Implementation Trial of Input-output Linearizing Control on an Industrial Evaporation Simulator

Kam, K. M.1, Tadé, M. O.1 and Le Page, G. P.2

1 School of Chemical Engineering, Curtin University of Technology, GPO Box U1987, PERTH, Western Australia 6845, Australia

2 Alcoa of Australia Ltd, P. O. Box 161, KWINANA, Western Australia 6167, Australia

Abstract

An evaporation simulator is being used as a real time simulation platform for the evaporation stage of the liquor burning process associated with the Bayer process for alumina production at Alcoa’s Wagerup refinery. The simulator consists of a falling film effect, three forced circulation effects and one super-concentrator effect. It is currently being used for on-site operator training, as well as for performing control studies and controller tunings for the evaporation section prior to actual plant implementation.

This paper is devoted to the implementation trial of multi-input multi-output (MIMO) globally linearizing control (GLC) structure of Kravaris and Soroush (1990) on the simulator. Implementation issues for the nonlinear control trial, such as handling of plant input’s hard constraints, linking the conventional industrial PI controllers and the nonlinear control algorithms, initialisation of industrial PI controllers during start-up of the nonlinear control algorithm are discussed.

Key words:

globally linearizing control; input-output linearization; liquor burning process; evaporation simulator; implementation trial

Implementation Trial of Input-output Linearizing Control on an Industrial Evaporation Simulator

Kam, K. M., Tadé, M. O. and Le Page, G. P.

1.0 Introduction

Nonlinear control using input-output linearization technique (Isidori, 1995; Slotine and Li, 1991) for chemical process control has received significant attention in recent years. In particular, numerous studies have been carried out, both by simulation and experimental, on the evaporator due to its importance in the chemical process industries. Montano et al (1991) demonstrated the globally linearizing control (GLC) of Kravaris and Chung (1987) on a simulated double-effect evaporator. To et al (1995) and To et al (1997) applied the multi-input multi-output (MIMO) GLC structure of Kravaris and Soroush (1990) on an industrial single-effect evaporator by simulation and plant implementation, respectively. The application of MIMO GLC structure on a simulated industrial triple-effect evaporator was shown by To (1996). Kam et al (1998a) demonstrated the MIMO GLC structure on a simulated five-effect evaporation system of an alumina refinery. Simulation and implementation results obtained by the various researchers indicate that the control performance of input-output linearizing control was superior to those of linear control using single-input single-output (SISO) proportional-integral (PI) controllers.

This paper is devoted to the implementation trial of MIMO GLC structure on a simulator of an industrial five-effect evaporation stage of the liquor burning process associated with the Bayer process for alumina production at Alcoa’s Wagerup alumina refinery. The implementation trial on the simulator was carried out to ascertain the control performance and robustness of the nonlinear control structure on the evaporation system prior to its implementation on site. The design and implementation procedures for the nonlinear control structure are discussed.

2.0 THE EVAPORATION SIMULATOR

The evaporation simulator is created by a group of connected basic unit operation modules from SACDA’s module library. The simulator consists of a falling film effect, 3 forced circulation effect and a super concentrator. Each effect is modelled by a combination of pressurised vessel modules representing flash tank volumes, the shell side of the heater and flash pots. Heater modules and thermal capacitance modules are used to model the heat exchanges between tubes and shells of heaters.

The simulator was developed to have a one-to-one correspondence to the five-effect evaporation stage of the liquor burning process associated with the Bayer process for alumina production at Alcoa’s Wagerup alumina refinery. Similar to the evaporation system on site, the simulator is connected to a Honeywell distributed control system (DCS). It has been used extensively for operator training and, studies of control techniques and tuning of controllers on the evaporator.

3.0 MIMO GLOBALLY LINEARIZING CONTROl

Consider a square minimum-phase MIMO nonlinear system with structurally non-singular characteristic matrix, C(x) as shown (Isidori, 1995; Slotine and Li, 1991),

( 1)

where f, g1, g2, …, gm are smooth vector fields on Rn, and h1, h2, …, hm(x) are scalar fields on Rn, x is n´ 1 dimensional state vector, and u = [u1 u2um]T and y = [y1 y2ym]T are m-dimensional input and output vector, respectively.

The modified MIMO GLC structure of Kravaris and Soroush (1990) consist of state feedback control laws and m external SISO PI controllers to achieve an overall desired closed-loop performance of the outputs, yi’s. The state feedback control laws take the form,

( 2)

where v is m´ 1 dimensional vector of external inputs, and A(x) and B(x) are m´ m matrix and m´ 1 vector given as,

( 3)

( 4)

where {b ij} are the design parameters of the state feedback control laws. If the process model is perfect, the state-feedback control laws yield a linear input-output map of the MIMO nonlinear system in Equation (1) of the form,

( 5)

The input-output sub-system in Equation (5) is decoupled since the output, yi is only affected by only one external input, vi.

An external SISO PI controller, as shown in Equation (6), is used to regulate each input-output pair, (vi-yi) of the sub-system in Equation (5),

( 6)

where ts is the sampling or scan rate for the PI controller. The overall closed-lop transfer function (CLTF) for ith output is given as (Kam and Tadé, 1998a),

( 7)

and the characteristic equation for the CLTF is approximated as,

 

( 8)

The design parameters, {b ij} and the PI settings are selected to ensure that all poles of the overall CLTF are on the left-half of the s-plane (ie. all roots of Equation (8) have negative real parts), so that the closed-loop feedback control system is stable.

4.0 MIMO GLC OF SIMULATOR

The modified MIMO GLC structure was applied to the first four evaporator effects of the simulator. The state feedback control laws for the simulator were formulated from the dynamic equations of the first four effects of evaporator model M2 of Kam et al (1998b) using Maple (Redfern, 1996) procedures io of Kam et al (1998c). The equations for the state feedback control laws are not given here due to space limitations and can be found in Kam and Tadé (1998d).

The state variables were the liquor levels, densities and temperatures of the flash tanks. The controlled variables were the liquor levels of the flash tanks and the liquor density of flash tank #4 while the manipulated variables were the liquor product flow-rates from the flash tanks and steam flow-rate to heater #4. The state, manipulated and controlled variables of the simulator are summarised in Table 1.

Table 1: State, manipulated and controlled variables for the simulator

States

Manipulated

Controlled

x1

h1

u1

QP1

y1

h1

x2

h2

u2

QP2

y2

h2

x3

h3

u3

QP3

y3

h3

x4

h4

u4

QP4

y4

h4

x5

r 4

u5

y5

r 4

x6

T1

       

x7

T2

       

x8

T3

       

x9

T4

       

x10

r 1

       

x11

r 2

       

x12

r 3

       

4.1 Implementation of State Feedback Control Laws and Tuning of PI Controllers

The state feedback control laws (eg. Equation (2)) for the simulator were implemented as a set of algebraic equations with the industrial PI controllers (eg. Equation (6)) in cascade arrangement as shown in Figure 1.

Figure 1

Implementation of State Feedback Control Laws

All equations for the state feedback control laws were implemented into the DCS using Honeywell control language (CL) code. The following digital PI controllers and the stat feedback control laws were implemented,

( 9)

( 10)

Due to the cascade arrangement in Figure 1, the outputs of the industrial PI controllers and the state feedback control laws are referred as the primary inputs and secondary inputs, respectively. The implementation algorithm for the nonlinear control structure in DCS is given in Figure 2.

Figure 2

Algorithm of Nonlinear Control for Simulator

Design parameters for the state feedback control laws and tuning parameters for the PI controllers used for the nonlinear control trial are given in Table 2. The design parameters b i0’s were selected so that the outputs of PI controllers, vi’s in Equation (6) have the same reference points (ie. same steady state values) as the manipulated variables, ui’s while b i1’s were selected to improve the condition of the decoupling matrix, A(x) in Equation (3) (Kam and Tadé, 1998a). The industrial PI tuning parameters were selected based on the results of linear control studies that were performed on the simulator.

Table 2

Design parameters and tuning parameters for the simulator implementation

 

Design parameters

PI settings

 

b i0

b i1

KCi = Kii (%/%)

t Ii (min)

y1 (h1)

20.5

1

5

75

y2 (h2)

11.2

1

5

75

y3 (h3)

9.8

1

5

75

y4 (h4)

8.7

1

5

75

y5 (r 4)

6.8

10

1

25

4.2 Anti-reset Windup for PI Controller

Anti-reset windup for the integral action of the industrial PI controllers were done by imposing the constraints of the secondary inputs to the primary inputs directly,

( 11)

This was possible since the steady state values of the primary inputs and the secondary inputs were the same due to the selection of design parameters, b i0’s as discussed previously. In order to impose upper limits and lower limits on the secondary inputs, the following saturation law was used for each secondary inputs (ie. manipulated variables),

( 12)

4.3 Initialisation of PI Controller

The PI controllers in Equation (6) were coded in the velocity form and need to be initialised once the nonlinear control algorithms were started up. Initialisation of the PI controllers were done by back transforming the current state and manipulated variables to the external inputs during start-up by using the inverse of the state feedback control law,

( 13)

where x0 and u0 are the state and input vector of the simulator during start-up of the nonlinear control algorithms.

Values of the external inputs, v’s and the manipulated inputs, u’s during the initialisation of the nonlinear control algorithms are given in Table 3. Note that the values are expressed in term of the % of the actual values (in engineering units) to their respective ranges, ie. ui*100%/(ui(max)- ui(min)). Since the design parameters, b i0’s were selected such that the primary and secondary inputs have the same steady state values, the start up values for the manipulated and external inputs should have been the same. The inconsistencies in their values were due to model uncertainties in model M2 that was used to generate the state feedback control laws. Consequently, the implementation tests also provided an indication of the robustness of the MIMO GLC structure against modelling errors on the five-effect evaporator.

Table 3

External and Manipulated Inputs at Start-up

External inputs (%)

Manipulated inputs (%)

v1

50.2

u1

55.8

v2

39.6

u2

46.4

v3

35.5

u3

40.0

v4

34.0

u4

35.6

v5

63.0

u5

54.5

6.0 CONCLUSIONS

Implementation of multi-input multi-output (MIMO) linearizing control (GLC) structure was carried out on an industrial five-effect evaporation simulator. Detail design and implementation procedures for the nonlinear control structure were discussed. It was shown that the MIMO GLC structure can be implemented in cascade arrangement to the conventional SISO PI controllers.

ACKNOWLEDGMENTS

The authors are grateful to Alcoa of Australia Ltd. for allowing the simulator to be used for the implementation trial and for allowing this paper to be presented. The first and second authors would like to thank Mr Graham P. Le Page for his tremendous effort and time in making the implementation trial a success. The support of an Australian Research Council (ARC) grant is also acknowledged.

NOMENCLATURE

Abbreviations

CL control language

CLTF close-loop transfer function

FT flash tank

GLC globally linearizing control

ITAE integral time weighted absolute error

MIMO multi-input multi-output

PI proportional-integral

SISO single-input single-output

Symbols

h liquor level

KC proportional gain

KI integral gain

steam flow-rate

QP liquor product flow-rate

T liquor temperature

ts sampling time or scan rate

ui ith manipulated input/secondary input

vi ith external input/primary input

yi ith controlled output

Greek symbols

b design parameters

r liquor density

t I integral time

 

REFERENCES

Isidori, A. (1995). Nonlinear Control Systems: An Introduction, 3rd ed., New York, Springer-Verlag.

Kam, K. M. and Tadé, M. O. (1998a). Simulated nonlinear control studies of five-effect evaporator models. Computers Chem. Engng. Submitted for review.

Kam, K. M., Tadé, M. O., To, L. C. and Le Page, G. P. (1998b). Dynamic modelling and differential geometric nonlinear control characteristics of an industrial multi-stage evaporator. J. Proc. Cont.. Submitted for review.

Kam, K. M., Tadé, M. O. and To, L. C. (1998c). Implementation of Maple procedures for simulating an industrial multi-stage evaporator. MapleTech. In print.

Kam, K. M. and Tadé, M. O. (1998d). Implementation of MIMO globally linearizing control structure on an industrial evaporation evaporator. Technical report 1/98 School of Chemical Engineering, Curtin University of Technology.

Kravaris, C. and Chung, C. B. (1987). Nonlinear state feedback synthesis by global input/output linearization. AIChE J., Vol. 33, No. 4, pp. 529-603.

Kravaris, C. and Soroush, M. (1990). Synthesis of multivariable nonlinear controllers by input/output linearization. AIChE J., Vol. 36, No. 2, pp. 249-264.

Montano, A., Silva, G. and Hernandez, V. (1991). Nonlinear control of a double effect evaporator. Proc. Advanced Control of Chemical Processes. pp. 167-172.

Redfern, D. (1996) The Maple Handbook: Maple V Release 4, New York, Springer-Verlag.

Slotine, J. J. E. and Li, W. (1991). Applied Nonlinear Control, New Jersey, Prentice-Hall.

To. L. C., Tadé, M. O. and Le Page, G. P. (1997). Implementation of input output linearization control technique on an industrial evaporative system. Proc. CHEMECA ’97 (CD Rom). pp. PC-2d.

To, L. C. (1996). Nonlinear control techniques in alumina refineries. Thesis (Ph.D) Curtin University of Technology.

To. L. C., Tadé, M. O., Kraetzl, M. and Le Page, G. P. (1995). Nonlinear control of a simulated industrial evaporation process. J. Proc. Cont., Vol. 5, No. 3. pp. 173-182.