1 –1.5 –2 –2.5 –3 y x3 switching signal
–3.5 5
10
15 20 time (s)
25
30
Fig. 3. The output trajectory under the uniform output feedback controller (20).
with which we obtain the following equality:
T1 Γ = T2 Γ =
0 0 1
1
γ
0 0
0 0 0
.
Consequently Theorem 4.1 is held, and the uniform controller can stabilize the output. The simulation results is depicted in Figs. 2 and 3. It is shown that the trajectory and the output of system (18) and (19) are stabilized via the uniform output feedback controller (20). 6. Conclusion This paper has given sufficient conditions for a class of linear switched systems, with which a uniform output feedback controller can be designed. After that, for a particular class of linear switched systems, we gave two methods to study the output stabilization problem via the proposed uniform output feedback controller. Then an illustrative example is given in order to highlight the proposed results. Nevertheless this paper is a preliminary work and other perspectives will be investigated in the future, such as the generalization of the proposed results to nonlinear systems, the deduction of less restrictive sufficient conditions for less restrictive class of linear switched systems than the one treated in this paper.
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Boutat, J.-P. Barbot, F. Kratz, A geometrical characterization of a class of 0-flat affine dynamical systems, in: Proc. of IEEE American Control Conference, 2009. [24] P.P. da, Silva Filho, Relative flatness and flatness of implicit systems, SIAM Journal on Control and Optimization 39 (6) (2001) 1929–1951. [25] K. Garg, Theory of Differentiation: A Unified Theory of Differentiation via New Derivate Theorems and New Derivatives, Wiley-Interscience, New York, 1998. [26] K.S. Narendra, J. Balakrishnan, A common Lyapunov function for stable lti systems with commuting a-matrices, IEEE Transactions on Automatic Control 39 (12) (1994) 2469–2471. [27] G. Zhai, B. Hu, K. Yasuda, A.N. Michel, Qualitative analysis of discrete-time switched systems, in: Proc. of IEEE American Control Conference, 2002, pp. 1880–1885. [28] J. Hespanha, A. Morse, Stability of switched systems with average dwell-time, in: Proc. of IEEE Conference on Decision and Control, 1999, pp. 2655–2660. [29] F. 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Contents lists available at ScienceDirect
Nonlinear Analysis: Hybrid Systems journal homepage: www.elsevier.com/locate/nahs
On uniform controller design for linear switched systems G. Zheng a,∗ , D. Boutat b , J.P. Barbot a,c a
INRIA Lille-Nord Europe, 40 avenue Halley, 59650 Villeneuve d’Ascq, France
b
LVR/ENSI, 10 Boulevard de Lahitolle, 18020 Bourges, France
c
ECS/ENSEA, 6 Avenue du Ponceau, 95014, Cergy-Pontoise, France
article
info
Article history: Received 3 April 2009 Accepted 11 September 2009 Keywords: Switched systems Feedback control Stability
abstract Traditional method to design a controller for each subsystem of switched systems will increase the complexity of the controller’s realization. Hence this paper gives sufficient conditions for designing a uniform output feedback controller for linear switched systems, and this common controller can be used for all subsystems of the switched systems. Then the output stabilization problem for a particular class of linear switched systems under this uniform output feedback controller has been studied. An illustrative example is given in order to highlight the proposed method. © 2009 Elsevier Ltd. All rights reserved.
1. Introduction The study of switched systems comes partly from the fact that switched systems have numerous applications in control of mechanical systems, process control, automotive industry, power systems, aircraft, traffic control, biology, network and many other fields (see for example [1–6]). The stabilization of switched systems is nowadays a widely investigated problem and many significant contributions were given with different techniques: A multiple Lyapunov approach [7,8], LaSalle invariant principle (linear [5] or nonlinear [9,10]), Lie algebra approach [11,12], smooth Lyapunov functions [13], small-gain theorem [14], generalized piecewise linear feedback [15] and switched systems with unknown time-varying delays [16]. For designing an output feedback controller, traditional method is to design different output feedback controllers associated to different subsystems of switched systems, and it is can be seen as designing a hybrid output feedback controller. Obviously, it will increase the complexity of the controller’s realization. In this paper, we just investigate the output stabilization problem of linear switched systems by common output for all subsystems, called uniform controller, with the concept of flat system [17]. The common output is used to overcome the difficulty to measure or at least observe [18] the discrete state, and this greatly simplifies the controller implementation. Hence sufficient conditions which guarantee the existence of a uniform output feedback controller should be studied. This paper is organized as follows. Motivations and problem statements are given in Section 2. In Section 3, we deduce sufficient conditions which guarantee the existence of a uniform output feedback controller for linear switched systems. Section 4 is devoted to studying the output stabilization problem for a particular class of linear switched systems via the uniform output feedback controller. This paper ends with an illustrative example for the purpose of highlighting the feasibility of the proposed results.
∗
Corresponding author. E-mail address: [email protected] (G. Zheng).
1751-570X/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.nahs.2009.09.008
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2. Problem statements Let us consider a class of switched systems as follows: x˙ = fσ (t ) (x) + gσ (t ) (x) u y = cσ (t ) (x)
(1)
where x ∈ Rn , y ∈ R, u ∈ R, fσ (t ) : Rn → Rn , gσ (t ) : Rn → Rn , cσ (t ) : Rn → R are sufficiently smooth where σ (t ) : R+ → Σ is a piecewise constant function representing the switching signal with Σ = {1, 2, . . . , m}. In order to design an output feedback controller, basic method is to design different output feedback controllers associated to different subsystems of (1), such as u = ϕi y, y˙ , . . . , y(n)
for i ∈ Σ .
(2)
By detecting the active mode of (1) from its output [19–22], we can then apply the pre-designed output feedback controller to achieve the control goal, such as stabilization of output. In (2), normally we have
ϕi 6= ϕj for i ∈ Σ and j ∈ Σ
(3)
which implies a hybrid controller should be synthesized, even if the desired output trajectory is the same for each subsystem. And this will absolutely increase the complexity of the controller’s structure. Motivated by this point, this paper considers the following problem: Is it possible to design an output feedback uniform controller for (1) to achieve the control goal? More precisely, we are looking for a controller for system (1) in the following form: u = ϕ y, y˙ , . . . , y(n)
∀i ∈ Σ
(4)
which means the functions ϕi and ϕj in (3) are equal to ϕ in (4). The interest of designing such a uniform output feedback controller can evidently simplify the structure of controller, since we do not need to design a sub-controller for each subsystem of (1). The structure of the controller (4) will keep unchanged no matter which subsystem of (1) is active. Moreover, unlike hybrid controller (2) which needs to use firstly an observer or estimator to detect which subsystem of (1) is active and then apply the associated controller, the uniform controller (4) does not need such kind of instrument. In fact, our main result is based on the following claim. Claim 2.1. For each subsystem of (1), if the following conditions are satisfied: 1. System (1) is 0-flat 1 for all i ∈ Σ ; 2. System (1) has a unique input–output relation for all i ∈ Σ ; then there exists a uniform output feedback controller for system (1).2 Indeed, the first condition of Claim 2.1 implies that, for each subsystem of (1), we can always obtain an input–output relation, such as u = Fi y, y˙ , . . . , y(n)
for all i ∈ Σ . The second condition implies that for all i ∈ Σ , we have Fi = F , which yields u = F y, y˙ , . . . , y(n) .
(5)
Hence for all i ∈ Σ , by well choosing a common function of the output and its successive derivatives as the output feedback control law, noted as u = G y, y˙ , . . . , y(n) , then (5) becomes G y, y˙ , . . . , y(n) = F y, y˙ , . . . , y(n)
which enables us to stabilize the output of system (1). In such a way, we can freely design a uniform output feedback controller for system (1) and it can be described in Fig. 1. This paper will treat the problems of unique input–output relation and stabilization of the switched systems by uniform output feedback controller only for linear switched systems. Based on Claim 2.1, this problem will be studied in two steps: the first step is to give sufficient conditions which guarantee the existence of a uniform output feedback controller for linear switched systems, and the second one is to study the output stabilization problem through the uniform controller.
1 0-flat means that the flat output depends only on the states [23,24]. 2 This claim ensures the same input–output behavior for each subsystem, but does not guarantee any stabilization property.
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Subsystem 1 Subsystem 2
Subsystem m
uniform controller Fig. 1. Illustrative diagram of uniform feedback controller design for switched systems.
3. Existence of a uniform output feedback controller for linear switched systems In this section, based on Claim 2.1 we will give sufficient conditions which guarantee the existence of a uniform output feedback controller for a class of linear switched systems. More precisely, let us consider the following dynamical system:
x˙ = Aσ (t ) x + Bσ (t ) u y = Cσ ( t ) x
(6)
where Aσ (t ) is n × n matrix, Bσ (t ) is n × 1 matrix and Cσ (t ) is 1 × n matrix for all σ (t ) ∈ Σ = {1, 2, . . . m} and our goal is to design a uniform output feedback controller in order to stabilize the output of system (6). It is obvious that, in order to guarantee the existence of a uniform output feedback controller for (6), one sufficient condition is the existence of a unique input–output relation for all i ∈ Σ . For this, let us note
C i Ci A i Ti = ...
.
Ci Ani −1
Theorem 3.1. If the following conditions are fulfilled: 1. Ci Aki Bi = 0 for all i ∈ Σ and 0 ≤ k ≤ n − 2; 2. Ci Ain−1 Bi = Cj Anj −1 Bj = γ 6= 0 for all i ∈ Σ and j ∈ Σ ; 3. C1 An1 , . . . , Cm Anm ∈ Im (T1 , . . . , Tm )
then we can deduce the following unique input–output relation: u=
1
γ
(n)
y
−
n −1 X
! lj y
(j)
(7)
j=0
where lj ∈ R, which implies there exists a uniform output feedback controller for (6). Proof. Let us consider system (6) for all i ∈ Σ , if conditions (1) and (2) of Theorem 3.1 are satisfied, then we have rank (Ti ) = n and the successive derivation of y gives y(j) = Ci Ai x j
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for 0 ≤ j ≤ n − 1 and y(n) = Ci Ani x + Ci Ani −1 Bi u
= Ci Ani x + γ u.
(8)
Moreover, if condition (3) of Theorem 3.1 is fulfilled, then there exists lj ∈ R such that C1 An1 , . . . , Cm Anm =
n −1 h X
j
lj C1 A1 , . . . , Cm Ajm
i
j =0
which implies, for all i ∈ Σ , we have Ci Ani x =
n −1 X
j
lj Ci Ai x =
j =0
n −1 X
lj y(j) .
j =0
Finally, (8) can be written as follows: u =
=
1
γ 1
γ
y(n) − Ci Ani x y(n) −
!
n −1 X
lj y(j)
j=0
which is a unique input–output relation independent of i ∈ Σ . Hence there exists a uniform output feedback controller for system (6). Remark 3.1. Conditions (1) and (2) are equivalent to the fact that each Ci x for i ∈ Σ is a flat output for the considered subsystem. Condition (3) is very restrictive and implies that the controller for each subsystem can be with the same structure. Remark 3.2. Theorem 3.1 gives only sufficient conditions to guarantee the existence of a uniform output feedback controller for system (6), and the deduced unique input–output relation is in the form of (7). In the next section, we will study the output stabilization problem for linear switched systems under the uniform output feedback control law. 4. Stability by the uniform output feedback control Let us reconsider system (6), for all σ (t ) ∈ Σ we note
eσ (t ),1 = y = Cσ (t ) x eσ (t ),j = e˙ σ (t ),j−1 = y(j−1)
for 2 ≤ j ≤ n and eσ (t ) = eσ (t ),1 , eσ (t ),2 , . . . , eσ (t ),n
T
.
Since Theorem 3.1 is satisfied, then the successive derivation of y gives y(n) = Cσ (t ) Anσ (t ) x + Cσ (t ) Anσ−(t1) Bσ (t ) u and finally we have 0
..
. e˙ σ (t ) = 0 0
1
.. .
0 0
··· .. . ··· ···
0
.. . eσ (t ) + 1 0
0
.. .
0 Cσ (t ) Anσ (t ) x + Cσ (t ) Aσn−(t1) Bσ (t ) u
(9)
for σ (t ) ∈ Σ . Remark 4.1. It should be noted that the output y is continuous for each subsystem, however discontinuous at the switching instant. Hence in this paper, we consider the Dini derivative [25] for y just after the switching instant tk , defined as follows: D+ y(tk ) = lim sup h→0+
y(tk + h) − y(tk ) h
and D+ y(tk ) = lim inf h→0+
y(tk + h) − y(tk ) h
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while in our case we have D+ y(tk ) = D+ y(tk ) = lim
y(tk + h) − y(tk ) h
h→0+
.
And for the control input, we keep u(tk ) = u(tk− ) at the switching instant. Let us assume that Theorem 3.1 is satisfied, then there exists a uniform feedback controller for system (6). By choosing the following control law: u=
1
γ
−
n −1 X
(j)
lj y
−
n X
j =0
! (j)
kj y
(10)
j =1
where kj ∈ R, which is chosen such that the polynomial k1 + k2 s + · · · + kn sn−1 is Hurwitz, (9) becomes: e˙ σ (t ) = Aeσ (t )
(11)
where
0
1
.
.. A= 0 −k1
.. .
0 −k2
··· .. . ··· ···
0
.. . .
(12)
1 −kn
Since eσ (t ) = Tσ (t ) x, hence the stability problem of (11) can be converted to study that of the following system: x˙ = Tσ−(1t ) ATσ (t ) x.
(13)
For system (13), some techniques are already developed to study the stability problem based on Lyapunov approaches. For example, under the asymptotic stability of each subsystem, a common Lyapunov function exists when the subsystems matrices are pairwise commutative [26,27]. In [11,12], the authors proposed a generalization of the commutativity notion, based on the solvability of the Lie algebra generated by the subsystems state matrices, i.e., state matrices are uppertriangularizable in a same reference frame. Based on the existing results, we note A¯ i = Ti−1 ATi for all i ∈ Σ . Let us remark that A defined in (12) can make the corresponding system exponentially stable with well chosen kj > 0 for 1 ≤ j ≤ n, hence A¯ i can also make (13) exponentially stable. If we note G the Lie algebra generated by A¯ i , the following proposition is obvious due to Theorem 3.1 of this paper and the results with respect to G stated in [11] and [12]. Proposition 4.1. Suppose that assumptions of Theorem 3.1 are satisfied. If there exists a common output feedback controller u in (10) such that all A¯ i for 1 ≤ i ≤ m are Hurwitz, and G the Lie algebra generated by A¯ i is solvable, then (13) is stabilizable via the uniform output feedback controller. A particular case of solvable Lie algebra is when the following commutative condition is satisfied:
[A¯ i , A¯ j ] = A¯ i A¯ j − A¯ j A¯ i = 0. It is obvious that it is not a difficult task to find a uniform output feedback controller in the form of (10) which makes all A¯ i for 1 ≤ i ≤ m exponentially stable, however it is generally difficult to guarantee all A¯ i satisfy the commutative condition. Hence, in the following, we will focus on a particular class of linear switched systems whose switching signal σ (t ) depends on its states as follows:
x˙ = A1 x + B1 u x˙ = A2 x + B2 u
and y = C1 x and y = C2 x
if Hx < 0 if Hx ≥ 0
(14)
where Ai is n × n matrix, Bi is n × 1 matrix, Ci and H are 1 × n matrices for all i ∈ Σ = {1, 2}. 4.1. Continuity of Lyapunov function on the commutative surface As discussed in the last section, after applying the uniform controller (10), we can deduce the following observation error: e˙ σ (t ) = Aeσ (t ) . Then let us consider the following manifold: S = x | Hx = 0, x ∈ Rn
which represents the commutation surface between the first and the second subsystem of (14). Then we can denote the projection matrix Γ to represent the projection of x on S.
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Theorem 4.1. Suppose that Theorem 3.1 is satisfied. If there exist two positive definitive symmetric matrices P, Q such that AT P + PA = −Q and if the following condition holds: T1 Γ = T2 Γ
where Γ is the projection matrix of x on S, Ti = CiT , (Ci Ai )T , . . . , Ci Ain−1
T T
for all i ∈ Σ , then the uniform controller (10)3
exponentially stabilizes the output of the linear switched systems (14). Proof. Without loss of generality, let us consider the following time sequence: t0 < t1 < · · · < ts , and suppose that when t ∈ [ti−1 , ti [ the first subsystem of (14) is active and when t ∈ [ti , ti+1 [ the second subsystem of (14) becomes active where ti is the switching instant from the first subsystem to the second one through the commutation surface S. Let us consider the following Lyapunov candidate function: Vσ (t ) (e) = eTσ (t ) Peσ (t ) with P a positive definitive symmetric matrix. Then when t ∈ [ti−1 , ti [, we have V˙ σ (t ) (e) = eTσ (t ) AT P + PA eσ (t )
= −eTσ (t ) Qeσ (t ) ≤ −α Vσ (t ) (e) with α = λmin { Q } /λmax {P } where λmax {Λ} (or λmin {Λ}) represents the maximal (or minimal) eigenvalue of Λ. And this leads V − (e) ≤ e−α (ti −ti−1 ) V + (e) σ t σ t i−1
i
+
−
where t (or t ) represents the instant just after (or before) the switching instant. Analogously, for t ∈ [ti , ti+1 [, with the same Lyapunov function, we obtain V˙ σ (t ) (e) ≤ −α Vσ (t ) (e) which yields V
σ ti− +1
(e) ≤ e−α(ti+1 −ti ) Vσ t + (e) . i
At the moment of switching instant t = ti , considering the projection of x on S, the Lyapunov function Vσ (t ) (e) becomes
T
P T1 Γ x ti−
V − (e) = T2 Γ x ti− σ t
T
P T2 Γ x ti−
= T2 Γ x ti+ = Vσ t + (e)
T
P T2 Γ x ti+
V − (e) = T1 Γ x ti− σ t i
.
If T1 Γ = T2 Γ , then i
i
which implies the continuity of the Lyapunov function V − and V + on S. Hence we have σ ti
V
σ ti+ +1
(e) = Vσ
− ti+1
σ ti
(e)
≤ e−α(ti+1 −ti ) Vσ t + (e) i
≤e
−α (ti+1 −ti−1 )
V
σ ti+ −1
(e) .
By induction, for t ∈ [tk , tk+1 [ with k tends to infinity, we have Vσ (t ) (e) ≤ e−α(t −t1 ) V + (e), which implies exponential σ t 0
convergence of y to zero. Remark 4.2. The equality T1 Γ = T2 Γ in Theorem 4.1 implies the output trajectory is continuous on the commutation surface, when switching from one active subsystem to another one. This condition is a little restrictive and it will be relaxed in the following.
3 Obviously if P and Q exist, this implies that k + k s + · · · + k sn−1 is a Hurwitz polynomial where k is given in (12). 1 2 n i
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4.2. Minimal dwell time between the commutations Similar to the above section, let us suppose that there exist two positive definitive symmetric matrices P, Q , such that AT P + PA = −Q
(15)
hence we can define α = λmin { Q } /λmax {P }. Then let us chose for each subsystem of (14) a Lyapunov function as follows: Vσ (t ) (e) = eTσ (t ) Peσ (t ) .
(16)
Denote βi = λmax TiT PTi , βi = λmin
n o TiT PTi for 1 ≤ i ≤ 2 and set β = max1≤i≤2 βi , β = max1≤i≤2 βi and µ = β/β .
Lemma 4.1. For each subsystem of (14), the chosen Lyapunov function (16) satisfies the following inequality: V + (e) ≤ µV − (e) σ ti σ ti
(17)
where ti ∈ R+ represents the switching instant. Proof. Consider the switching instant t = ti without generality we suppose that σ ti_ = 1 and σ ti+ = 2, then on the commutation surface S with the projection matrix Γ , we have
V + (e) = eT + Pe + σ ti
σ ti
σ ti
= xT ti Γ T T2T PT2 Γ x ti+ . Since β > β2 = λmax T2T PT2 and β < β1 = λmin T1T PT1 , we have V + (e) ≤ β2 xT ti+ Γ T Γ x ti+ σ ti ≤ β xT ti+ Γ T Γ x ti+ +
≤
β T − T T x ti Γ T1 PT1 Γ x ti− βi
≤
β T e Pe − = µV − (e) σ ti β σ ti− σ ti
and this ends the proof.
The inequality (17) means that the change of Lyapunov function (16) before and after the switching time is bounded, and Eq. (15) implies the stability of each subsystem of (14) via the uniform controller. Many methods are proposed to study the global stability of such type of linear switched systems. Roughly speaking, if we know the minimal dwell time for each subsystem of (14), then the well-known theorem based on dwell time [1,28,29] can be stated as follows: Theorem 4.2. If there exists a minimal dwell time τmin between the commutations, such that
τmin >
log µ
α
then there exists a uniform controller 4 of the form (10) which asymptotically stabilizes the output of the linear switched systems (14). Remark 4.3. Since system (14) is state dependent, alternative conditions on dwell time and poles placement for such type of system are also largely studied in the literature (see [1,30,31] for details). 5. Illustrative example Let us consider the following linear switched system:
x˙ =
−1
0 1
−β
y = 0
1 0
0 0 1
1
γ
βγ
!
1 0 u 0
!
x+
−γ x
4 which ensures the existence of P and Q .
if x3 < 0 (18)
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and
x˙ =
−1
0 1
−1
0 0 1
β
y = 0
1 0 u 0
!
0
!
x+
−γ −1 x. βγ
γ
if x3 ≥ 0 (19)
For systems (18) and (19) we have
0
C1 T1 = C1 A 1 = 0 2 C1 A 1 1
1
1
γ
βγ 1 − β γ β
0 0
and
0
C2 T2 = C2 A 2 = 0 C2 A22 1
1
−
γ
1
βγ 1 . β γ − β
0 0
Moreover it is easy to check that
C1 B1 = C2 B2 = 0 C1 A1 B1 = C2 A2 B2 = 0 C A2 B = C A2 B = 1 1 1 1 2 2 2 and
C1 A31 = −γ C2 A32 = −γ
−γ T1 −γ T2
−β − 1 −β − 1
which implies that C1 A31 , C2 A32 ∈ Im (T1 , T2 ) .
Hence Theorem 3.1 is satisfied and we can design a uniform controller for systems (18) and (19). According to Theorem 4.1, we design the following uniform output feedback controller: u = −γ y − (β + 1) y˙ − γ y¨ −
3 X
kj y(j)
(20)
j =1
where kj ∈ R for 1 ≤ j ≤ 3 is chosen such that the polynomial k1 + k2 s + k3 s2 is Hurwitz. For this example, we take β = γ = 1, k1 = 1, k2 = 0.1 and k3 = 1. Now let us check whether this uniform controller can stabilize the output or not. The straightforward calculation shows that the commutative condition of Proposition 4.1 is not satisfied, i.e.
[A¯ 1 , A¯ 2 ] 6= 0 where A¯ i = Ti−1 ATi with A =
0 0 −k1
1 0 −k2
0 1 −k3
. Hence Proposition 4.1 cannot be applied.
However, with the chosen kj for 1 ≤ j ≤ 3, based on the definition of A in (12), it is easy to find two positive definitive symmetric matrices P and Q , such that AT P + PA = −Q . For (18) and (19), it is obvious that the commutation surface between these two subsystems is defined as follows: S1 = x | x3 = 0, x ∈ Rn
which gives the projection matrix Γ of x on S1 as follows:
Γ =
1 0 0
0 1 0
0 0 0
!
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197
3 2 1 0 –1 –2 x1 x2 x3
–3 –4
0
5
10
15 20 time (s)
25
30
35
Fig. 2. The trajectory of x under the uniform output feedback controller (20).
1 0.5 0 –0.5 –1 –1.5 –2 –2.5 –3 y x3 switching signal
–3.5 5
10
15 20 time (s)
25
30
Fig. 3. The output trajectory under the uniform output feedback controller (20).
with which we obtain the following equality:
T1 Γ = T2 Γ =
0 0 1
1
γ
0 0
0 0 0
.
Consequently Theorem 4.1 is held, and the uniform controller can stabilize the output. The simulation results is depicted in Figs. 2 and 3. It is shown that the trajectory and the output of system (18) and (19) are stabilized via the uniform output feedback controller (20). 6. Conclusion This paper has given sufficient conditions for a class of linear switched systems, with which a uniform output feedback controller can be designed. After that, for a particular class of linear switched systems, we gave two methods to study the output stabilization problem via the proposed uniform output feedback controller. Then an illustrative example is given in order to highlight the proposed results. Nevertheless this paper is a preliminary work and other perspectives will be investigated in the future, such as the generalization of the proposed results to nonlinear systems, the deduction of less restrictive sufficient conditions for less restrictive class of linear switched systems than the one treated in this paper.
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