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Dynamics and Control of Maglev Vehicles with Parameter Uncertainties
Copyright © I FAC Control Science and Technology (8th Triennial World Congress) Kyoto , Japan , 1981
DYNAMICS AND CONTROL OF MAGLEV VEHICLES WITH PARAMETER UNCERTAINTIES W. Breinl and K. Popp Institut e of M echanics, T echnical University Munich, Munich, Federal R epublic of Germany
Abstract. The overall dynamics of Maglev vehicles is investigated , regarding vehicle , suspension control and guideway. The mathematical description results in a very complicated high order model containing uncertain parameters for the suspension magnets. However , since decentralized control is applied , a simple third order model can be used for the control system design . The controller and observer design with respect to low sensitivity against parameter variations is shown in detail . The corresponding closed- loop system is analyzed by numerical simulations as well as by laboratory experiments and shows very satisfactory results. Keywords . Control system syn t hesis ; robust control ; observers ; system analysis; Maglev vehicles .
I NTRODUCTION Magnetical l y levitated vehicles are under de velopment for rapid transit in high l y populated areas as well as for high speed transportation over large distances . The feas i bi l ity of elec t romagnetic guidance and cont ro l, particularly for high speed operations in connect i on with the use of linear induction mo tors for propulsion , has been shown by vari ous test-vehicle runs (Glatzel and others , 1980) . Speeds of more than 400 km/h already have been reached. Figure 1 shows the TRANSRAPID 05 , the first public Maglev vehicle which was presented at the International Transportation Exhibition in Hamburg , 1979. Due to the properties of electromagnets , the guidance and support magnets must be controlled. Since some of the magnet parameters can be measured only with large uncert ainties (e.g . inductivities) , a controller with low sensitivity against parameter variations is required . Furthermore , there is a strong dynamic coupling between vehicle motion and guideway vibration and also an interaction with the suspension control system . A large number of publications is devoted to the dynamics and control of Maglev vehicles . Short literature reviews may be found in Popp (1980a , 1980b), and an extensive review is given in Popp (1978).
CAR800Y SECONDARY SUSPENSION
D
51<10 GUIDANCE
MAGNET
Fig . 1. Front view at TRANSRAPID 05 and electromagnetic levitation principle . sensitivity study is carried out. In this paper, we confine ourselves and neglect stochastic disturbances , e . g . guideway irregularities and measurement noise . Thus , the control problem is deterministic with uncertainties in some of the parameters involved .
In the present paper the modeling of the openloop system is described at first. Then , the design philosophy is explained and the design of a control system with low sensitivity against parameter uncertainties is shown in de tail. At the end of the theoretical part, methods for the analysis of the closed- loop system are mentioned . The derived results are applied to a single magnet control and a
2431
MODELING OF THE OPEN-LOOP SYSTEM The vehicle - guideway system under consideration is a very complex system with many degrees of freedom. However, the motions in a vertical plane , i . e . heave, pitch and vertical bending are decoupled from the other
W. Breinl and K. Popp
2432
motions. Here, o nly these dominant motions are considered. Furthermore, we assume small displacements except the forward motion of the vehicle which may take place on a straight trac k with constant speed v. The entire openloo p system contains: • vehicle mo del, magnetic suspension model, and guideway model together with an estimation of the parameter uncertainties. The vehic les are usually modeled as mass point systems or multibody systems. The mathematical description is given by a system of o rdinary differential equations of second order (Popp , 1978, 1980b). The action of the suspension magnets is replaced by single forces. Although electromagnets are highly nonlinear elements, a linearized model described by an ordinary first order differ ential equation has proven itself well as shown by Gottzein, Lange (1975). Unfortunately, some of the magnet parameters (e.g. inductivities) are not known exactly. The guideway model consists of an infinite sequence of identical and uncoupled Bernoulli-Euler beamstructure elements of length L. The deflection of a guideway element is governed by the wellknown fourth order partial differential equation, which can be replaced by an infinite set of ordinary differential equations using modal expansion. For our purposes an approximate solution comprising a finite number of eigenmo des is sufficient . In order to keep the mathematical description of the entire guideway as simple as possible, on ly the minimum number of elements is regarded, which results in a periodic shifting of elements under the moving vehicle. Although the details of the mathematical des cription of the entire open-loop system depend on the chosen subsystem models, the resulting state equation E always has the same structure, (1)
E ~(t= V T+O)
= Ux(t= vT-O), v = 1,2 .... , x(+O) = ~o ,
(2)
where x(t) is the nx1-state vector and u(t) is the mxl-control vector; A( a , t) denotes the nx1 - system matrix , B( a ) th; nxm-control input matrix, i(t) the nXl-disturbance vector (e.g . moving static loads) and U the nxn-state transitio n matrix. The matric;s ~ und ~ depend on uncertain parameters a j' a j = a jo + ~aj' j = 1, ... , r , where a jo are the nominal values and 6a j are the unc ertainties . The uncertain parameters are combined to the rx1-parameter vec tor a . The system E is periodic time-variant, i.e.-~(t+T)=~(t) ,[~(t+T)=i(tU, T = L/v, because a periodically constructed guideway (element length L) and a constant speed v have been assumed Furthermore, it shows periodically jumping states, eq. (2), due to the fact that the vehicle front reaches a resting element at the end of each period. Thus , some guideway states become suddenly zero . Equation (2) describes also the shifting of the guideway elements. The dimensio n n of the state vector which corresponds t o the order of the system matrix is given by n = 2f + + 2nf, where f
m
and f represent the degrees of freedom of the vehicle and of one guideway element, respectively, n is the minimum number o f guideway elements within the system bounds and is the number of magnet forces . In a realistic system description the total system o rder n is quite large. Thus,the open-loop system can be mathematically described by a • linear, periodic high order state equation with periodically jumping states and parameter uncertainties . The state equation (1) has to be completed by the measurement equation
m
2(t) =
£ ~(t)
(3)
where the measurement vector 2(t) represents the real output of the system and C is the measurement matrix. In technical applications one usually measures the vertical accelerations and the airgaps of the suspension magnets.
CONTROL SYSTEM DESIGN Design Philosophy The most important technical design criteria for the control system are • asymptotic stabi lity of all motions, best possible safety and reliability, sufficient ride comfort, l ow power consumption . Since Maglev vehicles are unstable without active control of the primary suspension, the first criterion is essential to operate the system. There are different feasible control concepts . Most promising is a decentralized control regime where primarily the air gap between ve hicle and guideway is controlled, resulting in best safety conditions (Gottzein and others , 1977). The suspension magnets are decoupled from each other and flexibly mounted on the car body . Each magnet has an individual power supply, s e nsors and controller. These components result in a single unit which is sometimes called a magnetic wheel. Here, redundancy is realized by configuration. The control system design must start with lower order models. Thus, the sophisticated higher order model I , cL (1), (2), has to be simplified to a lower order model L • In the present case, this c an be done by assuming a rigid and straight guideway, which results immediately in a constant system matrix of reduced order, A = const, and vanishing disturbances, ~(t) ~ Q,-as-well as vanishing state jumps, ~ =~. Since a decentralized control concept has been chosen, it turns out that a vehicle model with only one single magnet is sufficient for the contro ller design. The simplified low order model can be stated as
X(t)
~(£)~(t)
(4)
+ b(a)u(t)
L
Y. (t)
£
~(t)
,
where x(t) is the nx1-state v ecto r, u(t) is the scalar control, ~ (~) is the nx1 - c o ntrol
(5)
Dynamics and Control of Maglev Vehicles input v ecto r and Z(t) is the mxl-measurement vector; ~~) denotes the nxn-system matrix, £ the mxn-measurement matrix and a characterizes the rxl-parameter vector with uncertainties. Thus, the system E can be described as a • linear, constant, reduced order state equation with single input and parameter uncertainties. Since the parameters of the suspension magnets are uncertain, in E and I the same vector ~ occurs. For a = a the controller and observer design is st~ightforward. However, here a control system design for a = a +6a is given which shows low sensitivitYOagainst parameter variations 6a.
2433
Starting with (9), where the nominal values A and b are used, a sensitivity analysis for tRe clo~d-loop system can be performed resulting in a low sensitive gain vector k, which is shown later. Extended sensitivity model including the observer. For the realization of the control law ~he state vector x(t) has to be replaced by the state estimation R(t), which can be found utilizing the wellknown Luenberger-observer ~ l(t)
+:!:. ~u(t) +
!:
Z(t) (l0)
x(t)
~l s.(t)
+ ~2 Z(t)
Low-Sensitivity-Control Controller design. We start with system E and assume that the pair (A ,b ) is completely controllable and (A ,C~iSOcompletely observable, where A =A(a~,-b =b(a ). To stabilize the system, aolin~r t~e-i~ariant control law is assumed,
Here, s.(t) denotes the qxl-observer vector and the various matrices f o llow from (lla) (llb) ~
DT
- LC, Re Ai
(~)
< 0, i=l, .. ,q, (llc)
(6)
where k is the constant gain vector. As wellknown, ~ can be calculated so that the corresponding closed-loop system is asymptotically stable, using optimization with respect to a quadratic performance index or pole assignment. However, to analyze the system E with respect t o sensitivity against uncertain parameters OC., j = l, ... ,r, a sensitivity model has to bJ derived. The sensitivity vector o . with respect to the j-th parameter is given)by ax(t,a)
o . (t,a )
-)
aa
=
-0
I ~
j
(7)
a
-0
The elements of o. are called sensitivity functions. The serlsitivity equation corresponding to (4) reads (Frank, 1978),
where I is the nxn-identity matrix and A. (D) is the~-th eigenvalue of ~ . The observe? (10) estimates asymptotically a subspace of the state space, ;(t)~ x(t), and ~(t) is an asymptotic estimation of the state,-x (t)+x (t) . Introducing the error ~(t+A= s.(t) - TX(t) and the control law u(t)= -k x(t) the closed-loop system corresponding to-(4) reads
In order to analyze the closed-loop system/including the observe~with respect to the sensitivity, in (8) the state estimation R(t) has to be regarded
a.
-)
aA(a)1 A(a ) o. (t,a )+-a-x(t,a)+ - -0 -) -0 a .!!o-0
a. (t, -0 a ) -)
j
a~(~)'1
+-aa . )
au(t,a)l u(t,a )+b(a ) - a -,
a
-0
--0
a.
o.
-
-) 0 -
--=:Q..
aa .
I
a
-0
)
=:O
-
A .
~)
=
(l4 )
'
(9)
A-b kTJ
~
I
-
a
J
~;
is used denoting
-0
the sensitivity error, which is a measure for the detuning between plant and observer in presence of parameter variations in the plant. The sensitivity-error follows from (10),
-O~
~)
I
a~(~) . _b aa . a ' ~j )
. (t) .
a)
.
a; (a)
Considering only the j-th sensitivity vector, eq. (8 ) combined with (4) and (6) yields
-)
u
a )
For shortness
[!.J [~ .~
-0
In general, the observer vector l(t) is a function of the system parameters. Thus, the derivative of the input becomes
(8)
•
o
. u (t) +b
-et)
(13)
u
ax
-et) -
~)
a
)
-0
x(t) +b
(t) =A o. (t) +A .
-0
a~(~) aa . )
I a'
-0
au(t'~)1 da . )
Thus,
(l3) becomes
a
-0
T
T
(A -b k S2C)O. + (A . -b .k Sl); + -0
~
-
-
-)
T
~)
-et)- -
T
+ (A .-b .k )S2Cx-b k Sl O ~ ~)
~J-
--~--.,
-
( 16)
w.
24 3 4
Breinl and K. Popp
For the entire closed-loop system consisting of the plant (4) (with nominal values A and b ), the observer (10) and the sensiti~ty iii8del (15), (1 6) , a sensitivity analysis can be carried o ut. Ho wev er, the system order is 2(n+q) . In the n ext step, a new design f o r an o bse r v e r with l o w sensitivity is shown. Obser v er with l o w sensitivity. The combinatio n o f system ( 12 ) (now, using the actual values ~,£) with the sensitivity model (15), (1 6) and the erro r ~ ~~ - ~ ~j yields
of the c o ntrol system. The remaining step is to determine the matrices D and L for an observer of minimal order q;n-m. For this, we use a coordinate transformation ~, so that ~ becomes £;~ ~[Q .!m]. NOW, if the matrix T has the special structure ~ ; [I -T~ Q], the-rank conditions (lla) and (19) ~ be satisfied. Thus, from condition (lib) one obtains
(21 )
To satisfy condition (11c), consider the transformed matrices A and b to be 9iven in submatrices corresponding to T
A
[~11
~12
~21
~22
~13] ~23 ,~;
~31
~32
~33
( 17 ) Fro m (1 7 ) it can be seen, that the algebraic decoupling of the controller and the observer (separation principle) is not given. Thus, separation requires
~ [~
- ~J ; 0
~
,
[£ -
~J
;Q
( 18)
-
CA-
;
0
-
. . s rank [ ~,~ T TJT ; n, < No w fro m condLtLon rank [A . b .] ; 1, it follows rank T - n - 1. Theref~J ~Jobserver with low sensitivity can only be of reduced order. Howe v er, if the system is in c omp anio n form, whi c h is the case f o r the considered Maglev v e hicle model, (19) is satisfied by TA . ; O. This results also in ~ £ ; Q. Thus~~e observer becomes even robust for disturbances of the input signal u(t). If conditio n (19) h o lds, from (17) it f o ll o ws, that the separatio n principle is valid also f o r p arameter variatio ns. Even more, the o rde r o f ( 17) can be reduced: The e equ a ti o n in ( 17) becomes ~ ; ~ e ; utilizing the initial c o nditio n e (0) =0 it~ollows -{) ~(t) = Q. Thus, th e ~-equati o n can be canceled and th e sensitivity model is given only by the G. -eq uatio n. This give s a wel come reduction of t~e system o rde r t o 2n+q and the entire system ( 17) reads
! :(
T* ~ ; ~11 - -2
D TIf - -2
11
~1 ,
(23b)
; L -2
(23c)
This system of matrix equations can be solved if and only if the pair of matrices (~11'~21) is observable. Or in other words, there must be measurements, so that the observer subspace corresponding to T is observable. The matrix D (23a) must be stable, cf. (11c). The stabilization is the same problem as in the controller design, thus, the same pro cedure s can be used. For an optimal design with respect to a quadratic perfo.rmance index, the covariance matrix of the initial state vector can be applied (Muller 1977, Breinl 1980). Now, the optimal observer with low sensitivity is ready for application. Closed-Loop System Description In the final step the control system obtained for the reduced order system E is introduced in the original high order system ~ , re sultLng Ln the closed-loop syst~ If the ideal control law (6), u(t); -~ ~(t) with a low-sensitive gain vector k is introduced in (1), (2), one obtains
r.
"3< (t);~ (t)
. . 1'_ -~~l
~23
(23a)
~21
+ ~12 - ~2 ~22
~13 - -2
( 19)
.
(11c) reads
TIf
In the sense o f a first orde r sensitivity the ory, (18) is equal to T
Thus,
[::]
(22)
~(t) +i (t),
! (t+T);! (t) , ~(t+T);i(t),
(24)
~(t; vT+O); ~~(t; vT-O), v;l, 2, .. ,~(+O) ;~, I'"
(25) (2 0) I n (2 0), the G. -equatio n is o nl y used to analyze the systeJ with respect t o sensitivity, but it is no t n e cessary for the realization
where the system order n remains unchanged. On the othef hand, if the realistic control law u(t); -k x(t) , is introduced in the same way, then the Zbserver has to be regarded and the system order is increased by the number of
2435
Dynamics and Control of Maglev Vehicles observer states mq I q=n-m . In either case, if a state feedback with respect to L is computed and implemented in the original higher order system E , then this results generally in an incomplete state feedback with respect to E and Qothing can be said about the stability of E or other properties. Thus, a pe~formance analysis of the closed-loop system L is required.
POI~T
MSS MODEL [1
/ /
/
/
SYSTEM ANALYSIS The dynamic analysis of the closed-loop system ~ is carried out starting with a stability in-
vestigation and followed by the steady-state response calculations. For the stability investigation the wellknown analytical methods for periodic systems based on Floquet theory have to be applied and extended to jumping states. Here only the main results are shown. All derivations and proofs can be found in Popp~(1978). Since asymptotic stability of system E is essential,the stability analysis can be carried out applying the Stability Theorem: The closed-loop system E is asymptotically stable if and only if all eigenvalues ~. of the growth matrix U~(T) have absoluteLvalues less than one , ~. < 1, i = 1, ... ,n , where ~ is the jump matrIx and the transition matrix ~ (T) follows from
I
.
~(T)
-
~
= A(T)
~(T)
--
~
(0)
.!n,
(26)
by numerical integration over one s~ngle period, 0 S T ~ T. In case of a system L without jumping states, i.e. ~ E this result is wellknown. For the steady-state-response calculations an ~proximatesolution shall be mentioned first. To check the vehicle behavior together with the suspension dynamics, a rigid guideway of sine-wave shape is often used. Then, the modified closed-loop system posesses a constant system matrix, a harmonic external excitation, and no jumping states. Thus, frequency-response methods can be applied. In case of an elastic guideway, however, either numerical integration of (24), (25) has to be carried out until steady state is reached or the analytical solution for steady state has to be used. The latter solution reads
.!n '
v->=
Fig. 2. System models rigid straight-line guideway and serves as a model for the control system design. The model L2 (n =7) contains a rigid sine-wave gui2 deway and can be considered as worst case model with respect to ride comfort , while model L3 (n =11) is traveling on an elastic guideway 3 and represents a worst case model with respect to guideway loading. In the following the controller and observer design for model L1 is shown. Then the resulting controller is implemented in the models L , L3 and a system ana2 lysis is carried out . Control-System Design for the Single-Mass Model Controller design. The vertical motion of the single-mass model L1 is described by the linear time invariant third-order system (4), (5), where (28)
l~
x= [:], A = 0 Z -0 KsR (Ks_KiKS) mL m mL o 0
APPLICATIONS Models Figure 2 shows two different types of Maglev vehicle models. The point mass model L1 of order n =3 characterizes a single magnet on a 1
b = -0
L 0
The nx1-state vector x comprises the gap z as well as and The inatrix tains the system parameters given in The control input vector contains of the system parameters .
z
z.
Eo
~
r1_ Ki ]. ~
mL
0
magnet A con~ble 1. also some
Table 1: System parameters of the magnet model Mass of the magnet Ohmic resistance Coefficient of gap z
The solution (27) requires only integrations over one single period.
~l'
_~.
m
R
16
kg
8 4 Sl
K =5.7·10 N/m s Coefficient 0z~ current i K.=K.= 114 N/A and velocity L s 0.5 VS/A Inductivity Lo
The system is completely controllable and, assuming the measurement of gap z and acceleration also completely observable. Here, (28) is already given in companion form. The gain . vector ~ has been optimized with respect to a quadratic performance index, where the weighting factor corresponding to z was varied and the results were in each step analyzed using (9) . The overall results are listed in Table 2.
z,
W. Breinl and K. Popp
2436
This cont~oller design shows low sensitivity against parameter variations as well as a reasonable transient behavio r of the closed-loop system. Table 2:
~ontrol
_ _ nomIna l
\
a)
o.s
--- .20-'. Abw •••• · 20·'.Abw
\
gains
Con trol gains k1 = - 9123 . 5; k2 = -470.2; k3 = -7.1 ~o.
Design of an observer with low sensitivity. The obse rver design shown earlier yields (dimension q = 1 )
-t~ ,
b)
O. ZS
.§.1
-0 . 50
·0.7S
1t 2 L = [ -t 2
l' /'
s
I
.
I
o.
Tj "I
d
1/ - I~
l
,'t
t.J
" "
"
o-:r -
-
I
I
,t
-0 . 2S
d =
__
----
Ei genva lue sof the closed-loop system A1 = - 53 . 07; A ,3 = - 31.89 ± i·18.92. 2
D -
o~----
I
I
o. )
a.
'"
i
I I
I
I\J
i
4
(29) l.S
where t~ = P2/ P 1 is identical with the obse rver e igenvalue A . Here, P , P are the esti2 1 mated maximal in~tial values, P ~ = 1 x. 12 . 1. 1.0 max To demonstrate the theorY,a parameter variation ~L = ±20% of the inductivity L is investigated. Some numerical simulationoresults are shown in Fig. 3, where the actual trajectories are plotted as well as the corresponding sensitivity functions. This results are in very good agreement with experimental results obtained by means of a laboratory model of a single suspension magnet. System Analysis for the Worst Case Models The linear controller (6) with the control gains shown in Table 2 is implemented in the worst case models L2 and L , cf. Fig. 2. Howev3 er, instead ofTstate vector (28) the vector x = Cs , S , 2 1] is fed back, where s denotes the air gap between vehicle and guideway and 21 is the acceleration of mass m . Since the en t1.re 1 ve hi cle mass m , m = m + m + m3 ' exceeds 1 2 the mass m of the point mass model, the number m/m of magnets are assumed to act in parallel in order to heave the vehic le. The air gap s depends on the guideway deflection, which acts as vehicle excitation. The excitation frequency f reads f = v/L ~ 4.4 Hz, where v=400 km/h and L = 25m have been assumed. Depending on the suspensio n design, characterized by f2 = = ~ /2n , the excitation frequency f is larger or smaller f2 ' i.e. above or below resonance of the secondary suspension. Previous investigations have shown (Engl, 1980) that the tuning f > f2 leads to better results than f < f . The aim of the analysis of systems L2 2 and L1 is to show the steady-state behavior of the iaeally controlled worst case vehicleguideway models with respect to deterministic disturbances. Here, the observer dynamics is neglected. The parameters of the vehicle are the frequencies f. = ~ / 2n and the damping ratios D. = a. /(2/~-:-E-:-') , i = 2 , 3 , 1. (f 2 -- 2 HZ, f 3 1._ - O.tlHz, D21. -_1. O. 5 , O. 3< -D ~ O. 7) . 3 The lowest natural frequency of the guiaeway element in L3 is f =6Hz. The consideration of two beam eigenmode~ proved itself sufficient. In either model asymptotic stability is pro-
cl l.0
,
~
\
O.S
\
,, SIG 21x10 2 ) __ , -I -,-' '\-?j1 ,, /sIGl
.~ -0.5
\ \ \ \
-3. 0
\
I
\,
-1. S
',SIG 31x IOl)
, ,,
-/f
_~--J
\
I
i
,- 0j)
a.
'"
I
1
Fig. 3. Transient behavior using an observer with low sensitivity ( A = -10, z =1 mm, Z =0,5.10- 2 m/ s, Z =0) Ba) magne~ gap ZO b) accelerationoz c) sensitivity functions for the gap z (SIG 1), velocity (SIG 2) and acceleration (S IG 3).
z
z
vided by the controller implemented. Figure 4 shows frequency responses for model L , which 2 is a worst case model with respect t o rid e comfort. As can be seen, the comfort standard ISO 2631 with four hour exposure time (dotted line) is kept. On the other hand, the mass m fol1 lows nearly perfectly the guideway. Similarly, Figure 5 shows the results for model L • Here, 3 an elastic guideway is assumed which results in a periodic time-variant state equation with jumping states. Thus, the analysis is more laborious than for L • However , the above men 2 tioned conclusions arawn for L2 can also be verified for the more realist1.c model L • 3
CONCLUSIONS Maglev vehicles are complex dynamical systems. The corresponding mathematical descriptio n results in a complicated high order state equation, where the parameters for the suspension magnets are uncertain. Thus, a control system with low sensitivity against parameter variations has been developed. Since decentralized control is applied, a simple third-order model can be used for the control system design. It is shown in detail how the sensitivity model can be applied to obtain a controller and an observer with low sensitivity. I n the present
Dynamics and control of Maglev vehicles case, the observer becomes even robust with respect to input signals. Thus, the sensitivity of the entire closed-loop system can be reduced considerably . The low-sensitivity control system is implemented in a worst case vehicle model. The steady-state responses to deterministic disturbances fulfill comfort and safety requirements. ISO-STAclDARD (
2437
Gottzein, E., Lange, B. (1975). Magnetic Suspension Control System for the German High Speed Train. Automatica, ~, 271 - 284 . Gottzein , E., Brock, K.H., Schneider, E., Pfefferl, J. (1977). Control Aspects of a Magnetic Levitation High Speed Test Vehicle. Automatica, !l, 201 - 223 .
IS C -SL~l':J.\.:C:; (.;.:-:) .............. .....................
Z3 ( ::'!llS) ::J / s? O. '5
J
6
2
3
iC
2
2
4.
6
3
2
10 :J_=0.5 )
------. -.-. C
2
< •
4
,~
c
2
.,.,
6
5
10
?:l.3C.~!CY 1Hz
Fig. 4. Frequency responses for model L • 2 REFERENCES Breinl, W. (1980) . Entwurf eines unempfindlichen Tragregelsystems fur ein Magnetschwebefahrzeug. Fortschr.-Ber. VDI-Z.,8 No 34, Dusseldorf. Engl, A. (1980). Storverhalten von MagnetSchwebefahrzeugen. Sem.-Arbeit, Lehrst.B Mech., TU Munchen. Frank, P.M. (1978). Introduction of System Sensitivity Theory. Academic Press, New York. Glatzel, K., Rogg, D., Schulz, H. (1980). Research and Development of Magnetically Suspended Transport System in the Federal Republic of Germany. Proc. IEEE Intern. Conf. Cybernetics and Society, Cambridge, Mass., 763-774.
Fig. 5. Results for model L • 3 Muller, P.C. (1977). Design of Optimal StateObservers and its Application to Maglev Vehicle Suspension Control. 4th IFAC Syrnp. Multivariable Technol. Syst., Fredericton, Canada, Paper No 25. Popp, K. (1978). Beitrage zur Dynamik von Magnetschwebebahnen auf gestanderten Fahrwegen. Fortschr.-Ber. VDI-Z., 12 No 35, Dusseldorf. Popp, K. (1980a). Dynamics of Maglev Vehicles on Elevated Guideways. Proc. IEEE Conf. Cybernetics and Society, Cambridge, Mass., 1018-1027. Popp, K. (1980b). Contributions to the Dynamic Analysis of Maglev Vehicles on Elevated Guideways. Shock and Vibration Bulletin, Part 3, 39-61.
DYNAMICS AND CONTROL OF MAGLEV VEHICLES WITH PARAMETER UNCERTAINTIES W. Breinl and K. Popp Institut e of M echanics, T echnical University Munich, Munich, Federal R epublic of Germany
Abstract. The overall dynamics of Maglev vehicles is investigated , regarding vehicle , suspension control and guideway. The mathematical description results in a very complicated high order model containing uncertain parameters for the suspension magnets. However , since decentralized control is applied , a simple third order model can be used for the control system design . The controller and observer design with respect to low sensitivity against parameter variations is shown in detail . The corresponding closed- loop system is analyzed by numerical simulations as well as by laboratory experiments and shows very satisfactory results. Keywords . Control system syn t hesis ; robust control ; observers ; system analysis; Maglev vehicles .
I NTRODUCTION Magnetical l y levitated vehicles are under de velopment for rapid transit in high l y populated areas as well as for high speed transportation over large distances . The feas i bi l ity of elec t romagnetic guidance and cont ro l, particularly for high speed operations in connect i on with the use of linear induction mo tors for propulsion , has been shown by vari ous test-vehicle runs (Glatzel and others , 1980) . Speeds of more than 400 km/h already have been reached. Figure 1 shows the TRANSRAPID 05 , the first public Maglev vehicle which was presented at the International Transportation Exhibition in Hamburg , 1979. Due to the properties of electromagnets , the guidance and support magnets must be controlled. Since some of the magnet parameters can be measured only with large uncert ainties (e.g . inductivities) , a controller with low sensitivity against parameter variations is required . Furthermore , there is a strong dynamic coupling between vehicle motion and guideway vibration and also an interaction with the suspension control system . A large number of publications is devoted to the dynamics and control of Maglev vehicles . Short literature reviews may be found in Popp (1980a , 1980b), and an extensive review is given in Popp (1978).
CAR800Y SECONDARY SUSPENSION
D
51<10 GUIDANCE
MAGNET
Fig . 1. Front view at TRANSRAPID 05 and electromagnetic levitation principle . sensitivity study is carried out. In this paper, we confine ourselves and neglect stochastic disturbances , e . g . guideway irregularities and measurement noise . Thus , the control problem is deterministic with uncertainties in some of the parameters involved .
In the present paper the modeling of the openloop system is described at first. Then , the design philosophy is explained and the design of a control system with low sensitivity against parameter uncertainties is shown in de tail. At the end of the theoretical part, methods for the analysis of the closed- loop system are mentioned . The derived results are applied to a single magnet control and a
2431
MODELING OF THE OPEN-LOOP SYSTEM The vehicle - guideway system under consideration is a very complex system with many degrees of freedom. However, the motions in a vertical plane , i . e . heave, pitch and vertical bending are decoupled from the other
W. Breinl and K. Popp
2432
motions. Here, o nly these dominant motions are considered. Furthermore, we assume small displacements except the forward motion of the vehicle which may take place on a straight trac k with constant speed v. The entire openloo p system contains: • vehicle mo del, magnetic suspension model, and guideway model together with an estimation of the parameter uncertainties. The vehic les are usually modeled as mass point systems or multibody systems. The mathematical description is given by a system of o rdinary differential equations of second order (Popp , 1978, 1980b). The action of the suspension magnets is replaced by single forces. Although electromagnets are highly nonlinear elements, a linearized model described by an ordinary first order differ ential equation has proven itself well as shown by Gottzein, Lange (1975). Unfortunately, some of the magnet parameters (e.g. inductivities) are not known exactly. The guideway model consists of an infinite sequence of identical and uncoupled Bernoulli-Euler beamstructure elements of length L. The deflection of a guideway element is governed by the wellknown fourth order partial differential equation, which can be replaced by an infinite set of ordinary differential equations using modal expansion. For our purposes an approximate solution comprising a finite number of eigenmo des is sufficient . In order to keep the mathematical description of the entire guideway as simple as possible, on ly the minimum number of elements is regarded, which results in a periodic shifting of elements under the moving vehicle. Although the details of the mathematical des cription of the entire open-loop system depend on the chosen subsystem models, the resulting state equation E always has the same structure, (1)
E ~(t= V T+O)
= Ux(t= vT-O), v = 1,2 .... , x(+O) = ~o ,
(2)
where x(t) is the nx1-state vector and u(t) is the mxl-control vector; A( a , t) denotes the nx1 - system matrix , B( a ) th; nxm-control input matrix, i(t) the nXl-disturbance vector (e.g . moving static loads) and U the nxn-state transitio n matrix. The matric;s ~ und ~ depend on uncertain parameters a j' a j = a jo + ~aj' j = 1, ... , r , where a jo are the nominal values and 6a j are the unc ertainties . The uncertain parameters are combined to the rx1-parameter vec tor a . The system E is periodic time-variant, i.e.-~(t+T)=~(t) ,[~(t+T)=i(tU, T = L/v, because a periodically constructed guideway (element length L) and a constant speed v have been assumed Furthermore, it shows periodically jumping states, eq. (2), due to the fact that the vehicle front reaches a resting element at the end of each period. Thus , some guideway states become suddenly zero . Equation (2) describes also the shifting of the guideway elements. The dimensio n n of the state vector which corresponds t o the order of the system matrix is given by n = 2f + + 2nf, where f
m
and f represent the degrees of freedom of the vehicle and of one guideway element, respectively, n is the minimum number o f guideway elements within the system bounds and is the number of magnet forces . In a realistic system description the total system o rder n is quite large. Thus,the open-loop system can be mathematically described by a • linear, periodic high order state equation with periodically jumping states and parameter uncertainties . The state equation (1) has to be completed by the measurement equation
m
2(t) =
£ ~(t)
(3)
where the measurement vector 2(t) represents the real output of the system and C is the measurement matrix. In technical applications one usually measures the vertical accelerations and the airgaps of the suspension magnets.
CONTROL SYSTEM DESIGN Design Philosophy The most important technical design criteria for the control system are • asymptotic stabi lity of all motions, best possible safety and reliability, sufficient ride comfort, l ow power consumption . Since Maglev vehicles are unstable without active control of the primary suspension, the first criterion is essential to operate the system. There are different feasible control concepts . Most promising is a decentralized control regime where primarily the air gap between ve hicle and guideway is controlled, resulting in best safety conditions (Gottzein and others , 1977). The suspension magnets are decoupled from each other and flexibly mounted on the car body . Each magnet has an individual power supply, s e nsors and controller. These components result in a single unit which is sometimes called a magnetic wheel. Here, redundancy is realized by configuration. The control system design must start with lower order models. Thus, the sophisticated higher order model I , cL (1), (2), has to be simplified to a lower order model L • In the present case, this c an be done by assuming a rigid and straight guideway, which results immediately in a constant system matrix of reduced order, A = const, and vanishing disturbances, ~(t) ~ Q,-as-well as vanishing state jumps, ~ =~. Since a decentralized control concept has been chosen, it turns out that a vehicle model with only one single magnet is sufficient for the contro ller design. The simplified low order model can be stated as
X(t)
~(£)~(t)
(4)
+ b(a)u(t)
L
Y. (t)
£
~(t)
,
where x(t) is the nx1-state v ecto r, u(t) is the scalar control, ~ (~) is the nx1 - c o ntrol
(5)
Dynamics and Control of Maglev Vehicles input v ecto r and Z(t) is the mxl-measurement vector; ~~) denotes the nxn-system matrix, £ the mxn-measurement matrix and a characterizes the rxl-parameter vector with uncertainties. Thus, the system E can be described as a • linear, constant, reduced order state equation with single input and parameter uncertainties. Since the parameters of the suspension magnets are uncertain, in E and I the same vector ~ occurs. For a = a the controller and observer design is st~ightforward. However, here a control system design for a = a +6a is given which shows low sensitivitYOagainst parameter variations 6a.
2433
Starting with (9), where the nominal values A and b are used, a sensitivity analysis for tRe clo~d-loop system can be performed resulting in a low sensitive gain vector k, which is shown later. Extended sensitivity model including the observer. For the realization of the control law ~he state vector x(t) has to be replaced by the state estimation R(t), which can be found utilizing the wellknown Luenberger-observer ~ l(t)
+:!:. ~u(t) +
!:
Z(t) (l0)
x(t)
~l s.(t)
+ ~2 Z(t)
Low-Sensitivity-Control Controller design. We start with system E and assume that the pair (A ,b ) is completely controllable and (A ,C~iSOcompletely observable, where A =A(a~,-b =b(a ). To stabilize the system, aolin~r t~e-i~ariant control law is assumed,
Here, s.(t) denotes the qxl-observer vector and the various matrices f o llow from (lla) (llb) ~
DT
- LC, Re Ai
(~)
< 0, i=l, .. ,q, (llc)
(6)
where k is the constant gain vector. As wellknown, ~ can be calculated so that the corresponding closed-loop system is asymptotically stable, using optimization with respect to a quadratic performance index or pole assignment. However, to analyze the system E with respect t o sensitivity against uncertain parameters OC., j = l, ... ,r, a sensitivity model has to bJ derived. The sensitivity vector o . with respect to the j-th parameter is given)by ax(t,a)
o . (t,a )
-)
aa
=
-0
I ~
j
(7)
a
-0
The elements of o. are called sensitivity functions. The serlsitivity equation corresponding to (4) reads (Frank, 1978),
where I is the nxn-identity matrix and A. (D) is the~-th eigenvalue of ~ . The observe? (10) estimates asymptotically a subspace of the state space, ;(t)~ x(t), and ~(t) is an asymptotic estimation of the state,-x (t)+x (t) . Introducing the error ~(t+A= s.(t) - TX(t) and the control law u(t)= -k x(t) the closed-loop system corresponding to-(4) reads
In order to analyze the closed-loop system/including the observe~with respect to the sensitivity, in (8) the state estimation R(t) has to be regarded
a.
-)
aA(a)1 A(a ) o. (t,a )+-a-x(t,a)+ - -0 -) -0 a .!!o-0
a. (t, -0 a ) -)
j
a~(~)'1
+-aa . )
au(t,a)l u(t,a )+b(a ) - a -,
a
-0
--0
a.
o.
-
-) 0 -
--=:Q..
aa .
I
a
-0
)
=:O
-
A .
~)
=
(l4 )
'
(9)
A-b kTJ
~
I
-
a
J
~;
is used denoting
-0
the sensitivity error, which is a measure for the detuning between plant and observer in presence of parameter variations in the plant. The sensitivity-error follows from (10),
-O~
~)
I
a~(~) . _b aa . a ' ~j )
. (t) .
a)
.
a; (a)
Considering only the j-th sensitivity vector, eq. (8 ) combined with (4) and (6) yields
-)
u
a )
For shortness
[!.J [~ .~
-0
In general, the observer vector l(t) is a function of the system parameters. Thus, the derivative of the input becomes
(8)
•
o
. u (t) +b
-et)
(13)
u
ax
-et) -
~)
a
)
-0
x(t) +b
(t) =A o. (t) +A .
-0
a~(~) aa . )
I a'
-0
au(t'~)1 da . )
Thus,
(l3) becomes
a
-0
T
T
(A -b k S2C)O. + (A . -b .k Sl); + -0
~
-
-
-)
T
~)
-et)- -
T
+ (A .-b .k )S2Cx-b k Sl O ~ ~)
~J-
--~--.,
-
( 16)
w.
24 3 4
Breinl and K. Popp
For the entire closed-loop system consisting of the plant (4) (with nominal values A and b ), the observer (10) and the sensiti~ty iii8del (15), (1 6) , a sensitivity analysis can be carried o ut. Ho wev er, the system order is 2(n+q) . In the n ext step, a new design f o r an o bse r v e r with l o w sensitivity is shown. Obser v er with l o w sensitivity. The combinatio n o f system ( 12 ) (now, using the actual values ~,£) with the sensitivity model (15), (1 6) and the erro r ~ ~~ - ~ ~j yields
of the c o ntrol system. The remaining step is to determine the matrices D and L for an observer of minimal order q;n-m. For this, we use a coordinate transformation ~, so that ~ becomes £;~ ~[Q .!m]. NOW, if the matrix T has the special structure ~ ; [I -T~ Q], the-rank conditions (lla) and (19) ~ be satisfied. Thus, from condition (lib) one obtains
(21 )
To satisfy condition (11c), consider the transformed matrices A and b to be 9iven in submatrices corresponding to T
A
[~11
~12
~21
~22
~13] ~23 ,~;
~31
~32
~33
( 17 ) Fro m (1 7 ) it can be seen, that the algebraic decoupling of the controller and the observer (separation principle) is not given. Thus, separation requires
~ [~
- ~J ; 0
~
,
[£ -
~J
;Q
( 18)
-
CA-
;
0
-
. . s rank [ ~,~ T TJT ; n, < No w fro m condLtLon rank [A . b .] ; 1, it follows rank T - n - 1. Theref~J ~Jobserver with low sensitivity can only be of reduced order. Howe v er, if the system is in c omp anio n form, whi c h is the case f o r the considered Maglev v e hicle model, (19) is satisfied by TA . ; O. This results also in ~ £ ; Q. Thus~~e observer becomes even robust for disturbances of the input signal u(t). If conditio n (19) h o lds, from (17) it f o ll o ws, that the separatio n principle is valid also f o r p arameter variatio ns. Even more, the o rde r o f ( 17) can be reduced: The e equ a ti o n in ( 17) becomes ~ ; ~ e ; utilizing the initial c o nditio n e (0) =0 it~ollows -{) ~(t) = Q. Thus, th e ~-equati o n can be canceled and th e sensitivity model is given only by the G. -eq uatio n. This give s a wel come reduction of t~e system o rde r t o 2n+q and the entire system ( 17) reads
! :(
T* ~ ; ~11 - -2
D TIf - -2
11
~1 ,
(23b)
; L -2
(23c)
This system of matrix equations can be solved if and only if the pair of matrices (~11'~21) is observable. Or in other words, there must be measurements, so that the observer subspace corresponding to T is observable. The matrix D (23a) must be stable, cf. (11c). The stabilization is the same problem as in the controller design, thus, the same pro cedure s can be used. For an optimal design with respect to a quadratic perfo.rmance index, the covariance matrix of the initial state vector can be applied (Muller 1977, Breinl 1980). Now, the optimal observer with low sensitivity is ready for application. Closed-Loop System Description In the final step the control system obtained for the reduced order system E is introduced in the original high order system ~ , re sultLng Ln the closed-loop syst~ If the ideal control law (6), u(t); -~ ~(t) with a low-sensitive gain vector k is introduced in (1), (2), one obtains
r.
"3< (t);~ (t)
. . 1'_ -~~l
~23
(23a)
~21
+ ~12 - ~2 ~22
~13 - -2
( 19)
.
(11c) reads
TIf
In the sense o f a first orde r sensitivity the ory, (18) is equal to T
Thus,
[::]
(22)
~(t) +i (t),
! (t+T);! (t) , ~(t+T);i(t),
(24)
~(t; vT+O); ~~(t; vT-O), v;l, 2, .. ,~(+O) ;~, I'"
(25) (2 0) I n (2 0), the G. -equatio n is o nl y used to analyze the systeJ with respect t o sensitivity, but it is no t n e cessary for the realization
where the system order n remains unchanged. On the othef hand, if the realistic control law u(t); -k x(t) , is introduced in the same way, then the Zbserver has to be regarded and the system order is increased by the number of
2435
Dynamics and Control of Maglev Vehicles observer states mq I q=n-m . In either case, if a state feedback with respect to L is computed and implemented in the original higher order system E , then this results generally in an incomplete state feedback with respect to E and Qothing can be said about the stability of E or other properties. Thus, a pe~formance analysis of the closed-loop system L is required.
POI~T
MSS MODEL [1
/ /
/
/
SYSTEM ANALYSIS The dynamic analysis of the closed-loop system ~ is carried out starting with a stability in-
vestigation and followed by the steady-state response calculations. For the stability investigation the wellknown analytical methods for periodic systems based on Floquet theory have to be applied and extended to jumping states. Here only the main results are shown. All derivations and proofs can be found in Popp~(1978). Since asymptotic stability of system E is essential,the stability analysis can be carried out applying the Stability Theorem: The closed-loop system E is asymptotically stable if and only if all eigenvalues ~. of the growth matrix U~(T) have absoluteLvalues less than one , ~. < 1, i = 1, ... ,n , where ~ is the jump matrIx and the transition matrix ~ (T) follows from
I
.
~(T)
-
~
= A(T)
~(T)
--
~
(0)
.!n,
(26)
by numerical integration over one s~ngle period, 0 S T ~ T. In case of a system L without jumping states, i.e. ~ E this result is wellknown. For the steady-state-response calculations an ~proximatesolution shall be mentioned first. To check the vehicle behavior together with the suspension dynamics, a rigid guideway of sine-wave shape is often used. Then, the modified closed-loop system posesses a constant system matrix, a harmonic external excitation, and no jumping states. Thus, frequency-response methods can be applied. In case of an elastic guideway, however, either numerical integration of (24), (25) has to be carried out until steady state is reached or the analytical solution for steady state has to be used. The latter solution reads
.!n '
v->=
Fig. 2. System models rigid straight-line guideway and serves as a model for the control system design. The model L2 (n =7) contains a rigid sine-wave gui2 deway and can be considered as worst case model with respect to ride comfort , while model L3 (n =11) is traveling on an elastic guideway 3 and represents a worst case model with respect to guideway loading. In the following the controller and observer design for model L1 is shown. Then the resulting controller is implemented in the models L , L3 and a system ana2 lysis is carried out . Control-System Design for the Single-Mass Model Controller design. The vertical motion of the single-mass model L1 is described by the linear time invariant third-order system (4), (5), where (28)
l~
x= [:], A = 0 Z -0 KsR (Ks_KiKS) mL m mL o 0
APPLICATIONS Models Figure 2 shows two different types of Maglev vehicle models. The point mass model L1 of order n =3 characterizes a single magnet on a 1
b = -0
L 0
The nx1-state vector x comprises the gap z as well as and The inatrix tains the system parameters given in The control input vector contains of the system parameters .
z
z.
Eo
~
r1_ Ki ]. ~
mL
0
magnet A con~ble 1. also some
Table 1: System parameters of the magnet model Mass of the magnet Ohmic resistance Coefficient of gap z
The solution (27) requires only integrations over one single period.
~l'
_~.
m
R
16
kg
8 4 Sl
K =5.7·10 N/m s Coefficient 0z~ current i K.=K.= 114 N/A and velocity L s 0.5 VS/A Inductivity Lo
The system is completely controllable and, assuming the measurement of gap z and acceleration also completely observable. Here, (28) is already given in companion form. The gain . vector ~ has been optimized with respect to a quadratic performance index, where the weighting factor corresponding to z was varied and the results were in each step analyzed using (9) . The overall results are listed in Table 2.
z,
W. Breinl and K. Popp
2436
This cont~oller design shows low sensitivity against parameter variations as well as a reasonable transient behavio r of the closed-loop system. Table 2:
~ontrol
_ _ nomIna l
\
a)
o.s
--- .20-'. Abw •••• · 20·'.Abw
\
gains
Con trol gains k1 = - 9123 . 5; k2 = -470.2; k3 = -7.1 ~o.
Design of an observer with low sensitivity. The obse rver design shown earlier yields (dimension q = 1 )
-t~ ,
b)
O. ZS
.§.1
-0 . 50
·0.7S
1t 2 L = [ -t 2
l' /'
s
I
.
I
o.
Tj "I
d
1/ - I~
l
,'t
t.J
" "
"
o-:r -
-
I
I
,t
-0 . 2S
d =
__
----
Ei genva lue sof the closed-loop system A1 = - 53 . 07; A ,3 = - 31.89 ± i·18.92. 2
D -
o~----
I
I
o. )
a.
'"
i
I I
I
I\J
i
4
(29) l.S
where t~ = P2/ P 1 is identical with the obse rver e igenvalue A . Here, P , P are the esti2 1 mated maximal in~tial values, P ~ = 1 x. 12 . 1. 1.0 max To demonstrate the theorY,a parameter variation ~L = ±20% of the inductivity L is investigated. Some numerical simulationoresults are shown in Fig. 3, where the actual trajectories are plotted as well as the corresponding sensitivity functions. This results are in very good agreement with experimental results obtained by means of a laboratory model of a single suspension magnet. System Analysis for the Worst Case Models The linear controller (6) with the control gains shown in Table 2 is implemented in the worst case models L2 and L , cf. Fig. 2. Howev3 er, instead ofTstate vector (28) the vector x = Cs , S , 2 1] is fed back, where s denotes the air gap between vehicle and guideway and 21 is the acceleration of mass m . Since the en t1.re 1 ve hi cle mass m , m = m + m + m3 ' exceeds 1 2 the mass m of the point mass model, the number m/m of magnets are assumed to act in parallel in order to heave the vehic le. The air gap s depends on the guideway deflection, which acts as vehicle excitation. The excitation frequency f reads f = v/L ~ 4.4 Hz, where v=400 km/h and L = 25m have been assumed. Depending on the suspensio n design, characterized by f2 = = ~ /2n , the excitation frequency f is larger or smaller f2 ' i.e. above or below resonance of the secondary suspension. Previous investigations have shown (Engl, 1980) that the tuning f > f2 leads to better results than f < f . The aim of the analysis of systems L2 2 and L1 is to show the steady-state behavior of the iaeally controlled worst case vehicleguideway models with respect to deterministic disturbances. Here, the observer dynamics is neglected. The parameters of the vehicle are the frequencies f. = ~ / 2n and the damping ratios D. = a. /(2/~-:-E-:-') , i = 2 , 3 , 1. (f 2 -- 2 HZ, f 3 1._ - O.tlHz, D21. -_1. O. 5 , O. 3< -D ~ O. 7) . 3 The lowest natural frequency of the guiaeway element in L3 is f =6Hz. The consideration of two beam eigenmode~ proved itself sufficient. In either model asymptotic stability is pro-
cl l.0
,
~
\
O.S
\
,, SIG 21x10 2 ) __ , -I -,-' '\-?j1 ,, /sIGl
.~ -0.5
\ \ \ \
-3. 0
\
I
\,
-1. S
',SIG 31x IOl)
, ,,
-/f
_~--J
\
I
i
,- 0j)
a.
'"
I
1
Fig. 3. Transient behavior using an observer with low sensitivity ( A = -10, z =1 mm, Z =0,5.10- 2 m/ s, Z =0) Ba) magne~ gap ZO b) accelerationoz c) sensitivity functions for the gap z (SIG 1), velocity (SIG 2) and acceleration (S IG 3).
z
z
vided by the controller implemented. Figure 4 shows frequency responses for model L , which 2 is a worst case model with respect t o rid e comfort. As can be seen, the comfort standard ISO 2631 with four hour exposure time (dotted line) is kept. On the other hand, the mass m fol1 lows nearly perfectly the guideway. Similarly, Figure 5 shows the results for model L • Here, 3 an elastic guideway is assumed which results in a periodic time-variant state equation with jumping states. Thus, the analysis is more laborious than for L • However , the above men 2 tioned conclusions arawn for L2 can also be verified for the more realist1.c model L • 3
CONCLUSIONS Maglev vehicles are complex dynamical systems. The corresponding mathematical descriptio n results in a complicated high order state equation, where the parameters for the suspension magnets are uncertain. Thus, a control system with low sensitivity against parameter variations has been developed. Since decentralized control is applied, a simple third-order model can be used for the control system design. It is shown in detail how the sensitivity model can be applied to obtain a controller and an observer with low sensitivity. I n the present
Dynamics and control of Maglev vehicles case, the observer becomes even robust with respect to input signals. Thus, the sensitivity of the entire closed-loop system can be reduced considerably . The low-sensitivity control system is implemented in a worst case vehicle model. The steady-state responses to deterministic disturbances fulfill comfort and safety requirements. ISO-STAclDARD (
2437
Gottzein, E., Lange, B. (1975). Magnetic Suspension Control System for the German High Speed Train. Automatica, ~, 271 - 284 . Gottzein , E., Brock, K.H., Schneider, E., Pfefferl, J. (1977). Control Aspects of a Magnetic Levitation High Speed Test Vehicle. Automatica, !l, 201 - 223 .
IS C -SL~l':J.\.:C:; (.;.:-:) .............. .....................
Z3 ( ::'!llS) ::J / s? O. '5
J
6
2
3
iC
2
2
4.
6
3
2
10 :J_=0.5 )
------. -.-. C
2
< •
4
,~
c
2
.,.,
6
5
10
?:l.3C.~!CY 1Hz
Fig. 4. Frequency responses for model L • 2 REFERENCES Breinl, W. (1980) . Entwurf eines unempfindlichen Tragregelsystems fur ein Magnetschwebefahrzeug. Fortschr.-Ber. VDI-Z.,8 No 34, Dusseldorf. Engl, A. (1980). Storverhalten von MagnetSchwebefahrzeugen. Sem.-Arbeit, Lehrst.B Mech., TU Munchen. Frank, P.M. (1978). Introduction of System Sensitivity Theory. Academic Press, New York. Glatzel, K., Rogg, D., Schulz, H. (1980). Research and Development of Magnetically Suspended Transport System in the Federal Republic of Germany. Proc. IEEE Intern. Conf. Cybernetics and Society, Cambridge, Mass., 763-774.
Fig. 5. Results for model L • 3 Muller, P.C. (1977). Design of Optimal StateObservers and its Application to Maglev Vehicle Suspension Control. 4th IFAC Syrnp. Multivariable Technol. Syst., Fredericton, Canada, Paper No 25. Popp, K. (1978). Beitrage zur Dynamik von Magnetschwebebahnen auf gestanderten Fahrwegen. Fortschr.-Ber. VDI-Z., 12 No 35, Dusseldorf. Popp, K. (1980a). Dynamics of Maglev Vehicles on Elevated Guideways. Proc. IEEE Conf. Cybernetics and Society, Cambridge, Mass., 1018-1027. Popp, K. (1980b). Contributions to the Dynamic Analysis of Maglev Vehicles on Elevated Guideways. Shock and Vibration Bulletin, Part 3, 39-61.