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 of LQR Optimum value of the cost function Extension of LQR design For cross-product term in cost function Rate of state minimization Rate of control minimization LQR design with prescribed degree of stability LQR for command tracking LQR for inhomogeneous systems ADVANCED CONROL SYSEM DESIGN
Stability and Robustness Properties Dr. Radhakant Padhi Asst. Professor Dept. of Aerospace Engineering Indian Institute of Science - Bangalore
LQR Design: Stability of Closed Loop System Closed loop system Lyapunov function V = X PX + X PX ) X = AX + BU = A BK X ) = V X X PX ) ) = A BK X PX + X P A BK X ) = X A BR B P P P A BR B P) + X = ) + + = X Q PBR B P X 1 X PA A P PBR B P Q Q PBR B P X ADVANCED CONROL SYSEM DESIGN 4
LQR Design: Stability of Closed Loop System For R >, R >. Also P > -1 So PBR B P > Also Q. ) -1 PBR B P Q) Hence, + > V X < Hence, the closed loop system is always asymptotically stable! ADVANCED CONROL SYSEM DESIGN 5
LQR Design: Minimum value of cost function 1 J = X QX + U RU) dt t 1 = X QX + R B PX) R R B PX) dt t 1 = X Q + PBR B P) X dt t 1 ) 1 1 = V dt [ V] X PX t = = t 1 1 = X PX X PX X PX = ) t ADVANCED CONROL SYSEM DESIGN 6
LQR Design: Robustness of Closed Loop System Gain Margin: Phase Margin: 6 Ref.: D. S. Naidu, Optimal Control Systems, CRC Press, 3.) ADVANCED CONROL SYSEM DESIGN 7
Extensions of LQR Design Dr. Radhakant Padhi Asst. Professor Dept. of Aerospace Engineering Indian Institute of Science - Bangalore
LQR Extensions: 1. Cross Product erm in P.I. 1 J = X QX + X WU + U RU) dt t Let us consider the expression: ) ) + + + ) X Q WR W X U R W X R U R W X = X QX + X WU + U RU ) = + + + X QX U RU U W X X WU ADVANCED CONROL SYSEM DESIGN 9
LQR Extensions: 1. Cross Product erm in P.I. 1 ) ) J = X Q WR W X U R W X R U R W X) + + + dt t Q1 U1 1 = X Q ) 1X + U1RU1 dt t X = AX + BU ) 1 1 ) = AX + B U R W X = A BR W X + BU = AX+ BU 1 1 Control Solution U = KX 1 U = U1 R W X ) = K + R W X ADVANCED CONROL SYSEM DESIGN 1
LQR Extensions:. Weightage on Rate of State 1 J = X QX + U RU + X SX ) dt t 1 = X QX U RU AX BU ) S AX BU ) + + + + dt t 1 X QX + U RU + X A S A X + X A SBU = dt t + U B S AX + U B SBU Q1 R1 W 1 = X Q A SA) X U R B SB) U X A SB) U dt + + + + t 1 = X Q1X + U RU 1 + ) X WU dt t Leads to a cross product case ADVANCED CONROL SYSEM DESIGN 11
LQR Extensions: 3. Weightage on Rate of Control 1 J = X QX + U RU + U RU ˆ ) dt X Let X =, V = U U 1 Q J V RV ˆ = X dt R X+ 1 ˆ J = X QX+ V RV ˆ ) dt ADVANCED CONROL SYSEM DESIGN 1
LQR Extensions: 3. Weightage on Rate of Control X = AX + BU, X ) = X U = V A B X = V Aˆ BV ˆ + = + X I X Aˆ { AB} Bˆ Note : 1) he dimension of the problem has increased from n to n+ m) ) If, is controllable, it can be shown that the new system is also controllable. ADVANCED CONROL SYSEM DESIGN 13
LQR Extensions: 3. Weightage on Rate of Control Solution : ˆ ˆ V = U = R B PˆX where Pˆ is the solution of ˆ ˆ ˆ ˆˆˆˆ A P + PA PBR B Pˆ + Qˆ = Hence ˆ ˆ ˆ P 1 11 P 1 1 [ ] ˆ ˆ ˆ X U = R I X = R P1 P ˆ P ˆ U 1 P = Rˆ Pˆ X Rˆ Pˆ U 1 ADVANCED CONROL SYSEM DESIGN 14
LQR Extensions: 3. Weightage on Rate of Control ˆ 1ˆ 1 However, U ˆ = R P ˆ 1 X R PU is a dynamic equation in U and hence is not easy for implementation. For this reason, we want an expression in the RHS only as a function of X and operations on it. State equation: his suggests: X = AX + BU + U = B X AX ) Note : his is only an approximate solution, unless m n) ADVANCED CONROL SYSEM DESIGN 15
LQR Extensions: 3. Weightage on Rate of Control ˆ 1ˆ 1 1 U ˆ ˆ + ˆ ˆ + = R P1 X R PB X R PB AX ˆ 1 ˆ ˆ + ) ˆ 1 R P ˆ + = 1 + PB A X R PB X 1 K = KX KX ) Integrating this expression both sides, U = K X K X z dz+ U Note : t 1 Proportional Initial condition Itegral U can be obtained using a performance index without the U term K 1 ADVANCED CONROL SYSEM DESIGN 16
LQR Extensions: 4. Prescribed Degree of Stability Condition: All the Eiganvalues of the closed loop system should lie to the left of line AB 1 αt J = e where, t X QX + U RU dt α 1 ) αt αt αt αt = e X Q e X e U R e U dt t + A 1 = X QX + U RU ) dt t αt Let X = e X Co-ordinate α transformation B αt U = e U ADVANCED CONROL SYSEM DESIGN jω 17 σ
LQR Extensions: 4. Prescribed Degree of Stability αt αt X = e X + αe X αt = + + αt e AX BU αe X ) αt ) αt ) αt α ) = A e X + B e U + e X X = A+ α I X + BU ) Control Solution: U αt e U U = KX = = KX αt Ke X ADVANCED CONROL SYSEM DESIGN 18
LQR Extensions: 4. Prescribed Degree of Stability Modified System: U = KX = ) + α = ) X A BK I X K ) Actual System: U = KX X A BK X is designed in such a way that eigenvalues of A BK + α I Hence, eigenvalues of will lie in the left-half plane. A BK ) will lie to the left of a line parallel to the imaginary axis, which is located away by distance α from the imaginary axis. ADVANCED CONROL SYSEM DESIGN 19
LQR Design for Command racking Dr. Radhakant Padhi Asst. Professor Dept. of Aerospace Engineering Indian Institute of Science - Bangalore
LQR Design for Command racking Problem: o design U such that a part of the state vector of the linear system X = AX + BU tracks a commanded reference signal. X i.e. X rc, where X = X N Solution: 1) Formulate a standard LQR problem. However, select the Q matrix properly. ) Implement the controller as U X K r c = X N Q ypically Q = ADVANCED CONROL SYSEM DESIGN 1
LQR Design for Command racking U B + X + A X LQ Regulation -K U B + X + X LQ racking -K A + X X N rc ) t - ADVANCED CONROL SYSEM DESIGN
LQR Design for Command racking with Integral Feedback Solution with integral controller): 1) Augment the system dynamics with integral states X A AN X B X N AN ANN X N B = + N U X I I X I ) Select the Q matrix properly should penalize only X and XI states) t 3) Control solution U = K X rc) XN X rc) dt ADVANCED CONROL SYSEM DESIGN 3
LQR Design for Inhomogeneous Systems Dr. Radhakant Padhi Asst. Professor Dept. of Aerospace Engineering Indian Institute of Science - Bangalore
LQR Design for Inhomogeneous Systems Reference ADVANCED CONROL SYSEM DESIGN 5
LQR Design for Inhomogeneous Systems o derive the state of a linear rather linearized) system X X = AX + BU + C to the origin by minimizing the following quadratic performance index cost function) 1 1 ) J = X ) fsfx f + X QX + U RU dt where S, Q psdf), R > pdf) f t f t ADVANCED CONROL SYSEM DESIGN 6
LQR Design for Inhomogeneous Systems Performance Index to minimize): 1 1 ) J = X ) fsf X f + X QX + U RU dt Path Constraint: t f t X = AX + BU+ C Boundary Conditions: X t f = X :Specified ) :Fixed, X t ) f :Free ADVANCED CONROL SYSEM DESIGN 7
LQR Design for Inhomogeneous Systems erminal penalty: Hamiltonian: State Equation: 1 ) ϕ X ) f = X fsf X f 1 H = X QX + U RU + AX + BU + C X = AX + BU + C ) λ ) Costate Equation: ) H / X QX A λ ) λ = = + Optimal Control Eq.: Boundary Condition: / ) = = H U U R B λ ϕ/ X ) λ = = S X f f f f ADVANCED CONROL SYSEM DESIGN 8
LQR Design for Inhomogeneous Systems Guess λ t) = P t) X t) + K t) λ = PX + PX + K = PX + P AX + BU + C) + K = PX + P AX BR B + PC + K λ ) )) ) QX + A PX + K = PX + P AX BR B PX + K + PC + K P PA A P PBR B P Q X + + + ) ) K A K PBR B P PC) + + = ADVANCED CONROL SYSEM DESIGN 9
LQR Design for Inhomogeneous Systems Riccati equation P PA A P PBR B P Q + + + = Auxiliary equation K + A PBR B K + PC = ) Boundary conditions ) + ) = is free) P t X K t S X X f f f f f f P t ) f = S f K t ) f = ADVANCED CONROL SYSEM DESIGN 3
LQR Design for Inhomogeneous Systems Control Solution: U = R B λ = + R B PX K ) = R BPX R BK Note: here is a residual controller even after X. his part of the controller offsets the continuous disturbance. ADVANCED CONROL SYSEM DESIGN 31
ADVANCED CONROL SYSEM DESIGN 3