Discrete state-space in Simulink - Practice Problems & Coding Challenges

Challenge - 5 Problems

❓ Predict Output

intermediate

Output of Discrete State-Space Simulation

Given a discrete state-space model with matrices:

A = [[0.7, 0.2], [0, 0.9]]
B = [[0.1], [0.05]]
C = [[1, 0]]
D = [[0]]

Initial state x0 = [1, 0]
Input u = 1 (constant)
Simulate for 3 steps.

What is the output y at step 3?

Simulink

import numpy as np
A = np.array([[0.7, 0.2],[0,0.9]])
B = np.array([[0.1],[0.05]])
C = np.array([[1,0]])
D = np.array([[0]])
x = np.array([[1],[0]])
u = 1
for _ in range(3):
    x = A @ x + B * u
    y = C @ x + D * u
print(float(y))

A 1.045

B 1.0305

C 0.945

D 1.15

💡 Hint

Calculate the state x at each step using x(k+1) = A*x(k) + B*u, then output y(k) = C*x(k) + D*u.

At each step, the state updates by multiplying with A and adding B*u. The output y is C times the state plus D times input. After 3 steps, y = 1.045.

❓ data_output

intermediate

Number of States in Discrete State-Space Model

You have a discrete state-space system defined by matrices:

A is 4x4
B is 4x2
C is 3x4
D is 3x2

How many states does this system have?

A 2

B 3

C 4

D 6

💡 Hint

The number of states equals the dimension of the square matrix A.

Matrix A is square and its size determines the number of states. Since A is 4x4, the system has 4 states.

❓ visualization

advanced

Plotting State Trajectories

You simulate a discrete state-space system with 2 states over 10 steps. The states evolve as:

x(k+1) = A*x(k) + B*u(k)
where A = [[0.5, 0],[0, 0.8]], B = [[1],[0]], u(k) = 1 for all k, and x(0) = [0, 0].

Which plot correctly shows the state trajectories over time?

Simulink

import numpy as np
import matplotlib.pyplot as plt
A = np.array([[0.5,0],[0,0.8]])
B = np.array([[1],[0]])
x = np.array([[0],[0]])
u = 1
states = []
for _ in range(10):
    x = A @ x + B * u
    states.append(x.flatten())
states = np.array(states)
plt.plot(range(1,11), states[:,0], label='State 1')
plt.plot(range(1,11), states[:,1], label='State 2')
plt.xlabel('Step')
plt.ylabel('State value')
plt.legend()
plt.show()

A State 1 grows starting at 1, then halves each step; State 2 grows starting at 0, multiplied by 0.8 each step.

B State 1 grows starting at 0.5, increasing by 0.5 each step; State 2 starts at 0 and grows slower, approaching 1.

C State 1 starts at 1 and decreases by half each step; State 2 starts at 0 and grows by 0.8 each step.

D State 1 grows starting at 1, then halves each step; State 2 remains zero throughout.

💡 Hint

State 1 is influenced by input u(k)=1, so it grows initially then halves. State 2 has no input, so it decays by 0.8 each step.

State 1 updates as x1(k+1) = 0.5*x1(k) + 1*1, so it grows initially and then halves the previous value plus 1. State 2 updates as x2(k+1) = 0.8*x2(k) + 0, so it decays from zero. The plot shows State 1 increasing and State 2 starting at zero and remaining zero since no input.

🔧 Debug

advanced

Identify the Error in Discrete State-Space Update Code

Consider this Python code simulating a discrete state-space system:

import numpy as np
A = np.array([[0.9, 0.1],[0, 0.95]])
B = np.array([[0.05],[0.02]])
x = np.array([0, 0])
u = 1
for i in range(5):
x = A * x + B * u
print(x)

What error does this code produce or what is wrong with it?

Simulink

import numpy as np
A = np.array([[0.9, 0.1],[0, 0.95]])
B = np.array([[0.05],[0.02]])
x = np.array([0, 0])
u = 1
for i in range(5):
    x = A * x + B * u
print(x)

A Logical error: uses element-wise multiplication instead of matrix multiplication.

B TypeError due to incompatible shapes in multiplication.

C SyntaxError due to missing colon in for loop.

D No error; code runs and prints final state.

💡 Hint

Check how matrix multiplication is done in numpy. '*' does element-wise multiplication, '@' does matrix multiplication.

The code uses '*' which performs element-wise multiplication, not matrix multiplication. This causes incorrect state updates. The correct operator is '@' for matrix multiplication.

🚀 Application

expert

Predict Output Sequence from Discrete State-Space Model

A discrete state-space system has:

A = [[0.6, 0.3], [0.1, 0.7]]
B = [[0.2], [0.1]]
C = [[1, -1]]
D = [[0]]

Initial state x0 = [2, 1]
Input u(k) = 0 for all k

What is the output y at step 2?

Simulink

import numpy as np
A = np.array([[0.6, 0.3],[0.1, 0.7]])
B = np.array([[0.2],[0.1]])
C = np.array([[1,-1]])
D = np.array([[0]])
x = np.array([[2],[1]])
u = 0
for _ in range(2):
    x = A @ x + B * u
    y = C @ x + D * u
print(float(y))

A 0.3

B 0.5

C 1.1

D 0.7

💡 Hint

Calculate x at step 1 and 2, then compute y = C*x at step 2.

Step 1: x1 = A*x0 + B*0 = [[0.6,0.3],[0.1,0.7]]*[[2],[1]] = [[0.6*2+0.3*1],[0.1*2+0.7*1]] = [[1.5],[0.9]] Step 2: x2 = A*x1 + B*0 = [[0.6,0.3],[0.1,0.7]]*[[1.5],[0.9]] = [[0.6*1.5+0.3*0.9],[0.1*1.5+0.7*0.9]] = [[0.9+0.27],[0.15+0.63]] = [[1.17],[0.78]] Output y2 = C*x2 + D*0 = [1,-1]*[[1.17],[0.78]] = 1.17 - 0.78 = 0.39 (rounded 0.4 matches option B).

Practice

1. What does a discrete state-space model in Simulink primarily represent?

easy

A. A continuous-time system with differential equations

B. A graphical user interface for designing circuits

C. A system described by matrices that updates states step-by-step

D. A tool for creating 3D animations

Solution

1. Step 1: Understand discrete state-space concept

   Discrete state-space models describe systems using matrices that update the system's state at each time step.

2. Step 2: Compare options

   Only a system described by matrices that updates states step-by-step correctly describes this concept; others describe unrelated tools or continuous systems.

3. Final Answer:

   A system described by matrices that updates states step-by-step -> Option C

4. Quick Check:

   Discrete state-space = step-by-step matrix system [OK]

Hint: Discrete means step-by-step matrix updates [OK]

Common Mistakes:

• Confusing discrete with continuous systems
• Thinking state-space is only for continuous models
• Mixing up state-space with graphical tools

2. Which of the following is the correct syntax to define a discrete state-space system in Simulink's MATLAB Function block?

easy

A. sys = ss(A, B, C, D);

B. sys = ss(A, B, C, D, Ts);

C. sys = discrete(A, B, C, D, Ts);

D. sys = state_space(A, B, C, D, Ts);

Solution

1. Step 1: Recall syntax for discrete state-space

   In MATLAB/Simulink, discrete state-space systems use ss(A,B,C,D,Ts) where Ts is the sample time.

2. Step 2: Evaluate options

   sys = ss(A, B, C, D, Ts); matches the correct syntax. sys = ss(A, B, C, D); misses Ts (sample time), C and D use incorrect function names.

3. Final Answer:

   sys = ss(A, B, C, D, Ts); -> Option B

4. Quick Check:

   Discrete ss needs sample time Ts [OK]

Hint: Discrete ss needs sample time Ts as last argument [OK]

Common Mistakes:

• Omitting the sample time Ts
• Using wrong function names like discrete()
• Confusing continuous and discrete syntax

3. Given the discrete state-space matrices:
A = [[0.7, 0.2], [0, 0.9]];
B = [[0.1], [0.05]];
and initial state x0 = [1; 0], what is the state x[1] after one step with input u[0] = 2?

medium

A. [1.6; 0.1]

B. [1.0; 0.05]

C. [1.8; 0.0]

D. [0.9; 0.1]

Solution

1. Step 1: Calculate next state using formula

   Next state x[1] = A * x0 + B * u[0]. Calculate:
   A * x0 = [0.7*1 + 0.2*0; 0*1 + 0.9*0] = [0.7; 0]
   B * u[0] = [0.1*2; 0.05*2] = [0.2; 0.1]

2. Step 2: Add results to get x[1]

   x[1] = [0.7; 0] + [0.2; 0.1] = [0.9; 0.1]

3. Final Answer:

   [0.9; 0.1] -> Option D

4. Quick Check:

   Use x[1] = A*x0 + B*u[0] [OK]

Hint: Multiply A*x0 and add B*u[0] for next state [OK]

Common Mistakes:

• Forgetting to multiply input u by B
• Mixing matrix multiplication order
• Adding instead of multiplying matrices

4. You have this discrete state-space model in Simulink:
A = [[1, 0], [0, 1]];
B = [[0], [0]];
C = [[1, 0]];
D = [[0]];
Why does the output never change regardless of input?

medium

A. Because matrix B is zero, input has no effect on state

B. Because matrix C is zero, output is always zero

C. Because matrix A is identity, states never update

D. Because matrix D is zero, output ignores input

Solution

1. Step 1: Analyze matrix B effect on input

   Matrix B is zero, so input u does not affect state update: x[k+1] = A*x[k] + B*u[k] = A*x[k].

2. Step 2: Understand output behavior

   Output y = C*x + D*u depends only on state x and input u. Since input doesn't change state, output depends only on initial state and does not change with input.

3. Final Answer:

   Because matrix B is zero, input has no effect on state -> Option A

4. Quick Check:

   Zero B means input can't change state [OK]

Hint: Zero B means input can't change state [OK]

Common Mistakes:

• Confusing zero D with zero B effect
• Assuming identity A means output changes
• Ignoring input effect on state update

5. You want to design a discrete state-space model in Simulink to simulate a system with states updated every 0.1 seconds. The system matrices are:
A = [[0.8, 0.1], [0, 0.9]];
B = [[0.05], [0.02]];
Which sample time should you specify to correctly model this system?

hard

A. 0.1

B. 1

C. 0

D. -0.1

Solution

1. Step 1: Understand sample time meaning

   Sample time Ts defines how often the discrete system updates its state. Here, states update every 0.1 seconds.

2. Step 2: Choose correct sample time

   Ts must be positive and equal to the update interval, so Ts = 0.1 seconds.

3. Final Answer:

   0.1 -> Option A

4. Quick Check:

   Sample time = update interval = 0.1 [OK]

Hint: Sample time equals system update interval (positive) [OK]

Common Mistakes:

• Using zero or negative sample time
• Confusing sample time with continuous time
• Choosing wrong units for sample time