DYNAMIC MODEL IN STATE SPACE

Dynamic & Control

INTRODUCTION

This sheet is to do a quick scan to the resonances of a desired transfer of a dynamic system via state space approach. From state space a bode-diagram
can be created with appropriate software.

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Equations of Motion (n-dimensional)

\underline M\ \underline{\" x}+\underline D\ \underline{\dot x}+\underline K\ \underline x+\underline F=\underline 0 with \underline x=[x_1 ... x_n]^T\rightarrow nx1

M=\left[\begin{matrix}m_1&0&0&0\\0&m_2&0&0\\0&0&{\ddots}&0\\0&0&0&m_n\end{matrix}\right]\rightarrow n\ x\ n

\underline K=\left[\begin{matrix}K_{11}&K_{12}&{\dots}&K_{1n}\\K_{21}&K_{22}&{\dots}&K_{2n}\\{\vdots}&{\vdots}&{\ddots}&{\vdots}\\K_{\mathit{n1}}&K_{n2}&{\dots}&K_{nn}\end{matrix}\right]\rightarrow n\ x\ n

K_{i,j}\rightarrow K_{mi,xj} = Sum of all c that work on m_i if
x_j is moved*

\underline D=\left[\begin{matrix}D_{11}&D_{12}&{\dots}&D_{1n}\\D_{21}&D_{22}&{\dots}&D_{2n}\\{\vdots}&{\vdots}&{\ddots}&{\vdots}\\D_{\mathit{n1}}&D_{\mathit{n2}}&{\dots}&D_{\mathit{nn}}\end{matrix}\right]\rightarrow n\ x\ n

D_{i,j}\rightarrow D_{mi,xj} = Sum of all d that work on m_i if x_j is moved*

* Check for K and D: If all elements of row or column I are summed the result is the stiffness or damping of mass I in relation to the fixed world (see example).

Furthermore these matrices are

\underline F= external force \rightarrow n\ x\ 1}

(not composed of stiffness/dampers)

State Space form

(SISO, n-dimensional, time independent)

\underline{\dot q}=\underline A\underline q+\underline Bu
y=\underline C\underline q+\underline Du

state vector: \underline q=[\begin{matrix}x_1&{\dots}&x_n\end{matrix}\ \begin{matrix}\dot x_1&{\dots}&\dot x_n\end{matrix}]^T\rightarrow 2n\ x\ 1

u= input \rightarrow 1\ x\ 1

u should be at least \frac {d}{dt}, so no x_i, always a flux or a multiplication of fluxes with parameters (see example).

This sheet assumes no feed forward so: \underline D=\underline 0

y=output \rightarrow 1\ x\ 1

y should be in the form x_i or \dot x_i, so no x_i and multiplications with parameters are possible but no double flux or higher fluxes (see example).

\underline A=\left[\begin{matrix}\underline 0_{nxn}&\underline I_{nxn}\\\underline M^{-1}\underline K&\underline M^{-1}\underline D\end{matrix}\right]\rightarrow 2n\ x\ 2n (system matrix)

\underline B\rightarrow 2n\ x\ 1 \ is the input matrix; composition see examples
\underline C\rightarrow 2n\ x\ 1 \ is the output matrix; composition see examples

This sheet assumes no feed forward so: \underline D=\underline 0

Equations of Motion (n-dimensional)
Dynamic model in state space - Block diagram representation
Example
Dynamic model in state space - Block diagram representation - Example

\underline x=[\begin{matrix}x_1&x_2\end{matrix}]^T so: n=2
\underline M=\left[\begin{matrix}m_1&0\\0&m_2\end{matrix}\right] with m_0=0
\underline K=\left[\begin{matrix}-c_1-c_2&c_2\\c_2&-c_2-c_3\end{matrix}\right]
\underline D=\left[\begin{matrix}-d_1-d_2-d_3&d_2\\d_2&-d_2\end{matrix}\right]
\underline F= [F_1-F_2]
\underline q=[\begin{matrix}x_1&x_2&\dot x_1&\dot x_2\end{matrix}]^T
\underline A=\left[\begin{matrix}\begin{matrix}0&0\\0&0\end{matrix}&\begin{matrix}1&0\\0&1\end{matrix}\\\underline M^{-1}\underline K&\underline M^{-1}\underline D\end{matrix}\right] with M^{-1}=\left[\begin{matrix}1/m_1&0\\0&1/m_2\end{matrix}\right]

3 examples of inputs:
u_1=\" x_1, u_2=F_1, u_3=\dot x_2-\dot x_1

Thus;
\underline B_1=[\begin{matrix}0&0&1&0\end{matrix}]^T
\underline B_2=[\begin{matrix}0&0&\frac {1}{m_1}&0\end{matrix}]^T
\underline B_3=[\begin{matrix}-1&1&0&0\end{matrix}]^T

3 examples of outputs:
y_1=\dot x_2, y_2=x_2-x_1, y_3=c_2(x_2-x_1)+d_2(\dot x_2-\dot x_1)

Thus;
\underline C_1=[\begin{matrix}0&0&0&1\end{matrix}]^T
\underline C_2=[\begin{matrix}-1&1&0&0\end{matrix}]^T
\underline C_3=[-\begin{matrix}c_2&c_2&-d_2&d_2\end{matrix}]^T

\underline D=\underline 0

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