Homework Set #5- Solutions
Spring 2013 Control Systems 1
Name: ______
Aux-25 For the two systems with the input r(t) and output y(t) that are described by the following differential equations, find the Laplace transfer function for each system
Aux_26- For problem P2.2 on page 141 in the text book (page 126 in 11th Edition), write the two equations describing the system in Laplace notation (in terms of “s”). Essentially you are equating the forces acting on mass M1, and M2.
Aux_27Position control systems often experience disturbance inputs that causes the output to experience unwanted perturbations.
The disturbance effects can often be cancelled by a sensor, G1(s) that senses the disturbance input and generates a signal that negates the disturbance effects. Find the appropriate gain, G1(s), so that the disturbance N(s) has no effect on C(s). It helps if you assume the input R(s) = 0.
Solution: The output C(s) is given by
C(s) = Gc(s)G(s)E(s) + Gn(s)N(s)
Where
E(s) = R(s) – C(s) + G1(s) N(s)
We obtain
C(s) = Gc(s)G(s)[R(s) – C(s)] + [Gc(s)G(s) G1(s) + Gn(s)]N(s)
This last equation gives the output C(s) in terms of )[R(s) – C(s)] and disturbance N(s).
G1(s) is determined such that the effects of N(s) are eliminated in the output C(s). What this really means is that the inputs into the second summing junction must be equal and opposite. Thus
Gn(s) = - G1(s) Gc(s)G(s)
Solving for G1(s) gives G1(s) = - Gn(s)/[Gc(s)G(s)]
Aux_28 For the transfer function below: (a) Using Direct Decomposition construct a Simulation Diagram (That is one with integrators). On the Simulation Diagram identify the two state variables x1(s), x2(s), (b) From the Simulation Diagram construct the State Equations including the A, B, and C Matrices.
Solution Aux_28
(a) Using direct decomposition:
(b)
Aux_29-Work problem P2.7 on page 142 in the 12th Edition ( page 126 11th Edition).
Solution Aux_29.
For an ideal op-amp, the voltage gain (as a function of frequency) is
Aux_30 Using partial fraction expansion method, (a) Compute the residue associated with each of the denominator factors, and (b) Find the inverse Laplace transform for the expanded transfer function. That is find the time response x(t).
Aux-30 Solution
Aux_31For the transfer function below, derive the A, B, and C vector-matrices.
A B C
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Last printed 2/6/2013 12:53 PM