### Fuzzy logic system

Fuzzy logic system includes singleton fuzzification, sum-product inference, and center off-sets defuzzification, which can be expressed by

$$ f(x)= \frac{\sum^{N}_{j=1}\theta _{j}\prod^{n}_{i=1}\mu _{F_{i}^{j}}(x_{i})}{ \sum^{N}_{j=1} [\prod^{n}_{i=1}\mu _{F_{i}^{j}}(x_{i}) ] }, $$

(5)

where *x* is the input, \(f(x)\) is the output. The membership of *j*th rule is \(\mu _{F_{i}^{j}}(x_{i})\), and the centroid of the *j*th consequent set is \(\theta _{j}\). Then (5) can be rewritten as follows:

$$ f(x)=\theta ^{T}\psi (x), $$

(6)

where \(\theta = [\theta _{1},\ldots,\theta _{N}]\), \(\psi (x)=[p_{1}(x),p_{2}(x),\ldots,p_{N}(x)]^{T}\) and the fuzzy basis function is \(p_{j}(x)= \frac{\prod^{n}_{i=1}\mu _{F_{i}^{j}}(x_{i})}{\sum^{N}_{j=1} [\prod^{n}_{i=1}\mu _{F_{i}^{j}}(x_{i}) ] }\).

### Lemma 1

([29])

*Suppose that* \(f(x)\) *is a continuous function and* \(x\in \Omega \), *where* Ω *is a compact set*. *For* (6), *there exists a fuzzy system such that*

$$ \sup_{x\in \Omega } \bigl\vert f(x)-\theta ^{T}\psi (x) \bigr\vert \leq \varepsilon , $$

(7)

*where* \(\varepsilon >0\).

### Controller design and stability analysis

Adaptive fuzzy control of the commensurate fractional-order model of financial risk chaotic system (4) is as follows:

$$ \textstyle\begin{cases} \frac{d^{q_{1}}x}{dt^{q_{1}}}=\delta (y-x)+yz+u_{1}, \\ \frac{d^{q_{2}}y}{dt^{q_{2}}}=rx-y-xz+u_{2}, \\ \frac{d^{q_{3}}z}{dt^{q_{3}}}=xy-bz+u_{3}. \end{cases} $$

(8)

Let \(f_{1}(x_{1})=\delta y+yz\), \(f_{2}(y_{2})=-xz+rx\), and \(f_{3}(z_{3})=xy\) be unknown as nonlinear functions, respectively. Then system (8) can be rewritten as

$$ \textstyle\begin{cases} \frac{d^{q_{1}}x}{dt^{q_{1}}}=-\delta x+f_{1}(x_{1})+u_{1}, \\ \frac{d^{q_{2}}y}{dt^{q_{2}}}=-y+f_{2}(y_{2})+u_{2}, \\ \frac{d^{q_{3}}z}{dt^{q_{3}}}=-bz+f_{3}(z_{3})+u_{3}. \end{cases} $$

(9)

Based on *Lemma **1*, the unknown functions can be respectively approximated by a fuzzy logic system as follows:

$$ \hat{f_{i}}(,\theta _{i})=\theta ^{T}_{i}\psi _{i},\quad i=1,2,3. $$

(10)

Let the optimal parameter estimation of fuzzy systems be \(\theta ^{\ast }_{i}=\min [\sup \vert f_{i}-\hat{f_{i}}(, \theta _{i}) \vert ]\), where \(\theta ^{\ast }_{i}\) is a constant.

Let the parameter error and optimal estimation error of the fuzzy system be respectively

$$\begin{aligned}& \tilde{\theta _{i}}=\theta _{i}-\theta ^{\ast }_{i}, \end{aligned}$$

(11)

$$\begin{aligned}& \varepsilon _{i}=f_{i}-\hat{f_{i}}\bigl(,\theta ^{\ast }_{i}\bigr). \end{aligned}$$

(12)

Based on [30], we can suppose that \(\vert \varepsilon _{i} \vert \leq \varepsilon ^{\ast }_{i}\), where \(\varepsilon ^{\ast }_{i}\) is a positive constant.

The estimation error of the unknown nonlinear function can be written as

$$ \begin{aligned} \hat{f}(,\theta )-f &= \hat{f}(,\theta )-\hat{f}\bigl(,\theta ^{\ast }\bigr)+ \hat{f}\bigl(,\theta ^{\ast }\bigr)-f \\ &=\hat{f}(,\theta )-\hat{f}\bigl(,\theta ^{\ast }\bigr)-\varepsilon \\ &=\theta ^{T}\psi -\theta ^{\ast T}\psi -\varepsilon \\ &=\tilde{\theta ^{T}}\psi -\varepsilon . \end{aligned} $$

(13)

Based on the above discussion, the controllers can be designed as

$$\begin{aligned}& u_{1}=-k_{1}x-\theta ^{T}_{1} \psi _{1}(x_{1})-\hat{\varepsilon }^{\ast }_{1} \operatorname{sign}(x), \end{aligned}$$

(14)

$$\begin{aligned}& u_{2}=-k_{2}y-\theta ^{T}_{2} \psi _{2}(y_{2})-\hat{\varepsilon }^{\ast }_{2} \operatorname{sign}(y), \end{aligned}$$

(15)

$$\begin{aligned}& u_{3}=-k_{3}y-\theta ^{T}_{3} \psi _{3}(z_{3})-\hat{\varepsilon }^{\ast }_{3} \operatorname{sign}(z), \end{aligned}$$

(16)

where \(k_{i}>0\), \(\hat{\varepsilon }^{\ast }_{i} \) is an estimate of the unknown constant \(\varepsilon ^{\ast }_{i}\) for \(i=1,2,3\).

In this subsection, we propose the fractional-order parameters adaptive laws as follows:

$$\begin{aligned}& \frac{d^{q}\hat{\theta _{1}}}{dt^{q}}=\mu _{1}x\psi _{1}(x_{1}), \end{aligned}$$

(17)

$$\begin{aligned}& \frac{d^{q}\hat{\varepsilon _{1}^{\ast }}}{dt^{q}}=\sigma _{1} \bigl\vert x^{T} \bigr\vert , \end{aligned}$$

(18)

$$\begin{aligned}& \frac{d^{q}\hat{\theta _{2}}}{dt^{q}}=\mu _{2}y\psi _{2}(y_{2}), \end{aligned}$$

(19)

$$\begin{aligned}& \frac{d^{q}\hat{\varepsilon _{2}^{\ast }}}{dt^{q}}=\sigma _{2} \bigl\vert y^{T} \bigr\vert , \end{aligned}$$

(20)

$$\begin{aligned}& \frac{d^{q}\hat{\theta _{3}}}{dt^{q}}=\mu _{3}z\psi _{3}(z_{3}), \end{aligned}$$

(21)

$$\begin{aligned}& \frac{d^{q}\hat{\varepsilon _{3}^{\ast }}}{dt^{q}}=\sigma _{3} \bigl\vert z^{T} \bigr\vert , \end{aligned}$$

(22)

where \(\mu _{i},\sigma _{i}>0\), \(i=1,2,3\).

To check the stability of the controlled system, some results of stability analysis of fractional-order systems are given in advance as follows.

### Lemma 2

([30])

*Let* \(V=\frac{1}{2}x^{2}+\frac{1}{2}y^{2}\), *where* \(x,y \in R\) *and* *x*, *y* *have a continuous first derivative*, *respectively*. *If there exists a constant* \(h>0\) *satisfying*

$$ \frac{d^{q}V}{dt^{q}}\leq -hx^{2}, $$

(23)

*then one has*

$$ x^{2}\leq 2V(0)E_{q}\bigl(-2ht^{q} \bigr), $$

(24)

*where* \(E^{q}(-2ht^{q})\) *is the Mittag*-*Leffler function*.

### Lemma 3

([30])

*Let* \(V=\frac{1}{2}x^{T}x+\frac{1}{2}y^{T}y\), *where* \(x,y \in R^{n}\) *and* *x*, *y* *have a continuous first derivative*, *respectively*. *There exists a constant* \(k>0\) *such that*

$$ \frac{d^{q}V}{dt^{q}}\leq -kx^{T}x. $$

(25)

*Then* \(\Vert x \Vert \), \(\Vert y \Vert \) *are bounded and* *x* *asymptotically approaches zero*, *where* \(\Vert \varepsilon \Vert \) *represents Euclid from*.

### Lemma 4

([31–34])

*If* \(x \in R^{n}\) *is a continuous differentiable function*, *one holds*

$$ \frac{1}{2}\frac{d^{q}x^{T}x}{dt{q}}\leq x^{T} \frac{d^{q}x}{dt^{q}}. $$

(26)

In order to facilitate, we write the fractional order of system (8) as *q*. From what has been discussed above we can obtain the following:

$$ \begin{aligned} \frac{d^{q}x}{dt^{q}} &= -\delta x+f_{1}(x_{1})+u_{1} \\ &=-\delta x+f_{1}(x_{1})-\hat{f}_{1}(x_{1}, \theta _{1})+\hat{f}_{1}(x_{1}, \theta _{1})+u_{1} \\ &=-\delta x-\tilde{\theta }^{T}_{1}\psi _{1}(x_{1})+\varepsilon _{1}(x_{1})-k_{1}x- \theta _{1}^{T}\psi _{1}(x_{1})-\hat{ \varepsilon }^{\ast }_{1}\operatorname{sign}(x)+ \theta ^{T}_{1}\psi _{1}(x_{1}) \\ &=-a_{1}x-\tilde{\theta }^{T}_{1}\psi _{1}(x_{1})+\varepsilon _{1}(x_{1})- \hat{\varepsilon }^{\ast }_{1}\operatorname{sign}(x), \end{aligned} $$

(27)

where \(a_{1}=\delta +k_{1}\).

Multiply both sides of the equation (26) by \(x^{T}\), one has

$$ \begin{aligned} x^{T}\frac{d^{q}x}{dt^{q}} &=-a_{1}x^{T}x-x^{T}\tilde{\theta }^{T}_{1} \psi _{1}(x_{1})+x^{T} \varepsilon _{1}(x_{1})-x^{T}\hat{\varepsilon }^{ \ast }_{1}\operatorname{sign}(x) \\ &\leq -a_{1}x^{T}x+\varepsilon ^{\ast }_{1} \bigl\vert x^{T} \bigr\vert - \hat{\varepsilon }^{\ast }_{1} \bigl\vert x^{T} \bigr\vert -x^{T}\tilde{\theta }^{T}_{1} \psi _{1}(x_{1}) \\ &=-a_{1}x^{T}x-\tilde{\varepsilon }^{\ast }_{1} \bigl\vert x^{T} \bigr\vert -x^{T} \tilde{\theta }^{T}_{1}\psi _{1}(x_{1}). \end{aligned} $$

(28)

Similar, we can obtain

$$\begin{aligned}& y^{T}\frac{d^{q}y}{dt^{q}}=-a_{2}y^{T}y- \tilde{\varepsilon }^{\ast }_{2} \bigl\vert y^{T} \bigr\vert -y^{T}\tilde{\theta }^{T}_{2}\psi _{2}(x_{2}), \end{aligned}$$

(29)

$$\begin{aligned}& z^{T}\frac{d^{q}z}{dt^{q}}=-a_{3}z^{T}z- \tilde{\varepsilon }^{\ast }_{3} \bigl\vert z^{T} \bigr\vert -z^{T}\tilde{\theta }^{T}_{3}\psi _{3}(z_{3}), \end{aligned}$$

(30)

where \(a_{2}=1+k_{2}\), \(a_{3}=b+k_{3}\).

### Theorem 1

*Under given initial conditions*, *the variables* *x*, *y*, *and* *z* *of fractional*-*order system* (8) *converge to zero under the action of adaptive controller* (14), (15), (16) *and fractional*-*order parameter adaptive laws* (17), (18), (19), (20), (21), (22), *and all variables in the closed*-*loop system are bounded*.

### Proof

Let the Lyapunov function be

$$ V_{1}=\frac{1}{2}x^{T}x+ \frac{1}{2\mu _{1}}\tilde{\theta }^{T}_{1} \tilde{\theta }_{1}+\frac{1}{2\sigma _{1}}\tilde{\varepsilon }^{\ast T}_{1} \tilde{\varepsilon }^{\ast }_{1}, $$

(31)

where \(\tilde{\theta }_{1}=\hat{\theta }_{1}-\theta ^{\ast }_{1}\) and \(\tilde{\varepsilon }^{\ast }_{1}=\hat{\varepsilon }^{\ast }_{1}- \varepsilon ^{\ast }_{1}\).

Based on *Lemma **4*, (17), (18), and (28), we can obtain

$$ \begin{aligned} \frac{d^{q}V_{1}}{dt^{q}} &= x^{T} \frac{d^{q}x}{dt^{q}}+ \frac{1}{\mu _{1}}\tilde{\theta }^{T}_{1} \frac{d^{q}\tilde{\theta }_{1}}{dt^{q}}+\frac{1}{\sigma _{1}} \tilde{\varepsilon }^{\ast T}_{1} \frac{d^{q}\tilde{\varepsilon }^{\ast }_{1}}{dt^{q}} \\ &\leq -a_{1}x^{T}x-\tilde{\varepsilon }^{\ast }_{1} \bigl\vert x^{T} \bigr\vert - x^{T} \tilde{\theta }^{T}_{1}\psi _{1}(x_{1})+x^{T} \tilde{\theta }^{T}_{1} \psi _{1}(x_{1})+ \tilde{\varepsilon }^{\ast T}_{1} \bigl\vert x^{T} \bigr\vert \\ &\leq -a_{1}x^{T}x, \end{aligned} $$

(32)

where \(a_{1}>0\). We know from *Lemma **3* that *x* asymptotically approaches zero, namely \(\lim_{t\to \infty } \Vert x \Vert =0\).

Choose the Lyapunov function as

$$ V_{2}=\frac{1}{2}y^{T}y+ \frac{1}{2\mu _{2}}\tilde{\theta }^{T}_{2} \tilde{\theta }_{2}+\frac{1}{2\sigma _{2}}\tilde{\varepsilon }^{\ast T}_{2} \tilde{\varepsilon }^{\ast }_{2}, $$

(33)

where \(\tilde{\theta }_{2}=\hat{\theta }_{2}-\theta ^{\ast }_{2}\) and \(\tilde{\varepsilon }^{\ast }_{2}=\hat{\varepsilon }^{\ast }_{2}- \varepsilon ^{\ast }_{2}\).

Based on *Lemma **4*, (19), (20), and (29), we can obtain

$$ \begin{aligned} \frac{d^{q}V_{2}}{dt^{q}} &= y^{T} \frac{d^{q}y}{dt^{q}}+ \frac{1}{\mu _{2}}\tilde{\theta }^{T}_{2} \frac{d^{q}\tilde{\theta }_{2}}{dt^{q}}+\frac{1}{\sigma _{2}} \tilde{\varepsilon }^{\ast T}_{2} \frac{d^{q}\tilde{\varepsilon }^{\ast }_{2}}{dt^{q}} \\ &\leq -a_{2}y^{T}y-\tilde{\varepsilon }^{\ast }_{2} \bigl\vert y^{T} \bigr\vert - y^{T} \tilde{\theta }^{T}_{2}\psi _{2}(y_{2})+y^{T} \tilde{\theta }^{T}_{2} \psi _{2}(y_{2})+ \tilde{\varepsilon }^{\ast T}_{2} \bigl\vert y^{T} \bigr\vert \\ &\leq -a_{2}y^{T}y, \end{aligned} $$

(34)

where \(a_{2}>0\). We know from *Lemma **3* that *y* asymptotically approaches zero, namely \(\lim_{t\to \infty } \Vert y \Vert =0\).

Consider the Lyapunov function

$$ V_{3}=\frac{1}{2}z^{T}z+ \frac{1}{2\mu _{3}}\tilde{\theta }^{T}_{3} \tilde{\theta }_{3}+\frac{1}{2\sigma _{3}}\tilde{\varepsilon }^{\ast T}_{3} \tilde{\varepsilon }^{\ast }_{3}, $$

(35)

where \(\tilde{\theta }_{3}=\hat{\theta }_{3}-\theta ^{\ast }_{3}\) and \(\tilde{\varepsilon }^{\ast }_{3}=\hat{\varepsilon }^{\ast }_{3}- \varepsilon ^{\ast }_{3}\).

Based on *Lemma **4*, (21), (22), and (30), we can obtain

$$ \begin{aligned} \frac{d^{q}V_{3}}{dt^{q}} &= z^{T} \frac{d^{q}z}{dt^{q}}+ \frac{1}{\mu _{3}}\tilde{\theta }^{T}_{3} \frac{d^{q}\tilde{\theta }_{3}}{dt^{q}}+\frac{1}{\sigma _{3}} \tilde{\varepsilon }^{\ast T}_{3} \frac{d^{q}\tilde{\varepsilon }^{\ast }_{3}}{dt^{q}} \\ &\leq -a_{3}z^{T}z-\tilde{\varepsilon }^{\ast }_{3} \bigl\vert z^{T} \bigr\vert - z^{T} \tilde{\theta }^{T}_{3}\psi _{3}(x_{3})+z^{T} \tilde{\theta }^{T}_{3} \psi _{3}(z_{3})+ \tilde{\varepsilon }^{\ast T}_{3} \bigl\vert z^{T} \bigr\vert \\ &\leq -a_{3}z^{T}z, \end{aligned} $$

(36)

where \(a_{3}>0\). We know from *Lemma **3* that *z* asymptotically approaches zero, namely \(\lim_{t\to \infty } \Vert z \Vert =0\). □

Noting that \(\tilde{\varepsilon }^{\ast }_{i} \in R\), \(i=1,2,3\), so it has \(\tilde{\varepsilon }^{\ast }_{i}=\tilde{\varepsilon }^{\ast T}_{i}\).

We know from *Lemma **2* that \(\tilde{\theta }_{i}\), \(\tilde{\varepsilon }^{\ast }_{i}\) are bounded, and \(\hat{\theta }_{i}\), \(\hat{\varepsilon }^{\ast }_{i}\) are also bounded. From the above proof, *x*, *y*, and *z* are bounded, and the construction of controllers (14), (15), and (16) shows that \(u_{i}\) is bounded for \(i=1,2,3\). So, all signals in a closed loop system (8) are bounded.

### Simulation studies

In this subsection, we choose fractional-order system (8) as an example.

Let the parameters of the financial risk chaotic system (8) be \(\delta =10\), \(r=28\), \(b=\frac{8}{3}\), with the initial conditions of system (8) being \((2.5,0.5,4)\) and \(q=0.95\). In the simulation, *x*, *y*, *z* are the inputs of the fuzzy systems. We choose four Gaussian membership functions on \([-3,3]\). \(k_{1}=15\), \(k_{2}=15\), \(k_{3}=10\) and \(\mu _{i}=700\), \(\sigma _{i}=0.8\) for \(i=1,2,3\). The estimated value of the approximation error of the fuzzy system is \(\hat{\varepsilon }^{\ast }_{1}(0)=1\), \(\hat{\varepsilon }^{\ast }_{2}(0)=1\), \(\hat{\varepsilon }^{\ast }_{3}(0)=1.5\). The simulation results are shown in Fig. 5, Fig. 6, and Fig. 7. In Fig. 5, the system variables have a rapid convergence. Figure 6 shows the smoothness of the control inputs, and Fig. 7 indicates the convergence of the fuzzy parameters under the proposed fractional-order adaptation laws. It has been shown that good control performance has been obtained.