Stability Of Linear Time-invariant Systems Research Articles

Parameter-Dependent Lyapunov Functions for Linear Systems With Constant Uncertainties

Robust stability of linear time-invariant systems with respect to structured uncertainties is considered. The small gain condition is sufficient to prove robust stability and scalings are typically used to reduce the conservatism of this condition. It is known that if the small gain condition is satisfied with constant scalings then there is a single quadratic Lyapunov function which proves robust stability with respect to all allowable time-varying perturbations. In this technical note we show that if the small gain condition is satisfied with frequency-varying scalings then an explicit parameter dependent Lyapunov function can be constructed to prove robust stability with respect to constant uncertainties. This Lyapunov function has a rational quadratic dependence on the uncertainties.

Stabilization of linear time-invariant systems with periodic communication constraints by output sample hold control

This article considers feedback control systems wherein the control loops are closed through a real-time network, and expresses the linear time-invariant system with the constraint in an input or output as a periodic discrete time system. It is shown that this system is stabilized by using output sample hold contol. This method has the merit that the capacity of a sensor-controller communication bus is small.

Stabilization of linear time-invariant systems with periodic communication constraints by output sample hold control

Stabilization of linear time-invariant systems with periodic communication constraints by output sample hold control

Hybrid Control: Implementing Output Feedback MPC with Guaranteed Stability Region 1

Abstract In this work, a hybrid control scheme that employs switching between bounded control and model predictive control (MPC) is proposed for the output feedback stabilization of linear time-invariant systems with input constraints. Initially, we design a bounded output feedback controller for which the region of constrained closed-loop stability is explicitly characterized and an MPC controller that minimizes a given performance objective subject to constraints. Switching laws are derived to orchestrate the transition between the two controllers in a way that reconciles their respective stability and optimally properties, and guarantees asymptotic closed-loop stability for all initial conditions within the stability region of the bounded controller. The hybrid scheme is shown to provide a safety net for the practical implementation of output feedback MPC by providing a priori knowledge, through off-line computations, of a large set of initial conditions for which closed-loop stability is guaranteed. The proposed hybrid control approach is illustrated through a simulation example.

Uniting bounded control and MPC for stabilization of constrained linear systems

In this work, a hybrid control scheme, uniting bounded control with model predictive control (MPC), is proposed for the stabilization of linear time-invariant systems with input constraints. The scheme is predicated upon the idea of switching between a model predictive controller, that minimizes a given performance objective subject to constraints, and a bounded controller, for which the region of constrained closed-loop stability is explicitly characterized. Switching laws, implemented by a logic-based supervisor that constantly monitors the plant, are derived to orchestrate the transition between the two controllers in a way that safeguards against any possible instability or infeasibility under MPC, reconciles the stability and optimality properties of both controllers, and guarantees asymptotic closed-loop stability for all initial conditions within the stability region of the bounded controller. The hybrid control scheme is shown to provide, irrespective of the chosen MPC formulation, a safety net for the practical implementation of MPC, for open-loop unstable plants, by providing a priori knowledge, through off-line computations, of a large set of initial conditions for which closed-loop stability is guaranteed. The implementation of the proposed approach is illustrated, through numerical simulations, for an exponentially unstable linear system.

A spectral characterization of exponential stability for linear time-invariant systems on time scales

We prove a necessary and sufficient condition for the exponential stability of time-invariant linear systems on time scales in terms of the eigenvalues of the system matrix. In particular, this unifies the corresponding characterizations for finite-dimensional differential and difference equations. To this end we use a representation formula for the transition matrix of Jordan reducible systems in the regressive case. Also we give conditions under which the obtained characterizations can be exactly calculated and explicitly calculate the region of stability for several examples.

Design Method for Robust Supervisor Controller

We suggest a method to design supervisor which guarantees robust stability for linear time-invariant systems in the presence of the uncertainty via output feedback. The system is a composite of continuous-time uncertain plant and switched controller. The controller is defined by a collection of given controllers whose state is shared. Furthermore the proposed supervisor controller determines the mode of the controller to guarantee robust stability of the total closed-loop system. A sufficient condition of robust stability is given in terms of an algebraic Riccati equation and bilinear matrix inequalities(BMI).

On the use of multi-loop circle criterion for saturating control synthesis

This paper is concerned with regional stabilization of linear time-invariant systems by dynamic output feedback controllers subject to known bounds on the magnitudes of the control inputs. Specifically, we consider the achievable region of attraction, i.e. the set of vectors with the following property: There exists a (nonlinear) controller such that any closed-loop state trajectory converges to the origin as long as the initial state belongs the set. Two subsets of such set are characterized. One is derived from the linear analysis that considers the behavior of the states in the linear (non-saturated) region only, while the other is based on the nonlinear analysis using the multi-loop circle criterion. The main result of this paper shows that the two sets are exactly the same. Thus we conclude that the circle criterion does not help, within our framework, to increase the size of the region of attraction in saturating control synthesis when compared with that resulting from the linear analysis.

A Characterization of Bounded-Input Bounded-Output Stability for Linear Time-Invariant Systems with Distributional Inputs

A Characterization of Bounded-Input Bounded-Output Stability for Linear Time-Invariant Systems with Distributional Inputs

Characterization of the stability radius via bifurcation techniques

Robustness of stability of linear time-invariant systems using the relationship between the structured complex stability radius and a parametrized algebraic Riccati equation is analysed. Our approach is based on the observation that the algebraic Riccati equation can be viewed as a bifurcation problem. It is proved that the stability radius is, under certain assumptions, associated with a turning point of the bifurcation problem given by the parametrized algebraic Riccati equation. As a byproduct, the stability radius can be computed via path following. Some numerical examples are presented.

On the design of stabilizing controllers for singularly perturbed systems

The robustness of output feedback control designs for singularly perturbed systems based on the reduced systems is discussed. A frequency-domain approach is presented to determine the conditions for the stability of linear time-invariant systems subject to neglected high-frequency dynamics. In contrast with the qualitative analyses for robust stability that have appeared in the literature, this approach gives an explicit, computable bound on unmodeled dynamics which does not destabilize the systems. Stability conditions for various feedback control schemes are presented. It is noted that the approach, extended to systems with mixed singular-regular perturbations, can be used to derive the robust stability condition. >

Phase margin of linear time invariant systems from Routh array

A simple method to determine the gain margin of linear time invariant systems (LTIS) employing Routh stability criterion was developed in an earlier paper. The term 'stability ratio' was defined connecting the marginal gain K/sub m/ determined from the Routh array and the actual gain K of the system. This established a direct link between the Routh criterion and the gain margin. It was also shown that the phase cross-over frequency omega /sub cp/ can be found from the auxiliary equation formed from the elements of the s/sup 2/-row of the Routh array. Empirical formulae for computing the gain cross-over frequency omega /sub cg/ of type 0 systems and of higher-type systems in terms of system gain K and the marginal gain K/sub m/ are suggested. The gain cross-over frequency and phase margin calculated using the empirical formulae are in close agreement with exact values, justifying the use of the empirical formulae. It is shown that the Routh stability criterion can be used to measure the relative stability of LTIS.

Robust stabilization of linear time-invariant systems

The problem of robustly stabilizing families of linear time-variant plants consisting of a nominal plant and its frequency domain neighborhood is discussed. Necessary and sufficient conditions are obtained for robust stabilizability of such families of plants. The set of robustly stabilizing controllers is parameterized and the largest achievable stability margin is characterized for a linear time-invariant plant. >

Proof of stability condition

It is a widely-used result that absolute integrability, or summability, of impulse response is both necessary and sufficient for bounded-input/bounded-output (BIBO) stability of linear time-invariant systems. However, for the case of realizable inputs (those which are not nonzero for arbitrarily large negative time), many elementary texts give proofs only of sufficiency, deferring necessity to more advanced treatments. The proof given here is believed to be the simplest one available for the troublesome but important realizable case. It is hoped the reader will find that this treatment satisfies Yuri Manin's maxim, “A good proof is one which makes us wiser”.

Stability of multiloop LQ regulators with nonlinearities--Part II: Regions of ultimate boundedness

This note investigates the closed-loop stability of linear time-invariant systems controlled by linear quadratic (LQ) regulators when there are nonlinearities in the loops which escape the ( <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0.5, \infty</tex> ) stability sector in a bounded region containing the origin. An estimate of the region of ultimate boundedness is obtained, which provides methods for selecting the performance function weights to get better regulator designs.

Stability of linear time-invariant distributed parameter single-loop feedback systems

Sufficient conditions for stability of linear time-invariant systems are shown also to be necessary. If the open-loop system can be broken up into either a series or parallel combination of stabilizable systems, the composite system is stable if the closed-loop transfer function is free of poles on the right-half plane.

Input-output stability of linear time-invariant systems

The Nyquist stability criterion and the notion of type n systems have been extended from lumped parameter systems to a larger class of linear single-loop single-input single-output time-invariant systems, with constraints presented in the text.

Stability, Instability, Invertibility and Causality

1. Introduction. A number of interesting stability criteria for feedback systems have recently appeared in the control theory literature. The procedures used in proving these criteria can roughly be divided into three classes; the first based on Popov-like methods, the second using Lyapunov theory with Lyapunov functions derived from spectral factorizations or Riccati-type algebraic matrix equations, and the third treating the stability problem from a functional analysis point of view. Each of these methods has relative merits, e.g., the Lyapunov methods seem to be the only ones which allow us to obtain an estimate of the domain of attraction in the case of nonglobal stability. However, the method based on functional analysis appears to be the more satisfactory one, in view of the essential simplicity, of the intuitive nature of the results (loop gain less than one, passivity conditions), and of the fact that it unifies the various criteria (as, e.g., the circle criterion and the Popov criterion). It therefore deserves more investigation and exposition than it has thus far been given. A peculiarity of this method, as presently employed, is that most of the analysis and estimates have to be made on extended spaces which, although derived from normed spaces, are themselves not normed. This entails in general rather cumbersome mathematical manipulations. One however suspects the

Stability of Linear Time-Invariant Systems

The stability of a single-input, single-output, singleloop, linear, time-invariant system is related to the properties of its open-loop gain. The impulse response of the open-loop system may be of the form g(t) = r + g_{a}(t) + \sum_{i=0}^{\infty} g_{i} \delta (t - t_{i}) where r is a nonnegative constant, g_{a} is integrable on [0, \infty) , and \sum_{i=0}^{\infty} |g_{i}| \infty . If the Nyquist diagram of the open-loop gain does not go through nor encircle the critical point, then the closed-loop system is inputoutput stable, in the several meanings explained in the paper.

Stability of linear time-invariant systems

It is well known that the stability of a linear time-invariant system can be determined using a Liapunov function of the quadratic form. However, for establishing the asymptotic stability of such a system for all linear negative feedback gains in a given range, a Liapunov function which yields both necessary and sufficient conditions has not been available so far. This correspondence presents such a function and affirms a recent conjecture of Narendra. The interest in this problem is due to its being a limiting case of the Lur'e problem.