Nonlinear Discrete-time Design Methods for Missile Flight Control Systems

Transcription

1 Presente at the 4 AIAA Guiance, Navigation, an Control Conference, August 6-9, Provience, RI. Nonlinear Discrete-time Design Methos for Missile Flight Control Systems P. K. Menon *, G. D. Sweriuk an S. S. Vai Optimal Synthesis Inc., Palo Alto, CA 9433 an E. J. Ohlmeyer Naval Surface Warfare Center, Dahlgren, VA 448 Discrete-time esigns of flight control systems are require for implementation on missile-borne computers. While extensive literature is available on linear iscrete-time control system esign methos, nonlinear iscrete-time control system esign techniques have not been iscusse to the same egree. This paper presents three nonlinear, iscretetime control system esign methos. These are the iscrete-time feeback linearization metho, iscrete-time state-epenent Riccatti equation metho, an the iscrete-time recursive backstepping technique. Nonlinear missile autopilot esign for a conventional missile an an integrate guiance-control system esign for a moving mass actuate missile are given as illustrative examples. I. Introuction esign methos for iscrete-time linear control systems have reache an avance level of maturity,. D However, the irect esign of nonlinear iscrete-time control systems remains to be fully evelope. Although textbooks are available on nonlinear control system esign 3-5, literature on iscrete-time nonlinear control system esign is rather sparse. From an applications point-of-view, iscrete-time esigns are important because most controllers are implemente using igital computers. Design techniques of interest in this paper are those that permit the synthesis of iscrete-time controllers for continuous-time nonlinear ynamic systems. The present work is motivate by the nee to implement nonlinear control system esigns synthesize using computer-aie esign techniques 6, 7 onboar missiles. Three ifferent iscrete-time control system esign techniques have been investigate in the present research. All of them are iscrete-time analogs of continuous-time nonlinear system esign techniques iscusse in the literature. The first approach is the iscrete-time version of the state-epenent Riccati equation (SDRE) technique iscusse in References 8 an 9. The secon esign technique is a iscrete-time version of the recursive backstepping methoology, an employs iscretize system ynamics. The thir technique is the iscrete-time version of the feeback linearization esign approach. In this last technique, the system ynamics is first transforme into a linear, time-invariant form through the efinition of state variable feeback. The transforme moel is then converte into iscrete-time form by efining sample-hols at the input an the outputs. The iscretize is linear moel is then use for control system esign. All three techniques have been employe for the esign of missile flight control systems. Section II will present each of the esign techniques in etail. Section III an IV will escribe the missile control system esign examples that illustrate the application of the esign techniques. Numerical simulation results will also be presente, together with a iscussion on real-time performance of the system on an off-the-shelf computer running the RT-Linux operating system. Conclusions will be given in Section V. * Chief Scientist, 868 San Antonio Roa. Associate Fellow, AIAA. Research Scientist, 868 San Antonio Roa. Member AIAA. Research Scientist, 868 San Antonio Roa. Senior Guiance an Control Engineer, Coe G3. Associate Fellow, AIAA. Optimal Synthesis Inc., 4. Publishe by American Institute of Aeronautics an Astronautics with Permission

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3 Presente at the 4 AIAA Guiance, Navigation, an Control Conference, August 6-9, Provience, RI. II. Discrete-time Nonlinear Control System Design Techniques The three nonlinear iscrete-time control esign techniques iscusse in this paper are (i) the State Depenent Riccatti Equation Metho (ii) Recursive Backstepping Technique an (iii) the Feeback Linearization Metho. A block iagram of the close loop systems incorporating a iscrete-time controller is given in Figure. The ynamics of the system uner control is given in the form of continuous-time nonlinear ifferential equations, while the controller computations occur at a specifie sample rate. As a first-step, sample an hol circuits are introuce at the system inputs an outputs, which will be realize in practice using the analog-to-igital converters at the inputs an igital-to-analog converters at the output of the controller. Zeroth-orer hols are assume in the present research. The continuous-time ynamics of the system is use as the basis for controller esign. The first step in the esign process is that of extracting a esign moel of the ynamic system. The esign moel shoul inclue all the important ynamic features of the system, excluing the ynamics of the sensors, an the actuators, if their ynamic responses are much faster than the system ynamics. Next, the variables in the system are separate into state an parameter vectors. The system parameter vector is mae up of those constants an slowly varying variables that are not being controlle. For instance, in missiles that o not allow the irect control of engines, Mach number may be consiere as a parameter. This separation also epens on the type of control system being esigne. In missile autopilot esign, the state vector may consist of angle of attack an pitch rate in the pitch axis, while the state vector in an integrate guiance-control system esign may inclue lateral position state as an aitional element. Other elements of the system parameter vector may consist of vehicle inertia properties, atmospheric ensity an the spee of soun. For the present research, the system ynamics is assume to be of the form: ( x,p) g( x,p)u & () x = f + where x is the state vector, u is the control vector an p is a parameter vector. The parameter vector consists of a set of constants or slowly varying states that are not controlle by the close-loop system. Note that the present evelopment assumes that the controls appear linearly in the state equations. If this is not the case, a ynamic compensator at the input can be use to transform the system ynamics into this form 7, 9. Next, the system performance requirements are generate. This may inclue the esire transient an steay state response characteristics. The controller sample rate can be chosen base on the expecte close-loop performance an the esign moel ynamic characteristics. The esign techniques iscusse in the following subsections can then be use to erive nonlinear controllers. Missile Missile Dynamics (Continuous-Time) Sample & Hol Digital Digital Controller (Discrete-Time) Figure. Discrete-time Close Loop Control System Sample & Hol A. Discrete-Time State Depenent Riccatti Equation Metho The evelopment in this section closely follows References 8 an 9. The first step in the SDRE metho is the transformation of the system ynamics into the state epenent coefficient (SDC) form. The SDC form is an instantaneous parameterization of the original nonlinear system into the form: Optimal Synthesis Inc., 4. Publishe by American Institute of Aeronautics an Astronautics with Permission

4 Presente at the 4 AIAA Guiance, Navigation, an Control Conference, August 6-9, Provience, RI. ( ) x g( x) x & = A x + u. () Since f(x) is a vector an A(x) is a matrix, an infinite number of such realizations exist. However, from a numerical stanpoint, only those parameterizations for which the pair [A(x), g(x)] is controllable at a given value of the state vector x shoul be use to set up the esign proceure. Reference 7 has escribe a general numerical proceure for constructing the SDC parameterization from a ynamic moel given in the form (). Next, the SDC matrices [A(x), g(x)] are transforme into a iscrete-time system using a sampling perio of T. Since the value of the feeback state vector remains constant over a sample interval, efine two new matrices: A A( x[ k ])T, B T e A( x[ k ]) = e = τ τ g ( x[ k ]) (3) The iscretize form of the system ynamics in SDC form then becomes: ( k ) = A ( k )x( k ) B ( k )u ( k ) x + + (4) The iscretize SDC form of the system ynamics is then use to cast the control problem as an infinite-horizon nonlinear regulator problem with the goal of minimizing the cost function: [ x t ( k ) Q( x[ k ])x( k) + u t ( k ) R( x[ k ])u( k )] J = (5) k= where the Q(x) matrix is require to be positive semi-efinite an R(x) must be positive efinite, for all x. The solution to this problem can be obtaine as 8, 9 : u[ k ] with the state-epenent feeback gain matrix compute as: ( x[ k ]) x[ k ] = K (6) K ( x[ k ]) = BT P ( x[ k ] ) B + R( x[ k ]) BT P ( x[ k ] ) A (7) The matrix P satisfies the state-epenent Riccati equation: P = ( x[ k ]) + AT ( x[ k ]) P( x[ k ]) I + B R ( x[ k ]) BT P ( x[ k ] ) A + Q( x[ k ]) (8) Note that every step in the above erivation can be numerically implemente. For the present research, algorithms have been built into the nonlinear system synthesis software 7 to irectly work with a numerical simulation moel. Using a perturbation vector, the SDC parameterization, moel iscretization an state-epenent gain computation are carrie out by the esign software. The next two sub-sections will iscuss recursive backstepping an feeback linearization techniques in the iscrete-time setting. However, the following remarks on the ifference between SDRE an the other two techniques are appropriate at this juncture. The only restriction the SDRE technique places on the esign moel is that the moel be controllable. On the other han, recursive backstepping an feeback linearization methos require the moel to have a specific triangular structure. B. Discrete-Time Recursive Backstepping Technique Unlike the iscrete-time SDRE metho, or the iscrete-time feeback linearization technique escribe in the next subsection, the iscrete-time recursive backstepping approach requires the iscretization of the nonlinear ynamic moel as the first step. The continuous time nonlinear moel can be converte into an approximate 3 Optimal Synthesis Inc., 4. Publishe by American Institute of Aeronautics an Astronautics with Permission

5 Presente at the 4 AIAA Guiance, Navigation, an Control Conference, August 6-9, Provience, RI. iscrete-time moel using numerical integration techniques such as the Euler s metho an the Aams-Bashforth metho. Moreover, the technique assumes that the system is given in a form: x [ k + ] = x [ k + ] = f ( x [ k ]) f ( x [ k ],x [ k ]) g ( x [ k ])x [ k ]... =... x i [ k + ] = f i ( x [ k ],x [ k ],...x i [ k ]) +... xn [ k + ] = fn( x [ k ],x [ k ],...xn [ k ]) g ( x [ k ],x [ k ])x 3 [ k ] g i ( x [ k ],x [ k ],...x i [ k ])x i + [ k ] gn( x [ k ],x [ k ],...xn [ k ])u[ k ] (9) It shoul be note that the ' f i s an ' g i s in the above state equations coul also be functions of a parameter vector p. In the case where m control variables are available, it is assume that the system ynamics can be partitione an arrange with respect to each of the control variables. The recursive backstepping technique procees by recursively constructing a Lyapunov function with respect to each control variable, in a recursive manner. As a first step, a quaratic Lyapunov function is assume for the first state as follows: V [ k ] = x [ k ] () Since the iscrete-time Lyapunov stability analysis requires the Lyapunov function to be a contraction mapping, This implies V [ k + ] = η V [ k ], where η [ ] () ( f g [ k ]x [ k ]) [ k ] + = η x [ k ] () Treating the secon state x as a control-like variable for x, the esire value of x can be compute as: f [ k ] ±η x x [ k ] c [ k ] = (3) g [ k ] The above equation offers two choices for the reference trajectory. Although both choices lea to a stable system, the following solution is aopte in this work f [ k ] +η x x [ k ] c [ k ] = (4) g [ k ] For an equivalent linear ynamic system, the above solution results in a close-loop pole location on the positive real axis. The other choice woul have resulte in a close pole location on the negative real axis. This is not esirable in iscrete-time ynamic systems, ue to the fact it places the system operation close to the Nyquist rate. It shoul be note that the choice η [ ], ensures the close loop pole of the equivalent linear system lies within the unit circle. The Lyapunov function at the secon stage of the backstepping process is chosen as: 4 Optimal Synthesis Inc., 4. Publishe by American Institute of Aeronautics an Astronautics with Permission

6 Presente at the 4 AIAA Guiance, Navigation, an Control Conference, August 6-9, Provience, RI. V [ k ] = ( x x c [ k ]) [ k ] (5) Invoking the stability analysis an treating the thir state as a control-like variable, an expression for x 3c [ k ] can be erive as follows: From which, Consequently, V [ k + ] = η V [ k ], where η [ ] (6) ( f ]) [ k ] x [ k ] + g [ k ]x 3c [ k ] x c [ k + = η ( x [ k ]) (7) c x 3c [ k ] = f [ k ] + x c [ k + ] + η ( x [ k ] x c [ k ]) g [ k ] (8) As in the previous stage of the recursive backstepping process, the solution corresponing to the positive real axis location of the close loop pole has been chosen. Note that the above solution requires x c [ k + ] at the k th time step. This can be obtaine by propagating Equation (4). x c [ k + ] = f [ k + ] + η x [ k + ] g [ k + ] (9) Since the right han sie epens only on x [ k ] an x [ k ], it can be evaluate at the k th time step. It shoul be note that the overall Lyapunov function for the first an secon states is V [ k ] + V [ k ]. It can be shown that for the above choice of η, η the overall Lyapunov function also is a contraction mapping. Proceeing along similar lines, the reference for the i th state in the control chain can be written as: x ic [ k ] = f i [ k ] + x (i )c [ k + ] + ηi ( x i [ k ] x (i )c [ k ]) g i [ k ] () An the control at the k th time step can be obtaine as: u[ k ] = f n [ k ] + x nc [ k + ] + ηn ( x n [ k ] x nc [ k ]) g n [ k ] () The overall Lyapunov function for the recursive backstepping-base close-loop system is: V [ k ] V [ k ] + V [ k ] +...V. = () n [ k ] The first term in the Lyapunov function is a quaratic term containing just the leaing state, an the other terms are: 5 Optimal Synthesis Inc., 4. Publishe by American Institute of Aeronautics an Astronautics with Permission

7 Presente at the 4 AIAA Guiance, Navigation, an Control Conference, August 6-9, Provience, RI. V i [ k ] ( x x ic [ k ]) i [ k ], i =,3...n =. (3) It may be observe that the recursive backstepping metho automatically constructs a Lyapunov function for the ynamic system uring the esign process. This can be use not only for control system esign, but also for stability an robustness analysis of the close-loop system. C. Discrete-Time Feeback Linearization Metho As in the recursive backstepping technique, the feeback linearization technique 3-7, 3 assumes that the system ynamics can be partitione with respect to each control variable, an arrange in a triangular structure, as follows: f ( x ) g ( x,x ) ( ) ( ) x & = + & (4) x = f x,x + g x,x,x 3 M ( x,x, L ) ( x,x, L )u x & n = fn,xn + gn The technique then uses repeate ifferentiation of the leaing state equation an substitutions to transform the moel into Brunovsky canonical form 4. In this form, the system ynamics appear as chains of integrators with all the system nonlinearities move to the input:...n x = f ( x,x,...x n ) + g( x,x,...x n )u (5) Such chains of integrators can be constructe for each of the control variables. The right han sie of equation (5) is then replace with a pseuo control variable v to yiel the transforme system:...n x = v = f ( x,x,...x n ) + g( x,x,...x n )u (6) This transformation makes the ynamic system linear an time-invariant with respect to the pseuo-control variables. Linear control techniques can then be use to esign controllers for the transforme moel. For the case of a single control input, the transforme moel can be written in state-space form as shown below: where,xn z & = Az + Bv (7) A = M M M L L O L L M, B = M (8) Next, sample an hol circuits are introuce at every control input an state to obtain a iscrete-time, linear ynamic system as: 6 Optimal Synthesis Inc., 4. Publishe by American Institute of Aeronautics an Astronautics with Permission

8 Presente at the 4 AIAA Guiance, Navigation, an Control Conference, August 6-9, Provience, RI. ( k ) = A z( k ) B v( k ) z + + (9) Discrete-time linear control techniques such as pole-placement an linear quaratic regulator theory 5 can be use to esign feeback control laws for the system. For instance, an optimal iscrete-time control law can be esigne as: ( k ) G( k ) z( k ) v = (3) where, the gain matrix G(k) is obtaine from the solution of a iscrete-time Riccati equation 5. The actual control variables can then be recovere from the pseuo-control variables using the relationship: u[ k ] [ z[ k ]) f ( x[ k ])] = g( x[ k ]) v( (3) If the system nonlinearities are known reasonably well an the sampling rate chosen appropriately, the resulting close-loop system will have ynamic properties close to the iscrete-time linear system. An interesting feature of the feeback linearization approach is that once the system ynamics is transforme into linear, time-invariant form, any control system esign technique can be applie to erive the controller. For instance, the H control technique 6 or the sliing moe control 3 technique can be applie to the transforme control problem. The next two sections will illustrate the application of these three iscrete-time nonlinear control system esign techniques to two missile flight control system esign examples. III. Missile Longituinal Autopilot Design The first esign example consiers an angle of attack regulation problem of a hypothetical tail-controlle missile moel from Reference 7. The equations of motion given in the following consists of angle of attack α, pitch rate q, Mach number M an flight path angle γ as the state variables an employs pitch fin eflection δ as the control input. & α =.48 Mα 3 cos ( α ).649 M α α cos( α ) M. M α cos( α ).43 Mcos( α ) δ 3 cos( γ ) q M 8M 49.8 M M 3.6 M q& = α α α + 7 α M δ -.M q (3) (33) M& =.48 M α 3 sin ( α ).649 M α α sin( α ). M.43 M sin M 3 α sin( α ).6 M ( α ) δ -.3 sin( γ ) (34) 7 Optimal Synthesis Inc., 4. Publishe by American Institute of Aeronautics an Astronautics with Permission

9 Presente at the 4 AIAA Guiance, Navigation, an Control Conference, August 6-9, Provience, RI. & γ =.48 Mα 3 cos ( α ) M α α cos( α ) +. M cos( γ ) -.3 M M 3 α cos( α ) +.43 M cos( α ) δ (35) The control objective is to esign an autopilot that stabilizes the missile airframe an tracks a commane angle of attack or normal acceleration. As is customary in missile flight control system esign 8, only the short-perio ynamics of the missile consisting of the angle of attack α an pitch rate q ynamics are use in the autopilot esign. Mach number M an flight path angle γ are treate as parameters in the short perio moel that are available as measurements. Moreover, eventhough the fin eflection appears explicitly on the right han sie of the angle of attack equation its effect is ignore in recursive backstepping metho an the feeback linearization approach. It is well known 8 that the effect of fin eflection on the angle of attack equation causes the nonminimum phase ynamic behavior in tail controlle missiles. Finally, since missiles are capable of generating several g s of lateral acceleration, the acceleration ue to gravity is neglecte. The three ifferent nonlinear iscrete-time controller esigns base on the above assumptions have been evelope for the longituinal missile ynamics moel. The results given in this section will illustrate the regulation of the angle of attack about zero. The initial conitions for the close loop simulation are chosen as α ( ) =., q( ) =., γ ( ) =. an M ( ) =. 5, an a sampling perio of.5 secons is use in all the control system esigns. The state weighting matrix Q an the control weighting R for the SDRE esign metho are chosen as constants. Figures an 3 show the close loop simulation results using SDRE metho: Q = R = (36) Angle of Attack(eg) Time(s) Figure. Angle of Attack History for the Discrete-Time SDRE Controller The iscrete-time recursive backstepping controller is esigne is obtaine by iscretizing the continuous time moel using Euler s metho, with the same sampling perio as the SDRE metho. The convergence rates use are η =.6 for angle of attack an η =. 3 for pitch rate. Figures 4 an 5 show the close loop simulation results using iscrete-time recursive backstepping autopilot. 8 Optimal Synthesis Inc., 4. Publishe by American Institute of Aeronautics an Astronautics with Permission

10 Presente at the 4 AIAA Guiance, Navigation, an Control Conference, August 6-9, Provience, RI. 4 Fin Deflection(eg) Time(s) Figure 3. Fin Deflection History for the Discrete-Time SDRE Controller α(eg) Time Figure 4. Angle of Attack History for the Discrete-Time Recursive Backstepping Controller 5 δ(eg) Time Figure 5. Fin Deflection History for the Discrete-Time Recursive Backstepping Controller 9 Optimal Synthesis Inc., 4. Publishe by American Institute of Aeronautics an Astronautics with Permission

11 Presente at the 4 AIAA Guiance, Navigation, an Control Conference, August 6-9, Provience, RI. The behavior of the iscrete-time Lyapunov function for the close loop system with the recursive backstepping controller is given in Figure 6. The constant convergence rate of the Lyapunov function can be observe. The thir autopilot esign is base on the iscrete-time feeback linearization methoology. For the present research, a pole placement controller was chosen for controlling the linear ynamics resulting after performing the feeback linearization. The iscrete-time close-loop pole locations are chosen as: [3.7486e- + i3.3e e- - i3.3e-]. The sampling perio of.5 secons is retaine for this technique as well. Figures 7 an 8 show the close loop simulation results for the iscrete-time feeback linearize autopilot. The autopilot performance in all three cases is qualitatively similar. It is important to note that these esigns coul be improve consierably by iterating on the esign parameters LF Time Figure 6. Time History of the Lyapunov Function for the Close-Loop System with Discrete-Time Recursive Back Stepping Controller Fin Deflection(eg) Time(s) Figure 7. Angle of Attack History for the Discrete-Time Feeback Linearize Autopilot Optimal Synthesis Inc., 4. Publishe by American Institute of Aeronautics an Astronautics with Permission

12 Presente at the 4 AIAA Guiance, Navigation, an Control Conference, August 6-9, Provience, RI. Fin Deflection(eg) Time(s) Figure 8. Fin Deflection History for the Discrete-Time Feeback Linearize Autopilot IV. Integrate Flight Control System Design for a Moving-Mass Actuate Interceptor Recently, internal mass movement has been propose as a control methoology for a kinetic warhea (KW) in atmospheric an exo-atmospheric engagements 9. As shown in Figure 9, the moving-masses positione by servos insie the vehicle changes the location of its center of mass relative to the external forces to generate the esire control moments. The moving-mass control concept works equally well in space when the KW is thrusting, or in the atmosphere, when the vehicle experiences aeroynamic forces. An avantage of this actuation technology is that it can be employe in kinetic warheas that have both atmospheric an exo- atmospheric interception capabilities. Moving Masses C.M After Moving the Masses X B Thrust Nominal Center of Mass (C.M) Y B Z B Figure 9. Moving-Mass Control Concept As iscusse in Reference 9, the moving mass actuator controlle kinetic warhea is a high-orer couple nonlinear ynamic system. The complexity of the moel along with the varying aeroynamic characteristics of the vehicle makes it essential to use computer-aie nonlinear control system esign techniques. The benefits of employing computer-aie nonlinear control system esign techniques for esign of continuous time control systems for the kinetic warhea target s interception were emonstrate in Reference 9. This section will illustrate the use of iscrete-time feeback linearization metho in conjunction with iscrete-time pole placement methoology for controlling the moving-mass KW. Optimal Synthesis Inc., 4. Publishe by American Institute of Aeronautics an Astronautics with Permission

13 Presente at the 4 AIAA Guiance, Navigation, an Control Conference, August 6-9, Provience, RI. As in Reference 9, the present kinetic warhea (KW) inclues two masses that can be move along the boyframe y an z irections. The task of interception is achieve by riving the line-of-sight rates & λ z, & λ y between the KW an the target to zero. The component of the relative motion vector along longituinal axis of the KW is uncontrollable. The control influence chains remain the same as they were in Reference 9. However, the control commans are position commans ( δ zc an δ yc ) to the actuators positioning the masses instea of the force commans. The moving-mass actuator proportional + erivative servos have a banwith of about 4 Hz. δ δ zc & δ z & yc δ y δ δ z q w & λz y r v & λ y (37) The close loop pole locations are chosen as { } for both the pitch an yaw channels. The roll channel oes not incorporate close-loop control. A sample frequency of Hz is use in the esign. This engagement scenario consiere in this section has the warhea an target initially at an altitue of 45, ft, an 5, ft. apart. Both vehicles have 5-egree flight path angles an are on reciprocal heaings, an both have an initial velocity of 6 ft/sec. In aition, the target has an initial offset of ft. in the east irection. The performance of the kinetic warhea is shown in Figures through 5. Comparison with the results in Reference 9 show that the iscrete-time feeback linearize controller performance is close to the continuous-time case. The miss-istance for this maneuver was.34(ft). warhea target 8 East Position (ft) North Position (ft) x 4 Figure. KW an Target Trajectories in the Horizontal Plane Optimal Synthesis Inc., 4. Publishe by American Institute of Aeronautics an Astronautics with Permission

14 Presente at the 4 AIAA Guiance, Navigation, an Control Conference, August 6-9, Provience, RI. 5. x 4 warhea target 5. 5 Altitue (ft) North Position (ft) x 4 Figure. KW an Target Trajectories in the Vertical Plane boy accelerations a z a y 8 6 Acceleration (g) Time (sec) Figure. Lateral Acceleration Components of the KW uring Interception 3 Optimal Synthesis Inc., 4. Publishe by American Institute of Aeronautics an Astronautics with Permission

15 Presente at the 4 AIAA Guiance, Navigation, an Control Conference, August 6-9, Provience, RI...5 LOS Angle Rates λ y,λ z (λ y )/t (λ z )/t. Angular Velocity (ra/sec) Time (sec) Figure 3. KW Line-of-Sight Rates During the Interception.5.4 Actual Comman Y Actuator.3.. y (ft) Figure 4. Actual an Commane Positions of the Moving Mass in the Yaw Axis 4 Optimal Synthesis Inc., 4. Publishe by American Institute of Aeronautics an Astronautics with Permission

16 Presente at the 4 AIAA Guiance, Navigation, an Control Conference, August 6-9, Provience, RI..5.4 Actual Comman Z Actuator.3.. z (ft) Figure 5. Actual an Commane Positions of the Moving Mass in the Pitch Axis As with classical proportional navigation guiance law, the presente integrate guiance-control system esign generates large amplitue maneuvers towars the en of the maneuver. Real-Time Evaluation of the Nonlinear Integrate Guiance-Control System The iscrete-time nonlinear control law escribe in the foregoing has been evaluate in a real-time computing environment. The computer-aie nonlinear control system synthesis software currently has the capability to automatically generate C coe from the user-specifie esign parameters. Freely available LAPACK linear algebra routines are use in this process. The guiance-control law was implemente in a Pentium III, 8 Mhz computer running the RTLinuxFree operating system. This real-time operating system is community-supporte an permits free open source istribution of the operating system uner the General Public License (GPL). Several runs have been mae, an the results inicate that the iscrete-time nonlinear guiance-control law can be safely operate at sample intervals of about ms. Moreover, further spee improvements appear feasible by reorganizing the C-coe moules. This experiment inicates that the integrate guiance-control law iscusse in this section can be implemente on state-of-the-art airborne computers. V. Conclusion This paper iscusse three ifferent iscrete-time nonlinear flight control system esign techniques. These were the iscrete-time SDRE technique, iscrete-time recursive backstepping technique, an the iscrete-time feeback linearization approach. These techniques have been integrate with computer-aie nonlinear control system esign software. The utility of these techniques was illustrate using two missile flight control system esign examples. The feasibility of real-time implementation on a commercial off-of-the-shelf computer was investigate for one of the esign examples. The results given here emonstrate that nonlinear iscrete-time control system esigns can be carrie out in a manner similar to the continuous-time esigns. Acknowlegments This research was supporte uner the MDA/Navy Contract No. N78-3-C Optimal Synthesis Inc., 4. Publishe by American Institute of Aeronautics an Astronautics with Permission

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