03.Transfer functions (the s-domain)
.pdfExercises for Chapter 3 |
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E3.13 Consider the circuit of Exercise E1.9 with the output of Exercise E2.18.
(a)Compute the transfer function for the system.
(b)Determine all open-circuit pairs (u(t), y(t)).
(c)Determine all short-circuit pairs (u(t), y(t)).
(d)Comment on the open/short-circuit behaviour in terms of controllability, observability, and zero dynamics.
E3.14 For the coupled mass system of Exercises E1.4 and E2.19 (assume no damping), take as input the case of α = 0 described in Exercise E2.19—thus the input is a force u(t) applied to the leftmost mass. Determine an output that renders the system unobservable.
E3.15 Using Theorem 3.15, determine the spectrum of the zero dynamics for the pendulum/cart system of Exercises E1.5 and E2.4 for each of the following linearisations:
(a)the equilibrium point (0, 0) with cart position as output;
(b)the equilibrium point (0, 0) with cart velocity as output;
(c)the equilibrium point (0, 0) with pendulum angle as output;
(d)the equilibrium point (0, 0) with pendulum angular velocity as output;
(e)the equilibrium point (0, π) with cart position as output;
(f)the equilibrium point (0, π) with cart velocity as output;
(g)the equilibrium point (0, π) with pendulum angle as output;
(h)the equilibrium point (0, π) with pendulum angular velocity as output.
E3.16 Consider the double pendulum of Exercises E1.6 and E2.5. In the following cases, use Theorem 3.15 to determine the spectrum of the zero dynamics:
(a)the equilibrium point (0, 0, 0, 0) with the pendubot input;
(b)the equilibrium point (0, π, 0, 0) with the pendubot input;
(c)the equilibrium point (π, 0, 0, 0) with the pendubot input;
(d)the equilibrium point (π, π, 0, 0) with the pendubot input;
(e)the equilibrium point (0, 0, 0, 0) with the acrobot input;
(f)the equilibrium point (0, π, 0, 0) with the acrobot input;
(g)the equilibrium point (π, 0, 0, 0) with the acrobot input;
(h)the equilibrium point (π, π, 0, 0) with the acrobot input.
In each case, use the angle of the second link as output.
E3.17 Determine the spectrum of the zero dynamics for the linearised coupled tank system of Exercises E1.11 and E2.6 for the following outputs:
(a)the height in tank 1;
(b)the height in tank 2;
(c)the di erence of the heights in the tanks.
E3.18 Given the SISO linear system (N, D) in input/output form with
D(s) = s3 + 4s2 + s + 1, N(s) = 3s2 + 1,
determine the canonical minimal realisation ΣN,D. Is the triple (A, b, c) you found complete? Was your first impulse to answer the previous question by doing calculations? Explain why these are not necessary.
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E3.19 |
State and prove a version of Theorem 3.20 that assigns to a SISO system (N, D) in |
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input/output form a SISO linear system Σ = (A, b, ct, D) so that A and c are in |
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observer canonical form. Also produce the block diagram analogous to Figure 3.8 in |
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this case. |
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E3.20 |
Suppose you are handed a SISO linear system (N, D) in input/output form, and are |
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told that it comes from a SISO linear system Σ = (A, b, ct, 01)—that is, you are told |
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that TΣ = TN,D. Is it possible for you to tell whether (A, b) is controllable? |
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Hint: Consider Example 2.19 and Theorem 3.20. |
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E3.21 |
For a SISO linear system (N, D) in input/output form, show that the transfer function |
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TN,D has the property that TN,D(¯s) = TN,D(s) for all s C. In particular, show that |
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s0 C is a zero or pole of TN,D if and only if s¯0 is a zero or pole, respectively. |
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E3.22 |
Verify Theorem 3.22 and Proposition 3.24 for Exercise E3.4 (recall that you had |
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computed the impulse response for this problem in Exercise E2.26). |
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E3.23 |
For the following SISO linear systems in input/output form, use Proposition 3.32 to |
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obtain the output corresponding to the given input and initial conditions. |
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(a) (N(s), D(s)) = (1, s + 3), u(t) = 1(t) (the unit step input), and y(0) = 1. |
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(b) |
(N(s), D(s)) = (1, s + 3), u(t) = 1(t)eat, a R, and y(0) = 0. |
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(c) |
(N(s), D(s)) = (s, s3 + s), u(t) = 1(t) cos t, and y(0) = 1, y˙(0) = 0, and y¨(0) = 0. |
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(d) (N(s), D(s)) = (1, s2), u(t) = 1(t), and y(0) = 0 and y˙(0) = 1. |
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E3.24 |
For the SISO linear systems in input/output form from Exercise E3.23 for which you |
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obtained the solution, apply Proposition 3.27 to obtain the same solution. |
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E3.25 |
For the following SISO systems in input/output form, use Proposition 3.40 to setup |
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the initial value problem for the step response, and use a computer package to plot |
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the step response. |
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(a)(N(s), D(s)) = (s + 1, s2 + s + 1).
(b)(N(s), D(s)) = (s2 + 2s + 1, s3 + 3s + 1).
(c)(N(s), D(s)) = (s − 1, s4 + 15s3 + 20s2 + 10s + 2).
(d)(N(s), D(s)) = (s3 + 1, s5 + 9s4 + 20s3 + 40s2 + 50s + 25).
E3.26 Consider the di erential equation
y¨(t) + 4y˙(t) + 8y(t) = 2u˙(t) + 3u(t), |
(E3.1) |
where u(t) is a specified function of time. If we define
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(t) = |
1 , t [0, ] |
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(0, otherwise, |
and let y (t) be the solution to the di erential equation (E3.1) when u = u , determine lim →0 y (t). (Note that in this problem although u is in U , u˙ is not in U . Thus to compute “directly” the solution to the di erential equation (E3.1) with u = u is not actually something you know how to do!)
E3.27 Consider the SISO linear system Σ = (A, b, ct, D) given by
A = 0 , b = 1 , c = 1 , D = 0 ,
and let u(t) = 1(t)et2 .
Exercises for Chapter 3 |
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(a)Show that if x(0) = 0 then the output for the input u is y(t) = erf is the error function given by
erf(t) = √π Z0 |
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dτ. |
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e−τ |
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√
2iπ erf(it), where
(b)Laplace transform techniques are always limited by one’s ability to compute the inverse transform. Are there limitations in this example beyond the di culty in determining the inverse transform?