- Email: [email protected]
Generalised Predictive Control (GPC) Integrates with Fuzzy Logic Control to Regulate Neuromuscular Blockade
Copyright © IFAC Modelling and Control in Biomedical Systems, Warwick, UK, 1997
GENERALISED PREDICTIVE CONTROL (GPq INTEGRATES WITH FUZZY LOGIC CONTROL TO REGULATE NEUROMUSCULAR BLOCKADE M.Mahfouf'!'. M.F.Abbod n and D.A.Linkens Q
'!' Manchester
School ofEngineering Division ofMechanical Engineering The University ofManchester Oxford Road. Manchester M13 9PL n Department ofAutomatic Control and Systems Engineering The University ofSheffield Mappin Street. Sheffield S1 3JD Tel: +44 (0) 1142225607 Fax: +44 (0) 1142731729 Email: [email protected]. uk
Abstract. Many synergisms have been descnOed in the past between intelligent control systems approaches such as Neural Networks, Fuzzy Logic, and Genetic Algorithms and have been shown not only to work well but also to add more robustness to the control system in question. In this paper, a new form of synergism is demonstrated between a mathematical model-based approach represented by the popular Generalised Predictive Control (GPC) algorithm and an intelligent control approach in the form of Self-Organising Fuzzy Logic Control (SOFL'C). The new algorithm named Predictive Self-Organising Fuzzy Logic Control (pSOFLC) is applied to the muscle relaxation process and is shown to perform better than the SOFLC in terms of set-point tracking, flexibility, and number of rules generated.
Kevwords: Muscle Relaxation, Predictive Control, Fuzzy Control, Self-Organising Control, Intelligent Control Structures, Synergy.
(Mahfouf et aI, 1992), self-organising fuzzy logic control (SOFLC) (Linkens and Hasnain, 1991)
1. INTRODUCTION
It is common knowledge that in the last 40 years, control engineers have relied heavily on mathematical models in order to design controllers, this being to learn about their behaviour and make provisions in the design. These controllers became known as Model-Based Controllers or Quantitative Controllers. One of the requirements of this model is that they should be good enough to satisfy the control objective(s). In other words the model should be parsimonious (economical) in structure and yet include enough information to complete successfully the design. Model-based control designs have been successful mostly in aerospace systems because of the availability of good dynamic models, small noise corruption in measurement, and
In order to help the surgeon perform an operation while minimising the risks of tissue damage, there is a need for the muscle to be relaxed (paralysed). This is achieved by administering a number of drugs which induce muscle relaxation. Muscle relaxant drugs are conventionally administered by anaesthetists who, based on their own experience, determine the adequate dose in order to achieve a pre-defined degree of paralysis. However, failure to maintain a steady level of paralysis prompted the use of closed-loop control which not only showed that it can be effective but also safe. Among the methods that have been investigated are PID controllers (Brown et aI, 1980). self-tuning GPC
187
considered to be adequate for a wide class of processes including those systems which exhibit large and varying time-delays.. The process model upon which GPC's strategy is based is of the following CARIMA form:
not many inaccessible variables. Most control designs have been based on linear models, despite the fact that all systems are inherently nonlinear, due to the difficulty associated with analysing nonlinear systems; one common approach has been to linearise the system around various operating points. Among the most popular model-based control approaches are three term PID controllers, Model-Based Predictive Control (MBPC), and robust control (Hoo) .
A(z
-1
-1 CC: ).y(t)=B(z ).ul(t-l)+
-1
X(t)
(1)
11 where y(t) is the measured output variable, u(t) the input variable, and~(t) is the disturbance.
In recent years, there has been a move among system engineers towards intelligent control with a qualitative dimension due to the widespread dissatisfaction with quantitative engineering. One of the main attractions of intelligent system design is the possibility of multivariable system control without the need for extensive dynamic models of the process. The main difficulty in the multivariable case is the interaction between variables and sensitivity to faults in various channels. Neural Networks (NN), Fuzzy Logic Control (FLC), and Genetic Algorithms (GA) have been at the forefront of such methodologies and have proved to be strong contenders for other forms of control.
A(z -I),B(z -1),C(z -1) are polynomials in the
backward shift operator z -1 with the following form: -1 -1 -2 -n A(z ) = I+alz +a2z + ...+anz h... -2 B(z -1 ) =b0 + bl z-1 +~.lz +...+bm-Iz -m+ 1
C(z-l) =CO +CIZ-1+'2z-2+...+cpZ-p+I
and f1 = l-z-1 The controller computes the vector of controls using optimisation of a function of the form:
Various synergisms have been described between fuzzy logic, neural networks and genetic algorithms which not only showed that these intelligent structures can interact together but also can make the overall structure, hence obtained, more robust against model uncertainties as well as disturbances. In this paper, we show that synergism between a mathematical model-based approach in the form of self-tuning Generalised Predictive Control (GPC) (Clarke et al. 1987) and Self-Organising Fuzzy Logic Control (SOFLC) (procyk., 1977) is possible with the former being used as a mechanism to make the latter adjust itself to improve the overall system's performance. First, the GPC and SOFLC algorithms are presented. Ne:-.."t, the new modified algorithm which consists of GPC teaching SOFLC is presented. Simulation results using the muscle relaxation processes associated ",ith the drug atracurium are presented. Finally, conclusions are drawn with regard to the new proposed scheme.
N2
J
= L
(P(z -I)y(t +}) -m(t + j=N1
NU
+ L A.U).(oul (t + ) }=l
1»
i»~ (2)
2
where NI is the minimum prediction horizon, N2 is the ma-amum prediction horizon, NU is the control horizon, CO is the future set-point usually presumed mown, A.(j) is the control weighting sequence,
and
P(z -1) is
an
inverse
model
with P(l) = 1. Furthermore, the CC:-1) polynomial in equation (1) is replaced by a fL"Xed polynomial T(z -1) known as the observer polynomial for the predictions (p(z-I).Y(t + }» (Clarke et al.. 1987). This also enables the controller to offset the effect of the 11 operator as a high-pass filter on the input-output data.
2. BACKGROUND BEIDND GPC AND
The minimisation of the cost function (2) leads to the following projected control increment:
SOFLC
l1ul (t)
=gT (C!) -
'1')
(3)
2.l. A model-based approach due to GPC
where
gT is the first row of the matrix
(G T G +A..l)-I Gd T and Gd is the dynamic d d (step-response) matrix of the form given in Clarke et al. (1987).
Generalised predictive control is a self-tuning approach which belongs to a category of Long Range Predictive Controllers (LRPC) which are
188
E(nT - mT) ~ CE(nT - mT)
2.2. A qualitative dimension due to SOFLC
~
(4)
U(nT-mT)+~(n1)
where
Similarly other intelligent control structures such as neural networks, fuzzy logic control has a long history. It stems from the theoretical work of Lotfi Zadeh (1965) who proposed the use offuzzy logic to mimic the human's ability to use imprecise statements to solve complex problems.
~(nT)
is issued by the performance index, E
is the error, and CE is its derivative.
3. A PREDICTIVE SELF-ORGANISING FUZZY LOGIC CON1ROL (pSOFLC) ALGORITHM:
The Fuzzy Logic Self..()rganising Controller (SOFLC) shown in fig. 1 consists of two levels; the first level which consists of a simple fuzzy controller, and a second level which consists of the self-organising mechanism acting as a monitor and an evaluator to the controller performance. In the first level, the input signal to the controller is taken at each sampling instant in the form of error and change-in-error. Each signal is mapped to its correspondent discrete level by using the error and change-in-error scaling factors respectively and sent to the Self-Organising Controller (SOC). The SOC, according to control rules issued by the second level, calculates the output with respect to the inputs. The output control signals are scaled to real values using the output scaling factors and sent to the process being controlled. The second level consists of four blocks: the performance index, the process reference model, the rules modifier, and the state buffer.
The idea behind the new algorithm is to replace the performance index table with a different mechanism that will allow the rules modifier block in fig. 1 to create, delete and alter rules in a similar way to a self-organising fuzzy logic control scheme. The performance index rules contained in a look-up table are standard and used for a wide range of systems and reflect characteristics of a system's time response with adequate damping, overshoot, and settling-time. Since the index used for the rules modifier algorithm is an incremental quantity which excites the rule generation process, we propose to replace it with increments generated by a selftuning algorithm based on prediction (GPC) whose parameters are adjusted via a parameter estimation algorithm, namely Recursive Least-Squares algorithm (RLS). In all the simulations we carried out a second order discrete-time model with no delay was sufficient. Moreover, in the same way in which the performance index output is scaled, the increment from GPC is also scaled to map the real world onto the fuzzy world. This is shown in Fig. 2 where the PSOFLC scheme is represented.
Further details on the design of a SOFLC can be found elsewhere (Mahfouf and Abbod, 1994) but suffice here to concentrate on the learning part. The self-organising controller is based on observation of the trajectory of the process to be controlled. Any deviation from the desired trajectory path should be corrected by modifying the rule or rules responsible for the undesired performance.
Hence, using the above scheme, equation (-+) will become: E(nT -m T) ~ CE(nT -m T) U (nT - m T) + G ·Ou1 (nT)
~
(5)
ou1
The performance index functions as an evaluation criterion of the controller performance, In general terms it measures the deviation from the desired trajectory and issues the appropriate correction to the rule that give the present behaviour. It is derived from linguistic conditional statements by means of using standard fuzzy operations and written in a look-up table form.
where DUI is the increment generated by the GPC algorithm, and this will represent the basis upon which the adaptation is carried out.
4. SWULATION RESULTS
As far as the rules modification procedure is concerned, it can be explained assuming that a process has a time-lag of m samples, this means that the control action at sample (nT-mY) has most contributed to the process performance at the sampling instance n T. Thus, if the present instant is nT, the modification is made to the c"lntroller output U, mT samples earlier, the rule to be included being:
A series of simulations were conducted using the new proposed algorithm (pSOFLC) together with the SOFLC and their performances were compared using the muscle relaxation process associated with the drug atracurium. The continuous model associated with the drug atracurium is highly nonlinear and identified to be
189
of the following Wiener structure (Mahfouf et ai, 1992):
s G(s)=!.L= ·(l+10.6s).eU (1 +4.Ss)(l + 34.4s)(1 + 3.Is)
demonstrates better set-point tracking properties and a steady level of control signal. Moreover, such performance was possible with a number of rules w~ch was lower than in the previous case (18 rules) as fig. 4b shows.
(6)
and the overall nonlinear model is obtained by combining equation (6) together with the following Hill equation:
E
1 efJ- 1+ (0.404)2.98
5. CONCLUSIONS
n)
Soft computing which includes the three intelligent systems approaches, neural networks, genetic algorithms, and fuzzy logic theory has been increasingly embraced by a wide section of systems engineers. Various synergisms are known to exist between the above three strands which can work well together by forming structures which can tackle complex problems in a robust way without the need for mathematical models. In this study, a new algorithm was proposed in which a synergism was shown to be posSlble between a mathematical model-based approach (self-tuning GPC) and an intelligent control system with a qualitative dimension (fuzzy logic control). This substitutes the GPC algorithm for the performance index table used in the standard SOFLC algorithm to modify the rules of the low level represented by the direct fuzzy controller. Whatever the structure of the process in question, a second order model was found to be sufficient for the GPC structure. This is compatible with the performance index table used in the standard SOFLC which reflects characteristics similar to a second order system with specific damping, overshoot, and settling-time. Using the muscle relaxation process as a test bed, the new proposed strategy named Predictive SOFLC (pSOFLC) was shown to be more flexible in terms of tailoring the output response to a particular application with the help of the GPC tuning parameters such as the prediction horizons, modelfollowing polynomial and observer polynomial. This in turn led to a better performance and to generation of a lower number of control rules. This is the focus of extensive research which aims at demonstrating that superior performance need not be linked with a large number of rules but rather to the quality of these rules. It is hoped to conduct future clinical trials using this new scheme which performed equally well with the binary distillation column and the CSTR system.
(XE)2.98
where U is the drug input, EefJ is the actual output (muscle relaxation or paralysis) and X E is the drug concentration. To simulate the above model, a fourth order RungeKutta method with fixed step length was used for integration together with a sampling period of one minute. A set-point profile of 80% then 70% changed every 100 minutes was used. For the SOFLC algorithm a selection of GE = 10/20.0, GC= 10/7.0, and GU = 100.0110 was chosen together with a delay in reward of 1 minute. For GPC a combination of (1, 10, 2, 0) was used for (NI' N 2 , NU).), plus a model polynomial of
P(z -1)
= 3.33(1- 0.7z -1) together
with
an
observer polynomial T(z -1) = 25(1- 0.8z -1) 2 . For the estimation routine a UD-factorisation method was used with an initial covariance matrix of P =100. I where I is the identity matrix and a forgetting factor of p = 0.995. In order to map the GPC increment
8u I into
the
fuzzy part, a scaling factor of G..
= 300.0
was
vu}
used (the input-output variables ,yere scaled by a factor of a 100.0). The first experiments involved using the SOFLC as described in Section 2.2. Figure 3a shows the result of a run. After an initial undershoot then an overshoot, the response displayed a series of undershoots following dO\\TIwards set-point changes, a feature which is undesirable in muscle rela"'(ation therapy. In turn, the associated control signal was oscillatory and took rather a long time to settle to a steady leveL It is possible to tailor the response further but this would require careful tuning of the fuzzy scaling factors. Figure 3b shows the evolution of the number of rules generated which reached a number of 20 rules at the end of the run. When the PSOFLC was used the performance obtained was that of fig. 4a which
6.ACKNO~DGEMENT
The first and last authors acknowledge financial support from a Leverhulme Trust Research Grant.
190
Predictive Control (GPC) in the -operating theatre, lEE Proc. PtO, 139, pp 404-420.
7. REFERENCES
Mahfouf, M. and Abbod, MF. A comparative study of generalised predictive control (GPC) and intelligent self-<>rganising fuzzy logic control (SOFLC) for multivariable anaesthesia (1994), Chapter 4 in Intelligent Control in Biomedicine, D.A Linkens · (ed.), Taylor and Francis Pub., London, ISBN 0-7484-0115-6.
Brown, B.H., Asbury, Al, Linkens, D.A, Perks, P. and Anthony, M (1980) Closed-loop control of muscle rela"
Procyk, T.l (1977) Self-Organising Control for Dynamic Processes, PhD Thesis, Queen Mary College, London.
Linkens, D.A, Hasnain, S.B. (1991) Selforganising fuzzy logic control and applications to muscle relaxant anaesthesia, Proc. of the IEE, Pt.D, 138, pp 274-284.
Zadeh L.A (1965) Fuzzy Sets, Information and Control, 8, pp 338-353.
Mahfouf, M, Linkens, D.A, Asbury, Al, Gray, W.M. and Peacock, lE. (1992) Generalised
Fig. 1 A Self-Organising Fuzzy Logic Controller (SOFLC)
GC P roc:e:ss
Fig. 2 Predictive Self-Organising Fuzzy Logic Control (pSOFLC)
191
100~--~----~--~----~--~~--~----~--~
Paralysis (%)
-·t·_·- ../--!':";-:.,.- - - - . . . -
.,-:........;.,------""'"
I,
-~
I~~'
(a)
I
50 ~.~ I
~
I
/.
I
• , ... 1
•
~.
_
.41
••
'
r
l ,. )ji!ut".b.. _'-'-' ,. ,i )j./.f''''-'-'--[T r\!r f jj. ! O~--~----~----~--~----~----~--~--~ o 300 350 400 50 100 150 200 250 I ,IA.-
i ;'
' _. ___ ._-1
,,1
,'.AN..... _._._._._\
Time (m in) Number of Rules
20 (b)
I
15 10 5 0
V 0
50
100
150
200
250
350
300
400
Time (min) Fig. 3 SOFLC of muscle relaxation; solid line: output; dash:reference; dash-dote: control signal.
,
100r----r----~--~----~--~~--~----r---~
Paralysis (%)
H...............
(a)
r!~ , it'1
-------..:..-
.~-----
..--.----....
... ~.
.. . \,...
50 i iji"
.Ij.~,l./. " ,1 'r', 1
( I' .........
iIi o .,,' o
i: J
50
rr .! .1,.-._ ._._._ ....t\ ." /'-'-'-'--
__ 1 - - . , :• • _._. ___ .J
:l 100
fi~ J
... I '
150
200
250
•
Fl 300
350
400
300
350
400
Time (min) Number of Rules
20 (b)
I
15 10 5
1I
0 0
50
100
150
200
250
Time (min) Fig. 4 PSOFLC of muscle relaxation.
192
GENERALISED PREDICTIVE CONTROL (GPq INTEGRATES WITH FUZZY LOGIC CONTROL TO REGULATE NEUROMUSCULAR BLOCKADE M.Mahfouf'!'. M.F.Abbod n and D.A.Linkens Q
'!' Manchester
School ofEngineering Division ofMechanical Engineering The University ofManchester Oxford Road. Manchester M13 9PL n Department ofAutomatic Control and Systems Engineering The University ofSheffield Mappin Street. Sheffield S1 3JD Tel: +44 (0) 1142225607 Fax: +44 (0) 1142731729 Email: [email protected]. uk
Abstract. Many synergisms have been descnOed in the past between intelligent control systems approaches such as Neural Networks, Fuzzy Logic, and Genetic Algorithms and have been shown not only to work well but also to add more robustness to the control system in question. In this paper, a new form of synergism is demonstrated between a mathematical model-based approach represented by the popular Generalised Predictive Control (GPC) algorithm and an intelligent control approach in the form of Self-Organising Fuzzy Logic Control (SOFL'C). The new algorithm named Predictive Self-Organising Fuzzy Logic Control (pSOFLC) is applied to the muscle relaxation process and is shown to perform better than the SOFLC in terms of set-point tracking, flexibility, and number of rules generated.
Kevwords: Muscle Relaxation, Predictive Control, Fuzzy Control, Self-Organising Control, Intelligent Control Structures, Synergy.
(Mahfouf et aI, 1992), self-organising fuzzy logic control (SOFLC) (Linkens and Hasnain, 1991)
1. INTRODUCTION
It is common knowledge that in the last 40 years, control engineers have relied heavily on mathematical models in order to design controllers, this being to learn about their behaviour and make provisions in the design. These controllers became known as Model-Based Controllers or Quantitative Controllers. One of the requirements of this model is that they should be good enough to satisfy the control objective(s). In other words the model should be parsimonious (economical) in structure and yet include enough information to complete successfully the design. Model-based control designs have been successful mostly in aerospace systems because of the availability of good dynamic models, small noise corruption in measurement, and
In order to help the surgeon perform an operation while minimising the risks of tissue damage, there is a need for the muscle to be relaxed (paralysed). This is achieved by administering a number of drugs which induce muscle relaxation. Muscle relaxant drugs are conventionally administered by anaesthetists who, based on their own experience, determine the adequate dose in order to achieve a pre-defined degree of paralysis. However, failure to maintain a steady level of paralysis prompted the use of closed-loop control which not only showed that it can be effective but also safe. Among the methods that have been investigated are PID controllers (Brown et aI, 1980). self-tuning GPC
187
considered to be adequate for a wide class of processes including those systems which exhibit large and varying time-delays.. The process model upon which GPC's strategy is based is of the following CARIMA form:
not many inaccessible variables. Most control designs have been based on linear models, despite the fact that all systems are inherently nonlinear, due to the difficulty associated with analysing nonlinear systems; one common approach has been to linearise the system around various operating points. Among the most popular model-based control approaches are three term PID controllers, Model-Based Predictive Control (MBPC), and robust control (Hoo) .
A(z
-1
-1 CC: ).y(t)=B(z ).ul(t-l)+
-1
X(t)
(1)
11 where y(t) is the measured output variable, u(t) the input variable, and~(t) is the disturbance.
In recent years, there has been a move among system engineers towards intelligent control with a qualitative dimension due to the widespread dissatisfaction with quantitative engineering. One of the main attractions of intelligent system design is the possibility of multivariable system control without the need for extensive dynamic models of the process. The main difficulty in the multivariable case is the interaction between variables and sensitivity to faults in various channels. Neural Networks (NN), Fuzzy Logic Control (FLC), and Genetic Algorithms (GA) have been at the forefront of such methodologies and have proved to be strong contenders for other forms of control.
A(z -I),B(z -1),C(z -1) are polynomials in the
backward shift operator z -1 with the following form: -1 -1 -2 -n A(z ) = I+alz +a2z + ...+anz h... -2 B(z -1 ) =b0 + bl z-1 +~.lz +...+bm-Iz -m+ 1
C(z-l) =CO +CIZ-1+'2z-2+...+cpZ-p+I
and f1 = l-z-1 The controller computes the vector of controls using optimisation of a function of the form:
Various synergisms have been described between fuzzy logic, neural networks and genetic algorithms which not only showed that these intelligent structures can interact together but also can make the overall structure, hence obtained, more robust against model uncertainties as well as disturbances. In this paper, we show that synergism between a mathematical model-based approach in the form of self-tuning Generalised Predictive Control (GPC) (Clarke et al. 1987) and Self-Organising Fuzzy Logic Control (SOFLC) (procyk., 1977) is possible with the former being used as a mechanism to make the latter adjust itself to improve the overall system's performance. First, the GPC and SOFLC algorithms are presented. Ne:-.."t, the new modified algorithm which consists of GPC teaching SOFLC is presented. Simulation results using the muscle relaxation processes associated ",ith the drug atracurium are presented. Finally, conclusions are drawn with regard to the new proposed scheme.
N2
J
= L
(P(z -I)y(t +}) -m(t + j=N1
NU
+ L A.U).(oul (t + ) }=l
1»
i»~ (2)
2
where NI is the minimum prediction horizon, N2 is the ma-amum prediction horizon, NU is the control horizon, CO is the future set-point usually presumed mown, A.(j) is the control weighting sequence,
and
P(z -1) is
an
inverse
model
with P(l) = 1. Furthermore, the CC:-1) polynomial in equation (1) is replaced by a fL"Xed polynomial T(z -1) known as the observer polynomial for the predictions (p(z-I).Y(t + }» (Clarke et al.. 1987). This also enables the controller to offset the effect of the 11 operator as a high-pass filter on the input-output data.
2. BACKGROUND BEIDND GPC AND
The minimisation of the cost function (2) leads to the following projected control increment:
SOFLC
l1ul (t)
=gT (C!) -
'1')
(3)
2.l. A model-based approach due to GPC
where
gT is the first row of the matrix
(G T G +A..l)-I Gd T and Gd is the dynamic d d (step-response) matrix of the form given in Clarke et al. (1987).
Generalised predictive control is a self-tuning approach which belongs to a category of Long Range Predictive Controllers (LRPC) which are
188
E(nT - mT) ~ CE(nT - mT)
2.2. A qualitative dimension due to SOFLC
~
(4)
U(nT-mT)+~(n1)
where
Similarly other intelligent control structures such as neural networks, fuzzy logic control has a long history. It stems from the theoretical work of Lotfi Zadeh (1965) who proposed the use offuzzy logic to mimic the human's ability to use imprecise statements to solve complex problems.
~(nT)
is issued by the performance index, E
is the error, and CE is its derivative.
3. A PREDICTIVE SELF-ORGANISING FUZZY LOGIC CON1ROL (pSOFLC) ALGORITHM:
The Fuzzy Logic Self..()rganising Controller (SOFLC) shown in fig. 1 consists of two levels; the first level which consists of a simple fuzzy controller, and a second level which consists of the self-organising mechanism acting as a monitor and an evaluator to the controller performance. In the first level, the input signal to the controller is taken at each sampling instant in the form of error and change-in-error. Each signal is mapped to its correspondent discrete level by using the error and change-in-error scaling factors respectively and sent to the Self-Organising Controller (SOC). The SOC, according to control rules issued by the second level, calculates the output with respect to the inputs. The output control signals are scaled to real values using the output scaling factors and sent to the process being controlled. The second level consists of four blocks: the performance index, the process reference model, the rules modifier, and the state buffer.
The idea behind the new algorithm is to replace the performance index table with a different mechanism that will allow the rules modifier block in fig. 1 to create, delete and alter rules in a similar way to a self-organising fuzzy logic control scheme. The performance index rules contained in a look-up table are standard and used for a wide range of systems and reflect characteristics of a system's time response with adequate damping, overshoot, and settling-time. Since the index used for the rules modifier algorithm is an incremental quantity which excites the rule generation process, we propose to replace it with increments generated by a selftuning algorithm based on prediction (GPC) whose parameters are adjusted via a parameter estimation algorithm, namely Recursive Least-Squares algorithm (RLS). In all the simulations we carried out a second order discrete-time model with no delay was sufficient. Moreover, in the same way in which the performance index output is scaled, the increment from GPC is also scaled to map the real world onto the fuzzy world. This is shown in Fig. 2 where the PSOFLC scheme is represented.
Further details on the design of a SOFLC can be found elsewhere (Mahfouf and Abbod, 1994) but suffice here to concentrate on the learning part. The self-organising controller is based on observation of the trajectory of the process to be controlled. Any deviation from the desired trajectory path should be corrected by modifying the rule or rules responsible for the undesired performance.
Hence, using the above scheme, equation (-+) will become: E(nT -m T) ~ CE(nT -m T) U (nT - m T) + G ·Ou1 (nT)
~
(5)
ou1
The performance index functions as an evaluation criterion of the controller performance, In general terms it measures the deviation from the desired trajectory and issues the appropriate correction to the rule that give the present behaviour. It is derived from linguistic conditional statements by means of using standard fuzzy operations and written in a look-up table form.
where DUI is the increment generated by the GPC algorithm, and this will represent the basis upon which the adaptation is carried out.
4. SWULATION RESULTS
As far as the rules modification procedure is concerned, it can be explained assuming that a process has a time-lag of m samples, this means that the control action at sample (nT-mY) has most contributed to the process performance at the sampling instance n T. Thus, if the present instant is nT, the modification is made to the c"lntroller output U, mT samples earlier, the rule to be included being:
A series of simulations were conducted using the new proposed algorithm (pSOFLC) together with the SOFLC and their performances were compared using the muscle relaxation process associated with the drug atracurium. The continuous model associated with the drug atracurium is highly nonlinear and identified to be
189
of the following Wiener structure (Mahfouf et ai, 1992):
s G(s)=!.L= ·(l+10.6s).eU (1 +4.Ss)(l + 34.4s)(1 + 3.Is)
demonstrates better set-point tracking properties and a steady level of control signal. Moreover, such performance was possible with a number of rules w~ch was lower than in the previous case (18 rules) as fig. 4b shows.
(6)
and the overall nonlinear model is obtained by combining equation (6) together with the following Hill equation:
E
1 efJ- 1+ (0.404)2.98
5. CONCLUSIONS
n)
Soft computing which includes the three intelligent systems approaches, neural networks, genetic algorithms, and fuzzy logic theory has been increasingly embraced by a wide section of systems engineers. Various synergisms are known to exist between the above three strands which can work well together by forming structures which can tackle complex problems in a robust way without the need for mathematical models. In this study, a new algorithm was proposed in which a synergism was shown to be posSlble between a mathematical model-based approach (self-tuning GPC) and an intelligent control system with a qualitative dimension (fuzzy logic control). This substitutes the GPC algorithm for the performance index table used in the standard SOFLC algorithm to modify the rules of the low level represented by the direct fuzzy controller. Whatever the structure of the process in question, a second order model was found to be sufficient for the GPC structure. This is compatible with the performance index table used in the standard SOFLC which reflects characteristics similar to a second order system with specific damping, overshoot, and settling-time. Using the muscle relaxation process as a test bed, the new proposed strategy named Predictive SOFLC (pSOFLC) was shown to be more flexible in terms of tailoring the output response to a particular application with the help of the GPC tuning parameters such as the prediction horizons, modelfollowing polynomial and observer polynomial. This in turn led to a better performance and to generation of a lower number of control rules. This is the focus of extensive research which aims at demonstrating that superior performance need not be linked with a large number of rules but rather to the quality of these rules. It is hoped to conduct future clinical trials using this new scheme which performed equally well with the binary distillation column and the CSTR system.
(XE)2.98
where U is the drug input, EefJ is the actual output (muscle relaxation or paralysis) and X E is the drug concentration. To simulate the above model, a fourth order RungeKutta method with fixed step length was used for integration together with a sampling period of one minute. A set-point profile of 80% then 70% changed every 100 minutes was used. For the SOFLC algorithm a selection of GE = 10/20.0, GC= 10/7.0, and GU = 100.0110 was chosen together with a delay in reward of 1 minute. For GPC a combination of (1, 10, 2, 0) was used for (NI' N 2 , NU).), plus a model polynomial of
P(z -1)
= 3.33(1- 0.7z -1) together
with
an
observer polynomial T(z -1) = 25(1- 0.8z -1) 2 . For the estimation routine a UD-factorisation method was used with an initial covariance matrix of P =100. I where I is the identity matrix and a forgetting factor of p = 0.995. In order to map the GPC increment
8u I into
the
fuzzy part, a scaling factor of G..
= 300.0
was
vu}
used (the input-output variables ,yere scaled by a factor of a 100.0). The first experiments involved using the SOFLC as described in Section 2.2. Figure 3a shows the result of a run. After an initial undershoot then an overshoot, the response displayed a series of undershoots following dO\\TIwards set-point changes, a feature which is undesirable in muscle rela"'(ation therapy. In turn, the associated control signal was oscillatory and took rather a long time to settle to a steady leveL It is possible to tailor the response further but this would require careful tuning of the fuzzy scaling factors. Figure 3b shows the evolution of the number of rules generated which reached a number of 20 rules at the end of the run. When the PSOFLC was used the performance obtained was that of fig. 4a which
6.ACKNO~DGEMENT
The first and last authors acknowledge financial support from a Leverhulme Trust Research Grant.
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Predictive Control (GPC) in the -operating theatre, lEE Proc. PtO, 139, pp 404-420.
7. REFERENCES
Mahfouf, M. and Abbod, MF. A comparative study of generalised predictive control (GPC) and intelligent self-<>rganising fuzzy logic control (SOFLC) for multivariable anaesthesia (1994), Chapter 4 in Intelligent Control in Biomedicine, D.A Linkens · (ed.), Taylor and Francis Pub., London, ISBN 0-7484-0115-6.
Brown, B.H., Asbury, Al, Linkens, D.A, Perks, P. and Anthony, M (1980) Closed-loop control of muscle rela"
Procyk, T.l (1977) Self-Organising Control for Dynamic Processes, PhD Thesis, Queen Mary College, London.
Linkens, D.A, Hasnain, S.B. (1991) Selforganising fuzzy logic control and applications to muscle relaxant anaesthesia, Proc. of the IEE, Pt.D, 138, pp 274-284.
Zadeh L.A (1965) Fuzzy Sets, Information and Control, 8, pp 338-353.
Mahfouf, M, Linkens, D.A, Asbury, Al, Gray, W.M. and Peacock, lE. (1992) Generalised
Fig. 1 A Self-Organising Fuzzy Logic Controller (SOFLC)
GC P roc:e:ss
Fig. 2 Predictive Self-Organising Fuzzy Logic Control (pSOFLC)
191
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Paralysis (%)
-·t·_·- ../--!':";-:.,.- - - - . . . -
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I,
-~
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~
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r
l ,. )ji!ut".b.. _'-'-' ,. ,i )j./.f''''-'-'--[T r\!r f jj. ! O~--~----~----~--~----~----~--~--~ o 300 350 400 50 100 150 200 250 I ,IA.-
i ;'
' _. ___ ._-1
,,1
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Time (m in) Number of Rules
20 (b)
I
15 10 5 0
V 0
50
100
150
200
250
350
300
400
Time (min) Fig. 3 SOFLC of muscle relaxation; solid line: output; dash:reference; dash-dote: control signal.
,
100r----r----~--~----~--~~--~----r---~
Paralysis (%)
H...............
(a)
r!~ , it'1
-------..:..-
.~-----
..--.----....
... ~.
.. . \,...
50 i iji"
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( I' .........
iIi o .,,' o
i: J
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rr .! .1,.-._ ._._._ ....t\ ." /'-'-'-'--
__ 1 - - . , :• • _._. ___ .J
:l 100
fi~ J
... I '
150
200
250
•
Fl 300
350
400
300
350
400
Time (min) Number of Rules
20 (b)
I
15 10 5
1I
0 0
50
100
150
200
250
Time (min) Fig. 4 PSOFLC of muscle relaxation.
192