Nonstandard Linear Quadratic Optimal Control

Huanshui Zhang is a Professor at Shandong University of Science and Technology, China. His research spans optimal LQ control, decentralized control, Stackelberg game control, time-delay systems, and stochastic systems. Professor Zhang is an IEEE Fellow. Juanjuan Xu is a Professor with the School of Control Science and Engineering at Shandong University, China. Her research covers stochastic systems, optimal control, cooperative multi-agent systems, game theory, and time-delay systems. She received the CAA Excellent Doctoral Dissertation Award in 2015. Hongxia Wang is a Professor with the School of Electrical and Automation Engineering at Shandong University of Science and Technology, Qingdao, China. Her research focuses on optimal control, H-infinity control, stochastic systems, and time-delay systems.

• List of Figures xviiList of Tables xixForeword xxiPreface xxviiAcknowledgments xxixAcronyms xxxiList of Symbols xxxiiiIntroduction xxxvPART I IRREGULAR LQ CONTROL1 Irregular LQ Optimal Control 31.1 Problem Formulation 31.2 Finite Horizon Case 41.2.1 Solution via the Maximum Principle Method 41.2.2 Solution via the Complete Square Method 81.3 Infinite Horizon Case 121.3.1 Regular Case 131.3.2 Irregular Case 141.4 Notes and Comments 172 Stochastic Irregular LQ Optimal Control with Additive Noise 192.1 Problem Formulation 192.2 Preliminaries 212.2.1 Maximum Principle 212.2.2 Regular Case 222.3 Solution to Problem 242.4 Notes and Comments 273 Stochastic Irregular LQ Control with Multiplicative Noise 293.1 Problem Formulation 293.1.1 Preliminaries 303.1.2 Solution to Problem 353.2 Numerical Examples 383.3 Notes and Comments 38PART II LQ CONTROL OF INFINITEDIMENSION SYSTEMS: TIMEDELAY SYSTEMS4 LQ Control for Stochastic Systems with a Single Input Delay 434.1 DiscreteTime Case 434.1.1 Problem Formulation 434.1.2 Finite Horizon Stochastic LQ control 454.1.3 Infinite Horizon Stochastic LQ control 504.1.4 Numerical Examples 634.2 ContinuousTime Case 644.2.1 Problem Formulation 644.2.2 Finite Horizon Stochastic LQ Control 654.2.3 Infinite Horizon Stochastic LQ Control 744.2.4 Numerical Examples 864.3 Notes and Comments 875 LQ Control for Stochastic Systems with Multiple Input Delays 915.1 Discretetime Case 915.1.1 Problem Formulation 915.1.2 Transform Delayed Systems into DelayFree Ones 935.1.3 Stabilization of the Stochastic System with Multiple Input Delays 1015.2 Continuoustime Case 1115.2.1 Finite Horizon Stochastic LQ Control 1115.2.2 Infinite Horizon Stochastic LQ Control 1175.3 Notes and Comments 1206 LQ Control Problem for It ˆo Systems with State Delays 1216.1 Problem Formulation 1216.2 Solution to Problem 1226.3 Notes and Comments 1247 Other DelayRelated Stochastic LQ Control 1257.1 Stochastic LQ Control for Delayed MeanField Systems: Discrete Time 1257.1.1 Finite Horizon Optimal Control 1267.1.2 Infinite Horizon Optimal Control 1287.2 Stochastic LQ Control for Delayed Meanfield Systems: Continuous Time 1297.2.1 Finite Horizon Optimal Control 1297.2.2 Infinite Horizon Optimal Control 1317.3 Stochastic LQ Control for Delayed Markov Systems 1337.3.1 Finite Horizon Optimal Control 1347.3.2 Infinite Horizon Optimal Control 1367.4 Stochastic LQ Control for Delayed Markov Systems with Multiplicative Noises 1387.4.1 Finite Horizon Optimal Control 1397.4.2 Infinite Horizon Optimal Control 1457.5 Notes and Comments 148PART III LQ CONTROL FOR INFINITEDIMENSION SYSTEMS: PARTIAL DIFFERENTIAL SYSTEM8 LQ Control for FirstOrder Hyperbolic Systems 1518.1 Problem Formulation 1518.2 Main Results 1528.3 A Continuous Method to Verify the Main Results 1558.4 Numerical Examples 1578.5 Notes and Comments 1589 LQ Control for SecondOrder Parabolic Systems 1599.1 Problem Formulation 1599.2 Main Results 1609.2.1 Discretization Problem 1609.2.2 Optimal Controller Design for Discretization Problem 1619.2.3 Optimal Controller Design for the Original Problem 1639.3 Numerical Examples 1649.4 Notes and Comments 166PART IV LQ CONTROL WITH ASYMMETRIC INFORMATION10 Decentralized LQ Control with Information Inclusion Structure 16910.1 Problem Formulation 16910.2 Optimal Control Design 17110.3 Solving the Coupled Forward and Backward Riccati Equations 17610.4 Numerical Examples 17810.5 Notes and Comments 18211 Decentralized LQG Control with dStep Delayed Information Sharing Pattern 18311.1 Problem Formulation 18311.2 Optimal Estimation 18611.3 Optimal Control Design 18911.4 Asymptotically Optimal Solution by Decoupling the Control Gain and Estimation Gain 19311.5 Numerical Examples 19411.6 Notes and Comments 19612 Decentralized LQ Control with Private Input and Measurement Information 19712.1 Problem Formulation 19712.2 Preliminaries 19812.3 Decentralized Control with Input Sharing Pattern 19912.4 Decentralized Control with Private Input Information 20012.5 Extension to the Case with Multiple Inputs and Application to MultiAgent Systems 20312.6 Numerical Examples 20612.7 Notes and Comments 207PART V LQ CONTROL WITH ASYMMETRIC POSITION13 OpenLoop Stackelberg Strategy 21113.1 Problem Formulation 21113.2 Main Results 21213.2.1 Optimization for the Follower 21213.2.2 Optimization for the Leader 21413.2.3 Solution to Problem 21913.2.4 Numerical Examples 22213.3 Notes and Comments 22214 OpenLoop Stackelberg Strategy with Time Delay 22314.1 Difference Game 22314.1.1 Problem Formulation 22314.1.2 Optimization for the Follower 22414.1.3 Optimization for the Leader 22614.1.4 Solution to Problem 22714.2 Differential Game 23414.2.1 Problem Formulation 23414.2.2 Optimization for the Follower 23514.2.3 Optimization for the Leader 23614.2.4 Solution to Problem 23714.3 Notes and Comments 24315 Linear ClosedLoop Stackelberg Strategy 24515.1 Problem Formulation 24515.2 Main Result 24615.2.1 Optimization for the Follower 24815.2.2 Optimization for the Leader 25215.2.3 Solution to Problem 25615.3 Numerical Examples 25815.4 Notes and Comments 259PART VI APPLICATIONS IN NETWORKED CONTROL SYSTEMS16 Stabilization Control for NCSs 26316.1 Problem Formulation 26316.2 Main Results 26516.2.1 Optimal LQ Control and Stabilization of Networked Control System 26516.2.2 Existence Theorem of the Maximum Packet Dropout Rate 26716.2.3 Maximum Packet Dropout Rate and Maximum Allowable Delay Bound 26916.3 Notes and Comments 27217 LQ Control with Remote and Local Controllers 27317.1 Finite Horizon Optimal Control 27317.1.1 Problem Formulation 27317.1.2 Solution to Problem 27417.2 Infinite Horizon Optimal Control 27917.2.1 Problem Formulation 27917.2.2 Solution to Problem 28017.3 Numerical Examples 28517.3.1 Finite Horizon Case 28617.3.2 Infinite Horizon Case 28717.4 Notes and Comments 28818 Optimal Consensus of MultiAgent Systems 29118.1 Problem Formulation and Preliminaries 29118.1.1 Problem Formulation 29218.1.2 Preliminaries 29218.2 Distributed Optimal Controller Design 29418.2.1 Consensus of MultiAgent System Based on Relative Error Feedback 29418.2.2 Comparison with Traditional Consensus Algorithms 29918.2.3 Special Case: Consensus of MultiAgent Systems via State Feedback Controller 29918.3 Numerical Simulations 30218.4 Notes and Comments 30519 Asymptotically Optimal Distributed Consensus of Heterogeneous MultiAgent Systems 30719.1 Problem Formulation 30719.2 Main Results 30819.2.1 State Consensus of Heterogeneous MultiAgent Systems 30819.2.2 Comparison with Traditional Consensus Algorithms 31419.2.3 Output Consensus of Heterogeneous MultiAgent Systems 31519.3 Numerical Simulations 31819.4 Notes and Comments 319A Classical LQR for Deterministic Systems 323B Classical LQR for Stochastic Systems 325C Solution to Linear DiscreteTime FBSDEs 327C.1 Problem Formulation 327C.2 The Explicit Solution to FBSDEs 329C.2.1 Solvability of FBSDEs 331C.2.2 Unique Solvability of FBSDEs 332D Solution to Linear DiscreteTime FBSDEs with Time Delay 335D.1 Problem Formulation 335D.2 The Explicit Solution to FBSDEs 336E Solution to Linear ContinuousTime FBSDEs with Time Delay 343E.1 Problem Formulation 343E.2 The Method of Discretization 344E.3 The Explicit Solution to FBSDEs 348F Solution to Linear ContinuousTime FBSDEs with State Delay 355F.1 Problem Formulation 356F.2 The Explicit Solution to FBSDEs 356F.2.1 Corresponding Discretetime FBSDEs 357F.2.2 Limitation of the Discretetime FBSDEs 361F.2.3 Proof of Theorem F.1 365Index