Control of MIMO Dead Time Processes Using Fuzzy Relational Models C.R.Edgar; B.E.Postlethwaite and B.A.Gormandy Process Cybernetics Group, Department of Chemical and Process Engineering James Weir Building, The University of Strathclyde, 75 Montrose St., Glasgow, G1 1XJ, Scotland, UK. Phone: +0141-548-2372, Fax: +0141-552-2302 E-mail:
[email protected]
ABSTRACT: A Fuzzy Internal Model Controller (FIMC) is proposed for the control of MIMO systems with dead time. The controller is based on the well-known Internal Model Control (IMC) formulation but utilises a non-linear crisp consequent Fuzzy Relational Model (ccFRM) at its core. The use of the ccFRM allows an exact inverse of the process model to be used in the design of the controller. The feasibility of the proposed control scheme is demonstrated on a simulated binary distillation column. Key Words: Fuzzy Control; Model Based Control; IMC; Dead time I.
Introduction
The majority of process industries are non-linear, Multi-Input Multi-Output (MIMO) systems. The control of these systems is met with a number of difficulties due to process interactions, dead time and process non-linearities [Marlin, (1995)]. Process interactions occur when more than one controlled output is affected by a particular manipulation. Dead time (or time-delay) is encountered due transportation lags or as pseudo-dead times in approximating higher order dynamics. d sp Process interactions can be dealt with effectively by utilising model-based controller techniques. In these + Controller Process techniques all process input and output measurements are Fs y G--1 + used simultaneously, by a dynamic model of the process, to determine the manipulated variables. Single-Input Single-Output (SISO) systems with dead time are handled well by the standard linear IMC (Figure 1). The + Model non-invertible parts (dead time and right-half plane G+ Gzeroes) are factored into G+and the controller block, C, is an inverse of the invertible part of the model that is left i.e. G-. Therefore, time delays are not included in the Fe model inverse, and hence are not taken into account when calculating controller action. The two low pass Figure 1.The Standard Linear IMC Structure. filters Fs and Fe can be used to make the controller more robust to model error. In MIMO systems with different dead times the manipulations have to be lagged relative to one another. The IMC structure is also limited by its reliance on a linear process model, resulting in poor control of highly non-linear processes. However, Fuzzy Relational Models (FRMs) placed in IMC (FIMC) form have been successfully used to model and control non-linear processes [Sing and Postlethwaite, 1997].
In this contribution a ccFRM, a modification of the FRM, is used as the process model. The ccFRM can be exactly inverted and used in the design of the controller. Also a method is outlined for appropriately selecting the manipulation lags relative to one another for MIMO systems with different dead times.
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The ccFRM The general form of a static SISO FRM is given as:
Y = R o X1 o K o X n
(1)
where, Y represents the fuzzy possibility vector of the model output; X1 to Xn are the possibility vectors of the model inputs; R is the relational array and o is the summated-product operator. If each variable Xi is partitioned into ri fuzzy sets and the output Y is partitioned into r fuzzy sets, then R has dimensions r by r1 by… by rn. However, if the inputs Xi are n
represented by column vectors then the shape of R can be altered to R’, an r by
∏r
i
matrix, yielding
i =1
Y = R i' j × kron (X1 L X n )
∀ j ∈ [1, r ]
(2)
where, kron is the Kronecker-tensor product of the model inputs. When the fuzzy mean method of defuzzification and fully overlapping triangular reference sets are used (2) is reduced to
(
y = R D i × kron X 1 L X n n
where, RD is a vector of length
∏ i =1
r
ri and RDi =
å i =1
)
é yˆ i × R j i ∀i ∈ ê1, ëê
(3) n
∏r
i
i =1
ù ú. ûú
RD is a reduced FRM (r times smaller than the normal FRM) and is termed a crisp consequent FRM (ccFRM). Strictly speaking it is no longer a true FRM as the mapping is now from fuzzy input sets to crisp output values. If there are, N samples of process data available then there will be N equations in the form of equation (3). The relational array, RD, is identified from N samples of past process data by a least-square problem.
FIMC Controller Design
For simplicity of notation, a MIMO system consisting of two inputs (n=2) and two outputs (m=2) is considered. The FIMC is obtained by placing the ccFRM into the IMC structure (Figure 1). However, ccFRM in (1) must be reformulated into a dynamic model. This is achieved by re-writing (1) as a standard auto-regressive First Order Plus Dead Time (FOPDT) model
y1 = R D1 × kron [Y1 (k − 1), X 1 (k − 1 − τ 11 ), X 2 (k − 1 − τ 12 )]
(4)
y 2 = R D 2 × kron [Y2 (k − 1), X 1 (k − 1 − τ 21 ), X 2 (k − 1 − τ 22 )]
(5)
where τij is the dead time between the jth input and ith output. Thus the problem of right-half plane zeros is avoided. And though it is not possible to obtain a general inverse of the ccFRM, it has been shown by Edgar (1998) that the ccFRM can be inverted at each sample time to determine manipulation values. The inputs for all the models are considered to be the same but there is no requirement for this to be the case. There is however a requirement for the number of manipulations to be the same as the number of outputs and also the number of fuzzy sets used to partition the manipulated variable must be the same for each sub-model. Also for the development of the inverse of the ccFRM the simplest case is used i.e. it is assumed that all the dead times are equal thus τij =τ. At any sample time, k, the aim of the FIMC is to determine values for the manipulations at that sample time. Since all τ’s are equal the models represented by (4) and (5) can be rearranged to
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( ) = R D1 × kron [Y1 (k + τ ), X1 (k ), X 2 (k ) ] y 2 (k + 1 + τ ) = R D 2 × kron [Y2 (k + τ ), X 1 (k ), X 2 (k ) ] y1 k + 1 + τ
(6) (7)
Now, y1(k+1+τ) and y2(k+1+τ) will be equated with the set point for output y1 and y2 during the inversion procedure. The future values y1(k+τ) and y2(k+τ), can be obtained by making a prediction using the ccFRM. Errors will be introduced by this prediction, but will be small if the ccFRM is reasonably accurate. At any particular sample time the y1(k+1+τ) and y2(k+1+τ) terms can be equated with the desired output set points. The entries in the relational arrays and any nonmanipulation inputs will also be known. Because fully overlapping triangular reference sets are used to partition the variable ranges, each variable will belong to a maximum of two fuzzy sets and these two sets will form a partition of unity. That is, if manipulation i is considered to belong to sets Ai and Bi, then: µ A i (x i ) + µ Bi (x i ) = 1,
∀ i ∈ [1 , n ]
(8) n
Now, Ai and Bi could appear in (ri-1) different locations in µ(xi) and thus solving (6) and (7) leads to a possible
∏ (r
i
− 1)
i =1
‘solutions’. However, the correct solution is constrained by µ A i (x i ) , µ Bi (x i ) ∈ [0,1],
∀ i ∈ [1 , n ]
(9)
The fuzzy mean defuzzification technique is employed to determine the manipulations. Therefore, when dead times are equal they are effectively discounted during the model inversion and determination of the manipulated variables. It has been stated by Garcia and Morari (1985) that, for MIMO systems with multiple different dead times the trivially optimal control response is one which yields the least possible delay in each output response. For a MIMO FIMC, the optimal control response in this sense derived from a model inverse can be found using the following procedure. The general form of the dynamic ccFRM for an m-output n-input system is given as y1 = R D1 o kron[ Y1 (k − 1) , X 1 (k − 1 − τ 11 ), L X n (k − 1 − τ 1n )] M
y m = R Dm o kron [ Y2 (k − 1) , X 1 (k − 1 − τ m1 ), L X n (k − 1 − τ mn )]
(10)
where the symbols have their usual meanings and the manipulations are X(1+ n -m), …, Xn. The manipulations can be obtained by the inversion procedure described above. However, this inversion does not take account of the lags in the model and compensation must be made. The maximum time delay for any particular fuzzy sub-model is given as
(
)
L j = ∨ τ ij + 1 ∀ i ∈ [1 + n − m, n ]
(11)
The appropriate lags for each manipulation, xi can be found by using :-
(
(
))
λ i = ∧ L j − τ i j +1
∀j ∈ [1, m]
(12)
Thus the suggested control action will be :x i (k + λ i ) = x i
(13)
This means that the ith manipulation will be lagged by λi sample units. Dynamic interactions will be caused by selecting an inappropriate lag for some outputs but the final stead state will not be affected.
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Control of a Simulated Distillation Column
The performance of the proposed FIMC structure was tested and compared with the industry standard of using multi- PI loops to control the Wood and Berry (1973) FOPDT transfer function model of a methanol-water distillation column:
é 12.8e − s ê é x D ( s)ù ê16.7 s + 1 = ê ú ê 6.6e −7 s x ( s ) ë B û ê ë10.9s + 1
− 18.9e −3s 21.0s + 1
ù ú é R( s)ù úê ú − 19.4e −3s ú ë S ( s ) û ú 14.4s + 1 û
é 3.8e −8 s ù ú ê 14.9 s + 1 ú +ê ê 4.9e −3s ú ú ê ë13.2 s + 1 û
(14)
where xD is the mole fraction of methanol in the tops; xB is the mole fraction of methanol in the bottoms; R is the reflux flow rate; S is the steam flow rate and F is the feed flow rate. The time for the simulation is measured in minutes and steady state values for all operating variables are given in Table 1. Variable Steady State Value XD 96.25 mol% methanol XB 0.5 mol% methanol S 1.71 lb/min R 1.95 lb/min F 2.49 lb/min Table 1: Steady state values of parameters distillation column
Manipulation Fe Fs XD 1.0 0.1 XB 1.0 0.1 Table 2: FIMC filter constants for the simulated distillation column
The ccFRM of the column was determined as: x D (k ) = R D1 × kron [x D (k − 1), R (k − 2), S(k − 4 ) ]
x B (k ) = R D 2 × kron [x B (k − 1), R (k − 8), S(k − 4) ]
(15) (16)
with two fuzzy sets for the input variable ranges. The controller filter constants were set to the values shown in Table 2. The two SISO PI loops [Garcia and Morari, (1985)] using a 1minute sampling interval were
æ 1 æ z +1 öö g 1 = 0.20çç1 + ç ÷ ÷÷ è 1.889 è z − 1 ø ø
and
æ 1 æ z +1öö g 2 = − 0.04çç1 + ç ÷ ÷÷ è 5.333 è z − 1 ø ø
(17)
where g1 represents the PI controller for the xD/R loop and g2 represents the PI controller for the xB/S loop.
Discussion
The non-linear modelling capabilities of the ccFRM offer the FIMC no advantage as the test system is linear. The improvement offered over the PI control scheme is purely a result of the ability of the FIMC to handle process dead time and interaction. The response to a set point change in the top composition and the bottom composition to 97 mol% and 1mol% methanol respectively is shown in figure 2. The response to a set point change in the top composition 97 mol% is shown in figure 3. Shorter settling time, less overshoot in the output and less oscillatory controller response are obtained by the FIMC using the ccFRM as the process model.
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Tops Comp (mol%)
Tops comp. (mol %)
9 7 .5 9 7 .0 9 6 .5 9 6 .0 0
50
100
9 7 .5 0 9 7 .0 0 9 6 .5 0 9 6 .0 0
150
0
50
t im e ( m in )
Bttms Comp (mol%)
Bttms comp (mol%)
2 .0 1 .5 1 .0 0 .5 0 .0 0
50
100
100
150
t im e ( m in )
1 .5 0 1 .0 0 0 .5 0 0 .0 0 0
150
50
100
150
t im e ( m in )
t im e ( m in )
Reflux flow (lb/min)
2 .2 0 Reflux flow (lb/min)
2 .2 0
2 .0 0
1 .8 0 0
50
100
2 .0 0
1 .8 0
150
0
50
1 .7 6 1 .7 4 1 .7 2 1 .7 0 1 .6 8 1 .6 6 1 .6 4 0
50
100
150
t im e ( m in )
Steam flow (lb/min)
Steam flow (lb/min)
t im e ( m in )
100
150
t im e ( m in )
Figure 2.Wood/Berry Column; response to set changes xDSP(96.25) xBsp(0.5) to xDSP(97) xBsp(1.0) FIMC (- ) and SISO PI loops (---).SISO
1 .8 2 1 .8 0 1 .7 8 1 .7 6 1 .7 4 1 .7 2 1 .7 0 0
50
100
150
t im e ( m in )
Figure 3. Wood/Berry column; response to set point changes xDSP(96.25) to xDSP(97), FIMC (- ) and SISO PI loops (---).
Conclusion
A FIMC structure with a non-linear ccFRM at its core has been proposed for the control of MIMO systems with dead time. It is observed that the proposed controller performs significantly better than multi-loop PID. References
1. 2. 3. 4. 5.
Edgar, C. R 1998. A novel Fuzzy Internal Model Controller (FIMC). PhD Thesis, Department of Chemical and Processing Engineering, University of Strathclyde, UK. Garcia, C.E and Morari, M. 1982. Internal Model Control 2. Design Procedure for Multivariable Systems. Ind. Eng. Chem. Process Des. Dev., Vol 21, 308-323. Marlin, T. E. 1995. Process Control. Designing Processes and Control Systems for Dynamic Performance. Mc Graw Hill. Sing, C.H. and Postlethwaite, B.E. 1997. pH control: Handling non-linearity and dead time with fuzzy relational model based control. IEE Proceedings on Control Theory Applications, Vol. 144, No.3, 263-268. Wood, R. K. and Berry, M. W. 1973. Terminal Composition Control of a Binary Distillation Column. Chem. Eng. Sci. Vol. 28, 1707-1717.
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