MAE143C: Digital Control of Dynamic Systems (2012)

MAE143C: Digital Control of Dynamic Systems (2012) Midterm Part A: Linear Systems Review 45 minutes, no electronics, closed book, 1 page of notes allowed. Recall from §17.5.2 that the fact that each sinusoidal input to a stable system eventually leads to a sinusoidal output at the same frequency, but at a different magnitude and phase, is clearly seen if we consider what happens if we put a complex input u1 (t) = eiωt into a SISO system. [This is not possible in a physical experiment, but can easily be done mathematically if we know the transfer function G(s) of the stable system under consideration.] In this case, by the abbreviated table of Laplace transforms below, U1 (s) = 1/(s − p0) where p0 = iω, and thus the partial fraction expansion of the output Y1 (s) may be written Y1 (s) = G(s)U1 (s) =

d0 + other terms s − p0

y1 (t) = d0 eiωt + other terms.

⇔

(1a)

The “other terms” in the partial fraction expansion of Y1 (s) all have their poles in the LHP, because G(s) is stable, and thus the “other terms” in its inverse Laplace transform, y1 (t), are all stable. Thus, the magnitude and phase shift of the persistent component of the output is given by the magnitude and phase of the complex coefficient d0 , given (see Appendix B) by: h i d0 = Y1 (s) · (s − p0 )

h = G(s)

s=iω

i 1 · (s − p0) = G(iω). s − p0 s=iω

(1b)

The magnitude and phase of the Bode plot at any given ω are thus simply the magnitude and phase of G(iω). 1a. Consider now what happens if we put the complex input u2 (t) = e−iωt into the system. Compute the magnitude and phase shift of the persistent sinusoidal component of the output, y2 (t) = c0 e−iωt , in an analogous fashion. 1b. Now consider the real input u3 (t) = [u1 (t) + u2(t)]/2. First, write u3 (t) in terms of sines and cosines instead of complex exponentials. Then, by superposition, compute the magnitude and phase of the persistent sinusoidal component of the corresponding output, y3 (t), again writing the output in terms of sines and cosines instead of complex exponentials. Is the output of the system real in this case? 2. Now consider the real input u4 (t) = sin(ωt). Without decomposing in terms of complex exponentials and appealing to superposition, but instead using directly the entries in the Laplace transform table below for cos(bt) and sin(bt), compute the coefficients f0 and g0 in the expression Y4 (s) = f0 /(s2 + ω2 ) + g0 s/(s2 + ω2 ) + other terms,

(2)

where both f0 and g0 are both constrained to be real [hint: to compute the coefficients of the partial fraction expansion, multiply (2) by (s2 + ω2 ), simplify, and evaluate the result in the case that s2 = −ω2 ]. Then determine an expression for the magnitude and phase shift of the corresponding output y4 (t). Is this answer consistent with that determined in problem 1b above? Discuss in detail.

f (t) (for t ≥ 0− )

F(s)

eat

1/(s − a)

at

1/(s − a)2

te

1

1/s

t

1/s2

δσ (t)

−−−→ 1

cos(bt)

σ→0 s/(s2 + b2)

sin(bt)

b/(s2 + b2 )

Some other random hints: G(−iω) = G(iω), where the overbar denotes complex conjugate. eiφ = cos(φ) + i sin(φ), sin(x + y) = sin x cos y + cosx sin y, cos(x + y) = cos x cos y − sin x sin y.