State-space block in Simulink - Interactive Code Practice

Practice - 5 Tasks

1fill in blank

easy

Complete the code to define the state-space matrices A, B, C, D.

Simulink

A = [1];
B = [0; 1];
C = [1 0];
D = 0;

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

A[0 1; -2 -3]

B[1 0; 0 1]

D[0 1; -1 -1]

Attempts:

3 left

💡 Hint

Common Mistakes

Confusing matrix A with B or C matrices.

Using incorrect matrix dimensions.

The matrix A defines the system dynamics. Here, A = [0 1; -2 -3] represents a second-order system.

2fill in blank

medium

Complete the code to create the state-space system using the ss() function.

Simulink

sys = ss([1], B, C, D);

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

AA

BB

CC

DD

Attempts:

3 left

💡 Hint

Common Mistakes

Passing B or C as the first argument instead of A.

Swapping the order of matrices.

The ss() function takes matrices A, B, C, D in order to create the state-space system. The first argument is A.

3fill in blank

hard

Fix the error in the code to simulate the system response to input u.

Simulink

[y, t, x] = lsim(sys, [1], t);

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

Ay

Bu

Cx

Dsys

Attempts:

3 left

💡 Hint

Common Mistakes

Passing the system object instead of input signal.

Using output y or state x as input.

The lsim function simulates the system response to input u over time t. The input signal is u.

4fill in blank

hard

Fill both blanks to create a state-space system and simulate its step response.

Simulink

sys = ss([1], [2], C, D);
step(sys);

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

AA

BB

CC

DD

Attempts:

3 left

💡 Hint

Common Mistakes

Swapping B and C matrices.

Omitting matrix B.

The ss() function requires matrices A and B as first two arguments to define the system before simulating the step response.

5fill in blank

hard

Fill all three blanks to create a state-space system, simulate its impulse response, and plot the output.

Simulink

sys = ss([1], [2], [3], D);
impulse(sys);
title('Impulse Response');

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

AA

BB

CC

DD

Attempts:

3 left

💡 Hint

Common Mistakes

Passing matrices in wrong order.

Forgetting to add title to the plot.

The ss() function requires A, B, C matrices in order before D. Then impulse() simulates the impulse response and title() adds a plot title.

Practice

1. What does the State-space block in Simulink primarily represent?

easy

A. a mathematical model using matrices to describe system dynamics

B. A graphical user interface for designing circuits

C. A tool to generate random numbers

D. A block to display system output only

Solution

1. Step 1: Understand the purpose of the State-space block

   The State-space block models systems using matrices that describe how states change over time.

2. Step 2: Compare options with the block's function

   Only a mathematical model using matrices to describe system dynamics correctly describes the use of matrices to represent system dynamics.

3. Final Answer:

   a mathematical model using matrices to describe system dynamics -> Option A

4. Quick Check:

   State-space = matrix model [OK]

Hint: State-space means matrix model of system changes [OK]

Common Mistakes:

• Confusing State-space with graphical tools
• Thinking it only displays output
• Assuming it generates random data

2. Which of the following is the correct way to specify the State-space block matrices in Simulink?

easy

A. A = [[0,1],[0,0]]; B = [[0],[1]]; C = [[1,0]]; D = [[0]]

B. A = [0 1; 0 0]; B = [0; 1]; C = [1 0]; D = [0]

C. A = {0,1;0,0}; B = {0;1}; C = {1,0}; D = {0}

D. A = (0,1),(0,0); B = (0),(1); C = (1,0); D = (0)

Solution

1. Step 1: Recall MATLAB/Simulink matrix syntax

   Matrices in MATLAB/Simulink use square brackets with semicolons separating rows and spaces or commas separating columns.

2. Step 2: Identify the correct syntax

   A = [0 1; 0 0]; B = [0; 1]; C = [1 0]; D = [0] uses correct MATLAB matrix syntax: [0 1; 0 0] etc., while others use invalid or incorrect formats.

3. Final Answer:

   A = [0 1; 0 0]; B = [0; 1]; C = [1 0]; D = [0] -> Option B

4. Quick Check:

   Correct matrix syntax = A = [0 1; 0 0]; B = [0; 1]; C = [1 0]; D = [0] [OK]

Hint: Use [rows; rows] with spaces or commas for columns [OK]

Common Mistakes:

• Using curly braces {} instead of brackets []
• Using parentheses () for matrices
• Separating rows with commas instead of semicolons

3. Given the State-space matrices:
A = [0 1; -2 -3],
B = [0; 1],
C = [1 0],
D = [0]
and input u(t) = 1, what is the initial output y(0) if the initial state x(0) = [0; 0]?

medium

A. 1

B. -3

C. 0

D. Undefined

Solution

1. Step 1: Calculate output y(0) using y = C*x(0) + D*u(0)

   Given x(0) = [0;0], u(0) = 1, y(0) = C*[0;0] + D*1 = 0 + 0 = 0.

2. Step 2: Confirm calculation

   Multiplying C by zero vector gives zero, D times input is zero, so output is zero.

3. Final Answer:

   0 -> Option C

4. Quick Check:

   y(0) = Cx0 + Du0 = 0 [OK]

Hint: Output at zero state = D times input only [OK]

Common Mistakes:

• Ignoring initial state and using input only
• Confusing matrix multiplication order
• Assuming output equals input directly

4. You entered the following matrices in a State-space block:
A = [0 1; -1 -1],
B = [0; 1],
C = [1; 0],
D = [0].
When simulating, you get an error about matrix dimensions. What is the likely cause?

medium

A. Matrix A must be square but is not

B. Matrix B has wrong number of rows compared to A

C. Matrix D must be a scalar zero, not a matrix

D. Matrix C has wrong number of columns compared to A

Solution

1. Step 1: Check matrix dimensions for State-space consistency

   Matrix A is 2x2, so B must be 2x1, C must be 1x2, D must be 1x1. Here B=[0;1] (2x1) ok, but C=[1; 0] is 2x1.

2. Step 2: Identify mismatch

   Matrix C has wrong number of columns (1 vs 2 for A).

3. Final Answer:

   Matrix C has wrong number of columns compared to A -> Option D

4. Quick Check:

   C columns must match A columns [OK]

Hint: C columns = A columns; B rows = A rows [OK]

Common Mistakes:

• Mixing rows and columns in matrices
• Entering C as a column vector instead of row
• Assuming D can be any size

5. You want to model a mechanical system with two states using a State-space block. The system has matrices:
A = [0 1; -1 -1],
B = [0; 1],
C = [1 0],
D = [0].
If the input force is a step input of magnitude 3, what will be the steady-state output y(t) as t approaches infinity?

hard

A. 3

B. 0

C. 1

D. Cannot determine without simulation

Solution

1. Step 1: Find steady-state state x_ss using (A)x_ss + (B)u_ss = 0

   At steady state, dx/dt=0, so 0 = A*x_ss + B*u_ss. With u_ss=3, solve for x_ss: x_ss = -inv(A)*B*3.

2. Step 2: Calculate output y_ss = C*x_ss + D*u_ss

   Calculate x_ss, then y_ss. For given matrices, y_ss = C*(-inv(A)*B*3) + 0 = 3 (after calculation).

3. Final Answer:

   3 -> Option A

4. Quick Check:

   Steady output = input magnitude for stable system [OK]

Hint: Steady output = input if system is stable [OK]

Common Mistakes:

• Ignoring steady-state calculation
• Assuming output equals zero without input
• Not inverting matrix A correctly