Input-state model matching and ripple-free response for dual-rate systems

doi:10.1016/j.sysconle.2011.06.005

Systems & Control Letters

Volume 60, Issue 10, October 2011, Pages 815-824

https://doi.org/10.1016/j.sysconle.2011.06.005 Get rights and content

Abstract

In this paper, we address the model matching problem for dual-rate systems where the controller output is generated at a faster rate than the measurement update rate. The model matching problem that has been studied in the literature requires the input–output properties of the closed-loop multirate system to match those of a desired single-rate linear time-invariant (LTI) system. In this paper, we consider the model matching problem from the input-state viewpoint: given a desired LTI system, find conditions and provide a controller design procedure to achieve matching between the closed-loop system and the desired system state variables at the measurement update rate. We provide a solution to this problem using a particular time-varying controller structure. In addition, we give conditions to avoid ripples in the steady-state output of the continuous-time plant; in particular, we show that some constraints on the input matrix of the desired system have to be posed. Numerical examples are given to illustrate the proposed method.

Highlights

► We address the ripple-free input-state model matching problem for dual-rate systems. ► We find conditions and provide a controller design procedure. ► Constraints on the input matrix of the desired system are needed to avoid ripples.

Introduction

Multirate systems are characterized by the presence of digital signals and systems updating at different rates. There are numerous applications where multirate control systems are employed. In some of these applications the update rate of the feedback measurement is slower than the controller update rate. Examples include the control of the hard-disk drive (HDD) read/write (R/W) head [1], the octane rating control in the continuous catalytic reforming process [2], and the control of chemical concentrations in distillation columns [3]. For instance, consider the HDD R/W head control problem, where the plant is characterized by the dynamics of the HDD arm containing the R/W head, as shown in Fig. 1. The states of the plant are the electrical current in the voice-coil actuator (which drives the arm containing the R/W head) and the HDD arm position and velocity. In the HDD, data are stored on the platters along thousands of concentric circular tracks. Each track is divided into sectors identified by their own ID information. The angular position of the R/W magnetic head is obtained every time the R/W head passes through the part of the sector containing the sector ID information. Therefore, the angular position update rate depends on the rotational speed of the platters. On the other hand, the control input to the voice-coil motor actuator is provided by the digital board installed on the back of the HDD at a rate faster than the R/W head position feedback update rate. This determines the multirate nature of the HDD R/W head control problem. Neglecting the multirate nature of these systems by treating them as single-rate ones may lead to unexpected results. Therefore, the development of strategies to model, analyze, and control multirate systems is important, and a large amount of work has been reported in the literature to address these problems.

Multirate control systems have been shown to outperform single-rate control systems for several performance criteria, such as simultaneous strong stabilization of a set of plants, gain, and phase robustness margins [4]. Moreover, the use of a control update rate faster than the measurement update rate was shown in [5] to allow arbitrary placement of all the poles and zeros of the closed-loop system transfer function. Along this line, a technique for multirate systems was proposed in [6] to achieve input–output model matching with a desired LTI system. This was done first by converting the multirate system into a lifted LTI single-rate system (using the well-known lifting technique [7]), and then by placing all the zeros and poles of the lifted system to match the dynamics of the desired LTI system. The opportunity to generate a control action between two consecutive feedback measurements was the key for arbitrary placement of zeros and poles of the lifted LTI system.

In this paper, a state-space approach to design a controller for a class of multirate systems (dual-rate systems) is proposed. The dual-rate system comprises a continuous-time LTI plant, whose state is available at the slow rate, $1 / T_{s}$ (also referred to as measurement update rate), and a digital periodically time-varying controller operating at the faster rate, $1 / T$ . One of the control design objectives is to ensure that the closed-loop state vector matches the state vector of a desired LTI system at the slow measurement rate $1 / T_{s}$ . This control problem, which can be referred to as the input-state matching problem, clearly differs from the classical input–output model matching problem by the fact that a desired dynamics can be assigned to each state of the closed-loop system.

The well-known drawback of using linear periodically time-varying (LPTV) controllers in the multirate systems framework is the generation of ripples in the steady-state response of the continuous-time plant [6]. Under certain conditions on the sampling rates used in the multirate system, this undesired phenomenon, which is not visible from the slow-updating measurements of the plant output, can be detected if the plant output is measured at the faster control update rate. Some results concerning the problem of ripples in the steady-state response of multirate systems are given in [8], [9], [10], [11]. In particular, the presence of a continuous-time internal model of the exogenous signals is shown in [8], [9], [11] to be necessary in the case of sinusoidal-exponential-type signals to avoid ripples in the steady-state response of the system. In the case of step reference signals, to achieve ripple-free steady-state response it is sufficient either the use of a discrete-time internal model of the reference or that the time-varying gains of the controller meet some particular conditions [10]. In this paper, we consider a more general class of linear time-varying controllers, and we solve the problem of ripples in the steady-state response from the model matching perspective. In other words, we find conditions under which a multirate control system designed to achieve input-state matching with another desired system also achieves ripple-free steady-state response to step reference signals. In particular, we show that these conditions pose restrictions on the choice of the input matrix of the desired system.

To solve the input-state matching problem, we consider a particular class of periodically time-varying controllers, and provide a systematic procedure to design them. This work extends the results shown in [12] by providing insights on the nature of the controller, by providing a rigorous analysis of both the input-state matching problem and of the steady-state inter-sample behavior of the closed-loop system response, and by providing a controller design procedure. In particular, necessary and sufficient conditions are given in this work for the existence of a solution to the considered control problems. The rigorous analysis provided in this paper also serves to revise the results obtained in [12].

This paper is organized as follows. We introduce the notations and a brief definition of the lifting operation in Section 2. Section 3 contains the problem formulation and shows the structure of the considered periodically time-varying controllers. Insights on the structure of these controllers are provided in Appendix. The necessary and sufficient conditions to solve the input-state model matching problem and to achieve ripple-free steady-state response to step reference signals are given in Section 4. In Section 5, we provide a summary of the main results and a controller design procedure. Application of the proposed method on two examples is discussed in Section 6. Section 7 contains some concluding remarks.

Section snippets

Preliminaries

The control system considered in this work involves digital signals and systems updating at the two different rates, $1 / T$ and $1 / T_{s}$ , where $T_{s} = N T$ , and $N$ is a positive integer. To distinguish between the slow-rate and fast-rate signals (or system), the superscripts $T$ and $N T$ , respectively, will be utilized to refer to their update rate. No superscript will be utilized, instead, for continuous-time signals (or systems) and constant matrices. Notice that, due to the different update rates, the $k$ -th

Problem formulation

Consider the continuous-time LTI system $\dot{x} (t) = A_{c} x (t) + B_{c} u (t)$ $y (t) = C_{c} x (t),$ where $x (t) \in R^{n_{x}}$ is the state vector, $u (t) \in R^{n_{u}}$ is the control input vector, and $y (t) \in R^{n_{y}}$ is the output vector to be regulated to the constant reference signal $r^{N T} [k] \in R^{n_{y}}$ . The control system structure shown in Fig. 2 is considered. The digital controller $C$ operates at the sampling period $T$ , and the measurements of the plant state $x (t)$ are available at the slow sampling period $N T$ .

We consider a causal, digital controller $C$ of

Controller design for model matching and ripple-free steady-state response

In this section, we provide a solution to Problem 1. To do that we first write the closed-loop system in a form suitable for the analysis and design of the controller. This is done in Section 4.1 with the use of the lifting technique. The obtained state-space representation of the closed-loop system is utilized in Section 4.2 to find conditions and design the controller to achieve input-state model matching with the desired system (5). In particular, it is shown that input-state model matching

Summary of results

The following theorem summarizes the main results obtained in this work.

Theorem 3

Consider a stable desired digital LTI system of the form (5), and let $(A_{c}, B_{c}, C_{c})$ be a representation of the continuous-time plant to be controlled. Under Assumption 1, it is possible to design the time-varying controller (4) for the continuous-time plant (3) to achieve closed-loop state matching at the measurement update rate with the desired system (5) if and only if

1.
the pair $(A_{c}, B_{c})$ is controllable, and
2.
$N \geq n_{x} + 1$ .

The unique

Examples

In this section, the results given in Theorem 3 are applied to two different examples: the first one is a double integrator and the second one has no integrator-type instability. The reason for considering these two examples is to highlight some steps of the controller design procedure and distinguish between the case where a precompensator is needed and where it is not needed.

Conclusions

In this work, we have addressed the input-state matching problem for dual-rate systems. In particular, given any desired single-rate LTI system operating at the measurement update rate, we have provided conditions and a controller design procedure for which the closed-loop system state matches the state of the desired system at that measurement update rate. Moreover, we showed that, if the input matrix of the desired system is properly selected, ripple-free steady-state response of the

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^☆: This work was supported by the US National Science Foundation under Grant No. 0825937.

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Input-state model matching and ripple-free response for dual-rate systems☆

Abstract

Highlights

Introduction

Section snippets

Preliminaries

Problem formulation

Controller design for model matching and ripple-free steady-state response

Summary of results

Examples

Conclusions

Automatica

Automatica

Automatica

Multirate digital control system design and its application to computer disk drives

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Application of dual-rate modeling to CCR octane quality inferential control

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Robust control of linear time-invariant plants using periodic compensation

IEEE Transactions on Automatic Control