PID Aaptive Control Design Base on Singular Perturbation Technique: A Flight Control Example Valery D. Yurkevich, Novosibirsk State Technical University, 20 K. Marx av., Novosibirsk, 630092, Russia (e-mail: yurkev@ac.cs.nstu.ru) Abstract: The paper treats a question of proportional-integral-erivative (PID) controller esign for aircraft pitch attitue in the presence of uncertain aeroynamics. The presente esign methoology guarantees esire pitch attitue transient performance inices by inucing of two-time-scale motions in the close-loop system where the controller ynamics is a singular perturbation with respect to the system ynamics. Stability conitions impose on the fast an slow moes an sufficiently large moe separation rate between fast an slow moes can ensure that the full-orer close-loop nonlinear system achieves the esire properties in such a way that the pitch attitue transient performances are esire an insensitive to external isturbances an variations of aeroynamic characteristics. The singular perturbation metho is use throughout the paper in orer to get explicit expressions for evaluation of the controller parameters. The high-frequency-gain online ientification an gain tuning are incorporate in the control loop in orer to maintain the stability of the fast-motion transients for a large range of aeroynamic characteristics variations. Numerical example an simulation results are presente. Keywors: flight control; aircraft pitch attitue control; aaptive control; singular perturbations. 1. INTRODUCTION Due to the complexity, variability, an uncertainties of aeroynamics, many ifferent control techniques are use in orer to esign of aircraft an missile control systems aime to maintain transient performances in the presence of uncertain aeroynamics. There are numerous approaches to flight control system esign, for instance, base on the Nonlinear Inverse Dynamics (NID) metho, feeback linearization, nonlinear H, linear matrix inequalities, l 1 optimal control, an LQR (see, for example, Petrov an Krutko (1981); Vukobratović an Stojić (1988); Lane an Stengel (1988); Wise (1995)). In particular, control systems with sliing moe iscusse by Schumacher (1994) an control systems with the highest erivative in feeback treate by B lachuta et al. (1997), Czyba an B lachuta (2003), are very powerful tools for aircraft an missile control system esign uner uncertainties. Note that the esign methoology iscusse by B lachuta et al. (1997), Czyba an B lachuta (2003) leas to the structure of proportional-integral-erivative (PID) controller in case of systems with relative egree equals two. It is well known that throughout the huge set of flight control applications, the PI (PID) controllers are extensively use. Problems of PI (PID) control system analysis an esign are iscusse, for instance, by O Dwyer (2003). This work was financially supporte by Russian Founation for Basic Research (RFBR) uner grant no. 08-08-00982-a. The majority of known esign proceures of PI (PID) controllers are applicable merely for stable linear systems. For example, the well known tuning rules suggeste by Ziegel an Nichols (1942), or its various moifications are wiely use for selection of controller parameters. In orer to fetch out the best PI an PID controllers in accorance with the assigne esign objectives, a set of tuning rules, ientification an aaptation schemes has been evelope (see Åström et al. (1993)). The main isavantage for the most part of the existing proceures for PI or PID controller esign is that the esire transient performances can not be guarantee in the presence of nonlinear plant parameter variations an unknown external isturbances. In orer to overcome this isavantage, the singular perturbation technique (see Kokotović et al. (1999), Naiu (2002)) may be use for PI or PID controller esign as was shown by Yurkevich (2004) where esire output transients are guarantee by inucing of two-time-scale motions in the close-loop system. Stability conitions impose on the fast an slow moes an sufficiently large moe separation rate between fast an slow moes can ensure that the full-orer close-loop nonlinear system achieves the esire properties in such a way that the output transient performances are esire an insensitive to external isturbances an parameter variations of the system. The stability of fast-motion transients in the close-loop system is provie by proper selection of controller parameters, while slow-motion transients correspon to the stable reference moel of esire mapping from reference input into controlle output.
The critical point of the workability of the singular perturbation esign methoology is that the controller parameters shoul be selecte in accorance with the requirement on fast-motion transients stability an, moreover, the esire egree of time-scale separation between the fast an slow moes in the close-loop system shoul be provie (see Yurkevich (2004)). However, the large variations of parameters of the system may give the isappearance of the two-time-scale structure of the trajectories in the close-loop system. In orer to maintain the two-timescale structure of the trajectories in the close-loop system an the esire amping of fast transients, an aaptive gain tuning scheme is propose in this paper. As a result, the esire pitch attitue transient performance inices can be provie for significantly large range of aeroynamic characteristics variations. The paper is organize as follows. First, a moel of the aircraft longituinal motion is efine. Secon, the singular perturbation esign methoology is highlighte for the purpose of aircraft pitch attitue control in the presence of uncertain aeroynamics. Thir, the utmost importance of high-frequency-gain online ientification an gain tuning in nonlinear control systems esigne via singular perturbation technique is shown. Fourth, a gain tuning proceure an high-frequency-gain online ientification proceure are introuce. Finally, simulation results of the aircraft pitch attitue aaptive control system are presente as well. 2. AIRCRAFT CONTROL PROBLEM 2.1 Aircraft Longituinal Motion Moel The iscusse approach to PID aaptive controller esign for aircraft pitch attitue angle is treate base on the aircraft longituinal motion moel represente by Wojciechowski et al. (1989): θ = q, u = wq g sin θ + a c m δ c + + 1 2m ρv2 a {S xc x (α, δ h ) cos α S z c z (α, δ h ) sin α}, ẇ = uq + g cos θ + (1) + 1 2m ρv2 a {S xc x (α, δ h ) sin α + S z c z (α, δ h ) cos α}, q = 1 2J y ρv 2 a L ys y m y (α, δ h ), where, in accorance with Fig. 1, we have that v a is an air velocity value, v a = (vax 2 + vaz) 2 1/2 ; v ax, v az are components of the air velocity, v ax = u v wx cos θ + v wz sin θ; v az = w v wx sin θ v wz cos θ; α is an angle of attack, α = tan 1 (v az /v ax ); m is an aircraft mass, m = 2000 kg; J y is a moment of inertia, J y = 5000 kg m 2 ; L y is an equivalent arm force in the flow coorinate system, L y = 0.5 m; S x, S y, S z are equivalent areas in the flow coorinate system, S x = 0.5 m 2, S y = 2.0 m 2, S z = 10.0 m 2 ; ρ is the air ensity, ρ = 1.2 kg/m 3 ; g is the acceleration, g = 9.81 m/s 2 ; a c is a coefficient,a c = 20.0 N/% ; θ is an aircraft pitch attitue angle, θ [ 0.3, 0.3] ra; u is a longituinal velocity, u [50, 300] m/s; w is a normal velocity, w [ 20, 20] m/s; q is an aircraft pitch rate, q [ 0.15, 0.15] ra/s; δ h is an angle of elevator eflection, δ h [ 0.7, 0.7] ra; δ c is a thrust coefficient, δ c [0, 400] % ; v wx, v wz are components of win velocity in the inertial system, v wx [ 20, 20] m/s, v wz [ 10, 10] m/s; We assume also that the functions c x (α, δ h ), c z (α, δ h ), m y (α, δ h ) have the following form: c x (α, δ h ) = c o x + cα2 x α2 + c h2 x (δ h) 2 c z (α, δ h ) = c o z + cα z α + ch z δ h m y (α, δ h ) = m α y α + m h yδ h where c o x = 0.2; cα2 x = 0.002; c h2 x = 0.002; c h z = 10 4 ; c o z = 0.15; cα z = 8.6; mα y = 0.057; mh y = 0.01. Fig. 1. Aircraft longituinal motion moel. 2.2 Problem of Aircraft Pitch Attitue Control The control problem is to provie the following conition: lim e θ(t) = 0 (2) t where e θ (t) is the error of the reference input realization; e θ (t) = θ (t) θ(t); θ (t) is the reference input. Moreover, the controlle transients e θ (t) 0 shoul have a esire behavior. These transients shoul not epen on the external isturbances an varying parameters of the aircraft moel (1). 2.3 Input-Output Mapping of Aircraft Moel From (1) it follows that the secon time erivative of θ(t) epens algebraically on the control variable δ h, that is where 2 θ/t 2 = f θ ( ) + b θ ( )δ h (3) f θ ( ) := 1 ρv 2J al 2 y S y m α y α, b θ ( ) := 1 ρv y 2J al 2 y S y m h y. y Hence, the secon time erivative of θ(t) is the relative highest erivative of the output variable θ(t). Note, the conition b θ ( ) < 0 hols for the above efine parameters. Remark 1. The expression (3) escribes the input-output mapping of the aircraft moel without taking into account the internal ynamics of aircraft (in particular case, that is zero-ynamics). It is assume through the text that the internal ynamics is stable or at least is boune. The analysis of the internal behavior of the system (1) was one by Yurkevich et al. (1991).
Remark 2. The parameter b θ is calle as the highfrequency gain of the input-output mapping efine by (3), where θ is consiere as the output variable, while δ h is consiere as the input variable. Remark 3. The high-frequency gain b θ may unergo extensive variations epening on operating point an aeroynamic characteristics, in particular, ue to the variation of the air velocity v a. 3. CONTROLLER DESIGN VIA TIME-SCALE SEPARATION 3.1 PID Controller Consier the controller given by µ 2 δ (2) h + 1µ δ (1) h = k[f (θ(1), θ, θ ) θ (2) ] (4) where µ is the small positive parameter, an F (θ (1), θ, θ ) := a 1 θ (1) a 0 [θ θ] where a 0 > 0 an a 1 > 0. The parameters a 0, a 1 are selecte such that the polynomial s 2 + a 1 s + a 0 (5) has the esire root istribution insie the left part of the s-plane, where roots of the polynomial (5) are efine by the requirements impose on the esire output transient performance inices of θ(t) in the system (1). Remark 4. The control law (4) can be expresse in terms of transfer functions, that is the structure of the conventional PID controller given by k { a0 } δ h (s) = µ(µs + 1 ) s [θ (s) θ(s)] (s + a 1 )θ(s) where the controller is proper an implemente without an ieal ifferentiation of θ(t) or θ (t). The replacement of θ (2) in (4) by the right member of (3) yiels the close-loop system equations in the form θ (2) = f θ ( ) + b θ ( )δ h (6) µ 2 δ (2) h + 1µ δ (1) h + kb θ( )δ h = k[f (θ (1), θ, θ ) f θ ( )]. Denote θ 1 = θ, θ 2 = θ (1), δ h1 = δ h, δ h2 = µ δ (1) h. Hence, from (6), the singularly perturbe ifferential equations θ 1 = θ 2, θ 2 = f θ ( ) + b θ ( )δ h1 µ δ h1 = δ h2, (7) µ δ h2 = kb θ ( )δ h1 1 δ h2 + k[f (θ 2, θ 1, θ ) f θ ( )] result as µ 0. If µ 0, then fast an slow moes are artificially force in the system (7) where the time-scale separation between these moes epens on the parameter µ. Hence, the properties of (7) can be analyze on basis of the twotime-scale technique an, as a result, slow an fast motion subsystems are erive in the next subsection of the paper. 3.2 Two-Time-Scale Motions Analysis Let us introuce the new fast time scale t 0 = t/µ. Hence, from (7), the close-loop system equations θ 1 = µθ 2, θ 2 = µ[f θ ( ) + b θ ( )δ h1 ] δ h1 = δ h2, (8) δ h2 = kb θ ( )δ h1 1 δ h2 + k[f (θ 2, θ 1, θ ) f θ ( )] result. If µ 0, then from (8) the FMS equations δ h1 = δ h2, (9) δ h2 = kb θ ( )δ h1 1 δ h2 + k[f (θ 2, θ 1, θ ) f θ ( )]. follow. Then, by returning to the primary time scale t = µt 0, we obtain the following FMS equations: µ δ h1 = δ h2, (10) µ δ h2 = kb θ ( )δ h1 1 δ h2 + k[f (θ 2, θ 1, θ ) f θ ( )] where θ 1 an θ 2 are treate as the constant values uring the transients in (10). This requirement can be easily satisfie for sufficiently small esign parameter µ. Finally, the FMS equations (10) may by rewritten as µ 2 δ (2) h + 1µ δ (1) h + kb θ( )δ h = k[f (θ (1), θ, θ ) f θ ( )](11) where f θ ( ) an b θ ( ) are treate as the constant values uring the transients in (11). Assume that the sign of k is selecte such that the conition kb θ ( ) > 0 hols an the esire fast amping of FMS transients is provie by selection of controller parameters 1, µ, an k. Then, the quasi-steay state for the FMS (11) yiels δ h (t) = δh i (t) where an δ i h δh i = b 1 θ ( )[F (θ(1), θ, θ ) f θ ( )] is exactly the inverse ynamics solution. Substitu- into (3) yiels the SMS equation tion of δ h = δ i h θ (2) + a 1 θ (1) + a 0 θ = a 0 θ. (12) By the another way, the SMS equation (12) can be irectly erive from (7), by taking µ = 0. In accorance with the main qualitative property of the singularly perturbe systems (see Klimushchev an Krasovskii (1962); Hoppensteat (1966); Kokotović et al. (1999); Naiu (2002)), we have, if an isolate equilibrium point of the FMS (11) exists an one is exponentially stable, then there exists µ > 0 such that for all µ (0, µ ) the trajectories of the singularly perturbe system (7) approximate to the trajectories of the SMS (12). So, if a sufficient time-scale separation between the fast an slow moes in the system (7) an exponential convergence of FMS transients to equilibrium are provie, then, after the amping of fast transients, the esire output behavior prescribe by (12) is fulfille espite that f θ ( ) an b θ ( ) are unknown complex functions. Thus, the output transient performance inices are insensitive to parameter variations of the nonlinear system an external isturbances, by that the solution of the iscusse control problem (2) is maintaine.
4. PID ADAPTIVE CONTROLLER 4.1 Problem of High-Frequency-Gain Online Ientification an Gain Tuning The critical point of the workability of the iscusse esign methoology via singular perturbation technique is that the controller parameters shoul be selecte in accorance with the requirement on FMS stability an, moreover, the esire egree of time-scale separation between the fast an slow moes in the close-loop system (7) shoul be provie. From (11), the FMS characteristic polynomial A fms (s) = µ 2 s 2 + 1 µ, s + γ (13) follows, where γ = kb θ ( ). The main isavantage is that the ecrease of timescale separation egree an loss of the FMS transient performances may occur in case of large variations of the high-frequency gain b θ ( ). In orer to overcome the isavantage cause by variations of b θ, two problems have to be solve. The first one is online ientification of b θ. The secon one is the aaptive gain tuning base on the knowlege of an estimate ˆb θ for b θ. Then, by taking k = k 1 k0 an k 1 0 = ˆb θ (14) the conition γ k 1 hols where k 1 > 0. As a result of gain tuning given by (14), the variations of the highfrequency gain b θ o not alter the FMS transient performance inices. Take, for example, the tuning rule in the following form k 0 t = α γ[γ 0 ˆγ 0], k 0 (0) = k 0 0 (18) where γ 0 is the esire value of γ 0 an ˆγ 0 is the estimate of γ 0 := kk 0 b θ ( ). It is clear, in the case of steay-state when k 0 /t = 0, from (18) the conition k 0 = γ 0 /( kˆb θ ( )) results where ˆb θ ( ) is an estimate of b θ ( ). 4.3 High-Frequency-Gain Online Ientification The high-frequency small oscillations are force in ˆδ h (t) an θ(t) ue to the high-frequency probing signal δh 0 sin(ωt) what was incorporate in the control system. Let be Aˆδh the amplitue of oscillations with frequency ω that are force in the control ˆδ h (t) an A θ be the amplitue of oscillations with frequency ω that are force in the output θ(t). From (3) an (15) it follows that A θ (ω) lim ω (ω) ω2 = γ 0. (19) Aˆδh Hence, the high-frequency-gain online ientification via relation (19) involves two amplitue etectors for A θ (ω) an (ω). For example, the amplitue etector base Aˆδh on the relationship A ξ = ξ 2 + ( ξ/ω) 2 can be use when ξ(t) = A ξ sin(ωt). The approach by the use of (19), which is aitionally supplemente by the low-pass an highpass filtering, is use in this paper in orer to get the estimates Âθ(ω) an ˆδh (ω), where the amplitue etector for ˆδh (ω) is escribe by 4.2 Aaptive Gain Tuning The block iagram of the propose aircraft pitch attitue control system with aaptive gain tuning is shown in Fig. 2. Take k 0 = kk 0 an δ h = kk 0δh ˆ (15) where δ ˆ h is the new control variable, k = sgn(b θ ) = 1, k 0 is the tuning gain, an k 0 > 0. τ 2 0 2 δh t 2 + 2τ δ h 0 t + 1 = τ 0 2 2ˆδh t 2 τf 2 2 u 1 t 2 + 2τ u 1 f t + 1 = δ h τf 2 2 u 2 t 2 + 2τ u 2 f t + 1 = δ h t = k ˆδh f u 2 1 + (u 2/ω) 2 k f = [1 (τ f ω) 2 ] 2 + (2τ f ω) 2 (20) The amplitue etector for Âθ(ω) is the same as (20). Then, with the help of the low-pass filter given by Fig. 2. Block iagram of the aircraft pitch attitue control system with aaptive gain tuning Consier the metho of online ientification base on a high-frequency probing signal (see, for example, Eykchoff (1974)). Let δˆ h (t) = δ h (t) + δh 0 sin(ωt) (16) where δh 0 sin(ωt) is the high-frequency probing signal with small value of amplitue δh 0. Next, let us rewrite the controller given by (4) as µ 2 δ(2) h + (1) 1µ δ h = k 1[F (θ (1), θ, θ ) θ (2) ] (17) τ 1 ˆγ 0 t + ˆγ 0 =  θ ˆδh + ɛ ω2, ˆγ 0 (0) = ˆγ 0 0, (21) the estimate ˆγ 0 results where ɛ is the small positive parameter in orer to avoi the singularity conition in the right member of (21). 5. SIMULATION RESULTS The simulation results of the close-loop system are base on the aircraft longituinal motion moel (1) with (15). The high-frequency probing signal is incorporate in the system as shown in (16). In accorance with (17), consier the feeback controller (C) in Fig. 2 in the following form: µ 2 δ(2) h + (1) 1µ δ h = k 1[ a 1 θ (1) a 0 [θ θ] θ (2) ]. (22) The aaptive gain tuning for k 0 is provie in accorance with the rule (18) while the high-frequency-gain online
ientification scheme consists of (20)-(21). The controller parameters are selecte as k 1 = 10, a 0 = 0.1145, a 1 = 0.4, an µ = 0.6609 s. The parameters of the aaptive gain tuning scheme an the high-frequency-gain online ientification scheme are selecte as γ0 = 1, ˆγ 0 (0) = 1, ɛ = 10 5. α γ = 0.2, k 0 (0) = 65, ω = 100 ra/s, A ω = 0.0003, τ f = 0.003 s, τ 0 = 0.01 s, τ 1 = 0.3 s, an δ c (t) = 150 % for all t. The simulation results of the close-loop system are shown in Figs. 3 12. Figure 3 contains the plots of θ (t) an θ(t) showing the output step response in the close-loop system where the air velocity v a (t) variations are isplaye in Figure 4. From Figure 9 it can be observe that the conition ˆγ 0 (t) γ0 is kept ue to the tuning of the gain k 0 (t) as seen in Figure 10. Fig. 6. Plot of the aircraft pitch rate q(t) (ra/s). Fig. 3. Plots of the aircraft pitch attitue θ(t) (ra) an θ (t) (ra). Fig. 7. Plot of the angle of attack α(t) (ra). Fig. 8. Plot of the elevator eflection δ h (t) (ra). Fig. 4. Plot of the air velocity v a (t) (m/s). Fig. 9. Plot of ˆγ 0 (t) where ˆγ 0 (t) is the estimate of γ 0 (t). 6. CONCLUSION Fig. 5. Plot of the normal velocity w(t) (m/s). The main avantage of the iscusse singular perturbation technique for control system analysis an esign is that the parameters of the PID aaptive controller for nonlinear aircraft moel can be analytically erive in
Fig. 10. Plot of the tuning gain k 0 (t). Fig. 11. Plot of win gusts v wx (t) (m/s). Fig. 12. Plot of win gusts v wz (t) (m/s). accorance with such inirect performance objectives as the esire root istribution of the reference moel characteristic polynomial, while the esire root istribution is efine by such irect aircraft pitch attitue performance objectives as the settling time an overshoot. The application of the singular perturbation technique in the presente esign methoology allows to get esire aircraft pitch attitue transient performance inices for nonlinear aircraft moel uner uncomplete knowlege about external isturbances an aeroynamic characteristics. It has been shown, the high-frequency-gain online ientification an gain tuning in the iscusse aaptive control system allow to maintain the two-time-scale structure of the trajectories of the close-loop system an esire transient performance inices of fast transients in case of significantly large range of aeroynamic characteristics variations. Simulation results show the effectiveness of the propose control laws. REFERENCES Åström, K. J., Hagglun, T., Hang,C. C., an Ho, W. K. (1993). Automatic tuning an aaptation for PID controllers - A survey, Contr. Eng. Pract., volume 1, no. 4, 699 714. B lachuta, M. J., Yurkevich, V. D., an Wojciechowski, K. (1997). Design of analog an igital aircraft flight controllers base on ynamic contraction metho. Proc. of the AIAA Guiance, Navigation, an Control Conference. New Orleans, LA, volume 3, 1719-1729. Czyba, R. an B lachuta, M. J. (2003). Robust longituinal flight control esign: ynamic contraction metho. Proc. of American Control Conf., Denver, Colorao, 1020 1025. O Dwyer, A. (2003). Hanbook of PI an PID Tuning Rules, Lonon, UK: Imperial College Press. Eykchoff, P. (1974). System Ientification, Wiley. Hoppensteat, F. C. (1966). Singular perturbations on the infinite time interval. Trans. of the American Mathematical Society, volume 123, 521 535. Klimushchev, A. I. an Krasovskii, N. N. (1962). Uniform asymtotic stability of systems of ifferential equations with a small parameter in the erivative terms. J. Appl. Math. Mech., volume 25, 1011 1025. Kokotović, P. V., Khalil, H. K., an O Reilly, J. (1999). Singular Perturbation Methos in Control: Analysis an Design, Philaelphia, PA: SIAM. Lane, S. an Stengel, R. (1988). Flight control esign using non-linear inverse ynamics, Automatica, volume 24, 471 483. Naiu, D. S. (2002). Singular perturbations an time scales in control theory an applications: An overview. Dynamics of Continuous, Discrete & Impulsive Systems (DCDIS), Series B: Applications & Algorithms, volume 9, no. 2, 233 278. Petrov, B. N. an Krutko, P. D. (1981). Inverse problems of controlle-flight ynamics: lateral motion, Sov. Phys.- Dokl., USA. volume 26, no. 2, 147 149. Schumacher, D. A. (1994). Rapi response missile pitch to trim flight control using sliing moes, Proc. of the 33r IEEE Conf. on Decision an Control, volume 4, 3854 3859. Vukobratović, M. an Stojić, R. (1988). Moern aircraft flight control. Lecture Notes in Control an Information Sciences, Springer-Verlag, volume 109. Wise, K. A. (1995). Applie controls research topics in the aerospace inustry, Proc. of the 34th Conf. on Decision an Control, New Orleans, LA, 751 756. Wojciechowski, K., Orys, A., an Polańska, J. (1989). Moel Przestrzennego Ruchu Samolotu a Celow Symulacji i Sterowania, Zeszyty Naukowe Politechniki Slaskiej, 1989. (in Polish) Yurkevich, V. D., Blachuta, M. J., an Wojciechowski, K. (1991). Stabilization system for the aircraft longituinal motion, Archiwum Automatyki i Robotyki, Warsaw, volume 36, no. 3-4, 517 535. (in Polish) Yurkevich, V. D. (2004). Design of Nonlinear Control Systems with the Highest Derivative in Feeback, Worl Scientific. Ziegel, G. an Nichols, N. B. (1942). Optimum settings for automatic controllers, Trans. ASME, volume 64, no. 8, 759 768.