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A Missile Autopilot Based on Feedback Linearization
European Journal of Control (2002)8:553±560 # 2002 EUCA
A Missile Autopilot Based on Feedback Linearization Emmanuel Devaud1,2 and Houria Siguerdidjane1 1 2
EÂcole SupeÂrieure d'EÂlectriciteÂ, Service Automatique, Plateau de Moulon, 91192 Gif-sur-Yvette, France and Aerospatiale, 2 rue BeÂranger, 92323 ChaÃtillon, France
This paper is dealing with a missile autopilot design. The objective is to seek the approaches which significantly enhance the ability of an aerospace control engineer to deal with a practical nonlinear flight control problem. For this purpose, two nonlinear methods are considered: ``approximate feedback'' and ``approximate dynamic feedback'' with two time-scale separation. Both methodologies have been applied in order to overcome the difficulties caused by the non-minimum phase phenomenon of the missile. For the sake of realism, the implementation of the controller includes a nonlinear observer. The second method, which incorporates the classical industrial know how, has been shown to be better among the two in terms of both performance and complexity of the control law.
In recent years, there has been a great interest in missile autopilot design. The autopilot design task amounts to finding a control law such that the acceleration of the missile center of mass tracks a given reference signal, generated by the guidance ± navigation system. The main difficulties in missile autopilot design arise from the uncertainties of the
aerodynamic parameters. Due to the very imprecise knowledge of the aerodynamic coefficients, the designed controller must not be sensitive to the variations of these coefficients. Moreover, the missile dynamics, are known to be non-minimum phase with respect to direct control, so then perfect tracking is not feasible. One may therefore, make further investigations to achieve system stabilization. As the system is not feedback linearizable, a wide literature has been concentrated in gain scheduling approaches [12,16], extended linearization [3] and input±output approximate feedback [4,7] or input±output feedback with output redefinition [2]. In [10], two time-scale separation has been applied to overcome the non-minimum phase phenomenon, but the authors have implicitly neglected the small coupling terms, as in approximate linearization. This has been detailed in [14], where an extra outer-loop has been added in order to achieve the robustness with regards to the model neglected terms. Some different control strategies have been proposed in [4,17], the latter one is based on the angle of attack normal form. In this paper, the main objective is to investigate several approaches to a practical nonlinear flight control problem, in order to improve the performance and robustness of the controlled system. The paper is organized as follows: Section 2 formulates the problem under consideration. Sections 3 and 4 are dedicated to the control designs which were supposed to fulfill the requirements. It was found that the approximate static feedback linearization leads to
Correspondence and offprint request to: H. Siguerdidjane, EÂcole SupeÂrieure d'EÂlectriciteÂ, Service Automatique, Plateau de Moulon, 91192 Gif-sur-Yvette, France.
Received 17 February 1999; Accepted in revised form 6 May 2002. Recommended by M. Tomizuka and M. Steinbuch.
Keywords: Approximate feedback linearization; Autopilot; Missile; Nonlinear control
1. Introduction
554
E. Devaud and H. Siguerdidjane
a little more complicated control law than a dynamic approximate feedback one, based on the classical industrial knowledge. Finally, Section 5 presents a nonlinear quadratic observer. Simulation results and a conclusion are given in Sections 6±7.
2. Problem Description The aim of a missile autopilot is to force a missile center of mass acceleration to track a given reference, generated by an outer loop, the so-called guidance loop. The performance of the controlled missile are commonly expressed in terms of the time to first peak ( 0.35 s), the overshoot (D < 20/%), and the accuracy (tracking error, 5%), under some constraints (the actuator is limited in position and speed). The model was first presented in [13] and has been used as a benchmark by several authors since. Consider the tail controlled pitch axis missile airframe depicted in Fig. 1: _ t K M2 Cn cos t=Vm q t,
1
_ Kq M2 Cm , q t
where denotes the angle of attack, q is the pitch rotational rate, Vm is the missile speed, is the tail fin deflection, Cn and Cm are the aerodynamic coefficients, the expressions of which depend on , and on the Mach number M: Cn sgn an j j3 bn j j2 cn 2
M=3j j dn ,
2
Cm sgn am j j3 bm j j2 cm 7 8M=3j j dm :
→
F
P G
→
F
3
The remaining parameters are constant. The tail fin actuator dynamics are described by a second order differential equation rather than the first order differential equation as proposed in [13] and used in [4], since it is more representative of a realistic actuator. As a matter of fact, many industrial flight simulators use second order differential actuator models (see [3]). So then, the actuator model is described by _1 2 , _2 2a !a 2
!2a 1 !2a c ,
4
where !a 150 rd/s is the corner frequency, a 0.7 is the damping ratio and c is the commanded tail-fin deflection.
3. Approximate Input±Output Feedback It has been shown in [17] that the zero dynamics of system (1) are unstable, and that the non-minimum phase phenomenon is caused by the location of the fin (one has to bear in mind that the considered missile is a tail-fin controlled missile). This implies that input± output feedback linearization cannot be straightforwardly applied (see for instance [8]), otherwise it would generate unobservable and uncontrollable modes which are unstable. The idea is then to design a linearizing controller on a new model, in order to decrease the size of the unobservable manifold without increasing the size of the system by a dynamic linearizing feedback, i.e., without increasing the complexity of the associated control law. Basically, this is the principle of the so-called ``approximate feedback'' [7]. For the undertaken missile problem, approximate feedback consists of designing a linearizing feedback on the following approximate system:
q_ Kq M2 Cm , , →
V
Fig. 1. Missile airframe.
t K M2 Cn t, t:
_ K M2 C0n cos t=Vm q,
The output of the system is the normal acceleration:
K M
2
5
C0n ,
where C0n is derived from Cn , : C0n Cn , 0. It can be easily verified that the relative degree of the output is 2, hence an input±output linearizing
555
Missile Autopilot Based on Feedback Linearization
control can be addressed for the approximate model: t 1 @F =@
F q
2 @ F @F @F F q @ 2 @ @ 6 !2 ! 2 r ,
@F M @ 2!_
where F K M2 C0n cos =Vm , M KqM2 Cm( , ), F K M2 C0n , KqM2dm r denotes the reference acceleration and and ! are the damping ratio and the corner frequency of the closed loop system, respectively. By applying, on the original system, the control law designed on the approximate one, it finally appears that, under conditions involving the smoothness of the input and the amplitude of the neglected terms, the tracking error is bounded and the output is stable (one may see [7] for a mathematical proof and additional discussions). In order to improve the system performance, several authors, especially in aircraft control [4,6,14], have added an outer loop upon the feedback linearization. However, even if this last one presents good margins and robustness, the original system (namely the missile airframe) could have bad margins and robustness properties. In the sequel, another idea was developed, which consists of designing a nonlinear controller based on the classical linear one.
according to Fig. 2. As a matter of fact, the feedback of the inner loop is q and the controller consists of proportional and integral actions, which mainly ensure the stability of the plant, especially in the case of a static unstable missile airframe. The primary control loop of the autopilot is the acceleration feedback, and has the role to ensure the tracking of the desired acceleration provided by the guidance loop. This can be generally obtained by a single gain. Improvements (gain scheduling, addition of filters) can herein be made on this basic structure but this is irrelevant here. It turns out that designing the classical controller is equivalent to looking for the four gains k0, k1, k2 and k3, which in turn is equivalent to ``fixing'' a certain dynamic to the closed loop system, defined by a time constant , a damping ratio and a frequency !, under some constraints, among which there is the bandwidth of the actuator (see [11]). Nevertheless, one can wonder why there is no integrator in the outer loop. The main reasons are the following: (a) In practice, two integrators in the loop might decrease the phase margin too much. (b) The outer loop integrator would be dif®cult to design because of the non-minimum phase phenomenon, which appears in the outer loop only, and which varies with the ¯ight conditions. Both reasons are based on industrial knowledge.
4. Nonlinear Dynamical Feedback with Two Time-scale Separation
4.2. Nonlinear Approximate Feedback with Two Time-scale Separation
4.1. Brief Recall on the Linear Design
In order to simplify the nonlinear design as well as to obtain a nonlinear autopilot based on the linear classical controller (hence on the classical aeronautical engineering know-how), we first try to split the system into two subsystems with two different
For the pitch axis model presented in Section 1, the sensor measurements used for classical feedback control are the missile rotational rate q and the normal acceleration . Two loops are thus designed [4]
c
k0
+ –
k1
+ +
k2
1 s
+
c
Actuator
Airframe
+ q k3
Fig. 2. Classical controller.
556
E. Devaud and H. Siguerdidjane
dynamics: a fast one and a slow one, according to singular perturbation theory (see, e.g., [5]). The choice of the slow and fast subsystems, i.e., of the slow and fast state variables, even if not so clear in the open loop system, is dictated by the behavior of the cell feedback with a linear autopilot. Hence, for the nonlinear design, the slow state variable is (reflecting the behavior of , according to Eq. (3), but it is more practical, since it schedules the nonlinearities of the model under consideration) and the fast state variable is q.
a static state feedback can be straightforwardly addressed:
The state variable is , the considered input is now q and the output is the normal acceleration . Since the system is non-minimum phase, one explicitly needs to neglect in the state space equation, changing the system _ cos K M Cn , =Vm q, K M2 Cn , ,
7
into _ cos K M2 Cn0 =Vm q,
8
K M2 Cn0 ,
by using the same notations as in the previous section. Since _ K M2 dC0n =d cos K M2 C0n = Vm q, the static state feedback is simply: cos K M2 Cn0 =Vm 1 1 2 s K M dCn0 =d
r :
9
These fixes a first order dynamic with the time constant s to the closed loop slow approximate subsystem. 4.2.2. Fast Dynamics For the fast subsystem, the state variable, which also represents the output, is q. The reference input is qr, which is the input of the slow subsystem determined by Eq. (9). Finally, the control input is . From the state equations q_ Kq M2 Cm , , y q,
10
qr ,
11
dCm0 Kq M2 dm _ dt
12
The following control ensures a second order behavior (with damping ratio and frequency !) to the closed loop approximate system:
2
1 q f Kq M2 dm
with qr defined above by Eq. (9). Since one wants to ``mimic'' the linear controller, it is necessary to consider the second time derivative of the subsystem output: q t Kq M2
4.2.1. Slow Dynamics
q
Cm0 =dm
2! q K q M 2 dm Z !2 q cos K M2 C0n =Vm K q M 2 dm 1 1 dt 13 r s K M2 dC0n =d Cm0 =dm
Moreover, taking advantage of the fact that _ cos K M2 C0n =Vm q, it may be interesting to equation, the expression by R replace, in the previous q cos K M2 C0n =Vm dt by . For real-world applications, uncertainties on Cn may average 20%, and such a modification can lead to significant improvements. Finally, it is found the following control law, which is rather simpler to be implemented than the previous one:
2! q C0m =dm K q M 2 dm Z !2 1 1 2 2 K q M dm s K M dC0n =d r dt 14
One may notice that the state space representation of the system by means of Eqs (8) and (10) is in an appropriate form for using the backstepping based approach [9], it may however be shown that the resulting control is a little bit more complicated than Eq. (11).
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Missile Autopilot Based on Feedback Linearization
5. Observer Design The measured variables are q and only, then cannot be used directly, so one needs an estimation of it. Let us first recall an important result about the separation of estimation and control [15]. Theorem 1. [15]. Let G be a non-anticipative nonlinear dynamical operator with finite incremental gain; let x^ be any estimate of x that is non divergent with finite gain. If the system is closed-loop bounded (finite gain stable) with feedback u Gx, then the system with feedback u G^ x is also closed-loop bounded (finite gain stable). The proof lies on the fact that the modification, due to the new feedback law, may be seen as an additive bounded perturbation for the initial system. In our case, we moreover determine a quadratic observer, i.e., it is possible to find a quadratic Lyapunov function. This observer is merely given by:
which implies the previous condition, because of the mean value theorem (see [1]). It is important to notice that the control law needs a slight modification in order to be incremental and thus, to satisfy the theorem assumptions. In fact, the integral structure must be replaced by a high gain control, namely 1 1 r , s K M2 dC0n =d 2! q Kq M2 C0m K q M 2 dm
x_ k
!2 xk , Kq M2 dm has to be replaced by x_ k
^ cos =V ^ ^_ K M2 Cn , m q K ^ y
^ , K M2 Cn ,
where " is sufficiently small. Hence, the corresponding scheme of the dynamic control law is given in Fig. 3. The results are discussed in the following section.
^ cos ^ K M =Vm Cn , 2 ^ KK M Cn , Cn , cos Cn , ^ < "
16
for some positive real number ", which guarantees exponential stability for the observer (choose V(x) x2 as a Lyapunov function). In practice, K is chosen in such a way that:
<
@ Cn , cos @ " M
r +
–
r ,
19
2
K =Vm
1 1 s K M2 dC0n =d 2! q Kq M2 C0m K q M 2 dm
"xk
!2 xk , Kq M2 dm
15
where K is chosen so as to ensure: ^
18
Slow controller
KK
@ Cn , @ 17
qf
Fast controller
6. Simulation Results The simulations are performed with the numerical values given in [13]. For each one, acceleration (in g) and control (in rad) are displayed versus time (see Figs 4, 5 and 6). Each control law has been tested on the nominal system (solid plot) and on the perturbed one (dashed plots). The perturbations considered here are multiplicative input uncertainties ranging from 0.4 to 0.4.
Slow subsystem Fast subsystem
Fig. 3. Two time-scale separation observer.
Nonlinear observer q
ˆ
558
E. Devaud and H. Siguerdidjane
(a)
2
(b)
150
1.5
100
1 0.5
c (rad)
(g)
50
0
0 –0.5 –1
–50 –1.5
–100
0
1
2
3 Time (s)
4
5
–2
6
0
1
2
3 Time (s)
4
5
6
Fig. 4. Approximate feedback linearization. (a) Acceleration (g) versus time (s); (b) Control (rad) versus time (s).
(a)
(b)
40
0.5 0.4
30
0.3
20 c (rad)
(g)
0.2 10 0
0.1 0
–10
–0.1
–20
–0.2
–30
–0.3 0
1
2
3 Time (s)
4
5
6
0
1
2
3 Time (s)
4
5
6
Fig. 5. Approximate dynamical feedback, with 2 time-scale separation. (a) Acceleration (g) versus time (s); (b) Control (rad) versus time (s).
Hence, from Figs 4±6, it is possible to compare each controller in terms of: nominal behavior of the closed-loop system, robustness of the closed-loop system with respect to speci®ed uncertainties. Figures 4 and 5 are obtained assuming that is measurable, namely, no observer is used, while Fig. 6 is obtained by using the quadratic observer described in Section 5. Figure 4 (Approximate feedback linearization) shows a relatively good nominal behavior:
the the the the
system is stable, settling time is about 0.25 s, overshoot is quite negligible, control remains within the bounded values.
Nevertheless, in presence of uncertainties (dashed plots) the degradation of the results in terms of settling
time and accuracy is so large than this solution must be discarded. Figure 5 (Approximate dynamic feedback linearization) shows significant improvements: ®rst, the settling time is approximately the same as the preceding case, while a slight overshoot is arising, nevertheless it remains below the given speci®cation (<20%), the control is still bounded. Robustness with respect to uncertainties appears to be quite good (dashed plots). However, the deviation between the nominal and the perturbed cases has nothing to do with the previous case. In both cases, the use of an observer does not significantly affect the previous results: the plots are quite the same, it is therefore not necessary to display them. Figure 6 shows the influence of the variation of the observer initial condition, on a shorter simulation
559
Missile Autopilot Based on Feedback Linearization
40
(b)
0.5
35
0.4
30
0.3
25
0.2
20
0.1
c (rad)
c (g)
(a)
15
eta (g)
0
10
–0.1
5
–0.2
0
–0.3
–5
0
0.2
0.4
(c)
0.6 Time (s)
0.8
–0.4 0
1
0.2
0.4
0.6 Time (s)
0.8
1
10 5
(rad)
0 –5 –10 –15 –20 –25 0
0.2
0.4
0.6 Time (s)
0.8
1
Fig. 6. Approximate dynamical feedback, with 2 time-scale separation with the quadratic observer. (a) Acceleration (g) versus time (s); (b) Control (rad) versus time (s); (c) Angle of attack versus time.
time. Let us point out that this set of plots is obtained by considering a very large offset of initial conditions (10 ) between the missile and the observer. The specifications are still met in the case of the dynamic controller combined with the quadratic observer.
includes a nonlinear observer. The second method, which incorporates the classical industrial know how, has been shown to be better among the two in terms of both performance and complexity of the control law.
7. Conclusion
References
In this paper, two different nonlinear approaches have been presented for controlling a single-axis missile: ``approximate feedback'' and ``approximate dynamic feedback'' with two time-scale separation. Both nonlinear methodologies are applied in order to increase missile performance (nonlinear controller). In addition, the difficulties caused by the non-minimum phase phenomenon of the missile are overcome. The choices presented here have been inspired by the industrial context. In order to meet the specifications, the implementation of the controller
1. AlexeÂev V, Tikhomirov V, Fomine S. Commande optimale. Editions Mir, Moscou 1982 2. Benvenuti L, Di Benedetto MD, Grizzle JW. Approximate output tracking for nonlinear non-minimum phase system with an application to flight control. Int J Robust Nonlinear Control 1994; 4: 397±414 3. Devaud E, Harcaut JP, Siguerdidjane H. Three-axes missile autopilot design from linear to nonlinear control strategies. AIAA J Guidance, Control, Dynamics 2000; 23(5): 1±8 4. Devaud E, Siguerdidjane H, Font S. Some control strategies of a high angle of attack missile autopilot. Control Eng Practice 2000; (8): 885±892.
560 5. Fossard AJ, Foisneau J, Hun Huynh T. Optimisation approcheÂe en boucle fermeÂe des systeÁmes non lineÂaires par la meÂthodologie des perturbations singulieÁres. SysteÁmes non lineÂaires, Tome III, AJ. Fossard and D. Normand-Cyrot, Ed Masson, 1993 6. Harcaut JP. Nonlinear control of a missile through a geometric approach, In: A Bensoussan, JL Lions (eds). Lecture notes in control and information sciences, vol. 144. Springer Verlag 1990, pp. 27±36 7. Hauser J, Sastry S, Meyer G. Nonlinear controller design for slightly non minimum phase system: application to V/STOL aircraft. Automatica 1992; 28(4): 665±679 8. Isidori A. Nonlinear control systems, Springer Verlag 1995 9. Krstic M, Kanellakopoulos I, Kokotovic. Nonlinear and adaptive control design, John Wiley & Sons, 1995 10. Menon PK, Yousefpor M. Design of nonlinear autopilots for high angle of attack missiles. In: Proceedings of AIAA Guidance Navigation and Control Conference, 1996, no 96-3913
E. Devaud and H. Siguerdidjane
11. Nesline FW, Nabbefeld NC. Design of digital autopilots for homing missiles. In: Proceedings of the AGARD conference, 1979 12. Nichols RA, Reichert RT, Rugh WJ. Gain Scheduling for H1 controllers: a flight control example. Proc IEEE Trans Control Sys Technol 1993; 1(2): 69±78 13. Reichert RT. Dynamic scheduling of modern-robustcontrol autopilot design for missiles. IEEE Control Systs Mag 1992; 35±42 14. Reiner J, Balas GJ, Garrard WL. Flight control design using robust dynamic inversion and time-scale separation. Automatica 1996; 32(11): 1493±1504 15. Safonov MG. Stability and robustness of multivariable feedback systems, MIT Press, Cambridge 1980 16. Shamma JS, Cloutier JR. Gain-scheduled missile autopilot design using Linear Parameter Varying transformation. AIAA J Guidance, Control, Dynamics 1993; 16(2): 256±263 17. Siguerdidjane H, Devaud E. Nonlinear missile autopilot design based angle of attack normal form. Eur J Control 2000; 6(2): 154±164
A Missile Autopilot Based on Feedback Linearization Emmanuel Devaud1,2 and Houria Siguerdidjane1 1 2
EÂcole SupeÂrieure d'EÂlectriciteÂ, Service Automatique, Plateau de Moulon, 91192 Gif-sur-Yvette, France and Aerospatiale, 2 rue BeÂranger, 92323 ChaÃtillon, France
This paper is dealing with a missile autopilot design. The objective is to seek the approaches which significantly enhance the ability of an aerospace control engineer to deal with a practical nonlinear flight control problem. For this purpose, two nonlinear methods are considered: ``approximate feedback'' and ``approximate dynamic feedback'' with two time-scale separation. Both methodologies have been applied in order to overcome the difficulties caused by the non-minimum phase phenomenon of the missile. For the sake of realism, the implementation of the controller includes a nonlinear observer. The second method, which incorporates the classical industrial know how, has been shown to be better among the two in terms of both performance and complexity of the control law.
In recent years, there has been a great interest in missile autopilot design. The autopilot design task amounts to finding a control law such that the acceleration of the missile center of mass tracks a given reference signal, generated by the guidance ± navigation system. The main difficulties in missile autopilot design arise from the uncertainties of the
aerodynamic parameters. Due to the very imprecise knowledge of the aerodynamic coefficients, the designed controller must not be sensitive to the variations of these coefficients. Moreover, the missile dynamics, are known to be non-minimum phase with respect to direct control, so then perfect tracking is not feasible. One may therefore, make further investigations to achieve system stabilization. As the system is not feedback linearizable, a wide literature has been concentrated in gain scheduling approaches [12,16], extended linearization [3] and input±output approximate feedback [4,7] or input±output feedback with output redefinition [2]. In [10], two time-scale separation has been applied to overcome the non-minimum phase phenomenon, but the authors have implicitly neglected the small coupling terms, as in approximate linearization. This has been detailed in [14], where an extra outer-loop has been added in order to achieve the robustness with regards to the model neglected terms. Some different control strategies have been proposed in [4,17], the latter one is based on the angle of attack normal form. In this paper, the main objective is to investigate several approaches to a practical nonlinear flight control problem, in order to improve the performance and robustness of the controlled system. The paper is organized as follows: Section 2 formulates the problem under consideration. Sections 3 and 4 are dedicated to the control designs which were supposed to fulfill the requirements. It was found that the approximate static feedback linearization leads to
Correspondence and offprint request to: H. Siguerdidjane, EÂcole SupeÂrieure d'EÂlectriciteÂ, Service Automatique, Plateau de Moulon, 91192 Gif-sur-Yvette, France.
Received 17 February 1999; Accepted in revised form 6 May 2002. Recommended by M. Tomizuka and M. Steinbuch.
Keywords: Approximate feedback linearization; Autopilot; Missile; Nonlinear control
1. Introduction
554
E. Devaud and H. Siguerdidjane
a little more complicated control law than a dynamic approximate feedback one, based on the classical industrial knowledge. Finally, Section 5 presents a nonlinear quadratic observer. Simulation results and a conclusion are given in Sections 6±7.
2. Problem Description The aim of a missile autopilot is to force a missile center of mass acceleration to track a given reference, generated by an outer loop, the so-called guidance loop. The performance of the controlled missile are commonly expressed in terms of the time to first peak ( 0.35 s), the overshoot (D < 20/%), and the accuracy (tracking error, 5%), under some constraints (the actuator is limited in position and speed). The model was first presented in [13] and has been used as a benchmark by several authors since. Consider the tail controlled pitch axis missile airframe depicted in Fig. 1: _ t K M2 Cn cos t=Vm q t,
1
_ Kq M2 Cm , q t
where denotes the angle of attack, q is the pitch rotational rate, Vm is the missile speed, is the tail fin deflection, Cn and Cm are the aerodynamic coefficients, the expressions of which depend on , and on the Mach number M: Cn sgn an j j3 bn j j2 cn 2
M=3j j dn ,
2
Cm sgn am j j3 bm j j2 cm 7 8M=3j j dm :
→
F
P G
→
F
3
The remaining parameters are constant. The tail fin actuator dynamics are described by a second order differential equation rather than the first order differential equation as proposed in [13] and used in [4], since it is more representative of a realistic actuator. As a matter of fact, many industrial flight simulators use second order differential actuator models (see [3]). So then, the actuator model is described by _1 2 , _2 2a !a 2
!2a 1 !2a c ,
4
where !a 150 rd/s is the corner frequency, a 0.7 is the damping ratio and c is the commanded tail-fin deflection.
3. Approximate Input±Output Feedback It has been shown in [17] that the zero dynamics of system (1) are unstable, and that the non-minimum phase phenomenon is caused by the location of the fin (one has to bear in mind that the considered missile is a tail-fin controlled missile). This implies that input± output feedback linearization cannot be straightforwardly applied (see for instance [8]), otherwise it would generate unobservable and uncontrollable modes which are unstable. The idea is then to design a linearizing controller on a new model, in order to decrease the size of the unobservable manifold without increasing the size of the system by a dynamic linearizing feedback, i.e., without increasing the complexity of the associated control law. Basically, this is the principle of the so-called ``approximate feedback'' [7]. For the undertaken missile problem, approximate feedback consists of designing a linearizing feedback on the following approximate system:
q_ Kq M2 Cm , , →
V
Fig. 1. Missile airframe.
t K M2 Cn t, t:
_ K M2 C0n cos t=Vm q,
The output of the system is the normal acceleration:
K M
2
5
C0n ,
where C0n is derived from Cn , : C0n Cn , 0. It can be easily verified that the relative degree of the output is 2, hence an input±output linearizing
555
Missile Autopilot Based on Feedback Linearization
control can be addressed for the approximate model: t 1 @F =@
F q
2 @ F @F @F F q @ 2 @ @ 6 !2 ! 2 r ,
@F M @ 2!_
where F K M2 C0n cos =Vm , M KqM2 Cm( , ), F K M2 C0n , KqM2dm r denotes the reference acceleration and and ! are the damping ratio and the corner frequency of the closed loop system, respectively. By applying, on the original system, the control law designed on the approximate one, it finally appears that, under conditions involving the smoothness of the input and the amplitude of the neglected terms, the tracking error is bounded and the output is stable (one may see [7] for a mathematical proof and additional discussions). In order to improve the system performance, several authors, especially in aircraft control [4,6,14], have added an outer loop upon the feedback linearization. However, even if this last one presents good margins and robustness, the original system (namely the missile airframe) could have bad margins and robustness properties. In the sequel, another idea was developed, which consists of designing a nonlinear controller based on the classical linear one.
according to Fig. 2. As a matter of fact, the feedback of the inner loop is q and the controller consists of proportional and integral actions, which mainly ensure the stability of the plant, especially in the case of a static unstable missile airframe. The primary control loop of the autopilot is the acceleration feedback, and has the role to ensure the tracking of the desired acceleration provided by the guidance loop. This can be generally obtained by a single gain. Improvements (gain scheduling, addition of filters) can herein be made on this basic structure but this is irrelevant here. It turns out that designing the classical controller is equivalent to looking for the four gains k0, k1, k2 and k3, which in turn is equivalent to ``fixing'' a certain dynamic to the closed loop system, defined by a time constant , a damping ratio and a frequency !, under some constraints, among which there is the bandwidth of the actuator (see [11]). Nevertheless, one can wonder why there is no integrator in the outer loop. The main reasons are the following: (a) In practice, two integrators in the loop might decrease the phase margin too much. (b) The outer loop integrator would be dif®cult to design because of the non-minimum phase phenomenon, which appears in the outer loop only, and which varies with the ¯ight conditions. Both reasons are based on industrial knowledge.
4. Nonlinear Dynamical Feedback with Two Time-scale Separation
4.2. Nonlinear Approximate Feedback with Two Time-scale Separation
4.1. Brief Recall on the Linear Design
In order to simplify the nonlinear design as well as to obtain a nonlinear autopilot based on the linear classical controller (hence on the classical aeronautical engineering know-how), we first try to split the system into two subsystems with two different
For the pitch axis model presented in Section 1, the sensor measurements used for classical feedback control are the missile rotational rate q and the normal acceleration . Two loops are thus designed [4]
c
k0
+ –
k1
+ +
k2
1 s
+
c
Actuator
Airframe
+ q k3
Fig. 2. Classical controller.
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E. Devaud and H. Siguerdidjane
dynamics: a fast one and a slow one, according to singular perturbation theory (see, e.g., [5]). The choice of the slow and fast subsystems, i.e., of the slow and fast state variables, even if not so clear in the open loop system, is dictated by the behavior of the cell feedback with a linear autopilot. Hence, for the nonlinear design, the slow state variable is (reflecting the behavior of , according to Eq. (3), but it is more practical, since it schedules the nonlinearities of the model under consideration) and the fast state variable is q.
a static state feedback can be straightforwardly addressed:
The state variable is , the considered input is now q and the output is the normal acceleration . Since the system is non-minimum phase, one explicitly needs to neglect in the state space equation, changing the system _ cos K M Cn , =Vm q, K M2 Cn , ,
7
into _ cos K M2 Cn0 =Vm q,
8
K M2 Cn0 ,
by using the same notations as in the previous section. Since _ K M2 dC0n =d cos K M2 C0n = Vm q, the static state feedback is simply: cos K M2 Cn0 =Vm 1 1 2 s K M dCn0 =d
r :
9
These fixes a first order dynamic with the time constant s to the closed loop slow approximate subsystem. 4.2.2. Fast Dynamics For the fast subsystem, the state variable, which also represents the output, is q. The reference input is qr, which is the input of the slow subsystem determined by Eq. (9). Finally, the control input is . From the state equations q_ Kq M2 Cm , , y q,
10
qr ,
11
dCm0 Kq M2 dm _ dt
12
The following control ensures a second order behavior (with damping ratio and frequency !) to the closed loop approximate system:
2
1 q f Kq M2 dm
with qr defined above by Eq. (9). Since one wants to ``mimic'' the linear controller, it is necessary to consider the second time derivative of the subsystem output: q t Kq M2
4.2.1. Slow Dynamics
q
Cm0 =dm
2! q K q M 2 dm Z !2 q cos K M2 C0n =Vm K q M 2 dm 1 1 dt 13 r s K M2 dC0n =d Cm0 =dm
Moreover, taking advantage of the fact that _ cos K M2 C0n =Vm q, it may be interesting to equation, the expression by R replace, in the previous q cos K M2 C0n =Vm dt by . For real-world applications, uncertainties on Cn may average 20%, and such a modification can lead to significant improvements. Finally, it is found the following control law, which is rather simpler to be implemented than the previous one:
2! q C0m =dm K q M 2 dm Z !2 1 1 2 2 K q M dm s K M dC0n =d r dt 14
One may notice that the state space representation of the system by means of Eqs (8) and (10) is in an appropriate form for using the backstepping based approach [9], it may however be shown that the resulting control is a little bit more complicated than Eq. (11).
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Missile Autopilot Based on Feedback Linearization
5. Observer Design The measured variables are q and only, then cannot be used directly, so one needs an estimation of it. Let us first recall an important result about the separation of estimation and control [15]. Theorem 1. [15]. Let G be a non-anticipative nonlinear dynamical operator with finite incremental gain; let x^ be any estimate of x that is non divergent with finite gain. If the system is closed-loop bounded (finite gain stable) with feedback u Gx, then the system with feedback u G^ x is also closed-loop bounded (finite gain stable). The proof lies on the fact that the modification, due to the new feedback law, may be seen as an additive bounded perturbation for the initial system. In our case, we moreover determine a quadratic observer, i.e., it is possible to find a quadratic Lyapunov function. This observer is merely given by:
which implies the previous condition, because of the mean value theorem (see [1]). It is important to notice that the control law needs a slight modification in order to be incremental and thus, to satisfy the theorem assumptions. In fact, the integral structure must be replaced by a high gain control, namely 1 1 r , s K M2 dC0n =d 2! q Kq M2 C0m K q M 2 dm
x_ k
!2 xk , Kq M2 dm has to be replaced by x_ k
^ cos =V ^ ^_ K M2 Cn , m q K ^ y
^ , K M2 Cn ,
where " is sufficiently small. Hence, the corresponding scheme of the dynamic control law is given in Fig. 3. The results are discussed in the following section.
^ cos ^ K M =Vm Cn , 2 ^ KK M Cn , Cn , cos Cn , ^ < "
16
for some positive real number ", which guarantees exponential stability for the observer (choose V(x) x2 as a Lyapunov function). In practice, K is chosen in such a way that:
<
@ Cn , cos @ " M
r +
–
r ,
19
2
K =Vm
1 1 s K M2 dC0n =d 2! q Kq M2 C0m K q M 2 dm
"xk
!2 xk , Kq M2 dm
15
where K is chosen so as to ensure: ^
18
Slow controller
KK
@ Cn , @ 17
qf
Fast controller
6. Simulation Results The simulations are performed with the numerical values given in [13]. For each one, acceleration (in g) and control (in rad) are displayed versus time (see Figs 4, 5 and 6). Each control law has been tested on the nominal system (solid plot) and on the perturbed one (dashed plots). The perturbations considered here are multiplicative input uncertainties ranging from 0.4 to 0.4.
Slow subsystem Fast subsystem
Fig. 3. Two time-scale separation observer.
Nonlinear observer q
ˆ
558
E. Devaud and H. Siguerdidjane
(a)
2
(b)
150
1.5
100
1 0.5
c (rad)
(g)
50
0
0 –0.5 –1
–50 –1.5
–100
0
1
2
3 Time (s)
4
5
–2
6
0
1
2
3 Time (s)
4
5
6
Fig. 4. Approximate feedback linearization. (a) Acceleration (g) versus time (s); (b) Control (rad) versus time (s).
(a)
(b)
40
0.5 0.4
30
0.3
20 c (rad)
(g)
0.2 10 0
0.1 0
–10
–0.1
–20
–0.2
–30
–0.3 0
1
2
3 Time (s)
4
5
6
0
1
2
3 Time (s)
4
5
6
Fig. 5. Approximate dynamical feedback, with 2 time-scale separation. (a) Acceleration (g) versus time (s); (b) Control (rad) versus time (s).
Hence, from Figs 4±6, it is possible to compare each controller in terms of: nominal behavior of the closed-loop system, robustness of the closed-loop system with respect to speci®ed uncertainties. Figures 4 and 5 are obtained assuming that is measurable, namely, no observer is used, while Fig. 6 is obtained by using the quadratic observer described in Section 5. Figure 4 (Approximate feedback linearization) shows a relatively good nominal behavior:
the the the the
system is stable, settling time is about 0.25 s, overshoot is quite negligible, control remains within the bounded values.
Nevertheless, in presence of uncertainties (dashed plots) the degradation of the results in terms of settling
time and accuracy is so large than this solution must be discarded. Figure 5 (Approximate dynamic feedback linearization) shows significant improvements: ®rst, the settling time is approximately the same as the preceding case, while a slight overshoot is arising, nevertheless it remains below the given speci®cation (<20%), the control is still bounded. Robustness with respect to uncertainties appears to be quite good (dashed plots). However, the deviation between the nominal and the perturbed cases has nothing to do with the previous case. In both cases, the use of an observer does not significantly affect the previous results: the plots are quite the same, it is therefore not necessary to display them. Figure 6 shows the influence of the variation of the observer initial condition, on a shorter simulation
559
Missile Autopilot Based on Feedback Linearization
40
(b)
0.5
35
0.4
30
0.3
25
0.2
20
0.1
c (rad)
c (g)
(a)
15
eta (g)
0
10
–0.1
5
–0.2
0
–0.3
–5
0
0.2
0.4
(c)
0.6 Time (s)
0.8
–0.4 0
1
0.2
0.4
0.6 Time (s)
0.8
1
10 5
(rad)
0 –5 –10 –15 –20 –25 0
0.2
0.4
0.6 Time (s)
0.8
1
Fig. 6. Approximate dynamical feedback, with 2 time-scale separation with the quadratic observer. (a) Acceleration (g) versus time (s); (b) Control (rad) versus time (s); (c) Angle of attack versus time.
time. Let us point out that this set of plots is obtained by considering a very large offset of initial conditions (10 ) between the missile and the observer. The specifications are still met in the case of the dynamic controller combined with the quadratic observer.
includes a nonlinear observer. The second method, which incorporates the classical industrial know how, has been shown to be better among the two in terms of both performance and complexity of the control law.
7. Conclusion
References
In this paper, two different nonlinear approaches have been presented for controlling a single-axis missile: ``approximate feedback'' and ``approximate dynamic feedback'' with two time-scale separation. Both nonlinear methodologies are applied in order to increase missile performance (nonlinear controller). In addition, the difficulties caused by the non-minimum phase phenomenon of the missile are overcome. The choices presented here have been inspired by the industrial context. In order to meet the specifications, the implementation of the controller
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