Discrete state-space in Simulink - Mini Project: Build & Apply

Discrete State-Space Model in Simulink

📖 Scenario: You are working on a control system project where you need to model a discrete-time system using state-space representation. This is common in digital control where systems are sampled at fixed intervals.

🎯 Goal: Build a discrete state-space model in Simulink that represents a system with given matrices and simulate its output for a given input sequence.

📋 What You'll Learn

Create the state-space matrices A, B, C, and D with exact values

Set the sample time Ts for the discrete system

Use the Discrete State-Space block in Simulink with the matrices and sample time

Simulate the system with a given input and observe the output

💡 Why This Matters

🌍 Real World

Discrete state-space models are used in digital control systems for robotics, aerospace, and industrial automation where systems are controlled by computers.

💼 Career

Understanding discrete state-space modeling is essential for control engineers and data scientists working on system identification, control design, and simulation.

1

Define the state-space matrices

Create the matrices A, B, C, and D in MATLAB workspace with these exact values: A = [0.7 0.2; 0 0.9], B = [0.1; 0.05], C = [1 0], and D = 0.

Simulink

% Your code here to define A, B, C, D matrices

Hint

Use square brackets to create matrices and separate rows by semicolons.

2

Set the sample time for the discrete system

Create a variable called Ts and set it to 0.1 to represent the sample time in seconds.

Simulink

A = [0.7 0.2; 0 0.9];
B = [0.1; 0.05];
C = [1 0];
D = 0;
% Define the sample time Ts

Hint

Sample time is the time interval between discrete steps.

3

Configure the Discrete State-Space block in Simulink

In Simulink, add a Discrete State-Space block and set its parameters to use the workspace variables A, B, C, D, and sample time Ts.

Simulink

A = [0.7 0.2; 0 0.9];
B = [0.1; 0.05];
C = [1 0];
D = 0;
Ts = 0.1;
% Configure Discrete State-Space block parameters in Simulink

Hint

Use the block's parameter dialog to enter variable names exactly as defined.

4

Simulate the system and display the output

Run the Simulink model with an input signal of your choice (for example, a step input) and display the output signal using a Scope block.

Simulink

A = [0.7 0.2; 0 0.9];
B = [0.1; 0.05];
C = [1 0];
D = 0;
Ts = 0.1;
% Run the simulation and observe output on Scope block

Hint

Use a Step block as input and connect the output of the Discrete State-Space block to a Scope block to visualize the output.

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