Stability analysis using differential approximation, is well-known as a base in control theory. The advantage of the contraction method is that, it yields *global* and *exact* results of stability analysis for nonlinear systems. This section presents a summary of the basic results of Bandyopadhyay and Kamal [29], to which reference can be made for more details.

Consider an autonomous dynamical system

$$\begin{array}{}{\displaystyle \dot{\mathrm{x}}=f\left(\mathrm{x}\left(t\right)\right)\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}x\left(0\right)={\mathrm{x}}_{0}}\end{array}$$(1)

where *f* is a nonlinear vector field and x(*t*) is an *n*-dimensional state vector. The dynamic system (1) can be stated in the fractional derivative form

$$\begin{array}{}{\displaystyle \dot{\mathrm{x}}={{}_{{t}_{0}}^{RL}D}_{t}^{1}\mathrm{x}=f(\mathrm{x}(t))}\end{array}$$

where
$\begin{array}{}{\displaystyle {{}_{{t}_{0}}^{RL}D}_{t}^{1}}\end{array}$
indicates the Riemann-Liouville (R-L) fractional derivative, which for the order *α ∈* ℝ^{+}, *m* – 1 < *α* < *m*, *m* ∈ ℕ, is defined as [29]:

$$\begin{array}{}{\displaystyle {{}_{{t}_{0}}^{RL}D}_{t}^{\alpha}f\left(t\right)={D}^{m}{D}^{-\left(m-\alpha \right)}f\left(t\right)}\\ {\displaystyle \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}=\frac{{d}^{m}}{d{t}^{m}}\left[\frac{1}{\mathit{\Gamma}\left(m-\alpha \right)}\underset{0}{\overset{t}{\int}}\frac{f\left(\tau \right)}{{\left(t-\tau \right)}^{\alpha -m+1}}d\tau \right].}\end{array}$$(2)

For simplicity, we omit the left superscript *RL* and subscript *t*_{0} and assume
$\begin{array}{}{\displaystyle {D}_{t}^{\alpha}}\end{array}$
to be the Riemann-Liouville *α*-order fractional derivative operator that exists and is continuous. It is clear that for the system (1), *δ* *ẋ* = *δf*(*x*(*t*)) =
$\begin{array}{}{\displaystyle \frac{\mathrm{\partial}f}{\mathrm{\partial}x}\delta x,}\end{array}$
the fractional virtual dynamics is represented by

$$\begin{array}{}{\displaystyle {\delta}^{\alpha}{D}_{t}^{1}x={\delta}^{\alpha}f\left(x\left(t\right)\right)={D}_{x}^{\alpha}f\left(x\left(t\right)\right){\left({D}_{x}^{\alpha}x\right)}^{-1}{\delta}^{\alpha}x,}\end{array}$$(3)

where *δ*^{α} *x* and *δ* ^{α} *ẋ* are termed as virtual displacement and virtual velocity respectively [29]. Now, consider an FOS:

$$\begin{array}{}{\displaystyle {D}_{t}^{\alpha}x\left(t\right)=f\left(x\left(t\right)\right).}\end{array}$$(4)

Taking
$\begin{array}{}{\displaystyle {D}_{t}^{1-\alpha}}\end{array}$
in both sides of the dynamical system (4), one can write (see [29] Page 195, [30] Page 29):

$$\begin{array}{}{\displaystyle {D}_{t}^{1}x\left(t\right)=\dot{x}\left(t\right)={D}_{t}^{1-\alpha}f\left(x\left(t\right)\right),}\end{array}$$

now by (3) we have

$$\begin{array}{}{\displaystyle {\delta}^{\alpha}{D}_{t}^{1}x={D}_{x}^{\alpha}{D}_{t}^{1-\alpha}f\left(x\left(t\right)\right){\left({D}_{x}^{\alpha}x\right)}^{-1}{\delta}^{\alpha}x.}\end{array}$$(5)

#### Theorem 1

*If the matrix*
$\begin{array}{}{\displaystyle {D}_{x}^{\alpha}{D}_{t}^{1-\alpha}f\left(x\left(t\right)\right){\left({D}_{x}^{\alpha}x\right)}^{-1}}\end{array}$
*is uniformly negative definite (UND)*, *all the solution trajectories of the system* (4) *converge exponentially to a single trajectory*, *discounting the initial conditions*.

#### Proof

The time derivative of squared distance between the two neighboring trajectories will be

$$\begin{array}{}{\displaystyle {D}_{t}^{1}\left({\left({\delta}^{\alpha}x\right)}^{T}{\delta}^{\alpha}x\right)}\\ {\displaystyle =2{\left({\delta}^{\alpha}x\right)}^{T}{D}_{x}^{\alpha}{D}_{t}^{1-\alpha}f\left(x\left(t\right)\right){({D}_{x}^{\alpha}x)}^{-1}{\delta}^{\alpha}x}\\ {\displaystyle \le 2{\lambda}_{m}(x,t){\left({\delta}^{\alpha}x\right)}^{T}{\delta}^{\alpha}x}\end{array}$$(6)

where *λ*_{m}(*x*,*t*) is the largest eigenvalue of the symmetric part of

$$\begin{array}{}{\displaystyle {D}_{x}^{\alpha}{D}_{t}^{1-\alpha}f\left(x\left(t\right)\right){\left({D}_{x}^{\alpha}x\right)}^{-1}.}\end{array}$$

Therefore, from (6) one can find

$$\begin{array}{}{\displaystyle \u2225{\delta}^{\alpha}x\u2225\le \u2225{\delta}^{\alpha}{x}_{0}\u2225{e}^{{\int}_{0}^{t}{\lambda}_{m}\left(x,t\right)dt}.}\end{array}$$(7)

If *λ *_{m}(*x*,*t*) is strictly uniformly negative then any infinitesimal length ∥*δ*^{α} x∥ converges exponentially to zero as *t* tends to infinity. Thus all the solution trajectories of the system (4) converge exponentially to a single trajectory, irrespective of the initial conditions. □

Note that by a matrix *A* being UND, we mean that

$$\begin{array}{}{\displaystyle \mathrm{\exists}\beta >0\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}such\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}that\phantom{\rule{thinmathspace}{0ex}}\frac{1}{2}\left(A+{A}^{T}\right)\le -\beta \mathrm{I}<0,}\end{array}$$

by convention, the above can be written as *A*≤ – βI < 0.

#### Example 1

Consider the following FOS:

$$\begin{array}{}{\displaystyle {\mathrm{D}}_{t}^{\alpha}\mathrm{x}=u.}\end{array}$$(8)

In order to design a controller *u* that is able to stabilize the system (8), the convergence condition of Theorem 1, must be stablished. Therefore,

$$\begin{array}{}{\displaystyle {D}_{x}^{\alpha}{D}_{t}^{1-\alpha}(u){\left({D}_{x}^{\alpha}x\right)}^{-1}}\end{array}$$

must be UND in the whole state space. There are many possible values of *u* which satisfies the above convergence condition. One can choose

$$\begin{array}{}{\displaystyle u={D}_{t}^{\alpha -1}{D}_{x}^{-\alpha}(-k{x}^{1-\alpha}),}\end{array}$$

where *k* > 0. Substituting the value of *u* in the convergence condition, one can get [29, 30]:

$$\begin{array}{}{\displaystyle {D}_{x}^{\alpha}{D}_{t}^{1-\alpha}(u){\left({D}_{x}^{\alpha}x\right)}^{-1}}\\ {\displaystyle ={D}_{x}^{\alpha}{D}_{t}^{1-\alpha}({D}_{t}^{\alpha -1}{D}_{x}^{-\alpha}(-k{x}^{1-\alpha})){\left({D}_{x}^{\alpha}x\right)}^{-1}}\\ {\displaystyle =\frac{-k{x}^{1-\alpha}}{\frac{{x}^{1-\alpha}}{\mathit{\Gamma}(2-\alpha )}}=-k\mathit{\Gamma}(2-\alpha )<0}\end{array}$$

Therefore, the proposed controller stabilizes the system (8).

This example shows the advantage of the contraction theory for analyzing the stability of fractional order systems and designing the fractional order controller.

#### Example 2

Consider the following nonlinear FOS:

$$\begin{array}{}{\displaystyle {D}_{t}^{\alpha}{x}_{1}=-{x}_{1}+{x}_{1}{x}_{2}}\\ {\displaystyle \phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{D}_{t}^{\alpha}{x}_{2}=-{x}_{1}^{2}-{x}_{2}}\end{array}$$

or, in the matrix equation form:

$$\begin{array}{}\left[\begin{array}{c}{D}_{t}^{\alpha}{x}_{1}\\ {D}_{t}^{\alpha}{x}_{2}\end{array}\right]=\left[\begin{array}{cc}-1& {x}_{1}\\ -{x}_{1}& -1\end{array}\right]\left[\begin{array}{c}{x}_{1}\\ {x}_{2}\end{array}\right].\end{array}$$

One can consider the following virtual *y*-system:

$$\begin{array}{}\left[\begin{array}{c}{D}_{t}^{\alpha}{y}_{1}\\ {D}_{t}^{\alpha}{y}_{2}\end{array}\right]=\left[\begin{array}{cc}-1& {x}_{1}\\ -{x}_{1}& -1\end{array}\right]\left[\begin{array}{c}{y}_{1}\\ {y}_{2}\end{array}\right].\end{array}$$

Since, the matrix

$$\begin{array}{}\left[\begin{array}{cc}-1& {x}_{1}\\ -{x}_{1}& -1\end{array}\right]\end{array}$$

is UND, according to Remark 1, the virtual system is contracting with two particular solutions, namely

$$\begin{array}{}\left[\begin{array}{c}{y}_{1}\\ {y}_{2}\end{array}\right]=\left[\begin{array}{c}0\\ 0\end{array}\right]and\left[\begin{array}{c}{y}_{1}\\ {y}_{2}\end{array}\right]=\left[\begin{array}{c}{x}_{1}\\ {x}_{2}\end{array}\right].\end{array}$$

Since the virtual system is contracting, all the trajectories and especially two particular solutions converge to each other. Therefore, the arbitrary
$\begin{array}{}\left[\begin{array}{c}{x}_{1}\\ {x}_{2}\end{array}\right]\end{array}$
tends to
$\begin{array}{}\left[\begin{array}{c}0\\ 0\end{array}\right]\end{array}$
and the original *x*-system is stable.

One can see that this stability analysis is intuitive and much simpler than finding lyapunov function.

#### Definition 1

Given the FOS
$\begin{array}{}{\displaystyle {D}_{t}^{\alpha}x(t)}\end{array}$
= *f*(*x*(*t*)), a region *Ω* ⊆ ℝ^{n} of the state space is entitled as a contraction (semi-contraction) region, if the matrix
$\begin{array}{}{\displaystyle {D}_{x}^{\alpha}{D}_{t}^{1-\alpha}f\left(x\left(t\right)\right){\left({D}_{x}^{\alpha}x\right)}^{-1}}\end{array}$
is UND (uniformly negative semi-definite) in that region.

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