A computational study of optimal control of Markov jump systems

This paper considers a class of optimal control problems governed by Markov jump systems. Our focus is to develop a computational method, based on the control parametrization approach, for solving this class of optimal control problems. Due to the existence of the continuous-time Markov chain, the optimal control problem under consideration is a stochastic optimal control problem, and hence the control parametrization technique cannot be applied directly. For this, a derandomization approach is introduced to obtain a representative deterministic optimal control problem. Then, the control parametrization method is applied to obtain an approximate finite dimensional optimization problem which can be computed numerically using the gradient-based optimization method. For this, the gradient formulas of the cost function and the constraint functions with respect to control variables are derived. Finally, a practical application involving a RLC circuit model is solved using the method proposed.     

Fuzzy Bang-Bang control problem under granular differentiability

In this paper, a class of fuzzy optimal control problem called fuzzy Bang-Bang (FBB) control problem is revisited. The FBB control problem aims at finding control inputs which transfer the states of an uncertain dynamical system to the origin in a minimum time. Since the conditions and/or parameters of the dynamical system are uncertain, the FBB control problem is a challenge task. To address the problem, we first introduce the concepts of the granular integral and derivative of fuzzy functions whose domain is uncertain. In addition, the notion of granular partial derivative of a multivariable fuzzy function whose variables are fuzzy functions with uncertain domains is presented. Then, we propose a theorem which is proved to be applicable to the FBB control problem. Moreover, taking the relative-distance-measure fuzzy interval arithmetic and horizontal membership functions into consideration, we further give complementary theorems to ensure that if the problem has a solution, then the controller assumes its boundary values. The simulated results confirm this theoretical conclusion. These findings may enrich our insight of the behavior of the uncertain dynamical system subjected to the FBB control problem, and guide to predict the uncertain trajectories.     

Trading performance for state constraint feasibility in stochastic constrained control: A randomized approach

Constrained control for stochastic linear systems is generally a difficult task due to the possible infeasibility of state constraints. In this paper, we focus on a finite control horizon and propose a design methodology where the constrained control problem is formulated as a chance-constrained optimization problem depending on some parameter. This parameter can be tuned so as to decide the appropriate trade-off between control cost minimization and state constraints satisfaction. An approximate solution is computed via a randomized algorithm. Precise guarantees about its feasibility for the original chance-constrained problem are provided. A numerical example shows the efficacy of the proposed methodology.     

A solution to the path planning problem via algebraic geometry and reinforcement learning

In this paper, the path planning problem for an unicycle-like mobile robot is considered. By using some results borrowed from algebraic geometry, a technique is given to determine a dynamical system that is affine in the input and whose trajectories tend to a chosen algebraic set independently of the control input. Since this does not guarantee that the corresponding paths of motion are collision free, an optimal control problem is formulated to enforce this behavior, and its approximate solution is determined via integral reinforcement learning. Finally, it is shown how such results can be used to derive a feedback control law for unicycle-like mobile robots.     

Optimal control strategies for an ecological model including infection and competition

This contribution is concerned with the application of optimal control theories and methods to an ecosystem. There are four major ingredients. The first ingredient is to present an optimal control problem for an ecosystem including infection and competition. The second ingredient is the existence results, which assert that in a given Hilbert space the controlled system admits a unique strong solution and the optimal control problem has at least one optimal pair. The third, which is the main new ingredient of this paper, is a maximum principle for the optimal control problem. The first-order necessary optimality condition for the optimal control is established by employing dual techniques. It provides a theoretical basis for the design of the optimal control strategy. The fourth ingredient is an example and its numerical simulation, which further confirm the results of this paper.     

Containment control of singular heterogeneous multi-agent systems

In this paper, we first consider the containment control problem of singular heterogeneous multi-agent systems, where all the followers converge to the convex hull spanned by the leaders. To solve this problem, we propose two distributed control laws: one is based on the state feedback control framework, which is suitable for the case that the full state information of each follower is accessible; and the other is based on the output regulation framework, where each follower only can access to its output. Furthermore, the distributed observers are designed for every follower to estimate the convex combination of the leader states which is determined by the communication graph. It should be noted that our results can also regard the non-singular multi-agent systems’ containment control problem as a special case. Finally, simulation results corroborate the effectiveness of our analytical results.     

Adaptive pseudospectral methods for solving constrained linear and nonlinear time-delay optimal control problems

In this paper, we first develop an adaptive shifted Legendre–Gauss (ShLG) pseudospectral method for solving constrained linear time-delay optimal control problems. The delays in the problems are on the state and/or on the control input. By dividing the domain of the problem into a uniform mesh based on the delay terms, the constrained linear time-delay optimal control problem is reduced to a quadratic programming problem. Next, we extend the application of the adaptive ShLG pseudospectral method to nonlinear problems through quasilinearization. Using this scheme, the constrained nonlinear time-delay optimal control problem is replaced with a sequence of constrained linear-quadratic sub-problems whose solutions converge to the solution of the original nonlinear problem. The method is called the iterative-adaptive ShLG pseudospectral method. One of the most important advantages of the proposed method lies in the case with which nonsmooth optimal controls can be computed when inequality constraints and terminal constraints on the state vector are imposed. Moreover, a comparison is made with optimal solutions obtained analytically and/or other numerical methods in the literature to demonstrate the applicability and accuracy of the proposed methods.     

Distributed bearing-based formation control of networked thrust-propelled vehicles

In this paper, the distributed bearing-based formation control problem of networked thrust-propelled vehicles (TPVs) is addressed, in which both the constant and time-varying velocity leaders are considered, respectively. By introducing a reference acceleration and adaptive control scheme for the followers, the mass knowledge is not necessary in contrast to the existing works. Based on the designed reference accelerations, distributed adaptive control laws are proposed for the networked TPVs. Then the stabilization conditions are presented and an inner-bearing prescribed formation can be achieved. Under the proposed control laws, the leader-follower formation maneuver problem for networked TPVs with system uncertainties can be solved. Finally, simulation results are provided to demonstrate the effectiveness of the proposed control laws.     

Bond graph formulation of an optimal control problem for linear time invariant systems

A recent communication has proposed a conjectural procedure for representing a category of optimal control problems in bond graph language [W. Marquis-Favre, B. Chereji, D. Thomasset, S. Scavarda, Bond graph representation of an optimal control problem: the dc motor example, in: ICBGM’05 International Conference of Bond Graph Modelling and Simulation, New Orleans, USA, January 23-27, 2005, pp. 239-244]. This paper aims at providing a fundamental theory for proving the effectiveness of this procedure. The class of problem that the procedure can deal with has been extended. Its application was formerly restricted to linear time invariant siso system. The systems considered now are linear time invariant mimo systems. The optimization objective is the minimization of dissipation and input. The developments concerning the optimal control problem are based on the Pontryagin maximum principle and the proof of the effectiveness of the procedure makes a broad use of the port-Hamiltonian concept. As a result, the bond graph representation of the given optimization problem enables the analytical system, which provides the optimal solution, to be derived. The work presented in this paper is the first step in research with perspectives towards formulating dynamic optimization problems in bond graph and, towards coupling this formulation with a sizing methodology using bond graph language and a state-space inverse model approach. This sizing methodology, however, is not the topic of this paper and thus is not presented here.     

Event-triggered control of leader-following and cluster consensus for MAS under directed graph with designable minimum event interval

This paper investigates the event-based consensus problems for linear multi-agent systems under directed network topology. First, a new event-triggered control method is proposed for the leader-following consensus problem of agents under directed graphs. Then this new method is applied to the cluster control problem under special topological conditions. The new event-based control scheme is better than some existing literature in the following aspects. 1) The graph only needs to contain a spanning tree instead of being required to be strongly connected graph or undirected, and the triggering function is state-dependent rather than time-dependent. 2) Some parameters are designable for the trade-off between the event interval and the performance of the controlled system. Besides, the optimization of some parameters is studied to reduce the trigger frequency. All the agents can achieve consensus with an exponential speed when communications among follower agents are intermittent, and Zeno behavior is excluded under the proposed method. 3) When applying this method to the cluster control problem, agents in the same cluster share the same form of triggering function. Cluster consensus can be achieved regardless of intra- and inter-cluster relative coupling strength under the event-triggered control framework.     

On the set-valued approach to optimal control of sliding mode processes

This paper is devoted to a theoretic framework for a general optimal control problem (OCP) associated with the classic sliding mode process. The sliding dynamic behavior is interpreted here as a special kind of additional constraints related to the main optimization problem. We are specially interested in the development of some adequate constructive approximations of the original OCPs. The mathematical approach based on the set-valued analysis allows to study the discontinuity of sliding mode dynamics in the abstract setting. Moreover, we also establish some sensitivity properties of the optimal solutions. The obtained results provide an universal analytical tool for the corresponding conceptual approximation schemes related to the original OCPs. The constructive approximations proposed in this paper are numerically stable and can be applied to various classes of optimal control processes governed by the affine control systems.     

Further results on stabilization for NCSs with packet losses and transmission delay: UDP case

The stabilization problem concerning the user datagram protocol (UDP) for networked control systems (NCSs) is investigated in this paper. More specifically, both the state and control signal can be lost during transmission, and the networked induced delay occurs while the control signal delivered to the actuator. On the other hand, no communication channel can be accessed from the actuator to the estimator (no acknowledge signal). The innovations of this paper include: (1) Based on the modified Riccati-ZXL equations, the necessary and sufficient stabilization conditions are developed for the NCSs with packet losses and transmission delay (UDP case); (2) We have verified that the separation principle holds for the finite horizon optimal output feedback control problem; (3) For 1-dimensional case, the maximal delay bound and the maximal packet losses rate are derived.     

Optimized control for human-multi-robot collaborative manipulation via multi-player Q-learning

In this paper, optimized interaction control is investigated for human-multi-robot collaboration control problems, which cannot be described by the traditional impedance controller. To realize global optimized interaction performance, the multi-player non-zero sum game theory is employed to obtain the optimized interaction control of each robot agent. Regarding the game strategies, Nash equilibrium strategy is utilized in this paper. In human-multi-robot collaboration problems, the dynamics parameters of the human arm and the manipulated object are usually unknown. To obviate the dependence on these parameters, the multi-player Q-learning method is employed. Moreover, for the human-multi-robot collaboration problem, the optimized solution is difficult to resolve due to the existence of the desired reference position. A multi-player Nash Q-learning algorithm considering the desired reference position is proposed to deal with the problem. The validity of the proposed method is verified through simulation studies.     

Robust suboptimal feedback control for a fed-batch nonlinear time-delayed switched system

The optimal control strategy constructed in the form of a state feedback is effective for small state perturbations caused by changes in modeling uncertainty. In this paper, we investigate a robust suboptimal feedback control (RSPFC) problem governed by a nonlinear time-delayed switched system with uncertain time delay arising in a 1,3-propanediol (1,3-PD) microbial fed-batch process. The feedback control strategy is designed based on the radial basis function to balance the two (possibly competing) objectives: (i) the system performance (concentration of 1,3-PD at the terminal time of the fermentation) is to be optimal; and (ii) the system sensitivity (the system performance with respect to the uncertainty of the time-delay) is to be minimized. The RSPFC problem is subject to the continuous state inequality constraints. An exact penalty method and a novel time scaling transformation approach are used to transform the RSPFC problem into the one subject only to box constraints. The resulting problem is solved by a hybrid optimization algorithm based on a filled function method and a gradient-based algorithm. Numerical results are given to verify the effectiveness of the developed hybrid optimization algorithm.     

汽车起动机保护策略

本文首先论述传统点火锁故障导致起动机烧毁的M题,然后阐述了一种通过发动机ECM接受起动请求信号,并控制和切断起动机完成发动机点火的策略。以保护起动机反拖烧毁。     

Direct solution of nonlinear optimal control problems using quasilinearization and Chebyshev polynomials

In this paper, a numerical method to solve nonlinear optimal control problems with terminal state constraints, control inequality constraints and simple bounds on the state variables, is presented. The method converts the optimal control problem into a sequence of quadratic programming problems. To this end, the quasilinearization method is used to replace the nonlinear optimal control problem with a sequence of constrained linear-quadratic optimal control problems, then each of the state variables is approximated by a finite length Chebyshev series with unknown parameters. The method gives the information of the quadratic programming problem explicitly (The Hessian, the gradient of the cost function and the Jacobian of the constraints). To show the effectiveness of the proposed method, the simulation results of two constrained nonlinear optimal control problems are presented.     

Minimum-norm control of linear systems with partially known initial conditions

The problem of controlling a linear system with initial conditions known to lie in a bounded set of the state space, and which is otherwise unknown, is treated in this paper by a modification of the techniques associated with the L-problem of moments. The set of initial conditions is assumed to be a convex polytope. The problem is reduced to devising a control function whose components are bounded functions over a given interval which maps the vertices of the polytope into a closed ball with the center at the origin, and has the least norm (sup-norm) among all control functions to do so. It is possible to apply a modified version of the techniques used to solve the L-problem of moments, and in this manner the optimal control, if it exists, is characterized and existence and uniqueness conditions are derived. A by-product of the analysis is an efficient computational method for optimal control. The procedure is shown by a numerical example.     

Output feedback gain scheduled control of actuator saturated linear systems with applications to the spacecraft rendezvous

The problem of stabilization of a linear system that is asymptotic null controllable with bounded control is studied in this paper. By combining the parametric Lyapunov equation approach and the gain scheduling technique, a new observer-based output feedback gain scheduling controller is proposed to solve the semi-global stabilization problem for a linear system subject to actuator saturation. By scheduling the design parameters online the convergence rate of the state can be improved. Numerical simulations for a spacecraft rendezvous system show the effectiveness of the proposed approaches.     

Control of multi-scroll Chen system

This paper investigates the chaos control problem of a new multi-scroll chaotic system. Two nonlinear control methods are studied, namely high-order and predictive types of control. The proposed methodologies offer the possibility of stabilizing unstable periodic orbits and unstable equilibrium points from the state equations. For this purpose, we apply the high-order control method for stabilizing a desired unstable periodic orbit, while the predictive control method is applied for stabilization problem of unstable equilibrium points. In particular, these approaches are effective and easy to be implemented since we only need to apply small perturbations to the system dynamics. The multi-scroll Chen system is used as representative example to show the working principle of these methods. Numerical simulation results indicate the potential engineering applications of the proposed control methods for various multi-scroll chaos-based practical applications.     

A direct approach using the finite element method for the solution of the linear optimal control problem with a quadratic criterion

The present paper proposes a numerical approach to a linear optimal control problem with a quadratic performance index. In this technique, the time interval is divided into a number of time segments and all of the unknown functions which appear in the performance index are either interpolated linearly with respect to time or assumed to be constant in each time segment. The augmented performance index is discretized within each time element through the ordinary finite element technique.The main advantage of the present method is as follows: all of the necessary conditions for the performance index to be stationary can be expressed in the form of algebraic equations and the performance sequence of the state variables can be eliminated. As a result, the optimal control problem is reduced to the simple one of finding the sequence of control variables alone, which minimizes the quadratic performance index.A general formulation of the method is given and simple numerical examples are shown to demonstrate the effectiveness of the technique.