Adaptive fuzzy sliding mode control of input-delayed uncertain nonlinear systems through output-feedback

Abstract

This paper deals with the design of stabilizing controllers for a class of uncertain nonlinear systems with unknown constant input time delay. A new control method is introduced which merges an adaptive fuzzy sliding mode predictor with an adaptive fuzzy sliding mode controller to circumvent the effects of the unknown time delay, nonlinearities, and unknown plant dynamics. The proposed predictor and controller can be implemented in the real time without recursive and tedious calculations and only by using output-feedback. The uniform ultimate boundedness of the closed-loop system is also guaranteed. Simulation and experiment studies on a nonlinear benchmark problem reveal the effectiveness of the proposed method.

Appendices

Appendix 1: Proof of Lemma 1

Proof

Select the Lyapunov candidate function for the predictor (13) as

$$\begin{aligned}&V_p \Big (e, \tilde{\mathbf {h}}, \tilde{\mathbf {q}}, \tilde{\Phi } \Big ) = \frac{1}{2} {e}^2 + \frac{1}{2 \gamma _1} {\tilde{\mathbf {h}}}^\mathrm{T} {\tilde{\mathbf {h}}}\nonumber \\&\quad +\, \frac{1}{2 \gamma _2} {\tilde{\mathbf {q}}}^\mathrm{T} {\tilde{\mathbf {q}}} + \frac{1}{2 \gamma _3} {\tilde{\Phi }}^2 \end{aligned}$$

(40)

$$\begin{aligned}&\dot{V}_p = e \ C (A - QC) {\tilde{\mathbf {x}}}_p (t+\bar{\tau }|t)\nonumber \\&\quad +\, \tilde{\mathbf {h}}^\mathrm{T} \Big [e \ CB \ {\mathbf {w}_f} (t+\bar{\tau }|t) \nonumber \\&\quad + \frac{1}{\gamma _1} \dot{\tilde{\mathbf {h}}} \Big ] + \tilde{\mathbf {q}}^\mathrm{T} \Big [e \ CB \ {\mathbf {w}_g} (t+\bar{\tau }|t) u(t) \nonumber \\&\quad + \frac{1}{\gamma _2} \dot{\tilde{\mathbf {q}}} \Big ] - e\ CB \ v_\mathrm{rb} + e \ CB \ {\varepsilon }_p + \frac{1}{\gamma _3} {\tilde{\Phi }} \dot{\tilde{\Phi }} \end{aligned}$$

(41)

By using adaptation algorithms (14), (15), and (17) and the robust term Definition (16), (41) is simplified as below.

$$\begin{aligned} \Rightarrow \dot{V}_p = e \ C (A - QC) {\tilde{\mathbf {x}}}_p (t+\bar{\tau }|t) + e \ CB \ {\varepsilon }_p. \end{aligned}$$

By using Youngs inequality $ab \le {\Big (a^2 + {k_1}^2 b^2 \Big )}/{2 k_1}, k_1 > 0$

$$\begin{aligned} k_1= & {} {\lambda }_{\min }[QC-A] \Rightarrow (CB{\varepsilon }_p)e\\\le & {} \frac{1}{2 {\lambda }_{\min }[QC-A]} CB{(CB)}^\mathrm{T} {{\varepsilon }_p}^2\\&+\frac{1}{2} {\lambda }_{\min }[QC-A] e^2\\\Rightarrow & {} {\dot{V}_p} \le - \frac{1}{2} {\lambda }_{\min }[QC - A] e^2\\&+ \frac{1}{2 {\lambda }_{\min }[QC-A]} CB{(CB)}^\mathrm{T} {{\varepsilon }_p}^2. \end{aligned}$$

Then, it can be deduced that outside of a bounded region, the Lyapunov derivative $\dot{V}_p$ is negative semi-definite as below.

$$\begin{aligned}&\frac{1}{({\lambda }_{\min }[QC-A])^2} CB{(CB)}^\mathrm{T} (C^\mathrm{T}C)^{-1}{{\varepsilon }_p}^2\\&\quad = |x_B| \le |{\tilde{\mathbf {x}}}_p| \Rightarrow {\dot{V}_p} \le 0 \end{aligned}$$

Thus, based on [15] because $V_p$ is positive definite, $\dot{V}_p$ is semi-negative definite outside a bounded region, and a class $\mathcal {K}$ function $\sigma (x)$ exists such that $\dot{V}_p \le -\sigma ({\tilde{\mathbf {x}}}_p)$, then ${\tilde{\mathbf {x}}}_p(t)$ and the fuzzy output vector errors $\tilde{\mathbf {h}}$, $\tilde{\mathbf {q}}$, and $\tilde{\Phi }$ are uniformly ultimately bounded.

Appendix 2: Proof of Lemma 2

Select the Lyapunov candidate function for the controller as

$$\begin{aligned}&{V_c}\left( {s\left( t \right) ,s\left( {t - \bar{\tau }} \right) ,\tilde{\mathbf {k}}{\text {,}}\tilde{\Psi }} \right) = \frac{1}{2}{s^2}(t)\nonumber \\&\quad + \int _{t - \bar{\tau }}^t {\left| g(\mathbf {x}) \right| {s^2}(r) dr} + \frac{1}{{2{\alpha _1}}}{{\tilde{\mathbf {k}}}^\mathrm{T}}{\tilde{\mathbf {k}}} + \frac{1}{{2{\alpha _2}}}{\tilde{\Psi } ^2}. \end{aligned}$$

(42)

Then, by time derivation of (42) with noting that $\bar{\tau }$ is constant, $\dot{V}_c$ is derived as

$$\begin{aligned}&\Rightarrow {\dot{V}_c} = s (t)\dot{s} (t)\\&\quad + \left[ {\left| g( \mathbf {x} (t)) \right| {s^2}(t) - \left| g(\mathbf {x} (t - \bar{\tau })) \right| {s^2}({t - \bar{\tau }})} \right] \\&\quad +\frac{1}{{{\alpha _1}}}{{\tilde{\mathbf {k}}}^\mathrm{T}}{\dot{\tilde{\mathbf {k}} }} + \frac{1}{{{\alpha _2}}}\tilde{\Psi } \dot{\tilde{\Psi }}. \end{aligned}$$

After implementing (6), (7), and (32), the derivative is rewritten as

$$\begin{aligned}&\Rightarrow {\dot{V}_c} = s (t) g(\mathbf {x} (t)) \left[ {u_\mathrm{eq}}^* - {\hat{u}}_\mathrm{fuz} - u_\mathrm{rb} \right] \\&\quad + \left[ {\left| g( \mathbf {x} (t)) \right| {s^2}(t) - \left| g(\mathbf {x} (t - \bar{\tau })) \right| {s^2}({t - \bar{\tau }})} \right] \\&\quad +\frac{1}{{{\alpha _1}}}{\tilde{ \mathbf {k}}^\mathrm{T}}{\dot{\tilde{\mathbf {k}}}} + \frac{1}{{{\alpha _2}}}\tilde{\Psi } \dot{\tilde{\Psi }} \end{aligned}$$

from the adaptive law (26) and the robust term (29), it is concluded that

$$\begin{aligned}&{\dot{V}_{c}} = {\tilde{\mathbf {k}}}^\mathrm{T} \left[ s(t) {{\mathbf {q}}}^\mathrm{T}{{\mathbf {w}}_g} {{\mathbf {w}}} + \frac{1}{{{\alpha _1}}}{\dot{\tilde{\mathbf {k}}}} \right] + \left| {{\mathbf {q}}}^\mathrm{T}{{\mathbf {w}}_g ({\mathbf {x} (t)}}) \right| {s^2}(t )\\&\quad + \left| {{\mathbf {q}}}^\mathrm{T}{{\mathbf {w}}_g ({\mathbf {x} ({t - \bar{\tau }})})} \right| {s^2}({t - \bar{\tau }})\\&\quad -\, s(t) \left| {{\mathbf {q}}}^\mathrm{T}{{\mathbf {w}}_g ({\mathbf {x} (t)})} \right| \Big ( \tilde{\Psi } \text {sgn}(s(t)) + s(t) \Big ) + \frac{1}{{{\alpha _2}}}\tilde{\Psi } \dot{\tilde{\Psi }}, \end{aligned}$$

by using the adaptive law (30) and the fuzzy approximation (10), the derivative is simplified as follows:

$$\begin{aligned} \Rightarrow {\dot{V}_c} = - \left| {{\mathbf {q}}}^\mathrm{T}{{\mathbf {w}}_g ({\mathbf {x} ({t - \bar{\tau }})})} \right| {s^2}({t - \bar{\tau }}). \end{aligned}$$

$\dot{V}_c$ is semi-negative definite, then the system is stable and s, $\tilde{\mathbf {k}}$, and $\tilde{\Psi }$ will remain bounded. Finally, because the overall Lyapunov function of system ($V = V_c + V_p$) has negative semi-definite derivative for $|x_B| \le |{\tilde{\mathbf {x}}}_p|$, and with respect to the previous stability results for the predictor and controller, the system is uniformly ultimately bounded.