Fastest recovery after feedback disruption

The problem of quickly reducing operating errors during recovery from a feedback disruption is considered. The objective is to design controllers that reduce operating errors as quickly as possible, once feedback has been restored. It is shown that robust optimal feedback controllers that achieve this objective do exist. Furthermore, it is shown that the performance of optimal controllers can be approximated as closely as desired by controllers that generate bang–bang input signals for the controlled system. Controllers that generate bang–bang signals are relatively easy to derive and implement, since bang–bang signals are characterised by a finite list of scalars – their switching times.     

Robust Optimal Control of Nonlinear Systems With System Disturbance During Feedback Disruption

In this paper, a robust optimal control problem of nonlinear systems with system disturbance during feedback disruption is considered. This is an extended work of previous time‐delay optimal control results, by adding external disturbance in the considered system. It is shown that there exists an optimal input signal which keeps the performance error within the specified bound for the longest time. Then, it is shown that such an optimal input signal can be approximated by an implementable bang‐bang input signal in terms of control performance. Two examples are given for illustration.     

Two-degree-of-freedom dead-beat control system with robustness: multivariable case

A design procedure for two-degree-of-freedom (TDF) dead-beat controllers for multivariable systems is presented. Based on the parametrization of all stabilizing TDF controllers, a minimal-time dead-beat control system with optimal robustness is constructed for an arbitrary reference input. The optimal robustness is achieved under the constraint of robust tracking. A comparison of a TDF system with a one-degree-of-freedom (ODF) system reveals that, under the same robustness criterion, the optimal robustness of the TDF system is always superior to that of the ODF system irrespective of its settling time. It is also shown that the optimal robustness of the ODF system becomes identical to that of the TDF only when the settling time of the ODF system goes to infinity. Some numerical examples are given to illustrate the theoretical results.     

Periodic sampling: maximising the sampling period

The process of periodic sampling is investigated for a class of nonlinear systems. The objective is to achieve the longest sampling period that is compatible with a specified error bound. It is shown that there are robust optimal controllers that achieve this objective. It is also shown that the performance of such optimal controllers can be approximated by bang-bang controllers – controllers that are relatively easy to design and implement.     

输入随机的不可靠柔性制造系统的反馈控制

对于具有随机输入和随机需求的一类不可靠柔性制造系统,利用转移率一致化技术和随机动态规划方法,给出了输入率和服务率分配的最优反馈控制策略,指出系统的最优控制具有bang-bang形式的天关结构,数值例子验证了文中的结果。     

Optimal control of batteries with fully and partially available rechargeability

Motivated by the increasing dependence of many systems on battery energy, we study the problem of optimally controlling how to discharge and recharge a non-ideal battery so as to maximize the work it can perform over a given time period and still maintain a desired final energy level. Modeling a battery as a dynamic system, we adopt a Kinetic Battery Model (KBM) and formulate a finite-horizon optimal control problem when recharging is always feasible under the constraint that discharging and recharging cannot occur at the same time. The solution is shown to be of bang–bang type with the property that the battery is always in recharging mode during the last part of the interval. When the length of the time horizon exceeds a critical value, we also show that the optimal policy includes chattering. Numerical results are included to illustrate our analysis. We then extend the problem to settings where recharging is only occasionally feasible and show that it can be reduced to a nonlinear optimization problem which can be solved at least numerically.     

一类模糊P I D 控制器的鲁棒优化设计

研究一类模糊 PID控制器的鲁棒设计。以小增益定理分析得到该模糊 PID控制系统稳定性条件。针对参数摄动系统的“最坏点”,用该稳定性条件作为约束 ,采用遗传算法对标称系统的性能进行优化 ,求得优化鲁棒控制器。以倒立摆为例进行鲁棒模糊 PID控制器的设计 ,实验结果表明了该方法的有效性     

Robust Quadratic-Optimal Control of TS-Fuzzy-Model-Based Dynamic Systems With Both Elemental Parametric Uncertainties and Norm-Bounded Approximation Error

This paper considers the design problem of the robust quadratic-optimal parallel-distributed-compensation (PDC) controllers for Takagi–Sugeno (TS) fuzzy-model-based control systems with both elemental parametric uncertainties and norm-bounded approximation error. By complementarily fusing the robust stabilizability condition, the orthogonal functions approach (OFA), and the hybrid Taguchi genetic algorithm (HTGA), an integrative method is presented in this paper to design the robust quadratic-optimal PDC controllers such that 1) the uncertain TS-fuzzy-model-based control systems can be robustly stabilized, and 2) a quadratic integral performance index for the nominal TS-fuzzy-model-based control systems can be minimized. In this paper, the robust stabilizability condition is proposed in terms of linear matrix inequalities (LMIs). By using the OFA and the LMI-based robust stabilizability condition, the robust quadratic-optimal PDC control problem for the uncertain TS-fuzzy-model-based dynamic systems is transformed into a static constrained-optimization problem represented by the algebraic equations with constraint of LMI-based robust stabilizability condition, thus greatly simplifying the robust optimal PDC control design problem. Then, for the static constrained-optimization problem, the HTGA is employed to find the robust quadratic-optimal PDC controllers of the uncertain TS-fuzzy-model-based control systems. Two design examples of the robust quadratic-optimal PDC controllers for an uncertain inverted pendulum system and an uncertain nonlinear mass–spring–damper mechanical system are given to demonstrate the applicability of the proposed integrative approach.      

Delay-dependent robust stability and H8 control of jump linear systems

In this paper we develop an optimality-based robust control framework for non-linear uncertain systems with structured parametric uncertainty. Specifically, using an optimal non-linear robust control framework, we develop a family of globally stabilizing robust backstepping controllers parametrized by the cost functional that is minimized. Furthermore, it is shown that the Lyapunov function for the closed-loop system guaranteeing robust stability over a prescribed range of structured system parametric uncertainty is a solution to the steady-state Hamilton-Jacobi-Bellman equation for the controlled system and thus guarantees robust stability and robust performance. The results are then used to design robust controllers for jet engine compression systems with uncertain system dynamics.     

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.     

Synthesis of minimax optimal controllers for uncertain time-delay systems with structured uncertainty

This paper is concerned with the design of robust state feedback controllers for a class of uncertain time-delay systems. The uncertainty is assumed to satisfy a certain integral quadratic constraint. The controller proposed is a minimax optimal controller in the sense that it minimizes the maximum value of a corresponding linear quadratic cost function over all admissible uncertainties. The controller leads to an absolutely stable closed loop uncertain system and is constructed by solving a finite dimensional parameter-dependent algebraic Riccati equation.     

Fastest recovery after feedback disruption: nonlinear delay-differential systems

The problem of reducing operating errors after a feedback disruption is considered for a class of nonlinear delay-differential systems. It is shown that there are optimal controllers that reduce such errors in minimal time, once feedback has been restored. It is further shown that optimal performance can be approximated by bang-bang controllers – controllers that are easy to design and implement.     

Optimal control of saturable linear plants for minimum integral-square-error and similar performance criteria†

The paper deals with the problem of minimizing integral–square–error and similar performance indices for linear single–variable plants with saturating control input. It is shown that the optimal control can be obtained by treating it as the limit as υ→∞ of the Sequence of controls which minimize the performance index on the class of all bang–bang controls containing not more than υ switches. These optimal υ switch controls are shown to satisfy a modified maximum principle.The method enables the computation of optimal controls from any given initial plant state. The particular problems of minimizing integral–modulus–error for a two–integrator plant and integral–square–error for a three–integrator plant are worked out in detail so as to illu3trata the techniques involved.     

Synthesis of fixed-structure robust controllers using a constrained particle swarm optimizer with cyclic neighborhood topology

The purpose of this paper is to develop design scheme based on an easy-to-use meta-heuristic approach with superior reliability and validity for fixed-structure robust controllers, satisfying both multiple control specifications and system stability conditions. For this purpose, a particle swarm optimizer is first developed, which reduces the probability of premature convergence to local optima in the PSO (particle swarm optimization) by exploiting the particle’s local social learning based on the idea of cyclic-network topology. Next, it is shown how to obtain a fixed-structure robust controller with constraints on multiple H_∞ specifications and system stability based on the developed PSO technique incorporated with a simple constraint handling method. Finally, typical numerical examples are studied to show the applicability of the proposed methodology to the synthesis of fixed-structure robust controllers. These examples clearly verify that the developed design scheme gives a novel and powerful impetus with remarkable reliability to fixed-structure robust controller syntheses.     

Loop‐Shaping Design Of Pid Controllers With Constant Ti/Td RATIO

This paper presents new tuning rules for PID controllers based on loop shaping. Previous research has shown that maximization of the integral gain subject to constraints on the maximum sensitivity is an efficient design method for PI controllers, but that additional constraints are needed for design of PID controllers. In this paper, an additional constraint is obtained by restricting the ratio between integral time and derivative time. It is shown that this gives a numerically efficient method that gives good results for a large class of processes.     

Optimality and stability in a class of bang–bang controlled biochemical reaction systems

This paper deals with the optimization of biochemical reaction systems of rank one. Two optimization problems are solved: the problem of optimal operation for maximum productivity in steady state and the problem of the start-up to the optimal steady state. Application of Pontryagin's maximum principle shows that the controller is of the bang–bang type, with no singular intervals. The determination of the optimal switching surface involves the solution of a two point boundary value problem. Solving such a difficult problem is avoided by choosing candidate switching surfaces on a heuristic basis. This study shows that switching on the stability boundary of the nominal operating point corresponding to the maximum dilution rate is the best choice. Here the value of the cost index is minimum amongst the various switching surfaces considered and the stability boundary satisfies the conditions imposed on a candidate switching surface for proper operation. Simple, robust algorithms are formulated for accurately estimating the system's stability boundary. The obtained results display the influence of feedback control on the stability of the set point. The bang–bang controller substantially increases the set point's region of attraction in state space as compared to the uncontrolled bioreactor.     

An optimal control approach to the decentralized robustservomechanism problem

A linear-quadratic optimal control approach is presented for designing linear time-invariant decentralized controllers to solve the robust servomechanism problem. It is assumed that not only the plant and the measurements but also the controllers may be subject to certain disturbances. A physically meaningful quadratic cost functional is defined, making it possible to assign desired weights to the errors, to their derivatives, to the system effort, and to the controller efforts. It is shown that the resulting controlled system has many properties like asymptotic tracking, stability, and robustness. The solution does not require much control effort at high frequencies, and the system is guaranteed not to have high frequency oscillations     

Mixed H2/H∞ control:a convex optimization approach

The problem of finding an internally stabilizing controller that minimizes a mixed H₂/H_∞ performance measure subject to an inequality constraint on the H_∞ norm of another closed-loop transfer function is considered. This problem can be interpreted and motivated as a problem of optimal nominal performance subject to a robust stability constraint. Both the state-feedback and output-feedback problems are considered. It is shown that in the state-feedback case one can come arbitrarily close to the optimal (even over full information controllers) mixed H₂/H_∞ performance measure using constant gain state feedback. Moreover, the state-feedback problem can be converted into a convex optimization problem over a bounded subset of (n×n and n ×q, where n and q are, respectively, the state and input dimensions) real matrices. Using the central H_∞ estimator, it is shown that the output feedback problem can be reduced to a state-feedback problem. In this case, the dimension of the resulting controller does not exceed the dimension of the generalized plant     

Optimal linear stochastic systems with independent controllers

The problem of synthesis of optimal controllers for linear stochastic systems with independent control channels is considered. In the considered systems two independent controllers with independent measurement devices are used. The equations for optimal cost function, covariance matrix of the state estimation error and for the controlled system output are derived. It is shown that the application of a well-known separation principle or certainty equivalence principle results in controllers which are not optimal and which can be far from the optimality. It is also shown that improvement of the estimation accuracy by means of manipulation of the controller parameters can significantly improve the control performance as a whole. The numerical examples of calculation of the optimal independent controllers are given to demonstrate the obtained theoretical results and superiorities over controllers based on the separation principle. It is also demonstrated that the optimal controllers have smaller control gains for the control loops with larger observation noises, which can be characterized as cautious properties of the controllers.