Exercise 10

10.1 Sampling of Systems

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

a) $G(s) = \frac{2}{s+3}$

b) $G(s) = \frac{2}{s+3}e^{-0.2s}$

10.2 Recovering the continuous time system

After identification of a sampled system (zero-order hold) with sampling period  $h=1$ 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

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

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

$y(t) = 0, t\leq 0$.

$y(1) = 1$
$y(2) = 1.7$
$y(3) = 1.99$
$y(4) = 1.853$

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

10.4 Identification in closed loop

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

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

10.5 Identification example

The file ex10_5.m Download 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 $H=1$ used on Lecture 9 with $H=\frac{1}{1+z^{-2}}$ and redo the identifications.

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

b) Which model structures gives correct estimation of $H$ ?

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 $a$ in

$y(t) = ay(t-1) + w(t)$

gives a bias if the true system is

$y(t) = ay(t-1) + e(t) + ce(t-1)$

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

Verify the claimed formula

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

 

 

Solutions to exercise 10 Download Solutions to exercise 10