Exercise10

  • Due No Due Date
  • Points None

10.1 Sampling of Systems

A system with input u and output y has transfer function LaTeX: G(s). Find the sampled system G(z) when the sampling period is h=0.1. Also find the coefficients in polynomials A(q) and B(q) so that A(q)y(t) = B(q)u(t).

a) LaTeX: G(s) = \frac{2}{s+3}

b) LaTeX: G(s) = \frac{2}{s+3}e^{-0.2s}

10.2 Recovering the continuous time system

 

After identification of a sampled system with sampling period  LaTeX: h=1 you have obtained the discrete time model

\(y(t) - 1.4y(t-1) +0.48 y(t-2) = 2u(t-1) - 1.4 u(t-2)\)

The continuous time system is of the form

\(G(s) = \frac{b_0s+b_1}{s^2+a_1s+a_2}\)

Find the values of the parameters \(b_0, b_1, a_1\) and \(a_2\).

Hint: \(\frac{2z-1.4}{z^2-1.4+0.48} = \frac{1}{z-0.6} + \frac{1}{z-0.8}\)

10.3 Identification and Prediction 

A second order (noise-free) system of the form

LaTeX: y(t) + a_1y(t-1) + a_2y(t-2) = b_1u(t-1) + b_2u(t-2)

has step response (LaTeX: u(t)=1 for LaTeX: t \geq 0 ) as

LaTeX: y(t) = 0, t\leq 0.

LaTeX: y(1) = 1
LaTeX: y(2) = 1.7
LaTeX: y(3) = 1.99
LaTeX: y(4) = 1.853

a) Identify the coefficients LaTeX: a_1,a_2,b_1,b_2.
b) Determine predictions LaTeX: \widehat{y}(5|4) \textrm{and } \widehat{y}(6|4) (the predictions at time 4 of LaTeX: y(5) and LaTeX: y(6)).

10.4 Identification in closed loop

We want to find parameters LaTeX: a and LaTeX: b in the system
LaTeX: y(t) + ay(t-1) = bu(t-1) + e(t).
where e is white noise. The system is controlled by a proportional controller. Can the parameters be identified if

a) LaTeX: u(t) = -K y(t),
b) LaTeX: u(t) = -K y(t-1)
c) LaTeX: u(t) = -Ky(t) + r(t), where LaTeX: r(t)=1

10.5 Identification example

The file ex10_5.m performs the parameter identification of the resonant system on Lecture 9 using the OE, ARX, ARMAX and BJ model structures.

Study the file and make sure you understand what happens. Then replace the noise dynamics LaTeX: H=1 used on Lecture 9 with LaTeX: H=\frac{1}{1+z^{-2}} and redo the identifications.

a) Which model structures give now correct estimation of the dynamics LaTeX: G ?

b) Which model structures gives correct estimation of LaTeX: H ?

c) Which identified model gives the best prediction results?

10.6 Bias problem by unmodeled noise dynamics

On the lecture we claimed that the least squares identification of the parameter LaTeX: a in

LaTeX: y(t) = ay(t-1) + w(t)

gives a bias if the true system is

LaTeX: y(t) = ay(t-1) + e(t) + ce(t-1)

(where LaTeX: e(t) is white noise, i.e. LaTeX: e(t_1) and LaTeX: e(t_2) are independent).

Verify the claimed formula

LaTeX: \widehat{a}_N \to a + \frac{c(1-a^2)}{1+2ac+c^2}, \qquad \textrm{as } N\to \infty

 

 

Solutions to exercise 10

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