A Brief Summary on Comparisons

The following Table 6.3 is made for the comparisons with other robust criterions. It shows the applied parametric robust Popov criterion can deal with fuzzy logic control systems with the uncertain interval plants and the constant reference inputs cases. The other three approaches: the robust Lur’e test, the robust circle criterion and the robust Popov criterion just can deal with the uncertain interval plants and the zero reference inputs cases. In previous demonstrated examples, the stability will crash due to reference input shift. On the other hand, the stability of the fuzzy control systems with uncertain interval plants can be assured under the interval range reference inputs by the applied parametric robust Popov criterion.

-2 -1 0 1 2 3 4 5 -6

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

Real

Imag

Fig. 6.1 The robust Lur’e test.

-2000 -1500 -1000 -500 0 500 1000 1500 2000

-1500 -1000 -500 0 500 1000 1500

e

uf

the control surface of the fuzzy logic controller the sector bound

Fig. 6.2 The sector bound from the robust Lur’e test and the control surface of the fuzzy logic controller.

Table 6.1

Parameters of fuzzy logic controller for the robust Lur’e test

nbe nme nse zre pse pme pbe

e -2000 -1025 -1000 0 1000 1025 2000

nbu nmu nsu zru psu pmu pbu

uf

-740 -350 100 0 100 350 740

0 100 200 300 400 500 600 700 800 900 1000

-0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015 0.02 0.025 0.03

time(sec)

y

Fig. 6.3 The time waveform of the stable test case respect to the robust Lur’e test.

0 100 200 300 400 500 600 700 800 900 1000 0

200 400

time(sec)

y

Fig. 6.4 The time waveform of the unstable test case respect to the robust Lur’e test.

-2 -1 0 1 2 3 4 5

-6 -5 -4 -3 -2 -1 0 1

Real

Imag

Fig. 6.5 Robust circle criterion.

-2000 -1500 -1000 -500 0 500 1000 1500 2000 -6000

-4000 -2000 0 2000 4000 6000

e

uf

the control surface of the fuzzy logic controller the sector bound

Fig. 6.6 The sector bound from the robust circle criterion and control surface of fuzzy logic controller.

Table 6.2

Parameters of fuzzy logic controller for the robust circle criterion

nbe nme nse zre pse pme pbe

e -2000 -1020 -1000 0 1000 1020 2000

nbu Nmu nsu zru psu pmu pbu

uf

-4158.4 -1630 -630 0 630 1630 4158.4

0 100 200 300 400 500 600 700 800 900 1000 -0.1

-0.05 0 0.05 0.1 0.15 0.2 0.25

time(sec)

y

Fig. 6.7 The time waveform of the stable test case respect to the robust circle criterion.

0 100 200 300 400 500 600 700 800 900 1000

-200 0 200 400 600 800 1000 1200 1400 1600 1800

time(sec)

y

Fig. 6.8 The time waveform of the unstable test case respect to the robust circle criterion.

-2.5 -2 -1.5 -1 -0.5 0 -2

-1.5 -1 -0.5 0 0.5

Real

Imag

Fig. 6.9 Robust Popov criterion.

-2000 -1500 -1000 -500 0 500 1000 1500 2000

-4000 -3000 -2000 -1000 0 1000 2000 3000 4000

e

uf

the control surface of the fuzzy logic controller the sector bound

Fig. 6.10 The sector bound from the robust Popov criterion and control surface of fuzzy logic controller.

-11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 -2

-1.5 -1 -0.5 0 0.5

Real

Imag

Fig. 6.11 Parametric robust Popov criterion for the reference inputs.

0 100 200 300 400 500 600 700 800 900 1000

-0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015 0.02 0.025 0.03

time(sec)

y

Fig 6.12 The time waveform of the stable test case respect to the parametric robust Popov criterion with a bounded pulse reference.

0 100 200 300 400 500 600 700 800 900 1000 0

50 100 150

time(sec)

y

Fig. 6.13 The time waveform of the stable test case respect to the parametric robust Popov criterion with the reference input r990.

0 100 200 300 400 500 600 700 800 900 1000

-150 -100 -50 0

time(sec)

y

Fig. 6.14 The time waveform of the stable test case respect to the parametric robust Popov

Table 6.3

The validity of the different robust stability tests Parametric

robust Popov criterion

Robust Lur’e test Robust circle criterion

Robust Popov criterion

Zero reference inputs

Yes Yes Yes Yes

Constant reference inputs

Yes No No No

Chapter 7

Application: Observer-Based Synchronization for a Class of Unknown Chaotic Systems with Adaptive Fuzzy-Neural Network

7.1 Overview

The study of the synchronization for a class of unknown chaotic systems with adaptive fuzzy-neural network is based on the concepts of AFNO, Brunowsky canonical form and Lur’e systems. The proposed synchronization system contains chaos master with the canonical form and the soft-computing slave with AFNO. The AFNO is composed of a FNN and a linear observer. In this design, the AFNO in the slave should synchronize with all states in the master by a scale transmitted signal only. The FNN in the AFNO is utilized to model the nonlinear function in the master end adaptively. The linear observer estimates the all states at the slave end with three inputs including a transmitted state, output of the FNN, and robust compensation input for counteracting the effect of the external disturbance. When all states in the master end are estimated at slave end, the synchronization is achieved.

Simulation results confirm that the AFNO is applied to chaos synchronization is valid.

7.2 Overall Structure of Adaptive Synchronization with Fuzzy-Neural Observer Design

7.2.1 Introduction of Overall Structure

Assume that the master and slave are all Lur’e type. Figure 7.1 illustrates the overall structure of adaptive synchronization with AFNO, which is synthesized with an FNN and a linear observer. In this design, only a scalar transmitted signal x_M1 is sent to the slave from the master. By the observed state x ,ˆ_S f x can be computed to approximate_S( )ˆ_S f_M(x_M) with FNN. The adaptive laws update the weights in FNN when the error exists between x_M1 and

ˆ_s1

x . The linear observer inputs are u_S f x_S( )ˆ_S , the transmission signal x , and the robust_M1 input u . The synchronization is achieved when_r x_M  .xˆ_S

7.2.2 Dynamics of the Master and Slave Ends

Master End:

0 1 0 0

C  C  ; d denotes an bounded external disturbance;

( 1) unknown (uncertain) but bounded continuous functions. [81,82]

Synchronization Error:

The synchronization error can be defined as:

syn M ˆS

e  x x , (69)

where e_syn[e_syn e _syn e_syn^{( 1)n}]^T[e_syn1 e_syn2  e_synn]^Tⁿ.

The master and slave achieve synchronization when all states are estimated at the slave.

7.3 Adaptive Fuzzy-Neural Network Observer Design

In this subchapter, AFNO is introduced. Under an assumption, the designed AFNO can estimate the master’s states to achieve synchronization. AFNO can then be synthesized by an

FNN and a linear observer.

7.3.1 Fuzzy-Neural Network [73,80]

The FNN is designed to model the nonlinear function f_M(x_M) with f x . The FNN_S( )ˆ_S depicted in Fig. 7.2 is utilized as an approximator to model the nonlinear functions such as f x( ). The FNN [83,84], which consists of fuzzy IF-THEN rules and a fuzzy inference engine, is adopted as a function approximator. The fuzzy inference engine employs the IF-THEN rules to generate a mapping from an input linguistic vector

1 2

[ _n]^{T n}

xx x  x to an output linguistic variabley x( ). Fuzzy IF-THEN rule ith is thus written as:

( )i

 , respectively. By using product inference, center-average, and singleton fuzzifier, output y x( ) from the fuzzy-neural approximator can be written as

1 1

 denotes the membership function value of fuzzy variable x_j; h is the total

number of IF-THEN rules, and yⁱ is the point at which i( ) 1

   denotes an adjustable parameter vector, and

1 2

[  ^{h T}]

 

 represents a fuzzy basic vector, where  is given byⁱ

1

By adjusting the parameter vector  in (70) with adaptive laws, the uncertain nonlinear_f

function f x( ) can be approximated by f x^ˆ( ) generated in (72). By using the fuzzy-neural approximator, the estimated functions f x^ˆ( ) can be determined from the outputs of the fuzzy-neural approximator, which is defined as follows:

ˆ( θ) θ ( )_f ^T_f

f x  x , (72)

where θ_f is an adjustable parameter vector.

In summary, (72) can describe the input-output relation of the FNN. The overall structure of the FNN is divided into four layers as shown in Fig. 7.2. The physical meanings of (72) can be interpreted by Fig. 7.2 in the following. The input nodes in Layer I represent input linguistic vectors. Nodes in Layer II denote values of the membership function of total linguistic variables. Each node in Layer III excuses a fuzzy rule. The output of Layer IV is the output signal modeling the nonlinear function. The connection parameters between layer III and layer IV are adjusted by using adaptive laws. The number of fuzzy rules can be dependent on complex level of nonlinear systems. In general, the more complex the systems are, the more numerous rules are demand. Of course, the computing load is heavy with more numerous rules. On the other hands, when the rules are less, the computing load is slight. This is a trade off problem.

7.3.2 Adaptive Fuzzy-Neural Network Observer

Assumption 7.1 [73,80]:

The master state vector x_M and the slave state vector ˆx belong to compact sets_S S_M and S respectively, where_S



^: M



and  and  are designed parameters.

The optimal parameter vector θ^_ff falls in some convex region with constant radius  . The_f convex region can be specified as shown in (75).



^:



f f

h

f f

R_   _ . (75)

The optimal parameter vector  can be described as:^*_f

,ˆ

Remark 7.1: The optimal  is possible in an ideal situation. In our applications, the^*_f

adaptive laws will be applied to tune  to approach_f  .^*_f

The adaptive fuzzy-neural nonlinear observer with respect to a class of nonlinear systems (67) can be designed under assumption 7.1. AFNO can be designed [73,80]:

1 1 dynamical systems, and u denotes the robust input to compensate the effect due to external_r disturbance and the approximated modeling error by FNN. Based on [73,80], u can be_r designed as follows:

The small gamma will cause large u to attenuate the effect of disturbance. Indeed, the_r better attenuation performance will be obtained when the small  is chosen. Additionally,

T 0

Q Q  will make the Riccati-like equation satisfied in stability and adaptive law derivation with Lyapunove function [80].

The adaptive laws in FNN are as follows:

H s L s into a proper strictly-positive real (SPR) transfer function, and  denotes the₁ designed parameter. The function H s( ) is represented as follows:

( ) _S( ( _{S o S})) 1 _S.

The design procedure, stability proof and adaptive laws (79) can be referred in [73,80]

7.4 Simulation Results

This subchapter verifies the feasibility of AFNO for synchronization using two examples.

7.4.1 Example 1

In this example, AFNO is applied to synchronize a master Chua’s circuit under modeling error, different initial conditions and external bounded disturbances .The results will demonstrate the adaptability and robustness of AFNO.

The master Chua’s circuit is reformed as a canonical form [85].

1 1

The adaptive laws tune FNN to approach f_M(x_M). The observer is designed to place poles of A_S K C_{o S}in -30 i.e. linear observer gain vector isK_o^T 



90 2700 27000



.

In this example, three states should be estimated, accounting for why the fuzzy rules in process are 343. The initially adjustable parameters in adaptive FNN are chosen to be

(0) 0

f  to demonstrating modeling error. The weights of FNN are turned by the adaptive laws to form f_M(x_M).

Different initial conditions of the master and slave are listed in Table 7.1. Furthermore, the distinct disturbances are listed in Table 7.2.

Figures 7.3~7.5 summarize the simulation results of different initial conditions for three states in AFNO. In Figs. 7.3~7.5, the distinct initial conditions for each state in AFNO are listed in Table 7.1 and a type of disturbance in the master end is set as Case 1 in Table 7.2.

Figure 7.3 illustrates that the first state xˆ_S1 in AFNO with three different initial conditions synchronizes x_M1 in Chua’s circuit. Figures 7.4 and 7.5 illustrate that xˆ_S2 and xˆ_S3 synchronize x_M2 and x , respectively. Although the initial conditions differ from each_M3 other, AFNO synchronizes with Chua’s circuit quickly, well, and adaptively. Moreover, the synchronization error approaches zero as time goes to infinity. The robustness of AFNO can be also specified from Figs. 7.6~7.8 with various intensity disturbances in the master end. In Figs. 7.6~7.8, the initial conditions of three states are selected as Case1 in Table 7.1 and the different disturbances are chosen as Table 7.2. Figure 7.6 demonstrates that the first state xˆ_S1 in the slave synchronizes x_M1 in the master end immediately and well under three different disturbances. Figures 7.7 and 7.8 reveal that xˆ_S2 and xˆ_S3 synchronize x_M2 and x ,_M3 individually. Even if the different disturbances are added in the master Chua’s circuit, AFNO synchronizes with the master robustly.

7.4.2 Example 2

Example 2 demonstrates the adaptability of the utilized method by switched master between Chua’s circuit and Rössler system as shown in Fig. 7.9. When the master is switched to another system, the slave follows to synchronize another chaotic system soon and well. The similar different initial conditions and disturbances listed in Tables 7.1 and 7.2 are considered in simulations for demonstrating the robustness of AFNO.

The original Rössler system can be presented as [62]:

1 2 1

z z az

2 1 3

The Rössler system is reformed as the canonical form with

2 2 2

The parameters of AFNO at the slave resemble those in Example 1. The initial condition of Rössler system is set [0 0 0]^T.

Figures 7.10~7.12 indicate the simulation results with respect to each state for diverse initial conditions in AFNO and switched masters. The distinct initial conditions for each state in AFNO are shown in Table 7.1 and a kind of disturbance in the master end is set as Case 1 in Table 7.2. Figure 7.10 illustrates that the first state xˆ_S1 in AFNO with three different initial conditions synchronizes x_M1 in the master end, even if the switched masters exist at the third second (Chua’s circuit to Rössler system) and the sixth second (Rössler system to Chua’s circuit). Figures 7.11 and 7.12 exhibit that xˆ_S2 and xˆ_S3 synchronize x_M2 and x ,_M3 respectively. Although the initial conditions differ from each other and the switched masters exist, AFNO synchronizes with the switched masters fast, well, and adaptively. On the other hand, simulation results in Figs. 7.13~7.15 verify the robustness of AFNO for the different disturbances and the switched systems in the master end. In Figs. 7.13~7.15, the initial

conditions of three states are chosen as Case1 in Table 7.1 and the different disturbances are selected as Table 7.2. Figure 7.13 displays that the first state xˆ_S1 synchronizes x_M1 immediately and well under three different disturbances, even thought the switched masters exist at the third second (Chua’s circuit to Rössler system) and the sixth second (Rössler system to Chua’s circuit). Figures 7.14 and 7.15 reveal that xˆ_S2 and xˆ_S3 synchronize x_M2 and x , separately. In spite of the different disturbances and the switched systems are_M3 considered in the master end, AFNO synchronizes with the master robustly.

It is noted that Figs 7.10~7.15 display the simulation results indicating AFNO synchronizes with Chua’s circuit at 0~3 sec. The Rössler system also runs dynamically from the initial condition. AFNO synchronizes with Rössler at 3~6 sec, while Chua’s circuit runs simultaneously.

From these simulation results, AFNO can synchronize with a class of unknown chaotic systems adaptively and robustly.

7.5 Conclusion Remarks

This work has applied AFNO for synchronization with respect to a class of unknown chaotic systems via a scalar transmitted signal only. Once the nonlinear chaotic systems could be transformed into the canonical form of Lur’e system type by the differential geometric method, the AFNO method can be utilized for synchronization. In this approach, the nonlinear term in the master end was modeled by the adaptive fuzzy-neural network (FNN) in AFNO on line. Furthermore, the states in the master end were observed from a scale transmitted signal by observer design. When states in the master and slave ends were identical, we said the synchronization was reached. By this scheme, the AFNO could estimate the unknown master’sstatesadaptively, even though the master was altered into another chaotic system.

disturbance to demonstrate its robustness advantage. Simulation results showed that the adaptive and robust AFNO was suitable for chaos synchronization with respect to a class unknown chaotic systems.

Fig. 7.1 The overall structure of synchronization with AFNO.

Layer I Layer II Layer III Layer IV x

1

x

2

x n



( ) y x y

1

y2

y

h



1



2



h

A

₁i

A₂i

i

A

n









Table 7.1

Three cases of the initial conditions

Cases Initial conditions

Case 1 x_M(0) [0 0 0] ^T, and xS⁽⁰⁾



^{1 1 1}



^T

Case 2 x_M(0) [0 0 0] ^T, and xS⁽⁰⁾



^{2 2 2}



^T

Case3 x_M(0) [0 0 0] ^T, and xS⁽⁰⁾



^{3 3 3}



^T

Note: In the simulations, the disturbances in the master end are set as Case 1 in Table 7.2 in three cases.

Table 7.2

Three cases of the disturbances

Cases Disturbance (d)

Case 1 0.5 with period 2

Case 2 0.8 with period 2

Case3 1 with period 2

Note: In the simulations, the initial conditions are chosen as Case1 in Table 7.1 in three cases.

0 0.5 1 1.5 2 2.5 3

FNO of initial condition case 1 FNO of initial condition case 2

FNO of initial condition case 3

Fig. 7.3 The first states x_M1 and xˆ_S1 in Chua’s circuit and AFNO under different initial conditions.

FNO of initial condition case 3 FNO of initial condition case 2

FNO of initial condition case 1 Chua's circuit

Fig. 7.4 The second states x_M2 and xˆ_S2 in Chua’s circuit and AFNO under different initial conditions.

0 0.5 1 1.5 2 2.5 3

FNO of initial condition case 1

FNO of initial condition case 2

FNO of initial condition case 3

Fig. 7.5 The third states x_M3 and xˆ_S3 in Chua’s circuit and AFNO under different initial conditions.

Chua's circuit of three disturbance cases

Chua's circuit and FNO of disturbance cases 1

Chua's circuit and FNO of disturbance cases 2

Chua's circuit and FNO of disturbance cases 3

Fig. 7.6 The first states x_M1 and xˆ_S1 in Chua’s circuit and AFNO under different disturbances.

0 0.5 1 1.5 2 2.5 3

FNO of three disturbance cases

Chua's circuit of three disturbance cases Chua's circuit of three disturbance case 1

Chua's circuit of three disturbance case 2

Chua's circuit of three disturbance case 3

Fig. 7.7 The second states x_M2 and xˆ_S2 in Chua’s circuit and AFNO under different disturbances.

FNO of three disturbance cases

Chua's circuit of three disturbance cases

Chua's circuit and FNO of three disturbance cases

Fig. 7.8 The third states x_M3 and xˆ_S3 in Chua’s circuit and AFNO under different

Fig. 7.9 The structure of synchronization with the switched masters.

0 1 2 3 4 5 6 7 8 9 10

FNO of three initial condition cases

(a)

FNO of initial condition case 3

FNO of initial condition case 2

FNO of initial condition case 1

(b)

Fig. 7.10 The first states in Chua’s circuit, Rössler system and AFNO under different initial conditions and switched masters: (a) actual figure size (b) enlarged figure size of local

0 1 2 3 4 5 6 7 8 9 10

FNO of three initial conditions cases

Fig. 7.11 The second states in Chua’s circuit, Rössler system and AFNO under different initial conditions and switched masters.

0 1 2 3 4 5 6 7 8 9 10

FNO of three initial condition cases

Rossler system

Chua's circuit

Fig. 7.12 The third states in Chua’s circuit, Rössler system and AFNO under different initial conditions and switched masters.

0 1 2 3 4 5 6 7 8 9 10

Chua's circuit of disturbance case 1

Chua's circuit of disturbance case 2

Chua's circuit of disturbance case 3

FNO of disturbance case 3 FNO of disturbance case 2 FNO of disturbance case 1 Rossler system of disturbance case 1

Rossler system of disturbance case 2

Rossler system of disturbance case 3

Fig. 7.13 The first states in Chua’s circuit, Rössler system and AFNO under different disturbances and switched masters.

Chua's circuit of disturbance case 1

Chua's circuit of disturbance case 2

Chua's circuit of disturbance case 3

FNO of three disturbance cases

Rossler system of disturbance case 3 Rossler system of disturbance case 2

Rossler system of disturbance case 1

Fig. 7.14 The second states in Chua’s circuit, Rössler system and AFNO under different

0 1 2 3 4 5 6 7 8 9 10 -10

-8 -6 -4 -2 0 2 4 6 8 10

time (sec)

thirdstatesofChua'scircuit,RosslersystemandFNO

FNO of disturbance case 1

FNO of disturbance case 2

FNO of disturbance case 3

Chua's circuit of disturbance case 3

Chua's circuit of disturbance case 2 Chua's circuit of disturbance case 1

Rossler system of disturbance case 1 Rossler system of disturbance case 2

Rossler system of disturbance case 3

Fig. 7.15 The third states in Chua’s circuit, Rössler system and AFNO under different disturbances and switched masters.

Chapter 8

Conclusions

In this dissertation, the parametric absolute stability in P and PD type fuzzy logic control systems with both certain and uncertain linear plants with parameters such as the reference input, actuator gain and interval plan have been analyzed. The adaptive AFNO has been also applied to synchronize a class of unknown chaotic systems via a scalar transmitted signal only.

In the stability analysis, for certain linear plants, the Popov and linearization methods are applied to analyze the stability in both P and PD type fuzzy control systems under different reference inputs and actuator gains. The steady state errors of the fuzzy control systems are also analyzed.Foruncertain plants,theparametricrobustPopov criterion based on theLur’e system is applied to the stability analysis of P and PD type fuzzy control systems. Moreover, a fuzzy current controlled RC circuit is designed to compare theoretical analyses with PSPICE simulation results. Furthermore, the oscillation phenomena in fuzzy control systems are interpreted from the point of view of the equilibriums in this simulation example. Finally, the parametric robust Popov criterion is compared with the other approaches to show the effectiveness respect to non-zero reference inputs.

About application with the fuzzy control system, AFNO has been applied for synchronization with respect to a class of unknown chaotic systems via a scalar transmitted signal only. Once the nonlinear chaotic systems could be transformed into the canonical form of Lur’e system type by the differential geometric method, the AFNO method can be utilized for synchronization. In this approach, the nonlinear term in the master end was modeled by

the adaptive fuzzy-neural network (FNN) in AFNO on line. Furthermore, the states in the master end were observed from a scale transmitted signal by observer design. When states in the master and slave ends were identical, we said the synchronization was reached. By this scheme, the AFNO could estimate the unknown master’sstatesadaptively, even though the master was altered into another chaotic system. On the other hand, AFNO could deal with the modeling error, and external bounded disturbance to demonstrate its robustness advantage.

the adaptive fuzzy-neural network (FNN) in AFNO on line. Furthermore, the states in the master end were observed from a scale transmitted signal by observer design. When states in the master and slave ends were identical, we said the synchronization was reached. By this scheme, the AFNO could estimate the unknown master’sstatesadaptively, even though the master was altered into another chaotic system. On the other hand, AFNO could deal with the modeling error, and external bounded disturbance to demonstrate its robustness advantage.

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