Nonlinear control of the salnikov model reaction

~

Computers and Chemical Engineering Supplement (1999) S28~S292 «:> 1999 Elsevier Science Ltd. All rights reserved PH: 5009 8-1354/99/00)21-6

Pergamon

Nonlinear Control of the Salnikov Model Reaction Bodil Recke Sten Bay Jergenson CAPEC, Department of Chemical Engineering, DTU May 1999

Abstract -T his pap er explores different nonlinear control schemes, applied to a simple model reaction. The model is the Salnikov model, consisting of two ordinary differential equations. The control strategies investigated are I/O-Iinearisation, Exact linearisation, exact linearisation combined with LQR and Control Lyapunov Functions (CLF's). The results show that based on the lowest possible cost function and shortest settling time, the exact linearisation perform s marginally better than the other methods. INTRODUCTION

with

The motivation behind this paper is to explore the differences between various types of nonlinear control, both with respect to qualitative control performance and to computational complexity in preparation and implementation. The Salnikov (1948) model describes a closed chemical system, with two sequential reactions. The rate of the precursor reaction is assumed to be zeroth order whereas the second reaction is first order and exothermic with Arrhenius dependent kinetics. The non-adiabatic system is described by two differential equations, a mass balance for the reactant of the second reaction and an energy balance. The nonlinear dynamic behaviour of this model (Scott (1991); Recke (1998)) results in at most three solutions that can coexist: a steady state and two periodic solutions. The steady state is stable as is one of the periodic solutions, while the other periodic solution is unstable. The highly nonlinear nature makes this system a good candidate for a comparative investigation of nonlinear control designs for a class of chemical reactors. METHODS When nonlin ear control theory is to be applied to the process it is convenient to write the model as an input affine (here SISO) system. i: = f(x) y = h(x)

+ g(x)u

x

= [a] 0

g(x) =

[~]

f(x) u

= [11 - «a(exp9 (1:.0)] )

(3)

= Ba

(4)

o exp

1+<9

-

h(x) = B

e

However some of the control methods that will be appli ed requires full state feedback and in this case full state information is assumed. The parameters used are /\, 0.0025, e 0.1745 and Jl = 0.02. IjO-linearisation Assumption one for applying I/O-linearisation [Kravaris and Arkun (1991); Recke (1998)) is that the relative order of the system is finite. This is fulfilled since the relative order is given by:

=

=

LgL~h(x)

= Lgh(x) = 1 t 0

Assumption two is that the system should be minimum phase, which corresponds to the system having stable zero dynamics. Since the model is already written as a normal form, the zero dynamics is simply the first equation with 0 considered as a constant. In order to determine the stability of this equation, it is differentiated with respect to a resulting in dO:

da

0 ) = -/\,exp ( 1 +gO

(1) Since /\, is a positive constant and the exponential (2) function always is positive as well, the equation is

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stable, and consequently the second assumption is The control to render the system feedback linear also fulfilled. Now the IfO-linearisation can be ap- is given by the following equation. plied. The control to be applied is.

u=

where r is the relative order of the system, and the p-constants are determined by the desired closed loop response of the system. Here it is selected that the closed loop system should be a simple first order system with a stable pole at -1. This choice results in Po 1 and PI 1. Now v can be viewed as a set-point of the reduced system, if v == eO the closed loop system will stabilise at the steady state of the original system. The open loop steady state «0'°,0°)=(0.2838 ,8.0)) is used in the following comparison of the control algorithms. Exact linearisaiion The first assumption for applying exact or feedback linearisation is that the system should be strongly accessible, that is the space Co must have the same dimension as the order of the system, where Co is defined by

=

Co

=

=span {g(x),ad}g(x), ... , ad/- g(x)} I

For the present system the dimension of Co = 2 as long as 0' -::J O. The only possibility of 0' = 0 at steady state occurs if the constant precursor concentration J1. = 0, which makes little sense. The second assumption is the involutivity condition, which however is trivially satisfied for a two dimensional system [Kravaris and Arkun (1991)). To shape the closed loop, an additional assumption on the coordinate transformation is required ¢(xO) = O. This corresponds to stabilisation of the equilibrium point of the transformed system at z = 0, where z = tP(x). The coordinate transform is determined from the partial differential equation,

Thus the transformation is a function of 0' only. The simpl est transformation, that at the same time respects that ¢(xO) = 0 is ¢I(x) = 0' - 0'0. Hence the full coordinate transformation is given by. %1

= ¢I(X) = a

%2

= 92(X) = L'¢l(X) =

_

(5)

0'0 1-10 -

xo exp

(_8_) 1+f.8

(6)

v - I:~=o 1'kL~¢(x) 1¢(x) 1'n Lg L

'F

(7)

Where the 1'-values are det ermined by the desired closed loop response. In this case it is desired that the closed loop poles should be placed as a double pole at -1. This is inserted into the control equation (7). The standard form of feedback linearisation as described in e.g. Isidori (1989), places all the closed loop poles at O. This corresponds to setting 1'n = 1 and 1'i == 0, for i = 1 ... n - 1 in equation (7). Thereby reducing the nonlinear system to a controllable linear system, where standard linear control design methods can be used. For the following comparison the feedback linearising coordinate transform is also combined with a standard LQR designed to minimise the cost function f z¥ + zi + v2 dt. CLF-design Both the above linearisation procedures may result in cancellation of nonlinearities that actually has a stabilising effect. Such cancellation is limited when the controller is designed based on a Control Lyapunov Function CLF (Sepulchre et al. (1997); Recke (1998)). An additional advantage is that this method can incorporate penalties on excessive usage of the actuator or large deviations of the states. In this respect the method is similar to conventional optimal control design, and the combination or feedback linearisation and LQR just described. The challenge with this method is to find a CLF. However a candidate CLF can be directly given if the system is feedback linearisable, i.e, the same conditions have to be satisfied as for exact linearisation. The additional condition on the coordinate transformation that ¢(XO) = 0 has to be fulfilled when the equilibrium point of the system is not at the origin. The method applied now is term ed inverse optimal control design and is described in e.g. Sepulchre et al. (1997). The first step is to find a candidate CLF. For systems that are feedback linearisable this can be done in a syst ematic way. First the following Riccati equation is solved.

AT P+ PA - PEET P+ Q = 0

(8)

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Computers and Chemical Engineer ing Supplement (1999) S289-S292

where A and B are the matrices that would result if the system was feedback linearised and Q ? O. Here Q = I is chosen. From the solution to the Riccati equation (8) the control Lyapunov function tP(xf P¢(x). This CLF is constructed as Vex) is used to construct the feedback control. In the present case the control Lyapunov function is:

=

I~':"":';~~-l---:--r-:--I Method I a cost I i,

I/O-lin.

n n + LQR eLF

10099 2737 3177 5057

153 17 21 18

cost

b:=J comments ~

49694 183109 216415 321015

173 23 22

pole at -1 poles at -1 cost function

20

Q=/

Table 1: Comparison of Nonlinear methods for two initial conditions (a):(a(O), 0(0) = (0.5,10) and where Pz y is the off diagonal elements of the so- (b):(a(O), 0(0) = (3,0.5) lution to the Riccati equation (8) which for the present objective function happens to be identical and equal to one. The numerical values of the remaining elements in the P-matrix are Pu = Pn = settling times. Also the cost function is larger for 1.732. The coordinate transform ¢ is the same as the I/O-linearisation when the starting point is in the case of exact linearisation (5)-(6). A possible close to the steady state, whereas it is considerably feedback control known as Sontags formula is given smaller than that obtained by any of the other by (e.g. Sepulchre et at. (1997)) methods when the starting point is far away from the steady state. (10) Exact linearisation has the smallest cost function in both cases, and consequently the cumbersome derivation of the control equation necessary for When V(x) given by equation (9) is inserted into these equations the control can be calculated. It the CLF-controller is not worth the effort in can be shown that (10) results in optimal con- this case. The approach to the steady state is trol, with cost function given by (Sepulchre et at. compared for the different control designs in figure 1. The I/O-linearisation only exhibits a small (1997)): increase in temperature compared to the other methods. This is however not very surprising since it is designed to approximate a first order response. The CLF-design with initial conditions (a) is the only design that results in an oscillatory transient. Oscillatory behaviour of the closed RESULTS AND DISCUSSION loop is unwanted because of the stable limit cycle The approach used in the .qualitative comparison existing in the open loop system. It is somewhat of the control methods described in this paper surprising that the CLF-design results in very is twofold. Both to give a rough estimate of the large increases in temperature, since these should performance and to evaluate the effort in actually be punished by the design. This problem can constructing the control equations. The rough however be eliminated by choosing a different CLF performance estimate is obtained by evaluation of for the system, by e.g. backstepping described in settling times t, for two different sets of initial con- e.g. Sepulchre et at. (1997). The backstepping ditions, and a cost function given by:J~' 02 + u 2dt. procedure is however non generic therefore the The cost function does not include a, since the procedure will be different from system to system. numerical value would not be available as a is The feedback linearisation with the linear LQR, considered unmeasured. The two initial points has similar problems as the CLF-design, but in are chosen as one relatively close (a) and one this case it is not possible to make a design change quite far away (b) from tpe steady state values. to avoid the problem. The design that performed The result of the comparison is listed in table 1. the best on the investigated system is the standard From table 1 it can be seen that I/O-linearisation feedback linearisation with the two poles placed at yield relatively long settling time in both cases -1. The reason for selecting this design as the best whereas the other methods have more similar is that it has the lowest cost function and one of

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Computers and Chemical Engineering Supplement (/999) S289-S292

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the smallest settling times with both initial condinumber 35, pages 3926-3931. IEEE. tions tested. Furthermore the temperature of the Isidori, A. (1989). Nonlinear Control Systems. Springer-Verlag, Heidelberg. system is not increased to as high values as for the CLF-design and the combined fl and LQR method. Kravaris, C. and Arkun, Y. (1991). Geometric nonlinear control - an overview. In Y. Arkun and The IjO-linearisation is discarded because of the W. H. Ray, editors, International Conference on long settling times, since this is considered to be Chemical Process Control (CPCIV), number 4, a disadvantage in the present case. An additional pages 477-515. disadvantage of the IjO-linearisation is that it can not as the name suggests be calculated based on Recke, B. (1998). Nonlinear Dynamics and Control outputs alone, but requires full state information. of Chemical Processes. Ph.D. thesis, Technical University of Denmark, Department of Chemical Engineering. in Press. I. Y. (1948). Themokinetic model of a Salnikov, CONCLUSIONS homogenous periodic reaction. Dokl. Akad. Nauk From controllers examined here it can be said SSSR, 60, 405-8. that the more conventional nonlinear controllers Scott, S. K. (1991). Chemical Chaos. Number 24 designed all have the disadvantage that the control in The International Series of Monographs on equations even for the simple system investigated Chemistry. Oxford University Press. here are quite complex. The qualitative compari- Sepulchre, R., Jankovic, !\L, and Kokotovic, son made in this paper suggests that exact lineariP. (1997). Constructive Nonlinear Control. sation gives the best results. This is based both on Springer-Verlag, London. the relative ease of construction of the equations Sontag, E. D. (1989). A 'universal' construction and the performance in the crude evaluation of the of Artstein's theorem on nonlinear stabilization. cost function. Finally it should be stressed that Systems & Control Letters, 13,117-123. none of the designs take into account disturbance rejection or modelling errors . REFERENCES Freeman, R. A. and Primbs, J . A. (1996). Control lyapunov functions: New ideas from an old source. In Conference on Decision and Control,