Team-optimal closed-loop Stackelberg strategies in hierarchical control problems

This paper is concerned with the derivation of closed-loop Stackelberg (CLS) solutions of a class of continuous-time two-player nonzero-sum differential games characterized by linear state dynamics and quadratic cost functionals. Explicit conditions are obtained for both the finite and infinite horizon problems under which the CLS solution is a representation of the optimal feedback solution of a related team problem which is defined as the joint minimization of the leader's cost function. First, a specific class of representations is considered which depend linearly on the current and initial values of the state, and then the results are extended to encompass a more general class of linear strategies that also incorporate the whole past trajectory. The conditions obtained all involve solutions of linear matrix equations and are amenable to computational analysis for explicit determination of CLS strategies.     

Efficient management of interconnected power systems: A game-theoretic approach

The optimal management over a one year planning horizon, of two interconnected hydro-thermal power systems is considered. The optimal production in each system is modelled as a stochastic control problem whose solution is searched in a particular class of control strategies. The efficient exchange of energy between the two systems and its pricing are viewed as a cooperative game and the Nash-Harsanyi bargaining solution is characterized. Various information structures for the exchange and price strategies are discussed and it is shown that, in all cases, the price strategy is equivalent to the definition of a compensatory side payment which equalizes the advantages accruing to each of the two players with respect to a status quo situation where no interconnection is available. A numerical illustration based on a typical European power system is presented to assess the potential gain when using a closed loop exchange strategy instead of an open loop one.     

On constrained infinite-time linear quadratic optimal control with stochastic disturbances

In this work, we study the infinite-time linear quadratic optimal control problem for systems with stochastic disturbances and constrained inputs. A number of stochastic problem formulations under the full state information (FSI) structure are considered with a particular focus on the subject of feedback structure and its impact on certainty equivalence. In particular, we clarify results concerning the open-loop hard constrained, closed-loop statistically constrained, and closed-loop hard constrained cases. Extension to the infinite-time framework provides a vehicle for interpreting these controllers and indicates that the last of the three is of most interest to regulation type applications. Additionally, the partial state information problem is considered, and conditions are given for which a separated configuration consisting of the optimal estimator cascaded with the FSI optimal controller remains optimal.     

Robust stochastic maximum principle for multi-model worst case optimization

This paper develops a version of the robust maximum principle applied to the minimax Mayer problem formulated for stochastic differential equations with a control-dependent diffusion term. The parametric families of first and second order adjoint stochastic processes are introduced to construct the corresponding Hamiltonian formalism. The Hamiltonian function used for the construction of the robust optimal control is shown to be equal to the sum of the standard stochastic Hamiltonians corresponding to each possible value of the parameter. The cost function is defined on a finite horizon and contains the mathematical expectation of a terminal term. A terminal condition, given by a vector function, is also considered. The optimal control strategies, adapted for available information, for the wide class of multi-model systems given by a stochastic differential equation with parameters from a given finite set are constructed. This problem belongs to the class of minimax stochastic optimization problems. The proof is based on the recent results obtained for deterministic minimax Mayer problem by Boltyanski and Poznyak as well as on the results of Zhou and of Yong and Zhou, obtained for stochastic maximum principle for non-linear stochastic systems with a single-valued parameter. Two illustrative examples, dealing with production planning and reinsurance-dividend management, conclude this study.     

Pole assignment in a specified disk

The problem of assigning all poles of a closed-loop system in a specified disk by state feedback is considered for both continuous and discrete systems. A state feedback control law is determined by using a discrete Riccati equation. This kind of pole assignment problem is namedD-pole assignment, and its relation to the optimal control problem and its robustness properties are discussed. The gain and phase margins for all closed-loop poles to stay inside the specified diskDare determined for the proposed control.     

Optimal open-loop feedback control for linear systems with unknown parameters

The problem of controlling a class of linear systems with unknown parameters is considered. The optimal open-loop strategies are obtained for linear systems with unknown parameters in the system matrix with a quadratic performance index. The method of solution is based upon the use of the Hamilton-Jacobi theory.     

Non-fragile robust optimal guaranteed cost control of uncertain 2-D discrete state-delayed systems

This paper is concerned with the problem of non-fragile robust optimal guaranteed cost control for a class of uncertain two-dimensional (2-D) discrete state-delayed systems described by the general model with norm-bounded uncertainties. Our attention is focused on the design of non-fragile state feedback controllers such that the resulting closed-loop system is asymptotically stable and the closed-loop cost function value is not more than a specified upper bound for all admissible parameter uncertainties and controller gain variations. A sufficient condition for the existence of such controllers is established under the linear matrix inequality framework. Moreover, a convex optimisation problem is proposed to select a non-fragile robust optimal guaranteed cost controller stabilising the 2-D discrete state-delayed system as well as achieving the least guaranteed cost for the resulting closed-loop system. The proposed method is compared with the previously reported criterion. Finally, illustrative examples are given to show the potential of the proposed technique.     

On the control of finite-dimensional mechanical systems withunilateral constraints

This paper focuses on the problem of the control of a class of mechanical systems with a finite number of degrees-of-freedom, subject to unilateral constraints on the position. Roughly speaking, those systems are described by a set of ordinary differential equations that represent smooth dynamics, together with an algebraic inequality condition F(q)⩾0 (where q is the vector of generalized coordinates) and an impact rule relating the interaction impulse and the velocity. Nonsmooth dynamics is at the core of the study of such systems. This implies that one can suitably define solutions and stability concepts that fit with the considered model. We then discuss the closed-loop control problem, and analyze various switching control strategies     

Singular linear-quadratic optimal control problem for a class of discrete singular systems with multiple time-delays

The singular linear-quadratic optimal control problem for a class of discrete-time linear singular systems with multiple time-delays is investigated. The equivalent relationship between this problem and the non-singular linear-quadratic problem for standard state-space discrete-time linear systems without time-delay is derived, and the necessary and sufficient condition guaranteeing this equivalent relationship is also given. The optimal control is synthesized as state feedback, and the closed-loop system is regular, impulse-free and without time-delay. An example is given to show the validity of the proposed method.     

The output control of linear time-invariant multivariable systems with unmeasurable arbitrary disturbances

Necessary and sufficient conditions are derived for a minimal order linear time-invariant differential feedback control system to exist for a linear time-invariant multivariable system with unmeasurable arbitrary disturbances of a given class occurring in it, such that the outputs of the system asymptotically become equal to preassigned functions of a given class of outputs, independent of the disturbances occurring in the system, and such that the closed-loop system is controllable. The feedback gains of the control system are obtained so that the dynamic behavior of the closed-loop system is specified by using either an integral quadratic optimal control approach or a pole assignment approach. The result may be interpreted as being a generalization of the single-input, single-output servomechanism problem to multivariable systems or as being a solution to the asymptotic decoupling problem.     

Stackelberg strategies in discrete linear quadratic problems

Linear closed-loop Stackelberg strategies in sequential decision-making problems for linear time-invariant discrete systems and trace criteria which may be interpreted as the expected value of quadratic criteria are considered. Necessary conditions for the solution are developed and an algorithm of the multi-level Stackelberg problem is presented. An example for a two-level problem is presented to illustrate the proposed numerical algorithm     

Hierarchical control of distributed-parameter systems using Stackelberg policies†

This paper treats the multilevel hierarchical control problem of multicontroller distributed-parameter systems, using the Stackelberg optimality criterion. Both continuous-time and discrete-time systems, linear as well as non-linear, are considered. Two-level hierarchies with one coordinator and many second-level controllers, and linear multilevel hierarchical structures are studied. Solution equations for open-loop, feedback, and closed-loop Stackelberg control strategies are derived and discussed.     

Adaptive control design under structured model information limitation: A cost-biased maximum-likelihood approach

Networked control strategies based on limited information about the plant model usually result in worse closed-loop performance than optimal centralized control with full plant model information. Recently, this fact has been established by utilizing the concept of competitive ratio, which is defined as the worst-case ratio of the cost of a control design with limited model information to the cost of the optimal control design with full model information. We show that an adaptive controller, inspired by a controller proposed by Campi and Kumar, with limited plant model information, asymptotically achieves the closed-loop performance of the optimal centralized controller with full model information for almost any plant. Therefore, there exists, at least, one adaptive control design strategy with limited plant model information that can achieve a competitive ratio equal to one. The plant model considered in the paper belongs to a compact set of stochastic linear time-invariant systems and the closed-loop performance measure is the ergodic mean of a quadratic function of the state and control input.     

基于神经动态优化的非线性系统近似最优跟踪控制

提出一种输入约束下一类连续时间非线性系统最优跟踪控制问题的近似求解方法.针对有限时间跟踪性能指标下一类单输入单输出非线性系统,利用所提出的最优跟踪控制方法实现目标系统所对应性能指标近似最优.首先将系统的性能指标沿时间泰勒展开,得到一个近似的性能指标;其次,在系统状态可观测条件下,将该问题进一步转化为以控制输入为决策变量的非线性规划问题;再次,利用神经动态优化方法,求解含不等式约束下的近似最优控制问题并给出相应的递归神经网络模块原理图;进而,针对整个闭环系统进行理论分析,证明在一定条件下闭环系统的稳定性;最后,通过两个实例仿真验证所提出方法的有效性.     

Closed-loop properties of the infinite-time linear-quadratic optimal regulator for systems with delays

The infinite-time linear-quadratic optimal-control problem for systems with delays is discussed from the viewpoint of closed-loop properties: the Kalman equation and the circle condition for the optimal closed-loop control law are derived, and the optimal closed-loop poles are characterized via the hamiltonian matrix.     

System equivalence in a class of nonlinear optimal control problems

In a family of nonlinear optimal control problems, equivalence classes of problems that yield identical optimal closed-loop systems are delineated. The equivalence relations are established as one-to-one correspondences between the solutions of the Hamilton-Jacobi equations associated with the problems. Two such equivalence relations are presented. These allow an arbitrary problem in the family to be reduced to an equivalent basic standard form of problem. Also, a new formal procedure for solving this basic problem is given that allows the determination of coefficients in the series expansion of the optimal closed-loop system. The set of algebraic equations to be solved is considerably simpler than that derived by other methods.     

Dynamic Output Feedback Control of Switched Linear Systems

This paper is devoted to stability analysis and control design of switched linear systems in both continuous and discrete-time domains. A particular class of matrix inequalities, the so-called Lyapunov--Metzler inequalities, provides conditions for open-loop stability analysis and closed-loop switching control using state and output feedback. Switched linear systems are analyzed in a general framework by introducing a quadratic in the state cost determined from a series of impulse perturbations. Lower bounds on the cost associated with the optimal switching control strategy are derived from the determination of a feasible solution to the Hamilton--Jacobi--Bellman inequality. An upper bound on the optimal cost associated with a closed-loop stabilizing switching strategy is provided as well. The solution of the output feedback problem is based on the construction of a full-order linear switched filter whose state variable is used by the mechanism for the determination of the switching rule. Throughout, the theoretical results are illustrated by means of academic examples. A realistic practical application related to the optimal control of semiactive suspensions in road vehicles is reported.     

A Lyapunov's direct method for the global robust stabilization of nonlinear cascaded systems

The global robust stabilization problem of cascaded systems with dynamic uncertainty has been approached by the small gain theorem. This method, however, does not produce an explicit Lyapunov function for the closed-loop system. In this paper, we develop a Lyapunov's direct method based recursive approach to solving the global robust stabilization problem for the mentioned systems. This method also produces an explicit Lyapunov function for the closed-loop system which is a superposition of those of individual subsystems. This Lyapunov function is indispensable when the adaptive control of the same class of systems is further considered.