On optimal control of non-autonomous switched systems with a fixed mode sequence

We consider differentiability with respect to the switch times of the value function of an optimal control problem for a non-autonomous switched system. The control variables are the switch times between the modes and the input in each mode. We provide a method to compute the derivative of the cost function given a nominal input. Then, we view the optimal control problem as a parametrized optimization problem in which the switch times are the parameters and the optimization is over the set of feasible inputs of each mode. From this point of view, we provide conditions under which the continuity and differentiability of the optimal value function, that is the cost function optimized over the inputs, can be guaranteed.     

A moving switching sequence approach for nonlinear model predictive control of switched systems with state‐dependent switches and state jumps

In this paper, we propose an approach for real‐time implementation of nonlinear model predictive control (NMPC) for switched systems with state‐dependent switches called the moving switching sequence approach. In this approach, the switching sequence on the horizon moves to the present time at each time as well as the optimal state trajectory and the optimal control input on the horizon. We assume that the switching sequence is basically invariant until the first predicted switching time reaches the current time or a new switch enters the horizon. This assumption is reasonable in NMPC for systems with state‐dependent switches and reduces computational cost significantly compared with the direct optimization of the switching sequence all over the horizon. We update the switching sequence by checking whether an additional switch occurs or not at the last interval of the present switching sequence and whether the actual switch occurs or not between the current time and the next sampling time. We propose an algorithm consisting of two parts: (1) the local optimization of the control input and switching instants by solving the two‐point boundary‐value problem for the whole horizon under a given switching sequence and (2) the detection of an additional switch and the reconstruction of the solution taking into account the additional switch. We demonstrate the effectiveness of the proposed method through numerical simulations of a compass‐like biped walking robot, which contains state‐dependent switches and state jumps.     

Domain decomposition and model reduction for the numerical solution of PDE constrained optimization problems with localized optimization variables

We introduce a technique for the dimension reduction of a class of PDE constrained optimization problems governed by linear time dependent advection diffusion equations for which the optimization variables are related to spatially localized quantities. Our approach uses domain decomposition applied to the optimality system to isolate the subsystem that explicitly depends on the optimization variables from the remaining linear optimality subsystem. We apply balanced truncation model reduction to the linear optimality subsystem. The resulting coupled reduced optimality system can be interpreted as the optimality system of a reduced optimization problem. We derive estimates for the error between the solution of the original optimization problem and the solution of the reduced problem. The approach is demonstrated numerically on an optimal control problem and on a shape optimization problem.     

Optimal almost-periodic control problem: a trigonometric approximation approach

The problem of finding an almost-periodic control that is optimal with respect to a certain time-averaged criterion for the dynamic system operated over a long period of time is considered. The existence of the optimal solution, spectral properties of which satisfy certain regularity conditions, is hypothesized. The problem is approximated by a sequence of finite-dimensional optimization problems containing trigonometric sums for the approximation of the state and control variables, and using a Fejér-Riesz type representation for a positive trigonometric sum to handle the instantaneous constraints for these variables. Sufficient conditions for the sequence of approximate optimal solutions of the discretized problems to be an approximately minimizing sequence for the basic problem are given. The constructive character of the proposed approach and its potential applications are pointed out both for dynamic systems affected by irregularly pulsating disturbances and for stationary systems, the non-linear dynamics of which can be exploited by a non-stationary control to improve the averaged system performance.     

An efficient sequential linear quadratic algorithm for solving nonlinear optimal control problems

We develop a numerically efficient algorithm for computing controls for nonlinear systems that minimize a quadratic performance measure. We formulate the optimal control problem in discrete-time, but many continuous-time problems can be also solved after discretization. Our approach is similar to sequential quadratic programming for finite-dimensional optimization problems in that we solve the nonlinear optimal control problem using sequence of linear quadratic subproblems. Each subproblem is solved efficiently using the Riccati difference equation. We show that each iteration produces a descent direction for the performance measure, and that the sequence of controls converges to a solution that satisfies the well-known necessary conditions for the optimal control.     

Design of optimal controllers for spatially invariant systems with finite communication speed

We consider the problem of designing optimal distributed controllers whose impulse response has limited propagation speed. We introduce a state-space framework in which all spatially invariant systems with this property can be characterized. After establishing the closure of such systems under linear fractional transformations, we formulate the H₂ optimal control problem using the model-matching framework. We demonstrate that, even though the optimal control problem is non-convex with respect to some state-space design parameters, a variety of numerical optimization algorithms can be employed to relax the original problem, thereby rendering suboptimal controllers. In particular, for the case in which every subsystem has scalar input disturbance, scalar measurement, and scalar actuation signal, we investigate the application of the Steiglitz–McBride, Gauss–Newton, and Newton iterative schemes to the optimal distributed controller design problem. We apply this framework to examples previously considered in the literature to demonstrate that, by designing structured controllers with infinite impulse response, superior performance can be achieved compared to finite impulse response structured controllers of the same temporal degree.     

On diffusion approximation of controlled queueing processes

Consider a queueing system that can be controlled by switching service rates. When there is a cost to switch the service rate, the control problem turns out to be a sequential decision problem, i.e. to find a sequence of optimal stopping times to switch the service rate. Under heavy traffic conditions, we show the optimal cost functions of a sequence of controlled rescaled queueing processes converge to that of a corresponding diffusion for a finite time and an infinite time with discount factor criterions.     

An integrated state space partition and optimal control method of multi-model for nonlinear systems based on hybrid systems

Multilinear model approach turns out to be an ideal candidate for dealing with nonlinear systems control problem. However, how to identify the optimal active state subspace of each linear subsystem is an open problem due to that the closed-loop performance of nonlinear systems interacts with these subspaces ranges. In this paper, a new systematic method of integrated state space partition and optimal control of multi-model for nonlinear systems based on hybrid systems is initially proposed, which can deal with the state space partition and associated optimal control simultaneously and guarantee an overall performance of nonlinear systems consequently. The proposed method is based on the framework of hybrid systems which synthesizes the multilinear model, produced by nonlinear systems, in a unified criterion and poses a two-level structure. At the upper level, the active state subspace of each linear subsystem is determined under the optimal control index of a hybrid system over infinite horizon, which is executed off-line. At the low level, the optimal control is implemented online via solving the optimal control of hybrid system over finite horizon. The finite horizon optimal control problem is numerically computed by simultaneous method for speeding up computation. Meanwhile, the model mismatch produced by simultaneous method is avoided by using the strategy of receding-horizon. Simulations on CSTR (Continuous Stirred Tank Reactor) confirm that a superior performance can be obtained by using the presented method.     

Stacking sequence optimization with genetic algorithm using a two-level approximation

We propose a new method for laminate stacking sequence optimization based on a two-level approximation and genetic algorithm (GA), and establish an optimization model including continuous size variables (thicknesses of plies) and discrete variables (0/1 variables that represent the existence of each ply). To solve this problem, a first-level approximate problem is constructed using the branched multipoint approximate (BMA) function. Since mixed-variables are involved in the first-level approximate problem, a new optimization strategy is introduced. The discrete variables are optimized through the GA. When calculating the fitness of each member in the population of GA, a second-level approximate problem that can be solved by the dual method is established to obtain the optimal thicknesses corresponding to the each given ply orientation sequence. The two-level approximation genetic algorithm optimization is performed starting from a ground laminate structure, which could include relatively arbitrarily discrete set of angles. The method is first applied to cylindrical laminate design examples to demonstrate its efficiency and accuracy compared with known methods. The capacity of the optimization strategy to solve more complex problems is then demonstrated using a design example. With the presented method, the stacking sequence in analytical tools can be directly taken as design variables and no intermediate variables need be adopted.     

Optimal control laws for lot-sizing and timing of jobs on a single production facility

The optimal control of a single machine processing a certain number of jobs and modeled as a discrete-event dynamic system is considered. The number of jobs and their sequence are fixed, whereas their timing and sizes represent the control variables of the system. The objective function to be optimized is a weighted sum of the quadratic earliness and tardiness of each job, and of the quadratic deviations of job lot sizes and actual machine service speeds from those specified by the production demand and by the regular machine speeds. An optimization problem with quadratic cost function and nonlinear constraints is stated and formalized as a multistage optimal control problem. Necessary conditions to be satisfied by an optimal control sequence are derived. A simpler model is also considered in which the machine speeds are fixed; in this case, the control problem is solved by a procedure making use of dynamic programming techniques. The optimal control laws at each stage are thus obtained.     

Optimal control of continuous-time switched affine systems

This paper deals with optimal control of switched piecewise affine autonomous systems, where the objective is to minimize a performance index over an infinite time horizon. We assume that the switching sequence has a finite length, and that the decision variables are the switching instants and the sequence of operating modes. We present two different approaches for solving such an optimal control problem. The first approach iterates between a procedure that finds an optimal switching sequence of modes, and a procedure that finds the optimal switching instants. The second approach is inspired by dynamic programming and identifies the regions of the state space where an optimal mode switch should occur, therefore providing a state feedback control law.     

A Novel Approach for Optimal Cost-Effective Design of Complex Repairable Systems

Almost every engineering and manufacturing system consists of several subsystems, which are in general nonidentical and are subjected to stochastic failures and repairs. The system success logic can be represented using a combinatorial reliability model in terms of the states of subsystems, where as the success logic of each subsystem can be represented using a k-out-of- n structure. The long run cost associated with the downtime can be lowered by adding additional spares in each subsystem, which in turn can increase the operational and maintenance costs. Thus, it is desirable to find the optimal number of components in each subsystem that minimizes the overall cost associated with the system. The main contributions of this paper are the following: 1) formulation of an average cost function of complex repairable systems and 2) development of a new method to obtain tighter bounds for the optimal number of spares for each subsystem. The tighter bounds are extremely useful to reduce the search space and hence improve the efficiency of the optimization algorithm. With the proposed bounds, for a series system consisting of m parallel subsystems, the computational complexity to find the near optimal solution, which is the optimal solution in most cases, is O(m)     

Optimal control problems with multiple characteristic time points in the objective and constraints

In this paper, we develop a computational method for a class of optimal control problems where the objective and constraint functionals depend on two or more discrete time points. These time points can be either fixed or variable. Using the control parametrization technique and a time scaling transformation, this type of optimal control problem is approximated by a sequence of approximate optimal parameter selection problems. Each of these approximate problems can be viewed as a finite dimensional optimization problem. New gradient formulae for the cost and constraint functions are derived. With these gradient formulae, standard gradient-based optimization methods can be applied to solve each approximate optimal parameter selection problem. For illustration, two numerical examples are solved.     

Time-domain decomposition of optimal control problems for the wave equation

We consider the problem of boundary optimal control of a wave equation with boundary dissipation by the way of time-domain decomposition of the corresponding optimality system. We develop an iterative algorithm which shows that the decomposed optimality system corresponds to local-in-time optimal control problems which can be treated in parallel. We show convergence of the algorithm. Finally, we provide a time discretization which is reminiscent of an instantaneous control scheme. We thereby also contribute to the problem of convergence of such schemes.     

Trigonometric approximation of optimal periodic control problems for distributed-parameter systems

The optimal periodic control problem for a system described by first order partial differential equations is approximated by a sequence of discretized optimization problems. Trigonometric polynomials in two variables are used in the latter problems to approximate the state trajectory, the control and functions appearing in differential equations and in the criterion of the basic problem. The state equations and the instantaneous constraints on the state and the control are taken into account by the mixed exterior-interior penalty function. Sufficient conditions are given for the convergence of solutions of discretized problems to the optimal solution of the basic problem. The possibility of applying the method to a class of optimal periodic control problems in chemical engineering is emphasized.     

Optimal control based on a preposteriori estimates of set-membership uncertainty

We consider the optimal control problem for a linear nonstationary dynamical system under set-membership uncertainty with a combined discrete closable loop. Our solution is based on an a preposteriori analysis of the surveillance and control subsystems. Based on the surveillance subsystem analysis, we introduce closures and construct an optimal closable program (a preposteriori analysis of the control subsystem) that yields a positional solution for the optimal control problem. We present an optimal control quasi-realization method with optimal estimators and a real-time controller. We illustrate our results with an example.     

Optimal disturbance rejection control for singularly perturbed composite systems with time‐delay

The optimal control problem for a class of singularly perturbed time‐delay composite systems affected by external disturbances is investigated. The system is decomposed into a fast linear subsystem and a slow time‐delay subsystem with disturbances. For the slow subsystem, the feedforward compensation technique is proposed to reject the disturbances, and the successive approximation approach (SAA) is applied to decompose it into decoupled subsystems and solve the two‐point boundary value (TPBV) problem. By combining with the optimal control law of the fast subsystem, the feedforward and feedback composite control (FFCC) law of the original composite system is obtained. The FFCC law consists of analytic state feedback and feedforward terms and a compensation term which is the limit of the adjoint vector sequence. The compensation term can be obtained from an iteration formula of adjoint vectors. Simulation results are employed to test the validity of the proposed design algorithm. Copyright © 2009 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Optimal tracking control for nonlinear large-scale systems with persistent disturbances

This paper studies the optimal control with zero steady-state error problem for nonlinear large-scale systems affected by external persistent disturbances. The nonlinear large-scale system is transformed into N nonlinear subsystems with interconnect terms. Based on the internal model principle, a disturbance compensator is constructed such that the ith subsystem with external persistent disturbances is transformed into an augmented subsystem without disturbances. According to the sensitivity approach, the optimal tracking control law for the ith nonlinear subsystem can be obtained. The optimal tracking control law for the nonlinear large-scale systems can be obtained. A numerical simulation shows that the method is effective.     

事件触发双模分布式预测控制

本文针对有界扰动作用下的线性离散大系统,提出了事件触发双模分布式预测控制设计方法.利用输入状态稳定性(input-to-state stability,ISS)理论建立了仅与子系统自身信息相关的事件触发条件.只有子系统满足相应的事件触发条件,才进行状态信息的传输和分布式预测控制优化问题的求解,并与邻域子系统交互最优解作用下的关联信息.当子系统进入不变集时,采用状态反馈控制律进行镇定,并与进入不变集的邻域子系统不再交互信息.分析了算法的递推可行性和系统的闭环稳定性,给出了扰动的上界.最后,通过车辆控制系统对算法进行仿真验证,结果表明,本文提出的方法能够有效降低优化问题的求解次数和关联信息的交互次数,节约计算资源和通信资源.     

Rapid distributed model predictive control design using singular value decomposition for linear systems

The issue of model predictive control design of distribution systems using a popular singular value decomposition (SVD) technique is addressed. Namely, projection to a set of conjugate structure is dealt with in this paper. The structure of the resulting predictive model is decomposed into small sets of subsystems. The optimal inputs can be separately designed at each subsystem in parallel without any interaction problems. The optimal inputs can be directly obtained and the communication among the subsystems can be significantly reduced. In addition, the design of distribution model predictive control (DMPC) with constraints using the SVD framework is also presented. The unconstraint inputs are checked in parallel in the conjugate space. Without solving the QP problem of each subsystem, the suboptimal solution can be quickly obtained by selecting the bigger singular values and discarding the small singular values in the singular value space. The convergence condition of the proposed algorithm is also proved. Two case studies are used to illustrate the distribution control systems using the suggested approach. Comparisons between the centralized model predictive control method and the proposed DMPC method are carried out to show the advantages of the newly proposed method.