Lyapunov–Krasovskii stable T2FNN controller for a class of nonlinear time-delay systems

Abstract

In this paper, a type-2 fuzzy neural network (T2FNN) controller has been designed for a class of nonlinear time-delay systems using the feedback error learning (FEL) approach. In the FEL strategy, the T2FNN controller is in the feedforward path to overcome the nonlinearity and time delay and a classical controller is in the feedback path to guarantee the stability of the controlled system. Using the Lyapunov–Krasovskii stability theorem, the adaptation rules for training of T2FNN controller have been achieved in a way that, in the presence of the unknown disturbance and time-varying delay, the tacking error becomes zero. In the proposed stability criteria and adaptation laws, since just the training error is utilized, i.e., the mathematical model of the system or its parameters is not needed, the overall training and control algorithm is computationally simple. In the present study, the effect of delay has been considered in tuning the T2FNN parameters and, therefore, the performance of the proposed controller has been improved. The proposed strategy has been applied to systems with time-varying input delay and measurement noise and compared with indirect type-1 fuzzy sliding controller. The effectiveness of the proposed controller is shown by some simulation results.

Appendix A

The time derivative of the Lyapunov–Krasovskii functional in Eq. (19) is derived as

$$\begin{aligned} \dot{V}(t)= & {} e_{tr} \dot{e}_{tr} +e_{tr} ^{2}- (1-\dot{\tau })e_{tr} (t-\tau )^{2}+\dot{e}_{tr} ^{2}\nonumber \\&- (1-\dot{\tau })\dot{e}_{tr} (t-\tau )^{2}+\tau _{\max } e_{tr} (t)^{2} \nonumber \\&-\int _{t+\theta }^t {e_{tr} (s)^{2}} \hbox {d}s +\frac{\dot{\eta }}{\gamma }(\eta -\eta ^{{*}}) \end{aligned}$$

(A-1)

and by considering the assumptions in Eqs. (2) and (5), we have:

$$\begin{aligned}&\dot{V}(t)\le e_{tr} \dot{e}_{tr} +e_{tr} ^{2}+\dot{e}_{tr} ^{2} +\tau _{\max } e_{tr} ^{2}+\frac{\dot{\eta }}{\gamma }(\eta -\eta ^{{*}}) \nonumber \\&\quad <e_{tr} (\beta -\dot{u}_{ff} )+(1+\tau _{\max } )e_{tr} ^{2}+\dot{e}_{tr} ^{2} +\frac{\dot{\eta }}{\gamma }(\eta -\eta ^{{*}})\nonumber \\ \end{aligned}$$

(A-2)

The Lyapunov–Krasovskii stability requires:

$$\begin{aligned} \dot{V}(t)< & {} e_{tr} (\beta -\dot{u}_{ff} )+(1+\tau _{\max } )e_{tr} ^{2}+\dot{e}_{tr} ^{2} \nonumber \\&+\,\frac{\dot{\eta }}{\gamma }(\eta -\eta ^{{*}})<0 \end{aligned}$$

(A-3)

Substituting the value from Eq. (18) we have:

$$\begin{aligned} \dot{V}(t)< & {} e_{tr} \beta +e_{tr} \left( -\frac{\partial u_{ff} }{\partial m_1 ^{ji}}\dot{m}_1 ^{ji}-\frac{\partial u_{ff} }{\partial \sigma _1 ^{ji}}\dot{\sigma }_1 ^{ji}\right. \nonumber \\&\left. - \frac{\partial u_{ff} }{\partial \sigma _2 ^{ji}}\dot{\sigma }_2 ^{ji}-\frac{\partial u_{ff} }{\partial c^{j}}\dot{c}^{j}-\frac{\partial u_{ff} }{\partial r}\dot{r}\right) \nonumber \\&+ (1+\tau _{\max } )e_{tr} ^{2}+\dot{e}_{tr} ^{2} +\frac{\dot{\eta }}{\gamma }(\eta -\eta ^{{*}}) \end{aligned}$$

(A-4)

Considering Eq. (17), we can get:

$$\begin{aligned} \dot{V}(t)< & {} e_{tr} \beta +\frac{\partial E}{\partial m_1 ^{ji}}\dot{m}_1 ^{ji}+\frac{\partial E}{\partial \sigma _1 ^{ji}}\dot{\sigma }_1 ^{ji}\nonumber \\&+ \frac{\partial E}{\partial \sigma _2 ^{ji}}\dot{\sigma }_2 ^{ji}+\frac{\partial E}{\partial c^{j}}\dot{c}^{j}-e_{tr} \frac{\partial u_{ff} }{\partial r}\dot{r}\nonumber \\&+(1+\tau _{\max } )e_{tr} ^{2}+\dot{e}_{tr} ^{2} +\frac{\dot{\eta }}{\gamma }(\eta -\eta ^{{*}}) \end{aligned}$$

(A-5)

Combining Eqs. (16) and (A-5) gives (with an absolute value of $e_{tr})$:

$$\begin{aligned} \dot{V}(t)< & {} \left| {e_{tr} } \right| \beta +\frac{\partial E}{\partial m_1 ^{ji}}\left( (\alpha -1)m^{ji}+\eta \frac{\partial E}{\partial m^{ji}}\right) \nonumber \\&+ \frac{\partial E}{\partial \sigma _1 ^{ji}}\quad \left( (\alpha -1)\sigma _1 ^{ji} +\eta \frac{\partial E}{\partial \sigma _1 ^{ji}}\right) \nonumber \\&+\frac{\partial E}{\partial \sigma _2 ^{ji}}\left( (\alpha -1)\sigma _2 ^{ji}+\eta \frac{\partial E}{\partial \sigma _2 ^{ji}}\right) \nonumber \\&+ \frac{\partial E}{\partial c^{j}}\left( (\alpha -1)c^{j} +\eta \frac{\partial E}{\partial c^{j}}\right) -e_{tr} \frac{\partial u_{ff} }{\partial r}\dot{r} \nonumber \\&+ \,(1+\tau _{\max } )e_{tr} ^{2}+\dot{e}_{tr} ^{2} +\frac{\dot{\eta }}{\gamma }(\eta -\eta ^{{*}}) \end{aligned}$$

(A-6)

It is considered that $\eta ^{{*}}$ is large as

$$\begin{aligned} \eta ^{{*}}>\beta \end{aligned}$$

(A-7)

By substituting the adaptation law for $\eta$ in Eq. (20) into Eq. (A-6) and rearranging that for satisfying the mentioned stability condition, we have:

$$\begin{aligned}&\dot{V}(t)<-(\overbrace{-\eta \left( \frac{\partial E}{\partial \sigma _1 ^{ji}}\right) ^{2}+(1-\alpha )\sigma _1 ^{ji}\frac{\partial E}{\partial \sigma _1 ^{ji}}-0.5\eta \left| {e_{tr} } \right| }^I) \nonumber \\&\quad -(\overbrace{-\eta \left( \frac{\partial E}{\partial \sigma _2 ^{ji}}\right) ^{2}+(1-\alpha )\sigma _2 ^{ji}\frac{\partial E}{\partial \sigma _2 ^{ji}}-0.5\eta \left| {e_{tr} } \right| }^{II}) \nonumber \\&\quad -(\overbrace{-\eta \left( \frac{\partial E}{\partial m^{ji}}\right) ^{2}+(1-\alpha )m^{ji}\frac{\partial E}{\partial m^{ji}}-\dot{e}_{tr} ^{2}}^{III} ) \nonumber \\&\quad -(\overbrace{-\eta \left( \frac{\partial E}{\partial c^{j}}\right) ^{2}+(1-\alpha )c^{j}\frac{\partial E}{\partial c^{j}}-(1+\tau _{\max } )e_{tr} ^{2}+e_{tr} \frac{\partial u_{ff} }{\partial r}\dot{r}}^{IV})\nonumber \\ \end{aligned}$$

(A-8)

Equation (A-8) is in the form of four quadratic equations. This equation satisfies the Lyapunov–Krasovskii stability criterion only when:

$$\begin{aligned} \dot{V}(t)< & {} -\left( \overbrace{\frac{\partial E}{\partial \sigma _1 ^{ji}}+A}^I\right) ^{2}-\left( \overbrace{\frac{\partial E}{\partial \sigma _2 ^{ji}}+B}^{II}\right) ^{2} \nonumber \\&-\left( \overbrace{\frac{\partial E}{\partial m^{ji}}+C}^{III} \right) ^{2} -\left( \overbrace{\frac{\partial E}{\partial c^{j}}+D}^{IV}\right) ^{2}<0 \end{aligned}$$

(A-9)

where A, B, C, and D are the solutions of the equations. It should be noted that if $b^{2}-4ac=0$ exists in Eq. (A-9), then A, B, C, and D are solved. (a, b, and c are the coefficients of the standard quadratic equation.) Therefore, $b^{2}-4ac=0$ is calculated for the equations as follows:

$$\begin{aligned}&((1-\alpha )\sigma _1 ^{ji})^{2}-4(-0.5\eta \left| {e_{tr} } \right| )(-\eta )=0 \end{aligned}$$

(A-10)

$$\begin{aligned}&((1-\alpha )\sigma _2 ^{ji})^{2}-4(-0.5\eta \left| {e_{tr} } \right| )(-\eta )=0 \end{aligned}$$

(A-11)

$$\begin{aligned}&((1-\alpha )m^{ji})^{2}-4(-\dot{e}_{tr} ^{2})(-\eta )=0 \end{aligned}$$

(A-12)

$$\begin{aligned}&((1-\alpha )c^{j})^{2}-4(-(1+\tau _{\max } )e_{tr} ^{2}+e_{tr} \frac{\partial u_{ff} }{\partial r}\dot{r})(-\eta )=0\nonumber \\ \end{aligned}$$

(A-13)

Each of Eqs. (A-10)–(A-13) has two solutions (related to the T2FNN parameters). We suppose that $\sigma _{1,1}$ and ${\sigma }_{1,2}$ are the solutions of $\sigma _{1}$, ${\sigma }_{2,1}$ and ${\sigma }_{2,2}$ are the solutions of $\sigma _{2}$, $m_{1}$ and $m_{2}$ are the solutions of m, and $c_{1}$ and $c_{2}$ are the solutions of c. All of the solutions satisfy Eq. (A-9) giving:

$$\begin{aligned} A_1= & {} \frac{(1-\alpha )\sigma _{1,1} }{2\eta } ,\quad A_2 =\frac{(1-\alpha )\sigma _{1,2} }{2\eta } \end{aligned}$$

(A-14)

$$\begin{aligned} B_1= & {} \frac{(1-\alpha )\sigma _{2,1} }{2\eta } ,\quad B_2 =\frac{(1-\alpha )\sigma _{2,2} }{2\eta } \end{aligned}$$

(A-15)

$$\begin{aligned} C_1= & {} \frac{(1-\alpha )m_1 }{2\eta } ,\quad C_2 =\frac{(1-\alpha )m_2 }{2\eta } \end{aligned}$$

(A-16)

$$\begin{aligned} D_1= & {} \frac{(1-\alpha )c_1 }{2\eta } ,\quad D_2 =\frac{(1-\alpha )c_2 }{2\eta } \end{aligned}$$

(A-17)

Choosing the values stated in Eqs. (21)–(24) makes $\dot{V}(t)$ more negative, and it can be concluded that $\dot{V}(t)<0$ and, therefore, the designed control system is Lyapunov–Krasovskii stable.