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.     

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.     

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.     

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.     

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.     

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Aiming at the complex mechanical and electrical products quality control and early warning problems, a performance analysis model of control chart, which combines the multivariate Bayesian statistical method with the economic performance analysis is constructed. In the solution model, a FT VSI strategy is used in the multivariate Bayesian control chart. If a small probability of random failure occurs, then a loose sampling scheme is selected. Otherwise, a strict sampling program is applied. To quantify the correlation between the economic and the statistical performance of the multivariate Bayesian control chart, a quality control model based on Monte Carlo simulation is used and the ANOSE (Average Number of Observations to Signals or End of the production run) is taken under different economic parameters, which performs the degree of influence of the statistical performance of the control chart. In addition, the relationship between the quality control cost and the false alarm rate of the multi-Bayesian control chart is explained. Finally, for instance, a multiple quality control process of the automatic transmission of the automobile is used to verify the performance evaluation and optimization of the multivariate FT VSI Bayesian control chart. The results show that the method has a better application.     

Continuity in the parameter of the minimum value of an integral functional over the solutions of an evolution control system

The problem of minimization of an integral functional with an integrand that is nonconvex with respect to the control is considered. We minimize our functional over the solution set of a nonlinear evolution control system with a time-dependent subdifferential operator in a Hilbert space. The control constraint is given by a nonconvex closed bounded set. The integrand, the control constraint, the initial conditions and the operators in the equation describing the control system all depend on a parameter. We consider, along with the original problem, the problem of minimizing an integral functional with an integrand convexified with respect to the control over the solution set of the same system, but now subject to the convexified control constraint. By a solution of the control system we mean a “trajectory–control” pair. We prove that for each value of the parameter the convexified problem has a solution, which is the limit of a minimizing sequence of the original problem, and the minimum value of the functional of the convexified problem is a continuous function of the parameter.     

Stabilizer control design for nonlinear systems based on the hyperbolic modelling

In this paper, a new approach to design a stabilizer control for nonlinear systems is presented. First, the general form of the nonlinear control systems is approximated with a hyperbolic model. Then, by a feedback technique, a stabilizer control for the hyperbolic system is obtained. It is shown that this control can stabilize the original nonlinear control system. Finally, by some illustrative examples, the efficiency of the presented approach is demonstrated.     

On ε-optimal continuous selectors and their application in discounted dynamic programming

A quadratic regulator problem for a class of nonlinear systems is considered in which the control cost is multiplied by a small parameter, which becomes a so-called cheap control problem. Conditions are found under which the minimum cost becomes zero (perfect regulation) and the linear part in the optimal control law becomes dominant as the small parameter goes to zero. Near optimality of control laws truncated from the optimal control law in series form is also found.     

Impulsive stabilization of complex-valued stochastic complex networks via periodic self-triggered intermittent control

The aim of this article is to research the stabilization issue of complex-valued stochastic Markovian switching complex networks with time delays and time-varying multi-links (CSMCTM) via periodic self-triggered intermittent impulsive control (PSIIC). Thereinto, PSIIC is designed for the first time by combining intermittent impulsive control with periodic self-triggered control for intermittent control. It is worth emphasizing that the triggered protocol is designed to be more flexible, and easier to implement than previously reported triggered protocol. Then, by means of impulsive control, intermittent control, event-driven control theory and stability analysis, a stabilization criterion of CSMCTM in the sense of exponential stability in mean square is obtained. Whereafter, the stability of a class of complex-valued inertial neural networks is researched as a practical application of our theoretical results. Ultimately, a numerical example gives a corresponding verification.     

Optimal control of backward stochastic heat equation with Neumann boundary control and noise

This paper considers a stochastic control problem in which the dynamic system is a controlled backward stochastic heat equation with Neumann boundary control and boundary noise and the state must coincide with a given random vector at terminal time. Through defining a proper form of the mild solution for the state equation, the existence and uniqueness of the mild solution is given. As a main result, a global maximum principle for our control problem is presented. The main result is also applied to a backward linear-quadratic control problem in which an optimal control is obtained explicitly as a feedback of the solution to a forward–backward stochastic partial differential equation.     

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.     

Maximum principle for the optimal control of a hyperbolic equation in one space dimension,part 1: Theory

A maximum principle is developed for a class of problems involving the optimal control of a damped-parameter system governed by a linear hyperbolic equation in one space dimension that is not necessarily separable. A convex index of performance is formulated, which consists of functionals of the state variable, its first- and second-order space derivatives, its first-order time derivative, and a penalty functional involving the open-loop control force. The solution of the optimal control problem is shown to be unique. The adjoint operator is determined, and a maximum principle relating the control function to the adjoint variable is stated. The proof of the maximum principle is given with the help of convexity arguments. The maximum principle can be used to compute the optimal control function and is particularly suitable for problems involving the active control of structural elements for vibration suppression.     

Optimal chattering modes in the problem of the control of a Timoshenko beam

The linear problem of the control in a plane of the motion of a Timoshenko beam, one end of which is clamped to a rotating disc is considered. The angular acceleration of the disc serves as the control. It is proved that, in the problem of the quenching of the first mode, the optimal control has an infinite number of switchings in a finite time interval (a chattering control). The construction of a suboptimal control with a finite number of switchings is described.     

考虑污染和种内关系的种群扩散最优控制模型

针对污染和种内关系均影响细菌种群扩散这一管理生态学问题,本文建立了基于非线性拟抛物方程的最优控制模型,将外界环境向细菌种群输入的毒素率作为控制变量,运用控制理论和方法探讨污染和种内关系双重影响下种群扩散系统的最优控制问题。利用Schauder不动点定理证明了该种群扩散系统的适定性;同时,通过建立新的Carleman型估计,给出了容许控制和最优控制的存在性。最后,通过数值算例分析了理论推导的结果,在算例中都找到一对时间最优控制,验证了种群扩散系统最优控制模型的有效性。该研究结果对现代传染病预防具有借鉴意义,也为有效控制瘟疫的爆发和流行提供理论参考。     

Research on mean square exponential stability of networked control systems with multi-step delay

A new control mode is proposed for networked control systems whose network-induced delay is longer than a sampling period. The proposed control mode can make full use of control information and improve the performance of the system. Under the control mode, the mathematical model of networked control systems is obtained. Markov characteristic of the transfer delay is discussed. Based on Markov chain theory, the infinite horizon controller is designed, which is shown to render corresponding networked control systems mean square exponentially stable. Simulation results show the validity of the proposed theory.     

Stochastic optimal time-delay control of quasi-integrable Hamiltonian systems

A nonlinear stochastic optimal time-delay control strategy for quasi-integrable Hamiltonian systems is proposed. First, a stochastic optimal control problem of quasi-integrable Hamiltonian system with time-delay in feedback control subjected to Gaussian white noise is formulated. Then, the time-delayed feedback control forces are approximated by the control forces without time-delay and the original problem is converted into a stochastic optimal control problem without time-delay. After that, the converted stochastic optimal control problem is solved by applying the stochastic averaging method and the stochastic dynamical programming principle. As an example, the stochastic time-delay optimal control of two coupled van der Pol oscillators under stochastic excitation is worked out in detail to illustrate the procedure and effectiveness of the proposed control strategy.     

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.     

Dynamic Robust Output Min–Max Control for Discrete Uncertain Systems

Min–max control is a robust control, which guarantees stability in the presence of matched uncertainties. The basic min–max control is a static state feedback law. Recently, the applicability conditions of discrete static min–max control through the output have been derived. In this paper, the results for output static min–max control are further extended to a class of output dynamic min–max controllers, and a general parametrization of all such controllers is derived. The dynamic output min–max control is shown to exist in many circumstances under which the output static min–max control does not exist, and usually allows for broader bounds on uncertainties. Another family of robust output min–max controllers, constructed from an asymptotic observer which is insensitive to uncertainties and a state min–max control, is derived. The latter is shown to be a particular case of the dynamic min–max control when the nominal system has no zeros at the origin. In the case where the insensitive observer exists, it is shown that the observer-controller has the same stability properties as those of the full state feedback min–max control.