Double integrator

In systems and control theory, the double integrator is a canonical example of a second-order control system.[1] It models the dynamics of a simple mass in one-dimensional space under the effect of a time-varying force input ${\textbf {u}}$ .

Differential equations

The differential equations which represent a double integrator are:

{\ddot {q}}=u(t)

y=q(t)

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

{\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 ${\textbf {u}}$ is the second derivative of the output ${\textbf {x}}$ . In the scalar form, the control input is the second derivative of the output $q$

State space representation

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

{\dot {\textbf {x}}}(t)={\begin{bmatrix}0&1\\0&0\\\end{bmatrix}}{\textbf {x}}(t)+{\begin{bmatrix}0\\1\end{bmatrix}}{\textbf {u}}(t)

{\textbf {y}}(t)={\begin{bmatrix}1&0\end{bmatrix}}{\textbf {x}}(t).

According to this model, the input ${\textbf {u}}$ is the second derivative of the output ${\textbf {y}}$ , 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

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

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

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References

Venkatesh G. Rao and Dennis S. Bernstein (2001). "Naive control of the double integrator" (PDF). IEEE Control Systems Magazine. Retrieved 2012-03-04.

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[Maxwell1867-1] Venkatesh G. Rao and Dennis S. Bernstein (2001). "Naive control of the double integrator" (PDF). IEEE Control Systems Magazine. Retrieved 2012-03-04.