A Step Beyond the State of the Art Robust Model Predictive Control - PowerPoint PPT Presentation

A Step Beyond the State of the Art Robust Model Predictive Control Synthesis Methods by c ∗ Saˇ sa V. Rakovi´ (based on recent collaborative research with B. Kouvaritakis, M. Cannon & C. Panos) ∗ ISR, University of Maryland www.sasavrakovic.com & svr@sasavrakovic.com Institute for Systems Research, University of Maryland, College Park, USA, February 13 th 2012 SVR’s FPTMPC Talk @ ISR, UMD, USA, February 13 , 2012 – p. 1

Papers • Parameterized Robust Control Invariant Sets for Linear Systems: Theoretical Advances and Computational Remarks, IEEE-TAC Regular Paper, (Published) (Rakovi´ c and Bari´ c), • Parameterized Tube MPC, IEEE-TAC Regular Paper, (Accepted) (Rakovi´ c, Kouvaritakis, Cannon, Panos and Findeisen), • Fully Parameterized Tube MPC, IFAC 2011 , (Published) (Rakovi´ c, Kouvaritakis, Cannon, Panos and Findeisen), • Fully Parameterized Tube MPC, IJRNC D. W. Clarke’s Special Issue Paper, (Accepted) (Rakovi´ c, Kouvaritakis, Cannon and Panos), SVR’s FPTMPC Talk @ ISR, UMD, USA, February 13 , 2012 – p. 2

§0 – Outlook §0 – Outlook SVR’s FPTMPC Talk @ ISR, UMD, USA, February 13 , 2012 – p. 3

Outlook • Setting & Objectives §1 • Earlier Robust Model Predictive Control Methods §2 • Fully Parameterized Tube Optimal & Model Predictive Control §3 • Comparative Remarks & Illustrative Examples §4 • Concluding Remarks §5 SVR’s FPTMPC Talk @ ISR, UMD, USA, February 13 , 2012 – p. 4

§1 – Setting & Objectives §1 – Setting & Objectives • System Description • Problem Description • Synthesis Objectives SVR’s FPTMPC Talk @ ISR, UMD, USA, February 13 , 2012 – p. 5

System Description – Min–Max Linear–PCP Case • Linear discrete time system x + = Ax + Bu + w , • Variables x ∈ R n , u ∈ R m , w ∈ R n and ( A, B ) ∈ R n × n × R n × m , • Constraints x ∈ X , u ∈ U and w ∈ W , • Sets X ∈ PolyPC( R n ) , U ∈ PolyPC( R m ) and W ∈ PolyC( R n ) , • Matrix pair ( A, B ) stabilizable, • Information is variable x so feedback rules u ( x ) : X → U . SVR’s FPTMPC Talk @ ISR, UMD, USA, February 13 , 2012 – p. 6

Brief Problem Description – Illustration SVR’s FPTMPC Talk @ ISR, UMD, USA, February 13 , 2012 – p. 7

Brief Problem Description • Given an integer N ∈ N + and x ∈ X , select (if possible): � State Tube X N := { X k } k ∈ N [0: N ] , � Control Tube U N − 1 := { U k } k ∈ N [0: N − 1] , and � Control Policy Π N − 1 := { π k ( · ) } k ∈ N [0: N − 1] such that x ∈ X 0 , ∀ k ∈ N N − 1 , X k ⊆ X , U k ⊆ U , ∀ y ∈ X k , Ay + Bπ k ( y ) ⊕ W ⊆ X k +1 , ∀ y ∈ X k , π k ( y ) ∈ U k , X N ⊆ X f ⊆ X , which optimize V N ( X N , U N − 1 ) := � k ∈ N N − 1 L ( X k , U k ) + V F ( X N ) . SVR’s FPTMPC Talk @ ISR, UMD, USA, February 13 , 2012 – p. 8

Synthesis Objectives & Key Ingredients • Equally Important Objectives Robust Constraint Satisfaction, � Robust Stability (Boundedness and Attractiveness), � Computational Practicability, � Optimized (Meaningful) Performance. � • Key Ingredients � Fully Parameterized Tubes, � Induced, More General, Non–Linear Control Policy, � Repetitive Online Implementation. SVR’s FPTMPC Talk @ ISR, UMD, USA, February 13 , 2012 – p. 9

§2 – Earlier Robust MPC Methods §2 – Earlier Robust Model Predictive Control Methods • Open–Loop Min–Max MPC × , × , × , • Feedback Min–Max MPC � , � , × , • Dynamic Programming Based Robust MPC � , × , × , • Tube MPC � , × , � , • Disturbance Affine Feedback RMPC � , � , � . SVR’s FPTMPC Talk @ ISR, UMD, USA, February 13 , 2012 – p. 10

Feedback Min–Max OC and MPC – Preview SVR’s FPTMPC Talk @ ISR, UMD, USA, February 13 , 2012 – p. 11

Feedback Min–Max OC – Basic Idea • The set w N d of extreme disturbance sequences w ( i,N − 1) := { w ( i,k ) } k ∈ N N − 1 , with w ( i,k ) ∈ Vertices( W ) , • A set u N d of extreme control sequences u ( i,N − 1) := { u ( i,k ) } k ∈ N N − 1 , • A set x N d of extreme state sequences x ( i,N ) := { x ( i,k ) } k ∈ N N , • A sensible decision making process for selecting u N d := { u ( i,N − 1) : i ∈ N [1: N d ] } , and x N d := { x ( i,N ) : i ∈ N [1: N d ] } . (here N d := q N , and q := Cardinality(Vertices( W )) .) SVR’s FPTMPC Talk @ ISR, UMD, USA, February 13 , 2012 – p. 12

Feedback Min–Max OC – Decision Making Process • Given N ∈ N + and x ∈ X , select (if possible) sets of extreme: � State Sequences x N d = { x ( i,N ) : i ∈ N [1: N d ] } and � Control Sequences u N d = { u ( i,N − 1) : i ∈ N [1: N d ] } such that ∀ i ∈ N [1: N d ] , ∀ k ∈ N N − 1 , x ( i,k +1) = Ax ( i,k ) + Bu ( i,k ) + w ( i,k ) , with x ( i, 0) = x, x ( i,k ) ∈ X , u ( i,k ) ∈ U , and x ( i,N ) ∈ X f , ∀ ( i 1 , i 2 ) ∈ N [1: N d ] × N [1: N d ] , ∀ k ∈ N N − 1 , x ( i 1 ,k ) = x ( i 2 ,k ) ⇒ u ( i 1 ,k ) = u ( i 2 ,k ) which minimize V N ( x N d , u N d ) := max { V ( i,N ) ( x N d , u N d ) : i ∈ N [1: N d ] } , where i � V ( i,N ) ( x N d , u N d ) := ℓ ( x ( i,k ) , u ( i,k ) ) + V f ( x ( i,N ) ) . k ∈ N N − 1 SVR’s FPTMPC Talk @ ISR, UMD, USA, February 13 , 2012 – p. 13

Feedback Min–Max MPC – Summarized • Repetitive Online Application of Feedback Min–Max OC, • Dimension of Decision Variable Proportional to N d = q N , • Number of Constraints Proportional to N d = q N , • Computation Exceedingly Demanding and Impracticable. SVR’s FPTMPC Talk @ ISR, UMD, USA, February 13 , 2012 – p. 14

Feedback Min–Max OC and MPC – Summarized SVR’s FPTMPC Talk @ ISR, UMD, USA, February 13 , 2012 – p. 15

Feedback Min–Max OC – Important Remarks • Feedback Min–Max OC Utilizes: � State Tubes X N := { X k } k ∈ N N , with X k := Convh( { x ( i,k ) : i ∈ N [1: N d ] } ) , and � Control Tubes U N − 1 := { U k } k ∈ N N − 1 , with U k := Convh( { u ( i,k ) : i ∈ N [1: N d ] } ) . � Induced Control Policy Π N − 1 := { π k ( · , X k , U k ) } k ∈ N N − 1 , with π k ( · , X k , U k ) : X k → U k . • Feedback Min–Max OC Indicates Weakness of Open Loop Min–Max OC: � Additional Constraints ∀ i ∈ N [1: N d ] , ∀ k ∈ N N − 1 , u ( i,k ) = u k . SVR’s FPTMPC Talk @ ISR, UMD, USA, February 13 , 2012 – p. 16

Disturbance Affine Feedback ROC and RMPC – Preview SVR’s FPTMPC Talk @ ISR, UMD, USA, February 13 , 2012 – p. 17

Disturbance Affine Feedback (DAF) ROC – Basic Idea • Control Parameterization u 0 = v 0 , u k = v k + � k − 1 j =0 M ( k,j ) w j , with M ( k,j ) ∈ R m × n , • State Parameterization x = x 0 = z 0 , x k = z k + � k − 1 j =0 T ( k,j ) w j , with T ( k,j ) ∈ R n × n , • A set M N − 1 of control matrices { M ( k,j ) : j ∈ N k − 1 , k ∈ N [1: N − 1] } , • A nominal control sequence v N − 1 := { v k } k ∈ N N − 1 , • A set T N of state matrices { T ( k,j ) : j ∈ N k − 1 , k ∈ N [1: N ] } , • A nominal state sequence z N := { z k } k ∈ N N , • A sensible decision making process for selecting M N − 1 , v N − 1 , T N , and z N . SVR’s FPTMPC Talk @ ISR, UMD, USA, February 13 , 2012 – p. 18

DAF ROC – Decision Making Process • Given N ∈ N + and x ∈ X , select (if possible) sets of: � State and Control Matrices T N and M N − 1 and � Nominal State and Control Sequences z N and v N − 1 such that ∀ k ∈ N N − 1 , z k +1 = Az k + Bv k , with z 0 = x ∈ X , v 0 = u 0 ∈ U , ∀ k ∈ N [1: N − 1] , k − 1 k − 1 N − 1 � � � z k ⊕ T ( k,j ) W ⊆ X , v k ⊕ M ( k,j ) W ⊆ U , and, z N ⊕ T ( N,j ) W ⊆ X f , j =0 j =0 j =0 ∀ j ∈ N k − 1 , T ( k +1 ,j ) = AT ( k,j ) + BM ( k,j ) with T ( k +1 ,k ) = I. which minimize a sensible cost � V N ( x N , u N − 1 , T N , M N − 1 ) := ℓ ( z k , v k , T k , M k ) + V f ( z N , T N ) . k ∈ N N − 1 SVR’s FPTMPC Talk @ ISR, UMD, USA, February 13 , 2012 – p. 19

Disturbance Affine Feedback RMPC – Summarized • Repetitive Online Application of Disturbance Affine Feedback ROC, • Dimension of Decision Variable Proportional to hN 2 , • Number of Constraints Proportional to hN 2 , • Computation Practicable. SVR’s FPTMPC Talk @ ISR, UMD, USA, February 13 , 2012 – p. 20

DAF ROC and RMPC – Summarized SVR’s FPTMPC Talk @ ISR, UMD, USA, February 13 , 2012 – p. 21

Disturbance Affine Feedback ROC – Important Remarks • Disturbance Affine Feedback ROC Utilizes: � State Tubes X N := { X k } k ∈ N N , with X k := z k ⊕ � k − 1 j =0 T ( k,j ) W , and � Control Tubes U N − 1 := { U k } k ∈ N N − 1 , with U k := v k ⊕ � k − 1 j =0 M ( k,j ) W . � Disturbance Affine Control Policy Π N − 1 := { π k ( · , X k , U k ) } k ∈ N N − 1 . • Disturbance Affine Feedback ROC Indicates Weakness of Open Loop Min–Max OC: � Additional Constraints M ( k,j ) = 0 and T ( k,j ) = A k − 1 (Problems for Unstable A ). SVR’s FPTMPC Talk @ ISR, UMD, USA, February 13 , 2012 – p. 22

§3 – Fully Parameterized Tube OC & MPC §3 – Fully Parameterized Tube Optimal & Model Predictive Control • Prediction Structure • Constraint Handling • Sensible Cost • FPT Optimal & Model Predictive Control • System Theoretic Properties SVR’s FPTMPC Talk @ ISR, UMD, USA, February 13 , 2012 – p. 23