Modelling and Gain Scheduled Control of a Tailless Fighter

Transcription

1 Modelling and Gain Scheduled Control of a Tailless Fighter Nico G. M. Rademakers a, Rini kmeliawati, Roin Hill, Cees Bil c, Henk Nijmeijer a a Department of Mechanical Engineering Mathematics and Statistics Department Eindhoven University of Technology RMIT University Eindhoven, The Netherlands Melourne, ustralia n.g.m.rademakers@student.tue.nl rini.akmeliawati@rmit.edu.au h.nijmeijer@tue.nl r.hill@rmit.edu.au c erospace Engineering Department RMIT University Melourne, ustralia c.il@rmit.edu.au stract This paper presents the nonlinear model and flight control system (FCS) of a tailless fighter aircraft. The presented nonlinear model consist of six coupled second order differential equations. gain scheduled (GS) controller is developed for the longitudinal dynamics of the aircraft. n approach is provided to select the operating points systematically, such that the staility roustness of the overall gain scheduled control is guaranteed a priori. Nonlinear simulation of the aircraft and flight control system is executed to analyze the performance of the flight control system. Introduction In 999, a conceptual design project of a new fighter aircraft, referred as the FX TIPN, was started at RMIT erospace Engineering, ased on a hypothetical specification aimed at replacing the rapidly ageing RF fleet of F- and F-8 aircraft with a single airframe. This has resulted in a design without a traditional vertical tailplane. In the aircraft was modelled and staility was studied. It appeared that the FX TIPN can e stailized with a Linear Quadratic Regulator (LQR) state feedack for a certain flight condition. However, the simple LQR technique to design a controller for a certain operating point, is not sufficient to ensure the controllaility of the aircraft over its entire flight envelope. In the present work, we further develop the model of the aircraft. Gain Scheduled controller is designed for the longitudinal dynamics of the FX TIPN. method is developed to automatically generate a grid of operating points around the nominal operating point, such that the staility in the entire control area is guaranteed. This method is ased on [] and uses the concept of the complex staility radius. Simulation of the flight control system and the nonlinear model of the longitudinal dynamics of the FX TIPN is executed to analyze the performance of the flight control system. The paper is organized as follows. In Section the aircraft that is considered in this paper, the FX- TIPN, is discussed. First, a description of the tailless aircraft is provided, after which the equations of motion and the force equations of the full six degree of freedom model are presented. In Section 3 Gain Scheduling is studied. The gain scheduled controller is applied for the longitudinal dynamics of the FX- TIPN in chapter 4. Finally, some conclusions are drawn in Section 5. The FX TIPN The FX TIPN is a tailless aircraft, which is designed to e a multirole aircraft. The airframe is a fighter-omer, which is capale to carry a large and diverse payload for the strike role. The aircraft is equipped with two JSF9-6 fterurning Turofans. These engines enale the use of a 3D thrust vectoring system in order to improve the manoeuvraility of the aircraft. The maximum pitch and yaw angle of the nozzles are fixed to. Moreover, the vectored thrust property enales additional control authority.

2 . Tailless configuration The application of thrust vectoring allowed the designers to consider a tailless design. The main advantages of a tailless configuration are: By removing the tail and shortening of the fuselage, the drag of the aircraft can e reduced y % - 35% [4]. The structural weight of the aircraft can e reduced y % [4]. Reduced structural inspection and maintenance costs [8]. The range for a given fuel load is increased ecause of the reduced drag [8]. H H! H? H + / H O B H >! G H?! G H > B O H > H? The stealth characteristics are improved ecause of the reduced surface area [8]. Greater flexiility in arranging different elements of the aircraft [8]. However, a tailless design also has several disadvantages: Tendency of autorotation and the spinning due to insufficient damping. Pitching tendency ecause of the asence of a downward force provided y the weight of the vertical tail plane. Because of these disadvantages, a flight control system is needed to stailize the aircraft.. ircraft Model In this section the equations of motion are derived and the force model is provided. The otained model is an improvement of the model derived in [8]... ssumptions: For the modelling of the aircraft dynamics, several assumptions are used. These assumptions follow from [8]. The aircraft is considered as a rigid ody with six degrees of freedom; three translational and three rotational degrees. The mass of the aircraft is constant. The Earth is flat and therefore the gloal coordinate system is fixed to it. Consequently, the Earth is considered as an inertia system and Newton s second law can e applied. OX and OZ are planes of symmetry for the aircraft, therefore: I xy = yz dm and I xy = xy dm are equal to zero and Ixz = I zx in the inertia tensor matrix. Figure : ircraft axes systems The atmosphere is fixed to the Earth and is at rest (no wind). The airflow around the aircraft is assumed quasisteady. This means that the aerodynamics forces and moments only depend on the velocities of the vehicle relative to the air mass. s the aerodynamics is quasi-steady, the forces and moments of the aircraft are considered to act with respect to the aircraft centre of gravity (CG). The vector thrust, aerodynamic forces and moments can e resolved into ody-axis components at any instant of time. The contriution to the inertial coupling of all portions of the control system other than the aerodynamic surfaces is neglected. Each control system has one degree of freedom relative to the ody axes and is frictionless. Each control system consists of a linkage of rigid elements, attached to a rigid airplane. The acceleration due to gravity is considered as constant with a value equal to 9.8 m/s. There are no gyroscopic effects acting on the aircraft... xes Systems: For the modelling of the aircraft, three primary axes systems are employed, which are depicted in figure. The first axes system is the Earth initial axes system ( e e ). e e is pointed towards the North, e e towards the East and e e 3 downward. The second axes system is the aircraft-carried inertial axes system ( e c ). Theodyaxessystem( e )

3 where: c = C ψ C θ S φ S θ C ψ C φ S ψ C φ S θ C ψ + S φ S ψ S ψ C θ S φ S θ S ψ + C φ C ψ C φ S θ S ψ S φ C ψ S θ S φ C θ C φ C θ L G M H = F > K 8 J (3) The angular velocity vector gives the relation etween the Tait-Bryant angles and their derivatives and the roll (p), pitch (q) and yaw (r) rates of the aircraft. c ω = p q r = ψ sin θ + φ θ cos φ + ψ sin φ cos θ θ sin φ + ψ cos φ cos θ (4) Figure : Body axes frame and velocities is also an aircraft-carried axes system, with e pointed towards the nose of the aircraft, e towards the right wing and e 3 to the ottom of the aircraft. The axis system is otained through successive rotations of the aircraft-carried inertial frame with Tait-Bryant angles ψ, θ and φ. The velocity vectors along these axes are u, v and w and the angular velocity vectors are respectively: roll rate p, pitch rate q and yaw rate r. These velocity vectors are shown in figure...3 Equations of Motion in Body xes: The motion of an aircraft considered as a rigid ody can e descried y a set of six coupled nonlinear second order differential equations. In general the model can e descried as ẋ = f(x,u) y = g(x,u) () where x R is the state vector, u R 7 is the input vector and y R is the output vector. x = [ x y z φ θ ψ u v w p q r ] T u = [ δ T p n y n T p n y n ] T y = [ x y z φ θ ψ u v w p q r ] T The state vector consists of x, y, z, which determine the position of the aircraft with respect to the earth inertial axes frame, the three Tait-Bryant angles φ, θ, ψ, the translational velocities u, v, w and the angular velocities p, q, r. δ is the aileron angle, T i the magnitude of the thrust force, p n,i the pitch angle of the nozzle and y n,i the yaw angle of the nozzle. The relation etween the velocities of the aircraft with respect to the inertia axis frame can e expressed in terms of the velocities with respect to the ody fixed axes frame using the direction cosine matrix. ẋ ẏ = c u v () ż w The nonlinear model of the tailless aircraft can e derived using Newton-Euler equations. The translational motion of the aircraft s center of gravity are descried y: ( ) d F (ṙ ) = m dt CG + c ω ṙ CG (5) where the superscript indicates that the vectors are defined with respect to the ody fixed axes system. F = [ F ] F F T 3 is the resultant of all external forces applied to the aircraft, m the mass of the aircraft and ṙ CG =[u v w]t the linear velocity vector of the center of gravity. c ω =[p q r] T is the angular velocity vector. This results in the following equations. F = m ( u + qw rv) F = m ( v + ru pw) (6) F 3 = m (ẇ + pv qu) Euler equations are applied for the rotational dynamics of the aircraft. The rotational dynamics are descried y: M = d ( J c dt CG ω ) + c ω JCG c ω (7) where M is the resultant of the external moments applied to a ody relative to its center of gravity. c ω = [p q r] T is the angular velocity vector. The inertia matrix is I JCG x I xz = I y I zx I z This results in the following equations. M = I x ṗ I xz ṙ + qr(i z I y ) I xz pq M = I y q + rq(i x I z )+I xz (p r ) M 3 = I xz ṗ + I z ṙ + pq(i y I x )+I xz qr (8)..4 Forces and Moments: The forces and moments at the center of gravity of the aircraft have components due to the gravitational effects, aerodynamics effect and the forces and moments due to the

4 3D thrust vectoring system. The force model is ased on the force model derived in [8]. The components of the gravitational forces in the aircraft center of gravity with respect to the ody axes are: FG = mg cos(θ) FG = mg sin(φ) cos(θ) (9) FG3 = mg cos(φ) cos(θ) The aerodynamic forces acting on the aircraft are asically the drag, lift, side-force and the aerodynamic rolling, pitching and yawing moment. F Drag = qsc D F Lift = qsc L F Side = qsc y M = qsc l M = qsmacc m M3 = qsc n () where q = ρv t is the dynamic pressure of the freestream, S the wing reference area, the wing span and mac the mean aerodynamic chord. The values of these parameters are determined in [4]. C D, C L, C Y, C l, C m and C n are the dimensionless coefficients of drag, lift, side-force, rolling-moment, pitching-moment and yawing-moment respectively. The aerodynamic coefficients are mainly a function of the forward and downward velocity. lthough the dependency of the aerodynamic coefficients to these parameters can e very nonlinear in high Mach numer region, a linear approximation is used. The aerodynamic coefficients are derivedin[4]. Since the drag force is defined in negative flight direction and the lift force perpendicular to the flight direction pointed to the top of the aircraft, the components of these aerodynamic forces in the ody fixed frame are: F, = F Lift sin(α) F Drag cos(α) F,3 () = F Lift cos(α) F Drag sin(α) The thrust forces are generated y a 3D thrust vectoring system. The thrust forces in ody axes frame are derived using goniometric relations FT = i= T i cos(p ni ) cos(y ni ) FT = i= T i sin(y ni ) FT 3 = () i= T i sin(p ni ) cos(y ni ) where i denote the numer of the engine, T the magnitude of the thrust force, p n the pitch angle of the nozzle and y n the yaw angle of the nozzle. The moments acting on the aircraft generated y the thrust forces are: MT = (FT 3, F T 3, )d MT = (FT 3, + F T 3, )X cg (3) MT 3 = (FT, F T, )d (F T, + F T, )X cg where d is the distance etween the nozzle exit center and the aircraft vertical plane and X cg the distance from the nozzle exit to the aircraft center of gravity. 3 Gain Scheduling The suject of this section is gain-scheduling. First, gain-scheduling is discussed in general. n approach is developed to construct automatically a regular grid of operating points. This approach is ased on [] and uses the concept of the staility radius. 3. Introduction Gain-scheduling is the main technique that is used in flight control systems. gain-scheduled controller can provide control over an entire flight envelope. In this paper gain scheduling will e considered as the continuously variation of the controller coefficients according to the current value of the scheduling signals. ccording to [] the design of a gain scheduled controller for a nonlinear plant can e descried with a four step procedure. The first step involves the computation of a linear parameter varying model for the plant. The most common approach is to linearize the nonlinear plant around a selection of equilirium points. This results in a family of operating points. The second step is to design a family of controllers for the linearized models in each operating point. Because of the linearization, simple linear controller design methods can e used to stailize the system around the operating point. The third step is the actual gain scheduling. Gain scheduling involves the implementation of the family of linear controllers such that the controller coefficients are scheduled according to the current value of the scheduling variales. The last step is the performance assessment. This can e done analytically or using extensive simulation. 3. Staility of the Gain Scheduled Controller Consider the nonlinear system ẋ = f (x,u) (4) Let Γ R q denote the set of GS-parameters, such that for every γ Γ there is an equilirium (x γ,u γ ) of (4). Consider the control law for system (4) ( ) u = u γ K γ x xγ (5) where K γ R mxn is a feedack gain that strictly stailizes the linearized plant at equilirium (x γ,u γ ).

5 The staility and the roustness of the nonlinear plant (4) with controller (5) as the system moves from one equilirium to another along an aritrary path is studied in []. Theorem 4. in [] proves that the control law (5) is roust with respect to perturations if the scheduling variales vary slowly. 3.3 Staility Radius In most gain scheduling applications, the operating points are selected heuristically. In [] a systematic way to select the operating points, such that the staility of the entire control area is guaranteed, is explored, ased on the idea of the complex staility radius. The automatic construction of a regular grid of operating points for the control area of the FX TIPN, is ased on this technique. Consider a linear system of the form: ẋ = x + Bu (6) where (,B) is assumed to e stailizale. pplying the state feedack stailizing controller 3.4 Selection of the operating points Using the concept of the complex staility radius, new operating points can e selected in such a way that the staility and performance roustness of the overall design is guaranteed. Given the complex staility radius, r c (γ i ), of operating point γ i, a sphere, B γi (α), can e constructed in which the state feedack gain matrix K γi can e used to stailize the aircraft. This methode is descried in [] and is depicted in figure 3 for the D case with scheduling variales. In this figure the construction of a circle around γ i, contained in the staility radius, is shown. Mathematically the construction of the sphere can e formulated as follows: E γ H = ( i B i K i ) ( γ B γ K i ) r(γ) = γ Γ i = {γ : r(γ) r c } B γi (α) = { γ γ i α} B γi (α) Γ i (6) can e written as u = Kx (7) ẋ = cl x (8) where cl = BK. Since cl is stale, the eigenvalues of cl are in the left half plane. Consider a perturation of the closed-loop system with a structured perturation E H, where E R nxp, H R qxn are scale matrices that define the structure of the perturation and R pxq is an unknown linear perturation matrix. The closed-loop pertured system then ecomes = * C E= M I C E / E = 9 K I ẋ =( cl + E H) x (9) ccording to [5], the complex structured staility radius of (9) is defined y r c ( cl,e,h) = inf{ ; R pxq, σ( cl + E H)C + } () where C + denotes the closed right half plane and the spectral norm (i.e. largest singular value) of. Further, in [5] it is proved that the determination of the complex staility radius () is equivalent to the computation of the H -norm G H of the associated transfer function =max G(iω) () ω R G(s) =H (si cl ) E () Figure 3: Construction of B γ,i (α) Susequently, a regular grid of operating points is constructed. The idea is to fill the control area with a grid of adjoining squares of the same size. The size of the squares is determined y the point in the control area for which the radius of the sphere, B γi (α), is the smallest. The centers of the squares are the new operating points. This approach is depicted in figure Linear Quadratic Regulator (LQR) State Feedack Design most effective and widely used technique of linear control systems design is the optimal Linear Quadratic Regulator (LQR). rief description of LQR state feedack design is given elow. For more details, see [6].

6 M I 4 Longitudinal Gain Scheduling Control In this section the gain scheduling technique will e applied for the longitudinal control of the FX TIPN. First, the longitudinal model is descried. regular grid of operating points is constructed according the technique of Section 3. For these operating points LQR controllers are designed. The resulting gloal gain scheduled controller is applied to the longitudinal flight control system of the FX TIPN. Figure 4: Neighourhood K I 4. Longitudinal dynamics The full model of the tailless aircraft is descried in Section. In this section only the longitudinal dynamics are considered. The longitudinal model of the FX-TIPN in ody axes are defined as Consider a linear time invariant system ẋ = x + Bu y = Cx + Du (3) with state vector, x(t) R n, input vector, u(t) R m and output vector y(t) R l. If all the states are measurale, the state feedack u = Kx (4) with state feedack gain matrix, K R mxn,cane applied to otain desirale closed loop dynamics ẋ = ( BK)x (5) For LQR control the following cost function is defined: J = [ x(t) T Qx(t)+u(t) T Ru(t) ] dt (6) The ojective of LQR control, is to find a state feedack gain matrix, K, such that the cost function (6) is minimized. In (6), the matrices Q R nxn and R R mxm are weighting matrices, which determine the closed-loop response of the system. Q should e selected to e positive semi-definite and R to e positive definite. Minimization of the cost function (6) is equivalent to solving the following Ricatti equation. T P + P+ Q PBR B T P = (7) The Riccati equation is a matrix quadratic equation, which can e solved for P given,b,q and R, provided that (,B) is controllale and (Q,) is oservale. In that case (7) has two solutions, there is one positive definite and one negative definite solution. The positive definite solution has to e selected and the optimal feedack is given as K = R B T P (8) ẋ lon = f lon (x lon,u lon ) y lon = g lon (x lon,u lon ) (9) where x lon R 4 = [θ u w q] T is the state vector, u lon R = [T p n ] T is the input vector and y lon R 4 =[θ u w q] T is the output vector. The longitudinal dynamics are considered for a side velocity v =,rollangleφ =andayawangleψ =. The angular velocities p and r are also equal to zero. The thrust force and pitch angle of oth engines are assumed to e equal and the yaw angle for the nozzle is considered to e zero. This results in the following longitudinal dynamics θ = q m ( u + qw) = T cos p n + qsc L sin θ +q SC D cos θ mg sin θ m (ẇ qu) = T sin p n qsc L cos θ q SC D sin θ + mg cos θ I yy q = T sin p n X CG + qsc m mac (3) 4. Scheduling Variales The first step in the design approach is the selection of the scheduling variales. In [] it was proved that the scheduling variales should slowly vary to maintain staility when the aircraft moves from one equilirium point to another. From [9] and [] it also follows that the scheduling variales should e slowly varying and capture the nonlinearities of the system. However, this is only a qualitative result. In this paper the forward velocity, u, and the downward velocity, w, are chosen as scheduling variales. Intuitively, these variales are slowly time varying and capture the dynamic nonlinearities. However, not all nonlinearities are captured. To capture all nonlinearities, the density of the air and the pitch angle should also e considered. In this paper the pitch angle and the density of the air are kept on a fixed value, such that the nonlinearities caused y these parameters not occur.

7 Tale : Equilirium states and remaining accelerations α State Value cc Value θ rad θ. rad/s u 66.5 m/s u.98 6 m/s w.758 m/s ẇ m/s q. rad/s q rad/s w [m/s] Selection of the Operating Points 4.3. Nominal operating point: In order to design a LQR controller, the nonlinear model (3) has to e linearized around a certain equilirium point. Because it is not possile to find an equilirium analytically, nonlinear optimization techniques are used to find values for the elements of the inputs T i and p ni such that the sum of the squared remaining acceleration is minimized for a given operating point. The ojective function O is: min O ( ) x opt = u +ẇ + q (3) x opt R n r c u [m/s] x 3 Figure 5: Determination of α Staility radius with x opt =[T i p ni ]. The upper and lower ound for the optimization parameters are:.5 N T i 3446 N p ni w [m/s] u [m/s] The ojective function (3) is used to determine the initial operating point as well. However, as optimization parameters oth states and inputs are used. The results are listed in tale. This nominal operating point is considered to e in the middle of the area for which the controller is developed. It can e oserved that the remaining accelerations on the aircraft are not zero. This means that no equilirium point is found Grid construction: For the nominal operating point, an LQR Controller is designed and the concept of the complex staility radius is used to determine the size of the staility all. The radius of the staility all of the nominal operating point is 3 m/s. This means that the controller designed for the nominal operating point is ale to stailize the aircraft for a maximum of 3 m/s variation around the nominal forward and downward velocities. For a regular grid of 5 points within this control area, the staility all is constructed. For the structure of the perturations matrices, E = diag ([ ]) and H = diag ([ ]) are selected ecause there is no uncertainty in the first equation of (3) and unstructured disturances is assumed. The singular values of G(s) =H (si cl ) E are computed etween rad/s and rad/s for all the points. In eight directions the value for the radius of Figure 6: Staility Radius for the control area the staility all, α, is determined such that B γi (α) Γ i. This is illustrated for the nominal operating point in figure 5. The maximal values for α are depicted in figure 6 for each of the 5 points. From this figure it appears that α = 3 m/s for all the points in the entire control area. This means that the distance etween the operating points ecomes 3 m/s. However, ecause only 5 points in the control area are chosen, a distance of.5 3 m/s is used. The grid of operating points is depicted in figure 7. The ranges of the scheduling variales for which the controller is developed are: 66. u w.763 It should e noted that this is only a very small area of the entire flight envelope of the FX TIPN. However, the principle of the selection of the operating points can e applied in the same way for the entire envelope.

8 .764 Operating Points ctual and desired states actual desired Θ d [rad] u d [m/s] w [m/s] x w d [m/s] q d [rad/s] u [m/s].755 Figure 7: Operating points Figure 8: Desired response and simulated plant response 4.4 Scheduling Now the operating points are selected, input optimization is applied to minimize the remaining accelerations and LQR controllers are designed for each of the operating points. The scheduling of the gains of these 5 controllers is performed using spline interpolation. Spline interpolation ensures continuous variation of the controller coefficients. The controller gain of a point in the scheduling space will e composed of the controller gains of the LQR controllers in conformance with the distance from this point to the operating points. The distance from the current point γ to operating point i is calculated as follows: d i = (u γ u i ) +(w γ w i ) (3) The equilirium input matrix and feedack gain matrix are computed using a percentage of the U and K matrices according to the distance. where K γ = U γ = ξ i = η η ξ i K i i= η ξ i U i i= ( η i= d ) i d i η i= d i (33) with η the numer of operating points. Using (33) the resulting controller ecomes: u lon = U γ K γ (x lon x des lon) (34) 4.5 Simulation 4.5. Flight path: In order to analyse the tracking capacities of the gain scheduled controller, a flight path is defined. The flightpath consists of three parts e Θ [rad].5 x Error: X X d e w [m/s] 3 4 x 3 e u [m/s] e q [rad/s].5.5 x 3 3 x 6 Figure 9: Error as function of time During the first seconds, the aircraft is kept at the initial operating point x lon, = [ ] T During the next seconds a step of 5 5 m/s acceleration is applied in forward direction and a step of 5 5 m/s in downward direction. fter seconds, the velocities are kept at a fixed value. This enales to study the steady state response Simulation Results: Using the nonlinear longitudinal model of the FX TIPN and the gain scheduled flight control system, simulations are executed. In figure 8 the desired and the simulated plant response are depicted during a 3 seconds simulation. From this figure it appears that the desired trajectory and the simulated plant response do not agree. The error, which is defined as the difference etween the simulated value of the states and the desired value

9 of the states, is depicted in figure 9. It can e oserved that a steady state error remains. The reason for the steady state error is that the operating points which are used are not equilirium points. From tale it is shown that for the initial operating points the acceleration of the aircraft is not equal to zero. Remaining accelerations for the operating points are equivalent to a constant disturance which acts on equilirium points. That is why a steady state error can e oserved in figure 9. The size of the steady state error is completely determined y the nonzero accelerations. The error is small enough such that the aircraft stays in the control area. Summarizing, it can e concluded that the gain scheduled controller has een successfully applied to the stailization of the aircraft. However, a steady state error remains. [6] Lewis, F.L., Syrmos, V.L., Optimal Control, second edition, John Wiley and sons Inc., 995 [7] Nelson, R.C., Flight Staility and utomatic Control, McGraw-Hill, 989 [8] Panella, I. pplication of modern control theory to the staility and manoeuvraility of a tailless fighter, RMIT University, Undergraduate Thesis in Bachelor of Engineering (erospace), [9] Rugh, W.J., nalytical Framework for Gain Scheduling, IEEE Control Systems Magazine, p , January 99 [] Rugh, W.J., Shamma, J.S., Research on gain scheduling, utomatica, Vol. 36, p. 4-45, 5 Conclusions In this paper the six degree of freedom dynamical model of the FX TIPN is derived. gain scheduled controller is developed for the longitudinal dynamics of the aircraft. The concept of the staility radius is used to determine the neighourhood of an operating point in which the LQR controller is ale to stailize the nonlinear system and an approach to construct a regular grid of operating points for the gain scheduled controller is provided to assure gloal staility of the entire control area. Simulation of the nonlinear longitudinal model and the gain scheduled controller have proved that the gain scheduled controller is ale to stailize the aircraft. However, ecause the control area does not consist of a link up of equilirium points, a steady state error can e oserved. References [] kmeliawati, R., Nonlinear Control for utomatic Flight Control Systems, University of Melourne, PhD Thesis, [] kmeliawati,. R., Panella, I., Hill, R., Bil., C., Nonlinear Modeling and Stailization of a Tailless Fighter, RMIT University, [3] kmeliawati,. R., Panella, I., Hill, R., Bil., C., Stailization of a Tailless Fighter using Thrust Vectoring, RMIT University, [4] Garth, J., Phillips, M., Turner J., FX TIPN, RMIT University, Undergraduate Thesis in Bachelor of Engineering (erospace), 999 [5] Hinrichsen, D., Kel, B., n lgoritm for the computation of the Structured Complex Staility Radius, utomatica, Vol. 5, No 5 p , 989