Linear Matrix Inequality Relaxations Research Articles

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.

Stability Analysis and Performance Design for Fuzzy-Model-Based Control System Under Imperfect Premise Matching

This paper investigates the stability analysis and performance design of nonlinear systems. To facilitate the stability analysis, the Takagi-Sugeno (T-S) fuzzy model is employed to represent the nonlinear plant. Under the imperfect premise matching in which T-S fuzzy model and fuzzy controller do not share the same membership functions, a fuzzy controller with enhanced design flexibility and robustness property is proposed to control the nonlinear plant. However, the nice characteristic given by the perfect premise matching, leading to conservative stability conditions, vanishes. In this paper, under the imperfect premise matching, information of membership functions of the fuzzy model and controller are considered in stability analysis. With the introduction of slack matrices, relaxed linear matrix inequality (LMI)-based stability conditions are derived using Lyapunov-based approach. Furthermore, LMI-based performance conditions are provided to guarantee system performance. Simulation examples are given to illustrate the effectiveness of the proposed approach.

Convergent LMI Relaxations for Quadratic Stabilizability and ${{\mathscr H}}_{\infty}$ Control of Takagi–Sugeno Fuzzy Systems

This paper investigates the quadratic stabilizability of Takagi-Sugeno (T-S) fuzzy systems by means of parallel distributed state feedback compensators. Using Finsler's lemma, a new design condition assuring the existence of such a controller is formulated as a parameter-dependent linear matrix inequality (LMI) with extra matrix variables and parameters in the unit simplex. Algebraic properties of the system parameters and recent results of positive polynomials are used to construct LMI relaxations that, differently from most relaxations in the literature, provide certificates of convergence to solve the control design problem. Due to the degrees of freedom obtained with the extra variables, the conditions presented in this paper are an improvement over earlier results based only on Polya's theorem and can be viewed as an alternative to the use of techniques based on the relaxation of quadratic forms. An extension to cope with guaranteed H infin attenuation levels is also given, with proof of asymptotic convergence to the global optimal controller under quadratic stability. The efficiency of the proposed approach in terms of precision and computational effort is demonstrated by means of numerical comparisons with other methods from the literature.

Delay-Dependent Robust $\hbox{\it H}_\infty$ Control for Uncertain Discrete-Time Fuzzy Systems With Time-Varying Delays

This paper deals with the robust H infin control problem for discrete-time Takagi-Sugeno (T-S) fuzzy systems with norm-bounded parametric uncertainties and interval time-varying delays. First, based on a new Lyapunov functional, we present a sufficient condition guaranteeing that the resulting closed-loop system is robustly stable and satisfies a prescribed H infin performance level. The Lyapunov functional used here depends on not only the fuzzy basis function but on the lower and upper bounds of the time-varying delay as well. Second, two classes of delay-dependent conditions for the existence of the concerned H infin fuzzy controllers are given in terms of relaxed linear matrix inequalities (LMIs), and a desired controller can be designed by using the solutions to these LMIs. Finally, a numerical example is provided to demonstrate the effectiveness of the proposed design method.

Relaxed Stability and Performance LMI Conditions for Takagi--Sugeno Fuzzy Systems With Polynomial Constraints on Membership Function Shapes

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> Most linear matrix inequality (LMI) fuzzy control results in literature are valid for any membership function, i.e., independent of the actual membership shape. Hence, they are conservative (with respect to other nonlinear control approaches) when specific knowledge of the shapes is available. This paper presents relaxed LMI conditions for fuzzy control that incorporate such shape information in the form of polynomial constraints, generalizing previous works by the authors. Interesting particular cases are overlap (product) bounds and ellipsoidal regions. Numerical examples illustrate the achieved improvements, as well as the possibilities of solving some multiobjective problems. The results also apply to polynomial-in-membership Takagi--Sugeno fuzzy systems. </para>

Analog circuit design by nonconvex polynomial optimization: Two design examples

AbstractWe present a framework for synthesizing low‐power analog circuits through global optimization over generally nonconvex multivariate polynomial objective function and constraints. Specifically, a nonconvex optimization problem is formed, which is then efficiently solved through convex programming techniques based on linear matrix inequality (LMI) relaxation. The framework allows both polynomial inequality and equality constraints, thereby facilitating more accurate device modelings and parameter tuning. Compared to traditional nonlinear programming (NLP), the proposed methodology exhibits superior computational efficiency, and guarantees convergence to a globally optimal solution. As in other physical design tasks, circuit knowledge and insight are critical for initial problem formulation, while the nonconvex optimization machinery provides a versatile tool and systematic way to locate the optimal parameters meeting design specifications. Two circuit design examples are given, namely, a nested transconductance(Gm)–capacitance compensation (NGCC) amplifier and a delta–sigma (ΔΣ) analog‐to‐digital converter (ADC), both of them being the key components in many electronic systems. Copyright © 2008 John Wiley &amp; Sons, Ltd.

Convergent LMI relaxations for robust analysis of uncertain linear systems using lifted polynomial parameter-dependent Lyapunov functions

This paper investigates the problems of checking robust stability and evaluating robust ℋ 2 performance of uncertain continuous-time linear systems with time-invariant parameters lying in polytopic domains. The novelty is the ability to check robust stability by constructing a particular parameter-dependent Lyapunov function which is a polynomial function of the uncertain system matrices, as opposed to a general polynomial function of the uncertain parameter. The degree of the polynomial is tied to a certain integer κ . The existence of such Lyapunov function can be proved by solving parameter-dependent Linear Matrix Inequalities (LMIs), which are guaranteed to be solvable for a sufficiently large yet finite value of κ whenever the system is robustly stable. Extensions to guaranteed ℋ 2 cost computation are also provided. Numerical aspects concerning the programming and the evaluations of the proposed tests are discussed and illustrated by examples.

Simulation of discrete linear time-invariant fuzzy dynamic systems

In this paper, a new approach to simulation of discrete linear time-invariant dynamic systems with parameters and initial conditions described by fuzzy numbers is proposed. This approach solves the problem of fuzzy simulation by determining at each iteration a nested family of intervals, corresponding to different levels of certainty ( α -cuts), that enclose the possible system states using optimisation. The originality of this approach consists in considering uncertain parameters as time-invariant during the simulation process. The optimisation problems associated with the approach proposed must be solved globally in order to guarantee that the minimum volume box enclosing the region of system states for each level of certainty has been derived. Then, a global optimisation algorithm, based on the theory of moments and linear matrix inequality relaxations suitable for the type of problems (polynomial programs) that appear in the fuzzy simulation algorithm, is used. Finally, two real engineering application examples allow assessing the performance of the approach proposed.

A Descriptor System Approach to Fuzzy Control System Design via Fuzzy Lyapunov Functions

There has been a flurry of research activities in the analysis and design of fuzzy control systems based on linear matrix inequalities (LMIs). This paper presents a descriptor system approach to fuzzy control system design using fuzzy Lyapunov functions. The design conditions are still cast in terms of LMIs but the proposed approach takes advantage of the redundancy of descriptor systems to reduce the number of LMI conditions which leads to less computational requirement. To obtain relaxed LMI conditions, new types of fuzzy controller and fuzzy Lyapunov function are proposed. A salient feature of the LMI conditions derived in this paper is to relate the feasibility of the LMIs to the switching speed of each linear subsystem (to be exact, to the lower bounds of time derivatives of membership functions). To illustrate the validity and applicability of the proposed approach, two design examples are provided. The first example shows that the LMI conditions based on the fuzzy Lyapunov function are less conservative than those based on a common (standard) Lyapunov function. The second example illustrates the utility of the fuzzy Lyapunov function approach in comparison with a piecewise Lyapunov function approach.

Delay-dependent stabilization for stochastic fuzzy systems with time delays

This paper is concerned with the delay-dependent stabilization problem for a class of time-delay stochastic fuzzy systems. The time delays are assumed to appear in both the state and the control input. The purpose is the design of a state-feedback fuzzy controller such that the resulting closed-loop system is asymptotically stable in the mean square. A delay-dependent condition for the solvability of this problem is obtained in terms of relaxed linear matrix inequalities (LMIs). By solving these LMIs, a desired controller can be obtained. Finally, a numerical example is given to demonstrate the effectiveness of the present results.

LMI relaxation to Riccati equations in structured control

In this paper we discuss structured control methods for large-scale interconnected systems. Based on a relaxation of Riccati equations, we derive some linear matrix inequality (LMI) conditions for sub-optimal controllers in which information structure can be imposed. In particular, we derive controllers by solving low-dimensional LMIs, which are decentralized except for the sharing information between neighbours, as determined by the plant interconnection; also we optimize a performance bound for each of the derived controllers.

Gain bandwidth optimization in two-pump fiber optical parametric amplifiers under bounded zero-dispersion wavelength fluctuations

We study gain bandwidth optimization in a two-pump fiber optical parametric amplifier (2P-OPA) with bounded zero-dispersion wavelength (ZDW) uncertainty. Maximum-bandwidth 2P-OPAs are designed ensuring positive parametric gain, tunable gain spectrum, and robustness against ZDW fluctuations. By recognizing the polynomial nature of the phase mismatch, the design task is formulated as a nonconvex multivariate polynomial optimization problem, which is then solved through the latest convex programming techniques based on linear matrix inequality relaxations. The proposed framework exhibits superior computational efficiency and guarantees, up to modeling accuracy, globally optimal wavelength assignment. It provides an analytical ground to assist experimental tunings in practice, and allows visualization of tradeoff between gain bandwidth and spectrum quality against ZDW uncertainty.

RELAXED STABILITY CONDITIONS FOR DELAYED RECURRENT NEURAL NETWORKS WITH POLYTOPIC UNCERTAINTIES

This paper investigates the problem of stability analysis for recurrent neural networks with time-varying delays and polytopic uncertainties. Parameter-dependent Lypaunov functionals are employed to obtain sufficient conditions that guarantee the robust global exponential stability of the equilibrium point of the considered neural network. The derived stability criteria are expressed in terms of a set of relaxed linear matrix inequalities, which can be easily tested by using commercially available software. Two numerical examples are provided to demonstrate the effectiveness of the proposed results.

Convergent Relaxations of Polynomial Matrix Inequalities and Static Output Feedback

Using a moment interpretation of recent results on sum-of-squares decompositions of nonnegative polynomial matrices, we propose a hierarchy of convex linear matrix inequality (LMI) relaxations to solve nonconvex polynomial matrix inequality (PMI) optimization problems, including bilinear matrix inequality (BMI) problems. This hierarchy of LMI relaxations generates a monotone sequence of lower bounds that converges to the global optimum. Results from the theory of moments are used to detect whether the global optimum is reached at a given LMI relaxation, and if so, to extract global minimizers that satisfy the PMI. The approach is successfully applied to PMIs arising from static output feedback design problems.

Robust H/sub /spl infin// control for fuzzy systems with Frobenius norm-bounded uncertainties

In this paper, we investigate the problem of robust H/sub /spl infin// performance and stabilization for a class of uncertain fuzzy systems with Frobenius norm-bounded parameter uncertainties in all system matrices. Both continuous- and discrete-time uncertain fuzzy systems are considered under a unified treatment called bounded real lemma for fuzzy systems. Unlike the bounded real lemma in the linear theory of robust H/sub /spl infin// control where necessary and sufficient conditions were obtained, only sufficient condition based on Lyapunov method is shown. Furthermore, connection between robust H/sub /spl infin// problems involving uncertainty and standard uncertainty-free H/sub /spl infin// problems is established via matrix algebra. As for controller synthesis, a state feedback fuzzy control law is designed via relaxed linear matrix inequality (LMI) formulations.

STOCHASTIC STABILITY OF A CLASS OF STOCHASTIC BILINEAR HYBRID SYSTEMS: CONVEX ANALYSIS AND SYNTHESIS

This paper deals with the problem of local exponential stochastic stabilization of continuous time Bilinear Active Fault Tolerant Control Systems with Markovian Parameters (BAFTCSMP). The design technique is based on a differential inclusion of the bilinear term for a restricted domain of the state space. The above problematic is addressed under a convex programming approach. Indeed, conditions for local stochastic stability of the BAFTCSMP are derived and formulated through some matrix inequalities. The implicit problem being to maximize the region in which the closed-loop stability can be ensured, some convex optimization problems with LMI (linear matrix inequalities) relaxations schemes are stated. A numerical example is presented to illustrate the theoretical results.

Matrix Sum-of-Squares Relaxations for Robust Semi-Definite Programs

We consider robust semi-definite programs which depend polynomially or rationally on some uncertain parameter that is only known to be contained in a set with a polynomial matrix inequality description. On the basis of matrix sum-of-squares decompositions, we suggest a systematic procedure to construct a family of linear matrix inequality relaxations for computing upper bounds on the optimal value of the corresponding robust counterpart. With a novel matrix-version of Putinar's sum-of-squares representation for positive polynomials on compact semi-algebraic sets, we prove asymptotic exactness of the relaxation family under a suitable constraint qualification. If the uncertainty region is a compact polytope, we provide a new duality proof for the validity of Putinar's constraint qualification with an a priori degree bound on the polynomial certificates. Finally, we point out the consequences of our results for constructing relaxations based on the so-called full-block S-procedure, which allows to apply recently developed tests in order to computationally verify the exactness of possibly small-sized relaxations.

SUM OF SQUARES RELAXATIONS FOR ROBUST POLYNOMIAL SEMI-DEFINITE PROGRAMS

A whole variety of robust analysis and synthesis problems can be formulated as robust Semi-Definite Programs (SDPs), i.e. SDPs with data matrices that are functions of an uncertain parameter which is only known to be contained in some set. We consider uncertainty sets described by general polynomial semi-definite constraints, which allows to represent norm-bounded and structured uncertainties as encountered in μ-analysis, polytopes and various other possibly non-convex compact uncertainty sets. As the main novel result we present a family of Linear Matrix Inequalities (LMI) relaxations based on sum-of-squares (sos) decompositions of polynomial matrices whose optimal values converge to the optimal value of the robust SDP. The number of variables and constraints in the LMI relaxations grow only quadratically in the dimension of the underlying data matrices. We demonstrate the benefit of this a priori complexity bound by an example and apply the method in order to asses the stability of a fourth order LPV model of the longitudinal dynamics of a helicopter.

Relaxations for Robust Linear Matrix Inequality Problems with Verifications for Exactness

Robust semidefinite programming problems with rational dependence on uncertainties are known to have a wide range of applications, in particular in robust control. It is well established how to systematically construct relaxations on the basis of the full block S-procedure. In general, such relaxations are expected to be conservative, but for concrete problem instances they are often observed to be tight. The main purpose of this paper is to suggest novel computationally verifiable conditions for when general classes of linear matrix inequality relaxations do not involve any conservatism. If the convex set of uncertainties is finitely generated, we suggest a novel sequence of relaxations which can be proved to be asymptotically exact. Finally, our results are applied to the particularly relevant robustness analysis problem for linear time-invariant dynamical systems affected by uncertainties that are full ellipsoidal or repeated and contained in intersections of disks or circles. This leads to extensions of known results on relaxation exactness for small block structures, including an elementary proof for tightness of standard structured singular value computations for three full complex uncertainty blocks.

Solving nonconvex optimization problems

It is well known that most analysis and design problems in robust and nonlinear control can be formulated as global optimization problems with polynomial objective functions and constraints. The objective of this article is to show how GloptiPoly can solve challenging nonconvex optimization problems in robust nonlinear control. GloptiPoly is a general-purpose software with a user-friendly interface. First, we describe several nonconvex optimization problems arising in control system analysis and design. These problems involve multivariable polynomial objective functions and constraints. We then review the theoretical background behind GloptiPoly. Next, an example is presented to illustrate the successive linear matrix inequality relaxations and general features of the GloptiPoly software. Finally, we apply GloptiPoly to several control-related problems.