Quadratic Performance Index Function Research Articles

A novel unified variational finite difference (VFD) solution method for optimal control problems

This article is aimed to propose a simple yet efficient unified numerical strategy for solving both linear and non-linear optimal control problems. To do so, the general form of quadratic performance index function and nonlinear state equations are considered first. Then, the idea of variational differential quadrature method is used to convert the integral/differential equations to the equivalent algebraic form. Since time is the only independent variable is this research, the finite difference method with an equally-spaced discretization scheme would be a more appropriate technique rather than the differential quadrature approach. So, the implemented numerical solution is called now as the variational finite difference method. The method of Lagrange multipliers is then utilized for minimization purpose and, as a result, the final set of nonlinear algebraic equations are obtained. Finally, the quadratic and triadic forms of non-linearity are considered and an explicit formulation is represented for the residual and Jacobian of the Newton-based iterative solution procedure. To demonstrate the accuracy and efficiency of the proposed approach in the quadratic optimal control area, several benchmark problems involving linear time-invariant, linear time-variant and nonlinear examples are successfully solved and the results are confirmed with those existed in the literature.

Contour error dynamic analysis and predictive control for multi-axis motion system

Contour tracking accuracy is the key performance index of a multi-axis motion control system. In the previous design of the contour error feedback controller, the independent design idea of the tracking controller and contour controller of each logic axis is often adopted, which is not easy to realize the optimal control of tracking accuracy and contour accuracy. The coupling increment of each controller is inevitable in motion control compensation, which impacts the control system’s stability. Here, we propose a predictive control method for contour error. At first, through the state space model of the feed system, the system dynamics are judged to predict the contour accuracy in the future finite time domain. Besides, the linear quadratic performance index function of contour error is established, and the performance of the control objective is continuously optimized during the limited time domain with the form of rolling optimization. A closed-loop solution is obtained, which is most suitable for the current motion situation and could add to the system to realize the optimal control considering the trajectory contour and the tracking stability of each servo axis in the subsequent few sampling cycles. Finally, utilizing comparative experiments on the three-axis motion control platform, it is verified that the proposed contour predictive control strategy not only effectively reduces the tracking error but also significantly improves the contour accuracy, which is suitable for application in a complex industrial environment.

Optimal Clutch Pressure Control in Shifting Process of Automatic Transmission for Heavy-Duty Mining Trucks

The optimal control of automatic transmission plays an important role in the shifting smoothness and fuel economy of heavy-duty mining trucks. In this paper, a dynamic model of the powertrain system is built to study the clutch pressure control during the shifting process. A linear-quadratic optimal regulator is used to achieve the optimum control pressure of clutches, where shifting jerk and clutch friction loss are chosen to a form quadratic performance index function. Besides, a detailed solution of the linear-quadratic problem with the disturbance matrix in the state equations is provided. This paper also carries out a software simulation and verification of the normal condition (no load without slope) and the extreme condition (full load with maximum slope). Compared with the preset reference trajectory control, the simulation results show that the proposed optimal clutch pressure control can effectively reduce jerk and friction loss during the shifting process and has good robustness to different operating conditions.

Sliding Mode Preview Control for a Class of Continuous-Time Linear Systems

In this paper, the sliding mode preview control problem for a class of continuous-time linear systems is studied. Firstly, an extended error system consisting of the error equation and the state equation of the original system is constructed, and the tracking problem is transformed into the regulation problem. Secondly, a quadratic performance index function is introduced for the state variables of the extended error system, and the preview sliding surface is designed based on the existing conclusions of the optimal preview control. Thirdly, the sliding mode controller with preview effect for the extended error system is designed by using the reaching law method; then, the desired sliding mode preview controller of the original system can be obtained through the integral. The numerical simulation verifies the feasibility of the proposed method.

Optimal control of discrete-time switched linear systems

This paper considers the optimal control problems of switched autonomous linear systems and switched non-autonomous linear systems in discrete-time, where the objective is that of minimizing a linear quadratic performance index function or multiple linear quadratic performance index function. The subsystems switching sequence and the number of switchings are free in this paper. By employing an embedding technique, we embed the origin switched autonomous or non-autonomous linear systems into a larger family of systems. Owing to the embedding technique, the optimal control problem of the origin switched linear systems can be relaxed into a traditional optimal control problem for the larger family systems. We thus derive the optimal switching control law and the optimal control input for the bang-bang type optimal solution of the relaxed optimal control problem simultaneously. The optimal solution of the origin switched optimal control problem can be obtained by the bang-bang type optimal solution of the relaxed optimal control problem. Finally, results of numerical simulation examples are given to demonstrate the remarkable effectiveness and superiority of the developed method.

An Optimal Fuzzy Control Method for Nonlinear Time-Delayed Batch Processes

An optimal fuzzy controller design scheme is proposed to address the influence of time delay and disturbance on the control performance of nonlinear batch processes. First, a two-dimensional (2D) equivalent Takagi-Suguno (T-S) fuzzy error model is formulated. By introducing a quadratic performance index function and adopting 2D Lyapunov-Krasovskii theory, the existence condition of the optimal fuzzy control law is given. Furthermore, its solvable condition, which depends on time-delay bounds, is constructed in terms of linear matrix inequalities, and its gain is obtained by using an optimization algorithm. This design has the advantages of faster tracking and better tracking performance. Finally, two different algorithms (with and without optimization) are used to control the water level of a triple-capacity water tank. The results show that the presented strategy is more effective and feasible.

Event-triggered communication scheme for stochastic systems in wireless sensor networks

With the development of sensors and network technology, wireless sensor networks are widely used in various scenes. However, in most applications, the energy of the sensor nodes cannot be supplemented. In addition, since the lifetime of the electronic switching element is measured by the number of trigger times, which makes the effective use of energy, extending the working life of the network becomes one of the key factors to be considered. This paper discusses an event-triggered control scheme for stochastic systems. The mechanism determines when the controller sends control data in order to balance system performance and drive frequency. In this paper, the control scheme is deduced by a class of quadratic performance index function. Then, the formula of performance index is designed based on state feedback and output feedback, and the upper bound of performance index is deduced and the conditions in the theorem can be constructed by solving a set of linear matrix inequalities, which can be easily tested with linear matrix inequality algorithms. Finally, a numerical example is given to illustrate the results.

Research on Typical Methods of S Surface Controller Parameter Self-Tuning for Underwater Vehicles

S surface control is a simple and operative motion control algorithm for underwater vehicles, but it has two parameters requiring to be adjusted manually. In order to enhance the adaptability of S surface controller, the research of S surface controller parameter self-tuning methods based on rules and models is carried out. Firstly, combined with fuzzy control, parameter self-tuning method based on fuzzy rules is presented. Then by means of predictive control theory, model-based parameter self-tuning method is proposed. By introducing the nonlinear autoregressive moving average model, the prediction model of underwater vehicles is established using parallel Elman neural network, and the optimal parameters of S surface controller is obtained by constructing quadratic performance index function. The results of simulation experiments show that the response speed of S surface controller with parameter self-tuning modules is improved, and the parameter self-tuning methods is demonstrated feasible and effective.

Guaranteed Cost Fault-tolerant Control of Networked Control Systems with Short Output Delay and Short Control Delay Based on State Observer

Supposing that the sensor and controller nodes were time-driven and the actuator node was event-driven, the problem of integrity against sensor failures for the networked control systems with short output delay and short control delay was discussed based on observer. The state observer of the system according to the time-delay compensation strategy was designed. Then, considering possible sensor failures, an augmented mathematic model for the networked control systems based on observer was developed. In terms of the given quadratic performance index function, the integrity condition of the system was given and the designs for guaranteed cost fault-tolerant controller and observer were presented respectively by using the cooperative design approach of the controller and observer and the approach of bilinear matrix inequalities. Finally, a numerical simulation example demonstrated the conclusions are feasible and effective. The proposed control method meets the requirements in industrial networked control systems.

Active control of a smart beam with time delay by Legendre wavelets

Active vibration control in a flexible beam is investigated by taking into account the time-delay which may occur in voltage controlled piezoelectric actuators. The voltage applied to a piezoelectric patch actuator to suppress the dynamic responses of the beam at a specified terminal time is computed by solving an optimal control problem. The quadratic performance index function consists of related potential energy and kinetic energy at the terminal time as well as control effort as a penalty term. The solution method is based on a combination of spatial modal expansion and direct approximation approaches. The spatial modal expansion approach is used to transfer the optimal control problem of distributed parameter system (DPS) into the optimal control problem of a linear time-invariant lumped parameter system (LPS). A direct state-control parametrization approach is formulated where Legendre wavelets are employed to solve time delayed LPS. The operational matrices of integration and delay are utilized to reduce the solution of linear time-varying delayed systems to the solution of algebraic equations. Illustrative examples are included to demonstrate the applicability and the efficiency of the proposed method and the numerical results indicate the effectiveness of the proposed control mechanism.

An Optimal Control Scheme for a Class of Discrete-time Nonlinear Systems with Time Delays Using Adaptive Dynamic Programming

Abstract In this paper, an optimal control scheme for a class of nonlinear systems with time delays in both state and control variables with respect to a quadratic performance index function is proposed using a new iterative adaptive dynamic programming (ADP) algorithm. By introducing a delay matrix function, the explicit expression of the optimal control is obtained using the dynamic programming theory and the optimal control can iteratively be obtained using the adaptive critic technique. Convergence analysis is presented to prove that the performance index function can reach the optimum by the proposed method. Neural networks are used to approximate the performance index function, compute the optimal control policy, solve delay matrix function, and model the nonlinear system, respectively, for facilitating the implementation of the iterative ADP algorithm. Two examples are given to demonstrate the validity of the proposed optimal control scheme.

Predictive Control Based on Information Fusion Optimal Estimation for Nonlinear Discrete System

For the quadratic optimal predictive control problem of the nonlinear discrete system,an iterative predictive control algorithm based on nonlinear information fusion optimal estimation is presented in this paper.By fusing the soft constraints information of the future reference trajectory and the control energy in the quadratic performance index function,and the hard constraints information supplied by the system state equation and the output equation,the optimal estimations of the co-state sequence and the control variable sequence are obtained.Simulation results of the shift control of a two-degree of freedom manipulator indicate the good stability and robustness of the proposed control algorithm.