WE-6-6
Local Vector Control of an AC Drive System Load Simulator R.E. Betz, H.B. Penfold and R.W. Newton Department of Electrical and Computer Engineering University of Newcastle, Callaghan, 2308, Australia. FAX: +6149 60 1712 email:
[email protected] Keywords: Nonlinear control, simulator, AC drive systems, dynamic systems
not allow the simulation of a variety of dynamic load characteristics.
Abstract
This paper w i l l present a novel approach to implementing a dynamic load dynamometez using the Local Vector (LV) control strategy. This conml suaregy has also been used successfully in a more conventional control situation (51.
This paper describes the application of the Local Vector (IV)non-linear control strategy to a load simulator for an AC drive system. The load simulator physically consists of a computer controlled DC machine which is connected to the AC machine under test by a rigid shaft. With appropriate control the DC machine load is able to simulate static loads, first order dynamic loads (i.e. loads with varying J, f), and second order dynamic loads (i.e. rotating loads connected to the test machine via a compliant shaft). This paper presents SIMULINK simulations of the proposed control system and discusses some implementation issues.
The remainder of the paper is organised as follows. A review of the LV control strategy is presented with emphasis on its digital implementation, followed by the mathematical formulation of the specific load simulation problem for the cases to be treated. SIMULINK simulation results are presented for the controllers developed followed by the conclusion.
THE LV CONTROL STRATEGY
INTRODUCTION
The Local Vector (LV) control algorithm -socalled because of its concern with phenomena at the system trajectory level, rather than with the underlying vector-field - operates by ensuring that. for each time-local step, the control action is such as to map the observed behaviour of the controlled system onto that computed for a dynamical reference system specified by the designer. In this sense it is model-reference control. but with the important distinction from the usual approach that the model-reference control action occurs independently of the system history at each control interval.
There are usually two main objectives to the testing of AC machine drive systems:* to test the actual machine itself
in order to determine its efficiency at different loading levels, its thermal performance, torque ripple, etc.,
for machines that are fed from variable frequency supplies, to test the control characteristics of the variable frequency supply/machine combination. In order to achieve this the test machine has to be loaded. Traditionally this was carried out using a DC machine controlled via a Ward-Leonard system. However, such systems were restricted to essentially static tests, and were not very useful for controller testing.
The topology of an LV system is different from a traditional control system. Rather than augmenting the plant with an explicit external dynamical system so that the closed-loop differential equations are those desired. it focuses on the alteration to the behaviour of the controlled system so that it marches the behaviour of the user-specified reference system at each control interval. Once this behaviour modification has been achieved a simple controller can used around the reference system to define its txajectory-following properties. The “twostep” control stxategy results in the design problem being reduced to that of choosing a controller for the reference model. The choice of this model is relatively kee -it must, however, be stable. It may, for example, be a 2nd-order linear stable system with appropriate transient response propexties. Figure 1
The use of power electronics coupled with modem computing offers the opportunity to implement sophisticated control system testing. The concept is that the torque of the load machine is controlled such that it creates the same shaft torque on the test machine as some desired load type. Similar systems have been built in the past. mainly it seems by motor vehicle manufacmers [l]. The papers published on these systems do not contain significant detail. Various other papers have appeared in relation to static dynamometen [2][3][4].but these do
72 1
0-7803-1872-2/94/$4.CKI0 1994 IEEE ~~
~
_
_
_
is a block diagram illusuating the general structure of the algorithm.
(i) From the required hirial conditions,and with
the appropriate ex*&
The LV control algorirhm addresses the control of systems having an inputloutput description of the form-
inpuL
simulate
one time step of each of systems R and T.For the f i m step the initial conditions of R and T are the same, but as described later, this w ill not necessarily be true for subsequent steps. We ensure that the system R (which has desirable dynamic properties) follows the system T (which generates a desirable mjeUory) by using proportionalfeedback K, .
where the integers m and n are gven, y (I) is a scalar output and u ( i ) is a scalar input variable. The notation y ( Q ( 1 ) refers to the i th derivative of y ( 1 ) with respect to time. No other seuctural assumptions are made except the requirement for stability of the inverse system and sufficient regularity to ensure existence and uniqueness of solutions over some h t e open interval. The generality of the above system will be clear. A similarly general reference model is employed although in practice a second-orderstable linear system is usually chosen.
kT,
uR (U)
=
kT,
uir + KR(
yy
-y:)
From past meaSurementS of the output of systems S, ys. up to and including time LT',. .w + 1) r,
, of the next value
form a prediction. ss
of 3s. A necessary assumption at this point is that the same control as was applied to S during the previous time step will again be applied in the next time step which begins at time kT, . In this step it is desirable to use a
The key idea is to reconstruct from available data numerical estimates of the values of the functions f( ...) and g (...) at e s h control time k T x . These estimates, f and 2, are then used for predichon of the system t x a ~ t o r over y the next time step and the calculation of a control of the "cancel-and-substitute" form. The theory defines the requirements on the sample period Ts such that the closed-loop system will have guaranteed mnsient performance with exponentially fast recovery from disturbances. Referred to [7J[8][9][10] for theoretical derails.
measurement sampling period somewhat shorter than the control interval so as to ensure that the relevant derivative "S are computed as close to kT, as possible. (iii) From the incremental trajectories in ys for the behaviour reference and the subject systems compute:
For 2nd order systems a simple dlgital implementation of the algorithm can be consuucted A further
(t+l)T,
AYR = YR
simplification is available by assuming that the gain term g ( ...) does not vary widely, and that irs effects can be largely absorbed in the drift t a m f( ...) . It is this simplified form of the algorithm which provided the results presented in this paper.
.(k+ 1)T,
AYs =
,kT, -YR
(3)
kT,
-3s
(4)
Note that in keeping with our principles of behaviour modification and ' t i - l o c a l ness". the system behaviour has to map to the reference behaviour only o v a the period k T s + (L+ l ) T , , the initial conditions on R
Digital implementation The specific computational steps of the digital form of the algorithm are as follows:
"S
Sysrcm. R
conrrdlcr
State Feedback Figure 1: Block Diagram of LV control strategy
722
and S at the step beginning at time kT, are in
tem in less than 50psecs [SI. For the purposes of this paper we shall be using a modified form of the standard LV conmila converted to a SIMULTNK block.
fact the same, and, ‘T, kT, YR = YS
(5)
DYNAMIC LOADS The control to be applied to the plant system, S,during the time step commencing at kTs is that which will reduce the incremental t r a ~ c tory mismatch between S and R to zero by mapping the solution mjectory of S at time (k + 1) T3 onto that of R. This yields the con-
The LV control seemed appropriate because of its behaviour modification property. Furthermore, it does not maaa whether the behaviour reference is non-linear. Therefore complex non-linear load effects such as backlash and stiction can be easily produced
trol:
where g is the estimate of the input gain term.In practice this term has to be greater than or equal to the true value of the input gain term.
Two main types of loads will be considered in this paper:
.
(i) First order loads consisting of an inertia. friction and fixed torque load connected to the drive machine via a stiff shaft.
(iv) At the time ( k+ 1) Ts from measuremenrs of ys and either measuremenrs or estimates
of
(ii) Second order compliant shaft load (i.e. the
ys ,set the initial conditions of the behaviour reference system, R, for the next step equal to the final conditions of the subject system, S. This resetting is done regardless of the accuracy with which the incremental trajectory for S actually mapped onto that for R. This implements a strid “time-localness” on R with respect to S.
shaft to the load has low stiffness) connected to the drive machine via 1:l gearing with backlash. It is assumed in the first case that both the drive and load machines are separately excited M3 machines with the following armature circuit model: diu
The initial conditions of the trajectory reference system, T, for the next step are simply the final conditions of the previous step; The subject system S,of course does this without our intervention.
dr
(k-l)T, +&(Y~-t(x+
1))
Ra.
1
La
La
--la+-(va-kOc)
(9)
using standard notation. The parameters chosen for the machine were Ra = 0.1R. La = 0.001H p , = 2 (pole pairs), and k = 1.35weben. In the second case the simulation results presented use the ideal load motor (with no machine dynamics).
.
The above algorithm is for the case where the plant to be controlled is such that the 2nd derivative of the system is directly influenced by the control input. Second order and higher order plants fall into this category. A 1st order plant on the other hand has the second derivative controlled by the derivative of the control input In this case a 1st order form of the controller can be defined The algorithm follows the same general set of steps for the 2nd order case above, except that now the control is: kT,
=
Te = p P X iO
Repeat the algorithm from the beginning, but for the next time step.
U
The dynamic load problem essentially involves controlling an electric load machine in a dynamometer so that it emulates different types of loads to the test machine. In this paper the control stmtegy for the load machine will be based on LV control. Conventional control can also be used successfully for the simple loads [6].
In all of the resulrs presented it has been assumed that the electromagnetic torque, Tc.for both the drive and load machines are known. Techniques for accurately measuring these are being investigated. The shaft measurements that are assumed to be available are the shaft torque, Tshafr (measured by an in-shaft torque wnsducer). and the shaft angle, eshaf,.The angular velocity and acceleration have to be estimated from eshafl,and are assumed to be available in the following control strategies.
(8)
= U
Case 1 - The First Order Load Case The above digital formulation of the LV control algorithm can be readily programmed into any high level language. For example, the first order form of the algorithm can be coded into approximately fourteen lines of ‘C‘ code which can be executed together with all the other code in the connol sys-
A linear fnst order rotating load is described by the following equation:-
JT(; + fTo = Te + TF where:
723
0
Stiff shaft
Drive or test machine
I
Load machine
I
Figure 2: Block diagram of the Structure of the load system o = angular velocity in rad/sec
troller based on the 8 version of (10).An enhancement over the controller in Figure 1 is the inclusion of o and 8 feedback between the behaviour reference and tra~ctorysystems.
f T z total fridion coefficient in Nm sec
JT = total rotational inertia in kg m
2
Simulation Results: The plots in Figure 3 show the performance of the LV controllex on the system shown the Figure 2.
T,E Electromagnetic torque in N m TF3 Fixed load torque in Nm The essential form of the system is shown in Figure 2. JT =
Jlood and fT
Jdrive
= fdrivc
+fiood
The parameters of the system simulated are as in Table 1 where the drive subscript refers to the
Table 1
'
There are two choices for the type of LV controller: Jdrrve
(i) a fist order controller based on controlling angular velocity.
-Io,= 0.3 kg-m
(ii) and a second order LV con&oller attempting to track the desired shaft angle.
Jd,
The temptation is to implement the fmt of these. since h s would mean that there are only first order derivatives being taken within the LV conuoller, with a consequent reduction in noise problems. However. the approximare nature of the behaviour reference &=rete propagation leads to small w errors which can integrate to a large €tJkofl emor.
-loOo
I
I
I
I
2
3
4
= 0.08 kg-m
fd,,,,
ful
= 0.01 Nm-sec
= 0.015 Nm-sec
f,, = 0.01 Nm-sec
dnve machine, the of means actual load machine values, and dl means the desired load machine values. As a n be seen from Table 1. the inertia of the load machine is considerably larger than the desired inenia. Therefore during the acceleration phases of
The simulation results are for the 2nd order LV con-
loa -
= 0.1 kg-m
5
lime (secs)
Eme(secs)
Figure 3: First order load angular velocity and angular error plots.
724
the simulation the controller has to produce torque in the same d i r e i o n as the drive machine torque so that the effective inertia of the load machine is lowered appropriately. Notice that the angular velocity mor between a simulation of the real plant, and that produced by the LV controlled DC machine is very S W .
-
Case 2 The Second Order Compliant Shaft Load with Backlash A compliant shaft conneaed load is a problem of great practical importance, since it commonly ocaxs in industry. For example, most rolling mill control problems are represented by this type of a model. Backlash also is of industrial relevance, since many systems are connected via gears which almost always have k k l a s h . The resultant deadbands can cause signikant control problems.
The differential equations for two rotating loads connected via a compliant shaft are:-
where the L and d s u b s u i p ~refer ~ to the load and drive machines. Obviously when the machines are coupled together the total system forms a 4th order plant The physical set up of the system is much the same as that shown in Figure 2, except for the compliant shaft which results in the possibility of different angular velocities (only in transient situations) and angles at either end of the shaft. The inclusion of backlash into the equations implies that when the angle difference B d - B L becomes less than the backlash value then the K (8, - eL) terms in both sections of the equation become zero - that is. borh the rotating masses become decoupled bom one another. It should be noted that the presence of com-
pliance in the shaft means that the system does not operate the same as a rigid shaft system in relation to backlash. With a rigid shaft the disconnection of the masses occurs immediately the angle difference between the masses at either end of the shaft starts to decrease (the maximum angle difference being equal to the backlash angle). Equation (11)can be modified to simulate the effect of backlash by the inclusion of a deadzone in the (0, - eL) term. SIMULINK has a standard block for a deadzone which faQliIates easy implementation. For the reasons cited above. the inbuilt backlash component included with SIMULINK should not be used for this case.
In a manner similar to Case 1 it is required that the behaviour reference model implement the desired dynamics of the compliant shaft - backlash load. Normally a LV controller is consnained KIa second order behaviour reference model (mainly due to the difficulties in dealing with higher order derivatives). However, in this case a 4th order behaviour reference model is required Fortunately, it is possible to gain the required 4th order dynamics in the behaviour reference model, whilst still only requiring 2nd order derivatives. This is achieved by making the fiKt expression in (11) (i.e. the drive part of the expression) have its input augmented as follows. Denote x as the output of the deadzone block in the trajectory system. The effeaive torque that should be produced in the shaft is K(0;-0;)
0,' = e d - x and 0, is the desired 0, from the trajectory model. Therefore, if the deadzone is active then the shaft torque is zero. else it is K x as required. The value of 0,' is fed into the behaviour model as shown in (12):
25
T i e (secs)
whae
Time (secs)
Figure 4: Angular veloaty and shaft torque plots for the compliant load with baddash case
A experimental system is cunently being constructed to implement a dynamic dynamometer using the above control strategy.
Where
sw
REFERENCES
= Oforr = 0 = 1fOrX#O
[I]
L. Koch and P. Zeller. ‘Test Stand for Dynamic Investigation of Combustion Engines”. Automobiltechnische Zeiuchrift vol. 89, r~).11. NOV.1987. p~ 585-586.589592.
Simulation Results As with Case 1 the t r a ~ c cory/behaviour reference loops have been implemented using o, and 8, feedback. The parameten used in the simulation are shown in Table 2
[21
C.R. Wasko. “5OOHP. 120 Hz Current-fed Field Oriented Control Inverter for Fuel Pump Test Stands”, Conference Record IEEE-IAS Annual Meexing, 1986, pp 314320.
Table 2
131
C.R. Wasko. *‘A Universal AC Dynamometer for Testing Motor Drive Systems”, Conference Record IEEE-IAS A M U ~Meeting, 1987. pp 409-412.
[4]
A.C. Williamson. “An Improved EngineTesting Dynamometer”, IEE 4th Intanational Conference on Electrical Machines and Drives, 1989. pp 374-378.
15)
R.E. I3etz. H.B. Penfold and R. Lagerquisf “LV Conuol of the Synchronous Reluctance Machine”. Proceedings IEAurt AUPEC’93. Wollongong, Australia. SeptOCs 1993.
[6]
R.W. Newton. R.E. Bee and R.H. Middleton, “Dynamic Dynamometer for Elecmcal Machine Ttsting”, Proceedings IEAuFt AUPEC’93,Wollongong. Australia, SeptOct 1993.
[7l
H.B. Penfold. Nonlinear Control: An Alfernafive Perspective, PhD dissertation. Depart-
Note that 9 switch function is required since €IL’ contains 8, andnot 8, .The €IL’ tamessentially contains the dynamics of the load, therefore (12) behaves as a the T u k d 4th order system.
Jd,iwe = 0.1kg-m2 Ja, = 0.15kg-mz
fdriVr = 0.001Nm-sec far
= 0.OISNm-sec
J,, = 0.08kg-m2
jdl = 0.100528Nm-sec
K = 12.6Ndtad
Badclash = 10’
A, = O.ooO1sec
Te = 5Nm
I
The backlash has been chosen at a high value so that the effecrs can be easily seen in the simulations. As with the Case 1 sunulation, the desired inertia has been chosen to be less than the intrinsic inertia of the load machine. The stiffness has been chosen to give a locked drive machine resonance of approximately 2 Hz. The 5 Nm ekctromagneac torque input step is applied to the system for 1 second and then removed. In Figure 4 notice the resonant response of the angular velocity. Note the flat areas in the torque plot when the backlash region is entered. These regions can also be seen as flat areas in the angular velocity plot For more detail see [Ill. A number of test simulations have been carried out with a variety of load machine, drive machine and stiffness parameters. In all cases the performance has been excellent. The cases shown are now being extended to include stiaion and double compliant shafts with gearboxes.
ment of Electrical and Computer Engineaing. University of Newcastle, Ausnalia. April 1990. [8]
H.B. Penfold and RJ. Evans, “Control Algorithm for Unknown Tiie-Varying Systems”, Inf. Jownal of Control, Vol. 50, 1989, pp. 1332.
[9]
I. M. Y. Marels. H. B. Penfold and R. J. Evans, “Controlling Nonlinear Systems Xa Euler Approximation”, Automatics, Vol. 28. No. 4, pp. 681-696. July 1992.
CONCLUSION
[lo] H. B. Penfold, I. M. Y. Mareels and R. J. Evans. “Adaptively Controlling Nonlinear systems Using Trajectory Approximations”. Inf. J. Adapriw Control and SigmI Processing, Special issue on Nonlinear Control, Vol. 6, No. 4, pp. 395411. July 1992.
This paper has developed several controllers based on the LV control strategy to allow a DC machine rigidly coupled to a drive machine to simulate the dynamics of both rigid shaft and compliant shaft loads (the laaer with backlash in the shaft). The behaviour reference propaty of the LV conmol algorirhm is exploited to achieve this. One very nice f a m e of the algorithm is its ability to handle both linear and non-linear load simulations with essentially the sitme conaol seategy.
[I11 R.E. Eke, H.B. Penfold and R.W. Newton. “LV Control of an AC Drive System Load Simulator”, Elec. Eng. Tech. Report EE9418. May 1994.
726
