An LMI solution to the robust synthesis problem for multi-rate sampled-data systems

In this paper we address the asynchronous multi-rate sampled-data H_∞ synthesis problem. Necessary and sufficient conditions are given for the existence of a controller achieving the desired performance, and the problem is shown to be equivalent to a convex optimization problem expressed in the form of linear operator inequalities. In the case where the sample and hold rates are synchronous, these operator inequalities reduce to linear matrix inequalities, for which standard numerical software is available.     

Non-fragile robust finite-time H ∞ control for nonlinear stochastic it? systems using neural network

This paper deals with the problem of non-fragile robust finite-time H _?? control for a class of uncertain nonlinear stochastic It? systems via neural network. First, applying multi-layer feedback neural networks, the nonlinearity is approximated by linear differential inclusion (LDI) under statespace representation. Then, a sufficient condition is proposed for the existence of non-fragile state feedback finite-time H _?? controller in terms of matrix inequalities. Furthermore, the problem of nonfragile robust finite-time H _?? control is reduced to the optimization problem involving linear matrix inequalities (LMIs), and the detailed solving algorithm is given for the restricted LMIs. Finally, an example is given to illustrate the effectiveness of the proposed method.     

H∞ controller synthesis for pendulum-like systems

This paper focus on a stabilization problem for a class of nonlinear systems with periodic nonlinearities, called pendulum-like systems. A notion of Lagrange stabilizability is introduced, which extends the concept of Lagrange stability to the case of controller synthesis. Based on this concept, we address the problem of designing a linear dynamic output controller which stabilizes (in the Lagrange sense) a pendulum-like system within the framework of the H_∞ control theory. Lagrange stabilizability conditions for uncertainty-free systems and systems with norm-bounded uncertainty in the linear part are derived, respectively. When these conditions are satisfied, the desired stabilization output feedback controller can be constructed via feasible solutions of a certain set of linear matrix inequalities (LMIs).     

Inverse system design for LPV systems using parameter-dependent Lyapunov functions

This paper addresses the design problem of gain-scheduled inverse systems (GSISs) for linear parameter-varying (LPV) systems, whose state-space matrices are represented as parametrically affine matrices, using parameter-dependent Lyapunov functions (PDLFs), and proposes a method for them via parametrically affine linear matrix inequalities (LMIs). Our method includes robust inverse system (RIS) design as a special case. For RIS design, our method theoretically encompasses the method using constant Lyapunov functions. A design example is included to illustrate our conclusions.     

AN ITERATIVE LMI APPROACH TO RFDF FOR LINEAR SYSTEM WITH TIME‐VARYING DELAYS

This paper deals with robust fault detection filter (RFDF) problem for a class of linear uncertain systems with time‐varying delays and model uncertainties. The RFDF design problem is formulated as an optimization problem by using L₂‐induced norm to represent the robustness of residual to unknown inputs and modelling errors, and the sensitivity to faults. A sufficient condition to the solvability of formulated problem is established in terms of certain matrix inequalities, which can be solved with the aid of an iterative linear matrix inequality (ILMI) algorithm. Finally, a numerical example is given to illustrate the effectiveness of the proposed method.     

A Semi‐Tensor Product Approach to Pseudo‐Boolean Functions with Application to Boolean Control Networks

Using the semi‐tensor product method, this paper investigates several fundamental problems of general pseudo‐Boolean functions with application to the optimal control of Boolean control networks, and establishes a new framework to deal with pseudo‐Boolean inequalities, the optimization problem and the best linear approximation of pseudo‐Boolean functions. First, the pseudo‐Boolean function is expressed in the algebraic form via constructing its unique structural matrix. Second, based on the matrix expression, solving pseudo‐Boolean inequalities is converted into finding solutions to algebraic inequalities, and a set of new formulas are presented. Third, the optimization problem and the linear approximation of the pseudo‐Boolean function are considered and several new results are established. Finally, as an application, we investigate the optimal control of Boolean control networks, and present a new optimal control design procedure. It is shown through the study of illustrative examples that the new results proposed in this paper work very well.     

一类线性离散时滞大系统的分散镇定

用一组线性矩阵不等式给出一类线性离散时滞大系统分散能镇定的一个充分条件,进而,通过建立和求解一个凸优化问题,提出了具有较反馈增益参数的分散稳定化状态反馈控制律的设计方法,所得到的控制器不仅使得闭环境系统是稳定的,而且还可以使得闭环系统状态具有给定的衰减度。     

Gain‐Scheduled Control via Switching of LTI Controllers and State Reset

This paper proposes a synthesis method of gain‐scheduled control systems that switch linear time‐invariant controllers according to hysteresis of the scheduling parameter. Stability and L₂‐gain analysis and synthesis methods for switched systems are applied to the switched gain‐scheduled control synthesis using reset of the controller state, where also the reset law is computed via linear matrix inequalities (LMIs). In addition to optimization of an upper bound of L₂‐gain, we reduce jumps of control input via an auxiliary optimization. Numerical examples are presented to illustrate the switched gain‐scheduled controller.     

Observer-based <Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript> guaranteed cost control for uncertain singular time-delay systems with input saturation

In this paper, the problem of observer-based H _∞ guaranteed cost control for uncertain singular timedelay systems with actuator saturation is concerned. A delay-dependent sufficient condition is proposed, which guarantees that the closed-loop system is admissible via Lyapunov theory and linear matrix inequality (LMI) approach. Then, with this condition, the estimation of stability region, the upper bound of cost function and the design method of observer-based H _∞ guaranteed cost controller are given by solving linear matrix inequalities and convex optimization problems. Finally, numerical examples are provided to demonstrate the effectiveness of the proposed method.     

Ellipsoidal sets for resilient and robust static output-feedback

The implementation of a controller, if not exact, may lead to the so-called fragility problem, i.e., the loss of expected closed-loop properties. In the present note, this difficult problem is dealt with considering robust static-output feedback (SOF) control for uncertain linear time-invariant systems. By analogy with robust analysis theory based on quadratic separation, a new formulation for the SOF design is shown to be a valuable way to tackle fragility issues. Indeed, the use of a quadratic separator for design purpose allows to define a whole resilient (nonfragile) set of SOF control laws. Results are formulated as matrix inequalities one of which is nonlinear. A numerical algorithm based on nonconvex optimization is provided ant its running is illustrated on classical examples from literature.     

Offline Robust Model Predictive Control for Lipschitz Non‐Linear Systems Using Polyhedral Invariant Sets

In this paper the concept of maximal admissible set (MAS) for linear systems with polytopic uncertainty is extended to non‐linear systems composed of a linear constant part followed by a non‐linear term. We characterize the maximal admissible set for the non‐linear system with unstructured uncertainty in the form of polyhedral invariant sets. A computationally efficient state‐feedback RMPC law is derived off‐line for Lipschitz non‐linear systems. The state‐feedback control law is calculated by solving a convex optimization problem within the framework of linear matrix inequalities (LMIs), which leads to guaranteeing closed‐loop robust stability. Most of the computational burdens are moved off‐line. A linear optimization problem is performed to characterize the maximal admissible set, and it is shown that an ellipsoidal invariant set is only an approximation of the true stabilizable region. This method not only remarkably extends the size of the admissible set of initial conditions but also greatly reduces the on‐line computational time. The usefulness and effectiveness of the method proposed here is verified via two simulation examples.     

Using the method of invariant ellipsoids for linear robust output stabilization of spacecraft

Consideration was given to the problem of robust output stabilization of the nonlinear system using the method of invariant ellipsoids on assumption that the system satisfies a Lipschitz-like condition and the system output is subjected to bounded external disturbances. It was reduced to a special optimization problem with solution based on the theory of linear matrix inequalities. The proposed approach was illustrated by an example of stabilization of a spacecraft with elastic dynamic elements. The designed control represents a combination of the “accelerate-decelerate“ algorithm and a linear feedback obtained using the method of invariant ellipsoids.     

Synthesis of robust discrete systems based on comparison with stochastic model

The problem of synthesis of the stabilizing control for a discrete linear system with uncertain parameters is considered. The stochastic model of comparison with parametric noise is constructed; the mean-square stability of this model implies the stability of the system with uncertain parameters varying in a given domain. For the case of the state feedback, the methods of synthesis of the robust stabilizing control based on the solution of either linear matrix inequalities or an auxiliary optimization problem with constraints in the form of linear matrix inequalities are proposed. The converging iterative algorithm of synthesis of the robust stabilizing control for the case of the output feedback is proposed. An example is given.     

一类离散非线性不确定互联系统的模糊分散控制

利用模糊控制方法研究一类离散非线性互联系统的分散控制问题.首先采用模糊(T-S)模型对离散非线性不确定互联系统进行模糊建模,应用并行分布补偿算法(PDC)给出状态反馈分散模糊控制方案,并基于李亚普诺夫函数方法证明了闭环系统的稳定性.然后当系统的状态不完全可测时,设计模糊分散观测器来估计各子系统的状态,从而给出基于观测器的状态反馈分散模糊控制设计的方法.因为该分散模糊控制设计问题是以线性矩阵不等式的形式给出,所以很容易用凸优化方法求解.仿真结果验证了所提出控制方法的有效性.     

Guaranteed cost control for saturated uncertain linear systems

In this paper, we discuss the problem of guaranteed cost control for uncertain linear systems subject to actuator saturation. Based on the quadratic and non-quadratic Lyapunov functions, sufficient conditions for the robust stability and performance are derived. Moreover, all the conditions can be expressed as linear matrix inequalities （LMIs） or bilinear matrix inequalities （BMIs） in terms of the feedback gain. Thus, the static controller can be effectively synthesized via convex optimization. A numerical example illustrates the effectiveness of the method.     

H∞ approach to monotonically convergent ILC for uncertain time-varying delay systems

This paper deals with iterative learning control (ILC) design for uncertain time-delay systems. Monotonic convergence of the resulting ILC process is studied, and a sufficient condition within an H_∞-based framework is developed. It is shown that under this framework, delay-dependent conditions can be obtained in terms of linear matrix inequalities (LMIs), together with formulas for gain matrices design. A numerical example is provided to illustrate the effectiveness of the robust H_∞-based approach to ILC designed via LMIs.     

Robust stabilization of the Takagi-Sugeno fuzzy model via bilinearmatrix inequalities

Quadratic stability has enabled, mainly via the linear matrix inequality framework, the analysis and design of a nonlinear control system from the local matrices of the system's Takagi-Sugeno (T-S) fuzzy model. It is well known, however, that there exist stable differential inclusions, hence T-S fuzzy models whose stability is unprovable by a globally quadratic Lyapunov function. At present, literature in the broader area of stability analysis suggests piecewise-quadratic stability as a means to avoid such conservatism. This paper generalizes the idea and proposes a framework that supports less conservative sufficient conditions for the stability of the T-S model by using piecewise-quadratic generalized Lyapunov functions. The advocated approach results in the formulation of the controller synthesis, which, herein, aims for robust stabilization, as a problem of bilinear rather than linear matrix inequalities. Simulation studies, which include an algorithm for solution of bilinear matrix inequalities, demonstrate the proposed method     

Almost Asymptotic Regulation of Markovian Jumping Linear Systems in Discrete Time

This paper considers the problems of almost asymptotic output regulation for discrete‐time Markovian jumping linear systems. Based on a stochastic Lyapunov‐Krasovskii functional framework, sufficient conditions for the extension of the regulation scheme to such stochastic systems are obtained via state feedback and via error feedback. Relying on a characterization of the feedback controllers, the almost asymptotic regulation is accomplished. The problem of guaranteeing stochastic stability and almost asymptotic tracking is achieved by solving linear matrix inequalities subject to a set of linear equality constraints. In order to ensure relaxed solutions of the regulation equations, a semi‐definite optimization approach via disciplined convex programming is outlined. Simulation results also are given to illustrate the effectiveness of the proposed design approach.     

A convex optimization approach to the mode acceleration problem

The purpose of this note is to introduce an alternative procedure to the mode acceleration method when the underlying structure model includes damping. We will show that the problem can be cast as a convex optimization problem that can be solved via linear matrix inequalities.