On the time transformation of mixed integer optimal control problems using a consistent fixed integer control function

Nonlinear control systems with instantly changing dynamical behavior can be modeled by introducing an additional control function that is integer valued in contrast to a control function that is allowed to have continuous values. The discretization of a mixed integer optimal control problem (MIOCP) leads to a non differentiable optimization problem and the non differentiability is caused by the integer values. The paper is about a time transformation method that is used to transform a MIOCP with integer dependent constraints into an ordinary optimal control problem. Differentiability is achieved by replacing a variable integer control function with a fixed integer control function and a variable time allows to change the sequence of active integer values. In contrast to other contributions, so called control consistent fixed integer control functions are taken into account here. It is shown that these control consistent fixed integer control functions allow a better accuracy in the resulting trajectories, in particular in the computed switching times. The method is verified on analytical and numerical examples.     

Galerkin projections for some wave and plate boundary control problems

In this paper we describe a method for projecting the solutions of a wide class of wave and plate control problems with one active boundary control. These projections are reduced to solving completely controllable finite dimensional linear control problems with a scalar control where the control is introduced through a fixed function defined on the boundary of the wave or plate.     

Simultaneous Optimal Controls for Non-Stationary Stokes Systems

This paper deal with optimal control problems for a non-stationary Stokes system. We study a simultaneous distributed-boundary optimal control problem with distributed observation. We prove the existence and uniqueness of a simultaneous optimal control and we give the first order optimality condition for this problem. We also consider a distributed optimal control problem and a boundary optimal control problem and we obtain estimations between the simultaneous optimal control and the optimal controls of these last ones. Finally, some regularity results are presented.     

Chaos control of Bonhoeffer–van der Pol oscillator using neural networks

This paper addresses the control of chaos using a neural network for a continuous time dynamical system. The neural network is trained on both the Ott–Grebogi–Yorke (OGY) control algorithm and the Pyragas's delayed feedback control algorithm. The system considered for this study is a Bonhoeffer–van der Pol (BVP) oscillator. A feed-forward backpropagating neural network is used for the control application. It is found that the control effected by the neural network trained on the OGY control algorithm results in smaller control transients than when the control is effected directly by the OGY algorithm itself. The control transients are of the same order in the case of the Pyragas method.     

Event-triggered impulsive control of lower-triangular large-scale nonlinear systems based on gain scaling technique

This paper investigates the event-triggered impulsive control problem for a class of large-scale nonlinear systems in lower-triangular form. Based on gain scaling technique and impulsive control theory, a novel decentralized event-triggered impulsive control strategy is first put forward by introducing a static scaling gain, where no control input exists between two consecutive triggering points. Moreover, when the large uncertainties exist in system nonlinearities, we further develop a new control strategy by introducing a time-varying scaling gain. It is proved that the proposed closed-loop control strategies exclude the Zeno behavior without sacrificing the global convergence of system states. Compared with the existing results, it is the first time to apply impulsive control to lower-triangular large-scale nonlinear systems, and the advantages of event-triggered impulsive control and gain scaling technique are subtly combined in the proposed control strategies. Finally, two simulation examples are used to demonstrate the effectiveness of the proposed schemes.     

Stabilization of nonlinear switched systems using control Lyapunov functions

In this paper, we study the stabilization of general nonlinear switched systems by using control Lyapunov functions. The concept of control Lyapunov function for nonlinear control systems is generalized to switched control systems. The first part of our contribution provides a necessary and sufficient condition of stabilization. The main idea is to use a common control Lyapunov function; this is achieved with the converse Lyapunov theorem dedicated to switched systems. In the second part, an explicit construction of a common control Lyapunov function is addressed with respect to a finite family of switched systems. The approach uses a family of control Lyapunov functions attached to the subsystems.     

A numerical method for an optimal control problem with minimum sensitivity on coefficient variation

In this paper, we consider a class of optimal control problem involving an impulsive systems in which some of its coefficients are subject to variation. We formulate this optimal control problem as a two-stage optimal control problem. We first formulate the optimal impulsive control problem with all its coefficients assigned to their nominal values. This becomes a standard optimal impulsive control problem and it can be solved by many existing optimal control computational techniques, such as the control parameterizations technique used in conjunction with the time scaling transform. The optimal control software package, MISER 3.3, is applicable. Then, we formulate the second optimal impulsive control problem, where the sensitivity of the variation of coefficients is minimized subject to an additional constraint indicating the allowable reduction in the optimal cost. The gradient formulae of the cost functional for the second optimal control problem are obtained. On this basis, a gradient-based computational method is established, and the optimal control software, MISER 3.3, can be applied. For illustration, two numerical examples are solved by using the proposed method.     

Necessary and sufficient existence conditions for a boundary control of vibrations of a string and a spherical layer for small times

We find necessary and sufficient conditions for the existence of a boundary control of vibrations of a string or a spherical layer for critical and subcritical times. We completely analyze the existence of a boundary control of vibrations of a spherical layer by a force on two spheres. We find necessary and sufficient existence conditions for the control. Along with the control problem for vibrations of a spherical layer, we consider a similar control problem for string vibrations.     

A computational method for free terminal time optimal control problem governed by nonlinear time delayed systems

This paper considers a free terminal time optimal control problem governed by nonlinear time delayed system, where both the terminal time and the control are required to be determined such that a cost function is minimized subject to continuous inequality state constraints. To solve this free terminal time optimal control problem, the control parameterization technique is applied to approximate the control function as a piecewise constant control function, where both the heights and the switching times are regarded as decision variables. In this way, the free terminal time optimal control problem is approximated as a sequence of optimal parameter selection problems governed by nonlinear time delayed systems, each of which can be viewed as a nonlinear optimization problem. Then, a fully informed particle swarm optimization method is adopted to solve the approximate problem. Finally, two free terminal time optimal control problems, including an optimal fishery control problem, are solved by using the proposed method so as to demonstrate its applicability.     

Relaxing mixed integer optimal control problems using a time transformation

Mixed integer control systems are used to model dynamical behavior that can change instantly, for example a driving car with different gears. Changing a gear corresponds to an instant change of the differential equation what is achieved in the model by changing the value of the integer control function. The optimal control of a mixed integer control system by a discretize-then-optimize approach leads to a mixed integer optimization problem that is not differentiable with respect to the integer variables, such that gradient based optimization methods can not be applied. In this work, differentiability with respect to all optimization variables is achieved by reformulating the mixed integer optimal control problem (MIOCP). A fixed integer control function and a time transformation are introduced. The combination of both allows to change the sequence of active differential equations by partially deactivating the fixed integer control function. In contrast to other works, here different fixed integer control functions are taken into account. Advantages of so called control consistent (CC) fixed integer control functions are discussed and confirmed on a numerical example. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Computation of Control for Linear Approximately Controllable System Using Tikhonov Regularization

Computation of control for a controlled partial differential equation is a di?cult task, especially when the control problem is ill posed. In this paper, we propose a method of computing the regularized control of a diffusion control system using Tikhonov regularization approach when the system is approximately controllable. The method proposed here for choosing regularization parameter guarantees the convergence of the proposed control.     

A simple method of chaos control for a class of chaotic discrete-time systems

In this paper, a simple method is proposed for chaos control for a class of discrete-time chaotic systems. The proposed method is built upon the state feedback control and the characteristic of ergodicity of chaos. The feedback gain matrix of the controller is designed using a simple criterion, so that control parameters can be selected via the pole placement technique of linear control theory. The new controller has a feature that it only uses the state variable for control and does not require the target equilibrium point in the feedback path. Moreover, the proposed control method cannot only overcome the so-called “odd eigenvalues number limitation” of delayed feedback control, but also control the chaotic systems to the specified equilibrium points. The effectiveness of the proposed method is demonstrated by a two-dimensional discrete-time chaotic system.     

Direct model reference adaptive control (MRAC) design and simulation for the vibration suppression of piezoelectric smart structures

The paper presents control system design based on a non-linear model reference adaptive control law (MRAC) used for the vibration suppression of a smart piezoelectric mechanical structure. Numerical simulation of the proposed control system is performed based on the finite element (FE) model of the structure, modally reduced in order to meet the requirements of the control system design. First the MRAC problem is defined and a direct control algorithm described in the paper is suggested as a solution to the control problem. The basic MRAC algorithm is modified by augmenting the integral term of the control law in order to provide the robustness of the control system with respect to the stability. This approach provides preserving the boundness of the system states and adaptive gains, with small trackings error over large ranges of non-ideal conditions and uncertainties. The efficiency of the suggested control for the vibration suppression is tested and shown through a numerical simulation of the funnel-shaped piezoelectric structure.     

Tracking control of nonlinear chaotic systems with dynamics uncertainties

This paper deals with the tracking control of nonlinear chaotic systems with dynamics uncertainties. A robust control strategy is developed to control a class of nonlinear chaotic systems with uncertainties. The proposed strategy is an input-output control scheme which comprises an uncertainty estimator and a linearizing-like feedback. The control time is explicitly computed. Computer simulations of the Duffing system are provided to verify the validity of the proposed control scheme.     

Existence and properties of solutions of a control system with hysteresis effect

We consider a control system described by two ordinary nonlinear differential equations subject to a control constraint given by a multivalued mapping with closed nonconvex values, which depends on the phase variables. One of the equations contains the subdifferential of the indicator function of a closed convex set depending on the unknown phase variable. The equation containing the subdifferential describes an input-output relation of hysteresis type.Along with the original control constraint, we also consider the convexified control constraint and the constraint consisting of the extremal points of the convexified control constraint.We prove the existence of solutions of our control system with various control constraints and establish certain relationships between corresponding solution sets.     

Global optimization of polynomial-expressed nonlinear optimal control problems with semidefinite programming relaxation

In this paper, we propose a new deterministic global optimization method for solving nonlinear optimal control problems in which the constraint conditions of differential equations and the performance index are expressed as polynomials of the state and control functions. The nonlinear optimal control problem is transformed into a relaxed optimal control problem with linear constraint conditions of differential equations, a linear performance index, and a matrix inequality condition with semidefinite programming relaxation. In the process of introducing the relaxed optimal control problem, we discuss the duality theory of optimal control problems, polynomial expression of the approximated value function, and sum-of-squares representation of a non-negative polynomial. By solving the relaxed optimal control problem, we can obtain the approximated global optimal solutions of the control and state functions based on the degree of relaxation. Finally, the proposed global optimization method is explained, and its efficacy is proved using an example of its application.     

A study on time-limited control of single-pest with stage-structure

Two kinds of time-limited pest control models of single-pest with stage-structure, which can be described by the boundary value problem of ordinary differential equation and impulsive differential equation, are presented according to the ways of artificial control (continuous control and impulsive control). The conditions under which the corresponding model has a solution are given. If the model has a solution, the corresponding aim of pest control can be achieved. The theoretical results show that both the mature and the immature pest should be controlled synchronously, otherwise the aims of pest control can not be achieved in a finite time. Finally, some discussions and numerical simulations show that the impulsive control is more practical than the continuous control.     

Construction of a terminal control for a second-order system with state constraints

We consider a terminal control problem with state constraints and additional constraints on the qualitative behavior of the terminal trajectory for a second-order system in two-dimensional Euclidean space under geometric constraints on control parameters. A class of control functions solving this control problem is proposed. Numerical results for the control system with model parameters are presented.     

基于组合智能控制的柔性臂微机控制系统研究

针对难以控制的柔性机械臂，设计了强鲁棒性、强自适应能力的模糊滑模控制策略；并针对控制对象与控制算法的复杂性，设计了基于DSP＋FPGA的控制系统，保证了系统的实时性和精确度。     

Necessary Optimality Conditions for Some Control Problems of Elliptic Equations with Venttsel Boundary Conditions

In this paper we derive a necessary optimality condition for a local optimal solution of some control problems. These optimal control problems are governed by a semi-linear Vettsel boundary value problem of a linear elliptic equation. The control is applied to the state equation via the boundary and a functional of the control together with the solution of the state equation under such a control will be minimized. A constraint on the solution of the state equation is also considered.