Chapter 8 Step 4.From H-P equation,we have (t)=-x(t)+(t) u(t)+2()=0 heory x(t)=-x(t)+u(t) Above equation set can be rewritten as Step 5.Solve the equation --1=0 A-n=-11- The characteristic is .2=2
Step 4. From H-P equation, we have ( ) ( ) ( ) ( ) ( ) 0 ( ) ( ) ( ) x t x t u t u t t t x t t Above equation set can be rewritten as ( ) ( ) 1 1 1 1 ( ) ( ) t x t t x t Step 5. Solve the equation 0 1 1 1 1 A I The characteristic is 1,2 2 Chapter 8
Chapter 8 Then the solution is w=222++(-火]20e-e} 2025-e+05-+5+】 The optimal control u*(t)is 0=-2m7对卡-g+40k55 where 20-2++柜-业 -e-27
Then the solution is x( t ) e e ( )( e e ) 2t 2t 2t 2t 2 1 2 1 0 2 2 1 t t t t t e e e e 2 2 2 2 (0) 2 1 2 1 2 2 1 ( ) The optimal control u*(t) is t t t t u t t e e e e 2 2 2 2 (0) 2 1 2 1 2 2 1 *( ) *( ) where T T T T e e e e 2 2 2 2 2 1 2 1 (0) Chapter 8
Chapter 8 8.2 Optimal Control Law for Linear System with Quadratic Performance Index 1.Description of the problem min JQX+RU s.t X(t)=A(t)X(t)+B(t)U(t) where J is The cost functional(or say performance functional) Q is a symmetric constant positive semi-definite weighting matrix R is a symmetric constant positive definite weighting matrix. Find optimal control law to minimize the cost functional J
8.2 Optimal Control Law for Linear System with Quadratic Performance Index 1.Description of the problem J (X (t)QX(t) U (t)RU(t))dt T T 2 0 1 s.t X(t) A(t)X(t) B(t)U(t) where J is The cost functional (or say performance functional). Q is a symmetric constant positive semi-definite weighting matrix R is a symmetric constant positive definite weighting matrix. Find optimal control law to minimize the cost functional J. min U Chapter 8