Nonnegativity, stability, and regularization of discrete-time descriptor systems (original) (raw)
Positive Realness and Absolute Stability Problem of Descriptor Systems
IEEE Transactions on Circuits and Systems I: Regular Papers, 2007
This paper considers a class of nonlinear descriptor systems described by a linear time-invariant descriptor system with feedback-connected sector-constrained nonlinearities. First, we discuss the positive realness problem of descriptor systems and present a new version of positive real lemma. Second, we define the notion of strongly absolute stability (SAB) which is equivalent to the linear part is regular and impulsive-free and the overall feedback system is exponential stable and a SAB criteria in frequency domain is derived. Then, we address the problem of designing a state feedback controller such that the closed-loop feedback-connected system is SAB. To achieve this, we give a linear matrix inequality (LMI)-based SAB criteria, and the above problem is converted into an LMI feasibility problem. Finally, some numerical examples are given to illustrate our approach.
Some invariants of discrete-time descriptor systems
Applied Mathematics and Computation, 2002
Many papers analyze the role of controllability and observability indices of normal linear systems in control theory and their use in the study of structural properties. This work introduces these indices for linear discrete-time descriptor systems. Such indices are studied in two different ways, by means of the state-space approach and using its transfer matrix. A canonical form of the state-space system and a special decomposition of the nonproper transfer matrix are considered in order to discuss the reachability and observability indices.
Input-to-state stability of a class of descriptor systems
International Journal of Robust and Nonlinear Control, 2014
This paper studies the input-to-state stability (ISS) of descriptor systems with exogenous disturbances. on the basis of the ISS theory of standard state-space nonlinear systems, a sufficient condition for a class of nonlinear descriptor system to be ISS is proved. Furthermore, a design method of the state feedback controllers is given to make the closed-loop system ISS. A numerical example is given to illustrate the effectiveness of the controller design.
Feedback Linearization and Control Design for Nonlinear Descriptor Systems
Anais do 14º Simpósio Brasileiro de Automação Inteligente, 2019
This paper deals with the stabilization problem of continuous-time nonlinear descriptor systems. The methodological contribution is to propose a state transformation based on a canonical controllable form, originally proposed for linear descriptor systems, such that a feedback linearizable nonlinear descriptor model can be achieved and, consequently, the control law design designed. The closed-loop stability is checked in the sense of the standard Lyapunov theory. Two examples are presented to illustrate details of implementation. The concluding remarks discuss about the effectiveness and drawbacks of the proposed strategy.
Dynamic Output Feedback Control of Constrained Descriptor Systems
This paper aims to present an approach for design of dynamic output feedback compensators for linear discrete-time descriptor systems subject to state and control constraints. To this end, output feedback controlled invariant polyhedra are constructed by taking a pair of polyhedral sets: a controlled invariant set and a conditioned invariant set. By defining an augmented system composed of the original system plus the dynamic compensator, a control action can be computed online, which optimizes the contraction rate of the augmented state trajectory and enforces the constraints. The results are illustrated through numerical examples, which show that the proposed dynamic compensators outperform static feedback controllers under the same conditions.
A classification of the solutions of linear, time invariant non-regular, discrete descriptor systems is given in terms of the structural invariants of the associated matrix pencil aE -A. The lack of conditionability (in the general case) implies a partitioning of the behavior and thus a classification of the solutions according to their boundary values. A generalization of the boundary mapping equation is also given.
Qualitative sign stability of linear time invariant descriptor systems
Acta Polytechnica
This article discusses assessing the instability of a continuous linear homogeneous timeinvariant descriptor system. Some necessary conditions and sufficient conditions are derived to establish the stability of a matrix pair by the fundamentals of qualitative ecological principles. The proposed conditions are derived using only the qualitative (sign) information of the matrix pair elements. Based on these conditions, the instability of a matrix pair can easily be determined, without any magnitude information of the matrix pair elements and without numerical eigenvalues calculations. With the proposed theory, Magnitude Dependent Stable, Magnitude Dependent Unstable, and Qualitative Sign Stable matrix pairs can be distinguished. The consequences of the proposed conditions and some illustrative examples are discussed.
A note on approximating the nearest stable discrete-time descriptor systems with fixed rank
Applied Numerical Mathematics, 2019
Consider a discrete-time linear time-invariant descriptor system Ex(k + 1) = Ax(k) for k ∈ Z +. In this paper, we tackle for the first time the problem of stabilizing such systems by computing a nearby regular index one stable systemÊx(k + 1) =Âx(k) with rank(Ê) = r. We reformulate this highly nonconvex problem into an equivalent optimization problem with a relatively simple feasible set onto which it is easy to project. This allows us to employ a block coordinate descent method to obtain a nearby regular index one stable system. We illustrate the effectiveness of the algorithm on several examples.
Minimum norm regularization of descriptor systems by mixed output feedback
Linear Algebra and its Applications, 1999
We study the regularization problem for linear, constant coecient descriptor systems i x ex fuY y 1 gxY y 2 C x by proportional and derivative mixed output feedback. Necessary and sucient conditions are given, which guarantee that there exist output feedbacks such that the closed-loop system is regular, has index at most one and i fqC has a desired rank, i.e., there is a desired number of dierential and algebraic equations. To resolve the freedom in the choice of the feedback matrices we then discuss how to obtain the desired regularizing feedback of minimum norm and show that this approach leads to useful results in the sense of robustness only if the rank of E is decreased. Numerical procedures are derived to construct the desired feedback gains. These numerical procedures are based on orthogonal matrix transformations which can be implemented in a numerically stable way.
Positivity of continuous time irregular linear descriptor systems
TURKISH JOURNAL OF MATHEMATICS, 2019
The positivity of continuous-time irregular linear descriptor systems is investigated in this study. The present method is introduced based on Weierstrass decomposition. We obtain necessary and sufficient conditions for the positivity of the linear irregular descriptor systems. The conditions are investigated by two examples with Simulink/MATLAB.