666 research outputs found
Structured backward errors for eigenvalues of linear port-Hamiltonian descriptor systems
When computing the eigenstructure of matrix pencils associated with the
passivity analysis of perturbed port-Hamiltonian descriptor system using a
structured generalized eigenvalue method, one should make sure that the
computed spectrum satisfies the symmetries that corresponds to this structure
and the underlying physical system. We perform a backward error analysis and
show that for matrix pencils associated with port-Hamiltonian descriptor
systems and a given computed eigenstructure with the correct symmetry structure
there always exists a nearby port-Hamiltonian descriptor system with exactly
that eigenstructure. We also derive bounds for how near this system is and show
that the stability radius of the system plays a role in that bound
A Characterization of all Solutions to the Four Block General Distance Problem
All solutions to the four block general distance problem which arises in H^∞ optimal control are characterized. The procedure is to embed the original problem in an all-pass matrix which is constructed. It is then shown that part of this all-pass matrix acts as a generator of all solutions. Special attention is given to the characterization of all optimal solutions by invoking a new descriptor characterization of all-pass
transfer functions. As an application, necessary and sufficient conditions are found for the existence of an H^∞ optimal controller. Following that, a descriptor representation of all solutions is derived
Generalizing Negative Imaginary Systems Theory to Include Free Body Dynamics: Control of Highly Resonant Structures with Free Body Motion
Negative imaginary (NI) systems play an important role in the robust control
of highly resonant flexible structures. In this paper, a generalized NI system
framework is presented. A new NI system definition is given, which allows for
flexible structure systems with colocated force actuators and position sensors,
and with free body motion. This definition extends the existing definitions of
NI systems. Also, necessary and sufficient conditions are provided for the
stability of positive feedback control systems where the plant is NI according
to the new definition and the controller is strictly negative imaginary. The
stability conditions in this paper are given purely in terms of properties of
the plant and controller transfer function matrices, although the proofs rely
on state space techniques. Furthermore, the stability conditions given are
independent of the plant and controller system order. As an application of
these results, a case study involving the control of a flexible robotic arm
with a piezo-electric actuator and sensor is presented
Strict Positive Realness of Descriptor Systems in State Space
In this paper we give necessary and sufficient
spectral conditions for various notions of strict positive realness
for single input single output, impulse free Descriptor Systems.
These conditions only require calculation of eigenvalues of a
single matrix. A characterization of a KYP-like lemma for
descriptor systems is also derived, and its implications for the
stability of a class of switched descriptor systems are briefly
discussed
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
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