The nonlinear state-space model of a spherical pendulum cannot be linearized and cannot be controlled?

Question

I am attempting to model and control a spherical pendulum. I have derived the equations as follows:

The positions of the mass m:

x = \[ScriptL] Sin[\[Theta][t]] Cos[\[Phi][t]];

y = \[ScriptL] Sin[\[Theta][t]] Sin[\[Phi][t]];

z = \[ScriptL] Cos[\[Theta][t]];

The velocities are hence:

{
\!\(\*OverscriptBox[\(x\), \(.\)]\), 
\!\(\*OverscriptBox[\(y\), \(.\)]\), 
\!\(\*OverscriptBox[\(z\), \(.\)]\)} = Table[D[q, t], {q, {x, y, z}}]

The Lagrangian is L=T-V and can now be constructed:

T = 1/2 m \[ScriptV]^2

V = -m g z

\[ScriptCapitalL] = T - V

The equations of motion are derived from the Euler-Lagrange formula:

eqns = Table[\!\(
\*SubscriptBox[\(\[PartialD]\), \(t\)]\(D[\[ScriptCapitalL], \(q'\)[t]]\)\) - 
     D[\[ScriptCapitalL], q[t]] == Subscript[\[ScriptCapitalT], 
    q], {q, {\[Theta], \[Phi]}}] /. {Subscript[\[ScriptCapitalT], \[Theta]] ->
     Subscript[\[ScriptCapitalT], 1][t], 
   Subscript[\[ScriptCapitalT], \[Phi]] -> Subscript[\[ScriptCapitalT], 2][t]}

The parameters of the system as a list of replacement rules:

pars = {g -> 9.8, m -> 0.5, \[ScriptL] -> 1};

The nonlinear state-space model:

spend = NonlinearStateSpaceModel[
  eqns, {\[Theta][t], \[Theta]'[t], \[Phi][t], \[Phi]'[
    t]}, {Subscript[\[ScriptCapitalT], 1][t], 
   Subscript[\[ScriptCapitalT], 2][t]}, {\[Theta][t], \[Phi][t]}, t]/.pars

Linearizing results in error:

ssm = StateSpaceModel[spend]

Curiously, the output response can be simulated:

OutputResponse[{spend, {0.1, 0.3, 0.6, 0.01}}, {0, 0}, {t, 0, 10}];
Plot[%, {t, 0, 10}, PlotRange -> All]

Can anyone help figure out what's wrong with this system? Thanks!

Answer 1

This is because there is no standard linear state-space representation for a spherical pendulum.

Compute the nonlinear Euler-Lagrange equations.

x = ℓ Sin[θ[t]] Cos[ϕ[t]];
y = ℓ Sin[θ[t]] Sin[ϕ[t]];
z = ℓ Cos[θ[t]];
\[ScriptV] = Table[D[q, t], {q, {x, y, z}}];
T = Simplify[1/2 m \[ScriptV] . \[ScriptV]];
V = -m g z;
ℒ = T - V;
eqns = Table[D[D[ℒ, Derivative[1][q][t]], t] - D[ℒ, q[t]] == 
       Subscript[\[ScriptCapitalT], q][t], {q, {θ, ϕ}}]

$$\left\{g m \ell \sin (\theta (t))+m \ell ^2 \theta ''(t)-m \ell ^2 \sin (\theta (t)) \cos (\theta (t)) \phi '(t)^2=\mathcal{T}_{\theta }(t),\\2 m \ell ^2 \theta '(t) \sin (\theta (t)) \cos (\theta (t)) \phi '(t)+m \ell ^2 \sin ^2(\theta (t)) \phi ''(t)=\mathcal{T}_{\phi }(t)\right\}$$

Look at the second equation. If we linearize it we are left with $0=\mathcal{T}_{\phi }(t)$.

If we convert it to a StateSpaceModel, we see the same equation is the last one in the descriptor state-space model.

StateSpaceModel[eqns, {θ[t], θ'[t], ϕ[t], ϕ'[t]}, {Subscript[\[ScriptCapitalT], θ][t], 
  Subscript[\[ScriptCapitalT], ϕ][t]}, {θ[t], ϕ[t]}, t] 

So at best, we can get this rather wanting linear descriptor state-space representation of the system. It cannot be converted to the standard form because the descriptor matrix is singular.

However, we can have a standard NonlinearStateSpaceModel representation of the system.

nssm = NonlinearStateSpaceModel[eqns, {θ[t], θ'[t], ϕ[t], ϕ'[t]}, {Subscript[\[ScriptCapitalT], θ][t], 
  Subscript[\[ScriptCapitalT], ϕ][t]}, {θ[t], ϕ[t]}, t] 

What happens if we try to linearize this to get a standard linear state-space representation as you have attempted to do? First the Jacobian is computed and then evaluated at the operating points. And you can see the source of the messages that you are encountering.

Normal[nssm][[1, 1]]
jacobian = D[%, {{θ[t], Subscript[\[FormalX], 1][t], ϕ[t], Subscript[\[FormalX], 2][t]}}];
jacobian /. Thread[{θ[t], Subscript[\[FormalX], 1][t], ϕ[t], Subscript[\[FormalX], 2][t]} -> 0]

In summary, a linear state-space representation is not adequate to model a nonlinear spherical pendulum.