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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

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

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

\(\dot{x_1} = 0x_1 + 1x_2 + 0u_1\) \(\dot{x_2} = -\frac{k}{m}x_1 - \frac{b}{m}x_2 + \frac{1}{m}u_1\)

rewrite as a matrix

\[ \begin{bmatrix} \dot{x_1} \\ \dot{x_2} \end{bmatrix} =\begin{bmatrix} 0 & 1 \\ -\frac{k}{m} & -\frac{b}{m} \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} + \begin{bmatrix} 0 \\ \frac{1}{m} \end{bmatrix} u_1 \]
\[\dot{x} = \begin{bmatrix} 0 & 1 \\ -\frac{k}{m} & -\frac{b}{m} \end{bmatrix} x + \begin{bmatrix} 0 \\ \frac{1}{m} \end{bmatrix} u\]
\[\dot{x} = A x + B u\]

Output

How is the ouput (x) depend and the system state and control input

in the example we want to find change in distance \(x(t)\)

  • \(y = x_1 = x(t)\)

\(y = Cx + Du\)

\[ y = 1x_1 + 0x_2 + 0u_1 \]
\[y = \begin{bmatrix} 1 & 0 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} + \begin{bmatrix} 0 \end{bmatrix} u_1\]

for example if we want to look at the system if the velocity is the output \(y = \dot(x) = x_2\)

\[ y = 0x_1 + 1x_2 + 0u_1 \]
\[y = \begin{bmatrix} 0 & 1 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} + \begin{bmatrix} 0 \end{bmatrix} u_1\]

Reference