Optimal and suboptimal control over bunches of trajectories of automaton-type deterministic systems

The problem of optimal control of automatic deterministic discrete systems in conditions of parametric uncertainty is considered. The state change (switchover) of a system is described by a recurrent equation in which instant multiple switchovers are admitted. The times and the number of switchovers are not specified in advance—they are found as a result of optimization of a functional. The initial state of the system is not known exactly, but the set of possible initial states is known; therefore, problems of average optimal control and guaranteed optimal control of bunches of trajectories are formulated. For solving these problems, it is suggested to use the principle of separation: namely, to control the bunch of trajectories by applying a control optimal for one specially chosen trajectory of the bunch. The resulting control of the bunch is suboptimal, but it may be satisfactory in practice. On the basis of sufficient conditions for optimality of a full-feedback control, an algorithm for constructing a suboptimal control in conditions of uncertainty is developed. Counterexamples of linear–quadratic problems of control of automaton-type systems in which the principle of separation fails are presented.     

Suboptimal control of bunches of trajectories of discrete deterministic automaton time-invariant systems

The optimal control problem for discrete deterministic time-invariant automaton systems under parametric uncertainty is considered. The changes in the state (switching) of the system are described by a recurrence equation. The switching times and their number are not specified in advance—they are found by optimization. The initial state of the system is not known exactly. For this reason, the problems of finding the optimal on the average and the optimal guaranteeing (minimax) controls of bunches of trajectories are formulated. It is proposed to solve these problems using the separation principle, which assumes that the bunch is controlled using the control that is optimal for a specially chosen individual trajectory in the bunch. Generally, this control is suboptimal. Algorithms for the synthesis of suboptimal control of bunches are developed. It is proved that, in the linear-quadratic time-invariant problems, the suboptimal bunch control is optimal on the average or is the optimal minimax control, depending on the problem being solved.     

Constrained optimal steady-state control for isolated traffic intersections

The steady-state or cyclic control problem for a simplified isolated traffic intersection is considered. The optimization problem for the green-red switching sequence is formulated with the help of a discrete-event max-plus model. Two steady-state control problems are formulated： optimal steady-state with green duration constraints, and optimal steady-state control with lost time. In the case when the criterion is a strictly increasing, linear function of the queue lengths, the steady-state control problems can be solved analytically. The structure of constrained optimal steady-state traffic control is revealed, and the effect of the lost time on the optimal solution is illustrated.     

Planning rigid body motions using elastic curves

This paper tackles the problem of computing smooth, optimal trajectories on the Euclidean group of motions SE(3). The problem is formulated as an optimal control problem where the cost function to be minimized is equal to the integral of the classical curvature squared. This problem is analogous to the elastic problem from differential geometry and thus the resulting rigid body motions will trace elastic curves. An application of the Maximum Principle to this optimal control problem shifts the emphasis to the language of symplectic geometry and to the associated Hamiltonian formalism. This results in a system of first order differential equations that yield coordinate free necessary conditions for optimality for these curves. From these necessary conditions we identify an integrable case and these particular set of curves are solved analytically. These analytic solutions provide interpolating curves between an initial given position and orientation and a desired position and orientation that would be useful in motion planning for systems such as robotic manipulators and autonomous-oriented vehicles.     

An application of model predictive control to the dynamic economic dispatch of power generation

Two formulations exist for the problem of the optimal power dispatch of generators with ramp rate constraints: the optimal control dynamic dispatch (OCDD) formulation based on control system models, and the dynamic economic dispatch (DED) formulation based on optimization. Both are useful for the dispatch problem over a fixed time horizon, and they were treated as equivalent formulations in literature. This paper first shows that the two formulations are in fact different and both formulations suffer from the same technical deficiency of ramp rate violation during the periodic implementation of the optimal solutions. Then a model predictive control (MPC) approach is proposed to overcome such a technical deficiency. Furthermore, it is shown that the MPC solutions, which are based on the OCDD framework, converge to the optimal solution of an extended version of the DED problem and they are robust under certain disturbances and uncertainties. Two standard examples are studied: the first one of a ten-unit system shows the difference between the OCDD and DED, and possible ramp rate violations, and the second one of a six-unit system shows the convergence and robustness of the MPC solutions, and the comparison with OCDD as well.     

Kinematic Control of Omni-directional Robots for Time-optimal Movement between Two Configurations

The main goal of this study is to investigate the time-optimal control problem of an omni-directional mobile robot between two configurations. In the proposed method, this problem is formulated and solved as a constrained nonlinear programming (NLP) one. During the optimization process, the count of control steps is fixed initially and the sampling period is treated as a variable to be determined. The goal is to minimize the sampling period such that it is below a specific minimum value, which is set in advance considering the accuracy of discretization. To generate initial feasible solutions of the NLP problem, a systematic approach is also proposed. Since different initial feasible solutions can be generated, the optimization process of the NLP problem can be started from many different points to find the optimal solution. To show the feasibility of the proposed method, simulation and experimental results are included for illustration.     

Dynamic start-up optimization of a plate reactor with uncertainties

In this paper, start-up of a plate reactor with an exothermic reaction is considered. Dynamic optimization is used to obtain start-up trajectories, while a feedback control system is designed for on-line control. The problem is challenging, since the process model is highly nonlinear, and subject to uncertainty. Limited bandwidth of the feedback control system requires that the optimization gives optimal solutions with low sensitivity to uncertainty. Special attention is given to the problem of formulating an optimal control problem based on physical insight to reduce the sensitivity of the solution, thus increasing the overall robustness of the closed loop system. The resulting solutions are evaluated in Monte Carlo simulations. The study shows that introducing concentration constraints on the injected chemical in combination with high frequency penalties on the control signals lead to less sensitive optimal solutions, thus yielding a more robust start-up. Automatic computer tools that greatly simplifies the task of formulating complex dynamic optimization problems have been used to obtain the start-up trajectories and are briefly presented.     

Supervisory control of switching control systems

In this paper, the problem of designing a switching policy for an adaptive switching control system is formulated as a problem of supervisory control of a discrete-event system (DES). Two important problems in switching control are then addressed using the DES formulation and the theory of supervisory control under partial observation. First, it is verified whether for a given set of controllers, a switching policy satisfying a given set of constraints on the transitions among controllers exists. If so, then a minimally restrictive switching policy is designed. Next, an iterative algorithm is introduced for finding a minimal set of controllers for which a switching policy satisfying the switching constraints exists. It is shown that in the supervisory control problem considered in this paper, limitations on event observation are the factors that essentially restrict supervisory control. In other words, once observation limitations are respected, limitations on control will be automatically satisfied. This result is used to simplify the proposed iterative algorithm for finding minimal controller sets.     

Embedding and Optimization of the Linear Systems

Provision of the given (desired) matrix transfer functions in the dynamic linear systems was considered. The possible nonuniqueness of the solution to this problem, which is characteristic of the multivariable systems, is described exhaustively by the system embedding technology. It was shown that an additional problem of optimal control can be formulated and solved on the set of the resulting equivalent solutions. At that, the main characteristic of the optimization problem lies in the specific parameterized representation of the set of the control laws from which the optimal solution is selected. Examples are presented.     

多速率分段线性系统预测控制的显式设计

针对输出采样周期是输入更新周期N倍的多速率分段线性(Piecewise linear, PWL)系统, 本文提出了保证稳定性的显式预测控制器设计方法. 首先, 基于动态规划原理将预测控制优化问题分解为多个单级优化问题; 然后, 根据分段线性系统各子模型以及目标函数的具体形式, 进一步将各单级优化问题分为若干个子问题, 再利用多参数二次规划(Multiparametric quadratic programming, MP-QP)方法求解;最后,通过比较各子问题的解从而得到系统的最优显式控制律. 在设计过程中, 将系统的最大正不变集作为优化问题的终端约束集, 从而保证了系统的稳定性. 仿真结果表明本文提出的显式预测控制方法能够有效降低多速率分段线性系统的在线计算时间, 在保证系统稳定性的同时, 满足其对输入更新速度的要求.     

Optimal control of linear differential-difference systems of neutral type

The optimal control of linear differential-difference systems of neutral type with respect to quadratic performance criteria over an infinite time interval is treated. The linear differential-difference systems with delays in coordinates, derivatives, and control actions are analysed. Dynamic programming is used to determine the optimization equations. A set of Riccati type equations is obtained. The closed-loop system synthesis problem is discussed. The approximate optimal problem is formulated. A connection is established between the original and approximate optimal problems. Special attention is paid to the suboptimal system synthesis problem.     

Reducing energy consumption through trajectory optimization for a metro network

The techniques of optimal control and optimization are applied to a practical problem of reducing energy consumed by the Montreal Metro (subway) system. The problem considered is the determination of tunnel trajectories in the "equivalent" vertical plane when trains traveling in both directions must follow the same trajectory. The problem is first formulated as a control problem with control and state constraints. Then, under certain simplifying assumptions, an heuristic method employing a direct search algorithm is presented and used in the trajectory optimization. The trajectories are optimized to reduce the sum of the energy consumed by the trains traveling in both directions on the trajectory. The results show an average reduction of 7.73 percent in energy consumption as compared with existing trajectories. The trajectories found using the method presented here will be followed in future tunnel construction.     

Discrete optimal control with multiple constraints I Constraint separation and transformation technique

The discrete optimal control problem with multiple inequality constraints on functions of the state and/or control variables is presented. A new transformation technique for dealing with problems in which the number of constraint functions does not exceed the number of control variables is derived. The application of this technique results in a set of control constrained system equations of the same dimensionality as the original system equations.This transformation technique is then extended by the introduction of the principle of constraint separation to cater for the more general problem in which the constraint functions outnumber the control variables. It is shown that the time interval may be split up into segments over which the number of dominant constraints does not exceed the number of control variables. The necessary conditions for extremals and the interpoint boundary conditions at the switchover points are derived and a numerical switching algorithm is presented. It is shown that the state and adjoint variables are single-valued at the switchover points and that all sub-optimal trajectories are feasible in that they will not violate the constraints imposed.Two practical examples are solved in which the constraint functions outnumber the control variables. The solution to the second example is compared with that obtained by using a penalty function approach.     

State waypoint approach to continuous‐time nonlinear optimal control problems

In this paper, we propose an optimal control technique for a class of continuous‐time nonlinear systems. The key idea of the proposed approach is to parametrize continuous state trajectories by sequences of a finite number of intermediate target states; namely, waypoint sequences. It is shown that the optimal control problem for transferring the state from one waypoint to the next is given an explicit‐form suboptimal solution, by means of linear approximation. Thus the original continuous‐time nonlinear control problem reduces to a finite‐dimensional optimization problem of waypoint sequences. Any efficient numerical optimization method, such as the interior‐reflection Newton method, can be applied to solve this optimization problem. Finally, we solve the optimal control problem for a simple nonlinear system example to illustrate the effectiveness of this approach. Copyright © 2009 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

一类优化问题的分解算法

讨论一类大规模系统的优化问题,提出一种递阶优化方法．该方法首先将原问题转化为多目标优化问题,证明了原问题的最优解在多目标优化问题的非劣解集中,给出了从多目标优化问题的解集中挑出原问题最优解的算法,建立了算法的理论基础．仿真结果验证了算法的有效性．     

Robust output‐based controller design for enlarging the region of attraction of input saturated linear systems

The aim of this study was to design a constrained robust output feedback control strategy for stabilizing linear systems affected by uncertainties and output perturbations. The input is constrained by a saturation function. A classical Luenberger observer reconstructed the states from output observations. The state feedback control employed the estimated states given by the observer. This work proposed a Barrier Lyapunov Function (BLF) to manage the saturation problem in the control input. Moreover, an extended version of the attractive ellipsoid method (AEM) characterized the zone of convergence due the presence of perturbations in the output. A convex optimization procedure, formulated as a set of matrix inequalities, yielded the control parameters and the region of attraction for the close‐loop system as well as a minimal ultimately bounded set for the system trajectories. Numerical simulations supported the theoretical results formulated in this study.     

Optimal control of discrete-time bilinear systems with applications to switched linear stochastic systems

This paper aims at characterizing the most destabilizing switching law for discrete-time switched systems governed by a set of bounded linear operators. The switched system is embedded in a special class of discrete-time bilinear control systems. This allows us to apply the variational approach to the bilinear control system associated with a Mayer-type optimal control problem, and a second-order necessary optimality condition is derived. Optimal equivalence between the bilinear system and the switched system is analyzed, which shows that any optimal control law can be equivalently expressed as a switching law. This specific switching law is most unstable for the switched system, and thus can be used to determine stability under arbitrary switching. Based on the second-order moment of the state, the proposed approach is applied to analyze uniform mean-square stability of discrete-time switched linear stochastic systems. Numerical simulations are presented to verify the usefulness of the theoretic results.     

Sensor scheduling in continuous time

Let N be the number of available sensor sources. Noisy observations of an underlying state process are available for these N sources. We consider the continuous time sensor scheduling problem in which N₁ of these N sources are to be chosen to collect data at each time point. This sensor scheduling problem (with switching costs and switching constraints) is formulated as a constrained optimal control problem. In this framework, the controls represent the sensors that are chosen at a particular time. Thus, the control variables are constrained to take values in a discrete set, and switchings between sensors can occur in continuous time. By incorporating recent results on discrete valued optimal control, we show that this problem can be transformed into an equivalent continuous optimal control problem. In this way, we obtain the sensor scheduling policy as well as the associated switching times.     

Consistent hierarchical economic NMPC for a class of hybrid systems using neighboring-extremal updates

A hierarchical two-layer control algorithm is developed for a class of hybrid (discrete-continuous dynamic) systems to support economically optimal operation of batch or continuous processes with a predefined production schedule. For this class of hybrid systems, the optimal control moves as well as the controlled switching times between two adjacent modes are determined online. In contrast to closely related schemes for integrated scheduling and control, the sequence of modes is not optimized. On the upper layer, the economic optimal control problem is solved rigorously by a slow hybrid economic model predictive controller at a low sampling rate. On the lower layer, a fast hybrid neighboring-extremal controller is based on the same economic optimal control problem as the slow controller to ensure consistency between both layers. The fast neighboring-extremal controller updates rather than tracks the optimal trajectories from the upper layer to account for disturbances. Consequently, the fast controller steers the process to its operational bounds under disturbances and the economic potential of the process is exploited anytime. The suggested two-layer control algorithm provides fully consistent control action on the fast and slow time-scale and thus avoids performance degradation and even infeasibilities which are commonly encountered if inconsistent optimal control problems are formulated and solved.