Steady-state error analysis in Simulink - Practice Problems & Coding Challenges

Challenge - 5 Problems

🧠 Conceptual

intermediate

Understanding steady-state error in control systems

In a Simulink model of a unity feedback control system, which of the following best describes the steady-state error?

A The difference between the input and output signals as time approaches infinity.

B The maximum overshoot of the system response during transient state.

C The time it takes for the system to reach the first peak after a step input.

D The initial difference between the input and output signals at time zero.

💡 Hint

Think about what happens to the error after the system settles.

Steady-state error is the difference between the desired input and the actual output after the system has settled and transient effects have died out.

❓ data_output

intermediate

Calculate steady-state error from Simulink step response data

You run a step input simulation in Simulink and get the output data array: [0, 0.5, 0.8, 0.95, 0.99, 1.0, 1.0]. What is the steady-state error if the input step magnitude is 1?

A 0.01

B 0.05

C 1.0

D 0.0

💡 Hint

Look at the last value of the output and subtract it from the input step value.

The steady-state output is 1.0 (final settled value), input is 1, so error = 1 - 1.0 = 0.0.

❓ visualization

advanced

Identify steady-state error from Simulink plot

Given a Simulink step response plot where the output settles at 0.8 for a step input of 1, what is the steady-state error?

A 0.8

B 0.2

C 1.0

D 0.0

💡 Hint

Subtract the steady output value from the input step value.

Steady-state error = input - output = 1 - 0.8 = 0.2.

❓ Predict Output

advanced

Simulink MATLAB script output for steady-state error

Consider this MATLAB code snippet used with Simulink output data:

y = [0 0.3 0.6 0.85 0.95 0.98 1.0];
input = 1;
steady_state_error = input - y(end-1);

What is the value of steady_state_error?

Simulink

y = [0 0.3 0.6 0.85 0.95 0.98 1.0];
input = 1;
steady_state_error = input - y(end-1);

A 0.02

B 0.0

C 0.05

D 0.98

💡 Hint

Check the second last value in the output array.

The second last value is 0.98, so error = 1 - 0.98 = 0.02.

🚀 Application

expert

Determine system type from steady-state error behavior

A Simulink model of a unity feedback system shows zero steady-state error for a step input but a finite steady-state error for a ramp input. What is the system type?

A Type 0 system

B Type 2 system

C Type 1 system

D Type 3 system

💡 Hint

Recall how system type relates to steady-state error for step and ramp inputs.

Type 1 systems have one integrator and zero steady-state error for step inputs but finite error for ramp inputs.

Practice

1. What does steady-state error in a Simulink control system represent?

easy

A. The initial delay before the system starts responding

B. The maximum output value during system response

C. The difference between the desired input and the actual output after the system settles

D. The time taken for the system to reach half of its final value

Solution

1. Step 1: Understand steady-state error meaning

   Steady-state error measures how close the system output is to the input after settling.

2. Step 2: Compare options with definition

   Only The difference between the desired input and the actual output after the system settles correctly describes this difference after settling.

3. Final Answer:

   The difference between the desired input and the actual output after the system settles -> Option C

4. Quick Check:

   Steady-state error = final input - final output [OK]

Hint: Steady-state error = input minus output after settling [OK]

Common Mistakes:

• Confusing steady-state error with response time
• Thinking it measures initial delay
• Mixing it with peak output value

2. Which Simulink block is commonly used to measure steady-state error in a control system?

easy

A. Gain block

B. Sum block

C. Scope block

D. Integrator block

Solution

1. Step 1: Identify block function

   The Sum block calculates the difference between signals, useful for error measurement.

2. Step 2: Match block to steady-state error

   Steady-state error is the difference between input and output, so Sum block is used.

3. Final Answer:

   Sum block -> Option B

4. Quick Check:

   Sum block calculates error difference [OK]

Hint: Use Sum block to find difference between signals [OK]

Common Mistakes:

• Choosing Scope block which only displays signals
• Selecting Gain block which scales signals
• Confusing Integrator block with error calculation

3. Given this MATLAB code snippet analyzing steady-state error in Simulink:

sys = tf([1], [1 3 2]);
y = step(sys, 10);
steady_error = abs(1 - y(end));
disp(steady_error);

What is the output of steady_error?

medium

A. 0.5

B. 1

C. 0

D. 2

Solution

1. Step 1: Analyze transfer function and step response

   The transfer function G(s) = 1/(s^2 + 3s + 2) has DC gain G(0) = 1/2 = 0.5. Poles at -1 and -2 ensure stability, step response settles at 0.5.

2. Step 2: Calculate steady-state error

   Steady-state error = abs(1 - final output) = |1 - 0.5| = 0.5.

3. Final Answer:

   0.5 -> Option A

4. Quick Check:

   DC gain = 1/2, error = |1 - 0.5| = 0.5 [OK]

Hint: Step response final value = DC gain for unit step [OK]

Common Mistakes:

• Miscalculating DC gain as 1 instead of 1/2
• Confusing transient response with steady-state
• Thinking all stable systems have zero steady-state error

4. You have this Simulink model code snippet to calculate steady-state error:

sys = tf([2], [1 2 1]);
y = step(sys, 10);
steady_error = 1 - y(end);

Why might this code give a wrong steady-state error value?

medium

A. Because the variable y is not defined

B. Because the step function is called with wrong time

C. Because the transfer function numerator is incorrect

D. Because it does not take the absolute value of the error

Solution

1. Step 1: Check error calculation formula

   The code calculates error as 1 - y(end) without absolute value, which can be negative.

2. Step 2: Understand steady-state error definition

   Steady-state error should be absolute difference to represent magnitude correctly.

3. Final Answer:

   Because it does not take the absolute value of the error -> Option D

4. Quick Check:

   Error magnitude needs absolute value [OK]

Hint: Always use absolute value for steady-state error [OK]

Common Mistakes:

• Ignoring sign of error
• Assuming step time affects error calculation
• Misdefining transfer function numerator

5. In a Simulink model, you want to reduce steady-state error for a system with transfer function G(s) = 1/(s+1). Which modification will most effectively reduce steady-state error to a step input?

hard

A. Add an integrator in the forward path

B. Increase the gain of the existing transfer function

C. Add a differentiator in the feedback path

D. Add a delay block before the output

Solution

1. Step 1: Understand steady-state error for type 0 system

   G(s) = 1/(s+1) is type 0, which has nonzero steady-state error to step input.

2. Step 2: Effect of adding integrator

   Adding an integrator (1/s) increases system type to 1, eliminating steady-state error for step inputs.

3. Final Answer:

   Add an integrator in the forward path -> Option A

4. Quick Check:

   Integrator reduces steady-state error to zero for step input [OK]

Hint: Add integrator to eliminate steady-state error for step input [OK]

Common Mistakes:

• Thinking gain increase alone removes steady-state error
• Confusing differentiator effect with error reduction
• Adding delay which does not affect steady-state error