S deals with the problem of building a mathematical model for

Transcription

1 JOURNAL OF GUIDANCE, CONTROL, AND DYNAMICS Vol. 18, NO. 4, July-August 1995 Identification of Linear Stochastic Systems Through Projection Filters Chung-Wen Chen* Geophysical and Environmental Research Corporation, Millbrook, New York Jen-Kuang Huangt Old Dominion Universio, Norfolk, Virginia and Jer-Nan Juang$ NASA Langley Research Centel; Hampton, Virginia A novel method is presented for identifying a state-space model and a state estimator for linear stochastic systems from input and output data. The method is primarily based on the relationship between the state-space model and the finite difference model of linear stochastic systems derived through projection filters. It is proved that least-squares identification of a finite difference model converges to the model derived from the projection filters. System pulse response samples are computed from the coefficients of the finite difference model. In estimating the corresponding state estimator gain, a z-domain method is used. First the deterministic component of the output is subtracted out, and then the state estimator gain is obtained by whitening the remaining signal. An experimental example is used to illustrate the feasibility of the method. Introduction YSTEM identification, sometimes also called system modeling, S deals with the problem of building a mathematical model for a dynamic system based on its input/output data. This technique is important in many disciplines such as economics, communication, and system dynamics and control.' The mathematical model allows researchers to understand more about the properties of the system, so that they can explain, predict, or control the behaviors of the system. Recently, a method has been introduced in Refs. 2 and 3 to identify a state-space model from a finite difference model. The difference model, called autoregressive with exogeneous input (ARX), is derived through Kalman filter theories. However, the method requires to use an ARX model of large order, which causes intensive computation in the embedded least-squares operation. In Ref. 4 a method is derived to obtain a state-space model from input/output data using the notion of state observers. This approach can use an ARX model with an order much smaller than that derived through the Kalman filter, but the derivation is based on a deterministic approach. In Ref. 5, it has been proved that, as the order of the ARX model increase to infinity, the observer identification converges to the Kalman filter identification. However, for a stochastic system and an ARX model of a small order, to what the least-squares identification of the ARX model will converge in a stochastic sense is not clear. This paper addresses the above-mentioned problems using a stochastic approach. The approach is primarily based on the relationship between the state-space model and the finite difference model via the projection filter.3 First, an ARX model is chosen, and then the ordinary least squares is used to estimate the coefficient matrices. Based on the relationship between the projection filter and the state-space model matrices, the system pulse response samples (Le., the system Markov parameters) can be calculated from the coefficients of the identified ARX model. The eigensystem realization algorithm (ERA)6 is used to decompose the Markov parameters into a state-space model. In contrast to the time-domain approaches used in Refs. 2 and 5, a different method is developed in this paper using a z-domain approach to compute the state estimator gain. After identifying a state-space model, the deterministic part of the output is subtracted out. The remaining signal represents the stochastic part. A movingaverage (MA) model is then introduced to describe the remaining signal. The MA model is computed by identifying the corresponding autoregressive (AR) model first and then inverting it. From the identified MA model, the state estimator gain is then calculated. Finally, identification of a 10-bay structure is used to illustrate the feasibility of the approach. Relationship Between Projection Filters and Finite Difference Models The projection filter is a linear transformation that projects (transforms) a finite number of input/output data of a system into its current state space. The filter is designed such that the mean-square estimation error is minimized; therefore, the image of the projection is an optimal estimate of the current state.3 Here, filter is a generic term referring to a data processing procedure that extracts desired information from data. To explain the relationship between the projection filter and the finite difference model of a linear system, we start from a simple case and then gradually move to some more general cases. Consider a finite-dimensional, linear, discrete-time, timeinvariant, noise-free dynamic system, represented by the state-space model Presented as Paper at the AIAAIASMEIASCEIAHSIASC 33rd X ~ + I = AX^ + Bu~ Structures, Structural Dynamics, and Materials Conference, Dallas, TX, April 13-15,1992; received May 18,1992; revision received June 14,1994; accepted for publication Oct. 10, by the American Yk = cxk DUk Institute of Aeronautics and Astronautics, Inc. All rights reserved. *Technical Engineering Advisor. Member AIAA. tassociate Professor, Department of Mechanical Engineering. Member AIAA. tprincipal Scientist, Structural Dynamics Branch. Fellow AIAA. 767 x is an n x 1 state vector, u an rn x 1 input vector, and y a p x 1 measurement or output vector. Matrices A, B, C, and D are respectively the state matrix, input matrix, output matrix, and (1) (2)

2 768 CHEN, HUANG, AND JUANG direct influence matrix. The integer k is the sample indicator. From Eqs. (1) and (2) it is easy to show that V:k = [v:,..., u~',+~]. Note that the unknown xk is still a deterministic variable in this case. Hence, by the theory of optimal parameter estimation for deterministic parameters from a linear equation with independent white noise; one can write the optimal estimate of xk the same as Eq. (6), but the filter F, becomes or in short, which is a weighted pseudoinverse of H,. In Eq. (lo), R = is the Kronecker product, R the covariance of the measurement noise, and I, the identity matrix of dimension q. The optimality is defined by the minimum variance of the state estimation error. To derive an ARX model using the relationship provided by the projection filter is similar to the previous case. One can form a onestep-ahead output prediction using the estimated state of the last step, that is, or in a normal form, and define Ek = CA2k-i + CBuk-1 + D u~ (1 1) q denotes the number of data stacked up in forming the equation. The meanings of other notations are self-evident. If xk is the unknown variable to solve, Eq. (5) contains n unknowns and p x q equations; however, there are only at most n independent equations. Therefore, for a number q that is sufficiently large to make H, fullcolumn ranked (p x q > n), the unique solution of xk is Yk = jk + ;)lk (12) ;)lk is the prediction error. Therefore, Yk = ca2k-1 4- CBUk-1 + DUk + qk = kcaf,iyk-i + Duk + (CB- CAF,lD)uk_l (7) is the pseudoinverse of Hq and also the projection filter in this case. If p x q = n (Le., H, is square), F, becomes H';'. The number q can be any integer larger than the integer qmin, which is the minimum number required to make H, full-column ranked. The solution 2, is identical with the true value xk in the noise-free case. To write a difference model of the system expressing the current output as a linear transformation of finite previous input/output data, one can combine Eqs. (l),(2), and (6) to have yk = C X + ~ Du~ = CAxk-1 + CBuk-1 + D u ~ = ACAF,;yk-; -C DUk + (CB - CAF,~D)U~-~ i =2 i =o N F,;(E Rnxp) and Gqi(e R(PX'l)Xm) are the ith partitions of F, and G,, respectively, defined as F,; and Gqi are defined in Eq. (9), but now F, is defined by Eq. (10) in this case. Next, consider a more general case of a system with both process and measurement noises. In state-space format the system can be modeled as Xkfl = A Xk + BUk + Wk Yk = CXk + DUk + vk (14) (15) the sequences (wk. and (uk] are the process noise and the measurement noise, respectively. Both are assumed to be Gaussian, zero mean, and white with constant covariance matrices Q and R, respectively. They are also assumed statistically uncorrelated with each other. Similarly, by writing the previous output in terms of the current state using Eqs. (14) and (15), one can derive Fq = [F,, i F,2 i...; F,,], Gq = [G,I i Gqz i... i G,,] (9) and Ai = CAFqi(i = 1,...,q); Bo = D, BI = CB - CAFqlD, Bi = C A F,Gqi (i = 2,..., q). The model described by Eq. (8) is an ARX model. Next, consider a system without process noise but with additive measurement noise. The output equation then becomes yk = CXk + DUk + vk vk represents the measurement noise. The measurement noise is assumed white, Gaussian, and zero mean and not correlated with the state variable. Similarly, one can derive a matrix equation HqXk = yq5k 4- GqUk - vq,k Equation (16) can be further simplified to HqXk = y:,k + &/,k (17) y;,k = yq,k + GqUk, tq,k = Mqwq,k - x1.k Note that the unknown xk is a random variable in this case. Now the overall noise vector &,k is still Gaussian and zero mean because

3 CHEN, HUANG., AND JUANG 769 wq,k and V4,k are Gaussian and zero mean, but it is correlated with the Xk because wq,k is correlated with Xk. Denote the covariance between xk and &,k by Px6 and the autocovariance of tyik by RE. For Eq. (17), given the mean of the current state i k and its variance Px, by the theory of random parameter e~timation?.~ the optimal estimate of xk can be obtained by.?k =?k + Fq (Yi,k - rq,,) (18) the overbar denotes the expectation value, Pi,k = H&, and F~ = (P,H,T+P,~)(H~P~H~+H,P,~+P~H~+R,)-~ (19) The matrix F4 is the projection filter in this case. The optimality is defined by the minimum variance of the state estimation error. Similarly, to derive an ARX model, one can use one-step-ahead output prediction as Eq. (1 1) and have fk = CAik-1 + CBUk-1 -t DUk = CAfXk-1 + Fq(Y(i,k-l -?i,k-1)1 + CBUk-1 + DUk = CAFYYg,k-l + CBUk-1 + CAFqGqUk-l + CA(In - F4Hq)&-1 + Duk = kcafqjyk-j + DUk + (CB - CAFqlD)Uk-I Identification of State-Space Model Three steps are required to identify a state-space model including the least-squares identification of an ARX model, computation of system Markov parameters from the identified ARX model, and identification of system matrices. Least-Squares Identification of ARX Model A general ARX model of a linear system can be written as i =O (41. q2) is the order of the model. Given a set of input and output data (yk,...,yo, uk,..., uo) of the system, one can us_e the 1ea;st-sguares method to find a set of matrix coefficients (AI,..., A,1, BO,..., BY2] that fit the equation and the data best under the sense of least-squares error of output prediction. The leastsquares method for a single-input, single-output ARX model (a scalar equation) can be found in many textbooks,, and the extension to multi-input, multi-output models is ~traightforward.~ The ARX models derived in the last section have an order (4.9). or just q for simplicity, which is a special case of the general ARX model. It can be proved 3 that, if one chooses an ARX model of order q and uses the least-squares method to identify the parameters of the model, the parameters will converge to the model derived through the projection filter. 0 L = I, - FqHY and Fqi and GYi are again defined as in Eq. (9) but FY is defined by Eq. (19) instead. Equation (20) represents the best prediction of yk one can make using q previous input/output data. If the prediction is made once and for all, namely, no prediction of previous state is made, the best value assigned to Xk is zero. However, if previous state estimation has been made, the best choice for n k is the a priori Kalman filter estimate. Note that for a Kalman filter Xk-1 = AXk-2 + AK(yk-2 - C2k-2 - Du~-2) + B~k-2 =... Computation of System Markov Parameters from Identified ARX Model There are some relationships between the system pulse response samples (i.e., the system Markov parameters) and the coefficient matrices of the ARX models derived in the previous section. Based on this relationship one can obtain the system Markov parameters from the ARX models. For noise-free systems, from Eq. (S), if the coefficient matrices of yk-j and uk-j are denoted by Aj and Bj, respectively, one can derive a relation 3 + AYik-q (21) A = A(In - KC) and K is the optimal steady-state Kalman filter gain. Based on the argument above, one can replace in Eq. (20) by Eq. (21) and obtain This equation allows one to calculate the system Markov parameters CAj B ( j = 1,..., q - 1) recursively from the coefficient matrices of an ARX model of order q (note that Bo = D and C B = B, + AID). Note that CAI B (j = q. q + 1,...) can also be recursively calculated by Yk = f k + qk = CAF41yk-j + k CA(F.i + LA - AK)yk-j + DUk i=2 i =O 4; = v k +CAq.&. Note that if q is not large enough, [q;) is not white. Equations (S), (13), and (22) represent the ARX models of linear systems in several different noise situations. The equation in each case provides a best prediction of the output measurement at time k in the sense of minimum state error at time k - 1 using q previous input and output data. Though derived from noise-free systems, it is interesting to see that Eq. (24) also holds for systems with both process and measurement noise.3 However, Eiq. (25) does not hold for the systems with process noise.3 Hence, for an ARX model of order q, only q terms of the system Markov parameters can be obtained. A system of dimension n has only n independent Markov parameters; all the rest are the linear combination of these n independent ones. Therefore, the order chosen for the ARX model q should be greater than or equal to n. In general, more Markov parameters can improve the accuracy of the identified state-space model in the latter procedure; however, the trade-off is the increase in computation. Identification of System Matrices To decompose the identified Markov parameters into the statespace model matrices [A, B, C], one can use the ERA.6 The ERA is an algorithm for identification of linear systems from pulse response samples. The realized model is not unique, although the Markov parameters are unique.

4 770 CHEN, HUANG, AND JUANG Identification of a State Estimator Gain After obtaining a state-space model via the ERA, the corresponding state estimator gain can be computed as follows. This method is enlightened by the Kalman filter theory. From Kalman filter formulations, the filter can be represented by the innovation model'": i; is the optimal prediction made by the Kalman filter based on all of the data prior to the moment k and Kk is the Kalman filter gain; the quantity &k, called the residual, is the difference between the true output yk and the predicted output jk (= Ci; + Duk). In steady state the filter gain is constant and its subscript can be dropped. Equations (26) and (27) are called the innovation model because the quantity sk, called the "innovation," contains new information that cannot be obtained from previous data. A useful property of {&k} is that, for an optimal Kalman filter, it is a white sequence."' From the innovation model it can be seen that the Kalman filter is driven by the deterministic input uk through B and by the stochastic input E~ through A K. Hence, the filter state and output can be decomposed into two parts: one caused by the deterministic input and the other caused by the stochastic input. Accordingly, the innovation model can be divided into two models: K. If the MA model is known, the filter gain can be computed from its coefficients. The problem encountere din estimating the MA model in Eq. (35) is that the white sequence { E ~ is } not available; therefore, the frequently used least-squares methods in estimating the coefficients of linear equations cannot be applied. However, one can estimate a corresponding AR model first and then invert it to obtain an approximation of the original MA model." To explain this point, we take z transform of both sides of Eq. (35) to become a i=o M(z-') is a polynomial matrix in z-' (a matrix whose entries are polynomials in z-l). Matrix M(z-') can be regarded as a filter that receives E~ and its delayed versions as input and yields sk as output. Iftheinverse filter N(z-') of M(z-') [Le., N(z-')M(z-') = I,] can be found, premultiplying Eq. (36) by N(z-'), one has (37) The inverse matrix N (z-') in general is an infinite-ordered polynomial matrix. In Eq. (37), N(z-') can be considered as a whitening filter that receives sk and its delayed versions as input and yields white sequence { E ~ as ) output. To obtain a whitening filter for the signal sk, one can write an AR model of sk with order Y in time domain as and i=o NiSk-i = ck (38) Yk,2 = Cii2 + Ek A-,,?;+' = X ~ + ~, ~ and yk = yk,l + Yk.2. Expanding Eqs. (29) and (31) based on Eqs. (28) and (30), respectively, one can derive Combining the above two equations yields NO = I,, and estimate the AR parameters N1,..., N,.9 Comparing Eq. (37) with Eq. (38), it can be seen that the infinite-ordered polynomial matrix N (z-') is approximated by a finite-ordered polynomial matrix ET=,, N,z-'. The parameter estimation of the AR model can be accomplished using the ordinary least-squares method. After obtaining an identified N(z-'), one can invert it to approximate M(z-'). Inverting a square polynomial matrix is similar to inverting an ordinary scalar square matrix; the result is the adjoint matrix divided by the determinant of the matrix. In operation, polynomial multiplication is equivalent to convoluting the coefficient sequences of the two polynomials, and polynomial division is equivalent to deconvoluting the coefficient sequence of the numerator polynomial over that of the denominator, expanding to as long as desired. Note that another way of computing the MA model in the discrete-time domain is presented in Ref. 12. After obtaining the estimated MA model and collecting q1 coefficients, one can form a matrix Equation (34) clearly shows the two parts of which the output is composed. If the state-space parameters [A, B, C, D] are known accurately, the deterministic part yk,l can be calculated, and one can subtract this part out from the output Yk; that is, by defining sk E yk.2 = Yk - Yk, 1 I and the least-squares solution of K is (39) Co = I,, Ci = CA'K. The remaining signal sk represents the stochastic component driven by the sequence { E~}. For a stable system there exists a number q such that all the terms CA'K are negligibly small when i > q. Therefore, when k is large, the upper limit of the summation of Eq. (35) can be replaced by q. In Eq. (35) sk is described as a linear transformation of a white sequence { ~ k } This. equation is called an MA model,'" and the coefficient matrices C1,..., Cq are called the MA parameters. The term moving average arises because sk can be regarded as a weighted average of Ek,..., ~ k - Note ~. that the MA parameters are expressed in terms of state-space parameters A, C and steady-state Kalman filter gain i=o H = [(CA)T, (CA2)T,..., (CAql)T]T (41) Matrix H is an observability-like matrix, which is full-column ranked for an observable system; H+ is the pseudoinvery of H. Because of the approximation used in the process, K is not a real optimal Kalman filter gain; however, it represents an identified state estimator gain (or suboptimal Kalman filter gain). The quality of the identified gain relies on the accuracy of the identified statespace model and the order of the whitening filter r. If the identified state-space model is accurate and the order r chosen is large enough, the identified gain will converge to the optimal steady-state Kalman filter gain.

5 CHEN, HUANG, AND JUANG 771 Experimental Example An experimental example is used to demonstrate the feasibility of the integrated system identification and state estimation method developed above. A IO-bay structure as shown in Fig. l is c~nsidered.'~ The truss is one of the structures built at NASA Langley Research Center for experiments in studies of control and structure interaction (CSI). It is 100 in. long, with a square cross section of IO x IO in. All the tubing (longerons, battens, and diagonals) and ball joints are made of aluminum. The structure is in a vertical configuration attached from the top using an L-shaped fixture to a backstop. Two cold air thrusters acting in the same direction are placed at the tip. The thrusters used for excitation and control have a maximum thrust of 2.2 lb each. A mass of approximately 20 lb is attached at the beam tip to lower the fundamental frequency of the truss. Two servo accelerometers located at a corner of the square cross section provide the in-plan tip acceleration. The structure was excited using random inputs to both thrusters for 30 s. The input signals were filtered to concentrate the energy in the low-frequency range. A total of 3750 data points at sampling rate 125 Hz is taken. The two output acceleration signals were filtered using a three-pole Bessel filter with a break frequency of 20 Hz. From the output one can tell the dominant mode is about 5-6 Hz. The order of the ARX model is set to 100. Figure 2 shows the identified system Markov parameters CA'-'B(i = I,..., 100). By the ERA three modes are identified and the identified modal frequencies and dampings are listed in Table 1. The corresponding state-space model matrices in normalized 1, modal format are [ = diag { [ B= ]} L D=[ The identified state-space model matrices are used to estimate the corresponding Kalman filter gain. The identified Markov Wall T Table 1 Identified frequencies and dampings Mode Frequency, Hz Damping, % zl\ Power index (i) Fig. 3 Identified Kalman filter Markov parameters CA'K [the (1, 1) element]. 7 5, I I a) First output 4 0, I I Time kec) b) Second output Fig. 4 Comparison of true and estimated outputs. parameters' CA'K (i = I,..., 100) are shown in Fig. 3, and the estimated state estimator gain is *=[ I lip Mass Fig. 1 Ten-bay truss structure test Configuration. I -1; ' Power Index (i) Fig. 2 Identified system Markov parameters CA'-'B [the (1, 1) element]. To evaluate the quality of the system identification and state estimation, the estimated outputs calculated based on the estimated state are compared to the true outputs. Because the true state is not available, output comparison is the only way to validate the results. The comparison of the first output is shown in Fig. 4, one can see the estimated and the true outputs are in good agreement. The covariance of the error is less than 1.5% of the covariance of the output. Concluding Remarks In contrast to most existing system identification methods of which the great majority use deterministic approach, the method developed in this paper is derived under the stochastic framework, taking into account the effects of process noise as well as measurement noise. The use of a projection filter to derive a state-space model provides stochastic insight into the model. The accuracy of the identified state estimator gain relies on the accuracy of the identified state-space model and the order of the whitening filter. The order

6 772 CHEN, HUANG, AND JUANG of the whitening filter is not necessary to be equal to the order of the system. The larger the filter order is, the whiter the residual will be. If the identified model is accurate and the order of the whitening filter is sufficiently large, the identified gain converges to the optimal steady-state Kalman filter gain. An experimental example shows the feasibility of the method. References Juang, J.-N., Applied System Identijication, Prentice-Hall, Englewood Cliffs, NJ, Chen, C.-W., Huang, J.-K., Phan, M., and Juang, J.-N., Integrated System Identification and Modal State Estimation for Control of Flexible Space Structures, Journal of Guidance, Control, and Dynamics, Vol. 15, No. 1, 1992, pp Chen, C.-W., Integrated System Identification and State Estimation for Control of Flexible Structures, Ph.D. Dissertation, Old Dominion Univ., Norfolk, VA, Phan, M.,Horta,L. G., Juang, J.-N., andl0ngman.r. W., Linear System Identification via an Asymptotically Stable Observer, Journal QfOptimization Theory and Application, Vol. 79, No. 1, 1993, pp Juang, J.-N., Phan, M., Horta, L. G., and Longman, R. W., Identification of ObservedKalman Filter Markov Parameters: Theory and Experiments, Journal of Guidance, Control, and Dynamics, Vol. 16, No. 2, 1993, pp Juang, J.-N., and Pappa, R. S., An Eigensystem Realization Algorithm for Modal Parameter Identification and Model Reduction, Journal ofguidance, Control, and Dynamics, Vol. 8, 1985, pp Elbert, T. F., Estimation and Control ofsystems, Van Nostrand Reinhold, New York, Hsia, T. C., System Identification: Least-Squures Methods, Heath, Lexington, MA, aykin, S., Adaptive Filter Theory, Prentice-Hall, Englewood Cliffs, NJ, Goodwin, G. C., and Sin, K. S., Adaptive Filtering, Prediction und Control, Prentice-Hall, Englewood Cliffs, NJ, Chen, C.-W., and Huang, J.-K., Estimation of Optimal Kalman Filter Gain from Non-Optimal Filter Residuals, Journal qf Dynumic Systems, Meusurement, and Control, Vol. 116, No. 3, 1994, pp Juang, J.-N., Chen, C.-W., and Phan, M., Estimation of Kalman Filter Gain from Output Residuals, Journal qf Guidance, Control, and Dynamics, Vol. 16, No. 5, 1993, pp Horta, L. G., Phan, M., Juang, J.-N., Longman, R. W., and Sulla, J., Frequency Weighted System Identification and Linear Quadratic Controller Design, Journal ofguidance, Control, and Dynamics, Vol. 16, No. 2,1993, pp