Systems & Control Letters. controllability of a class of bimodal discrete-time piecewise linear systems including

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1 Systems & Control Letters 6 (3) Contents lists available at SciVerse ScienceDirect Systems & Control Letters journal homepage: wwwelseviercom/locate/sysconle Controllability of a class of bimodal discrete-time piecewise linear systems E Yurtseven ad MK Camlibel bc WPMH Heemels a a Eindhoven University of Technology Department of Mechanical Engineering Hybrid and Networked Systems group The Netherlands b University of Groningen Johann Bernoulli Inst for Mathematics and Computer Science Groningen The Netherlands c Dogus University Department of Electronics and Communication Engineering Istanbul Turkey d SKF Regional Sales and Service System and Product Development Nieuwegein The Netherlands a r t i c l e i n f o a b s t r a c t Article history: Received 6 August Received in revised form 6 December Accepted 7 January 3 Available online 8 February 3 Keywords: Bimodal systems Piecewise linear systems Controllability Reachability Hybrid systems Non-convex input constraint set In this paper we will provide full algebraic necessary and sufficient conditions for the controllability/reachability/null controllability of a class of bimodal discrete-time piecewise linear systems including several instances of interest that are not covered by existing works which focus primarily on the planar case In particular the class is characteried by a continuous right-hand side a scalar input and a transfer function from the control input to the switching variable with at most two eroes whereas the state can be of any dimension To prove the main result we will make use of geometric control theory for linear systems and a novel result on controllability for input-constrained linear systems with non-convex constraint sets 3 Elsevier BV All rights reserved Introduction Controllability has always played an important role in modern control theory Kalman and Hautus studied this notion for linear systems and gave complete characteriations in algebraic forms In the presence of input constraints characteriations for the controllability of discrete-time linear systems were given in [ 3] In case of hybrid dynamical systems such as piecewise linear systems such complete characteriations of null controllability reachability or controllability are hard to come by Indeed it is known from [4] that certain controllability problems for discrete-time piecewise linear systems in a general setting are undecidable However several results were obtained on the controllability of different subclasses of piecewise linear systems In [56] Xu and Xie give characteriations for the controllability of discrete-time planar bimodal piecewise linear systems Brogliato obtains necessary and sufficient conditions for the controllability of a class of continuoustime piecewise linear systems in [7] but his results only apply to This work was partially supported by the European Commission through project MOBY-DIC Model-based synthesis of digital electronic circuits for embedded control (FP7-INFSO-ICT-48858) Corresponding author at: SKF Regional Sales and Service System and Product Development Nieuwegein The Netherlands addresses: evrenyurtseven@gmailcom evrenyurtseven@skfcom (E Yurtseven) mkcamlibel@rugnl kcamlibel@dogusedutr (MK Camlibel) mheemels@tuenl (WPMH Heemels) planar systems as well Bemporad et al [8] propose an algorithmic approach based on optimiation tools Although this approach makes it possible to check controllability of a given discrete-time system it does not allow for drawing conclusions about any general class of systems Arapostathis and Broucke give a fairly complete treatment of stability and controllability of continuous-time planar conewise linear systems in [9] which are piecewise linear systems for which the regions are convex cones In [] Lee and Arapostathis provide a characteriation of controllability for a class of continuous-time piecewise linear systems but they assume among other things that the number of inputs in each subsystem is one less than the number of states In [] Camlibel et al give algebraic necessary and sufficient conditions for the controllability of a class of continuous-time conewise linear systems while stabiliability characteriations for bimodal piecewise linear systems are given in [] In this paper we present algebraic necessary and sufficient conditions for the controllability reachability and null controllability of a class of bimodal discrete-time piecewise linear systems The class is characteried by the property that the dynamics is continuous across the switching plane and that the input is scalar We impose no restriction on the dimension of the state variable as opposed to the earlier mentioned works [5 79] that study the controllability of planar piecewise linear systems Also we do not adopt the assumption in [] namely that the number of inputs is one less than the number of states However we assume that 67-69/$ see front matter 3 Elsevier BV All rights reserved doi:6/jsysconle36

2 E Yurtseven et al / Systems & Control Letters 6 (3) the transfer function from the control input to the switching variable ie the variable that determines the active mode has at most two eroes This seemingly rather odd assumption forms a complexity boundary in solving this problem for larger classes of PWL systems Allowing three or more eroes in the transfer function necessitates solving the controllability problem for discrete-time linear systems with non-convex input constraint sets which is known to be a very hard problem In spite of this assumption our results are the first to provide algebraic necessary and sufficient conditions for the controllability for the indicated class of piecewise linear systems that contains several instances of interest not covered by the existing results At the end of the paper we will demonstrate this fact with an example whose controllability could indeed not have been checked by earlier works The paper is organied as follows We lay the groundwork to solve the controllability problem in Section and we define the class of systems we are interested in Section 3 In Sections 4 6 we consider two different problems whose solutions are needed to tackle the main problem In Section 7 we present our main results Conclusions are presented in Section 8 Definitions Definitions Consider the discrete-time system x[k + ] = g(x[k] u[k]) y[k] = h(x[k]) () where x[k] R n u[k] R m y[k] R p are the state the input and the output variable respectively at discrete time k N Here g : R n R m R n and h : R n R p are given functions Given an initial state x R n and an input sequence u = {u[] u[n]} with N N we will denote the state trajectory corresponding to u and the initial state x[] = x by x x u Likewise we will denote the corresponding output sequence by y x u Definition Consider the system () Let y = {y[] y[n]} be a sequence with y[k] R p k = N Given an initial state x R n we say that x is compatible with y if there exists an input sequence u = {u[] u[n ]} such that y x u[k] = y[k] for k = N Likewise given an input sequence ū = {ū[] ū[n ]} we say that ū is compatible with y if there exists an initial state x R n such that y x ū[k] = y[k] for k = N Definition Consider the system () We say that () is null controllable if for all x R n there exist an N N and an input sequence u = {u[] u[n ]} such that x x u[n] = reachable if for all x f R n there exist an N N and an input sequence u = {u[] u[n ]} such that x u [N] = x f controllable if for all x x f R n there exist an N N and an input sequence u = {u[] u[n ]} such that x x u[n] = x f Classical results Consider the linear system x[k + ] = Ax[k] + Bu[k] () with x[k] R n being the state and u[k] R m being the input together with the input constraint u[k] U k N (3) where U R m is a solid polyhedral closed cone ie there exists a matrix U R l m for some l N such that U = {u R m Uu } and U has a nonempty interior The inequalities in Uu are to be interpreted component-wise The definitions of null controllability reachability and controllability as in Definition are similar for the constrained system () (3) with the understanding that (3) should hold for the input sequence Definition 3 Let U R m be a nonempty set We define the dual cone U of U as U := {v R m v u u U} The following lemma can be based on [3] Lemma 4 Let U be a solid polyhedral closed cone The constrained system () (3) is null controllable if and only if the following implications hold: λ C \ {} C n A = λ = B = (4a) λ ( ) R n A = λ B U = (4b) reachable/controllable if and only if the following implications hold: λ C C n A = λ = (5a) B = λ [ ) R n A = λ B U = (5b) 3 Problem definition In this paper we are interested in the null controllability/ reachability/controllability of bimodal piecewise linear (PWL) systems that are of the form A x[k] + B x[k + ] = u[k] y[k] (6) A x[k] + B u[k] y[k] y[k] = C x[k] where x[k] R n is the state u[k] R is the scalar input y[k] R is a variable determining the active mode at discrete time k N and the matrices A A R n n B B C R n are given We assume C and that the right-hand side of (6) is continuous This is equivalent to the existence of a vector E R n such that A = A + EC and B = B =: B (7) To show that this controllability problem is far from being trivial and that the controllability of (6) cannot be inferred from the controllability properties of the subsystems only we would like to give a motivating example Example 3 Consider the PWL system given by x [k] x x [k + ] = [k] x [k] x [k] + x [k] x [k] x [k + ] = u[k] (8a) (8b) In this example both of the subsystems are controllable as linear systems as is easily verified However the piecewise linear system is not controllable as x [k] x [] k N This shows that controllability of a PWL system cannot be characteried only in terms of the controllability of its subsystems

3 34 E Yurtseven et al / Systems & Control Letters 6 (3) Define the transfer functions G i () = C (I A i ) B for i = It follows from (7) that σ is a ero of G () if and only if it is a ero of G () and that G () if and only if G () In the rest of the paper we will assume that G () as otherwise the system (6) would not be controllable and that it has at most two eroes We will provide insights on the need of this rather odd requirement later Let V i be the largest (A i B)- invariant subspace contained in ker C for i = ie V i is the largest of the subspaces V i that satisfy (A i + BF i )V i V i ker C for some F i R n i = We will denote the set {F i R n (A i + BF i )V i V i ker C } by F i Likewise let Z i be the smallest (C A i )-invariant subspace that contains im B for i = ie Z i is the smallest of the subspaces Z i that satisfy (A i + K i C )Z i Z i and im B Z i for some K i R n i = See [4 6] for a detailed discussion on these particular subspaces Since G i () i = G i () is invertible as a rational function Therefore R n admits the decomposition R n = V i Z i for i = See [4] for the proof of this implication Due to (7) it holds that V = V =: V Z = Z =: Z and F = F =: F First we apply the pre-compensating state feedback u[k] = Fx[k] + v[k] to (6) with F F Due to (7) we have that (A + BF) V = (A + BF) V Then we apply the similarity transformation x = T x with T = T T where im T = V and im T = Z For ease of exposition we will not use a different symbol for the new state vector and we will denote v[k] by u[k] Then we obtain the following representation of (6) that is easier to deal with for characteriing null controllability reachability and controllability: x [k + ] = Hx [k] + x [k + ] = y[k] = c x [k] g y[k] y[k] g y[k] y[k] J x [k] + bu[k] y[k] J x [k] + bu[k] y[k] (9a) (9b) in which x [k] g g R n x [k] b c R n for k N and H R n n J J R n n where n = dim V n = dim Z and n + n = n Due to the assumption we made on the number of eroes of G () we have that n Obviously controllability properties are invariant under precompensating feedbacks and similarity transformations Hence null controllability/reachability/controllability of (9) is equivalent to null controllability/reachability/controllability of (6) We will now study particular properties of the subsystems (9a) and (9b) that are useful for the main developments The subsystem (9b) belongs to a special class of systems which we will introduce and analye in the next section In Section 6 we will analye the subsystem (9a) which we call a push pull type of system Note that Gi () = c (I J i ) b for i = Let Ṽ i denote the largest (J i b)-invariant subspace contained in ker c for i = Then Ṽ = Ṽ = {} Let Z i denote the smallest (c J i )-invariant subspace that contains im b Then Z = Z = Rn Due to (7) we also have that J = J + e c and g = g + e () where e e = T E and C T = c We would like to underline that what made this decomposition possible is the continuity assumption (7) As we will see in Section 7 the main results follow from the results we derive on the subsystems (9a) and (9b) Hence the continuity assumption is one of the cornerstones of this work Its omittance would require an entirely different treatment of the subject than the one in this paper which is currently an open interesting problem 4 Observability and invertibility of a special class of bimodal discrete-time piecewise linear systems Consider the discrete-time piecewise linear system A x[k] + Bu[k] y[k] x[k + ] = A x[k] + Bu[k] y[k] y[k] = C x[k] (a) (b) where x[k] R n u[k] y[k] R for k N and A A R n n B C R n We will study a special class of the system () as we will need these results later in proving our main result In particular we adopt the following assumptions Assumption 4 The following statements hold The transfer functions G i () = C (I A i ) B for i = ; The right-hand side of (a) is continuous ie A = A + EC for some vector E R n ; 3 The largest (A i B)-invariant subspace contained in ker C V i is {} for i = 4 The smallest (C A i )-invariant subspace containing im B Z i is R n for i = We would like to point out that the system (9b) belongs to this special class of discrete-time piecewise linear systems Corollary 4 Consider the system () and suppose Assumption 4 holds Then the following statements hold (C A i ) is observable (A i B) is controllable 3 C B = C A i B = = C A n i B = 4 C A n i B for i = Definition 43 We say that () is both observable and invertible if for any output sequence y = {y[] y[] } of infinite length there exist a unique initial state x R n and a unique input sequence of infinite length {u[] u[] } that are both compatible with y Proposition 44 Consider the system () for which Assumption 4 holds Then it is both observable and invertible In particular the first n entries of the output sequence {y[] y[n ]} uniquely determine the initial state x Proof Let y[k] > i k = y[k] k N Then one can write for the system () y[] C y[] C A i = Qx[] with Q = y[n ] C A in A i A i Based on the second statement in Assumption 4 one can factorie Q in the following way C Q = f 3 C A i f n f nn C n A i

4 E Yurtseven et al / Systems & Control Letters 6 (3) The left factor is a lower triangular matrix whose off-diagonal terms vary based on the values of i i n The right factor is the observability matrix generated by (C A i ) Since the pair (C A i ) is observable it follows that Q is nonsingular Thus for any sequence {y[] y[n ]} there exists a unique x[] that produces this output sequence Furthermore y[n] = C A in A in A i x[] + Su[] where S = C A in A in A i B Based on Corollary 4 we have that S = C A n i B Since x[] is uniquely determined by {y[] y[n ]} u[] is uniquely determined by {y[] y[n]} By repeating the same argument u[] is uniquely determined by {y[] y[n + ]} and so on Thus () is both observable and invertible 5 Insights on the complexity of the controllability problem Lemma 5 The system (6) for which (7) holds with G i () = C (I A i ) B i = is controllable if and only if the push pull system (9a) is controllable Proof Since the controllability of (6) is equivalent to the controllability of (9) the proof will be complete when we show (9) is controllable if and only if (9a) is controllable The necessity of this condition is clear To show its sufficiency we take two arbitrary states (x x ) R n and (x f x f ) R n We would like to demonstrate how to steer (x x ) to (x f x f ) From (9b) we produce an output sequence {y[] y[n ]} that is compatible with x and apply it to (9a) Let x m = x [n ] denote the state the system (9a) is steered to upon the application of this sequence Since (9a) is controllable there must exist r n such that x [n ] = x m and x [r] = x f for some sequence {y[n ] y[r ]} Then we extend the output sequence with {y[r] y[r + ] } which is chosen to be compatible with x f Due to Proposition 44 there is a unique input sequence {u[] u[] } compatible with {y[] y[] } for which it necessarily holds that x [] = x x [r] = x f Thus we have constructed an input that satisfies x [r] = x f x [r] = x f with x [] = x x [] = x which completes the proof Corollary 5 The system (6) for which (7) holds with G i () = C (I A i ) B i = is null controllable if and only if the push pull system (9a) is null controllable Proof The proof is obtained in exactly the same way as in the proof of Lemma 5 by taking (x f x f ) = Now that we have established the equivalency of the controllabilities of (6) and (9a) we embark upon the task of characteriing the controllability of the push pull system (9a) Although we will give results on the controllability of push pull systems under some conditions in the next section first we would like to discuss the complexities inherent in this problem We start with the framework under which we will discuss the controllability of push pull systems A bimodal discrete-time push pull system with a scalar input is given by B+ u[k] u[k] x[k + ] = Ax[k] + () B u[k] u[k] where x[k] R n u[k] R for k N and A R n n B + B R n Note that (9a) fits in this framework with y considered as the input The system () can easily be written as the following equivalent input-constrained linear system x[k + ] = Ax[k] + [B + B ]v[k] with v[k] Ũ (3) where Ũ = [ ) {} {} [ ) Ũ is clearly a non-convex set Therefore solving the controllability problem for the PWL system (6) is equivalent to solving the controllability problem for the discrete-time linear system (3) with a particular non-convex input constraint set Even though controllability of continuous-time linear systems with non-convex input constraint sets can be solved by convexification of the input constraint set through incorporating all the Filippov solutions (see eg [7]) clearly this cannot be done for discrete-time systems Therefore we need to resort to another method In the next section we will show how we obtained the controllability characteriation for (3) under the state dimension restriction n { } through a case-by-case analysis Solving the controllability problem for (3) without imposing any restriction on the dimension of the state would allow us to remove the assumption we made on the maximum number of eroes of G () However at higher dimensions n 3 case-by-case analysis is a very complicated if not impossible task Thus a structural approach in characteriing controllability of (3) is indispensable if one wants to solve the controllability problem for (6) in a broader generality that is without limitation on the maximum number of eroes of G () as long as the number of eroes is finite But to the best of our knowledge controllability of discrete-time linear systems with non-convex input constraint sets has not been solved so far and it is known to be a very hard problem That is why in this paper we only deal with systems of the form (6) having at most two eroes Despite this our results are still the first in characteriing controllability for the class of systems described in Section 3 and this class is not captured by any of the works in the literature Now we are ready to state our results on the controllability of push pull systems of the form () for which n = holds We would like to stress that these results are novel findings on the controllability of discrete-time linear systems with non-convex input constraint sets 6 Controllability of bimodal discrete-time push pull systems Theorem 6 Consider the system () with n { } The following statements are equivalent (i) () is reachable (ii) The linear system given by x[k + ] = Ax[k] + [B + B ]v[k] with v[k] Ū (4) where Ū = R + is reachable Proof (i) (ii): Recall that () and (3) are equivalent systems Since Ũ Ū the implication follows (ii) (i): First we consider the n = case Note that in this case the matrices A B + B are scalars Based on Lemma 4 reachability of the input-constrained linear system (4) dictates that if A then B + B > and if A < then either B + or B Now we will show that these conditions are sufficient for the reachability of (3) For the A case it is easy to see that if B + B > then any state can be reached in one step by using the appropriate control input For the A < case since either B + or B the sign of the terms in one of the sequences {B + AB + A B + } and {B AB A B } is alternating which is sufficient to conclude that any state can be reached in at most two steps Now we turn our attention to the case where n = Note that under similarity transformation reachability properties of (3) and (4) remain invariant First we apply a similarity transformation to the system (4) to obtain [k + ] = M[k] + [N N ]v[k] with v[k] Ū (5) where M is in Jordan form Since (4) is reachable (5) is reachable as well Thus the implications in Lemma 4 hold for (5) We will

5 34 E Yurtseven et al / Systems & Control Letters 6 (3) show that these implications are sufficient for the reachability of the system [k + ] = M[k] + [N N ]v[k] with v[k] Ũ (6) This will imply the reachability of (3) and thus the reachability of () We will make a case-by-case analysis of the implications in Lemma 4 Based on the Jordan form of M we distinguish three possible main cases We adopt the following notation N = [p p ] N = [q q ] We point out that the state of the system (6) at the kth time instant starting from ero initial condition is given by [k] = [N N MN MN M k N M k N ] v k where v k Ũ k It is easy to see that for reachability of (6) it suffices to find a finite set of cones generated by vector pairs {M i N l M j N l } with i j and (l l) { } { } such that their union is R In the rest of this proof we will employ this idea to show reachability of (6) M = λ λ with λ λ R (a) λ = λ In this case we have from implications (5) that the system (5) can never be reachable It is easy to see that (6) can never be reachable in this case either Hence this case is irrelevant (b) λ > and λ < From implication (5a) we have that either p or q From implication (5b) we have p q < We consider two situations in this case p = q and p q In the former the union of the three cones generated by the vector pairs {N MN } {N M N } {MN M N } is R For the latter we consider two possible situations p q > and p q < If p q < then the union of the four cones generated by the vector pairs {N MN } {N MN } {N MN } {N MN } is R If p q > and λ < λ then the union of the five cones generated by {N MN } {MN M 3 N } {M 3 N N } {N M N } {M N N } is R If p q > and λ λ the union of the four cones generated by {N MN } {M N M 3 N } {N M N } {MN M 3 N } is R (c) λ > λ > and λ λ From implication (5b) we have p q < and p q < We consider two possible configurations; rank([n N ]) = and rank([n N ]) = In the former the union of the cones generated by the vector pairs {N MN } {N MN } {N MN } and{n MN } is R In the latter one can always find sufficiently large j l N such that the union of the four cones generated by the vector pairs {N M j N } {N M j N } {N M l N } and {N M l N } is R (d) λ = λ > From implication (5b) we have p q < and p q < Under these conditions the union of the four cones generated by the vector pairs {N MN } {N MN } {N MN } {N MN } is R (e) λ = λ < From implication (5b) we get p q < and from implication (5a) we get either p or q We consider two situations p q = and p q In the former the union of the four cones generated by {N MN } {N MN } {N M N } {N M N } is R In the latter we again consider two situations p q > and p q < If p q > then the union of the four cones generated by {N MN } {N MN } {N M N } {N M N } is R If p q < the union of the four cones generated by {N MN } {N MN } {N MN } {N MN } is R (f) λ = λ < From implication (5a) we get rank[n N ] = Under this condition the union of the four cones generated by {N MN } {N MN } {MN M 3 N } {N M N } is R (g) λ < λ < and λ λ From implication (5a) we have either p or q and either p or q First we consider the case where p q p = q = In this case the union of the four cones generated by {N MN } {N MN } {N M N } {MN M 3 N } is R The case where p q and p = q = can be treated in a similar way Next we take p p and diverge from our usual approach here to proceed as follows Consider the following system where w[k] is the scalar non-negative input [k + ] = M[k] + N w[k] w[k] (7) By invoking Lemma 4 under the conditions p p we can immediately conclude that (7) is reachable We can recast (7) as a two-input system by constraining the second input to be ero as follows M = [k + ] = M[k] + [N N ]v[k] v[k] U (8) where U = [ ) {} Note that (7) and (8) are equivalent in terms of reachability Comparing the systems (6) and (8) we see that since U Ũ and (8) is reachable (6) is reachable as well λ λ (a) λ From implication (5b) we have p q < When λ = the union of the four cones generated by {N MN } {N MN } {N MN } {N MN } is R For λ > and p = q = the union of the four cones generated by {N MN } {N MN } {N MN } {N MN } is R To analye the cases in which λ > and either p or q we observe the evolutions of M k N and M k N which are given as λ k λp + kp λp and λ k λq + kq λq respectively It becomes clear from this formulation that one can always find large enough j l N such that the union of the four cones generated by {N M l N } {N M l N } {N M j N } {N M j N } is R (b) λ < From implication (5a) we have either p or q We take p and consider the system (7) Note that the M matrix here is different than the one in the subcase (g) By invoking Lemma 4 under the condition p we can immediately conclude that (7) is reachable By following the same reasoning as in the subcase (g) we can arrive at the conclusion that (6) is reachable 3 M = σ ω ω σ with ω This case corresponds to M having a pair of complex conjugate eigenvalues σ jω From implication (5a) we have that rank([n N ]) We take N and consider the system (7) Note that the M matrix here is different than the one in the subcase (g) By invoking Lemma 4 under the condition N we immediately conclude that (7) is reachable By following the same reasoning as in the subcase (g) we deduce that (6) is reachable Corollary 6 Consider the system () with n { } The following statements are equivalent (i) The system () is null controllable (ii) The system (4) is null controllable with the input constraint v[k] Ū := R +

6 E Yurtseven et al / Systems & Control Letters 6 (3) Proof (i) (ii): This implication is obvious (ii) (i): First we consider the case n = If A = then the system (3) is clearly always null controllable We will now consider the reverse-time versions of (3) and (4) given by x[k] = A x[k + ] + [ A B + A B ]v[k] v[k] Ũ (9) and x[k] = A x[k + ] + [ A B + A B ]v[k] v[k] Ū () respectively Whenever A exists these reverse-time versions of (3) and (4) exist If (4) is null controllable then () is reachable From Theorem 6 we deduce that (9) is reachable as well Hence its forward-time version (3) is null controllable Thus we have established the null controllability of (3) for the case A and hence for the case n = Now we turn our attention to the case n = The cases where A is nonsingular are immediately covered by the reasoning above For the remaining cases we will build a case-by-case analysis based on the transformed system (6) M = λ λ with λ λ R (a) λ = λ = In this case the system (6) is clearly always null controllable (b) λ = and λ We can expand the systems (5) and (6) as [k + ] = + [N [k + ] N ]v[k] () λ [k] [k] with v[k] Ū and [k + ] [k] = [k + ] λ [k] + [N N ]v[k] () with v[k] Ũ respectively Next we extract the dynamics of these systems as [k + ] = λ [k] + [p q ]v[k] v[k] Ū (3) and [k + ] = λ [k] + [p q ]v[k] v[k] Ũ (4) respectively If (4) is null controllable (5) and () are null controllable Hence (3) is null controllable as well Based on the analysis for the n = case we can conclude (4) is null controllable Considering the system () since (4) is null controllable we can first steer [] to the origin with the appropriate control input and apply ero input afterwards to steer to the origin Therefore we can steer any [] = [ [] []] to the origin which shows () and (6) are null controllable M = λ λ = the system (6) is always null controll- (a) λ = As M able Corollary 63 Consider the system () with n { } The following statements are equivalent (i) The system () is controllable (ii) The system (4) is controllable with the input constraint v[k] Ū := R + Proof (i) (ii): This is obvious (ii) (i): Since (4) is controllable it is reachable and null controllable From Theorem 6 and Corollary 6 we have that () is reachable and null controllable Therefore () is controllable 7 Main results We are now in a position to present the main result of this paper Theorem 7 Consider the system (6) for which (7) holds with G i () = C (I A i ) B and G i () having at most two eroes i = Then the following statements hold The system (6) is null controllable if and only if the following implications hold: λ C \ {} C n A i = λ i = = (5a) B = λ ( ) R n w w i A i = λi B w i C = = (5b) The system (6) is controllable if and only if the following implications hold λ C C n A i = λ i = = (6a) B = λ [ ) R n w w i A i = λi B w i C = = (6b) Proof We will prove the second statement only The proof of the first can be obtained in a similar way and is therefore omitted From Lemma 5 we know that the controllability of the push pull system (9a) is equivalent to the controllability of (9) and thus to the controllability of (6) Using Lemma 4 and Corollary 63 we can give a characteriation of the controllability of (9a) which is as follows λ C C n H = λ g = g = λ [ ) R n H = λ g g = = (7a) (7b) Our aim is now to show the equivalence of (7) and (6) Observe that the condition (7a) is equivalent to the controllability of (H [g g ]) as a linear system without constraints Due to (7) (6a) is equivalent to the controllability of (A [B E]) as a linear system without constraints Therefore we need to demonstrate that controllability of (H [g g ]) and controllability of (A [B E]) as linear systems without constraints are equivalent For this purpose we will use the Hautus test which states that the controllability of (A [B E]) as a linear system without constraints is equivalent to the implication λ C C n [A λi B E] = = (8) Condition (8) can also be written for the system (9) since the controllability properties do not change under similarity transformation and pre-compensating feedback Thus instead of (8) we will use the following implication λ C C n H λi g c e = (9) = J λi b e First we take = ( ) then observe that when H λi g c e = (3) J λi b e

7 344 E Yurtseven et al / Systems & Control Letters 6 (3) for some λ C it holds that g c = (3) J λi b We have that the largest (J b)-invariant subspace contained in ker c Ṽ is {} Then it follows from Proposition II3 in [] that g = and = Substituting = into (3) we obtain e = thus g = due to () Hence assuming that (7a) holds it now dictates that = and hence = Therefore (A [B E]) is controllable as a linear system if (H [g g ]) is controllable as a linear system To show how the controllability of (A [B E]) implies the controllability of (H [g g ]) we take = ( ) = ( ) in (3) Then assuming that (9) holds we recover the implication (7a) To prove the equivalence of the implications (6b) and (7b) we will again use the transformed system (9) We take = ( ) Then the equality in the condition (6b) w i H λi g ic J i λi b = i = (3) c can be rewritten as H = λ [ w + g J λi ] [ w + g J λi ] = b c b c = (33) Since the largest (J i b)-invariant subspace contained in ker c Ṽ i is {} for i = it follows from Proposition II3 in [] that = g = w and g = w Due to (7b) = hence = We have shown that (7b) implies (6b) Conversely if (6b) holds by taking = w = g and w = g in (6b) for the transformed system (9) the implication (7b) follows Thus we have proven the equivalence of the implications (6b) and (7b) which completes the proof Clearly due to the complexity boundary indicated in Section 5 the necessary and sufficient conditions for controllability apply to a particular class of piecewise linear systems To indicate that this class is not captured by the previous works we provide an example below We would like to stress that the example below is the simplest one that could have been chosen within the class of systems our results apply to and beyond the application range of the existing results Example 7 Consider the PWL system (6) given by the matrices A = A = B = B = C = This system has three states and one input thus it does not fall under the framework of the results in [5] Note that the transfer functions G i () = C (I A i ) B i = have no eroes which means that this PWL system is covered by the results in this paper Indeed using Theorem 7 it is easily verified that this PWL system is controllable Now we will show with hand calculation that this system is controllable First we take x[] = and we want to steer the state to some arbitrary x f in finite time say r N ie x[r] = x f It is easy to see that x[] = u[] u[] u[] As y[] x[] is deterministically given as x[] = u[] + u[] u[] u[] + u[] Since y[] we have x[3] = u[] u[] + u[] u[] u[] + u[] + u[] We can write this last equality as u[] u[] = x[3] (34) u[] Due to the invertibility of the 3 3 matrix in the above equation for any given x[3] = x f one can find the appropriate input sequence {u[] u[] u[]} to steer the state of the system from the origin to x f Thus we have established that this PWL system is reachable By a similar reasoning and hand calculation one can also show null controllability of this system Since this system is reachable and null controllable we conclude that it is controllable 8 Conclusion In this paper we presented algebraic characteriations of the controllability reachability and null controllability of a class of discrete-time bimodal piecewise linear systems By using geometric control theory we showed that the controllability problem is equivalent to the controllability of a particular subsystem of the original system which is in the form of a so-called push pull system By studying the controllability of linear systems with a particular non-convex input constraint set we derived conditions for the controllability of push pull systems up to a certain dimension hence leading to our main result Future research in this topic will be the characteriation of controllability for discretetime push pull systems of arbitrary dimension References [] M Evans The convex controller: controllability in finite time Int J Syst Sci 6 (985) 3 47 [] K Nguyen On the null-controllability of linear 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