1,265 research outputs found
Quadratic Lyapunov Functions for Mechanical Systems
The “mechanical systems” define a large and important class of highly nonlinear dynamical equations which, for example, encompasses all robots. In this report it is shown that a strict Lyapunov Function suggested by the simplest examplar of the class - a one degree of freedom linear time invariant dynamical system - may be generalized over the entire class. The report lists a number of standard but useful consequences of this discovery. The analysis suggests that the input-output properties of the entire class of nonlinear systems share many characteristics in common with those of a second order, phase canonical, linear time invariant differential equation.
For more information: Kod*La
Piecewise Linear Control Systems
This thesis treats analysis and design of piecewise linear control systems. Piecewise linear systems capture many of the most common nonlinearities in engineering systems, and they can also be used for approximation of other nonlinear systems. Several aspects of linear systems with quadratic constraints are generalized to piecewise linear systems with piecewise quadratic constraints. It is shown how uncertainty models for linear systems can be extended to piecewise linear systems, and how these extensions give insight into the classical trade-offs between fidelity and complexity of a model. Stability of piecewise linear systems is investigated using piecewise quadratic Lyapunov functions. Piecewise quadratic Lyapunov functions are much more powerful than the commonly used quadratic Lyapunov functions. It is shown how piecewise quadratic Lyapunov functions can be computed via convex optimization in terms of linear matrix inequalities. The computations are based on a compact parameterization of continuous piecewise quadratic functions and conditional analysis using the S-procedure. A unifying framework for computation of a variety of Lyapunov functions via convex optimization is established based on this parameterization. Systems with attractive sliding modes and systems with bounded regions of attraction are also treated. Dissipativity analysis and optimal control problems with piecewise quadratic cost functions are solved via convex optimization. The basic results are extended to fuzzy systems, hybrid systems and smooth nonlinear systems. It is shown how Lyapunov functions with a discontinuous dependence on the discrete state can be computed via convex optimization. An automated procedure for increasing the flexibility of the Lyapunov function candidate is suggested based on linear programming duality. A Matlab toolbox that implements several of the results derived in the thesis is presented
Preservation of Common Quadratic Lyapunov Functions and Padé Approximations
It is well known that the bilinear transform,
or first order diagonal Padé approximation to the matrix
exponential, preserves quadratic Lyapunov functions
between continuous-time and corresponding discrete-time
linear time invariant (LTI) systems, regardless of the
sampling time. It is also well known that this mapping
preserves common quadratic Lyapunov functions between
continuous-time and discrete-time switched systems. In this
note we show that while diagonal Padé approximations do
not in general preserve other types of Lyapunov functions
(or even stability), it is true that diagonal Padé approximations
of the matrix exponential, of any order and sampling
time, preserve quadratic stability. A consequence of this
result is that the quadratic stability of switched systems is
robust with respect to certain discretization methods
Quadratic Lyapunov Functions for Systems with State-Dependent Switching
In this paper, we consider the existence of quadratic Lyapunov functions for certain
types of switched linear systems. Given a partition of the state-space, a set of matrices
(linear dynamics), and a matrix-valued function A(x) constructed by associating these
matrices with regions of the state-space in a manner governed by the partition, we ask
whether there exists a positive definite symmetric matrix P such that A(x)T P +PA(x)
is negative definite for all x(t). For planar systems, necessary and sufficient conditions
are given. Extensions for higher order systems are also presented
Joint Spectral Radius and Path-Complete Graph Lyapunov Functions
We introduce the framework of path-complete graph Lyapunov functions for
approximation of the joint spectral radius. The approach is based on the
analysis of the underlying switched system via inequalities imposed among
multiple Lyapunov functions associated to a labeled directed graph. Inspired by
concepts in automata theory and symbolic dynamics, we define a class of graphs
called path-complete graphs, and show that any such graph gives rise to a
method for proving stability of the switched system. This enables us to derive
several asymptotically tight hierarchies of semidefinite programming
relaxations that unify and generalize many existing techniques such as common
quadratic, common sum of squares, and maximum/minimum-of-quadratics Lyapunov
functions. We compare the quality of approximation obtained by certain classes
of path-complete graphs including a family of dual graphs and all path-complete
graphs with two nodes on an alphabet of two matrices. We provide approximation
guarantees for several families of path-complete graphs, such as the De Bruijn
graphs, establishing as a byproduct a constructive converse Lyapunov theorem
for maximum/minimum-of-quadratics Lyapunov functions.Comment: To appear in SIAM Journal on Control and Optimization. Version 2 has
gone through two major rounds of revision. In particular, a section on the
performance of our algorithm on application-motivated problems has been added
and a more comprehensive literature review is presente
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