geromel, controle linear de sistemas dinâmicos
TRANSCRIPT
CAPÍTULO 2
2.7. Transfer function:
12800 s^2 + 51200 s
-----------------------------------------
s^4 + 58 s^3 + 2352 s^2 + 30080 s + 51200
2.8. Transfer function:
4 s^2 + 40 s + 100
------------------
s^2 + 12 s + 20
2.23-24.
a) Transfer function:
13
--------------
s^2 + 4 s + 13
b) Transfer function:
130
-------------------------
s^3 + 14 s^2 + 53 s + 130
c) Transfer function:
39
-----------------------
s^3 + 7 s^2 + 25 s + 39
d) Transfer function:
340 s + 340
--------------------------
s^3 + 20 s^2 + 134 s + 340
e) Transfer function:
50 s + 200
-----------------------------------
s^4 + 10 s^3 + 54 s^2 + 160 s + 200
f) Transfer function:
50 s^2 + 300 s + 250
------------------------
s^3 + 10 s^2 + 41 s + 50
g) Transfer function:
65.28 s^2 + 522.2 s + 1632
--------------------------------------
s^4 + 19 s^3 + 232 s^2 + 1096 s + 1632
h) Transfer function:
-3.25 s^2 - 9.75 s + 13
-----------------------
s^3 + 7 s^2 + 19 s + 13
CAPÍTULO 3
3.1
a)
Transfer function:
s^2 + 12 s + 35
---------------
s^2 + 4 s + 3
b) Transfer function:
s^2 + 6 s + 25
-----------------
s^3 + 2 s^2 + 5 s
c)
Transfer function:
s + 3
-------------------------
s^4 + 5 s^3 + 9 s^2 + 5 s
d)
Transfer function:
s + 2
----------------------------
s^4 + 10 s^3 + 26 s^2 + 80 s
3.2.
a) Transfer function:
s + 2
------
s + 10
b)
Transfer function:
1
---------------------
s^3 + 5 s^2 + 8 s + 4s + 10
c)
Transfer function:
s^2 + 11 s + 10
--------------------------------------
s^4 + 106 s^3 + 612 s^2 + 1208 s + 800
d)
Transfer function:
s + 1
------------
s^3 + 10 s^2
e)
Transfer function:
s + 1
---------
s^2 + 2 s
f)
Transfer function:
1
----------------------
s^3 + 2 s^2 + 9 s + 18
g)
Transfer function:
10 s + 1
----------------------------------------
2000 s^4 + 2600 s^3 + 645 s^2 + 46 s + 1
h)
Transfer function:
s - 1
---------
s^3 + s^2
i)
Transfer function:
s
---------------
s^2 + 12 s + 20
j)
Transfer function:
s^2 - s
---------------
s^2 + 12 s + 20
3.4.
b) Transfer function:
1
-----------------
s^3 + 2 s^2 + 2 s
3.8.
a) %Solução pelo Matlab
%LYAP Solve continuous-time Lyapunov equations.
%
% X = LYAP(A,Q) solves the Lyapunov matrix %equation:
%
% A*X + X*A' + Q = 0
>> A = [0 -7 -7;1 -1 -1;-7 -1 10 ]
A =
0 -7 -7
1 -1 -1
-7 -1 10
>> Q = [1 0 0; 0 1 0; 0 0 1]
Q =
1 0 0
0 1 0
0 0 1
>> X = LYAP(A,Q)
X =
4.3240 0.1075 -0.0361
0.1075 0.6091 -0.0015
-0.0361 -0.0015 -0.0500
3.10.
a) Transfer function:
1
-------------
s^2 + 6 s + 5
b)
Transfer function:
s^2 + 8 s + 20
-----------------------
s^3 + 6 s^2 + 15 s + 14
c)
Transfer function:
10 s - 10
-----------------
s^3 + 4 s^2 + 4 s
d) Transfer function:
1
-----------------------
s^3 + 7 s^2 + 12 s + 10
e)
Transfer function:
s + 2
-----------------------
s^3 + 7 s^2 + 16 s + 10
f)
Transfer function:
5
-----------------------------
s^4 + 16 s^3 + 93 s^2 + 180 s
g)
Transfer function:
s^2 - 4 s + 3
-------------
s^2 + 5 s
h) Transfer function:
s^2 + 16 s + 73
-----------------------
s^3 + 9 s^2 + 49 s + 41
i)
Transfer function:
2 s^2 + 12 s + 180
------------------
s^3 + 4 s^2 + 13 s
j)
Transfer function:
s + 10
-------------
s^2 + 2 s - 8
k)
Transfer function:
s^2
-------------------------------------
s^4 + 27 s^3 + 250 s^2 + 900 s + 1000
l) Transfer function:
s^2 - 2 s
--------------
s^2 + 8 s + 15
3.11
a)
>> alpha = a(1)
alpha =
0.5000
>> f = tf([1 2],[1 (alpha + 2) (2*alpha + 1) alpha])
Transfer function:
s + 2
-------------------------
s^3 + 2.5 s^2 + 2 s + 0.5
>> rlocus(f)
>>
>> alpha = a(2)
alpha =
1.5000
>> f = tf([1 2],[1 (alpha + 2) (2*alpha + 1) alpha])
Transfer function:
s + 2
-------------------------
s^3 + 3.5 s^2 + 4 s + 1.5
>> rlocus(f)
>>
>> alpha = a(3)
alpha =
5
>> f = tf([1 2],[1 (alpha + 2) (2*alpha + 1) alpha])
Transfer function:
s + 2
----------------------
s^3 + 7 s^2 + 11 s + 5
>> rlocus(f)
>>
b)
>> a = [0.5 3/2 5]
a =
0.5000 1.5000 5.0000
>> b = a(1)
b =
0.5000
>> f = tf([1 2],[(1) (b + 2) (2*b + 5) 5])
Transfer function:
s + 2
-----------------------
s^3 + 2.5 s^2 + 6 s + 5
>> rlocus(f)
>>
>> b = a(2)
b =
1.5000
>> f = tf([1 2],[(1) (b + 2) (2*b + 5) 5])
Transfer function:
s + 2
-----------------------
s^3 + 3.5 s^2 + 8 s + 5
>> rlocus(f)
>>
>> b = a(3)
b =
5
>> f = tf([1 2],[(1) (b + 2) (2*b + 5) 5])
Transfer function:
s + 2
----------------------
s^3 + 7 s^2 + 15 s + 5
>> rlocus(f)
>>
c) >> a = [1/2 3/2 5]
a =
0.5000 1.5000 5.0000
>> c = a(1)
c =
0.5000
>> f = tf([1 c],[1 8 20 16])
Transfer function:
s + 0.5
-----------------------
s^3 + 8 s^2 + 20 s + 16
>> rlocus(f)
>>
>> c = a(2)
c =
1.5000
>> f = tf([1 c],[1 8 20 16])
Transfer function:
s + 1.5
-----------------------
s^3 + 8 s^2 + 20 s + 16
>> rlocus(f)
>>
>> c = a(3)
c =
5
>> rlocus(f)
>>
d)
>> a = [1/2 3/2 5]
a =
0.5000 1.5000 5.0000
>> c = a(1)
c =
0.5000
>> f = tf([1 c],[1 8 29 52])
Transfer function:
s + 0.5
-----------------------
s^3 + 8 s^2 + 29 s + 52
>> rlocus(f)
>>
>> c = a(2)
c =
1.5000
>> f = tf([1 c],[1 8 29 52])
Transfer function:
s + 1.5
-----------------------
s^3 + 8 s^2 + 29 s + 52
>> rlocus(f)
>>
>> c = a(3)
c =
5
>> f = tf([1 c],[1 8 29 52])
Transfer function:
s + 5
-----------------------
s^3 + 8 s^2 + 29 s + 52
>> rlocus(f)
>>
3.12.
>> a = [1 4]
a =
1 4
>> b = a(1)
b =
1
>> f = tf([1],[1 10 (b + 32) (2*b + 32)])
Transfer function:
1
------------------------
s^3 + 10 s^2 + 33 s + 34
>> rlocus(f)
>>
>> b = a(2)
b =
4
>> f = tf([1],[1 10 (b + 32) (2*b + 32)])
Transfer function:
1
------------------------
s^3 + 10 s^2 + 36 s + 40
>> rlocus(f)
>>
3.13.
a) Transfer function:
2048 s
--------------------------------------
s^4 + 21 s^3 + 148 s^2 + 1152 s + 1024
c)
d)
3.14.
a) Transfer function:
s^2 + 15 s + 50
G = ---------------
s^3 + 2 s^2
b)
F = (C*G)/(C*G + 1)
Transfer function:
26.6 s^5 + 452.2 s^4 + 2128 s^3 + 2660 s^2
F = ------------------------------------------------
s^6 + 30.6 s^5 + 456.2 s^4 + 2128 s^3 + 2660 s^2
c) Transfer function:
s^2 + 15 s + 50
G = ---------------
s^4 + 2 s^3
>> Instable for any value of
1
C = -
s
F = (C*G)/(C*G + 1)
Transfer function:
s^6 + 17 s^5 + 80 s^4 + 100 s^3
-----------------------------------------------
s^8 + 4 s^7 + 5 s^6 + 17 s^5 + 80 s^4 + 100 s^3
d) s + 20
Co = ------
s
Transfer function:
s^3 + 35 s^2 + 350 s + 1000
F = G*Co = ---------------------------
s^4 + 2 s^3
C = 67.3*Co
F = (C*G)/(C*G + 1)
Transfer function:
67.3 s^7 + 2490 s^6 + 28266 s^5 + 114410 s^4 + 134600 s^3
F = ---------------------------------------------------------------
s^8 + 71.3 s^7 + 2494 s^6 + 28266 s^5 + 114410 s^4 + 134600 s^3
d) Transfer function:
s^2 + 15 s + 50
---------------
s^3
CAPÍTULO 4
04.01.
a) Transfer function:
5 s + 5
-------------
s^2 + 4 s + 5
b) Transfer function:
s + 20
-----------------
4 s^2 + 16 s + 20
c) Transfer function:
40
-----------------------
s^3 + 5 s^2 + 44 s + 40
d)
Transfer function:
400
-------------------------
s^3 + 14 s^2 + 80 s + 400
e)
f)
g)
h)
4.05
a) Transfer function:
1
--------------------------
s^3 + 21 s^2 + 120 s + 100
>> step(108*G)
b) >> C = 108*tf([1 1],[1 0])
Transfer function:
108 s + 108
-----------
s
>> G
Transfer function:
1
--------------------------
s^3 + 21 s^2 + 120 s + 100
>> C*G
Transfer function:
108 s + 108
------------------------------
s^4 + 21 s^3 + 120 s^2 + 100 s
>> sisotool(G*C)
d)
%Calculo do polinômio P(s)%
>> f = conv([1 4],[1 4])
f =
1 8 16
>> g = [1 16]
g =
1 16
>> f = conv(f , g)
f =
1 24 144 256
>> f = conv(f , g)
f =
1 40 528 2560 4096
>> f = conv(f , g)
f =
1 56 1168 11008 45056 65536
>> f = conv(f , g)
f =
1 72 2064 29696 221184 786432 1048576
%Vetor correspondente aos coeficientes de P(s)%
>> f = conv(f , g)
f =
1 88 3216 62720 696320 4325376 13631488 16777216
%Após a solução da Eqaução Diofantina, o controlador desejado encontrado foi o
%seguinte :
>> C = tf([85609 1863156 11720388 16777216],[1 67 1689 19111 0])
Transfer function:
85609 s^3 + 1.863e006 s^2 + 1.172e007 s + 1.678e007
---------------------------------------------------
s^4 + 67 s^3 + 1689 s^2 + 19111 s
% Função de tranferência da Planta
>> G = tf([1],[1 21 120 100])
Transfer function:
1
--------------------------
s^3 + 21 s^2 + 120 s + 100
%Função de malha fechada%
>> F = feedback(C*G,1)
Transfer function:
85609 s^3 + 1.863e006 s^2 + 1.172e007 s + 1.678e007
------------------------------------------------------------------------------------------
s^7 + 88 s^6 + 3216 s^5 + 62720 s^4 + 696320 s^3 + 4.325e006 s^2 + 1.363e007 s + 1.678e007
%Polos da função de malha fechada%
>> pole(F)
ans =
-16.0221 + 0.0161i
-16.0221 - 0.0161i
-15.9916 + 0.0260i
-15.9916 - 0.0260i
-15.9727
-4.0000
-4.0000