1 Use of Multiple Integration and Laguerre Models for System Identification: Methods Concerning Practical Operating Conditions Yu-Chang Huang ( 黃宇璋 ) Department of Chemical and Materials Engineering National Kaohsiung University of Applied Sciences 2010/09/24

2 Outline Introduction and available identification methods Identification of continuous-time SISO systems Use of multiple integration Two-stage algorithms Identification of discrete-time MIMO systems Use of a single Laguerre model Augmented order to deal with unknown disturbances Identification of discrete-time MIMO systems Use of double Laguerre models Suited to obtaining a process model of reduced order Conclusions and future work

3 Introduction (I) System identification finding process and disturbance models based on input-output testing is often faced with practical operating conditions as follows: unsteady and unknown initial states load disturbances of unknown dynamics and unpredicted nature stochastic disturbances unknown model structure (order and delay) and parameters constraints on the input signal to a test experiment continuous-time or discrete-time

4 Introduction (II) Available identification methods for linear systems Diamessis (1965) assumed implicitly that all initial conditions were zero – multiple integration Lecchini and Gevers (2004) delivered a Laguerre analysis under zero initial conditions and no disturbances Hang et al. (1993), Shen et al. (1996) and Park et al. (1997) resolved static load disturbances but not slow and periodic disturbances – relay tests Hwang and Wang (2003) developed a time- and frequency- weighted method to deal with non-static disturbances Hwang and Lai (2004) and Liu and Gao (2008) presented methods based on specified test signals

5 Identification Method for Continuous- Time SISO Systems Use of multiple integration to avoid time derivatives of the input-output signals A sequential least-squares method that identifies a parametric model using a two-segment test signal (first complicated and then simple) in face of the practical difficulties A convenient technique to determine the model structure based on the same test data The method is robust with respect to unsteady initial states, unknown load disturbances, noise, and model structure mismatch

6 Nth-Order Continuous System y(t) and u(t): output and input signals n and m: system orders d: time delay a i, b i ： model parameters (t): unknown disturbance

7 Multiple Integration To avoid time derivatives of a signal x(t), we define a multiple integral filter as

8 Underlying Identification Model – unsteady initial states are unknown – static disturbance (an offset) – f i accounts for the effects of the nonzero initial states and the unknown offset c – the number of parameters to be estimated is high

9 Sequential Algorithms Based on Two-Segment Testing First segment gives the estimation of d and b i Second segment gives the estimation of n and a i Treat the identification problem as two sub- problems sequentially Two-segment testing signal

10 The Input Signal for Plant Tests u 2 = 0 → a pulse test with arbitrary shape u 2 → a simple combination of a step and a sinusoid

11 First-Stage of Estimation The intermediate parameters g i consist of f i and those resulting from the two input functions A goodness-of-fit criterion E n can be developed to determine the best value of n t Applying the ordinary least-squares gives rise to estimates of the model parameters a i

12 Second-Stage of Estimation (I) d L where Applying the ordinary least-squares gives rise to estimates of the model parameters b i

13 Second Stage of Estimation (II) A goodness-of-fit criterion E d can be developed to determine the best value of d Order m can be set to n – 1 or specified by users The number of parameters to be estimated at each stage is minimized A complicated input can be employed at this stage to enhance estimation accuracy

14 Rejection of Slow Disturbances Modifying the regression equation for the first-stage estimation as In practice, p = n+1 or n+2

15 Rejection of Periodic Disturbances Employ pulse testing, i.e. Modifying the regression equation for the first-stage estimation as : frequency of the periodic disturbance

16 Cancellation of Measurement Noise The use of the integral filter could eliminate the effect of measurement noise to a certain extent In the presence of severe noise, it is better to employ the wavelets de-noising procedure based on multi-level decomposition and reconstruction of the output signal (Mallat, 1989)

17 Fitting Model Predictions to Output Measurements response of the model assuming the zero conditions Model verification Once f i are calculated, the model predictions can be obtained as effects of the nonzero initial states and disturbance

18 Simulation Study subjected to Case I: a static disturbance (offset) & NSR = 10% Case II: a slowly changing load (drift) & NSR = 5% Case III: a periodic disturbance & NSR = 5%

19 Input test signalModel predictions for Cases I and II

20 Identification input-output data for Case III

21 The goodness-of-fit function E n versus order n Finding the Model Order

22 Estimated parameters under different test conditions

23 Identification Method Based on a Single Laguerre Model for Discrete-Time MIMO Systems Use of a Laguerre ARX model with a time-scaling factor in face of unpredicted load and unknown stochastic disturbances The idea of augmented order is introduced to account for the MISO process and distinct load dynamics Three error criteria are developed to find the best values for the time-scaling factor, load entering time, and process delays Not suited to finding a process model of reduced order Persistent excitation for the input is required

24 Discrete Laguerre expansions (I)  : time-scaling factor g i : Laguerre coefficients T: sample time Laguerre IIR (infinite impulse response) model (Wahlberg, 1991)

25 Discrete Laguerre expansions (II) Laguerre FIR (finite impulse response) model Laguerre ARX (autoregressive with an exogenous input) model, : Laguerre coefficients

26 Conversion Relationships between, and,

27 SISO Identification Model (I) Y(z), U D (z): z–transforms of the output y(k) and delayed input u D (k) = u(k -  ) G I (z): initial states G L1 (z), G L2 (z),  L : first and second load disturbances and load entering time G V (z): stochastic disturbance

28 SISO Identification Model (II) A P (z): denominator polynomial for process of order n P A L1 (z), A L2 (z): distinct load dynamics A L (z): monic polynomial of degree n L, the least common multiple of A L1 (z), A L2 (z)

29 SISO Identification Model (IV) Applying the Laguerre ARX model gives Regression equation in time domain

30 Recovery of Augmented Model Simulated outputs,. Applying the least–squares estimation leads to the construction of

31 MIMO Identification Model (I) For a w outputs, m inputs system, the lth MISO subsystem can be expressed by Augmented order

32 MIMO Identification Model (II)

33 Two Error Criteria for Identification under Deterministic Disturbances Find the best values for  l,  L,l, and  lj The first is the output error criterion:

34 The second is the relative error criterion the process–only outputs predicted by the Laguerre ARX models of true order outputs predicted by the Laguerre FIR models

35 Error Criterion for Identification under Stochastic Disturbances The filtered output error criterion where

36 MIMO study: Example 2,, T = 1 Wood and Berry (1973) The exact discrete model

37 Load Disturbances: Cases A and B Case A: Case B: ― both subjected to measurement noise of NSR = 5%

38 Identification results for Case A

39 Identification results for Case B

40 Comparison of the actual outputs and load disturbances with those predicted by the identified models for Case A

41 Comparison of the actual outputs and load disturbances with those predicted by the identified models for Case B

42 Type I:, Type II:, Type III: Three types of noise characteristics:, Stochastic Disturbances

43 Mean of time-scaling factor versus NSR

44 Effect of NSR on identification reliability

45 Identification Method Based on Double Laguerre Models for Discrete-Time MIMO Systems A good reduced-order model for the process is sometimes desired for controller design Use of double Laguerre ARX models to account for the process and distinct load dynamics separately Two different time-scaling factors  and  need to be sought for each MISO subsystem

46 SISO Identification Model (I) Assume n = n P and multiply the above equation by A(z) = A P (z) yields

47 SISO Identification Model (II) Applying double Laguerre ARX models gives Regression equation in time domain

48 Two Error Criteria for MIMO Systems

49 MIMO Example under Load Disturbances,, ― subjected to measurement noise of NSR = 5%

50 Comparison of the actual outputs and load disturbances with those predicted by the identified models for the two subsystems

51 Comparison of the actual and identified disturbances by virtue of Nyquist plots

52 Conclusions (I) We have developed three effective methods to deal with system identification based on plant tests under practical operating conditions The first method using multiple integration and a sequential algorithm can identify a continuous-time SISO process from a relatively simple test experiment

53 Conclusions (II) The second method based on a single Laguerre model with an adjustable time- scaling factor can identify a discrete-time MIMO process if the process order is not too high The third method based on double Laguerre models with different time-scaling factors can identify a good reduced-order model for a discrete-time MIMO process

54 Future Work Extend the first method to the Identification of continuous-time MIMO systems Consider the use of other orthogonal functions for system identification Extend the second and third methods to the identification of nonlinear processes

55 Thanks for your attention!