On the inverse optimization problem for periodic systems

Self Cite

Quadratic optimization algorithms for discrete-time linear periodic systems are discussed. Consideration is given to the conventional linear-quadratic problem statement where the optimal controller is found by solving the discrete algebraic Riccati equation and to input-feedback controller design problems. Considerable attention is focused on the synthesis of a reliable controller with a guaranteed stability margin. The following inverse problem is solved: given plant and controller matrices, find the weight matrices of the functional being optimized. The efficiency of the algorithms is demonstrated by way of examples Keywords: periodic Riccati equation, output feedback, stabilization problem, inverse problem 1. Introduction. Optimization problems for linear periodic systems have a wide range of applications, including the synthesis of stabilization systems for walking and hopping robots [4,[37][38][39][40]42]. This is why these problems are still of interest [14, 22-25, 30, 36, 56-58, etc.]. To solve such problems, use is made of various approaches that somewhat generalize the approaches to the analysis of stationary systems to periodic systems. Noteworthy are the algorithms involving the search for the stabilizing solution of the Riccati equation [1,5,6,18,52] and the algorithms based on linear matrix inequalities (LMIs) [41, 44, 47, etc.]. Attempts are usually made to generalize a synthesis algorithm for a stationary system to a periodic system.We will briefly discuss the optimization algorithms for periodic systems developed at the Institute of Mathematics (Ukraine), Institute of Mechanics (Ukraine), and the Institute of Applied Mathematics of the Baku State University (Azerbaijan). Some of these results were obtained within the framework of Project INTAS 04-77-6902.The present paper is organized as follows. In Sec. 2, the quadratically optimal controller for a discrete periodic system and a controller for a hybrid periodic system (described by both differential and finite-difference equations) are synthesized by finding the stabilizing solution of the matrix discrete algebraic Riccati equation (DARE) [5] (see [56] for a comparison of the algorithms [5] with similar algorithms). Sections 3-5 detail algorithms for solving DARE. It is shown [53] how the SDA-1 algorithm [59] can be obtained from [5]. Section 6 outlines algorithms for solving the discrete-time periodic Riccati equation (DPRE) and demonstrates how one of these algorithms can be implemented based on MATLAB arbitrary-precision arithmetic. Sections 7-9 address algorithms for the output feedback stabilization of a periodic system. Sections 10-15 discuss the synthesis of a reliable controller with guaranteed stability margin. Section 16 addresses the inverse periodic-optimization problem: given the matrices of the plant and controller, find the matrices of the functional being optimized. The algorithms are illustrated by examples. Note that some MATLAB programs implementing the algorithms discussed here are available at the Internet address indicated ...

Section: Example Of a Reliable Controller With A Given Stability Margmentioning

confidence: 99%

Optimization problems for periodic systems

Aliev

Larin

2009

Self Cite

“…For fixed C and µ −1 , arbitrary time intervals T h , h ≥ 0, are specified as in [21][22][23][24], where T T 0 = for h = 0. Definition 1.…”

Section: Introductionmentioning

confidence: 99%

“…x on a given finite time interval T I ⊂ if the following condition is satisfied along the perturbed solutions x t ( )of the Cauchy problem (12)-(18), (22) for the measure ρ( ) x associated with the initial process:…”

Section: Introductionmentioning

confidence: 99%

“…where the value ρ( ) x 0 of the measure ρ is defined on the initial values (22), and the preset constant γ = > const 0 and bounded function η( ) t satisfy conditions (21) on the interval T I ⊂ . …”

Section: Introductionmentioning

confidence: 99%

“…The problem to be formulated below will be solved using the comparison and direct Lyapunov methods and is independent of the sliding-mode conditions on the switching hyperplane in the phase space. The present study is based on the results reported in [14,[21][22][23][24][25].1. Problem Formulation.…”

mentioning

confidence: 99%

See 2 more Smart Citations

Technical stability of stationary automatic control systems of variable structure with switched filters

Matviichuk

2006

Sufficient conditions for the technical stability in measure of a nonstationary control system with variable structure are established. The controller of the system has feedback-switched filters functioning together with shaper and actuator. It is assumed that the nonstationary parameters of the system vary within given ranges, at a finite rate, with appropriate control laws, with adjustment against mismatch signal, its derivatives of finite order, and all variable parameters of the filter. The parameters of the switching hyperplane remain constant. This approach for analysis of technical stability does not involve sliding mode conditions. Criteria of technical instability in measure for the control system under consideration are formulated using the properties of systems of comparison from below. The general criteria of technical stability and instability are applied to nonstationary filtered-control systems of variable structure of the third order. The comparison method based on normalized Lyapunov functions is used Keywords: technical stability and instability, nonstationary control systems with variable structure, switched filters, normalized Lyapunov functions, comparison method Introduction. Among various control systems that allow optimal control, of widespread use are nonstationary automatic-control systems of variable structure with a control filter subjected to internal feedback and functioning together with a shaper and an actuator [1, 3, 8, etc.]. Control actions are discontinuous functions of the phase coordinates and external actions [4,17,19]. Filters introduced into the control loop increase the amount of information needed to design controllers [2, 21, 22, etc.]. Such variable-structure systems with filters make it possible to gain the maximum effect, including optimality, from the use of automatics [9, 11, 12, etc.]. Under certain conditions, these systems may give rise to a specific motion, so-called sliding mode [8,9,19].The present study is concerned with the technical stability and instability in measure of nonstationary variable-structure automatic-control systems with filters in the control loop. The problem to be formulated below will be solved using the comparison and direct Lyapunov methods and is independent of the sliding-mode conditions on the switching hyperplane in the phase space. The present study is based on the results reported in [14,[21][22][23][24][25].1. Problem Formulation. Let us consider a nonstationary automatic-control system of variable structure that includes a feedback-switched filter to provide high-quality control of real processes [1, 9, etc.]. Let the following differential equation of the nth order describe the operation of the basic part of the system-the actuator and the control plant, which have time-dependent parameters [8,19]:

On static output-feedback stabilization of a periodic system

Larin

2006

Self Cite

Relations are derived that allow standard MATLAB routines to be used to solve the static output-feedback control problem for a periodic discrete-time system. The efficiency of the approach proposed to design optimal static output-feedback controllers is demonstrated by examples Keywords: periodic discrete-time system, static output feedback, objective function Introduction. The optimization problem for linear periodic systems has a wide range of applications [2, 3, 6, 9-12, 14, 15, 18, 19, 30-32]. One of such is the synthesis of stabilization systems for walking and hopping robots [20][21][22][23][24][25][27][28][29]. It is known (see, e.g., [13]) that for a nonstationary linear system whose phase vector can only partially be observed, the feedback design procedure reduces to solving two matrix differential Riccati equations.One of these equations describes a filter that generates an estimate of the whole phase vector, and the other equation produces a controller matrix that relates this estimate and the control. This is the so-called case of dynamic feedback design. Designing static feedback is a more complicated problem because it requires determining constant matrices that form the control directly from the observable portion of the phase vector. Even if the system is stationary, this problem is very complicated [16,33,35]. To solve it, numerical algorithms based on the gradient of the objective function were proposed in [7,8,[33][34][35].Moreover, if the system is unstable, these algorithms additionally require an initial approximation, i.e., a stabilizing matrix. Forming this matrix is an independent problem [33]. Various approaches to its solution were proposed in [4,8,26,33,36].Here, we outline an algorithm for design of the optimal (i.e., minimizing a quadratic performance criterion) static output-feedback controller for a periodic discrete(-time) system. This algorithm generalizes the approach from [4,26] to periodic discrete-time systems, which substantially simplifies the procedure of selecting an initial approximation. The relations describing the objective function and its gradient are also generalized appropriately. We will show that standard MATLAB routines can be used to implement this algorithm.1. Problem Formulation. Consider a p-periodic discrete-time system described by the following difference relations: