Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userRobust output regulation of 2 x 2 hyperbolic systems part I: Control law and Input-to-State Stability
58
0
0.0
(
0
)
Authors
Pierre-Olivier Lamare
Ask ChatGPT about the research
No Arabic abstract
We consider the problem of output feedback regulationfor a linear first-order hyperbolic system with collocatedinput and output in presence of a general class of disturbancesand noise. The proposed control law is designed through abackstepping approach incorporating an integral action. Toensure robustness to delays, the controller only cancels partof the boundary reflection by means of a tunable parameter.This also enables a trade-off between disturbance and noisesensitivity.We show that the boundary condition of the obtainedtarget system can be transformed into a Neutral DifferentialEquation (NDE) and that this latter system is Input-to-StateStable (ISS). This proves the boundedness of the controlledoutput for the target system. This extends previous worksconsidering an integral action for this kind of system [16], andconstitutes an important step towards practical implementationof such controllers. Applications and practical considerations,in particular regarding the systems sensitivity functions arederived in a companion paper.
rate research
Read More
This paper presents a systematic method to analyze stability and robustness of uncertain Quantum Input-Output Networks (QIONs). A general form of uncertainty is introduced into quantum networks in the SLH formalism. Results of this paper are built up on the notion of uncertainty decomposition wherein the quantum network is decomposed into nominal (certain) and uncertain sub-networks in cascade connection. Sufficient conditions for robust stability are derived using two different methods. In the first approach, a generalized small-gain theorem is presented and in the second approach, robust stability is analyzed within the framework of Lyapunov theory. In the second method, the robust stability problem is reformulated as feasibility of a Linear Matrix Inequality (LMI), which can be examined using the well-established systematic methods in the literature.
In linear systems theory its a well known fact that a regulator given by the cascade of an oscillatory dynamics, driven by some regulated variables, and of a stabiliser stabilising the cascade of the plant and of the oscillators has the ability of blocking on the steady state of the regulated variables any harmonics matched with the ones of the oscillators. This is the well-celebrated internal model principle. In this paper we are interested to follow the same design route for a controlled plant that is a nonlinear and periodic system with period T : we add a bunch of linear oscillators, embedding n o harmonics that are multiple of 2$pi$/T , driven by a regulated variable of the nonlinear system, we look for a stabiliser for the nonlinear cascade of the plant and the oscillators, and we study the asymptotic properties of the resulting closedloop regulated variable. In this framework the contributions of the paper are multiple: for specific class of minimum-phase systems we present a systematic way of designing a stabiliser, which is uniform with respect to n o , by using a mix of high-gain and forwarding techniques; we prove that the resulting closed-loop system has a periodic steady state with period T with a domain of attraction not shrinking with n o ; similarly to the linear case, we also show that the spectrum of the steady state closed-loop regulated variable does not contain the n harmonics embedded in the bunch of oscillators and that the L 2 norm of the regulated variable is a monotonically decreasing function of n o. The results are robust, namely the asymptotic properties on the regulated variable hold also in presence of any uncertainties in the controlled plant not destroying closed-loop stability.
We consider the effect of parametric uncertainty on properties of Linear Time Invariant systems. Traditional approaches to this problem determine the worst-case gains of the system over the uncertainty set. Whilst such approaches are computationally tractable, the upper bound obtained is not necessarily informative in terms of assessing the influence of the parameters on the system performance. We present theoretical results that lead to simple, convex algorithms producing parametric bounds on the $mathcal{L}_2$-induced input-to-output and state-to-output gains as a function of the uncertain parameters. These bounds provide quantitative information about how the uncertainty affects the system.
In this paper, we continue to consider the generalized Liouville system: $$ Delta_g u_i+sum_{j=1}^n a_{ij}rho_jleft(frac{h_j e^{u_j}}{int h_j e^{u_j}}- {1} right)=0quadtext{in ,}M,quad iin I={1,cdots,n}, $$ where $(M,g)$ is a Riemann surface $M$ with volume $1$, $h_1,..,h_n$ are positive smooth functions and $rho_jin mathbb R^+$($jin I$). In previous works Lin-Zhang identified a family of hyper-surfaces $Gamma_N$ and proved a priori estimates for $rho=(rho_1,..,rho_n)$ in areas separated by $Gamma_N$. Later Lin-Zhang also calculated the leading term of $rho^k-rho$ where $rhoin Gamma_1$ is the limit of $rho^k$ on $Gamma_1$ and $rho^k$ is the parameter of a bubbling sequence. This leading term is particularly important for applications but it is very hard to be identified if $rho^k$ tends to a higher order hypersurface $Gamma_N$ ($N>1$). Over the years numerous attempts have failed but in this article we overcome all the stumbling blocks and completely solve the problem under the most general context: We not only capture the leading terms of $rho^k-rhoin Gamma_N$, but also reveal new robustness relations of coefficient functions at different blowup points.
We consider an abstract class of infinite-dimensional dynamical systems with inputs. For this class, the significance of noncoercive Lyapunov functions is analyzed. It is shown that the existence of such Lyapunov functions implies norm-to-integral input-to-state stability. This property in turn is equivalent to input-to-state stability if the system satisfies certain mild regularity assumptions. For a particular class of linear systems with unbounded admissible input operators, explicit constructions of noncoercive Lyapunov functions are provided. The theory is applied to a heat equation with Dirichlet boundary conditions.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
