Internal Model Control for Nonlinear Systems: Application to an Induction Motor

Copyright © IFAC 12th Triennial World Congress. Sydney. Australia. 1993

INTERNAL MODEL CONTROL FOR NONLINEAR SYSTEMS: APPLICA TION TO AN INDUCTION MOTOR 1 Jalme Alvarez·, Joaquln Alvarez·· and R.J. Herrera** *Seccion de ConJrol AUlomalico. Deplo . de Ingenieria EJeclrica. CINVESTAV. Apdo. PoslaI14-740. 07000 Mexico . D.F. Mexico **Deplo. de Eleclronica y Telecomunicaciones. Apdo. Posla12732. 22830 Ensenada. B.C . Mexico

Abstract. An internal model control (I MC) technique for multiple-input multiple-output nonlinear systems is proposed. The IMC structure provides a practical approach to the design of robust controllers. A practical procedure is proposed to quantify the robust stability of the closed-loop system. This procedure has been applied, in simulation. to an induction motor. Keywords: Nonline.ar systems . Iinearization techniques. robust control, motor controls.

l. INTRODUCTION.

the model and disturbance signals on the plant are present. The procedure is evaluated by an application to a nonlinear model of an induction motor.

A procedure often used to obtain simple but adequate models for control design purposes, is the linearization of the original nonlinear system about a suitable operating point. However, even when the plant is linear, the model is not free from being inaccurate. In consequence, the performance objectives of the control system are far from being obtained.

2. IMC STRUCTURE FOR LINEAR SYSTEMS . In an IMC scheme the process model is an explicit part of the controller (Fig. 1). P is the plant, P its nominal model, Q the controller and r, U, d, y, and y are the reference, control input, disturbance, plant and model output, respectively.

Another problem in process control is that disturbance signals are usually present in a physical plant. In the linear case, the Hoo approach gives a powerful procedure for robust design. Some preliminary results about an extension of this technique to nonlinear systems (NLS) were presented by Van der Schaft (1991). Some other control schemes are based on a heuristic approach (Richalet et ai, 1978; Cutler and Ramaker, 1979), but have suitably incorporated some modem robustness ideas (Morari and Zafrriou, 1989). The Internal Model Control (lMC) scheme presented by Morari and Zafrriou (1989) has given good results in several applications, and a mismatch between the plant and the model can be addressed if a filter is properly designed .

d

Q

u

p

p

y

Fig. l. Internal model control structure.

The design procedure for the linear !MC uses a frequency domain approach, while NLS are usually described by nonlinear state-space models. Despite this difficulty, several nonlinear techniques incorporating ideas from linear !MC schemes have been developed (Economou et ai, 1986; Calver and Arkun, 1988). Unfortunately, those techniques are not well suited extensions of the IMC approach to NLS (Henson and Seborg, 1991).

Definition 1. The sensitivity E is a function that relates the inputs rand d to the error e =y - r by the expression e

=(/ - PQ) (/ + (P -

p)Q r ' (d - r) =£(d - r) .

(1)

When P = P, the IMC is internally stable if and only if P and Q are stable. For the controller works well on the real plant it is necessary that the closed-loop system fulfills the specifications of robust stability and robust performance. A precise specification for robust stability requires having a description of the type of uncertainty associated to the process model. If a multiplicative output uncertainty is considered, the robust stability condition is specified by Morari and Zafrriou (1989):

In this paper a procedure to apply the IMC technique to a certain class of NLS is presented. It is shown that by employing this technique it is possible to maintain the control system performance within reasonable limits when uncertainties in the parameter values of I Work partially supported by the CONACYT. Mexico.

1063

(2)

The inverse model can be constructed by the method of Hirschorn (1979). However, in this work it is proposed to linearize flfst the model by standard feedback linearization techniques, then to design an IMC, following the procedure described in the previous section.

where 10 is a bound on the output uncertainty (maximum singular value of Le = (P - p)p-I ). Often what we is needed is robust performance. This specification can be given by 11

WfiWll1 ~ < 1,

'iP

E

I1

Modellinearization. The relative degree r/. is defined as the smallest integer so that at least one 01 the inputs

(3)

appears in Y? (the 'rth time derivative of y). That is

where I1 is the set of all plants with bounded uncertainties, W I and W 2 are performance weights (Morari and Zafrriou, 1989). A two-step procedure is proposed by Morari and Zafrriou (1989) to generate good engineering solutions to the robust control problem.

lj = L?ii .(x) + J

L.L'l'j- lii .ui

(7)

J

where qii stand for the a-th Lie derivative of ii with respect to 1. Let A(.) be the matrix

Step 1: Nominal performance. A controller Q is selected to yield a good system response for the inputs of interest, without regarding for constraints and model uncertainties. Several objective functions may be considered, for example min[(v I) + .. . + (v")],

_ [Li'L? - liil

L'm L? _Iiil ] ....

A (x) - . . . ' - 1-

where (v) = 11 Well ~= 11 W(I -PQ)vll~, {yi, i=I, ... ,n} is the set of input vectors applied to the system, and W is a weighting factor.

(8)

r-l_

L,IL/ h m

Q

L'm L/ h m

Then, when A(.) is nonsingular, the state feedback law

Step 2: Robust stability and performance. The controller Qobtained in step 1 is detuned to satisfy the robustness requirements. For that purpose Q is augmented by a filter F, which is often selected as Q = QF. Q = p-I fulfiUs the objective (5) for the class of systems considered in this paper.

u(x) = -A -I(X)

L, hi ;. ~ + A -I(X)V

[ "-]

(9)

L, mhm

yields the closed-loop linear system

l'=v

Regarding the filter design, it is recommended to have a diagonal structure with a diagonal element

i,

(10)

(i=I, ... ,m)

where {Vi' i=I, ...,m} is a set of new inputs (Isidori, define the transformations 1989). If we
(4)

J;(s)

i

i=I' /

J

J

11'1 + '2 + ... +rm + I' .. ·,11" as new coordinates in such a way to complete a diffeomorphism, then the normal form can be obtained. Let us denote J!._( I 'I I 11.'2 ",I 'mY d

where VI = mol + k; rI'Ioi is the largest multiplicity of the nominal plant pole at the origin, A is the tuning filter parameter, k is the number of unstable poles of P and V the pole-zero excess for F(s) .

'" - ,t)), ... , tPl ,q,2'" e, '+'2' · ··' 't'"p

•• • ,

4»m)

an

11=(11'I+ .. +rm+I, ... ,11nl SO that system (6) can be rewritten in the new set of coordinates as

~I =~2

3. IMC STRUCTIJRE FOR A CLASS OF NLS. The plant (P) considered in this work has the form

(11)

m

X = I(x) + i

I. gi(X)U i =I

(S.a)

~rl+l =~'1+2

(S.b) wherexE R",UiE R'Yi E R,andf,gi,hiareassumed smooth. The model (P) is assumed to have the form

~'I + +rm = bm(/;, 11) + Am(~' 11)U

m

:i = j(:i) + I. i/i)ui i

=1

Tt = l(~, 11) + p(~, 11)u

(6.a)

T

where u=(ul, ... ,UJ , M~, 11) is L?iil and Ai(~' 11) is the i-th row of A (x), both in the new set of coordinates (~,11). The outputs are

(6.b)

where x ER", Ui E R, Yi ER and 1, ii and iii are assumed smooth. If the model is perfect, Q can be easily obtained as the right inverse of the model (Henson and Seborg, 1991): Q = p- I =p-I . This controller can achieve perfect control, that is y(t)=r(t), for all t>0.

YI=/;I'

If

zero

Y2=/;,,+I"'"

is

an

iil(O) = ... = iim(O) = 0,

Ym=~'I+ ... +rm _I+1 eqUilibrium of then the

(12)

(6) and subspace

{~= 0, 11 E R" -',- -'m} is rendered invariant by the control

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(13)

Modellinearization. The relative degree of the model is equal to ('I , ' 2} = (2, 2} if x 2 or X3 are not equal to zero. Because + < n, a linear subsystem of fourth order can be obtained when the feedback given by equation (9) is applied. By imposing an additional feedback to the linear subsystem is possible to arrive to the transfer matrix

'I '2

The zero dynamics of (6) are defined to be the dynamics of Tl, which is described by (Isidori, 1989) ~

=1(0, Tl) - p(O, l1)A -1(0, l1)b(O, 11) =q(O, 11)

(14)

The change of coordinates and the feedback control law (9) yield a system whose transfer matrix is given

(17)

by P(s) = diag{ 11s ''} ;i = I, ... , m} . This transfer matrix corresponds to a decoupled MIMO linear system . It is possible to impose a new feedback law to assign a specific set of poles to the linearized system. Then, P(s) can be transformed to the following expression 1 P(S)=dia g{ ,

;i=I , ...

, _1

+ ", + Ci. IS+C i. O

s ' +Ci.,, _ IS'

Analysis of the zero dynamics. Let us define the following change of coordinates: ~ I = fil(x), ~2 =Ljil(x),

,m}

(18) which is a pure integrator. However, 11 is bounded between zero and 21t. Moreover, the states x2 and x3 , which define 11, evolve in such a way that x/ + x/ = constant, so they are bounded. Therefore, the zero dynamics are stable.

4. APPLICATION TO AN A.C. MOTOR.

Model. The model presented here is based on the two-phase equivalent induction motor representation (Marino and Valighi, 1991) .

[MC design. Following the procedure described in section 2 a causal IMC controller can be obtained: Q(s)

npA/ . . TL = JL (
7

,

~=Lji2(X),

By applying the control law (9), the zero dynamics of the model (5) will be

Now the design of the IMC can be made by following the procedure described in section 2 (for the case P = P) . If '1+'2+ ' ''+'m
0)

~3 =fi ix),

11 = tan-I(x/ x 2) mod 21t.

= Q(s)F(s) =

. {S 2+ = dlag

CI S +co s 2+dl s + do}

(}"IS + 1)2

(15)

(19)

,--~--=

(Azs + 1)2

Analysis of the robust stability. It is not possible to show analytically that condition given by Eq. (2) is verified, because it is not possible to know a bound on 10 , Instead, a numerical approximation for 11 PQIdl ~ is proposed. (i) Determine the uncertain parameters of the plant

and an estimate of the minimal and maximal bounds on them. (ii) In the cube defined in the parameter space, choose

where 0) is the angular speed; i,
,

a number of points {SJ equally spaced such that they cover the cube. (iii) For each Sj substituted in the model, obtain an input-output response for the "perturbed plant" linearized by state feedback, putting as input an appropriate signal for identification, and obtain a linear model P. (iv) For each linear model P, obtain the Nyquist plot of PQ . This will give an estimate of the family of perturbed plants.

crL,

The subscripts' and s will be dropped since only rotor fluxes and stator currents are used. The state and input vectors are chosen as x=(O),
(v) Check that condition (2) is verified for the family of perturbed plants (see below).

= (u a , ubl = (u l , U~T. The outputs to be controlled are the rotor speed and some quantity related to the torque. If the objective is to regulate the system around a constant reference, then it is advantageous to define as outputs the following expressions • - 2 . ,2 • (16) YI=XI-Yl' Y2=X 2+ .... 3 -Y2

It can be shown (Morari and Zafiriou, 1989) that the maximum value of the magnitude of the difference between each PQ(jO) and PQ(jO) obtained at each frequency is the radius of a disk centered at PQ(jO) . The disk includes all functions PQ's of the family of perturbed plants. Its radius is equal to IPQII o , so the maximum value of this radius for all frequencies gives an approximation of 11 PQIo lI ~. For our example we

U

1065

_ ._ - - - - - -- _ . _ - - - - - - ------,

have obtained 11 PQloL '" 0.573, so the controller Q we have designed satisfies the condition given by (2). Figure 2 shows the Nyquist plots obtained for the induction motor.

OUTP UT RESPONSE

>40 '00

!

Simulation. The parameter values have been set to R,=0.18 .0, L,=0 .0699 H, R,=0.15 .0, L,=0.0699 H, M=0.068 H, J=0.0586 Kgm 2, np= 1, TL=50 N-m. Figure 3 shows the controlled outputs when they evolve from the initial conditions YI (O) = .v1(0) = 0, YiO) = ylO) = 1, to the required reference values

t--~--~============~

200 220

~ 180

"';; 140

;::

s:: 100 00 20

...

-2g. 0=0--0~.':-:-0-"-:0:-:.'::0 "-:0:-':: . :-" 0 --= 07.60:-"::0."::' 70::-:0.'~0~1.0:-::0-,:-".2:-::0 --:,:-".,-=o -1.'5'0 tlm. Cu e)

= 320[1 - e- (l + lOt)] rad/sec, and 20 I Y2d = 2[1 - e- (l + 20t)] (V.S)2. A disturbance of 20% IOI

Yi d

(a)

around the nominal value of the load torque has been introduced, also a parameter uncertainty of 25 % around the rotor resistance nominal value. It has been observed from the simulation results that the internal dynamics remains stable when the IMC structure is tested. The coefficients in the linearizing control laws (ci , di) have been chosen such that the linear subsystems obtained have their poles at -100, AI and Az were equal to 0.0l. Condition (2) for robust stability is always satisfied for all 10such that 111011 00 :5: l. Moreover, in a large frequency range but in a small band, the magnitude of the sensitivity function E was less than one.

OUTPUT RESPONSE

2 .00

r~~--=============l

1. 1:»

., .50

11.20 ~, . oo

~O.7~ ~

0 .50 0 .25 0 .6

0 .8 tlm . ( •• c)

1 .0

1. 2

1 .~

(b)

Fig . 3 Output response (l') and reference (l'iJ: Output I (a), Output 2 (b). 6. REFERENCES

NYQU IST P LOT OF PO l l Ow}

Calvet, 1. and Y. Arkun. (1988) . Feedforward and feedback linearization of nonlinear systems and its implementation using internal model control (!MC). lnd. Eng. Chem. Res. 27, 1822.

-0 .1 -0 . 2

~ -O.3

~ -0.'

Cutler, C . R. and B. L. Ramaker. (1979). Dynamic matrix control: a computer control algorithm. AlChE Natiorwl Mtg. Houston, TX.

- O.!I

-0 . 6 -0.7 L..-~~~_~_~_~~~_"---' 0 .0 0 .1 0 .2 0 . .3 0 .4 c.!) 0 .8 0 .7 0 .8 0 .9 1 .0 R ea l pal ' Ow}

Economou, C. G., M. Morari, and B. O . Palsson. (1986) . Internal model control. 5. Extension to nonlinear systems. lnd. Eng. Chem. Proc. Des. And Dev. 25,403.

(a)

Henson, M. A. and D. E. Seborg. (1991). An internal model control strategy for nonlinear systems. AIChE 1. , 37, 1065.

NYQUIST FlL OT Of F'022(jw) oo~~_~

- 0 .1

- 0 .2 . - 0.3

""~ - 0.' .§

-o . ~

- 0 .6 - 0 .7

______

~

__

~

__

~-,~

{(~~;;;;~~~

Hirschorn, R. M. (1979). Invertibility of nonlinear control systems. SIAM 1. Contr. Optimiz. 17,289 . Isidori, A. (1989) . Nonlinear control systems: an introduction. 2nd. Ed. Springer-Verlag, NY.

-0.80.'-=-0- :0:"-.':---=0"':'. 2--='0."'-,'-:0"".'--'-::' 0.""0 - :0:"-.•=---=0"':'. ] --=' 0."'. '-:0="=.'~ 1.""0 ....J

Marino, R. and P. VaJighi. (1991). Nonlinear control of induction motors: a simulation study. Proc. ECC9I. European Control COnference. Grenoble, France.

Real P022(jw)

(b)

Fig. 2. Nyquist plots of components ii of PQUro) , for different plants P: PQII (a), PQ22 (b).

Morari, M . and E. Zaflriou. (1989) . Robust process control. Prentice-Hall, Englewood Cliffs, NJ.

5. CONCLUSIONS.

Richalet, J. A. , A. Rault, J . L. Testud, and J. Papon. (1978). Model predictive heuristic control: applications lo an industrial process. Automatica, 14, 4l3-428.

The control scheme proposed here is an extension of the IMC strategy for linear systems to a class ofMIMO minimum phase NLS . The proposed procedure for quantifying the robust stability of the closed-loop system ha,> been used for evaluating the performance of the IMC scheme when it is applied in simulation to an induction motor. The results obtained in simulation show that the !MC strategy provides a robust performance when disturbances and parameter uncertainties are present in the nonlinear plant analyzed.

Van der Schaft, A. J. (1991). A state - space approach to nonlinear Hoo control. Systems and Control Letters, 16, 1-8.

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