Discrete state-space in Simulink - Cheat Sheet & Quick Revision

Recall & Review

beginner

What is a discrete state-space model?

A discrete state-space model describes a system using equations that update the system's state at fixed time steps. It uses matrices to represent how the current state and inputs affect the next state and output.

beginner

What are the main matrices in a discrete state-space model?

The main matrices are:
• A: State transition matrix (how current state affects next state)
• B: Input matrix (how inputs affect next state)
• C: Output matrix (how state affects output)
• D: Feedthrough matrix (how inputs affect output directly)

beginner

How does Simulink represent a discrete state-space system?

Simulink uses the 'Discrete State-Space' block where you enter matrices A, B, C, D and specify the sample time. This block updates the system state at each time step and outputs the system response.

intermediate

Why use discrete state-space models instead of continuous ones?

Discrete models work with digital computers and controllers that operate in steps. They are easier to simulate and implement in software, especially when the system is sampled at regular intervals.

intermediate

What is the role of the sample time in a discrete state-space model?

Sample time defines how often the system updates its state. A smaller sample time means more frequent updates and can capture faster system changes, but requires more computation.

Which matrix in a discrete state-space model shows how inputs affect the next state?

A A matrix

B D matrix

C C matrix

D B matrix

The B matrix shows how the input affects the next state in the system.

In Simulink, what must you specify when using the Discrete State-Space block?

A Matrices A, B, C, D and sample time

B Only sample time

C Initial state only

D Only matrices A and B

You must specify all four matrices (A, B, C, D) and the sample time for the block to work correctly.

What happens if you choose a very large sample time in a discrete state-space model?

A The system updates more frequently

B The system becomes continuous

C The system updates less frequently

D The system ignores inputs

A larger sample time means the system updates less often, which can miss fast changes.

Which matrix relates the current state to the output in a discrete state-space model?

A C matrix

B B matrix

C A matrix

D D matrix

The C matrix maps the current state to the output.

Why are discrete state-space models preferred for digital controllers?

A Because they work with continuous signals

B Because they update states at fixed time steps

C Because they do not use matrices

D Because they ignore inputs

Digital controllers operate in steps, so discrete models that update at fixed intervals fit naturally.

Explain the role of each matrix (A, B, C, D) in a discrete state-space model.

Think about how the system state and output depend on current state and inputs.

Describe how sample time affects the behavior of a discrete state-space system in Simulink.

Consider what happens if sample time is very small or very large.

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