Systems & Control Letters. controllability of a class of bimodal discrete-time piecewise linear systems including
|
|
- Berniece Small
- 6 years ago
- Views:
Transcription
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 discrete-time systems with restrained controls J Optim Theory Appl 5 (986) [3] M Evans D Murthy Controllability of discrete-time systems with positive controls IEEE Trans Automat Control (977) [4] V Blondel J Tsitsiklis Complexity of stability and controllability of elementary hybrid systems Automatica 35 (999) [5] J Xu L Xie Controllability and reachability of discrete-time planar bimodal piecewise linear systems in: Proc of the American Control Conference 6 pp [6] J Xu L Xie Null controllability of discrete-time planar bimodal piecewise linear systems Internat J Control 78 (5) [7] B Brogliato Some results on the controllability of planar variational inequalities Systems Control Lett 55 (5) 65 7 [8] A Bemporad G Ferrari-Trecate M Morari Observability and controllability of piecewise affine and hybrid systems IEEE Trans Automat Control 45 () [9] A Arapostathis M Broucke Stability and controllability of planar conewise linear systems Systems Control Lett 56 (7) 5 58 [] K Lee A Arapostathis On the controllability of piecewise linear hypersurface systems Systems Control Lett 55 (987) 65 7 [] M Camlibel W Heemels J Schumacher Algebraic necessary and sufficient conditions for the controllability of conewise linear systems IEEE Trans Automat Control 53 (8) [] M Camlibel W Heemels J Schumacher Stabiliability of bimodal piecewise linear systems with continuous vector field Automatica 44 (8) 6 67 [3] W Heemels M Camlibel Null controllability of discrete-time linear systems with input and state constraints in: 47th IEEE Conference on Decision and Control 8 pp [4] HL Trentelman AA Stoorvogel M Hautus Control Theory for Linear Systems in: Communications and Control Engineering Series Springer [5] W Wonham Linear Multivariable Control: A Geometric Approach third ed Springer 985 [6] G Basile G Marro Controlled and Conditioned Invariants in Linear System Theory Prentice Hall Englewood Cliffs NJ 99 [7] M Camlibel W Heemels J Schumacher On the controllability of bimodal piecewise linear systems in: Hybrid Systems; Computation and Control in: Lecture Notes in Computer Science vol 993 Springer Berlin/Heidelberg 4 pp 57 64
Null Controllability of Discrete-time Linear Systems with Input and State Constraints
Proceedings of the 47th IEEE Conference on Decision and Control Cancun, Mexico, Dec. 9-11, 2008 Null Controllability of Discrete-time Linear Systems with Input and State Constraints W.P.M.H. Heemels and
More informationControllability of Linear Systems with Input and State Constraints
Proceedings of the 46th IEEE Conference on Decision and Control New Orleans, LA, USA, Dec. 12-14, 2007 Controllability of Linear Systems with Input and State Constraints W.P.M.H. Heemels and M.K. Camlibel
More informationPopov Belevitch Hautus type tests for the controllability of linear complementarity systems Camlibel, M. Kanat
University of Groningen Popov Belevitch Hautus type tests for the controllability of linear complementarity systems Camlibel, M. Kanat Published in: Systems & Control Letters IMPORTANT NOTE: You are advised
More information3118 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 57, NO. 12, DECEMBER 2012
3118 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 57, NO 12, DECEMBER 2012 ASimultaneousBalancedTruncationApproachto Model Reduction of Switched Linear Systems Nima Monshizadeh, Harry L Trentelman, SeniorMember,IEEE,
More informationDiscussion on: Measurable signal decoupling with dynamic feedforward compensation and unknown-input observation for systems with direct feedthrough
Discussion on: Measurable signal decoupling with dynamic feedforward compensation and unknown-input observation for systems with direct feedthrough H.L. Trentelman 1 The geometric approach In the last
More informationGeometric Control Theory for Hybrid Systems
Geometric Control Theory for Hybrid Systems Evren Yurtseven H&NS 2010.03 Master s thesis Coach(es): Committee: prof.dr.ir. W.P.M.H. Heemels (supervisor) dr. M.K. Camlibel dr. M.K. Camlibel prof.dr.ir.
More informationObservability of Switched Linear Systems in Continuous Time
University of Pennsylvania ScholarlyCommons Departmental Papers (ESE) Department of Electrical & Systems Engineering March 2005 Observability of Switched Linear Systems in Continuous Time Mohamed Babaali
More informationEquivalence of dynamical systems by bisimulation
Equivalence of dynamical systems by bisimulation Arjan van der Schaft Department of Applied Mathematics, University of Twente P.O. Box 217, 75 AE Enschede, The Netherlands Phone +31-53-4893449, Fax +31-53-48938
More informationPredictive control of hybrid systems: Input-to-state stability results for sub-optimal solutions
Predictive control of hybrid systems: Input-to-state stability results for sub-optimal solutions M. Lazar, W.P.M.H. Heemels a a Eindhoven Univ. of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands
More informationZero controllability in discrete-time structured systems
1 Zero controllability in discrete-time structured systems Jacob van der Woude arxiv:173.8394v1 [math.oc] 24 Mar 217 Abstract In this paper we consider complex dynamical networks modeled by means of state
More informationCANONICAL LOSSLESS STATE-SPACE SYSTEMS: STAIRCASE FORMS AND THE SCHUR ALGORITHM
CANONICAL LOSSLESS STATE-SPACE SYSTEMS: STAIRCASE FORMS AND THE SCHUR ALGORITHM Ralf L.M. Peeters Bernard Hanzon Martine Olivi Dept. Mathematics, Universiteit Maastricht, P.O. Box 616, 6200 MD Maastricht,
More informationElementary Linear Algebra
Matrices J MUSCAT Elementary Linear Algebra Matrices Definition Dr J Muscat 2002 A matrix is a rectangular array of numbers, arranged in rows and columns a a 2 a 3 a n a 2 a 22 a 23 a 2n A = a m a mn We
More informationAn Observation on the Positive Real Lemma
Journal of Mathematical Analysis and Applications 255, 48 49 (21) doi:1.16/jmaa.2.7241, available online at http://www.idealibrary.com on An Observation on the Positive Real Lemma Luciano Pandolfi Dipartimento
More informationChapter 2: Linear Independence and Bases
MATH20300: Linear Algebra 2 (2016 Chapter 2: Linear Independence and Bases 1 Linear Combinations and Spans Example 11 Consider the vector v (1, 1 R 2 What is the smallest subspace of (the real vector space
More informationA proof of the Jordan normal form theorem
A proof of the Jordan normal form theorem Jordan normal form theorem states that any matrix is similar to a blockdiagonal matrix with Jordan blocks on the diagonal. To prove it, we first reformulate it
More informationPattern generation, topology, and non-holonomic systems
Systems & Control Letters ( www.elsevier.com/locate/sysconle Pattern generation, topology, and non-holonomic systems Abdol-Reza Mansouri Division of Engineering and Applied Sciences, Harvard University,
More informationDiscrete Applied Mathematics
Discrete Applied Mathematics 157 (2009 1696 1701 Contents lists available at ScienceDirect Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam Riordan group involutions and the -sequence
More informationFurther results on Robust MPC using Linear Matrix Inequalities
Further results on Robust MPC using Linear Matrix Inequalities M. Lazar, W.P.M.H. Heemels, D. Muñoz de la Peña, T. Alamo Eindhoven Univ. of Technology, P.O. Box 513, 5600 MB, Eindhoven, The Netherlands,
More informationNecessary and Sufficient Conditions for Reachability on a Simplex
Necessary and Sufficient Conditions for Reachability on a Simplex Bartek Roszak a, Mireille E. Broucke a a Edward S. Rogers Sr. Department of Electrical and Computer Engineering, University of Toronto,
More informationStabilization, Pole Placement, and Regular Implementability
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 47, NO. 5, MAY 2002 735 Stabilization, Pole Placement, and Regular Implementability Madhu N. Belur and H. L. Trentelman, Senior Member, IEEE Abstract In this
More information1 Invariant subspaces
MATH 2040 Linear Algebra II Lecture Notes by Martin Li Lecture 8 Eigenvalues, eigenvectors and invariant subspaces 1 In previous lectures we have studied linear maps T : V W from a vector space V to another
More informationMemoryless output feedback nullification and canonical forms, for time varying systems
Memoryless output feedback nullification and canonical forms, for time varying systems Gera Weiss May 19, 2005 Abstract We study the possibility of nullifying time-varying systems with memoryless output
More informationDisturbance Attenuation Properties for Discrete-Time Uncertain Switched Linear Systems
Disturbance Attenuation Properties for Discrete-Time Uncertain Switched Linear Systems Hai Lin Department of Electrical Engineering University of Notre Dame Notre Dame, IN 46556, USA Panos J. Antsaklis
More informationThe Cyclic Decomposition of a Nilpotent Operator
The Cyclic Decomposition of a Nilpotent Operator 1 Introduction. J.H. Shapiro Suppose T is a linear transformation on a vector space V. Recall Exercise #3 of Chapter 8 of our text, which we restate here
More informationMath 350 Fall 2011 Notes about inner product spaces. In this notes we state and prove some important properties of inner product spaces.
Math 350 Fall 2011 Notes about inner product spaces In this notes we state and prove some important properties of inner product spaces. First, recall the dot product on R n : if x, y R n, say x = (x 1,...,
More informationWe simply compute: for v = x i e i, bilinearity of B implies that Q B (v) = B(v, v) is given by xi x j B(e i, e j ) =
Math 395. Quadratic spaces over R 1. Algebraic preliminaries Let V be a vector space over a field F. Recall that a quadratic form on V is a map Q : V F such that Q(cv) = c 2 Q(v) for all v V and c F, and
More informationObserver design for a general class of triangular systems
1st International Symposium on Mathematical Theory of Networks and Systems July 7-11, 014. Observer design for a general class of triangular systems Dimitris Boskos 1 John Tsinias Abstract The paper deals
More informationNP-hardness of the stable matrix in unit interval family problem in discrete time
Systems & Control Letters 42 21 261 265 www.elsevier.com/locate/sysconle NP-hardness of the stable matrix in unit interval family problem in discrete time Alejandra Mercado, K.J. Ray Liu Electrical and
More informationOn at systems behaviors and observable image representations
Available online at www.sciencedirect.com Systems & Control Letters 51 (2004) 51 55 www.elsevier.com/locate/sysconle On at systems behaviors and observable image representations H.L. Trentelman Institute
More informationApplications of Controlled Invariance to the l 1 Optimal Control Problem
Applications of Controlled Invariance to the l 1 Optimal Control Problem Carlos E.T. Dórea and Jean-Claude Hennet LAAS-CNRS 7, Ave. du Colonel Roche, 31077 Toulouse Cédex 4, FRANCE Phone : (+33) 61 33
More informationFoundations of Matrix Analysis
1 Foundations of Matrix Analysis In this chapter we recall the basic elements of linear algebra which will be employed in the remainder of the text For most of the proofs as well as for the details, the
More informationonly nite eigenvalues. This is an extension of earlier results from [2]. Then we concentrate on the Riccati equation appearing in H 2 and linear quadr
The discrete algebraic Riccati equation and linear matrix inequality nton. Stoorvogel y Department of Mathematics and Computing Science Eindhoven Univ. of Technology P.O. ox 53, 56 M Eindhoven The Netherlands
More information4F3 - Predictive Control
4F3 Predictive Control - Lecture 2 p 1/23 4F3 - Predictive Control Lecture 2 - Unconstrained Predictive Control Jan Maciejowski jmm@engcamacuk 4F3 Predictive Control - Lecture 2 p 2/23 References Predictive
More informationON CHATTERING-FREE DISCRETE-TIME SLIDING MODE CONTROL DESIGN. Seung-Hi Lee
ON CHATTERING-FREE DISCRETE-TIME SLIDING MODE CONTROL DESIGN Seung-Hi Lee Samsung Advanced Institute of Technology, Suwon, KOREA shl@saitsamsungcokr Abstract: A sliding mode control method is presented
More informationControl for Coordination of Linear Systems
Control for Coordination of Linear Systems André C.M. Ran Department of Mathematics, Vrije Universiteit De Boelelaan 181a, 181 HV Amsterdam, The Netherlands Jan H. van Schuppen CWI, P.O. Box 9479, 19 GB
More informationOn the Stabilization of Neutrally Stable Linear Discrete Time Systems
TWCCC Texas Wisconsin California Control Consortium Technical report number 2017 01 On the Stabilization of Neutrally Stable Linear Discrete Time Systems Travis J. Arnold and James B. Rawlings Department
More informationBare-bones outline of eigenvalue theory and the Jordan canonical form
Bare-bones outline of eigenvalue theory and the Jordan canonical form April 3, 2007 N.B.: You should also consult the text/class notes for worked examples. Let F be a field, let V be a finite-dimensional
More informationEducation in Linear System Theory with the Geometric Approach Tools
Education in Linear System Theory with the Geometric Approach Tools Giovanni Marro DEIS, University of Bologna, Italy GAS Workshop - Sept 24, 2010 Politecnico di Milano G. Marro (University of Bologna)
More informationSTABILITY OF 2D SWITCHED LINEAR SYSTEMS WITH CONIC SWITCHING. 1. Introduction
STABILITY OF D SWITCHED LINEAR SYSTEMS WITH CONIC SWITCHING H LENS, M E BROUCKE, AND A ARAPOSTATHIS 1 Introduction The obective of this paper is to explore necessary and sufficient conditions for stability
More informationFINITE HORIZON ROBUST MODEL PREDICTIVE CONTROL USING LINEAR MATRIX INEQUALITIES. Danlei Chu, Tongwen Chen, Horacio J. Marquez
FINITE HORIZON ROBUST MODEL PREDICTIVE CONTROL USING LINEAR MATRIX INEQUALITIES Danlei Chu Tongwen Chen Horacio J Marquez Department of Electrical and Computer Engineering University of Alberta Edmonton
More informationLecture 9 Monotone VIs/CPs Properties of cones and some existence results. October 6, 2008
Lecture 9 Monotone VIs/CPs Properties of cones and some existence results October 6, 2008 Outline Properties of cones Existence results for monotone CPs/VIs Polyhedrality of solution sets Game theory:
More informationLecture 2: Linear Algebra Review
EE 227A: Convex Optimization and Applications January 19 Lecture 2: Linear Algebra Review Lecturer: Mert Pilanci Reading assignment: Appendix C of BV. Sections 2-6 of the web textbook 1 2.1 Vectors 2.1.1
More informationApplied Mathematics Letters. Comparison theorems for a subclass of proper splittings of matrices
Applied Mathematics Letters 25 (202) 2339 2343 Contents lists available at SciVerse ScienceDirect Applied Mathematics Letters journal homepage: www.elsevier.com/locate/aml Comparison theorems for a subclass
More informationElementary linear algebra
Chapter 1 Elementary linear algebra 1.1 Vector spaces Vector spaces owe their importance to the fact that so many models arising in the solutions of specific problems turn out to be vector spaces. The
More informationChapter 1. Preliminaries
Introduction This dissertation is a reading of chapter 4 in part I of the book : Integer and Combinatorial Optimization by George L. Nemhauser & Laurence A. Wolsey. The chapter elaborates links between
More informationASIGNIFICANT research effort has been devoted to the. Optimal State Estimation for Stochastic Systems: An Information Theoretic Approach
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 42, NO 6, JUNE 1997 771 Optimal State Estimation for Stochastic Systems: An Information Theoretic Approach Xiangbo Feng, Kenneth A Loparo, Senior Member, IEEE,
More informationRank Tests for the Observability of Discrete-Time Jump Linear Systems with Inputs
Rank Tests for the Observability of Discrete-Time Jump Linear Systems with Inputs Ehsan Elhamifar Mihály Petreczky René Vidal Center for Imaging Science, Johns Hopkins University, Baltimore MD 21218, USA
More informationOn the projection onto a finitely generated cone
Acta Cybernetica 00 (0000) 1 15. On the projection onto a finitely generated cone Miklós Ujvári Abstract In the paper we study the properties of the projection onto a finitely generated cone. We show for
More informationarxiv: v1 [cs.sy] 12 Oct 2018
Contracts as specifications for dynamical systems in driving variable form Bart Besselink, Karl H. Johansson, Arjan van der Schaft arxiv:181.5542v1 [cs.sy] 12 Oct 218 Abstract This paper introduces assume/guarantee
More informationChapter 7. Linear Algebra: Matrices, Vectors,
Chapter 7. Linear Algebra: Matrices, Vectors, Determinants. Linear Systems Linear algebra includes the theory and application of linear systems of equations, linear transformations, and eigenvalue problems.
More informationV (v i + W i ) (v i + W i ) is path-connected and hence is connected.
Math 396. Connectedness of hyperplane complements Note that the complement of a point in R is disconnected and the complement of a (translated) line in R 2 is disconnected. Quite generally, we claim that
More informationAuxiliary signal design for failure detection in uncertain systems
Auxiliary signal design for failure detection in uncertain systems R. Nikoukhah, S. L. Campbell and F. Delebecque Abstract An auxiliary signal is an input signal that enhances the identifiability of a
More informationALGEBRA QUALIFYING EXAM PROBLEMS LINEAR ALGEBRA
ALGEBRA QUALIFYING EXAM PROBLEMS LINEAR ALGEBRA Kent State University Department of Mathematical Sciences Compiled and Maintained by Donald L. White Version: August 29, 2017 CONTENTS LINEAR ALGEBRA AND
More informationMath 341: Convex Geometry. Xi Chen
Math 341: Convex Geometry Xi Chen 479 Central Academic Building, University of Alberta, Edmonton, Alberta T6G 2G1, CANADA E-mail address: xichen@math.ualberta.ca CHAPTER 1 Basics 1. Euclidean Geometry
More informationA basic canonical form of discrete-time compartmental systems
Journal s Title, Vol x, 200x, no xx, xxx - xxx A basic canonical form of discrete-time compartmental systems Rafael Bru, Rafael Cantó, Beatriz Ricarte Institut de Matemàtica Multidisciplinar Universitat
More informationLinear Algebra: Lecture Notes. Dr Rachel Quinlan School of Mathematics, Statistics and Applied Mathematics NUI Galway
Linear Algebra: Lecture Notes Dr Rachel Quinlan School of Mathematics, Statistics and Applied Mathematics NUI Galway November 6, 23 Contents Systems of Linear Equations 2 Introduction 2 2 Elementary Row
More informationObservability of Switched Linear Systems
Observability of Switched Linear Systems Mohamed Babaali and Magnus Egerstedt Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USA {babaali,magnus}@ece.gatech.edu
More informationA linear algebra proof of the fundamental theorem of algebra
A linear algebra proof of the fundamental theorem of algebra Andrés E. Caicedo May 18, 2010 Abstract We present a recent proof due to Harm Derksen, that any linear operator in a complex finite dimensional
More informationRiccati difference equations to non linear extended Kalman filter constraints
International Journal of Scientific & Engineering Research Volume 3, Issue 12, December-2012 1 Riccati difference equations to non linear extended Kalman filter constraints Abstract Elizabeth.S 1 & Jothilakshmi.R
More informationA linear algebra proof of the fundamental theorem of algebra
A linear algebra proof of the fundamental theorem of algebra Andrés E. Caicedo May 18, 2010 Abstract We present a recent proof due to Harm Derksen, that any linear operator in a complex finite dimensional
More informationVector Spaces. Addition : R n R n R n Scalar multiplication : R R n R n.
Vector Spaces Definition: The usual addition and scalar multiplication of n-tuples x = (x 1,..., x n ) R n (also called vectors) are the addition and scalar multiplication operations defined component-wise:
More informationNull controllable region of LTI discrete-time systems with input saturation
Automatica 38 (2002) 2009 2013 www.elsevier.com/locate/automatica Technical Communique Null controllable region of LTI discrete-time systems with input saturation Tingshu Hu a;, Daniel E. Miller b,liqiu
More informationBoundedly complete weak-cauchy basic sequences in Banach spaces with the PCP
Journal of Functional Analysis 253 (2007) 772 781 www.elsevier.com/locate/jfa Note Boundedly complete weak-cauchy basic sequences in Banach spaces with the PCP Haskell Rosenthal Department of Mathematics,
More informationGeneralized eigenspaces
Generalized eigenspaces November 30, 2012 Contents 1 Introduction 1 2 Polynomials 2 3 Calculating the characteristic polynomial 5 4 Projections 7 5 Generalized eigenvalues 10 6 Eigenpolynomials 15 1 Introduction
More informationPartial Eigenvalue Assignment in Linear Systems: Existence, Uniqueness and Numerical Solution
Partial Eigenvalue Assignment in Linear Systems: Existence, Uniqueness and Numerical Solution Biswa N. Datta, IEEE Fellow Department of Mathematics Northern Illinois University DeKalb, IL, 60115 USA e-mail:
More informationFINITE CONNECTED H-SPACES ARE CONTRACTIBLE
FINITE CONNECTED H-SPACES ARE CONTRACTIBLE ISAAC FRIEND Abstract. The non-hausdorff suspension of the one-sphere S 1 of complex numbers fails to model the group s continuous multiplication. Moreover, finite
More informationLecture notes: Applied linear algebra Part 1. Version 2
Lecture notes: Applied linear algebra Part 1. Version 2 Michael Karow Berlin University of Technology karow@math.tu-berlin.de October 2, 2008 1 Notation, basic notions and facts 1.1 Subspaces, range and
More information2 Sequences, Continuity, and Limits
2 Sequences, Continuity, and Limits In this chapter, we introduce the fundamental notions of continuity and limit of a real-valued function of two variables. As in ACICARA, the definitions as well as proofs
More informationEquality: Two matrices A and B are equal, i.e., A = B if A and B have the same order and the entries of A and B are the same.
Introduction Matrix Operations Matrix: An m n matrix A is an m-by-n array of scalars from a field (for example real numbers) of the form a a a n a a a n A a m a m a mn The order (or size) of A is m n (read
More informationIrreducible subgroups of algebraic groups
Irreducible subgroups of algebraic groups Martin W. Liebeck Department of Mathematics Imperial College London SW7 2BZ England Donna M. Testerman Department of Mathematics University of Lausanne Switzerland
More informationThis article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and
This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and education use, including for instruction at the authors institution
More informationOctober 25, 2013 INNER PRODUCT SPACES
October 25, 2013 INNER PRODUCT SPACES RODICA D. COSTIN Contents 1. Inner product 2 1.1. Inner product 2 1.2. Inner product spaces 4 2. Orthogonal bases 5 2.1. Existence of an orthogonal basis 7 2.2. Orthogonal
More informationControllability of Linear Discrete-time Systems with Time-delay in State and Control 1
Controllability of Linear Discrete-time Systems with Time-delay in State and Control 1 Peng Yang, Guangming Xie and Long Wang Center for Systems and Control, Department of Mechanics and Engineering Science,
More informationSome Results Concerning Uniqueness of Triangle Sequences
Some Results Concerning Uniqueness of Triangle Sequences T. Cheslack-Postava A. Diesl M. Lepinski A. Schuyler August 12 1999 Abstract In this paper we will begin by reviewing the triangle iteration. We
More informationDS-GA 1002 Lecture notes 0 Fall Linear Algebra. These notes provide a review of basic concepts in linear algebra.
DS-GA 1002 Lecture notes 0 Fall 2016 Linear Algebra These notes provide a review of basic concepts in linear algebra. 1 Vector spaces You are no doubt familiar with vectors in R 2 or R 3, i.e. [ ] 1.1
More informationSmall Gain Theorems on Input-to-Output Stability
Small Gain Theorems on Input-to-Output Stability Zhong-Ping Jiang Yuan Wang. Dept. of Electrical & Computer Engineering Polytechnic University Brooklyn, NY 11201, U.S.A. zjiang@control.poly.edu Dept. of
More informationA note on continuous behavior homomorphisms
Available online at www.sciencedirect.com Systems & Control Letters 49 (2003) 359 363 www.elsevier.com/locate/sysconle A note on continuous behavior homomorphisms P.A. Fuhrmann 1 Department of Mathematics,
More informationMATH 117 LECTURE NOTES
MATH 117 LECTURE NOTES XIN ZHOU Abstract. This is the set of lecture notes for Math 117 during Fall quarter of 2017 at UC Santa Barbara. The lectures follow closely the textbook [1]. Contents 1. The set
More informationOptimal Polynomial Control for Discrete-Time Systems
1 Optimal Polynomial Control for Discrete-Time Systems Prof Guy Beale Electrical and Computer Engineering Department George Mason University Fairfax, Virginia Correspondence concerning this paper should
More informationApplied Mathematics Letters
Applied Mathematics Letters 25 (2012) 974 979 Contents lists available at SciVerse ScienceDirect Applied Mathematics Letters journal homepage: www.elsevier.com/locate/aml On dual vector equilibrium problems
More informationEIGENVALUES AND EIGENVECTORS 3
EIGENVALUES AND EIGENVECTORS 3 1. Motivation 1.1. Diagonal matrices. Perhaps the simplest type of linear transformations are those whose matrix is diagonal (in some basis). Consider for example the matrices
More informationMeans of unitaries, conjugations, and the Friedrichs operator
J. Math. Anal. Appl. 335 (2007) 941 947 www.elsevier.com/locate/jmaa Means of unitaries, conjugations, and the Friedrichs operator Stephan Ramon Garcia Department of Mathematics, Pomona College, Claremont,
More informationSolution. That ϕ W is a linear map W W follows from the definition of subspace. The map ϕ is ϕ(v + W ) = ϕ(v) + W, which is well-defined since
MAS 5312 Section 2779 Introduction to Algebra 2 Solutions to Selected Problems, Chapters 11 13 11.2.9 Given a linear ϕ : V V such that ϕ(w ) W, show ϕ induces linear ϕ W : W W and ϕ : V/W V/W : Solution.
More informationReview of Controllability Results of Dynamical System
IOSR Journal of Mathematics (IOSR-JM) e-issn: 2278-5728, p-issn: 2319-765X. Volume 13, Issue 4 Ver. II (Jul. Aug. 2017), PP 01-05 www.iosrjournals.org Review of Controllability Results of Dynamical System
More informationCONTROLLABILITY OF NONLINEAR DISCRETE SYSTEMS
Int. J. Appl. Math. Comput. Sci., 2002, Vol.2, No.2, 73 80 CONTROLLABILITY OF NONLINEAR DISCRETE SYSTEMS JERZY KLAMKA Institute of Automatic Control, Silesian University of Technology ul. Akademicka 6,
More informationarxiv: v1 [cs.sy] 2 Apr 2019
On the Existence of a Fixed Spectrum for a Multi-channel Linear System: A Matroid Theory Approach F Liu 1 and A S Morse 1 arxiv:190401499v1 [cssy] 2 Apr 2019 Abstract Conditions for the existence of a
More informationCONTROL SYSTEMS, ROBOTICS AND AUTOMATION - Vol. VII - System Characteristics: Stability, Controllability, Observability - Jerzy Klamka
SYSTEM CHARACTERISTICS: STABILITY, CONTROLLABILITY, OBSERVABILITY Jerzy Klamka Institute of Automatic Control, Technical University, Gliwice, Poland Keywords: stability, controllability, observability,
More informationarxiv: v1 [math.na] 5 May 2011
ITERATIVE METHODS FOR COMPUTING EIGENVALUES AND EIGENVECTORS MAYSUM PANJU arxiv:1105.1185v1 [math.na] 5 May 2011 Abstract. We examine some numerical iterative methods for computing the eigenvalues and
More information2. Intersection Multiplicities
2. Intersection Multiplicities 11 2. Intersection Multiplicities Let us start our study of curves by introducing the concept of intersection multiplicity, which will be central throughout these notes.
More informationLyapunov Stability of Linear Predictor Feedback for Distributed Input Delays
IEEE TRANSACTIONS ON AUTOMATIC CONTROL VOL. 56 NO. 3 MARCH 2011 655 Lyapunov Stability of Linear Predictor Feedback for Distributed Input Delays Nikolaos Bekiaris-Liberis Miroslav Krstic In this case system
More informationLECTURE 7: STABLE RATIONALITY AND DECOMPOSITION OF THE DIAGONAL
LECTURE 7: STABLE RATIONALITY AND DECOMPOSITION OF THE DIAGONAL In this lecture we discuss a criterion for non-stable-rationality based on the decomposition of the diagonal in the Chow group. This criterion
More informationCHAPTER 7. Connectedness
CHAPTER 7 Connectedness 7.1. Connected topological spaces Definition 7.1. A topological space (X, T X ) is said to be connected if there is no continuous surjection f : X {0, 1} where the two point set
More informationThe Skorokhod reflection problem for functions with discontinuities (contractive case)
The Skorokhod reflection problem for functions with discontinuities (contractive case) TAKIS KONSTANTOPOULOS Univ. of Texas at Austin Revised March 1999 Abstract Basic properties of the Skorokhod reflection
More informationMAT 445/ INTRODUCTION TO REPRESENTATION THEORY
MAT 445/1196 - INTRODUCTION TO REPRESENTATION THEORY CHAPTER 1 Representation Theory of Groups - Algebraic Foundations 1.1 Basic definitions, Schur s Lemma 1.2 Tensor products 1.3 Unitary representations
More informationStat 159/259: Linear Algebra Notes
Stat 159/259: Linear Algebra Notes Jarrod Millman November 16, 2015 Abstract These notes assume you ve taken a semester of undergraduate linear algebra. In particular, I assume you are familiar with the
More informationOHSx XM511 Linear Algebra: Solutions to Online True/False Exercises
This document gives the solutions to all of the online exercises for OHSx XM511. The section ( ) numbers refer to the textbook. TYPE I are True/False. Answers are in square brackets [. Lecture 02 ( 1.1)
More informationIRREDUCIBLE REPRESENTATIONS OF SEMISIMPLE LIE ALGEBRAS. Contents
IRREDUCIBLE REPRESENTATIONS OF SEMISIMPLE LIE ALGEBRAS NEEL PATEL Abstract. The goal of this paper is to study the irreducible representations of semisimple Lie algebras. We will begin by considering two
More informationThe Jordan canonical form
The Jordan canonical form Francisco Javier Sayas University of Delaware November 22, 213 The contents of these notes have been translated and slightly modified from a previous version in Spanish. Part
More informationRose-Hulman Undergraduate Mathematics Journal
Rose-Hulman Undergraduate Mathematics Journal Volume 17 Issue 1 Article 5 Reversing A Doodle Bryan A. Curtis Metropolitan State University of Denver Follow this and additional works at: http://scholar.rose-hulman.edu/rhumj
More informationObservability of Linear Hybrid Systems
Observability of Linear Hybrid Systems René Vidal 1, Alessandro Chiuso 2, Stefano Soatto 3, and Shankar Sastry 1 1 Department of EECS, University of California, Berkeley, CA 94720, USA, Phone: (510) 643
More information