Optimal control with a cost on changing control

In this paper, we consider a class of optimal control problems in which the cost functional is the sum of the terminal cost, the integral cost, and the full variation of control. The term involving the full variation of control is to measure the changes on the control action. A computational method based on the control parametrization technique is developed for solving this class of optimal control problems. This computational method is supported by a convergence analysis. For illustration, two numerical examples are solved using the proposed method.This project was partially supported by an Australian Research Grant.This paper is dedicated to Professor L. Cesari on the occasion of his 80th birthday.     

Dynamically similar control systems and a globally minimum gain control technique: IMSC

The concept of dynamically similar control systems is introduced. The necessary and sufficient conditions to minimize a quadratic modal gain measure are given for dynamically similar closed-loop control systems. The globally minimum modal gain is obtained when the independent modal space control (IMSC) is used. Corollaries of the results for the control of infinite-dimensional structural distributed parameter systems (DPS) are given. Based on the results, a modal interaction parameter (MIP) is defined for all control systems. The minimum value of MIP is zero and uniquely corresponds to the IMSC. A nonzero value of MIP corresponds to all other coupled control (CC) designs and implies suboptimality relative to the IMSC design. The relative optimality of the real-space gain matrices of the IMSC and the CC designs depends on the actuator locations for the IMSC. Based on this, a real-space interaction parameter (RIP) is defined. A positive value of RIP renders IMSC optimal in its real-space gain matrix. The MIP and RIP are indications of suboptimality of a particular control technique and can be used to tune-up the control design via actuator locations. Actuator distribution criteria are suggested for both CC and IMSC designs, based on the values of MIP and RIP, respectively.This work was supported by the National Science Foundation, Grant No. MEA-82-04920.     

Optimal control asymptotics of a magnetohydrodynamic flow

An optimal multiplicative control problem is considered for a one-dimensional magnetohydrodynamic flow between parallel planes (Hartman flow). The external magnetic field is used as a control function. An optimality system is derived, and the asymptotics of an optimal control with respect to a regularization parameter and the Reynolds number are constructed.     

Optimal control of a large turboalternator

The optimal torque and voltage control for a large turbogenerator is found by using the minimum norm formulation. It should be noted that the model used is highly nonlinear. Numerical results are presented.This work was supported in part by the National Research Council of Canada, Grant No. A4146.     

Decentralized optimal control of a group of dynamical objects

The problem of optimal control of a group of coupled dynamical objects is considered. The cases are examined in which the centralized control of a group of objects is impossible. Fast real-time optimal control algorithms of each of the dynamical systems are described that use information exchanged between group members in the course of control. The proposed methods supplement the earlier developed real-time optimal control methods for an individual dynamical system. The results are illustrated using optimal control of two coupled mathematical pendulums as an example.     

An optimal control problem with a random stopping time

This paper deals with a stochastic optimal control problem where the randomness is essentially concentrated in the stopping time terminating the process. If the stopping time is characterized by an intensity depending on the state and control variables, one can reformulate the problem equivalently as an infinite-horizon optimal control problem. Applying dynamic programming and minimum principle techniques to this associated deterministic control problem yields specific optimality conditions for the original stochastic control problem. It is also possible to characterize extremal steady states. The model is illustrated by an example related to the economics of technological innovation.This research has been supported by NSERC-Canada, Grants 36444 and A4952; by FCAR-Québec, Grant 88EQ3528, Actions Structurantes; and by MESS-Québec, Grant 6.1/7.4(28).     

Fuzzy predictive temperature control for a class of metallurgy lime kiln models

The combustion temperature and progress control problems are key factors to ensure the production quality of metallurgy lime kiln. The combustion process of lime kiln is a nonlinear and large time‐delay thermal process, so it is difficult to achieve satisfactory results by the traditional proportional integral derivative control, fuzzy control, or predictive control. This article analyses physics and chemistry mechanism of the combustion process and expounds the complex nonlinear, multivariable and large time‐delay characteristics, and the control target of the production system. Then, the mathematical model of combustion control system is deduced in view of the requirements of simulation. Based on these, the fuzzy predictive control scheme is employed. Through simulation, the control algorithm is verified to be effective. Finally, the industrial sleeve kiln as a practical example is used to demonstrate the effectiveness and feasibility of the control algorithm. © 2016 Wiley Periodicals, Inc. Complexity 21: 249–258, 2016     

Exact minimum-time control of a distributed system using a traveling wave formulation

In this paper, the minimum-time control problem for rest-to-rest translation of a one-dimensional second-order distributed parameter system by means of two bounded control inputs at the ends is solved. A traveling wave formulation allows the problem to be solved exactly, i.e., without modal truncation. It is found that the minimum-time control is not bang-bang, as it is for systems with a finite number of degrees of freedom. Rather, it is bang-off-bang, where a period of control inactivity in the middle of the control time interval is required for synchronization with waves propagated through the system.This research was supported in part by AFOSR Grant No. AFOSR-90-0297. The helpful suggestions of the referees are gratefully acknowledged.     

Real-time distributed control: A fuzzy and model predictive control approach for a nonlinear problem

Nowadays the study of faults and their consequences has become an issue for highly safety critical computer network systems. How to bound the effects of a fault and how to tackle them in a dynamic system is still an open field. Here, an approach to studying this problem is presented as a hybrid strategy. The use of a multi-model technique from the point of view of real-time systems and intelligent control structure is described, in order to accomplish one challenge: to overcome the problem of the presence of local faults and the respective time delays within a real-time distributed system. This approach is pursued as a reconfigurable strategy according to communication time delays within a real-time distributed system. In fact, it is pursued as a reconfigurable strategy according to communication delays and local faults where the control strategy is modified from several perspectives.     

Numerical method for solving a nonlinear time-optimal control problem with additive control

Nonlinear systems whose right-hand sides are divided by the state and control and are linear in control are considered. An iterative method is proposed for solving time-optimal control problems for such systems. The method is based on constructing finite sequences of adjacent simplexes with their vertices lying on the boundaries of reachability sets. For a controllable system, it is proved that the minimizing sequence converges to an ?-optimal solution in a finite number of iterations.     

Optimal control of a bioreactor

This paper is concerned with the analysis of a control problem related to the optimal management of a bioreactor. This real-world problem is formulated as a state-control constrained optimal control problem. We analyze the state system (a complex system of partial differential equations modelling the eutrophication processes for non-smooth velocities), and we prove that the control problem admits, at least, a solution. Finally, we present a detailed derivation of a first order optimality condition - involving a suitable adjoint system - in order to characterize these optimal solutions, and some computational results.     

Optimal control of a fine structure

An optimal control problem for a multivalued system governed by a nonconvex variational problem, involving a regularization parameter >0, is proposed and studied. The solution to the variational problem exhibits typically rapid oscillations (a so-called fine structure) corresponding to a multiphase state of the material. We want to control only this fine structure. Existence of an optimal control is proved. Its convergence with 0 is studied by means of an optimal control problem for a relaxed variational problem involving (suitably generalized) Young measures. The uniqueness of the solution to the relaxed variational problem, which is nontrivial but is very important in the context of optimal control, is studied in special cases. A finite-element approximation is proposed.The second author gratefully acknowledges support for this research by the Alexander von Humboldt Foundation during his stay at the Institute for Mathematics of the University of Augsburg.     

Feedback control in LQCP with a terminal inequality constraint

This paper considers the linear-quadratic control problem (LQCP) for systems defined by evolution operators with a terminal state inequality constraint. It is shown that, under suitable assumptions, the optimal control exists, is unique, and has a closed-loop structure. The synthesis of the feedback control requires one to solve the integral Riccati equation for the unconstrainted LQCP and a linear integral equation whose solution depends on a real parameter satisfying an additional condition.This work was completed while the author was visiting the Control Theory Centre, University of Warwick, Coventry, England.     

Adaptive fuzzy backstepping control for a class of MIMO switched nonlinear systems with unknown control directions

In this article, an adaptive fuzzy output tracking control approach is proposed for a class of multiple‐input and multiple‐output uncertain switched nonlinear systems with unknown control directions and under arbitrary switchings. In the control design, fuzzy logic systems are used to identify the unknown switched nonlinear systems. A Nussbaum gain function is introduced into the control design and the unknown control direction problem is solved. Under the framework of the backstepping control design, fuzzy adaptive control and common Lyapunov function stability theory, a new adaptive fuzzy output tracking control method is developed. It is proved that the proposed control approach can guarantee that all the signals in the closed‐loop system are bounded and the tracking error remains an adjustable neighborhood of the origin. A numerical example is provided to illustrate the effectiveness of the proposed approach. © 2015 Wiley Periodicals, Inc. Complexity 21: 155–166, 2016     

Optimal control in a macroeconomic problem

The Pontryagin maximum principle is used to develop an original algorithm for finding an optimal control in a macroeconomic problem. Numerical results are presented for the optimal control and optimal trajectory of the development of a regional economic system. For an optimal control satisfying a certain constraint, an invariant of a macroeconomic system is derived.     

Remarks on bang-bang control in a Hilbert space

The control of a linear system, whose performance index is the sum of a linear term and a quadratic term, is considered. A necessary and sufficient condition is given for the optimal control to be bang-bang, and this is used to extend and clarify the results of Refs. 1–2. As an illustration, an application to an elliptic boundary-value problem is given.This research was supported by the SFB 72 of the DFG, West Germany.     

Deterministic equivalent for a continuous-time linear-convex stochastic control problem

We consider a finite-horizon control model with additive input. There are two convex functions which describe the running cost and the terminal cost within the system. The cost of input is proportional to the input and can take both positive and negative values. It is shown that there exists a deterministic control problem whose optimal cost is the same as the one in the stochastic control problem. The optimal policy for the stochastic problem consists of keeping the process as close to the optimal deterministic trajectory as possible.This research is supported by NSERC Grant A4619, MRCO, NSF Grant DMS-86-01510, and AFOSR Grant 87-0278.     

A virtual control concept for state constrained optimal control problems

A linear elliptic control problem with pointwise state constraints is considered. These constraints are given in the domain. In contrast to this, the control acts only at the boundary. We propose a general concept using virtual control in this paper. The virtual control is introduced in objective, state equation, and constraints. Moreover, additional control constraints for the virtual control are investigated. An error estimate for the regularization error is derived as main result of the paper. The theory is illustrated by numerical tests.     

Complicated version of a robust control scheme

A new robust control design is introduced. The worst cases of controlled system performance and control magnitude are both investigated. Their comparison with early counterparts is also made.     

Optimal control of harvesting coupled with boundary control in a predator—prey system

We consider boundary control and control via harvesting in a parabolic predator—prey system for a bounded region. The boundary control depicts the relationship between the boundary environment and the possibly harmful species. In addition, a proportion of the predator is harvested for profit. We choose to maximize the objective functional which incorporates the amount of the prey and the revenue of harvesting of the predator less the economic cost of sustaining a satisfactory boundary habitat and the cost due to the harvesting component. Moreover, we characterize the unique optimal control in terms of the solution to the optimality system, which is the state system coupled with the adjoint system.