Question

3 +3 Consider the following TF G(s) - 13 l. Realize a state space(s)formulation using MATLAB. Then use Forward EueriFE) to compute the discrete SS for the system, choose Ts 0.1 Then find the TF of the discrete SS system description. Use FE directly on G(s) to arrive at a discrete TF, G(z). Realize the discrete TF in SS Are the two discrete TF from part 1 and 2 the same? What about the ss formulations? Comment on your observations. 2. 3.

Answer 1

Given transfer function,

G(s)--s+3

and sampling time, Ts=0.1

MATLAB code:

Num=[1 3]; Den=[1 2 1];

[A B C D]=tf2ss(Num,Den)

Result:

A=[-2 -1;1 0]

B=[1;0]

C=[1 3]

D=0

The state space in continuous form is:

tA(t) Bu(t) and y(t)C(t) D

1. Using Forward Euler in the state space form:

According to Forward Euler approximation, the derivative term can be rewritten as:

. Since the sampling is done in the integer multiples of T, with T being the smallest value,

(t) =

a small change in x(t) can be written as $\dot{x}(kT)=\frac{x(kT+T)-x(kT)}{T}$

Upon substituting the above expression, the continuous can system can be rewritten as:

+)(k)Ar(k)Bu(k) and yk) Cx(k) since D 0

Upon solving we get,

r(k + 1)-(1 + AT)x(k) + TBu(k) and y(k) = Cx(k) since D = 0

substituting the values of A,B and C we get the discretized state space model as:

「 101一2-1 0 1+100 0.1)(k)0.2 Bu(k) and y(k) = Cx(k) 0.8-0.1 01 k)( 0.1

Applying z-transform on both sides we get,

0.8 -0.x(01) -0.1 z-1 ~ -0.97 G(s)- or 22-1.8z 0.81 0.81~-2-1.82-1 1 81:-2-180,-I + 100 , -10.97 2 G(z) = or (z) =一100-1-97:- G

2. Directly using Forward Euler on G(s):

Substituting s=(z-1)/T in the given transfer function, we get,

G(s) =- 7-0. land s = G(z) = s+1 T. 10(z 1) +3 (102 10z 1) 1 10-7 100 * (~2-2c + 1) + 20: _ 20 + 1 or 100:2-180: + 81 10:-1-7:-2 G(z) = 81-2- 180z-1 100

The state space form is given by:

Ad=[1.8 -0.81;1 0]; Bd=[1;0]; Cd=[0.1 -0.07]

3. The two discrete transfer functions are not same. However, their characteristic equations are same but differ in the location of zero. Similarly their respective state space matrices also differ.