ROBUST STABILITY ANALYSIS OF UNCERTAIN INTERCONNECTION IN THE BEHAVIORAL FRAMEWORK

This paper presents a robust control synthesis method for linear time-invariant (LTI) differential systems with parametric uncertainty. The uncertain dynamics is described by a parametric kernel representation. We use parametric quadratic differential forms (QDF) to represent the dissipativity properties (both storage function and supply rate) of the uncertain system. The control design utilizes sum-of-squares (SOS) programming to search for a QDF supply rate for the controller such that the supply rate of the closed-loop system satisfies robust stability and performance conditions. Such a supply rate is then applied for control synthesis based J-factorization.

We consider the algebraic stability criterion for linear dynamical systems described by the high order differential equations in a behavioral framework. We derive our new stability criterion by using quadratic differential forms and two variable polynomial matrices. The result in this paper allows one to evaluate the asymptotic stability from the view point of original coefficient matrices.

A matrix-operator representation of parameter sensitivities is used to provide an algebraic "structural" analysis of local parameter identifiability in linear time-invariant ordinary differential equation systems. Necessary conditions for identifiability depend only upon the system matrices, no integrals must be computed, and arbitrary parametrization may be used. Relations to insensitivity are discussed, and design techniques are suggested which use the nonidentifiable (insensitive) subspace to systematically reduce the number of exciting parameters in individually designed parameter identification experiments. Finally, sufficiency conditions for zero-state identifiability are examined in terms of control inputs which continually excite the natural modes of the parameter sensitivities.

In this note, a new result that extends the scaled small gain theorem is presented. As is well known, the scaled small gain theorem gives necessary and sufficient conditions for robust stability of a nominal linear time-invariant system in feedback with structured operators of norm less than or equal to unity. We propose alternative linear matrix inequality conditions that give necessary and sufficient conditions for robust stability against the class of structured unitary operators (invertible operators of exactly unit norm). It is shown that this result, besides being a less conservative version of the scaled small gain theorem, has connections to several recent results on the control of spatially interconnected systems and serves to unify and quantify the conservatism of those results.

We study the robustness of minimax controllers, originally designed for nominal linear or nonlinear systems, to unknown static nonlinear perturbations in the state dynamics, measurement equation, and performance index. When the nominal system is linear, we consider both perfect state measurements and general imperfect state measurements; in the case of nominally nonlinear systems, we consider perfect state measurements only. Using a differential game theoretic approach, we show for the former class that, as the perturbation parameter (say, e>0) approaches zero, the optimal disturbance attenuation level for the overall system converges to the optimal disturbance attenuation level for the nominal system if the nonlinear structural uncertainties satisfy certain prescribed growth conditions. We also show that anH∞-controller, designed based on a chosen performance level for the nominal linear system, achieves the same performance level when the parameter e is smaller than a computable threshold, except for the finite-horizon imperfect state measurements case. For that case, we show that the design of the nominal controller must be based on a decreased confidence level of the initial data, and a controller thus designed again achieves a desired performance level in the face of nonlinear perturbations satisfying a computable norm bound. In the case of nominally nonlinear systems, and assuming that the nominal system is solvable, we obtain sufficient conditions such that the nominal controller achieves a desired performance in the face of perturbations satisfying computable norm bounds. In this way, we provide a characterization of the class of uncertainties that are tolerable for a controller designed based on the nominal system. The paper also presents two numerical examples; in one of these, the nominal system is linear; in the other one, it is nonlinear.

Generation of multi-wing chaotic attractor in fractional order system

A simple necessary and sufficient LMI condition for the strong delay-independent stability of LTI systems with single delay

We study the robustness of H/sup /spl infin// controllers to unknown static nonlinear perturbations in the state dynamics, the measurement equation, and the performance index. When the nominal system is linear, we consider both perfect and general imperfect state measurements, and in the case of nominally nonlinear systems, we consider perfect state measurements only. Using a differential game theoretic approach, we show for the former class that as the perturbation parameter (say, /spl epsiv/>0) approaches zero, the optimal disturbance attenuation level for the overall system converges to a value that is bounded above by the optimal disturbance attenuation level for the nominal system if the nonlinear structural uncertainties satisfy prescribed growth conditions. In particular, the overall optimal disturbance attenuation level converges to the nominal optimal disturbance attenuation level under perfect state measurements. We also show that the H/sup /spl infin//-optimal controller designed for a chosen performance level for the nominal linear system achieves the same performance level when the parameter |/spl epsiv/| is smaller than a computable threshold, except for the finite-horizon imperfect state measurements case. For that case, we show that the design of the nominal controller must be based on a decreased confidence level of the initial data, and a controller thus designed again achieves a desired performance level in the face of nonlinear perturbations satisfying a, computable norm bound. In the case of nominally nonlinear systems, and assuming that the nominal system is solvable, we obtain sufficient conditions such that the nominal controller achieves a desired performance in the face of perturbations satisfying computable norm bounds. In this way, we provide a characterization of the class of uncertainties that is tolerable for a controller designed based on the nominal system. >

Systems possessing symmetries often admit robust heteroclinic cycles that persist under perturbations that respect the symmetry. In previous work, we began a systematic investigation into the asymptotic stability of such cycles. In particular, we found a sufficient condition for asymptotic stability, and we gave algebraic criteria for deciding when this condition is also necessary. These criteria are satisfied for cycles in R3.Field and Swift, and Hofbauer, considered examples in R4 for which our sufficient condition for stability is not optimal. They obtained necessary and sufficient conditions for asymptotic stability using a transition-matrix technique.In this paper, we combine our previous methods with the transition-matrix technique and obtain necessary and sufficient conditions for asymptotic stability for a larger class of heteroclinic cycles. In particular, we obtain a complete theory for ‘simple’ heteroclinic cycles in R4 (thereby proving and extending results for homoclinic cycles that were stated without proof by Chossat, Krupa, Melbourne and Scheel). A partial classification of simple heteroclinic cycles in R4 is also given. Finally, our stability results generalize naturally to higher dimensions and many of the higher-dimensional examples in the literature are covered by this theory.

This paper studies stability analysis of linear time-invariant time-delay feedback systems consisting of a finite-dimensional system and a pure delay. The continuous-time state transition is described by the monodromy operator in discrete-time through the lifting technique, and a stability condition in terms of its spectral radius is first restated in terms of the norm of a power of the operator. Quasi-finite-rank approximation is then applied on the power of the operator, and by evaluating the effect of the approximation error on the norm, a new necessary and sufficient stability condition is given finally.

Stabilization of Discrete-Time Interconnected Systems Under Control Constraints

This paper develops some dissipativity conditions for linear time-invariant (LTI) systems in the behavioural framework. The behaviour of a system is characterised by its persistently exciting trajectories. For the dissipativity conditions, both the supply rate and the storage function are represented using quadratic difference forms (QdFs) using past steps. We show that it is possible to define an LTI system of arbitrary length using trajectories with low order of excitation. The system can be defined in a similar way as an image representation and the dissipativity conditions can hence be derived using a similar logic. The conditions are presented in the form of linear matrix inequalities (LMIs).

Asymptotic stability independent of the delay of linear differential delay systems with commensurate and non-commensurate delays is analysed using 2-D (two-dimensional) and n-D (n-dimensional) state-space models. Sufficient conditions for stability are obtained in terms of the frequency dependent 1-D (one-dimensional) Lyapunov equations. 2-D and n-D Lyapunov equations with constant parameters are derived using the discrete positive real lemma and the continuous bounded real lemma.

Multirate control: A new approach

Recently, the idea of using ecological sign stability approach for designing robust controllers for engineering systems has attracted attention with promising results. In this paper, continued research on this topic is presented. It is well known that, in the field of control systems, key to a good controller design is the choice of the appropriate nominal system. Since it is assumed that the perturbations are about this nominal, the extent of allowed perturbation to maintain the stability and/or performance very much depends on this dasianominalpsila system. In this paper, we propose that the stability robustness measures for parameter perturbation are considerably improved if the dasianominalpsila system is taken (or driven) to be a dasiasign stablepsila system. Motivated by this observation, a new method for designing a robust controller for linear uncertain time invariant state space systems using ecological sign stability approach is presented along with conditions under which such a controller exists. The resulting controller design method is illustrated with examples in the flight control area including aircraft lateral flight dynamics control and satellite formation flying control.