Geometric Control Theory for Hybrid Systems · Geometric control theory for linear time-invariant (LTI) systems has a long and rich history, as is evidenced by the availability of

Geometric Control Theory for HybridSystems

Evren Yurtseven

H&NS 2010.03

Master’s thesis

Coach(es): prof.dr.ir. W.P.M.H. Heemels (supervisor)dr. M.K. Camlibel

Committee: dr. M.K. Camlibelprof.dr.ir. W.P.M.H. Heemelsprof.dr. J.M. Schumacherdr. S. Weiland

Eindhoven University of TechnologyDepartment of Mechanical EngineeringHybrid & Networked Systems

Eindhoven, August, 2010

TABLE OF CONTENTS

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2. Disturbance Decoupling of Switched Linear Systems . . . . . . . . 4

3. Controllability of Bimodal Discrete-timePiecewise Linear Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

CHAPTER 1

INTRODUCTION

Geometric control theory for linear time-invariant (LTI) systems has a long and rich history,as is evidenced by the availability of various textbooks on the topic [H.L02, BM92, Won85].For solving certain problems such as disturbance decoupling problems (DDPs), character-ization of sets of particular interest including reachable and unobservable sets, geometriccontrol theory proves to be an extremely powerful tool. Solutions to various problemsfor smooth nonlinear systems, based on a geometric approach, are also available in theliterature [IKGGN81, NVdS82]. Interestingly, geometric control theory for hybrid systemsis largely absent in the literature, despite the enourmous attention this class of dynamicalsystems has received over the past decades. This thesis focuses on the development ofgeometric control theory for certain classes of hybrid systems. For this purpose, geometriccontrol theory will be used to solve DDPs and controllability problems for switched linearand piecewise linear systems, respectively.

1.1 Problem Formulation

Although many geometric control theoretic results are available for linear and smoothnonlinear systems, when one leaves the realm of linear and smooth nonlinear systemsand looks into hybrid dynamical systems, the availability and usage of geometric controltheory become quite limited. There are, however, some results available for interestingproblems for certain classes of hybrid systems, that are based on a geometric approach. In[JvdS02] the largest controlled invariant set for switched linear systems (SLSs) is studiedin which both the switching (discrete control input) and the continuous input can bemanipulated as control inputs. In the context of linear parameter-varying (LPV) systems

CHAPTER 1. INTRODUCTION

various parameter-varying (controlled and conditioned) invariant subspaces are introducedin [BBS03] and various algorithms are presented to compute them. Based on [BBS03] firstresults in the direction of applying these concepts to solve DDP in LPV systems are givenin [SBS03] using parameter-dependent state feedback. Recently, in [Ots10] DDP for SLSsusing mode-dependent state feedback control is solved. In [ZCL05] local versions of DDPwith respect to continuous disturbances in switched nonlinear systems are solved using bothmode-dependent and mode-independent state feedback. It should be pointed out that theworks mentioned so far prove their main results by first extending fundamental conceptsof geometric control theory for LTI systems such as A-invariance, controlled invarianceand so on, to the class of hybrid systems they are interested in and then using these newextended concepts in their solutions to the problems.

There exist also studies in the literature that provide solutions to certain problems forparticular classes of hybrid systems, that heavily make use of geometric control theory forLTI systems in proving their main results. In [CHS08a], algebraic necessary and sufficientconditions for the controllability of continuous-time conewise linear systems are given. Thiswork solves a seemingly very complicated problem for an important class of hybrid systemsby first transforming the hybrid system of interest into a much more manageable form usinggeometric control theory for LTI systems and proving the controllability of the originalsystem to be equivalent to the controllability of a particular subsystem of the originalsystem. Also in solving other problems defined for hybrid systems, the usage of geometriccontrol theory for LTI systems offers attractive solutions, see e.g. [CHS08b, Cam07].

In the light of these works, this thesis will focus on the following problems in which geo-metric control theory will play an essential role:

• Find conditions under which a switched linear system’s output is or can be madedecoupled from

(i) the continuous disturbances

(ii) the switching signal

(iii) both the continuous disturbances and the switching signal

• Find conditions under which a bimodal discrete-time piecewise linear system is

(i) null controllable

(ii) reachable

(iii) controllable

2

CHAPTER 1. INTRODUCTION

1.2 Contributions of This Thesis

The first contribution of this thesis will be to give complete characterizations of distur-bance decoupling in switched linear systems, that is under what conditions is a given SLSdisturbance decoupled or can it be made disturbance decoupled by using, either mode-dependent or mode-independent, static state feedback. We do so by first extending thebuilding blocks of geometric control theory for LTI systems such as A-invariance, controlledinvariance, conditioned invariance et cetera, to switched linear systems. Then, making useof these new concepts we present the main results and the algorithms with which our resultscan be put to use. We consider three different versions of disturbance decoupling: withrespect to the continuous disturbances; with respect to the discrete switching signal, i.e.the mode of the SLS; and with respect to both. While the first version is a well-motivatedand well-known problem, the second and the third are considered for the first time in thisthesis. Disturbance decoupling with respect to the switching signal is a desirable propertywhen the switching signal cannot be manipulated and each mode of the SLS correspondsto a certain failure scenario and particular outputs of the system are desired not to beinfluenced by these scenarios. Disturbance decoupling with respect to both the switchingsignal and the continuous disturbances is a property that we will show to be strongly re-lated to disturbance decoupling of piecewise linear systems with respect to the continuousdisturbances. Before concluding this part of the thesis, we will also provide geometric nec-essary and sufficient conditions for the solvability of the disturbance decoupling problemwith respect to the continuous disturbances by mode-dependent dynamic measurementfeedback. These contributions are presented in Chapter 2. An abridged version of thechapter is accepted for publication in the proceedings of the IEEE Conference on Decisionand Control 2010 in Atlanta, USA. The complete contents will further be sent for journalpublication.

The second part of this thesis is devoted to providing algebraic necessary and sufficientconditions for the null controllability, reachability and controllability of a class of bimodaldiscrete-time piecewise linear systems. The class is characterized by continuous right-handside and scalar input whereas the state can be of any dimension. By using geometric controltheory for LTI systems we will show that the controllability problem is equivalent to thecontrollability of a particular subsystem of the original system, which is in the form of a so-called push-pull system. Exploiting the results on the controllability of input-constrainedlinear systems we will derive conditions for the controllability of push-pull systems leadingto our main results. These contributions are explained in detail in Chapter 3. The resultswill be submitted for journal publication as well.

3

CHAPTER 2

DISTURBANCE DECOUPLING OF

SWITCHED LINEAR SYSTEMS

DisturbanceDecouplingof Switched

LinearSystems

AtYarragi

Essegin ziki

Abstract—In this paper we consider disturbance decou-pling problems for switched linear systems. We will providenecessary and sufficient conditions for three differentversions of disturbance decoupling, which differ basedon which signals are considered to be the disturbance.In the first version the continuous exogenous input isconsidered as the disturbance, in the second the switchingsignal and in the third both of them are considered asdisturbances. The latter instance of the problem is relevantfor disturbance decoupling of piecewise linear systems, aswe will show. The solutions of the three disturbance decou-pling problems will be based on geometric control theoryfor switched linear systems and will entail both mode-dependent and mode-independent static state feedback.In addition, a mode-dependent dynamic measurementfeedback based solution of the disturbance decouplingproblem with respect to the continuous disturbances willbe provided.

Index Terms—Disturbance decoupling, switched linearsystems, invariant subspaces, geometric control theory

I. INTRODUCTION

Geometric control theory for linear time-invariantsystems has a long and rich history, as is evidencedby the availability of various textbooks on the topic[2, 6, 18]. In particular, for solving disturbancedecoupling problems (DDPs) for linear systemsthe usage of geometric theory turned out to beextremely powerful. Also solutions to various prob-

lems for smooth nonlinear systems using a nonlineargeometric approach are available in the literature,see, e.g., [3, 7, 11]. However, outside the contextof linear or smooth nonlinear control systems, thenumber of results on DDPs is rather limited. This isspecifically surprising for hybrid dynamical systemsor subclasses such as switched systems [9], as theyhave been studied extensively over the last twodecades.

Only a few results are available on geometriccontrol theory and solutions to DDPs for switchedsystems. In [8] the largest controlled invariant setfor switched linear systems (SLSs) is studied inwhich both the switching (discrete control input)and the continuous input can be manipulated ascontrol inputs. In the context of linear parameter-varying (LPV) systems various parameter-varying(controlled and conditioned) invariant subspaces areintroduced in [1] and various algorithms are pre-sented to compute them. Based on [1] first re-sults in the direction of applying these conceptsto DDPs with respect to continuous disturbancesare given in [14] using parameter-dependent statefeedback. Recently, in [12] the DDP for switchedlinear systems using mode-dependent state feedbackcontrol is solved and combined with results onquadratic stabilizability [4, 17]. Also for reachability

CHAPTER 2. DISTURBANCE DECOUPLING OF SWITCHED LINEAR SYSTEMS

problems for SLSs invariant subspaces played animportant role. In particular, in [15, 16] it wasshown that the reachable set of an SLS is equalto the smallest controlled invariant set containingthe subspace spanned by all the input matrices ofthe individual subsystems. For switched nonlinearsystems, the only work known to the authors is[19]. In [19] local versions of DDP with respectto continuous disturbances are solved using bothmode-dependent and mode-independent static statefeedback.

The objectives of this paper are to provide com-plete answers to DDPs for SLSs using variousnew controlled and conditioned invariant subspacesfor SLSs. We first assume that the control inputis absent and analyze the disturbance decouplingproperties of a SLS. In contrast with the abovementioned references, which only study disturbancedecoupling (DD) with respect to continuous exoge-nous disturbances, we consider three variants of DDas will be formally defined in Section II, namely DDwith the disturbances being either (i) the exogenouscontinuous disturbances, (ii) the switching signal,or (iii) both the continuous disturbances and theswitching signal. We will show that the latter in-stance of the problem is relevant for disturbance de-coupling of piecewise linear systems. In Section IIIwe will fully characterize these three DD proper-ties. In Section IV we will add continuous controlinputs to the problem and solve the DDP usingstate feedback controllers that may be both mode-dependent and mode-independent. We will allow fordirect feedthrough terms of the control input into theto-be-decoupled output variable, a situation that wasnot considered in the aforementioned references.Note also that variant (ii) and (iii) are in the presentpaper for the first time. In Section V we provide al-gorithms to compute the largest common controlledinvariant subspaces using both mode-dependent andmode-independent feedback, which can be used toverify the characterizations of the solvability of theDDPs provided in Section IV and also to constructfeedbacks solving the DDPs. Before stating theconclusions, we will provide also the solution to theDDP using mode-dependent output-based dynamicfeedback controllers.

II. PROBLEM FORMULATION

A switched linear system (SLS) without controlinputs is described by the following equations

x(t) = Aσ(t)x(t) + Eσ(t)d(t) (1a)

z(t) = Hσ(t)x(t) (1b)

where x(t) ∈ Rnx , d(t) ∈ Rnd and z(t) ∈ Rnz

denote the state variable, the exogenous input andoutput, respectively, at time t ∈ R+ := [0,∞). Foreach i ∈ {1, . . . ,M}, Ai ∈ Rnx×nx , Ei ∈ Rnx×nd

and Hi ∈ Rnz×nx are matrices describing a linearsubsystem. Switching between subsystems (modes)is orchestrated by the switching signal σ. We assumethat σ lies in the set S of right-continuous functionsR+ → {1, . . . ,M} that are piecewise constant witha finite number of discontinuities in a finite lengthinterval. Particular switching signals are the constantones σi ∈ S, i = 1, . . . ,M , which are definedas σi(t) = i for all t ∈ R+. We assume thatthe exogenous signal d is locally integrable, i.e.d ∈ Lloc1 (R+,Rnd). Clearly, the SLS (1) has for eachd ∈ Lloc1 (R+,Rnd), initial condition x(0) = x0 ∈Rnx and switching signal σ ∈ S a unique solutionxx0,σ,d and a corresponding output zx0,σ,d.

We will consider the following three variantsof disturbance decoupling, which differ based onwhich signals are considered to be the disturbance.

Definition II.1 The SLS (1) is called disturbancedecoupled (DD) with respect to d if

zx0,σ,d1 = zx0,σ,d2 (2)

for all x0 ∈ Rnx , σ ∈ S and d1, d2 ∈ Lloc1 (R+,Rnd).

Definition II.2 The SLS (1) is called disturbancedecoupled (DD) with respect to σ if

zx0,σ1,d = zx0,σ2,d (3)

for all x0 ∈ Rnx , σ1, σ2 ∈ S and d ∈ Lloc1 (R+,Rnd).

Definition II.3 The SLS (1) is called disturbancedecoupled (DD) with respect to both σ and d if

zx0,σ1,d1 = zx0,σ2,d2 (4)

for all x0 ∈ Rnx , σ1, σ2 ∈ S and d1, d2 ∈Lloc1 (R+,Rnd).

5


Remark II.4 DD with respect to continuous distur-bances d is commonly studied and well motivatedwithin the context of linear systems [2, 6, 18] andnonlinear systems [3, 7, 11]. DD with respect toσ or both σ and d is typical for switched systemsas studied here. These variants of DD are relevantin situations where the switching signal σ is un-controlled and we would like to design a (closed-loop) system in which σ does not influence certainimportant performance variables z. In particular,when σ models certain faults in the system such asbreakage of pipes, actuators, sensors, etc. and thuseach mode i ∈ {1, . . . ,M} corresponds to one ofthese discrete fault scenarios, it would be desirableto decouple z from σ (and possibly other continuousdisturbances d). Hence, as such these variants ofDD are fundamental problems in the area of fault-tolerant control [10]. Another motivation for DDwith respect to both σ and d are DD problems forpiecewise linear systems, see Section III-D below.

III. DISTURBANCE DECOUPLINGCHARACTERIZATIONS

To provide characterizations for the above men-tioned DD properties, we need to introduce someconcepts and a technical lemma. We call a sub-space V ∈ Rnx A-invariant for A ∈ Rnx×nx ,if AV ⊆ V . We call a subspace {A1, . . . , AM}-invariant Ai ∈ Rnx×nx , i = 1, . . . ,M , if AiV ⊆ Vfor all i = 1, . . . ,M . Given a matrix A ∈ Rnx×nx

and a subspace W ∈ Rnx , let 〈A|W〉 denote thesmallest A−invariant subspace that containsW , i.e.,

〈A|W〉 =W + AW + . . .+ Anx−1W (5)

For a set of matrices {A1, . . . , AM} and a sub-space W , the smallest {A1, . . . , AM}-invariant sub-space that contains W , denoted by Vs(W), isuniquely defined by the following three properties:1) W ⊆ Vs(W);2) Vs(W) is {A1, . . . , AM}-invariant;3) For any subspace V being {A1, . . . , AM}-

invariant withW ⊆ V , it holds that Vs(W) ⊆ V .Calculation of Vs(W) can be done using the recur-rence relation

V1 =W ; Vi+1 =M∑

j=1

〈Aj|Vi〉

Since Vi ⊆ Vi+1 for i = 1, 2, . . . and Vp = Vp+1

implies Vq = Vp for all q ≥ p, it holds that Vq =Vs(W) for all q ≥ nx, see, e.g., [1, 16].

The reachable set of (1) is defined as R :={x0,σ,d(T ) | T ∈ R+, σ ∈ S and d ∈ Lloc1 (R+,Rnd)}being the set of states that can be reached from theorigin in finite time for some σ and d.

Lemma III.1 For the SLS (1),

R = Vs(M∑

i=1

im Ei)

See [15, 16] for the proof of this lemma.

A. Disturbance decoupling with respect to d

In this section we consider DD with respect to d.

Theorem III.2 The SLS (1) is DD with respect to dif and only if an {A1, . . . , AM}-invariant subspaceV exists such that

M∑

i=1

im Ei ⊆ V ⊆ ker

H1...

HM

(6)

Proof: Once σ is fixed the SLS (1) reducesto a linear time-varying system. Using the resultinglinearity properties shows that (17) is equivalent to

z0,σ,d = 0 ∀σ ∈ S ∀d ∈ Lloc1 (R+,Rnd) (7)

Necessity: Since each x ∈ R can be reached forsome σ and d, i.e. x = x0,σ,d(T ) for some T ∈ R+,

we can take σi(t) =

{σ(t) 0 ≤ t < T

σi(t) t ≥ Tfor i =

1, . . . ,M in (7) to get that Hix = z0,σi,d(T ) = 0.Hence (6) holds for V = R due to Lemma III.1.

Sufficiency: Since x0,σ,d(t) ∈ R for all t ∈ R+,it follows from Lemma III.1 that x0,σ,d(t) ∈ V for

all t ∈ R+ as R = Vs(M∑

i=1

im Ei) ⊆ V . Hence, due

to (6), z0,σ,d(t) = Hσ(t)x0,σ,d(t) = 0 and thus (7) issatisfied.

6


B. Disturbance decoupling with respect to σ

In this section we consider DD with respect to σin which we will use Ni being the unobservablesubspace corresponding to the pair (Hi, Ai), i.e.,Ni = kerHi ∩ kerHiAi ∩ . . . ∩ kerHiA

nx−1i . Note

that Ni is also the largest Ai-invariant subspacecontained in kerHi, i ∈ {1, . . . ,M}.

Theorem III.3 The SLS (1) is DD with respect toσ if and only if for all (i, j) ∈ {1, . . . ,M} ×{1, . . . ,M} the following conditions hold

(i) Hi = Hj;(ii) Ni = Nj;

(iii) im(Ai − Aj) ⊆ Ni;(iv) im(Ei − Ej) ⊆ Ni.

Proof: Necessity: To prove the necessity of theconditions, take σ1 = σi and σ2 = σj . Let d(t) = d0

for all t ∈ R+. Then, one gets due to DD withrespect to σ that

zx0,σi,d = zx0,σj ,d

and hence

HieAitx0 + (

∫ t0Hie

Ai(t−s)Ei ds)d0

qHje

Ajtx0 + (∫ t

0Hje

Aj(t−s)Ej ds)d0

(8)

for all t ∈ R+. Since x0 and d0 are both arbitrary,one gets

HiAki = HjA

kj (9)

HiAkiEi = HjA

kjEj (10)

for all k ≥ 0 by differentiating (8) with respect totime and evaluating at t = 0. Condition (i) followsfrom (9) for k = 0, and (ii) from k = 0, 1, . . . , nx−1. To see that (iii) holds, note that

HiAì(Ai − Aj) = HiA

`+1i −HiA

ìAj

(9)= HiA

`+1j −HiA

ìAj

= Hi(A`j − Aì)Aj

(9)= 0

for all `. For condition (iv), observe that

HiAìEi

(10)= HjA

`jEj

(9)= HiA

ìEj

and hence HiAì(Ei − Ej) = 0 for all ` ≥ 0.

Sufficiency: First, we will show that

HiAki = HjA

kj (11)

for all i, j ∈ {1, 2, . . . ,M}×{1, 2, . . . ,M} and k ≥0 by induction. Note that (11) holds for k = 0 dueto condition (i). Assume that it holds for some k.Then

HiAk+1i −HjA

k+1j = HiA

k+1i −HiA

kiAj

= HiAki (Ai − Aj)

(iii)= 0.

Also note that

HiAkiEi −HjA

kjEj

(11)= HiA

kiEi −HiA

kiEj

= HiAki (Ei − Ej)

(iv)= 0

for all k ≥ 0. Thus, we get

HiAkiEi = HjA

kjEj (12)

for all i, j ∈ {1, 2, . . . ,M}×{1, 2, . . . ,M} and k ≥0. Now, let

z = zx0,σ1,d − zx0,σ2,d (13)x = xx0,σ1,d − xx0,σ2,d (14)

for some x0, σ1, σ2, and d. It follows from (11) fork = 0 that

z(t) = Hσ1(t)x(t).

By differentiating and using (11) for k = 1 and (12)for k = 0, one gets

˙z(t) = Hσ1(t)Aσ1(t)x(t)

for all t ∈ R+. Repeating the same argument, oneobtains

z(k)(t) = Hσ1(t)Akσ1(t)x(t)

for all k ≥ 0 and for all t ∈ R+. Due to (11) thisyields

z(k)(t) = HiAki x(t)

for all t ∈ R+, in which i ∈ {1, . . . ,M} canbe selected arbitrarily. Then, there exists a nonzeropolynomial p(λ) (e.g., the characteristic polynomialof Ai for any i) such that

p(d

dt)z = 0

7


by the Cayley-Hamilton theorem. Since x(0) = 0,one has z(k)(0) = 0 for all k ≥ 0. Therefore, z(t) =0 for all t ∈ R+. In other words,

zx0,σ1,d = zx0,σ2,d

for all x0, σ1, σ2, and d.Based on Theorem III.3, we can also derive an

alternative characterization of DD with respect to σ,which is more geometric in nature.

Corollary III.4 The SLS (1) is DD with respectto σ if and only if there exists an {A1, . . . , AM}-invariant subspace V such that for all (i, j) ∈{1, . . . ,M} × {1, . . . ,M}

(i) Hi = Hj

(ii) im (Ai − Aj) ⊆ V ⊆ kerHi

(iii) im (Ei − Ej) ⊆ V

Proof: Necessity: The necessity of these condi-tions follows directly from Theorem III.3 by takingV = Ni.

Sufficiency: As Ni is the largest Ai-invariantsubspace that is contained in kerHi for alli ∈ {1, . . . ,M}, condition (ii), together with the{A1, . . . , AM}-invariance of V , implies that V ⊆Ni. This fact together with (i) implies that condition(i), (iii) and (iv) of Theorem III.3 are satisfied. Toshow that also (ii) of Theorem III.3 is satisfied, weobserve that statement (ii) of this theorem impliesthat HiAi = HiAj for all (i, j) ∈ {1, . . . ,M} ×{1, . . . ,M}. Due to im (Ai − Aj) ⊆ V and Vbeing {A1, . . . , AM}-invariant, it must hold thatAki im (Ai − Aj) ⊆ V ⊆ kerHi for all k ∈ N andthus HiA

k+1i = HiA

kiAj for all i, j. We will now

prove that

HiAki = HjA

kj (15)

for all k ∈ N using induction. Clearly, it holds fork = 0. Suppose it holds for k, then

HjAk+1j

(15)= HiA

kiAj = HiA

k+1i

Hence, (15) holds for all k ∈ N and thus Ni = Njfor all i, j. As such, we recovered the conditions ofTheorem III.3, which completes the proof.

C. Disturbance decoupling with respect to σ and d

Using the above results, we will characterize DDwith respect to σ and d now.

Lemma III.5 The SLS (1) is DD with respect toboth σ and d if and only if it is DD with respect toσ and DD with respect to d.

Proof: The necessity of the conditions followsby taking, first σ1 = σ2 in (4), which yields (17)and secondly, d1 = d2 in (4), which gives (3). Thesufficiency follows from

zx0,σ1,d1 = zx0,σ1,0 + z0,σ1,d1

(17)= zx0,σ1,0 + z0,σ1,d2

(3)= zx0,σ2,0 + z0,σ2,d2

= zx0,σ2,d2

where in the first and the last equalities we used thelinearity properties of the SLS (1) for a fixed σ.

Theorem III.6 The following statements are equiv-alent:

1) The SLS (1) is DD with respect to σ and d.2) The conditions

(i) Hi = Hj = H ,(ii) Ni = Nj = N ,

(iii) im(Ai − Aj) ⊆ N , and(iv) imEi ⊆ N

hold for all (i, j) ∈ {1, . . . ,M}×{1, . . . ,M}.3) There exists an {A1, . . . , AM}-invariant sub-

space V such that(i) Hi = Hj = H ,

(ii) im(Ai − Aj) ⊆ V ⊆ kerH , and(iii) imEi ⊆ Vfor all (i, j) ∈ {1, . . . ,M} × {1, . . . ,M}.Proof: (1) ⇒ (2): According to Lemma III.5,

the hypotheses of Theorem III.2 and Theorem III.3are necessarily true when the SLS (1) is DD withrespect to σ and d. For all x ∈ V and for all i =1, . . . ,M one can write

HAki x = 0

for all k ∈ N. This shows that x ∈ N . ThereforeV ⊆ N . Combining the conditions of Theorem III.2and Theorem III.3 gives (2).

8


(2) ⇒ (3): Since Ni is an Ai-invariant subspacecontained in kerH and Ni = Nj = N , one canwrite

AiN ⊆ N

for all i = 1, . . . ,M . Thus, it follows that N isan {A1, . . . , AM}-invariant subspace contained inkerH .

(3) ⇒ (1): It follows from (3) thatM∑

i=1

im Ei ⊆

V ⊆ kerH . This recovers the condition of The-orem III.2. One can also recover the conditionsof Theorem III.3 by following the steps shown inthe sufficiency part of Corollary III.4. Based onLemma III.5, (1) follows.

D. Disturbance decoupling of piecewise linear sys-tems

Based on the above results we can show theimportance of DD with respect to σ and d fordisturbance decoupling of piecewise linear (PWL)systems of the form

x(t) = Aix(t) + Eid(t) if x(t) ∈ Xi (16a)

z(t) = Hx(t) (16b)

in which Xi ⊂ Rnx , i ∈ {1, . . . ,M}, are polyhedralregions with non-empty interiors that form a parti-tioning of the state space Rnx , i.e.,

⋃Mi=1Xi = Rnx

and the interiors of different regions have an emptyintersection. Since the right-hand side of a PWLsystem can be discontinuous, solutions will be inter-preted in the sense of Filippov [5], which includespossible sliding motions at the boundaries of theregions. See [5] for more details and exact defini-tions of Filippov solutions. To avoid any ambiguityin the definition of the output as in (16b) duringsliding motions we assumed that the output matrixH is independent of i. Note that this is a necessarycondition for the SLS (1) corresponding to (16) tobe DD with respect to both σ and d.

Definition III.7 The PWL system (16) is calleddisturbance decoupled (DD) with respect to d if forall x0 ∈ Rnx , d1, d2 ∈ Lloc1 (R+,Rnd), it holds that

z1 = z2 (17)

where zj = Hxj , j = 1, 2 and xj , j = 1, 2, is aFilippov solution to (16) for initial state x(0) = x0

and disturbance input dj , j = 1, 2.

Note that due to possible non-uniqueness of Fil-ippov solutions multiple solutions might correspondto one initial condition and one disturbance input.

Theorem III.8 The PWL (16) system is DD withrespect to d, if the corresponding SLS (1) is DDwith respect to σ and d.

Proof: Let xj , j = 1, 2, be Filippov solutionsto (16) for x0 and dj , j = 1, 2, and zj = Hxj , j =1, 2. Consider the absolutely continuous functionsx = x1 − x2 and z = z1 − z2. Due to the definitionof Filippov solutions we have that for almost allt ∈ R+ there exist αi, βj depending on t such thatM∑

i=1

αi = 1,M∑

j=1

βj = 1, αi, βj ≥ 0 for all i ∈

{1, . . . ,M}, j ∈ {1, . . . ,M} and

˙x(t) = x1(t)− x2(t) =M∑

i=1

αi(Aix1(t) + Eid1(t))

−M∑

j=1

βj(Ajx2(t) + Ejd2(t))

(18)

Equation (18) can be rewritten as

˙x(t) =M∑

i=1

βiAi(x1(t)− x2(t)) +M∑

i=1

αiEid1(t)+

−M∑

i=1

βjEjd2(t) +M∑

j=1

M∑

i=1

δij(Ai − Aj)x1(t)

(19)

with

αi − βi =M∑

j=1

(δij − δji) (20)

For convenience, we take δii = 0 for all i =1, . . . ,M . To show that (20) holds, we write (20)in vector notation using

δ =(δ12 . . . δ1M δ21 . . . δ2M . . . δM1 . . .

)>

9


which results in the following system of linearequations.

Qδ =

α1 − β1

...αM − βM

= v (21)

where Q ∈ RM×(M−11) is a suitably defined matrix.For a solution to exist to (21), it must hold thatimv ⊆ imQ ⇔ (imQ)⊥ = kerQ> ⊆ (imv)⊥.

By construction of Q, kerQ> = span{

11...1

}.

SinceM∑

i=1

(αi − βi) = 0, span{

11...1

} ⊆ (imv)⊥.

Therefore a solution for δ always exists, whichproves that equation (18) can always be written inthe form of (19).

Since the SLS (1) is DD with respect to d and σ,we have that

Akl im(Ai − Aj) ⊆ kerH

imEi ⊆ kerH

HAki = HAkj

for all (i, j, l) ∈ {1, . . . ,M} × {1, . . . ,M} ×{1, . . . ,M} and for all k ∈ N. Therefore, multi-plying (19) with H gives for almost all t ∈ R+

˙z(t) = H ˙x(t) = HAix(t) (22)

Differentiating (22) with respect to t and repeatingthe arguments above, we obtain

z(k)(t) = HAki x

for all k ∈ N. Similar to the arguments used inthe proof of Theorem III.3, one can conclude thatz(t) = 0 for all t ∈ R+.This theorem demonstrates the relevance of DDwith respect to σ and d for SLS in the contextof DD to d for PWL systems. As such, whencontrol inputs u are present, disturbance decouplingproblems (DDPs), i.e., designing controllers thatrender the closed-loop SLS DD with respect to σand d, can be used also for solving DDPs of PWL

systems with respect to d. DDPs for SLSs willbe considered in the next section. However, beforedoing so, we would like to show that although DDof the SLS with respect to σ and d is a sufficientcondition for DD of a PWL system to d, it is notnecessary.

Example III.9 Consider the PWL system

x(t) =

[−1 00 0

]x(t) +

[01

]d(t) x1(t) ≥ 0

x(t) =

[−2 00 0

]x(t) +

[02

]d(t) x1(t) < 0

z =[1 0

]x

It is clear that the corresponding SLS is not dis-turbance decoupled with respect to σ and d. Yet,the PWL system described above is disturbancedecoupled with respect to d.

IV. DDP BY STATE FEEDBACK

In the previous section we provided full character-izations of DD properties. Now we will consider ifand how we should choose control inputs in order torender a SLS disturbance decoupled in some sense.In order to do so, consider the SLS

x(t) = Aσ(t)x(t) +Bσ(t)u(t) + Eσ(t)d(t) (23a)

z(t) = Hσ(t)x(t) + Jσ(t)u(t) (23b)

where we included now a control input u(t) ∈ Rnu

at time t ∈ R+. As before we denote the solu-tion corresponding to x0 ∈ Rnx , σ ∈ S, d ∈Lloc1 (R+,Rnd) and u ∈ Lloc1 (R+,Rnu) by xx0,σ,d,uand the corresponding output by zx0,σ,d,u. We arenow interested in finding conditions under whichcontrollers can be found such that the closed-loopsystem is DD with respect to d, to σ, or to both.We start with static state feedback controllers.

A. Solution of DDP with respect to d by mode-dependent state feedback

Problem IV.1 The disturbance decoupling problemwith respect to d (DDPd) by mode-dependent statefeedback for SLS (23) amounts to finding Fi ∈Rnu×nx , i = 1, . . . ,M such that

x(t) = (Aσ(t) +Bσ(t)Fσ(t))x(t) + Eσ(t)d(t) (24a)

10


z(t) = (Hσ(t) + Jσ(t)Fσ(t))x(t) (24b)

is DD with respect to d.

Note that the SLS (24) results from putting thesystem (23) in closed loop with u(t) = Fσ(t)x(t),which requires knowledge of the active mode σ(t)at time t ∈ R+.

Definition IV.2 Consider the SLS (23) withd = 0. A subspace V is called output-nulling{(A1, B1), . . . , (AM , BM)}-invariant if for anyx0 ∈ V and σ ∈ S there exists a control inputu ∈ Lloc1 (R+,Rnu) such that xx0,σ,0,u(t) ∈ V andzx0,σ,0,u(t) = 0 for all t ∈ R+.

Sometimes an output-nulling {(A1, B1), . . . ,(AM , BM)}-invariant subspace is called a commonoutput-nulling controlled invariant subspace for(23).

Theorem IV.3 Consider the SLS (23) with d = 0.Let V be a subspace of Rnx and V0 ∈ Rnx × Rnz

denote the extended subspace V×{0}. The followingstatements are equivalent.

(i) V is common output-nulling controlled invari-ant.

(ii)[AjHj

]V ⊆ V0 +im

[Bj

Jj

]for all j = 1, . . . ,M .

(iii) There exist Fj ∈ Rnu×nx , j = 1, . . . ,M , suchthat(Aj + BjFj)V ⊆ V ⊆ ker (Hj + JjFj) for allj = 1, . . . ,M .

Proof: The proof is similar in nature to the(more complicated) proof of Theorem IV.9, whichwe will provide below, and is therefore omitted.

Let {Vj | j ∈ J } be a collection of commonoutput-nulling controlled invariant subspaces forthe SLS (23). It follows from Definition IV.2 that∑

j∈JVj is common output-nulling controlled invari-

ant. Therefore, the set of all common output-nullingcontrolled invariant subspaces admits a largest ele-ment. The largest common output-nulling controlledinvariant subspace for a given SLS plays a crucialrole in the solution of DDPd by mode-dependentfeedback.

Definition IV.4 Consider the SLS (23) with d = 0.We define V∗md as the largest common output-nulling

controlled invariant subspace for the SLS (23) thatis

(i) V∗md is common output-nulling controlled in-variant;

(ii) if V is a common output-nulling controlledinvariant subspace for the SLS (23), then V ⊆V∗md.

Theorem IV.5 Consider the SLS (23). DDPd bymode-dependent feedback is solvable if and only if

M∑

i=1

im Ei ⊆ V∗md (25)

Proof: The proof of the theorem follows di-rectly from Theorem III.2, Theorem IV.3, and Def-inition IV.4.

In Section V we will provide an algorithm tocompute V∗md for a given SLS.

Remark IV.6 For the special case that Ji = 0,i = 1, . . . ,M , this problem was solved also in[12]. In [12] the DDP with respect to d by mode-dependent state feedback was combined with thequestion of quadratic stability. Sufficient conditionswere given exploiting known results for quadraticstabilization as in [4, 17]. These stability conditionscan also be added to the theorems that we presenthere, but unfortunately, they are, just as in [12],not so trivial to verify, certainly for a high numberof subsystems. Indeed, the sufficient conditions thatguarantee solvability of DDP with quadratic sta-bility (DDPQS) according to Theorem 3.2 in [12]are the existence of Fj , j = 1, . . . ,M , such that(Aj + BjFj)V∗md ⊆ V∗md ⊆ ker (Hj + JjFj) for allj = 1, . . . ,M and there exists a convex combination∑αj(Aj + BjFj) with

∑Mj=1 αj = 1 and αj ≥ 0,

j = 1, . . . ,M , being a Hurwitz matrix. Since aparameterization of all Fj , j = 1, . . . ,M , satisfying(Aj + BjFj)V∗md ⊆ V∗md ⊆ ker (Hj + JjFj) is hardto come by in the first place, and one has to searchfor both αj and Fj , j = 1, . . . ,M , which is a non-convex problem, these conditions are not easy toverify. Also for the satisfaction of the stabilizationcondition it is unclear if using V∗md instead ofanother common output-nulling controlled invari-ant subspace containing

∑Mi=1 im Ei is introducing

conservatism into the conditions. Another difference

11


with respect to [12], is that DDP with respectto d by mode-independent state feedback was notconsidered in [12], while we treat this problem inthe next section.

B. Solution of DDP with respect to d by mode-independent state feedbackProblem IV.7 The disturbance decoupling problemwith respect to d (DDPd) by mode-independentfeedback for SLS (23) amounts to finding F ∈Rnu×nx such that

x(t) = (Aσ(t) +Bσ(t)F )x(t) + Eσ(t)d(t) (26a)

z(t) = (Hσ(t) + Jσ(t)F )x(t) (26b)

is DD with respect to d.

Note that the SLS (26) results from putting thesystem (23) in closed loop with u(t) = Fx(t).The latter state feedback controller does not requireknowledge of the active mode σ(t) at time t ∈ R+.

Definition IV.8 Consider the SLS (23) withd = 0. A subspace V is called output-nulling{(A1, B1), . . . , (AM , BM)}-invariant under mode-independent control if for any x0 ∈ V thereexists a control input u ∈ Lloc1 (R+,Rnu) such thatxx0,σ,0,u(t) ∈ V and zx0,σ,0,u(t) = 0 for all σ ∈ Sand for all t ∈ R+.

Sometimes a subspace that is output-nulling{(A1, B1), . . . , (AM , BM)}-invariant under mode-independent control is called a common output-nulling controlled invariant subspace under mode-independent control for (23).

Theorem IV.9 Consider the SLS (23) with d = 0.Let V be a subspace of Rnx and V0 ∈ Rnx×Rnz de-note the extended subspace V×{0}. Define the ma-trices As ∈ RM(nx+nz)×nx and Bs ∈ RM(nx+nz)×nu

As =

A1

H1...

AMHM

, Bs =

B1

J1...

BM

JM

(27)

and V0M as V0M =

M times︷︸︸︷V0 × V0 × . . .× V0. The fol-

lowing statements are equivalent.

(i) V is common output-nulling controlled invari-ant under mode-independent control.

(ii) AsV ⊆ V0M + imBs.(iii) There exists F ∈ Rnu×nx such that (Aj +

BjF )V ⊆ V ⊆ ker (Hj + JjF ) for allj = 1, . . . ,M .

Proof: : (i) ⇒ (ii). Assume that σ(t) = σj

for some j ∈ {1, . . . ,M}. Let x0 ∈ V and u be acontrol input such that xx0,σj ,0,u(t) = xj(t) ∈ V andzx0,σj ,0,u(t) = zj(t) = 0 for all j = 1, . . . ,M andt ∈ R+. Then one can write

xx0,σ1,0,u(t)

...xx0,σM ,0,u(t)

=

x0...x0

+

∫ t

0

(

A1 0

. . .0 AM

x1(s)

...xM(s)

+

B1...

BM

u(s))ds

(28)

Furthermore, zx0,σj ,0,u(t) = Hjx(t)+Jju(t) = 0 forall j = 1, . . . ,M and for all t ∈ R+. Thus, one canalso write

∫ t

0

zx0,σ1,0,u(s)

...zx0,σM ,0,u(s)

ds = 0 (29)

Combining (28) with (29) we get

A1 0H1

. . .AM

0 HM

∫ t

0

x1(s)

...xM(s)

ds =

xx0,σ1,0,u(t)− x0

0...

xx0,σM ,0,u(t)− x0

0

︸︷︷︸∈V0M

−

B1

J1...

BM

JM

∫ t

0

u(s)ds

︸︷︷︸∈imBs

(30)

12


Now we divide the left-hand side of (30) by t andtake the limit to obtain

limt→0

1

t

A1 0H1

. . .AM

0 HM

∫ t

0

x1(s)

...xM(s)

ds =

A1 0H1

. . .AM

0 HM

x0...x0

=

A1

H1...

AMHM

x0 (31)

Equality (31) holds because xx0,σj ,0,u(t) is continu-ous for all j = 1, . . . ,M . The rest of the proof isclear from here on.

(ii) ⇒ (iii). Choose a basis {q1, . . . , qnv , qnv+1,. . . , qnx} for Rnx such that {q1, . . . , qnv} is a basisfor V . For k = 1, . . . , nv there exist vectors qj,k ∈V and uk ∈ Rnu such that Ajqk = qj,k + Bjukand Hjqk = Jjuk for all j = 1, . . . ,M . For k =1, . . . , nv define Fqk = −uk and for k = nv +1, . . . , nx let Fqk be arbitrary vectors in Rnx . Thenfor (j, k) ∈ {1, . . . ,M}×{1, . . . , nv} we have (Aj+BjF )qk = qj,k ∈ V and (Hj + JjF )qk = 0. Hence(Aj + BjF )V ⊆ V ⊆ ker (Hj + JjF ) for all j =1, . . . ,M .

(iii) ⇒ (i). Let x0 ∈ V and apply the feedbacklaw u(t) = Fx(t) to obtain the closed loop systemx(t) = (Aσ(t)+Bσ(t)F )x(t), z(t) = (Hj+JjF )x(t).Clearly, xx0,σ,0,u(t) ∈ V and zx0,σ,0,u(t) = 0 for allt ∈ R+. Thus, (i) follows.

Definition IV.10 Consider the SLS (23) with d =0. We define V∗mi as the largest common output-nulling controlled invariant subspace under mode-independent control for the SLS (23), that is

1) V∗mi is common output-nulling controlled in-variant under mode-independent control;

2) if V is common output-nulling controlled in-variant under mode-independent control for theSLS (23), then V ⊆ V∗mi.

Theorem IV.11 Consider the SLS (23). DDPd bymode-independent feedback is solvable if and only

ifM∑

i=1

im Ei ⊆ V∗mi (32)

Proof: The proof of the theorem directly fol-lows from Theorem III.2, Theorem IV.9 and Defi-nition IV.10.

In Section V we will present an algorithm tocompute the largest common output-nulling con-trolled invariant subspace using mode-independentfeedback for a given SLS.

C. Solution of DDP with respect to σ by mode-dependent state feedback

Consider the SLS (23).

Problem IV.12 The disturbance decoupling prob-lem with respect to σ (DDPσ) by mode-dependentstate feedback amounts to finding Fi ∈ Rnu×nx ,i = 1, . . . ,M such that the SLS (24) is DD withrespect to σ.

Before giving the theorem for the solvability ofProblem IV.12 we need to introduce the followinglemma.

Lemma IV.13 Consider the SLS (23). DDPσ bymode-dependent feedback is solvable if and onlyif there exist a common output-nullling con-trolled invariant subspace V and Fk ∈ Rnu×nx ,k = 1, . . . ,M , with (Ak + BkFk)V ⊆ V ⊆ker (Hk + JkFk), k = 1, . . . ,M , such that thefollowing three conditions hold for all (i, j) ∈{1, . . . ,M} × {1, . . . ,M}:

(i) Hi + JiFi = Hj + JjFj ,(ii) im (Ai +BiFi − Aj −BjFj) ⊆ V ,

(iii) im (Ei − Ej) ⊆ V .

Proof: The proof of the lemma directly followsfrom Corollary III.4 and Theorem IV.3.

Theorem IV.14 Consider the SLS (23). DDPσ bymode-dependent feedback is solvable if and only ifthere exist Gi ∈ Rnu×nx , i = 1, . . . ,M , such thatfor all (i, j) ∈ {1, . . . ,M} × {1, . . . ,M} it holdsthat

(i) Hi + JiGi = Hj + JjGj ,(ii) im (Ai +BiGi − Aj −BjGj) ⊆ V∗md,

(iii) im (Ei − Ej) ⊆ V∗md.

13


Proof: Necessity: For a subspace V we will usethe notation

Fi(V) = {F |(Ai +BiF )V ⊆ V ⊆ ker (Hi + JiF )}If DDPσ is solvable, then as a result ofLemma IV.13, there exists a common output-nullingcontrolled invariant subspace, V , such that Hi +JiF

′i = Hj + JjF

′j , F

′k ∈ Fk(V) k = 1, . . . ,M ,

for all (i, j) ∈ {1, . . . ,M} × {1, . . . ,M}. It alsoholds that

im (Ai+BiF′i−Aj−BjF

′j) ⊆ V ⊆ ker (Hi + JiF

′i )

im (Ei − Ej) ⊆ VLet Fi ∈ Fi(V∗md) for all i = 1, . . . ,M . SinceV ⊆ V∗md, one can choose a basis {v1, v2, . . . , vnx}for Rnx such that {v1, . . . , vp} is a basis for V and{v1, . . . , vq} is a basis for V∗md. Define

Givk =

{Fivk k ∈ {1, 2, . . . , q}F ′ivk k ∈ {q + 1, q + 2, . . . , nx}

Clearly, Gi ∈ Fi(V∗md) and Hi + JiGi = Hj + JjGj

for all i, j. We claim that im (Ai + BiGi − Aj −BjGj) ⊆ V∗md. To see this, note that

[(Ai +BiGi)− (Aj +BjGj)]vk ∈{V∗md k ∈ {1, 2, . . . , q}V k ∈ {q + 1, q + 2, . . . , nx}

Since V ⊆ V∗md, it immediately follows thatim (Ai +BiGi − Aj −BjGj) ⊆ V∗md. Note that

im (Ei − Ej) ⊆ V ⊆ V∗mdand thus also (iii) holds.

Sufficiency: Let {v1, v2, . . . , vnx} be a basis forRnx such that {v1, . . . , vq} is a basis for V∗md.Furthermore, let Fi ∈ Fi(V∗md), i = 1, . . . ,M .Define

Fivk =

{Fivk k ∈ {1, 2, . . . , q}Givk k ∈ {q + 1, q + 2, . . . , nx}

Note that Fi ∈ Fi(V∗md) for all i. It is easy to seethat im (Ai +BiFi −Aj −BjFj) ⊆ V∗md and Hi +JiFi = Hj + JjFj for all i, j. Thus we recoveredthe conditions of Lemma IV.13, thereby completingthe proof.

D. Solution of DDP with respect to d and σ(DDPdσ) by mode-dependent state feedback

Consider the SLS (23).

Problem IV.15 The disturbance decoupling prob-lem with respect to d and σ (DDPdσ) by mode-dependent state feedback amounts to finding Fi ∈Rnu×nx , i = 1, . . . ,M such that the SLS (24) is DDwith respect to d and σ.

Theorem IV.16 Consider the SLS (23). DDPdσ bymode-dependent feedback is solvable if and only ifthere exist Gi ∈ Rnu×nx , i = 1, . . . ,M such that forall (i, j) ∈ {1, . . . ,M} × {1, . . . ,M} it holds that

(i) Hi + JiGi = Hj + JjGj ,(ii) im (Ai +BiGi − Aj −BjGj) ⊆ V∗md,

(iii) im Ei ⊆ V∗md.Proof: The proof of the theorem is obtained

along similar lines as in the proof of Theorem IV.14.

Remark IV.17 In Theorem IV.14 andTheorem IV.16, if V∗md is replaced with V∗mi andGi = Gj for all (i, j) ∈ {1, . . . ,M} ×{1, . . . ,M},then the solutions to DDPσ and DDPdσ bymode-independent feedback, respectively, areobtained.

V. ALGORITHMS TO TEST THE HYPOTHESES OFTHE SOLUTIONS TO THE DDP

A. Algorithm for Theorem IV.5

In this subsection we will present an algorithm tofind the largest common output-nulling controlledinvariant subspace for the SLS (23).

Algorithm V.1

V0 = Rnx ; (33a)

Vi+1 =M⋂

j=1

{x | ∃u Ajx+Bju ∈ Vi, Hjx+Jju = 0}

(33b)

From this recurrence relation it follows that Vi+1 ⊆Vi for all i = 0, 1, . . ., and if Vk = Vk+1 for somek, then Vi = Vk for all i ≥ k. Let q be the smallestk ∈ N such that Vk = Vk+1. Obviously, q ≤ nx. We

14


claim that Vq = V∗md. For space reasons we omit theproof.

Theorem V.2 Consider the SLS (23) and Algo-rithm V.1. Then, Vq = V∗md with q := min{k ∈N | Vk = Vk+1} ≤ nx.

Proof: We will first show Vi+1 ⊆ Vi for i ∈ Nby induction. It is obvious that V1 ⊆ V0. Supposenow that Vi+1 ⊆ Vi.

Vi+2 =M⋂

j=1

{x|∃u Ajx+Bju ∈ Vi+1, Hjx+ Jju = 0} ⊆

M⋂

j=1

{x|∃u Ajx+Bju ∈ Vi, Hjx+ Jju = 0} = Vi+1

from which Vi+2 ⊆ Vi+1 follows. Therefore, Vi+1 ⊆Vi for i ∈ N.

Suppose for some k, Vk+1 = Vk. Then

Vk+2 =M⋂

j=1

{x|∃u Ajx+Bju ∈ Vk+1, Hjx+ Jju = 0}

=M⋂

j=1

{x|∃u Ajx+Bju ∈ Vk, Hjx+ Jju = 0}

= Vk+1

Hence, Vi = Vk for all i ≥ k. Since Vk ⊆ Vk−1 ⊆. . . ⊆ V0 , we have Vq = Vq+1 for some q ≤ nx.

Vq =M⋂

j=1

{x | ∃u Ajx+Bju ∈ Vq, Hjx+ Jju = 0}

This shows that[AjHj

]Vq ⊆ V0

q + im

[Bj

Jj

]

for all j = 1, . . . ,M . By Theorem IV.3, Vq is indeedcommon output-nulling controlled invariant.

To show that Vq is the largest common output-nulling controlled invariant subspace, we considera common output-nulling controlled invariant sub-space V for the SLS (23).

V ⊆ V0, thus the following:

V =M⋂

j=1

{x ∈ W|Ajx ∈ V + im Bj} ⊆

M⋂

j=1

{x | ∃u Ajx+Bju ∈ V0, Hjx+ Jju = 0} = V1

Hence, V ⊆ V1. Repeating the procedure above wearrive at the conclusion V ⊆ Vq.B. Algorithm for Theorem IV.11

In this subsection we will present the algorithmto find the largest common output-nulling controlledinvariant under mode-independent control for theSLS (23).

Algorithm V.3 Define the matrices As and Bs andthe extended subspace V0M in the same way as inTheorem IV.9.

V0 = Rnx ; Vi+1 = {x | Asx ∈ V0iM

+ im Bs}(34)

As above, it holds that Vi+1 ⊆ Vi for all i = 0, 1, . . .and if Vk = Vk+1 for some k, then Vi = Vk for alli ≥ k. Let q be the smallest k ∈ N such that Vk =Vk+1. Obviously, q ≤ nx. We claim that Vq = V∗miin the next theorem of which the proof is omitteddue to space reasons.

Theorem V.4 Consider the SLS (23) and Algo-rithm V.3. Then, Vq = V∗mi with q := min{k ∈ N |Vk = Vk+1} ≤ nx.

Note that by defining As as in (27) and redefiningBs ∈ RMnx×Mnu as

Bs =

B1 0 . . . 0

0. . . 0

...0 . . . 0 BM

algorithm V.3 yields the largest common output-nulling controlled invariant subspace (under mode-dependent control).

Remark V.5 The algorithm of Section V-A to ob-tain the largest common controlled invariant sub-space (with mode-dependent feedback) for a SLS

15


inside another subspace is related to the algorithmpresented before in [1, 14] for LPV systems forthe special case that Ji = 0, i = 1, . . . ,M .The algorithm in Section V-B for mode-independentfeedback was not presented in the literature before.

VI. DDP BY MEASUREMENT FEEDBACK

Previously, we assumed that the full state feed-back was available for feedback, which is hardly thecase in practice. In this section we will thereforeshow how the DDP with respect to d can be solvedusing dynamic measurement feedback.

A. Solution of DDP with respect to d by mode-dependent measurement feedback

Consider the SLS

x(t) = Aσ(t)x(t) +Bσ(t)u(t) + Eσ(t)d(t) (35a)

y(t) = Cσ(t)x(t) (35b)

z(t) = Hσ(t)x(t) + Jσ(t)u(t) (35c)

and the mode-dependent dynamic measurementfeedback controller given by the mode-dependentcontroller

w(t) = Kσ(t)w(t) + Lσ(t)y(t) (36a)

u(t) = Mσ(t)w(t) +Nσ(t)y(t) (36b)

which results in the closed-loop system(x(t)w(t)

)= Ae,σ(t)

(x(t)w(t)

)+

(Eσ(t)

0

)d(t) (37a)

z(t) =(Hσ(t) + Jσ(t)Nσ(t)Cσ(t) Jσ(t)Mσ(t)

)(x(t)w(t)

)

(37b)

where Ae,i =

(Ai +BiNiCi BiMi

LiCi Ki

), i =

1, . . . ,M . The problem we would like to tackle inthis section is formulated as follows.

Problem VI.1 The disturbance decoupling prob-lem with respect to d by mode-dependent mea-surement feedback amounts to finding Ki ∈Rnw×nw , Li ∈ Rnw×ny , Mi ∈ Rnu×nw , Ni ∈ Rnu×ny ,i = 1, . . . ,M such that (37) is DD with respect tod.

Removing the output-nulling requirement in Def-inition IV.2 and using Theorem IV.3 lead to thefollowing definition.

Definition VI.2 A subspace V ⊆ Rnx is called{(A1, B1), . . . , (AM , BM)}-invariant, if there existFj, j = 1, . . . ,M such that (Aj + BjFj)V ⊆ V forall j = 1, . . . ,M .

To tackle the problem as introduced in this sectionwe need the following concept that is the dual ofcommon controlled invariant subspaces.

Definition VI.3 A subspace Z ⊆ Rnx is called{(C1, A1) , . . . , (CM , AM)}-invariant, if there existGj, j = 1, . . . ,M such that (Aj +GjCj)Z ⊆ Z forall j = 1, . . . ,M .

Sometimes a {(C1, A1), . . . , (CM , AM)}-invariantsubspace is called a common conditionedinvariant subspace for (35). Similarly, a{(A1, B1), . . . , (AM , BM)}-invariant subspaceis sometimes called a common controlled invariantsubspace. The collection of all sets {G1, . . . , GM}that satisfies (Ai + GiCi)Z ⊆ Z , i = 1, . . . ,M , isdenoted by Gmd(Z). Likewise, the collection of allsets {F1, . . . , FM} that satisfies (Ai +BiFi)V ⊆ V ,i = 1, . . . ,M , is denoted by Fmd(V). It isclear that (Ai + BiFi)V ⊆ V if and only if(ATi + F T

i BTi )V⊥ ⊆ V⊥, i = 1, . . . ,M . Here,

V⊥ denotes the orthogonal complement of V , i.e.,V⊥ = {w | w>v = 0 for all v ∈ V}. Thus, V⊥ is{(B>1 , A>1 ), . . . , (B>M , A

>M)} invariant if and only

if V is {(A1, B1), . . . , (AM , BM)}-invariant. Thisduality can be used to prove the following.

Theorem VI.4 The following statements are equiv-alent.

(i) Z is common conditioned invariant,(ii) Aj(Z ∩ kerCj) ⊆ Z for all j = 1, . . . ,M .

We have the following corollary of Theorem III.2.

Corollary VI.5 The SLS (37) is disturbance decou-pled with respect to d if and only if there exists an{Ae1, . . . , AeM}-invariant subspace Ve ∈ Rnx×Rnw

such thatM∑

i=1

im

[Ei0

]⊆ Ve ⊆

ker

H1 + J1N1C1 J1M1...

HM + JMNMCM JMMM

(38)

16


We are now interested in relating {Ae1, . . . , AeM}-invariance of a subspace to geometric proper-ties of the original SLS (35). As an exten-sion of the linear case [13], the concept of{(C1, A1, B1), . . . , (CM , AM , BM)}-pairs will be ofimportance.

Definition VI.6 Consider the SLS (35).A pair of subspaces (Z,V) is called a{(C1, A1, B1), . . . , (CM , AM , BM)}-pair, if Zis a common conditioned invariant subspace for(35), V is a common controlled invariant subspacefor (35) and Z ⊆ V . Ceteris paribus, if V isa common output-nulling controlled invariantsubspace then (Z,V) is called an output-nulling{(C1, A1, B1), . . . , (CM , AM , BM)}-pair.

Since Ve as in Corollary VI.5 lies in the extendedstate space Rnx×Rnw , it can be projected onto Rnx

by

p(Ve) = {x ∈ Rnx|∃w(xw

)∈ Ve}

We can also define the intersection of Ve and Rnx×{0} as

i(Ve) = {x ∈ Rnx|(x0

)∈ Ve}

Theorem VI.7 Consider the controlled SLS(37) and let Ve denote an {Ae1, . . . , AeM}-invariant subspace. Then, (i(Ve), p(Ve)) is a{(C1, A1, B1), . . . , (CM , AM , BM)}-pair. If Vesatisfies

Ve ⊆ ker

H1 + J1N1C1 J1M1...

HM + JMNMCM JMMM

(39)

in addition, then (i(Ve), p(Ve)) is an output-nulling{(C1, A1, B1), . . . , (CM , AM , BM)}-pair.

Proof: Taking σ = σj , the proof followsalong similar lines as the proof of Theorem 6.2(making the necessary modifications to include thefeedthrough terms Ji, i = 1, . . . ,M ) in [6].

Vice versa we have the following result.

Theorem VI.8 Consider the SLS(35). Let (Z,V) be an output-nulling{(C1, A1, B1), . . . , (CM , AM , BM)}-pair. Then

there exist a mode-dependent controller (36)and an {Ae1, . . . , AeM}-invariant subspaceVe ∈ Rnx × Rnw satisfying (39) such thati(Ve) = Z , p(Ve) = V .

Proof: Let {F1, . . . , FM} ∈ Fmd(V) and{G1, . . . , GM} ∈ Gmd(Z). Construct Nj such that(Aj + BjNjCj)Z ⊆ V by applying Lemma 6.3 in[6] for all j = 1, . . . ,M . Take the mode-dependentcontroller and Ve such that Kj = Aj + BjFj +GjCj−BjNjCj , Lj = BjNj−Gj, Mj = Fj−NjCj ,j = 1, . . . ,M and

Ve =

{(x1

0

)+

(x2

x2

) ∣∣∣∣x1 ∈ Z, x2 ∈ V}

{Ae1, . . . , AeM}-invariance property of Ve as con-structed above can be proven mutatis mutandisas in the proof of Theorem 6.4 in [6]. Here wewill only show that Ve satisfies (39). If x1 ∈ Z ,(x>1 , 0)> ∈ Ve. We claim that Hjx1 +JjNjCjx1 = 0for all j = 1, . . . ,M . Since Nj is chosen such that(Aj +BjNjCj)Z ⊆ V , our claim is true. If x2 ∈ V ,(x>2 , x

>2 )> ∈ Ve. It must then hold that (Hj +

JjNjCj)x2 +Jj(Fj−NjCj)x2 = (Hj+JjFj)x2 = 0for all j = 1, . . . ,M . Since x2 ∈ V , this is alsotrue. Thus we have shown that Ve satisfies (39). Theproof of the two remaining claims, i(Ve) = Z andp(Ve) = V , can also be obtained from the proofTheorem 6.4 in [6] with minor modifications.

Theorem VI.9 Consider the SLS (35). The dis-turbance decoupling problem by mode-dependentmeasurement feedback is solvable if and only ifthere exists an output-nulling {(C1, A1, B1), . . . ,(CM , AM , BM)}-pair (Z,V) such that

M∑

i=1

im Ei ⊆ Z

Proof: The proof follows from Theorem VI.7,Theorem VI.8 and Corollary VI.5.

Before concluding the paper we will show howa common conditioned invariant subspace that is ofparticular interest to us can be computed.

Definition VI.10 Let W ⊆ Rnx be a subspace.Consider the SLS (35) with d = 0. We defineZ∗md(W) as the smallest common conditioned in-variant subspace that contains W for the SLS (35)that is

17


(i) W ⊆ Z∗md(W)(ii) Z∗md(W) is common conditioned invariant;

(iii) if Z is a common conditioned invariant sub-space for the SLS (35) and W ⊆ Z , thenZ∗md(W) ⊆ Z .

Let {Zj | j ∈ J } be a collection of commonconditioned invariant subspaces for the SLS (35)such that Zj ⊇ W for all j ∈ J whereW is a givensubspace. It follows from Theorem VI.4 that

⋂

j∈JZj

is common conditioned invariant and it containsW .Thus, the set of all common conditioned invariantsubspaces containing a given subspace W admits asmallest element.

Remark VI.11 The orthogonal complement of thelargest {(A>1 , C>1 ), . . . , (A>M , C

>M)}-invariant sub-

space contained in W⊥ is equal to the small-est {(C1, A1), . . . , (CM , AM)} - invariant subspacecontaining a subspace W .

The following algorithm produces the smallest com-mon conditioned invariant subspace that contains asubspace W . It can be used to check the conditionof Theorem VI.9. It is derived from the previousremark and it is a special case of Algorithm V.1where the condition Hjx + Jju = 0 is ignored andV0 =W for some subspace W .

Algorithm VI.12

V0 =W⊥; Vi+1 =M⋂

j=1

{x ∈ W⊥ | A>j x ∈ Vi+im C>j }

Z∗md(W) = V⊥nx

VII. CONCLUSIONS

In this paper three different disturbance decou-pling (DD) properties for switched linear systemswere analyzed. The difference between the threeproperties is induced by which signals are consid-ered as the disturbances: (i) the continuous exoge-nous signal, (ii) the switching signal, or (iii) boththe continuous exogenous signal and the switchingsignal. The latter variant of DD is relevant in thecontext of fault-tolerant control and piecewise linearsystems, as we motivated in the paper. In particular,DD of a switched linear system with respect to

the switching signal and the continuous disturbancesignal implies DD of corresponding piecewise lin-ear systems with respect to the continuous dis-turbance signal. Complete geometric characteriza-tions for these properties were given, which wereused to solve also disturbance decoupling prob-lems (DDPs) by suitable choice of controllers. Bothmode-dependent and mode-independent controllerswere considered. The case of static state feedbackcontrollers was solved for the three instances of theDDP whereas the case of dynamic measurementfeedback controller was solved only for the DDPd.We used common controlled and common con-ditioned invariant subspaces to characterize theseproperties. Algorithms to compute these subspaceswere provided as well, so that these results can beapplied by straightforward computations.

REFERENCES

[1] G. Balas, J. Bokor, and Z. Szabo. InvariantSubspaces for LPV Systems and Their Ap-plication. IEEE Transactions on AutomaticControl, 48:2065–2069, 2003.

[2] G.B. Basile and G. Marro. Controlled andConditioned Invariants in Linear System The-ory. Prentice Hall, Englewood Cliffs, NJ,1992.

[3] C.I. Byrnes. Feedback decoupling of rota-tional distributions for spherically constrainedsystems. In Proc. 23rd IEEE Conf. on Deci-sion and Control, pages 421–426, Las Vegas,Nevada, USA, 1984.

[4] E. Feron. Quadratic stablizibility of switchedsystems via state and output feedback. Cicsp-468, MIT Technical Report, 1996.

[5] A.F. Filippov. Differential Equations withDiscontinuous Righthand Sides. Mathematicsand its Applications. Kluwer, The Netherlands,1988.

[6] H.L. Trentelman and A.A. Stoorvogel and M.Hautus. Control Theory for Linear Systems.Communications and control engineering se-ries. Springer, 2002.

[7] A. Isidori, A.J. Krener, C. Gori-Giorgi,and S. Nonaco. Nonlinear decoupling viafeedback: a different geometric approach.26(2):331–345, IEEE Trans. Autom. Control.

18


[8] A.A. Julius and A.J. van der Schaft. TheMaximal Controlled Invariant Set of SwitchedLinear Systems. In Proc. of the 41st IEEEConf. on Decision and Control, pages 3174–3179, Las Vegas, Nevada USA, 2002.

[9] D. Liberzon. Switching in Systems and Con-trol. Systems and Control: Foundations andApplications. Birkhauser, Boston, MA, 2003.

[10] J. Lunze and J. H. Richter. Reconfigurablefault-tolerant control: A tutorial introduction.European Journal of Control, 14(5):359–386,2008.

[11] H. Nijmeijer and A.J. Van der Schaft. Con-trolled invariance for nonlinear systems. IEEETrans. Autom. Control, 27(4):904–914, 1982.

[12] N. Otsuka. Disturbance decoupling withquadratic stability for switched linear systems.Systems and Control Letters, 2010, to appear.

[13] J.M. Schumacher. Compensator synthesis us-ing (C,A,B)-pairs. IEEE Trans. Aut. Contr.,25:1133–1138, 1980.

[14] G. Stikkel, J. Bokor, and Z. Szabo. Distur-bance Decoupling Problem with Stability forLPV Systems. In European Control Confer-ence, Cambridge, U.K., 2003.

[15] Zhendong Sun and S.S. Ge. Switched LinearSystems: Control and Design. Communica-tions and control engineering series. Springer,2005.

[16] Zhendong Sun, S.S. Ge, and T.H. Lee.Controllability and Reachability Criteria forSwitched Linear Systems. Automatica,38:775–786, 2002.

[17] M. Wicks, P. Peleties, and R. DeCarlo.Switched controller synthesis for the quadraticstabilization of a pair of unstable linear sys-tems. Eur. J. Contr., 4:140–147, 1998.

[18] W.M. Wonham. Linear multivariable control:A geometric approach. Springer, 3rd edition,1985.

[19] L. Zhang, D. Cheng, and C. Li. Disturbancedecoupling of switched nonlinear systems. IEEProceedings, 152(1):49–54, 2005.

19

CHAPTER 3

CONTROLLABILITY OF BIMODAL DISCRETE-TIME

PIECEWISE LINEAR SYSTEMS

Controllability ofBimodal

Discrete-TimePiecewise

LinearSystems

AT yarragiessek ziki

Abstract—In this paper we study the controllabilityproperties of a class of bimodal discrete-time piecewiselinear systems. In particular, the class is characterized bya continuous right-hand side and scalar input whereasthe state can be of any dimension. We will providefull algebraic necessary and sufficient conditions for thecontrollability/reachability/null controllability for this classof systems. We will make use of geometric control theoryfor linear systems and classical results on controllabilityfor input-constrained linear systems.

Index Terms—Bimodal systems, piecewise linear sys-tems, controllability, null controllability, hybrid systems,push-pull systems

I. INTRODUCTION

Controllability has always played an importantrole in modern control theory. Kalman and Hautusstudied this property for linear systems and gavecomplete characterizations in algebraic forms. Inthe presence of input constraints, characterizationsfor the controllability of discrete-time linear sys-tems were given in [6, 10, 15]. In case of classesof hybrid dynamical systems, such as piecewiselinear systems, such complete characterizations ofnull controllability, reachability or controllability arehard to come by, Indeed, it is known from [4] thatcontrollability problems for discrete-time piecewiselinear systems in a general setting are undecidable.

However, several results were obtained on the con-trollability of different classes of piecewise linearsystems. In [17, 18], Xu and Xie give characteriza-tions for the controllability of discrete-time planarbimodal piecewise linear systems. Brogliato obtainsnecessary and sufficient conditions for the controlla-bility of a class of continuous-time piecewise linearsystems in [7], but his results only apply to planarsystems as well. Bemporad et al. [3] propose analgorithmic approach based on optimization tools.Although this approach makes it possible to checkcontrollability of a given discrete-time system, itdoes not allow drawing conclusions about any gen-eral class of systems. Arapostathis and Broucke givea fairly complete treatment of stability and control-lability of continuous-time planar conewise linearsystems in [1], which are piecewise linear systemsfor which the regions are convex cones. In [14], Leeand Arapostathis provide a characterization of con-trollability for a class of continuous-time piecewiselinear systems but they assume, among other things,that the number of inputs in each subsystem is oneless than the number of states. In [9], Camlibel etal give algebraic necessary and sufficient conditionsfor the controllability of continuous-time conewiselinear systems.

In this paper we present algebraic necessary and

CHAPTER 3. CONTROLLABILITY OF BIMODAL DISCRETE-TIMEPIECEWISE LINEAR SYSTEMS

sufficient conditions for the controllability of a classof bimodal discrete-time piecewise linear systems.The class is characterized by the property that thedynamics is continuous across the switching planeand that the input is scalar. We impose no restrictionon the dimension of the state variable as opposedto the earlier mentioned works [1, 7, 17, 18] thatstudy the controllability of planar piecewise linearsystems. Also, we do not adopt the assumptionsin [14], namely that the number of inputs is oneless than the number of states. In this sense, ourresults are the first to provide algebraic necessaryand sufficient conditions for the controllability forthe indicated class of piecewise linear systems.

In proving the main result of this paper some ofthe main lines of reasoning follow similar steps asin the continuous-time case, which was provided in[9]. However, at various points different argumentshave to be used including the study of observabilityand invertibility properties of certain classes ofpiecewise linear systems, which were not neededfor the continuous-time result. In addition, deriv-ing null controllability, reachability and control-lability conditions for so-called push-pull systemsrequires essentially different ingredients than in thecontinuous-time case.

The paper is organized as follows. We lay thegroundwork for the solvability of the controllabilityproblem in Section II and we define the class ofsystems we are interested in in Section III. In Sec-tion IV and V we consider two different problemsthat will be used to tackle the main problem. InSection VI we give our main results.

II. PRELIMINARY RESULTS

A. DefinitionsConsider the discrete-time system

x[k + 1] = g(x[k], u[k]) (1a)

y[k] = h(x[k]) (1b)

where x[k] ∈ Rn, u[k] ∈ Rm, y[k] ∈ Rp are thestate, the input and the output variable, respectively,at discrete time k ∈ N. Here, g : Rn × Rm → Rn

and h : Rn → Rp are given functions. Given aninitial state x0 ∈ Rn and an input sequence u ={u[0], . . . , u[N ]} with N ∈ N, we will denote thestate trajectory corresponding to u and the initial

state x[0] = x0 by xx0,u. Likewise, we will denotethe corresponding output sequence by yx0,u.

Definition II.1 Consider the system (1). Let y ={y[0], . . . , y[N ]} be a sequence with y[k] ∈ Rp,k = 0, . . . , N . Given an initial state x0 ∈ Rn, wesay that x0 is compatible with y, if there exists aninput sequence u = {u[0], . . . , u[N − 1]} such thatyx0,u[k] = y[k] for k = 0, . . . , N . Likewise, givenan input sequence u = {u[0], . . . , u[N − 1]}, wesay that u is compatible with y, if there exists aninitial state x0 ∈ Rn such that yx0,u[k] = y[k] fork = 0, . . . , N .

Definition II.2 We say that (1) is null controllableif for all x0 ∈ Rn there exists an N ∈ N and aninput sequence u = {u[0], . . . , u[N − 1]} such thatxx0,u[N ] = 0.

We say that (1) is reachable if for all xf ∈ Rn

there exists an N ∈ N and an input sequenceu = {u[0], . . . , u[N − 1]} such that x[0] = 0 andxx0,u[N ] = xf .

We say that (1) is controllable if for all x0, xf ∈Rn there exists an N ∈ N and an input sequenceu = {u[0], . . . , u[N − 1]} such that x[0] = x0 andxx0,u[N ] = xf .

B. Classical ResultsConsider the linear system

x[k + 1] = Ax[k] +Bu[k] (2)

with x[k] ∈ Rn being the state variable andu[k] ∈ Rm being the input, together with the inputconstraint

u[k] ∈ U , k ∈ N (3)

where U ⊆ Rm is a solid polyhedral closed cone,i.e. there exists a matrix U ∈ Rl×m for some l ∈ Nsuch that U = {u ∈ Rm | Uu ≥ 0} and U has a non-empty interior. The inequalities in Uu ≥ 0 are to beinterpreted component-wise. The definitions of nullcontrollability, reachability and controllability as inDefinition II.2 are similar for the constrained system(2)-(3) with the understanding that (3) should holdfor the input sequence.

Definition II.3 Let U ⊆ Rm be a set. We define thedual cone of U , U∗, to be

U∗ = {v ∈ Rm | v>u ≥ 0 ∀u ∈ U}

21


The following lemma can be found in [10, 12,15].

Lemma II.4 The constrained system (2)-(3) is nullcontrollable if, and only if, the following implica-tions hold:

λ ∈ C \ {0}, z ∈ Cn,z>A = λz>, z>B = 0

}⇒ z = 0 (4a)

λ ∈ (0,∞), z ∈ Rn,z>A = λz>, B>z ∈ U∗

}⇒ z = 0 (4b)

The constrained system (2)-(3) isreachable/controllable if, and only if, the followingimplications hold:

λ ∈ C, z ∈ Cn,z>A = λz>, z>B = 0

}⇒ z = 0 (5a)

λ ∈ [0,∞), z ∈ Rn,z>A = λz>, B>z ∈ U∗

}⇒ z = 0 (5b)

III. PROBLEM DEFINITION

In this paper we are interested in the null con-trollability/reachability/controllability of bimodalpiecewise linear systems that are of the form

x[k + 1] =

{A1x[k] +B1u[k] y[k] ≥ 0

A2x[k] +B2u[k] y[k] ≤ 0(6)

y[k] = C>x[k]

where x[k] ∈ Rn is the state, u[k] ∈ R is thescalar input, y[k] ∈ R is a variable determining theactive mode at discrete time k ∈ N and the matricesA1, A2 ∈ Rn×n, B1, B2, C ∈ Rn×1 are given.

We assume that the right-hand side of (6) iscontinuous. This is equivalent to the existence ofa vector E ∈ Rn such that

A2 = A1 + EC> and B1 = B2 =: B (7)

Define the transfer functions Gi(z) = C>(zI −Ai)−1B for i = 1, 2. It follows from (7) that

G1(z) ≡ 0 if and only if G2(z) ≡ 0. In the restof the paper we will assume that G1(z) 6≡ 0, asotherwise the system (6) cannot be controllable.Let V∗i be the largest (Ai, B)-invariant subspacecontained in kerC> for i = 1, 2, i.e. V∗i is the largestof the subspaces, Vi, that satisfy (Ai + BFi)Vi ⊆

Vi ⊆ kerC> for some Fi ∈ R1×n, i = 1, 2. Wewill denote the set {Fi ∈ R1×n | (Ai + BFi)V∗i ⊆V∗i ⊆ kerC>} by F∗i . Likewise, let Z∗i be thesmallest (C>, Ai)-invariant subspace that containsimB for i = 1, 2, i.e. Z∗i is the smallest of thesubspaces, Zi, that satisfy (Ai + GiC

>)Zi ⊆ Zi

and imB ⊆ Zi for some Gi ∈ Rn×1, i = 1, 2.See [2, 13, 16] for a detailed discussion on theseparticular subspaces. Since Gi(z) 6≡ 0, i = 1, 2, itis invertible as a rational function. Therefore, Rn

admits the following decomposition

Rn = V∗i ⊕Z∗ifor i = 1, 2. See [11] for the proof of this implica-tion. Due to (7), the following identities hold.

V∗1 = V∗2 =: V∗

Z∗1 = Z∗2 =: Z∗

F∗1 = F∗2 =: F∗

First, we apply the pre-compensating state feedbacku[k] = Fx[k] + v[k] to (6) with F ∈ F∗. Due to (7)we have that

(A1 +BF )|V∗ = (A2 +BF )|V∗

Then, we apply the similarity transformation, x =T−1x.

T =[T1 T2

]

with im T1 = V∗ and im T2 = Z∗. For ease ofexposition, we will not use a different symbol forthe new state vector and we will denote v[k] withu[k]. Then, we obtain the following representationof (6) that is easier to deal with for characterizingnull controllability, reachability and controllability:

x1[k + 1] = Hx1[k] +

{g1y[k] y[k] ≥ 0

g2y[k] y[k] ≤ 0(8a)

x2[k + 1] =

{J1x2[k] + bu[k] y[k] ≥ 0

J2x2[k] + bu[k] y[k] ≤ 0(8b)

y[k] = c>x2[k] (8c)

in which x1[k], g1, g2, f ∈ Rn1 , x2[k], b, c ∈ Rn2 fork ∈ N and H ∈ Rn1×n1 , J1, J2 ∈ Rn2×n2 wheren1 = dimV∗, n2 = dimZ∗ and n1 + n2 = n.

22


Obviously, controllability properties are invari-ant under pre-compensating feedbacks and sim-ilarity transformations. Hence null controllabil-ity/reachability/controllability of (8) is equivalentto null controllability/reachability/controllability of(6). We will now study particular properties ofthe subsystems (8a) and (8b)-(8c) that are usefulfor the main developments. In the next section wewill analyze the subsystem (8a), which is calleda push-pull type of system. The subsystem (8b)-(8c) belongs to a special class of systems which wewill introduce and analyze in Section V. Note thatGi(z) = c>(zI−Ji)−1b 6≡ 0 for i = 1, 2. Let V∗i de-note the largest (Ji, b)-invariant subspace containedin ker c> for i = 1, 2. Then, V∗1 = V∗2 = {0}. LetZ∗i denote the smallest (c>, Ji)-invariant subspacethat contains im b. Then, Z∗1 = Z∗2 = Rn2 . Due to(7), we also have that

g2 = g1 + e1 (9)

J2 = J1 + e2c>

where(e1

e2

)= T−1E as C>T =

[0 c>

].

IV. CONTROLLABILITY OF BIMODALDISCRETE-TIME PUSH-PULL SYSTEMS

A bimodal discrete-time push-pull system with ascalar input is given by

x[k + 1] = Ax[k] +

{B+u[k] u[k] ≥ 0

B−u[k] u[k] ≤ 0(10)

where x[k] ∈ Rn, u[k] ∈ R for k ∈ N and A ∈Rn×n, B+, B− ∈ Rn×1. Note that (8a) is a specialcase of (10) with y being the “input”. Sometimeswe will use the notation

f(u[k]) =

{B+u[k] u[k] ≥ 0

B−u[k] u[k] ≤ 0

Theorem IV.1 Consider the system (10). The fol-lowing statements are equivalent.

(i) (10) is controllable.(ii) (10) is reachable.

(iii) z>AkB+ ≥ 0 and z>AkB− ≤ 0 for all k ≥0⇒ z = 0

(iv) The linear system given by (A, [B+ −B−]) iscontrollable with the input constraint U = R2

+.

Proof:(i)⇒ (iv): Define the discrete-time linear system

x[k + 1] = Ax[k] + [B+ −B−]u[k] (11)

where x[k] ∈ Rn and u[k] ∈ R2+ for k ∈ N. If the

system (10) is controllable, then for any x0, xf ∈ Rn

one can write

xf = Apx0 +

p−1∑

k=0

Akf(u[p− k − 1]) (12)

for some p ∈ N. Define

u[k] =

[|u[k]| 0

]>u[k] ≥ 0

[0 |u[k]|

]>u[k] ≤ 0

Based on this definition one can write (12) as

xf = Apx0 +

p−1∑

k=0

Ak[B+ −B−]u[p− k − 1] (13)

It is clear that u[k] ∈ R2+ for all k ∈ N. One

can observe that (13) is the response of the system(11) to the initial state x0 and the input sequence{u[0], . . . , u[p − 1]}. Thus, if the system (10) iscontrollable, so is the system (11).

(iv)⇒ (iii): If the statement (iv) holds, then forany xf ∈ Rn one can write

xf = B+u1[p]−B−u2[p] + . . .

+Ap−1B+u1[0]− Ap−1B−u2[0]

for some p ∈ N where the sequence{u1[0], u2[0], . . . , u1[p], u2[p]} consists of non-negative entries. Assume that for some z,

z>AkB+ ≥ 0 and z>AkB− ≤ 0

for all k ∈ N. Therefore, it must hold that z>xf ≥ 0for all xf ∈ Rn. Thus, z = 0.

(iii) ⇒ (ii): Let U be the set containing inputsequences, v, of infinite length that satisfy

v = {u[0], u[1], . . .}with u[k] ≥ 0 when k is even and u[k] ≤ 0 whenk is odd. Clearly, U is a convex set. Let R bethe set of states that can be reached from 0 forthe system (10) using input sequences in U. Dueto U being a convex cone, one can observe thatR is a convex cone. Assume that R 6= Rn. Since

23


R 6= Rn andR is a cone, it must hold that the originis in the boundary of R. Note that the origin isalways a reachable state. Thus, due to the supportinghyperplane theorem (see [5]) there exists a z 6= 0such that z>xf ≥ 0 for all xf ∈ R. Let v1, v2 ∈ Ube the input sequences

v1 = {1, 0, 0, 0, . . .}

v2 = {0,−1, 0, 0 . . .}Upon the application of v1, the set of reachablestates is given by

X 1 =⋃

k∈N{AkB+}

Likewise for v2,

X 2 =⋃

k∈N{−AkB−}

Clearly, X 1,X 2 ⊆ R. Therefore, z>AkB+ ≥ 0 andz>AkB− ≤ 0 for all k ≥ 0. However, statement(iii) dictates that z = 0. Thus, we get a contra-diction and hence R = Rn.

Before showing how (ii) implies (i), we wouldlike to demonstrate how (ii) implies (iv), as we willuse this result to complete the proof.

(ii) ⇒ (iv): Following the steps in the proof ofthe implication (i)⇒ (iv), it can be seen that when(10) is reachable, so is (11). Lemma II.4 shows thatwhen (11) is reachable, it is also controllable.

(ii)⇒ (i): Using the real Jordan decomposition,one can transform (10) into[x0[k + 1]x1[k + 1]

]=

[A0 00 A1

] [x0[k]x1[k]

]+

[B0

+

B1+

]u[k] u[k] ≥ 0

[B0−

B1−

]u[k] u[k] ≤ 0

(14)

where A0 ∈ Rn0×n0 has only zero eigenvalues andA1 ∈ Rn1×n1 has only non-zero eigenvalues andis therefore invertible. Obviously, the controllabilityand reachability properties of (10) have not changedunder this similarity transformation. We would like

to show that (14) is null controllable. Once wedo so, we will combine this with the fact that(14) is reachable, which in turn will imply thecontrollability of (14) and hence the controllabilityof (10), thereby completing the proof. To provenull controllability, we will first look into the nullcontrollability of the subsystems of (14) separately,which are given by

x0[k + 1] = A0x0[k] +

{B0

+u[k] u[k] ≥ 0

B0−u[k] u[k] ≤ 0

(15)

x1[k + 1] = A1x1[k] +

{B1

+u[k] u[k] ≥ 0

B1−u[k] u[k] ≤ 0

(16)

Since (14) is reachable, the push-pull system (16)must also be reachable as it is a subsystem of (14).Since we have already shown the equivalence ofthe statements (ii), (iii) and (iv), we have that theconstrained linear system given by

x1[k + 1] = A1x1[k] + [B1+ −B1

−]u[k] (17)

with u[k] ∈ R2+, k ∈ N, is reachable.

To be able to show the null controllability of (16),we first write its time-reversed version,

x1[k] = (A1)−1x1[k+1]−

{(A1)

−1B1

+u[k] u[k] ≥ 0

(A1)−1B1−u[k] u[k] ≤ 0

(18)for which we would like to establish reachability.As before, we have that the reachability of (18)is equivalent to the reachability of the constrainedlinear system given by

x[k] = (A1)−1x[k+1] + [−(A1)

−1B+ (A1)

−1B−]u[k]

(19)with u[k] ∈ R2

+, k ∈ N. Note that (19) is thetime-reversed version of (17). Based on Lemma II.4one can show that when (17) is reachable, (19)is reachable as well. This implies that (18) isreachable. Therefore, (16) is null controllable. Thus,

for any(x0

0

x10

)∈ Rn there exist a q ∈ N and an

input sequence {u[0], . . . , u[q − 1]} such that the

state trajectory satisfies(x0[q]x1[q]

)=

(x0m

0

)for some

x0m ∈ Rn0 . It is clear that for some positive integer l,

(A0)l= 0. Therefore, extending the input sequence

24


with zero values for an additional l steps, we get(x0[q + l]x1[q + l]

)=

(00

), which completes the proof.

Corollary IV.2 The following statements are equiv-alent.

(i) The system (10) is null controllable.(ii) The linear system given by (A, [B+ − B−])

is null controllable with the input constraintU = R2

+.

Proof: (i)⇒ (ii): This follows from the samearguments used in the corresponding part of theproof of Theorem IV.1.

(ii) ⇒ (i): By using the real Jordan decompo-sition we can separate the zero and the non-zeroeigenvalues of A, thus obtaining the subsystems

x0[k + 1] = A0x1[k] + [B0+ −B0

−]u[k] (20)

x1[k + 1] = A1x1[k] + [B1+ −B1

−]u[k] (21)

Since A1 has no zero eigenvalues we can concludebased on Lemma II.4 that when (21) is null control-lable with the input constraint u[k] ∈ R2

+, k ∈ N, itis controllable with the input constraint u[k] ∈ R2

+,k ∈ N. According to Theorem IV.1, this implies(16) is controllable, hence null controllable. Since(A0)

l= 0 for some positive integer l, using the

same arguments as in the proof of Theorem IV.1we conclude that (14) is null controllable and hence(10) is null controllable.

V. OBSERVABILITY AND INVERTIBILITY OF ASPECIAL CLASS OF BIMODAL DISCRETE-TIME

PIECEWISE LINEAR SYSTEMS

Consider the discrete-time piecewise linear sys-tem

x[k + 1] =

{A1x[k] +Bu[k] y[k] ≥ 0

A2x[k] +Bu[k] y[k] ≤ 0(22a)

y[k] = C>x[k] (22b)

where x[k] ∈ Rn, u[k], y[k] ∈ R for k ∈ N andA1, A2 ∈ Rn×n, B,C ∈ Rn×1. We will study aspecial class of the system (22) as we will needthese results later in proving our main result. Inparticular, we adopt the following assumptions.

Assumption V.1 The following statements hold.

1) The transfer functions Gi(z) = C>(zI −Ai)−1B 6≡ 0 for i = 1, 2;

2) The right-hand side of (22) is continuous, i.e.A2 = A1 + EC> for some vector E ∈ Rn;

3) The largest (Ai, B)-invariant subspace con-tained in kerC>, V∗i , is {0} for i = 1, 2.

4) The smallest (C>, Ai)-invariant subspace con-taining im B, Z∗i , is Rn for i = 1, 2.

We would like to point out that the system (8b)-(8c) belongs to this special class of discrete-timepiecewise linear systems.

Corollary V.2 Consider the system (22) and sup-pose Assumption V.1 holds. Then, the followingstatements hold.

1) (C>, Ai) is observable,2) (Ai, B) is controllable,3) C>B = C>AiB = . . . = C>An−2

i B = 0,4) C>An−1

i B 6= 0for i = 1, 2.

Definition V.3 We say that (22) is both observ-able and invertible if for any output sequencey = {y[0], y[1], . . .} of infinite length there existsa unique initial state x0 ∈ Rn and a unique inputsequence of infinite length, {u[0], u[1], . . .}, that areboth compatible with y.

Proposition V.4 Consider the system (22) forwhich Assumption V.1 holds. Then, it is both observ-able and invertible. In particular, the first n elementsof the output sequence {y[0], . . . , y[n−1]} uniquelydetermine the initial state x0.

Proof: Let

ik =

{1 y[k] > 0

2 y[k] ≤ 0k ∈ N

Then one can write for the system (22)

y[0]y[1]

...y[n− 1]

= Qx[0] (23)

where

Q =

C>

C>Ai0...

C>Ain−2 . . . Ai1Ai0

25


Based on the second statement in Assumption V.1one can factorize Q in the following way.

Q =

1 0 0 . . . 00 1 0 . . . 00 f3,2 1 . . . 0... . . .0 fn,2 . . . fn,n−1 1

C>

C>Ai0...

C>Ain−10

The left factor is a lower triangular matrix whoseoff-diagonal terms vary based on the values ofi0, . . . , in−2. The right factor is the observabil-ity matrix generated by (C>, Ai1). Since the pair(C>, Ai1) is observable it follows that Q is nonsin-gular. Thus, for any sequence {y[0], . . . , y[n − 1]}there exists a unique x[0] that produces this outputsequence.

Furthermore, y[n] = C>Ain−1Ain−2 . . . Ai0x[0]+Su[0] where S = C>Ain−1Ain−2 . . . Ai1B. Basedon Corollary V.2, one can show that S =C>Ai1

n−1B 6= 0. Since x[0] is uniquely deter-mined by {y[0], . . . , y[n − 1]}, u[0] is uniquelydetermined by {y[0], . . . , y[n]}. By repeating thesame argument, u[1] is uniquely determined by{y[0], . . . , y[n + 1]} and so on. Thus, (22) is bothobservable and invertible.

VI. MAIN RESULTS

Lemma VI.1 The system (8) is controllable if andonly if the push-pull system (8a) is controllable.

Proof: The necessity of this condition is clear.To show its sufficiency we take two arbitrary

states, (x>10, x>20)> ∈ Rn and (x>1f , x

>2f )> ∈ Rn. We

would like to demonstrate how to steer (x>10, x>20)>

to (x>1f , x>2f )>. From (8b)-(8c) we produce an output

sequence {y[0], . . . , y[n2 − 1]} that is compatiblewith x20 and apply it to (8a). Let x1m = x1[n2]denote the state the system (8a) is steered to uponthe application of this sequence. Since (8a) iscontrollable there must exist r ≥ n2 such thatx1[n2] = x1m and x1[r] = x1f for some sequence{y[n2], . . . , y[r − 1]}. Then, we extend the outputsequence with {y[r], y[r + 1], . . .}, which is chosento be compatible with x2f . Due to Proposition V.4,there is a unique input sequence {u[0], u[1], . . .}compatible with {y[0], y[1], . . .} and necessarily,x2[0] = x20, x2[r] = x2f . Thus we have a uniqueinput that satisfies x1[r] = x1f , x2[r] = x2f with

x1[0] = x10, x2[0] = x20, which completes theproof.

Corollary VI.2 The system (8) is null controllableif and only if the push-pull system (8a) is nullcontrollable.

Proof: The proof is obtained in exactly thesame way as in the proof of Lemma VI.1 by taking(x>1f , x

>2f )> = 0.

We are now in a position to present the mainresult of this paper.

Theorem VI.3 Consider the system (6) such thatC>(zI − A1)−1B1 6≡ 0. Then the following state-ments hold.

1) The system (6) is null controllable if and onlyif the following implications hold:

λ ∈ C \ {0}, z ∈ Cn,z>Ai = λz>, z>B = 0

i = 1, 2

⇒ z = 0 (24a)

λ ∈ (0,∞), z ∈ Rn,[z> wi

] [Ai − λI BC 0

]= 0

w1 ≤ 0, w2 ≥ 0, i = 1, 2

⇒ z = 0

(24b)2) The system (6) is controllable if and only if the

following implications hold.

λ ∈ C, z ∈ Cn,z>Ai = λz>, z>B = 0

i = 1, 2

⇒ z = 0 (25a)

λ ∈ [0,∞), z ∈ Rn,[z> wi

] [Ai − λI BC 0

]= 0

w1 ≤ 0, w2 ≥ 0, i = 1, 2

⇒ z = 0

(25b)

Proof: We will show the proof of the secondstatement only. The proof of the first is obtained ina similar way.

From Lemma VI.1, we know that the controllabil-ity of the push-pull system (8a) is equivalent to thecontrollability of (8) and thus to the controllabilityof (6). Using Lemma II.4 and Theorem IV.1 we can

26


give a characterization of the controllability of (8a),which is as follows

λ ∈ C, z1 ∈ Cn1 ,z>1 H = λz>1 , z

>1 g1 = z>1 g2 = 0

}⇒ z1 = 0

(26a)λ ∈ [0,∞), z1 ∈ Rn1 ,

z>1 H = λz>1 , z>1 g1 ≥ 0, z>1 g2 ≤ 0

}⇒ z1 = 0

(26b)Our aim is now to show the equivalence of (26) and(25). Observe that the condition (26a) is equivalentto the controllability of (H, [g1 g2]) as a linear sys-tem without constraints. Due to (7), (25a) is equiva-lent to the controllability of (A1, [B E]) as a linearsystem without constraints. Therefore we need todemonstrate that controllability of (H, [g1 g2]) andcontrollability of (A1, [B E]), as linear systemswithout constraints, are equivalent. For this purposewe will use the Hautus’ test, which states that thecontrollability of (A1, [B E]) as a linear systemwithout constraints is equivalent to the implication

λ ∈ C, z ∈ Cn z>[A1 − λI B E] = 0⇒ z = 0(27)

Condition (27) can also be written for the system (8)since the controllability properties do not change un-der similarity transformation and pre-compensatingfeedback. Thus, instead of (27) we will use thefollowing implication

λ ∈ C, z ∈ Cn

z>[H − λI g1c

> 0 e1

0 J1 − λI b e2

]= 0

⇒ z = 0

(28)First we take z> = (z>1 , z

>2 ), then observe that when

[z>1 z>2

] [H − λI g1c> 0 e1

0 J1 − λI b e2

]= 0 (29)

for some λ ∈ C, it holds that[z>1 g1 z>2

] [ c> 0J1 − λI b

]= 0 (30)

We have that the largest (J1, b)-invariant subspacecontained in ker c>, V∗1 , is {0}. Then it follows fromProposition II.3 in [9] that z>1 g1 = 0 and z>2 = 0.Substituting z>2 = 0 into (29), we obtain z>1 e1 = 0,thus z>1 g2 = 0 due to (9). Hence, assuming that(26a) holds, it now dictates that z1 = 0 and hencez = 0. Therefore, (A1, [B E]) is controllable asa linear system if (H, [g1 g2]) is controllable as

a linear system. To show how the controllabil-ity of (A1, [B E]) implies the controllability of(H, [g1 g2]), we take z> = (z>1 , z

>2 ) = (z>1 , 0) in

(29). Then, assuming that (28) holds, we recoverthe implication (26a).

To prove the equivalence of the implications(25b) and (26b) we will again use the transformedsystem (8). We take z> = (z>1 , z

>2 ). Then, the

equality in the implication (25b),

[z>1 z>2 wi

]H − λI gic

> 00 Ji − λI b0 c> 0

= 0, i = 1, 2

(31)can be rewritten as

z>1 H = λz>1

[z>2 w1 + z>1 g1]

[J1 − λI bc> 0

]= 0

[z>2 w2 + z>1 g2]

[J2 − λI bc> 0

]= 0

Since the largest (Ji, b)-invariant subspace con-tained in ker c>, V∗i , is {0} for i = 1, 2, it followsfrom Proposition II.3 in [9] that z2 = 0, z>1 g1 =−w1 ≥ 0 and z>1 g2 = −w2 ≤ 0. Due to (26b),z1 = 0, hence z = 0. We have shown that (26b)implies (25b). Vice versa, if (25b) holds, by takingz2 = 0, w1 = −z>1 g1 ≤ 0 and w2 = −z>1 g2 ≥ 0 in(25b) for the transformed system (8) the implication(26b) follows. Thus, we have proven the equiva-lence of the implications (25b) and (26b), whichcompletes the proof.

VII. CONCLUSION

In this paper we presented algebraic characteriza-tions of the controllability of a class of discrete-timebimodal piecewise linear systems. By using geomet-ric control theory we showed that the controllabilityproblem is equivalent to the controllability of aparticular subsystem of the original system, whichis in the form of a so-called push-pull system.Exploiting the results on the controllability of input-constrained linear systems we derived conditions forcontrollability of push-pull systems leading to ourmain result. Future research in this topic could bethe characterization of controllability for discrete-time piecewise linear systems with multiple inputsand more than two modes.

27


REFERENCES

[1] A. Arapostathis and M.E. Broucke. Stabilityand controllability of planar, conewise linearsystems. Syst. Control Lett., 56(12):150–158,2007.

[2] G.B. Basile and G. Marro. Controlled andConditioned Invariants in Linear System The-ory. Prentice Hall, Englewood Cliffs, NJ,1992.

[3] A. Bemporad, G. Ferrari-Trecate, andM. Morari. Observability and controllabilityof piecewise affine and hybrid systems. IEEETrans. Automat. Control, 45(10):1864–1876,2000.

[4] V.D. Blondel and J.N. Tsitsiklis. Complexityof stability and controllability of elemantaryhybrid systems. Automatica, 35(3):479–490,1999.

[5] S. Boyd and L. Vandenberghe. Convex Opti-mization. Cambridge University Press, 2004.

[6] R.F. Brammer. Controllability in linear au-tonomous systems with positive controllers.SIAM J. Control, 10(2):329–353, 1972.

[7] B. Brogliato. Some results on the controllabil-ity of planar variational inequalities. Systemsand Control Letters, 55(1):65–71, 2005.

[8] M.K. Camlibel, W.P.M.H. Heemels, and J.M.Schumacher. On the controllability of bimodalpiecewise linear systems. In Hybrid Systems;Computation and Control, volume 2993 ofLecture Notes in Computer Science, pages571–614. Springer Berlin / Heidelberg, 2004.

[9] M.K. Camlibel, W.P.M.H. Heemels, and J.M.Schumacher. Algebraic necessary and suf-ficient conditions for the controllability ofconewise linear systems. IEEE Transactionson Automatic Control, 53(3):762–774, 2008.

[10] M.E. Evans. The convex controller: controlla-bility in finite time. International Journal ofSystems Science, 16(1):31–47, 1985.

[11] M.L.J. Hautus and L.M. Silverman. Systemstructure and singular control. J. OptimizationTheory Appl., 50(2):313–329, 1986.

[12] W.P.M.H. Heemels and M.K. Camlibel. Nullcontrollability of discrete-time linear systemswith input and state constraints. In 47th IEEECDC Conference, pages 3487–3492, 2008.

[13] H.L. Trentelman and A.A. Stoorvogel and M.Hautus. Control Theory for Linear Systems.Communications and control engineering se-ries. Springer, 2002.

[14] K.K. Lee and A. Arapostathis. On the control-lability of piecewise linear hypersurface sys-tems. Syst. Control Lett., 55(1):65–71, 1987.

[15] K.S. Nguyen. On the null-controllability oflinear discrete-time systems with restrainedcontrols. J. Optimization Theory Appl.,50(2):313–329, 1986.

[16] W.M. Wonham. Linear Multivariable Control:A Geometric Approach. Springer, 3rd edition,1985.

[17] J. Xu and L. Xie. Null controllability ofdiscrete-time planar bimodal piecewise linearsystems. International Journal of Control,78(18):1486–1496, 2005.

[18] J. Xu and L. Xie. Controllability and reach-ability of discrete-time planar bimodal piece-wise linear systems. In Proc. of the AmericanControl Conference, pages 4387–4392, 2006.

28

BIBLIOGRAPHY

[BBS03] G. Balas, J. Bokor, and Z. Szabo. Invariant Subspaces for LPV Systems andTheir Application. IEEE Transactions on Automatic Control, 48:2065–2069,2003.

[BM92] G.B. Basile and G. Marro. Controlled and Conditioned Invariants in LinearSystem Theory. Prentice Hall, Englewood Cliffs, NJ, 1992.

[Cam07] M.K. Camlibel. Popov–belevitch–hautus type controllability tests for linearcomplementarity systems. Systems and Control Letters, 56(5):381–387, 2007.

[CHS08a] M.K. Camlibel, W.P.M.H. Heemels, and J.M. Schumacher. Algebraic neces-sary and sufficient conditions for the controllability of conewise linear systems.IEEE Transactions on Automatic Control, 53(3):762–774, 2008.

[CHS08b] M.K. Camlibel, W.P.M.H. Heemels, and J.M. Schumacher. A full characteri-zation of stabilizability of bimodal piecewise linear systems with scalar inputs.Automatica, 44(5):1261–1267, 2008.

[H.L02] H.L. Trentelman and A.A. Stoorvogel and M. Hautus. Control Theory forLinear Systems. Communications and control engineering series. Springer,2002.

[IKGGN81] A. Isidori, A.J. Krener, C. Gori-Giorgi, and S. Nonaco. Nonlinear decouplingvia feedback: a different geometric approach. IEEE Trans. Autom. Control,26(2):331–345, 1981.

[JvdS02] A.A. Julius and A.J. van der Schaft. The Maximal Controlled Invariant Setof Switched Linear Systems. In Proc. of the 41st IEEE Conf. on Decision andControl, pages 3174–3179, Las Vegas, Nevada USA, 2002.

BIBLIOGRAPHY

[NVdS82] H. Nijmeijer and A.J. Van der Schaft. Controlled invariance for nonlinearsystems. IEEE Trans. Autom. Control, 27(4):904–914, 1982.

[Ots10] N. Otsuka. Disturbance decoupling with quadratic stability for switched linearsystems. Systems and Control Letters, 2010.

[SBS03] G. Stikkel, J. Bokor, and Z. Szabo. Disturbance Decoupling Problem withStability for LPV Systems. In European Control Conference, Cambridge, U.K.,2003.

[Won85] W.M. Wonham. Linear multivariable control: A geometric approach. Springer,3rd edition, 1985.

[ZCL05] L. Zhang, D. Cheng, and C. Li. Disturbance decoupling of switched nonlinearsystems. IEE Proceedings, 152(1):49–54, 2005.

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