PROBLEMS ON FEEDBACK CONTROL OF UNCERTAIN SYSTEMS: A THEORETICAL INTERPRETATION†

Abstract

The main problem in control theory is the definition of the optimal control law that must be applied to a physical system in order to obtain a desired behaviour. A general approach to this problem is developed, in order to explain the consideration of Rapoport that: “.feedback loops suggest that information is somehow fed into the system, or that the system obtains information about itself.” To this extent, the meanings of “system” and approximating “model” are properly clarified. The correct use of an observer, as the device which measures the uncertainty about the internal structure of the system and about the environmental influences on the system itself, is defined. The concept of “equilibrium” of a controlled system is introduced and some necessary and sufficient conditions in order that such an equilibrium may be reached, are stated.     

Adaptive output tracking of linear uncertain discrete-time networked systems over switching topologies

ABSTRACT

In this paper, we study the asymptotic output tracking problem for a class of minimum phase uncertain linear discrete-time multi-agent systems over jointly connected digraphs. As the systems contain arbitrarily large uncertain parameters, the robust feedback control technique does not work. We develop a distributed adaptive control law composed of a distributed observer and a purely decentralised adaptive control law. We first establish a stability result for a class of linear switched systems, which extends the existing results. This stability result leads to a novel distributed observer for the leader system which estimates the leader's signal using the output of the leader system locally and passes the estimated leader's signal to each follower. We then further show that the distributed adaptive control law solves our problem under some mild conditions. An example is used to illustrate the effectiveness and generality of our approach.     

Stochastic maximum principle for delayed doubly stochastic control systems and their applications

ABSTRACT

In this paper, we investigate the optimal control problems for delayed doubly stochastic control systems. We first discuss the existence and uniqueness of the delayed doubly stochastic differential equation by martingale representation theorem and contraction mapping principle. As a necessary condition of the optimal control, we deduce a stochastic maximum principle under some assumption. At the same time, a sufficient condition of optimality is obtained by using the duality method. At the end of the paper, we apply our stochastic maximum principle to a class of linear quadratic optimal control problem and obtain the explicit expression of the optimal control.     

Maximizing biogas production from the anaerobic digestion

This paper presents an optimal control law policy for maximizing biogas production of anaerobic digesters. In particular, using a simple model of the anaerobic digestion process, we derive a control law to maximize the biogas production over a period T using the dilution rate as the control variable. Depending on initial conditions and constraints on the actuator (the dilution rate D(·)), the search for a solution to the optimal control problem reveals very different levels of difficulty. In the present paper, we consider that there are no severe constraints on the actuator. In particular, the interval in which the input flow rate lives includes the value which allows the biogas to be maximized at equilibrium. For this case, we solve the optimal control problem using classical tools of differential equations analysis. Numerical simulations illustrate the robustness of the control law with respect to several parameters, notably with respect to initial conditions. We use these results to show that the heuristic control law proposed by Steyer et al., 1999 [20] is optimal in a certain sense. The optimal trajectories are then compared with those given by a purely numerical optimal control solver (i.e. the “BOCOP” toolkit) which is an open-source toolbox for solving optimal control problems. When the exact analytical solution to the optimal control problem cannot be found, we suggest that such numerical tool can be used to intuiter optimal solutions.     

Suppression of nonlinear regenerative chatter in milling process via robust optimal control

During the milling process, self-excited vibration or chatter adversely affects tool life, surface quality and productivity rate. In this paper, nonlinear cutting forces of milling process are considered as a function of chip thickness with a complete third order polynomial (instead of the common linear dependency). An optimal control strategy is developed for chatter suppression of the system described through nonlinear delay differential equations. Counterbalance forces exerted by actuators in x and y directions are the control inputs. For optimal control problem, an appropriate performance index is defined such that the regenerative chatter is suppressed while control efforts are minimized. Optimal control law is determined based on variation of extremals algorithm. Results show that under unstable machining conditions, regenerative chatter is suppressed effectively after applying the optimal control strategy. In addition, optimal controller guarantees robust performance of the process in the presence of model parametric uncertainties.     

Hybrid Solution Method for Dynamic Programming Equations for MDOF Stochastic Systems

An optimal control problem is considered for a multi-degree-of-freedom (MDOF) system, excited by a white-noise random force. The problem is to minimize the expected response energy by a given time instantT by applying a vector control force with given bounds on magnitudes of its components. This problem is governed by the Hamilton-Jacobi-Bellman, or HJB, partial differential equation. This equation has been studied previously [1] for the case of a single-degree-of-freedom system by developing a hybrid solution. Specifically, an exact analitycal solution has been obtained within a certain outer domain of the phase plane, which provides necessary boundary conditions for numerical solution within a bounded in velocity inner domain, thereby alleviating problem of numerical analysis for an unbounded domain. This hybrid approach is extended here to MDOF systems using common transformation to modal coordinates. The multidimensional HJB equation is solved explicitly for the corresponding outer domain, thereby reducing the problem to a set of numerical solutions within bounded inner domains. Thus, the problem of bounded optimal control is solved completely as long as the necessary modal control forces can be implemented in the actuators. If, however, the control forces can be applied to the original generalized coordinates only, the resulting optimal control law may become unfeasible. The reason is the nonlinearity in maximization operation for modal control forces, which may lead to violation of some constraints after inverse transformation to original coordinates. A semioptimal control law is illustrated for this case, based on projecting boundary points of the domain of the admissible transformed control forces onto boundaries of the domain of the original control forces. Case of a single control force is considered also, and similar solution to the HJB equation is derived.     

基于特征结构配置的二阶线性系统鲁棒容错控制设计

研究基于特征结构配置的二阶线性系统鲁棒容错控制设计问题,目的是重新设计状态反馈控制律,使得故障闭环系统和正常闭环系统具有相同的特征值.两闭环系统的特征向量依最小二乘法接近,而且能通过极小化灵敏度指标提高系统的鲁棒性.基于状态反馈特征结构配置的参数化结果,将系统灵敏度指标优化问题转化为含有约束条件的优化问题,并提出了鲁棒容错控制设计方法.数值算例及其仿真结果验证了所提出设计方法的有效性.     

A frequency‐constrained geometric Pontryagin maximum principle on matrix Lie groups

We present a geometric discrete‐time Pontryagin maximum principle (PMP) on matrix Lie groups that incorporates frequency constraints on the control trajectories in addition to pointwise constraints on the states and control actions directly at the stage of the problem formulation. This PMP gives first‐order necessary conditions for optimality and leads to two‐point boundary value problems that may be solved by numerical techniques to arrive at optimal trajectories. We demonstrate our theoretical results with numerical simulations on the optimal trajectory generation of a wheeled inverted pendulum and an attitude control problem of a spacecraft on the Lie group SO(3).     

拓扑切换下IT2 T-S模糊非线性多智能体系统全局逆最优控制

针对连续非线性多智能体系统的全局最优协同控制问题,本文提出了模糊输出反馈和逆最优方法的分布式一致性最优控制律和相应的控制策略.首先,通过一种区间2型T-S (interval type 2 Takagi-Sugeno IT2 T-S)模糊模型将非线性系统等价转化为线性系统.其次,基于逆最优方法设计了全局最优协同控制律和相应的模糊输出反馈控制策略,智能体间仅仅通过局部通信,即可实现拓扑切换下非线性多智能体系统的二次性能全局最优控制,且系统的收敛速度大大提高.基于局部稳定性理论给出了全局逆最优控制的充要条件.最后,通过MATLAB算例验证所提方法的正确性和可行性.     

Analytic Controlling Reorientation of a Spacecraft Using a Combined Criterion of Optimality

We consider the problem of optimally controlling the reorientation of a spacecraft (SC) from an arbitrary initial angular state into a given final angular position. We study the case when the minimized functional joins, in the given proportion, the time spent and the integral of the squared modulus of the angular momentum on the reorientation of a SC. The problem is solved in a kinematic setting. We consider two versions of the problem of the optimal rotation of a SC, with bounded and unbounded control. Using the necessary optimality conditions in the form of the Pontryagin maximum principle and the quaternion method for solving control problems on the motion of spacecrafts, we obtain an analytical solution of the posed problem. The solution of the problem is based on the quaternionic differential equation relating the angular momentum vector of a SC with the orientation quaternion of the related coordinate system. We present formalized equations and give computational expressions for constructing the optimal control program. We state the control law as an explicit dependence of the control variables on the phase coordinates. Using the transversality condition as a necessary optimality condition, we determine the maximal value of the modulus of the angular momentum for the optimal motion. For a dynamically symmetric SC, the problem of reorientation in space is solved completely: we obtain the dependences for the optimal law of the change of the angular momentum vector as explicit time functions. We give the results of the mathematical modeling of the motion for optimal control which demonstrate the practical realizability of designed algorithm for controlling the spatial orientation of a SC.     

Numerical Solution of Optimal Control Problems with Constant Control Delays

We investigate a class of optimal control problems that exhibit constant exogenously given delays in the control in the equation of motion of the differential states. Therefore, we formulate an exemplary optimal control problem with one stock and one control variable and review some analytic properties of an optimal solution. However, analytical considerations are quite limited in case of delayed optimal control problems. In order to overcome these limits, we reformulate the problem and apply direct numerical methods to calculate approximate solutions that give a better understanding of this class of optimization problems. In particular, we present two possibilities to reformulate the delayed optimal control problem into an instantaneous optimal control problem and show how these can be solved numerically with a state-of-the-art direct method by applying Bock’s direct multiple shooting algorithm. We further demonstrate the strength of our approach by two economic examples.      

Optimal control with constrained total variance for Markov jump linear systems with multiplicative noises

We consider in this paper the stochastic optimal control problem of discrete-time linear systems subject to Markov jumps and multiplicative noises (MJLS-mn for short). Our objective is to present an optimal policy for the problem of maximising the system's total expected output over a finite-time horizon while restricting the weighted sum of its variance to a pre-specified upper-bound value. We obtain explicit conditions for the existence of an optimal control law for this problem as well as an algorithm for obtaining it, extending previous results in the literature. The paper is concluded by applying our results to a portfolio selection problem subject to regime switching.     

A perspective-based convex relaxation for switched-affine optimal control

We consider the switched-affine optimal control problem, i.e., the problem of selecting a sequence of affine dynamics from a finite set in order to minimize a sum of convex functions of the system state. We develop a new reduction of this problem to a mixed-integer convex program (MICP), based on perspective functions. Relaxing the integer constraints of this MICP results in a convex optimization problem, whose optimal value is a lower bound on the original problem value. We show that this bound is at least as tight as similar bounds obtained from two other well-known MICP reductions (via conversion to a mixed logical dynamical system, and by generalized disjunctive programming), and our numerical study indicates it is often substantially tighter. Using simple integer-rounding techniques, we can also use our formulation to obtain an upper bound (and corresponding sequence of control inputs). In our numerical study, this bound was typically within a few percent of the optimal value, making it attractive as a stand-alone heuristic, or as a subroutine in a global algorithm such as branch and bound. We conclude with some extensions of our formulation to problems with switching costs and piecewise affine dynamics.     

Optimal stabilization of bodies in electromagnetic suspensions without measurements of their location

The optimal stabilization problem is considered for bodies in electromagnetic suspensions. To solve this problem, we form a linear stationary control law for the linearized system. This law is based on the feedback principle and uses the measuring of the current intensity in the circuit of the electromagnet, while the location of the body and its velocity are not measured. The optimality criterion is the generalized H _∞-norm of the linearized system: it characterizes the extinguishing level for perturbations generated by external actions and unknown initial conditions. To compute the feedback parameters, the technique of linear matrix inequalities is applied. We provide mathematical simulation examples for the dynamics of a body in an electromagnetic suspension.     

Optimal Control for Partially Observed Nonlinear Deterministic Systems with Fuzzy Parameters

In this paper we consider the control problem for a class of partially observed deterministic systems governed by nonlinear differential equations with fuzzy parameters. Using Takagi–Sugeno fuzzy model, we propose a linear (fuzzy) controller, driven by the output process, for controlling the system. Further, using calculus of variations, we have developed a set of necessary conditions on the basis of which optimal control can be determined. Based on these necessary conditions we have proposed a numerical algorithm for computing optimal control along with some numerical simulations to illustrate the effectiveness of the proposed (fuzzy) control scheme.     

Time-Optimal Boundary Control for Systems Defined by a Fractional Order Diffusion Equation

We consider the optimal control problem for a system defined by a one-dimensional diffusion equation with a fractional time derivative. We consider the case when the controls occur only in the boundary conditions. The optimal control problem is posed as the problem of transferring an object from the initial state to a given final state in minimal possible time with a restriction on the norm of the controls. We assume that admissible controls belong to the class of functions L_∞[0, T ]. The optimal control problem is reduced to an infinite-dimensional problem of moments. We also consider the approximation of the problem constructed on the basis of approximating the exact solution of the diffusion equation and leading to a finitedimensional problem of moments. We study an example of boundary control computation and dependencies of the control time and the form of how temporal dependencies in the control dependent on the fractional derivative index.     

Classical necessary optimality conditions in discrete optimal control problems with nonlocal conditions

We consider an optimal control problem in which the system’s state is described by a system of difference equations with nonlocal (two-point) conditions; this problem includes, as particular cases, the initial value problem (Cauchy problem) and different types of boundary value problems. It is assumed that the admissible controls take values from an open set. The first and second functional variations are calculated; these variations are used to express first and second order necessary optimality conditions in the classical sense for discrete optimal control problems.     

Optimal control theory for ambient pollution

We study optimal control of systems of distributed parameters applied to problems of ambient pollution. The model consists of a parabolic partial differential equation that models the transport of a pollutant in an incompressible viscous fluid, with boundary conditions and initial value, that is in our model we consider the velocity that the pollutant propagates in the environment. The developed mathematical modelling allows us to calculate the pollution concentration that is poured in a region of the space in such a way that at time t?=?T, the pollution concentration is as close as possible to the acceptable maximum concentration in the environment. We characterise the optimal control to obtain an optimality system that allows the numerical calculation of the problem and show the convergence of the method. As application, we study the case of the contamination of a river with mercury (Hg) in water without movement and with movement. We present some numerical experiments.     

Optimal control of mean-field backward doubly stochastic systems driven by Itô-Lévy processes

ABSTRACT

In this paper, we introduce a new class of backward doubly stochastic differential equations (in short BDSDE) called mean-field backward doubly stochastic differential equations (in short MFBDSDE) driven by Itô-Lévy processes and study the partial information optimal control problems for backward doubly stochastic systems driven by Itô-Lévy processes of mean-field type, in which the coefficients depend on not only the solution processes but also their expected values. First, using the method of contraction mapping, we prove the existence and uniqueness of the solutions to this kind of MFBDSDE. Then, by the method of convex variation and duality technique, we establish a sufficient and necessary stochastic maximum principle for the stochastic system. Finally, we illustrate our theoretical results by an application to a stochastic linear quadratic optimal control problem of a mean-field backward doubly stochastic system driven by Itô-Lévy processes.     

Infinite time optimal control for a class of stochastic systems

Optimization results developed by the author (Brandeberry and Wu 1970) for a class of stochastic regulator problems will be extended from the finite time interval to the infinite time interval. Conditions are given for the existence and stability of the infinite constant feedback of the system state; necessary conditions will also be obtained for the inverse problem (i.e. conditions for a linear constant control law to be optimal for some cost functional).