In this simulation, a Daimler Benz 0305 bus [18] is adopted. Its linearized system with
actuator input δ =steering angle rate, and output y=displacement of front antenna, has the
following transfer function
The controller used is taken as given by
2
and was determined by Muench [18].
Case 1 : 2D GM/PM Analysis in q1−q Plane2 . Consider the system parameter q=[ ,q q1 2]
with an uncertain parameter region S as in Fig. 3.3 for studying GM/PM performances.
The S−parameter region is
and the closed-loop characteristic polynomials is as in (3.6). By substituting s= jω into the
numerator of the above polynomials and by lengthy computation, the coefficients of the real
part polynomial U( , , , )ω q k θ with a specific c in (3.8) are (6669992.4 9.1087 10 ) sin( ),
16828125 21000 3375 10 14293
r q k
(4.2505 10 5716875 1.1316 10 ) sin( ), 262500 4218750000 6669992.4 cos( ) 9.1087 10 cos( ) ,
15625 840 1346250 135 10 ,5
0, for q by varying k and θ , and the stable boundary representation curves for gain and
phase margins are shown as in Figs. 3.4 and 3.5 in the q1−q2 plane, respectively. We are
only interested in positive solutions q1 >0 and q2 >0 for practical reasons. The GM of the
perturbed control system with the domain region S is -4.3dB and its PM is 19.336○ as seen
in Figs. 3.4 and 3.5, respectively. In general, the specifications on the stability robustness
point of view are GM≥3dB and PM≥30○ , which the system with the original controller
(3.21) doesn’t satisfy. A new controller is designed in the following section and its
performance is improved significantly.
The gain boundary curves associated with different gains shown in Fig. 3.4 reveal that
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 curve. But it is less at a point on the other side.
Similarly in Fig. 3.5, the phase boundary curves show that 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
curve. But it is less at a point on the other side. At the point A (( ,q q1 2)=(20, 32)) in both
Figs.3.4 and 3.5 the system has the minimal GM and PM of all the points within the entire S
region.
Case 2 : 3D GM/PM Analysis in m v u− − Space.
Select q=[ ,q q q1 2, 3] [ , , ]= m v u in the block diagram of the closed system in Fig. 3.2. The
perturbed parameter space R as in Fig. 3.6 as follows
24 / 32
Gain and phase boundary curves in the m v u− − parameter space are generated from
the solutions for q to (3.12) and (3.16), respectively. Those curves corresponding to
different k and θ by varying the frequency ω are shown in Figs. 3.7 and 3.8. A specific
gain k (dB) corresponding to a boundary curve which is tangent to the perturbed region R
at a point on the edge EF of R is defined as the GM of the system. It is also the minimal GM of the perturbed control system within R . Its PM is defined in the same way. The
system with uncertain parameters within the R−space has GM=-4.3dB and PM =19.336○.
3.4.2 Controller Design
The system parameter q=[ ,q q1 2] within S is considered for the controller design.
Assume the controller to be designed is given as
2
where c c0, 1 and c are the controller coefficients to be designed under the user-specified 2
constraints and the system parameter domain is within the region S as in Fig. 3.3. Equation (3.21) is a special case of (3.27) with c0 =9375, c1=10938 and c2 =2344.
1) Controller Design for GM ≥3dB and PM ≥300
The design problem of interest is to find all the controller coefficients c c and 0, 1 c2
a coefficient region in c−space is to be found out by the use of gain and phase boundary
curves associated with different k and θ .
By solving (3.10), the coefficients of the real part of the characteristic polynomial ( , , , , )
(388600 609.8 48280 ) cos( ), (609.8 388600 ) sin( ),
16828125 21000 3375 10 , 609.8 cos( ),
( , , , ) 0 ( , , , ) 0 U c k V c k
ω θ
ω θ
⎧ =
⎨ =
⎩ . (3.30) Two controller coefficients of c c and 0, 1 c2 are chosen as adjustable parameters and the
other one is fixed for this design. By solving (3.30), a shaded area is determined by gain and phase boundary curves from the solutions for ( , )c c pairs with 0 1 c2 =2344 under GM and
PM specifications given as above in c0−c1plane, as shown in Fig. 3.9.
For the vertices A,B,C and D of S as in Fig. 3.3, stability boundary curves are plotted
to determine the qualified shaded area. Two gain boundary curves are obtained associated
with k=0dB and 3dB given θ =0○ for each vertex. In a similar way, two phase
boundary ones are also generated corresponding to θ =0○ and θ =30○ with k=1.
Let c0 =9375. Select c1 and c2 as adjustable coefficients. Gain and phase stability
curves are generated in the same way in c1−c2 plane and the shaded region within which
1 and 2
c c satisfy specified constraints is founded, as shown in Fig. 3.10.
In Figs. 3.9 and 3.10 the desired controller coefficients can be chosen according to the
specified gain and phase constraints. The controller coefficient is selected from the above
shaded region so that the whole system with the chosen controller has the desired
specifications. With the designed controller, Tables 3.1 and 3.2 show the GM and PM of the
system operating at several points within the region S . The Bode plots of magnitude and
phase are provided in Figs. 3.11 and 3.12.
2) The constant-sensitivity loci constant and i=0,1. Gain and phase boundary curves in Fig. 3.11 are plotted with the system
operating at the point A in the region S . If the specified sensitivity locus passes through the
shaded area as in Fig. 3.9, a point on the locus is chosen and the controller at this location in
0 1
c − plane is desired. The point c Q1 on the sensitivity locus with the constraint
0 1
performance on stability has been improved.
Let c0 =9375. The solutions to the equality H j( ) 12
i s j
Sc ω s
ω
= = , where i=1, 2, give a plot of the constant-sensitivity loci in c1−c2 plane, as shown in Fig. 3.14. Choose the point
Q2 in Fig. 3.14 with c1 =410 and c1 =6000 on the sensitivity locus
1
( ) H j
c s j
S ω
ω
= =
2
( ) 7
H j 10
c s j
S ω
ω
−
= = and the system operating at the point B in S has GM=6.08dB and PM=31○.
3.5 Concluding Remarks
This chapter introduces a new method on performance analysis and controller design by
frequency domain approach for a perturbed control system. Based on the parameter space
method and robust stability criteria, the performances of a perturbed vehicle control system
are analyzed in graphical portrayals. With the help of gain and phase boundary curves
resulting from the roots of the system characteristic polynomial equation, the GM and PM
have been obtained. In controller design, a methodology is proposed for portraying regions in
a selected controller coefficient plane so that the designed controller is to meet the specified
requirements on GM, PM and sensitivity. Simulation results demonstrate the objectives have
been achieved as desired.
( , ) G s q ( )
+ C s
( ) s −
δ y s ( )
Fig. 3.1 The perturbed vehicle control system with uncertain parameter q .
+ j
ke − θ C s ( ) G s q ( , ) ( ) s −
δ y s ( )
Fig. 3.2 The perturbed vehicle control system in series with a gain-phase tester.
8 10 12 14 16 18 20 22 24 26
q2(mass tons) A (20,32)
B(20,24) C(12,24)
D(12,32)
The perturbed region S
Fig. 3.3 The parameter domain region S in q1-q2 plane.
8 10 12 14 16 18 20 22 24 26
q2(mass tons) A (20,32)
B(20,24)
Fig. 3.4 Gain boundary curves by varying k with GM=-4.3dB.
8 10 12 14 16 18 20 22 24 26
q2(mass tons) A (20,32)
B(20,24)
10
Fig. 3.7 Gain boundary curves in 3D with by varying k with GM=-4.3dB.
10
Fig. 3.8 Phase boundary curves in 3D with PM=19.336○.
Fig. 3.9 The controller coefficient region for GM≥3dB and PM≥300 as indicated in the shaded area in c0− plane with c1 c2 =2344.
Fig. 3.10 The controller coefficient region for GM≥3dB and PM≥300 as indicated in the shaded area in c1−c2 plane with c0 =9375
Table 3.1 The GM and PM of the system with c2 =2344
0 180.7, 1 18.83
c = c =
Location inside S GM PM
Point A(20,32) 4.13dB 37.10
Point B(20,24) 3.18dB 350
Point C(12,24) 9.76dB 67.10
Point D(12,32) 10.3dB 64.50
Point(17,30) 5.73dB 46.50
Table 3.2 The GM and PM of the system with co =9375
1 410, 2 6000
c = c =
Location inside S GM PM
Point A(20,32) 8.48dB 570
Point B(20,24) 6.08dB 310
Point C(12,24) 6.29dB 55.70
Point D(12,32) 8.64dB 67.70
Point(14,26) 6.87dB 57.90
100 101 102 vertices of the perturbed region S .
100 101 102 vertices of the perturbed region S .
140 150 160 170 180 190 200 210 220 230 240 based on the control system at the vertex A (20, 32) of the perturbed parameter region S .
100 200 300 400 500 600 700 800
5600 based on the control system at the vertex B (20, 24) of the perturbed parameter region S .
Chapter 4
Gain-Phase Margin Analysis of Nonlinear Perturbed Vehicle Control Systems for Limit Cycle Prediction
4.1 Overview
The chapter is concentrated on the subject of predicting the limit cycle of a nonlinear
perturbed vehicle control system under specific gain-phase margin (GM/PM) constraints. A
gain-phase margin tester is included in series with the perturbed vehicle system to perform the
GM/PM analysis. GM and PM are determined from the gain and phase values of the
gain-phase margin tester at which the undesirable limit cycle caused by nonlinearities of the
system with uncertain parameters occurs. The nonlinear elements in this system are linearized
by the method of the conventional describing functions. By the use of the parameter space
method, describing function method and stability criteria, a concise and clear way will be
given in the geometric representation in the parameter coordinate to show the gain-phase
margin performances for a nonlinear vehicle control system with uncertain parameters which
are the velocity, road friction and car weight of the plant. The proposed method is applied to a
car model and simulation results are presented to illustrate the GM and PM performances for
the limit cycle.