Stochastic affine quadratic regulator with applications to tracking control of quantum systems

This paper deals with the regulator theory of stochastic affine systems. Applying a tensor formal power series method, stochastic bilinear quadratic regulator is solved numerically. An example about tracking control of quantum systems is given to show the usefulness of the developed theory.     

Output regulation of time-varying systems

In this paper the output regulation problem for linear time-varying systems is considered. Replacing the regulator equation by a regulator differential equation we give a necessary and sufficient condition for the problem to be solvable. As in the time-invariant case we first solve the output regulation problem by state feedback and obtain the required condition. Then with the aid of observers we show that the same condition solves the general problem with measurement feedback. We then consider the classes of almost periodic and periodic systems and refine the main results. A simple example of an almost periodic system and simulation results are given to illustrate the theory.     

The linear periodic output regulation problem

The problem of asymptotic output regulation for linear systems driven by time-varying, T-periodic exosystems is considered in this paper. Necessary and sufficient condition for its solvability based on the existence of periodic solutions of differential Sylvester equations are derived. These conditions constitute a generalization to the periodic case of the celebrated algebraic regulator equations of Francis. A general algorithm for the synthesis of an error-feedback regulator is given. For the case of minimum-phase systems, it is shown that the regulator design can be carried out without the knowledge of the Floquet decomposition of the exosystem, thus extending significantly the applicability of the general result. The more challenging issue of robust regulation by error feedback is also addressed, and solved under a stronger observability condition.     

离散系统鲁棒容错线性调节器设计

提出一种离散系统鲁棒容错线性调节器的设计方法，给出了鲁棒容错线性调节器的设计步骤，并用算例说明了该方法的有效性。     

Global nonlinear output regulation: Convergence-based controller design

In this paper we present output-feedback controllers solving the global output regulation problem for a class of nonlinear systems. The proposed controllers are based on the notion of convergent systems. The presented solution extends well-established results on the linear output regulation problem and the local nonlinear output regulation problem to the global case. For Lur’e systems, which are not necessarily in the output-feedback form, the proposed controllers can be found by solving the regulator equations and certain linear matrix inequalities. For systems in the output-feedback form with uncertain parameters and uncertain nonlinearities we provide a robust regulator that does not rely on the minimum phaseness assumption on the system, which is crucial in the previous regulator designs for output-feedback systems. The results are illustrated by examples.     

非线性不确定系统的自适应观测器设计

非线性状态观测器可改善过程控制性能和故障诊断,针对一类参数不确定非线性系统提出了自适应观测器设计方法。通过微分同胚变换,将非线性系统转换为仅依赖原系统输入、输出的自适应观测器规范形式。利用自适应调节器估计未知参数,用构造的观测器实现状态的重构。Lyapunov稳定性理论分析了状态观测误差动态方程的稳定性,用来证明所设计的自适应观测器为全局渐近收敛的,既实现了系统状态的渐近重构又确保了在持续激励条件下未知参数估计以指数快速收敛到真值,并通过仿真试验。仿真结果表明提出方法的有效性。     

Analysis of an iterative scheme for approximate regulation for nonlinear systems

This paper concerns the analysis of an iterative scheme delivering approximate control laws for the tracking regulation problems for nonlinear systems. The procedure can be applied to finite‐ and infinite‐dimensional systems, and the underlying methodology derives from the geometric methods, which have been developed for both linear and nonlinear systems. In the nonlinear case, the main tool is the center manifold theorem. Indeed, in the geometric methodology, under the assumption that the signals to be tracked are generated by a finite‐dimensional exo‐system, the desired control is obtained by solving a pair of operator equations called the regulator equations. In this paper, we extend the concept of regulator equations to what we refer to as the dynamic regulator equations. Just as it is generally quite difficult to solve the regulator equations, it can be equally difficult to solve the dynamic regulator equations. As the authors have already shown in the linear case, a straightforward attempt to solve the dynamic regulator equations leads to a singular system, which can be regularized to obtain an iterative scheme that provides approximate control laws that provide accurate tracking with very a small tracking error after only a couple of iterations. In this paper, we generalize the iterative scheme to nonlinear systems and provide error estimates for the first 3 iterations. Both finite‐ and infinite‐dimensional examples are given to validate the estimates. We comment that the method has also been applied to a wide range of nonlinear distributed parameter examples described in the references.     

On the stability of continuous-time adaptive controllers for pure delay systems

We analyse the stability properties of the linear retarded differential-difference equation (RDDE) that arises in the study of adaptive control of pure delay systems. Three different results are given. Firstly, using Floquet theory necessary and sufficient conditions for stability are established for the case of periodic signals of specified frequencies. Secondly, the theory of averaging is used to derive stability-instability conditions under slow adaptation. Finally, a globally stable modified adaptive regulator for systems with known, possibly time-varying, time delay is presented. Two alternative proofs of the latter results are given. One based on the method of Lyapunov functionals and the second one using a Razumikhin-type theorem.     

On the optimal tuning of a robust controller for parabolic distributed parameter systems

Optimal tuning of robust multivariable controllers for stable parabolic distributed parameter systems is discussed. In order to evaluate the behaviour of the closed loop system, a quadratic criterion is selected. Then it is proved that the system may be decomposed into slow and fast subsystems, and that the value of the criterion will mainly depend on the slow subsystem. Tuning of the controller, for this subsystem, turns out to be a standard finite-dimensional quadratic regulator problem. Two examples are given to show the improvement due to the new selection of the control parameters.     

Sequential stability and optimization of large scale decentralized systems

The notion of sequential stability in the synthesis of decentralized control for large scale systems is introduced in this paper. This notion is concerned with the property of a synthesis technique which allows the decentralized controllers of a large scale system to be connected to the systems one at a time (in a sequential way) such that the controlled system remains stable at all times. The motivation for introducing this constraint is that in practical terms, it is generally impossible to connect all decentralized controllers to a system simultaneously (due to the difficulties of communication etc.). A practical design procedure for the synthesis of a decentralized robust regulator for the servomechanism problem, based on a sequential approach to system design, is then given. The design procedure proceeds in two stages: (1) decentralized controllers are initially connected to the system in a sequential way to guarantee stability; (2) the parameters of the decentralized controllers are then sequentially adjusted, in a way to guarantee stability, so as to optimize a given performance index for the system. Applications of the above procedure are then made to the synthesis of centralized multivariable controllers and to the decentralized robust control of unknown systems.A simple example is given to illustrate the design synthesis.     

Robust output regulation for containment control of heterogeneous discrete-time nonlinear multi-agent systems

ABSTRACT

In existing researches on containment control of heterogeneous multi-agent systems (MASs), the solution is usually dependent on the solvability of regulator equations. However, the closed-form solution of many nonlinear regulator equations of systems is rarely obtained. Towards this end, in this paper the containment control problem of heterogeneous discrete-time nonlinear MASs subject to parameter uncertainties is considered, and the power series approach is adopted to solve complex regulator equations by decomposing them into a series of solvable linear equations. Then, a distributed robust control law based on internal model principle is presented by utilising the solution of the linear equations. Theoretical analysis shows that under certain assumptions asymptotic containment control is achieved for the heterogeneous discrete-time nonlinear MASs with sufficiently small parameter perturbations. Finally, a numerical simulation is implemented to verify the proposed control law.     

Sufficient and necessary conditions for the solvability of the state feedback regulation problem

In this paper, we discuss the state feedback output regulation problem (SFRP) for infinite‐dimensional linear control systems with infinite‐dimensional exosystems. Under the polynomial stabilizability assumption, sufficient and necessary conditions are given for the solvability of the SFRP. The solvability of this problem is characterized in terms of the solvability of a pair of linear regulator equations. An application of the solvability of the SFRP for polynomial stable SISO system is given. Copyright © 2013 John Wiley & Sons, Ltd.     

Discrete-time decentralized linear quadratic control for linear time-varying systems

This article addresses the problem of designing a decentralized control solution for a network of agents modeled by linear time-varying (LTV) dynamics, in a discrete-time framework. A general scheme is proposed, in which the problem is formulated as a classical linear quadratic regulator problem, for the global system, subject to a given sparsity constraint on the gain, which reflects the decentralized nature of the network. A method able to compute a sequence of well-performing stabilizing regulator gains is presented and validated resorting to simulations of two randomly generated LTV systems, one stable and the other unstable. Moreover, a tracking solution is developed, building on the solution to the regulator problem. Both methods rely on a closed-form solution, thus they can be computed very rapidly. Similarly to the centralized solution, both the presented methods require that a window of the future system dynamics is known. Both methods are validated resorting to simulations of: (i) a nonlinear network of four interconnected tanks; and (ii) a large-scale nonlinear network of interconnected tanks. When implemented to a nonlinear network, approximated by an LTV system, the proposed methods are able to compute well-performing gains that track the desired output. Finally, both algorithms are scalable, being adequate for implementation in large-scale networks.     

Measurement dynamic feedback output regulation in hybrid linear systems with state jumps

This work investigates the problem of designing a hybrid dynamic feedback regulator that forces the output of a hybrid linear system to asymptotically converge to the reference generated by a hybrid exogenous system, asymptotically rejects the exogenous disturbance, and attains global asymptotic stability of the compensated hybrid linear dynamics. The class of hybrid linear systems addressed exhibits a continuous‐time linear behavior except at isolated points of the time axis, where the state is subject to discontinuities that are caused by a jump behavior. In the presence of possibly unequally spaced state jumps, under the only constraint that the minimum time between any two consecutive jumps is no smaller than a given positive real constant, both implicit and explicit sufficient conditions for the existence of a solution to the stated problem are shown. The explicit condition is constructive, in the sense that it outlines the algorithmic procedure for the synthesis of the hybrid feedback regulator, provided that a certain output‐nulling hybrid controlled invariant subspace be known. Then, a necessary and sufficient condition for the existence of such subspace is proven, so that a computational means to derive it, if any exists, is given. Finally, the devised approach is applied to a numerical example borrowed from the literature, with the twofold aim of illustrating its implementation and making a comparison with the available method.     

Partial eigenstructure assignment problem and its application to the constrained linear problem

This article presents a new partial eigenstructure assignment method. This technique keeps the open-loop stable eigenvalues and the corresponding eigenspace unchanged. The remaining undesirable eigenvalues are replaced by other chosen values. This methodology is easy and permits to overcome some limitations encountered in the previous methods. Furthermore, our method is applied to solve the constrained control problem for linear invariants continuous-time systems. Indeed, the problem of finding a stabilising regulator matrix gain taking into account the asymmetrical control constraints is transformed to a Sylvester equation resolution. Examples are given to illustrate the obtained results.     

Stability tests and stabilization for piecewise linear systems based on poles and zeros of subsystems

This paper provides several stability tests for piecewise linear systems and proposes a method of stabilization for bimodal systems. In particular, we derive an explicit and exact stability test for planar systems, which is given in terms of coefficients of transfer functions of subsystems. Restricting attention to the bimodal and planar case, we show simple stability tests. In addition, we drive a necessary stability condition and a sufficient stability condition for higher-order and bimodal systems. They are given in terms of the eigenvalue loci and the observability of subsystems. All the stability tests provided in this paper are computationally tractable, and our results are applied to the stabilizability problem. We confirm the exactness and effectiveness of our approach by illustrative examples.     

A unified approach to output synchronization of heterogeneous multi-agent systems via L2-gain design

In this paper, a unified design procedure is given for output synchronization of heterogeneous multi-agent systems (MAS) on communication graph topologies, using relative output measurements from neighbors. Three different control protocols, namely, full-state feedback, static output-feedback, and dynamic output-feedback, are designed for output synchronization. It is seen that a unified design procedure for heterogeneous MAS can be given by formulation and solution of a suitable local $\mathcal{L}{_2}$-gain design problem. Sufficient conditions are developed in terms of stabilizing the local agents'' dynamics, satisfying a certain small-gain criterion, and solving the output regulator equations. Local design procedures are presented for each agent to guarantee that these sufficient conditions are satisfied. The proposed control protocols require only one copy of the leader''s dynamics in the compensator, regardless of the dimensions of the outputs. This results in lower-dimensional compensators for systems with high-order outputs, compared to the $p$-copy internal model approach. All three proposed control protocols are verified using numerical simulations.     

离散系统状态反馈控制中的跟踪问题

针对离散系统，本文讨论状态调节器的输出跟踪问题，指出了现有的离散状态调节器存在的问题，提出了一种新型的离散状态PI调节器．     

全状态输出调节系统的结构

本文讨论了全状态输出调节系统的结构,得到了全状态输出调节器的结构特征,给出了开 环系统存在全状态输出调节器的充分必要条件.     

The time-invariant linear-quadratic optimal control problem

This paper provides a review of one of the basic problems of systems theory—the general time-invariant optimal control problem involving linear systems and quadratic costs. The problem includes on one hand the regulator problem of optimal control and on the other, the theory of linear dissipative systems, itself central to network theory and to the stability theory of feedback systems. The theory is developed using simple properties of dynamical systems and involves a minimum of ‘hard’ analysis or algebra. It includes a full existence theory of the matrix quadratic equation, of interest in its own right.