Sufficient Conditions For Exponential Stability Research Articles

Necessary and Sufficient Conditions for Exponential Stability and Ultimate Boundedness of Systems Governed by Stochastic Partial Differential Equations

Consider the following infinite dimensional stochastic evolution equation over some Hilbert space H with norm ∣·∣: X t = x 0 + ∫ 0 t f ( X s , s ) d s + ∫ 0 t g ( X s , s ) d W s , t ⩾ 0 , P almost surely It is proved that under certain mild assumptions, the strong solution Xt(x0)∈V↪H↪V*, t ⩾ 0, is mean square exponentially stable if and only if there exists a Lyapunov functional Λ(·, ·):H×R+→R1 which satisfies the following conditions: (i) c 1 | x | 2 − k 1 e − μ 1 t ⩽ Λ ( x , t ) ⩽ c 2 | x | 2 + k 2 e − μ 2 t ; (ii) ℒ Λ ( x , t ) ⩽ − c 3 Λ ( x , t ) + k 3 e − μ 3 t , ∀ x ∈ V , t ⩾ 0 ; where L is the infinitesimal generator of the Markov process Xt and ci, ki, μi, i = 1, 2, 3, are positive constants. As a by-product, the characterization of exponential ultimate boundedness of the strong solution is established as the null decay rates (that is, μi = 0) are considered.

Exponential stability of singularly perturbed stochastic systems

The sufficient conditions of exponential stability of singularly perturbed, nonlinear stochastic systems are established. The excitations are assumed to be parametric white noises. In this case, the objective is to analyze the full-order system in their lower order subsystems, i.e., the reduced-order system and the boundary-layer system, and in terms of their interconnecting structure and the perturbation parameter varepsilon. The exponential bounds depend on the moments of norms of trajectories given for the "slow" and "fast" components of the full-order system. The estimation of the rate of convergence of the full-order system is also shown.

Prediction-based control of chaos

Pyragas proposes a practically useful control method, called delayed feedback control, for control of chaos. Conditions for (local) stabilization by the delayed feedback control, however, are more restricted than those by the OGY method. In order to overcome this problem, we propose a novel control method, called a prediction-based feedback control, for discrete-time chaotic systems. Moreover, we give necessary and sufficient conditions for exponential stabilization of fixed points by the proposed method.

Lyapunov transformation and stability of differential equations in banach spaces

A sufficient condition of exponential stability of regular linear systems with bifurcation on a Banach space is proved.

On exponential stability of nonlinear time-varying differential equations

Within the Liapunov framework, a sufficient condition for exponential stability of ordinary differential equations is proposed. Unlike with classical Liapunov theory, the time derivative of the Liapunov function, taken along solutions of the system, may have positive and negative values. Verification of the conditions of the main theorem may be harder than in the classical case. It is shown that the proposed conditions are useful for the investigation of the exponential stability of fast time-varying systems. This sets the stability study by means of averaging in a Liapunov context. In particular, it is established that exponential stability of the averaged system implies exponential stability of the original fast time-varying system. A comparison of our work with results taken from the literature is included.

Sufficient conditions for exponential stability and dichotomy of evolution equations

We present several sufficient conditions for exponential stability and dichotomy of solutions of the evolution equation u′(t) = A(t)u(t) (∗) on a Banach space X. Our main theorem says that if the operators A(t) generate analytic semigroups on X having exponential dichotomy with uniform constants and A(·) has a sufficiently small Holder constant, then (∗) has exponential dichotomy. We further study robustness of exponential dichotomy under time dependent unbounded Miyadera-type perturbations. Our main tool is a characterization of exponential dichotomy of evolution families by means of the spectra of the so-called evolution semigroup on C0(R, X) or L(R, X).

A Note on Exponential Stability of Partially Slowly Time-Varying Nonlinear Systems

Consider a system x˙=(x,t,tβ) with a time-varying vectorfield which contains a regular and a slow time scale (β3 large). Assume there exist an α(τ) and a K(τ) such that ‖xτ(t,t0,x0)‖≤k(τ)‖x0‖eα(τ)(t−t0)where xτ(t,t0,x0) is the solution of the system x˙=f(x,t,τ) with initial state Xo at β formula We show that for β3 sufficiently large, x˙=(x,t,tβ) is exponentially stable when ‘on average’ α(τ) formula is negative. This result can be used to extend the circle criterion Le. to obtain a sufficient condition for exponential stability of a feedback interconnection of a slowly time-varying linear system and a sector nonlinearity. An example is included which shows that the technique can be used to obtain an exponential stability result for a pendulum with a nonlinear partially slowly time-varying friction attaining positive and negative values.

Synthesis of a Stable Discrete-Time Repetitive Controller for MIMO Systems

This paper presents some new results on synthesis and stability analysis of a class of repetitive controllers for discrete-time multi-input multi-output (MIMO) linear time-invariant (LTI) systems. In particular, the necessary and sufficient conditions for exponential stability of the resulting closed-loop systems are derived. A simulation case study featuring a flexible mechanical system is also presented.

On the exponential stability of discrete-time systems with applications in observer design

Exponential stability of discrete-time systems, stability domains, and state bounds are obtained with the use of vector norm Lyapunov functions. For linear systems, necessary and sufficient conditions are given for asymptotic stability, and exponential bounds are derived. For nonlinear systems, sufficient conditions for exponential stability also yield a stability domain and exponential bounds. Applications are demonstrated in observer design for uncertain discrete-time nonlinear systems. >

Stochastic stability analysis for continuous-time fault tolerant control systems

Active fault tolerant control systems are feedback control systems that reconfigure the control law in real time based on the response from an automatic failure detection and identification (FDI) scheme. The dynamic behaviour of such systems is characterized by stochastic differential equations because of the random nature of the failure events and the FDI decisions. The stability analysis of these systems is addressed in this paper using stochastic Lyapunov functions and supermartingale theorems. Both exponential stability in the mean square and almost-sure asymptotic stability in probability are addressed. In particular, for linear systems where the coefficients of the closed loop system dynamics are functions of two random processes with markovian transition characteristics (one representing the random failures and the other representing the FDI decision behaviour), necessary and sufficient conditions for exponential stability in the mean square are developed.

Exponentially stable linear discrete systems

The exponential stability of linear shift-variant discrete systems is investigated. A discrete system is exponentially stable if there exists constants k > 0 and C > 0 such that the solution ‖X(n)‖→ exp (-nk)‖X(0)‖C → 0 as n → ∞. Some new sufficient conditions for exponential stability are provrn and a procedure is given to calculate a bound on the allowable variations in the system matrix such that the system remains exponentially stable.

On the design of stabilizing compensators via semi-infinite optimization

The authors derive a modified Nyquist stability test which makes possible the use of semi-infinite optimization for the design of stabilizing compensators. The test provide a simple necessary and sufficient condition of exponential stability and is computationally well-conditioned. Computational aspects of the derived stability test are discussed and a design example is given.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Exponential stabilization of non-linear stochastic systems

We consider the stabilization of non-linear systems whose parameters are subjected to “white noise”. For stochastic systems with non-linear feedback, we derive sufficient (frequency-domain) conditions of exponential stabilization by a controller that uses information about the system output (incomplete system state information). The problem of the stabilization of linear stochastic systems has been studied in some detail /1–3/. Yet for non-linear stochastic systems we only have general theorems that reduce the stabilization problem to finding a stochastic Lyapunov function /4, 5/. In this paper, we derive sufficient conditions of exponential stabilization by methods of the theory of absolute stochastic stability. The advantages of these methods are well-known: the specific Lyapunov function is not required, and its existence in the class of functions “quadratic form plus integrals over non-linearities” is easily checked /6/. The latest results of this theory for stochastic systems /7/ make it possible to solve the stabilization problem for a wide class of non-linear systems with parametric disturbances.

Circuits, k-ports, hidden modes, and stability of interconnected k-ports

This paper exclusively considers lumped <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</tex> -ports and circuits which contain linear time-invariant elements, independent sources, and controlled sources. The <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</tex> -ports are represented by their hybrid matrices. Tableau equations of circuits are used as a special case of polynomial matrix description. The hidden modes of a circuit are determined by inspection from its tableau equations in the Hermite row form. The same form is used also to determine the exponential stability of the circuit and that of the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</tex> -port. Finally, necessary and sufficient conditions for exponential stability of interconnected <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</tex> -ports are given; the hidden modes of the interconnection are studied. The paper is self-contained.

Sufficient conditions for the exponentialP-stability andP-stabilizability of linear stochastic systems

Sufficient conditions for exponential stability and instability in pth mean are given for linear stochastic systems. Applying this stability result to the stabilization problem of linear control systems subject to state dependent noise, we obtain a. sufficient condition for p-stabilizability (P≥2). This condition is close to the controllability condition for the systems with no noise term.

Complementary energy principles in dissipative fluids

Necessary and sufficient conditions for exponential stability of equilibria in dissipative fluids are discussed on the basis of energy principles. First, a known maximum principle is reformulated in a manner which is more appropriate for evaluation of actual growth rates. Secondly, complementary variational principles are presented. The latter are quite useful for qualitative estimates and numerical computations since they lead to upper and lower bounds for the exact growth rates. The results are applicable to ideal magnetohydrodynamic (MHD) cases as well as resistive plasmas and also solitary waves.

A note on the exponential asymptotic properties of linear stochastic systems

In a previous paper, sufficient conditions for exponential stability and instability in mean square of linear stochastic systems subject to state dependent disturbances were shown. It was also conjectured that these conditions are close to necessary and sufficient conditions for exponential stability and instability. In this note, however, it is shown that this conjecture does not hold precisely except for a special class of systems. Examples which disprove the necessity of these conditions are given.

Stability of stochastic partial differential equation

Stochastic partial differential equations such as occur in vibration problems for mechanical structures subjected to random loading are modelled as infinite dimensional stochastic Itô differential equations using a semigroup approach. Sufficient conditions for exponential stability of the expected energy of the system, as well as for the exponential decay of the sample paths of the displacement and velocity, are given. Under these same conditions it is shown that the zero solution is pathwise asymptotically stable relative to finite dimensional initial conditions. Illustrative examples are included.

Stability criterion for a linear differential-difference system

A linear differential-difference system of the form \dot{x}(t)= A_{1}x(t - \tau), x(t)\in R^{n} is considered, and necessary and sufficient conditions for exponential stability with a given decay rate are derived. The conditions are given in terms of eigenvalues of the matrix A 1 and extend the well-known result for the scalar case.

Exponential stability of linear equations arising in adaptive identification

The stability, properties are examined of various time-varying linear differential equations which arise in model reference adaptive identification schemes. Necessary and sufficient conditions for exponential stability are presented.