Family Of Linear Systems Research Articles

Uniform solvability for families of linear systems on time scales

We give explicit criteria of solvability for families of linear systems on time scales. We introduce a new method of embedding a time scale into a non-autonomous system of ODEs.This will be the first step to implementing the structural stability result obtained by one of the co-authors together with V. A. Pliss to time scale dynamics.

Uniform and Lq-ensemble reachability of parameter-dependent linear systems

In this paper, we consider families of linear systems (linear ensembles) defined by matrix pairs (A(θ),B(θ)) depending on a parameter θ∈P that is varying over a compact subset P of the complex plane. In particular, we investigate the existence of open-loop controls which are independent of the parameter θ∈P and steer a given family of initial states x0(θ) arbitrarily close to a desired family of terminal states f(θ) in finite time. Here, the maps θ↦x0(θ) and θ↦f(θ) are assumed to lie in a common appropriately chosen Banach space Xn(P) of Cn-valued functions. If this task is solvable for all initial and terminal states, the pair (A(θ),B(θ)) is called (completely) ensemble controllable with respect to Xn(P).Using a well-known infinite-dimensional version of the Kalman rank condition for systems on Banach spaces, we derive sufficient conditions for cascade and parallel connections of linear ensembles. Moreover, we prove an abstract decomposition theorem which results from a spectral splitting of the matrix family A(θ). Based on these findings as well as on cyclicity conditions for multiplication operators and approximation theory, we obtain necessary and sufficient conditions for ensemble controllability (reachability) with respect to the space of continuous functions and the space of Lq-functions. In the last section, results on averaged controllability (reachability) for linear families (A(θ),B(θ),C(θ)) are presented.

Control of a small helicopter with linear matrix inequality-based design assuring stability and performance

This article focuses on linear matrix inequality-based controller designs that can achieve stabilization and reference tracking for a small unmanned helicopter at various flight conditions. A nonlinear mathematical model of a small-scale helicopter is constructed. Then trim conditions are found and linearized around different equilibrium points. Local [Formula: see text] controllers are designed at trim conditions based on the local linear models. The pointwise controllers achieve local stability and performance, but fail at stabilization and tracking over the full envelope. A scheduling controller is built by blending the local controller outputs. In addition, grid-based [Formula: see text] controllers are designed at each operating point with common Lyapunov function. This allows controller scheduling between the adjacent design points with guaranteed stability and performance across the design envelope. Based on the family of linear systems which are obtained from the nonlinear model, an affine parameter-dependent model is built to exploit the approximate linear parameter dependency. Then, a parameter-dependent linear parameter varying controller is synthesized for the affine parameter-dependent model. Although local performance is satisfactory for all given design methods, local [Formula: see text] controllers and affine parameter-dependent controller cannot yield satisfactory performance over the full flight envelope apart from the grid-based controller with common Lyapunov function approach.

Sensor placement for distinguishability and single structure observer design in continuous timed Petri nets

This work is concerned with the sensor placement problem for distinguishability and the observer design in continuous timed Petri nets with infinite servers semantics (ContPN). The ContPN are represented by a family of linear systems (LS) switching among them. Depending on the ContPN marking, a LS evolves. Thus the distinguishability problem deals with the possibility of determining which LS is actually evolving, and the sensor placement for distinguishability problem deals with selecting which places must be measured to ensure distinguishability. On the other hand, the observer’s design problem is concerned with developing a mathematical entity devoted to compute the actual state of the ContPN by means of the knowledge of the marking in the places which have a sensor.This work presents three main contributions. The first one shows that there exist cases where the classical theorems in switched linear systems cannot be used in ContPN, since these theorems could establish that the ContPN is not distinguishable, even when, using the ContPN structure, the distinguishability is guaranteed. The second contribution is an algorithm devoted to place sensors (to build the output map) in such a way that the ContPN becomes distinguishable. The input to the algorithm is an output map S, that can be computed as in a previous work, which guarantees that every LS of the ContPN representation is observable. Then new sensors are included to gain distinguishability between every pair of LS. Thus the distinguishability and observability of the ContPN is guaranteed. Finally, the third main contribution is a single structure observer for the bounded and observable ContPN. This strategy is relevant to avoid the need of computing one observer per LS in the family that represents the ContPN, which in practice becomes unfeasible.

Generalized Linear Systems on Curves and Their Weierstrass Points

Let C be a projective Gorenstein curve over an algebraically closed field of characteristic 0. A generalized linear system on C is a pair (ℐ, ε) consisting of a torsion-free, rank-1 sheaf ℐ on C, and a map of vector spaces ε: V → Γ(C, ℐ). If the system is nondegenerate on every irreducible component of C, we associate to it a 0-cycle W, its Weierstrass cycle. Then we show that for each one-parameter family of curves C t degenerating to C, and each family of linear systems (ℒ t , ε t ) along C t , with ℒ t invertible, degenerating to (ℐ, ε), the corresponding Weierstrass divisors degenerate to a subscheme whose associated 0-cycle is W. We show that the limit subscheme contains always an “intrinsic” subscheme, canonically associated to (ℐ, ε), but the limit itself depends on the family ℒ t .

Sensor Placement for Distinguishability in Continuous Timed Petri Nets

This work is concerned with the sensor placement problem for distinguishability in continuous timed Petri nets with infinite servers semantics (ContPN). The ContPN are considered to be composed by a family of linear systems (LS) switching among them. Depending on the ContPN marking, a LS is chosen to evolve. Thus the distinguishability problem deals with the possibility of determining which LS is evolving, and the sensor placement for distinguishability problem deals with selecting which places must be measured to ensure distinguishability. This work presents two main contributions. The first one shows that there exist cases where the classical theorems in switched linear systems cannot be used in ContPN, since these theorems could establish that the ContPN is not distinguishable, even when, using the ContPN structure, the distinguishability is guaranteed. The second contribution is an algorithm devoted to place sensors (to build the output map) in such a way that the ContPN becomes distinguishable. The input to the algorithm is the output map S, computed as in a previous work, which guarantees that every LS of the ContPNis observable. Then new sensors are included to gain distinguishability between ever pair of LS. Thus the distinguishability and observability of the ContPN is guaranteed.

Stabilisability of nonlinear systems by means of time-dependent switching rules

Given a family of linear systems which admit an asymptotically stable convex combination, the existence of stabilising time-dependent switching rules can be proved by using the Baker–Campbell–Hausdorff formula for exponentials. The control laws obtained in this way are periodic, fast switching and independent of the initial state. We prove that under a similar assumption, the approach can be extended to provide stabilising time-dependent switching rules for the families of nonlinear vector fields, as well. However, the resulting control law, in general, is not periodic, not fast switching and it may depend on the initial state.

Output Regulation Problem for Differentiable Families of Linear Systems

Given a family of linear systems depending on a parameter varying in a differentiable manifold, we obtain sufficient conditions for the existence of a (global or local) differentiable family of controllers solving the output regulation problem for the given family. Moreover, we construct it when these conditions hold.

Transformée en échelle de signaux stationnaires

Using the scale transform of a discrete time signal we define a new family of linear systems. We focus on a particular case related to function theory in the bidisk. To cite this article: D. Alpay, M. Mboup, C. R. Acad. Sci. Paris, Ser. I 347 (2009).

A Preconditioned Recycling GMRES Solver for Stochastic Helmholtz Problems

We present a parallel Schwarz type domain decomposition preconditioned recycling Krylov subspace method for the numerical solution of stochastic indefinite elliptic equations with two random coefficients. Karhunen-Loeve expansions are used to represent the stochastic variables and the stochastic Galerkin method with dou- ble orthogonal polynomials is used to derive a sequence of uncoupled deterministic equations. We show numerically that the Schwarz preconditioned recycling GMRES method is an effective technique for solving the entire family of linear systems and, in particular, the use of recycled Krylov subspaces is the key element of this successful approach.

OBSERVABILITY OF A CLASS OF SWITCHED LINEAR SYSTEMS

In this work, the problem of observability in switched linear systems (SLS) where the continuous part is represented by a family of linear systems (LS) and the discrete part is represented by an interpreted Petri net (IPN), is addressed. Here, the final state of the evolving linear system is mapped to the initial state of the next linear system selected during the switching using state commutation functions. These functions enlarge the class of SLS that can be analyzed. In addition, the observability characterization for SLS exhibiting the observability property is presented. Finally, an asymptotic observer design for SLS is presented.

Computation of a Stabilizing Set of Feedback Matrices bf a Large-scale Nonlinear Musculoskeletal Dynamic Model

The purpose of this study is to present a general mathematical framework to compute a set of feedback matrices which stabilize an unstable nonlinear anthropomorphic musculoskeletal dynamic model. This method is activity specific and involves four fundamental stages. First, from muscle activation data (input) and motion degrees-of-freedom (output) a dynamic experimental model is obtained using system identification schemes. Second, a nonlinear musculoskeletal dynamic model which contains the same number of muscles and degrees-of-freedom and best represents the activity being considered is proposed. Third, the nonlinear musculoskeletal model (anthropomorphic model) is replaced by a family of linear systems, parameterized by the same set of input/ output data (nominal points) used in the identification of the experimental model. Finally, a set of stabilizing output feedback matrices, parameterized again by the same set of nominal points, is computed such that when combined with the anthropomorphic model, the combined system resembles the structural form of the experimental model. The method is illustrated in regard to the human squat activity.

Input-output linearization by velocity-based gain-scheduling

The velocity-based analysis framework provides direct support for divide and conquer design approaches, such as the gain-scheduling control design methodology, whereby the design of a non-linear system is decomposed into the design of an associated family of linear systems. The velocity-based gain-scheduling approach is quite general and directly supports the design of feedback configurations for which the closed-loop dynamics are non-linear. However, the present paper concentrates on the velocity-based design of controllers which, when combined with a non-linear plant, attain linear dynamics. The resulting approach is a direct generalization to non-linear systems of classical frequency-domain pole-zero inversion which is, in many ways, complementary to the Input-Output Linearization approach. In particular the former is dynamic and reduces to open-loop inversion in the linear case whilst the latter is essentially static and utilizes full state feedback.

Redundancy in linear inequality system

This paper considers three types of redundant constraints in systems of an arbitrary number of inequalities posed on real locally convex spaces. The properties of each class of redundant constraints are analized and different classification criteria are given, some of them extending well-known redundancy tests for ordinary linear systems. The paper also provides results for more paticular families of linear systems, for instance, systems with either finitely many unknowns (constraints systems in semi-infinite optimization) or finitely many inequalities (constraints systems in certain moment problems)

A general approach to input-output pseudolinearization for nonlinear systems

A nonlinear system is said to be input-output pseudolinearized if its family of linearizations about constant operating points has input-output behavior that is independent of the particular operating point. The problem of constructing static state feedback laws that achieve input-output pseudolinearization for a general class of nonlinear systems is considered. A generalization of the well-known structure algorithm to the case of parameterized families of linear systems plays an important role.

On exponential stability of linear systems and Hurwitz stability of characteristic quasipolynomials

A family of linear systems with uncertainties in parameters and delays, and its corresponding family of characteristic quasipolynomials, are considered. Both commensurate and non-commensurate delays are considered. It is shown that for a large class of systems, namely those which comply with a simple condition ensuring no degree reduction, Hurwitz stability implies also exponential stability.

Exponential stability of linear systems with commensurate time-delays

In this paper we consider a family of linear systems with commensurate time-delays. Each coefficient of the characteristic quasipolynomial and the delay is allowed to vary in prespecified intervals. The main result of the paper is a necessary and sufficient condition for stability of the family of quasipolynomials. This yields a computationally tractable numerical procedure which determines whether or not the family is exponentially stable.

On separatrix curves of a family of linear systems with pulse influence

For a family of linear systems of differential equations with pulse influence, we establish conditions of existence of separatrix curves and present a method for finding these curves.

Synthesis of Robust Controllers with Few Degrees of Freedom for Systems with Structured Real Parametric Uncertainty

The focal point of this paper is the problem of synthesizing a controller which achieves "robust performance" for a given family of linear systems. This family is obtained as a result of structured real uncertainty. The vector of uncertain parameters corresponds to plant coefficients and is restricted to a sphere. Within this framework, we show that the problem of synthesizing a controller which achieves robust performance is equivalent to an interpolation problem involving a frequency-dependent "fourth order set" in the complex plane. For the special case when the controller has a fixed structure, this interpolation problem is transferred to the space of controller parameters. For controllers described by two or three parameters (for example, lead, lag, PI, PO and PID), a complete solution to the synthesis problem is obtained.

An exact quadratic stability condition of uncertain linear systems

It is pointed out that in order to check the quadratic stability of a family of linear systems with a stable nominal system, the positive definiteness constraint can be dropped from the quadratic Lyapunov function. >