Piecewise Quadratic Lyapunov Functions for Piecewise Affine Time-Delay Systems

Transcription

1 Piecewise Quadratic Lyapunov Functions for Piecewise Affine ime-delay Systems Vishwesh Kulkarni Myungsoo Jun João Hespanha Abstract We investigate some particular classes of hybrid systems subect to a class of time delays; the time delays can be constant or time varying For such systems, we present the corresponding classes of piecewise continuous Lyapunov functions Index erms Lyapunov functions, hybrid systems, stability I INRODUCION Construction of Lyapunov functions is a fundamental problem in system theory its importance stems from the fact that the internal stability of a system is concluded if an associated Lyapunov function is shown to exist his paper concerns such a construction for a class of systems that are hybrid in the sense that the state traectory evolution is governed by different dynamical equations over different polyhedral partitions X i of the state-space X; ie, the system is modelled by an ensemble of subsystems, each of which is a valid representation of the system over a set of such partitions A motivating application for the study of such systems is described in 6 Conceptually, perhaps the simplest solution is a common quadratic Lyapunov function, ie a quadratic function which is a global Lyapunov function for the subsystems comprising the hybrid system 3 However, the construction of such a Lyapunov function is an N P-hard problem even when the subsystems are linear time invariant Furthermore, the existence of such a function is, in principle, an overly restrictive requirement to deduce the stability 4, Section IV Conservatism introduced by a global Lyapunov function V can be reduced by searching for a set {V i } of local Lyapunov functions and by ensuring that the Lyapunov functions match in the sense that the values of Lyapunov functions V i and V are equal when the state traectory leaves a cell X i and enters a cell X, where V i is a local Lyapunov function in the cell X i and V is a local Lyapunov function in the cell X (see 2 and 7) In this context, an elegant result has been recently derived by 4 to construct Dr V Kulkarni is with the Laboratory for Information and Decision Systems, Massachusetts Institute of echnology, Cambridge, MA vishwesh@mitedu Research supported in part by the NSF Grant and the DCI-MI Grant Dr M Jun is with the Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, NY , USA m73@cornelledu Research sponsored by AFOSR Grant F Prof J Hespanha is with the Department of Electrical and Computer Engineering, University of California, Santa Barbara hespanha@eceucsbedu Lyapunov functions when the subsystem dynamics are known to be affine time invariant; an independent interpretation of this result is given in 3 For some practical applications, however, the piecewise affine structure must be modified to address modelling uncertainties and time delays 6 For such systems, consequently, the stability conditions laid down by 4 get modified as we will demonstrate he paper is organized as follows he notation and the key relevant concepts are introduced in Section II he problems are formulated in Section III and the relevant prior art is described in Section IV Our main results are presented in Section V and discussed in Section VI he paper is concluded in Section VII Formal proofs are presented in the Appendix II PRELIMINARIES he notation is introduced as and when necessary Capital letter symbols, such as F and G, denote operators whereas small letter symbols, such as x and y, denote real signals which may possibly be vector valued or matrix valued he set of all real (complex) numbers is denoted R (C) and the set of all integers is denoted Z he notation stands for defined as he inner product x, y y(t) x(t) dt he Euclidean norm x x, x he vector space of signals for which the Euclidean norm exists is denoted L n 2 he vector space L n 2 is generally referred to as L 2 Fourier transform of x is denoted x Conugate transpose of a vector or matrix ( ) is denoted ( ) ; its transpose is denoted ( ) and ( ( ) 2) is denoted ( ) 2 Given z R n n, z implies that every element of z is nonnegative he (i, )-th element of a matrix ( ) is denoted as either ( ) i, or ( ) i, depending on the ease of reading ime derivative of the signal x is denoted ẋ Definition (Piecewise Affine Systems, 4): he class S H of hybrid systems is defined by a family of ordinary differential equations as: ẋ(t) A i x(t) + a i, where A i R n n, a i R n, and {X i } i I R n is a partition of the state-space into a finite number of closed, and possibly unbounded, polyhedral cells with pairwise disoint interior he set of cells that include the origin is denoted I, ie a i, i I ; its compliment is denoted I Definition 2 (Piecewise Affine ime-delay Systems S τc ): he class S τc of hybrid systems is defined by a family of

2 retarded ordinary differential equations as: ẋ(t) A i x(t) + A di x(t τ) + a i, where A i, A di R n n, a i R n, < τ R and {X i } i I R n is a partition of the state-space as in S he set of cells that include the origin is denoted I, ie a i, i I ; its compliment is denoted I Definition 3 (Piecewise Affine ime-delay Systems S τcl ): he class S τcl is obtained from the S τc by replacing the term A di x(t τ) with the term A dil x(t τ l ) where A dil R n n, < τ l R, and < L Z Definition 4 (Piecewise Affine ime-delay Systems S τv ): he class S τv of hybrid systems is defined by a family of retarded ordinary differential equations as: ẋ(t) A i x(t) + A di x(t τ(t)) + a i, where the time varying time delay is constrained as τ(t) h, τ(t) d < t R, for some h, d R, A i, A di R n n, a i R n, and {X i } i I R n is a partition of the state-space as in S he set of cells that include the origin is denoted I, ie a i, i I ; its compliment is denoted I Definition 5 (Piecewise Affine ime-delay Systems S τvl ): he class S τvl is obtained from the S τv by replacing the term A di x(t τ(t)) with the term A dil x(t τ l (t)) where the time varying time delay is constrained as τ l (t) h l, τ l (t) d l < t R, A dil R n n, < τ l (t) R, and < L Z III PROBLEM FORMULAION Problem : Determine a set of computationally tractable analytical conditions under which S τc is stable Problem 2: Determine a set of computationally tractable analytical conditions under which S τcl is stable Problem 3: Determine a set of computationally tractable analytical conditions under which S τv is stable Problem 4: Determine a set of computationally tractable analytical conditions under which S τvl is stable IV PRIOR AR An elegant result on the stability analysis of S s given by 4 Briefly speaking, the development is as follows Denote Ai a Ā i i Ei e i, Fi, where f i Fi ei Let Ē i f i such that x Ē i, x F i x x X i, i I;, i I,, x X i X, i, I() Lemma (heorem, 4): Consider symmetric matrices, U i, and W i such that U i and W i have non negative entries while P i F i F i, for all i I, and P F F, for all I, satisfy A i P i + P i A i + E i U i E i < (2) P i Ei W i E i > (3) Ā P + P Ā + Ē U Ē < (4) P Ē W Ē > (5) for all i I and for all I hen, every piecewise continuous traectory of S H tends to zero exponentially Remark : An independent interpretation, and a slight improvement, of this result is given in 3 Remark 2: o ensure that the local Lyapunov functions match on the cell boundaries, 4 takes the predetermined matrices F i and F as the given variables, the predetermination being as given by (), and uses the elements of the matrix as the free variables Now, the condition () allows for a number of choices of Fi and F which might violate the matching condition, thereby incurring an unnecessarily high cost of computation his can be avoided by working directly with the local Lyapunov functions P i and P as the unknown variables and by stipulating that P i P 2 herm (F i K i ), i, where the elements K i are known variables V MAIN RESULS It is not possible to consider an aggregate state ζ(t) x(t) x(t τ) and apply the arguments of 4 in a straightforward manner to the system of dynamical equations described in terms of ζ his is so because, in general, it is difficult to deduce the cell containing x() given that a particular cell contains x(t) and, hence, it is difficult to state the correct matching conditions for the local Lyapunov functions We now present solutions to Problem and Problem 2 Denote Ad Ā d Lemma 2 (Solution to Problem ): Consider symmetric matrices, U i and W i such that U i and W i have nonnegative entries while P i F i F i, for all i I, and P F F, for all I, satisfy the following inequalities: τp i τa i A di RA2 di τp i τr < τa 2 di RA dia i τa 2 di RA2 di Q (6) P i Ei W ie i >, Q >, R > H τ P τā Ā d RĀ2 d τ P τ R < τā2 d RĀdĀ τā2 d RĀ2 d Q (7) P Ē W Ē >, Q >, R >

3 for all i I and all I where à i Ai + A di,  Ā + Ād, à i P i + P i à i + Q + τa i A dira di A i + E i U i E i, H  P + P  + Q + τā Ā d RĀdĀ + Ē U Ē τ 2 hen, every piecewise continuous traectory of S τc tends to zero exponentially Proof: See the proof in the Appendix section Remark 3: Lemma may be derived as a special case of our heorem by setting τ, A di, Q his is so because the Lyapunov function used by 4 can be derived as a special of our Lyapunov function, given by (A), by setting the V 2 ( ) and V 3 ( ) terms to zero Remark 4: A conservative delay-independent condition is formulated as follows: A i P i + P i A i + Q + Ei U ie i P i A di A di P < i Q (8) P i Ei W ie i >, Q > Ā P + P Ā + Q + Ē U Ē P Ā d Ā P d Q < (9) P Ē W Ē >, Q > x 2 x x (a) τ 2 for all i I and I Remark 5: A further conservative condition, stated by the small gain theorem, is obtained by setting Q I Remark 6: A lower bound on the maximum delay τ for which the system S τ is stable can be obtained by checking whether the conditions laid down by heorem are satisfied as τ increases, starting with τ : the least value τ for which the conditions laid down by heorem are not satisfied, is a conservative estimate of the maximum delay τ under which the system S τ is stable Example : Consider the following piecewise linear time-delay system ẋ(t) A i x(t) + A di x(t τ) with the cell decomposition expressed by E i x, E E 3, E 2 E 4 he system matrices are given by A A 3, A 2 A 4, 5 A d A d3, A d2 A d4 5 he system is reduced to Example in 4 when τ It can be verified from Eq (8) that the system is not stable regardless of delay By applying Lemma 2, the estimated delay margin is τ 42 We can observe from simulations that the system becomes unstable with time-delay between 2 and 2 with initial value x 2 See Figure Remark 7: By applying the delay-dependent condition in 5 and 8, the same procedure as in Lemma 2 yields x (b) Fig State traectories of the system in Example with (a) τ 2, and (b) τ 2 the condition τp i A di A i τp i A 2 di τa i A di P i τq < () τa 2 di P i τr where à i P i+p i à i +τq+τr+ei U ie i Application of the condition to the above example shows the estimated delay margin is τ 36, which is more conservative than the conditions in Lemma 2 heorem (Solution to Problem 2): Consider symmetric matrices, U i and W i such that U i and W i have nonnegative entries while P i F i F i satisfy the condition () for all i I where X l A dil R l A dil, à i Ai + A dil, à i P i + P i à i + Q + τ l A i X l A i + Ei U i E i he conditions for I is formulated similarly hen, every piecewise continuous traectory of S τcl tends to zero exponentially Proof: See the proof in the Appendix section Lemma 3 (Solution to Problem 3): Consider symmetric matrices, U i and W i such that U i and W i have nonnegative entries while P i F i F i, for all i I,

4 L τ P i τ L P i τ l A i X l A di τ l A i X l A dil τ P i τ R τ L P i τ L R L < τ l A dix l A i τ l A dix l A di Q τ l A dix l A dil τ l A dilx l A i τ l A dilx l A di τ l A dilx l A dil Q L P i Ei W ie i >, Q l >, R l >, l,, L () and P F F, for all I, satisfy the following inequalities: hp i ha i A di RA2 di hp i hr < ha 2 di RA dia i ha 2 di RA2 di + (d )Q P i Ei W ie i >, Q >, R > H h P hā Ā d RĀ2 d h P h R < hā2 d RĀdĀ hā2 d RĀ2 d + (d ) Q P Ē W iē >, Q >, R > for all i I and all I where à i Ai + A di,  Ā + Ād, à i P i + P i à i + Q + ha i A dira di A i + E i U i E i, H  P + P  + Q + hā Ā d RĀdĀ + Ē U Ē hen, every piecewise continuous traectory of S τv tends to zero exponentially Proof: he proof follows on the lines of the proof of Lemma 2 by replacing τ by τ(t) in Eq (A) and applying Leibniz rule heorem 2 (Solution to Problem 3): Consider symmetric matrices, U i and W i such that U i and W i have nonnegative entries while P i F i F i satisfy the condition (3) for all i I he conditions for I is formulated similarly hen, every piecewise continuous traectory of S τvl tends to zero exponentially Proof: he proof follows on the lines of the proof of Lemma 3 and heorem VI DISCUSSION An application of this theory is the design of an advanced hazard warning system for highway transportation safety he problem of designing a decentralized advance hazard warning system for highway transportation systems entails the development of efficient switching controllers It so turns out that the vehicle dynamics can be represented by a finite number of modes, each of which is represented by a low order transfer function and a constant time delay he problem of highway safety analysis then gets translated into that of the stability analysis of a time delay hybrid system Effectively, the mode changes partition the state space into cells that share, at most, only each other s boundaries, and the hybrid system has a piecewise affine form in each of the cells A detailed case study is given in 6 VII CONCLUSION We have derived classes of piecewise continuous Lyapunov functions for classes of time-delay hybrid systems inspired by a highway safety application described in 6 Our heorem and heorem 2 extend the well known 4, heorem VIII ACKNOWLEDGMEN We gratefully acknowledge the discussions with Dr Vincent Fromion (INRA, Montpellier), Prof Srinivasa Salapaka (UIUC), and Prof Alex Megretski (MI) Research supported in part by the NSF Grant and the DCI- MI Grant REFERENCES V Blondel and J sitsiklis NP-hardness of some linear control design problems SIAM Journal on Control and Optimization, 35(6):28 227, M Branicky Multiple Lyapunov functions and other analysis tools for switched and hybrid systems IEEE ransactions on Automatic Control, 43: , J Hespanha Stabilization through hybrid control UNESCO Encyclopaedia of Life Support Systems, Nov 22 4 M Johansson and A Rantzer Computation of piecewise quadratic Lyapunov functions for hybrid systems IEEE rans on Automatic Control, 44():99 93, Oct V Kolmanovskii, S Niculescu, and J Richard On the Liapunov- Krasovskii functionals for stability analysis of linear delay systems International Journal of Control, 72(4): , 999

5 L h P i h L P i h l A i X l A di h l A i X l A dil h P i h R h L P i h L R L < h l A dix l A i h l A dix l A di + (d )Q h l A dix l A dil h l A dilx l A i h l A dilx l A di h l A dilx l A dil + (d L )Q L P i Ei W ie i >, Q l >, R l >, l,, L (3) 6 V Kulkarni Optimal mode changes for highway transportation safety In MILCOM, Boston, MA, Oct 23 (accepted) 7 P Peleties and R DeCarlo Asymptotic stability of m-switched systems using Lyapunov like functions In Proc of American Control Conference, pages , June 99 8 J Richard ime-delay systems: an overview of some recent advances and open problems Automatica, 39: , 23 A Proof of Lemma 2 APPENDIX FORMAL PROOFS Consider the Lyapunov function where V (x, t, τ) V (x, t) + V 2 (x, t, τ) + V 3 (x, t, τ) V (x, t) x(t) P i x(t), V 2 (x, t, τ) V 3 (x, t, τ) τ x(ξ) Qx(ξ)dξ, t+ζ Ψ(ξ) A dira di Ψ(ξ)dξ dζ, Ψ(ξ) A i x(ξ) + A di x(ξ τ) (A) he term V 3 is to account for the delay dependency Let Π Hi + τp i R P i Ei U ie i τa i A di RA2 di τa 2 di RA dia i Q + τa 2 di RA2 di It can be easily verified that V (x, t, τ) is continuous in x and t, piecewise continuously differentiable in t, and α x(t) V (x, t, τ) β x(t), t for some α > and β > Now, note that < x(t) Ei U i E i x(t),, (A2) 2a ( b inf a Xa + b X b ), X> ẋ(t) Ãix(t) ( A di Ai x(ξ) + A di x(ξ τ) ) dξ Hence, it may be verified, by using(6), (A2) and Schur complement, that V t 2x(t) P i à i x(t) 2x(t) P i A di + x(t) Qx(t) x(t τ) Qx(t τ) + τψ(t) A dira di Ψ(t) Ψ(ξ) dξ Ψ(ξ) A dira di Ψ(ξ)dξ x(t) ( à i P i + P i à i + Q + τp i R P i ) x(t) x(t τ) Qx(t τ) + τψ(t) A dira di Ψ(t) x(t) x(t) Π x(t τ) x(t τ) < Hence the proof B Proof of heorem Proof: Choosing the Lyapunov function V (x, t, τ) V (x, t) + V 2 (x, t, τ) + V 3 (x, t, τ) (A3) with V (x, t) x(t) P i x(t), V 2 (x, t, τ) x(ξ) Q l x(ξ)dξ, l V 3 (x, t, τ) Ψ(ξ) τ l A i x(ξ) + t+ζ Ψ(ξ) A dilr l A dil Ψ(ξ)dξ dζ, A dil x(ξ τ l ), the proof follows on the lines of the proof of Lemma 2 (A4)