State space representation

  • 24 janv. 2024
  • 22 nov. 2024
  • 2 min de lecture
  • llms.txt

See also sides

time-domain technique

x˙=Ax+Bu y=Cx+Du\begin{align} \dot{x} &= Ax + Bu \\\ y &= Cx + Du \end{align}
  • Linearly independent
  • State vector: x=[x1,x2,…,xn]Tx = [x_{1},x_{2},\ldots, x_{n}]^{T}

transfer function to a state space representation

controller form

Given

G(s)=∑i=1n−1bisi+b0sn+∑i=1n−1aisi+a0=Y(s)U(s)G(s) = \frac{\sum_{i=1}^{n-1}b_is^i + b_{0}}{s^n + \sum_{i=1}^{n-1}a_is^{i} + a_{0}} = \frac{Y(s)}{U(s)}

We get controller canonical state space form:

x˙(t)=[010⋯00 001⋯00 ⋮⋮⋮⋱⋮⋮ 000⋯10 000⋯01 −a0−a1−a2⋯−an−2−an−1]x(t)+[0 0 ⋮ 0 0 1]u(t) y(t)=[b0b1⋯bn−2bn−1]x(t).\begin{aligned} \dot{x}(t) &= \begin{bmatrix} 0 & 1 & 0 & \cdots & 0 & 0 \\\ 0 & 0 & 1 & \cdots & 0 & 0 \\\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\\ 0 & 0 & 0 & \cdots & 1 & 0 \\\ 0 & 0 & 0 & \cdots & 0 & 1 \\\ -a_0 & -a_1 & -a_2 & \cdots & -a_{n-2} & -a_{n-1} \end{bmatrix} x(t) + \begin{bmatrix} 0 \\\ 0 \\\ \vdots \\\ 0 \\\ 0 \\\ 1 \end{bmatrix} u(t) \\\ y(t) &= \begin{bmatrix} b_0 & b_1 & \cdots & b_{n-2} & b_{n-1} \end{bmatrix} x(t). \end{aligned}

observer form

We get observer canonical state space form:

x˙(t)=[−an−110⋯00−an−201⋯00⋮⋮⋮⋱⋮⋮−a200⋯10−a100⋯01−a000⋯00]x(t)+[bn−1bn−2⋮b2b1b0]u(t)y(t)=[10⋯00]x(t).\begin{aligned} \dot{x}(t) &= \begin{bmatrix} -a_{n-1} & 1 & 0 & \cdots & 0 & 0 \\ -a_{n-2} & 0 & 1 & \cdots & 0 & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ -a_2 & 0 & 0 & \cdots & 1 & 0 \\ -a_1 & 0 & 0 & \cdots & 0 & 1 \\ -a_0 & 0 & 0 & \cdots & 0 & 0 \end{bmatrix} x(t) + \begin{bmatrix} b_{n-1} \\ b_{n-2} \\ \vdots \\ b_2 \\ b_1 \\ b_0 \end{bmatrix} u(t) \\ y(t) &= \begin{bmatrix} 1 & 0 & \cdots & 0 & 0 \end{bmatrix} x(t). \end{aligned}

Liens retour

Tout ce qu'il faut savoir sur la conception des systèmes de contrôle
See also source for code and jupyter notebook Book: ISBN: 978-1-119-47422-7 and pdf Note sp.Heaviside(t) is u(t) snippets import sympy import sympy as sp from symbol import symbols, ...

State space representation

See also sides

time-domain technique

x˙=Ax+Bu y=Cx+Du\begin{align} \dot{x} &= Ax + Bu \\\ y &= Cx + Du \end{align}
  • Linearly independent
  • State vector: x=[x1,x2,…,xn]Tx = [x_{1},x_{2},\ldots, x_{n}]^{T}

transfer function to a state space representation

controller form

Given

G(s)=∑i=1n−1bisi+b0sn+∑i=1n−1aisi+a0=Y(s)U(s)G(s) = \frac{\sum_{i=1}^{n-1}b_is^i + b_{0}}{s^n + \sum_{i=1}^{n-1}a_is^{i} + a_{0}} = \frac{Y(s)}{U(s)}

We get controller canonical state space form:

x˙(t)=[010⋯00 001⋯00 ⋮⋮⋮⋱⋮⋮ 000⋯10 000⋯01 −a0−a1−a2⋯−an−2−an−1]x(t)+[0 0 ⋮ 0 0 1]u(t) y(t)=[b0b1⋯bn−2bn−1]x(t).\begin{aligned} \dot{x}(t) &= \begin{bmatrix} 0 & 1 & 0 & \cdots & 0 & 0 \\\ 0 & 0 & 1 & \cdots & 0 & 0 \\\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\\ 0 & 0 & 0 & \cdots & 1 & 0 \\\ 0 & 0 & 0 & \cdots & 0 & 1 \\\ -a_0 & -a_1 & -a_2 & \cdots & -a_{n-2} & -a_{n-1} \end{bmatrix} x(t) + \begin{bmatrix} 0 \\\ 0 \\\ \vdots \\\ 0 \\\ 0 \\\ 1 \end{bmatrix} u(t) \\\ y(t) &= \begin{bmatrix} b_0 & b_1 & \cdots & b_{n-2} & b_{n-1} \end{bmatrix} x(t). \end{aligned}

observer form

We get observer canonical state space form:

x˙(t)=[−an−110⋯00−an−201⋯00⋮⋮⋮⋱⋮⋮−a200⋯10−a100⋯01−a000⋯00]x(t)+[bn−1bn−2⋮b2b1b0]u(t)y(t)=[10⋯00]x(t).\begin{aligned} \dot{x}(t) &= \begin{bmatrix} -a_{n-1} & 1 & 0 & \cdots & 0 & 0 \\ -a_{n-2} & 0 & 1 & \cdots & 0 & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ -a_2 & 0 & 0 & \cdots & 1 & 0 \\ -a_1 & 0 & 0 & \cdots & 0 & 1 \\ -a_0 & 0 & 0 & \cdots & 0 & 0 \end{bmatrix} x(t) + \begin{bmatrix} b_{n-1} \\ b_{n-2} \\ \vdots \\ b_2 \\ b_1 \\ b_0 \end{bmatrix} u(t) \\ y(t) &= \begin{bmatrix} 1 & 0 & \cdots & 0 & 0 \end{bmatrix} x(t). \end{aligned}