Análise de Controlabilidade e Observabilidade

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Problem 7.29PP
The state-space model for a certain application has been given to us with the following state
description matrices;
'0.174 0 0 0 O' ■-0.207'
0.1S7 0.645 0 0 0 -0.005
0 1 0 0 0 4 B = 0
0 0 1 0 0 0
0 0 0 1 0 0
C = [ 1 0 0 0 0 ].
(a) Draw a block diagram of the realization with an integrator for each statevariable.
(b)A student has computed det C = 2.3x 10-7 and claims that the system is uncontrollable. Is 
the student right or wrong? Why?
(b)A student has computed det C = 2.3x io -7 and claims that the system is uncontrollable. Is 
the student right or wrong? Why?
(c) Is the realization observable?
Step-by-step solution
step 1 of 7
(a)
Write the general form of state space equation.
x = ...... (1)
y = C x+D ...... (2)
Consider the matrix A.
0.174 0 0 0
0.157 0.645 0 0 
0 1 0 0
0 0 1 0
0 0 0 1
Consider the matrix B.
’-0.207*
■(3)
B s
-0.005
0
0
0
(4)
Consider the matrix C.
C = [l 0 0 0 0] (5)
Consider the matrix D.
O = [0] (6)
Write the state equation in control canonical form from equations (1), (3) and (4).
-*»
■«4
■(7)
*0.174 0 0 0 o' ■-0.207'
0.157 0.645 0 0 0 ■*1 -0.005
= 0 1 0 0 0 + 0
0 0 1 0 0 0
0 0 0 1 0̂ 0
0.174 0 0 0 0* -0.207
0.157 0.645 0 0 0 -0.005
0 1 0 0 0 JC4 0 U ■■■■
0 0 0 0 0
0 0 0 1 0 0
Write the output equation in control canonical form from equations (2), (5) and (6).
;- = [l 0 0 0 0] +0
^ = [1 0 0 0 0 ] * + 0 .........(8)
step 2 of 7 ^
Draw the block diagram for controller canonical form from equations (7) and (8).
Figure 1
step 3 of 7
Hence, the block diagram of realization with an integrator for each state variable is IdrawnI -
Step 4 of 7
(b)
The general fonnula for controllability matrix, C is.
C = [ b AB ... A - ’b ] ..... (9)
Calculate AB^rom equations (3) and (4).
■-0.207*
A B s
A B s
0.174 0 0 0 0
0.157 0.645 0 0 0 
0 1 0 0 0
-0.036018
-0.035724
-0.005
0
0
0 1 0
( 10)
-0.005
0
Step 5 of 7
Calculate equations (3) and (10).
A'B = A B x A ......(11)
Substitute equations (3) and (10) in equation (11).
A’B-
0.174 0 0
0.157 0.645 0 
0 1 0
0 0 
0 0 
0 0
A'B =
I 0 0 
0 1 0
-0.036018
-0.035724
-0.005
0
0
( 12)
-6.267x10’
-0.0287 
-0.035724 
-0.005 
0
Calcuiate a ^BIiiiiii equations (3) and (12).
A’B = A‘BxA ..... (13)
Substitute equations (3) and (12) in equation (13).
A*B =
0.174 0 0 0 0
0.157 0.645 0 0 0
0 1
0 0
0 0
0 0 0 
I 0 0 
0 I 0
-6.267x10’’
-0.0287
-0.035724
-0.005
0
A*B = .(14)
-1.09x10’’
-0.0195 
-0.0287 
-0.035724 
-0.005
Calculate a ^B ^^^ equations (3) and (14).
A’B = A’BxA ..... (15)
Substitute equations (3) and (14) in equation (15).
A*B =
-1.8966x10"*
-0,01275 
-0.0195 
-0.0287 
-0.035724
Substitute equations (4), (10). (12). (14) and (16) in equation (9).
0.174 0 0 0 O' -1.09x10’’
0.157 0.645 0 0 0 -0.0195
0 1 0 0 0 -0.0287
0 0 1 0 0 -0.035724
0 0 0 1 0 -0.005
A^B> .(16)
-0.207 -0.036018 -6.267x10’’ -1.09x10’’ -1.8%6xl0-*
-0.005 -0.035724 -0.0287 -0.0195 -0.01275
0 -0.005 -0.035724 -0.0287 -0.0195
0 0 -0.005 -0.035724 -0.0287
0 0 0 -0.005 -0.035724
Write the MATLAB program to find detennination of Matrix C- 
A=[-.207 -0.036018 -6.267*1 -1.09*10^-3 -1.8966*10''-4;
-0.005 -0.035724 -0.0287 -0.0195 -0.01275;
0 -0.005 -0.035724 -0.0287 -0.0195 ;
0 0 -0.005 -0.035724 -0.0287;
0 0 0 -0.005 -0.035724;]; 
det(A)
The output of the MATLAB program is given below, 
ans =
-2.2828e-07
Hence, value of |det|C| = -2.282xlQ-’ | . This value is not equal to zero. So this is controllable 
system. The student computation is wrong.
Step 6 of 7
(C)
Determine the system observabiiity O- 
Write the observabie condition from description matrices. 
C 
CA
.(17)CA’
CA’
CA’
Step 7 of 7
Calcuiate CAftoni equations (3) and (5).
0.174 0 0
C A -[1 0 0 0 0]
0.157 0.645 
0 1
0 0
0 0
0 0 
0 00
0 0 0 
1 0 0 
0 1 0
CA = [0.174 0 0 0 0 ] ...... (18)
Calculate equations (5) and (18).
’0.174 0
CA’ =[0.174 0 0 0 0]
0 0 0 
0.157 0.645 0 0 0 
0 1
0 0
0 0
0 0 0 
1 0 0
CA’ = [0.030276 0 0 0 O] (19) 
Calculate equations (5) and (19).
*0.174
CA’ =[0.030276 0 0 0 0]
0 0 0 
0.157 0.645 0 0 
0 1
0 0
0 0
0 0
CA’ =[5.268x10"’ 0 0 0 o ] .....(20)
Calculate C A ^^°^ equations (5) and (20).
CA’ =[5.268x10” 0 0 0 O]
0.174 0 0 0 0
0.157 0.645 0 0 0
0 1
0 0
0 0
0 0 0 
1 0 0 
0 1 0
CA’ = [9.166x10’ * 0 0 0 o ].....(21)
Substitute equations (5), (18). (19). (20) and (21) in equation (17).
0 =
1
0.174
0.030276
5.268x10’’
O.lOOxlO”
0 0 0 0 
0 0 0 0 
0 0 0 0 
0 0 0 0 
0 0 0 0
. (22 )
Determine the determination ^matrix from equation (22).
det|O|=0
Hence, value of det|O| = 0- So this is (not observable system).