System stability

Explain in your own words what it means for an automatic system to be stable and give two examples: one of a stable system (a pendulum with friction) and one of an unstable system (a broom balanced vertically on the palm of a hand).

Compare the stability of a ball at the bottom of a bowl, a ball on top of a dome, and a ball on a flat horizontal surface. Relate each case to a stable system, an unstable system, and a critically stable system (it neither returns nor rolls away), and give a technological example of each.

Explain what a sustained oscillation is (it neither grows nor dies away) and why it is the boundary between a stable and an unstable system. In which devices is it intentionally desired (electronic oscillators, clocks)?

Distinguish between the two states in the time response of a system: transient state and steady state. In the example of a heater being turned on, which part of the temperature curve corresponds to each?

Explain what step and ramp test inputs are and what information each provides about a system (step: speed, overshoot and steady-state error; ramp: ability to track setpoints that change over time). Give a real-world example of each type of setpoint.

A first-order system responds by rising gradually without oscillations, like the temperature of a heater. Explain what its time constant $\tau$ is (time to reach $63\,\%$ of the total change) and calculate how long a system with $\tau = 20\ \text{s}$ takes to practically reach its final value (about $4\tau$ to $5\tau$).

The figure shows the response of three systems to the same step input. Classify each response (A, B and C) as stable overdamped, stable underdamped (damped oscillatory) or unstable, and justify your choice based on how each curve evolves over time.

Define the steady-state error of a control system and explain, using a heating system as an example, what it means for it to be non-zero (the room settles half a degree below the setpoint). What type of controller eliminates it completely?

Explain what overshoot and settling time are in a step response, and comment on the usual trade-off between speed and overshoot: why does a very fast system tend to oscillate more?

Find examples of deliberately unstable systems or systems in sustained oscillation that are useful: the oscillator of a digital watch, the multivibrator of a car indicator, the acoustic feedback howl at a concert. Explain for each why it is designed this way and what prevents it from 'blowing up' (non-linearities, saturation).

A data centre uses simulators to test its control systems before installing them. Explain the advantages of experimenting with simulation (safety, cost, repeatability, ability to force extreme situations) and what fundamental limitation a simulated model has.

In a simulator, the step response of a system takes $2\ \text{s}$ to reach $63\,\%$ of the total change and does not oscillate. a) Estimate its time constant and its first-order model. b) Predict its output at $4\ \text{s}$ and $10\ \text{s}$ after a step of $100\,\%$. c) Explain how you would verify in the simulator that it is indeed first order.

The operator of a thermal power plant takes $5\ \text{s}$ to notice any change (pure measurement delay). Explain why pure delay is the worst enemy of a control loop (the correction arrives when the error has already changed) and what consequences it has on the maximum usable gain before oscillations appear.

A prototype of a self-balancing two-wheeled transport vehicle (Segway type) uses a gyroscope and accelerometers to keep the mast vertical by acting on the motor torque at the wheels (a system that can be modeled as an inverted pendulum).

During closed-loop calibration tests, three anomalous situations are observed:

1. a) With a very low proportional gain ($K_p = 4$), upon an external step disturbance (a push) the vehicle barely reacts and slowly falls to the ground.
2. b) When increasing the gain to a very high value ($K_p = 85$) seeking an energetic response, the vehicle does not stabilize: it enters a violent oscillation of increasing frequency and ends up tipping over.
3. c) A developer proposes solving the oscillation in b) by introducing a digital filter into the sensor readings that adds a time delay of $120\text{ ms}$.

Physically justify the behavior of a) and b) in terms of dynamics and closed-loop stability. Predict with reasoning the effect that the developer's proposal will have on the stability of the vehicle and propose which basic control action (integral or derivative) you would add to damp the oscillations without destabilizing the system, justifying your choice.

During flight tests of a quadcopter drone on a test bench, the roll angle (horizontal tilt) stabilization loop is subjected to a sudden gust of lateral wind acting as a constant disturbance. The control team tests three different tunings of the proportional controller:

• Configuration A: The drone tilts $14^\circ$ and remains banked at that constant angle while the wind lasts, without recovering the horizontal.

• Configuration B: After the wind, the drone oscillates indefinitely around the horizontal with a pure sinusoidal oscillation of constant amplitude of $\pm 6^\circ$ at $3\text{ Hz}$ that neither grows nor damps out.

• Configuration C: The drone begins to oscillate with increasing amplitudes ($\pm 3^\circ, \pm 8^\circ, \pm 22^\circ\dots$) until the safety system disconnects the motors to prevent damage.

Interpret the three behaviors by relating each case to the loop gain and the system stability condition (asymptotically stable with error, at the boundary of stability, or unstable). Justify what physical limitation of the P controller causes the failure in configuration A and what gain adjustment you must make in configuration C to return the system to the stable region.