Verification and Control of Safety-Factor Profile in Tokamaks

Transcription

1 Verification and Control of Safety-Factor Profile in Tokamaks Aditya Gahlawat, Matthew M. Peet, and Emmanuel Witrant Illinois Institute of Technology Chicago, Illinois International Federation of Automatic Control World Congress Università Cattolica del Sacro Cuore Milan, Italy September, 2

2 Nuclear Fusion A Renewable Energy Source Fusion energy is the potential energy difference between particles in free state and particles bound together by the strong nuclear force. E from the strong nuclear force for the 2 H + 3 H to 4 He + n reaction is 3.5 MeV/nucleon. E from the Coulomb Barrier for the 2 H + 3 H to 4 He + n reaction is +.MeV/nucleon Nuclear Fission of U 235 only releases.85 MeV/nucleon Unfortunately.MeV/nucleon means a temperature of 2 6 K. Temperature at center of sun is K. From Maxwell-Boltzmann distribution, we only need = 6 K for a statistically significant reaction rate A. Gahlawat, M. Peet, E. Witrant SOS for Tokamaks: Introduction 2 / 28

3 Tokamaks Magnetic Confinement of Plasma Magnetic Confinement At high temperature, atoms ionize. Hydrogen 2 H ion + electron. Charged particles oscillate in a uniform magnetic field. But a uniform field must eventually end. Particles will eventually escape. Tokamaks loop the field back on itself. Particles rotate indefinitely. Inertial Confinement Compress the fuel quickly Plasma does not have time to expand spatially before creating additional reactions. Similar to a hydrogen bomb. A. Gahlawat, M. Peet, E. Witrant SOS for Tokamaks: Introduction 3 / 28

4 Magnetic Confinement of Plasma in Tokamaks Poloidal and Toroidal Fields The plasma is contained through the combined action of toroidal and solenoid field coils. The toroidal coils produce a magnetic field, B φ. Field lines are orthogonal to the Z-axis. The solenoid produces a plasma current which produces a poloidal magnetic field, B θ. Field lines in the R Z plane. A. Gahlawat, M. Peet, E. Witrant SOS for Tokamaks: Introduction 4 / 28

5 The Safety-Factor and Safety-Factor Profile A Useful Heuristic The Safety Factor, q is the number toroidal field rotations for every poloidal rotation. Triggers internal transport barriers which increase energy confinement The higher the safety-factor, the better the plasma is contained. The Safety-Factor Profile is the distribution of the safety-factor along an idealized radius. q(x,t) = φ(x,t)/ x ψ(x,t)/ x = B φ a 2 x ψ(x,t)/ x, where x = normalized radius B φ = toroidal magnetic field at the plasma center a = radius of the last closed magnetic surface (LCMS) φ = magnetic flux of the toroidal field ψ = magnetic field of the poloidal field To control q(x,t), we control ψ x (x,t) = ψ(x,t)/ x. A. Gahlawat, M. Peet, E. Witrant SOS for Tokamaks: Introduction 5 / 28

6 The Dynamics of the Poloidal Flux Gradient To Control the Safety-Factor Profile, we regulate ψ x (x,t) = x ψ(x,t). ψ x (x,t) = ( ) η (x,t) t µ a 2 x x x (xψ ( x(x,t)) +R η (x,t)j ni (x,t) ). x where R = magnetic center location µ = permeability of free space η (x,t) = plasma resistivity j ni (x,t) = non-inductive current density with the boundary conditions ψ x (,t) = and ψ x (,t) =. () The dynamics are coupled to electron temperature via Plasma Resistivity, η. Depends on dynamics of temperature, density, etc. Nonlinear coupling Assume a separation of time-scales A. Gahlawat, M. Peet, E. Witrant SOS for Tokamaks: Introduction 6 / 28

7 Dynamical System Representation Lets represent this PDE as an abstract differential system where A and B are the operators ẋ(t) = Ax(t)+Bu(t) (Aψ)(x) := µ a 2 x (Bj ni )(x) := x We ignore the bootstrap current. ( η (x) ( η (x)j eni (x) ) ) x (xψ(x)) This System generates a strongly continuous semigroup on X = L 2 [,] with domain x D A = {y L 2 [,] : y,y x,y xx L 2 [,],y() = y() = }. A. Gahlawat, M. Peet, E. Witrant SOS for Tokamaks: Introduction 7 / 28

8 Linear Operator Inequalities The Lyapunov Inequality For linear dynamical systems, we have the following characterization of stability. Suppose the operator A generates a strongly continuous semigroup on Hilbert space X with domain D A. Theorem. The system ẋ(t) = Ax(t) is stable if and only if there exist a positive operator P L(D A D A ) such that x,(a P +PA)x X < x 2 X for all x D A. Stability is equivalent to a feasibility problem with operator-valued variables linear inequality constraints A. Gahlawat, M. Peet, E. Witrant SOS for Tokamaks: Control of PDEs 8 / 28

9 Linear Operator Inequalities Controlling Linear PDEs The Variable Substitution Trick: Theorem 2. The system ẋ(t) = Ax(t)+Bu(t) is stabilizable via full-state feedback if and only if there exist operators P > and Z such that PA +AP +BZ +Z B < Then K = ZP. Here the inequality PA +AP +BZ +Z B < means for all x = P y, y D A. x,(pa +AP +BZ +Z B)x X If P : D A D A, then this is no harder than the Lyapunov inequality. A. Gahlawat, M. Peet, E. Witrant SOS for Tokamaks: Control of PDEs 9 / 28

10 Tractable or Intractable? Convex Optimization Problem: max bx subject to Ax C The problem is convex optimization if C is a convex cone. b and A are affine. Computational Tractability: Convex Optimization over C is, in general, tractable if The set membership test for y C is in P. x is finite dimensional. A. Gahlawat, M. Peet, E. Witrant SOS for Tokamaks: Control of PDEs / 28

11 The Stabilization Problem is Convex Optimization Problem: Find P L(Z) and Z L(D A ) such that PA +AP +BZ +Z B < P > Inequality represents the convex cone of positive operators on D A with inner product X. Composition and adjoint are linear operations. Convex combinations of positive operators are positive. Problems The space of operators is infinite-dimensional. Verifying positivity of an operator is hard. A. Gahlawat, M. Peet, E. Witrant SOS for Tokamaks: Control of PDEs / 28

12 Solving Linear Operator Inequalities A Finite-Dimensional Subspace Question: How to parameterize the set of operators? Later, we will enforce positivity. Classes of Operators: x R C[ τ,] (Ax)(s) = M(s)x(s)+ N(s, t)x(t)dt M(s) is the multiplier of a Multiplier Operator. N(s,t) is the kernel of an Integral Operator. Question: How to parameterize multiplier and integral operators We consider polynomial multipliers and kernels M(s) = c T D(s) D(s) is a monomial basis c is a vector of decision variables. For a finite basis, the set of operators is finite-dimensional Now, how do we enforce positivity on D A? A. Gahlawat, M. Peet, E. Witrant SOS for Tokamaks: Control of PDEs 2 / 28

13 Optimization of Polynomials Problem: max b T x n subject to A (y)+ x ia i(y) y i= The A i are matrices of polynomials in y. e.g. Using multi-index notation, A i(y) = α A i,α y α Computationally Intractable The problem: Is p(x) for all x R n? (i.e. p R + [x]? ) is NP-hard. A. Gahlawat, M. Peet, E. Witrant SOS for Tokamaks: Control of PDEs 3 / 28

14 Sum-of-Squares (SOS) Programming Problem: max b T x n subject to A (y)+ x i A i (y) Σ s i= Definition 3. Σ s R + [x] is the cone of sum-of-squares matrices. If S Σ s, then for some G i R[x], r S(y) = G i (y) T G i (y) i= Computationally Tractable: S Σ s is an SDP constraint. A. Gahlawat, M. Peet, E. Witrant SOS for Tokamaks: Control of PDEs 4 / 28

15 SOS Programming: Why is M Σ s an SDP? Let Zd n (x) be the vector of monomial bases in dimension n of degree d or less. e.g., if x R 2, then Z2(x) T = [ ] x x 2 x x 2 x 2 x 2 2 and [ [ Z 2 x x (x)t = 2 Z = (x) x x2] Z (x) ] Feasibility Test: Lemma 4. Suppose M is polynomial of degree 2d. M Σ s iff there exists some Q such that M(x) = Z d (x) T QZ d (x). A. Gahlawat, M. Peet, E. Witrant SOS for Tokamaks: Control of PDEs 5 / 28

16 Problem: Optimizing Locally Positive Functions Solution: Positivstellensatz Results Let X := { x : p } i(x) i =,...,k q j (x) = j =,...,m Theorem 5 (Putinar). Suppose X is compact+ and v(x) for x X. Then there exist s i Σ s and t i R[x] such that v(x) k m s i (x)p i (x)+ t i (x)q i (x) s = i= i= A. Gahlawat, M. Peet, E. Witrant SOS for Tokamaks: Control of PDEs 6 / 28

17 Control of Tokamaks Choosing Our Operators Recall that for the Tokamak problem, we have (Aψ)(x) := µ a 2 x (Bj ni )(x) := x and X = L 2 [,] with domain ( η (x) ( η (x)j eni (x) ) ) x (xψ(x)) D A = {y L 2 [,] : y,y x,y xx L 2 [,],y() = y() = }. We parameterize our operator P simply using a multiplier as (Px)(s) = M(s)x(s) An important choice is that of the controller: K : D A X (Kψ)(x) = K (x)ψ(x)+ d dx (Z 2(x)ψ(x)) The structure of K and P imposes a structure on Z = KP: (Zψ)(x) = (KPψ)(x) = Z (x)ψ(x)+ d dx (Z 2(x)ψ(x)) A. Gahlawat, M. Peet, E. Witrant SOS for Tokamaks: Positivity of Operators 7 / 28

18 Control of Tokamaks Enforcing Positivity First Constraint: Enforcing positivity of P is easy. if and only if x,px = x(s)m(s)x(s)ds M(s) for alls [,] Second Constraint: Enforcing negativity of PA +AP +BZ +Z B can be reformulated as x,(pa +AP+BZ+Z B)x = x(s)r (s)x(s)ds+ ẋ(s)r 2 (s)ẋ(s)ds where R and R 2 are linear in variables Z, Z 2, and M (Next Slide). We require both R (s) and R 2 (s) for all s [,]. A. Gahlawat, M. Peet, E. Witrant SOS for Tokamaks: Positivity of Operators 8 / 28

19 Enforcing Positivity As promised: R (s) := µ a 2b where b (x, ( x, ( d )M(x)+b 2 x, dx ( d )Z (x)+b 3 x, dx ) d Z 2 (x) dx R 2 (s) := µ a 2c (x)m(x)+c 2 (x)z 2 (x). ) d ( η,x = f(x) dx x η ) ( x 2 +f (x) η ) x +η,x +f (x)η + f(x)η ) x d b 2 (x, = f (x)+f(x) d dx dx, ) b 3 (x, d dx d dx +( ) d 2 f(x)η +f(x)η,x dx 2, = η,x f (x)+η f (x)+η,x f(x) d dx +η f(x) d2 dx 2, c (x) = η f(x),c 2 (x) = 2η f(x) and f(x) = x 2 ( x). A. Gahlawat, M. Peet, E. Witrant SOS for Tokamaks: Positivity of Operators 9 / 28

20 Simulation safety factor profile safety factor error q time x.5 Figure: Time evolution of the safety factor profile or the q-profile. q q ref time.5 x Figure: Time evolution of the q-profile Error, q(x,t) q ref (x). Here x is the normalized spatial variable. A. Gahlawat, M. Peet, E. Witrant SOS for Tokamaks: Positivity of Operators 2 / 28

21 Simulation Note: Although not discussed, we also constraint j ni 3MA We use finite difference methods to obtain the numerical solution of the closed loop system. To simulate the controller under realistic scenarios we use the plasma resistivity η (x,t) data from the Tore Supra Tokamak. The other data used from the Tore Supra Tokamak are: I p (plasma current) =.6MA B φ (toroidal magnetic field at the plasma center) =.9T a(radius of the last closed magnetic surface) =.72m R (magnetic center location) = 2.38m. From this data the boundary conditions for ψ x (x,t) are calculated to be ψ x (,t) = and ψ x (,t) =.285. A. Gahlawat, M. Peet, E. Witrant SOS for Tokamaks: Positivity of Operators 2 / 28

22 Simulation ψ x profile ψ x error ψ x sim.5.5 time.5 x Figure: Time evolution of ψ x-profile. ψ x ψ x,ref time.5 x Figure: ψ x-profile error, ψ x ψ x,ref. Here ψ x,ref is obtained from the reference q-profile, q ref. A. Gahlawat, M. Peet, E. Witrant SOS for Tokamaks: Positivity of Operators 22 / 28

23 Simulation Control effort x 6 4 j eni time x.5 Figure: External non-inductive current deposit, j eni(x,t). A. Gahlawat, M. Peet, E. Witrant SOS for Tokamaks: Positivity of Operators 23 / 28

24 Ongoing Work The Non-Inductive Source Term We can improve our model of actuator control ( R η (x,t)j ni (x,t) ) x The control is via the Non-Inductive Source Term, j ni. Spatially-distributed A sum of Gaussians j ni (x,t) = a e (x b )2 c +a 2 e (x b 2 )2 c 2 +a 3 e (x b 3 )2 c 3 We can parameterize a Gaussian as a e (x b ) 2 c = ap a (x)+bp b (x)+cp c (x) = [ p a (x) p b (x) p c (x) ] We can look for a controller as u = K(x)ψ x (x)+ d dx (K 2(x)ψ(x))dx a b = P(x) T u c A. Gahlawat, M. Peet, E. Witrant SOS for Tokamaks: Positivity of Operators 24 / 28

25 Ongoing Work: Observing PDE systems Heat Equation Example Problem: Feedback requires a knowledge of the heat distribution. Sensors can only measure heat at a single point. Consider the dynamics of heat flux. w t (x,t) = w xx (x,t) Suppose we only observe at a single point, y(t) = w(,t) and control the gradient at the same point: w z (,t) = u(t) To know the state, we want Luenberger Observer, L: ẑ(t) = (A+LC +BF)ẑ(t) Ly(y) with both A+LC and A+BF stable. Then ẑ(s,t) z(s,t). A. Gahlawat, M. Peet, E. Witrant SOS for Tokamaks: Positivity of Operators 25 / 28

26 Ongoing Work: Observer-Based Controller The Heat Equation Observer state estimate Estimation error.5.5 w obs (z,t).5.5 w(z,t) w o (z,t) time.2.4 z time.5 z Figure: Estimate of the State Figure: Error in the Observed State A. Gahlawat, M. Peet, E. Witrant SOS for Tokamaks: Positivity of Operators 26 / 28

27 Ongoing Work: Observer-Based Controller The Heat Equation System solution Control error 2 2 w(z,t) w(z,t) w ref (z,t) time.2.4 z time.2.4 z.6.8 Figure: Effect of Observer-Based Boundary Control Figure: Error in Observer-Based Boundary Control A. Gahlawat, M. Peet, E. Witrant SOS for Tokamaks: Positivity of Operators 27 / 28

28 Concluding Remarks: Research Directions Directions: Theory of Linear Operator Inequalities Duality Optimal H Control Other Research: Immunology/Cancer Identify Feedback Mechanisms Decentralized Control Parallel Algorithms for SOS/Polya GPU Computing Analysis/Synthesis Tokamaks Observers Temperature/Density Coupling RF-Heating Some algorithms are available for download at: Thanks for Listening A. Gahlawat, M. Peet, E. Witrant SOS for Tokamaks: Conclusion 28 / 28