Output feedback optimal guaranteed cost control of uncertain piecewise linear systems

This paper proposes the output feedback optimal guaranteed cost controller design method for uncertain piecewise linear systems based on the piecewise quadratic Lyapunov functions technique. By constructing piecewise quadratic Lyapunov functions for the closed‐loop augmented systems, the existence of the guaranteed cost controller for closed‐loop uncertain piecewise linear systems is cast as the feasibility of a set of bilinear matrix inequalities (BMIs). Some of the variables in BMIs are set to be searched by genetic algorithm (GA), then for a given chromosome corresponding to the variables in BMIs, the BMIs turn to be linear matrix inequalities (LMIs), and the corresponding non‐convex optimization problem, which minimizes the upper bound on cost function, reduces to a semidefinite programming (SDP) which is convex and can be solved numerically efficiently with the available software. Thus, the output feedback optimal guaranteed cost controller can be obtained by solving the non‐convex optimization problem using the mixed algorithm that combines GA and SDP. Numerical examples show the effectiveness of the proposed method. Copyright © 2008 John Wiley & Sons, Ltd.     

Piecewise linear quadratic optimal control

The use of piecewise quadratic cost functions is extended from stability analysis of piecewise linear systems to performance analysis and optimal control. Lower bounds on the optimal control cost are obtained by semidefinite programming based on the Bellman inequality. This also gives an approximation to the optimal control law. An upper bound to the optimal cost is obtained by another convex optimization problem using the given control law. A compact matrix notation is introduced to support the calculations and it is proved that the framework of piecewise linear systems can be used to analyze smooth nonlinear dynamics with arbitrary accuracy     

A team algorithm for robust stability analysis and control design of uncertain time‐varying linear systems using piecewise quadratic Lyapunov functions

A team algorithm based on piecewise quadratic simultaneous Lyapunov functions for robust stability analysis and control design of uncertain time‐varying linear systems is introduced. The objective is to use robust stability criteria that are less conservative than the usual quadratic stability criterion. The use of piecewise quadratic Lyapunov functions leads to a non‐convex optimization problem, which is decomposed into a convex subproblem in a selected subset of decision variables, and a lower‐dimensional non‐convex subproblem in the remaining decision variables. A team algorithm that combines genetic algorithms (GA) for the non‐convex subproblem and interior‐point methods for the solution of linear matrix inequalities (LMI), which form the convex subproblem, is proposed. Numerical examples are given, showing the advantages of the proposed method. Copyright © 2001 John Wiley & Sons, Ltd.     

时变不确定系统鲁棒控制器设计的新方法

基于包含两个二次项的分段Lyapunov函数,研究了线性时变不确定系统的鲁棒控制器设计问题.所考虑的系统由两个矩阵的凸组合构成,通过引入一个附加矩阵,推导出鲁棒控制器存在的充分条件.该控制器的状态反馈增益的求解问题可以转化为一组带有两个比例参数的线性矩阵不等式的凸优化问题.最后的数值示例说明了该设计方法的可行性.     

Piecewise Fuzzy Anti-Windup Dynamic Output Feedback Control of Nonlinear Processes With Amplitude and Rate Actuator Saturations

In this paper, a novel anti-windup dynamic output compensator is developed to deal with the robust H _infin output feedback control problem of nonlinear processes with amplitude and rate actuator saturations and external disturbances. Via fuzzy modeling of nonlinear systems, the proposed piecewise fuzzy anti-windup dynamic output feedback controller is designed based on piecewise quadratic Lyapunov functions. It is shown that with sector conditions, robust output feedback stabilization of an input-constrained nonlinear process can be formulated as a convex optimization problem subject to linear matrix inequalities. Simulation study on a strongly nonlinear continuously stirred tank reactor (CSTR) benchmark plant is given to show the performance of the proposed anti-windup dynamic compensator.     

Piecewise quadratic stability of fuzzy systems

Presents an approach to stability analysis of fuzzy systems. The analysis is based on Lyapunov functions that are continuous and piecewise quadratic. The approach exploits the gain-scheduling nature of fuzzy systems and results in stability conditions that can be verified via convex optimization over linear matrix inequalities. Examples demonstrate the many improvements over analysis based on a single quadratic Lyapunov function. Special attention is given to the computational aspects of the approach and several methods to improve the computational efficiency are described     

Augmented Lagrange Hopfield network initialized by quadratic programming for economic dispatch with piecewise quadratic cost functions and prohibited zones

This paper proposes a method based on quadratic programming (QP) and augmented Lagrange Hopfield network (ALHN) for solving economic dispatch (ED) problem with piecewise quadratic cost functions and prohibited zones. The ALHN method is a continuous Hopfield neural network with its energy function based on augmented Lagrange function which can properly deal with constrained optimization problems. In the proposed method, the QP method is firstly used to determine the fuel cost curve for each unit and initialize for the ALHN method, then a heuristic search is used for repairing prohibited zone violations, and the ALHN method is finally applied for solving the problem if any violations found. The proposed method has been tested on different systems and the obtained results are compared to those from many other methods in the literature. The result comparison has indicated that the proposed method has obtained better solution quality than many other methods. Therefore, the proposed QP-ALHN method could be a favorable method for solving the ED problem with piecewise quadratic cost functions and prohibited zones.     

A convex approach for NMPC based on second order Volterra series models

In model predictive control (MPC), the input sequence is computed, minimizing a usually quadratic cost function based on the predicted evolution of the system output. In the case of nonlinear MPC (NMPC), the use of nonlinear prediction models frequently leads to non‐convex optimization problems with several minimums. This paper proposes a new NMPC strategy based on second order Volterra series models where the original performance index is approximated by quadratic functions, which represent a lower bound of the original performance index. Convexity of the approximating quadratic cost functions can be achieved easily by a suitable choice of the weighting of the control increments in the performance index. The approximating cost functions can be globally minimized by convex optimization techniques in order to compute the input sequence. The minimization of the performance index is carried out by an iterative optimization procedure, which guarantees convergence to the solution. Furthermore, for a nominal prediction model, asymptotic stability for the proposed NMPC strategy can be shown. In the case of considering an estimation error in the prediction model, input‐to‐state practical stability is assured. The control performance of the NMPC strategy is illustrated by experimental results. Copyright © 2014 John Wiley & Sons, Ltd.     

Convex spline interpolants with minimal curvature

The problem of finding convex spline interpolants with minimal mean curvature leads to a quadratic optimization problem of special structure. In the present note a corresponding dual problem without constraints is derived. Its objective function is piecewise quadratic and therefore admits an effective numerical treatment.     

Quadratic approximate dynamic programming for input‐affine systems

We consider the use of quadratic approximate value functions for stochastic control problems with input‐affine dynamics and convex stage cost and constraints. Evaluating the approximate dynamic programming policy in such cases requires the solution of an explicit convex optimization problem, such as a quadratic program, which can be carried out efficiently. We describe a simple and general method for approximate value iteration that also relies on our ability to solve convex optimization problems, in this case, typically a semidefinite program. Although we have no theoretical guarantee on the performance attained using our method, we observe that very good performance can be obtained in practice.Copyright © 2012 John Wiley & Sons, Ltd.     

Distributed Multi-Parametric Quadratic Programming

One of the fundamental problems in the area of large-scale optimization is to study locality features of spatially distributed optimization problems in which the variables are coupled in the cost function as well as constraints. Such problems can motivate the development of fast and well-conditioned distributed algorithms. In this paper, we study spatial locality features of large-scale multi-parametric quadratic programming (MPQP) problems with linear inequality constraints. Our main application focus is receding horizon control of spatially distributed linear systems with input and state constraints. We propose a new approach for analysis of large-scale MPQP problems by blending tools from duality theory with operator theory. The class of spatially decaying matrices is introduced to capture couplings between optimization variables in the cost function and the constraints. We show that the optimal solution of a convex MPQP is piecewise affine- represented as convolution sums. More importantly, we prove that the kernel of each convolution sum decays in the spatial domain at a rate proportional to the inverse of the corresponding coupling function of the optimization problem.     

Absolute stability analysis of discrete-time systems with composite quadratic Lyapunov functions

A generalized sector bounded by piecewise linear functions was introduced in a previous paper for the purpose of reducing conservatism in absolute stability analysis of systems with nonlinearity and/or uncertainty. This paper will further enhance absolute stability analysis by using the composite quadratic Lyapunov function whose level set is the convex hull of a family of ellipsoids. The absolute stability analysis will be approached by characterizing absolutely contractively invariant (ACI) level sets of the composite quadratic Lyapunov functions. This objective will be achieved through three steps. The first step transforms the problem of absolute stability analysis into one of stability analysis for an array of saturated linear systems. The second step establishes stability conditions for linear difference inclusions and then for saturated linear systems. The third step assembles all the conditions of stability for an array of saturated linear systems into a condition of absolute stability. Based on the conditions for absolute stability, optimization problems are formulated for the estimation of the stability region. Numerical examples demonstrate that stability analysis results based on composite quadratic Lyapunov functions improve significantly on what can be achieved with quadratic Lyapunov functions.     

The piecewise parametric optimal control of uncertain linear quadratic models

The optimal control of linear quadratic model is given in a feedback form and determined by the solution of a Riccati equation. However, the control-related Riccati equation usually cannot be solved analytically such that the form of optimal control will become more complex. In this paper, we consider a piecewise parametric optimal control problem of uncertain linear quadratic model for simplifying the form of optimal control. By introducing a piecewise control parameter, a piecewise parametric optimal control model is established. Then we present a parametric optimisation method for solving the optimal piecewise control parameter. Finally, an uncertain inventory-promotion optimal control problem is discussed and a comparison is made to show the effectiveness of proposed piecewise parametric optimal control model.     

Output feedback controller design for uncertain piecewise linear systems

This paper proposes output feedback controller design methods for uncertain piecewise linear systems based on piecewise quadratic Lyapunov function. The α-stability of closed-loop systems is also considered. It is shown that the output feedback controller design procedure of uncertain piecewise linear systems with α-stability constraint can be cast as solving a set of bilinear matrix inequalities （BMIs）. The BMIs problem in this paper can be solved iteratively as a set of two convex optimization problems involving linear matrix inequalities （LMIs） which can be solved numerically efficiently. A numerical example shows the effectiveness of the proposed methods.     

Simulation and Optimization by Quantifier Elimination

We present a highly optimized method for the elimination of linear variables from a Boolean combination of polynomial equations and inequalities. In contrast to the basic method described earlier, the practical applicability of the present method goes far beyond academic examples. The optimization is achieved by various strategies to prune superfluous branches in the elimination tree constructed by the method.The main application concerns the simulation of large technical networks of (electric, mechanical or hydraulic) components, whose characteristic curves are piecewise linear (or quadratic) in the variables to be eliminated. Typical goals are the computation of admissible ranges for certain variables and the detection of a malfunction of a network component. The algorithms are currently used in a commercial software system for industrial applications.Moreover, we extend the author's elimination method for parametric linear programming to the non-convex case by allowing arbitraryand–orcombinations of parametric linear inequalities as constraints. We present a new strategy for finding smaller elimination sets and thus smaller elimination trees for parametric linear programming. Some benchmark examples from thenetliblibrary oflpproblems show the significance of this strategy even for convex linear programming problems.     

Computation of piecewise quadratic Lyapunov functions for hybridsystems

This paper presents a computational approach to stability analysis of nonlinear and hybrid systems. The search for a piecewise quadratic Lyapunov function is formulated as a convex optimization problem in terms of linear matrix inequalities. The relation to frequency domain methods such as the circle and Popov criteria is explained. Several examples are included to demonstrate the flexibility and power of the approach     

二次参数曲线拟合平面点的一种新方法

对平面上给定的一组数据点进行了研究,提出了构造参数曲线拟合数据点的一种新方法。所构造的拟合参数曲线是C′连续的分段二次参数曲线。本文以实例对新方法与二次插值样条曲线进行了比较。     

Passive fault-tolerant control of discrete time piecewise affine systems against actuator faults

In this article, we propose a new method for passive fault-tolerant control of discrete time piecewise affine systems. Actuator faults are considered. A reliable piecewise linear quadratic regulator state feedback is designed such that it can tolerate actuator faults. A sufficient condition for the existence of a passive fault-tolerant controller is derived and formulated as the feasibility of a set of linear matrix inequalities (LMIs). The upper bound on the performance cost can be minimised using a convex optimisation problem with LMI constraints which can be solved efficiently. The approach is illustrated on a numerical example and a two degree of freedom helicopter.     

Designing parametric linear quadratic regulators for parametric LTI systems via LMIs

This paper addresses the problem of determining parametric linear quadratic regulators (LQRs) for continuous-time linear-time invariant systems affected by parameters through rational functions. Three situations are considered, where the sought controller has to minimise the best cost, average cost, and worst cost, respectively, over the set of admissible parameters. It is shown that candidates for such controllers can be obtained by solving convex optimisation problems with linear matrix inequality (LMI) constraints. These candidates are guaranteed to approximate arbitrarily well the sought controllers by sufficiently increasing the size of the LMIs. In particular, the candidate that minimises the average cost approximates arbitrarily well the true LQR over the set of admissible parameters. Moreover, conditions for establishing the optimality of the found candidates are provided. Some numerical examples illustrate the proposed methodology.     

A feasible-side globally convergent modifier-adaptation scheme

In the context of static real-time optimization (RTO) of uncertain plants, the standard modifier-adaptation scheme consists in adding first-order correction terms to the cost and constraint functions of a model-based optimization problem. If the algorithm converges, the limit is guaranteed to be a KKT point of the plant. This paper presents a general RTO formulation, wherein the cost and constraint functions belong to a certain class of convex upper-bounding functions. It is demonstrated that this RTO formulation enforces feasible-side global convergence to a KKT point of the plant. Based on this result, a novel modifier-adaptation scheme with guaranteed feasible-side global convergence is proposed. In addition to the first-order correction terms, quadratic terms are added in order to convexify and upper bound the cost and constraint functions. The applicability of the approach is demonstrated on a constrained variant of the Williams–Otto reactor for which standard modifier adaptation fails to converge in the presence of plant-model mismatch.