Optimal Feedback for Structures Controlled by Hydraulic Semi-active Dampers

On the stochastic linear quadratic control problem with piecewise constant admissible controls

This article presents a unified approach to simultaneous solution of optimization and optimal control problems in batch distillation, operating under different modes of operation: variable, constant, or optimal reflux. The simplified, computationally efficient short‐cut method and a novel algorithm to solve the optimal control problems in batch distillation is the basis of this unified approach. The short‐cut method identifies the feasible region of operation essential for optimization and optimal control problems, and provides analytical partial derivatives of the model parameters crucial to the solution.The new algorithm for the solution of optimal combination of the maximum principle and NLP optimization techniques. It circumvents the problems associated with the maximum principle approach (iterative solution of a two‐point boundary value problem, undounded control variables, and inability to handle the simultaneous optimization and optimal control problem), and the coupled ODE discretization‐NLP optimization scheme for nonlinear models (higher system nonlinearities, multiplicity of solutions, sensitivity of convergence to initial guesses). This algorithm reduces the dimensionality of the problem, and the nature of the algorithm allows a common platform to optimal solutions of different operating conditions. This article also shows that different categories of the optimal control problems in batch distillation essentially involve the solution of the maximum distillate problem.

1 A Survey on Computational Optimal Control.- Issues in the Direct Transcription of Optimal Control Problems to Sparse Nonlinear Programs.- Optimization in Control of Robots.- Large-scale SQP Methods and their Application in Trajectory Optimization.- Solving Optimal Control and Pursuit-Evasion Game Problems of High Complexity.- 2 Theoretical Aspects of Optimal Control and Nonlinear Programming.- Continuation Methods In Boundary Value Problems.- Second Order Optimality Conditions for Singular Extremals.- Synthesis of Adaptive Optimal Controls for Linear Dynamic Systems.- Control Applications of Reduced SQP Methods.- Time Optimal Control of Mechanical Systems.- 3 Algorithms for Optimal Control Calculations.- Second Order Algorithm for Time Optimal Control of a Linear System.- An SQP-type Solution Method for Constrained Discrete-Time Optimal Control Problems.- Numerical Methods for Solving Differential Games, Prospective Applications to Technical Problems.- Construction of the Optimal Feedback Controller for Constrained Optimal Control Problems with Unknown Disturbances.- Repetitive Optimization for Predictive Control of Dynamic Systems under Uncertainty.- Optimal Control of Multistage Systems Described by High-Index Differential-Algebraic Equations.- A New Class of a High Order Interior Point Method for the Solution of Convex Semiinfinite Optimization Problems.- A Structured Interior Point SQP Method for Nonlinear Optimal Control Problems.- 4 Software for Optimal Control Calculations.- Automated Approach for Optimizing Dynamic Systems.- ANDECS: A Computation Environment for Control Applications of Optimization.- Application of Automatic Differentiation to Optimal Control Problems.- OCCAL: A mixed symbolic-numeric Optimal Control CALculator.- 5 Applications of Optimal Control.- A Robotic Satellite with Simplified Design.- Nonlinear Control under Constraints of a Biological System.- An Object-Oriented Approach to Optimally Describe and Specify a SCADA System Applied to a Power Network.- Near-Optimal Flight Trajectories Generated by Neural Networks.- Performance of a Feedback Method with Respect to Changes in the Air-Density during the Ascent of a Two-Stage-To-Orbit Vehicle.- Linear Optimal Control for Reentry Flight.- Steady-State Modelling of Turbine Engine with Controllers.- Shortest Paths for Satellite Mounted Robot Manipulators.- Optimal Control of the Industrial Robot Manutec r3.

This article addresses the optimal tracking control problem with prescribed performance for uncertain nonlinear systems subject to input constraint and unknown disturbances. First, a fixed‐time monotonic convergence function is introduced to restrain tracking error, and a nonlinear mapping technique is employed to transform the constrained error into an unconstrained variable, then the fixed‐time output tracking issue is boiled down to the boundedness problem of the transformed variable. With the aid of a nonquadratic cost function, the input constraint is encoded into the optimization problem. To solve the unknown disturbances, an auxiliary system and an auxiliary disturbance policy are constructed, and the optimal control problem is formulated as a two‐player zero‐sum game. Moreover, a Hamilton–Jacobi–Isaacs (HJI) equation associated with this nonquadratic zero‐sum game is established to give the optimal control and the worst‐case disturbance policy solution. Subsequently, to avoid using knowledge of the system dynamics, three neural network approximators, namely, actor, critic, and disturbance, which are tuned online and simultaneously for approximating the solution of HJI, are constructed based on the integral reinforcement learning algorithm. Theoretical analysis shows that the reconstructed error system states and the weight estimation errors are semi‐globally uniformly ultimately bounded. Finally, the simulation study further tests the availability of the proposed control strategy.

A robust algorithm for a class of optimal control problems subject to regional stability constraints and disturbances

In the optimal control theory, the Hamiltonian formalism is a famous one to find an optimal solution. However, when the performance index is complicated or for a degenerate case with a non-convexity of the Hamiltonian function with respect to the control force the Hamiltonian method does not work to find the solution. In this paper we will address this important issue via a quite different approach, which uses the optimal control problem of nonlinear Duffing oscillator as a demonstrative example. The optimally controlled vibration problem of nonlinear oscillator is recast into a nonlinear inverse problem by identifying the unknown heat source in a nonlinear parabolic partial differential equation (PDE). Then through a semi-discretization of the resultant PDE, the inverse problem is further reformulated to be a system of n-dimensional ODEs with n unknown pointwise sources, which allows a Lie-group adaptive method (LGAM) to recover the point-wise sources. The present method has three-fold advantages: it can easily minimize a complicated performance index to find an optimal control force of the nonlinear vibration system, it is effective for highly nonlinear optimal control problem, and it does not resort on the classical Hamiltonian formulation, which provides only a necessary condition, but not a sufficient condition, for the optimality of the control law. Numerical examples show that the LGAM may find a better performance than the classical one.

Beyond Lack of Compactness and Lack of Stability of a Coupled Parabolic-Hyperbolic Fluid-Structure System.- A Continuous Adjoint Approach to Shape Optimization for Navier Stokes Flow.- Recent Advances in the Analysis of State-constrained Elliptic Optimal Control Problems.- Fast and Strongly Localized Observation for a perturbed Plate Equation.- Representations, Composition, and Decomposition of C 1,1-hypersurfaces.- On Some Nonlinear Optimal Control Problems with Vector-valued Affine Control Constraints.- Weak Solutions to a Model for Crystal Growth from the Melt in Changing Magnetic Fields.- Lavrentiev Prox-regularization Methods for Optimal Control Problems with Pointwise State Constraints.- Nonlinear Feedback Solutions for a Class of Quantum Control Problems.- Optimal Feedback Synthesis for Bolza Control Problem Arising in Linearized Fluid Structure Interaction.- Single-step One-shot Aerodynamic Shape Optimization.- Shape Differentiability of Drag Functional for Compressible Navier-Stokes Equations.- Null-controllability for a Coupled Heat-Finite-dimensional Beam System.- Feedback Modal Control of Partial Differential Equations.- Optimization Problems for Thin Elastic Structures.- A New Non-linear Semidefinite Programming Algorithm with an Application to Multidisciplinary Free Material Optimization.- How to Check Numerically the Sufficient Optimality Conditions for Infinite-dimensional Optimization Problems.- Hidden Boundary Shape Derivative for the Solution to Maxwell Equations and Non Cylindrical Wave Equations.

This paper builds up two equivalence theorems for different kinds of optimal control problems of internally controlled Schrodinger equations. The first one is an equivalence of the minimal norm and the minimal time control problems where the target sets are the origin of the state space. (The minimal time control problems are also called the first type of optimal time control problems.) The second one is an equivalence of three optimal control problems: optimal target control problems, optimal norm control problems, and the second type of optimal time control problems.

Previous article Next article The Reduction of Certain Control Problems to an “Ordinary Differential” TypeJ. WargaJ. Wargahttps://doi.org/10.1137/1010036PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout Previous article Next article FiguresRelatedReferencesCited byDetails Stationarity conditions in optimal control problems class related to dynamical objects group control Cross Ref On the First- and Second-Order Conditions in Optimal Control Problems Related to Autonomous Objects Group lifecycle Cross Ref Optimal Control Problems with Time Delays: Constancy of the HamiltonianRichard B. Vinter25 July 2019 | SIAM Journal on Control and Optimization, Vol. 57, No. 4AbstractPDF (525 KB)On obtaining the stationarity conditions in an optimal control problem for a trajectory with a smooth contact with the phase boundary Cross Ref On one Method to Obtain Stationarity Conditions for an Optimal Control Problem Trajectory With a Smooth Boundary ContactIFAC-PapersOnLine, Vol. 51, No. 32 Cross Ref The Maximum Principle for Optimal Control Problems with Time DelaysA. Boccia and R. B. Vinter19 September 2017 | SIAM Journal on Control and Optimization, Vol. 55, No. 5AbstractPDF (407 KB)Problems with relaxed shifted controls and the pointwise maximum principleNonlinear Analysis: Theory, Methods & Applications, Vol. 75, No. 3 Cross Ref The Instability of Optimal Control Problems to Time DelayAlexey S. Matveev26 July 2006 | SIAM Journal on Control and Optimization, Vol. 43, No. 5AbstractPDF (340 KB)On the Characterization of Properly Relaxed Delayed ControlsJournal of Mathematical Analysis and Applications, Vol. 209, No. 1 Cross Ref Approximation of strongly relaxed minimizers with ordinary delayed controlsApplied Mathematics & Optimization, Vol. 32, No. 1 Cross Ref Jack Warga: In Appreciation14 July 2006 | SIAM Journal on Control and Optimization, Vol. 33, No. 2AbstractPDF (809 KB)Minimizing approximate original solutions for commensurate delayed controlsApplied Mathematics Letters, Vol. 7, No. 5 Cross Ref How to Properly Relax Delayed Controls Cross Ref A proper relaxation of shifted and delayed controlsJournal of Mathematical Analysis and Applications, Vol. 169, No. 2 Cross Ref Optimal controls and their discontinuities in quadratic problems of delay systemsIEEE Transactions on Automatic Control, Vol. 30, No. 7 Cross Ref Optimal Controls with PseudodelaysJ. Warga18 July 2006 | SIAM Journal on Control, Vol. 12, No. 2AbstractPDF (1227 KB)A conservative bound on the estimation error covariance matrix in the presence of correlated driving noise and correlated discrete measurement noiseJournal of Mathematical Analysis and Applications, Vol. 43, No. 2 Cross Ref References Cross Ref Volume 10, Issue 2| 1968SIAM Review History Submitted:18 October 1967Published online:18 July 2006 InformationCopyright © 1968 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1010036Article page range:pp. 219-222ISSN (print):0036-1445ISSN (online):1095-7200Publisher:Society for Industrial and Applied Mathematics

In this paper we investigate the use of optimal control techniques to improve Functional Electrical Stimulation (FES) for drop foot correction on hemiplegic patients. A model of the foot and the tibialis anterior muscle, the contraction of which is controlled by electrical stimulation has been established and is used in the optimal control problem. The novelty in this work is the use of the ankle accelerations and shank orientations (so-called external states) in the model, which have been measured on hemiplegic patients in a previous experiment using Inertial Measurement Units (IMUs). The optimal control problem minimizes the square of muscle excitations which serves the overall goal of reducing energy consumption in the muscle. In a first step, an offline optimal control problem is solved for test purposes and shows the efficiency of the FES optimal control for drop foot correction. In a second step, a Nonlinear Model Predictive Control (NMPC) problem - or online optimal control problem, is solved in a simulated environment. While the ulitmate goal is to use NMPC on the real system, i.e. directly on the patient, this test in simulation was meant to show the feasibility of NMPC for online drop foot correction. In the optimization problem, a set of fixed constraints of foot orientation was applied. Then, an original adaptive constraint taking into account the current ankle height, was introduced and tested. Comparisons between results under fixed and adaptive constraints highlight the advantage of the adaptive constraints in terms of energy consumption, where its quadratic sum of controls, obtained by NMPC, was three times lower than with the fixed constraint. This feasibility study was a first step in application of NMPC on real hemiplegic patients for online FES-based drop foot correction. The adaptive constraints method presents a new and efficient approach in terms of muscular energy consumption minimization.

Existing methods for nonlinear robust control often use scenario‐based approaches to formulate the control problem as large nonlinear optimization problems. The optimization problems are challenging to solve due to their size, especially if the control problems include time‐varying uncertainty. This paper draws from local reduction methods used in semi‐infinite optimization to solve robust optimal control problems with parametric and time‐varying uncertainty. By iteratively adding interim worst‐case scenarios to the problem, methods based on local reduction provide a way to manage the total number of scenarios. We show that the local reduction method for optimal control problems consists of solving a series of simplified optimal control problems to find worst‐case constraint violations. In particular, we present examples where local reduction methods find worst‐case scenarios that are not on the boundary of the uncertainty set. We also provide bounds on the error if local solvers are used. The proposed approach is illustrated with two case studies with parametric and additive time‐varying uncertainty. In the first case study, the number of scenarios obtained from local reduction is 101, smaller than in the case when all extreme scenarios are considered. In the second case study, the number of scenarios obtained from the local reduction is two compared to 512 extreme scenarios. Our approach was able to satisfy the constraints both for parametric uncertainty and time‐varying disturbances, whereas approaches from literature either violated the constraints or became computationally expensive.

A stochastic minimax optimal time-delay state feedback control strategy for uncertain quasi-integrable Hamiltonian systems is proposed. First, a stochastic optimal state feedback control problem of uncertain quasi-integrable Hamiltonian system with time delay in feedback control subjected to Gaussian white noise is formulated. Then, the time-delayed state feedback control forces are approximated by the control forces without time delay and the original problem is converted into a stochastic optimal state feedback control problem of an uncertain system without time delay. After that, by following a procedure based on the stochastic averaging method and stochastic differential game, the worst-case disturbances and the optimal controls are obtained from solving a Hamiltonian–Jacobi–Isaacs (HJI) equation. As an example, the stochastic minimax optimal state feedback control of a Duffing oscillator with parametric disturbances and time-delayed feedback control is worked out in detail to illustrate the procedure and effectiveness of the proposed control strategy.

In the present survey paper, we discuss applications of differential and operator inclusions to some optimization and optimal control problems. The Filippov implicit function lemma is considered and its application to the optimization of a feedback control system governed by a semilinear differential equation in a Banach space is presented. We describe the construction of the oriented coincidence degree for a compact multivalued perturbation of a nonlinear Fredholm operator and apply it to an optimal control problem induced by an ordinary differential equation with the Hopf boundary condition. We study also an optimal feedback control problem for a mathematical model of the motion of weakly concentrated water polymer solutions.

Optimal semistable control for continuous-time linear systems

In this paper we define ``embedded optimal control problems" which prescribe parametrized families of well defined associated optimal control problems. We show that the extremal generating Hamiltonian equations for an embedded optimal control problem and any associated optimal control problem are simply related by a projection. Furthermore normal extremals project to normal extremals and similarly for abnormal extremals. An interesting class of embedded optimal control problems consists of Clebsch optimal control problems. We provide necessary conditions for a Clebsch optimal control problem to describe a variational problem and thereby a mechanical system. There may be many advantages to analyzing an embedded optimal control problem instead of a particular associated optimal control problem, for example the former being defined on a linear space and the latter on a nonlinear space. The continuous analysis is paralleled by a similar discrete analysis. We define a discrete embedded/Clebsch optimal control problem along with associated discrete optimal control problems and we show results that are analogous to the continuous results. We apply the theory, both in the continuous and the discrete setting, to two example systems: mechanical systems on matrix Lie groups and mechanical systems on $n$-spheres.