24 research outputs found
Singular Switched Systems in Discrete Time: Solvability, Observability, and Reachability Notions
Discrete-time singular (switched) systems, also known as(switched) difference-algebraic equations and discrete-time (switched)descriptor systems, have in general three solvability issues:inconsistent initial values, nonexistence ornonuniqueness of solutions, and noncausalities, which are generallynot desired in applications. To deal with those issues, newsolvability notions are proposed in the study, and the correspondingnecessary and sufficient conditions have been derived with the help of(strictly) index-1 notions. Furthermore, surrogate (switched)systems--ordinary (switched) systems that have equivalentbehavior--have also been established for solvable systems. Byutilizing those surrogate systems, fundamental analysis includingobservability, determinability, reachability, and controllability has also beencharacterized for singular linear (switched) systems. The solvabilitystudy has been extended to singular nonlinear (switched) systems, andmoreover, Lyapunov and incremental stability analyses have beenderived via single and switched Lyapunov function approaches
Reduced realizations and model reduction for switched linear systems:a time-varying approach
In the last decades, switched systems gained much interest as a modeling framework in many applications. Due to a large number of subsystems and their high-dimensional dynamics, such systems result in high complexity and challenges. This motivates to find suitable reduction methods that produce simplified models which can be used in simulation and optimization instead of the original (large) system. In general, the study aims to find a reduced model for a given switched system with a fixed switching signal and known mode sequence. This thesis concerns first the reduced realization of switched systems with known mode sequence which has the same input-output behavior as original switched systems. It is conjectured that the proposed reduced system has the smallest order for almost all switching time duration. Secondly, a model reduction method is proposed for switched systems with known switching signals which provide a good model with suitable thresholds for the given switched system. The quantitative information for each mode is carried out by defining suitable Gramians and, these Gramians are exploited at the midpoint of the given switching time duration. Finally, balanced truncation leads to a modewise reduction. Later, a model reduction method for switched differential-algebraic equations in continuous time is proposed. Thereto, a switched linear system with jumps and impulses is constructed which has the identical input-output behavior as original systems. Finally, a model reduction approach for singular linear switched systems in discrete time is studied. The choice of initial/final values of the reachability and observability Gramians are also investigated
Model Reduction of Descriptor Systems
Model reduction is of fundamental importance in many control applications. We consider model reduction methods for linear continuous-time descriptor systems. The methods are based on balanced truncation techniques and closely related to the controllability and observability Gramians and Hankel singular values of descriptor systems. The Gramians can be computed by solving the generalized Lyapunov equations with special right-hand sides. The numerical solution of generalized Lyapunov equations is also discussed. A numerical example is given
Balanced truncation model reduction for semidiscretized Stokes equation
We discuss model reduction of linear continuous-time descriptor systems that arise in the control of semidiscretized Stokes equations. Balanced truncation model reduction methods for descriptor systems are presented. These methods are closely related to the proper and improper controllability and observability Gramians and Hankel singular values of descriptor systems. The Gramians can be computed by solving projected generalized Lyapunov equations. Important properties of the balanced truncation approach are that the asymptotic stability is preserved in the reduced order system and there is a priori bound on the approximation error. We demonstrate the application of balanced truncation model reduction to the semidiscretized Stokes equation
Stability, stochastic stationarity, and generalized Lyapunov equations for two-point boundary-value descriptor systems
Cover title.Includes bibliographical references.Supported by the National Science Foundation. ECS-8700903 Supported by the Air Force Office of Scientific Research. AFOSR-88-0032 Supported by the Army Research Office. DAAL03-86-K-0171Ramine Nikoukhah, Bernard C. Levy, and Alan S. Willsky
Descriptor Variable and Generalized Singularly Perturbed Systems: A Geometric Approach
Marked with alternative report numbers UILU-80-2214 and UILU-80-2258U.S. Air Force / AFOSR-78-3633National Science Foundation / NSF ECS-79-19396Joint Services Electronics Program / N00014-79-C-0424Ope
Stability, stochastic stationarity and generalized Lyapunov equations for two-point boundary-value descriptor systems
Bibliography: p. 48-50.Supported in part by a grant from the National Science Foundation. ECS-8700903 Supported in part by a grant from the Air Force Office of Scientific Research. AFOSR-88-0032 Supported in part by a grant from the Army Research Office. DAAL03-86-K-0171Ramine Nikoukhah, Bernard C. Levy and Alan S. Willsky
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Algebraic and Geometric Methods and Problems for Implicit Linear Systems
This thesis investigates a number of problems of the implicit linear systems framework.
First, the problem of realisations of nonproper transfer functions is considered. The main result obtained here is the generalisation of the realisation method from MFDs to the case of the nonproper transfer functions. The obtained realisations are singular systems. The method treats both finite and infinite frequency behaviour in a unified way and generalises the results related to the minimality of the realisation and coprimeness and column reducedness of the MFD. Furthermore, it displays transparently the relation between the extended MacMillan degree of the transfer function and the minimal realisation.
The next problem considered is the problem of canonical forms of minimal singular systems under restricted system equivalence transformations. For systems with outputs a canonical form is obtained and it is shown that it is directly related to the echelon form of the composite matrix of an MFD of the transfer function of the system. This result is a direct generalisation of the results of Popov and Forney for strictly proper systems. The canonical form obtained is of Popov type and may be considered as a direct generalisation of the well known form for strictly proper systems. The second canonical form is for systems without outputs. A Popov type canonical form for a class of these systems is obtained. This class is that of systems with equal reachability indices. For both canonical forms, the sequence of the transformations yielding the canonical description is described in detail. In the general case of systems without outputs a semi canonical Popov type form is obtained.
Another problem considered in the thesis is the problem of first order realisations of autoregressive equations within the external equivalence framework. An alternative to the existing methods is provided; in fact, the proposed method is simpler than the existing ones and allows the derivation of the realisation by inspection of the autoregressive equations. A generalisation of the observability indices is proposed for nonsquare descriptor systems and their connection to the autoregressive equations is established.
The problem of model matching for implicit systems is considered next. This is a generalisation of the model matching problem for systems described by transfer functions. Here a controller is interconnected to the given plant such that the overall system has a desired external behaviour. The problem is studied within the framework of external and A-external (input-output) equivalence. Necessary as well as sufficient conditions for the solvability of the problem are derived and the equations of the controllers are found in a constructive way.
The last problem considered here is the generalised dynamic cover problem of geometric theory i.e. the problem of finding the family of (A, B)-invariant subspaces covering a given subspace. This problem is formulated here by using the matrix pencil approach of the geometric theory. This approach allows the unification of the problem for state-space and nonsquare descriptor systems. An extension of the problem to the case of infinite spectrum spaces is also obtained. The solution of the above problems is reduced to the solution of appropriately defined systems of linear equations. Finally, an alternative method for the solution involving systems of multilinear equations is proposed using the mathematical tool of Groebner bases
A geometric framework for constraints and data:from linear systems to convex processes
In part one of this thesis we develop the theory of analyis of convex processes. The results of this development can be directly applied to the analysis of discrete time, linear, time-invariant mathematical systems with conic constraints. Such constraints arise from physical properties of natural phenomena, and hence it is important that these are considered in the mathematical models thereof. In part two we focus determining whether a system has a given system theoretic property on the basis of measured data. For this, we develop the informativity framework, which allows us to consider and resolve a large number of such problems
Stability results for constrained dynamical systems
Differential-Algebraic Equations (DAE) provide an appropriate framework to model and
analyse dynamic systems with constraints. This framework facilitates modelling of the
system behaviour through natural physical variables of the system, while preserving the
topological constraints of the system. The main purpose of this dissertation is to investigate
stability properties of two important classes of DAEs. We consider some special cases of
Linear Time Invariant (LTI) DAEs with control inputs and outputs, and also a special class of
Linear switched DAEs. In the first part of the thesis, we consider LTI systems, where we focus
on two properties: passivity and a generalization of passivity and small gain theorems called
mixed property. These properties play an important role in the control design of large-scale
interconnected systems. An important bottleneck for a design based on the aforementioned
properties is their verification. Hence we intend to develop easily verifiable conditions to
check passivity and mixedness of Single Input Single Output (SISO) and Multiple Input
Multiple Output (MIMO) DAEs. For linear switched DAEs, we focus on the Lyapunov stability
and this problem forms the basis for the second part of the thesis. In this part, we try
to find conditions under which there exists a common Lyapunov function for all modes
of the switched system, thus guaranteeing exponential stability of the switched system.
These results are primarily developed for continuous-time systems. However, simulation and
control design of a dynamic system requires a discrete-time representation of the system
that we are interested in. Thus, it is critical to establish whether discrete-time systems,
inherit fundamental properties of the continuous-time systems from which they are derived.
Hence, the third part of our thesis is dedicated to the problems of preserving passivity,
mixedness and Lyapunov stability under discretization. In this part, we examine several
existing discretization methods and find conditions under which they preserve the stability
properties discussed in the thesis