We consider the problem of designing distributed controllers for a class of systems which can be obtained from the interconnection of a number of identical subsystems. If the state space matrices of these systems satisfy a certain structural property, then it is possible to derive a procedure for designing a distributed controller which has the same interconnection pattern as the plant. This procedure is basically a multiobjective optimization under linear matrix inequality constraints, with system norms as performance indices. The explicit expressions for computing these controllers are given for both H infin or H 2 performance, and both for static state feedback and dynamic output feedback (in discrete time). At the end of the paper, two application examples illustrate the effectiveness of the approach.

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Distributed Control for Identical Dynamically Coupled Systems: A Decomposition Approach

Subspace identification of circulant systems

The recent progress in technology, as in miniaturization and microtechnologies, is now forcing control engineers to confront themselves with systems of incredibly high dimensionality, with an ever growing number of input and output channels. For such systems, which we call "large scale systems", it is necessary to take a new approach in order to solve control problems in a reasonable time, as well as for being able to design controllers which can be realized in a physically implementable way. This thesis concerns a class of linear time-invariant large scale systems which we call "decomposable systems". Decomposable systems describe systems made of a set of identical subsystems (or agents) that are interacting with each other, and they can be considered as an example of homogeneous systems with arbitrary interconnections. This means that each subsystem interacts only with a limited set of the others, and the interconnection pattern does not have to stick to a special structure or lattice. This class of systems describes very well a number of physical systems of interest, such as formations of vehicles or mechanical elements made of identical subparts. Decomposable systems are interesting under the point of view of the theory as they prove to be amenable to a kind of modal decomposition that depends only on the interconnection pattern and not on the specific system; this property, or "decomposition theorem", is at the basis of all the results shown in this thesis. The first part of this work concerns the problem of synthesizing distributed controllers for decomposable systems. By "distributed", we mean that the controller can be implemented as a set of simple, local controllers interacting with each other, each of them commanding a limited set of neighboring subsystems. This approach is demanded when the number of subsystems is very high: in this case it is not feasible to implement a centralized controller that reads all the outputs and decides all the control inputs. The decomposition property is exploited to convert the problem of controller synthesis for the global decomposable system into the problem of synthesizing controllers for the "modal" systems making up its decomposed version; such modal subsystems have the same order as a single agent. Then, by using techniques from robust control as well as a few results from graph theory, it is possible to cast the distributed controller synthesis problem as an optimization problem under Linear Matrix Inequality constraints. This leads to methods allowing performance-based synthesis of controllers in a variety of cases (e.g. H-2 or H-infinity performance, continuous or discrete time, state or output feedback). The methods only offer suboptimality results, which can be considered as the price to be paid in exchange for the distributed structure of the controller. The distributed controller methods are then used in simulation for two examples of relevant technological application. The first application is the distributed H-2 control of a deformable mirror for adaptive optics; as future Earth-based telescopes will feature deformable mirrors with actuators and sensors in the order of the thousands, the independence of the computational complexity from the system size makes the methods of this thesis very attractive. The second application considered is satellite formation flying, for which an extension of the H-infinity methods to the time-varying case is proposed. The controller is evaluated on two examples of space missions involving non-Keplerian orbits. The last part of the thesis concerns a problem that is complementary to the one of control; namely, it investigates the possibility of identifying models of decomposable systems from data. This possibility is useful in case such models are not available from first principles. It is shown that the decomposition property can be used in this case as well. The problem is first treated for a special case, namely for the class of circulant systems, and then examined in the general case. An approach based on subspace identification is proposed.

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Decomposition methods for distributed control and identification

This paper studies structured control synthesis for structured systems, with a focus on block circulant systems corresponding to the structure of a topological ring. We show that several classic control synthesis problems are amenable to a structured synthesis. Moreover, the findings suggest that structured systems naturally admit structured controllers.

Block circulant control: A geometric approach

Engineering systems are often architected to consist of a number of interconnected parts that interact in distinct patterns. Because most control design methods only provide general, unpatterned control laws, a compelling open question is how to synthesise distributed control laws that adhere to a system's interconnection pattern. This paper addresses patterned control synthesis for systems with interconnection patterns. The pattern is encoded algebraically through commuting relationships of the system's state space matrices. We show that several classic control problems are amenable to a patterned synthesis. Moreover, we show that these patterned control problems have the same solvability conditions as their unpatterned counterparts. That is, a patterned control law can be found whenever any control law can be found. Our findings suggest that patterned systems naturally admit patterned controllers.

Control of a class of patterned systems

The high dimensionality of large systems can cause conventional control techniques fail to give reasonable solutions with reasonable computational efforts. Many large systems encountered in practice are composed of subsystems with similar dynamics interconnected symmetrically. The analysis and control of a large system with such features can take advantage of the structural properties to achieve computational simplifications of the overall problem. The focus of this paper is the development of computational techniques for systems with circulant symmetry using H/sub /spl infin// synthesis.

N. Denis

Papers

H/sub /spl infin// controller design for systems with circulant symmetry