Abstract
Understanding and controlling the behavior of dynamical distributed systems, especially biological ones, represents a challenging task. Such systems, in fact, are characterized by a complex web of interactions among their composing elements or subsystems. A typical pattern observed in these systems is the emergence of complex behaviors, in spite of the local nature of the interaction among elements in close spatial proximity. Yet, we point out that each element is a proper system, with its inputs, its outputs and its internal behavior. Moreover, such elements tend to implement feedback control or regulation strategies, where the outputs of a subsystem A are fed as inputs to another subsystem B and so on until, eventually, A itself is influenced. Such complex feedback loops are understood only by considering, at the same time, low- and high-level perspectives, i.e., by regarding such systems as a collection of systems and as a whole, emerging entity. In particular, dynamical distributed systems show nontrivial robustness properties, which are, from one side, inherent to the each subsystem and, from another, depend on the complex web of interactions. In this chapter, therefore, we aim at characterizing the robustness of dynamical distributed systems by using two coexisting levels of abstraction: first, we discuss and review the main concepts related to the robustness of systems, and the relation between robustness, model and control; then, we decline these concepts in the case of dynamical distributed systems as a whole, highlighting similarities and differences with standard systems. We conclude the chapter with a case study related to the chemotaxis of a colony of E. Coli bacteria. We point out that the very reason of existence of this chapter is to make accessible to a vast and not necessarily technical audience the main concepts related to control and robustness of dynamical systems, both traditional and distributed ones.
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Notes
- 1.
For the sake of simplicity, here we are assuming that one can control a system and steer its condition towards a desired one; this is not true in general and, for this to happen, the system must be controllable. Similarly, we are assuming that we can implement a successful feedback based on the available measures of the system’s outputs; again, this is not true in general and the available measures must ensure that the system is observable. The interested reader can find detailed information about this issue on books about basic control engineering , such as (Luenberger 1979).
- 2.
When K = 100 you would need to rotate of about α = 2000 degrees, i.e., make about 5.56 complete rotations! Interestingly, the more “cautious”, yet successful approach we have in our everyday usage of the shower’s handle suggests that we are unconsciously taking into account the delays occurring between the movement of the handle and the change of temperature , and we are limiting the magnitude of our control action accordingly.
- 3.
For simplicity, here we are presenting relations like y = Fu as static, while in a rigorous discussion we should consider the relation between the output y(t) and the input u(t) at time t. Also, we are implicitly limiting our scope to linear and stationary systems, which, in a rough way, can be defined as systems where the ratio between the output and the input is constant over time. Considering linear and time-invariant systems, the response of a system is y = Fu only if we are considering a frequency representation (e.g., Laplace transform), while if we are considering their temporal evolution, the signals and functions discussed in this example undergo the mathematical operation known as convolution. See (Luenberger 1979) for an exhaustive and formal discussion.
- 4.
The open-loop solution to this problem would be to set u = y desired /F, so that y = Fy desired /F = y desired. This approach, however, is very fragile, and unless F has been calculated exactly, it is deemed to fail.
- 5.
Note that, although appealing, one can not select arbitrarily high values of K for several reasons: the actuation systems might not be able to produce stimuli that are too high in magnitude, or it might be onerous and thus impractical for both artificial and biological systems. Each system has typically an upper limit for K over which the system might experience saturations, instability or disruption of some of its composing elements.
- 6.
This result applies as presented only to the narrow class of linear and stationary systems under positive feedback; for general systems this is true only to some extent (see Morari and Zafiriou (1989) for more details).
- 7.
The interested reader is invited to verify that, when y = u + 0.1u 3 and u is chosen as in Eq. (10.2), then y becomes more and more similar to y desired as K grows.
- 8.
The objectives of having y = y desired or being unaffected by disturbances might never be completely achieved; in this case these goals are obtained only asymptotically, i.e., they are achieved completely only when time goes to infinity. In practice, however, there is a time after which such results are almost achieved, which is sufficient for most practical applications.
- 9.
In the case of linear and stationary systems it can be shown that the ability to generate a constant signal is mathematically equivalent to an integration operation (for more details see basic control books such as Luenberger 1979).
- 10.
The same considerations hold true for artificial distributed systems such as swarms of autonomous mobile robots .
- 11.
By synchronization, we mean the fact that the different elements have all the same state or behavior, which may change over time in the same way over all systems. Examples in this sense include the internal clocks in a network for computers, the flashing of fireflies, the clapping of hands of people at a concert, etc.
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Menci, M., Oliva, G. (2018). Robustness vs. Control in Distributed Systems. In: Bertolaso, M., Caianiello, S., Serrelli, E. (eds) Biological Robustness. History, Philosophy and Theory of the Life Sciences, vol 23. Springer, Cham. https://doi.org/10.1007/978-3-030-01198-7_10
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