32 research outputs found
Local Nash Realizations
In this paper we investigate realization theory of a class of non-linear
systems, called Nash systems. Nash systems are non-linear systems whose vector
fields and readout maps are analytic semi-algebraic functions. In this paper we
will present a characterization of minimality in terms of observability and
reachability and show that minimal Nash systems are isomorphic. The results are
local in nature, i.e. they hold only for small time intervals. The hope is that
the presented results can be extended to hold globally.Comment: 8 pages, extended conference pape
Realization Theory for LPV State-Space Representations with Affine Dependence
In this paper we present a Kalman-style realization theory for linear
parameter-varying state-space representations whose matrices depend on the
scheduling variables in an affine way (abbreviated as LPV-SSA representations).
We deal both with the discrete-time and the continuous-time cases. We show that
such a LPV-SSA representation is a minimal (in the sense of having the least
number of state-variables) representation of its input-output function, if and
only if it is observable and span-reachable. We show that any two minimal
LPV-SSA representations of the same input-output function are related by a
linear isomorphism, and the isomorphism does not depend on the scheduling
variable.We show that an input-output function can be represented by a LPV-SSA
representation if and only if the Hankel-matrix of the input-output function
has a finite rank. In fact, the rank of the Hankel-matrix gives the dimension
of a minimal LPV-SSA representation. Moreover, we can formulate a counterpart
of partial realization theory for LPV-SSA representation and prove correctness
of the Kalman-Ho algorithm. These results thus represent the basis of systems
theory for LPV-SSA representation.Comment: The main difference with respect to the previous version is as
follows: typos have been fixe
Towards Efficient Maximum Likelihood Estimation of LPV-SS Models
How to efficiently identify multiple-input multiple-output (MIMO) linear
parameter-varying (LPV) discrete-time state-space (SS) models with affine
dependence on the scheduling variable still remains an open question, as
identification methods proposed in the literature suffer heavily from the curse
of dimensionality and/or depend on over-restrictive approximations of the
measured signal behaviors. However, obtaining an SS model of the targeted
system is crucial for many LPV control synthesis methods, as these synthesis
tools are almost exclusively formulated for the aforementioned representation
of the system dynamics. Therefore, in this paper, we tackle the problem by
combining state-of-the-art LPV input-output (IO) identification methods with an
LPV-IO to LPV-SS realization scheme and a maximum likelihood refinement step.
The resulting modular LPV-SS identification approach achieves statical
efficiency with a relatively low computational load. The method contains the
following three steps: 1) estimation of the Markov coefficient sequence of the
underlying system using correlation analysis or Bayesian impulse response
estimation, then 2) LPV-SS realization of the estimated coefficients by using a
basis reduced Ho-Kalman method, and 3) refinement of the LPV-SS model estimate
from a maximum-likelihood point of view by a gradient-based or an
expectation-maximization optimization methodology. The effectiveness of the
full identification scheme is demonstrated by a Monte Carlo study where our
proposed method is compared to existing schemes for identifying a MIMO LPV
system
Identification of Piecewise Linear Models of Complex Dynamical Systems
The paper addresses the realization and identification problem or a subclass
of piecewise-affine hybrid systems. The paper provides necessary and sufficient
conditions for existence of a realization, a characterization of minimality,
and an identification algorithm for this subclass of hybrid systems. The
considered system class and the identification problem are motivated by
applications in systems biology
Balanced truncation for linear switched systems
In this paper, we present a theoretical analysis of the model reduction
algorithm for linear switched systems. This algorithm is a reminiscence of the
balanced truncation method for linear parameter varying systems. Specifically
in this paper, we provide a bound on the approximation error in L2 norm for
continuous-time and l2 norm for discrete-time linear switched systems. We
provide a system theoretic interpretation of grammians and their singular
values. Furthermore, we show that the performance of bal- anced truncation
depends only on the input-output map and not on the choice of the state-space
representation. For a class of stable discrete-time linear switched systems (so
called strongly stable systems), we define nice controllability and nice
observability grammians, which are genuinely related to reachability and
controllability of switched systems. In addition, we show that quadratic
stability and LMI estimates of the L2 and l2 gains depend only on the
input-output map.Comment: We have corrected a number of typos and inconsistencies. In addition,
we added new results in Theorem
Minimal realizations of input-output behaviors by LPV state-space representations with affine dependency
The paper makes the first steps towards a behavioral theory of LPV
state-space representations with an affine dependency on scheduling, by
characterizing minimality of such state-space representations. It is shown that
minimality is equivalent to observability, and that minimal realizations of the
same behavior are isomorphic.Finally, we establish a formal relationship
between minimality of LPV state-space representations with an affine dependence
on scheduling and minimality of LPV state-space representations with a dynamic
and meromorphic dependence on scheduling
PAC bounds of continuous Linear Parameter-Varying systems related to neural ODEs
We consider the problem of learning Neural Ordinary Differential Equations
(neural ODEs) within the context of Linear Parameter-Varying (LPV) systems in
continuous-time. LPV systems contain bilinear systems which are known to be
universal approximators for non-linear systems. Moreover, a large class of
neural ODEs can be embedded into LPV systems. As our main contribution we
provide Probably Approximately Correct (PAC) bounds under stability for LPV
systems related to neural ODEs. The resulting bounds have the advantage that
they do not depend on the integration interval.Comment: 12 page
Model Predictive Control Of Water Levels In A Navigation Canal Affected By Resonance Waves
In order to operate navigation canals several requirements need to be met: keeping minimum ecological flow, flood protection, but also for safe operation the water level has to be kept within a certain range around the normal navigation level. The water level is disturbed by several factors: known (measured tributaries) and unknown (unknown tributaries, rain) inputs. However, the most important one is the operation of the locks. If the navigation reach is bounded by locks that overcome large elevation differences, their operation can create big disturbances. These locks should be operated fast enough to allow the crossing of several boats, however the faster they are operated the bigger waves they create. These waves can lead to large deviations from the normal navigation level. Moreover, they can travel several times back and forth before they attenuate, especially in cases of low base flow, high water level, and smooth surface – these are typical characteristics of a lot of navigation canals. Therefore, when the water level is controlled actively (e.g. by the gates located next to the locks) the effect of these waves should be taken into account. In this paper we present a method for a centralized control of water levels. This method decreases the effect of the waves. The method is presented on the example of the Cuinchy-Fontinettes case study