Discrete state-space in Simulink

Introduction

Discrete state-space helps us understand how systems change step-by-step over time using simple math. It breaks down complex systems into small, easy-to-follow parts.

Modeling how a robot moves in steps based on commands.

Predicting the temperature changes in a room every minute.

Controlling a car's speed using digital signals.

Simulating financial data that updates daily.

Designing digital filters for audio signals.

Syntax

Simulink

x[k+1] = A * x[k] + B * u[k]
y[k] = C * x[k] + D * u[k]

x[k] is the state at step k (current state).

u[k] is the input at step k.

y[k] is the output at step k.

A, B, C, D are matrices that define system behavior.

Examples

Simple one-step update with scalar values.

Simulink

A = [0.7];
B = [0.2];
C = [1];
D = [0];
x[0] = 0;
u[0] = 1

x[1] = A * x[0] + B * u[0]
y[0] = C * x[0] + D * u[0]

Two-dimensional state example for position and velocity.

Simulink

A = [[1, 0.1], [0, 1]]
B = [[0], [0.1]]
C = [1, 0]
D = [0]
x[0] = [0, 0]
u[0] = 1

x[1] = A * x[0] + B * u[0]
y[0] = C * x[0] + D * u[0]

Sample Program

This code simulates a discrete state-space system for 5 steps. It shows how the state and output change with each input.

Simulink

import numpy as np

# Define system matrices
A = np.array([[0.9, 0.1], [0, 0.8]])
B = np.array([[0.1], [0.2]])
C = np.array([[1, 0]])
D = np.array([[0]])

# Initial state
x = np.array([[0], [0]])

# Input sequence (5 steps)
u = np.array([[1], [0], [1], [0], [1]])

# Lists to store states and outputs
states = [x]
outputs = []

for k in range(len(u)):
    x = A @ x + B @ u[k]
    y = C @ x + D @ u[k]
    states.append(x)
    outputs.append(y)

# Print results
for i, (state, output) in enumerate(zip(states[1:], outputs)):
    print(f"Step {i+1} state:\n{state.flatten()}\nOutput: {output.flatten()[0]:.2f}\n")

Important Notes

Make sure matrix sizes match to avoid errors.

Discrete means the system updates at fixed steps, like frames in a video.

Initial state is important to start the simulation correctly.

Summary

Discrete state-space models systems step-by-step using matrices.

It helps predict system behavior over time with inputs and states.

Useful for digital control and simulations in many fields.

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