State-space block in Simulink - Mini Project: Build & Apply

Using the State-space Block in Simulink

📖 Scenario: You are working on a control system project where you need to model a simple dynamic system using Simulink. The system is described by state-space equations, and you want to simulate its behavior over time.

🎯 Goal: Build a Simulink model using the State-space block to represent the system with given matrices, configure the input, and observe the output response.

📋 What You'll Learn

Create matrices A, B, C, and D with specified values

Add a State-space block and configure it with these matrices

Add an input source block to provide input to the system

Add an output sink block to visualize the system output

💡 Why This Matters

🌍 Real World

State-space models are used in engineering to represent and simulate dynamic systems like robots, vehicles, and electrical circuits.

💼 Career

Understanding state-space representation and simulation is important for control engineers, system modelers, and data scientists working with time-series data.

1

Create the state-space matrices

Create four matrices called A, B, C, and D with these exact values:
A = [[0, 1], [-2, -3]]
B = [[0], [1]]
C = [[1, 0]]
D = [[0]]

Simulink

# Your code here

Hint

Use Python list of lists to create matrices exactly as shown.

2

Configure the State-space block parameters

Create a variable called state_space_params as a dictionary with keys 'A', 'B', 'C', and 'D' and assign the matrices A, B, C, and D respectively.

Simulink

A = [[0, 1], [-2, -3]]
B = [[0], [1]]
C = [[1, 0]]
D = [[0]]
# Your code here

Hint

Create a dictionary named state_space_params with the four matrices as values.

3

Simulate the system response using the State-space block

Write a function called simulate_state_space that takes state_space_params and an input list u and returns the output list y. Use a simple discrete-time simulation for 5 time steps assuming zero initial state. Use the state-space equations:
x[k+1] = A x[k] + B u[k]
y[k] = C x[k] + D u[k]
Use for loop with variable k from 0 to 4.

Simulink

A = [[0, 1], [-2, -3]]
B = [[0], [1]]
C = [[1, 0]]
D = [[0]]
state_space_params = {'A': A, 'B': B, 'C': C, 'D': D}
# Your code here

Hint

Use a loop from 0 to 4, update state vector x, compute output y at each step.

4

Print the output response for a given input

Create a list called input_signal with values [1, 0, 0, 0, 0]. Then call simulate_state_space(state_space_params, input_signal) and print the result.

Simulink

A = [[0, 1], [-2, -3]]
B = [[0], [1]]
C = [[1, 0]]
D = [[0]]
state_space_params = {'A': A, 'B': B, 'C': C, 'D': D}

def simulate_state_space(params, u):
    A, B, C, D = params['A'], params['B'], params['C'], params['D']
    x = [0.0, 0.0]  # initial state vector
    y = []
    for k in range(5):
        yk = C[0][0]*x[0] + C[0][1]*x[1] + D[0][0]*u[k]
        y.append(yk)
        x_next_0 = A[0][0]*x[0] + A[0][1]*x[1] + B[0][0]*u[k]
        x_next_1 = A[1][0]*x[0] + A[1][1]*x[1] + B[1][0]*u[k]
        x = [x_next_0, x_next_1]
    return y

# Your code here

Hint

Create the input list exactly as [1, 0, 0, 0, 0], call the function, and print the output.

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