Steady-state error analysis in Simulink - Interactive Code Practice

Practice - 5 Tasks

1fill in blank

easy

Complete the code to calculate the steady-state error for a unit step input.

Simulink

ess = 1 / (1 + [1]);

Drag options to blanks, or click blank then click option'

ATs

BKi

CKd

DKp

Attempts:

3 left

💡 Hint

Common Mistakes

Using Ki or Kd instead of Kp.

Confusing sampling time Ts with error constants.

The steady-state error for a unit step input depends on the position error constant Kp.

2fill in blank

medium

Complete the code to calculate the steady-state error for a ramp input.

Simulink

ess = 1 / [1];

Drag options to blanks, or click blank then click option'

AKi

BKp

CKv

DKd

Attempts:

3 left

💡 Hint

Common Mistakes

Using Kp or Ki instead of Kv.

Forgetting that the error is inversely proportional to Kv.

For a ramp input, the steady-state error is the inverse of the velocity error constant Kv.

3fill in blank

hard

Fix the error in the code to compute the steady-state error for a parabolic input.

Simulink

ess = 1 / ([1]);

Drag options to blanks, or click blank then click option'

AKa

BKv

CKp

DKi

Attempts:

3 left

💡 Hint

Common Mistakes

Using Kv or Kp instead of Ka.

Misunderstanding the type of input signal.

The steady-state error for a parabolic input is the inverse of the acceleration error constant Ka.

4fill in blank

hard

Fill both blanks to create a dictionary of error constants for steady-state error analysis.

Simulink

error_constants = {'position': [1], 'velocity': [2]

Drag options to blanks, or click blank then click option'

AKp

BKv

CKa

DTs

Attempts:

3 left

💡 Hint

Common Mistakes

Mixing Ka or Ts in place of Kp or Kv.

Swapping the constants for position and velocity.

The dictionary maps 'position' to Kp and 'velocity' to Kv, which are used for step and ramp inputs respectively.

5fill in blank

hard

Fill all three blanks to create a function that returns steady-state error based on input type.

Simulink

def steady_state_error(input_type):
    constants = {'step': [1], 'ramp': [2], 'parabolic': [3]
    return 1 / constants.get(input_type, 1)

Drag options to blanks, or click blank then click option'

AKp

BKv

CKa

DTs

Attempts:

3 left

💡 Hint

Common Mistakes

Using Ts instead of error constants.

Mixing up the constants for different input types.

The function uses Kp for step, Kv for ramp, and Ka for parabolic inputs to calculate steady-state error.

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