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 \(x_1, x_2, u_1\)
\(\dot{x_1} = 0x_1 + 1x_2 + 0u_1\)
\(\dot{x_2} = -\frac{k}{m}x_1 - \frac{b}{m}x_2 + \frac{1}{m}u_1\)
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 \(y = \dot(x) = x_2\)
\[
y = 0x_1 + 1x_2 + 0u_1
\]
\[y =
\begin{bmatrix}
0 & 1
\end{bmatrix}
\begin{bmatrix}
x_1 \\
x_2
\end{bmatrix}
+
\begin{bmatrix}
0
\end{bmatrix}
u_1\]