Control System Stability Exercises

These exercises practise the stability tools used in introductory control systems: pole locations, Routh-Hurwitz tests, gain ranges, second-order response, steady-state error constants, state-space eigenvalues, discrete-time poles, gain and phase margins, and delay margin.

All problems assume linear time-invariant models unless stated otherwise. Stability conclusions apply to the mathematical model given in the exercise. A real control system also requires checking saturation, sensor noise, actuator limits, sampling, delay, and unmodelled dynamics.

How to use these exercises

Work each problem in two passes. First, solve the algebraic stability condition exactly from the model. Second, interpret what the result would mean for a physical controller. A gain range, damping ratio, phase margin, or pole location is not only a number; it is a clue about robustness, transient response, noise sensitivity, and how close the design is to a stability boundary.

When an answer gives a marginal case, treat it as a mathematical boundary rather than a practical operating point. Real systems need margin because parameters drift, sensors filter signals, actuators saturate, computations sample in time, and unmodelled dynamics add phase lag. The exercises therefore practise both calculation and engineering judgement.

Solution Checks for Engineering Use

After each exercise, check whether the conclusion would still make sense for a real loop. A stable polynomial does not prove acceptable performance if the damping is low, the bandwidth is beyond actuator capability, the sampling rate is too slow, or the controller relies on exact cancellation of uncertain dynamics.

Useful checks include pole location, damping, gain and phase margin, delay sensitivity, actuator saturation, sensor noise, quantization, model order, and operating point. For discrete-time examples, remember that stability depends on pole magnitude relative to the unit circle, not on the sign of the pole alone.

The habit to build is simple: solve the mathematical test, then ask what assumption would break the result first in hardware, software, or process operation.

Exercise 1: Identify second-order stability and response

A closed-loop system has characteristic equation:

1. Is the system stable?
2. Find the natural frequency \omega_n.
3. Find the damping ratio \zeta.
4. Estimate percent overshoot.
5. Estimate the 2 percent settling time.
6. Estimate peak time.

Solution

Compare the denominator with the standard second-order form:

The constant term gives:

so:

The coefficient of s gives:

Substitute \omega_n = 5:

Because \zeta > 0 and \omega_n > 0, the poles have negative real parts. The system is stable and underdamped.

Percent overshoot is:

Substitute \zeta = 0.4:

The 2 percent settling-time approximation is:

The damped natural frequency is:

Peak time is:

Exercise 2: Routh-Hurwitz test with a gain parameter

A closed-loop characteristic polynomial is:

Find the range of K for closed-loop stability.

Solution

For a cubic polynomial:

the stability conditions are:

and:

Here:

The positive coefficient condition gives:

The cubic inequality gives:

Therefore the stability range is:

At K = 20, the system is marginal in the ideal model. At K > 20, at least one pole lies in the right half-plane.

Exercise 3: Closed-loop gain range for a unity-feedback plant

A unity-feedback system has open-loop transfer function:

Find the range of K for closed-loop stability.

Solution

For unity negative feedback, the characteristic equation is:

Substitute G(s):

Multiply by the denominator:

Expand:

Use the cubic Routh-Hurwitz conditions:

First:

Second:

Therefore:

This gain range is based on the ideal transfer function. In real hardware, unmodelled delay and high-frequency dynamics usually require a more conservative gain.

Exercise 4: Steady-state error for step and ramp inputs

A unity-feedback system has open-loop transfer function:

Assume the closed-loop system is stable.

1. What is the system type?
2. What is the steady-state error for a unit step input?
3. What is the steady-state error for a unit ramp input?

Solution

The open-loop transfer function has one pole at the origin because of the factor s in the denominator. Therefore it is a Type 1 system.

For a Type 1 unity-feedback system, the steady-state error to a unit step input is zero:

For a ramp input, use the velocity error constant:

Substitute:

Cancel s:

The ramp steady-state error is:

The result shows why integral action improves step tracking but does not automatically eliminate ramp error unless the system type is increased again.

Exercise 5: State-space stability from eigenvalues

A continuous-time system has state matrix:

Is the unforced system stable?

Solution

The eigenvalues of A are the roots of:

Compute:

The determinant is:

Set it equal to zero:

The roots are:

Both eigenvalues have negative real part. Therefore the continuous-time system is asymptotically stable.

The response is oscillatory because the eigenvalues are complex, and it decays because the real part is negative.

Exercise 6: Discrete-time pole stability

A discrete-time closed-loop system has poles:

Is the system asymptotically stable?

Solution

A discrete-time linear system is asymptotically stable if every closed-loop pole lies strictly inside the unit circle:

Check each pole:

Since z_3 lies outside the unit circle, the system is unstable.

The pole at 1.05 may look close to the stability boundary, but repeated multiplication by 1.05 grows with time. Small instability still matters.

Exercise 7: Phase margin and gain margin from frequency data

A loop transfer function L(s) has:

• gain crossover at \omega_{gc} = 12\ \text{rad/s}, where \angle L(j\omega_{gc}) = -135^\circ;
• phase crossover at \omega_{pc} = 30\ \text{rad/s}, where the loop magnitude is 20\log_{10}|L(j\omega_{pc})| = -8\ \text{dB}.

Find the phase margin and gain margin.

Solution

Phase margin is:

Substitute:

Gain margin in decibels is:

If the magnitude at phase crossover is already given in decibels as -8\ \text{dB}, then:

As a ratio:

So the loop could tolerate approximately a 2.51 times gain increase before reaching the classical gain-margin boundary, assuming the standard single-loop interpretation is valid.

Exercise 8: Maximum added delay from phase margin

A loop has gain crossover frequency:

and phase margin:

Estimate the maximum additional pure time delay that would reduce the phase margin to zero.

Solution

A pure delay T adds phase:

At crossover, the approximate delay that consumes the phase margin is:

Convert 35^\circ to radians:

Then:

The estimate assumes the crossover frequency does not shift significantly after adding delay. In a detailed design, the loop should be re-analysed with the delay included.

Exercise 9: Stability with proportional control

A plant is:

It is placed in unity negative feedback with proportional controller C(s) = K.

1. Find the closed-loop characteristic equation.
2. Find the range of K for stability.

Solution

The characteristic equation for unity feedback is:

Substitute:

Multiply by the denominator:

The characteristic equation is:

For a second-order continuous-time polynomial:

stability requires all coefficients to have the same positive sign:

Here a_2 = 1 and a_1 = 3, so those are positive. The remaining condition is:

Therefore:

In most practical proportional-control contexts, K would also be constrained to be positive so that the feedback sign remains physically meaningful. Under that usual design restriction, all K > 0 are stable for this simplified plant.

Exercise 10: Sampling and continuous-to-discrete pole mapping

A continuous-time closed-loop pole is:

The controller is sampled with period:

Find the corresponding discrete-time pole magnitude.

Solution

The continuous-to-discrete pole relation is:

Substitute:

The magnitude is:

Since |e^{0.2j}| = 1:

The discrete pole lies inside the unit circle, so this pole corresponds to a decaying discrete-time mode.

Exercise 11: Interpreting a marginal pole

A continuous-time closed-loop model has poles:

Is the system asymptotically stable?

Solution

For continuous-time asymptotic stability, every pole must lie strictly in the open left half-plane:

The first two poles satisfy this:

The third pole is:

This pole is on the imaginary axis, not in the open left half-plane. Therefore the system is not asymptotically stable.

Depending on the complete system and pole multiplicity, it may be marginally stable or unstable in the bounded-input bounded-output sense. For control design, a pole at the origin requires careful interpretation. It may represent intentional integral action, but closed-loop asymptotic stability still requires the final closed-loop poles to be placed appropriately for the controlled variables of interest.

Exercise 12: Conceptual robustness check

A controller is tuned on a simplified plant model. The simulated step response has 5 percent overshoot and settles in 1 second. On the real plant, the response oscillates and the actuator command repeatedly saturates.

Name three likely causes and three corrective actions.

Solution

Likely causes include:

• the real plant has additional delay or phase lag not included in the model;
• the actuator saturation invalidates the linear simulation;
• the controller gain or integral action is too aggressive;
• a flexible mode or unmodelled resonance is being excited;
• sensor noise or filtering is changing the effective loop dynamics;
• the actuator has rate limits that were not modelled.

Corrective actions include:

• include actuator saturation and rate limits in simulation;
• reduce loop bandwidth or controller gains;
• add anti-windup logic for integral action;
• identify the plant more accurately near the operating condition;
• include sensor filtering and computational delay in the model;
• add notch filtering or redesign the controller to avoid exciting resonant modes;
• test with smaller commands and increase authority gradually.

The main lesson is that stability is not only a polynomial calculation. Real closed-loop behaviour depends on the plant, controller, sensor, actuator, software, and operating limits together.