Problem Solution

boundary curves will be generated from these q values in the q -parameter space by

varying ω given specific k and A with θ =0^○ . Phase boundary curves will be

generated by varying ω given specific θ and A with k =1. The gain and phase margins

of the perturbed vehicle system for the limit cycle will be analyzed from boundary curves

geometrically in 2 and 3 dimensions. A specific gain or phase value corresponding to the

boundary curve which is tangent to the perturbed parameter region is defined as the gain or

phase margin for predicting the limit cycles, respectively.

4.3 Problem Solution

The block diagram of the perturbed vehicle system with nonlinear elements N₁ and

N2 is illustrated as in Fig. 4.2. The transfer function of a vehicle control model used for the

investigations with the input δ_F=the front wheel deflection angle and the output r =the yaw

(4 10 56500 ) 587225 1.9932

r F

The steering actuator is modeled as a linear dynamic system with the actuator bandwidth

ωa = 4π ,

The closed loop characteristic polynomial is

1 2

part of the characteristic polynomial are

10 2 2

The coefficients of the imaginary part of the polynomial are

10 2 2

is illustrated in Fig. 4.3.

7 30

that a limit cycle with a specific amplitude is generated and the gain boundary curve

corresponding to k_minwhich is tangent to the perturbed parameter region Q is observed.

Phase margin is the minimal phase θ of the gain-phase tester (_min ke^−jθ ) with k=1 such that

a limit cycle is generated and the phase boundary curve corresponding to θ which is _min

tangent to the perturbed parameter region Q is observed.

Based on the previous analysis, some limit cycle loci with the gain k=1 and the phase

θ =0^○ of the gain-phase tester are depicted as in Fig. 4.3. In Figs. 4.4 and 4.5, the gain and

phase boundary curves are generated. It is obviously observed that the gain and phase margins

of the perturbed vehicle system are 0.772dB and 9.4126 deg, respectively. The time

response shown in Fig. 4.6 has demonstrated the consistence with the results in Fig. 4.4 and

Fig. 4.5.

4.3.2 Case2: Gain-phase Margin Analysis in v− − Space µ m

The perturbed parameter q=[ ,q q q_{1 2}, ₃] [ , , ]= m µ v are considered for analyzing gain and

phase margins. The system parameter q₁=m is the car mass, q₁ =µ is the road friction and

q3 = is the vehicle velocity. ν

The perturbed parameter space R in the µ ν− − coordinate is illustrated in Fig. 4.7. m 7 30

650 1 region

5 70

1730Kg m 2330Kg

ν µ

ν

⎧ + ≤ ≤

⎪⎪⎨ ≤ ≤

⎪⎪ ≤ ≤

⎩

R (4.14)

weights are also shown in Figs.4.7 and 4.8. For example, the gain and phase margins for the vehicle weight equal to 1730Kg are 0.922dBand 12.12 deg, respectively.

4.4 Concluding Remarks

In this chapter, some effective techniques are presented involving describing function

methods, parameter space methods, and a gain-phase margin tester. The methods in previous

studies are extended to analyze GM and PM performances of a vehicle plant with three

parameters in a perturbed space for predicting the limit cycle occurred by using a gain-phase

tester and 3D graphical representations are also provided to give a concise and clear way to

study the robustness stability of the system with nonlinearities. The method proposed here

would further be extended to be used in a system with more than three perturbed parameters.

( )t

δ

F

r t ( )

ke

⁻ j

^{θ G s q N N ( , ,}

¹

^,

²

⁾

+ −

Fig. 4.1 The block diagram of a nonlinear control system with a gain-phase margin tester.

1 S

1 ( ) S

Ga s ( )

Gp s ( )t

δ

F

( ) r t

R1 R2

R2 1 −

−R N1

N2

x

2

x 1

+ −

Fig. 4.2 The block diagram of the perturbed nonlinear system.

0 10 20 30 40 50 60 70 80 90 100

Fig. 4.3 The limit cycle loci in the parameter plane.

0 10 20 30 40 50 60 70 80 90 100

Fig. 4.4 Gain boundary curves with the vehicle weight 1830Kg (GM=0.772dB).

0 10 20 30 40 50 60 70 80 90 100 0

0.2 0.4 0.6 0.8 1 1.2

u(road friction)

0 deg 30 deg

9.4126 deg(PM) Q-region

Asymptotically stable

region Limit cycle region

v(velocity m/s)

Fig. 4.5 Phase boundary curves with the vehicle weight 1830Kg (PM=9.4126 deg).

1600

u(road friction) mass(kg)

region

R

Fig. 4.7 Gain boundary curves in 3-dimension.

160018002000220024002600 0.5 0

Fig. 4.8 Phase boundary curves in 3-dimension.

Chapter 5

Parameter Plane Analysis of Fuzzy Vehicle Steering Control Systems

5.1 Overview

The main purpose of this chapter is to analyze the robust stability for a fuzzy vehicle

steering control system. In general, fuzzy control system is a nonlinear control system.

Therefore, the fuzzy controller may be linearized by the use of describing function first. After

then, parameter plane method is then applied to determine the conditions of robust stability

when the system has perturbed or adjustable parameters. A systematic procedure is proposed

to solve this problem. The effects of plant parameters and control factors are both considered

here. Furthermore, the problem of relative stability by using a gain-phase margin tester is also

addressed. The limit cycles provided by a static fuzzy controller can be easily suppressed if

the control factors are chosen properly. Simulation results show the efficiency of our

approach.

Fig. 5.1 shows the single track vehicle model and the related symbols are listed in Table

5.1. The equations of motion are [22]

( ) _{f r}

The numerical data are listed in Table 5.2. According to the above analysis of a single track vehicle model, the transfer function from the input of front deflection angle δ to the _f

output of yaw rate r can be obtained as

In addition, the steering actuator is modeled as

In our study, a fuzzy vehicle control system is presented in Fig. 5.3. The open loop

transfer function G s_O( ) is defined as

( , , ) ( ) ( , , )

O a r f

G s µ v =G s G _δ s µ v (5.8)

The control factors k_p, k_d and k_u can be determined by the designer. By transferring Fig.

5.3 to Fig. 5.4, the overall open loop transfer function can be obtained as

( , , , , , ) ( , , )

5.3 Describing Function of Static Fuzzy Controller

The describing function N₁ of static fuzzy controller shown in Fig. 5.4 can be obtained,

which depends only on the amplitude of A and is independent of the frequency of ω, and

can be expressed as follows [32]:

1 1

of fuzzy membership functions (Φ_i’s) as follows:

5.4 Stability Analysis of Fuzzy Vehicle Control Systems

If the gain-phase margin tester Ke^−jθ is added in the open loop of Fig. 5.4, the closed

loop transfer function is

1

Case 1: Perturbed Plant Parameters

Arrange (5.12), the following characteristic equation is obtained.

2 2 2 2

1.4621 10 ( 17.7688 157.9137)

2.1818 10 ( )

2.2345 10 ( )

1.5271 10 ( 17.7668 157.9137) 9.4384 10 ( 17.7668 157.9137) 5.656 10 ( 17.7668 15

Let s= jω, K =0 dB and θ = . Equation (5.13) is divided into two stability equations 0

1.0064 10 1.6776 10 (1.5179 10

2.3999 10 ) 2.2345 10 (2.5986

10 2.1818 10 ) ,

10 (2.3091 10 1.4621 10

2.1818 10 ) .

In order to obtain the solution of µ and v, the following equation is solved 0

also changed, the solutions of µ and v called limit cycle loci can be displayed in the

parameter plane.

Case 2: Control Factors

After some simple manipulations, the characteristic equation of (5.12) can be obtained as ( , _p, _d, _u, , , , ) _{p d} 0

10 2 12 2 1

2 6 2 2 9

10 2 8 2

(2.1818 10 2.2345 10 )

1.414 ( 17.7688 157.9137)(6.675 10 1.0746 10 4.0045 10 1.034 10 )

with real part and imaginary part of characteristic equation

( , _p, _d, _u, , ) _{R I} 0

Case 3: Gain-phase Margin Analysis

The gain-phase margin tester can be expressed as

cos sin

and Then, the characteristic equation can be written as

( , _p, _d, _u, , , _R, _I) _{R I} 0

1.414 ( 17.7688 157.9137)(6.675 10

1.0746 10 4.0045 10 1.034 10 )

Let s= jω, (5.28) is divided into two stability equations with real part and imaginary part of

characteristic equation

and

1 2 2 1

1 2 2 1

I ,

K W U W U

U V U V

⋅ − ⋅

= ⋅ − ⋅ (5.34)

5.5 Simulation Results

In our work, five fuzzy rules and parameters are adopted and listed in Tables 5.3 and 5.4,

respectively. Fig. 5.5 shows the premise triangle membership functions of fuzzy controller.

The consequent parts are singletons. Fig. 5.6 shows the control surface of fuzzy controller.

If k_p =0.2, k_d =0.3 and k_u =0.2 are selected first, (5.14) can be solved when A is

fixed and ω is changed from 0 to ∞. Fig. 5.7 shows the stability boundary and some limit

cycle loci in the µ- v parameter plane. Two stability regions including asymptotically stable

and limit cycle are divided. In order to verify the accuracy of Fig. 5.7, four operating points

Q1-Q3 (limit cycle region) and Q4 (asymptotically stable region) are illustrated for testing.

Fig. 5.8 shows the time responses of input signal x t( ). It is obvious that the results shown in

Fig. 5.8 consist with the predicted results in Fig. 5.7. For examples, if Q1(µ= and 1 v=70)

is chosen, the limit cycle occurs and the amplitude is 0.0465. Besides, if Q4 (µ= and 1

5

v= ) is chosen, the system is stable and no limit cycle happens. On the other hand, if

p 0.1

k = , k_d =0.27 and k_u =0.1 are selected, Fig. 5.9 shows the stability boundary. We

can find that no limit cycle will occur in the overall operating region Q.

If k_u =0.2, µ= and 1 v=70 are selected, (5.19) and (5.20) can be solved in the

k -p k parameter plane when _d A is fixed and ω is changed from 0 to ∞. Fig. 5.10 shows the stability boundary and some limit cycle loci. Four testing points Q5-Q8 are illustrated.

If Q8 (k_p =0.1, k_d =0.1, k_u =0.2, µ= , 1 v=70) in Fig. 5.10 is selected, (5.23) and

(5.24) can be solved in the K -_R K parameter plane when _I A is fixed and ω is changed

from 0 to ∞ . Because Q8 is in asymptotically stable region, the gain-phase margin tester can

be viewed as a compensator to generate the limit cycle (from stable region to limit cycle

region). For example, if A=0.05 is expected, the related gain margin (Q9: GM=3.2,

θ= ) and phase margin (Q10: 0 PM=46.6 , K = ) to generate limit cycles can be easily 1

obtained in Fig. 5.11. On the other hand, when the original system is in limit cycle region like

Q5-Q7, the related gain margin and phase margin to suppress limit cycle could be also

obtained in the parameter plane.

5.6 Concluding Remarks

Based on the parameter plane approach, the complete stability analysis of a fuzzy vehicle

steering control system is proposed in this chapter. A systematic procedure is presented to deal

with this problem. In addition, the effects of control factor and gain-phase margins are also

considered. Simulation results show that more information can be obtained by this approach.

Fig. 5.1 Single track vehicle model.

Table 5.1 Vehicle system quantities

f, r

F F lateral wheel force at front and rear wheel

r yaw rate

β side slip angle at center of gravity (CG)

v velocity

af lateral acceleration

f,r

l l distance from front and rear axis to CG

f r

l= +l l wheelbase

δf front wheel steering angle m mass

Table5.2Vehicle system parameters

0

cf 5000 N/rad

cro 100000 N/rad

m 1830Kg

lf 1.51 m

lr 1.32 m

Fr Ff

r

CG

v

af

δf

lr l_f

β

Fig. 5.2 Operating Range.

Fig. 5.3 Block diagram of a fuzzy vehicle control system.

Fig. 5.4 Block diagram of a fuzzy vehicle control system.

0 10 20 30 40 50 60 70 80 90 100

Table 5.3 Rules of fuzzy controller

Table 5.4 Parameters of fuzzy controller

Fig. 5.5 Membership functions of fuzzy controller.

Fig. 5.6 Control surface.

PBU

0 10 20 30 40 50 60 70 80

Fig. 5.7 Limit cycle loci.

0 5 10

0 10 20 30 40 50 60 70 80

Fig. 5.9 Stability boundary.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Fig. 5.10 Stability boundary.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 -2

-1.5 -1 -0.5 0 0.5 1 1.5 2

KR

KI

Q9(3.2,0) Q10(0.7,0.74)

GM=3.2 PM=46.6deg.

A=0.05 w=7.1

w=12.8

Fig. 5.11 GM and PM analysis.

Chapter 6

Robust Design for Perturbed Phase-Locked Loops

6.1 Overview

A control algorithm is presented in this chapter for phase-locked loop (PLL) design with

perturbed parameters satisfying frequency-domain specifications. By the use of a gain-phase

tester, the parameter plane method and robust stability criteria, the range of the designed

parameters of PLL is determined based on specified constraints of gain and phase margins

(GM and PM) on frequency domain with uncertain parameters perturbed in some intervals.

The PLL model used in this design is assumed to be a linearized one if in the locked state. The

proposed method is applied to a PLL model with first and second order low-pass filters as

examples. With the help of stability boundary curves, the area in the selected designed

parameters of the corresponding coordinate plane is found out such that the whole PLL

system with the desired parameters in that area will meet given conditions. Simulation results

are provided to illustrate the design technique based on GM and PM, and the resulted PLL is

to robustly meet the specified constraints as expected with uncertain parameters varying in

intervals.

6.2 Basic Concept of PLL

PLL is an electronic circuit which causes a output signal to keep track of the input

reference signal applied to it and the output signal keeps synchronization with the input one.

Three basic functional blocks, a phase detector (PD), a loop filter (LF) and a voltage

controlled oscillator (VCO), are contained in a PLL depicted in Fig. 6.1.

Assume the reference input ( )v t_i =A sin(ω_it+θ_i) and the VCO output

( ) Bsin( o )

o o

v t = ω t+θ , where ω_i and ω_o are angular frequencies, θ_i and θ_o are phases,

A and B are amplitudes of v t and _i( ) v t , respectively. _o( )

In a PLL, the function of the PD is to measure the phase difference between v t and _i( )

0( )

v t and produces an output voltage v t proportional to the phase error of ( )_d( ) v t and _i

0( )

v t . Assume the PD be a linear multiplier in linear PLL (LPLL) through this chapter.

The LF is a low-pass filter and is used to suppress noise and high-frequency signal

components which are unwanted signals. The lower frequency and dc parts are passed

through the LF and delivered to control the frequency of the VCO output.

The VCO is an oscillator which produces a periodic signal with the frequency that is

proportional to the dc voltage from the LF.

When the PLL is locked, the frequencies ω_i and ω_o are identical. The PD is linear

and the LF output voltage is proportional to the phase error. The linearized mathematical model of the PLL is shown in Fig. 6.2 if the phase difference θ θ_i− _o is very small. As seen

in this figure, the PLL structure is in fact a feedback control mechanism. The phase transfer function ( )Γ s that relates the phase θ_i of the reference input to the phase θ_o of the VCO

transfer function of the LF.

6.3 Stability Boundary Analysis

Consider a gain-phase tester ke^−jθ included in series with the original control system

as in Fig. 6.3, and its transfer function is given by

F s q is the transfer function of the LF with the perturbed parameter vector q .

1 2

[ , ,...., _r]

m= m m m is a designed parameter vector of PLL in the m1−m2−....−mr coordinate

space under user-defined specifications.

The closed-loop characteristic polynomial is P s q m k( , , , , )θ and

6.3.1 Gain Boundary Curves

Let 0θ = ^Ο and q be a specific perturbed parameter. Equation (6.4) is rewritten into

and a region of the designed parameters in m−space is to be found out so that the whole

PLL system with the designed parameters chosen from the above determined region will

satisfy the GM condition. The GM of the control system at a point on one side of a specific

gain boundary curve is greater than that at a point on the boundary curve. But it is less at the

points on the other side.

6.3.2 Phase Boundary Curves

Given k=1 and a specific q , (6.4) is written into the form ( , , ) 0

( , , ) 0

U m

V m

ω θ ω θ

⎧ =

⎨ =

⎩ . (6.6) Phase boundary curves are developed under the PM specification in a similar way. They

are generated in m−space from the solutions m of (6.6) by varying ω for every θ .

Given a specific θ , a phase boundary curve will be generated and a region of the designed

parameters in m−space is to be found out so that the whole PLL system with the designed

parameters chosen from the above determined region will satisfy the PM condition. The PM

of the control system at a point on one side of a specific phase boundary curve is greater than

that at a point on the boundary curve. But it is less at a point on the other side.

6.3.3 PLL Robust Design

In physical systems, uncertainties usually exist in system parameters. The LF is usually

connected to a PLL IC externally and implemented by the designer under the specified

constraints. The system parameters are separated into the designed parameters, which are the

parameters of PD and VCO, and the perturbed parameters, which are the ones of the LF. The

designed parameters are the parameters the range of which is to be determined so that the

performance of the whole PLL system can meet the specified conditions under the perturbed

parameters varying in a region as long as the designed parameters are within the determined

range.

In this chapter, the LFs with different order are used as examples to demonstrate the

proposed design method of PLL robust design. The robust design is based on gain and phase

boundary curves with respect to k and θ of a gain-phase margin tester and they are drawn

in m−space from the locations of the roots of (6.5) and (6.6) with respect to different k

and θ , respectively. The range of the designed parameters is going to be found out in

space

m− under the constraints of specified GM and PM.

Based on the discussions mentioned above, the design algorithm is as the followings:

Step (1) Set up user-defined specifications on GM and PM.

Step (2) For every system parameter q at the vertices of the perturbed system parameter

region in q -plane, draw the gain boundary curves corresponding to the specified

GM in m-plane by solving (6.5).

Step (3) For every q at the vertices of the perturbed system parameter region in q -plane,

draw the phase boundary curves corresponding to the specified PM in m-plane by

solving (6.6).

Step (4) Determine a gain region in m−space with the help of the gain boundary curves as

in step (2)

so that the designed parameters with the coefficients in that region satisfy the

specified GM constraints.

Step (5) Determine a phase-region in m−space with the help of the phase boundary curves

as in step (3) so that the designed parameters with the coefficients in that region

satisfy the specified PM constraints.

Step (6) Find out the common region of the determined gain and phase ones as in steps (4)

and (5). The perturbed PLL system with the designed parameters in that region is the

desired one satisfying the specified GM and PM conditions.

6.4 Simulation Results of PLL Design for GM≥3dB and PM≥30^○

6.4.1 The First Order LF

The transfer function F s₁( ) of the first order filter as in Fig. 6.4 is given by

1

1 2

( ) 1 1 F s s

s τ τ

= +

+ , (6.7) where τ₁=R C_{1 1} and τ₂ =(R₁+R )C_{2 1}. R₁ and C₁ are perturbed parameters inside the

S region as in Fig. 6.6, where the S region is and its closed-loop transfer function is

1

By solving (6.5) and (6.6) for every specific k and θ, gain and phase boundary curves are

generated to determine the designed parameters m. The m-desired region is determined

solely by phase boundary curves because of the GM of the closed PLL system is infinity. Let

θ =30^○. The phase boundary curves corresponding to PM=30^o are drawn with the perturbed

parameters at the vertices S1, S2, S3 and S4 of the region S in Fig. 6.6. The shaded area in

-R2

d v

k k plane, seen in Fig. 6.7, is the desired one so that the designed parameters

R2 and k k in the determined area cause the PLL system to meet the phase requirement _{d v} PM≥30^○.

Choose 13000 rad/(sec volt)k_v = × which signifies that the frequency created by the

VCO changes about 20KHz if the input signal v t of the VCO in Fig. 1 changes by 1 volt. _f( )

The shaded area in k_d-R₂ plane in Fig. 6.8 is found. The point Q₁=(k R_d, ₂)=(1.8,14K

ohm) is selected as an example point in this area and the PMs at the vertices and other points

of S are listed in Table 6.1. The corresponding bode plots are also shown in Fig. 9. The

simulation results are achieved as desired.

Choose designed parameters m=[ ,k R_{v 2}] with k_d =0.6. In a similar way, the desired

For the closed PLL system, the transfer function is given by

3 4

4 5

The coefficients of the real part of the characteristic polynomial are

0

and the coefficients of the real part of the characteristic polynomial are

0 R is depicted in 3D-coordinate in Fig. 6.12.

The 3D perturbed space R is defined by

chart of phase boundary curves is developed in Figs. 6.14 and 6.15 in k_d-R₂ plane. Choose

=

corresponding bode plots and the PMs are shown and listed in Fig. 6.16 and Table 6.4.

Assume m=[ ,k R_{v 2}] and k_d =0.8. The phase boundary curves are shown in Fig. 6.17

and the desired shaded area are found in Fig. 6.18 in the k_v-R₂ plane.

4 2

Q (k_v=50000, R =45Kohm) is the selected point. The bode plots and the PMs at the

vertices and other points in ℜ are depicted and listed in Fig. 6.19 and Table 6.5,

respectively.

6.5 Concluding Remarks

This chapter introduces a new method on PLL design by frequency domain approach for

a perturbed PLL control system. Based on parameter space method and robust stability

criteria, the desired system parameters of PLLs in the selected coordinate plane are

determined in graphical portrayals. With the help of gain and phase boundary curves resulting

from the roots of characteristic polynomial equation in the closed PLL system, a methodology

is proposed for portraying regions in a selected designed parameter plane so that the

performance of the whole PLL system can meet the specified requirements on

frequency-domain constraints. Simulation results have demonstrated the objectives have been

achieved as desired.

i ( )

v t v t

_d

( ) v

f

^{( )} t

o

( ) v t output signal input

signal ^phase _detector (PD)

Voltage Controlled Oscillator (VCO)

Loop Filter (LF)

Fig. 6.1 The functional block diagram of PLL.

k d ^{k v}

( ) s

i

F s

θ + θ _e v _e v _c θ

_o

−

Fig. 6.2 The linearized mathematical model of PLL.

( , , ) G s q m

+ − ^{ke − j θ}

Fig. 6.3 The closed feedback system with a gain-phase margin tester ke^−jθ.

R

1

R

2

C

1

Fig. 6.4 The first order loop filter.

R

1

R

2

C

1

C

2

Fig. 6.5 The second order filter.

0 100 200 300 400 500 600 700 800 900 1000

Fig.6.7 The designed-parameter shaded area in k k_{d v}-R₂ plane meeting the phase specifications PM≥30^○ with the first order LF.

Fig.6.7 The designed-parameter shaded area in k k_{d v}-R₂ plane meeting the phase specifications PM≥30^○ with the first order LF.

在文檔中 參數空間法用於擾動控制系統之分析與設計 (頁 57-0)