Adaptive Control Of Nonaffine Systems With Applications To Flight Control

Chapter 2Literature Review2.1Short Period DynamicsThe continuing development of flight control design methods in parallel to advances in nonlinear control theory have made controllability at high angles of attack near and beyond stallincreasingly relevant. The main challenge encountered at high angles of attack, is that theshort-period aircraft dynamics appear to be highly uncertain and nonlinear both in the statesand in the control signal [6]. Hence, traditional linearization and gain scheduling methodsare insufficient for modelling and controlling the system at high-angles of attack. To obtaina better control design and model, various methods have been explored and discussed. Onesuch method is to re-write the system linearly by expanding the output and control vectorsinto a truncated Taylor series expansion in order to apply optimal control design [7]. Nonlinear control methods for affine-in-control systems using feedback linearization via dynamicinversion are found in [8] - [11]. The authors of [8] have utilized the natural time-scale separation between the faster pitch rate dynamics and the slower angle-of-attack dynamics andassumed that the control effectors have no effect on α so that α can be pseudo-controlledthrough the pitch rate. Without any modelling uncertainties, this controller was valid for anangle of attack range of up to 55 . In [9], two nonlinear controllers are implemented by separating the 6 DOF system dynamics into two time scales: angular rates which are fast, andangles of attack and sideslip, which are slow. The controller for the fast dynamics are foundby inversion of the momentum equations, and the slow dynamics controller is obtained by inverting the force equations. The controller performance is tested in high-fidelity simulations,valid up to angles of attack of 30 . The problem with feedback linearization is that it has4

Amanda YoungChapter 2. Literature Review5a low tolerance twoards modelling errors. Neural network approximations are introduced in[10] and [11] in order to compensate for modelling uncertainties. [10] uses an off-line neuralnetwork configuration to make an initial approximation of the nonlinear system dynamicsand a second on-line neural network approximation to estimate the inversion error. In [11],the neural network approximation captures the nonlinear dynamics of the system for anangle of attack range of α [ 5 , 21 ], and the inverse dynamics are computed by differentiating the output with respect to time until a term appears that contains the input u. Theauthors of [12] differentiate the nonaffine-in-control system with respect to time to obtain ahigher-order system, in which the derivative of the control input appears linearly. They thenapply linear control methods and integrate to obtain the structure of the controller. Thismethod results in a large control signal and would be susceptible to system saturation inrealistic systems. Direct projection-based adaptive control is then easily implemented andthe system is shown to be stable using a standard Lyapunov argument.In the first part of this thesis, we apply the theory developed in [13] to a nonaffine fighteraircraft like the F-16 performing at or near the stall angle in the presence of system uncertainties. The method in [13] introduces a specialized set of radial basis functions (RBF) thatretains the monotonic property with respect to control effectiveness of the original aircraftdynamics. It further defines the adaptive controller via fast dynamics and achieves timescale separation between the system dynamics and the controller dynamics. The stabilityand tracking results follow from Tikhonov’s theorem in singular perturbations theory.2.2Dutch-Roll DynamicsThe Dutch-Roll is a type of lightly damped oscillatory lateral/directional motion betweensideslip, roll, and yaw that consists of an out-of-phase combination of ”tail-wagging” androcking from side to side that occurs when an aircraft is disturbed laterally from equilibriumflight. This type of motion occurs when a statically stable aircraft tries to re-establish lateraland directional equilibrium. Dutch roll is usually the most troublesome of the natural modesassociated with the dynamics of an aircraft in free flight due to its sensitive stability balance,which could easily result in instability [16]. Hence, it is important to model the Dutch-Rollsystem as accurately as possible. Traditional Dutch-Roll approximations are made by linearizing the full nonlinear equations of motion so that the dynamic modes can be decoupledand evaluated [17], [18]. Additionally, the aerodynamic stability and control derivatives aretaken from wind-tunnel data and the eigenvalues of the linearized equations can be numeri-

Amanda YoungChapter 2. Literature Review6cally derived, so that a linear controller can be designed [15], [19] - [24]. Typically, these typesof approximations neglect the rolling rate, bank angle, rolling momentum and aerodynamiccoupling, which decreases the accuracy of the models [25]. An improved closed-loop formis derived in [16] using Taylor series expansion to approximate the roll-damping derivative.The effects of inertial, gyroscopic, and aerodynamic coupling are included for a linear modelin [26]. The problem with linearization approaches, is that although they are accurate forlow angles of attack, they becomes less than ideal for aggressive aircraft maneuvers and highangle of attack flight regimes when the nonlinear effects become significant enough to drivethe system to instability.In the second part of this thesis, we extend the method developed in [14] to uncertainnonaffine-in-control multi-input multi-output (MIMO) systems, which describe the lateral/directionaldynamics of an F-16 at different angles of sideslip in the presence of system uncertainties.

Chapter 3Mathematical PreliminariesIn this chapter, we recall some facts that will be used for derivation of main results. First,some results from nonlinear systems theory is given. Next, we summarize the main resultfrom [2] on approximation of monotonic functions. Lastly, Tikhonov’s theorem is statedfrom [1], which enables definition of the adaptive controller via fast dynamics.3.1Preliminaries on Nonlinear Systems TheoryLemma 1 ([3], p.116) A time-varying linear systemẋ A(t)x,(3.1)is globally exponentially stable if1. for any time t 0, the eigenvalues of A(t) have negative real parts, that is, β 0, t 0, λ(A(t)) β,(3.2)where λ(A(t)) denotes the eigenvalues of A(t),2. the matrix A(t) remains bounded:Z A (t)A(t)dt .07(3.3)

Amanda Young8Chapter 3. Mathematical PreliminariesLemma 2 (Comparison Lemma [1], p.102) Consider the scalar differential equationẋ f (t, x),x(t0 ) x0 ,(3.4)where f (t, u) is continuous in t and locally Lipshitz in x, for all t 0. Let [t0 , T ) (whereT could be infinity) be the maximal interval of existence of the solution x(t). Let v(t) be acontinuous function which satisfies the differential inequalityv̇(t) f (t, v(t)),v(t0 ) x0 .(3.5)Then, v(t) x(t) for all t [t0 , T ).Lemma 3 ([1], p.368) Consider the systemż f (z h(α), α) , g(z, α)and suppose g(z, α) is continuously differentiable and the Jacobian matrices g(z, α) L1 , z g(z, α) L2 kzk α(3.6) g zand g αsatisfy(3.7)for all (z, α) D Γ, where Γ Rm and D {z Rn kzk r}. Let k1 , λ1 , and γ0be positive constants with r0 kr1 , and define D0 {z Rn kzk r0 }. Assume that thetrajectories of the system satisfykz(t)k k1 kz(0)ke λ1 t , z(0) D0 , α Γ, t 0.(3.8)Then, there is a function V : D0 Γ R that satisfiesc1 kzk2 V (z, α) c2 kzk2 Vg(z, α) c3 kzk2 z V c4 kzk z V c5 kzk2 α(3.9)(3.10)(3.11)(3.12)for positive constants c1 , c2 , c3 , c4 , c5 and all z D, α Γ. Moreover, if all the assumptionshold globally (in z), then V (z, α) is defined and satisfies (3.9) through (3.12) on Rn Γ.

Amanda Young3.2Chapter 3. Mathematical Preliminaries9Preliminaries on Approximation TheoryIn this section, we recall the main result from [2]. To follow the notations in [2], let N ,R, and Rr denote the set of natural numbers, the set of real numbers, and the set of realr-vectors, respectively. Let Lp (Rr ), L (Rr ), C(Rr ), Cc (Rr ), denote the usual spaces of Rvalued maps f : Rr R such that f is pth power integrable, essentially bounded, continuous,rrrand continuous with compact support. Let further Lp (Rr ), L (R ), C (R ), Cc (R ) be thespaces of positive valued maps f : Rr R respectively. Since Cc (Rr ) is dense in Lp (Rr )( [4], p. 69), then for every f Lp (Rr ) there exists an fc Cc (Rr ) such that fc f p ε,where k · kp denotes the p-norm. The main result from [2] is summarized via the followingtheorem:Theorem 1 Let K : RZr R be an integrable bounded function such that K is continuousK(x)dx 6 0. Then the family SK of functions q : Rr R definedalmost everywhere andRras MXx ziwi Kq(x) ,(3.13)δi 1where M N , δ 0, wi R, and zi Rr , is dense in Lp (Rr ) for every p [1, ).The proof of this theorem in [2] gives an explicit expression for the coefficients wi in (3.13)as: r12T1R,(3.14)wi r fc (αi )δnK(x)dxRrwhere the set {αi Rr : i 1, ., nr } consists of all points in [ T, T ]r of the form[ T (2i1 T /n), ., T (2ir T /n)],i1 , i2 , ., ir 1, 2., n,in which T is a number such that sup(fc ) [ T, T ]r , where sup(·) is defined as thesupremum or least upper bound. This consequently implies that if f Lp (Rr ), then fc Cc (Rr ), and consequently the coefficients wi in (3.14) are positive, i.e. wi 0. Let SG bethe subclass of functions from SK with positive wi ’s. Then SG is dense in Lp (Rr ). We alsonotice that the Gaussians given by (x xci )2 x xc iσ2iφ e,σiwhere xci is the center, while σi is the width parameter, represent one particular choice ofK.

Amanda Young3.310Chapter 3. Mathematical PreliminariesPreliminaries on Singular Perturbations TheoryConsider the problem of solving the system(ẋ(t) f (t, x(t), u(t), ε),Σ0 :εu̇(t) g(t, x(t), u(t), ε),x(0) ξ(ε)u(0) η(ε),(3.15)where ξ : ε 7 ξ(ε) and η : ε 7 η(ε) are smooth. Additionally, assume that f and g arecontinuously differentiable in their arguments for (t, x, u, ε) [0, ] Dx Du [0, ε0 ],where Dx Rn and Du Rm are domains, ε0 0. In addition, let Σ0 be in standard form,that is,0 g(t, x, u, 0)(3.16)has k 1 isolated real roots u hi (t, x), i {1, . . . , k} for each (t, x) [0, ] Dx .Choosing a particular i and keeping it fixed, the subscript i is dropped. Let ν(t, x) u h(t, x).In singular perturbations theory, the system given byΣ00 : ẋ(t) f (t, x(t), h(t, x(t)), 0),x(0) ξ(0),(3.17)is called the reduced system, and the system given byΣb :dν g(t, x, ν h(t, x), 0),dτν(0) η0 h(0, ξ0 )(3.18)is called the boundary layer system, where η0 η(0) and ξ0 ξ(0), (t, x) [0, ) Dx aretreated as fixed parameters. The new time scale τ is related to the original time t via therelationship τ εt . The following result is due to Tikhonov [1], [5]:Theorem 2 Consider the singular perturbation system Σ0 given in (3.15) and let u h(t, x)be an isolated root of (3.16). Assume that the following conditions are satisfied for all [t, x, u h(t, x), ǫ] [0, ) Dx Dv [0, ǫ0 ] for some domains Dx Rn and Dv Rm , which containtheir respective origins:A1. On any compact subset of Dx Dv , the functions f , g, their first partial derivatives withrespect to (x, u, ǫ), and the first partial derivative of g with respect to t are continuous g and bounded, h(t, x) and u(t, x, u, 0) have bounded first derivatives with respect to (t, x, h(t, x)) is Lipschitz in x, uniformly in t, and the initial datatheir arguments, f xgiven by ξ and η are smooth functions of ǫ.

Amanda YoungChapter 3. Mathematical Preliminaries11A2. The origin is an exponentially stable equilibrium point of the reduced system Σ00 givenby equation (3.17). There exists a Lyapunov function V : [0, ) Dx [0, ) thatsatisfiesW1 (x) V (t, x) W2 (x) V t(t, x) V(t, x)f (t, x, h(t, x), 0) x W3 (x)for all (t, x) [0, ) Dx , where W1 , W2 , W3 are continuous positive definite functionson Dx , and let c be a nonnegative number such that {x Dx W1 (x) c} is a compactsubset of Dx .A3. The origin is an equilibrium point of the boundary layer system Σb given by equation(3.18), which is exponentially stable uniformly in (t, x).dvLet Rv Dv denote the region of attraction of the autonomous system dτ g(0, ξ0 , v h(0, ξ0 ), 0), and let Ωv be a compact subset of Rv . Then for each compact set Ωx {x Dx W2 (x) ρc, 0 ρ 1}, there exists a positive constant ǫ such that for all t 0,ξ0 Ωx , η0 h(0, ξ0 ) Ωv and 0 ǫ ǫ , Σ0 has a unique solution xǫ on [0, ) andxǫ (t) x00 (t) O(ǫ)holds uniformly for t [0, ), where x00 (t) denotes the solution of the reduced system Σ00in (3.17).The following Remarks of [1] will be useful for verifying Assumption A3:Remark 1 Verification of Assumption A3 can be done via a Lyapunov argument. If thereexists a Lyapunov function V : [0, ) Dx Dv that satisfiesc1 kνk2 V (t, x, v) c2 kνk2(3.19) Vg(t, x, ν h(t, x)) c3 kνk2 ,(3.20) νfor positive constants c1 , c2 , c3 , c4 and for all (t, x, v) [0, ) Dx Dv , then AssumptionA3 is satisfied.Remark 2 Assumption A3 can be locally verified by linearization. Let ϕ denote the mapν 7 g(t, ξ, ν h(t, ξ), ε). It can be shown that if there exists ω0 0 such that the Jacobianmatrix ϕsatisfies the eigenvalue condition ν ϕRe λ(t, x, h(t, x), 0) ω0 0(3.21) ν

Amanda YoungChapter 3. Mathematical Preliminaries12for all (t, x) [0, ) Dx , then Assumption A3 is satisfied.Proof. The boundary layer system (3.18) can be written as a perturbation of its linearizationat the equilibrium ν 0,dν A(t, x)ν (g(t, x, ν h(t, x), 0) A(t, x)ν),{z} dτ(3.22)ψ(t,x,ν)whereA(t, x) g(t, x, h(t, x), 0). ν(3.23) g(t, x, ν h(t, x), 0) and g(t, x, ν h(t, x), 0)From Assumption A1 of Tikhonov’s theorem, νare Lipshitz in ν so that the perturbation term ψ(t, x, ν) is also Lipshitz and satisfieskψ(t, x, ν)kp kkνk2p , (t, x) [0, ) Dx ,(3.24)where k · kp denotes the p-norm. Assumption A1 implies that A(t, x) is bounded for alltime and (3.21) implies that the eigenvalues of A(t, x) have negative real parts, so it can beconcluded from Lemma 1 that the nominal systemdν A(t, x)νdτ(3.25)is globally exponentially stable for all (t, x) [0, ) Dx and the trajectories of the lineartime-varying system (3.25) satisfykv(t)kp k1 kv(0)kp e λ1 t , v(0) Dν , x Dx , t 0(3.26)for positive constants k1 , λ1 .The partial derivatives of A(t, x)ν with respect to its arguments are bounded as a resultof Assumption A1 of Tikhonov’s theorem and the time-varying nominal system (3.25) isglobally exponentially stable. Hence, it follows then from applying Lemma 3, that thereexists a Lyapunov function V : [0, ) Dx Dν R that satisfies the inequalitiesc1 kνk2p V (t, x, ν) c2 kνk2p VA(t, x)ν c3 kνk2p ν V c4 kνkp(t, x, ν) νp(3.27)

Amanda YoungChapter 3. Mathematical Preliminaries13for some positive constants c1 , c2 , c3 , and c4 . Taking the τ derivative of the Lyapunov functionV along the trajectory of (3.18) yields VV̇ (t, x, ν) (t, x, ν) g(t, x, ν h(t, x), 0) ν V(t, x, ν) (A(t, x)ν ψ(t, x, ν)) ν c3 kνk2p c4 kkνk3p c3 kνk2pfor kνkp c3.c4 kThis in turn impliesV̇ (t, x, ν) c3V (t, x, ν),c2(3.28)and, consequently,c c3 τV (τ ) V (0) e2.Using Lemma 2 (Comparison Lemma), it follows thatrrcc2c1c3 2c3 τkν(0)kp e 2 , kν(0)k , x Dx , τ 0kν(τ )kp c1c4 k c2(3.29)(3.30)for positive constants c1 , c2 , c3 , c4 , k. Thus, it can be concluded that Assumption A3 ofTikhonov’s theorem on exponential stability of the time-varying boundary layer system atthe origin can be verified locally by linearization.

Chapter 4Adaptive Controller for NonaffineSingle Input System4.1Problem Formulation for Short Period DynamicsNeglecting the influence of gravity and thrust, the short-period dynamics of a rigid aircraftperforming high angle of attack maneuvers can be given as Lδe (α0 )Lα (α0 ) α̇ V α q V δe(4.1) q̇ M0 (α) Mq (α)q Mδe (α, δe ),where q is the pitch rate, α is the angle of attack, α0 is the trim angle, δe is the incremental elevator deflection, Lα (α0 ) is the known lift curve slope at α0 , Lδe (α0 ) is the knownlift effectiveness due to elevator deflection, V is the trimmed airspeed, Mq (α) is the pitchdamping, and M0 (α) and Mδe are the pitching moment components. From (4.1), it follows that the pitch dynamics depend on Mδe , which is nonlinear with respect to α and δe .Hence, the short-period dynamics of an aircraft at high angles of attack are nonlinear andnonaffine-in-control.In general, the terms M0 (α), Mq (α), and Mδe (α, δe ) are unknown. However, some partialknowledge of aerodynamic stability and control derivatives are usually available from windtunnel experiments and theoretical predictions. Incorporating prior known data, the short-14

Amanda YoungChapter 4. Adaptive Controller for Nonaffine Single Input System15period dynamics of (4.1) can be rewritten as: L (α )α̇ LαV(α0 ) α q δeV 0 δe q̇ M (α )α M (α )q M (α , 0)δ α0q0δe0e(4.2) (M (α) Mα (α0 )α) (Mq (α) Mq (α0 )) q (Mδe (α, δe ) Mδe (α0 , 0), δe ), {z} {z} 0 {z} M0 (α) Mδe (α,δe ) Mq (α)where it is assumed that Mα (α0 ), Mq (α0 ), and Mδe (α0 , 0) are the known constant stabilityand control derivatives. Additionally, the lift derivative Lδe (α0 ) is known to be small withrespect to airspeed V , so that for control design purposes, one can assumeLδe (α0 ) 0,V(4.3)which leads to the following model description: α̇ LVα α q (4.4) q̇ Mα α Mq q Mδe δe Mδe ( M0 (α) Mq (α)q Mδe (α, δe )) . {z} f (α,δe )This can be rewritten in state space form## ""#"# "α(t)0α̇(t) LVα 1 (δe (t) f (α(t), δe (t))), q(t)Mδeq̇(t)Mα Mq{z} {z } {z } Ay(t) x(t)(4.5)b[1 0] x(t), {z }c where f (α, δe ) is unknown to the controller. Based on wind-tunnel experiments and usingcurve-fitting methods, f (α, δe ) is known to have the structure 2 α2f (α, δe ) (1 C0 ) e 2σ C0 (tanh (δe h) tanh (δe h) 0.01δe ) , (4.6)α2which is well-defined for all α, δe R and continuously differentiable, where e 2σ2 is aGaussian function with width σ and C0 , h are positive constan

Due to the nonlinear-in-control nature of the dynamical system in aggressive flight maneuvers, well-known dynamic inversion methods cannot be applied to determine the explicit form of the control law. Additionally, . linear control methods for affine-in-control systems using feedback linearization via dynamic inversion are found in [8] - [11 .