On the unimprovability of full-memory strategies in problems of guaranteed result optimization

A problem of guaranteed result optimization is considered for a control system described by an ordinary differential equation and for a performance functional that depends continuously on the trajectory of the system. The values of the control and of the disturbance satisfy compact geometric constraints. It is also assumed that the realization of the disturbance is subject to an unknown functional constraint from a given set of constraints that are compact in a Lebesgue space. It is shown that the optimal guaranteed result in the class of full-memory strategies in this problem coincides with the value of the optimal guaranteed result in the class of quasi-strategies.     

On a minimax control problem for a positional functional under geometric and integral constraints on control actions

Within the game-theoretical approach, we consider a minimax feedback control problem for a linear dynamical system with a positional quality index, which is the norm of the deviation of the motion from given target points at given times. Control actions are subject to both geometric and integral constraints. A procedure for the approximate calculation of the optimal guaranteed result and for the construction of a control law that ensures the result is developed. The procedure is based on the recursive construction of upper convex hulls of auxiliary program functions. Results of numerical simulations are presented.     

Optimal persistent disturbance attenuation control for linear hybrid systems

In this paper, the disturbance attenuation properties for a class of linear hybrid systems are investigated, and a hybrid optimal persistent disturbance attenuation control problem is studied. First, a procedure is developed to determine the minimal l^∞$l^{\infty }$ induced gain of linear hybrid systems. However, for general hybrid systems, the termination of the procedure is not guaranteed. Then, the decidability issues are discussed. The termination of the procedure in a finite number of steps is shown for a subclass of hybrid systems with simplified discrete event dynamics, called switched linear systems. Finally, the optimal persistent disturbance attenuation controller synthesis problem is studied. It is shown that the optimal performance level can be achieved by a piecewise linear state feedback control law, and a systematic approach is proposed to design such feedback control.     

The optimization of a guaranteed result with a delay in the control

The problem of feedback control with a quality index in the form of the Euclidean norm of a set of deviations of the trajectory of motion of the system from the origin of coordinates at specified instants of time is considered for a linear dynamical system that is subjected to the effects of noise and contains a delay in the control element. A procedure for calculating the value of the optimal guaranteed result based on the recursive construction of upper convex hulls of auxiliary functions is given. A method for shaping the controlling actions that guarantee the achievement of this result is described.     

Nonlinear positional differential game in the class of mixed strategies

The feedback control problem is considered for a nonlinear dynamic system under lack of information on disturbances. The problem on minmax-maxmin of the guaranteed result for a given positional quality index is formalized as an antagonistic two-player differential game in the framework of the concept developed in the Sverdlovsk (now Yekaterinburg) school on the theory of differential games. The problem is solved in the class of mixed strategies. The existence of the value of the game and a saddle point is established. The solution to the problem is based on the application of appropriate leader models, the method of extremal shift to the accompanying points and the method of upper convex hulls. The results of the study are applied to a realistic control model. The simulation outputs are presented.     

State estimation with guaranteed performance for switching-type fuzzy neural networks in presence of sensor nonlinearities

This paper investigates the state estimation with guaranteed performance for a class of switching fuzzy neural networks. A switching-type fuzzy neural networks (STFNNs) model is proposed which captures external disturbances, sensor nonlinearities, and mode switching phenomenon of the fuzzy neural networks without the Markovian process assumption. For such a model, a state estimation problem is formulated to achieve the guaranteed performance: the estimation error system is exponentially stable with certain decay rate and a prescribed H_∞ disturbance attenuation level. A novel sufficient condition for this problem is established using the Lyapunov functional method and the average dwell time approach, and the estimator parameters are explicitly given. A numerical example is presented to show the effectiveness of the developed results.     

Robust Chaotic Synchronization for a Class of Disturbed Nonlinear Systems with Multiple Equilibria

This study concerns with the robust H _∞ synchronization problem for a class of nonlinear feedback control systems, which are subject to a vector-valued periodic nonlinearity in the feedback path. Under such synchronization configuration, the master system is assumed to be subject to an energy bounded input disturbance, and the slave one is under control. Sufficient conditions for controller design are proposed in terms of linear matrix inequalities by respectively utilizing the output feedback control and the dynamic output control strategies, such that the master system robustly synchronizes the slave one with a guaranteed H _∞ performance. The derived methods can be applied to the robust H _∞ synchronization of many practical systems, and effectiveness of the obtained results are demonstrated through a concrete example of phase-locked loops (PLL).     

Mathematical modeling and trajectory planning of mobile manipulators with flexible links and joints

This paper is concerned with mathematical modeling and optimal motion designing of flexible mobile manipulators. The system is composed of a multiple flexible links and flexible revolute joints manipulator mounted on a mobile platform. First, analyzing on kinematics and dynamics of the model is carried out then; open-loop optimal control approach is presented for optimal motion designing of the system. The problem is known to be complex since combined motion of the base and manipulator, non-holonomic constraint of the base and highly non-linear and complicated dynamic equations as a result of the flexible nature of both links and joints are taken into account. In the proposed method, the generalized coordinates and additional kinematic constraints are selected in such a way that the base motion coordination along the predefined path is guaranteed while the optimal motion trajectory of the end-effector is generated. This method by using Pontryagin’s minimum principle and deriving the optimality conditions converts the optimal control problem into a two point boundary value problem. A comparative assessment of the dynamic model is validated through computer simulations, and then additional simulations are done for trajectory planning of a two-link flexible mobile manipulator to demonstrate effectiveness and capability of the proposed approach.     

Robust H∞ output dynamic observer-based control of uncertain time-delay systems

In this paper, the robustness and H_∞ control problems of output dynamic observer-based control for a class of uncertain linear systems with time delay are considered. Under no disturbance input, the asymptotic stabilization for uncertain time-delay systems will be guaranteed. Linear matrix inequality (LMI) optimization approach is used to design three classes of the H_∞ output dynamic controls. Based on the results of this paper, the constraint of matrix equality is not necessary for designing the observer-based controls.     

On some properties of the control problem under a program disturbance in a formalization based on the minimax risk (regret) criterion

A control problem under uncertainty for a system described by an ordinary differential equation with a terminal performance index is considered. The control and disturbance are subject to geometric constraints. The problem is formalized in classes of nonanticipating control strategies and program disturbances with the use of constructive ideal motions and the Savage minimax risk (regret) criterion. The properties of the used motion bundles are described and a number of relations characterizing the optimal risk function, which is an element of the formalization, are presented.     

Problem of guaranteed guidance by measuring part of the state vector coordinates

We study the problem of guaranteed guidance of an elliptic control system to a given objective set at a given time under the assumption that the system is subjected to an unknown disturbance. The case in which only part of the state coordinates are measured is considered. For this problem, we suggest a solution algorithm based on a combination of dynamic inversion theory and the extremal shift principle.     

Minimax and viscosity solutions in optimization problems for hereditary systems

For a dynamical system with discrete and distributed time delays, a control problem under disturbance or counteraction is considered. The problem is formalized in the context of the game-theoretical approach in the class of control strategies with memory. The problem is associated with a functional Hamilton-Jacobi type equation with coinvariant derivatives. The minimax and viscosity approaches to a generalized solution to this equation are discussed. It is shown that, under the same condition at the right endpoint, the minimax and viscosity solutions coincide, thereby uniquely defining the functional of optimal guaranteed result in the control problem.     

Optimal Guaranteed Cost Control of Linear Systems with Mixed Interval Time-Varying Delayed State and Control

This paper deals with the problem of optimal guaranteed cost control for linear systems with interval time-varying delayed state and control. The time delay is assumed to be a continuous function belonging to a given interval, but not necessary to be differentiable. A linear–quadratic cost function is considered as a performance measure for the closed-loop system. By constructing a set of augmented Lyapunov–Krasovskii functional combined with Newton–Leibniz formula, a guaranteed cost controller design is presented and sufficient conditions for the existence of a guaranteed cost state-feedback for the system are given in terms of linear matrix inequalities (LMIs). Numerical examples are given to illustrate the effectiveness of the obtained result.     

Positioning control for a linear actuator with nonlinear friction and input saturation using output‐feedback control

This article deals with the positioning control problem via the output feedback scheme for a linear actuator with nonlinear disturbances. In this study, the proposed controller accounts for not only the nonlinear friction, force ripple, and external disturbance but also the input saturation problem. In detail, the energy consumption for conquering friction and disturbance rejection is estimated and used as compensation based on the hybrid controller including and sliding‐mode‐based adaptive algorithms, which ensures the tracking performance and robustness of electromechanical servo system. Moreover, to confront the input saturation, a saturation observer and an anti‐windup controller are designed. The global robustness of the controller is guaranteed by an output feedback robust law. Theoretically, the designed controller can guarantee a favorable tracking performance in the presence of various disturbance forces and input saturation, which is essential for high accuracy motion plant in industrial application. The simulation results verify the robustness and effectiveness for the motion system with the proposed control strategy under various operation conditions. © 2016 Wiley Periodicals, Inc. Complexity 21: 191–200, 2016     

Novel Optimal Guaranteed Cost Control of Uncertain Discrete Systems with Both State and Input Delays

The problem of the guaranteed cost control (GCC) for a class of uncertain discrete-time systems with both state and input delays is considered in this paper. A novel LMI-based approach is proposed for the existence of a state feedback controller which guarantees not only the asymptotic stability of the closed-loop system, but also an adequate performance bound over all the possible parameter uncertainties. A convex optimization algorithm is given to design the state feedback controller which minimizes a bound on a quadratic performance index. The result exhibits some favorable features in computation as shown by a numerical example. This work was supported by the National Science Foundation of China under Grants 60504012 and 60774039.     

An Optimal Control Problem for a Singular System of Solid–Liquid Phase Transition

In this article, we deal with a control problem for a singular system regarding a phase-field model which describes a solid–liquid transition by the Ginzburg–Landau theory. The purpose is to control the system by the means of the heat supply r able to guide it into a certain state with a solid (or liquid) part in a prescribed subset Ω₀ of the space domain Ω, and maintain it in this state during a period of time. The transition is described by a nonlinear differential system of two equations for the phase field and temperature. The control problem is set for some expressions of the cost functional which might reveal cases of physical interest. An approximating control problem is introduced and the existence of at least an optimal pair is proved. The first-order optimality conditions for the approximating problem are determined and a convergence result is given.     

Constructive Solution of Inverse Parametric Linear/Quadratic Programming Problems

Parametric convex programming has received a lot of attention, since it has many applications in chemical engineering, control engineering, signal processing, etc. Further, inverse optimality plays an important role in many contexts, e.g., image processing, motion planning. This paper introduces a constructive solution of the inverse optimality problem for the class of continuous piecewise affine functions. The main idea is based on the convex lifting concept. Accordingly, an algorithm to construct convex liftings of a given convexly liftable partition will be put forward. Following this idea, an important result will be presented in this article: Any continuous piecewise affine function defined over a polytopic partition is the solution of a parametric linear/quadratic programming problem. Regarding linear optimal control, it will be shown that any continuous piecewise affine control law can be obtained via a linear optimal control problem with the control horizon at most equal to 2 prediction steps.     

组合的分组测试算法的近似控制标准

对于给定的一个集合,分组测试问题是通过一系列的测试去确定这个集合的一个子集. 在文中, 作者首先运用动态规划的理论与方法, 建立了一个近似控制标准, 目的是对分组测试算法的构建过程进行有效控制, 使所构建的算法达到最优. 其次, 应用该近似控制标准研究了在n个硬币集合中确定一个伪硬币的最小平均测试数的问题. 文中所涉及的近似控制问题, 给出了在一个给定集合中去确定这个集合的一个子集的最优分组测试算法, 该最优分组测试算法是在平均测试步骤最少意义下的最优分组测试算法.     

Optimal control of Maxwell’s equations with regularized state constraints

This paper is devoted to an optimal control problem of Maxwell??s equations in the presence of pointwise state constraints. The control is given by a divergence-free three-dimensional vector function representing an applied current density. To cope with the divergence-free constraint on the control, we consider a vector potential ansatz. Due to the lack of regularity of the control-to-state mapping, existence of Lagrange multipliers cannot be guaranteed. We regularize the optimal control problem by penalizing the pointwise state constraints. Optimality conditions for the regularized problem can be derived straightforwardly. It also turns out that the solution of the regularized problem enjoys higher regularity which then allows us to establish its convergence towards the solution of the unregularized problem. The second part of the paper focuses on the numerical analysis of the regularized optimal control problem. Here the state and the control are discretized by Nédélec??s curl-conforming edge elements. Employing the higher regularity property of the optimal control, we establish an a priori error estimate for the discretization error in the $\boldsymbol{H}(\bold{curl})$ -norm. The paper ends by numerical results including a numerical verification of our theoretical results.     

Optimal synthesis in the problem of impulsive correction of motion

A minimax problem of correcting the motion of a dynamic system acted upon by perturbing forces of restricted magnitude is considered. The corrective action is carried out in the form of impulsive control, with a restriction on the total magnitude of the impulses and their number. This formulation models the problem of the impulsive correction of motion of an aircraft acted upon by external perturbations. The problem represents, in fact, a differentially-impulsive game /1/ in which the player controlling the correction aims to secure for himself a guaranteed minimum of the terminal functional. Following the methods used in /2/ we construct, for the problem with isotropic dynamics, an optimal synthesis of the correction instances. The present paper touches on the work done in /3–5/.