A unified method for the design of nonovershooting linear multivariable state-feedback tracking controllers

doi:10.1016/j.automatica.2009.11.018

Automatica

Volume 46, Issue 2, February 2010, Pages 312-321

https://doi.org/10.1016/j.automatica.2009.11.018 Get rights and content

Abstract

We consider the use of linear multivariable feedback control to achieve a nonovershooting step response. A method is given for designing a linear time invariant state feedback controller to asymptotically track a constant step reference with zero overshoot and arbitrarily small rise time, under some mild assumptions. We present a unified design method that can be applied to continuous and discrete time systems, square and non-square systems, minimum and nonminimum phase systems, and also strictly proper and nonstrictly proper systems.

Introduction

The problem of ensuring that a linear time invariant (LTI) plant has a nonovershooting step response has been studied for the past few decades. The problem is of great importance in some applications such as manufacturing processes, where overshoot can compromise tolerances and damage the product.

Papers giving analytic results include the following. In Lin and Fang (1997) third-order continuous time single-input single-output (SISO) systems were considered, and necessary and sufficient conditions were given in terms of the closed-loop poles for which the step response is nonovershooting. In Stewart and Davison (2006) it was shown that, for a continuous time SISO system with two nonminimum phase real zeros (right-hand complex plane), the step response must overshoot if the settling time is sufficiently small. Some recent papers have considered the problem of designing a suitable closed-loop feedback controller to achieve a nonovershooting response. For continuous time systems, in Darbha and Bhattacharyya (2003) it was shown how to design a two-parameter feedback controller for an LTI plant that renders the step response nonovershooting. In Kim, Keel, and Bhattacharyya (2003) a method of characteristic ratio assignment was used to achieve arbitrarily small overshoot, for minimum phase plants. In Bement and Jayasuriya (2004) an eigenstructure assignment method was proposed to obtain a nonovershooting LTI state feedback controller for plants with one nonminimum phase zero. In Darbha (2003) conditions are given for the existence of a controller to achieve a sign invariant impulse response, and hence also a nonovershooting step response. Corresponding conditions for discrete systems are given in Darbha and Bhattacharyya (2002). Recently, in Krstic and Bement (2006) strict feedback nonlinear systems were considered, and a backstepping approach was exploited to convert the system to a nonovershooting linear system. However, the problem of determining the existence of a nonovershooting controller for a general LTI plant is still open, even for single-input single-output systems.

A common feature of these papers is that they considered strictly proper SISO systems, and that the system state is assumed to be initially at rest. Moreover, with the exception of Darbha and Bhattacharyya (2003), these papers assumed the system to be of minimum phase, or else to contain at most one nonminimum phase zero. In this paper, we consider multiple-input multiple-output (MIMO) systems, and use linear state feedback control to design a nonovershooting controller for a step reference. The design methods proposed here make use of the combined eigenvalue and eigenvector placement methods given in Moore (1976), and are applicable to both continuous time and discrete time systems. The design method is applicable to both square and non-square systems, minimum phase and nonminimum phase systems, strictly proper and nonstrictly proper systems, and do not assume that the initial state of the system is at rest. Conditions are given under which a linear state feedback controller can be obtained to asymptotically track a step reference with guaranteed no overshoot, from any initial condition. The controller can be readily chosen to achieve any desired convergence rate (settling time).

This paper is organized as follows. First, in Section 2, the nonovershooting control problem considered in this paper is formulated precisely. The details of our design method for square systems are outlined in Section 3. In the first part of this section, Moore’s algorithm is recalled and its use in connection with the nonovershooting control problem is shown first in the simple case where the system has at least $n - p$ stable invariant zeros (where $n$ and $p$ are, respectively, the numbers of states and outputs). Moore’s algorithm allows us to render the $n - p$ modes corresponding to the stable invariant zeros invisible on the tracking error, and to distribute the remaining $p$ modes (exponentials in the continuous time case and powers in the discrete time case) evenly with one mode per component of the tracking error. Zero overshoot is achieved for all initial conditions since in this way each component of the tracking error does not change sign.

The assumption on the number of stable invariant zeros is then progressively relaxed. Systems with $n - 2 p$ stable invariant zeros are next considered. We characterize the region of the state space containing all the initial conditions for which a nonovershooting step response is achieved. Lastly, the general case of systems with $n - l p$ stable invariant zeros is considered, for arbitrary integer $l$ . For the case $l = 3$ , we give necessary and sufficient conditions under which a given initial condition yields a nonovershooting step response. For $l \geq 4$ , obtaining such necessary conditions becomes very complex, and we are only able to offer sufficient conditions for a given initial condition to yield a nonovershooting response.

In Section 4 we show how the design method may be adapted to non-square systems. We first consider systems with fewer control outputs than control inputs, and consider how this additional control freedom may be employed to design a linear state feedback controller that is nonovershooting from all initial conditions. Importantly, in contrast with square systems, here it is not necessary to make any assumptions about the number of stable zeros. Lastly we consider systems with more outputs than control inputs. In this case, reference tracking may not be achievable for all target references. For those targets that are trackable, we show how to achieve a step response that is nonovershooting in $m - 1$ output components, where $m$ is the number of control inputs.

The design method is substantially the same whether the underlying system evolves in continuous or discrete time. Accordingly, we discuss the design method for the two cases simultaneously, noting the minor differences when they appear. The time index set of any signal is denoted by $T$ , on the understanding that this represents either $R^{+}$ in the continuous time case or $N$ in the discrete time case. The symbol $C_{g}$ denotes either the open left-half complex plane $C^{-}$ for continuous time systems, or the open unit disc $C^{\circ}$ for discrete time systems. A complex number is said to be stable if it belongs to $C_{g}$ , and a square matrix is said to be stable if all its eigenvalues belong to $C_{g}$ .

Section snippets

Problem formulation

Consider the LTI system $Σ$ governed by $Σ : {\begin{matrix} ρ x (t) = A x (t) + B u (t), & x (0) = x_{0} \in R^{n}, \\ y (t) = C x (t) + D u (t), \end{matrix}$ where, for all $t \in T$ , $x (t) \in R^{n}$ is the state, $u (t) \in R^{m}$ is the control input, $y (t) \in R^{p}$ is the output, and $A$ , $B$ , $C$ and $D$ are appropriate dimensional constant matrices. The operator $ρ$ denotes either the time derivative in the continuous time case, i.e., $ρ x (t) = \dot{x} (t)$ , or the unit time shift in the discrete time case, i.e., $ρ x (t) = x (t + 1)$ . We assume that $B$ has full column rank and $C$ has full row rank.

In this paper we are

Design of nonovershooting feedback controllers for square systems

Here we consider systems subject to the following assumption:

Assumption 3.1

System $Σ$ is square ( $p = m$ ).

Assumption 2.1, Assumption 3.1 imply that

Σ

is invertible.

Design of nonovershooting feedback controllers for non-square systems

Firstly, we consider systems with more control inputs than outputs.

Assumption 4.1

System $Σ$ is such that $p < m$ .

The additional control inputs can be exploited to achieve a globally nonovershooting step response with any desired convergence rate (settling time). We begin by augmenting the system

Σ

by adding one additional row vector

C_{p + 1}

and

D_{p + 1}

to matrices

C

and

D

as follows:

\bar{C} = [\begin{matrix} C \\ C_{p + 1} \end{matrix}], \bar{D} = [\begin{matrix} D \\ D_{p + 1} \end{matrix}] .

This yields the augmented system

Σ_{aug} : {\begin{matrix} ρ x (t) = A x (t) + B u (t), & x_{0} \in R^{n}, \\ \bar{y} (t) = \bar{C} x (t) + \bar{D} u (t) . \end{matrix}

The row vectors

C_{p + 1}

and

D_{p + 1}

Examples

Example 5.1

Let $Σ_{1}$ be the continuous time system $A = [\begin{matrix} - 9 & - 9 & 5 & 0 & - 3 \\ - 8 & 0 & 0 & - 7 & 0 \\ - 10 & - 9 & - 5 & 0 & - 8 \\ - 10 & 0 & 8 & - 5 & 0 \\ 1 & 0 & 0 & 0 & - 7 \end{matrix}], B = [\begin{matrix} 0 & 0 \\ 6 & 0 \\ 9 & 0 \\ 2 & - 10 \\ 0 & 0 \end{matrix}],$ $C = [\begin{matrix} 10 & 0 & 0 & 0 & - 1 \\ 0 & 0 & 0 & - 1 & - 4 \end{matrix}], D = [\begin{matrix} 1 & 2 \\ 0 & - 1 \end{matrix}] .$ Assume that the tracking target is $r = {[8 - 8]}^{⊤}$ . We see that the system has 5 invariant zeros at $14.42 \pm 3.08 i$ , $- 17.18 \pm 11.98 i$ and −6.48. The system is square, invertible and of nonminimum phase with $n - p = 3$ stable invariant zeros. Thus Assumption 2.1, Assumption 3.1, Assumption 3.2 hold. Following the design procedure outlined above, we choose closed-loop poles $λ_{1} = - 17.18 + 11.98 i$

Conclusion

A unified design method for a linear state feedback tracking controller to achieve a nonovershooting step response for MIMO systems has been introduced. The method may be applied to both continuous and discrete time systems, strictly and nonstrictly proper systems, square and non-square systems, as well as minimum and nonminimum phase systems. In each case the transient response is shaped by a suitable application of Moore’s algorithm. To the best of the authors’ knowledge, this is the first

Acknowledgements

The authors would like to thank Ben Chen from the National University of Singapore for bringing the paper by Moore (1976) to their attention. The authors would also like to thank the anonymous reviewers for several helpful comments, and for bringing the classical paper by Laguerre (1883) to their attention.

Robert Schmid received his Honours B.Sc. degree from Latrobe University in 1986 and his M.Sc. degree from the University of Melbourne in 1992, both in Mathematics. In 2003 he received his Ph.D. in Electrical Engineering from the University of Melbourne. From 2000 he has been a Faculty Member of the Department of Electrical and Electronic Engineering at the University of Melbourne. Since 2007 he has been an Associate Editor for Systems & Control Letters. His research interests include periodic

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Lorenzo Ntogramatzidis received his “Laurea” degree, cum laude, in Computer Engineering in 2001 from the University of Bologna, Italy. He subsequently received his Ph.D. degree in Control and Operations Research in 2005.

From 2005 to 2008, he was a post-doctoral Research Fellow in the Department of Electrical and Electronic Engineering, University of Melbourne, Australia. Since 2009, he has been with the Department of Mathematics and Statistics at Curtin University of Technology, Perth, Australia, where he is currently an ARC APD Research Fellow.

His research interests lie in the broad area of systems and control theory.

^☆: This work was supported in part by The University of Melbourne (MRGS) and the Australian Research Council. The material in this paper was partially presented at 17th IFAC World Congress, July 2008, Seoul. This paper was recommended for publication in revised form by Associate Editor Didier Henrion under the direction of Editor Roberto Tempo.

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A unified method for the design of nonovershooting linear multivariable state-feedback tracking controllers☆

Abstract

Introduction

Section snippets

Problem formulation

Design of nonovershooting feedback controllers for square systems

Design of nonovershooting feedback controllers for non-square systems

Examples

Conclusion

Acknowledgements

Automatica

Systems and Control Letters

Systems and Control Letters

Use of state feedback to achieve a nonovershooting step response for a class of nonminimum phase systems

Journal of Dynamical Systems, Measurement and Control

On the synthesis of controllers for a nonovershooting step response

IEEE Transactions on Automatic Control

Controller synthesis for sign invariant impulse response

IEEE Transactions on Automatic Control

Robust pole assignment in linear state feedback

International Journal of Control

Transient response control via characteristic ratio assignment

IEEE Transactions on Automatic Control