Accelerated iterative distributed controller synthesis with a Barzilai-Borwein step size

This tutorial paper provides an introduction to the evolving field of life extending control (LEC) where the objective is to achieve a desired trade-off between structural durability and dynamic performance of complex dynamical systems. The paper also presents a procedure for the synthesis of fuzzy-logic-based LEC systems via feedback of online predictions of damage and damage rate. The proposed control system has a two-tier structure: linear sampled-data controller(s) tracks a reference trajectory vector in the lower tier, while a fuzzy-logic-based damage controller at the upper tier makes a trade-off between system performance and structural damage in critical plant components. The above LEC concept and several different design methodologies were tested by simulation experiments on a reusable rocket engine, and the airframe structures of rotorcraft and tactical aircraft.

This paper attempts to show how complexity can defeat the analysis, prediction and control of large-scale dynamic systems. Some characteristics and facets of complexity are explored, and three research strategies are discussed which lead to a satisfactory treatment of complexity in dynamic systems. The research strategies relate to (i) the control-theoretic, hierarchical approach, (ii) the design-oriented, algebraic approach, (iii) models using catastrophe theory. The virtues of these approaches are compared, and some practical difficulties in dealing with complex systems are discussed.

A method is proposed to obtain a model of a dynamic system with a state vector of high dimension. The model is derived by aggregating the original system state vector into a lower-dimensional vector. Some properties of the aggregation method are investigated in the paper. The concept of aggregation, a generalization of that of projection, is related to that of state vector partition and is useful not only in building a model of reduced dimension, but also in unifying several topics in the control theory such as regulators with incomplete state feedback, characteristic value computations, model controls, and bounds on the solution of the matrix Riccati equations, etc. Using the quantitative definition of weak coupling proposed by Milne, a suboptimal control policy for the weakly coupled system is derived. Questions of performance degradation and of stability of such suboptimally controlled systems are also answered in the paper.

The Barzilai-Borwein (BB) method is a popular and efficient tool for solving large-scale unconstrained optimization problems.Its search direction is the same as for the steepest descent (Cauchy) method, but its stepsize rule is different.Owing to this, it converges much faster than the Cauchy method.A feature of the BB method is that it may generate too long steps, which throw the iterates too far away from the solution.Moreover, it may not converge, even when the objective function is strongly convex.In this paper, a stabilization technique is introduced.It consists in bounding the distance between each pair of successive iterates, which often allows for decreasing the number of BB iterations.When the BB method does not converge, our simple modification of this method makes it convergent.For strongly convex functions with Lipschits gradients, we prove its global convergence, despite the fact that no line search is involved, and only gradient values are used.Since the number of stabilization steps is proved to be finite, the stabilized version inherits the fast local convergence of the BB method.The presented results of extensive numerical experiments show that our stabilization technique often allows the BB method to solve problems in a fewer iterations, or even to solve problems where the latter fails.

Distributed controller design for a class of sparse singular systems with privacy constraints

<p style='text-indent:20px;'>In various applications, such as the computation of energy excitation states of electrons and molecules, and the analysis of interstellar clouds, the linear response eigenvalue problem, which is a special type of the Hamiltonian eigenvalue problem, is frequently encountered. However, traditional eigensolvers may not be applicable to this problem owing to its inherently large scale. In fact, we are usually more interested in computing some of the smallest positive eigenvalues. To this end, a trace minimum principle optimization model with orthogonality constraint has been proposed. On this basis, we propose an unconstrained surrogate model called trace minimization via penalty, and we establish its equivalence with the original constrained model, provided that the penalty parameter is larger than a certain threshold. By avoiding the orthogonality constraint, we can use a gradient-type method to solve this model. Specifically, we use the gradient descent method with Barzilai–Borwein step size. Moreover, we develop a restarting strategy for the proposed algorithm whereby higher accuracy and faster convergence can be achieved. This is verified by preliminary experimental results.</p>

The adaptive cyclic Barzilai-Borwein (BB) method [DZ05] for unconstrained optimization is extended to bound constrained optimization. Using test problems from the CUTE library [BCGT95], performance is compared with SPG2 (a BB method), GENCAN (a BB/conjugate gradient scheme), and L-BFGS-B (limited BFGS for bound constrained problems).

The centralized paradigm of a single controller and a single plant upon which modern control theory is built is no longer applicable to modern cyber-physical systems of interest, such as the power-grid, software defined networks or automated highways systems, as these are all large-scale and spatially distributed. Both the scale and the distributed nature of these systems has motivated the decentralization of control schemes into local sub-controllers that measure, exchange and act on locally available subsets of the globally available system information. This decentralization of control logic leads to different decision makers acting on asymmetric information sets, introduces the need for coordination between them, and perhaps not surprisingly makes the resulting optimal control problem much harder to solve. In fact, shortly after such questions were posed, it was realized that seemingly simple decentralized optimal control problems are computationally intractable to solve, with the Wistenhausen counterexample being a famous instance of this phenomenon. Spurred on by this perhaps discouraging result, a concerted 40 year effort to identify tractable classes of distributed optimal control problems culminated in the notion of quadratic invariance, which loosely states that if sub-controllers can exchange information with each other at least as quickly as the effect of their control actions propagates through the plant, then the resulting distributed optimal control problem admits a convex formulation. The identification of quadratic invariance as an appropriate means of convexifying distributed optimal control problems led to a renewed enthusiasm in the controller synthesis community, resulting in a rich set of results over the past decade. The contributions of this thesis can be seen as being a part of this broader family of results, with a particular focus on closing the gap between theory and practice by relaxing or removing assumptions made in the traditional distributed optimal control framework. Our contributions are to the foundational theory of distributed optimal control, and fall under three broad categories, namely controller synthesis, architecture design and system identification. We begin by providing two novel controller synthesis algorithms. The first is a solution to the distributed H∞ optimal control problem subject to delay constraints, and provides the only known exact characterization of delay-constrained distributed controllers satisfying an H∞ norm bound. The second is an explicit dynamic programming solution to a two player LQR state-feedback problem with varying delays. Accommodating varying delays represents an important first step in combining distributed optimal control theory with the area of Networked Control Systems that considers lossy channels in the feedback loop. Our next set of results are concerned with controller architecture design. When designing controllers for large-scale systems, the architectural aspects of the controller such as the placement of actuators, sensors, and the communication links between them can no longer be taken as given -- indeed the task of designing this architecture is now as important as the design of the control laws themselves. To address this task, we formulate the Regularization for Design (RFD) framework, which is a unifying computationally tractable approach, based on the model matching framework and atomic norm regularization, for the simultaneous co-design of a structured optimal controller and the architecture needed to implement it. Our final result is a contribution to distributed system identification. Traditional system identification techniques such as subspace identification are not computationally scalable, and destroy rather than leverage any a priori information about the system's interconnection structure. We argue that in the context of system identification, an essential building block of any scalable algorithm is the ability to estimate local dynamics within a large interconnected system. To that end we propose a promising heuristic for identifying the dynamics of a subsystem that is still connected to a large system. We exploit the fact that the transfer function of the local dynamics is low-order, but full-rank, while the transfer function of the global dynamics is high-order, but low-rank, to formulate this separation task as a nuclear norm minimization problem. Finally, we conclude with a brief discussion of future research directions, with a particular emphasis on how to incorporate the results of this thesis, and those of optimal control theory in general, into a broader theory of dynamics, control and optimization in layered architectures.

In this paper, we present a numerical reconstruction method for quantitative photoacoustic tomography (QPAT), based on the radiative transfer equation (RTE), which models light propagation more accurately than diffusion approximation (DA). We investigate the reconstruction of absorption coefficient and scattering coefficient of biological tissues. An improved fixed-point iterative method to retrieve the absorption coefficient, given the scattering coefficient, is proposed for its cheap computational cost; the convergence of this method is also proved. The Barzilai–Borwein (BB) method is applied to retrieve two coefficients simultaneously. Since the reconstruction of optical coefficients involves the solutions of original and adjoint RTEs in the framework of optimization, an efficient solver with high accuracy is developed from Gao and Zhao (2009 Transp. Theory Stat. Phys. 38 149–92). Simulation experiments illustrate that the improved fixed-point iterative method and the BB method are competitive methods for QPAT in the relevant cases.

In analysis and control of large-scale nonlinear dynamical systems, a distributed approach is often an attractive option due to its computational tractability and usually low communication requirements. Success of the distributed control design relies on the separability of the network into weakly interacting subsystems such that minimal information exchange between subsystems is sufficient to achieve satisfactory control performance. While distributed analysis and control design for dynamical network have been well studied, decomposition of nonlinear networks into weakly interacting subsystems has not received as much attention. In this article we propose a vector Lyapunov functions based approach to quantify the energy-flow in a dynamical network via a model of a comparison system. Introducing a notion of power and energy flow in a dynamical network, we use sum-of-squares programming tools to partition polynomial networks into weakly interacting subsystems. Examples are provided to illustrate the proposed method of decomposition.

For a massive multiple-input multiple-output (MI-MO) system, how to cope with the detection difficulties brought by increasing antennas is intractable. Linear methods such as zero-forcing (ZF) and minimum mean square error (MMSE) can achieve sub-optimal performance while suffer from high complexity because of large-scale matrix inversion. Recently, some iterative detectors such as steepest descent (SD) and conjugate gradient (CG) have been proposed to balance the complexity and performance. However, their fast convergence would not maintain when the system loading factor increases. To address the aforementioned issues, this paper 1) introduces a new iterative algorithm called Barzilai-Borwein (BB) that outperforms SD with inexpensive operations and 2) proposes its improved form entitled SDBB to accelerate the convergence even in bad conditions. Both theoretical and numerical results have demonstrated its advantages over the state-of-the-art ones. More specifically, SDBB can surpass the existing split pre-conditioned conjugate gradient (SPCG) detector by more than 2 dB at the bit error rate (BER) of 103 when the number of users is relatively large, and reach a complexity reduction of 10%.

Extension of Kamada-Kawai algorithm, which was designed for calculating layouts of simple undirected graphs, is presented in this paper. Graphs drawn by Kamada-Kawai algorithm exhibit symmetries, tend to produce aesthetically pleasing and crossing-free layouts for planar graphs. Minimization of Kamada-Kawai algorithm is based on Newton-Raphson method, which needs Hessian matrix of second derivatives of minimized node. Disadvantage of Kamada-Kawai embedder algorithm is computational requirements. This is caused by searching of minimal potential energy of the whole system, which is minimized node by node. The node with highest energy is minimized against all nodes till the local equilibrium state is reached. In this paper with Barzilai-Borwein (BB) minimization algorithm, which needs only gradient for minimum searching, instead of Newton-Raphson method, is worked. It significantly improves the computational time and requirements.

The importance of robust proton treatment planning to mitigate the impact of uncertainty is well understood. However, its computational cost grows with the number of uncertainty scenarios, prolonging the treatment planning process. We developed a fast and scalable distributed optimization platform that parallelizes the robust proton treatment plan computation over the uncertaintyscenarios. We modeled the robust proton treatment planning problem as a weighted least-squares problem. To solve it, we employed an optimization technique called the alternating direction method of multipliers with Barzilai-Borwein step size (ADMM-BB). We reformulated the problem in such a way as to split the main problem into smaller subproblems, one for each proton therapy uncertainty scenario. The subproblems can be solved in parallel, allowing the computational load to be distributed across multiple processors (e.g., CPU threads/cores). We evaluated ADMM-BB on four head-and-neck proton therapy patients, each with 13 scenarios accounting for 3 mm setup and 3.5% range uncertainties. We then compared the performance of ADMM-BB with projected gradient descent (PGD) applied to the sameproblem. For each patient, ADMM-BB generated a robust proton treatment plan that satisfied all clinical criteria with comparable or better dosimetric quality than the plan generated by PGD. However, ADMM-BB's total runtime averaged about 6 to 7 times faster. This speedup increased with the number ofscenarios. ADMM-BB is a powerful distributed optimization method that leverages parallel processing platforms, such as multicore CPUs, GPUs, and cloud servers, to accelerate the computationally intensive work of robust proton treatment planning. This results in (1) a shorter treatment planning process and (2) the ability to consider more uncertainty scenarios, which improves planquality.

In this paper, we present the integration of controller synthesis techniques in the environment through the description of a tool dedicated to the incremental construction of reactive controllers. The plant is specified in Signal and the control synthesis is performed on a logical abstraction of this program, named polynomial dynamical system (PDS) over \{-1,0,+1\}. The control of the plant is performed by restricting the controllable input values with respect to the control objectives. These restrictions are obtained by incorporating new algebraic equations into the initial system. This theory sets the basis for the verification and the controller synthesis tool, Signal. Moreover, we present a tool developed around the Signal environment allowing the visualization of the synthesized controller by an interactive simulation of the controlled system. In a first stage, the user specifies in Signal both the physical model and the control objectives to be ensured. A second stage is performed by the Signal compiler which translates the initial Signal program into a PDS, and the control objectives in terms of polynomial relations/operations. The controller is then synthesized using Signal. The result is a controller coded by a polynomial and then by a Ternary Decision Diagram (TDD). Finally, in a third stage, the obtained controller and some simulation processes are automatically included in the initial Signal program. It is then sufficient for the user to compile the resulting Signal program which generates executable code ready for simulation. Different academic examples are used to illustrate the application of the tool.

One-dimensional simulations of nonlinear burn control in ITER