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1 Topics in Model Based Control with Application to the Thermo Mechanical Pulping Process Dr. ing. thesis David Di Ruscio f Department of Engineering Cybernetics The Norwegian Institute of Technology 1993 June 27, 1994 Report W Department of Engineering Cybernetics The Norwegian Institute of Technology N-7034 Trondheim, Norway

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3 Acknowledgement I would like to gratefully acknowledge the guidance and valuable discussions with my advisor, Professor Jens Glad Balchen at the Department of Engineering Cybernetics, the Norwegian Institute of Technology. I would also acknowledge the discussions with Alfred Holmberg at Norske Skog Teknikk. Additionally, Professor Rolf Henriksen at the Department of Engineering Cybernetics and Dr. Ing. Dag Ljungquist at Hydro Aluminium are acknowledged for their contributions to the work on realization of time series. This research has been sponsored by Norske Skog and the Royal Norwegian Council for Scientic and Industrial Research (NTNF). The nancial support is gratefully acknowledged.

4 ii ACKNOWLEDGEMENT

5 Contents Acknowledgement i Introduction 1 I REALIZATION OF TIME SERIES 3 1 State Space Model Realization from Input-Output Time Series Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : Problem denition and preliminaries : : : : : : : : : : : : : Method 1 : : : : : : : : : : : : : : : : : : : : : : : : : : : : Correlation analysis of the output y : : : : : : : : : Covariance analysis of the output y : : : : : : : : : : Cross-correlation analysis of y and u : : : : : : : : : Cross-covariance analysis of y and u : : : : : : : : : Determination of the noise covariance matrix : : : : Discussion : : : : : : : : : : : : : : : : : : : : : : : : Method 2 : : : : : : : : : : : : : : : : : : : : : : : : : : : : Numerical examples : : : : : : : : : : : : : : : : : : : : : : Concluding remarks : : : : : : : : : : : : : : : : : : : : : : References : : : : : : : : : : : : : : : : : : : : : : : : : : : : Appendix: Tables for the numerical examples : : : : : : : : 25

6 iv CONTENTS 2 A Method for the Identication of State Space Models Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : Problem denition and preliminaries : : : : : : : : : : : : : Main results : : : : : : : : : : : : : : : : : : : : : : : : : : : Result 1 : : : : : : : : : : : : : : : : : : : : : : : : : Result 2 : : : : : : : : : : : : : : : : : : : : : : : : : Discussion : : : : : : : : : : : : : : : : : : : : : : : : Determination of C and : : : : : : : : : : : : : : : : : : : Numerical examples : : : : : : : : : : : : : : : : : : : : : : Concluding remarks : : : : : : : : : : : : : : : : : : : : : : References : : : : : : : : : : : : : : : : : : : : : : : : : : : : Appendix: Tables for the numerical example : : : : : : : : 39 3 A Solution to the Problem of Constructing a State Space Model from Time Series Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : Problem denition and preliminaries : : : : : : : : : : : : : Results I : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Results II : : : : : : : : : : : : : : : : : : : : : : : : : : : : Numerical examples : : : : : : : : : : : : : : : : : : : : : : Conclusions : : : : : : : : : : : : : : : : : : : : : : : : : : : References : : : : : : : : : : : : : : : : : : : : : : : : : : : : Appendix : : : : : : : : : : : : : : : : : : : : : : : : : : : : 54 II ANALYSIS AND DESIGN OF LQ SYSTEMS 55 4 A Note on a Necessary Condition for Optimality Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : Theory : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Conclusion : : : : : : : : : : : : : : : : : : : : : : : : : : : 60

7 CONTENTS v 4.4 References : : : : : : : : : : : : : : : : : : : : : : : : : : : : 60 5 Maximal Imaginary Eigenvalues in Optimal Systems Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : Problem formulation : : : : : : : : : : : : : : : : : : : : : : Main results : : : : : : : : : : : : : : : : : : : : : : : : : : : Numerical examples : : : : : : : : : : : : : : : : : : : : : : Example 1 : : : : : : : : : : : : : : : : : : : : : : : : Example 2 : : : : : : : : : : : : : : : : : : : : : : : : Example 3 : : : : : : : : : : : : : : : : : : : : : : : : Concluding remarks : : : : : : : : : : : : : : : : : : : : : : References : : : : : : : : : : : : : : : : : : : : : : : : : : : : Appendix : : : : : : : : : : : : : : : : : : : : : : : : : : : : 75 6 On the Location of LQ-Optimal Closed-Loop Poles Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : Problem formulation : : : : : : : : : : : : : : : : : : : : : : Main results : : : : : : : : : : : : : : : : : : : : : : : : : : : Imaginary parts of the closed loop eigenvalues : : : : Magnitude of the closed loop eigenvalues : : : : : : : Real parts of the closed loop eigenvalues : : : : : : : Concluding remarks : : : : : : : : : : : : : : : : : : : : : : References : : : : : : : : : : : : : : : : : : : : : : : : : : : : Appendix : : : : : : : : : : : : : : : : : : : : : : : : : : : : 88 7 A Schur Method for Designing LQ-Optimal Systems with Prescribed Eigenvalues Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : Linear algebra review : : : : : : : : : : : : : : : : : : : : : Basic denition : : : : : : : : : : : : : : : : : : : : : : : : : Solution by block triangulization : : : : : : : : : : : : : : : 97

8 vi CONTENTS 7.5 The second order problem : : : : : : : : : : : : : : : : : : : The n-dimensional problem : : : : : : : : : : : : : : : : : : Numerical examples : : : : : : : : : : : : : : : : : : : : : : Example 1 : : : : : : : : : : : : : : : : : : : : : : : : Example 2 : : : : : : : : : : : : : : : : : : : : : : : : Example 3 : : : : : : : : : : : : : : : : : : : : : : : : Example 4 : : : : : : : : : : : : : : : : : : : : : : : : Concluding remarks : : : : : : : : : : : : : : : : : : : : : : References : : : : : : : : : : : : : : : : : : : : : : : : : : : : Appendix : : : : : : : : : : : : : : : : : : : : : : : : : : : : An Algorithm for Design of Decentralized Suboptimal Controllers with a Specied Structure Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : Problem denition : : : : : : : : : : : : : : : : : : : : : : : Solution algorithm : : : : : : : : : : : : : : : : : : : : : : : The output feedback design and the stabilization problem : Design of robust controllers : : : : : : : : : : : : : : : : : : Numerical examples : : : : : : : : : : : : : : : : : : : : : : Example 1, distillation column. : : : : : : : : : : : : Example 2, unstable plant : : : : : : : : : : : : : : : Example 3, robust controller : : : : : : : : : : : : : Concluding remarks : : : : : : : : : : : : : : : : : : : : : : References : : : : : : : : : : : : : : : : : : : : : : : : : : : : Appendix : : : : : : : : : : : : : : : : : : : : : : : : : : : : A Method for the Stabilization of Linear Feedback Systems Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : Problem denition and preliminaries : : : : : : : : : : : : : Main results : : : : : : : : : : : : : : : : : : : : : : : : : : : Concluding remarks : : : : : : : : : : : : : : : : : : : : : : 142

9 CONTENTS vii 9.5 References : : : : : : : : : : : : : : : : : : : : : : : : : : : : Measures for Stability Robustness in Linear Quadratic Systems Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : Problem formulation and preliminaries : : : : : : : : : : : : Robustness measures : : : : : : : : : : : : : : : : : : : : : : Results from time domain conditions : : : : : : : : : Results from frequency domain conditions : : : : : : Numerical examples : : : : : : : : : : : : : : : : : : : : : : Concluding remarks : : : : : : : : : : : : : : : : : : : : : : References : : : : : : : : : : : : : : : : : : : : : : : : : : : : Appendix : : : : : : : : : : : : : : : : : : : : : : : : : : : : 156 III MODEL BASED CONTROL OF TMP PROCESS The Thermo Mechanical Pulping Process Description of the TMP process : : : : : : : : : : : : : : : : Main process input variables : : : : : : : : : : : : : Main process measurements : : : : : : : : : : : : : : Problem denition : : : : : : : : : : : : : : : : : : : : : : : Control system design : : : : : : : : : : : : : : : : : Modeling the TMP process : : : : : : : : : : : : : : References : : : : : : : : : : : : : : : : : : : : : : : : : : : : Basic Rener Modeling Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : Modeling the screw feeding system : : : : : : : : : : : : : : Modeling the conical screw : : : : : : : : : : : : : : Modeling the cylindrical screw : : : : : : : : : : : : Summary of the chips feeding model : : : : : : : : : 180

10 viii CONTENTS 12.3 The basic conservation equations : : : : : : : : : : : : : : : Total conservation of mass : : : : : : : : : : : : : : : The consistency : : : : : : : : : : : : : : : : : : : : : Total conservation of energy : : : : : : : : : : : : : : Steady state analysis : : : : : : : : : : : : : : : : : : Summary of the basic rener model : : : : : : : : : : : : : Concluding remarks : : : : : : : : : : : : : : : : : : : : : : Appendix: Least squares equations for saturated steam : : A State Space Model for the TMP Process Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : Modeling of ber breakage : : : : : : : : : : : : : : : : : : : Two dimensional particle geometry : : : : : : : : : : One dimensional particle geometry : : : : : : : : : : Radial ber size distribution : : : : : : : : : : : : : : : : : : Model development : : : : : : : : : : : : : : : : : : : Boundary and initial conditions : : : : : : : : : : : : Simplied ber distribution models : : : : : : : : : : : : : : Static model without diusion : : : : : : : : : : : : Dynamic model without diusion : : : : : : : : : : : Determination of breakage rates from experiments : : : : : Examples : : : : : : : : : : : : : : : : : : : : : : : : : : : : Example 1 : : : : : : : : : : : : : : : : : : : : : : : : Example 2 : : : : : : : : : : : : : : : : : : : : : : : : Example 3 : : : : : : : : : : : : : : : : : : : : : : : : Concluding remarks : : : : : : : : : : : : : : : : : : : : : : References : : : : : : : : : : : : : : : : : : : : : : : : : : : : Appendix : : : : : : : : : : : : : : : : : : : : : : : : : : : : Power Consumption in the TMP Process 223

11 CONTENTS ix 14.1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : Modeling the power : : : : : : : : : : : : : : : : : : : : : : Basic equations : : : : : : : : : : : : : : : : : : : : : Shear stress equation : : : : : : : : : : : : : : : : : : Proposed relation for : : : : : : : : : : : : : : : : The rate of debration : : : : : : : : : : : : : : : : : Summary of the power model : : : : : : : : : : : : : : : : : The Power Model : : : : : : : : : : : : : : : : : : : : Simplied power models : : : : : : : : : : : : : : : : Concluding remarks : : : : : : : : : : : : : : : : : : : : : : References : : : : : : : : : : : : : : : : : : : : : : : : : : : : An Estimator for the TMP Process The process model and the measurements : : : : : : : : : : The measurements : : : : : : : : : : : : : : : : : : : The state and disturbance model : : : : : : : : : : : The estimator : : : : : : : : : : : : : : : : : : : : : : : : : : Concluding remarks : : : : : : : : : : : : : : : : : : : : : : A Model Based Control Strategy for the TMP Process Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : A system theoretic description of the model : : : : : : : : : Process model for control : : : : : : : : : : : : : : : Denition of model variables : : : : : : : : : : : : : Linearized model : : : : : : : : : : : : : : : : : : : : Controllability analysis : : : : : : : : : : : : : : : : A system theoretic description of the control : : : : : : : : Description of the control strategy : : : : : : : : : : Linear control system design : : : : : : : : : : : : : Discussion : : : : : : : : : : : : : : : : : : : : : : : : 261

12 x CONTENTS 16.4 References : : : : : : : : : : : : : : : : : : : : : : : : : : : : Appendix : : : : : : : : : : : : : : : : : : : : : : : : : : : : 264 References 267

13 Introduction This thesis is basically a collection of papers. These papers are listed in the reference list on page 267. The thesis is divided into the following three parts: Part I Part II Realization of time series. Analysis and design of LQ systems. Part III Model based control of the TMP process.

14 2 INTRODUCTION

15 Part I REALIZATION OF TIME SERIES

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17 Chapter 1 State Space Model Realization from Input-Output Time Series 1 Abstract. Methods are presented that are based on known system input and output time series for discrete realization of linear, time invariant state space models on innovations form. New methods are compared with an existing method and found to give improved results. The methods work for systems which have nonminimum phase behavior from the input to the output time series. 1.1 Introduction Aoki (1987), (1990) has presented a method for the realization of state space linear discrete stochastic models on innovations form. This method has many interesting numerical properties. It is based on factorization of the Hankel matrix, constructed from covariance matrices of the output time series, by singular value decomposition. The states are presented relative to an "internally balanced" coordinate system, which has some interesting properties when dealing with model reduction. The denition of the "internally balanced" coordinate system is due to Moore (1981). See also Silverman and Bettayeb (1980), and Laub (1980). The above mentioned method has given very good results in the state space modeling of economic time series. However, it cannot handle the combined 1 This chapter is based on the paper Di Ruscio and Ljungquist (1992).

18 6 State Space Model Realization deterministic/stochastic problem, i.e. the problem of model realization of stochastic systems which have manipulable input variables. A modication of the above method to take input variables into account is presented in Aoki (1991), and Ostermark and Aoki (1992). Although this last method is reported to give promising results, a simulation study has pointed out that the method in some cases fails to give satisfactory results. The contribution of this paper is to present some modied methods which is shown by numerical simulations of known systems to give improved results. The rest of the paper is organized as follows. Section 1.2 presents the problem denitions. The main results are presented in Sections 1.3, 1.4 and 1.5. A method based on system analysis by correlation and covariance estimation is presented in Section 1.3. Algorithms based on instrumental variables are presented in Section 1.4. The methods are compared with numerical examples in Section 1.5, and some concluding remarks follow in Section Problem denition and preliminaries We assume that a system can be illustrated as shown in Figure 1.2. y 2 < m u? y - System - e Figure 1.1: Dynamic system with inputs and outputs is the system output which is measured, u 2 < r is the system input which can be manipulated, e 2 < m is an unknown innovations process of white noise, assumed to be serially uncorrelated, covariance stationary, with zero mean and E(e i e T i ) =. We also assume that the system can be described by the discrete, time invariant, linear innovations state space model x i+1 = Ax i + Bu i + Ce i (1.1) y i = Dx i + Eu i + e i (1.2)

19 1.3 Method 1 7 where i 0 is discrete time, ie. an integer, x 2 < n is the state vector with initial value x 0. A, B, C, D and E are constant matrices of appropriate dimensions, where (D; A) is an observable pair. Moreover, the transfer function from e to y, h s (z), is assumed to be invertible and h?1 s (z) is assumed to be stable. This means that h s (z) has all its zeros inside the unit circle. The problem investigated in this paper is to estimate the state space model matrices A, B, C, D, E and from known input output time series. The following notation is used throughout this paper: (A) The eigenvalues of a square matrix A. h d (z) The transfer function from u to y; D(Iz? A)?1 B + E. h d (1) The steady state gain of h d (z); D(I? A)?1 B + E. h d (0) The gain at time zero of h d (z);?da?1 B + E. p d The zeros of h d (z). ~h d (z) The transfer function from u to y when E 6= 0 is left out; D(Iz? A)?1 B. ~h d (1) The steady state gain of h ~ d (1); D(I? A)?1 B. ~p d The zeros of h ~ d (z). h s (z) The transfer function from e to y; D(Iz? A)?1 C + I m. I m m m identity matrix. h s (1) The steady state gain of h s (z); D(I? A)?1 C + I m. h s (0) The gain at time zero of h s (z);?da?1 C + I m. p s The zeros of h s (z). 1.3 Method 1 The algorithms presented in this section are based on system analysis by correlation and covariance estimation. The methods have some restrictions for the input sequence. However, the presentation is also valuable for the understanding of the general problem, ie. when there are no restrictions on the input type. For simplicity, E is assumed to be zero throughout this section. The algorithms presented are extensions of the method by Aoki (1987) Ch. 9 to take input variables into account. Simulation experiments show that the system matrices A and D should be estimated from correlation matrices E(y i+k yi T ) of the output time series, rather than from cross-correlation matrices E(y i+k u T i ) between the output and the input time series, provided the dynamic system is satised, ie. the

20 8 State Space Model Realization correlation matrices must satisfy Equations (1.6) and (1.7) presented in Section The justication for this statement is also explained as follows: Assume that process noise is present in the system. In this case the correlation matrices of the outputs reect the dynamics better than cross-correlation between outputs and inputs, because the excitations from the noise in addition to the input excitations are reected in the correlation matrices. This is also easily seen in the case when the input sequences are zero, and the system dynamics is only excited by process noise. In this case the cross-correlation matrices are zero, but the correlation matrices contain information about the system dynamics. Another fact is that the correlation and the cross-correlation matrices give the same information of system dynamics in the deterministic case, ie. when no process noise is present, provided the dynamic system given by Equations (1.6) and (1.7) is satised. However, the cross-correlation matrices between the input and output time series give important information about the input matrix B. This section is divided into six subsections. Sections and show how A, D and h d (1) can be computed. In addition some useful relations to the noise covariance matrix are presented. Determination of B is discussed in Sections and 1.3.4, while Section shows how C and can be computed. Finally, a discussion of the general case, ie. when there are no restrictions on the type of input, follows in Section Correlation analysis of the output y Dene the correlation and cross-correlation matrices k = E(y i+k y T i ) 2 < mm (1.3) S k = E(u i+k y T i ) 2 < rm (1.4) Z k = E(x i+k y T i ) 2 < nm (1.5) where k 0. These matrices are assumed to satisfy the following dynamic system, which is obtained by shifting time index with i =: i + k, post multiply with yi T in Equations (1.1) and (1.2) and take expectations where Z k+1 = AZ k + BS k 8 k 1 (1.6) k = DZ k (1.7) Z 1 = AZ 0 + BS 0 + C (1.8)

21 1.3 Method = DZ 0 + (1.9) Assume that the input is a step change. In this case the correlation S k+1 = S ns 8 k ns, where ns depends on the time instant the step change is applied to the system and on the number of inputs. For example, if r = 1 and the step is applied at time instant i, then S 0 = S 1 =, and we can choose ns = 0. Dene the matrices which satisfy the relation d p = k+p? k+p?1 (1.10) dz k+1 = Z k+1? Z k (1.11) d p = DA p?1 dz k+1 8 p = 1; 2; ; 2K k = ( 1 if ns 1 ns if ns > 1 (1.12) when S k+1 = S ns 8 k ns (1.13) The correlation dierence matrices d p 8 p = 1; ; 2K are sucient for the construction of the block Hankel matrix, K 2 < KmKm, and the shifted block Hankel matrix, A 2 < KmKm. The structure of these matrices are given by. 2 3 d 1 d K 6 K = = OC (1.14) d K d 2K?1 A = d 2 d K = OAC (1.15) d K+1 d 2K where O 2 < Kmn is the observability matrix for the pair (D; A) and C 2 < nkm is the controllability matrix for the pair (A; dz k+1 ). The order of the dynamic system, n, and the system matrices D and dz k+1 are estimated by factorization of the block Hankel matrix, K, into the product OC by singular value decomposition. The system matrix A is determined from the shifted block Hankel matrix by solving A = OAC. See Silverman and Bettayeb (1980), Moore (1981) and Aoki (1987) ch. 9.

22 10 State Space Model Realization We must ensure that K n to obtain a proper estimate. K = n is the theoretical minimum in the deterministic case, ie. when e i = 0 in Equations (1.1) and (1.2). More system information can be deduced at this stage. We will rst nd an equation for h d (1), and then an equation for the gain of the stochastic part of the system at time zero, h s (0). Finally the problem of determining h s (1) is illustrated. These results are presented in the following three remarks. Remark 1 The formula, relating h d (1) to the correlation matrices, is determined as follows dz k+1 =?(I? A)Z k + BS k (1.16) Pre-multiplication with D(I? A)?1 and substitution for k give k + D(I? A)?1 dz k+1 = D(I? A)?1 BS k (1.17) Equation (1.17) shows that h d (1) is identiable if S k is invertible, because k, A, D and dz k+1 are known. Note that the condition for S k always is satised for single input single output systems, provided the input is a step at time instant i. However, this condition can be removed as shown in Section 1.3.3, Equations (1.43) and (1.44). See also Sections and for the computation of h d (0). This section has not presented any information which can be directly used to estimate B. The B matrix should be estimated from cross-correlation or cross-covariance analysis of y and u. This will be shown in Sections and B can therefore be considered known in Remarks 2 and 3. Remark 2 In order to simplify the presentation we will assume that A, D and dz 2 are determined. The nesting formula from the known matrix dz k+1 to dz 2 is given by X k?1 dz 2 = A?(k?1) dz k+1? A?j B(S j? S j?1 ) (1.18) j=1 An expression for h s (0) is derived as follows. We have dz 2 = AZ 1 + BS 1? (AZ 0 + BS 0 + C) (1.19) Pre-multiplication by DA?1 and substituting for 1 and 0 give 0? 1 + DA?1 dz 2? DA?1 B(S 1? S 0 ) = (?DA?1 C + I) (1.20)

23 1.3 Method 1 11 Equation (1.20) shows that h s (0) times the noise correlation matrix is i- dentiable. Remark 3 At this stage we have not presented any information that can be used to derive h s (1). The following equations will illustrate the problem. For k = 0 we have (I? A)?1 dz 1 =?Z 0 + (I? A)?1 BS 0 + (I? A)?1 C (1.21) Pre-multiplication by D and substitution for 0 give 0 + D(I? A)?1 Z 1 = D(I? A)?1 Z 0 + D(I? A)?1 BS 0 +D(I? A)?1 C + (1.22) where Z 1 is related to known information by Z 1 =?(I? A)?1 dz 2 + (I? A)?1 BS 1 (1.23) Equation (1.22) shows that the steady state gain for the stochastic part only can be determined if the initial statistics Z 0 is known. Note that Z 0 = X 0 D T where X 0 = E(x i x T i ). X 0 and a relation for the steady state gain must be developed from X k = E(x i+k x T i+k ), see Section Covariance analysis of the output y Dene the covariance and cross-covariance matrices k = E[(y i+k? y)(y i? y) T ] 2 < mm (1.24) S k = E[(u i+k? u)(y i? y) T ] 2 < rm (1.25) Z k = E[(x i+k? x)(y i? y) T ] 2 < nm (1.26) where k 0, y, u and x are the mean values of the time series y i, u i and x i, respectively. By assumption, the innovations process e i has zero mean, and hence, =. Assume that the covariance matrices of the input satisfy S k = 0 8 k ns (1.27) where ns 1. The covariance matrices are assumed to satisfy the dynamic system, ie. Z 1 = A Z0 + B S0 + C (1.28) 0 = D Z0 + (1.29) Z k = A Zk?1 + B Sk?1 2 ns k (1.30) p+k?1 = DA p?1 Zk p = 1; :::; 2K k = ns (1.31)

24 12 State Space Model Realization The matrices A, D and Zk can be estimated from the block Hankel matrix and the shifted block Hankel matrix constructed from the sequence given by Equation (1.31), in the same way as described in Section 1.3.1, Equation (1.15). The unused covariance matrices, k 8 k = 0; 1; :::; ns? 1, can be used to determine B in the same way as shown in Section if Sk is invertible. The method is found to be very robust in the detection of A and D and the gain at time zero. The method can detect nonminimum phase behavior in the deterministic part of the system when the signal to noise ratio is large, but will have problems when the data are noisy, ie. when the signal to noise ratio is small. The covariance matrices will probably not satisfy the dynamic system in this case. Here we will discuss the case when ns = 1. Note that dz 1 6= Z 1, where dz 1 is dened in Section We have the following relation?da?1 Z DA?1 B S0 = (?DA?1 C + I) (1.32) The last equation is important because it gives the h s (0) times the noise covariance matrix. When the noise is unknown, only the product is identi- able Cross-correlation analysis of y and u Dene the correlation and cross-correlation matrices s k = E(y i+k u T i ) 2 < mr (1.33) U k = E(u i+k u T i ) 2 < rr (1.34) z k = E(x i+k u T i ) 2 < nr (1.35) where k 0. The matrices, Equations (1.33)-(1.35), are assumed to satisfy the dynamic system which is obtained by shifting time index with i =: i+k, post-multiply with u T i in Equations (1.1) and (1.2) and take expectations z k+1 = Az k + BU k 8 k 0 (1.36) s k = Dz k (1.37) Assume that the input is a step change. In this case the correlation U k+1 = U ns 8 k ns, where ns depends on the time instant the step change is applied to the system and on the number of inputs. For example, if r = 1 and the step is applied at time instant i, then, U 0 = U 1 =, and ns = 0.

25 1.3 Method 1 13 Dene the matrices which satisfy the dynamic system when ds p = s k+p? s k+p?1 (1.38) dz k+1 = z k+1? z k (1.39) ds p = DA p?1 dz k+1 8 p = 1; 2; ; 2K k = ns (1.40) U k+1 = U ns 8 k ns (1.41) The matrices A, D and dz k+1 can be determined from the block Hankel matrix and the shifted block Hankel matrix constructed from Equation (1.40). See Section However, when process noise is present, A and D should be determined as described in Section Assume that A and D are known. dz k+1 is then determined by dz k+1 = (O T O)?1 O T ds 1 ds 2. ds n where O is the observability matrix for the system (D; A). A formula, relating the steady state gain to the data is (1.42) s k + D(I? A)?1 dz k+1 = D(I? A)?1 BU k (1.43) which gives the steady state gain if U k is invertible. Nesting back to k = 0 gives s 0 + D(I? A)?1 dz 1 = D(I? A)?1 BU 0 (1.44) A fundamental condition for h d (1) to be identiable is that U 0 is invertible. Note that the nesting formula from the known matrix dz k+1 to dz 1 is given by dz 1 = A?k dz k+1? kx j=1 A?j B(U j? U j?1 ) (1.45) An expression which can be used to compute h d (1) and a solution for B will be presented in the next section.

26 14 State Space Model Realization Cross-covariance analysis of y and u Dene the covariance and cross-covariance matrices s k = E[(y i+k? y)(u i? u) T ] 2 < mr (1.46) U k = E[(u i+k? u)(u i? u) T ] 2 < rr (1.47) z k = E[(x i+k? x)(u i? u) T ] 2 < nr (1.48) Assume that the covariance matrices of the input satisfy U k = 0 8 k ns (1.49) where ns 1. The matrices, Equations (1.46)-(1.48), are assumed to satisfy and z k = Az k?1 + B Uk?1 1 ns k (1.50) s k?1 = Dz k?1 (1.51) s p+k?1 = DA p?1 z k p = 1; :::; 2K k = ns (1.52) As usual, A, D and z k can be estimated from Hankel matrix factorization of the sequence given by Equation (1.52). See Section Assume that A and D are known, then we must determine z k in the same coordinate system. We have from Equation (1.52) z k = (O T O)?1 O T s k s k+1. s n+k? (1.53) where O is the observability matrix for the system (D; A). z k can now be substituted into the unused cross-covariance matrices. We have s i = DA?(k?i) z k? kx j=1 DA?(k?i?j) B Uj?1 0 i ns? 1 (1.54) To illustrate the value of this formula, assume ns = 3, then ~s 0 ~s 1 5 =? ~s D DA DA A?3 B U2 (1.55)

27 1.3 Method 1 15 which can be solved for B as 2 3 B =?A 3 (O T O)?1 O T 6 4 ~s 0 7 ~s 1 5 U?1 2 (1.56) ~s 2 where O is the observability matrix for the pair (D; A), and ~s 2 = s 2? DA?1 z 3 (1.57) ~s 1 = s 1? DA?2 z 3? ~s 2U?1 2 U 1 (1.58) ~s 0 = s 0? DA?3 z 3? ~s 1 U?1 2 U 1? ~s 2U?1 2 U 0 (1.59) This is an identication method which can be used if an "arbitrary" input signal is applied to the process from time zero to time instant i, and then switch to a signal satisfying Equation (1.49), say a step change. It is also possible to do two experiments on the process. The rst experiment is used to determine A and D, the second to determine B. Note that the cross-correlation dierence dz k dened in Section is related to the covariance matrix z k in the single input case by z k = dz k (1.60) Assume that ns = 1, in this case z 1 is known. An equation for h d (0) is then given by (s 0? DA?1 z 1 ) U?1 0 =?DA?1 B (1.61) Some alternatives to Equation (1.56), for the determination of B are discussed below. Case 1 Combining Equations (1.44) and (1.61) gives the following equation # B = ( O ~ T O) ~?1 O ~ T (1.62) " (s0 + D(I? A)?1 dz 1 )U?1 0 (s 0? DA?1 z 1 ) U?1 0 which denes B for the special case when O ~ T O ~ is invertible, where O ~ is an equivalent observability matrix for the control input matrix B, given by ~O = " # D(I? A)?1?DA?1 (1.63) This means that B is determined to satisfy the transfer function in steady state and at time zero. Moreover, h d (0) is needed to detect non-minimum phase behavior.

28 16 State Space Model Realization We have shown that the steady state gain and the gain at time zero are identiable without any assumptions about initial statistics, z 0 and z 0. Case 2 It can be shown that z 1 = z 0 when N is large and x 0 =0. In this case, an equation for B is simply B = (I? A)dz 1 U?1 0 (1.64) which denes B if U0 is non-singular. This method is not recommended when noise is present. Case 3 The solution for B in the case when D is non-singular is simply z 0 = D?1 s 0 (1.65) B = (z 1? AD?1 s 0 ) U?1 0 (1.66) Case 4 The matrix B can be constructed from the estimate of h d (1): x i+1 = Ax i + Bu i (1.67) y i = Dx i (1.68) B = (I? A)D T (DD T )?1 h d (1) (1.69) This model can be viewed as an approximately dynamic model with an exact steady state gain. The model has the same spectrum of eigenvalues as the underlying process, but may have a dierent phase behavior, ie. the model is approximately dynamic. This is probably a good approach when the process is known to be minimum phase and the data are noisy, because in this case the estimation of A, D and h d (1) is robust. Case 5 Assume that the input u i is white noise. In this case z 0 = 0, and A, B and D satisfy the normalized sequence s 0U?1 0 = E (1.70) s ku?1 0 = DA k?1 B 8 k 1 (1.71) The matrices A, B and D can be determined by Hankel matrix factorization or determined from a possibly known observability matrix for the pair (D; A).

29 1.3 Method Determination of the noise covariance matrix The model, Equations (1.1) and (1.2), can be separated as and x s i+1 = Ax s i + Ce i x s 0 = x 0 (1.72) y s i = Dx s i + e i (1.73) x d i+1 = Ax d i + Bu i x d 0 = 0 (1.74) y d i = Dx d i (1.75) y i = y d i + y s i (1.76) Once the the matrices A, B, and D are determined, the deterministic part of the model, Equations (1.74) and (1.75), can be simulated for yi d and ys i computed from (1.76). The method by Aoki (1987) can now be applied to determine and C. From weak stationarity, X s 0 = E(x s i xst i ) = E(x s i+1 xst i+1 ), the following Riccati type equation can be solved for X0. s X s 0 = AX s 0 AT + (Z s 1? AXs 0 DT )( s 0? DXs 0 DT )?1 (Z s 1? AXs 0 DT ) T (1.77) This equation can be solved iteratively. X s 0 = 0 is a good choice as initial value for the iterations. The Schur method by Laub (1979) is also recommended. For very ill-conditioned systems, the solution from the Schur method should be used as the initial value for the iterative method, to rene the solution further. C and are determined from X s 0 and Z s 1 as = s 0? DX s 0D T (1.78) C = (Z s 1? AXs 0 DT )?1 (1.79) Note that Z s 1 satises the sequence s k = DAk?1 Z s 1 k 1, where s k = E(y s i+k (ys i )T ). When A and D are known, Z s 1 should be determined as shown in Equation (1.53) Discussion We have discussed four methods in Sections to These can be used to estimate A and D by factorization of the Hankel matrix determined from the sequence given by Equations (1.12), (1.31), (1.40) or (1.52). These methods work well if the correlation or covariance matrices for the input

30 18 State Space Model Realization satisfy, or approximately satisfy, the conditions given by Equations (1.13), (1.27), (1.41) or (1.49). However, the following combination of the above mentioned equations is found to give satisfactory results in the general case, ie. when there are no restrictions to the input type: Dene dh k = s k U?1 0? k S?1 0 8 k = 1; :::; 2K (1.80) where s k, U 0, k and S 0 are given by Equations (1.33), (1.34), (1.3) and (1.4). We have assumed that m = r, and that U 0 and S 0 are invertible. The matrices dh k 8 k = 1; :::; 2K are sucient for the construction of the block Hankel matrix and the shifted block Hankel matrix, in the same way as illustrated in Section 1.3.1, Equation (1.15). The order of the dynamic system, n, and the system matrices A and D are estimated from the block Hankel matrix and the shifted Hankel matrix, see Section No major dierences are found if covariance matrices are used in place of correlation matrices in Equation (1.80), ie. by using d Hk = s k U?1 0? k S?1 0 8 k = 1; :::; 2K (1.81) for the construction of the block Hankel matrix and the shifted block Hankel matrix. s k, U0, k and S0 are given by Equations (1.46), (1.47), (1.24) and (1.25). The matrix B can be estimated as illustrated in Sections and However, a more interesting approach seems to be to combine this method for the determination of A and D with the instrumental variable method, presented in Section 1.4. The idea is to use the instruments to estimate the initial value z 0, Equation (1.47), and then, with A and D known, solve Equations (1.50) and (1.51) for B. This procedure is under investigation. 1.4 Method 2 A modication of Aoki's original algorithm (Aoki, 1987) to take input (exogenous) variables into account, is outlined in Aoki, This algorithm, which will be referred to as Algorithm A1, can be used to generate a statespace representation for input-output data from an underlying process. The basic idea of this algorithm is to construct a set of Hankel matrices from auto-covariance matrices and cross-covariance matrices for the input-output data and utilize singular value decomposition. The input variables are incorporated using them as a part of the instruments as will be claried below.

31 1.4 Method 2 19 Although Algorithm A1 is reported to give promising results ( Ostermark and Aoki, 1992), a simulation study has pointed out that the algorithm fails to give satisfactory results when the input variables are changing rapidly. However, the algorithm can easily be modied to give improved results. Two alternative algorithms are briey outlined below. (1991), we introduce the notation y? i?1 = " yi?1 y? i?2 # Following Aoki (1.82) which denes y? i?1 recursively. Denote the covariance matrix of this vector by R? = cov(y? i?1 ; y? ) and dene the matrix = i?1 cov(x i; y? ). i?1 These matrices are constant if weak stationarity of the data is assumed. The state vector of the lter is then given by z i = E(x i jy? i?1 ) (1.83) z i = R?1? y? i?1 (1.84) The state vector z cannot be computed from Equation (1.84) since the time series of x i is unknown. However, by using z i = 2 R??1 y? i?1 as instruments, for some 2, Algorithm A1 can be developed as described in Aoki (1991) Section 8 and Ostermark and Aoki (1992) Section 2.1. Remark: The reason why the notation cov( ; ) is used instead of the expectation operator E( ) is twofold. First, using the cov operator will simplify the notation. Second, the algorithms presented in this section will work if either of the denitions cov(y i ; u i ) = E[y i u T i ] or cov(y i; u i ) = E[(y i? y i )(u i? u i ) T ] is used. Strictly speaking the second denition is the correct one and it results in a better estimate of the model order, although errors may be introduced in the estimated zeros and steady-state gain. A modied version of A1 is obtained by using the instruments which corresponds to the lter z i = 2 R?1? y? i (1.85) z i = E(x i jy? i ) (1.86) The resulting algorithm, denoted A2, is summarized in Table 1.1 where the following notation is used:

32 20 State Space Model Realization y + i y? i = [y 1;i y 2;i : : : y m;ijy 1;i+1 y 2;i+1 : : : y m;i+1j : : : jy 1;i+K y 2;i+K : : : y m;i+k] T = [y 1;i y 2;i : : : y m;ijy 1;i?1 y 2;i?1 : : : y m;i?1j : : : jy 1;i?K+2 y 2;i?K+2 : : : y m;i?k+2] T y? i?1 u? i = [y1;i?1 y2;i?1 : : : ym;i?1jy1;i?2 y2;i?2 : : : ym;i?2j : : : jy1;i?k+1 y2;i?k+1 : : : ym;i?k+1]t = [u 1;i u 2;i : : : u r;iju 1;i?1 u 2;i?1 : : : u r;i?1j : : : ju 1;i?L+2 u 2;i?L+2 : : : u r;i?l+2] T The integers K and L are cuto factors for the output and input time series respectively. Moreover, y k;i denotes element number k of y at the time instant i. Table 1.1: The Algorithm A2. Step 1. 1 = cov(u i ; y? ) i?1 2 <rkm H = cov(y i + ; y? ) i?1 2 <KmKm H u = cov(y i + ; u? i ) 2 <KmLr R? = cov(y? i ; y? i ) 2 <KmKm Y = cov(y? i ; u? i ) 2 <KmLr U = cov(u i ; u? i ) 2 <rlr U + = U T (UU T )?1 2 < Lrr Step 2. O 2 2 = (H? H u U + 1 )(I Km? R?1? Y U + 1 )?1 2 < KmKm Step 3. Compute the singular value decomposition of O 2 2 = UV T and assign O 2 = U 1=2 2 < Kmn, 2 = 1=2 V T 2 < nkm where n is the chosen model order. Step 4. O 1 = (H u? O 2 2 R?1? Y )U + 2 < Kmr Step 5. ^E = [Im 0 : : : 0]O 1 2 < mr ^D = [I m 0 : : : 0]O 2 2 < mn Step 6. ^z i = 2 R?1? y? i ^e i = y i? ^D^zi? ^Eui [ ^A ^B] = [cov(^zi+1 ; ^z i ) cov(^z i+1 ; u i )] ^C = cov(^z i+1? ^A^zi? ^Bui ; ^e i )cov(^e i ; ^e i )?1 " #?1 cov(^zi ; ^z i ) cov(^z i ; u i ) cov(u i ; ^z i ) cov(u i ; u i ) In Table 1.1, 0 denotes a matrix of zeros with appropriate dimensions. = cov(e i ; e i ) can be estimated in Step 6 of Algorithm A2. Simulations show that the dierence between the responses from the underlying process and the generated model is signicantly reduced by extending Step 6 by

33 1.5 Numerical examples 21 the following iteration: where superscript stable. ^z (2) = (2) i+1 ^A^z i + ^Bui + ^C^ei ^e (2) (2) i = y i? ^D^z i? ^Eui (1.87) ^ = cov(^e (2) i ; ^e (2) i ) ^C (2) = cov(^z (2) i+1? (2) ^A^z i? ^Bui ; ^e (2) i ) ^?1 (2) denotes iterated quatities and ^A is assumed to be In many dynamical systems the E-matrix in Equation (1.2) will be zero. If E = 0 is known, A2 can be simplied to an algorithm denoted A3. The simplications which result in Algorithm A3 are summarized in Table 1.2. Table 1.2: Algorithm A3 described by simplications of A2. In Step 1 only H and R? need to be computed. Step 2 is replaced by O 2 2 = H. Step 4 is omitted. Step 5 is replaced by ^E = 0 2 < mr, ^D = [Im 0 : : : 0]O 2 2 < mn. Steps 3 and 6 are kept unchanged. 1.5 Numerical examples Results from a simulation study of two examples are presented and discussed in this section. Both examples are single-input single-output (SISO). SISO systems are chosen to simplify the presentation of the results, and the discussed algorithms can be directly applied to multiple-input multipleoutput systems. The rst example is of the second order, and the transfer function from e to y, h s (z) has both zeros outside the unit circle. This nonminimum phase phenomenon cannot be estimated and is the reason for the assumptions made on h s (z) in the problem denition in Section 1.2. The second example is of the third order, and the transfer function from u to y has one zero outside the unit circle. The system is chosen to illustrate that this nonminimum phase phenomenon can be estimated. The following input sequences were tried out in the simulations.

34 22 State Space Model Realization u 1 A step change from 0 to 2 u 2 u i = 0:2( sin(i=25) + sin(i=10) + sin(i=5) + sin(i) ) u 3 A white noise sequence, E(u i u T k ) = 1:0 ik Example Ex1 The model, Equations (1.1) and (1.2), is simulated with dierent values of the noise covariance matrix. The following model matrices are specied. " # " # " # 1:5 1:0 2:0 0:5 A = B = C = (1.88)?0:7 0?1:3 1:5 h i D = 1 0 E = 0 The simulation results obtained by Method 1 with 4000 samples are shown in Table 1.3. For the input type u 1, applied at time instant i = 1, A and D were estimated from Equation (1.12), the steady state gain for the deterministic part, ^h d (1), was estimated from Equation (1.17) and B was estimated from Equation (1.62). For the input type u 3, A, B and D were estimated from Equation (1.71). Finally, C and were determined as shown in Section The simulation results when Algorithms A1, A2 and A3 (Method 2) were applied to 201 samples with the mean value subtracted are summarized in Tables 1.5, 1.6 and 1.7, where the step change for u 1 was applied at time instant i = 20. In Algorithms A2 and A3 Step 6 was extended by the iteration in Equation (1.87). Only 201 samples were used to demonstrate that satisfactory results can be obtained with such a low number of datapoints. It should be noted that the results were not signicantly improved if the number of samples was increased. Some important properties of the underlying system are summarized in the last row of the tables. Example Ex2 The model, Equations (1.1) and (1.2), is simulated with dierent values on the noise covariance matrix. The following model matrices are specied :8 0:3 0:1 6 A = 4 7?0:3 0:8 0:1 5 6 B = C = 4 0:1 7 0:6 5 (1.89) 0 0 0: h i D = 3 0?0:6 E = 0 The simulation results obtained by Method 1 with 4000 samples are shown

35 1.5 Numerical examples 23 in Table 1.4. A and D were estimated from Equation (1.52) and B from Equation (1.56). The step change was applied at time instant i = 3. Table 1.8 summarizes some simulation results from Algorithms A1, A2 and A3 based on 201 samples with the mean value subtracted. Only the results obtained for the input sequence which gave the best results (u 1 ) for all the algorithms are given. In Table 1.8 \No" denotes the algorithm which was used in the simulation corresponding to that row. Discussion The simulation results show that nonminimum phase phenomena in h s (z) cannot be estimated. For h d (z), however, zeros outside the unit circle can be estimated. This is in agreement with theoretical considerations. Tables 1.3 and 1.4 show that Method 1 works very well when the input is either a step change or approximately a white noise sequence. Moreover, a simulation study (not presented here) has shown that Method 1 gives satisfactory results even when the number of datapoints is low (200 samples for SISO systems) and that Method 1 gives results that are very close to those obtained by Method 2 for a general input sequence when A and D are generated according to Equation (1.80). The following conclusions can be made regarding Method 2 based on Tables 1.5 to 1.8: The realizations obtained by Algorithm A1 have eigenvalues that are satisfactory close to that of the true system for all the simulations considered. However, a large deviation in ^E will frequently result in poor estimates of the zeros and the steady-state gain of the transfer function from the inputs to the measurements. This is especially so when the inputs are changing rapidly, see the results obtained when the input is approximately white noise (measured) in Table 1.5. The problem connected to a bad ^E estimate is signicantly reduced when Algorithm A2 is applied. More specically A2 gives satisfactory results also when the input is white noise. All the algorithms give a rather large deviation in the steady-state gain when the input is white noise. This is due to the well-known fact that the slow dynamics are poorly identiable for such an input, so the result is not specic for the algorithms under consideration. The deterministic part of the realized models is satisfactory even under very small signal to noise ratios (0.043 when u 2 is applied to Ex1

36 24 State Space Model Realization with = 1:0). However, further research is needed to modify the algorithms to give improved estimates of ^C and ^. This can probably be done by using a strategy that is more similar to that used by Aoki (1987) for systems without input (exogenous) variables. Algorithm A3 does not seem to be able to estimate the nonminimum phase in Example Ex2. It should be noted that the nonminimum phase in the realizations obtained by A1 and A2 is due to the estimate ^E 6= 0, compare the ~p d and p d columns in Table 1.8. Algorithm A2 seems to have better overall performance for the two examples than A1 and A Concluding remarks The method presented in Section 1.3 is found to be very robust in the determination of A, D, and the steady state gain for the deterministic part of the system. It gives an exact result in the deterministic case, provided the correlation or covariance matrices are exactly determined, and it works for systems which have nonminimum phase behavior from the input to the output. An important feature is that the method is constructed to give exact results when certain restrictions on the input sequence are satised. Moreover, the concept can be modied to obtain an algorithm which is general with respect to the input sequence. The method presented in Section 1.4 is also general with respect to the input sequence. These algorithms do not have the property of giving an exact result in the deterministic case, and further research is needed to improve the estimated stochastic part of the model. Both methods, presented in Sections 1.2 and 1.4, are found to give satisfactory results even when the number of datapoints is low (200 samples for SISO systems). Method 1 is found to give better results compared to method 2 in the case that the input is a step change. Therefore, Method 1 should be chosen in the case that the input is a step change and Method 2 should be chosen in the case of other input signals, irrespective of the number of datapoints. Although satisfactory realizations were obtained by the methods presented in this paper, it is believed that the presented results can be combined and developed further to obtain a general algorithm with even better overall performance.

37 1.7 References References Aoki M. (1987). State Space Modeling of Time Series. Springer-Verlag Berlin, Heidelberg. Aoki M. (1990). State Space Modeling of Time Series. Second, Revised and Enlarged Edition. Springer-Verlag Berlin, Heidelberg. Aoki M. (1991). Two Complimentary Representations of Multiple Time Series in State Space Innovation Forms. Invited paper presented at Seminar on Recent Advances on Time Series Analysis and their Impact on Economic Forecasting, Madrid, Spain, December Laub, A. J. (1979). A Schur method for solving algebraic Riccati equations. IEEE Trans. on Automatic Control, Vol. AC-24, pp Laub, A. J. (1980). Computation of "balancing" transformations. Proc. JACC, San Francisco, FA8-E. Moore, B. C. (1981). Principal Component Analysis in Linear Systems: Controllability, Observability, and Model Reduction. IEEE Trans. on Automatic Control, Vol. AC-26, pp Silverman L. M. and M. Bettayeb (1980). Optimal approximation of linear systems. Proc. JACC, San Francisco, FA8-A. Ostermark, R. and M. Aoki (1992) Time Series Evidence of Impacts of The U.S. Economy on The Scandinavian Economy (by State Space Modeling). IFAC workshop on economic time series analysis and system identication, Vienna July Appendix: Tables for the numerical examples

38 26 State Space Model Realization Table 1.3: Results obtained by Method 1 on Ex1. System Algorithm Model Realization u K ^h d (1) ( ^A) ~p d hd ~ (1) p s h s(1) ^ u i u i 0.644i u i 0.652i u i 0.662i u i u i 0.595i u i 0.421i True System i 1.396i Table 1.4: Results obtained by Method 1 on Ex2. System Algorithm Model Realization u K ^h d (1) ( ^A) ~p d ~ hd (1) p s h s(1) ^ u , i True System 0.850, i