to Solutions Skill-Assessment Exercises
To Accompany
Control Systems Engineering rd 3 Edition
By Norman S. Nise
John Wiley & Sons
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Solutions to Skill-Assessment Exercises
Chapter 2 2.1. The Laplace transform of t is F(s) = 2.2. Expanding F(s) by partial fractions yields: F(s) = where, A B C D + + + 2 s s + 2 (s + 3) (s + 3) 1 . (s + 5)2
1 using Table 2.1, Item 3. Using Table 2.2, Item 4, s2
A=
10 ( s + 2)(s + 3)2
=
S0
5 10 B= s(s + 3)2 9
= -5 C =
S -2
10 10 = , and s(s + 2) S -3 3
D = (s + 3)2
40 dF(s) = ds s -3 9
Taking the inverse Laplace transform yields,
f (t) =
2.3.
5 10 40 - 5e -2t + te -3t + e -3t 9 3 9
Taking the Laplace transform of the differential equation assuming zero initial conditions yields: s3C(s) + 3s2C(s) + 7sC(s) + 5C(s) = s2R(s) + 4sR(s) + 3R(s) Collecting terms,
(s 3 + 3s 2 + 7s + 5)C(s) = (s 2 + 4s + 3)R(s)
Thus,
2
Solutions to Skill-Assessment Exercises
s 2 + 4s + 3 C(s) = 3 R(s) s + 3s 2 + 7s + 5
2.4. G(s) = 2s + 1 C(s) = 2 R(s) s + 6s + 2
Cross multiplying yields, dc dr d 2c + 6 + 2c = 2 + r 2 dt dt dt 2.5. C(s) = R(s)G(s) = where 1 s 1 A B C * = = + + 2 s (s + 4)(s + 8) s(s + 4)(s + 8) s (s + 4) (s + 8)
A=
1 1 1 1 1 1 = B= = - , and C = = (s + 4)(s + 8) S0 32 s(s + 8) S -4 16 s(s + 4) S -8 32
Thus,
c(t) =
2.6.
1 1 1 - e -4t + e -8t 32 16 32
Mesh Analysis Transforming the network yields,
Now, writing the mesh equations,
Chapter 2
3
(s + 1)I1 (s) - sI2 (s) - I3 (s) = V(s) -sI1 (s) + (2s + 1)I2 (s) - I3 (s) = 0 -I1 (s) - I2 (s) + (s + 2)I3 (s) = 0
Solving the mesh equations for I2(s),
(s + 1) V(s) -s -1 I2 (s) = (s + 1) -s -1 0 0 -s -1 -1 (s + 2) -1 = (s 2 + 2s + 1)V(s) s(s 2 + 5s + 2)
(2s + 1) -1 -1 (s + 2)
But, V L (s) = sI2 (s) Hence,
V L (s) = (s 2 + 2s + 1)V(s) (s 2 + 5s + 2)
or
V L (s) s 2 + 2s + 1 = V(s) s 2 + 5s + 2
Nodal Analysis Writing the nodal equations, 1 ( + 2)V1 (s) - V L (s) = V(s) s 2 1 -V1 (s) + ( + 1)V L (s) = V(s) s s Solving for V L (s) ,
1 ( + 2) V(s) s 1 V(s) -1 (s 2 + 2s + 1)V(s) s V L (s) = = 1 (s 2 + 5s + 2) ( + 2) -1 s 2 -1 ( + 1) s
or
V L (s) s 2 + 2s + 1 = V(s) s 2 + 5s + 2
4
Solutions to Skill-Assessment Exercises
2.7. Inverting
Z2 (s) -100000 = = -s Z1 (s) (10 5 / s) Noninverting G(s) = -
10 5 + 10 5 ) ( [Z1 (s) + Z2 (s)] G(s) = = s 5 = s +1 10 Z1 (s) ( ) s 2.8. Writing the equations of motion, (s 2 + 3s + 1)X1 (s) - (3s + 1)X2 (s) = F(s) -(3s + 1)X1 (s) + (s 2 + 4s + 1)X2 (s) = 0 Solving for X2 (s) ,
(s 2 + 3s + 1) F(s) -(3s + 1) 0 (3s + 1)F(s) X2 (s) = 2 = 3 s(s + 7s 2 + 5s + 1) (s + 3s + 1) -(3s + 1) -(3s + 1) (s 2 + 4s + 1)
Hence, X2 (s) (3s + 1) = 3 F(s) s(s + 7s 2 + 5s + 1) 2.9. Writing the equations of motion,
(s 2 + s + 1)1 (s) - (s + 1) 2 (s) = T(s) -(s + 1)1 (s) + (2s + 2) 2 (s) = 0
where 1 (s) is the angular displacement of the inertia. Solving for 2 (s) ,
(s 2 + s + 1) T(s) -(s + 1) 0 (s + 1)F(s) 2 (s) = 2 = 3 (s + s + 1) -(s + 1) 2s + 3s 2 + 2s + 1 -(s + 1) (2s + 2)
From which, after simplification,
Chapter 2
5
2 (s) =
2.10.
1 2s + s + 1
2
Transforming the network to one without gears by reflecting the 4 N-m/rad spring to the left and multiplying by (25/50)2, we obtain,
T(t)
1(t) 1 kg
1 N-m-s/rad
a(t) 1 N-m/rad
Writing the equations of motion,
(s 2 + s)1 (s) - s a (s) = T(s) -s1 (s) + (s + 1) a (s) = 0
where 1 (s) is the angular displacement of the 1-kg inertia. Solving for a (s) ,
(s 2 + s) T(s) -s 0 sT(s) a (s) = 2 = 3 2 s +s +s (s + s) -s -s (s + 1)
From which,
a (s) 1 = 2 T(s) s + s + 1
But, 2 (s) = Thus,
1 a (s) . 2
2 (s) 1/ 2 = 2 T(s) s + s + 1
2.11. First find the mechanical constants. 1 1 1 Jm = Ja + J L ( * )2 = 1 + 400( )=2 5 4 400 1 1 1 )=7 Dm = Da + DL ( * )2 = 5 + 800( 5 4 400
6
Solutions to Skill-Assessment Exercises
Now find the electrical constants. From the torque-speed equation, set m = 0 to find stall torque and set T m = 0 to find no-load speed. Hence,
T stall = 200
no-load = 25
which, Kt T stall 200 = = =2 100 Ra Ea Kb =
no-load
Ea
=
100 =4 25
Substituting all values into the motor transfer function,
m (s) 1 = = 1 KT K b 15 Ea (s) s(s + (Dm + ) s(s + ) Jm Ra 2
where m (s) is the angular displacement of the armature. Now L (s) =
KT Ra Jm
1 m (s) . Thus, 20
L (s) 1 / 20 = ) Ea (s) s(s + 15 ) 2
2.12. Letting
1 (s) = 1 (s) / s 2 (s) = 2 (s) / s
in Eqs. 2.127, we obtain (J1s + D1 + K K ) 1 (s) - 2 (s) = T(s) s s K K - 1 (s) + (J2 s + D2 + ) 2 (s) s s
From these equations we can draw both series and parallel analogs by considering these to be mesh or nodal equations, respectively.
Chapter 2
7
Series analog
Parallel analog 2.13. Writing the nodal equation,
C
dv + ir - 2 = i(t) dt
But, C =1 v = vo + v ir = e vr = e v = e vo + v Substituting these relationships into the differential equation,
d(vo + v) + e vo + v - 2 = i(t) dt
We now linearize e v . The general form is
f (v) - f (vo ) df v dv vo
(1)
Substituting the function, f (v) = e v , with v = vo + v yields, e vo + v - e vo de v dv
v
vo
Solving for e vo + v ,
8
Solutions to Skill-Assessment Exercises
e vo + v = e vo +
de v dv
v = e vo + e vo v
vo
Substituting into Eq. (1)
d v + e vo + e vo v - 2 = i(t) (2) dt
Setting i(t) = 0 and letting the circuit reach steady state, the capacitor acts like an open circuit. Thus, vo = vr with ir = 2 . But, ir = e vr or vr = ln ir . Hence, vo = ln 2 = 0.693 . Substituting this value of vo into Eq. (2) yields
d v + 2 v = i(t) dt
Taking the Laplace transform,
(s + 2) v(s) = I(s)
Solving for the transfer function, we obtain
1 v(s) = I(s) s + 2
or V(s) 1 = about equilibrium. I(s) s + 2
9
Chapter 3 3.1. Identifying appropriate variables on the circuit yields
Writing the derivative relations = iC1 dt di L L = vL dt dvC2 C2 = iC2 dt C1 dvC1
(1)
Using Kirchhoff's current and voltage laws,
iC1 = iL + iR = iL + vL = -vC1 + vi iC2 = iR = 1 (vL - vC2 ) R 1 (vL - vC2 ) R
Substituting these relationships into Eqs. (1) and simplifying yields the state equations as
dvC1 dt
=-
1 1 1 1 vC1 + iL - vC2 + vi RC1 C1 RC1 RC1
1 1 diL = - vC1 + vi L L dt dvC2 1 1 1 =- vC1 - vC2 vi RC2 RC2 RC2 dt
where the output equation is
vo = vC2
Putting the equations in vector-matrix form,
10
Solutions to Skill-Assessment Exercises
1 - 1 RC C 1 11 x= - 0 L - 1 0 RC2 y = [0 0 1]x
1 1 RC RC1 11 vi (t) 0 x + L 1 1 - RC2 RC2 -
3.2. Writing the equations of motion
(s 2 + s + 1)X1 (s)
- sX2 (s)
= F(s) - X3 (s) = 0
- sX1 (s) + (s 2 + s + 1)X2 (s)
- X2 (s) + (s 2 + s + 1)X3 (s) = 0
Taking the inverse Laplace transform and simplifying,
x1 = -x1 - x1 + x2 + f x2 = x1 - x2 - x2 + x3 x3 = - x3 - x3 + x 2
Defining state variables, zi,
z1 = x1 ; z2 = x1 ; z3 = x2 ; z4 = x2 ; z5 = x3 ; z6 = x3
Writing the state equations using the definition of the state variables and the inverse transform of the differential equation,
z1 = z2 z2 = x1 = -x1 - x1 + x2 + f = -z2 - z1 + z4 + f z3 = x2 = z4 z4 = x2 = x1 - x2 - x2 + x3 = z2 - z4 - z3 + z5 z5 = x3 = z6 z6 = x3 = - x3 - x3 + x2 = -z6 - z5 + z3
The output is z5 . Hence, y = z5 . In vector-matrix form,
Chapter 3
11
0 1 0 0 0 0 0 -1 -1 0 1 0 0 1 0 0 0 1 0 0 0 z= z + f (t); y = [0 0 0 0 1 0]z 0 1 -1 -1 1 0 0 0 0 0 0 0 1 0 0 0 1 0 -1 -1 0
3.3. First derive the state equations for the transfer function without zeros. X(s) 1 = 2 R(s) s + 7s + 9 Cross multiplying yields
(s 2 + 7s + 9)X(s) = R(s)
Taking the inverse Laplace transform assuming zero initial conditions, we get
x + 7 x + 9x = r
Defining the state variables as,
x1 = x x2 = x
Hence,
x1 = x2 x2 = x = -7 x - 9x + r = -9x1 - 7x2 + r
Using the zeros of the transfer function, we find the output equation to be,
c = 2 x + x = x1 + 2x2
Putting all equation in vector-matrix form yields,
0 1 0 x= x + 1 r -9 -7 c = [1 2]x
3.4. The state equation is converted to a transfer function using
G(s) = C(sI - A)-1 B
where
(1)
12
Solutions to Skill-Assessment Exercises
-4 -1.5 2 A= , B = , and C = [1.5 0.625] . 0 0 4
Evaluating (sI - A) yields
s + 4 1.5 (sI - A) = s -4
Taking the inverse we obtain
(sI - A)-1 =
1 s -1.5 s 2 + 4s + 6 4 s + 4
Substituting all expressions into Eq. (1) yields
G(s) =
3.5.
3s + 5 s + 4s + 6
2
Writing the differential equation we obtain d2x + 2x 2 = 10 + f (t) dt 2 (1)
Letting x = xo + x and substituting into Eq. (1) yields d 2 (xo + x) + 2(xo + x)2 = 10 + f (t) 2 dt Now, linearize x 2 . (xo + x)2 - xo 2 = from which d(x 2 ) x = 2xo x dx x o (2)
(xo + x)2 = xo 2 + 2xo x
(3)
Substituting Eq. (3) into Eq. (1) and performing the indicated differentiation gives us the linearized intermediate differential equation, d 2 x + 4xo x = -2xo 2 + 10 + f (t) (4) dt 2 The force of the spring at equilibrium is 10 N. Thus, since F = 2x 2 ,
10 = 2xo 2
from which
xo = 5
Chapter 3
13
Substituting this value of xo into Eq. (4) gives us the final linearized differential equation. d 2 x + 4 5 x = f (t) dt 2 Selecting the state variables,
x1 = x x2 = x
Writing the state and output equations
x1 = x2 x2 = x = -4 5x1 + f (t) y = x1
Converting to vector-matrix form yields the final result as
1 0 0 x + f (t) x= -4 5 0 1 y = [1 0]x
14
Chapter 4
4.1. For a step input C(s) 10(s ) 4)(s ) 6) A B C D E = + + + + s(s ) 1)(s ) 7)(s ) 8)(s ) 10) s s + 1 s + 7 s + 8 s + 10
Taking the inverse Laplace transform,
c(t) = A + Be -t + Ce -7t + De -8t + Ee -10t
4.2. Since a = 50 , Tc = 4.3. a. Since poles are at 6 j19.08, c(t) = A + Be -6t cos(19.08t + ) . b. Since poles are at 78.54 and 11.46, c(t) = A + Be -78.54t + Ce -11.4t . c. Since poles are double on the real axis at 15 c(t) = A + Be -15t + Cte -15t . d. Since poles are at j25, c(t) = A + Bcos(25t + ) . 4.4. a. n = 400 = 20 and 2 n = 12; = 0.3 and system is underdamped. b. n = 900 = 30 and 2 n = 90; = 1.5 and system is overdamped. c. n = 225 = 15 and 2 n = 30; = 1 and system is critically damped. d. n = 625 = 25 and 2 n = 0; = 0 and system is undamped. 4.5.
1 1 4 4 2.2 2.2 = = 0.02 s; T s = = = 0.08 s; and Tr = = = 0.044 s. a 50 a 50 a 50
n = 361 = 19 and 2 n = 16; = 0.421.
Now, T s =
4 = 0.5 s and T p = = 0.182 s. n n 1 - 2
From Figure 4.16, n T r = 1.4998. Therefore, Tr = 0.079 s .
-
Finally, %os = e
1 - 2
*100 = 23.3%
Chapter 4
15
4.6. a. The second-order approximation is valid, since the dominant poles have a real part of 2 and the higher-order pole is at 15, i.e. more than five-times further. b. The second-order approximation is not valid, since the dominant poles have a real part of 1 and the higher-order pole is at 4, i.e. not more than five-times further. 4.7. a. Expanding G(s) by partial fractions yields G(s) =
1 0.8942 1.5918 0.3023 . + - - s s + 20 s + 10 s + 6.5
But 0.3023 is not an order of magnitude less than residues of second-order terms (term 2 and 3). Therefore, a second-order approximation is not valid. b. Expanding G(s) by partial fractions yields G(s) =
1 0.9782 1.9078 0.0704 . + - - s s + 20 s + 10 s + 6.5
But 0.0704 is an order of magnitude less than residues of second-order terms (term 2 and 3). Therefore, a second-order approximation is valid. 4.8. See Figure 4.31 in the textbook for the Simulink block diagram and the output responses. 4.9.
1 s + 5 2 s -2 a. Since sI - A = , (sI - A)-1 = 2 . Also, s + 5s + 6 -3 s 3 s + 5 0 BU(s) = . 1 / (s + 1)
The state vector is X(s) = (sI - A)-1[x(0) + BU(s)] = 2(s 2 + 7s + 7) 1 . (s + 1)(s + 2)(s + 3) s 2 - 4s - 6
5s 2 + 2s - 4 0.5 12 17.5 =- - + . The output is Y(s) = [1 3]X(s) = s +1 s + 2 s + 3 (s + 1)(s + 2)(s + 3)
Taking the inverse Laplace transform yields y(t) = -0.5e -t - 12e -2t + 17.5e -3t . b. The eigenvalues are given by the roots of sI - A = s 2 + 5s + 6 , or 2 and 3.
16
Solutions to Skill-Assessment Exercises
4.10.
1 s + 5 2 s -2 a. Since (sI - A) = , (sI - A)-1 = 2 . Taking the Laplace s + 5s + 4 -2 s 2 s + 5
transform of each term, the state transition matrix is given by
4 e -t - 1 e -4t 3 (t) = 3 2 -t 2 -4t - e + e 3 3 2 -t 2 -4t e - e 3 3 1 -t 4 -4t . - e + e 3 3 2 -(t - ) 2 -4(t - ) e - e 0 3 3 and Bu( ) = -2 , 1 4 e - e -(t - ) + e -4(t - ) 3 3
4 e -(t - ) - 1 e -4(t - ) 3 b. Since (t - ) = 3 2 -(t - ) 2 -4(t - ) + e - e 3 3 2 e - e -t - 2 e 2 e -4t 3 (t - )Bu( ) = 31 4 2 -4t . - -t - e e + e e 3 3
10 e -t - e -2t - 4 e -4t 3 Thus, x(t) = (t)x(0) + (t - )Bu( )d = 35 8 -4t . -t -2t 0 - e + e + e 3 3
t
c. y(t) = [2 1]x = 5e -t - e -2t
17
Chapter 5 5.1. Combine the parallel blocks in the forward path. Then, push pickoff point.
s
1 to the left past the s
R( s)
+
s +
2
1 s
1 s
C(s)
-
s
Combine the parallel feedback paths and get 2s. Then, apply the feedback s3 + 1 formula, simplify, and get, T ( s) = 4 . 2s + s2 + 2s 5.2. Find the closed-loop transfer function, T(s) = where G(s) = G(s) 16 = 2 , 1 + G(s)H(s) s + as + 16
16 and H(s) = 1. Thus, n = 4 and 2 n = a , from which s(s + a)
a = . But, for 5% overshoot, = 8
a = 5.52 . 5.3. Label nodes.
% ) a 100 = 0.69. Since, = , % 8 2 + ln 2 ( ) 100 - ln(
18
Solutions to Skill-Assessment Exercises
N1 ( s)
N2 ( s)
N3 ( s )
N4 ( s)
N5 ( s)
N6 ( s)
N7 ( s)
Draw nodes.
R( s ) N1 ( s) N2 ( s) N5 ( s) N7 ( s) N3 ( s ) N6 ( s) N4 ( s) C ( s)
Connect nodes and label subsystems.
-1
R(s ) 1 N ( s) N ( s) 1 s s 2
1 N3( s) N4 ( s) s 1
C ( s)
1 -1 N5 (s) 1 s N6 ( s)
1
N ( s) 7
s
Eliminate unnecessary nodes.
-1
R(s)
1
s
s
1 s
C(s)
1 s -s
5.4. Forward-path gains are G1G2 G3 and G1G3 .
Chapter 5
19
Loop gains are -G1G2 H1 , -G2 H2 , and -G3 H3 . Nontouching loops are [-G1G2 H1 ][-G3 H3 ] = G1G2 G3 H1 H3 and [-G2 H2 ][-G3 H3 ] = G2 G3 H2 H3 . Also, = 1 + G1G2 H1 + G2 H2 + G3 H3 + G1G2 G3 H1 H3 + G2 G3 H2 H3 . Finally, 1 = 1 and 2 = 1 .
C(s) = Substituting these values into T(s) = R(s)
T(s) =
T
k k
k
yields
G1 (s)G3 (s)[1 + G2 (s)] [1 + G2 (s)H2 (s) + G1 (s)G2 (s)H1 (s)][1 + G3 (s)H3 (s)]
5.5. The state equations are,
x1 = -2x1 + x2 x2 = -3x2 + x3 x3 = -3x1 - 4x2 - 5x3 + r y = x2
Drawing the signal-flow diagram from the state equations yields
1
1 s 1 s
r
1
x
3
1
x
2
1
1 s
x
1
y
-5
-3
-2
-4 -3
5.6. From G(s) =
100(s + 5) we draw the signal-flow graph in controller canonical s 2 + 5s + 6
form and add the feedback.
20
Solutions to Skill-Assessment Exercises
100
r
1 500 y -5 -6
-1
Writing the state equations from the signal-flow diagram, we obtain
. -105 -506 1 x= x + 0 r 0 1 y = [100 500]x
5.7. From the transformation equations,
3 -2 P -1 = 1 -4
Taking the inverse,
0.4 -0.2 P= 0.1 -0.3
Now,
3 0.4 -0.2 6.5 -8.5 3 -2 1 P -1AP = = 1 -4 -4 -6 0.1 -0.3 9.5 -11.5 3 -2 1 -3 P -1B = = 1 -4 3 -11 0.4 -0.2 CP = [1 4] = [0.8 -1.4] 0.1 -0.3
Therefore,
6.5 -8.5 -3 z= z + u 9.5 -11.5 -11 y = [0.8 -1.4]z
Chapter 5
21
5.8. First find the eigenvalues. I - A = 0 3 - 1 -3 0 1 = 2 + 5 + 6 - -4 -6 = 4 +6
From which the eigenvalues are 2 and 3. Now use Ax i = x i for each eigenvalue, . Thus,
3 x1 x1 1 x = x -4 -6 2 2
For = -2,
3x1 + 3x2 = 0 -4x1 - 4x2 = 0
Thus x1 = -x2 For = -3
4x1 + 3x2 = 0 -4x1 - 3x2 = 0
Thus x1 = -x2 and x1 = -0.75x2 ; from which we let
0.707 -0.6 P= -0.707 0.8
Taking the inverse yields
5.6577 4.2433 P -1 = 5 5
Hence,
3 0.707 -0.6 -2 0 5.6577 4.2433 1 D = P -1 AP = -4 -6 -0.707 0.8 = 0 -3 5 5 5.6577 4.2433 1 18.39 = P -1B = 5 3 20 5 0.707 -0.6 CP = [1 4] = [ -2.121 2.6] -0.707 0.8
22
Solutions to Skill-Assessment Exercises
Finally,
-2 0 18.39 z= u z + 0 -3 20 y = [ -2.121 2.6]z
23
Chapter 6 6.1. Make a Routh table. s7 s6 s5 s4 s3 s2 s1 s0 3 9 4.666666667 -4.35714286 12.90163934 10.17026684 -1.18515742 6 6 4 4.333333333 8 6.426229508 6 0 0 7 8 0 6 0 0 0 0 2 6 0 0 0 0 0 0
Since there are four sign changes and no complete row of zeros, there are four right half-plane poles and three left half-plane poles. 6.2. Make a Routh table. We encounter a row of zeros on the s 3 row. The even polynomial is contained in the previous row as -6s 4 + 0s 2 + 6 . Taking the derivative yields -24s 3 + 0s . Replacing the row of zeros with the coefficients of the derivative yields the s 3 row. We also encounter a zero in the first column at the s 2 row. We replace the zero with and continue the table. The final result is shown now as s6 s5 s4 s3 s2 s1 s0 1 1 -6 -24 144/ 6 -6 0 0 0 6 0 0 -1 -1 6 0 0 0 0 6 0 0 0 ROZ 0 0 0
There is one sign change below the even polynomial. Thus the even polynomial (4th order) has one right half-plane pole, one left half-plane pole, and 2 imaginary axis poles. From the top of the table down to the even polynomial yields one sign change. Thus, the rest of the polynomial has one right half-plane root, and one left
24
Solutions to Skill-Assessment Exercises
half-plane root. The total for the system is two right half-plane poles, two left half-plane poles, and 2 imaginary poles. 6.3. Since G(s) = K(s + 20) G(s) K(s + 20) = 3 , T(s) = 2 s(s + 2)(s + 3) 1 + G(s) s + 5s + (6 + K)s + 20K
(6 + K)
Form the Routh table. s3 s2 s1 1 5
20K
30 - 15K 5
s0 20K
From the s1 row, K < 2 . From the s0 row, K > 0. Thus, for stability, 0 < K < 2 . 6.4. First find -1 -1 s 0 0 2 1 1 (s - 2) 0 s 0 - 1 7 1 = -1 sI - A = (s - 7) -1 = s 3 - 4s 2 - 33s + 51 3 -4 (s + 5) 0 0 s -3 4 -5 Now form the Routh table. s3 s2 S1 S0 1 -4 -20.25 51 -33 51
There are two sign changes. Thus, there are two rhp poles and one lhp pole.
25
Chapter 7 7.1. a. First check stability.
T(s) = G(s) 10s 2 + 500s + 6000 10(s + 30)(s + 20) = 3 = 2 1 + G(s) s + 70s + 1375s + 6000 (s + 26.03)(s + 37.89)(s + 6.085)
Poles are in the lhp. Therefore, the system is stable. Stability also could be checked via Routh-Hurwitz using the denominator of T(s) . Thus, 15u(t): estep () = 15 15 = =0 1 + lim G(s) 1 +
s0
15tu(t): eramp () =
15 15 = = 2.1875 10 * 20 * 30 lim sG(s) s0 25 * 35 15 30 30 15t 2u(t): e parabola () = = = , since L [15t 2 ] = 3 2 lim s G(s) 0 s
s0
b. First check stability.
T(s) = G(s) 10s 2 + 500s + 6000 = 5 1 + G(s) s + 110s 4 + 3875s 3 + 4.37e04s 2 + 500s + 6000
=
10(s + 30)(s + 20) (s + 50.01)(s + 35)(s + 25)(s 2 - 7.189e - 04s + 0.1372)
From the second-order term in the denominator, we see that the system is unstable. Instability could also be determined using the Routh-Hurwitz criteria on the denominator of T(s) . Since the system is unstable, calculations about steadystate error cannot be made. 7.2. a. The system is stable, since T(s) = G(s) 1000(s + 8) 1000(s + 8) = = 2 and is of 1 + G(s) (s + 9)(s + 7) + 1000(s + 8) s + 1016s + 8063
Type 0. Therefore,
K p = lim G(s) =
s0
1000 * 8 = 127; Kv = lim sG(s) = 0; and Ka = lim s 2 G(s) = 0 s0 s0 7*9
b. estep () =
1 1 = = 7.8e - 03 1 + lim G(s) 1 + 127
s0
26
Solutions to Skill-Assessment Exercises
eramp () =
1 1 = = lim sG(s) 0
s0
e parabola () =
7.3.
1 1 = = 2 lim s G(s) 0
s0
System is stable for positive K. System is Type 0. Therefore, for a step input
estep () =
12K 1 ; from = 0.1. Solving for K p yields K p = 9 = lim G(s) = s0 14 *18 1 + Kp
which we obtain K = 189 . 7.4. System is stable. Since G1 (s) = 1000, and G2 (s) =
eD () = -
(s + 2) , (s + 4)
1 1 =- = -9.98e - 04 1 2 + 1000 lim + lim G1 (s) s0 G (s) s0 2
7.5. System is stable. Create a unity-feedback system, where He (s) = The system is as follows:
1 -s . -1 = s +1 s +1
R(s) -
+ -
Ea(s)
100 s +4
C(s)
-s s +1
Thus, 100 G(s) 100(s + 1) (s + 4) Ge (s) = = = 2 100s 1 + G(S)He (s) 1 - S - 95s + 4 (s + 1)(s + 4) Hence, the system is Type 0. Evaluating K p yields
Chapter 7
27
Kp =
100 = 25 4
1 1 = = 3.846e - 02 1 + K P 1 + 25
The steady-state error is given by
estep () =
7.6. Since G(s) = 1 1 10 K(s + 7) = = , e() = . 1 + K p 1 + 7K 10 + 7K s + 2s + 10 10
2
Calculating the sensitivity, we get
Se:K =
K e K 7K (-10)7 = =- 2 e K 10 (10 + 7K) 10 + 7K 10 + 7K
7.7. Given
1 0 1 0 A= ; B = 1 ; C = [1 1]; R(s) = s . -3 -6
Using the final value theorem, s -1 0 estep () = lim sR(s)[1 - C(sI - A) B] = lim[1 - [1 1] ] s0 s0 3 s + 6 1 s + 6 1 -3 s 0 s 2 + 5s + 2 2 ] = lim 2 = lim[1 - [1 1] 2 = s0 s + 6s + 3 1 s0 s + 6s + 3 3
-1 -1
Using input substitution,
0 1 step () = 1 + CA B = 1 - [1 1] -3 -6 -6 -1 3 0 0 = 1 + [1 1] 1 = 1 + [1 3
-1
-1
0 1
- 1 2 1] 3 = 0 3
28
Chapter 8
8.1. a. F(-7 + j9) = (-7 + j9 + 2)(-7 + j9 + 4)0.0339 (-5 + j9)(-3 + j9) = (-7 + j9)(-7 + j9 + 3)(-7 + j9 + 6) (-7 + j9)(-4 + j9)(-1 + j9) = (-66 - j72) = -0.0339 - j0.0899 = 0.096 < -110.7o (944 - j378)
j (-7+j9)
b. The arrangement of vectors is shown as follows:
s-plane M1 M5
M2
M3 M4
-7
X -6
-5
-4
X -3
-2
-1
X 0
From the diagram,
F(-7 + j9) = M2 M4 (-3 + j9)(-5 + j9) = M1 M3 M5 (-1 + j9)(-4 + j9)(-7 + j9)
= 8.2.
(-66 - j72) = -0.0339 - j0.0899 = 0.096 < -110.7o (944 - j378)
a. First draw the vectors.
Chapter 8
29
j
X
j3
j2
s-plane j1
-3 -2 -1 0
-j1
-j2
X
-j3
From the diagram,
angles = 180
o
-3 -3 - tan -1 - tan -1 = 180o - 108.43o + 108.43o = 180o . -1 1
b. Since the angle is 1800, the point is on the root locus.
pole lengths = c. K = zero lengths
8.3.
(
12 + 32
)(
1
12 + 32
) = 10
First, find the asymptotes.
a = a =
poles - zeros = (-2 - 4 - 6) - (0) = -4
# poles-# zeros 3-0 (2k + 1) 5 = , , 3 3 3
Next draw root locus following the rules for sketching.
30
Solutions to Skill-Assessment Exercises
5
4
3
2
1 Imag Axis
0
-1
-2
-3
-4
-5 -8
-7
-6
-5
-4
-3 -2 Real Axis
-1
0
1
2
3
8.4. a.
j
j3
X
s-plane
O -2
0 2
-j3
X
b. Using the Routh-Hurwitz criteria, we first find the closed-loop transfer function. T(s) = K(s + 2) G(s) = 2 1 + G(s) s + (K - 4)s + (2K + 13)
Using the denominator of T(s), make a Routh table.
Chapter 8
31
s2 s1 s0
1 K-4 2K+13
2K+13 0 0
We get a row of zeros for K = 4. From the s2 row with K = 4, s2 + 21 = 0. From which we evaluate the imaginary axis crossing at c. From part (b), K = 4. d. Searching for the minimum gain to the left of 2 on the real axis yields 7 at a gain of 18. Thus the break-in point is at 7. e. First, draw vectors to a point close to the complex pole.
21 .
At the point close to the complex pole, the angles must add up to zero. Hence, angle from zero angle from pole in 4th quadrant angle from pole in 1st quadrant 3 = 1800, or tan -1 - 90o - = 180o . Solving for the angle of departure, = 4 233.1.
32
Solutions to Skill-Assessment Exercises
8.5. a.
= 0.5
j j4
s-plane
X
o
-3 0 2
o
4
X
-j4
b. Search along the imaginary axis and find the 1800 point at s = j4.06 . c. For the result in part (b), K = 1. d. Searching between 2 and 4 on the real axis for the minimum gain yields the break-in at s = 2.89 . e. Searching along = 0.5 for the 1800 point we find s = -2.42 + j4.18 . f. For the result in part (e), K = 0.108. g. Using the result from part (c) and the root locus, K < 1. 8.6. a.
= 0.591 s-plane j
X -6
X -4
X -2
0
Chapter 8
33
b. Searching along the = 0.591 (10% overshoot) line for the 1800 point yields - 2.028+j2.768 with K = 45.55. c. T s =
4 4 = = 1.97 s; T p = = = 1.13 s; Re 2.028 Im 2.768
n Tr = 1.8346 from the rise-time chart and graph in Chapter 4. Since n is the
radial distance to the pole, n = 2.0282 + 2.7682 = 3.431. Thus, Tr = 0.53 s; since the system is Type 0, K p =
estep () = 1 = 0.51. 1 + Kp
K 45.55 = = 0.949 . Thus, 2*4*6 48
d. Searching the real axis to the left of 6 for the point whose gain is 45.55, we find 7.94. Comparing this value to the real part of the dominant pole, -2.028, we find that it is not five times further. The second-order approximation is not valid. 8.7. Find the closed-loop transfer function and put it the form that yields pi as the root locus variable. Thus,
100 2 G(s) 100 100 T(s) = = = = s + 100 1 + G(s) s 2 + pi s + 100 (s 2 + 100) + pi s 1 + pi s s 2 + 100
Hence, KG(s)H(s) =
pi s . The following shows the root locus. s + 100
2
34
Solutions to Skill-Assessment Exercises
j s-plane X j10
O 0
X -j10
8.8. Following the rules for plotting the root locus of positive-feedback systems, we obtain the following root locus:
j
s-plane
o -4
X
X
X
-3
-2
-1
0
Chapter 8
35
8.9. The closed-loop transfer function is T(s) = denominator with respect to K yields K(s + 1) . Differentiating the s + (K + 2)s + K
2
2s
s s s + (K + 2) + (s + 1) = (2s + K + 2) + (s + 1) = 0 K K K
s -(s + 1) K s -K(s + 1) s = = , we get . Thus, Ss:K = . s K s(2s + K + 2) K (2s + K + 2) K
-10(s + 1) . s(s + 11)
Solving for
Substituting K = 20 yields Ss:K =
Now find the closed-loop poles when K = 20 . From the denominator of T(s) , s1,2 = -21.05, - 0.95, -when K = 20 . For the pole at 21.05,
s = s(Ss:K )
K -10(-21.05 + 1) = -21.05 0.05 = -0.9975 . -21.05(-21.05 + 11) K K -10(-0.95 + 1) = -0.95 0.05 = -0.0025 . -0.95(-0.95 + 11) K
For the pole at 0.95,
s = s(Ss:K )
36
Chapter 9
9.1. a. Searching along the 15% overshoot line, we find the point on the root locus at 3.5 + j5.8 at a gain of K = 45.84. Thus, for the uncompensated system, Kv = lim sG(s) = K / 7 = 45.84 / 7 = 6.55 .
s0
Hence, eramp _ uncompensated () = 1 / Kv = 0.1527 . b. Compensator zero should be 20x further to the left than the compensator pole. Arbitrarily select Gc (s) = (s + 0.2) . (s + 0.01)
c. Insert compensator and search along the 15% overshoot line and find the root locus at 3.4 + j5.63 with a gain, K = 44.64. Thus, for the compensated system, Kv = d.
1 44.64(0.2) = 0.0078 . = 127.5 and eramp _ compensated () = Kv (7)(0.01)
eramp _ uncompensated 0.1527 = = 19.58 0.0078 eramp _ compensated
9.2. a. Searching along the 15% overshoot line, we find the point on the root locus at 3.5 + j5.8 at a gain of K = 45.84. Thus, for the uncompensated system,
Ts = 4 4 = = 1.143 s . Re 3.5
b. The real part of the design point must be three times larger than the uncompensated pole's real part. Thus the design point is 3(-3.5) + j 3(5.8) = -10.5 + j17.4. The angular contribution of the plant's poles and compensator zero at the design point is 130.80. Thus, the compensator pole must contribute 1800 130.80 = 49.20. Using the following diagram,
Chapter 9
37
j
j17.4 s-plane 49.20 -pc -10.5
we find
17.4 = tan 49.2o , from which, pc = 25.52. Adding this pole, we find p c - 10.5
the gain at the design point to be K = 476.3. A higher-order closed-loop pole is found to be at 11.54. This pole may not be close enough to the closed-loop zero at 10. Thus, we should simulate the system to be sure the design requirements have been met. 9.3. a. Searching along the 20% overshoot line, we find the point on the root locus at 3.5 + 6.83 at a gain of K = 58.9. Thus, for the uncompensated system,
Ts = 4 4 = = 1.143 s . Re 3.5
s0
b. For the uncompensated system, Kv = lim sG(s) = K / 7 = 58.9 / 7 = 8.41. Hence,
eramp _ uncompensated () = 1 / Kv = 0.1189 .
c. In order to decrease the settling time by a factor of 2, the design point is twice the value, uncompensated or 7 + j13.66. Adding the angles from the plant's poles and the compensator's zero at 3 to the design point, we obtain 100.80. Thus, the compensator pole must contribute 1800 100.80 = 79.20. Using the following diagram,
38
Solutions to Skill-Assessment Exercises
j j13.66
s-plane 79.20 -pc -7
we find
13.66 = tan 79.2o , from which, pc = 9.61. Adding this pole, we find the pc - 7
gain at the design point to be K = 204.9. Evaluating Kv for the lead-compensated system:
Kv = lim sG(s)Glead = K(3) / [(7)(9.61)] = (204.9)(3) / [(7)(9.61)] = 9.138 .
s0
Kv for the uncompensated system was 8.41. For a 10x improvement in steadystate error, Kv must be (8.41)(10) = 84.1. Since lead compensation gave us Kv = 9.138, we need an improvement of 84.1/9.138 = 9.2. Thus, the lag compensator zero should be 9.2x further to the left than the compensator pole. Arbitrarily select Gc (s) = (s + 0.092) . (s + 0.01)
Using all plant and compensator poles, we find the gain at the design point to be K = 205.4. Summarizing the forward path with plant, compensator, and gain yields Ge (s) = 205.4(s + 3)(s + 0.092) . s(s + 7)(9.61)(s + 0.01)
Higher-order poles are found at 0.928 and 2.6. It would be advisable to simulate the system to see if there is indeed pole-zero cancellation. 9.4. The configuration for the system is shown in the figure below.
Chapter 9
39
R(s) + K -
+ -
1 s( s + 7)(s + 10)
C(s)
Kf s
Minor-Loop Design: For the minor loop, G(s)H(s) =
Kf . Using the following diagram, we (s + 7)(s + 10)
find that the minor-loop root locus intersects the 0.7 damping ratio line at 8.5 + j8.67. The imaginary part was found as follows: = cos-1 = 45.570. Hence,
Im = tan 45.570 , from which Im = 8.67. 8.5
= 0.7 (-8.5 + j8.67) j
Im
s-plane X -7
X -10
-8.5
The gain, K f , is found from the vector lengths as
K f = 1.52 + 8.672 1.52 + 8.672 = 77.42
Major-Loop Design: Using the closed-loop poles of the minor loop, we have an equivalent forwardpath transfer function of Ge (s) = K K = . 2 s(s + 8.5 + j8.67)(s + 8.5 - j8.67) s(s + 17s + 147.4)
40
Solutions to Skill-Assessment Exercises
Using the three poles of Ge (s) as open-loop poles to plot a root locus, we search along = 0.5 and find that the root locus intersects this damping ratio line at 4.34 + j7.51 at a gain, K = 626.3. 9.5. a. An active PID controller must be used. We use the circuit shown in the following figure:
where the impedances are shown below as follows:
Matching the given transfer function with the transfer function of the PID controller yields
1 R (s + 0.1)(s + 5) s + 5.1s + 0.5 0.5 C RC Gc (s) = = = s + 5.1 + = - 2 + 1 + R2 C1s + 1 2 s s s s R1 C2 Equating coefficients
2
1 = 0.5 R1C2
R2 C1 = 1
(1) (2)
R2 C1 + = 5.1 (3) R1 C2 In Eq. (2) we arbitrarily let C1 = 10 -5 . Thus, R2 = 10 5 . Using these values along with Eqs. (1) and (3) we find C2 = 100 F and R1 = 20 k .
Chapter 9
41
b. The lag-lead compensator can be implemented with the following passive network, since the ratio of the lead pole-to-zero is the inverse of the ratio of the lag pole-to-zero:
Matching the given transfer function with the transfer function of the passive laglead compensator yields
1 1 s + s + R2 C2 R1C1 (s + 0.1)(s + 2) (s + 0.1)(s + 2) Gc (s) = = 2 = (s + 0.01)(s + 20) s + 20.01s + 0.2 1 1 1 1 + + s2 + s + R1 R2 C1C2 R1C1 R2 C2 R2 C1 Equating coefficients
1 = 0.1 R1C1 1 =2 R2 C2
(1) (2)
1 1 1 + + = 20.01 (3) R1C1 R2 C2 R2 C1 Substituting Eqs. (1) and (2) in Eq. (3) yields
1 = 17.91 R2 C1
(4)
Arbitrarily letting C1 = 100 F in Eq. (1) yields R1 = 100 k . Substituting C1 = 100 F into Eq. (4) yields R2 = 558 k . Substituting R2 = 558 k into Eq. (2) yields C2 = 900 F .
42
Chapter 10 10.1. a.
G(s) =
1 1 ; G( j ) = 2 (s + 2)(s + 4) (8 - ) + j6
M( ) = (8 - 2 )2 + (6 )2
6 For < 8, ( ) = -tan -1 . 8-2
6 For > 8, ( ) = - + tan -1 8 - 2 .
b.
Bode Diagrams
0 -20 -40
Phase (deg); Magnitude (dB)
-60 -80 -100 0 -50 -100 -150 -200 -1 10
10
0
10 Frequency (rad/sec)
1
10
2
Chapter 10
43
c.
Nyquist Diagrams
0.08 0.06 0.04
Imaginary Axis
0.02 0 -0.02 -0.04 -0.06 -0.08
-0.05
0
0.05 Real Axis
0.1
0.15
0.2
10.2.
Asymptotic
-40
20 log M
-60 -80 -100 -120 0.1
Actual
-20 dB/dec -40 dB/dec -20 dB/dec -40 dB/dec
1
10 Frequency (rad/s)
100
1000
-45o/dec
Phase (degrees)
-50 -100 -150
-90o/dec -45o/dec -90o/dec -45o/dec Asymptotic Actual -45o/dec
100 1000
-200 0.1 1 10 Frequency (rad/s)
44
Solutions to Skill-Assessment Exercises
10.3. The frequency response is 1/8 at an angle of zero degrees at = 0 . Each pole rotates 900 in going from = 0 to = . Thus, the resultant rotates 1800 while its magnitude goes to zero. The result is shown below.
Im
= 0
=0 1 8
Re
10.4. a. The frequency response is 1/48 at an angle of zero degrees at = 0 . Each pole rotates 900 in going from = 0 to = . Thus, the resultant rotates 2700 while its magnitude goes to zero. The result is shown below.
Im
= 6.63 = 0 - 1 480
=0 1 48
Re
b. Substituting j into G(s) =
1 1 = 3 and 2 (s + 2)(s + 4)(s + 6) s + 12s + 44s + 48
(48 - 12 2 ) - j(44 - 3 ) simplifying, we obtain G( j ) = 6 . The Nyquist + 56 4 + 784 2 + 2304
Chapter 10
45
diagram crosses the real axis when the imaginary part of G( j ) is zero. Thus, the Nyquist diagram crosses the real axis at 2 = 44 , or = 44 = 6.63 rad/s. At this frequency G( j ) = - 10.5. If K = 100, the Nyquist diagram will intersect the real axis at 100/480. Thus,
1 . Thus, the system is stable for K < 480 . 480
GM = 20 log
480 = 13.62 dB. From Skill-Assessment Exercise Solution 10.4, the 100
1800 frequency is 6.63 rad/s. 10.6. a.
-60 -80 -100
20 log M
-120 -140 -160 -180 1 10 Frequency (rad/s) 100 1000
0 -50
Phase (degrees)
-100 -150 -200 -250 -300 1 10 Frequency (rad/s) 100 1000
b. The phase angle is 180 at a frequency of 36.74 rad/s. At this frequency the gain is 99.67 dB. Therefore, 20 log K = 99.67 , or K = 96,270 . We conclude that the system is stable for K < 96,270 . c. For K = 10,000 , the magnitude plot is moved up 20 log10,000 = 80 dB. Therefore, the gain margin is 99.67- 80 = 19.67 dB. The 1800 frequency is 36.7
0
46
Solutions to Skill-Assessment Exercises
rad/s. The gain curve crosses 0 dB at = 7.74 rad/s, where the phase is 87.10. We calculate the phase margin to be 1800 87.10 = 92.90. 10.7. Using =
-ln(% / 100)
+ ln 2 (% / 100)
2
, we find = 0.456 , which corresponds to 20%
4 (1 - 2 2 ) + 4 4 - 4 2 + 2 = 5.79 rad/s. T s
overshoot. Using T s = 2 , BW = 10.8. For both parts find that G( j ) =
160 (6750000 - 101250 2 ) + j1350( 2 - 1350) . For a range of * 27 6 + 2925 4 + 1072500 2 + 25000000
values for , superimpose G( j ) on the a. M and N circles, and on the b. Nichols chart. a.
Im 3 G-plane
F = 20 o
2
M = 1.3 M = 1.0 1.4 1.5
40
1.6 1.8 2.0
30 o
25 o
o
1
70
50 o o
0.6 0.5 0.4
M = 0.7
Re
-70 -1
o o -40 o -30 o -25 o -20 o
-50
-2
-3 -4 -3 -2
-1
1
2
Chapter 10
47
b.
Nichols Charts
0.25 dB 0.5 dB 1 dB 3 dB 6 dB 0 dB -1 dB -3 dB -6 dB -12 dB -20 dB -40 dB
0
-50
Open-Loop Gain (dB)
-60 dB -80 dB
-100
-100 dB -120 dB -140 dB
-150
-160 dB -180 dB
-200
-200 dB -220 dB -240 dB
-350
-300
-250
-200
-150
-100
-50
0
Open-Loop Phase (deg)
Plotting the closed-loop frequency response from a. or b. yields the following plot:
48
Solutions to Skill-Assessment Exercises
0 -20
20 log M
-40 -60 -80 -100 -120 1 0 -50 10 Frequency (rad/s) 100 1000
Phase (degrees)
-100 -150 -200 -250 -300 1 10 Frequency (rad/s) 100 1000
10.9. The open-loop frequency response is shown in the following figure:
Chapter 10
49
Bode Diagrams
40 20 0 Phase (deg); Magnitude (dB) -20 -40
-100 -120 -140 -160
10
-1
10
0
10 Frequency (rad/sec)
1
10
2
The open-loop frequency response is 7 at = 14.5 rad/s. Thus, the estimated bandwidth is WB = 14.5 rad/s. The open-loop frequency response plot goes through zero dB at a frequency of 9.4 rad/s, where the phase is 151.980. Hence, the phase margin is 1800 151.980 = 28.020. This phase margin corresponds to
= 0.25 . Therefore, %OS = e
Ts = 4
- / 1- 2
(
) x100 = 44.4% ,
BW
(1 - 2 2 ) + 4 4 - 4 2 + 2 = 1.64 s and
Tp =
BW
1- 2
(1 - 2 2 ) + 4 4 - 4 2 + 2 = 0.33 s
10.10. The initial slope is 40 dB/dec. Therefore, the system is Type 2. The initial slope intersects 0 dB at = 9.5 rad/s. Thus, K a = 9.52 = 90.25 and K p = Kv = .
50
Solutions to Skill-Assessment Exercises
10.11. a. Without delay, G( j ) = 10 10 = , from which the zero dB j ( j + 1) (- + j) 10
frequency is found as follows: M =
2 +1
= 1. Solving for ,
2 + 1 = 10 , or after squaring both sides and rearranging, 4 + 2 - 100 = 0 .
Solving for the roots, 2 = -10.51, 9.51 . Taking the square root of the positive root, we find the 0 dB frequency to be 3.08 rad/s. At this frequency, the phase angle, = -(- + j) = -(-3.08 + j) = -162o . Therefore the phase margin is 1800 1620 = 180. b. With a delay of 3 s,
= -(- + j) - T = -(-3.08 + j) - (3.08)(3) = -162o - 9.24o = -171.24o .
Therefore the phase margin is 1800 171.240 = 8.760. c. With a delay of 7 s,
= -(- + j) - T = -(-3.08 + j) - (3.08)(7) = -162o - 21.56o = -183.56o .
Therefore the phase margin is 1800 183.560 = -3.560. Thus, the system is unstable. 10.12. Drawing judicially selected slopes on the magnitude and phase plot as shown below yields a first estimate.
Chapter 10
51
Experimental 20 10
Gain(dB) Phase(deg)
0 -10 -20 -30 -40 -45 -50 -55 -60 -65 -70 -75 -80 -85 -90 -95
1
2
3 4 5 6 7 8 10
20
30 40 50 70 100
200 300
500
1000
Frequency (rad/sec)
We see an initial slope on the magnitude plot of 20 dB/dec. We also see a final 20 dB/dec slope with a break frequency around 21 rad/s. Thus, an initial estimate is G1 (s) = 1 . s(s + 21)
Subtracting G1 (s) from the original frequency response yields the frequency response shown below.
52
Solutions to Skill-Assessment Exercises
Experimental Minus 1/s(s+21) 90 80 Gain(dB) Phase(deg) 70 60 50 40 100 80 60 40 20 0
1
2
3
4 5 6 7 8 10
20 30 40 50 70 100 Frequency (rad/sec)
200 300
500
1000
Drawing judicially selected slopes on the magnitude and phase plot as shown yields a final estimate. We see first-order zero behavior on the magnitude and phase plots with a break frequency of about 5.7 rad/s and a dc gain of about 44 dB = 20 log(5.7K) , or K = 27.8. Thus, we estimate G2 (s) = 27.8(s + 7) . Thus, G(s) = G1 (s)G2 (s) = 27.8(s + 5.7) . It is interesting to note that the original s(s + 21) 30(s + 5) . s(s + 20)
problem was developed from G(s) =
53
Chapter 11
11.1. The Bode plot for K = 1 is shown below.
Bode Diagrams
-60 -80 -100 -120 Phase (deg); Magnitude (dB) -140 -160 -180
-100
-150
-200
-250 10
-1
10
0
10
1
10
2
10
3
Frequency (rad/sec)
A 20% overshoot requires =
% - log 100 = 0.456 . This damping ratio 2 2 % + log 100
implies a phase margin of 48.10, which is obtained when the _ = -1800 + 48.10 = 131.90. This phase angle occurs at = 27.6 rad/s. The magnitude at this frequency is 5.15 x 10-6. Since the magnitude must be unity K =
1 = 194,200 . 5.15x10 -6
54
Solutions to Skill-Assessment Exercises
11.2. To meet the steady-state error requirement, K = 1,942,000. The Bode plot for this gain is shown below.
Bode Diagrams
60 40 20
Phase (deg); Magnitude (dB)
0 -20 -40
-100
-150
-200
-250 10
-1
10
0
10
1
10
2
10
3
Frequency (rad/sec)
A 20% overshoot requires =
% - log 100 = 0.456 . This damping ratio 2 2 % + log 100
implies a phase margin of 48.10. Adding 100 to compensate for the phase angle contribution of the lag, we use 58.10. Thus, we look for a phase angle of 1800 + 58.10 = -129.90. The frequency at which this phase occurs is 20.4 rad/s. At this frequency the magnitude plot must go through zero dB. Presently, the magnitude plot is 23.2 dB. Therefore draw the high frequency asymptote of the lag compensator at 23.2 dB. Insert a break at 0.1(20.4) = 2.04 rad/s. At this frequency, draw 20 dB/dec slope until it intersects 0 dB. The frequency of intersection will be the low frequency break or 0.141 rad/s. Hence the
Chapter 11
55
compensator is Gc (s) = Kc
(s + 2.04) , where the gain is chosen to yield 0 dB at (s + 0.141)
low frequencies, or Kc = 0.141 / 2.04 = 0.0691 . In summary, Gc (s) = 0.0691 11.3. A 20% overshoot requires = (s + 2.04) 1,942,000 and G(s) = . (s + 0.141) s(s + 50)(s + 120)
% - log 100 = 0.456 . The required 2 2 % + log 100
4 (1 - 2 2 ) + 4 4 - 4 2 + 2 = 57.9 T s
bandwidth is then calculated as BW =
rad/s. In order to meet the steady-state error requirement of Kv = 50 =
K , (50)(120)
we calculate K = 300,000 . The uncompensated Bode plot for this gain is shown below.
Bode Plot for K = 300000
40 20 0 Phase (deg); Magnitude (dB) -20 -40 -60
-100
-150
-200
-250 10
-1
10
0
10
1
10
2
10
3
Frequency (rad/sec)
56
Solutions to Skill-Assessment Exercises
The uncompensated system's phase margin measurement is taken where the magnitude plot crosses 0 dB. We find that when the magnitude plot crosses 0 dB, the phase angle is -144.80. Therefore, the uncompensated system's phase margin is -1800 + 144.80 = 35.20. The required phase margin based on the required damping ratio is M = tan -1
2 -2 2 + 1 + 4 4 = 48.1o . Adding a 100 correction factor, the
required phase margin is 58.10. Hence, the compensator must contribute max = 58.10 - 35.20 = 22.90. Using max = sin -1
1 - sin max 1- = 0.44 . The ,= 1 + sin max 1+
1 = 1.51. Now find the
compensator's peak magnitude is calculated as Mmax =
frequency at which the uncompensated system has a magnitude 1/ Mmax , or 3.58 dB. From the Bode plot, this magnitude occurs at max = 50 rad/s. The compensator's zero is at zc = compensator's pole is at pc =
1 1 . But, max = . Therefore, zc = 33.2 . The T T
1 z = c = 75.4. The compensator gain is chosen to T
yield unity gain at dc. Hence, Kc = 75.4 / 33.2 = 2.27 . Summarizing, Gc (s) = 2.27 11.4. A 10% overshoot requires = 300,000 (s + 33.2) , and G(s) = . s(s + 50)(s + 120) (s + 75.4)
% - log 100 = 0.591 . The required bandwidth 2 2 % + log 100
is then calculated as BW =
Tp
1- 2
(1 - 2 2 ) + 4 4 - 4 2 + 2 = 7.53 rad/s.
K , we (8)(30)
In order to meet the steady-state error requirement of Kv = 10 =
calculate K = 2400 . The uncompensated Bode plot for this gain is shown below.
Chapter 11
57
Bode Diagrams
40 20 0 -20 Phase (deg); Magnitude (dB) -40 -60 -80 -100
-100
-150
-200
-250 10
-1
10
0
10
1
10
2
10
3
Frequency (rad/sec)
Let us select a new phase-margin frequency at 0.8 BW = 6.02 rad/s. The required phase margin based on the required damping ratio is M = tan -1
2 -2 + 1 + 4
2 4
= 58.6o . Adding a 50 correction factor, the
required phase margin is 63.60. At 6.02 rad/s, the new phase-margin frequency, the phase angle is which represents a phase margin of 1800 138.30 = 41.70. Thus, the lead compensator must contribute max = 63.60 41.70 = 21.90. Using
max = sin -1
1 - sin max 1- = 0.456 . , = 1 + sin max 1+
We now design the lag compensator by first choosing its higher break frequency one decade below the new phase-margin frequency, that is, zlag = 0.602 rad/s. The lag compensator's pole is plag = zlag = 0.275 . Finally, the lag compensator's gain is Klag = = 0.456 .
58
Solutions to Skill-Assessment Exercises
Now we design the lead compensator. The lead zero is the product of the new phase margin frequency and plead =
, or zlead = 0.8 BW = 4.07. Also,
zlead 1 = 8.93. Finally, Klead = = 2.19. Summarizing, (s + 0.602) (s + 4.07) ; Glead (s) = 2.19 ; and K = 2400 . (s + 0.275) (s + 8.93)
Glag (s) = 0.456
59
Chapter 12 12.1. We first find the desired characteristic equation. A 5% overshoot requires =
% - log 100 = 0.69 . Also, n = = 14.47 rad/s. Thus, the Tp 1 - 2 2 2 % + log 100
characteristic equation is s 2 + 2 n s + n 2 = s 2 + 19.97s + 209.4 . Adding a pole at 10 to cancel the zero at 10 yields the desired characteristic equation,
(s 2 + 19.97s + 209.4)(s + 10) = s 3 + 29.97s 2 + 409.1s + 2094 . The compensated system
1 0 0 . The 0 0 1 matrix in phase-variable form is A - BK = -(k1 ) -(36 + k2 ) -(15 + k3 ) characteristic equation for this system is
sI - (A - BK)) = s 3 + (15 + k3 )s 2 + (36 + k2 )s + (k1 ) . Equating coefficients of this
equation with the coefficients of the desired characteristic equation yields the gains as K = [ k1 12.2. k2 k3 ] = [2094 373.1 14.97].
2 1 1 The controllability matrix is CM = B AB A B = 1 4 -9 . Since CM = 80 , 1 -1 16
[
2
]
CM is full rank, that is, rank 3. We conclude that the system is controllable. 12.3. First check controllability. The controllability matrix is
CMz
1 0 0 0 1 -17 . Since C = -1 , C is full rank, that is, rank = B AB A B = Mz Mz 1 -9 81
[
2
]
3. We conclude that the system is controllable. We now find the desired characteristic equation. A 20% overshoot
60
Solutions to Skill-Assessment Exercises
requires =
% - log 100 4 = 4.386 rad/s. Thus, the = 0.456 . Also, n = Ts 2 2 % + log 100
characteristic equation is s 2 + 2 n s + n 2 = s 2 + 4s + 19.24 . Adding a pole at 6 to cancel the zero at 6 yields the resulting desired characteristic equation,
(s 2 + 4s + 19.24)(s + 6) = s 3 + 10s 2 + 43.24s + 115.45 .
Since G(s) = s+6 (s + 6) = 3 , we can write the phase2 (s + 7)(s + 8)(s + 9) s + 24s + 191s + 504
1 0 0 0 0 ; B = 0 ; C = 6 1 0 . variable representation as A p = 0 1 ] p p [ -504 -191 -24 1
The compensated system matrix in phase-variable form is 0 1 0 . The characteristic equation for A p - Bp K p = 0 0 1 -(504 + k1 ) -(191 + k2 ) -(24 + k3 ) this system is sI - (A p - BpK p )) = s 3 + (24 + k3 )s 2 + (191 + k2 )s + (504 + k1 ) . Equating coefficients of this equation with the coefficients of the desired characteristic equation yields the gains as K p = [ k1 k2 k3 ] = [ -388.55 -147.76 -14] .
We now develop the transformation matrix to transform back to the z-system.
CMz = [Bz
A z Bz
1 0 0 A z 2 Bz ] = 0 1 -17 and 1 -9 81 1 0 0 0 1 -24 . A p Bp = 1 -24 385
2
CMp = Bp
[
A p Bp
]
Therefore, P = CMz CMx Hence,
-1
1 191 24 1 1 0 0 0 0 0 1 -17 24 1 0 = 7 1 0 = 1 -9 81 1 0 0 56 15 1
Chapter 12
61
0 0 1 -7 1 0 = -40.23 62.24 -14 . K z = K p P = [-388.55 -147.76 -14] ] [ 49 -15 1
-1
12.4. -(24 + l1 ) 1 0 For the given system e x = (A - LC)e x = -(191 + l2 ) 0 1 e x . The characteristic -(504 + l3 ) 0 0
polynomial is given by [sI - (A - LC) = s 3 + (24 + l1 )s 2 + (191 + l2 )s + (504 + l3 ) . Now we find the desired characteristic equation. The dominant poles from Skill-Assessment Exercise 12.3 come from (s 2 + 4s + 19.24) . Factoring yields (-2 + j3.9) and (-2 - j3.9). Increasing these poles by a factor of 10 and adding a third pole 10 times the real part of the dominant second-order poles yields the desired characteristic polynomial,
(s + 20 + j39)(s + 20 - j39)(s + 200) = s 3 + 240s 2 + 9921s + 384200 . Equating
coefficients of the desired characteristic equation to the system's characteristic
216 equation yields L = 9730 . 383696
12.5.
6 8 C 4 CA = -64 -80 -78 , where The observability matrix is OM = 2 CA 674 848 814 25 28 32 A = -7 -4 -11 . The matrix is of full rank, that is, rank 3, since OM = -1576 . 77 95 94
2
Therefore the system is observable. 12.6. The system is represented in cascade form by the following state and output equations:
-7 1 0 0 z = 0 -8 1 z + 0 u 0 0 -9 1 y = [1 0 0]z
62
Solutions to Skill-Assessment Exercises
The observability matrix is OMz
0 0 Cz 1 = Cz A z = -7 1 0 , where Cz A z 2 49 -15 1
49 -15 1 1 1 = 3 , we A = 0 64 -17 . Since G(s) = 2 (s + 7)(s + 8)(s + 9) s + 24s + 191s + 504 0 0 81
2 z
can write the observable canonical form as
-24 x = -191 -504 y = [1 0
1 0 0 0 1 x + 0 u 1 0 0 0]x
The observability matrix for this form is OMx
0 0 Cx 1 = Cx A x = -24 1 0 , where Cx A x 2 385 -24 1
-24 1 385 4080 -191 0 . A = 12096 -504 0
2 x
We next find the desired characteristic equation. A 10% overshoot requires =
% - log 100 4 = 67.66 rad/s. Thus, the = 0.591 . Also, n = Ts 2 2 % + log 100
characteristic equation is s 2 + 2 n s + n 2 = s 2 + 80s + 4578.42 . Adding a pole at 400, or 10 times the real part of the dominant second-order poles, yields the resulting desired characteristic equation,
(s 2 + 80s + 4578.42)(s + 400) = s 3 + 480s 2 + 36580s + 1.831x10 6 .
For the system represented in observable canonical form -(24 + l1 ) 1 0 e x = (A x - L x Cx )e x = -(191 + l2 ) 0 1 e x . The characteristic polynomial is given -(504 + l3 ) 0 0
by [sI - (A x - L x Cx ) = s 3 + (24 + l1 )s 2 + (191 + l2 )s + (504 + l3 ) . Equating coefficients of the desired characteristic equation to the system's characteristic equation yields
Chapter 12
63
456 L x = 36,389 . 1,830, 496
Now, develop the transformation matrix between the observer canonical and cascade forms. P = OMz -1OMx 0 0 1 0 0 1 0 0 1 0 0 1 -7 1 0 -24 1 0 = 7 1 0 -24 1 0 = 49 -15 1 385 -24 1 56 15 1 385 -24 1
-1
0 0 1 -17 1 0 . = 81 -9 1 0 0 456 456 456 1 Finally, L z = PL x = -17 1 0 36,389 = 28,637 28,640 . 81 -9 1 1,830, 496 1,539,931 1,540,000
12.7. We first find the desired characteristic equation. A 10% overshoot requires
=
% - log 100 = 0.591 2 2 % + log 100 .
Also, n =
Tp
= 1.948 rad/s. Thus, the characteristic equation is 1- 2
s 2 + 2 n s + n 2 = s 2 + 2.3s + 3.79 . Adding a pole at 4, which corresponds to the
original system's zero location, yields the resulting desired characteristic equation,
(s 2 + 2.3s + 3.79)(s + 4) = s 3 + 6.3s 2 + 13s + 15.16 .
x (A - BK) BKe x 0 x Now, = + r; and y = [C 0] , 0 x N 1 xN x N -C
where
0 1 0 A - BK = - [ k1 -7 -9 1
0 1 0 k2 ] = - -7 -9 k1
0 0 1 = -(7 + k ) -(9 + k ) k2 1 2
C = [ 4 1]
64
Solutions to Skill-Assessment Exercises
0 0 Bke = ke = 1 ke
x 0 1 0 x1 x1 1 -(7 + k ) -(9 + k ) k x + 0 r ; y = 4 1 0 x . Thus, x2 = [ ] 2 1 2 e 2 1 xN -1 0 xN x N -4
Finding the characteristic equation of this system yields 0 1 0 s 0 0 (A - BK) BKe - -(7 + k ) -(9 + k ) k sI - = 0 s 0 1 2 e 0 -C -1 0 0 0 s -4 -1 0 s (7 + k ) s + (9 + k ) -k = s 3 + (9 + k )s 2 + (7 + k + k )s + 4k = 1 2 e 2 1 e e 4 1 s Equating this polynomial to the desired characteristic equation,
s 3 + 6.3s 2 + 13s + 15.16 = s 3 + (9 + k2 )s 2 + (7 + k1 + ke )s + 4ke
Solving for the k's,
K = [2.21 -2.7] and ke = 3.79 .
65
Chapter 13 13.1.
f (t) = sin( kT); f * (t) = sin( kT) (t - kT);
k =0
F * (s) = sin( kT)e - kTs =
k =0
(e jkT - e - jkT )e - kTs 1 T ( s - j ) - k = ) - (e T ( s + j )- k (e 2 j k =0 2j k =0
But,
x
k =0
-k
=
1 1 - x -1
Thus,
F * (s) =
1 1 1 e -Ts e jT - e -Ts e - jT = 1 - -T (s - j ) -T (s + j ) 1 - (e -Ts e jT - e -Ts e - jT ) + e -2Ts 2 j 1 - e 1- e 2j
-Ts
=e
13.2. F(z) =
sin( T ) z -1 sin( T ) 1 - e -Ts 2 cos( T ) + e -2Ts = 1 - 2z -1 cos( T ) + z -2
z(z + 1)(z + 2) (z - 0.5)(z - 0.7)(z - 0.9)
(z + 1)(z + 2) F(z) = (z - 0.5)(z - 0.7)(z - 0.9) z
= 46.875 F(z) = 46.875
1 1 1 - 114.75 + 68.875 z - 0.5 z - 0.7 z - 0.9
z z z , - 114.75 + 68.875 z - 0.5 z - 0.7 z - 0.9
f (kT) = 46.875(0.5)k - 114.75(0.7)k + 68.875(0.9)k
13.3. Since G(s) = (1 - e -Ts ) 8 , s(s + 4)
B z - 1 2 2 8 z -1 A G(z) = (1 - z -1 )z z + z + = . = z s s + 4 z s s + 4 s(s + 4)
Let G2 (s) =
2 2 . Therefore, g2 (t) = 2 - 2e -4t , or g2 (kT) = 2 - 2e -4kT . + s s+4
2z 2z 2z(1 - e -4T ) - = . z - 1 z - e -4T (z - 1)(z - e -4T )
Hence, G2 (z) =
66
Solutions to Skill-Assessment Exercises
Therefore, G(z) = For T = 13.4.
z -1 2(1 - e -4T ) G2 (z) = . z (z - e -4T )
1.264 1 s, G(z) = . z - 0.3679 4
Add phantom samplers to the input, feedback after H(s) , and to the output. Push
G1 (s)G2 (s) , along with its input sampler, to the right past the pickoff point and
obtain the block diagram shown below.
Hence, T(z) = 13.5. Let G(s) =
G1G2 (z) . 1 + HG1G2 (z)
G(s) 20 4 4 20 = = - . Let G2 (s) = . Taking the inverse s s(s + 5) s s + 5 s+5
4z 4z 4z(1 - e -5T ) - = . z - 1 z - e -5T (z - 1)(z - e -5T )
Laplace transform and letting t = kT , g2 (kT) = 4 - 4e -5kT . Taking the z-transform yields G2 (z) = Now, G(z) =
z -1 4(1 - e -5T ) G(z) 4(1 - e -5T ) G2 (z) = = . Finally, T(z) = . z (z - e -5T ) 1 + G(z) z - 5e -5T + 4
The pole of the closed-loop system is at 5e -5T - 4 . Substituting values of T , we find that the pole is greater than 1 if T > 0.1022 s. Hence, the system is stable for 0 < T < 0.1022 s. 13.6. Substituting z =
s +1 into D(z) = z 3 - z 2 - 0.5z + 0.3 , we obtain s -1
D(s) = s 3 - 8s 2 - 27s - 6 . The Routh table for this polynomial is shown below.
Chapter 13
67
s3 s2 s1 s0
1 -8 -27.75 -6
-27 -6 0 0
Since there is one sign change, we conclude that the system has one pole outside the unit circle and two poles inside the unit circle. The table did not produce a row of zeros and thus, there are no j poles. The system is unstable because of the pole outside the unit circle. 13.7. Defining G(s) as G1 (s) in cascade with a zero-order-hold,
1/ 4 2/5 (s + 3) -Ts 3 / 20 G(s) = 20(1 - e -Ts ) = 20(1 - e ) s + (s + 4) - (s + 5) . s(s + 4)(s + 5)
Taking the z-transform yields 5(z -1) 8(z -1) (3 / 20)z (1 / 4)z (2 / 5)z G(z) = 20(1 - z -1 ) z -1 + z - e -4T - z - e -5T = 3 + z - e -4T - z - e -5T . Hence for T = 0.1 second, K p = lim G(z) = 3 , K v =
z1
1 lim(z -1)G(z) = 0 , and T z1
Ka =
1 lim(z -1)2 G(z) = 0 . Checking for stability, we find that the system is 2 z1 T
1.5z - 1.109 G(z) = 2 has poles 1 + G(z) z + 0.222z - 0.703
stable for T = 0.1 second, since T(z) =
inside the unit circle at 0.957 and +0.735. Again, checking for stability, we find that the system is unstable for T = 0.5 second, since T(z) = 3.02z - 0.6383 G(z) = 2 has poles inside and outside 1 + G(z) z + 2.802z - 0.6272
the unit circle at +0.208 and 3.01, respectively. 13.8. Draw the root locus superimposed over the = 0.5 curve shown below. Searching along a 54.30 line, which intersects the root locus and the = 0.5 curve, we find the point 0.58754.3o = (0.348+j0.468) and K = 0.31.
68
Solutions to Skill-Assessment Exercises
z-Plane Root Locus
1.5
1
0.5
Imag Axis
(0.348+j0.468) K=0.31 54.30
0
-0.5
-1
-1.5
-3
-2.5
-2
-1.5
-1
-0.5 Real Axis
0
0.5
1
1.5
2
13.9. Let Ge (s) = G(s)Gc (s) = 100K 342720(s + 25.3) 2.38(s + 25.3) = . s(s + 36)(s + 100) (s + 60.2) s(s + 36)(s + 100)(s + 60.2)
The following shows the frequency response of Ge ( j ) .
Chapter 13
69
Bode Diagrams
40 20 0 Phase (deg); Magnitude (dB) -20 -40 -60
-100
-150
-200
-250 10
-1
10
0
10
1
10
2
10
3
Frequency (rad/sec)
We find that the zero dB frequency, M , for Ge ( j ) is 39 rad/s. Using Astrom's guideline the value of T should be in the range, 0.15 / M = 0.0038 second to
0.5 / M = 0.0128 second. Let us use T = 0.001 second.
Now find the Tustin transformation for the compensator. Substituting s = into Gc (s) = Gc (z) = 2.34 13.10.
G c (z) = X(z) 1899z 2 - 3761z + 1861 = 2 . Cross-multiply and obtain E(z) z - 1.908z + 0.9075
2(z - 1) T(z + 1)
2.38(s + 25.3) with T = 0.001 second yields (s + 60.2) (z - 0.975) . (z - 0.9416)
(z 2 - 1.908z + 0.9075)X(z) = (1899z 2 - 3761z + 1861)E(z) . Solve for the highest
power of z operating on the output, X(z) , and obtain
z 2 X(z) = (1899z 2 - 3761z + 1861)E(z) - (-1.908z + 0.9075)X(z) . Solving for
70
Solutions to Skill-Assessment Exercises
X(z) on the left-hand side yields
X(z) = (1899 - 3761z -1 + 1861z -2 )E(z) - (-1.908z -1 + 0.9075z -2 )X(z) . Finally, we
implement this last equation with the following flow chart: