The norm observer is a tool which allows to estimate an upper bound to the norm of each system state variable. This is performed by means of the upper bounds of the system parameters and the available state variables. This section was based on [107].
For instance, consider a scalar uncertain nonlinear system described by:
˙
x(t) = αx(t) +βz(t) +γq(t)r(t)s(t), (101)
where q ∈ R, r ∈ R, s ∈ R and z ∈ R are input signals and x ∈ R is the unmeasured state variable. We assume that the parameters are uncertain such that α ∈ R−, β ∈ R and γ ∈R, with known upper bounds satisfying
¯
α≥α, β¯≥ |β| e γ¯≥ |γ|. (102) The description of a norm observer for an unmeasured variable depends on the amount of available inputs. When all the system inputs in (101) are known, the norm observer is a dynamic system which can be described by:
˙ˆ
x(t) = ¯αˆx(t) + ¯β|z(t)|+ ¯γ|q(t)||r(t)||s(t)|. (103)
If only one of the inputs is unknown, namely q(t), it is necessary to know the respective state-norm estimate ˆq(t) for it. Thus, applying the Young’s inequality [76], p.395, in equation (103), the norm observer to the system (101) can be obtained as
˙ˆ
x(t) = ¯αˆx(t) + ¯β|z(t)|+γ¯
µ|ˆq(t)|p+ ¯γµp−11 (|r(t)||s(t)|)p−1p . (104) In case of two or more entrances are unknown, it is necessary that the respective observers to be available. Suppose all the inputs q, r, s and z are unavailable. In this case, the Young’s inequality must be recursively applied in equation (103) so that the norm observer to the system (101) can be constructed by cascading the norm estimates of the unknown
inputs as follows
˙ˆ
x(t) = ¯αˆx(t) + ¯β|ˆz(t)|+γ¯
µ|q(t)|ˆ p+ ¯γµ
2−p
p−1|ˆr(t)| p
2 p−1
+ ¯γµp−12 |ˆs(t)|(p−1p )2, (105) where ˆq∈R, ˆr∈R, ˆs ∈Rand ˆz ∈Rare the norm-estimate variables and the parameters µand p came from the application of the Young’s inequality.
Definition 1. The Young’s inequality [76], p.395, is given by:
vw≤ 1
µvp+µp−11 wp−1p valid ∀ v ≥0, w≥0, µ >0 and p > 1.
The following lemma describes how to obtain the norm observers (103)–(105) for the dynamic system (101).
Lemma 1. Consider the scalar nonlinear system (101) with state variable x ∈ R and the norm observers candidates (103)–(105), with state variable xˆ ∈ R and input signals q∈ R, r ∈R, s ∈R, z ∈R. If α,¯ β¯ and γ¯ are upper bounds to the parameters α, β and γ satisfying the inequalities (102), then the absolute value of the state variable x satisfies:
|x(t)| ≤x(t) + eˆ αt¯ (|x(0)|+|ˆx(0)|) , ∀t≥0. (106) Proof 3.: Define the quadratic function V :=x2. From the differential equation (101) and the upper bounds in (102), it can be concluded the time-derivative of V satisfies
V˙ = 2x(αx+βz+γqrs)
= 2(αx2+βxz+γqrsx)
≤2(α|x|2+|β||x||z|+|γ||q||r||s||x|)
≤2( ¯α|x|2+ ¯β|x||z|+ ¯γ|q||r||s||x|)
= 2|x|( ¯α|x|+ ¯β|z|+ ¯γ|q||r||s|). (107)
Using the Young’s inequality, we have
V˙ ≤2|x| ( ¯α|x|+ ¯β|z|+γ¯
µ|q|p+ ¯γµp−11 (|r||s|)p−1p ), (108) and applying the Young’s inequality once again, one can write
V˙ ≤2|x|
α|x|¯ + ¯β|z|+γ¯
µ|q|p+ ¯γµ2−pp−1|r| p
2 p−1
+ ¯γµp−12 |s|(p−1p )2
. (109)
Now, defining
˜ x:=√
V =|x|, (110)
the inequality (107) can be rewritten as
˙˜
x≤α¯x˜+ ¯β|z|+ ¯γ|q||r||s|, ∀t ≥0. (111) From the Comparison Lemma [76], pp.84-87, an upper bound x¯ for x˜ is the solution of the differential equation
x˙¯= ¯α˜x+ ¯β|z|+ ¯γ|q||r||s|, ∀t≥0, (112)
provides the initial condition x(0) = ˜¯ x(0) ≥ 0. The solution of the differential equation (112) is given by
¯
x(t) = eαt¯ x(0) + e¯ αt¯ ∗( ¯β|z|+ ¯γ|q||r||s|), t ≥0, (113) which is an upper bound for |x(t)|, where x(0) =¯ |x(0)| and ∗ denotes the convolution operator. On the other hand, the solution of the differential equation of the norm observer (103) is given by
ˆ
x(t) = eαt¯ x(0) + eˆ αt¯ ∗( ¯β|z|+ ¯γ|q||r||s|), t ≥0. (114) By subtracting (113) from (114), we have
¯
x(t)−x(t) = eˆ αt¯ (|x(0)| −x(0))ˆ , t ≥0. (115)
Finally, since |x(t)| ≤ x(t), the inequality (106) holds, with¯ x(t)ˆ governed by (103). The process to obtain the norm observer from inequalities (108) and (109) is analogous to the one done for (107). The observers described by equations (104) and (105) satisfy (115) since q,ˆ rˆ and ˆs are at least norm bounds for q, r and s, respectively. Therefore, in all these cases, (106) is also verified.
If the parameter ¯α < 0, then the state-norm observers (103)–(105) and, conse- quently, the linear time-variant system (101) are ISS. The concept of minimum phase to linear time-invariant system can be generalized to nonlinear systems assuming that the zero dynamics (112) is stable.
Using Lemma 1, the dynamic equation that governs the norm observer for (1) with norm estimate ˆx1 follows the same steps taken to obtain (103). The norm observer ˆx2
falls in the case described by the equation (104). At last, the norm observers with norm bounds ˆx3 and ˆx4 for x3 and x4 are referred to (105). Thereby, it is possible to obtain the following cascade norm observers for the subsystem (92)–(95) from the measurable signalsx5, x6 and x7:
˙ˆ
x1 = ¯α1xˆ1+ ¯β1|x5|+ ¯γ1|x5||x6||x7| (116)
˙ˆ
x2 = ¯α2xˆ2+ ¯β2|x6|+γ¯2
µ2|ˆx1|p2+ ¯γ2µ
1 p2−1
2 (|x6||x7|)
p2
p2−1 (117)
˙ˆ
x3 = ¯α3xˆ3+ ¯β3|x5|+ γ¯3
µ3|ˆx2|p3 + ¯γ3µ
2−p3 p3−1
3 |ˆx1|
p2 3
p3−1 + ¯γ3µ
2 p3−1
3 |x7|
p
3 p3−1
2
(118)
˙ˆ
x4 = ¯α4xˆ4+ ¯β4|ˆx1|+ γ¯4 µ4
|ˆx3|p4 + ¯γ4µ
2−p4 p4−1
4 |ˆx1|
p2 4
p4−1 + ¯γ4µ
2 p4−1
4 |ˆx2|
p
4 p4−1
2
, (119)
where −15≤ α¯1 <0, −0.5 ≤ α¯2 < 0, −15 ≤ α¯3 < 0, −15≤ α¯4 <0, ¯β1 ≥ 15, ¯β2 ≥ 25, β¯3 ≥ 15, ¯β4 ≥10, ¯γ1 ≥5, ¯γ2 ≥1, ¯γ3 ≥1, ¯γ4 ≥1, p2 >1, p3 >1, p4 > 1, µ2 >0, µ3 >0 and µ4 >0.
2.4 Stabilization via Output Feedback
For the controller design, the output vector y is described by y = [y1 y2 y3]T = [x5 x6 x7]T. In addition, the control signal u has been injected into the fifth differential equation (96). These choices were motivated by the need to ensure the ISS properties in order to construct the norm observers for the following state variables: x1, x2, x3 and x4. Moreover, this input-output selection restricts the control problem to the case of
relative degree one [76]. The unitary relative degree facilitates the design of a sliding mode controller via output feedback, making it amenable by pure Lyapunov function.
Theorem 3. Consider the hyperchaotic system described by (92)–(99) and the norm observers (116)–(119). If the following sliding mode control law via output feedback is applied
u=−ρ(t) sgn(x5(t)), (120)
with modulation function ρ satisfying
ρ(t) = ¯α5|x5|+ ¯β5|x7|+ ¯γ5|xˆ2||ˆx3||ˆx4|+ ¯du(t) +δ, (121) where δ > 0 is any arbitrary constant and |du(t)| ≤ d¯u(t), ∀t ≥ 0, then the ideal sliding mode y1 =x5 = 0 occurs in finite time. Moreover, the equilibrium point of the closed-loop system
[x1 x2 x3 x4 x5 x6 x7 xˆ1 xˆ2 xˆ3 xˆ4]T = [0 0 · · · 0 0]T is globally asymptotically stable.
Proof 4.: From Lemma 1, the state variable xi can be majorized by
|xi(t)| ≤xˆi(t) +πi(t), ∀t ≥0, (122)
where i= 1,2,3 and 4; xˆi(t) is the norm observer state and πi(t) = eα¯it(|xi(0)|+|ˆxi(0)|) decays exponentially. Consider the following Lyapunov function candidate
V = 1
2x52, (123)
with time-derivative V˙ =x5x˙5. Hence, V˙ =x5x˙5
=x5[−15x5+ 10x7−x2x3x4+du+u]
=−15x25+10x5x7−x2x3x4x5+x5du−ρx5sgn(x5)
≤α¯5|x5|2+ ¯β5|x5||x7|+ ¯γ5|x2||x3||x4||x5| +|x5||du| −ρ|x5|
≤ |x5|
¯
α5|x5|+ ¯β5|x7|+ ¯γ5|ˆx2+π2||ˆx3 +π3||ˆx4+π4|+ ¯du−ρ
≤ |x5|
¯
α5|x5|+ ¯β5|x7|+ ¯γ5|ˆx2||ˆx3||ˆx4|+|ˆx2||ˆx3|π4 +|ˆx2||ˆx4|π3+|ˆx2|π3π4+|ˆx3||ˆx4|π2+|xˆ3|π2π4
+|ˆx4|π2π3+π2π3π4+ ¯du−ρ
(124)
Applying the Young’s inequality, with µ = 1 and p = 2, into the terms multiplied by πi, we have :
V˙ ≤|x5|
¯
α5|x5|+ ¯β5|x7|+ ¯γ5|ˆx2||ˆx3||ˆx4|+ ˆx22xˆ23+π24 + ˆx22xˆ24+π23+ ˆx22+π23π42+ ˆx23xˆ24+π22+ ˆx23 +π22π24+ ˆx24+π22π32+π2π3π4+ ¯du−ρ
≤ |x5|
¯
α5|x5|+ ¯β5|x7|+ ¯γ5|ˆx2||ˆx3||ˆx4|+ ˆx22xˆ23 + ˆx22xˆ24+ ˆx22+ ˆx23xˆ24+ ˆx23+ ˆx24+π22+π32+π42 +π22π23+π22π24+π32π24+π2π3π4+ ¯du−ρ
Therefore, using a modulation function that satisfies (121), it can be concluded that
V˙ ≤[−δ+ Π(t)]|y1|, t ≥0, (125)
whereΠ(t) =π22+π32+π24+π22π32+π22π24+π23π42+π2π3π4 is a term which decays exponentially.
It is possible to rewrite (125) with y˜1(t) :=√
2V =|y1(t)| such that
˙˜
y1 ≤ −δ+ Π(t), t≥0. (126)
Using the Comparison Theorem [106], there exists an upper boundy¯1(t)fory˜1(t)satisfying the differential equation
˙¯
y1 =−δ+ Π(t), y¯1(0) = ˜y1(0)≥0, t≥0. (127) Integrating both sides of the above equation, we arrive at the solution
¯
y1(t)−y¯1(0) =−δt+ Z t
0
Π(τ)dτ t≥0. (128)
Since the function Π(t) is positive and decays exponentially, its integral converges to some constant which depends on the initial state, which is denoted byC(x(0)). In this sense, the following inequality is valid:
¯
y1(t) = −δt+ Z t
0
Π(τ)dτ + ¯y1(0)
≤ −δt+ Z +∞
0
Π(τ)dτ + ¯y1(0)
=−δt+C(x(0)) + ¯y1(0), t≥0. (129) Considering that y¯1 ≥ 0 is continuous, then y¯1(t) becomes identically zero ∀t ≥ t1 = δ−1[C(x(0)) + ¯y1(0)]. Thus, it can be concluded that there exists a finite time 0< ts≤t1, where the sliding mode starts such that y1(t) =x5 = 0, ∀t ≥ts.
It is well known that the chaotic orbits remain at least within a bounded region of the state space, even when x5 = 0, since the system (92)–(98) has two positive Lyapunov exponents [105]. It ensures the trajectories are confined in a compact set corresponding to the hyperchaotic atractor a priori. Therefore, the cross terms xixjxk, for i 6= j 6= k, disappear if one of the variables eventually converges to zero.
Now, the stabilization result can be easily verified from equations (92)–(98). With x5 in sliding motion (x5 = 0), it can be claimed that: x1 and x6 converge exponentially to zero because (92) and (97) are ISS with respect tox5. Since the dynamics of x4 in (95) is ISS with respect to x1, so if x1 converges to zero, x4 decays exponentially. In the case of the state variable x2, the ISS property of (97) is verified concerning the variablex6. Since x6 decays exponentially,x2 goes to zero exponentially too. The subsystem that governs x7
is ISS with respect to the inputsx2, x4, x5 and x6. Since all inputs converge to zero, thus
x7 decays exponentially. The state variable x3 in (94) also decays exponentially, since x1, x2, x5 and x7 tend to zero. In a nutshell, it can be concluded that x5 goes to zero by the direct action of the controller, while the other state variables of the hyperchaotic system converge exponentially, due to the ISS property of each one of the subsystems in (92)–(98).
Since the terms π1(t), π2(t), π3(t) and π4(t) converge asymptotically to zero as well as x1(t), ..., x7(t), then, we conclude that the state variables of the norm observers in (116)–(119), namely,xˆ1(t), xˆ2(t), xˆ3(t)and xˆ4(t), also converge asymptotically to zero
by using inequality (122).
2.5 Simulations
In the following simulation results for the hyperchaotic system (92)–(98), the para- meters of the norm observer (116)–(119) have been chosen as ¯α1 =−13, ¯β1 = 20, ¯γ1 = 10,
¯
α2 =−0.5, ¯β2 = 25, ¯γ2 = 1, ¯α3 =−1, ¯β3 = 16, ¯γ3 = 0.15, ¯α4 = −14, ¯β4 = 11, ¯γ4 = 1.5, µ2 =µ3 =µ4 = 1 andp2 =p3 =p4 = 2.
The input disturbance in (96) is du(t) = 0.02 cos (4t) and its upper bound used in the modulation signalρwas ¯du = 0.05, so that was met the hypothesis that the disturbance d was uniformly bounded, as was said in equation (100). In addition, the parameter δ= 0.001 was chosen to implement (120)–(121), where δ is a sufficiently small constant.
The initial conditions are given by: x1(0) = −1, x2(0) = 1, x3(0) = −0.5, x4(0) = 0.5, x5(0) = 0,x6(0) =−0.25, x7(0) = 0.25, ˆx1(0) = 0.5 and ˆx2(0) = ˆx3(0) = ˆx4(0) = 1.
Figure 19 and Figure 20 present the trajectories of |x1(t)| and its upperbound ˆ
x1(t) given by the norm observer (116) in open-loop and closed-loop operation modes, respectively. Similar curves can be obtained for the duos (|x2|, ˆx2), (|x3|, ˆx3) and (|x4|, ˆ
x4). As expected, the signal ˆx1(t) majorizes |x1(t)| after a short transient due to the system initial conditions.
Figure 21 and Figure 22 show the performance of the proposed output-feedback sliding mode controller (120)–(121). The state of the closed loop system is forced to reach the equilibrium point as shown in Figure 21.
In Table 2, it is shown a class of chaotic systems with ISS properties for which the methodology presented here can be directly applied. Moreover, it is presented the cascade norm observers and the required control laws for each case.
0 5 10 15 20 0
10 20 30 40 50
t(s)
|x1| ˆ x1
Figure 19: Open-loop system: |x1| and its upperbound ˆx1.
0 0.5 1 1.5 2
0 0.2 0.4 0.6 0.8 1 1.2
t(s)
|x1| ˆ x1
Figure 20: Closed-loop system: |x1|and its upperbound ˆx1.
0 2 4 6 8 10 12
−1
−0.5 0 0.5 1 1.5
t(s)
x1 x2 x3 x4 x5 x6 x7
Figure 21: Convergence for the state variables of the controlled hyperchaotic system.
2.6 Synchronization Applied to Secure Communication via Equivalent/Average Control Here, a strategy to synchronize two seven dimensional chaotic systems is presented.
It is considered that only the output variables of each system (master and slave) are available. Norm observers are designed such that global stability properties of the closed-
0 2 4 6 8 10 12
−40
−20 0 20 40
t(s)
u
Figure 22: Control signal.
loop system are preserved. Moreover, it is worth to mention that syncronizing such chaotic systems is nothing but reaching the stabilization of the error dynamics. Therefore, the results presented in Section 2.4 are utilized here.
The chaotic transmitter dynamics is driven by
˙
xm1 =−15xm1+ 15xm5−5xm5x6xm7, (130)
˙
xm2 =−0.5xm2−25xm6+xm1xm6xm7, (131)
˙
xm3 =−15xm3+ 15xm5−0.1xm1xm2xm7, (132)
˙
xm4 =−15xm4+ 10xm1−xm1xm2xm3, (133)
˙
xm5 =−15xm5+ 10xm7−xm2xm3xm4+m , (134)
˙
xm6 =−10xm6+ 10xm5+xm2xm3xm5, (135)
˙
xm7 =−5xm7+ 4xm2−1.5xm1xm5xm6, (136) ym = [ym1 ym2 ym3]T := [xm5 xm6 xm7]T , (137)
while in receptor, the dynamics is given by
˙
xs1 =−15xs1+ 15xs5−5xs5xs6xs7, (138)
˙
xs2 =−0.5xs2−25xs6+xs1xs6xs7, (139)
˙
xs3 =−15xs3+ 15xs5−0.1xs1xs2xs7, (140)
˙
xs4 =−15xs4+ 10xs1−xs1xs2xs3, (141)
˙
xs5 =−15xs5+ 10xs7−xs2xs3xs4+u5, (142)
˙
xs6 =−10xs6+ 10xs5+xs2xs3xs5+u6, (143)
˙
xs7 =−5xs7+ 4xs2−1.5xs1xs5xs6+u7, (144) ys = [ys1 ys2 ys3]T := [xs5 xs6 xs7]T , (145)
The error dynamics is defined as e=xs−xm, then
˙
e1 =−15e1+ 15e5−5e5xs6xs7+ 5xm5e6xs7+
+ 5xm5xm6e7, (146)
˙
e2 =−0.5e2−25e6+e1xs6xs7+ 5xm1e6xs7+
+ 5xm1xm6e7, (147)
˙
e3 =−15e3+ 15e5−0.1e1xs2xs7−0.15xm1e2xs7+
−0.1xm1xm2e7, (148)
˙
e4 =−15e4+ 10e1−e1xs2xs3−xm1e2xs7+
−xm1xm2e3, (149)
˙
e5 =−15e5+ 10e7−e2xs3xs4−xm2e3xs4+
−xm2xm3e4−m(t) +u5, (150)
˙
e6 =−10e5+ 10e5+e2xs3xs5+xm2e3xs5+
+xm2xm3e5+u6, (151)
˙
e7 =−5e7+ 4e2−1.5e1xs5xs6−1.5xm1e5xs6+
−1.5xm1xm5e6+u7. (152)
At this point, norm observers are applied in order to provide norm upper bounds
for the unmeasured state variables. The design follows the steps presented in Section 2.3.
In this sense, upper bounds for the unmeasured master variables are obtained through
˙ˆ
xm1 =−13.5ˆxm1+ 17.5|xm5|+ 6|xm5xm6xm7|, (153)
˙ˆ
xm2 =−0.45ˆxm2+ 27.5|xm6|+ 1.1|ˆxm1xm6xm7|, (154)
˙ˆ
xm3 =−13.5ˆxm3+ 11|xm5|+ 0.15|ˆxm1xˆm2xm7|, (155)
˙ˆ
xm4 =−13.5ˆxm4+ 11|ˆxm1|+ 1.1|ˆxm1xˆm2xˆm3|, (156) and, for slave system,
˙ˆ
xs1 =−13.5ˆxs1+ 17.5|xs5|+ 6|xs5xs6xs7|, (157)
˙ˆ
xs2 =−0.45ˆxs2+ 27.5|xs6|+ 1.1|ˆxs1xs6xs7|, (158)
˙ˆ
xs3 =−13.5ˆxs3+ 11|xs5|+ 0.15|ˆxs1xˆs2xs7|, (159)
˙ˆ
xs4 =−13.5ˆxs4+ 11|ˆxs1|+ 1.1|ˆxs1xˆs2xˆs3|, . (160) Basically, the adopted strategy consists in driving the output errorye = [e5, e6, e7] to zero in finite-time, so that the remaining error variables converge exponentially due to the system ISS properties [76]. Therefore, the inputsu5,u6andu7are applied to equations (146), (147) and (152), respectively. When the system is in sliding mode (e5 ≡0,e6 ≡0 and e7 ≡0) we can state that the error dynamics in (146)–(149) becomes
˙
e1 =−15e1, (161)
˙
e2 =−0.5e2+e1xs6xs7, (162)
˙
e3 =−15e3−0.1e1xs2xs7−0.15xm1e2xs7, (163)
˙
e4 =−15e4+ 10e1−e1xs2xs3−xm1e2xs7+
−xm1xm2e3. (164)
Thus, it is clear that the error variablese1,e2,e3 ande4 tend to zero exponentially due to ISS relationship withe5, e6 and e7. In order to reach this condition, the following output-feedback control laws are applied:
u5 = [17(|xs5|+|xm5|) + 12(|xs7|+|xm7|)+
+|ˆxs2xˆs3xˆs4|+|ˆxm2xˆm3xˆm4|+δ5] sgn(e5), (165) u6 = [11(|xs6|+|xm6|) + 12(|xs5|+|xm5|)+
+|ˆxs2xˆs3xs5|+|ˆxm2xˆm3xm5|+δ6] sgn(e6), (166) u7 = [7(|xs7|+|xm7|) + 6(|ˆxs2|+|ˆxm2|)+
+1.5|ˆxs1xs5xs6|+ 1.5|ˆxm1xm5xm6|+δ7] sgn(e7). (167)
Note the messagem(t) appears as a perturbation in e5 dynamics. Thus, by using the equivalent control theory [104], it is possible to recover the message m(t) in receptor side through an approximation by average control. When the system reachs the sliding mode, ˙e5 = 0 andu5 =ueq5, then from (150)
−15e5+ 10e7−e2xs3xs4−xm2e3xs4−xm2xm3e4
| {z }
→0
−m(t) +ueq5 = 0, (168)
Although, in theory, the message could be recovered through ueq5, this signal is not available in practice. Nevertheless, it is possible to find an approximate signal by using the equivalent control theory [104]. According to this theory, an approximation uav for ueq is obtained by filtering the signal u. In our context, the dynamics of uav is given by
τu˙5av(t) =−u5av(t) +u5(t), (169) whereτis an appropriate small constant, such that 0< τ <<1. Once in sliding mode, one can state thatu5av ≈u5eq, then|u5av−u5eq| ≤O(τ)→0. Thus, u5av(t)≈u5eq(t) = m(t).
In the following simulation results, the parameters are chosen as: δ5 = 3,δ6 = 0.01, δ7 = 0.01. The initial conditions are given by: xs(0) = [0.1,0.2,0.3,0.4,0.5,0.6,0.7]T, xm(0) = [−0.1,−0.2,−0.3,−0.4,−0.5,−0.6,−0.7]T and ˆxs(0) = ˆxm(0) = [0,0,0,0]T.
It can be seen in Figure 23 that the state of the error system (146)–(152) in the synchronization problem is driven to the origin by the output-feedback controller in (165)–(167). Thus, the master (130)–(137) and the slave (138)–(145) oscillators become synchronized.
0 5 10 15 20 -0.5
0 0.5 1 1.5
Figure 23: State error signals in the synchronization system.
0 0.2 0.4 0.6
-0.5 0 0.5 1 1.5
The messagemis incorporated toxm5 dynamics. Hence, it is worth remarking that in the secure communication scheme proposed, the signalm is not directely transmitted to the receptor device. This fact, makes our approach more secure than other possible strategies found in literature [103], [102]. However, when the systems are synchronized, the message can be recovered through extended equivalent control theory. The Figure 24 depicts the recovering process of the message for different τ values. The smaller the τ values, the better is the equivalence between uav5 and ueq5. Consequently, the better is also the equivalence betweenuav5 and the original message.
0 0.5 1 1.5 2 -5
0 5
(a)
0 0.5 1 1.5 2
-5 0 5
(b)
0 0.5 1 1.5 2
-5 0 5
(c)
0 0.5 1 1.5 2
-5 0 5
(d)
Figure 24: Control signal (u5), average control (uav5) and original message (m).
Table 2 presents a class of chaotic systems with ISS properties to which the metho- dology presented here holds. The cascade norm observers and the control laws demanded in each case are also presented.
Table 2: ISS Chaotic Systems
System Norm Observers Control Law
Liu et al., (2004) [96]:
˙
x1=−10x1+ 10x2 x˙ˆ1= ¯α1ˆx1+ ¯β1|x2| u=−ρsgn(y)
˙
x2= 40x1−x1x3+u x˙ˆ3= ¯α3ˆx3+ ¯β3xˆ21 ρ= ¯β2|ˆx1|+ ¯γ2|ˆx1ˆx3|+δ
˙
x3=−2.5x3+ 4x21 −10≤α¯1<0 β¯1≥10 β¯2≥40 γ¯2≥1
y=x2 −2.5≤α¯3<0 β¯3≥4
Li. et al.,(2010) [108]:
˙
x1=−12x1+ 12x2 x˙ˆ1= ¯α1ˆx1+ ¯β1|x2| u=−ρsgn(y)
˙
x2= 23x1−x1x3−x2+x4+u x˙ˆ3= ¯α3ˆx3+ ¯γ3(ˆx21+x22) ρ= ¯β2|ˆx1|+ ¯γ2|ˆx1ˆx3|+ ¯α2|x2|+ ¯ξ2|ˆx4|+δ
˙
x3=x1x2−2.1x3 x˙ˆ4= ¯α4ˆx4+ ¯β4|x2| β¯2≥23 ¯γ2≥1
˙
x4=−0.2x4−6x2 −12≤α¯1<0 β¯1≥12 α¯2≥1 ξ¯2≥1
y=x2 −2.1≤α¯3<0 ¯γ3≥1
−0.2≤α¯4<0 β¯4≥6 Liu et al., (2009) [109]:
˙
x1=−15x1+ 15x2 x˙ˆ1= ¯α1ˆx1+ ¯β1|x2| u=−ρsgn(y)
˙
x2=−x1x3+ 10x2+x4+u x˙ˆ3= ¯α3ˆx3+ ¯γ3(ˆx21+x22) ρ= ¯β2|ˆx4|+ ¯γ2|ˆx1ˆx3|+ ¯α2|x2|+δ
˙
x3=−5x3+x1x2 x˙ˆ4= ¯α4ˆx4+ ¯β4xˆ2 α¯2≥10 β¯2≥1
˙
x4=−x4+x3 −15≤α¯1<0 β¯1≥15 γ¯2≥ −1
y=x2 −5≤α¯3<0 ¯γ3≥1
−1≤α¯4<0 β¯4≥1 Elabbasy et al., (2006) [110]:
˙
x1=−36x1+ 36x2 x˙ˆ1= ¯α1ˆx1+ ¯β1|x2| u=−ρsgn(y)
˙
x2=−x1x3+ 20x2+u x˙ˆ3= ¯α3ˆx3+ ¯γ3(ˆx21+x22) ρ= ¯γ2|ˆx1xˆ3|+ ¯α2|x2|+δ
˙
x3=−3x3+x1x2 −36≤α¯1<0 β¯1≥36 α¯2≥1 ¯γ2≥20
y=x2 −3≤α¯3<0 ¯γ3≥1
Liu et al., (2014) [111]:
˙
x1=−8x1+ 8x2 x˙ˆ1= ¯α1ˆx1+ ¯β1|x2| u=−ρsgn(y)
˙
x2= 40x1−1.5x1x3−1.2x4+u x˙ˆ3= ¯α3ˆx3+ ¯β3|x2|+ ¯γ3(ˆx21+x22) ρ= ¯β2|ˆx1|+ ¯γ2|ˆx1ˆx3|+ ¯ξ2|ˆx4|+δ
˙
x3= 1.5x1x2−103x2−4x3 x˙ˆ4= ¯α4ˆx4+ ¯β4xˆ1 β¯2≥40 γ¯2≥1.5
˙
x4=−2x1−2x4 x˙ˆ5= ¯α5ˆx5+ ¯γ5(ˆx23+x24) ξ¯2≥1.2
˙
x5=−2x3x4−2x5 −8≤α¯1<0 β¯1≥8 y=x2 −4≤α¯3<0 ¯β3≥103 ˆγ3≥1.5
−2≤α¯4<0 β¯4≥2
−2≤α¯5<0 ¯γ5≥2 L¨u and Chen, (2002a) [112]:
˙
x1=−30x1+ 30x2 x˙ˆ1= ¯α1ˆx1+ ¯β1|x2| u=−ρsgn(y)
˙
x2=−x1x3+ 22.2x2+u x˙ˆ3= ¯α3ˆx3+ ¯γ3(ˆx21+x22) ρ= ¯α2|x2|+ ¯γ2|ˆx1ˆx3|+δ
˙
x3=−8.83x3+x1x2 −30≤α¯1<0 β¯1≥30 α¯2≥22.2 ¯γ2≥1
y=x2 −8.83 ≤α¯3<0 γ¯3≥1
L¨u and Chen, (2002b) [113]:
˙
x1= (25φ+ 10)(−x1+x2) x˙ˆ1= ¯α1ˆx1+ ¯β1|x2| u=−ρsgn(y)
˙
x2= (28−35φ)x1−x1x3+ (29φ−1)x2+u x˙ˆ3= ¯α3ˆx3+ ¯γ3(ˆx21+x22) ρ= ¯β2|ˆx1|+ ¯α2|x2|+ ¯γ2|ˆx1xˆ3|+δ
˙
x3=−(8+φ)3 x3+x1x2 −83≤α¯3<0 −10≤α¯1<0 α¯2≥28 γ¯2≥1
y=x2 ¯γ3≥1 β¯1≥35 β¯2≥28
Chen and Ueta, (1999) [114]:
˙
x1=−35x1+ 35x2 x˙ˆ1= ¯α1ˆx1+ ¯β1|x2| u=−ρsgn(y)
˙
x2=−7x1−x1x2+ 28x2+u x˙ˆ3= ¯α3ˆx3+ ¯γ3(ˆx21+x22) ρ= ¯α2|x2|+ ¯γ2|ˆx1ˆx3|+ ¯β2|ˆx1|+δ
˙
x3=−3x3+x1x2 −35≤α¯1<0 β¯1≥35 α¯2≥28 β¯2≥7
y=x2 −3≤α¯3<0 ¯γ3≥1 γ¯2≥1
Lorenz, (1963) [88]:
˙
x1=−10x1+ 10x2 x˙ˆ1= ¯α1ˆx1+ ¯β1|x2| u=−ρsgn(y)
˙
x2= 28x1−x1x3−x2+u x˙ˆ3= ¯α3ˆx3+ ¯γ3(ˆx21+x22) ρ= ¯α2|x2|+ ¯γ2|ˆx1ˆx3|+ ¯β2|ˆx1|+δ
˙
x3=−83x3+x1x2 −10≤α¯1<0 β¯1≥10 α¯2≥1 β¯2≥28
y=x2 −83≤α¯3<0 ¯γ3≥1 γ¯2≥1
Liu et al., (2009) [109]:
˙
x1=−x1−x22 x˙ˆ1= ¯α1ˆx1+ ¯β1x22 u=−ρsgn(y)
˙
x2= 2.5x2−4x1x3 x˙ˆ3= ¯α3ˆx3+ ¯γ3(ˆx21+x22) ρ= ¯α2|x2|+ ¯γ2|ˆx1ˆx3|+δ
˙
x3=−5x3+ 4x1x2 −1≤α¯1<0 β¯1≥1 α¯2≥2.5 γ¯1≥4
y=x2 −5≤α¯3<0 ¯γ3≥4
CONCLUSION
In Chapter 1, two different control strategies based on sliding mode control were used in order to safely regulate the blood glucose concentration of a T1DM patient. From a control point of view, this is a challenging problem since the process is represented by a nonlinear and time-varying model, where the parameters were considered uncertain. Meals of varying carbohydrate content are understood as unmatched disturbances. The output has relative degree two with regard to control signal, and since the model adopted describes a biological process, the control problem must be treated as a strictly positive system.
Such challenges were overcome by using some upper bounds for the plant para- meters and disturbances as well as a bihormonal actuator formulation with the sliding mode control theory. A rigorous stability analysis through Lyapunov functions and nu- merical simulations show that the proposed control strategy was efficient in the glycemic regulation task.
In Chapter 2, exploring the ISS properties from a class of hyperchaotic systems (see Table 2), control laws based on sliding modes and norm observers in cascade can be developed to the stabilization problem. Global stability properties have been demons- trated using only output feedback, even considering that all the parameters of a seventh dimension system were uncertain. Indeed, the proposed controller was shown to be robust to parametric uncertainties and exogenous disturbances.
In addition, the proposed sliding mode controller seems to be an attractive metho- dology for synchronization, since the convergence of the output signal can be achieved in finite time and the convergence of the other state variables is exponential, being also robust to exogenous disturbances. The application of the synchronization approach to secure communication systems was successfully evaluated. This method provides reliable cryptography using an equivalent control strategy.
As future works, we propose the experimental validation of the presented control- lers. Moreover, in an eventual application, we could replace the fixed-gain differentiator presented here by a variable-gain differentiator [115] in order to mitigate the sensitivity of the control system in closed-loop with respect to the measurement error/noise. At last, we suggest the assessment of different sliding mode control techniques aiming to achieve the euglycemia for patients with T1DM.
Publications, Participation in Events and Awards
In the course of my master course that has lead to this dissertation, a plenty of cientific works were published in conferences. Among them,
• DANSA, M. ; RODRIGUES, V. H. P. ; OLIVEIRA, T. R. . Best Paper Presen- tation in the Session “Epidemic Modeling and Control” of the 22nd International Conference on System Theory, Control and Computing (ICSTCC 2018), IEEE - Institute of Electrical and Electronics Engineers, Sinaia, Romania, October 2018.
• DANSA, M. ; RODRIGUES, V. H. P. ; OLIVEIRA, T. R. . Controlador Bi- hormonal por Modos Deslizantes Aplicado `a Regula¸c˜ao de Glicemia em Pacientes Diab´eticos Tipo 1. In: XXII Congresso Brasileiro de Autom´atica (CBA 2018), Jo˜ao Pessoa. Setembro de 2018.
• DANSA, M. ; RODRIGUES, V. H. P. ; OLIVEIRA, T. R. . Bihormonal Sliding Mode Controller Applied to Blood Glucose Regulation in Patients with Type 1 Diabetes. In: 22nd International Conference on System Theory, Control and Com- puting (ICSTCC 2018), Sinaia. October 2018. p. 79-86.
• DANSA, M. ; RODRIGUES, V. H. P. ; OLIVEIRA, T. R. . Blood Glucose Regula- tion Through Bihormonal Non-Singular Terminal Sliding Mode Controller. In: 15th International Conference on Control, Automation, Robotics and Vision (ICARCV 2018), Downtown Core. November 2018. p. 474-479.
• DANSA, M. ; RODRIGUES, V. H. P. ; OLIVEIRA, T. R. . Blood Glucose Regula- tion in Patients with Type 1 Diabetes by means of Output-Feedback Sliding Mode Control. In: Ahmad Taher Azar. (Org.). Control Applications for Biomedical Engineering Systems. XXed.: Elsevier, 2019, p. 1–30.
• DANSA, M. ; RODRIGUES, V. H. P. ; OLIVEIRA, T. R. . Regula¸c˜ao de Gli- cemia Atrav´es de um Controlador Bi-hormonal por Modos Deslizantes Terminal N˜ao-Singular. In: XIV Simp´osio Brasileiro de Automa¸c˜ao Inteligente (SBAI 2019), Ouro Preto. Outubro de 2019.
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