Digitally networked control systems

Digitally networked control systems are spatially distributed systems in which the components (sensors, controllers and actuators) communicate over a shared digital communication network. Examples can be found in automated traffic control, control of unmanned underwater vehicles for surveillance or rescue tasks, in space exploration and many other application areas. In digitally networked systems, the analog plant outputs have to be coded into finite bit strings which are then transmitted over the communication network. Realistic models of networked systems therefore violate the standard assumption of classical control theory that controllers and actuators have access to state information of arbitrarily high precision. Therefore, the characterization of a system property such as stabilizability involves not only characteristics of the dynamical system but also characteristics of the communication network.

Main research areas

Stabilization over digital communication channels

We consider a control system with a controller that receives its state information over a discrete communication channel. Here the question arises how much information per unit of time needs to be transmitted so that the controller can stabilize the system. This problem has been investigated under different assumptions on the system, the communication channel and the concrete stabilization objective. A classical result, known as the data-rate theorem, says that for linear systems the necessary data rate is given by the logarithm of the unstable determinant of the open-loop system. In the discrete-time case, this is the sum of the logarithms of all unstable eigenvalues. If one wants to generalize this result to nonlinear systems, first the stabilization objective needs to be specified, because in the nonlinear case it makes a big difference if the goal is local stabilization (to an equilibrium point), semi-local stabilization (to a set) or global stabilization (e.g., in a stochastic sense). If the goal is to keep the state in a fixed compact subset Q of the state space, the associated minimal average data rate is described by the invariance entropy of Q. This intrinsic quantity of the open-loop system is, in general, hard to compute or estimate. Under certain dynamical and control-theoretic assumptions it is, however, possible to make precise statements. Roughly speaking, the invariance entropy of Q turns out to be the difference of two quantities, the first one of which can be regarded as a measure for the total instability of the system on Q, while the second one describes the portion of the instability which does not lead to exit from Q. Both of these quantities are well-studied and understood in the theory of dynamical systems.

State estimation over digital communication channels

Consider a dynamical system and a discrete communication channel over which information is transmitted at certain time instances. At the other end of the channel, the transmitted information is used to produce an estimate of the state which should have a prescribed level of precision. In this context, it is of interest to determine the smallest rate of data transmission above which an arbitrarily precise estimate of the state can be generated by an appropriate choice of a communication protocol. Under mild assumptions, one obtains that the smallest data rate is given by the topological entropy of the system. For several reasons, this result is not satisfying. For instance, the quantities needed to implement the communication protocol are hard to compute or estimate. Furthermore, topological entropy depends discontinuously on system parameters, which can lead to wrong results. Finally, to guarantee a maximal estimation error of &epsilon, the initial error at time 0 has to be much smaller than &epsilon, in general. All of these problems can be solved by introducing another entropy-like quantity whose definition is guided by the goal to restore an arbitrarily small initial error after a certain time. This is the notion of restoration entropy. One can show that all of the mentioned problems can be resolved by using this quantity as the figure-of-merit for the data rate. The price one has to pay is that the minimal data rate increases, since (roughly speaking) restoration entropy measures the maximal instability while topological entropy measures the average instability of a system.

Hyperbolic dynamics and control

Hyperbolic dynamics and control theory are two largely separated fields. However, there are some very interesting relations. For example, the technique of shadowing of pseudo-orbits in hyperbolic dynamics can be used to prove controllability on certain subsets of the state space of a control system. In a certain sense, the shadowing property is also dual to local controllability. For instance, if (x0,u0) is an equilibrium pair with controllable linearization, then one can, by local controllability, assign to each point x in a neighborhood of x0 a control u, that steers this point in a given time T back to itself. By application of the control u to x0, one obtains a pseudo-orbit which is shadowed by a unique orbit. This can only be the orbit starting in x. In this way, shadowing yields a one-sided inverse to the "local controllability operator". Further examples for the applicability of methods from hyperbolic dynamics to control problems can be found in some of my works about stabilization under communication constraints.

Scale-free methods for large networks

Some of the standard methods for stability analysis and stabilization of networked systems do not scale well with the size of the network. A promising idea to overcome this problem is to over-approximate large-but-finite networks by infinite networks and to develop methods for the stability analysis and stabilization of these infinite networks. An important notion of stability in control is input-to-state stability (ISS) which in the simplest case guarantees that the norm of solutions can be estimated by the norm of the input and that of the initial state. This, for instance, guarantees that bounded inputs do not lead to unbounded trajectories. For very large systems (respectively, networks of systems) it is often difficult to verify the ISS property directly. One approach to verify ISS consists in verifying ISS for each single subsystem, thereby considering the influence of other subsystems as part of the input, and then checking a so-called small-gain condition which ensures that the influence of other subsystems on an arbitrary but fixed subsystem is small enough to guarantee stability of the whole network.

Research problems

Invariance entropy for hyperbolic sets in discrete time

Prove a tight achievability result for set-invariance of uniformly hyperbolic sets (see Control of Chaos for details). A related question: How can one stabilize a system around an Axiom A basic set with minimal data rate?

Stabilization to hyperbolic sets in continuous time

Generalize the theory developed in Control of Chaos for local stabilization to a uniformly hyperbolic set to continuous-time systems (where a one-dimensional center bundle is present).

Product formula for invariance entropy

Prove or disprove the product formula for invariance entropy: Given two control systems Σ1 and Σ2, let Σ be their direct product. Let Q1 and Q2 be compact controlled invariant sets for Σ1 and Σ2, respectively. Is it true that the invariance entropy of the direct product of Q1 and Q2 is the sum of the individual entropies?

Control over noisy channels

Assume that a control task cannot be solved over a noiseless channel of capacity C. Is it possible that the same control task can be solved over a noisy channel (with or without feedback) of the same capacity?

Stochastic stabilization problems I

For data-rate theorems related to stochastic stabilization problems, linearization techniques are not very useful, because the problems are of global nature and apparently cannot be localized (in general). Can global linearization techniques, e.g. a Koopman operator approach, be used to study such problems?

Stochastic stabilization problems II

For stabilization and observation of deterministic nonlinear systems over rate-limited channels, it is in many cases understood which classical quantities determine the minimal data rate. In particular, topological entropy, measure-theoretic entropy, escape rates and different types of Lyapunov exponents appear in this context. In contrast, for stochastic systems and different types of stochastic stabilization, no relations to classical quantities of simular nature have been discovered yet. Hence, it is an important subject of investigation to find such relations.

Numerical computation of topological entropy

The paper FJO by Froyland, Junge and Ochs offers a quite general algorithm for the numerical estimation of topological entropy, based on set-valued analysis and symbolic dynamics. If appropriate assumptions are satisfied, this algorithm produces guaranteed upper bounds of topological entropy. However, it has not been clarified in which cases these upper bounds are tight (i.e., approximate the entropy). It can be shown that this is the case if the employed partitions of the state space have the so-called small-boundary property, which is however rarely satisfied. Hence, the problem is to find more general conditions.



    The paper "A small-gain theory for infinite networks via infinite-dimensional gain operators" is available on arXiv. (link)