Invertibility and decoupling of linear time-varying systems

Before the advent of non-linear control theory, the methods employed to invert and decouple linear time-varying systems were generalizations of those used for linear time-invariant models. This paper demonstrates that the non-linear theory gives simple and concise expressions for relative order and differential orders when a linear time-varying system is transformed to a linear analytic system.     

Input-output decoupling of linear time-varying singular systems

The problem of input-output decoupling by state feedback is considered for linear time-varying singular systems. Assumptions are introduced, and an algorithm is developed such that the system can be expressed in a simple normal form. A time-varying feedback law is then constructed which ensures, under appropriate conditions, that the closed-loop system is regular, impulse-free, and noninteractive     

Disturbance decoupling of linear time-varying singular systems

The problem of disturbance decoupling by state feedback is defined for a linear time-varying singular system. It is required that the closed-loop system has a unique impulse-free solution and its output is not affected by disturbances. An algorithm, namely disturbance decoupling algorithm, is proposed. It is proved that the feasibility of the disturbance decoupling algorithm is invariant under any regular feedback control law. Based on the disturbance algorithm, a constructive method is provided to design a disturbance decoupling feedback control law. Sufficient conditions for the solvability of the disturbance decoupling problem are derived. It is proved that one of the sufficient conditions is also necessary provided that other conditions are satisfied     

On linear quadratic optimal control of linear time-varying singular systems

Linear time-varying singular systems are treated in this paper. We focus on systems with constant-rank E matrices. It is shown that the existence of state feedback for impulse elimination is both sufficient and necessary for the existence of linear-quadratic optimal control. Also optimal control exists if and only if the corresponding fast subsystem is impulse-controllable. The results obtained are extensions of the existing time-invariant theory.     

Output feedback uniform decoupling problem of linear time-varying analytic systems

A new approach to the input-output uniform decoupling problem of linear time-varying analytic systems via proportional (static) output feedback is introduced. A major feature of the proposed approach is that it reduces the solution of the uniform decoupling problem to that of solving a linear algebraic system of equations, This system of equations greatly facilitates the solution of the three major aspects of the decoupling problem: the necessary and sufficient conditions; the general analytical expressions for the controller matrices; and the structure of the uniformly decoupled closed-loop system.     

具有时变时滞的不确定时滞系统的输入-输出能量解耦

提出一种针对具有时变时滞的范数有界不确定线性时滞系统的能量解耦方法,即从输入-输出的能量关系上实现解耦,使得任何一个输入的能量主要控制对应的一个输出的能量,对其它输出能量的影响尽可能小.所有结论均由线性矩阵不等式描述,求解非常方便.     

Model-matching and decoupling for continuous- and discrete-time linear time-varying systems

Solutions to the exact model-matching and block-decoupling problems for both continuous- and discrete-time linear time-varying systems are presented. The parametrisation of the whole class of proper solutions is given. For the decoupling, the minimal delay problem is also considered in a time-varying setting. The approach is algebraic and based on the Smith–MacMillan form at infinity of a transfer matrix of a time-varying system which has been recently introduced in systems theory. This avoids the difficulties related to the inversion of the transfer matrices with entries in non-commutative fields over which the determinants (of Dieudonné or Ore type) are much more complicated. The solutions presented here involve only standard matrix computations excluding direct matrix inversions and are thus easy to implement in practice. Examples are treated in detail to illustrate the theoretical results and the way in which the computations are done and a physical example is also shown.     

On control for linear systems with interval time-varying delay

This paper deals with the problem of delay-dependent robust H_∞ control for linear time-delay systems with norm-bounded, and possibly time-varying, uncertainty. The time-delay is assumed to be a time-varying continuous function belonging to a given interval, which means that the lower and upper bounds for the time-varying delay are available, and no restriction on the derivative of the time-varying delay is needed, which allows the time-delay to be a fast time-varying function. Based on an integral inequality, which is introduced in this paper, and Lyapunov–Krasovskii functional approach, a delay-dependent bounded real lemma (BRL) is first established without using model transformation and bounding techniques on the related cross product terms. Then employing the obtained BRL, a delay-dependent condition for the existence of a state feedback controller, which ensures asymptotic stability and a prescribed H_∞ performance level of the closed-loop systems for all admissible uncertainties, is proposed in terms of a linear matrix inequality (LMI). A numerical example is also given to illustrate the effectiveness of the proposed method.     

Adaptive control of linear time-varying systems

Single-input single-output uncertain linear time-varying systems are considered, which are affected by unknown bounded additive disturbances; the uncertain time-varying parameters are required to be smooth and bounded but are neither required to be sufficiently slow nor to have known bounds. The output, which is the only measured variable, is required to track a given smooth bounded reference trajectory. The undisturbed system is assumed to be minimum-phase and to have known and constant relative degree, known sign of the ‘high frequency gain’, known upper bound on the system order. An adaptive output feedback control algorithm is designed which assures: (i) boundedness of all closed-loop signals; (ii) arbitrarily improved transient performance of the tracking error; (iii) asymptotically vanishing tracking error when parameter time derivatives are L₁ signals and disturbances are L₂ signals.     

Adaptive control of time-varying linear systems

The authors present a global bounded-input-bounded-state stability theory for a class of continuously adapting controllers applied to time-varying linear systems. This gives theoretical support to the application of many of the algorithms used in practice. A key feature of the analysis is that no persistence of excitation requirement is needed     

Sensitive decoupling control of linear disturbed-parameter systems

The input-output decoupling control problem for a generalized multivariable linear model, covering most practical situations, is treated for the case where the model involves disturbances in the parameters. A sensitive decoupling controller is derived which maintains the decoupling conditions to first order in the parameter variations, about their nominal values.     

Uniformly optimal control of linear time-varying systems

An optimal control problem is formulated in the context of linear, discrete-time, time-varying systems. The cost is the supremum, over all exogenous inputs in a weighted ball, of the sum of the weighted energies of the plant's input and output. The controller is required to be causal and to achieve internal stability. Existence of an optical controller is proved and a formula for the minimum cost is derived.     

Input-output block decoupling of linear time-varying singularsystems

The input-output block decoupling problem by state feedback is studied for linear time-varying singular systems. First, an algorithm, the regularization algorithm, is developed such that the system can be made by state feedback to have a unique impulse-free solution. Second, another algorithm, the block decoupling algorithm, is proposed, which provides sufficient conditions for the solvability of the input-output block decoupling problem. Then, a decoupling feedback law is constructed such that the corresponding closed-loop system is regular, impulse-free, and noninteractive. Finally, an example is given to illustrate the applicability of the algorithms     

Closed-form control laws for linear time-varying systems

Closed-form control laws are developed for continuous, linear, time-varying (LTV) systems based on approximate solutions to a receding-horizon control problem. These control laws can be derived in the first- or higher order closed forms. Once obtained, the control laws need no explicit gain-scheduling or online integrations to implement. The notion of practical stability is used, and practical or uniform asymptotic stability of the closed-loop system, depending on conditions imposed on the system, is established     

Optimal overtaking control for linear time-varying systems

The problem of tracking given bounded signals is considered for time-varying linear systems. The cost functional is quadratic and optimal overtaking controls are sought. Both deterministic and stochastic cases are considered. Under stabilizability and detectability unique optimal controls are obtained.     

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.     

Uniform disturbance localization with simultaneous uniform decoupling for linear time-varying analytic systems

The problem of uniform disturbance localization with simultaneous uniform decoupling for linear time-varying analytic systems via static state feedback is investigated, for the first time. The approach presented has the main feature that it reduces the problem to that of solving a homogeneous algebraic system of equations. On the basis of this basic system of equations, the following aspects of the problem are resolved: the necessary and sufficient solvability conditions, the general analytical expressions of all admissible controllers and the morphological characteristics of the closed-loop system     

线性多变量系统有限时间最优解耦控制

针对线性多变量系统, 将前馈解耦控制与有限时间最优跟踪控制相结合, 提出一种新的最优解耦控制方法. 首先, 将关于状态的微分方程转化成关于输出的微分方程, 将系统内部矩阵和控制输入矩阵分别分解成对角矩阵和对角元为零的矩阵; 然后, 通过引入中间虚拟变量, 采用前馈和输出反馈的方法对系统进行解耦; 最后, 采用有限时间最优跟踪控制方法实现系统对任意参考输入的跟踪. 仿真结果表明了所提出方法的有效性和优越性.     

线性定常系统的Petri网解耦控制

将Petri网与现代控制理论相结合,应用于连续系统的性能分析如可控性、可观性和稳定性等已日益普遍,但Petri网应用于系统的解耦控制研究很少.提出了广义连续自控网系统的形式化定义,描述了线性定常系统的广义连续自控网系统模型并分析了广义连续自控网系统模型与状态空间描述的等效性.基于状态反馈动态解耦的基本原理,探讨了利用Petri网模型结构实现线性定常系统解耦控制的新方法.该方法采用图的遍历算法,可有效的判断系统的可解耦性以及实现解耦控制律,避免了传统解耦控制方法中计算所需的大量矩阵运算.最后给出了两个具体的应用实例.