Team Metalau

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Section: Scientific Foundations

Exotic systems

Hybrid dynamical systems

Originally motivated by problems encountered in modeling and simulation of failure detection systems, the objective of this research is the development of a solid formalism for efficient modeling of hybrid dynamical systems.

A hybrid dynamical system is obtained by the interconnection of continuous time, discrete time and event driven models. Such systems are common in most control system design problems where a continuous time model of the plant is hooked up to a discrete time digital controller.

The formalism we develop here tries to extend methodologies from Synchronous languages to the hybrid context. Motivated by the work on the extension of Signal language to continuous time, we develop a formalism in which through a generalization of the notion of event to what we call activation signal , continuous time activations and event triggered activations can co-exist and interact harmoniously. This means in particular that standard operations on events such as subsampling and conditioning are also extended and operate on activation signals in general paving the way for a uniform theory.

The theoretical formalism developed here is the backbone of the modeling and simulation software Scicos. Scicos is the place where the theory is implemented, tested and validated. But Scicos has become more than just an experimental tool for testing the theory. Scicos has been successfully used in a number of industrial projects and has shown to be a valuable tool for modeling and simulation of dynamical systems.

Encouraged by the interest in Scicos, expressed both by the academia and industry, beyond the theoretical studies necessary to ensure that the bases of the tool are solid, the project-team has started to invest considerable effort on improving its usability for real world applications. Developing a robust user-friendly Scicos has become one of the objectives of the project-team.

It turns out that the Scicos formalism and the Modelica language share many common features, and are in many respects complementary. Scicos formalism provides a solid ground for modeling discrete-time and event dynamics, in a hybrid framework, based on the theory of synchronous languages, and Modelica is a powerful language for the construction of continuous-time models. We work closely with Modelica association and other actors in the Modelica community to make sure Modelica remains consistent with Scicos. We do this in particular by proposing new discrete-time extensions to Modelica inspired by Scicos formalism.

Maxplus Algebra, Discrete Event Systems and Dynamic Programming

In the modeling of human activities, in contrast to natural phenomena, quite frequently only the operations max (respectively min) and + are needed (this is the case in particular of some queuing or storage systems, synchronized processes encountered in manufacturing, traffic systems, when optimizing deterministic dynamic processes, etc.).

The set of real numbers endowed with the operation max (respectively min) denoted $ \oplus$ and the operation + denoted $ \otimes$ is a nice mathematical structure that we may call an idempotent semi-field. The operation $ \oplus$ is idempotent and has the neutral element $ \varepsilon$ = -$ \infty$ but it is not invertible. The operation $ \otimes$ has its usual properties and is distributive with respect to $ \oplus$ . Based on this set of scalars we can build the counterpart of a module and write the general (n, n) system of linear maxplus equations:

Ax$ \oplus$b = Cx$ \oplus$d,

using matrix notation where we have made the natural substitution of $ \oplus$ for + and of $ \otimes$ for × in the definition of the matrix product.

A complete theory of such linear system is still not completely achieved. In recent development we try to have a better understanding of image and kernel of maxplus matrices.

System theory is concerned with the input (u )-output (y ) relation of a dynamical system (Im16 $\#119982 $ ) denoted y = S(u) and by the improvement of this input-output relation (based on some engineering criterium) by altering the system through a feedback control law u = F(y, v) . Then the new input (v )-output (y ) relation is defined implicitly by y = S(F(y, v)) . Not surprisingly, system theory is well developed in the particular case of linear shift-invariant systems. Similarly, a min-plus version of this theory can also be developed.

In the case of SISO (single-input-single-output) systems, u and y are functions of time. In the particular case of a shift-invariant linear system, S becomes an inf-convolution:

Im17 ${y=h\#9633 u\mover ={def}\munder infs{[h{(s)}+u{(·-s)}]}}$

where h is a function of time called the impulse response of system Im16 $\#119982 $ . Therefore such a system is completely defined by its impulse response. Elementary systems are combined by arranging them in parallel, series and feedback. These three engineering operations correspond to adding systems pointwise ($ \oplus$ ), making inf-convolutions ($ \otimes$ ) and solving special linear equations (y = h$ \otimes$(f1$ \otimes$y$ \oplus$f2$ \otimes$v) ) over the set of impulse responses. Mathematically we have to study the algebra of functions endowed with the two operations $ \oplus$ and $ \otimes$ and to solve special classes of linear equations in this set, namely when A = E in the notation of the first part.

An important class of shift-invariant min-plus linear systems is the process of counting events versus time in timed event graphs (a subclass of Petri nets frequently used to represent manufacturing systems). A dual theory based on the maxplus algebra allows the timing of events identified by their numbering.

The Fourier and Laplace transforms are important tools in automatic control and signal processing because the exponentials diagonalize simultaneously all the convolution operators. The convolutions are converted into multiplications by the Fourier transform. The Fenchel transform (Im18 $\#8497 $ ) defined by:

Im19 ${{[\#8497 {(f)}]}{(p)}=\munder supx{[px-f{(x)}]},}$

plays the same role in the min-plus algebra context. The affine functions diagonalize the inf-convolution operators and we have:

Im20 ${\#8497 (f\#9633 g)=\#8497 (f)+\#8497 (g).}$

A general inf-convolution is an operation too complicated to be used in practice since it involves an infinite number of operations. We have to restrict ourselves to convolutions that can be computed with finite memory. We would like that there exists a finite state x representing the memory necessary to compute the convolution recursively. In the discrete-time case, given some h , we have to find (C, A, B) such that hn = CAnB , and Im21 ${y=h\#9633 u}$ is then `realized' as

xn + 1 = Axn$ \oplus$Bun, yn = Cxn.

SISO systems (with increasing h) which are realizable in the min-plus algebra are characterized by the existence of some $ \lambda$ and c such that for n large enough:

hn + c = c×$ \lambda$ + hn.

If h satisfies this property, it is easy to find a 3-tuple (A, B, C) .

This beautiful theory is difficult to apply because the class of linear systems is not large enough for realistic applications. Generalization to nonlinear maxplus systems able to model general Petri nets is under development.

Dynamic Programming in the discrete state and time case amounts to finding the shortest path in a graph. If we denote generically by n the number of arcs of the paths, the dynamic programming equation can be written linearly in the min-plus algebra:

Xn = A$ \otimes$Xn-1,

where the entries of A are the lengths of the arcs of the graph and Xn denotes the matrix of the shortest lengths of paths with n  arcs joining any pair of nodes. We can consider normalized matrices defined by the fact that the infimum in each row is equal to 0. Such kind of matrices can be viewed as the min-plus counterpart of transition matrices of a Markov chain.

The problem

Im22 ${v_x^n=\munder minu~\mfenced o=[ c=] \munderover \#8721 {i=n}{N-1}\#966 {(u_i)}+\#968 {(x_N)}\#8739 x_n=x~,~x_{i+1}=x_i-u_i}$

may be called dynamic programming with independent instantaneous costs ($ \varphi$ depends only on u and not on x ). Clearly v satisfies the linear min-plus equation:

Im23 ${v^n=\#966 \#9633 v^{n+1},~~v^N=\#968 }$

(the Hamilton-Jacobi equation is a continuous version of this problem).

The Cramer transform (Im24 ${\#119966 \mover ={def}\#8497 \#8728 log\#8728 \#8466 }$ ), where Im25 $\#8466 $ denotes the Laplace transform, maps probability measures to convex functions and transform convolutions into inf-convolutions:

Im26 ${\#119966 (f*g)=\#119966 (f)\#9633 \#119966 (g).}$

Therefore it converts the problem of adding independent random variables into a dynamic programming problem with independent costs.

These remarks suggest the existence of a formalism analogous to probability calculus adapted to optimization that we have developed.

The theoretical research in this domain is currently done in the MAXPLUS project-team. In the METALAU project-team we are more concerned with applications to traffic systems of this theory.


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