Parameter-dependent Quadratic Lyapunov Functions Research Articles

Polynomially parameter dependent exponential stabilization of sampled-data LPV systems

This paper proposes a polynomially sampled parameter dependent controller design for sampled-data linear parameter varying (LPV) systems. The goal of this study is to design a controller which exponentially stabilizes the system with a larger maximum sampling interval. To achieve this, we utilize the information of the parameters in the controller part. The proposed controller is dependent on the polynomials with respect to the sampled parameters, which is more effective to stabilize the system with a larger maximum sampling interval. To design the controller, an exponential stabilization condition is derived from a new Polynomially Parameter Dependent Quadratic Lyapunov Function with Looped-Functionals (PPDQLLF) which is dependent on the parameters of a plant. The derived condition is represented in terms of the linear matrix inequality (LMI) and it is formulated as sum of squares (SOS) conditions to obtain feasible solutions. The effectiveness of the proposed method is shown by the simulation results of numerical examples.

Event-triggered Dynamic Output Feedback Controller for Discrete-time LPV Systems with Constraints

This paper examines an event-triggered control design approach for discrete-time linear parameter-varying (LPV) systems under control constraints. A parameter-dependent dynamic output-feedback controller with an event-triggering condition is designed to ensure the regional asymptotic stability of the closed-loop system while minimizing a quadratic cost function. Sufficient conditions are derived in terms of linear matrix inequalities (LMIs) thanks to the use of a parameter-dependent quadratic Lyapunov function. Also, convex optimization schemes are proposed using these conditions to minimize an upper bound of a given cost function or to maximize the size of the region of closed-loop stability. Examples illustrate the proposal.

Development of Dynamic Model Based Gain Scheduling Pid Controller for Linear Parameter Varying System Using Modified Spiral Dynamic Algorithm

In chemical industries such as food industries, pharmaceuticals and so on, the Evaporative crystallization plays a major important role. The crystal growth is affected by factors such as the operating temperature, pressure and concentration. In this paper, gain scheduling PID controller has implemented with the Linear Parameter Varying (LPV) system methods in that the Parameter Dependent Quadratic Lyapunov Function (PDQLF) are utilized on behalf of controlling the factors of evaporative crystallizer operative temperature, absolute pressure and concentration. Here we also introduced the ModifiedSpiral Dynamic Algorithm (M-SDA) for the minimization of error percentage and improvement of accuracy at 94 %.

Stability and stabilisation for time-varying polytopic quadratic systems

ABSTRACTThis paper develops the problems of local stability and stabilisation for time-varying polytopic quadratic systems. The state-space data is assumed to be dependent on parameters that are measurable in real time and vary in a compact set with bounded variation rates. These problems are solved by utilising the parameter-dependent quadratic Lyapunov function and S-procedure approach. Sufficient conditions for local stability and stabilisation are first formulated as optimisation problems with ‘quasi’ linear matrix inequalities. Simulation examples are then provided to confirm the effectiveness of the given approach.

On homogeneous parameter-dependent quadratic Lyapunov function for robust fault detection filter design and application

This paper is concerned with the problem of the fault detection filter design for discrete-time system with polytopic uncertainties. The proposed fault detection filter framework consists of the fault effect on the residual signal is maximized and the disturbance effects on the residual signals is minimized. Since model uncertainties may result in significant changes in the residual, homogeneous parameter-dependent quadratic Lyapunov function method is used to suppress the effect of model uncertainties. Moreover, it is more effective compared with common Lyapunov function method. Lastly, simulation results obtained in inductance resistance capacitor circuit system and numerical examples are used to illustrate the effectiveness of our method.

FIR-type robust H 2 and H ∞ control of discrete linear time-invariant polytopic systems via memory state-feedback control laws

In this paper, robust H 2 and H ∞ control problems for discrete linear time-invariant (LTI) systems with polytopic uncertainties are addressed. The so-called finite impulse response (FIR) controller incorporating the states over several samples from the past to the present is adopted to design robust control laws with improved performances. For the closed-loop stability, parameter-dependent quadratic Lyapunov functions (PD-QLFs) are employed. Sufficient controller synthesis conditions are derived in the form of linear matrix inequalities (LMIs). Finally, examples are given to demonstrate the usefulness of the proposed methods.

Estimation of Attractive Stability Region via Homogeneous Parameter-Dependent Quadratic Lyapunov Function for Impulsive Switched Linear System with Saturated Control Input

A class of homogeneous polynomial functions is used to estimate the attractive stability region for an impulsive switched linear system with saturated control input. Under the arbitrary switching rule and the dwell time switching rule, local stabilization conditions are obtained in terms of linear matrix inequalities. The corresponding optimization problems are formulated to obtain a larger attractive stability region. Using the simple ellipsoid method to estimate the attractive stability region produces very conservative results, because the attractive stability region is normally irregular. To solve this problem, a polyhedron constructed using a level set of the homogeneous parameter-dependent quadratic Lyapunov function is used to estimate the attractive stability region. The polyhedron is closer to the attractive stability region than the simple ellipsoid. Finally, numerical examples are given to demonstrate the effectiveness of the proposed method.

On Homogeneous Parameter-Dependent Quadratic Lyapunov Function for RobustH∞Filtering Design in Switched Linear Discrete-Time Systems with Polytopic Uncertainties

This paper is concerned with the problem of robustH∞filter design for switched linear discrete-time systems with polytopic uncertainties. The condition of being robustly asymptotically stable for uncertain switched system and less conservativeH∞noise-attenuation level bounds are obtained by homogeneous parameter-dependent quadratic Lyapunov function. Moreover, a more feasible and effective method against the variations of uncertain parameter robust switched linear filter is designed under the given arbitrary switching signal. Lastly, simulation results are used to illustrate the effectiveness of our method.

Robust LPV autopilot design for hypersonic reentry vehicle

Purpose – The purpose of this paper is to design a robust angle-of-attack (AOA) tracking control system for the hypersonic reentry vehicle (HRV) based on the linear parameter varying (LPV) theory, as the aerodynamic coefficients of the hypersonic vehicle vary quickly during the reentry phase. Design/methodology/approach – First, longitudinal moment trim is done along the desired flight trajectory. The linearized system at each trim point is built and the dynamic characteristics analysis is made. Then the LPV control law with parameter-dependent quadratic Lyapunov function (PDQLF-LPV) is applied to design the AOA tracking autopilot at each trim point. Frequency performance of the autopilot is assessed and the step response simulation is conducted to validate the effectiveness of the control system. Finally, actual AOA command tracking simulations based on the time-varying nonlinear model are carried out to test the correctness and robustness of the PDQLF-LPV autopilot. Findings – Analysis results demonstra...

Enlarging the guaranteed region of attraction in nonlinear systems with bounded parametric uncertainty

A novel approach to enlarge the guaranteed region of attraction in nonlinear systems with bounded parametric uncertainties based on the design of a nonlinear controller is proposed. The robust domain of attraction (RDA) is estimated using the parameter-dependent quadratic Lyapunov function and enlarged by the optimal controlling parameters. The problem of extending the RDA is indicated in a form of three-layer optimization problem. Some examples illustrate the efficiency of the proposed strategy in enlarging RDA.

Worst-Case Mahler Measure in Polytopic Uncertain Systems

The Mahler measure provides a way to quantify the unstable and plays a key role in stabilization problems. This technical brief addresses the computation of the worst-case Mahler measure in systems depending polynomially on uncertain parameters constrained in a polytope. A sufficient condition for establishing an upper bound of the worst-case Mahler measure is provided in terms of linear matrix inequality (LMI) feasibility tests, where a homogeneous parameter-dependent quadratic Lyapunov function (HPD-QLF) is searched for. Moreover, it is shown that the best upper bound guaranteed by this condition can be obtained by solving generalized eigenvalue problems. Then, the conservatism of this methodology is investigated, showing that the upper bound is monotonically nonincreasing with the degree of the HPD-QLF, and that there exists a degree for which the upper bound is guaranteed to be tight. Some numerical examples illustrate the proposed results.

Lyapunov-based exact stability analysis and synthesis for linear single-parameter dependent systems

We propose a class of polynomially parameter-dependent quadratic (PPDQ) Lyapunov functions for assessing the stability of single-parameter-dependent linear, time-invariant, (s-PDLTI) systems in a non-conservative manner. It is shown that the stability of s-PDLTI systems is equivalent to the existence of a PPDQ Lyapunov function. A bound on the degree of the polynomial dependence of the Lyapunov function in the system parameter is given explicitly. The resulting stability conditions are expressed in terms of a set of matrix inequalities whose feasibility over a compact and connected set can be cast as a convex, finite-dimensional optimisation problem. Extensions of the approach to state-feedback controller synthesis are also provided.

Time-invariant uncertain systems: A necessary and sufficient condition for stability and instability via homogeneous parameter-dependent quadratic Lyapunov functions

This paper investigates linear systems with polynomial dependence on time-invariant uncertainties constrained in the simplex via homogeneous parameter-dependent quadratic Lyapunov functions (HPD-QLFs). It is shown that a sufficient condition for establishing whether the system is either stable or unstable can be obtained by solving a generalized eigenvalue problem. Moreover, this condition is also necessary by using a sufficiently large degree of the HPD-QLF.

Stabilization of systems with variable and uncertain sampling period and time delay

Stability and control issues of systems with uncertain and time-varying sampling period and time-delay are addressed in this paper. These arise, e.g., in embedded or networked control applications where limited computation or communication resources have to be scheduled. The uncertain and time-varying sampling period and time delay are transformed into polytopic and additive norm-bounded uncertainties in the discretized system description. Control design and stability analysis methods are given in the form of LMIs applying switched parameter-dependent quadratic Lyapunov functions. Two reduction algorithms are proposed in order to decrease the amount of LMIs necessary for stability analysis. The control design and stability analysis methods as well as the reduction algorithms are illustrated by an example.

On the non-conservatism of a novel LMI relaxation for robust analysis of polytopic systems

It is known that robust analysis of polytopic systems, i.e. linear system with polytopic state space uncertainty, is a difficult problem. Recently we have proposed for this problem, in both continuous-time and discrete-time cases, the use of homogeneous polynomially parameter-dependent quadratic Lyapunov functions (HPD–QLFs) by introducing a novel LMI relaxation which provides a sufficient condition via a convex optimization without requiring the use of auxiliary multipliers of unknown degree. In this paper we show that this condition is not only sufficient but it is also necessary.

Stability and Control of Systems with Uncertain Time-Varying Sampling Period and Time Delay

This paper addresses stability and control issues of systems with uncertain and time-varying sampling period and time-delay. These arise e.g. in embedded or networked control applications where limited computation or communication resources have to be scheduled. The uncertain and time-varying sampling period and time delay are transformed into polytopic and additive norm-bounded uncertainties in the discretized system description. Control design and stability analysis methods are given in the form of LMIs applying switched parameter-dependent quadratic Lyapunov functions. A reduction algorithm is proposed in order to decrease the amount of LMIs necessary for stability analysis. The control design and stability analysis methods as well as the reduction algorithm are illustrated by an example.

Establishing tightness in robust [formula omitted] analysis via homogeneous parameter-dependent Lyapunov functions

Several methods have been proposed to provide upper bounds to the robust H ∞ performance of a linear system with polytopic state space uncertainty. A common problem of these methods is the lack of a test for establishing if the found upper bound is tight or not, which is essential to determine if the search of the sought performance is finished or should continue. This paper provides a necessary and sufficient condition for establishing the tightness of the upper bound provided by homogeneous polynomially parameter-dependent quadratic Lyapunov functions (HPD-QLFs) which recently have been shown to be a powerful tool for computing these upper bounds. This condition is based on the existence of particular vectors in suitable linear spaces. Techniques for finding these vectors are also provided.

H∞ control of parameter-dependent state-delayed systems using polynomial parameter-dependent quadratic functions

This paper presents the issue of robust disturbance attenuation and robust asymptotic stability problem for finite-dimensional linear parameter-dependent state-delayed systems. The use of polynomial parameter-dependent quadratic Lyapunov functions and linear matrix inequalities (LMIs) formulations for robust H ∞ control are considered. It is shown that the state feedback control can be determined to guarantee the stability of the closed-loop system independently of the time-delay. We present an illustrative example to demonstrate the applicability of the proposed design approach.

Polynomially parameter-dependent Lyapunov functions for robust stability of polytopic systems: an LMI approach

In this note, robust stability of state-space models with respect to real parametric uncertainty is considered. Specifically, a new class of parameter-dependent quadratic Lyapunov functions for establishing stability of a polytope of matrices is introduced, i.e., the homogeneous polynomially parameter-dependent quadratic Lyapunov functions (HPD-QLFs). The choice of this class, which contains parameter-dependent quadratic Lyapunov functions whose dependence on the uncertain parameters is expressed as a polynomial homogeneous form, is motivated by the property that a polytope of matrices is stable if and only there exists an HPD-QLF. The main result of the note is a sufficient condition for determining the sought HPD-QLF, which amounts to solving linear matrix inequalities (LMIs) derived via the complete square matricial representation (CSMR) of homogeneous matricial forms and the Lyapunov matrix equation. Numerical examples are provided to demonstrate the effectiveness of the proposed approach.

A Convex Approach to Robust Stability for Linear Systems with Uncertain Scalar Parameters

In this paper, robust stability for linear systems with several uncertain (complex and/or real) scalar parameters is studied. A countable family of conditions sufficient for robust stability is given, in terms of solvability of some simple linear matrix inequalities (LMIs). These conditions are of increasing precision, and it is shown conversely that robust stability implies solvability of these LMIs from a certain rank and beyond. This result constitutes an extension of the characterization by solvability of Lyapunov inequality of the asymptotic stability for usual linear systems. It is based on the search of parameter-dependent quadratic Lyapunov functions, polynomial of increasing degree in the parameters.