Structured adaptive control, or how to solve LMIs with Simulink Alexandru - Razvan LUZI Dimitri PEAUCELLE IEIIT-CNR Torino, october 2012 1/21
Introduction Introduction ■ Direct adaptive control: Adaptation of control gains done directly based on measurements. ▲ � = Indirect adaptive control: Estimator of model parameters + scheduled control gain ■ Feedback-loop stabilizing gains, MRAC not considered ■ Lyapunov based stability proofs, not gradient approximation ‘MIT rule’ ■ Framework initiated by V.A. Yakubovich in the late 1960’s ● Contributions: new adaptive control law with asymptotic structure + may solve LMIs 2/21
Plan Plan Passivity-based adaptive control 1 LMIs are strict-passifiable systems 2 Structured adaptive control 3 Numerical Example 4 3/21
Passivity-based adaptive control Passivity-based adaptive control of LTI systems Theorem The following two conditions are equivalent: ➊ There exists a static control u ( t ) = Fy ( t ) + w ( t ) for the system x ( t ) = A x ( t ) + B u ( t ) , ˙ y ( t ) = C x ( t ) , z ( t ) = y ( t ) that makes the closed-loop strictly passive (with respect to w / z). ➋ For all Γ ≻ 0 the following adaptive control ˙ K ( t ) = − y ( t ) y T ( t )Γ u ( t ) = K ( t ) y ( t ) + w ( t ) , makes the closed-loop globally strictly-passive. 4/21
Passivity-based adaptive control ● Strict-passivity includes asymptotic stability of x = 0 ● Adaptive control converges to K ( ∞ ): strictly-passifying static gain ▲ Theorem for square systems - extensions exist for non-square systems ▲ Not all stabilizable systems are strictly-passifiable - modified adaptive laws exist for stabilizable systems ● Condition ➊ also reads in terms of matrix inequalities as ( A + B F C ) T Q + Q ( A + B F C ) ≺ 0 , Q B = C T ∃ Q ≻ 0 : It happens to be an LMI constraint! A T Q + Q A + C T ( F T + F ) C ≺ 0 , Q B = C T ∃ Q ≻ 0 : ■ Finding F solution to the LMI is equivalent to simulating the system with the adaptive control law and taking F = K ( ∞ ). 5/21
LMIs are strict-passifiable systems All LMIs define strict-passifiable systems ● Let us consider an example: LMIs for an upper bound on the H ∞ norm � A T P + PA + C T C PB + C T D � P = P T ≻ 0 . ≺ 0 , B T P + D T C − γ 2 1 + D T D 6/21
LMIs are strict-passifiable systems All LMIs define strict-passifiable systems ● Let us consider an example: LMIs for an upper bound on the H ∞ norm � A T P + PA + C T C PB + C T D � P = P T ≻ 0 . ≺ 0 , B T P + D T C − γ 2 1 + D T D ▲ All LMI constraints can be gathered in one A T P + PA + C T C PB + C T D 0 ≺ 0 , B T P + D T C − γ 2 1 + D T D P = P T 0 0 0 − P 6/21
LMIs are strict-passifiable systems ▲ All LMI constraints can be gathered in one A T P + PA + C T C PB + C T D 0 ≺ 0 , B T P + D T C − γ 2 1 + D T D P = P T 0 − P 0 0 7/21
LMIs are strict-passifiable systems ▲ All LMI constraints can be gathered in one A T P + PA + C T C PB + C T D 0 ≺ 0 , B T P + D T C − γ 2 1 + D T D P = P T 0 − P 0 0 ▲ Can be decomposed in a sum with elementary matrix variables 0 P 0 A + B T B P + B T γ ( − γ 2 1 ) B γ ≺ 0 , P = P T P 0 0 P 0 0 − P C T C C T D A B 0 0 D T C D T D , B P = , B γ = � � A = 0 1 0 0 0 1 0 0 0 0 0 0 1 7/21
LMIs are strict-passifiable systems ▲ Can be decomposed in a sum with elementary matrix variables 0 P 0 A + B T B P + B T γ ( − γ 2 1 ) B γ ≺ 0 , P = P T P 0 0 P 0 0 − P 8/21
LMIs are strict-passifiable systems ▲ Can be decomposed in a sum with elementary matrix variables 0 P 0 A + B T B P + B T γ ( − γ 2 1 ) B γ ≺ 0 , P = P T P 0 0 P 0 0 − P ▲ Or equivalently when gathering all variables in a block-diagonal matrix A + B T F B ≺ 0 , � � B = B P B γ with the structural equality constraints � F P 0 P 0 � 0 F γ = − γ 2 1 P = P T F = , F P = P 0 0 , , F γ 0 − P 0 0 8/21
LMIs are strict-passifiable systems ▲ Can be decomposed in a sum with elementary matrix variables 0 P 0 A + B T B P + B T γ ( − γ 2 1 ) B γ ≺ 0 , P = P T P 0 0 P 0 0 − P ▲ Or equivalently when gathering all variables in a block-diagonal matrix A + B T F B ≺ 0 , � � B = B P B γ with the structural equality constraints � F P 0 P 0 � 0 F γ = − γ 2 1 P = P T F = , F P = P 0 0 , , F γ 0 − P 0 0 ● The constraint A + B T F B ≺ 0 holds iff ( A , B , C = B T ) is strictly-passifiable by F . 8/21
LMIs are strict-passifiable systems ▲ Can be decomposed in a sum with elementary matrix variables 0 P 0 A + B T B P + B T γ ( − γ 2 1 ) B γ ≺ 0 , P = P T P 0 0 P 0 0 − P ▲ Or equivalently when gathering all variables in a block-diagonal matrix A + B T F B ≺ 0 , � � B = B P B γ with the structural equality constraints � F P 0 P 0 � 0 F γ = − γ 2 1 P = P T F = , F P = P 0 0 , , F γ 0 − P 0 0 ● The constraint A + B T F B ≺ 0 holds iff ( A , B , C = B T ) is strictly-passifiable by F . ■ LMI converted to strict-passification problem, with equality constraints. 8/21
LMIs are strict-passifiable systems ■ Procedure applies to any LMI: concludes with search of passifying F 1 0 ... F = 0 F N for a (symmetric) system ( A , B , C = B T ) with additional structural equality constraints that can be compacted in vec( F i ) = S i x i ⇔ U i vec( F i ) = 0 where vec( F i ) is the vector composed of stacked columns of F i , x i are vectors of independent scalar decision variables and U i = S ⊥ i . 9/21
LMIs are strict-passifiable systems ■ Procedure applies to any LMI: concludes with search of passifying F 1 0 ... F = 0 F N for a (symmetric) system ( A , B , C = B T ) with additional structural equality constraints that can be compacted in vec( F i ) = S i x i ⇔ U i vec( F i ) = 0 where vec( F i ) is the vector composed of stacked columns of F i , x i are vectors of independent scalar decision variables and U i = S ⊥ i . ● When starting from the canonical representation L 0 + � j ˆ x j L j ≺ 0 , then the structural constraints are all of the type � ˆ x j 1 r j 1 � 0 F j = 0 − ˆ x j 1 r j 2 and r j 1 + r j 2 can be very large and U j is huge. ● F and U i s expected to be smaller when matrix representation. 9/21
Structured adaptive control Block-diagonal adaptive control with asymptotic structure Theorem The following two conditions are equivalent: ➊ There exists a decentralized static control u i ( t ) = F i y i ( t ) + w i ( t ) satisfying the structural constraints U i vec( F i ) = 0 for the system � x ( t ) = A x ( t ) + ˙ B i u i ( t ) , y i ( t ) = C i x ( t ) , z ( t ) = y ( t ) that makes the closed-loop strictly passive (with respect to w / z). ➋ For all Γ i ≻ 0 , α i > 0 the following adaptive control u i ( t ) = K i ( t ) y i ( t ) + w i ( t ) , ˙ K i ( t ) = − y i ( t ) y T U T � � i ( t )Γ i − α i · mat i U i · vec( K i ( t )) Γ i makes the closed-loop globally strictly-passive. ‘mat’ is the function such that mat(vec( F )) = F . 10/21
Structured adaptive control Proof of ➊ ⇒ ➋ ● ➊ reads as ( A + B F C ) T Q + Q ( A + B F C ) < 0 , Q B = C T , ∃ F , ∃ Q ≻ 0 : (1) � � F = diag · · · F i · · · , U i · vec( F i ) = 0 11/21
Structured adaptive control Proof of ➊ ⇒ ➋ ● ➊ reads as ( A + B F C ) T Q + Q ( A + B F C ) < 0 , Q B = C T , ∃ F , ∃ Q ≻ 0 : (1) � � F = diag · · · F i · · · , U i · vec( F i ) = 0 ● Let the Lyapunov function for the non-linear system (with adaptive law) � ( K i − F i )Γ − 1 ( K i − F i ) T �� V ( x , K ) = 1 � � x T Qx + Tr 2 i 11/21
Structured adaptive control Proof of ➊ ⇒ ➋ ● ➊ reads as ( A + B F C ) T Q + Q ( A + B F C ) < 0 , Q B = C T , ∃ F , ∃ Q ≻ 0 : (1) � � F = diag · · · F i · · · , U i · vec( F i ) = 0 ● Let the Lyapunov function for the non-linear system (with adaptive law) � ( K i − F i )Γ − 1 ( K i − F i ) T �� V ( x , K ) = 1 � � x T Qx + Tr 2 i ● After manipulations and using Q B = C T , U i · vec( F i ) = 0, we get: ˙ � V ( x , K ) = x T ( A + B F C ) T Qx + w T z − α i ( U i · vec( K i )) T ( U i · vec( K i )) . i 11/21
Structured adaptive control Proof of ➊ ⇒ ➋ (continued) ˙ � V ( x , K ) = x T ( A + B F C ) T Qx + w T z − α i ( U i · vec( K i )) T ( U i · vec( K i )) . i 12/21
Structured adaptive control Proof of ➊ ⇒ ➋ (continued) ˙ � V ( x , K ) = x T ( A + B F C ) T Qx + w T z − α i ( U i · vec( K i )) T ( U i · vec( K i )) . i ▲ First term is strictly negative due to (1), until x = 0, ▲ Last term is strictly negative, until U i · vec( K i ) = 0 ■ If no perturbations ( w = 0) the system converges to the attractor A = { ( x , K ) : x = 0 , U i · vec( K i ) = 0 } ■ On the attractor ˙ K i = 0: the gains K i ( ∞ ) are constant. 12/21
Structured adaptive control Proof of ➊ ⇒ ➋ (continued) ˙ � V ( x , K ) = x T ( A + B F C ) T Qx + w T z − α i ( U i · vec( K i )) T ( U i · vec( K i )) . i 13/21
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