Canonical H∞ state-space parametrization

Sage, H. G.; Mathelin, M.F. de

doi:10.1016/s0005-1098(00)00015-7

Cited by 3 publications

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“…The decay rate of the Hankel singular values is studied by introducing a new set of input/output system invariants, namely the quantities p(s) q * (s) , where q(s) * = q(−s), evaluated at the poles of G(s). These results are expected to yield light into the structure of the above problem (6). Another paper of interest especially for the suboptimal approximation case, is [6].…”

Section: Available Results and Possible Solution Pathsmentioning

confidence: 88%

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Problem 8.1 H∞-norm approximation

Antoulas¹,

Astolfi²

2009

Unsolved Problems in Mathematical Systems and Control Theory

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Let RH m ∞ be the (Hardy) space of real-rational scalar 1 transfer functions of order m, bounded on the imaginary axisand analytic into the right-half complex plane. The optimal approximation problem in the H ∞ norm can be statedas follows.For such a problem, lettwo further problems can be posed.1 Similar considerations can be done for the nonscalar case. 2 By find we mean find an exact solution or an algorithm converging to the exact solution.

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Section: Available Results and Possible Solution Pathsmentioning

confidence: 88%

“…These results are expected to yield light into the structure of the above problem (6). Another paper of interest especially for the suboptimal approximation case, is [6]. In this paper the set, of all systems whose H ∞ norm is less than some positive number γ is parameterized.…”

Section: Available Results and Possible Solution Pathsmentioning

confidence: 99%

Problem 8.1 H∞-norm approximation

Antoulas¹,

Astolfi²

2009

Unsolved Problems in Mathematical Systems and Control Theory

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H2 approximation of multiple input/output delay systems

Zhang

Huang

2004

Journal of Process Control

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Preservation of solvability conditions in Riccati equations when applying SPR0 substitutions

Fernández‐Anaya

Martínez-García

Aguilar-George

Proceedings of the 41st IEEE Conference on Decision and Control, 2002.

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e m a i l daguilar@ctrl.cinvestau.~ A b s t r a c tIn this paper we are concerned by algebraic Riccati equations associated to robust control problems. We tackle the preservation of solvability conditions of algebraic Riccati equations when applying substitutions of the Laplace variable s by some class of strictly positive real functions in the corresponding singleinput and singleoutput transfer functions. In particular, we study the preservation of solvability conditions associated to the existence ofstabilizing solutions of algebraic Riccati equations characterized by an H"-norm constraint on a particular linear time-invariant system. We illustrate our results with the study of the closed-loop stability of a family of linear time-invariant systems, affected by timevarying uncertainty, controlled via robust sliding mode control. 1 I n t r o d u c t i o n As is pointed out in [15] and [16], the concept of positive realness of a transfer function plays a central role in Stability Theory. The definition of rational Positive Real functions (PR functions) arose in the context of Circuit Theory. In fact, the driving point impedance of a passive network is rational and positive real. If the network is dissipative (due to the presence of resistors), the driving point impedance of the network is a Strictly Positive Real transfer function (SPR function). Thus, positive real systems, also called passive systems, are systems.that do not generate energy. The celebrated Kalman-Yakubovich-Popov ( K Y P ) lemma established the key role that strict positivity realness plays in the obtention of Lyapunov functions associated to the stability analysis of a particular class of nonlinear systems, i.e., Linear Time Invariant systems (LTI systems) with .a single memoriless nonlinearity. In fact; positive realness has been extensively studied by the Automatic Control community, see for instance the studies concerning: absolute stability 1111; characterization and construction of robust strictly positive real systems [3]; relationship between time domain and frequency domain conditions for strict positive realness [Zl]; relationships between positivity realness of proper and stable LTI systems and stabilizing solutions of Riccati equations 1221; stability of adaptive control schemes based on parameter adaptation algorithms [l]; passive filters [a]. As far as the frequency-described continuous LTI systems are concerned, the study of control-oriented properties (like stability) resulting from the substitution of the complex Laplace variable s by rational transfer functions have been little studied by the Automatic Control community. However, some interesting results have recently been published. Concerning the study of the so-called uniform systems, i.e., LTI systems consisting of identical components and amplifiers, it was established in [17] a general criterion for robust stability for functions of the form D (f (s)), where D (s) is a polynomial and f (s) is a rational transfer function. By applying such a criterium, it gave a generalisa...

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Canonical H∞ state-space parametrization

Cited by 3 publications

References 10 publications

Problem 8.1 H∞-norm approximation

Problem 8.1 H∞-norm approximation

H2 approximation of multiple input/output delay systems

Preservation of solvability conditions in Riccati equations when applying SPR0 substitutions

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