TITLE: Analytical calculation of control input for given system
QUESTION [0 upvotes]: In the equations given below is it possible to calculate $u$ analytically to drive system from the initial condition of $x_{1} = 5$ ,$x_{2} = 0$ to the equilibrium point in the origin $x_{1} = 0$ ,$x_{2} = 0$?
$$(1) \qquad \dfrac{\mathrm{d} x_{1}}{\mathrm{d} t} = x_{2}$$
$$(2) \qquad \dfrac{\mathrm{d} x_{2}}{\mathrm{d} t} = u$$

REPLY [1 votes]: Your system is
$$\dot{\boldsymbol{x}}=\begin{bmatrix}0 & 1\\0 & 0 \end{bmatrix}\boldsymbol{x}+\begin{bmatrix}0\\1 \end{bmatrix}u,$$
in which $\boldsymbol{x}=\left[x_1,x_2 \right]^T$
Using full state feedback with $u = -k_1x_1 -k_2x_2$ we obtain the closed loop system
$$\dot{\boldsymbol{x}}=\begin{bmatrix}0 & 1\\-k_1 & -k_2 \end{bmatrix}\boldsymbol{x}.$$
The eigenvalue equation of the closed-loop system is given by
$$\chi(s)=s^2+k_2s+k_1.$$
By the Hurwitz criterion for quadratic polynomials, we can guarantee asymptotic stability of the closed-loop system if $k_1$ and $k_2$ are both positive but not zero. Independent from your initial conditions the system will converge to the equilibrium point in the origin for $t\to \infty$. Only if we start at the origin we will not take infinitely long to reach the origin. Another way to arrive at the origin is to use a nonlinear control law (see time optimal control and the theorem of Feldbaum for time optimal control of linear systems with constraints on the control inputs. This is sometimes called Bang-Bang-Control)
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