Robust Model Predictive Control

Transcription

1 Robust Model Predictive Control Motivation: An industrial C3/C4 splitter: MPC assuming ideal model: MPC considering model uncertainty

2 Robust Model Predictive Control Nominal model of the plant: 1 = G () z D () z N () z Norm-bounded uncertainty: Gz ( ) = G( z) + [ ] Gz ( ) = G( z) 1+ 1 [ ] [ ] Gz ( ) = D( z) + N( z) + D N Uncertainty is measured through the H infinity norm

3 Closed Loop with Model Uncertainty K(z) stabilizes the nominal plant G (z) Additive uncertainty: Gz ( ) = G ( z) + ( ) 1 M ( z) = K( z) I + G ( z) K( z) Small gain theorem: σ jωt ( M e ) ( ) < 1 ( ) 1 jωt jωt jωt σ Ke ( ) I+ Ke ( ) G( e ) < 1

4 Closed Loop Stability Multiplicative Uncertainty Multiplicative uncertainty: Gz ( ) = G( z) [ 1+ ] ( ) 1 M( z) = K( z) G ( z) I + KG ( Z) Small gain theorem: σ jωt ( M e ) ( ) < 1 ( ) 1 jωt jωt jωt jωt σ Ke ( ) G( e ) I+ Ke ( ) G( e ) < 1

5 Robust Stability of DMC Question: How much model uncertainty a well tuned DMC can tolerate?

6 Robust Stability of the Unconstrained DMC Predicting Model { } [ y] = A[ y] + B u k + K y C[ y] ( 1) F k k/ k k 1/ k 1 k/ k 1 [ ] [ ] y = A y + B u( k 1) k/ k 1 k 1/ k 1 y= y( k) y( k+ 1)! y( k+ n h ) T T T Iny! S1 Iny Iny! S I 2 ny A = " " " # ", B = ", KF = ",! Iny S nh Iny I! ny S nh + 1 I ny T S i S1,1, i S1,2, i! S1, nu, i S2,1, i S2,2, i! S 2, nu, i = " " # " Sny,1, i Sny,2, i! Sny, nu, i [ ] / yk ( ) = C y k k C = C = I ny!

7 Robust Stability of the Unconstrained DMC True Plant Model [ ] [ ] y = A y + B u ( k ) k/ k k 1/ k 1 DMC control law [ ] ( ) { T } 1 T ( ) [ ] { } { [ ] } m T m T m T sp sp k s n n n k/ k MPC k/ k u = N S Q QS + R R S Q QN y y = K y y [ u] = u( k) u( k + 1)! u( k + m 1) k T T T T N N I = ny. n ( ny. n) ( ny.( nh 1) ) I = ( ) (.( 1) ) s nu nu nu m S m n S1! S2 S1! = " " # " Sm Sm 1! S1 " " " Sn Sn 1! Sn m+ 1

8 Robust Stability of the Unconstrained DMC Combining predicting model and plant y A y B uk ( 1) y = K CA ( I K C) A y + B+ K C( B B) k/ k F F k 1/ k 1 F sp sp y y y y uk ( ) = K = K [ ] MPC sp sp y y y y k/ k k / k Additive uncertainty B = B+ S = T T T S S1 S! 2 S nh + 1 T

9 Robust Stability of the Unconstrained DMC Closed loop with DMC [ $ ] φ [ $ ] β 1 β 2 y = y + uk ( ) + uk ( ) k+ 1/ k+ 1 k/ k A = KFCA ( I KFC) A φ 1 B β = B sp y y uk ( ) = K y sp y I ' β2 = S = 2 KFC β S k/ k ' 1 2 M( z) = K I + ( zi φ) β K ( zi φ) β

10 Robust Stability of the Unconstrained DMC Example 1 Gs () s+ 1 5s+ 1 = s s + 1 DMC parameters: T=5, n=12, m=5, Q=diag(1,1), R=diag(.3,.3) Question: What is the max size of 11, 12, 21 and 22 such that DMC will remain stable? Response: DMC will be stable for S not larger than σ ( ) = min 1 S ω ωti σ M( e ) ' 1 2 M( z) = K I + ( zi φ) β K ( zi φ) β

11 Example 1 DMC with: T=5, n=12, m=5, Q=diag(1,1), R=diag(.3,.3) min 1 ωti = ω σ M( e ) The approximate bounds are: S.13 S S +.13 i= 1,2; j= 1,2; l = 1,,12 i, j, l i, j, l i, j, l Or:.13 i, j +.13 i= 1, 2; j = 1, 2

12 Example 1 Stability bounds are very conservative S.13 S S +.13 i= 1,2; j= 1,2; l = 1,,12 i, j, l i, j, l i, j, l s+ 1 5s+ 1 Gs () = s s + 1

13 Example 1 Stability bounds are very conservative S.3 S S +.3 i= 1,2; j= 1,2; l = 1,,12 i, j, l i, j, l i, j, l s+ 1 5s+ 1 Gs () = s s + 1

14 Example 1 Stability bounds are very conservative s+ 1 5s+ 1 Gs () = s s + 1

15 Robust Stability - Multiplicative gain uncertainty y A y B uk ( 1) y = K CA ( I K C) A y + B+ K C( B B) k/ k F F k 1/ k 1 F B ( 1, ) = B + ij ij i j B = B+ S1,1! S1,2!! S1, nu! S2,1 S2,2 S2, nu!!!! S = " " # " " " # "! " " # "! Sny,1! Sny,2!! Sny, nu = S$ N S K u S 1! 1,1 " "! 1 #! " ny,1 1! # " "! " 1, nu 1! 1 #! ny, nu " "! "! 1 & '('') nu [ $ ] φ [ $ ] β 1 β 2 y = y + uk ( ) + uk ( ) k+ 1/ k+ 1 k/ k A = KFCA ( I KFC) A B φ β1 = B β2 = K Nu = β2 KNu KFCB$ B$ $

16 Robust Stability - Multiplicative gain uncertainty = $ 2 M( z) = Nu K I + ( zi φ) β K ( zi φ) β

17 Example 1 DMC with: T=5, n=12, m=5, Q=diag(1,1), R=diag(.3,.3) 1.77(1 + 11) 5.88(1 + 12) 6s+ 1 5s+ 1 Gs () = 4.41(1 + 21) 7.2(1 + 22) 44s s + 1 min 1 ωti =.112 ω σ M( e ) The approximate bounds are: = 1, 2; = 1, 2 i, j i j

18 Example 1 Gain uncertainty - Stability bounds are not so conservative = 1, 2; = 1, 2 i, j i j 1.77(1.11) 5.88(1 +.11) 6s+ 1 5s+ 1 Gs () = 4.41(1 +.11) 7.2(1.11) 44s s + 1

19 Example 1 Gain uncertainty - Stability bounds are not so conservative but misses some particular systems = 1, 2; = 1, 2 i, j i j 1.77(1 +.5) s+ 1 5s+ 1 Gs () = s s + 1

20 Robust Stability with H Approach -The ranges of general unstructured model uncertainties where DMC is guaranteed to be stable are quite conservative. Less conservative bounds can be obtained by taking into account the structure of the uncertainty. However, the results are still not useful for practical applications. -The procedure does not account for constraints on the inputs and input moves. When a constraint becomes active, the closed loop system may become unstable, even if uncertainty is smaller than the maximum uncertainty tolerated by the unconstrained DMC controller.

21 Robust MPC with Infinite Horizon IHMPC (regulator case): x( k + 1) = Ax( k) + Bu( k) yk ( ) = Cxk ( ) Control objective: min uk ( ), uk ( + 1),, uk ( + m 1) J k T m 1 J = xk ( + i) Qxk ( + i) + uk ( + j) Ruk ( + j) k min i= j= max u u( k + j) u ; j =,1,!, m 1 T

22 Robust MPC with Infinite Horizon IHMPC (regulator case) for stable systems: min uk ( ), uk ( + 1),, uk ( + m 1) min m J max u u( k + j) u ; j =,1,!, m 1 k m 1 T T T J = x( k + j) Q x( k + j) + x( k + m) Px( k + m) + u( k + j) Ru( k+ j) k j= j= T A PA P = Q Advantage: The closed-loop is stable for any set of tuning parameters Drawback: Cannot be applied to the set point tracking case Model is written in terms of u u ss and y y ss

23 Robust MPC with Infinite Horizon Robust IHMPC (regulator case): x( k + 1) = Ax( k) + Bu( k) yk ( ) = Cxk ( ) θ = Multi plant system: Ω *{ θ θ θ },,, L 1 2 ( A, B) True plant: θ = ( A, B ) θ Ω T T T T Nominal plant: θ = ( A, B ) θ Ω N N N N m m 1 T T T k, θi θi θi θi θi θi j= j= J = xk ( + j) Qxk ( + j) + xk ( + m) Pxk ( + m) + uk ( + j) Ruk ( + j) T θi θi θi θi A PA P = Q

24 Robust MPC with Infinite Horizon Robust IHMPC (regulator case) is obtained from the solution to: min uk ( ), uk ( + 1),..., uk ( + m 1) Subject to: J k, i k, min J k, θn Jˆ θ θ θ i Ω i max u u( k + j) u ; j =,1,!, m 1 * * * uk = u ( k) u ( k + 1) u ( k + m 1)! k J ˆ ( i = 1,..., L ) k, θi is obtained with the following control sequence: * * uˆ k = u ( k) u ( k + m 2)! k 1 Robust IHMPC (regulator) cannot be applied to the set-point tracking: ˆ Jk, θ i is not bounded for all plants θi Ω with different gains.

25 Extended Infinite Horizon MPC Process model in the incremental form: s s s x ( k+ 1) I ny x ( k) B = + uk ( ) d d d x ( k+ 1) F x ( k) B yk ( ) s x ( k) = I Ψ ny uk ( ) = uk ( ) uk ( 1) d x ( k) Control objective: m 1 T T T k = (( + ) δ k) (( + ) δ k) + ( + ) ( + ) +δk δk j= j= V e k j Q e k j u k j R u k j S ek ( + j) = yk ( + j) y sp

26 Extended IHMPC Problem P2: min u k, δ k V k Subject to: m T k ( k) ( k) d T d j= V = e( k + j) δ Q e( k + j) δ + x ( k + m) Qx ( k + m) m-1 T T + uk ( + j) R uk ( + j) +δ Sδ j= T T T Q F QF = F Ψ QΨF $ B$ s = [ B s! B s ] s s e ( k) + B u k δ k = min max u u( k + j) u max k &''('') j max! i= u u( k 1) + u( k + i) u ; j =,1,, m 1 k m k [ ( ) ( 1)! ( 1) ] u = u k u k + u k + m k

27 Extended Robust IHMPC Problem P3: Subject to: min Vk, θn, uk δ k, θi, i= 1,..., L s e ( k) + B$ u δ = i = 1,..., L s i k k, θ i min max u u( k + j) u max j max! i= u u( k 1) + u( k + i) u ; j =,1,, m 1 V Vˆ i = 1,..., L ˆ Vk, θ k, θ k, θ i i i δ ˆk k, θi is computed with: u and * * uˆ = ( ) ( + 2) k u k! u k m k 1 s e ( k) + B$ uˆ δ ˆ = i= 1,..., L s i k k, θi ˆ

28 Robust Extended IHMPC Example 2 : The high purity distillation column y1( s) (1 + γ1).6147(1 +γ2) u1( s) = y ( s) 22.98s (1 +γ ).982(1 +γ ) u ( s) Ω.4 γ1, γ2.4 Ω [(,) (.4,.4) (.4,.4) (.4,.4) (.4,.4)] Tuning parameters of RE-IHMPC: T = 2, m = 3, Q = diag(1, 1), R = diag(1-2,1-2 ), S = diag(1 2,1 2 ), u max =.25, u max = 1.5, u min = -1.5

29 Example 2 Nominal model: ( γ, γ ) = (,) 1 2 Case 1: true plant is ( γ1, γ 2) = (,) Case 2: true plant is ( γ1, γ 2) = (.4,.4) Case 3: true plant is ( γ, γ ) = (.4,.4) 1 2 ( ) (----) ( )

30 Example 2 Nominal model: ( γ, γ ) = (,) 1 2 Case 1: true plant is ( γ1, γ 2) = (,) Case 2: true plant is ( γ1, γ 2) = (.4,.4) Case 3: true plant is ( γ, γ ) = (.4,.4) 1 2 ( ) (----) ( )

31 Example 2 RE-IHMPC with input saturation: Case 2: with u max =.5

32 Robust Stability with IHMPC Advantages: -Quite large ranges of model uncertainties can be considered in the MPC control law computation. -The structure of MPC is preserved (constraints are included) and can be implemented in practice. -Performance is expected to deteriorate if uncertainty is too large but better than the controller with nominal model. Disadvantages: -Computer effort tends to be larger than in the nominal MPC. Should be applied only in subsystems where uncertainty is critical. -Still lacking a practical solution to integrating and unstable systems.