In search of the unreachable setpoint

Transcription

1 In search of the unreachable setpoint Adventures with Prof. Sten Bay Jørgensen James B. Rawlings Department of Chemical and Biological Engineering June 19, 2009 Seminar Honoring Prof. Sten Bay Jørgensen CAPEC Technical University of Denmark Rawlings Adventures with Sten Bay 1 / 32

2 Outline 1 Unreachable setpoints in feedback control 2 Model identification 3 Conclusions Rawlings Adventures with Sten Bay 2 / 32

3 The first paper Rawlings Adventures with Sten Bay 3 / 32

4 A brief history of this paper John Jørgensen visits Madison. January August 2000 Rawlings Adventures with Sten Bay 4 / 32

5 A brief history of this paper John Jørgensen visits Madison. January August 2000 Dennis Bonné visits Madison. February August 2002 Rawlings Adventures with Sten Bay 4 / 32

6 A brief history of this paper John Jørgensen visits Madison. January August 2000 Dennis Bonné visits Madison. February August 2002 Paper is submitted to IEEE TAC. July 2007 Rawlings Adventures with Sten Bay 4 / 32

7 A brief history of this paper John Jørgensen visits Madison. January August 2000 Dennis Bonné visits Madison. February August 2002 Paper is submitted to IEEE TAC. July 2007 Paper is submitted again. November 2007 Rawlings Adventures with Sten Bay 4 / 32

8 A brief history of this paper John Jørgensen visits Madison. January August 2000 Dennis Bonné visits Madison. February August 2002 Paper is submitted to IEEE TAC. July 2007 Paper is submitted again. November 2007 First reviews come back. February 2008 Rawlings Adventures with Sten Bay 4 / 32

9 A brief history of this paper John Jørgensen visits Madison. January August 2000 Dennis Bonné visits Madison. February August 2002 Paper is submitted to IEEE TAC. July 2007 Paper is submitted again. November 2007 First reviews come back. February 2008 Revised paper submitted. February 2008 Rawlings Adventures with Sten Bay 4 / 32

10 A brief history of this paper John Jørgensen visits Madison. January August 2000 Dennis Bonné visits Madison. February August 2002 Paper is submitted to IEEE TAC. July 2007 Paper is submitted again. November 2007 First reviews come back. February 2008 Revised paper submitted. February 2008 Paper is accepted. April 2008 Rawlings Adventures with Sten Bay 4 / 32

11 A brief history of this paper John Jørgensen visits Madison. January August 2000 Dennis Bonné visits Madison. February August 2002 Paper is submitted to IEEE TAC. July 2007 Paper is submitted again. November 2007 First reviews come back. February 2008 Revised paper submitted. February 2008 Paper is accepted. April 2008 Paper appears. October 2008 Rawlings Adventures with Sten Bay 4 / 32

12 A brief history of this paper John Jørgensen visits Madison. January August 2000 Dennis Bonné visits Madison. February August 2002 Paper is submitted to IEEE TAC. July 2007 Paper is submitted again. November 2007 First reviews come back. February 2008 Revised paper submitted. February 2008 Paper is accepted. April 2008 Paper appears. October /2 years since John first arrived in Madison Rawlings Adventures with Sten Bay 4 / 32

13 Setpoints and unreachable setpoints Rawlings Adventures with Sten Bay 5 / 32

14 Setpoints and unreachable setpoints Consider the steady state of a dynamic model with state x, controlled input u, and disturbance w x(k + 1) = Ax(k) + Bu(k) + B d w(k) Rawlings Adventures with Sten Bay 5 / 32

15 Setpoints and unreachable setpoints Consider the steady state of a dynamic model with state x, controlled input u, and disturbance w x(k + 1) = Ax(k) + Bu(k) + B d w(k) x s = (I A) 1 B u }{{} s + (I A) 1 B d w }{{} s G d s x s = Gu s + d s Rawlings Adventures with Sten Bay 5 / 32

16 Steady states unconstrained system x s x s = Gu s + d s d s1 = 1 d s2 = 0 x sp G d s3 = 1 u s1 u s2 us3 u s For an unconstrained system with G 0, any setpoint x sp with any disturbance d s has a corresponding u s. Rawlings Adventures with Sten Bay 6 / 32

17 Constraints and unreachable setpoints x s x s = Gu s + d s 0 u s 1 d s1 = 1 d s2 = 0 x sp G d s3 = 1 0 u s1 1 u s2 u s3 u s For a constrained system, the setpoint x sp may be unreachable for a given disturbance d s. MPC is method of choice for this situation. Rawlings Adventures with Sten Bay 7 / 32

18 Constraints and unreachable setpoints x s x s = Gu s + d s d s G 0 u s 1 x sp 0 d s G 0 1 u s d s 0 As the estimated disturbance changes with time, the setpoint may change between reachable and unreachable. Rawlings Adventures with Sten Bay 8 / 32

19 What closed-loop behavior is desirable? Fast tracking x(0) Q R (fast tracking) x sp x x x(0) k Rawlings Adventures with Sten Bay 9 / 32

20 What closed-loop behavior is desirable? Slow tracking x(0) Q R (slow tracking) x sp x x x(0) k Rawlings Adventures with Sten Bay 10 / 32

21 What closed-loop behavior is desirable? Asymmetric tracking x(0) Q R (fast tracking) x sp x x x(0) k Rawlings Adventures with Sten Bay 11 / 32

22 Why analysis? Unexpected closed-loop behavior A finite horizon objective function may not give a stable controller! Rawlings Adventures with Sten Bay 12 / 32

23 Why analysis? Unexpected closed-loop behavior A finite horizon objective function may not give a stable controller! How is this possible? Rawlings Adventures with Sten Bay 12 / 32

24 Why analysis? Unexpected closed-loop behavior A finite horizon objective function may not give a stable controller! How is this possible? x 2 0 k x 1 Rawlings Adventures with Sten Bay 12 / 32

25 Why analysis? Unexpected closed-loop behavior A finite horizon objective function may not give a stable controller! How is this possible? x 2 0 k + 1 k x 1 Rawlings Adventures with Sten Bay 12 / 32

26 Why analysis? Unexpected closed-loop behavior A finite horizon objective function may not give a stable controller! How is this possible? x 2 0 k + 2 k + 1 k x 1 Rawlings Adventures with Sten Bay 12 / 32

27 Why analysis? Unexpected closed-loop behavior A finite horizon objective function may not give a stable controller! How is this possible? x 2 0 closed-loop trajectory k + 2 k + 1 k x 1 Rawlings Adventures with Sten Bay 12 / 32

28 Unreachable case challenges for analyzing closed-loop behavior Sequence of optimal costs is not monotone decreasing Rawlings Adventures with Sten Bay 13 / 32

29 Unreachable case challenges for analyzing closed-loop behavior Sequence of optimal costs is not monotone decreasing Infinite horizon cost is unbounded for all input sequences Rawlings Adventures with Sten Bay 13 / 32

30 Unreachable case challenges for analyzing closed-loop behavior Sequence of optimal costs is not monotone decreasing Infinite horizon cost is unbounded for all input sequences Optimal cost is not a Lyapunov function for the closed-loop system Rawlings Adventures with Sten Bay 13 / 32

31 Unreachable case challenges for analyzing closed-loop behavior Sequence of optimal costs is not monotone decreasing Infinite horizon cost is unbounded for all input sequences Optimal cost is not a Lyapunov function for the closed-loop system Standard nominal MPC stability arguments do not apply Rawlings Adventures with Sten Bay 13 / 32

32 Unreachable case challenges for analyzing closed-loop behavior Sequence of optimal costs is not monotone decreasing Infinite horizon cost is unbounded for all input sequences Optimal cost is not a Lyapunov function for the closed-loop system Standard nominal MPC stability arguments do not apply Simulations indicate the closed loop is stable Rawlings Adventures with Sten Bay 13 / 32

33 Unreachable case challenges for analyzing closed-loop behavior Sequence of optimal costs is not monotone decreasing Infinite horizon cost is unbounded for all input sequences Optimal cost is not a Lyapunov function for the closed-loop system Standard nominal MPC stability arguments do not apply Simulations indicate the closed loop is stable How can we be sure? Rawlings Adventures with Sten Bay 13 / 32

34 Unreachable case theoretical result Theorem (Asymptotic Stability of Terminal Constraint MPC) The optimal steady state is the asymptotically stable solution of the closed-loop system under terminal constraint MPC. Its region of attraction is the steerable set. Rawlings Adventures with Sten Bay 14 / 32

35 Some funny s during the long journey From: "James B. Rawlings" To: John Bagterp Date: Wed, 2 Aug :10: (CDT) Rawlings Adventures with Sten Bay 15 / 32

36 Some funny s during the long journey From: "James B. Rawlings" To: John Bagterp Date: Wed, 2 Aug :10: (CDT) p.s. If Sten doesn t get excited about the approach pretty soon, you should check his food; maybe he s on some medication. Rawlings Adventures with Sten Bay 15 / 32

37 Dennis comes to Madison Let no one ignorant of geometry enter Engraved over the door of Plato s Academy Rawlings Adventures with Sten Bay 16 / 32

38 Dennis takes a crack On 5-Aug-2002, Dennis Bonne <db@olivia.kt.dtu.dk> wrote: Rawlings Adventures with Sten Bay 17 / 32

39 Dennis takes a crack On 5-Aug-2002, Dennis Bonne <db@olivia.kt.dtu.dk> wrote: Now, I am 90% convinced that the "no movement" property is a consequence of the saddle point geometry of the Hamilton problem, BUT I fail to see why...?..or at least how to utilize it in a proof. Rawlings Adventures with Sten Bay 17 / 32

40 Dennis takes a crack On 5-Aug-2002, Dennis Bonne <db@olivia.kt.dtu.dk> wrote: Now, I am 90% convinced that the "no movement" property is a consequence of the saddle point geometry of the Hamilton problem, BUT I fail to see why...?..or at least how to utilize it in a proof. It is very frustrating being so close and not realizing what is probably painstakingly obvious and simple... Rawlings Adventures with Sten Bay 17 / 32

41 Dennis takes a crack On 5-Aug-2002, Dennis Bonne <db@olivia.kt.dtu.dk> wrote: Now, I am 90% convinced that the "no movement" property is a consequence of the saddle point geometry of the Hamilton problem, BUT I fail to see why...?..or at least how to utilize it in a proof. It is very frustrating being so close and not realizing what is probably painstakingly obvious and simple... "No one ignorant of geometry shall"... come to Madison. Rawlings Adventures with Sten Bay 17 / 32

42 The end is finally in sight Subject: ieee article proofs From: "James B. Rawlings" To: John Bagterp Jorgensen Sten Bay Jorgensen Dennis Bonn Date: Tue, 16 Sep :45: Rawlings Adventures with Sten Bay 18 / 32

43 The end is finally in sight Subject: ieee article proofs From: "James B. Rawlings" To: John Bagterp Jorgensen Sten Bay Jorgensen Dennis Bonn Date: Tue, 16 Sep :45: Guys, Gentle reminder: we don t want to let the grass grow under our feet on this one. Jim Rawlings Adventures with Sten Bay 18 / 32

44 All s well that ends well Subject: Re: forwarded message from jsinay@ieee.org From: sbj@kt.dtu.dk To: "James B. Rawlings" <rawlings@engr.wisc.edu> Date: Thu, 18 Sep :34: (CEST) Dear Jim, I do hope the mower is not out yet! Rawlings Adventures with Sten Bay 19 / 32

45 All s well that ends well Subject: Re: forwarded message from jsinay@ieee.org From: sbj@kt.dtu.dk To: "James B. Rawlings" <rawlings@engr.wisc.edu> Date: Thu, 18 Sep :34: (CEST) Dear Jim, I do hope the mower is not out yet! This is indeed a fine paper! Thanks for you stamina in making this come through! Sincerely Sten Rawlings Adventures with Sten Bay 19 / 32

46 Example 1. Single input single output system G(s) = s s + 1 Sample time T = 10 sec Input constraint, 1 u 1 Setpoint y sp = 0.25 Q y = 10, R = 0, S = 1, Q = C Q y C I 2 Horizon length N = 80 Periodic state disturbance d x = [ ] which is estimated from the measurements Rawlings Adventures with Sten Bay 20 / 32

47 Disturbance estimation As the estimated disturbance changes with time, the setpoint changes between reachable and unreachable. x s d s G x sp 0 d s G 0 0 u s 1 1 u s d s 0 Rawlings Adventures with Sten Bay 21 / 32

48 Disturbance estimation As the estimated disturbance changes with time, the setpoint changes between reachable and unreachable. x s d s G x sp x sp 0 d s G x (k) 0 0 u s 1 1 d s 0 u s 0 ˆd(k) k Rawlings Adventures with Sten Bay 21 / 32

49 y Time (sec) setpoint target (y ) y(sp-mpc) y(targ-mpc) u Time (sec) target (u ) u(sp-mpc) u(targ-mpc) Rawlings Adventures with Sten Bay 22 / 32

50 Summary of Example 1 Performance targ-mpc sp-mpc (index)% Measure V u V y V Rawlings Adventures with Sten Bay 23 / 32

51 Optimizing economics: Current industrial practice Planning and Scheduling Steady State Optimization Validation Model Update Reconciliation Two layer structure Drawbacks Controller Plant Rawlings Adventures with Sten Bay 24 / 32

52 Optimizing economics: Current industrial practice Planning and Scheduling Steady State Optimization Validation Controller Plant Model Update Reconciliation Two layer structure Drawbacks Inconsistent models Re-identify linear model as setpoint changes Time scale separation may not hold Economics unavailable in dynamic layer Rawlings Adventures with Sten Bay 24 / 32

53 An economics controller Profit Input (u) State (x) Rawlings Adventures with Sten Bay 25 / 32

54 An economics controller Profit Input (u) State (x) Rawlings Adventures with Sten Bay 25 / 32

55 The second paper Rawlings Adventures with Sten Bay 26 / 32

56 Obtaining Q and R from Data Xf, Ff X D X 1 X 2 Condenser y 1 Model discretized with t k = k t: 2 3 d dt 6 4 X D. X B 7 5 {z } x(t)» y 1 (t k ) = y 2 = F (x(t), u(t) ) {z} X f,f f» 1 0 x(t 0 1 k ) X N 1 X B Reboiler Measurements are only X D, X B at the discretization times y 2 Rawlings Adventures with Sten Bay 27 / 32

57 Obtaining Q and R from Data X D X 1 X 2 Condenser y 1 Model discretized with t k = k t: x k+1 = f (x k, u k ) + g(x k, u k )w k [ ] [ ] y y 2 = x 0 1 k k Xf, Ff w X N 1 X B Reboiler Measurements are only X D, X B at the discretization times Noise w k affects all the states y 2 Rawlings Adventures with Sten Bay 27 / 32

58 Obtaining Q and R from Data X D X 1 X 2 Condenser v 1 y 1 Model discretized with t k = k t: x k+1 = f (x k, u k ) + g(x k, u k )w k [ ] [ ] [ ] y v 1 y 2 = x 0 1 k + v 2 k k Xf, Ff w v 2 y 2 X N 1 X B Reboiler Measurements are only X D, X B at the discretization times Noise w k affects all the states Noise v k corrupts the measurements Rawlings Adventures with Sten Bay 27 / 32

59 Motivation for Using Autocovariances Xf, Ff w X D X 1 X 2 X N 1 X B Condenser v 1 y 1 Idea of Autocovariances The state noise w k gets propagated in time The measurement noise v k appears only at the sampling times and is not propagated in time Taking autocovariances of data at different time lags gives covariances of w k and v k v 2 Reboiler y 2 Rawlings Adventures with Sten Bay 28 / 32

60 Motivation for Using Autocovariances Xf, Ff w X D X 1 X 2 X N 1 X B Condenser v 1 y 1 Idea of Autocovariances The state noise w k gets propagated in time The measurement noise v k appears only at the sampling times and is not propagated in time Taking autocovariances of data at different time lags gives covariances of w k and v k v 2 y 2 Reboiler Let w k, v k have zero means and covariances Q and R Rawlings Adventures with Sten Bay 28 / 32

61 Mathematical Formulation of the ALS Linear State-Space Model: x k+1 = Ax k + Gw k w k N(0, Q) y k = Cx k + v k v k N(0, R) Model (A, C, G) known from the linearization, finite set of measurements: {y 0,..., y k } given. Only unknowns are noises w k and v k. Rawlings Adventures with Sten Bay 29 / 32

62 Mathematical Formulation of the ALS Linear State-Space Model: x k+1 = Ax k + Gw k w k N(0, Q) y k = Cx k + v k v k N(0, R) Model (A, C, G) known from the linearization, finite set of measurements: {y 0,..., y k } given. Only unknowns are noises w k and v k. y k = Cx k y k+1 = CAx k + CGw k y k+2 = CA 2 x k +CAGw k +CGw k+1 Rawlings Adventures with Sten Bay 29 / 32

63 Mathematical Formulation of the ALS Linear State-Space Model: x k+1 = Ax k + Gw k w k N(0, Q) y k = Cx k + v k v k N(0, R) Model (A, C, G) known from the linearization, finite set of measurements: {y 0,..., y k } given. Only unknowns are noises w k and v k. y k = Cx k + v k y k+1 = CAx k + CGw k + v k+1 y k+2 = CA 2 x k +CAGw k +CGw k+1 +v k+2 Rawlings Adventures with Sten Bay 29 / 32

64 Mathematical Formulation of the ALS Linear State-Space Model: x k+1 = Ax k + Gw k w k N(0, Q) y k = Cx k + v k v k N(0, R) Model (A, C, G) known from the linearization, finite set of measurements: {y 0,..., y k } given. Only unknowns are noises w k and v k. y k = Cx k + v k y k+1 = CAx k + CGw k + v k+1 E[y k y T k ] = R y k+2 = CA 2 x k +CAGw k +CGw k+1 +v k+2 Rawlings Adventures with Sten Bay 29 / 32

65 Mathematical Formulation of the ALS Linear State-Space Model: x k+1 = Ax k + Gw k w k N(0, Q) y k = Cx k + v k v k N(0, R) Model (A, C, G) known from the linearization, finite set of measurements: {y 0,..., y k } given. Only unknowns are noises w k and v k. y k = Cx k + v k y k+1 = CAx k + CGw k + v k+1 y k+2 = CA 2 x k +CAGw k +CGw k+1 +v k+2 E[y k y T k ] = R E[y k+2 y T k+1 ] = CAGQG T C T Rawlings Adventures with Sten Bay 29 / 32

66 The Autocovariance Least-Squares (ALS) Problem Skipping a lot of algebra, we can write: Autocovariance Least Squares [ ] Φ = min Q,R A (Q)s N ˆb (R) s 2 1 A least-squares problem in a vector of unknowns, Q, R Rawlings Adventures with Sten Bay 30 / 32

67 The Autocovariance Least-Squares (ALS) Problem Skipping a lot of algebra, we can write: Autocovariance Least Squares [ ] Φ = min Q,R A (Q)s N ˆb (R) s 2 1 A least-squares problem in a vector of unknowns, Q, R 2 Form A N from known system matrices Rawlings Adventures with Sten Bay 30 / 32

68 The Autocovariance Least-Squares (ALS) Problem Skipping a lot of algebra, we can write: Autocovariance Least Squares [ ] Φ = min Q,R A (Q)s N ˆb (R) s 2 1 A least-squares problem in a vector of unknowns, Q, R 2 Form A N from known system matrices 3 ˆb is a vector containing the estimated correlations from data ˆb = 1 y T k y T k T. k=1 y k+n 1 yk T s Rawlings Adventures with Sten Bay 30 / 32

69 State estimation using (Q, R) from ALS Efficient software developed for ALS Rawlings Adventures with Sten Bay 31 / 32

70 State estimation using (Q, R) from ALS Efficient software developed for ALS Data requirements for (Q, R) identification are reasonable Rawlings Adventures with Sten Bay 31 / 32

71 State estimation using (Q, R) from ALS Efficient software developed for ALS Data requirements for (Q, R) identification are reasonable Industrial testing underway Rawlings Adventures with Sten Bay 31 / 32

72 State estimation using (Q, R) from ALS Efficient software developed for ALS Data requirements for (Q, R) identification are reasonable Industrial testing underway Preliminary results from Eastman chemical cracking furnace and Aspentech evaluation are encouraging Rawlings Adventures with Sten Bay 31 / 32

73 Things that I learned from Sten A researcher needs to have good taste Rawlings Adventures with Sten Bay 32 / 32

74 Things that I learned from Sten A researcher needs to have good taste... and have faith Rawlings Adventures with Sten Bay 32 / 32

75 Things that I learned from Sten A researcher needs to have good taste... and have faith Choosing the right problems is the most important task in research Rawlings Adventures with Sten Bay 32 / 32

76 Things that I learned from Sten A researcher needs to have good taste... and have faith Choosing the right problems is the most important task in research The graduate students in Sten s group are very talented! Rawlings Adventures with Sten Bay 32 / 32

77 Things that I learned from Sten A researcher needs to have good taste... and have faith Choosing the right problems is the most important task in research The graduate students in Sten s group are very talented! The research climate at DTU and CAPEC is first rate Rawlings Adventures with Sten Bay 32 / 32