Nonlinear Control of a Simulated Industrial Evaporation System Using

Ind. Eng. Chem. Res. 1999, 38, 2995-3006

2995

Nonlinear Control of a Simulated Industrial Evaporation System Using a Feedback Linearization Technique with a State Observer Kiew M. Kam and Moses O. Tade´ * School of Chemical Engineering, Curtin University of Technology, GPO Box U 1987, Perth, WA 6845, Australia

This paper demonstrates the application of a feedback linearization control technique with a state observer on an industrial multiple-effect evaporation system of an alumina refinery. The nonlinear controller consists of a state feedback controller, single-input single-output (SISO) proportional-integral (PI) controllers, and a reduced-order observer. The simulation study serves to verify the performance of the nonlinear controller on the evaporation system by comparing the existing multiloop PI control system. Closed-loop simulation results showed that the nonlinear controller provides smoother operation for the evaporation despite significant model mismatches between the controller model and the simulated evaporator. A computer algorithm, consisting of the sequential execution code for the state feedback control law, SISO PI controllers, and a reduced-order observer, has been developed for implementation on the evaporator on-site. 1. Introduction Evaporator, a common and energy-intensive unit operation in the process industries, has attracted significant research into modeling of its dynamic behavior and control.1-5 To et al. described an evaporator model, which was used for the synthesis of a state feedback control law for a single-effect evaporation system in an alumina refinery.6 The nonlinear model-based controller was later shown to deliver superior control performance to the multiloop proportional-integral (PI) controllers through real-time implementation.7 Superiority of the feedback linearization technique for controlling a multiple-effect evaporator in the alumina refinery has been demonstrated both by a computer simulation study8 and an implementation study on an industrial evaporation simulator9 with complete state information. However, not all states of the evaporator are measured on-line in an actual plant situation. This paper extends the previous study into the case with incomplete state information, i.e., where all states of the evaporator are not measurable for state feedback. The reduced-order observer design methodology,10 by combining the multi-input multi-output (MIMO) globally linearizing control (GLC) structure,11 is adopted to design a nonlinear output regulatory controller for an industrial four-effect evaporation system. This method is chosen because of its computation simplicity and ease of implementation in real time.10 The performance of the nonlinear controller for the evaporation system is verified through computer simulation and compared to the control performance of the existing multiloop PI control. This paper presents an approach to initialize and implement the nonlinear controller without excessive computational requirements for application to the evaporator. In addition, the paper also demonstrates the use of two evaporator models to examine the ability of the nonlinear controller to deal with model uncertainties, without any assumptions on the bound and struc* Author to whom correspondence is addressed. E-mail: [email protected]. Tel.: +61 8 9266 7704. Fax: +61 8 9266 3554.

ture of model errors. Furthermore, a simulated nonlinear control study of this kind for an industrial-scale four-effect evaporator has not been reported in the open literature. The paper is organized as follows: (a) Section 2 provides a brief review of the design methodology of MIMO GLC structure with a reduced-order observer. (b) A description and the differential geometric nonlinear control characteristics of the evaporation system are given in section 3. (c) Implementation issues and algorithms for the MIMO GLC structure are discussed in section 5. (d) Simulation results, with their implications for real-time implementation of the control structure to the evaporation system, are discussed in section 6. (e) Section 7 provides conclusions for the paper. 2. A Review of MIMO GLC with a Reduced-Order Observer The methodology presented in this section is the application of a standard input-output linearization technique for nonlinear control system designs. Advanced topics in the control of input-output linearizable nonlinear systems such as input constraints handling,12-16 time delay compensation,17 and robust control18-21 can be found in the open literature. Consider a minimum-phase,22,23 nonlinear square (equal numbers of inputs and outputs), MIMO process described by a state-space model of the following form: m

x3 ) f(x(t)) +

gj(x(t))uj ∑ j)1

yi ) hi(x(t)), i ) 1, ..., m

(1)

where x ) [x1...xn]T, u ) [u1...um]T, and y ) [y1...ym]T are the n × 1 vector of state variables and m × 1 vectors of manipulated inputs and controlled outputs, respectively. All of the variables are actual variables rather than their deviations. The functions g1(x(t)), ..., gm(x(t)) and f(x(t)) are n × 1 smooth vector fields on Rn, while h(x(t)) ) [h1(x(t)) ... hm(x(t))]T is a m × 1 vector of the

10.1021/ie980798j CCC: $18.00 © 1999 American Chemical Society Published on Web 07/03/1999

{

2996 Ind. Eng. Chem. Res., Vol. 38, No. 8, 1999

output maps that are smooth scalar fields on Rn. Smoothness of vectors implies that the vectors are infinitely differentiable with respect to the state variables and their derivatives are bounded.24 Furthermore, it is assumed that the nonlinear model in (1) has a welldefined relative order22,23,25 and the characteristic matrix22,25 is invertible for all x. In addition, it is assumed that the entries of the characteristic matrix cannot change signs.24 The class of nonlinear model considered in this paper is control-affine; i.e., the nonlinear model is linear in the manipulated inputs. Analyses and designs of nonlinear control systems for a general nonlinear model are beyond the scope of this paper and can be found elsewhere.26-29 If there are s additional on-line measurements other than the controlled outputs, which are algebraic functions of the state variables

Y ) H(x)

(2)

[ ]

xˆ˘ 1 ) F1(xˆ 1,...,xˆ n-s-m,Y,y) + m

G1j(xˆ 1,...,xˆ n-s-m,Y,y) uj ∑ j)1

· · · xˆ˘ n-s-m ) Fn-s-m(xˆ 1,...,xˆ n-s-m,Y,y) + m

G(n-s-m)j(xˆ 1,...,xˆ n-s-m,Y,y) uj ∑ j)1

if the subsystem in (6) is locally stable.10 The variables x1, ..., xn-s-m are the estimates of the state variables that are not measured on-line. The differential geometric22,23,25 nonlinear control characteristics for the nonlinear model in (5) are then computed. The relative order of the ith output (i.e., the (n - m + i)th element of the new coordinate) with respect to the jth input is defined as10

that satisfy the rank condition

rij )

∂H(x) ∂x rank )s+m ∂h(x) ∂x

(3)

[] []

if G(n-m+i)j * 0 if G(n-m+i)j ) 0

(7)

where r*ij is the smallest positive integer such that

(4)

x1 · · d · ) F(x1...,xn-s-m,Y,y) + xn-s-m dt Y y

(8)

Let ri be the relative order in the major diagonal position of the ith row of the relative-order matrix Mr;12 then the ijth element of the m × m dimensional characteristic matrix C(xˆ 1,...,xn-s-m,Y,y) is given as10

Cij(xˆ 1,...,xˆ n-s-m,Y,y) )

{

G(n-m+i)j(xˆ 1,...,xˆ n-s-m,Y,y)

ri ) 1

LGjLrFi-2Fn-m+i(xˆ 1,...,xˆ n-s-m,Y,y)

ri > 1

(9)

The input-output linearizing and decoupling controller of MIMO GLC structure11 is designed in terms of the controlled outputs, the secondary outputs, and the estimated state variables as follows:

The process model in the above coordinate is given as10

u ) A-1{v - B}

(10)

where A is the m × m decoupling matrix, which is dependent on the characteristic matrix of the nonlinear system in (5)

m

Gj(x1,...,xn-s-m,Y,y) uj ∑ j)1

(5)

yi ) yi, i ) 1, ..., m where

) )

∂F (x) f(x) F(x1,...,xn-s-m,Y,y) ) ∂x Gj(x1,...,xn-s-m,Y,y) )

1, r*ij > 1,

/

x1 x1 · · · · · · ) F (x) ) xn-s-m xn-s-m H(x) Y y h(x)

( (

{

LGjLrF ij-2F(n-m+i)(xˆ 1,...,xn-s-m,Y,y)

then these variables, Y ) [Y1...Ys]T, can be used as the secondary outputs for the reduced-order observer to estimate the unmeasured state variables.10 When it is assumed that the state vector transformation is invertible,10 the coordinate of the nonlinear differential equations in (1) can be transformed into the following new coordinate:10

[]

(6)

∂F (x) gj(x) ∂x

The first (n - s - m) state variables, which are unmeasured, are estimated from on-line simulation of the first (n - s - m) nonlinear differential equations in (5)10

[

βˆ 1r1

0 A) · · · 0

{

]

0

· · ·

0

βˆ 2r2

· · ·

· · · 0

··

0 C(xˆ 1,...,xˆ n-s-m,Y,y) (11) · · · βˆ mrm

·

· · ·

and B is a m × 1 vector whose ith entry is given as

βˆ i1Fn-m+i(xˆ 1,...,xˆ n-m-s,Y,y) + βi0yi

ri ) 1

ri

Bi ) βˆ ik

ˆ 1,...,xˆ n-m-s,Y,y) + ∑ Lk-1 F Fn-m+i(x k)2

βi1Fn-m+i(xˆ 1,...,xˆ n-m-s,Y,y) + βi0yi

ri > 1 (12)

where {βˆ i} are the design parameters for the ith output

Ind. Eng. Chem. Res., Vol. 38, No. 8, 1999 2997

Figure 1. MIMO GLC structure with a reduced-order observer.10

Figure 2. Schematic of the five-effect evaporator.

and the ν’s are the external inputs that are determined by the PI controllers

[

sp νi ) βˆ i0ysp i + KPi (yi - yi) +

]

∫0t(yspi - yi) dt ,

1 τIi

i ) 1, ..., m (13)

The overall schematic for the MIMO GLC structure with the reduced-order observer is given in Figure 1.10 3. Evaporation System The evaporation system considered here is the first stage of the liquor burning process for organic impurities removal associated with the Bayer process for alumina production. It consists of a falling film evaporator, three vertical forced circulation evaporators, and a superconcentrator. The five-effect evaporation system is shown in Figure 2. In this paper, the first four effects are considered in the simulation study because the state feedback control technique was previously implemented on the first four effects of the evaporation simulator.9 The spent liquor is fed to the first effect while live steam is used as the heat source for the vertical heat exchangers of the third and last effects (heaters 3 and 4). The amount of steam to heater 3 is set in a ratio to the amount of steam to heater 4. Liquor is concentrated by evaporating off water in each flash tank under a high recycle rate. The flashed vapor from flash tanks 3 and 4 is combined and flows uncontrolled to heater 2, while the flashed steam from flash tank 2 flows uncontrolled to heater 1. All flashed vapor from flash tank 1 is

condensed in a cooling water contact condenser. The primary objective for process control is to regulate the product liquor density of flash tank 4 (F4) by using the steam to heater 4 (m ˘ S4), while the liquor levels of the flash tanks (h1, h2, h3, and h4) are maintained by manipulating the respective liquor product flows (QP1, QP2, QP3, and QP4). All of the controlled outputs (i.e., F4, h1, h2, h3, and h4) are measured on-line. In addition to the on-line measurements of the controlled outputs, liquor temperatures of all flash tanks (i.e., T1, T2, T3, and T4) are also measured on-line. However, on-line measurements for the product liquor densities of the first three flash tanks (i.e., F1, F2, and F3) are not available. Therefore, a state observer is required to estimate the unmeasured liquor densities so that the input-output linearizing and decoupling controller can be implemented. Two evaporator models, models M1 and M2 as shown in the appendix, are used to describe the evaporation system. Both models are represented by 12 nonlinear differential equations with 5 input and output variables. The state, output, and input variables for the evaporator models are given in Table 1. It has been shown that model M1 has a more complex model structure and is control-nonaffine while model M2 has a simpler model structure and is control-affine.31 Consequently, model M2 is used for the synthesis of the input-output linearizing and decoupling controller because of its simpler model structure, while model M1 is used to generate the state trajectories of the evaporation system in the simulation study. The simulation study also reviews the robustness of the nonlinear controller in Figure 1 on the evaporation system because there are


From (9), the characteristic matrix of the transformed model M2 is given as

Table 1. State, Input, and Output Variables for Evaporator Models states

outputs

inputs

secondary outputs

h1, x1 h2, x2 h3, x3 h4, x4 F4, x5 T1, x6 T2, x7 T3, x8 T4, x9 F1, x10 F2, x11 F3, x12

h1, y1 h2, y2 h3, y3 h4, y4 F4, y5

QP1, u1 QP2, u2 QP3, u3 QP4, u4 m ˘ S4, u5

T1, Y1 T2, Y2 T3, Y3 T4, Y4

G81(Fˆ 1,T1) G82(Fˆ 2,T2)

4. Synthesis of the Nonlinear Controller In order to synthesize the input-output linearizing controller and the reduced-order observer in Figure 1 for the evaporation system, the model equation of model M2 is transformed into the new coordinate of (4). Because the controlled outputs are also the state variables (i.e., y1 ) h1 ) x1, etc.), coordinate transformation is achieved simply by rearranging the state vector T

F (x) ) [Fˆ 1, Fˆ 2, Fˆ 3, T1, T2, T3, T4, h1, h2, h3, h4, F4]

(14) The evaporator model M2 in the above coordinate is given in the appendix. The first three differential equations of the transformed model M2 are used as the reduced-order observer to estimate the unmeasured liquor densities (i.e., F1, F2, and F3)

{

dt

)

0

G91(Fˆ 1,T1) G92(Fˆ 2,T2)

G84(T4,F4)

G85

G94(T4,F4)

G95

0 G113(T4,F4)

G105

0

0 G102(Fˆ 2,T2) G103(Fˆ 3,T3)

0

0

G113(Fˆ 3,T3)

0

G123(Fˆ 3,T3,h4,F4) G123(T4,h4,F4) G125(h4,F4)

0

G115

]

(17)

From (10)-(12), the state feedback controller for the evaporator is given as

significant structural uncertainties between models M1 and M2 that are evident from the difference in model equations in the appendix.

dFˆ 1

[

C(Fˆ 1,Fˆ 2,Fˆ 3,T1,...,T4,h1,...,h4,F4) )

1 (-4.2Fˆ 1 + 5.66 + 0.187 × h1

10-6Fˆ 1T1(Fˆ 1 - 1)QP1 + 0.206 × 10-6Fˆ 2T2(Fˆ 1 - 1)QP2 -4

1.80 × 10 F4T4(Fˆ 1 - 1)QP4 + 0.385(Fˆ 1 - 1)m ˘ S4) dFˆ 2 1 -2 (0.125(0.142 × 10 Fˆ 1T1(Fˆ 2 - 1) ) dt h2 Fˆ 2 Fˆ 1 - 1 QP1 + 0.393(Fˆ 2 - 1)m ˘ S4 0.211 × Fˆ 1

( )) -6

10 Fˆ 2T2(Fˆ 2 - 1)QP2 - 1.84 × 10 F4T4(Fˆ 2 - 1)QP4) dFˆ 3 1 ) (0.125(0.145 × 10-4Fˆ 2T2(Fˆ 3 - 1) dt h3 Fˆ 3 Fˆ 2 - 1 QP2 + 0.288(Fˆ 3 - 1)m ˘ S4 - 1.85 × Fˆ 2

0 βˆ 21 0 0 0

0 0 βˆ 31 0 0

0 0 0 βˆ 41 0

0 0 0 0 βˆ 51

{

( ))

Y ) [T1 T2 T3 T4]T, y ) [h1 h2 h3 h4 F4]T are the secondary and the controlled output variable vectors of the evaporator, respectively. The external inputs, ν’s are each determined by the PI controller action in (13).

[ [ [ [

∞ ∞ 1 1 1

1 1 ∞ 1 1

1 1 1 1 1

]

(16)

] ] ] ]

∫0t(hsp1 - h1) dt

sp ν1 ) βˆ 10hsp 1 + KP1 (h1 - h1) +

1 τI1

sp ν2 ) βˆ 20hsp 2 + KP2 (h2 - h2) +

1 τI2

sp ν3 ) βˆ 30hsp 3 + KP3 (h3 - h3) +

1 τI3

[

It was shown that the reduced-order observer in (15) is locally stable.32 This implies that the system in (15) is stable in the close vicinity of the nominal values of the unmeasured liquor densities. For the transformed model M2 in the appendix, the relative-order matrix is

(18)

B1 ) 4.189βˆ 11 + βˆ 10h1 B2 ) βˆ 20h2 B3 ) βˆ 30h3 B4 ) βˆ 40h4 B5 ) βˆ 50F4

sp ν5 ) βˆ 50Fsp 4 + KP5 (F4 - F4) +

1 1 1 ∞ ∞

] {[ ] [ ]} ν1 B1 ν2 B2 ν3 - B3 ν4 B4 ν5 B5

-1

sp ν4 ) βˆ 40hsp 4 + KP4 (h4 - h4) +

10-4Fˆ 3T3(Fˆ 3 - 1)QP3)

[

where

βˆ 11 0 0 0 0

(15)

-4

1 1 Mr ) ∞ ∞ ∞

[] [

QP1 QP2 QP3 ) [C(Fˆ 1,Fˆ 2,Fˆ 3,Y,y)]-1 QP4 m ˘ S4

∫0t(hsp2 - h2) dt

∫1τ

I3

(hsp 3 - h3) dt

∫0t(hsp4 - h4) dt

1 τI4

]

∫0t(Fsp4 - F4) dt

1 τI5

(19)

The values of βˆ i0’s are selected such that the bias values of the PI controllers in (19) are equal to the values of the inputs at steady state while the values of βˆ i1’s are chosen to improve the condition for inversion in (18). The design parameters and PI tuning parameters are given in Table 2. The observer in (15), the state feedback controller in (18), and the PI controllers in (19) form the overall nonlinear control scheme for the evaporator. 5. Implementation of the Nonlinear Controller The simulation of the nonlinear control scheme for the evaporator is implemented in the computer algebra


tanks 1-3 that correspond to the start-up values of the inputs, secondary outputs, and controlled outputs are calculated from the linearized reduced-order observer. The start-up values of the inputs, secondary outputs, and controlled outputs are the measured values when the nonlinear controller is switched on (i.e., T1(t0), T2(t0), etc.). The linear reduced-order observer is obtained by local linearization of (15),

Table 2. Design and Tuning Parameters for the Evaporator outputs

βˆ i0

βˆ i1

Kpi

τIi (h)

h1 h2 h3 h4 F4

21.8 12.3 10.6 9.62 1.55

1 1 1 1 10

15 15 15 15 3

6.25 6.25 6.25 6.25 5.00

system, MAPLE V release 4.33 In the simulation, the discrete forms of the state feedback controller in (18), the reduced-order observer in (15), and the PI controllers in (19) are used.

[ ]

QP1(tk) QP2(tk) QP3(tk) ) Ψ(v(tk),Fˆ 1(tk),Fˆ 2(tk),Fˆ 3(tk),Y(tk),y(tk)) (20) QP4(tk) m ˘ S4(tk)

{

Fˆ 1(tk) - Fˆ 1(tk-1)

) Φ(u(tk),Fˆ 1(tk),Fˆ 2(tk),Fˆ 3(tk),Y(tk),y(tk)) ∆t Fˆ 2(tk) - Fˆ 2(tk-1) ) Φ2(u(tk),Fˆ 1(tk),Fˆ 2(tk),Fˆ 3(tk),Y(tk),y(tk)) ∆t Fˆ 3(tk) - Fˆ 3(tk-1) ) Φ3(u(tk),Fˆ 1(tk),Fˆ 2(tk),Fˆ 3(tk),Y(tk),y(tk)) ∆t

ν1(tk) ) ν1(tk-1) +

(21)

βˆ 10(hsp 1 (tk)

[

-

hsp 1 (tk-1))

+

KP1 (e1(tk) - e1(tk-1)) +

]

∆t e (t ) τI1 1 k

sp ν2(tk) ) ν2(tk-1) + βˆ 20(hsp 2 (tk) - h2 (tk-1)) +

[

KP2 (e2(tk) - e2(tk-1)) +

]

∆t e (t ) τI2 2 k


[

KP3 (e3(tk) - e3(tk-1)) +

]

∆t e (t ) τI3 3 k


[

KP4 (e4(tk) - e4(tk-1)) +

]

∆t e (t ) τI4 4 k

sp ν5(tk) ) ν5(tk-1) + βˆ 50(Fsp 4 (tk) - F4 (tk-1)) + ∆t KP5 (e5(tk) - e5(tk-1)) + e5(tk) (22) τI5

[

[ ][ ] [ ][ ] [ ][ ] [ ][ ]

[]

δFˆ 1 ∂Φ2 ∂Φ2 δFˆ 2 ) · · · ∂Fˆ 1 ∂Fˆ 3 δFˆ 3 ∂Φ3 ∂Φ3 · · · ∂Fˆ 1 ∂Fˆ 3 ∂Φ1 ∂Φ1 · · · ∂h1 ∂F4 ∂Φ2 ∂Φ2 · · · ∂h1 ∂F4 ∂Φ3 ∂Φ3 · · · ∂h4 ∂F4

∂Φ1 ∂Φ1 · · · ∂T1 ∂T4

δFˆ 1 ∂Φ2 ∂Φ2 δFˆ 2 + · · · ∂T1 ∂T4 δFˆ 3 ∂Φ3 ∂Φ3 · · · ∂T4 ∂T4

∂Φ1 ∂Φ1 · · · δh1 ∂QP1 ∂QP4 δh2 ∂Φ2 ∂Φ2 δh3 + · · · ∂QP1 ∂QP4 δh4 ∂Φ3 ∂Φ3 δF4 · · · ∂QP1 ∂QP4

δT1 δT2 + δT3 δT4

∂Φ1 ∂m ˘4 ∂Φ2 ∂m ˘4 ∂Φ3 ∂m ˘4

δQP1 δQP2 δQP3 δQP4 δm ˘ S4

(23)

where δ represents deviation from the steady-state value. Note that all entries of the Jacobian matrices are evaluated at the steady-state values of the unmeasured liquor densities, the manipulated inputs, and secondary and controlled outputs. The initialization values for the reduced-order observer (i.e., Fˆ 1(t0), Fˆ 2(t0), and Fˆ 1(t0)) are obtained by setting the differential terms in (23) equal to zeroes and solving for the unmeasured liquor densities forced by the start-up values of the inputs, controlled outputs, and secondary outputs

( [ ]) {[ ] [ ] [ ][ ] [ ][ ]}

∂Φ1 ∂Φ1 · · · ∂Fˆ 1 ∂Fˆ 3 δFˆ 1(t0) ∂Φ2 ∂Φ2 δFˆ 2(t0) ) · · · ∂Fˆ 1 ∂Fˆ 3 δFˆ 3(t0) ∂Φ3 ∂Φ3 · · · ∂Fˆ 1 ∂Fˆ 3

-1

[ ]

]

where ∆t is the sampling period. The equations in (22) are the velocity form of the digital PI controllers. They are used in the simulation because the PI actions in the industrial control system are coded in velocity forms. It should be noted that the implementation algorithms of the nonlinear controller to be discussed are equally applicable to its real-time implementation on the distributed control system (DCS) on-site. The overall structure for the implementation includes the initialization and execution phases of the nonlinear controller and is shown in Figure 3. 5.1. Initialization of the Nonlinear Controller. The PI controllers in (22) and the reduced-order observer in (21) need to be initialized when the nonlinear controller is switched on at t0. To initialize the reducedorder observer, the unmeasured liquor densities of flash

∂Φ1 ∂Φ1 · · · ∂Fˆ 1 ∂Fˆ 3

∂Φ1 ∂Φ1 · · · ∂h1 ∂F4 δT1(t0) ∂Φ2 ∂Φ2 δT2(t0) + · · · δT3(t0) ∂h1 ∂F4 ∂Φ3 ∂Φ3 δT4(t0) · · · ∂h4 ∂F4 ∂Φ1 ∂Φ1 · · · ∂QP1 ∂QP4 ∂Φ2 ∂Φ2 · · · ∂QP1 ∂QP4 ∂Φ3 ∂Φ3 · · · ∂QP1 ∂QP4

and

∂Φ1 ∂Φ1 · · · ∂T1 ∂T4 ∂Φ2 ∂Φ2 · · · × ∂T1 ∂T4 ∂Φ3 ∂Φ3 · · · ∂T4 ∂T4 δh1(t0) δh2(t0) δh3(t0) + δh4(t0) δF4(t0)

∂Φ1 ∂m ˘4 ∂Φ2 ∂m ˘4 ∂Φ3 ∂m ˘4

δQP1(t0) δQP2(t0) δQP3(t0) δQP4(t0) δm ˘ S4(t0)

(24)


[ ] {[

inputs in (18) is used. ν1 ν2 ν3 ) ν4 ν5

βˆ 11 0 0 0 0

0 βˆ 21

0 0 βˆ 31

0 0 0

0 0

0 0 0 βˆ 41 0

] }[ ] [ ]

0 0 0 C(Fˆ 1,Fˆ 2,Fˆ 3,Y,y) 0 βˆ 51

QP1 B1 QP2 B2 QP3 + B3 QP4 B4 m ˘ S4 B5

(26)

Again, the PI controllers are initialized by a set of five algebraic equations forced by the start-up values of the inputs and secondary and controlled outputs and the initialization values of the reduced-order observer (i.e., estimates of unmeasured liquor densities correspond at start-up of a nonlinear controller) from (24) and (25)

νi(t0) ) φi(Fˆ 1(t0),Fˆ 2(t0),Fˆ 3(t0),Y(t0),y(t0),u(t0)), i ) 1, ..., 5 (27) where Y(t0), y(t0), and u(t0) are the start-up value vectors for the secondary outputs, controlled outputs, and manipulated inputs of the evaporator, respectively. 5.2. Execution of the Nonlinear Controller. Once the reduced-order observer and the PI controllers are initialized, the nonlinear control scheme is executed at every sampling period ∆t. At time tk, the nonlinear controller is executed sequentially as follows: 1. Perform sampling of the inputs and secondary and controlled outputs, i.e., u(tk), Y(tk), and y(tk), respectively. 2. Calculate the outputs of PI controllers, ν(tk)’s, from (22). 3. Calculate the actual ν(tk)’s from the saturation laws of the PI controller outputs,

{

νi(max)

νi(tk) ) νi(tk) νi(min)

Figure 3. Structure for implementation of the nonlinear controller for the evaporator.

[ ][ ][] Fˆ 1(t0) δFˆ 1(t0) Fˆ 1s Fˆ 2(t0) ) δFˆ 2(t0) + Fˆ 2s Fˆ 3s δFˆ 3(t0) Fˆ 3(t0)

(25)

The solutions to (24) and (25) are a set of three algebraic equations that can be easily implemented into any digital computer, and the initialization values can be obtained with minimal computation time. However, it is acknowledged that the estimates of the initialization values for the observer may not be optimal. Optimality for the estimates of the initialization values for the reduced-order observer is not necessary in this case because it will be shown in the simulation results that the estimation errors converge to zeros as time progresses, t f ∞. Once the reduced-order observer is initialized, the initialization values for the unmeasured liquor densities (i.e., Fˆ 1(t0), Fˆ 2(t0), and Fˆ 1(t0)), together with the start-up values of the inputs and secondary and controlled outputs, are used to initialize the PI controllers (i.e., to obtain v(t0)). In order to initialize the PI controllers, the relation between the external inputs and manipulated

νi(tk) g νi(max) νi(min) < νi(tk) < νi(max) νi(tk) e νi(min)

(28)

The upper and lower limits of the PI controller outputs are obtained by mapping the constraints on the manipulated inputs, i.e., umax and umin through the v-u relation in (10). As mentioned previously, the design parameters βˆ 10, ..., βˆ m0 are chosen such that the bias values of the PI actions in (9) are equal to the steadystate values of their respective manipulated variables, i.e., βˆ i0ysp i ) ui(t0), i ) 1, ..., m. In essence, the PI controllers determine the changes in manipulated inputs (i.e., u1, ..., um) with respect to the errors of the controlled outputs. Conceptually, the bounds on u1, ..., um, can be equally applied to the outputs of the PI controllers, i.e., ν1, ..., νm. As such, the upper and lower limits of the PI controllers are the same as those of the manipulated inputs in the simulation study. The constraints on the primary inputs are required to prevent the PI controllers from delivering values that are outside the operating ranges of the manipulated inputs. It is acknowledged that the constraint transformation is not proper. Rigorous constraint mapping approaches can be found in the open literature.12-16,34 However, their approaches are proposed for SISO nonlinear systems, and their validity for MIMO nonlinear systems need to be addressed. Furthermore, it is shown in the simulation results that the method is appropriate for the evaporator.


Figure 4. Closed-loop response of the controlled outputs.

4. Obtain the estimates of the unmeasured liquor densities Fˆ 1(tk), Fˆ 2(tk), and Fˆ 1(tk) from (21) by using Euler’s method of integration

{

Fˆ 1(tk) ) Φ1(u(tk),Fˆ 1(tk-1),Fˆ 2(tk-1),Fˆ 3(tk-1),Y(tk),y(tk)) ∆t + Fˆ 1(tk-1) Fˆ 2(tk) ) Φ2(u(tk),Fˆ 1(tk-1),Fˆ 2(tk-1),Fˆ 3(tk-1),Y(tk),y(tk)) ∆t + Fˆ 2(tk-1)

This step is required to impose limits on the outputs of the state feedback controller within the ranges of the manipulated inputs. Unlike in the case for the SISO nonlinear system, where saturation on ν will guarantee nonviolation of the manipulated input u,13 this condition is not guaranteed for the MIMO nonlinear system as demonstrated in the appendix. Furthermore, the validity of the technique has been demonstrated through successful implementation on the evaporation simulator.9

Fˆ 3(tk) ) Φ3(u(tk),Fˆ 1(tk-1),Fˆ 2(tk-1),Fˆ 3(tk-1),Y(tk),y(tk)) 6. Simulation Results and Discussions

∆t + Fˆ 3(tk-1)

(29)

It should be noted that more accurate estimates of unmeasured liquor densities can be obtained by other numerical integration methods such as the 4th-order Runge-Kutta method. However, a simple Euler’s method was desired to allow ease of implementation of state estimation and to minimize the computational load and the time for digital computer (i.e., the DCS). 5. Execute the state feedback controller (20) to 3 S4(tk). calculate QP1(tk), QP2(tk), QP3(tk), QP4(tk), and m 6. Perform saturation tests on the calculated inputs in (5) so that their operating ranges are not exceeded.

{

ui(max) ui(tk) ) ui(tk) ui(min)

ui(tk) gui(max) ui(min) < ui(tk) < ui(max) ui(tk) e ui(min)

(30)

The performance of the nonlinear controller on the evaporator is investigated by making a step change of +2 m3/h in the liquor feed flow to the first effect of the evaporator when the closed-loop simulation of the nonlinear controller on evaporator model M1 is initiated. The closed-loop responses of the evaporator model M1 subject to the same disturbance while under multiloop PI control, with the same tuning parameters in Table 2, are also simulated to provide a benchmark for comparison. Multiloop PI controllers are chosen as the benchmark for comparison because the existing evaporation system is operated by multiloop PI controllers. The use of other methods such as a linear model control (MPC) (which is not currently in place) will be fair. The PI tuning parameters are similar to those that were obtained from previous simulation studies on the simulator of the evaporation system. The closed-loop re-


Figure 5. Closed-loop response of the secondary outputs.

Figure 6. Closed-loop response of the manipulated inputs.

sponses of the controlled outputs, secondary outputs, and manipulated inputs are given in Figures 4-6, respectively. Figure 7 represents the comparison between the estimated liquor densities and the actual

liquor densities of the flash tanks 1-3. It should be noted that the actual liquor densities of flash tanks 1-3 of the evaporator were recovered from model M1 during the simulation study purely for comparison, and they


Figure 7. Closed-loop response of the unmeasured liquor densities.

cannot be obtained in the actual plant situation. The process outputs (i.e., the controlled and secondary outputs) are expressed in percent of deviation from their respective steady-state values while the manipulated inputs are expressed in percent of their operating ranges due to the proprietary operating conditions of the evaporator. It is noted from Figure 4 that the increase in the liquor feed flow results in large perturbations of the controlled outputs under multiloop PI control when compared to those of nonlinear control. For example, the maximum magnitude of the perturbations of the liquor levels of flash tanks 1-4 and the liquor density of the flash tank 4 under the multiloop PI control are 10, 50, 3000, 1000, and 2.5 times larger than those under nonlinear control, respectively. This is due to the fast initial response of the nonlinear controller to the disturbance as can be seen from Figure 6. This indicates that the nonlinear controller results in smoother operation (i.e., smaller perturbations) of the evaporator than the multiloop PI control when the closed-loop system is disturbed. Despite its ability to deliver smoother operation, the nonlinear controller requires relatively longer time to completely reject the offsets in the outputs. Furthermore, offset is evident for the liquor density of flash tank 4 (i.e., Figure 4e). The slower disturbance rejection and output offset is attributed to model mismatches between models M2 and M1. The assumption that the evaporation system is minimum-phase is validated by the stability of the internal variables of the evaporator (i.e., F1, F2, F3, T1, T2, T3, and T4), as shown in Figures 5 and 7. Furthermore, it can be seen from Figure 7 that the observer errors converge to zero as time progresses (i.e., F1(tk) - Fˆ 1(tk) f 0 as k f ∞), despite the initial large estimation errors. This indicates that optimality in the state estimation is not necessarily required, in particular, for the four-effect evaporator. Overall, the nonlinear controller for the evaporator is robust to unmeasured disturbance and model uncertainties as shown from the simulation results. The robustness of the proposed method to the real evaporation system is yet to be tested on the

evaporation simulator or on the actual evaporation system on-site. It is also shown that the computer algorithm for the nonlinear controller is easy to implement, requires minimal computational load and time, and is feasible for implementation on the evaporation system. 7. Conclusions A nonlinear control system consisting of state feedback controller, SISO PI controllers, and a reducedorder observer has been designed for an industrial foureffect evaporator. The reduced-order observer is used to estimate the unmeasured state variables of the evaporator for state feedback. A closed-loop simulation study indicated that the nonlinear control system provides smoother operation for the evaporator to multiloop PI controllers despite the presence of significant model uncertainties between the controller model and the simulated evaporator. A computer algorithm for the nonlinear controller has been designed for the evaporator with minimal computational requirements. Efforts are continuing to implement the proposed method on-site. Acknowledgment The authors acknowledge the support of an Australian Research Council (ARC) large grant. The authors are also grateful to Alcoa of Australia Ltd. for providing the operating information on the evaporator. Appendix Evaporator Models M1 and M2. Each evaporator model consists of 12 nonlinear differential equations. Because of space limitations, only the differential equations for the liquor level, density, and temperature of flash tank 4 are given:


Model M1:

Liquor density

Liquor level

dF4

dh4 1 1 ) QP3 - QP4 dt 8 8 ˘ S4 + 3320QP3F3T3 - 3410QP4F4T4 2.132 × 106m 0.125 -2543T4 + 2.611 × 105F4 + 0.218 × 107

dt

(

Liquor density

Liquor temperature dT4 1 ) 17.320(-4.349 × 102 + 4.094T4) QP3 dt 8 1 ˘ S4 - 1.850 × 10-3QP3F3T3 + Q - 0.119m 8 P4 8.865 1.900 × 10-3QP4F4T4 / 242.5 + T4

dF4 dt

(

)

1 × 8h4

)

1 × 8h4

(2.132 × 106m ˘ S4 + 3320QP3F3T3 - 3410QP4F4T4)(F4 - 1) 2.244 × 106

-

( ))

QP3F3

F4 -1 F3

(

6

(2.132 × 10 m ˘ S4 + 3320QP3F3T3 - 3410QP4F4T4)(F4 - 1) - 2543T4 + 2.611 × 105F4 + 0.218 × 107

( ))

QP3F3

[

Liquor temperature

-

F4 -1 F3

[

a2 dT4 - 1184(3628 + QP1F1) T4 ) 3.473a4 dt a1

(

)

3250(3628 + QP1F1)

]

3250(3628 + QP1F1)

(

/a1 +

]

)

a2 1 1 e(18.30-3816/a3) QP3 - QP4 - 0.125 8 8 a1 2.304 / a4 (36 - 8h4) 3.696e(18.30-3816/a3)

( ))

QP3F3

(

)(

1 3816 a2(F4 - 1) a4 a1 a2

](

3

(

)

[][ ][ ] [ ][ ] [ ][ ]

G11(F1,T1,h1) F1 F1(F1,h1) G21(F1,F2,T1,h2) F2 0 F3 0 0 T1 G41(F1T1,h1) F4(T1,h1) T2 G51(F1,T1,T2,h1) 0 d T3 0 ) + 0 QP1 + dt T4 0 0 h1 F8 G81(F1,T1) h2 0 G91(F1,T1) h3 0 0 h4 0 0 F4 0 0

e(-297/3628+QP1F1) -

0.118 × 108T2 + 3290QP1F1T1

)

-434.9 + 4.094T4 (36 - 8h4)(242.5 + T4) 2.165 (242.5 + T4)2

It can be noted from the differential equations that there are significant differences in model structures of the evaporator models, especially for the liquor temperature. Transformed Evaporator Model. The transformed evaporator model M2 is given as

102.7F4 + 128.8 - T4 - 102.7F4 + 128.8 0.118 x 108T2 + 3290QP1F1T1

) ((

))

F4 3816 1 - 1 /(a4h4) / e(18.30-3816/a3) 2 F3 a a3 4

where

a1 ) -2543T4 + 2.611 × 105F4 + 0.218 × 107 a2 ) -2.132 × 106m ˘ S4 + 3320QP3F3T3 -3410QP4F4T4 a3 ) 355.8 + T4 - 102.7F4 a4 ) 401.9 + T4 - 102.7F4 Model M2: Liquor level dh4 1 1 ) QP3 - QP4 - 0.119m ˘ S4 - 0.185 × dt 8 8 10-3QP3F3T3 + 0.190 × 10-3QP4F4T4

G12(F1,F2,T1,h1) G22(F2,T2,h2) G32(F2,F3,T2,h3) G42(F2,T1,T2,h1) G52(F2,T2,h2) G62(F2,T2,T3,h3) QP2 + 0 G82(F2,T2) G92(F2,T2) G102(F2,T2) 0 0

0 0 G33(F3,T3,h3) 0 0 G63(F3,T3,h3) G73(F3,T3,T4,h4) QP3 + 0 0 G103(F3,T3) G113(F3,T3) G123(F3,T3,h4,F4)

G14(F1,T4,h1,F4) G24(F2,T4,h2,F4) 0 G44(T1,T4,h1,F4) G54(T2,T4,h2,F4) 0 QP4 + G74(T4,h4,F4) G84(T4,F4) G94(T4,F4) 0 G114(T4,F4) G124(T4,h4,F4)

G15(F1,h1) G25(F2,h2) G35(F3,h3) G45(T1,h1) G55(T2,h2) G65(T3,h3) m ˘ S4 G75(T4,h4) G85 G95 G105 G115 G125(h4,F4)


where G85, G95, G105, G115, and F8 are constants. The entries of the vectors can be found in Kam and Tade´.31 Constraints Mapping for Input-Output Linearizable Nonlinear Systems. Constraints Mapping for SISO Nonlinear Systems. To map the constraints of the original inputs to the input constraints of the input-output linearized system, the state-dependent relation between ν and u that is derived from the input-output linearizng control law can be used.13

ν1(max) ) -0.125u1(min) - 0.266 × 10-4u2(min) + 0.375 × 10-1u4(max) - 0.385u5(min) + b1 ν1(min) ) -0.125u1(max) - 0.266 × 10-4u2(max) + 0.375 × 10-1u4(min) - 0.385u5(max) + b1 ν2(max) ) 0.109u1(max) - 0.125u2(min) + 0.382 × 10-1u4(max) - 0.393u5(min) + b2 ν2(min) ) 0.109u1(min) - 0.125u2(max) + 0.382 × 10-1u4(min) - 0.393u5(max) + b2

ν(k) ) Lfrh(x(k)) + LgLfr-1h(x(k)) u(k) Note that the state-dependent relation is the inverse of the input-output linearizing control law. The mapping of the original input’s (u) constraint to the transformed input’s (ν) constraint requires optimization of the statedependent relation subject to the constraint on u. Proposed constraint mapping approaches through the state-dependent relation13 for SISO nonlinear systems will guarantee that saturation on ν results in nonviolation of constraints on u. Constraints Mapping for MIMO Nonlinear Systems. Constraint mapping through the use of a statedependent relation requires that the following optimization problems be solved for each external inputs at each time interval:

ν3(max) ) 0.102u2(max) - 0.894 × 10-1u3(min) 0.287u5(min) + b3 ν3(min) ) 0.102u2(min) - 0.894 × 10-1u3(max) 0.287u5(max) + b3 ν4(max) ) 0.894 × 10-1u3(max) - 0.855 × 10-1u4(min) 0.118u5(min) + b4 ν4(min) ) 0.894 × 10-1u3(min) - 0.855 × 10-1u4(max) 0.118u5(max) + b4 ν5(max) ) 0.575 × 10-1u3(max) - 0.948 × 10-1u4(min) + 0.285u5(max) + b5

vi(max)(k) )

max Ai(x(k))u(k) + Bi(x(k)), umin e u e umax i ) 1, ..., m

ν5(min) ) 0.575 × 10-1u3(min) - 0.948 × 10-1u4(max) + 0.285u5(min) + b5

vi(min)(k) )

min Ai(x(k))u(k) + Bi(x(k)), umin e u e umax i ) 1, ..., m

The input-output linearizing and decoupling control law that is derived from the state-dependent relation is given as

To simplify the discussions that follow, the constraints are held constant over the control horizon.13 At each time interval, the entries of the ith row of the decoupling matrix A and vector B are evaluated using the available state variables. As such, the upper and lower limits on νi are obtained by solving the optimization problem using linear programming. To illustrate that saturation on the transformed inputs does not necessarily guarantee nonviolation of the constraints on u’s for the evaporator, let us consider the state-dependent relation for the evaporator as

[][

ν1 -0.125 ν2 0.109 ν3 ) 0 ν4 0 ν5 0

-0.000 026 6 -0.125 0.102 0 0

0 0 -0.0895 0.0895 0.0575

0.0375 0.0382 0 -0.0855 -0.0948

] [][]

[] [

u1 u2 u3 ) u4 u5 -7.797 -6.589 -5.693 -5.162 -0.568

0.233 -7.556 -6.529 -5.920 -0.652

0.284 0.542 -8.003 -7.256 -0.799

5.256 9.947 15.059 5.659 -1.156

where b’s are real numbers. It can be seen that the saturation limits on ν’s are

{[ ] [ ]} ν1 b1 ν2 b2 ν3 - b3 ν4 b4 ν5 b5

-0.385 -0.393 -0.287 × -0.119 0.285 u1 b1 u2 b2 u3 + b3 u4 b4 u5 b5

]

-7.730 -14.622 -18.465 × -20.080 0.555

If all the external inputs are at their upper limits, manipulated inputs that are calculated from the control law are

[][ ] u1 -6.6 u2 -70.1 u3 ) -99.1 u4 -139.5 u5 -11.8


The manipulated inputs are outside their constraints (i.e., 0). This implies that saturation laws need to be imposed on both the external inputs and the manipulated inputs for the four-effect evaporator. Literature Cited (1) Driscoll, R. H.; Ng, S.; Chuaprasert, S. Evaporator Process Control Using Computer Models. Food Aust. 1995, 47, 27. (2) Montano, A.; Silva, G.; Hernandez, V. Nonlinear Control of A Double Effect Evaporator. Advanced Control of Chemical ProcessessADCHEM ‘91, Toulouse, France, Oct 1991; Pergamon Press, Inc.: New York; p 167. (3) Vanwijck, M. P. C. M.; Quaak, P.; Vanharan, J. J. Multivariable Supervisory Control of a 4-effect Falling Film Evaporator. Food Control 1994, 5, 83. (4) Wang, F. Y.; Cameron, I. I. Control Studies on a model Evaporation ProcesssConstrained State Driving with Conventional and Higher Relative Degree System. J. Process Control 1994, 4, 59. (5) Young, B. R.; Allen, R. M. MIMO Identification of a Pilot Plant Climbing Film Evaporator. Control Eng. Pract. 1995, 3, 1067. (6) To, L. C.; Tade´, M. O.; Kraetzl, M.; Le Page, G. P. Nonlinear Control of a Simulated Industrial Evaporation Process. J. Process Control 1995, 5, 173. (7) To, L. C.; Tade´, M. O.; Le Page, G. P. Implementation of a differential geometric nonlinear controller on an industrial evaporator system. Control Eng. Pract. 1998, 6, 1309. (8) Kam, K. M.; Tade´, M. O. Simulated Control Studies of Fiveeffect Evaporator Models. Comput. Chem. Eng. 1998, submitted for publication. (9) Kam, K. M.; Tade´, M. O.; Le Page, G. P. Implementation Trial of Input-output Linearizing Control on an Industrial Evaporation Simulator. 5th International Alumina Quality Workshop, Bunbury, Western Australia, March 1998; Paper 23. (10) Soroush, M.; Kravaris, C. Multivariable Nonlinear Control of a Continuous Polymerization Reactor: an Experimental Study. AIChE J. 1993, 39, 1920. (11) Kravaris, C.; Soroush, M. Synthesis of Multivariable Nonlinear Controllers by Input/Output Linearization. AIChE J. 1990, 36, 249. (12) Kendi, T. A.; Doyle, F. J. An anti-windup scheme for multivariable nonlinear systems. J. Process Control 1997, 7, 329. (13) Kurtz, M. J.; Henson, M. A. Input-output Linearization Control of Constrained Nonlinear Processes. J. Process Control 1997, 7, 3. (14) Kurtz, M. J.; Henson, M. A. Feedback linearizing control of discrete-time nonlinear systems with input constraints. Int. J. Control 1998, 70, 603. (15) Soroush, M.; Soroush, H. M. Input-output linearizing nonlinear model predictive control. Int. J. Control 1997, 68, 1449. (16) Valluri, S.; Soroush, M. Analytical control of siso nonlinear processes with input constraints. AIChE J. 1998, 44, 116. (17) Chou, Y. S.; Tsai, S. L. Time delay compensation for a class of nonlinear systems. J. Chin. Inst. Chem. Eng. 1998, 29, 437.

(18) Aloliwi, B.; Khalil, H. K. Adaptive output feedback regulation of a class of nonlinear systemssconvergence and robustness. IEEE Trans. Autom. Control 1997, 42, 1714. (19) Qu, Z. H.; Kamen, E. W.; Dorsey, J. F. Continuous i/o robust control of siso time-varying systems. Automatica 1997, 33, 533. (20) To, L. C.; Tade, M. O.; Kraetzl, M. An uncertainty vector adjustment for process modelling error compensation. J. Process Control 1998, 8, 265. (21) Wang, W. H.; Soh, C. B.; Chai, T. Y. Robust stabilization of mimo nonlinear time-varying mismatched uncertain systems. Automatica 1997, 33, 2197. (22) Isidori, A. Nonlinear Control Systems: An Introduction; Springer-Verlag: New York, 1995. (23) Kravaris, C.; Kantor, C. J. Geometric Methods for Nonlinear Process Control: Parts 1 & 2. Ind. Eng. Chem. Res. 1990, 29, 2295. (24) Henson, M. A.; Seborg, D. E. An Internal Model Contol Strategy for Nonlinear Systems. AIChE J. 1991, 37, 1065. (25) Henson, M. A.; Seborg, D. E. A Critique of Differential Geometric Control Strategies for Process Control. J. Process Control 1991, 1, 122. (26) Guay, M.; McLellan, P. J. Input-Output Linearization of General Nonlinear Control Systems. 1994 American Control Conference, Baltimore, MD, June 1994; American Automatic Control Council: Green Valley, AZ; p 2695. (27) Guay, M.; McLellan, P. J.; Bacon, D. W. A condition for dynamic feedback linearization of control-affine nonlinear systems. Int. J. Control 1997, 68, 87. (28) Henson, M. A.; Seborg, D. E. Input-Output Linearization of General Nonlinear Processes. AIChE J. 1990, 36, 1753. (29) Lin, W. Feedback stabilization of general nonlinear control systemssa passive system approach. Syst. Control Lett. 1995, 25, 41. (30) Daoutidis, P.; Kumar, A. Structural Analysis and Output Feedback Control of Nonlinear Multivariable Processes. AIChE J. 1994, 40, 647. (31) Kam, K. M.; Tade´, M. O. Mathematical Modelling and Differential Geometric Analysis of an Industrial Multi-stage Evaporation System; School of Chemical Engineering, Curtin University of Technology: Perth, Australia, 1997. (32) Ekaputra, R. Nonlinear Control of a Simulated Multi-stage Evaporation Unit with State Estimation. Undergraduate Chemical Engineering Thesis, School of Chemical Engineering, Curtin University of Technology, Perth, Australia, 1998. (33) Redfern, D. Maple 5 Release 4: The Maple Handbook; Springer-Verlag: New York, 1996. (34) Kendi, T. A.; Doyle, F. J., III. Nonlinear Internal Model Control for Systems with Measured Disturbances and Input Constraints. Ind. Eng. Chem. Res. 1998, 37, 489.

Received for review December 30, 1998 Revised manuscript received May 10, 1999 Accepted May 19, 1999 IE980798J

Nonlinear Control of a Simulated Industrial Evaporation System Using

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