Simulation Results

Therefore, we conclude that the nonlinear system (6-1) satisfies assumptions 1-3 and the total control law is

d s

f w u u

u

u= (eˆ| )+ + (6-36) with the state observer (6-9) and the RGA with the fitness function in (6-27). Then all signals in the closed-loop system are bounded, and e₁(t)→0 as t→∞.

Design Algorithm:

Step 1: Select the feedback and observer gain vectors K_c,K_o such that the matrices

To summarize, Fig. 6-1 shows the overall scheme of the GA-based output feedback direct adaptive fuzzy-neural control system proposed in this paper.

6.3 Simulation Results

Example 1: Consider the Duffing forced oscillation system

.

It is assumed that the external disturbance d(t) is a square wave having an amplitude

±1 with a period of ²π. The control objective is to control the state x of the system ₁ to track the reference trajectory y , under the condition that only the system output _m ^y is measurable. The design parameters are selected as V =0.005 and ρ=20. The

The initial states are chosen to be x₁(0)=3,x₂(0)=3,xˆ₁(0)=−1,xˆ₂(0)=−1 and )

0 ( ˆ ) 0 ( ) 0

ˆ( y x

e = m − . Simulation results are provided for two cases with different reference trajectories, i.e.,ym =0 (Case 1), y_m =sint (Case 2)。

As shown in Figs. 6-2 (case 1) and 6-3 (case 2), the trajectory of the estimation state ˆx ₁ catches up to the trajectory of the system state x very quickly and well for case 1 and ₁ case 2, respectively. Moreover, the tracking performance is very good as shown in Fig.

6-3 (case 1) and Fig 6-6 (case 2), in which y_m is the reference trajectory andx₁is the system output. As shown in Figs. 6-4 (case 1) and 6-7 (case 2), a chattering effect on the control input (due to u_d +u_s) for these cases appears in the initial searching. In this period, the RGA searches the neighborhood for the optimal parameters of the GODAF controller. After this initial searching, the RGA has almost found the optimal parameters and the chattering effect on the control input (due to u_d +u_s) disappears.

Fig. 6-1 Overall scheme of the proposed GODAF controller.

∑

∑

∑

The adaptive law θ^ in (18)

Plant：

x y

d u xⁿ

=

+ +

=f( ) g( )

)

( x x

K0

( )

^eˆ

ϕ

T

f w

u =

Let w be adjusted by the RGA with (4-27)

u_d =v

1 0

ˆ ~ )

ˆ (A BK^T_c e K e

e= − +

ˆe1

e1 1

~e

eˆ CT

1

~e

uf

u

u d

+ +

+

_ _

+

y

ym

=1 0 I

= I

us

+

0 0 . 1 0 . 2 0 . 3 0 .4 0 .5 0 .6 0 . 7 0 . 8 0 . 9 1 -1

-0 . 5 0 0 . 5 1 1 . 5 2 2 . 5 3 3 . 5

x

1

ˆx

1

Time(sec)

0 0 . 1 0 . 2 0 . 3 0 .4 0 .5 0 .6 0 . 7 0 . 8 0 . 9 1

-1 -0 . 5 0 0 . 5 1 1 . 5 2 2 . 5 3 3 . 5

x

1

ˆx

1

Time(sec)

^Time(sec)

Fig. 6-2 Trajectories of the state x₁ and the estimation state ˆx of case 1 in example ₁

0 2 4 6 8 1 0 1 2 1 4 1 6 1 8 2 0

-0 . 5 0 0 . 5 1 1 . 5 2 2 . 5 3 3 . 5

x

1

Time(sec)

0 2 4 6 8 1 0 1 2 1 4 1 6 1 8 2 0

-0 . 5 0 0 . 5 1 1 . 5 2 2 . 5 3 3 . 5

x

1

Time(sec)

^Time(sec)

Fig. 6-3. Trajectories of the output x1 and ym =0 of case 1 in example 1.

0 2 4 6 8 1 0 1 2 1 4 1 6 1 8 2 0 -1 5 0

-1 0 0 -5 0 0 5 0 1 0 0 1 5 0

u(t)

Time(sec)

0 2 4 6 8 1 0 1 2 1 4 1 6 1 8 2 0

-1 5 0 -1 0 0 -5 0 0 5 0 1 0 0 1 5 0

u(t)

Time(sec)

^Time(sec)

Fig. 6-4 Control input u of case 1 in example 1.

u(t)

0 0 .1 0. 2 0 .3 0 .4 0 .5 0 .6 0. 7 0 .8 0. 9 1

-1 -0 .5 0 0 .5 1 1 .5 2 2 .5 3 3 .5

x

1

ˆx

1

Time(sec)

0 0 .1 0. 2 0 .3 0 .4 0 .5 0 .6 0. 7 0 .8 0. 9 1

-1 -0 .5 0 0 .5 1 1 .5 2 2 .5 3 3 .5

x

1

ˆx

1

Time(sec)

^Time(sec)

Fig. 6-5 Trajectories of the state x₁ and the estimation state ˆx of ₁ case 2 in example 1.

0 2 4 6 8 1 0 1 2 1 4 1 6 1 8 2 0 -1

-0 . 5 0 0 . 5 1 1 . 5 2 2 . 5 3 3 . 5

x

1

y

m

Time(sec)

0 2 4 6 8 1 0 1 2 1 4 1 6 1 8 2 0

-1 -0 . 5 0 0 . 5 1 1 . 5 2 2 . 5 3 3 . 5

x

1

y

m

Time(sec)

^Time(sec)

Fig. 6-6 Trajectories of the output x1 and y_m =sint of case 2 in example

0 2 4 6 8 1 0 1 2 1 4 1 6 1 8 2 0

-1 5 0 -1 0 0 - 5 0 0 5 0 1 0 0 1 5 0

u(t)

Time(sec)

0 2 4 6 8 1 0 1 2 1 4 1 6 1 8 2 0

-1 5 0 -1 0 0 - 5 0 0 5 0 1 0 0 1 5 0

u(t)

Time(sec)

Fi g. 6-7 Control input u of Case 2 in example 1.

Time(sec) u(t)

Example 2: Consider the inverted pendulum system as (5-16)

It is assumed that the external disturbance d(t) is a square wave having an amplitude

±1 with a period of 2π. The control objective is to control the state x of the system ₁ to track the reference trajectory y , under the condition that only the system output _m ^y is measurable. The design parameters are selected as V =0.005 and _ρ=20. The feedback and observer gain vectors are given as K_c =[144 24]^T and K_o=[60 900]^T, respectively. The filter L⁻¹(s) is given as L⁻¹(s)=1(s+2) . The membership functions for eˆ_j, j=1,2 are the same as those in example 1. The initial states are

chosen to be ,ˆ (0) 0

2 ) 1 0 ˆ ( , 0 ) 0 ( 60, )

0

( _{2 1 2}

1 =− x = x =− x =

x π

and eˆ(0)=ym(0)−xˆ(0). Simulation results are provided for two cases with different reference trajectories, i.e.,ym =0 (Case 1), y_m =sint (Case 2). As shown in Figs. 6-8 (case 1) and 6-9 (case 2), the trajectory of the estimation state ˆx catches up to the trajectory of the ₁ system state x very quickly and well for case 1 and case 2, respectively. Moreover, ₁ the tracking performance is very good as shown in Fig. 6-9 (case 1) and Fig 6-12 (case 2), in which y_m is the reference trajectory andx₁is the system output. As shown in Fig.

6-10 (case 1) and Fig. 6-12 (case 2), a chattering effect on the control input (due to

s

d u

u + ) for these cases appears in the initial searching. In this period, the SGA searches the neighborhood for the optimal parameters of the GODAF controller. After this initial searching, the SGA has almost found the optimal parameters and the chattering effect on the control input (due to u_d +u_s) disappears.

0 0 .1 0 .2 0 .3 0 .4 0 .5 0 .6 0 .7 0 .8 0 . 9 1

-0 . 5 -0 . 4 -0 . 3 -0 . 2 -0 . 1 0 0 . 1 0 . 2

x

1

ˆx

1

Time(sec)

0 0 .1 0 .2 0 .3 0 .4 0 .5 0 .6 0 .7 0 .8 0 . 9 1

-0 . 5 -0 . 4 -0 . 3 -0 . 2 -0 . 1 0 0 . 1 0 . 2

x

1

ˆx

1

Time(sec)

Fig. 6-8. Trajectories of the state x1 and the estimation state ˆx of ₁ case 1 in example 2.

x

1

0 2 4 6 8 10 12 14 16 18 20

-0.25 -0.2 -0.15 -0.1 -0.05 0 0.05

-- 0.2 - 0.15 -0.1 -0.05 0 0.05

Time(sec)

Fig. 6-9. Trajectories of the output x₁ and ym =0 of case 1 in example 2.

0 2 4 6 8 1 0 1 2 1 4 1 6 1 8 2 0

-1 5 0 -1 0 0 - 5 0 0 5 0 1 0 0 1 5 0

u(t)

Time(sec)

0 2 4 6 8 1 0 1 2 1 4 1 6 1 8 2 0

-1 5 0 -1 0 0 - 5 0 0 5 0 1 0 0 1 5 0

u(t)

Time(sec)

^Time(sec)

Fig. 6-10 Control input u of case 1 in example 2.

u(t)

0 0 . 1 0 . 2 0 . 3 0 .4 0 .5 0 .6 0 . 7 0 . 8 0 . 9 1

-0 . 6 -0 . 4 -0 . 2 0 0 . 2 0 . 4 0 . 6 0 . 8

x

1

ˆx1

Time(sec)

0 0 . 1 0 . 2 0 . 3 0 .4 0 .5 0 .6 0 . 7 0 . 8 0 . 9 1

-0 . 6 -0 . 4 -0 . 2 0 0 . 2 0 . 4 0 . 6 0 . 8

x

1

ˆx1

Time(sec)

^Time(sec)

Fig. 6-11 Trajectories of the state x1 and the estimation state ˆx ₁ of case 2 in example 2.

0 2 4 6 8 1 0 1 2 1 4 1 6 1 8 2 0

-1 . 5 -1 -0 . 5 0 0 . 5 1 1 . 5

x

1

y

m

Time(sec)

0 2 4 6 8 1 0 1 2 1 4 1 6 1 8 2 0

-1 . 5 -1 -0 . 5 0 0 . 5 1 1 . 5

x

1

y

m

Time(sec)

^Time(sec)

Fig. 6-12 Trajectories of the output x1 and y_m =sint of case 2 in example

0 2 4 6 8 1 0 1 2 1 4 1 6 1 8 2 0 -1 5 0

-1 0 0 - 5 0 0 5 0 1 0 0 1 5 0

Time(sec)

u(t)

0 2 4 6 8 1 0 1 2 1 4 1 6 1 8 2 0

-1 5 0 -1 0 0 - 5 0 0 5 0 1 0 0 1 5 0

Time(sec)

u(t)

^u(t)

Time(sec) Fig. 6-13 Control input u of case 2 in example 2.

Chapter 7 Conclusions

This paper proposes a simplified genetic algorithm (SGA), which can be successfully applied in BMF fuzzy-neural network to search for the optimal parameters, in spite of a vast number of adjustable parameters, including both the weightings of the neural fuzzy networks and the control points of the BMFs. The SGA described in this paper is characterized by three features: the population size is fixed and can be reduced to a minimum size of 4; the crossover operator is simplified to be a single point crossover; only one chromosome in the population is selected for mutation. A search method, SSCP, was proposed to determine the crossover point for the single gene crossover. Off-line learning for the BMF fuzzy-neural network is first considered to show the learning ability of the SGA with the SSCP method with satisfactory performance. From the simulation results, the SGA proposed in this paper provides a well-suited way of learning for BMF fuzzy-neural network.

Secondly, SIAFC and GODAF are proposed in this plan. The free parameters of the adaptive fuzzy-neural controller can be successfully tuned on-line via the SGA approach, instead of solving complicated mathematical equations. The SIAFC and GODAF guarantee the stability of the resulting closed-loop system. Moreover, the proposed design algorithm has been successfully applied to control an inverted pendulum system and a Duffing forced-oscillation system to track a reference signal trajectory. The simulation results show that the SIAFC and GODAF perform on-line tracking successfully.

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