# A Passivity Measure Of Systems In Cascade Based On Passivity Indices

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2 Definition (Supply Rate [8]) The supply rate ω(t)= ω(u(t),y(t)) is a real valued function defined on U Y, such that for any u(t) U and x X and y(t)=h(φ(t,t, x,u)), ω(t) satisfies t t ω(τ) dτ< () Definition (Dissipative System [8]) System H with supply rateω(t) is said to be dissipative if there exists a nonnegative real function V(x):X R +, called the storage function, such that, for all t t, x X and u U, V(x ) V(x ) t t ω(τ)dτ (3) where x =φ(t,t, x,u), andr + is a set of nonnegative real numbers Definition 3(Passive System [8]) System H is said to be passive if there exists a storage function V(x) such that V(x ) V(x ) if V(x) isc, then we have t t u(τ) T y(τ)dτ, (4) V(x) u(t) T y(t), t (5) One can see that passive system is a special case of dissipative system with supply rateω(t)=u(t) T y(t) Definition 4(Excess/Shortage of Passivity [6]) System H is said to be: Input Feed-forward Passive (IFP) if it is dissipative with respect to supply rateω(u,y)=u T y νu T uforsomeν R, denoted as IFP(ν) Output Feedback Passive (OFP) if it is dissipative with respect to the supply rateω(u,y)=u T y ρy T yforsome ρ R, denoted as OFP(ρ) Input Feed-forward Output Feedback Passive (IF-OFP) if it is dissipative with respect to the supply rateω(u,y)= u T y ρy T y νu T uforsomeρ R andν R, denoted as IF-OFP(ν,ρ) A positiveνorρmeans that the system has an excess of passivity; otherwise, the system is lack of passivity In the case when ν> orρ>, the system is said to be input strictly passive(isp) or output strictly passive(osp) respectively Definition 5(Zero-State Observability and Detectability [6]) Consider the system H with zero input, that is ẋ= f (x,), y= h(x,), and let Z R n be its largest positively invariant set contained in{x R n y=h(x,)=} WesayH is zero-state detectable(zsd) if x = is asymptotically stable conditionally to Z if Z={}, we say that H is zero-state observable (ZSO) III Secant Criterion and Some Facts on Diagonally Stable Matrices and Quasi-Dominant Matrices The connection between diagonal stability and the secant criterion has been shown in []-[] We briefly summarize these results here Definition 4 (Diagonal Stability [7]) AmatrixA:= (a ij ) is said to belong to the class of Hurwitz diagonally stable matrix if there exists a diagonal matrix D> such that A T D+ DA< (6) Definition 5 (Quasi-dominant matrix [7]) A square matrix is a quasi-dominant matrix, or in the class of diagonally rowsum or column-sum quasi-dominant matrices if there exists a positive diagonal matrix P=diag{p, p,,p n } such that a ii p i j i a ij p j, i, (respectively) a jj p j i j a ji p i, j If these inequalities are strict, the matrix is referred to as strictly row-sum (respectively column-sum) quasi-dominant If P can be chosen as the identity matrix, then the matrix is called row- or column- diagonally dominant Theorem (Secant Criterion []) A matrix of the form: α β n β α A= β α 3 α i >,β i >, (7) β n α n i=,,n, is diagonally stable, that is, it satisfies (5) for some diagonal matrix D>, if and only if the secant criterion β β n < sec(π/n) n = α α n cos( π (8) n )n holds; here we assume n> Corollary [9] Every symmetric quasi-dominant matrix is positive definite Lemma [7] If a matrix A is diagonally stable, then A T is also diagonally stable In the subsequent sections, we will show how we use the secant criterion and the properties of quasi-dominant matrix to measure the shortage/excess of passivity for cascade systems IV Main Results In this section, we show passivity measures for three classes of cascade systems, which include output strictly passive systems(ofp(ρ) with ρ>), input strictly passive systems(ifp(ν) with ν>) and input feed-forward output feedback passive systems(if-ofp(ν, ρ) with ρ, ν R) One should notice that while the first and the second classes assume that each interconnected subsystem is passive, the third class applies to the more general case of dissipative systems We also show sufficient conditions under which those cascade systems can be stabilized via output feedback A Passivity measure of a cascade of OSP systems Proposition Consider the cascade interconnection shown in Fig, where n Let each block be OFP(ρ i ) withρ i >, namely there existsc storage function V i for each subsystem, such that V i ρ i y T i y i+ u T i y i, (9) 87

6 Choosing V (x)= x (t) as the storage function for H,we can obtain V = u (t)y (t) 75u (t)+y (t), (5) so the passivity indices for H areρ = andν = 75 We can see that H is ZSD but unstable H admits a storage function given by V (x)= 6 x,and V = y (t)u (t) 4u (t) 8 y (t), (5) so the passivity indices for H areρ = 8 andν = 4 We can see that H is stable and ZSD H 3 admits a storage function given by V 3 (x)= 8 x4 3 (t)+ x 3,and V 3 = u 3 (t)y 3 (t) 5u 3 (t)+5y 3 (t), (5) so the passivity indices for H 3 areρ 3 = 5 andν 3 = 5 H 3 is neither ZSD or stable Then the A matrix defined in Proposition 6 is given by ρ ν Since A= ρ ν 3 (53) ρ 3 ν ν +ρ = 39>,ν +ρ 3 = 7>,ν 3 +ρ = 4 9 > (54) A is row/column diagonally dominant Moreover, since {y = y = y 3 = } is contained in the invariant set for which V= 3 i= V i =, according to Proposition 6, the cascade of H, H and H 3 could be stabilized directly via unity output feedback The simulation results for the closed-loop system with r= (seefig4)areshowninfig6-fig7 VI Conclusions In this paper, we introduce a way to measure the degree of passivity for cascaded ISP/OSP systems We further proposed a method for passivity measure of cascaded IF- OFP systems(dissipative systems), and study the conditions under which the cascade interconnection can be stabilized via output feedback VII Acknowledgments The support of the National Science Foundation under Grant No CCF is gratefully acknowledged References [] E D Sontag, Passivity gains and the secant condition for stability, Systems & Control Letters, Volume 55, Issue 3, March 6, Pages [] M Arcak, E D Sontag, Diagonal stability of a class of cyclic systems and its connection with the secant criterion, Automatica, Volume 4, Issue 9, September 6, Pages [3] J J Tyson, H G Othmer, The dynamics of feedback control circuits in biochemical pathways, in: R Rosen, F M Snell (Eds), in: Progress in Theoretical Biology, vol5, Academic Press, New York, 978, pp-6 [4] C D Thron, The secant condition for instability in biochemical feedback control-parts I and II Bulletin of Mathematical Biology 53 (99) pp outputs time(s) states Fig 6: Outputs of H,H and H Y Y Y 3 X 3 X time(s) Fig 7: States of H,H and H 3 [5] C I Byrnes, A Isidori, and J C Willems, Passivity, feedback equivalence and the global stabilization of minimum phase nonlinear systems, IEEE Transactions on Automatic Control, vol36, no, pp8-4, Nov99 [6] R Sepulchre, M Jankovic, and P V Kokotovic, Constructive Nonlinear Control, Springer-Verlag, 997 [7] E Kaszkurewicz, A Bhaya, Matrix Diagonal Stability in Systems and Computation,Birkhauser Boston, 999 [8] J C Willems, Dissipative dynamical systems part I: General theory, Archive for Rational Mechanics and Analysis, Springer Berlin, Volume 45, Number 5, Page 3-35, January, 97 [9] J T Wen, Time domain and frequency domain conditions for strict positive realness, IEEE Trans on Automatic Control, vol33, no, pp988-99, 988 [] J T Wen, Robustness analysis based on passivity, In Proceedings of American Control Conference, page 7-, Atlanta, 988 [] V M Popov, Absolute stability of nonlinear systems of automatic control, vol, no8, pp , Aug, 96 [] V M Popov, Hyperstability of Control Systems, Springer-Verlag, 973 [3] C I Byrnes, A Isidori, New results and examples in nonlinear feedback stabilization, Systems & Control Letters Volume, Issue 5, June 989, Pages [4] P V Kokotovic and H J Sussmann, A positive real condition for global stabilization of nonlinear systems, Systems & Control Letters, Volume 3, Issue, August 989, Pages 5-33 [5] D J Hill and P J Moylan, Stability results for nonlinear feedback systems, Automatica, Vol 3, pp , July 977 [6] J P LaSalle, Some extensions of Lyapunovs second method, IRE Transactions on Circuit Theory, CT-7, pp5-57, 96 [7] M Jankovic, R Sepulchre, P V Kokotovic, Constructive Lyapunov stabilization of nonlinear cascade systems, IEEE Transactions on Automatic Control, Volume 4, Issue, Dec 996, Page [8] R Sepulchre, M Jankovic, P V Kokotovic, Integrator Forwarding: A New Recursive Nonlinear Robust Design, Automatica, Volume 33, Number 5, May 997, pp [9] O Taussky, A recrurring theorem on determinants, American Mathematical Monthly, 56:67-676, 949 X X 9

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