Parameter-dependent Lyapunov Function Research Articles

Gain-scheduled ℋ 2 and ℋ ∞ control of discrete-time polytopic time-varying systems

This study presents ℋ2 and ℋ∞ performance analysis and synthesis procedures for the design of both gain-scheduled and robust static output feedback controllers for discrete-time linear systems with time-varying parameters. The obtained controllers guarantee an upper bound on the ℋ2 or ℋ∞ performance of the closed-loop system. As an immediate extension, the mixed ℋ2/ℋ∞ guaranteed cost control problem is also addressed. The scheduling parameters vary inside a polytope and are assumed to be a priori unknown, but measured in real-time. If bounds on the rate of parameter variation are known, they can be taken into account, providing less conservative results. The geometric properties of the polytopic domain are exploited to derive finite sets of linear matrix inequalities (LMIs) based on the existence of a parameter-dependent Lyapunov function. An application of the methodology to a realistic vibroacoustic problem, with experimentally obtained data, illustrates the benefits of the proposed approach and shows that the techniques can be used for real engineering problems.

Robust filtering for 2-D discrete-time linear systems with convex-bounded parameter uncertainty

This paper is concerned with the problems of robust H ∞ and H 2 filtering for 2-dimensional (2-D) discrete-time linear systems described by a Fornasini–Marchesini second model with matrices that depend affinely on convex-bounded uncertain parameters. By a suitable transformation, the system is represented by an equivalent difference-algebraic representation. A parameter-dependent Lyapunov function approach is then proposed for the design of 2-D stationary discrete-time linear filters that ensure either a prescribed H ∞ performance or H 2 performance for all admissible uncertain parameters. The filter designs are given in terms of linear matrix inequalities. Numerical examples illustrate the effectiveness of the proposed filter design methods.

Stabilization of Discrete-time 2-D T-S Fuzzy Systems Based on New Relaxed Conditions

This paper is concerned with the problem of stabilization of the Roesser type discrete-time nonlinear 2-D system that plays an important role in many practical applications. First, a discrete-time 2-D T-S fuzzy model is proposed to represent the underlying nonlinear 2-D system. Second, new quadratic stabilization conditions are proposed by applying relaxed quadratic stabilization technique for 2-D case. Third, for sake of further reducing conservatism, new non-quadratic stabilization conditions are also proposed by applying a new parameter-dependent Lyapunov function, matrix transformation technique, and relaxed technique for the underlying discrete-time 2-D T-S fuzzy system. Finally, a numerical example is provided to illustrate the effectiveness of the proposed results.

Decentralized Robust Control of Linear Uncertain Systems with Reduced Conservatism

In this paper a novel decentralized robust control design scheme is proposed, considering the structured uncertain system matrix with a constant part (typical case for a system with dynamic controller as PID); the resulting robust control design is based on developed robust stability LMI condition which is less conservative than those from literature. In the following we use parameter dependent Lyapunov function and structured auxiliary matrices to receive “structured robust stability condition” which is the base for robust controller design. All developments are performed in general formulation including both discrete-time and continuous-time systems.

Robust ℋ ∞ networked control for systems with uncertain sampling rates

This study investigates the problem of controller design for systems with uncertain sampling rates. The system is controlled through a communication network. The sampling period, within a given interval, is assumed to be time-varying and a simplified framework for the network-induced delay is considered. The overall system is thus described by an uncertain discrete-time model with time-varying parameters inside a polytope whose vertices are obtained by means of the Cayley–Hamilton theorem. A digital robust controller that minimises an upper bound to the ℋ∞ performance of the closed-loop networked control system (NCS) is determined. The design conditions rely on a particular parameter-dependent Lyapunov function and are expressed as bilinear matrix inequalities (BMIs) in terms of extra matrix variables, which may be explored in the search for a better system behaviour. Numerical examples illustrate the results.

Delay-dependent and Parameter-dependent Robust Stability for Discrete-time Delayed Uncertain Singular Systems

The problem of delay-dependent and parameter-dependent robust stability condition for discrete-time uncertain singular systems with polytopic uncertainty and interval time-varying delay is considered. A new robust stability condition based on parameter-dependent Lyapunov function is derived in terms of LMI (linear matrix inequality). Moreover, the proposed robust stability condition is a general condition for both singular and non-singular systems. A numerical example is presented to demonstrate the effectiveness of the proposed method.

Robust Filtering for Linear Polytopic Discrete-Time Periodic Systems

This paper is concerned with the problems of robust H∞ and guaranteed variance filtering for linear discrete-time periodic systems with polytopic-type parameter uncertainty in the matrices of the system state-space model. Filtering methods are derived for designing linear periodic asymptotically stable filters with either a prescribed upper-bound on the l2-gain from the noise signals to the estimation error, in the H∞ case, or a guaranteed upper-bound on the average steady-state variance of the estimation error variance, for the guaranteed variance filtering, in spite of the parameter uncertainty. The proposed methods are based on parameter-dependent Lyapunov functions and are given in terms of linear matrix inequalities. The potentials of the proposed filtering methods are demonstrated by an example.

Robust analysis of Demeter benchmark via quadratic separation

AbstractThe purpose of this paper is to give numerical illustrations of recent results obtained in the area of robust analysis of uncertain parametric LTI systems. The idea is to highlight theoretical as well as numerical issues pertaining to real life applications of sophisticated robust analysis of spaceborne control systems. The framework of quadratic separation applied to interconnected implicit linear transformations and uncertain operators as defined in Peaucelle et al. (2007) is renewed. Improved robust stability tests involving more complex parameter-dependent Lyapunov functions are applied on a benchmark from the space industry. It is based on the problem of attitude control of a flexible satellite developed at CNES and it is used to analyze the relevance as well as some shortcomings of the proposed approach.

Linear parameter-varying control of a large blended wing body flexible aircraft

AbstractIn this paper a linear parameter-varying (LPV) controller design approach is applied to the longitudinal dynamics of a large blended wing body aircraft. The method is based on parameter-dependent Lyapunov functions utilizing the information given by bounds for the maximum parameter rate of variation. The entire parameter space is approximated by a set of linearized models in trimmed operating points which leads to a finite-dimensional convex optimization problem. Typical design goals for flexible aircraft control such as handling qualities, loads and vibration reduction were considered with the Mach number as scheduling parameter. The obtained LPV controller is extensively tested, and the obtained results demonstrate the high potential of the methodology for flexible aircraft control.

Robust Model Predictive Control Using Polytopic Description of Input Constraints

In this paper, we propose a less conservative a linear matrix inequality (LMI) condition for the constrained robust model predictive control of systems with input constraints and polytopic uncertainty. Systems with input constraints are represented as perturbed systems with sector bounded conditions. For the infinite horizon control, closed-loop stability conditions are obtained by using a parameter dependent Lyapunov function. The effectiveness of the proposed method is shown by an example.

A descriptor system approach to robust H∞ control for linear systems with time-varying uncertainties

This article addresses the problems of robust H ∞ performance analysis and synthesis for continuous-time linear systems with time-varying uncertainties using parameter-dependent Lyapunov functions. An H ∞ performance criterion in terms of linear matrix inequalities is derived via employing a descriptor model transformation of linear systems with polytopic uncertainties (polytopic systems). This H ∞ performance criterion can be applied to robust controller synthesis and filtering design. The descriptor system approach is also taken to explore robust H ∞ performance analysis and controller synthesis of affine parameter systems. A flight control system and two numerical examples are given to illustrate the effectiveness of the proposed analysis and synthesis procedures.

Poly‐quadratic stability of discrete‐time nonlinear systems in Takagi‐Sugeno's form

AbstractThis paper studies poly‐quadratic stability of nonlinear systems represented by a Takagi‐Sugeno (T‐S) discrete fuzzy model. A system is poly‐quadratically stable when its stability can be verified by a parameter dependent Lyapunov function. There are already some results in this area. This paper, however, presents less conservative results by adopting more slack‐matrix‐variables in the linear matrix inequalities (LMIs). The parallel distributed compensation (PDC) law is utilized in both the analysis and the synthesis problems. The effectiveness of the new results in this paper is validated by a simulation example. Copyright © 2009 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society

Robust H ∞ static output feedback control of discrete-time switched polytopic linear systems with average dwell-time

This paper investigates the problem of robust exponential H ∞ static output feedback controller design for a class of discrete-time switched linear systems with polytopic-type time-varying parametric uncertainties. The objective is to design a switched static output feedback controller guaranteeing the exponential stability of the resulting closed-loop system with a minimized exponential H ∞ performance under average dwell-time switching scheme. Based on a parameter-dependent discontinuous switched Lyapunov function combined with Finsler’s lemma and Dualization lemma, some novel conditions for exponential H ∞ performance analysis are first proposed and in turn the static output feedback controller designs are developed. It is shown that the controller gains can be obtained by solving a set of linear matrix inequalities (LMIs), which are numerically efficient with commercially available software. Finally, a simulation example is provided to illustrate the effectiveness of the proposed approaches.

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.

Finite frequency [formula omitted] filtering for uncertain discrete-time switched linear systems

This paper is concerned with the problem of robust H ∞ filtering for discrete-time switched linear systems with polytopic uncertainties in the finite frequency domain. Based on the generalized Kalman–Yakubovich–Popov (GKYP) lemma and switched parameter-dependent Lyapunov functions, a switched full-order filter is designed such that the corresponding filtering error system is asymptotically stable and satisfies a prescribed finite frequency H ∞ performance index. Compared with the existing full frequency approaches, the proposed finite frequency one receives better results for the cases in which the frequency ranges of noises are known. A numerical example is given to illustrate the effectiveness of the proposed method.

Time-varying discrete-time linear systems with bounded rates of variation: Stability analysis and control design

This paper investigates the problems of robust stability analysis and state feedback control design for discrete-time linear systems with time-varying parameters. It is assumed that the time-varying parameters lie inside a polytopic domain and have known bounds on their rate of variation. A convex model is proposed to represent the parameters and their variations as a polytope and linear matrix inequality relaxations that take into account the bounds on the rates of parameter variations are proposed. A feasible solution provides a parameter-dependent Lyapunov function with polynomial dependence on the parameters assuring the robust stability of this class of systems. Extensions to deal with robust control design as well as gain-scheduling by state feedback are also provided in terms of linear matrix inequalities. Numerical examples illustrate the results.

Min-max model predictive control for constrained nonlinear systems via multiple LPV embeddings

A min-max model predictive control strategy is proposed for a class of constrained nonlinear system whose trajectories can be embedded within those of a bank of linear parameter varying (LPV) models. The embedding LPV models can yield much better approximation of the nonlinear system dynamics than a single LTV model. For each LPV model, a parameter-dependent Lyapunov function is introduced to obtain poly-quadratically stable control law and to guarantee the feasibility and stability of the original nonlinear system. This approach can greatly reduce computational burden in traditional nonlinear predictive control strategy. Finally a simulation example illustrating the strategy is presented.

Formula omitted] filtering for discrete-time linear systems with bounded time-varying parameters

In this paper, the problem of linear parameter varying (LPV) filter design for time-varying discrete-time polytopic systems with bounded rates of variation is investigated. The design conditions are obtained by means of a parameter-dependent Lyapunov function and extra variables for the filter design, expressed as bilinear matrix inequalities. An LPV filter, which minimizes an upper bound to the H ∞ performance of the estimation error, is obtained as the solution of an optimization problem. A convex model to represent the parameters and their variations as a polytope is proposed in order to provide less conservative design conditions. Robust filters for time-varying polytopic systems can be obtained as a particular case of the proposed method. Numerical examples illustrate the results.

New characterisations of positive realness and static output feedback control of discrete-time systems

This article is concerned with the problems of positive real analysis and control synthesis for discrete-time systems. New linear matrix inequality (LMI) characterisations of positive realness are derived, which enable one to check the positive realness by using parameter-dependent Lyapunov function. The relationship between the proposed characterisations and the existing ones are clarified, which shows that our new results are of less conservatism for characterising the positive realness of discrete-time systems with polytopic uncertainty. In addition, sufficient conditions for static output feedback positive real controller design are given in terms of solutions to a set of linear matrix inequalities. Numerical examples are included for illustration.

Stability analysis of T–S fuzzy control systems using parameter-dependent Lyapunov function

The system stability of T–S fuzzy-model-based control systems with a parameter-dependent Lyapunov function (PDLF) is investigated. As PDLF approach includes information of the membership function (time derivatives of membership functions), it has been reported that relaxed stability conditions can be achieved compared to the parameter-independent Lyapunov function (PILF). To investigate the system stability, the non-linear plant is represented by a T–S fuzzy model. Various non-parallel distribution compensation (PDC) fuzzy controllers, which can better utilise the characteristic of the PDLF, are proposed to close the feedback loop. To relax the stability conditions, an improved PDLF is employed. Some inequalities are proposed to relate the membership functions and its time derivatives, which allow the introduction of some slack matrices to facilitate the stability analysis. Stability conditions in terms of linear matrix inequalities are derived to aid the design of stable fuzzy-model-based control systems. Simulation examples are given to illustrate the effectiveness of the proposed non-PDC fuzzy control schemes.