Parameter-dependent Lyapunov Function Research Articles

Gain-scheduled H ∞ filtering of parameter-varying discrete-time systems via parameter-dependent Lyapunov functions

This paper is concerned with the problem of gain-scheduled H∞ filter design for a class of parameter-varying discrete-time systems. A new LMI-based design approach is proposed by using parameter-dependent Lyapunov functions. n k y ∈ℜ is the measured output; z n k z ∈ℜ is the signal to be estimated; m k α ∈ℜ and m k β ∈ℜ are introduced variables; s k η ∈ℜ and is allowed to vary in the unit simplex

Constrained robust feedback model predictive control for uncertain systems with polytopic description

For the constrained uncertain systems with polytopic description, a new design method of robust feedback model predictive control is proposed. By using a sequence of feedback control laws and designing the parameter-dependent Lyapunov function for each model, the design method provides more degrees of freedom and then can improve the control performance and enlarge the feasible region. Based on the characteristic property of the design method, an offline design algorithm is developed to reduce the MPC controller's online computation burden. The numerical examples verify the effectiveness of the results presented in this article.

Stability analysis and gain-scheduled state feedback control for continuous-time systems with bounded parameter variations

The problems of robust stability analysis and state feedback control based on gain-scheduling for continuous-time systems with time-varying parameters that have bounded rates of variation and lie inside a polytope are addressed in this article. With respect to previous results in the literature, two main contributions of the article are: (i) the robust stability analysis conditions are less conservative and demand less computational effort than the existing ones; (ii) the conditions can be extended to cope with the problem of control design for this class of system. Parameter-dependent linear matrix inequality (LMI) conditions are given for the existence of a parameter-dependent Lyapunov function quadratic in the state and homogeneous polynomially of arbitrary degree in the parameter assuring robust stability. Two convex procedures based on LMIs exhibiting distinct complexities are proposed to solve the problem of robust stability. An extension to deal with the computation of a stabilising parameter-dependent state feedback gain for this class of time-varying systems is also provided, as a sequence of LMIs of increasing precision. Examples illustrate the results, including comparisons with other techniques from the literature.

Synchronization analysis of a complex network family

This paper is concerned with a meaningful and challenging problem: Is it possible to establish synchronization criteria for a network family, the members of which have different network structure? Based on the parameter-dependent Lyapunov function, some synchronization criteria are proposed to address this problem. These simple yet generic criteria can lead to the advantage of easy synchronization verification of every member from the network family. On the other hand, the problem mentioned above can be regarded as a synchronization problem for a network with structure uncertainty, which is also investigated in this paper. Numerical examples are given to demonstrate the effectiveness of the derived criteria.

An Asymptotically Exact Approach to Robust Semidefinite Programming Problems with Function Variables

This technical note provides an approximate approach to a semidefinite programming problem with a parameter-dependent constraint and a function variable. This problem covers a variety of control problems including a robust stability/performance analysis with a parameter-dependent Lyapunov function. In the proposed approach, the original problem is approximated by a standard semidefinite programming problem through two steps: first, the function variable is approximated by a finite-dimensional variable; second, the parameter-dependent constraint is approximated by a finite number of parameter-independent constraints. Both steps produce approximation error. On the sum of these approximation errors, this technical note provides an upper bound. This bound enables quantitative analysis of the approach and gives an efficient way to reduction of the approximation error. Moreover, this technical note discusses how to verify that an optimal solution of the approximate problem is actually optimal also for the original problem.

Enhanced H ∞ filtering for continuous-time state-delayed systems

The H∞ filtering problem for continuous-time polytopic uncertain time-delay systems is investigated. Attention is focused on the design of full-order filters guaranteeing a prescribed H∞ attenuation level for the filtering error system. First, a simple alternative proof is given for an improved linear matrix inequality (LMI) representation of H∞ performance. Then, based on the performance criterion which keeps Lyapunov matrices out of the product of system dynamic matrices, a sufficient condition for the existence of robust estimators is formulated in terms of LMIs, and the corresponding filter design is cast into a convex optimization problem which can be efficiently handled by using standard numerical algorithms. It is shown that the proposed design strategy allows the use of parameter-dependent Lyapunov functions and hence it is less conservative than some earlier results. A numerical example is employed to demonstrate the feasibility and advantage of the proposed design.

Robust model-following controller design for LTI systems affected by parametric uncertainties: a design example for aircraft motion

This article addresses the design problem of a robust model-following controller (MFC) which minimises the error between plant controlled output and model output for a linear time-invariant (LTI) plant system affected by parametric uncertainties and an LTI target model. To design such an MFC, a previously proposed MFC scheme, whose applicability has already been demonstrated with flight controller design, is adopted in this article. Our design procedure is as follows: first, a basic MFC is designed using the nominal LTI plant model and the LTI target model while a structured free matrix in the MFC is not assigned; second, model-following performance of the MFC for appropriately defined disturbance input and model input for the parametric uncertain plant model and the LTI target model is minimised using the structured free matrix; and finally, a robust MFC is obtained using the basic MFC and the optimal structured matrix. In the second step, an iterative design method for robust H 2 controllers for LTI parameter-dependent (LTIPD) systems using parameter-dependent Lyapunov functions (PDLFs), which is also proposed in this article, is applied. Two MFCs for the lateral-directional (L/D) motions of a research aircraft, which has been developed for an in-flight simulator, for two different real aircraft models, i.e. a Boeing 747 model and a Lockheed Jetstar model, are designed and their performance is confirmed via numerical simulations and flight tests.

Clearance of flight control laws via parameter-dependent Lyapunov functions

Lyapunov-based analysis is a widely used tool to assess robust stability of dynamic systems affected by parametric uncertainties. When the uncertain parameters are constant, it is well known that parameter-dependent Lyapunov functions allow one to reduce conservatism of sufficient conditions for robust stability. In this paper, we consider several different techniques for robustness analysis of systems with LFR uncertainty structure, based on parameter-dependent Lyapunov functions. Such techniques are applied to LFR systems derived from real-world flight control schemes, with the aim of finding the largest region in the uncertain parameter space and/or in the flight envelope, for which robust stability is guaranteed. The trade off between conservatism and computational burden is analyzed for each considered technique.

Model predictive control for linear parameter varying systems using path-dependent Lyapunov functions

This paper is concerned with the design of Model Predictive Control (MPC) for Linear Parameter Varying (LPV) discrete-time systems. Sufficient linear matrix inequality (LMI) conditions are provided for the existence of a path-dependent Lyapunov function which generalizes previous results based on affine parameter-dependent Lyapunov functions. At each sampling time the control law is obtained from a convex optimization problem under LMI constraints. As illustrated by examples, the proposed approach yields less conservative results than other available methods for MPC.

Robust H∞ Filtering for 2-D Discrete-Time Linear Systems with Convex-Bounded Parameter Uncertainty

This paper is concerned with the robust H∞ filtering problem for 2-dimensional (2-D) discrete-time linear systems described by a Fornasini-Marchesini second model with convex-bounded parameter uncertainty. By a suitable transformation, the system is converted into an equivalent difference-algebraic representation. A parameter-dependent Lyapunov function approach is then proposed for the design of a 2-D stationary discrete-time linear filter that ensures a prescribed upper-bound on the H∞-norm of the transfer function of the estimation error system for all admissible uncertain parameters. The filter design is given in terms of linear matrix inequalities.

Formula omitted]-stability and [formula omitted]-stabilization of linear discrete time-delay systems with polytopic uncertainties

Sufficient conditions for D -stability and D -stabilization of linear discrete time-delay systems with polytopic uncertainties are proposed in this paper. The results are based on parameter-dependent Lyapunov functions obtained from the feasibility of a set of linear matrix inequalities defined at the vertices of polytopic systems. These conditions guarantee all the poles of a corresponding system in a specified circular region. Finally, an illustrative example shows that the proposed conditions are effective to check the D -stability and to design the D -stabilizing controller for linear discrete time-delay systems.

Polynomial LPV synthesis applied to turbofan engines

Results on polynomial fixed-order controller design are extended to SISO gain-scheduling with guaranteed stability and H ∞ performance over the whole scheduling parameter range. Salient features of the approach are: (a) the use of polynomials as modeling objects; (b) the use of flexible linear matrix inequalities (LMI) conditions allowing polynomial dependence of the open-loop system and controller transfer functions in the scheduling parameters; and (c) the decoupling in the LMI conditions between the Lyapunov variables and the controller variables, allowing both parameter-dependent Lyapunov functions and fixed-order controller design. The synthesis procedure is integrated into the ATOL framework developed by the manufacturer of aircraft and space engines Snecma to systematically design reduced complexity gain-scheduled control laws for aircraft turbofan engines.

Parameter-dependent robust [formula omitted] filtering for uncertain discrete-time systems

This paper is concerned with the problem of parameter-dependent H ∞ filtering for discrete-time systems with polytopic uncertainties. The uncertain parameters are supposed to reside in a polytope. Being different from previous results in the quadratic framework, the parameter-dependent Lyapunov function is used in this paper. Both full- and reduced-order filters are designed, which guarantee the asymptotic stability and a prescribed H ∞ performance level. The filter parameters can be obtained from the solution of convex optimization problems in terms of linear matrix inequalities, which can be solved via efficient interior-point algorithms. Numerical examples are presented to illustrate the feasibility and less conservativeness of the proposed method.

Robust reliable H-infinity control for nonlinear uncertain stochastic time-delay systems with Markovian jumping parameters

This paper deals with the problems of robust reliable exponential stabilization and robust stochastic stabilization with H-infinity performance for a class of nonlinear uncertain time-delay stochastic systems with Markovian jumping parameters. The time delays are assumed to be dependent on the system modes. Delay-dependent conditions for the solvability of these problems are obtained via parameter-dependent Lyapunov functionals. Furthermore, it is shown that the desired state feedback controller can be designed by solving a set of linear matrix inequalities. Finally, the simulation is provided to demonstrate the effectiveness of the proposed methods.

Generalized internal model architecture for gain-scheduled control

A two-degree-of-freedom controller architecture and its design strategy for linear parameter-varying (LPV) systems, where the dependent parameters are assumed to be measurable, are proposed in the generalized internal model control (GIMC) framework. First, coprime factorization and Youla parameterization for LPV systems are introduced based on a parameter-dependent Lyapunov function. Then, the GIMC architecture for linear time-invariant systems is extended to LPV systems with these descriptions. Based on this architecture, good tracking performance and good robustness (disturbance rejection) are compatibly achieved by a nominal controller and a conditional controller, respectively. Furthermore, each controller design problem is formulated in terms of linear matrix inequalities related to each L2-gain performance. Finally, a simple design example is illustrated.

A New Fault Detection Scheme for Networked Control Systems Subject to Uncertain Time-Varying Delay

Fault detection of networked control systems (NCS) subject to uncertain time-varying delay is studied in this paper. For the convenience of residual generator design, influence caused by network-induced delay is first transformed into time-varying polytopic uncertainty, which greatly facilitates further manipulation. Then design of the optimal residual generator is formulated as a model matching problem, i.e., to design a residual generator best matching the optimal residual generator of NCS in the delay free case. This procedure is assisted by parameter-dependent Lyapunov function matrix based bounded real lemma, which has been proved to be much better than single constant Lyapunov function matrix based results. This approach not only can be used in conditions that the variation part of the delay is less than one sampling period, but also can be applied to scenarios where the variation part of the delay is larger than one sampling period. Simulation results are also given to illustrate effectiveness of the proposed method.

Robust root-clustering analysis in a union of subregions for descriptor systems

A robust matrix root-clustering analysis for descriptor systems is considered. It states a necessary and sufficient condition for a descriptor system to be regular, impulse free and to have its finite poles in a specified region of the complex plane. This region is a union of convex and possibly disjoint and non-symmetric subregions. A sufficient condition to preserve those properties in the presence of polytopic and norm-bounded uncertainties affecting the state matrix is also established. This condition is based upon the implicit derivation of parameter-dependent Lyapunov functions. All the conditions are expressed in terms of linear matrix inequalities.

Robust H∞ performance using lifted polynomial parameter-dependent Lyapunov functions

This paper investigates the robust H ∞ performance of time-invariant linear uncertain systems where the uncertainty is in polytopic domains. Robust H ∞ is checked by constructing a quadratic parameter-dependent Lyapunov function. The matrix associated with this quadratic Lyapunov function is a polynomial function of the uncertain parameters, expressed as a particular polynomial matrix involving κ powers of the dynamic matrix of the system and one symmetric matrix to be determined. The degree of this polynomial matrix function is arbitrary. Finsler's Lemma is used to lift the obtained stability conditions into a larger space in which sufficient stability tests can be developed in the form of linear matrix inequalities. As κ increases, less conservative H ∞ evaluations are obtained. Both continuous and discrete-time systems are investigated. Numerical examples illustrate the method and compare the present results with similar works in the literature.

Output feedback control of saturated discrete-time linear systems using parameter-dependent Lyapunov functions

In this paper, we present a new output feedback control approach for discrete-time linear systems subject to actuator saturations using parameter-dependent Lyapunov functions. The saturation level indicator serves as a scheduling parameter. The resulting nonlinear controller is expressed in a quasi-LPV (linear parameter-varying) form, and the stabilization and disturbance attenuation problems are formulated and solved as finite-dimensional linear matrix inequality (LMI) optimization problems. Our approach is less conservative than a single quadratic Lyapunov function method. Specifically, the proposed output feedback control law asymptotically stabilizes the open loop system with a larger domain of attraction and achieves better disturbance attenuation under energy and magnitude bounded disturbances.

On switching [formula omitted] controllers for nuclear steam generator water level: A multiple parameter-dependent Lyapunov functions approach

Poor control of the steam generator (SG) water level in the secondary circuit of a nuclear power plant can lead to frequent reactor shutdowns. From the viewpoint of economic operation, it is critical to keep the water level of SG in well control. This paper presents a new framework for addressing this problem based on switching control to linear parameter varying (LPV) systems. A family of LPV controllers are designed, and each of them is suitable for a specific operating range. Under the admissible switching logic, the closed-loop system remains stable and achieves prescribed H ∞ performance in the entire operating range. The approach is based on multiple parameter-dependent Lyapunov functions, which is less conservative than the employment of a single Lyapunov function. The switching LPV control scheme is then applied to the level control problem.