Sensitivity computation based on CasADi for dynamic optimization problems

Transcription

1 Sensitivity computation based on CasADi for dynamic optimization problems 16 th European Workshop on Automatic Differentiation 8 9 December 2014, Jena, Germany Evgeny Lazutkin, Abebe Geletu, Siegbert Hopfgarten, Pu Li Technische Universität Ilmenau Simulation and Optimal Processes Group evgeny.lazutkin@tu-ilmenau.de Page 1

2 Motivation Content Combined Multiple Shooting and Collocation (CMSC) method Application of the symbolic computations Sensitivity analysis First Order Sensitivity (FOS) Path constraints Summary and current work Page 2

3 1. Motivation Why? Safe operations Fast and efficient What? startup / shutdown High dimensional, dynamical, and nonlinear systems with constraints Where? Power plants Robotics Petrochemical industry Power plant¹ Petrochemical industry³ Robotics² ¹ ² ³ Page 3

4 1. Motivation Easy application Object-oriented modelling for optimization Solution methods Advanced numerical algorithms Nonlinear programming Combined multiple shooting and collocation Computational implementation Automatic and symbolic differentiation Complex control system FOS are required Fast analysis and solution Functions, sparsity, etc. Sensitivities Approximation Analytical Control system Offline solution Online solution (Nonlinear Model Predictive Control (NMPC)) Example Modelica models⁴ Example: Automatic differentiation⁵ ⁴ ⁵ Page 4

5 2. CMSC method [1,2,5] Original NOCP Multiple Shooting* min x(t),u(t) s.t. J(x t, u(t)) x t 0 = x 0, x t = f x t, u t, t, x mmm x t x mmm, u mmm u t u mmm. Collocation** Ipopt Sensitivity calculation Solution *Bock, H. G., Plitt, K. J. A Multiple Shooting Algorithm for Direct Solution of Optimal Control Problems, Prepr. 9th IFAC World Congress, Budapest, 1984, pp **Cuthrell, J. E., Biegler, L. T.: Simultaneous optimization and solution methods for batch reactor control profiles. Comput. Chem. Eng. 13(1989),pp Page 5

6 2. CMSC method [1,2,5] Original NOCP Multiple Shooting* min x(t),u(t) s.t. J(x t, u(t)) x t 0 = x 0, x t = f x t, u t, t, x mmm x t x mmm, u mmm u t u mmm. Collocation** Ipopt Sensitivity calculation Solution *Bock, H. G., Plitt, K. J. A Multiple Shooting Algorithm for Direct Solution of Optimal Control Problems, Prepr. 9th IFAC World Congress, Budapest, 1984, pp **Cuthrell, J. E., Biegler, L. T.: Simultaneous optimization and solution methods for batch reactor control profiles. Comput. Chem. Eng. 13(1989),pp Page 6

7 2. CMSC method [1,2,5] Original NOCP Multiple Shooting* Collocation** Ipopt Sensitivity calculation Solution *Bock, H. G., Plitt, K. J. A Multiple Shooting Algorithm for Direct Solution of Optimal Control Problems, Prepr. 9th IFAC World Congress, Budapest, 1984, pp **Cuthrell, J. E., Biegler, L. T.: Simultaneous optimization and solution methods for batch reactor control profiles. Comput. Chem. Eng. 13(1989),pp Page 7

8 2. CMSC method [1,2,5] Original NOCP Multiple Shooting* Collocation** Ipopt Sensitivity calculation Solution *Bock, H. G., Plitt, K. J. A Multiple Shooting Algorithm for Direct Solution of Optimal Control Problems, Prepr. 9th IFAC World Congress, Budapest, 1984, pp **Cuthrell, J. E., Biegler, L. T.: Simultaneous optimization and solution methods for batch reactor control profiles. Comput. Chem. Eng. 13(1989),pp Page 8

9 2. CMSC method [1,2,5] Original NOCP Multiple Shooting* Collocation** Ipopt Sensitivity calculation Solution *Bock, H. G., Plitt, K. J. A Multiple Shooting Algorithm for Direct Solution of Optimal Control Problems, Prepr. 9th IFAC World Congress, Budapest, 1984, pp **Cuthrell, J. E., Biegler, L. T.: Simultaneous optimization and solution methods for batch reactor control profiles. Comput. Chem. Eng. 13(1989),pp Page 9

10 2. CMSC method [1,2,5] Original NOCP Multiple Shooting* Collocation** Ipopt Sensitivity calculation Solution *Bock, H. G., Plitt, K. J. A Multiple Shooting Algorithm for Direct Solution of Optimal Control Problems, Prepr. 9th IFAC World Congress, Budapest, 1984, pp **Cuthrell, J. E., Biegler, L. T.: Simultaneous optimization and solution methods for batch reactor control profiles. Comput. Chem. Eng. 13(1989),pp Page 10

11 2. CMSC method ADVANTAGES Combines advantages of multiple shooting method and collocation method Reduction of the NLP variables in optimizer (Ipopt) Precision approximation of the state trajectories Can be applied for NMPC Allows parallel computation due to independence of each interval The symbolic and automatic pre-calculation of the derivatives Use Modelica models Implementation on JModelica.org and CasADi platform for wider public availability Page 11

12 3. Application of symbolic computations Original model formulation Modelica model formulation model batch Real x1(start=1, fixed=true, min=0,max=1); Real x2(start=0, fixed=true, min=0,max=1); input Real u1; equation der(x1) = -(u *u1*u1)*x1; der(x2) = u1*x1; end batch; optimization batch_ocp(objective= -x2(finaltime), starttime=0, finaltime=1) extends batch(u1(free=true,min=0,max=5)); end batch_ocp; Modelica model can be transferred into Python through JModelica.org software for further manipulations Page 12

13 3. Symbolic computations with CasADi Transferred ODE system in symbolic form SX_ODE = [(der(x1)+((u1+((0.5*u1)*u1))*x1)), (der(x2)-(u1*x1))] Inputs for ODE system INPUTS = [Matrix<SX>(der(x1)), Matrix<SX>(der(x2)), Matrix<SX>(x1), Matrix<SX>(x2), Matrix<SX>(u1)] SXFunction instance SX_ODE_FUNCTION = SXFunction([vertcat(INPUTS)], [SX_ODE]) SX_ODE_FUNCTION.init() Features Evaluation and AD Symbolically Numerically Required variables States at collocation points Parameterized states Parameterized controls state = ssym("s",n_int*(n_d N_CP)) => [s_0,s_1,s_2,s_3,s_4,s_5] p_state = ssym("ps",(n_int+1) N_D) p_ctrl = ssym("c",n_int N_C) Advantages Check results Problem independent Simple to implement Page 13

14 3. Application of symbolic computations Applied method for SXFunction: eval RES = SX_ODE_FUNCTION.eval([vertcat([der_expr, diff, ctrl])])[0] Time discretization by CMSC method: 1 intervals, 3 collocation points [((((( *s_0)+(1.0328*s_2))+( *s_4))+( *ps_0))+((c+((0.5*c)*c))*s_0)), ((((( *s_1)+(1.0328*s_3))+( *s_5))+( *ps_1))-(c*s_0))] [((((( *s_0)+(1.7746*s_2))+( *s_4))+(2.6619*ps_0))+((c+((0.5*c)*c))*s_2)), ((((( *s_1)+(1.7746*s_3))+( *s_5))+(2.6619*ps_1))-(c*s_2))] [((((( *s_0)+( *s_2))+( *s_4))+( *ps_0))+((c+((0.5*c)*c))*s_4)), ((((( *s_1)+( *s_3))+( *s_5))+( *ps_1))-(c*s_4))] Pack this system to new symbolic function G and create new SXFunction with only symbolical inputs The same system will be solved in a Newton solver to obtain state trajectories NewtonImplicitSolver KinsolSolver How to calculate required FOS? Page 14

15 4. Sensitivity analysis: FOS X ii X ii X ii = G X ii Sensitivity w.r.t. parameterized state Matrix equations X ii X ii U = G U Sensitivity w.r.t. parameterized control,, can be constructed using AD! X ii X ii U Function = SXFunction([vertcat([state, p_state, p_ctrl])],[vertcat([g])]) Function.init() full_jacobian=function.jac() dgdx_sym = SXMatrix(SYSTEM_INDEX,SYSTEM_INDEX) dgdx_sym = full_jacobian[range(system_index), range(system_index)] dgdx_sx = SXFunction([c_s,vertcat([p_s, p_c])],[dgdx_sym]) dgdx_sx.init() Step 0 Step 1.1 A = msym('a', dgdx_sym.sparsity()) Step 2.1 X.1 means matrix number SYSTEM_INDEX is a number of collocation variables per interval Page 15

16 4. Sensitivity analysis: FOS X ii X ii X ii = G X ii Sensitivity w.r.t. parameterized state Matrix equations,, can be constructed using AD! X ii X ii U X ii X ii U = G U Sensitivity w.r.t. parameterized control dgdxp_sym = SXMatrix(SYSTEM_INDEX,N_D) self.dgdxp_sym = full_jacobian[system_index:system_index+n_d]] dgdxp_sx = SXFunction([d_s,vertcat([p_s, p_c])],[dgdxp_sym]) dgdxp_sx.init() Step 1.2 B = msym( B', dgdxp_sym.sparsity()) SOLVER_X = CSparse(A.sparsity(), shape(b)[1]) SOLVER_X.init() SOLVER_X_SOL = SOLVER_X.solve(A,B) Step 2.2 Step 3 Create symbolical MXFunction to obtain sensitivities Page 16

17 4. Sensitivity analysis: Path constraints X ii = G X ii X ii X ii Sensitivity w.r.t. parameterized state Matrix equations x 1 2 x 0,1, x 0,2, u + x 2 2 x 0,1, x 0,2, u 1 0 X ii X ii U = G U Sensitivity w.r.t. parameterized control How to calculate sensitivity for optimizer automatically? pc_function = SXFunction([vertcat([ORIGINAL_STATES])],[vertcat([PATH])]) pc_function.init() GRADINET=pc_function.grad() x 1 [2 x 1 x 0,1, x 0,2, u, 2 x 2 (x 0,1, x 0,2, u)] x 0,1 x 2 AND x 1 u Automatic approach via CasADi (even for complicated path constraints) x 0,1 Page 17

18 5. Summary and current work Summary Pre-calculated derivatives and their symbolic representation Sensitivities are automatically available Using Modelica model formulation (limited to CasADi 1.8.2) CasADi package (not only for AD) Can handle complicated expressions Symbolical and numerical evaluation / Solvers Documentation via API, examples, Google group, manual Limited to explicit form of sensitivities Implementation time Current work Investigation of the CPU/GPU parallelization Analytic Hessian matrix Page 18

19 References [1] Tamimi, J.: Development of the Efficient Algorithms for Model Predictive Control of Fast Systems. PhD Thesis, Technische Universität Ilmenau, VDI Verlag, [2] Tamimi, J., Li, P.: A Combined approach to nonlinear model predictive control of fast systems. J. Process Control, 20(2010)9, pp [3] Andersson, J., Åkesson, J., Diehl, M.: CasADi: A Symbolic Package for Automatic Differentiation and Optimal Control. Lecture Notes in Computational Science and Engineering, Vol. 87, Springer, 2012, pp [4] Wächter, A., Biegler, L.T.: On the Implementation of a Primal-Dual Interior Point Filter Line Search Algorithm for Large-Scale Nonlinear Programming, Mathematical Programming, Ser. A, 106(2006)1, pp [5] E. Lazutkin, A. Geletu, S. Hopfgarten, P. Li: Modified Multiple Shooting Combined with Collocation Method in JModelica.org with Symbolic Calculations, Proceedings of the 10th International Modelica Conference, DOI /ECP , March, 2014, Lund, Sweden, pp Page 19

20 Acknowledgments This work has been supported by Model Driven Physical Systems Operation project (MODRIO) by ITEA2, No , and by the German BMBF (BMBF Förderkennzeichen: 01IS12022H). Page 20

21 DISCUSSION Thank you very much for your attention! Page 21