Fixed-order robust controller design with the polynomial toolbox 3.0 (original) (raw)
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Overcoming non-convexity in polynomial robust control design
When developing efficient and reliable computer-aided control system design (CACSD) tools for low-order robust control systems analysis and synthesis, the main issue faced by theoreticians and practitioners is the non-convexity of the stability domain in the space of polynomial coefficients, or equivalently, in the space of design parameters. In this paper, we survey some of the recently developed techniques to overcome this non-convexity, underlining their respective pros and cons. We also enumerate some related open research problems which, in our opinion, deserve particular attention.
Positive polynomials and robust stabilization with fixed-order controllers
IEEE Transactions on Automatic Control, 2003
Recent results on positive polynomials are used to obtain a convex inner approximation of the stability domain in the space of coefficients of a polynomial. An application to the design of fixed-order controllers robustly stabilizing a linear system subject to polytopic uncertainty is then proposed, based on linear matrix inequality optimization. The key ingredient in the design procedure resides in the choice of the central polynomial. Several numerical examples illustrate the relevance of the approach.
Robust stability and performance with fixed-order controllers
Automatica, 1999
This paper deals with the problem of synthesizing or designing a feedback controller of xxed dynamic order for a linear time-invariant plant for a "xed plant, as well as for an uncertain family of plants containing parameter uncertainty, so that stability, robust stability and robust performance are attained. The desired closed-loop speci"cations considered here are given in terms of a target performance vector representing a desired closed-loop design. In general, these point targets are unattainable with a "xed-order controller. By enlarging the target from a "xed-point set to an interval set the solvability conditions are relaxed and a solution is enabled. Results from the parametric robust control literature can be used to design the interval target family so that the poles and zeros of the closed-loop system are guaranteed to remain in a prescribed region of the complex plane, and closed-loop performance, measured for instance in the H norm are attained, even when plant uncertainty is present. It is shown that it is possible to devise a computationally simple linear programming approach that attempts to meet the desired closed-loop speci"cations. The approach developed here gives the entire set of controllers attaining the speci"cations as a convex set and also can be recast to give the lowest-order controller attaining speci"cations. 0005-1098/99/$ -see front matter 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 5 -1 0 9 8 ( 9 9 ) 0 0 0 8 0 -1
IFAC Proceedings Volumes, 2011
In this paper, the computation of robust stability and robust performance margins for linear time invariant systems under polynomial parameter uncertainty is considered. The characteristic equations of such systems appear as polynomial functions of uncertain parameters. The problem is formulated as a polynomial optimization problem using the zero-exclusion theorem. Then, two global optimization techniques are used to solve the optimization problems. One of the algorithms is based on the Reformulation-Linearization Technique (RLT), and the other uses the Linear Matrix Inequalities (LMI) scheme. The two methods are applied to two case studies and the results are compared. The outcomes show the superiority of the RLT methodology in converging to a global solution, and in computational time.
Robust Control Design with Matlab
Robustness is of crucial importance in control systems design, because real engineering systems are vulnerable to external disturbance and measurement noise, and there are always discrepancies between mathematical models used for design and the actual system in practice. Typically a control engineer is required to design a controller which will make the closed loop system stable and achieve certain performance levels in the presence of disturbance signals, noise interference, unmodeled plant dynamics and plant parameter variations. The purpose of this book is to help post-graduate students and control engineers learn how to use well-developed, advanced robust control system design methods and the state-of-the-art MATLAB® tools in practical design cases.
Robust controller design based
2001
This paper presents a method for designing a ®xed controller, which is able to control in a stable fashion a family of linear time-invariant n-th order plants with arbitrary relative degree. This plant family is de®ned in terms of a nominal transfer function of rational type and bounded variations of the coef®cients of numerator and denominator polynomials. The design method is based on the algebraic relationship existing in the model reference adaptive control technique between the true plant parameters, the ideal controller parameters and the model reference parameters. While applying the proposed method, the resulting plant families are broader if compared with other techniques used to design robust controllers.
Robust controller synthesis via non-linear matrix inequalities
International Journal of Control, 1999
Over the last several years xed-structure multiplier versions of MSSV theory have been developed and have led to the development of LMI's for the analysis of robust stability and performance. These LMI's have in turn led to the development of BMI's for the synthesis of robust controllers. The BMI formulation in practice requires the multiplier to lie in the span of a stable basis, potentially introducing signi cant conservatism. This paper uses the LMI approach to MSSV analysis to develop an approach to robust controller synthesis that is based on the stable factors of the multipliers and does not require the multipliers to be restricted to a basis. It is shown that this approach requires the solution of nonlinear matrix inequalities. A continuation algorithm is presented for the solution of NMI's. The primary computational burden of the continuation algorithm is the solution of a series of LMI's.
Fixed, Low-Order Controller Design with Time Response Specifications Using Non-Convex Optimization
2007
In this paper, we present a new algorithm for designing a fixed, low-order controller with time response specifications for a linear time invariant (LTI), single input single output (SISO) plant. For a two-parameter feedback configuration, the problem of finding a fixed or low-order controller to meet the desired time response specification is reduced to the least square estimation (LSE) in the sense of partial model matching (PMM), which minimizes a quadratic cost function. The closed-loop stability condition imposed on the controller parameters is formulated by the polynomial matrix inequality (PMI) constraint associated with the cost function. When the cascade feedback structure is considered, the zeros of the controller may be a substantial obstacle when designing a controller that has a good time response. This problem can also be formulated using polynomial constraints. Consequently, it is shown that the total problem here can be formulated as an optimization problem with a quadratic objective function and several polynomial constraints in the controller parameter space. We show that the SeDuMi with YALMIP interface [Löfberg J. YALMIP: A toolbox for modeling and optimization in MATLAB, in: Proceedings of the IEEE symposium on computer aided control systems design 2004. p. 284-9. http://control.ee.ethz.ch/˜joloef/yalmip.php\] can be used for solving this problem. Finally, several illustrative examples are given.
Benchmark problems for robust control design (1992 ACC version)
American Control Conference, 1992, 1992
1. Iroduction Simple, yet meaiul, control problems to hig lht is sues in robust control design and to provide a forum for thie applicaion of a variety of robust control desg methodolo-gies are formlated in is paper. Such problm have been studied by several rsarchers under a variety of ...
Systems & Control Letters, 2008
This paper investigates the problems of checking robust stability and evaluating robust H 2 performance of uncertain continuous-time linear systems with time-invariant parameters lying in polytopic domains. The novelty is the ability to check robust stability by constructing a particular parameter-dependent Lyapunov function which is a polynomial function of the uncertain system matrices, as opposed to a general polynomial function of the uncertain parameter. The degree of the polynomial is tied to a certain integer κ. The existence of such Lyapunov function can be proved by solving parameter-dependent Linear Matrix Inequalities (LMIs), which are guaranteed to be solvable for a sufficiently large yet finite value of κ whenever the system is robustly stable. Extensions to guaranteed H 2 cost computation are also provided. Numerical aspects concerning the programming and the evaluations of the proposed tests are discussed and illustrated by examples.