Nonlinear Dynamic Control of a Non-Minimum-Phase CSTR

Copyright © IFAC NonIinear Control Systems Design, Tahoe City, California, USA, 1995

Nonlinear Dynamic Control of a Non-Minimum-Phase CSTR F. Svaricek* "Department of Measurement and Contro l, (Prof. Dr.-Ing . H . S chwarz) , Faculty of Mechanical Engineering, University of Duisburg, D-J,70J,8 Duisburg , Germany , sv@risc .uni-duisburg.de

Abstract. In this paper a nonlinear control scheme for a non-mini mum-phase model of a continuous stirred tank reactor (CSTR) is developed . The design is based on a linear algebraic framework for the analysis of nonlinear systems which is largely inspired by the differential algebraic approach of Fliess and makes use of an approximate input-output decoupling. Simulation results illustrate the advantages of the proposed method. Key Words. Nonlinear systems analysis , zeros at infinity, reactor control, input-output decoupling.

1. INTRODUCTION

2. NOTATION AND PRELIMINARIES Consider a nonlinear control system

Recently a detailed model of a continuous stirred tank reactor (CSTR) has been presented by Engell and Klatt (1993). Such reactors are central components of many plants in the chemical and biochemical industry and exhibit highly nonlinear dynamics, especially if consecutive and side reactions are present.

:i;

y

= =

L

of the form

!(x)+g(x)u

(1)

hex)

(2)

where x(t) ERn , u(t) E R17l, yet) E RP, and !(.) , the columns of g( -) and the rows of h(·) are meromorphic functions of X j that is, they are elements of the fraction field :F of the ring of functions of the variable x which are analytic on a domain VCR n •

The reaction mechanism considered by Engell and Klatt can be written as

Following Di Benedetto et al. (1989), we associate to a chain of vector spaces over the field K of meromorphic functions of x , u, ... , u(n-1) defined as follows: Let E denote the vector space spanned over K by {dx , du , ... , du(n-1)}. One defines the subspaces [0 C .. . C En of E by

L

2A~D where A (cyclopentadiene) is the educt, B (cyclopentenol) the desired product , C (cyclopentadiene) and D (dicyclopentadiene) the unwanted by-products.

span {dx,dY, ... , dy( k)}

This process can be described by a fourth order nonlinear dynamical MIMO system which has unstable zero dynamics and thus cannot be controlled by using the well-known exact linearization and decoupling techniques (Isidori 1989).

and the associated list of dimensions Po Pk

:s; . .. :s;

(3) Pn

by

(4)

By using this list of dimensions the infinite zeros or structure at infinity can be defined as follows :

Recently a new "linear algebraic" framework for the analysis of nonlinear systems has been introduced by Di Benedetto et . al (1989). This approach is centred around the study of a finite chain of ordinary vector spaces consisting of differentials of functions constructed from the output of a nonlinear system. Moreover, the work of Di Benedetto et al. is largely inspired by the differential algebraic approach of Fliess (1986) which has enabled a fundamentally new understanding of system theory.

Definition 2.1 (Moog 1988, Martin 1993) The integer IJk := Pk - Pk-1 ;::: 0 is the number of zeros at infinity of order less than or equal to k . IJn is the total number of zeros at infinity. The structure at infinity is given by the list {/L1, . .. , /Lu n } of indices k for which IJk - IJk-1 =I 0, where IJ -1 := 0, these indices being repeated IJk - IJk-1 times (i.e. the structure at infinity is the list of the orders of the zeros at infinity).

This algebraic framework is used to develop in several but simple steps a nonlinear dynamic control scheme which is based on an approximate decoupling and linearization of the nonlinear system. Simulation results illustrate that the given nontrivial control problem can be solved by using the proposed method.

In the input-output decoupling problem another list of integers called the essential orders /Le;, i = 1, . .. , p plays a crucial role. These integers correspond to the output components Yi in the following way :

191

Definition 2.2 (Glumineau and Moog 1989) Consider a right-invertible nonlinear system (1,2), i.e. Un = p. The essential order J.Lei of the output Yi of (2) is defined as the smallest k E {I, .. . , n} for which dy~k) is an essential vector of [no

in the range 0.

{J.Le" . .. , J.Lep}'

m

i=l

i=}

(5)

X2

=

For the determination of the structure at infinity the dimensions of the subspaces (3) must be calculated. The dimension Po of the subspace [0 is given by the order of the process model to Po = 4. The next dimension

k l (X3)XI - k2(X3)X2 - X2UI

X

dim

PI

XJ)UI

-Cl' I L:.HR AB k l(X3)XI - Cl'1L:.HRBC

X3

X4

-

[I

=

dim (span {dx,dy})

X

(7) (13)

- X3) It follows that

The values and a detailed description of the parameters of this model can be found in (Engell and Klatt 1993) .

span {dXI,dx2,dx3,dx4,dYI,dY2}

=

span {dx , -x 2duJ)

=

dim (span {dx , -x 2dud) = 5.

PI

From the difference

¥

kJ On'

(8)

=

It is assumed that the concentration X2 and the temperature X3 are available from measurements so that the outputs of the system are

The control of YI

= [~ o

= X2

1

0 1

UI

of the dimensions

dim (span [I) PI -

h(x)

(14)

and

The control inputs UI, U2 are the normalised inflow and the amount of heat removed by the coolant. They can vary in the following ranges:

-8500

(12)

can be obtained by considering the dUi dependent terms in the differentials

k2(X3)X2 - Cl'1L:.HRADk3(X3)xI

+ (~o - X3)UI + Cl'2(X4 = Cl'3U2 + Cl'4(X3 - X4) .

y

(ll)

4.1. Structure at Infinity

The reactor can be described by a fourth order nonlinear state space model with XI, X2 the concentrations of the educt A and the desired product B, X3 the temperature of the contents of the reactor and X4 the temperature of the reactor coolant:

+ (CAo

mol S; 5.7 -1-

In this section the linear algebraic framework briefly recalled in Section 2 will be applied to investigate whether the model of the CSTR can be decoupled by regular static state feedback .

3. BRIEF DESCRIPTION OF PROCESS AND CONTROL PROBLEM

-kl (X3)XI - k3(X 3)XI

CA o

4. ALGEBRAIC ANALYSIS

(6)

Xl

(10)

It is assumed that this disturbance CAo cannot be measured . Changes of the set point of X2 should lead to well damped smooth transient responses which reach the new steady-state value as fast as possible, at most after 15 minutes. The temperature X3 should be in a small range around the value at the main operation point .

On the other hand, if for an invertible system with m in- and outputs this statement is not fulfilled then the minimal order n emin of a dynamic state feedback for solving the decoupling problem is given by (Huijberts et al. 1992) m

mol S; X2(t) S; 0.95 -1-

- mol < 4 . i) 1 -

Theorem 1 (Glumineau and Moog 1989) The system 1; can be decoupled by regular static state feedback if and only if the ordered lists of infinite zeros and essential orders satisfy:

=

1

can be attained without steady-state error if the concentration CAo of the educt in the inflow varies in the range

At first with these two lists of integers a simple criterion for the solvability of the decoupling problem with regular static state feedback can be given.

{J.LI , .•• , J.Lp}

7 mol

po

=

PI

and po

dim (span [0)

5 - 4

=

1

(15)

(16)

it can be obtained that the model has a zero at infinity of order 1. For the calculation of the subspace [2 the second derivations of the output components must be considered. In this derivations are only such terms of interest which are functions of the first derivative of

(9)

is desired such that any value

192

Ul

or the second input U2:

gives the result that at least one integrator has to attach to one of the inputs. From the fact that an equality between the two list of integers can only be achieved if the modified system has no first order zeros at infinity directly follows that this integrator has to be placed for the input Ul. The reader can easily check that this extended system satisfies the static decoupling condition (5).

(17)

Hence , the subspace (18) (19) has the dimension P2 = 7. The difference P2 - PI

=

7 - 5

=

2

5. STATIC FEEDBACK DESIGN

(20) In this section , an extended decouplable model of the CSTR is considered. This model has a further state

is now equal to the number m of the input and output components, respectively. Since the integers (7 i cannot be greater than m (cf. Di Benedetto et al. 1989) a calculation of the subspaces £3 and £4 is not necessary. Thus, from (71 , (72 and the Definition 2.1 the structure at infinity of the CSTR model is given by {ttl , tt2} =

(24) with the simple state equation

Xe(t)

{I , 2} .

X2

= =

kl(X3)Xl - k2(X3)X2 - X2X S

X3

=

-crlLiHRABkl(X3)XI - crI LiHRBC X

Xl

According to Theorem 1 the CSTR model can only be decoupled by regular static state feedback if the ordered list of the essential orders coincides with the list of the infinite zeros . A closer look on the subspace

X4

shows that neither dYl nor dY2 is a essential vector of £2 since the dimension of this subspace cannot be changed by removing only one of these vectors. Therefore , the essential orders tte l and tt e 2 of the CSTR model must be greater than 1.

cr(X)

(22 )

1

(27)

,B(x)

=

D-I(x) (28)

can be found which solves the decoupling problem. By using the notation established in this paper and avoiding the normally used (Isidori 1989, Schwarz 1991) Lie products the decoupling matrix D of the extended CSTR model can be written as () (1"<1 )

_y_l__

D(x )

=

=

By using these lists of integers and equation (6) a simple calculation 2 - 1 - 2

+ ,B(x)w

-D-1(x)f(x),

At this point, it is clear that the condition (5) of Theorem 1 is not fulfilled. Therefore, the CSTR model cannot be decoupled by regular static state feedback .

+

cr(X)

with

It is easy to see that the differentials dY I and dY2 are both essential for the dimension of this subspace. In other words, the essential orders of the CSTR model are given by tt el = tt e, = 2.

2

- X3)

= xe(t) for a clear notation .

vet) ] [ U2(t)

In contrast to this, the situation is totally different for the subspace

=

(26)

It is well known (Isidori 1989) , that for a static decouplable system a feedback of the form

span {dx, -x2duI , COo-x3)duI}

nemin

+

v,

if one sets Xs(t) (21)

- xI)xs

k2(X3)X2 - cr 1LiHR AD k 3(X3)xf

+ (190 - X3)XS + cr2(X4 = cr3U2 + cr4(X3 - X4)

Xs

span {dXl , dx2 , dx 3, dx4 , dYl,dY2}

+ (CAo

-kl(X3)Xl - k3(X3)xf

X

4.2. Essential Orders and Dynamic Decoupling

span {dx , dYl , dY2, dih , dih}

(25)

where vet) is the new first control variable. Hence, the extend model is described by

The total number (7n = (74 = (73 = (72 = 2 of the infinite zeros corresponds to the differential output rank p. of Fliess (Di Benedetto et al. 1989). From this follows that the nonlinear model of the CSTR is invertible and at least a dynamic solution for the decoupling problem can be found (Descusse and Moog 1988) .

£2

vet) ,

[ [

{)v

ay~"') {)U2

1 (29)

2) _() Y_2(l" __ e

{)v

() ( 1"<2 )

_Y_2__ {)U2

-X2

(19 0 - x3)

cr Ocr ] 2 3

(30)

The rank of this matrix depends on X2 and is therefore not invertible when the reaction process starts , i.e. for

(23)

193

1~r--~--~--~-~--~-~

X(1)

1~~----------------------~

3.5

y(2). x(3)

132 §:130

~ 128

1,26

-------------- --- -- ~I,

,,

...

124 1.5

--

,,

122

,

y(1). x(2)

Tme[min)

,

120

....

O.50~---":'0---:'20""'----:30~---":40---:'SO""'---!60

_----

~(~_/

1180:-----;';;,0,----::2O;;----;30~---40-:::----;'SO;;-----:!60· Tmolmin)

Fig. 1: Behavior of Xl and X2 for a constant disturbance at the input v

Fig. 2: Behavior of X3 and X4 for a constant disturbance at the input v

300r--_r--~--~--~--~-~

·5000

250 ·7000 ~ ·8000

'"~ 150

l!E.

~ .9000

e ·10000 §

() ·11000 ·12000

50

·13000

%~-~'~O--~20""'---3~0---4~0---:'50""'--~60

.,4ooo0'--~,~0----~20---3~0----:4~0---=50~---:'60

Time [min]

Time (min]

Fig. 3: Behavior of the control

X2(t)

Ul

=

Fig. 4: Behavior of the control

Xs

= O. Hence, by using the inverse

operation point (see Engell and Klatt 1993)

_1_

x sp

-X 2

D-l(x) [

(-19 0

-

(31)

X 3)

and

f(x)

(32)

+

(t9 0

-

X3)Wl

X2Q2Q 3

+

X2

= [ 1.235 0.9 134.14 128.96 18.83 ] T. (34)

After 30 minutes a small constant disturbance is added to the input v. It is easy to see that this change of the first input has no influence on the second output. But due to the fact that at this operation point the zero dynamics of the CSTR model is unstable the internal states Xl and X4 show a step response instead of a double integrated behavior.

X2Q 2 Q 3

the desired decoupling control law for by

U2

Obviously the zero dynamics are not globally unstable since the states Xl and X4 achieve a new set point at Xl ~ 3.3 and X4 ~ 120 for which the behavior is as expected. Unfortunately, the controls in Figure 3 and 4 exceed the allowed ranges (8). Therefore, the exact decoupling law (33) cannot be realized in practice.

:j:. 0 is given

6. CONTROLLER DESIGN VIA APPROXIMATE INPUT-OUTPUT DECOUPLlNG

_1- w2 . Q2Q 3

It is well-known that the zero dynamics of a linear system , i.e. the number and the values of the finite zeros, can be altered by connecting a feedforward compensator parallel to the system. In the following a similar approach is used to change the unstable zero dynamics of the nonlinear CSTR model. Alternative approaches for tackling nonlinear systems with nonminimum-phase behavior can be found in (Hauser et al. 1992) and (L. Benvenuti et al. 1993). Here the

5.1. Simulation Results In the Figures 1 - 2 the simulated behavior of the extended CSTR model (26) with the applied control law (33) is shown if the reactor works at his main

194

1 . 4',--~--~---~--~--~~

__

135;~----,-----r---,------,----,----

x(l)

1~'r---------------------

1.3

y(2)

= x(3)

133

,.1.2

1c

£

~1.1

~C>.

~ 132

.2 C

...~ 131

11o

()

~ o.9'r-------------

130

0.80<-.---1'-0--~20---3-'-0---4.... 0 - - - - ' 5 0 - - - - " 60

128'0:-------::10=----2:':0-----=3':-0- - - - ' 4 0 , - - - -..... 50-----1 60 TIme [minI

1~

Time (minJ

Fig. 5: Behavior of Xl and X2 for a constant disturbance at the input v

- - - - - - - - - __________ _

---

x(4)

Fig. 6: Behavior of X3 and X4 for a constant disturbance at the input v l~'r---__,_----r---r---__,_--~--~

1.25r---__,_----r---,----~--~---

1.2

~

135 x(l)

1~t------

1.15

y(2)

.,

= x (3)

£133

~

132

~

,!

131

130

0.95 y(l)

= x(2)

----

0.91-- - - - - - - - - - - - -_ _ __

1~

0.850!------:'10,----2:':0-----=3.,,0--~40:------:5,..,0-----,J60 Time

[minI

- - - - - - - - - - _________ - -

X(4)

128'0:-----'10,----2:':0-----=3'0 - - - - ' 40= -----:-50-----1 60 Time [minI

Fig. 7: Behavior of Xl and X2 for a constant disturbance at the input U2

Fig. 8: Behavior of X3 and X4 for a constant disturbance at the input U2

output of a first order compensator is added to the output YI of the CSTR (cf. Fig. 11) where the compensator input coincides with the first control input of the CSTR

sator parameters K and T . An investigation of the Jacobian linearization of the compensated moael shows that the zero dynamics are stable if the compensator parameters fulfill the conditions

K .1 l +.T 1

I X6(t)

K

1

T

CSTR U2

I

(t)

I

Y2 (t)

Fig. 11. CSTR with compensator

The compensator in Fig. 11 can be described by a linear differential equation of the form

2:6 (t)

1

T

> kl (19)

(37)

It is easy to see that the compensated and the original model have the same infinite zero structures and the same essential orders. Therefore , the compensated model of order 6 must also be decouplable by regular static state feedback . The corresponding control law looks very similar to the already computed law (33) of the uncompensated model.

fir (t)

v (t)

und

A compensator with a ratio KIT according to (37) has not only the effect of stabilizing the zero dynamics but yields also a control law which is well defined in the whole region of interest.

(35)

with the gain K and the time lag T . Hence , the compensated model has a new fictive output

The Figures 5 and 6 show simulated results of the compensated CSTR model where the compensator parameters are chosen to T = 0,5 hand K = 1. In contrast to the results in the Figures 1 and 2 a small constant disturbance added to the input v after 30 minutes yields no heavy reactions in the state responses. Furthermore, Fig. 6 shows that the second output is

(36) and a changed zero dynamics which order has been increased to 2. However, these dynamics can now be stabilized by an appropriated choice of the compen-

195

1.4

' 35,.----~--~--~--~---,..._-~

1.3

1~r------------------~

y(2) = x(3) w(2) 1.2

\ x(l)

133 1

2:

,. ~

I I I

, I

0.7

I

w(l)

- --

...~ 131

, ,

, , ,

0.8

130

I

, ,

1- __

10

y(l)

15

20

X(4)

1 2 9 r - - -_ _,

= x(2)

25

12801----;5--~,0;;---~1-;;5----;::20;;----..::;2;:5==:::;!30

30

Time [min)

Fig. 9: Responses for changes of the

132

~ Q.

1

Time [mini X2

set point

Fig. 10: Responses for changes of the

decoupled from the first input as desired.

X2

set point

Prof. Dr.-Ing. H. Schwarz for the important suggestions about the treatment of unstable zero dynamics of nonlinear systems.

However, the simulation results in Fig. 7 show that the real first output Y2 = X2 is not exact decoupled if a small disturbance is added to the second input. This result is valid since the control law has only the task to decouple the fictive output ih (t) = X2(t) + X6(t).

8. REFERENCES Benvenuti, L., M.D. Di Benedetto and J.W. Grizzle (1993) . Trajectory control of an aircraft using approximate output tracking. Proc. Second European Control Conference, Groningen, 16381643. Descusse, J . and C .H. Moog (1987) . Dynamic decoupling for right-invertible nonlinear systems. Systems f3 Control Letters, 8, 345-349. Di Benedetto, M.D ., J .W . Grizzle and C.H. Moog (1989) . Rank invariants of nonlinear systems. SIAM J. Control and Optimization, 27, 658-672 . Engell, S. , K.-U . Klatt (1993) . Gain scheduling control of a non-minimum-phase CSTR. Proc. Second European Control Conference , Groningen, 2323-2328. Fliess, M. (1986) . A new approach to the structure at infinity of nonlinear systems . Systems f3 Control Letters, 7, 419-421. Glumineau, A. and C .H. Moog (1989) . The essential orders and the nonlinear decoupling problem . Int. J. Control, 50, 1825-1834. Hauser , J ., S. Sastry and P. Kokotovic (1992). Nonlinear control via approximate input-output linearization: The ball and beam example. IEEE Trans. Autom. Control, 37, 392-398. Huijberts, H.J .C ., R. Nijmeijer and L.L.M. van der Wegen (1992). Minimality of dynamic inputoutput decoupling for nonlinear systems. Systems f3 Control Letters, 18, 435-443 . Nonlinear Control Systems. Isidori, A. (1989) . Springer, Berlin. Martin , P. (1993). A intrinsic sufficient condition for regular decoupling. Systems f3 Control Letters , 20, 383-391. Moog, C .R. (1988) . Nonlinear decoupling and structure at infinity. Math. Control Signals Systems, 1, 257-268. Schwarz H. (1991) . Nichtlineare Regelungssysteme. Oldenbourg, Munchen . Vardulakis, A.I.G. (1980) . On infinite zeros. Int. J. Control, 32, 849-866 .

Finally, to solve the control problem given in Section 3 the two approximately linearized and decoupled SISO systems are closed by using classical PD controllers with the transfer function (38) in the feed backs. The coefficients of the PD controllers are chosen in such a form , that for a given Tv and a chain of two integrators the closed-loop systems has a damping ratio equal to 1. The figures 9 and 10 show simulated results for the main operation point given by Engell and Klatt (1993) if the set point is changed to X 2 = 0.7 after 9 minutes and to X2 = 0.95 after 21 minutes (cf. Section 3). The parameters of the compensator are chosen to T = 1 hand K = 2 and the coefficients of the PD controllers to Tv, = 0.022 h and Tv, = 0.03 h , respectively. Fig. 9 shows results which are much better as these presented by Engell and Klatt (1993) . The new steady-state values are reached in a better damped fashion and in only the half of time . Similar results can be obtained for step changes of the disturbance CAo'

7. CONCLUSION In this paper , a multi-step procedure was proposed to design a controller for a nonlinear non-minimumphase system . This was applied to control the reactions in a continuous stirred tank reactor (CSTR) . The proposed method is based on a linear algebraic framework for nonlinear systems and consists of some but simple design steps . Simulation results show the advantages of the proposed method .

Acknowledgements:

The author would like to thank Dipl.-Ing . T . Wey for helpful discussions and

196