[SOLVED] EE6203 HW1 ASSIGNMENT

1. A system is described by the following difference equation

c k( 3) 3 (c k 2) 5 (c k 1) 7 ( )c k 9 ( )u k

where the output y k( ) c k( ). Define the state variables as

x k₁( ) c k x k( ); ₂( ) c k( 1);x k₃( ) c k( 2)

Obtain a state-space representation for the system.

2. Consider the two systems connected as shown below.

The respective state-space representations are given as follow:

System S₁ :

0.1 (u k 1) 0.2 ( )u k 0.3 ( )e k

System S₂ :

x k₁( 1) 0.4 0.5x k₁( ) 0.8

u k( )

x k₂( 1) 0.6 0.7x k₂( ) 0.9

x k₁( )

( )y k 1 2

x k₂( )

If x k₃( ) u k( ), give a state-space representation for the overall system,

x k₁( )

x(k 1) Ax( )k Be k( ); x( )k x k₂( )

x k₃( )

( )y k Cx( )k de k ( )

3. Given the state equation of a linear system as

x( )t Ax( )t Bu t( )

The ZOH equivalent, with a sampling period of T seconds, is of the following form:

x(k 1) A x_d ( )k B_du k( )

If

0 1 0

A4 5;B 1 ;T 0.5 sec

(i) Find A_d and B_d .

X( )z (ii) Find the transfer function .

U z( )

(iii) Determine the characteristic equation of the discretised system and obtain the eigenvalues of A_d .

4. A discrete-time system is given by

1 2 3

x(k 1) x( )k u k( )

1 2 4

( )y k 5 6x( )k

• Determine a co-ordinate transformation, i.e. find Q in the following

w_Q( )k =Q x¹ ( )k

that transforms the system into the observable canonical form (OCF). Hence, using Q, determine a state-space representation which is in the OCF form.

• Determine a co-ordinate transformation, i.e. find P in the following

w_P( )k =P x¹ ( )k

that transforms the system into the controllable canonical form (CCF). Hence, using P, determine a state-space representation which is in the CCF form.

5. A continuous-time system is as shown below. Let M 625,K_S 10.

• Obtain a state-space representation for the continuous-time system with the state variables as indicated and the output variable y t( ) x t₁( ).
• The system is sampled with a zero-order hold and the sampling period is 0.5 second. Obtain a zero-order hold equivalent of the continuous-time system.

• Find the deadbeat control law of the following form u k( ) Kx( )k

Show that the response is indeed deadbeat.