Decentralized supervisory control with communicating controllers

The decentralized control problem for discrete-event systems addressed in this paper is that of several communicating supervisory controllers, each with different information, working in concert to exactly achieve a given legal sublanguage of the uncontrolled system's language model. A novel information structure model is presented for dealing with this class of problems. Existence results are given for the cases of when controllers do and do not anticipate future communications, and a synthesis procedure is given for the case when controllers do not anticipate communications. Several conditions for optimality of communication policies are presented, and it is shown that the synthesis procedure yields solutions, when they exist for this class of controllers, that are optimal with respect to one of these conditions     

Discrete time mean-field stochastic linear-quadratic optimal control problems

This paper firstly presents necessary and sufficient conditions for the solvability of discrete time, mean-field, stochastic linear-quadratic optimal control problems. Secondly, the optimal control within a class of linear feedback controls is investigated using a matrix dynamical optimization method. Thirdly, by introducing several sequences of bounded linear operators, the problem is formulated as an operator stochastic linear-quadratic optimal control problem. By the kernel-range decomposition representation of the expectation operator and its pseudo-inverse, the optimal control is derived using solutions to two algebraic Riccati difference equations. Finally, by completing the square, the two Riccati equations and the optimal control are also obtained.     

Robust static and dynamic output-feedback stabilization: Deterministic and stochastic perspectives

Three parallel gaps in robust feedback control theory are examined: sufficiency versus necessity, deterministic versus stochastic uncertainty modeling, and stability versus performance. Deterministic and stochastic output-feedback control problems are considered with both static and dynamic controllers. The static and dynamic robust stabilization problems involve deterministically modeled bounded but unknown measurable time-varying parameter variations, while the static and dynamic stochastic optimal control problems feature state-, control-, and measurement-dependent white noise. General sufficiency conditions for the deterministic problems are obtained using Lyapunov's direct method, while necessary conditions for the stochastic problems are derived as a consequence of minimizing a quadratic performance criterion. The sufficiency tests are then applied to the necessary conditions to determine when solutions of the stochastic optimization problems also solve the deterministic robust stability problems. As an additional application of the deterministic result, the modified Riccati equation approach of Petersen and Hollot is generalized in the static case and extended to dynamic compensation.     

Finite dimensional approximations for a class of infinite dimensional time optimal control problems

ABSTRACT

In this work, we study the numerical approximation of the solutions of a class of abstract parabolic time optimal control problems with unbounded control operator. Our main results assert that, provided that the target is a closed ball centered at the origin and of positive radius, the optimal time and the optimal controls of the approximate time optimal problems converge (in appropriate norms) to the optimal time and to the optimal controls of the original problem. In order to prove our main theorem, we provide a nonsmooth data error estimate for abstract parabolic systems.     

Optimal periodic control problem: a modified trigonometric approximation and quasi-stationarity conditions

The constrained optimal periodic control problem for a system described by differential equations and endowed with inertial controllers is considered, A sequence of discretized problems using trigonometric polynomials is proposed to approximate the problem. Instantaneous constraints for the state and control are handled by a new and more precise approach that imposes only a small number of non-linear but easily computable constraints. The convergence conditions for a sequence of optimal solutions of discretized problems are derived. The inclusion in the approximating scheme of various quasi-stationarity conditions for the control and state variables is analysed. Extension of a new approximating approach for inertialess and smooth problems is also discussed.     

Pareto suboptimal controllers against coalitions of disturbances

We consider a multi-criteria problem of suppressing disturbances with linear feedback with respect to the state or output measured with noise. We assume that the system has N potentially possible inputs for disturbances from given classes, and the criteria are induced norms of operators generated by the system from the corresponding input to the common target output. We obtain necessary Pareto optimality conditions. We show that based on scalar optimization of the suppression level for the disturbances that act on all inputs we can synthesize Pareto suboptimal controllers whose relative losses compared to Pareto optimal controllers do not exceed 1 ? \(\sqrt N /N\). Our results generalize to the case when disturbances from different classes may form coalitions.     

Investigation of Commutative Properties of Discontinuous Galerkin Methods in PDE Constrained Optimal Control Problems

The aim of this paper is to investigate commutative properties of a large family of discontinuous Galerkin (DG) methods applied to optimal control problems governed by the advection-diffusion equations. To compute numerical solutions of PDE constrained optimal control problems there are two main approaches: optimize-then-discretize and discretize-then-optimize. These two approaches do not always coincide and may lead to substantially different numerical solutions. The methods for which these two approaches do coincide we call commutative. In the theory of single equations, there is a related notion of adjoint or dual consistency. In this paper we classify DG methods both in primary and mixed forms and derive necessary conditions that can be used to develop new commutative methods. We will also derive error estimates in the energy and L ² norms. Numerical examples reveal that in the context of PDE constrained optimal control problems a special care needs to be taken to compute the solutions. For example, choosing non-commutative methods and discretize-then-optimize approach may result in a badly behaved numerical solution.     

Trigonometric approximation of optimal periodic control problemsfor systems with inertial controllers

A constrained optimal periodic control (OPC) problem for nonlinear systems with inertial controllers is considered. A sequence of approximate problems containing trigonometric polynomials for approximation of the state, control, and functions in the state equations and in the optimality criterion is formulated. Sufficient conditions for a sequence of nearly optimal solutions of approximate problems to be norm-convergent to the basic problem optimal solution are derived. It is pointed out that the direct approximation approach in the space of state and control combined with the finite-dimensional optimization methods such as the space covering and gradient-type methods makes probable the finding of the global optimum for OPC problems     

Some nonlinear optimal control problems with closed‐form solutions

Optimal controllers guarantee many desirable properties including stability and robustness of the closed‐loop system. Unfortunately, the design of optimal controllers is generally very difficult because it requires solving an associated Hamilton–Jacobi–Bellman equation. In this paper we develop a new approach that allows the formulation of some nonlinear optimal control problems whose solution can be stated explicitly as a state‐feedback controller. The approach is based on using Young's inequality to derive explicit conditions by which the solution of the associated Hamilton–Jacobi–Bellman equation is simplified. This allows us to formulate large families of nonlinear optimal control problems with closed‐form solutions. We demonstrate this by developing optimal controllers for a Lotka–Volterra system. Copyright © 2001 John Wiley & Sons, Ltd.     

正规和奇异H∞控制器的统一表达式

给出了一种新的H∞控制器设计方法,通过引入设计时可选的非奇异实数阵,取消了控制器设计时对D矩阵的秩限制.适用于正规的H∞控制问题和奇异的H∞控制问题.对状态反馈等四种典型问题和输出反馈控制问题,给出了控制器存在的充分必要条件.控制器通过Riccati方程的解,用参数化方法表示.输出反馈控制器,通过解两个Riccati方程得到.讨论了控制器的相关特性.     

A characterization of convex problems in decentralized control

We consider the problem of constructing optimal decentralized controllers. We formulate this problem as one of minimizing the closed-loop norm of a feedback system subject to constraints on the controller structure. We define the notion of quadratic invariance of a constraint set with respect to a system, and show that if the constraint set has this property, then the constrained minimum-norm problem may be solved via convex programming. We also show that quadratic invariance is necessary and sufficient for the constraint set to be preserved under feedback. These results are developed in a very general framework, and are shown to hold in both continuous and discrete time, for both stable and unstable systems, and for any norm. This notion unifies many previous results identifying specific tractable decentralized control problems, and delineates the largest known class of convex problems in decentralized control. As an example, we show that optimal stabilizing controllers may be efficiently computed in the case where distributed controllers can communicate faster than their dynamics propagate. We also show that symmetric synthesis is included in this classification, and provide a test for sparsity constraints to be quadratically invariant, and thus amenable to convex synthesis.     

A characterization of convex problems in decentralized Control/sup */

We consider the problem of constructing optimal decentralized controllers. We formulate this problem as one of minimizing the closed-loop norm of a feedback system subject to constraints on the controller structure. We define the notion of quadratic invariance of a constraint set with respect to a system, and show that if the constraint set has this property, then the constrained minimum-norm problem may be solved via convex programming. We also show that quadratic invariance is necessary and sufficient for the constraint set to be preserved under feedback. These results are developed in a very general framework, and are shown to hold in both continuous and discrete time, for both stable and unstable systems, and for any norm. This notion unifies many previous results identifying specific tractable decentralized control problems, and delineates the largest known class of convex problems in decentralized control. As an example, we show that optimal stabilizing controllers may be efficiently computed in the case where distributed controllers can communicate faster than their dynamics propagate. We also show that symmetric synthesis is included in this classification, and provide a test for sparsity constraints to be quadratically invariant, and thus amenable to convex synthesis.     

Weakly monotone solutions of the Hamilton-Jacobi inequality and optimality conditions with positional controls

We obtain necessary global optimality conditions for classical optimal control problems based on positional controls. These controls are constructed with classical dynamical programming but with respect to upper (weakly monotone) solutions of the Hamilton-Jacobi equation instead of a Bellman function. We put special emphasis on the positional minimum condition in Pontryagin formalism that significantly strengthens the Maximum Principle for a wide class of problems and can be naturally combined with first order sufficient optimality conditions with linear Krotov’s function. We compare the positional minimum condition with the modified nonsmooth Ka?kosz-Lojasiewicz Maximum Principle. All results are illustrated with specific examples.     

Necessary and sufficient conditions for optimality inlinear-quadratic problems of optimal control

Linear-quadratic Bolza problems of optimal control with variable end points are considered. Under the strengthened Legendre condition, necessary and sufficient optimality conditions are established, and it is shown that the linear-quadratic Bolza problem of optimal control can be reduced to a quadratic minimization problem in a finite-dimensional space. Simple simulations where solutions of a nonlinear problem can be recovered from solutions of the accessory linear-quadratic problem are indicated. Conjectures regarding sufficient conditions for optimality in nonlinear Bolza problems are included     

Decentralized state-feedback stabilization and robust control of uncertain large-scale systems with integrally constrained interconnections

This paper is concerned with a problem of stabilization and robust control design for interconnected uncertain systems. A new class of uncertain large-scale systems is considered in which interconnections between subsystems as well as uncertainties in each subsystem are described by integral quadratic constraints. The problem is to design a set of local (decentralized) controllers which stabilize the overall system and guarantee robust disturbance attenuation in the presence of the uncertainty in interconnections between subsystems as well as in each subsystem. The paper presents necessary and sufficient conditions for the existence of such a controller. The proposed design is based on recent absolute stabilization and minimax optimal control results and employs solutions of a set of game-type Riccati algebraic equations arising in H_∞ control.     

Optimal semistable control for continuous-time linear systems

In this paper, we develop a new H₂ semistability theory for linear dynamical systems. Specifically, necessary and sufficient conditions based on the new notion of weak semiobservability for the existence of solutions to the semistable Lyapunov equation are derived. Unlike the standard H₂ optimal control problem, a complicating feature of the H₂ optimal semistable control problem is that the semistable Lyapunov equation can admit multiple solutions. We characterize all the solutions using matrix analysis tools. With this theory, we present a new framework to design H₂ optimal semistable controllers for linear coupled systems by converting the original optimal control problem into a convex optimization problem.     

On robust asymptotic tracking: perturbations on coprime factors andparameterization of all solutions

The robust asymptotic tracking problem is analyzed in this paper relative to unstructured perturbations on each coprime factor of the transfer functions from the control input to the measured and controlled outputs. In each case, necessary and sufficient conditions for the existence of solutions are presented which are explicitly given in terms of problem data. Under such conditions, explicit parameterizations are given of all controllers which achieve robust asymptotic tracking, in terms of free, rational proper, and stable matrices     

A fuzzy self-tuning parallel genetic algorithm for optimization

The genetic algorithm (GA) is now a very popular tool for solving optimization problems. Each operator has its special approach route to a solution. For example, a GA using crossover as its major operator arrives at solutions depending on its initial conditions. In other words, a GA with multiple operators should be more robust in global search. However, a multiple operator GA needs a large population size thus taking a huge time for evaluation. We therefore apply fuzzy reasoning to give effective operators more opportunity to search while keeping the overall population size constant. We propose a fuzzy self-tuning parallel genetic algorithm (FPGA) for optimization problems. In our test case FPGA there are four operators—crossover, mutation, sub-exchange, and sub-copy. These operators are modified using the eugenic concept under the assumption that the individuals with higher fitness values have a higher probability of breeding new better individuals. All operators are executed in each generation through parallel processing, but the populations of these operators are decided by fuzzy reasoning. The fuzzy reasoning senses the contributions of these operators, and then decides their population sizes. The contribution of each operator is defined as an accumulative increment of fitness value due to each operator's success in searching. We make the assumption that the operators that give higher contribution are more suitable for the typical optimization problem. The fuzzy reasoning is built under this concept and adjusts the population sizes in each generation. As a test case, a FPGA is applied to the optimization of the fuzzy rule set for a model reference adaptive control system. The simulation results show that the FPGA is better at finding optimal solutions than a traditional GA.     

Tuning of PID controller based on a multiobjective genetic algorithm applied to a robotic manipulator

Most controllers optimization and design problems are multiobjective in nature, since they normally have several (possibly conflicting) objectives that must be satisfied at the same time. Instead of aiming at finding a single solution, the multiobjective optimization methods try to produce a set of good trade-off solutions from which the decision maker may select one. Several methods have been devised for solving multiobjective optimization problems in control systems field. Traditionally, classical optimization algorithms based on nonlinear programming or optimal control theories are applied to obtain the solution of such problems. The presence of multiple objectives in a problem usually gives rise to a set of optimal solutions, largely known as Pareto-optimal solutions. Recently, Multiobjective Evolutionary Algorithms (MOEAs) have been applied to control systems problems. Compared with mathematical programming, MOEAs are very suitable to solve multiobjective optimization problems, because they deal simultaneously with a set of solutions and find a number of Pareto optimal solutions in a single run of algorithm. Starting from a set of initial solutions, MOEAs use iteratively improving optimization techniques to find the optimal solutions. In every iterative progress, MOEAs favor population-based Pareto dominance as a measure of fitness. In the MOEAs context, the Non-dominated Sorting Genetic Algorithm (NSGA-II) has been successfully applied to solving many multiobjective problems. This paper presents the design and the tuning of two PID (Proportional–Integral–Derivative) controllers through the NSGA-II approach. Simulation numerical results of multivariable PID control and convergence of the NSGA-II is presented and discussed with application in a robotic manipulator of two-degree-of-freedom. The proposed optimization method based on NSGA-II offers an effective way to implement simple but robust solutions providing a good reference tracking performance in closed loop.     

State-space approach to discrete-time 2-optimal control with a causality constraint

Controller design with a causality constraint arises in periodic or multirate control systems. In this paper complete state-space solutions to the optimal and suboptimal ₂ control problems are developed with a causality constraint on controller feedthrough terms. Explicit formulas for the controllers are given in terms of solutions of two Riccati equations. The results are more implementable than existing frequency-domain solutions.