上海交通大学：《现代控制理论》课程教学资源（讲稿）Chapter 8 Fundamentals of Optimal Control

Chapter 8 2 Chapter 8 Fundamentals of Optimal Control Objectives: Optimal control from Hamilton-Pontryagin Equation Optimal Control Law for Linear System with Quadratic Performance Index Design procedure and examples

Chapter 8 Fundamentals of Optimal Control Objectives: • Optimal control from Hamilton-Pontryagin Equation • Optimal Control Law for Linear System with Quadratic Performance Index • Design procedure and examples Chapter 8 2

Chapter 8 8.1 Optimal Control based on Hamilton-Pontryagin Equation 1.The optimal control problem for feedback control systems. To minimize J=L(X.U)d subject to X(t)=f(X,U) TU X)e∈R" state vector U(t)∈R' the control vector f(X,U) the function vector(Rn+r-R) L(X,U) Rnr→Rn,J is a function 2.Target: Euler-Lagrange Equation>Hamilton-Pontryagin Equation

8.1 Optimal Control based on Hamilton-Pontryagin Equation 1.The optimal control problem for feedback control systems. To minimize   T J L X U dt 0 ( , ) subject to X(t)  f(X,U)  state vector n X(t)R r U(t)R f(X, U) L(X,U) the control vector the function vector(Rn+r→Rn ) Rn+r→Rn , J is a function 2.Target: Euler-Lagrange Equation Hamilton  -Pontryagin Equation Chapter 8

Chapter 8 3.The design approach Step 1.Rewrite the state equation -X(t)+f(X(t),U(t)=0 Step 2.Set up the augment functional J[L(X,U)+A"(f(X,U)-x)]dt =L(X,U)+A"f-A"x)]dt Define: F=L(X,U)+Af(X,U)-A'X Step 3.Define a scalar function-Hamiltonian H(X,U,A)=L(X,U)+A'f(X,U) then F=H(X,U,A)-A"X

3.The design approach Step 1. Rewrite the state equation  X(t) f(X(t),U(t))  0  Step 2. Set up the augment functional         T T T T T L dt J L dt 0 0 [ ( , ) )] [ ( , ) ( ( , ) )] X U Λ f Λ X X U Λ f X U X   Define： X U Λ T f X U Λ T X  F  L( , )  ( , )  Step 3. Define a scalar function —Hamiltonian (X,U, Λ) (X,U) Λ f(X,U) T H  L  then F H X U Λ Λ T X   ( , , )  Chapter 8

Chapter 8 Step 4.Applying Euler-Lagrange Equation to this problem OF d oF 0 dt ox OF d =00 aU d而 OF d oF =0 an dt a Step 5.By substituting into the equation above aH(X,A,U)d=0 Furthermore OH 1(0= OX dt ox aH(X,A,U) 0 OH =0 aU aU F(&,A,U,X-0 X=fX,U) aA Hamilton-Pontryagin equation------The necessary condition for optimizing control systems

Step 4. Applying Euler-Lagrange Equation to this problem                            0 0 0 Λ Λ U U X X    F dt F d F dt F d F dt F d Step 5. By substituting into the equation above                    0 ( , , , ) 0 ( , , ) 0 ( , , ) Λ X Λ U X U X Λ U Λ X X Λ U F  H dt H d Furthermore                  X f(X,U) U H X H Λ(t)   0 Hamilton-Pontryagin equation------The necessary condition for optimizing control systems. Chapter 8

Chapter 8 Example Consider the optimal control problem Min J+(dr s.t (t)=-x(t)+u(t) boundary condition x(0)=1,x(T)=0 Our task:To find the optimal control law u*(t). Step 1.Rewrite the equation as -x(t)+（-x(t)+u(t)=0 Step 2.Set up the augment functional J-r++ex*- eory Step 3.Write Hamiltonian 2+2-x+ H=x 2 2

Example Consider the optimal control problem Min    T J x t u t dt 0 2 2 ( ( ) ( )) 2 1 s.t x (t)  x(t)  u(t) boundary condition : x(0)=1, x(T)=0 Our task: To find the optimal control law u*(t). Step 1. Rewrite the equation as x (t)  x(t)  u(t)  0 Step 2. Set up the augment functional        T J x u x u x dt 0 2 2 ( )] 2 1 2 1 [   Step 3. Write Hamiltonian H  x  u  x  u 2 2 2 1 2 1 Chapter 8

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

Chapter 8 2 Design approach------Riccati Differential Equation Construct an augment functional JXX+URU(AX+BU()-d The Hamiltonian HA-OURU+U According to Hamilton-Pontryagin Equation,get the optimizing condition 入(0= HX,A(X,=-OX(-AA(t) Ox HX,A,心=RU0+B'(O0=0→U@=-RB(0M( au(t) Since x(t)=A(t)X(t)+B(t)U(t) so X(t)=A(t)X(t)-B(t)R-B'(t)A(t)

2 Design approach------Riccati Differential Equation Construct an augment functional J [(X QX U RU) Λ (t)(A(t)X(t) B(t)U(t)) Λ (t)X(t)]dt T T T T         0 2 1 The Hamiltonian 1 1 2 2 T T T T H(X,Λ,U) X QX U RU     Λ AX Λ BU According to Hamilton-Pontryagin Equation, get the optimizing condition Λ( ) H(X,Λ(X, QX(t) A (t)Λ(t) X t T         ( ) ( ) ( ) 0 ( ) ( )       t t t t H T RU B Λ U X, ,U X(t)  A(t)X(t) B(t)U(t) Since  so ( ) ( ) ( ) ( ) ( ) ( ) 1 t t t t t t T X A X B R B Λ     U(t) R B (t)Λ(t) 1 T   Chapter 8

Chapter 8 Suppose A(t)=P(t)X(t) P(t)Eh(n,n) From H-P equation,we have P(t)X(t)+P(t)X(t)=-QX(t)-A"(t)P(t)X(t) U(t)=-R-B'(t)P(t)X(t) X(t)=A(t)X(t)-B(t)R-B"(t)P(t)X(t) Through substituting {P()+P(t)[A(t)-B()R-B'()P(OX)=(-Q-AT(IP(t)X() The Riccati Differential Equation -P(t)=P(t)A(t)+A"(t)P(t)-P(t)B(t)RB'(t)P(t)+

Suppose Λ(t)  P(t)X(t) P(t)(n, n) From H-P equation, we have (t) (t) (t) (t) (t) (t) (t) (t) T P  X P X   QX  A P X ( ) ( ) ( ) ( ) 1 t t t t T U R B P X    (t) (t) (t) (t) (t) (t) (t) T X A X B R B P X 1    Through substituting  (t) (t)[ (t) (t) (t) (t) (t) ( (t) (t)) (t) T T P P A B R B P X Q A P X 1        P t P t A t A t P t P t B t R B t P t Q T T       ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )  1 The Riccati Differential Equation Chapter 8