Historically, feedback has been used in control system engineering as a means for satisfying design constraints requiring:

- Stabilization of insufficiently stable systems.
- Reduction of system response to noise.
- Realization of specic transient and frequency response properties.
- Improvement of robustness against parameter variations, unmodelled dynamics or nonlinearity.

An impressive wealth of theories and approaches exists today that enable to tackle the above problems on a more or less case-to-case basis. The classical approach to the problem of controlling a dynamical system is to first identify an appropriate class of systems into which our particularproblem falls and then, once a model has been established, to devise a specific controller. This very common approach is based, of course, on the choice of the modeling strategy, and therefore, on a set of asumptions that often are disuptable. As an example, imagine a high order system in which one wishes to neglect the presence of the high order modes. The process, known as model reduction, produces what is known as a ROM (Reduced Order Model) which, given its normally modest dimension, enables to invoke the various popular control design mechanisms available today. The ROM generation process introduces evidently further elements of uncertainty as far as the formulation of the control problem is concerned. There are of course many model reduction techniques such as balancing, based on the observability and controllability Grammians, or decomposition into modal costs which may be chosen. The dilemma is: which of these methods is best ? It is clear that a universal method does not exist and one must basically rely on proper experience.

The fundamental problem in model order reduction is the size of the model one wants to end up with and this is where the problems start. It is possible that one method of order reduction suggest to neglect the n last states while another approach eliminates, say, n – 1. Which one is right ? Needless to say that the modern order reduction methods actually take into account the truncated states (or modes) and reflect their presence in the ROM. This, as one might expect, leads to a different ROM each time a different order reduction mechanism is employed. Different models of the same system lead to different controllers, different performance and also to different costs. It is therefore clear that the problem of controlling a dynamical system, or process, is, to a large extent, a problem that may be approached in a wide variety of manners given that a unique description of the underlying plant and environment does not exist.

Given the non-unique nature of the problem of modelling, truncation (order reduction) and controller design, which cannot be separated from the model synthesis, one may ask the fundamental question: do we really need a model? Is it not possible to observe and control a dynamic system without actually resorting to some form of model? Nature is full of examples where this is possible. Take for example the highly complex problem of controlling the dynamics of a human body while running. Imagine also that the individual in question is avoiding obstacles, jumping, changing suddenly direction and, maybe, at the same time optimising his own trajectory so as to select the fastest route to the desired destination. If this problem is posed and approached on classical model-controller grounds it appears simply as unsolvable. It is clear that a running individual does not perform a real-time simulation with his own dynamical model (which would evidently be extremely complex and large) in order to optimize his trajectory, to maintain certain minimum levels of dynamic stability and to limit, at the same time, the waste of energy. It is evident that no person is in possession of his/her own dynamic model and that in the majority of the cases, even if such a model were available to everyone, only a small number of individuals would know how to handle it. Finally, the real-time management of such a problem would require a rigorous access to memory and very rigid and reliable computational mechanisms that are rarely found in humans. It is clear that in reality such a model does not exist, neither for human beings nor for animals. However, it is at the same time evident that living organisms do possess remarkable control capability which, if sustained by classical model-controller paradigms, would quickly surpass the limits of the fastest computers. The fantastic control capability of living organisms is based in fact on a totally different approach, namely on the model-free paradigm. Model-free methods rely exclusively on the existence of an input-output relationship (function) which may be obtained by special training techniques, such as in the case of neural networks, or by empirical or semi-empirical means. Artificial Neural Networks (ANNs) are in fact a good example of how a collection of a relatively large number of simple computational cells (neurons) may be trained to perform a complex task without actually any knowledge of what that task is. This strong analogy with living organisms is strengthened by the fact that this training is expensive and lengthy. In the case of humans, for example, the capability to read, speak or to ride a bicycle, requires months or even years. However, it is clear that a child, for example, once in the conditions to maintain a bicycle stable, has no knowledge of the complex dynamics of the system bicycle-body, which, incidentally, is intrisically unstable. How is it then that the model-free approach works? How can a system be controlled without the controller having any knowledge of the system’s order, degree of nonlinearity or time-variability? The answer, not at all straightfoward and simple, relies most probably on two basic components, namely the sensors and some global performance quantities. In the case of humans and other complex living organisms, the sensorial system is extremely evolved and sophisticated and is actually able to “compute”, or synthesize, additional “measurements” such as, for example, the amount of equilibrium. Actuation of the various (hundreds) of muscles is then “simply” a matter of applying the acquired (learnt) rules to these measurements (or hyper-measurements) without paying any attention to the mass, inertial of stiffness properties of the hundreds of bones and organs composing the body. It is therefore evident that the control of complex living organisms is based heavily on the actual outputs (sensor readings) in a more or less direct output feed-back context (without any intermediate state estimation for example). The amount of sensors that an organism can dispose of for the performance of a certain task is, of course, directly linked to the importance that this particulr function plays for the organism itself. 3D vision, equilibrium, temperature and sound are just simple examples of complex sensorial mechanisms that feed the input-output relationship and, as time progresses, are of paramount importance as far as its further tuning is concerned.

The complex control mechanisms of living organisms have in part inspired

the work described herein. The concept of lambda-orbits, which may also be viewed as some sort of blend of eigenvalues and phase-portraits, is one possible global system descriptor that can furnish a unifying picture of the state of a system. This globalness is due to the fact that all the characteristics of the system under consideration are reflected in the lambda-orbit regardless of the systems type, size, class, stability properties, time-dependence, etc. Moreover, the lambda-orbit descriptor is based exclusively on the real measurements and, of course, does not depened on any underlying model. However, probably the most remarkable property of lambda-orbits is that they provide a fusion mechanism for the system and the environment it operates in. An informed reader will certainly recall the modelling-control design inseparability principle. The same fundamental concept may and should be applied to the system-environment pair. Since a system interacts with its environment and, given that the inverse is also true, it is clear that both in influence each other. Therefore, any artificial isolation of a system from its environment, in order to facilitate the generation of the system’s model, necessarily brakes and neglects this interaction mechanism. Similarly, the known disturbance accomodation techniques normally impose on the engineer particular disturbance models that result easily integrable with the plant model. This somewhat artificial augmentation of the plant with the disturbance model results in a hybrid dynamical entity that seldom yields a unique description of the situation. Lambda-orbits, on the other hand, do not rely on this artifical separation of the plant from its environment and, consqeuently, provide an undistorted view of the situation. This complete picture may be used for two purposes. First of all, lambda-orbits may be used as an efficient means of system monitoring providing information on the global stability of the plant-environment pair, the frequency content and, last but not least, on the evolutionary nature of the system under consideration. This last feature is particularly attractive in systems which have an unknown structure and order and where a reasonably safe extrapolation of a global tendency is required. A second attractive field of applicability of lambda-orbits lies in a novel formulation of the control problem. These orbits possess their own dynamics that resemble strongly the phase-portraits of nonlinear systems. It has been shown that the control of the orbit dynamics leads to the control of the underlying plant or process.

Examples of lambda-orbits pertaining to nonlinear oscillators.

Eigenvalue, or Lambda-Orbits: A novel approach to the problem of characterizing and controlling generic dynamic systems without a-priori knowledge of their structure. The approach stems from the concept of Eingenvalue Orbit, introduced by J. Marczyk in 1999, which provides a new means of viewing the general transient and stability properties of dynamic systems. Eigenvalue Orbits are a generalization of the concept of eigenvalue. The theory of Eigenvalue Orbits has been published in the *Journal of Guidance, Control and Dynamics*, a peer-reviewed publication of the American Institute of Aeronautics and Astronautics, see here.

For more information on lambda-orbits click here.

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