Simulation Results

In this section, two examples are proposed. Example 1 is second-order unknown nonlinear system. Example 2 is third-order unknown nonlinear system. Theorem 1 is used to design the ideal backstepping controller and theorem 2 is used to design the B-spline adaptive backstepping controller.

2.1) Example: Consider the second-order unknown nonlinear system

) ( )) cos(

2 ( ) cos(

12 1

.

0 _{2 1}³ _{1 1}

2 2 1

t u x x

x x x

x x

+ + +

−

−

=

= (2.43)

By using theorem 1, the ideal backstepping controller can be expressed

follow equation _{1 1 1 2 2 2 1}

1 2

1 ( ( , )

( , )

ideal m

u y c z f x x c z z )

g x x

= − − − − . Because

) , (x₁ x₂

g and f(x₁,x₂) are unknown functions, the adaptive B-spline neural

network identifier controller can be designed as uˆ =θˆ^Tξˆ(q,Tˆ). By using theorem 2, the adaptive laws are θ~ θˆ Φ^TTˆ

2 1z γ

−

=

−

= and T~ Tˆ Φθˆ

2 2z γ

=

−

= .

Robust controller is ²₂ 2ρ

u_a =− z . Reference signal is y_m =sin(t). Learning

rates are γ₁ =5 ,γ₂ =0.5 . Initial condition point is (0.5,0.5). Damping constants are c₁=2.5,c₂ =2.5. B-spline order is k =3. The knot vectors

are t₀ =−5,t₁ =−4,t₂ =−3, ,t₁₂ =6 . The simulation results of ideal backstepping controller are show in Fig. 2.3-Fig. 2.5. The simulation results of adaptive B-spline neural network controller forρ =0.1 are shown in Fig.

2.6-Fig. 2.8, ρ =0.2 are shown in Fig. 2.9-Fig. 2.11, ρ=0.3 are shown in Fig. 2.12-Fig. 2.14. The mean square errors of ρ =0.1, ρ=0.2 and

3 .

=0

ρ are shown in Table 2.1. By simulation results, the better tracking performance can be achieved as ρ is chosen smaller. The mean square error (MSE) is shown as follow:

∑

=

−

= ^m

i mi mi

MSE m

1

)2

( ˆ

1 (2.44)

where i is the index of the m points over which the MES is computed, m _i is the actual output of the system, and mˆ is the estimated output of the _i system.

Table 2.1 Three cases of ρ =0.1ρ =0.2andρ =0.3 Attenuation constant (ρ ) Mean

square error 1

.

=0

ρ 0.0007412 2

.

=0

ρ 0.0016021 3

.

=0

ρ 0.004711

0 5 10 15 20 -1.5

-1 -0.5 0 0.5 1 1.5

time

y ym

(sec)

Fig. 2.3 Output y follow reference signaly with ideal _m backstepping controller

0 5 10 15 20

-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6

time

error

(sec)

Fig. 2.4. Errory_m− with ideal backstepping controller y

0 5 10 15 20 -7

-6.5 -6 -5.5 -5 -4.5 -4 -3.5 -3 -2.5 -2

time

u

(sec)

Fig. 2.5 Control input u with ideal backstepping controller

0 2 4 6 8 10

-1 -0.5 0 0.5 1 1.5

time

y ym

(sec)

Fig. 2.6 Output y follow reference signaly with B-spline _m adaptive backstepping controller (ρ =0.1)

0 2 4 6 8 10 0

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

time

error

(sec)

Fig. 2.7 Errory_m− with B-spline adaptive y backstepping controller (ρ =0.1)

0 2 4 6 8 10

-250 -200 -150 -100 -50 0 50 100

time

u

(sec)

Fig. 2.8 Control input u with B-spline adaptive backstepping controller (ρ =0.1)

0 2 4 6 8 10 -1

-0.5 0 0.5 1 1.5

time

y ym

(sec)

Fig. 2.9 Output y follow reference signaly with B-spline adaptive _m backstepping controller (ρ =0.2)

0 2 4 6 8 10

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

time

error

(sec)

Fig. 2.10. Errory_m − with B-spline adaptive y backstepping controller (ρ =0.2)

0 2 4 6 8 10 -1

-0.5 0 0.5 1 1.5

time

y ym

(sec)

Fig. 2.12 Output y follow reference signaly with B-spline adaptive _m backstepping controller (ρ =0.3)

0 2 4 6 8 10

-60 -50 -40 -30 -20 -10 0 10

time

u

(sec)

Fig. 2.11 Control input u with B-spline adaptive backstepping controller (ρ =0.2)

0 2 4 6 8 10 0

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

time

error

(sec)

Fig. 2.13 Errory_m− with B-spline adaptive y backstepping controller (ρ =0.3)

0 2 4 6 8 10

-25 -20 -15 -10 -5 0 5

time

u

(sec)

Fig. 2.14. Control input u with B-spline adaptive backstepping controller (ρ =0.3)

2.2) Example: Consider the third-order unknown nonlinear system

By using theorem 2.1, the ideal backstepping controller can be expressed

follow equation 1 _{1 1 2 2 3 3 1 2} results of ideal backstepping controller are shown in Fig. 2.15-Fig. 2.17. The simulation results of adaptive B-spline neural network controller for ρ =0.2 are shown in Fig. 2.18~Fig. 2.20, ρ =0.3 are shown in Fig. 2.21-Fig. 2.23,

0 5 10 15 20 -0.1

0 0.1 0.2 0.3 0.4 0.5 0.6

time

error

(sec)

Fig. 2.16 Errory_m− with ideal backstepping controller y

0 5 10 15 20

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

time

u

(sec)

Fig. 2.17 Control input u with ideal backstepping controller

0 2 4 6 8 10 -1.5

-1 -0.5 0 0.5 1 1.5

time

y ym

(sec)

Fig. 2.18 Output y follow reference signaly with B-spline _m adaptive backstepping controller (ρ =0.2)

0 2 4 6 8 10

-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6

time

error

(sec)

Fig. 2.19 Errory_m− with B-spline adaptive backstepping y controller (ρ=0.2)

0 2 4 6 8 10 -120

-100 -80 -60 -40 -20 0 20 40

time

u

(sec)

Fig. 2.20 Control input u with B-spline adaptive backstepping controller (ρ =0.2)

0 2 4 6 8 10

-1.5 -1 -0.5 0 0.5 1 1.5

time

y ym

(sec)

Fig. 2.21 Output y follow reference signaly with B-spline adaptive _m backstepping controller (ρ =0.3)

0 2 4 6 8 10 -0.1

0 0.1 0.2 0.3 0.4 0.5 0.6

time

error

(sec)

Fig. 2.22 Errory_m− with B-spline adaptive y backstepping controller (ρ=0.3)

0 2 4 6 8 10

-50 -40 -30 -20 -10 0 10 20 30

time

u

(sec)

Fig. 2.23 Control input u with B-spline adaptive backstepping controller (ρ =0.3)

0 2 4 6 8 10 -1.5

-1 -0.5 0 0.5 1 1.5

time

y ym

(sec)

Fig. 2.24 Output y follow reference signaly with B-spline _m adaptive backstepping controller (ρ =0.4)

0 2 4 6 8 10

-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6

time

error

(sec)

Fig. 2.25 Errory_m− with B-spline adaptive y backstepping controller (ρ =0.4)

0 2 4 6 8 10 -30

-25 -20 -15 -10 -5 0 5 10 15

time

u

(sec)

Fig. 2.26 Control input u with B-spline adaptive backstepping controller (ρ =0.4)

Chapter 3

B-spline Adaptive Backstepping Controllers for Nonaffine Nonlinear Systems

In this chapter, B-spline adaptive backstepping scheme for nonaffine nonlinear systems is proposed. The control scheme incorporates the backstepping design technique with the mean-estimation B-spline neural networks which are utilized to estimate the system dynamics. The mean-estimation B-spline neural networks use a mean estimation technique to develop the update laws for the design of online adaptive controllers. In doing so, differentiating B-spline basis functions is not required, and therefore the computation burden can be mitigated. In addition, two kinds of robust controllers are used to compensate unmodeling dynamics. Finally, computer simulation results are shown to demonstrate the applicability of the proposed scheme, and comparison between the two robust controllers is given by the simulation results.

This chapter is organized as follows. The system problem is formulated in section 3.1. The ideal backstepping controller and virtual controller is described in section 3.2. In section 3.3, a B-spline adaptive backstepping controller is described. Computer simulation results are shown in section 3.4. Finally, some conclusions are given in section 3.5.

3.1 Problem Formulation

Consider the nonaffine nonlinear systems expressed in or transformed into a state-space form

( , )

xn = f x u (3.1) where x=[ ,x x_{1 2}, x_n]^T =[ , ,x x x⁽ⁿ⁻¹⁾]^T ∈Rⁿ is the state vector, and u∈R is the control input. The system function f( , )x u is unknown continuous

function, and it is assumed that 0< ∂ ∂ < ∞ . The control objective is to f u design a controller for system (3.1) such that all the signals in the closed-loop system are stable and the state x₁can track a bound reference signal y _m arbitrarily closely.

3.2 Design of Backstepping Controllers for Known Systems

Consider the nth-order nonlinear system (3.1) where the system function ( , )

f x u is known. By using backstepping technique, an ideal backstepping controller can be expressed follow:

Step 1) Define a tracking error as

z₁ =x₁−y_m (3.2) Then, differentiating z can be expressed as ₁

z₁=x₁−y_m (3.3) Define a virtual controller as

α₁ = y_m −c₁z₁ (3.4) where c₁ >0 is a design parameter. From (3.3) and (3.4), if α₁= , x₁ then we have lim ₁ 0

t z

→∞ → , that is, the state trajectory x can ₁ asymptotically track the reference signal y_m. Thus, define an error state as z₂ =x₁−α₁ =x₂ −α₁. Then, our next goal is to force the error state z to decay to zero. By using (3.3) and the fact that ₂

1 2 1

x =z + , Eq. (3.3) can be rewritten as α

z₁=z₂−c z_{1 1} (3.5) Step 2) Differentiating z can be expressed as ₂

z₂ =x₂−α₁ =x₃−(−c₁z₁+y_m) (3.6) Likewise, define a virtual controller as

α₂ = y_m−c₁z₁−c₂z₂ −z₁ (3.7) where c₂ >0 is a design parameter. Moreover, define an error state

2 3 3 = x −α

z . Then, by using (3.6) and the fact that x₂ = +z₃ α₂, equation (3.6) can be rewritten as

z₂ = −z₃ c z_{2 2} − (3.8) z₁ Step 3) Let k be a positive integer. Define an error state as z_k =x_k −α_k−1.

Then, differentiating z is _k

z_k =x_k −α_k−1 (3.9) where 3≤ ≤ − . Define a virtual controller as k n 1

1

( ) ( ) ( 1 )

1 1

k k

k k i k j

k m i i j

i j

y c z z

α ^{− − − −}

= =

= −

∑

−

∑

^{, where ci} > is a design parameter. ⁰ Moreover, define an error state as z_k+1 =x_k+1−α_k. Then, by using the

fact that x_k = z_k+1+α_k, equation (3.9) can be rewritten as z_k = z_k+1−c z_{k k} −z_k−1 (3.10)

Step4) Differentiating z_n can be expressed as

−1

−

= _{n n}

n x

z α (3.11) Defineϕ= f(x,u)+c_nz_n −α_n−1+z_n−1. By the assumption that

0< ∂ ∂ < ∞ and implicit function theorem, there exists a ideal f u backstepping controller u_ideal such that ϕ(x,u_ideal)=0. By adding and subtracting the term c z_{n n} −α_n−1+z_n−1 in (3.1) and using the fact that

( ,u_ideal) 0

ϕ x = , we obtain x_n = −c z_{n n} −z_n−1+α_n−1. As a result, Eq.

(3.11) can be rewritten as

n n n n 1

z = −c z −z ₋ (3.12)

Step5) Consider the Lyapunov function as follows

∑

=

= ⁿ

i

zi

V

1 2

2

1 (3.13) By differentiating (3.13) and using (3.5), (3.8), (3.10) and (3.12), we have

² _{1 1}²

1 1

n n

i i i i

i i

V z z c z c z

= =

=

∑

= −

∑

≤ − (3.14) From (3.13) and (3.14), we can conclude that z_i is bounded.

Moreover, from (3.5), z₁ is also bounded. Integrating (3.14), we get

2

1 1

0^∞z ( )τ τd = −( ( )V ∞ −V(0)) c

∫

(3.15) Because of the fact that the right side of (3.15) is bounded, we have

1 2

z ∈ . According to Barbalat’s Lemma, we have L lim ₁ 0

t z

→∞ = , that is, the state trajectory x₁ can asymptotically track the reference signal y . _m

3.3 Design of B-spline Adaptive Backstepping Controllers for Unknown Systems

Because the system function is unknown, the ideal backstepping controller can not be precisely obtained. To solve this problem, the B-spline adaptive backstepping controller is proposed for the unknown nonlinear system (3.1), instead of the ideal backstepping controller.

The network structure and features of the b-spline neural network are shown in section 2.3. According to the universal approximation theorem, let

σ

σ = +

+

= θ^∗Tξ(q,T^∗)

ideal u

u ^* , where σ denotes an approximation error, T^∗ is an optimal knot vector, θ^∗ is an optimal parameter vector. According to the definition of the B-spline neural networks, define the B-spline neural control input as

) T ξ(q, θˆ^T ˆ ˆ=

u (3.16) where uˆ, andT are the estimation of ˆ u^∗, and T^∗, respectively. In the following, the update laws of uˆ, andT will be developed for the B-spline neural control ˆ input (2.16) by using the mean estimation technique, and such B-spline neural networks with the mean estimation technique is called the mean estimation B-spline neural networks.

By using the mean value theorem, there exist a point ˆT - δ between T and ^* T , such that ˆ

ˆ ˆ ˆ

( )− ( )= ∂ ( )

∂

* *

T=T-δ

ξ q,T ξ q,T ξ | T - T

T (3.17) Define ξ ξ q T= ( , ^*)−ξ q T , ( , )ˆ T T= ^*−T , and ˆ = ∂ _ˆ

∂ ^T=T-δ Φ ξ |

T . Then, by adding and subtracting the term Φ T ΦT , we obtain ^* + ˆ

= ˆ + +

ξ ΦT ΦT d (3.18) where Φ Φ= ^*−Φ , ˆ d=(Φ Φ T . Therefore, according to (3.18), let − ^*)

u u

u~= _ideal −ˆ. By substituting u_ideal =θ ξ(q,T )^{∗T ∗} +σ and uˆ=θˆ^Tξ(q,Tˆ) in u

u

u~= _ideal −ˆ, and adding and subtracting the term θ ξ q,T , we obtain ˆ^T ( ^*)

* ˆ

( )

T T

u =θ ξ q,T +θ ξ+σ (3.19) where θ~=θ*−θˆ. Then, by adding and subtracting the term θ ξ q,T , Eq. ^T ( ˆ) (3.19) can be rewritten as

ˆ ( ˆ)

T T T

u =θ ξ θ ξ θ ξ q,T+ + +σ (3.20) By substituting ξ ΦT ΦT d in (3.20) and after some manipulations, we = ˆ + + obtain

1 1

ˆ ˆ

ˆ ˆ ˆ ˆ ˆ

( ( ) )

m n

T T T

ij j i

i j

u Φ Tθ ε

= =

=^{θ ξ q,T} −^{Φ T} +^{T Φθ}−

∑∑

+ (3.21) where ε =θ~_TΦ~_TT~+θˆ_TΦ~_TT^*+θ^*d+σ

, and ^* ˆ

ij ij ij

Φ =Φ −Φ , which are the elements in the ith row and jth column of the matrix Φ.

Next, utilizing (3.21) to derive the update law for the mean estimation B-spline neural networks and developing the B-spline adaptive backstepping

controller for the nonaffine systems (3.1) are given as follows.

By using mean value theorem, the function ϕ = f( , )x u +c z_{n n} −α_n−1+z_n−1 can be rewritten as ( )ϕ u =ϕ(u_ideal)+g( )(x u u− _ideal), where u_ideal < < and u u

( ) |_{u u}

g x = ∂ ∂ϕ u ₌ . Because of the fact thatϕ⁽u_ideal⁾=⁰, we have ( )u g( )(u u_ideal)

ϕ = x − (3.22) For developing the B-spline adaptive backstepping controller for the

nonaffine systems (3.1), we assume that there exist positive constants ofg_uand glsuch thatg_l ≤ g(x)≤g_u.

From (3.22), Eq. (1) can be rewritten as

x_n =g( )(x u u− _ideal)−c z_{n n} +α_n−1−z_n−1 (3.23) Define the B-spline adaptive backstepping controller as

ua

u

u = ˆ+ (3.24) where u is a robust controller. By substituting (3.11) into (3.23) _a

( ) ( ) 1

n a n n n

z = −g x u g+ x u −c z −z ₋ (3.25) On the basis of the above discussion, the following theorem can be obtained.

Theorem3.1 : Consider the nonlinear nonaffine systems (3.1). Suppose that the update laws of the mean estimation B-spline neural networks are

1

2

3

ˆ ( ( ˆ) ˆ ˆ) ˆ

ˆ ˆ

ˆ ˆ ˆ

T n

n

ij ij n j i

z z

z T γ

γ

γ θ

= − = −

= − = Φ = −Φ = −

θ θ ξ q,T Φ T

T T Φθ (3.26)

where γ1, γ2, γ3are the positive learning rates. The B-spline adaptive

backstepping controller with the update laws (3.26) is designed as (3.24), where the B-spline neural control input ˆu is given as (3.16). Then, according to the design of the robust controller, the following properties are guaranteed.

(a). Suppose that

2ρ2ⁿ a

u −z

= (3.27) where ρ is a prescribed attenuation constant. Then, all the signals in

the closed-loop are bounded, and the following inequality is satisfied. where α is a constant described later. Then, all the signals in the closed-loop are bounded, and lim 1

( )

0

t z t

→∞ = .

▲ Proof of (a): Define a Lyapunov function as

2

By differentiating (3.30) with respect to time and using (3.5) (3.8) (3.10) and (3.23), and after some manipulations, we obtain

2 Substituting (3.26) into the above equation, we obtain

2

. By substituting (3.37) into (3.33), using the fact

that g( )x g_l ≥1, and completing the squares, we obtain

2 2 manipulations, we obtain

2 by using projection method to avoid the parameter drift phenomenon, the assumption that ζ ≤ is reasonable. Let ω α ≥ω. By substituting (3.29) into

According to (3.34) and Barbalat’s lemma, we can easily prove that all the closed-loop signals are bounded and lim 1

( )

0

t z t

⎪⎪

⎩

⎪⎪

⎨

⎧

<

−

≤

>

=

β β β

β

n n n

n

n

z z z

z z

sat

, 1

, , 1 )

( (3.38)

where β > . The design procedure of the B-spline adaptive backstepping 0 controller is summarized in the following.

Step 5) Select the positive design parameters c_i > , i=1,2,…,n. 0 Step 6) Choose a constant α in (3.29) or an appropriate value for ρ in

(3.27), and appropriate values for γ₁ and γ₂in (3.26), respectively.

Step 7) Determine the order of the B-spline function and the number of knots for T . Then, compute the basis vector ξ in (3.16).

Step 8) Obtain the update laws (3.26), and the control law, including the B-spline neural control input in (3.16) and the robust controller in (3.24) or (3.25).

3.4 Simulation Results

Example3.1: Consider the second-order unknown nonlinear system

1 2

3

2 0.1 2 1 12 cos( )1 (2 cos( )) ( )1

x x

x x x x x u t

=

= − − + + + (3.39)

The adaptive laws are θˆ = −γ_{1 2}z ( (ξ q,Tˆ)−Φ T ,ˆ ˆ^T ) Tˆ = −γ_{2 2}zΦθ , ˆ ˆ ˆ _{3 2}ˆ ˆ

ij γ θz Ti j

Φ = .

The reference signal is given as y_m =sin(t). Letγ₁ =5, γ₂ =5,c₁ =5,c₂ =5, and

3 =5

c . The initial states are set as x(0)=[0.5, 0.5]. The order of the B-spline neural network approximator is selected as k=3, and the number of its knot points is twelve. The robust controller is selected as u_a =−αsat(z_n), where

=3

β and α =5, and in this case, the simulation results are shown in Fig. 3.1.

The mean square errors (MSE) of u_a =−αsat(z_n) is 0.0029445. In addition, the robust control is selected as ₂

2

n a

u z

ρ

= − , and in this case, simulation results

are shown in Figs. 3.2-3.4, and the MSE of the tracking output error for {0.1, 0.3, 0.5}

ρ = are shown in Table 3.1. Figs. 3.2-3.4 show the results of 1

.

=0

ρ , ρ =0.3, andρ =0.5, respectively. From simulation results, as ρ is chosen smaller, the better tracking performance can be achieved at the expense of the larger control input u.

Table 3.1 Three cases ofρ =0.1, ρ =0.3 and ρ =0.5 Attenuation constant (^ρ) Mean square error

1 .

=0

ρ 0.0026799 3

.

=0

ρ 0.0033743 5

.

=0

ρ 0.0051655

Fig. 3.1. The simulation results when u_a =−αsat(z_n).

0 5 10

-2 -1 0 1 2

time(sec)

0 5 10

-40 -20 0 20

time(sec)

u

0 5 10

-0.2 0 0.2 0.4 0.6

time(sec)

z1

0 5 10

-1 0 1 2

time(sec)

z2

y

ym

Fig. 3.2. The simulation results when 2ρ2ⁿ

a

u =−z , where ρ=0.1

0 5 10

-1 0 1 2

time(sec)

0 5 10

-150 -100 -50 0 50

time(sec)

u

0 5 10

0 0.2 0.4 0.6 0.8

time(sec)

z1

0 5 10

-1 0 1 2

time(sec)

z2

y

ym

Fig. 3.3. The simulation results when 2ρⁿ2

a

u =−z , where ρ =0.3

0 5 10

-2 -1 0 1 2

time(sec)

0 5 10

-15 -10 -5 0 5

time(sec)

u

0 5 10

-0.2 0 0.2 0.4 0.6

time(sec)

z1

0 5 10

-1 0 1 2

time(sec)

z2

y

ym

Fig. 3.4. The simulation results when 2ρ2ⁿ

a

u =−z , where ρ=0.5

0 5 10

-1 0 1 2

time(sec)

0 5 10

-5 0 5

time(sec)

u

0 5 10

-0.2 0 0.2 0.4 0.6

time(sec)

z1

0 5 10

-2 -1 0 1 2

time(sec)

z2

y

ym

Example3.2: Consider the third-order unknown nonlinear system

) 5 . 0 1 sin(

2 _{1 2}

2 1 3

2 1 3

3 2

2 1

u x x x

u x x x x x

x x

x x

+ +

=

=

=

(3.40)

The adaptive law θˆ = −γ_{1 3}z ( (ξ q,Tˆ)−Φ T ,ˆ ˆ^T ) Tˆ = −γ_{2 3}zΦθ , ˆ ˆ ˆ _{3 3}ˆ ˆ

ij γ θz Ti j

Φ = . The

reference signal is given as y_m =sin(0.5 )t +cos( )t . Letγ₁= , 2

2 2

γ = ,c₁=1.5,c₂ = , and1 c₃ = . The initial states are set as (0) [0.5,0.5]1 x = . The order of the B-spline neural network approximator is selected as k=3, and the number of its knot points is twelve. The robust control is selected as

) ( _n

a sat z

u =−α , where β =3 and α =20, and in this case, the simulation results are show in Fig. 3.5. The MSE of u_a =−αsat(z_n) is 0.0093624. In addition, the robust control is selected as ₂

2

n a

u z

ρ

= − , and in this case,

simulation results are shown in Figs. 3.6-3.8, and the MSE of the tracking output error for ρ ={0.15, 0.175, 0.2} are shown in Table 3. 2. Figs.

3.6-3.8 show the results of ρ =0.15, ρ =0.175and ρ =0.2, respectively.

From simulation results, as ρis chosen smaller, the better tracking

performance can be achieved at the expense of the larger control input u.

Table 3.2 Three cases of ρ =0.15, ρ =0.175and ρ =0.2 Attenuation constant (^ρ) Mean square error

15 .

=0

ρ 0.0092555 175

.

=0

ρ 0.0096391 2

.

=0

ρ 0.0178290

Fig. 3.5. The simulation results when ua =−αsat⁽zn⁾.

0 2 4 6 8 10 12 14 16 18 20

-5 0 5

time(sec)

0 2 4 6 8 10 12 14 16 18 20

-50 0 50

time(sec)

u

0 2 4 6 8 10 12 14 16 18 20

-0.5 0 0.5

time(sec)

z1

ym

y

Fig. 3.6. The simulation results

when ₂

2ρⁿ

a

u =−z , where ρ=0.15

0 2 4 6 8 10 12 14 16 18 20

-5 0 5

time(sec)

0 2 4 6 8 10 12 14 16 18 20

-20 0 20

time(sec)

u

0 2 4 6 8 10 12 14 16 18 20

-0.5 0 0.5

time(sec)

z1

ym

y

Fig. 3.7. The simulation results

when ₂

2ρⁿ

a

u =−z , where ^ρ=0.175.

0 2 4 6 8 10 12 14 16 18 20

-2 0 2

time(sec)

0 2 4 6 8 10 12 14 16 18 20

-10 0 10

time(sec)

u

0 2 4 6 8 10 12 14 16 18 20

-0.5 0 0.5

time(sec)

z1

ym

y

Fig. 3.8. The simulation results

when ₂

2ρⁿ

a

u =−z , where ρ=0.2

0 2 4 6 8 10 12 14 16 18 20

-2 0 2

time(sec)

0 2 4 6 8 10 12 14 16 18 20

-10 0 10

time(sec)

u

0 2 4 6 8 10 12 14 16 18 20

-0.5 0 0.5

time(sec)

z1

ym

y

Chapter 4

B-spline Adaptive Backstepping Controllers for Nonaffine Nonlinear Systems with First Order Filter

In this chapter, B-spline adaptive backstepping scheme for nonaffine nonlinear systems with first order filters are proposed. The control scheme incorporates the backstepping design technique with the mean-estimation B-spline neural networks which are utilized to estimate the system dynamics.

The backstepping design scheme has the explosion of complexity problem as the order n of system increases. This explosion of complexity is caused virtual controller include basis function and every step of backstepping technique need differentiate the basis of virtual controller. In order to overcome this problem, this chapter uses first order filter at each step of the backstepping design.

This chapter is organized as follows. The system problem is formulated in section 4.1. In section 4.2, a B-spline adaptive backstepping controller with first order filters is described. The stability analysis is shown in section 4.3.

Computer simulation results are shown in section 4.4.

4.1 Problem Formulation

Consider the nonaffine nonlinear systems expressed in or transformed into a state-space form

) , (

) , (

) (

3 2 1 2 2

2 1 1 1

u f x

x x x f x

x x f x

n

n = x

+

=

+

=

(4.1)

where x = [x₁, x₂ x_n ]∈ R ⁿ is the state vector, and u∈ is the R control input. The system functions f₁,f₂, ,f_n are unknown continuous functions, and it is assumed that 0<∂f_n /∂u<∞. The control objective is to design a controller for system (4.1) such that all the signals in the closed-loop system are stable and the statex₁can track a bound reference

signaly arbitrarily closely. _m

4.2 Design of B-spline Adaptive Backstepping Controllers with First Order Filter

In this section, we use basckstepping technique to design a B-spline neural network controller for system (4.1).

Step 1: At this step, we consider the first equation of system (4.1)

2 1 1

1 f (x) x

x = + (4.2) Define the tracking error as

ym

x

z₁ = ₁− (4.3) Then, differentiating z can be expressed as ₁

2 1

1

1 f (x ) y x

z = − _m + (4.4) Define the virtual controller as

1 1 1

1

2 f (x) y cz

X =− + _m − (4.5) where c₁ >0 is a design parameter. From (4.4) and (4.5), if X₂ = , then x₂ we have lim ₁ 0

t z

→∞ → , that is, the state trajectory x can asymptotically track ₁ the reference signal y . _m

According to the universal approximation theorem, let

1 1 1

* 1 1

* 1

1(x)= f +σ =θ^Tξ (q,T^*)+σ

f . An estimate function is defined as

) T (q, ξ θ_{1 1 1}

1 ˆ ˆ

ˆ =

f , where ˆf , ₁ ˆθ , and ₁ T are the estimations ofˆ₁ f , ₁^* θ , and ₁^* T . ₁^*

Let ˆf pass through a first-order filter and introduces a new state variable ₁ expressed as

h f

The estimated virtual controller is

1 First, the illustration of the mean estimation technique is shown in Fig. 2.1.

By using the mean value theorem, there exist a point Tˆ₁− between δ₁ By substituting (4.10) in (4.12) and after some manipulations, we obtain

∑∑

= =

Finally, we define

2 2

2 x Xˆ

z = − (3.14) Subtracting (4.14) into (4.4), we obtain

2

Step 2: At this step, we consider the second equation of system (4.1)

3 Then, differentiating z can be expressed as ₂

2 Define the virtual controller as

1

Because the system function and X are unknown, the virtual controller can ˆ₂ not be precisely obtained. To solve this problem, we assume

2

f = − . According to the universal approximation theorem, let

2

f . An estimate function is defined as

) expressed as

2

Then the estimated virtual controller is

1

By using the mean value theorem, there exist a point Tˆ₂− between δ₂ Subtracting (4.25) into (4.17), we obtain

1 Then, differentiating z can be expressed as _i

i Define the virtual controller as

1

Because the system function and Xˆ are unknown, the virtual controller _i

can not be precisely obtained. In order to solve this problem, we assume Let fˆ passes through a first-order filter and introduces a new state variable _i

Fˆ . i

ˆ_i ˆ_i/ _i 1

F = f h s+ (4.31) where s is a Laplace variable, h is a Laplace constant. Define the filter _i error between fˆ and _i Fˆ is _i e_i =Fˆ_i − fˆ_i. Then, differentiation e can be _i expressed as

i

Then the estimated virtual controller is

1

by adding and subtracting the term ˆ

i i + i i Subtracting (4.37) into (4.38), we obtain

1

Step n: The control laws will be derived in final step. We consider final equation of the system (4.1)

) Then, differentiating z can be expressed as _n

n

we have ∂ϕ∂u>0. According to implicit function theorem, there exists an ideal backstepping controller u_ideal such that ϕ(x,u_ideal)=0.

Because the system function is unknown, the ideal backstepping controller can not be precisely obtained. According to the universal approximation theorem, let u_ideal =u^* +σ =θ_n^*Tξ_n(q,T_n^*)+σ_n, where σ_ndenotes the approximation error, T is an optimal knot vector, _n^* θ is an optimal ^*_n

adaptive parameter vector. In fact, the optimal parameter vectors are difficult to obtain. An estimate B-spline neural network identifier function is defined as

Define ~ξ_n =ξ_n(q,T_n*)−ξ_n(q,Tˆ_n), T~_n =T_n*−Tˆ_n, and

By using mean value theorem, the functionϕcan be rewritten

asϕ(u)=ϕ(u_ideal)+g(x)(u−u_ideal), whereu_ideal <u <uandg(_x)=∂ϕ∂u|_u=u. Because of the fact thatϕ(u_ideal)=0, above equation can be rewritten

asϕ(u)=g(x)(u−u_ideal). For design the controller of system (1), we make the following assumption.

Assumption 4-1: There exist positive constants of g and _u g such _l that g_l ≤g(x)≤g_u.

By using above equation, (4.41) can be rewritten as

1

Define the controller

ˆ _a

u u u= + (4.47) where u Is robust controller and substituting into (4.46) _a

) 1 On the basis of the above discussion, the stability analysis well discuss in next section.

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

∑

2q_i >h_if_up2. 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 = , σ_i =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 y_m =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 u_a = −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

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

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