Design of B-spline Adaptive Backstepping Controllers for Unknown

=

= 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

在文檔中 非線性系統之倒階適應性類神經控制器設計 (頁 45-58)