the disturbance decoupling problem with stability for switching

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Systems & Control Letters 70 (2014) 1–7

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Systems & Control Letters

journal homepage: www.elsevier.com/locate/sysconle

The disturbance decoupling problem with stability for switchingdynamical systems

G. Conte a, A.M. Perdon a,∗, N. Otsuka b

a Dipartimento di Ingegneria dell’Informazione, Università Politecnica delle Marche, Ancona, Italyb Division of Science, School of Science and Engineering, Tokyo Denki University, Hatoyama-Machi, Hiki-Gun, Saitama 350-0394, Japan

a r t i c l e i n f o

Article history:Received 8 December 2012

Received in revised form

18 March 2014

Accepted 17 May 2014

Available online 5 June 2014

Keywords:

Switching systems

Disturbance decoupling

Geometric approach

a b s t r a c t

The disturbance decoupling problem with stability is dealt with by means of the geometric approach forswitching systems. The existence of feedbacks which decouple the disturbance and, at the same time, as-

sure stabilityis difficultto characterize,since theactionof thefeedbackcouples with thatof theswitching

law. Under suitable conditions,it is shown thatthe above requirement can be dealt with in separate ways

and this allows us to state a checkable necessary condition and, on that basis, also a sufficient condition

for solvability of the problem.

© 2014 Elsevier B.V. All rights reserved.

1. Introduction

The geometric approach developed by Basile and Marro [1] and

by Wonham [2] in the early 1970 has been, since then, successfullyapplied to a large class of control problems in thefield of linear dy-

namical systems. The approach has been then extended to otherclasses of dynamical systems, providing solutions to many non-

interacting control problems, regulation problems and observation

problems.More recently, geometric concepts have been applied to the

study of control problems which involve switching systems. In

many control, regulation or observation problems, solutions arecharacterized by structural conditions, concerning existence and

properties of subspaces of the state space, and by qualitative con-

ditions, in general concerning stability of specific subsystems. Thegeometric approach has proven to be well suited for dealing with

structural issues and, therefore, considering a switching system

Σ σ , it is quite natural to employ it in analyzing structural proper-ties of the modes Σ i, i ∈ I that, in an appropriate sense, do not

depend on the variation of the index i ∈ I . On the other hand,

switching largely affects qualitative properties, like stability, and

therefore the fulfillment of qualitative conditions is mainly relatedto properties of σ , both when σ can be appropriately chosen to

control the system and when it cannot be chosen. Although it is

not always possible to decouple structural properties of the modes

∗ Corresponding author. Tel.: +39 0712204598.

E-mail addresses: [email protected] (G. Conte), [email protected]

(A.M. Perdon), [email protected] (N. Otsuka).

from qualitative properties of the switching system, this approach

provides valuable insight into many cases, where it gives com-

plete or partial characterization of solutions to problems of above-

mentioned kinds. Previous examples along this line are the resultsof [3–6] where the problem of decoupling a disturbance from the

output of a switching system, under various conditions, has been

considered, and those of [7–9] where a regulation problem for a

switching system has been considered.

In this paper, we revise the formulation of the Disturbance De-

coupling Problem with Stability (DDPS) by means of state feed-

back in the case in which the switching rule depends only on time

(differently from [4,5], where state-dependent switching rules are

considered) and can be conveniently chosen. The problem is stud-

ied by using geometric concepts and by introducing a new charac-

terization of specific properties of time-dependent switching rules.

The existence of solutions is characterized, under suitable hypoth-

esis, by means of structural geometric conditions and of quali-

tative conditions that are not coupled, but that can be fulfilledindependently one from the other. This makes possible to get, first,

a necessary condition for the existence of solutions and, then, a

sufficient condition that, when the necessary one is verified, is

practically checkable and constructive. These results improve and

deepen those already proved in [3].

The paper is organized as follows. In Section 2, we present and

discuss the geometric notions of controlled invariance and con-

ditioned invariance for switching systems. Structural decomposi-

tion with respect to controlled invariance is one of the fundamen-

tal concepts we introduce. The problem of decoupling a distur-

bance from the output by means of state feedback is discussed in

Section 3. We analyze the case in which decoupling has to be

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2 G. Conte et al. / Systems & Control Letters 70 (2014) 1–7

achieved forarbitrary choiceof theswitchingrule (DisturbanceDe-

coupling Problem under Arbitrary Switching, or DDPAS) and then,

adding the requirement of stability, we concentrate on the case in

which both decoupling and stability have to be achieved by choos-

ing, in addition to the state feedback, a specific switching rule.

Motivations for taking into account this way of stating the prob-

lem are discussed in Remark 1. Specific properties of the switching

rule, namely exhaustiveness and essentiality, are defined and stud-

ied in order to give, in Proposition 7, a first necessary condition for

the solvability of the DDPS. In Section 4, the necessary condition

for solvability is restated, under slightly more restrictive hypothe-

ses, showing that stability does not depend on the choice of the

state feedback which fulfills the structural requirements. The in-

sight provided in this way makes possible to state,in Proposition 8,

a necessaryconditionfor solvability of theDDPS that exploits prop-

erties of switching systemswhose modes arecharacterized by nor-

mal dynamic matrices. Section 5 contains conclusions and it out-

lines a possible way to investigate further and to ameliorate the

sufficient condition.

2. Preliminaries

Let R and R+ denote respectively the field of real numbers andthe subset of nonnegative ones. We consider the switching linear

system Σ σ defined by the equations

Σ σ ≡

˙ x(t ) = Aσ(t ) x(t ) + Bσ(t )u(t )

y(t ) = C σ(t ) x(t ) (1)

where t ∈ R+ is the time variable; x ∈ X = R

n is the state; u ∈U = R

m is the input; y ∈ Y = R p is the output; σ is a function,

representing a switching rule, which takes values in the set I = {1,. . . , N } and that, in our framework, is assumed to depend on time

only, that isσ : R+ → I , and, finally, for any value i ∈ I taken byσ , Ai, Bi, C i are matrices of suitable dimensions with real entries.

Without loss of generality, we will assume that Bi is full column

rank for all i ∈ I .In other terms, a switching system Σ σ consists of an indexed

family Σ = {Σ i}i∈I of continuous-time, linear systems of the form

Σ i ≡

˙ x(t ) = Ai x(t ) + Biu(t )

y(t ) = C i x(t ) for i = 1, . . . , N , (2)

called modes of Σ σ , and of a supervisory switching rule σ , whose

value σ (t ) specifies the mode which is active at time t . A stan-

dard requirement on σ is that it generates only a finite number of

switches on anytime interval of finitelength, so to exclude chatter-

ing phenomena. This will be guaranteed by asking that the length

of the time interval between any two consecutive switchings is not

smaller thana constant τ σ , called the dwell-time of σ , with τ σ > 0.

According to applications, interest can be in studying properties

of Σ σ which hold for any choice of the switching rule σ , as wellas in investigating the existence of (restricted classes of) switch-

ing rules that guarantee the fulfillment of specific requirement (see

e.g. [10]).

In theabove-definedframework, we introducea numberof geo-

metric notions and results forswitchingsystems of form (1). Proofs

of the statements in this Section can be obtained by applying stan-

dard geometric techniques in thesame way as described in [1,2] in

the corresponding situations or can be found in the quoted refer-

ences.

2.1. Controlled invariance and conditioned invariance

Definition 1 ([11,12]). Given a family Σ = {Σ i}i∈I of linear sys-

tems of the form (2), a subspaceV ⊆ X is called a robust controlledinvariant subspace for Σ , o r a robust ( Ai∈I , Bi∈I )-invariant subspace,

if AiV ⊆ V + Im Bi for all i = 1, . . . , N . If Σ σ is a switching lin-ear system of form (1) defined by the elements of Σ , any robustcontrolled invariant subspace V for Σ is said to be a controlled in-variant subspace for Σ σ .

Proposition 1. Given a family Σ = {Σ i}i∈I of linear systems of the form (2), a subspace V ⊆ X is a robust controlled invariant for Σ if and only if there exists a family F = {F i, i = 1, . . . , N } of feedbacks

F i : X → U , with i = 1, . . . , N, such that ( Ai + BiF i)V ⊆ V for alli = 1, . . . , N. Any family F of that kind is called a family of friendsof V .

If K ⊆ X is a subspace, the set V ( Ai∈I , Bi∈I ,K) of all robustcontrolled invariant subspaces contained in K forms a semi-lattice with respect to inclusion and sum of subspaces; there-fore V ( Ai∈I , Bi∈I ,K) has a maximum element, usually denoted byV∗( Ai∈I , Bi∈I ,K) or simply by V∗ if no confusion arises. An algo-rithm to computeV∗( Ai∈I , Bi∈I ,K) is reported in [4] and the samewas already given in [12] and proved to hold, under suitable hy-pothesis, also in the case of infinite families of systems.

Definition 2. Given a family Σ = {Σ i}i∈I of linear system of theform (2), a subspace S ⊆ X is called a robust conditioned invari-ant subspace for Σ , or a robust ( Ai∈I , C i∈I )-invariant subspace, if

Ai(S ∩ Ker C i) ⊆ S for all i = 1, . . . , N . If Σ σ is a switching lin-ear system of form (1) defined by the elements of Σ , any robustconditioned invariant subspace S for Σ is said to be a conditionedinvariant subspace for Σ σ .

If M ⊆ X is a subspace,the set S( Ai∈I , C i∈I ,M) of allrobust con-ditioned invariant subspaces containing M forms a semi-latticewith respect to inclusion and intersection of subspaces; there-fore S( Ai∈I , C i∈I ,M) has a minimum element, usually denoted byS∗( Ai∈I , C i∈I ,M) or simply by S∗ if no confusion arises. An algo-rithm to compute S∗( Ai, C i,M) can easily be obtained by dualityfrom the corresponding algorithm for V∗( Ai, Bi,K).

Notations. In the rest of the paper, given a family Σ = {Σ i}i∈I of linear systems of the form (2), we will denote, respectively, by Kthe subspace K = 1,...,N Ker C i and by B the subspace B =

1,...,N Im Bi. Accordingly,V∗ and S∗ will usually be understood, if no confusion arises, to denote the maximum robust controlled in-variant subspace for Σ contained inK and, respectively, the min-imum conditionally invariant subspace for Σ containingB.

Given a subspaceM ⊆ V∗ , we can consider the set V (M) of allthe robust controlled invariant subspaces containing M and con-tained inK . A key property of V (M), when (V∗

S∗) ⊆ M holds,

is given below.

Proposition 2. Given a family Σ = {Σ i}i∈I of linear systems of the form (2), with the same notation as above, let M ⊆ X be a subspacesuch that the following condition holds.

V∗

S∗

⊆ M ⊆ V∗. (3)

Then, the set V (M) of all robust controlled invariant subspaces for Σ containing M and contained inK is a lattice with respect to inclusion,sum and intersection of subspaces. As a consequence, V (M) has aminimum element.

Proof. It is enough to show that the intersection of two elementsof V (M) is a robust controlled invariant subspace and this followsfrom [13, Theorem 2.2] using (3).

In the hypothesis of Proposition 2, we denote by V∗(M) theminimum element of the set V (M). It is known that V∗(M) =V∗ S∗( Ai∈I , C i∈I ,M +B).

Proposition 3. Given a family Σ = {Σ i}i∈I of linear systems of the form (2), let M ⊆ X be a subspace such that condition (3) holds andlet V ⊆ V∗ be a robust controlled invariant subspace for Σ with

M ⊆ V . Then, any family F of friends of V is a family of friend also of V∗(M).



G. Conte et al. / Systems & Control Letters 70 (2014) 1–7 3

2.2. Structural decomposition via controlled invariance

Given a family of system Σ = {Σ i}i∈I of the form (2), let

V(⊆V∗) and (S∗ ⊆) S be, respectively, a robust controlled

invariant subspace and a robust conditioned invariant subspace for

Σ . Then, combining the properties of V∗ and of S∗, it can easily

be shown that the subspace V S is a robust controlled invariant

subspace for Σ contained inK . In addition, any family F = {F i}

i∈I of friends of V is a family of friends of VS.

Then, given a robust controlled invariant subspace V , assume

that T = [T 1 T 2 T 3] is a column-block matrix such that the columns

of T 1 form a basis of V S∗, the columns of [T 1 T 2] forma basis of V

and the columns of T form a basis of X. Without loss of generality,

we canassume that T 3 is such thatS∗ is contained into span{T 1, T 3}.

If F = {F i}i∈I is a family of friends of V , in the basis defined by

T , the matrices which describe the compensated system Σ F i , for

i = 1, . . . , N , take the form

˜ Ai = T −1( Ai + BiF i)T =

Ai11 Ai

12 Ai13

0 Ai22 Ai

23

0 0 Ai33

;

B̃i = T −1Bi =

Bi

1

0

Bi3

; C̃ i = C iT =

0 0 C i3

.

(4)

Note that the second row-block Bi2 is 0 because Im Bi ⊆ B ⊆ S∗ ⊆

span {T 1, T 3}. Since the subspace V is invariant for the closed loop

dynamics ( ˜ Ai + B̃i F̃ i), any other family of feedbacks F̃ = {F̃ i =

F i1 F i2 F i3

}i∈I is a family of friends of V if and only if B i3F i1 =

Bi3F i2 = 0. This implies that in thedynamic matrix ˜ Ai of thecompen-

sated systemΣ F i , the block Ai

22 doesnot dependon thechoice ofthe

family F of friends of V . In this way, we have proved the following

proposition.

Proposition 4. With the above notation, the dynamics Σ

F

i|Z inducedby the compensated system Σ F

i on the quotient subspace Z = V/VS∗ is invariant with respect to any family F of friends of V and

it is characterized by the eigenstructure of Ai22, i = 1, . . . , N.

The familyof lineardynamicsΣ i|Z, that are inducedby the com-

pensated systemΣ F i onthe quotient spaceZ = V/V

S∗ and are

described, for i = 1, . . . , N , by ˙ z (t ) = Ai22 z (t ), defines a switch-

ing linear system Σ F σ |Z, where σ takes values in I and whose state

space is Z. Note that, by Proposition 4, Σ F σ |Z does not depend on

the choice of the family F .

2.3. Disturbance Decoupling Problem under Arbitrary Switching (DDPAS)

Given a switching linearsystemΣ σ d defined by equationsof the

form

Σ σ d ≡

˙ x(t ) = Aσ(t ) x(t ) + Bσ(t )u(t ) + Dσ(t )d(t )

y(t ) = C σ(t ) x(t ) (5)

where d ∈ Rq is a disturbance and D is a matrix of suitable dimen-

sions with real entries, the Disturbance Decoupling Problem under

Arbitrary Switching (DDPAS) consists in finding, if possible, a fam-

ily F = {F i}i∈I of feedbacks such that, for any switching rule σ , the

output of the compensated switching system

Σ F σ d ≡

˙ x(t ) = ( Aσ(t ) + Bσ(t )F σ(t )) x(t ) + Bσ(t )u(t ) + Dσ(t )d(t ) y(t ) = C σ(t ) x(t )

(6)

is not influenced by the disturbance d(t ).

Theorem 3.1 of [4] states that the DDPAS is solvable if and only

if, lettingD =

i=1,...,N Im Di, the condition

D ⊆ V∗ (7)

is satisfied. Remark that each solution of theDDPAS, if any exists,isgiven by a family F = {F i}i∈I of friends (not necessarily of V∗, but)

of a controlled invariant subspace V ⊆ K (possibly strictly con-

tained in V∗

) such thatD ⊆ V .

2.4. Disturbance Decoupling Problem with Stability (DDPS)

In applying a feedback for solving a decoupling problem, one

is usually interested also in achieving stability of the closed loopdynamics, possibly by selecting specific switching laws. From thispoint of view, given a switching linear system Σ σ d of the form (5),

the Disturbance Decoupling Problem with Stability (DDPS) consistsin finding, if possible,a familyF = {F i}i∈I of feedbacks anda switch-ing rule s : R+ → I such that the feedback compensated, switch-

ing system (6) is asymptotically stable for σ = s and its output isnot influenced by the disturbance d(t ).

Remark 1. Note that by letting σ be time dependent, we can as-sure that, by decoupling the disturbance from the output in ev-

ery mode and at the occurrence of switches, the overall influenceof the disturbance on the output of the system is null. In case wechose to consider switching rules which depend explicitly on the

state, which may be a natural approach for constructing stabilizingswitching rules, it is not sufficient to decouple the disturbance in

each mode and at the occurrence of switches for annihilating itsinfluence on the output of the system. The disturbance can in factalter the state, hence it can cause a switch and, in this way, it can

affect the output.

Note that each solution of the DDPS in the above formulation

consists of a pair (F , s), where F is a family of feedbacks and s is aswitching rule. It can be remarked that the component s of a solu-

tion (F , s) to the DDPS is not defined in a feedback manner, but, onthe contrary, its choice appears to be performed in an open loopway. This, however, does not increase the sensitiveness of the con-

trol loop to disturbances, if one assumes that disturbances do notaffect the measure of time or the value s(t ). The first assumption isusually understood and taken for granted in dealing with dynami-

calcontrol systems having no delays, while the second is quite nat-ural and not too restrictive in a deterministic context where the

codomain of s is a finite and discreteset. In such case, in fact, an er-ror on s(t ) would be better viewed as a failure in the controlsystemthat cannot in general be counteracted even by classic feedback,

rather than viewed as the effect of a disturbance.

2.5. Exhaustive and essential switching rules

In dealing with the DDPS, decoupling and stability depend onboth components of the pair (F , s) in a coupled manner. On one

side, the family F can be chosen insucha way to guarantee stabiliz-ability by mean of a suitable s, in addition to achieving disturbancedecoupling in any mode and at the occurrence of switches. On the

other side,the choice of theswitching rule can be used to avoid theactivation of some mode or to impose/avoid specific sequences of modesin such a way to facilitate decoupling, in addition to achiev-

ing stability. In order to take into account and to investigate thisaspect of the problem, we find useful to introduce the following

concepts.

Definition 3. Given a subset J ⊆ I , a switching rule s : R+ → I is said J -exhaustive if for any t 0 ≥ 0 and for any i ∈ J ⊆ I , there

exists t 1 ≥ t 0 with s(t 1) = i. If s is J -exhaustive with J = Im(s),then it is said auto-exhaustive.




In practical terms, auto-exhaustiveness means that any modewhich is activated by s in the interval [0, +∞) is also activated in

the interval [t 0, +∞) for any t 0.

Definition 4. Given a switching system Σ σ of the form (6) and asystem theoretic property P , a switching rule s : R+ → I is saidP -essential if

1. Σ σ has property P for σ = s;2. there is no switching rule s′ : R+ → I with Im(s′) ⊆ Im(s)

such that Σ σ has property P for σ = s′.

In practical terms, P -essentiality means that property P cannot

be achieved by activating only a proper subset of the set of modeswhich are activated by s. Note that property P could be achievedby activating a subset of modes different from (but not strictlycontained in) that of the modes which are activated by s.

Clearly, if a property P can be achieved by a suitable choice of the switching rule, it can be achieved by a P -essential switchingrule s(t ). This means that in studying conditions under whichthere exist switching rules that guarantee property P one can limithimself to consider only P -essential switching rules.

In the following, we denote by P asyst the property of Σ σ of being

asymptotically stable.

Proposition 5. Given a switching system Σ σ of the form (1), anyP asyst-essential switching rule s is auto-exhaustive.

Proof. Let s be P asyst-essential and let assume that, for some t 0 ≥ 0and for some i ∈ Im(s) ⊆ I , we have s(t 1) ̸ = i for all t 1 ≥ t 0. Forany x ∈ X, using the time-reversibility of each mode of Σ σ andthe fact that only a finite number of modes has been activated by

s on the interval [0, t 0], we can find x0 ∈ X such that the motion

starting at x0 at time 0 arrives in x at t 0. Then, due to asymptoticstability, that motion goes to 0 as t goes to ∞. This shows that fors′(t ) = s(t + t 0) any state x of Σ σ , for σ = s′, goes to 0 as t goesto ∞ and hence that s′ provides asymptotic stability. Since Im(s′)is strictly contained into Im(s), this contradicts the fact that s(t ) is

P asyst-essential.

Proposition 6. Given a switching system Σ σ d of the form (5), thecorresponding DDPS is solvable only if there exists a subset J ⊆ I for which

i∈ J

Im Di ⊆ V∗( Ai∈ J , Bi∈ J ,K′) (8)

withK ′ =

i∈ J Ker C i.

Proof. If the DDPS is solvable, it is solvable by a family F = {F i}i∈I

of feedbacks and a P asyst-essential switching rule s : R+ → I . By

Proposition 5, s is auto-exhaustive and we can denote by J the set

of indices of the modes which are activated by s, i.e. J = Im(s). As-

sume that the sequence of indices defined by s is i1, i2, . . . , ik, . . .

and consider the sequence of subspaces Vk defined by

V1 = ⟨ Ai1 + Bi1F i1|ImDi1⟩

= ImDi1 + ( Ai1 + Bi1F i1)ImDi1 · · · + ( Ai1 + Bi1F i1)n−1ImDi1

V2 = ⟨ Ai2 + Bi2F i2|(ImDi2 + V1)⟩= (ImDi2 + V1) + ( Ai2 + Bi2F i2)(ImDi2 + V1) · · ·

+ ( Ai2 + Bi2F i2)n−1(ImDi2 + V1)

· · · · · ·

Vk = ⟨ Aik + BikF ik|(ImDik + Vk−1)⟩= (ImDik + Vk−1) + ( Aik + BikF ik)(ImDik + Vk−1) · · ·

+ ( Aik + BikF ik)n−1(ImDik + Vk−1)

· · · · · · .

We have ImDi1 ⊆ V1 ⊆ V1 + ImDi2 ⊆ V2 ⊆ · · · ⊆ Vk ⊆ . . . and,since X is finite dimensional, there exists h such that Vh = Vh+n

for any n. Let [t h1, t h2) be the interval on which s(t ) = ih and let ibe any index in J . By exhaustiveness, there exists t ′ ≥ t h2 such thats(t ′) = i and, therefore, ( Ai + BiF i)Vh ⊆ Vh. This implies that Vh

is a robust controlled invariant subspace for the family of systemsΣ id with i ∈ J and, since the disturbance is decoupled, that Vh ⊆K ′ =

i∈ J Ker C i. As a consequence, Vh ⊆ V∗( Ai∈ J , Bi∈ J ,K

′) and,

sincei∈I Im Di ⊆ Vh we geti∈I Im Di ⊆ V∗( Aiv∈ J , Bi∈ J ,K′).

Although elementary, the above proposition is important, to-gether with Proposition 4, in deriving more useful solvabilityconditions. Note that the necessary condition that it states can bepractically checked by constructingV∗( Ai∈ J , Bi∈ J ,K

′) foreach sub-set J ⊆ I . Moreover, it implies, in particular, that, if J ⊆ I is suchthat

i∈ J Im Di ⊆ V∗( Ai∈ J , Bi∈ J ,K

′) with K ′ =

i∈ J Ker C i, theDDPS can be solved if there exists a family F = {F i}i∈ J of friends of V∗( Ai∈ J , Bi∈ J ,K

′) and a switching rule s : R+ → J such that the

resulting feedback compensated, switching system Σ F σ d is asymp-

totically stable for σ = s.

3. Solvability condition for the DDPS

Given thelinearswitching systemΣ σ d of theform (5), subject tothe disturbance inputd, assumethat thenecessarycondition (8) forsolvability of the corresponding DDPSexpressedin Proposition 6 issatisfied for a subset J ⊆ I . Without loss of generality, we can thenconsider the possibility of solving the DDPS by means of a switch-ing rule s with Im(s) = J . Therefore, in the rest of this section wewill take into account only the set of modes of Σ σ d whose indicesare in J and allthe considered controlled and conditioned invariantsubspaces will pertain to the family of systems {Σ id}i∈ J .

Let us consider, now, the following condition

V∗S∗ ⊆ D =

i∈ J

Im Di. (9)

Remark that (9) holds, in particular, if

Im BiV∗ = {0}, for all i ∈ J (10)

holds. In turn, (10) is akin to, but weaker than, a condition of theform Im Bi

V∗

i = {0} for all i ∈ J , where V∗i is the maximum

( Ai, Bi)-invariant subspace contained in Ker C i, which would implyleft invertibility of each system of the family Σ = {Σ i}i∈ J .

Conditions (8) and (9) can be jointly expressed by

V∗S∗ ⊆ D ⊆ V∗ (11)

andthis,by Proposition 2, implies that there existsa minimum con-trolled invariant subspace V∗(D) which contains D . By Proposi-tion 3, it is clear that, in such case, any family F = {F i}i∈ J of feed-backs that may be used to solve the DDPS is, in particular, a familyof friends of V∗(D). Exploiting this remark, we can now state thefollowing result.

Proposition 7. Giventhe switchinglinear systemΣ σ d of the form (5),subject to the disturbance input d(t ), assume that condition(11) holds.Then, the DDPS is solvable only if, denoting by V∗(D) the minimumcontrolled invariant subspace of Σ σ d contained inV∗ and containing D and letting F = {F i}i∈ J be any one of the possible families of friendsof V∗(D), there existsa switchingrule s such that the switchinglinear system Σ F

σ |Z induced onZ = V∗(D)/V∗(D)S∗ by the switching

compensated system Σ F σ d is asymptotically stable for σ = s.

Proof. By Proposition 4 and the comments which follow it, Σ F σ |Z

does not depends on thechoice of the family F of friends of V∗(D).Then, we can choose any F and decompose ( Ai + BiF i), for i ∈ J ,in block triangular form as in (4) by means of a suitable changeof basis x = Tz in X. Asymptotic stability of Σ F

σ d for σ = s im-plies asymptotic stability of the switching dynamicsdefined by the

block of indices 2, 2 in the triangular decomposition and hence of the switching linear system Σ F s|Z.




Remark 2. The above proposition states that the existence of a

switching rule s that asymptotically stabilizes the switching lin-

ear system Σ F σ |Z does not depend on the choice of F and it is an

obstruction to the solvability of the DDPS. We can then conclude

that, under the mild hypothesis that (11) holds, the geometric as-

pect of the DDPS and that related to stability can be dealt with in a

disjoint way.

The content of the above remark provides an insight into theDDPS that simplify the study of the problem and makes possible,

by using additional result, to state a sufficient solvability condition.

In fact, if the necessary condition of Proposition 7 is satisfied for

some family of feedbacks F = {F i}i∈ J , it is clear, using (4) and the

consideration that follow it, that any solution of the DDPS, if any

exists, is of the form

F̄ =

F i + [U i1 U i2]

F i11 F i12 F i13

0 0 F i23

T −1

i∈ J

(12)

where T is the change of basis used to get (4), [U i1 U i2] is a change

of basis inU such that

Im(B̃iU i1) = ImB̃i V∗(D)

Im(B̃i[U i1 U i]) = Im

Bi

11 Bi12

0 0

0 Bi32

= ImB̃i

(13)

and F i11, F i13, F i13, F i32 are arbitrary.

In order to proceed, let us briefly recall that a linear dynam-

ical system, say, in particular, the mode Σ i described by (2), is

called normal if its dynamics matrix is normal, that is, if AT i Ai =

Ai AT i . Switching systems with normal, asymptotically stable modes

have been shown to be asymptotically stable for any choice of the

switching law σ in [14].

Definition 5 (See [15]). A linear dynamical systemΣ i of form (2) is

called N -stabilizable if there exists a feedback matrix F i such thatthe compensated dynamic matrix( Ai +BiF i) is normaland Hurwitz

stable.

N -stabilizability has been studied in [15], where a simple

characterization of that property in terms of the entries of Ai and Bi

can be found (see [15] Proposition 1). Using the notions and results

just recalled, we can now state the following proposition.

Proposition 8. Given the switching linear system Σ σ d of form (5) ,subject to the disturbance input d(t ), assume that condition(11) holdsand that Σ F

σ |Z is uniformly asymptotically stable for some friend F of

V∗(D) and some switching law σ = s. Then, the DDPS is solvable if,decomposing Σ F

σ as in (4), (13) and using the corresponding nota-

tions, the control subsystems defined respectively by the pair ( Ai

11

,

Bi11) and ( Ai

33, Bi32) are N-stabilizable for any i ∈ J .

Proof. Apply the feedback F and decompose Σ F σ as in (4) and in

(13) by suitable changes of base x = Tz and u = Uu′. Chose F i11 and

F i23 (F i12 and F i13 are arbitrary) in such a way that ( Ai11 + Bi

11F i11) and

( Ai33 + Bi

32F i23) are normal and Hurwitz stable for any i ∈ J and, as

a consequence, the corresponding switching subsystems Σ F 1σ and

Σ F 3σ are asymptotically stable, in particular, forσ = s. Denoting by

Σ ′σ the resulting compensated system, it remains to show asymp-

totic stability of the time-varying system Σ ′s , whose dynamic ma-

trix can be represented as:

A

′

(t ) = A

s(t )11 + B

s(t )11 F

s(t )11 ∗ ∗

0 A

s(t )

22 A

s(t )

230 0 A

s(t )33 + B

s(t )32 F

s(t )23

. (14)

For any initial value z 3(0), the component z 3(t ) of the state z (t ) =[ z 1(t ) z 2(t ) z 3(t )]T of Σ ′s goes to 0 as t goes to ∞. The uniform

asymptotic stability of the subsystem defined by As(t )22 implies the

BIBO stability of the control system ˙ z 2(t ) = As(t )22 z 2(t ) + A

s(t )23 z 3(t )

(see [16]), and therefore, we have that, for any initial value z 2(0)

and z 3(0), both the components z 2(t ) and z 3(t ) of the state z (t ) go

t o 0 a s t goes to ∞. Finally, note that N -stability of Ai11 implies uni-

form asymptotic stability of As(t )11 by Eq. (2) in [15]. Then, repeatingthe same argument as above, we get that for any initial value z (0),

all the components of z (t ) go to 0 as t goes to ∞. The DDPS is then

solved by the feedback

F̃ = F + U

F i11 ∗ ∗

0 0 F i23

i∈I

T −1. (15)

4. Example

For sake of illustration, let us consider the switching linear

system Σ σ d defined by the equations

˙ x(t ) = Aσ(t ) x(t ) + Bσ(t )u(t ) + Dσ(t )d(t ) y(t ) = C σ(t ) x(t ).

The switching rule σ takes values in the set I = {1, 2, 3} and the

modes of Σ σ d are defined by the matrices ( Ai, Bi, Di, C i), for i ∈ I ,where

A1 =

−28

38 −34 −94 −46 −18

7/6 1 7/2 15 1 7/2

−17

31/4 −9/4 −3/2 1 −3/4

29

2−7

61

283 41

33

2

−

65

6 7 −

47

2 −63 −37 −23/2

−71

63/4 −

73

4−

111

2−18 −

55

4

B1 =

1 −5

1 1

−1 1

0 3

1 −2

−2 −2

, D1 =

1

2

1

−1

2

−3

,

C 1 =

−1 0 −1 −3 −2 −1

2 −1/2 5/2 7 4 3/2

;

A2 =

−11/2 7 −8 −17 −52 −1

−9/2 −1 −5 −4 −44 1

27

2−

41

4

71

451

63

2

45

4−2 −5/2 −3/2 4 2 3/2

−17/2 9 −12 −44 −26 −21/2

12 −21

4

59

427

115

29/4

B2 =

1 0

1 3

0 2

0 2

0 −3

−1 −3

, D2 =

−4

−3

−1

1

−1

4

,

C 2 =

−1 0 −1 −3 −2 −12 −1/2 5/2 7 4 3/2

;




A3 =

15/2 −3/2 13/2 24 13 13/2

−29

24 −22 −53 −29 −13

3/2 −5/2 1/2 1 4 −1

−4 3/2 −15/2 −16 −8 −9/2

−11/2 3 −3 −14 −11 −3/2

12 −13/241

246 25 10

B3 =

1 0

1 3

0 2

0 2

0 −3

−1 −3

, D3 =

−4

−3

−1

1

−1

4

,

C 3 =

−1 0 −1 −3 −2 −1

2 −1/2 5/2 7 4 3/2

.

As described in Section 3, our aim is to find a subset J ⊆ I , a family

F = {F i}i∈ J of feedbacks and a switching rule s : R+ → J such that

the compensated systemΣ F s is asymptotically stable andits output

is not influenced by the disturbance d(t ). Since condition (9) turns

out to be verified, let us start by checking the necessary condition

of Proposition 7. Standard computations show that Ker C 1 =Ker C 2 = KerC 3 = V∗ = V∗(D) and that F = {F 1, F 2, F 3}, where

F 1 =

1/3 −2 2 1 5 0

3 1 −1 −2 2 1

F 2 =

0 −4 2 −1 41 0

1 1 1 −1 −1 −1

F 3 =

1 1 2 −1 0 1

0 0 1 1 0 1

is a family of friend of V∗(D) (recall that at this stage we can pick

any family of friends). Applying F and the change of basis x = Tz

with

T =

1 0 −2 0 0 −2

1 0 0 1 0 1

0 −1 1 2 0 1

0 0 0 −1 1 1

0 1 0 1 −1 −1

−1 −1 1 −1 −2 0

,

we have that the compensated modes Σ F i , for i = 1, 2, 3, take the

form

˜ A1 =

0 −1 0 0 5 0

0 −3 0 1 −3 2

0 0 1 −1 1 1

0 0 0 −3/2 −1 −1

0 0 0 0 −4 2

0 0 0 0 −6 7/2

,

B̃1 =

1 −1

1 1

0 0

0 0

0 1

0 2

, D̃1 =

1

1

0

1

0

0

,

C̃ 1 =

0 0 0 0 1 0

0 0 0 0 0 1

;

˜ A2 =

−3/2 −4 1 2 1 0

2 −1/2 1/2 1/2 0 −2

0 0 −3/2 1 1 1

0 0 0 1 −1 −1

0 0 0 0 0 2

0 0 0 0 2 −1

,

B̃2 =

1 2

0 −1

0 0

0 0

0 1

0 1

, D̃2 =

−2

0

1

−1

0

0

,

C̃ 2 =

0 0 0 0 1 0

0 0 0 0 0 1

;

˜ A3 =

1/2 −3 1 1 −3 −3

2 −1/2 1/2 1/2 2 0

0 0 1 1 1 1

0 0 0 1 −1 −1

0 0 0 0 −2 0

0 0 0 0 0 −3

,

B̃3 =

1 2

0 −1

0 0

0 0

0 1

0 1

, D̃3 =

−2

0

1

−1

0

0

,

C̃ 3 =

0 0 0 0 1 0

0 0 0 0 0 1

.

The modes of the switching system Σ F

σ |Z, that is induced on Z =V∗D/V∗D

S∗ = span{ z 1, z 2, z 3, z 4}/span{ z 1, z 2} by Σ F

σ , are

characterized by the unstable dynamic matrices

˜ A122 =

1 −1

0 −3/2

and ˜ A2

22 =

−3/2 1

0 1

and

˜ A322 =

−3/2 1

0 1

.

Therefore, there is no stabilizing switching lawif J = {1} or J = {2}or J = {3}. Also, there is no stabilizing switching lawif J = {1, 3} or

J = {2, 3}, since, in those cases, either the component z 3 or the

component z 4 of the state z (t ), for z (0) = (0 0 z 3(0) 0 0 0)T or res-

pectively for z (0) = (0 0 0 z 4(0) 0 0)T with z 3(0) ̸ = 0 and z 4(0) ̸ =

0, go to ∞ as t goes to ∞. On the other hand, the auto-exhaustiveswitching rule s : R+ → {1, 2} defined by




s(t ) =

1 t = 0

1 2k < t ≤ 2k + 1

2 2k + 1 < t ≤ 2k + 2

for any integer k ≥ 0

is easily seen to be such that Σ F s is uniformly asymptotically stable,

because, for any integer k ≥ 0, we have

[ z 3(2k) z 4(2k)]T

=

e˜ A2

22 · e˜ A1

22k

[ z 3(0) z 4(0)]T

=

e ∗

0 e−3/2

e−3/2 ∗

0 e

k

[ z 3(0) z 4(0)]T

=

e−1/2 ∗

0 e−1/2

k

[ z 3(0) z 4(0)]T .

Hence, the hypothesis of Proposition 8 is verified. Now, using the

technique described in [15], it is possible to check that the subsys-

tems defined by the blocks

˜ A111 =

0 1

0 −3

, B̃1

11 =

1

1

and

˜ A211 =

−3/2 −4

2 −1/2

, B̃2

12 =

1

0

are N -stabilizable and to construct practically two feedbacks, e.g.

F 111 = [−1 2] and F 211 = [1 2], which make the compensated dy-

namic matrices normal and Hurwitz stable. The same holds for the

subsystems defined by the blocks

˜ A133 =

−4 2

−6 7/2

, B̃1

32 =

1

2

and

˜ A233 =

0 2

2 −1

, B̃2

32 =

1

1

and the feedbacks, e.g., F 123 = [3 − 2] and F 223 = [−1 − 1]. By

applying, e.g., the feedbacks

F̃ 1 =

−1 2 0 0 0 0

0 0 0 0 3 −2

,

F̃ 2 =

1 2 0 0 0 0

0 0 0 0 −1 −1

,

in fact, we finally get the compensated modes

AA1 =

−1 1 0 0 1 1

−1 −1 0 1 −1 −1

0 0 1 −1 1 1

0 0 0 −3/2 −1 −1

0 0 0 0 −1 0

0 0 0 0 0 −1/2

,

B̃1, C̃ 1, D̃1

AA2 =

−1/2 −2 1 1 −1 −1

2 −1/2 1/2 1/2 1 −1

0 0 −3/2 1 1 1

0 0 0 1 −1 −1

0 0 0 0 −1 1

0 0 0 0 1 −2

,

B̃2, C̃ 2, D̃2.

The original DDPS is therefore solved by the feedbacks of the

family F ′ = {F 1 + F̃ 1T −1, F 2 + F̃ 2T −1} and the switching law s : R+

→ {1, 2}.

5. Conclusions

The problem of disturbance decoupling with stability has been

considered for switching systems. By exploiting the geometry of

the problem, under mildly restrictive hypothesis, the structural re-

quirements can be dealt with separately from the qualitative ones.

So, stability can, possibly, be achieved after decoupling has been

assured. This approach makes possible to point out, in terms of a

necessary condition, a fundamental obstruction to the solvability

of the problem and also to obtain a sufficient condition for it. More

generally, if thenecessarycondition is satisfied,we canremark that

the existence of solutions can be investigated by studying the sta-

bilizability of a time-varying system, whose dynamic matrix and

input matrix have a block diagonal structure as in (4) and in (13).

Future work can search for other sufficient conditions, that possi-

bly apply to situations in which that of Proposition 8 does not hold,

along the same line, using analysis and synthesis tools for time-

varying systems.

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