State-space block in Simulink - Practice Problems & Coding Challenges

Challenge - 5 Problems

🧠 Conceptual

intermediate

Understanding the State-Space Block Inputs and Outputs

In Simulink, the State-space block models a system using matrices A, B, C, and D. Which of the following correctly describes the inputs and outputs of this block?

A Input is the control vector u, output is the state vector x.

B Input is the state vector x, output is the control vector u.

C Input is the control vector u, output is the system output y.

D Input is the system output y, output is the control vector u.

💡 Hint

Think about what the State-space block represents: it takes inputs that affect the system and produces outputs based on the system's state.

The State-space block takes the control input vector u and produces the system output y based on the state equations. The state vector x is internal and not an input or output port.

❓ Predict Output

intermediate

Output of a State-Space System Simulation

Given the state-space system defined by matrices:

A = [[0,1],[ -2,-3]], B = [[0],[1]], C = [[1,0]], D = [[0]]

and input u(t) = 1 for t ≥ 0, what is the output y at t=0?

Simulink

import numpy as np
A = np.array([[0,1],[-2,-3]])
B = np.array([[0],[1]])
C = np.array([[1,0]])
D = np.array([[0]])
u0 = 1
x0 = np.array([[0],[0]])
y0 = C @ x0 + D * u0
print(y0[0,0])

A 0.0

B -1.0

C 1.0

D None (error)

💡 Hint

At time zero, the state vector x is zero, so output depends on x0 and input u0.

Output y = Cx + Du. Since x0 = 0 and D = 0, y0 = 0.

❓ data_output

advanced

State Vector Evolution After One Time Step

Using the same system as before, with A, B, and input u=1, initial state x0 = [0, 0], and time step dt=1, what is the state vector x after one step using Euler integration?

Simulink

import numpy as np
A = np.array([[0,1],[-2,-3]])
B = np.array([[0],[1]])
u = 1
x0 = np.array([[0],[0]])
dt = 1
x1 = x0 + dt * (A @ x0 + B * u)
print(x1.flatten().tolist())

A [0, 1]

B [1, -3]

C [0, 0]

D [-1, 0]

💡 Hint

Euler integration: x_next = x_current + dt * (A*x_current + B*u).

Calculate A*x0 + B*u = [0,1], then multiply by dt=1 and add to x0 = [0,0], so x1 = [0,1].

❓ visualization

advanced

Plotting the Output Response of a State-Space System

Which code snippet correctly plots the output y(t) of the state-space system with matrices A, B, C, D and step input u(t)=1 over 5 seconds using Euler integration with dt=0.1?

A

import numpy as np
import matplotlib.pyplot as plt
A = np.array([[0,1],[-2,-3]])
B = np.array([[0],[1]])
C = np.array([[1,0]])
D = np.array([[0]])
dt = 0.1
x = np.array([[0],[0]])
y = []
for _ in range(50):
    u = 1
    y.append((C @ x + D * u)[0,0])
    x = x + dt * (B * u)
plt.plot(np.arange(0,5,dt), y)
plt.xlabel('Time (s)')
plt.ylabel('Output y')
plt.title('Output response')
plt.show()

B

import numpy as np
import matplotlib.pyplot as plt
A = np.array([[0,1],[-2,-3]])
B = np.array([[0],[1]])
C = np.array([[1,0]])
D = np.array([[0]])
dt = 0.1
x = np.array([[0],[0]])
y = []
for _ in range(50):
    u = 1
    x = x + dt * (A @ x + B * u)
    y.append((C @ x + D * u)[0,0])
plt.plot(np.arange(0,5,dt), y)
plt.xlabel('Time (s)')
plt.ylabel('Output y')
plt.title('Output response')
plt.show()

C

import numpy as np
import matplotlib.pyplot as plt
A = np.array([[0,1],[-2,-3]])
B = np.array([[0],[1]])
C = np.array([[1,0]])
D = np.array([[0]])
dt = 0.1
x = np.array([[0],[0]])
y = []
for _ in range(50):
    u = 1
    y.append((C @ x + D * u)[0,0])
    x = x + dt * (A @ x)
plt.plot(np.arange(0,5,dt), y)
plt.xlabel('Time (s)')
plt.ylabel('Output y')
plt.title('Output response')
plt.show()

D

import numpy as np
import matplotlib.pyplot as plt
A = np.array([[0,1],[-2,-3]])
B = np.array([[0],[1]])
C = np.array([[1,0]])
D = np.array([[0]])
dt = 0.1
x = np.array([[0],[0]])
y = []
for _ in range(50):
    u = 1
    y.append((C @ x + D * u)[0,0])
    x = x + dt * (A @ x + B * u)
plt.plot(np.arange(0,5,dt), y)
plt.xlabel('Time (s)')
plt.ylabel('Output y')
plt.title('Output response')
plt.show()

💡 Hint

Remember to update output y before updating state x in each loop iteration.

Option D correctly computes output y before updating state x using Euler integration including input u. Other options either update x before y or omit input u in state update.

🔧 Debug

expert

Identifying the Cause of Incorrect Output in State-Space Simulation

A user simulates a state-space system but notices the output y remains zero for all time steps despite a non-zero input u. Which code snippet below contains the bug causing this behavior?

A

for _ in range(50):
    u = 0
    y.append((C @ x + D * u)[0,0])
    x = x + dt * (A @ x + B * u)

B

for _ in range(50):
    u = 1
    y.append((C @ x + D * u)[0,0])
    x = x + dt * (A @ x)

C

for _ in range(50):
    u = 1
    x = x + dt * (A @ x + B * u)
    y.append((C @ x + D * u)[0,0])

D

for _ in range(50):
    u = 1
    y.append((C @ x + D * u)[0,0])
    x = x + dt * (A @ x + B * u)

💡 Hint

Check if the input u affects the state update in each option.

Option B updates state x without adding the input term B*u, so input u has no effect, causing output y to remain zero.

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