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Control Systems 2003 [Dr. Illingworth]

This experimental lecture examines some of the control mechanisms used by living organisms, and compares them with closed loop control systems constructed by engineers, business people and politicians. All such systems are governed by similar rules. The comparison is sometimes instructive, indicating the selection pressures that have shaped the biological systems, and also suggesting ways in which some human social systems might be changed.

Topics

The first recorded example of an automatic control system was in ancient Greece, and over the last century electrical and mechanical engineers have learned to design control systems with impressive accuracy and stability. We take their achievements for granted, expecting that compact disks will deliver realistic music recordings, that vehicle power steering will be accurate, and that space probes will successfully navigate to Neptune, without realising that teams of engineers spent many years designing the systems to achieve these objectives. The behaviour of artificial control systems is often analysed using the Laplace transform, and there are innumerable text books on control theory that adopt a rigorous mathematical approach. [See, for example, Dorf, RC & Bishop, RH Modern Control Systems 8th edn. Addison-Wesley 1999; Kuo, BC Automatic Control Systems 7th edn. Prentice Hall 1994. Leeds University Library has these and similar titles.]

Biological and social control systems are no less complicated, but (except for economic models) they are rarely analysed mathematically. This is partly because the numerical parameters are difficult to estimate, and partly because living organisms contain numerous interlocking control loops, where it is difficult to study one system in isolation. Some of the most detailed analysis has been performed on the cardiovascular system, and on the proprioceptive systems controlling voluntary muscles. [See, for example, Handbook of Physiology American Physiological Society 1981, or other advanced physiology texts.] Metabolic pathways have been extensively modelled using computers [for some examples see: Cornish-Bowden, A & Cardenas, ML (eds) Control of Metabolic Processes Plenum Press 1990] but the analysis of their control behaviour is often superficial. Relatively few good books have been published in this area, but these include Segel, LA [ed] Mathematical models in molecular and cellular biology Cambridge University Press 1980, and Goldbeter, A Biochemical oscillations and cellular rhythms Cambridge University Press 1996. The explanations in this lecture are based in part on Phelan, RM Automatic Control Systems Cornell University Press 1977, which has the great advantage of being easy to read! All these books are in the Library.

It is possible to have open-loop control systems, and biological examples abound. Under normal circumstances, cellular control mechanisms lead to the transcription and translation of the gene for the CFTR protein in pulmonary epithelial cells, and this usually results in the insertion of optimal amounts of a cAMP-regulated chloride channel into the plasma membrane. But no mechanism checks whether this has actually taken place, and in individuals homozygous for the delF508 mutation, a defective protein is assembled and directed to the wrong cellular compartment, with disasterous consequences for the person concerned. The distinguishing feature of closed loop control systems is that a check is made on the outcome, and corrective measures are initiated if the result differs from the original plan.

Certain fundamental features are common to all closed-loop control systems, which are illustrated in figure 1. Every closed loop system keeps a controlled variable C as close as possible to some reference value R despite interference by an external load L which disturbs the result. In order to achieve its objective the control system subtracts C from R so as to generate an error signal, E. This error signal regulates the flow of material or energy M into the controlled system so as to minimise E and compensate for the effects of the external load.

Every closed loop system needs a reference value which provides a target to aim for. This is true even for biological control systems, although sometimes the targets are obscure. There is no requirement for the target to stay constant, although they often do. Biological reference values may be genetically determined, for example through the amino acid sequences of regulatory proteins, which define their binding constants for allosteric effectors. Behavioural targets for an organism might also reflect the genetically programmed "wiring diagram" for the central nervous system.

Living organisms have innumerable feedback loops and it seems self-evident that they must be advantageous. However, it may not be obvious where these benefits accrue. Can you specify any biological advantages arising from the regulation of phosphofructokinase and glycogen phosphorylase by 5'AMP? Here are two suggestions.


The overall performance of any closed loop control system can be judged in terms of the following criteria:

1) Accuracy: How closely does it approach the target value?

2) Stability: Is it free from overshoots and oscillations?

3) Resilience: How well does it cope with abnormal loads?

4) Speed: How quickly does it respond to a transient load, or to a change in the target value?

Real systems must compromise between these four conflicting requirements, since the features which make for good performance in one area may well have deleterious effects elsewhere.

It is sometimes necessary to distinguish between the control system proper and the final control elements which actually deliver energy to the controlled system. The control system itself (e.g. nerves, electronic amplifiers, office procedures) may exhibit ideal behaviour, but the final control elements (e.g. muscles, motorised valves and bloody-minded employees) may suffer from a variety of imperfections. It may be necessary to build a secondary control system around the final control elements to ensure that they obey their instructions with reasonable fidelity.

Biological systems, in particular, abound with these inner control loops. Apparently simple actions, such as reaching for a glass of water, may themselves form part of a larger mechanism for the long-term regulation of salt and water balance. Even the simplest muscular actions depend on a vast network of neural feedback loops which respond to visual, proprioceptive and tactile error signals until the immediate objective has been achieved. All the individual nerves and muscles involved in the action are themselves dependent on thousands of intracellular control mechanisms which regulate every aspect of their metabolism, and ensure that they will respond quickly and accurately to the instructions received from the higher centres.

It is often convenient to divide closed loop systems into the standard building blocks shown in figure 2. H, G1, G2 and G3 are the transfer functions which define the relationship between the input and output signals for each of the blocks. Transfer functions in general are mathematical operators (analogous to multiplication, differentiation etc) but they may sometimes be simple numerical constants. If the blocks function independently (i.e. have zero output impedance and infinite input impedance) then the various transfer functions may be combined in order to calculate composite functions for the complete system. For example:

M1 = G1 E'  M3 = G1 G2 (R - H C) + L

The symbol E' is used above in preference to the error E because the output signal C might be modified within the feedback loop. These considerations lead eventually to the conclusion:

   G    G3   
C=-------------R+-------------L  (equation 1)
  (1 + G H)     (1 + G H)   

where G = G1 G2 G3 ... etc. The composite operator G is the resultant transfer function for the whole of the forward loop while H is the resultant transfer function for the feedback loop. Equation 1 is the operational equation for the system.

C   G
--- = -------------
R   (1 + G H)
  [at constant L]    
C   G3
--- = -------------
L   (1 + G H)
  [at constant R]

In an ideal world, C/R would be constant and C/L would be zero: the output should exactly follow the reference signal and be independent of the load. This requires that the loop gain G H should be as large as possible when traversing both the forward and the feedback routes. The ratio C/R is variously known as the closed loop transfer function or the closed loop gain. Unfortunately, electrical and mechanical engineers sometimes use different names and symbols for the same basic concepts, although the underlying theory is identical:

mechanical term  electrical term
forward loop transfer functionG  amplifier open loop gainA
feedback loop transfer functionH  feedback fractionB
closed loop transfer functionC/R  closed loop gainVout/Vin

It is clear from the relationships above that when the amplifier open loop gain is very large, the closed loop gain comes to depend only on the properties of the feedback network and not on those of the amplifier itself. This conclusion is of enormous practical importance to electrical engineers, because it proves very easy to design relatively imperfect, high-gain operational amplifiers, and then correct them with a "perfect" feedback loop. What is more, it is possible to create a range of feedback loops with a number of desirable properties (logarithmic, integral or derivative transfer functions for instance) and thereby use the same basic amplifier to achieve a wide variety of different objectives.

All this may seem a far cry from biology, until one reflects that voltage-gated ion channels, hormone receptors and protein kinase signalling cascades are all examples of high-gain biological amplifiers, that commonly form part of negative feedback systems. They obey the same rules as their electronic counterparts.

It is instructive to examine the properties of the simple mechanical system shown below. Very similar considerations apply to complex electrical, biological, economic and political systems, although these may be more difficult to visualise:

In our example, a force M is applied to a mass of inertia I attached to a spring with elastic modulus k. The motion of the system is restrained by a damper which exerts a drag proportional to the velocity of the mass. This arrangement is very similar to the springs and shock absorbers on a motor vehicle suspension. If the damper were omitted the mass would execute simple harmonic motion on the end of the spring. It is desired to stabilise the mass and to control its displacement C from the equilibrium length of the spring.

Since   force = mass * acceleration   we can write down a differential equation:

M - a dC/dt - k C = I d2C/dt2   (equation 2)

where "a" is the viscous damping coefficient. [The left hand side of this equation is simply the applied force M minus the contributions from the damper and the spring.]

Some of the terms may be negligible in real systems. If we regard M as the restoring force generated by the control system in response to an error signal E, we can distinguish various "orders" of control system, viz.

zero order control systems - when a and I are very small, the elastic component k may dominate. Such systems have a natural equilibrium position which they adopt in the absence of a control input, M. Some intracellular feedback systems fall into this category.

first order control systems - the inertia I is negligible but some damping is present, and possibly the spring as well. In the simplest case of pure viscous damping with no spring the system lacks a unique resting position when the control input is removed. Resistive losses provide the damping term in electrical circuits, while in thermal or chemical control systems the transfer of heat (or the material needed to change a chemical concentration) gives rise to analogous behaviour. In all such cases, the effect on C depends on the duration of the restoring input M, and a larger M is necessary if C is required to change quickly.

second order control systems - inertia is present, and possibly the damper and the spring as well. Such systems are the most difficult to control, and once pushed may remain in motion for considerable periods of time. Inductance and capacitance are responsible for similar effects in electrical networks. It is hard to visualise momentum and inertia in purely chemical systems. Where a cell is committed to a process which takes time to complete (for example, mRNA translation) there will be unfinished product in the pipeline with some inertial properties. The real inertias of arms and legs are beautifully compensated by the superb control mechanisms responsible for neuromuscular coordination within the central nervous system.

The complete solution to equation (2) is the sum of the particular integral (which depends mainly on the forcing function "M") plus the complementary function, which is the solution of the corresponding homogeneous equation. The homogeneous equation is obtained by setting M = 0 after which it is possible to write down an auxiliary equation:

I s2 + a s + k = 0     (equation 3)

This is sometimes referred to as the characteristic equation, since the two roots S1 and S2 define the principal characteristics of the controlled system. Using the usual formula to find the roots of a quadratic equation,

The complementary function is then:

c = B1 . eS1.t + B2 . eS2.t

where B1 and B2 are both arbitrary constants and t represents time. In the absence of a control input M, both roots must always be negative and c will eventually settle to some finite value as t increases.

The response of the controlled system to a sudden load or a change in target value depends on the square root term in the roots of the auxiliary equation. If a2 < 4kI then there will be a pair of complex roots and we have an under-damped system which will repeatedly overshoot the final value whenever it is disturbed. Some time may elapse before things settle down. If a2 > 4kI then we have an over-damped system. There will be no overshoot, but the system will still approach the final position very slowly if the damping term a is large. If a2 = 4kI then the square root term will vanish from the solution, and we have a critically damped system which will stabilise at the final position faster than either of the other two alternatives.

The situation is more complicated when there is an energy input "M". There is now a possibility of sustained oscillations. The characteristic equation has a similar form to equation (3) but the coefficients will depend on the feedback network, and there might be additional terms. The system will be stable provided that all the roots are negative. When there are pairs of complex roots then the real parts must be negative or the system will be unstable.

There are a number of different strategies which can be adopted to minimise the errors within a closed loop control system. The easiest technique is to apply a fixed restoring force whenever the error exceeds some pre-set bound. This method of on/off control is used by an ordinary room thermostat and may be reasonably satisfactory for first order systems with heavy damping under a fairly constant load. On/off control is not appropriate for pure zero order systems (why?) and will cause second order systems to oscillate uncontrollably if any hysteresis is present.

LIVE LECTURE

ILLUSTRATION

AT THIS POINT

It is difficult to achieve precise results with on/off control and a widely varying load, because the restoring force that produces a rapid response after a large excursion will be too coarse as the set point is approached. In this case the obvious policy is to make use of proportional control where the energy or material input M is directly proportional to the size of the error signal E. It is necessary to pay due regard to the arithmetical sign of the error in order to ensure that the restoring force is applied in the correct direction!


Oscillations may be caused by phase-shifts within the control system. The signal applied by the control system should be out of phase with the error, so that we eat when we are hungry and stop when we are fed. This is readily achieved for slowly varying signals, but as the signal frequency increases the increasing phase shifts within the system will eventually convert negative feedback into positive feedback. The controller now responds to out-of-date information and adopts a perverse strategy, making matters worse instead of better. This can cause sustained oscillations. Some high frequency "noise" is present in all control systems and this can trigger oscillations at high loop gains, even when the controller is initially on target. The ONLY solution is to reduce the loop gain at higher frequencies, as illustrated by the Bode Plot in figure 6.

The Bode plot shows gain and phase shift versus frequency. The gain and frequency scales are usually logarithmic. The example above has a single curve for phase shift: this system will be stable if the loop gain follows the continuous sloping line, but will oscillate if the loop gain follows the dotted line.

We must be clear about the terms defined earlier. The loop gain (GH) is the composite transfer function when traversing the entire system including both the forward and the feedback pathways. Do not confuse this with either the open loop gain G for the error amplifier acting alone, or with the closed loop gain G/(1 + GH) which defines the relationship between output and input with the feedback loop connected. Phase shifts are also measured over the entire system, and include contributions from the error amplifier and the feedback loop.

The open loop gain for an error amplifier acting alone normally declines at high signal frequencies, while the internal phase shifts increase. This was first recognised for electronic circuits, where it can be analysed in terms of stray capacitances, but it applies to all transducers and information processing systems, which ultimately fail to keep pace with rapid changes in the input signal. "Rapid changes" for a video amplifier might mean 20MHz: much lower values apply to biological and social feedback systems, where response times may be measured in years. The crucial stability criterion for all control systems is whether the loop gain for the error amplifier plus the feedback loop falls below 1.0 at the frequency where the phase shift reaches 180°. If the loop gain is less than one, any perturbation will be reduced in size after traversing both sections of the loop, and will eventually die away whatever the phase shift. However, if the loop gain at this critical frequency is greater than one then the system will always oscillate uncontrollably.

It is important to appreciate the generality of this conclusion, which applies to all electronic circuits, mechanical controllers, metabolic pathways, genetic regulators, and to the whole gamut of human activities from traffic management to international finance. Engineers define two "safety factors": the phase margin (PM on the diagram) by which the phase shift falls short of 180° when the gain is 1.0, and the gain margin (GM) whereby the gain is less than 1 when the phase shift is 180°. It is instructive to apply this analysis to the British political system, which combines the worst features of on/off control, high loop gains and massive internal phase shifts, and consistently fails to work in exactly the fashion that theory would predict.

Phase shifts are very common in biological systems. Gene expression involves a series of delays (transcription, RNA processing, protein synthesis, post-translational modification) and these must limit the "high" frequency loop gains (GH) which can be achieved in control systems based solely on induction and repression. Low loop gains imply a poor resistance to external loads (C/L = G3/(1 + GH)) in a rapidly changing environment, although the low frequency performance may be unaffected. Living cells can circumvent this restriction by introducing additional, higher-speed control systems based on allosteric enzymes or covalent modification, operating in parallel with the genetic controls. Allosteric enzymes and gated ion channels can respond in milliseconds, protein kinases typically take a few seconds, and changes in eukaryotic gene expression may require hours, or even days. One effect of attenuation on prokaryotic gene expression is to improve the speed of the response. Metabolic pathways may also introduce substantial phase shifts when they have numerous intermediates, or the pool sizes are large. Figure 7 compares the phase shifts in a metabolic delay line with its electrical and mechanical equivalents.

The need to minimise phase shifts in order to maintain stability may explain why some enzyme activities are many times larger than the observed metabolic flux. Selection pressure must favour high enzyme activities and small metabolite pools when the response times are important. Notice how cells prefer low-concentration metabolites (such as NADH, 5'AMP or calcium) for signalling and control purposes and generally avoid the use of high-concentration intermediates like NAD and ATP where the feedback will be much slower. Some biological systems are required to oscillate (e.g. heartbeat, peristalsis, reproductive cycles) and in such cases the opposite considerations might apply.

Negative voltage feedback reduces the output impedance of electronic amplifiers, and analogous effects are expected whenever metabolic pathways are controlled by a feedback loop which responds to the final product concentration. The product from such a regulated pathway can be withdrawn in large and variable amounts for bio-synthetic processes without significant alterations in its chemical potential (i.e. the pathway has a low chemical output impedance) but much larger changes would be expected if the regulation were missing or defective. The existence of a feedback system may also make entry into a metabolic pathway independent of the precursor concentration (i.e. the pathway exhibits a high chemical input impedance) in a striking parallel to the properties of the non-inverting voltage amplifier shown above.

Proportional control can be used for both zero order and first order control systems, but it will always lead to oscillations if it is applied to a second order system where inertia is present. There will always be a residual error when proportional control is applied to a zero order system, or to a first order system under load, because the controller must settle a short distance away from the set point so as to generate the restoring force which can oppose the action of the spring or the external load. With an integral control system the restoring force M is proportional to the integral of the error signal. When subjected to a constant reference input and load these systems stabilise with a non-zero integrator output. This permits the final control elements to apply a constant restoring force to the controlled system after the error signal has been eliminated.

Integral control systems produce results similar to figure 9 when challenged with a sudden load. The spikes represent the time taken to charge the integrator up when the load is first applied, and to discharge the integrator contents after the load is removed. This graph recurs whenever integral controllers are in use: you will find it in published data sheets for voltage regulator chips and in the biological responses to injected hormones or drugs. Integral control systems are very common in living organisms, where they are often based on covalent modification of an enzyme, for example by a protein kinase.

Integration is inherently tolerant to noisy signals from the measuring system (because it tends to average them out) but it suffers from a serious problem if the control system is unable to cope with the load, and attempts to drive the final control elements outside the region where they are capable of a linear response. In this case, the integrator will accumulate a massive total as a result of the continuing errors while the system is overloaded, and this total will lead to a subsequent over-correction when control is eventually restored. The only way for the integrator to wipe out the mistake is for it to make a series of errors with the opposite sign, and this will seriously degrade the system performance.

The answer to this problem of integrator wind-up is to limit the integrator output to the linear operating region for the final control elements. This is no problem for the biological version, where the extent of any covalent modification is inherently limited. It is readily achieved in a digital system by setting upper and lower bounds for the integrator total. Used in this way, integral controllers yield the best results with zero order systems, but additional measures are required for stability in first and second order control systems.

When the lower order components are missing there is no natural resting point for the controlled system, and no reason for any oscillations to die away. Once the set point has been achieved, the restoring force required from the final control elements depends only on the external load. Under these conditions an integral control system is neutrally stable and will oscillate gently about the set point, making alternate positive and negative errors which neither increase nor decrease with time. The best way to stop this happening is to differentiate either the output signal or the error signal, and feed this back as well.

Historically, differentiation was first introduced within the forward loop and this was eventually developed into the famous proportional-integral-derivative (PID) control algorithm. This is a three-term control strategy where the restoring force M includes components proportional to the raw error signal, to the integral of the error and to the first derivative of the error. PID controllers perform reasonably well and are used in many industrial systems.

Professor Phelan (who wrote the text-book mentioned in the introduction) parted company from most of his colleagues, and developed his own personal preference for "pseudo-derivative" [PDF] control systems, claiming that these are the optimal solution for all classes of control problem. PDF controllers integrate the error signal as previously described, but also subtract a proportion of the overall output "C" from the integrator output signal before feeding the result M1 to the final control elements. It can be shown mathematically that this is equivalent to feeding back the first derivative of the output signal. In addition, higher derivatives might be calculated explicitly. (Note that these derivatives are calculated directly from the output signal "C", not from the error signal "E" as in a PID controller. This makes a difference if the reference signal is variable.) It must be acknowledged that his idea has not achieved widespread popularity among control engineers. On the other hand, some biological control systems resemble the pseudo-derivative model.

Many important biological systems exploit covalent modification of enzymes to integrate their error signals over time. They often simultaneously feedback the raw output data, and they generate first derivative information at an early stage of the signalling process. In bacteria, the methyl-accepting chemotaxis proteins respond to changes in their ligand concentrations. In voluntary muscles, the type II afferent nerves from the sensory muscle spindles signal the degree of stretch, but the type Ia afferents signal changes in the muscle length. Many tissues responding to hormonal stimulation react strongly to an increase or decrease in the effector concentration, but the responses are attenuated when the stimulation is prolonged. When people take part in political activities, they tend to ignore their absolute position on the social scale, and consider whether their own immediate circumstances are getting better or worse. Variations in their environment are significant for living organisms, and this is reflected in their control systems. There may be some biological advantages to the PDF strategy, so at present the jury on Professor Phelan's hypothesis is still out.


Summary

  1. There is a common theory that applies to ALL control systems: subcellular, physiological, electronic, mechanical and social. Time scales and energy inputs may be very different, but the underlying mechanisms are the same.

  2. "Open loop" control systems do not check the outcome to see whether it corresponds to the expected effect. Much eukaryotic gene expression is regulated in this way: if a gene is defective, or the gene dosage is wrong, then the cells do very little to ameliorate the adverse effects.

  3. "Closed loop" control systems compare the current output with some reference value (which is not necessarily constant) in order to calculate an error signal. The error is used to adjust the flow of energy into the controlled system in order to bring the output closer to the target value.

  4. Biological reference values often depend on genetically determined ligand binding constants to proteins. Biological error amplifiers always involve allosteric proteins, and especially ligand-gated ion channels and enzyme cascades.

  5. Control systems differ considerably in their performance. Poor design or execution will impair accuracy, stability, resilience and speed.

  6. Practical control systems must cope with changing circumstances. Signals varying at different frequencies traverse the feedback loop at different speeds. This means that the output is delayed (i.e. phase-shifted) with respect to the input, and these delays vary with the signal frequency.

  7. The greatest initial accuracy and resilience in the face of varying loads is achieved with a high gain error amplifier, and a large amount of negative feedback. Such systems are invariably unstable unless the negative feedback is very quick. Delayed feedback arrives late (i.e. phase-shifted) when the situation has already changed. This causes the feedback to be applied in the wrong direction, making the position worse instead of better.

  8. Amplifier gains normally decline with increasing frequency and the internal phase shifts increase. A correct relationship between these factors is necessary for stability. In particular, the high-frequency gain must "roll-off" to be less than unity at the lowest frequency where the phase-shift on traversing the complete forward and feedback loops first reaches 180 degrees.

  9. If the feedback is inherently slow (as will be the case for most control systems based solely on gene expression) then the resulting phase shifts limit the maximum usable gain and reduce the performance of the control system. High frequency gain and phase shifts must be correctly optimised for success.

  10. Living cells can enormously improve the performance of sluggish genetic control systems by including faster local feedback loops based on protein kinases and allosteric enzymes. It is advantageous to use signalling molecules that are naturally present in very low concentrations (e.g. 5' AMP) because their concentrations change quickly and minimise undesirable phase shifts within the feedback loop.

  11. Control systems are normally expected to approach their target values as quickly and accurately as possible. This requires the complete system to be critically damped. If thecontrolled system on its own is naturally over- or under-damped, then the control system must supply the missing factors to achieve critical damping.

  12. Very simple control systems (such as room thermostats) operate by on/off control, but most effective control systems operate with smooth feedback that is proportional to the magnitude of the error.

  13. Purely proportional control systems cannot hit their target exactly when exposed to a constant load, because they have to make an error in order to generate the control output that will oppose the load. It is necessary to integrate the errors in order to overcome this problem. Protein kinases can do this very well.

  14. If the controlled system has inertia, then the control system must also calculate the first derivative of the error (or its mathematical equivalent) in order to achieve stability. Arms and legs have inertia, and the most obvious biological examples are the muscle spindles that sense muscle length and shortening velocity.

  15. "Off the shelf" engineering controllers are often based on the proportional, integral & derivative (PID) algorithm, where the controller output is proportional to (a) the error, (b) the integral of the previous errors and (c) the first derivative of the error. The relative weighting of these three components can be adjusted to suit particular circumstances.

  16. Biological control systems may use "pseudo-derivative" feedback instead of PID, but we are not yet completely sure.