Output feedback stabilization for a class of nonlinear systems with hysteresis input and sensor uncertainties

Abstract

In this paper, an output feedback scheme is proposed for a class of nonlinear systems with hysteresis input and sensor uncertainties. A hysteresis inverse function is applied to counteract the effect of the hysteresis input. The system nonlinearities and sensor uncertainties are dealt with by a dynamic gain, based on which a reduced-order observer and an output feedback controller are developed. It shows that the proposed scheme can globally stabilize the system. Finally, the effectiveness of the proposed scheme is verified by simulation.

Appendix

Proof of (33):

• 1. Let

  $$\begin{aligned} V_1 = \frac{1}{2}{z_1^2} + \frac{1}{2l}{z_1^2}. \end{aligned}$$

  (A.1)

  Thus,

  $$\begin{aligned} \dot{V}_1&\le -{\rho _s}{\beta }{\vert z_1 \vert ^2} + 2 {\dot{\rho }_s}{\rho _s^{-1}}{\vert z_1 \vert ^2} + 2{\rho _s}{z_1}{z_2} + 2{l}{a_1}{\rho _s}{z_1^2} \nonumber \\&\quad + 2{\rho _s}{\varphi _1}{z_1} + 2\left( {\rho _s}{\varepsilon _1}-{\dot{\rho }_s}{\rho _s^{-1}}{f_s}+{\dot{f}_s}\right) {z_1} \nonumber \\&\quad \,- \frac{\dot{l}}{2l^2}{|z_1|^2} \nonumber \\&\le -{\frac{1}{2}}{\beta }{\vert z_1 \vert ^2} + \left( \frac{{\alpha _1}}{2}+\frac{45}{2\sigma _1}+{\frac{1}{d_1}}+3c+4M\right) {\vert z_1 \vert ^2} \nonumber \\&\quad \, + {\frac{1}{5}}{\sigma _1}{l}{\Vert \bar{e} \Vert ^2} + {\frac{1}{5}}{\sigma _1}{l}{\Vert \bar{\eta } \Vert ^2} - \frac{\alpha _2(y)}{2l}{|z_1|^2} + {d_2} \nonumber \\&\quad + {c}{|f_s|^2}, \end{aligned}$$

  (A.2)

  where

  $$\begin{aligned}&\!\!\!2{\rho _s}{z_1}{z_2} + 2{l}{a_1}{\rho _s}{z_1^2} \le 2{l^{\frac{1}{2}}}{\rho _s}{z_1}({\bar{e}_1}+\bar{\eta }_1) \le \frac{45}{2\sigma _1}{\vert z_1 \vert ^2} \nonumber \\&\quad + {\frac{1}{5}}{l}{\sigma _1}{\Vert \bar{e} \Vert ^2} + {\frac{1}{5}}{l}{\sigma _1}{\Vert \bar{\eta } \Vert ^2}, \nonumber \\&2{\rho _s}{\varphi _1}{z_1} \le 2{c}{|z_1|^2} + 2{c}{|f_s|}{|z_1|} \le 3{c}{|z_1|^2} + {c}{|f_s|^2}, \nonumber \\&2\left( {\rho _s}{\varepsilon _1}-{\dot{\rho }_s}{\rho _s^{-1}}{f_s}+{\dot{f}_s}\right) {z_1} \le {\frac{1}{d_1}}{\vert z_1 \vert ^2} + {d_2}, \nonumber \\&-\frac{\dot{l}}{2l}{|z_1|^2} \le \frac{{\alpha _1}}{2}{|z_1|^2} - \frac{\alpha _2(y)}{2l}{|z_1|^2}. \end{aligned}$$

  (A.3)

• 2. Let

  $$\begin{aligned} V_2 = {\sigma _3}{\bar{e}^\textrm{T}}{P_1}{\bar{e}}. \end{aligned}$$

  (A.4)

  So,

  $$\begin{aligned} \dot{V}_2&\le -{\sigma _3}{\sigma _1}{l}{\Vert \bar{e} \Vert ^2} + 2{\sigma _3}{l^{\frac{1}{2}}}{\bar{\rho }}{\bar{e}^\textrm{T}}{P_1}{H}{z_2}+ 2{\sigma _3}{l^{\frac{3}{2}}}{a_1}{\bar{\rho }}{\bar{e}^\textrm{T}} \nonumber \\&\quad \quad \,\cdot {P_1}{H}{z_1}- 2{\sigma _3}{l^{\frac{1}{2}}}{\bar{\rho }}{\beta }{\bar{e}^\textrm{T}}{P_1}{H}{z_1} + 2{\sigma _3}{\bar{e}^\textrm{T}}{P_1}{L_3^{-1}}{\phi } \nonumber \\&\quad \quad \, - {\sigma _3}{\frac{\dot{l}}{l}}{\bar{e}^\textrm{T}}\left( {P_1}{D}+{D}{P_1}+ {P_1}\right) {\bar{e}} \nonumber \\&\le -\left( {\sigma _3}{\sigma _1}-{\sigma _3}{\sigma _2}{\alpha _1}{\Vert P_1 \Vert }-\frac{13}{10}{\sigma _1}\right) {l}{\Vert \bar{e} \Vert ^2} \nonumber \\&\quad \quad + {\frac{1}{5}}{\sigma _1}{l}{\Vert \bar{\eta } \Vert ^2} + \left( {\frac{5}{\sigma _1}}{\sigma _3^2}{\Vert P_1H \Vert ^2}+{\varPsi _3}\right) {|z_1|^2} \nonumber \\&\quad \quad + {\varPsi _1(y)}{\Vert \bar{e} \Vert ^2} + {\varPsi _2(y)}{\Vert \bar{\eta } \Vert ^2} - \frac{1}{2}{\sigma _3}{\lambda _1}{\alpha _2(y)}{\Vert \bar{e} \Vert ^2} \nonumber \\&\quad \quad \, + {d_3}, \end{aligned}$$

  (A.5)

  where

  $$\begin{aligned}&2{\sigma _3}{l^{\frac{1}{2}}}{\bar{\rho }}{\bar{e}^\textrm{T}}{P_1}{H}{z_2} + 2{\sigma _3}{l^{\frac{3}{2}}}{a_1}{\bar{\rho }}{\bar{e}^\textrm{T}}{P_1}{H}{z_1} \nonumber \\&\quad \, \le 2{\sigma _3}{l}{\bar{\rho }}{\bar{e}^\textrm{T}}{P_1}{H}{(\bar{e}_1+\bar{\eta }_1)} \le 2{\sigma _3}{l}{|\bar{\rho }|}{\Vert P_1H \Vert }{\Vert \bar{e} \Vert ^2} \nonumber \\&\quad \quad + 2{\sigma _3}{l}{|\bar{\rho }|}{\Vert \bar{e} \Vert }{\Vert P_1H \Vert }{\Vert \bar{\eta } \Vert } \le {\frac{3}{5}}{\sigma _1}{l}{\Vert \bar{e} \Vert ^2} \nonumber \\&\quad \quad + {\frac{1}{5}}{\sigma _1}{l}{\Vert \bar{\eta } \Vert ^2}, -2{\sigma _3}{l^{\frac{1}{2}}}{\bar{\rho }}{\beta }{\bar{e}^\textrm{T}}{P_1}{H}{z_1} \nonumber \\&\qquad \le {\frac{1}{5}}{\sigma _1}{l}{\Vert \bar{e} \Vert ^2}+ {\frac{5}{\sigma _1}}{\sigma _3^2}{\Vert P_1H \Vert ^2}{|z_1|^2}, \nonumber \\&|\phi _2| \le {c}{\rho _s^{-1}}{|z_1|} + {c}{\rho _s^{-1}}{|f_s|} + {\gamma _2(y)}{l^{\frac{1}{2}}}{|\bar{e}_1+\bar{\eta }_1|} \nonumber \\&\quad \quad \, + {\gamma _2(y)}{\beta }{|z_1|} + {l}{a_1}{c}{|z_1|} + {l}{a_1}{c}{|f_s|} + {|\varepsilon _2|} \nonumber \\&\quad \quad \, + {l}{a_1}{\rho _s^{-1}}{|\dot{\rho }_s|}{|z_1|} + {l}{a_1}{\vert {{\rho _s}{\varepsilon _1}}-{{\dot{\rho }_s}{\rho _s^{-1}}{f_s}}+{\dot{f}_s}\vert } \nonumber \\&\quad \quad \, + {\rho _s}{\beta ^2}{|z_1|} + {\beta }{|\dot{\rho }_s|}{\rho _s^{-1}}{|z_1|} + {l^{\frac{1}{2}}}{\rho _s}{\beta }{|\bar{e}_1+\bar{\eta }_1|} \nonumber \\&\quad \quad + {c}{\rho _s}{\beta }{|z_1|} + {\beta }{\vert {{\rho _s}{\varepsilon _1}}-{{\dot{\rho }_s}{\rho _s^{-1}}{f_s}}+{\dot{f}_s}\vert }, \nonumber \\&|\phi _{i+1}| \le {c}{\rho _s^{-1}}{|z_1|} + {c}{\rho _s^{-1}}{|f_s|} + {\gamma _i(y)}\sum _{j=1}^{i}{l^{j-\frac{1}{2}}}{|\bar{e}_j+\bar{\eta }_j|} \nonumber \\&\quad + {\gamma _i(y)}{\beta }{|z_1|} + {l^i}{a_i}{c}{|z_1|} +{l^i}{a_i}{c}{|f_s|} + {|\varepsilon _{i+1}|} \nonumber \\&\quad \, + {l^i}{a_i}{\rho _s^{-1}}{|\dot{\rho }_s|}{|z_1|} + {l^i}{a_i}{\vert {{\rho _s}{\varepsilon _1}}-{{\dot{\rho }_s}{\rho _s^{-1}}{f_s}}+{\dot{f}_s}\vert }, \nonumber \\&2{\sigma _3}{\bar{e}^\textrm{T}}{P_1}{L_3^{-1}}{\phi } \le 2{\sigma _3}{\Vert P_1 \Vert }{\Vert \bar{e} \Vert }{\sum _{i=1}^{n-1}}{l^{-i+\frac{1}{2}}}{\phi _{i+1}} \nonumber \\&\quad \le {\varPsi _1(y)}{\Vert \bar{e} \Vert ^2} + {\varPsi _2(y)}{\Vert \bar{\eta } \Vert ^2} + {\varPsi _3}{|z_1|^2} + \frac{1}{2}{\sigma _1}{l}{\Vert \bar{e} \Vert ^2} \nonumber \\&\qquad + {d_3}, \nonumber \\&\quad -{\sigma _3}{\frac{\dot{l}}{l}}{\bar{e}^\textrm{T}}\left( {P_1}{D}+{D}{P_1}+ {P_1}\right) {\bar{e}} \nonumber \\&\quad \le {\sigma _3}{\sigma _2}{\alpha _1}{l}{\bar{e}^\textrm{T}}{P_1}{\bar{e}} - \frac{1}{2}{\sigma _3}{\alpha _2(y)}{\bar{e}^\textrm{T}}{P_1}{\bar{e}} \nonumber \\&\quad \le {\sigma _3}{\sigma _2}{\alpha _1}{l}{\Vert P_1 \Vert }{\Vert \bar{e} \Vert ^2} - \frac{1}{2}{\sigma _3}{\lambda _1}{\alpha _2(y)}{\Vert \bar{e} \Vert ^2}, \end{aligned}$$

  (A.6)

  and $\lambda _1$ is the minimum eigenvalue of $P_1$.

• 3. Let

  $$\begin{aligned} V_3 = {\bar{\eta }^\textrm{T}}{P_2}{\bar{\eta }}. \end{aligned}$$

  (A.7)

Therefore,

$$\begin{aligned}&\dot{V}_3 \le -2{\sigma _1}{l}{\Vert \bar{\eta } \Vert ^2} + 2{l}{\bar{\eta }^\textrm{T}}{P_2}{H}{\bar{e}_1} - 2{l^{\frac{1}{2}}}{\bar{\rho }}{\bar{\eta }^\textrm{T}}{P_2}{H}{z_2} \nonumber \\&\quad \,- 2{l^{\frac{3}{2}}}{a_1}{\bar{\rho }}{\bar{\eta }^\textrm{T}}{P_2}{H}{z_1} + 2{l^{\frac{1}{2}}}{\bar{\rho }}{\beta }{\bar{\eta }^\textrm{T}}{P_2}{H}{z_1} \nonumber \\&\quad + 2{l^{\frac{1}{2}}}{\dot{\rho _s}}{\rho _s^{-1}}{\bar{\eta }^\textrm{T}}{P_2}{H}{z_1} + 2{l^{\frac{1}{2}}}{\rho _s}{\bar{\eta }^\textrm{T}}{P_2}{H}{\varphi _1} \nonumber \\&\quad \, + 2{l^{\frac{1}{2}}}{\bar{\eta }^\textrm{T}}{P_2}{H}{\left( {\rho _s}{\varepsilon _1} -{\dot{\rho }_s}{\rho _s^{-1}}{f_s}+{\dot{f}_s}\right) } \nonumber \\&\quad \,- {\frac{\dot{l}}{l}}{\bar{\eta }^\textrm{T}}\left( {P_2}{D}+{D}{P_2}+{P_2}\right) {\bar{\eta }} \nonumber \\&\le -\left( \frac{3}{5}{\sigma _1}-{\sigma _2}{\alpha _1}{\Vert {P_2} \Vert }\right) {l}{\Vert \bar{\eta } \Vert ^2} + \left( {\frac{1}{5}}{\sigma _1} \right. \nonumber \\&\quad \,\left. +{\frac{5}{\sigma _1}}{\Vert {P_2}{H} \Vert ^2}\right) {l}{\Vert {\bar{e}} \Vert ^2} + \left( {\frac{5}{\sigma _1}{\Vert {P_2}{H} \Vert ^2}} \right. \nonumber \\&\quad \, \left. + \frac{20}{\sigma _1}{M^2}{\Vert P_2H \Vert ^2} + {\frac{10}{\sigma _1}}{c^2}{\Vert {P_2}{H} \Vert ^2}\right) {\vert z_1 \vert ^2} \nonumber \\&\quad \, - \frac{1}{2}{\alpha _2(y)}{\lambda _2}{\Vert {\bar{\eta }} \Vert ^2} + {d_4} + {\frac{10}{\sigma _1}}{c^2}{\Vert {P_2}{H} \Vert ^2}{\vert f_s \vert ^2}, \end{aligned}$$

(A.8)

where

$$\begin{aligned}&2{l}{\bar{\eta }^\textrm{T}}{P_2}{H}{\bar{e}_1} \le \frac{1}{5}{\sigma _1}{l}{\Vert {\bar{\eta }} \Vert ^2} + {\frac{5}{\sigma _1}}{l}{\Vert {P_2}{H} \Vert ^2}{\Vert {\bar{e}} \Vert ^2}, \nonumber \\&-2{l^{\frac{1}{2}}}{\bar{\rho }}{\bar{\eta }^\textrm{T}}{P_2}{H}{z_2} - 2{l^{\frac{3}{2}}}{a_1}{\bar{\rho }}{\bar{\eta }^T}{P_2}{H}{z_1} \nonumber \\&\quad \,\le -2{l}{\bar{\rho }}{\bar{\eta }^\textrm{T}}{P_2}{H}{({\bar{e}_1}+\bar{\eta }_1)} \le 2{l}{|\bar{\rho }|}{\Vert P_2H \Vert }{\Vert \bar{\eta } \Vert ^2} \nonumber \\&\quad \, + 2{l}{|\bar{\rho }|}{\Vert \bar{\eta } \Vert }{\Vert P_2H \Vert }{\Vert \bar{e} \Vert } \le \frac{2}{5}{\sigma _1}{l}{\Vert {\bar{\eta }} \Vert ^2} + \frac{1}{5}{\sigma _1}{l}{\Vert {\bar{e}} \Vert ^2}, \nonumber \\&2{l^{\frac{1}{2}}}{\bar{\rho }}{\beta }{\bar{\eta }^\textrm{T}}{P_2}{H}{z_1} \le {\frac{1}{5}}{\sigma _1}{l}{\Vert {\bar{\eta }} \Vert ^2} + \frac{5}{\sigma _1}{\Vert P_2H \Vert ^2}{\vert z_1 \vert ^2}, \nonumber \\&2{l^{\frac{1}{2}}}{\dot{\rho _s}}{\rho _s^{-1}}{\bar{\eta }^\textrm{T}}{P_2}{H}{z_1} \le 4{l^{\frac{1}{2}}}{M}{\Vert \bar{\eta } \Vert }{\Vert P_2H \Vert }{|z_1|} \nonumber \\&\quad \,\le {\frac{1}{5}}{\sigma _1}{l}{\Vert {\bar{\eta }} \Vert ^2} + \frac{20}{\sigma _1}{M^2}{\Vert P_2H \Vert ^2}{\vert z_1 \vert ^2}, \nonumber \\&2{l^{\frac{1}{2}}}{\rho _s}{\bar{\eta }^\textrm{T}}{P_2}{H}{\varphi _1} \le 2{l^{\frac{1}{2}}}{c}{\Vert \bar{\eta } \Vert }{\Vert P_2H \Vert }{|z_1|} \nonumber \\&\quad \, + 2{l^{\frac{1}{2}}}{c}{\Vert \bar{\eta } \Vert }{\Vert P_2H \Vert }{|f_s|} \le \frac{1}{5}{\sigma _1}{l}{\Vert {\bar{\eta }} \Vert ^2} \nonumber \\&\quad \, + {\frac{10}{\sigma _1}}{c^2}{\Vert {P_2}{H} \Vert ^2}{\vert z_1 \vert ^2} + {\frac{10}{\sigma _1}}{c^2}{\Vert {P_2}{H} \Vert ^2}{\vert f_s \vert ^2}, \nonumber \\&2{l^{\frac{1}{2}}}{\bar{\eta }^\textrm{T}}{P_2}{H}{\left( {\rho _s}{\varepsilon _1}-{\dot{\rho }_s} {\rho _s^{-1}}{f_s}+{\dot{f}_s}\right) } \le \frac{1}{5}{\sigma _1}{l}{\Vert {\bar{\eta }} \Vert ^2} + {d_4}, \nonumber \\&-{\frac{\dot{l}}{l}}{\bar{\eta }^\textrm{T}}\left( {P_2}{D}+{D}{P_2}+{P_2}\right) {\bar{\eta }} \nonumber \\&\quad \,\le {\sigma _2}{\alpha _1}{l}{\bar{\eta }^\textrm{T}}{P_2}{\bar{\eta }} - \frac{1}{2}{\alpha _2(y)}{\bar{\eta }^\textrm{T}}{P_2}{\bar{\eta }} \nonumber \\&\quad \,\le {\sigma _2}{\alpha _1}{l}{\Vert {P_2} \Vert }{\Vert {\bar{\eta }} \Vert ^2} - \frac{1}{2}{\alpha _2(y)}{\lambda _2}{\Vert {\bar{\eta }} \Vert ^2}, \end{aligned}$$

(A.9)

and $\lambda _2$ is the minimum eigenvalue of $P_2$.

Combining (A.2), (A.5) and (A.8), (33) can be obtained.