Nonlinear Predictive Control by Inversion

Copyright © IFAC Nonlinear Control Systems Design, Capri, Italy 1989

NONLINEAR PREDICTIVE CONTROL BY INVERSION S. Abu el Ata-Doss* and M. Fliess** *ADERSA, 7 bd. du lHareehal Juin , F-91371 Verrieres- le-BuissulI Cedex , Franee **Laboratoire des Signal/x et S),stemes, CNRS-ESE , Platea u du Mouloll, 91192 Gif-sur-h'ette Cedex, France

Abstract. A psedictive control technique using system inversion is presented in this paper. It is a natural extension to the existing predictive control techniques but while these techniques are essentially developed for linear models, the new technique can be applied to nonlinear models as well. In this last case, the proposed technique does not require any type of local or global linearizing transformations. The most essential feature of the new technique is that it advantageously profits of both the maturity of the predictive control approach and the recent progress of the theory of system inversion particularly in the nonlinear case. Simulation results are given to illustrate the proposed technique performances. Keywords. Predictive control; system inversion; nonlinear control.

INTRODUCTIOO

'IHE TECHNIQUE DESCRIPTIOO

The control of nonlinear systems is a very large

We consider for Simplicity the single input single output case. The real-time control problem can be set as follows : at each instant, the process output is knOIf.ll'1
ar£i rich field but i t is also a very difficult one (Isidori, 1985). This explains the fact that many

research groups have concentrated their efforts on this subject from already many years ago. Fortunately, a lot of work has been done, many problems have been lightened and some tools exist. But a consirerable anount of investigation has still to be oone and, what is extremely if1l)ortant, particular attention has to be oriented towards reveloping tools of practical applicability to concrete situations.

Set point

A new predictive control techniQJe is introduced

in this paper. Although original, the approach is natural 'ehra, 1982 ; Bruijn and Verbruggen, 1984; Ydstie, 1984 ; De Keyser and Van Cauwenberghe, 1985 ; Clarke and co-workers, 1987 ; ~t:Dennott, 1987) and the theory of system inversion which has recently known a real progress particularly in the nonlinear case (Silvenran, 1969 ; Hirschorn, 1979 a, 1979 b ; Singh, 1981, 1982 ; Nijmeijer, 1982 ; Tsinias and Kalouptsidis, 1983 ; Fliess, 1986, 1989 ; !>bog, 1987). The proposed techniQ..le makes direct use of the nonlinear dynamic model of the process without need to any local or global linearizing transformations.

..... /.".,.,.~ ....··/~ferenoe

Process output .... /

trajectory

Centrol variable

Past <--- Present ---> Future Fig. 1

The solution for this control problem wruld then consists to detennine a future sequence of the centrol variable to be applied at the process input in orrer to obtain the reference trajectory as output.

The organisation of the paper is as follows : the techniQJe description is first given, then sane remarks en system inversien are presented and finally simulation examples are treated to illustrate the applicability of the new approach.

Thus fornulated, at each instant, the centrol problem is a left inversion problem : we have a system with initial cenditions, given by the process state at the considered instant an:! we

Ilg

120

S. Abu el Ala-Doss and M. Fliess

went to detennine the control variable sequence which yields a desired future output trajectory defined by the reference trajectory. '!he theory of system inversion Cal thus provide a

sol ution to the above prescribed problem. Of course, left invertibility of the process roodel has to be guaranteed. Also, the reference trajectory has to satisfy some compat ibili ty condi tions wi th respec t to the initial condi tions (Hirschorn, 1979). Now, following the predictive control principles only the first value of the determined future control sequence will be actually applied to the process, the whole procedure being repeated at the next control sample instant and so on. Consequently, the above stated inversion problem will not be completely solved on the whole prediction horizon, only the first value of the control variable has to be calculated at each sillfl)le instant.

Computing control variable by inversion makes the difference between the proposed technique and the other known predictive control techniques in which the control calculation is done throJgh : - a structuration of the future control variable the most general case considers a linear combination of a set of pre-specified functions (Richalet and co-workers, 1987, 1988). - the minimisation of some I di stance I between the reference trajectory and the output predicted through the roodel by using the above mentioned structuration. Al though applicable

to both linear and nonlinear systems, the predictive control technique by inversion is roore interesting for the nonlinear case : in fact, in the linear case, it just adds a new member to the family of the existing predictive control techniques but in the nonlinear case, it introduces a new approach which extends the predictive control principles to nonlinear systellE. What is attractive in the new technique is that it can profit of all existing predictive control tools developed in the various techniques. For example, in the new technique, we have incorporated the self- compensat ion princ iple (Richalet and co-workers, 1988) which allows to take into account, the difference between the process and the roodel outputs; this difference can be due to a mismatch between the process and the model and/or to perturbations. The selfc anpens at ion procedure ccnsists to extrapol ate this difference in the future using observed past values CI"ld to correct accordingly the set point trajectory for the model. In fact, in these condi tions, the reference trajectory has to be initialised on the model output since the inversion concerns the model. When no self-corrpensation is considered, at each instant, the set point trajectory is corrected assuning the difference remains constant in the future.

sew:

REMARKS CN INVERSICN

Vvhile the inversicn of linear systems (Silverma1, 1969) and nonlinear single input - single output systems (Hirschorn, 1979) has been well understood from a long time ago, it is only recently, with Differential Algebra, that the inversion of nonlinear rrultivariable systellE has been apprehended (Fliess 1986, 1989). To situate the frare in this last case, we surnnarize it by the following, let : u

Urn)

y

Up)

be differential quantities that design respectively the input and output vectors. Let 1R (resp. 1R, 1R
the extension of differential fields 1R!1R(u) is differentially algebraic, Le. each component of y satisfies a differential equation with coefficients in 1R. Left inversion of the system is possible if the extension R/IR is differentially algebraic which is a CCJlTq)act way to set the existence of the inverse system. This is verified if the differential output rank, defined by the differential transcendence degree of the extent ion 1R/lR , is equal to the rumber m of the input corrponents. Recall that this rank, which is the generalisation in the nonlinear case, of the transfer matrix rank, in the linear case, can be computed by an algorithm (Grizzle and co-workers, 1987).

The two first examples treated in the next paragraph are limited to systems for which the inverse has no dynamics. In the field theory language, this means that 1R = 1R Le. every component of u can be given as a rational expression in function of the corrponents of y and their derivatives. This simple but irrportant case, has been approached in (Isidori and co-workers, 1986) through a different formalism. The third example gives an inversicn with dynamics. E)W-lPLES

The sirrulation results of the applicaticn of the predicti ve control technique by inversion are given here for three examples. The first and third exanples are academic ones ald the second example concerns a particular aircraft guidance problem. Example 1 : a Single input - single output nonlinear case. ""e consider the system

This self-canpensation principle increases the robustness of the designed control. Nevertheless, the procedure used actually has been developed for linear condi tions (a polynomial is used for the extrapolation) ; it must be analysed for suitable extension to nonlinear conditions. Concerning the irrplementation, it is clear that the proposed technique is of great sirrplici ty : few on-line calculations are required and tuning of the characteristics of the reference trajectories is made in relation to the desired closed-loop process behaviour.

with: (2)

and the initial conditions xl(O)

= X2(O) = 0 (3)

u(O) = 0

Nonlinear Predictive Control by Inversion To apply the technique, we have to verifY the invertibility of system (1). Differentiating (2), we get:

Exanple 2 : a rrultivariable bilinear case. The considered aircraft guidance problem, is presented in (Abu el Ata-l):)ss and co-workers, 1989). It is forrrulated by the following system:

i.e. (4)

which shows that the system is left invertible on the whole space.

For this example, the compatibility conditions to be satisfied by the reference trajectory are continui ty conditions on y a-!d Y at each instant. The reference trajectory is of the second order type, it is defined by the equation:

= x2 + u1 x3

x2

=

u1 x4 - u2 (6)

- u1 Xl + x4 u1(x2 + xS)

x4

Xs = u2 and

Yi

= Xi

i

= 1,

.. ,

S

(7)

wi th specific initial conditions on x and u. The objective is expressed by a constant set point equal to 0 for Y1 and Y2' This objective induces the following set points for :

(s) where ER denotes the difference between point and reference trajectory on the horizon. With this type of trajectory, the loop time response and the overshooting specified.

Xl x3

y in (4) will be replaced by the corresponding reference trajectory, then u is calculated from (1) a-!d (4).

121

the set future closedcan be

Y3

a ramp

Y4

a constant

Ys

a constant

slope of this ramp

Some constraints have to be respected on The results illustrated in Fig.2 below correspond to set point trajectory : c

1 + t 2,

=

sirrulation sClq)ling period period = 0.01 sec.,

- the closed-loop titre response

control action

reference trajectory parameters : t; = 0.9 and w = 10 giving a closed-loop time response equal to O.S sec. and negligeable overshooting.

---,------,----,-,--....,----,,----r---, l .

. - . - . - ..... . - . - . -. ...... -.-.- . - j -.-.- . - . ~ . -.- . -

•.

· -·-· -· t·- ·-· -·~- -· -- ---:- · -·---- ;--- · -

,

,

..... -.-.-.. . . -.-._.

-

,

c

._. y

f--......;.~=·-~·-,~'-=-:::-:::-:-:-:;:r- · - · - · - - ~· -- - -. ~ . - . - - -.~ . - . - . -. -: .~

i-

!

r

... !

..

, !

J

.. !

·-·-·-· ... .....-._._... . . _._.-·-1 _._ ._ .'!' . _ __ . _ . .l. _ __ . _ _ !_. _ __ ._ .L ._ ._ ._ ."!" ._ ._ ._ . .J _ __ __ __ ! ·-·- · - · ~ · -- . - · -I- · - - - . - · ~ . - . - . -

.e ,'

I

....

I

I

I

I

I

- the length of the control sampling period. Concerning the invertibility of system (6), it is easily verified that the differential output rank is equal to two, in fact, at most two from the five output components are independent. This number being equal to that of the input components, the system is left invertible. The set point trajectories being specified on Y1 and Y2. it is necessary to define reference trajectories corresponding to these two output co~nents. The control variables will thus be expressed in function of Y1' Y2 and their derivatives. The calculations give:

I

_._._.~

u

:~~:::i':==~:::~i~:::·i=:i·:::;~: :~i' :::~ : =· =: =: r-. - . =:~. - · - · - · -:-

...

- the variation of Xs with respect to its initial value

-._._ -'-'1"-'---, '-'-'-'-,-'- '- -r -- '- '- ''''- - _..., - - - -, ....... _._._._1_._ ._ ._.1-._ ._ ._ ..... _._._.-1._.- _._1

.: :t--:±:::;_·_- -!:==:~::::~:::~:::::! ~

the Clq)li tude of u1 and u2

-_._.:._._.-: . -. - . - . ~ _._._.-: Fig. 2

It is shown that the set point trajectory is tracked without error ( e: = c-y tends to 0) and that the closed-loop time response specification is respected.

(8)

For this example, only the first derivative of the output is necessary for inversion. Thus no compatibility condition is needed concerning y and the two reference trajectories are bcth chosen of the first order type i.e. satisfying each an e~ation of the type (9)

where the choice of (l Ca-! specifY the closed-loop time response 1( Cl 'V 1 /3).

122

S. Abu el Ala-Doss and M. F1iess

Figures 3 and 4 give the results of application of the control technique by inversion to the considered example with: simulation sampling period - control sampling period

=

= 0.1

sec.

~~~~l=:-t-

1 sec.

tine responses for the two referen:::e trajectories 10 sec.

~.

. ,::: : . .

I

/..

I~.

l~.

I

I

I

Figure 4 shows the five output variables (or state variables) all objectives are attained, specifications realized and constraints respected. In order to test the robustness of the control technique, a mismatch is considered between the IIDcie 1 given by (6) and the process which we assume to be represented by the system : Xl

x2 + u1 x3

x2

0.9 u1 x4 - u2

x3

- u1 Xl + x4

.

x4 x5

U1(x2 + x5)

)~ .

)'. .

I

I

I

J ____ i ____ L ___ 1. ___ ...! ____ ! I

I

I

::~~ 1

i

I

I

I

I

I

- i-- -= +===~ ====; ====~===+ ===~ ====:

': ~-- --[----:f --1--~:- -- -j Fig. 4

(10)

--+ ---;

- ----·T'- ·- ·- -., - '- '- -- l- '- '- '- -r-- ' - '- ' T'- ' - - --, _.- - -! I

Figure 3 illustrates the control variables and the errors El and E2 between the output cooponents Y1 and Y2 and their respective set points. I t is clear that no asymptotic error is obtained.

T.. .

w.

;

,J-~:IL.......;

: 1--.

Adapted case

i

~.~ -----_ ' ,

-i

1.2 u2

In the absence of self-canpensation procedure which considers an extrapolation of the difference between the process and model outputs, the technique application yields a constant tracking error for the output Y1 as shown in Fig. 5 and 6, where e1 and e2 denote the difference between the process and model outputs for Y1 and Y2 respectively. It is to be noted that e1 tends to a rarp and e2 to a constant. IntrodUCing the self-compensation procedure leads to the suppression of the tracking error as can be seen in Fig. 7 and 8. Fig. 5 Non-adapted case (without self-compensation)

--~~-'--~--~-- r ,

I

I

- - -., - . - . - -,- ' - -- -- r- -- ' - ' - ' T - _ . _ . ., I

,

I

I

I

L-=::::::t:::=::::i===i.===,*,=.==;;===~==:!

_ ._ .- .

._ .- - ""j '- -- ' - ' -j- - - - ,- - - - -, _ ._ ._ --; _ ._ . _ .-

Fig. 3

. nu .

t

Adapted case

I

Fig. 6 Non-adapted case (without self-compensation)

Nonlinear Predictive Control by Inversion

123

The reference trajectories are chosen as follows : for Y1 : a first order type with compatibility condi tion on Y1 for Y2 : a 2nd order. type with compatibility condi tions on Y2 cnd Y2 The resul ts shown in Fi g. 9 correspond to : initial conditions: Xl

5

constant set points

6 for Y1 2.5 for Y2

Cl

c2

x2 = 2

o

silllllation sampling period = control action period : 0.05 sec. time responses for the two reference trajectories: 1.5 sec. Fig. 7 Ncn-adapted case (with self-compensation)

,

'_'_'-:'-:':r-'-'-'--'---- '- '- ';-- - -- -- ---r--- '-'-'- r- '-'-'--

t...

."

i

i

i

i

i

· -.~ · - · -f-· - · -· -·4· - · -· -·-:+-·-- -·- · - ·i- · - · -· - · ~-·- · -·- ·

'

! i ! i l.~,L . ~_ ___.JL-_ _..L_ _-,l-_ ____.J_ _ _-!-__--!

.,.

•.• ;---:.;----",=-...:;.-----.::;---7---....:;------'1

.

'-r- '

l'

__ ___ ...!._._. _ ._ I_._._ ._ .L._ __ ._.1. ._ ._ ._ ....!._ . __ ._1 I ! I I ! ! ~

~ .

''' .

I' .

U .

/' .

'. .

~--~----+----~----~---~---. i i I i i ! . j

l~.

: : : .f~====i====~===f-==~~===i •

L

,\.

lJ

.. ~

.

h' .

!

, . ~I:..-----.J---..L---L-----.J---,,L---..J

··'r---~IL---~i----'~,--~'L---71--~1

~: ~~:=:=:=:~:=:=:=.=:t·=:=.=.=·~: =:=·=:~=:=-=:=:J

)S.

....

·r·--_;..L--L-·T·-----.::'~·--_;··L----"T·--_1

I

i

i

I

i

r

:::~:=:=:=l=:=:=:=:~:=:=:==: t.:==:=:=~:==:=:~=:=:=:=: ·· ·F~·-·-·-1·-·-·-·-·i----r----r---·....

i

--,·----,--·--I----·r--·- ·r --·--,·-·--I

_ . __ .l . _ __ __ ....l,_ __ .__ I_ .___ L_._ ._ ."!'._.__ ..J. ____ ._! J ! ! ! I I I i-'---~;':'--~'---":';':'i--:r----'-;-'---",.'----'" . I I I I ! I I

s.

+---+--I----r---+---1---1

Fig. 9 No tracking error is observed and the specifica-

Fig. 8 Nco-adapted case (with self-compensation)

tions are respected.

ExcrIJ?le 3 : a Illlltivariable bilinear case (inverse wi th clynarnic s) • This example is taken from (M::>og, system is given by : xl

xl u1

x2

x3 - x3 u1

x3

u2

1987).

The

(11)

with i

Yi = xi

=

1, 2

A new predictive control technique has been introduced in this paper. It results from matching together the predictive control approach and the theory of system inversion. In the linear case, the proposed techniq..Ie ackls a member to the family of existing predi c tive control techniques while in the nonlinear case, i t introduces a new control approach which is expected to be applicable to a large class of nonlinear systems.

(12)

This system is left invertible and we get

(13) 1

with singularities (14) 1

crnCWSIrn

System inversion has allowed the extension of the predictive control principles to nonlinear models. Thus the recent progress of the theory wi th Differential Algebra will be certainly very profitable for the new technique. en the other side, the maturity and professionalism of the available predictive control tools will help for the adaptabili ty of the techniQ..le to real istic si tuations. The most attractive feature in the predictive control by inversion is the great sifll)licity of the technique ~lerrentation no matter the rodel is linear or nonlinear : few on-line calculations and easy tuning.

S. Abu el Ata-Doss a nd M. Fliess

124

Predictive control by inversion, particularly in the nonlinear case, is promising. Its applicability has been shown through the examples treated. In the near future, the effort will be oriented to establish the conditions and the class of nonlinear systems for which the technique is most efficient.

ACKNCMLEIXJEMENTS This work has been partly financed by DRET c ontract nO 87/ 1226.

REFERENCES

Abu el Ata-Doss, S ., A. corc, and S. Chavy (1989). Commande predictive non-lineaire par inversion. Application ~ un probleme de guidage d'avion. COlloque S.M.A.I. "L'Automatique pour l'Aeronautique et 1 'Espace" , Paris, March. Bruijn, P., and H.B. Verbruggen (1984). Model Algori thmic Contro l using impulse response models. Jrumal A, vo!. 25, nO 2. Clarke, D.W., C. lbhtadi, and P.S. Tuffs (1987). Generalised Predic tive Control. Part 1 : The basic algorithm. Part II : Extensions and interpretations. Automatic a, vol. 23, nO 2. Cutler, C.R., and B.C. Ramaker ( 1980). Dynamic Matrix Control. A computer control algori thm. YACC, San Francisco. De Keyser, R.M.C., and A. Van Cauwenberghe (1985). Extended Predictive Se If-Adaptive Control. Proc. of 7th IFAC snnsium on Iden. & Syst. Param. Estim., York U.K.). Fliess, M. (1986). A note on the invertibility of nonlinear input-rutput differential systems. Syst. Contr. Lett. 8, 2. Fliess, M. (1989). Automatique et corps differentiels. Forum Math. 1, Paris. Grizzle, J .W., M.D. Di Benedetto, and C.H. /bog (1987). Corrputing the differential rutput rank of a nonlinear system. Proc. 26th CDC, Los Angeles. Hi rschorn, R. M. (1979 a). Invertibi lity of lll.lltivariable nonlinear control systems. IEEE Trans. Aut. Control, AC-24. -Hirschorn, R.M. (1979 b). Invertibility of nonlinear control systems. SIAM J. Control & Opt., vo!. 17. Isidori, A. ( 1985). Nonlinear control systems : an introduction. Lecture Notes in Control and Information Sciences nO 72. Springer Verlag. Isidori, A., C.H. /bog, and A. De L\.ca (1986). A sufficient condition for full linearization via dynanic state feedback. Proc. 25th CDC, Athens. M:Dennott, P.£' (1987). Adaptive rrultivariable optimal predictive control. Int. J. of Adapt. Contr. and Sig. Proc., vol. 1. Moog, C.H. (1987) Inversion, decouplage, poursuite de modele des systemes non-lineaires. TheSis, University of Nantes (France). --Nijrreijer, H. (1982). Invertibility of affine nonlinear control systems : a geometric approach. Syst. Contr. Lett. 2, 3. Richalet, J., A. Rault, J.L. Testud, and J. Papon (1978). M:xiel Predictive Heuristic Control : applications to industrial processes. Automatica, vol.14. Richalet, J., S. Abu el Ata-DosS, and C. Arber (1987). Predictive Functional Control. Application to fast and accurate robcts. 10th IFAC World Congress, M..nich.

Richalet, J., S. Abu el Ata-Doss, J.L. Estival, and M. Abi Karam (1988). Model Based Predicti ve Control. Part I : A qualitative description. Part II : Predictive Functional Control, a particular algorithm. Submitted to Automa.tica. Rouhani, H., R.K. Mehra (1982). M:xiel Algorithmic Control. Basic theoretical properties. Automatic a, Vol. 18. Silverman, L.M. (1969). Inversion of rrultivariable linear systems. IEEE Trans. Aut. Cont., AC-14. Singh, S.N. (1981). A modified algorithm for invertibility in nonlinear systems. IEEE !rang. Aut. Cont., AC-26. -Singh, S.N. (1982). Invertibility of observable rTllltivariable nonlinear systems. IEEE Trans. Aut. Cont., AC-27. Tsinias, J., N. Kalouptsidis (1983). Invertibility of nonlinear systems. IEEE !rans. Aut. Contr. , AC-28. Ydst-re,-B. (1984). Extended Horizon Adaptive Control. 9th IFAC World Congress, Budapest.