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The problem of adaptive fuzzy observer-based tracking control for nonlinear singular systems is considered in this paper. The nonlinear singular systems are composed of two kinds of subsystems, being called the differential subsystem and algebraic subsystem, which are coupled with each other. The systems can be non-strict feedback structures. Through designing a new state observer and a linear controller, an error system with a tunable parameter is obtained. By constructing new one-sided Lipschitz conditions, the regularization and impulse-free conditions are proposed for the error system. With the help of the tunable parameter in the observer, we design an output feedback controller for the nonlinear singular systems to ensure that all states of the closed-loop system are bounded and the tracking and observer errors remain in a neighborhood of the origin. Two examples are provided to illustrate the effectiveness of the presented method.

In this paper we show the existence and multiplicity of positive solutions using the sub-supersolution method and Mountain Pass Theorem in a general singular system which the operator is not homogeneous neither linear.

Singular systems play an important role in many fields, and some new fractional operators, which are general, have been proposed recently. Therefore, singular systems on the basis of the mixed derivatives including the integer order derivative and the generalized fractional operators are studied. Firstly, Lagrange equations within mixed derivatives are established, and the primary constraints are presented for the singular systems. Then the constrained Hamilton equations are constructed by introducing the Lagrange multipliers. Thirdly, Noether symmetry, Lie symmetry and the corresponding conserved quantities for the constrained Hamiltonian systems are investigated. And finally, an example is given to illustrate the methods and results.

In this article, we focus on a class of singular linear systems of fractional differential equations with given nonconsistent initial conditions (IC). Because the nonconsistency of the IC can not lead to a unique solution for the singular system, we use two optimization techniques to provide an optimal solution for the system. We use two optimization techniques to provide the optimal solution for the system because a unique solution for the singular system cannot be obtained due to the non-consistency of the IC. These two optimization techniques involve perturbations to the non-consistent IC, specifically, an l2 perturbation (which seeks an optimal solution for the system in terms of least squares), and a second-order optimization technique at an l1 minimum perturbation, (which includes an appropriate smoothing). Numerical examples are given to justify our theory. We use the Caputo (C) fractional derivative and two recently defined alternative versions of this derivative, the Caputo-Fabrizio (CF) and the Atangana-Baleanu (AB) fractional derivative. Science Foundation Ireland

This paper investigates the problem of a state bounding estimation for a linear continuous-time singular system with time-varying delay. By employing the maximal Lyapunov–Krasovskii functional and applying the new free-matrix-based integral inequality, some proper conditions are derived in terms of LMIs and a bounding estimation lemma and set are obtained for the studied singular system.

Dimirovski, Georgi M. (Dogus Author) -- Conference full title: 36th Chinese Control Conference (CCC); Dalian; China; 26 July 2017 throgh 28 July 2017. Aiming at singular property for a class of link constrained manipulator, the design of soft sliding mode controller is considered by disturbance compensation in this paper. First, a singular system model of a three-link manipulator is established by the constraint equations and the force descriptions. Then, a dynamic switching function is given and a novel discrete singular exponential reaching law with dynamic disturbance compensation is designed. Finally, soft sliding mode controller for three-link constrained manipulator with uncertainties is discussed by the proposed discrete reaching law. Moreover, it can drive the manipulator system to reach the switching band in finite time and remain on the switching plane. The reason of system chattering will be analyzed and the width of the switching band will be decreased by the proposed controller.

International audience; This paper is concerned with the fixed-time observer (FTO) design for discrete-time singular systems subject to unknown inputs. Firstly, the observer matching condition (OMC) and the strongly observability condition (SOC) are given. Then, a new FTO design framework with a reduced-order is given which is free from the influences of the unknown inputs. It is proven that the SOC together with the OMC guarantees the existence of a FTO which is able to provide an exact state estimation in an arbitrarily pre-defined time step. Another advantage of the FTO is that both the estimation accuracy and the convergence time step can be guaranteed regardless of whatever the initial values of the observer are. Finally, a numerical simulation validates the effectiveness of the proposed result.

We consider a classical control problem: the infinite horizon singular LQ problem, i.e., some inputs are unpenalized in the quadratic performance index. In this case, it is known that the slow dynamics is constrained to be in a proper subspace of the state-space, with the optimal input for the slow dynamics implementable by feedback. In this technical note we show that both the fast dynamics and the slow dynamics can be implemented by a feedback controller. Moreover, we show that the feedback controller cannot be a static feedback controller but can be PD, i.e.,, proportional + differentiate exactly once, in the state. We show that the closed loop system is a singular descriptor state space system and we also characterize the conditions on the system/performance index for existence of inadmissible initial conditions, i.e., initial conditions that cause impulsive solutions. There are no inadmissible initial conditions in the controlled system if and only if in the strictly proper transfer matrix from the unpenalized inputs to the penalized states, there exists at least one maximal minor of relative degree equal to the number of unpenalized inputs. In addition to the above, we prove solvability of the infinite horizon singular LQ problem under milder assumptions than in the literature.

In this work we study conditions for guaranteeing the nonnegativity of a discrete-time singular control system. A first approach can be found in the literature for general systems, using the whole coefficient matrices. Also, the particular case of matrices of index 1 has been treated by using a block decomposition and the group-projector of the matrix that gives the singularity to the system. In order to complete this study, an analysis of the nonnegativity of a singular control system for matrices having arbitrary index is done by means of the core nilpotent decomposition. This technique allows us to reduce the size of the original matrices, improving the results where the whole coefficients are involved., This paper was partially supported by the Ministry of Education of Spain (grant DGI MTM2010-18228).

The fuzzy matrix equations $A\tilde{X}=\tilde{Y}$ is called a singular fuzzy matrix equations while the coefficients matrix of its equivalent crisp matrix equations be a singular matrix. The singular fuzzy matrix equations are divided into two parts: consistent singular matrix equations and inconsistent fuzzy matrix equations. In this paper, the inconsistent singular fuzzy matrix equations is studied and the effect of generalized inverses in finding minimal solution of an inconsistent singular fuzzy matrix equations are investigated.

The problem of robust stability of a singular time-delay system is investigated. A novel Lyapunov–Krasovskii functional (LKF) is introduced which is a singular-type complete quadratic Lyapunov–Krasovskii functional with polynomial parameters. Stability conditions are derived in the form of linear matrix inequalities. Numerical examples are given to illustrate the effectiveness and lower conservatism of the new proposed stability criterion.

In this paper we consider the numerical differentiation of functions specified by noisy data. A new approach, which is based on an integral equation of the first kind with a suitable compact operator, is presented and discussed. Since the singular system of the compact operator can be obtained easily, TSVD is chosen as the needed regularization technique and we show that the method calls for a discrete sine transform, so the method can be implemented easily and fast. Numerical examples are also given to show the efficiency of the method.

The present paper is concerned with the convergence problem of Newton's method to solve singular systems of equations with constant rank derivatives. Under the hypothesis that the derivatives satisfy a type of weak Lipschitz condition, a convergence criterion based on the information around the initial point is established for Newton's method for singular systems of equations with constant rank derivatives. Applications to two special and important cases: the classical Lipschitz condition and the Smale's assumption, are provided; the latter, in particular, extends and improves the corresponding result due to Dedieu and Kim in [J.P. Dedieu, M. Kim, Newton's method for analytic systems of equations with constant rank derivatives, J. Complexity 18 (2002) 187–209].

We have proposed an approach to derive a continuous system of differential algebraic equations (DAE) of index one that is dynamically equivalent to a discontinuous index one DAE system. This involves augmenting the convex combination of the ordinary differential equations with the algebraic equations from individual models. This result is proved by using the implicit function theorem. This procedure is illustrated with the help of an ideal gas-liquid system in which the algebraic variables can be expressed as explicit functions of differential variables. It is also demonstrated with an example from a soft-drink manufacturing process, in which, it is difficult to express the algebraic variables as explicit functions of differential variables. Through computer simulation, it is shown that the equivalent dynamic DAE system and the discontinuous DAE system have identical solutions. The proposed method is several orders of magnitude more efficient than the procedure that works with the discontinuous system of DAEs. (c) 2006

Recently, Calvetti et al. have published an interesting paper [Linear Algebra Appl. 316 (2000) 157–169] concerning the least-squares solution of a singular system by using the so-called range restricted GMRES (RRGMRES) method. However, one of the main results (cf. [loc. cit., Theorem 3.3]) seems to be incomplete. As a complement of paper [loc. cit.], in this note we first make an example to show the incompleteness of that theorem, then we give a modified result.

The partial realization problem under consideration consists in finding, for a given sequence s =(s k ) 0 N−1 of blocks, matrices (A,E,B,C) of appropriate size such that s i =CE N−1−i A i B and the identity matrix is a linear combination of A and E. We discuss the question whether there is always a realization of this form for which the state space dimension is equal to the maximal rank of the underlying Hankel matrices. We show that this question has an affirmative answer if the block size is less than or equal to 2 and some other cases but not in general. The paper strengthens results obtained by Manthey et al. [cf. W. Manthey, U. Helmke, D. Hinrichsen, in: U. Helmke et al. (Eds.), Operators, Systems, and Linear Algebra, Teubner, Stuttgart, 1997, pp. 138–156]. The main tools are the results of the authors obtained in connection with Vandermonde factorization of block Hankel matrices. Finally, an interpretation of the problem in periodic discrete-time systems is given.

The Ostrowski–Reich theorem gives the necessary and sufficient condition of convergence of the SOR method for hermitian and positive definite matrices. Ortega and Plemmons have generalized the theorem to non-hermitian matrices. For a general nonsingular matrix A, they have given necessary and sufficient conditions of convergence of splitting methods, and have studied the convergence of the SOR method. This note is a generalization of the Ortega and Plemmons theorems to singular matrices. Some necessary and sufficient conditions of semi-convergence for singular matrices are given.

Krylov subspace methods have been recently considered to solve singular linear systems Ax = b . In this paper, we derive the necessary and sufficient conditions guaranteeing that a Krylov subspace method converges to a vector ADb+Px0, where AD is the Drazin inverse of A and P is the projection P = I−A D A . Let k be the index of A. We further show that A D b+Px 0 , x 0 ∈ R (A k−1 )+ N (A) , is a generalized least-squares solution of Ax = b in R (A k )+ N (A) . Finally, we present the convergence bounds for the quasi-minimal residual algorithm (QMR) and transpose-free quasi-minimal residual algorithm (TFQMR). The index k of A in this paper can be arbitrary, which extends to the main results of Freund and Hochbruck (Numer. Linear Algebra Appl. 1 (1994) 403–420) that only considers the case k = 1 .

We consider switched singular linear systems and determine the set of reachable/controllable states. We derive necessary and sufficient conditions for such a system to be reachable/controllable when an “equisingularity condition” holds.