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Differential equations

The differential equations which represent a double integrator are:

q ¨ = u ( t ) {\displaystyle {\ddot {q}}=u(t)} y = q ( t ) {\displaystyle y=q(t)}

where both q ( t ) , u ( t ) ∈ R {\displaystyle q(t),u(t)\in \mathbb {R} } Let us now represent this in state space form with the vector x(t) = [ q q ˙ ] {\displaystyle {\textbf {x(t)}}={\begin{bmatrix}q\\{\dot {q}}\\\end{bmatrix}}}

x ˙ ( t ) = d x d t = [ q ˙ q ¨ ] {\displaystyle {\dot {\textbf {x}}}(t)={\frac {d{\textbf {x}}}{dt}}={\begin{bmatrix}{\dot {q}}\\{\ddot {q}}\\\end{bmatrix}}}

In this representation, it is clear that the control input u {\displaystyle {\textbf {u}}} is the second derivative of the output x {\displaystyle {\textbf {x}}} . In the scalar form, the control input is the second derivative of the output q {\displaystyle q} .

State space representation

The normalized state space model of a double integrator takes the form

x ˙ ( t ) = [ 0 1 0 0 ] x ( t ) + [ 0 1 ] u ( t ) {\displaystyle {\dot {\textbf {x}}}(t)={\begin{bmatrix}0&1\\0&0\\\end{bmatrix}}{\textbf {x}}(t)+{\begin{bmatrix}0\\1\end{bmatrix}}{\textbf {u}}(t)} y ( t ) = [ 1 0 ] x ( t ) . {\displaystyle {\textbf {y}}(t)={\begin{bmatrix}1&0\end{bmatrix}}{\textbf {x}}(t).}

According to this model, the input u {\displaystyle {\textbf {u}}} is the second derivative of the output y {\displaystyle {\textbf {y}}} , hence the name double integrator.

Transfer function representation

Taking the Laplace transform of the state space input-output equation, we see that the transfer function of the double integrator is given by

Y ( s ) U ( s ) = 1 s 2 . {\displaystyle {\frac {Y(s)}{U(s)}}={\frac {1}{s^{2}}}.}

Using the differential equations dependent on q ( t ) , y ( t ) , u ( t ) {\displaystyle q(t),y(t),u(t)} and x(t) {\displaystyle {\textbf {x(t)}}} , and the state space representation:

References

1. Venkatesh G. Rao and Dennis S. Bernstein (2001). "Naive control of the double integrator" (PDF). IEEE Control Systems Magazine. Retrieved 2012-03-04. http://www-personal.umich.edu/~dsbaero/others/25-DoubleIntegrator.pdf ↩