Totally dissipative systems

In a totally dissipative behavior, all non-trivial trajectories dissipate energy. A characterization of such behaviors is given in terms of properties of the one- and two-polynomial matrices associated with the supply rate and with their kernel- and image representation.     

Conserved- and zero-mean quadratic quantities in oscillatory systems

We study quadratic functionals of the variables of a linear oscillatory system and their derivatives. We show that such functionals are partitioned in conserved quantities and in trivially- and intrinsic zero-mean quantities. We also state an equipartition of energy principle for oscillatory systems.     

A two-variable approach to solve the polynomial Lyapunov equation

A two-variable polynomial approach to solve the one-variable polynomial Lyapunov equation is proposed. Lifting the problem from the one-variable to the two-variable context allows to use Faddeev-type recursions in order to solve the polynomial Lyapunov equation in an iterative fashion. The method is especially suitable for applications requiring exact or symbolic computation.     

The Kalman-Yakubovich-Popov Lemma in a behavioural framework

The classical Kalman-Yakubovich-Popov Lemma provides a link between dissipativity of a system in state-space form and the solution to a linear matrix inequality. In this paper we derive the KYP Lemma for linear systems described by higher-order differential equations. The result is an LMI in terms of the original coefficients in which the dissipativity problem is posed. Subsequently we study the connection between dissipativity and spectral factorization of polynomial matrices. This enables us to derive a new algorithm for polynomial spectral factorization in terms of an LMI in the coefficients of a polynomial matrix.     

On the existence of a continuous storage function for dissipative systems

Conditions for nonlocal existence of a continuous storage function for nonlinear dissipative system are presented. More precisely, it is shown that under the local w-uniform reachability assumption at one point x, the required supply function is continuous on the set of points reachable from x. Conditions for the local w-uniform reachability based on the local controllability properties of the system are provided.     

The subspace Nevanlinna interpolation problem and the most powerful unfalsified model

A generalization of the tangential Nevanlinna interpolation problem will be studied from a behavioral point of view. Necessary and sufficient conditions for its solvability and a characterization of all its solutions are derived. These results are obtained by associating to the interpolation data a behavior that enjoys a special structure.     

Quadratic stability and stabilization of uncertain linear discrete-time systems with state delay

This paper deals with the problem of quadratic stability analysis and quadratic stabilization for uncertain linear discrete-time systems with state delay. The system under consideration involves state time delay and time-varying norm-bounded parameter uncertainties appearing in all the matrices of the state-space model. Necessary and sufficient conditions for quadratic stability and quadratic stabilization are presented in terms of certain matrix inequalities, respectively. A robustly stabilizing state feedback controller can be constructed by using the corresponding feasible solution of the matrix inequalities. Two examples are presented to demonstrate the effectiveness of the proposed approach.     

The strict dissipativity synthesis problem and the rank of the coupling QDF

The problem of existence of a controlled behavior that is strictly dissipative with respect to a quadratic supply rate is studied. The relation between strictness and the rank of a suitable coupling condition that combines the dissipativity properties of the hidden behavior and the orthogonal complement of the plant behavior is analyzed.     

New Algorithms for Polynomial J-Spectral Factorization

In this paper new algorithms are developed for J-spectral factorization of polynomial matrices. These algorithms are based on the calculus of two-variable polynomial matrices and associated quadratic differential forms, and share the common feature that the problem is lifted from the original one-variable polynomial context to a two-variable polynomial context. The problem of polynomial J-spectral factorization is thus reduced to a problem of factoring a constant matrix obtained from the coefficient matrices of the polynomial matrix to be factored. In the second part of the paper, we specifically address the problem of computing polynomial J-spectral factors in the context of H _∞ control. For this, we propose an algorithm that uses the notion of a Pick matrix associated with a given two-variable polynomial matrix. Date received: January 1, 1998. Date revised: October 15, 1998.     

Identification and data-driven model reduction of state-space representations of lossless and dissipative systems from noise-free data

We illustrate procedures to identify a state-space representation of a lossless or dissipative system from a given noise-free trajectory; important special cases are passive systems and bounded-real systems. Computing a rank-revealing factorization of a Gramian-like matrix constructed from the data, a state sequence can be obtained; the state-space equations are then computed by solving a system of linear equations. This idea is also applied to perform model reduction by obtaining a balanced realization directly from data and truncating it to obtain a reduced-order model.     

Systems with sector bound nonlinearities: A behavioral approach

The class of nonlinear dynamical systems obtained by interconnection of a linear, time-invariant dynamical system and a memoryless, time-invariant, sector-bound nonlinearity has been widely studied in literature. In this paper it is shown that many important results for this class of systems can be elegantly unified using behavioral theory of dynamical systems. Systems with slope-restricted nonlinearities are also considered.     

Feedback stabilization of dissipative impulsive dynamical systems

In this paper, we derive conditions under which a dissipative impulsive dynamical system is asymptotically stabilizable by a feedback controller. Specializing the obtained results to the case of dissipative linear impulsive dynamical systems with the quadratic supply rate, we establish the corresponding sufficient conditions. Finally, simulation results are given to demonstrate the effectiveness of our results.     

Passivity and stability of switched systems: A multiple storage function method

This paper presents a concept of passivity for switched systems using multiple storage functions. This passivity property is invariant under compatible feedback interconnection. Branicky's stability theorem of multiple Lyapunov functions is generalized by relaxing the non-increasing condition on values of Lyapunov-like functions. Using this result we show that a passive switched system is stable in the sense of Lyapunov. Moreover, asymptotic stability is reached if all subsystems are asymptotically detectable.     

Conditions for the existence of continuous storage functions for nonlinear dissipative systems

The problem of existence of continuous storage functions for dissipative nonlinear systems is considered. It is shown that, if a nonlinear system is dissipative in the state x_*, then, under certain assumptions, a continuous storage function can be constructed on a set of points accessible from x_* by concatenation of a finite number of forward and backward motions of the system. Most of these assumptions are weaker than certain controllability-type properties and can be checked using similar tests.     

Eigenstructure assignment in state covariance control

It is quite common in stochastic control problems that performance objectives are expressed in terms of the covariance of the system.state. In this research, the design freedom of the state covariance controller is exploited to achieve the given eigenstructure with desired eigenvalues. Necessary and sufficient conditions for the existence of achievable eigenstructures are discussed. Gain matrices that simultaneously achieve the specified state covariance and the desired eigenstructure H are also given. A numerical example is provided to demonstrate the effectiveness of the design procedure.     

On the quadratic stability of switched linear systems associated with symmetric transfer function matrices

In this paper we give necessary and sufficient conditions for weak and strong quadratic stability of a class of switched linear systems consisting of two subsystems, associated with symmetric transfer function matrices. These conditions can simply be tested by checking the eigenvalues of the product of two subsystem matrices. This result is an extension of the result by Shorten and Narendra for strong quadratic stability, and the result by Shorten et al. on weak quadratic stability for switched linear systems. Examples are given to illustrate the usefulness of our results.     

Stability robustness for linear state space models - a Lyapunov mapping approach

Some Pythagorean-like conditions for the existence of a common Lyapunov solution for two given matrices are presented. These conditions are set on the Lyapunov mappings associated with given matrices. The key is the semipositiveness of linear mappings defined on symmetric matrices. From the viewpoint of the existence of a common Lyapunov solution, some known stability robustness results are reviewed and refined.     

Decentralized robust control of multiple static var compensators via Hamiltonian function method

Using the energy-based Hamiltonian function method, this paper investigates the decentralized robust nonlinear control of multiple static var compensators (SVCs) in multimachine multiload power systems. First, the uncertain nonlinear differential algebraic equation model is constructed for the power system. Then, the dissipative Hamiltonian realization of the system is completed by means of variable transformation and prefeedback control. Finally, based on the obtained dissipative Hamiltonian realization, a decentralized robust nonlinear controller is put forward. The proposed controller can effectively utilize the internal structure and the energy balance property of the power system. Simulation results verify the effectiveness of the control scheme.     

Piecewise-affine state feedback for piecewise-affine slab systems using convex optimization

This paper shows that Lyapunov-based state feedback controller synthesis for piecewise-affine (PWA) slab systems can be cast as an optimization problem subject to a set of linear matrix inequalities (LMIs) analytically parameterized by a vector. Furthermore, it is shown that continuity of the control inputs at the switchings can be guaranteed by adding equality constraints to the problem without affecting its parameterization structure. Finally, it is shown that piecewise-affine state feedback controller synthesis for piecewise-affine slab systems to maximize the decay rate of a quadratic control Lyapunov function can be cast as a set of quasi-concave optimization problems analytically parameterized by a vector. Before casting the synthesis in the format presented in this paper, Lyapunov-based piecewise-affine state feedback controller synthesis could only be formulated as a bi-convex optimization program, which is very expensive to solve globally. Thus, the fundamental importance of the contributions of the paper relies on the fact that, for the first time, the piecewise-affine state feedback synthesis problem has been formulated as a convex problem with a parameterized set of LMIs that can be relaxed to a finite set of LMIs and solved efficiently to a point near the global optimum using available software. Furthermore, it is shown for the first time that, in some situations, the global can be exactly found by solving only one concave problem.