Unconstrained optimal control of regular languages

This paper formulates an unconstrained optimal policy for control of regular languages realized as deterministic finite state automata (DFSA). A signed real measure quantifies the behavior of controlled sublanguages based on a state transition cost matrix and a characteristic vector as reported in an earlier publication. The state-based optimal control policy is obtained by selectively disabling controllable events to maximize the measure of the controlled plant language without any further constraints. Synthesis of the optimal control policy requires at most n iterations, where n is the number of states of the DFSA model. Each iteration solves a set of n simultaneous linear algebraic equations. As such, computational complexity of the control synthesis is polynomial in n.     

Optimal supervisory control of finite state automata

This paper presents optimal supervisory control of dynamical systems that can be represented by deterministic finite state automaton (DFSA) models. The performance index for the optimal policy is obtained by combining a measure of the supervised plant language with (possible) penalty on disabling of controllable events. The signed real measure quantifies the behaviour of controlled sublanguages based on a state transition cost matrix and a characteristic vector as reported in earlier publications. Synthesis of the optimal control policy requires at most n iterations, where n is the number of states of the DFSA model generated from the unsupervised plant language. The computational complexity of the optimal control synthesis is polynomial in n. Syntheses of the control algorithms are illustrated with two application examples.     

Signed real measure of regular languages for discrete event supervisory control

This paper reviews, expands, and clarifies the underlying concepts of a signed real measure of regular languages, which has been used as a novel tool for synthesis of discrete event supervisory control systems. The language measure is constructed upon the principles of automata theory and real analysis. It allows total ordering of a set of partially ordered sublanguages of a regular language for quantitative evaluation of the supervised behaviour of deterministic finite state automata (DFSA) under different supervisors. In the setting of the language measure, a supervisor's performance is superior if the supervised plant is more likely to terminate at a good marked state and/or less likely to terminate at a bad marked state. The computational complexity of the language measure algorithm is polynomial in the number of DFSA states.     

Least costly identification experiment for control

All approaches to optimal experiment design for control have so far focused on deriving an input signal (or input signal spectrum) that minimizes some control-oriented measure of plant/model mismatch between the nominal closed-loop system and the actual closed-loop system, typically under a constraint on the total input power. In practical terms, this amounts to finding the (constrained) input signal that minimizes a measure of a control-oriented model uncertainty set. Here we address the experiment design problem from a “dual” point of view and in a closed-loop setting: given a maximum allowable control-oriented model uncertainty measure compatible with our robust control specifications, what is the cheapest identification experiment that will give us an uncertainty set that is within the required bounds? The identification cost can be measured by either the experiment time, the performance degradation during experimentation due to the added excitation signal, or a combination of both. Our results are presented for the situation where the control objective is disturbance rejection only.     

Relations between uncertainty structures in identification for robust control

Various techniques of system identification exist that provide a nominal model and an uncertainty bound. An important question is what the implications are for the particular choice of the structure in which the uncertainty is described when dealing with robust stability/performance analysis of a given controller and when dealing with robust synthesis. It is shown that an amplitude-bounded (circular) uncertainty set can equivalently be described in terms of an additive, Youla parameter and ν-gap uncertainty. As a result, the choice of structure does not matter provided that the identification methods deliver optimal uncertainty sets rather than an uncertainty bound around a prefixed nominal model. Frequency-dependent closed-loop performance functions based on the uncertainty sets are again bounded by circles in the frequency domain, allowing for analytical expressions for worst-case performance and for the evaluation of the consequences of uncertainty for robust design. The results can be used to tune optimal experimental conditions in view of robust control design and in the further development of experiment-based robust control design methods.     

Robust reliability method for non-fragile guaranteed cost control of parametric uncertain systems

The problem of non-fragile guaranteed cost control of uncertain systems is studied from a new point of view of reliability against uncertainties. An efficient robust reliability method for the analysis and design of non-fragile guaranteed cost controller of parametric uncertain systems is presented systematically. By the method, a robust reliability measure of an uncertain control system satisfying required robust performance can be obtained, and the robustness bounds of uncertain parameters such that the control cost of a system is guaranteed can be provided. The optimal non-fragile guaranteed cost controller obtained in the paper may possess optimal guaranteed cost performance satisfying the precondition that the system is robustly reliable with respect to uncertainties occurring in both the controlled plant and controller gain. The presented formulations are in the framework of linear matrix inequality and thus can be carried out conveniently. The presented method provides an essential basis for the tradeoff between reliability and control cost in controller design of uncertain systems. Two numerical examples are provided to demonstrate the efficiency and feasibility of the presented method. It is shown that the coordination and simultaneous realization of the system performance, control cost, and robust reliability in control design of uncertain systems are significant.     

Robust stability and performance via fixed-order dynamic compensation with guaranteed cost bounds

A feedback control-design problem involving structured plant parameter uncertainties is considered. Two robust control-design issues are addressed. The Robust Stability Problem involves deterministic bounded structured parameter variations, while the Robust Performance Problem includes, in addition, a quadratic performance criterion averaged over stochastic disturbances and maximized over the admissible parameter variations. The optimal projection approach to fixed-order, dynamic compensation is merged with the guaranteed cost control approach to robust stability and performance to obtain a theory of full- and reduced-order robust control design. The principle result is a sufficient condition for characterizing dynamic controllers of fixed dimension which are guaranteed to provide both robust stability and performance. The sufficient conditions involve a system of modified Riccati and Lyapunov equations coupled by an oblique projection and the uncertainty bounds. The full-order result involves a system of two modified Riccati equations and two modified Lyapunov equations coupled by the uncertainty bounds. The coupling illustrates the breakdown of the separation principle for LQG control with structured plant parameter variations. Supported in part by the Air Force Office of Scientific Research under Contract F49620-86-C-0002.     

A robust two degree-of-freedom controller for systems with both model and measurement uncertainty

This paper presents a design method for robust two degree-of-freedom (DOF) controllers that optimize the control performance with respect to both model uncertainty and signal measurement uncertainty. In many situations, non-causal feedforward is a welcome control addition when closed loop feedback bandwidth limitations exist due to plant dynamics such as: delays, non-minimum phase zeros, poorly placed zeros and poles (Xie, Alleyne, Greer, and Deneault (2013); Xie (2013), etc. However, feedforward control is sensitive to both model uncertainty and signal measurement uncertainty. The latter is particularly true when the feedforward is responding to pre-measured disturbance signals. The combined sensitivity will deteriorate the feedforward controller performance if care is not taken in design. In this paper a two DOF design is introduced which optimizes the performance based on a given estimate of uncertainties. The controller design uses H_∞ tools to balance the controlled system bandwidth with increased sensitivity to signal measurement uncertainties. A successful case study on an experimental header height control system for a combine harvester is shown as an example of the approach.     

不确定性机械系统的最优鲁棒控制设计: 模糊法

针对不确定性机械系统,提出了一种新的最优鲁棒控制方法.本文用模糊法去描述机械系统中的不确定性.机械系统的性能要求是确定的(保证最低要求),同时也是模糊的(成本控制里用到).所提出的控制方法是确定的,而不是基于假设的规则.经过严格的理论证明,控制系统最终可达到理想的性能指标.基于模糊信息,本文设计了一个性能指标(综合成本,包括系统的平均模糊性能和控制成本).通过最小化此性能指标,可解决控制的最优设计问题.这种最优设计方法可得到唯一的解析形式的最优解.总的来说,这种最优鲁棒控制方法较为系统,能够保证确定的系统性能得以实现,同时控制成本最小.最后,本文选了一个机械系统作为例子.     

服务率不确定的单站点传送带给料加工站系统鲁棒优化控制

研究了服务率不确定情况下的单站点传送带给料加工站(CSPS)系统的鲁棒优化控制问题。在仅知服务率区间的条件下,以CSPS系统的前视距离作为控制变量,将鲁棒优化控制问题建模成不确定参数的半马尔可夫决策过程(SMDP)的极大极小优化问题,在状态相关的情况下,给出全局优化算法进行鲁棒控制策略求解。首先,运用遗传算法求解固定策略下的最差性能值;其次,根据求解得到的最差性能值,运用模拟退火算法求解最优鲁棒控制策略。仿真结果表明,服务率不确定的CSPS系统的最优鲁棒性能代价与服务率固定为区间中值系统的最优性能代价相差不大,并且随着不确定区间的缩小,两者的差值越小,说明了全局优化算法的有效性。     

Robust nonlinear feedback control for uncertain linear systems with nonquadratic performance criteria

In this paper we develop a unified framework to address the problem of optimal nonlinear robust control for linear uncertain systems. Specifically, we transform a given robust control problem into an optimal control problem by properly modifying the cost functional to account for the system uncertainty. As a consequence, the resulting solution to the modified optimal control problem guarantees robust stability and performance for a class of nonlinear uncertain systems. The overall framework generalizes the classical Hamilton–Jacobi–Bellman conditions to address the design of robust nonlinear optimal controllers for uncertain linear systems. © 1998 Elsevier Science B.V.     

Robust nonlinear feedback control for uncertain linear systems with nonquadratic performance criteria

In this paper we develop a unified framework to address the problem of optimal nonlinear robust control for linear uncertain systems. Specifically, we transform a given robust control problem into an optimal control problem by properly modifying the cost functional to account for the system uncertainty. As a consequence, the resulting solution to the modified optimal control problem guarantees robust stability and performance for a class of nonlinear uncertain systems. The overall framework generalizes the classical Hamilton–Jacobi–Bellman conditions to address the design of robust nonlinear optimal controllers for uncertain linear systems. © 1998 Elsevier Science B.V.     

Depth control of autonomous underwater vehicles using indirect robust control method

In this article, we propose a robust depth control design scheme for autonomous underwater vehicles (AUVs) in the presence of hydrodynamic parameter uncertainties and disturbances. The controller is designed via a new indirect robust control method that handles the uncertainties by formulating the uncertainty bounds into the cost functional and then transforming the robust control problem into an equivalent optimal control problem. Both robust asymptotic stability and optimality can be achieved and proved with this new formulation. The θ-D method is utilised to solve the resultant nonlinear optimal control problem such that an approximate closed-form feedback controller can be obtained and thus is easy to implement onboard without intensive computation load. Simulation results demonstrate that robust depth control is accomplished under the system parameter uncertainties and disturbances with small control fin deflection requirement.     

An optimal effective controller for discrete event systems

An approach to the online synthesis of an optimal effective controller for discrete event systems is presented. The optimal effective controller can achieve the prescribed (cumulative) effectiveness measure while minimizing the total cost incurred for the execution of events. This approach is constructed over a generalized control framework for automata‐based discrete event systems, which allows event enforcement in addition to the (original) event disablement/enablement as the control mechanism. The optimal effective control policy generated by this approach is proved to be the least restrictive among all the possible optimal effective control policies for the given online expansion tree of the system behavior. Copyright © 2008 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Undecidable problems of decentralized observation and control on regular languages

We introduce a decentralized observation problem, where the system under observation is modeled as a regular language L over a finite alphabet Σ and n subsets of Σ model distributed observation points. A regular language K⊆L models a set of distinguished behaviors, say, correct behaviors of the system. The objective is to check the existence of a function which, given the n observations corresponding to a behavior ρ∈L, decides whether ρ is in K or not. We prove that checking the existence of such a function is undecidable. We then use this result to show undecidability of a decentralized supervisory control problem in the discrete event system framework.     

Iterative learning control for uncertain systems: Noncausal finite time interval robust control design

In this paper, we present a novel robust Iterative Learning Control (ILC) control strategy that is robust against model uncertainty as given by an additive uncertainty model. The design methodology hinges on ??_∞ optimization, but formulated such that the obtained ILC controller is not restricted to be causal, and inherently operates on a finite time interval. Optimization of the robust ILC (R‐ILC) solution is accomplished for the situation where any information about structure in the uncertainty is discarded, and for the situation where the information about the structure in the uncertainty is explicitly taken into account. Subsequently, the convergence and performance properties of resulting R‐ILC controlled system are analyzed. On an experimental set‐up, we show that the presented R‐ILC control strategy can outperform an existing linear‐quadratic norm‐optimal ILC approach and an existing causal R‐ILC approach based on frequency domain ??_∞ synthesis. Copyright © 2010 John Wiley & Sons, Ltd.     

Language-measure-theoretic optimal control of probabilistic finite-state systems

Supervisory control theory for discrete event systems, introduced by Ramadge and Wonham, is based on a non-probabilistic formal language framework. However, models for physical processes inherently involve modelling errors and noise-corrupted observations, implying that any practical finite-state approximation would require consideration of event occurrence probabilities. Building on the concept of signed real measure of regular languages, this paper formulates a comprehensive theory for optimal control of finite-state probabilistic processes. It is shown that the resulting discrete-event supervisor is optimal in the sense of elementwise maximizing the renormalized langauge measure vector for the controlled plant behaviour and is efficiently computable. The theoretical results are validated through several examples including the simulation of an engineering problem.     

A Novel Robust Constraint Force Servo Control for Under‐actuated Manipulator Systems: Fuzzy and Optimal

We consider the control design for under‐actuated manipulator systems. The task is to drive the system to be close to a prescribed constraint. The system contains uncertainty. It is bounded where the bounding information is prescribed by a fuzzy set (e.g., the bound is close to 1). The initial condition is also prescribed by a fuzzy set. A class of robust control is proposed, which guarantees a deterministic performance. On top of that, the choice of a control design parameter is cast into a fuzzy‐theoretic setting. A performance index, consisting of accumulated fuzzy‐based system performance and control cost, is proposed. The optimal control design parameters, which minimize the performance index, can be obtained by solving two algebraic quartic (fourth‐order) equations. As a result, the control design problem, which addresses both fuzzy and optimal characteristics, is completely solved.