Initial Tuning of Predictive Controllers by Reverse Engineering

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1 Initial Tuning of Predictive Controllers by Reverse Engineering Edward Hartley Monday 11th October 17h-18h, ESAT.57, KU Leuven Cambridge University Engineering Department

2 Outline Introduction Design decisions Structure Partition of dynamics Cost function Examples Single axis attitude control system Robustness to a class of failures Subsystems sharing saturated actuators Avoiding artificial cross coupling Cross coupling Conclusions 2/47

3 Introduction Model predictive control Prediction model Objective function Constraints Disturbances Optimisation problem Optimisation algorithm Plant Online tasks State estimator 3/47

4 Introduction Model predictive control Prediction model Objective function Constraints Disturbances Optimisation problem Optimisation algorithm Plant Online tasks State estimator 3/47

5 Introduction Model predictive control Constraints Daisychaining Online reconfiguration 4/47

6 Introduction Model predictive control Constraints Computation Daisychaining Tuning Online reconfiguration K 4/47

7 Introduction Model predictive control Constraints Computation Daisychaining Tuning Online reconfiguration K 4/47

8 Introduction Model predictive control Constraints Computation Daisychaining Tuning Online reconfiguration K 4/47

9 Reverse engineering Basic principles K G K Original controller 5/47

10 yk uk + Σ ŷ k k 1 B Kf + + Σ ˆx k k + + Σ ˆxk+1 k A z 1 C D ˆx k k 1 Reverse engineering Basic principles K Σ + + G K c G K Obs Original controller Observer-based realisation (Which one?) 5/47

11 yk uk + Σ ŷ k k 1 B Kf + + Σ ˆx k k + + Σ ˆxk+1 k A z 1 C D ˆx k k 1 Reverse engineering Basic principles K Σ + + G K c G MPC G K Original controller Obs Observer-based realisation (Which one?) Obs MPC controller (Cost function?) 5/47

12 Outline Introduction Design decisions Structure Partition of dynamics Cost function Examples Single axis attitude control system Robustness to a class of failures Subsystems sharing saturated actuators Avoiding artificial cross coupling Cross coupling Conclusions 6/47

13 Outline Introduction Design decisions Structure Partition of dynamics Cost function Examples Single axis attitude control system Robustness to a class of failures Subsystems sharing saturated actuators Avoiding artificial cross coupling Cross coupling Conclusions 7/47

14 Reverse engineering: choice of structure Continuous or discrete? G(z), K (z) (un)discretise G(s), K (s) Re-cast Discrete time observer and feedback Inverse optimality Re-cast Continuous time observer and feedback Inverse optimality Discrete time LQR controller MPC Controller Discrete equivalent cost Sample Continuous time LQR controller 8/47

15 Reverse engineering: choice of structure Continuous or discrete? G(z), K Continuous (z) Time Route V Continuous time observer Re-cast V Added modes not visible (un)discretise XDiscrete Not antime exact match Xobserver Not minimising and the same cost feedback X No feedthrough term in observer Inverse optimality G(s), K (s) Re-cast Continuous time observer and feedback Inverse optimality Discrete time LQR controller MPC Controller Discrete equivalent cost Sample Continuous time LQR controller 1 1 J. M. Maciejowski. Reverse engineering existing controllers for MPC design. In IFAC Symposium on System Structure and Control, Iguassu Falls, Brazil, October /47

16 Reverse engineering: choice of structure Continuous or discrete? G(z), K (z) Equivalent Cost Route V Continuous time observer Re-cast X Added modes can be visible XDiscrete Not antime exact match Vobserver Better match and than before though X Minimising feedback equivalent cost Discrete time LQR controller (un)discretise X No feedthrough Inverse optimality term in observer MPC Controller Discrete equivalent cost G(s), K (s) Re-cast Continuous time observer and feedback Inverse optimality Sample Continuous time LQR controller 1 1 J. M. Maciejowski. Reverse engineering existing controllers for MPC design. In IFAC Symposium on System Structure and Control, Iguassu Falls, Brazil, October /47

17 Reverse engineering: choice of structure Continuous or discrete? G(z), K (z) Re-cast Discrete time observer and feedback Inverse optimality Discrete time LQR controller (un)discretise Discrete Time RouteG(s), K (s) V Discrete time observer Re-cast V Exact match to discretised system V Feedthrough sometimes possible Continuous time observer observer and X (But only in some cases) feedback V New error modes invisible Inverse optimality when unconstrained Continuous time MPC Controller LQR controller Sample 1 Discrete equivalent cost 1 E. N. Hartley and J. M. Maciejowski. Initial tuning of predictive controllers by reverse engineering. In Proceedings of the European Control Conference, pages , August /47

18 Reverse engineering: choice of structure Observer structure Predictor x(k k 1) A K f C B K f I u(k) y(k) X No feedthrough V ˆx(k k 1) available before time k Loop shifting or delay A B C Σ A B C A B C G(z) K (z) A K B K C K D K G(z) D K K (z) MPC A K B K C K or z 1 I G(z) K (z) MPC A K C K B K D K 9/47

19 Reverse engineering: choice of structure Observer structure Filter ^x(k k) Add a dipole A AK f C B AK f I K f C K f A B C G(z) u(k) y(k) V Can only reproduce controllers if K () = X ˆx(k k) available after time k inevitable delay A B C G(z) K (z) A K B K C K D K Wz Wz 1 I K (z) MPC A K C K B K D K 1/47

20 Reverse engineering: choice of structure Assertion that K () = for filter-form observer controller u(k) K c ^x(k k) A AK f C B AK f u(k) I K f C K f y(k) Figure: Observer-based controller (stand-alone) [ Ao C o B o D o ] [ A + BKc (I K = f C) AK f C K c K c K f C AK f + BK c K f K c K f ] K obs (z) = D o + C o (zi A o ) 1 B o K obs () = K c K f K c (I K f C)(I K f C) 1 (A + BK c ) 1 (A + BK c )K f = 11/47

21 Outline Introduction Design decisions Structure Partition of dynamics Cost function Examples Single axis attitude control system Robustness to a class of failures Subsystems sharing saturated actuators Avoiding artificial cross coupling Cross coupling Conclusions 12/47

22 Reverse engineering: partition of dynamics Observer based realisation Transformation Original controller is an observer of some function of the plant state 2 : x K = T ˆx Theory revisited by Alazard 3 for controllers of higher order than plant using Youla parameter and used as basis for cross standard form 4 for obtaining (unconstrained) H 2 and H inverse optimal controllers. We use similar ideas for the MPC implementation but we do not use the Youla parameter 2 D. J. Bender and R. A. Fowell. Computing the estimator controller form of a compensator. International Journal of Control, 41(6): , D. Alazard and P. Apkarian. Exact observer-based structures for arbitrary compensators. International Journal of Robust and Nonlinear Control, 9(2):11 118, February F. Delmond, D. Alazard, and C. Cumer. Cross standard form: a solution to improve a given controller with H2 or H-infinity specifications. International Journal of Control, 79(4): , April 26 13/47

23 Reverse engineering: partition of dynamics Disturbance model Integral action will be inherited even without modification If K exhibits integral action, the plant model should be augmented with a disturbance model anyway A B C A B B I C Otherwise state estimates will be heavily biased poor constraint handling Disturbance model can also be used to bring the plant up to order Angle (measured) Input 1 (Nm) Error on state 1 Error on state Time (s) Time (s).4.2 Input 2 (Nm) Angular Velocity (unmeasured) 15 x Time (s).1 Basic model Augmented model Time (s) Figure: States and inputs Time (sec) Time (sec) Error on state Basic model Augmented model Time (sec) Figure: Observer error 14/47

24 Reverse engineering: partition of dynamics Observer based realisation Predictor form gains K c = C K T K f = T B K Filter form gains K c = C K T + D K C K f = A 1 ( BD K T B K ) Multiple solutions for T (through non-symmetric Riccati equation) [ ] [ ] [ ] I A + BDK C BC T I Acl = (e.g.) A T cl = K B K C T = T + + T X is a right inverse of T Choice of T determines partition of closed loop poles between observer and plant A K 15/47

25 Reverse engineering: partition of dynamics Distribution of fixed closed loop poles Observer Feedback 16/47

26 Reverse engineering: partition of dynamics Distribution of original closed loop poles Distribution governed by choice of T Slowest observer mode should be as fast as possible Unlike the conventional controller, with constraints the state estimate is interpreted literally Poles from a dipole to get zeros at z = should go in the observer (More about this later) The usual stuff Uncontrollable plant modes must remain in state feedback Uncontrollable error modes must remain in observer dynamics Conjugate pairs must not be separated 17/47

27 Reverse engineering: partition of dynamics Distribution of fixed closed loop poles Solving for T Use method of invariant subspaces (Macfarlane-Potter method) Can use Schur decomposition rather than eigenvalue decomposition to get a set of nested subspaces 5 [ ] [ ] [ ] A + BDK C BC K U1 U1 = Λ B K C A K U 2 U 2 [ ] [ ] [ ] A + BDK C BC K I U1 B K C A K U 2U 1 = ΛU U 2 [ ] [ ] [ ] A + BDK C BC K I I B K C A K U 2U 1 = 1 U 2U 1 U 1ΛU A. J. Laub. A Schur method for solving algebraic Riccati equations. IEEE Transactions on Automatic Control, AC-24(6): , /47

28 Reverse engineering: partition of dynamics Distribution of fixed closed loop poles Solving for T [ A + BDK C B K C ] [ ] [ ] BC K I I A K U 2 U 1 = 1 U 2 U 1 U 1 ΛU Let T = U 2 U 1 1 and premultiply both sides by [ T I ], the non-symmetric Riccati equation is exactly obtained: [ ] [ ] [ ] A + BD T I K C BC K I = B K C A K T Closed loop poles therefore chosen by choice of invariant subspace: A + BD K C + BC K T = A + BK c = U 1 ΛU /47

29 Reverse engineering: partition of dynamics Allocation of free poles Free poles When K is of lower order than G, n n k poles in the observer do not correspond to the poles of the original closed loop system. When constraints are not imposed, these are in the nullspace of K c Observer T T T n k n n k Fixed by A cl Free Observer of higher order than K K is observer of Tx Error modes corresponding to T T x are not visible through K c but freely assignable through choice of T (i.e. how much of T is added) Due to piecewise affine nature of linearly constrained quadratic/linear MPC 6, they do affect the constrained solution 6 A. Bemporad, M. Morari, V. Dua, and E. N. Pistikopoulos. The explicit linear quadratic regulator for constrained systems. Automatica, 38:3 2, 22 2/47

30 Outline Introduction Design decisions Structure Partition of dynamics Cost function Examples Single axis attitude control system Robustness to a class of failures Subsystems sharing saturated actuators Avoiding artificial cross coupling Cross coupling Conclusions 21/47

31 Reverse engineering: cost function Quadratic MPC formulation Zero-Cost Stage Cost l(k) = (û(k) K c ˆx(k)) T R (û(k) K c ˆx(k)) = [ ˆx T (k) û T (k) ] [ Kc T RK c Kc T R RK c R ] [ ˆx(k) û(k) ] Quadratic Finite Horizon Cost ( N 1 ) J(m) = l(m + k) + x(m + N) T Px(m + N) k= Dynamics: x(k + 1) = Ax(k) + Bu(k) Constraints: M u u(k) c u, M x x(k) c x 22/47

32 Reverse engineering: cost function Quadratic MPC formulation Zero-Cost Stage Cost ( P = ) l(k) = (û(k) K c ˆx(k)) T R (û(k) K c ˆx(k)) = [ ˆx T (k) û T (k) ] [ Kc T RK c Kc T R RK c R ] [ ˆx(k) û(k) ] Quadratic Finite Horizon Cost J(m) = ( N 1 ) l(m + k) k= Dynamics: x(k + 1) = Ax(k) + Bu(k) Constraints: M u u(k) c u, M x x(k) c x 22/47

33 Reverse engineering: cost function Alternative MPC formulation State trajectory matching Input matching is a means to an end really want to match outputs not inputs Therefore penalise predicted deviation from closed loop with original controller J(m) = ˆQ ( x(1) A 1 fb x()) ˆQ ( x(2) A 2 fb x()). ˆQ ( x(n) A N fb x()) + ˆRu() ˆRu(1). ˆRu(N 1) 1 where A fb = (A + BK c ) ˆQ and ˆR chosen to prioritise trajectory error minimisation and then prioritise order in which actuators are used 23/47

34 Reverse engineering: cost function Another possible MPC formulation Finding ordinary LQR gains Q and R Sometimes if the K c obtained is in the domain of LQR control gains, it is possible to find Q, R (and P) that exactly reproduce it using an LMI formulation. 7 Not always possible Sometimes you might be lucky and be able to find standard LQR gains Q, R (and P) that approximately reproduce K c (and this may be good enough). 8 Alternatively can try varying Q(k) and R(k) matrices throughout prediction horizon to exactly match K c. 9 7 S. Boyd, L. E. Ghaoui, E. Feron, and V. Balakrishnan. Linear Matrix Inequalities in System and Control Theory. Society for Industrial and Applied Mathematics, E. N. Hartley. Model predictive control for spacecraft rendezvous. PhD thesis, University of Cambridge, 3 April 21 9 S. Di-Cairano and A. Bemporad. Model predictive controller matching: Can MPC enjoy small signal properties of my favorite linear controller? In Proceedings of the European Control Conference, pages , 29 24/47

35 Outline Introduction Design decisions Structure Partition of dynamics Cost function Examples Single axis attitude control system Robustness to a class of failures Subsystems sharing saturated actuators Avoiding artificial cross coupling Cross coupling Conclusions 25/47

36 Outline Introduction Design decisions Structure Partition of dynamics Cost function Examples Single axis attitude control system Robustness to a class of failures Subsystems sharing saturated actuators Avoiding artificial cross coupling Cross coupling Conclusions 26/47

37 Reverse engineering: example Attitude control Single axis satellite attitude control system (with extra torque pair) G (z) = K (z) K (z) Ts=.25 = Baseline classical controller taken from texbook on Spacecraft control 1 1 M. J. Sidi. Spacecraft dynamics and control: A practical engineering approach. Cambridge University Press, /47

38 Reverse engineering: example Attitude control Continuous or discrete? K (z) already discrete time Discrete time Observer structure? D K large Unit delay not tolerated Filter introduce dipole Augmented model? K (z) exhibits integral action Yes Fixed poles? Dipole around z = Max speed of observer poles, ensure pole near zero in observer Free poles? None Dipole brings up to order Do nothing 28/47

39 Reverse engineering: example (Quadratic cost unconstrained) Angle (measured) Time (s) Angular Velocity (unmeasured) 15 x Time (s).5.5 Input 1/Nm.1 Input 2/Nm Time (s) Time (s) 29/47

40 Reverse engineering: example (Quadratic cost constrained, one actuator) Angle (measured) Time (s) Angular Velocity (unmeasured) 15 x Time (s).5.5 Input 1/Nm.1 Input 2/Nm Time (s) Time (s) 29/47

41 Reverse engineering: example (Quadratic cost constrained, second actuator allowed) Angle (measured) Time (s) Angular Velocity (unmeasured) 15 x Time (s).5.5 Input 1/Nm.1 Input 2/Nm Time (s) Time (s) 29/47

42 Reverse engineering: example (Trajectory matching constrained, second actuator allowed) Angle (measured) Time (s) Angular Velocity (unmeasured) 15 x Time (s).5.5 Input 1/Nm.1 Input 2/Nm Time (s) Time (s) 29/47

43 ch observer-based realisation of the original controller has different robustness propties. By keeping the pole near the origin in the observer (or rather, by not placing it Reverse engineering: example Effects of parasitic delays the pure state feedback dynamics), the elements of K c are kept relatively small and us the open loop gain from u(k 1) to u(k) is kept relatively small. In doing this, the ssover frequency is kept relatively low, allowing a relatively large delay margin. G(z) Zero order hold K c when unconstrained MPC Obs Figure 5.1: Reverse engineered controller with parasitic delays and broken loop gure 5.11 shows Previously the Bodeasserted: plot for the if aopen dipole loopislinserted, u u for different its poles realisations must go of in the iginal controller observer (noting the introduction of the negative sign because K c is calculated stabilise when in positive feedback). The bode magnitude plot crosses the db 3/47

44 Reverse engineering: example Effects of parasitic delays Angle (measured) 15 x Angular Velocity (unmeas.) Time (s) Input 1 (Nm) Time (s) Time (s) Input 2 (Nm) Time (s) Figure: Constrained 31/47

45 Reverse engineering: example Effects of parasitic delays Angle (measured) 15 x Angular Velocity (unmeas.) Time (s) Input 1 (Nm) Time (s) Time (s) Input 2 (Nm) Time (s) Figure: Unconstrained (!!!) 31/47

46 Reverse engineering: example Effects of parasitic delays Magnitude (db) Phase (deg) Phase (deg) Bode Diagram Frequency (rad/sec) Realisation 1 Realisation 2 Bode Diagram Realisation 3 Realisation 1 Realisation 2 Realisation Realisation OBS FB FB.5653 OBS FB OBS.98 ±.1214i FB OBS OBS.9785 OBS OBS FB 1 FB FB FB Crossover freq (rad/s) Delay margin (steps) Frequency (rad/sec) Delay margin largest for 1st realisation Figure 5.11: Loop margins for observer-based realisations Robustness to actuator failures The previous example demonstrates how MPC can exploit the redundancy present 32/47

47 Outline Introduction Design decisions Structure Partition of dynamics Cost function Examples Single axis attitude control system Robustness to a class of failures Subsystems sharing saturated actuators Avoiding artificial cross coupling Cross coupling Conclusions 33/47

48 Reverse engineering: example Robustness to a class of failures An off failure for the first torque pair is simulated at t = 1 s..8.6 Angle (measured).15.1 Angular Velocity (unmeas.) K (z) Actual response Time (s).5 Input 1 (Nm) K (z) Time (s).5 Input 2 (Nm).5 Controller request Actual response Time (s) Time (s) The reverse-engineered MPC controller, when given knowledge of the second actuator exhibits a degree of robustness to failures. 34/47

49 Outline Introduction Design decisions Structure Partition of dynamics Cost function Examples Single axis attitude control system Robustness to a class of failures Subsystems sharing saturated actuators Avoiding artificial cross coupling Cross coupling Conclusions 35/47

50 Reverse engineering: example Subsystems sharing saturated actuators Reverse engineer QFT RDV controller from an ESA simulator Combine with a reverse engineered LQR controller Implement in a different simulator (i.e. slightly out of context) as joint attitude/translation controller. Prediction model for translational dynamics: Clohessy-Wiltshire equations of relative dynamics 11 Prediction mode for attitude dynamics: Small angle attitude model M. J. Sidi. Spacecraft dynamics and control: A practical engineering approach. Cambridge University Press, W. Fehse. Introduction to Automated Rendezvous and Docking of Spacecraft. Cambridge University Press, 23 36/47

51 CONTROL Reverse engineering: example Subsystems 2.1 Autonomous sharing saturatedrendezvous actuators and capture scenario In this dissertation we will focus primarily on the final controlled approach of the chaser craft towards the target from a distance of a few hundred metres. A chaser craft must be controlled in order to approach and capture a passive target in a circular orbit, 5 km above the surface of Mars (Figure 2.1). Target Chaser Surface of Mars Figure 2.1: Rendezvous and capture scenario (not to scale) The primary sensor used for relative navigation is LIDAR (ESA, 26). This measures the distance to the target, and the target s azimuth and elevation in relation to the LIDAR pointing direction. The primary actuators for position and attitude control are gas thrusters. The spacecraft also has reaction wheels for attitude control but these are not used in the final approach. 37/47

52 Reverse engineering: example Subsystems sharing saturated actuators Target Chaser x lof zlof y lof x lof (V ) y lof z lof (R) Passive target (b) Local coordinate system (a) Orbit 37/47

53 Reverse engineering: example Subsystems sharing saturated actuators 2.1 Autonomous rendezvous and capture scenario Visibility cone V No-thrust zone 7.67 cm Orbiter module Passive target LIDAR R Figure 2.5: LIDAR visibility cone and no-thrust zone (not to scale) 37/47 an appropriately conforming velocity and to take the overshoot when responding to

54 Reverse engineering: example Subsystems sharing saturated actuators Estimator Attitude setpoint + Low-level regulation Mission planning Feed-forward Actuator management Plant Trajectory planning Setpoint calculation + Low-level regulation + + Estimator 38/47

55 Reverse engineering: example Subsystems sharing saturated actuators 5 2 Position x 5 1 Position z Time Time 2 Angle x (degrees) 1 1 Angle z (degrees) Time Time Figure: Closed loop trajectory without constraints 39/47

56 Reverse engineering: example Subsystems sharing saturated actuators Position x Time.8 Position z Time.5 Angle x (degrees) Angle z (degrees) Time Time Figure: Closed loop trajectory with constraints 39/47

57 Outline Introduction Design decisions Structure Partition of dynamics Cost function Examples Single axis attitude control system Robustness to a class of failures Subsystems sharing saturated actuators Avoiding artificial cross coupling Cross coupling Conclusions 4/47

58 Outline Introduction Design decisions Structure Partition of dynamics Cost function Examples Single axis attitude control system Robustness to a class of failures Subsystems sharing saturated actuators Avoiding artificial cross coupling Cross coupling Conclusions 41/47

59 Avoiding artificial cross coupling What do I mean by cross coupling? G(z) = A 1 B A m B m C C m A K1 B K A K(z) = Km C K B Km C Km where A 1 = A 2 =... = A m, B 1 = B 2 =... = B m etc. Would expect: K c =. Kc1.. K cm K f = f 1. K.. K fm. Not always the case! In some cases the systems are coupled by the observer, and then decoupled again by the control gain 42/47

60 Avoiding artificial cross coupling Avoiding it! General rule: Ensure subspaces corresponding to a particular repeated eigenvalue are either included completely in [ ] U1 T U2 T T or not at all! 43/47

61 Outline Introduction Design decisions Structure Partition of dynamics Cost function Examples Single axis attitude control system Robustness to a class of failures Subsystems sharing saturated actuators Avoiding artificial cross coupling Cross coupling Conclusions 44/47

62 Conclusions Conclusions Unconstrained reverse engineered MPC closely matches an unconstrained existing controller Constraints can be added, as with any other MPC controller... but some care must be taken as to which observer-based realisation the MPC controller is based on The special properties of MPC are not lost by using this technique Reverse Engineering could prove a useful technique for initial tuning of a constrained MPC controller when there already exists an almost good enough linear controller if a designer wishes to start a new design by using classical techniques before exploiting constrained optimisation using MPC 45/47

63 Acknowledgements Prof. J. M. Maciejowski Dr. Samir Bennani Dr. Marc Dinh 46/47

64 The End D. Alazard and P. Apkarian. Exact observer-based structures for arbitrary compensators. International Journal of Robust and Nonlinear Control, 9(2):11 118, February A. Bemporad, M. Morari, V. Dua, and E. N. Pistikopoulos. The explicit linear quadratic regulator for constrained systems. Automatica, 38:3 2, 22. D. J. Bender and R. A. Fowell. Computing the estimator controller form of a compensator. International Journal of Control, 41(6): , S. Boyd, L. E. Ghaoui, E. Feron, and V. Balakrishnan. Linear Matrix Inequalities in System and Control Theory. Society for Industrial and Applied Mathematics, F. Delmond, D. Alazard, and C. Cumer. Cross standard form: a solution to improve a given controller with H2 or H-infinity specifications. International Journal of Control, 79(4): , April 26. S. Di-Cairano and A. Bemporad. Model predictive controller matching: Can MPC enjoy small signal properties of my favorite linear controller? In Proceedings of the European Control Conference, pages , 29. W. Fehse. Introduction to Automated Rendezvous and Docking of Spacecraft. Cambridge University Press, 23. E. N. Hartley. 47/47

DISTURBANCES MONITORING FROM CONTROLLER STATES

DISTURBANCES MONITORING FROM CONTROLLER STATES Daniel Alazard Pierre Apkarian SUPAERO, av. Edouard Belin, 3 Toulouse, France - Email : alazard@supaero.fr Mathmatiques pour l Industrie et la Physique, Université