Finite-dimensional Linear Matrix Inequalities Research Articles

An oracle for the discrete-time integral quadratic constraint problem

In the theory of integral quadratic constraints (IQCs), the condition for robust performance is given in terms of a frequency-domain inequality. This inequality can be transformed into an equivalent finite-dimensional linear matrix inequality by the application of the Kalman–Yakubovich–Popov (KYP) lemma and the introduction of an auxiliary matrix variable. Thus, the IQC problem can be expressed as a semi-definite program (SDP) and solved using convex optimization tools such as general-purpose SDP solvers based on interior point methods. However, by introducing the auxiliary variable, the size of the IQC problem significantly increases, and the aforementioned solvers may fail due to the high computational cost of the SDP. To address this issue, specialized SDP solvers have been developed to efficiently solve the SDP by exploiting its structure. An alternative approach consists of using cutting plane methods to directly solve the frequency-domain inequality without resorting to the KYP lemma. Cutting plane methods require an oracle that confirms if a given candidate solution satisfies the frequency-domain inequality. If the candidate solution is not feasible, the oracle generates a hyperplane that separates the candidate solution from the set of feasible solutions. In this paper, the analytic center cutting plane method is applied to the discrete-time IQC problem. The oracle for the discrete-time IQC problem is based on the spectrum of an extended symplectic matrix pencil and does not require an invertibility assumption that holds in continuous-time but not necessarily in discrete-time. An illustrative example is provided that shows the efficacy of the proposed method and points out directions for future research.

Non-algorithmic Robust Static Output Feedback Control Designs for Parametric Uncertain Systems

This paper proposes novel static output feedback (SOF) control designs for the continuous-time systems under parametric uncertainty and linear time-invariant systems. The convex solutions based on the finite-dimensional linear matrix inequality are improved without any algorithmic approaches, thanks to the slack variable such that the solutions are achieved by isolation Lyapunov matrix from the system and controller matrix. In addition, the method is extended to H2 SOF control design for the systems. Moreover, less conservative results are provided via the isolation in terms of the robust stability against the uncertainties. Finally, several numerical examples are presented to demonstrate the applicability of the proposed controller design.

Gain-scheduled static output feedback controller synthesis for discrete-time LPV systems

ABSTRACTThis study is concerned with the problem of gain-scheduled static output feedback H∞ controller synthesis for discrete-time linear parameter varying systems. The scheduling parameters, which may affect all the system matrices, and their deviations are supposed to lie in a known hyper-rectangle. A design procedure is presented in terms of sequentially solving three parameter-dependent linear matrix inequality (LMI) optimisation problems. The parameter-dependent LMIs are solved through finite-dimensional LMI relaxations exploiting homogeneous polynomial matrices of arbitrary degree. One of the advantages of the new method lies in its less conservatism which is theoretically shown in comparison with some available approaches. By an iterative scheme, the performance of the method can be improved. Numerical examples reveal the effectiveness of the new method.

LPV control of a tensegrity-membrane system

This article presents the details of designing a linear parameter-varying (LPV) controller for a tensegrity-membrane system. The major control objective is to deploy the system from its initial packaged configuration to its final deployed configuration. Since LPV synthesis naturally leads to an infinite-dimensional linear matrix inequality (LMI) problem, a gridding method is used to reduce it to a finite-dimensional LMI problem and the control design is performed using a control-oriented model based on the linearized system dynamics at a set of system equilibriums. To test the performance of the resulting controller, a nonlinear finite element model and the control-oriented model are used in numerical simulations. To test the robustness of the controller, sensor noise is considered in another nonlinear finite element simulation and the results are compared with the simulation results given by the nominal system.

Stability and Disturbance Attenuation for Markov Jump Linear Systems with Time-Varying Transition Probabilities

We address stability and disturbance attenuation for Markov jump linear systems where transition probabilities vary in a finite set. The time variation of the transition probabilities may be a priori known or unknown. Necessary and sufficient conditions for uniform stochastic stability and uniform stochastic disturbance attenuation are reported. In both cases, conditions are expressed as a set of finite-dimensional linear matrix inequalities that can be solved efficiently.

A General Integral Quadratic Constraints Theorem with Applications to a Class of Sampled-Data Systems

In the framework of integral quadratic constraints a more general stability theorem is derived by relaxing the assumptions on the uncertainty. In contrast to existing results, input-output relations are only required to hold on a subspace of the full signal space. This result is employed to incorporate uncertainties exhibiting sampling behavior into the framework of integral quadratic constraints. The arising infinite dimensional stability test is exactly reduced to a finite dimensional linear matrix inequality. Advantages of this approach are illustrated for the specific case of an interconnection including a pulse-width modulator.

Convex Computation of the Region of Attraction of Polynomial Control Systems

We address the long-standing problem of computing the region of attraction (ROA) of a target set (e.g., a neighborhood of an equilibrium point) of a controlled nonlinear system with polynomial dynamics and semialgebraic state and input constraints. We show that the ROA can be computed by solving an infinite-dimensional convex linear programming (LP) problem over the space of measures. In turn, this problem can be solved approximately via a classical converging hierarchy of convex finite-dimensional linear matrix inequalities (LMIs). Our approach is genuinely primal in the sense that convexity of the problem of computing the ROA is an outcome of optimizing directly over system trajectories. The dual infinite-dimensional LP on nonnegative continuous functions (approximated by polynomial sum-of-squares) allows us to generate a hierarchy of semialgebraic outer approximations of the ROA at the price of solving a sequence of LMI problems with asymptotically vanishing conservatism. This sharply contrasts with the existing literature which follows an exclusively dual Lyapunov approach yielding either nonconvex bilinear matrix inequalities or conservative LMI conditions. The approach is simple and readily applicable as the outer approximations are the outcome of a single semidefinite program with no additional data required besides the problem description. The approach is demonstrated on several numerical examples.

Sampling method for semidefinite programmes with non-negative Popov function constraints

An important class of optimisation problems in control and signal processing involves the constraint that a Popov function is non-negative on the unit circle or the imaginary axis. Such a constraint is convex in the coefficients of the Popov function. It can be converted to a finite-dimensional linear matrix inequality via the Kalman–Yakubovich–Popov lemma. However, the linear matrix inequality reformulation requires an auxiliary matrix variable and often results in a very large semidefinite programming problem. Several recently published methods exploit problem structure in these semidefinite programmes to alleviate the computational cost associated with the large matrix variable. These algorithms are capable of solving much larger problems than general-purpose semidefinite programming packages. In this paper, we address the same problem by presenting an alternative to the linear matrix inequality formulation of the non-negative Popov function constraint. We sample the constraint to obtain an equivalent set of inequalities of low dimension, thus avoiding the large matrix variable in the linear matrix inequality formulation. Moreover, the resulting semidefinite programme has constraints with low-rank structure, which allows the problems to be solved efficiently by existing semidefinite programming packages. The sampling formulation is obtained by first expressing the Popov function inequality as a sum-of-squares condition imposed on a polynomial matrix and then converting the constraint into an equivalent finite set of interpolation constraints. A complexity analysis and numerical examples are provided to demonstrate the performance improvement over existing techniques.

Convex Computation of the Maximum Controlled Invariant Set For Polynomial Control Systems

We characterize the maximum controlled invariant (MCI) set for discrete- as well as continuous-time nonlinear dynamical systems as the solution of an infinite-dimensional linear programming problem. For systems with polynomial dynamics and compact semialgebraic state and control constraints, we describe a hierarchy of finite-dimensional linear matrix inequality (LMI) relaxations whose optimal values converge to the volume of the MCI set; dual to these LMI relaxations are sum-of-squares (SOS) problems providing a converging sequence of outer approximations to the MCI set. The approach is simple and readily applicable in the sense that the approximations are the outcome of a single semidefinite program with no additional input apart from the problem description. A number of numerical examples illustrate the approach.

Necessary and sufficient condition for robust stability of discrete-time descriptor polytopic systems

This study investigates the robust stability of discrete-time descriptor polytopic systems. Necessary and sufficient conditions for robust stability expressed as parameterised linear matrix inequalities (LMIs) are obtained. Furthermore, the parameterised LMIs are reduced to parameter-independent and finite-dimensional LMIs. Finally, two numerical examples are given, illustrating the effectiveness of the proposed method.

Linear matrix inequality tests for frequency domain inequalities with affine multipliers

This paper revisits an alternative formulation of the Kalman–Yakubovich–Popov (KYP) Lemma, relating an infinite dimensional Frequency Domain Inequality (FDI) to a pair of finite dimensional Linear Matrix Inequalities (LMI). It is shown that this alternative formulation allows a certain class of the coefficient matrix of the FDI to be frequency dependent without introducing conservatism. The construction provided in the present paper is helpful in other problems where system augmentation is commonly used, such as those involving rational or polynomial multipliers for stability and performance analysis.

An alternative Kalman–Yakubovich–Popov lemma and some extensions

This paper introduces an alternative formulation of the Kalman–Yakubovich–Popov (KYP) Lemma, relating an infinite dimensional Frequency Domain Inequality (FDI) to a pair of finite dimensional Linear Matrix Inequalities (LMI). It is shown that this new formulation encompasses previous generalizations of the KYP Lemma which hold in the case the coefficient matrix of the FDI does not depend on frequency. In addition, it allows the coefficient matrix of the frequency domain inequality to vary affinely with the frequency parameter. One application of this results is illustrated in an example of computing upper bounds to the structured singular value with frequency-dependent scalings.

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.

Optimal Output Regulation for Discrete-Time Switched and Markovian Jump Linear Systems

First, three different but related output regulation performance criteria for the linear time-varying system are defined in the discrete-time domain, namely, the peak impulse response, peak output variance, and average output variance per unit time. Then they are extended for switched linear systems and Markovian jump linear systems, and characterized by an increasing union of finite-dimensional linear matrix inequality conditions. Finally, the infinite-horizon suboptimal LQG control problem, which aims to maintain the average output variance below a given level subject to the uniform exponential stability of the closed-loop system, is solved for both switched linear systems and Markovian jump linear systems; the solution is given by a dynamic linear output feedback controller that not only perfectly observes the present mode but also recalls a finite number of past modes.

Dynamic output feedback control of quasi-LPV mechanical systems

A methodology for controlling nonlinear mechanical systems expressed in non-singular parametric descriptor quasi-linear parameter varying form is developed. For mechanical systems, this form is naturally obtained through the Lagrangian formulation. Use of this form avoids the explicit inversion of the mass matrix, thus eliminating the additional resulting linear fractional parameter dependence complexity. Parametric Lyapunov functions are used for the design. Sufficient conditions for the stability and dynamic output feedback control with guaranteed induced ${\cal L}_2$-norms from the disturbance to the controlled output are given. The conditions for the reduction of the solvability conditions to finite-dimensional linear matrix inequalities are provided. The results are shown on a two-link flexible manipulator example.

Absolute Stability via Boundary Control of a Semilinear Parabolic PDE

We consider the problem of achieving global absolute stability of an unstable equilibrium solution of a semilinear dissipative parabolic partial differential equation (PDE) through boundary control. The state space of the system is extended in order to write the action of the boundary control as an unbounded operator in an abstract evolution equation. Absolute stability via boundary control is accomplished by analyzing a control Lyapunov function based on the infinite-dimensional dynamics and applying a finite-dimensional linear quadratic regulator (LQR) controller. Sufficient conditions for absolute stability of the infinite-dimensional system are established by the feasibility of two finite-dimensional linear matrix inequalities (LMIs). Numerical results are presented for a Dirichlet boundary controlled system, however the analysis in this work applies to Nuemann and Robin type boundary controllers as well.

H2 gain scheduled state feedback for LPV system with new LMI formulation

The H2 gain scheduled state feedback problem is considered by using the parameter-dependent Lyapunov function for linear parameter varying system, in which the dependent parameter is assumed to be measurable on-line in a polytope space. First, a new linear matrix inequality (LMI) representation provides a type of H2 cost computation for this system. With the introduction of the extra variable, this condition provides a decoupling between Lyapunov function matrix and system matrices and will be useful for a synthesis problem study. Secondly, using a polytopic characteristic of the dependent parameter, based on this representation gain scheduled state, the feedback synthesis problem can be transformed into finite-dimensional LMIs formulation. In addition, by using the parameter-dependent Lyapunov function, it can reduce the conservatism that occurred before in the H2 performance synthesis problem with a fixed Lyapunov function. Finally, a numerical example is included to illustrate the efficiency of the proposals.

Distributed Control Design for Systems Interconnected Over an Arbitrary Graph

We consider the problem of synthesizing a distributed dynamic output feedback controller achieving H/sub /spl infin// performance for a system composed of different interconnected sub-units, when the topology of the underlying graph is arbitrary. First, using tools inspired by dissipativity theory, we derive sufficient conditions in the form of finite-dimensional linear matrix inequalities when the interconnections are assumed to be ideal. These inequalities are coupled in a way that reflects the spatial structure of the problem and can be exploited to design distributed synthesis algorithms. We then investigate the case of lossy interconnection links and derive similar results for systems whose interconnection relations can be captured by a class of integral quadratic constraints that includes constant delays.

Exponential stability assignment of neutral delay-differential systems by a class of linear–quadratic regulators

A method to construct a simple feedback law for neutral delay-differential systems is proposed. The feedback gain is determined with a solution of a matrix inequality, which can be solved numerically via the technique of solving finite-dimensional linear matrix inequalities. It is shown that the resulting closed-loop system is asymptotically stable, and if some transformation of variables is introduced, it is assured to have a pre-assigned degree of exponential stability. Moreover, it belongs to a class of linear–quadratic regulators. The design procedure is demonstrated and the effectiveness of the feedback is evaluated with a numerical simulation.

Insensitivity of a class of LQ regulators of neutral systems

It is well known that the linear quadratic regulators of finite dimensional linear systems have the insensitivity property. In this paper, as a particular class of infinite dimensional systems, a neutral delay-differential system is considered as a plant. A class of linear quadratic regulators is constructed for the plant using a simple feedback law, which does not require real-time either integral or derivative operations. The feedback gain is calculated by solving a finite dimensional linear matrix inequality. First, it is shown that the regulator satisfies the circle condition. Then, its sensitivity against the parameter variations is evaluated. In the single-input case, it is done by calculating the absolute value of the sensitivity function. In the multi-input case, it is done by the so-called “comparison sensitivity” method where the sensitivities of the closed-loop system and the open-loop system are compared based on certain sensitivity indices. Procedures used for the investigation of the regulator's properties are based on natural extension of methods proposed by Perkins and Cruz for the finite dimensional case. As a result, it is found that the class of linear quadratic regulators of neutral systems, similarly to finite dimensional systems, has the insensitivity property. © 2004 Wiley Periodicals, Inc. Electr Eng Jpn, 150(2): 28–34, 2005; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/eej.20013