170 research outputs found
A Generalized LMI Formulation for Input-Output Analysis of Linear Systems of ODEs Coupled with PDEs
In this paper, we consider input-output properties of linear systems
consisting of PDEs on a finite domain coupled with ODEs through the boundary
conditions of the PDE. This framework can be used to represent e.g. a lumped
mass fixed to a beam or a system with delay. This work generalizes the
sufficiency proof of the KYP Lemma for ODEs to coupled ODE-PDE systems using a
recently developed concept of fundamental state and the associated
boundary-condition-free representation. The conditions of the generalized KYP
are tested using the PQRS positive matrix parameterization of operators
resulting in a finite-dimensional LMI, feasibility of which implies prima facie
provable passivity or L2-gain of the system. No discretization or approximation
is involved at any step and we use numerical examples to demonstrate that the
bounds obtained are not conservative in any significant sense and that
computational complexity is lower than existing methods involving
finite-dimensional projection of PDEs
PIETOOLS: A Matlab Toolbox for Manipulation and Optimization of Partial Integral Operators
In this paper, we present PIETOOLS, a MATLAB toolbox for the construction and
handling of Partial Integral (PI) operators. The toolbox introduces a new class
of MATLAB object, opvar, for which standard MATLAB matrix operation syntax
(e.g. +, *, ' e tc.) is defined. PI operators are a generalization of bounded
linear operators on infinite-dimensional spaces that form a *-subalgebra with
two binary operations (addition and composition) on the space RxL2. These
operators frequently appear in analysis and control of infinite-dimensional
systems such as Partial Differential equations (PDE) and Time-delay systems
(TDS). Furthermore, PIETOOLS can: declare opvar decision variables, add
operator positivity constraints, declare an objective function, and solve the
resulting optimization problem using a syntax similar to the sdpvar class in
YALMIP. Use of the resulting Linear Operator Inequalities (LOIs) are
demonstrated on several examples, including stability analysis of a PDE,
bounding operator norms, and verifying integral inequalities. The result is
that PIETOOLS, packaged with SOSTOOLS and MULTIPOLY, offers a scalable,
user-friendly and computationally efficient toolbox for parsing, performing
algebraic operations, setting up and solving convex optimization problems on PI
operators
Delay-robust stabilization of an n + m hyperbolic PDE-ODE system
International audienceIn this paper, we study the problem of stabilizing a linear ordinary differential equation through a system of an n + m (hetero-directional) coupled hyperbolic equations in the actuating path. The method relies on the use of a backstepping transform to construct a first feedback to tackle in-domain couplings present in the PDE system and then on a predictive tracking controller used to stabilize the ODE. The proposed control law is robust with respect to small delays in the control signal
Duality and -Optimal Control Of Coupled ODE-PDE Systems
In this paper, we present a convex formulation of -optimal
control problem for coupled linear ODE-PDE systems with one spatial dimension.
First, we reformulate the coupled ODE-PDE system as a Partial Integral Equation
(PIE) system and show that stability and performance of the PIE
system implies that of the ODE-PDE system. We then construct a dual PIE system
and show that asymptotic stability and performance of the dual
system is equivalent to that of the primal PIE system. Next, we pose a convex
dual formulation of the stability and -performance problems using
the Linear PI Inequality (LPI) framework. LPIs are a generalization of LMIs to
Partial Integral (PI) operators and can be solved using PIETOOLS, a MATLAB
toolbox. Next, we use our duality results to formulate the stabilization and
-optimal state-feedback control problems as LPIs. Finally, we
illustrate the accuracy and scalability of the algorithms by constructing
controllers for several numerical examples
Decentralized sliding mode control and estimation for large-scale systems
This thesis concerns the development of an approach of decentralised robust control and estimation for large scale systems (LSSs) using robust sliding mode control (SMC) and sliding mode observers (SMO) theory based on a linear matrix inequality (LMI) approach. A complete theory of decentralized first order sliding mode theory is developed. The main developments proposed in this thesis are: The novel development of an LMI approach to decentralized state feedback SMC. The proposed strategy has good ability in combination with other robust methods to fulfill specific performance and robustness requirements. The development of output based SMC for large scale systems (LSSs). Three types of novel decentralized output feedback SMC methods have been developed using LMI design tools. In contrast to more conventional approaches to SMC design the use of some complicated transformations have been obviated. A decentralized approach to SMO theory has been developed focused on the Walcott-Żak SMO combined with LMI tools. A derivation for bounds applicable to the estimation error for decentralized systems has been given that involves unknown subsystem interactions and modeling uncertainty. Strategies for both actuator and sensor fault estimation using decentralized SMO are discussed.The thesis also provides a case study of the SMC and SMO concepts applied to a non-linear annealing furnace system modelderived from a distributed parameter (partial differential equation) thermal system. The study commences with a lumped system decentralised representation of the furnace derived from the partial differential equations. The SMO and SMC methods derived in the thesis are applied to this lumped parameter furnace model. Results are given demonstrating the validity of the methods proposed and showing a good potential for a valuable practical implementation of fault tolerant control based on furnace temperature sensor faults
Mixed H2/H∞ control for infinite dimensional systems
The class of infinite dimensional systems often occurs when dealing with distributed
parameter models consisting of partial differential equations. Although forming a
comprehensive description, they mainly become manageable by finite dimensional approximations
which likely neglect important effects, but underlies a certain structure.
In contrast to common techniques for controlling infinite dimensional systems, this
work focuses on using robust control methods. Thus, the uncertainty structure that
occurs due to the discretization shall be taken into account particularly. Additionally,
optimal performance measures can be included into the design process. The mixed
H2/H∞ control approach handles the inclusion of disturbances and inaccuracies while
guaranteeing specified energy or magnitude bounds.
In order to include various of these system requirements, multi-objective robust control
techniques based on the linear matrix inequality framework are utilized. This offers
great flexibility concerning the formulation of the control task and results in convex
optimization problems which can be solved numerically efficient by semi-definite programming.
A flexible robot arm structure serves as the major application example during this
work. The model discretization leads to an LTI system of specified order with an uncertainty
model which is obtained by considering the concrete approximation impact
and frequency domain tests. A structural analysis of the system model relates the
neglected dynamics to a robust characterization. For the objective selection, stability
shall be ensured under all expected circumstances while the aspects of optimal H2 performance,
passive behavior and optimal measurement output selection are included.
The undesirable spillover effect is thoroughly investigated and thus avoided.Tesi
Stability analysis of linear ODE-PDE interconnected systems
Les systèmes de dimension infinie permettent de modéliser un large spectre de phénomènes physiques pour lesquels les variables d'états évoluent temporellement et spatialement. Ce manuscrit s'intéresse à l'évaluation de la stabilité de leur point d'équilibre. Deux études de cas seront en particulier traitées : l'analyse de stabilité des systèmes interconnectés à une équation de transport, et à une équation de réaction-diffusion. Des outils théoriques existent pour l'analyse de stabilité de ces systèmes linéaires de dimension infinie et s'appuient sur une algèbre d'opérateurs plutôt que matricielle. Cependant, ces résultats d'existence soulèvent un problème de constructibilité numérique. Lors de l'implémentation, une approximation est réalisée et les résultats sont conservatifs. La conception d'outils numériques menant à des garanties de stabilité pour lesquelles le degré de conservatisme est évalué et maîtrisé est alors un enjeu majeur. Comment développer des critères numériques fiables permettant de statuer sur la stabilité ou l'instabilité des systèmes linéaires de dimension infinie ? Afin de répondre à cette question, nous proposons ici une nouvelle méthode générique qui se décompose en deux temps. D'abord, sous l'angle de l'approximation sur les polynômes de Legendre, des modèles augmentés sont construits et découpent le système original en deux blocs : d'une part, un système de dimension finie approximant est isolé, d'autre part, l'erreur de troncature de dimension infinie est conservée et modélisée. Ensuite, des outils fréquentiels et temporels de dimension finie sont déployés afin de proposer des critères de stabilité plus ou moins coûteux numériquement en fonction de l'ordre d'approximation choisi. En fréquentiel, à l'aide du théorème du petit gain, des conditions suffisantes de stabilité sont obtenues. En temporel, à l'aide du théorème de Lyapunov, une sous-estimation des régions de stabilité est proposée sous forme d'inégalité matricielle linéaire et une sur-estimation sous forme de test de positivité. Nos deux études de cas ont ainsi été traitées à l'aide de cette méthodologie générale. Le principal résultat obtenu concerne le cas des systèmes EDO-transport interconnectés, pour lequel l'approximation et l'analyse de stabilité à l'aide des polynômes de Legendre mène à des estimations des régions de stabilité qui convergent exponentiellement vite. La méthode développée dans ce manuscrit peut être adaptée à d'autres types d'approximations et exportée à d'autres systèmes linéaires de dimension infinie. Ce travail ouvre ainsi la voie à l'obtention de conditions nécessaires et suffisantes de stabilité de dimension finie pour les systèmes de dimension infinie.Infinite dimensional systems allow to model a large panel of physical phenomena for which the state variables evolve both temporally and spatially. This manuscript deals with the evaluation of the stability of their equilibrium point. Two case studies are treated in particular: the stability analysis of ODE-transport, and ODE-reaction-diffusion interconnected systems. Theoretical tools exist for the stability analysis of these infinite-dimensional linear systems and are based on an operator algebra rather than a matrix algebra. However, these existence results raise a problem of numerical constructibility. During implementation, an approximation is performed and the results are conservative. The design of numerical tools leading to stability guarantees for which the degree of conservatism is evaluated and controlled is then a major issue. How can we develop reliable numerical criteria to rule on the stability or instability of infinite-dimensional linear systems? In order to answer this question, one proposes here a new generic method, which is decomposed in two steps. First, from the perspective of Legendre polynomials approximation, augmented models are built and split the original system into two blocks: on the one hand, a finite-dimensional approximated system is isolated, on the other hand, the infinite-dimensional truncation error is preserved and modeled. Then, frequency and time tools of finite dimension are deployed in order to propose stability criteria that have high or low numerical load depending on the approximated order. In frequencies, with the aid of the small gain theorem, sufficient stability conditions are obtained. In temporal, with the aid of the Lyapunov theorem, an under estimate of the stability regions is proposed as a linear matrix inequality and an over estimate as a positivity test. Our two case studies have been treated with this general methodology. The main result concerns the case of ODE-transport interconnected systems, for which the approximation and stability analysis using Legendre polynomials leads to exponentially fast converging estimates of stability regions. The method developed in this manuscript can be adapted to other types of approximations and exported to other infinite-dimensional linear systems. Thus, this work opens the way to obtain necessary and sufficient finite-dimensional conditions of stability for infinite-dimensional systems
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