Numerical Synthesis of Pontryagin Optimal Control Minimizers Using Sampling-Based Methods

Transcription

1 Numerical Synthesis of Pontryagin Optimal Control Minimizers Using Sampling-Based Methods Humberto Gonzalez Department of Electrical & Systems Engineering Washington University in St. Louis April 7th, 2017 Sampling-Based Optimal Control Synthesis ESE, Washington Univ. in St. Louis 1

2 Critical Point Characterization Synthesis of Optimal Control Inputs Simulation: Optimal HVAC Operation Optimal Control Applications x x1 4 5 x2 2 Path Planning (Vasudevan et al., 2013) Sampling-Based Optimal Control Synthesis ESE, Washington Univ. in St. Louis 2

3 Critical Point Characterization Synthesis of Optimal Control Inputs Simulation: Optimal HVAC Operation Optimal Control Applications x x1 4 5 x2 2 Dynamic Game Verification (Margellos & Lygeros, 2011) Sampling-Based Optimal Control Synthesis ESE, Washington Univ. in St. Louis 2

4 Critical Point Characterization Synthesis of Optimal Control Inputs Simulation: Optimal HVAC Operation Optimal Control Applications x x1 4 5 x2 2 Robust & Ensemble Control (Zlotnik & Li, 2012) Sampling-Based Optimal Control Synthesis ESE, Washington Univ. in St. Louis 2

5 General Optimal Control Formulation min u( ), x 0, T T 0 L ( x(s), u(s) ) ds + ϕ ( x(t ) ), subject to: ẋ(t) = f ( x(t), u(t) ), t [0, T ], x(0) = x 0, x(t) X, t [0, T ], u(t) U, t [0, T ], ( x0, x(t ) ) S. Sampling-Based Optimal Control Synthesis ESE, Washington Univ. in St. Louis 3

6 From Theory to Practice... Pros: Optimal control framework is flexible. Decades of accumulated literature. Particular cases can be efficiently solved. Cons: General numerical solvers are prone to converge to non-minimizers. Computation time can be very long, even for small problems. Sampling-Based Optimal Control Synthesis ESE, Washington Univ. in St. Louis 4

7 From Theory to Practice... Pros: Optimal control framework is flexible. Decades of accumulated literature. Particular cases can be efficiently solved. Cons: General numerical solvers are prone to converge to non-minimizers. Computation time can be very long, even for small problems. Sampling-Based Optimal Control Synthesis ESE, Washington Univ. in St. Louis 4

8 From Theory to Practice... Pros: Optimal control framework is flexible. Decades of accumulated literature. Particular cases can be efficiently solved. Cons: General numerical solvers are prone to converge to non-minimizers. Computation time can be very long, even for small problems. Sampling-Based Optimal Control Synthesis ESE, Washington Univ. in St. Louis 4

9 From Theory to Practice... Pros: Optimal control framework is flexible. Decades of accumulated literature. Particular cases can be efficiently solved. Cons: General numerical solvers are prone to converge to non-minimizers. Computation time can be very long, even for small problems. Sampling-Based Optimal Control Synthesis ESE, Washington Univ. in St. Louis 4

10 Summary of this Talk We developed a new class of numerical optimal control synthesis method. Trade-off between accuracy and efficiency can be easily configured. Aim to solve large-scale problems. Our algorithm avoids many undesirable stationary points. Enable real-time applications. R. He and H. Gonzalez, Numerical Synthesis of Pontryagin Optimal Control Minimizers Using Sampling-Based Methods,, arxiv: Sampling-Based Optimal Control Synthesis ESE, Washington Univ. in St. Louis 5

11 Summary of this Talk We developed a new class of numerical optimal control synthesis method. Trade-off between accuracy and efficiency can be easily configured. Aim to solve large-scale problems. Our algorithm avoids many undesirable stationary points. Enable real-time applications. R. He and H. Gonzalez, Numerical Synthesis of Pontryagin Optimal Control Minimizers Using Sampling-Based Methods,, arxiv: Sampling-Based Optimal Control Synthesis ESE, Washington Univ. in St. Louis 5

12 Summary of this Talk We developed a new class of numerical optimal control synthesis method. Trade-off between accuracy and efficiency can be easily configured. Aim to solve large-scale problems. Our algorithm avoids many undesirable stationary points. Enable real-time applications. R. He and H. Gonzalez, Numerical Synthesis of Pontryagin Optimal Control Minimizers Using Sampling-Based Methods,, arxiv: Sampling-Based Optimal Control Synthesis ESE, Washington Univ. in St. Louis 5

13 Summary of this Talk We developed a new class of numerical optimal control synthesis method. Trade-off between accuracy and efficiency can be easily configured. Aim to solve large-scale problems. Our algorithm avoids many undesirable stationary points. Enable real-time applications. R. He and H. Gonzalez, Numerical Synthesis of Pontryagin Optimal Control Minimizers Using Sampling-Based Methods,, arxiv: Sampling-Based Optimal Control Synthesis ESE, Washington Univ. in St. Louis 5

14 1. Critical Point Characterization 2. Synthesis of Optimal Control Inputs 3. Simulation: Optimal HVAC Operation Sampling-Based Optimal Control Synthesis ESE, Washington Univ. in St. Louis 6

15 Simplified Optimal Control Formulation Let us define the original optimal control problem: where: (P o ) min u( ) ϕ( x(t ) ), x 0 R n is given subject to: ẋ(t) = f ( x(t), u(t) ), t [0, T ], x(0) = x 0, U R m is compact and convex. u(t) U, t [0, T ]. E. Polak, Optimization: Algorithms and Consistent Approximations, ser. Applied Mathematical Sciences. Springer, 1997, Chapter Sampling-Based Optimal Control Synthesis ESE, Washington Univ. in St. Louis 7

16 Relaxed Optimal Control Formulation Now consider the relaxed optimal control problem: (P r ) min {µ t} t [0,T ] ϕ ( x(t ) ), subject to: ẋ(t) = R m f ( x(t), u ) dµ t (u), t [0, T ], x(0) = x 0, supp(µ t ) U, t [0, T ]. where, for each t [0, T ], µ t is a unitary Borel measure, i.e.: R m dµ t (u) = 1. J. Warga, Optimal Control of Differential and Functional Equations. Academic Press, 1972 Sampling-Based Optimal Control Synthesis ESE, Washington Univ. in St. Louis 8

17 Original-Relaxed Equivalence Given û(t): [0, T ] U, let µ t (u) = 1[u = û(t)] for each t. Then both original and relaxed trajectories generated by û and µ are equivalent. Thus, Feas(P o ) Feas(P r ), and Value(P o ) > Value(P r ). Proposition Value(P o ) = Value(P r ). The proof follows using Berkovitz s Chattering Lemma for switched systems and the fact that L 2 convergence of inputs implies uniform convergence of the trajectories. L. D. Berkovitz, Optimal Control Theory, ser. Applied Mathematical Sciences. Springer, 1974 Sampling-Based Optimal Control Synthesis ESE, Washington Univ. in St. Louis 9

18 Original-Relaxed Equivalence Given û(t): [0, T ] U, let µ t (u) = 1[u = û(t)] for each t. Then both original and relaxed trajectories generated by û and µ are equivalent. Thus, Feas(P o ) Feas(P r ), and Value(P o ) > Value(P r ). Proposition Value(P o ) = Value(P r ). The proof follows using Berkovitz s Chattering Lemma for switched systems and the fact that L 2 convergence of inputs implies uniform convergence of the trajectories. L. D. Berkovitz, Optimal Control Theory, ser. Applied Mathematical Sciences. Springer, 1974 Sampling-Based Optimal Control Synthesis ESE, Washington Univ. in St. Louis 9

19 Original-Relaxed Equivalence Given û(t): [0, T ] U, let µ t (u) = 1[u = û(t)] for each t. Then both original and relaxed trajectories generated by û and µ are equivalent. Thus, Feas(P o ) Feas(P r ), and Value(P o ) > Value(P r ). Proposition Value(P o ) = Value(P r ). The proof follows using Berkovitz s Chattering Lemma for switched systems and the fact that L 2 convergence of inputs implies uniform convergence of the trajectories. L. D. Berkovitz, Optimal Control Theory, ser. Applied Mathematical Sciences. Springer, 1974 Sampling-Based Optimal Control Synthesis ESE, Washington Univ. in St. Louis 9

20 Optimality Functions Definition (Polak, 1997) Consider the following optimization problem: min{ψ(x) x X }. We say that θ : X R is an optimality function of this problem if: θ(x) 0 for each x X ; and, if x is a minimizer of ψ, then θ(x) = 0. For example, if X R n then: θ(x) = min ψ(x), h, and θ(x) = min ψ(x), h + h 2 2, h 1 h are optimality functions. Sampling-Based Optimal Control Synthesis ESE, Washington Univ. in St. Louis 10

21 Optimality Functions Definition (Polak, 1997) Consider the following optimization problem: min{ψ(x) x X }. We say that θ : X R is an optimality function of this problem if: θ(x) 0 for each x X ; and, if x is a minimizer of ψ, then θ(x) = 0. For example, if X R n then: θ(x) = min ψ(x), h, and θ(x) = min ψ(x), h + h 2 2, h 1 h are optimality functions. Sampling-Based Optimal Control Synthesis ESE, Washington Univ. in St. Louis 10

22 Original Problem Optimality Functions Let u 0 : [0, T ] U and x 0 be its associated trajectory. Let: ϕ( θ o,l (x 0, u 0 ) = min x0 (T ) ) δx(t ), δu( ) x subject to: δẋ(t) = f ( x0 (t), u 0 (t) ) δx(t)+ x + f ( x0 (t), u 0 (t) ) δu(t), u δx(0) = 0, u 0 (t) + δu(t) U, t [0, T ]. We say that θ o,l is the linearized, or variational, optimality function of P o. Sampling-Based Optimal Control Synthesis ESE, Washington Univ. in St. Louis 11

23 Original Problem Optimality Functions Let u 0 : [0, T ] U and x 0 be its associated trajectory. Let: ϕ( θ o,l (x 0, u 0 ) = min x0 (T ) ) δx(t ), δu( ) x subject to: δẋ(t) = f ( x0 (t), u 0 (t) ) δx(t)+ x + f ( x0 (t), u 0 (t) ) δu(t), u δx(0) = 0, u 0 (t) + δu(t) U, t [0, T ]. We say that θ o,l is the linearized, or variational, optimality function of P o. Sampling-Based Optimal Control Synthesis ESE, Washington Univ. in St. Louis 11

24 Original Problem Optimality Functions Also, let: θ o,h (x 0, u 0 ) = min u( ) subject to: T 0 p 0 (t) T ( f ( x 0 (t), u(t) ) f ( x 0 (t), u 0 (t) )) dt, ṗ 0 (t) = f T ( x0 (t), u 0 (t) ) p 0 (t), x p 0 (T ) = ϕ ( ) x(t ), x u(t) U, t [0, T ]. We say that θ o,h is the Pontryagin optimality function of P o. Proposition θ o,h (x 0, u 0 ) = 0 θ o,l (x 0, u 0 ) = 0. Sampling-Based Optimal Control Synthesis ESE, Washington Univ. in St. Louis 12

25 Original Problem Optimality Functions Also, let: θ o,h (x 0, u 0 ) = min u( ) subject to: T 0 p 0 (t) T ( f ( x 0 (t), u(t) ) f ( x 0 (t), u 0 (t) )) dt, ṗ 0 (t) = f T ( x0 (t), u 0 (t) ) p 0 (t), x p 0 (T ) = ϕ ( ) x(t ), x u(t) U, t [0, T ]. We say that θ o,h is the Pontryagin optimality function of P o. Proposition θ o,h (x 0, u 0 ) = 0 θ o,l (x 0, u 0 ) = 0. Sampling-Based Optimal Control Synthesis ESE, Washington Univ. in St. Louis 12

26 Original Problem Optimality Functions Also, let: θ o,h (x 0, u 0 ) = min u( ) subject to: T 0 p 0 (t) T ( f ( x 0 (t), u(t) ) f ( x 0 (t), u 0 (t) )) dt, ṗ 0 (t) = f T ( x0 (t), u 0 (t) ) p 0 (t), x p 0 (T ) = ϕ ( ) x(t ), x u(t) U, t [0, T ]. We say that θ o,h is the Pontryagin optimality function of P o. Proposition θ o,h (x 0, u 0 ) = 0 θ o,l (x 0, u 0 ) = 0. Sampling-Based Optimal Control Synthesis ESE, Washington Univ. in St. Louis 12

27 Relaxed Problem Optimality Functions Let: θ r,l (x 0, µ 0 ) = min {δµ t} t ϕ( x0 (T ) ) δx(t ), x ( ( x0 (t), u ) ) dµ 0,t (u) δx(t)+ f subject to: δẋ(t) = R m x + f ( x 0 (t), u ) dδµ t (u), R m δx(0) = 0, supp(µ 0,t + δµ t ) U, t [0, T ]. We say that θ r,l is the linearized, or variational, optimality function of P r. Sampling-Based Optimal Control Synthesis ESE, Washington Univ. in St. Louis 13

28 Relaxed Problem Optimality Functions Let: θ r,l (x 0, µ 0 ) = min {δµ t} t ϕ( x0 (T ) ) δx(t ), x ( ( x0 (t), u ) ) dµ 0,t (u) δx(t)+ f subject to: δẋ(t) = R m x + f ( x 0 (t), u ) dδµ t (u), R m δx(0) = 0, supp(µ 0,t + δµ t ) U, t [0, T ]. We say that θ r,l is the linearized, or variational, optimality function of P r. Sampling-Based Optimal Control Synthesis ESE, Washington Univ. in St. Louis 13

29 Relaxed Problem Optimality Functions Also, let: θ r,h (x 0, µ 0 ) = min {δµ t} t T 0 p 0 (t) ( R T f ( x 0 (t), u ) ) dδµ t (u) dt, m ( f T ( subject to: ṗ 0 (t) = x0 (t), u ) dµ 0 (u)) p 0 (t), R m x p 0 (T ) = ϕ ( ) x(t ), x supp(µ 0,t + δµ t ) U, t [0, T ], dδµ t (u) = 0, t [0, T ]. R m We say that θ r,h is the Pontryagin optimality function of P r. Proposition θ r,h (x 0, u 0 ) = θ r,l (x 0, u 0 ), and so are their minimizing arguments. Sampling-Based Optimal Control Synthesis ESE, Washington Univ. in St. Louis 14

30 Relaxed Problem Optimality Functions Also, let: θ r,h (x 0, µ 0 ) = min {δµ t} t T 0 p 0 (t) ( R T f ( x 0 (t), u ) ) dδµ t (u) dt, m ( f T ( subject to: ṗ 0 (t) = x0 (t), u ) dµ 0 (u)) p 0 (t), R m x p 0 (T ) = ϕ ( ) x(t ), x supp(µ 0,t + δµ t ) U, t [0, T ], dδµ t (u) = 0, t [0, T ]. R m We say that θ r,h is the Pontryagin optimality function of P r. Proposition θ r,h (x 0, u 0 ) = θ r,l (x 0, u 0 ), and so are their minimizing arguments. Sampling-Based Optimal Control Synthesis ESE, Washington Univ. in St. Louis 14

31 Relaxed Problem Optimality Functions Also, let: θ r,h (x 0, µ 0 ) = min {δµ t} t T 0 p 0 (t) ( R T f ( x 0 (t), u ) ) dδµ t (u) dt, m ( f T ( subject to: ṗ 0 (t) = x0 (t), u ) dµ 0 (u)) p 0 (t), R m x p 0 (T ) = ϕ ( ) x(t ), x supp(µ 0,t + δµ t ) U, t [0, T ], dδµ t (u) = 0, t [0, T ]. R m We say that θ r,h is the Pontryagin optimality function of P r. Proposition θ r,h (x 0, u 0 ) = θ r,l (x 0, u 0 ), and so are their minimizing arguments. Sampling-Based Optimal Control Synthesis ESE, Washington Univ. in St. Louis 14

32 Idea Solve the relaxed problem using (tried and tested) variational-based methods, where at each step we compute θ r,h to check for optimality. 2. Once we find a minimizer, synthesize an input signal that approximates the optimal relaxed trajectory. Sampling-Based Optimal Control Synthesis ESE, Washington Univ. in St. Louis 15

33 Idea Solve the relaxed problem using (tried and tested) variational-based methods, where at each step we compute θ r,h to check for optimality. 2. Once we find a minimizer, synthesize an input signal that approximates the optimal relaxed trajectory. Sampling-Based Optimal Control Synthesis ESE, Washington Univ. in St. Louis 15

34 1. Critical Point Characterization 2. Synthesis of Optimal Control Inputs 3. Simulation: Optimal HVAC Operation Sampling-Based Optimal Control Synthesis ESE, Washington Univ. in St. Louis 16

35 Computing θ r,h Note that θ r,h is a convex optimization problem. In fact, it is a linear infinite-dimensional program. Also note that we can rewrite θ r,h as: T θ r,h (x 0, µ 0 ) = min p 0 (t) T f {µ 1,t } t 0 ( R ( x 0 (t), u ) dµ 1,t (u)+ m f ( x 0 (t), u ) ) dµ 0,t (u) dt, R m subject to: supp(µ 1,t ) U, t [0, T ]. Sampling-Based Optimal Control Synthesis ESE, Washington Univ. in St. Louis 17

36 Computing θ r,h Note that θ r,h is a convex optimization problem. In fact, it is a linear infinite-dimensional program. Also note that we can rewrite θ r,h as: T θ r,h (x 0, µ 0 ) = min p 0 (t) T f {µ 1,t } t 0 ( R ( x 0 (t), u ) dµ 1,t (u)+ m f ( x 0 (t), u ) ) dµ 0,t (u) dt, R m subject to: supp(µ 1,t ) U, t [0, T ]. Sampling-Based Optimal Control Synthesis ESE, Washington Univ. in St. Louis 17

37 Computing θ r,h Proposition Let x R n. Then: { } f(x, u) dµ(u) supp(µ) U = co{f(x, u) u U}. R m Hence: T θ r,h (x 0, µ 0 ) = min p 0 (t) (z(t) T f ( x 0 (t), u ) ) dµ 0,t (u) dt, {µ 1,t } t 0 R m subject to: z(t) co { f ( x 0 (t), u ) u U }. Computing Pontryagin-optimal points is as hard as finding a good representation for the convex hull of f(x, ). Sampling-Based Optimal Control Synthesis ESE, Washington Univ. in St. Louis 18

38 Computing θ r,h Proposition Let x R n. Then: { } f(x, u) dµ(u) supp(µ) U = co{f(x, u) u U}. R m Hence: T θ r,h (x 0, µ 0 ) = min p 0 (t) (z(t) T f ( x 0 (t), u ) ) dµ 0,t (u) dt, {µ 1,t } t 0 R m subject to: z(t) co { f ( x 0 (t), u ) u U }. Computing Pontryagin-optimal points is as hard as finding a good representation for the convex hull of f(x, ). Sampling-Based Optimal Control Synthesis ESE, Washington Univ. in St. Louis 18

39 Computing θ r,h Proposition Let x R n. Then: { } f(x, u) dµ(u) supp(µ) U = co{f(x, u) u U}. R m Hence: T θ r,h (x 0, µ 0 ) = min p 0 (t) (z(t) T f ( x 0 (t), u ) ) dµ 0,t (u) dt, {µ 1,t } t 0 R m subject to: z(t) co { f ( x 0 (t), u ) u U }. Computing Pontryagin-optimal points is as hard as finding a good representation for the convex hull of f(x, ). Sampling-Based Optimal Control Synthesis ESE, Washington Univ. in St. Louis 18

40 Sampling-Based Synthesis Let {u i } N i=1 U be a set of samples. Then for each t [0, T ]: co { f ( x 0 (t), u ) u U } { N ω i (t) f ( ) N } x 0 (t), u i ω i (t) = 1, ω i (t) 0. i=1 After sampling, our classical dynamical system becomes a switched hybrid system. Hence, we can use the PWM-based projection method in Vasudevan et al. (SICON, 2013) to synthesize input signals. i=1 R. Vasudevan, H. Gonzalez, R. Bajcsy, and S. S. Sastry, Consistent Approximations for the Optimal Control of Constrained Switched Systems Part 1: A Conceptual Algorithm, Siam journal on control and optimization, vol. 51, no. 6, pp , 2013 R. Vasudevan, H. Gonzalez, R. Bajcsy, and S. S. Sastry, Consistent Approximations for the Optimal Control of Constrained Switched Systems Part 2: An Implementable Algorithm, Siam journal on control and optimization, vol. 51, no. 6, pp , 2013 Sampling-Based Optimal Control Synthesis ESE, Washington Univ. in St. Louis 19

41 Sampling-Based Synthesis Let {u i } N i=1 U be a set of samples. Then for each t [0, T ]: co { f ( x 0 (t), u ) u U } { N ω i (t) f ( ) N } x 0 (t), u i ω i (t) = 1, ω i (t) 0. i=1 After sampling, our classical dynamical system becomes a switched hybrid system. Hence, we can use the PWM-based projection method in Vasudevan et al. (SICON, 2013) to synthesize input signals. i=1 R. Vasudevan, H. Gonzalez, R. Bajcsy, and S. S. Sastry, Consistent Approximations for the Optimal Control of Constrained Switched Systems Part 1: A Conceptual Algorithm, Siam journal on control and optimization, vol. 51, no. 6, pp , 2013 R. Vasudevan, H. Gonzalez, R. Bajcsy, and S. S. Sastry, Consistent Approximations for the Optimal Control of Constrained Switched Systems Part 2: An Implementable Algorithm, Siam journal on control and optimization, vol. 51, no. 6, pp , 2013 Sampling-Based Optimal Control Synthesis ESE, Washington Univ. in St. Louis 19

42 Sampling-Based Synthesis Algorithm 1. Compute samples of U. 2. Iteratively solve the relaxed optimal control problem using a gradient-based method. 3. Synthesize an approximation of the optimal relaxed input using PWM-based projection. Sampling-Based Optimal Control Synthesis ESE, Washington Univ. in St. Louis 20

43 Numerical Considerations Note that if f(x, u k ) lays strictly in the relative interior of co{f(x, u) u {u i } i }, then its computation is irrelevant. Use qhull to reduce the number of samples. In large-scale problems most of the computation time is spent calculating f ( x 0 (t), u i ) for each of the samples. Use an l 1 regularizer to force sparsity on the set {ω i (t)} i. If smoothness is desirable, then use an l 2 regularizer and a large number of samples. Sampling-Based Optimal Control Synthesis ESE, Washington Univ. in St. Louis 21

44 Numerical Considerations Note that if f(x, u k ) lays strictly in the relative interior of co{f(x, u) u {u i } i }, then its computation is irrelevant. Use qhull to reduce the number of samples. In large-scale problems most of the computation time is spent calculating f ( x 0 (t), u i ) for each of the samples. Use an l 1 regularizer to force sparsity on the set {ω i (t)} i. If smoothness is desirable, then use an l 2 regularizer and a large number of samples. Sampling-Based Optimal Control Synthesis ESE, Washington Univ. in St. Louis 21

45 Numerical Considerations Note that if f(x, u k ) lays strictly in the relative interior of co{f(x, u) u {u i } i }, then its computation is irrelevant. Use qhull to reduce the number of samples. In large-scale problems most of the computation time is spent calculating f ( x 0 (t), u i ) for each of the samples. Use an l 1 regularizer to force sparsity on the set {ω i (t)} i. If smoothness is desirable, then use an l 2 regularizer and a large number of samples. Sampling-Based Optimal Control Synthesis ESE, Washington Univ. in St. Louis 21

46 Numerical Considerations Note that if f(x, u k ) lays strictly in the relative interior of co{f(x, u) u {u i } i }, then its computation is irrelevant. Use qhull to reduce the number of samples. In large-scale problems most of the computation time is spent calculating f ( x 0 (t), u i ) for each of the samples. Use an l 1 regularizer to force sparsity on the set {ω i (t)} i. If smoothness is desirable, then use an l 2 regularizer and a large number of samples. Sampling-Based Optimal Control Synthesis ESE, Washington Univ. in St. Louis 21

47 Numerical Considerations Note that if f(x, u k ) lays strictly in the relative interior of co{f(x, u) u {u i } i }, then its computation is irrelevant. Use qhull to reduce the number of samples. In large-scale problems most of the computation time is spent calculating f ( x 0 (t), u i ) for each of the samples. Use an l 1 regularizer to force sparsity on the set {ω i (t)} i. If smoothness is desirable, then use an l 2 regularizer and a large number of samples. Sampling-Based Optimal Control Synthesis ESE, Washington Univ. in St. Louis 21

48 1. Critical Point Characterization 2. Synthesis of Optimal Control Inputs 3. Simulation: Optimal HVAC Operation Sampling-Based Optimal Control Synthesis ESE, Washington Univ. in St. Louis 22

49 Comprehensive Building Operation Besides controlling the HVAC unit in a building, we can make suggestions to the occupants regarding door configuration to improve efficiency and comfort. d 3 h 2 s 2 h 4 d 4 s 3 d 2 d 1 s 1 h 3 h 1 Let θ {0, 1} 4, where θ i = 1 if i-th door is open. Sampling-Based Optimal Control Synthesis ESE, Washington Univ. in St. Louis 23

50 CFD Model Heat convection-diffusion: T e t (x, t) x (κ(x, θ) x T e (x, t))+ + u(x) x T e (x, t) = g Te (x, t), Stationary imcompressible Navier-Stokes: 1 Re xu(x) + ( u(x) x ) u(x)+ + x p(x) + α(x, θ) u(x) = g u (x), u = 0 Initial condition: T e (x, 0) = π 0, Boundary conditions: T e (t, x) = 0, and u(x) = 0, x Ω R. He and H. Gonzalez, Zoned HVAC Control via PDE-Constrained Optimization, in Proceedings of the 2016 american control conference, arxiv: R. He and H. Gonzalez, Gradient-Based Estimation of Air Flow and Geometry Configurations in a Building Using Fluid Dynamic Adjoint Equations, in Proceedings of the 4th international high performance buildings conference at purdue, arxiv: Sampling-Based Optimal Control Synthesis ESE, Washington Univ. in St. Louis 24

51 Critical Point Characterization Synthesis of Optimal Control Inputs Simulation: Optimal HVAC Operation Single Zone Control Closed doors: Open doors: 4.8 d3 h4 d4 h2 d2 d h3 4.8 d3 4.0 target h4 d4 d2 0.8 h3 target ms Sampling-Based Optimal Control Synthesis 4.0 d h1 h2 1.6 h ms ESE, Washington Univ. in St. Louis 25

52 Critical Point Characterization Synthesis of Optimal Control Inputs Simulation: Optimal HVAC Operation Dual Zone Control Closed doors: target h4 d4 d3 h3 h2 d2 d1 target h1 0.5 ms Sampling-Based Optimal Control Synthesis Open doors: target h4 d4 d3 h3 h2 d2 d1 target h ms ESE, Washington Univ. in St. Louis 26

53 There is nothing so practical as a good theory Kurt Lewin Sampling-Based Optimal Control Synthesis ESE, Washington Univ. in St. Louis 27