The paper deals with the nonlinear adaptive digital control of a synchronous motor using a nonlinear mod- elization in the (d,q) frame. It is shown that a mul-.
-
FP5 16:20 NONLINEAR MULTIRATEADAPTIVE CONTROL
OF A S Y N C H R O N O U S
MOTOR
G. Georgiout, A. Chelouaht, S. Monaco$ and D. Normand-Cyrott tLaboratoire des Signaux & Systgmes CNRS/ESE Plateau de Moulon 91192 Gif-sur-Yvette Cedex, France
SDipartimento di Informatica e Sistemistica Universiti di Roma “La Sapienza” via Eudossiana 18, 00184 Roma, Italy
Abstract
with decreasing gain, in order to guarantee the stability of the whole system and the asymptotic fullfilment of the control objectives. A related problem has been studied in [6] for the restrictive case of systems fully linearizable with the same number of inputs as states. The paper is organized as follows: in section 2 and on the basis of a nonlinear continuous time model of the synchronous motor in the ( d , q ) frame, a multirate control law is proposed. In the case of parametric uncertainty, the action of a certainty equivalence version of the digital control law enables to define a discrete time error equation on the basis of which a discrete time adaptation scheme is discussed in section 3. The stability of the overall system is studied and assured in terms of the asymptotic hyperstability theorem. Simulation results are reported in section 4, where we show that the control objectives of maintaining the current id to zero and assigning a constant reference value to the speed s1,are satisfied.
The paper deals with the nonlinear adaptive digital control of a synchronous motor using a nonlinear modelization in the ( d , q ) frame. It is shown that a multirate control strategy provides an appropriate framework in order to achieve speed regulation, ensuring the stability of the whole control system. When parametric uncertainties on the resistance of the stator windings and the load torque occur, this scheme is completed with an adaptation law deduced from hyperstability concepts. This results in the asymptotic satisfaction of the control objectives at the sampling instants. Simulation results are presented.
1
Introduction
The application of nonlinear control strategies in the electrotechnology domain has been directed during the last years particularly to the continuous time case, as in [7, 1 , 2, 41,while the digital implementation, which is crucial for applications, is not yet fully understood. On the other hand, the development of nonlinear adaptive control schemes for continuous time systems ([16], [9] and references therein) has made it possible to cope with parametric uncertainties in the case of motor models [9, 31. The objectives of the present paper is to show that an adaptive version of a multirate control strategy can be proposed for the speed control of a synchronous motor. Nonlinear multirate control strategies have been introduced in [14, 151 in order t o maintain, when digital implementation of continuous time control schemes is considered, input-output performances and stability properties. In the presenc,e of parametric uncertainties we suggest to combine such an approach with discrete time adaptive schemes based on hyperstability concepts [lo, 121. More precisely, a certainty equivalence version of the multirate control is put together with a discrete time least squares adaptation algorithm
2 2.1
The nonlinear continuous time control with perfect parameter knowledge
The state-space model which will be used in the sequel (called the ( d , q ) model), is deduced from the application of the Park transformation [13]. The nonlinear equations which describe the evolution of the electrical variables and the mechanical s eed, in the case of the synchronous motor with smoot! poles ( L d = L q ) , may be written in the following compact form & YI Y2
=
=
=
f(.)
111(.)
+
h(.)
QlUl(t)
+g z m ( t )
(2.1)
where tT = ( ~ 1 ~ 2 2 = ~ ~( i d3 , i)q , R ) E M an open subset of R3 is the measurable state, uT = (211, u2) = ( q , v q ) E R2the input andyT = (yl,y2) = ( i d , Q ) E R2the output. The analytic vector fields f,g1 and g2
‘This work was financially supported by the E.E.C., Italian A.S.I. and French C.N.R.S. grants.
CH3229-2/92/0000-3523$1 .OO 0 1992 IEEE
Multirate control of the synchronous motor
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on M , are given by
91
o)T*
=
92
=
(0.
l T gl 0)
(2.3)
where i, v , $2 denote the current, the voltage and the mechanical speed respectively, 0
@isj
0
p is the number of pairs of poles,
0
R, is the resistance of the stator windings,
0
Ld = L, are the inductances on the d - q axes,
0
J is the inertia,
0
the flux created by the permanent magnets,
of a continuous time control law making use of zeroorder-holder circuits, is not satisfactory with respect to both control performances and stability properties. This necessitates the design of digital controllers based on a sampled version of the continuous time process. In order to reproduce, at the sampling instants, the input-output behavior of a closed-loop continuous time system, it has been shown in [15] that multirate control techniques offer an efficient conceptual framework. This approach consists in sampling the inputs more rapidly than the state measurements, so introducing enough degrees of freedom on the controls in order to match, at the sampling instants, the control objectives. Specifying the method on the basis of the equation (2.1), we assume that in a sampled-data control scheme the state is measured at a rate 5 , while the inputs of the continuous time system are held constant over subintervals of 5 , according to
,:C is load torque,
0
u1 ( t )
=
for t E [k6, ( k
up
+ l)6[
(2.7)
f is the viscous friction coefficient
Easy computations show that the relative degrees associated with the outputs y1 and y2, are equal to = 1 (L,,hl # 0) and r2 = 2 (L,,hz = L,,hz = L,, Lf h z = 0 and L,, L j h z # 0) respectively. Following the classic input-output linearizing and decoupling procedures [a], one easily computes the feedback law
For the sake of convenience, we denote by X ’ and X 2 the Lie operators defined as follow x’= L f +U?&, + ‘ZlYLg, (i = 1 , 2 ) (2.9) Following [15], the corresponding multirate sampled model, is described by the equations ZD(k
+ 1 ) = F6(.D(k),.~,’.,”,’.,”))
=
e,3x1 ,BX*
(2.10)
(ld)lzD(k)
where e D ( k )E M , 8 = $, I d denotes the identity function, I denotes the identity operator and “0” denotes composition of functions and the exponential Lie operator is defined by the series expansion with the decoupling matrix A ( x ) given by
P
-(Lf
(2.5)
which enables to maintain the direct current y1 = id to y1, = 0, and to regulate the mechanical speed R to a constant value yzq = Q,. The coefficients ml,nizl and m22 are chosen in such a manner that s rial and s2 m21s nt22 are Hurwitz polynomials. It can be verified that after the coordinates transformation CT = y1, y2, $2), and the feedback action (2.4), the equations escribing the output error dynamics, are given by ; ~ ( t+ ) inlel(t) = 0
+
+
+
6
&(t)
+ m z i d z ( t ) + mzzez(t)
=
0
(2.6)
P!
Multirate control with perfect parameter knowledge
It is well known, in a linear context too, that a state feedback action and the sampling procedure do not commute. In particular, the digital implementation
(2.11)
The sampling period 5 €10, b o [ , with 6, small enough to assure the convergence of the series (2.11). Under piecewise constant inputs (2.7), (2.13) and for the same initializations, the state evolution of the continuous model 2.1) is matched, at the sampling instants t = k6, by t e state of the multirate sampled model (2.10). In order to ensure the control objectives (2.6) at the sampling instants, the discretization of these linear behaviors, using the classic sampling procedure, is needed. One obtains
‘h
where e i ( t ) = yi(t) - yi,, i = 1 , 2 represents the difference between the output yi(t) and its reference trajectory yiR.
2.2
+ 6 ( L f + u D L g )+ . . . + + u D L g ) O P+ 0 ( 6 ( P t 1 ) ) ] ( . )
ea(LItuDLg)(-) = [I
with \
\
/
Because of (2.10), the digital feedback law which achieves the exact reproduction of (2.12) should satisfy the following equality of series Y2(k
+ 1)
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e'x' e6x'
0
where an estimate e ( k ) is used at the place of 0 in the expressions of $, 6 and 1. The application of (2.22) and (2.23) leads to a discrete time error equation of the form
o e ' x 2 ( / i 1)Iz(k) e 6 X 2 ( / i z ) l z ( k ) - yz,
,ax1 ,ax, ( Lf
(2.14)
h)lz( I ; )
The existence of a soluticm to the equation (2.14) is a consequence of the implicit function theorem and the definition of the relative dqgree. Considering an expansion in powers of 6 of the exact solution, that is
{
up
1uy
2
D
u2
= =
=
upo +$upl + O ( P ) 11':;
2
+s
'Uy'
Cl+62 U20
uy,
+ O(62)
+ O(62)
+ 1) = GX(k) - WT(k)$(k)
x(k
(2.24)
with xT(I;) = [id(k),R ( k ) - R,, f i ( k ) ] , $(k) = 8(k) - 8, and the regressor W ( k )being a func,tion of measurable quantities a t the instant k . In (2.24), the remaining, higher order, 8 dependent terms appearing in (2.16), are not taken into account.
(2.15)
By substituting the multirate control thus defined in the righthand side of the relation (2.14) one obtains, after regrouping terms of the same power in 6
The multirate discrete time adaptive scheme
3
A discrete time adaptive control based on hyperstability concepts inspired from [lo, 111 is discussed hereafter.
The adaptive control law
3.1
A least squares adaptmationalgorithm initialized at 6 ( 0 ) , of the form e(k
+ 1) = J ( k ) + P ( k ) W ( k ) v ( k+ 1)
(3.1)
is used. The decreasing gain matrix is updated as follows P ( k + I)
(with P ( 0 ) lemma
P ( k ) - P ( k ) W ( k )[ R ( k ) + WT (k)P ( I;) w (I;)] - 1 w ( k )P( k )
=
'
> 0)
(3.2)
or equivalently, using the inversion
P - ' ( k + 1) = P - l ( k )
+ W(k)R-'(k)W'(k)
(3.3)
Choosing R ( k ) > 0 V k , and noting that P ( k ) remains always positive definite, we assure the invertibility of the matrices in (3.2), (3.3). Defining the a posteriori quantity g ( k ) as ?(k
+ 1) +
f#J(k 1 )
= =
+
(3,4)
>0
(3.5)
G i ( k ) - WT(k)d;(k 1) e^(k 1) - 8, j j ( 0 ) = x ( 0 )
+
one sets, according to [IO]
2.3
u(k
The error equation in the presence of parametric uncertainties
It has been shown in the above para raph, that in the case of a erfect parameter knowlecfge, the multirate feedback f12.15), (2,20), (2.21) can reproduce the desired control objectives up to any prefixed order of approximation in 6. When parametric uncertainties occur in R, and C,. (the load torque is typically an unknown parameter, assumed to be piecewise constant) , one has to consider an approximated certainty equivalence version of (2.15), (2.20), (2.2 I),namely
;
,
' i
8
2
,
S
1
!
I
.
:
, ,
I
I
*
\
02
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