Free Response Characterization Via Flow Invariance (original) (raw)

On the componentwise stability of linear systems

International Journal of Robust and Nonlinear Control, 2005

The componentwise asymptotic stability (CWAS) and componentwise exponential asymptotic stability (CWEAS) represent stronger types of asymptotic stability, which were first defined for symmetrical bounds constraining the flow of the state-space trajectories, and then, were generalized for arbitrary bounds, not necessarily symmetrical. Our paper explores the links between the symmetrical and the general case, proving that the former contains all the information requested by the characterization of the CWAS/ CWEAS as qualitative properties. Complementary to the previous approaches to CWAS/CWEAS that were based on the construction of special operators, we incorporate the flow-invariance condition into the classical framework of stability analysis. Consequently, we show that the componentwise stability can be investigated by using the operator defining the system dynamics, as well as the standard e À d formalism. Although this paper explicitly refers only to continuous-time linear systems, the key elements of our work also apply, mutatis mutandis, to discrete-time linear systems.

Robustness analysis of componentwise asymptotic stability

The componentwise asymptotic stability represents a particular type of asymptotic stability characterized by a supplementary requirement constraining the state-space trajectories, which is derived from the concept of flowinvariance. The paper addresses the robustness of this property with respect to both unstructured and structured perturbations acting upon a linear, continuous or discrete-time dynamic system. For structured perturbations, results are formulated in terms of an arbitrary matrix norm, whereas the approach to structured perturbations yields spectral radius based expressions. Relevant aspects of the robustness analysis are illustrated by a numerical example.

The Stability of Linear Systems

2018

This paper examines the relation of the exponential dichotomy and the stability concepts for systems of linear differential equations. We are going to show some relationship between the studied concepts, more precisely we are presenting how the stability of a linear non-autonomous system is investigated with the help of the exponential dichotomy. Furthermore we are going to show how the stable and unstable subspace of an exponentially dichotomic system can be specified using the definition of the exponential dichotomy.

Stability of linear infinite-dimensional systems revisited

International Journal of Control, 1988

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Contributions to time-varying linear control systems

I 0 1 Contents References 160 Subject Index 156 Symbol Index 16t Analytic rpproech Concerning exponential stability of rystems of the form n$) = A(t)z(t), ,>0 (0.3) thre mpects ue etudied in this thesie: for short, sufrcient conditions for exponential stabilitg; sufacient and necessuy conditions fior the stabilizability of systens (0.1) by state fedback; mbuslnece of stability. It is well-known that if, for all , > 0, the spectrum of A(t) is lying in the oper left half plane and the parameter variation of A(t) is "slow enough", then (0.3) is exponentially stable. see e.g. Rosendmlc (1963), coppel (1978). However, these results are qualitative. h llchmann, owens and Prdtzel-wolters (1987b) we derived quantitatire rcsulte, Thie means, upper bounds for the eigenvalues md for the rate of change of.4(r) which engure expotrential stability of (0.3) ae determined. This is presented in Section 4.1 of the present thesis. Ikeda, Maeclc and Kodama (1972) md (1975) rtudied the problem to etabilize a timevarying system (0.1) by state feedback. F\rthermore they gave a sufrcieat condition which guaantees that (0.1) is stabilizable by deterministic state estimation fedback. lt Ilchmann anil Kern (1987) these problems were analysed in case that the eystem (0.3) possessee ai exponential dichotomy. when this is msumed the concept of controllability into eubspaces, introdued in Section 1.2, is the appropriate tool to give necessary and sufrcient conditions for stabilizability. These results are presented in Section 4.2. In the remainder of the "malytic chapter" some robuslnesa iseuee concerning the stability of (0.3) are studied. For time-invarimt systems there exist two funda.mental approachea concerning stability: the successful r/€-approrch (see Zome{1981) and Francis and zames (rg$)) baaed on transform techniques and the state space approuh (ee Einrichsen and pritchard (1gg6a,b)) based on the concept of "stability radius". It is not clea.r how to extend transform techniquee to tim*vmying systems, whereu there are natural stensione in the state Bpace 6etup. Einichsen and Pitchard(7986a) defined the (complex) stability mdiusof A € ex" as the digta^nce of A from the set of unstable matrices in the Euclidean topologr, In Einrichsen and Pritclrord (lg86b) they also treated structured perturbations of the form BDC (B,c a^re known ecaling matrices) and showed that the associated stmctutl stability radiusr a(A;B,c) can be determined by the norm of a certain convolution operator ("p.r'turhtion operctor"). ueing optimization techniques they proved thar r2c(AiB,c) is the maximal paameter p € IR for which the algebrric Ricuti equat;on A'P + PAoc.c-pBB.p = o has an Hermitian solution. rn Hinichsen, Ilchmann and Pitchard (1g82) these results were patially extended to timevarying systems. A new class of time-larying coordinate transformations (BohI tmwfomatiow') was introduced md a lower bound for the stability radiue r s(u{; B, c) in termr of tbe norm of perturbation operator wu given. Exigtence of muimal bounded llermitia,n rclutionr of the tlifrerential Riccati equationparametrized by p € IR P(,)+A(r)'.P(r)+p(r),{(4pc(t).c(t)-,P(4^B(r)r(rrp(4 =0, r> 0 was chuacterized via the norn of the perturbation operator. Thie ie preentcd in Sectiou .l.B to 4.8 of this thesis. Each chapter hu an own detailed introduction. A suDject ao d eymbol index can bc fould rt thr end.