# Computational Complex Dynamics of
$\displaystyle{f_{\alpha,\beta,\gamma,\delta}(z)=\frac{\alpha z+\beta}{\gamma
z^{2}+\delta z}}$

Sk. Sarif Hassan
_Department of Mathematics_
_College of Engineering Studies_
_University of Petroleum and Energy Studies_
_Bidholi, Dehradun, India_
Email<EMAIL_ADDRESS>

###### Abstract

The dynamics of the family of maps
$\displaystyle{f_{\alpha,\beta,\gamma,\delta}(z)=\frac{\alpha z+\beta}{\gamma
z^{2}+\delta z}}$ in complex plane is investigated computationally. This
dynamical system $z_{n+1}=f_{\alpha,\beta,\gamma,\delta}(z_{n})=\frac{\alpha
z_{n}+\beta}{\gamma z_{n}^{2}+\delta z_{n}}$ has periodic solutions with
higher periods which was absent in the real line scenario. It is also found
that there are chaotic fractal and non-fractal like solutions of the dynamical
systems. A few special cases of parameters are also have been taken care.

Keywords: Family of maps, Local asymptotic stability, Periodicity, Chaotic and
Fractal-like trajectory.

Mathematics Subject Classification: 39A10 & 39A11.

##  1 Brief Review and Results

Consider the family of rational maps
$f_{\alpha,\beta,\gamma,\delta}:\mathbb{C}\rightarrow\mathbb{C}$ of four
complex parameters $\alpha$, $\beta$, $\gamma$ and $\delta$ as

$\displaystyle{f_{\alpha,\beta,\gamma,\delta}(z)=\frac{\alpha z+\beta}{\gamma
z^{2}+\delta z}}$ (1)

The corresponding discrete dynamical system is

$\displaystyle{z_{n+1}=f_{\alpha,\beta,\gamma,\delta}(z_{n})=\frac{\alpha
z_{n}+\beta}{\gamma z_{n}^{2}+\delta z_{n}}}$ (2)

The dynamics of this family of maps Eq.(1) has been studied in [1] when the
parameters and initial condition are non-negative real numbers. Also there are
many works on first, second and third order rational difference equations are
done in [2], [3] [4], [5] and [6]. The main results obtained in the real line
of the dynamical system Eq.(2) are given below:

  * •

$\maltese$ Let $\alpha<\gamma$ and $\beta>\delta$, then the following happens
…

$\spadesuit$ There are infinitely many solutions, such that for each one of
its subsequence $(z_{2n})_{n}$ and $(z_{2n-1})_{n}$, converges to zero and
other diverges to infinity.

$\spadesuit$ There exist solutions which …

$\clubsuit$ converges to zero if $\gamma+\delta>\alpha+\beta$;

$\clubsuit$ diverges to infinity if $\gamma+\delta<\alpha+\beta$;

$\clubsuit$ are constant if $\gamma+\delta=\alpha+\beta$;

  * •

$\maltese$ Let $\alpha=\gamma$ and $\beta>\delta$ then for each positive
solution $(z_{n})_{n}$, one of the subsequences $(z_{2n})_{n}$ and
$(z_{2n-1})_{n}$ diverges to infinity and other to a positive number that can
be arbitrarily large depending on initial values. Further there, are positive
initial values for which the corresponding solution increases monotonically to
infinity.

  * •

$\maltese$ Let $\alpha<\gamma$ and $\beta=\delta$ then for each positive
solution $(z_{n})_{n}$, one of the subsequences $(z_{2n})_{n}$ and
$(z_{2n-1})_{n}$ diverges to zero and other to a nonnegative number. Further
there, are positive initial values for which the corresponding solution
decreases monotonically to zero.

Let us make a note that in the real line set-up, there exists neither any
periodic nor chaotic solutions of the dynamical system of the family of maps.
We shall now see if such kind of solutions exist of the same dynamical system
Eq.(2) in the complex plane.

Here our main aim to investigate computationally the dynamics of the family of
maps (1) under the assumption that the parameters and initial condition are
arbitrary complex numbers. Similar works on rational maps are made in [7] and
[8].

The key dynamics what we have achieved in the complex plane are summarized
here …

  * •

There are three fixed points of the dynamical systems (2) and there are
certain parameters $\alpha,\beta,\gamma$ and $\delta$ (examples are given)
such that those fixed points are stable (sink) and unstable (source).

  * •

The dynamical system (2) possess higher order periodic solutions too (few
examples are given).

  * •

The dynamical system (2) has chaotic (fractal like and fractal-unlike)
solutions.

In the following sections, a detail dynamics of the dynamical system (2) is
characterized.

##  2 Local Asymptotic Stability of the Fixed Points

The fixed points of the family of maps (1) are the solutions of the cubic
equation

$\bar{z}=\frac{\alpha\bar{z}+\beta}{\gamma\bar{z}^{2}+\delta\bar{z}}$

The map (1) has the three fixed points $\bar{z}_{1,2,3}$ …

$-\frac{\delta}{3\gamma}-\frac{2^{1/3}\left(-3\alpha\gamma-\delta^{2}\right)}{3\gamma\left(27\beta\gamma^{2}-9\alpha\gamma\delta-2\delta^{3}+\sqrt{4\left(-3\alpha\gamma-\delta^{2}\right)^{3}+\left(27\beta\gamma^{2}-9\alpha\gamma\delta-2\delta^{3}\right)^{2}}\right)^{1/3}}+\frac{\left(27\beta\gamma^{2}-9\alpha\gamma\delta-2\delta^{3}+\sqrt{4\left(-3\alpha\gamma-\delta^{2}\right)^{3}+\left(27\beta\gamma^{2}-9\alpha\gamma\delta-2\delta^{3}\right)^{2}}\right)^{1/3}}{3\
2^{1/3}\gamma}$,
$-\frac{\delta}{3\gamma}+\frac{\left(1+i\sqrt{3}\right)\left(-3\alpha\gamma-\delta^{2}\right)}{3\
2^{2/3}\gamma\left(27\beta\gamma^{2}-9\alpha\gamma\delta-2\delta^{3}+\sqrt{4\left(-3\alpha\gamma-\delta^{2}\right)^{3}+\left(27\beta\gamma^{2}-9\alpha\gamma\delta-2\delta^{3}\right)^{2}}\right)^{1/3}}-\frac{\left(1-i\sqrt{3}\right)\left(27\beta\gamma^{2}-9\alpha\gamma\delta-2\delta^{3}+\sqrt{4\left(-3\alpha\gamma-\delta^{2}\right)^{3}+\left(27\beta\gamma^{2}-9\alpha\gamma\delta-2\delta^{3}\right)^{2}}\right)^{1/3}}{6\
2^{1/3}\gamma}$,
$-\frac{\delta}{3\gamma}+\frac{\left(1-i\sqrt{3}\right)\left(-3\alpha\gamma-\delta^{2}\right)}{3\
2^{2/3}\gamma\left(27\beta\gamma^{2}-9\alpha\gamma\delta-2\delta^{3}+\sqrt{4\left(-3\alpha\gamma-\delta^{2}\right)^{3}+\left(27\beta\gamma^{2}-9\alpha\gamma\delta-2\delta^{3}\right)^{2}}\right)^{1/3}}-\frac{\left(1+i\sqrt{3}\right)\left(27\beta\gamma^{2}-9\alpha\gamma\delta-2\delta^{3}+\sqrt{4\left(-3\alpha\gamma-\delta^{2}\right)^{3}+\left(27\beta\gamma^{2}-9\alpha\gamma\delta-2\delta^{3}\right)^{2}}\right)^{1/3}}{6\
2^{1/3}\gamma}$ respectively.

The linearized equation of dynamical system Eq.(2) with respect to the a fixed
point $\bar{z_{i}}$ for $i=1,2$ and $3$ is

$\displaystyle{z_{n+1}=f^{\prime}_{\alpha,\beta,\gamma,\delta}(z_{n}),n=0,1,\ldots}$
(3)

######  Theorem 2.1.

The fixed points $\bar{z_{i}}$ for $i=1,2$ and $3$ of the Eq.(2) is …
_locally asymptotically stable_ if

$\left\lvert
f^{\prime}_{\alpha,\beta,\gamma,\delta}(\bar{z_{i}})\right\rvert<1$

_unstable_ if

$\left\lvert
f^{\prime}_{\alpha,\beta,\gamma,\delta}(\bar{z_{i}})\right\rvert>1$

_non-hyperbolic_ if

$\left\lvert
f^{\prime}_{\alpha,\beta,\gamma,\delta}(\bar{z_{i}})\right\rvert=1$

######  Remark 2.1.

It is observed that the minimum value of the $\left\lvert
f^{\prime}_{\alpha,\beta,\gamma,\delta}(\bar{z_{1}})\right\rvert$ is
$6.57147\times 10^{-9}$ which is less than $1$ when the parameters are taken
as
$\alpha=0.816885-0.0738146i,\beta=-0.0245487+0.0678909i,\gamma=-0.360452-0.788031i$
and $\delta=1.89324+0.104191i$.
This numerical observation makes a guarantee that there is parameters
$\alpha$, $\beta$, $\gamma$ and $\delta$ such that $\left\lvert
f^{\prime}_{\alpha,\beta,\gamma,\delta}(\bar{z_{1}})\right\rvert<1$ holds
good. Therefore existence of parameters is ensured for local asymptotic
stability (sink) of the fixed point $\bar{z_{1}}$ of the dynamical system
Eq.(2).

######  Example 2.1.

Let the parameters be
$\alpha=0.92735+0.9174938i,\beta=0.713574+0.618337i,\gamma=0.343287+0.9360273i$
and $\delta=0.124774+0.7305853i$, then the fixed points of the dynamical
system Eq.(2) are
$\\{\bar{z}_{1}=-0.708428+0.171918i\\},\\{\bar{z}_{2}=1.1079-0.305049i\\}$ and
$\\{\bar{z}_{3}=-1.13054-0.00168788i\\}$.
It is noted that the $\left\lvert\alpha\right\rvert=1.3045$,
$\left\lvert\beta\right\rvert=0.9442$, $\left\lvert\gamma\right\rvert=0.9970$,
$\left\lvert\delta\right\rvert=0.7412$,
$\left\lvert\alpha+\beta\right\rvert=2.2475$ and
$\left\lvert\gamma+\delta\right\rvert=1.7311$ and hence
$\left\lvert\alpha\right\rvert>\left\lvert\gamma\right\rvert$,
$\left\lvert\beta\right\rvert>\left\lvert\delta\right\rvert$,
$\left\lvert\alpha+\beta\right\rvert>\left\lvert\gamma+\delta\right\rvert$.
Here $\left\lvert
f^{\prime}_{\alpha,\beta,\gamma,\delta}(\bar{z_{1}})\right\rvert$ is $3.51012$
which is greater than $1$. Hence the fixed point is _unstable (source)_.

The trajectory plots of $10000$ and $40000$ iterations with an initial value
have been figures in Fig. $1$.

---

Figure 1: Trajectory Plots.

It is seen in the Fig. 1 that the trajectory plots are very unstable in nature
while it is converging to the fixed point $-0.708428+0.171918i$.

######  Remark 2.2.

It is observed that the maximum value of the $\left\lvert
f^{\prime}_{\alpha,\beta,\gamma,\delta}(\bar{z_{2}})\right\rvert$ is
$2.70948\times 10^{11}$ which is much greater than $1$ when the parameters are
taken as
$\alpha=-0.0000775242-0.0000567123i,\beta=-1.66869+0.581375i,\gamma=1.33603\times
10^{-6}-6.20472\times 10^{-7}i$ and $\delta=-5.94147-1.73645i$.
This numerical observation confirms that there is parameters $\alpha$,
$\beta$, $\gamma$ and $\delta$ such that $\left\lvert
f^{\prime}_{\alpha,\beta,\gamma,\delta}(\bar{z_{2}})\right\rvert>1$ holds
good. Therefore existence of parameters is ensured for unstable (source)
solution at the fixed point $\bar{z_{2}}$ of the dynamical system Eq.(2).

######  Example 2.2.

Consider $\alpha\to 0.27481+0.24150174i,\beta\to 0.243145+0.154159i,\gamma\to
0.956416+0.935661i,\delta\to 0.818714+0.728261i$ then the fixed points are
$\\{\bar{z_{1}}\to
0.515402-0.0307232i\\},\\{\bar{z_{2}}\to-0.484732+0.0782783i\\},\\{\bar{z_{3}}\to-0.848703-0.00872243i\\}$.
It is noted that the $\left\lvert\alpha\right\rvert=0.3658$,
$\left\lvert\beta\right\rvert=0.2879$, $\left\lvert\gamma\right\rvert=1.3380$,
$\left\lvert\delta\right\rvert=1.0957$,
$\left\lvert\alpha+\beta\right\rvert=0.6518$ and
$\left\lvert\gamma+\delta\right\rvert=2.4330$ and hence
$\left\lvert\alpha\right\rvert<\left\lvert\gamma\right\rvert$,
$\left\lvert\beta\right\rvert<\left\lvert\delta\right\rvert$,
$\left\lvert\alpha+\beta\right\rvert<\left\lvert\gamma+\delta\right\rvert$.
Here $\left\lvert
f^{\prime}_{\alpha,\beta,\gamma,\delta}(\bar{z_{1}})\right\rvert$ is
$0.991066$ which is lesser than $1$. Hence the fixed point is _locally
asymptotically stable (sink)_.

---

Figure 2: Trajectory Plots.

In Fig.$2$, the trajectory plot for an initial value of $1000$ iterations is
given. it is seen in the Fig.$2$ that the trajectory plot is very stable in
nature while it is converging to the fixed point $0.515402-0.0307232i$.

##  3 Verification of Different Criteria

In this section, we shall explore the convergence, divergence and constant
solution conditions for the dynamical system Eq.(2) which are mentioned in the
section $1$.

###  3.1 Verification of Convergence Criteria

Consider the parameters of the dynamical system Eq.(2)
$\alpha=0.27481+0.241501i$ ($\left\lvert\alpha\right\rvert=0.3658$),
$\beta=1.2431+0.1542i$ ($\left\lvert\beta\right\rvert=1.2526$),
$\gamma=0.956416+0.935661i$ ($\left\lvert\gamma\right\rvert=1.3380$) and
$\delta=0.818714+0.728261i$ ($\left\lvert\delta\right\rvert=1.0957$),
$\left\lvert\alpha+\beta\right\rvert=1.5686$ and
$\left\lvert\gamma+\delta\right\rvert=2.4330$.
Here $\left\lvert\alpha\right\rvert<\left\lvert\gamma\right\rvert$,
$\left\lvert\beta\right\rvert<\left\lvert\delta\right\rvert$,
$\left\lvert\alpha+\beta\right\rvert<\left\lvert\gamma+\delta\right\rvert$
which are the necessary conditions to have convergent solution which converge
to zero as stated in the section $1$. But as it can be seen that the solution
in this case in the complex plane is unbounded. The trajectory plots for $20$
different initial values are given in Fig.$3$.

---

Figure 3: Trajectory Plot of Unbounded Solution.

Here none of the twenty trajectories is convergent and converges to zero, all
are unbounded indeed. This does not nullify the result which was in the real
line but it encounters trajectories on contrary in the complex plane.

###  3.2 Verification of Divergence Criteria

Consider the parameters of the dynamical system Eq.(2)
$\alpha=0.917193+0.2858390i$ ($\left\lvert\alpha\right\rvert=0.9607$),
$\beta=1.13764+1.32155i$ ($\left\lvert\beta\right\rvert=1.7438$),
$\gamma=0.993047+0.33978i$ ($\left\lvert\gamma\right\rvert=1.0496$) and
$\delta=0.7572+0.753729i$ ($\left\lvert\delta\right\rvert=1.0684$),
$\left\lvert\alpha+\beta\right\rvert=2.6088$ and
$\left\lvert\gamma+\delta\right\rvert=2.063$.
Here it is found that
$\left\lvert\alpha\right\rvert<\left\lvert\gamma\right\rvert$,
$\left\lvert\beta\right\rvert<\left\lvert\delta\right\rvert$,
$\left\lvert\alpha+\beta\right\rvert>\left\lvert\gamma+\delta\right\rvert$
which are the necessary conditions to have a divergent solution and which
diverges to infinity as stated in the section $1$. Here as it can be seen that
the solution in the complex plane is converging to infinity. The trajectory
plots for $50$ different initial values are given in Fig.$4$.

---

Figure 4: Trajectory Plot of Unbounded Solution.

Here all the fifty trajectories is divergent and diverges to infinity. This
does justify the truth of result in the complex plane.

###  3.3 Verification of Constant Criteria

Consider the parameters of the dynamical system Eq.(2) $\alpha=0.9172+0.2858i$
($\left\lvert\alpha\right\rvert=0.9607$), $\beta=1.1376+1.3216i$
($\left\lvert\beta\right\rvert=1.7438$), $\gamma=3.1376+1.3216i$
($\left\lvert\gamma\right\rvert=3.4046$) and $\delta=-1.0828+0.2858i$
($\left\lvert\delta\right\rvert=1.1199$),
$\left\lvert\alpha+\beta\right\rvert=\left\lvert\gamma+\delta\right\rvert=2.6088$.
Here $\left\lvert\alpha\right\rvert<\left\lvert\gamma\right\rvert$,
$\left\lvert\beta\right\rvert<\left\lvert\delta\right\rvert$,
$\left\lvert\alpha+\beta\right\rvert=\left\lvert\gamma+\delta\right\rvert$
which are the necessary conditions to have a constant solution as stated in
the section $1$.
But here it is seen that the solution in this case in the complex plane is
divergent and diverges to infinity. The trajectory plots for $50$ different
initial values are given in Fig.$5$.

---

Figure 5: Trajectory Plot of Unbounded Solution.

Here none of the fifty trajectories is constant. This does not nullify the
result in the complex plane which was in real case but it encounters
trajectories on its contrary.

##  4 Periodic of Solutions

A solution $\\{z_{n}\\}_{n}$ of a dynamical system is said to be _globally
periodic_ of period $t$ if $z_{n+t}=z_{n}$ for any given initial conditions. A
solution $\\{z_{n}\\}_{n}$ is said to be _periodic with prime period_ $p$ if p
is the smallest positive integer having this property.

As mentioned already that in the real line, there is no mention of periodic
solutions of the dynamical system Eq.(2). Here a computational attempt has
been made in the complex plane scenario to search for the periodic solutions
if it has any.

######  Theorem 4.1.

There does not exists any period two solutions (non-trivial) of the dynamical
system Eq.(2).

###### Proof.

Suppose $\phi$ an $\psi$ are two period 2-cycle solution of the dynamical
system Eq.(2). Therefore by definition of the period point, $\phi$ and $\psi$
satisfy two equations

$\phi=\frac{\beta+\alpha\psi}{\delta\psi+\gamma\phi^{2}},\psi=\frac{\beta+\alpha\phi}{\delta\phi+\gamma\psi^{2}}$

It i trivial to observe that except the fixed points of the map
$f_{\alpha,\beta,\gamma,\delta}(z)=\frac{\alpha z+\beta}{\gamma z^{2}+\delta
z}$, there is no other solution of these system of two linear equations.
Hence there does not exists any non-trivial period $2$-cycle solution of the
Eq.(2). ∎

###  4.1 Example Cases of Periodic Solutions of Different Periods

Example 4.1: Consider the parameters
$\alpha=0.655098003973841+0.162611735194631i$
($\left\lvert\alpha\right\rvert=0.674978$),
$\beta=0.118997681558377+0.498364051982143i$
($\left\lvert\beta\right\rvert=0.512374$), $\gamma=\beta$ and $\delta=\alpha$,
$\left\lvert\alpha+\beta\right\rvert=\left\lvert\gamma+\delta\right\rvert=1.0179$.
Here $\left\lvert\alpha\right\rvert>\left\lvert\gamma\right\rvert$,
$\left\lvert\beta\right\rvert<\left\lvert\delta\right\rvert$,
$\left\lvert\alpha+\beta\right\rvert=\left\lvert\gamma+\delta\right\rvert$.
This set up of parameters produce a periodic solution of period $5$. One of
the periodic 5-cycles is
$\\{0.14044571-0.62692799i,0.02078534+0.517795i,2.994135-1.828805i,0.27873451-0.0515247i,1.8686505+1.9830857i$
and $0.14044571-0.62692799i\\}$. The trajectory plots for five different
initial values are given in the Fig.$6$.

---

Figure 6: Periodic trajectory plot (from 800 to 850 iterations of total 1000
iterations) of period 5.

Example 4.2: Consider the parameters $\alpha=0.1909+0.4283i$
($\left\lvert\alpha\right\rvert=0.4689$), $\beta=0.4820+0.1206i$
($\left\lvert\beta\right\rvert=0.4969$), $\gamma=0.5895+0.2262i$
($\left\lvert\gamma\right\rvert=0.6314$) and $\delta=0.3846+0.5830i$,
($\left\lvert\delta\right\rvert=0.6984$)
$\left\lvert\alpha+\beta\right\rvert=0.8684$
$\left\lvert\gamma+\delta\right\rvert=1.2664$. Here
$\left\lvert\alpha\right\rvert<\left\lvert\gamma\right\rvert$,
$\left\lvert\beta\right\rvert<\left\lvert\delta\right\rvert$,
$\left\lvert\alpha+\beta\right\rvert<\left\lvert\gamma+\delta\right\rvert$.
This set up of parameters produce a periodic solution of period $8$. One of
the periodic 8-cycles is
$\\{0.1245073+0.0238456i,3.7977682-3.81535i,0.0328615+0.1531786i,-1.76641-3.292791i,-0.296000+0.07258225i,-1.224311+1.8140937i,0.08601284-0.129181i,4.8712818+1.8954348i$
and $0.1245073+0.0238456i\\}$. The trajectory plots for five different initial
values are given in the Fig.$7$.

---

Figure 7: Periodic trajectory plot (from 900 to 1000 iterations of total 1000
iterations) of period 8.

Example 4.3: Consider the parameters $\alpha=0.1557+0.8190i$
($\left\lvert\alpha\right\rvert=0.8337$), $\beta=0.6249+0.7386i$
($\left\lvert\beta\right\rvert=0.9675$), $\gamma=0.8051+0.0672i$,
($\left\lvert\gamma\right\rvert=0.8079$) and $\delta=0.9508+0.4976i$,
($\left\lvert\delta\right\rvert=1.0731$)
$\left\lvert\alpha+\beta\right\rvert=1.74224$
$\left\lvert\gamma+\delta\right\rvert=1.8445$.
Here $\left\lvert\alpha\right\rvert>\left\lvert\gamma\right\rvert$,
$\left\lvert\beta\right\rvert<\left\lvert\delta\right\rvert$,
$\left\lvert\alpha+\beta\right\rvert<\left\lvert\gamma+\delta\right\rvert$.
This set up of parameters produce a periodic solution of period $8$. One of
the periodic 13-cycles is
$\\{0.4995-1.9204i,-0.4405+0.5198i,-0.2696-0.6103i,-1.6184+1.5608i,0.1910-0.1647i,1.3640+3.7100i,0.2073-0.0035i,3.7982+2.2357i,0.1717+0.1174i,4.0798-0.3778i,0.0933+0.2274i,2.4590-2.0608i,-0.0609+0.3453i,0.4995-1.9204i\\}$.
The trajectory plots for five different initial values are given in the
Fig.$8$.

---

Figure 8: Periodic trajectory plot (from 1450 to 1500 iterations of total 1500
iterations) of period 13.

Example 4.4: Consider the parameters $\alpha=0.6228+0.7966i$
($\left\lvert\alpha\right\rvert=1.0112$), $\beta=0.7459+0.1255i$
($\left\lvert\beta\right\rvert=0.7564$), $\gamma=0.8224+0.0252i$
($\left\lvert\gamma\right\rvert=0.8228$) and $\delta=0.4144+0.7314i$,
($\left\lvert\delta\right\rvert=0.8407$)
$\left\lvert\alpha+\beta\right\rvert=1.6504$
$\left\lvert\gamma+\delta\right\rvert=1.4499$.
Here $\left\lvert\alpha\right\rvert>\left\lvert\gamma\right\rvert$,
$\left\lvert\beta\right\rvert<\left\lvert\delta\right\rvert$,
$\left\lvert\alpha+\beta\right\rvert>\left\lvert\gamma+\delta\right\rvert$.
This set up of parameters produce a periodic solution of period $8$. One of
the periodic $21$-cycles is
$\\{0.1508-0.2674i,4.2660+1.6796i,0.2370+0.0609i,2.6997-2.6189i,0.0772+0.3937i,-0.3141-1.3571i,-2.2634+2.1519i,0.0920-0.2518i,4.6679+2.2400i,0.2107+0.0451i,3.0966-2.9892i,0.0554+0.3390i,-0.5706-1.5730i,-2.0809+0.6391i,0.1141-0.3835i,3.5201+2.3141i,0.2519+0.0117i,3.2825-2.0879i,0.1469+0.3346i,-0.0137-1.7473i,-0.8858+1.3368i$
and $0.1508-0.2674i\\}$. The trajectory plot for five different initial values
is given in the Fig.$9$.

---

Figure 9: Periodic trajectory plot (from 1970 to 2000 iterations of total 2000
iterations) of period 21.

In the above four different examples, we found different periodic
trajectories/solutions of the dynamical system Eq.(2).
In the _Example 4.1_ , we saw the parameters are obeying
$\left\lvert\alpha\right\rvert>\left\lvert\gamma\right\rvert$,
$\left\lvert\beta\right\rvert<\left\lvert\delta\right\rvert$ and
$\left\lvert\alpha+\beta\right\rvert=\left\lvert\gamma+\delta\right\rvert$
conditions which did not arise in real scenario.
In _Example 4.2_ , the parameters are satisfying
$\left\lvert\alpha\right\rvert<\left\lvert\gamma\right\rvert$,
$\left\lvert\beta\right\rvert<\left\lvert\delta\right\rvert$ and
$\left\lvert\alpha+\beta\right\rvert<\left\lvert\gamma+\delta\right\rvert$
which condition did not appear in the real set up.
In _Example 4.3_ , the parameters are satisfying
$\left\lvert\alpha\right\rvert>\left\lvert\gamma\right\rvert$,
$\left\lvert\beta\right\rvert<\left\lvert\delta\right\rvert$ and
$\left\lvert\alpha+\beta\right\rvert<\left\lvert\gamma+\delta\right\rvert$
which condition did not appear in the real set up.
In _Example 4.4_ , the parameters are satisfying
$\left\lvert\alpha\right\rvert>\left\lvert\gamma\right\rvert$,
$\left\lvert\beta\right\rvert<\left\lvert\delta\right\rvert$ and
$\left\lvert\alpha+\beta\right\rvert>\left\lvert\gamma+\delta\right\rvert$
which condition did not appear in the real set up.

It is to be noted that in all the above four examples, the parameters satisfy
the $\left\lvert\alpha\right\rvert>\left\lvert\gamma\right\rvert$ or
$\left\lvert\beta\right\rvert<\left\lvert\delta\right\rvert$ conditions which
did not arise in real scenario [1].

##  5 Slow and Fast Convergence

Here we encounter two solutions of the Eq.(2), one of which is very slow in
converging to the fixed point and another one is very fast inc converging to
the fixed point of the dynamical system.
Here the parameters
$\alpha=0.9274+0.9175i,(\left\lvert\alpha\right\rvert=1.3045)$,
$\beta=0.7136+0.6183i,(\left\lvert\beta\right\rvert=0.9442)$,
$\gamma=0.3433+0.9360i,(\left\lvert\gamma\right\rvert=0.9970)$,
$\delta=0.1248+0.7306i,(\left\lvert\delta\right\rvert=0.7412)$ and
$\left\lvert\alpha+\beta\right\rvert=2.247$ and
$\left\lvert\gamma+\delta\right\rvert=1.7310$. That is the parameters are
satisfying $\left\lvert\alpha\right\rvert<\left\lvert\gamma\right\rvert$,
$\left\lvert\beta\right\rvert>\left\lvert\delta\right\rvert$ and
${\left\lvert\alpha+\beta\right\rvert>\left\lvert\gamma+\delta\right\rvert}$
which was the necessary creation for convergence to _zero_ in the real set up
but here under this set of parameters the solution are convergent for any
initial values to $1.10789-0.305048i$ (_not zero_!) with about $10^{5}$ number
of iterations which is slow indeed.
The trajectory plots with ten different arbitrary initial values are given in
the following figure Fig.$10$.

---

Figure 10: Slow convergent trajectory plot.

Here it is noted that the trajectory is unstable for any initial values since
it the condition is $\left\lvert
f^{\prime}_{\alpha,\beta,\gamma,\delta}(\bar{z_{i}})\right\rvert=7.45235>1$
satisfied.

Here the parameters
$\alpha=0.2748+0.2415i,(\left\lvert\alpha\right\rvert=0.3658)$,
$\beta=0.2431+0.1542i,(\left\lvert\beta\right\rvert=0.2879)$,
$\gamma=0.9564+0.9357i,(\left\lvert\gamma\right\rvert=1.3380)$,
$\delta=0.8187+0.7283i,(\left\lvert\delta\right\rvert=1.0957)$ and
$\left\lvert\alpha+\beta\right\rvert=0.6517$ and
$\left\lvert\gamma+\delta\right\rvert=2.433$. That is the parameters are
satisfying $\left\lvert\alpha\right\rvert>\left\lvert\gamma\right\rvert$,
$\left\lvert\beta\right\rvert<\left\lvert\delta\right\rvert$ and
${\left\lvert\alpha+\beta\right\rvert<\left\lvert\gamma+\delta\right\rvert}$
which condition did not arise at all in real scenario but here under this set
of parameters, the solutions are convergent for any initial values to
$0.515394-0.03072$ with about $10^{3}$ number of iterations which is fast
indeed. The trajectory plots with ten different arbitrary initial values are
given in the following figure Fig.$11$.

---

Figure 11: Fast convergent trajectory plot.

Here it is noted that the all the ten trajectories are unstable since it the
condition is $\left\lvert
f^{\prime}_{\alpha,\beta,\gamma,\delta}(\bar{z_{i}})\right\rvert=4.85602>1$
satisfied.

It is interesting to note that in both cases above, the trajectories are
unstable as $\left\lvert
f^{\prime}_{\alpha,\beta,\gamma,\delta}(\bar{z_{i}})\right\rvert>1$ but in one
case the convergence is slow and in another case is fast.

##  6 Chaotic Solutions

Here we have explored the existence of chaotic solutions of the dynamical
system Eq.(2). Computationally, some chaotic solutions of the dynamical system
Eq.(2) for some parameters $\alpha$, $\beta$, $\gamma$ and $\delta$ which are
given in the following Table $1$ are fetched.
The largest Lyapunov exponent is calculated for all such solutions of the
dynamical system Eq.(1) numerically [9] to show the trajectories are chaotic.
From computational evidence, it is arguable that for complex parameters
$\alpha$, $\beta$, $\gamma$ and $\delta$ which are stated in the following
Table $1$, the solutions are chaotic for any initial values.

Parameters |  Parameters Conditions |  Lyapunav exponent (Fractal like/unlike
---|---|---
$\alpha=0.6849+0.2083i$, $\beta=0.6082+0.3262i$, $\gamma=0.8808+0.1334i$, $\delta=0.1024+0.9591i$, $\left\lvert\alpha\right\rvert=0.7158$, $\left\lvert\beta\right\rvert=0.6901$, $\left\lvert\gamma\right\rvert=0.8909$, $\left\lvert\delta\right\rvert=0.9646$, $\left\lvert\alpha+\beta\right\rvert=1.3991$ & $\left\lvert\gamma+\delta\right\rvert=1.4698$ |  $\left\lvert\alpha\right\rvert<\left\lvert\gamma\right\rvert$, $\left\lvert\beta\right\rvert<\left\lvert\delta\right\rvert$, $\left\lvert\alpha+\beta\right\rvert<\left\lvert\gamma+\delta\right\rvert$ |  Lyapunav Exponent:$1.3342$
(Fig.12: Fractal-unlike) |  |
$\alpha=0.8491+0.9340i$, $\beta=0.6787+0.7577i$, $\gamma=0.7431+0.3922i$, $\delta=0.6555+0.1712i$, $\left\lvert\alpha\right\rvert=1.2623$, $\left\lvert\beta\right\rvert=1.0173$, $\left\lvert\gamma\right\rvert=0.8403$, $\left\lvert\delta\right\rvert=0.6775$, $\left\lvert\alpha+\beta\right\rvert=2.2795$ & $\left\lvert\gamma+\delta\right\rvert=1.5078$ |  $\left\lvert\alpha\right\rvert>\left\lvert\gamma\right\rvert$, $\left\lvert\beta\right\rvert>\left\lvert\delta\right\rvert$, $\left\lvert\alpha+\beta\right\rvert>\left\lvert\gamma+\delta\right\rvert$ |  Lyapunav Exponent: $1.4765$, Fractal dimension:$1.3435$
(Fig.13: Fractal-like) |  |
$\alpha=0.9322+0.8351i$, $\beta=0.8954+0.5825i$, $\gamma=0.5827+0.8549i$, $\delta=0.0349+0.8854i$, $\left\lvert\alpha\right\rvert=1.2515$, $\left\lvert\beta\right\rvert=1.0682$, $\left\lvert\gamma\right\rvert=1.0346$, $\left\lvert\delta\right\rvert=0.8861$, $\left\lvert\alpha+\beta\right\rvert=2.3130$ & $\left\lvert\gamma+\delta\right\rvert=1.8467$ |  $\left\lvert\alpha\right\rvert>\left\lvert\gamma\right\rvert$, $\left\lvert\beta\right\rvert>\left\lvert\delta\right\rvert$, $\left\lvert\alpha+\beta\right\rvert>\left\lvert\gamma+\delta\right\rvert$ |  Lyapunav Exponent: $1.6225$, Fractal dimension:$1.4235$
(Fig.14: Fractal-like) |  |
$\alpha=0.5078+0.5856i$, $\beta=0.7629+0.0830i$, $\gamma=0.6616+0.5170i$, $\delta=0.1710+0.9386i$, $\left\lvert\alpha\right\rvert=0.7751$, $\left\lvert\beta\right\rvert=0.7674$, $\left\lvert\gamma\right\rvert=0.8396$, $\left\lvert\delta\right\rvert=0.9540$, $\left\lvert\alpha+\beta\right\rvert=1.4359$ & $\left\lvert\gamma+\delta\right\rvert=1.6769$ |  $\left\lvert\alpha\right\rvert<\left\lvert\gamma\right\rvert$, $\left\lvert\beta\right\rvert<\left\lvert\delta\right\rvert$, $\left\lvert\alpha+\beta\right\rvert<\left\lvert\gamma+\delta\right\rvert$ |  Lyapunav Exponent: $1.4872$
(Fig.15: Fractal-unlike) |  |

Table 1: Chaotic trajectories of the equation Eq.(2) for different choice of
parameters and ten set of initial values.

The chaotic trajectory plots including corresponding complex plots of four
examples whose parameters are given in Table $1$ starting from top row are
given the following Fig.$12$, Fig.$13$, Fig.$14$ and Fig. $15$ respectively.

---

Figure 12: Chaotic Trajectories of the equation Eq.(2) (Refer Row-1 of
Table-1) with 10 set of initial values (10 different colors of trajectories).

---

Figure 13: Chaotic Trajectories of the equation Eq.(2) (Refer Row-2 of
Table-1) with 10 set of initial values (10 different colors of trajectories).

---

Figure 14: Chaotic Trajectories of the equation Eq.(2) (Refer Row-3 of
Table-1) with 10 set of initial values (10 different colors of trajectories).

---

Figure 15: Chaotic Trajectories of the equation Eq.(2) (Refer Row-4 of
Table-1) with 10 set of initial values (10 different colors of trajectories).

It is needless to mention that the above fractal-like and fractal-unlike
trajectories are chaotic as they have largest Lyapunav exponent as positive
which are shown in the Table $1$. The fractal-like trajectories are proven to
be fractals by the fractal dimension which are fractional.

The fractal-unlike chaotic trajectories are plotted for $10$ different initial
values in the Fig. $12$, and Fig.$15$. The fractal-like chaotic trajectories
are plotted for ten initial values in the Fig.$13$ and Fig. $14$. It is
observed that in the in the case of fractal-like trajectories, the parameters
follow the following conditions:
$\left\lvert\alpha\right\rvert<\left\lvert\gamma\right\rvert$,
$\left\lvert\beta\right\rvert<\left\lvert\delta\right\rvert$,
$\left\lvert\alpha+\beta\right\rvert<\left\lvert\gamma+\delta\right\rvert$ and
in fractal unlike trajectories, the parameters follow the conditions:
$\left\lvert\alpha\right\rvert>\left\lvert\gamma\right\rvert$,
$\left\lvert\beta\right\rvert>\left\lvert\delta\right\rvert$,
$\left\lvert\alpha+\beta\right\rvert>\left\lvert\gamma+\delta\right\rvert$. We
also perform a set of 500 such chaotic trajectories are studied and it is
found that the parameters corresponding to the fractal-like chaotic
trajectories are following the condition:
$\left\lvert\alpha\right\rvert<\left\lvert\gamma\right\rvert$,
$\left\lvert\beta\right\rvert<\left\lvert\delta\right\rvert$,
$\left\lvert\alpha+\beta\right\rvert<\left\lvert\gamma+\delta\right\rvert$ and
the same for fractal-unlike chaotic trajectories are following the condition:
$\left\lvert\alpha\right\rvert>\left\lvert\gamma\right\rvert$,
$\left\lvert\beta\right\rvert>\left\lvert\delta\right\rvert$,
$\left\lvert\alpha+\beta\right\rvert>\left\lvert\gamma+\delta\right\rvert$
(data not shown). From these two computational observations, the following
conjecture has been made.

######  Conjecture 6.1.

The chaotic trajectories Eq.(2) are fractal-like only if the parameters
satisfy the conditions:
$\left\lvert\alpha\right\rvert<\left\lvert\gamma\right\rvert$,
$\left\lvert\beta\right\rvert<\left\lvert\delta\right\rvert$,
$\left\lvert\alpha+\beta\right\rvert<\left\lvert\gamma+\delta\right\rvert$ and
fractal-unlike only if
$\left\lvert\alpha\right\rvert>\left\lvert\gamma\right\rvert$,
$\left\lvert\beta\right\rvert>\left\lvert\delta\right\rvert$,
$\left\lvert\alpha+\beta\right\rvert>\left\lvert\gamma+\delta\right\rvert$

##  7 A Few Special Cases of Parameters

Here we shall explore a few case study by considering some restrictions on the
parameters $\alpha$, $\beta$, $\gamma$ and $\delta$ as follows.

###  7.1 $\alpha=\beta,\gamma=\delta$

Consider $\alpha=\beta$ and $\gamma=\delta$ in the dynamical system Eq.(2) and
then it becomes

$\displaystyle{z_{n+1}=f_{\alpha,\gamma}(z_{n})=\frac{\alpha(z_{n}+1)}{\gamma(z_{n}^{2}+z_{n})}}=\frac{\alpha}{\gamma
z_{n}}$ (4)

The Eq.(4) has two fixed points $z_{\pm}^{*}=$
$\pm\sqrt{\frac{\alpha}{\gamma}}$ and both of which are neither stable nor
unstable since the $\left\lvert f^{\prime}(z_{\pm}^{*})\right\rvert=1$. The
Eq.(4) has periodic solutions of period $2$ only and they are
$\phi=\frac{\alpha}{\gamma\psi}$ and $\psi=\frac{\alpha}{\gamma\phi}$. It is
noted that there is no nontrivial periodic points of higher periods. For $50$
different initial values the trajectories are plotted in the following Fig.
$16$.

---

Figure 16: 50 Period 2 Trajectories of the equation Eq.(4).

###  7.2 $\alpha=-\beta,\gamma=\delta$

Consider $\alpha=-\beta$ and $\gamma=\delta$ in the dynamical system Eq.(2)
and then it becomes

$\displaystyle{z_{n+1}=f_{\alpha,\gamma}(z_{n})=\frac{\alpha(z_{n}-1)}{\gamma(z_{n}^{2}+z_{n})}}$
(5)

For arbitrary any initial value, the trajectory is convergent and converge to
$\infty$ which is a member of the extended complex plane $\mathbb{C}^{*}$
which is shown in the Fig.$17$.

---

Figure 17: Trajectory plots of the equation Eq.(5).

###  7.3 $\gamma=\alpha,\delta=\beta$

Consider $\gamma=\alpha$ and $\delta=\beta$ in the dynamical system Eq.(2) and
then it becomes

$\displaystyle{z_{n+1}=f_{\alpha,\beta}(z_{n})=\frac{\alpha
z_{n}+\beta}{\alpha z_{n}^{2}+\beta z_{n}}}$ (6)

The Eq.(6) has two fixed points $z_{\pm}^{*}=$ $\pm 1$ and both of which is
neither stable nor unstable since the $\left\lvert
f^{\prime}(z_{\pm}^{*})\right\rvert=1$ but the Eq.(6) possess the period $2$
solution which are reciprocal to each other. For any initial values (except
$-\frac{\beta}{\alpha}$) all solutions of the Eq.(6) are convergent and
converge to the period two solution.

###  7.4 $\gamma=\beta,\delta=\alpha$

Consider $\gamma=\beta$ and $\delta=\alpha$ in the dynamical system Eq.(2) and
then it becomes

$\displaystyle{z_{n+1}=f_{\alpha,\beta}(z_{n})=\frac{\alpha z_{n}+\beta}{\beta
z_{n}^{2}+\alpha z_{n}}}$ (7)

The Eq.(7) has three fixed points $z^{*}=1$,
$z^{*}=\frac{-0.5\alpha-0.5\beta-0.5\sqrt{\alpha^{2}+2\alpha\beta-3\beta^{2}}}{\beta}$
and
$z^{*}=\frac{0.5\left(-\alpha-\beta+\sqrt{\alpha^{2}+2\alpha\beta-3\beta^{2}}\right)}{\beta}$.The
local asymptotically stability of the fixed point $1$ depends on $\left\lvert
f^{\prime}(z^{*})\right\rvert=\frac{2\beta}{\alpha+\beta}$. So it turns out
that if
$2\left\lvert\beta\right\rvert<\left\lvert\alpha+\beta\right\rvert<\left\lvert\alpha\right\rvert+\left\lvert\beta\right\rvert$
i.e. $\left\lvert\beta\right\rvert<\left\lvert\alpha\right\rvert$ then the
fixed point $1$ is a sink and otherwise it is a source. If
$\left\lvert-2-\frac{\alpha}{\beta}\right\rvert<1$ i.e.
$\left\lvert\alpha+2\beta\right\rvert<\left\lvert\beta\right\rvert$ then the
fixed points
$z^{*}=\frac{-0.5\alpha-0.5\beta-0.5\sqrt{\alpha^{2}+2\alpha\beta-3\beta^{2}}}{\beta}$
and
$z^{*}=\frac{0.5\left(-\alpha-\beta+\sqrt{\alpha^{2}+2\alpha\beta-3\beta^{2}}\right)}{\beta}$
would be a sink. Beside the regular dynamics the Eq.(7) possess chaotic
solutions. Here is an example of chaotic solutions for $10$ different initial
values. Here $\alpha=\delta=28+68i$ and $\beta=\gamma=66+17i$ and
$\left\lvert\alpha\right\rvert=73.5391$,
$\left\lvert\beta\right\rvert=68.1542$ and
$\left\lvert\alpha+\beta\right\rvert=\left\lvert\gamma+\delta\right\rvert=126.7320$
i.e. $\left\lvert\alpha\right\rvert>\left\lvert\gamma\right\rvert$
$\left\lvert\beta\right\rvert<\left\lvert\delta\right\rvert$ and
$\left\lvert\alpha+\beta\right\rvert=\left\lvert\gamma+\delta\right\rvert$.
The chaotic solution is plotted in the Fig. $18$.

---

Figure 18: Chaotic Trajectory plots of the equation Eq.(7).

##  8 Future Endeavours

It is seen that the dynamical system Eq.(2) exhibits a phenomenal jump in
terms of dynamical behaviour from real line to complex plane. In continuation
of the present work the study of the dynamical system with time delay $l$,
${z_{n+1}=\frac{\alpha_{n}z_{n-l}+\beta_{n}}{\gamma_{n}z_{n-l}^{2}+\delta_{n}z_{n-l}}}$
where $\alpha_{n}$, $\beta_{n}$, $\gamma_{n}$ and $\delta_{n}$ are all
convergent sequence of complex numbers and converges to $\alpha$, $\beta$,
$\gamma$ and $\delta$ respectively would indeed be an interesting problem and
which we would like to pursue further.

## References

  * [1] D. M. Chan, C. M. Kent, and N. L. Ortiz-Robinson, (2009) _Convergence Results on a Second-Order Rational Difference Equation with Quadratic Terms_ , Advances in Difference Equations, Article ID 985161, 7 pages.
  * [2] Saber N Elaydi, Henrique Oliveira, José Manuel Ferreira and João F Alves, (2007) _Discrete Dyanmics and Difference Equations, Proceedings of the Twelfth International Conference on Difference Equations and Applications_ , World Scientific Press.
  * [3] W. A. Kosmala, M. R. S. Kulenovic, G. Ladas, and C. T. Teixeira, (2000), On the Recursive Sequence $y_{n+1}=\frac{p+y_{n-1}}{qy_{n}+y_{n-1}}$, _Journal of Mathematical Analysis and Applications_ 251, 571-58.6
  * [4] E. Camouzis, E. Chatterjee, G. Ladas and E. P. Quinn, (2004) On Third Order Rational Difference Equations – Open Problems and Conjectures, _Journal of Difference Equations and Applications_ , 10, 1119 – 1127.
  * [5] R. Abo-Zeid, (2015) On the oscillation of a third order rational difference equation, _Journal of the Egyptian Mathematical Society_ , 23, 62-66.
  * [6] M.R.S. Kulenovi$\acute{c}$ and G. Ladas, (2001) _Dynamics of Second Order Rational Difference Equations; With Open Problems and Conjectures_ , Chapman & Hall/CRC Press.
  * [7] Sk. S. Hassan, E. Chatterjee, (2015) Dynamics of the equation $\displaystyle{z_{n+1}=\frac{\alpha+\beta z_{n}}{A+z_{n-1}}}$ in the Complex Plane, _Cogent Mathematics, Taylor and Francis_ $2$, $1-12$.
  * [8] Sk. S. Hassan, (2016) _Dynamics of $\displaystyle{z_{n+1}=\frac{\alpha+\alpha z_{n}+\beta z_{n-1}}{1+z_{n}}}$ in Complex Plane_, Communicated.
  * [9] A. Wolf, J. B. Swift, H. L. Swinney and J. A. Vastano, (1985) Determining Lyapunov exponents from a time series _Physica D_ , 126, 285-317.


# Sliding Mode Observer for Set-valued Lur’e Systems and Chattering Removing

Samir Adly Ba Khiet Le Laboratoire XLIM, Université de Limoges, 123 Avenue
Albert Thomas, 87060 Limoges CEDEX, France.Email:
<EMAIL_ADDRESS>Research Group, Faculty of Mathematics and
Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam.E-mail:
<EMAIL_ADDRESS>
###### Abstract

In this paper, we study a sliding mode observer for a class of set-valued
Lur’e systems subject to uncertainties. We show that our approach has obvious
advantages than the existing Luenberger-like observers. Furthermore, we
provide an effective continuous approximation to eliminate the chattering
effect in the sliding mode technique.

Keywords. Sliding mode observer; set-valued Lur’e systems; chattering effect

AMS Subject Classification. 28B05, 34A36, 34A60, 49J52, 49J53, 93D20

##  1 Introduction

Hybrid systems are a class of dynamic systems that may present both continuous
and discrete behavior. They are characterized by the presence of continuous
state variables, inputs/outputs, as well as discrete state variables,
inputs/outputs. On the other hand, Lur’e-type nonlinear systems [21] are
represented by the combination of a linear time-invariant system and a
memoryless nonlinearity through a feedback connection. The link between Lur’e
systems and hybrid systems lies in the coupling of the continuous and discrete
dynamics through the nonlinear state equation and the linear output equation.
This coupling makes Lur’e systems well-suited for modeling complex physical,
biological, or social systems that exhibit both continuous and discrete
behavior, and have found applications in areas such as control systems, signal
processing, and system identification. For a comprehensive guide to hybrid
dynamical systems from the modeling, stability and robustness point of view,
we refer to [14]. In this paper we focus on Lur’e set-valued dynamical systems
where the nonlinear feedback is given by a set-valued relation. The Lur’e set-
valued dynamical system is a widely studied model in applied mathematics and
control theory. In recent decades, it has received significant attention, as
evidenced by the numerous studies found in references such as [1, 2, 3, 7, 8,
9, 10, 11, 13, 15, 16, 17, 18, 19, 20, 26]. Despite advancements in the
understanding of the system’s existence and uniqueness, stability and
asymptotic analysis, the design of control and observers still poses many
interesting open questions. Specifically, most observers for Lur’e set-valued
systems follow the Luenberger design, which has limitations when the system is
subjected to uncertainty, as discussed in references such as [15, 16, 17, 26].
Recently, using the powerful sliding mode technique, B. K. Le in [20] proposed
a sliding mode observer for a general class of set-valued Lur’e systems
subject to uncertainties as follows

$\left\\{\begin{array}[]{l}\dot{x}(t)=Ax(t)+B\lambda(t)+Eu(t)+G\xi(t,u,x)\;{\rm
for\;a.e.}\;t\in[0,+\infty),\\\ \\\ w(t)=Cx(t)+D\lambda(t),\\\ \\\
\lambda(t)\in-\mathcal{F}_{t}(w(t)),\;t\geq 0,\\\ \\\ y(t)=Fx(t),\\\ \\\
x(0)=x_{0},\end{array}\right.$ (1)

where $A\in{\mathbb{R}}^{n\times n},B\in{\mathbb{R}}^{n\times
m},C\in{\mathbb{R}}^{m\times n},D\in{\mathbb{R}}^{m\times
m},E\in{\mathbb{R}}^{n\times l},F\in{\mathbb{R}}^{p\times
n},G\in{\mathbb{R}}^{n\times k}$ are given matrices, $\lambda$ and $w$ are two
connecting variables, $u$ is the control input, $y$ is the physically
measurable output and $\xi$ is some uncertainty. The set-valued operator
$\mathcal{F}_{t}:{\mathbb{R}}^{m}\rightrightarrows{\mathbb{R}}^{m}$ is time-
dependent and assumed to be maximal monotone. Note that $(1)$ can be rewritten
into a first order time-dependent differential inclusion as follows

$\dot{x}\in Ax-B(\mathcal{F}_{t}^{-1}+D)^{-1}Cx+Eu+G\xi,\;\;x(0)=x_{0}.$ (2)

If $D=0$ and $\mathcal{F}_{t}=\mathcal{F}$, we obtain the following classical
case

$\dot{x}\in Ax-B\mathcal{F}Cx+Eu+G\xi.$

The proposed sliding mode observer for $(1)$ is

$\left\\{\begin{array}[]{l}\dot{\tilde{x}}\in
A\tilde{x}+B\tilde{\lambda}-Le_{y}+Eu-P^{-1}F^{T}\Psi(e_{y}),\\\ \\\
\tilde{w}=C\tilde{x}+D\tilde{\lambda},\\\ \\\
\tilde{\lambda}\in-\mathcal{F}(\tilde{w}+Ke_{y}),\\\ \\\
\tilde{y}=F\tilde{x},\end{array}\right.$ (3)

where

$e=\tilde{x}-x,e_{y}=\tilde{y}-y=Fe,$
$\displaystyle\Psi(e_{y})=\sigma_{1}e_{y}+(\|J\|\ \rho(t,u,y)+\sigma_{2}){\rm
Sign}(e_{y})$

and $\rho$ is a bound of the uncertainty for some suitable matrix $J$ and real
number $\sigma_{1}\geq 0,\sigma_{2}>0$. Then under some mild conditions, the
observer state converges to the original state asymptotically and the
observation error converges to zero in finite time (we refer to [20] for more
details).

In this paper, our first contribution is to provide a sliding mode observer
for the system considered in [15] as follows

$\left\\{\begin{array}[]{lll}\;\dot{x}=Ax+B\omega+f_{1}(x,u)+f_{2}(x,u)\theta,\\\
&&\\\ \;\omega\in-\mathcal{F}(Cx),\\\ &&\\\ \;y=Fx,\end{array}\right.$ (4)

where $A\in{\mathbb{R}}^{n\times n},B\in{\mathbb{R}}^{n\times
m},C\in{\mathbb{R}}^{m\times n},F\in{\mathbb{R}}^{p\times n}$ are given
matrix, $x\in{\mathbb{R}}^{n}$ is the state,
$\mathcal{F}:{\mathbb{R}}^{m}\rightrightarrows{\mathbb{R}}^{m}$ is a maximal
monotone operator, $u\in{\mathbb{R}}^{r}$ is the control input and
$y\in{\mathbb{R}}^{p}$ is the measurable output. The functions $f_{1},f_{2}$
are known smooth while $\theta\in{\mathbb{R}}^{l}$ is an unknown constant
parameter vector. An adaptive Luenberger-like observer was proposed by Huang
et al in [15]. In our paper, the parameter $\theta$ may not always be
constant, such as in the case of perturbations. Our sliding mode approach is
efficient and straightforward, without the need to solve an additional
ordinary differential equation, as seen in [15]. Our results show that the
convergence of the observer state to the actual state is exponential, while
the rate of convergence in [15] is unknown. Additionally, if the matrix
function $f_{2}$ is bounded, the conditions for solvability of the associated
LMI are significantly improved, leading to finite time convergence of the
observation error under more favorable assumptions than in [20]. Our
contributions also extend to the reduced-order case, where we have improved
conditions. Lastly, we propose a new smooth approximation of the sliding mode
technique that reduces the chattering effect while still effectively managing
uncertainty.

The paper is organized as follows. In Section 2, we revisit some key concepts.
In Section 3, we introduce a sliding mode observer for (4) and demonstrate
that it holds more benefits compared to the Luenberger-like observer discussed
in [15]. Section 4 focuses on the reduced-order observer. In Section 5, we
present a new and refined version of the sliding mode technique. Numerical
examples to support the results are provided in Section 6. Finally, Section 7
concludes the paper and highlights future possibilities.

##  2 Notations and mathematical background

We denote the scalar product and the corresponding norm of Euclidean spaces by
$\langle\cdot,\cdot\rangle$ and $\|\cdot\|$ respectively. A matrix
$P\in{\mathbb{R}}^{n\times n}$ is called positive definite, written $P>0$, if
there exists $a>0$ such that

$\langle Px,x\rangle\geq a\|x\|^{2},\;\;\forall\;x\in{{\mathbb{R}}^{n}}.$

The Sign function in ${\mathbb{R}}^{m}$ is defined by

${\rm Sign}(x)=\left\\{\begin{array}[]{l}\frac{x}{\|x\|}\;\;\;\;{\rm
if}\;\;\;\;x\neq 0\\\ \\\ \mathbb{B}\;\;\;\;\;\;\;{\rm
if}\;\;\;\;\;x=0,\end{array}\right.$

where $\mathbb{B}$ denotes the unit ball in ${\mathbb{R}}^{m}$.

Figure 1: Sign function in ${\mathbb{R}}$

A set-valued mapping
$\mathcal{F}:{\mathbb{R}}^{m}\rightrightarrows{\mathbb{R}}^{m}$ is called
$\it{monotone}$ if for all $x,y\in{\mathbb{R}}^{m}$ and
$x^{*}\in\mathcal{F}(x),y^{*}\in\mathcal{F}(y)$, one has

$\langle x^{*}-y^{*},x-y\rangle\geq 0.$

Furthermore, $\mathcal{F}$ is called $\it{maximal\;monotone}$ if there is no
monotone operator $\mathcal{G}$ such that the graph of $\mathcal{F}$ is
contained strictly in the graph of $\mathcal{G}.$

######  Lemma 1.

[20] Let $F\in{\mathbb{R}}^{p\times n}$ be a full row rank matrix ($p\leq n$)
and $P\in{\mathbb{R}}^{n\times n}$ be a symmetric positive definite matrix. If
$x\in{\rm im}(P^{-1}F^{T})$ then

$F^{T}(FP^{-1}F^{T})^{-1}Fx=Px,$ (5)

where ${\rm im}(A)$ denotes the range of $A$.

##  3 Exponential convergence of the sliding mode observer

In this section, we will propose a sliding mode observer for the system (4)
under the following assumptions:

Assumption 1:  The set-valued operator
$\mathcal{F}:{\mathbb{R}}^{n}\rightrightarrows{\mathbb{R}}^{n}$ is a monotone,
upper semi-continuous with non-empty, closed convex and bounded values.

Assumption 2:  The functions
$f_{1}:{\mathbb{R}}^{n}\times{\mathbb{R}}^{r}\to{\mathbb{R}}^{n}$ and
$f_{2}:{\mathbb{R}}^{n}\times{\mathbb{R}}^{r}\to{\mathbb{R}}^{n\times l}$ are
Lipschitz continuous w.r.t $x$, i. e., there exist $L_{1}>0$ and $L_{2}>0$
such that for all $x_{1},x_{2}\in{\mathbb{R}}^{n},u\in{\mathbb{R}}^{r}$, we
have

$\|f_{1}(x_{1},u)-f_{2}(x_{2},u)\|\leq L_{1}\|x_{1}-x_{2}\|\;\;{\rm
and}\;\;\|f_{2}(x_{1},u)-f_{2}(x_{2},u)\|\leq L_{2}\|x_{1}-x_{2}\|.$

Assumption 3:  The unknown $\theta(t,x,u)$ is bounded by $L_{3}>0$.

Assumption 4:  Let $\gamma=L_{1}+L_{2}L_{3}$. There exist $\epsilon>0$,
$P\in{\mathbb{R}}^{n\times n}>0$, $L\in{\mathbb{R}}^{n\times p}$,
$K\in{\mathbb{R}}^{m\times p}$ and the matrix function
$h:{\mathbb{R}}^{n}\times{\mathbb{R}}^{r}\to{\mathbb{R}}^{l\times p}$ such
that

$\displaystyle P(A-LF)+(A-LF)^{T}P+\gamma P^{2}+\gamma I+\epsilon I\leq 0,$
(6) $\displaystyle B^{T}P=C-KF,$ (7) $\displaystyle f_{2}^{T}(x,u)P=h(x,u)F.$
(8)

######  Remark 1.

From $(\ref{f2p})$, we have $f_{2}(x,u)=P^{-1}F^{T}h^{T}(x,u)$. It means that
${\rm im}(f_{2}(x,u))\in{\rm im}(P^{-1}F^{T})$ for all $(x,u)$.

The proposed sliding mode observer for (4) is

$\left\\{\begin{array}[]{lll}\;\dot{\tilde{x}}=A\tilde{x}+B\tilde{\omega}-Le_{y}+f_{1}(\tilde{x},u)-\beta
P^{-1}F^{T}\|h(\tilde{x},u)\|{\rm Sign}(e_{y}),\\\ &&\\\
\;\tilde{\omega}\in-\mathcal{F}(C\tilde{x}-Ke_{y}),\\\ &&\\\
\;\tilde{y}=F\tilde{x},\end{array}\right.$ (9)

where $\beta\geq L_{3}$.

######  Remark 2.

Under the conditions outlined in Assumptions 1 to 4, it has been established
that solutions for both the original system and the observer system exist.
This is due to the fact that the right-hand side operators in the
corresponding differential inclusions are upper semi-continuous and have non-
empty, closed, convex, and bounded values (see for instance [5, 12, 15]).The
following result confirms the exponential convergence of the observer state to
the original state.

######  Theorem 2.

Let Assumptions 1–4 hold. Then the observer state $\tilde{x}$ of (9)
exponentially converges to the original state $x$ of (4).

###### Proof.

Let $e=\tilde{x}-x,e_{y}=\tilde{y}-y=Fe$. From (4) and (9), we have

$\displaystyle\dot{e}$ $\displaystyle\in$
$\displaystyle(A-LF)e-B(\tilde{\omega}-{\omega})+f_{1}(\tilde{x},u)-f_{1}({x},u)$
(10) $\displaystyle-$ $\displaystyle\beta P^{-1}F^{T}\|h(\tilde{x},u)\|{\rm
Sign}(e_{y})-f_{2}(x,u)\theta.$

Consider the Lyapunov function $V(e)=\langle Pe,e\rangle$. Then the orbital
derivative of $V$ is

$\displaystyle\dot{V}(e)$ $\displaystyle=$ $\displaystyle 2\langle
P\dot{e},e\rangle=2\langle P(A-LF)e-PB(\tilde{\omega}-{\omega}),e\rangle$ (11)
$\displaystyle+$ $\displaystyle 2\langle
P(f_{1}(\tilde{x},u)-f_{1}({x},u)),e\rangle-2\beta\|h(\tilde{x},u)\|\|e_{y}\|-2\langle
Pf_{2}(x,u)\theta,e\rangle.$

From (7) and the monotonicity of $\mathcal{F}$, we have

$\displaystyle\langle
PB(\tilde{\omega}-{\omega}),e\rangle=\langle\tilde{\omega}-{\omega}),B^{T}Pe\rangle=\langle\tilde{\omega}-{\omega},(C-KF)e\rangle\leq
0.$ (12)

On the other hand

$\displaystyle 2\langle P(f_{1}(\tilde{x},u)-f_{1}({x},u)),e\rangle\leq
2L_{1}\|e\|\|Pe\|\leq{L_{1}}(\|e\|^{2}+\|Pe\|^{2})$ (13)

and

$\displaystyle Y$ $\displaystyle:=$
$\displaystyle-2\beta\|h(\tilde{x},u)\|\|e_{y}\|-2\langle
Pf_{2}(x,u)\theta,e\rangle$ (14) $\displaystyle=$
$\displaystyle-2\beta\|h(\tilde{x},u)\|\|e_{y}\|-2\langle
P(f_{2}(x,u)-f_{2}(\tilde{x},u))\theta,e\rangle-2\langle
Pf_{2}(\tilde{x},u)\theta,e\rangle$ $\displaystyle\leq$
$\displaystyle-2\beta\|h(\tilde{x},u)\|\|e_{y}\|+2L_{2}L_{3}\|e\|\|Pe\|-2\langle\theta,h(\tilde{x},u)e_{y}\rangle$
$\displaystyle\leq$ $\displaystyle{L_{2}L_{3}}(\|e\|^{2}+\|Pe\|^{2}).$

Let $\alpha_{max}$ and $\alpha_{min}$ be the largest eigenvalue and the
smallest eigenvalues of $P$, respectively. From (6), (11), (12), (13) and
(14), we have

$\frac{dV}{dt}\leq-\epsilon\|e\|^{2}\leq-\frac{\epsilon}{\alpha_{max}}V(t).$

Using Gronwall’s inequality, we obtain

${\alpha_{min}\|e\|^{2}}\leq V(t)\leq{\rm exp}({\frac{-\epsilon
t}{\alpha_{max}}})V(t_{0}).$

Hence, $\|e\|\leq{\rm exp}({\frac{-\epsilon
t}{2\alpha_{max}}})\sqrt{\frac{V(t_{0})}{{\alpha_{min}}}}$ and the conclusion
follows. ∎

######  Remark 3.

(i) According to [15], under similar conditions, our implementation of a
sliding mode observer leads to an exponential convergence of the observer
state, whereas the adaptive observer in the same study only achieves
convergence without a defined rate. Furthermore, our method avoids the need to
solve a high-dimensional additional ODE, which can be computationally
expensive, as seen in [15].
(ii) In Assumption 4, if $h$ is bounded, then $\gamma=L_{1}+L_{2}L_{3}$ can be
replaced by just $L_{1}$, which is usually much smaller than $\gamma$. This
makes it easier to satisfy (6)-(8) and enables us to solve a wider class of
set-valued Lur’e systems that cannot be tackled using the method in [15].
Furthermore, our technique also ensures finite time convergence of the
observation error $e_{y}$ to zero. A bounded matrix function $h$ can be
guaranteed if $f_{2}$ is bounded and $F$ is full row rank, as one can always
choose a full row rank matrix $F$ for the output $y=Fx$ without losing any
information.

######  Lemma 3.

Suppose that $F\in{\mathbb{R}}^{p\times n}$ be a full row rank matrix. If
$f_{2}$ is bounded then $h$ is also bounded.

###### Proof.

From Remark 1, we imply that $h(x,u)=(FP^{-1}F^{T})^{-1}Ff_{2}(x,u)$ and the
condition follows. ∎

######  Theorem 4.

If $h(x,u)$ is bounded by some constant $L_{4}>0$ then $\gamma$ in Assumption
4 can be reduced to $L_{1}$ to still obtain the exponential convergence of the
original state by using the observer

$\left\\{\begin{array}[]{lll}\;\tilde{x}=A\tilde{x}+B\tilde{\omega}-Le_{y}+f_{1}(\tilde{x},u)+\beta
P^{-1}F^{T}{\rm Sign}(e_{y}),\\\ &&\\\
\;\tilde{\omega}\in-\mathcal{F}(C\tilde{x}-Ke_{y}),\\\ &&\\\
\;\tilde{y}=F\tilde{x}\end{array}\right.$ (15)

where $\beta>L_{3}L_{4}$. In addition if $F\in{\mathbb{R}}^{p\times n}$ be a
full row rank matrix and ${\rm im}(B)\subset{\rm im}(P^{-1}F)$ then the
observation error $e_{y}$ converges in explicitly finite time.

###### Proof.

Under the new assumption, in (14) we have

$\displaystyle-\beta\|e_{y}\|-\langle
Pf_{2}(x,u)\theta,e\rangle=-\beta\|e_{y}\|-\langle\theta,h({x},u)e_{y}\rangle\leq-(\beta-
L_{3}L_{4})\|e_{y}\|\leq 0$

and similarly one obtains the exponential convergence of the observer state.

To attain the finite time convergence of the observation error, we consider
the new Lyapunov function $W=\frac{1}{2}\langle
e_{y},(FP^{-1}F^{T})^{-1}e_{y}\rangle$. Then

$\frac{dW}{dt}=\langle(FP^{-1}F^{T})^{-1}\dot{e}_{y},e_{y}\rangle.$

From (10), we have

$\displaystyle\dot{e}_{y}$ $\displaystyle=$ $\displaystyle
F(A-LF)e-FB(\tilde{\omega}-{\omega})+F\Big{(}f_{1}(\tilde{x},u)-f_{1}({x},u)\Big{)}$
(16) $\displaystyle-$ $\displaystyle\beta FP^{-1}F^{T}{\rm
Sign}(e_{y})-Ff_{2}(x,u)\theta.$

Since ${\rm im}(B)\subset{\rm im}(P^{-1}F)$, using Lemma 1, we have

$F^{T}(FP^{-1}F^{T})^{-1}FB=PB$

and hence

$\displaystyle\langle(FP^{-1}F^{T})^{-1}FB(\tilde{\omega}-{\omega}),e_{y}\rangle=\langle
F^{T}(FP^{-1}F^{T})^{-1}FB(\tilde{\omega}-{\omega}),e\rangle$ $\displaystyle=$
$\displaystyle\langle
PB(\tilde{\omega}-{\omega}),e\rangle=\langle\tilde{\omega}-{\omega},B^{T}Pe\rangle=\langle\tilde{\omega}-{\omega},(C-KF)e\rangle\leq
0.$

On the other hand, from (8) we deduce that $f_{2}=P^{-1}F^{T}h^{T}$ and thus

$(FP^{-1}F^{T})^{-1}Ff_{2}(x,u)\theta=(FP^{-1}F^{T})^{-1}(FP^{-1}F^{T})h^{T}(x,u)\theta=h^{T}(x,u)\theta.$
(17)

Since $\|e\|\leq{\rm exp}({\frac{-\epsilon
t}{2\alpha_{max}}})\sqrt{\frac{V(t_{0})}{{\alpha_{min}}}}$ as in the proof of
Theorem 2, we can find some $t_{1}>0$ such that for all $t\geq t_{1}$, one has

$\|(FP^{-1}F^{T})^{-1}\Big{(}F(A-LF)e+F\big{(}f_{1}(\tilde{x},u)-f_{1}({x},u)\big{)}\Big{)}\|\leq\frac{\sigma}{2}$

where $\sigma=\beta-L_{3}L_{4}>0$. Indeed, we can choose

$t_{1}:=\frac{2\alpha_{max}}{\epsilon}\ln\Big{(}\frac{2\|(FP^{-1}F^{T})^{-1}F\|(\|A-LF\|+L_{1})}{\sigma}\sqrt{\frac{V(t_{0})}{{\alpha_{min}}}}\Big{)}.$

Then for all $t\geq t_{1}$, we have

$\displaystyle\frac{dW}{dt}$ $\displaystyle\leq$
$\displaystyle\frac{\sigma}{2}\|e_{y}\|-(\beta-\|h^{T}(x,u)\theta\|)\|e_{y}\|\leq\frac{\sigma}{2}\|e_{y}\|-(\beta-
L_{3}L_{4})\|e_{y}\|$ $\displaystyle=$
$\displaystyle-\frac{\sigma}{2}\|e_{y}\|\leq-\kappa\sqrt{W(t)},$

where $\kappa:=\sigma/\sqrt{2\gamma_{max}}$ and $\gamma_{max}$ is the largest
eigenvalue of $(FP^{-1}F^{T})^{-1}$ since
$\|e_{y}\|\geq\sqrt{\frac{2W(t)}{{\mu_{max}}}}.$ Suppose that $W(t)>0$ for all
$t\geq t_{1}$ then we have

$\frac{W^{\prime}}{2\sqrt{W}}\leq-\frac{\kappa}{2}$

and thus

$\sqrt{W(t)}-\sqrt{W(t_{1})}\leq-\frac{\kappa}{2}(t-t_{1})\to-\infty\;\;{\rm
as}\;t\to\infty,$

a contradiction. Let $t_{f}\geq t_{1}$ be the first time such that
$W(t_{f})=0$, then we deduce that $W(t)=0$ for all $t\geq t_{f}$ since $W$ is
non-negative and decreasing. It means that $e_{y}$ converges to $0$ in finite
time. Similarly as above we have

$-\sqrt{W(t_{1})}=\sqrt{W(t_{f})}-\sqrt{W(t_{1})}\leq-\frac{\kappa}{2}(t_{f}-t_{1})$

and hence

$t_{f}\leq t_{1}+2\sqrt{W(t_{1})}/\kappa.$

∎

######  Remark 4.

(i) Based on the observation (17), we can have a significantly better
estimation for the gain $\beta$ than in [20]. Indeed, in the current paper the
gain $\beta$ does not change to obtain the finite time convergence of the
observation error from the exponential convergence of the observer state. In
addition, we can also provide an explicit estimation for $t_{f}$.
(ii) If Assumption 1 is substituted by the maximal monotone property of the
set-valued $\mathcal{F}$, similar results can still be obtained. The existence
of solutions for the original system (4) remains guaranteed, as shown in
references such as [5, 12]. Existence of solutions to the observer systems
(9), (15) can be also obtained by approximating the Sign set-valued operator
with smooth functions. It’s worth mentioning that the normal cone operator
$N_{C}$, which is associated to a nonempty closed convex set $C$, is a crucial
maximal monotone set-valued mapping in mechanical and electrical engineering
that does not satisfy the boundedness requirement in Assumption 1.

##  4 Reduced-order observer

Suppose that the given matrices and matrix-functions can be decomposed as
follows

$x=\left(\begin{array}[]{ccc}x_{1}\\\ \\\
x_{2}\end{array}\right),\;\;A=\left(\begin{array}[]{cc}A_{11}&\;\;A_{12}\\\
\\\ A_{21}&\;\;A_{22}\end{array}\right),B=\left(\begin{array}[]{ccc}B_{1}\\\
\\\ B_{2}\end{array}\right),C=(C_{1}\;\;C_{2}),F=(F_{q}\;\;0)$
$P=\left(\begin{array}[]{cc}P_{11}&\;\;P_{12}\\\ \\\
P_{21}&\;\;P_{22}\end{array}\right),\;\;f_{1}(x,u)=\left(\begin{array}[]{ccc}f_{11}(x,u)\\\
\\\
f_{12}(x,u)\end{array}\right),\;\;f_{2}(x,u)=\left(\begin{array}[]{ccc}f_{21}(x,u)\\\
\\\ f_{22}(x,u)\end{array}\right),$

where $F_{q}\in R^{q\times q}$ is an invertible matrix and the following is
satisfied:

Assumption 4’:  There exist $\epsilon>0$,
$Q\in{\mathbb{R}}^{(n-q)\times(n-q)}>0$,
$P_{22}\in{\mathbb{R}}^{(n-q)\times(n-q)}$ invertible,
$P_{21}\in{\mathbb{R}}^{(n-q)\times q}$ such that

$\displaystyle
Q(A_{22}+KA_{12})+(A_{22}+KA_{12})^{T}Q+L_{1}Q(KK^{T}+I_{n-q})Q$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+(L_{1}+\epsilon)I_{n-q}\leq
0,$ (18) $\displaystyle(B_{2}+KB_{1})^{T}Q=C_{2},$ (19)
$\displaystyle\Big{(}P_{21}\;\;P_{22}\Big{)}f_{2}(x,u)=0,$ (20)

where $K=P_{22}^{-1}P_{21}$.

Note that (4) can be rewritten as follows

$\left\\{\begin{array}[]{lll}\;\dot{x}_{1}=A_{11}x_{1}+A_{12}x_{2}+B_{1}\omega+f_{11}\Big{(}\left(\begin{array}[]{ccc}x_{1}\\\
\\\
x_{2}\end{array}\right),u\Big{)}+f_{21}\Big{(}\left(\begin{array}[]{ccc}x_{1}\\\
\\\ x_{2}\end{array}\right),u\Big{)}\theta\\\
\;\dot{x}_{2}=A_{21}x_{1}+A_{22}x_{2}+B_{2}\omega+f_{12}\Big{(}\left(\begin{array}[]{ccc}x_{1}\\\
\\\
x_{2}\end{array}\right),u\Big{)}+f_{22}\Big{(}\left(\begin{array}[]{ccc}x_{1}\\\
\\\ x_{2}\end{array}\right),u\Big{)}\theta\\\ &&\\\
\;\omega\in-\mathcal{F}(C_{1}x_{1}+C_{2}x_{2})\\\ &&\\\
\;y=F_{q}x_{1}.\end{array}\right.$ (21)

Using (20), we have

$\left\\{\begin{array}[]{lll}\;\dot{z}&=&(A_{22}+KA_{12})z+(B_{2}+KB_{1})\omega+[(A_{21}+KA_{11})-(A_{22}+KA_{12})K]x_{1}\\\
&+&(K\;\;I_{n-q})f_{1}\Big{(}\left(\begin{array}[]{ccc}x_{1}\\\ \\\
z-Kx_{1}\end{array}\right),u\Big{)}\\\ &&\\\
\;\omega&\in&-\mathcal{F}(C_{2}z+(C_{1}-C_{2}K)x_{1}),\\\ &&\\\
\;x_{2}&=&z-Kx_{1}.\end{array}\right.$ (22)

The adaptive observer is

$\left\\{\begin{array}[]{lll}\;\dot{\tilde{z}}&=&(A_{22}+KA_{12})\tilde{z}+(B_{2}+KB_{1})\tilde{\omega}+[(A_{21}+KA_{11})-(A_{22}+KA_{12})K]x_{1}\\\
&+&(K\;\;I_{n-q})f_{1}\Big{(}\left(\begin{array}[]{ccc}x_{1}\\\ \\\
\tilde{z}-Kx_{1}\end{array}\right),u\Big{)}\\\ &&\\\
\;\tilde{\omega}&\in&-\mathcal{F}(C_{2}\tilde{z}+(C_{1}-C_{2}K)x_{1}),\\\
&&\\\ \;\tilde{x}_{2}&=&\tilde{z}-Kx_{1}.\end{array}\right.$ (23)

######  Theorem 5.

Let Assumptions 1, 2, 3, 4’ hold. Then (23) is a reduced-order observer of
$(\ref{sysh})$, i.e.,
$\displaystyle{\lim_{t\to\infty}(x_{2}(t)-\tilde{x}_{2}(t))}=0$.

###### Proof.

Let $e_{z}=\tilde{z}-{z}$. Then we have

$\left\\{\begin{array}[]{lll}\;\dot{e}_{z}&=&(A_{22}+KA_{12})e_{z}+(B_{2}+KB_{1})(\tilde{\omega}-\omega)\\\
&+&(K\;\;I_{n-q})\Big{[}f_{1}\Big{(}\left(\begin{array}[]{ccc}x_{1}\\\ \\\
\tilde{z}-Kx_{1}\end{array}\right),u\Big{)}-f_{1}\Big{(}\left(\begin{array}[]{ccc}x_{1}\\\
\\\ {z}-Kx_{1}\end{array}\right),u\Big{)}\Big{]}\\\ &&\\\
\;{\omega}&\in&-\mathcal{F}(C_{2}{z}+(C_{1}-C_{2}K)x_{1}).\\\ &&\\\
\;\tilde{\omega}&\in&-\mathcal{F}(C_{2}\tilde{z}+(C_{1}-C_{2}K)x_{1}).\end{array}\right.$
(24)

Let us consider the Lyapunov function $W(e_{z})=\langle Qe_{z},e_{z}\rangle$,
then $\dot{W}(e_{z})=2\langle Q\dot{e}_{z},e_{z}\rangle$. From (19) and the
monotonicity of $\mathcal{F}$, we have

$\langle
Q(B_{2}+KB_{1})(\tilde{\omega}-\omega),e_{z}\rangle=\langle\tilde{\omega}-\omega,(B_{2}+KB_{1})^{T}Qe_{z}\rangle=\langle\tilde{\omega}-\omega,C_{2}e_{z}\rangle\leq
0.$

On the other hand

$\displaystyle\langle
2Q(K\;\;I_{n-q})\Big{[}f_{1}\Big{(}\left(\begin{array}[]{ccc}x_{1}\\\ \\\
\tilde{z}-Kx_{1}\end{array}\right),u\Big{)}-f_{1}\Big{(}\left(\begin{array}[]{ccc}x_{1}\\\
\\\ {z}-Kx_{1}\end{array}\right),u\Big{)}\Big{]},e_{z}\rangle$
$\displaystyle\leq$ $\displaystyle 2L_{1}\|Q(K\;\;I_{n-q})e_{z}\|\|e_{z}\|\leq
L_{1}e_{z}^{T}Q(K\;\;I_{n-q})(K\;\;I_{n-q})^{T}Qe_{z}+L_{1}e_{z}^{T}e_{z}$
$\displaystyle\leq$ $\displaystyle
L_{1}e_{z}^{T}Q(KK^{T}+I_{n-q})Qe_{z}+L_{1}e_{z}^{T}e_{z}.$

Combining with (18), similarly as in the proof of Theorem 2, we have
$\dot{W}(e_{z})\leq-\epsilon\|e_{z}\|^{2}$. It deduces that $e_{z}$ converges
to zero exponentially and the conclusion follows. ∎

######  Remark 5.

(i) Note that if we have (8), then $Pf_{2}(x,u)=F^{T}h(x,u)$ and thus
$\Big{(}P_{21}\;\;P_{22}\Big{)}f_{2}(x,u)=0$. It would be interesting to
improve or remove the condition (20).
(ii) When the matrices are decomposable, it is more effective to provide
assumptions directly with the new lower-dimension matrices. Note that
Assumption 4’ is strictly weaker than Assumption 4 even when $Q=P_{22}$ [15].
It is remarkable that $Q>0$ can be different from $P$ and it is unnecessary to
require that $P>0$ but only invertible $P_{22}$.

##  5 A new continuous approximate of the sliding mode technique

Although the sliding mode method is effective, it has a persistent issue: the
chattering effect caused by the discontinuity of the Sign function. To
eliminate this issue, the Sign function in ${\mathbb{R}}$ is typically
approximated by the “sigmoid function” (as noted in [22])

${\rm Sign(x)}\approx\frac{x}{|x|+\epsilon}$

or by [23]

${\rm Sign(x)}\approx\frac{x}{\sqrt{x^{2}+\epsilon}}$

for some small fixed $\delta>0$. Other continuous approximates can be also
found in [23].

Figure 2: Sign function in ${\mathbb{R}}$ and its approximations

In this paper, we provide a new smooth approximate of the set-valued function
Sign by using a time-dependent guiding function. Given a continuous guiding
function $\delta:{\mathbb{R}}^{+}\to{\mathbb{R}}^{+}$, we define the function
${\rm
Sign}_{\delta}:{\mathbb{R}}^{+}\times{\mathbb{R}}^{n}\to{\mathbb{R}}^{n}$ as
follows

${\rm
Sign}_{\delta}(t,x)=\left\\{\begin{array}[]{l}\frac{x}{\|x\|}-\frac{1-{\|x\|}/{\delta(t)}}{(1+M\|x\|)^{N}}\frac{x}{\|x\|}\;\;{\rm
if}\;\;x\neq 0\;{\rm and}\;\|x\|\leq\delta(t),\\\ \\\
\frac{x}{\|x\|}\;\;\;\;{\rm if}\;\;\;\;\|x\|>\delta(t),\\\ \\\
0\;\;\;\;\;\;\;\;{\rm if}\;\;\;\;\;x=0,\end{array}\right.$ (26)

for some $M,N>0$ and the function $\delta(\cdot)$ is decreasing which
converges to zero when $t\to\infty$. For example, we can choose
$\delta(t)={\rm exp}(-k_{1}t-k_{2})$ for some $k_{1}>0,k_{2}>0$. It can be
seen that the ${\rm Sign}_{\delta}(t,x)$ function is a continuous and smooth
function with respect to both time $t$ and state $x$. When the magnitude of
$x$ is greater than $\delta(t)$, the function ${\rm Sign}_{\delta}(t,x)$
becomes equal to ${\rm Sign}(x)$. This property makes it effective in reducing
the chattering effect while still handling uncertainty. In comparison, the
norm of the sigmoid function $\frac{x}{|x|+\epsilon}$ is always less than $1$
which does not deal with the uncertainty entirely. The sigmoid function only
leads to convergence of the state to an approximate region of the sliding
surface, as reported, e.g., in [22]. The suitable choice of the guiding
function $\delta$ can completely remove the chattering effect. This can be
seen in Example 1, Figure 3.

##  6 Numerical Examples

In this section we provide some numerical examples to show the effectiveness
of our approach.

######  Example 1.

To test the effectiveness of our new approximation method, we will analyze a
simple one-dimensional system: $\dot{x}\in\mu-L{\rm Sign}(x)$, where $\mu$ is
an uncertainty with a constraint $|\mu|<L$. As an example, let us set
$\mu=3\sin x$ and $L=4$. It is known that the state variable $x$ will converge
to zero in a finite time. To evaluate the numerical simulations, we use the
explicit scheme with an initial value of $x_{0}=0.1$ (see Figure 3). The Sign
function creates a chattering effect (as seen in the state $x_{1}$). However,
by replacing the ${\rm Sign}$ function with ${\rm Sign}_{\delta}(t,x)$ defined
in equation (26), using a guiding function $\delta(t)=e^{-0.5t}$ and
$M=1,N=3$, the chattering effect can be eliminated and the convergence to zero
is achieved (as seen in the state $x_{3}$). The performance of this
approximation is better than using the sigmoid function with
$\epsilon=10^{-3}$ or $\epsilon=10^{-6}$ (as seen in the state $x_{2}$). It is
important to note that the implicit scheme can only remove the chattering
effect numerically.

Figure 3: An effective continuous approximation of Sign function in
${\mathbb{R}}$

######  Example 2.

Next we consider the system (4) with the following data

$A=\left(\begin{array}[]{ccc}-1&\;\;5&\;\;0\\\ \\\ 9&\;\;-0.9&\;\;0\\\ \\\
0&\;\;0&\;\;-1\end{array}\right),B=\left(\begin{array}[]{ccc}2\\\ \\\ -3\\\
\\\ 4\end{array}\right),f_{1}(x,u)=\left(\begin{array}[]{ccc}3u+0.8\sin
x_{2}\\\ \\\ 2u+0.9\cos x_{1}\\\ \\\ -u+0.8\sin x_{3}\end{array}\right)$

$f_{2}(x,u)=\left(\begin{array}[]{ccc}3\sin x_{2}\\\ \\\ 0\\\ \\\
0\end{array}\right),C=\left(\begin{array}[]{ccc}5&\;\;-3&\;\;4\end{array}\right),F=\left(\begin{array}[]{ccc}1&\;\;0&\;\;0\end{array}\right).$

Suppose that the unknown $\theta=3\sin t$, the control input $u=8\cos t$ and

$\mathcal{F}(x)=\left\\{\begin{array}[]{lll}{\rm sign}(x)(2|x|+5)&\mbox{ if
}&x\neq 0,\\\ &&\\\ \;[-5,5]&\mbox{ if }&x=0.\end{array}\right.$

Then $L_{1}=0.8,L_{2}=3,L_{3}=3$ and $\gamma=9.8$. We cannot find the matrices
$P,K,L$ such that the LMIs in Assumption 4 (and also the LMIs in [15]) are
satisfied but Theorem 4 can be applied with

$P=\left(\begin{array}[]{ccc}1&\;\;0&\;\;0\\\ \\\ 0&\;\;1&\;\;0\\\ \\\
0&\;\;0&\;\;1\end{array}\right),\;\;\;L=\left(\begin{array}[]{ccc}0\\\ \\\
14\\\ \\\ 0\end{array}\right),\;\;\;\epsilon=0.2,\;\;\;K=3.$

Using the sliding mode observer (15) with the gain $\beta=10$ and initial
point $x_{0}=(3\;2\;1)^{T},\;\tilde{x}_{0}=(15\;27\;16)^{T}$, the convergence
of the error $e=\tilde{x}-{x}$ to zero can be seen in the Figure 4.

Figure 4: The convergence of the sliding mode observer (15)

##  7 Conclusion and perspectives

In this paper, the advantages of sliding mode observers are demonstrated for
set-valued Lur’e dynamical systems that are faced with uncertainties. The
robustness of this observer technique is a significant factor in the analysis
and control of such systems. We also present a new and efficient continuous
approximation of the sliding mode technique, which provides improved
performance compared to conventional methods. However, the traditional
condition $(\ref{f2p})$ is a limiting factor in the applicability. Further
research is needed to explore the possibility of relaxing or removing this
condition to increase the range of systems that can be analyzed. This is an
area that merits further investigations and has the potential to enhance the
performance of sliding mode observers for set-valued Lur’e dynamical systems.

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