Observability of the discrete state for dynamical piecewise hybrid systems

In this paper, we deal with the observability of piecewise-affine hybrid systems. Our aim is to give sufficient conditions to observe the discrete and continuous states, in terms of algebraic and geometrical conditions. Firstly, we will give the algebraic conditions to observe the discrete state based on the switch function reconstruction for linear hybrid systems. Secondly, we will give a geometrical condition based on the transversality concept for nonlinear hybrid systems. Throughout this paper, we illustrate our propositions with examples and simulations.     

Observability of nonlinear systems

New necessary and sufficient conditions are presented for the observability of systems described by nonlinear ordinary differential equations with nonlinear observations. The conditions are based on extension of the necessary and sufficient conditions for observability of time-varying linear systems to the linearized trajectory of the nonlinear system. The result is that the local observability of any initial condition can be readily determined, and the observability of the entire initial domain can be computed. The observability of constant parameters appearing in the differential equations is also considered. Examples are presented to illustrate the theory.This research was supported by the National Science Foundation, Grant No. NSF GK-10136.     

Observability of coupled systems

By applying the theory of semigroups, we generalize an earlier result of Komornik and Loreti [5] on the observability of compactly perturbed systems. As an application, we answer a question of the same authors concerning the observability of weakly coupled linear distributed systems. This revised version was published online in June 2006 with corrections to the Cover Date.     

Theory of oscillations of discrete-continuous systems with periodic properties

The oscillations of discrete-continuous systems of lumped masses with elastic inertial couplings are studied. A numerical determination of the monodromy matrix and its multipliers is compared with an exact solution of the problem for the special case of piecewise constant parameters. The possibility of smoothing the system parameters at their points of discontinuity is analyzed. Analytic dependences are obtained for low frequency oscillations. Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 29, pp. 122–126, 1999.     

Hybrid discrete-continuous systems: I. stability and stabilizability

For hybrid discrete-continuous linear stationary systems, we consider the basic problems of qualitative control theory, namely, stability and stabilization. For such systems, we obtain parametric criteria for asymptotic and exponential stability ensuring a prescribed stability exponent and a rank criterion for stabilizability. We consider the problem of finding the minimum number of inputs for which the considered system is stabilizable. We suggest an effective algorithm for constructing the matrix describing the structure of the input device of a minimum-input stabilizable system. An example illustrating the results is given.     

Hybrid discrete-continuous systems: II. Controllability and reachability

For hybrid discrete-continuous linear stationary systems, we consider basic problems of qualitative control theory such as controllability and reachability. For these systems, we obtain parametric criteria for relative and total controllability and relative and total reachability. We consider the problem of the evaluation of the minimum number of inputs for which the system has some controllability or reachability property. An example illustrating obtained results is considered.     

Observability of linear differential systems defined in differential rings

We study the observability of linear differential systems in sufficiently arbitrary differential rings; observability is treated in the classical sense, i.e., as the injectivity of the “initial condition-output vector” mapping. A number of observability conditions is obtained. We define observability by finitely many measurements, where a measurement is the value of a homomorphism into the ring of constants on the output variable. We show that observability is equivalent to observability by finitely many measurements.     

Observability for systems with more outputs than inputs and asymptotic observers

Both authors are partly supported by: NSERC GRANT OGP 0036498     

Weighted o-minimal hybrid systems

We consider weighted o-minimal hybrid systems, which extend classical o-minimal hybrid systems with cost functions. These cost functions are “observer variables” which increase while the system evolves but do not constrain the behaviour of the system. In this paper, we prove two main results: (i) optimal o-minimal hybrid games are decidable; (ii) the model-checking of WCTL, an extension of CTL which can constrain the cost variables, is decidable over that model. This has to be compared with the same problems in the framework of timed automata where both problems are undecidable in general, while they are decidable for the restricted class of one-clock timed automata.     

Robust viability of hybrid systems

Robust viability of hybrid systems is examined based on the controllability operator subject to transition dynamics uncertainty. Two modifications to the controllability operator are introduced, these being the uncertain controllability operator and the uncertainty operator. It is shown how existing fixed-point approximation algorithms can be generalized and applied to compute robustly viable sets for hybrid systems. The three-tank control problem is considered. Robust viable sets are computed for this system subject to transition dynamics uncertainties.     

Geometric control of hybrid systems

In this paper, we present a geometric approach for computing controlled invariant sets for hybrid control systems. While the problem is well studied in the ellipsoidal case, this family is quite conservative for constrained or switched linear systems. We reformulate the invariance of a set as an inequality for its support function that is valid for any convex set. This produces novel algebraic conditions for the invariance of sets with polynomial or piecewise quadratic support functions.     

Solving the discrete-continuous project scheduling problem via its discretization

In this paper a discrete-continuous project scheduling problem is considered. In this problem activities simultaneously require discrete and continuous resources. The processing rate of each activity depends on the amount of the continuous resource allotted to this activity at a time. All the resources are renewable ones. The activities are nonpreemtable and the objective is to minimize the makespan. Discretization of this problem leading to a classical (i.e. discrete) project scheduling problem in the multi-mode version is presented. A simulated annealing (SA) approach to solving this problem is described and tested computationally in two versions: with and without finding an optimal continuous resource allocation for the final schedule. In the former case a nonlinear solver is used for solving a corresponding convex programming problem. The results are compared with the results obtained using SA for the discrete-continuous project scheduling problem where the nonlinear solver is used for exact solving the continuous part in each iteration. The results of a computational experiment are analyzed and some conclusions are included.     

Frequency characteristics of flexibly vibration discrete-continuous mechatronic system

The purpose of this paper is formulating of problem of flexibly vibrating mechatronic system. The main approach of the subject was to formulate the problem in the form of set of differential equation of motion and state equation of considered mechatronic model of object. The considered flexibly vibrating mechanical system is a continuous beam, clamped at one of its end. Integral part of mechatronic system is a transducer, extorted by harmonic voltage. In the paper the linear mechanical subsystem and linear electric subsystem of mechatronic system has been considered. The methods of analysis and obtained results can be base on design and investigation for this type of mechatronic systems. The mechatronic system formed from mechanical and electric subsystems with electromechanical bondage has been considered. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)