Abstract. In this paper, we consider the specifications for the automatic steering of a city bus based on the IFAC benchinark problem and we design a robust ...
Proceedings of the 1998 IEEE International Conference on Control Applications Trieste, Italy 1-4 September 1998
WM04
Robust Controller Design for Automatic Car Steering Problem with Nonlinear Parametric Uncertainties
WI. Abrishamchian, NI. R. Modabbernia Department of Electrical Engineering E(.N.Toosi, University of Technology
P.O.Box 16315-1355 Tehran, Iran mmabrishQrose.ipm.ac.ir
with a nonconservative formulation of the uncertainty block. A, is possible. This is essential in achieving the optimal results for the problem under consideration in this paper. Without the approach in this paper, and simply by taking any combination of parametric nonlinear uncertainties and overbounding them with new linear uncertain variables the resulting P - K - A becomes too conservative for designing the desired robust controller. This paper is organized as follows: In next section, we discuss the model of a city bus system (0305) based on the IFAC benchmark problem [l]. In Section 3 , we show how to reconfigure the system into the structure required for p synthesis in a nonconservative manner. In Section 4.we discuss our design of a robust controller based on D - K iteration procedure for complex p-synthesis. In Section 5, we compare our results with the linear controller design of Ackermann e t al. [2] by considering the original nonlinear niodel of the system.
Abstract In this paper, we consider the specifications for the automatic steering of a city bus based on the IFAC benchinark problem and we design a robust controller for proper automatic steering using p-synthesis. The success of our design lies on our nonconservative modeling of the nonlinear uncertain parameters of the system into the required P - K - A configuration. Furthermore, by considering several typical maneuvers of the bus, we make a comparison between our p-synthesis based controller and existing applicable controllers. This comparison demonstrates that our controller performs well for the given maneuvers. The siniulation analysis is based on the original nonlinear model of the syst>em.
1. Introduction Our main task in this paper is to design a cont.roller for proper automatic steering based o n the performance criteria specified in the IF,4c' benchinark example> [ 11. Furtherniore, t.he resulting closed loop system should be robust with respect to nonlinear parametric uncertainties resulting from large variations in velocity and mass of the vehicle and the adhesion between tires and road surface. For automatic car steering, Parameter Space Approach and Nonlinear Sliding Mode Method [a] for designing robust controllers have responded successfully to several typical driving maneuvers. Furthermore, Robust control law is achieved by yaw rate feedback which helps the driver to steer the car in the presence of output noises such as crosswind [ 3 ] . In this paper. we use structured singular value ( p ) based robust controller design to synthesize and analyze the robust performance and robust stability of the automatic steering system. But the fundamental question is "How can we transform the system with nonlinear parametric uncertainties into a p-synthesis and analysis framework (P - K - A configuration) Lvith inininial or no conservatism in the uncertainty A I>lock'?'' For automatic steering problem, we show that, indeed by proper consideration of the nonliiiear uncert,ainties a minimal P - K - A configuration
2 . A Model for Automatic Car Steering The vehicle steering dynamics in a horizontal plane which has been called t h e sangle-track (two wheels) by Riekert and Shunk [2] is shown in Figure 1. This F,
Figure 1: Automatic car steering model figure shows car steering based on the path tracking ( p r r j = &). lateral motions and car deviation from the reference path. The basic assumptions of this model are that the center of gravity (C'G) is on the street level and forces
0-7803-4104-X/98/$10.0001998 IEEE 258
on C G do not cause any motions of rolling or pitching. Under these assumptions, the right and left front and rear wheels can be lumped into one wheel in the cent'er line of the car. We now define the variables in Figure 1 and of interest in describing the state space equat,ions of the system: Sf is steering angle of the front wheel: L s ( L , ) is the distance of CG from the L,. is the based wheel; front(rear) axle; L = L f is the velocity vector at C'G with a magnitude of V - ; 4 is the sideslip angle between the vehicle center line and the velocity vector; F f ( F , ) is the lateral force that is generated by the front (rear) tire; F, is lateral forces disturbance, i.e. wind force; L , is the distance between CG and the point of effect (F,); L s is the distance between CG and location of the sensor which measures the displacement from the guidline; A* is the angle between centerline of the vehicle and t,angent of the guideline; yaw rate) is t8hedeviation of A @ related to time, e.g. T = A$; the cornering stiffness equals to p C f for front axle and pC,. for rear axle, where p is a common road adhesion factor (with p = 1 for dry road and p = 0.5 for wet road). FLUthermore! Cf and C, are the cornering stiffness of front and rear; the mass of the vehicle M is normalized by p , i.e: rl/l = M which is a '' Virtual Mass ".
3. Modeling of t h e Uncertainties of the System Nonconservative modeling of the uncertainties of a system plays an important role in solving the associated robust synthesis problem. For t.he case of under consideration in this paper, by looking on the uncerta.in mass M and velocity V I it is obvious t h a t the uncertain parameters are entering state matrix in a nonlinear way; e.g., see parameter q 2 . The simple way of overbounding these uncertainties with a linear uncertain parameters results in a conservative modeling of the uncertainty block and hence obtaiiiing a suitable controller is out of reach. Furthermore, because the uncertain parameters do not enter the state matrix affinely, therefore, usual formulation of converting the uncerta,iiity block to a structure suitable for applying p synthesis, the so called M - A structure is not applicable. In this part, a step by step method is proposed which allows us to pull the nonlinear uncertain variables out of the state space equations. This is done in a manner to obtain the required M - A structure in a nonconservative way.
+
The moment of inertia ( J ) is normalized as J- =
Step 1.
J
L.
According to Equations (2) and (3), two uncertain variables interfere in automatic bus steering:
The curvature p r e f = 1 of the guideline appears R,, f as a reference input to the system. It is assumed that the vehicle is steered only by the front wheel; i.e., (& = 0). Therefore, the front wheel steering actuator is modeled as a n integrator:
0
0
6f = Uf (1) where Uf is the controller output. Due to the above explanations, the automatic car steering is described bv the fifth order model.
Vehicle speed V . Vehicle mass M and common road adhesion factor p that have the same effect and are presented by virtual mass lk in the model of the system.
M and V are described in terms of uncertain variables
6~ and S ~ where J j b ~5 l 1 and 16~15 1. M =
where (C, i"f)
all=
--
bll =
MTi
(C,.L,
-c
al? = - 1 +
f
L
'
&I
i,
b21 =
"21 =
(CrLr - CfLf)
(3)
n,,
=
MV
a22 =
-
(C, L?
+ C fL:' 7V
'21
=
LVJ J
(5)
y.
J
1
J
nir,(l+bMM)
e,
-
M 112
=
V = G"+SVV = V,(l+SvV) (6) In Equations 5 and 6, MOand are nominal values of virtual mass and speed respectively with Go = 20975 ky and CO = 10.5 M and V are their deviation measures froin their nominal values. Note = and V = Furthermore, According to that VO the Equation 3, we can define the following uncertain functions
-
C fL f
) f
c:
M,+s,nn/r
The systems output are the distance from the guideline (y ) and raw rate (?-). The system parameters are given below based on the model for City-Bus 0 305 [a].
L f = 3.67 In L,, = 6.12 m C f = SY8000 V E [l 201 7 p E [0.511
L , = 1.93 m L , = 0.565 m
5
C, = 470000 n/l E [9950 S6000] k y i f E [9950 320001 k g
(7) (4) f5(6V) =
1 -. V
Step 2. All of the uncertainty functions discussed above are converted to the general M - A structure by apply-
and J = i2M a i d i2 = 10.85 m2. The parameter ,i2 is called Inertial Radius.
259
transient state and 0.02 m for the steady state; the m lateral acceleration is assumed to be maximum 2 S m and for the tip-over the limit is set for 4 T ;the nat-
iiig lower Linear-Fractional-Transformation (LFT). Step 3. After transforming all of the nonlinear uncertainties to their L F T configuration, we can draw a general block diagram of the system with Mk - Ak blocks. Figure 2 shows this general block diagram.
S
ural frequency of the lateral motion must be at most 1.2H z . For applying p-synthesis, we include a fictitious and a weighting function uncertainty block A WP(S) = d s a g [ W p , ( s p W p , ( s ) ] such that connects the error output vector, e t o the disturbance input vector d. Note that 2 = [P, d I T , Uf is the input of the actuator or the controller output, Y = [y rIT and W = [Po e]'. Furthermore, the generalized plant consists of the plant as decribed before and the weighting function matrix W p (s) and it is also based on considering the modified uncertainty block which is described by
A
(8)
= (dtag(GMI3 6 ~ 1 5
where
&ME
c; 6 v E c;
Ap E
c2x2
and C n x n is the set of complex matrices. Automatic bus steering system has three poles on the origin. Therefore, certain assumptions that are needed to solve the standard H m synthesis problem are being violated. A simple bilinear transformation has been proposed by Chiang and Safanov [4] that overcomes this problem. To begin, we first transform the system in s-domain to S-domain by letting s = To keep this transformed system similar to
F. ,+1
the original one, p l should have a sufficiently sinal1 negative value and p 2 should have a sufficiently large negative value. We, now, design a controller by considering all the factors discussed above. In this stage, we design a controller for the transformed system in S domain with the help of H m theory. Then, we transform this controller to s-domain by letting S = *. P2
Figure 2: General block diagram of uncertain system with Mk - Ak blocks
In this section, by choosing the required p l and p 2 and the proper stable weighting functions W p , ( s ) ,W p , (s), we design two controllers for the automatic bus steering.
Step 4. General block diagram in Figure 2 consists of basic inputs Uf, p r e f , Fw and outputs y , r of the original system, Pzk E P,inputs and Pok E Po outputs k = 1 , 2 , . . ,8plus ail error output e. By writing the state space equations of the block diagrams, we can convert the model of the system to the P - K - A structure suitable for p synthesis aiid analysis.
Design of Controller K l ( s ) : First of all, we assume that W p ,( s ) and W p , (s) are unity functions. By choosing p 1 = -0.01 and p 2 = -100, we achieve a fifth order controller in the first D - K iteration. This controller has two inputs y , ~ and one output Ufgiven by K l ( s ) = [ K l I ( s ) K12(s)] where K l l ( s ) is stable and minimum-phase, but K l 2 ( s ) has two zeros in the right half plane. If we concentrate on bode plot of K 1 2 ( s ) , we see that the controller's gain is genrally very low. Therefore, if we disperse the yaw rate r loop, the system should maintain its performance. Simulation results and p value for robust stability without yaw rate loop prove this claim.
4. Design of a Robust Controller Using pSynthesis: The perforniance criteria for the automatic bus steering based on the IFAC benchmark problem (for example, see [2])are described below: The steering angle and its rate are limited to / S f /5 40 dey and /Uf/ 5 23 respectively; the displacement from the guideline is maximum 0.15 m for the
%
260
Design of Controller K ~ ( s ) :
Maneuver 3: In this maneuver, bus has the speed of 20 and the virtual mass of 32000 k g and is in transient state from manual steering to the automatic one. The bus is assumed to drive in a distance of g = 0.15 m parallel to the guideline when the control action is needed. Figure 5 shows that f c z ( s ) results in an unstable system for this maneuver, but K ~ ( s and ) f c l ( s ) both show good deviation, but their steering angle rate U f is more than the desired limit. Lateral acceleration is admissible.
Based on our observation in the previous part, we choose the the weighting functions 0.002s 1 w p ,( s ) = w p ,( s ) = 20s 100 which create sniall weights on the output E with respect to the input disturbance. Furthermore, we let p 1 = -0.2 a,nd p 2 = -100 to shift the poles on the origin. We now come to a controller in the second step of t,he D - IC iteration. According to our conclusion in previous part, we again ornit the yaw rate T feedback and we consider the controller as a SISO block. With these assuniptioiis, we obtain a fifth order S E 0 controller. K ~ ( S ) .
+
+
Maneuver
y.
5. Siinulations
Iii this section, for simulation purpose, we use the original nonlinear model of t,he automat,ic car deering. Four different maneuvers for automatic bus steering are considered. In these cases, the motion specifications considering the yaw rate, the lateral acceleration and the deviation from designated path for three different controllers are considered and compared. These cont,rollers are the second controller which is desigiied based 011 p synthesis, na,mely K ~ ( s ) and the controllers f c l ( s ) and .fc.(s) which were designed hy Ackerinann e t al. [a] based on parameter space approach. In all t,he simulations, we specify K % ( s )with solid lines, f,l(.s) with dotted and f c 2 ( s ) with clashed lines. 0
4:
In this maneuver, wind attacks the bus from the lateral direction. It is assumed that the wind velocity is based on Vw(t) = 20(l _e ot.) According to Figure 6, the deviat,ion from guideline for K 2 ( s ) is less than the fcl(s) but more than fc.(s). The Controller .fc2(s) performs better than the other controllers in this maneuver. But as it was discussed before it does not demonstrate admissible performance for maneuvers 1 and 3.
6. Conclusion
In this paper, we performed a lower L F T on the automatic steering system of a city bus 0 305 model with nonlinea,r uncertainties due to variations of mass and velocity parameters of the system. This resulted in a P - K - A configuration suitable for applying p-synthesis to the automatic steering problem. Although the system uncertainties enter into state space equation in a nonlinear form, but we were able to pull them out into the uncertainty block A in a nonconservat,ive inanner. The simulation analysis based on the original nonlinear model of the system demonstrates t,hat our controller performs well for the given maneuvers.
Mrineuuer 1: In this maneuver, it is assurned that the bus enters a circular arc with a radius of 400 m . and its virtual inass The bus speed is 20 is 32000 k g . Figure 3 shows t,liat the deviat,ioii from guideline for K ~ ( sis) less than controller f c : l ( s ) and more than f C 2 ( s ) . Furthermore, we see that the steering angle rate U f for f c 2 ( s ) is more than 23 9 ( 0 . 4 *). The lateral acceleration is desirable for three controllers. Therefore, I T z ( s ) provides good perforinances and satisfies all the limitations as described in Section 4.
References [l] J. Acl
