- Email: [email protected]
On the Controllability and Stabilizability of Linear Complementarity Systems
7th IFAC Symposium on System Structure and Control 7th IFAC Symposium on System Structure and Control Sinaia, Romania, September 9-11,Structure 2019 7th Symposium on and 7th IFAC IFAC Symposium on System System and Control Control Sinaia, Romania, September 9-11,Structure 2019 Available online at www.sciencedirect.com Sinaia, Romania, September 9-11,Structure 2019 7th IFAC Symposium on System and Control Sinaia, Romania, September 9-11, 2019 Sinaia, Romania, September 9-11, 2019
ScienceDirect
On On On On
IFAC PapersOnLine 52-17 (2019) 1–6
the the Controllability Controllability and and Stabilizability Stabilizability the Controllability and Stabilizability Linear Complementarity Systems the Controllability and Stabilizability Linear Complementarity Systems Linear Complementarity Systems Linear Complementarity Systems M.K. Camlibel ∗∗
of of of of
M.K. Camlibel ∗∗ M.K. M.K. Camlibel Camlibel M.K. Camlibel ∗ Bernoulli Institute of Mathematics, Bernoulli Institute of Mathematics, Computer Computer Science, Science, and and Artificial Artificial Bernoulli Institute University of Mathematics, Mathematics, Computer Science, and Artificial Artificial Intelligence, of Groningen, The Netherlands Bernoulli Institute of Computer Science, and Intelligence, University of Groningen, The Netherlands ∗ Bernoulli Institute of Mathematics, Computer Science, and Artificial Intelligence, University of Groningen, Groningen, The Netherlands (e-mail: [email protected]). Intelligence, University of The Netherlands (e-mail: [email protected]). Intelligence, University of Groningen, The Netherlands (e-mail: [email protected]). (e-mail: [email protected]). (e-mail: [email protected]). Abstract: This This paper paper studies studies controllability controllability and and stabilizability stabilizability of of linear linear complementarity complementarity Abstract: Abstract: This paper studies controllability and stabilizability of linear complementarity systems that can be cast as continuous piecewise affine dynamical systems. Under aa certain certain Abstract: This paper studies controllability and stabilizability of linear complementarity systems that can be caststudies as continuous piecewise dynamical Under Abstract: controllability andaffine stabilizability ofsystems. linear complementarity systems that can be piecewise affine dynamical systems. Under aa certain right-invertibility assumption, we aa la necessary and sufficient conditions for systems thatThis can paper be cast cast as as continuous continuous affine dynamical Under certain right-invertibility assumption, we present presentpiecewise la Hautus Hautus necessary andsystems. sufficient conditions for systems that can assumption, beand caststabilizability. as continuous affinenecessary dynamical Under a certain right-invertibility assumption, we present presentpiecewise a la la Hautus Hautus necessary andsystems. sufficient conditions for both controllability right-invertibility we a and sufficient conditions for both controllability and stabilizability. right-invertibility assumption, we present a la Hautus necessary and sufficient conditions for both and both controllability controllability and stabilizability. stabilizability. Copyright © 2019. The Authors. Published by Elsevier Ltd. All rights reserved. both controllability and stabilizability. Keywords: Controllability, Controllability, stabilizability, stabilizability, linear linear complementarity complementarity systems, systems, continuous continuous Keywords: Keywords:affine Controllability, stabilizability, linear complementarity complementarity systems, systems, continuous continuous piecewise dynamical systems Keywords: Controllability, stabilizability, linear piecewise dynamical systems Keywords: Controllability, linear complementarity systems, continuous piecewise affine affine dynamical stabilizability, systems piecewise affine dynamical systems piecewise affine dynamical systems 1. INTRODUCTION INTRODUCTION affine systems systems and and the the results results in in Thuan Thuan and and Camlibel Camlibel 1. affine 1. INTRODUCTION INTRODUCTION affine systems and the results in Thuan Thuan necessary and Camlibel Camlibel (2014) in order to present tailor-made and 1. affine systems and the results in and (2014) in order to present tailor-made necessary and 1. INTRODUCTION affine systems and the results in Thuan and Camlibel (2014) in order to present tailor-made necessary and sufficient conditions for controllability and stabilizability (2014) in order to present tailor-made necessary and Controllability is is not not only only a fundamental fundamental system-theoretic system-theoretic sufficient conditions for controllability and stabilizability Controllability (2014) order to for present tailor-made and sufficient conditions for controllability andnecessary stabilizability of linear linearincomplementarity complementarity systems. conditions controllability and stabilizability Controllability is not not only aa a fundamental fundamental system-theoretic property but also a prerequisite virtually for all all control control sufficient of systems. Controllability is only system-theoretic property but also a prerequisite virtually for sufficient conditions for controllability and stabilizability of linear complementarity systems. of linear complementarity systems. Controllability is not only a and fundamental system-theoretic property but also also prerequisite virtually for for all controlcontrol The organization of the paper is as follows. In Section 2, we design problems. problems. Kalman’s Hautus’ tests property but aa prerequisite virtually for all control design Kalman’s and Hautus’ tests for controlof linear complementarity systems. The organization of the paper is as follows. In Section 2, we property butamong alsoKalman’s athe prerequisite virtually for all controlcontrol design problems. Kalman’s and Hautus’ Hautus’ tests for controllability are classical results of linear system The The organization of the of paper is as as follows. In In Section Section 2, we we design problems. and tests for will introduce introduce theof class linear complementarity systems. organization the paper is follows. 2, lability are among the classical results of linear system will the class of linear complementarity systems. design problems. Kalman’s and Hautus’ tests for controllability are among the classical results of linear system theory. Apart from finite-dimensional linear time-invariant The organization of the paper is as follows. In Section 2, we will introduce the class of linear complementarity systems. lability are among the classical results of linear system Section 3 is devoted to continuous piecewise affine dynamwill introduce the class of linear complementarity systems. theory. Apart from finite-dimensional linear time-invariant Section 3 is devoted to continuous piecewise affine dynamlability are among the classical results of time-invariant linear system will theory. Apart from finite-dimensional linear time-invariant systems, there are only which easily introduce theHautus-like class of linear complementarity Section is devoted devoted to continuous continuous piecewise affine affinesystems. dynamtheory. linear ical systems systems and controllability/stabilizability 33 is to piecewise dynamsystems,Apart therefrom are finite-dimensional only a a few few cases cases for for which easily verver- Section ical and Hautus-like controllability/stabilizability theory. Apart linear time-invariant systems, therefrom are finite-dimensional only few cases cases for which easily necver- Section ifiable controllability controllability conditions exist. For instance, 3these is devoted to continuous piecewise affine dynamical systems and Hautus-like controllability/stabilizability systems, there are only aa few for which easily vertests for systems. Subsequently, Section 4 will discuss ical systems and Hautus-like controllability/stabilizability ifiable conditions exist. For instance, nectests for these systems. Subsequently, Section will discuss systems, there are only a few cases for which easily ver- ical ifiable controllability conditions exist. For instance, necessary controllability and sufficient conditions that are linear in nature nature systems Hautus-like controllability/stabilizability tests for these theseand systems. Subsequently, Section 444 will willsystems discuss ifiable conditions exist. For instance, necthe relationships between linear complementarity tests for systems. Subsequently, Section discuss essary and sufficient conditions that are linear in the relationships between linear complementarity systems ifiable controllability conditions exist. For instance, nec- tests essary and sufficient conditions that are linear in nature nature exist for for linear switched systemsthat withare state-independent for these systems. Subsequently, Section discuss the relationships between linear systems. complementarity systems essary and sufficient conditions linear in and relationships continuous piecewise affine This4 will be folfolthe between linear complementarity systems exist linear switched systems with state-independent and continuous piecewise affine systems. This will be essary and sufficient conditions that are linear in nature exist for linear switched systems with state-independent switching Sun et al. (2002). This result relies on the relationships between linear complementarity systems and continuous piecewise affine systems. Thisinwill will be folfolexist for linear systems with state-independent lowed by the main results and their proofs Section 5. and continuous piecewise affine systems. This be switching Sun etswitched al. (2002). This result relies on the fact fact the lowed by the main results and their proofs in Section 5. exist for switching linear systems with state-independent switching Sun et etswitched al.signal (2002). This result reliesason onanthe the fact and that the the canThis be considered considered extercontinuous piecewise affine systems. Thisin will be followed bythe thepaper main results and their proofs in Section 5. switching Sun al. (2002). result relies fact Finally, closes with the conclusions in Section 6. lowed by the main results and their proofs Section 5. that switching signal can be as an exterFinally, the paper closes with the conclusions in Section 6. switching Sun al.signal (2002). result relies fact lowed that the switching switching signal canThis be for considered asonan anthe external input. input. Yet et another instance which controllability thepaper maincloses results and proofs in 5. Finally,bythe the paper closes with thetheir conclusions in Section Section 6. 6. that the can be considered as exterFinally, with the conclusions in Section nal Yet another instance for which controllability that the switching signal can be considered as an external input. Yet another instance for which controllability admits a simple characterization is the case of continuous Finally, the paper closes with the conclusions in Section 6. nal input. Yet another instance for which controllability 2. LINEAR COMPLEMENTARITY SYSTEMS admits a simple characterization is the case of continuous 2. LINEAR COMPLEMENTARITY SYSTEMS nal input. Yet another instance for which admits simple characterization is the the case controllability of continuous continuous piecewise affine systems Thuan and Camlibel admits aa simple characterization is case of 2. LINEAR LINEAR COMPLEMENTARITY COMPLEMENTARITY SYSTEMS SYSTEMS 2. piecewise affine dynamical dynamical systems Thuan and Camlibel admits simple characterization is the case and of continuous piecewise affine dynamical dynamical systems Thuan and Camlibel Consider (2014). aContrary Contrary to linear linear systems switched systems, piecewise piecewise affine Thuan Camlibel 2. LINEAR COMPLEMENTARITY SYSTEMS the linear system (2014). to switched systems, piecewise Consider the linear system piecewise affine dynamical systems Thuan and piecewise Camlibel (2014). Contrary to linear linear switched systems, piecewise affine dynamical dynamical systems exhibit state-dependent switch- Consider (2014). Contrary to switched systems, Consider the the linear linear system system affine systems exhibit state-dependent switch(2014). Contrary to linear switched systems, piecewise affine dynamical systems exhibit state-dependent switchx(t) ˙ = Ax(t) (1a) ing phenomenon that makes the analysis much harder. affine dynamical systems exhibit state-dependent switchConsider the linear system x(t) ˙ = Ax(t) + + Bu(t) Bu(t) + + Ez(t) Ez(t) (1a) ing phenomenon that makes the analysis much harder. x(t) ˙ = Ax(t) + Bu(t) + Ez(t) (1a) affine dynamical systems exhibit state-dependent switching phenomenon that makes the analysis much harder. Nevertheless, Hautus-like tests can be devised by exploitx(t) ˙ = Ax(t) + Bu(t) + Ez(t) (1a) ing phenomenon that makes much harder. Nevertheless, Hautus-like teststhe can analysis be devised by exploitw(t) = Cx(t) + Du(t) + F z(t) (1b) w(t) = Ax(t) Cx(t) + Bu(t) Du(t) + + Ez(t) F z(t) (1b) x(t) ˙ (1a) ing phenomenon that makes the much harder. Nevertheless, Hautus-like tests can analysis be devised devised by exploitexploitthe structure imposed by continuity continuity of the the vector field. Nevertheless, Hautus-like tests can be by w(t) = = Cx(t) Cx(t) + + Du(t) Du(t) + +F F z(t) z(t) (1b) ing the structure imposed by of vector field. w(t) (1b) n m p+p Nevertheless, Hautus-like tests can be devised by exploiting the structure structure imposed by by continuity of oftest the also vector field. where x ∈ Rnw(t) Moreover, this Hautus-like controllability extends ing the imposed continuity the vector field. , u ∈ R , and (z, w) ∈ R . When the m p+p = Cx(t) + Du(t) + F z(t) (1b) Moreover, this Hautus-like controllability test also extends where x ∈ R n , u ∈ Rm m , and (z, w) ∈ Rp+p p+p . When the ing the structure imposed by continuity oftest the also vector field. where Moreover, this Hautus-like Hautus-like controllability test also extends n to stabilizability. where x ∈ R , u ∈ R , and (z, w) ∈ R . When the Moreover, this controllability extends external variables (z, w) satisfy the so-called complemenx ∈ R , u ∈ R , and (z, w) ∈ R . When the to stabilizability. external variables (z, w)m satisfy the so-called complemenMoreover, this Hautus-like controllability test also extends where to stabilizability. stabilizability. x variables ∈ Rn , u (z, ∈ R ,satisfy and (z, w)so-called ∈ Rp+pcomplemen. When the external w) the to tarity relations external variables (z, w) satisfy the so-called complemenIn stabilizability. this paper, paper, we we study study a particular particular class class of of continuous continuous tarity relations to In this external variables (z, w) satisfy the so-called complementarity relations In this paper, paper, we study aa a systems, particular class linear of continuous continuous piecewise affine dynamical namely comple- tarity relations In this we study particular class of 00 (1c) piecewise affine dynamical systems, namely linear completarity relations z(t) z(t) ⊥ ⊥ w(t) w(t) 00 (1c) In this paper, we study a systems, particular class linear of are continuous piecewise affine dynamical systems, namely linear complementarity systems. Complementarity systems encounpiecewise affine dynamical namely comple z(t) z(t) ⊥ ⊥ w(t) w(t) 00 (1c) 00 (1c) mentarity systems. Complementarity systems are encounand ⊥ piecewise dynamical systems, namely linear complementarity systems. Complementarity systems are encoun- where tered in in aaaffine multitude of applications applications in various disciplines mentarity systems. Complementarity systems are encoun0 are z(t)componentwise ⊥ w(t) 0 (1c) where the the inequalities inequalities are componentwise and ⊥ denotes denotes tered multitude of in various disciplines T the componentwise and ⊥ denotes orthogonality, that is zzare ⊥ w if and only if zz T w 0. We mentarity systems. Complementarity encoun- where tered in aa and multitude of applications applications insystems variousare disciplines where the inequalities inequalities are componentwise and ⊥= denotes of science engineering including electrical/mechanical tered in multitude of in various disciplines orthogonality, that is ⊥ w if and only if w = 0. We T ⊥ denotes of science and engineering including electrical/mechanical T where the inequalities are componentwise and orthogonality, that is z ⊥ w if and only if z w = 0. We call the overall system (1) a linear complementarity system tered in a multitude of applications in various disciplines of science and engineering including electrical/mechanical orthogonality, that is z ⊥ w if and only if z w = 0. We engineering and operations research van der Schaft and of science and engineering the overall system (1) a linear complementarity system engineering and operationsincluding researchelectrical/mechanical van der Schaft and call orthogonality, is z(1) if and only if z T w = system 0. We call the overall overall that system (1)⊥aawlinear linear complementarity system (LCS). of science and engineering including engineering and operations researchelectrical/mechanical van der derHeemels Schaft and call the system complementarity Schumacher (1998, 2000); Brogliato (2003); engineering and operations research van Schaft (LCS). Schumacher (1998, 2000); Brogliato (2003); Heemels and the overall system (1) a linear complementarity system (LCS). engineering and operations research van derHeemels Schaft and Schumacher (1998, 2000); Brogliato Brogliato (2003); Heemels and call (LCS). Brogliato (2003); (2003); Camlibel et al. al. (2003); (2004); Schumacher Schumacher (1998, 2000); Brogliato Camlibel et (2004); Schumacher (LCS). Schumacher (1998, 2000); Brogliato (2003); Heemels and Brogliato (2003); Camlibel et al. (2004); Schumacher 3. (2004). We We(2003); refer to to van van der Schaft Schaft and(2004); Schumacher (1996); Brogliato Camlibel et al. Schumacher 3. CONTINUOUS CONTINUOUS PIECEWISE PIECEWISE AFFINE AFFINE SYSTEMS SYSTEMS (2004). refer der and Schumacher (1996); 3. CONTINUOUS PIECEWISE AFFINE SYSTEMS SYSTEMS Brogliato (2003); Camlibel et al. (2004); Schumacher (2004). We refer to van der Schaft and Schumacher (1996); Heemels et al. (2000); Camlibel (2001); Camlibel and 3. CONTINUOUS PIECEWISE AFFINE (2004). van der Schaft and Schumacher (1996); HeemelsWe etrefer al. to (2000); Camlibel (2001); Camlibel and 3.order CONTINUOUS PIECEWISE AFFINEaffine SYSTEMS (2004). We van der Schaft and Schumacher (1996); Heemels etrefer al. to (2000); Camlibel (2001); Camlibel and Schumacher (2002); Camlibel et al. (2003); Heemels et al. In to introduce continuous piecewise Heemels et al. (2000); Camlibel (2001); Camlibel and In order to introduce continuous piecewise affine dynamidynamiSchumacher (2002); Camlibel et al. (2003); Heemels et al. Heemels et (2002); al. (2000); Camlibel Camlibel Schumacher (2002); Camlibel etCamlibel al. (2001); (2003); Heemels etand al. In In order order to introduce introduce continuous piecewiseaffine affinefunctions dynami(2002); Camlibel et al. (2002); (2007); Camlibel cal systems, we first begin with piecewise Schumacher Camlibel et al. (2003); Heemels et al. to continuous piecewise affine dynamical systems, we first begin with piecewise affine functions (2002); Camlibel et al. (2002); Camlibel (2007); Camlibel Schumacher (2002); Camlibel etCamlibel al. the (2003); Heemels et al. In order toproperties. introduce continuous piecewiseaffine affinefunctions dynami(2002); Camlibel et for al. (2002); Camlibel (2007); Camlibel cal systems, we first begin with piecewise et al. (2008b, 2014) the work on analysis of general and their (2002); Camlibel et al. (2002); (2007); Camlibel cal systems, we first begin with piecewise affine functions and their properties. et al. (2008b, 2014) for the work on the analysis of general (2002); Camlibel et al. (2002); Camlibel (2007); Camlibel cal systems, we first begin with piecewise affine functions et al. (2008b, 2014) for the work on the analysis of general and their properties. LCSs. et al. (2008b, 2014) for the work on the analysis of general and their properties. ν LCSs. R is said to be affine if ψ et al. (2008b, 2014) for the work on the analysis of general A and their properties. LCSs. → if there there A function function ψ :: R Rννν → LCSs. is said to be affine ×νR → is said to be affine if A function ψ : R Under certain conditions, a linear complementarity system and a vector g ∈ R that exist a matrix H ∈ R ×νR such → R is said to be affine if there there A function ψ : R LCSs. and a vector g ∈ R such that exist a matrix H ∈ν R×ν Under certain conditions, a linear complementarity system ν ×νR φsuch isRaasaid tofunction be affine if called there A function ψ :H Under certain conditions, linear complementarity system ψ(x) and gg ∈ R that exist aa=matrix ∈ can cast a piecewise affine Hx gg R for all x ∈ A is ν .vector Under aa linear complementarity system and vector ∈ R such that exist ∈R R→ can be becertain cast as asconditions, a continuous continuous piecewise affine dynamical dynamical ψ(x) =matrix Hx + +H for all x ∈ R . A function φ is called ×ν ν ν Under a linear complementarity system piecewise and g family ∈ Rφ that exist a= ∈ there Rall can becertain cast asconditions, aexploit continuous piecewise affine dynamical ψ(x) =matrix Hx +H g iffor for all x xexists ∈ R Ra a..vector Afinite function φsuch is called called system. We will the structure of such piecewise affine of affine can be cast as a continuous piecewise affine dynamical ψ(x) Hx + g ∈ A function is piecewise affine if there exists ν a finite family of affine system. We will exploit the structure of such piecewise can be cast as aexploit continuous piecewise of ψ(x) = Hx + g if all xexists ∈ R aa. Afinite function is called system. We will will exploit the structure structure ofaffine suchdynamical piecewise piecewise piecewise affine ifforthere there exists finite familyφ of of affine system. We the such piecewise affine family affine system. will ©exploit the structure of such piecewise piecewise affine if there exists a finite family of affine 2405-8963We Copyright 2019. The Authors. Published by Elsevier Ltd. All Copyright © 2019 IFAC 19rights reserved. ∗ ∗ ∗ ∗
Copyright © under 2019 IFAC 19 Control. Peer review responsibility of International Federation of Automatic Copyright © 2019 2019 IFAC IFAC 19 Copyright © 19 10.1016/j.ifacol.2019.11.017 Copyright © 2019 IFAC 19
2019 IFAC SSSC 2 Sinaia, Romania, September 9-11, 2019
M.K. Camlibel / IFAC PapersOnLine 52-17 (2019) 1–6
functions ψi : Rν → R with i = 1, 2, . . . , r such that φ(x) ∈ {ψ1 (x), ψ2 (x), . . . , ψr (x)} for all x ∈ Rν .
assumption is that the number of inputs should be at least as the number of outputs. Whenever this is the case, right invertibility is generically satisfied.
Continuous piecewise affine functions admit useful geometrical representations that will be used later on this paper. To elaborate, we first need to introduce some terminology. A set of Rν is called a polyhedron if it is the intersection of a finite family of closed half-spaces or hyperplanes in Rν . If P ⊆ Rν is a polyhedron, then there exist a matrix P ∈ Rk×ν and a vector q ∈ Rk such that P = {x ∈ Rν | P x q}. A finite collection Ξ of polyhedra of Rν is said to be a polyhedral subdivision of Rν if (see Facchinei and Pang (2002) for more details)
A complete characterization of controllability and stabilizability of CPASs are established in Thuan and Camlibel (2014). In order to quote the results of Thuan and Camlibel (2014), we need to introduce the following terminology. For a nonempty convex set S ⊆ Rn , S − denotes the dual cone S − := {ξ ∈ Rn | ξ T x 0 for all x ∈ S},
S ⊥ denotes orthogonal complement in Cn , and Cr (S) denotes the recession cone
i. the union of all polyhedra in Ξ is equal to R , ii. each polyhedron in Ξ is of dimension ν, and iii. the intersection of any two polyhedra in Ξ is either empty or a common proper face of both polyhedra ν
Cr (S) := {ξ ∈ Rn | z + λx ∈ Λ for all λ 0 and z ∈ S}. Necessary and sufficient conditions for controllability of CPASs are stated next. Proposition 4. (Thuan and Camlibel (2014),Thm. 3.4]). Consider a CPAS (2) satisfying Assumption 3. Suppose that the origin is contained in the closure of the convex hull of the image of the function Φ. Let Ξ = {Y 1 , Y 2 , . . . , Y h } be a polyhedral subdivision of Rp induced by Φ. Also let
As stated next, polyhedral subdivisions appear in the description of continuous piecewise affine functions. Proposition 1. (Facchinei and Pang (2002), Prop. 4.2.1). A continuous function ψ : Rν → R is piecewise affine if and only if there exist a polyhedral subdivision Ξ = {Ξ1 , Ξ2 , . . . , Ξr } of Rν and a finite family of affine functions ψi : Rν → R with i = 1, 2, . . . , r such that ψ coincides ψi on Ξi for every i = 1, 2, . . . , r.
Φ(y) = H i y − g i whenever y ∈ Y i .
Then, the system (2) is controllable if and only if the following four implications hold:
By a continuous piecewise affine system (CPAS), we mean a dynamical system of the form x(t) ˙ = Ax(t) + Bu(t) + Φ(y(t))
(2a)
y(t) = Cx(t) + Du(t)
(2b)
i. λ ∈ C, z ∈ Cn , z ∗ g i = 0, and z ∗ A + H i C − λI B + H i D = 0 ∀ i =⇒ z = 0.
ii. λ ∈ R, z ∈ Rn , (z, wi ) ∈ ({g i } × Y i )− , and T A + H i C − λI B + H i D z = 0 ∀ i =⇒ z = 0. i w C D
where x ∈ R is the state, u ∈ R is the input, y ∈ Rp is the output, all involved matrices are of appropriate sizes, and Φ : Rp → Rn is a continuous piecewise affine function. n
(3)
m
Since the piecewise affine function Φ is continuous, the right-hand side of (2a) is globally Lipschitz (see e.g. (Facchinei and Pang, 2002, Proposition 4.2.2)). Thus, it follows from the theory of ordinary differential equations that the system (2) must admit a unique solution for each initial state x0 and locally integrable input u. We denote this solution by xu (t; x0 ), and denote the corresponding output by y u (t; x0 ).
iii. λ ∈ C, Re(λ) = 0, z ∈ Cn , wi ∈ Cr (Yi )⊥ , and ∗ A + H i C − λI B + H i D z = 0 ∀ i =⇒ z = 0. wi C D iv. λ ∈ R, λ = 0, z ∈ Rn , wi ∈ Cr (Yi )− , and T A + H i C − λI B + H i D z = 0 ∀ i =⇒ z = 0. wi C D
Next, we define controllability and stabilizability for CPASs. Definition 2. We say that the system (2) is
Stabilizability of CPASs can also be characterized by similar spectral conditions. Proposition 5. ((Thuan and Camlibel, 2014, Thm. 3.6)). Consider a CPAS (2) satisfying Assumption 3. Suppose that Φ(0) = 0. Let Ξ = {Y 1 , Y 2 , . . . , Y h } be a polyhedral subdivision of Rp induced by Φ. Also let
• controllable if for any two states x0 , xf there exist T > 0 and a locally integrable input u such that xu (T ; x0 ) = xf . • stabilizable if for any initial state x0 there exists a locally integrable input u such that lim xu (t; x0 ) = 0. t→∞
Φ(y) = H i y − g i whenever y ∈ Y i .
In this paper, the following blanket assumption will be in force. Assumption 3. The transfer matrix D + C(sI − A)−1 B is right invertible as a rational matrix.
(4)
Then, the system (2) is stabilizable if and only if the following four implications hold: i. λ ∈ C, Re(λ) 0, z ∈ Cn , z ∗ g i = 0 and z ∗ A + H i C − λI B + H i D = 0 ∀ i =⇒ z = 0.
Similar invertibility assumptions are common in the analysis of linear systems with output constraints (see e.g. Saberi et al. (2002); Shi et al. (2003); Saberi et al. (2003); Heemels and Camlibel (2008)). The main restriction of the
ii. λ ∈ R, λ 0, z ∈ Rn , (z, wi ) ∈ ({g i } × Y i )− , and 20
2019 IFAC SSSC Sinaia, Romania, September 9-11, 2019
z wi
T
M.K. Camlibel / IFAC PapersOnLine 52-17 (2019) 1–6
A + H i C − λI B + H i D = 0 ∀ i =⇒ z = 0. C D
−1 H α = −E•α Fαα yα ,
and
α
(a) The system (2) is controllable if and only if the following two implications hold:
A linear complementarity system of the form (1) can be cast as a CPAS of the form (2) when F is a P -matrix, that is a matrix whose principal minors are all positive. It is well-known that every positive definite matrix is a P -matrix (see for instance (Cottle et al., 1992, Thm. 3.1.6 and Thm. 3.3.7)). In particular, P -matrices play an important role in the context of linear complementarity problem, i.e. the problem of finding a p-vector z satisfying
i. λ ∈ C, z ∈ Cn , and −1 −1 z ∗ A − E•α Fαα Cα• − λI B − E•α Fαα Dα• = 0 for all α ⊆ {1, 2, . . . , p} implies z = 0.
ii. λ ∈ R, z ∈ Rn , wα ∈ (Y α )− , and T −1 −1 Cα• − λI B − E•α Fαα Dα• A − E•α Fαα z =0 α w C D
0 z ⊥ q + Fz 0 for a given p-vector q and a p × p matrix F . We denote the linear complementarity problem for a given vector q and a given matrix F by LCP(q, F ). If F is a P matrix, LCP(q, F ) admits a unique solution z(q) for every q ∈ Rp (see (Cottle et al., 1992, Thm. 3.3.7)). In addition, one can verify that for every q there exists an index set α ⊆ {1, 2, . . . , p} such that −(Fαα )−1 0|α|×|αc | qα 0 (5a) qα c −Fαc α (Fαα )−1 I|αc |
for all α ⊆ {1, 2, . . . , p} implies z = 0. (b) The system (2) is stabilizable if and only if the following two implications hold: i. λ ∈ C, Re(λ) 0, z ∈ Cn , and −1 −1 z ∗ A − E•α Fαα Cα• − λI B − E•α Fαα Dα• = 0 for all α ⊆ {1, 2, . . . , p} implies z = 0.
ii. λ ∈ R, λ 0, z ∈ Rn , wα ∈ (Y α )− , and T −1 −1 Cα• − λI B − E•α Fαα Dα• A − E•α Fαα z =0 wα C D
and the unique solution z of the LCP(q, F ) is given by (5b)
Here αc denotes the set {1, 2, . . . , p} \ α. Furthermore, the map q → z(q) is known to be Lipschitz continuous (Cottle et al., 1992, Lem. 7.3.10).
for all α ⊆ {1, 2, . . . , p} implies z = 0. To prove this proposition, we need the following auxiliary result. Lemma 7. Let C ⊆ Rp be a polyhedral cone given by C = {ξ ∈ Rp | M ξ 0} for some matrix M ∈ Rq×p . Then, C − = {η ∈ Rp | η = M T ζ for some 0 ζ ∈ Rq }. Moreover, if ker M T = {0}, then C ⊥ = {0}.
By using the Lipschitz continuity, it can be shown that Camlibel (2001) for every initial state x0 ∈ Rn and locally integrable input u, there exists a unique locally absolutely continuous x : R+ → Rn such that x(0) = x0 , (1a) is satisfied for almost all t 0, and (1b)-(1c) are satisfied for all t 0. Moreover, we can employ the parametrization given in (5) to conclude that there is oneto-one correspondence between the trajectories of the LCS (1) and those of the following conewise linear system (see Camlibel et al. (2008a) for more details) x(t) ˙ = Ax(t) + Bu(t) + Ψ y(t) (6a) y(t) = Cx(t) + Du(t)
Proof. The first statement readily follows from Farkas’ Lemma (see e.g. (Bertsekas, 2009, Prop. 2.3.1)). To prove the second statement, note that C ⊥ = −C − ∩ C − . Therefore, η ∈ C ⊥ if and only if there exist ζ1 0 and ζ2 0 such that η = M T ζ1 = M T ζ2 . This means that ζ1 − ζ2 ∈ ker M T and hence ζ1 = ζ2 = 0. Consequently, η = 0.
(6b)
where Ψ is a Lipschitz continuous function given by Ψ(y) = H α yα if y ∈ Y α
In this section, our aim is first to streamline the conditions in Proposition 4 and Proposition 5. Proposition 6. Consider an LCS of the form (1) satisfying Assumption 3. Suppose that F is a P -matrix. Then, the following statements hold:
4. FROM LCS TO CPAS
z(q)αc = 0.
p
5. MAIN RESULTS
iv. λ ∈ R, λ > 0, z ∈ Rn , wi ∈ Cr (Yi )− , and T A + H i C − λI B + H i D z = 0 ∀ i =⇒ z = 0. wi C D
and
(8)
0|α|×|αc | yα −(Fαα )−1 0}. (9) Y = {y ∈ R | yα c −Fαc α (Fαα )−1 I|αc |
iii. λ ∈ C+ , z ∈ Cn , wi ∈ Cr (Yi )⊥ , and ∗ A + H i C − λI B + H i D z = 0 ∀ i =⇒ z = 0. wi C D
z(q)α = −(Fαα )−1 qα
3
Now, we are in a position to prove Proposition 6.
(7)
Proof of Proposition 6: The idea of the proof is to apply Proposition 4 for controllability and Proposition 5
where α ⊆ {1, 2, . . . , p}, 21
2019 IFAC SSSC 4 Sinaia, Romania, September 9-11, 2019
M.K. Camlibel / IFAC PapersOnLine 52-17 (2019) 1–6
(i) Let λ ∈ C and z ∈ Cn . Then, for all α ⊆ {1, 2, . . . , p} −1 −1 z ∗ A − E•α Fαα Cα• − λI B − E•α Fαα Dα• = 0
for stabilizability. First note that, we have Ψ(0) = 0 due to (7)-(9). Therefore, hypotheses of both propositions are satisfied.
if and only if
It follows from (9) that Y α is a polyhedral cone for all α ⊆ {1, 2, . . . , p}. As such, we have Cr (Y α ) = Y α . Moreover, it follows from Lemma 7 that Y ⊥ = {0} for all α ⊆ {1, 2, . . . , p} since the matrix 0|α|×|αc | −(Fαα )−1 −Fαc α (Fαα )−1 I|αc |
z ∗ [A − λI B E] = 0.
(ii) Let λ ∈ R and z ∈ Rn . Then, for all α ⊆ {1, 2, . . . , p} there exists wα ∈ (Y α )− such that T −1 −1 z Cα• − λI B − E•α Fαα Dα• A − E•α Fαα =0 wα C D
is nonsingular.
if and only if there exists w 0 such that T T z A − λI B E z = 0 and 0. w C D F w
As (8) does not contain translation term, that is g α = 0, we see that the condition (iv) of Proposition 4 is implied by the condition (ii) of Proposition 4 whereas the condition (iii) of Proposition 4 is implied by the condition (i) of Proposition 4. Moreover, the same reasoning holds if we replace Proposition 4 by Proposition 5 above.
Proof. For the ‘only if’ part of (i), note that the choice α = ∅ results in
Note that α
H C=
−1 −E•α Fαα Cα•
and
α
H D=
z ∗ [A − λI B] = 0.
−1 −E•α Fαα Dα• .
(10)
Therefore, it remains to show that z ∗ E = 0. Let α = {1, 2, . . . , p}. Then, we have z ∗ A − EF −1 C − λI B − EF −1 D = 0.
Therefore, we can conclude that (a) is implied by Proposition 4 and (b) by Proposition 5.
By using (10), we further obtain 0 = z ∗ EF −1 C EF −1 D = z ∗ EF −1 [C D] .
Proposition 6 requires checking the involved conditions for all index sets α ⊆ {1, 2, . . . , p}. It turns out that the structure of LCSs can be exploited to obtain more streamlined conditions that can be checked much more easily as stated next. Theorem 8. Consider an LCS of the form (1) satisfying Assumption 3. Suppose that F is a P -matrix. Then, the following statements hold:
For the ‘if’ part of (i), note that z ∗ E = 0 implies that z ∗ E•α = 0 for all α ⊆ {1, 2, . . . , p}. Therefore,
(a) The system (2) is controllable if and only if the following two implications hold:
implies
Note that [C D] is full row rank due to Assumption 3. Therefore, it follows from (11) that z ∗ E = 0.
z ∗ [A − λI B E] = 0
−1 −1 z ∗ A − E•α Fαα Cα• − λI B − E•α Fαα Dα• = 0
i. (A, [B E]) is controllable. ii. λ ∈ R, z ∈ Rn , w 0, and T A − λI B z = 0 and C D w
(11)
for all α ⊆ {1, 2, . . . , p}.
For the ‘only if’ part of (ii), we claim that w∅ 0, T z A − λI B = 0, C D w∅
T E z 0 F w
implies z = 0 and w = 0.
(12)
and
(b) The system (2) is stabilizable if and only if the following two implications hold: i. (A, [B E]) is stabilizable.
z w∅
T E 0. F
(13)
Since (12) readily follows from the choice of α = ∅, it remains to show that w∅ 0 and (13) holds. To do so, we first note that Y ∅ = Rp+ due to (9). Since w∅ ∈ (Y ∅ )− , ¯ = {1, 2, . . . , p} and we see that w∅ 0. For the rest, let α w ¯ = wα¯ . Then, we have T z A − EF −1 C − λI B − EF −1 D = 0. w ¯ C D
ii. λ ∈ R, λ 0, z ∈ Rn , w 0, and T T A − λI B z E z = 0 and 0 C D w F w implies z = 0 and w = 0. The proof of this theorem will rely on the following intermediate result. Lemma 9. Suppose that (A, B, C, D) satisfies Assumption 3 and F is a P -matrix. Then, the following statements hold:
By using (12), we obtain T z −EF −1 C −EF −1 D = 0. w ¯ − w∅ C D 22
2019 IFAC SSSC Sinaia, Romania, September 9-11, 2019
M.K. Camlibel / IFAC PapersOnLine 52-17 (2019) 1–6
Hence, we see that T w ¯ − (w∅ )T − z T EF −1 [C D] = 0.
η ∈ (Y α )− ⇐⇒ η =
Y α¯ = {y | −F −1 y 0}
Proof of Theorem 8: In view of Lemma 9, the conditions in the statement (a) of Proposition 6 are equivalent with the conditions in the statement of Theorem 8. This proves the controllability claims of Theorem 8. The claims about stabilizability also follows from Lemma 9 in a similar fashion.
(Y α¯ )− = {η | η = (F T )−1 ζ for some 0 ζ}.
Therefore, w ¯ = (F T )−1 ζ¯ for some ζ¯ 0. Note that w ¯ T F = ζ¯T 0.
6. CONCLUSIONS In this paper, we studied controllability and stabilizability of linear complementarity systems. The main results are Hautus-like necessary and sufficient conditions for both controllability and stabilizability. By working under the assumption that the transfer matrix of the underlying linear system from inputs to one of the complementarity variable is right invertible, we first showed that the existing tests for controllability and stabilizability of continuous piecewise affine systems can be streamlined. Based on these streamlined tests, we presented tailor-made tests for linear complementarity systems.
(14)
(15)
For all α ⊆ {1, 2, . . . , p}, define wα by wααc = wαc
and
−1 (16) (wαα )T = (wα )T + z T E•α Fαα
With this definition, we have T −1 −1 z Dα• Cα• − λI B − E•α Fαα A − E•α Fαα wα C D T −1 −1 z A − E•α Fαα Cα• − λI B − E•α Fαα Dα• = wαα Cα• Dα• α w αc C αc • D αc • T A − λI B z = 0. = C D w
Future research directions include relaxing the rightinvertibility assumption on the one hand and studying linear complementarity systems that cannot be cast as continuous piecewise affine systems on the other hand. REFERENCES Bertsekas, D.P. (2009). Convex Optimization Theory. Athena Scientific. Brogliato, B. (2003). Some perspectives on the analysis and control of complementarity systems. IEEE Transactions on Automatic Control, 48(6), 918–935. Camlibel, M.K. (2001). Complementarity Methods in the Analysis of Piecewise Linear Dynamical Systems. Dissertation Series of Center for Economic Research. Tilburg University, The Netherlands. (ISBN: 90 5668 079 X). Camlibel, M.K. (2007). Popov-Belevitch-Hautus type controllability tests for linear complementarity systems. Systems and Control Letters, 56, 381–387. Camlibel, M.K., Heemels, W.P.M.H., and Schumacher, J.M. (2002). On linear passive complementarity systems. European Journal of Control, 8(3). Camlibel, M.K., Heemels, W.P.M.H., and Schumacher, J.M. (2008a). Algebraic necessary and sufficient conditions for the controllability of conewise linear systems. IEEE Transactions on Automatic Control, 53(3), 762– 774. Camlibel, M.K., Heemels, W.P.M.H., and Schumacher, J.M. (2008b). A full characterization of stabilizability
It remains to show that wα ∈ (Y α )− . From (15), we see that z T E•α + wαT Fαα + wαTc Fαc α 0.
By using (16), we get −1 z T E•α + (wαα )T − z T E•α Fαα Fαα + (wααc )T Fαc α 0. This leads to
(wαα )T Fαα + (wααc )T Fαc α 0.
(17)
In addition, we have wααc 0
Now, we are in a position to prove Theorem 8.
due to (9), it follows from Lemma 7 that
T E z 0. F w
(Fαα )−T (Fαα )−T (Fαc α )T ζ. 0 −I
Therefore, (19) implies that wα ∈ (Y α )− .
As such, it is enough to show that w ¯ T F 0. Note that α ¯ − w ¯ ∈ (Y ) . Since
and
for some ζ 0. Note that (Fαα )−T (Fαα )−T (Fαc α )T (Fαα )T (Fαc α )T = I. 0 −I 0 −I
Since [C D] is full row rank due to Assumption 3, we obtain T z E =w ¯ T F. ∅ F w
For the ‘if’ part of (ii), w be such that w 0, T z A − λI B = 0, w C D
5
(18)
since w 0. By combining (17) and (18), we can write α (Fαα )T (Fαc α )T wα 0. (19) wααc 0 −I It follows from (9) and Lemma 7 that 23
2019 IFAC SSSC 6 Sinaia, Romania, September 9-11, 2019
M.K. Camlibel / IFAC PapersOnLine 52-17 (2019) 1–6
of bimodal piecewise linear systems with scalar inputs. Automatica, 44(5), 1261–1267. Camlibel, M.K., Heemels, W.P.M.H., van der Schaft, A.J., and Schumacher, J.M. (2003). Switched networks and complementarity. IEEE Transactions on Circuits and Systems I, 50(8), 1036–1046. Camlibel, M.K., Iannelli, L., and Vasca, F. (2004). Modelling switching power converters as complementarity systems. In Proc. of the 43th IEEE Conference on Decision and Control. Paradise Islands (Bahamas). Camlibel, M.K., Iannelli, L., and Vasca, F. (2014). Passivity and complementarity. Mathematical Programming, 145(1-2), 531–563. Camlibel, M.K. and Schumacher, J.M. (2002). Existence and uniqueness of solutions for a class of piecewise linear dynamical systems. Linear Algebra and its Applications, 351-352, 147–184. Cottle, R.W., Pang, J.S., and Stone, R.E. (1992). The Linear Complementarity Problem. Academic Press, Inc., Boston. Facchinei, F. and Pang, J.S. (2002). Finite Dimensional Variational Inequalities and Complementarity Problems. Springer. Heemels, W.P.M.H. and Brogliato, B. (2003). The complementarity class of hybrid dynamical systems. European Journal of Control, 26(4), 651–677. Heemels, W.P.M.H. and Camlibel, M.K. (2008). Null controllability of discrete-time linear systems with input and state constraints. In 47th IEEE Conference on Decision and Control. Cancun (Mexico). Heemels, W.P.M.H., Camlibel, M.K., and Schumacher, J.M. (2002). On the dynamic analysis of piecewiselinear networks. IEEE Trans. on Circuits and Systems– I: Fundamental Theory and Applications, 49(3), 315– 327. Heemels, W.P.M.H., Schumacher, J.M., and Weiland, S. (2000). Linear complementarity systems. SIAM J. Appl. Math., 60(4), 1234–1269. Saberi, A., Han, J., and Stoorvogel, A.A. (2002). Constrained stabilization problems for linear plants. Automatica, 38, 639–654. Saberi, A., Stoorvogel, A.A., Shi, G., and Sannuti, P. (2003). Output regulation of linear plants subject to constraints. International Journal of Control, 76(2), 149–164. Schumacher, J.M. (2004). Complementarity systems in optimization. Mathematical Programming Series B, 101, 263–295. Shi, G., Saberi, A., Stoorvogel, A.A., and Sannuti, P. (2003). Output regulation of discrete-time linear plants subject to state and input constraints. International Journal of Robust and Nonlinear Control, 13, 691–713. Sun, Z., Ge, S., and Lee, T. (2002). Controllability and reachability criteria for switched linear systems. Automatica, 38(5), 775–786. Thuan, L.Q. and Camlibel, M.K. (2014). Controllability and stabilizability of piecewise affine dynamical systems. SIAM Journal on Control and Optimization, 52(3), 1914–1934. van der Schaft, A.J. and Schumacher, J.M. (1996). The complementary-slackness class of hybrid systems. Mathematics of Control, Signals and Systems, 9, 266–301.
van der Schaft, A.J. and Schumacher, J.M. (1998). Complementarity modelling of hybrid systems. IEEE Transactions on Automatic Control, 43(4), 483–490. van der Schaft, A.J. and Schumacher, J.M. (2000). An Introduction to Hybrid Dynamical Systems. SpringerVerlag, London.
24
ScienceDirect
On On On On
IFAC PapersOnLine 52-17 (2019) 1–6
the the Controllability Controllability and and Stabilizability Stabilizability the Controllability and Stabilizability Linear Complementarity Systems the Controllability and Stabilizability Linear Complementarity Systems Linear Complementarity Systems Linear Complementarity Systems M.K. Camlibel ∗∗
of of of of
M.K. Camlibel ∗∗ M.K. M.K. Camlibel Camlibel M.K. Camlibel ∗ Bernoulli Institute of Mathematics, Bernoulli Institute of Mathematics, Computer Computer Science, Science, and and Artificial Artificial Bernoulli Institute University of Mathematics, Mathematics, Computer Science, and Artificial Artificial Intelligence, of Groningen, The Netherlands Bernoulli Institute of Computer Science, and Intelligence, University of Groningen, The Netherlands ∗ Bernoulli Institute of Mathematics, Computer Science, and Artificial Intelligence, University of Groningen, Groningen, The Netherlands (e-mail: [email protected]). Intelligence, University of The Netherlands (e-mail: [email protected]). Intelligence, University of Groningen, The Netherlands (e-mail: [email protected]). (e-mail: [email protected]). (e-mail: [email protected]). Abstract: This This paper paper studies studies controllability controllability and and stabilizability stabilizability of of linear linear complementarity complementarity Abstract: Abstract: This paper studies controllability and stabilizability of linear complementarity systems that can be cast as continuous piecewise affine dynamical systems. Under aa certain certain Abstract: This paper studies controllability and stabilizability of linear complementarity systems that can be caststudies as continuous piecewise dynamical Under Abstract: controllability andaffine stabilizability ofsystems. linear complementarity systems that can be piecewise affine dynamical systems. Under aa certain right-invertibility assumption, we aa la necessary and sufficient conditions for systems thatThis can paper be cast cast as as continuous continuous affine dynamical Under certain right-invertibility assumption, we present presentpiecewise la Hautus Hautus necessary andsystems. sufficient conditions for systems that can assumption, beand caststabilizability. as continuous affinenecessary dynamical Under a certain right-invertibility assumption, we present presentpiecewise a la la Hautus Hautus necessary andsystems. sufficient conditions for both controllability right-invertibility we a and sufficient conditions for both controllability and stabilizability. right-invertibility assumption, we present a la Hautus necessary and sufficient conditions for both and both controllability controllability and stabilizability. stabilizability. Copyright © 2019. The Authors. Published by Elsevier Ltd. All rights reserved. both controllability and stabilizability. Keywords: Controllability, Controllability, stabilizability, stabilizability, linear linear complementarity complementarity systems, systems, continuous continuous Keywords: Keywords:affine Controllability, stabilizability, linear complementarity complementarity systems, systems, continuous continuous piecewise dynamical systems Keywords: Controllability, stabilizability, linear piecewise dynamical systems Keywords: Controllability, linear complementarity systems, continuous piecewise affine affine dynamical stabilizability, systems piecewise affine dynamical systems piecewise affine dynamical systems 1. INTRODUCTION INTRODUCTION affine systems systems and and the the results results in in Thuan Thuan and and Camlibel Camlibel 1. affine 1. INTRODUCTION INTRODUCTION affine systems and the results in Thuan Thuan necessary and Camlibel Camlibel (2014) in order to present tailor-made and 1. affine systems and the results in and (2014) in order to present tailor-made necessary and 1. INTRODUCTION affine systems and the results in Thuan and Camlibel (2014) in order to present tailor-made necessary and sufficient conditions for controllability and stabilizability (2014) in order to present tailor-made necessary and Controllability is is not not only only a fundamental fundamental system-theoretic system-theoretic sufficient conditions for controllability and stabilizability Controllability (2014) order to for present tailor-made and sufficient conditions for controllability andnecessary stabilizability of linear linearincomplementarity complementarity systems. conditions controllability and stabilizability Controllability is not not only aa a fundamental fundamental system-theoretic property but also a prerequisite virtually for all all control control sufficient of systems. Controllability is only system-theoretic property but also a prerequisite virtually for sufficient conditions for controllability and stabilizability of linear complementarity systems. of linear complementarity systems. Controllability is not only a and fundamental system-theoretic property but also also prerequisite virtually for for all controlcontrol The organization of the paper is as follows. In Section 2, we design problems. problems. Kalman’s Hautus’ tests property but aa prerequisite virtually for all control design Kalman’s and Hautus’ tests for controlof linear complementarity systems. The organization of the paper is as follows. In Section 2, we property butamong alsoKalman’s athe prerequisite virtually for all controlcontrol design problems. Kalman’s and Hautus’ Hautus’ tests for controllability are classical results of linear system The The organization of the of paper is as as follows. In In Section Section 2, we we design problems. and tests for will introduce introduce theof class linear complementarity systems. organization the paper is follows. 2, lability are among the classical results of linear system will the class of linear complementarity systems. design problems. Kalman’s and Hautus’ tests for controllability are among the classical results of linear system theory. Apart from finite-dimensional linear time-invariant The organization of the paper is as follows. In Section 2, we will introduce the class of linear complementarity systems. lability are among the classical results of linear system Section 3 is devoted to continuous piecewise affine dynamwill introduce the class of linear complementarity systems. theory. Apart from finite-dimensional linear time-invariant Section 3 is devoted to continuous piecewise affine dynamlability are among the classical results of time-invariant linear system will theory. Apart from finite-dimensional linear time-invariant systems, there are only which easily introduce theHautus-like class of linear complementarity Section is devoted devoted to continuous continuous piecewise affine affinesystems. dynamtheory. linear ical systems systems and controllability/stabilizability 33 is to piecewise dynamsystems,Apart therefrom are finite-dimensional only a a few few cases cases for for which easily verver- Section ical and Hautus-like controllability/stabilizability theory. Apart linear time-invariant systems, therefrom are finite-dimensional only few cases cases for which easily necver- Section ifiable controllability controllability conditions exist. For instance, 3these is devoted to continuous piecewise affine dynamical systems and Hautus-like controllability/stabilizability systems, there are only aa few for which easily vertests for systems. Subsequently, Section 4 will discuss ical systems and Hautus-like controllability/stabilizability ifiable conditions exist. For instance, nectests for these systems. Subsequently, Section will discuss systems, there are only a few cases for which easily ver- ical ifiable controllability conditions exist. For instance, necessary controllability and sufficient conditions that are linear in nature nature systems Hautus-like controllability/stabilizability tests for these theseand systems. Subsequently, Section 444 will willsystems discuss ifiable conditions exist. For instance, necthe relationships between linear complementarity tests for systems. Subsequently, Section discuss essary and sufficient conditions that are linear in the relationships between linear complementarity systems ifiable controllability conditions exist. For instance, nec- tests essary and sufficient conditions that are linear in nature nature exist for for linear switched systemsthat withare state-independent for these systems. Subsequently, Section discuss the relationships between linear systems. complementarity systems essary and sufficient conditions linear in and relationships continuous piecewise affine This4 will be folfolthe between linear complementarity systems exist linear switched systems with state-independent and continuous piecewise affine systems. This will be essary and sufficient conditions that are linear in nature exist for linear switched systems with state-independent switching Sun et al. (2002). This result relies on the relationships between linear complementarity systems and continuous piecewise affine systems. Thisinwill will be folfolexist for linear systems with state-independent lowed by the main results and their proofs Section 5. and continuous piecewise affine systems. This be switching Sun etswitched al. (2002). This result relies on the fact fact the lowed by the main results and their proofs in Section 5. exist for switching linear systems with state-independent switching Sun et etswitched al.signal (2002). This result reliesason onanthe the fact and that the the canThis be considered considered extercontinuous piecewise affine systems. Thisin will be followed bythe thepaper main results and their proofs in Section 5. switching Sun al. (2002). result relies fact Finally, closes with the conclusions in Section 6. lowed by the main results and their proofs Section 5. that switching signal can be as an exterFinally, the paper closes with the conclusions in Section 6. switching Sun al.signal (2002). result relies fact lowed that the switching switching signal canThis be for considered asonan anthe external input. input. Yet et another instance which controllability thepaper maincloses results and proofs in 5. Finally,bythe the paper closes with thetheir conclusions in Section Section 6. 6. that the can be considered as exterFinally, with the conclusions in Section nal Yet another instance for which controllability that the switching signal can be considered as an external input. Yet another instance for which controllability admits a simple characterization is the case of continuous Finally, the paper closes with the conclusions in Section 6. nal input. Yet another instance for which controllability 2. LINEAR COMPLEMENTARITY SYSTEMS admits a simple characterization is the case of continuous 2. LINEAR COMPLEMENTARITY SYSTEMS nal input. Yet another instance for which admits simple characterization is the the case controllability of continuous continuous piecewise affine systems Thuan and Camlibel admits aa simple characterization is case of 2. LINEAR LINEAR COMPLEMENTARITY COMPLEMENTARITY SYSTEMS SYSTEMS 2. piecewise affine dynamical dynamical systems Thuan and Camlibel admits simple characterization is the case and of continuous piecewise affine dynamical dynamical systems Thuan and Camlibel Consider (2014). aContrary Contrary to linear linear systems switched systems, piecewise piecewise affine Thuan Camlibel 2. LINEAR COMPLEMENTARITY SYSTEMS the linear system (2014). to switched systems, piecewise Consider the linear system piecewise affine dynamical systems Thuan and piecewise Camlibel (2014). Contrary to linear linear switched systems, piecewise affine dynamical dynamical systems exhibit state-dependent switch- Consider (2014). Contrary to switched systems, Consider the the linear linear system system affine systems exhibit state-dependent switch(2014). Contrary to linear switched systems, piecewise affine dynamical systems exhibit state-dependent switchx(t) ˙ = Ax(t) (1a) ing phenomenon that makes the analysis much harder. affine dynamical systems exhibit state-dependent switchConsider the linear system x(t) ˙ = Ax(t) + + Bu(t) Bu(t) + + Ez(t) Ez(t) (1a) ing phenomenon that makes the analysis much harder. x(t) ˙ = Ax(t) + Bu(t) + Ez(t) (1a) affine dynamical systems exhibit state-dependent switching phenomenon that makes the analysis much harder. Nevertheless, Hautus-like tests can be devised by exploitx(t) ˙ = Ax(t) + Bu(t) + Ez(t) (1a) ing phenomenon that makes much harder. Nevertheless, Hautus-like teststhe can analysis be devised by exploitw(t) = Cx(t) + Du(t) + F z(t) (1b) w(t) = Ax(t) Cx(t) + Bu(t) Du(t) + + Ez(t) F z(t) (1b) x(t) ˙ (1a) ing phenomenon that makes the much harder. Nevertheless, Hautus-like tests can analysis be devised devised by exploitexploitthe structure imposed by continuity continuity of the the vector field. Nevertheless, Hautus-like tests can be by w(t) = = Cx(t) Cx(t) + + Du(t) Du(t) + +F F z(t) z(t) (1b) ing the structure imposed by of vector field. w(t) (1b) n m p+p Nevertheless, Hautus-like tests can be devised by exploiting the structure structure imposed by by continuity of oftest the also vector field. where x ∈ Rnw(t) Moreover, this Hautus-like controllability extends ing the imposed continuity the vector field. , u ∈ R , and (z, w) ∈ R . When the m p+p = Cx(t) + Du(t) + F z(t) (1b) Moreover, this Hautus-like controllability test also extends where x ∈ R n , u ∈ Rm m , and (z, w) ∈ Rp+p p+p . When the ing the structure imposed by continuity oftest the also vector field. where Moreover, this Hautus-like Hautus-like controllability test also extends n to stabilizability. where x ∈ R , u ∈ R , and (z, w) ∈ R . When the Moreover, this controllability extends external variables (z, w) satisfy the so-called complemenx ∈ R , u ∈ R , and (z, w) ∈ R . When the to stabilizability. external variables (z, w)m satisfy the so-called complemenMoreover, this Hautus-like controllability test also extends where to stabilizability. stabilizability. x variables ∈ Rn , u (z, ∈ R ,satisfy and (z, w)so-called ∈ Rp+pcomplemen. When the external w) the to tarity relations external variables (z, w) satisfy the so-called complemenIn stabilizability. this paper, paper, we we study study a particular particular class class of of continuous continuous tarity relations to In this external variables (z, w) satisfy the so-called complementarity relations In this paper, paper, we study aa a systems, particular class linear of continuous continuous piecewise affine dynamical namely comple- tarity relations In this we study particular class of 00 (1c) piecewise affine dynamical systems, namely linear completarity relations z(t) z(t) ⊥ ⊥ w(t) w(t) 00 (1c) In this paper, we study a systems, particular class linear of are continuous piecewise affine dynamical systems, namely linear complementarity systems. Complementarity systems encounpiecewise affine dynamical namely comple z(t) z(t) ⊥ ⊥ w(t) w(t) 00 (1c) 00 (1c) mentarity systems. Complementarity systems are encounand ⊥ piecewise dynamical systems, namely linear complementarity systems. Complementarity systems are encoun- where tered in in aaaffine multitude of applications applications in various disciplines mentarity systems. Complementarity systems are encoun0 are z(t)componentwise ⊥ w(t) 0 (1c) where the the inequalities inequalities are componentwise and ⊥ denotes denotes tered multitude of in various disciplines T the componentwise and ⊥ denotes orthogonality, that is zzare ⊥ w if and only if zz T w 0. We mentarity systems. Complementarity encoun- where tered in aa and multitude of applications applications insystems variousare disciplines where the inequalities inequalities are componentwise and ⊥= denotes of science engineering including electrical/mechanical tered in multitude of in various disciplines orthogonality, that is ⊥ w if and only if w = 0. We T ⊥ denotes of science and engineering including electrical/mechanical T where the inequalities are componentwise and orthogonality, that is z ⊥ w if and only if z w = 0. We call the overall system (1) a linear complementarity system tered in a multitude of applications in various disciplines of science and engineering including electrical/mechanical orthogonality, that is z ⊥ w if and only if z w = 0. We engineering and operations research van der Schaft and of science and engineering the overall system (1) a linear complementarity system engineering and operationsincluding researchelectrical/mechanical van der Schaft and call orthogonality, is z(1) if and only if z T w = system 0. We call the overall overall that system (1)⊥aawlinear linear complementarity system (LCS). of science and engineering including engineering and operations researchelectrical/mechanical van der derHeemels Schaft and call the system complementarity Schumacher (1998, 2000); Brogliato (2003); engineering and operations research van Schaft (LCS). Schumacher (1998, 2000); Brogliato (2003); Heemels and the overall system (1) a linear complementarity system (LCS). engineering and operations research van derHeemels Schaft and Schumacher (1998, 2000); Brogliato Brogliato (2003); Heemels and call (LCS). Brogliato (2003); (2003); Camlibel et al. al. (2003); (2004); Schumacher Schumacher (1998, 2000); Brogliato Camlibel et (2004); Schumacher (LCS). Schumacher (1998, 2000); Brogliato (2003); Heemels and Brogliato (2003); Camlibel et al. (2004); Schumacher 3. (2004). We We(2003); refer to to van van der Schaft Schaft and(2004); Schumacher (1996); Brogliato Camlibel et al. Schumacher 3. CONTINUOUS CONTINUOUS PIECEWISE PIECEWISE AFFINE AFFINE SYSTEMS SYSTEMS (2004). refer der and Schumacher (1996); 3. CONTINUOUS PIECEWISE AFFINE SYSTEMS SYSTEMS Brogliato (2003); Camlibel et al. (2004); Schumacher (2004). We refer to van der Schaft and Schumacher (1996); Heemels et al. (2000); Camlibel (2001); Camlibel and 3. CONTINUOUS PIECEWISE AFFINE (2004). van der Schaft and Schumacher (1996); HeemelsWe etrefer al. to (2000); Camlibel (2001); Camlibel and 3.order CONTINUOUS PIECEWISE AFFINEaffine SYSTEMS (2004). We van der Schaft and Schumacher (1996); Heemels etrefer al. to (2000); Camlibel (2001); Camlibel and Schumacher (2002); Camlibel et al. (2003); Heemels et al. In to introduce continuous piecewise Heemels et al. (2000); Camlibel (2001); Camlibel and In order to introduce continuous piecewise affine dynamidynamiSchumacher (2002); Camlibel et al. (2003); Heemels et al. Heemels et (2002); al. (2000); Camlibel Camlibel Schumacher (2002); Camlibel etCamlibel al. (2001); (2003); Heemels etand al. In In order order to introduce introduce continuous piecewiseaffine affinefunctions dynami(2002); Camlibel et al. (2002); (2007); Camlibel cal systems, we first begin with piecewise Schumacher Camlibel et al. (2003); Heemels et al. to continuous piecewise affine dynamical systems, we first begin with piecewise affine functions (2002); Camlibel et al. (2002); Camlibel (2007); Camlibel Schumacher (2002); Camlibel etCamlibel al. the (2003); Heemels et al. In order toproperties. introduce continuous piecewiseaffine affinefunctions dynami(2002); Camlibel et for al. (2002); Camlibel (2007); Camlibel cal systems, we first begin with piecewise et al. (2008b, 2014) the work on analysis of general and their (2002); Camlibel et al. (2002); (2007); Camlibel cal systems, we first begin with piecewise affine functions and their properties. et al. (2008b, 2014) for the work on the analysis of general (2002); Camlibel et al. (2002); Camlibel (2007); Camlibel cal systems, we first begin with piecewise affine functions et al. (2008b, 2014) for the work on the analysis of general and their properties. LCSs. et al. (2008b, 2014) for the work on the analysis of general and their properties. ν LCSs. R is said to be affine if ψ et al. (2008b, 2014) for the work on the analysis of general A and their properties. LCSs. → if there there A function function ψ :: R Rννν → LCSs. is said to be affine ×νR → is said to be affine if A function ψ : R Under certain conditions, a linear complementarity system and a vector g ∈ R that exist a matrix H ∈ R ×νR such → R is said to be affine if there there A function ψ : R LCSs. and a vector g ∈ R such that exist a matrix H ∈ν R×ν Under certain conditions, a linear complementarity system ν ×νR φsuch isRaasaid tofunction be affine if called there A function ψ :H Under certain conditions, linear complementarity system ψ(x) and gg ∈ R that exist aa=matrix ∈ can cast a piecewise affine Hx gg R for all x ∈ A is ν .vector Under aa linear complementarity system and vector ∈ R such that exist ∈R R→ can be becertain cast as asconditions, a continuous continuous piecewise affine dynamical dynamical ψ(x) =matrix Hx + +H for all x ∈ R . A function φ is called ×ν ν ν Under a linear complementarity system piecewise and g family ∈ Rφ that exist a= ∈ there Rall can becertain cast asconditions, aexploit continuous piecewise affine dynamical ψ(x) =matrix Hx +H g iffor for all x xexists ∈ R Ra a..vector Afinite function φsuch is called called system. We will the structure of such piecewise affine of affine can be cast as a continuous piecewise affine dynamical ψ(x) Hx + g ∈ A function is piecewise affine if there exists ν a finite family of affine system. We will exploit the structure of such piecewise can be cast as aexploit continuous piecewise of ψ(x) = Hx + g if all xexists ∈ R aa. Afinite function is called system. We will will exploit the structure structure ofaffine suchdynamical piecewise piecewise piecewise affine ifforthere there exists finite familyφ of of affine system. We the such piecewise affine family affine system. will ©exploit the structure of such piecewise piecewise affine if there exists a finite family of affine 2405-8963We Copyright 2019. The Authors. Published by Elsevier Ltd. All Copyright © 2019 IFAC 19rights reserved. ∗ ∗ ∗ ∗
Copyright © under 2019 IFAC 19 Control. Peer review responsibility of International Federation of Automatic Copyright © 2019 2019 IFAC IFAC 19 Copyright © 19 10.1016/j.ifacol.2019.11.017 Copyright © 2019 IFAC 19
2019 IFAC SSSC 2 Sinaia, Romania, September 9-11, 2019
M.K. Camlibel / IFAC PapersOnLine 52-17 (2019) 1–6
functions ψi : Rν → R with i = 1, 2, . . . , r such that φ(x) ∈ {ψ1 (x), ψ2 (x), . . . , ψr (x)} for all x ∈ Rν .
assumption is that the number of inputs should be at least as the number of outputs. Whenever this is the case, right invertibility is generically satisfied.
Continuous piecewise affine functions admit useful geometrical representations that will be used later on this paper. To elaborate, we first need to introduce some terminology. A set of Rν is called a polyhedron if it is the intersection of a finite family of closed half-spaces or hyperplanes in Rν . If P ⊆ Rν is a polyhedron, then there exist a matrix P ∈ Rk×ν and a vector q ∈ Rk such that P = {x ∈ Rν | P x q}. A finite collection Ξ of polyhedra of Rν is said to be a polyhedral subdivision of Rν if (see Facchinei and Pang (2002) for more details)
A complete characterization of controllability and stabilizability of CPASs are established in Thuan and Camlibel (2014). In order to quote the results of Thuan and Camlibel (2014), we need to introduce the following terminology. For a nonempty convex set S ⊆ Rn , S − denotes the dual cone S − := {ξ ∈ Rn | ξ T x 0 for all x ∈ S},
S ⊥ denotes orthogonal complement in Cn , and Cr (S) denotes the recession cone
i. the union of all polyhedra in Ξ is equal to R , ii. each polyhedron in Ξ is of dimension ν, and iii. the intersection of any two polyhedra in Ξ is either empty or a common proper face of both polyhedra ν
Cr (S) := {ξ ∈ Rn | z + λx ∈ Λ for all λ 0 and z ∈ S}. Necessary and sufficient conditions for controllability of CPASs are stated next. Proposition 4. (Thuan and Camlibel (2014),Thm. 3.4]). Consider a CPAS (2) satisfying Assumption 3. Suppose that the origin is contained in the closure of the convex hull of the image of the function Φ. Let Ξ = {Y 1 , Y 2 , . . . , Y h } be a polyhedral subdivision of Rp induced by Φ. Also let
As stated next, polyhedral subdivisions appear in the description of continuous piecewise affine functions. Proposition 1. (Facchinei and Pang (2002), Prop. 4.2.1). A continuous function ψ : Rν → R is piecewise affine if and only if there exist a polyhedral subdivision Ξ = {Ξ1 , Ξ2 , . . . , Ξr } of Rν and a finite family of affine functions ψi : Rν → R with i = 1, 2, . . . , r such that ψ coincides ψi on Ξi for every i = 1, 2, . . . , r.
Φ(y) = H i y − g i whenever y ∈ Y i .
Then, the system (2) is controllable if and only if the following four implications hold:
By a continuous piecewise affine system (CPAS), we mean a dynamical system of the form x(t) ˙ = Ax(t) + Bu(t) + Φ(y(t))
(2a)
y(t) = Cx(t) + Du(t)
(2b)
i. λ ∈ C, z ∈ Cn , z ∗ g i = 0, and z ∗ A + H i C − λI B + H i D = 0 ∀ i =⇒ z = 0.
ii. λ ∈ R, z ∈ Rn , (z, wi ) ∈ ({g i } × Y i )− , and T A + H i C − λI B + H i D z = 0 ∀ i =⇒ z = 0. i w C D
where x ∈ R is the state, u ∈ R is the input, y ∈ Rp is the output, all involved matrices are of appropriate sizes, and Φ : Rp → Rn is a continuous piecewise affine function. n
(3)
m
Since the piecewise affine function Φ is continuous, the right-hand side of (2a) is globally Lipschitz (see e.g. (Facchinei and Pang, 2002, Proposition 4.2.2)). Thus, it follows from the theory of ordinary differential equations that the system (2) must admit a unique solution for each initial state x0 and locally integrable input u. We denote this solution by xu (t; x0 ), and denote the corresponding output by y u (t; x0 ).
iii. λ ∈ C, Re(λ) = 0, z ∈ Cn , wi ∈ Cr (Yi )⊥ , and ∗ A + H i C − λI B + H i D z = 0 ∀ i =⇒ z = 0. wi C D iv. λ ∈ R, λ = 0, z ∈ Rn , wi ∈ Cr (Yi )− , and T A + H i C − λI B + H i D z = 0 ∀ i =⇒ z = 0. wi C D
Next, we define controllability and stabilizability for CPASs. Definition 2. We say that the system (2) is
Stabilizability of CPASs can also be characterized by similar spectral conditions. Proposition 5. ((Thuan and Camlibel, 2014, Thm. 3.6)). Consider a CPAS (2) satisfying Assumption 3. Suppose that Φ(0) = 0. Let Ξ = {Y 1 , Y 2 , . . . , Y h } be a polyhedral subdivision of Rp induced by Φ. Also let
• controllable if for any two states x0 , xf there exist T > 0 and a locally integrable input u such that xu (T ; x0 ) = xf . • stabilizable if for any initial state x0 there exists a locally integrable input u such that lim xu (t; x0 ) = 0. t→∞
Φ(y) = H i y − g i whenever y ∈ Y i .
In this paper, the following blanket assumption will be in force. Assumption 3. The transfer matrix D + C(sI − A)−1 B is right invertible as a rational matrix.
(4)
Then, the system (2) is stabilizable if and only if the following four implications hold: i. λ ∈ C, Re(λ) 0, z ∈ Cn , z ∗ g i = 0 and z ∗ A + H i C − λI B + H i D = 0 ∀ i =⇒ z = 0.
Similar invertibility assumptions are common in the analysis of linear systems with output constraints (see e.g. Saberi et al. (2002); Shi et al. (2003); Saberi et al. (2003); Heemels and Camlibel (2008)). The main restriction of the
ii. λ ∈ R, λ 0, z ∈ Rn , (z, wi ) ∈ ({g i } × Y i )− , and 20
2019 IFAC SSSC Sinaia, Romania, September 9-11, 2019
z wi
T
M.K. Camlibel / IFAC PapersOnLine 52-17 (2019) 1–6
A + H i C − λI B + H i D = 0 ∀ i =⇒ z = 0. C D
−1 H α = −E•α Fαα yα ,
and
α
(a) The system (2) is controllable if and only if the following two implications hold:
A linear complementarity system of the form (1) can be cast as a CPAS of the form (2) when F is a P -matrix, that is a matrix whose principal minors are all positive. It is well-known that every positive definite matrix is a P -matrix (see for instance (Cottle et al., 1992, Thm. 3.1.6 and Thm. 3.3.7)). In particular, P -matrices play an important role in the context of linear complementarity problem, i.e. the problem of finding a p-vector z satisfying
i. λ ∈ C, z ∈ Cn , and −1 −1 z ∗ A − E•α Fαα Cα• − λI B − E•α Fαα Dα• = 0 for all α ⊆ {1, 2, . . . , p} implies z = 0.
ii. λ ∈ R, z ∈ Rn , wα ∈ (Y α )− , and T −1 −1 Cα• − λI B − E•α Fαα Dα• A − E•α Fαα z =0 α w C D
0 z ⊥ q + Fz 0 for a given p-vector q and a p × p matrix F . We denote the linear complementarity problem for a given vector q and a given matrix F by LCP(q, F ). If F is a P matrix, LCP(q, F ) admits a unique solution z(q) for every q ∈ Rp (see (Cottle et al., 1992, Thm. 3.3.7)). In addition, one can verify that for every q there exists an index set α ⊆ {1, 2, . . . , p} such that −(Fαα )−1 0|α|×|αc | qα 0 (5a) qα c −Fαc α (Fαα )−1 I|αc |
for all α ⊆ {1, 2, . . . , p} implies z = 0. (b) The system (2) is stabilizable if and only if the following two implications hold: i. λ ∈ C, Re(λ) 0, z ∈ Cn , and −1 −1 z ∗ A − E•α Fαα Cα• − λI B − E•α Fαα Dα• = 0 for all α ⊆ {1, 2, . . . , p} implies z = 0.
ii. λ ∈ R, λ 0, z ∈ Rn , wα ∈ (Y α )− , and T −1 −1 Cα• − λI B − E•α Fαα Dα• A − E•α Fαα z =0 wα C D
and the unique solution z of the LCP(q, F ) is given by (5b)
Here αc denotes the set {1, 2, . . . , p} \ α. Furthermore, the map q → z(q) is known to be Lipschitz continuous (Cottle et al., 1992, Lem. 7.3.10).
for all α ⊆ {1, 2, . . . , p} implies z = 0. To prove this proposition, we need the following auxiliary result. Lemma 7. Let C ⊆ Rp be a polyhedral cone given by C = {ξ ∈ Rp | M ξ 0} for some matrix M ∈ Rq×p . Then, C − = {η ∈ Rp | η = M T ζ for some 0 ζ ∈ Rq }. Moreover, if ker M T = {0}, then C ⊥ = {0}.
By using the Lipschitz continuity, it can be shown that Camlibel (2001) for every initial state x0 ∈ Rn and locally integrable input u, there exists a unique locally absolutely continuous x : R+ → Rn such that x(0) = x0 , (1a) is satisfied for almost all t 0, and (1b)-(1c) are satisfied for all t 0. Moreover, we can employ the parametrization given in (5) to conclude that there is oneto-one correspondence between the trajectories of the LCS (1) and those of the following conewise linear system (see Camlibel et al. (2008a) for more details) x(t) ˙ = Ax(t) + Bu(t) + Ψ y(t) (6a) y(t) = Cx(t) + Du(t)
Proof. The first statement readily follows from Farkas’ Lemma (see e.g. (Bertsekas, 2009, Prop. 2.3.1)). To prove the second statement, note that C ⊥ = −C − ∩ C − . Therefore, η ∈ C ⊥ if and only if there exist ζ1 0 and ζ2 0 such that η = M T ζ1 = M T ζ2 . This means that ζ1 − ζ2 ∈ ker M T and hence ζ1 = ζ2 = 0. Consequently, η = 0.
(6b)
where Ψ is a Lipschitz continuous function given by Ψ(y) = H α yα if y ∈ Y α
In this section, our aim is first to streamline the conditions in Proposition 4 and Proposition 5. Proposition 6. Consider an LCS of the form (1) satisfying Assumption 3. Suppose that F is a P -matrix. Then, the following statements hold:
4. FROM LCS TO CPAS
z(q)αc = 0.
p
5. MAIN RESULTS
iv. λ ∈ R, λ > 0, z ∈ Rn , wi ∈ Cr (Yi )− , and T A + H i C − λI B + H i D z = 0 ∀ i =⇒ z = 0. wi C D
and
(8)
0|α|×|αc | yα −(Fαα )−1 0}. (9) Y = {y ∈ R | yα c −Fαc α (Fαα )−1 I|αc |
iii. λ ∈ C+ , z ∈ Cn , wi ∈ Cr (Yi )⊥ , and ∗ A + H i C − λI B + H i D z = 0 ∀ i =⇒ z = 0. wi C D
z(q)α = −(Fαα )−1 qα
3
Now, we are in a position to prove Proposition 6.
(7)
Proof of Proposition 6: The idea of the proof is to apply Proposition 4 for controllability and Proposition 5
where α ⊆ {1, 2, . . . , p}, 21
2019 IFAC SSSC 4 Sinaia, Romania, September 9-11, 2019
M.K. Camlibel / IFAC PapersOnLine 52-17 (2019) 1–6
(i) Let λ ∈ C and z ∈ Cn . Then, for all α ⊆ {1, 2, . . . , p} −1 −1 z ∗ A − E•α Fαα Cα• − λI B − E•α Fαα Dα• = 0
for stabilizability. First note that, we have Ψ(0) = 0 due to (7)-(9). Therefore, hypotheses of both propositions are satisfied.
if and only if
It follows from (9) that Y α is a polyhedral cone for all α ⊆ {1, 2, . . . , p}. As such, we have Cr (Y α ) = Y α . Moreover, it follows from Lemma 7 that Y ⊥ = {0} for all α ⊆ {1, 2, . . . , p} since the matrix 0|α|×|αc | −(Fαα )−1 −Fαc α (Fαα )−1 I|αc |
z ∗ [A − λI B E] = 0.
(ii) Let λ ∈ R and z ∈ Rn . Then, for all α ⊆ {1, 2, . . . , p} there exists wα ∈ (Y α )− such that T −1 −1 z Cα• − λI B − E•α Fαα Dα• A − E•α Fαα =0 wα C D
is nonsingular.
if and only if there exists w 0 such that T T z A − λI B E z = 0 and 0. w C D F w
As (8) does not contain translation term, that is g α = 0, we see that the condition (iv) of Proposition 4 is implied by the condition (ii) of Proposition 4 whereas the condition (iii) of Proposition 4 is implied by the condition (i) of Proposition 4. Moreover, the same reasoning holds if we replace Proposition 4 by Proposition 5 above.
Proof. For the ‘only if’ part of (i), note that the choice α = ∅ results in
Note that α
H C=
−1 −E•α Fαα Cα•
and
α
H D=
z ∗ [A − λI B] = 0.
−1 −E•α Fαα Dα• .
(10)
Therefore, it remains to show that z ∗ E = 0. Let α = {1, 2, . . . , p}. Then, we have z ∗ A − EF −1 C − λI B − EF −1 D = 0.
Therefore, we can conclude that (a) is implied by Proposition 4 and (b) by Proposition 5.
By using (10), we further obtain 0 = z ∗ EF −1 C EF −1 D = z ∗ EF −1 [C D] .
Proposition 6 requires checking the involved conditions for all index sets α ⊆ {1, 2, . . . , p}. It turns out that the structure of LCSs can be exploited to obtain more streamlined conditions that can be checked much more easily as stated next. Theorem 8. Consider an LCS of the form (1) satisfying Assumption 3. Suppose that F is a P -matrix. Then, the following statements hold:
For the ‘if’ part of (i), note that z ∗ E = 0 implies that z ∗ E•α = 0 for all α ⊆ {1, 2, . . . , p}. Therefore,
(a) The system (2) is controllable if and only if the following two implications hold:
implies
Note that [C D] is full row rank due to Assumption 3. Therefore, it follows from (11) that z ∗ E = 0.
z ∗ [A − λI B E] = 0
−1 −1 z ∗ A − E•α Fαα Cα• − λI B − E•α Fαα Dα• = 0
i. (A, [B E]) is controllable. ii. λ ∈ R, z ∈ Rn , w 0, and T A − λI B z = 0 and C D w
(11)
for all α ⊆ {1, 2, . . . , p}.
For the ‘only if’ part of (ii), we claim that w∅ 0, T z A − λI B = 0, C D w∅
T E z 0 F w
implies z = 0 and w = 0.
(12)
and
(b) The system (2) is stabilizable if and only if the following two implications hold: i. (A, [B E]) is stabilizable.
z w∅
T E 0. F
(13)
Since (12) readily follows from the choice of α = ∅, it remains to show that w∅ 0 and (13) holds. To do so, we first note that Y ∅ = Rp+ due to (9). Since w∅ ∈ (Y ∅ )− , ¯ = {1, 2, . . . , p} and we see that w∅ 0. For the rest, let α w ¯ = wα¯ . Then, we have T z A − EF −1 C − λI B − EF −1 D = 0. w ¯ C D
ii. λ ∈ R, λ 0, z ∈ Rn , w 0, and T T A − λI B z E z = 0 and 0 C D w F w implies z = 0 and w = 0. The proof of this theorem will rely on the following intermediate result. Lemma 9. Suppose that (A, B, C, D) satisfies Assumption 3 and F is a P -matrix. Then, the following statements hold:
By using (12), we obtain T z −EF −1 C −EF −1 D = 0. w ¯ − w∅ C D 22
2019 IFAC SSSC Sinaia, Romania, September 9-11, 2019
M.K. Camlibel / IFAC PapersOnLine 52-17 (2019) 1–6
Hence, we see that T w ¯ − (w∅ )T − z T EF −1 [C D] = 0.
η ∈ (Y α )− ⇐⇒ η =
Y α¯ = {y | −F −1 y 0}
Proof of Theorem 8: In view of Lemma 9, the conditions in the statement (a) of Proposition 6 are equivalent with the conditions in the statement of Theorem 8. This proves the controllability claims of Theorem 8. The claims about stabilizability also follows from Lemma 9 in a similar fashion.
(Y α¯ )− = {η | η = (F T )−1 ζ for some 0 ζ}.
Therefore, w ¯ = (F T )−1 ζ¯ for some ζ¯ 0. Note that w ¯ T F = ζ¯T 0.
6. CONCLUSIONS In this paper, we studied controllability and stabilizability of linear complementarity systems. The main results are Hautus-like necessary and sufficient conditions for both controllability and stabilizability. By working under the assumption that the transfer matrix of the underlying linear system from inputs to one of the complementarity variable is right invertible, we first showed that the existing tests for controllability and stabilizability of continuous piecewise affine systems can be streamlined. Based on these streamlined tests, we presented tailor-made tests for linear complementarity systems.
(14)
(15)
For all α ⊆ {1, 2, . . . , p}, define wα by wααc = wαc
and
−1 (16) (wαα )T = (wα )T + z T E•α Fαα
With this definition, we have T −1 −1 z Dα• Cα• − λI B − E•α Fαα A − E•α Fαα wα C D T −1 −1 z A − E•α Fαα Cα• − λI B − E•α Fαα Dα• = wαα Cα• Dα• α w αc C αc • D αc • T A − λI B z = 0. = C D w
Future research directions include relaxing the rightinvertibility assumption on the one hand and studying linear complementarity systems that cannot be cast as continuous piecewise affine systems on the other hand. REFERENCES Bertsekas, D.P. (2009). Convex Optimization Theory. Athena Scientific. Brogliato, B. (2003). Some perspectives on the analysis and control of complementarity systems. IEEE Transactions on Automatic Control, 48(6), 918–935. Camlibel, M.K. (2001). Complementarity Methods in the Analysis of Piecewise Linear Dynamical Systems. Dissertation Series of Center for Economic Research. Tilburg University, The Netherlands. (ISBN: 90 5668 079 X). Camlibel, M.K. (2007). Popov-Belevitch-Hautus type controllability tests for linear complementarity systems. Systems and Control Letters, 56, 381–387. Camlibel, M.K., Heemels, W.P.M.H., and Schumacher, J.M. (2002). On linear passive complementarity systems. European Journal of Control, 8(3). Camlibel, M.K., Heemels, W.P.M.H., and Schumacher, J.M. (2008a). Algebraic necessary and sufficient conditions for the controllability of conewise linear systems. IEEE Transactions on Automatic Control, 53(3), 762– 774. Camlibel, M.K., Heemels, W.P.M.H., and Schumacher, J.M. (2008b). A full characterization of stabilizability
It remains to show that wα ∈ (Y α )− . From (15), we see that z T E•α + wαT Fαα + wαTc Fαc α 0.
By using (16), we get −1 z T E•α + (wαα )T − z T E•α Fαα Fαα + (wααc )T Fαc α 0. This leads to
(wαα )T Fαα + (wααc )T Fαc α 0.
(17)
In addition, we have wααc 0
Now, we are in a position to prove Theorem 8.
due to (9), it follows from Lemma 7 that
T E z 0. F w
(Fαα )−T (Fαα )−T (Fαc α )T ζ. 0 −I
Therefore, (19) implies that wα ∈ (Y α )− .
As such, it is enough to show that w ¯ T F 0. Note that α ¯ − w ¯ ∈ (Y ) . Since
and
for some ζ 0. Note that (Fαα )−T (Fαα )−T (Fαc α )T (Fαα )T (Fαc α )T = I. 0 −I 0 −I
Since [C D] is full row rank due to Assumption 3, we obtain T z E =w ¯ T F. ∅ F w
For the ‘if’ part of (ii), w be such that w 0, T z A − λI B = 0, w C D
5
(18)
since w 0. By combining (17) and (18), we can write α (Fαα )T (Fαc α )T wα 0. (19) wααc 0 −I It follows from (9) and Lemma 7 that 23
2019 IFAC SSSC 6 Sinaia, Romania, September 9-11, 2019
M.K. Camlibel / IFAC PapersOnLine 52-17 (2019) 1–6
of bimodal piecewise linear systems with scalar inputs. Automatica, 44(5), 1261–1267. Camlibel, M.K., Heemels, W.P.M.H., van der Schaft, A.J., and Schumacher, J.M. (2003). Switched networks and complementarity. IEEE Transactions on Circuits and Systems I, 50(8), 1036–1046. Camlibel, M.K., Iannelli, L., and Vasca, F. (2004). Modelling switching power converters as complementarity systems. In Proc. of the 43th IEEE Conference on Decision and Control. Paradise Islands (Bahamas). Camlibel, M.K., Iannelli, L., and Vasca, F. (2014). Passivity and complementarity. Mathematical Programming, 145(1-2), 531–563. Camlibel, M.K. and Schumacher, J.M. (2002). Existence and uniqueness of solutions for a class of piecewise linear dynamical systems. Linear Algebra and its Applications, 351-352, 147–184. Cottle, R.W., Pang, J.S., and Stone, R.E. (1992). The Linear Complementarity Problem. Academic Press, Inc., Boston. Facchinei, F. and Pang, J.S. (2002). Finite Dimensional Variational Inequalities and Complementarity Problems. Springer. Heemels, W.P.M.H. and Brogliato, B. (2003). The complementarity class of hybrid dynamical systems. European Journal of Control, 26(4), 651–677. Heemels, W.P.M.H. and Camlibel, M.K. (2008). Null controllability of discrete-time linear systems with input and state constraints. In 47th IEEE Conference on Decision and Control. Cancun (Mexico). Heemels, W.P.M.H., Camlibel, M.K., and Schumacher, J.M. (2002). On the dynamic analysis of piecewiselinear networks. IEEE Trans. on Circuits and Systems– I: Fundamental Theory and Applications, 49(3), 315– 327. Heemels, W.P.M.H., Schumacher, J.M., and Weiland, S. (2000). Linear complementarity systems. SIAM J. Appl. Math., 60(4), 1234–1269. Saberi, A., Han, J., and Stoorvogel, A.A. (2002). Constrained stabilization problems for linear plants. Automatica, 38, 639–654. Saberi, A., Stoorvogel, A.A., Shi, G., and Sannuti, P. (2003). Output regulation of linear plants subject to constraints. International Journal of Control, 76(2), 149–164. Schumacher, J.M. (2004). Complementarity systems in optimization. Mathematical Programming Series B, 101, 263–295. Shi, G., Saberi, A., Stoorvogel, A.A., and Sannuti, P. (2003). Output regulation of discrete-time linear plants subject to state and input constraints. International Journal of Robust and Nonlinear Control, 13, 691–713. Sun, Z., Ge, S., and Lee, T. (2002). Controllability and reachability criteria for switched linear systems. Automatica, 38(5), 775–786. Thuan, L.Q. and Camlibel, M.K. (2014). Controllability and stabilizability of piecewise affine dynamical systems. SIAM Journal on Control and Optimization, 52(3), 1914–1934. van der Schaft, A.J. and Schumacher, J.M. (1996). The complementary-slackness class of hybrid systems. Mathematics of Control, Signals and Systems, 9, 266–301.
van der Schaft, A.J. and Schumacher, J.M. (1998). Complementarity modelling of hybrid systems. IEEE Transactions on Automatic Control, 43(4), 483–490. van der Schaft, A.J. and Schumacher, J.M. (2000). An Introduction to Hybrid Dynamical Systems. SpringerVerlag, London.
24