Optimal Control with Information Pattern Constraints

Despite the abundance of available literature that starts with the seminal paper of Wang and Davison almost forty years ago, when dealing with the problem of decentralized control for linear dynamical systems, one faces a surprising lack of general design methods, implementable via computationally tractable algorithms. This is mainly due to the fact that for decentralized control configurations, the classical control theoretical framework falls short in providing a systematic analysis of the stabilization problem, let alone cope with additional optimality criteria. Recently, a significant leap occurred through the theoretical machinery developed in Rotkowitz and Lall, IEEE-TAC, vol. 51, 2006, pp. 274-286 which unifies and consolidates many previous results, pinpoints certain tractable decentralized control structures, and outlines the most general known class of convex problems in decentralized control. The decentralized setting is modeled via the structured sparsity constraints paradigm, which proves to be a simple and effective way to formalize many decentralized configurations where the controller feature a given sparsity pattern. Rotkowitz and Lall propose a computationally tractable algorithm for the design of H2 optimal, decentralized controllers for linear and time invariant systems, provided that the plant is strongly stabilizable. The method is built on the assumption that the sparsity constraints imposed on the controller satisfy a certain condition (named quadratic invariance) with respect to the plant and that some decentralized, strongly stablizable, stabilizing controller is available beforehand. For this class of decentralized feedback configurations modeled via sparsity constraints, so called quadratically invariant, we provided complete solutions to several open problems. Firstly, the strong stabilizability assumption was removed via the so called coordinate free parametrization of all, sparsity constrained controllers. Next we have addressed the unsolved problem of stabilizability/stabilization via sparse controllers, using a particular form of the celebrated Youla parametrization. Finally, a new result related to the optimal disturbance attenuation problem in the presence of stable plant perturbations is presented. This result is also valid for quadratically invariant, decentralized feedback configurations. Each result provides a computational, numerically tractable algorithm which is meaningful in the synthesis of sparsity constrained optimal controllers

A convex parameterization of all stabilizing controllers for non-strongly stabilizable plants, under quadratically invariant sparsity constraints

This paper addresses the design of controllers, subject to sparsity constraints, for linear and timeinvariant plants. Prior results have shown that a class of stabilizing controllers, satisfying a given sparsity constraint, admits a convex representation of the Youla–type, provided that the sparsity constraints imposed on the controller are quadratically invariant with respect to the plant and that the plant is strongly stabilizable. Another important aspect of the aforementioned results is that the sparsity constraints on the controller can be recast as convex constraints on the Youla parameter, which makes this approach suitable for optimization using norm-based costs. In this paper, we extend these previous results to non-strongly stabilizable plants. Our extension also leads to a Youla-type representation for the class of controllers, under quadratically invariant sparsity constraints. In our extension, the controller class also admits a representation of the Youla–type, where the Youla parameter is subject to only convex constraints

H2/H∞ controller design for input-delay and preview systems based on state decomposition approach

This thesis concentrates on the efficient solution methods of H2/H∞ optimal control problems for input-delay and preview systems. Although the problems can be reformulated to the ones for delay-free systems by augmenting the state space of the controlled systems, the numerical solution of the Riccati/KYP (Kalman-Yakubovich-Popov) equations for the augmented systems requires special efforts, and complicates controller tuning. On the other hand, it is known that the optimal control laws for certain classes of time-delay systems can be constructed without solving the augmented Riccati/KYP equations. Such design problems are called reduced-order construction problems in this thesis. The solutions of the reducedorder construction problems are still limited in theoretical and practical perspectives. The main purpose of the thesis is to propose a new approach for the reduced-order construction problems, which enables to derive the optimal output feedback controllers for input-delayed and preview systems in a unified manner. We focus on the internal dynamics of the overall systems, and decompose it toward the H^2 and H^∞ performance objectives. The fundamental idea of our approach is first introduced for the discrete-time inputdelayed H^2/H^∞ control problems. The state decomposition enables to solve the output feedback problem through the simpler ones, namely, the full information and output estimation problems. The discrete-time optimal controllers are obtained in the Smith predictor form. They are constructed from the Riccati/KYP equations for the delay-free systems. The solution procedure is further extended to the continuous-time preview H^2/H^∞ control problems in an output feedback setting. The optimal utilization of the preview information is exploited at the full information and output estimation problems. The clear structures of the optimal controllers are revealed as the combination of the finite-dimensional observers and preview-feedforward compensation. In the H^∞ control problems for the input-delayed and preview systems, the J-spectral factorization techniques in the literature are employed. Their interconnection to the augmented Riccati/KYP equations is clarified by reviewing the techniques from a view point of the internal state dynamics.首都大学東京, 2014-03-25, 博士(工学), 甲第440号首都大学東

Normalizer Circuits and Quantum Computation

(Abridged abstract.) In this thesis we introduce new models of quantum computation to study the emergence of quantum speed-up in quantum computer algorithms. Our first contribution is a formalism of restricted quantum operations, named normalizer circuit formalism, based on algebraic extensions of the qubit Clifford gates (CNOT, Hadamard and $\pi/4$-phase gates): a normalizer circuit consists of quantum Fourier transforms (QFTs), automorphism gates and quadratic phase gates associated to a set $G$, which is either an abelian group or abelian hypergroup. Though Clifford circuits are efficiently classically simulable, we show that normalizer circuit models encompass Shor's celebrated factoring algorithm and the quantum algorithms for abelian Hidden Subgroup Problems. We develop classical-simulation techniques to characterize under which scenarios normalizer circuits provide quantum speed-ups. Finally, we devise new quantum algorithms for finding hidden hyperstructures. The results offer new insights into the source of quantum speed-ups for several algebraic problems. Our second contribution is an algebraic (group- and hypergroup-theoretic) framework for describing quantum many-body states and classically simulating quantum circuits. Our framework extends Gottesman's Pauli Stabilizer Formalism (PSF), wherein quantum states are written as joint eigenspaces of stabilizer groups of commuting Pauli operators: while the PSF is valid for qubit/qudit systems, our formalism can be applied to discrete- and continuous-variable systems, hybrid settings, and anyonic systems. These results enlarge the known families of quantum processes that can be efficiently classically simulated. This thesis also establishes a precise connection between Shor's quantum algorithm and the stabilizer formalism, revealing a common mathematical structure in several quantum speed-ups and error-correcting codes.Comment: PhD thesis, Technical University of Munich (2016). Please cite original papers if possible. Appendix E contains unpublished work on Gaussian unitaries. If you spot typos/omissions please email me at JLastNames at posteo dot net. Source: http://bit.ly/2gMdHn3. Related video talk: https://www.perimeterinstitute.ca/videos/toy-theory-quantum-speed-ups-based-stabilizer-formalism Posted on my birthda