Mark A. Pinsky

Stability impulse control of faulted nonlinear systems

This paper presents the impulse control approach intended to urgently return to the stability basin the system states affected by abrupt changes in certain system coefficients on a short time interval. Because of its short duration, the modeling of both the fault and controller involves /spl delta/-functions significantly simplifying analysis and control of fault phenomena. The design of an impulse controller is based on the technique for computing fault-induced jumps of the system states, which is described in the paper. A sample impulse controller instantaneously returning states of a Van-der-Pol system to the stability basin is designed.

Exponential stability and solution bounds for systems with bounded nonlinearities

Estimation of Lyapunov exponents of systems with bounded nonlinearities plays an essential part in their robust control. Known results in this field are based on the Gronwall inequality yielding relatively conservative bounds for Lyapunov exponents. In this note, we obtained more accurate upper bounds for the general Lyapunov exponent for systems consisting of a known linear time-varying part and an unknown nonlinear component with a bounded Lipschitz constant at zero. Consequently, a sufficient condition for exponential stability of this system is formulated. The systems are indicated for which the obtained bound is precise, i.e., cannot be improved without additional information on the nonlinear term. In the presence of a persisting perturbation, an upper bound for the solution norm is derived and expressed in the norm of the solution of the corresponding linear system. Using the obtained results, a condition for exponential stability of a linear time-varying control system with a nonlinear feedback is derived. Numerical results are obtained for a second-order time-varying system and for the Lienard equation; in the latter case they are favorably compared with stability conditions previously obtained using the Lyapunov function method.

Delay-Independent Stability Conditions for Time-Varying Nonlinear Uncertain Systems

A new stability criterion for time-varying systems consisting of linear and norm bounded nonlinear terms with uncertain time-varying delays is formulated. An explicit delay-independent sufficient stability condition is formulated in the terms of the transition matrix of the given linear part without delay and the bounds for the uncertain terms. The obtained condition turns out to be also necessary if the matrix of the linear part is time-invariant and symmetric; it is shown that these systems satisfy the well-known Aizermans conjecture. The obtained criterion is contrasted by some of stability estimates available in the literature for these kinds of systems; in all cases the proposed criterion provides less conservative stability bounds

On impulse and continuous observation control design in Kalman filtering problem

Abstract This paper develops the observation control method for refining the Kalman–Bucy estimates, which is based on impulsive modeling of the transition matrix in an observation equation, thus engaging discrete-continuous observations. The impulse observation control generates on-line computable jumps of the estimate variance from its current position towards zero and, as a result, enables us to instantaneously obtain the estimate, whose variance is closer to zero. The filtering equations over impulse-controlled observations are obtained in the Kalman–Bucy filtering problem. The method for feedback design of control of the estimate variance is developed. First, the pure impulse control is used, and, next, the combination of the impulse and continuous control components is employed. The considered examples allow us to compare the properties of these control and filtering methodologies.

General Solution of Stability Problem for Plane Linear Switched Systems and Differential Inclusions

Characterization and control of stability of switched dynamical systems and differential inclusions have attracted significant attention in the recent past. The most of the current results for this problem are obtained by application of the Lyapunov function method which provides sufficient but frequently over conservative stability conditions. For planar systems, practically verifiable necessary and sufficient conditions are found only for switched systems with two subsystems. This paper provides explicit necessary and sufficient conditions for asymptotic stability of switched systems and differential inclusions with arbitrary number of subsystems; these conditions turned out to be identical for the both classes of systems. A precise upper bound for the number of switching points in a periodic solution, corresponding to the break of stability, is found. It is shown that, for a switched system, the break of stability may also occur on a solution with infinitely fast switching (chattering) between some two subsystems.

Absolute Stability Criteria for a Generalized Lur'e Problem with Delay in the Feedback

A nonautonomous linear system controlled by a nonlinear sector-restricted feedback with a time-varying delay is considered. Delay-independent sufficient conditions for absolute stability and instability (expressed in the transfer function of the linear part and the sector bounds) are established. For a system with an exponentially stable linear part, an upper bound for the Lyapunov exponent is found. It is shown that if the transfer function is sign-constant, asymptotic stability of the system with the margin-linear feedback guarantees absolute stability of the considered system; thus, such systems satisfy the known Aizerman conjecture. They include, in particular, a closed-loop system consisting of any number of time-varying first order links and feedback with arbitrary delay. Under some additional condition (which is certainly true for a time-invariant linear block), the obtained stability criterion is precise. The approach employed in the proofs is based on a direct analysis of the corresponding Volterra equation which contains only the transfer function of the linear block and, therefore, embraces a wide range of control systems. As an example, a second order system is considered; it is shown that here the obtained stability bound is reached for a linear feedback with a periodic delay function.

Minimal periods of periodic solutions of some Lipschitzian differential equations

A problem of finding lower bounds for periods of periodic solutions of a Lipschitzian differential equation, expressed in the supremum Lipschitz constant, is considered. Such known results are obtained for systems with inner product norms. However, utilizing the supremum norm requires development of a new technique, which is presented in this paper. Consequently, sharp bounds for equations of even order, both without delay and with arbitrary time-varying delay, are found. For both classes of system, the obtained bounds are attained in linear differential equations.

ANALYSIS AND CONTROL OF BIFURCATION PHENOMENA IN AIRCRAFT FLIGHT

This paper addresses a theoretical framework for a unified methodology which allows analysis of nonlinear stability and efficient control of high-dimensio nal nonlinear plants modeling aircraft flight. It is shown that analysis of nonlinear transition phenomena (bifurcations) is central to revealing the limitation of robust control (i.e., an accurate estimate of the basin of stability). Omitting transition behavior causes over control and provides a very local stabilization. Analysis and control of bifurcations of aircraft flight are given in the spirit of the generalized normal forms method, which provides one with the nonreducible system that preserves stability characteristics of the initial plant. Stabilization of a plants bifurcations is then given in terms of the resonance control methodology. Efficiency of the developed methodology is demonstrated by analyzing and controlling an unstable nonlinear plant relevant to the lateral dynamics of an aircraft. Whereas the initial plant is governed by a number of coupled nonlinear equations, the reduced system (the resonance normal form) turns out to be much easier to analyze and even integrable in many cases. Analysis of bifurcations of the resonance normal forms may shape efficient control actions which a pilot may undertake to ensure stability of an aircraft in a prescribed neighborhood of a trim condition and also can furnish a design of a flights automatic control.

Analysis and Resonance Stabilization of Bifurcation Phenomena in Aircraft Flight

This paper addresses a theoretical framework for a unified methodology which allows analysis of nonlinear stability and efficient control of high-dimensional nonlinear plants modeling aircraft flight. It is shown that analysis of nonlinear transition phenomena (bifurcations) is central to revealing the limitation of robust control (i.e., an accurate estimate of the basin of stability). Omitting transition behavior causes over control and provides a very local stabilization. Analysis and control of bifurcations of aircraft flight are given in the spirit of the generalized normal forms method which provides one with the nonreduceable system preserving stability characteristics of the initial plant. Stabilization of a plants bifurcations is then given in terms of the resonance control methodology.

Brief paper: Optimal exponential feedback stabilization of planar systems

This paper solves the problem of finding an optimal feedback control ensuring the maximal rate of convergence of system solutions to the origin for a general class of planar control systems including switched, bilinear systems and ones described by differential inclusions, etc. The prescribed control set is assumed to be compact but not necessarily convex. The developed approach is based on finding the minimal Lyapunov exponent of the system with an open loop control which provides an upper bound for the optimal convergence rate of the closed loop system. Then an optimal feedback controller is constructed for which the obtained bound is attained.