Continuous

StateSpace Block

State-space realization: x'=Ax+Bu, y=Cx+Du with A,B,C,D matrices

# StateSpace Block Documentation

Overview

The StateSpace block implements a general state-space linear system described by matrices A, B, C, D:

x' = Ax + Bu    (state equation)
y  = Cx + Du    (output equation)

This is the most general form of linear time-invariant (LTI) systems and is fundamental to modern control theory.

Mathematical Model

ẋ = Ax + Bu
y = Cx + Du

where:

x ∈ ℝⁿ: state vector (n states)
u ∈ ℝᵐ: input vector (m inputs)
y ∈ ℝᵖ: output vector (p outputs)
A: n×n state matrix
B: n×m input matrix
C: p×n output matrix
D: p×m feedthrough matrix

Where:

A: n×n state matrix
B: n×m input matrix
C: p×n output matrix
D: p×m feedthrough matrix

Inputs

Input(s): Multiple numeric inputs (u₁, u₂, ...)

Outputs

Output(s): Multiple numeric outputs (y₁, y₂, ...)

Parameters

A Matrix (State)

Description: The n×n state transition matrix
Format: MATLAB-style matrix string
- Rows separated by semicolons (;)
- Elements separated by spaces
- Examples: "1 2; 3 4" or "[1 0; 0 -2]"
Default: "-1" (1×1 single pole at -1)

B Matrix (Input)

Description: The n×m input coupling matrix
Format: MATLAB-style matrix string
Default: "1"

C Matrix (Output)

Description: The p×n output matrix
Format: MATLAB-style matrix string
Default: "1"

D Matrix (Feedthrough)

Description: The p×m direct feedthrough matrix (usually 0 for proper systems)
Format: MATLAB-style matrix string
Default: "0"

Initial State

Description: Initial condition for the state vector
Format: Comma-separated values
Default: "0"

Matrix Format Examples

2×2 State Matrix

1 -2
0 -3

2×1 Input Matrix (2 states, 1 input)

0
1

1×2 Output Matrix (1 output, 2 states)

1 0

Examples

Example 1: First-Order System

A simple exponential decay system with time constant τ = 1:

Parameters:

A: "-1"
B: "1"
C: "1"
D: "0"

This represents: ẋ = -x + u, y = x (or any state)

Example 2: Second-Order Mass-Spring System

m·ẍ + c·ẋ + k·x = u

Let x₁ = position, x₂ = velocity. Then:

ẋ₁ = x₂
ẋ₂ = -k/m·x₁ - c/m·x₂ + 1/m·u
y = x₁ (output position)

Parameters (m=1, c=2, k=1):

A: "0 1; -1 -2"
B: "0; 1"
C: "1 0"
D: "0"

Example 3: MIMO System (2 inputs, 2 outputs)

A: "0 1 0; 0 0 1; -2 -3 -1"
B: "0 0; 1 0; 0 1"
C: "1 0 0; 0 1 0"
D: "0 0; 0 0"

Common Use Cases

1. Control system design: Implement controllers, observers, compensators 2. Filter design: Digital and analog filters 3. Simulation of mechanical systems: Robot dynamics, vehicle models 4. Power systems: Electrical machine models 5. Aerospace: Aircraft dynamics, satellite attitude control

Numerical Integration

The block uses forward Euler integration for simplicity and stability in most cases:

x[k+1] = x[k] + ẋ[k] · dt

For better accuracy with large sample times, consider using a continuous transfer function approach or a more sophisticated integrator.

Dimension Requirements

The matrices must satisfy strict dimension constraints:

A: Must be square n×n (n = number of states)
B: Must be n×m (n states, m inputs)
C: Must be p×n (p outputs, n states)
D: Must be p×m (p outputs, m inputs)

If dimensions are inconsistent, the block outputs 0 and skips computation. Verify matrix dimensions match before running the simulation.

Stability Constraints (Forward Euler)

The block uses Forward Euler integration. For stability:

All eigenvalues of A must satisfy |1 + λ·Δt| < 1
For a real eigenvalue at -a: requires Δt < 2/a
Rule of thumb: Δt ≤ 1/(5 × |largest eigenvalue|)
If the simulation diverges, reduce the sample time

Performance Considerations

Computational cost: O(n²) for matrix-vector operations (n = number of states)
Stability: Ensure eigenvalues of A are in the left-half plane (continuous) for stability
Numerical precision: Works well for n < 20 states; higher orders may accumulate numerical errors

Remarks

Dimension consistency: Matrix dimensions must satisfy A(n×n), B(n×m), C(p×n), D(p×m) or the block outputs zero
Euler stability: Forward Euler integration requires Δt < 2/|λ_max|; reduce sample time if simulation diverges
MIMO support: Full multi-input, multi-output capability; use comma-separated initial states for multi-state systems
State order: For n > 20 states numerical precision may degrade; consider model reduction
Feedthrough: If D ≠ 0 the output depends directly on the input (instantaneous response), which can create algebraic loops