Outline. Model Predictive Control: Current Status and Future Challenges. Separation of the control problem. Separation of the control problem

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1 Otline Model Predictive Control: Crrent Stats and Ftre Challenges James B. Rawlings Department of Chemical and Biological Engineering University of Wisconsin Madison UCLA Control Symposim May, 6 Overview MPC at the Small Scale MPC to replace PID 3 MPC at the Large Scale Large, networked systems 4 MPC and State Estimation 5 Conclsions Rawlings (UW) MPC: Stats and Ftre UCLA Control Symposim / 5 Rawlings (UW) MPC: Stats and Ftre UCLA Control Symposim / 5 Separation of the control problem Separation of the control problem inpt process otpt y Inpt/otpt description inpt process otpt y inpt R m state x R n otpt y R p process sensor State description estimate ˆx estimator Estimation problem inpt process reglator state x Reglation problem reglator inpt process estimator otpt y Control problem ˆx Rawlings (UW) MPC: Stats and Ftre UCLA Control Symposim 3 / 5 Rawlings (UW) MPC: Stats and Ftre UCLA Control Symposim 4 / 5

2 The Reglation Problem Models and constraints Linear dynamics and constraints Model and constraints Objective fnction 3 Feedback dx = Ax + B dt x j+ = Ax j + B j y = Cx y j = Cx j D d D j d Hx h Hx j h x Rawlings (UW) MPC: Stats and Ftre UCLA Control Symposim 5 / 5 Rawlings (UW) MPC: Stats and Ftre UCLA Control Symposim 6 / 5 Process models (cont.) Nonlinear dynamics and constraints Objective Fnction (t) dx dt = f (x, ) x j+ = f (x j, j ) y = g(x) y j = g(x j ) x(t) U x X j U x j X x k Past k Present k + k + Ftre Controller objective fnction t vale of control objective Φ(x, (t)) = L(x k, k ) k= L(x, ) = x Qx + R, qadratic measre common Rawlings (UW) MPC: Stats and Ftre UCLA Control Symposim 7 / 5 Rawlings (UW) MPC: Stats and Ftre UCLA Control Symposim 8 / 5

3 Feedback Feedback One techniqe for obtaining a feedback controller synthesis from knowledge of open-loop controllers is to measre the crrent control process state and then compte very rapidly for the open-loop control fnction. The first portion of this fnction is then sed dring a short time interval, after which a new measrement of the process state is made and a new open-loop control fnction is compted for this new measrement. The procedre is then repeated. k Past k Present k + k + Ftre x(t) t (t) vale of control objective Lee and Marks (967) Fondations of Optimal Control Theory min k Φ(x, ) Everything has been thoght of before, bt the problem is to think of it again. Goethe s.t. x k+ = Ax k + B k x = ˆx k x k X k U Rawlings (UW) MPC: Stats and Ftre UCLA Control Symposim 9 / 5 Rawlings (UW) MPC: Stats and Ftre UCLA Control Symposim / 5 Unexpected closed-loop behavior A finite horizon objective fnction may not even stabilize! How is this possible? Terminal constraint soltion Adding a terminal constraint ensres stability May case infeasibility Open-loop predictions not eqal to closed-loop behavior x x Φ k+ Φ k+ L(x k+, k+ ) closed-loop trajectory k + k + k k + k + k Φ k+ Φ k L(x k, k ) Rawlings (UW) MPC: Stats and Ftre UCLA Control Symposim / 5 Φ k Rawlings (UW) MPC: Stats and Ftre UCLA Control Symposim / 5

4 Infinite horizon soltion The infinite horizon ensres stability Open-loop predictions eqal to closed-loop behavior May be difficlt to implement Fll Enmeration Unconstrained soltion: LQ reglator (?) = Kx x Φ k+ = Φ k+ L(x k+, k+ ) Constrained soltion: MPC = K i x + b i k + k + k Φ k+ = Φ k L(x k, k ) Φ k Rawlings (UW) MPC: Stats and Ftre UCLA Control Symposim 3 / 5 in which i enmerates different possible active sets for the ineqality constraints (?) There are 3 mn different active sets... m k k =,..., N Rawlings (UW) MPC: Stats and Ftre UCLA Control Symposim 4 / 5 The active set table Example First order pls time delay = K i x + b i N = N = 4 i constraint set K i b i {, } {, } 3 {, } 4 {, } K 4 b 4 5 {, } K 5 b 5 6 {, } K 6 b 6 7 {, } 8 {, } 9 {, } i constraint set K i b i {,,, } {,,, } 3 {,,, } 4 {,,, } K 4 b 4 4 {,,, } K 4 b 4 4 {,,, } K 4 b 4 79 {,,, } 8 {,,, } 8 {,,, } The first example is a first order pls time delay (FOPTD) system (?) G (s) = e s s + sampled with T s =.5 The inpt is assmed to be constrained.5 The control horizon is N = 4 Rawlings (UW) MPC: Stats and Ftre UCLA Control Symposim 5 / 5 Rawlings (UW) MPC: Stats and Ftre UCLA Control Symposim 6 / 5

5 Setpoint change and load distrbances FOPTD system: nominal case In all simlations the setpoint is changed from to at time zero At time 5 a load distrbance passing throgh the same dynamics as the plant of magnitde.5 enters the system At time 5 the distrbance magnitde becomes (which makes the setpoint nreachable) Finally at time 75 the distrbance magnitde becomes.5 again. Controlled variable CLQ CLQ PID PID Setpoint 8 Maniplated variable CLQ CLQ PID PID 8 Rawlings (UW) MPC: Stats and Ftre UCLA Control Symposim 7 / 5 Rawlings (UW) MPC: Stats and Ftre UCLA Control Symposim 8 / 5 FOPTD system: noisy case FOPTD system: effect of plant/model mismatch. Controlled variable CLQ PID Setpoint 8 Maniplated variable CLQ PID 8 Performance index (gain mismatch) - CLQ (gain) PID (gain) CLQ (delay) PID (delay) Relative mismatch Performance index (delay mismatch) Rawlings (UW) MPC: Stats and Ftre UCLA Control Symposim 9 / 5 Rawlings (UW) MPC: Stats and Ftre UCLA Control Symposim / 5

6 Comptation time for (complete) enmeration Electrical power distribtion The comptational brden of CLQ is comparable to that of PID. The CPU is a.7 GHz Athlon PC rnning Octave average maximm CPU time (ms) CPU time (ms) PID.5. CLQ..55 Rawlings (UW) MPC: Stats and Ftre UCLA Control Symposim / 5 Rawlings (UW) MPC: Stats and Ftre UCLA Control Symposim / 5 Chemical plant integration MPC at the Large Scale Material flow Most large-scale systems consist of networks of interconnected/interacting sbsystems Chemical plants, electrical power grids, water distribtion networks,... Energy flow Traditional approach: Decentralized control Wealth of literatre from the early 97 s on improved decentralized control (???) Well-known that poor performance may reslt if the interconnections are not negligible Rawlings (UW) MPC: Stats and Ftre UCLA Control Symposim 3 / 5 Rawlings (UW) MPC: Stats and Ftre UCLA Control Symposim 4 / 5

7 MPC at the Large Scale Nomenclatre: Consider Two Interacting Units Steady increase in available comptational power has provided the opportnity for centralized control Most practitioners view centralized control of large, networked systems as impractical and nrealistic Centralized control law grows exponentially with system size Difficlt to tailor a centralized controller to meet operational objectives A divide and conqer strategy is essential for control of large, networked systems (?) Centralized control: A benchmark control framework for comparing and assessing other control formlations Objective fnctions Φ (, ), Φ (, ) and Φ(, ) = w Φ (, ) + w Φ (, ) decision variables for nits Ω, Ω Decentralized Control Commnication-based Control (Nash eqilibrim) Cooperation-based Control (Pareto optimal) Centralized Control (Pareto optimal) min Φ ( ) Ω min Φ (, ) Ω min Φ(, ) Ω min Φ ( ) Ω min Φ (, ) Ω min Φ(, ) Ω min Φ(, ), Ω Ω Rawlings (UW) MPC: Stats and Ftre UCLA Control Symposim 5 / 5 Rawlings (UW) MPC: Stats and Ftre UCLA Control Symposim 6 / 5 Noninteracting systems Weakly interacting systems - - b Φ () n, d, p Φ () - - a Φ () b Φ () p a n, d Rawlings (UW) MPC: Stats and Ftre UCLA Control Symposim 7 / 5 Rawlings (UW) MPC: Stats and Ftre UCLA Control Symposim 8 / 5

8 Moderately interacting systems Strongly interacting (conflicting) systems b Φ () p Φ () n a d Φ () b p Φ () a d Rawlings (UW) MPC: Stats and Ftre UCLA Control Symposim 9 / 5 Rawlings (UW) MPC: Stats and Ftre UCLA Control Symposim 3 / 5 Strongly interacting (conflicting) systems Application chemical plant 6 4 n D, x Ad, x Bd MPC MPC MPC 3 F prge F, x A F, x A Φ () Φ () H r A B B C F r, x Ar, x Br Hm A B B C F m, x Am, x Bm H b Q F b, x Ab, x Bb, T (?), (?) Rawlings (UW) MPC: Stats and Ftre UCLA Control Symposim 3 / 5 Rawlings (UW) MPC: Stats and Ftre UCLA Control Symposim 3 / 5

9 Two Reactor Chain with Nonadiabatic Flash Performance of different MPC frameworks The conditional density fnction H m setpoint Cent-MPC Comm-MPC FC-MPC ( iterate) H b setpoint Cent-MPC Comm-MPC FC-MPC ( iterate) For the linear, time invariant model with Gassian noise, x(k + ) = Ax + B + Gw y = Cx + v w N(, Q) v N(, R) x() N(x, Q ).4... We can compte the conditional density fnction exactly F Cent-MPC Comm-MPC FC-MPC ( iterate) D Cent-MPC Comm-MPC FC-MPC ( iterate) p x Y (x Y (k )) = N( x, P ) p x Y (x Y (k)) = N( x, P) (before y(k)) (after y(k)) Rawlings (UW) MPC: Stats and Ftre UCLA Control Symposim 33 / 5 Rawlings (UW) MPC: Stats and Ftre UCLA Control Symposim 34 / 5 Large R, ignore the measrement, trst the forecast Medim R, blend the measrement and the forecast 8 x() 8 x() x 6 4 x() y() x 6 4 x() y() Rawlings (UW) MPC: Stats and Ftre UCLA Control Symposim 35 / 5 Rawlings (UW) MPC: Stats and Ftre UCLA Control Symposim 36 / 5

10 Small R, trst the measrement, override the forecast Large R, y measres only 8 x() 8 x() x 6 4 x() y() x 6 4 x() y() Rawlings (UW) MPC: Stats and Ftre UCLA Control Symposim 37 / 5 Rawlings (UW) MPC: Stats and Ftre UCLA Control Symposim 38 / 5 Medim R, y measres only Small R, y measres only 8 x() 8 x() x 6 4 x() y() x 6 4 x() y() Rawlings (UW) MPC: Stats and Ftre UCLA Control Symposim 39 / 5 Rawlings (UW) MPC: Stats and Ftre UCLA Control Symposim 4 / 5

11 The challenge of nonlinear estimation Fll information estimate of trajectory Linear Estimation Nonlinear Estimation The trajectory of states X (T ) := {x(),... x(t )} Estimation Possibilities: one state is the optimal estimate infinitely many states are optimal estimates (nobservable) Estimation Possibilities: one state is the optimal estimate infinitely many states are optimal estimates (nobservable) 3 finitely many states are locally optimal estimates Maximizing the conditional density fnction max p X Y (X (T ) Y (T )) X (T ) Rawlings (UW) MPC: Stats and Ftre UCLA Control Symposim 4 / 5 Rawlings (UW) MPC: Stats and Ftre UCLA Control Symposim 4 / 5 Eqivalent optimization problem Arrival cost and moving horizon estimation Using the model and taking logarithms T min V (x ) + L w (w j ) + X (T ) j= T L v (y j h(x j )) j= Most recent N states X (T N : T ) := {x T N... x T } Optimization problem sbject to x(j + ) = F (x, ) + w min V T N(x T N ) X (T N:T ) }{{} arrival cost + T j=t N L w (w j ) + T j=t N L v (y j h(x j )) L w (w) := log(p w (w)) V (x) := log(p x (x)) L v (v) := log(p v (v)) sbject to x(j + ) = F (x, ) + w. Rawlings (UW) MPC: Stats and Ftre UCLA Control Symposim 43 / 5 Rawlings (UW) MPC: Stats and Ftre UCLA Control Symposim 44 / 5

12 Arrival cost approximation Seqential Monte Carlo Sampling The statistically correct choice for the arrival cost is the conditional density of x T N Y (T N ) V T N (x) = log p xt N Y (x Y (T N )) Arrival cost approximations (?) niform prior (and large N) EKF covariance formla MHE smoothing Represent distribtion at time k via N samples (or particles), and weights, Particles, x i (T ), i =,..., N Weights, q i (T ), i =,..., N Any moment can be approximated as, q i (T ) E(f (x(t ))) N i q i (T )f (x i (T )) Point estimate, Highest Posterior Density regions, etc. may be compted from {x i (T ), q i (T )} Captring system dynamics and measrements reqires efficient algorithm for propagating particles and weights over time, {x i (T ), q i (T )} {x i (T ), q i (T )} Combine Seqential Monte Carlo Sampling with Importance Sampling x i (T ) Rawlings (UW) MPC: Stats and Ftre UCLA Control Symposim 45 / 5 Rawlings (UW) MPC: Stats and Ftre UCLA Control Symposim 46 / 5 Particle Filtering Methodology Conclsions A convenient importance fnction is π(x i (T ) x i (T ), y(t )) = p(x(t ) x i (T )) x i (T ) Posterior at T p(x(t ) Y (T )) State Eqation x k = f (x k, ω k ) x i (T ) q i (T ) x i (T ) x i (T ) Posterior at T p(x(t ) Y (T )) MPC is finding new application on small-scale, fast loops as well as large-scale, networked systems. State estimation is an integral component of MPC and remains a crrent research challenge. MHE and particle filtering are high-qality soltions for nonlinear models. They reqire more ser experience to set p properly and more comptational resorces to execte. The payoff can be sbstantial, however. Measrement Eqation y(t ) = h(x(t ), ν(t ) y(t ) Rawlings (UW) MPC: Stats and Ftre UCLA Control Symposim 47 / 5 Rawlings (UW) MPC: Stats and Ftre UCLA Control Symposim 48 / 5

13 Ftre Challenges Acknowledgments MPC of large-scale systems Develop identification methods for minimal modeling of the nit interactions. Exciting applications in many fields! State estimation in MPC Process systems are typically nobservable or ill-conditioned, i.e. nearby measrements do not imply nearby states. We mst decide on the sbset of states to reconstrct from the data an additional part to the modeling qestion. Nonlinear systems prodce mlti-modal densities. We need better soltions for handling these mlti-modal densities in real time. Aswin Venkat, Mrali Rajamani John Eaton for table coding advice Bhavik Bakshi, Ohio State Tom Badgwell, Aspentech Financial spport from NSF grant #CTS and Texas Wisconsin Modeling and Control Consortim (TWMCC) members Rawlings (UW) MPC: Stats and Ftre UCLA Control Symposim 49 / 5 Rawlings (UW) MPC: Stats and Ftre UCLA Control Symposim 5 / 5 Frther Reading I Rawlings (UW) MPC: Stats and Ftre UCLA Control Symposim 5 / 5