1. Introduction

Fractional calculus is a generalization of differential and integral calculus which involves generalized functions. The first to work this new branch of mathematics was Leibniz. Due to the growing interest in the applications of fractional calculation, in this work we obtain conditions that guarantee the tracking of trajectories of nonlinear systems generated by differential equations of fractional order which we will call plants (This term is widely used in engineering), which in our case will be a mechanical arm, a helicopter, a plane or limbs of a humanoid, all of fractional order.

The problem of tracking control of trajectories is very important, since the control function allows the non-linear system to carry out a previously assigned

#### Figure 1.

Adaptive recurrent control diagram.

task, work or trajectory, for example, a mechanical arm and its objective is to cut a piece with a previously generated form, or the coupling of two aircraft in space.

In this chapter we use adaptive recurrent neural networks with time delay, since its use allows us to work with systems whose mathematical model is unknown and with the presence of uncertainties, this is a well-known problem of robust control.

We include mathematical models with time delay, since the processing and transmission of information is important in this type of systems, which depending on the delay, these systems can generate undesirable oscillatory or chaotic dynamics, and cause instability in the mathematical model that describes the trajectory tracking error.

The chapter is organized as follows: first, the general mathematical model of non-linear systems is proposed, as a second part, the Neural Network is proposed that will adapt to the non-linear system and the reference signal that both must follow, as a third part obtains the dynamics of the tracking error between the nonlinear system and the reference, after obtaining conditions in the laws of adaptation of weights in the Neural Network and obtaining the control law that guarantees that the tracking error converges to zero, so that the non-linear system will follow the indicated reference signal, which is what was wanted to be demonstrated. Finally simulations are presented, which illustrate the theoretical results previously demonstrated. The proposed new control scheme is applied via simulations to control of a 4-DOF Biped Robot [1].

We use the scheme of Figure 1 to indicate the procedure used in the obtaining of the laws of adaptation of weights and the laws of control that guarantee that the tracking error between the non-linear system, the neural network and the reference signal converges to zero.

#### 2. Time-delay adaptive neural network and the reference

#### 2.1 List of variables

W<sup>∗</sup> is the matrix weights. e ¼ xp � xr, error between the plant and the reference. W^ is part of the approach, given by W<sup>∗</sup> : Ωu1, Ωu2, up, un are the controls PI<sup>λ</sup> <sup>D</sup><sup>α</sup> <sup>¼</sup> Kpe tðÞþ KiaD�<sup>λ</sup> <sup>t</sup> e tðÞþ KdaD<sup>α</sup> <sup>t</sup> e tð Þ control law τ is time delay

Trajectory Tracking Using Adaptive Fractional PID Control of Biped Robots with Time-Delay… DOI: http://dx.doi.org/10.5772/intechopen.90901

tr aD<sup>α</sup> <sup>t</sup> <sup>W</sup><sup>~</sup> <sup>T</sup> <sup>W</sup><sup>~</sup> n o ¼ �eTW<sup>~</sup> <sup>σ</sup>ð Þ x tð Þ � <sup>τ</sup> , learning law from the neural network weights Ðt <sup>t</sup>�<sup>τ</sup> <sup>Ø</sup><sup>T</sup> <sup>σ</sup> ð Þ<sup>s</sup> <sup>W</sup>^ <sup>T</sup> <sup>W</sup>^ <sup>Ø</sup>σð Þ<sup>s</sup> h ids, Lyapunov-Krasovskii Function Dqt ð Þþ ð Þ€q tð Þ Cqt ð Þ ð Þ, q t \_ð Þ q t \_ðÞþ Gqt ð Þ¼ ð Þ Bτð Þt , dynamics of the bipedal robot Dqt ð ð Þ) is the inertia matrix Cqt ð Þ ð Þ, q t \_ð Þ is the matrix of Coriolis and centripetal forces Gqt ð Þ ð Þ represents a matrix of gravitational effects B defines the input matrix

There are several ways to define the fractional calculation, in this research we will use the well-known derivative of Caputo, which has the following notation:

$$aD\_t^af(t) = \frac{1}{\Gamma(a-n)} \int\_a^t \frac{f^{(n)}(\tau)}{(t-\tau)^{a-n+1}}d\tau \tag{1}$$
 
$$\text{For } (n-1 < a < n).$$

The nonlinear system, Eq. (2), which is forced to follow a reference signal:

$$aD\_t^a \mathbf{x}\_p = f\_p\left(\mathbf{t}, \mathbf{x}\_p(\mathbf{t}) + \mathbf{x}\_p(\mathbf{t} - \tau)\right), \mathbf{t} \in [0, T], 0 < a \le 1,\tag{2}$$

$$\mathbf{x}\_p(\mathbf{t}) = \mathbf{g}(\mathbf{t})$$

$$\mathbf{x}\_p, f\_p \in \mathbb{R}^n, u \in \mathbb{R}^m, \mathbf{g}\_p \in \mathbb{R}^{n \times n}.$$

The differential equation will be modeled by the neural network [2]:

$$\mathfrak{a}D\_t^a \mathfrak{x}\_p = \mathbf{A}(\mathfrak{x}) + \mathbf{W}^\* \Gamma\_\mathbf{z}(\mathfrak{x}(t-\tau) + \mathfrak{Q}\_\mathbf{u})$$

The tracking error between these two systems:

$$
\omega\_{per} = \mathbf{x} - \mathbf{x}\_p \tag{3}
$$

We use the next hypotheses.

$$
\hbar \mathcal{D}\_t^a w\_{per} = -k w\_{per} \tag{4}
$$

In this research we will use <sup>k</sup> <sup>¼</sup> 1, so that, Eq. (5), aD<sup>α</sup> <sup>t</sup> wper <sup>¼</sup> aD<sup>α</sup> <sup>t</sup> <sup>x</sup> � aD<sup>α</sup> <sup>t</sup> xp, so:

$$aD\_t^a \mathbf{x}\_p = aD\_t^a \mathbf{x} + w\_{per}$$

The nonlinear system is [3]:

$$
\rho \mathbf{d} D\_t^a \mathbf{x}\_p = \mathbf{d} D\_t^a \mathbf{x} + \mathbf{w}\_{per} = \mathbf{A}(\mathbf{x}) + \mathbf{W}^\* \boldsymbol{\Gamma}\_\mathbf{z} [\mathbf{x}(t-\tau)] + \mathbf{w}\_{per} + \boldsymbol{\Omega}\_\mathbf{u} \tag{5}
$$

where the W<sup>∗</sup> is the matrix weights.

### 3. Tracking error problem

In this part, we will analyze the trajectory tracking problem generated by

$$
\mu D\_t^a \mathfrak{x}\_r = f\_r(\mathfrak{x}\_r, \mathfrak{u}\_r), \mathfrak{w}\_r, \mathfrak{x}\_r \in \mathbb{R}^n \tag{6}
$$

are the state space vector, input vector and fr, is a nonlinear vectorial function. To achieve our goal of trajectory tracking, we propose the error between the plant

and the reference as: e ¼ xp � xr ¼ xp � xn <sup>þ</sup> ð Þ¼ xn � xr xp � <sup>x</sup> <sup>þ</sup> ð Þ <sup>x</sup> � xr . Let ep ¼ xp � x, and en ¼ x � xr, be the trajectory tracking error and e ¼ ep þ er

$$e\_n = \mathfrak{x} - \mathfrak{x}\_r \tag{7}$$

The time derivative of the error is:

$$aD\_t^a e\_\mathfrak{n} = aD\_t^a \mathbf{x} - aD\_t^a \mathbf{x}\_r = \mathbf{A}(\mathbf{x}) + \mathbf{W}^\* \Gamma\_\mathbf{z}[\mathbf{x}(t-\tau)] + \boldsymbol{w}\_{per} + \boldsymbol{\Omega}\_\mathbf{u} - f\_r(\mathbf{x}\_r, \boldsymbol{u}\_r) \tag{8}$$

Eq. (8), can be rewritten as follows, adding and subtracting, the next terms <sup>W</sup>^ <sup>Γ</sup>zð Þ xrx tð Þ � <sup>τ</sup> , <sup>α</sup><sup>r</sup> t,W^ , Ae and wper <sup>¼</sup> <sup>x</sup> � xp, then,

$$aD\_t^\mathbf{u} e = \mathbf{A}(\mathbf{x}) + \mathbf{W}^\* \Gamma\_\mathbf{z} (\mathbf{x}(t-\tau)) + \mathbf{x} - \mathbf{x}\_p + \boldsymbol{\Omega}\_\mathbf{u} - f\_r(\mathbf{x}\_r, \boldsymbol{u}\_r) + \hat{\mathcal{W}} \Gamma\_\mathbf{z} (\mathbf{x}\_r(t-\tau))$$

$$- \hat{\mathcal{W}} \Gamma\_\mathbf{z} (\mathbf{x}\_r(t-\tau)) + \boldsymbol{\Omega} a\_r(\mathbf{t}, \dot{\mathcal{W}}) - \boldsymbol{\Omega} a\_r(\mathbf{t}, \dot{\mathcal{W}}) + A e - A e$$

$$a D\_t^\mathbf{z} e = \mathbf{A}e + \mathbf{W}^\* \Gamma\_\mathbf{z} (\mathbf{x}(t-\tau)) + \boldsymbol{\Omega} a\_\mathbf{u} - f\_r(\mathbf{x}\_r, \boldsymbol{u}\_r) + \dot{\mathcal{W}} \Gamma\_\mathbf{z} (\mathbf{x}\_\mathbf{}(t-\tau)) \tag{9}$$

$$+ \boldsymbol{\Omega} a\_r(\mathbf{t}, \dot{\mathcal{W}}) - \boldsymbol{\Omega} a\_r(\mathbf{t}, \dot{\mathcal{W}}) - e - \mathbf{x}\_r - A e + \mathbf{x} + A(\mathbf{x}) - \dot{\mathcal{W}} \Gamma\_\mathbf{z} (\mathbf{x}\_r(t-\tau))$$

The unknown plant will follow the fractional order reference signal, if:

$$A\boldsymbol{\kappa}\_r + \hat{W}\boldsymbol{\Gamma}\_\mathbf{z}(\boldsymbol{\kappa}\_r(t-\tau)) + \boldsymbol{\kappa}\_r - \boldsymbol{\kappa}\_p + \Omega a\_r(\mathbf{t}, \hat{W}) = \boldsymbol{f}\_r(\boldsymbol{\kappa}\_r, \boldsymbol{\mu}\_r),$$

where

$$
\Omega a\_r(\mathbf{t}, \hat{\mathcal{W}}) = f\_r(\mathbf{x}\_r, \boldsymbol{\mu}\_r) - A\mathbf{x}\_r - \hat{\mathcal{W}}\Gamma\_\mathbf{z}(\mathbf{x}\_r(t-\tau)) - \mathbf{x}\_r + \mathbf{x}\_p \tag{10}
$$

$$
a D\_t^a e = \mathbf{A}e + \mathbf{W}^\* \Gamma\_\mathbf{z}(\mathbf{x}(t-\tau)) - \hat{\mathcal{W}}\Gamma\_\mathbf{z}(\mathbf{x}\_r(t-\tau)) - \mathbf{A}e + (A+I)(\mathbf{x}-\mathbf{x}\_r)
$$

$$
+ \boldsymbol{\Omega}(\boldsymbol{u} - a\_r(\mathbf{t}, \hat{\mathcal{W}})) \tag{11}
$$

Now, W^ is part of the approach, given by W<sup>∗</sup> . Eq. (11) can be expressed as Eq. (12), adding and subtracting the term <sup>W</sup>^ <sup>Γ</sup>zð Þ x tð Þ � <sup>τ</sup> and if <sup>Γ</sup>zð Þ¼ x tð Þ � <sup>τ</sup> Γð Þ zxt ð Þ� ð Þ � τ z xð Þ <sup>r</sup>ð Þ t � τ

$$aD\_t^a e = \mathbf{A}e + \left(\mathbf{W}^\* - \hat{W}\right)\Gamma\_\mathbf{z}(\mathbf{x}(t-\tau)) + \hat{W}\Gamma(\mathbf{z}(\mathbf{x}(t-\tau)) - \mathbf{z}(\mathbf{x}\_r(t-\tau)))$$

$$+ (\mathbf{A} + I)(\mathbf{x} - \boldsymbol{\varkappa}\_r) - \mathbf{A}e + \boldsymbol{\Omega}(\boldsymbol{u} - a\_r(\mathbf{t}, \hat{W})) \tag{12}$$

If

$$
\tilde{\mathcal{W}} = \mathcal{W}^\* - \hat{\mathcal{W}} \text{ and } \tilde{u} = u - a\_r(\mathfrak{t}, \hat{\mathcal{W}}) \tag{13}
$$

And by replacing Eq. (13) in Eq. (12), we have:

$$\begin{aligned} aD\_t^a e &= \mathbf{A}e + \tilde{W}\Gamma\_z(\mathbf{x}(t-\tau)) + \hat{W}\Gamma(z(\mathbf{x}(t-\tau)) - z(\mathbf{x}\_r(t-\tau))) \\ &+ (\mathbf{A} + I)(\mathbf{x} - \mathbf{x}\_r) - \mathbf{A}e + \Omega\tilde{u} \end{aligned}$$

$$\begin{array}{l} \text{s\!D}\_{t}^{a}e = \text{A}e + \tilde{W}\Gamma\_{\text{z}}(\text{x}(t-\tau)) \\ \quad + \hat{W}\Gamma\left(\text{z}(\text{x}(t-\tau)) - \text{z}\left(\mathbf{x}\_{p}(t-\tau)\right) + \mathbf{z}\left(\mathbf{x}\_{p}(t-\tau)\right) - \mathbf{z}\left(\mathbf{x}\_{r}(t-\tau)\right)\right) \\ \quad + (\mathcal{A}+I)\left(\mathbf{x}-\mathbf{x}\_{p}+\mathbf{x}\_{p}-\mathbf{x}\_{r}\right) - \mathbf{A}e + \Omega\tilde{\tilde{u}} \end{array} \tag{14}$$

Trajectory Tracking Using Adaptive Fractional PID Control of Biped Robots with Time-Delay… DOI: http://dx.doi.org/10.5772/intechopen.90901

And:

$$
\tilde{u} = u\_1 + u\_2 \tag{15}
$$

So, the result for Ωu<sup>1</sup> is

$$
\Delta \mathfrak{U}\_1 = -\hat{W}\Gamma\left(z(\mathfrak{x}(t-\tau)) - z(\mathfrak{x}\_p(t-\tau))\right) - (A+I)\left(\mathfrak{x}-\mathfrak{x}\_p\right) \tag{16}
$$

and Eq. (14), is simplified:

$$\begin{aligned} \mathbf{a}D\_t^a e &= \mathbf{A}e + \check{W}\Gamma\_\mathbf{z}(\mathbf{x}(t-\tau)) + \hat{W}\Gamma\left(\mathbf{z}(\mathbf{x}\_p(t-\tau)) - \boldsymbol{z}(\mathbf{x}\_r(t-\tau))\right) \\ &+ (\boldsymbol{A} + I)(\boldsymbol{\chi}\_p - \boldsymbol{\chi}\_r) - \mathbf{A}e + \Omega\check{u} \end{aligned}$$

Taking into account that e ¼ xp � xr, shortening notation a little bit by setting <sup>σ</sup> <sup>¼</sup> <sup>Γ</sup>z, and defining Øσð Þ¼ <sup>t</sup> � <sup>τ</sup> <sup>σ</sup> xpð Þ <sup>t</sup> � <sup>τ</sup> � <sup>σ</sup>ð Þ xrð Þ <sup>t</sup> � <sup>τ</sup> , the equation for aD<sup>α</sup> <sup>t</sup> e is

$$
\sigma D\_t^a \mathbf{e} = (A + I)\mathbf{e} + \tilde{W}\sigma(\mathbf{x}(t-\tau)) + \hat{W}\mathcal{O}\_\sigma(t-\tau) + \Omega\mu\_2 \tag{17}
$$

Now, the problem is to find the control law Ωu2, which it stabilizes to the system Eq. (20). The control law, we will obtain using the fractional order Lyapunov-Krasovskii methodology.

#### 4. Study of trajectory tracking error

Our mathematical model of the dynamics in the tracking error is described in (17). In this equation we can see that an equilibrium state of this system is <sup>e</sup>,W^ <sup>¼</sup> 0.

Without loss of generality we can assume that the matrix A is given A ¼ �λI, λ>0, where I is the identity matrix of order nxn.

For the study of the stability of the tracking error we propose the following PID control law [4], widely used in science and engineering.

We will determine conditions in the parameters that guarantee that the tracking error converges to zero, and we will also use the following control law [5].

$$
\Delta \mathfrak{U}\_2 = K\_p e + K\_i a D\_t^{-a} e + K\_v a D\_t^a e - \gamma \left(\frac{1}{2} + \frac{1}{2} ||\mathring{W}||^2 L\_\phi^2\right) e \tag{18}
$$

We also include the following control law, PI<sup>λ</sup> D<sup>α</sup> [6]:

$$u(t) = K\_p e(t) + K\_i a D\_t^{-\\\lambda} e(t) + K\_d a D\_t^a e(t)$$

Substituting Eq. (18) in Eq. (17):

$$\begin{aligned} aD\_t^a e &= (A+I)e + \ddot{W}\sigma(\varkappa(t-\tau)) + \dot{W}\mathcal{O}\_\sigma(t-\tau) \\ &+ K\_p e + K\_i a D\_t^{-a} e + K\_v a D\_t^a e - \gamma \left(\frac{1}{2} + \frac{1}{2}||\dot{W}||^2 L\_\phi^2\right) e, \end{aligned}$$

then

$$\begin{aligned} (\mathbf{1} - K\_v)aD\_t^a e &= (A + I)e + \bar{W}\sigma(\mathbf{x}(t - \tau)) + \hat{W}\mathcal{O}\_\sigma(t - \tau) \\ &+ K\_p e + K\_i a D\_t^{-a} e - \gamma \left(\frac{1}{2} + \frac{1}{2}||\hat{W}||^2 L\_\phi^2\right) e. \end{aligned}$$

If a ¼ ð Þ 1 � Kv , then

$$aD\_t^a e = \frac{1}{a}(A+I)e + \frac{1}{a}\ddot{W}\sigma(\mathbf{x}(t-\tau)) + \frac{1}{a}\hat{W}\mathcal{O}\_\sigma(t-\tau) + \frac{1}{a}K\_pe + \frac{1}{a}K\_i a D\_t^{-a}e$$

$$-\frac{\gamma}{a}\left(\frac{1}{2} + \frac{1}{2}||\dot{W}||^2 L\_\phi^2\right)e\tag{19}$$

$$aD\_t^a e = \frac{-\mathbf{1}}{a} (\lambda - \mathbf{1} + K\_p) e + \frac{\mathbf{1}}{a} \ddot{W} \sigma(\mathbf{x}(t-\tau)) + \frac{\mathbf{1}}{a} \dot{W} \mathcal{O}\_{\sigma}(t-\tau) + \frac{\mathbf{1}}{a} K\_i a D\_t^{-a} e$$

$$-\frac{\gamma}{a} \left(\frac{\mathbf{1}}{2} + \frac{\mathbf{1}}{2} ||\dot{W}||^2 L\_\phi^2 \right) e \tag{20}$$

And if <sup>w</sup> <sup>¼</sup> <sup>1</sup> aKiaD�<sup>α</sup> <sup>t</sup> e, then aD<sup>α</sup> <sup>t</sup> <sup>w</sup> <sup>¼</sup> <sup>1</sup> <sup>a</sup>Kie tð Þ, [7], then Eq. (20) we rewrite as:

$$aD\_t^a e\_n = \frac{-1}{a} \left(\lambda - 1 + K\_p\right) e + \frac{1}{a} \ddot{W} \sigma(\varkappa(t-\tau)) + \frac{1}{a} \dot{W} \mathcal{O}\_\sigma(t-\tau) + w$$

$$-\frac{\gamma}{a} \left(\frac{1}{2} + \frac{1}{2} \left\|\dot{W}\right\|^2 L\_\phi^2\right) e \tag{21}$$

We will show, the new state ð Þ en, w <sup>T</sup> is asymptotically stable, and the equilibrium point is ð Þ ene, w <sup>T</sup> <sup>¼</sup> ð Þ 0, 0 <sup>T</sup>, when <sup>W</sup><sup>~</sup> <sup>σ</sup>ð Þ¼ xrð Þ <sup>t</sup> � <sup>τ</sup> 0, as an external disturbance. Let V be, the next candidate Lyapunov function as [8, 9]:

$$V = \frac{1}{2} \left( e\_n^{-T}, w^T \right) (e\_n, w)^T + \frac{1}{2a} tr \left\{ \vec{W}^T \vec{W} \right\} \tag{22}$$

$$+ \frac{1}{a} \int\_{t-\tau}^t \left[ \mathcal{O}\_\sigma^T(s) \hat{W}^T \hat{W} \mathcal{O}\_\sigma(s) \right] ds$$

The fractional order time derivative of (22) along the trajectories of Eq. (21) is:

$$\begin{aligned} aD\_t^a V &= e^T a D\_t^a e + w^T a D\_t^a w + \frac{1}{a} tr \left\{ a D\_t^a \hat{W}^T \ddot{W} \right\} \\ &+ \frac{1}{a} \left[ \mathcal{O}\_\sigma^T(t) \hat{W}^T \hat{W} \mathcal{O}\_\sigma(t) - \mathcal{O}\_\sigma^T(t-\tau) \hat{W}^T \hat{W} \mathcal{O}\_\sigma(t-\tau) \right] \end{aligned} \tag{23}$$

$$\begin{split}d\boldsymbol{D}\_{t}^{a}\boldsymbol{V} &= \boldsymbol{e}^{T} \left(\frac{-1}{a} \left(\boldsymbol{\lambda} - \boldsymbol{1} + \boldsymbol{K}\_{p} \right) \boldsymbol{e} + \frac{1}{a} \boldsymbol{\tilde{W}} \boldsymbol{\sigma} (\boldsymbol{x}(t-\tau)) + \frac{1}{a} \boldsymbol{\hat{W}} \boldsymbol{\mathcal{O}}\_{\sigma} (t-\tau) + \boldsymbol{w} \\ &- \frac{\gamma}{a} \left(\frac{1}{2} + \frac{1}{2} ||\boldsymbol{\dot{W}}||^{2} \boldsymbol{L}\_{\phi}^{2} \right) \boldsymbol{e} \right) + \frac{1}{a} \boldsymbol{\tilde{W}}^{T} \boldsymbol{K}\_{t} \boldsymbol{e} + \frac{1}{a} \boldsymbol{tr} \Big{\boldsymbol{\left} a \boldsymbol{D}\_{t}^{a} \boldsymbol{\tilde{W}}^{T} \boldsymbol{\dot{W}} \right\} \\ &+ \frac{1}{a} \left[\boldsymbol{\mathcal{O}}\_{\sigma}^{T} (t) \boldsymbol{\hat{W}}^{T} \boldsymbol{\hat{W}} \boldsymbol{\mathcal{O}}\_{\sigma} (t) - \boldsymbol{\mathcal{O}}\_{\sigma}^{T} (t-\tau) \boldsymbol{\hat{W}}^{T} \boldsymbol{\hat{W}} \boldsymbol{\mathcal{O}}\_{\sigma} (t-\tau) \right] \end{split} \tag{24}$$

In this part, we select the next learning law from the neural network weights as in [10, 11]:

$$\text{tr}\left\{a\mathbf{D}\_t^a\tilde{\mathbf{W}}^T\tilde{\mathbf{W}}\right\} = -\mathbf{e}^T\tilde{\mathbf{W}}\sigma(\mathbf{x}(t-\tau))\tag{25}$$

Trajectory Tracking Using Adaptive Fractional PID Control of Biped Robots with Time-Delay… DOI: http://dx.doi.org/10.5772/intechopen.90901

Then Eq. (24) is reduced to

$$\begin{split} aD\_t^a V &= \frac{-1}{a} \left( \lambda - 1 + K\_p \right) e^T e + \frac{e^T}{a} \hat{W} \mathcal{O}\_\sigma(t - \tau) \\ &+ \left( 1 + \frac{K\_i}{a} \right) e^T w - \frac{\gamma}{a} \left( \frac{1}{2} + \frac{1}{2} ||\hat{W}||^2 L\_\phi^2 \right) e^T e \\ &+ \frac{1}{a} \left[ \mathcal{O}\_\sigma^T(t) \hat{W}^T \hat{W} \mathcal{O}\_\sigma(t) - \mathcal{O}\_\sigma^T(t - \tau) \hat{W}^T \hat{W} \mathcal{O}\_\sigma(t - \tau) \right] \end{split} \tag{26}$$

Next, let us consider the following inequality proved in [12]

$$X^T Y + Y^T X \le X^T \Lambda X + Y^T \Lambda^{-1} Y \tag{27}$$

Which holds for all matrices <sup>X</sup>, <sup>Y</sup> <sup>∈</sup> nxkand <sup>Λ</sup><sup>∈</sup> nxn with <sup>Λ</sup> <sup>¼</sup> <sup>Λ</sup><sup>T</sup> <sup>&</sup>gt;0. Applying (27) with <sup>Λ</sup> <sup>¼</sup> <sup>I</sup> to the term <sup>e</sup><sup>T</sup> <sup>a</sup> <sup>W</sup>^ <sup>Ø</sup>σð Þ <sup>t</sup> � <sup>τ</sup> from Eq. (26), where

$$\frac{e^T}{a}\hat{W}\mathcal{O}\_\sigma(t-\tau) \le \frac{1}{a} \left[e^T e + \mathcal{O}\_\sigma^T(t-\tau)\hat{W}^T\hat{W}\mathcal{O}\_\sigma(t-\tau)\right]$$

we get

$$\begin{split} aD\_t^a V \leq & \frac{-1}{a} \left( \lambda - \mathbf{1} + K\_p \right) e^T e + \frac{\mathbf{1}}{a} \left( \frac{e^T e}{2} + \frac{\mathbf{1}}{2} \left\| \hat{W} \right\|^2 L\_\phi^2 \right) e^T e \\ & + \left( \mathbf{1} + \frac{K\_i}{a} \right) e^T w - \frac{\chi}{a} \left( \frac{\mathbf{1}}{2} + \frac{\mathbf{1}}{2} \left\| \hat{W} \right\|^2 L\_\phi^2 \right) e^T e \end{split} \tag{28}$$

Here, we select 1 <sup>þ</sup> Ki a � � <sup>¼</sup> 0 and Kv <sup>¼</sup> Ki <sup>þ</sup> 1, with Kv <sup>≥</sup>0 then Ki <sup>≥</sup> � 1, with this selection of the parameters from Eq. (28) is reduced to:

$$aD\_t^a V \le \frac{-1}{a} \left(\lambda - 1 + K\_p\right) e^T e - \frac{(\gamma - 1)}{a} \left(\frac{1}{2} + \frac{1}{2} \left\|\hat{W}\right\|^2 L\_\phi^2\right) e^T \tag{29}$$

From the previous inequality, we need to guarantee that Eq. (29) is less than zero, for which we select,

<sup>λ</sup> � <sup>1</sup> <sup>þ</sup> Kp <sup>&</sup>gt;0, <sup>a</sup>>0, ð Þ <sup>γ</sup> � <sup>1</sup> <sup>&</sup>gt;0, so that: aD<sup>α</sup> <sup>t</sup> <sup>V</sup> <sup>≤</sup>0,∀e, <sup>w</sup>,W^ 6¼ 0, <sup>e</sup> 6¼ 0, is wanted to be demonstrate.

The control law is given by Eq. (30)

$$\begin{aligned} u\_n &= \Omega^\dagger \left[ -\hat{W}\Gamma\left( \mathbf{z}(\mathbf{x}\_n(t-\tau)) - \mathbf{z}\left(\mathbf{x}\_p(t-\tau)\right) \right) \right] \\ &- (A+I)\left(\mathbf{x} - \mathbf{x}\_p\right) + K\_p e + K\_i a D\_t^{-a} e + K\_r a D\_t^a e \\ &- \Gamma\left(\frac{1}{2} + \frac{1}{2} \left||\hat{W}||^2 L\_\phi^2\right| e\_n + f\_r(\mathbf{x}\_r, u\_r) - A\mathbf{x}\_r \\ &- \hat{W}\Gamma\_\mathbf{z}(\mathbf{x}\_r(t-\tau)) - \mathbf{x}\_r + \mathbf{x}\_p \right] \end{aligned} \tag{30}$$

Theorem: The control law Eq. (30) and the neuronal adaptation law given by Eq. (25) guarantee that the fractional tracking error converges to zero, by which the tracking of trajectories of the non-linear system is guaranteed Eq. (5).

Corollary 2: If aD<sup>α</sup> <sup>t</sup> V ≤ �<sup>1</sup> <sup>a</sup> λ � 1 þ Kp � � en <sup>T</sup> � �ð Þ� en ð Þ <sup>γ</sup>�<sup>1</sup> a 1 <sup>2</sup> <sup>þ</sup> <sup>1</sup> <sup>2</sup> <sup>W</sup>^ � � � � 2 L2 ϕ � � en <sup>T</sup> � �ð Þ en <sup>&</sup>lt; 0, <sup>∀</sup><sup>e</sup> 6¼ 0, <sup>∀</sup>W^ , where <sup>V</sup> is decreasing and bounded from below by <sup>V</sup>ð Þ <sup>0</sup> , and:

$$\mathcal{V} = \frac{1}{2} \left( e\_n^T, w^T \right) \left( e\_n, w \right)^T + \frac{1}{2a} tr \left\{ \bar{\boldsymbol{W}}^T \boldsymbol{\bar{W}} \right\} + \frac{1}{a} \int\_{t-\tau}^t \left[ \mathcal{O}\_\sigma^T(s) \boldsymbol{\hat{W}}^T \boldsymbol{\hat{W}} \mathcal{O}\_\sigma(s) \right] ds,$$

then we conclude that e, W~ ∈ L1; this means that the weights remain bounded.
