# Achieving asymptotic tracking of a nonlinear system with bounded input

I have the following nonlinear, continuous-time ODE

$$\dot{x}=K-Lq-q^2u,$$ where the constant values $K$ and $L$ are strictly positive real numbers, the state $q$ and the input variable $u$ evolve in $\mathbb{R}$. Consider a periodic reference $X_{ref}$ signal given whose Fourier decomposition is given by \begin{equation*} X_{ref}(t)=\sum_{i=1}^Na_i\cos(i\omega t)+b_i\sin(i\omega t),\quad\forall t\in\mathbb{R} \end{equation*} where the constant values $a_i,b_i$ are real numbers, the constant value $N$ is a strictly positive integer.

To achieve asymptotic tracking, I defined the error variable $e=x-X_{ref}$. Taking the derivative of the error variable, it yields \begin{equation*} \begin{array}{rcl} \dot{e}&=&\dot{x}-\dot{X}_{ref}(t)\\ &=&K-Lq-\dot{X}_{ref}(t)-q^2u \end{array} \end{equation*} which is a time-varying nonlinear system affine in the input variable u. To design a controller for such system, usual approaches are based on Control-Lyapunov functions (CLF) and Sontag's formula [1].

Assuming that the constant values $K$ and $L$ satisfy

1. $K>\sup\{|\dot{r}(t)|t\in\mathbb{R}\}$;
2. $K-\sup\{|\dot{r}(t)|t\in\mathbb{R}\}<L\inf\{|r(t)|:t\in\mathbb{R}\}+\inf\{|\dot{r}(t)|:t\in\mathbb{R}\}$ the function $\mathbb{R} \ni e\mapsto V(e)=e^2/2\in\mathbb{R}_+$ is a CLF for the error system.

Now my question follows: is it possible to use the CLF for the error system to design a feedback law $\phi:\mathbb{R}\to\mathbb{R}$ that asymptotically stabilizes the origin and such that given two constant values $0<u_{\min}<u_{\max}$, \begin{equation*} \forall (t,e)\in\mathbb{R}\times\mathbb{R},\quad u_{\min}<\phi(t,e)<u_{\max}. \end{equation*}

I am aware that there exists a result for time-invariant systems [2,3]. However, I am not sure if such result could be generalized for time-variant. Does someone know how proceed in this case?

References

[1] Z. Jiang, Y. Lin, Y. Wang, "Stabilization of nonlinear time-varying systems: a control-Lyapunov function approach", Jrl Syst Sci & Complexity (2009) 22: 683–696 [2] Y. Lin, E. D. Sontag, "A universal formula for stabilization with bounded controls", System and Control Letters, 16 (1991) 393-397 [3] Y. Lin, E. D. Sontag, "Control-Lyapunov Universal Formuals for Restricted Inputs", Control-Theory and Advanced Technology, Vol. 10, No. 4, 1995