Research Topics (Baotic, Bemporad, Borrelli, Ferrari-Trecate, Geyer, Grieder, Mignone, Torrisi, Morari)

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1 Research Topics (Baotic, Bemporad, Borrelli, Ferrari-Trecate, Geyer, Grieder, Mignone, Torrisi, Morari) Analysis Reachability/Verification Stability Observability Synthesis Control (MPC) Explicit PWA MPC controllers Control with performance (LMI) State estimation (MHE)/fault detection MLD/PWA Hybrid Systems Modeling Computational Issues HYSDEL Identification Mixed Integer Programming Polyhedral computation Multiparametric Programming

2 Observability (Sontag, 1996) Sets of admissible initial states X (0) and inputs U The MLD system is observable in T steps on X(0) and if there exists a scalar w >0 such that, { u(t) } Tà1 t=0 min P Tà1 t=0 ky 1 (t) à y 2 (t) k à wkx 1,0 à x 2,0 k 1 õ 0 U w.r.t. x 1,0,x 2,0 X(0) equations + constraints. and u(t) U and subj. to the MLD Observability is undecidable (Sontag, 1996)

3 Practical Observability min P Tà1 t=0 ky 1 (t) à y 2 (t) k à wkx 1,0 à x 2,0 k 1 õ 0 1. w>0 is a sensitivity indicator Require w >w min 2. is an observability index Require T ô T max T min 1+2 Practical observability Practical observability is decidable

4 Observability of HS is Complex PWA system with unobservable components may be locally observable. ô x 1 x 2 õ ( t + 1 ) = y ( t ) = ú ô õ 1 0 ô 1 1 õ ô ô x 1 x 2 x 1 x 2 x 1 ( t ) if x 1 ( t ) > x 2 ( t ) x 2 ( t ) if x 1 ( t ) ô x 2 ( t ) õ õ ( t ) if x 1 ( t ) > x 2 ( t ) ( t ) if x 1 ( t ) ô x 2 ( t ) X(0) ú Sector 1 Sector 2 is observable X(0) ú Sector 3 Sector 4 is unobservable : x (t +1)à x (t) normalized vector field

5 Controllability of HS is Complex For a given PWA system and an initial state x 0, the problem of determining if there exists a control that drives x 0 to the origin is undecidable. For a given PWA system and an initial state x 0, the problem of determining if there exists a control that drives x 0 to the origin in at most k step is NP-complete

6 Research Topics (Baotic, Bemporad, Borrelli, Ferrari-Trecate, Geyer, Grieder, Mignone, Torrisi, Morari) Analysis Reachability/Verification Stability Observability Synthesis Control (MPC) Explicit PWA MPC controllers Control with performance (LMI) State estimation (MHE)/fault detection MLD/PWA Hybrid Systems Modeling Computational Issues HYSDEL Identification Mixed Integer Programming Polyhedral computation Multiparametric Programming

7 Reachability Analysis/Verification X(0) X Z1 (0) Z 1 X Z2 (0) Z 2 Given: X ZL (0) Σ A hybrid system A set of initial conditions X(0) Target sets Z 1, Z 2,..., Z L (disjoint) Time horizon T max Problem: Z i X(0) t ô T max Z L Is reachable from in steps? If yes, from which subset X Zi (0) of X(0)? Disturbance/input sequences driving to. X Zi (0) Z i

8 Reachability Analysis x 1 X 0 t =1 t =4 t =6 t =2 t =3 t =5 t =4 C j C i x 2 Reach set is expressed as a polyhedron Switchings are detected using MIFT New intersections are described by union of hyper-rectangles Fathoming criteria

9 Applications Safety ( Z 1, Z 2,..., Z L = unsafe sets) Liveness ( Z 1 =set to be reached within a finite time) Robust Simulation Scheduling Performance Assessment of Model Predictive Control Z 1 Z 2 ( =invariant set around the origin, =set of infeasible states) Z 1 Stability ( = invariant set around the origin)

10 Research Topics (Baotic, Bemporad, Borrelli, Ferrari-Trecate, Geyer, Grieder, Mignone, Torrisi, Morari) Analysis Reachability/Verification Stability Observability Synthesis Control (MPC) Explicit PWA MPC controllers Control with performance (LMI) State estimation (MHE)/fault detection MLD/PWA Hybrid Systems Modeling Computational Issues HYSDEL Identification Mixed Integer Programming Polyhedral computation Multiparametric Programming

11 State Estimation Problem x (t +1)=Ax(t) y (t) =Cx(t) E 2 î(t) +E 3 z(t) ô E 4 x (t) +E 5 ô w c w l õ E 6 w(t) ô E 7 w(t) = (t) is aconstrained input noise, v(t) is the output noise Knowledge of the state x(t) + B 2 î(t) +B 3 z(t) + D 2 î(t) +D 3 z(t) Regulator design Fault detection + w(t) + v(t) (Bemporad, Mignone, Morari, ACC99) Problem: When the state is not measurable it has to be reconstructed from the output measurements y(k),k =0,...,T

12 Moving Horizon Estimation (MHE) (Michalska, Mayne, Morari, Muske, Rao, Rawlings...) M>0 : fixed time horizon Time T à 1 y(t à M),...,y(T à 1) are available 1) Solve Θ ã T =min x(tàm),w P Tà1 k=tàm subj. to MLD dynamics + constraints kw(k)k 2 Q + kv(k)k 2 R + Γ TàM ( x(t à M) ) and compute the estimates xê(t à M T),..., xê(t T) Initial penalty (summarizing the neglected data) 2) At time T collect y(t), shift the data window and cycle...

13 Two major results: MHE: Conclusions Theory: Sufficient conditions for the convergence of MHE for hybrid systems in the MLD form Practice: Algorithm for computing quadratic initial penalties that are upper bounded by the approximate arrival cost. - Implementation of MHE as MIQPs Future research: - Noise rejection property - Application of MHE to fault detection in a noisy environment - Probabilistic analysis of MHE

14 Research Topics (Baotic, Bemporad, Borrelli, Ferrari-Trecate, Geyer, Grieder, Mignone, Torrisi, Morari) Analysis Reachability/Verification Stability Observability Synthesis Control (MPC) Explicit PWA MPC controllers Control with performance (LMI) State estimation (MHE)/fault detection MLD/PWA Hybrid Systems Modeling Computational Issues HYSDEL Identification Mixed Integer Programming Polyhedral computation Multiparametric Programming

15 Model orders n a, n b fixed X Identification Dataset: S = {(x(k),y(k)), k =1,...,N} models input/output constraints Examples: ë ô u(k) ô ì y(k +1)à y(k) ôí The shape of X is known The number s of submodels is known X i If the regions are known the identification problem amounts to the identification of ARX models The switching law is assumed unknown: Both the submodels and the shape of the regions must be estimated from the dataset s

16 Part Two Hybrid Systems Hybrid systems and MLD Models Research overview Hysdel Observability/Controllability Reachability/Verification* State estimation Identification Extension 4: Constrained optimal control of Hybrid/PWA Systems State feedback solution of COC Characterization of the solution Computation of the solution Examples

17 Extension 4: Hybrid/PWA Systems min U Px(N) p + P k=0 Nà1 Qx(k) p + Ru(k) p subj.to x(t +1)=A i x(t)+b i u(t)+f i if [x(t),u(t)] X i, i =1,...,s Ex(k)+Lu(k) ô M, k =0,...,Nà 1 x(n) X f x(k) R n, u(k) R m, U,{u(0),u(1),...,u(N à 1)} u * (0) is used in a Receding Horizon fashion for infinite time control Feedback control law?

18 Translation into Mixed Integer Program min U N J(UN,x(0)), kpx(n)k p + P k=0 Nà1 kqx(k)kp + kru(k)k p x(n) X f x(k +1)=Ax(k)+B subj. to 1 u(k)+b 2 î(k)+b 3 z(k) y(k) =Cx(k)+D 1 u(k)+d 2 î(k)+d 3 z(k) E 2 î(k) + E 3 z(k) ô E 1 u(k) + E 4 x(k) + E 5 Equivalent MLD representation of the PWA system Mixed Integer Program min ï ï T Hï +(f T + x(0) T F)ï s.t. Gï ô w + Fx(0)

19 Multiparametric Mixed Integer Program min ï ï T Hï +(f T + x(0) T F)ï is solved for all x(0) by using s.t. Gï ô w + Fx(0) p= 1/ Multiparametric mixed integer linear programming (H=0) p=2 Multiparametric mixed integer quadratic programming (H 0) to compute u * (x(0))=f PWA (x(0))

20 Part Two Hybrid Systems Hybrid systems and MLD Models Research overview Hysdel Observability/Controllability Reachability/Verification* State estimation Identification Extension 4: Constrained optimal control of Hybrid/PWA Systems State feedback solution of COC Characterization of the solution Computation of the solution Examples

21 Characterization of the Solution (p=2) The solution to the optimal control problem is a time varying PWA state feedback control law of the form (Sontag 1981, Mayne 2001) u ã (x(k)) = F k i x(k)+gk i if x(k) P k i,{x : x0 L k i (j)x + Mk i (j)x ô Kk i (j)} {P k i } N k i=1 partition of the set of feasible states x(k). X ã k

22 Characterization of the Solution: Proof x(t +1)=A i x(t) +B i u(t) +f i y(t) =C i x(t) +g i for ô x (t) u (t) õ X i Fix a switching sequence of length N to obtain constrained linear time variant system Solve finite time optimal control to obtain state-space polyhedral partition and corresponding PWA input and PWQ value function v={1,3,4,4} Obtain optimal control law by comparing value functions on polyhedron of "multiple feasibility"

23 Polyhedron of "multiple feasibility" v 1 ={1,2,3,4} v 2 ={1,2,3,3}

24 Polyhedron of "multiple feasibility" v 1 ={1,2,3,4} v 2 ={1,2,3,3} Polyhedron of multiple feasibility: switch v 1 and v 2 both admissible

25 Polyhedron of "multiple feasibility" v 1 ={1,2,3,4} v 2 ={1,2,3,3}

26 Polyhedron of "multiple feasibility" v 1 ={1,2,3,4} 1 v 2 ={1,2,3,3}

27 Characterization of the Solution (p=2) When is the Partition Polyhedral? If If U * * (x(0)) is is unique for for all all x(0), x(0), then then the the solution to to the the optimal control problem is is u ã (x(k)) = F k i x(k)+gk i if x(k) P k i,{x : Mk i (j)x ô Kk i (j)} {P k i } N k i=1 is is a partition of of the the set set of of feasible states x(k). x(k).

28 Polyhedral Partition: Proof Possible intersections of the value functions Case 1 Case 2 Case 3 Continuity of the PWA optimal control law excludes Case 3

29 Characterization of the Solution (p=1, ) The solution to the optimal control problem is a PWA state feedback control law of the form u ã (x(k)) = F k i x(k)+gk i if x(k) P k i,{x : Mk i (j)x ô Kk i (j)} {P k i } N k i=1 X ã k is a partition of the set of feasible states x(k).

30 Part Two Hybrid Systems Hybrid systems and MLD Models Research overview Hysdel Observability/Controllability Reachability/Verification* State estimation Identification Extension 4: Constrained optimal control of Hybrid/PWA Systems State feedback solution of COC Characterization of the solution Computation of the solution Examples

31 mp-miqp Solver based on (Borrelli, Baotic, Bemporad, Morari, 2002) Dynamic programming recursion Multiparametric quadratic program solver (mp-qp) Basic polyhedral manipulation (intersection and union) Special data structure for storing the solution

32 mp-milp Efficient algorithm based on branch and bound (Dua, Pistikopoulos, 1999) Successfully applied to a Traction Control problem Practical disadvantages of 1/ norms satisfactory performance only with long time-horizons performance may not depend smoothly on the weights in the performance index Cannot be extended to mp-miqp

33 Multiparametric Program Solvers mp-qp Bemporad, Morari, Dua, Dua, Pistikopoulos, Godwin, De De Dona, Tondel, Johansen, Bemporad, mp-lp Gal, Gal, Borrelli, Bemporad, Morari, mp-milp Dua, Dua, Pistikopoulos, mp-miqp Borrelli, Baotic, Bemporad, Morari, Dua, Dua, Pistikopoulos,

34 Optimal Control Problems: Summary Constrained Linear System Solution: u ã (k) =F k i x(k)+gk if x(k) D k i i where polyhedra D k, controller F ik, G ik are found from i mp-lp (linear performance index) mp-qp (quadratic performance index) Constrained PWA System Solution: u ã (k) =F k i x(k)+gk i if x(k) D k i Uncertain System Solution: D k i where sets, controller F ik, G ik are found from mp-milp (linear performance index) mp-miqp (quadratic performance index) u ã (k) =F k x(k) i +Gk if x(k) D k i i where polyhedra D k, controller F ik, G ik are found from i mp-lp (linear performance index)

35 Part Two Hybrid Systems Hybrid systems and MLD Models Research overview Hysdel Observability/Controllability Reachability/Verification* State estimation Identification Extension 4: Constrained optimal control of Hybrid/PWA Systems State feedback solution of COC Characterization of the solution Computation of the solution Examples

36 Applications Traction control (Ford Research Center ) Gas supply system (Kawasaki Steel ) Batch evaporator system (Esprit Project ) Anesthesia (Hospital Bern ) Hydroelectric power plant ( ) Power generation scheduling ( ) Integrated management of the power-train ( ) Gear shift operation on automotive vehicles ( )

37 Simple Example Furnaces

38 Alternate Heating of Two Furnaces (Hedlund and Rantzer, CDC1999) Objective: Control the temperature to a given set-point T 1, T 2 ô õ Tç 1 Tç 2 ô õ 1 0 ô õ B i = 0 ô 1 õ 0 0 ô = à 1 0 õô T1 0 à 2 T 2 if first furnace heated if second furnace heated if no heating õ + B i u 0 Constraints: Only three operation modes: 1- Heat only the first furnace 2- Heat only the second furnace 3- Do not heat any furnaces Amount of heating power is constant u 0

39 Alternate Heating of Two Furnaces MLD system [ 1 0 0] if first furnace heated u(t)= [ 0 1 0] if second furnace heated [ 0 0 1] if no heating mp-milp optimization problem minn oj(v 2,x(t)), P 2 kr(v(k +1)àv(k))k 0 k=0 + kq(x(k t) à x e )k v 2 0 Sampling time = 0.08 s State x(t) Input u(t) Aux. binary vector δ(t) Aux. continuous vector z(t) 2 variables 3 variables 0 variables 9variables to be solved in the region à 1 ô x 1 ô 1 à 1 ô x 2 ô 1 0 ô u 0 ô 1 parameterized! Computational complexity of mp-milp linear constraints 168 continuous variables 33 binary variables 9 parameters 3 time to solve the mp-milp 5min number of regions 105

40 u 0 =0.4 mp-milp Solution u 0 =0.8 No Heat No Heat T 2 Heat 1 T 2 Heat 1 Heat 2 Heat 2 T 1 T 1 X 2 X 2 Set point cannot be reached X 1 X 1 Set point is reached

41 Example Traction Control

42 Hybrid Control Design for Traction Control Francesco Borrelli Alberto Bemporad Manfred Morari Mike Fodor Davor Hrovat Mitch McConnell