Skip to content

State Space Model

Robotics / control

A State Space Model is linear representation of dynamic system,

Good Video - Intro to Control - 6.1 State-Space Model Basics

\[\dot{x} = A x + B u\]

Where:

  • x = state vector (what describes the system right now) (\(R^{n}\))
  • u = control input (what you apply) (\(R^{m}\))
  • \(\dot{x}\) = how the state is changing
  • A = system (physics) matrix (\(R^{n*n}\))
  • B = input (control) matrix (\(R^{n*m}\))

A matrix

It describes the natural physics of the system.

A tells how the system changes by itself, without any control input.

Demo: mass on a spring system

\[ u(t) - kx(t)-b\dot{x}(t) = m\ddot{x}(t) \]
  • u(t) : Input force
  • kx(t): Spring force
  • \(b\dot{x}\): frication
  • \(m\ddot{x}(t)\): ma

Named variables

  • \(x(t) = x_{1}\)
  • \(\dot{x}(t) = x_{2} = \dot{x_1}\)
  • \(u(t) = u_1\)
\[ u_1 - kx_1 - bx_2 = m\dot{x_2} \]

find dynamic

  • \(\dot{x_1}= ?\)
  • \(\dot{x_2}= ?\)

\(\dot{x_1} = x_2\)

\(\dot{x_2} = \frac{1}{m}u_1 - \frac{k}{m}x_1 - \frac{b}{m}x_2\)

rewrite the equation

related to \(x_1, x_2, u_1\)

Equation 1 \(\dot{x_1} = 0x_1 + 1x_2 + 0u_1\)

Equation 2 \(\dot{x_2} = -\frac{k}{m}x_1 - \frac{b}{m}x_2 + \frac{1}{m}u_1\)

rewrite as a matrix

Matrix A

$A = \begin{bmatrix} \text{coeff of } x_1 \text{ in eq1} & \text{coeff of } x_2 \text{ in eq1} \ \text{coeff of } x_1 \text{ in eq2} & \text{coeff of } x_2 \text{ in eq2} \end{bmatrix} $

so:

$A = \begin{bmatrix} 0 & 1 \ -\frac{k}{m} & -\frac{b}{m} \end{bmatrix} $

Matrix B

$B = \begin{bmatrix} \text{coeff of } u \text{ in eq1} \ \text{coeff of } u \text{ in eq2} \end{bmatrix} $

so:

$B = \begin{bmatrix} 0 \ \frac{1}{m} \end{bmatrix} $

Final

\[\dot{x} = A x + B u\]

$ \begin{bmatrix} \dot{x_1} \ \dot{x_2} \end{bmatrix} =\begin{bmatrix} 0 & 1 \ -\frac{k}{m} & -\frac{b}{m} \end{bmatrix} \begin{bmatrix} x_1 \ x_2 \end{bmatrix} + \begin{bmatrix} 0 \ \frac{1}{m} \end{bmatrix} u_1 $

Write as a vector state

$\dot{x} = \begin{bmatrix} 0 & 1 \ -\frac{k}{m} & -\frac{b}{m} \end{bmatrix} x + \begin{bmatrix} 0 \ \frac{1}{m} \end{bmatrix} u $

Output

State-space has two parts

\[\dot{x} = Ax + Bu \quad \text{(system dynamics)}\]
\[y = Cx + Du \quad \text{(what you measure / output)}\]

👉 A, B → physics 👉 C, D → what you choose to observe

Demo: Mass String system

State:

\[ x = \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} =\begin{bmatrix} position \\ velocity \end{bmatrix} \]

Case 1: Output = position

Tip

C is just a matrix that picks which part of the state you want to observe.

Step1

I want output = position

\[y = x_1\]

but the state vector \(\(x = \begin{bmatrix} x_1 \\ x_2 \end{bmatrix}\)\)

Step 2

Build C matrix as selector

\[y = \begin{bmatrix} 1 & 0 \end{bmatrix} x\]
Step 3

Define D

Ask Chat

Position depends on control input, so shouldn’t $ D\ne0$ ?”

\[D = 0\]
Final
\[y = Cx + Du$$ $$y = \begin{bmatrix} 1 & 0 \end{bmatrix} x + 0 \cdot u\]

Reference