Self-Organizing Systems pp 351-397 | Cite as

# Control Paradigms and Self-Organization in Living Systems

## Abstract

This chapter describes control of systems characterized by numerically valued variables. It opens with a qualitative description of the fundamental concepts behind open-loop and feedback control methods. The chief aim of feedback control is to cause a system’s output to maintain some desired relation to an input, in spite of disturbances or deviations in plant dynamics.

Classical linear control theory is quantitatively described, using the standard, frequency-domain (jω) notation and Fourier transform method. The open-loop and closed-loop (feedback) equations are each shown for a continuous, linear system with time-invariant parameters and structure and single input with single output. In some cases it is necessary to achieve independent control of each of multiple disturbances or to control more than a single plant output. Or there may be more than a single control input available with which to control the plant’s outputs. One may then employ multiloop control or more complex control arrangements for multivariable control.

Classical linear control concepts have been extended in modern optimal control theory. The new approach is far more general, and it emphasizes the state-space description of dynamic systems. It is general enough to encompass time-varying and nonlinear conditions. Using the general formulation of the deterministic, optimal control problem, open-loop control is reexamined from the point of view of the Pontryagin Maximum Principle. Some similarities between the optimal control problem and Hamiltonian mechanics are pointed out. It is concluded that optimal control theory in this form represents a significant conceptual generalization of the variational theory of classical mechanics to a much wider class of problems. In this sense, optimal control theory overlaps and transcends ordinary physical theory and cannot properly be considered to be a simple consequence of the known physical laws of mechanics.

The optimal control approach is next generalized to include stochastic processes appearing as unpredictable perturbations. The mathematical difficulty increases, but such difficulties do not detract from the underlying rich structure of optimal control theory and its substantial connections with and differences from the variational theory of mechanics.

The equations of stochastic optimal control do not arise in statistical mechanics because standard statistical mechanics does not deal with concepts that involve estimating the state of dynamic systems for purposes of control. Neither does physical theory appear to deal with any problems analogous to or embedded in the stochastic optimal control problem. Again, one is forced to the conclusion that control theory cannot, in general, be considered to be a part of, or a simple extension of, known physical theory.

Finally, this chapter addresses control in living systems, and asks whether or not the principles of technological control, as outlined in the first part, apply. Living systems are dominated by regulatory processes. The history of our discovery of some of these processes is summarized. Enzyme-catalyzed reactions and feedback control of enzyme levels by genetic repression-derepression constitute interesting sample cases for control theory analysis. The analysis shows that control theory provides a basis for showing what experiments and data are necessary to validate the claims of the biochemists that the closed, inhibitory pathways they discover (usually *in vitro*) actually have physiological relevance. This validation has yet to be made. It is concluded that some of the concepts of classical control theory do illuminate some biochemical control processes, but that the richness of modern, optimal control theory cannot be brought to bear on the richness of modern genetic control of protein biosynthesis. —The Editor

## Keywords

Feedback Control Control Theory Optimal Control Problem Optimal Control Theory Sensor Noise## Preview

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