The design of state feedback gain-scheduled controllers for linear parameter-varying systems with saturating actuators is addressed in the paper. The parameters can vary arbitrarily fast inside a polytope with known …
The design of state feedback gain-scheduled controllers for linear parameter-varying systems with saturating actuators is addressed in the paper. The parameters can vary arbitrarily fast inside a polytope with known vertices. Sufficient conditions for the existence of gain-scheduled controllers assuring asymptotic stability for initial conditions inside a region of the state space are provided in terms of parameter-dependent linear matrix inequalities. A complete characterization of the solutions of these inequalities is given in terms of homogeneous polynomially parameter-dependent matrices of arbitrary degree that can be obtained from finite linear matrix conditions, written in terms of the vertices of the polytope. A procedure based on an extension of Polya's Theorem produces a sequence of sufficient conditions which tend to the necessity as the level of relaxation increases. The scheduled controller is a homogeneous polynomially parameter-dependent state feedback gain of arbitrary degree that quadratically stabilizes the closed-loop system and provides an estimate of the domain of attraction of the origin, as illustrated by numerical examples.
This paper synthesizes a gain-scheduled controller to stabilize all possible Linear Parameter-Varying (LPV) plants that are consistent with measured input/state data records. Inspired by prior work in data informativity and …
This paper synthesizes a gain-scheduled controller to stabilize all possible Linear Parameter-Varying (LPV) plants that are consistent with measured input/state data records. Inspired by prior work in data informativity and LTI stabilization, a set of Quadratic Matrix Inequalities is developed to represent the noise set, the class of consistent LPV plants, and the class of stabilizable plants. The bilinearity between unknown plants and `for all' parameters is avoided by vertex enumeration of the parameter set. Effectiveness and computational tractability of this method is demonstrated on example systems.
This letter synthesizes a gain-scheduled controller to stabilize all possible Linear Parameter-Varying (LPV) plants that are consistent with measured input/state data records. Inspired by prior work in data informativity and …
This letter synthesizes a gain-scheduled controller to stabilize all possible Linear Parameter-Varying (LPV) plants that are consistent with measured input/state data records. Inspired by prior work in data informativity and LTI stabilization, a set of Quadratic Matrix Inequalities is developed to represent the noise set, the class of consistent LPV plants, and the class of stabilizable plants. The bilinearity between unknown plants and 'for all' parameters is avoided by vertex enumeration of the parameter set. Effectiveness and computational tractability of this method is demonstrated on example systems.
In this paper, we propose a method to establish the robust performance based on a parameter dependent controller under the parameter estimation errors.We first propose a model of the parameter …
In this paper, we propose a method to establish the robust performance based on a parameter dependent controller under the parameter estimation errors.We first propose a model of the parameter which represents the estimation error less conservatively. A sufficient condition for the existence of a controller is derived based on the scaled maximum singular value condition, and it is given by LMIs (Linear Matrix Inequalities) with a BMI (Bilinear Matrix Inequality) constraint. The relation of estimation accuracy and performance level is also investigated by a numerical example. Furthermore, we introduce a bilinear transformation to represent the situation that the accuracy of the estimation depends on the value of the parameter, and it is shown that the controller can be constructed by a simple modification for the plant.
This paper concerns with a new class of adaptive gain-scheduled H∞ control of linear parameter-varying (LPV) systems. The plants in this manuscript are assumed to be polytopic LPV systems, but …
This paper concerns with a new class of adaptive gain-scheduled H∞ control of linear parameter-varying (LPV) systems. The plants in this manuscript are assumed to be polytopic LPV systems, but the time-varying parameters in those plants are not available for measurement, and thus, the conventional gain-scheduled control strategy cannot be applied. In the proposed adaptive schemes, the estimates of those unknown parameters are obtained recursively, and the current estimates are fed to the controllers to stabilize the plants and to attain H∞ control performance adaptively. Stability analysis of the adaptive control systems is carried out by utilizing Lyapunov approaches based on linear matrix inequalities in the bounded real lemma.
This paper deals with the problem of H∞ reduced order dynamic output feedback control for discrete-time linear systems affected by time-varying parameters with polynomial dependency and norm-bounded terms. The main …
This paper deals with the problem of H∞ reduced order dynamic output feedback control for discrete-time linear systems affected by time-varying parameters with polynomial dependency and norm-bounded terms. The main motivation comes from recent discretization methods for uncertain systems, that produce polynomially parameter-dependent discretized systems of arbitrary degree with norm-bounded terms. The design conditions are provided in terms of sufficient parameter-dependent LMI conditions combined with scalar searches, being capable to synthesize robust or gain-scheduled controllers. The approach is also particularized to handle the popular class of time-varying polytopic systems, having as novelty no requirement of special treatment for the output measured matrix. Numerical examples are provided to illustrate the potentialities of the approach to cope with discretized systems and the efficiency of the relaxations when compared with the existing methods for gain-scheduled or robust stabilization of polytopic time-invariant and time-varying systems.
This paper demonstrates the design of H∞ loop-shaping controller for a linear time invariant (LTI) system with input saturation constraint. The design problem has been formulated in the four-block H∞ …
This paper demonstrates the design of H∞ loop-shaping controller for a linear time invariant (LTI) system with input saturation constraint. The design problem has been formulated in the four-block H∞ synthesis framework, which is equivalent to normalized coprime factor robust stabilization problem. The shaped plant is represented as a polytopic linear parameter varying (LPV) system while saturation nonlinearity is considered. For a polytopic model, the LTI H∞ loop-shaping controllers have been designed at each vertex of the polytope using linear matrix inequalities, and subsequently controllers are scheduled by adopting a certain interpolation procedure. The proposed controller ensures the stability and robust L2-performance of the closed-loop system due to vertex property of the polytopic LPV shaped plant. The effectiveness of the design method has been illustrated through a numerical example.
This paper is concerned with a new characterization of fixed-order controllers based on the LMI approach. We give necessary and sufficient conditions for the existence of both reduced and higher-order …
This paper is concerned with a new characterization of fixed-order controllers based on the LMI approach. We give necessary and sufficient conditions for the existence of both reduced and higher-order controllers which satisfy various control specifications such as H2 or H performance, and characterize all such controllers. A computation method is given in order to obtain reduced-order controllers, with a numerical example.
We present a direct data-driven approach to synthesize robust control invariant (RCI) sets and their associated gain-scheduled feedback control laws for linear parameter-varying (LPV) systems subjected to bounded disturbances. A …
We present a direct data-driven approach to synthesize robust control invariant (RCI) sets and their associated gain-scheduled feedback control laws for linear parameter-varying (LPV) systems subjected to bounded disturbances. A data-set consisting of a single state-input-scheduling trajectory is gathered from the system, which is directly utilized to compute polytopic RCI set and controllers by solving a semidefinite program. The proposed method does not require an intermediate LPV model identification step. Through a numerical example, we show that the proposed approach can generate RCI sets with a relatively small number of data samples when the data satisfies certain excitation conditions.
A unified approach to the strong stabilisation problem and the H∞ strong stabilisation problem is presented. New sufficient conditions for the existence of strongly stabilising controllers and stable H∞ controllers …
A unified approach to the strong stabilisation problem and the H∞ strong stabilisation problem is presented. New sufficient conditions for the existence of strongly stabilising controllers and stable H∞ controllers are derived, in a unified manner, in terms of the solvability of a positive real controller synthesis problem and a multi-objective control problem, respectively. A linear matrix inequality (LMI) technique developed by Scherer et al. is adopted to make the most use of its power to deal with the general case of the problems. Several advantages brought by the adopted LMI technique are explored. New parameterisations of stable controllers for both the problems are discussed. In particular, the parameterisations are independent of a particular method for solving strong stabilisation problems. Explicit state-space synthesis algorithms are given and numerical examples are provided to demonstrate the potential of the proposed methods.
This paper is concerned with the problem of suboptimal stable mixed H 2 / H X control for linear time-invariant systems. The designed controller with the same order as that …
This paper is concerned with the problem of suboptimal stable mixed H 2 / H X control for linear time-invariant systems. The designed controller with the same order as that of the considered system is required to satisfy a prescribed H X performance bound or a prescribed degree of stability. By reducing the stable controller synthesis problem to a multiobjective state feedback control problem for two different state models, sufficient conditions for the solvability of the considered problem are given in terms of solutions to algebraic Riccati equations and matrix inequalities. An LMI-based iterative algorithm is developed to solve the stable controller synthesis problem. The proposed algorithm is shown to be convergent. Examples are given to illustrate the proposed methods.
This Paper present an approach to order reduction of parameter varying controller . First, we find faesible solutions which satisfy parameter varying Lyapunovinequqlitie for constructing balanced parameter varying controllers. Next …
This Paper present an approach to order reduction of parameter varying controller . First, we find faesible solutions which satisfy parameter varying Lyapunovinequqlitie for constructing balanced parameter varying controllers. Next a singular perturbation methode of the time invariant systems is generalized to reduce the balanced conntroller order. Futhermorem , we will show that the reduced-order of the parameter varying closed loop system with the full-order controller is equivalent to the parameter varying closed loop system with bthe reduced- order controller. We also show that the reduced-order parameter varyingb controller guarantees closed looop stability and performence. The effectiveness of the proposed method and that found by balanced truncation method is compared
This article addresses the problem of robust stabilization of uncertain linear systems by static state- and output-feedback control laws. The uncertainty is supposed to belong to a polytope and both …
This article addresses the problem of robust stabilization of uncertain linear systems by static state- and output-feedback control laws. The uncertainty is supposed to belong to a polytope and both continuous- and discrete-time systems are investigated. Contrarily to the main stream of the linear matrix inequality (LMI)-based robust stabilization methods available in the literature, where the product between the Lyapunov (or slack variable) and control gain matrices is transformed in a new variable, this article proposes a change of paradigm, avoiding the standard change of variables and providing synthesis conditions that deal directly with the control gain as an optimization variable. The design procedure is formulated in terms of a locally convergent iterative algorithm based on LMIs, having as main novelties the following points: Both the Lyapunov and the closed-loop dynamic matrices appear affinely in the conditions; the iterative scheme involves the slack variables only, avoiding the classic alternation between the Lyapunov matrix and the control gain; an extra degree of freedom is created in terms of an additional scalar variable, which also appears affinely in the conditions and represents a scaling on the closed-loop dynamic matrix; a smart termination for the algorithm through a stability analysis condition. Numerical experiments based on exhaustive simulations show that all these features combined provide a robust stabilization technique capable to outperform the best available methods in the literature in terms of effectiveness, being specially suitable to address static output-feedback and decentralized control problems.
This paper develops a stability and performance preserving controller order reduction method for linear time-invariant continuous-time single-input, single-output systems. In this method, the error between the complementary sensitivity functions of …
This paper develops a stability and performance preserving controller order reduction method for linear time-invariant continuous-time single-input, single-output systems. In this method, the error between the complementary sensitivity functions of the nominal closed-loop system and closed-loop system using the reduced-order controller is converted to a frequency-weighted error between the Youla parameters of the full-order and reduced-order controllers and then the H/sub /spl infin// norm of this error, subject to a set of linear matrix inequality constraints, is minimized. The main ideas of order reduction and stability preservation are contained in the constraints of the optimization problem. However, since this minimization problem is nonconvex, the Youla parameter of the reduced-order controller is obtained by solving a suboptimal linear matrix inequality problem, that is convex and readily solved using existing semi-definite programming solvers. It is shown that the resulting reduced-order controller preserves the stability and performance of the nominal closed-loop system in disturbance rejection and input tracking.
This paper presents a new algorithm for the design of linear controllers with special structural constraints imposed on the control gain matrix. This so called SLC (structured linear control) problem …
This paper presents a new algorithm for the design of linear controllers with special structural constraints imposed on the control gain matrix. This so called SLC (structured linear control) problem can be formulated with linear matrix inequalities (LMI's) with a nonconvex equality constraint. This class of problems includes fixed order output feedback control, multi-objective controller design, decentralized controller design, joint plant and controller design, and other interesting control problems. Our approach includes two main contributions. The first is that many design specifications are described by a similar matrix inequality. A new matrix variable is introduced to give more freedom to design the controller. Indeed this new variable helps to find the optimal fixed-order output feedback controller. The second contribution is to propose a linearization algorithm to search for a solution to the nonconvex SLC problems. This has the effect of adding a certain potential function to the nonconvex constraints to make them convex. The convexified matrix inequalities will not bring significant conservatism because they will ultimately go to zero, guaranteeing the feasibility of the original nonconvex problem. Numerical examples demonstrate the performance of the proposed algorithms and provide a comparison with some of the existing methods.
Gain scheduling based on plant linearizations about equilibria implicitly imposes a linearization requirement on the controller. This paper presents an approach for gain scheduling that yields controllers that automatically possess …
Gain scheduling based on plant linearizations about equilibria implicitly imposes a linearization requirement on the controller. This paper presents an approach for gain scheduling that yields controllers that automatically possess the appropriate linearization property and that can also assume a linear parameter-varying (LPV) form. The approach is a modification of the so-called /spl Dscr/ method described in the literature wherein the need for time-derivatives of certain controller inputs is avoided. A gain scheduled controller is designed for the ball and beam experiment to illustrate the technique.
The problem of reduced-order LQG (linear quadratic Gaussian) optimization is addressed in a finite-horizon, linear time-varying system setting. First-order necessary conditions for local optimality in the parameter space are provided …
The problem of reduced-order LQG (linear quadratic Gaussian) optimization is addressed in a finite-horizon, linear time-varying system setting. First-order necessary conditions for local optimality in the parameter space are provided in terms of four coupled matrix differential equations. This result provides a transparent generalization of the optimal projection equations of Hyland and Bernstein for the optimal, steady-state compensation of linear time-invariant plants.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">></ETX>
Gain scheduling has proven to be a successful design methodology in many engineering applications. In the absence of a sound theoretical analysis, these designs come with no guarantees of the …
Gain scheduling has proven to be a successful design methodology in many engineering applications. In the absence of a sound theoretical analysis, these designs come with no guarantees of the robustness, performance, or even nominal stability of the overall gain-scheduled design. An analysis is presented for two types of nonlinear gain-scheduled control systems: (1) scheduling on a reference trajectory, and (2) scheduling on the plant output. Conditions which guarantee stability, robustness, and performance properties of the global gain schedule designs are given. These conditions confirm and formalize popular notions regarding gain scheduled designs, such as that the scheduling variable should vary slowly, and capture the plant's nonlinearities.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">></ETX>