Gain scheduled state feedback control of discrete-time systems with time-varying uncertainties: An LMI approach (original) (raw)
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Gain-scheduled ℋ2 and ℋ∞ control of discrete-time polytopic time-varying systems
IET Control Theory & Applications, 2010
This study presents H 2 and H 1 performance analysis and synthesis procedures for the design of both gain-scheduled and robust static output feedback controllers for discrete-time linear systems with time-varying parameters. The obtained controllers guarantee an upper bound on the H 2 or H 1 performance of the closed-loop system. As an immediate extension, the mixed H 2 /H 1 guaranteed cost control problem is also addressed. The scheduling parameters vary inside a polytope and are assumed to be a priori unknown, but measured in real-time. If bounds on the rate of parameter variation are known, they can be taken into account, providing less conservative results. The geometric properties of the polytopic domain are exploited to derive finite sets of linear matrix inequalities (LMIs) based on the existence of a parameter-dependent Lyapunov function. An application of the methodology to a realistic vibroacoustic problem, with experimentally obtained data, illustrates the benefits of the proposed approach and shows that the techniques can be used for real engineering problems.
2008 American Control Conference, 2008
In this paper, the problem of gain scheduling for time-varying systems with time delays is investigated. By using a memory at the feedback loop, a discrete gain scheduled controller which minimizes an upper bound to the H ∞ performance of the closed loop system is determined. The design conditions, expressed in terms of bilinear matrix inequalities, are obtained from the Finsler's Lemma combined with the Lyapunov theory. The extra variables introduced by the Finsler's Lemma represent an alternative way in the search of better system behavior. The time-varying uncertainties are modeled using polytopic domains. The controller is obtained by the solution of an optimization problem formulated only in terms of the vertices of the polytope. No grids in the parametric space are used. Numerical examples illustrate the efficiency of the proposed approach.
Gain-scheduled H ∞ control via parameter-dependent Lyapunov functions
International Journal of Systems Science, 2013
Synthesizing a gain-scheduled output feedback H∞ controller via parameter-dependent Lyapunov functions for linear parameter-varying (LPV) plant models involves solving an infinite number of linear matrix inequalities (LMIs). In practice, for affine LPV models, a finite number of LMIs can be achieved using convexifying techniques. This paper proposes an alternative approach to achieve a finite number of LMIs. By simple manipulations on the bounded real lemma inequality, a symmetric matrix polytope inequality can be formed. Hence, the LMIs need only to be evaluated at all vertices of such a symmetric matrix polytope. In addition, a construction technique of the intermediate controller variables is also proposed as an affine matrix-valued function in the polytopic coordinates of the scheduled parameters. Computational results on a numerical example using the approach were compared with those from a multi-convexity approach in order to demonstrate the impacts of the approach on parameter-dependent Lyapunov-based stability and performance analysis. Furthermore, numerical simulation results show the effectiveness of these proposed techniques.
A BMI approach for ℋ︁∞ gain scheduling of discrete time-varying systems
International Journal of Robust and Nonlinear Control, 2009
The problem of gain-scheduled state feedback control for discrete-time linear systems with time-varying parameters is considered in this paper. The time-varying parameters are assumed to belong to the unit simplex and to have bounded rates of variation, which depend on the values of the parameters and can vary from slow to arbitrarily fast. An augmented state vector is defined to take into account possible time-delayed inputs, allowing a simplified closed-loop analysis by means of parameter-dependent Lyapunov functions. A gain-scheduled state feedback controller that minimizes an upper bound to the H ∞ performance of the closed-loop system is proposed. No grids in the parametric space are used. The design conditions are expressed in terms of bilinear matrix inequalities (BMIs) due to the use of extra variables introduced by the Finsler's lemma. By fixing some of the extra variables, the BMIs reduce to a convex optimization problem, providing an alternate semi-definite programming algorithm to solve the problem. Robust controllers for time-invariant uncertain parameters, as well as gain-scheduled controllers for arbitrarily time-varying parameters, can be obtained as particular cases of the proposed conditions. As illustrated by numerical examples, the extra variables in the BMIs can provide better results in terms of the closed-loop H ∞ performance.
52nd IEEE Conference on Decision and Control, 2013
In this paper, a new approach to fixed-order H∞ and H2 output feedback control of MIMO discrete-time systems with polytopic uncertainty is proposed. The main idea of this approach is based on the definition of SPR-pair matrices and the use of some instrumental matrices which operates as a tool to overcome the original non-convexity of fixed-order controller design. Then, stability condition as well as H∞ and H2 performance constraints are presented by a set of linear matrix inequalities with linearly parameter dependent Lyapunov matrices. An iterative algorithm for update on the instrumental matrices is developed, that monotonically converges to a suboptimal solution. Simulation results show the effectiveness of the proposed approach.
Automatica, 2013
This paper is concerned with gain-scheduled control of two-dimensional discrete-time linear parametervarying systems described by a Roesser state-space model with matrices depending affinely on timevarying scheduling parameters. The parameter admissible values and variations are assumed to belong to given intervals. Linear matrix inequality based methods are devised for designing static state feedback gain-scheduled controllers with either an H ∞ or quadratic regulator-type performance. The control designs build on quadratically parameter-dependent Lyapunov functions and allow for incorporating information on available bounds on the parameters variation. The proposed controller gain can be independent, affine, quadratic, or a matrix fraction of quadratic polynomial matrices in the scheduling parameters.
LMI approach for ℋ∞ linear parameter-varying state feedback control
IEE Proceedings - Control Theory and Applications, 2005
Linear matrix inequality conditions are given for the existence of a stabilising linear parameter dependent state feedback gain for continuous time-varying systems in convex polytopic domains. Although there exist several results dealing with this problem in the literature, up to now all approaches assume that some matrices describing the system must be constant and/or must satisfy structural constraints. Here, all the system matrices are assumed to be affected by timevarying uncertainties and there are no structural constraints. The strategy proposed is much simpler than standard gain scheduling techniques, being specially adequate for systems with parameters that have unbounded or a priori unknown rates of variation, for instance, switched systems. Moreover, the conditions can also assure a guaranteed H 1 attenuation level for the closed-loop system under arbitrarily fast parameter variations significantly improving the results based on a fixed gain obtained through quadratic stabilisability conditions. Numerical examples illustrate the use of the proposed control design with applications to two physical systems: a linear model of a helicopter subject to actuator failures and an electrical circuit used as a lowpass filter in the output stage of power converters.
Gain-Scheduled H∞ control for tensor product type polytopic plants
2014
A tensor product (TP) model transformation is a recently proposed technique for transforming a given linear parameter-varying (LPV) model into polytopic model form for which there are many methods that can be used for controller design. This paper proposes an alternative approach to design a gain-scheduled output feedback H∞ controller with guaranteed L2-gain parameter-dependent performance for a class of TP type polytopic models using parameter-dependent Lyapunov functions where the linear matrix inequalities (LMIs) need only to be evaluated at all vertices of the system state-space model matrices and the variation rate of the scheduled parameters. In addition, a construction technique of the intermediate controller variables is also proposed as a matrix-valued function in the polytopic coordinates of the scheduled parameters. The performance of the proposed approach is tested on a missile autopilot design problem. Furthermore, nonlinear simulation results show the effectiveness of these proposed techniques.
Advances in Difference Equations, 2012
This article addresses the robust stability for a class of nonlinear uncertain discrete-time systems with convex polytopic of uncertainties. The system to be considered is subject to both interval time-varying delays and convex polytopic-type uncertainties. Based on the augmented parameter-dependent Lyapunov-Krasovskii functional, new delay-dependent conditions for the robust stability are established in terms of linear matrix inequalities. An application to robust stabilization of nonlinear uncertain discrete-time control systems is given. Numerical examples are included to illustrate the effectiveness of our results. MSC: 15A09; 52A10; 74M05; 93D05
Discrete-time H∞ control of linear parameter-varying systems
International Journal of Control, 2018
We introduce new conditions for the H ∞ synthesis of discrete-time Linear Parameter Varying (LPV) systems in the form of Linear Matrix Inequalities (LMIs). A distinctive feature of the proposed conditions is the ability to handle variation in both the dynamics and the input matrices without resorting to dynamic augmentation or iterative procedures. We show that this new condition contains the poly-quadratic H ∞ synthesis result of Daafouz and Bernussou (2001) as a particular case. We also derive a corollary which shows improvement even in the stronger case of quadratic H ∞ synthesis. Additionally, we show that, surprisingly, a dynamic gain-scheduled quadratic H ∞ controller can result in inferior performance compared to a static robust controller. Numerical examples illustrate our results.