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On Padé approximations, quadratic stability and discretization of switched linear systems

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In this note we consider the stability preserving properties of diagonal Padé approximations to the matrix exponential. We show that while diagonal Padé approximations preserve quadratic stability when going from continuous-time to discrete-time, the
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  On Padé Approximations, Quadratic Stability andDiscretization of Switched Linear Systems R. Shorten 1 , M. Corless 2 , S. Sajja 1, ∗ , S. Solmaz 1 Abstract In this note we consider the stability preserving properties of diagonal Padé ap-proximations to the matrix exponential. We show that while diagonal Padé ap-proximations preserve quadratic stability when going from continuous-time todiscrete-time, the converse is not true. We discuss the implications of this resultfor discretizing switched linear systems. We also show that for continuous-timeswitched systems which are exponentially stable, but not quadratically stable, aPadé approximation may not preserve stability. Keywords: Padé Approximations, Quadratic Stability, Switched Linear Systemsand Discretization. 1. Introduction The Diagonal Padé approximations to the exponential function are known to maptheopenleft halfofthecomplexplanetotheopeninterioroftheunitdisk[3]. Thisgives rise to a correspondence between continuous-time stable LTI (linear timeinvariant) systems and their discrete-time stable counterparts (a fact that is oftenexploitedin the systemsand control community[6]). Perhaps the best known mapofthiskindisthefirst orderdiagonalPadéapproximant(alsoknownasthebilinearor Tustin map [3]). The bilinear map is known not only to preserve stability, butalso preserve quadratic Lyapunov functions. That is, a positive definite matrix P satisfying A ∗ c P + PA c < 0 will also satisfy A ∗ d  PA d  − P < 0 where A d  is the mapping ∗ Corresponding Author  Email address: ×ÙÖÝº×º¾¼¼ÒÙѺ  (S. Sajja) URL: ØØÔ»»ÛÛÛºÑÐØÓÒº»  (S. Sajja) 1 The Hamilton Institute, National University of Ireland, Maynooth, Co. Kildare, Ireland. 2 School of Aeronautics and Astronautics, Purdue University, West Lafayette, IN, USA. Preprint submitted to Systems and Control Letters November 16, 2010  of  A c under the bilinear transform [6] with some sampling time h [1]. This makesit extremely useful when transforming a continuous-time switching system:˙  x = A c ( t  )  x , A c ( t  ) ∈ A   c (1)into an approximate discrete-time counterpart 1 ,  x ( k  + 1 ) = A d  ( k  )  x ( k  ) , A d  ( k  ) ∈ A   d  (2)because, the existence of a common positive definite matrix P satisfying A ∗ c P + PA c < 0 for all all A c ∈ A   c implies that the same P satisfies A ∗ d  PA d  − P < 0 for all  A d  ∈ A   d  . Thus quadratic stability of the continuous-time switching system im-plies quadratic stability of the discrete-time counterpart. This property is usefulin obtaining results in discrete-time from their continuous-time counterparts [6],and in providinga robust method to obtain a stablediscrete-timeswitching systemfrom a continuous-time one.Our objective in this present note is to determine whether this property is pre-served by higher order (more accurate) Padé approximants. From the point of view of discretization, low order approximants are not always satisfactory, andoneoften chooses higher order Padéapproximationsin real applications. Later wepresentanexampleofaexponentiallystablecontinuous-timeswitchingsystemforwhich a discretisation based on a first order Pade approximation is unstable, but,discretizations based on second order approximations are stable for any samplingtime. Also, it is well known that the first order Padé approximation (the bilinearapproximation) can map a negative real eigenvalue to a negative eigenvalue if thesampling time is large. In such situations, while stability is preserved, qualita-tive behavior is not preserved even for LTI systems; a non-oscillatory continuousmode is transformed into an oscillatory discrete-time mode. In this context weestablish the following facts concerning general diagonal Padé approximations.(i) Consider an LTI system Σ c : ˙  x = A c  x and let Σ d  : x ( k  + 1 ) = A d   x ( k  ) be anydiscrete-time system obtained from Σ c using any diagonal Padé approxima-tion and any sampling time. If  V  is any quadratic Lyapunov function for Σ c then, V  is a quadratic Lyapunov function for Σ d  .(ii) The converse of the statement in (i) is only true for first order Padé approx-imations. 1 Discretization error is zero, only at sampling instants. 2  (iii) Consider a switched system Σ sc : ˙  x = A sc ( t  )  x , A sc ( t  ) ∈ {  A c 1 ,...,  A cn } andlet Σ d  : x ( k  + 1 ) = A sd  ( k  )  x ( k  ) , A sd  ( k  ) ∈ {  A d  1 ,...,  A dn } be a discrete-timeswitched systemobtained from Σ sc using any diagonal Padé approximationsand any sampling times. If  V  is any quadratic Lyapunov function for Σ sc then, V  is a quadratic Lyapunov function for Σ sd  .(iv) The converse of the statement in (iii) is only true for first order Padé ap-proximations.(v) Consider an exponentiallystableswitched system Σ sc : ˙  x =  A sc ( t  )  x , A sc ( t  ) ∈{  A c 1 ,...,  A cn } . Let Σ d  : x ( k  + 1 ) = A sd  ( k  )  x ( k  ) , A sd  ( k  ) ∈{  A d  1 ,...,  A dn } be adiscrete-timeswitchedsystemobtainedfrom Σ sc usinga p ’th order diagonalPadé approximation. Then, Σ ds may be unstable, even when p = 1.These results are quite subtle, but we believe that they are important for a numberor reasons. Discretization of switched systems is a relatively new research direc-tion in the control systems community. To the best of our knowledge, few papersexist on this topic; for example see [7]. In the context of such studies, our resultssay that quadratic stability is robust with respect to diagonal Padé approximations.That is, quadratic stability is always preserved, even when the sampling time ispoorly chosen. This is an important fact when building simulators of switchedlinear systems. Our results also indicate that Padé approximations do not, in gen-eral, preservethestabilitypropertiesofexponentially(butnotquadratically)stablesystems. In such cases, building a (stability preserving) discrete-time simulationmodel of such systems that preserve stability is non-trivial and remains an openquestion.The consequences of our observations go beyond numerical simulation. In manyapplications one converts a continuous-time switched system to a discrete-timeequivalent before embarking on control design. Our results indicate that one mustexhibit extreme caution in discretizing a continuous-time switched system model.In particular, care is needed in assuming that properties of thesrcinal continuous-timeproblem are inherited from properties of the-discrete timeapproximation [4].In fact, stability of the discrete-time model does not necessarily imply stability of thecontinuous-timeone: even fordiscrete-timesystemsthatare quadraticallysta-ble . Our results also pose questions for model order reduction of switched linearsystems. Again this is a relatively new area of study of considerable interest in theVLSI community. In such applications, where the ultimate objective is numeri-cal simulation, stability may be preserved in the reduction of the continuous-time3  model to another lower order continuous-time model, only for it to be lost in thediscretization step. 2. Mathematical Preliminaries The following definitions and results are useful in developing the main result,Theorem 1, which is given in Section 3. Notation : A square matrix A c is said to be ÀÙÖÛØÞ×ØÐ  if all of its eigenvalueslie in the open left-half of the complex plane. A square matrix A d  is said to be ËÙÖ×ØÐ  if all its eigenvalues lie in the open interior of the unit disc. The no-tation M  ∗ is used to denote the complex conjugate transpose of a general squarematrix M  ; M  is hermitian if  M  ∗ = M  . A hermitian matrix P is said to be positive(negative) definite if  x ∗ Px > 0 (  x ∗ Px < 0) for all non-zero x and we denote this by P > 0 ( P < 0). In all of the following definitions, P = P ∗ > 0.A matrix P is a ÄÝÔÙÒÓÚÑØÖÜ  for a Hurwitz stable matrix A c if  A ∗ c P + PA c < 0. In this case, V  (  x ) = x ∗ Px is a ÕÙÖØÄÝÔÙÒÓÚÙÒØÓÒ´ÉÄµ  for thecontinuous-time LTI system ˙  x ( t  ) = A c  x ( t  ) . A matrix P is a ËØÒÑØÖÜ  for aSchur stable matrix A d  if  A ∗ d  PA d  − P < 0. In this case, V  (  x ) = x ∗ Px is a ÕÙÖØ ÄÝÔÙÒÓÚÙÒØÓÒ  for the discrete-time LTI system x ( k  + 1 ) = A d   x ( k  ) .Given a finite set of Hurwitz stable matrices A   c a matrix P is a ÓÑÑÓÒÄÝÔÙÒÓÚ ÑØÖÜ´Äŵ  for A   c if   A ∗ c P + PA c < 0 forall A c in A   c . In thiscase, wesay that thecontinuous-timeswitching system (1) is ÕÙÖØÐÐÝ×ØÐ´É˵  with Lyapunovfunction V  (  x ) = x ∗ Px and V  is a ÓÑÑÓÒÕÙÖØÄÝÔÙÒÓÚÙÒØÓÒ´ÉÄµ  for A   c .Given a finite set of Schur stable matrices A   d  a matrix P is a ÓÑÑÓÒËØÒÑ¹ ØÖÜ´Ëŵ  for A   d  if  A ∗ d  PA d  − P < 0 for all A d  in A   d  . In this case, we say thatthe discrete-timeswitching system (2) is ÕÙÖØÐÐÝ×ØÐ´É˵  with Lyapunovfunction V  (  x ) = x ∗ Px and V  is a ÓÑÑÓÒÕÙÖØÄÝÔÙÒÓÚÙÒØÓÒ´ÉÄµ  for A   d  .Our primary interest in this note is to examine the invariance of quadratic Lya-punov functions under diagonal Padé approximations to the matrix exponential.Recall the definition of the diagonal Padé approximations to the exponential func-tion.4  Definition 1. (Diagonal Padé Approximations) [3][12]: The p th order diagonalPadé approximation to the exponential function e s is the rational function C   p de- fined byC   p ( s ) = Q  p ( s ) Q  p ( − s ) (3) whereQ  p ( s ) =  p ∑ k  = 0 c k  s k  and c k  =( 2  p − k  ) !  p ! ( 2  p ) ! k  ! (  p − k  ) ! . (4)Thus the p th order diagonal Padé approximation to e  A c h , the matrix exponentialwith sampling time h , is given by C   p (  A c h ) = Q  p (  A c h ) Q − 1  p ( −  A c h ) (5)where Q  p (  A c h ) = ∑  pk  = 0 c k  (  A c h ) k  .Much is known about diagonal Padé maps in the context of LTI systems. In par-ticular, the fact that such approximations map the open left half of the complexplane to the interior of the unit disc is widely exploited in systems and control.This implies the well known fact that these maps preserve stability of LTI systemsas stated formally in the following lemma. Lemma 1. [3] ( Preservation of stability) Suppose that A c is a Hurwitz stablematrix and, for any sampling time h > 0  , let A d  = C   p (  A c h ) be a diagonal Padé approximation of e  A c h of any order p. Then A d  is Schur stable. A special diagonal Padé approximation is the first order approximation. This isalso sometimes referred to as the bilinear (or Tustin) transform. Definition 2. (Bilinear transform) [3][12]: The first order diagonal Padé ap- proximation to the matrix exponential with sampling time h is defined by:C  1 (  A c h ) =   I  +  A c h 2   I  −  A c h 2  − 1 . (6)This approximation is known to not only preserve stability, but also to preservequadratic Lyapunov functions [1, 2, 6]; namely if  P is a Lyapunov matrix for A c then it is also aStein matrixfor A d  = C  1 (  A c h ) . Theconverse statementis also true.Actually, we have the following known result which is a special case of Lemma 3below.5
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