Luca Benvenuti

Design of Observers for Hybrid Systems

A methodology for the design of dynamical observers for hybrid plants is proposed. The hybrid observer consists of two parts: a location observer and a continuous observer. The former identifies the current location of the hybrid plant, while the latter produces an estimate of the evolution of the continuous state of the hybrid plant. A synthesis procedure is offered when a set of properties on the hybrid plant is satisfied. The synthesized hybrid observer identifies the current location of the plant after a finite number of steps and converges exponentially to the continuous state.

A tutorial on the positive realization problem

This paper is a tutorial on the positive realization problem, that is the problem of finding a positive state-space representation of a given transfer function and characterizing existence and minimality of such representation. This problem goes back to the 1950s and was first related to the identifiability problem for hidden Markov models, then to the determination of internal structures for compartmental systems and later embedded in the more general framework of positive systems theory. Within this framework, developing some ideas sprang in the 1960s, during the 1980s, the positive realization problem was reformulated in terms of a geometric condition which was recently exploited as a tool for finding the solution to the existence problem and providing partial answers to the minimality problem. In this paper, the reader is carried through the key ideas which have proved to be useful in order to tackle this problem. In order to illustrate the main results, contributions and open problems, several motivating examples and a comprehensive bibliography on positive systems organized by topics are provided.

Positive and compartmental systems

When dealing with compartmental systems, an important question is: given an experiment, i.e., an input-output sequence, and supposing there is no error in the data, is the sequence compatible with the compartmental assumption? If the process under analysis is linear, then the previous question is obviously equivalent to asking whether a given transfer function is that of a compartmental system. In this note we provide an answer to the latter question giving necessary and sufficient conditions for a transfer function to be that of a compartmental system of some finite order (i.e., number of compartments). Another problem tackled in this note originates from the observation that in many cases one wants to determine the number of compartments involved in the process. In this note we report a step toward the solution of this fundamental problem by proving necessary and sufficient conditions for a given third order transfer function with real poles to be that of a compartmental system with three compartments.

Positive realizations of linear systems

Abstract This paper deals with the realization problem of linear systems in which a sign pattern is imposed on the realization{ A b c T }. This problem arises in the context of positive linear systems, i.e. systems in which state variables and the output take nonnegative values whenever initial states and inputs are nonnegative. Using a unitary framework for continuous-time and discrete-time systems we derive necessary and sufficient conditions for positive realizability by means of convex analysis. Moreover, the complete solution in the case of a transfer function of McMillan degree n ⩽ 3 is obtained.

Minimal positive realizations of transfer functions with positive real poles

A standard result of linear-system theory states that a SISO rational nth-order transfer function always has an nth-order realization. In some applications, one is interested in having a realization with nonnegative entries (i.e., a positive system) and it is known that a positive system may not be minimal in the usual sense. In this paper, we give an explicit necessary and sufficient condition for a third-order transfer function with distinct real positive poles to have a third-order positive realization. The proof is constructive so that it is straightforward to obtain a minimal positive realization.

Individual cylinder characteristic estimation for a spark injection engine

Engine control policies are mostly based on the assumption that all injectors have the same behavior independent of location and aging. In reality, injectors do vary and age. To contain variations around a nominal value, tight tolerances are imposed on the manufacturing process. Even if the manufacturing process is tightly controlled, the air-to-fuel (A/F) ratio needed to satisfy emission constraints is difficult to achieve due to aging and even slight mismatch among different injectors. To devise control policies that take into account behavior differences among injectors, we need to estimate injector characteristics from measurements that are taken on the engine during its life time. In this paper, we present an estimation technique for injector characteristics based on a set of measurements that can be carried out using the sensors present in the car, i.e., intake manifold pressure, crank-shaft speed, throttle-valve plate angle, injection timings and exhaust A/F ratio, which is measured by a single UEGO sensor placed at the exhaust pipe output.

Eigenvalue regions for positive systems

Positive systems are systems in which the state variables take only nonnegative values for all times. In this paper, we derive eigenvalue regions for discrete and continuous-time positive linear systems by resorting to the immediately available information on the values of the main diagonal entries of the system matrix.

Filtering through combination of positive filters

The linear filters characterized by a state-variable realization given by matrices with nonnegative entries (called positive filters) are heavily restricted in their achievable performance. Nevertheless, such filters are the only choice when dealing with the charged coupled device MOS technology of charge routing networks (CRNs), since nonnegativity is a consequence of the underlying physical mechanism. In order to exploit the advantages offered by this technology, the authors try to overcome the above-mentioned limitation by realizing an arbitrary transfer function as a difference of two positive filters.

A contract-based formalism for the specification of heterogeneous systems

We present the mathematical formalism and the verification methodology of the contract-based model developed in the framework of the SPEEDS project. SPEEDS aims at developing methods and tools to support ldquospeculative designrdquo, a design methodology in which distributed designers develop different aspects of the overall system, in a concurrent but controlled way. Our generic mathematical model of contract supports this style of development. This is achieved by focusing on behaviors, by supporting the notion of ldquorich componentrdquo where functional and non-functional aspects of the system can be considered and combined, by representing rich components via their set of associated contracts, and by formalizing the process of component composition.

Maximal Safe Set Computation for Idle Speed Control of an Automotive Engine

The specification for the idle control problem for automotive engines is to maintain the crankshaft speed within a given range in the presence of load changes. A new cycle-detailed hybrid model of the engine that captures well the interactions between the discrete phenomena of torque generation and spark ignition, and the continuous evolution of the power-train and air dynamics, is proposed. The idle control problem is formalized as a safety specification problem on the hybrid system. The Tomlin-Lygeros-Sastry procedure [12] is applied to compute the maximal controlled invariant set that satisfies the safety specification.