Adaptive neural fault-tolerant control for uncertain MIMO nonlinear systems with actuator faults and coupled interconnections

Abstract

Handling both intermittent actuator faults and coupled interconnections in uncertain multiple-input–multiple-output nonlinear system is still a challenge in the control community. In this paper, to address this issue, an adaptive neural fault-tolerant control scheme is developed. Firstly, neural networks with random hidden nodes are used to approximate unknown functions, and an inequality is introduced to construct controllers such that the singularity problem of the controllers can be circumvented. Secondly, a projection algorithm is adopted to update online the estimated parameters in the controllers such that the boundedness of estimated parameters is ensured. In particular, the boundedness of estimate of unknown fault parameters with intermittent jumps can be definitely guaranteed. Due to the effects of intermittency jumps of unknown parameters on the system stability during operation, a modified Lyapunov function is developed to prove the system stability. It is proved that the system stability only depends on the jumping amplitude of Lyapunov function and the minimum fault time interval and is not affected by the total number of faults. Thirdly, a root mean square type of bound for the tracking error is established by using iterative calculation to illustrate that the system transient performance in the sense of the tracking error is adjustable by proper choice of design parameters. Finally, simulation studies are carried out to verify the effectiveness of the theoretical results.

Appendix

1.1 Proof of Lemma 2

Proof

Integrating both sides of (31) over $[t_{i}^{q_i},t_{i}^{q_i+1})$ and using $V_{i}(t_{i}^{(q_i+1)-})=V_{i}(t_{i}^{(q_i+1)+})-\Delta {V}_{i}^{q_i}$ obtained from (32), we have

$$\begin{aligned} V_{i}\left( t_{i}^{(q_i+1)+}\right)\le & V_{i}\left( t_{i}^{q_i+}\right) {\mathrm{e}}^{-\chi _{i}\left( {t}_{i}^{q_i+1}-{t}_{i}^{q_i}\right) }+\Delta {V}_{i}^{q_i}\nonumber \\&+\,{\mathrm{e}}^{-\chi _{i}\left( {t}_{i}^{q_i+1}-{t}_{i}^{q_i}\right) }\int _{t_{i}^{q_i}}^{t_{i}^{q_i+1}}\delta _{i}{\mathrm{e}}^{\chi _{i}\left( t-{t}_{i}^{q_i}\right) }\mathrm {d}t, \end{aligned}$$

(49)

We suppose that $N_i(0,T)$ denotes the total number of actuator faults for the $i\hbox {th}$ subsystem during [0, T). Then, according to (49), in the intervals $\left[ 0,t_{i}^{1+}\right] ,\ldots ,\left[ t_{i}^{\left( N_i(0,T)-1\right) +},t_{i}^{(N_i(0,T))+}\right]$, the following inequalities hold:

$$\begin{aligned} V_{i}\left( t_{i}^{1+}\right)\le & V_{i}\left( 0\right) {\mathrm{e}}^{-\chi _{i}t_{i}^1}+\Delta {V}_{i}^1\nonumber \\&+\,{\mathrm{e}}^{-\chi _{i}t_{i}^1}\int _{0}^{t_{i}^1}\delta _{i}{\mathrm{e}}^{\chi _{i}t}\mathrm {d}t, \end{aligned}$$

(50)

$$\begin{aligned}&\vdots&\nonumber \\ V_{i}\left( t_{i}^{N_i(0,T)+}\right)\le & V_{i}\left( t_{i}^{(N_i(0,T)-1)+}\right) {\mathrm{e}}^{-\chi _{i}\left( t_{i}^{N_i(0,T)}-t_{i}^{N_i(0,T)-1}\right) }\nonumber \\&+\,\Delta {V}_{i}^{N_i(0,T)}\nonumber \\&+\,{\mathrm{e}}^{-\chi _{i}\left( t_{i}^{N_i(0,T)}-t_{i}^{N_i(0,T)-1}\right) }\nonumber \\&\int _{t_{i}^{N_i(0,T)-1}}^{t_{i}^{N_i(0,T)}}\delta _{i}{\mathrm{e}}^{\chi _{i}\left( t-t_{i}^{N_i(0,T)-1}\right) }\mathrm {d}t. \end{aligned}$$

(51)

Considering (50) and (51), in the time interval [0, T), we have

$$\begin{aligned} V_{i}(T)\le & V_{i}\left( t_{i}^{N_i(0,T)+}\right) {\mathrm{e}}^{-\chi _{i}\left( T-t_{i}^{N_i(0,T)}\right) }\nonumber \\&+\,{\mathrm{e}}^{-\chi _{i}\left( T-t_{i}^{N_i(0,T)}\right) }\int _{t_{i}^{N_i(0,T)}}^{T}\delta _{i}{\mathrm{e}}^{\chi _{i}\left( t-t_{i}^{N_i(0,T)}\right) }\mathrm {d}t\nonumber \\\le & V_{i}(0){\mathrm{e}}^{-\chi _{i}T}+\sum _{q=1}^{N_i(0,T)}\Delta {V}_{i}^{q_i}{\mathrm{e}}^{\chi _{i}\left( t_{i}^{q_i}-T\right) }\nonumber \\&+\,{\mathrm{e}}^{-\chi _{i}T}\int _{0}^{T}\delta _{i}{\mathrm{e}}^{\chi _{i}t}\mathrm {d}t. \end{aligned}$$

(52)

According to [15], we can easily obtain $N_i(t_{i}^{q_i},T)\le (T-t_{i}^{q_i})/T_i^*$ where $T_i^*$ is defined in (34), which implies that $t_{i}^{q_i}-T\le -N_i(t_{i}^{q_i},T)T_i^*$. Thus, the term $\sum _{q=1}^{N_i(0,T)}{\mathrm{e}}^{\chi _{i}(t_{i}^{q_i}-T)}\le (1-{\mathrm{e}}^{-\chi _{i}T_i^*N_i(0,T)})/(1-{\mathrm{e}}^{-\chi _{i}T_i^*})$. Using the fact $N_i(0,T)\le {T}/T_i^*$ and $|\Delta {V}_{i}^{q_i}|\le \zeta _i$ from (33), (52) satisfies

$$\begin{aligned} V_{i}(T)\le & V_{i}(0){\mathrm{e}}^{-\chi _{i}T}\nonumber \\&+\,\left[ \frac{\delta _{i}}{\chi _{i}}+\frac{\zeta _i}{1-{\mathrm{e}}^{-\chi _{i}T_i^*}}\right] \left( 1-{\mathrm{e}}^{-\chi _{i}T}\right) . \end{aligned}$$

(53)

Using the fact that $\lim _{T\rightarrow \infty }(1-{\mathrm{e}}^{-\chi _{i}T})=1$, the inequality (34) can be obtained. $\square$

1.2 Proof of Theorem 2

Proof

Revisiting (31), the time derivative of $V_i$ can also satisfy

$$\begin{aligned} {\dot{V}}_{i}(t)\le -c_i\Vert z_i\Vert ^2+\delta _{i}, \end{aligned}$$

(54)

where $\delta _{i}$ is given in (31), $c_i=\min _{j_i=1,\ldots ,n_i}\{c_{i,j_i}\}$, $z_i=[z_{i,1},\ldots ,z_{i,n_i}]^{\mathrm {T}}$.

Integrating both sides of (54) over $[t_{i}^{q_i},t_{i}^{q_i+1})$, we can obtain the following iteration formula by using $V_{i}(t_{i}^{(q_i+1)-})=V_{i}(t_{i}^{(q_i+1)+})-\Delta {V}_{i}^{q_i}$ obtained from (32)

$$\begin{aligned} \int _{t_{i}^{q_i}}^{t_{i}^{q_i+1}}\Vert z_i(t)\Vert ^2\mathrm {d}t\le & \frac{1}{c_i}\bigg [V_{i}\left( t_{i}^{q_i+}\right) +\Delta {V}_{i}^{q_i}-V_{i}\left( t_{i}^{(q_i+1)+}\right) \nonumber \\&+\,\int _{t_{i}^{q_i}}^{t_{i}^{q_i+1}}\delta _{i}\mathrm {d}t\bigg ]. \end{aligned}$$

(55)

For an arbitrary time instant $T>0$, $N_i(0,T)$ denotes the total number of actuator faults for $i\hbox {th}$ subsystem during [0, T). So, using iteration formula (55), in the intervals $\left[ 0,t_{i}^{1+}\right] ,\ldots ,\left[ t_{i}^{(N_i(0,T)-1)+},t_{i}^{N_i(0,T)+}\right]$, we have

$$\begin{aligned} \int _{0}^{t_{i}^{1}}\Vert z_i(t)\Vert ^2\mathrm {d}t\le & \frac{1}{c_i}\Bigg [V_{i,n_i}(0)+\Delta {V}_{i}^{1}-V_{i}\left( t_{i}^{1+}\right) \nonumber \\&+\,\int _{0}^{t_{i}^1}\delta _{i}\mathrm {d}t\Bigg ], \end{aligned}$$

(56)

$$\begin{aligned}&\vdots&\nonumber \\ \int _{t_{i}^{N_i(0,T)}}^{T}\Vert z_i(t)\Vert ^2\mathrm {d}t\le & \frac{1}{c_i}\Bigg [V_{i}\left( t_{i}^{(N_i(0,T)-1)+}\right) +\Delta {V}_{i}^{N_i(0,T)}\nonumber \\&-\,V_{i}\left( t_{i}^{N_i(0,T)+}\right) +\int _{t_{i}^{N_i(0,T)-1}}^{t_{i}^{N_i(0,T)}}\delta _{i}\mathrm {d}t\Bigg ]. \end{aligned}$$

(57)

From (56), (57) and $|\Delta {V}_{i}^{q_i}|\le \zeta _i$, $q_i=1,\ldots ,N_i(0,T)$, obtained from (33), in the time interval [0, T), we have

$$\begin{aligned}&\frac{1}{T}\int _{0}^{T}\Vert z_i(t)\Vert ^2\mathrm {d}t\nonumber \\&\quad \le \frac{1}{c_i}\bigg [\frac{|V_{i}(0)-V_{i}(T)|}{T}+\frac{N_{i}(0,T)\zeta _i}{T}+\delta _{i}\bigg ] \end{aligned}$$

(58)

On the other hand, from (53), we have

$$\begin{aligned} \frac{\big |V_{i}(0)-V_{i}(T)\big |}{T}\le & \frac{1-{\mathrm{e}}^{-\chi _{i}T}}{T}\Big (V_{i}(0)\nonumber \\&+\,\frac{\delta _{i}}{\chi _{i}}\Big )+\frac{\zeta _i}{T\left( 1-{\mathrm{e}}^{-\chi _{i}T_i^*}\right) }\nonumber \\\le & \chi _{i,n_i}V_{i,n_i}(0)+\delta _{i,n_i} \end{aligned}$$

(59)

where we have used the fact that $(1-{\mathrm{e}}^{-\chi _{i}T})/T\le \chi _{i}$ and $\lim _{T\rightarrow \infty }{\zeta _i}/({T(1-{\mathrm{e}}^{-\chi _{i}T_i^*})})=0$. By substituting (59) into (58) and using the fact $\chi _{i}/c_i\le 2$, we have

$$\begin{aligned} \frac{1}{T}\int _{0}^{T}\Vert z_i(t)\Vert ^2\mathrm {d}t\le 2V_{i}(0)+\frac{2}{c_i}\delta _{i}+\frac{N_{i}(0,T)\zeta _i}{T} \end{aligned}$$

(60)

For the whole system, considering (60), we obtain

$$\begin{aligned} \int _{0}^{T}\Vert z_i(t)\Vert ^2\mathrm {d}t\le & 2V(0)+\frac{2\sum _{i=1}^N\delta _{i}}{c_0}\nonumber \\&+\,\frac{1}{c_0}\sum _{i=1}^N\frac{N_{i}(0,T)\zeta _i}{T} \end{aligned}$$

(61)

where $c_0=\min _{i=1,\ldots ,N}\{c_i\}$. $V(0)=\sum _{i=1}^NV_{i}(0)$ and

$$\begin{aligned} V_{i}(0)& = \frac{1}{2}\sum _{j_i=1}^{n_i}\Big [z_{i,j_i}^2(0)+{\gamma _{\theta {i,j_i}}^{-1}}\Vert {\tilde{\theta }}_{i,j_i}(0)\Vert ^2\nonumber \\&+\,{\gamma _{\mathrm {d}{i,j_i}}^{-1}}{\tilde{d}}_{i,j_i}^2(0)+{\gamma _{\mathrm {b}{i,j_i}}^{-1}}{\underline{g}}_{i,j_i}{\tilde{b}}_{i,j_i}^2(0)\Big ]\nonumber \\&+\,{\gamma _{\beta {i}}^{-1}}{\tilde{\beta }}_{i}^2(0) \end{aligned}$$

(62)

By using trajectory initialization technique [1, 3], we can set $z_{i,j_i}(0)$ to zero, i.e., let $z_{i,1}(0)=y_{\mathrm {d}i}(0)$, $z_{i,j_i}(0)=\alpha _{i,j_i-1}(0)$. Then using the fact $N_i(0,T)T_i^*\le {T}$ and substituting (62) and $\delta _{i}$ into (61), the bound (36) is established. $\square$