Linear Quadratic Regulator (LQR) II

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1 Optimal Control, Guidance and Estimation Lecture 11 Linear Quadratic Regulator (LQR) II Pro. Radhakant Padhi Dept. o Aerospace Engineering Indian Institute o Science - Bangalore Outline Summary o LQR design Stability o closed-loop system in LQR Optimum value o the cost unction Extension o LQR design For cross-product term in cost unction Rate o state minimization Rate o control minimization LQR design with prescribed degree o stability LQR or command tracking LQR or inhomogeneous systems OPIMAL CONROL, GUIDANCE AND ESIMAION 2

2 Summary o LQR Design Pro. Radhakant Padhi Dept. o Aerospace Engineering Indian Institute o Science - Bangalore LQR Design: Problem Statement Perormance Index (to minimize): Path Constraint: t 1 1 J = ( S ) + 2 Q + U RU dt t 2 ϕ( ) L(, U ) ɺ = A + BU Boundary Conditions: t = : Speciied ( ) : Fixed, t : Free OPIMAL CONROL, GUIDANCE AND ESIMAION 4

3 LQR Design: Necessary Conditions o Optimality 1 erminal penalty: ϕ ( ) = ( S ) 2 1 Hamiltonian: H = Q + U RU + A + BU 2 State Equation: ɺ = A + BU Costate Equation: Optimal Control Eq.: Boundary Condition: λ ( H / ) ( Q A λ ) ɺ λ = = + ( / ) = = H U U R B λ ( ϕ / ) λ = = S OPIMAL CONROL, GUIDANCE AND ESIMAION 5 LQR Design: Derivation o Riccati Equation Guess λ ( t) = P ( t) ( t) ɺ λ = P ɺ + Pɺ = P ɺ + P A + BU = P ɺ + P( A BR B λ ) = P ɺ + P( A BR B P ) ( Q + A P ) = ( Pɺ + PA PBR B P) 1 ( Pɺ + PA + A P PBR B P + Q) = OPIMAL CONROL, GUIDANCE AND ESIMAION 6

4 LQR Design: Derivation o Riccati Equation Riccati equation ɺ = P PA A P PBR B P Q Boundary condition ( ) = ( is ree) P t S ( ) P t = S OPIMAL CONROL, GUIDANCE AND ESIMAION 7 LQR Design: Ininite ime Regulator Problem heorem (By Kalman) As t, or constant Q and R matrices, Pɺ t Algebraic Riccati Equation (ARE) PA A P PBR B P Q Final Control Solution: + + = U = R B P = K OPIMAL CONROL, GUIDANCE AND ESIMAION 8

5 Stability o closed-loop loop system in LQR Pro. Radhakant Padhi Dept. o Aerospace Engineering Indian Institute o Science - Bangalore LQR Design: Stability o Closed Loop System Closed loop system Lyapunov unction Vɺ = ɺ P + Pɺ ɺ = A + BU = A BK V = P = A BK P + P A BK = + A BR B P P P A BR B P = PA A P PBR B P Q Q PBR B P Q PBR B P = OPIMAL CONROL, GUIDANCE AND ESIMAION 1

6 LQR Design: Stability o Closed Loop System For R >, R >. Also P > -1 So PBR B P > Also Q. -1 ( PBR B P Q) Hence, + > V ɺ < Hence, the closed loop system is always asymptotically stable! OPIMAL CONROL, GUIDANCE AND ESIMAION 11 LQR Design: Minimum value o cost unction 1 J = ( Q + U RU ) dt 1 = Q + ( R B P ) R ( R B P ) dt 1 = ( Q + PBR B P) dt 1 1 = ( Vɺ ) dt [ V ] P t = = = P P P 2 = 2 OPIMAL CONROL, GUIDANCE AND ESIMAION 12

7 Extensions o LQR Design Pro. Radhakant Padhi Dept. o Aerospace Engineering Indian Institute o Science - Bangalore LQR Extensions: 1. Cross Product erm in P.I. J = ( Q 2 WU U RU ) dt + + Let us consider the expression: ( ) + ( + ) ( + ) Q WR W U R W R U R W = Q + U RU + U W + WU = Q + 2 WU + U RU OPIMAL CONROL, GUIDANCE AND ESIMAION 14

8 LQR Extensions: 1. Cross Product erm in P.I. J = ( Q WR W ) ( U R W ) R( U R W ) dt Q1 U1 = ( Q1 + U1 RU1) dt t ɺ = A + BU Control Solution = A + B U R W U = K ( 1 ) ( A BR W ) BU1 = + = A + BU U = U1 R W = K + R W OPIMAL CONROL, GUIDANCE AND ESIMAION 15 LQR Extensions: 2. Weightage on Rate o State J = ( Q + U RU + ɺ Sɺ ) dt = Q U RU ( A BU ) S ( A BU ) dt Q + U RU + A S A + A SBU = dt + U B S A + U B SBU Q1 R1 W = ( Q A SA) U ( R B SB) U 2 ( A SB) U dt = ( Q1 + U RU ) WU dt Leads to a cross product case OPIMAL CONROL, GUIDANCE AND ESIMAION 16

9 LQR Extensions: 3. Weightage on Rate o Control ( ɺ ɺ ) 1 J = Q + U RU + U RU ˆ dt Let =, V = Uɺ U 1 Q J V RV ˆ = dt + R ( ) 1 J = Qˆ + V RV ˆ dt OPIMAL CONROL, GUIDANCE AND ESIMAION 17 LQR Extensions: 3. Weightage on Rate o Control ɺ = A + BU, = Uɺ = V A B ɺ = V Aˆ BV ˆ + = + I Note : Aˆ { A B} Bˆ (1) he dimension o the problem has increased rom n to ( n + m) (2) I, is controllable, it can be shown that the new system is also controllable. OPIMAL CONROL, GUIDANCE AND ESIMAION 18

10 LQR Extensions: 3. Weightage on Rate o Control Solution : ˆ ˆ V = Uɺ = R B Pˆ where Pˆ is the solution o ˆ A P PA PBR B P Qˆ ˆ ˆ ˆ ˆ ˆ ˆ + ˆ + = Hence ˆ ˆ ˆ P 1 11 P 12 1 [ ] ˆ ˆ ˆ Uɺ = R I = R P12 P 22 ˆ P ˆ U 12 P 22 = Rˆ Pˆ Rˆ Pˆ U OPIMAL CONROL, GUIDANCE AND ESIMAION 19 LQR Extensions: 3. Weightage on Rate o Control However, Uɺ = Rˆ Pˆ Rˆ Pˆ U is a dynamic equation in U and hence is not easy or implementation. For this reason, we want an expression in the RHS only as a unction o and operations on it. State equation: ɺ = A + BU + his suggests: U = B ɺ A ( Note : his is only an approximate solution, unless m n) OPIMAL CONROL, GUIDANCE AND ESIMAION 2

11 LQR Extensions: 3. Weightage on Rate o Control Uɺ = Rˆ Pˆ Rˆ Pˆ B ɺ Rˆ Pˆ B Aɺ ˆ 1 ˆ ˆ + ˆ 1 R P ˆ + = 12 + P22 B A R P22 B ɺ 1 2 K2 = K ɺ K K1 Integrating this expression both sides, U = K K z dz + U t 1 2 Proportional Initial condition Itegral Note : U can be obtained using a perormance index without the Uɺ term OPIMAL CONROL, GUIDANCE AND ESIMAION 21 LQR Extensions: 4. Prescribed Degree o Stability Condition: All the Eiganvalues o the closed loop system should lie to the let o line AB 2αt J = where, e t Q + U RU dt α αt αt αt αt = ( e Q e e U R e U ) dt t + A = ( ɶ Qɶ + Uɶ RUɶ ) dt t αt Let ɶ = e Co-ordinate α transormation B αt Uɶ = e U OPIMAL CONROL, GUIDANCE AND ESIMAION jω 22 σ

12 LQR Extensions: 4. Prescribed Degree o Stability ɺɶ ɺ αt αt = e + αe αt = + + αt e A BU αe αt αt αt α = A e + B e U + e ɺɶ = A + I ɶ + BUɶ ( α ) Control Solution: Uɶ = K ɶ αt e U U = = K αt K e OPIMAL CONROL, GUIDANCE AND ESIMAION 23 LQR Extensions: 4. Prescribed Degree o Stability Modiied System: Uɶ = K ɶ ɺɶ = ( ) + α ɶ ɺ = ( ) A BK I K Actual System: U = K A BK is designed in such a way that eigenvalues o A BK + α I Hence, eigenvalues o will lie in the let-hal plane. ( A BK ) will lie to the let o a line parallel to the imaginary axis, which is located away by distance α rom the imaginary axis. OPIMAL CONROL, GUIDANCE AND ESIMAION 24

13 LQR Design or Command racking Pro. Radhakant Padhi Dept. o Aerospace Engineering Indian Institute o Science - Bangalore LQR Design or Command racking Problem: o design U such that a part o the state vector o the linear system ɺ = A + BU tracks a commanded reerence signal. i.e. rc, where = N Solution: 1) Formulate a standard LQR problem. However, select the Q matrix properly. r c 2) Implement the controller as U = K N Q ypically Q = OPIMAL CONROL, GUIDANCE AND ESIMAION 26

14 LQR Design or Command racking U B + ɺ + A LQ Regulation - K U B + ɺ + LQ racking - K A + N rc ( t) - OPIMAL CONROL, GUIDANCE AND ESIMAION 27 LQR Design or Command racking with Integral Feedback Solution (with integral controller): 1) Augment the system dynamics with integral states ɺ A AN B ɺ N AN ANN N B = + N U ɺ I I I 2) Select the Q matrix properly (should penalize only and states) t 3) Control solution U = K ( rc ) N ( rc ) dt I OPIMAL CONROL, GUIDANCE AND ESIMAION 28

15 LQR Design or Inhomogeneous Systems Pro. Radhakant Padhi Dept. o Aerospace Engineering Indian Institute o Science - Bangalore LQR Design or Inhomogeneous Systems Reerence OPIMAL CONROL, GUIDANCE AND ESIMAION 3

16 LQR Design or Inhomogeneous Systems o derive the state o a linear (rather linearized) system ɺ = A + BU + C to the origin by minimizing the ollowing quadratic perormance index (cost unction) 1 1 J = ( S ) + ( Q + U RU ) dt 2 where S, Q (psd), R > (pd) t OPIMAL CONROL, GUIDANCE AND ESIMAION 31 LQR Design or Inhomogeneous Systems Perormance Index (to minimize): 1 1 J = ( S ) + ( Q + U RU ) dt 2 Path Constraint: t ɺ = A + BU + C Boundary Conditions: t = : Speciied ( ) : Fixed, t : Free OPIMAL CONROL, GUIDANCE AND ESIMAION 32

17 LQR Design or Inhomogeneous Systems 1 erminal penalty: ϕ ( ) = ( S ) 2 1 Hamiltonian: H = Q + U RU + A + BU + C 2 State Equation: ɺ = A + BU + C Costate Equation: Optimal Control Eq.: Boundary Condition: λ ( H / ) ( Q A λ ) ɺ λ = = + ( / ) = = H U U R B λ ( ϕ / ) λ = = S OPIMAL CONROL, GUIDANCE AND ESIMAION 33 LQR Design or Inhomogeneous Systems Guess λ ( t) = P( t) ( t) + K ( t) ɺ λ = P ɺ + Pɺ + Kɺ = P ɺ + P A + BU + C + Kɺ = ɺ + ( λ ) + + ( ) ɺ ( ) P P A BR B PC Kɺ Q + A P + K = P + P A BR B P + K + PC + Kɺ 1 ( Pɺ + PA + A P PBR B P + Q) 1 ( Kɺ A K PBR B P PC) = OPIMAL CONROL, GUIDANCE AND ESIMAION 34

18 LQR Design or Inhomogeneous Systems Riccati equation ɺ = P PA A P PBR B P Q Auxiliary equation Boundary conditions Kɺ + A PBR B K + PC = ( ) + ( ) = ( is ree) P t K t S ( ) P t = S K ( t ) = OPIMAL CONROL, GUIDANCE AND ESIMAION 35 LQR Design or Inhomogeneous Systems Control Solution: U = R B λ = + R B P K = R B P R B K Note: here is a residual controller even ater. his part o the controller osets the continuous disturbance. OPIMAL CONROL, GUIDANCE AND ESIMAION 36

19 hanks or the Attention.!! OPIMAL CONROL, GUIDANCE AND ESIMAION 37

Linear Quadratic Regulator (LQR) Design II

Linear Quadratic Regulator (LQR) Design II Lecture 8 Linear Quadratic Regulator LQR) Design II Dr. Radhakant Padhi Asst. Professor Dept. of Aerospace Engineering Indian Institute of Science - Bangalore Outline Stability and Robustness properties