Optimality of nonlinear design techniques: A converse HJB approach

The issue of optimality in nonlinear controller design is confronted by using the converse HJB approach to classify dynamics under which certain design schemes are optimal. In particular, the techniques of Jacobian linearization, pseudo-Jacobian linearization, and feedback linearization are analyzed. Finally, the conditions for optimality are applied to the 2-D nonlinear oscillator, where simple, nontrivial examples are produced in which the various design techniques are optimal

Kuhn-Tucker-based stability conditions for systems with saturation

This paper presents a new approach to deriving stability conditions for continuous-time linear systems interconnected with a saturation. The method presented can be extended to handle a dead-zone, or in general, nonlinearities in the form of piecewise linear functions. By representing the saturation as a constrained optimization problem, the necessary (Kuhn-Tucker) conditions for optimality are used to derive linear and quadratic constraints which characterize the saturation. After selecting a candidate Lyapunov function, we pose the question of whether the Lyapunov function is decreasing along trajectories of the system as an implication between the necessary conditions derived from the saturation optimization, and the time derivative of the Lyapunov function. This leads to stability conditions in terms of linear matrix inequalities, which are obtained by an application of the S-procedure to the implication. An example is provided where the proposed technique is compared and contrasted with previous analysis methods

Constrained Finite Receding Horizon Linear Quadratic Control

Issues of feasibility, stability and performance are considered for a finite horizon formulation of receding horizon control (RHC) for linear systems under mixed linear state and control constraints. It is shown that for a sufficiently long horizon, a receding horizon policy will remain feasible and result in stability, even when no end constraint is imposed. In addition, offline finite horizon calculations can be used to determine not only a stabilizing horizon length, but guaranteed performance bounds for the receding horizon policy. These calculations are demonstrated on two examples

Constrained nonlinear optimal control: a converse HJB approach

Extending the concept of solving the Hamilton-Jacobi-Bellman (HJB) optimization equation backwards [2], the so called converse constrained optimal control problem is introduced, and used to create various classes of nonlinear systems for which the optimal controller subject to constraints is known. In this way a systematic method for the testing, validation and comparison of different control techniques with the optimal is established. Because it naturally and explicitly handles constraints, particularly control input saturation, model predictive control (MPC) is a potentially powerful approach for nonlinear control design. However, nonconvexity of the nonlinear programs (NLP) involved in the MPC optimization makes the solution problematic. In order to explore properties of MPC-based constrained control schemes, and to point out the potential issues in implementing MPC, challenging benchmark examples are generated and analyzed. Properties of MPC-based constrained techniques are then evaluated and implementation issues are explored by applying both nonlinear MPC and MPC with feedback linearization

A receding horizon generalization of pointwise min-norm controllers

Control Lyapunov functions (CLFs) are used in conjunction with receding horizon control to develop a new class of receding horizon control schemes. In the process, strong connections between the seemingly disparate approaches are revealed, leading to a unified picture that ties together the notions of pointwise min-norm, receding horizon, and optimal control. This framework is used to develop a CLF based receding horizon scheme, of which a special case provides an appropriate extension of Sontag's formula. The scheme is first presented as an idealized continuous-time receding horizon control law. The issue of implementation under discrete-time sampling is then discussed as a modification. These schemes are shown to possess a number of desirable theoretical and implementation properties. An example is provided, demonstrating their application to a nonlinear control problem. Finally, stronger connections to both optimal and pointwise min-norm control are proved

On receding horizon extensions and control Lyapunov functions

Control Lyapunov functions (CLFs) are used in conjunction with receding horizon control (RHC) to develop a new class of control schemes. In the process, strong connections between the seemingly disparate approaches are revealed, leading to a unified picture that ties together the notions of pointwise min-norm, receding horizon, and optimal control. This framework is used to develop a control Lyapunov function based receding horizon scheme, of which a special case provides an appropriate extension of a variation on Sontag's formula. These schemes are shown to possess a number of desirable theoretical and implementation properties. An example is provided, demonstrating their application to a nonlinear control problem

Nonlinear Games: examples and counterexamples

Popular nonlinear control methodologies are compared using benchmark examples generated with a “converse Hamilton-Jacobi-Bellman” method (CoHJB). Starting with the cost and optimal value function V, CoHJB solves HJB PDEs “backwards” algebraically to produce nonlinear dynamics and optimal controllers and disturbances. Although useless for design, it is great for generating benchmark examples. It is easy to use, computationally tractable, and can generate essentially all possible nonlinear optimal control problems. The optimal control and disturbance are then known and can be used to study actual design methods, which must start with the cost and dynamics without knowledge of V. This paper gives a brief introduction to the CoHJB method and some of the ground rules for comparing various methods. Some very simple examples are given to illustrate the main ideas. Both Jacobian linearization and feedback linearization combined with linear optimal control are used as “strawmen” design methods

共和分性に基づく最適ペアトレード

一般に，株式価格はランダムウォークに従い，将来の価格を予測することはできない．一方，同業同規模の企業の銘柄など，株式価格が一定の差(スプレッド) を維持しながら推移するような値動きが，市場において観測されることがある．このような株式価格における現象は，共和分性として特徴付けることができる．株式価格のペアが共和分する場合，スプレッドは平均回帰性をもつ．従って，一時的にスプレッドが平均を上回る(あるいは下回る) ような状況が生じても，いずれは平均的な水準に収束することが期待される．さらに，共和分ペアが複数存在すれば，これらを利用したポートフォリオ最適化問題を考えることができる．本論文では，このような共和分性をもつ株式価格のペアを抽出し，複数のスプレッドを利用して最適ポートフォリオを構築する手法について検討する．最適ポートフォリオについては，離散時間設定における条件付き平均・分散最適化問題, および，連続時間設定における動的最適化問題の2 つの問題設定の下で計算し，実データを用いてシミュレーションを行う．さらに，離散時間設定の最適ポートフォリオに対し，取引コストやパラメータ推定期間の影響について考察する

The natural resources of Morro Bay

The primary purpose of this report, then, is to document the natural resources of Morro Bay and their values; point out significant problems regarding their use and to make recommendations for preservation of these resources to planners, administrators and interested citizens. A secondary purpose of this report is to pull together into one source, all data and references on the biological resources of Morro Bay. Up to now these data have been widely distributed amongst letters, reports, papers, etc., to which few have access. At the request of Senate Resolution No. 176, 1966 First Extraordinary Session, the Department completed in December, 1966 a report entitled, "Report of the Natural Resources of Morro Bay and Proposal for Comprehensive Area Plan." Based largely upon the recommendations of that report, the San Luis Obispo County Board of Supervisors appointed a task force to prepare a comprehensive area plan for the Morro Bay area and its watershed. Hence, the information herein is presented in order that the natural resources of Morro Bay will be given adequate consideration, based on the best data available, in the proposed comprehensive area plan and in other plans such as one being prepared by the Coastal Zone Conservation Commission. (148ppp.

Creative destruction in science

Drawing on the concept of a gale of creative destruction in a capitalistic economy, we argue that initiatives to assess the robustness of findings in the organizational literature should aim to simultaneously test competing ideas operating in the same theoretical space. In other words, replication efforts should seek not just to support or question the original findings, but also to replace them with revised, stronger theories with greater explanatory power. Achieving this will typically require adding new measures, conditions, and subject populations to research designs, in order to carry out conceptual tests of multiple theories in addition to directly replicating the original findings. To illustrate the value of the creative destruction approach for theory pruning in organizational scholarship, we describe recent replication initiatives re-examining culture and work morality, working parents\u2019 reasoning about day care options, and gender discrimination in hiring decisions. Significance statement It is becoming increasingly clear that many, if not most, published research findings across scientific fields are not readily replicable when the same method is repeated. Although extremely valuable, failed replications risk leaving a theoretical void\u2014 reducing confidence the original theoretical prediction is true, but not replacing it with positive evidence in favor of an alternative theory. We introduce the creative destruction approach to replication, which combines theory pruning methods from the field of management with emerging best practices from the open science movement, with the aim of making replications as generative as possible. In effect, we advocate for a Replication 2.0 movement in which the goal shifts from checking on the reliability of past findings to actively engaging in competitive theory testing and theory building. Scientific transparency statement The materials, code, and data for this article are posted publicly on the Open Science Framework, with links provided in the article