Semidefinite programming duality and linear time-invariant systems

Several important problems in control theory can be reformulated as semidefinite programming problems, i.e., minimization of a linear objective subject to linear matrix inequality (LMI) constraints. From convex optimization duality theory, conditions for infeasibility of the LMIs, as well as dual optimization problems, can be formulated. These can in turn be reinterpreted in control or system theoretic terms, often yielding new results or new proofs for existing results from control theory. We explore such connections for a few problems associated with linear time-invariant systems.     

《自动控制原理(多学时)》课程中线性系统校正部分教学研究

对《自动控制原理(多学时)》课程中线性系统校正部分教学进行了探索研究。列举了线性系统校正部分教学中存在的一些问题,并分析了产生问题的原因,提出了线性系统校正部分教学的一些方法。     

The time-invariant linear-quadratic optimal control problem

This paper provides a review of one of the basic problems of systems theory—the general time-invariant optimal control problem involving linear systems and quadratic costs. The problem includes on one hand the regulator problem of optimal control and on the other, the theory of linear dissipative systems, itself central to network theory and to the stability theory of feedback systems. The theory is developed using simple properties of dynamical systems and involves a minimum of ‘hard’ analysis or algebra. It includes a full existence theory of the matrix quadratic equation, of interest in its own right.     

Controller design with multiple objectives

In this paper we study multi-objective control problems that give rise to equivalent convex optimization problems. We develop a uniform treatment of such problems by showing their equivalence to linear programming problems with equality constraints and an appropriate positive cone. We present some specialized results on duality theory, and we apply them to the study of three multi-objective control problems: the optimal l₁ control with time-domain constraints on the response to some fixed input, the mixed H₂/l₁ -control problem, and the l₁ control with magnitude constraint on the frequency response. What makes these problems complicated is that they are often equivalent to infinite-dimensional optimization problems. The characterization of the duality relationship between the primal and dual problem allows us to derive several results. These results establish connections with special convex problems (linear programming or linear matrix inequality problems), uncover finite-dimensional structures in the optimal solution, when possible, and provide finite-dimensional approximations to any degree of accuracy when the problem does not appear to have a finite-dimensional structure. To illustrate the theory and highlight its potential, several numerical examples are presented     

The infimum principle

This paper reports a reasonably complete theory of necessary and sufficient conditions for a control to be superior with respect to a nonscalar-valued performance criterion. The latter maps into a finite-dimensional integrally closed directed partially ordered linear space. The applicability of the theory to the analysis of dynamic vector estimation problems and to a class of uncertain optimal control problems is demonstrated.     

Uniform approach for solving some classical problems on a lineararray

It is shown that a number of classical problems from linear algebra and graph theory, including instances of the algebraic path problem, matrix multiplication, matrix triangularization, and matrix transpose, can be solved using the same basic recurrence. A simple mapping of the recurrence onto a unidirectional linear array is discussed. Qualitative advantages to programming linear arrays using this approach include uniformity of design, simplicity of programming, and scalability to larger problems. The major disadvantage is that the resulting algorithms are not necessarily optimal     

On the generalized Sylvester mapping and matrix equations

General parametric solution to a family of generalized Sylvester matrix equations arising in linear system theory is presented by using the so-called generalized Sylvester mapping which has some elegant properties. The solution consists of some polynomial matrices satisfying certain conditions and a parametric matrix representing the degree of freedom in the solution. The results provide great convenience to the computation and analysis of the solutions to this family of equations, and can perform important functions in many analysis and design problems in linear system theory. It is also expected that this so-called generalized Sylvester mapping tool may have some other applications in control system theory.     

A new gradient-based neural network for solving linear andquadratic programming problems

A new gradient-based neural network is constructed on the basis of the duality theory, optimization theory, convex analysis theory, Lyapunov stability theory, and LaSalle invariance principle to solve linear and quadratic programming problems. In particular, a new function F(x, y) is introduced into the energy function E(x, y) such that the function E(x, y) is convex and differentiable, and the resulting network is more efficient. This network involves all the relevant necessary and sufficient optimality conditions for convex quadratic programming problems. For linear programming and quadratic programming (QP) problems with unique and infinite number of solutions, we have proven strictly that for any initial point, every trajectory of the neural network converges to an optimal solution of the QP and its dual problem. The proposed network is different from the existing networks which use the penalty method or Lagrange method, and the inequality constraints are properly handled. The simulation results show that the proposed neural network is feasible and efficient.     

Distributed Multi-Parametric Quadratic Programming

One of the fundamental problems in the area of large-scale optimization is to study locality features of spatially distributed optimization problems in which the variables are coupled in the cost function as well as constraints. Such problems can motivate the development of fast and well-conditioned distributed algorithms. In this paper, we study spatial locality features of large-scale multi-parametric quadratic programming (MPQP) problems with linear inequality constraints. Our main application focus is receding horizon control of spatially distributed linear systems with input and state constraints. We propose a new approach for analysis of large-scale MPQP problems by blending tools from duality theory with operator theory. The class of spatially decaying matrices is introduced to capture couplings between optimization variables in the cost function and the constraints. We show that the optimal solution of a convex MPQP is piecewise affine- represented as convolution sums. More importantly, we prove that the kernel of each convolution sum decays in the spatial domain at a rate proportional to the inverse of the corresponding coupling function of the optimization problem.     

Computational structural mechanics. Optimal control and semi-analytical method for PDE

Based on the generalized variational principle the analysis of a substructural chain is considered, and the 1:1 relationship between the structural analysis problem and the linear quadratic optimal control problem is then introduced. Hence, the algebraic Riccati equation can be solved in two ways; the upper-bound and lower-bound iterative methods. The theory and methods of structural analysis problems can then be transferred to the linear quadratic optimal control problems. As to the continuous coordinate, and/or continuous-time problems, it can be shown that the linear quadratic control problem also corresponds to the semi-analytical method of the elliptic partial differential equation. It is hoped that the unified method of these disciplines will lead to further progress.     

Idempotent Interval Analysis and Optimization Problems

Many problems in optimization theory are strongly nonlinear in the traditional sense but possess a hidden linear structure over suitable idempotent semirings. After an overview of "Idempotent Mathematics" with an emphasis on matrix theory, interval analysis over idempotent semirings is developed. The theory is applied to construction of exact interval solutions to the interval discrete stationary Bellman equation. Solution of an interval system is typically NP-hard in the traditional interval linear algebra; in the idempotent case it is polynomial. A generalization to the case of positive semirings is outlined. This revised version was published online in August 2006 with corrections to the Cover Date.     

基于LMI的高超鲁棒控制及仿真

研究了高超声速飞行器鲁棒飞行控制器设计以及飞行控制系统的仿真验证问题,通过选择适当的加权函数矩阵,确定广义受控对象.线性矩阵不等式(LMI)技术是控制领域中研究问题的有效工具,控制器的分析与综合等问题可转化为LMI问题的求解,采用基于线性矩阵不等式的H∞控制器设计方法,设计了鲁棒控制器.仿真结果表明所设计的飞行控制系统具有鲁棒性,能有效地抗飞行过程中存在各种各样的干扰及摄动,很好地满足了飞行控制系统性能指标.     

连续混沌系统的混沌同步控制

给出了一种实现混沌系统混沌同步的控制方法．通过引入一待定的控制项,将两系统的混沌同步问题转化为讨论与其对应的线性系统的0解渐近稳定性问题,然后根据线性系统控制理论确定此控制项,以实现两混沌系统的同步目的．该方法简单易行,可有效的实现两个混沌系统的混沌同步,且其同步是全局渐近稳定的．     

Proper invariant realizations of singular system problems

The notion of proper invariant realization of a triple for a specially defined linear space is introduced and its properties and construction are investigated. This notion of realization allows the reduction of the study of dynamic characterization of subspaces of singular systems to standard problems of linear systems theory. The proof of the results is given     

Confidence regions of two-stage stochastic linear complementarity problems

Two-stage stochastic linear complementarity problems (TSLCP) model a large class of equilibrium problems subject to data uncertainty, and are closely related to two-stage stochastic optimization problems. The sample average approximation (SAA) method is one of the basic approaches for solving TSLCP and the consistency of the SAA solutions has been well studied. This paper focuses on building confidence regions of the solution to TSLCP when SAA is implemented. We first establish the error-bound condition of TSLCP and then build the asymptotic and nonasymptotic confidence regions of the solutions to TSLCP by error-bound approach, which is to combine the error-bound condition with central limit theory, empirical likelihood theory, and large deviation theory.     

Continuous-time and discrete-time design of water flow and water level regulators

A thorough analysis of the water flow and water level control problems arising in several industrial systems is provided using classical and sampled data linear control theory. The analysis is based on simplified linear models of the processes under control. Resulting design rules enable engineers to choose the correct controller both in the continuous-time and in the discrete-time case. A practical application example is included.     

On linear systems and noncommutative rings

This paper studies some problems appearing in the extension of the theory of linear dynamical systems to the case in which parameters are taken from noncommutative rings. Purely algebraic statements of some of the problems are also obtained.Through systems defined by operator rings, the theory of linear systems over rings may be applied to other areas of automata and control theory; several such applications are outlined.This research was supported in part by US Army Research Grant DA-ARO-D-31-124-72-G114 and by US Air Force Grant 72-2268 through the Center for Mathematical System Theory, University of Florida, Gainesville, FL 32611, USA.