Organizations of the Dissertation

This thesis is organized as follows. Chapter 1 is an introduction. Chapter 2 describes the P and PD type fuzzy control systems. Chapter 3 analyzes the equilibrium points and stability in P type fuzzy control system. Chapter 4 then performs the same analyses in a PD type fuzzy control system. Chapter 5 provides simulation results with Matlab and PSPICE simulators. In Chapter 6, the comparisons are made to show the superiority of the applied analysis method.

Furthermore, in Chapter 7, the observer-based synchronization for a class of unknown chaotic systems with adaptive fuzzy-neural network is presented. Finally, some conclusions are given in Chapter 8.

Chapter 2

The Fuzzy Logic Control Systems

2.1 Fuzzy Logic Controller

Both P and PD type fuzzy logic control systems include a linear plant with time-invariant uncertainty, adjustable actuator gain and reference input. Moreover, the fuzzy logic controllers are the cores of systems. An FLC can be taken as multiple bends of piecewise linear functions, since it has singleton and specific membership functions. Hence, a fuzzy logiccontrolsystem can betreated asaLur’etype system.

Consider the fuzzy logic control system in Fig. 2.1. The IF-THEN rules in single input fuzzy logic controller can be described as:

i:

Rule If e is M , then_i u_f is u ,_i (1) where e is the control error and M and_i u denote fuzzy sets. If a singleton is applied in a_i fuzzifier, then the product inference and center average are formulated in the inference engine and defuzzifier, respectively. The output of the fuzzy logic controller can be represented as

f i( ) i

For simplification, this study uses the fuzzy rules and membership functions listed in Table 2.1 [27] and Fig. 2.2 are adopted in this thesis, respectively. Table 2.2 presents the fuzzy controller parameters. Figure 2.3 shows the control function of the fuzzy controller, which can

be described as: for n multiple bends of a control function. The control output of the static fuzzy system is given by: The control function  satisfies

 

²

ˆ ˆ ˆ

0e (e e ) ( )e k e e( ) ,  e ,  eˆ , (5) where (0) 0 , k0and  indicates some neighborhood of e0.

2.2 P Type Fuzzy Logic Control System

Figure 2.1 illustrates a P type fuzzy control system with a fuzzy logic controller, a parametric linear time-invariant system and adjustable parameters, which include actuator

linear function, and is depicted in Fig. 2.3. vector p exists in a compact and simple connected region ^l.

The transfer functionG s p K( , , ) with amplifier gain K can be stated as

 

¹

( , , ) ( ) ( ) ( , )

G s p K C p sI A p ^B p K (7) where B p K( , )KB p( ) , and^n1 K. The overall static fuzzy logic control system in Fig. 2.1 can be described as:

( ) ( , ) _f xA p x B p K u ,

( )

y C p x , (8)

where the control input u_f ( )e ; the control error e r y  , xⁿ, e and y; the reference input r is a constant value, and r is a constant value, and r.

The closed loop system is given by

 

( ) ( , ) ( )

x A p x B p K  r C p x . (9)

The error equilibrium points and relative stability under the influence of parameters including actuator gain K, reference input r and time invariant uncertainty in linear plants are addressed. The parameter vector is defined as ( , , )r p K .

2.3 PD Type Fuzzy Logic Control System

This subchapter discusses the PD type SFLC depicted in Fig. 2.4. The SFLC’s output u_f is proportional to a negative signed distance D . Additionally, the number of the fuzzy rules,_s

as shown in Table 2.3 [52], is significantly reduced into 1-D space, as in Table 2.4, owing to the single input and skew-symmetric property. Due to the skew-symmetric property of the rule table, (e ,e) can be split into five regions. Figure 2.5 illustrates an example of this division of(e ,e). The reduced 1-D rules improve the efficiency of the controller by saving time cost for a look up rule table, although it also adds the calculation time of signed distance.

Therefore, the SFLC is suitable for implementation in circuit control. The SFLC is introduced in this subchapter for further equilibrium points and stability analysis in the following subchapters.

2.3.1 Calculation of signed distance

The control error in SFLC is defined as

d( )

e t   .y r (10)

The switching line s as shown in Fig. 2.5 is given by_l

: 0

l d d

s ee  . (11)

The signed perpendicular distance D of general point_S Q e e( , )_d  to a switching line is_d calculated as follows:

The control output u_f (D_S)is defined according to the control rule in SFLC as given in Table 2.4 and Fig. 2.4.

2.3.2 The presentation of the SFLC system

The SFLC system can be described as:

( ) ( , ) _f xA p x B p K u ,

( )

y C p x , (13)

where the control input u_f (D_s).

2.3.3 The analytic representation of the SFLC system

If Tables 2.2, 2.4 and Fig. 2.2 are applied into the controller in SFLC, then the control function ( ) of the fuzzy controller is as displayed in Fig. 2.6. The surface of the fuzzy controller in SFLC is typically oddly symmetrical; therefore, the control force is given by

( ) ( ) ( )

f s S

u D   D  , (14)

where

1 2 s

e e

D 

 

  



 .

In the following analysis, this representation as illustrated in Fig. 2.7 is applied to PD type analysis. In Chapter 3, the SFLC system is reformatted as a special P type fuzzy control system, and is employed to analyze the equilibrium

Fig. 2.1 The P type fuzzy control system.

e

(a)

u

f (b)

Fig.2.2 The membership functions of the fuzzy logic controller.

Table 2.1

Rules of the fuzzy logic controller

e NBE NME NSE ZRE PSE PME PBE

uf NBU NMU NSU ZRU PSU PMU PBU

Table 2.2

Parameters of the fuzzy logic controller

NBE NME NSE ZRE PSE PME PBE

e

a3

 a₂ a_{1 0} a₁ a₂ a₃

NBU NMU NSU ZRU PSU PMU PBU

uf

Fig. 2.3 The control function of the fuzzy logic controller.

y

Fig. 2.4 The single-input fuzzy logic control system.

Table 2.3

Rules of conventional FLC with control error defined as e_d ed

Fig. 2.5 The skew-symmetric property in (e ,e) and the calculation of signed distance.

Ds

a

1 a₂ a₃ a1

2 

a a3



b1

b2

b3

b1



b

2

 b

3



( ) D

_s



Fig. 2.6 The control function of the fuzzy logic controller in SFLC.

y

Fig. 2.7 The transition formation in the transformation.

Chapter 3

Equilibrium Points and Stability Analysis in P and PD Type Fuzzy Control Systems

3.1 Equilibrium Point Analysis for P Type Fuzzy Control Systems with Linear Plants

This subchapter presents the analysis of error equilibrium points and stability in P type fuzzy control systems. The equilibrium point in fuzzy control systems can be derived when equilibrium points can be solved. Moreover, the stability of the equilibrium point can be judged with the linearizing system around the equilibrium or the Popov criterion in the following subchapter. If the error equilibrium points of the overall system are stable, then the steady state error can be derived from this result.

By (9), let x0, then

 

( ) 0

Ax B K r Cx  . (15)

If A^1 exists, then (16) is obtained.

1 ( ) ( ) 0

x A B K ^ e  , (16)

where e r Cx  .

Multiply the result by C in (16), and let Cx r e , then

1 ( ) ( ) 0

e r CA B K  ^ e  . (17)

The state equilibrium points represented as x^e, and the error equilibrium points denoted as

ee, can be determined from (16) and (17), respectively.

Assumption 3.1: The unique solution exists in (17). In other words, an error equilibrium

point uniquely exists.

Under Assumption 3.1, the error equilibrium points can be solved from (18) by replacing (4) in each segment.

One of these error equilibrium points is the unique point of the overall system. The unique point is identified by checking whether e^e is located in its own error region.

3.2. Stability Analysis for P Type Fuzzy Control Systems with a Certain Linear Plant

In the certain linear plant case, the stability can be determined by the time or frequency domain approaches proposed in [51]. In the time domain approach, the eigenvalues of the linearizied system (8) can be applied to determine the stability. In the frequency domain, the Popov criterion is utilized to test stability.

3.2.1 Frequency domain approach

Consider the error dynamic system for a given parameter vector ( , , )r p K . ˆ ( )ˆ ( ) (ˆ ( ) )ˆ

The error equilibrium point of the P type fuzzy control system is given by

( , , ) ( ) ( , , )

e e

e r p K  r C p x r p K . (20)

The error dynamic system is also of Lur’e type. The function  satisfies the followingˆ sector condition if e r p K^e( , , ) . system exists, then the stability can be determined from the linearization of (9) near the state equilibrium point.

Remark 3.1: If the unique state equilibrium is stable, then the steady state error in fuzzy control systems can be obtained from the state equilibrium by e^e  r Cx^e.

3.3 Stability Analysis for P Type Fuzzy Control Systems with an Uncertain Linear Plant

In this subchapter, the parametric absolute stability can be tested using the parametric robust Popov criterion incorporated with Kharitonov theorem, when the parameter vector

( , , )r p K R_ref   , where R_ref [ , ]r r .

The value of e r p K^e( , , ) is difficult to calculate from the results in the previous

subchapter, because fuzzy control function ( ) is sometimes impossible to obtain mathematically, and parameters ( , , )r p K vary in a range in real application. Therefore, the stability analysis by the parametric robust Popov criterion in [51] is adopted to handle this situation.

Applying Theorem 1 in [51], let’sconsider the uncertain P type fuzzy control system (9) satisfying the following conditions. Then, the P type fuzzy control system is parametric absolute stable.

(1) If the fuzzy controller is continuous, and for some neighborhood  of e0 satisfies referred in the Lemma 1 of [51].

(3) If for a given region R_ref of r and for any p , the condition ^e_R( )p is satisfied, and a real number v_o v r p K_o( , , ) exists such that the following inequality holds

Re[(1 ) ( , , )] 1 0

The P type fuzzy control system is then parametric absolute stable. [51]

Remark 3.3:

(1) This test can be extended to the general P type fuzzy control functions design.

(2) The assumption in Remark 3.2 does not lose generality, since most systems have (0, , ) 0

G p K  .

(3) The effect of Kcan be combined into plant parameters p.

The existence of v_o v p_o( ) for every p should be guaranteed in (28). This is generally a difficult problem. Therefore, the parametric robust Popov criterion incorporated with Kharitonov [51], [53], [54] for interval Lur’e systems is introduced into a parametric absolute stable analysis.

Consider the following as a family of interval plants ( , , ) ( )



^{( ): ( ) 0 1 , i i, i , 0, , ,}

^

defined as the 16 plants of the following set,

 

A P type fuzzy control system is absolutely stable in sector

 

^{0, k for all}

( ) ( , , )

G s G s p K , if a real v can be obtained by verifying the robust Popov condition for_o ( ) _K( )

G s G s to satisfy inequality (28).

Remark 3.4:

(1) The previous descriptions imply that only 16 Popov plots need to be drawn from family

K( )

G s to check that the P type fuzzy logic control system is stable when the robust

Popov condition (28) holds for the whole familyG s( ).

(2) The P type fuzzy control systems of Lur’e type can be tested by the parametric robust Popov criterion. By [51], [53], [54], the criterion incorporated with Kharitonov for interval Lur’e systems can be considered here for parametric absolute stability analysis of P type fuzzy control systems.

3.4 Transformation SFLC from PD to P Type

In the following, the SFLC is transformed from PD to P type, so that the equilibrium point and stability can be analyzed by the transformed special P type fuzzy logic control system.

From Fig. 2.4, the factor

2

1

1 of SFLC is integrated into both the proportional and derivative factors. The  and  in Fig. 2.7 are then defined as

2 ,

Assumption 3.2: CB0.

According to Assumption 3.2 and Fig 2.7, the following derivation can be obtained.

.

e r y r Cx    (33)

By differentiating both sides, then

( _f) .

e Cx C Ax Bu CAx (34) From (33) and (34), then

( ) ( ) 1 ,

e e r Cx CAx r C x

           (35)

where C₁(CCA), and rr.

After transformation, the transformed plant in Fig. 3.1 can be obtained

1

( , , ) 1( )[ ( )] ( , ).

GPD s p K C p sI A p ^B p K (36)

From Fig. 3.1, the special P type transformation from the SFLC system can be described as:

The transfer function H_PD( , )s p of the transformed plant in Fig. 3.1 can be described as

 

¹

( , ) 1( ) ( ) ( )

HPD s p C p sI A p ^B p , (38)

3.5 Equilibrium Point Analysis for PD Type Fuzzy Control Systems with Linear Plants

From Fig. 3.1, the equilibrium point can be analyzed ( ) ( , ) ( ).

By multiplying the result of (40) by Cand using (35), then

( ) ( ) 1( ) ( , ) ( ) 0

Remark 3.5: The error equilibrium point of the PD type fuzzy control system is

ee ^e_d^e. (44)

3.6 Stability Analysis for PD Type Fuzzy Control Systems with Linear Plants

The transformed P type of SFLC in Fig. 3.1 can be employed to analyze the stability of SFLC for a given ( , , )r p K .

3.6.1 Frequency domain approach

Consider the error dynamic system in Fig. 3.1 for the given parameter vector( , , )r p K .

( ) ( , ) ( 1( ) )

x A p x B p K  C p x, (45) where x x x r p K ^e( , , ), (C p x₁( ) ) C p x e r p K₁( )^e( , , )   e r p K^e( , , ),

and e r p K^e( , , ) rC p x r p K₁( ) ( , , )^e . (46) The error dynamic system is also of Lur’e type. The function _satisfies the following sector condition, if e r p K^e( , , ).

Consider an arbitrary parameter vector ( , , )r p K in SFLC. Suppose that an equilibrium state x r p K^e( , , ) of the system exists. The stability can be determined by the linearization of

(37) near the error equilibrium point.

3.7 Stability Analysis for PD Type Fuzzy Control Systems with Uncertain Linear Plants

Since the transformed SFLC is a special P type fuzzy control system as shown in Fig. 3.1, the parametric Popov criterion [51] incorporated with Kharitonov theorem is adopted to analyze the stability of PD type fuzzy control systems with uncertainties.

y

r r   ^{ y }

s I

^1 ( )

B p

x C p

₁

( ) ( )

Ap

u

f

( , , ) G s p K PD

 ( ) 

( , ) B p K

( , ) H s p

PD

Fig. 3.1 The transformed SFLC with the special P type fuzzy control system formation.

Chapter 4

Fuzzy Current Control RC Circuit System Design

The temperature control is an important issue in many industrial processes or medical applications. The temperature controls systems are analogous to RC electrical circuits and are governed by the following third-order equation (49) [75]. In our design, FLC is applied to control the RC electrical circuits to reach the specified output voltage. In other words, it is similar to regulate the temperature to desired set point. This chapter specifies fuzzy current control RC circuit systems of P and PD types for verifying the theoretical analysis using PSPICE simulation.

In this chapter, the circuit structure is specified first. The fuzzy logic controller is then designed to construct the fuzzy control function, which is mapping I/O relation of the fuzzy controller. Finally, some components of the overall structure of the fuzzy logic control system are introduced.

4.1 The Block Diagram of the Fuzzy Current Control RC Circuit System

Figure 4.1 depicts the block diagram of a fuzzy current control RC circuit. The control objective of this system is to track a dc constant reference voltage r. To avoid the loading

effect from the circuit of the next stage, the voltage buffer is utilized to feed the output voltage v back into the controller to generate the control error voltage₃ v . The core of this_e system is the fuzzy controller. Both P and PD type fuzzy controllers are designed in the circuit system. The control voltage v_of is transformed into the control current i_ovc with a voltage controlled current circuit.

Finally, the amplified current u t( ) from the current amplifier is injected into circuit plant to let output voltage v to track a reference voltage r.₃

4.2 Circuit Plant

The circuit plant in Fig. 4.2 [75] is composed of RC circuits and external current source control input u t( ).The output voltage is v . Consider the transfer function of circuit plant₃

( ) 3 1

4.3 Fuzzy Logic Controller Circuit

The circuit of a fuzzy logic controller is shown in Fig. 4.3. This circuit is designed to construct the control function of the fuzzy controller. Figure 4.4 illustrates the relationship between the circuit parameters and the control function [76], [77].

4.4 The Overall Design Circuit

Figure 4.3 shows the overall design circuit. For simplification, the voltage controlled

current circuit, current amplifier and PD type signal generator are introduced in [78].

4.4.1 Voltage controlled current circuit

Fig. 4.3 displays the voltage controlled current circuit. If the following equalities (50) stand, then

The current amplifier is designed to normalize the signal from voltage controlled current circuit and amplifies it. The control input u t( ) from the current amplifier for the circuit plant is given by

4.4.3 PD type signal generation

The derivative and proportional signals are generated by OP amplifier differentiator and OP inverting amplifier as illustrated in Fig. 4.3.

The OP amp differentiator is designed as

12 4 e d

v R C dv

 dt . (53)

The value R C is chosen to meet_{12 4} .

Conversely, the OP inverting amplifier is given by

10 8

p e

v R v

R . (54)

where ¹⁰

8

R

R .

In Fig. 4.3, a P type fuzzy control system is chosen when two switches open at P positions.

Conversely, a PD type fuzzy control system is selected when two switches close at PD.

Fig. 4.1 The block diagram of a fuzzy current control RC circuit system.

y

R1 R₂ R₃

C1 C₂ C₃

( ) u t

Fig. 4.2 The RC circuit plant [75].

Differentiator circuit

Fig. 4.3 The designed fuzzy current control RC circuit system.

0 2

Fig. 4.4 The control function of a fuzzy controller with circuit design parameters.

Chapter 5

Simulation Results

In this chapter, a fuzzy control RC circuit plant as shown in Fig. 4.2 is utilized to investigate the parametric equilibrium points and stability when the circuit plant is certain or uncertain with P and PD type fuzzy logic controllers, respectively. The varying parameters include reference input r , an adjustable parameter K and an interval circuit plant parameters p.

For the analysis of certain plants, the equilibrium points under the ( , )r K parameter space with stable notation are given. The phase plane and time waveforms are given to verify the analytical results. The design circuit with PSPICE simulation is also provided to check theoretical analysis. On the other hand, the parametric robust Popov criterion is employed to test the stability of the parameter vector ( , , )r p K R_ref  P . From this point of view, the effect of Kcan be combined into plant parameters by the previous introduction.

Let R₁   , andR₂ R₃ 1 C₁  C₂ C₃ 1F in (49), the third-order transfer with form

From Fig. 2.1, combining the adjustable parameter K, the transfer function is given by

0

The state space representation for G s K( , )can be derived

0 3 1 3 2 3

The fuzzy rules are adapted in this simulation as follows:

1:

Figure 5.1 illustrates the membership functions. Table 5.1 shows the fuzzy control system parameters. Fig. 2.3 shows the control function, where k₀  ,6 k₁4 / 9 and c₁5 / 9. Consider the following simulation with K 1 ~ 20, r1 ~ 1 and the initial condition

 

(0) 0 0 0

x  . Table 5.2 lists the circuit components in Fig. 4.3. For practical considerations, the parameters of the fuzzy controller are selected as Table 5.2 in order to approach the ideal control function depicted in Fig. 5.2.

5.1 P Type Example Demonstrations 5.1.1 Certain linear circuit plant

Under Assumptions 3.1, the equilibrium points of the fuzzy control systems in each segment can be calculated using (18).

 

Equation (60) can be solved by linearizing (9) and using (57)

0 1 0 function, and  is determined by e^e from (18). In (18), the reference r and actuator gain K affect e^e. Figure 5.3 depicts the analysis of the stability of equilibrium points. The reason for the formation of unstable oscillations is discussed in the following subchapter. Figures 5.4 and 5.5 display the verification of the analysis in Fig. 5.3, with respect to P1 (unstable) and P2 (stable point).

5.1.2 Mechanism of oscillations in the fuzzy control system

In this example, the P type fuzzy control system is a piecewise-linear system with three segments. An equilibrium ( ,e e^e ^e 0) exists in every segment for a specific ( , )r K pair.

Figure 5.4 (a) shows the three error equilibriums of every piecewise segment in the phase plane of ( , )e e when ( , ) (0.2,5)r K  . Three equilibrium points are represented as ‘*’(stable equilibrium point for segment 1), x (unstable equilibrium point for segment 2) and ‘▽’

(stable equilibrium point for segment 3), for segments 1–3, respectively. Assume that ( , )e elocates in segment 1 initially. ( , )e e is pulled into the equilibrium point ‘*’of segment 1 located in segment 3. When ( , )e e enters segment 2, ( , )e eis pushed away from

equilibrium point x of segment 2. After ( , )e e is pushed away from segment 2 and enters segment 3, ( , )e e is pulled back to the equilibrium point ‘▽’of segment 3. The limit cycle is formulated by pushing and pulling.

Conversly ( , )e e crosses the segments 1, 2, and 3, is all pulled into equilibrium points and finally ( , )e e achieves the global equilibrium point of segment 2. The authors discuss in detail the stability under different design parameters [79].

5.1.3 Alternative control function

In Fig. 5.3, the effect of reference for stability is not obvious. Therefore, the different fuzzy controllers in Table 5.3 are designed with different control functions. The results in Fig. 5.6 specify how the different controllers will influence the equilibrium points and stability besides

r and K.

5.1.4 Uncertain linear circuit plant

In this part, the stability of the fuzzy control system with interval plant is checked by (28) incorporated with Kharitonov theorem. In the following simulations, r [ 1,1], K 2,

1 ~ 3

R R and C₁ ~C in circuit plant listed in Table 5.2 with tolerance₃ 5% and k_R^* 6 in (28) are selected. The plant (56) for P type fuzzy control system can be rewritten as

0 0

By (28) incorporated with Kharitonov theorem, the absolute stability can be tested as

are plotted enough to indicate the stability in such a case.

5.2 PD Type Example Demonstrations

In the following simulation, 10 is selected in PD type fuzzy control system.

5.2.1 Certain linear circuit plant

In this subchapter, Fig. 3.1 demonstrates the PD type fuzzy control system. Under the Assumptions 3.1, and 3.2, the error equilibrium points of the fuzzy control systems in every segment can be obtained by (43).

 

By linearizing (39) and using (57), (63) can be carried out, and Fig. 5.8 can be obtained.

( , , ) ( , , ) ( ) 1

In the following, Figs. 5.9 and 5.10 verify the analysis in Fig. 5.8 with respect to P1 (unstable) and P2 (stable point).

5.2.2 Alternative control function

The alternative controller in Table 5.3 obviously influences the equilibrium point and

stability, when the reference is varying. Figure 5.11 shows the analytical results.

5.2.3 Uncertain linear circuit plant

In this subchapter, Fig. 3.1 is adopted to demonstrate the parametric stability of the PD type fuzzy control system. Following transformation, the analytic new plant for PD type fuzzy systems is given by (38):

2 listed in Table 5.2 with tolerance 5% and k^*_R 6 in (28), are specified to evaluate the stability of a PD type fuzzy control system. From (36), the analytic new plant for PD type fuzzy control system can be recast as

1 1 0 0 sixteen Popov curves illustrated in Fig. 5.12 are plotted to verify that the PD type fuzzy control system is stable according to (28) incorporated with Kharitonov theorem.

Table 5.1

Parameters of the fuzzy logic controller in simulations

NBE NSE ZRE PSE PBE

e (or)

1 0.1 0 0.1 1

NBU NSU ZRU PSU PBU

uf

1 0.6 0 0.6 1

(or )

e 

(a)

u

f

(b)

Fig. 5.1 The membership functions of the fuzzy control system.

Fig. 5.2 The fuzzy control function with PSPICE simulation by Table 5.2 parameters.

Table 5.2

Parameters of the fuzzy current control RC circuit system Circuit voltage controlled current circuit design.

P type design: Power source VCC=15V, VEE=-15V, VCC1=8V, VEE1=-8V,

VCC2=30V, and VEE2=-30V.

Operational amplifiers in

design

P type design:

OP amps 1~6 with OPA602, and OP amps 7~8 with LM675 (Power op amp).

PD type design:

OP amps 1~6 with OPA602, OP amps 7 with OPA501 (Power op amp) and OP amps 8 with LM675.

-1

Fig. 5.3 The equilibrium stability of the P type fuzzy control system by Table 5.1 for ( , )r K ,

Fig. 5.3 The equilibrium stability of the P type fuzzy control system by Table 5.1 for ( , )r K ,

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