Copyright © IFAC Control in Power Electronics and Electrical Drives, Lausanne, Switzerland, 1983
COMPARISON AMONG VARIOUS APPROACHES FOR THE OPTIMAL FEEDBACK CONTROL OF INDUCTION MOTOR DRIVES G. Figalli*, L. Tomasi*, G. Ulivi* and M. La Cava** *Istituto di Automatica, University of Rome, Italy * *System Department, University of Calo.bria, Arcavacata (CS), Italy
Abstract. The application of optimal control techniques to drives using as an actuator a frequency controlled induction motor makes it possible to impose suitable restrictions on the stator current and flux, so to have a good operation bo th of the motor and of the static converter supplying the motor. An approach to this problem consists of its subdivision in two control loops: an outer one giving the required value of the motor torque and an inner one that forces the electromagnetic circuits of the motor to produce the desired torque. This second control loop is the more difficult to be synthetized. The paper takes into consideration several different approaches to the optimal tracking problem for the electromagnetic state variables of the motor. The main peculiarities of the different approaches are described, the obtained con trol solutions are presented and their performances and computational diffi culties are compared by simulation. Also the implementation problems of each solution on a microcomputer-based system are evaluated. Keywords. Optimal control; multivariable control systems; tracking systems; time-varying systems; electric drives; a.c. motors; microprocessors.
of imposing the input variables of the machine so as to obtain the value of the motor torque given by the solution of the first subproblem.
I NTRODUCTI ON
In drives using a frequency controlled induction motor as an actuator, the control device should be designed so as to impose suitable restrictions on the stator currents and fluxes. In this way, it is possible to have a good operation both of the motor and of the static converter supplying the machine.
The first subproblem can be greatly simplified by supposing that the solution of the second subproblem makes it possible to obtaininstantaneously the desired value of the motor torque .
In fact, in order to have a satisfactory operation of the motor and to assure a good matching of its mathematical model, it isnecessary to impose that the motor flux never exceeds a properly assigned value. Moreover, to reduce the size of the inverter, it is necessary to avoid that the stator current has , also during very short time-intervals, values appreciabily higher than those required to produce the maximum desired value of the motor torque .
This hypothesis is justified by considering that the dynamics of the motor electromagne~c circuits is quite faster than the dynamics of the mechanical variable. The second subproblem can be subdivided intwo steps: the first step consists of determining the desired values of the state vector components in order to obtain the desired value of the motor torque, the second one consists of determining the input vector in order to impose that the state variables optimally track their desired shapes.
In a previous paper (Bellini, 1981), an approach was proposed, which allowed the application of optimal control techniques, yet taking into account the above-mentioned restrictions on the stator currents and fluxes.
The second suboroblem can be solved by means of different approaches depending on the choice of the control vector and on the model of the controlled system.
Such an approach implies the subdivision of the optimization problem in two subproblems, the solution of which is less difficult than that of the whole problem . The first subproblem consists of determining the desired shape of the motor torque; the second one consists
Assuming as elements of the control vector the components of the motor supply voltage and the angular slip frequency, the model describing the behaviour of the motor electromagnetic
This work is partially supported by National Ministry of Education, "Fondi 40%".
95
96
G. Figalli et al.
circuits is bilinear. The solution can be obtained either by applying control methodologies for bilinear systems or by considering the variation model around a prefixed operating point. In addition, taking into consideration the possibility of imposing directly the value of the angular slip frequency on the basis of the desired value of the motor torque, the model of the motor electromagnetic circuits becomes linear with slowly varying parameters. In this case the problem can be solved taking into account either the full range variations of the state and input variables or their small variations around an operating point. This paper, at first, describes the considered approaches showing for each of them the main peculiarities of their formulation. Then, the different approaches are compared both as for the obtained performances and as for the computational difficulty of their solutions and the implementation problems of the solutions on a microcomputer-based control system. The comparison is effected by simulation for some induction motor drives of different power and for different operating conditions. Finally as an example, the most significant results related to a particular drive and a selected operating condition are presented.
flux modulus and desired value tically equal to zero, i.e.:
is iden-
~ qs
~ ds
~D
(1 )
~ qs
o
(2 )
Then, ~ D is assumed cons tant and equa 1 to the highest value admissible for a good operation of the motor, since in the usual motor operating conditions a reduction of the stator flux involves an increase of the stator current necessary to give the desired value TD of the motor torque. Moreover, it is convenient to select as control variables, besides the components of the supply voltage (Vds,V qs ) according to the reference axes, the slip angular frequency ws· Therefore, the model of the motor electromagnetic circuits is described by the following system of bilinear differential equations, whose input variables are Vds,Vqs, ws and motor angular speed w: x = Alx + A2 . wX + N . wsx + B v
(3)
where i ds
x
iqs
Al
~ ds
- (Cl +S)
0
S/ Ls
0
0
-( a+S)
0
S/Ls
0
0
0
0
0
-ao Ls
-ao Ls
0
~ qs
FORMULATION OF THE CONTROL PROBLEM As previously said, the control problem can be solved by subdividing it in two subproblems without regard to its formulation (Bel1i ni, 1981). The detailed formulation of the first subproblem and its solution on the basis of the optimization theory of steady-state linear servosystems with quadratic performance index is given in the previously mentioned reference. The second subproblem can be subdivided in two steps: the first step consists of determining a suitable set of state variables (desired values) which assures that the torque assumes its desired value; the second step consists of determining the input vector to impose that the state variables optimally track their desired shapes.
A2
0
0
0
-l /o Ls
0
;N= -1
0
0
0
0
0
1
0
0
0
0
0
0
0
0 -1
-1 l/oLS
(b l : b2 )
0
1 0 0
0
, B
l/ oLs
0
0
0
l /(l Ls
1
0
0
1
+;0
l-K s Kr' Ks
0 0
vds v Vqs
and R R S= a = _s· oL s ' o r
L sr., K r
-r;
Lsr -Lr
where Rs and Rr are the stator and rotor resistances; Ls and Lr are the stator and rotor self-inductances and Lsr is the mutual inductance.
In order to simplify both the computation of the desired values of the state variables and the in-line determination of their values (Bellini, 1979), it is convenient to assume as state variables the components ids,iqs,~ds and ~qs of the stator current and flux according to two orthogonal rotating axes d and q; axis d is assumed coincident with the desired direction of the stator flux representative vector ~s.
Since, as already mentioned, the dynamics of the motor angular speed is considerably slower than the dynamics of the state variables, the effect of w on the equations of the electromagnetic circuits can be considered as a parametric variation.
In this way, the desired value ~ds of ~ ds is equal to the desired value ~D of the stator
Therefore, eq.(3) become: x = A'( w)x + Nws . x + B v
(4 )
Comp a rison Among Various Approaches
where A' (w) = A1 + AZt.' To obtain the desired value TD of the motor torque, the desired va lue s of the stator current components and the reference values of the input variables must be: TD i qs (5 ) (6 )
P6(1- o)t Z- 1 pZ e 2(1_ o )2 ~4_4 i3 2 0 2L 2TZ o
sr =
w
2cLsTD
J( ,: , u)=
(8)
(9)
(9) .
The choice of using the desired value of the motor torque to determine both the desired values of the state variables and the reference values of the input variables i s justified by the great difference between the response speed of the two contro l l oops, inner and outer. Denoting by symbol 6 the variations of the variables appearing in eq.(4 ) around the defined steady-state operating point, the following model i s obtained: 6X=A ' (W)6X+Nwsr6x +Nx 6w s +N 6w s 6x+B 6V
(10)
the block diagram of the considered control problem is shown in fig.l. T
D
:.·,xTQ .'. x+·uTR:.u:dt
(11)
x
A' (w)x + N("! + Bv s
SOLUTI ON OF THE CO NTROL PROBL EM As prev iou sly mentioned, the so lution of the optimal control problem for t he t rac king of the desired valu es of the state variables of the motor electromagneti c c i rcu its ha s been obtained by so l ving a r egu lati on problem with an inte gral performance i ndex wit h infinite time horizon, The different ways, in wh i ch the model of the system t o be contro lled i s chosen, define different subop timal so luti ons of the general optim ization problem f or the considered case.
The controlled sys t em mode l i s give n by the followinC] equation: ~~
s
!:::.x
As f or the com par i son of differe nt control s trategie s for th e solution of the regulat ion problem, four cases have been taken into cons iderati on. To t hi s a i m, at f i rs t the control vector is obtained for each of the considered case s on the bas i s of the partic ular model chosen to descri be th e dynamic behaviour of the motor electromagne ti c circuits. In addition, the control vector , which i s opt i ma l with re spect to eac h cons idered mod el, is applied to the comp l ete model given byeq.(4). In this sense, the obta in ed co ntrol vector re presents a subopti mal so luti on of the general optimization problem , giv en by the minimization of in dex (11 ) wit h t he dynamic cons traint (10).
Cas e 1 - Bilinear mode l
w
+
x
Fig. 1.
I
in wh i ch ( ~ )T re presents the transpose of (.), Q i s a 4 x 4 symme tri c pos itive definite matrix and R i s a 3 x 3 symme tric positive semidefinite matrix.
(7 )
To carry out the model of the control system, it i s convenient to c0n ~ ider a variation model of the motor electromagnetic circuits around the operating point, giving the desired value of the motor torque, characterized by the values of the state and input variables given by eqs.(l), (2), (5), (6) and (7), (8),
!:::.v,!:::.r.w
The s tructure of bl ock C2 i s derived by the application of the me thodologies of the optimal control. For t he co ns idered prob lem an optimal trac ki ng problem for the state variables can be formulated, wh i ch in this particular case coincides wit h a regulation prob l em . The refore , a general performan ce inde x i s considered hav ing the form: '0
v dsr =o.oLs i ds vqsr =aoLs i qs + (w+w ) 1>0 s being p the number of pole pairs.
+
In the fi gure, block Cl gives the desired values of t he s t ate var ia bl es and the referen ce values of the input variables of the motor accordinC] to eqs.(l), (2), (5) , (6) and (7) , (8) , (9) . Block C2 represents the system which gives the feedba ck control variables.
' + -..:
s [)
D
97
Block diagra m of the control system
-
= A~x +
N~ - s : x
+ B: u
(12 )
in which
-
I
A=A'( .) +N' sr
_
B= S;NX
T T' ; .:. u = ~v : ~~s I
with the foll ow in g constraints on t he control variables ~ :-: 1':"':
· s'::"':" Z
,' V1':".J v'::":" V2
( 13)
G. Figalli et al .
98
In this case, the system from which the control vector is obtained is coincident with the system described by eq.(lO). It may be shown that, if the performance index I( ~ u) is chosen so that:
- Q
,a,Tp
+
~x
= A~x
( 19)
B~v
+
In this case, the optimization problem must be slightely modified, since the performance index (11) becomes the following:
PA
where P is a symmetric positive definite matrix and R is a diagonal matrix given by: R
quency ws is directly given by the desired value of the motor torque TD , the model of the controlled system is given by:
o o
(20)
The solution of the optimization problem with performance inde x (20) and constraint (19) is given by the following equation: ~ v'
for for for
dt
in which Rl is a 2 x 2 positive semidefinite matrix and it is obtained from matrix R by eliminating its third row and third column.
then, performance inde x I( ~ u) is minimized by control ~ u' (Figalli, 1982), the components of which are : rl ~ Vl .::.-fd~Tl ~ V2 -f d'::' ~ V1r 1
( 14a)
-f d'::'~ V2r l
for for for
-f q'::' ~ V2r2
for for for
rl .::.-f w.::.r 3 ~ r2 -fw'::'~ S"l lr3 -fw'::'~ S"l2r 3
-fq,::, ~ Vlr2
r3
~
x
(21)
T
- -T
B Kl-K l A-A
Kl = Q
(22)
Case 4 - Quasi-linear model (14b)
(14c)
componen~of
vec-
T
(15 )
A different problem may the control ing to eqs.
formulation of the optimization be obtained supposing that only variable ~ w s is chosen accord(14) and (15)- i.e.:
, = - -1 ~ x T QN x (23) s r3 if quantity f'= ~x TQNx assumes a value belonging to range h S"l 1,r3S"l21; while ~w ~ is equal to S"l l or S"l2 respectively if -f~ lS lower than ~S"l 1·r3 or higher than ~S"l2·r3. ~w
In this case, in fact, the model of the controlled system is given by: (24)
Case 2 - Full linearized model A fully linearized model can be obtained from the bilinear one neglecting the term N· ~w s· ~ x. In this case, the model of the controlled system is given by: (16 )
-
where matrices A and B are those defined in the previous paragraph. The solution of the optimization problem for index (11) and constraint (16) is given by the following equation: ~u = - R-1S T K ~x (17) where K is the solution of the following algebraic Riccati equation: KSR-lgTK - KA - ATK = Q
T
B Kl
where Kl is the solution of the following algebraic Riccati equation: -1
~
Q bl ~xT Q b2 ~x T Q Nx
-1
= - Rl
Kl B Rl
r 2 ~ V1'::'- f q.::.r 2 ~ V2
where, fd,fg and f w are the tor f deflned as: ~x
Il( ~v)= J~+oo [~xTQ~X+6VTR1 ~vJ
(18)
Case 3 - Linearized model with constant slip - -.- - frequency Assuming that the value of angular slip fre-
The solution of the optimization problem for index (20) and constraint (24) is given by the following equation: • -1 T ~ v = - Rl B K2 ~ x (25) where matri x Rl is the same of the previous paragraph and matri x K2 is given by the following algebraic Riccati equation: -1 T T K2 B Rl B K2 + K2 Al + Al K2 = Q (26) In this case, the control vector by: ~u * =
I ~ v • T': ~w ' I T , s
~ u*
is given ( 27)
The model given by eq.(24) is obtained according to the following procedure: i) an optimal control problem is solved with performance index (20) assuming the value of ws equal to its rated value Wsr (Bellini,1981); ii)the control vector is obtained by adjoining to the elements of vector ~ *v a third control variable obtained an the basis of
Comparison Among Various Approaches
99
eqs.(14) and (15)(0). <1>,
PERFORMANCES OF THE PROPOSED SOLUTION APPROAC~ES The performances of the control system, obtained by the proposed approaches, have been analyzed by means of simulation taking into account a drive using as an actuator an induction motor characterized by the following rated values: 280 Kt~ Power Vo 1tage 2300 V Frequency 50 Hz Po 1e pa i rs 3 The parameters of the equivalent two-phase machine assume the following values: Cl
= 27.232; 8 = 17.697; Ls = 0.179H; a = 0.064
s
D
TDM
I
T s
M
o
T
Do
o
The mechanical load, referred to the motor axis, is characterized by the following values of friction F and inertia J F = 1.3 Nm ·s ec; J = 50 Nm.sec 2 The desired values of the stator current and flux have been assumed equal to 150 A and 7.3 V·sec.
T, I
0.1
Fig. 2. <1>,
T, I
Waveform of
<1>,
T and
I~(Case
1)
s
T
T
DM
The comparison among them is made by considering a torque transient which starts from a given torque (and, therefore, from the related values of the state and input variables) and which stops at the maximum value of the torque. During the transient, obviously, the motor angular speed is assumed constant. Assuming a value of the motor angular speed equal to 200 rdd/sec, the desired value of the motor torque has been varied from TDO= =83.4 N·m to TDW 2888 . N·m, consequently the modulus of the stator varies from 10=40.9 A to IW150A .
s
I
o
T
Do L-~~~-----r----~-----'r-----~--
o
Figs.2,3,4,5 show the shapes of the stator current modulus IS, of the motor torque T and of the stator flux modulus ~ obtained by applying the aforesaid solicitation to the four control systems. In each case the parameters appearing in the performance index have been adjusted in order to obtain a good tracking of the state variables without requiring high values of theco~ trol variables.
0.1
Fi~. ,
3.
Waveform of
T, I
<1>,
T and Is (Case 2)
s
T
T
DM
I
s M
In particular, the figures respectively nfer to the control systems denoted as "Bilinear model","Full linearized model","Linearized model with ws constant" and "Quasi-linear model" . I
(0) In this case eqs. (14) and (15), obviously,
play the role of arbitrary relations and are not the solutions of an optimization problem.
o
TOo
o Fig. 5.
Waveform of
<1>,
T and
s
0.1 (Case 4)
G. Figalli et al.
100
Kind of control
Tgg (s)
Torque overshoot (T = 2888) MD
Current overshoot (I = 150)
M
Neverthless, the performances of the modem16 bits microcomputer make it possible to tackle these kinds of problems. On the other hand, also the inverter operation imposes heavylimitations.
D
(%)
M
Case 1
.040
0.0
0.0
Case 2
.085
5.38
0.052
Case 3
.270
1. 33
0.0
6
Case 4
.081
2.95
0.015
2
The implementation of the whole control scheme can be subdivided in the same way in which the general problem has been subdivided.
3 12
Therefore, a first microcomputer can be used to implement the outer control loop and eventually the state observer. It outputs the reference values of the input v~riables (vr and wsr) and the desired value x of the state vector. So, the second microcomputer has only to ~rk out the computations required to implement block C2 of fig.l, which is the only one that is interested by the choice among the descri~ ed approaches.
TABLE 1 Comparative summary of the performances
The control system obtained by applying the "Bilinear model" is the fastest of the considered ones . However, to obtain this speed unrealistic sampling times are necessary. Therefore, it must be made slower and its performances become close to those of the other systems . As for the other approaches, it could be remarked how apparently they present the same computational complexity, since they require the solution of an algebraic Riccati equation However, taking into account also the closure of the outer feedback loop for the control of the mechanical variable w, the approach based on the "Full linearization" presents very high implementation difficulties. In fact, in this case the Riccati matrix must be computed in-line, since it depends not only on the value of the motor angular speed w, but also on the reference value of the slip angular frequency Wsr'
An analysis of the shapes of the considered quantities in the four cases shows that the control strateoies denoted as "Bilinear mode 1" and "Quas i: li near model" present a behaviour much more satisfactory than that of the remaining ones both as for the response speed and as for the absence of oscillations. Therefore, using as control variable also the slip angular frequency ws (determined on the basis of eq.(14c) or (23)), makes it possible to improve the performances as regards the tracking of the state variables. Table 1 shows the most significant parameters of the transient responses obtained by means of simulation, where T99 represents the time i nterva 1 (expressed in seconds) necessary to torque T to reach a value equal to 0.99 TDM and 6~ ~frepresents the maximum absolute value of the percentual deviation of the stator flux modulus around its desired value ~D .
On the contrary, for the other two approaches the Riccati matrix does not depend on the slip angular frequency. Only some elements of it, all having the same value, are linearly dependent on w (Bellini, 1981, 1982). This result is due to the particular structure both of the matrices of the dynamic model and of of the matrices chosen to charac-
All the control circuits that can be synthetized by the described approaches are very difficult to be implemented on a microcomputer, since a rather large number of operations is required and small sampling times a re a 11 owed. D
T
DM
T
s
o
T
Do
0.1
t (s)
~----~----'-----'-----'-----.-----r-----r---~.---~'-----
Fig. 4.
Waveform of
<1>,
T and
"
(Case 3)
Comparison Among Various Approaches
terize the performance index. As a Cunsequence, the varying elements can be out-line computed in correspondence with the minimum and maximum value of w. To evaluate their value during the operation, only an in-line product of a constant by w is needed. CONCLUSIONS The paper deals with the problem of the closed loop tracking for the state of the electromagnetic circuits of an induction motor. For the solution of this problem several approaches based on optimal control techniques are taken into consideration. The various approaches differ one from another for the choice of the model on the basis of which the control variables have been obtained . Simulation results have confirmed the validity of all the proposed approaches. They pr~ sent, however, significant differencies as for the difficulty of implementation on a microcomputer. Current studies are devoted to a detailed analysis of the implementation problems of
101
the proposed approaches. In addition, the authors are considering also similar aoproaches based on the utilization of discrete time models with the aim of simplifyina the implementation on a digital computation system. REFERENCES Bellini, A., Figalli, G. and Ulivi, G. (1981). A simplified approach for the in-line optimal control of induction motor drives. In Conf. Rec.-lEEE Trans. Ind. Appl., Philade1phla, pp.572-577. Bellini, A. , Fi']alli, G. and Ulivi, G. (1979). Realization of a bilinear observer of the induction machine. In Conf . Rec . -2nd Int. Conf. on Electrical Variable Speed Drives, London, pp.175-178. Figalli, G., La Cava, M. and Tomasi, L. (1982). An optimum feedback control for a bilinear model of induction motor drives . In Conf. Rec.-2nd lASTED Symposium on Modeling~ dentification and Control, Davos, pp.169173. Bellini, A., Figalli, G. and Ulivi, G. (1982). State feedback control of a.c. drives. In Conf. Rec.-ETG /GMR Conference, Darmstadt.