1 Modern Control Theory Optimal Control ( An undergraduate optional course ) Hui PENG Department of Automation School of Information Science Stochastic optimal control; Adaptive optimal control; Optimal control of large-scale systems; Suboptimal control; Optimal control sensitivity; Multi-goal optimal control; Differential games; 1.1 Overview 9 Applied fields of optimal control Control engineering; Space technology; System engineering; Economic management; Financial engineering; 1.1 Overview 10 Some basic courses for studying optimal control Advanced math Linear algebra Automatic control theory (classical) Linear control system (state-space method) 1.1 Overview References (books) Optimal Control, F. L. Lewis and V. L. Syrmos, John Wiley 3) (Quasi-) Newton method; 4) Gauss-Newton method; 5) Levenberg-Marquardt method; 6) Sequential quadratic programming; 7) Goal attainment method (for multiobjective optimization); 8) Genetic or evolutionary programming algorithm (Shi et al. 1999 ); 9) Hybrid algorithm (McLoone et al. 1998 ); 10) Variable separation method (Golub 11) Structured nonlinear parameter optimization method (Peng et al. 2002, 2003). 21 Contents Chapter 1 Introduction Chapter 2 Static Optimization Chapter 3 Variational Methods Chapter 4 The Pontryagin Minimum Principle Chapter 5 Discrete-Time Optimal Control Chapter 6 Dynamic Programming Chapter 7 Minimum Time Optimal Control Chapter 8 Linear-Quadratic Optimal Control 22 Chapter 2 Static Optimization (Function Extremum) 2.1 Problems without constraints 2.2 Problems with equality constraints 2.3 Problems with inequality constraints 2.4 Convex sets and functions 23 Problems without constraints 2.1 Problems without constraints Performance index: 12( ), parameter vectornnJF xRx xxRx Assuming that F(x) has first and second partial derivatives everywhere, then F(x) can be approximated in the neighborhood of by the first three terms of the Taylor series: The problem is to find x that minimizes J. 0023 0000021( )()2TTxxFFF xF xxxxxxxxxxx0x24 The gradient of F(x): 2.1 Problems without constraints 121grad ( )( )xnnF xFFxF xF xFxF x The Hessian of F(x): 22211121222 221222222212nxxnnnnnn nFFF x xx xx xFFFFFxxxxxxxFFF xxxxxx 25 Necessary condition for an extremum: 2.1 Problems without constraints 002000021( )()2TTxxFFF xF xxxxxxxxxgrad ( )( )0xFF xF xFx Sufficient condition for a local minimum: 220, is positive definite (all eigenvalues positive)xxxxFFFxSufficient condition for a local maximum: 220, is negtive definite (all eigenvalues negtive)xxxxFFFxTaylor expansion: 26 Example: unconstraint extremum in one variable 2.1 Problems without constraints 27 Chapter 2 Static Optimization (Function Extremum) 2.1 Problems without constraints 2.2 Problems with equality constraints 2.3 Problems with inequality constraints 2.4 Convex sets and functions 28 Problems with equality constraints 2.2 Problems with equality constraints Performance index: ( , ) . . ( , )0, , , mnnJF x u s tg x uFRxRuRgR The method of Lagrange multipliers: The problem is to find x and u that minimizes J subject to g(x,u)=0. Necessary condition for an extremum: 00TTFFdFdxduxu ggdgdxduxu 29 The Jacobian assumption: 2.2 Problems with equality constraints From necessary condition for an extremum: 1111222212120nnnnnnn nggg xxxggggxxxxggg xxx 11100 0TTT TTTTTTFFdFdxduxuFggFFFggduuxxuuxuxggdxduxuFFg xx 1110 00, 0TTTTTTT TTT T TgFFggdxxxxxxxFg FFFFFgxx Fgxuxxux uu Assuming that: 30 The method of Lagrange multipliers 2.2 Problems with equality constraints Define the Lagrange function: 00( , )0T TTT TTLFg xxx LFg uuu Lg x u Suppose performance index is ( , )s.t. ( , )=0, , , nnmJF x u g x uFRxRuRgR( , )( , ), Lagrange multipliersnTLF x ug x uR, ,min( , ) then min( , )( , ) s.t. ( , )=0Tx ux uJF x u LF x ug x u g x uNecessary condition for a extremum: 31 The method of Lagrange multipliers in general case 2.2 Problems with equality constraints Define the Lagrange function: 0( )0T TTLFg xxx Lg xSuppose performance index is ( ) s.t. ( )=0, , , nmJF x g xFRxRgRnm( )( ), Lagrange multipliersTmLF xg xR,min( )then min( )( ) s.t. ( )=0TxxJF xLF xg x g xNecessary condition for a extremum: 32 Example (1): constrained maximization for two variables and one constrain