INVERSE PROBLEMS IN AERODYNAMICS AND CONTROL THEORY

Transcription

1 INVERSE PROBLEMS IN AERODYNAMICS AND CONTROL THEORY Antny Jamesn Department f Aernautics and Astrnautics Stanfrd University, CA Internatinal Cnference n Cntrl, PDEs and Scientific Cmputing Dedicated t J. L. Lins Beijing, China September 10-13, 2004

2 SOLUTION OF THE INVERSE PROBLEM FOR AIRFOIL DESIGN IN IDEAL FLOW

3 Slutin f the Inverse Prblem fr Airfil Design in Ideal Flw Lighthill-James-Jamesn Methd D D C C 1a: z-plane. 1b: σ-plane. Transfrm the prfile t a unit circle by a cnfrmal mapping.

4 Slutin f the Inverse Prblem fr Airfil Design in Ideal Flw On the prfile the velcity is q = 1 h φ, where φ is the ptential fr circulatry flw past a circle, and h is the mdulus f the transfrmatin, dz h = dσ. (1) Nw set q = q t, where q t is the target velcity, and slve fr h, h = φ q t = q c q t (2) where q c is the velcity ver the circle. Since the functin defining the mapping is analytic, the knwledge f h n the bundary is sufficient t define the mapping cmpletely

5 Slutin f the Inverse Prblem fr Airfil Design in Ideal Flw Because the mapping shuld be ne-t-ne at infinity, lg dz 0 at (3) dσ Thus it can be defined by a Laurent series with inverse pwers On C this can be expanded as lg dz dσ = n= c n σn, (4) lg ds (α dθ i θ π ) = (a n cs(nθ) b n sin(nθ) i (b n cs(nθ) a n sin(nθ)) (5) 2 where α is the surface tangent angle and s is the arc length. Als φ = r 1 cs θ Γ θ is knwn (6) r 2π where Γ is the circulatin. Nw a n and b n can be determined as the Furier cefficients f lg h = lg q c q t

6 Slutin f the Inverse Prblem fr Airfil Design in Ideal Flw Cnstraints n the target velcity T preserve q, Hence, Als, integratin arund a circuit gives Clsure c 1 = 0 dz dσ z = 1 at. c 0 = 0. dz dσ dσ = 2πic 1 Thus, we require lg(qt )dθ = 0 lg(qt ) cs(θ)dθ = 0 lg(qt ) sin(θ)dθ = 0 (7)

7 Slutin f the Inverse Prblem fr Airfil Design in Ideal Flw Here we need q t (θ) as a functin f the angle arund the circle. Hwever, we can determine φ as φ = q t ds where the integral is ver the arc length f the desired prfile. This implicitly defines q t as a functin f φ. Since φ is knwn, ne can find q t as a functin f θ by a Newtn iteratin. This step, missing frm the riginal Lighthill methd, was prvided by James and Jamesn. Cp W AIRFOIL ALPHA CL CD CM GRID 80 Inverse calculatin, recvering Whitcmb airfil

8 AIRFOIL DESIGN FOR TRANSONIC POTENTIAL FLOW VIA CONTROL THEORY

9 Airfil Design fr Transnic Ptential Flw via Cntrl Thery Frmulatin f the Inverse Prblem as an Optimizatin Prblem Because a shape des nt necessarily exist fr an arbitrary pressure distributin the inverse prblem may be ill psed if ne tried directly t enfrce a specified pressure as a bundary cnditin. This difficulty is avided by frmulating the inverse prblem as an ptimizatin prblem in which ne seeks a shape which minimized a cst functin such as I = 1 2 (p pt ) 2 ds where p and p t are the actual and target pressures.

10 Airfil Design fr Transnic Ptential Flw via Cntrl Thery Shape Design Based n Cntrl Thery Regard the wing as a device t generate lift (with minimum drag) by cntrlling the flw Apply thery f ptimal cntrl f systems gverned by PDEs (Lins) with bundary cntrl (the wing shape) Merge cntrl thery and CFD

11 Airfil Design fr Transnic Ptential Flw via Cntrl Thery Autmatic Shape Design via Cntrl Thery Apply the thery f cntrl f partial differential equatins (f the flw) by bundary cntrl (the shape) Find the Frechet derivative (infinite dimensinal gradient) f a cst functin (perfrmance measure) with respect t the shape by slving the adjint equatin in additin t the flw equatin Mdify the shape in the sense defined by the smthed gradient Repeat until the perfrmance value appraches an ptimum

12 Airfil Design fr Transnic Ptential Flw via Cntrl Thery Aerdynamic Shape Optimizatin: Gradient Calculatin Fr the class f aerdynamic ptimizatin prblems under cnsideratin, the design space is essentially infinitely dimensinal. Suppse that the perfrmance f a system design can be measured by a cst functin I which depends n a functin F(x) that describes the shape,where under a variatin f the design δf(x), the variatin f the cst is δi. Nw suppse that δi can be expressed t first rder as where G(x) is the gradient. Then by setting ne btains an imprvement δi = G(x)δF(x)dx δf(x) = λg(x) δi = λ G 2 (x)dx unless G(x) = 0. Thus the vanishing f the gradient is a necessary cnditin fr a lcal minimum.

13 Airfil Design fr Transnic Ptential Flw via Cntrl Thery Symblic Develpment f the Adjint Methd Let I be the cst (r bjective) functin where The first variatin f the cst functin is I = I(w, F) w = flw field variables F = grid variables δi = I T δw I T δf w F The flw field equatin and its first variatin are R(w, F) = 0 δr = 0 = R δw w R F δf

14 Airfil Design fr Transnic Ptential Flw via Cntrl Thery Symblic Develpment f the Adjint Methd (cnt.) Intrducing a Lagrange Multiplier, ψ, and using the flw field equatin as a cnstraint δi = I T δw I T δf ψ T R δw R δf w F w F = T I ψ T R w w T δw I ψ T F By chsing ψ such that it satisfies the adjint equatin we have R w T ψ = I w, δf R F δf T I δi = ψ T R F F This reduces the gradient calculatin fr an arbitrarily large number f design variables at a single design pint t One Flw Slutin One Adjint Slutin

15 Airfil Design fr Transnic Ptential Flw via Cntrl Thery The Adjint Equatin fr Transnic Ptential Flw Cnsider the case f tw-dimensinal cmpressible inviscid flw. In the absence f shck waves, an initially irrtatinal flw will remain irrtatinal, and we can assume that the velcity vectr q is the gradient f a ptential φ. In the presence f weak shck waves this remains a fairly gd apprximatin. D C D C z-plane Figure 1: Cnfrmal Mapping. σ-plane Let p, ρ, c, and M be the pressure, density, speed-f-sund, and Mach number q/c. Then the ptential flw equatin is (ρ φ) = 0, (8) where the density is given by while ρ = 1 γ 1 2 M 2 ( 1 q 2 ) (γ 1), (9) p = ργ, c 2 = γp γm 2 ρ. (10) Here M is the Mach number in the free stream, and the units have been chsen s that p and q have a value f unity in the far field. 1

16 Airfil Design fr Transnic Ptential Flw via Cntrl Thery Suppse that the dmain D exterir t the prfile C in the z-plane is cnfrmally mapped n t the dmain exterir t a unit circle in the σ-plane as sketched in Figure 1. Let R and θ be plar crdinates in the σ-plane, and let r be the inverted radial crdinate 1 R. Als let h be the mdulus f the derivative f the mapping functin Nw the ptential flw equatin becmes h = dz dσ. (11) θ (ρφ θ) r r (rρφ r) = 0 in D, (12) where the density is given by equatin (9), and the circumferential and radial velcity cmpnents are while u = rφ θ h, v = r2 φ r h, (13) q 2 = u 2 v 2. (14) The cnditin f flw tangency leads t the Neumann bundary cnditin v = 1 φ = 0 n C. (15) h r In the far field, the ptential is given by an asympttic estimate, leading t a Dirichlet bundary cnditin at r = 0.

17 Airfil Design fr Transnic Ptential Flw via Cntrl Thery Suppse that it is desired t achieve a specified velcity distributin q d n C. Intrduce the cst functin I = 1 2 (q q C d) 2 dθ, The design prblem is nw treated as a cntrl prblem where the cntrl functin is the mapping mdulus h, which is t be chsen t minimize I subject t the cnstraints defined by the flw equatins (8 15). A mdificatin δh t the mapping mdulus will result in variatins δφ, δu, δv, and δρ t the ptential, velcity cmpnents, and density. The resulting variatin in the cst will be where, n C, q = u. Als, δi = C (q q d) δq dθ, (16) δu = r δφ θ h uδh h, δv = r2δφ r h vδh h, while accrding t equatin (9) It fllws that δφ satisfies where L θ ρ Lδφ = θ 1 u2 c 2 ρ u = ρu c 2, θ ρuv ρm 2 φ θ δh h ρ v = ρv c 2. r r c r 2 r r r ρ ρm 2 rφ r δh h 1 v2 c 2 r r ρuv c 2 θ. (17)

18 Airfil Design fr Transnic Ptential Flw via Cntrl Thery Then, if ψ is any peridic differentiable functin which vanishes in the far field, ψ D r 2L δφ ds = ρm 2 φ ψ δh ds, (18) D h where ds is the area element r dr dθ, and the right hand side has been integrated by parts. Nw we can augment equatin (16) by subtracting the cnstraint (18). The auxiliary functin ψ then plays the rle f a Lagrange multiplier. Thus, δi = (q q C d) q δh h dθ δφ q q d dθ ψ C θ h D r 2Lδφ ds ρm 2 φ ψ δh D h ds. Nw suppse that ψ satisfies the adjint equatin with the bundary cnditin Then, integrating by parts, ψ r = 1 ρ θ Lψ = 0 in D (19) q q d h n C. (20) ψ D r 2Lδφ ds = ρψ C rδφ dθ, and δi = (q q C d) q δh h dθ ρm 2 φ ψ δh ds. D h (21) Here the first term represents the direct effect f the change in the metric, while the area integral represents a crrectin fr the effect f cmpressibility. When the secnd term is deleted the methd reduces t a variatin f Lighthill s methd.

19 Airfil Design fr Transnic Ptential Flw via Cntrl Thery Equatin (21) can be further simplified t represent δi purely as a bundary integral because the mapping functin is fully determined by the value f its mdulus n the bundary. Set where lg dz dσ = F iβ, and F = lg dz dσ = lg h, δf = δh h. Then F satisfies Laplace s equatin F = 0 in D, and if there is n stretching in the far field, F 0. Als δf satisfies the same cnditins.

20 Airfil Design fr Transnic Ptential Flw via Cntrl Thery Intrduce anther auxiliary functin P which satisfies P = ρm 2 ψ ψ in D, (22) and P = 0 n C. Then, the area integral in equatin (21) is and finally D P δf ds = C P δf r dθ P δf ds, D δi = C G δf dθ, where F c is the bundary value f F, and This suggests setting G = P r (q q d) q. (23) δf c = λg s that if λ is a sufficiently small psitive quantity δi = C λg2 dθ < 0.

21 Airfil Design fr Transnic Ptential Flw via Cntrl Thery Arbitrary variatins in F cannt, hwever, be admitted. The cnditin that F 0 in the far field, and als the requirement that the prfile shuld be clsed, imply cnstraints which must be satisfied by F n the bundary C. Suppse that lg ( ) dz dσ is expanded as a pwer series lg dz = c n dσ n= σn, (24) where nly negative pwers are retained, because therwise ( ) dz dσ wuld becme unbunded fr large σ. The cnditin that F 0 as σ implies c = 0. Als, the change in z n integratin arund a circuit is z = dz dσ dσ = 2πi c 1, s the prfile will be clsed nly if c 1 = 0. In rder t satisfy these cnstraints, we can prject G nt the admissible subspace fr F c by setting c = c 1 = 0. (25) Then the prjected gradient G is rthgnal t G G, and if we take it fllws that t first rder δf c = λ G, δi = C λg G dθ = C λ ( G G G) G dθ = C λ G 2 dθ < 0.

22 Airfil Design fr Transnic Ptential Flw via Cntrl Thery If the flw is subsnic, this prcedure shuld cnverge tward the desired speed distributin since the slutin will remain smth, and n unbunded derivatives will appear. If, hwever, the flw is transnic, ne must allw fr the appearance f shck waves in the trial slutins, even if q d is smth. Then q q d is nt differentiable. This difficulty can be circumvented by a mre sphisticated chice f the cst functin. Cnsider the chice I = 1 2 C λ 1 Z 2 λ dz 2 dθ 2 dθ, (26) where λ 1 and λ 2 are parameters, and the peridic functin Z(θ) satisfies the equatin Then λ 1 Z d dθ λ dz 2 dθ = q q d. (27) δi = dz d λ C 1 Z δz λ 2 dθ dθ δz dθ = Z λ C 1 δz d dθ λ d 2 dθ δz dθ = C Z δq dθ.

23 Airfil Design fr Transnic Ptential Flw via Cntrl Thery Thus, Z replaces q q d in the previus frmulas, and if ne mdifies the bundary cnditin (20) t ψ r = 1 ρ θ the frmula fr the gradient becmes Z n C, (28) h G = P Zq (29) r instead f equatin (23). Smthing can als be intrduced directly in the descent prcedure by chsing δf c t satisfy δf c θ β θ δf c = λg, (30) where β is a smthing parameter. Then t first rder 1 G δf = δf 2 c δf c λ θ β θ δf c dθ = 1 δfc 2 β 2 λ θ δf c dθ < 0. The smthed crrectin shuld nw be prjected nt the admissible subspace.

24 Airfil Design fr Transnic Ptential Flw via Cntrl Thery The final design prcedure is thus as fllws. Chse an initial prfile and crrespnding mapping functin F. Then: 1. Slve the flw equatins (8 15) fr φ, u, v, q, ρ. 2. Slve the rdinary differential equatin (27) fr Z. 3. Slve the adjint equatin (17 and 19) r ψ subject t the bundary cnditin (28). 4. Slve the auxiliary Pissn equatin (22) fr P. 5. Evaluate G by equatin (29) 6. Crrect the bundary mapping functin F c by δf c calculated frm equatin (30), prjected nt the admissible subspace defined by (25). 7. Return t step 1.

25 THREE DIMENSIONAL TRANSONIC INVERSE DESIGN USING THE EULER EQUATIONS

26 Three Dimensinal Transnic Inverse Design using The Euler Equatins Design using the Euler Equatins The three-dimensinal Euler equatins may be written as where w = ρ ρu 1 ρu 2 ρu 3 ρe w t f i x i = 0 in D, (31), f i = and δ ij is the Krnecker delta functin. Als, ρu i ρu i u 1 pδ i1 ρu i u 2 pδ i2 ρu i u 3 pδ i3 ρu i H ( u 2 i ) p = (γ 1) ρ E 1 2, (33) and ρh = ρe p (34) where γ is the rati f the specific heats. (32)

27 Three Dimensinal Transnic Inverse Design using The Euler Equatins Design using the Euler Equatins In rder t simplify the derivatin f the adjint equatins, we map the slutin t a fixed cmputatinal dmain with crdinates ξ 1, ξ 2, ξ 3 where and K ij = x i, J = det (K), K 1 ξ j S = JK 1. ij = The elements f S are the cfactrs f K, and in a finite vlume discretizatin they are just the face areas f the cmputatinal cells prjected in the x 1, x 2, and x 3 directins. Using the permutatin tensr ɛ ijk we can express the elements f S as ξ i x j S ij = 1 2 ɛ jpqɛ irs x p ξ r x q ξ s. (35),

28 Three Dimensinal Transnic Inverse Design using The Euler Equatins Design using the Euler Equatins Then ξ i S ij = 1 2 ɛ 2 x p x q jpqɛ irs x p 2 x q ξ r ξ i ξ s ξ r ξ s ξ i = 0. (36) Als in the subsequent analysis f the effect f a shape variatin it is useful t nte that S 1j = ɛ jpq x p ξ 2 x q ξ 3, S 2j = ɛ jpq x p ξ 3 x q ξ 1, S 3j = ɛ jpq x p ξ 1 x q ξ 2. (37)

29 Three Dimensinal Transnic Inverse Design using The Euler Equatins Design using the Euler Equatins Nw, multiplying equatin(31) by J and applying the chain rule, J w R (w) = 0 t (38) where f j R (w) = S ij = (S ij f j ), ξ i ξ i (39) using (36). We can write the transfrmed fluxes in terms f the scaled cntravariant velcity cmpnents U i = S ij u j as F i = S ij f j = ρu i ρu i u 1 S i1 p ρu i u 2 S i2 p ρu i u 3 S i3 p ρu i H.

30 Three Dimensinal Transnic Inverse Design using The Euler Equatins Design using the Euler Equatins Fr simplicity, it will be assumed that the prtin f the bundary that underges shape mdificatins is restricted t the crdinate surface ξ 2 = 0. Then equatins fr the variatin f the cst functin and the adjint bundary cnditins may be simplified by incrprating the cnditins n 1 = n 3 = 0, n 2 = 1, db ξ = dξ 1 dξ 3, s that nly the variatin δf 2 needs t be cnsidered at the wall bundary. The cnditin that there is n flw thrugh the wall bundary at ξ 2 = 0 is equivalent t U 2 = 0, s that δu 2 = 0 when the bundary shape is mdified. Cnsequently the variatin f the inviscid flux at the bundary reduces t 0 0 δf 2 = δp S 21 S 22 p δs 21 δs 22. (40) S 23 δs

31 Three Dimensinal Transnic Inverse Design using The Euler Equatins Design using the Euler Equatins In rder t design a shape which will lead t a desired pressure distributin, a natural chice is t set I = 1 2 (p p B d) 2 ds where p d is the desired surface pressure, and the integral is evaluated ver the actual surface area. In the cmputatinal dmain this is transfrmed t where the quantity I = 1 2 B w (p p d ) 2 S 2 dξ 1 dξ 3, S 2 = S 2j S 2j dentes the face area crrespnding t a unit element f face area in the cmputatinal dmain.

32 Three Dimensinal Transnic Inverse Design using The Euler Equatins Design using the Euler Equatins In the cmputatinal dmain the adjint equatin assumes the frm C T i ψ ξ i = 0 (41) where f j C i = S ij w. T cancel the dependence f the bundary integral n δp, the adjint bundary cnditin reduces t ψ j n j = p p d (42) where n j are the cmpnents f the surface nrmal n j = S 2j S 2.

33 Three Dimensinal Transnic Inverse Design using The Euler Equatins Design using the Euler Equatins This amunts t a transpiratin bundary cnditin n the c-state variables crrespnding t the mmentum cmpnents. Nte that it impses n restrictin n the tangential cmpnent f ψ at the bundary. We find finally that δi = ψ T δs D ij f j dd ξ i (δs B 21 ψ 2 δs 22 ψ 3 δs 23 ψ 4 ) p dξ 1 dξ 3. (43) W Here the expressin fr the cst variatin depends n the mesh variatins thrughut the dmain which appear in the field integral. Hwever, the true gradient fr a shape variatin shuld nt depend n the way in which the mesh is defrmed, but nly n the true flw slutin. In the next sectin we shw hw the field integral can be eliminated t prduce a reduced gradient frmula which depends nly n the bundary mvement.

34 Three Dimensinal Transnic Inverse Design using The Euler Equatins The Reduced Gradient Frmulatin Cnsider the case f a mesh variatin with a fixed bundary. Then, δi = 0 but there is a variatin in the transfrmed flux, δf i = C i δw δs ij f j. Here the true slutin is unchanged. Thus, the variatin δw is due t the mesh mvement δx at each mesh pint. Therefre δw = w δx = w x j δx j (= δw ) and since δf i = 0, ξ i it fllws that (δs ij f j ) = (C i δw ). ξ i ξ i (44) It has been verified by Jamesn and Kim that this relatin hlds in the general case with bundary mvement. * Reductin f the Adjint Gradient Frmula in the Cntinuus Limit, A.Jamesn and S. Kim, 41 st AIAA Aerspace Sciences Meeting & Exhibit, AIAA Paper , Ren, NV, January 6-9, 2003.

35 Three Dimensinal Transnic Inverse Design using The Euler Equatins The Reduced Gradient Frmulatin Nw Here n the wall bundary D φt δrdd = D φt ξ i C i (δw δw ) dd = B φt C i (δw δw ) db D φ T ξ i C i (δw δw ) dd. (45) C 2 δw = δf 2 δs 2j f j. (46) Thus, by chsing φ t satisfy the adjint equatin and the adjint bundary cnditin, we reduce the cst variatin t a bundary integral which depends nly n the surface displacement: δi = B W ψ T (δs 2j f j C 2 δw ) dξ 1 dξ 3 B W (δs 21 ψ 2 δs 22 ψ 3 δs 23 ψ 4 ) p dξ 1 dξ 3. (47)

36 Three Dimensinal Transnic Inverse Design using The Euler Equatins Sblev Gradient Key issue fr successful implementatin f the Cntinuus adjint methd. Define the gradient with respect t the Sblev inner prduct Set δi = < ḡ, δf > = (ḡδf ɛḡ δf ) dx δf = λḡ, δi = λ < ḡ, ḡ > This apprximates a cntinuus descent prcess df dt = ḡ The Sblev gradient ḡ is btained frm the simple gradient g by the smthing equatin ḡ x ɛ ḡ x = g Cntinuus descent path

37 Three Dimensinal Transnic Inverse Design using The Euler Equatins Outline f the Design Prcess The design prcedure can finally be summarized as fllws: 1. Slve the flw equatins fr ρ, u 1, u 2, u 3, p. 2. Slve the adjint equatins fr ψ subject t apprpriate bundary cnditins. 3. Evaluate G and calculate the crrespnding Sblev gradient Ḡ. 4. Prject Ḡ int an allwable subspace that satisfies any gemetric cnstraints. 5. Update the shape based n the directin f steepest descent. 6. Return t 1 until cnvergence is reached. Flw Slutin Adjint Slutin Gradient Calculatin Repeat the Design Cycle until Cnvergence Sblev Gradient Shape & Grid Mdificatin Design cycle

38 Three Dimensinal Transnic Inverse Design using The Euler Equatins NACA 0012 WING TO ONERA M6 TARGET NACA 0012 WING TO ONERA M6 TARGET Mach: Alpha: CL: CD: CM: Design: 0 Residual: E01 Grid: 193X 33X 49 Cp = -2.0 NACA 0012 WING TO ONERA M6 TARGET Mach: Alpha: CL: CD: CM: Design: 100 Residual: E-01 Grid: 193X 33X 49 Cp = -2.0 Tip Sectin: 92.3% Semi-Span Cl: Cd: Cm: Tip Sectin: 92.3% Semi-Span Cl: Cd: Cm: Cp = -2.0 Cp = -2.0 Cp = -2.0 Cp = -2.0 Rt Sectin: 6.2% Semi-Span Cl: Cd: Cm: Mid Sectin: 49.2% Semi-Span Cl: Cd: Cm: Rt Sectin: 6.2% Semi-Span Cl: Cd: Cm: Mid Sectin: 49.2% Semi-Span Cl: Cd: Cm: Starting wing: NACA 0012 Target wing: ONERA M6 This is a difficult prblem because f the presence f the shck wave in the target pressure and because the prfile t be recvered is symmetric while the target pressure is nt.

39 The pressure distributin f the final design match the specified target, even inside the shck. Starting Cp Target Cp GRID 192X32 NDES 0 RES0.598E-03 GMAX 0.100E-05 GRID 192X32 NDES 100 RES0.128E-05 GMAX 0.100E-05 CL CD CM CL CD CM MACH ALPHA Z 0.11 MACH ALPHA Z 0.11 NACA 0012 WING TO ONERA M6 TARGET NACA 0012 WING TO ONERA M6 TARGET Cp 0.1E01 0.8E00 0.4E00 -.1E E00 -.8E00 -.1E01 -.2E01 -.2E01 Cp 0.1E01 0.8E00 0.4E00 -.1E E00 -.8E00 -.1E01 -.2E01 -.2E01 Pressure Prfiles at 11% span Three Dimensinal Transnic Inverse Design using The Euler Equatins

40 The pressure distributin f the final design match the specified target, even inside the shck. Starting Cp Target Cp GRID 192X32 NDES 0 RES0.598E-03 GMAX 0.100E-05 GRID 192X32 NDES 100 RES0.128E-05 GMAX 0.100E-05 CL CD CM CL CD CM MACH ALPHA Z 0.30 MACH ALPHA Z 0.30 NACA 0012 WING TO ONERA M6 TARGET NACA 0012 WING TO ONERA M6 TARGET Cp 0.1E01 0.8E00 0.4E00 -.1E E00 -.8E00 -.1E01 -.2E01 -.2E01 Cp 0.1E01 0.8E00 0.4E00 -.1E E00 -.8E00 -.1E01 -.2E01 -.2E01 Pressure Prfiles at 30% span Three Dimensinal Transnic Inverse Design using The Euler Equatins

41 The pressure distributin f the final design match the specified target, even inside the shck. Starting Cp Target Cp GRID 192X32 NDES 0 RES0.598E-03 GMAX 0.100E-05 GRID 192X32 NDES 100 RES0.128E-05 GMAX 0.100E-05 CL CD CM CL CD CM MACH ALPHA Z 0.48 MACH ALPHA Z 0.48 NACA 0012 WING TO ONERA M6 TARGET NACA 0012 WING TO ONERA M6 TARGET Cp 0.1E01 0.8E00 0.4E00 -.1E E00 -.8E00 -.1E01 -.2E01 -.2E01 Cp 0.1E01 0.8E00 0.4E00 -.1E E00 -.8E00 -.1E01 -.2E01 -.2E01 Pressure Prfiles at 48% span Three Dimensinal Transnic Inverse Design using The Euler Equatins

42 The pressure distributin f the final design match the specified target, even inside the shck. Starting Cp Target Cp GRID 192X32 NDES 0 RES0.598E-03 GMAX 0.100E-05 GRID 192X32 NDES 100 RES0.128E-05 GMAX 0.100E-05 CL CD CM CL CD CM MACH ALPHA Z 0.67 MACH ALPHA Z 0.67 NACA 0012 WING TO ONERA M6 TARGET NACA 0012 WING TO ONERA M6 TARGET Cp 0.1E01 0.8E00 0.4E00 -.1E E00 -.8E00 -.1E01 -.2E01 -.2E01 Cp 0.1E01 0.8E00 0.4E00 -.1E E00 -.8E00 -.1E01 -.2E01 -.2E01 Pressure Prfiles at 67% span Three Dimensinal Transnic Inverse Design using The Euler Equatins

43 The pressure distributin f the final design match the specified target, even inside the shck. Starting Cp Target Cp GRID 192X32 NDES 0 RES0.598E-03 GMAX 0.100E-05 GRID 192X32 NDES 100 RES0.128E-05 GMAX 0.100E-05 CL CD CM CL CD CM MACH ALPHA Z 0.86 MACH ALPHA Z 0.86 NACA 0012 WING TO ONERA M6 TARGET NACA 0012 WING TO ONERA M6 TARGET Cp 0.1E01 0.8E00 0.4E00 -.1E E00 -.8E00 -.1E01 -.2E01 -.2E01 Cp 0.1E01 0.8E00 0.4E00 -.1E E00 -.8E00 -.1E01 -.2E01 -.2E01 Pressure Prfiles at 86% span Three Dimensinal Transnic Inverse Design using The Euler Equatins

44 SOLUTION OF THE INVERSE PROBLEM OF LINEAR OPTIMAL CONTROL WITH POSITIVENESS CONDITIONS AND RELATION TO SENSITIVITY

45 Slutin f the Inverse Prblem f Linear Optimal Cntrl with Psitiveness Cnditins and Relatin t Sensitivity Frmulatin Let ẋ = Ax Bu x R m ; u R n, (48) u = Dx, (49) be a given cntrl. It is desired t find a perfrmance index J = x T (t f )F x(t f ) t f t 0 (x T Qx u T Ru)dt, (50) with R = R T > 0, Q = Q T 0, which is minimized by u. The slutin f this prblem withut psitiveness cnditins n Q is given in references,. If a perfrmance index (50) exists which is minimized by (49), then where P is a symmetric matrix satisfying RD = B T P, (51) P = A T P P A D T RD Q, P (t f ) = F. (52) * Optimality f Linear Cntrl Systems (with E. Kreindler), IEEE Trans. n Autmatic Cntrl, Vl. AC-17, 1972, pp Inverse Prblem f Linear Optimal Cntrl (with E. Kreindler), SIAM J. n Cntrl, Vl. 11, 1973, pp

46 Slutin f the Inverse Prblem f Linear Optimal Cntrl with Psitiveness Cnditins and Relatin t Sensitivity Frmulatin If (52) is multiplied n the left by x T and n the right by x, then substituting frm (48) and (51) Integrating frm t 1 t t 2 x T Qx u T Ru = d dt (xt P x) (u Dx) T R(u Dx) (53) t 2 t 1 (x T Qxu T Ru)dtx T (t 2 )P (t 2 )x(t 2 ) = x T (t 1 )P (t 1 )x(t 1 ) t 2 t 1 ((u Dx) T R(u Dx))dt. (54) Setting t 1 = t 0, t 2 = t f, P (t f ) = F, it is seen that since the final term is nn-negative. J x T (t 0 )P (t 0 )x(t 0 ), (55) Nte als that n setting t 2 = t f and u = Dx it fllws that if Q C and F 0, then P (t 1 ) 0 fr all t 1 < t f because the left side is nn-negative. Als multiplying (51) n the right by B, the symmetry f P is seen t imply the symmetry f RDB.

47 Slutin f the Inverse Prblem f Linear Optimal Cntrl with Psitiveness Cnditins and Relatin t Sensitivity Frmulatin The cnditin fr the existence f R = R T > 0 and P = P T > 0 satisfying (51) are given in reference. They are: DB has n independent real eigen-vectrs (A1) The eigen-values f DB are nn-psitive (A2) r DB = r D, where r D dentes the rank f D, etc. (A3) Fr P > 0 A3 is replaced by r DB = r D = r B (A3 ) If R = R T > 0 is given, then the cnditins fr a slutin P 0 t (51) are RDB is symmetric (B1) RDB 0 (B2) r RDB = r RD (B3) Fr P > 0 B3 is replaced by r RDB = r RD = r B (B3 ) Here (B3) and (B3 ) are simply restatements f (A3) and (A3 ), but (B3) and (B3 ) wuld still be needed fr a case where R is nly nn-negative. * Inverse Prblem f Linear Optimal Cntrl (with E. Kreindler), SIAM J. n Cntrl, Vl. 11, 1973, pp

48 Slutin f the Inverse Prblem f Linear Optimal Cntrl with Psitiveness Cnditins and Relatin t Sensitivity Sensitivity Inequality The prperty that the feedback cntrl minimizes sme perfrmance index (50) with Q 0 is f cnsiderable interest because f its cnnectin with the ability f the cntrl t reduce the sensitivity f the system fr parameter variatins Let x c be the trajectry deviatin resulting frm plant variatins, A and B, when the feedback cntrl (49) is used, and let x be the deviatin when (49) is replaced by an pen lp cntrl which wuld give the same trajectry in the absence f parameter deviatins. Als let A = ɛδa, B = ɛδb, (56) and define δx = lim ɛ 0 x ɛ. Using the equivalence f cntrls when δa = δb = 0 we have δẋ c = (A BD)δx c (δa δbd)x, (57) δẋ 0 = Aδx (δa δbd)x. (58) Whence where δx c = δx δ (59) δ = Aδ BDδx c (60) * On Criteria fr Clsed Lp Sensitivity Reductin (with E. Kreindler), J. Math. Analysis and Applicatins, Vl. 37, 1972, pp

49 Slutin f the Inverse Prblem f Linear Optimal Cntrl with Psitiveness Cnditins and Relatin t Sensitivity Sensitivity Inequality Then t δx T t 0 c DT RDδx c dt t δx T t 0 0 DT RDδx dt t δ T Y δdt, (61) t 0 fr all t if the fllwing cnditin is satisfied C: The sensitivity inequality S y (t) = { t t (u Dx) T R(u Dx) u T Ru } dt t x T Y xdt 0 (62) 0 t 0 hlds, where x is the slutin f (48) with x(t 0 ) = 0 under an arbitrary input u. This fllws n setting u = Dδx c and interpreting x as δ. Nw setting t 1 = t 0, x(t 0 ) = 0, and t 2 = t in (54) it is seen that C hlds with Y = Q when the cntrl (49) minimizes the perfrmance index (50), prvided that P (t) 0. This is in turn ensured if Q 0 and F 0. The nn-negativeness f Q and F thus guarantees a reductin in sensitivity t parameter variatins in the sense f (61).

50 Slutin f the Inverse Prblem f Linear Optimal Cntrl with Psitiveness Cnditins and Relatin t Sensitivity Slutin f the Inverse Prblem with Q 0 It was remarked earlier that Q 0 implies P 0. If R is nt specified, cnditins (A1), (A2), and (A3) are necessary fr a slutin f the inverse prblem. Als (A1), (A2), and (A3 ) are necessary and sufficient fr P > 0, and it will nw be shwn that the existence f a slutin P > 0 t 51) is sufficient fr a slutin with Q 0. We thus have Therem: Cnditins (A1),(A2) and (A3 ) are necessary fr a slutin f the inverse prblem with Q 0. Cnditins (A1), (A2) and (A3 ) are als sufficient. Prf Necessity has already been established. T prve sufficiency bserve that frm (48) d dt (xe αt ) = (A αi)xe αt Bue αt. (63) Thus the cntrl (49) minimizes the perfrmance index If (51) hlds tgether with J = t f t 0 e 2αt (x T Q x u T R u)dt e 2αt f x T (t f )F x(t f ). (64) P = (A αi) T P P (A αi) D T R D Q, P (t f ) = F. (65)

51 Slutin f the Inverse Prblem f Linear Optimal Cntrl with Psitiveness Cnditins and Relatin t Sensitivity Slutin f the Inverse Prblem with Q 0 Under cnditins (A1), (A2) and(a3 ) it is pssible t cnstruct R = R T > 0 and P = P T > 0 satisfying (51). Then Q may be cnstructed as where Q = Q 1 2αP, (66) Q 1 = D T R D A T P P A P. (67) Since P > 0 and Q 1 is a fixed functin f t, it is always pssible t chse α > 0 sufficiently large that Q 0. Als cmparisn f (65) and (66) with (65) shws that the cntrl (49) minimizes (64) and hence (50) n setting Q = Q e 2αt, R = R e 2αt, F = F e 2αt f. If the cntrl (49) satisfies cnditins A1, A2 and A3* then it satisfies the criterin (61) fr sensitivity reductin fr sme Y 0. Observe that the prcedure f the abve Therem gives a jint slutin fr Q and R, but cannt be used t cnstruct Q 0 when R is given. In particular, if the system is cnstant it leads t a perfrmance with an expnential time weighting factr e 2αt, α > 0, and establishes the sensitivity criterin (61) with a similar factr.

52 Slutin f the Inverse Prblem f Linear Optimal Cntrl with Psitiveness Cnditins and Relatin t Sensitivity Slutin f the Inverse Prblem with Q 0 fr given R We nw cnsider methds f slving fr Q when R is a given matrix satisfying cnditins (B1), (B2) and (B3). T find additinal requirements n R fr Q 0 multiply (52) n the left by B T. Then using (51) But differentiating (51) Thus where B T P = B T A T P RDA B T D T RD B T Q. (68) B T P ḂT P = d (RD). (69) dt B T Q = L, (70) L = B T D T RD RDA (B T A T ḂT )P d (RD). (71) dt Als multiplying (71) n the right by B and again using (51) where B T QB = M, (72) M = B T D T RDB RDAB B T A T D T R d dt (RDB) RDḂ ḂT D T R. (73)

53 Slutin f the Inverse Prblem f Linear Optimal Cntrl with Psitiveness Cnditins and Relatin t Sensitivity Slutin f the Inverse Prblem with Q 0 fr given R Since M depends nly n the system matrices and R, and B T QB 0 if Q 0, the necessary cnditin fr Q 0 is: M 0 where M is defined in (73) (B4) In rder t cnstruct Q 0 we shall assume the strnger cnditin: M > 0 (B4 ) As lng as (51) hlds B T L T = M. (74) Thus if (70) is regarded as an equatin fr Q it has a symmetric slutin Mrever if Q is any ther slutin then Thus the general symmetric slutin fr Q is where Y is any symmetric matrix such that Q = L T M 1 L. (75) B T (Q Q ) = 0. (76) Q = L T M 1 L Y, (77) B T Y = 0 (78)

54 Slutin f the Inverse Prblem f Linear Optimal Cntrl with Psitiveness Cnditins and Relatin t Sensitivity Slutin f the Inverse Prblem with Q 0 fr given R Cnditin (B4 ) ensures that Q 0 if Y 0. Als let x = (I BM 1 L)z, (79) where z is an arbitrary vectr. Then x T Qx = z T Y z. (80) Thus Y 0 is als necessary fr Q 0.

55 Slutin f the Inverse Prblem f Linear Optimal Cntrl with Psitiveness Cnditins and Relatin t Sensitivity Slutin f the Inverse Prblem with Q 0 fr given R Cnsider the differential equatin that is btained when (77) is substituted fr Q in (52): P = A T P P A D T RD L T M 1 L Y. (81) Since M is given and L is linear in P, this is a Ricatti equatin which can be integrated t determine P and hence L. We shall verify that (81) has slutins that satisfy (51). Define K by K = B T P RD (82) s that K = 0 if (51) hlds. Then when (51) is n lnger assumed t hld (71) and (73) yield instead f (74). Als using (81) B T L T = M K(AB Ḃ) (83) K = B T P ḂT P d (RD) (84) dt = B T A T P B T P A B T D T RD B T L T M 1 L B T Y ḂT P d (RD), (85) dt whence in view f (71), (78) and (83) K = L KA [ M K(AB Ḃ)] M 1 L (86) = K [ A (AB Ḃ)M 1 L ].

56 Slutin f the Inverse Prblem f Linear Optimal Cntrl with Psitiveness Cnditins and Relatin t Sensitivity Slutin f the Inverse Prblem with Q 0 fr given R Under cnditins B1, B2 and B3 it is pssible t chse P (t f ) 0 satisfying (51). Then K(t f ) = 0 and the slutin f (86) when integrated backwards is K = 0. The crrespnding slutin f (81) therefre satisfies (51). Therem Given R = R T > 0, cnditins (B1 4) are necessary fr a slutin f the inverse prblem with Q 0. Cnditins (B1 3) are sufficient fr a slutin ver sme finite time interval. Every slutin with Q 0 is then given by the slutin f (71) and (81) fr sme Y 0 satisfying (78)

57 Slutin f the Inverse Prblem f Linear Optimal Cntrl with Psitiveness Cnditins and Relatin t Sensitivity Slutin f the Inverse Prblem with Q 0 fr given R If equatin (86) is unstable when integrated backwards then the integratin f (81) wuld tend t drift away frm satisfying (51). T vercme this difficulty we may intrduce instead f L and M the matrices L 1 = B T D T RD RDA 1 B T P (A 1 A) (B T A T ḂT )P d (RD), (87) dt and M 1 = B T L 1 (88) where A 1 is a matrix t be selected. Then L 1 = L K(A 1 A), (89) and frm (83) M 1 = B T L T B T (A T 1 A T )K T (90) = M K(AB Ḃ) (BT A T B T A T 1 )K T. (91)

58 Slutin f the Inverse Prblem f Linear Optimal Cntrl with Psitiveness Cnditins and Relatin t Sensitivity Slutin f the Inverse Prblem with Q 0 fr given R Thus L 1 = L and M 1 = M when K = 0. We nw integrate Then frm (78) and (88) whence (87) yields P = A T P P A D T RD L 1 M 1 1 L 1 Y (92) K = B T A T P B T P A B T D T RD L 1 ḂT P d (RD) (93) dt K = KA 1 (94) Thus if A 1 is chsen as any stable matrix, backward integratin f (92) will preserve K = 0 withut danger f drift. But then L 1 = L and M 1 = M, s that alng the integratin path M 1 = M T 1 and under cnditin B4*, M 1 > 0.

59 Slutin f the Inverse Prblem f Linear Optimal Cntrl with Psitiveness Cnditins and Relatin t Sensitivity Slutin f the Inverse Prblem with Q 0 fr given R While cnditins B1-B3 and B4* establish the existence f Q 0 ver sme finite time interval, and every Q can then be cnstructed frm equatin (77) r (92), these cnditins d nt establish the existence f Q 0 ver an arbitrarily large time interval, because equatin (81) r (92) which fllws the same integratin path, may have a finite escape time. In particular, cnditins B1-B3 and B4* are nt sufficient fr the existence f a slutin with cnstant Q 0 and R > 0, when A, B and D are cnstant. This is easily seen frm an example. Cnsider the system ẋ = x 1 0 u, (95) with u = [ 2 1 ] 2 x, (96) where it is desired t find a perfrmance index which is minimized by u. RDB and M are scalars, J = 0 (xt Qx u 2 )dt, (97) RDB = 2, (98) and M = (B T D T ) 2 2DAB = 5. (99)

60 Slutin f the Inverse Prblem f Linear Optimal Cntrl with Psitiveness Cnditins and Relatin t Sensitivity Slutin f the Inverse Prblem with Q 0 fr given R S cnditins B1 - B3 and B4* all hld. On the ther hand (51) may be slved fr P t give P = P 22, (100) where P 22 is the nly undetermined element in P. Then substituting fr P in (52) with P = 0 gives Q = D D A T P P A 5 1 P = 22 1 P s that it is nt pssible t btain Q 0 by chice f P 22 when P is cnstant.,

61 Slutin f the Inverse Prblem f Linear Optimal Cntrl with Psitiveness Cnditins and Relatin t Sensitivity Slutin f the Inverse Prblem with Q 0 fr given R This example indicates the need t examine the cnditins under which (81) r (92) can be integrated ver an arbitrary interval. Since (92) fllws the same path as (81) when integrated frm a final value f P which satisfies (51), it suffices t cnsider (81). Substituting frm (71), it may be written as P 1 = A T P 1 P 1 A D T RD Y D T 1 MD 1, (101) where and Let P 1 = P, (102) D 1 = M 1 (B T A T P 1 ḂT D T RD RDA d dt (RD) ḂT P 1 ) (103) ẋ 1 = Ax 1 ABu 1 Ḃu 1. (104) Then equatins (101) and (103) are equatins fr determining the cntrl which minimizes J 1 = t f t 1 u 1 = D 1 x 1, (105) { x T 1 (D T RD Y )x 1 (106) 2u T 1 (B T D T RD RDA d dt (RD))x 1 u T 1 Mu 1 dt x T 1 (t f)p 1 (t f )x 1 (t f ).

62 Slutin f the Inverse Prblem f Linear Optimal Cntrl with Psitiveness Cnditins and Relatin t Sensitivity Slutin f the Inverse Prblem with Q 0 fr given R Substituting fr M frm (73) and using (78), (106) becmes Set J 1 = t f t 1 (x 1 Bu 1 ) T (D T RD Y )(x 1 Bu 1 )dt (107) 2 t f t 1 t f t 1 u T 1 Then x satisfies (48) where u 1 = u, r u T 1 (RDA d dt (RD))(x 1 Bu 1 )dt d dt (RDB)u 1dt x T 1 (t f )P 1 (t f )x 1 (t f ). x = x 1 Bu 1. (108) u 1 = t f t 1 udt, (109) there being n cnstant if (104) and (48) are bth t be in equilibrium with zer cntrl. The secnd and third terms in (107) becme 2 t f t 1 = t f t 1 u T 1 (RDA d dt (RD))(x)dt t f t 1 u T 1 d dt (RDB)u 1dt (110) d dt (RDB)u 1 dt (111) 2uT 1 RDẋ 2u T d 1 dt (RD)x 2uT 1 RDBu 1 u T 1 = [ 2u T 1 RDx u T 1 RDBu ] t f 1 2 t f urdxdt. (112) t 1 t 1

63 Slutin f the Inverse Prblem f Linear Optimal Cntrl with Psitiveness Cnditins and Relatin t Sensitivity Slutin f the Inverse Prblem with Q 0 fr given R Als using (51) and (102) Thus (107) becmes where S y is defined by (62) and x T 1 P 1 x 1 2u T 1 RDx 1 u T 1 RDBu 1 = x T P 1 x = x T P x. (113) J 1 = S y x T (t f )P (t f )x(t f ) I(t 1 ), (114) I = 2u 1 RDx 1 u 1 RDBu 1. (115) Nw (109) shws that a nn-zer value f u 1 at t = t 1 crrespnds t an impulse at t = t 1 in u u = u 1 (t 1 )δ(t t 1 ). (116) But under such an impulse x is shifted frm x 0 t x 0 = x 0 Bu 1 (t 1 ) with a cntributin t S y exactly equal t I(t 1 ), since x 1 is cntinuus s that x 1 (t 1 ) = x 0. Thus the first and third terms in J 1 equal S y evaluated frm t 1 in case f an initial impulse in u. It fllws that x T (t 1 )P (t 1 )x(t 1 ) is the minimum value f S y when x(t 1 ) = x 0. Suppse that this quantity is nt bunded. Then since the system is cntrllable, it can be brught frm x(t 0 ) = 0 t x(t 1 ) = x 0 with a finite cntributin t S y, and hence ver the interval (t, t f ) cnditin C wuld be vilated. We deduce that cnditin C is sufficient fr the existence f a slutin t (81).