Parameter-dependent Lyapunov Function Research Articles

Using unknown input observers for robust adaptive fault detection in vector second-order systems

The purpose of this manuscript is to construct natural observers for vector second-order systems by utilising unknown input observer (UIO) methods. This observer is subsequently used for a robust fault detection scheme and also as an adaptive detection scheme for a certain class of actuator faults wherein the time instance and characteristics of an incipient actuator fault are detected. Stability of the adaptive scheme is provided by a parameter-dependent Lyapunov function for second-order systems. Numerical example on a mechanical system describing an automobile suspension system is used to illustrate the theoretical results.

Formula omitted] guaranteed cost computation by means of parameter-dependent Lyapunov functions

A linear matrix inequality approach to compute H ∞ guaranteed costs by means of parameter-dependent Lyapunov functions is presented in this paper. The uncertain linear time-invariant systems are supposed to belong to convex bounded domains (polytope-type uncertainty). The conditions proposed are less conservative than other parameter-dependent based methods from the literature, providing better estimates of worst-case H ∞ norms, in both continuous- and discrete-time systems, as illustrated by means of numerical examples.

Robust gain-scheduled control for vibration suppression

Limited control authority is a key issue in the field of structural control and is a major research area since most of the practical control problems are dominated by constraints on the control signal. The paper presents a simple and practical gain-scheduled controller design procedure for active vibration suppression of a three-storey flexible structure. First, system identification experiments are performed and the plant’s uncertainty is derived. Next, robust controller design with constraint on the control signal is presented. For a better trade-off between control performance and control constraint a gain-scheduling approach is investigated. Stability analysis of the gain-scheduled controller is analysed using a parameter-independent Lyapunov function (quadratic stability) as well as a parameter-dependent Lyapunov function (biquadratic stability). Finally, the gain-scheduled controller is tested experimentally when the flexible structure is excited with a scaled historical earthquake record (1940 El Centro record). Successful experimental results show that the proposed robust gain-scheduled control approach offers good performance in the case of control authority limitation.

Robust H∞ control for discrete-time polytopic uncertain systems with linear fractional vertices

The robust H∞ control problem for discrete-time uncertain systems is investigated in this paper. The uncertain systems are modelled as a polytopic type with linear fractional uncertainty in the vertices. A new linear matrix inequality (LMI) characterization of the H∞ performance for discrete systems is given by introducing a matrix slack variable which decouples the matrix of a Lyapunov function candidate and the parametric matrices of the system. This feature enables one to derive sufficient conditions for discrete uncertain systems by using parameter-dependent Lyapunov functions with less conservativeness. Based on the result, H∞ performance analysis and controller design are carried out. A numerical example is included to demonstrate the effectiveness of the proposed results.

Natural second-order observers for second-order distributed parameter systems

The aim of this manuscript is to present an alternative method for designing state observers for second-order distributed parameter systems without resorting to a first-order formulation. This method has the advantage of utilizing the algebraic structure that second-order systems enjoy with the obvious computational savings in observer gain calculations. The proposed scheme ensures that the derivative of the estimated position is indeed the estimate of the velocity component and to achieve such a result, a parameter-dependent Lyapunov function was utilized to ensure the asymptotic convergence of the state estimation error.

Stability of Linear Parameter Varying and Linear Switching Systems

AbstractWe consider stability of families of linear time‐varying systems, that are determined by a set of time‐varying parameters which adhere to certain rules. The conditions are general enough to encompass on the one hand stability questions for systems that are frequently called linear parameter varying systems in the literature and on the other hand also linear switching systems, in which parameter variations are allowed to have discontinuities. Combinations of these two sets of assumptions are also possible within the framework studied here. Under the assumption of irreducibility of the sets of system matrices, we show how to construct parameter dependent Lyapunov functions for the systems under consideration that exactly characterize the exponential growth rate. It is clear that such Lyapunov functions do not exist in general. But every system of our class can be reduced to a finite number of subsystems for which irreducibility holds.

A synthesis approach of on-line constrained robust model predictive control

A new method to address constrained robust model predictive control (MPC) for systems with polytopic description (RPC–SPD) is proposed. This is motivated by two previous techniques. One is the well-known robust MPC that parameterizes the infinite horizon control moves into a single linear state feedback law. The other is a nominal on-line MPC that splits the infinite horizon control moves into a set of free control moves over a fixed horizon and a state feedback law in the terminal region. The basic idea here is to adopt the rationale of the second technique to improve feasibility and optimality of the first one. A parameter-dependent Lyapunov function is developed for establishing the closed-loop stability. The performance of the controller is demonstrated by an example.

H∞ and L2 - to - L∞ gain control of linear parameter-varying systems with parameter-varying delays

The analysis and state-feedback synthesis problems for linear parameter-varying systems with parameter-varying time delays are addressed. It is assumed that the state-space data and the time delays are dependent on parameters that are measurable in real-time and vary in a compact set with bounded variation rates. We explore the stability, L2 gain performance and L2 - to- L∞ gain performance for these systems using parameter-dependent Lyapunov functionals. In addition, the design of parameter-dependent state-feedback controllers that guarantee stability and desired induced norm performance are examined. Both analysis and synthesis conditions are formulated in terms of linear matrix inequalities that can be solved via efficient interior-point algorithms.

Parametric Lyapunov function approach to H2 analysis and control of linear parameter-dependent systems

The problems of H/sub 2/ analysis and control synthesis for linear continuous-time parameter-dependent systems are addressed. The admissible values of the parameters and their rates of variation are assumed to belong to a given polytope. Linear matrix inequality-based methods of H/sub 2/ performance analysis and state feedback control are developed. Both the design of gain-scheduled and robust controllers are addressed. The proposed methods have the feature that stability as well as the guaranteed H/sub 2/ performance cost are based on a parameter-dependent Lyapunov function which is quadratic on the system parameters. The controller implementation does not require online knowledge of the rate of variation of the parameters

Robust Output Feedback Control Synthesis: LMI Approach

The paper addresses the problem of robust output feedback controller design with a guaranteed cost and parameter dependent Lyapunov function quadratic stability for linear continuous time polytopic systems. The proposed design methods lead to a non-iterative LMI based algorithm. Numerical example is given to illustrate the design procedure.

Static H2 and H∞ Output-Feedback of Linear Systems with Stochastic Uncertainties

A linear parameter dependent approach for designing a constant output feedback controller for a linear time-invariant system with stochastic multiplicative Wiener-type noise is introduced, that achieves a minimum bound on either the stochastic H2 or the H∞ performance level. A solution is achieved also for the case where in addition to the stochastic parameters, the system matrices reside in a given polytope. In this case, a parameter dependent Lyapunov function is introduced which enables the derivation of the required constant gain via a solution of a set of linear matrix inequalities that correspond to the vertices of the uncertainty polytope.The stochastic uncertainties appear in both the dynamic and the measurement matrices of the system. The problems are solved using the expected value of the standard performance index over the stochastic parameters.

H∞ Guaranteed Cost Computation by Means of Parameter Dependent Lyapunov Functions

A linear matrix inequality approach to compute H∞ guaranteed costs by means of parameter dependent Lyapunov functions is presented in this paper. The uncertain linear systems are supposed to belong to convex bounded domains (polytope type uncertainty). Less conservative evaluations are obtained, in both continuous-time and discrete-time systems, as illustrated by means of numerical examples.

Robust Output Feedback Controller Design: LMI Approach

The paper addresses the problem of robust output feedback controller design with a guaranteed cost and quadratic, and parameter dependent Lyapunov function quadratic stability for linear continuous time affine systems. The proposed design methods lead to a non-iterative LMI based algorithm. Numerical examples are given to illustrate the design procedure.

Robust Output Feedback Design of Linear Discrete-Time Systems - LMI Approach

The problem of robust output feedback stabilization is studied. The new robust stability LMI condition including guaranteed cost and the respective simple control design procedure is developed. Though the proposed condition is sufficient, it is supposed not to be overly conservative. To reduce the conservatism parameter-dependent Lyapunov function is employed and an extra degree of freedom is included by using previous results of (Oliveira et al., 1999) and introducing other additional slack matrix and scalar parameters. The example is given to analyse and compare the obtained results with other ones.

Robust control of state delayed systems with polytopic type uncertainties via parameter-dependent Lyapunov functionals

This paper considers the problem of robust stability and robust stabilization for linear systems with a constant time-delay in the state and subject to real convex polytopic uncertainty. First of all, for robust stability problem, we exploit new matrix inequalities characterization of delay-dependent quadratic stability results, demonstrate that it allows the use of parameter-dependent Lyapunov functionals, and develop control design methods based on linear matrix inequalities (LMIs) for solving the robust control problem. Next, the problem of determining the maximum time-delay under which the system remains stable is cast into a generalized eigenvalue problem and thus solved by LMI techniques. Finally, illustrative examples are given to demonstrate the advantage of these new representations.

An lpd approach to robust H/sub 2/ and H∞ static output-feedback design

A linear parameter dependent approach for designing a constant output feedback controller for a linear time-invariant system with uncertain parameters that achieves a minimum bound on either the H/sub 2/ or H/sub /spl infin// performance level is introduced. Assuming that the uncertain parameters reside in a given polytope, a parameter dependent Lyapunov function is described which enables the derivation of the required constant gain via a solution of a set of linear matrix inequalities that correspond to the vertices of the uncertainty polytope.

An improved lmi condition for robust D-stability of uncertain polytopic systems

A sufficient condition for robust D-stability of linear systems with polytope type uncertainties is proposed. The result is based on a linear parameter-dependent Lyapunov function obtained from the feasibility test of a set of linear matrix inequalities (LMIs) defined at the vertices of the polytope. This improved LMI condition encompasses previous results based on additional inequalities, as well as results based on extra variables.

Robust matrix 𝒟U‐stability analysis

AbstractIn this paper, the problem of robust matrix root‐clustering is addressed. The studied matrices are subject to both polytopic and unstructured uncertainties. An original point is the large choice of clustering regions enabled by the proposed approach since these regions can be unions of possibly disjoint and non‐symmetric subregions of the complex plane. The precise purpose is, considering a specified polytope, to determine the greatest robustness bound on the unstructured uncertainty such that robust matrix root‐clustering is ensured. To reduce conservatism in the derivation of the bound, the reasoning relies on a framework based upon parameter‐dependent Lyapunov functions. The bound value is computed by solving an ℒ︁ ℳ︁ ℐ problem. Copyright © 2003 John Wiley & Sons, Ltd.

Robust energy-to-peak filtering with improved LMI representations

The authors revisit the problem of robust energy-to-peak filtering for linear systems with parametric uncertainty residing in a polytope. Based on two results that appeared recently, they derive new L/sub 2/-L/sub /spl infin// performance criteria which allow the use of parameter-dependent Lyapunov functions for analysis and synthesis problems. Robust L/sub 2/-L/sub /spl infin// filters are then designed upon the new conditions and by means of the linear matrix inequality (LMI) technique, with the result that less conservativeness is achieved compared with earlier results that are based on a quadratic framework. Both continuous and discrete-time cases are considered and numerical examples illustrate the feasibility and advantage of the proposed designs.

Computing a robust D-stability bound using a parameter-dependent Lyapunov approach

The problem of robust matrix root-clustering against additive structured uncertainty is addressed. A bound on the size of the uncertainty domain preserving matrix D-stability is derived from an LMI approach. A recently proposed sufficient condition for robust matrix D-stability with respect to convex polytopic uncertainty is used. It is relevant to the framework dealing with parameter-dependent Lyapunov functions. Using this condition, the problem of computing the robustness bound is formulated as a generalised eigenvalue problem, that enables the bound value to be maximised.