On the Safe Stabilization Problem

2009

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: Optimization Procedurementioning

confidence: 99%

“…Example 5[21,67]. The matrices A and B appearing in (be given matrices B 1 and B 2 derived by zeroing the first and second column of B, respectively: Q I = , R I = , N = 0 in (3.2).…”

mentioning

confidence: 99%

Optimization problems for periodic systems

Aliev

2009

“…Adding (3.1) to the relations (2.8) and determining the matrices Q and W corresponding to the minimum value of λ, we obtain a controller K WQ = −1 closest to K 0 , i.e., the solution of the reliable-stabilization problem for the system (1.1) with a stability margin ρ µ ≤ 1/ . Let us demonstrate the algorithm by an example from [5,30]. Example [5,30].…”

Section: Reliable Stabilization Of a System With A Given Stability Mamentioning

confidence: 99%

“…Let us demonstrate the algorithm by an example from [5,30]. Example [5,30]. The matrices A and B appearing in (1.…”

Section: Reliable Stabilization Of a System With A Given Stability Mamentioning

confidence: 99%

See 1 more Smart Citation

On Reliable Stabilization of Linear Periodic Systems

2005

An algorithm is proposed to synthesize a reliable controller with a given stability margin for linear and periodic systems optimized with respect to a quadratic performance criterion. A reliable controller synthesized by the algorithm guarantees the stability margin and is close to the linear-quadratic requlator. The importance of ensuring the stability margin is demonstrated. The proposed algorithm is based on methods of linear matrix inequalities and can be implemented using standard MATLAB routines. As an example, a reliable controller that stabilizes the program motion of a hopping machine is synthesized Keywords: reliable stabilization, linear matrix inequality, stability margin, periodic systems Introduction. The stabilization problem for linear periodic systems has a wide variety of applications [9, 13] among which, in particular, is the synthesis of a stabilization system for walking and hopping robots [17][18][19][20][21][24][25][26][27]. This kind of problems is solved using different approaches (generalized to periodic systems) intended to analyze stationary systems. Noteworthy are synthesis algorithms based on a stabilizing solution of the Riccati equation [2,6,7,12,16,28], algorithms based on linear matrix inequalities (LMIs) [1,3,8,22], etc. Note that researchers usually try to generalize a synthesis algorithm for stationary systems to periodic systems. In this connection, we will solve the stabilization problem for periodic systems using an approach developed to synthesize robust controllers with a guaranteed stability margin for stationary systems [29]. As indicated in [31], the first formulations of robust H 2 -optimization problems were to ensure a stability margin of synthesized controllers. The difficulties faced then gave rise to problems of synthesis of robust controllers in terms of H ∞ -norm optimization. Such approaches, however, give less attention to other important characteristics that are more naturally described in terms of H 2 -norms. This is why robust controllers were synthesized using various combinations of H 2 -and H ∞ -norms. Alternatively, it was proposed, e.g. in [29], to minimize a quadratic performance criterion ensuring a certain stability margin. The approach from [3,23] is modified to apply to the synthesis of a reliable controller with a guaranteed stability margin. The reliable-stabilization problem [4,5] is formulated as follows: given a plant P, find controllers C 1 and C 2 such that the closed-loop system is stable when either both controllers C 1 and C 2 or only one of them operates. In Sect. 3, this problem for a stationary system is formalized so that it can easily be generalized to a periodic system (Sect. 4).The proposed approach for stationary systems is detailed in the first part of the paper (Sects. 1-3) and is generalized to periodic systems in the second part (Sects. 4-6). The importance of ensuring a given stability margin for a synthesized reliable controller will be demonstrated by examples. Synthesis of a Controller with Feedback Gain Constraints and ...

On static output-feedback stabilization of a periodic system

2006

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: