CH5230 - System Identification - Engineering Assignment Help

Assignment Task

Exercises 

1. (a) Verify that for the 3-parameter model example in class, the input u[k] = cos2(ω0k) results in loss of identifiability. In general, using linear algebra principles, derive the condition that the input u[k] should satisfy. 

(b) Examine the identifiability of a first-order LTI system y[k]+a1y[k−1] = u[k]+a1u[k−1] for two different conditions (i) non-zero u[k] and zero y[0] (forced response) and (ii) zero u[k] but non-zero y[0] (free response). 

2. The impulse response of a d.t. system is given by g[k] = 2(0.6)k−1 + (0.4)k−1, k ≥ 1. 

a) Plot the impulse and step responses of the system. Is the system causal and stable? b) Arrive at the frequency response function G(ejω) of the system. Plot the magnitude and phase response of the system. 

c) What is an appropriate FIR approximation of the system? 

3. A deterministic d.t. LTI system is governed by the following difference equation: y[k] − 1.3y[k − 1] + 0.24y[k − 2] = 3u[k − 1] (1) 

a) What is the delay, order and gain of the system? 

b) Determine the IR of the system by solving (1). Assume a relaxed system. c) Express (1) as two first-order difference equations (introduce artificial variables) change? 

4. With reference to the liquid level case study1, generate a fresh flow-level response data by changing the measurement noise from white to filtered noise by placing a discrete-time fil ter block in the disturbance channel path, in the SIMULINK block diagram. Use the filter coefficients 

num = 1, den = [1 − 0.9] 

and set the SNR to 10. For this purpose, switch off the noise, simulate and compute the variance of the true response. Subsequently, turn the white-block noise on and adjust the variance parameter in the block such that the desired SNR is achieved. 

(a) Using the obtained data, determine the best OE model (based on the two tests of corre lation between inputs and residuals, and errors in parameter estimates) still manages to explain the deterministic process correctly.

(b) How does the ARX model (of a suitably high-order) fare in the above situation, i.e., data generated with colored noise?

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