Ph.D. received on: 7/7/1997

E-mail: balluchi@parades.rm.cnr.it

Tutor: Prof. Aldo Balestrino, Dipartimento di Sistemi Elettrici e Automazione, Università degli Studi di Pisa 

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Decentralization, robustness and optimality in variable structure control   ___________________________________________________________________________________________________________

Advisor:

Prof.  Aldo Balestrino, Dipartimento di Sistemi Elettrici e Automazione, Università degli Studi di Pisa 

Referee:

Prof. Antonio Bicchi, Dipartimento di Sistemi Elettrici e Automazione, Università degli Studi di Pisa

Summary:

The techniques developed in the framework of Variable Structure (VS) systems provide powerful tools for the solution of problems of different nature and complexity both in systems analysis and control laws design. The capabilities of variable structure techniques are illustrated considering three examples taken from the topics of robust decentralized control, optimal control and differential geometric control.
In the first example the synthesis of a robust decentralized controller for output tracking in large scale systems is considered. Assuming that the designer has identified a structure of the system composed of a number of interconnected subsystems, an equivalent model of the large scale system is proposed, where interactions are represented by input disturbances. The robustness property of variable structure systems allows us to design a decentralized control which cancels the effects of such disturbances. This is possible only in the case where a suitable dominance condition, obtained both for SISO and MIMO interacting subsystems, is verified. Then, the proposed solution is reformulated in the general framework of robust control, enlightening the analogies.  In this application the robustness of VSC with respect to bounded input disturbances and unmodeled  dynamics have been employed. This allowed us to adopt a decentralized technique to design the tracking dynamics.
The second area investigated is that of optimal control, with application to path tracking for mobile vehicle. The stabilization of the Dubins’ car along a circular path is considered. The optimal problem is to find a synthesis of the minimum length paths from an initial Dubins’ car configuration to a desired circle. The optimal synthesis, obtained from Pontryagin Maximum Principle, is implemented by a variable structure control law defined in the reduced state space of the lateral distance from the reference path and orientation error. Such VS control guarantees optimal stabilization and satisfies the curvature constraint. Thanks to the robustness property of VSC, the optimal VS control guarantees also stabilization along a generic path. In this application it is shown how from a regular synthesis, which is a solution to a given optimal problem, a variable structure control can be obtained. In the solution of optimal problems, the introduction of optimal sliding surfaces and, subsequently, the study of VS controls that provide optimal reaching can simplify the analysis.
The last example is concerned with trajectory analysis for nonholonomic systems with drift and bounded inputs. Following the variable structure design approach, an algorithm for trajectory planning and reachable set analysis is devised. The drift term implies that the amount of available torque and time horizon are significant in the planning problem. From the theory of trajectory analysis we have that, under particular conditions, the class of bang-bang input signals is a sufficient class for the reachable problem. Then, the planning problem is addressed choosing a sliding surface which corresponds to a bang-bang input and contains the desired state. Planning is solved providing a VS control which produces convergence to such sliding surface. The same method allows us to devise an algorithm for bounded time reachable sets investigation. Variable structure techniques are here used to the purpose of decomposing a difficult problem into a number of sub-problems of lower dimension.
 

 
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