Adaptive control with optimal robust tracking

The problem of optimal robust tracking in two-parameter adaptive control systems under non-linear time-varying unmodelled dynamics is examined. A new robust stability criterion is derived for analysing the robustness of adaptive control systems with non-linear time-varying model errors. Based on the concept of excess robustness and the theory of the minimum H^∞norm, a simple and feasible design algorithm is presented to synthesize a two-parameter adaptive controller which ensures that adaptive control systems can achieve the object of optimal robust tracking in the presence of non-linear time-varying unmodelled dynamics. Simulation results that demonstrate features of the two-parameter adaptive controller with optimal robust tracking in the light of the design algorithm are included.     

Globally optimal robust process control

A computational approach is developed for designing a globally optimal controller which is robust to time-varying nonlinear perturbations in the plant. This controller design problem is formulated as an optimization with bilinear matrix inequality (BMI) constraints, and is solved to optimality by a branch and bound algorithm. The algorithm is applied to a reactive ion etcher, and provides superior performance while providing robustness to nonlinear plant/model mismatch. The algorithm is also applied to a well known benchmark problem.     

Maximizing a state convex lagrange functional in optimal control

We consider the maximization problem for an integral functional with a state-convex integrand function along a standard control system. We show necessary and sufficient global optimality conditions related to the Pontryagin’s maximum principle. We study the properties of these conditions and their relations with optimal control theory. We also illustrate the efficiency of the resulting conditions on specific examples.     

Observer-based robust optimal control for helicopter with uncertainties and disturbances

Unknown model uncertainties and external disturbances widely exist in helicopter dynamics and bring adverse effects on control performance. Optimal control techniques have been extensively studied for helicopters, but these methods cannot effectively handle flight control problems since they are sensitive to uncertainties and disturbances. This paper proposes an observer-based robust optimal control scheme that enables a helicopter to fly optimally and reduce the influence of unknown model uncertainties and external disturbances. A control Lyapunov function (CLF) is firstly constructed using the backstepping method, then Sontag's formula is utilized to design an inverse optimal controller to stabilize the nominal system. Furthermore, it is stressed that the radial basis function (RBF) neural network is introduced to establish an observer with adaptive laws, approximating and compensating for the unknown model uncertainties and external disturbances to enhance the robustness of the closed-loop system. The uniform ultimate boundedness of the closed-loop system is ensured using the presented control approach via Lyapunov stability analysis. Finally, simulation results are presented to demonstrate the effectiveness of the proposed control strategy.     

An optimal control approach to robust control design

We propose an optimal control approach to robust control design. Our goal is to design a state feedback to stabilize a system under uncertainty. We translate this robust control problem into an optimal control problem of minimizing a cost. Because the uncertainty bound is reflected in the cost, the solution to the optimal control problem is a solution to the robust control problem. Our approach can deal with both linear and non-linear systems. Furthermore it can handle both matched and unmatched uncertainties. It can also handle uncertainty in the control input matrix.     

Selection of a representative value function in robust multiple criteria sorting

We introduce the concept of a representative value function in robust ordinal regression applied to multiple criteria sorting problems. The proposed approach can be seen as an extension of UTADIS^GMS, a new multiple criteria sorting method that aims at assigning actions to p pre-defined and ordered classes. The preference information supplied by the decision maker (DM) is composed of desired assignments of some reference actions to one or several contiguous classes—they are called assignment examples. The robust ordinal regression builds a set of general additive value functions compatible with the assignment examples and results in two assignments: necessary and possible. The necessary assignment specifies the range of classes to which the action can be assigned considering all compatible value functions simultaneously. The possible assignment specifies, in turn, the range of classes to which the action can be assigned considering any compatible value function individually. In this paper, we propose a way of selecting a representative value function among the set of compatible ones. We identify a few targets which build on results of the robust ordinal regression and could be attained by a representative value function. They concern enhancement of differences between possible assignments of two actions. In this way, the selected function highlights the most stable part of the robust sorting, and can be perceived as representative in the sense of robustness preoccupation. We envisage two possible uses of the representative value function in decision support systems. The first one is an explicit exhibition of the function along with the results of the UTADIS^GMS method, in order to help the DM to understand the robust sorting. The other is an autonomous use, in order to supply the DM with sorting obtained by an example-based procedure driven by the chosen function. Three case studies illustrating the use of a representative value function in real-world decision problems are presented. One of those studies is devoted to the comparison of the introduced concept of representativeness with alternative procedures for determining a single value function, which we adapted to sorting problems, because they were originally proposed for ranking problems.     

Linear convex stochastic optimal control with applications in production planning

Linear stochastic systems with convex performance criteria and convex, compact control regions are studied. The admissible control region is assumed to be a continuous function of the (perfectly) observed state. Optimal feedback controls are shown to exist within the class of Borel measurable functions of past states. In fact, they are shown to be continuous functions of the present state. Using dynamic programming the optimal return function is shown to be convex. Asymptotic results for stable systems are derived. These results are then used to explore several problems in aggregate production and work-force planning. Computational aspects of the results in the context of the smoothing problem are discussed.     

Identification and robust control for systems with ellipsoidal parametric uncertainty by convex optimization

In this paper, sufficient conditions for robust output feedback controller design for systems with ellipsoidal parametric uncertainty are given in terms of solutions to a set of linear matrix inequalities (LMIs) Performance specifications are in terms of combined pole placement with sensitivity function shaping in the H₂ or H_∞ norm. Furthermore, an optimal input design technique for parameter estimation that is integrated into the robust control design is employed in this paper. This means that performance specifications on the closed‐loop transfer functions are translated into the requirements on the input signal spectrum. The simulation results show the effectiveness of the proposed method. Copyright © 2011 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Control relevant identification for robust optimal control

This paper presents an approach to designing the input signal for an identification experiment, in which the process model estimate is to be used to formulate and solve for a robust (in a worst case sense) optimal controller. The input signal is designed to contain the information that is relevant for the end use of the model, that is for control purposes. The proposed approach uses sensitivity analysis to determine the input signal frequencies that are important with respect to a certain measure of achievable controller performance in conjunction with a frequency sampling filter model of the process. Based on the sensitivity analysis, an iterative experimental design methodology is suggested.     

Scoring procedures for multiple criteria decision aiding with robust and stochastic ordinal regression

We propose several scoring procedures for transforming the results of robustness analysis to a univocal recommendation. We use a preference model in form of an additive value function, and assume the Decision Maker (DM) to provide pairwise comparisons of reference alternatives. We adapt single- and multi-stage ranking methods to select the best alternative or construct a complete ranking by exploiting four types of outcomes: (1) necessary preference relation, (2) pairwise outranking indices, (3) extreme ranks, and (4) rank acceptability indices. In each case, a choice or ranking recommendation is obtained without singling out a specific value function. We compare the proposed scoring procedures in terms of their ability to suggest the same recommendation as the one obtained with the Decision Maker׳s assumed “true” value function. To quantify the results of an extensive simulation study, we use the following comparative measures (including some newly proposed ones): (i) hit ratio, (ii) normalized hit ratio, (iii) Kendall׳s τ, (iv) rank difference measure, and (v) rank agreement measure. Their analysis indicates that to identify the best “true” alternative, we should refer to the acceptability indices for the top rank(s), whereas to reproduce the complete “true” ranking it is most beneficial to focus on the expected ranks that alternatives may attain or on the balance between how much each alternative outranks and is outranked by all other alternatives.     

Analytical solution of a linear quadratic optimal control problem with control value constraints

Optimal control problem for time varying linear systems with fixed endpoints of the trajectories and quadratic functional is considered. A method for constructing feedback controls with regard to constraints imposed on them is proposed. The problem is solved using a special form of Lagrange multipliers.     

A perspective-based convex relaxation for switched-affine optimal control

We consider the switched-affine optimal control problem, i.e., the problem of selecting a sequence of affine dynamics from a finite set in order to minimize a sum of convex functions of the system state. We develop a new reduction of this problem to a mixed-integer convex program (MICP), based on perspective functions. Relaxing the integer constraints of this MICP results in a convex optimization problem, whose optimal value is a lower bound on the original problem value. We show that this bound is at least as tight as similar bounds obtained from two other well-known MICP reductions (via conversion to a mixed logical dynamical system, and by generalized disjunctive programming), and our numerical study indicates it is often substantially tighter. Using simple integer-rounding techniques, we can also use our formulation to obtain an upper bound (and corresponding sequence of control inputs). In our numerical study, this bound was typically within a few percent of the optimal value, making it attractive as a stand-alone heuristic, or as a subroutine in a global algorithm such as branch and bound. We conclude with some extensions of our formulation to problems with switching costs and piecewise affine dynamics.     

不确定时滞系统的全局鲁棒最优滑模控制

针对一类不确定性时滞系统, 研究线性二次型最优调节器的鲁棒性设计问题. 首先基于级数近似方法, 将原标称时滞系统的最优调节器问题转化为迭代求解一族不含时滞的两点边值问题, 从而获得标称时滞系统最优控制的近似解. 然后将滑模控制理论应用于最优调节器的设计, 使得系统对于不确定性具有全局的鲁棒性, 并且其理想滑动模态与标称系统的最优闭环控制系统相一致, 从而实现了全局鲁棒最优滑模控制. 仿真示例将所提出的方法与相应的二次型最优控制进行比较, 验证了该方法的有效性和优越性.     

鲁棒与最优控制在伺服系统中的应用

随着现代科技的迅速发展,高性能需求对伺服控制技术提出了更高的要求.鲁棒与最优控制技术在雷达天线、数控机床、机械臂、移动机器人和硬盘驱动等伺服系统的应用中表现出极大的优势,在伺服控制领域扮演着越来越重要的角色.结合伺服控制的应用背景回顾了鲁棒与最优控制理论中的一些主要问题与方法;以车载天线系统为例,对伺服控制设计的主要环节(指标定义、模型辨识、鲁棒分析和控制设计)进行讨论,强调了状态控制方法与经典频域法结合的重要性;对主流鲁棒与最优控制算法(回路成型、H_∞、mu综合、$H_2$、混合$H_2/H_{\infty     

Application of multiple shooting to the numerical solution of optimal control problems with bounded state variables

Algorithms for the numerical solution of optimal control problems with bounded state variables are developed. Two main cases are considered: either the control variable appears nonlinearly or the control variable appears linearly. In the first case, an extremal are touching the boundary or containing a boundary arc, is shown to satisfy a suitable two-point boundary value problem. In the second case, a numerical idea for solving the problem in statespace is presented which dispenses with the Lagrange-multipliers. Three numerical examples are discussed illustrating the efficiency of the different algorithms. The encountered two-point boundary value problems are solved with the method of multiple shooting.     

The optimal robust control policy for uncertain semi-Markov control processes

The optimization problems of Markov control processes (MCPs) with exact knowledge of system parameters, in the form of transition probabilities or infinitesimal transition rates, can be solved by using the concept of Markov performance potential which plays an important role in the sensitivity analysis of MCPs. In this paper, by using an equivalent infinitesimal generator, we first introduce a definition of discounted Poisson equations for semi-Markov control processes (SMCPs), which is similar to that for MCPs, and the performance potentials of SMCPs are defined as solution of the equation. Some related optimization techniques based on performance potentials for MCPs may be extended to the optimization of SMCPs if the system parameters are known with certainty. Unfortunately, exact values of the distributions of the sojourn times at some states or the transition probabilities of the embedded Markov chain for a large-scale SMCP are generally difficult or impossible to obtain, which leads to the uncertainty of the semi-Markov kernel, and thereby to the uncertainty of equivalent infinitesimal transition rates. Similar to the optimization of uncertain MCPs, a potential-based policy iteration method is proposed in this work to search for the optimal robust control policy for SMCPs with uncertain infinitesimal transition rates that are represented as compact sets. In addition, convergence of the algorithm is discussed.     

An iterative procedure for solving convex optimal control problems

A doubly iterative procedure for computing optimal controls in linear systems with convex cost functionals is presented. The procedure is based on an algorithm due to Gilbert [3] for minimizing a quadratic form on a convex set. Each step of the procedure makes use of an algorithm due to Neustadt and Paiewonsky [1] to solve a strictly linear optimal control problem.     

On a class of optimal control problems with an almost cost-free solution

For a class of linear singular optimal control problems with a nonunique singular arc, the solution of the corresponding nearly singular problem is analyzed and a limit solution based on formal singular perturbations is derived. A rigorous proof of the correctness of the result is given by constructing a convergent power series satisfying the Riccati equation of the nearly singular problem.