Optimal multi‐interval control of a cantilever beam by a recursive control algorithm

The optimal‐distributed control of a transversely vibrating cantilever beam is studied with the objective of minimizing the deflection and velocity in a given period of time with the minimum possible expenditure of force. The beam undergoes transient vibrations and is subject to given displacement and velocity initial conditions. The control is exercised by means of a transversely distributed force referred to as the control force. In the present study, a multi‐interval optimal control method is developed with the application of a maximum principle. The method consists of dividing the control duration into several intervals and using the maximum principle to obtain the optimality conditions at each interval. The explicit solutions for a cantilever beam are obtained by a recursive algorithm that takes the final conditions of the last interval as the initial conditions of the next interval. The formulation and the method of solution are suitable and convenient for digital computation. Numerical results are given, which compare the deflections, velocities and the control force under the optimal multi‐interval control with those under the optimal single‐interval control. Copyright © 2008 John Wiley & Sons, Ltd.     

Solving multi‐objective dynamic optimization problems with fuzzy satisfying method

This article proposes a novel algorithm integrating iterative dynamic programming and fuzzy aggregation to solve multi‐objective optimal control problems. First, the optimal control policies involving these objectives are sequentially determined. A payoff table is then established by applying each optimal policy in series to evaluate these multiple objectives. Considering the imprecise nature of decision‐maker's judgment, these multiple objectives are viewed as fuzzy variables. Simple monotonic increasing or decreasing membership functions are then defined for degrees of satisfaction for these linguistic objective functions. The optimal control policy is finally searched by maximizing the aggregated fuzzy decision values. The proposed method is rather easy to implement. Two chemical processes, Nylon 6 batch polymerization and Penicillin G fed‐batch fermentation, are used to demonstrate that the method has a significant potential to solve real industrial problems. Copyright © 2003 John Wiley & Sons, Ltd.     

An efficient scheme for minimax solutions of multiple linear‐quadratic control

Optimal control is one of the most important methodologies for studies of dynamic systems in many areas of sciences, engineering and economics. Minimax optimal control is a special topic in the general framework of multiple optimal control problems. Minimax optimal control can be considered as a dynamic game with multiple players under the same system. In this paper, we develop a fast search for a minimax solution of multiple linear‐quadratic control problems. The algorithm improves the existing solution scheme by adjusting the multiple weighting coefficients in each iteration and also including updates for step‐size control. Copyright © 2005 John Wiley & Sons, Ltd.     

Closed‐form minimax time‐delay filters for underdamped systems

This paper derives closed‐form solutions for the parameters of a time‐delay filter designed to be robust to uncertainties in frequencies to be cancelled. It is shown that the slope of the magnitude plot of the two time‐delay filter is zero at the nominal frequency indicating that it is a local maximum. This information is used for deriving the solution of the parameters of the time‐delay filter in closed form. Three time‐delay filters are also designed which force a zero of the filter to be located at the nominal frequency of the system. Uniform and non‐uniform distributions of the penalty over the uncertain regions are permitted in this formulation. The applicability of the proposed technique for the control of multi‐mode systems is also illustrated. Copyright © 2006 John Wiley & Sons, Ltd.     

Numerical method for weights adjustment in minimax multi‐model LQ‐control

The minimax linear quadratic problem, where ‘max’ is taken over a finite set of indices (models) and ‘min’ is taken over the set of admissible controls, is considered. The solution is obtained by the robust optimal control application. The control turns out to be a linear combination of the controls optimal for each individual model. This paper develops a numerical method for the optimal weights adjustment. An example shows a quick convergence of the proposed procedure. Copyright © 2007 John Wiley & Sons, Ltd.     

Low‐order multi‐rate linear time‐invariant decentralized trackers using the new observer‐based sub‐optimal method for unknown sampled‐data nonlinear time‐delay system with closed‐loop decoupling

This paper presents the low‐order multi‐rate linear time‐invariant decentralized trackers using the new observer‐based sub‐optimal method for a class of unknown sampled‐data nonlinear time‐delay system with closed‐loop decoupling. For the unknown sampled‐data nonlinear time‐delay system, we assume that the inner time delay is clearly known. Under this prerequisite, the appropriate (low‐) order decentralized linear observer for the unknown sampled‐data nonlinear system is determined by the off‐line observer/Kalman filter identification (OKID) method with artificial delay input and actual delay output. Then, the above observer has been further improved based on the proposed new observer‐based sub‐optimal approach. Sequentially, the decentralized tracker with the high gain property is proposed, so that the closed‐loop system has the decoupling property. The proposed approach constructs complete mathematics method including the concept of optimal control theory and state‐matching digital redesign technique and is quite useful for the complicated interconnected large‐scale sampled‐data nonlinear time‐delay system with unknown system equation. Copyright © 2010 John Wiley & Sons, Ltd.     

Linear quadratic regulator control of multi‐agent systems

This paper considers a collection of agents performing a shared task making use of relative information communicated over an information network. The designed suboptimal controllers are state feedback and static output feedback, which are guaranteed to provide a certain level of performance in terms of a linear quadratic regulator (LQR) cost. Because of the convexity of the LQR performance region, the suboptimal LQR control problem with state feedback is reduced to the solution of two inequalities, with the minimum and maximum eigenvalues of the Laplacian matrix as the coefficients. The advantage of the method is that the LQR control problem of network multi‐agent systems can be converted into the LQR control of a set of single‐agent systems, and the structure constraint on the feedback gain matrix can be eliminated. It can be shown that the size of the LQR control problem will not increase according to the number of the node in the fairly general framework. The method can be extended to the synthesis of the static output feedback, which is derived from the weighting matrices in LQR. Through some coordinate transformation and the augmentation of the output matrix, the LQR synthesis is provided on the basis of the output measurements of the adjacent agents. Numerical examples are presented to illustrate the proposed method. Copyright © 2013 John Wiley & Sons, Ltd.     

Finite‐time convergence results in robust model predictive control

Robust asymptotic stability (asymptotic attractivity and ?‐δ stability) of equilibrium regions under robust model predictive control (MPC) strategies was extensively studied in the last decades making use of Lyapunov theory in most cases. However, in spite of its potential application benefits, the problem of finite‐time convergence under fixed prediction horizon has not received, with some few exceptions, much attention in the literature. Considering the importance in several applications of having finite‐time convergence results in the context of fixed horizon MPC controllers and the lack of studies on this matter, this work presents a new set‐based robust MPC (RMPC) for which, in addition to traditional stability guarantees, finite‐time convergence to a target set is proved, and moreover, an upper bound on the time necessary to reach that set is provided. It is remarkable that the results apply to general nonlinear systems and only require some weak assumptions on the model, cost function, and target set.     

Robust reliability method and reliability‐based performance optimization for non‐fragile robust control design of dynamic system with bounded parametric uncertainties

An efficient robust reliability method for non‐fragile robust control design of dynamic system with bounded parametric uncertainties is presented systematically, in which the uncertainties existing in the controlled plant and controller realization are taken into account simultaneously in an integrated framework. Reliability‐based design optimization of non‐fragile robust control for parametric uncertain systems is carried out by optimizing the H₂ and H_∞ performances of the closed‐loop system, with the constraints on robust reliabilities. The non‐fragile robust controller obtained by the presented method may possess a coordinated optimum performance satisfying the precondition that the system is robustly reliable with respect to the uncertainties existing in controlled plant and controller. Moreover, the robustness bounds of uncertain parameters can be provided. The presented formulations are within the framework of linear matrix inequality and thus can be carried out conveniently. It is demonstrated by a numerical example that the presented method is effective and feasible. Copyright © 2016 John Wiley & Sons, Ltd.     

Optimal control of linear multi‐delay systems based on a multi‐interval decomposition scheme

This paper presents a novel approximation scheme to the numerical treatment of linear time‐varying multi‐delay systems with a quadratic performance index. A direct approach based on a hybrid of block‐pulse functions and Chebyshev polynomials is successfully developed. The operational matrix of delay associated to multi‐delay systems is constructed by an efficient manner. The excellent properties of hybrid functions together with the operational matrices of integration, delay, and product are then used to transform the optimal control problem into a mathematical optimization problem whose solution is much more easier than the original one. The procedure described in the current paper can be regarded as a multi‐interval decomposition scheme. The convergence of the proposed method is verified numerically. A wide variety of multi‐delay systems are investigated to demonstrate the effectiveness and computational efficiency of the proposed numerical scheme. The method has a simple structure, is easy to implement, and provides very accurate solutions. Copyright © 2015 John Wiley & Sons, Ltd.     

A novel projected Fletcher‐Reeves conjugate gradient approach for finite‐time optimal robust controller of linear constraints optimization problem: Application to bipedal walking robots

For finite‐time optimal robust control problem of bipedal walking robot, a class of global and feasible projected Fletcher‐Reeves conjugate gradient approach is proposed based on an online convex optimization algorithm. The optimal robust controllers are solved by projected Fletcher‐Reeves conjugate gradient approach. The approach can rapidly converge to a stable gait cycle by selecting an initial gait. Under some suitable conditions, we provide a rigorous proof of global convergence and well‐defined properties for projected Fletcher‐Reeves conjugate gradient approach. To demonstrate the effectiveness of the bipedal walking robot, we will conduct numerical simulations on the model of 3‐link robot with nonlinear, impulsive, and underactuated dynamics. Furthermore, to indicate the availability of high‐dimensional robotic system, the main result is illustrated on a nonlinear impulsive model of a bipedal walking robot through simulations via finite‐time optimal robust controller. Numerical results show that the projected Fletcher‐Reeves conjugate gradient approach is feasible and effective for bipedal walking robots. Therefore, it is reasonable to infer that the optimal robust control approach can be used in practical systems.     

Closed‐form solutions of a fishery harvesting model with state constraint

We derive a closed‐form solution for a well‐known fisheries harvesting model with an additional state constraint. The problem is linear in the control and previous solutions appearing in the literature have been numerical in nature. The so‐called direct adjoining approach is used in our derivation and the optimal solutions turn out to be a mixture of bang‐bang and boundary arcs. Copyright © 2013 John Wiley & Sons, Ltd.     

Optimal pattern of technology adoptions under embodiment: A multi‐stage optimal control approach

By deriving the necessary conditions for a multi‐stage optimal control problem where the endogenous switching instants appear as an argument of the state equation, we analyze the optimal pattern of technology adoptions under embodiment with a finite planning horizon. We show that the optimal pattern of technology adoptions depends crucially on how the growth rate advantage compares to the adjustment and the obsolescence costs inherent to embodiment. We obtain non‐stationary lifetimes for the adopted technologies due to finite planning horizon. We analyze numerically the effects of planning horizon, speed of adjustment to the new technology, growth rate of technology and the impatience rate on the optimal pattern. Copyright © 2010 John Wiley & Sons, Ltd.     

Robust observer‐based control for uncertain discrete time‐delay systems with nonlinearities electric‐hydraulic system under Hölder condition

In this paper, the robustness and control problems of output dynamic observer‐based control for uncertain discrete time‐delay systems with nonlinearities under a class of Hölder condition are considered. The parameter uncertainties enter into all the system matrices, the time‐varying delay is unknown with given lower and upper bounds, and the nonlinearities are described by satisfying a class of α Hölder condition. The asymptotic stabilization for uncertain time‐delay nonlinear system will be guaranteed. Linear matrix inequality optimization approach is used to design the robust observer‐based output dynamic controls. The feedback control and observer gains are got from linear matrix inequality optimization feasible solution. Electric‐hydraulic system is described and used to illustrate the method.     

Robust reliable guaranteed cost control for uncertain T‐S fuzzy neutral systems with interval time‐varying delay and linear fractional perturbations

The robust reliable guaranteed cost control for Takagi–Sugeno fuzzy systems with interval time‐varying delay is considered in this paper. Some free weighting matrices and non‐negative terms are provided to improve the conservativeness of our main results. An LMI optimization approach is applied to solve the problems of robust reliable guaranteed cost control and minimization of cost function. Copyright © 2014 John Wiley & Sons, Ltd.     

Analysis of the inertially symmetric rigid spacecraft time‐optimal three‐axis reorientation

This paper investigates the classical time‐optimal rest‐to‐rest three‐axis reorientation of the inertially symmetric rigid spacecraft. First‐order necessary optimality conditions are derived from the Pontryagin's maximum principle. Then, control structures (i.e., switching times and control torques) for the time‐optimal solution with five, six, and seven switches are given. For any five‐switch, six‐switch, or seven‐switch time‐optimal solution, a finite number of control structures exist, and relations among the control structures and their associated time‐optimal solutions are analytically derived. By utilizing the control structure, efficient numerical optimization algorithm based on multiple‐interval Radau pseudospectral method is proposed. Numerical results show that, after rounding to integer, five‐switch and six‐switch time‐optimal solutions exist for rotation angles on the interval [1,180] deg, and s es on the interval [1,72] deg. Finally, time‐optimal solutions for typical rotation angles are given to illustrate and validate the new findings. Copyright © 2016 John Wiley & Sons, Ltd.     

Multi‐model LQ‐constrained min–max control

This paper deals with the designing of a min–max controller that provides the minimum value of maximal (among a finite number of linear models) quadratic functional under a simple constraint for a control amplitude. Using the Lagrange multipliers approach, we show that the consideration of this constraint implies the existence of a new adjoint variable (treated as a time‐varying Lagrange multiplier), providing the closed‐form solutions for the considered multi‐model LQ‐constrained min–max control problem. The method is illustrated by three numerical examples. Copyright © 2015 John Wiley & Sons, Ltd.     

On the optimal control for fractional multi‐strain TB model

In this paper, optimal control of a general nonlinear multi‐strain tuberculosis (TB) model that incorporates three strains drug‐sensitive, emerging multi‐drug resistant and extensively drug‐resistant is presented. The general multi‐strain TB model is introduced as a fractional order multi‐strain TB model. The fractional derivatives are described in the Caputo sense. An optimal control problem is formulated and studied theoretically using the Pontryagin maximum principle. Four controls variables are proposed to minimize the cost of interventions. Two simple‐numerical methods are used to study the nonlinear fractional optimal control problem. The methods are the iterative optimal control method and the generalized Euler method. Comparative studies are implemented, and it is found that the iterative optimal control method is better than the generalized Euler method. Copyright © 2016 John Wiley & Sons, Ltd.     

A convex relaxation for the time‐optimal trajectory planning of robotic manipulators along predetermined geometric paths

In this paper, we deal with the problem of time‐optimal trajectory planning and feedforward controls for robotic manipulators along predetermined geometric paths. We propose a convex relaxation to generate time‐optimal trajectories and feedforward controls that are dynamically feasible with respect to the complete nonlinear dynamic model, considering both Coulomb friction and viscous friction. Even though the effects of viscous friction for time‐optimal motions become rather significant due to the required large speeds, in previous formulations, viscous friction was ignored. We present a strategic formulation that turns out non‐convex because of the consideration of viscous friction, which nonetheless leads naturally to a convex relaxation of the referred non‐convex problem. In order to numerically solve the proposed formulation, a discretization scheme is also developed. Importantly, for all the numerical instances presented in the paper, focusing on applying the algorithm results to a six‐axis industrial manipulator, the proposed convex relaxation solves exactly the original non‐convex problem. Through simulations and experimental studies on the resulting tracking errors, torque commands, and accelerometer readings for the six‐axis manipulator, we emphasize the importance of penalizing a measure of total jerk and of imposing acceleration constraints at the initial and final transitions of the trajectory. Copyright © 2016 John Wiley & Sons, Ltd.     

Multi‐player nonzero‐sum Nash differential game: variation and pseudo‐spectral method

A novel method is presented to solve the nonzero‐sum multi‐player Nash differential game. It combines the use of the variation and Legendre pseudo‐spectral methods. By the variation method, the original game is converted into a regular optimal control problem, avoiding the need to solve the associated Hamilton–Jacobi equation. Then the latter problem is converted into a common nonlinear programming problem via the Legendre pseudo‐spectral method, by which the saddle‐point for the original game can be achieved accurately. As an illustration, the air combat between two pursuers and an evader is formulated as a nonzero‐sum differential game. The simulation results show that numerical solutions can converge to the saddle‐points from different initial conditions, which demonstrates the feasibility and validity of the proposed method. Because the solution process requires little computational time, this method will allow for the development of a real time air combat control strategy. In addition, the simulations show that if the initial states of the two pursuers are fixed, there is an optimal initial heading angle for the evader to delay the interception time most effectively. Copyright © 2016 John Wiley & Sons, Ltd.