Chapter 8: System Properties, Poles and Zeroes, System Response and Stability

Transfer Function Limitations

The transfer function
\( H(s) = \frac{Y(s)}{U(s)} \),
derived via Laplace transforms from a system’s differential equations, assumes that the system is:

- Linear – Follows the principle of superposition.
- Time-Invariant – System properties do not change with time.
- Causal – Output depends only on current and past inputs.

These assumptions do not hold for many real systems, which may be nonlinear, time-varying, or even non-causal in simulations. Transfer functions therefore serve as linearized approximations of the actual dynamics, valid only around an equilibrium point or within a certain input range.

Linearity

A system is linear if it satisfies both:

- Homogeneity:
\( u(t) \rightarrow y(t) \Rightarrow \alpha u(t) \rightarrow \alpha y(t) \)

- Additivity:
\( u_1(t) \rightarrow y_1(t), \ u_2(t) \rightarrow y_2(t) \Rightarrow u_1 + u_2 \rightarrow y_1 + y_2 \)

Linear System:
\( T[x(t)] = 3x(t) \)

This system satisfies both homogeneity and additivity:

- Homogeneity:
\( T[\alpha x(t)] = 3 \alpha x(t) = \alpha T[x(t)] \)

- Additivity:
\( T[x_1(t) + x_2(t)] = 3(x_1(t) + x_2(t)) = 3x_1(t) + 3x_2(t) = T[x_1(t)] + T[x_2(t)] \)

Nonlinear System:
\( T[x(t)] = x(t)^2 \)

This system violates both:

- Homogeneity:
\( T[\alpha x(t)] = (\alpha x(t))^2 = \alpha^2 x(t)^2 \ne \alpha x(t)^2 = \alpha T[x(t)] \)

- Additivity:
\( T[x_1(t) + x_2(t)] = (x_1(t) + x_2(t))^2 \ne x_1(t)^2 + x_2(t)^2 = T[x_1(t)] + T[x_2(t)] \)

Linear System – Mass-Spring-Damper
\( m\ddot{x} + c\dot{x} + kx = F(t) \)

This is linear since the system response scales and sums appropriately for linear combinations of inputs.

Nonlinear System – Friction or Saturation
\( m\ddot{x} + c\dot{x} + kx + \mu \sin(\dot{x}) = F(t) \)

This is nonlinear due to the discontinuous and non-scaling nature of the friction term.

Time Invariance

A system is time-invariant if a time shift in the input causes an identical shift in the output. That is:
\( u(t) \rightarrow y(t) \Rightarrow u(t - t_0) \rightarrow y(t - t_0) \)

Invariant System:
\( T[x(t)] = 3x(t) \)

\[
T[x(t - t_0)] = 3x(t - t_0) \\
y(t - t_0) = 3x(t - t_0)
\]

So the time-shifted output equals the output of the time-shifted input — time invariance holds.

Time-Variant System:
\( T[x(t)] = t \cdot x(t) \)

\[
T[x(t - t_0)] = t \cdot x(t - t_0) \\
y(t - t_0) = (t - t_0) \cdot x(t - t_0)
\]

These two expressions are not equal — time invariance is violated.

Real-World Example:
Driving a car – as the car consumes fuel, the total mass changes, making it a time-varying system. Transfer function models become approximate in such cases.

Causality

A causal system does not rely on future inputs.
Additionally, the degree of the denominator must be greater than or equal to the numerator:

\[
H(s) = \frac{b_0 + b_1 s + \cdots + b_m s^m}{a_0 + a_1 s + \cdots + a_n s^n}
\quad \text{Causal if } n \geq m
\]

Example:

\[
H(s) = \frac{s^2 + 3}{s^3 + 2s + 5} \quad \text{Causal}
\quad \quad
H(s) = \frac{s^3 + 2}{s + 4} \quad \text{Non-Causal}
\]

Poles and Zeroes of Transfer Functions

Given a transfer function:

$$
H(s) = \frac{(s - z_1)(s - z_2)\cdots(s - z_m)}{(s - p_1)(s - p_2)\cdots(s - p_n)}
\]

Zeroes: values of \( s \) that make \( H(s) = 0 \) (roots of the numerator)
Poles: values of \( s \) that make \( H(s) \to \infty \) (roots of the denominator)

Poles determine:
- Stability (location in complex plane)
- Natural frequency
- Damping
- Long-term behavior

Zeroes affect:
- Response shape
- Peak amplitudes
- Transient effects

Let's consider a plant function with a delta function input. Let the plant function be:

$$
P(s) = \frac{1}{s+\sigma}
\]

The system has a pole at:

$$
s = -\sigma
\]

Taking the inverse Laplace transform:

$$
y(t) = \mathcal{L}^{-1}(P(s)) = e^{-\sigma t}
\]

If \( \sigma > 0 \), this corresponds to exponential decay. A larger \( \sigma \) causes faster decay.

We can visualize pole locations in the complex \( s \)-plane, which has real and imaginary axes.

Now let's return to the mass-spring-damper system:

This system is governed by the differential equation:

$$
m\ddot{x}+c\dot{x}+kx = u(t)
\]

Assuming impulse input \( u(t) = \delta(t) \), the plant transfer function is:

$$
P(s) = \frac{X(s)}{U(s)} = \frac{1}{m s^2 + c s + k}
\]

Solving the characteristic equation gives poles:

$$
s_{1,2} = \frac{-c \pm \sqrt{c^2 - 4mk}}{2m}
\]

Cases:
- \( c^2 - 4mk > 0 \): Two distinct real poles (Overdamped)
- \( c^2 - 4mk = 0 \): Repeated real pole (Critically damped)
- \( c^2 - 4mk < 0 \): Complex conjugate poles (Underdamped)

Example 1: Let \( m = 1, c = 3, k = 2 \). Then:

Poles: \( s = -1, -2 \)
This yields an overdamped response:

$$
x(t) = e^{-2t}(1 - e^t)
\]

Example 2: Let \( m = 1, c = 1, k = 2 \). Poles are complex conjugates.
Response includes oscillations with decay envelope set by the real part, and frequency determined by the imaginary part.

Example 3: Undamped case, \( c = 0 \):

$$
P(s) = \frac{1}{ms^2 + k}
\]

Response:

$$
x(t) = \frac{1}{\sqrt{mk}} \sin\left(\sqrt{\frac{k}{m}} t \right)
\]

Poles are imaginary: \( s = \pm j \sqrt{\frac{k}{m}} \)

Now consider a frictionless sliding mass:

Equation:

$$
u = m \dot{v}
\Rightarrow
P(s) = \frac{V(s)}{U(s)} = \frac{1}{ms}
\]

Pole: \( s = 0 \)
Response:

$$
v(t) = \frac{1}{m}
\]

This is a constant velocity response.

We can now map system responses to pole locations in the \( s \)-domain.

Summary from the figure:
- Left-half real poles → exponential decay
- Complex conjugate poles → underdamped oscillations
- Pure imaginary poles → sinusoidal oscillation
- Right-half poles → unstable growth

Design Insight:

- Pole placement is critical for shaping system response.
- Zero placement can fine-tune transient performance.
- Adding zeroes near poles can help flatten or sharpen responses.

Summary of Concepts

- Transfer functions represent LTI (Linear Time-Invariant), causal systems.
- Poles determine the dominant modes of system behavior.
- Real poles lead to exponential decay.
- Complex poles lead to oscillations.
- Zeroes shape response profiles, especially during transients.
- Pole-zero maps offer intuitive visual insight into system stability and dynamics.