Optimal control of nonstructured nonlinear descriptor systems

Transcription

1 Optimal control of nonstructured nonlinear descriptor systems TU Berlin DFG Research Center Institut für Mathematik MATHEON Workshop Elgersburg joint work with Peter Kunkel

2 Overview Applications Why DAEs and not ODEs A crash course in DAE theory Optimal control for ODEs Examples where the theory does not work for DAEs Optimality conditions for linear DAEs Differential-algebraic Riccati equations General nonlinear problems Numerical example

3 Optimal control of DAEs Optimal control problem: t J (x, u) = M(x(t)) + K(t, x, u) dt = min! t subject to a DAE constraint x state, u input, y output. F(t, x, u, ẋ) = 0, x(t) = x.

4 Linear quadratic optimal control Cost functional: J (x, u) = 1 2 x(t)t Mx(t) t t (x T Wx + 2x T Su + u T Ru) dt, W = W T C 0 (I, R n,n ), S C 0 (I, R n,l ), R = R T C 0 (I, R l,l ), M = M T R n,n. Constraint: E(t)ẋ = A(t)x + B(t)u + f, x(t) = x, E C 1 (I, R n,n ), A C 0 (I, R n,n ), B C 0 (I, R n,l ), f C 0 (I, R n ), x R n. Here: Determine optimal controls u U = C 0 (I, R l )., More general spaces and also nonsquare E, A are possible.

5 Drop size distributions in stirred liquid/liquid systems with M. Kraume from Chemical Engineering (S. Schlauch)

6 Drop size distributions in stirred liquid/liquid systems with M. Kraume from Chemical Engineering (S. Schlauch) simulation & control

7 Technological Application, Tasks Chemical industry: pearl polymerization and extraction processes Modelling of coalescence and breakage in turbulent flow. Numerical methods for simulation of coupled system of population balance equations/fluid flow equations. Development of optimal control methods for large scale coupled systems Model reduction and observer design. Control of real configurations via stirrer speed. Ultimate goal: Achieve specified average drop diameter and small standard deviation for distribution by real time-control of stirrer-speed.

8 Mathematical system components Navier Stokes equation (flow field) ( Sonjas Film). Population balance equation (drop size distribution). One or two way coupling. Initial and boundary conditions. Space discretization leads to an extremely large control system of nonlinear DAEs.

9 Active flow control with F. Tröltzsch (M. Navier Stokes equations (3D + boundary conditions speaker microphones input output Schmidt)

10 Technological Application, Tasks Control of detached turbulent flow on airline wing Test case (backward step to compare experiment/numerics.) Modelling of turbulent flow. Development of control methods for large scale coupled systems. Model reduction and observer design. Optimal control of real configurations via blowing and sucking of air in wing. Ultimate goal: Force detached flow back to wing.

11 Simulated flow

12 Controlled flow Movement of recirculation bubble following reference curve. 9 8 x ref (t) x r (t) time t 4 2 u(t) time t

13 Model based control of high-tech automatic gearboxes with P. Hamann Daimler/Chrysler ( DC Film)

14 Technological Application, Tasks Decrease full consumption, improve smoothness of switching Modelling of multi-model system, including elasticity, hydraulics Development of control methods for coupled system. Model reduction and observer design. Real time optimal control of real configuration. Ultimate goal: Increase efficiency, reduce costs Space discretization leads to a large hybrid, switched control system of nonlinear DAEs.

15 DAE control systems After space discretization all these problems are DAE control systems or in the linear case F(t, x, ẋ, u) = 0, G(t, x, u) = y, E(t)ẋ(t) = A(t)x(t) + B(t)u(t) + f (t), y(t) = C(t)x(t) + D(t)u(t) + g(t). Using a behavior approach, i.e., forming z(t) = (x, u, y) we obtain general non-square DAEs F(t, z, ż) = 0, E(t)ż = A(t)z. The behavior approach allows a uniform mathematical treatment of simulation and control problems!

16 Why DAEs and not ODEs? DAEs provide a unified framework for the analysis, simulation and control of coupled dynamical systems (continuous and discrete time). Automatic modelling leads to DAEs. (Constraints at interfaces). Conservation laws lead to DAEs. (Conservation of mass, energy, momentum). Coupling of solvers leads to DAEs (discrete time). Control problems are DAEs (behavior).

17 How does one solve such complex problems today? Simplified models. Space discretization with very coarse meshes. Identification and realization of black box models. Model reduction (mostly based on heuristic methods). Coupling of simulation packages. Use of standard optimal control techniques for simplified mathematical model. But do they work for these models?

18 Is there anything to do? Why not just apply the Pontryagin maximum principle? For linear ODEs the initial value problem has a unique solution x C 1 (I, R n ) for every u U, every f C 0 (I, R n ), and every initial value x R n. DAEs, where E(t) may be singular, may not be (uniquely) solvable for any u U and also the initial conditons may be restricted. Furthermore, we need solutions x X, where X usually is a larger space than C 1 (I, R n ).

19 Previous work Linear constant coefficient index 1 case, Bender/Laub 87, Campbell 87 M. 91, Geerts 93. Regularization to index 1, Bunse-Gerstner/M./Nichols 92, 94, Byers/Geerts/M. 97, Byers/Kunkel/M. 97. Linear variable coefficients index 1 case, Kunkel./M. 97. Semi-explicit nonlinear index 1 case, maximum principle, De Pinho/Vinter 97, Devdariani/Ledyaev 99. Semi-explicit index 2, 3 case Roubicek/Valasek 02. Linear index 1, 2 case with properly stated leading term, Balla/März, 02,04, Balla/Linh 05, Kurina/März 04, Backes 06. Multibody systems (structured index 3), Büskens/Gerdts 00, Gerdts 03,06.

20 A crash course in DAE Theory For numerical methods and for the design of controllers, we use derivative arrays (Campbell 1989). We assume that derivatives of original functions are available or can be obtained via computer algebra or automatic differentiation. Linear case: We put E(t)ẋ = A(t)x + f (t) and its derivatives up to order µ into a large DAE for z k = (x, ẋ,..., x (k) ). M 2 = M k (t)ż k = N k (t)z k + g k (t), k N 0 E 0 0 A Ė E 0 Ȧ 2Ë A Ė E, N 2 = A 0 0 Ȧ 0 0 Ä 0 0, z 2 = ẋ x ẍ.

21 Theorem, Kunkel/M Under some constant rank assumptions, for a linear DAE there exist integers µ, a, d and v such that: 1. corank M µ+1 (t) corank M µ (t) = v. 2. rank M µ (t) = (µ + 1)m a v on I, and there exists a smooth matrix function Z 2,3 (left nullspace of M µ ) with Z T 2,3 M µ(t) = The projection Z 2,3 can be partitioned into two parts: Z 2 (left nullspace of (M µ, N µ )) so that the first block column Â2 of Z 2 N µ(t) has full rank a and Z 3 N µ(t) = 0. Let T 2 be a smooth matrix function such that Â2T 2 = 0, (right nullspace of Â2). 4. rank E(t)T 2 = d = l a v and there exists a smooth matrix function Z 1 of size (n, d) with rank Ê1 = d, where Ê1 = Z T 1 E.

22 Reduced problem The quantity µ is called the strangeness-index. It describes the smoothness requirements for forcing or input functions. It generalizes the usual differentiation index to general DAEs (and counts slightly differently). We obtain a numerically computable condensed form Ê 1 (t)ẋ = Â1(t)x + ˆf 1 (t), ddifferential equations 0 = Â2(t)x + ˆf 2 (t), a algebraic equations 0 = ˆf3 (t), v consistency equations where Â1 = Z T 1 A, ˆf 1 = Z T 1 f, and ˆf 2 = Z T 2 g µ, ˆf 3 = Z T 3 g µ. The reduced system has the same solution set as the orignal problem but now it has strangeness-index 0. Remodeling!

23 Calculus of variations for ODEs (E=I) Introduce Lagrange multiplier function λ(t) and couple constraint into cost function, i.e. minimize J (x, u, λ) = 1 2 x(t)t Mx(t) t + λ T (ẋ Ax + Bu + f ) dt. t (x T Wx + 2x T Su + u T Ru) Consider x + δx, u + δu and λ + δλ. For a minimum the cost function has to go up in the neighborhood, so we get optimality conditions (Euler-Lagrange equations):

24 Optimality system Theorem If (x, u) is a solution to the optimal control problem, then there exists a Lagrange multiplier function λ C 1 (I, R n ), such that (x, λ, u) satisfy the optimality boundary value problem (a) ẋ = Ax + Bu + f, x(t) = x, (b) λ = Wx + Su A T λ, λ(t) = Mx(t), (c) 0 = S T x + Ru B T λ.

25 Naive Idea for DAEs Replace the identity in front of x by E(t) and then do the analysis in the same way. For DAEs the formal optimality system then could be (a) Eẋ = Ax + Bu + f, x(t) = x d (b) dt (E T λ) = Wx + Su A T λ, (E T λ)(t) = Mx(t), (b) 0 = S T x + Ru B T λ. This works if the system has strangeness-index µ = 0 as a free system with u = 0 but not in general.

26 What are the difficulties In the proof one has to guarantee that the resulting adjoint equation for λ has a unique solution. One needs a density argument in the solution space. But, the formal adjoint equation may not have a (unique) solution. The formal boundary conditions may not be consistent. The solution of the optimality systems may not exist or may not be unique.

27 Example Consider J (x, u) = (x u 2 )dt = min! subject to the differential-algebraic system [ ] [ ] [ ] [ ] [ 0 1 ẋ1 1 0 x1 1 = ẋ2 0 1 x 2 0 A simple calculation yields the optimal solution ] [ ] f1 u +. f 2 x 1 = u = λ 1 = 1 2 (f 1 + ḟ2), x 2 = f 2, λ 2 = 0.

28 In the formal optimality system we get x 1 = u = λ 1 = 1 2 (f 1 + ḟ2), x 2 = f 2, λ 2 = 1 2 (ḟ1 + f 2 ) without using the initial condition λ 1 (1) = 0. Depending on the data, this initial condition may be consistent or not. This initial condition should not be present. Moreover, λ 2 requires more smoothness of the inhomogeneity than in the optimal solution. Further examples, see Dissertation Backes 06.

29 Solution space To derive optimality conditions for DAEs, we need the right solution space for x: X = C 1 E + E(I, R n ) = { x C 0 (I, R n ) E + Ex C 1 (I, R n ) }, where E + denotes the Moore-Penrose inverse of the matrix valued function, which is the unique matrix function that satisfies the Penrose axioms. EE + E = E, E + EE + = E +, (EE + ) T = EE +, (E + E) T = E + E U is usually a set of piecewise continuous functions or even distributions.

30 Necessary optimality condition Theorem Consider the linear quadratic optimal control problem with a consistent initial condition. Suppose that the system has µ = 0 as a behavior system and that Mx(t) cokernel E(t). If (x, u) X U is a solution to this optimal control problem, then there exists a Lagrange multiplier function λ C 1 E + E (I, Rn ), such that (x, λ, u) satisfy the optimality boundary value problem E d dt (E + Ex) = (A + E d dt (E + E))x + Bu + f, (E + Ex)(t) = x, E T d dt (EE + λ) = Wx + Su (A + EE +Ė)T λ, (EE + λ)(t) = E + (t) T Mx(t), 0 = S T x + Ru B T λ.

31 Sufficient condition For this we need some kind of convexity. Theorem Consider the optimal control problem with a consistent initial condition and suppose that in the cost functional we have that [ ] W S S T, M R are (pointwise) positive semidefinite. If (x, u, λ) satisfies the formal optimality system then for any (x, u) satisfying the constraint we have J (x, u) J (x, u ).

32 Remarks If a minimum exists, then it satisfies the optimality system. If a unique solution to the formal optimality system exists, then x, u are the same as from the optimality system, λ may be different. To get a sufficient condition, we need some extra convexity. The optimality DAE may have µ > 0. Then it is numerically difficult to solve and further consistency conditions or smoothness requirements arise. The condition that the original system has µ = 0 as a behavior system is not necessary if the cost function is chosen appropriately, so that the resulting optimality system has µ = 0.

33 Differential-algebraic Riccati equations If R in the cost functional is invertible, and if the system has µ = 0 as a free system with u = 0, then one can (at least in theory) apply the usual Riccati approach to E d dt (E + Ex) = (A + E d dt (E + E))x + Bu + f, (E + Ex)(t) = x, E T d dt (EE + λ) = Wx + Su (A + EE +Ė)T λ, (EE + λ)(t) = E + (t) T Mx(t), 0 = S T x + Ru B T λ. If µ > 0 or R is singular, then the Riccati approach may not work, even if the boundary value problem has a unique solution.

34 Nonlinear problems For nonlinear systems F(t, x, ẋ) = 0 one considers nonlinear derivative arrays: We set 0 = F k (t, x, ẋ,..., x (k+1) ) = F (t, x, ẋ) F(t, x, ẋ)... d dt d k dt k F (t, x, ẋ). M k (t, x, ẋ,..., x (k+1) ) = F k;ẋ,...,x (k+1)(t, x, ẋ,..., x (k+1) ), N k (t, x, ẋ,..., x (k+1) ) = (F k;x (t, x, ẋ,..., x (k+1) ), 0,..., 0), z k = (t, x, ẋ,..., x (k+1) ).

35 Hypothesis: There exist integers µ, r, a, d, and v such that L = Fµ 1 ({0}). We have rank F µ;t,x,ẋ,...,x (µ+1) = rank F µ;x,ẋ,...,x (µ+1) = r, in a neighborhood of L such that there exists an equivalent system F(z µ ) = 0 with a Jacobian of full row rank r. On L we have 1. corank F µ;x,ẋ,...,x (µ+1) corank F µ 1;x,ẋ,...,x (µ+1) = v. 2. corank F x,ẋ,...,x (µ+1) = a and there exist smooth matrix functions Z 2 (left nullspace of M µ ) and T 2 (right nullspace of Â2 = F x ) with Z2 T F x,ẋ,...,x (µ+1) = 0 and Z2 T Â2T 2 = rank FẋT 2 = d, d = m a v, and there exists a smooth matrix function Z 1 with rank Z1 T F ẋ = d.

36 Theorem Kunkel/M The solution set L forms a (smooth) manifold of dimension (µ + 2)n + 1 r. The DAE can locally be transformed (by application of the implicit function theorem) to a reduced DAE of the form ẋ 1 = G 1 (t, x 1, x 3 ), (d differential equations), x 2 = G 2 (t, x 1, x 3 ), (a algebraic equations), 0 = 0 (v redundant equations). The variables x 3 represent undetermined components (controls).

37 Optimality conditions Assume that µ = 0 for the system in behavior form with z = (x, u), then in terms of the reduced DAE, the local optimality system is (a) ẋ 1 = L(t, x 1, u), x 1 (t) = x 1, (b) x 2 = R(t, x 1, u), (c) λ1 = K x1 (t, x 1, x 2, u) T L x1 (t, x 1, x 2, u) T λ 1 R x1 (t, x 1, u) T λ 1, λ 1 (t) = M x1 (x 1 (t), x 2 (t)) T (d) 0 = K x2 (t, x 1, x 2, u) T + λ 2, (e) 0 = K u (t, x 1, x 2, u) T L u (t, x 1, u) T λ 1 R u (t, x 1, u) T λ 2, (f) γ = λ 1 (t) Here λ 1, λ 2 are Lagrange multipliers associated with x 1, x 2 and γ is associated with the initial value constraint.

38 Remarks These are local results. All the results can be generalized to general nonsquare nonlinear systems. End point conditions for x can be included. Input and state constraints can be included to give a maximum principle.

39 Numerical Methods The linear optimality system that we would have to solve numerically has the form (a) Ê 1 ẋ = Â1x + ˆB 1 u + ˆf 1, (Ê + 1 Ê1x)(t) = x (b) 0 = Â2x + ˆB 2 u + ˆf 2, d (c) dt (Ê 1 T λ 1) = Wx + Su ÂT 1 λ 1 ÂT 2 λ 2, λ 1 (t) = [ Ê + 1 (t)t 0 ]Mx(t), (d) 0 = S T x + Ru ˆB 1 T λ 1 ˆB 2 T λ 2. where E i, A i, B i are obtained by projection with smooth orthogonal projections Z i from the derivative array. An analogous structure arises locally in the nonlinear case.

40 Numerical Problems In the implementation of the numerical integration codes we use nonsmooth projectors Z T 1, Z T 2. It would be too expensive to carry smooth projectors along. For numerical forward (in time) simulation, it is enough that we know the existence of smooth projectors. Integration methods like Runge-Kutta or BDF don t see the nonsmooth behavior. But the adjoint variables (Lagrange multipliers) depend on these projections and their derivatives. However, even if Z T 1, Z T 2 are nonsmooth, Z 1Z T 1 and Z 2Z T 2 are smooth.

41 Smooth optimality system Choose Ê T 1 λ 1 = E T Z 1 λ 1 = E T Z 1 Z T 1 Z 1 λ 1 = E T Z 1 Z T 1 ˆλ 1. With ˆλ 1 = Z 1 λ 1 we obtain smooth coefficients for ˆλ 1. However, we have to add the condition that ˆλ 1 range Z 1 to the system. If Z i completes Z i to a full orthogonal matrix (we compute these anyway when doing a QR or SVD computation) then these conditions can be expressed as Z i T ˆλi = 0, i = 1, 2

42 New linear optimality system For the numerical solution we use the optimality system. (a) Ê 1 ẋ = Â1x + ˆB 1 u + ˆf 1, (Ê + 1 Ê1x)(t) = x, (b) 0 = Â2x + ˆB 2 u + ˆf 2, (c) d dt (E T Z 1 Z1 T ˆλ 1 ) = Wx + Su A T ˆλ1 [ I n ]Nµ T ˆλ 2, (Z1 T ˆλ 1 )(t) = [ Ê + 1 (t)t 0 ]Mx(t), (d) 0 = S T x + Ru B T ˆλ1 [ 0 I l ]Nµ T ˆλ 2 (e) 0 = Z 1 T ˆλ1, (f) 0 = Z 2 T ˆλ 2. All quantities are available for all time steps. An analoguous system can be derived for each Gauss-Newton step in the nonlinear case.

43 Numerical Example A motor controlled pendulum with a motor in the origin shall be driven into its equilibrium with minimal costs, ex. from Büskens/Gerdts y m 3 l x mg

44 Model problem The model problem for this reads J(x, u) = 3 0 u(t) 2 dt = min! s.t. ẋ 1 = x 3, x 1 (0) = 2 1 2, g = 9.81 ẋ 2 = x 4, x 2 (0) = 2 1 2, ẋ 3 = 2x 1 x 5 + x 2 u, x 3 (0) = 0, ẋ 4 = g 2x 2 x 5 x 1 u, x 4 (0) = 0, 0 = x1 2 + x 2 2 1, x 5(0) = 1 2 gx 2(0).

45 Properties The DAE satisfies the Hypothesis with µ = 2, a = 3, d = 2, and v = 0. Hence, only two scalar initial values are sufficient to describe the initial state. We chose them to be x 2 (0) = and x3 (0) = 0. Similarly, x 1 (3) = 0 and x 3 (0) = 0 are sufficient to describe the equilibrium at the end point. As initial trajectory we took x 1 (t) = t, x3 (t) = 0, x 2 (t) = 1 x 1 (t) 2, x 4 (t) = 0, x 5 (t) = 1 2 gx 2(t). with all other unknowns set to zero on a equidistant grid of 60 intervals.

46 Gauss-Newton method Tolerance for the Gauß-Newton method was Let k count the iterations and w k 2 denote the Euclidian norm of the corresponding Gauß-Newton correction. k w k D D D D D-11 Initial bad convergence is due to a bad initial guess. Final value of cost function is J opt = 3.82 which is correct up to discretization and roundoff errors.

47 What has been achieved? Theoretical analysis (solvability) for general over and under-determined linear and nonlinear DAEs of arbitrary index (blue book) Optimal control and maximum principle. Model verification, model reduction and removal of redundancies is possible in a numerically stable way. General nonlinear (regular) DAE boundary value problem solver, Kunkel/M./Stöver 2004.

48 Thank you very much for your attention.