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.     

Optimum control of bobsled steering

Minimum-time and smooth-steering control algorithms are developed for bobsled optimal control. Numerical solutions are obtained both for one-curve optimal control and whole-course piecewise optimal control with application to realistic three-dimensional track surface shapes. Specific results are calculated for the Lillehammer Olympic Track.     

SQP-methods for solving optimal control problems with control and state constraints: adjoint variables, sensitivity analysis and real-time control

Parametric nonlinear optimal control problems subject to control and state constraints are studied. Two discretization methods are discussed that transcribe optimal control problems into nonlinear programming problems for which SQP-methods provide efficient solution methods. It is shown that SQP-methods can be used also for a check of second-order sufficient conditions and for a postoptimal calculation of adjoint variables. In addition, SQP-methods lead to a robust computation of sensitivity differentials of optimal solutions with respect to perturbation parameters. Numerical sensitivity analysis is the basis for real-time control approximations of perturbed solutions which are obtained by evaluating a first-order Taylor expansion with respect to the parameter. The proposed numerical methods are illustrated by the optimal control of a low-thrust satellite transfer to geosynchronous orbit and a complex control problem from aquanautics. The examples illustrate the robustness, accuracy and efficiency of the proposed numerical algorithms.     

Chaos synchronization and chaos control of quantum-CNN chaotic system by variable structure control and impulse control

In this paper, we derive some less stringent conditions for the exponential and asymptotic stability of impulsive control systems with impulses at fixed times. These conditions are then used to design an impulsive control law for the Quantum Cellular Neural Network chaotic system, which drives the chaotic state to zero equilibrium and synchronizes two chaotic systems. An active sliding mode control method is synchronizing two chaotic systems and controlling chaotic state to periodic motion state. And a sufficient condition is drawn for the robust stability of the error dynamics, and is applied to guiding the design of the controllers. Finally, numerical results are used to show the robustness and effectiveness of the proposed control strategy.     

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.     

Inverse optimal control and construction of control Lyapunov functions

In this paper, the construction of CLFs for nonlinear systems and a new inverse optimal control law are presented. The construction of a CLF for an affine nonlinear system is reduced to the construction of a CLF for a simpler system, and a new L _g V type control law with respect to a CLF is provided. This control law is a generalization of Sontag’s formula and contains a design parameter. Tuning this parameter gives many suboptimal solutions for the optimization problem. Also, the gain margin and sector margin of the control law are calculated. Examples are provided to illustrate the main theoretical results of the paper. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 61, Optimal Control, 2008.     

Feedback control and adaptive control of the energy resource chaotic system

In this paper, the problem of control for the energy resource chaotic system is considered. Two different method of control, feedback control (include linear feedback control, non-autonomous feedback control) and adaptive control methods are used to suppress chaos to unstable equilibrium or unstable periodic orbits. The Routh–Hurwitz criteria and Lyapunov direct method are used to study the conditions of the asymptotic stability of the steady states of the controlled system. The designed adaptive controller is robust with respect to certain class of disturbances in the energy resource chaotic system. Numerical simulations are presented to show these results.     

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.     

Error estimates for parabolic optimal control problems with control and state constraints

The numerical approximation to a parabolic control problem with control and state constraints is studied in this paper. We use standard piecewise linear and continuous finite elements for the space discretization of the state, while the dG(0) method is used for time discretization. A priori error estimates for control and state are obtained by an improved maximum error estimate for the corresponding discretized state equation. Numerical experiments are provided which support our theoretical results.     

A simple adaptive feedback control method for chaos and hyper-chaos control

This paper investigates the problem of chaos and hyper-chaos control, and proposes a simple adaptive feedback control method for chaos control under a reasonable assumption. In comparison with previous methods, the present control technique is simple both in the form of the controller and its application. Several illustrative examples with numerical simulations are studied by using the results obtained in this paper. Study of examples shows that our control method works very well in chaos control.     

Characterization and computation of control invariant sets for linear impulsive control systems

Impulsive control systems are suitable to describe and control a venue of real-life problems, going from disease treatment to aerospace guidance. The main characteristic of such systems is that they evolve freely in-between impulsive actions, which makes it difficult to guarantee its permanence in a given state-space region. In this work, we develop a method for characterizing and computing approximations to the maximal control invariant sets for linear impulsive control systems, which can be explicitly used to formulate a set-based model predictive controller. We approach this task using a tractable and non-conservative characterization of the admissible state sets, namely the states whose free response remains within given constraints, emerging from a spectrahedron representation of such sets for systems with rational eigenvalues. The so-obtained impulsive control invariant set is then explicitly used as a terminal set of a predictive controller, which guarantees the feasibly asymptotic convergence to a target set containing the invariant set. Necessary conditions under which an arbitrary target set contains an impulsive control invariant set (and moreover, an impulsive control equilibrium set) are also provided, while the controller performance are tested by means of two simulation examples.     

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.     

Fuzzy control with limited control opportunities and response delay––a production-inventory control scenario

This paper examines the utility of fuzzy control over crisp in situations where the control opportunities are limited and the system response to control actions is delayed. Such situations are often encountered in production systems where limited resources restrict the control opportunities and the operation time delays the response. The performance of a real-time production-inventory control system is studied with fuzzy control strategy and compared with a corresponding crisp control and no-control strategy. The system consists of a production shop having a number of identical processing machines which produce two products. The output goes into two bins whose inventory is required to be controlled at desired level by varying the number of machines allocated to the products. Real-time inventory variation, output, average inventory and machine usage, number of setups and stock-outs are used as performance measures. The simulation results of the system with various configurations show that the capability of fuzzy control is seriously inhibited by limited opportunities and response delay although fuzzy has distinct advantage over crisp. As control opportunities increase fuzzy control becomes increasingly efficient with diminishing effect of response delay.     

Identification and adaptive control: An application to flight control systems

This paper presents the first results obtained in the application of stochastic control theory to flight control problems. It involves identification and adaptive control of an aircraft which operates over a wide range of environmental conditions that affect its dynamic characteristics. The bulk of the paper deals with theidentification problem of estimating stability derivatives in the presence of turbulence. Simulation results are presented both for identification and control (windgust alleviation and desired response to pilot input). While no practical implementation is reported, the implications for such implementation appear to be promising.The author is indebted to Mr. K. Iliff for all the simulation results using the IBM 360-91 at UCLA. The many helpful discussions with Dr. H. Rediess are gratefully acknowledged. Thanks are also due to Mr. Jiri Ruzicka for lending a helping hand throughout.     

Convergence of the control parameterization Ritz method for nonlinear optimal control problems

This paper studies the local convergence properties of the control parameterization Ritz method in which the control variable is approximated over a finite-dimensional subspace. The nonlinear free-endpoint optimal control problem is considered, and error bounds are derived for both the cost functional and state-control convergence. Explicit error bounds are obtained for the particular case of approximations over spline spaces. On specializing the general results to the linear-quadratic regulator problem, global convergence results are obtained. Computational results supporting the theoretically derived error bounds are presented.This research was supported by the University Grants Committee of New Zealand.     

Solving elliptic control problems with interior point and SQP methods: control and state constraints

We study optimal control problems for semilinear elliptic equations subject to control and state inequality constraints. Both boundary control and distributed control problems are considered with boundary conditions of Dirichlet or Neumann type. By introducing suitable discretization schemes, the control problem is transcribed into a nonlinear programming problem. Necessary conditions of optimality are discussed both for the continuous and the discretized control problem. It is shown that the recently developed interior point method LOQO of [35] is capable of solving these problems even for high discretizations. Four numerical examples with Dirichlet and Neumann boundary conditions are provided that illustrate the performance of the algorithm for different types of controls including bang–bang controls.     

Multiple shooting methods for parabolic optimal control problems with control constraints

In the context of ordinary differential equations, shooting techniques are a state-of-the-art solver component, whereas their application in the framework of partial differential equations (PDE) is still at an early stage. We present two multiple shooting approaches for optimal control problems (OCP) governed by parabolic PDE. Direct and indirect shooting for PDE optimal control stem from the same extended problem formulation. Our approach reveals that they are structurally similar but show major differences in their algorithmic realizations. In the presented numerical examples we cover a nonlinear parabolic optimal control problem with additional control constraints. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A neighboring optimal feedback control scheme for systems using discontinuous control

The calculation and implementation of the neighboring optimal feedback control law for multiinput, nonlinear dynamical systems, using discontinuous control, is the subject of this paper. The concept of neighboring optimal feedback control of systems with continuous, unbounded control functions has been investigated by others. The differentiating features between this class of problems and that considered here are the control discontinuities and the inherent system uncontrollability during the latter stages of the control-law operating time.The neighboring control law is determined by minimizing the second-order terms in the expansion of the performance index about an optimal nominal path. The resulting gains are a function of the states associated with the nominal trajectory. The development of a feedback control scheme utilizing these gains requires a technique for choosing the gains appropriate for each neighboring state. Such a technique is described in this paper. The technique combines abootstrap algorithm for determining the number of neighboring switch times and the initial and final controls with a scheme based ontime-to-go along the nominal and neighboring paths until the next predicted switch time or the predicted final time. This scheme requires that the nominal state, which is used to specify the feedback gains, be chosen such that the predicted time-to-go from the neighboring state be identical to the time-to-go from the nominal state. This technique for choosing feedback gains possesses minimal storage requirements and readily leads to a real-time feedback implementation of the neighboring control law.The optimal feedback control scheme described in this paper is utilized to solve the minimum-time satellite attitude-acquisition problem. The action of the neighboring control scheme when applied to states which do not lie in an immediate neighborhood of the nominal path is investigated. For this particular problem, the neighboring control scheme performs quite well despite the fact that, when the state perturbations are finite, the terminal constraints can never be satisfied exactly.This research was sponsored by the National Aeronautics and Space Administration under Research Grant No. NGL-05-020-007 and is a condensed version of the investigation described in Ref. 1. The authors are indebted to Professor Arthur E. Bryson, Jr., for suggesting the topic and providing stimulating discussions.     

Study of an optimal control problem related to the Solow control model

We study the Solow model of optimal economic growth. The statement of the optimal control problem under consideration contains mixed constraints on the control. A special change of the control is used to reduce the problem to the study of a new controlled object with a usual geometric constraint. The reachable set of the original control system is found, an existence theorem for the optimal control is proved, and sufficient conditions for the absence of singular regimes are obtained.     

Avoidance control

We consider dynamical systems subject to control by two agents, one of whom desires that no trajectory of the system, emanating from outside a given set, intersects the set, no matter what the admissible actions of the other agent. Conditions are given whose satisfaction assures that a given control results in avoidance. Furthermore, these conditions are constructive in that they yield an avoidance feedback control. Some examples are presented.