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Table of Contents
Intro
Preface
Contents
Symbols and Abbreviations
Part I Stochastic Optimization Methods
1 Optimal Control Under Stochastic Uncertainty
1.1 Stochastic Control Systems
1.1.1 Differential and Integral Equations Under Stochastic Uncertainty
1.1.1.1 Parametric Representation of the Differential/Integral Equation Under Stochastic Uncertainty
1.1.2 Objective Function
1.1.2.1 Optimal Control Under Stochastic Uncertainty
1.2 Control Laws
1.3 Computation of Expectations by Means of Taylor Expansions
1.3.1 Complete Taylor Expansion
1.3.2 Inner or Partial Taylor Expansion
1.4 Taylor Approximation of Control Problems Under Stochastic Uncertainty: General Procedure
1.5 Control Problems with Linear and Sublinear Cost Functions
1.6 Stochastic Optimal Open-Loop Feedback Control of Tracking Systems
1.6.1 Approximation of the Expected Costs: Expansions of 1st Order
1.6.2 Approximate Computation of the Fundamental Matrix
References
2 Stochastic Optimization of Regulators
2.1 Introduction
2.2 Regulator Design Under Stochastic Uncertainty
2.3 Optimal Feedback Functions Under Stochastic Uncertainty
2.3.1 Quadratic Cost Functions
2.3.1.1 Computation of the Expectation by Taylor Expansion
2.3.1.2 Approximation of the Expectation of the Total Cost Function
2.4 Calculation of the Tracking Error Rates (Sensitivities)
2.4.1 Partial Derivative with Respect to pD
2.4.2 Partial Derivative with Respect to q0
2.4.3 Partial Derivative with Respect to 0
2.4.4 Partial Derivative with Respect to e0
2.4.4.1 Partial Derivative with Respect to eq
2.5 The Approximate Regulator Optimization Problem
2.6 Active Structural Control Under Stochastic Uncertainty
2.6.1 Example
5.3.1 Characterization of Safe States
5.4 Computation of the State Function for Concrete Cases
5.4.1 Mechanical Structures Under Stochastic Uncertainty
5.4.1.1 Trusses
5.4.1.2 Elastic-Plastic Mechanical Structures
5.4.2 Linear-Quadratic Problems with Scalar Response Function
5.4.3 Approximation of the General Operating Condition
5.4.4 Two-Sided Constraints for the Response Functions
5.5 Systems/Structures with Parameter-Dependent States
5.5.1 Dynamic Control Systems
5.5.2 Variational Problems
5.5.3 Example to Systems with Control and Variational Problems
Preface
Contents
Symbols and Abbreviations
Part I Stochastic Optimization Methods
1 Optimal Control Under Stochastic Uncertainty
1.1 Stochastic Control Systems
1.1.1 Differential and Integral Equations Under Stochastic Uncertainty
1.1.1.1 Parametric Representation of the Differential/Integral Equation Under Stochastic Uncertainty
1.1.2 Objective Function
1.1.2.1 Optimal Control Under Stochastic Uncertainty
1.2 Control Laws
1.3 Computation of Expectations by Means of Taylor Expansions
1.3.1 Complete Taylor Expansion
1.3.2 Inner or Partial Taylor Expansion
1.4 Taylor Approximation of Control Problems Under Stochastic Uncertainty: General Procedure
1.5 Control Problems with Linear and Sublinear Cost Functions
1.6 Stochastic Optimal Open-Loop Feedback Control of Tracking Systems
1.6.1 Approximation of the Expected Costs: Expansions of 1st Order
1.6.2 Approximate Computation of the Fundamental Matrix
References
2 Stochastic Optimization of Regulators
2.1 Introduction
2.2 Regulator Design Under Stochastic Uncertainty
2.3 Optimal Feedback Functions Under Stochastic Uncertainty
2.3.1 Quadratic Cost Functions
2.3.1.1 Computation of the Expectation by Taylor Expansion
2.3.1.2 Approximation of the Expectation of the Total Cost Function
2.4 Calculation of the Tracking Error Rates (Sensitivities)
2.4.1 Partial Derivative with Respect to pD
2.4.2 Partial Derivative with Respect to q0
2.4.3 Partial Derivative with Respect to 0
2.4.4 Partial Derivative with Respect to e0
2.4.4.1 Partial Derivative with Respect to eq
2.5 The Approximate Regulator Optimization Problem
2.6 Active Structural Control Under Stochastic Uncertainty
2.6.1 Example
5.3.1 Characterization of Safe States
5.4 Computation of the State Function for Concrete Cases
5.4.1 Mechanical Structures Under Stochastic Uncertainty
5.4.1.1 Trusses
5.4.1.2 Elastic-Plastic Mechanical Structures
5.4.2 Linear-Quadratic Problems with Scalar Response Function
5.4.3 Approximation of the General Operating Condition
5.4.4 Two-Sided Constraints for the Response Functions
5.5 Systems/Structures with Parameter-Dependent States
5.5.1 Dynamic Control Systems
5.5.2 Variational Problems
5.5.3 Example to Systems with Control and Variational Problems