A novel adaptive fuzzy prescribed performance congestion control for network systems with predefined settling time

Abstract

The adaptive predefined settling time performance tracking control for a class of uncertain nonlinear network systems with fast time-varying or even abrupt reference signals is studied. In this paper, an improved performance function with time-varying boundary (following the change of reference signal) is designed to overcome the problem that the traditional prescribed performance control (PPC) can only converge to a constant in steady state. Compared with the existing adaptive finite time prescribed performance congestion control for tracking fixed value reference signals, an adaptive predefined settling time prescribed performance congestion control strategy is proposed by means of some new recursive construction (introducing nonautonomous differential equations (NDE)) and analysis innovation. In addition, the network system considered in this paper is more general. Under the framework of backstepping, NDE is used iteratively to control the approximation error with boundary inequality. It is proved theoretically that all signals of the closed-loop system are predefined settling time bounded. Finally, the effectiveness of the proposed control strategy is verified by simulations.

Proof of Theorem 1

Proof. Let $\ell = {\kappa _1}(\zeta )$, then ${\kappa _2}(\Psi ) < \ell$ and ${\kappa _2}(\left\| {x({t_0})} \right\| ) \le \ell$. Let $\Upsilon = {\kappa _2}(\Psi )$ and define ${\Xi _{t,\Upsilon }} = \{ x \in {{\Omega _\zeta }}\vert V(t,x) \le \Upsilon \}$ and ${\Xi _{t,\ell }} = \{ x \in {\Omega _\zeta }\vert V(t,x) \le \ell \}$, then ${\Omega _\Psi } \subset {\Xi _{t,\Upsilon }} \subset \{ {\kappa _1}\left\| x \right\| \le \Upsilon \} \subset \{ {\kappa _1}\left\| x \right\| \le \ell \} = {\Omega _\zeta } \subset D$ and ${\Xi _{t,\Upsilon }} \subset {\Xi _{t,\ell }} \subset {\Omega _\zeta } \subset D$. The sets ${\Xi _{t,\ell }}$ and ${\Xi _{t,\Upsilon }}$ own the property that a solution starting in either set cannot leave it because $\dot{V}(t,x)<0$ is on the boundary. Since ${\kappa _2}(\left\| {x({t_0})} \right\| ) \le \ell \Rightarrow x({t_0}) \in {\Xi _{{t_0},\ell }}$, it can show that $x(t) \in {\Xi _{t,\ell }}$ for $\forall t \ge {t_0}$. A solution starts in ${\Xi _{t,\ell }}$ and must go into ${\Xi _{t,\Upsilon }}$ in predefined time since in $\{ {\Xi _{t,\ell }} - {\Xi _{t,\Upsilon }}\}$, ${\dot{V}}$ satisfies

$$\begin{aligned} \dot{V} \le - (\gamma - \varsigma )(1 - {e^{ - V}}){({t_f} - t)^\varpi }, \forall t \in [{t_0},\omega {t_f}) \end{aligned}$$

(A1)

Then, inequality (A1) insinuates that

$$\begin{aligned} V(t,x(t)) \le \textrm{ln}\left( {1 + \upsilon } \right) \end{aligned}$$

(A2)

where $\upsilon = {\rho _0}{e^{\frac{{(\gamma - \varsigma ){{({t_f} - t)}^{\varpi + 1}}}}{{\varpi + 1}}}}$ and ${\rho _0} = ({e^{V({x_0})}} - 1){e^{ - \frac{{(\gamma - \varsigma ){{({t_f} - {t_0})}^{\varpi + 1}}}}{{\varpi + 1}}}}$, demonstrating that V(t, x(t)) decreases to $\Upsilon$ within $t \in [{t_0},\omega {t_f})$. For a solution starting inside ${\Xi _{t,\Upsilon }}$, $\left\| {x(t)} \right\| \le k_1^{ - 1}({k_2}(\Psi )),\forall t \ge \omega {t_f}$ is satisfied for all $t \ge {t_0}$ since ${\Xi _{t,\Upsilon }} \subset \{ {\kappa _1}(\left\| x \right\| ) \le {\kappa _2}(\Psi )\}$. For a solution starting inside ${\Xi _{t,\ell }}$, but it is outside ${\Xi _{t,\Upsilon }}$, let $\omega {t_f}$ be the first time, this solution enters ${\Xi _{t,\Upsilon }}$, $\forall t \in [{t_0},\omega {t_f})$, then

$$\begin{aligned} \dot{V} \le - {\nu _3}(x) \le - {\kappa _3}(\left\| x \right\| ) \le - {\kappa _3}(\kappa _2^{ - 1}(V))\mathop = \limits ^{def} - \kappa (V) \end{aligned}$$

(A3)

where $\kappa = {\kappa _3} \circ \kappa _2^{ - 1}$ is a class K function defined on $[0,\zeta ]$. We assume that $\kappa$ is locally Lipschitz without loss of generality. Also, ${\phi }$ meets the autonomous first-order differential equation, then

$$\begin{aligned} \dot{\phi } = - \kappa (\phi ),\phi ({t_0}) = V({t_0},x({t_0})) \ge 0 \end{aligned}$$

(A4)

It can be demonstrated such a class KL function $\hbar$ that

$$\begin{aligned} V(t,x(t)) \le \hbar (V({t_0},x({t_0})),t - {t_0}),\forall t \in [{t_0},\omega {t_f}) \end{aligned}$$

(A5)

Defining ${\kappa _l}(\zeta ,\Im ) = \kappa _1^{ - 1}(\hbar ({\kappa _2}(\zeta ),\Im ))$, we obtain

$$\begin{aligned} \left\| {x(t)} \right\| \le {\kappa _l} (\left\| {x({t_0})} \right\| ,t - {t_0}),\forall t \in [{t_0},\omega {t_f}) \end{aligned}$$

(A6)