Geometric Control Theory for Hybrid Systems

Transcription

1 Geometric Control Theory for Hybrid Systems Evren Yurtseven H&NS Master s thesis Coach(es): Committee: prof.dr.ir. W.P.M.H. Heemels (supervisor) dr. M.K. Camlibel dr. M.K. Camlibel prof.dr.ir. W.P.M.H. Heemels prof.dr. J.M. Schumacher dr. S. Weiland Eindhoven University of Technology Department of Mechanical Engineering Hybrid & Networked Systems Eindhoven, August, 2010

2 TABLE OF CONTENTS 1. Introduction Disturbance Decoupling of Switched Linear Systems Controllability of Bimodal Discrete-time Piecewise Linear Systems Bibliography

3 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), characterization of sets of particular interest including reachable and unobservable sets, geometric control theory proves to be an extremely powerful tool. Solutions to various problems for smooth nonlinear systems, based on a geometric approach, are also available in the literature [IKGGN81, NVdS82]. Interestingly, geometric control theory for hybrid systems is largely absent in the literature, despite the enourmous attention this class of dynamical systems has received over the past decades. This thesis focuses on the development of geometric control theory for certain classes of hybrid systems. For this purpose, geometric control theory will be used to solve DDPs and controllability problems for switched linear and piecewise linear systems, respectively. 1.1 Problem Formulation Although many geometric control theoretic results are available for linear and smooth nonlinear systems, when one leaves the realm of linear and smooth nonlinear systems and looks into hybrid dynamical systems, the availability and usage of geometric control theory become quite limited. There are, however, some results available for interesting problems 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 studied in which both the switching (discrete control input) and the continuous input can be manipulated as control inputs. In the context of linear parameter-varying (LPV) systems

4 CHAPTER 1. INTRODUCTION various parameter-varying (controlled and conditioned) invariant subspaces are introduced in [BBS03] and various algorithms are presented to compute them. Based on [BBS03] first results in the direction of applying these concepts to solve DDP in LPV systems are given in [SBS03] using parameter-dependent state feedback. Recently, in [Ots10] DDP for SLSs using mode-dependent state feedback control is solved. In [ZCL05] local versions of DDP with respect to continuous disturbances in switched nonlinear systems are solved using both mode-dependent and mode-independent state feedback. It should be pointed out that the works mentioned so far prove their main results by first extending fundamental concepts of geometric control theory for LTI systems such as A-invariance, controlled invariance and so on, to the class of hybrid systems they are interested in and then using these new extended concepts in their solutions to the problems. There exist also studies in the literature that provide solutions to certain problems for particular classes of hybrid systems, that heavily make use of geometric control theory for LTI systems in proving their main results. In [CHS08a], algebraic necessary and sufficient conditions for the controllability of continuous-time conewise linear systems are given. This work solves a seemingly very complicated problem for an important class of hybrid systems by first transforming the hybrid system of interest into a much more manageable form using geometric control theory for LTI systems and proving the controllability of the original system to be equivalent to the controllability of a particular subsystem of the original system. Also in solving other problems defined for hybrid systems, the usage of geometric control 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 geometric control theory will play an essential role: Find conditions under which a switched linear system s output is or can be made decoupled 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

5 CHAPTER 1. INTRODUCTION 1.2 Contributions of This Thesis The first contribution of this thesis will be to give complete characterizations of disturbance decoupling in switched linear systems, that is under what conditions is a given SLS disturbance decoupled or can it be made disturbance decoupled by using, either modedependent or mode-independent, static state feedback. We do so by first extending the building blocks of geometric control theory for LTI systems such as A-invariance, controlled invariance, conditioned invariance et cetera, to switched linear systems. Then, making use of these new concepts we present the main results and the algorithms with which our results can be put to use. We consider three different versions of disturbance decoupling: with respect 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-motivated and well-known problem, the second and the third are considered for the first time in this thesis. Disturbance decoupling with respect to the switching signal is a desirable property when the switching signal cannot be manipulated and each mode of the SLS corresponds to a certain failure scenario and particular outputs of the system are desired not to be influenced by these scenarios. Disturbance decoupling with respect to both the switching signal and the continuous disturbances is a property that we will show to be strongly related to disturbance decoupling of piecewise linear systems with respect to the continuous disturbances. Before concluding this part of the thesis, we will also provide geometric necessary and sufficient conditions for the solvability of the disturbance decoupling problem with respect to the continuous disturbances by mode-dependent dynamic measurement feedback. These contributions are presented in Chapter 2. An abridged version of the chapter is accepted for publication in the proceedings of the IEEE Conference on Decision and Control 2010 in Atlanta, USA. The complete contents will further be sent for journal publication. The second part of this thesis is devoted to providing algebraic necessary and sufficient conditions for the null controllability, reachability and controllability of a class of bimodal discrete-time piecewise linear systems. The class is characterized by continuous right-hand side and scalar input whereas the state can be of any dimension. By using geometric control theory for LTI systems we will show that the controllability problem is equivalent to the controllability of a particular subsystem of the original system, which is in the form of a socalled push-pull system. Exploiting the results on the controllability of input-constrained linear systems we will derive conditions for the controllability of push-pull systems leading to our main results. These contributions are explained in detail in Chapter 3. The results will be submitted for journal publication as well. 3

6 CHAPTER 2 Disturbance Decoupling of Switched Linear Systems At Yarragi Essegin ziki DISTURBANCE DECOUPLING OF SWITCHED LINEAR SYSTEMS Abstract In this paper we consider disturbance decoupling problems for switched linear systems. We will provide necessary and sufficient conditions for three different versions of disturbance decoupling, which differ based on which signals are considered to be the disturbance. In the first version the continuous exogenous input is considered as the disturbance, in the second the switching signal and in the third both of them are considered as disturbances. The latter instance of the problem is relevant for disturbance decoupling of piecewise linear systems, as we will show. The solutions of the three disturbance decoupling problems will be based on geometric control theory for switched linear systems and will entail both modedependent and mode-independent static state feedback. In addition, a mode-dependent dynamic measurement feedback based solution of the disturbance decoupling problem with respect to the continuous disturbances will be provided. Index Terms Disturbance decoupling, switched linear systems, invariant subspaces, geometric control theory I. INTRODUCTION Geometric control theory for linear time-invariant systems has a long and rich history, as is evidenced by the availability of various textbooks on the topic [2, 6, 18]. In particular, for solving disturbance decoupling problems (DDPs) for linear systems the usage of geometric theory turned out to be extremely powerful. Also solutions to various problems for smooth nonlinear systems using a nonlinear geometric approach are available in the literature, see, e.g., [3, 7, 11]. However, outside the context of linear or smooth nonlinear control systems, the number of results on DDPs is rather limited. This is specifically surprising for hybrid dynamical systems or subclasses such as switched systems [9], as they have been studied extensively over the last two decades. Only a few results are available on geometric control theory and solutions to DDPs for switched systems. In [8] the largest controlled invariant set for switched linear systems (SLSs) is studied in which both the switching (discrete control input) and the continuous input can be manipulated as control inputs. In the context of linear parametervarying (LPV) systems various parameter-varying (controlled and conditioned) invariant subspaces are introduced in [1] and various algorithms are presented to compute them. Based on [1] first results in the direction of applying these concepts to DDPs with respect to continuous disturbances are given in [14] using parameter-dependent state feedback. Recently, in [12] the DDP for switched linear systems using mode-dependent state feedback control is solved and combined with results on quadratic stabilizability [4, 17]. Also for reachability

7 CHAPTER 2. DISTURBANCE DECOUPLING OF SWITCHED LINEAR SYSTEMS problems for SLSs invariant subspaces played an important role. In particular, in [15, 16] it was shown that the reachable set of an SLS is equal to the smallest controlled invariant set containing the subspace spanned by all the input matrices of the individual subsystems. For switched nonlinear systems, the only work known to the authors is [19]. In [19] local versions of DDP with respect to continuous disturbances are solved using both mode-dependent and mode-independent static state feedback. The objectives of this paper are to provide complete answers to DDPs for SLSs using various new controlled and conditioned invariant subspaces for SLSs. We first assume that the control input is absent and analyze the disturbance decoupling properties of a SLS. In contrast with the above mentioned references, which only study disturbance decoupling (DD) with respect to continuous exogenous disturbances, we consider three variants of DD as will be formally defined in Section II, namely DD with the disturbances being either (i) the exogenous continuous disturbances, (ii) the switching signal, or (iii) both the continuous disturbances and the switching signal. We will show that the latter instance of the problem is relevant for disturbance decoupling of piecewise linear systems. In Section III we will fully characterize these three DD properties. In Section IV we will add continuous control inputs to the problem and solve the DDP using state feedback controllers that may be both modedependent and mode-independent. We will allow for direct feedthrough terms of the control input into the to-be-decoupled output variable, a situation that was not considered in the aforementioned references. Note also that variant (ii) and (iii) are in the present paper for the first time. In Section V we provide algorithms to compute the largest common controlled invariant subspaces using both mode-dependent and mode-independent feedback, which can be used to verify the characterizations of the solvability of the DDPs provided in Section IV and also to construct feedbacks solving the DDPs. Before stating the conclusions, we will provide also the solution to the DDP using mode-dependent output-based dynamic feedback controllers. II. PROBLEM FORMULATION A switched linear system (SLS) without control inputs is described by the following equations ẋ(t) = A σ(t) x(t) + E σ(t) d(t) z(t) = H σ(t) x(t) (1a) (1b) where x(t) R nx, d(t) R n d and z(t) R n z denote the state variable, the exogenous input and output, respectively, at time t R + := [0, ). For each i {1,..., M}, A i R nx nx, E i R nx n d and H i R nz nx are matrices describing a linear subsystem. Switching between subsystems (modes) is orchestrated by the switching signal σ. We assume that σ lies in the set S of right-continuous functions R + {1,..., M} that are piecewise constant with a finite number of discontinuities in a finite length interval. Particular switching signals are the constant ones σ i S, i = 1,..., M, which are defined as σ i (t) = i for all t R +. We assume that the exogenous signal d is locally integrable, i.e. d L loc 1 (R +, R n d ). Clearly, the SLS (1) has for each d L loc 1 (R +, R n d ), initial condition x(0) = x0 R nx and switching signal σ S a unique solution x x0,σ,d and a corresponding output z x0,σ,d. We will consider the following three variants of disturbance decoupling, which differ based on which signals are considered to be the disturbance. Definition II.1 The SLS (1) is called disturbance decoupled (DD) with respect to d if z x0,σ,d 1 = z x0,σ,d 2 (2) for all x 0 R nx, σ S and d 1, d 2 L loc 1 (R +, R n d ). Definition II.2 The SLS (1) is called disturbance decoupled (DD) with respect to σ if z x0,σ 1,d = z x0,σ 2,d (3) for all x 0 R nx, σ 1, σ 2 S and d L loc 1 (R +, R n d ). Definition II.3 The SLS (1) is called disturbance decoupled (DD) with respect to both σ and d if z x0,σ 1,d 1 = z x0,σ 2,d 2 (4) for all x 0 R nx, σ 1, σ 2 S and d 1, d 2 L loc 1 (R +, R n d ). 5

8 CHAPTER 2. DISTURBANCE DECOUPLING OF SWITCHED LINEAR SYSTEMS Remark II.4 DD with respect to continuous disturbances d is commonly studied and well motivated within the context of linear systems [2, 6, 18] and nonlinear systems [3, 7, 11]. DD with respect to σ or both σ and d is typical for switched systems as studied here. These variants of DD are relevant in situations where the switching signal σ is uncontrolled and we would like to design a (closedloop) system in which σ does not influence certain important performance variables z. In particular, when σ models certain faults in the system such as breakage of pipes, actuators, sensors, etc. and thus each mode i {1,..., M} corresponds to one of these discrete fault scenarios, it would be desirable to decouple z from σ (and possibly other continuous disturbances d). Hence, as such these variants of DD are fundamental problems in the area of faulttolerant control [10]. Another motivation for DD with respect to both σ and d are DD problems for piecewise linear systems, see Section III-D below. III. DISTURBANCE DECOUPLING CHARACTERIZATIONS To provide characterizations for the above mentioned DD properties, we need to introduce some concepts and a technical lemma. We call a subspace V R nx A-invariant for A R nx nx, if AV V. We call a subspace {A 1,..., A M }- invariant A i R nx nx, i = 1,..., M, if A i V V for all i = 1,..., M. Given a matrix A R nx nx and a subspace W R nx, let A W denote the smallest A invariant subspace that contains W, i.e., A W = W + AW A nx 1 W (5) For a set of matrices {A 1,..., A M } and a subspace W, the smallest {A 1,..., A M }-invariant subspace that contains W, denoted by V s (W), is uniquely defined by the following three properties: 1) W V s (W); 2) V s (W) is {A 1,..., A M }-invariant; 3) For any subspace V being {A 1,..., A M }- invariant with W V, it holds that V s (W) V. Calculation of V s (W) can be done using the recurrence relation V 1 = W; V i+1 = A j V i j=1 Since V i V i+1 for i = 1, 2,... and V p = V p+1 implies V q = V p for all q p, it holds that V q = V s (W) for all q n x, see, e.g., [1, 16]. The reachable set of (1) is defined as R := {x 0,σ,d (T ) T R +, σ S and d L loc 1 (R +, R n d )} being the set of states that can be reached from the origin in finite time for some σ and d. Lemma III.1 For the SLS (1), R = V s ( im E i ) i=1 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 d if and only if an {A 1,..., A M }-invariant subspace V exists such that H 1 im E i V ker. (6) H M i=1 Proof: Once σ is fixed the SLS (1) reduces to a linear time-varying system. Using the resulting linearity properties shows that (17) is equivalent to z 0,σ,d = 0 σ S d L loc 1 (R +, R n d ) (7) Necessity: Since each x R can be reached for some σ and d, i.e. x = { x 0,σ,d (T ) for some T R +, we can take σ i σ(t) 0 t < T (t) = for i = σ i (t) t T 1,..., M in (7) to get that H i x = z 0,σ i,d(t ) = 0. Hence (6) holds for V = R due to Lemma III.1. Sufficiency: Since x 0,σ,d (t) R for all t R +, it follows from Lemma III.1 that x 0,σ,d (t) V for all t R + as R = V s ( im E i ) V. Hence, due i=1 to (6), z 0,σ,d (t) = H σ(t) x 0,σ,d (t) = 0 and thus (7) is satisfied. 6

9 CHAPTER 2. DISTURBANCE DECOUPLING OF SWITCHED LINEAR SYSTEMS B. Disturbance decoupling with respect to σ In this section we consider DD with respect to σ in which we will use N i being the unobservable subspace corresponding to the pair (H i, A i ), i.e., N i = ker H i ker H i A i... ker H i A nx 1 i. Note that N i is also the largest A i -invariant subspace contained in ker H i, i {1,..., M}. Theorem III.3 The SLS (1) is DD with respect to σ if and only if for all (i, j) {1,..., M} {1,..., M} the following conditions hold (i) H i = H j ; (ii) N i = N j ; (iii) im(a i A j ) N i ; (iv) im(e i E j ) N i. Proof: Necessity: To prove the necessity of the conditions, take σ 1 = σ i and σ 2 = σ j. Let d(t) = d 0 for all t R +. Then, one gets due to DD with respect to σ that and hence z x0,σ i, d = z x0,σ j, d H i e Ait x 0 + ( t H 0 ie Ai(t s) E i ds)d 0 H j e Ajt x 0 + ( (8) t H 0 je Aj(t s) E j ds)d 0 for all t R +. Since x 0 and d 0 are both arbitrary, one gets H i A k i = H j A k j (9) H i A k i E i = H j A k j E j (10) for all k 0 by differentiating (8) with respect to time and evaluating at t = 0. Condition (i) follows from (9) for k = 0, and (ii) from k = 0, 1,..., n x 1. To see that (iii) holds, note that H i A l i(a i A j ) = H i A l+1 i H i A l ia j (9) = H i A l+1 j H i A l ia j = H i (A l j A l i)a j (9) = 0 for all l. For condition (iv), observe that H i A l ie i (10) = H j A l je j (9) = H i A l ie j and hence H i A l i(e i E j ) = 0 for all l 0. Sufficiency: First, we will show that H i A k i = H j A k j (11) for all i, j {1, 2,..., M} {1, 2,..., M} and k 0 by induction. Note that (11) holds for k = 0 due to condition (i). Assume that it holds for some k. Then H i A k+1 i H j A k+1 j = H i A k+1 i H i A k i A j = H i A k i (A i A j ) Also note that (iii) = 0. H i A k i E i H j A k j E j (11) = H i A k i E i H i A k i E j = H i A k i (E i E j ) (iv) = 0 for all k 0. Thus, we get H i A k i E i = H j A k j E j (12) for all i, j {1, 2,..., M} {1, 2,..., M} and k 0. Now, let z = z x0,σ 1,d z x0,σ 2,d (13) x = x x0,σ 1,d x x0,σ 2,d (14) for some x 0, σ 1, σ 2, and d. It follows from (11) for k = 0 that z(t) = H σ1(t) x(t). By differentiating and using (11) for k = 1 and (12) for k = 0, one gets z(t) = H σ1(t)a σ1(t) x(t) for all t R +. Repeating the same argument, one obtains z (k) (t) = H σ1(t)a k σ 1(t) x(t) for all k 0 and for all t R +. Due to (11) this yields z (k) (t) = H i A k i x(t) for all t R +, in which i {1,..., M} can be selected arbitrarily. Then, there exists a nonzero polynomial p(λ) (e.g., the characteristic polynomial of A i for any i) such that p( d dt ) z = 0 7

10 CHAPTER 2. DISTURBANCE DECOUPLING OF SWITCHED LINEAR SYSTEMS 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, z x0,σ 1,d = z x0,σ 2,d for all x 0, σ 1, σ 2, 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 respect to σ if and only if there exists an {A 1,..., A M }- invariant subspace V such that for all (i, j) {1,..., M} {1,..., M} (i) H i = H j (ii) im (A i A j ) V ker H i (iii) im (E i E j ) V Proof: Necessity: The necessity of these conditions follows directly from Theorem III.3 by taking V = N i. Sufficiency: As N i is the largest A i -invariant subspace that is contained in ker H i for all i {1,..., M}, condition (ii), together with the {A 1,..., A M }-invariance of V, implies that V N i. This fact together with (i) implies that condition (i), (iii) and (iv) of Theorem III.3 are satisfied. To show that also (ii) of Theorem III.3 is satisfied, we observe that statement (ii) of this theorem implies that H i A i = H i A j for all (i, j) {1,..., M} {1,..., M}. Due to im (A i A j ) V and V being {A 1,..., A M }-invariant, it must hold that A k i im (A i A j ) V ker H i for all k N and thus H i A k+1 i = H i A k i A j for all i, j. We will now prove that H i A k i = H j A k j (15) for all k N using induction. Clearly, it holds for k = 0. Suppose it holds for k, then H j A k+1 j (15) = H i A k i A j = H i A k+1 i Hence, (15) holds for all k N and thus N i = N j for all i, j. As such, we recovered the conditions of Theorem III.3, which completes the proof. C. Disturbance decoupling with respect to σ and d Using the above results, we will characterize DD with respect to σ and d now. Lemma III.5 The SLS (1) is DD with respect to both σ and d if and only if it is DD with respect to σ and DD with respect to d. Proof: The necessity of the conditions follows by taking, first σ 1 = σ 2 in (4), which yields (17) and secondly, d 1 = d 2 in (4), which gives (3). The sufficiency follows from z x0,σ1,d 1 = z x0,σ1,0 + z 0,σ1,d1 (17) = z x0,σ1,0 + z 0,σ1,d2 (3) = z x0,σ2,0 + z 0,σ2,d2 = z x0,σ2,d 2 where in the first and the last equalities we used the linearity properties of the SLS (1) for a fixed σ. Theorem III.6 The following statements are equivalent: 1) The SLS (1) is DD with respect to σ and d. 2) The conditions (i) H i = H j = H, (ii) N i = N j = N, (iii) im(a i A j ) N, and (iv) ime i N hold for all (i, j) {1,..., M} {1,..., M}. 3) There exists an {A 1,..., A M }-invariant subspace V such that (i) H i = H j = H, (ii) im(a i A j ) V ker H, and (iii) ime i V for all (i, j) {1,..., M} {1,..., M}. Proof: (1) (2): According to Lemma III.5, the hypotheses of Theorem III.2 and Theorem III.3 are necessarily true when the SLS (1) is DD with respect to σ and d. For all x V and for all i = 1,..., M one can write HA k i x = 0 for all k N. This shows that x N. Therefore V N. Combining the conditions of Theorem III.2 and Theorem III.3 gives (2). 8

11 CHAPTER 2. DISTURBANCE DECOUPLING OF SWITCHED LINEAR SYSTEMS (2) (3): Since N i is an A i -invariant subspace contained in ker H and N i = N j = N, one can write A i N N for all i = 1,..., M. Thus, it follows that N is an {A 1,..., A M }-invariant subspace contained in ker H. (3) (1): It follows from (3) that im E i i=1 V ker H. This recovers the condition of Theorem III.2. One can also recover the conditions of Theorem III.3 by following the steps shown in the sufficiency part of Corollary III.4. Based on Lemma III.5, (1) follows. D. Disturbance decoupling of piecewise linear systems Based on the above results we can show the importance of DD with respect to σ and d for disturbance decoupling of piecewise linear (PWL) systems of the form ẋ(t) = A i x(t) + E i d(t) if x(t) X i z(t) = Hx(t) (16a) (16b) in which X i R nx, i {1,..., M}, are polyhedral regions with non-empty interiors that form a partitioning of the state space R nx, i.e., M i=1 X i = R nx and the interiors of different regions have an empty intersection. Since the right-hand side of a PWL system can be discontinuous, solutions will be interpreted in the sense of Filippov [5], which includes possible sliding motions at the boundaries of the regions. See [5] for more details and exact definitions of Filippov solutions. To avoid any ambiguity in the definition of the output as in (16b) during sliding motions we assumed that the output matrix H is independent of i. Note that this is a necessary condition for the SLS (1) corresponding to (16) to be DD with respect to both σ and d. Definition III.7 The PWL system (16) is called disturbance decoupled (DD) with respect to d if for all x 0 R nx, d 1, d 2 L loc 1 (R +, R n d ), it holds that z 1 = z 2 (17) where z j = Hx j, j = 1, 2 and x j, j = 1, 2, is a Filippov solution to (16) for initial state x(0) = x 0 and disturbance input d j, j = 1, 2. Note that due to possible non-uniqueness of Filippov solutions multiple solutions might correspond to one initial condition and one disturbance input. Theorem III.8 The PWL (16) system is DD with respect to d, if the corresponding SLS (1) is DD with respect to σ and d. Proof: Let x j, j = 1, 2, be Filippov solutions to (16) for x 0 and d j, j = 1, 2, and z j = Hx j, j = 1, 2. Consider the absolutely continuous functions x = x 1 x 2 and z = z 1 z 2. Due to the definition of Filippov solutions we have that for almost all t R + there exist α i, β j depending on t such that α i = 1, β j = 1, α i, β j 0 for all i i=1 j=1 {1,..., M}, j {1,..., M} and x(t) = ẋ 1 (t) ẋ 2 (t) = α i (A i x 1 (t) + E i d 1 (t)) i=1 β j (A j x 2 (t) + E j d 2 (t)) j=1 Equation (18) can be rewritten as x(t) = β i A i (x 1 (t) x 2 (t)) + α i E i d 1 (t)+ with i=1 i=1 j=1 i=1 i=1 (18) β j E j d 2 (t) + δ ij (A i A j )x 1 (t) α i β i = (19) (δ ij δ ji ) (20) j=1 For convenience, we take δ ii = 0 for all i = 1,..., M. To show that (20) holds, we write (20) in vector notation using δ = ( δ δ 1M δ δ 2M... δ M1... ) 9

12 CHAPTER 2. DISTURBANCE DECOUPLING OF SWITCHED LINEAR SYSTEMS which results in the following system of linear equations. α 1 β 1 Q δ =. = v (21) α M β M where Q R M (M 11) is a suitably defined matrix. For a solution to exist to (21), it must hold that imv imq (imq) = ker Q (imv). 1 By construction of Q, ker Q = span{ 1. }. 1 1 Since (α i β i ) = 0, span{ 1. } (imv). i=1 1 Therefore a solution for δ always exists, which proves that equation (18) can always be written in the form of (19). Since the SLS (1) is DD with respect to d and σ, we have that A k l im(a i A j ) ker H ime i ker H HA k i = HA k j for all (i, j, l) {1,..., M} {1,..., M} {1,..., M} and for all k N. Therefore, multiplying (19) with H gives for almost all t R + z(t) = H x(t) = HA i x(t) (22) Differentiating (22) with respect to t and repeating the arguments above, we obtain z (k) (t) = HA k i x for all k N. Similar to the arguments used in the proof of Theorem III.3, one can conclude that z(t) = 0 for all t R +. This theorem demonstrates the relevance of DD with respect to σ and d for SLS in the context of DD to d for PWL systems. As such, when control inputs u are present, disturbance decoupling problems (DDPs), i.e., designing controllers that render the closed-loop SLS DD with respect to σ and d, can be used also for solving DDPs of PWL systems with respect to d. DDPs for SLSs will be considered in the next section. However, before doing so, we would like to show that although DD of the SLS with respect to σ and d is a sufficient condition for DD of a PWL system to d, it is not necessary. Example III.9 Consider the PWL system [ ] [ ẋ(t) = x(t) + d(t) 0 0 1] x 1 (t) 0 [ ] [ ẋ(t) = x(t) + d(t) 0 0 2] x 1 (t) < 0 z = [ 1 0 ] x It is clear that the corresponding SLS is not disturbance decoupled with respect to σ and d. Yet, the PWL system described above is disturbance decoupled with respect to d. IV. DDP BY STATE FEEDBACK In the previous section we provided full characterizations of DD properties. Now we will consider if and how we should choose control inputs in order to render a SLS disturbance decoupled in some sense. In order to do so, consider the SLS ẋ(t) = A σ(t) x(t) + B σ(t) u(t) + E σ(t) d(t) z(t) = H σ(t) x(t) + J σ(t) u(t) (23a) (23b) where we included now a control input u(t) R nu at time t R +. As before we denote the solution corresponding to x 0 R nx, σ S, d L loc 1 (R +, R n d ) and u L loc 1 (R +, R nu ) by x x0,σ,d,u and the corresponding output by z x0,σ,d,u. We are now interested in finding conditions under which controllers can be found such that the closed-loop system 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 modedependent state feedback Problem IV.1 The disturbance decoupling problem with respect to d (DDPd) by mode-dependent state feedback for SLS (23) amounts to finding F i R nu nx, i = 1,..., M such that ẋ(t) = (A σ(t) + B σ(t) F σ(t) )x(t) + E σ(t) d(t) (24a) 10

13 CHAPTER 2. DISTURBANCE DECOUPLING OF SWITCHED LINEAR SYSTEMS z(t) = (H σ(t) + J σ(t) F σ(t) )x(t) is DD with respect to d. (24b) Note that the SLS (24) results from putting the system (23) in closed loop with u(t) = F σ(t) x(t), which requires knowledge of the active mode σ(t) at time t R +. Definition IV.2 Consider the SLS (23) with d = 0. A subspace V is called output-nulling {(A 1, B 1 ),..., (A M, B M )}-invariant if for any x 0 V and σ S there exists a control input u L loc 1 (R +, R nu ) such that x x0,σ,0,u(t) V and z x0,σ,0,u(t) = 0 for all t R +. Sometimes an output-nulling {(A 1, B 1 ),..., (A M, B M )}-invariant subspace is called a common output-nulling controlled invariant subspace for (23). Theorem IV.3 Consider the SLS (23) with d = 0. Let V be a subspace of R nx and V 0 R nx R nz denote the extended subspace V {0}. The following statements are equivalent. (i) V is common output-nulling controlled invariant. [ ] [ ] Aj (ii) V V H 0 Bj + im for all j = 1,..., M. j J j (iii) There exist F j R nu nx, j = 1,..., M, such that (A j + B j F j )V V ker (H j + J j F j ) for all j = 1,..., M. j J Proof: The proof is similar in nature to the (more complicated) proof of Theorem IV.9, which we will provide below, and is therefore omitted. Let {V j j J } be a collection of common output-nulling controlled invariant subspaces for the SLS (23). It follows from Definition IV.2 that V j is common output-nulling controlled invariant. Therefore, the set of all common output-nulling controlled invariant subspaces admits a largest element. The largest common output-nulling controlled invariant subspace for a given SLS plays a crucial role in the solution of DDPd by mode-dependent feedback. Definition IV.4 Consider the SLS (23) with d = 0. We define Vmd as the largest common output-nulling controlled invariant subspace for the SLS (23) that is (i) Vmd is common output-nulling controlled invariant; (ii) if V is a common output-nulling controlled invariant subspace for the SLS (23), then V Vmd. Theorem IV.5 Consider the SLS (23). DDPd by mode-dependent feedback is solvable if and only if im E i Vmd (25) i=1 Proof: The proof of the theorem follows directly from Theorem III.2, Theorem IV.3, and Definition IV.4. In Section V we will provide an algorithm to compute Vmd for a given SLS. Remark IV.6 For the special case that J i = 0, i = 1,..., M, this problem was solved also in [12]. In [12] the DDP with respect to d by modedependent state feedback was combined with the question of quadratic stability. Sufficient conditions were given exploiting known results for quadratic stabilization as in [4, 17]. These stability conditions can also be added to the theorems that we present here, but unfortunately, they are, just as in [12], not so trivial to verify, certainly for a high number of subsystems. Indeed, the sufficient conditions that guarantee solvability of DDP with quadratic stability (DDPQS) according to Theorem 3.2 in [12] are the existence of F j, j = 1,..., M, such that (A j + B j F j )Vmd V md ker (H j + J j F j ) for all j = 1,..., M and there exists a convex combination αj (A j + B j F j ) with M j=1 α j = 1 and α j 0, j = 1,..., M, being a Hurwitz matrix. Since a parameterization of all F j, j = 1,..., M, satisfying (A j + B j F j )Vmd V md ker (H j + J j F j ) is hard to come by in the first place, and one has to search for both α j and F j, j = 1,..., M, which is a nonconvex problem, these conditions are not easy to verify. Also for the satisfaction of the stabilization condition it is unclear if using Vmd instead of another common output-nulling controlled invariant subspace containing M i=1 im E i is introducing conservatism into the conditions. Another difference 11

14 CHAPTER 2. DISTURBANCE DECOUPLING OF SWITCHED LINEAR SYSTEMS with respect to [12], is that DDP with respect to d by mode-independent state feedback was not considered in [12], while we treat this problem in the next section. B. Solution of DDP with respect to d by modeindependent state feedback Problem IV.7 The disturbance decoupling problem with respect to d (DDPd) by mode-independent feedback for SLS (23) amounts to finding F R nu nx such that ẋ(t) = (A σ(t) + B σ(t) F )x(t) + E σ(t) d(t) z(t) = (H σ(t) + J σ(t) F )x(t) is DD with respect to d. (26a) (26b) Note that the SLS (26) results from putting the system (23) in closed loop with u(t) = F x(t). The latter state feedback controller does not require knowledge of the active mode σ(t) at time t R +. Definition IV.8 Consider the SLS (23) with d = 0. A subspace V is called output-nulling {(A 1, B 1 ),..., (A M, B M )}-invariant under modeindependent control if for any x 0 V there exists a control input u L loc 1 (R +, R nu ) such that x x0,σ,0,u(t) V and z x0,σ,0,u(t) = 0 for all σ S and for all t R +. Sometimes a subspace that is output-nulling {(A 1, B 1 ),..., (A M, B M )}-invariant under modeindependent control is called a common outputnulling controlled invariant subspace under modeindependent control for (23). (i) V is common output-nulling controlled invariant under mode-independent control. (ii) A s V V 0M + imb s. (iii) There exists F R nu nx such that (A j + B j F )V V ker (H j + J j F ) for all j = 1,..., M. Proof: : (i) (ii). Assume that σ(t) = σ j for some j {1,..., M}. Let x 0 V and u be a control input such that x x0,σ j,0,u(t) = x j (t) V and z x0,σ j,0,u(t) = z j (t) = 0 for all j = 1,..., M and t R +. Then one can write t 0 x x0,σ 1,0,u(t) x 0. =. + x x0,σ M,0,u(t) x 0 A 1 0 x 1 (s) B 1 ( u(s))ds 0 A M x M (s) B M (28) Furthermore, z x0,σ j,0,u(t) = H j x(t)+j j u(t) = 0 for all j = 1,..., M and for all t R +. Thus, one can also write t 0 z x0,σ 1,0,u(s). ds = 0 (29) z x0,σ M,0,u(s) Combining (28) with (29) we get Theorem IV.9 Consider the SLS (23) with d = 0. Let V be a subspace of R nx and V 0 R nx R nz denote the extended subspace V {0}. Define the matrices A s R M(nx+nz) nx and B s R M(nx+nz) nu A 1 H 1 A s =. A, B s = M H M B 1 J 1. B M J M (27) M times {}}{ and V 0M as V 0M = V 0 V 0... V 0. The following statements are equivalent. A 1 0 H 1... A M t 0 H M x x0,σ 1,0,u(t) x 0 0. x x0,σ M,0,u(t) x 0 } 0 {{ } V 0 M 0 x 1 (s). ds = x M (s) B 1 J 1. B M J M t 0 u(s)ds } {{ } imb s (30) 12

15 CHAPTER 2. DISTURBANCE DECOUPLING OF SWITCHED LINEAR SYSTEMS Now we divide the left-hand side of (30) by t and take the limit to obtain A 1 0 H 1 1 lim. t x 1 (s) t 0 t... ds = A 0 M x M (s) 0 H M A 1 0 A 1 H 1 x. 0 H 1... =. A M x 0 A x 0 (31) M 0 H M H M Equality (31) holds because x x0,σ j,0,u(t) is continuous for all j = 1,..., M. The rest of the proof is clear from here on. (ii) (iii). Choose a basis {q 1,..., q nv, q nv+1,..., q nx } for R nx such that {q 1,..., q nv } is a basis for V. For k = 1,..., n v there exist vectors q j,k V and u k R nu such that A j q k = q j,k + B j u k and H j q k = J j u k for all j = 1,..., M. For k = 1,..., n v define F q k = u k and for k = n v + 1,..., n x let F q k be arbitrary vectors in R nx. Then for (j, k) {1,..., M} {1,..., n v } we have (A j + B j F )q k = q j,k V and (H j + J j F )q k = 0. Hence (A j + B j F )V V ker (H j + J j F ) for all j = 1,..., M. (iii) (i). Let x 0 V and apply the feedback law u(t) = F x(t) to obtain the closed loop system ẋ(t) = (A σ(t) +B σ(t) F )x(t), z(t) = (H j +J j F )x(t). Clearly, x x0,σ,0,u(t) V and z x0,σ,0,u(t) = 0 for all t R +. Thus, (i) follows. Definition IV.10 Consider the SLS (23) with d = 0. We define V mi as the largest common outputnulling controlled invariant subspace under modeindependent control for the SLS (23), that is 1) V mi is common output-nulling controlled invariant under mode-independent control; 2) if V is common output-nulling controlled invariant under mode-independent control for the SLS (23), then V V mi. Theorem IV.11 Consider the SLS (23). DDPd by mode-independent feedback is solvable if and only if im E i Vmi (32) i=1 Proof: The proof of the theorem directly follows from Theorem III.2, Theorem IV.9 and Definition IV.10. In Section V we will present an algorithm to compute the largest common output-nulling controlled invariant subspace using mode-independent feedback for a given SLS. C. Solution of DDP with respect to σ by modedependent state feedback Consider the SLS (23). Problem IV.12 The disturbance decoupling problem with respect to σ (DDPσ) by mode-dependent state feedback amounts to finding F i R nu nx, i = 1,..., M such that the SLS (24) is DD with respect to σ. Before giving the theorem for the solvability of Problem IV.12 we need to introduce the following lemma. Lemma IV.13 Consider the SLS (23). DDPσ by mode-dependent feedback is solvable if and only if there exist a common output-nullling controlled invariant subspace V and F k R nu nx, k = 1,..., M, with (A k + B k F k )V V ker (H k + J k F k ), k = 1,..., M, such that the following three conditions hold for all (i, j) {1,..., M} {1,..., M}: (i) H i + J i F i = H j + J j F j, (ii) im (A i + B i F i A j B j F j ) V, (iii) im (E i E j ) V. Proof: The proof of the lemma directly follows from Corollary III.4 and Theorem IV.3. Theorem IV.14 Consider the SLS (23). DDPσ by mode-dependent feedback is solvable if and only if there exist G i R nu nx, i = 1,..., M, such that for all (i, j) {1,..., M} {1,..., M} it holds that (i) H i + J i G i = H j + J j G j, (ii) im (A i + B i G i A j B j G j ) V md, (iii) im (E i E j ) V md. 13

16 CHAPTER 2. DISTURBANCE DECOUPLING OF SWITCHED LINEAR SYSTEMS Proof: Necessity: For a subspace V we will use the notation F i (V) = {F (A i + B i F )V V ker (H i + J i F )} If DDPσ is solvable, then as a result of Lemma IV.13, there exists a common output-nulling controlled invariant subspace, V, such that H i + J i F i = H j + J j F j, F k F k(v) k = 1,..., M, for all (i, j) {1,..., M} {1,..., M}. It also holds that im (A i +B i F i A j B j F j) V ker (H i + J i F i ) im (E i E j ) V Let Fi F i (Vmd ) for all i = 1,..., M. Since V Vmd, one can choose a basis {v 1, v 2,..., v nx } for R nx such that {v 1,..., v p } is a basis for V and {v 1,..., v q } is a basis for Vmd. Define { Fi v k k {1, 2,..., q} G i v k = F i v k k {q + 1, q + 2,..., n x } Clearly, G i F i (Vmd ) and H i + J i G i = H j + J j G j for all i, j. We claim that im (A i + B i G i A j B j G j ) Vmd. To see this, note that [(A i + B i G i ) (A j + B j G j )]v k { V md k {1, 2,..., q} V k {q + 1, q + 2,..., n x } Since V Vmd, it immediately follows that im (A i + B i G i A j B j G j ) Vmd. Note that im (E i E j ) V V md and thus also (iii) holds. Sufficiency: Let {v 1, v 2,..., v nx } be a basis for R nx such that {v 1,..., v q } is a basis for Vmd. Furthermore, let Fi F i (Vmd ), i = 1,..., M. Define F i v k = { Fi v k k {1, 2,..., q} G i v k k {q + 1, q + 2,..., n x } Note that F i F i (Vmd ) for all i. It is easy to see that im (A i + B i F i A j B j F j ) Vmd and H i + J i F i = H j + J j F j for all i, j. Thus we recovered the conditions of Lemma IV.13, thereby completing the 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 problem with respect to d and σ (DDPdσ) by modedependent state feedback amounts to finding F i R nu nx, i = 1,..., M such that the SLS (24) is DD with respect to d and σ. Theorem IV.16 Consider the SLS (23). DDPdσ by mode-dependent feedback is solvable if and only if there exist G i R nu nx, i = 1,..., M such that for all (i, j) {1,..., M} {1,..., M} it holds that (i) H i + J i G i = H j + J j G j, (ii) im (A i + B i G i A j B j G j ) V md, (iii) im E i 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 and Theorem IV.16, if Vmd is replaced with V mi and G i = G j for all (i, j) {1,..., M} {1,..., M}, then the solutions to DDPσ and DDPdσ by mode-independent feedback, respectively, are obtained. V. ALGORITHMS TO TEST THE HYPOTHESES OF THE SOLUTIONS TO THE DDP A. Algorithm for Theorem IV.5 In this subsection we will present an algorithm to find the largest common output-nulling controlled invariant subspace for the SLS (23). Algorithm V.1 V i+1 = V 0 = R nx ; (33a) M {x u A j x+b j u V i, H j x+j j u = 0} j=1 (33b) From this recurrence relation it follows that V i+1 V i for all i = 0, 1,..., and if V k = V k+1 for some k, then V i = V k for all i k. Let q be the smallest k N such that V k = V k+1. Obviously, q n x. We 14

17 CHAPTER 2. DISTURBANCE DECOUPLING OF SWITCHED LINEAR SYSTEMS claim that V q = Vmd. For space reasons we omit the proof. Theorem V.2 Consider the SLS (23) and Algorithm V.1. Then, V q = Vmd with q := min{k N V k = V k+1 } n x. Proof: We will first show V i+1 V i for i N by induction. It is obvious that V 1 V 0. Suppose now that V i+1 V i. V i+2 = M {x u A j x + B j u V i+1, H j x + J j u = 0} j=1 M {x u A j x + B j u V i, H j x + J j u = 0} = V i+1 j=1 from which V i+2 V i+1 follows. Therefore, V i+1 V i for i N. Suppose for some k, V k+1 = V k. Then V k+2 = M {x u A j x + B j u V k+1, H j x + J j u = 0} j=1 = M {x u A j x + B j u V k, H j x + J j u = 0} j=1 = V k+1 Hence, V i = V k for all i k. Since V k V k 1... V 0, we have V q = V q+1 for some q n x. V q = M {x u A j x + B j u V q, H j x + J j u = 0} j=1 This shows that [ ] [ ] Aj V H q Vq 0 Bj + im j J j for all j = 1,..., M. By Theorem IV.3, V q is indeed common output-nulling controlled invariant. To show that V q is the largest common outputnulling controlled invariant subspace, we consider a common output-nulling controlled invariant subspace Ṽ for the SLS (23). Ṽ = Ṽ V 0, thus the following: M {x W A j x Ṽ + im B j} j=1 M {x u A j x + B j u V 0, H j x + J j u = 0} = V 1 j=1 Hence, Ṽ V 1. Repeating the procedure above we arrive at the conclusion Ṽ V q. B. Algorithm for Theorem IV.11 In this subsection we will present the algorithm to find the largest common output-nulling controlled invariant under mode-independent control for the SLS (23). Algorithm V.3 Define the matrices A s and B s and the extended subspace V 0M in the same way as in Theorem IV.9. V 0 = R nx ; V i+1 = {x A s x V 0 i M + im B s } (34) As above, it holds that V i+1 V i for all i = 0, 1,... and if V k = V k+1 for some k, then V i = V k for all i k. Let q be the smallest k N such that V k = V k+1. Obviously, q n x. We claim that V q = V mi in the next theorem of which the proof is omitted due to space reasons. Theorem V.4 Consider the SLS (23) and Algorithm V.3. Then, V q = V mi with q := min{k N V k = V k+1 } n x. Note that by defining A s as in (27) and redefining B s R Mnx Mnu as B B s = B M algorithm V.3 yields the largest common outputnulling controlled invariant subspace (under modedependent control). Remark V.5 The algorithm of Section V-A to obtain the largest common controlled invariant subspace (with mode-dependent feedback) for a SLS 15

18 CHAPTER 2. DISTURBANCE DECOUPLING OF SWITCHED LINEAR SYSTEMS inside another subspace is related to the algorithm presented before in [1, 14] for LPV systems for the special case that J i = 0, i = 1,..., M. The algorithm in Section V-B for mode-independent feedback was not presented in the literature before. VI. DDP BY MEASUREMENT FEEDBACK Previously, we assumed that the full state feedback was available for feedback, which is hardly the case in practice. In this section we will therefore show how the DDP with respect to d can be solved using dynamic measurement feedback. A. Solution of DDP with respect to d by modedependent measurement feedback Consider the SLS ẋ(t) = A σ(t) x(t) + B σ(t) u(t) + E σ(t) d(t) y(t) = C σ(t) x(t) z(t) = H σ(t) x(t) + J σ(t) u(t) (35a) (35b) (35c) and the mode-dependent dynamic measurement feedback controller given by the mode-dependent controller ẇ(t) = K σ(t) w(t) + L σ(t) y(t) u(t) = M σ(t) w(t) + N σ(t) y(t) (36a) (36b) which results in the closed-loop system (ẋ(t) ) ( ) ( ) x(t) Eσ(t) = A ẇ(t) e,σ(t) + d(t) (37a) w(t) 0 z(t) = ( ) ( ) x(t) H σ(t) + J σ(t) N σ(t) C σ(t) J σ(t) M σ(t) w(t) ( ) (37b) Ai + B where A e,i = i N i C i B i M i, i = L i C i K i 1,..., M. The problem we would like to tackle in this section is formulated as follows. Problem VI.1 The disturbance decoupling problem with respect to d by mode-dependent measurement feedback amounts to finding K i R nw nw, L i R nw ny, M i R nu nw, N i R nu ny, i = 1,..., M such that (37) is DD with respect to d. Removing the output-nulling requirement in Definition IV.2 and using Theorem IV.3 lead to the following definition. Definition VI.2 A subspace V R nx is called {(A 1, B 1 ),..., (A M, B M )}-invariant, if there exist F j, j = 1,..., M such that (A j + B j F j )V V for all j = 1,..., M. To tackle the problem as introduced in this section we need the following concept that is the dual of common controlled invariant subspaces. Definition VI.3 A subspace Z R nx is called {(C 1, A 1 ),..., (C M, A M )}-invariant, if there exist G j, j = 1,..., M such that (A j + G j C j )Z Z for all j = 1,..., M. Sometimes a {(C 1, A 1 ),..., (C M, A M )}-invariant subspace is called a common conditioned invariant subspace for (35). Similarly, a {(A 1, B 1 ),..., (A M, B M )}-invariant subspace is sometimes called a common controlled invariant subspace. The collection of all sets {G 1,..., G M } that satisfies (A i + G i C i )Z Z, i = 1,..., M, is denoted by G md (Z). Likewise, the collection of all sets {F 1,..., F M } that satisfies (A i + B i F i )V V, i = 1,..., M, is denoted by F md (V). It is clear that (A i + B i F i )V V if and only if (A T i + Fi T Bi T )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 {(B1, A 1 ),..., (BM, A M )} invariant if and only if V is {(A 1, B 1 ),..., (A M, B M )}-invariant. This duality can be used to prove the following. Theorem VI.4 The following statements are equivalent. (i) Z is common conditioned invariant, (ii) A j (Z ker C j ) Z for all j = 1,..., M. We have the following corollary of Theorem III.2. Corollary VI.5 The SLS (37) is disturbance decoupled with respect to d if and only if there exists an {A e1,..., A em }-invariant subspace V e R nx R nw such that im i=1 [ ] Ei V 0 e H 1 + J 1 N 1 C 1 J 1 M 1 ker. H M + J M N M C M J M M M (38) 16

19 CHAPTER 2. DISTURBANCE DECOUPLING OF SWITCHED LINEAR SYSTEMS We are now interested in relating {A e1,..., A em }- invariance of a subspace to geometric properties of the original SLS (35). As an extension of the linear case [13], the concept of {(C 1, A 1, B 1 ),..., (C M, A M, B M )}-pairs will be of importance. Definition VI.6 Consider the SLS (35). A pair of subspaces (Z, V) is called a {(C 1, A 1, B 1 ),..., (C M, A M, B M )}-pair, if Z is a common conditioned invariant subspace for (35), V is a common controlled invariant subspace for (35) and Z V. Ceteris paribus, if V is a common output-nulling controlled invariant subspace then (Z, V) is called an output-nulling {(C 1, A 1, B 1 ),..., (C M, A M, B M )}-pair. Since V e as in Corollary VI.5 lies in the extended state space R nx R nw, it can be projected onto R nx by ( x p(v e ) = {x R nx w V w) e } We can also define the intersection of V e and R nx {0} as ( x i(v e ) = {x R nx V 0) e } Theorem VI.7 Consider the controlled SLS (37) and let V e denote an {A e1,..., A em }- invariant subspace. Then, (i(v e ), p(v e )) is a {(C 1, A 1, B 1 ),..., (C M, A M, B M )}-pair. If V e satisfies H 1 + J 1 N 1 C 1 J 1 M 1 V e ker. H M + J M N M C M J M M M (39) in addition, then (i(v e ), p(v e )) is an output-nulling {(C 1, A 1, B 1 ),..., (C M, A M, B M )}-pair. Proof: Taking σ = σ j, the proof follows along similar lines as the proof of Theorem 6.2 (making the necessary modifications to include the feedthrough terms J i, 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 {(C 1, A 1, B 1 ),..., (C M, A M, B M )}-pair. Then there exist a mode-dependent controller (36) and an {A e1,..., A em }-invariant subspace V e R nx R nw satisfying (39) such that i(v e ) = Z, p(v e ) = V. Proof: Let {F 1,..., F M } F md (V) and {G 1,..., G M } G md (Z). Construct N j such that (A j + B j N j C j )Z V by applying Lemma 6.3 in [6] for all j = 1,..., M. Take the mode-dependent controller and V e such that K j = A j + B j F j + G j C j B j N j C j, L j = B j N j G j, M j = F j N j C j, j = 1,..., M and V e = {( x1 0 ) + ( ) } x2 x x 1 Z, x 2 V 2 {A e1,..., A em }-invariance property of V e as constructed above can be proven mutatis mutandis as in the proof of Theorem 6.4 in [6]. Here we will only show that V e satisfies (39). If x 1 Z, (x 1, 0) V e. We claim that H j x 1 +J j N j C j x 1 = 0 for all j = 1,..., M. Since N j is chosen such that (A j + B j N j C j )Z V, our claim is true. If x 2 V, (x 2, x 2 ) V e. It must then hold that (H j + J j N j C j )x 2 +J j (F j N j C j )x 2 = (H j +J j F j )x 2 = 0 for all j = 1,..., M. Since x 2 V, this is also true. Thus we have shown that V e satisfies (39). The proof of the two remaining claims, i(v e ) = Z and p(v e ) = V, can also be obtained from the proof Theorem 6.4 in [6] with minor modifications. Theorem VI.9 Consider the SLS (35). The disturbance decoupling problem by mode-dependent measurement feedback is solvable if and only if there exists an output-nulling {(C 1, A 1, B 1 ),..., (C M, A M, B M )}-pair (Z, V) such that im E i Z i=1 Proof: The proof follows from Theorem VI.7, Theorem VI.8 and Corollary VI.5. Before concluding the paper we will show how a common conditioned invariant subspace that is of particular interest to us can be computed. Definition VI.10 Let W R nx be a subspace. Consider the SLS (35) with d = 0. We define Zmd (W) as the smallest common conditioned invariant subspace that contains W for the SLS (35) that is 17