Optimal fractional-order adaptive fuzzy control on inverted pendulum model

Abstract

This paper shows the ability of Fractional-Order (FO) dynamics in adaptation laws of indirect adaptive fuzzy logic control (AFLC). The parameters of the FO-adaptive laws are optimized using Particle Swarm Optimization algorithm without the presence of the disturbance. The optimal FO-AFLC (OFO-AFLC) is introduced in this paper to overcome the drawbacks of AFLC. The complexity in AFLC is due to using the projection algorithm to keep the adapted parameters bounded. Also the AFLC uses a complex control to guarantee the error convergence. It has proven that the OFO-AFLC can guarantee the error convergence without using neither the projection algorithm nor complex control. The applicability and efficiency of the proposed method is compared to the ordinary AFLC, and demonstrated through simulations made on inverted pendulum model. Furthermore, simulation results show that the control signal in OFO-AFLC is smoother with less oscillations.

Appendix

FO Lyapunov notes

Some basic definitions related to Lyapunov stability will be introduced according to [56,57,58]. Consider a FO non-linear system is considered in autonomous form as follows:

$$\begin{aligned} {{_{0}^{c}D_{t}^{\alpha }\underline{{x}}(t)} ={\underline{f}}\left( {\varvec{{\underline{x}}(t)}}\right) ,\;{{\underline{x}}(t)}, \;{{\underline{f}}(\cdot )}\in {\mathfrak {R}^n}} \end{aligned}$$

(35)

where $\alpha \in (0,\;1)$.

Definition 1

(Equilibrium point) The equilibrium state of the system (35) is ${\underline{x}}^{*}$, if ${\underline{x}}(t)$ just reaches ${\underline{x}}^{*}$, it would stay there at ${\underline{x}}^{*}$. Whereas the equilibrium point ${\underline{x}}^{*}$ mathematically satisfies ${\underline{f}}({\varvec{{\underline{x}}^{*}}})=0$. Usually ${{\underline{x}}^{*}}$ is taken at the origin without loss of the generality.

Theorem 1

(FO Lyapunov Theorem [58]) Consider the FO non-linear system in (35) with ${{\underline{x}}(0)}={\underline{x}}_0$, and an equilibrium point ${\underline{x}}^{*}=0$ which yields the following case [59]:

• “uniformly asymptotically stable” , if there is a state ${\underline{x}}\in {M}$, and continuous scalar function $V\left( {\underline{x}},t\right)$ satisfies $V\left( {\underline{x}},t\right) >0$, decrescent, and $_{0}^{c}D_{t}^{{\alpha }}{V}{\left( {\underline{x}},t\right) }<{0}$, $\forall {{\underline{x}}\ne {0}}$.

• “asymptotically stable” , the equilibrium state of the system (35) at ${\underline{x}}^{*}=0$ is usually called “asymptotically stable” if and only if, it is Lyapunov stable,and there is a positive function $\delta$, and

  $$\begin{aligned} {||{{\underline{x}}(0)||}<\delta ,} \end{aligned}$$

  (36)

  then $\begin{aligned} {\lim _{t\rightarrow \infty }{{\underline{x}}(t)}=0.} \end{aligned}$