Frequency-domain parameter estimation of general multi-rate systems

Abstract

This paper studies the parameter estimation of a general multi-input, multi-output multi-rate system in the frequency-domain. Two methods, named dividing to subsystems and input extension, are introduced for dealing with multi-rate systems and the later method is easily used to convert a multi-input, multi-output multi-rate system to several sub-problems with fast input updating and slow output sampling. Then all the frequency-domain parameter estimation methods can be applied. Here a least-square parameter estimation method is generalized for parameter estimation in the multi-input, multi-output case. Several examples, including one with real industrial data, are given to show the effectiveness of the methods proposed.

Introduction

In a conventional sampled-data control system, the plant input updating and output sampling are at the same rate. However, it is not always possible to update the control input and sample the output at the same rate due to various limitations such as the cost of fast-rate sensors and actuators. Also sometimes the plant dynamics are such that it is not economical and also not useful to sample the different plant signals at the same rate. As a result, a multi-rate sampling scheme should be considered for such cases. Of course this multi-rate sampling scheme introduces the complication of mixed time steps.

Fig. 1 shows a general multi-input, multi-output (MIMO), multi-rate system, where every input has its own updating rate and every output is sampled at its own rate. Continuous arrows are used for continuous signals and dotted arrows for discrete signals. Here, $P_{\text{c}}$ is a continuous-time plant, $H$ is a multi-rate zero-order hold, and $S$ is a multi-rate output sampling device which can be defined as follows:

$$H=\left(\begin{array}{ccc}{H}_{p_{1}h}\hfill & \hfill & \hfill \\ \hfill & \ddots \hfill & \hfill \\ \hfill & \hfill & H_{p_{m}h}\hfill \end{array}\right)\text{,}S=\left(\begin{array}{ccc}{S}_{q_{1}h}\hfill & \hfill & \hfill \\ \hfill & \ddots \hfill & \hfill \\ \hfill & \hfill & S_{q_{n}h}\hfill \end{array}\right)\text{.}$$

These correspond to holding m input channels of u with periods $p_{i}h\text{,}i=1\text{,}\dots \text{,}m$, and sampling n channels of output $y_{\text{c}}$ with periods $q_{j}h\text{,}j=1\text{,}\dots \text{,}n$, respectively. Here, $p_i$ and $q_j$ are different integers and $h$ is a real number called the base period. If we partition the signals accordingly

$$y_{\text{c}}=\left(\begin{array}{c}\hfill y_{{\text{c}}_1}\hfill \\ \hfill ⋮\hfill \\ \hfill y_{{\text{c}}_n}\hfill \end{array}\right)\text{,}y=\left(\begin{array}{c}\hfill y_1\hfill \\ \hfill ⋮\hfill \\ \hfill y_n\hfill \end{array}\right)\text{,}$$

$$u_{\text{c}}=\left(\begin{array}{c}\hfill u_{{\text{c}}_1}\hfill \\ \hfill ⋮\hfill \\ \hfill u_{{\text{c}}_m}\hfill \end{array}\right)\text{,}u=\left(\begin{array}{c}\hfill u_1\hfill \\ \hfill ⋮\hfill \\ \hfill u_m\hfill \end{array}\right)\text{,}$$

then

$$u_{{\text{c}}_i}(t)=u_i(k)\text{,}k{p}_{i}h\le t<(k+1)p_{i}h\text{,}i=1\text{,}\dots \text{,}m\text{,}{y}_j(k)=y_{c_j}(k{q}_{j}h)\text{,}j=1\text{,}\dots \text{,}n\text{.}$$

We use u to denote the fictitious case that all inputs are at the fast-rate, that is $p_i=1\text{,}i=1\text{,}\dots \text{,}m$. Similarly, y denotes the output when all outputs are sampled at the fast-rate, or $q_j=1\text{,}j=1\text{,}\dots \text{,}n$.

Such systems especially arise in the chemical process industry. For example, in polymer reactors (Oshima, Hashimoto, Takeda, Yoneyama, & Goto, 1992), the composition and density measurements are typically obtained after several minutes of analysis, whereas the control inputs can be applied at relatively fast-rate. For another example, we can consider an industrial bleaching process (Han, Shah, Pakpahan, Patwardhan, & Robson, 2004) that is a chemical process applied to cellulose material to increase their brightness and usefulness; in this process, some output variables, like brightness, need laboratory analysis and are in slow rate and are irregularly sampled, while inputs can be applied at relatively fast-rate. One of the problems for such a system is to find the estimation of system parameters and also the output at those time instants when the measurements are not available.

One motivation for this work is output monitoring at fast-rate. Another interesting application is the use of output estimates to run an inferential control scheme, as most inferential control algorithms need the parameters of fast single-rate models, which are not usually available. Some work has been done in this area. The existing multi-rate identification methods can be divided in two main categories: state-space identification and frequency-domain identification. Li, Shah, and Chen (2001) studied the identification of a multi-rate sampled-data system consisting of a continuous-time process with or without time delay, a sampler with period $nT$ and a zero-order hold with period $mT(m<n)$ and the problem of identifying a fast-rate model with sampling period $mT$; the method used is state-space based, employing the lifting technique. Their work is continued by Wang, Chen, and Huang (2004) where a fast-rate model with sampling period $T$ is extracted. Lu and Fisher, 1988, Lu and Fisher, 1989, Lu and Fisher, 1996 and Lu, Fisher, and Shah (1990) studied the parameter and output estimation of dual-rate systems in the frequency-domain; they proposed least-square and projection based algorithms for dual-rate noise-free systems. Ding and Chen (2003) studied the problem for the dual-rate case for stochastic systems.

In this paper, we introduce two simple methods for dealing with general multi-rate systems. These methods are named dividing to subsystems and input extension and are useful in the frequency-domain. In the first method, a multi-rate system can be divided into some dual-rate subsystems and existing estimation methods can be used for parameter estimation of each subsystem; then, parameters of the original system can be extracted. In the second method, a multi-rate system can easily be converted to a dual-rate system with all input updating at fast-rate. A least-square parameter estimation algorithm is derived for such systems.

When system parameters are estimated, we can use it for different applications like inter-sample output estimation as shown in Fig. 2. Here, the process input(s) and sampled output(s) are fed to a parameter estimation engine that produces estimates of the parameters of the assumed model at the fast-rate. Based on estimated parameters and known inputs, output can be estimated at the fast-rate. In Fig. 2, $\hat{\theta }$ and $ŷ$ are used to show the estimates of parameters and y.

This paper is organized as follows. In Section 2, we consider the problem of transforming a general multi-rate MIMO system to some dual-rate MISO (multi-input, single-output) systems, where each subsystem has all inputs at the fast-rate, and study the parameter estimation of these sub-systems in the frequency-domain. In Section 3, we study the method of extracting a fast-rate model from estimated parameters. Some examples for both SISO and MIMO cases, including a real MIMO example with industrial data, are presented in Section 4. Finally, concluding remarks are given in Section 5.