Steady-state error analysis in Simulink

Introduction

Steady-state error analysis helps us understand how close a system's output stays to the desired value over time. It shows if the system can reach and keep the target without constant mistakes.

Checking if a temperature control system keeps the room at the set temperature without drifting.

Verifying if a motor speed controller maintains the desired speed steadily.

Testing if an automatic water level controller keeps the water at the set height.

Evaluating if a robotic arm reaches and holds a position accurately.

Ensuring a voltage regulator keeps output voltage stable despite input changes.

Syntax

Simulink

1. Build your control system model in Simulink.
2. Add a step input or desired reference signal.
3. Run the simulation for enough time to reach steady state.
4. Use scopes or 'To Workspace' blocks to capture output data.
5. Calculate steady-state error as: error = reference value - output value at steady state.

Steady-state error is usually measured after the system output stops changing significantly.

You can use MATLAB commands to analyze simulation data exported from Simulink.

Examples

Shows a simple calculation of steady-state error from simulation results.

Simulink

Step input: 1 unit
Output at steady state: 0.95 units
Steady-state error = 1 - 0.95 = 0.05

Shows how to calculate steady-state error using MATLAB after simulation.

Simulink

Use 'To Workspace' block to save output as 'y'
In MATLAB: steady_state_error = 1 - y(end);

Sample Program

This code simulates a simple system responding to a step input and calculates the steady-state error by comparing the final output value to the input.

Simulink

%% Simulink steady-state error analysis example
% This script simulates a simple control system and calculates steady-state error

% Define simulation time
simTime = 10; % seconds

% Define step input value
ref = 1;

% Simulate a first-order system with transfer function: G(s) = 1/(s+1)
% Using MATLAB's lsim function for simplicity

sys = tf(1, [1 1]);

t = linspace(0, simTime, 1000);
input = ref * ones(size(t));

output = lsim(sys, input, t);

% Calculate steady-state error as difference between input and output at last time
steady_state_error = ref - output(end);

% Display result
fprintf('Steady-state error: %.4f\n', steady_state_error);

Important Notes

Steady-state error depends on system type and input type (step, ramp, etc.).

In Simulink, you can use scopes or data logging to observe output behavior over time.

Small steady-state error means the system output closely matches the desired input after settling.

Summary

Steady-state error shows how well a system holds its target value over time.

It is calculated by comparing the final output to the input after the system settles.

Simulink models and MATLAB analysis help find and understand steady-state error easily.

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