Name of this control strategy Assume that we have a nonlinear dynamical system on $\mathbb{R}^n$ of the form
$$
\dot{x}(t) = f(x(t),u),
$$
where $f(0)=0$, i.e. the origin is an equilibrium. My question is, is there a particular control strategy which aims to stabilize such an equilibrium point with two controls $A(t)\in\mathbb{R}^{n\times n},\;b(t)\in\mathbb{R}^n$ passing to the system
$$ \dot{x} = f(x,\tilde{u}(t,x))=\tilde{f}(A(t)x+b(t))? $$
For example, if $f(x,u) = M(x+u) := \tilde{f}(x+u)$, for a matrix $M$, then I would just set $\tilde{u}=A(t)x+b(t)-x$, i.e. I get
$$f(x,\tilde{u}) = f(x,A(t)x+b(t)-x)=\tilde{f}(A(t)x+b(t))=M(A(t)x+b(t)).$$
More precisely, is there a known control strategy based on the idea of combining the nonlinearity with a time-dependent affine transformation of the state variable? I would just like the name of this strategy, if it exists, then I will figure out where to understand and study it.
It would still be interesting to know if a similar strategy as
$$\dot{x} = f(A(t)x)+b(t)$$
is used.
No, I don't think so. Saying something does not exist, is always a big stretch. I don't know every method ever conceived, but let me explain why I think such a strategy does not exist:
The progression of the dynamical system only indirectly depends on $t$. Directly it depends only on your state $x$ and your control input $u$ (&quot;Markov Property&quot;). So why should I inject a time-dependency into my controller, if the system itself does not care about the time?
Now, there is one important part of your problem missing: Optimal control always looks to minimise a cost function. What is your cost function? That is a modelling assumption, you have to make! You write, you want $x$ to stay close to the origin, so maybe we put something like $J(x, u, t) = \int_{t_0}^{t_1}||x||^2$.
Now, if you drop the time dependency for your controller $u$, and assume $f$ is linear, you'd arrive at a Linear-quadratic regulator. I think that would be a reasonable controller for the problem you describe. Or at least a good point to start, to find more sophisticated control algorithms.
No, I don't think so. Saying something does not exist, is always a big stretch. I don't know every method ever conceived, but let me explain why I think such a strategy does not exist:
The progression of the dynamical system only indirectly depends on $t$. Directly it depends only on your state $x$ and your control input $u$ (&quot;Markov Property&quot;). So why should I inject a time-dependency into my controller, if the system itself does not care about the time?
Now, there is one important part of your problem missing: Optimal control always looks to minimise a cost function. What is your cost function? That is a modelling assumption, you have to make! You write, you want $x$ to stay close to the origin, so maybe we put something like $J(x, u, t) = \int_{t_0}^{t_1}||x||^2$.
Now, if you drop the time dependency for your controller $u$, and assume $f$ is linear, you'd arrive at a Linear-quadratic regulator. I think that would be a reasonable controller for the problem you describe. Or at least a good point to start, to find more sophisticated control algorithms.