Solutions of Exercise 1.0

Solutions of Exercise 1.0#

Consider a simple mechanical system described by a mass \(m\), a spring constant \(k\), and a damping coefficient \(c\). The system is subjected to an external force \(u(t)\), and its position \(y(t)\) evolves according to the following linear ordinary differential equation (ODE):

\[ \ddot{y}(t) + 3 \dot{y}(t) + 2 y(t) = u(t) \]

  1. Compute the transfer function \(G(s)\) between the input \(u\) and the output \(y\).

  2. Compute the poles and zeros of the system.

  3. What is the dominant pole and its corresponding time-constant?

  4. Given this dominant pole, how long does it take before the output is in a 1% band of its final value?

  5. Compute the static gain of the system by setting \(s=0\).

Solution#

Question 1#

To find the transfer function \(G(s) = \frac{Y(s)}{U(s)}\), we apply the Laplace transform to the ODE, assuming zero initial conditions (\(\dot{y}(0) = y(0) = 0\)).

Recall that \(\mathcal{L}[\dot{y}(t)] = s Y(s)\) and \(\mathcal{L}[\ddot{y}(t)] = s^2 Y(s)\).

The ODE becomes:

\[ s^2 Y(s) + 3 s Y(s) + 2 Y(s) = U(s) \]

Factor out \(Y(s)\):

\[ Y(s) (s^2 + 3s + 2) = U(s) \]

Solving for the ratio \(Y(s)/U(s)\):

\[ G(s) = \frac{1}{s^2 + 3s + 2} \]

Question 2#

Zeros: The roots of the numerator. Here, the numerator is \(1\) (a constant), so there are no zeros.

Poles: The roots of the denominator \(D(s) = s^2 + 3s + 2\). We solve \(s^2 + 3s + 2 = 0\).

We can factor the polynomial as: $\( (s+1)(s+2) = 0 \)$

Thus, the poles are: $\( p_1 = -1, \quad p_2 = -2 \)$

Question 3#

The dominant pole is the pole closest to the imaginary axis (i.e., the one with the smallest absolute real part, or largest time constant).

Comparing \(p_1 = -1\) and \(p_2 = -2\), \(p_1\) is closer to \(0\).

Dominant pole: \(p_d = -1\).

The time constant \(\tau\) is defined as \(\tau = \frac{1}{|Re(p)|}\).

\[ \tau = \frac{1}{|-1|} = 1 \text{ second} \]

Question 4#

For a first-order system (or a system approximated by its dominant pole), the step response is proportional to \(1 - e^{-t/\tau}\).

A common empirical rule for estimating the settling time is:

  • \(T_{s, 5\%} \approx 3\tau\) (for a 5% band)

  • \(T_{s, 1\%} \approx 4.6\tau\) (for a 1% band)

In this case, we want to find \(t\) such that the error \(e^{-t/\tau} \leq 0.01\) (1%):

\[ e^{-t/\tau} = 0.01 \implies -\frac{t}{\tau} = \ln(0.01) \approx 4.6 \]

Given \(\tau = 1\)s: $\( t_{1\%} \approx 4.6 \cdot \tau = 4.6 \text{ seconds} \)$

Question 5#

The static gain (or DC gain) is found by evaluating the transfer function at \(s=0\).

\[ K_{static} = G(0) = \frac{1}{0^2 + 3(0) + 2} = \frac{1}{2} = 0.5 \]

We can verify these results using MATLAB.

s = tf("s");
G = 1 / (s^2 + 3*s + 2);

p = pole(G);
z = zero(G);
mu = dcgain(G);

disp(sprintf(['Poles: %s,\n Zeros: %s,\nStatic gain: %.2f'], mat2str(p), mat2str(z), mu));

Poles: [-2;-1],
 Zeros: zeros(0,1),
Static gain: 0.50
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