11

H∞ Control of Stochastic Switched Systems

with Dwell Time

11.1 Introduction

In this chapter [62] we address the problems of stabilization and H∞ state-feedback control of continuous-time, linear switched systems with state-multiplicative noise and dwell time constraint.

The stability analysis of deterministic linear switched systems with dwelltime has been vastly investigated in the past [3], [139]. While the literature onswitching of deterministic systems has been concentrated on the case whereswitching can occur immediately, the realistic case is where there is a minimumtime period, the dwell time, during which no switching occurs. Switching,without dwell time, produces dynamics that is a combination of the dynamicsof the subsystems. This leads to some switched systems that are unstablefor switching without dwell time and stable with (short) dwell time [110]. In[90] it is shown that the worst case (most destabilizing) switching law doesobey some dwell time constraint, at least for second order systems. Arbitrarilyfast switching may also cause large state transients at the switching points.A dwell time may thus be required for these transients to subside. This isprobably one of the reasons why the filed of switched systems with dwell timeis becoming increasingly popular.

Switching of stochastic systems has been considered in the literature inthe context of stochastic Markov jumps [23], [70], [84], [109] (see also [86] forthe deterministic case and for stochastic stability of jump systems see [30]).Switching of stochastic state-multiplicative systems, and switching with dwelltime constraint in particular, have hardly been investigated in the past.

In this chapter we apply a recently developed method which deals withthe switching of deterministic systems and which has been shown to yieldimproved results in the uncertain case [3]. Based on the latter, we address theproblem of stability of stochastic linear switched systems with dwell time andpolytopic type parameter uncertainties. The best stability result attained sofar for deterministic systems, that is both efficiently computable and yieldsa sufficient result, is the one of [43] (it is also used in [21]). This result is

E. Gershon & U. Shaked: Advanced Topics in Control & Estimation, LNCIS 439, pp. 177–191.DOI: 10.1007/978-1-4471-5070-1_11 c© Springer-Verlag London 2013

178 11 H∞ Control of Stochastic Switched Systems with Dwell Time

based on a quadratic LF. It is obtained for the case without uncertainty(namely for a nominal system), and it is derived using LMIs. These LMIsare not affine in the system matrices making it difficult to generalize to theuncertain case. In [14], a piecewise linear in time, quadratic form LF is used.Similarly to [3], we apply in the present work such a LF to stochastic switchedsystems with dwell time. The LF applied is non-increasing at the switchinginstants and is separately assigned to each subsystem. During the dwell time,this function varies piecewise linearly in time after switching occurs and itbecomes time invariant afterwards. Our choice of the LF, allows derivation ofsufficient conditions for robust stochastic stability of the switched system interms of LMIs. In the deterministic case, these conditions have been shownto be less conservative than those obtained by bounding the matrix exponentof [43] by a scalar matrix.

This chapter is organized as follows: Following the problem formulationof Section 11.2, the stochastic stability is investigated in Section 11.3. Theconditions obtained for the stability of the switched system are extended inSection 11.4 to find a robust bound on the L2-gain of the system. The resultsare first derived there for the nominal case and they are then extended tocope with polytopic uncertainty. Section 11.4 also presents an alternative,less conservative, condition for the bound on the stochastic L2-gain of theuncertain systems. The latter is applied, in Section 11.5, to the solution ofthe robust stochastic state-feedback control problem. The theory developedis applied in two examples, in Section 11.6 and Chapter 13, Section 13.3 . Inthe first example, a bound is obtained for the L2-gain of a switched systemwith uncertainties for various values of dwell time. The second example (inChapter 13) is one where a switched state-feedback controller is derived forrobust H∞control of the F4E fighter aircraft.

11.2 Problem Formulation

We consider the following linear stochastic state-multiplicative switched sys-tem [62]:

dx(t) = Aσ(t)x(t)dt+B σ(t)w(t)dt+ Fσ(t)x(t)dβ(t) + Gσ(t)x(t)dν(t), x(0) = 0,

z(t) = C σ(t)x(t) +D σ(t)w(t),(11.1)

which is defined for all t ≥ 0, where x(t) ∈ Rn is the system state vector,w(t) ∈ Rq is the exogenous disturbance in L2

Ft([0,∞);Rq) and z(t) ∈ Rz is

the objective vector. The variables β(t) and ν(t) are zero-mean real scalarWiener processes that satisfy:

E{dβ(t)2}=dt, E{dν(t)2}=dt, E{dβ(t)dν(t)} = 0. (11.2)

The switching rule σ(t), for each t ≥ 0, is such that Aσ(t) ∈ {A1, ..., AM},Fσ(t) ∈ {F1, ..., FM}, Bσ(t) ∈ {B1, ..., BM}, Gσ(t) ∈ {G1, ..., GM} and

11.3 Stochastic Stability 179

Cσ(t) ∈ {C1, ..., CM}, Dσ(t) ∈ {D1, ..., DM}, where Ai ∈ Rn×n, i = 1, ...Mis a Hurwitz matrix. The matrices Ai, Bi, Ci and Di are assumed to residewithin the following polytope:

Ωi =

Ni∑j=1

ηj(t)Ω(j)i ,

Ni∑j=1

ηj(t) = 1, ηj(t) ≥ 0 (11.3)

where:

Ωi =

[Ai Bi Fi

Ci Di Gi

]and Ω

(j)i =

[A

(j)i B

(j)i F

(j)i

C(j)i D

(j)i G

(j)i

], i = 1, ...M, j = 1, ...N.

(11.4)

The above model naturally imposes discontinuity in Aσ(t) since this matrixjumps instantaneously from Ai1 to Ai2 for some i1 �= i2 at the switchinginstant. The latter property also applies to all the system matrices.

For the stochastic state-feedback control problem, we consider the follow-ing system:

dx(t) = [Aσ(t)x(t) +Bσ(t)w(t) +B2,σ(t)u(t)]dt+ Fσ(t)x(t)dβ(t)

+Gσ(t)u(t)dν(t), x(0) = 0,

z(t) = Cσ(t)x(t) +Dσ(t)w(t) +D12,σ(t)u(t),

(11.5)

where x(t), w(t), β(t), ν(t) and σ(t) are defined above, u(t) ∈ Rl is thecontrol signal, and Ai, Bi, Ci, B2,i, Di, and D12,i are assumed to reside inthe following polytope:

Ωi =

Ni∑j=1

ηj(t)Ω(j)i ,

Ni∑j=1

ηj(t) = 1, ηj(t) ≥ 0 (11.6)

where:

Ωi =

[Ai Bi B2,i

Ci Di D12,i

]andΩ

(j)i =

[A

(j)i B

(j)i B

(j)2,i

C(j)i D

(j)i D

(j)12,i

], i = 1, ...M, j = 1, ...N.

(11.7)

Note that in the system (11.5a) Ai is no longer required to be Hurwitz matrixand that in the polytope description of (11.6) and (11.7) we require that ineach subsystem i = 1, ...M the matrices F and G are constant matrices.

11.3 Stochastic Stability

We start by investigating the stability of the stochastic switched system(11.1a,b) where w(t) ≡ 0, under a constraint on the rate of allowed

180 11 H∞ Control of Stochastic Switched Systems with Dwell Time

commutations (dwell time analysis), which means that if the dwell time isT, and the switching instants are τ1, τ2, ..., then τh+1 − τh ≥ T, ∀h ≥ 1.

We seek sufficient conditions for the system to be globally asymptoticallystable in probability where the latter is defined as follows.

Definition 11.1 The solution x(t) of (11.1a,b), where w(t) ≡ 0, is saidto be globally asymptotically stable in probability if, for any scalar ε > 0,limx→0

P{supt≥0

‖x(t)‖ > ε} = 0, and if P{ limt→∞x(t) = 0} = 1 for any initial state

x0 ∈ Rn.A well known sufficient condition for global stability in probability is the

following.

Lemma 11.3.1 ([9]) Assume that there exists a positive Lyapunov functionV (x, t) ∈ Cn,1, with V (0, t) = 0. Let L(x, t) be the infinitesimal generatorof the process x so that E{(LV )(x, t)} < 0 for all x ∈ Rn and for all t ≥ 0.Assume also that inf

t>0V (x, t) → ∞ as ‖x‖ → ∞. Then, the system of (11.1a,b)

is globally asymptotically stable in probability.

We introduce a time-varying LF. For this purpose the following result,inspired by [14], is applied.

Lemma 11.3.2 Assume that for some time interval t ∈ [t0, tf ], where δ =tf−t0, there exist two symmetric matrices P1 and P2 of compatible dimensionsthat satisfy the following:

P1, P2 > 0, P2−P1

δ + P1A+ATP1 + FTP1F + GTP1G < 0,

P2−P1

δ + P2A+ATP2 + FTP2F + GTP2G < 0.

(11.8)

Then, for the system dx(t) = Ax(t)dt+Fx(t)dβ+ Gx(t)dν the expected valueof the infinitesimal generator, denoted by L [9], of the following LF:

V (t, x(t)) = xT (t)P (t)x(t), with P (t) = P1 + (P2 − P1)t− t0δ

,

is strictly negative over the time interval t ∈ [t0, tf ].

Proof: Applying Ito formula [9] to V (t, x(t)) and taking expectation wefind that

E{LV (t, x(t))} = E{xT (t)P (t)x(t) + 2xT (t)P (t)Ax(t) + xT (t)FTP (t)Fx(t)

+xT (t)GTP (t)Gx(t)}

= E{xT (t)(P2−P1

δ + P (t)A+ATP (t) + FTP (t)F + GTP (t)G)x(t)},

where use was made of (11.2). Noting that

11.3 Stochastic Stability 181

P2−P1

δ + P (t)A+ATP (t) + FTP (t)F + GTP (t)G = λ1[P2−P1

δ + P1A+ATP1

+FTP1F + GTP1G] + λ2[P2−P1

δ + P2A+ATP2 + FTP2F + GTP2G]

where λ1 = 1− t−t0δ , λ2 = t−t0

δ , it follows from (11.8b,c) that E{LV (t)} < 0in the time interval specified above. ��The above proof is for a nominal system. The extension to the polytopicuncertainty case is, however, immediate choosing the same P1 and P2 for allthe vertices of the uncertainty polytope Ωi.

We next present sufficient conditions for the stability of nominal stochasticlinear switched systems. These conditions are given in terms of LMIs whichare affine in the systems matrices, and they can thus be easily extended to thepolytopic uncertainty case. Choosing the integer K according to the allowedcomputational complexity, we obtain the following:

Theorem 11.1. The nominal system (11.1a,b) with Bσ(t) = 0 is globallyasymptotically stable for any switching law with dwell time greater thanor equal to T > 0 if there exist a collection of positive definite matricesPi,k, i = 1, ...M, k = 0, ...K of compatible dimensions, such that, for alli = 1, ...M the following holds:

KPi,k+1−Pi,k

T + Pi,kAi +ATi Pi,k + FT

i Pi,kFi + GTi Pi,kGi < 0,

KPi,k+1−Pi,k

T + Pi,k+1Ai +ATi Pi,k+1 + F

Ti Pi,k+1Fi + G

Ti Pi,k+1Gi < 0,

k = 0, ...K − 1

Pi,KAi +ATi Pi,K + FT

i Pi,KFi + GTi Pi,KGi < 0, ∀ l = 1, ...i− 1, i+ 1, ...M,

Pi,K − Pl,0 ≥ 0.(11.9)

Proof: We define τh,k = τh + k TK and τh,0 = τh and note that the dwell time

constraint implies τh,K ≤ τh+1,0 = τh+1. We consider the LF, V (t, x(t)) =xT (t)Pσ(t)(t)x(t), where Pσ(t)(t) is given by:

Pi(t)=

⎧⎨⎩Pi,k + (Pi,k+1 − Pi,k)

K(t−τh,k)T t ∈ [τh,k, τh,k+1)

Pi,K t ∈ [τh,K , τh+1,0)Pi0,K t ∈ [0, τ1)

, k = 0, ...K − 1, h = 1, 2, ...

(11.10)

and where i = σ(t) is the index of the subsystem that is active at time t.Assuming that at some switching instant τh the systems switches

from the i-th subsystem to the l-th subsystem, we have V (τ−h , x(τh)) =x(τh)

TPi,Kx(τh) and V (τh, x(τh)) = x(τh)TPl,0x(τh). Therefore, for the LF

to be non increasing at the switching instant we demand Pi,K ≥ Pl,0

which is the condition (11.9d). After the dwell time, and before the next

182 11 H∞ Control of Stochastic Switched Systems with Dwell Time

switching occurs, we have that V (t, x(t)) = x(t)TPi,Kx(t), where dx(t) =Aix(t)dt + Fix(t)dβ(t) + Gix(t)dν(t). Therefore,

E{(LV )(t, x(t))} = E{x(t)T [Pi,KAi +ATi Pi,K + FT

i Pi,KFi + GTi Pi,KGi]x(t)}

and (11.9c) guarantees then that this expression is negative for any x(t) �= 0.During the dwell time we consider the time intervals t ∈ [τh,k, τh,k+1) whereTK = τh,k+1− τh,k. The matrix P (t) changes then linearly from Pi,k to Pi,k+1,and conditions (11.9a-c) guarantee that V (t, x(t)) is decreasing during thistime interval, according to Lemma 11.3.2. ��We note that the LMIs in Theorem 11.1 are affine in the system matricesand thus they can be readily applied to the stability problem of polytopicuncertain systems. Robust stability in the case of uncertain subsystems isthus given by the following corollary where K is a prechosen integer:

Corollary 11.3.1 Consider the system (11.1a,b) with Bσ(t) = 0 and (11.3).Assume that for some dwell time T > 0 there exist a collection of positivedefinite matrices Pi,k, i = 1, ...M, k = 0, ...K of compatible dimensions, suchthat, for all i = 1, ...M , and j = 1, ...N the following holds:

KPi,k+1−Pi,k

T + Pi,kA(j)i +A

(j)T

i Pi,k + F(j)T

i Pi,kF(j)i + G

(j)T

i Pi,kG(j)i < 0,

KPi,k+1−Pi,k

T + Pi,k+1A(j)i +A

(j)T

i Pi,k+1 + F(j)T

i Pi,k+1F(j)i

+G(j)T

i Pi,k+1G(j)i < 0, k = 0, ...K − 1,

Pi,KA(j)i +A

(j)T

i Pi,K + F(j)T

i Pi,KF(j)i + G

(j)T

i Pi,KG(j)i < 0,

Pi,K − Pl,0 ≥ 0, ∀ l = 1, ...i− 1, i+ 1, ...M.

Then, the system (11.1a,b) is globally asymptotically stable for any switchinglaw with dwell time greater than or equal to T .

11.4 Stochastic L2-Gain

We seek a sufficient condition for the mean square stability of the system(11.1a-c) and for the following inequality to hold, given a prescribed scalarγ > 0,

J = E{∫ ∞

0

(zT z − γ2wTw)dt} ≤ 0, ∀ w ∈ L2Ft([0,∞);Rq). (11.11)

Let i0 = σ(0), and let τ1, τ2, ... be the switching instances, where τh+1 − τh ≥T, ∀h = 1, 2, ... We define τh,k = τh + kT

K for k ≥ 1, and τh,0 = τh and notethat the dwell time constraint implies τh,K ≤ τh+1,0 = τh+1. We choose thefollowing LF.

11.4 Stochastic L2-Gain 183

V (t, x(t)) = xT (t)Pσ(t)x(t), (11.12)

with Pi(t) as defined in (11.10) where K is an integer that is chosen a priori,according to the allowed computational complexity. We also denote:

J = limt→∞E{V (t, x(t))} + E{∫ t

0

(zT z − γ2wTw)ds}

Since V (t, x(t)) ≥ 0 ∀t, we have that J ≤ J . Taking into account thatLV (t, x(t))dt exists for all t, except for the switching instances, and thatx(0) = 0, we write

limt→∞E{V (t, x(t))} =

E{∞∑h=0

∫ τh+1

τh

LV (t, x(t))dt +∞∑h=1

(V (τh, x(τh))− V (τ−h , x(τ−h )))}

where τ0 = 0. If the conditions for non increasing V (t, x(t)) at the switchinginstances are satisfied we find that V (τh, x(τh))−V (τ−h , x(τ

−h )) ≤ 0 ∀ h > 0,

and then:

limt→∞E{V (t, x(t))} ≤ E{∞∑h=0

∫ τh+1

τh

LV (t, x(t))dt}. (11.13)

Denoting

J = E{∞∑h=0

∫ τh+1

τh

LV (t, x(t))dt +

∫ ∞

0

(zT z − γ2wTw)dt}, (11.14)

we have from (11.13) that: J ≤ J ≤ J . Consequently, if J ≤ 0 and the aboveLF does not increase at the switching instants, the stochastic L2-gain of thesystem will be less than γ. We apply below the following LMI condition of[53] for the L2-gain of stochastic state-multiplicative linear systems (withoutswitching):

P > 0,

⎡⎢⎢⎢⎢⎣

P + PA+ATP PB CT FTP GTP∗ −γ2Iq DT 0 0∗ ∗ −Iz 0 0∗ ∗ ∗ −P 0∗ ∗ ∗ ∗ −P

⎤⎥⎥⎥⎥⎦ < 0, (11.15)

where A, B, C, F, G and D are as in (11.1a–c). Applying the method of[14] on the latter LMI for the switched system (11.1a–c), the following resultis obtained for a prescribed integer K:

184 11 H∞ Control of Stochastic Switched Systems with Dwell Time

Theorem 11.2. The stochastic L2-gain of the nominal system (11.1a–c) isless than a prescribed γ > 0 for a dwell time of T ≥ T if there exists a col-lection of positive definite matrices Pi,k, i = 1, ...M, k = 0, ...K of compatibledimensions such that, for all i = 1, ...M the following holds:

⎡⎢⎢⎢⎢⎣

KPi,k+1−Pi,k

T + Pi,kAi +ATi Pi,k Pi,kBi C

Ti FT

i Pi,k GTi Pi,k

∗ −γ2Iq DTi 0 0

∗ ∗ −Iz 0 0∗ ∗ ∗ −Pi,k 0∗ ∗ ∗ ∗ −Pi,k

⎤⎥⎥⎥⎥⎦ < 0,

⎡⎢⎢⎢⎢⎣

KPi,k+1−Pi,k

T + Pi,k+1Ai +ATi Pi,k+1 Pi,k+1Bi C

Ti FT

i Pi,k+1 GTi Pi,k+1

∗ −γ2Iq DTi 0 0

∗ ∗ −Iz 0 0∗ ∗ ∗ −Pi,k+1 0∗ ∗ ∗ ∗ −Pi,k+1

⎤⎥⎥⎥⎥⎦ < 0,

k = 0, ...K − 1

⎡⎢⎢⎢⎢⎣

Pi,KAi+ATi Pi,K Pi,KBi C

Ti FT

i Pi,K GTi Pi,K

∗ −γ2Iq DTi 0 0

∗ ∗ −Iz 0 0∗ ∗ ∗ −Pi,K 0∗ ∗ ∗ ∗ −Pi,K

⎤⎥⎥⎥⎥⎦ < 0, (11.16)

Pi,K−Pl,0≥0, ∀ l=1, ...i−1, i+1, ...M.

Proof: The theorem is proved by applying the result of Lemma 11.3.2 andP (t) of (11.10) to (11.15a,b). We first show that the above conditions guaran-tee the asymptotic stability of the system and that V does not increase at theswitching points. We then turn to (11.14) showing that the integrand thereis strictly negative. Condition (11.16d) means that once a switching occurs,P (t) switches in such a way that it does not increase at the switching instantof time. From this time on, P (t) becomes piecewise linear in time, where overthe time interval t ∈ [τh + k T

K , τh + (k + 1) TK ] , it changes linearly from Pi,k

to Pi,k+1, where i is the index of the active subsystem at t ∈ [τh, τh+1].Before the first switching occurs, P (t) is constant and by (11.16c) it guar-

antees that V (t) decreases while the system remains in the same subsystem. Itfollows then from (11.16a,b) that V (t) is strictly decreasing during the dwelltime, the LMIs (11.16c) guarantees that the LF is strictly decreasing for anyt ∈ [τh,K , τh+1,0), and (11.16d) guarantees that the LF is non-increasing atthe switching instances. Since the switching points are distinct, the system isasymptotically stable according to Lasalle’s invariance principle [68]. The firstdiagonal block of the LMIs (11.16a-c) corresponds to LV (t, x(t)) of (11.14).The second and the third columns and rows in these LMIs correspond to wTwand zT z there, respectively. ��

11.4 Stochastic L2-Gain 185

The LMIs in Theorem 11.2 are all affine in the system matrices and theresult there can thus be readily applied to the uncertain case where the systemparameters lie in the polytope of Ωi. One has then to replace Ai, Bi, Ci andDi by A

ji , B

ji , C

ji and Dj

i , respectively, and solve the resulting LMIs for allthe vertices in Ωi. The resulting LMIs are not suitable, however, for state-feedback control synthesis. For the latter we derive, in the sequel, a similarresult which will be suitable for the state-feedback control.

Let Q(t) = P−1(t). We depart from the assumption of linearity in time ofP (t) and assign this linearity to Q(t), namely

Q(t)=

⎧⎨⎩Qi,k + (Qi,k+1 −Qi,k)

K(t−τh,k)T t ∈ [τh,k, τh,k+1)

Qi,K t ∈ [τh,K , τh+1,0)Qi0,K t ∈ [0, τ1)

, h = 1, 2, ...

Multiplying (11.15b) by diag{Q, Iq, Iz, Q,Q} from the left and the right andapplying the method of Lemma 11.3.2 the following alternative LMI conditionsare obtained for the L2-gain bound of the system (11.1a-c):

Qi,k > 0,

⎡⎢⎢⎢⎢⎣

−KQi,k+1−Qi,k

T +AiQi,k +Qi,kATi Bi Qi,kC

Ti Qi,kF

Ti Qi,kG

Ti

∗ −γ2Iq DTi 0 0

∗ ∗ −Iz 0 0∗ ∗ ∗ −Qi,k 0∗ ∗ ∗ ∗ −Qi,k

⎤⎥⎥⎥⎥⎦ < 0,

⎡⎢⎢⎢⎢⎣

Υi,k Bi Qi,k+1CTi Qi,k+1F

Ti Qi,k+1G

Ti

∗ −γ2Iq DTi 0 0

∗ ∗ −Iz 0 0∗ ∗ ∗ −Qi,k+1 0∗ ∗ ∗ ∗ −Qi,k+1

⎤⎥⎥⎥⎥⎦ < 0,

Υi,k = −KQi,k+1−Qi,k

T +AiQi,k+1 +Qi,k+1ATi , k = 0, ...K − 1,

⎡⎢⎢⎢⎢⎣

AiQi,K +Qi,KATi Bi Qi,KC

Ti Qi,KF

Ti Qi,KG

Ti

∗ −γ2Iq DTi 0 0

∗ ∗ −Iz 0 0∗ ∗ ∗ −Qi,K 0∗ ∗ ∗ ∗ −Qi,K

⎤⎥⎥⎥⎥⎦ < 0,

Qi,K −Ql,0 ≤ 0, ∀ l = 1, ...i− 1, i+ 1, ...M.

(11.17)

The latter conditions can be readily applied to design a state-feedback con-troller for the switched systems. To reduce the conservatism entailed in theconditions of (11.17a–e) when applied to the uncertain case, the followingresult is obtained.

186 11 H∞ Control of Stochastic Switched Systems with Dwell Time

Corollary 11.4.1 The stochastic L2-gain of the system (11.1a–c) is less thanγ for a dwell time T ≥ T if there exists a collection of symmetric matricesQi,k > 0, Gi,k, Hi,k, i = 1, ...M , k = 0, ...,K of compatible dimensions, suchthat, for all i = 1, ...M, the following holds:

⎡⎢⎢⎢⎢⎢⎢⎣

Ψ11 Qi,k −GTi,k +AiHi,k G

Ti,kC

Ti Bi Qi,kF

Ti Qi,kG

Ti

∗ −Hi,k −HTi,k HT

i,kCTi 0 0 0

∗ ∗ −γ2Iq Di 0 0∗ ∗ ∗ −Iz 0 0∗ ∗ ∗ ∗ −Qi,k 0∗ ∗ ∗ ∗ ∗ −Qi,k

⎤⎥⎥⎥⎥⎥⎥⎦< 0,

⎡⎢⎢⎢⎢⎢⎢⎣

Ψ11 Qi,k+1−GTi,k+1+AiHi,k+1 G

Ti,k+1C

Ti Bi Qi,k+1F

Ti Qi,k+1G

Ti

∗ −Hi,k+1 −HTi,k+1 HT

i,k+1CTi 0 0 0

∗ ∗ −γ2Iq Di 0 0∗ ∗ ∗ −Iz 0 0∗ ∗ ∗ ∗ −Qi,k+1 0∗ ∗ ∗ ∗ ∗ −Qi,k+1

⎤⎥⎥⎥⎥⎥⎥⎦< 0,

k = 0, ...K − 1,

Ψ11 = −KQi,k+1−Qi,k

T +GTi,kA

Ti +AiGi,k,

Ψ11 = −KQi,k+1−Qi,k

T +GTi,k+1A

Ti +AiGi,k+1,⎡

⎢⎢⎢⎢⎢⎢⎣

GTi,KA

Ti +AiGi,K Qi,K−GT

i,K+AiHi,K GTi,KC

Ti Bi Qi,KF

Ti Qi,KG

Ti

∗ −Hi,K −HTi,K HT

i,KCTi 0 0 0

∗ ∗ −γ2Iq Di 0 0∗ ∗ ∗ −Iz 0 0∗ ∗ ∗ ∗ −Qi,K 0∗ ∗ ∗ ∗ ∗ −Qi,K

⎤⎥⎥⎥⎥⎥⎥⎦< 0,

Qi,K −Ql,0 ≤ 0, ∀ l = 1, ...i− 1, i+ 1, ...M.

(11.18)

Proof: Assuming that (11.18a–d) are feasible, we multiply (11.18a–c) by ΥT

from the left and by Υ from the right, where ΥΔ= diag{

[I 0 0AT

i I CTi

0 0 I

], I, I, I},

and readily retrieve the corresponding LMIs of (11.17a–c). ��

The result of Corollary 11.4.1 is readily extended to the uncertain case. Re-placing the system matrices Ai, Bi, Ci and Di by Aj

i , Bji , C

ji , and D

ji ,

respectively, and solving the resulting LMIs for all the vertices in Ωi, a solu-tion to the uncertain case is obtained.

11.5 H∞ State-Feedback Control 187

11.5 H∞ State-Feedback Control

Given the system (11.5a–c) with the uncertainty that is described in (11.6),a state-feedback controller is sought that stabilizes the system, and satisfiesthe performance criterion (11.11).

We distinguish between two cases: The first is characterized by a switchingsignal σ(t) that is measured on-line and different state-feedback gains cantherefore be applied to different subsystems. In the second case, it is assumedthat the switching signal is unknown, and thus the same state-feedback gainshould be applied to all subsystems.

We begin with the first case. Replacing the system matrices Ai, Bi, Ci,and Di in (11.17) by Aj

i , Bji , C

ji and Dj

i of (11.5a,c) respectively, where

Aji = Aj

i +Bj2,iKi,k, Cj

i = Cji +D

j12,iKi,k and denoting Yi,k = Ki,kQi,k, we

obtain the following result for a prescribed integer K:

Theorem 11.3. There exists a state feedback gain Kσ(t) that stabilizes (11.5a-c), with uncertainty as in (11.6) and a dwell time of T ≥ T , and achievesclosed-loop stochastic L2-gain bound that is less than a prescribed positivescalar γ, if there exists a collection of matrices: Yi,k, Qi,k > 0, i = 1, ...M, k =0, ...K of compatible dimensions such that, for all i = 1, ...M , and j = 1, ...Nthe following holds:

⎡⎢⎢⎢⎢⎣

Υ11,k B(j)i Qi,kC

(j),Ti + Yi,kD

(j),T12,i Qi,kF

Ti Y T

i,kGTi

∗ −γ2Iq D(j)i 0 0

∗ ∗ −Iz 0 0∗ ∗ ∗ −Qi,k 0∗ ∗ ∗ ∗ −Qi,k

⎤⎥⎥⎥⎥⎦ < 0,

where

Υ11,k=−KQi,k+1−Qi,k

T+Qi,kA

(j),Ti +A

(j)i Qi,k+B

(j)2,iYi,k + Y T

i,kB(j),T2,i ,

⎡⎢⎢⎢⎢⎣

Υ11,k+1 B(j)i Qi,k+1C

(j),Ti +Yi,k+1D

(j),T12,i Qi,k+1F

Ti Y T

i,k+1GTi

∗ −γ2Iq D(j)i 0 0

∗ ∗ −Iz 0 0∗ ∗ ∗ −Qi,k+1 0∗ ∗ ∗ ∗ −Qi,k+1

⎤⎥⎥⎥⎥⎦<0,

k = 0, ...K − 1,

where

Υ11,k+1=−KQi,k+1−Qi,k

T+Qi,k+1A

(j),Ti +A

(j)i Qi,k+1+B

(j)2,i Yi,k+1+Y

Ti,k+1B

(j),T2,i ,

188 11 H∞ Control of Stochastic Switched Systems with Dwell Time

⎡⎢⎢⎢⎢⎣

Υ11,K B(j)i Qi,KC

(j),Ti + Yi,KD

(j),T12,i Qi,KF

Ti Y T

i,KGTi

∗ −γ2Iq D(j)i 0 0

∗ ∗ −Iz 0 0∗ ∗ ∗ −Qi,K 0∗ ∗ ∗ ∗ −Qi,K

⎤⎥⎥⎥⎥⎦ < 0,

Υ11,K = Qi,KA(j),Ti +A

(j)i Qi,K +B

(j)2,i Yi,K + Y T

i,KB(j),T2,i

and

Qi,K −Ql,0 ≤ 0,∀ l = 1, ...i− 1, i+ 1, ...M.

(11.19)

The state-feedback gain is then given by:

Kσ(t)(t) =

⎧⎪⎪⎨⎪⎪⎩

Yσ(t)[Qσ,k +K(Qσ,k+1 −Qσ,k)t−τh,k

T ]−1 t ∈ [τh,k, τh,k+1),

Yσ,KQ−1σ,K t ∈ [τh,K , τh+1,0)

Yi0,KQ−1i0,K

t ∈ [0, τ1).

where in the upper row k = 0, 1, ..K− 1 and Yσ(t) = Yσ,k +t−τh,k

T K(Yσ,k+1−Yσ,k).

Theorem 11.3 provides a time-varying state-feedback gain that may bedifficult to implement, especially for large K. The solution there may also bequite conservative due to the fact that Qi,k are the same for all the points inΩi. This conservatism is reduced by applying the result of Corollary 11.4.1which allows Qi,k to be vertex dependent. However, in order to remain inthe linear arena, applying the latter result of Corollary 11.4.1 can be doneonly by taking Gσ(t)(t) = 0 in (11.5a). We thus obtain the following lessconservative solution to the problem of state-feedback control of (11.5), withuncertainty as in (11.6), for a prescribed integer K and where we denoteYi,k = Ki,kGi,k. Note that unlike the result of Corollary 11.4.1 which refersto the stochastic L2-gain, in the design of the state-feedback controller, oneis compelled to take Hi,k = ηGi,k, η > 0 with η as a tuning scalar parameter,in order to arrive at a feasible solution.

Corollary 11.5.1 For a dwell time T ≥ T , there exists a state feedback gainKσ(t) that stabilizes the system of (11.5a–c) and (11.6), and achieves, for agiven scalar η > 0, closed-loop stochastic L2-gain that is less than a prescribedpositive scalar γ, if there exists a collection of matrices Gi,k, Yi,k, Qi,k >0, i = 1, ...M, k = 0, ...K of compatible dimensions such that, for all i =1, ...M , and j = 1, ...N the following holds.

11.5 H∞ State-Feedback Control 189

⎡⎢⎢⎢⎢⎢⎣

Υ1,k Υ2,k GTi,kC

(j),Ti + Y T

i,kD(j)T12,i B

(j)i Q

(j)i,kF

Ti

∗ −η(Gi,k +GTi,k) ηG

Ti,kC

(j),Ti + ηY T

i,kD(j)T12,i 0 0

∗ ∗ −γ2Iq D(j)i 0

∗ ∗ ∗ −Iz 0

∗ ∗ ∗ ∗ −Q(j)i,k

⎤⎥⎥⎥⎥⎥⎦< 0,

Υ1,k = −KQ(j)

i,k+1−Q

(j)

i,k

T +GTi,kA

(j),Ti +A

(j)i Gi,k +B

(j)2,i Yi,k + Y T

i,kB(j),T2,i ,

Υ2,k = Q(j)i,k −GT

i,k + ηA(j)i Gi,k + ηB

(j)2,i Yi,k,

⎡⎢⎢⎢⎢⎢⎣

Υ1,k+1 Υ2,k+1 GTi,k+1C

(j),Ti +Y T

i,k+1D(j)T12,i B

(j)i Q

(j)i,k+1F

Ti

∗ −η(Gi,k+1 +GTi,k+1) ηG

Ti,k+1C

(j),Ti +ηY T

i,k+1D(j)T12,i 0 0

∗ ∗ −γ2Iq D(j)i 0

∗ ∗ ∗ −Iz 0

∗ ∗ ∗ ∗ −Q(j)i,k+1

⎤⎥⎥⎥⎥⎥⎦< 0,

Υ1,k+1 = −KQ(j)

i,k+1−Q(j)

i,k

T +GTi,k+1A

(j),Ti +A

(j)i Gi,k+1+B

(j)2,iYi,k+1+Y

Ti,k+1B

(j),T2,i ,

k = 0, ...K − 1,

Υ2,k+1 = Q(j)i,k+1 −GT

i,k+1 + ηA(j)i Gi,k+1 + ηB

(j)2,i Yi,k+1,

⎡⎢⎢⎢⎢⎢⎣

Υ1,K Υ2,K GTi,KC

(j),Ti + Y T

i,KD(j)T12,i B

(j)i Q

(j)i,KF

Ti

∗ −η(Gi,K +GTi,K) ηGT

i,KC(j),Ti + ηY T

i,KD(j)T12,i 0 0

∗ ∗ −γ2Iq D(j)i 0

∗ ∗ ∗ −Iz 0

∗ ∗ ∗ ∗ −Q(j)i,K

⎤⎥⎥⎥⎥⎥⎦< 0,

Υ1,K = GTi,KA

(j),Ti +A

(j)i Gi,K +B

(j)2,i Yi,K + Y T

i,KB(j),T2,i ,

Υ2,K = Q(j)i,K −GT

i,K + ηA(j)i Gi,K + ηB

(j)2,i Yi,K ,

Q(j)i,K −Q(j)

l,0 ≤ 0, ∀ l = 1, ...i− 1, i+ 1, ...M.

(11.20)

Remark 11.1. The result of Corollary 11.5.1 only improves the one of Theo-rem 11.3 where the switching strategy of σ(t) is measured on line. The samemethod can also be applied to a case where time-invariant controller Ki issought. One has then to choose the same Gi and Yi for all the time intervals.One may also choose Yi and Gi,k, k = 1, ...,K and obtain then a linearlydependent time-varying controller. In the case where the switching strategyis not known, one is obliged to choose the same constant G and Y for all thesubsystems which leads to a constant controller gain K.

190 11 H∞ Control of Stochastic Switched Systems with Dwell Time

11.6 Example – Stochastic L2-Gain Bound

We consider the system of (11.1a–c) with the following matrices:

A1 =

[0 1

−10(1 + δ1) −1

], F1 =

[0 0.060 0

], B1 =

[01

], C1 =

[0.8715 0

],

A2 =

[0 1

−0.1(1+δ1) −0.5

], F2 = F1, B2= B1, C2 =

[0 0.335

],

D1 = −0.8715, D2 = 0.335, G1 =

[0 0.020 0

], G2 = G1

where δ1 ∈ [−0.2 0.2]. The deterministic version of this system appearedin [21] without uncertainty. Table 11.1 below shows the minimal disturbanceattenuation level γ that is achieved for various dwell times and two values ofK by applying the robust version of Theorem 11.2 and the less conservative,robust version of Corollary 11.4.1. In Table 11.1, ‘n.f’ means ‘non feasible’.

Table 11.1 Values of γ for the example

Dwell time (sec) 3.7 5 6 7.8 8.3 10 15 30

Theorem 11.2,K = 10 n.f n.f n.f 164.32 24.19 7.81 4.54 4.06

Theorem 11.2, K = 6 n.f n.f n.f n.f 374.05 10.41 4.91 4.10

Corollary 11.4.1,K = 10 22.86 3.45 2.69 2.33 2.29 2.21 2.16 2.16

The results of the table clearly show the superiority of Corollary 11.4.1over Theorem 11.2. They also demonstrate the role of K in achieving smallerbounds for the L2-gain. We note that the Matlab codes of the results that aredescribed in Table 11.1 are given in Appendix C. One can readily produce,with these codes, additional results by taking different values of Td and K.

11.7 Conclusions

A new method for analyzing the mean square stability of stochastic state-multiplicative linear switched systems, using a switching dependent LF, isintroduced. The method is applied to both nominal and uncertain polytopic-type systems. The stability result is extended to solve the stochastic L2-gainproblem via a set of LMIs. Based on the latter solution, the H∞state-feedbackcontrol problem is solved where a time-varying switched state-feedback gain isderived. Constant and linear-in-time feedback gains are obtained then as spe-cial cases. In the case of polytopic type uncertainties, the results of the present

11.7 Conclusions 191

work can also be generalized, to situations where the uncertainty polytopesof the subsystems have different number of vertices. Other immediate exten-sions are to stochastic systems with a different dwell time for each subsystem,and systems whose switching is limited, in the sense that each subsystem canonly be switched to a given subset of subsystems (a case which is encounteredin modeling large uncertainties). The theory presented can also be easily ex-tended to include norm-bounded uncertainties. The example in Section 11.6and in Chapter 13, Section 13.3 clearly demonstrate the tractability and ap-plicability of the present work to real engineering systems. In both examples,a significant reduction of the bound on the attenuation level was obtained byapplying the less conservative results of Corollaries 11.4.1 and 11.5.1.