O R I G I N A L A R T I C L E
Limit cycle prediction of a neurocontrol vehicle system based
on gain-phase margin analysis
Jau-Woei Perng• Li-Shan Ma•Bing-Fei Wu
Received: 21 March 2009 / Accepted: 15 October 2009 / Published online: 31 October 2009 Ó Springer-Verlag London Limited 2009
Abstract Based on some useful frequency domain meth-ods, this paper proposes a systematic procedure to address the limit cycle prediction of a neural vehicle control system with adjustable parameters. A simple neurocontroller can be linearized by using describing function method firstly. According to the classical method of parameter plane, the stability of linearized system with adjustable parameters is then considered. In addition, gain margin and phase margin for limit cycle generation are also analyzed by adding a gain-phase margin tester into open loop system. Computer sim-ulations show the efficiency of this approach.
Keywords Neural network Describing functions Gain-phase margin Vehicle
1 Introduction
The traditional method of analyzing the amplitude and frequency of a limit cycle is to linearize the nonlinear elements according to the describing function method
[1–9]. Recently, this method has been extended to analyze the stability of fuzzy [10,11] and neural systems [12,13]. Uncertain parameters in a linear control system can be robustly analyzed by the parameter plane method or the parameter space method [14–17]. A designer must care-fully consider the range of safe operation of a system since varying parameters and phase lag always impact practical control systems. Gain margin (GM) and phase margin (PM) are two important specifications in the analysis and design of practical control systems. Methods of analyzing the gain-phase margin of linear control systems [18–20] and nonlinear systems [21–24] with adjustable parameters have been developed.
This work describes a systematic strategy for analyzing the limit cycles of a neural vehicle control system [9] with adjustable parameters. The vehicle model is a single track model from steering angle to yaw rate. A simple method is then presented to evaluate the gain-phase margins for limit cycle prediction after a gain-phase margin tester is added to the forward open loop of a linearized vehicle control system. After doing this, the relationship between the amplitude of limit cycles and stability margins can be easily figured out.
2 Basic approach
In this section, some useful frequency domain approaches including parameter plane, gain-phase margin tester and describing function of neurocontroller are considered for limit cycle prediction of vehicle control systems.
2.1 Parameter plane method
A general linearized system shown in Fig. 1with multiple nonlinear elements is considered, where Gðs; N1R; N1I;
J.-W. Perng (&)
Department of Mechanical and Electro-Mechanical Engineering, National Sun Yat-sen University, 70, Lienhai Road,
Kaohsiung 80424, Taiwan, ROC e-mail: [email protected] L.-S. Ma
Department of Electronic Engineering, Chienkuo Technology University, No. 1, Chieh Shou N. Rd, Changhua City, Taiwan, ROC L.-S. Ma B.-F. Wu
Institute of Electrical Control Engineering,
National Chiao Tung University, No. 1001, Ta Hsueh Road, Hsinchu 300, Taiwan, ROC
. . .; NmR; NmIÞ is the open loop transfer function. N1R; . . .; NmR and N1I; . . .; NmI are real parts and imaginary parts of the describing function (Ni) of n1; n2; . . .; nm,
respectively, which can be expressed as the following equation [2,3]
NiðA; xÞ ¼ NiRðA; xÞ þ jNiIðA; xÞ; i¼ 1; . . .; m; ð1Þ where A and x are the amplitude and frequency of sinu-soidal input to one of the nonlinearities.
The characteristic equation of this equivalent linear system can be expressed as
1þ KejhGðs; N1R; N 1I; . . .; NmR; . . .; NmIÞ ¼ 1 þ KejhNðs; N1R; N1I; . . .; NmR; . . .; NmIÞ Dðs; N1R; N1I; . . .; NmR; . . .; NmIÞ¼ 0; ð2Þ which is equivalent to fðsÞ D Dðs; N1R; N1I; . . .; NmR; . . .; NmIÞ þ KejhNðs; N1R; N1I; . . .; NmR; . . .; NmIÞ ¼ 0 ð3Þ Let s = jx, one has
fðjxÞ ¼ f ða; b; c; . . .; K; h; jxÞ ¼ 0; ð4Þ where a; b; c; . . . are variables which consist of the items (NiR; NiI) of describing functions and/or adjustable parameters of the linear portion of the system. Notice that the designer can define these variables arbitrarily in order to analyze the effect of system parameters. When only two parameters a and b are chosen to concern, (4) is arranged as the following equation
fðjxÞ ¼ f ða; b; c; . . .; K; h; jxÞ ¼ X a þ Y b þ Z ¼ 0 ð5Þ where X, Y and Z are functions of c; . . .; K; h and jx. Let Eq. (5) be partitioned into two stability equations with real part (fR) and imaginary part (fI) and written in the following
[14,15]
fRða; b; c; . . .; K; h; xÞ ¼ X1 a þ Y1 b þ Z1¼ 0 ð6Þ and
fIða; b; c; . . .; K; h; xÞ ¼ X2 a þ Y2 b þ Z2¼ 0 ð7Þ where X1, Y1, Z1and X2, Y2, Z2are real and imaginary parts
of X, Y and Z. Therefore, a and b are solved from linear functions of Eqs. (6) and (7), one has
a¼Y1 Z2 Y2 Z1
D ð8Þ
and
b¼Z1 X2 Z2 X1
D ; ð9Þ
where D¼ X1 Y2 X2 Y1: Note that if Eqs. (6) and (7) are not linear, but independent with a and b, they can be solved theoretically.
2.2 Gain-phase margin analysis
Let h¼ 0; Eq. (4) is rearranged as follows.
fðjxÞ ¼ f ða; b; c; . . .; K; jxÞ ¼ E K þ F ¼ 0: ð10Þ Partitioning Eq. (10) into real and imaginary parts yields fRða; b; c; . . .; K; xÞ ¼ E1 K þ F1¼ 0; ð11Þ and
fIða; b; c; . . .; K; xÞ ¼ E2 K þ F2¼ 0; ð12Þ where E1, E2, F1and F2are functions of a; b; c; . . . and x.
Thus, K can be determined directly from Eqs. (11) and (12), which yield K¼F1 E1 D K 0 ð13Þ and, K¼F2 E2 D K 00: ð14Þ If K0¼ K00¼ Kifor A = A
i, the values of Aiand Kirelated
to xi can be found by varying A from 0 to ?. For many
values of x, a set (GM) of desired values of A and K can be obtained. Alternatively, let K = 0 dB; Eq. (4) is rearranged as follows.
fðjxÞ¼f ða;b;c;...;h;jxÞ¼U coshþV sinhþW ¼0: ð15Þ Also partitioning Eq. (15) into real and imaginary parts yields
fRða; b; c; . . .; h; xÞ ¼ U1 cos h þ V1 sin h þ W1¼ 0 ð16Þ and
fIða; b; c; . . .; h; xÞ ¼ U2 cos h þ V2 sin h þ W2¼ 0; ð17Þ where U1, V1, W1, U2, V2and W2are functions of a; b; c; . . .;
and x. Hence, h can be determined directly from Eqs. (16) and (17), which yield
h¼ cos1 V1 W2 V2 W1 U1 V2 U2 V1 Dh0 ð18Þ and h¼ sin1 U1 W2 U2 W1 U1 V2 U2 V1 Dh00: ð19Þ θ j Ke− ) , ,..., , , ( ) , ,..., , , ( ) , ,..., , , ( 1 1 1 1 1 1 mI mR I R mI mR I R mI mR I R N N N N s D N N N N s N N N N N s G = + -) (s C ) (s R
Fig. 1 Block diagram of a general linearized nonlinear control system
If h0¼ h00¼ hi for A = Ai, Ai and hi related to xi can be
found by varying A from 0 to ?. For many values of x, a set (PM) of desired values for A and h can be obtained. 2.3 Describing function of a neurocontroller
The static neural network (SNN) shown in Fig.2 can be used as a controller (neurocontroller). The network structure is 1-m-1 and does not have bias weights [12,
13]. The parameters gk and hk are the neural network
weights and m is the number of hidden neurons. Based on the stability analysis in [12,13], the describing function of neurocontroller with sigmoid function tanh may be rep-resented as N1ðAÞ ¼X m k¼1 gk hk 1g 2 k A2 6 ð20Þ
where A is the amplitude of limit cycle.
3 Vehicle dynamic systems
In the section, the classical linearized single track vehicle model is given first [9]. The vehicle with yaw rate feedback is now considered for design. The transfer function from the input of front deflection angle (df) to the output of yaw
rate (r) can be obtained as follows.
Figure3shows the single track vehicle model, and the related symbols are listed in Table1. The equations of motion are mvð _b þ rÞ mlflr_r ¼ Ff þ Fr Fflf Frlr ð21Þ
The tire force can be expressed as
FfðafÞ ¼ lcf 0af; FrðarÞ ¼ lcr0ar ð22Þ with the tire cornering stiffnesses cf0, cr0, the road adhesion
factor l and the tire side slip angles af ¼ df b þlf vr ;ar¼ b lr vr ð23Þ The state equation of vehicle dynamics with b and r can be represented as _b _r ¼ lðcf 0þcr0Þ mv 1 þ lðcr0lrcf 0lfÞ mv2 lðcr0lrcf 0lfÞ mlflr lðcf 0l2fþcr0l2rÞ mlflrv 2 4 3 5 b r þ lcf 0 mv lcf 0 mlr df ð24Þ Hence, the transfer function from dfto r is
The numerical data in this paper are listed in Table2. According to the earlier mentioned analysis of a single track vehicle model, the transfer function from the input of front deflection angle dfto the output of yaw rate r can be
obtained as k g hk 1 = k m k= e u
Fig. 2 Static neural network (SNN)
Gr=d f ¼ cf 0mlflv2sþ cf 0cr0ll2v lflrm2v2s2þ lðcr0lrþ cf 0lfÞmlvs þ cf 0cr0l2l2þ ðcr0lr cf 0lfÞmlv2 ð25Þ r F Ff r CG v f a f δ r l lf β
Fig. 3 Single track vehicle model
Table 1 Vehicle system quantities
Ff, Fr Lateral wheel force at front and rear wheel
r Yaw rate
b Side slip angle at center of gravity (CG)
v Velocity
af Lateral acceleration
lf, lr Distance from front and rear axis to CG
l = lf? lr Wheelbase
df Front wheel steering angle
The operating range Q of the uncertain parameters l and v is depicted in Fig.4. In addition, the steering actuator is modeled as GAðsÞ ¼ x 2 n s2þpffiffiffi2xnsþ x2 n ð27Þ
where xn¼ 4p. The open loop transfer function is defined as
Goðs; l; vÞ ¼ Gr=d
fðs; l; vÞ GAðsÞ ð28Þ
4 Simulation results
Consider the static neural control system shown in Fig.5
and Eq. (28), the open loop transfer function can be obtained as Gðs; kp; kd; ku;l; vÞ ¼ kpþ s kd ffiffiffi 2 p ku s GOðs; l; vÞ: ð29Þ Combining with a gain-phase margin tester, Kejh and a static neural controller, N1, the closed loop transfer
function is
KejhN1Gðs; kp; kd; ku;l; vÞ 1þ KejhN
1Gðs; kp; kd; ku;l; vÞ
¼ 0 ð30Þ
Assume the input signal of N1 is xðtÞ ¼ A sin xt; the
describing function of static neural controller can be expressed as Eq. (20). After some manipulations, the characteristic equation of Eq. (30) is
fðs; kp; kd; ku;l; vÞ ¼ X a þ Y b þ Z ¼ 0 ð31Þ where a = kp and b = kd are two adjustable parameters
and X¼ KejhN 1kuð2:1818 1010lv2sþ 2:2345 1012l2vÞ Y ¼ KejhN1kusð2:1818 1010 lv2sþ 2:2345 1012 l2vÞ
Table 2 Vehicle system
parameters Cf0 50,000 N/rad cr0 100,000 N/rad m 1,830 kg lf 1.51 m lr 1.32 m 0 10 20 30 40 50 60 70 80 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 v (m/sec) mu Q
Fig. 4 Operating range
Gr=d fðs; l; vÞ ¼ ð1:382 108lv2sþ 1:415 1010l2vÞ 6:675 106v2s2þ 1:08 109lvsþ ð1:034 107lv2þ 4 1010l2Þ ð26Þ Vehicle Dynamics R r r e + -u s k 2 1 p k d k dt d + + Normalization Neurocontroller x ( , , ) o G sμv c δ
Fig. 5 The block diagram of a neural vehicle control system
Z¼1:414sðs2þ 17:7688sþ 157:9137Þð6:675 106v2s2 þ 1:0746 109lvsþ 4:0045 1010l2þ 1:034 108lv2Þ Substituting s = jx into Eq. (31) enables a and b to be determined from Eqs. (6)–(9) by varying x from 0 to ?. Then, the stability boundary (K = 0 dB, h¼ 0) can be plotted in the kp versus kd plane. Choosing ku= 0.2,
l = 1, v = 70 (the highest velocity and road adhesion factor) and the weights gkand hkare assumed as follows:
g1¼ g2¼ g3¼ 5; h1¼ h2¼ h3¼ 1:
Then, the describing function N1of neural controller can be
obtained by using Eq. (20). Figure6 shows some limit cycle loci. In order to test the accuracy of Fig.6, two points Q1(0.3, 0.15) (asymptotically stable region: A = 0) and
Q2(1, 0.52) (limit cycle region: A = 0.3) are selected. The
input signal x(t) shown in Fig.7can be obtained by using the famous tool, MATLAB/Simulink. We can clearly find that the amplitude of x(t) operating at two points Q1and Q2
in Fig.7is matched with the predicted results in Fig.6. On the other hand, if ku is changed from 0.1 to 0.4 and let
A = 0.3, then the limit cycle loci can be plotted in the kp kd ku parameter space. Figure8 shows the results. Four operating points Q3–Q6are illustrated for testing. The
input signals x(t) are obtained in Fig.9, which consist the results in Fig.8.
Due to analyzing the gain-phase margins for limit cycle prediction, the operating point Q1 is chosen. Firstly, let
h¼ 0. Equation (31) can be arranged as
fðsÞ ¼ E K þ F ¼ 0 ð32Þ
where
E¼ N1kuðkpþ kdsÞð2:1818 1010lv2sþ 2:2345 1012l2vÞ
F¼1:414sðs2þ17:7688sþ157:9137Þð6:675106v2s2 þ1:0746109lvsþ4:00451010l2þ1:034108lv2Þ By utilizing Eqs. (11)–(14), a set of GM can be obtained and plotted in Fig.10. If GM = 10 dB is selected, the predicted amplitude of limit cycle is 0.32. In Fig. 11, the input signal x(t) is depicted, and the amplitude of limit cycles conforms to the results in Fig.10. On the other hand, let K = 0dB. Equation (31) can be also arranged as fðsÞ ¼ U cos h þ V sin h þ W ¼ 0; ð33Þ where U¼ N1kuðkpþ kdsÞð2:1818 1010lv2sþ 2:2345 1012l2vÞ V¼ jN1kuðkpþ kdsÞð2:1818 1010 lv2s þ 2:2345 1012l2vÞ 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 kp kd A=0.2 A=0.3 A=0.4 Stability boundary
Limit cycle region
Asymptotically stable region Q1(0.3,0.15)
Q2(1,0.52)
Fig. 6 Limit cycle loci
0 2 4 6 8 10 12 14 16 18 20 -0.1 0 0.1 0.2 0.3 x(t) Q1 0 2 4 6 8 10 12 14 16 18 20 -0.5 0 0.5 1 Time (sec) x(t) Q2 (a) (b)
Fig. 7 Input signals x(t)
-300 -200 -100 0 100 -5 0 5 10 0.1 0.2 0.3 0.4 0.5 kp kd ku A=0.3 Q3(2,1.05,0.1) Q4(1,0.52,0.2) Q5(0.8,0.32,0.3) Q6(0.6,0.24,0.4)
W¼1:414sðs2þ17:7688sþ157:9137Þð6:675106v2s2 þ1:0746109lvsþ4:00451010
l2þ1:034108 lv2Þ By utilizing Eqs. (16)–(19), a set of PM can be obtained and plotted in Fig.12. If PM = 45° is selected, the pre-dicted amplitude of limit cycle is 0.31. In Fig.13, the input signal x(t) is also depicted and the amplitude of limit cycles conforms to the results in Fig.12.
Remark 1 The proposed method here can be easily applied to analyze the phenomena of limit cycles even if the control
system has multiple nonlinearities, like saturation, relay, hysteresis, backlash and different operating points.
5 Conclusions
In this paper, the limit cycle prediction of a neural vehicle control system with adjustable parameters is achieved by utilizing the frequency domain approaches with describing function, parameter plane and gain-phase margin tester.
0 5 10 15 20 -0.5 0 0.5 1 1.5 2 x(t) Q3 0 5 10 15 20 -0.5 0 0.5 1 1.5 2 Q4 0 5 10 15 20 -0.5 0 0.5 1 1.5 2 Time (sec) x(t) Q5 0 5 10 15 20 -0.5 0 0.5 1 1.5 2 Time (sec) Q6
Fig. 9 Input signals x(t)
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0 5 10 15 20 25 A GM (dB) GM=10dB
Fig. 10 Gain margin (GM) and amplitude of limit cycle
0 2 4 6 8 10 12 14 16 18 20 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Time (sec) x(t)
In addition, a simple method is also proposed to point out the gain margin and phase margin when limit cycles can occur in an operating point. A single track vehicle model is then illustrated to study. Finally, computer simulations show that more information about the characteristics of limit cycles could be acquired by this work.
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Fig. 12 Phase margin (PM) and amplitude of limit cycle
0 2 4 6 8 10 12 14 16 18 20 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Time (sec) x(t)
