System dynamics 2021 Spring Final Examination PDF

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System dynamics 2021 Spring Final Examination...

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System Dynamics Final Spring 2021 Instructor: J. Koo

Note :

I. The energy variables should be used for the states. Use the provided variables when you write your final state equations. You could add your own subscripts to them if needed. Type

Mechanical Translation

Mechanical Rotation

Electrical

Hydraulic

Momentum

P

h

λ

P

Displacement

y or x

θ

Q

V

II. Canonical form of a state space equation, x˙

=

Ax + Bu

y

=

Cx + Du

(1)

1. For a mechanical system shown in Figure 1 where F (t) is a force input and output y(t) is the mass position.

Figure 1: A mkb system

(a) Construct a set of state equation in the form of equation (1). (b) Derive the transfer function G(s) relating the mass position y(t) and the input force F (t). (c) With the plant G(s), a feedback system that controls y(t) for a desired reference r(t) is constructed as shown in Figure 2 where the controller K (s) is organized as, K (s) = Kp + Kd · s +

Ki . s

Derive the closed-loop transfer function using equation (2). (d) Determine the characteristic equation of the closed loop and the one of open loop G(s) without control. 1

(2)

(e) For a unit step input r(t) = 1, determine the steady state outputs of y(t) for both plant G(s) only and the closed loop. (f) Discuss on the system steady state performance for the unit step input if it operates without integral control?

Figure 2: A closed-loop system

2. A system g(t)’s transfer function is, G(s) =

s+2 s (s2 + 2s + 5)

(3)

This is not a stable system. Explain the reason without any calculation! 3. A system g(t)’s transfer function is, G(s) =

s2 + K . s4 + 7s3 + 15s2 + (25 + K )s + 2K

(4)

Determine the stability of the system. If it is not stable, can this system be stabilized with a proper selection of K? If yes, determine the range of K for stability. 4. For the two mechanical systems A and B, shown in Figure 3, determine which system provides a better high frequency isolating performance when they are used as a low pass filter relating flows (velocities) input vi (t) and output vo (t). Justify your decision using state equations in the form of equation (1) and corresponding transfer functions followed by a frequency analysis. Build electrical Resistance-Inductance-Capacitance (RLC) systems which are equivalent to those given mechanical systems A and B respectively. Input and output should be clearly marked in those equivalent electrical systems. 5. A first order system is controlled with an integral controller as shown in Figure 4. Determine the controller gain K and system time constant τ to produce the closed-loop system unit step response as shown in Figure 5. 6. For a hydraulic tank shown in Figure 6, the height h(t) should be controlled for the flow input Qin (t). If a proportional control Kp is used for the control, (a) Determine the steady state height hss(t) of the open loop system for a unit step input of Qin (t). (b) Design a controller to make the system follow the unit step input. And show your system block diagram. And prove the controller works well for the task.

2

Figure 3: Two mechanical systems

Figure 4: A closed-loop system

3

Figure 5: System response for a unit step input

Figure 6: A hydraulic system

4...