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SET induced COMParison principles for COMPlex systems

Final Report Summary - SETCOMP (SET induced COMParison principles for COMPlex systems)

Control theory aims on the one hand to understand and predict dynamic behaviors of systems in nature and engineering and on the other hand to provide methods for optimizing the performance and guaranteeing the stability of the systems under study. The research in control is focused on developing efficient, sustainable and low-cost solutions to control problems related to systems which are nonlinear, they are described by a large number of states or parameters and are subject to constraints. The research conducted during the period 2012-2014 in the Marie Curie IEF project SetComp aimed at establishing new theoretical tools for stability analysis and controller synthesis using set-induced comparison principles. Although comparison methods (which include Lyapunov functions) and invariant set theory have been utilized to deal with stability analysis and synthesis problems for many years, a limited number of tractable and constructive methods for complex systems exist. Also, the set theoretic methods, which are non-conservative, often fail to provide tractable solutions. The research in the SetComp project focused on tackling these challenges by introducing new set-induced tractable stability analysis and synthesis methods. The main results of the research can be categorized in the following two groups.

1. Construction of invariant/contractive sets and regions of stabilizability for complex systems.

- Reachability analysis. The problem of estimating the region of attraction and the region of stabilizability was addressed, by proposing a new type of reachability mappings, named directional reachability mappings. Compared to conventional techniques, the mapping exploits the geometric properties of the set involved in addition to the system dynamics. Consequently, a new iterative systematic method of obtaining monotonically enlarging sequences of invariant sets is now available. The sequences of sets converge to the domain of attraction for the autonomous case and the domain of stabilizability for the non-autonomous case. Additionally, directional expansion and construction of non-convex invariant sets is now possible for linear systems.

- Invariant sets under complexity specifications. The complexity of the shape of the invariant set is inherited to the computational complexity of the corresponding stability analysis (e.g. terminal set in the model predictive control schemes) and controller synthesis (e.g. vertex interpolation laws). To address this problem, a new method of computing sequences of polytopic invariant sets which are monotonically increasing and respect complexity specifications was proposed, making the construction of low complexity invariant sets with non-trivial size possible.

2. Stability analysis and controller synthesis of systems with a large number of states and / or parameters.

- Switched systems. New, alternative necessary and sufficient stability conditions were established for discrete-time switched systems. Although stability analysis of switched linear discrete-time systems is known to be an NP-hard problem, the conditions induce alternative, computationally appealing, iterative numerical methods for stability analysis. Additionally, and although it might appear counter intuitive, similar results were established for the case of linear switched systems whose switching patterns are defined by directed graphs.

- Large-scale systems. A new method of constructing safe sets and stabilizing controllers for large scale, possibly constrained, discrete-time linear systems was established. The developed controller is scalable with respect to the system dimensions while the synthesis procedure can be performed in a distributed fashion. The method can be applied, with modifications, to the control of homogeneous systems, which is a much broader class than the linear case.

- Time-delay systems. By introducing a relaxed notion of invariant sets in the time-delay space, a new stabilization method for time-delay linear discrete-time systems was obtained. The required computations for the controller synthesis and implementation can be carried out now in the original time-delay state space, rendering the method scalable with respect to the maximum delay.

- Periodic systems. A new, alternative stability analysis theorem for nonlinear periodic discrete-time systems, based on periodic Lyapunov functions, was proposed. The new developed conditions offer a trade-off between conservatism and complexity of the stability test, while they yield a tractable stabilizing controller synthesis method.

- Bilinear systems. Using polyhedral Lyapunov functions that correspond to a comparison system, a new iterative procedure for simultaneously computing stabilizing control laws and approximating the domain of attraction of the closed loop systems was proposed.

The practicability of the theoretical results has been already revealed in three non-trivial control applications, namely in (i) the control of DC-DC power converters, (ii) the admission control and resource allocation of clusters of web servers, and (iii) the attitude control of a magnetic satellite. Moreover, the theoretical results obtained have the potential of being applied to a wide variety of systems in engineering applications such in water management, and in systems found in biology.

The outcome of the research have been published/accepted for publication in three journal papers (including Automatica and IEEE Transactions on Circuits and Systems), while four more journal papers are already submitted or under preparation and will be submitted until early 2015. Additionally, the results were disseminated in twelve articles in international conferences (including CDC, ECC, IFAC World Congress), in three national control conferences, while the researcher has given four invited talks to control-oriented workshops and during academic visits. A part of the research results has been embedded in a new PhD course in the DISC (Dutch Institute of Systems and Control) winter programme in 2014.

Contact Details:

Scientist in Charge:
Dr. Mircea Lazar
Assistant Professor,
Control Systems Group
Electrical Engineering Department,
Eindhoven University of Technology,
The Netherlands
E-mail: m.lazar@tue.nl

Fellow:
Dr. Nikolaos Athanasopoulos
Post-doctoral research associate,
Control Systems Group
Electrical Engineering Department,
Eindhoven University of Technology,
The Netherlands
E-mail:n.athanasopoulos@tue.nl
Website: http://www.tue.nl/en/employee/ep/e/d/ep-uid/20114706/ep-tab/4/