Chapter 6: Nonparametric Time- and Frequency-Domain Methods. Problems presented by Uwe
|
|
- Aldous Hodges
- 5 years ago
- Views:
Transcription
1 System Identification written by L. Ljung, Prentice Hall PTR, 1999 Chapter 6: Nonparametric Time- and Frequency-Domain Methods Problems presented by Uwe System Identification Problems Chapter 6 p. 1/33
2 Outline Introduction Problem 6G.3 Problem 6E.1 Problem 6E.3 System Identification Problems Chapter 6 p. 2/33
3 Transient-Response Analysis Impulse-Response: If a strictly stable LTI system is subjected to a pulse input { α, t = 0 y(t) = G 0 (q)u(t) + v(t), u(t) = 0, t 0 System Identification Problems Chapter 6 p. 3/33
4 Transient-Response Analysis Impulse-Response: If a strictly stable LTI system is subjected to a pulse input { α, t = 0 y(t) = G 0 (q)u(t) + v(t), u(t) = 0, t 0 the output will be y(t) = k=1 g(k)q k u(t) + v(t) = k=1 g(k)u(t k) + v(t) = αg 0 (t) + v(t) System Identification Problems Chapter 6 p. 3/33
5 Transient-Response Analysis Impulse-Response: If a strictly stable LTI system is subjected to a pulse input { α, t = 0 y(t) = G 0 (q)u(t) + v(t), u(t) = 0, t 0 the output will be y(t) = αg 0 (t) + v(t) If the noise level is low, estimates for the impulse-response coefficients can be determined by ĝ 0 (t) = y(t) α with the errors ε(t) = v(t) α System Identification Problems Chapter 6 p. 3/33
6 Transient-Response Analysis Step-Response: If a strictly stable LTI system is subjected to a step input { α, t 0 y(t) = G 0 (q)u(t) + v(t), u(t) = 0, t < 0 System Identification Problems Chapter 6 p. 4/33
7 Transient-Response Analysis Step-Response: If a strictly stable LTI system is subjected to a step input { α, t 0 y(t) = G 0 (q)u(t) + v(t), u(t) = 0, t < 0 the output will be y(t) = k=1 g(k)q k u(t) + v(t) = k=1 g(k)u(t k) + v(t) = α t k=1 g 0(k) + v(t) System Identification Problems Chapter 6 p. 4/33
8 Transient-Response Analysis Step-Response: If a strictly stable LTI system is subjected to a step input { α, t 0 y(t) = G 0 (q)u(t) + v(t), u(t) = 0, t < 0 the output will be y(t) = α t k=1 g 0(k) + v(t) Since the step response is the integral of the impulseresponse, the estimates for g 0 (k) can be determined as y(t) y(t 1) v(t) v(t 1) ĝ 0 (t) = with ε(t) = 1 α 1 α System Identification Problems Chapter 6 p. 4/33
9 Constraints in Transient-Response Analysis Many physical processes do not allow pulse / step inputs of such an amplitude that the error is insignificant compared to the impulse-response coefficients. However, the step-response is suitable to determine some model characteristics, such as delay time, static gain, and dominating time constants. System Identification Problems Chapter 6 p. 5/33
10 Correlation Analysis Considering a strictly stable LTI system, where the input is a quasi-stationary sequence with R u (τ) = E {u(t)u(t τ)} R uv (τ) = E {u(t)v(t τ)} 0 (open-loop!) System Identification Problems Chapter 6 p. 6/33
11 Correlation Analysis Considering a strictly stable LTI system, where the input is a quasi-stationary sequence with R u (τ) = E {u(t)u(t τ)} R uv (τ) = E {u(t)v(t τ)} 0 (open-loop!) The input-output covariance function will be E {y(t)u(t τ)} = G 0 (q)e {u(t)u(t τ)} +E {v(t)u(t τ)} R yu (τ) = G 0 (q)r u (τ) = k=1 g 0(k)R u (k τ) (time domain) System Identification Problems Chapter 6 p. 6/33
12 Correlation Analysis (cont d) The input-output covariance function will be R yu (τ) = k=1 g 0(k)R u (k τ) If u(t) is chosen as white noise, it follows for the estimate R yu (τ) = λ g 0 (τ) = ĝ 0 (τ) = ˆR N yu (τ)/λ where ˆR N yu (τ) = 1 N N t=τ y(t)u(t τ). System Identification Problems Chapter 6 p. 7/33
13 Correlation Analysis (cont d) The input-output covariance function will be R yu (τ) = k=1 g 0(k)R u (k τ) If u(t) is chosen as white noise, it follows for the estimate R yu (τ) = λ g 0 (τ) = ĝ 0 (τ) = ˆR N yu (τ)/λ where ˆR N yu (τ) = 1 N N t=τ y(t)u(t τ). If the input is not white noise, we may estimate ˆR N u (τ) = 1 N N t=τ and solve u(t)u(t τ) ˆR N yu (τ) = M k=1 ĝ0(k) ˆR N u (k τ) = ĝ 0(k) System Identification Problems Chapter 6 p. 7/33
14 Frequency-Response Analysis Applying an input sinusoid to the strictly stable LTI system allows to determine y(t) = G 0 (q)u(t) + v(t) the amplitude α ĜN(e jω ) and phase shift ˆϕ N = argĝ N (e jω ) of the resulting output sinusoid for a number of frequencies. The influence of the noise component v(t) can be eliminated by the correlation method or filtering. System Identification Problems Chapter 6 p. 8/33
15 Empirical Transfer-Function Estimate (ETFE) In the frequency domain, the transfer-function estimate is Ĝ N (e jω ) = Y N(ω) U N (ω) based on data over the interval 1 t N, where Y N (ω) = 1 N N t=1 U N (ω) = 1 N N t=1 y(t)e jωt u(t)e jωt System Identification Problems Chapter 6 p. 9/33
16 Properties of the ETFE Given the strictly stable LTI system y(t) = G 0 (q)u(t) + v(t) it follows for the data interval 1 t N Y N (ω) = G 0 (e jω )U N (ω) + ρ 1 (N) + V N (ω), System Identification Problems Chapter 6 p. 10/33
17 Properties of the ETFE Given the strictly stable LTI system y(t) = G 0 (q)u(t) + v(t) it follows for the data interval 1 t N Y N (ω) = G 0 (e jω )U N (ω) + ρ 1 (N) + V N (ω), where ρ 1 (N) shows up as rest term due to the finite summation ρ 1 (N) C 1 N. If the input is periodic with period N, ρ 1 (N) is zero. System Identification Problems Chapter 6 p. 10/33
18 Properties of the ETFE (cont d) Ĝ N (e jω ) = Y N(ω) U N (ω) = G 0(e jω ) + ρ 1(N) U N (ω) + V N(ω) U N (ω) System Identification Problems Chapter 6 p. 11/33
19 Properties of the ETFE (cont d) Ĝ N (e jω ) = Y N(ω) U N (ω) = G 0(e jω ) + ρ 1(N) U N (ω) + V N(ω) U N (ω) Mean value (most expected value): Since the disturbance v(t) is assumed to have zero mean, E {V N (ω)} = 0, ω where ρ 1 (N) C 1. N } E {Ĝ N (e jω ) = G 0 (e jω ) + ρ 1(N) U N (ω) That means, the ETFE is asymptotically unbiased. System Identification Problems Chapter 6 p. 11/33
20 Properties of the ETFE (cont d) The covariance function of the disturbance can be computed as R v (τ) = E {v(t)v(t τ)} = E {V N (ω)v N ( ξ)} System Identification Problems Chapter 6 p. 12/33
21 Properties of the ETFE (cont d) The covariance function of the disturbance can be computed as R v (τ) = E {v(t)v(t τ)} = E {V N (ω)v N ( ξ)} E {V N (ω)v N ( ξ)} = Φ v (ω) + ρ 2 (N), = ρ 2 (N), if ξ = ω if ξ ω = 2πk N k = 1, 2,..., N 1 where ρ 2 (N) C 2. N System Identification Problems Chapter 6 p. 12/33
22 Properties of the ETFE (cont d) Covariance of the estimation error: ]} E {[Ĝ N (e jω ) G 0 (e )] [Ĝ jω N (e jξ ) G 0 (e jξ ) 1 [Φ U N (ω) 2 v (ω) + ρ 2 (N)], if ξ = ω = 1 U N (ω)u N ( ξ) ρ 2(N), where ρ 2 (N) C 2. N = if ξ ω = 2πk N k = 1, 2,..., N 1 That means, the ETFE has a variance of Φ v (ω)/ U N (ω) 2 at each frequency, but the estimates at different frequencies are asymptotically uncorrelated. System Identification Problems Chapter 6 p. 13/33
23 Problems with the ETFE The variance of the ETFE does not decay with increasing N. We determine as many independent estimates as we have data points. Linearity of the true system is the only assumption we made, i.e. the systems properties at different frequencies may be totally unrelated. Assuming that the values of the true transfer function at different frequencies are related will improve the poor variance properties. The result will be data and information compression. System Identification Problems Chapter 6 p. 14/33
24 Spectral Analysis Smoothing the ETFE The ETFE Ĝ N (e jω ) represents unbiased and uncorrelated estimates of G 0 (e jω ) for ω = 2πk/N with the variance Φ v (ω)/ U N (ω) 2. System Identification Problems Chapter 6 p. 15/33
25 Spectral Analysis Smoothing the ETFE The ETFE Ĝ N (e jω ) represents unbiased and uncorrelated estimates of G 0 (e jω ) for ω = 2πk/N with the variance Φ v (ω)/ U N (ω) 2. Provided that the frequency distance 2π/N is small compared to how quickly G 0 (e jω ) changes, we can assume G 0 (e jω ) to be constant over the interval 2πk 1 N = ω 0 ω < ω < ω 0 + ω = 2πk 2 N Forming a weighted average of the "measurements" for this interval, will estimate the constant G 0 (e jω 0 ). System Identification Problems Chapter 6 p. 15/33
26 Spectral Analysis Smoothing the ETFE (cont d) Weighting according to the inverse variance gives Ĝ N (e jω 0 ) = ω0 + ω ω 0 ω α(ξ)ĝ N (e jξ )dξ ω0 + ω ω 0 ω α(ξ)dξ, α(ξ) = U N (ξ) 2 Φ v (ξ) System Identification Problems Chapter 6 p. 16/33
27 Spectral Analysis Smoothing the ETFE (cont d) Weighting according to the inverse variance gives Ĝ N (e jω 0 ) = ω0 + ω ω 0 ω α(ξ)ĝ N (e jξ )dξ ω0 + ω ω 0 ω α(ξ)dξ, α(ξ) = U N (ξ) 2 Φ v (ξ) If the transfer function G 0 (e jω ) is not constant over the interval, additional weighting close to ω 0 is required. Ĝ N (e jω 0 ) = π π W γ(ξ ω 0 ) U N (ξ) 2 Ĝ N (e jξ )dξ π π W γ(ξ ω 0 ) U N (ξ) 2 dξ W γ (ξ) is a frequency window centered in ξ = 0, with γ as shape parameter. Φ v (ξ) is assumed constant around ω 0. System Identification Problems Chapter 6 p. 16/33
28 Spectral Analysis Smoothing the ETFE Example y(t) 1.5y(t 1)+0.7y(t 2) = u(t 1)+0.5u(t 2)+e(t) System Identification Problems Chapter 6 p. 17/33
29 Spectral Analysis Smoothing the ETFE Example System Identification Problems Chapter 6 p. 18/33
30 Relation of the ETFE to the Spectral Estimates Provided that u(t) and v(t) are independent, the cross spectrum between input and output is defined as Φ yu (ω) = G(e jω )Φ u (ω) This formula can also be applied to the smoothed ETFE Ĝ N (e jω 0 ) = ˆΦ N yu (ω 0) ˆΦ N u (ω 0) with the corresponding spectral estimates ˆΦ N yu (ω 0) = π π W γ(ξ ω 0 ) U N (ξ) 2 Ĝ N (e jξ )dξ ˆΦ N u (ω 0) = π π W γ(ξ ω 0 ) U N (ξ) 2 dξ System Identification Problems Chapter 6 p. 19/33
31 Estimating the Disturbance Spectrum Analogous to the input spectrum, we can write for the disturbance spectrum ˆΦ N v (ω) = π π W γ (ξ ω) V N (ξ) 2 dξ If v(t) is not measurable, an estimate can be used by ˆΦ N v (ω) = π π ˆv(t) = y(t) Ĝ N (q)u(t) W γ (ξ ω) Y N (ξ) Ĝ N (e jξ )U N (ξ) 2 dξ System Identification Problems Chapter 6 p. 20/33
32 Properties of the estimates Ĝ N (e jω 0 ) and Φ N v (ω) The properties are dependent on the shaping parameter γ: large γ generates a narrow window with small bias but high variance small γ generates a wide window with high bias but small variance System Identification Problems Chapter 6 p. 21/33
33 Problem 6G.3 The amplitude of the ETFE appears to be systematically larger than the true amplitude, despite the fact that the ETFE is unbiased (cf. slide 11). However, Ĝ being an unbiased estimate of G 0 does not imply that Ĝ is an unbiased estimate of G 0. In fact, prove that { E Ĝ N (e jω ) 2} = G0 (e jω ) 2 + Φ v(ω) asymptotically for large N. U N (ω) 2 System Identification Problems Chapter 6 p. 22/33
34 Problem 6G.3 The ETFE is defined as (cf. slide 11) Ĝ N (e jω ) = Y N(ω) U N (ω) = G 0(e jω ) + ρ 1(N) U N (ω) + V N(ω) U N (ω) The magnitude of the ETFE can be determined by Ĝ N (e jω ) 2 = Ĝ N (e jω )Ĝ N (e jω ) ( ) = G 0 (e jω ) + ρ 1(N) U N (ω) + V N(ω) U N ( (ω) ) G 0 (e jω ) + ρ 1(N) U N ( ω) + V N( ω) U N ( ω) System Identification Problems Chapter 6 p. 23/33
35 Problem 6G.3 The ETFE is defined as (cf. slide 11) Ĝ N (e jω ) = Y N(ω) U N (ω) = G 0(e jω ) + ρ 1(N) U N (ω) + V N(ω) U N (ω) The magnitude of the ETFE can be determined by Ĝ N (e jω ) 2 = Ĝ N (e jω )Ĝ N (e jω ) ( ) = G 0 (e jω ) + ρ 1(N) U N (ω) + V N(ω) U N ( (ω) G 0 (e jω ) + ρ 1(N) = G0 (e jω ) 2 ( ) + G 0 (e jω ) ρ1 (N) U N ( ω) + V N( ω) U N ( ω) U N ( ω) + V N( ω) U N ( ω) + ρ 1(N) U N (ω) G 0(e jω ) + ρ2 1(N) + ρ 1(N)V N ( ω) U N (ω) 2 U N (ω) 2 + V N(ω) U N (ω) G 0(e jω ) + ρ 1(N)V N (ω) + V N(ω)V N ( ω) U N (ω) 2 U N (ω) 2 ) System Identification Problems Chapter 6 p. 23/33
36 Problem 6G.3 Taking expectation with respect to v(t) (cf. slide 12): { E Ĝ N (e jω ) 2} = G0 (e jω ) 2 + G 0(e jω )ρ 1 (N) + G 0(e jω )ρ 1 (N) U N (ω) + Φ v(ω)+ρ 2 (N) U N (ω) 2 U N ( ω) + ρ2 1(N) U N (ω) 2 System Identification Problems Chapter 6 p. 24/33
37 Problem 6G.3 Taking expectation with respect to v(t) (cf. slide 12): { E Ĝ N (e jω ) 2} = G0 (e jω ) 2 + G 0(e jω )ρ 1 (N) + G 0(e jω )ρ 1 (N) U N (ω) + Φ v(ω)+ρ 2 (N) U N (ω) 2 U N ( ω) + ρ2 1(N) U N (ω) 2 For large N, the function becomes asymptotically to { E Ĝ N (e jω ) 2} = G0 (e jω ) 2 + Φ v(ω) U N (ω) 2 It means that the magnitude of the ETFE Ĝ N (e jω ) is an biased estimate of G0 (e jω ), where Φ v (ω)/ U N (ω) 2 represents the variance. System Identification Problems Chapter 6 p. 24/33
38 Problem 6E.1 Determine an estimate for G 0 (e jω ) based on the impulse-response estimates ĝ 0 (t) (cf. slide 3). Show that the estimate coincides with the ETFE. System Identification Problems Chapter 6 p. 25/33
39 Problem 6E.1 Determine an estimate for G 0 (e jω ) based on the impulse-response estimates ĝ 0 (t) (cf. slide 3). Show that the estimate coincides with the ETFE. Estimates for the impulse-response coefficients are ĝ 0 (t) = y(t) α The transfer function G 0 (q) is defined as G(q) = g(k)q k = G(e jω ) = k=1 k=1 g(k)e jωk which corresponds to G(e jω ) in the frequency domain. System Identification Problems Chapter 6 p. 25/33
40 Problem 6E.1 The transfer function estimate for the data interval [1, N] will be N Ĝ N (e jω ) = g(k)e jωk k=1 Using the impulse-response estimates, it follows N N Ĝ N (e jω ) = ĝ(k)e jωk y(k) = α e jωk k=1 = 1 α N k=1 y(k)e jωk k=1 System Identification Problems Chapter 6 p. 26/33
41 Problem 6E.1 Since the pulse input is defined as { α, t = 0 u(t) = 0, t 0 the ETFE will be Ĝ N (e jω ) = Y N(ω) U N (ω) = = N 1 k=0 y(k)e jωk α 1 N 1 N k=0 y(k)e jωk 1 N 1 N k=0 u(k)e jωk which is the same as the transfer function estimate based on the estimated impulse-response coefficients. System Identification Problems Chapter 6 p. 27/33
42 Problem 6E.3 Let ω k, k = 1,..., M, be independent random variables, all with mean values 1 and variances E { (ω k 1) 2} = λ k. Consider the weighted sum ω = M k=1 α k ω k Determine α k, k = 1,..., M, so that (a) E {ω} = 1. (b) E { (ω 1) 2} is minimized. System Identification Problems Chapter 6 p. 28/33
43 Problem 6E.3 The mean values of ω k are E {ω k } = 1 According to (a), it follows for the expectation of ω M M E {ω} = α k E {ω k } = α k = 1 k=1 k=1 System Identification Problems Chapter 6 p. 29/33
44 Problem 6E.3 The mean values of ω k are E {ω k } = 1 According to (a), it follows for the expectation of ω M M E {ω} = α k E {ω k } = α k = 1 k=1 k=1 The variance of the weighted sum ω equals to E { { (ω 1) 2} [ M = E k=1 α kω k M { [ M ] } 2 = E k=1 α k (ω k 1) k=1 α k ] 2 } System Identification Problems Chapter 6 p. 29/33
45 Problem 6E.3 Opening the bracket, it follows E { (ω 1) 2} { M = E k=1 α2 k (ω k 1) 2 +2 M 1 k=1 α k(ω k 1) } M i=k+1 α i(ω i 1) System Identification Problems Chapter 6 p. 30/33
46 Problem 6E.3 Opening the bracket, it follows E { (ω 1) 2} { M = E k=1 α2 k (ω k 1) 2 +2 M 1 k=1 α k(ω k 1) } M i=k+1 α i(ω i 1) Applying the variances E { (ω k 1) 2} = λ k and the covariances E {(ω k 1)(ω i 1)} = 0, for k i provided that ω k are independent, we obtain E { (ω 1) 2} M = α 2 k λ k k=1 System Identification Problems Chapter 6 p. 30/33
47 Problem 6E.3 The minimization problem, regarding the variance for the weighted sum ω, will be E { (ω 1) 2} M = α 2 k λ k min k=1 where the following condition has to be fulfilled: M E {ω} = α k = 1 k=1 System Identification Problems Chapter 6 p. 31/33
48 Problem 6E.3 The minimization problem, regarding the variance for the weighted sum ω, will be E { (ω 1) 2} M = α 2 k λ k min k=1 where the following condition has to be fulfilled: M E {ω} = α k = 1 k=1 The minimum can be found using Lagrangian multipliers ( M M ) L = α 2 k λ k + γ α k 1 k=1 k=1 System Identification Problems Chapter 6 p. 31/33
49 Problem 6E.3 The Lagrangian L leads to the system of equations L α k = 2α k λ k + γ = 0 L γ = M k=1 α k 1 = 0 System Identification Problems Chapter 6 p. 32/33
50 Problem 6E.3 The Lagrangian L leads to the system of equations L α k = 2α k λ k + γ = 0 L γ = M k=1 α k 1 = 0 where the solutions can be computed by α k = γ 1 2 λ k solving the following equation for γ M α k = 1 = γ 2 k=1 M k=1 1 λ k System Identification Problems Chapter 6 p. 32/33
51 Problem 6E.3 For example if M = 3, the coefficients α k are computed as γ = α 1 = α 2 = α 3 = λ λ λ 3 1 λ 1 λ λ λ λ 2 λ λ λ 3 1 λ 3 1 λ λ λ 3 System Identification Problems Chapter 6 p. 33/33
Non-parametric identification
Non-parametric Non-parametric Transient Step-response using Spectral Transient Correlation Frequency function estimate Spectral System Identification, SSY230 Non-parametric 1 Non-parametric Transient Step-response
More information6.435, System Identification
System Identification 6.435 SET 3 Nonparametric Identification Munther A. Dahleh 1 Nonparametric Methods for System ID Time domain methods Impulse response Step response Correlation analysis / time Frequency
More informationECE 636: Systems identification
ECE 636: Systems identification Lectures 7 8 onparametric identification (continued) Important distributions: chi square, t distribution, F distribution Sampling distributions ib i Sample mean If the variance
More informationEL1820 Modeling of Dynamical Systems
EL1820 Modeling of Dynamical Systems Lecture 10 - System identification as a model building tool Experiment design Examination and prefiltering of data Model structure selection Model validation Lecture
More informationIntroduction to system identification
Introduction to system identification Jan Swevers July 2006 0-0 Introduction to system identification 1 Contents of this lecture What is system identification Time vs. frequency domain identification Discrete
More informationSystem Identification
System Identification Lecture 4: Transfer function averaging and smoothing Roy Smith 28-- 4. Averaging Multiple estimates Multiple experiments: u r pk, y r pk, r,..., R, and k,..., K. Multiple estimates
More informationf-domain expression for the limit model Combine: 5.12 Approximate Modelling What can be said about H(q, θ) G(q, θ ) H(q, θ ) with
5.2 Approximate Modelling What can be said about if S / M, and even G / G? G(q, ) H(q, ) f-domain expression for the limit model Combine: with ε(t, ) =H(q, ) [y(t) G(q, )u(t)] y(t) =G (q)u(t) v(t) We know
More informationEL1820 Modeling of Dynamical Systems
EL1820 Modeling of Dynamical Systems Lecture 9 - Parameter estimation in linear models Model structures Parameter estimation via prediction error minimization Properties of the estimate: bias and variance
More informationSystem Modeling and Identification CHBE 702 Korea University Prof. Dae Ryook Yang
System Modeling and Identification CHBE 702 Korea University Prof. Dae Ryook Yang 1-1 Course Description Emphases Delivering concepts and Practice Programming Identification Methods using Matlab Class
More informationEE531 (Semester II, 2010) 6. Spectral analysis. power spectral density. periodogram analysis. window functions 6-1
6. Spectral analysis EE531 (Semester II, 2010) power spectral density periodogram analysis window functions 6-1 Wiener-Khinchin theorem: Power Spectral density if a process is wide-sense stationary, the
More informationTime series models in the Frequency domain. The power spectrum, Spectral analysis
ime series models in the Frequency domain he power spectrum, Spectral analysis Relationship between the periodogram and the autocorrelations = + = ( ) ( ˆ α ˆ ) β I Yt cos t + Yt sin t t= t= ( ( ) ) cosλ
More informationIdentification, Model Validation and Control. Lennart Ljung, Linköping
Identification, Model Validation and Control Lennart Ljung, Linköping Acknowledgment: Useful discussions with U Forssell and H Hjalmarsson 1 Outline 1. Introduction 2. System Identification (in closed
More informationFrequency-Domain C/S of LTI Systems
Frequency-Domain C/S of LTI Systems x(n) LTI y(n) LTI: Linear Time-Invariant system h(n), the impulse response of an LTI systems describes the time domain c/s. H(ω), the frequency response describes the
More informationLinear Approximations of Nonlinear FIR Systems for Separable Input Processes
Linear Approximations of Nonlinear FIR Systems for Separable Input Processes Martin Enqvist, Lennart Ljung Division of Automatic Control Department of Electrical Engineering Linköpings universitet, SE-581
More informationFrequency-Domain Robust Control Toolbox
Frequency-Domain Robust Control Toolbox Alireza Karimi Abstract A new frequency-domain robust control toolbox is introduced and compared with some features of the robust control toolbox of Matlab. A summary
More informationStochastic Process II Dr.-Ing. Sudchai Boonto
Dr-Ing Sudchai Boonto Department of Control System and Instrumentation Engineering King Mongkuts Unniversity of Technology Thonburi Thailand Random process Consider a random experiment specified by the
More informationEECE Adaptive Control
EECE 574 - Adaptive Control Recursive Identification in Closed-Loop and Adaptive Control Guy Dumont Department of Electrical and Computer Engineering University of British Columbia January 2010 Guy Dumont
More informationFurther Results on Model Structure Validation for Closed Loop System Identification
Advances in Wireless Communications and etworks 7; 3(5: 57-66 http://www.sciencepublishinggroup.com/j/awcn doi:.648/j.awcn.735. Further esults on Model Structure Validation for Closed Loop System Identification
More informationSubspace-based Identification
of Infinite-dimensional Multivariable Systems from Frequency-response Data Department of Electrical and Electronics Engineering Anadolu University, Eskişehir, Turkey October 12, 2008 Outline 1 2 3 4 Noise-free
More informationEECE Adaptive Control
EECE 574 - Adaptive Control Basics of System Identification Guy Dumont Department of Electrical and Computer Engineering University of British Columbia January 2010 Guy Dumont (UBC) EECE574 - Basics of
More informationA summary of Modeling and Simulation
A summary of Modeling and Simulation Text-book: Modeling of dynamic systems Lennart Ljung and Torkel Glad Content What re Models for systems and signals? Basic concepts Types of models How to build a model
More information8. Filter Bank Methods Filter bank methods assume that the true spectrum φ(ω) is constant (or nearly so) over the band [ω βπ, ω + βπ], for some β 1.
8. Filter Bank Methods Filter bank methods assume that the true spectrum φ(ω) is constant (or nearly so) over the band [ω βπ, ω + βπ], for some β 1. Usefull if it is not known that spectrum has special
More informationChapter 5 Frequency Domain Analysis of Systems
Chapter 5 Frequency Domain Analysis of Systems CT, LTI Systems Consider the following CT LTI system: xt () ht () yt () Assumption: the impulse response h(t) is absolutely integrable, i.e., ht ( ) dt< (this
More informationOn Moving Average Parameter Estimation
On Moving Average Parameter Estimation Niclas Sandgren and Petre Stoica Contact information: niclas.sandgren@it.uu.se, tel: +46 8 473392 Abstract Estimation of the autoregressive moving average (ARMA)
More informationSystem Identification & Parameter Estimation
System Identification & Parameter Estimation Wb3: SIPE lecture Correlation functions in time & frequency domain Alfred C. Schouten, Dept. of Biomechanical Engineering (BMechE), Fac. 3mE // Delft University
More informationLTI Approximations of Slightly Nonlinear Systems: Some Intriguing Examples
LTI Approximations of Slightly Nonlinear Systems: Some Intriguing Examples Martin Enqvist, Lennart Ljung Division of Automatic Control Department of Electrical Engineering Linköpings universitet, SE-581
More informationII. Nonparametric Spectrum Estimation for Stationary Random Signals - Non-parametric Methods -
II. onparametric Spectrum Estimation for Stationary Random Signals - on-parametric Methods - - [p. 3] Periodogram - [p. 12] Periodogram properties - [p. 23] Modified periodogram - [p. 25] Bartlett s method
More informationLecture 1: Introduction to System Modeling and Control. Introduction Basic Definitions Different Model Types System Identification
Lecture 1: Introduction to System Modeling and Control Introduction Basic Definitions Different Model Types System Identification What is Mathematical Model? A set of mathematical equations (e.g., differential
More informationMassachusetts Institute of Technology Department of Electrical Engineering and Computer Science : Discrete-Time Signal Processing
Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.34: Discrete-Time Signal Processing OpenCourseWare 006 ecture 8 Periodogram Reading: Sections 0.6 and 0.7
More informationChapter 5 Frequency Domain Analysis of Systems
Chapter 5 Frequency Domain Analysis of Systems CT, LTI Systems Consider the following CT LTI system: xt () ht () yt () Assumption: the impulse response h(t) is absolutely integrable, i.e., ht ( ) dt< (this
More informationExplanations and Discussion of Some Laplace Methods: Transfer Functions and Frequency Response. Y(s) = b 1
Engs 22 p. 1 Explanations Discussion of Some Laplace Methods: Transfer Functions Frequency Response I. Anatomy of Differential Equations in the S-Domain Parts of the s-domain solution. We will consider
More informationIdentification of Linear Systems
Identification of Linear Systems Johan Schoukens http://homepages.vub.ac.be/~jschouk Vrije Universiteit Brussel Department INDI /67 Basic goal Built a parametric model for a linear dynamic system from
More informationStochastic Dynamics of SDOF Systems (cont.).
Outline of Stochastic Dynamics of SDOF Systems (cont.). Weakly Stationary Response Processes. Equivalent White Noise Approximations. Gaussian Response Processes as Conditional Normal Distributions. Stochastic
More informationLecture 9 Infinite Impulse Response Filters
Lecture 9 Infinite Impulse Response Filters Outline 9 Infinite Impulse Response Filters 9 First-Order Low-Pass Filter 93 IIR Filter Design 5 93 CT Butterworth filter design 5 93 Bilinear transform 7 9
More informationIMPROVEMENTS IN MODAL PARAMETER EXTRACTION THROUGH POST-PROCESSING FREQUENCY RESPONSE FUNCTION ESTIMATES
IMPROVEMENTS IN MODAL PARAMETER EXTRACTION THROUGH POST-PROCESSING FREQUENCY RESPONSE FUNCTION ESTIMATES Bere M. Gur Prof. Christopher Niezreci Prof. Peter Avitabile Structural Dynamics and Acoustic Systems
More informationSolution: K m = R 1 = 10. From the original circuit, Z L1 = jωl 1 = j10 Ω. For the scaled circuit, L 1 = jk m ωl 1 = j10 10 = j100 Ω, Z L
Problem 9.9 Circuit (b) in Fig. P9.9 is a scaled version of circuit (a). The scaling process may have involved magnitude or frequency scaling, or both simultaneously. If R = kω gets scaled to R = kω, supply
More informationAdaptive beamforming. Slide 2: Chapter 7: Adaptive array processing. Slide 3: Delay-and-sum. Slide 4: Delay-and-sum, continued
INF540 202 Adaptive beamforming p Adaptive beamforming Sven Peter Näsholm Department of Informatics, University of Oslo Spring semester 202 svenpn@ifiuiono Office phone number: +47 22840068 Slide 2: Chapter
More informationLecture 8 Finite Impulse Response Filters
Lecture 8 Finite Impulse Response Filters Outline 8. Finite Impulse Response Filters.......................... 8. oving Average Filter............................... 8.. Phase response...............................
More informationAspects of Continuous- and Discrete-Time Signals and Systems
Aspects of Continuous- and Discrete-Time Signals and Systems C.S. Ramalingam Department of Electrical Engineering IIT Madras C.S. Ramalingam (EE Dept., IIT Madras) Networks and Systems 1 / 45 Scaling the
More informationChapter 6 THE SAMPLING PROCESS 6.1 Introduction 6.2 Fourier Transform Revisited
Chapter 6 THE SAMPLING PROCESS 6.1 Introduction 6.2 Fourier Transform Revisited Copyright c 2005 Andreas Antoniou Victoria, BC, Canada Email: aantoniou@ieee.org July 14, 2018 Frame # 1 Slide # 1 A. Antoniou
More informationTime domain identification, frequency domain identification. Equivalencies! Differences?
Time domain identification, frequency domain identification. Equivalencies! Differences? J. Schoukens, R. Pintelon, and Y. Rolain Vrije Universiteit Brussel, Department ELEC, Pleinlaan, B5 Brussels, Belgium
More informationIV. Covariance Analysis
IV. Covariance Analysis Autocovariance Remember that when a stochastic process has time values that are interdependent, then we can characterize that interdependency by computing the autocovariance function.
More informationCONTROL SYSTEMS, ROBOTICS, AND AUTOMATION - Vol. V - Prediction Error Methods - Torsten Söderström
PREDICTIO ERROR METHODS Torsten Söderström Department of Systems and Control, Information Technology, Uppsala University, Uppsala, Sweden Keywords: prediction error method, optimal prediction, identifiability,
More informationIDENTIFICATION FOR CONTROL
IDENTIFICATION FOR CONTROL Raymond A. de Callafon, University of California San Diego, USA Paul M.J. Van den Hof, Delft University of Technology, the Netherlands Keywords: Controller, Closed loop model,
More informationFrequency estimation by DFT interpolation: A comparison of methods
Frequency estimation by DFT interpolation: A comparison of methods Bernd Bischl, Uwe Ligges, Claus Weihs March 5, 009 Abstract This article comments on a frequency estimator which was proposed by [6] and
More informationStochastic Processes. M. Sami Fadali Professor of Electrical Engineering University of Nevada, Reno
Stochastic Processes M. Sami Fadali Professor of Electrical Engineering University of Nevada, Reno 1 Outline Stochastic (random) processes. Autocorrelation. Crosscorrelation. Spectral density function.
More informationELEC2400 Signals & Systems
ELEC2400 Signals & Systems Chapter 7. Z-Transforms Brett Ninnes brett@newcastle.edu.au. School of Electrical Engineering and Computer Science The University of Newcastle Slides by Juan I. Yu (jiyue@ee.newcastle.edu.au
More informationRational Implementation of Distributed Delay Using Extended Bilinear Transformations
Rational Implementation of Distributed Delay Using Extended Bilinear Transformations Qing-Chang Zhong zhongqc@ieee.org, http://come.to/zhongqc School of Electronics University of Glamorgan United Kingdom
More information7.2 Relationship between Z Transforms and Laplace Transforms
Chapter 7 Z Transforms 7.1 Introduction In continuous time, the linear systems we try to analyse and design have output responses y(t) that satisfy differential equations. In general, it is hard to solve
More informationStatistical signal processing
Statistical signal processing Short overview of the fundamentals Outline Random variables Random processes Stationarity Ergodicity Spectral analysis Random variable and processes Intuition: A random variable
More information14 - Gaussian Stochastic Processes
14-1 Gaussian Stochastic Processes S. Lall, Stanford 211.2.24.1 14 - Gaussian Stochastic Processes Linear systems driven by IID noise Evolution of mean and covariance Example: mass-spring system Steady-state
More informationProcess Control & Instrumentation (CH 3040)
First-order systems Process Control & Instrumentation (CH 3040) Arun K. Tangirala Department of Chemical Engineering, IIT Madras January - April 010 Lectures: Mon, Tue, Wed, Fri Extra class: Thu A first-order
More informationReview of Linear Time-Invariant Network Analysis
D1 APPENDIX D Review of Linear Time-Invariant Network Analysis Consider a network with input x(t) and output y(t) as shown in Figure D-1. If an input x 1 (t) produces an output y 1 (t), and an input x
More informationik () uk () Today s menu Last lecture Some definitions Repeatability of sensing elements
Last lecture Overview of the elements of measurement systems. Sensing elements. Signal conditioning elements. Signal processing elements. Data presentation elements. Static characteristics of measurement
More informationVIII. Coherence and Transfer Function Applications A. Coherence Function Estimates
VIII. Coherence and Transfer Function Applications A. Coherence Function Estimates Consider the application of these ideas to the specific problem of atmospheric turbulence measurements outlined in Figure
More informationLTI Systems (Continuous & Discrete) - Basics
LTI Systems (Continuous & Discrete) - Basics 1. A system with an input x(t) and output y(t) is described by the relation: y(t) = t. x(t). This system is (a) linear and time-invariant (b) linear and time-varying
More informationExam in Automatic Control II Reglerteknik II 5hp (1RT495)
Exam in Automatic Control II Reglerteknik II 5hp (1RT495) Date: August 4, 018 Venue: Bergsbrunnagatan 15 sal Responsible teacher: Hans Rosth. Aiding material: Calculator, mathematical handbooks, textbooks
More informationANNEX A: ANALYSIS METHODOLOGIES
ANNEX A: ANALYSIS METHODOLOGIES A.1 Introduction Before discussing supplemental damping devices, this annex provides a brief review of the seismic analysis methods used in the optimization algorithms considered
More informationImproving Convergence of Iterative Feedback Tuning using Optimal External Perturbations
Proceedings of the 47th IEEE Conference on Decision and Control Cancun, Mexico, Dec. 9-, 2008 Improving Convergence of Iterative Feedback Tuning using Optimal External Perturbations Jakob Kjøbsted Huusom,
More informationProbability Space. J. McNames Portland State University ECE 538/638 Stochastic Signals Ver
Stochastic Signals Overview Definitions Second order statistics Stationarity and ergodicity Random signal variability Power spectral density Linear systems with stationary inputs Random signal memory Correlation
More informationTimbral, Scale, Pitch modifications
Introduction Timbral, Scale, Pitch modifications M2 Mathématiques / Vision / Apprentissage Audio signal analysis, indexing and transformation Page 1 / 40 Page 2 / 40 Modification of playback speed Modifications
More informationEAS 305 Random Processes Viewgraph 1 of 10. Random Processes
EAS 305 Random Processes Viewgraph 1 of 10 Definitions: Random Processes A random process is a family of random variables indexed by a parameter t T, where T is called the index set λ i Experiment outcome
More informationLecture 3 January 23
EE 123: Digital Signal Processing Spring 2007 Lecture 3 January 23 Lecturer: Prof. Anant Sahai Scribe: Dominic Antonelli 3.1 Outline These notes cover the following topics: Eigenvectors and Eigenvalues
More informationECEN 420 LINEAR CONTROL SYSTEMS. Lecture 2 Laplace Transform I 1/52
1/52 ECEN 420 LINEAR CONTROL SYSTEMS Lecture 2 Laplace Transform I Linear Time Invariant Systems A general LTI system may be described by the linear constant coefficient differential equation: a n d n
More information-Digital Signal Processing- FIR Filter Design. Lecture May-16
-Digital Signal Processing- FIR Filter Design Lecture-17 24-May-16 FIR Filter Design! FIR filters can also be designed from a frequency response specification.! The equivalent sampled impulse response
More informationOn Identification of Cascade Systems 1
On Identification of Cascade Systems 1 Bo Wahlberg Håkan Hjalmarsson Jonas Mårtensson Automatic Control and ACCESS, School of Electrical Engineering, KTH, SE-100 44 Stockholm, Sweden. (bo.wahlberg@ee.kth.se
More informationSystems and Control Theory Lecture Notes. Laura Giarré
Systems and Control Theory Lecture Notes Laura Giarré L. Giarré 2017-2018 Lesson 7: Response of LTI systems in the transform domain Laplace Transform Transform-domain response (CT) Transfer function Zeta
More informationChap 2. Discrete-Time Signals and Systems
Digital Signal Processing Chap 2. Discrete-Time Signals and Systems Chang-Su Kim Discrete-Time Signals CT Signal DT Signal Representation 0 4 1 1 1 2 3 Functional representation 1, n 1,3 x[ n] 4, n 2 0,
More informationFRF parameter identification with arbitrary input sequence from noisy input output measurements
21st International Symposium on Mathematical Theory of Networks and Systems July 7-11, 214. FRF parameter identification with arbitrary input sequence from noisy input output measurements mberto Soverini
More informationPractical Spectral Estimation
Digital Signal Processing/F.G. Meyer Lecture 4 Copyright 2015 François G. Meyer. All Rights Reserved. Practical Spectral Estimation 1 Introduction The goal of spectral estimation is to estimate how the
More informationGeneralizing the DTFT!
The Transform Generaliing the DTFT! The forward DTFT is defined by X e jω ( ) = x n e jωn in which n= Ω is discrete-time radian frequency, a real variable. The quantity e jωn is then a complex sinusoid
More informationCorrelator I. Basics. Chapter Introduction. 8.2 Digitization Sampling. D. Anish Roshi
Chapter 8 Correlator I. Basics D. Anish Roshi 8.1 Introduction A radio interferometer measures the mutual coherence function of the electric field due to a given source brightness distribution in the sky.
More informationReliability Theory of Dynamically Loaded Structures (cont.)
Outline of Reliability Theory of Dynamically Loaded Structures (cont.) Probability Density Function of Local Maxima in a Stationary Gaussian Process. Distribution of Extreme Values. Monte Carlo Simulation
More information7. Find the Fourier transform of f (t)=2 cos(2π t)[u (t) u(t 1)]. 8. (a) Show that a periodic signal with exponential Fourier series f (t)= δ (ω nω 0
Fourier Transform Problems 1. Find the Fourier transform of the following signals: a) f 1 (t )=e 3 t sin(10 t)u (t) b) f 1 (t )=e 4 t cos(10 t)u (t) 2. Find the Fourier transform of the following signals:
More informationSignals & Systems. Lecture 5 Continuous-Time Fourier Transform. Alp Ertürk
Signals & Systems Lecture 5 Continuous-Time Fourier Transform Alp Ertürk alp.erturk@kocaeli.edu.tr Fourier Series Representation of Continuous-Time Periodic Signals Synthesis equation: x t = a k e jkω
More informationProperties of Open-Loop Controllers
Properties of Open-Loop Controllers Sven Laur University of Tarty 1 Basics of Open-Loop Controller Design Two most common tasks in controller design is regulation and signal tracking. Regulating controllers
More informationThe Local Polynomial Method for nonparametric system identification: improvements and experimentation
The Local Polynomial Method for nonparametric system identification: improvements and experimentation Michel Gevers, Rik Pintelon and Johan Schoukens Abstract The Local Polynomial Method (LPM) is a recently
More informationAtmospheric Flight Dynamics Example Exam 1 Solutions
Atmospheric Flight Dynamics Example Exam 1 Solutions 1 Question Figure 1: Product function Rūū (τ) In figure 1 the product function Rūū (τ) of the stationary stochastic process ū is given. What can be
More information8.2 Harmonic Regression and the Periodogram
Chapter 8 Spectral Methods 8.1 Introduction Spectral methods are based on thining of a time series as a superposition of sinusoidal fluctuations of various frequencies the analogue for a random process
More informationIdentification of ARX, OE, FIR models with the least squares method
Identification of ARX, OE, FIR models with the least squares method CHEM-E7145 Advanced Process Control Methods Lecture 2 Contents Identification of ARX model with the least squares minimizing the equation
More informationMassachusetts Institute of Technology
Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.011: Introduction to Communication, Control and Signal Processing QUIZ, April 1, 010 QUESTION BOOKLET Your
More informationReview of Frequency Domain Fourier Series: Continuous periodic frequency components
Today we will review: Review of Frequency Domain Fourier series why we use it trig form & exponential form how to get coefficients for each form Eigenfunctions what they are how they relate to LTI systems
More informationImproving performance and stability of MRI methods in closed-loop
Preprints of the 8th IFAC Symposium on Advanced Control of Chemical Processes The International Federation of Automatic Control Improving performance and stability of MRI methods in closed-loop Alain Segundo
More information2A1H Time-Frequency Analysis II Bugs/queries to HT 2011 For hints and answers visit dwm/courses/2tf
Time-Frequency Analysis II (HT 20) 2AH 2AH Time-Frequency Analysis II Bugs/queries to david.murray@eng.ox.ac.uk HT 20 For hints and answers visit www.robots.ox.ac.uk/ dwm/courses/2tf David Murray. A periodic
More informationECE 636: Systems identification
ECE 636: Systems identification Lectures 3 4 Random variables/signals (continued) Random/stochastic vectors Random signals and linear systems Random signals in the frequency domain υ ε x S z + y Experimental
More informationA system that is both linear and time-invariant is called linear time-invariant (LTI).
The Cooper Union Department of Electrical Engineering ECE111 Signal Processing & Systems Analysis Lecture Notes: Time, Frequency & Transform Domains February 28, 2012 Signals & Systems Signals are mapped
More informationReview of Fourier Transform
Review of Fourier Transform Fourier series works for periodic signals only. What s about aperiodic signals? This is very large & important class of signals Aperiodic signal can be considered as periodic
More informationLaboratory Project 2: Spectral Analysis and Optimal Filtering
Laboratory Project 2: Spectral Analysis and Optimal Filtering Random signals analysis (MVE136) Mats Viberg and Lennart Svensson Department of Signals and Systems Chalmers University of Technology 412 96
More informationExpressions for the covariance matrix of covariance data
Expressions for the covariance matrix of covariance data Torsten Söderström Division of Systems and Control, Department of Information Technology, Uppsala University, P O Box 337, SE-7505 Uppsala, Sweden
More informationHomework 4. May An LTI system has an input, x(t) and output y(t) related through the equation y(t) = t e (t t ) x(t 2)dt
Homework 4 May 2017 1. An LTI system has an input, x(t) and output y(t) related through the equation y(t) = t e (t t ) x(t 2)dt Determine the impulse response of the system. Rewriting as y(t) = t e (t
More informationX. Chen More on Sampling
X. Chen More on Sampling 9 More on Sampling 9.1 Notations denotes the sampling time in second. Ω s = 2π/ and Ω s /2 are, respectively, the sampling frequency and Nyquist frequency in rad/sec. Ω and ω denote,
More informationOn Input Design in System Identification for Control MÄRTA BARENTHIN
On Input Design in System Identification for Control MÄRTA BARENTHIN Licentiate Thesis Stockholm, Sweden 2006 TRITA-EE 2006:023 ISSN 1653-5146 ISBN 91-7178-400-4 KTH School of Electrical Engineering SE-100
More informationENT 315 Medical Signal Processing CHAPTER 2 DISCRETE FOURIER TRANSFORM. Dr. Lim Chee Chin
ENT 315 Medical Signal Processing CHAPTER 2 DISCRETE FOURIER TRANSFORM Dr. Lim Chee Chin Outline Introduction Discrete Fourier Series Properties of Discrete Fourier Series Time domain aliasing due to frequency
More informationEE538 Final Exam Fall :20 pm -5:20 pm PHYS 223 Dec. 17, Cover Sheet
EE538 Final Exam Fall 005 3:0 pm -5:0 pm PHYS 3 Dec. 17, 005 Cover Sheet Test Duration: 10 minutes. Open Book but Closed Notes. Calculators ARE allowed!! This test contains five problems. Each of the five
More informationEE 565: Position, Navigation, and Timing
EE 565: Position, Navigation, and Timing Kalman Filtering Example Aly El-Osery Kevin Wedeward Electrical Engineering Department, New Mexico Tech Socorro, New Mexico, USA In Collaboration with Stephen Bruder
More informationA METHOD OF ADAPTATION BETWEEN STEEPEST- DESCENT AND NEWTON S ALGORITHM FOR MULTI- CHANNEL ACTIVE CONTROL OF TONAL NOISE AND VIBRATION
A METHOD OF ADAPTATION BETWEEN STEEPEST- DESCENT AND NEWTON S ALGORITHM FOR MULTI- CHANNEL ACTIVE CONTROL OF TONAL NOISE AND VIBRATION Jordan Cheer and Stephen Daley Institute of Sound and Vibration Research,
More information7. Line Spectra Signal is assumed to consists of sinusoidals as
7. Line Spectra Signal is assumed to consists of sinusoidals as n y(t) = α p e j(ω pt+ϕ p ) + e(t), p=1 (33a) where α p is amplitude, ω p [ π, π] is angular frequency and ϕ p initial phase. By using β
More information2A1H Time-Frequency Analysis II
2AH Time-Frequency Analysis II Bugs/queries to david.murray@eng.ox.ac.uk HT 209 For any corrections see the course page DW Murray at www.robots.ox.ac.uk/ dwm/courses/2tf. (a) A signal g(t) with period
More informationProblem Sheet 1 Examples of Random Processes
RANDOM'PROCESSES'AND'TIME'SERIES'ANALYSIS.'PART'II:'RANDOM'PROCESSES' '''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''Problem'Sheets' Problem Sheet 1 Examples of Random Processes 1. Give
More informationA New Subspace Identification Method for Open and Closed Loop Data
A New Subspace Identification Method for Open and Closed Loop Data Magnus Jansson July 2005 IR S3 SB 0524 IFAC World Congress 2005 ROYAL INSTITUTE OF TECHNOLOGY Department of Signals, Sensors & Systems
More information