EE6203  - assignment-1 - Solved

            

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.

                                                                                                                               (10 marks)

 

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 0.5x k1( ) 0.8

                                                               u k( )

                               x k2( 1) 0.6 0.7x 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 ( )

                                                                                                                            (15 marks) 

 

 

 

 

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 xd ( )k Bdu k( )

 

If  

                                                  0    1        0

                    A4 5;B  1 ;T  0.5 sec  

(i)  Find  Ad and  Bd .

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

U z( )

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

                                                                                                        (30 marks)

4.       A discrete-time system is given by    

 

                                                                  1  2         3

                                            x(k 1)      x( )k  u k( )

                                                                  1  2         4      

                                                 ( )y k 5 6x( )k

 

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

 

wQ( )k =Q x1 ( )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.

  

(b)   Determine a co-ordinate transformation, i.e. find P in the following

 

wP( )k =P x1 ( )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.

                                                                                                              (20 marks)

 

 

 

 

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

       

(a)   Obtain a state-space representation for the continuous-time system with the state variables as indicated and the output variable y t( )  x t1( ).  

 

(b)   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.

 

(c)   Find the deadbeat control law of the following form                u k( ) Kx( )k  

                  Show that the response is indeed deadbeat.

                                                                                                                                  (25 marks)