find the local equilibrium point of system satisfying constraints - Maple Help

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DynamicSystems[EquilibriumPoint] - find the local equilibrium point of system satisfying constraints

 Calling Sequence EquilibriumPoint(eqs, u, opts)

Parameters

 eqs - equation, expression, or set or list of equations or expressions. If you specify an expression expr, it is interpreted as the equation $\mathrm{expr}=0$ u - list or set; input variables opts - (optional) equation(s) of the form option = value; specify options for the EquilibriumPoint command

Description

 • The EquilibriumPoint command finds the equilibrium point of eqs such that constraints, if specified by the constraints option, are satisfied, and returns four lists of equations specifying the values of states, derivatives of states, inputs, and outputs at the equilibrium point, respectively.
 • The equilibrium point of a system is a point at which derivatives of the states vanish. The EquilibriumPoint command performs a local search and returns an equilibrium point closest to the initial point, either specified by the initialpoint parameter or chosen randomly.
 • If the EquilibriumPoint command cannot find a point at which derivatives are zero, it returns a point that minimizes the derivatives. It is possible to prescribe a non-zero value to the derivatives using the optional parameter constraints.

Examples

 > $\mathrm{with}\left(\mathrm{DynamicSystems}\right):$
 > $\mathrm{sys1}:=\left[\frac{ⅆ}{ⅆt}{x}_{1}\left(t\right)={{x}_{2}\left(t\right)}^{2}-4,\frac{ⅆ}{ⅆt}{x}_{2}\left(t\right)={x}_{1}\left(t\right)-1+u\left(t\right),y\left(t\right)={x}_{1}\left(t\right)+{x}_{2}\left(t\right)\right]$
 ${\mathrm{sys1}}{:=}\left[\frac{{ⅆ}}{{ⅆ}{t}}{}{{x}}_{{1}}{}\left({t}\right){=}{{{x}}_{{2}}{}\left({t}\right)}^{{2}}{-}{4}{,}\frac{{ⅆ}}{{ⅆ}{t}}{}{{x}}_{{2}}{}\left({t}\right){=}{{x}}_{{1}}{}\left({t}\right){-}{1}{+}{u}{}\left({t}\right){,}{y}{}\left({t}\right){=}{{x}}_{{1}}{}\left({t}\right){+}{{x}}_{{2}}{}\left({t}\right)\right]$ (1)
 > $\mathrm{EquilibriumPoint}\left(\mathrm{sys1},\left[u\left(t\right)\right],\mathrm{constraints}=\left[0<{x}_{1}\left(t\right)\right],\mathrm{initialpoint}=\left[u\left(t\right)=0,{x}_{1}\left(t\right)=2,{x}_{2}\left(t\right)=4\right]\right)$
 $\left[{{x}}_{{1}}{}\left({t}\right){=}{1.49999999768708}{,}{{x}}_{{2}}{}\left({t}\right){=}{-}{2.00000001053627}\right]{,}\left[\frac{{ⅆ}}{{ⅆ}{t}}{}{{x}}_{{1}}{}\left({t}\right){=}{4.21450963017378}{}{{10}}^{{-8}}{,}\frac{{ⅆ}}{{ⅆ}{t}}{}{{x}}_{{2}}{}\left({t}\right){=}{-}{4.62584648364128}{}{{10}}^{{-9}}\right]{,}\left[{u}{}\left({t}\right){=}{-}{0.500000002312923}\right]{,}\left[{y}{}\left({t}\right){=}{-}{0.500000012849197}\right]$ (2)
 > sys2 := {piecewise(x[1](t)<0, x[1](t), x[2](t) + x[1](t)^2) * piecewise(u(t)<0, cos(y(t)), sin(y(t))) = sin(x[1](t)^2) + 5 * y(t) + diff(x[1](t), t, t), y(t) - x[1](t)^2 + u(t)*x[1](t), diff(x[2](t), t) = f(x[1](t), u(t))}; user_function := [     f,     [float, float],     float,     proc(x, y)     local d1, d2;         d1 := cos(x)+x^2;         d2 := y*d1 + y^2;         return d1*x+d2*y- exp(d1);     end proc     ];
 ${\mathrm{sys2}}{:=}\left\{{y}{}\left({t}\right){-}{{{x}}_{{1}}{}\left({t}\right)}^{{2}}{+}{u}{}\left({t}\right){}{{x}}_{{1}}{}\left({t}\right){,}\left({{}\begin{array}{cc}{{x}}_{{1}}{}\left({t}\right)& {{x}}_{{1}}{}\left({t}\right){<}{0}\\ {{x}}_{{2}}{}\left({t}\right){+}{{{x}}_{{1}}{}\left({t}\right)}^{{2}}& {\mathrm{otherwise}}\end{array}\right){}\left({{}\begin{array}{cc}{\mathrm{cos}}{}\left({y}{}\left({t}\right)\right)& {u}{}\left({t}\right){<}{0}\\ {\mathrm{sin}}{}\left({y}{}\left({t}\right)\right)& {\mathrm{otherwise}}\end{array}\right){=}{\mathrm{sin}}{}\left({{{x}}_{{1}}{}\left({t}\right)}^{{2}}\right){+}{5}{}{y}{}\left({t}\right){+}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{t}}^{{2}}}{}{{x}}_{{1}}{}\left({t}\right){,}\frac{{ⅆ}}{{ⅆ}{t}}{}{{x}}_{{2}}{}\left({t}\right){=}{f}{}\left({{x}}_{{1}}{}\left({t}\right){,}{u}{}\left({t}\right)\right)\right\}$
 ${\mathrm{user_function}}{:=}\left[{f}{,}\left[{\mathrm{float}}{,}{\mathrm{float}}\right]{,}{\mathrm{float}}{,}{\mathbf{proc}}\left({x}{,}{y}\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathbf{local}}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathrm{d1}}{,}{\mathrm{d2}}{;}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathrm{d1}}{:=}{\mathrm{cos}}{}\left({x}\right){+}{x}{^}{2}{;}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathrm{d2}}{:=}{y}{*}{\mathrm{d1}}{+}{y}{^}{2}{;}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathbf{return}}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathrm{d1}}{*}{x}{+}{\mathrm{d2}}{*}{y}{-}{\mathrm{exp}}{}\left({\mathrm{d1}}\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathbf{end proc}}\right]$ (3)
 > $\mathrm{EquilibriumPoint}\left(\mathrm{sys2},\left[u\left(t\right)\right],\mathrm{functions}=\left[\mathrm{user_function}\right],\mathrm{initialpoint}=\left[{x}_{1}\left(t\right)=1,{x}_{2}\left(t\right)=1,u\left(t\right)=1\right]\right)$
 $\left[{{x}}_{{1}}{}\left({t}\right){=}{0.853420831346895}{,}{{x}}_{{2}}{}\left({t}\right){=}{0.687832129312234}\right]{,}\left[\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{t}}^{{2}}}{}{{x}}_{{1}}{}\left({t}\right){=}{2.43748043970982}{}{{10}}^{{-9}}{,}\frac{{ⅆ}}{{ⅆ}{t}}{}{{x}}_{{1}}{}\left({t}\right){=}{0.}{,}\frac{{ⅆ}}{{ⅆ}{t}}{}{{x}}_{{2}}{}\left({t}\right){=}{-}{1.14152598484907}{}{{10}}^{{-9}}\right]{,}\left[{u}{}\left({t}\right){=}{1.07055911392119}\right]{,}\left[{y}{}\left({t}\right){=}{-}{0.185310333631790}\right]$ (4)