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1. A computer-implemented method for adaptive control of a physicals system, the method comprising: modeling a physical system using a nonlinear plant as {dot over (x)}=Ax+B[u+f(x)], where x(t):┌0,∞)→Rn is a state vector, u(t): 0,∞)→Rp is a control vector, A∈Rn×n and B∈Rn×p are known such that the pair (A,B) is controllable, and f(x):Rn→Rp is a matched uncertainty;modeling at least one of an unstructured uncertainty and a structured uncertainty, wherein the unstructured uncertainty f(x) is modeled as f(x)=Σi=1nθ*iΦi(x)+ε(x)=Θ*TΦ(x)+ε(x), where Θ*∈Rm×p is...

1. A computer-implemented method for adaptive control of a physicals system, the method comprising: modeling a physical system using a nonlinear plant as {dot over (x)}=Ax+B[u+f(x)], where x(t):┌0,∞)→Rn is a state vector, u(t): 0,∞)→Rp is a control vector, A∈Rn×n and B∈Rn×p are known such that the pair (A,B) is controllable, and f(x):Rn→Rp is a matched uncertainty;modeling at least one of an unstructured uncertainty and a structured uncertainty, wherein the unstructured uncertainty f(x) is modeled as f(x)=Σi=1nθ*iΦi(x)+ε(x)=Θ*TΦ(x)+ε(x), where Θ*∈Rm×p is an unknown constant ideal weight matrix that represents a parametric uncertainty, Φ(x):Rn→Rm is a vector of chosen basis functions that are continuous and differentiable in C1, andε(x):Rn→Rp is an approximation error which can be made small on a compact domain x(t)∈D⊂Rn by a suitable selection of basis functions, andthe structured uncertainty f(x) is modeled as f(x)=Θ*TΦ(x), where Φ(x):Rn→Rm is a vector of known basis functions that are continuous and differentiable in C1;using a feedback controller specified by u=−Kxx+Krr−uad, where r(t):0,∞)→Rp∈L∞ is a command vector, Kx∈Rp×n is a stable gain matrix such that A−BKx is Horwitz, Kr∈Rp×p is a gain matrix for r(t), and uad(t)∈Rp is a direct adaptive signal which estimates the parametric uncertainty in the plant such that uad=ΘTΦ(x), where Θ(t)∈Rm×p is an estimate of the parametric uncertainty Θ*;using a reference model specified as {dot over (x)}m=Amxm+Bmr, where Am∈Rn×n and Bm∈Rn×p are given by Am=A−BKx and Bm=BKr;modeling an estimation error of a parametric uncertainty by {tilde over (Θ)}=Θ−Θ* in a tracking error equation ė=Ame+B[{tilde over (Θ)}TΦ(x)−ε(x)], where e(t)=xm(t)−x(t) is the tracking error;estimating a parametric uncertainty by an optimal control modification adaptive law specified by at least one of {dot over (Θ)}=−ΓΦ(x)[eTP−νΦT(x)ΘBTPAm−1]B, where ν>0∈R is a modification parameter, Γ=ΓT>0∈Rm×m is an adaptive gain matrix, and P=PT>0∈Rn×n solves PAm+AmTP=−Q, where Q=QT>0∈Rn×n is a positive-definite weighting matrix, and{dot over (Θ)}=−ΓΦ(x)[eTPB+νΦT(x)ΘR], where R=RT>0∈Rp×p is a positive-definite weighting matrix. 2. A computer-implemented method for adaptive control of an aircraft, the method comprising: modeling an aircraft using a nonlinear plant as {dot over (x)}=A11x+A12z+B1U+f1(x,z) and ż=A21x+A22z+B2u+f2(x,z), where Aij and Bi, i=1,2, j=1,2 are nominal aircraft matrices which are assumed to be known, x=[p q r]T is a state vector of roll, pitch, and yaw rates, z=[ΔΦ Δα Δβ ΔV Δh Δθ]T is a state vector of aircraft attitude angles, airspeed, and altitude, and u=[Δδa Δδe Δδr]T is a control vector of aileron, elevator, and rudder deflections, and fi(x,z), i=1,2 is an uncertainty;modeling at least one of an unstructured uncertainty and a structured uncertainty, wherein the unstructured uncertainty fi(x,z) is modeled as fi(x,z)=Θ*iTΦ(x,z,u(x,z))+ε(x,z), where Θ*i is an unknown, constant ideal weight matrix that represents a parametric uncertainty, Φ(x,z,u(x,z)) is a vector of chosen basis functions that are continuous and differentiable in C1, and ε(x,z) is an approximation error which can be made small by a suitable selection of basis functions, andthe structured uncertainty fi(x,z) is modeled as fi(x,z)=Θ*iTΦ(x,z,u(x,z)), where Φ(x,z,u(x,z)) is a vector of known basis functions that are continuous and differentiable in C1;modeling a second-order reference model that specifies desired handling qualities with good damping and natural frequency characteristics in the roll axis by at least one of (s2+2ζpωps+ωp2)Φm=gpδlat and (s+ωp)pm=gpδlat, in the pitch axis by (s2+2ζqωqs+ωq2)θm=gqδlon, and in the yaw axis by (s2+2ζrωrs+ωr2)βm=grδrud, where Φm, θm, and βm are the reference bank, pitch, and sideslip angles, ωp, ωq, and ωr are the natural frequencies for desired handling qualities in the roll, pitch, and yaw axes, ζp, ζq, and ζr are the desired damping ratios, δlat, δlon, and δrud are the lateral stick input, longitudinal stick input, and rudder pedal input, and gp, gq, and gr are the input gains;representing the reference model in a state-space form as {dot over (x)}m=−Kpxm−Ki∫01xmdτ+Gr, where xm=[pm qm rm]T=[{dot over (Φ)}m {dot over (θ)}m−{dot over (β)}m]T, Kp=diag(2ζpωp,2ζqωq,2ζrωr), Ki=diag(ωp2,ωq2,ωr2)=Ω2, G=diag(gp,gq,gr), and r=[δlat δlon δrud]T;using a proportional-integral (PI) dynamic inversion feedback controller with adaptive augmentation control to improve aircraft rate response characteristics specified by at least one of u=B1−1({dot over (x)}m−A11x−A12z+ue−uad), where ue=Kp(xm−x)+Ki∫0t(xm−x)dτ is a nominal PI error compensator, and uad=Θ1TΦ(x,z,u) is an adaptive augmentation controller, and u=B1T(B1B1T)−1({dot over (x)}m−A11x−A12z+ue−uad);modeling a tracking error as e=[∫01(xm−x)dτ xm−x]T in a tracking error equation ė=Ame+B[Θ1TΦ(x,z)−f1(x,z)], where Am=[0I-Ki-Kp]andB=[0I];estimating the parametric uncertainty Θ*1 by an optimal control modification adaptive law for a nominal PI feedback controller specified by at least one of {dot over (Θ)}1=−Γ[Φ(x,z)eTP−νΦ(x,z)ΦT(x,z)Θ1BTPAm−1]B,{dot over (Θ)}1=−Γ[Φ(x,z)eTPB+cνΦ(x,z)ΦT(x,z)Θ1Ki−2], where c>0∈R is a weighting constant, and {dot over (Θ)}1=−Γ[Φ(x,z)eTPB+cνΦ(x,z)ΦT(x,z)Θ1Ω−4]. 3. The computer-implemented method of claim 2, further comprising: using a proportional-integral-derivative (PID) dynamic inversion feedback controller with ue=Kp(xm−x)+Ki∫0t(xm−x)dτ+Kd({dot over (x)}m−{dot over (x)}), where Kd=diag(kdp,kdq,kdr);modeling a tracking error as e=[∫01(xm−x)dτ xm−x]T in a tracking error equation ė=Ame+B[Θ1TΦ(x,z)−f1(x,z)], where Am=[0I-(I+Kd)-1Ki-(I+Kd)-1Kp]andB=[0(I+Kd)-1];estimating the parametric uncertainty Θ*1 by an optimal control modification adaptive law for a nominal PID feedback controller specified by at least one of Θ.1=-Γ[Φ(x,z)eTP-vΦ(x,z)ΦT(x,z)Θ1BTPAm-1]B,Θ.1=-Γ[Φ(x,z)eTP(I+Kd)-1B+cvΦ(x,z)ΦT(x,z)Θ1Ki-2],whereP(I+Kd)-1B=c[Ki-1Kp-1[I+(I+Kd)Ki-1]],andΘ.1=-Γ[Φ(x,z)eTP(I+Kd)-1B+cvΦ(x,z)ΦT(x,z)Θ1Ω-4]. 4. The computer-implemented method of claim 2, further comprising: modeling a first-order reference model in a state-space form {dot over (x)}m=−Kpxm+Grusing a proportional dynamic inversion feedback controller with ue=Kp(xm−x);modeling a tracking error as xe=xm−x in a tracking error equation {dot over (x)}e=−Kpxe+Θ1TΦ(x,z)−f1(x,z);estimating the parametric uncertainty Θ*1 by an optimal control modification adaptive law for a nominal proportional feedback controller specified by at least one of {dot over (Θ)}1=−Γ[Φ(x,z)xeTP+νΦ(x,z)ΦT(x,z)Θ1PKp−1], and {dot over (Θ)}1=−cΓ[Φ(x,z)xcTKp−1+νΦ(x,z)ΦT(x,z)Θ1Kp−2]. 5. The computer-implemented methods of claim 1, farther comprising: accessing an adaptive gain matrix Γ allowed to be time-varying;calculating, with a processor, optimal control modification adaptive laws with time-varying adaptive gain Γ(t);using a covariance adaptive gain adjustment method described by at least one of {dot over (Γ)}=−ηΓΦ(x)ΦT(x)Γ, where 0≦η0∈Rm×m is an adjustment matrix whose elements are the adjustment parameters for each individual element of the adaptive gain matrix such that 0≦λmax(η)0 is a positive-definite normalization matrix, {dot over (Γ)}=βΓ−ηΓΦ(x)ΦT(x)Γ, where β=βT>0 is the forgetting factor matrix, Γ.ij={[βΓ-ηΓΦ(x)ΦT(x)Γ]ijifΓij≤Γij(t0)0otherwise,Γ.=βΓ-ηΓΦ(x)ΦT(x)Γ1+ΦT(x)RΦ(x),andΓ.ij={[βΓ-ηΓΦ(x)ΦT(x)Γ1+ΦT(x)RΦ(x)]ijifΓij≤Γij(t0)0otherwise;using any of the aforementioned covariance adaptive gain adjustment methods with a resetting mechanism described by Γ.={Γ.withΓ(ti)=Γit≥tiwhenthresholdisexceededatt=tiΓ.withΓ(0)=Γ0otherwise, where Γi is a properly chosen new initial condition for the covariance adaptive gain adjustment method for t≧ti when a new uncertainty becomes present, and Γi is phased in by a first-order filter {dot over (Γ)}=−λ(Γ−Γi) or any order filter so chosen for t∈[ti,ti+nΔt], where λ>0, Γ=Γ(t) as t=ti−Δt computed from the previous time step and nΔt is the duration of transition from some previous adaptive gain value to the new initial condition Γi where n can be selected appropriately to ensure Γ→Γi to within a specified tolerance. 6. The computer-implemented methods of claim 2, further comprising: accessing an adaptive gain matrix Γ allowed to be time-varying;calculating, with a processor, optimal control modification adaptive laws with a time-varying adaptive gain Γ(t);using a covariance adaptive gain adjustment method described by at least one of {dot over (Γ)}=−ηΓΦ(x)ΦT(x)Γ, where 0≦η0∈Rm×m is an adjustment matrix whose elements are the adjustment parameters for each individual element of the adaptive gain matrix such that 0≦λmax(η)0 is a positive-definite normalization matrix, {dot over (Γ)}=βΓ−ηΓΦ(x)ΦT(x)Γ, where β=βT>0 is the forgetting factor matrix, Γ.ij={[βΓ-ηΓΦ(x)ΦT(x)Γ]ijifΓij≤Γij(t0)0otherwise,Γ.=βΓ-ηΓΦ(x)ΦT(x)Γ1+ΦT(x)RΦ(x),andΓ.ij={[βΓ-ηΓΦ(x)ΦT(x)Γ1+ΦT(x)RΦ(x)]ijifΓij≤Γij(t0)0otherwise;using any of the aforementioned covariance adaptive gain adjustment methods with a resetting mechanism described by Γ.={Γ.withΓ(ti)=Γit≥tiwhenthresholdisexceededatt=tiΓ.withΓ(0)=Γ0otherwise, where Γi is a properly chosen new initial condition for the covariance adaptive gain adjustment method for t≧ti when a new uncertainty becomes present, and Γi is phased in by a first-order filter {dot over (Γ)}=−λ(Γ−Γi) or any order filter so chosen for t∈[ti,ti+nΔt], where λ>0, Γ=Γ(t) as t=ti−Δt computed from the previous time step and nΔt is the duration of transition from some previous adaptive gain value to the new initial condition Γi where n can be selected appropriately to ensure Γ→Γi to within a specified tolerance. 7. The computer-implemented methods of claim 3, further comprising: accessing an adaptive gain matrix Γ allowed to be time-varying;calculating, with a processor, optimal control modification adaptive laws with time-varying adaptive gain Γ(t);using a covariance adaptive gain adjustment method described by at least one of {dot over (Γ)}=−ηΓΦ(x)ΦT(x)Γ, where 0≦η0∈Rm×m is an adjustment matrix whose elements are the adjustment parameters for each individual element of the adaptive gain matrix such that 0≦λmax(η)0 is a positive-definite normalization matrix, {dot over (Γ)}=βΓ−ηΓΦ(x)ΦT(x)Γ, where β=βT>0 is the forgetting factor matrix, Γ.ij={[βΓ-ηΓΦ(x)ΦT(x)Γ]ijifΓij≤Γij(t0)0otherwise,Γ.=βΓ-ηΓΦ(x)ΦT(x)Γ1+ΦT(x)RΦ(x),andΓ.ij={[βΓ-ηΓΦ(x)ΦT(x)Γ1+ΦT(x)RΦ(x)]ijifΓij≤Γij(t0)0otherwise;using any of the aforementioned covariance adaptive gain adjustment methods with a resetting mechanism described by Γ.={Γ.withΓ(ti)=Γit≥tiwhenthresholdisexceededatt=tiΓ.withΓ(0)=Γ0otherwise, where Γi is a properly chosen new initial condition for the covariance adaptive gain adjustment method for t≧ti when a new uncertainty becomes present, and Γi is phased in by a first-order filter {dot over (Γ)}=−λ(Γ−Γi) or any order filter so chosen for t∈[ti,ti+nΔt], where λ>0, Γ=Γ(t) as t=ti−Δt computed from the previous time step and nΔt is the duration of transition from some previous adaptive gain value to the new initial condition Γi where n can be selected appropriately to ensure Γ→Γi to within a specified tolerance. 8. The computer-implemented methods of claim 4, further comprising: accessing an adaptive gain matrix Γ allowed to be time-varying;calculating, with a processor, optimal control modification adaptive laws with time-varying adaptive gain Γ(t);using a covariance adaptive gain adjustment method described by at least one of {dot over (Γ)}=−ηΓΦ(x)ΦT(x)Γ, where 0≦η0∈Rm×m is an adjustment matrix whose elements are the adjustment parameters for each individual element of the adaptive gain matrix such that 0≦λmax(η)0 is a positive-definite normalization matrix, {dot over (Γ)}=βΓ−ηΓΦ(x)ΦT(x)Γ, where β=βT>0 is the forgetting factor matrix, Γ.ij={[βΓ-ηΓΦ(x)ΦT(x)Γ]ijifΓij≤Γij(t0)0otherwise,Γ.=βΓ-ηΓΦ(x)ΦT(x)Γ1+ΦT(x)RΦ(x),andΓ.ij={[βΓ-ηΓΦ(x)ΦT(x)Γ1+ΦT(x)RΦ(x)]ijifΓij≤Γij(t0)0otherwise;using any of the aforementioned covariance adaptive gain adjustment methods with a resetting mechanism described by Γ.={Γ.withΓ(ti)=Γit≥tiwhenthresholdisexceededatt=tiΓ.withΓ(0)=Γ0otherwise, where Γi is a properly chosen new initial condition for the covariance adaptive gain adjustment method for t≧ti when a new uncertainty becomes present, and Γi is phased in by a first-order filter {dot over (Γ)}=−λ(Γ−Γi) or any order filter so chosen for t∈[ti,ti+nΔt], where λ>0, Γ=Γ(t) at t=ti−Δt computed from the previous time step and nΔt is the duration of transition from some previous adaptive gain value to the new initial condition Γi where n can be selected appropriately to ensure Γ→Γi to within a specified tolerance. 9. The computer-implemented method of claim 6, further comprising: accessing an adaptive gain matrix Γ allowed to be time-varying;calculating, with a processor, optimal control modification adaptive laws with time-varying adaptive gain Γ(t);using an adaptive gain normalization method described by Γ(t)=Γ1+ΦT(x)RΦ(x), where Γ on the right hand side is a constant adaptive gain matrix and R=RT>0∈Rm×m is a positive-definite normalization matrix such that 0≦R0∈Rm×m is a positive-definite normalization matrix such that 0≦R1; modeling a reference model using a transfer function as ym=Wm(s)r=kmZm(s)Rm(s)r, where km is a high-frequency gain, and Zm(s) and Rm(s) are monic Hurwitz polynomials degrees mm and nm, respectively, and nm−mm≧1; specifying np−mp>nm−mm, so that the strictly positive real (SPR) condition is no longer possible to ensure tracking of the reference model;using an adaptive controller u=kyy+krr with the optimal control modification adaptive laws {dot over (k)}y=γy(ye−νx2ky) and {dot over (k)}r=γr(re−νr2kr);using the linear asymptotic property whereupon {dot over (k)}yb →0 and {dot over (k)}r→0 as t→∞ or whereupon γy−1{dot over (k)}y→0 and γr−1{dot over (k)}r→0 under fast adaptation as γy→∞ and γr→∞ to establish the equilibrium value of the adaptive controller as u_=2ym-2yv;using the linear asymptotic property whereupon {dot over (k)}y→0 and {dot over (k)}r→0 as t→∞ or whereupon γy−1{dot over (k)}y→0 ad γr−1{dot over (k)}r→0 under fast adaptation as γy→∞ and γr→∞ to establish the linear feedback control system in the limit as y_=Wc(s)r=2kmkpZp(s)Zm(s)Rm(s)(vRp(s)+2kpZp(s))r, where the modification parameter ν can be established such that he linear asymptotic closed-loop transfer function Wc(s) is stable. 18. The computer-implemented method for adaptive control of an aircraft of claim 17, further comprising: modeling an aircraft using a linear plant as [p.qq.]= [-(ωp+θp)100010-(ωp2+θq1)-(2ζpωp+θq2)][p∫0tqⅆτq]+ [100001][up-σpuq-σq],whereθp,θq1, and θq2 the uncertain parameters, and σp(t) and σq(t) are scalar, time-varying uncertain disturbances; modeling the pitch axis reference model as a second order reference model and the roll axis reference model as a first order reference model as [p.mqmqm.]=[-ωp100010-ωq2-2ζqωq][pm∫0tqmⅆτqm]+[100001][rprq], where the pilot commands rp(t) and rq(t) are computed from the pilot stick inputs δlat(t) and δlon(t) according to [rprq]=[(rmaxα)kpωpδlatkqωq2[δlon+La∫0tδlonⅆτ]];using a nonlinear dynamic inversion controller as δ=Bδ−1└I{dot over (x)}cmd+Ω×/Ω−{circumflex over (f)}A(y)┘+δ0, where the vector of estimated aerodynamic moments {circumflex over (f)}A(y) is calculated from the on-board aerodynamic lookup tables, while the angular rates Ω are measured using aircraft sensors, I is the inertia matrix of the aircraft, δ0 are pre-determined trim surface commands appropriate for the test flight condition, and Bδ−1 is a weighted pseudo-inverse of the control effectiveness derivatives with respect to the surface positions weighted by a control allocation matrix computed as Bδ−1=W−1BδT(BδW−1BδT)−1;modeling the angular acceleration commands x.cmd(t)=x.ref+x.c+x.a=[p.ref+p.c+p.aq.ref+q.c+q.ar.ref+r.c] as the sum of the desired reference dynamics {dot over (x)}ref(t) produced by the nonlinear dynamic inversion reference model, the output {dot over (x)}c(t) of the error compensator, and adaptive control augmentation {dot over (x)}a(t); using the nonlinear dynamic inversion adaptive flight controller to cancel out the effects of the uncertain parameters θp, θq1, θq2, σp(t), and σq(t) by the adaptive augmentation controller [upuq]=[rp+p.arp+q.a],where[p.aq.a]=[θ^pp+σ^pθ^q1∫0tqⅆτ+θ^q2q+σ^q];estimating the uncertain parameters by the optimal control modification adaptive laws with adaptive gain normalization as θ^.p=Γθp1+Nθpp2(pp~PpBmp+vθpp2θ^pBmpTPpAmp-1Bmp) and θ^.q=Γθq1+xqTNθqxp(xqx~qTPqBmq+vθqxqxqTθ^qBmqTPqAmq-1Bmq),whereθ^q=[θ^q1θ^q2]T,p~=pm-p,xq=[∫0tqⅆτq]T,andx~p=[∫0t(qm-q)ⅆτqm-q]T;rejecting disturbances by the optimal control modification adaptive laws with adaptive gain normalization plus disturbance rejection as θ^.p=Γσp1+Nσpp2(p~PpBmp+vσpθ^pBmpTPpAmp-1Bmp) and θ^.q=Γσq1+xqTNσqxp(x~qTPqBmq+vσqσ^qBmqTPqAmq-1Bmq).