[Solved] Control System Homework2

1. Simulate spring-mass-damper system of unforced response with damping ratio equal to 0.3, 0.6, and 1 (see Fig. 2.46)

0.150.100.050

• 05
• 10
• 15

FIGURE 2.46

Spring-mass- _0.20

damper unforced 0 1 2 3 4 5 6 7 8

response. Time (s)

2. Simulate Example 2.19

-_-;AAMPLE 2. i Multiloop reduction

A rnultiloop feedback system is shown in Figure 2.26. Our objective is to compute the closed-loop transfer function, T(s), with

1 1

G4(S)

^{3(s) =} S^2______ + 4s + 4 = s

s + 1

Fl_i(s) = H₂(s) = 2, and //₃(s) = 1.

s +

For this example, a five-step procedure is followed:

Ul Step 1. Input the system transfer functions,

• Step 2. Move H₂(s) behind G4(s)-
• Step 3. Eliminate the G((S) G4 (S) (S)

F Step 4. Eliminate the loop containing 11₂(s).

• Step 5. Eliminate the remaining loop and calculate T(s).

The five steps are utilized in Figure 2.64, and the corresponding block diagram reduction is shown in Figure 2.27. The result of executing the commands is

s⁵ + 4s⁴ + 6s³ + 6s² + 5s + 2

sys = ____________________________________________________

123⁶ + 205s⁵ + 1066s⁴ + 2517s³ + 31.283² + 2196s + 712^.

We must be careful in calling this the closed-loop transfer function. The transfer function is defined as the inputoutput relationship after polezero cancellations. If we compute the poles and zeros of T(s), we find that the numerator and denominator polynomials have (s + 1) as a common factor. This must be canceled before we can claim we have the closed-loop transfer function. To assist us in the polezero cancellation, we will use the minreal function. The minreal function, shown in Figure 2.65, removes common polezero factors of a transfer function. The final step in the block reduction process is to cancel out the common factors, as shown in Figure 2.66. After the application of the minreal function, we find that the order of the denominator polynomial has been reduced from six to five, implying one polezero cancellation.

H2(s) G₄(s)

(;1(s) 020

GO)

H₃(s) rt____

(a)

______ 62(s)

_____ H₃(s)

( b

s O. GI(s)

1 lei4(s )il (s) + 62(.003(s)H2(s) IT

P

I 6₃(s)G₄C.011₁(.0+ Gi(s)G .(N)14(N) + G _i(s). ₂0 ),G ₃006 _k(s

G i(N)G 2(.0G 3( .0G 4i.-F)

G2(S)G3(5)G4(5)

(c

(d)

FIGURE 2.26 Block diagram reduction of the system of Figure 2.25_

FIGURE 2.27 Signal-flow graph of the DC motor.

FIGURE 2.64

block

reduction.

FIGURE 2.65 The minreal function.

FIGURE 2.66

Application of the minreal function.

3. Simulate Example 2.20

EXAMPLE 2.20 Electric traction motor control

Finally, let us reconsider the electric traction motor system from Example 2.13. The block diagram is shown in Figure 2.43(c). The objective is to compute the closed-loop transfer function and investigate the response of co(s) to a commanded w_d(s). The first step, as shown in Figure 2.67, is to compute the closed-loop transfer function co(s)/t_o_d(s) = T(s). The closed-loop characteristic equation is second order with cia_ii = 52 and = 0.012. Since the damping is low, we expect the response to be highly oscillatory. We can investigate the response w(t) to a reference input, co_d(t), by utilizing the step function. The step function, shown in Figure 2.68, calculates the unit step response of a linear system. The step function is very important, since control system performance specifications are often given in terms of the unit step response.

If the only objective is to plot the output, At), we can use the step function without left-hand arguments and obtain the plot automatically with axis labels. If we need y(t) for any purpose other than plotting, we must use the step function with left-hand arguments, followed by the plot function to plot y(t). We define t as a row vector containing the times at which we wish the value of the output variable y(I). We can also select t

_ = _ t

_final, which results in a step response from 1 = to t = t111 ₁ and the number of intermediate points are selected automatically.

EXAMPLE 2.13 Electric traction motor control

The electric motor drive is shown in block diagram form in Figure 2.43(a), incorporating the necessary control. The goal of the design is to obtain a system model and the closed-loop transfer function of the system, w(s)/co_d(s), select appropriate resistors 11, R2, R3, and R4, and then predict the system response.

The first step is to describe the transfer function of each block. We propose the use of a tachometer to generate a voltage proportional to velocity and to connect that voltage, v,, to one input of a difference amplifier, as shown in Figure 2.430). The power amplifier is nonlinear and can be approximately represented by v₂ (t) = 2e³v₁(t) = g(2,₁), an exponential function with a normal operating point, vto = 1.5 V. We then obtain a linear model

40)2(0 =	d(n)		vi(t) = 6e3vt)Av1(t) = 540 Avi(t). (2.120)

Taking the Laplace transform, yields

V₂(s) = 540AVi(s).

H Poweramplifier 1 Sonsor _____		A rrnatu re-cOnliCoilCdMAGI

v2= 2diri vio= 1.5

FIGURE 243 Speed control of an electric traction motor.

FIGUR E 2.67Electric traction motor block reduction.

Siep _₁1 System H _Output

input G(s)

{