3. B-spline Adaptive Backstepping Controllers for Nonaffine Nonlinear
4.3 Stability Analysis
For the convenience of stability analysis, the following lemma be presented in [29] of bounded-input and bounded-output property for stable dynamic inequalities.
Lemma 4-1: Let function V(t)≥0 be continuous function defined ∀t∈R+ and V(0) bounded, and κ(t)∈ L∞ be real valued function. If the following inequality holds:
)
The following theorem shows the stability and control law of the system (4.1)
Theorem 4-1: Consider the nonlinear nonaffine systems (4.1).
(a) If the update laws of mean estimation B-spline neural networks in step1− − are n 1
The B-spline neural controller with the update laws (4.51) is designed as (4.47),
where the B-spline neural control input is given as (4.42). The robust controller
By differentiating (4.52) with respect to time and using (4.32) (4.36) (4.38) and (4.51), and after some manipulations, we obtain
ˆ )
(4.55)
Substituting (4.54), (4.55), (4.56) and (4.57) into (4.53), we obtain
2 )
(4.60) become
∑
2qi >hifup2. Then (4.61) become
Define Lyapunov function as
∑∑
= =By differentiating (4.64) with respect to time and
∑∑
Using (4.45) ,(4.51), and after some manipulations, we obtain
2
. Using lemma 1, we can conclude
that
∑
− using the Barbalat’s lemma, it implies that lim =0∞
→ n
t Z . i.e., the system output
x can track the reference signal 1 y . This completes the proof. m
4.4 Simulation Results
In this section, we present a simulation example. Consider the third-order uncertain nonlinear system
) , , , (
) , (
) (
3 2 1 3 3
3 2 1 2 2
2 1 1 1
u x x x f x
x x x f x
x x f x
=
+
=
+
=
(4.71)
where f1 = , x12 f2 =cos(x1+x2), f3= − −x2 x12+ +x3 2u+sin(ux3). Using following design procedure
Step 1) Select the positive design parameters are c1=c2 =c3 =15.5.
Step 2) Choose an appropriate value for
ρ
=0.1 appropriate values for,1 ,2 ,3 1.5
i i i
γ =γ =γ = , σi =0.15 in (4.50) where i=1,2 and
,1 ,2 ,3 0.5
n n n
γ =γ =γ = in (4.51).
Step 3) Determine the order of the B-spline function is select as k=3 and the number of knots points is twelve.
Step 4) Determine the parameter h1 = h2 =0.005 of first-order filter.
Step 5) Obtain the update laws (4.50) and virtual controllers are
1 1 1
2 ˆ
ˆ F y c z
X =− + m− and Xˆ3 =−Fˆ2 −c2z2 −z1 . The reference signal is ym =sin(t).
Step 6) Obtain the update laws (4.51), and the control law, including the B-spline neural control input in (4.42). The robust controller is
2 3
2ρ ua = −z .
In this case, the simulation results are show in Fig. 4.1. - Fig. 4.3. For
compare the tracking performance and control output, we assume f , 1 f are 2 known function, and c1 =c2 =c3 =20.5. The simulation results are shown in Fig. 4.4. - Fig. 4.6.
Fig. 4.1. Output y tracking reference signal y m
0 2 4 6 8 10 12 14 16 18 20
-1 -0.5 0 0.5 1 1.5
time(sec)
y
ym
Fig. 4.2. Control output u
0 2 4 6 8 10 12 14 16 18 20
-6000 -5000 -4000 -3000 -2000 -1000 0 1000 2000
time(sec)
u
Fig. 4.3. Error function z 1
0 2 4 6 8 10 12 14 16 18 20
-0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
time(sec)
z1
Fig. 4.4. Output y tracking reference signal y m with know function f and 1 f 2
0 2 4 6 8 10 12 14 16 18 20
-1.5 -1 -0.5 0 0.5 1 1.5
time(sec)
ym
y
Fig. 4.5. Control output u with know function f and 1 f 2
0 2 4 6 8 10 12 14 16 18 20
-14000 -12000 -10000 -8000 -6000 -4000 -2000 0 2000 4000 6000
time(sec)
u
Fig. 4.6. Error function z with unknown function 1 f and 1 f 2
0 2 4 6 8 10 12 14 16 18 20
-0.2 0 0.2 0.4 0.6 0.8 1 1.2
time(sec)
z1
Chapter 5 Conclusion
Three control laws using B-spline adaptive backstepping controller have been proposed in this thesis, one for affine nonlinear systems, one for nonaffine nonlinear systems and one for nonaffine nonlinear systems with first order filters for deriving the virtual controller.
In chapter 2, an adaptive law to tune the knot vectors of B-spline basis functions was proposed with the mean value theorem expansion replacing the Taylor expansion linearization to get rid of high order terms. Theorem 2-1 proves that, for the ideal backstepping controller and virtual controller, all closed-loop signals are bounded and the state x tracks the bounded reference 1 signal arbitrarily closely. Theorem 2-2 proves the same for the B-spline
adaptive backstepping controller. Finally, the simulation results support the theoretical arguments about the tracking performance of both controllers.
In chapter 3, the B-spline adaptive backstepping controller for nonaffine nonlinear systems was proposed. It was comprised of a mean estimation B-spline neural network and a robust controller. The B-spline basis functions are recursively defined, and it was difficult to differentiate them. The mean estimation technique avoids this difficulty. In addition, two different robust controllers are designed to compensate for unmodelled dynamics. Computer simulation shows that the proposed controllers still provide good tracking performance even though the derivatives of the B-spline basis functions are unknown. A comparative study between the two robust controllers was given.
In chapter 4, tackles the explosion of complexity problem for a class of nonaffine nonlinear systems. We add a first order filter at each virtual controller of the controller derived in chapter 3. Computer simulation shows that the
proposed controller has good transient response and steady-state tracking performance.
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