 Double integrator

The double integrator is the linear time-invariant system in which one control input is integrated into velocity, and velocity into position. It is a mathematical idealization of one-dimensional motion under commanded acceleration; for a constant mass $m$ and net force $F$, Newton's second law gives $u=F/m$. The worked examples here use normalized dimensionless quantities and assume no control disturbances, rather than describing a laboratory record. ^[ref]

Writing $x=(p,v{)}^T$ givesThe Kalman rank criterion therefore gives controllability in every $T>0$ for unrestricted real control inputs. More concretely, choose constants $a,b$ on the first and second halves. Starting at $x_0=(p_0,v_0{)}^T$, direct integration givesThe endpoint map has determinant $T^3/8\ne 0$, so these two amplitudes can supply any desired endpoint displacement. For a rest-to-rest move of distance $d$, take $a=4d/T^2$, $b=-4d/T^2$. At $T=2$ and $d=1$ these are the two commands in the first figure. No impulse is needed; shorter deadlines require larger finite commands.

A double integrator is a mathematical model that describes a system where the output is the second integral of the input. In foundational terms, it is often used in control theory and dynamics to represent the motion of an object under constant acceleration. Mathematically, the double integrator can be expressed with the following set of equations: 1. \( \dot{x}(t) = v(t) \) (the first integrator: velocity is the first integral of position) 2.