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Saturn, slime molds, and life support

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Robust control theory has some surprising applications, and software can ease the process, writes Brian Cogan

Modern control theory began with the study of regulation and stability. In 1857, James Clerk Maxwell published his analysis of why the rings of Saturn were stable. This was generalised by Henri Poincaré a quarter of a century later, to cover the stability of the solar system and the general problem of the stability of motion.

One crucial concept is that of 'robustness': a system is robust if its behaviour persists despite changes in the system parameters or some unexpected disturbance. Harry Nyquist in 1932 gave the first measures of robustness and, since the 1980s, very sophisticated techniques have been developed for the study and design of robust control systems - many of these are now available in software packages.

Recently, the techniques of robust control theory have been used in understanding regulation and control in biological systems. Regulation and robustness in man-made systems can be achieved by feedback - sending an error in the output of a system back to the system's regulator so that the regulator will reduce the error in the output. Alternatively, some function of the error, such as the integral (I) or its derivative (D) could be fed back to the regulator. More than 90 per cent of controllers in industry (so called PID controllers) use a combination of proportional (P), integral (I), and differential (D) feedback.

Over the millennia, natural selection has led to the development of robust control mechanisms in many species, including in our own bodies. These robust systems share the same essential characteristic - a persistence of the essential functioning of the organism, in spite of wide variations in the environment. Robust stability has evolved because it is essential for the survival of species. The networks that regulate cell function seem at first sight to be grossly over-engineered. For example, experiments indicate that about 90 per cent of the genes in the bacterium E. coli are not necessary for its day-to-day functioning. It seems that the cause of this complexity is the presence of regulatory networks that ensure robustness, but are not necessary for minimal functionality.

However, there are differences between the ways engineers and biologists use control theory. A control engineer designs a control system from known components to perform a certain well-defined function. A biologist must practice 'reverse engineering' and infer the structure and characteristics of a system from a study of the inputs and outputs. Another difference is in the terminology used by biologists and engineers. Behaviour described as 'robust perfect adaptation' by biologists is called 'asymptotic tracking' or 'step disturbance rejection' by control engineers.

A central problem in post-genomic biology is the study of the dynamics and control of intracellular networks of genes and proteins. One recent success for the application of robust control theory, in the analysis of cell signalling pathways, is in chemotaxis. Chemotaxis is the directed movement of a cell or organism in response to a specific chemical concentration gradient. Movement can be towards a higher concentration gradient (positive chemotaxis) or towards a lower concentration gradient (negative chemotaxis). Some recent observations seem to indicate that, in humans, the egg and sperm communicate prior to fertilisation by means of chemotaxis. The egg or its surrounding cells secrete a factor that is detected by the sperm cells. Also, many insects and animals rely on the sense of smell to convey information. Chemotaxis also plays a key role in inflammation, arthritis, and asthma.

A widely-studied example of chemotaxis is found in the behaviour of dictyostelium discoideum (also known as slime molds). These are social amoebae that feed on bacteria and live in soil. These organisms exhibit an interesting response to the disappearance of their food supply. They synthesise and release a chemoattractant, and up to 100,000 of them aggregate to form multicellular structures. This behaviour is essential for survival of the organism. These amoebae move by combination of random tumbling, followed by movement forward in some favoured direction. While tumbling the amoeba is in a sort of 'search mode', looking for a chemical gradient.

Research now indicates that the internal molecular network of signalling during chemotaxis uses six intracellular proteins, along with a transmembrane receptor that binds the stimulatory chemotactic ligand. From the viewpoint of a control engineer, one could say that the input to this system is the concentration of the stimulatory chemical, and the output is the frequency of tumbling.

Experiments have shown that the output (tumbling frequency) at a steady state after a transient disturbance (change in stimulus concentration) is exactly equal to the pre-stimulus value, regardless of the size of the disturbance. Indeed, up to a hundred-fold change in chemical concentration is accommodated by the signalling pathway. This robustness seems to be necessary for the organism to respond to stimulus concentration gradients. This network appears to use integral feedback control, where the integral of the error is fed back to the regulator. Integral feedback is necessary as schemes lacking this (such as P or PD control) yield a new steady-state value after a disturbance - a state that differs from the pre-disturbance value.

The maintenance of some physiological parameter within a range of values is called 'homeostasis'. Walter Cannon coined the term in his book Wisdom of the Body (Norton, New York, 1932). The study of homeostasis provides a multitude of examples of robustness in natural systems. The control of body core temperature, blood pressure, blood sugar level, and respiration are just a few examples of this.

Several computer packages are available for the analysis and design of robust control systems. These modern controllers are robust in the sense that they minimise certain performance specifications, usually measures of the size of signals in Hardy space.

MatLab, for example, distributes several excellent robust-controller design packages. The Control System Toolbox 5.2.1 provides tools for designing and analysing controllers for closed-loop control of dynamic systems in the aerospace, automotive, industrial equipment, mass storage, and electronics industries. Within the overall suite of control-system software, MatLab's Robust Control Toolbox provides tools for the design and analysis of multivariable control systems where robustness is a concern. This includes systems where there may be modelling errors, dynamics that are not completely known, or parameters that can vary during the lifespan of the product.

Designing complex control systems using the Mathworks suite of products (click to see a larger version of the image)

Wolfram Research has developed Control System Professional 2 and Advanced Numerical Methods 1 packages, which were reviewed in Scientific Computing World in May/June 2003.) Loehle Enterprises are now distributing an extension to CSP2 that may be used for robust controller analysis and design. This package, Advanced Modern Control for Mathematica, provides commands for designing general purpose, state-of-the-art, robust controllers.

Loehle's extension to CSP2 is easy to use; only the details of the (industrial) plant and the order of the controller are needed to get started. The documentation is thorough, comprehensive, and helpful. The designer has taken care to ensure that the presentation of tutorials is clear and logical, and there are many references to very recent literature. The stability and limits of performance of these algorithms are discussed.

Closed-loop automatic controllers have important applications beyond industry - for example, in modern medicine where accurate and reliable regulation of physiological variables is vital. Anaesthesia, insulin infusion, respiratory control, and heart-rate monitors are just some examples of the hundreds of areas where closed-loop automatic controllers may be found. Control theory techniques have also found applications in diverse areas such as the estimation of HIV/AIDS parameters, control of a genital herpes epidemic, and the analysis of chemotherapy regimes. Having software available to make control theory more accessible may, quite literally, be a matter of life or death.