Introduction to system identification

Transcription

1 Introduction to system identification Jan Swevers July

2 Introduction to system identification 1 Contents of this lecture What is system identification Time vs. frequency domain identification Discrete time representation of continuous time systems Frequency domain interpretation of sampled data signals Discrete time input-output models for linear time-invariant systems Linear least squares parameter estimation Example Conclusion

3 Introduction to system identification 2 What is system identification Definition of system identification : Selection of a model for a process (i.e. studied system or device under test (DUT)), using a limited number of measurements of the input and outputs, which may be disturbed by noise, and a priori system knowledge. Definition of parameter estimation : The experimental determination of values of parameters that govern the dynamic behaviour, assuming that the structure of the process model is known 3 basic steps: 1. collecting useful data, 2. choosing a convenient model set, 3. computing the (best) model within the model set, possibly following a certain identification criterion, i.e. parameter estimation.

4 Introduction to system identification 3 The data record determination of inputs and outputs choice of input signals: normal operation or specifically designed identification experiments persistency of excitation optimal excitation : maximally informative, minimal uncertainty data sampling rate: Nyquist frequency, aliasing/frequency folding, filtering: 10 time higher than highest frequency of interest

5 Introduction to system identification 4 The model set specify within which collection of models we are going to look for a suitable one very important but often difficult a priori available information : certain physical laws are known to hold true for the system preliminary data analysis: step or frequency response black box or trial and error

6 Introduction to system identification 5 The parameter estimation step determine within the set of models, the model that is the best approximation or provides the best explanation of the observed data we need a criterion to measure the model quality : the estimation of the model parameters corresponds to the minimization of the chosen criterion the choice of the criterion depends on the available information about and the purpose of the model. conflicting requirements: a model that is as simple as possible, and that explains as much as possible of the observed data

7 Introduction to system identification 6 Model validation how do you know if the model is satisfactory: use it and check if it serves its purpose this is often too dangerous: model validation criteria to get some feeling on the accuracy, confidence on its value simulate and compare with different sets of measurements compare parameter estimates with expectations or values found using other measuring techniques (if available)

8 Introduction to system identification 7 The identification loop

9 Introduction to system identification 8 Time vs. frequency domain identification Time domain identification: use measured data directly to estimate model parameters. Frequency domain identification: first transform data to frequency domain using DFT, then, estimate model parameter in the frequency domain.

10 Introduction to system identification 9 Pros and cons of FDI and TDI Frequency Domain Estimator Time Domain Estimator PROS - convenient frequency domain noise properties - usually linear in the parameters - compact data set - no leakage problems - clear insight into the effect of nonlinear distortions - on-line estimation is possible - applicable to discrete time as well as continuous time systems - measurements in time domain as well as frequency domain CONS - non-linear in the parameters - large data set - limited set of input signals (cf. leakage errors) - exact representation of continuous time systems? - often off-line processing only - correlated time domain noise? - nonlinear distortions? - time domain measurements only

11 Introduction to system identification 10 Two possible measurement situations

12 Introduction to system identification 11 Discrete time representation of continuous time systems Output of a linear time-invariant system for a given input u(t) and impulse response: y(t) = τ=0 g(τ)u(t τ)dτ The Laplace transform of the impulse response {g(τ)} τ=0 is called the transfer function G(s): G(s) = c 0s n c + c 1 s (n c 1) + + c nc d 0 s n d + d1 s (n d 1) + + d nd, with n c n d (due to the causality condition). We consider the output only at discrete times t k = kt, for k = 1, 2,...: y(kt) = τ=0 g(τ)u(kt τ)dτ

13 Introduction to system identification 12 Due to ZOH condictions, the input u(t) is kept constant between the sampling instants: This yields: u(t) = u k, kt t < (k + 1)T y(kt) = = τ=0 g(τ)u(kt τ)dτ = ( lt l=1 τ=(l 1)T l=1 lt ) g(τ)dτ u k l = τ=(l 1)T g(τ)u(kt τ)dτ g T (l)u k l l=1 where we define the (discrete) impulse response of that system: g T (l) = lt τ=(l 1)T g(τ)dτ

14 Introduction to system identification 13 We omit T: y(t) = g(k)u(t k), for t = 1, 2,... k=1 The z-transform of the discrete impulse response {g(k)} k=1 is called the discrete time transfer function G(z): G(z) = b 0z n b + b 1 z (n b 1) + + b nb a 0 z n a + a1 z (n a 1) + + a na, with n b < n a (due to the strict causality condition, i.e. g(k) = 0, for k = 0).

15 Introduction to system identification 14 The relationship between the parameters of the transfer function of continuous time system and the parameters of its zoh-discrete time equivalent can be calculated using published tables or CACSD (MATLAB) software.

16 Introduction to system identification 15 Example Consider: G(s) = s 2 + 2ζω n s + ωn 2 Assume that ζ < 1. This corresponds to the continuous time transfer function with: n c = 0 n d = 2 ω 2 n c 0 = w 2 n d 0 = 1 d 1 = 2ζω n d 2 = w 2 n

17 Introduction to system identification 16 Example (2) The zoh-discrete time equivalent of this transfer function equals : with G(z) = b 0 z + b 1 z 2 + a 1 z + a 2 a 1 = 2e σt cos(ωt) a 2 = e σt b 0 = 1 e σt (cos(ωt) + γ sin(ωt)) b 1 = e σt (e σt cos(ωt) + γ sin(ωt)) and with T is the sampling period. σ = ζω n, ω = ω n 1 ζ 2 γ = ζ 1 ζ 2

18 Introduction to system identification 17 Example (3) The relation between the continuous-time and discrete-time poles is: λ d = e λ ct The numbers of transfer zeros differ, and their relation is much more complex than the relation between the poles.

19 Introduction to system identification 18 Introduction of disturbances We assume that the disturbances and noise can be lumped into an additive term v(t) at the output: y(t) = g(k)u(t k) + v(t) k=1 Sources of disturbances: Measurement noise: the sensors that measure the signals are subject to noise and drift. Uncontrollable inputs: the system is subject to signals that have the character of inputs, but are not controllable by the user. This model does not consider input disturbances, for example noise on the measurements of the input data sequence.

20 Introduction to system identification 19 Introduction of disturbances (2) Time domain identification approach to model disturbances: v(t) = h(k)e(t k) k=0 where e(t) is a sequence of independent (identically distributed) random variables with a certain probability density function, and h(0) = 1. The mean value of e is equal to zero, yielding: E{v(t)} = h(k)e{e(t k)} = 0 k=0

21 Introduction to system identification 20 Shorthand notation qu(t) = u(t + 1) q 1 u(t) = u(t 1) y(t) = g(k)u(t k) + h(k)e(t k) = = k=1 g(k)q k u(t) + k=1 k=0 h(k)q k e(k)r k=0 [ ] [ ] g(k)q k u(t) + h(k)q k e(t) k=1 k=0 = G(q)u(t) + H(q)e(t)

22 Introduction to system identification 21 Shorthand notation (2) G(q) = H(q) = g(k)q k k=1 h(k)q k k=0 G(q) is the transfer operator or transfer function of the linear system. H(q) is the transfer function of the disturbance process.

23 Introduction to system identification 22 Frequency domain interpretation of sampled data signals Periodograms of signals over finite intervals Transformation of periodograms

24 Introduction to system identification 23 Periodograms of signals over finite intervals Consider the finite sequence of a signal z(t), t = 0, 1,,N 1. DFT: Z N (ω) = 1 N 1 N t=0 z(t)e jωt = 1 N 1 N t=0 z(t)w kt N with ω = 2πk/N, for k = 0, 1,,N 1, and W N = e j2π/n. Inverse DFT: Remark that: z(t) = 1 N 1 N k=0 Z N (ω) is periodic with period 2π: Z N ( 2πk N )ej2πkt/n Z N (ω + 2π) = Z N (ω) since z(t) is real: Z N ( ω) = Z N(ω)

25 Introduction to system identification 24 Periodograms of signals over finite intervals (2) This inverse DFT represents the signal z(t) as a linear combination of e jω for N different frequencies ω. The number Z N ( 2πk N ) tells us the weight that the frequency ω = 2πk/N carries in the decomposition of the sequence z(t). Z N ( 2πk N ) 2 is therefore a measure of the energy contribution of this frequency to the signal. Z N ( 2πk N ) 2 is known as the periodogram of the signal z(t), t = 0, 1,..., N 1.

26 Introduction to system identification 25 Transformation of periodograms Let {s(t)} and {w(t)} be related by a stable system G(q). s(t) = G(q)w(t) The input w(t) for t 0 is unknown, but obeys w(t) C w for all t. Define S N (ω) = W N (ω) = N 1 1 N t=0 N 1 1 N t=0 s(t)e jωt w(t)e jωt

27 Introduction to system identification 26 Transformation of periodograms (2) Then where S N (ω) = G(e jω )W N (ω) + R N (ω) R N (ω) 2C w k=1 k g(k) N If {w(t)} is periodic with period N, then R N (ω) is equal to zero for ω = 2πk/N.

28 Introduction to system identification 27 Discrete time input-output models for linear time-invariant systems Different parametrizations of G(q) and H(q) which are finite. Rather than h(k) and g(k) : an infinite number of parameters. Correspond to those used in the Matlab System Identification Toolbox. Extend the model with a parameter vector θ: y(t) = G(q, θ)u(t) + H(q, θ)e(t) In this course, we restrict ourselves to the most simple parametrization: the ARX model structure

29 Introduction to system identification 28 ARX model structure The most simple input-output model: a linear difference equation y(t)+a 1 y(t 1)+...+a na y(t n a ) = b 1 u(t n k )+...+b nb u(t n k n b +1)+e(t) The white-noise term e(t) enters as a direct error in the difference equation: equation error model. The model parameters: [ θ = a 1 a 2... a na b 1... b nb ] T If we introduce: A(q, θ) = 1 + a 1 q a na q n a B(q, θ) = b b nb q n b+1 we get: A(q, θ)y(t) = B(q, θ)u(t n k ) + e(t)

30 Introduction to system identification 29 ARX model structure (2) This corresponds to: G(q, θ) = q n k B(q, θ) A(q, θ), H(q, θ) = 1 A(q, θ) ARX model: AR refers to the autoregressive part A(q, θ)y(t) and X to the extra input B(q, θ)u(t n k ) (called the exogeneous variable in econometrics).

31 Introduction to system identification 30 Linear regressors One-step-ahead prediction of the output: predict the output at time t, using only input-output data prior to t, ŷ(t θ) = a 1 y(t 1) a 2 y(t 2)... a na y(t n a ) + b 1 u(t n k )... + b nb u(t n k n b + 1) = B(q, θ)u(t n k ) + [1 A(q, θ)]y(t) e(t) is not known, so we take the most likely value, i.e. E{e(t)} = 0.

32 Introduction to system identification 31 Linear regressors (2) Now introduce the vector: [ ϕ(t) = y(t 1)... y(t n a ) u(t n k )... u(t n k n b + 1) ] T ŷ(t θ) = θ T ϕ(t) = ϕ T (t)θ Model is linear in the parameters: linear regression model. The vector ϕ(t) is known as the regression vector.

33 Introduction to system identification 32 Linear least squares parameter estimation Section overview Minimizing prediction errors Frequency domain interpretation Implementation of the least squares estimate (LSE) Properties of the LSE Example Frequency domain interpretation of the result How to improve the result Identification of a partly known system Data filtering to improve the LSE

34 Introduction to system identification 33 Minimizing prediction errors Given: Sampled data sequence of inputs u(t) and outputs y(t): u(t) : t = 1,...,N y(t) : t = 1,...,N Define Z N = [y(1), u(1), y(2), u(2),...,y(n), u(n)] T

35 Introduction to system identification 34 Minimizing prediction errors (2) Determine the model parameters such that it provides the best approximation or the the best explanation of the observed data Therefore define the prediction error: ε(t, θ) = y(t) ŷ(t θ) Estimate the best set of model parameters θ such that, for the given data set Z N, the output prediction errors are minimal according to the least square cost function. V N (θ, Z N ) = 1 N N t=1 1 2 ε2 (t, θ) Then the estimate ˆθ N corresponds to: ˆθ N = arg min θ V N (θ, Z N ) This approach is called a prediction error identification method (PEM)

36 Introduction to system identification 35 Frequency domain interpretation ε(t, θ) = H 1 (q, θ) [y(t) G(q, θ)u(t)] = A(q, θ)y(t) B(q, θ)u(t) = e(t) Take the DFT of the prediction error and apply Parseval s theorem: V N (θ, Z N ) = 1 N = 1 N 1 2 N 1 k=0 N 1 k=0 E N (2πk/N, θ) ĜN(e j2πk/n ) G(e j2πk/n, θ) 2 Q N (e j2πk/n, θ) + R N with Q N (2πk/N, θ) = U N(e j2πk/n ) 2 H(e j2πk/n, θ) 2 R N C N

37 Introduction to system identification 36 Frequency domain interpretation (2) ĜN(e j2πk/n ) is called the empirical transfer function estimate (ETFE) and equals: Ĝ N (e j2πk/n ) = Y N(e j2πk/n ) U N (e j2πk/n )

38 Introduction to system identification 37 Implementation of the least squares estimate (LSE) ε(t, θ) = y(t) ŷ(t θ) and ŷ(t θ) = ϕ T (t)θ with ϕ(t) = [ y(t 1)... y(t n a ) u(t n k )... u(t n k n b + 1) ] T Criterion: (least squares) solution V N (θ, Z N ) = ˆθ LS N = arg min θ V N (θ, Z N ) = N t=1 [ N [ y(t) ϕ T (t)θ ] 2 ] 1 N 1 ϕ(t)ϕ T (t) N t=1 N ϕ(t)y(t) t=1

39 Introduction to system identification 38 Implementation of the least squares estimate (LSE) (2) Introduce the following N-dimensional column vectors: [ ŷ(θ) = ŷ(1 θ) ŷ(2 θ)... ŷ(n θ) [ ] T y = y(1) y(2)... y(n) ] T Introduce following N d matrix: Φ = ϕ T (1) ϕ T (2).. ϕ T (2) The vector containing the output predictions then equals: ŷ(θ) = Φθ

40 Introduction to system identification 39 Implementation of the least squares estimate (LSE) (3) The prediction errors: The LSE criterion: ε(θ) = y ŷ(θ) V N (θ, Z N ) = 1 2 ε(θ)t ε(θ) The LSE : ˆθ LS N = The the output predictions for the LSE: ŷ(ˆθ LS N ) = Φ [ ] 1 Φ T Φ Φ T y } {{ } Φ + [ Φ T Φ] 1 Φ T y Calculate the pseudo inverse using the singular value decomposition.

41 Introduction to system identification 40 Properties of the LSE The LSE gives an unbiased estimate if e(t) is a sequence of independent (identically distributed) zero-mean Gaussian random variables (i.e. e(t) N(0, σ 2 )). What if e(t) does not satisfy these conditions? BIAS.

42 Introduction to system identification 41 Example Consider the following model with output measurement noise e(t) (not an ARX model!!!): y(t) = B(q) u(t 1) + e(t) A(q) with which can be rewritten as: A(q) = 1 + a 1 q 1, B(q) = b1 + b 2 q 1 y(t) + a 1 y(t 1) = b 1 u(t 1) + b 2 u(t 2) + e(t) a 1 e(t 1) It can be written as: y(t) = ϕ T (t)θ + ν 0 (t), with: [ ϕ(t) = y(t 1) u(t 1) u(t 2) ν 0 (t) = e(t) a 1 e(t 1) ] T The sequence ν 0 (t) is not a sequence of independent random variables!!

43 Introduction to system identification 42 Example (2) In matrix form: y(3) y(4). =. y(n 2) }{{} y y(2) u(2) u(1) y(3) u(3) u(2) } y(n 1) u(n 1) {{ u(n) } Φ a 1 b 1 b 2 }{{} θ + ν 0 (3) ν 0 (4).. ν 0 (N 2) LSE: ˆθ LS N = Φ + y

44 Introduction to system identification 43 Frequency domain interpretation of the result with V N (θ, Z N ) 1 N N 1 k=0 1 2 ĜN(e j2πk/n ) B(ej2πk/N, θ) A(e j2πk/n, θ) 2 Q N (e j2πk/n, θ) Q N (2πk/N, θ) = U N (e j2πk/n ) 2 A(e j2πk/n, θ) 2 and Ĝ N (e j2πk/n ) = Y N(e j2πk/n ) U N (e j2πk/n )

45 Introduction to system identification 44 How to improve the result The frequency domain representation of the LS error clearly shows that the difference between the empirical transfer function estimate and the frequency response function of the model is weighted by two terms: periodogram of the input and the frequency response of the denominator of the model. Remedy 1: Excite only in the frequency band that interests you, i.e. limit periodogram of the input to the frequency band of interest Remedy 2: Try to compensate the frequency weighting introduced by the denominator of the model: this is a high-pass characteristic that emphasizes the higher frequencies. This is unwanted because in most cases: more noise is present at high frequencies we are interested in a good model at low frequencies So compensate this using a low-pass data filter.

46 Introduction to system identification 45 How to improve the result (2) Filtering the input and output data with a (digital) low pass filter W(z), and using the filtered input and output data for the LSE yields the following weighting function in the frequency domain expression of the least square criterion: Q N (2πk/N, θ) = W(e j2πk/n ) 2 U N (e j2πk/n ) 2 A(e j2πk/n, θ) 2 The filter W(z) must be selected to compensate the high-frequency emphasis caused by the denominator of the estimated model.

47 Introduction to system identification 46 Identification of a partly known system Assume that part of the system model is known: G(q, θ) = G k (q) }{{} known = q n k B k(q) A k (q) Then we have in our ARX model structure: G u (q, θ) }{{} unknown A u (q, θ) A u (q, θ) A k (q)a u (q, θ)y(t) = B k (q)b u (q, θ)y(t) + e(t) or y(t) = B k(q)b u (q, θ) A k (q)a u (q, θ) u(t) + 1 A k (q)a u (q, θ) e(t)

48 Introduction to system identification 47 Identification of a partly known system (2) Define a new input signal: u (t) = B k(q) A k (q) u(t) and perform the identification using u (t) and y(t) instead of u(t) and y(t), i.e. restricting the identification to the unknown part of the model y(t) = B u(q, θ) A u (q, θ) u (t) + 1 A k (q)a u (q, θ) e(t)

49 Introduction to system identification 48 Identification of a partly known system (3) Several alternative exist: Filter y(t) with inverse of known model: y (t) = A k(q) B k (q) y(t) and perform the identification with u(t) and y (t). or filter both y(t) and u(t): y (t) = A k (q)y(t), u (t) = B k (q)u(t)

50 Introduction to system identification 49 Identification of a partly known system (4) These different alternative approaches yield different error sequences: A u (q, θ)y(t) = B u (q, θ)u (t) + 1 A k (q) e(t) A u (q, θ)y (t) = B u (q, θ)u(t) + 1 B k (q) e(t) A u (q, θ)y (t) = B u (q, θ)u (t) + e(t)

51 Introduction to system identification 50 Identification of a partly known system (5) Frequency domain interpretation of all these different approaches: V N (θ, Z N ) 1 N with and N 1 k=0 1 2 ĜN(e j2πk/n ) B u(e j2πk/n, θ) A u (e j2πk/n, θ) 2 Q N (e j2πk/n, θ) Q N (2πk/N, θ) = U N(e j2πk/n ) 2 A u (e j2πk/n, θ) 2, Ĝ N (e j2πk/n ) = Y N (ej2πk/n ) U N (ej2πk/n ) for the first scheme : u = (B k /A k )u and y = y, for the second scheme: u = u and y = (A k /B k )y, and for the third scheme: u = B k u and y = A k y.

52 Introduction to system identification 51 Data filtering to improve the LSE Data filtering is required to improve the accuracy of the LSE: compensate the high frequency weighting of A(e j2πk/n, θ) or A u (e j2πk/n, θ). In some situations, also low frequency distortions are present (DC-offset and/or drift on measurements), also on inputs! These can be amplified by the pre-filtering with the known part of the model, e.g. if the system contains a pure integration or differentiation... Then, apply band-pass filters to remove these low frequency distortions. Perform data filtering on both input and output data (except to remove the effects of the known parts of the system), using the same digital filters in Matlab.

53 Introduction to system identification 52 Example: linearization and identification Section overview System description and linearization Discrete time model Available data sets Time domain identification for θ 0 = 0 Time domain identification for θ 0 = 45 Comparison of models for θ 0 = 0 and 45

54 Introduction to system identification 53 System description and linearization Consider the following pendulum: Nonlinear system with the following dynamics: T c = I θ + c θ + mgl sin(θ) with I = ml 2, l = 0.5m, m = 1kg, c = 0.05Nms/rad, and the input is a driving torque T c [Nm]. Problem: identify a linear model for this system that is a linear approximation of the system dynamics around θ 0 = 0 and θ 0 = 45.

55 Introduction to system identification 54 System description and linearization (2) Linearize the system dynamics around an equilibrium point θ 0 : define θ = θ 0 + θ and T c = T c0 + T c, and This yields: sin(θ) = sin(θ 0 + θ) sin(θ 0 ) + cos(θ 0 ) θ θ + c I θ + g l (sin(θ 0) + cos(θ 0 ) θ) = T c0 I + T c I Subtract equilibrium conditions mlg sin(θ 0 ) = T c0 yielding: θ + c I θ + g l cos(θ 0) θ = T c I

56 Introduction to system identification 55 System description and linearization (3) In Laplace domain: 1 θ(s) T c (s) = ml 2 s 2 + c ml s + g cos(θ 0) 2 l This is a second order system with an undamped natural frequency at g cos(θ0 ) ω n = l which corresponds to a resonance frequency of 0.70Hz around θ 0 = 0 and of 0.59Hz around θ 0 = 45.

57 Introduction to system identification 56 Discrete time model Discrete time model structure (orders of numerator and denominator) can be derived in several ways: published tables, via Matlab. Matlab: use random numbers: B = [0 0 1]; A = [1 2 3]; Ts=1/10; [Bd,Ad]=c2dm(B,A,Ts, zoh ) Bd = Ad = yielding or B(q) A(q) = b 1q 1 + b 2 q a 1 q 1 + a 2 q 2 θ(t) + a 1 θ(t 1) + a 2 θ(t 2) = b 1 T c (t 1) + b 2 T c (t 2)

58 Introduction to system identification 57 Available data sets We excited the system with a broad band random signal with an appropriate DC value to keep the system at θ 0 = 0 and 45, and measure the response. Sampling frequency is f s = 10Hz. N = Excitation and measured responses: 1 Excitation at 0 degrees 4.5 Excitation at 45 degrees Input [Nm] Input [Nm] Output [rad] Samples Samples Output [rad] Samples Samples

59 Introduction to system identification 58 Time domain identification for θ 0 = 0 Identification with noise-free measurements: θ(2) θ(1) T c (2) T c (1) θ(3) θ(2) T c (3) T c (2) Φ =.... θ(3999) θ(3998) T c (3999) T c (3998) [ ] T Y = θ(3) θ(4)... θ(4000) ˆθ = Φ + Y A = 1 + ˆθ(1)q ˆθ(2)q B = ˆθ(3)q ˆθ(4)q

60 Introduction to system identification 59 Time domain identification for θ 0 = 0 (2) In Matlab: phi = [-theta(2:end-1) -theta(1:end-2) Tc(2:end-1) Tc(1:end-2)]; Y = theta(3:end); mod = pinv(phi)*y; A1 = [1 mod(1:2) ]; B1 = [0 mod(3:4) ];

61 Introduction to system identification 60 Time domain identification for θ 0 = 0 (3) Model validation: compare FRFs of the identified model and of the exact continuous time model (presented above): 20 amplitude [db] continuous time model identified discrete time model continuous time model identified discrete time model phase [degrees] frequency [Hz] Remark: error because the data is obtained in a ZOH situation.

62 Introduction to system identification 61 Time domain identification for θ 0 = 0 (4) Model validation: compare FRFs of the identified model and of the exact discrete time model (derived using zoh equivalent): discrete time model identified discrete time model amplitude [db] discrete time model identified discrete time model phase [degrees] frequency [Hz] Remark: Very accurate model because no noise...

63 Introduction to system identification 62 Time domain identification for θ 0 = 0 (5) Repeat the identification with noisy output data (variance of noise is 10% of the RMS of signal): Model validation: compare FRFs of the identified model and of the exact discrete time model: discrete time model identified discrete time model amplitude [db] discrete time model identified discrete time model phase [degrees] frequency [Hz] Remark: inaccurate model: improve the model accuracy by filtering the input/output data!

64 Introduction to system identification 63 Time domain identification for θ 0 = 0 (6) Repeat the identification with same noisy output data, but first filter the data: [Bf,Af]=butter(4,2/fs); Tcf = filter(bf,af,tc); thetaf = filter(bf,af,theta);

65 Introduction to system identification 64 Time domain identification for θ 0 = 0 (7) Model validation: compare FRFs of the identified model and of the exact discrete time model: discrete time model identified discrete time model amplitude [db] discrete time model identified discrete time model phase [degrees] frequency [Hz] Remark: model is again very accurate...

66 Introduction to system identification 65 Time domain identification for θ 0 = 0 (8) Model validation: compare FRFs of the identified model and of the exact discrete time model, with the empirical transfer function estimate: discrete time model identified discrete time model estimated FRF amplitude [db] discrete time model identified discrete time model estimated FRF phase [degrees] frequency [Hz]

67 Introduction to system identification 66 Time domain identification for θ 0 = 0 (9) Time domain model validation: compare measured output with predicted output (using model identified with filtered noisy measurement) measured predicted Output [rad] Samples Prediction error [rad] Samples Remark: another data set is used (only short section is shown)!

68 Introduction to system identification 67 Time domain identification for θ 0 = 45 Repeat the whole procedure with the other data set: first identification with noise-free measurements Model validation: compare FRFs of the identified model and of the exact discrete time model: discrete time model identified discrete time model amplitude [db] discrete time model identified discrete time model phase [degrees] frequency [Hz] Remark: bad results due to DC-offset in both input and output measurements

69 Introduction to system identification 68 Time domain identification for θ 0 = 45 (2) Remove the DC-offset of both signals and repeat the identification with noise-free data Model validation: compare FRFs of the identified model and of the exact discrete time model: discrete time model identified discrete time model amplitude [db] discrete time model identified discrete time model phase [degrees] frequency [Hz]

70 Introduction to system identification 69 Time domain identification for θ 0 = 45 (3) Identification with noisy output data: filter i/o-data with same low-pass filter, remove the DC-offset of both signals and repeat the identification Model validation: compare FRFs of the identified model and of the exact discrete time model: discrete time model identified discrete time model amplitude [db] discrete time model identified discrete time model phase [degrees] frequency [Hz]

71 Introduction to system identification 70 Time domain identification for θ 0 = 45 (4) Model validation: compare FRFs of the identified model and of the exact discrete time model, with the empirical transfer function estimate: amplitude [db] discrete time model identified discrete time model estimated FRF phase [degrees] discrete time model identified discrete time model estimated FRF frequency [Hz]

72 Introduction to system identification 71 Time domain identification for θ 0 = 45 (5) Time domain model validation: compare measured output with predicted output (using model identified with filtered noisy measurement) measured predicted Output [rad] Samples Prediction error [rad] Samples Again another data set is used (only short section is shown)!

73 Introduction to system identification 72 Comparison of models for θ 0 = 0 and amplitude [db] model at 0 degrees model at 45 degrees phase [degrees] model at 0 degrees model at 45 degrees frequency [Hz]

74 Introduction to system identification 73 Conclusion Presented time domain identification method is only one example of many existing methods Simple to use (only measurement data is required), but it has bad properties Simple tricks to improve model accuracy are presented You have to be able to perform a time domain identification for systems similar to the one presented here...