Output Feedback H ∞ Control for Uncertain Piecewise Linear Systems

This paper proposes the output feedback optimal guaranteed cost controller design method for uncertain piecewise linear systems based on the piecewise quadratic Lyapunov functions technique. By constructing piecewise quadratic Lyapunov functions for the closed‐loop augmented systems, the existence of the guaranteed cost controller for closed‐loop uncertain piecewise linear systems is cast as the feasibility of a set of bilinear matrix inequalities (BMIs). Some of the variables in BMIs are set to be searched by genetic algorithm (GA), then for a given chromosome corresponding to the variables in BMIs, the BMIs turn to be linear matrix inequalities (LMIs), and the corresponding non‐convex optimization problem, which minimizes the upper bound on cost function, reduces to a semidefinite programming (SDP) which is convex and can be solved numerically efficiently with the available software. Thus, the output feedback optimal guaranteed cost controller can be obtained by solving the non‐convex optimization problem using the mixed algorithm that combines GA and SDP. Numerical examples show the effectiveness of the proposed method. Copyright © 2008 John Wiley & Sons, Ltd.

International audience

In this paper, we develop a systematic Lyapunov approach to the regional stability and performance analysis of saturated systems in a general feedback configuration. The only assumptions we make about the system are well-posedness of the algebraic loop and local stability. Problems to be considered include the estimation of the domain of attraction, the reachable set under a class of bounded energy disturbances and the nonlinear L2 gain. The regional analysis is established through an effective treatment of the algebraic loop and the saturation/deadzone function. This treatment yields two forms of differential inclusions, a polytopic differential inclusion (PDI) and a norm-bounded differential inclusion (NDI) that contain the original system. Adjustable parameters are incorporated into the differential inclusions to reflect the regional property. The main idea behind the regional analysis is to ensure that the state remain inside the level set of a certain Lyapunov function where the PDI or the NDI is valid. With quadratic Lyapunov functions, conditions for stability and performances are derived as linear matrix inequalities (LMIs). To obtain less conservative conditions, we use a pair of conjugate non-quadratic Lyapunov functions, the convex hull quadratic function and the max quadratic function. These functions yield bilinear matrix inequalities (BMIs) as conditions for stability and guaranteed performance level. The BMI conditions cover the corresponding LMI conditions as special cases, hence the BMI results are guaranteed to be as good as the LMI results. In most examples, the BMI results are significantly better than the LMI results

This paper presents a nonlinear control design method for robust stabilization and robust performance of linear differential inclusions. A recently introduced non-quadratic Lyapunov function, the convex hull quadratic function, is used for the construction of nonlinear state feedback laws. Design objectives include stabilization with maximal convergence rate, disturbance rejection with minimal reachable set and least /spl Lscr//sub 2/ gain. Conditions for stabilization and performances are derived in terms of bilinear matrix inequalities (BMIs), which cover the existing linear matrix inequality (LMI) conditions as special cases. Optimization problems with BMI constraints are formulated and solved effectively by combining the path-following algorithm and the direct iterative algorithm. The design results are guaranteed to be at least as good as the existing results obtained from LMI conditions. In most examples, significant improvements on system performances have been achieved, which demonstrate the advantages of using nonlinear feedback control over linear feedback control for linear differential inclusions. It is also observed through numerical computation that nonlinear control strategies may help to reduce control effort substantially.

Positive real control for 2-D discrete delayed systems via output feedback controllers

This paper deals with the problem of static output feedback (SOF) control design for a class of diffusion partial differential equation (PDE) and ordinary differential equation (ODE) cascades, where the ODE model is used to describe the dynamics of the multi-input and multi-output (MIMO) plant and the diffusion PDE model is employed to represent the dynamics of actuators. The objective of this paper is to develop a simple as well as effective SOF controller via the Lyapunov's direct method such that the resulting closed-loop system is globally exponentially stable. By constructing a quadratic Lyapunov function, the sufficient condition on the globally exponential stability of the closed-loop cascaded system is presented in terms of linear matrix inequality (LMI). Then, an LMI-based design method of the SOF controller is developed on the basis of the obtained stability analysis result. Finally, two numerical examples are provided to illustrate the effectiveness of the proposed design method.

In this paper, the output feedback controller is designed for the 2D discrete (T-S) fuzzy systems expressed by Fornasini-Marchesini II Model. Our aim is to develop static output feedback controller (SOF) so that the addressed closed loop system globally asymptotically stable. By using the fuzzy-basis-dependent Lyapunov function only, sufficient conditions are given in form of linear matrix inequalities (LMIs) to guarantee the 2D closed loop system to be globally asymptotically stable. The main advantage is design a SOF controller is that no constraints are imposed on system matrices. Illustrative example is given to show the potential and usefulness of the proposed results.

Considers decentralized H/sub /spl infin// controller design problems for multi-channel linear time-invariant systems. Our interest is focused on dynamic output feedback, and our discussion concerns both continuous-time systems and discrete-time ones. The decentralized H/sub /spl infin// control problems are expressed as feasibility problems of bilinear matrix inequalities (BMIs) with respect to variables of a coefficient matrix defining the controller and a Lyapunov matrix. To deal with the BMIs, we propose to set the Lyapunov matrix as block diagonal appropriately according to the desired order of the controller so that the BMIs; are reduced to linear matrix inequalities (LMIs) which are sufficient to the BMIs but much more tractable.

International audience

This paper investigates the problems of finite-time stability and finite-time stabilization for nonlinear quadratic systems with jumps. The jump time sequences here are assumed to satisfy some given constraints. Based on Lyapunov function and a particular presentation of the quadratic terms, sufficient conditions for finite-time stability and finite-time stabilization are developed to a set containing bilinear matrix inequalities (BLIMs) and linear matrix inequalities (LMIs). Numerical examples are given to illustrate the effectiveness of the proposed methodology.

This paper addresses stability analysis and stabilization for Takagi-Sugeno (T-S) fuzzy systems with parametric uncertainties via a so-called fuzzy Lyapunov function which is a multiple Lyapunov function. The fuzzy Lyapunov function is defined by fuzzily blending quadratic Lyapunov functions. First, the Takagi-Sugeno (T-S) fuzzy model with parametric uncertainties is used as the model for the uncertain nonlinear system. Based on the fuzzy Lyapunov function approach and a parallel distributed compensation (PDC) scheme, we give stabilization conditions for closed-loop fuzzy systems with parametric uncertainties. Second, all the conditions are formulated in the format of linear matrix inequalities (LMIs) and contain upper bounds of the time derivative of premise membership functions as LMI variables. Finally, the T-S fuzzy model of the chaotic Lorenz system, which has complex nonlinearity, is developed as a test bed. A numerical example of the chaotic Lorenz system is given to illustrate the utility of the fuzzy Lyapunov function approach.

In this paper, the problem of finite-time stability is studied for a class of networked predictive control systems with parameter uncertainties. Firstly, both the modelling uncertainties and the network-induced delays are considered. A networked predictive control scheme is proposed to actively compensate the network-induced delays. Secondly, a sufficient condition is given to ensure the finite-time stability of the augmented closed-loop system with time delay, where both bilinear matrix inequality and linear matrix inequality conditions are derived with the help of Lyapunov stability theorem. At the same time, an alternate algorithm is designed to obtain the gain matrices of observer and controller from bilinear matrix inequality and linear matrix inequality. Finally, a numerical example is given to confirm the feasibility of the proposed results.

T–S fuzzy-model-based piecewise [formula omitted] output feedback controller design for networked nonlinear systems with medium access constraint

This article concerns the finite-time robust guaranteed cost control problem for a class of linear continuous-time singular systems with norm-bounded uncertainties. In this study, the problem is to design a state feedback controller such that the closed-loop system is finite-time stable, and the closed-loop cost function value is not more than a specified upper bound for all admissible uncertainties. By constructing an appropriate Lyapunov function, a sufficient condition for the finite-time robust stability of the system based on linear matrix inequality (LMI) is established. Furthermore, the sufficient condition for the existence of the guaranteed cost controller is formulated in terms of LMIs, which can make the closed-loop uncertain singular system finite-time robust stable. Finally, two numerical examples are given for illustration of the proposed theoretical results.

This paper presents a systematic procedure to design a decentralized wide area damping controller for the wind farms and Flexible AC Transmission System (FACTS) controllers in a smart grid considering uncertainties in the load parameters. The effect of uncertainties in load parameters on damping of inter-area modes of oscillations is investigated through the probability collocation method. A robust H1 output feedback controller is, then, designed for the uncertain fuzzy system by satisfying Linear Matrix Inequalities (LMIs), which are obtained from the original Bilinear Matrix Inequalities (BMIs), derived from the Lyapunov based stability criterion. The input and output signals for the proposed controller are selected based on a joint observability/controllability index. The input signals are wide area signals obtained from the Phasor Measurement Units (PMUs) and the output signals are fed to wind farm generators and the FACTS controllers. The input signal delay compensation is addressed in this work. The performance of the controller has been tested on New England 39 bus system developed on Real Time Digital Simulator (RTDS) and on a practical large Indian power system, under various operating scenarios and faults. Its effectiveness is also compared with a conventional power system stabilizer, local damping controllers, and an existing robust centralized wide area controller designed based on mixed H2/ H1 output feedback synthesis.