- Email: [email protected]
Robust switching of discrete-time switched linear systems
Automatica 48 (2012) 239–242
Contents lists available at SciVerse ScienceDirect
Automatica journal homepage: www.elsevier.com/locate/automatica
Technical communique
Robust switching of discrete-time switched linear systems✩ Zhendong Sun 1 Center for Control and Optimization, College of Automation Science and Engineering, South China University of Technology, Guangzhou 510640, China
article
info
Article history: Received 22 June 2011 Accepted 6 September 2011 Available online 10 December 2011 Keywords: Switched linear systems Switching signal Switching distance Robustness
abstract In this work, we address the problem of robust switching design, which seeks a switching signal that makes the switched system exponentially stable and robust against switching perturbations. To properly capture the sensitivity of a switching signal undergoing various switching perturbations, we define the relative distance between state-feedback path-wise switching signals. We establish that any stabilizing state-feedback path-wise switching signal is robust with respect to switching perturbations. A lower bound of the robustness margin is explicitly presented. © 2011 Elsevier Ltd. All rights reserved.
1. Introduction In this communique, we address the problem of robust switching design for the discrete-time switched linear system described by x(t + 1) = Aσ (t ) x(t ),
(1) def
where x(t ) ∈ Rn is the system state, σ (t ) ∈ M ={1, . . . , m} is the switching signal, and Ai ∈ Rn×n , i ∈ M are known constant matrices. For the switched linear system, much effort has been paid to the problem of stabilization by means of proper switching (Bacciotti & Mazzi, 2010; Lee & Khargonekar, 2009; Sun & Peng, 2010; Zhang, Abate, Hu, & Vitus, 2009). Furthermore, the problem of robust switching design, which is to seek proper switching mechanisms that make the system stable and attenuate possible system perturbations, has also been attracting much attention (Lin & Antsaklis, 2007; Sun, 2004). For switched systems with perturbations, the perturbations could be either structural or unstructural, and can enter into either the subsystems or the switching signal. When the perturbations are imposed on the subsystems, robust switching design schemes were developed in Lin and Antsaklis (2004, 2007), among many others. It was revealed that both the periodic switching law and the hysteresis state-feedback switching law could attenuate small subsystem
✩ This work was supported by the National Natural Science Foundation of China
under grants 60925013, 60736024, and U0735003. The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Maurice Heemels, under the direction of Editor André L. Tits. E-mail address: [email protected]. 1 Tel.: +86 20 87114550; fax: +86 20 8711 4256. 0005-1098/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2011.10.004
perturbations, albeit at the expense of arbitrarily fast switching (Sun, 2004). For switched systems with switching perturbations, however, the robust design relies on an appropriate definition of the distance between the nominal and the perturbed switching laws. When a periodic switching signal undergoes a constant delay, the distance could be understood in an intuitive way, and the robust switching design problem was addressed by several researchers (Ji, Guo, Xu, & Wang, 2007; Sun & Ge, 2006; Xie & Wang, 2005; Xie & Wu, 2008). On the other hand, when the switching signal is state-dependent and defined over an infinite time horizon, even a slight switching perturbation (for example, an inexact measure of the state, or a mismatch of switching paths) could result in an infinite deviation of the perturbed switching signal from the nominal one. It is thus interesting to properly introduce a switching distance that both sensibly captures the real difference between the switching signals and enables a rigorous robustness analysis based on it. In this work, based on the state-feedback path-wise switching signal proposed in our early work (Sun, 2009), we introduce the definition of relative distance between two switching signals, which measures the average (absolute) distance in time over an infinite horizon. By establishing a qualitative connection between the relative switching distance and the norm deviation of the state transition matrix, we prove that any stabilizable switched linear system is also robust with respect to switching perturbations, and we present a lower bound of the robustness margin. 2. Preliminaries 2.1. Stabilizability by means of state-feedback path-wise switching Switched linear system (1) is said to be exponentially stabilizable, if there exists a switching signal that steers the system exponentially stable.
240
Z. Sun / Automatica 48 (2012) 239–242
Suppose that k is a natural number, Ωi , i = 1, . . . , k are cones in Rn satisfying ∪ki=1 Ωi = Rn and Ωi ∩ Ωj = ∅ for all i ̸= j, and θi : [0, si ) → M, i = 1, . . . , k are switching paths. The state-feedback Ω path-wise switching signal, denoted by ∧ki=1 θi si i or ∧ki=1 θi Ωi in short, is the switching signal that concatenates the switching paths {θ1 , . . . , θk } via {Ω1 , . . . , Ωk }. The reader is referred to Sun (2009) for details. Denote by φ(t ; t0 , x0 , σ ) the state of system (1) at time t with initial condition x(t0 ) = x0 and switching signal σ . Lemma 1 (Sun, 2009). Suppose that switched system (1) is exponentially stabilizable. Then, there exist a state-feedback path-wise switchΩ ing signal ∧ki=1 θi si i , and a real number µ ∈ [0, 1), such that the switched system is exponentially stable with
‖φ(si ; 0, x, θi )‖ ≤ µ‖x‖,
∀ x ∈ Ωi , i = 1 , . . . , k .
(2)
A design procedure was presented in Sun (2009) for computing a switching signal satisfying Lemma 1. 2.2. Distance between state-feedback path-wise switching signals In this subsection, we are to define the distance between two switching signals. For this, we first recall the following definition of distance between two switching paths. Definition 1 (Sun & Ge, 2006). For any switching paths p1 and p2 , the (absolute) distance between them is defined as d(p1 , p2 ) =
min
(2|p3 |) − |p1 | − |p2 |,
p3 ∈CP (p1 ,p2 )
(3)
where | · | is the length of the path, and CP (p1 , p2 ) is the set of common parents paths of p1 and p2 . For more details of the definition, the reader is referred to Sun and Ge (2006). Next, we are to define the relative distance between the nominal and perturbed switching signals with respect to the stateΩ feedback path-wise switching ∧ki=1 θi si i . In the light of the recursion expression of the state-feedback path-wise switching, we re-write the nominal switching signal as
σn =
∞
θli : xi ∈ Ωli ,
xi+1 = φ(sli ; 0, xi , θli ),
(4)
i=0
where xi is the nominal current relay state. For comparison, we rewrite the perturbed switching signal as
σp =
∞
pi : yi ∈ Ωji ,
yi+1 = φ(τi ; 0, yi , pi ),
(5)
i=0
where yi is the perturbed current relay state with y0 = x0 , and pi : [0, τi ) → M is the perturbed implementation of θji . It can be seen that a deviation occurs when pi ̸= θji .
(2) For an initial state x0 , the relative distance between σnx and σpx at x0 is defined as RDx0 (σp , σn ) = lim sup N →∞
1 N
DNx0 (σp , σn ).
(7)
(3) The supremal relative distance between σnx and σpx is defined as SRD(σp , σn ) = sup RDx0 (σp , σn ).
(8)
x0 ∈Rn
Remark 1. The N-distance measures the absolute distance between the nominal switching and the perturbed switching over the first N-concatenating periods. It is the summation of the distances between the nominal and perturbed switching paths. The relative distance, on the other hand, measures the average distance in time over an infinite horizon. It should be stressed that the relative distance is more subtle than the (absolute) N-distance in characterizing the distance between two switching signals. In fact, for any two switching signals, the absolute ∞-distance must be infinite if the relative distance is positive, but the reverse is not necessarily true. 3. Main result Suppose that switched linear system (1) is exponentially stabilizable, and the state-feedback path-wise switching signal σ = ∧ki=1 θi Ωi makes the system exponentially stable. Let us consider the situation that the system undergoes switching perturbations. A description of the perturbed system can be given as x(t + 1) = Aσ¯ x(t ), where σ¯ is the perturbed switching signal. The objective of this section is to analyze the robustness of the switched linear system with respect to the switching perturbations. For this, we need a technical lemma that estimates the deviation of the state in terms of the switching distance. Lemma 2. For switched linear system (1), suppose that p1 and p2 are switching paths defined over intervals [0, τ1 ) and [0, τ2 ), respectively, and x is an arbitrarily given state. Let ζ = d(p1 , p2 ). Then, we have
‖φ(τ2 ; 0, x, p2 ) − φ(τ1 ; 0, x, p1 )‖ ≤ η2 η1τ −1 ζ ‖x‖, (9) m where η1 = max 1, maxm i=1 ‖Ai ‖ , η2 = maxi=1 ‖Ai − I ‖, and τ = (τ1 + τ2 + ζ )/2. Proof. By the definition of the distance between two switching paths, there is a common parent path p of p1 and p2 , such that d(p, p1 ) + d(p, p2 ) = ζ . Denote ζ1 = d(p, p1 ), and ζ2 = d(p, p2 ). Suppose for instance that h
h
φ(τ1 ; 0, x, p1 ) = A22 A11 x
Definition 2. For a state-feedback path-wise switching signal Ω ∧ki=1 θi i , let σn and σp be the nominal switching signal and perturbed switching signal, respectively.
and
(1) For a natural number N and an initial state x0 , the N-distance between σnx and σpx at x0 is defined as
Other cases can be treated in exactly the same way. Simple computation yields
h
h
h
h
φ(τ ; 0, x, p) = A22 A44 A11 A33 x.
h
DNx0
(σp , σn ) =
N −1 −
d(pi , θji ),
h
h
h
h
h
‖φ(τ ; 0, x, p) − φ(τ1 ; 0, x, p1 )‖ ≤ ‖A22 A44 A11 A33 x − A22 A11 x‖ (6)
i =0
where d(·, ·) is the distance between two switching paths as defined in (3).
h
h
h
h
h
h
h
≤ ‖A22 (A44 − I )A11 A33 x‖ + ‖A22 A11 (A33 − I )x‖ h +h2 +h3
≤ η1 1
τ −1
≤ η2 η1
h −1
‖A4 − I ‖η14
ζ1 ‖x‖,
h +h2
h4 ‖x ‖ + η 1 1
h −1
‖A3 − I ‖η13
h3 ‖x ‖
Z. Sun / Automatica 48 (2012) 239–242 h where the relationships τ = i=1 hi , ζ1 = h3 + h4 , and ‖A − I ‖ ≤ h−1 ‖A − I ‖‖A‖ h for any A and h, have been used. In a similar way, we can prove that
∑4
‖φ(τ2 ; 0, x, p2 ) − φ(τ ; 0, x, p)‖ ≤ η2 η1τ −1 ζ2 ‖x‖.
241
between the perturbed switching and the nominal switching is upper bounded by N (γ + ϵ). For any N ≥ l, define N1 = # {i ≤ N − 1: ‖yi+1 ‖ > µ‖ ¯ yi ‖} ,
N2 = N − N1 ,
Combining the above facts gives
where # denotes the cardinality of a set. By the definition of Ndistance, we have
‖φ(τ2 ; 0, x, p2 ) − φ(τ1 ; 0, x, p1 )‖ ≤ η2 η1τ −1 ζ ‖x‖.
N1 ≤ ⌈N ϖ ⌉,
This completes the proof.
where ⌈a⌉ (⌊a⌋) denotes the smallest (largest) integer equal to or greater (less) than a. Based on the above facts, routine calculation gives
As a main result, we establish that, the path-wise state-feedback switching signal is robust against switching perturbations. For this, denote 1+µ s = max{s1 , . . . , sk }, µ ¯ = 2 and
1−µ , ϑ = min 1, 2s 2η1 η2
λ = η2 η12s−1 +
µ . ϑ
− ln µ ¯ 2(ln(λ/µ) ¯ + 2ϑ ln η1 + 2ϑ/e) 1
ϑϖ .
Ω
Theorem 1. Suppose that the switching signal ∧ki=1 θi i exponentially stabilizes the nominal system (1). Then, for any perturbed switched linear system x(t + 1) = Aσ¯ (t ) x(t )
(10)
with SRD(σ¯ , σ ) ≤ γ , the perturbed switching signal exponentially stabilizes the perturbed system. Proof. Let x0 be any given state, and σ x0 and σ¯ x0 be the nominal and perturbed switching signals with respect to initial state x(0) = x0 , respectively. Recall that SRD(σ¯ , σ ) ≤ γ means that RDx0 (σ¯ x0 , σ x0 ) ≤ γ , which further means that, for any given ϵ > 0, the N-distance between the perturbed switching and the nominal switching is upper bounded by N (γ + ϵ) for sufficiently large N. Re-write the perturbed switching signal as in (5): ∞
pi : yi+1 = φ(τi ; 0, yi , pi ),
di >ϑ
≤e λ η1 e ‖y 0 ‖ ≤ exp (N (−ν1 + ϖ (ν1 + ν2 + 2ν3 ϑ + 2ϑ/e)) + ν1 + ν2 ) ‖y0 ‖,
‖yN ‖ ≤
2 It is clear that γ is a positive real number that relies on µ, s, and Ai , i = 1, . . . , m.
σp =
d
λη1i di
−ν1 (N (1−ϖ )−1) N ϖ +1 2N ϖ ϑ 2N ϖ ϑ/e
def
and
γ =
∏
def
(12)
def
¯ , ν2 = ln λ, ν3 = ln η1 , and the fact that where ν1 = − ln µ x a maxx ax = e e was used. Then, it follows from (12) that
Furthermore, let
ϖ =
‖yN ‖ ≤ ‖y0 ‖µ ¯ N2
N2 ≥ ⌊N (1 − ϖ )⌋,
y0 = x0 .
ν λ 1 exp − N ‖y0 ‖, µ ¯ 2
∀ N ≥ l,
which means that the sequence yl , yl+1 , . . . is exponentially convergent with rate ν1 /2. This completes the proof. Remark 2. Theorem 1 reveals that the state-feedback pathwise switching law is robust with respect to (small) switching perturbations, which means that the switching law is fault tolerant. This is a very important property from the practical point of view. Moreover, as γ is a guaranteed lower bound of the allowable robustness margin, this indicates a way for designing a stabilizing switching law with a good robustness property. Indeed, suppose that we have a set of candidate stabilizing laws, from the viewpoint of robustness, we would choose the one with the largest robustness margin bound. 4. Conclusion In this work, we proved that, for a stabilizable discretetime switched linear system, any exponentially stabilizing statefeedback path-wise switching signal is robust against small switching perturbations. A lower bound for the robustness margin was explicitly presented.
i=0
We are to prove that the state sequence y0 , y1 , . . . is exponentially convergent, which implies the exponential stability of the perturbed system. For any non-negative integer i, let ji ∈ M be an index such that yi ∈ Ωji . It is clear that
‖φ(sji ; 0, yi , θji )‖ ≤ µ‖yi ‖. Applying Lemma 2 yields
‖yi+1 ‖ = ‖φ(τi ; 0, yi , pi )‖ ≤ ‖φ(τi ; 0, yi , pi ) − φ(sji ; 0, yi , θji )‖ + ‖φ(sji ; 0, yi , θji )‖ 2s+d −1 ≤ η2 η1 i di + µ ‖yi ‖ µ‖ ¯ yi ‖ if di ≤ ϑ ≤ (11) d λη1i di ‖yi ‖ otherwise, where di = d(pi , θji ). Now choose ϵ = γ . Let l be a (sufficiently large) natural number such that, for any N ≥ l, the N-distance
Acknowledgments Part of this work was done when the author was a Visiting Professor with the Department of Mechanical and Automation Engineering, The Chinese University of Hong Kong, and the author is grateful to Jie Huang for his hospitality. The author would like to thank the anonymous reviewers and the AE for their helpful and insightful comments for further improving the quality of this work. References Bacciotti, A, & Mazzi, L. 2010. A discussion about stabilizing periodic and nearperiodic switching signals. In preprints of the 8th IFAC symposium on nonlinear control systems (pp. 250–255). Ji, Z., Guo, X., Xu, X., & Wang, L. (2007). Stabilization of switched linear systems with time-varying delay in switching occurrence detection. Circuits Systems and Signal Processing, 26(3), 361–377. Lee, J.-W., & Khargonekar, P. P. (2009). Detectability and stabilizability of discretetime switched linear systems. IEEE Transactions on Automatic Control, 54(3), 424–437.
242
Z. Sun / Automatica 48 (2012) 239–242
Lin, H., & Antsaklis, P. J. 2004. Persistent disturbance attenuation properties for networked control systems. In proceedings of 43rd IEEE conference on decision and control (pp. 953–958). Lin, H., & Antsaklis, P. J. (2007). Switching stabilizability for continuous-time uncertain switched linear systems. IEEE Transactions on Automatic Control, 52(4), 633–646. Sun, Z. (2004). A robust stabilizing law for switched linear systems. International Journal of Control, 77(4), 389–398. Sun, Z. (2009). Stabilizing switching design for switched linear systems: a statefeedback path-wise switching approach. Automatica, 45(7), 1708–1714. Sun, Z., & Ge, S. S. (2006). On stability of switched linear systems with perturbed switching paths. Journal of Control Theory & Applications, 4(1), 18–25.
Sun, Z., & Peng, Y. (2010). Stabilizing design for switched linear control systems: a constructive approach. Transactions of the Institute of Measurement and Control, 32(6), 706–735. Xie, G., & Wang, L. (2005). Stabilization of switched linear systems with timedelay in detection of switching signal. Journal of Mathematical Analysis and Applications, 305(1), 277–290. Xie, D., & Wu, Y. (2008). Stabilisability of switched linear systems with time-varying delay in the detection of switching signals. IET Electric Power Applications, 3(4), 404–410. Zhang, W., Abate, A., Hu, J., & Vitus, M. (2009). Exponential stabilization of discretetime switched linear systems. Automatica, 45(11), 2526–2536.
Contents lists available at SciVerse ScienceDirect
Automatica journal homepage: www.elsevier.com/locate/automatica
Technical communique
Robust switching of discrete-time switched linear systems✩ Zhendong Sun 1 Center for Control and Optimization, College of Automation Science and Engineering, South China University of Technology, Guangzhou 510640, China
article
info
Article history: Received 22 June 2011 Accepted 6 September 2011 Available online 10 December 2011 Keywords: Switched linear systems Switching signal Switching distance Robustness
abstract In this work, we address the problem of robust switching design, which seeks a switching signal that makes the switched system exponentially stable and robust against switching perturbations. To properly capture the sensitivity of a switching signal undergoing various switching perturbations, we define the relative distance between state-feedback path-wise switching signals. We establish that any stabilizing state-feedback path-wise switching signal is robust with respect to switching perturbations. A lower bound of the robustness margin is explicitly presented. © 2011 Elsevier Ltd. All rights reserved.
1. Introduction In this communique, we address the problem of robust switching design for the discrete-time switched linear system described by x(t + 1) = Aσ (t ) x(t ),
(1) def
where x(t ) ∈ Rn is the system state, σ (t ) ∈ M ={1, . . . , m} is the switching signal, and Ai ∈ Rn×n , i ∈ M are known constant matrices. For the switched linear system, much effort has been paid to the problem of stabilization by means of proper switching (Bacciotti & Mazzi, 2010; Lee & Khargonekar, 2009; Sun & Peng, 2010; Zhang, Abate, Hu, & Vitus, 2009). Furthermore, the problem of robust switching design, which is to seek proper switching mechanisms that make the system stable and attenuate possible system perturbations, has also been attracting much attention (Lin & Antsaklis, 2007; Sun, 2004). For switched systems with perturbations, the perturbations could be either structural or unstructural, and can enter into either the subsystems or the switching signal. When the perturbations are imposed on the subsystems, robust switching design schemes were developed in Lin and Antsaklis (2004, 2007), among many others. It was revealed that both the periodic switching law and the hysteresis state-feedback switching law could attenuate small subsystem
✩ This work was supported by the National Natural Science Foundation of China
under grants 60925013, 60736024, and U0735003. The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Maurice Heemels, under the direction of Editor André L. Tits. E-mail address: [email protected]. 1 Tel.: +86 20 87114550; fax: +86 20 8711 4256. 0005-1098/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2011.10.004
perturbations, albeit at the expense of arbitrarily fast switching (Sun, 2004). For switched systems with switching perturbations, however, the robust design relies on an appropriate definition of the distance between the nominal and the perturbed switching laws. When a periodic switching signal undergoes a constant delay, the distance could be understood in an intuitive way, and the robust switching design problem was addressed by several researchers (Ji, Guo, Xu, & Wang, 2007; Sun & Ge, 2006; Xie & Wang, 2005; Xie & Wu, 2008). On the other hand, when the switching signal is state-dependent and defined over an infinite time horizon, even a slight switching perturbation (for example, an inexact measure of the state, or a mismatch of switching paths) could result in an infinite deviation of the perturbed switching signal from the nominal one. It is thus interesting to properly introduce a switching distance that both sensibly captures the real difference between the switching signals and enables a rigorous robustness analysis based on it. In this work, based on the state-feedback path-wise switching signal proposed in our early work (Sun, 2009), we introduce the definition of relative distance between two switching signals, which measures the average (absolute) distance in time over an infinite horizon. By establishing a qualitative connection between the relative switching distance and the norm deviation of the state transition matrix, we prove that any stabilizable switched linear system is also robust with respect to switching perturbations, and we present a lower bound of the robustness margin. 2. Preliminaries 2.1. Stabilizability by means of state-feedback path-wise switching Switched linear system (1) is said to be exponentially stabilizable, if there exists a switching signal that steers the system exponentially stable.
240
Z. Sun / Automatica 48 (2012) 239–242
Suppose that k is a natural number, Ωi , i = 1, . . . , k are cones in Rn satisfying ∪ki=1 Ωi = Rn and Ωi ∩ Ωj = ∅ for all i ̸= j, and θi : [0, si ) → M, i = 1, . . . , k are switching paths. The state-feedback Ω path-wise switching signal, denoted by ∧ki=1 θi si i or ∧ki=1 θi Ωi in short, is the switching signal that concatenates the switching paths {θ1 , . . . , θk } via {Ω1 , . . . , Ωk }. The reader is referred to Sun (2009) for details. Denote by φ(t ; t0 , x0 , σ ) the state of system (1) at time t with initial condition x(t0 ) = x0 and switching signal σ . Lemma 1 (Sun, 2009). Suppose that switched system (1) is exponentially stabilizable. Then, there exist a state-feedback path-wise switchΩ ing signal ∧ki=1 θi si i , and a real number µ ∈ [0, 1), such that the switched system is exponentially stable with
‖φ(si ; 0, x, θi )‖ ≤ µ‖x‖,
∀ x ∈ Ωi , i = 1 , . . . , k .
(2)
A design procedure was presented in Sun (2009) for computing a switching signal satisfying Lemma 1. 2.2. Distance between state-feedback path-wise switching signals In this subsection, we are to define the distance between two switching signals. For this, we first recall the following definition of distance between two switching paths. Definition 1 (Sun & Ge, 2006). For any switching paths p1 and p2 , the (absolute) distance between them is defined as d(p1 , p2 ) =
min
(2|p3 |) − |p1 | − |p2 |,
p3 ∈CP (p1 ,p2 )
(3)
where | · | is the length of the path, and CP (p1 , p2 ) is the set of common parents paths of p1 and p2 . For more details of the definition, the reader is referred to Sun and Ge (2006). Next, we are to define the relative distance between the nominal and perturbed switching signals with respect to the stateΩ feedback path-wise switching ∧ki=1 θi si i . In the light of the recursion expression of the state-feedback path-wise switching, we re-write the nominal switching signal as
σn =
∞
θli : xi ∈ Ωli ,
xi+1 = φ(sli ; 0, xi , θli ),
(4)
i=0
where xi is the nominal current relay state. For comparison, we rewrite the perturbed switching signal as
σp =
∞
pi : yi ∈ Ωji ,
yi+1 = φ(τi ; 0, yi , pi ),
(5)
i=0
where yi is the perturbed current relay state with y0 = x0 , and pi : [0, τi ) → M is the perturbed implementation of θji . It can be seen that a deviation occurs when pi ̸= θji .
(2) For an initial state x0 , the relative distance between σnx and σpx at x0 is defined as RDx0 (σp , σn ) = lim sup N →∞
1 N
DNx0 (σp , σn ).
(7)
(3) The supremal relative distance between σnx and σpx is defined as SRD(σp , σn ) = sup RDx0 (σp , σn ).
(8)
x0 ∈Rn
Remark 1. The N-distance measures the absolute distance between the nominal switching and the perturbed switching over the first N-concatenating periods. It is the summation of the distances between the nominal and perturbed switching paths. The relative distance, on the other hand, measures the average distance in time over an infinite horizon. It should be stressed that the relative distance is more subtle than the (absolute) N-distance in characterizing the distance between two switching signals. In fact, for any two switching signals, the absolute ∞-distance must be infinite if the relative distance is positive, but the reverse is not necessarily true. 3. Main result Suppose that switched linear system (1) is exponentially stabilizable, and the state-feedback path-wise switching signal σ = ∧ki=1 θi Ωi makes the system exponentially stable. Let us consider the situation that the system undergoes switching perturbations. A description of the perturbed system can be given as x(t + 1) = Aσ¯ x(t ), where σ¯ is the perturbed switching signal. The objective of this section is to analyze the robustness of the switched linear system with respect to the switching perturbations. For this, we need a technical lemma that estimates the deviation of the state in terms of the switching distance. Lemma 2. For switched linear system (1), suppose that p1 and p2 are switching paths defined over intervals [0, τ1 ) and [0, τ2 ), respectively, and x is an arbitrarily given state. Let ζ = d(p1 , p2 ). Then, we have
‖φ(τ2 ; 0, x, p2 ) − φ(τ1 ; 0, x, p1 )‖ ≤ η2 η1τ −1 ζ ‖x‖, (9) m where η1 = max 1, maxm i=1 ‖Ai ‖ , η2 = maxi=1 ‖Ai − I ‖, and τ = (τ1 + τ2 + ζ )/2. Proof. By the definition of the distance between two switching paths, there is a common parent path p of p1 and p2 , such that d(p, p1 ) + d(p, p2 ) = ζ . Denote ζ1 = d(p, p1 ), and ζ2 = d(p, p2 ). Suppose for instance that h
h
φ(τ1 ; 0, x, p1 ) = A22 A11 x
Definition 2. For a state-feedback path-wise switching signal Ω ∧ki=1 θi i , let σn and σp be the nominal switching signal and perturbed switching signal, respectively.
and
(1) For a natural number N and an initial state x0 , the N-distance between σnx and σpx at x0 is defined as
Other cases can be treated in exactly the same way. Simple computation yields
h
h
h
h
φ(τ ; 0, x, p) = A22 A44 A11 A33 x.
h
DNx0
(σp , σn ) =
N −1 −
d(pi , θji ),
h
h
h
h
h
‖φ(τ ; 0, x, p) − φ(τ1 ; 0, x, p1 )‖ ≤ ‖A22 A44 A11 A33 x − A22 A11 x‖ (6)
i =0
where d(·, ·) is the distance between two switching paths as defined in (3).
h
h
h
h
h
h
h
≤ ‖A22 (A44 − I )A11 A33 x‖ + ‖A22 A11 (A33 − I )x‖ h +h2 +h3
≤ η1 1
τ −1
≤ η2 η1
h −1
‖A4 − I ‖η14
ζ1 ‖x‖,
h +h2
h4 ‖x ‖ + η 1 1
h −1
‖A3 − I ‖η13
h3 ‖x ‖
Z. Sun / Automatica 48 (2012) 239–242 h where the relationships τ = i=1 hi , ζ1 = h3 + h4 , and ‖A − I ‖ ≤ h−1 ‖A − I ‖‖A‖ h for any A and h, have been used. In a similar way, we can prove that
∑4
‖φ(τ2 ; 0, x, p2 ) − φ(τ ; 0, x, p)‖ ≤ η2 η1τ −1 ζ2 ‖x‖.
241
between the perturbed switching and the nominal switching is upper bounded by N (γ + ϵ). For any N ≥ l, define N1 = # {i ≤ N − 1: ‖yi+1 ‖ > µ‖ ¯ yi ‖} ,
N2 = N − N1 ,
Combining the above facts gives
where # denotes the cardinality of a set. By the definition of Ndistance, we have
‖φ(τ2 ; 0, x, p2 ) − φ(τ1 ; 0, x, p1 )‖ ≤ η2 η1τ −1 ζ ‖x‖.
N1 ≤ ⌈N ϖ ⌉,
This completes the proof.
where ⌈a⌉ (⌊a⌋) denotes the smallest (largest) integer equal to or greater (less) than a. Based on the above facts, routine calculation gives
As a main result, we establish that, the path-wise state-feedback switching signal is robust against switching perturbations. For this, denote 1+µ s = max{s1 , . . . , sk }, µ ¯ = 2 and
1−µ , ϑ = min 1, 2s 2η1 η2
λ = η2 η12s−1 +
µ . ϑ
− ln µ ¯ 2(ln(λ/µ) ¯ + 2ϑ ln η1 + 2ϑ/e) 1
ϑϖ .
Ω
Theorem 1. Suppose that the switching signal ∧ki=1 θi i exponentially stabilizes the nominal system (1). Then, for any perturbed switched linear system x(t + 1) = Aσ¯ (t ) x(t )
(10)
with SRD(σ¯ , σ ) ≤ γ , the perturbed switching signal exponentially stabilizes the perturbed system. Proof. Let x0 be any given state, and σ x0 and σ¯ x0 be the nominal and perturbed switching signals with respect to initial state x(0) = x0 , respectively. Recall that SRD(σ¯ , σ ) ≤ γ means that RDx0 (σ¯ x0 , σ x0 ) ≤ γ , which further means that, for any given ϵ > 0, the N-distance between the perturbed switching and the nominal switching is upper bounded by N (γ + ϵ) for sufficiently large N. Re-write the perturbed switching signal as in (5): ∞
pi : yi+1 = φ(τi ; 0, yi , pi ),
di >ϑ
≤e λ η1 e ‖y 0 ‖ ≤ exp (N (−ν1 + ϖ (ν1 + ν2 + 2ν3 ϑ + 2ϑ/e)) + ν1 + ν2 ) ‖y0 ‖,
‖yN ‖ ≤
2 It is clear that γ is a positive real number that relies on µ, s, and Ai , i = 1, . . . , m.
σp =
d
λη1i di
−ν1 (N (1−ϖ )−1) N ϖ +1 2N ϖ ϑ 2N ϖ ϑ/e
def
and
γ =
∏
def
(12)
def
¯ , ν2 = ln λ, ν3 = ln η1 , and the fact that where ν1 = − ln µ x a maxx ax = e e was used. Then, it follows from (12) that
Furthermore, let
ϖ =
‖yN ‖ ≤ ‖y0 ‖µ ¯ N2
N2 ≥ ⌊N (1 − ϖ )⌋,
y0 = x0 .
ν λ 1 exp − N ‖y0 ‖, µ ¯ 2
∀ N ≥ l,
which means that the sequence yl , yl+1 , . . . is exponentially convergent with rate ν1 /2. This completes the proof. Remark 2. Theorem 1 reveals that the state-feedback pathwise switching law is robust with respect to (small) switching perturbations, which means that the switching law is fault tolerant. This is a very important property from the practical point of view. Moreover, as γ is a guaranteed lower bound of the allowable robustness margin, this indicates a way for designing a stabilizing switching law with a good robustness property. Indeed, suppose that we have a set of candidate stabilizing laws, from the viewpoint of robustness, we would choose the one with the largest robustness margin bound. 4. Conclusion In this work, we proved that, for a stabilizable discretetime switched linear system, any exponentially stabilizing statefeedback path-wise switching signal is robust against small switching perturbations. A lower bound for the robustness margin was explicitly presented.
i=0
We are to prove that the state sequence y0 , y1 , . . . is exponentially convergent, which implies the exponential stability of the perturbed system. For any non-negative integer i, let ji ∈ M be an index such that yi ∈ Ωji . It is clear that
‖φ(sji ; 0, yi , θji )‖ ≤ µ‖yi ‖. Applying Lemma 2 yields
‖yi+1 ‖ = ‖φ(τi ; 0, yi , pi )‖ ≤ ‖φ(τi ; 0, yi , pi ) − φ(sji ; 0, yi , θji )‖ + ‖φ(sji ; 0, yi , θji )‖ 2s+d −1 ≤ η2 η1 i di + µ ‖yi ‖ µ‖ ¯ yi ‖ if di ≤ ϑ ≤ (11) d λη1i di ‖yi ‖ otherwise, where di = d(pi , θji ). Now choose ϵ = γ . Let l be a (sufficiently large) natural number such that, for any N ≥ l, the N-distance
Acknowledgments Part of this work was done when the author was a Visiting Professor with the Department of Mechanical and Automation Engineering, The Chinese University of Hong Kong, and the author is grateful to Jie Huang for his hospitality. The author would like to thank the anonymous reviewers and the AE for their helpful and insightful comments for further improving the quality of this work. References Bacciotti, A, & Mazzi, L. 2010. A discussion about stabilizing periodic and nearperiodic switching signals. In preprints of the 8th IFAC symposium on nonlinear control systems (pp. 250–255). Ji, Z., Guo, X., Xu, X., & Wang, L. (2007). Stabilization of switched linear systems with time-varying delay in switching occurrence detection. Circuits Systems and Signal Processing, 26(3), 361–377. Lee, J.-W., & Khargonekar, P. P. (2009). Detectability and stabilizability of discretetime switched linear systems. IEEE Transactions on Automatic Control, 54(3), 424–437.
242
Z. Sun / Automatica 48 (2012) 239–242
Lin, H., & Antsaklis, P. J. 2004. Persistent disturbance attenuation properties for networked control systems. In proceedings of 43rd IEEE conference on decision and control (pp. 953–958). Lin, H., & Antsaklis, P. J. (2007). Switching stabilizability for continuous-time uncertain switched linear systems. IEEE Transactions on Automatic Control, 52(4), 633–646. Sun, Z. (2004). A robust stabilizing law for switched linear systems. International Journal of Control, 77(4), 389–398. Sun, Z. (2009). Stabilizing switching design for switched linear systems: a statefeedback path-wise switching approach. Automatica, 45(7), 1708–1714. Sun, Z., & Ge, S. S. (2006). On stability of switched linear systems with perturbed switching paths. Journal of Control Theory & Applications, 4(1), 18–25.
Sun, Z., & Peng, Y. (2010). Stabilizing design for switched linear control systems: a constructive approach. Transactions of the Institute of Measurement and Control, 32(6), 706–735. Xie, G., & Wang, L. (2005). Stabilization of switched linear systems with timedelay in detection of switching signal. Journal of Mathematical Analysis and Applications, 305(1), 277–290. Xie, D., & Wu, Y. (2008). Stabilisability of switched linear systems with time-varying delay in the detection of switching signals. IET Electric Power Applications, 3(4), 404–410. Zhang, W., Abate, A., Hu, J., & Vitus, M. (2009). Exponential stabilization of discretetime switched linear systems. Automatica, 45(11), 2526–2536.