Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userLog-Linear Dynamical Systems
109
0
0.0
(
0
)
Authors
Steven Diamond
Ask ChatGPT about the research
No Arabic abstract
We present log-linear dynamical systems, a dynamical system model for positive quantities. We explain the connection to linear dynamical systems and show how convex optimization can be used to identify and control log-linear dynamical systems. We illustrate system identification and control with an example from predator-prey dynamics. We conclude by discussing potential applications of the proposed model.
rate research
Read More
Semi-tensor product(STP) or matrix (M-) product of matrices turns the set of matrices with arbitrary dimensions into a monoid $({cal M},ltimes)$. A matrix (M-) addition is defined over subsets of a partition of ${cal M}$, and a matrix (M-) equivalence is proposed. Eventually, some quotient spaces are obtained as vector spaces of matrices. Furthermore, a set of formal polynomials is constructed, which makes the quotient space of $({cal M},ltimes)$, denoted by $(Sigma, ltimes)$, a vector space and a monoid. Similarly, a vector addition (V-addition) and a vector equivalence (V-equivalence) are defined on ${cal V}$, the set of vectors of arbitrary dimensions. Then the quotient space of vectors, $Omega$, is also obtained as a vector space. The action of monoid $({cal M},ltimes)$ on ${cal V}$ (or $(Sigma, ltimes)$ on $Omega$) is defined as a vector (V-) product, which becomes a pseudo-dynamic system, called the cross-dimensional linear system (CDLS). Both the discrete time and the continuous time CDLSs have been investigated. For certain time-invariant case, the solutions (trajectories) are presented. Furthermore, the corresponding cross-dimensional linear control systems are also proposed and the controllability and observability are discussed. Both M-product and V-product are generalizations of the conventional matrix product, that is, when the dimension matching condition required by the conventional matrix product is satisfied they coincide with the conventional matrix product. Both M-addition and V-addition are generalizations of conventional matrix addition. Hence, the dynamics discussed in this paper is a generalization of conventional linear system theory.
We show how to efficiently compute the derivative (when it exists) of the solution map of log-log convex programs (LLCPs). These are nonconvex, nonsmooth optimization problems with positive variables that become convex when the variables, objective functions, and constraint functions are replaced with their logs. We focus specifically on LLCPs generated by disciplined geometric programming, a grammar consisting of a set of atomic functions with known log-log curvature and a composition rule for combining them. We represent a parametrized LLCP as the composition of a smooth transformation of parameters, a convex optimization problem, and an exponential transformation of the convex optimization problems solution. The derivative of this composition can be computed efficiently, using recently developed methods for differentiating through convex optimization problems. We implement our method in CVXPY, a Python-embedded modeling language and rewriting system for convex optimization. In just a few lines of code, a user can specify a parametrized LLCP, solve it, and evaluate the derivative or its adjoint at a vector. This makes it possible to conduct sensitivity analyses of solutions, given perturbations to the parameters, and to compute the gradient of a function of the solution with respect to the parameters. We use the adjoint of the derivative to implement differentiable log-log convex optimization layers in PyTorch and TensorFlow. Finally, we present applications to designing queuing systems and fitting structured prediction models.
The well-known Deficiency One Theorem gives structural conditions on a chemical reaction network under which, for any set of parameter values, the steady states of the corresponding mass action system may be easily characterized. It is also known, however, that mass action systems are not uniquely associated with reaction networks and that some representations may satisfy the Deficiency One Theorem while others may not. In this paper we present a mixed-integer linear programming framework capable of determining whether a given mass action system has a dynamically equivalent or linearly conjugate representation which has an underlying network satisfying the Deficiency One Theorem. This extends recent computational work determining linearly conjugate systems which are weakly reversible and have a deficiency of zero.
We present a dynamical system framework for understanding Nesterovs accelerated gradient method. In contrast to earlier work, our derivation does not rely on a vanishing step size argument. We show that Nesterov acceleration arises from discretizing an ordinary differential equation with a semi-implicit Euler integration scheme. We analyze both the underlying differential equation as well as the discretization to obtain insights into the phenomenon of acceleration. The analysis suggests that a curvature-dependent damping term lies at the heart of the phenomenon. We further establish connections between the discretized and the continuous-time dynamics.
This paper studies model order reduction of multi-agent systems consisting of identical linear passive subsystems, where the interconnection topology is characterized by an undirected weighted graph. Balanced truncation based on a pair of specifically selected generalized Gramians is implemented on the asymptotically stable part of the full-order network model, which leads to a reduced-order system preserving the passivity of each subsystem. Moreover, it is proven that there exists a coordinate transformation to convert the resulting reduced-order model to a state-space model of Laplacian dynamics. Thus, the proposed method simultaneously reduces the complexity of the network structure and individual agent dynamics, and it preserves the passivity of the subsystems and the synchronization of the network. Moreover, it allows for the a priori computation of a bound on the approximation error. Finally, the feasibility of the method is demonstrated by an example.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
