Integral Control Action and Separated Feedback Control

Copyright © IFAC Theory and Application of Digital Control New Delhi , India 1982

INTEGRAL CONTROL ACTION AND SEPARATED FEEDBACK CONTROL O. L. R. Jacobs Department of Engineering Science, University of Oxford, Parks Road, Oxford, UK

Abstract. Integral control action is identified with the function of state-estimation in separated stochastic control. It follows that the separated controller structure could provide a basis for designing practical sub-optimal controllers which could be implemented using available on-line digital computers. Three case studies are summarised which confirm that separated controllers can reproduce integral action and can lead to substantial improvements in the control of difficult processes. The case studies include practical work on a laboratory-scale water-level control system, and simulation studies in connection with industrial collaborative work on pH control and with clinical collaborative work on feedback control of post-operative pain. Keywords. Bayes methods; control engineering computer applications; Kalman filter; level control; nonlinear filtering; pain control; pH control; state estimation; stochastic control. INTRODUCTION Integral control action is a feature of feedback control whose practical value is much more widely appreciated than is its theoretical significance. There has not hitherto been any well established and natural derivation of integral action as a result in optimal control theory. The resulting gap between theory and practice has had two unfortunate consequences: (i) to discredit theory in the view of some practioners and, more seriously, (ii) to deny designers of practical systems access to some useful insights offered by theory. This paper aims to clarify the theoretical significance of integral control action and to show how the resulting insight can be exploited in the design of practical control sys terns, especially computer-controlled syst~ms . It will be argued that the relevant optimal theory is stochastic control theory and that integral action can be identified with the estimator of the separated optimal feedback control structure shown in Fig. I. Other authors lJ.·a veconsidered the relationship between integral action and deterministic theory: for example Johnson (1968) and Athans (1971) used special cost functions in which rate of change of control action replaced the conventional control action term u representing cost of control. A similar approach is found in the, otherwise stochastic, context of practical adaptive control where it has been suggested (AstrBm, 1980; Clarke, 1981) that integral action be enforced by using a control law designed to generate increments in control action. The stochastic viewpoint

u

45

has been used by Fuller (1976); but he worked exclusively in the frequency domain and was thus unable to reveal the separated structure of state space theory. It is this structure which offers possibilities for exploitation in designing practical systems. The practical usefulness of the separated controller structure is well known, both in principle, see for example Athans ed (1971) or Kwakernaak and Sivan (1972), and in practice, see for example Choquette, Noton and Watson (1970) or Balchen and co-workers (1980). What seems not previously to have been emphasised is the role of this structure in generating integral action and its significance as a basis for sub-optimal designs. The possibilities are indicated here by summarising three case studies, each requiring some integral action, in which performance under conventional proportional plus integral (PI) control is compared to what can be achieved by a control designed according to the separated structure of Fig. I. For processes which are easy to control the performance using a separated structure may be only marginally better than under PI control, but for difficult processes it can be significantly better. 'Difficulty' in the case studies took the form of non-linearity, but it could also be associated with high-order plant dynamics, dead-times or multiplicity of inputs and outputs. The three cases are : (i) A computer-controlled laboratory-scale water-level system which was fairly easy to control, having only mild non-linearity. This study confirms that separated control does indeed reproduce integral action. (ii) A simulation study undertaken as part

O. L. R. Jacobs

46

of a continuing collaborative project where an on-line digital computer is used to regulate pH of effluent from a full-scale industrial process. This problem is wellknown to be difficult because of the logarithmic non-linearity inherent to pH processes. The separated control was much better than the PI. (iii) Another simulation study undertaken as part of a collaborative clinical project where a microcomputer is used to control infusion of pain-killer for relief of postoperative pain experienced by patients recovering from major surgery. The demand system by which the patient indicates pain but not comfort is so severely non-linear that integral action cannot be provided by conventional PI control. The separated controller provides the necessary action and gives control which is significantly better than what can otherwise be achieved. The simulation studies can do no more than indicate practical ways of exploiting the theoretical insight of the separated structure. It is hoped that some full-scale industrial and clinical results will be available at the Symposium. The presentation falls into two sections. Section 2 of the paper summarises the theoretical background of LQG optimal control theory and shows that controller dynamics arise within the estimator, not the control law, of Fig. 1. Examples are given to show how problems of disturbance rejection, of target tracking and of stabilisation all confirm this conclusion. Integral action is usually no more than a sub-optimal approximation to what would be optimal controller dynamics; but in many cases, including disturbance rejection, it leads to good suboptimal control. Section 3 presents the case studies. These show that an on-line computer generating explicit estimates of disturbances can give sub-optimal performance which, for difficult control problems, is significantly better than can be achieved by conventional PI control. INTEGRATORS IN OPTIMAL STOCHASTIC CONTROL

prohibitively corrupted by noise. It is well known that the solution to the stochastic optimal control problem is a separated controller having the structure of Fig. 1 in which estimation is done by a Kalman filter and in which the control law is certainty-equivalent and linear. The Kalman filter generates an estimate met) of the current state x(t) according to an equation such as

m= Am

+ Bu + ~ (y-Cm)

( 2)

where A, B,C are matrices of the controlled process. The control law is u

= -Kcm,

(3)

The gains KE and KC are in general timevarying but commonly converge to steady-state values. It is this steady-state solution which corresponds to the linear fixedparameter controllers of conventional control. Figure 2 is a block diagram showing such a steady-state optimal LQG controller. The only dynal!lics in the controller are those of the Kalman filter: they are the integrators, represented by the block 'l/s', whose outputs generate the estimate m of equation (2). This figure is the basis for the conclusion that controller dynamics in general, and integral action in particular, are associated with the estimation part of stochastic control. The conclusion carries over to discrete-time LQG problems and also to the generality of non-linear control problems (Bertsekhas, 1976) whose optimal controllers have the separated structure of Fig. I, with the control law being an instantaneous function of a conditional probability distribution for the current state, and with dynamics associated with the estimator which generates the probability distribution. Practical controllers are not necessarily optimal but they frequently include dynamic operations which can now be identified as sub-optimal versions of the integrators in the Kalman filter of Fig. 2. Some examples are as follows:Disturbance rejection

The relevant control theory for the purpose of this paper is LQG optimal control theory (Athans ed, 1971). Stochastic, rather than deterministic, theory is needed because it is uncertainty in the control problem which requires feedback in the controller. Deterministic versions of the linear quadratic (LQ) problem lead to solutions which can be expressed either in the form of prespecified future control action u(t) or in the form of a linear feedback control vet)

Kx(t)

( 1)

where x(t) is the current value of state. This feedback version (1) is of little use for practical purposes because the states x are not usually all available for measurement. They are frequently defined as successive timederivatives whose direct measurement would be

Figure 3 shows a feedback control system where integral action would be needed in the controller H(s) in order to reject the disturbance z. To pose this type of problem in terms of optimal control theory the disturbance would be modelled by some linear differential equation. This might be

z=0

( 4a)

which would represent a constant disturbance, or its stochastic equivalent

z = E;

(4b)

which is Brownian motion. A Kalman filter to estimate z from the measured output y would take the form

m=

K(y-Cm)

(5)

47

Integral Control Action of an integrator driven by the innovation y - Cm. This estimate would provide the basis for control action u to nullify the disturbance. It can be concluded that integrators used for disturbance rejection can be identified as, possibly sub-optimal, estimators of the disturbances. It is this type of integral action, disturbance rejection based on explicit estimates of the disturbances, which is illustrated inthe case studies of Section 3. Target tracking Practical feedback controllers for tracking moving targets Xo are usually designed to include integrators up to a number, known as the 'type number', not less than the degree of a polynomial representing the target motion. Constants, ramps or parabolic target motions are regarGed, for purposes of optimal control theory,as satisfying differential equations x•

0

'x ... x 0

0

0

for constant x

0

for ramp Xo

0

for parabolic x

(6a)

0

(6b) 0

(6c)

In stochastic control theory the target motion would be represented as the output of a linear system driven by white noise and having a transfer function corresponding to one of equations (6). The integrators which make up the type number are identified here as parts of a Kalman filter which estimates the states of this system modelling the target motions. A tutorial example of how controller dynamics arise in the optimal solution to a problem of stochastic target tracking can be found elsewhere (Jacobs, 1974). Stabilisation Many practical controllers include dynamics, in the form of 'phase advance', 'open-loop zeros' or 'derivative feedback', whose purpose is to stabilise closed-loop performance. Such controller dynamics are needed when the re is only a single measurement y from which to obtain estimates about current values of states of a controlled process of order greater than one. The states of such a process are usually successive time-derivatives of the output y and their estimation requires Kalman filters. The practical controllers mentioned above can be regarded as sub-optimal approximations to these Kalman filters. CASE STUDIES Water-level control system Figure 4 shows a laboratory-scale control system in which the inflow xI of water to a tank is adjusted by a signal u from an online micro computer (380z) so as to regulate the level X2 against unpredictable disturbances in outflow z which can be introduced by manually adjusting the position x3 of a

valve. Both inflow and outflow are subject to non-linearity as shown in the block diagram of Fig. 5. The double-edged block generating v :u) represents non-linearity between inflow and valve-stem position; the noise ~I represents uncertainty about that position, mainly due to stiction. The out f.low depends on x3 but is slightly affected by level x2 according to a relationship which is approximated by z

= x3

+ 0.0035 x3 (x 2 - 50) + ~3

(7)

where all variables are scaled to the range 0-100 and ~3 is a noise representing modelling error. The time constants TI and T2 represent respectively the response times of the inflow valve and of level to flow rate. The measured output y is corrupted by noise n having variance 2 which increases with . . fl ow accord~ng ~n ton a relationship approximated by 2 2 (8) on = 0.02 (I + x I / 100) •

°

This process is sufficiently easy to control, in that its non-linearities are mild and its order (two) is low, that a conventional PID controller can give satisfactory performance. Figure 6 shows a time-history of level under well-tuned (Ziegler Nichols) PID control during a test sequence which demonstrates rejction of disturbances due to large manual changes of outflow at times tl and t4 and shows target tracking to follow changes of set point at times t 2 , t3, t5' Disturbance rejection is achieved by the integral component of PID control. A separated controller having the structure of Figs. I and 2 was implemented in the microcomputer. It was based on a discretetime model of the tank having three state equations. xI (i+l)

(1-ll/TI)x I +

(ll/TI)(v(u)+~I)

(9a)

x (i+l) x + (A/T ) (xl-z) (9b) 2 2 2 x (i+1) ( 9c) 3 where A is the sampling period and i is an integer representing discrete time: for simplicity of notation here and subsequently the index (i) is omitted from current values of variables on the right hand side of equations (9). The controller included a look-up table to take out the non-linearity v(u) in the flow-valve; this was followed by an extended Kalman filter based on equations (7)-(9) cascaded with a minimimal-variance control law designed to set to zero the expected value of error two steps ahead in time by choosing u(i) to satisfy m2 (i+2I i ) xo' (10) The Kalman filter used an adaptive feature to detect changes in position x3 of the outflow valve. The innovation sequence was normalised and integrated to provide an indicator which should normally be close to zero: if this indicator exceeded a threshold value the increase was interpreted as being due to a change in x3 to some new constant value; the

48

O. L. R. Jacobs

relevant column of the covariance matrix in the filter was then automatically increased so as to initiatere-estimation of x 3 .

is conventionally achieved by integral action.

Conventional PID control from y to u here is usually unsatisfactory because the 'gain'of Figure 7 shows a time-history of level under the pH non-linearity changes by an order of the separated control during a test sequence magnitude for every unit of pH. The consimilar to that of Fig. 6. There is little sequence of this is that PID control would to choose between the quality of performance have to be severely detuned to achieve stability under the two controllers. This confirms and then could not achieve good steady-state that PID control is close to optimal for the performance. It has been shown (Jacobs, water-level process and that the stochastic Hewkin and While 1980) that significant improvements can be achieved by using an oncontrol theory does lead to controllers having the known nice properties of estab~ished line computer programmed to accept and process feed forward signals about the upstream practice. variables xI and z and provided with a lookIn other applications, such as the more diffup table to take out the logarithmic nonicult case studies of Sections 3.2 and 3.3, linearity. As part of continuing collaborative the separated structure achieves significant research into on-line computer control of pH improvements by exploiting the on-line computing in industrial processes a simulation study power which makes it possible to implement has been made of the performance of the look-up tables, extended Kalman filters and process of Fig. 9 under the separated control complex control laws. One constraint on of Figs. I and 2. Results of the study what can be done would be the speed and size are summarised here. of available computers, but this has not been investigated here. The 380z was The simulated system had three state variables: xI representing inflow in m3 min- I , programmed in interpretive CONTROL BASIC x2 representing ionic concentration difference (Clarke and Frost, 1979) which is not fast. The sampling time for the separated controller of effluent in units of normality, and x3 was therefore representing the ionic concentration z of influent, also in units of normality. Ionic t, = 2.19 s concentration difference, defined by and the same sample time was used for the x = [OH-] - [H+] , (12a) PID control of Fig. 6. Much faster sample times could be achieved with PID control but is a convenient measure of acidity because these gave only marginal improvements in when combined with the definition of pH performance. + pH = - 10gl0 [H ] (12b) Simulation of pH control Figure 8 is a diagram of a typical industrial system for neutralising effluent. The inflow u of reagent to a continuous stirred tank reactor (CSTR) is adjusted so as to regulate the pH of effluent flowing through the tank, against unpredictable upstream disturbances to inflow xl and ionic concentration z. Feedback control of pH is a notoriously difficult problem. The difficulty is caused by severe logarithmic nonlinearity between pH, which is measured, and concentration which is controlled. Figure 9 is a block diagram of a simple model of thee on trolled process in such a pH system: it shows an idealised version of the pH-nonlinearity and includes the time constant T and gain K of the mixing tank, both of which depend on the inflow xI (assumed here to be acidic) according to

it gives the symmetrical non-linearity sketched in Fig. 9. The simulated discretetime state equations were xI(i+l)

glx1+al+~1

x2(i+l)

X -(t,x /V) (x +x )+(t,[NaOH]/V)u

2

l

(13a) 2

3

( 13b) ( 13c)

with the stochastic upstream variables xI and x3 re presented by Wiener processes having rate const2nt gj' offsets aj and noises ~j of variance OJ related respect~vely to continuous time constants T., mean values ~j and variances Var(xi) ac~ordin2 to (for j=l 3) gj=(I-t,/Tj); aj=ij/(I-gj); Oj=var(j)/(I-g ij ) (14)

These states are observable provided that (15)

( I I) K = [NaOH] /xI The separated controller included a novel where V is the volume of the mixing tank and Bayesian estimator (to be reported elsewhere) [NaOH] is the concentration of caustic soda to handle the pH non-linearity as well as used as neutralising reagent. Equations (11) an extended Kalman filter to estimate the represent additional non-linearity, similar states of equations (13). No feedforward to that of equation (7) in the water-level was included. The resulting estimates were system. Non-linearities in the reagent flow cascaded with a control law, designed (Hewkin, valve are assumed to have been eliminated 1979) to mini~se errors in pH by choosing by a cascaded flow-control loop using a measureu(i) to satisfy ment of reagent flow: no such measurement was used in the water-level system. The measured ( 16) m2(i+11i) = x +I.2s 2 (i+lli) o output y is corrupted by noise n. The upstream concentration z enters Fig. 9 as the type of where !2 is predicted standard deviation. disturbance shown in Fig. 3 whose rejection

49

Integral Control Action The simulation included modelling errors so that simulated va~ues of such pl1nt parameters as V, [NaOH], Tj' Xj' Var (Xj), on could be randomised away from the nominal values on which the controller was based. Figure 10 shows a typical time-history of simulated pH regulated against stochastic upstream disturbances by the separated controller: it also shows the simulated performance under conventional PID control with the same target value Xo and the same sequence of random variables. The superiority of the separated control is very marked and was found to be maintained in other simulations. It is concluded that for a difficult process such as pH-control, the separated controller should give much better performance than can plain PID control. Integral action is present in the separated controller in the form of the explicit estimate m3 of upstream ionic concentration. The program of continuing collaborative industrial work on pH-control will include full-scale trials of the simulated separated control. It will also include investigation of the extent to which the separated controller's estimates of upstream variables are adequate substitutes for true feedforward signals. Simulation of pain control Figure II is a diagram of a clinical feedback control system, known as 'demand analgesia' , which relieves pain experienced by patients recovering from major surgery. The infusion rate u of a pain-killing drug is automatically controlled by an on-line microcomputer (380z) in accordance with demands which the patient makes by pressing a button. Clinical trials have already demonstrated (Jacobs and co-workers, 1981) that such an automatic system can give better pain control than can conventional nursing practice. These trials have so far only used simple proportional control equivalent to giving a predetermined dose (bolus) of painkiller, subject to safety checks, for each demand. Demand systems like this include a non-linearity wh~ch makes it impossible to introduce tne integral action which is needed for further improvements in performance. As part of continuing collaborative research a simulation study has been made of what could be achieved by using the separated control of Figs. I and 2. Results of this study are summarised here. Figure 12 is a block diagram representing the patient as a controlled process. It models the pain experienced by the patient as the difference between discomfort and comfort and shows, in the double-edged block on the right, the non-linearity which arises because the patient makes no demand when comfort exceeds discomfort. Only for positive pain can the output y be assumed proportional to pain. This output y is a demand rate in button presses per unit time.

The amount of pain corresponding to a demand rate of Is-I is called one 'pang' and variables such as pain, discomfort and comfort are all expressed in these units. The comfort here depends on the concentration xI of pain-killer in brain tissue according to neuophysiological and psychological factors about which little is known: this dependance is represented by a gain x2 called 'relief'. The concentration xI depends on infusion rate u according to pharmacokinetics which were modelled by a transfer function G(s) of the form G(s) =

K

l+sT

( 17)

The discomfort is modelled as the sum of a random white noise ~3 and an exponentially decaying term x3 which starts from an initial condition representing the pain due to surgery and is forced by another white noise ~I. Healing is represented by the time constant T~ with which x3 decays; the variance of ~3 ~s assumed to decay with the same time constant. Justification for the model of Fig. 12 is given elsewhere (Reasbeck, 1982) • The discomfort in Fig. 12 is the type of disturbance shown in Fig. 3 whose rejection is conventionally achieved by integral action. However, the demand non-linearity makes it impracticable to use ordinary PI control because negative error signals are never generated and so there is no way in which the integral component of controller output could ever be decreased. One conunercial system (White, Pearce and Norman, 1979) overcomes the difficulty by what amounts to introducing a positive non-zero desired value of pain. As part of continuing research into feedback control of post-operative pain a simulation study has been made of the performance 0 f the process of Fig. 12 under the separated control of Figs. I and 2. Results of the study are summarised here. The simulated sys tern had three state variab les: one to represent the pharmacokinetics G(s), one to represent the relief x2 and one to represent the exponentially decaying discomfort x3. There was provision for introducing modelling errors and for repeated simulations using different control laws with the same sequences of random numbers, as in the pH study. A separated controller was simulated which used the Bayesian estimator to handle the demand non-linearity together with an extended Kalman filter to estimate all three states. The estimates were used in a control law, similar to those of equations (10) and (16) but designed to make the expected comfort one step ahead in time greater than the expected discomfort by an amount proportional to the nominal standard deviation of the unpredictable discomfort ~3 mI (i + 11 i ) m2 ( i + 11 i) +° I 2 ( I + I[ i) =m3 ( i + I I1. ) +ko:3 (1+ I)

( 18)

50

O. L. R. Jacobs

The covariance term ffl2 on the left of (18) arises as part of the expected value of the comfort XjX 2 . A value of about 1.0 for the constant of proportionality k on the right of (18) was found to give good performance. Figure 13 simulated separated of random control.

shows typical time-histories of indicated pain y under the control and, for the same sequence variables, under simple proportional

practical experience indicates integrations?' W. A. Badran assisted with the pH simulation results and M. P. Reasbeck with the pain control results; they are both supported by S.R.C. REFERENCES

~trBm, K.J. (1980). Design principles for self-tuning regulators. In H. Unbehauen (Ed.) Methods and applications in adaptive control, Springer, Berlin. pp. 1-20. The total number of demands made by the Athans, M. (1971). On the design of PID patient provides a convenient performance regulators using optimal linear regulator criterion: the smaller the better. The theory. Automatica, 7, pp. 643-647. superiority of separated control is sigAthans, M. (Ed.) (1971). -The linear quadratic nificant when judged by this criterion. A gaussian problem, IEEE Trans. Autom. secondary criterion is that the amount of Control, AC-16. pain-killing drug administered should be Balchen, J.G., N.A. Jensen, E. Mathisen, and as little as possible consistent with Saelid, (1980). A dynamics positioning comfort. This amount is greater under system based on Kalman filtering and separated control than under proportional optimal control, Model. Ident. and control, in order to achieve the greater Control, I, pp. 135-164. comfort, but not dangerously so. Bertsekhas, D-:-P. (1976). Dynamic programming and stochastic control, Academic Press, It is concluded that for this difficult New York. process, where there is the severe nonChoquette, P.,A.R.M. No ton, and C.A.G. Watson linearity of the demand characteristic as (1970). Remote computer control of a~ well as uncertainty about the relief x2' the industrial process, Proc. IEEE, 58, separated controller can give much better pp. 10-16. performance than can conventional proporClarke, D.W., (1981). In C.J. Harris, and tional control. Integral action, which S. Billings (Eds.), The theory and cannot be satisfactorily introduced by conapplication of self-tuning control, ventional control, is present in the separated P. Peregrinus. controller in the form of the explicit Clarke, D.W. and P.J. Frost, (1979). Control estimate m3 of discomfort. BASIC for microcomputers. In lEE Conf. Pub 172 Trends in on-line computer control The program of continuing collaborative systems, pp. 53-57. clinical work on pain-control will include Fuller, A.T., (1976). Feedback control live trials of the simulated separated systems with low-frequency stochastic control. disturbances, Int. J. Control, 24, pp. 165-207. CONCLUSIONS Hewkin, P.F., (197Y). Control of pH using modern algorithms and on-line computers, It has been argued that, in feedback control D.Phil thesis, University of Oxford. systems, controller dynamics in general and Jacobs, O.L.R., (19i4). IntrOduction to integral action in particular can be regarded control theory, pp. 283-288, OUP, Oxford. as sub-optimal approximations to Kalman filters Jacobs, O.L.R., P.F. Hewkin, and C.While, which perform the estimation function in (1980). On-line computer control of pH separated stochastic control. This suggests in an industrial process, Proc. IEE -D, that the separated controller structure could 197, pp. 161-168. provide a useful basis for designing Jacobs, O.L.R., R.E.S. Bullingham, W.L. Davies practical sub-optimal controllers, in which and M.P. Reasbeck, (1981). Feedback the functions of estimation and of control control of post-operative pain, In IEE are explicitly separated and may be sub-optimal. Coni. pub. 194, Control and its applicaSuch controllers can be impletrented using tions, pp. 52-56. available on-line mini and microcomputers. Johnson, C.D. (1968). Optimal control of Case studies have been summarised which the linear regulator with con9tan~ disindicate that such controllers could lead to turbances, IEEE Trans. Autom. Control, substantially improved performance in the AC-13, pp. 416-421. (1970). Further control of difficult processes. Kwakernaak, H. and R. Sivan, (1972). Linear optimal control systems, Wiley, N;W---ACKNOWLEDGEMENTS York. Reasbeck, M. (1982). Modelling and control A. T. Fuller first drew my attention, in of post operative pain, D.Phil. thesis, about 1960, to the question which University of Oxford. motivated this paper: 'Why is it that White, W.D., D.J. Pearce, and M. Norman, (deterministic)optimal control theory seems (1979). Post operative analgesia, to indicate that control should be a BMJ, ~, pp. 166-167. function of derivatives of the output, whereas

Integral Control Action

OUTPVTy

INPUT u

_I

Fig . 4 Water-level control system

1--------------------, 1

1

1 I

ESTIMATES

OF

1 1

£O~T.!!.O.!±~

x

______________ __ 1I

Fi g. I Sepa rated optimal stoch a stic feedb ack c ontrol u

rSTIMATOR - - - - - IIKALMAN FILTER)

B

1 : 1+ I

1 -

-I CONTROL 1 1 LAW

1

1 1 1 1

1 I I Iu

1

1

Fi R. 5 Block diagram of water-level controlled process

L.£~l!!O.!c.L~ ____ ~_l ____ : Fi g . 2 Optimal LQG controller

70 50 30

DISTURBANCE

Fi r. 6 Wa t e r-level under PlO c ontrol

TARGET VAlUE "0

+

Fi g . 3 Feedback system needing integral action

MORE

t

LEVEL

FLOW

70 50 30

TIME t,

Fi g. 7 Water-level under separated stochastic control

o.

52

L. R. Jacobs

+

REAGENT

_u ~CI============~~ INFLUENT FLOW xl

U

ro~.

•

~:;:;,;.T;,;.R_ _~f=,.;D

1 ...'_'i_.• . _. •. _. •..;• •

·IF IT HURTS, PRESS THE BUTTON'

•.. _.•. . _i •.

Fi g . 8 pH control system

Fig. 11 Demand analges i a

Fi g . 9 Block diagram of pH controlled process

DISCOMFORT pH (PlO CONTROL)

Fi g . 12 Block diagram of patient as controlled process TIME pH (SEPARATED CONTROL)

PANGS (PROPORTIONAL CONTROL) 395 DEMANDS

0 .1

TiME

Fi g . 10 Simulated pH control (with 10?, modelling errors)

PANGS (SEPARATED CONTROL) 94 DEMANDS

0 .1

Fig. 13 Simulated pain control (with 20i. modelling errors)