Mihaela-Hanako Matcovschi

Max-type copositive Lyapunov functions for switching positive linear systems

The paper aims to expand the stability analysis framework based on copositive Lyapunov functions (CLFs) for arbitrarily switching positive systems with continuous- or discrete-time dynamics. The first part focuses on max-type CLFs. It provides an algebraic characterization (in terms of weak quasi-linear inequalities), as well as an existence criterion relying on a concrete construction procedure (in terms of Perron-Frobenius eigenstructure). The second part explores the connections between max-type and two other classes of CLFs, namely linear and quadratic-diagonal. New qualitative and quantitative features are revealed for the CLFs belonging to the two mentioned classes.

Modeling of a pressure reducing valve actuator for automotive applications

Significant research effort has been directed towards developing vehicle systems that reduce the energy consumption of an automobile and because pressure control valves are used as actuators in many control applications for automotive systems, a proper dynamic model is necessary. Starting from equations found in literature, where a single stage pressure reducing valve is modelled, in this paper, the concept of modeling a real three land three way solenoid valve actuator for the clutch system in the automatic transmission is presented. Two simulators for an input-output model and a state-space model were developed and these were validated with data provided from experiments with the real valve actuator on a test bench.

Flow-Invariance and Stability Analysis for a Class of Nonlinear Systems with Slope Conditions

New results of qualitative analysis are presented for a class of dynamical systems (including the Hopfield neural networks) whose nonlinearities satisfy certain slope conditions. The main instrument of this analysis consists in the individual monitoring of the statetrajectories by considering time-dependent rectangular sets that are forward invariant with respect to the dynamics of the investigated systems. Particular requirements for the rectangular sets approaching the equilibrium point allow a componentwise exploration of the stability properties, offering additional information with respect to the traditional framework (that expresses a global knowledge, built in terms of norms). Within this context, we are able to point out some important dynamical aspects that remained hidden for other works relying only on the standard tools of stability analysis. The refinement induced by the componentwise point of view is also revealed by two numerical examples.

Linear time-variant systems: Lyapunov functions and invariant sets defined by Hölder norms

Abstract For linear time-variant systems x ˙ ( t ) = A ( t ) x ( t ) , we consider Lyapunov function candidates of the form V p ( x , t ) = | | H ( t ) x | | p , with 1 ≤ p ≤ ∞ , defined by continuously differentiable and non-singular matrix-valued functions, H ( t ) : R + → R n × n . We prove that the traditional framework based on quadratic Lyapunov functions represents a particular case (i.e. p=2) of a more general scenario operating in similar terms for all Holder p-norms. We propose a unified theory connecting, by necessary and sufficient conditions, the properties of (i) the matrix-valued function H(t), (ii) the Lyapunov function candidate Vp(x,t) and (iii) the time-dependent set X p ( t ) = { x ∈ R n | | | H ( t ) x | | p ≤ e − rt } , with r≥0. This theory allows the construction of four distinct types of Lyapunov functions and, equivalently, four distinct types of sets which are invariant with respect to the system trajectories. Subsequently, we also get criteria for testing stability, uniform stability, asymptotic stability and exponential stability. For all types of Lyapunov functions, the matrix-valued function H(t) is a solution to a matrix differential inequality (or, equivalently, matrix differential equation) expressed in terms of matrix measures corresponding to Holder p-norms. Such an inequality (or equation) generalizes the role played by the Lyapunov inequality (equation) in the classical case when p=2. Finally, we discuss the diagonal-type Lyapunov functions that are easier to handle (including the generalized Lyapunov inequality) because of the diagonal form of H(t).

Petri Net Toolbox in control engineering education

The Petri Net Toolbox (PN Toolbox) for MATLAB is a software package that offers instruments for the simulation, analysis and design of untimed, deterministic and stochastic P-/T-timed and stochastic Petri nets (PNs). The facilities available in this toolbox are appropriate for studying the dynamics of many classes of discrete-event systems. The Petri Net Simulink Block (PNSB) allows the modeling and analysis of hybrid systems whose event-driven part(s) is (are) modeled based on the PN formalism. The current paper focuses on the exploitation of the PN Toolbox for illustrating the usage of the PN theory in Control Engineering from the pedagogic point of view, the discussion being supported by example problems and comments on the teaching goals. The PN Toolbox is included in the Connections Program of The MathWorks Inc., as a third party product.

Absolute componentwise stability of interval hopfield neural networks

The componentwise stability is a special type of asymptotic stability which ensures the individual monitoring of each state-space variable of a dynamical system. For an interval Hopfield neural network (IHNN), sufficient conditions are provided to analyze the absolute componentwise stability with respect to a class of activation functions (CAF). Both continuous- and discrete-time dynamics are considered. The conditions are formulated in terms of Hurwitz/Schur stability of a test matrix built from the information about the CAF and the interval matrices defining the IHNN. Some interesting results are derived as particular cases, which allow comparisons with several other works.

Computation of angular velocity and acceleration tensors by direct measurements

SummaryThe present paper investigates the properties of the angular velocity tensor ϕ and the angular acceleration tensor Ψ for rigid body motion. Three vectorial invariants of rigid body kinematics are presented. In case of tensor Ψ being non-singular, its inverse Ψ−1 is inferred. A novel procedure for automatic computation of tensor ϕ and Ψ based on measured velocity and acceleration data is developed. Depending of the type of available data, three algorithms are suggested. Numerical examples show the application of the method.

Comments on "Assessing the Stability of Linear Time-Invariant Continuous Interval Dynamic Systems

The first part of this note identifies and analyzes two weak points of the commented paper . i) Example 1 reports an incorrect result for which we provide a complete explanation and correction. ii) The practical use of Theorem 1 reveals major drawbacks such as a severe limitation in the applicability (due to the restrictive conditions requested by the hypothesis) and a relatively low accuracy of the results (i.e., rough values for the right outer bounds of the eigenvalue ranges of interval matrices). The second part of the note presents three methods that circumvent the use of Theorem 1 and provide the following information about the eigenvalue range of an interval matrix: a right outer bound-relying on an inequality proved by J. Rohn (see References); the right end point-by solving a constrained global maximization problem; a right outer bound (for arbitrary interval matrices) and the right end point (for some classes of interval matrices)-by calculating the spectral abscissa of a constant matrix that majorizes the considered interval matrix. For illustration by numerical examples, throughout the note we use three interval matrices related to Example 1 from the commented paper.

DIAGONALLY-INVARIANT EXPONENTIAL STABILITY

Abstract The diagonally-invariant exponential stability (DIES) is introduced as a special type of exponential stability (ES) which incorporates information about the sets invariant with respect to the state-space trajectories. DIES is able to unify, at the conceptual level, issues in stability analysis that have been separately addressed by previous researches Unlike ES, DIES is a norm-dependent property and its study requires appropriate instruments. These instruments are derived in terms of matrix measures from the characteristics of the system trajectories; their convenient exploitation in practice is ensured by methods based on matrix comparisons. The developed framework presents a noticeable generality and its applicability is illustrated for several classes of linear and non-linear systems. This framework can be simply adapted to discrete-time systems.

A generalized Hertz-type approach to the eigenvalue bounds of complex interval matrices

The paper proposes a methodology for estimating the eigenvalue bounds of complex interval matrices. The theoretical development is based on the use of matrix measures induced by arbitrary absolute and monotone vector norms, as well as the weighting of the matrix norms (implicitly of the matrix measures) by diagonal matrices with positive entries. Thus, for each norm and each diagonal matrix, one obtains a set of bounds (lower and upper for the real and for the imaginary part) expressed as sums of weighted matrix measures. The practical approach focuses on the 2-norm and, for each sum mentioned above, the optimal values of the weighted measures are looked for. The numerical tractability is ensured by global optimization; in our research we have used the MathWorks “ga” solver. By taking equal weights in our practical approach we get tighter bounds than the ones formulated by Hertz (see References), and we therefore consider that our framework generalizes Hertzs. Finally we illustrate the proposed methodology by several examples that yield relevant comparisons with Hertzs procedure and with another technique recently published.