Hierarchical Identification of Lifted State-Space Models for General ...

IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 52, NO. 6, JUNE 2005

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Hierarchical Identification of Lifted State-Space Models for General Dual-Rate Systems Feng Ding and Tongwen Chen

Abstract—This paper is motivated by practical consideration that the input updating and output sampling rates are often limited due to sensor and actuator speed constraints. In particular, for general dual-rate systems with different updating and sampling periods, we derive the lifted state-space models (mapping relations between available dual-rate input-output data), and, by using a hierarchical identification principle, present combined parameter and state estimation algorithms for identifying the canonical lifted models based on the given dual-rate input-output data, taking into account the causality constraints of the lifted systems. Finally, we give an illustrative example to indicate that the proposed algorithm is effective. Index Terms—Hierarchical identification principle, Kalman filtering, least squares, multirate systems, parameter estimation, state-space model, stochastic approximation, system identification.

I. INTRODUCTION

M

ANY conventional algorithms are used to estimate system parameters, see, e.g., [1]–[4], but most of them assume that input-output data are available at every sampling instant. For multirate sampled-data systems, these algorithms cannot be applied directly. For decades, the study of multirate optimal control systems has been very active, including [5]–[8], adaptive control [9]–[11], and so on; application examples include distillation columns [9], polymer reactors [12], fermentation processes [13], and bioreactors [14], [15]. In the process control area, Sheng et al. proposed the generalized predictive control scheme for multirate systems [16]; Li et al. applied dual-rate modeling to Octane quality inferential control and did the performance analysis [17], [18]. In the process identification literature, Lu and Fisher used projection and least-squares algorithms for estimating intersample outputs [19], [20], but their algorithms handle only noise-free dual-rate systems; Li et al. assumed that the system states were available and used them and multirate input-output data to estimate the parameters of lifted state-space models for multirate systems [21]. (However, the system states are usually unavailable in Manuscript received April 15, 2004; revised October 16, 2004. This work was supported by the Natural Sciences and Engineering Research Council of Canada and the National Natural Science Foundation of China. This paper was recommended by Associate Editor C. Hadjicostis. F. Ding is with the Control Science and Engineering Research Center, Southern Yangtze University, Wuxi 214122, China, and also with the University of Alberta, Edmonton, AB T6G 2V4, Canada (e-mail: [email protected]; [email protected]). T. Chen is with the Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AB T6G 2V4, Canada (e-mail: [email protected]). Digital Object Identifier 10.1109/TCSI.2005.849144

practice.) This lifting conversion from a time-varying system into a time-invariant one is similar to the construction presented in [22]. Recently, subspace state-space identification (4SID for short) methods have been developed for state-space models [23]–[31], the basic idea is to determine the extended observability matrix from the singular value decomposition or RQ factorization of an information matrix consisting of given input-output data, and then to compute the system parameter matrices. But, as the size of the information matrix grows, the difficulty and complexity in computation increase. Though these 4SID methods are popular and effective, the main disadvantage is that they cannot deal with the causality constraints generated by the lifted multirate systems [5]–[7], [16], [21]. For related work, see recent studies on identification of dual-rate systems by using the polynomial transformation technique [32], [33] and by using an auxiliary model identification method [34], [35], and also a survey on various identification schemes for multirate systems [36]. This paper focuses on the modeling and identification problem for general dual-rate systems. The key idea is the so-called hierarchical identification, and is inspired by the hierarchical control based on the decomposition-coordination principle for large-scale systems [37]–[39]. Hierarchical identification uses subsystem decomposition in identification, and is also called bootstrap identification. We tackle the causality problem by using the hierarchical identification principle [40]–[43], and directly identify the parameters of the canonical lifted models of dual-rate systems. The approach here differs from the one in [44] which identified the parameters of the noncanonical lifted models of dual-rate systems. Earlier work on conventional multiple-input–multiple-output (MIMO) system identification exists: some methods estimated the parameters of the difference equation models obtained from state-space models, e.g., [45]–[49]; others estimated the parameters of state-space models by using the two-stage bootstrap identification technique [3], [4]. They all assumed that the system observability indexes were known. In this paper, we extend this two-stage approach to identify the parameters of dual-rate sampled-data systems with causality constraints. The paper is organized as follows. Section II describes the modeling issues related to dual-rate systems. Section III derives the lifted state-space models of dual-rate systems. Section IV discusses the identification problem of the lifted models. Section V presents an illustrative example for the proposed algorithms in this paper. Finally, concluding remarks are given in Section VI.

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with the sampling period to get , and assume that state-space model of the form (see, e.g., [50])

has a

(3) Fig. 1. General dual-rate systems (T =

ph; T

=

qh)

.

where the two matrices

and

are as follows:

II. PROBLEM FORMULATION The focus of this paper is a class of multirate systems—the general dual-rate systems with additive disturbance [21]—as is a continuous-time process, the input depicted in Fig. 1, to is produced by a zero-order hold with period , processing a discrete-time signal ; the output of , which is corrupted by the noise , is sampled by with period , yielding a discrete-time signal a sampler . If , then we obtain a conventional discrete-time system, i.e., single-rate system. Without loss of generality, we and and are two coprime assume that integers, for otherwise, we can absorb any common factor of and into is a positive real number called the base period. s, s, then , For example, if and s. For such a dual-rate sampled-data system, the input–output . Since data available are the zero-order hold is used, we have for

For tractability, the lifting technique is used to transform a periodic time-varying system into a time-invariant one with a larger (i.e., the least common multiple of and ), period known as the frame period. However, if we group every input values and every output samples together, respectively, then we obtain a LTI system operating over period prois ; this is the idea of lifting [17], [18], vided that and [21]. to in Fig. 1, we For the dual-rate system from intend to lift the input and output to get a lifted system, which is ; and a single-rate system with underlying frame period and by , where represents hence, we need to lift by the -fold lifting operator [21]. The lifted process is denoted by from to , i.e., . For convenience, we introduce the following notation:

(1)

is a linear time-inThroughout the paper, we assume variant (LTI) continuous-time process with the following statespace representation: (2) is the state vector, the control input where the output vector, a stochastic vector, , and are matrices noise vector with zero mean, and of appropriate dimensions. is LTI, the system from to Although in Fig. 1 is linear periodically time-varying due to different updating and sampling periods [5]–[7], [21]. For such a timevarying dual-rate system, our objective is two-fold. • First, by using the lifting technique, to establish time-invariant models of the system from to ; That is, to find the mapping relationship (lifted state-space models) between the available input and output data. • Second, by means of the hierarchical identification principle, to develop recursive algorithms for combined parameter and state estimation based on the given dual-rate , measurement data taking causality constraints into consideration. III. LIFTED STATE-SPACE MODELS In order to derive the state-space model for the dual-rate system, we discretize via the zero-order hold method

Define the lifted input vector and lifted noise vector as

, lifted output vector

.. .

.. .

.. . Theorem 1: For the dual-rate system in Fig. 1, the coprime, ness of integers and implies that for every and such that there exist integers

DING AND CHEN: HIERARCHICAL IDENTIFICATION OF LIFTED STATE-SPACE MODELS

Then, the lifted state-space model is given by

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Thus (4)

where

.. .

or

the equation at the bottom of the page, and the following equations are true:

(5) Similarly, we have

is

is of dimensions . Proof: Replacing , we have

is in (3) with

is and noting that

Hence, it is not difficult to get

or

Using (1), we have

for

.. .

(6)

..

.

..

.

.. . .. . .. .

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Combining (5) and (6) leads directly to the results of Theorem 1.

.. .

IV. IDENTIFICATION ALGORITHMS For convenience, let

, and , then (4) can be rewritten as .. .

(7) Noting that is a (block) lower triangular matrix, i.e., the are zero; this represents the upper triangular blocks in so-call causality constraint. Since the models in (7) are derived from the continuous-time models, the causality constraint is automatically satisfied. However, we need to deal with the causality constraint for the lifted system identification. Identifiability of a system depends on its controllability and observability. Therefore, it is important if the lifted state-space models in (7) are controllable and observable. What conditions are required for the lifted models in (7) to be controllable and observable? To answer this question, we assume that the stateis conspace model in (3) is a minimal realization, i.e., trollable and is observable. Note that this is guaranteed is controllable as long as the continuous-time process and is observable and if the sampling period is nonpathological [50]. In the same way, for the nonpathological samis controllable and observpling period able. Controllability and observability of (7) can be achieved under a mild condition. Lemma 1: points 1) If for every eigenvalue of , none of the is an eigenimplies that value of , then controllability of of each , which in turn implies controllability of the model in (7). of , none of the 2) If for every eigenvalue points is an eigenvalue of , controllability of implies that of each , which in turn implies controllability of the model in (7); observability of implies that of each , which in turn implies observability of the model in (7). The proof is omitted here, but can be obtained in a similar way as that given in [51]. The following is to apply the hierarchical identification principle to study the identification problem of the lifted model. Let

.. . Any controllable and observable state-space model is equivalent to an observability canonical form that has the least number of parameters. Assume that (3) is observable and the conditions of Lemma 1 hold. Then, there exists a nonsingular matrix such that the transform makes the model in (7) become the following observability canonical form: (8) where

.. .

.. .

.. .

.. .

..

.. .

.. .

.. .

.. . .. .

.. .

.. .

.. . .


The transform matrix is given by ( the matrix )

represents the th row of

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Then, (9) is decomposed into

susbsystems:

or

.. . .. .

.. .

.. .

..

.. . Here, the observability structure indexes are assumed to . (If not, they can be known and satisfy be determined by using a similar approach as in [46].) Our goal is to estimate the unknown parameters in the system and unknown state parameter matrices of the canonical model in (8) by utilizing the lifted vector . input-output data From the structure of , and , a question arises: whether the number of parameters in the lifted system is less or or not if the original continuous-time system the discrete-time system is assumed to be canonical? The answer is negative. The assumption of or being canonical would not help, because there is no guarantee that the lifted system would be anonical; moreis canonical does not imply is canonical, and over, that is an visa versa. For example, if one assumes that observability canonical form, i.e., is a companion matrix and is a matrix consisting of unit row vectors, the lifted is not a companion matrix; hence, the transformation is also required. In order to estimate the unknown parameters in using the idea of decomposition identification [3], [4], [45], [47]–[49] as in the bootstrap algorithm, we divide (8) into the following two systems:

.. . .

.. .

.. .

.. .

(11)

.. .

.. .

(12) th row of Define the parameter matrix

and information vector

as

From (11), we have

(13)

(9) (10) According to the hierarchical identification principle, we first estimate the parameter matrices and state vector of the system in (9) from the available multirate inputoutput data, and then the parameter matrices of the system in (10) from the obtained state estimate and lifted input-output data. Let

.. .

.. .

Let be a forward shift operator: the th equation of (13) by summation is

, multiplying , and their

(14) From (12) and (14), we have (15) Here, we can see that this identification expression contains both the unknown state vector in and the unknown parameter vector . In order to identify them, according to the hierarchical identification principle, when recursive estimating the parameter vector , the unknown state vector in is replaced by its corresponding estimate , and by ; in the same way, when estimating the state vector , the unknown parameter vectors are also replaced by their estimates at time . This lead to the following hierarchical

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identification algorithms consisting of the parameter estimation and state estimation one. Parameter Estimation Algorithm: (30) (16) (17)

(18) are the estimate of where given later. of State Estimation Algorithm:

at time

is the estimate

where the estimates of are formed by using the entries of the obtained or . Here, the parameter estimator is derived by using the state estimates based on the least squares principle, and the state estimator is derived by using the obtained parameter estimates based on the Kalman filtering principle. But the state estimation algorithm in (19)–(21) based on the Kalman filtering involves heavy computational efforts; for computational efficiency, a stochastic approximation algorithm may be used for state estimation

(19) (20)

or

(21) .. .

.. .

.. .

.. .

..

(22)

.. .

(31) where

For example, we may take (23)

.

is the convergence factor satisfying

or

(32)

The following is to discuss the identification problem of the of the system in (10). Let parameter matrices .. .

.. .

(24)

So the system in (10) may be equivalently written as

.. .

(25)

From this, all the entries in the matrix the simple least squares algorithm

.. .

(26)

.. .

(27)

can be estimated by

or (28)

and (29)

for large However, since the upper triangular blocks in the zero entries of this upper triangular block in

are zero, i.e., need not be


identified, in order to tackle the so-called causality constraint for the lifted model, we decompose the system in (10) into subsystems

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V. EXAMPLE For a system depicted in Fig. 1, take the process model to be

and s, , hence s, s, and s. The identification procedure is summarized as follows: First, we generate a random signal sequence with zero mean and unit ; Second, we lift by and variance as the input signal to form the lifted signals lift by

where

and then the corresponding lifted state-space model is given by

.. . th block row of

Then, for large , the estimate of

is obtained from

is taken as a white noise sequence with zero mean and variance . Next, take , based on the lifted signals, we apply the proposed algorithm in (16)–(18) and (31)–(34) to estimate the parameters of the lifted model of the canonical form

.. .

The estimate

may be also recursively computed by

(33)

(34) The combined parameter and state estimation algorithms in (16)–(30) and (33)–(34) form the hierarchical identification algorithms of the lifted state-space model for dual-rate systems based on the least squares and Kalman filtering principles; and (16)–(18) and (31)–(34) form the hierarchical identification algorithms based on the least squares and stochastic approximation principles. To obtain consistent parameter estimation, the input signals are required to be persistently exciting from the formulation of the information vectors above. To initialize the above algorithms, we take , and with normally a large positive number (e.g., ), and and , some small real vector, e.g., , and so on, with being an -dimensional vector whose elements are all 1.

The true parameters and their estimates with different data lengths are shown in Table I, and the parameter versus is shown in Fig. 2, where estimation error represents the true parameter vector, the estimate of . The estimated lifted model is given by

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TABLE I PARAMETERS AND THEIR ESTIMATES ( = 0:10 )

Fig. 3 compares step responses of the actual system and the estimated lifed model. From Table I and Figs. 2 and 3, we can see that the parameter estimation error is becoming smaller (in general) as increases, and the step response of the estimated . This inmodel is very close to that of the actual system dicates that the estimated lifted model can capture the process dynamics very well, and can achieve satisfactory results. VI. CONCLUSION

Fig. 2. Parameter estimation error versus k ( = 0:10 ).

This paper presents hierarchical state-space model identification algorithms for general dual-rate stochastic systems. Although the algorithms are developed for output error models, namely, output measurement containing additive white noise disturbances, the method adopted can be easily extended to other cases, for example, dual-rate stochastic systems with colored noises and general multirate multivariable systems with each of input and output channels having different sampling periods, and with colored noises. Under what conditions the parameter estimation by the proposed algorithms is convergent still requires further investigation. REFERENCES

Fig. 3. Step responses of the actual system and the estimated model.

The estimated lifted model is very close to the true lifted canonical model

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Feng Ding was born in Guangshui, China. He received the B.Sc. degree in electrical engineering from the Hubei University of Technology, Wuhan, China, in 1984, and the M.Sc. and Ph.D. degrees in automatic control both from Tsinghua University, Beijing, China, in 1991 and 1994, respectively. From 1984 to 1988, he was an Electrical Engineer at the Hubei Pharmaceutical Factory, Hubei, China. He was with Department of Automation, Tsinghua University. He is now a Professor in the Control Science and Engineering Research Center, Southern Yangtze University, Wuxi, China, and has been a Research Associate at the University of Alberta, Edmonton, AB, Canada, since 2002. His current research interests include model identification and adaptive control. He co-authored the book Adaptive Control Systems (Beijing, China: Tsinghua University Press, 2002), and published over 80 papers on modeling and identification as the first author.

Tongwen Chen received the B.Sc. degree from Tsinghua University, Beijing, China, in 1984, and the M.A.Sc. and Ph.D. degrees from the University of Toronto in 1988 and 1991, respectively, all in electrical engineering. From 1991 to 1997, he was an Assistant/Associate Professor in the Department of Electrical and Computer Engineering, University of Calgary, Calgary, AB, Canada. Since 1997, he has been with the Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AB, Canada, and is presently a Professor of Electrical Engineering. He held visiting positions at the Hong Kong University of Science and Technology, Tsinghua University, and Kumamoto University. His current research interests include process control, multirate systems, robust control, network based control, digital signal processing, and their applications to industrial problems. He co-authored with B.A. Francis the book Optimal Sampled-Data Control Systems (New York: Springer, 1995). Dr. Chen received a University of Alberta McCalla Professorship for 2000/2001, and a Fellowship from the Japan Society for the Promotion of Science for 2004. He was an Associate Editor of the IEEE TRANSACTIONS ON AUTOMATIC CONTROL during 1998–2000. Currently he is an Associate Editor of Automatica, Systems and Control Letters, and DCDIS Series B. He is a registered Professional Engineer in Alberta, Canada.