Automatica L2 gain analysis for a class of switched systems$
Automatica 45 (2009) 965–972
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Brief paper
L2 gain analysis for a class of switched systemsI Liang Lu a , Zongli Lin b,∗ , Haijun Fang c a
Department of Automation, Shanghai Jiao Tong University, Shanghai, 200240, China
b
Charles L. Brown Department of Electrical and Computer Engineering, University of Virginia, P.O. Box 400743, Charlottesville, VA 22904-4743, USA
c
MKS Instrument, 100 Highpower Road, Rochester, NY 14623, USA
article
info
Article history: Received 24 February 2008 Received in revised form 4 August 2008 Accepted 25 October 2008 Available online 7 January 2009 Keywords: Disturbance rejection Disturbance tolerance L2 gain Actuator saturation Switched systems Set invariance
a b s t r a c t This paper considers the problem of disturbance tolerance/rejection for a family of linear systems subject to actuator saturation and L2 disturbances. For a given set of linear feedback gains, a given switching scheme and a given bound on the L2 norm of the disturbances, conditions are established in terms of linear or bilinear matrix inequalities under which the resulting switched system is bounded state stable, that is, trajectories starting from a bounded set will remain inside the set or a larger bounded set. With these conditions, both the problem of assessing the disturbance tolerance/rejection capability of the closed-loop system and the design of feedback gain and switching scheme can be formulated and solved as constrained optimization problems. Disturbance tolerance is measured by the largest bound on the disturbances for which the trajectories from a given set remain bounded. Disturbance rejection is measured by the restricted L2 gain over the set of tolerable disturbances. In the event that all systems in the family are identical, the switched system reduces to a single system under a switching feedback law. It will be shown that such a single system under a switching feedback law has stronger disturbance tolerance/rejection capability than a single linear feedback law can achieve. © 2008 Elsevier Ltd. All rights reserved.
The literature on analysis and design of switched systems has been growing rapidly in recent years (see, for example, Branicky (1994), Cheng (2005), DeCarlo, Branicky, Pettersson, and Lennartson (2000), Liberzon and Morse (1999), Pettersson (1999), Pettersson and Lennartson (2001), Sun and Ge (2005), Wicks, Peleties, and DeCarlo (1998) and Xi, Feng, Jiang, and Cheng (2003) and the references therein). Motivated by the results reported in this literature, we consider in this paper the following family of linear systems subject to input saturation and disturbances, x˙ = Ai x + Bi sat(u) + Ei w, z = Ci x, i ∈ IN := {1, 2, . . . , N },
(1)
where x ∈ Rn , u ∈ Rm , z ∈ Rp are respectively the state, input and output of the system, w ∈ Rq represents the disturbances, and sat : Rm → Rm is the vector valued standard saturation
I An abridged version of this paper was presented at the 17th IFAC World Congress, Seoul, Korea, July 6–11, 2008. This paper was recommended for publication in revised form by Associate Editor Ben M. Chen under the direction of Editor Ian R. Petersen. ∗ Corresponding author. Tel: +1 434 924 6342; fax: +1 434 924 8818. E-mail addresses: [email protected] (L. Lu), [email protected] (Z. Lin), [email protected] (H. Fang).
0005-1098/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2008.10.021
T
function sat(u) = sat(u1 ) sat(u2 ) · · · sat(um ) , sat(ui ) = sign(ui ) min{|ui |, 1}. A switched system then results by defining a controller/supervisor which chooses one of the systems at each time instant based on the measurement of the state and according to an index function, say, i = σ (x). A typical form of the index function is σ (x) = i for x ∈ Ωi with ∪Ni=1 Ωi = Rn . Thus, the control design involves the construction of both feedback gains for individual systems and the index function so that the resulting switched system possesses certain desired performances. In the absence of the disturbances w , a basic design objective is the local asymptotic stability of the resulting switched system with as large a domain of attraction as possible. By utilizing some techniques in dealing with actuator saturation (Hu & Lin, 2001) and the form of the largest region index function proposed by Pettersson (2003, 2004, 2005), we recently proposed a method for the design of the individual feedback gains and the index function that result in a locally asymptotically stable switched system Lu and Lin (2008). The design is formulated and solved as a constrained optimization problem with the objective of enlarging the domain of attraction of the resulting stable equilibrium at the origin. It was shown by numerical examples that such a design may result in a domain of attraction larger than that of a switched system, designed without taking actuator saturation into account. In this paper, we will carry out an analysis of, and design for, the disturbance tolerance/rejection capability of the switched system resulting from the family of systems (1). We will restrict ourselves
1. Introduction
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to a class of disturbances whose energies are bounded by a given value, i.e.,
W 2α := w : R+ → Rq :
∞
Z
wT (t )w(t )dt ≤ α ,
(2)
0
for some positive number α . For a given set of linear feedback gains, a given index function and a given value of α , conditions will be established in terms of linear or bilinear matrix inequalities under which the resulting switched system is bounded state stable. A system is said to be bounded state stable if its trajectories starting from a bounded set will remain inside the set or a larger bounded set. With these conditions, both the problem of assessing the disturbance tolerance/rejection capability of the closed-loop system and the design of feedback gain and switching scheme can be formulated and solved as constrained optimization problems. Disturbance tolerance is measured by the largest bound on the energy of the disturbance, α ∗ , for which the trajectories from a given set remain bounded. Disturbance rejection is measured by the restricted L2 gain over W 2α ∗ . An interesting special class of the systems we consider in this paper is the case when all the systems in (1) are identical. In this case, the switched system reduces to a single system under a switching linear feedback law. It will be shown that for a single linear system of the form (1), a switching feedback law will result in stronger disturbance tolerance/rejection capability than a single linear feedback law of Fang, Lin, and Hu (2004) and Fang, Lin, and Shamash (2006). The L2 gain analysis and design for linear systems under actuator saturation has been studied by several authors. A small sample of their works include Chitour, Liu, and Sontag (1995), Fang et al. (2004, 2006), Hindi and Boyd (1998), Hu and Lin (2001), Lin (1997), Nguyen and Jabbari (1999) and Xie, Wang, Hao, and Xie (2004). In particular, in our recent work (Fang et al., 2004, 2006), we considered the L2 gain analysis and design for a linear system under actuator saturation. The disturbance tolerance capability of the closed-loop system under a given feedback law was assessed, and the linear feedback law that results in a minimized restricted L2 gain was designed. The remainder of this paper is organized as follows. In Section 2, we state our problem and recall some preliminary materials that will be needed in the development of the results of this paper. Section 3 establishes bounded state stability conditions. Disturbance tolerance and disturbance rejection are addressed in Sections 4 and 5, respectively. Simulation results are presented in Section 6. Section 7 concludes the paper. 2. Problem statement and preliminaries For the family of systems (1), we would like to design a linear feedback law for each individual system in the family and an index function such that the resulting switched system possesses a high degree of disturbance tolerance and a high level of disturbance rejection capabilities. We will adopt the switching strategy of Pettersson (2003, 2004). Such a switching strategy is defined based on some appropriately chosen symmetric matrices Qi ∈ Rn×n , i ∈ IN . More specifically, at a given state x, the subsystem i will be activated if the quadratic function xT Qi x is greater or equal to any other xT Qj x, j 6= i. More specifically, this switching scheme is defined by the following index function Pettersson (2003, 2004), referred to as the largest region function,
i(x) = arg max xT Qi x . i∈IN
(3)
Based on the matrices Qi ’s, we define the following sets
Ωi = {x ∈ Rn |xT Qi x ≥ 0}, i ∈ IN , Ωi,j = {x ∈ Rn |xT Qj x = xT Qi x ≥ 0},
i ∈ IN , j ∈ IN .
Then, a well-defined switched system must satisfy the following properties:
• Covering property: Ω1 ∪ Ω2 ∪ · · · ∪ ΩN = Rn ; • Switching property: Ωi,j ⊆ Ωi ∩ Ωj , i ∈ IN , j ∈ IN . The first condition says that there are no regions in the state space where none of the subsystem is activated. The second condition, which is automatically satisfied by this choice of Ωi and Ωi,j , means that a switch from subsystem i to j occurs only for states where the regions Ωi and Ωj are adjacent. Consequently, switching occurs on the switching surface xT Qi x = xT Qj x. The following regarding the covering property was established in Pettersson (2003, 2004). Lemma 1 (Covering property). If for every x ∈ Rn ,
θ1 xT Q1 x + θ2 xT Q2 x + · · · + θN xT QN x ≥ 0,
(4)
where θi > 0, i ∈ IN , then Ω1 ∪ Ω2 ∪ · · · ∪ ΩN = Rn . 3. Bounded state stability We recall a tool from Hu and Lin (2001) for expressing a saturated linear feedback u = sat(Fx) on the convex hull of a mixture of the unsaturated control inputs and the auxiliary inputs. For an F ∈ Rm×n , let L(F ) = x ∈ Rn : |fi x| ≤ 1, i ∈ Im , where fi represents the ith row of matrix F . We note that L(F ) represents the region in Rn where F x does not saturate. Also, let V be the set of m × m diagonal matrices whose diagonal elements are either 1 or 0. There are 2m elements in V . Suppose these elements of V are labeled as Ds , s ∈ I2m . Denote D− s = I − Ds . Clearly, D− ∈ V if D ∈ V . The following lemma is adopted from Hu s s and Lin (2001). Lemma 2. Let F , H ∈ Rl×n . Then, for any x ∈ L(H ), m , sat(Fx) ∈ co Ds Fx + D− s Hx, s ∈ I2
where co stands for the convex hull. For a positive definite P ∈ R n×n and a scalar ρ > 0, matrix n T we define E (P , ρ) := x ∈ R : x Px ≤ ρ . The following theorem characterizes the bounded state stability of the switched system that results from the family of systems (1) and the switching scheme (3). Theorem 1. Consider system (1). If there exist Pi > 0, ξ > 0, Qi = QiT , Fi ∈ Rm×n , Hi ∈ Rm×n , ϑi ≥ 0, θi > 0 and ηi,j such that 1 T T − 1. (Ai + Bi (Ds Fi + D− s Hi )) Pi + Pi (Ai + Bi (Ds Fi + Ds Hi ))+ ξ Pi Ei Ei Pi + ϑi Qi ≤ 0, s ∈ I2m , i ∈ IN , 2. Pi = Pj + ηi,j (Qj − Qi ), i ∈ IN , j ∈ IN , 3. θ1 Q1 + θ2 Q2 + · · · + θN QN ≥ 0,
and E (Pi , 1 + αξ ) ∩ Ωi ⊂ L(Hi ), i ∈ IN , then every trajectory of the closed-loop system that starts from inside of ∩Ni=1 (E (Pi , 1) ∩ Ωi ) will remain inside of ∩Ni=1 (E (Pi , 1 + αξ ) ∩ Ωi ) for every w ∈ W 2α , as long as no sliding motion occurs or sliding motions only occur along switching surfaces with the corresponding ηi,j ≥ 0. If the condition E (Pi , 1 +αξ )∩ Ωi ⊂ L(Hi ) is replaced with E (Pi , αξ )∩ Ωi ⊂ L(Hi ), then any trajectory starting from the origin will remain inside the region ∩Ni=1 (E (Pi , αξ ) ∩ Ωi ) for every w ∈ W 2α as long as no sliding motion occurs or sliding motions only occur along switching surfaces with the corresponding ηi,j ≥ 0.
L. Lu et al. / Automatica 45 (2009) 965–972
Proof. By Lemmas 1 and 2, Condition 3 of the theorem and the largest region function strategy (3) guarantee that the resulting switched system is well defined. We will define the energy of the system as its trajectory evolves in the state space. In each region Ωi , we will measure its energy as Vi (x) = xT Pi x. The derivative of Vi (x) along the trajectory of the subsystem i is then given by V˙ i = 2xT Pi (Ai x + Bi sat(Fi x) + Ei w). By Lemma 2, for every x ∈ E (Pi , 1 + αξ ) ∩ Ωi (or x ∈ E (Pi , α) ∩ Ωi ⊂ L(Hi )),
⊂ L(Hi )
m . sat(Fi x) ∈ co Ds Fi x + D− s Hi x, s ∈ I2
It follows that
∩Ni=1 (E (Pi , 1 + αξ ) ∩ Ωi ), and any trajectory that starts from x(0) = 0 will remain inside ∩Ni=1 (E (Pi , αξ ) ∩ Ωi ). We next consider the situation when sliding motion does occur. Denote A¯ i,s = Ai + Bi (Es Fi + Es− Hi ), i ∈ IN , s ∈ I2m , and A¯ j,s = Aj + Bj (Es Fj + Es− Hj ), j ∈ IN , s ∈ I2m . A sliding motion occurs along the hyper surface xT Qi x = xT Qj x ≥ 0, between two neighboring regions xT Qi x ≥ 0 and xT Qj x ≥ 0, if vector fields A¯ i,s x, s ∈ I2m , all point into region xT Qj x ≥ 0 and vector fields A¯ j,s , s ∈ I2m , all point into region xT Qi x ≥ 0 (Pettersson, 2005). A sliding motion may lead to either stable or unstable dynamics along the switching surface according to Filippov’s convex combination (Filippov, 1988) x˙ = λ
" m 2 X
m . Ai x + Bi sat(Fi x) ∈ co Ai x + Bi (Ds Fi + D− s H i ) x , s ∈ I2
2xT Pi Ei w ≤
# ρi,s (x)A¯ i,s x + Ei w
s=1
Noting that,
+ (1 − λ) 1
ξ
xT Pi Ei EiT Pi x + ξ w T w,
967
" m 2 X
# ρj,s (x)A¯ j,s x + Ej w ,
∀0 ≤ λ ≤ 1,
s=1
∀ξ > 0,
where ρi,s (x)
we have
>
P2m
= 1, and ρj,s (x) s=1 ρi,s (x) ρ ( x ) = 1, are such that (see Lemma 2) s = 1 j ,s
P2m
V˙ i = 2xT Pi (Ai x + Bi sat(Fi x) + Ei w)
≤ max 2xT Pi (Ai x + Bi (Ds Fi + D− s Hi ))x
Ai x + Bi sat(Fi x) =
s∈I2m
0,
2m X
0,
ρi,s (x)A¯ i,s x,
s=1
1
+ xT Pi Ei EiT Pi x + ξ w T w. ξ
Aj x + Bj sat(Fj x) =
V˙ i ≤ −ϑi xT Qi x + ξ w T w,
∀x ∈ E (Pi , 1 + αξ ) ∩ Ωi (or ∀x ∈ E (Pi , αξ ) ∩ Ωi ).
Recall that system i is activated in the region E (Pi , 1 + αξ ) ∩ Ωi (or E (Pi , αξ ) ∩ Ωi ). In this region, xT Qi x ≥ 0. Thus, V˙ i ≤ ξ w T w, ∀x ∈ E (Pi , 1 + αξ ) ∩ Ωi
(or ∀x ∈ E (Pi , αξ ) ∩ Ωi ).
Let V (x) = Vi(x) (x) be the overall energy of the switched system for use as a Lyapunov function candidate. Such a function is, in general, not differentiable along the boundaries between Ωi ’s, i.e., Ωi,j , i ∈ IN , j ∈ IN , where the switch from subsystem i to subsystem j occurs and where Vi (x) = Vj (x) due to Condition 2 of the theorem. But, for any state not on the boundaries, we have V˙ ≤ ξ w T w,
∀x ∈ ∩Ni=1 (E (Pi , 1 + αξ ) ∩ Ωi )
(or ∀x ∈ ∪Ni=1 (E (Pi , αξ ) ∩ Ωi )).
2m X
ρj,s (x)A¯ j,s x.
s=1
It then follows from Condition 1 in Item 1 of the theorem that
(5)
Consequently, if sliding mode does not occur on any of the boundaries, then let tj , j = 1, 2, . . . , Nt , be the times when the trajectory crosses the boundaries of Ωi ’s before time t. Then, integrating both sides of the above inequality from 0 to t results in
We will show that a sliding motion will not destroy the closedloop property established above, if it occurs on a switching surface with corresponding ηi,j ≥ 0. Indeed, if ηi,j = 0, Condition 2 of the theorem implies that Pi = Pj , which result in a common Lyapunov function for dynamics in both Ωi and Ωj . That is, the Lyapunov function V (x) is differentiable along Ωi,j and (5) is also valid for any x ∈ Ωi,j . We next consider the case of ηi,j > 0. According to the analysis of sliding motions in Pettersson (2005), the sliding motion occurring along the surface xT Qi x = xT Qj x ≥ 0 implies that
2m X xT ρi,s (x)A¯ Ti,s (Qi − Qj )x s = 1 2m X T ρi,s (x)A¯ i,s x + 2xT (Qi − Qj )Ei w < 0, + x (Qi − Qj ) m
s =1
2 X T x ρj,s (x)A¯ Tj,s (Qi − Qj )x s=1 2m X + xT (Qi − Qj ) ρj,s (x)A¯ j,s x + 2xT (Qi − Qj )Ej w > 0. s=1
Nt −1
V (x(t )) ≤ V (x(0)) + ξ
+ξ
Z
tj+1
XZ j =0
Using Condition 2 and noting that ηi,j > 0, we have
w T (τ )w(τ )dτ
tj
2m X xT ρi,s (x)A¯ Ti,s (Pj − Pi )x s=1 2m X T ρi,s (x)A¯ i,s x + 2xT (Pj − Pi )Ei w < 0, + x (Pj − Pi )
t
w (τ )w(τ )dτ T
Nt
= V (x(0)) + ξ
t
Z
w T (τ )w(τ )dτ 0
1 + αξ ,
≤
>
αξ ,
m
if x(0) ∈ ∪ if x(0) = 0,
N i =1
(E (Pi , 1) ∩ Ωi ) ,
(6)
where we have set t0 = 0. This shows that any trajectory that starts from ∩Ni=1 (E (Pi , 1) ∩ Ωi ) will remain inside
s=1
2 X T ρj,s (x)A¯ Tj,s (Pj − Pi )x x s=1 2m X + xT (Pj − Pi ) ρj,s (x)A¯ j,s x + 2xT (Pj − Pi )Ej w > 0, s=1
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L. Lu et al. / Automatica 45 (2009) 965–972
which is equivalent to
2m 2m X X T T T ¯ x ρ ( x ) A P x + x P ρi,s (x)A¯ i,s x + 2xT Pj Ei w i ,s j i,s j s=1 s=1 2m 2m X X T T T <x ¯ ρ ( x ) A P x + x P ρi,s (x)A¯ i,s x + 2xT Pi Ei w, i,s i i ,s i s=1
s=1
s=1
s=1
2m 2m X X T T T ¯ x ρj,s (x)Aj,s Pi x + x Pi ρj,s (x)A¯ j,s x + 2xT Pi Ej w s=1 s=1 2m 2m X X T T T ¯ <x ρj,s (x)Aj,s Pj x + x Pj ρj,s (x)A¯ j,s x + 2xT Pj Ej w.
Hence, by Condition 1 of the theorem, we have
2m 2m X X T T T ¯ x ρi,s (x)Ai,s Pj x + x Pj ρi,s (x)A¯ i,s x + 2xT Pj Ei w s=1 s=1 < −ϑi xT Qi x + ξ w T w ≤ ξ w T w, 2m 2m X X T T T ¯ x ρj,s (x)Aj,s Pi x + x Pi ρj,s (x)A¯ j,s x + 2xT Pi Ej w s=1 s=1 < −ϑj xT Qj x + ξ w T w ≤ ξ w T w, from which and Condition 1 of the theorem, we have that, for all 0 ≤ λ ≤ 1,
! " X 2m 2m X T T ρi,s (x)A¯ i,s Pi + Pi ρi,s (x)A¯ i,s x λ s =1 s=1 !# 2m 2m X X T ¯ ¯ ρj,s (x)Aj,s Pi + Pi ρj,s (x)Aj,s x + (1 − λ) s =1 s=1 +"2λxT Pi Ei w + 2(1 − λ)xT Pi Ej w < ξ wT w, ! 2m 2m X X T T ¯ ¯ x λ ρj,s (x)Aj,s Pj + Pj ρj,s (x)Aj,s s =1 s =1 !# 2m 2m X X T + (1 − λ) ρi,s (x)A¯ i,s Pj + Pj ρi,s (x)A¯ i,s x s =1 s=1 + 2λxT Pj Ej w + 2(1 − λ)xT Pj Ei w < ξ wT w. This, in turn, implies that
!T 2m 2m X X T ρi,s (x)A¯ i,s + (1 − λ) ρj,s (x)A¯ j,s Pi x λ s = 1 s = 1 !# 2m 2m X X + Pi λ ρi,s (x)A¯ i,s + (1 − λ) ρj,s (x)A¯ j,s x s = 1 s = 1 T T + 2x Pi λEi w + (1 − λ)Ej w < ξ w w, !T 2m 2m X X T x λ ρi,s (x)A¯ i,s + (1 − λ) ρj,s (x)A¯ j,s Pj s =1 s =1 !# 2m 2m X X + Pj λ ρi,s (x)A¯ i,s + (1 − λ) ρj,s (x)A¯ j,s x s =1 s=1 + 2xT Pj λEj w + (1 − λ)Ei w < ξ wT w. Therefore, the inequality (5), and hence (6), is still valid for any x ∈ Ωi,j . This completes the proof. 4. Disturbance tolerance A fundamental problem to be addressed before determining the restricted L2 gain is the assessment of the disturbance tolerance capability of the closed-loop system. The disturbance tolerance capability is measured by the largest bound on the energy of
the disturbances, say α ∗ , under which the closed-loop trajectories starting from the origin or a given set of initial conditions remain bounded. As the restricted L2 gain will be defined with zero initial conditions, we will only assess disturbance tolerance with zero initial conditions in this section. Disturbance tolerance with a given set of initial conditions can be dealt with similarly by resorting to the first part of Theorem 1. As established in Theorem 1, under the three itemized conditions and the condition that E (Pi , αξ ) ⊂ L(Hi ), i ∈ IN , the trajectories of the closed-loop system that start from origin will remain inside the region ∩Ni=1 (E (Pi , αξ ) ∩ Ωi ) for every w ∈ W 2α , if no sliding motion occurs or sliding motions only occur along switching surfaces with the corresponding ηi,j ≥ 0. Without loss of generality, we can assume that ξ = 1 in the above mentioned result. If ξ 6= 1, we can multiply both sides of Condition 1 of Theorem 1 with 1/ξ to obtain Pi T Pi + (Ai + Bi (Ds Fi + D− (Ai + Bi (Ds Fi + D− s Hi )) s Hi ))
ξ
Pi
Pi Ei EiT
≤ 0, s ∈ I2m , i ∈ IN . ξ Let P˜i = Pi /ξ and Q˜ i = Qi /ξ . Then, P˜i and Q˜ i satisfy all conditions of Theorem 1 with ξ = 1 and E (P˜i , α) = E (Pi , αξ ). As a result, +
ξ
ξ
+ ϑi
ξ
Qi
the disturbance tolerance capability of the closed-loop system under zero initial conditions can be assessed through solving the following optimization problem,
α,
sup
(7)
Pi >0,Qi =QiT ,ηi,j ,δi >0,ϑi ≥0,Hi ,θi >0 T − (a) (Ai + Bi (Ds Fi + D− s Hi )) Pi + Pi (Ai + Bi (Ds Fi + Ds Hi ))
s.t.
+ Pi Ei EiT Pi + ϑi Qi ≤ 0, s ∈ I2m , i ∈ IN , (b) Pi = Pj + ηi,j (Qj − Qi ), i ∈ IN , j ∈ IN , (c) θ1 Q1 + θ2 Q2 + · · · + θN QN ≥ 0, (d) E (Pi , α) ∩ Ωi ⊂ L(Hi ), i ∈ IN . Let ν = 1/α . Then, Constraint (d) is implied by hi,k (Pi − δi Qi )−1 hTi,k ≤ ν, k ∈ Im , which, by Schur complements, is equivalent to,
ν
hi,k
≥ 0, k ∈ Im , (Pi − δi Qi ) where δi > 0 and hi,k denotes the kth row of Hi . hTi,k
(8)
Consequently, the optimization problem (7) can be written as the following BMI problem, inf
Pi >0,Qi =QiT ,ηi,j ,δi >0,ϑi ≥0,Hi ,θi >0
s.t.
ν,
T − (a)(Ai + Bi (Ds Fi + D− s Hi )) Pi + Pi (Ai + Bi (Ds Fi + Ds Hi ))
(9)
+ Pi Ei EiT Pi + ϑi Qi ≤ 0, s ∈ I2m , i ∈ IN , (b) Pi = Pj + ηi,j (Qj − Qi ), i ∈ IN , j ∈ IN , (c) θ1 Q1 + θ2 Q2 + · · · + θN QN ≥ 0, ν h i ,k (d) ≥ 0, k ∈ Im , i ∈ IN . T hi,k (Pi − δi Qi ) If we define PN = P and ηi = ηj,i , i ∈ IN −1 , then, Constraint (b)
simplifies to
Pi = P + ηi (QN − Qi ),
i ∈ IN − 1 , and consequently, Constraint (a) simplifies to T (a) (Ai + Bi (Ds Fi + D− s Hi )) (P + ηi (QN − Qi ))
+ (P + ηi (QN − Qi ))(Ai + Bi (Ds Fi + D− s Hi )) + (P + ηi (QN − Qi ))Ei EiT (P + ηi (QN − Qi )) + ϑi Qi ≤ 0, s ∈ I2m , i ∈ IN −1 , T (AN + BN (Ds FN + D− s HN )) P T + P (AN + BN (Ds FN + D− s HN )) + PEN EN P + ϑN QN ≤ 0, s ∈ I2m .
L. Lu et al. / Automatica 45 (2009) 965–972
In case of switching between only two subsystems, we can set Q1 = Q and Q2 = −Q , where Q is a symmetric matrix. Furthermore, we can, without loss of generality, scale θ1 = θ2 = 1. This implies that Constraint (c) in (9) is automatically satisfied. The optimization problem (9) then simplifies to inf
P >0,P −2ηQ >0,η,δ1 >0,δ2 >0,Q =Q T ,ϑ1 ≥0,ϑ2 ≥0,H1 ,H2
ν,
(10)
+ (P − 2ηQ )E1 E1T (P − 2ηQ ) + ϑ1 Q ≤ 0, s ∈ I2m , T (A2 + B2 (Ds F2 + D− s H2 )) P + P (A2 + B2 (Ds F2 − T m + Ds H2 )) + PE2 E2 P − ϑ2 Q ≤ 0, s ∈ I2 , ν h1,k (b) ≥ 0, k ∈ Im , hT1,k P − 2ηQ − δ1 Q ν h2,k ≥ 0, k ∈ Im . hT2,k P + δ2 Q
The problem of verifying the existence of the unknown variables solving the optimization problem (10), is a bilinear matrix inequality (BMI) problem, which is NP-hard and difficult to solve. However, many algorithms for BMI problems have been proposed on the basis of approximations, heuristics, branch & bound, or local search. For example, PENOPT offers a commercial solver PENBMI for solving the optimization problems with bilinear matrix inequality constraints. We will use PENBMI to obtain all our numerical results in Section 6. While the conservativeness of these numerical results is not clear, they do demonstrate the effectiveness of the proposed method. The above optimization problem can be adapted for the design of feedback gains Fi ’s. This can be readily done by viewing Fi ’s as additional optimization parameters. 5. L2 gain analysis The restricted L2 gain of the closed-loop system is defined over a set of tolerable disturbances, say, W 2α , α ∈ (0, α ∗ ], as
kz kL2 , kwkL2 x(0)=0,w∈Wα2 \{0} sup
where k · kL2 is the L2 norm of a signal. Thus, the closed-loop system has a restricted L2 gain over W 2α less than or equal to γ if, for x(0) = 0, t
Z
z (τ )z (τ )dτ ≤ γ T
0
2
By Lemma 2, for every x ∈ E (Pi , α) ∩ Ωi ⊂ L(Hi ), m . sat(Fi x) ∈ co Ds Fi x + D− s H i x , s ∈ I2
It follows that m . Ai x + Bi sat(Fi x) ∈ co Ai x + Bi (Ds Fi + D− s Hi )x, s ∈ I2
Recall that system i is activated in the region E (Pi , α), where xT Pi x ≥ 0. Thus, in view of Condition 1 of the theorem, and noting that, 2xT Pi Ei w ≤ xT Pi Ei EiT Pi x + w T w, we have V˙ i = 2xT Pi (Ai x + Bi sat(Fi x) + Ei w) T T T ≤ max 2xT Pi (Ai x + Bi (Ds Fi + D− s Hi ))x + x Pi Ei Ei Pi x + w w, s∈I2m
1
≤−
γ2
=−
γ2
1
xT CiT Ci x + w T w z T z + w T w, ∀x ∈ E (Pi , α) ∩ Ωi .
Now, let V (x) = Vi(x) (x) be the overall energy of the switched system for use as a Lyapunov function candidate. Such a function is in general not differentiable along the boundaries between Ωi ’s, i.e., Ωi,j ’s, where the switch from subsystem i to subsystem j occurs and where Vi (x) = Vj (x) due to Condition 2 of the theorem. However, for any state not on these boundaries, we have V˙ ≤ −
1
γ2
z T z + w T w,
∀x ∈ ∩Ni=1 (E (Pi , α) ∩ Ωi ) .
w (τ )w(τ )dτ , T
0
∀t ≥ 0, ∀w ∈ W α . 2
T T − 1. (Ai + Bi (Ds Fi + D− s Hi )) Pi + Pi (Ai + Bi (Ds Fi + Ds Hi )) + Pi Ei Ei Pi + 1 T C C + ϑi Qi ≤ 0, s ∈ I2m , i ∈ IN , γ2 i i
2. Pi = Pj + ηi,j (Qj − Qi ), i ∈ IN , j ∈ IN , 3. θ1 Q1 + θ2 Q2 + · · · + θN QN ≥ 0,
and E (Pi , α) ∩ Ωi ⊂ L(Hi ), i ∈ IN , then the restricted L2 gain from w to z over W 2α is less than or equal to γ , if no sliding motion occurs or sliding motions only occur along switching surfaces with the corresponding ηi,j ≥ 0.
tj+1
1 − 2 z T (τ )z (τ ) + wT (τ )w(τ ) dτ γ tj j=0 Z t 1 + − 2 z T (τ )z (τ ) + w T (τ )w(τ )d dτ γ Nt Z t 1 = V (x(0)) + − 2 z T (τ )z (τ ) + wT (τ )w(τ ) dτ , γ 0
V (x(t )) ≤ V (x(0)) +
The following theorem characterizes the conditions under which the switched linear system has a restricted L2 gain less than or equal to γ . Theorem 2. Consider system (1) and an α ∈ (0, α ∗ ]. If there exist Pi > 0, Qi = QiT , Fi ∈ Rm×n , Hi ∈ Rm×n , ϑi ≥ 0, ηi,j and θi > 0 such that
(11)
It was shown in the proof of Theorem 1, the set ∩Ni=1 (E (Pi , α) ∩ Ωi ) is an invariant set. Consequently, if sliding mode does not occur on any of the boundaries, then let tj , j = 1, 2, . . . , Nt , be the times when the trajectory crosses the boundaries of Ωi ’s before time t. Then, integrating both sides of the above inequality from 0 to t results in Nt −1
t
Z
Proof. By Lemmas 1 and 2, Condition 3 of the theorem and the largest region function strategy (3) guarantee that the resulting switched system is well defined. To continue with the proof, we define the energy of the system as its trajectory evolves in the state space. In each region Ωi , we will measure its energy as Vi (x) = xT Pi x. The derivative of Vi (x) along the trajectory of the subsystem i is then given by V˙ i = 2xT Pi (Ai x + Bi sat(Fi x) + Ei w).
T s.t (a) (A1 + B1 (Ds F1 + D− s H1 )) (P − 2η Q ) + (P − 2ηQ )(A1 + B1 (Ds F1 + D− s H1 ))
γ∗ =
969
XZ
∀x ∈ ∩Ni=1 (E (Pi , α) ∩ Ωi ) ,
(12)
where we have set t0 = 0. Noting that V (x(t )) ≥ 0 and V (0) = 0, we have that, for x(0) = 0, t
Z
z T (τ )z (τ )dτ ≤ γ 2 0
t
Z
wT (τ )w(τ )dτ ,
(13)
0
which indicates that, if no sliding motion occurs, the restricted L2 gain from w to z over W 2α is less than or equal to γ . We next examine the situation when a sliding motion occurs. We will show that the restricted L2 gain from w to z over W 2α is still less than or equal to γ , if a sliding motion occurs on a switching
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L. Lu et al. / Automatica 45 (2009) 965–972
surface with corresponding ηi,j ≥ 0. Indeed, if ηi,j = 0, Condition 2 of the theorem implies that Pi = Pj , resulting in a common Lyapunov function for the dynamics in both Ωi and Ωj . That is, the Lyapunov function V (x) is differentiable along Ωi,j and (12) is also valid for x ∈ Ωi,j . For the case when ηi,j > 0, as has been shown in the proof of Theorem 1, the sliding motion occurring along the surface xT Qi x = xT Qj x ≥ 0 implies (3), from which and Condition 1 of this theorem, we have
CiT 0 ≤ 0. −γ 2 I
(14)
(b) (c) (d)
ν ν
−5
1 ,
C1 = 1
0
0.1 E1 = , 0.1
0 B1 = , 1
0 , 1
B2 =
1 ,
F2 = 5
1 ,
F1 = 0
1 0
C2 = 1
,
0 , 1
E2 =
0.1 , 0.1
0 .
To design a switching law that maximizes the disturbance tolerance capacity of the resulting switched system, we solve the optimization problem (10) and obtain
Thus, based on Theorem 2, the restricted L2 gain can be estimated by solving the following optimization problem,
s.t. (a)
s ∈ I2m ,
As with the optimization problems for assessing disturbance tolerance, the optimization problems for estimating the restricted L2 gain can also be adapted for the design of feedback gains Fi ’s by simply viewing Fi ’s as additional optimization parameters.
A2 =
T − Ms,i = (Ai + Bi (Ds Fi + D− s Hi )) Pi + Pi (Ai + Bi (Ds Fi + Ds Hi )) + ϑi Qi .
(16)
Msi Pi Ei CiT EiT Pi −I 0 ≤ 0 , s ∈ I2 m , i ∈ IN , Ci 0 −γ 2 I Pi = Pj + ηi,j (Qj − Qi ), i ∈ IN , j ∈ IN , θ Q + θ2 Q2 + · · · + θN QN ≥ 0, "1 11 # hi,k ≥ 0, k ∈ Im , i ∈ IN . α hTi,k (Pi − δi Qi )
0
PE2 C2T −I 0 ≤ 0, s ∈ I2m , 0 −γ 2 I h1,k ≥ 0, k ∈ Im , P − 2ηQ − δ1 Q h 2 ,k ≥ 0, k ∈ Im , P + δ2 Q
hT1,k
1 A1 = 0
(15)
γ 2,
C1T 0 ≤ 0, −γ 2 I
Example 1. Let us consider system (1) with w ∈ W 2α and
where
inf
(17)
6. Numerical examples
Pi >0,Qi =QiT ,ηi,j ,δi >0,ϑi ≥0,Hi ,θi >0
(P − 2ηQ )E1 −I
Ms,1 (P − 2ηQ ) C1
hT2,k
By Schur complements, Condition 1 of Theorem 2 is equivalent Pi Ei −I 0
s.t (a)
(b)
to Ms,i EiT Pi Ci
E1T
which shows that (12) is also valid for x ∈ Ωi,j . This completes the proof of the theorem.
Ms,2 E2T P C2
This, together with Condition 1 of this theorem, imply that, for all 0 ≤ λ ≤ 1,
ν,
inf
P >0,P −2ηQ >0,η,δ1 >0,δ2 >0,Q =Q T ,ϑ1 ≥0,ϑ2 ≥0,H1 ,H2
X 2m 2m X T T T ¯ x ρi,s (x)Ai,s Pj x + x Pj ρi,s (x)A¯ i,s x + 2xT Pj Ei w s = 1 s = 1 1 < − 2 z T z + w T w, γ 2m 2m X X T T T x ¯ ρ ( x ) A P x + x P ρj,s (x)A¯ j,s x + 2xT Pi Ej w j ,s i j ,s i s=1 s=1 < − 1 z T z + w T w, γ2
! " 2m 2m X X T T x λ ρi,s (x)A¯ i,s Pi + Pi ρi,s (x)A¯ i,s s =1 s=1 !# 2m 2m X X T ¯ ¯ + (1 − λ) ρj,s (x)Aj,s Pi + Pi ρj,s (x)Aj,s x. s =1 s=1 1 + 2λxT Pi Ei w + 2(1 − λ)xT Pi Ej w < − 2 z T z + w T w, γ ! " 2m 2m X X xT λ ρj,s (x)A¯ Tj,s Pj + Pj ρj,s (x)A¯ j,s s =1 s=1 !# 2m 2m X X T ¯ ¯ + (1 − λ) ρi,s (x)Ai,s Pj + Pj ρi,s (x)Ai,s x. s =1 s =1 + 2λxT Pj Ej w + 2(1 − λ)xT Pj Ei w < − 1 z T z + w T w, γ2
without loss of generality, that θ1 = θ2 = 1. This implies that Constraint (c) in (16) is automatically satisfied. The optimization problem (16) then simplifies to
As we have done in Section 4, in case of switching between only two subsystems, we can set P2 = P, Q1 = Q and Q2 = −Q , then we have P1 = P − 2ηQ , where Q is a symmetric matrix, and assume,
ν ∗ = 0.3409, α ∗ = 2.9336, H1 = −0.3094 0.1259 , H2 = 3.3443 −1.3757 , −0.0422 0.0466 , Q1 = −Q2 = Q = 0.0466 0.0422 11.9804 9.9189 32.8817 −13.1624 P1 = , P2 = . 9.9189 28.0196 −13.1624 7.1183 Plotted in Fig. 1 are the ellipsoids E (P1 , α ∗ ) and E (P2 , α ∗ ), along with a trajectory starting from the origin and under a pulse disturbance of duration 0.2s and with a maximum energy α ∗ . A zoom in plot of this trajectory is shown in Fig. 2. We next estimate the restricted L2 gain of the resulting switched system over W 2α , α ∈ (0, α ∗ ]. This can be done by solving the optimization problem (17). Plotted in Fig. 3 is the obtained γ ∗ over different values of α . Example 2. Consider system (1) with w ∈ W 2α and 0.6 0.8
−0.8 , 0.6
A1 = A2 =
0.1 , 0.1
E1 = E2 =
F1 = 1.2231
B1 = B2 = 1 ,
C1 = C2 = 1
−2.2486 ,
2 , 4
F2 = 0.8396
−1.7221 .
We note that two subsystems result from the two different stabilizing feedback gains. For each of these two subsystems, we
L. Lu et al. / Automatica 45 (2009) 965–972
971
which results in a switched system with an estimated disturbance tolerance capability of α ∗ = 632.8927. We next proceed to design a switching law that minimizes the restricted L2 gain of the resulting switched system. To this end, let us set α = 500. Solving the optimization problem (17), we obtain a switching law characterized by
−433.9681 Q1 = −Q2 = Q = −10.2274
−10.2274 , 433.9681
with 822.9329 P1 = −688.0880
−688.0880 , 677.0671 1.0000 −0.6839 P2 = 103 × , −0.6839 0.5000 H1 = 10−51 × −0.9690 −0.1154 , H2 = 10−43 × −0.0459 −0.4108 .
Fig. 1. Ellipsoids E (P1 , α ∗ ) and E (P2 , α ∗ ) and a trajectory under a pulse disturbance with energy α ∗ .
This switching law results in a switched system with an estimated restricted L2 gain of γ ∗ = 0.1187. It is clear that γ ∗ < min{γ1∗ , γ2∗ }. On the other hand, when the actuator saturation does not occur, the closed-loop system resulting from the two individual feedback gains, each behaves as a linear system for which the restricted L2 gain can be exactly determined as its H∞ norm. They are γ1∗ = 0.1253 and γ2∗ = 0.1183. However, in the absence of actuator saturation, the L2 gain of the switched system can be estimated as γ ∗ = 0.1072. This demonstrates the effectiveness of the proposed switching scheme in reducing the L2 gain from the disturbance to the system output, as it results in an L2 gain that is smaller than the H∞ norm of each of the closed-loop system under the two individual feedback gains.
Fig. 2. Zoom in plot of the trajectory shown in Fig. 1.
7. Conclusions This paper considered the problem of disturbance tolerance/rejection of a switched system resulting from a family of linear systems subject to actuator saturation and L2 disturbances. Design algorithms for both feedback gains for individual systems and the switching scheme were developed. Several examples were worked out to illustrate the effectiveness of the proposed design method and indeed the power of switching control itself.
Acknowledgments The first and second author’s work was supported in part by National Natural Science Foundation of China under the Young Investigator Program (Class B: Overseas Collaboration). Fig. 3. The restricted L2 gain of the switched system over W 2α .
can use the algorithm in Fang et al. (2004) to estimate the restricted L2 gain. In particular, for α = 500, and the algorithm of Fang et al. (2004) results in γ1∗ = 0.1783 and γ2∗ = 0.1746. We next consider the design of a switching law to reduce the restricted L2 gain. To begin with, we solve the optimization problem (10). We obtain a switching law characterized by 0.3822 Q1 = −Q2 = Q = 10 × −1.1216 3
−1.1216 , −0.3822
References Branicky, M. S. (1994). Stability of switched and hybrid systems. In Proc. of the 33rd IEEE conf. decision and control (pp. 3498–3503). FL: Lake Buena Vista. Cheng, D. (2005). Controllability of switched bilinear systems. IEEE Transactions on Automatic Control, 50(4), 511–515. Chitour, Y., Liu, W., & Sontag, E. (1995). On the continuity and incremental-gain properties of certain saturated linear feedback loops. International Journal of Robust and Nonlinear Control, 65, 13–440. DeCarlo, R., Branicky, M., Pettersson, S., & Lennartson, B. (2000). Perspectives and results on the stability and stabilizability of hybrid systems. Proceedings of the IEEE, 88(7), 1069–1082.
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Fang, H., Lin, Z., & Hu, T. (2004). Analysis and control design of linear systems in the presence of actuator saturation and L2 -disturbances. Automatica, 40(7), 1229–1238. Fang, H., Lin, Z., & Shamash, Y. (2006). Disturbance tolerance and rejection for linear systems with imprecise knowledge of the actuator input output characteristics. Automatica, 42(9), 1523–1530. Filippov, A. F. (1988). Differential equations with discontinuous right-hand sides. Kluwer: Academic Press. Hindi, H., & Boyd, S. (1998). Analysis of linear systems with saturation using convex optimization. In Proc. 37th IEEE conf. dec. and contr. (pp. 903–908). Hu, T., & Lin, Z. (2001). Control systems with actuator saturation: Analysis and design. Boston: Birkhäuser. Liberzon, D., & Morse, A. S. (1999). Basic problems in stability and design of switched systems. IEEE Control System Magazine, 19(5), 59–70. Lin, Z. (1997). H∞ -almost disturbance decoupling with internal stability for linear systems subject to input saturation. IEEE Transactions on Automatic Control, 42(7), 992–995. Lu, L., & Lin, Z. (2008). Design of switched linear systems in the presence of actuator saturation. IEEE Transactions on Automatic Control, 53(6), 1536–1542. Nguyen, T., & Jabbari, F. (1999). Disturbance attenuation for systems with input saturation: an LMI approach. Institute of Electrical and Electronic Engineers. Transactions on Automatic Control, 44(4), 852–857. Pettersson, S. (2003). Synthesis of switched linear systems. In Proc. of the 42nd IEEE conf. decision and control (pp. 5283–5288). Pettersson, S. (2004). Controller design of switched linear systems. In Proc. of the 2004 American control conference (pp. 3869–3874). Pettersson, S. (2005). Synthesis of switched linear systems handling sliding motions. 2005 International symposium on intelligent control (pp. 18-23). Pettersson, S. (1999). Analysis and design of hybrid systems. Ph.D. thesis, Control engineering laboratory. Chalmers University of Technology. Pettersson, S., & Lennartson, B. (2001). Stabilization of hybrid systems using a minprojection strategy. In Proc. the American control conference (pp. 223–228). Sun, Z., & Ge, S. (2005). Switched linear systems: Control and design. London: Springer. Wicks, M., Peleties, P., & DeCarlo, R. A. (1998). Switched controller synthesis for the quadratic stabilisation of a pair of unstable linear systems. European Journal of Control, 4(2), 140–147. Xi, Z., Feng, G., Jiang, Z. P., & Cheng, D. (2003). A switching algorithm for global exponential stabilization of uncertain chained systems. IEEE Transactions on Automatic Control, 48(10), 1793–1798. Xie, D., Wang, L., Hao, F., & Xie, G. (2004). LMI approach to L2 -gain analysis and control synthesis of uncertain switched systems. IEE Proceedings of the Control Theory and Applications, 151(1), 21–28.
Liang Lu was born in Shenyang, China on May 16, 1980. He received his B.S. and M.S. degrees in Automatic Control from Shenyang University, Shenyang, China, and Huaqiao University, Quanzhou, China, respectively. He is currently working toward his Ph.D. degree in the Department of Automation, Shanghai Jiao Tong University and the Department of Electrical and Computer Engineering, University of Virginia.
Zongli Lin is a professor of Electrical and Computer Engineering at University of Virginia. He received his B.S. degree in mathematics and computer science from Xiamen University, Xiamen, China, in 1983, his Master of Engineering degree in automatic control from Chinese Academy of Space Technology, Beijing, China, in 1989, and his Ph.D. degree in electrical and computer engineering from Washington State University, Pullman, Washington, in 1994. His current research interests include nonlinear control, robust control, and control applications. He was an Associate Editor of the IEEE Transactions on Automatic Control and has served on the operating committees and program committees of several conferences. He currently serves on the editorial boards of several journals and book series, including Automatica, Systems & Control Letters, IEEE/ASME Transactions on Mechatronics and IEEE Control Systems Magazine. He is an elected member of the Board of Governors of the IEEE Control Systems Society. He is a Fellow of the IEEE. Haijun Fang received his Bachelor and Master of Science degrees in Automation from Tsinghua University, Beijing, China, in 1999 and 2001, respectively, and his Ph.D. degree in electrical engineering from University of Virginia in 2006. Dr. Fang is currently a control engineer at MKS Instrument, Rochester, New York. His current research interests include nonlinear control, robust control and control applications in semi-conductor manufacturing systems.
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Automatica journal homepage: www.elsevier.com/locate/automatica
Brief paper
L2 gain analysis for a class of switched systemsI Liang Lu a , Zongli Lin b,∗ , Haijun Fang c a
Department of Automation, Shanghai Jiao Tong University, Shanghai, 200240, China
b
Charles L. Brown Department of Electrical and Computer Engineering, University of Virginia, P.O. Box 400743, Charlottesville, VA 22904-4743, USA
c
MKS Instrument, 100 Highpower Road, Rochester, NY 14623, USA
article
info
Article history: Received 24 February 2008 Received in revised form 4 August 2008 Accepted 25 October 2008 Available online 7 January 2009 Keywords: Disturbance rejection Disturbance tolerance L2 gain Actuator saturation Switched systems Set invariance
a b s t r a c t This paper considers the problem of disturbance tolerance/rejection for a family of linear systems subject to actuator saturation and L2 disturbances. For a given set of linear feedback gains, a given switching scheme and a given bound on the L2 norm of the disturbances, conditions are established in terms of linear or bilinear matrix inequalities under which the resulting switched system is bounded state stable, that is, trajectories starting from a bounded set will remain inside the set or a larger bounded set. With these conditions, both the problem of assessing the disturbance tolerance/rejection capability of the closed-loop system and the design of feedback gain and switching scheme can be formulated and solved as constrained optimization problems. Disturbance tolerance is measured by the largest bound on the disturbances for which the trajectories from a given set remain bounded. Disturbance rejection is measured by the restricted L2 gain over the set of tolerable disturbances. In the event that all systems in the family are identical, the switched system reduces to a single system under a switching feedback law. It will be shown that such a single system under a switching feedback law has stronger disturbance tolerance/rejection capability than a single linear feedback law can achieve. © 2008 Elsevier Ltd. All rights reserved.
The literature on analysis and design of switched systems has been growing rapidly in recent years (see, for example, Branicky (1994), Cheng (2005), DeCarlo, Branicky, Pettersson, and Lennartson (2000), Liberzon and Morse (1999), Pettersson (1999), Pettersson and Lennartson (2001), Sun and Ge (2005), Wicks, Peleties, and DeCarlo (1998) and Xi, Feng, Jiang, and Cheng (2003) and the references therein). Motivated by the results reported in this literature, we consider in this paper the following family of linear systems subject to input saturation and disturbances, x˙ = Ai x + Bi sat(u) + Ei w, z = Ci x, i ∈ IN := {1, 2, . . . , N },
(1)
where x ∈ Rn , u ∈ Rm , z ∈ Rp are respectively the state, input and output of the system, w ∈ Rq represents the disturbances, and sat : Rm → Rm is the vector valued standard saturation
I An abridged version of this paper was presented at the 17th IFAC World Congress, Seoul, Korea, July 6–11, 2008. This paper was recommended for publication in revised form by Associate Editor Ben M. Chen under the direction of Editor Ian R. Petersen. ∗ Corresponding author. Tel: +1 434 924 6342; fax: +1 434 924 8818. E-mail addresses: [email protected] (L. Lu), [email protected] (Z. Lin), [email protected] (H. Fang).
0005-1098/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2008.10.021
T
function sat(u) = sat(u1 ) sat(u2 ) · · · sat(um ) , sat(ui ) = sign(ui ) min{|ui |, 1}. A switched system then results by defining a controller/supervisor which chooses one of the systems at each time instant based on the measurement of the state and according to an index function, say, i = σ (x). A typical form of the index function is σ (x) = i for x ∈ Ωi with ∪Ni=1 Ωi = Rn . Thus, the control design involves the construction of both feedback gains for individual systems and the index function so that the resulting switched system possesses certain desired performances. In the absence of the disturbances w , a basic design objective is the local asymptotic stability of the resulting switched system with as large a domain of attraction as possible. By utilizing some techniques in dealing with actuator saturation (Hu & Lin, 2001) and the form of the largest region index function proposed by Pettersson (2003, 2004, 2005), we recently proposed a method for the design of the individual feedback gains and the index function that result in a locally asymptotically stable switched system Lu and Lin (2008). The design is formulated and solved as a constrained optimization problem with the objective of enlarging the domain of attraction of the resulting stable equilibrium at the origin. It was shown by numerical examples that such a design may result in a domain of attraction larger than that of a switched system, designed without taking actuator saturation into account. In this paper, we will carry out an analysis of, and design for, the disturbance tolerance/rejection capability of the switched system resulting from the family of systems (1). We will restrict ourselves
1. Introduction
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to a class of disturbances whose energies are bounded by a given value, i.e.,
W 2α := w : R+ → Rq :
∞
Z
wT (t )w(t )dt ≤ α ,
(2)
0
for some positive number α . For a given set of linear feedback gains, a given index function and a given value of α , conditions will be established in terms of linear or bilinear matrix inequalities under which the resulting switched system is bounded state stable. A system is said to be bounded state stable if its trajectories starting from a bounded set will remain inside the set or a larger bounded set. With these conditions, both the problem of assessing the disturbance tolerance/rejection capability of the closed-loop system and the design of feedback gain and switching scheme can be formulated and solved as constrained optimization problems. Disturbance tolerance is measured by the largest bound on the energy of the disturbance, α ∗ , for which the trajectories from a given set remain bounded. Disturbance rejection is measured by the restricted L2 gain over W 2α ∗ . An interesting special class of the systems we consider in this paper is the case when all the systems in (1) are identical. In this case, the switched system reduces to a single system under a switching linear feedback law. It will be shown that for a single linear system of the form (1), a switching feedback law will result in stronger disturbance tolerance/rejection capability than a single linear feedback law of Fang, Lin, and Hu (2004) and Fang, Lin, and Shamash (2006). The L2 gain analysis and design for linear systems under actuator saturation has been studied by several authors. A small sample of their works include Chitour, Liu, and Sontag (1995), Fang et al. (2004, 2006), Hindi and Boyd (1998), Hu and Lin (2001), Lin (1997), Nguyen and Jabbari (1999) and Xie, Wang, Hao, and Xie (2004). In particular, in our recent work (Fang et al., 2004, 2006), we considered the L2 gain analysis and design for a linear system under actuator saturation. The disturbance tolerance capability of the closed-loop system under a given feedback law was assessed, and the linear feedback law that results in a minimized restricted L2 gain was designed. The remainder of this paper is organized as follows. In Section 2, we state our problem and recall some preliminary materials that will be needed in the development of the results of this paper. Section 3 establishes bounded state stability conditions. Disturbance tolerance and disturbance rejection are addressed in Sections 4 and 5, respectively. Simulation results are presented in Section 6. Section 7 concludes the paper. 2. Problem statement and preliminaries For the family of systems (1), we would like to design a linear feedback law for each individual system in the family and an index function such that the resulting switched system possesses a high degree of disturbance tolerance and a high level of disturbance rejection capabilities. We will adopt the switching strategy of Pettersson (2003, 2004). Such a switching strategy is defined based on some appropriately chosen symmetric matrices Qi ∈ Rn×n , i ∈ IN . More specifically, at a given state x, the subsystem i will be activated if the quadratic function xT Qi x is greater or equal to any other xT Qj x, j 6= i. More specifically, this switching scheme is defined by the following index function Pettersson (2003, 2004), referred to as the largest region function,
i(x) = arg max xT Qi x . i∈IN
(3)
Based on the matrices Qi ’s, we define the following sets
Ωi = {x ∈ Rn |xT Qi x ≥ 0}, i ∈ IN , Ωi,j = {x ∈ Rn |xT Qj x = xT Qi x ≥ 0},
i ∈ IN , j ∈ IN .
Then, a well-defined switched system must satisfy the following properties:
• Covering property: Ω1 ∪ Ω2 ∪ · · · ∪ ΩN = Rn ; • Switching property: Ωi,j ⊆ Ωi ∩ Ωj , i ∈ IN , j ∈ IN . The first condition says that there are no regions in the state space where none of the subsystem is activated. The second condition, which is automatically satisfied by this choice of Ωi and Ωi,j , means that a switch from subsystem i to j occurs only for states where the regions Ωi and Ωj are adjacent. Consequently, switching occurs on the switching surface xT Qi x = xT Qj x. The following regarding the covering property was established in Pettersson (2003, 2004). Lemma 1 (Covering property). If for every x ∈ Rn ,
θ1 xT Q1 x + θ2 xT Q2 x + · · · + θN xT QN x ≥ 0,
(4)
where θi > 0, i ∈ IN , then Ω1 ∪ Ω2 ∪ · · · ∪ ΩN = Rn . 3. Bounded state stability We recall a tool from Hu and Lin (2001) for expressing a saturated linear feedback u = sat(Fx) on the convex hull of a mixture of the unsaturated control inputs and the auxiliary inputs. For an F ∈ Rm×n , let L(F ) = x ∈ Rn : |fi x| ≤ 1, i ∈ Im , where fi represents the ith row of matrix F . We note that L(F ) represents the region in Rn where F x does not saturate. Also, let V be the set of m × m diagonal matrices whose diagonal elements are either 1 or 0. There are 2m elements in V . Suppose these elements of V are labeled as Ds , s ∈ I2m . Denote D− s = I − Ds . Clearly, D− ∈ V if D ∈ V . The following lemma is adopted from Hu s s and Lin (2001). Lemma 2. Let F , H ∈ Rl×n . Then, for any x ∈ L(H ), m , sat(Fx) ∈ co Ds Fx + D− s Hx, s ∈ I2
where co stands for the convex hull. For a positive definite P ∈ R n×n and a scalar ρ > 0, matrix n T we define E (P , ρ) := x ∈ R : x Px ≤ ρ . The following theorem characterizes the bounded state stability of the switched system that results from the family of systems (1) and the switching scheme (3). Theorem 1. Consider system (1). If there exist Pi > 0, ξ > 0, Qi = QiT , Fi ∈ Rm×n , Hi ∈ Rm×n , ϑi ≥ 0, θi > 0 and ηi,j such that 1 T T − 1. (Ai + Bi (Ds Fi + D− s Hi )) Pi + Pi (Ai + Bi (Ds Fi + Ds Hi ))+ ξ Pi Ei Ei Pi + ϑi Qi ≤ 0, s ∈ I2m , i ∈ IN , 2. Pi = Pj + ηi,j (Qj − Qi ), i ∈ IN , j ∈ IN , 3. θ1 Q1 + θ2 Q2 + · · · + θN QN ≥ 0,
and E (Pi , 1 + αξ ) ∩ Ωi ⊂ L(Hi ), i ∈ IN , then every trajectory of the closed-loop system that starts from inside of ∩Ni=1 (E (Pi , 1) ∩ Ωi ) will remain inside of ∩Ni=1 (E (Pi , 1 + αξ ) ∩ Ωi ) for every w ∈ W 2α , as long as no sliding motion occurs or sliding motions only occur along switching surfaces with the corresponding ηi,j ≥ 0. If the condition E (Pi , 1 +αξ )∩ Ωi ⊂ L(Hi ) is replaced with E (Pi , αξ )∩ Ωi ⊂ L(Hi ), then any trajectory starting from the origin will remain inside the region ∩Ni=1 (E (Pi , αξ ) ∩ Ωi ) for every w ∈ W 2α as long as no sliding motion occurs or sliding motions only occur along switching surfaces with the corresponding ηi,j ≥ 0.
L. Lu et al. / Automatica 45 (2009) 965–972
Proof. By Lemmas 1 and 2, Condition 3 of the theorem and the largest region function strategy (3) guarantee that the resulting switched system is well defined. We will define the energy of the system as its trajectory evolves in the state space. In each region Ωi , we will measure its energy as Vi (x) = xT Pi x. The derivative of Vi (x) along the trajectory of the subsystem i is then given by V˙ i = 2xT Pi (Ai x + Bi sat(Fi x) + Ei w). By Lemma 2, for every x ∈ E (Pi , 1 + αξ ) ∩ Ωi (or x ∈ E (Pi , α) ∩ Ωi ⊂ L(Hi )),
⊂ L(Hi )
m . sat(Fi x) ∈ co Ds Fi x + D− s Hi x, s ∈ I2
It follows that
∩Ni=1 (E (Pi , 1 + αξ ) ∩ Ωi ), and any trajectory that starts from x(0) = 0 will remain inside ∩Ni=1 (E (Pi , αξ ) ∩ Ωi ). We next consider the situation when sliding motion does occur. Denote A¯ i,s = Ai + Bi (Es Fi + Es− Hi ), i ∈ IN , s ∈ I2m , and A¯ j,s = Aj + Bj (Es Fj + Es− Hj ), j ∈ IN , s ∈ I2m . A sliding motion occurs along the hyper surface xT Qi x = xT Qj x ≥ 0, between two neighboring regions xT Qi x ≥ 0 and xT Qj x ≥ 0, if vector fields A¯ i,s x, s ∈ I2m , all point into region xT Qj x ≥ 0 and vector fields A¯ j,s , s ∈ I2m , all point into region xT Qi x ≥ 0 (Pettersson, 2005). A sliding motion may lead to either stable or unstable dynamics along the switching surface according to Filippov’s convex combination (Filippov, 1988) x˙ = λ
" m 2 X
m . Ai x + Bi sat(Fi x) ∈ co Ai x + Bi (Ds Fi + D− s H i ) x , s ∈ I2
2xT Pi Ei w ≤
# ρi,s (x)A¯ i,s x + Ei w
s=1
Noting that,
+ (1 − λ) 1
ξ
xT Pi Ei EiT Pi x + ξ w T w,
967
" m 2 X
# ρj,s (x)A¯ j,s x + Ej w ,
∀0 ≤ λ ≤ 1,
s=1
∀ξ > 0,
where ρi,s (x)
we have
>
P2m
= 1, and ρj,s (x) s=1 ρi,s (x) ρ ( x ) = 1, are such that (see Lemma 2) s = 1 j ,s
P2m
V˙ i = 2xT Pi (Ai x + Bi sat(Fi x) + Ei w)
≤ max 2xT Pi (Ai x + Bi (Ds Fi + D− s Hi ))x
Ai x + Bi sat(Fi x) =
s∈I2m
0,
2m X
0,
ρi,s (x)A¯ i,s x,
s=1
1
+ xT Pi Ei EiT Pi x + ξ w T w. ξ
Aj x + Bj sat(Fj x) =
V˙ i ≤ −ϑi xT Qi x + ξ w T w,
∀x ∈ E (Pi , 1 + αξ ) ∩ Ωi (or ∀x ∈ E (Pi , αξ ) ∩ Ωi ).
Recall that system i is activated in the region E (Pi , 1 + αξ ) ∩ Ωi (or E (Pi , αξ ) ∩ Ωi ). In this region, xT Qi x ≥ 0. Thus, V˙ i ≤ ξ w T w, ∀x ∈ E (Pi , 1 + αξ ) ∩ Ωi
(or ∀x ∈ E (Pi , αξ ) ∩ Ωi ).
Let V (x) = Vi(x) (x) be the overall energy of the switched system for use as a Lyapunov function candidate. Such a function is, in general, not differentiable along the boundaries between Ωi ’s, i.e., Ωi,j , i ∈ IN , j ∈ IN , where the switch from subsystem i to subsystem j occurs and where Vi (x) = Vj (x) due to Condition 2 of the theorem. But, for any state not on the boundaries, we have V˙ ≤ ξ w T w,
∀x ∈ ∩Ni=1 (E (Pi , 1 + αξ ) ∩ Ωi )
(or ∀x ∈ ∪Ni=1 (E (Pi , αξ ) ∩ Ωi )).
2m X
ρj,s (x)A¯ j,s x.
s=1
It then follows from Condition 1 in Item 1 of the theorem that
(5)
Consequently, if sliding mode does not occur on any of the boundaries, then let tj , j = 1, 2, . . . , Nt , be the times when the trajectory crosses the boundaries of Ωi ’s before time t. Then, integrating both sides of the above inequality from 0 to t results in
We will show that a sliding motion will not destroy the closedloop property established above, if it occurs on a switching surface with corresponding ηi,j ≥ 0. Indeed, if ηi,j = 0, Condition 2 of the theorem implies that Pi = Pj , which result in a common Lyapunov function for dynamics in both Ωi and Ωj . That is, the Lyapunov function V (x) is differentiable along Ωi,j and (5) is also valid for any x ∈ Ωi,j . We next consider the case of ηi,j > 0. According to the analysis of sliding motions in Pettersson (2005), the sliding motion occurring along the surface xT Qi x = xT Qj x ≥ 0 implies that
2m X xT ρi,s (x)A¯ Ti,s (Qi − Qj )x s = 1 2m X T ρi,s (x)A¯ i,s x + 2xT (Qi − Qj )Ei w < 0, + x (Qi − Qj ) m
s =1
2 X T x ρj,s (x)A¯ Tj,s (Qi − Qj )x s=1 2m X + xT (Qi − Qj ) ρj,s (x)A¯ j,s x + 2xT (Qi − Qj )Ej w > 0. s=1
Nt −1
V (x(t )) ≤ V (x(0)) + ξ
+ξ
Z
tj+1
XZ j =0
Using Condition 2 and noting that ηi,j > 0, we have
w T (τ )w(τ )dτ
tj
2m X xT ρi,s (x)A¯ Ti,s (Pj − Pi )x s=1 2m X T ρi,s (x)A¯ i,s x + 2xT (Pj − Pi )Ei w < 0, + x (Pj − Pi )
t
w (τ )w(τ )dτ T
Nt
= V (x(0)) + ξ
t
Z
w T (τ )w(τ )dτ 0
1 + αξ ,
≤
>
αξ ,
m
if x(0) ∈ ∪ if x(0) = 0,
N i =1
(E (Pi , 1) ∩ Ωi ) ,
(6)
where we have set t0 = 0. This shows that any trajectory that starts from ∩Ni=1 (E (Pi , 1) ∩ Ωi ) will remain inside
s=1
2 X T ρj,s (x)A¯ Tj,s (Pj − Pi )x x s=1 2m X + xT (Pj − Pi ) ρj,s (x)A¯ j,s x + 2xT (Pj − Pi )Ej w > 0, s=1
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L. Lu et al. / Automatica 45 (2009) 965–972
which is equivalent to
2m 2m X X T T T ¯ x ρ ( x ) A P x + x P ρi,s (x)A¯ i,s x + 2xT Pj Ei w i ,s j i,s j s=1 s=1 2m 2m X X T T T <x ¯ ρ ( x ) A P x + x P ρi,s (x)A¯ i,s x + 2xT Pi Ei w, i,s i i ,s i s=1
s=1
s=1
s=1
2m 2m X X T T T ¯ x ρj,s (x)Aj,s Pi x + x Pi ρj,s (x)A¯ j,s x + 2xT Pi Ej w s=1 s=1 2m 2m X X T T T ¯ <x ρj,s (x)Aj,s Pj x + x Pj ρj,s (x)A¯ j,s x + 2xT Pj Ej w.
Hence, by Condition 1 of the theorem, we have
2m 2m X X T T T ¯ x ρi,s (x)Ai,s Pj x + x Pj ρi,s (x)A¯ i,s x + 2xT Pj Ei w s=1 s=1 < −ϑi xT Qi x + ξ w T w ≤ ξ w T w, 2m 2m X X T T T ¯ x ρj,s (x)Aj,s Pi x + x Pi ρj,s (x)A¯ j,s x + 2xT Pi Ej w s=1 s=1 < −ϑj xT Qj x + ξ w T w ≤ ξ w T w, from which and Condition 1 of the theorem, we have that, for all 0 ≤ λ ≤ 1,
! " X 2m 2m X T T ρi,s (x)A¯ i,s Pi + Pi ρi,s (x)A¯ i,s x λ s =1 s=1 !# 2m 2m X X T ¯ ¯ ρj,s (x)Aj,s Pi + Pi ρj,s (x)Aj,s x + (1 − λ) s =1 s=1 +"2λxT Pi Ei w + 2(1 − λ)xT Pi Ej w < ξ wT w, ! 2m 2m X X T T ¯ ¯ x λ ρj,s (x)Aj,s Pj + Pj ρj,s (x)Aj,s s =1 s =1 !# 2m 2m X X T + (1 − λ) ρi,s (x)A¯ i,s Pj + Pj ρi,s (x)A¯ i,s x s =1 s=1 + 2λxT Pj Ej w + 2(1 − λ)xT Pj Ei w < ξ wT w. This, in turn, implies that
!T 2m 2m X X T ρi,s (x)A¯ i,s + (1 − λ) ρj,s (x)A¯ j,s Pi x λ s = 1 s = 1 !# 2m 2m X X + Pi λ ρi,s (x)A¯ i,s + (1 − λ) ρj,s (x)A¯ j,s x s = 1 s = 1 T T + 2x Pi λEi w + (1 − λ)Ej w < ξ w w, !T 2m 2m X X T x λ ρi,s (x)A¯ i,s + (1 − λ) ρj,s (x)A¯ j,s Pj s =1 s =1 !# 2m 2m X X + Pj λ ρi,s (x)A¯ i,s + (1 − λ) ρj,s (x)A¯ j,s x s =1 s=1 + 2xT Pj λEj w + (1 − λ)Ei w < ξ wT w. Therefore, the inequality (5), and hence (6), is still valid for any x ∈ Ωi,j . This completes the proof. 4. Disturbance tolerance A fundamental problem to be addressed before determining the restricted L2 gain is the assessment of the disturbance tolerance capability of the closed-loop system. The disturbance tolerance capability is measured by the largest bound on the energy of
the disturbances, say α ∗ , under which the closed-loop trajectories starting from the origin or a given set of initial conditions remain bounded. As the restricted L2 gain will be defined with zero initial conditions, we will only assess disturbance tolerance with zero initial conditions in this section. Disturbance tolerance with a given set of initial conditions can be dealt with similarly by resorting to the first part of Theorem 1. As established in Theorem 1, under the three itemized conditions and the condition that E (Pi , αξ ) ⊂ L(Hi ), i ∈ IN , the trajectories of the closed-loop system that start from origin will remain inside the region ∩Ni=1 (E (Pi , αξ ) ∩ Ωi ) for every w ∈ W 2α , if no sliding motion occurs or sliding motions only occur along switching surfaces with the corresponding ηi,j ≥ 0. Without loss of generality, we can assume that ξ = 1 in the above mentioned result. If ξ 6= 1, we can multiply both sides of Condition 1 of Theorem 1 with 1/ξ to obtain Pi T Pi + (Ai + Bi (Ds Fi + D− (Ai + Bi (Ds Fi + D− s Hi )) s Hi ))
ξ
Pi
Pi Ei EiT
≤ 0, s ∈ I2m , i ∈ IN . ξ Let P˜i = Pi /ξ and Q˜ i = Qi /ξ . Then, P˜i and Q˜ i satisfy all conditions of Theorem 1 with ξ = 1 and E (P˜i , α) = E (Pi , αξ ). As a result, +
ξ
ξ
+ ϑi
ξ
Qi
the disturbance tolerance capability of the closed-loop system under zero initial conditions can be assessed through solving the following optimization problem,
α,
sup
(7)
Pi >0,Qi =QiT ,ηi,j ,δi >0,ϑi ≥0,Hi ,θi >0 T − (a) (Ai + Bi (Ds Fi + D− s Hi )) Pi + Pi (Ai + Bi (Ds Fi + Ds Hi ))
s.t.
+ Pi Ei EiT Pi + ϑi Qi ≤ 0, s ∈ I2m , i ∈ IN , (b) Pi = Pj + ηi,j (Qj − Qi ), i ∈ IN , j ∈ IN , (c) θ1 Q1 + θ2 Q2 + · · · + θN QN ≥ 0, (d) E (Pi , α) ∩ Ωi ⊂ L(Hi ), i ∈ IN . Let ν = 1/α . Then, Constraint (d) is implied by hi,k (Pi − δi Qi )−1 hTi,k ≤ ν, k ∈ Im , which, by Schur complements, is equivalent to,
ν
hi,k
≥ 0, k ∈ Im , (Pi − δi Qi ) where δi > 0 and hi,k denotes the kth row of Hi . hTi,k
(8)
Consequently, the optimization problem (7) can be written as the following BMI problem, inf
Pi >0,Qi =QiT ,ηi,j ,δi >0,ϑi ≥0,Hi ,θi >0
s.t.
ν,
T − (a)(Ai + Bi (Ds Fi + D− s Hi )) Pi + Pi (Ai + Bi (Ds Fi + Ds Hi ))
(9)
+ Pi Ei EiT Pi + ϑi Qi ≤ 0, s ∈ I2m , i ∈ IN , (b) Pi = Pj + ηi,j (Qj − Qi ), i ∈ IN , j ∈ IN , (c) θ1 Q1 + θ2 Q2 + · · · + θN QN ≥ 0, ν h i ,k (d) ≥ 0, k ∈ Im , i ∈ IN . T hi,k (Pi − δi Qi ) If we define PN = P and ηi = ηj,i , i ∈ IN −1 , then, Constraint (b)
simplifies to
Pi = P + ηi (QN − Qi ),
i ∈ IN − 1 , and consequently, Constraint (a) simplifies to T (a) (Ai + Bi (Ds Fi + D− s Hi )) (P + ηi (QN − Qi ))
+ (P + ηi (QN − Qi ))(Ai + Bi (Ds Fi + D− s Hi )) + (P + ηi (QN − Qi ))Ei EiT (P + ηi (QN − Qi )) + ϑi Qi ≤ 0, s ∈ I2m , i ∈ IN −1 , T (AN + BN (Ds FN + D− s HN )) P T + P (AN + BN (Ds FN + D− s HN )) + PEN EN P + ϑN QN ≤ 0, s ∈ I2m .
L. Lu et al. / Automatica 45 (2009) 965–972
In case of switching between only two subsystems, we can set Q1 = Q and Q2 = −Q , where Q is a symmetric matrix. Furthermore, we can, without loss of generality, scale θ1 = θ2 = 1. This implies that Constraint (c) in (9) is automatically satisfied. The optimization problem (9) then simplifies to inf
P >0,P −2ηQ >0,η,δ1 >0,δ2 >0,Q =Q T ,ϑ1 ≥0,ϑ2 ≥0,H1 ,H2
ν,
(10)
+ (P − 2ηQ )E1 E1T (P − 2ηQ ) + ϑ1 Q ≤ 0, s ∈ I2m , T (A2 + B2 (Ds F2 + D− s H2 )) P + P (A2 + B2 (Ds F2 − T m + Ds H2 )) + PE2 E2 P − ϑ2 Q ≤ 0, s ∈ I2 , ν h1,k (b) ≥ 0, k ∈ Im , hT1,k P − 2ηQ − δ1 Q ν h2,k ≥ 0, k ∈ Im . hT2,k P + δ2 Q
The problem of verifying the existence of the unknown variables solving the optimization problem (10), is a bilinear matrix inequality (BMI) problem, which is NP-hard and difficult to solve. However, many algorithms for BMI problems have been proposed on the basis of approximations, heuristics, branch & bound, or local search. For example, PENOPT offers a commercial solver PENBMI for solving the optimization problems with bilinear matrix inequality constraints. We will use PENBMI to obtain all our numerical results in Section 6. While the conservativeness of these numerical results is not clear, they do demonstrate the effectiveness of the proposed method. The above optimization problem can be adapted for the design of feedback gains Fi ’s. This can be readily done by viewing Fi ’s as additional optimization parameters. 5. L2 gain analysis The restricted L2 gain of the closed-loop system is defined over a set of tolerable disturbances, say, W 2α , α ∈ (0, α ∗ ], as
kz kL2 , kwkL2 x(0)=0,w∈Wα2 \{0} sup
where k · kL2 is the L2 norm of a signal. Thus, the closed-loop system has a restricted L2 gain over W 2α less than or equal to γ if, for x(0) = 0, t
Z
z (τ )z (τ )dτ ≤ γ T
0
2
By Lemma 2, for every x ∈ E (Pi , α) ∩ Ωi ⊂ L(Hi ), m . sat(Fi x) ∈ co Ds Fi x + D− s H i x , s ∈ I2
It follows that m . Ai x + Bi sat(Fi x) ∈ co Ai x + Bi (Ds Fi + D− s Hi )x, s ∈ I2
Recall that system i is activated in the region E (Pi , α), where xT Pi x ≥ 0. Thus, in view of Condition 1 of the theorem, and noting that, 2xT Pi Ei w ≤ xT Pi Ei EiT Pi x + w T w, we have V˙ i = 2xT Pi (Ai x + Bi sat(Fi x) + Ei w) T T T ≤ max 2xT Pi (Ai x + Bi (Ds Fi + D− s Hi ))x + x Pi Ei Ei Pi x + w w, s∈I2m
1
≤−
γ2
=−
γ2
1
xT CiT Ci x + w T w z T z + w T w, ∀x ∈ E (Pi , α) ∩ Ωi .
Now, let V (x) = Vi(x) (x) be the overall energy of the switched system for use as a Lyapunov function candidate. Such a function is in general not differentiable along the boundaries between Ωi ’s, i.e., Ωi,j ’s, where the switch from subsystem i to subsystem j occurs and where Vi (x) = Vj (x) due to Condition 2 of the theorem. However, for any state not on these boundaries, we have V˙ ≤ −
1
γ2
z T z + w T w,
∀x ∈ ∩Ni=1 (E (Pi , α) ∩ Ωi ) .
w (τ )w(τ )dτ , T
0
∀t ≥ 0, ∀w ∈ W α . 2
T T − 1. (Ai + Bi (Ds Fi + D− s Hi )) Pi + Pi (Ai + Bi (Ds Fi + Ds Hi )) + Pi Ei Ei Pi + 1 T C C + ϑi Qi ≤ 0, s ∈ I2m , i ∈ IN , γ2 i i
2. Pi = Pj + ηi,j (Qj − Qi ), i ∈ IN , j ∈ IN , 3. θ1 Q1 + θ2 Q2 + · · · + θN QN ≥ 0,
and E (Pi , α) ∩ Ωi ⊂ L(Hi ), i ∈ IN , then the restricted L2 gain from w to z over W 2α is less than or equal to γ , if no sliding motion occurs or sliding motions only occur along switching surfaces with the corresponding ηi,j ≥ 0.
tj+1
1 − 2 z T (τ )z (τ ) + wT (τ )w(τ ) dτ γ tj j=0 Z t 1 + − 2 z T (τ )z (τ ) + w T (τ )w(τ )d dτ γ Nt Z t 1 = V (x(0)) + − 2 z T (τ )z (τ ) + wT (τ )w(τ ) dτ , γ 0
V (x(t )) ≤ V (x(0)) +
The following theorem characterizes the conditions under which the switched linear system has a restricted L2 gain less than or equal to γ . Theorem 2. Consider system (1) and an α ∈ (0, α ∗ ]. If there exist Pi > 0, Qi = QiT , Fi ∈ Rm×n , Hi ∈ Rm×n , ϑi ≥ 0, ηi,j and θi > 0 such that
(11)
It was shown in the proof of Theorem 1, the set ∩Ni=1 (E (Pi , α) ∩ Ωi ) is an invariant set. Consequently, if sliding mode does not occur on any of the boundaries, then let tj , j = 1, 2, . . . , Nt , be the times when the trajectory crosses the boundaries of Ωi ’s before time t. Then, integrating both sides of the above inequality from 0 to t results in Nt −1
t
Z
Proof. By Lemmas 1 and 2, Condition 3 of the theorem and the largest region function strategy (3) guarantee that the resulting switched system is well defined. To continue with the proof, we define the energy of the system as its trajectory evolves in the state space. In each region Ωi , we will measure its energy as Vi (x) = xT Pi x. The derivative of Vi (x) along the trajectory of the subsystem i is then given by V˙ i = 2xT Pi (Ai x + Bi sat(Fi x) + Ei w).
T s.t (a) (A1 + B1 (Ds F1 + D− s H1 )) (P − 2η Q ) + (P − 2ηQ )(A1 + B1 (Ds F1 + D− s H1 ))
γ∗ =
969
XZ
∀x ∈ ∩Ni=1 (E (Pi , α) ∩ Ωi ) ,
(12)
where we have set t0 = 0. Noting that V (x(t )) ≥ 0 and V (0) = 0, we have that, for x(0) = 0, t
Z
z T (τ )z (τ )dτ ≤ γ 2 0
t
Z
wT (τ )w(τ )dτ ,
(13)
0
which indicates that, if no sliding motion occurs, the restricted L2 gain from w to z over W 2α is less than or equal to γ . We next examine the situation when a sliding motion occurs. We will show that the restricted L2 gain from w to z over W 2α is still less than or equal to γ , if a sliding motion occurs on a switching
970
L. Lu et al. / Automatica 45 (2009) 965–972
surface with corresponding ηi,j ≥ 0. Indeed, if ηi,j = 0, Condition 2 of the theorem implies that Pi = Pj , resulting in a common Lyapunov function for the dynamics in both Ωi and Ωj . That is, the Lyapunov function V (x) is differentiable along Ωi,j and (12) is also valid for x ∈ Ωi,j . For the case when ηi,j > 0, as has been shown in the proof of Theorem 1, the sliding motion occurring along the surface xT Qi x = xT Qj x ≥ 0 implies (3), from which and Condition 1 of this theorem, we have
CiT 0 ≤ 0. −γ 2 I
(14)
(b) (c) (d)
ν ν
−5
1 ,
C1 = 1
0
0.1 E1 = , 0.1
0 B1 = , 1
0 , 1
B2 =
1 ,
F2 = 5
1 ,
F1 = 0
1 0
C2 = 1
,
0 , 1
E2 =
0.1 , 0.1
0 .
To design a switching law that maximizes the disturbance tolerance capacity of the resulting switched system, we solve the optimization problem (10) and obtain
Thus, based on Theorem 2, the restricted L2 gain can be estimated by solving the following optimization problem,
s.t. (a)
s ∈ I2m ,
As with the optimization problems for assessing disturbance tolerance, the optimization problems for estimating the restricted L2 gain can also be adapted for the design of feedback gains Fi ’s by simply viewing Fi ’s as additional optimization parameters.
A2 =
T − Ms,i = (Ai + Bi (Ds Fi + D− s Hi )) Pi + Pi (Ai + Bi (Ds Fi + Ds Hi )) + ϑi Qi .
(16)
Msi Pi Ei CiT EiT Pi −I 0 ≤ 0 , s ∈ I2 m , i ∈ IN , Ci 0 −γ 2 I Pi = Pj + ηi,j (Qj − Qi ), i ∈ IN , j ∈ IN , θ Q + θ2 Q2 + · · · + θN QN ≥ 0, "1 11 # hi,k ≥ 0, k ∈ Im , i ∈ IN . α hTi,k (Pi − δi Qi )
0
PE2 C2T −I 0 ≤ 0, s ∈ I2m , 0 −γ 2 I h1,k ≥ 0, k ∈ Im , P − 2ηQ − δ1 Q h 2 ,k ≥ 0, k ∈ Im , P + δ2 Q
hT1,k
1 A1 = 0
(15)
γ 2,
C1T 0 ≤ 0, −γ 2 I
Example 1. Let us consider system (1) with w ∈ W 2α and
where
inf
(17)
6. Numerical examples
Pi >0,Qi =QiT ,ηi,j ,δi >0,ϑi ≥0,Hi ,θi >0
(P − 2ηQ )E1 −I
Ms,1 (P − 2ηQ ) C1
hT2,k
By Schur complements, Condition 1 of Theorem 2 is equivalent Pi Ei −I 0
s.t (a)
(b)
to Ms,i EiT Pi Ci
E1T
which shows that (12) is also valid for x ∈ Ωi,j . This completes the proof of the theorem.
Ms,2 E2T P C2
This, together with Condition 1 of this theorem, imply that, for all 0 ≤ λ ≤ 1,
ν,
inf
P >0,P −2ηQ >0,η,δ1 >0,δ2 >0,Q =Q T ,ϑ1 ≥0,ϑ2 ≥0,H1 ,H2
X 2m 2m X T T T ¯ x ρi,s (x)Ai,s Pj x + x Pj ρi,s (x)A¯ i,s x + 2xT Pj Ei w s = 1 s = 1 1 < − 2 z T z + w T w, γ 2m 2m X X T T T x ¯ ρ ( x ) A P x + x P ρj,s (x)A¯ j,s x + 2xT Pi Ej w j ,s i j ,s i s=1 s=1 < − 1 z T z + w T w, γ2
! " 2m 2m X X T T x λ ρi,s (x)A¯ i,s Pi + Pi ρi,s (x)A¯ i,s s =1 s=1 !# 2m 2m X X T ¯ ¯ + (1 − λ) ρj,s (x)Aj,s Pi + Pi ρj,s (x)Aj,s x. s =1 s=1 1 + 2λxT Pi Ei w + 2(1 − λ)xT Pi Ej w < − 2 z T z + w T w, γ ! " 2m 2m X X xT λ ρj,s (x)A¯ Tj,s Pj + Pj ρj,s (x)A¯ j,s s =1 s=1 !# 2m 2m X X T ¯ ¯ + (1 − λ) ρi,s (x)Ai,s Pj + Pj ρi,s (x)Ai,s x. s =1 s =1 + 2λxT Pj Ej w + 2(1 − λ)xT Pj Ei w < − 1 z T z + w T w, γ2
without loss of generality, that θ1 = θ2 = 1. This implies that Constraint (c) in (16) is automatically satisfied. The optimization problem (16) then simplifies to
As we have done in Section 4, in case of switching between only two subsystems, we can set P2 = P, Q1 = Q and Q2 = −Q , then we have P1 = P − 2ηQ , where Q is a symmetric matrix, and assume,
ν ∗ = 0.3409, α ∗ = 2.9336, H1 = −0.3094 0.1259 , H2 = 3.3443 −1.3757 , −0.0422 0.0466 , Q1 = −Q2 = Q = 0.0466 0.0422 11.9804 9.9189 32.8817 −13.1624 P1 = , P2 = . 9.9189 28.0196 −13.1624 7.1183 Plotted in Fig. 1 are the ellipsoids E (P1 , α ∗ ) and E (P2 , α ∗ ), along with a trajectory starting from the origin and under a pulse disturbance of duration 0.2s and with a maximum energy α ∗ . A zoom in plot of this trajectory is shown in Fig. 2. We next estimate the restricted L2 gain of the resulting switched system over W 2α , α ∈ (0, α ∗ ]. This can be done by solving the optimization problem (17). Plotted in Fig. 3 is the obtained γ ∗ over different values of α . Example 2. Consider system (1) with w ∈ W 2α and 0.6 0.8
−0.8 , 0.6
A1 = A2 =
0.1 , 0.1
E1 = E2 =
F1 = 1.2231
B1 = B2 = 1 ,
C1 = C2 = 1
−2.2486 ,
2 , 4
F2 = 0.8396
−1.7221 .
We note that two subsystems result from the two different stabilizing feedback gains. For each of these two subsystems, we
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which results in a switched system with an estimated disturbance tolerance capability of α ∗ = 632.8927. We next proceed to design a switching law that minimizes the restricted L2 gain of the resulting switched system. To this end, let us set α = 500. Solving the optimization problem (17), we obtain a switching law characterized by
−433.9681 Q1 = −Q2 = Q = −10.2274
−10.2274 , 433.9681
with 822.9329 P1 = −688.0880
−688.0880 , 677.0671 1.0000 −0.6839 P2 = 103 × , −0.6839 0.5000 H1 = 10−51 × −0.9690 −0.1154 , H2 = 10−43 × −0.0459 −0.4108 .
Fig. 1. Ellipsoids E (P1 , α ∗ ) and E (P2 , α ∗ ) and a trajectory under a pulse disturbance with energy α ∗ .
This switching law results in a switched system with an estimated restricted L2 gain of γ ∗ = 0.1187. It is clear that γ ∗ < min{γ1∗ , γ2∗ }. On the other hand, when the actuator saturation does not occur, the closed-loop system resulting from the two individual feedback gains, each behaves as a linear system for which the restricted L2 gain can be exactly determined as its H∞ norm. They are γ1∗ = 0.1253 and γ2∗ = 0.1183. However, in the absence of actuator saturation, the L2 gain of the switched system can be estimated as γ ∗ = 0.1072. This demonstrates the effectiveness of the proposed switching scheme in reducing the L2 gain from the disturbance to the system output, as it results in an L2 gain that is smaller than the H∞ norm of each of the closed-loop system under the two individual feedback gains.
Fig. 2. Zoom in plot of the trajectory shown in Fig. 1.
7. Conclusions This paper considered the problem of disturbance tolerance/rejection of a switched system resulting from a family of linear systems subject to actuator saturation and L2 disturbances. Design algorithms for both feedback gains for individual systems and the switching scheme were developed. Several examples were worked out to illustrate the effectiveness of the proposed design method and indeed the power of switching control itself.
Acknowledgments The first and second author’s work was supported in part by National Natural Science Foundation of China under the Young Investigator Program (Class B: Overseas Collaboration). Fig. 3. The restricted L2 gain of the switched system over W 2α .
can use the algorithm in Fang et al. (2004) to estimate the restricted L2 gain. In particular, for α = 500, and the algorithm of Fang et al. (2004) results in γ1∗ = 0.1783 and γ2∗ = 0.1746. We next consider the design of a switching law to reduce the restricted L2 gain. To begin with, we solve the optimization problem (10). We obtain a switching law characterized by 0.3822 Q1 = −Q2 = Q = 10 × −1.1216 3
−1.1216 , −0.3822
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Liang Lu was born in Shenyang, China on May 16, 1980. He received his B.S. and M.S. degrees in Automatic Control from Shenyang University, Shenyang, China, and Huaqiao University, Quanzhou, China, respectively. He is currently working toward his Ph.D. degree in the Department of Automation, Shanghai Jiao Tong University and the Department of Electrical and Computer Engineering, University of Virginia.
Zongli Lin is a professor of Electrical and Computer Engineering at University of Virginia. He received his B.S. degree in mathematics and computer science from Xiamen University, Xiamen, China, in 1983, his Master of Engineering degree in automatic control from Chinese Academy of Space Technology, Beijing, China, in 1989, and his Ph.D. degree in electrical and computer engineering from Washington State University, Pullman, Washington, in 1994. His current research interests include nonlinear control, robust control, and control applications. He was an Associate Editor of the IEEE Transactions on Automatic Control and has served on the operating committees and program committees of several conferences. He currently serves on the editorial boards of several journals and book series, including Automatica, Systems & Control Letters, IEEE/ASME Transactions on Mechatronics and IEEE Control Systems Magazine. He is an elected member of the Board of Governors of the IEEE Control Systems Society. He is a Fellow of the IEEE. Haijun Fang received his Bachelor and Master of Science degrees in Automation from Tsinghua University, Beijing, China, in 1999 and 2001, respectively, and his Ph.D. degree in electrical engineering from University of Virginia in 2006. Dr. Fang is currently a control engineer at MKS Instrument, Rochester, New York. His current research interests include nonlinear control, robust control and control applications in semi-conductor manufacturing systems.
