A new Lagrangian dual global optimization algorithm for solving bilinear matrix inequalities

A new global optimization algorithm for solving bilinear matrix inequalities (BMI) problems is developed. It is based on a dual Lagrange formulation for computing lower bounds that are used in a branching procedure to eliminate partition sets in the space of complicating variables. The advantage of the proposed method is twofold. First, the lower bound computations reduce to solving easily tractable linear matrix inequality (LMI) problems. Secondly, the lower bounding procedure guarantees global convergence of the algorithm when combined with an exhaustive partitioning of the space of complicating variables. A rigorous proof of this fact is provided. Another important feature is that the branching phase takes place in the space of complicating variables only, hence limiting the overall cost of the algorithm. Also, an important point in the method is that separated LMI constraints are encapsulated into an augmented BMI for improving the lower bound computations. Applications of the algorithm to robust structure/controller design are considered. Copyright © 2000 John Wiley & Sons, Ltd.     

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.     

Multiobjective control via successive over‐bounding of quadratic terms

This paper addresses less conservative control design for multiple design specifications. Problems are described by a set of linear matrix inequalities (LMIs) and solved with non‐common LMI solutions in order to reduce the conservatism arising from seeking a common LMI solution in the past results. Noticing that completing the square can split two variables in bilinear matrix inequality (BMI) terms into two different LMI ones, we propose an iterative algorithm in which non‐positive quadratic terms are successively replaced by their upper bounds. An illustrated example is included. Copyright © 2005 John Wiley & Sons, Ltd.     

Robust control via concave minimization local and global algorithms

This paper is concerned with the robust control problem of linear fractional representation (LFT) uncertain systems depending on a time-varying parameter uncertainty. Our main result exploits a linear matrix inequality (LMI) characterization involving scalings and Lyapunov variables subject to an additional essentially nonconvex algebraic constraint. The nonconvexity enters the problem in the form of a rank deficiency condition or matrix inverse relation on the scalings only. It is shown that such problems, but also more generally rank inequalities and bilinear constraints, can be formulated as the minimization of a concave functional subject to LMI constraints. First of all, a local Frank and Wolfe (1956) feasible direction algorithm is introduced in this context to tackle this hard optimization problem. Exploiting the attractive concavity structure of the problem, several efficient global concave programming methods are then introduced and combined with the local feasible direction method to secure and certify global optimality of the solutions. Computational experiments indicate the viability of our algorithms, and in the worst case, they require the solution of a few LMI programs     

Control design in the time and frequency domain using nonsmooth techniques

Significant progress in control design has been achieved by the use of nonsmooth and semi-infinite mathematical programming techniques. In contrast with LMI or BMI approaches, these new methods avoid the use of Lyapunov variables, which gives them two major strategic advances over matrix inequality methods. Due to the much smaller number of decision variables, they do not suffer from size restrictions, and they are much easier to adapt to structural constraints on the controller. In this paper, we further develop this line and address both frequency- and time-domain design specifications by means of a nonsmooth algorithm general enough to handle both cases. Numerical experiments are presented for reliable or fault-tolerant control, and for time response shaping.     

Fixed‐order H∞ control design via a partially augmented Lagrangian method

In this paper we develop an augmented Lagrangian method to determine local optimal solutions of the reduced‐ and fixed‐order H_∞ synthesis problems. We cast these synthesis problems as optimization programs with a linear cost subject to linear matrix inequality (LMI) constraints along with nonlinear equality constraints representing a matrix inversion condition. The special feature of our algorithm is that only equality constraints are included in the augmented Lagrangian, while LMI constraints are kept explicitly in order to exploit currently available semi definite programming (SDP) codes. The step computation in the tangent problem is based on a Gauss–Newton model, and a specific line search and a first‐order Lagrange multiplier update rule are used to enhance efficiency. A number of computational results are reported and underline the strong practical performance of the algorithm. Copyright © 2003 John Wiley & Sons, Ltd.     

Optimization of static output feedback using substitutive LMI formulation

This note proposes a new design tool for optimizing static output feedback using a linear matrix inequality (LMI) formula called substitutive LMI. A matrix inequality derived from static output feedback is not usually linear. Adding a positive definite term including auxiliary variables, the matrix inequality is transformed into an LMI with respect to the positive definite matrix and the static output feedback gain. An iterative calculation algorithm is given to solve the substitutive LMI. In this note, designs of the static output feedback gain are shown in the frame of H/sub /spl infin// and H/sub 2/ syntheses. A numerical example is shown to demonstrate the effectiveness of the proposed technique.     

线性系统静态输出反馈镇定的LMI方法

利用无约束条件的线性矩阵不等式（LMI）研究了W－问题和P－问题,而后者的解可用来设计静态输出反馈（SOF）镇定控制,作为一个应用考虑了不确定系统的静态输出反馈问题,给出了依赖于LMI条件的SOF设计方法。     

Global optimization for robust control synthesis based on the Matrix Product Eigenvalue Problem

In this paper, we propose a new formulation for a class of optimization problems which occur in general robust control synthesis, called the Matrix Product Eigenvalue Problem (MPEP): Minimize the maximum eigenvalue of the product of two block‐diagonal positive‐definite symmetric matrices under convex constraints. This optimization class falls between methods of guaranteed low complexity such as the linear matrix inequality (LMI) optimization and methods known to be NP‐hard such as the bilinear matrix inequality (BMI) formulation, while still addressing most robust control synthesis problems involving BMIs encountered in applications. The objective of this paper is to provide an algorithm to find a global solution within any specified tolerance ε for the MPEP. We show that a finite number of LMI problems suffice to find the global solution and analyse its computational complexity in terms of the iteration number. We prove that the worst‐case iteration number grows no faster than a polynomial of the inverse of the tolerance given a fixed size of the block‐diagonal matrices in the eigenvalue condition. Copyright 2001 © John Wiley & Sons, Ltd.     

Static output feedback control for continuous-time T-S fuzzy systems: An LMI approach

This paper presents a design method of static output feedback control for continuous-time T-S fuzzy systems. Based on parallel distributed compensation (PDC), a static output feedback control is utilized. A new sufficient condition for the existence of static output feedback gains is represented in terms of linear matrix inequalities (LMIs). The sufficient condition does not need any transformation matrices, equality constraints, and block diagonal assumption of positive definite matrices in order to convert a bilinear matrix inequality (BMI) problem to an LMI one.     

含仿射不确定参数的多项式非线性系统控制器设计

针对含仿射时变不确定参数的多项式非线性系统，提出了基于多项式分解的控制方法. 多项式分解方法主要思想是将多项式系统转化成带自由变量的系数矩阵，从而偶次多项式的非负性验证问题可转化成线性矩阵不等式或双线性矩阵不等式求解问题. 文中多项式系统控制器综合基于 Lyapunov 稳定定理. 构造 Lyapunov 函数以及寻找反馈控制器可由所给的算法通过计算机程序自动完成. 对于多维系统相对高阶的控制器，由多项式全基构造的控制器将有很多项单项式. 为克服这一问题，文中算法给出含最少单项式的简约型控制器设计方法，并提出针对最小代价性能目标优化的增益受约次优控制. 数值仿真例子表明，文中所给的控制方法取得良好性能.     

含仿射不确定参数的多项式非线性系统控制器设计

针对含仿射时变不确定参数的多项式非线性系统，提出了基于多项式分解的控制方法. 多项式分解方法主要思想是将多项式系统转化成带自由变量的系数矩阵，从而偶次多项式的非负性验证问题可转化成线性矩阵不等式或双线性矩阵不等式求解问题. 文中多项式系统控制器综合基于 Lyapunov 稳定定理. 构造 Lyapunov 函数以及寻找反馈控制器可由所给的算法通过计算机程序自动完成. 对于多维系统相对高阶的控制器，由多项式全基构造的控制器将有很多项单项式. 为克服这一问题，文中算法给出含最少单项式的简约型控制器设计方法，并提出针对最小代价性能目标优化的增益受约次优控制. 数值仿真例子表明，文中所给的控制方法取得良好性能.     

General quadratic performance analysis and synthesis of differential algebraic equation (DAE) systems

In this paper, control of linear differential-algebraic-equation systems, subject to general quadratic constraints, is considered. This setup, especially, includes the H_∞ control problem and the design for strict passivity. Based on linear matrix inequality (LMI) analysis conditions, LMI synthesis conditions for the existence of linear output feedback controllers are derived by means of a linearizing change of variables. This approach is constructive: a procedure for the determination of controller parameterizations is given on the basis of the solution of the LMI synthesis conditions. A discussion of the possible applications of the presented results concludes the paper.     

一类非线性广义马尔可夫跳变系统的耗散控制

将耗散理论的二次型供给率中的矩阵Q推广到正定的情况.进而研究了在状态转移概率未知的情况下一类连续时间非线性广义马尔可夫跳变系统的严格耗散控制问题.在应用范围更广的Willems耗散性定义的基础上,首先基于一类Lyapunov函数,给出了相应的随机容许的条件,然后设计导数比例反馈控制器,通过一系列的矩阵构造和合同变换,将双线性矩阵不等式(BMI)转化为可用LMI工具箱解决的线性矩阵不等式(LMI).最后通过数值算例并结合Matlab给出实例,证明其可行性.     

An augmented Lagrangian method for a class of LMI-constrained problems in robust control theory

We present a new approach to a class of non-convex LMI-constrained problems in robust control theory. The problems we consider may be recast as the minimization of a linear objective subject to linear matrix inequality (LMI) constraints in tandem with non-convex constraints related to rank deficiency conditions. We solve these problems using an extension of the augmented Lagrangian technique. The Lagrangian function combines a multiplier term and a penalty term governing the non-convex constraints. The LMI constraints, due to their special structure, are retained explicitly and not included in the Lagrangian. Global and fast local convergence of our approach is then obtained either by an LMI-constrained Newton type method including line search or by a trust-region strategy. The method is conveniently implemented with available semi-definite programming (SDP) interior-point solvers. We compare its performance to the wellknown D - K iteration scheme in robust control. Two test problems are investigated and demonstrate the power and efficiency of our approach.     

Output feedback model predictive control for LPV systems based on quasi-min–max algorithm

This paper proposes a robust output feedback model predictive control (MPC) scheme for linear parameter varying (LPV) systems based on a quasi-min–max algorithm. This approach involves an off-line design of a robust state observer for LPV systems using linear matrix inequality (LMI) and an on-line robust output feedback MPC algorithm using the estimated state. The proposed MPC method for LPV systems is applicable for a variety of systems with constraints and guarantees the robust stability of the output feedback systems. A numerical example for an LPV system subject to input constraints is given to demonstrate its effectiveness.     

Cooperative control of discrete-time linear multi-agent systems with fixed information structure

In this paper,a cooperative control problem was investigated for discrete-time linear multi-agent systems with fixed information structure and without communication delays.Based on the bilinear matrix inequality(BMI),the sufficient condition was obtained for the stabilization of multi-agent systems composed of N agents.Then,the design problems of cooperative controllers were converted into the optimization problems with BMI constraints.To solve these problems,an optimization algorithm was proposed.Finally,numerical examples were provided to demonstrate the reduced conservatism of the proposed condition.     

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

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.      

Saturated linear quadratic regulation of uncertain linear systems: Stability region estimation and controller design

This paper considers the problems of estimating the stability region (domain of attraction) and controller design for uncertain linear continuous-time systems with input saturation when linear quadratic (LQ) optimal controller is used. By exploiting the structure of the LQ controller and the property of saturation functions, it is established that the estimation of stability region can be obtained by solving linear matrix inequality (LMI) problems. Moreover, an iterative LMI (ILMI) algorithm is presented to design an LQ controller such that the largest estimated stability region can be obtained. Two examples are given to compare our results with existing ones.     

分数阶系统线性矩阵不等式稳定判据的改进与鲁棒镇定: 0<α<1的情况

针对确定的分数阶系统,通过发掘其稳定区域与线性矩阵不等式(LMI)之间的联系,提出改进的LMI稳定判据.与已有结果相比,该判据形式简洁,求解变量少,且不含复矩阵变量,能够直接应用于控制器设计而不带来任何保守性,克服了原判据存在的主要缺点.然后基于该判据,进一步研究了当分数阶系统具有多胞型不确定时的鲁棒镇定控制器的设计方法.最后,仿真结果表明该方法的有效性.