Linear Multivariable Discrete-time Systems Research Articles

An LMI approach to robust iterative learning control with initial state learning

This paper presents a new robust convergence condition of iterative learning control with initial state learning (ILC-ISL) for linear multivariable discrete-time systems in the presence of iteration-varying uncertainty. This method is based on linear matrix inequality (LMI) and provides fixed learning gains during time and iteration. Since the convergence of the ILC algorithm may change due to uncertainty in the parameters of a system, and the ILC algorithm is incapable of dealing with iteration-related challenges, it is a major challenge to reject the effect of iteration varying uncertainty. Besides, in the basic ILC algorithm, the initial state is constant in each iteration and consequently always leads to a tracking error. In this paper, first, a convergence condition of the ILC-ISL algorithm is designed based on closed-loop system stability in the iteration domain, and second, a new robust convergence condition is achieved by the LMI approach. Finally, the effectiveness of the proposed robust convergence scheme is evaluated through a numerical example and a mechanical system, respectively.

Restricted structure predictive control for linear and nonlinear systems

ABSTRACT An optimal predictive control algorithm is introduced for the control of linear and nonlinear discrete-time multivariable systems. The controller is specified in a ‘restricted structure’ form involving a set of given linear transfer-functions and a set of gains that minimise a Generalised Predictive Control (GPC) cost-index. The set of functions can be chosen as proportional, integral and derivative terms, however, a wide range of controller structures is possible. This is referred to as Restricted-Structure GPC control. The multi-step predictive control cost-function is novel, since it includes weightings on the ‘low-order’ controller gains and the rate of change of gains. This considerably improves the numerical computations ensuring critical inverse computations cannot lead to a singular matrix. It also provides the option of adding soft or hard constraints on the controller gains which provides additional flexibility for control design. The ability to include a plant model that can include a general nonlinear operator is also new for restricted structure control solutions. The low-order controller provides a potential improvement in robustness, since it is often less sensitive to plant uncertainties. The simple controller structure also enables relatively unskilled staff to retune the system using familiar tuning terms, and provides a potentially simpler QP problem for the constrained case.

Zonotopic guaranteed state estimation for uncertain systems

This paper presents a new approach for guaranteed state estimation based on zonotopes for linear discrete-time multivariable systems with interval multiplicative uncertainties, in the presence of bounded state perturbations and noises. At each sample time, the presented approach computes a zonotope which contains the real system state. A P-radius-based criterion is minimized in order to decrease the size of the zonotope at each sample time and to obtain an increasingly accurate state estimation. The proposed approach allows one to efficiently handle the trade-off between the complexity of the computation and the accuracy of the estimation. An illustrative example is analyzed in order to highlight the advantages of the proposed state estimation technique.

Geometric Insight and Structure Algorithms for Unknown-State, Unknown-Input Reconstruction in Linear Multivariable Systems

AbstractAn algebraic approach to the synthesis of a dynamic system that reconstructs the generic inaccessible input of a discrete-time linear multivariable system with unknown initial state is discussed. The method devised exploits geometric properties of key subspaces for the original system and algebraic properties of the Moore-Penrose inverse of Toeplitz matrices related to the algorithms for computing those subspaces. Nonminimum-phase invariant zeros are taken into account implicitly with the proposed techniques, while minimum-phase invariant zeros require that a filter be inserted between the original system and the reconstructor. The procedure applies to either strictly-proper or non-strictly-proper systems.

Tracking Performance Limitations of Random Signal in Multivariable Discrete-time Systems

摘要: 研究线性时不变、多变量、离散系统对随机信号的跟踪性能极限问题, 所考虑的随机参考输入信号为布朗运动序列. 研究结果表明此类系统的跟 踪性能极限完全由被控对象的结构特征和参考输入的统计特征决定, 其中, 结构特征指被控对象的非最小相位零点和不稳定极点的位置和方向. 作为特殊情形, 本文给出了参考输入为一致随机信号以及被控对象仅含有单个非最小相位零点和单个不稳定极点时系统跟踪性能极限问题的解. 最后, 给出了两自由度补偿器跟踪系统对随机信号的跟踪性能极限. 关键词: 多变量离散系统 / 随机信号 / 跟踪性能极限 / 非最小相位零点 / 不稳定极点

Fundamental limit of discrete-time systems in tracking multi-tone sinusoidal signals

This paper studies the tracking performance of linear time-invariant multi-variable discrete-time systems. The specific problem under consideration is to track a multi-tone sinusoidal reference signal consisting of linear combinations of a step and several sinusoidal signals, whereas the tracking performance is measured by the energy of the error response between the output of the plant and the reference signal. Our purpose is to find the fundamental limit for the best attainable performance, under any control structures and parameters, and we seek to determine this limit analytically in terms of the given plant and reference characteristics. Both the full-information and partial-information tracking schemes are formulated and investigated to address these goals, which are concerned with whether or not the reference information is fully available for tracking. Analytical expressions are developed in full generality under full-information tracking, and for a more specialized case under partial-information scheme. In addition, an optimal cheap control design is constructed to show that the performance limit can be attained asymptotically in the full-information case. The results show that in general plant nonminimum phase zeros and reference modes can interact to fundamentally constrain a system's tracking ability. They also show that absence of full reference information can degrade the tracking performance, thus demonstrating an intrinsic trade-off between the tracking objective and the availability of the reference information.

Improving transient performance in tracking control for linear multivariable discrete-time systems with input saturation

In this paper, we present a composite nonlinear feedback (CNF) control technique for linear discrete-time multivariable systems with actuator saturation. The CNF control law serves to improve the transient performance of the closed-loop system by adding an additional nonlinear feedback. The linear feedback can be designed to yield a quick response at the initial stage, then the nonlinear feedback is introduced to smooth out overshoots when the system output approaches the target reference. As such, the resulting closed-loop system typically has very fast transient response and small overshoots. The goal of this work is to complete the theory for general discrete-time systems. The technique is applied to a magnetic-tape-drive servo system design and yields a huge improvement in settling time compared to that of a purely linear controller.

Time-optimal control of disturbance-rejection tracking systems for discrete-time time-delayed systems by state feedback

In this paper, we solve the disturbance-rejection and tracking problem for linear multivariable discrete-time systems with time-delayed controlled inputs. A set of necessary and sufficient conditions under which the proposed problem is controllable is defined. Also, the nilpotency properties of such systems is established and used as the basis of a comprehensive design procedure. This general procedure is illustrated by designing a time-optimal disturbance rejection tracking system for a stirred-tank with time-delayed control inputs.

2-d analysis for iterative learning controller for discrete-time systems with variable initial conditions

In this work, an iterative learning controller applying to linear discrete-time multivariable systems with variable initial conditions is investigated based on two-dimensional (2-D) system theory. The work first introduces a 2-D tracking error system and shows the effect of tracking errors against variable initial conditions. The sufficient conditions for the convergence of the learning control rules are derived and discussed. Based on the proposed iterative learning control (ILC) rule, we have shown that the convergence of the learning rule is guaranteed with less restriction. An improved ILC rule is proposed. As a result, the convergence is robust with respect to small perturbations of the system parameters. Two numerical simulation examples are used to validate the effectiveness of the proposed methodologies.

Stability analysis of shift-invariant multidimensional systems

In a state-space model of a class of linear discrete-time shift-invariant multivariable multidimensional dynamical systems, the problem of internal stability of a system is studied from various aspects. A sequence of equivalent statements is presented to characterize the necessary and sufficient conditions for the internal stability of a multidimensional dynamics. These statements are generalized to further enhance ones for meeting the stability of a mixed multidimensional-and-multicircular dynamic, while they degenerate into the stability condition of a circulant matrix when the underlining structure entirely degenerates. As a related topic, a model degree reduction problem is studied by the balancing realization method in a class of linear shift-invariant multivariable multidimensional systems.

Convolution profiles for right inversion of multivariable non-minimum phase discrete-time systems

The problem of the non-causal inversion of linear multivariable discrete-time systems is analyzed in the geometric approach framework and is solved through the computation of convolution profiles which guarantee perfect tracking under the assumption of infinite-length preaction and postaction time intervals. It is shown how the shape of the convolution profiles is related to both the relative degree and the invariant zeros of the plant. A computational setting for the convolution profiles is derived by means of the standard geometric approach tools. Feasibility constraints are also taken into account. A possible implementation scheme, based on a finite impulse response system acting on a stabilized control loop, is provided.

Robustness of discrete-time state feedback control systems

Abstract This paper presents stability robustness results for linear multivariable discrete-time systems with uncertainties. Robust stability of the full-state feedback linear quadratic regulator in the presence of perturbations of the system parameters is investigated. These results are based on recently developed bounds on structured perturbations of a stable linear discrete-time system. The paper applies these results to a paper-machine head box system example.

Robust and perfect tracking of discrete-time systems

We study in this paper the problem of robust and perfect tracking for discrete-time linear multivariable systems. By robust and perfect tracking, we mean the ability of a controller to track a given reference signal with arbitrarily small settling time in the face of external disturbances and initial conditions. A set of necessary and sufficient conditions under which the proposed problem is solvable is obtained and, under these conditions, constructive algorithms are given that yield the required solutions. As is general to discrete-time systems, the solvability conditions to the above problem are quite restrictive. To relax these conditions, we propose an almost perfect tracking scheme, which is capable of tracking references precisely after certain initial steps.

Linear Periodic Control for ρ-Stabilization and Asymptotic Tracking Under Unbounded Output Multiplicative Perturbations

In this paper the problem of the asymptotic tracking and stabilization with infinite gain margin, is addressed for linear time-invariant discrete-time multivariable systems in the case when unknown different scalar gains act on the outputs. Necessary and sufficient conditions for the solvability of the problem by means of a lmear penodlc discrete-time error feedback dynamic controller are derived. A procedure is given for designing the proposed periodic controller (containing an internal model of the reference signals)

Asymptotic Tracking with Infinite Gain Margin Through Linear Periodic Control

In this paper the problem of the asymptotic tracking and stabilization with infinite gain margin, with the additional requirement of a prescribed rate of convergence of the free responses, is addressed for linear time-invariant discrete-time multivariable systems. Sufficient conditions for the solvability of the problem by means of a linear periodic discrete-time output feedback dynamic compensator are derived. A procedure is given for designing the proposed periodic controller, whose structure is based on the internal model principle.

A discrete-time multivariable neuro-adaptive control for nonlinear unknown dynamic systems

First, we assume that the controlled systems contain a nonlinear matrix gain before a linear discrete-time multivariable dynamic system. Then, a forward control based on a nominal system is employed to cancel the system nonlinear matrix gain and track the desired trajectory. A novel recurrent-neural-network (RNN) with a compensation of upper bound of its residue is applied to model the remained uncertainties in a compact subset /spl Omega/. The linearly parameterized connection weight for the function approximation error of the proposed network is also derived. An e-modification updating law with projection for weight matrix is employed to guarantee its boundedness and the stability of network without the requirement of persistent excitation. Then a discrete-time multivariable neuro-adaptive variable structure control is designed to improve the system performances. The semi-global (i.e., for a compact subset /spl Omega/) stability of the overall system is then verified by the Lyapunov stability theory. Finally, simulations are given to demonstrate the usefulness of the proposed controller.

Identification of complex systems with unknown feedback subsystems

A novel identification algorithm for a linear discrete-time multivariable closed-loop system is proposed based on an output over-sampling scheme. If the feedback subsystem is operating with period T. the structure and parameters of forward and feedback subsystems can be determined simultaneously by using the over-sampled output data. It can be used in closed-loop identification even though neither the knowledge of the forward and feedback subsystems nor the observation of input to the forward subsystem can be available.

Perturbation analysis of the discrete-time H∞ optimal control problem

The paper deals with the sensitivity of the standard H∞ optimization problem for discrete-time linear multivariable systems. Both local and non-local bounds for the perturbation in the solution are obtained. A numerical example demonstrates the effectiveness of the estimates proposed.

Iterative Learning Control of Linear Discrete-Time Multivariable Systems

The authors present an iterative learning control rule for linear discrete-time multivariable systems. The control rule assures a zero output error for the whole desired trajectory after only one learning iteration. The robustness of the control rule is studied and two numerical examples are used to validate the new control rule.

Perfect regulation of linear discrete-time systems: A low-gain-based design approach

This paper is concerned with the design of state feedback controllers in order to achieve an objective called perfect regulation (p.r.). Necessary and sufficient conditions for the existence of state feedback controllers that achieve p.r. are derived for general linear multivariable time-invariant discrete-time systems. More importantly, an explicit eigenstructure assignment procedure by state feedback that achieves p.r. is developed.