[試題] 101上 王富正 自動控制

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課程名稱︰自動控制
課程性質︰必修
課程教師︰王富正
開課學院：工學院
開課系所︰機械系
考試日期（年月日）︰101/10/30
考試時限（分鐘）：180
是否需發放獎勵金：是
（如未明確表示，則不予發放）

試題 :
Automatic Control
1st Midterm Exam, 30/Oct/2012

1. (5%) Given f(t) = e^-at sin(ωt), derive its Laplace Transform L[f(t)].
        (必須有推導過程)

2. (20%) Refer to Fig.2,
   (1) (5%) Derive the two dynamic equations of the system.
   (2) (5%) Find the transfer function of X(s)/U(s).
   (3) (5%) Define the system state z = [ x xdot y ydot ]^T , find the
            state-space model of the system.
   (4) (5%) Draw the corresponding block diagram of the state-space
            representation of (3).

                                               ┌─┤├─┐
                                               │    C2  │
      ██████                             ├─══─┤
         ︴   │                               │┌┐R2  │
      k1 ︴    ︴                              ││└┐  │
     M1 □     ︴k2              ●─══─┤├┴┤- └┐│
        ︱     ︴                     R1    C1   │    ├┴───●
    b1│┴│   ︴                V1            ┌┤+ ┌┘
      └┬┘  │                               ││┌┘          V2
      ┌┴──┴┐                             │└┘
      └─┬──┘M2             ●──────┼────────●
          ↓u(t)                               ┴

         Fig.2                                  Fig.3

3. (15%) Refer to Fig.3,
   (1) (5%) Find the transfer function T = V2(s)/V1(s).
   (2) (5%) Find the sensitivity of T to R1, i.e., S(T)(R1).
   (3) (5%) Suppose R1 = 1 kΩ, R2 = 2 kΩ, C1 = 1 mF, C2 = 0.1 mF,
            find V2(t) to a step input V1(t) = 1, t ≧ 0

4. (20%) Consider a transfer function
         G(s) = ( s - 2 ) / ( s^3 + 6s^2 + 11s + 6 ),
   (1) (5%) Derive its controller canonical form, and draw the corresponding
            block diagram.
   (2) (5%) Derive its observer canonical form, and draw the corresponding
            block diagram.
   (3) (5%) Derive its Jordan form, and draw the corresponding block diagram.
   (4) (5%) Find the state response of (3) the the initial state [ 1 -1 1 ]^T.


5. (20%) Consider the closed-loop system of Fig.5, with
         G(s) = ( s + 1 ) / ( s^2 + 5s + 6 )
   (1) (5%) Using a proportional (P) control K = Kp = constant, find the
            steady-state error to a step input R.
   (2) (5%) Using a proportional-integral (PI) control K = Kp + Ki/s,
            where Kp, Ki are constant, find the steady-state error to a step
            input R, and the steady-state error to a step disturbance Td.
   (3) (5%) If the disturbance is a sinusoidal signal Td = sin(ωt), the output
            response will be oscillatory with the above controllers. However,
            we can eliminate the oscillation by putting the unstable mode at
            the controller (it is so-called internal model principle) as
            K = α / ( s^2 + ω^2 ), where α = constant. Find the steady-state
            error to Td = sin(ωt).
   (4) (5%) Design a controller such that (i) the steady-state error to a step
            input R is zero; and (ii) the steady-state error to a sinusoidal
            disturbance Td = sin(60t) is zero.

                ┌─────┐
                │Controller│    Td(s)
        +       ├─────┤   +↓
 R(s) ─→○─→│    K     ├─→○──┬→ Y(s)
         -↑    └─────┘  +       │
          └──────────────┘

                    Fig.5

6. (20%) Consider the closed-loop system of Fig.5, with
         G(s) = 1 / ( s^2 + 3s + 2 ) and a proportional-integral-derivative
         (PID) control K = Kp + Ki/s + Kds, where Kp, Ki, Kd are constant.
   (1) (5%) Find the sensitivity of the closed-loop system to Ki i.e., S(T)(Ki),
            where T = T(R→Y).
   (2) (5%) To make the steady-state error to a unit ramp input (R(t)=t,t≧0)
            to be less than 0.1, find the minimal value of Ki.
   (3) (5%) Set the desired closed-loop poles at s = -3, -4, -5, find the
            required Kp,Ki,Kd values.
   (4) (5%) Suppose Kp = 17 , Ki = 12 , Kd = 5 , find the output response Y(t)
            to a unit step R(t)=1 , t≧0.

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