Compensation of Actuator Dynamics in Nonlinear ... - Semantic Scholar
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IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 12, NO. 4, JULY 2004
Compensation of Actuator Dynamics in Nonlinear Missile Control Dongkyoung Chwa, Jin Young Choi, and Jin H. Seo
Abstract—This brief presents an approach to the compensation of the actuator dynamics in nonlinear missile control. In general, actuator dynamics are not considered in nonlinear missile control systems. Here, we analyzed the influences of actuator dynamics on the nonlinear missile controller and found that the second-order actuator dynamics showed risks of potentially destabilizing the overall system. Thus, to accommodate for the influence of the actuator dynamics, a compensator is proposed where the information of the nonlinear control input and the actual fin deflection is incorporated. We also conducted performance and stability analysis for the proposed control system, including actuator dynamics. We were able to confirm the validity of the proposed approach based on simulation results for the first- and second-order actuator dynamics. Index Terms—Actuator dynamics, compensation, nonlinear missile control.
Fig. 1.
Missile control system including actuator dynamics.
The nonlinear missile controller in this brief adopts the method proposed by the authors in [9]. Analysis on the influence of the actuator dynamics shows that the reduced parametric-affine missile model is not valid when the actuator dynamics are not fast enough. Thus, to accommodate for the influence of the actuator, a compensator using a backstepping technique in [12] is designed. The simulation results show that the proposed compensator can significantly improve the performance compared to the approach without compensation.
I. INTRODUCTION
S
EVERAL methods on autopilot designs have been studied to deal with the nonminimum phase problems in missiles with highly nonlinear characteristics [1]–[11]. The output-redefinition method for the acceleration control of tail-controlled skid-to-turn (STT) missiles was carried out in [1] and [2]. Also, the model inversion control using time-scaling separation and the robust nonlinear control were applied to the missile-autopilot system in [3]–[5] and [6]–[8], respectively. This brief proposed a nonlinear control approach [9] using a partial-linearization method [10] and a singular perturbation-like method [11]. Most approaches of nonlinear autopilot design assume that actuator dynamics are fast enough to be negligible. However, actuators show limited performance in real situations and, accordingly, the control performance would be severely degraded if the actuator dynamics are neglected. In this brief, we analyze the influence of actuator dynamics on the parametric-affine missile model proposed in [9] and design a controller considering actuator dynamics. The overall structure of the control-loop system in this brief consists of nonlinear controller, actuator compensator, actuator, is pitch and missile dynamics as shown in Fig. 1. Here, is pitch acceleration output; acceleration command and is control input from nonlinear controller and is control input from actuator compensator; and and are angle and angular rate of actual fin deflection. Manuscript received January 14, 2003; revised September 15, 2003. Manuscript received in final form October 31, 2003. Recommended by Associate Editor J. M. Buffington. This work was supported by the Agency for Defense Development, Automatic Control Research Center, Seoul National University, and Brain Korea 21 Project. The authors are with the School of Electrical Engineering and Computer Science, Seoul National University, Kwanak, Seoul 151-742, Korea (e-mail: [email protected]; [email protected]; [email protected]). Digital Object Identifier 10.1109/TCST.2004.825046
II. NONLINEAR CONTROLLER WITHOUT ACTUATOR COMPENSATION The controller is designed in accordance with the approach in [9] for the pitch missile dynamics where the aerodynamic coefficients in [13] are used. As discussed in [9], we assume that , and are missile mass , and the moments of inertia constants; the missile is symmetric in the plane; angular ; and linear rates of roll and pitch dynamics are zero velocities in and -directions are constants . Then, the pitch dynamics can be given by
(2.1)
where and are linear and angular velocities in -direction, is dynamic pressure, is aerodynamic is angle of attack, and aerodynamic reference area, and are decomposed coefficients and in [13] into where and are constants. Here, we introduce the new aerodynamic function as independent of
(2.2) where namics in (2.1) can be expressed as
1063-6536/04$20.00 © 2004 IEEE
. Since -dy-
(2.3)
IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 12, NO. 4, JULY 2004
and becomes
holds from
in assumptions, (2.1)
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theoretical analysis. Actuator dynamics in Fig. 1 are considered for the following two cases: the first-order low-pass filter (3.1a)
(2.4) and the second-order low-pass filter In the following, we design the controller without considering . To make the above system the actuator dynamics, i.e., almost linear, the control law is designed to satisfy
(3.1b)
A. Analysis of Actuator Influence (2.5) Then, (2.4) becomes (2.6)
It is assumed that the performance of the controller designed in the previous section may be degraded if the actuator dynamics in (3.1a) or (3.1b) are included. We will now discuss the validity of the reduction of -dynamics to (2.7) depending on the type of actuator dynamics. 1) First-Order Actuator Dynamics: When the first-order actuator dynamics are used, the actual fin deflection becomes
Since has a physically large value, angular rate can be assumed to converge to its steady-state value quickly via singular perturbation technique [11]
(3.2) unlike (2.5). Then, -dynamics in (2.3) becomes
(2.7) Then, the reduced missile model (2.8) becomes minimum phase from to system output , which makes it possible to design the feedback linearizing controller. Differentiation of the output in (2.8) yields and the choice yields . When of is generated by
(3.3) which can be expressed in state space form as
(3.4)
(2.9) the transfer function from the acceleration command to the acceleration output becomes the second-order linear system
Here, the eigenvalues of istic polynomial
become the roots of the character-
(3.5) (2.10) and are design parameters, and Here, and can be adjusted to obtain the desired response. The result of the nonlinear controller in (2.5) is effective when the response of the actuator dynamics is sufficiently fast, which is not necessarily so in actual situations. Thus, further analysis of the control system including actuator dynamics is required.
The positive guarantees the stability and the sufficiently large leads to the quick stabilization of -dynamics, which implies that singular perturbation method remains valid. 2) Second-Order Actuator Dynamics: When the secondorder actuator dynamics are used, -dynamics along with the actual control input become
III. INFLUENCE OF ACTUATOR DYNAMICS ON CONTROL PERFORMANCE In this section, the influence of actuator dynamics on control performance is analyzed by numerical simulations as well as
(3.6)
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This can be expressed in state–space form as
(3.7) and the eigenvalues of
are the roots of (3.8)
Even though the roots of (3.5) always have the negative real parts, the roots of (3.8) can have positive real parts for small . In this case, -dynamics do not converge to the steady-state value, which implies that singular perturbation method is no longer valid.
Fig. 2. Acceleration (solid: output, dotted: command) and fin deflection of the nonlinear controller in (2.5) for first-order actuator in (3.1a).
B. Numerical Verification of Actuator Influence Next, we will attempt to verify the above analysis through numerical experiments. The tracking performance is evaluated with respect to square wave acceleration commands for m/s. 1) First-Order Actuator Dynamics: The actuator model is and the design parameters of selected as (3.1a) with , and the control law in (2.9) are (rad/s). As the roots of (3.5) are -dynamics converge to its steady state within a short time as shown in Fig. 2, where the desired response can be obtained for the given acceleration commands. 2) Second-Order Actuator Dynamics: The actuator model , and is selected as (3.1b) with (rad/s). The roots of (3.8) change with as in Fig. 3. Two of the roots among the three remain in unstable less than (rad/s), in which case -dynamics region for cannot converge to its steady-state value. Thus, in this case, should be greater than (rad/s) to ensure the stable -dynamics, but this is not a realistic step. The roots of (3.8) , two of which have positive become real parts. As shown in Fig. 4, the control input as well as the system output shows chattering phenomenon. Of course, is greater than (rad/s), the results similar to Fig. 2 if could be obtained. As a result, it is necessary to modify the control law in (2.5) to accommodate for the influence of the actuator dynamics. IV. COMPENSATION OF ACTUATOR DYNAMICS IN NONLINEAR MISSILE CONTROL
Fig. 3. Root locus of (3.8) with respect to !
.
Fig. 4. Acceleration (solid: output, dotted: command) and fin deflection of the nonlinear controller in (2.5) for second-order actuator in (3.1b).
A. Compensator Design for the Second-Order Actuator A compensator is developed to reduce the difference between the nonlinear control input and the actual fin-deflection angle and this compensator can maintain the validity of the non-
linear control law in Section II. The structure of the compensation scheme is described in Fig. 1 in Section I. Here, and are assumed to be available.
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in the actuator dynamics (3.1b) is changed into
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as
(4.1) where is a control input from the actuator compensator. As the desired performance of the nonlinear controller can be maintained when becomes , the error is defined as
Fig. 5. Modified nonlinear control system with filtering without actuator dynamics.
or, in state–space form
(4.2) Taking the time derivative, we have (4.3) By backstepping procedure [12], the virtual input is designed , where is a positive constant. Then, we as which corresponds to the case of the firsthave order actuator dynamics in (3.1a) for the constant set point . Following the backstepping procedure, the variable:
(4.12) where the eigenvalues of
(4.13)
(4.4) is further introduced and its time derivative becomes
(4.5)
are the roots of
Here, the roots of (4.13) always have negative real parts. Thus, by increasing , the -dynamics converge to the steady-state value sufficiently fast. Now, instead of (2.8), we have
Now, we choose the Lyapunov function as
(4.14) (4.6) or
and we also take its time derivative to obtain
(4.15) (4.7) Choosing the control input to be (4.8)
where asymptotically converges to zero. Also, using (2.9), we can to becomes the show that the transfer function from second-order linear system
(4.7) becomes (4.16)
(4.9) which can satisfy the control objective by appropriately choosing . Substituting the control input in (4.8) into the actuator dynamics in (4.1), -dynamics can be given by
(4.10) . Then, using in (2.5), we can show that where -dynamics along with -dynamics become
which asymptotically becomes (2.10). Thus, the compensator in (4.8) can guarantee the expected control performance. The implementation of the control law in (4.8), however, requires the information of and , which may incur large control energy and also excite the unmodeled dynamics. Thus, we modify the compensator to reduce the high frequency components in and . When the actuator dynamics are neglected, the control input in (2.5) can be modified as as Fig. 5. Here, is filtered through the second-order low-pass filter
(4.17) where , and (rad/s). The previous section shows that can when (rad/s) holds. Thus, be used instead of to obtain
(4.11)
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Fig. 6. Modified backstepping compensation for actuator dynamics.
by substituting spectively) into input as
and and
(or and , rein (4.8), we can obtain a compensator
(4.18) and are used instead where of (4.2) and (4.4), respectively. Next, we will attempt to validate the compensation structure in Fig. 6. Substituting the compensator input in (4.18) into the actuator dynamics in (4.1), -dynamics become
Fig. 7. Acceleration (solid: output, dotted: command) and fin deflection of the compensator in (4.18) for actuator in (4.1).
(4.19) where
as in (4.10). Then, -dynamics become Fig. 8. Second-order actuator dynamics with amplitude and rate saturation.
(4.20) or, along with (4.17) and (4.19) in state–space form
(4.21) where
and the eigenvalues of
B. Numerical Simulation for the Compensator Performance The performance of the compensator in (4.18) was evaluated through simulations. The design parameters in (4.17) and (4.18) (rad/s) and . Simulations were selected as are performed by using several types of actuators. Firstly, the actuator was given by (4.1) with , and (rad/s). The tracking performance of the compensator in (4.18) was deemed satisfactory as shown in Fig. 7, whereas the controller in (2.5) without compenless than (rad/s), as shown sator showed chattering for in Fig. 4. We also used the actuator dynamics of the first-order ], third-order [(3.1a) cascaded with (4.1)], [(3.1a) with and fourth-order [(4.1) cascaded with (4.1)] and we could see good performance as in Fig. 7, which are omitted here. Additional simulation was conducted for the case where the actuator had amplitude and rate saturation in control fin deflection as in [14] and [15], where the high-performance fighter aircraft and a rate and amplitude limited second-order actuator was employed. Instead of (4.1), the actuator can be re-expressed like Fig. 8 as
are the roots of (4.23)
(4.22) where As shown in the previous section, every root of (4.22) has a negative part for (rad/s). This shows that the compensation structure in Fig. 6 with the control input (4.18) can guarantee the stability of the overall missile control system.
(4.24)
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Fig. 9. Acceleration (solid: output, dotted: command) and fin deflection of the nonlinear controller in (2.5) without compensator for actuator in (4.23) with _ = 150 deg/s.
j j
and
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Fig. 10. Acceleration (solid: output, dotted: command) and fin deflection = of the compensator in (4.25) for actuator in (4.23) with _ 150 deg/s.
j j
and are rate and amplitude saturation limits. Since and were used in the compensator instead of , (4.18) becomes
(4.25) (rad/s) and The parameters are shosen as deg/s. As shown in Fig. 9, the controller (2.5) without actuator compensator becomes more severely degraded than in Fig. 4 due to the rate saturation of the actuator. The compensator in (4.25), however, shows good performance in the rate saturated situation, as shown in Fig. 10. For accel, the amplitude saturation with eration command deg arises as in Figs. 11 and 12 for nonlinear controller (2.5) without and with actuator compensator in (4.25), respectively. The proposed compensator was found to be able to retain good tracking performance even with rate and amplitude saturation. From the above simulations, we could verify that the proposed method is valid.
Fig. 11. Acceleration (solid: output, dotted: command) and fin deflection of the nonlinear controller in (2.5) without compensator for actuator in (4.23) with = 15 deg.
V. CONCLUSION In this brief, the influence of actuator dynamics on nonlinear missile control system has been analyzed. We found that the first-order actuator dynamics does not influence the approach of the reduction of -dynamics via singular perturbation, but the second-order actuator dynamics may potentially destabilize the -dynamics, which means that the singular perturbation approach is no longer valid. To reduce the influence of actuator dynamics, we have developed a compensator using the backstepping technique. The proposed approach has been verified through simulations as well as analytic methods. Simulations showed that amplitude and rate saturations greatly influence the stability in the form of chattering. Thus, further analysis of saturations on the stability could be a subject for future research.
Fig. 12. Acceleration (solid: output, dotted: command) and fin deflection of = 15 deg. the compensator in (4.25) for actuator in (4.23) with
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REFERENCES [1] J. H. Ryu and M. J. Tahk, “Plant inversion control of tail-controlled missiles,” in Proc. AIAA Guidance, Navigation, and Control Conf., 97-3766, New Orleans, LA, Aug. 1997, pp. 1691–1696. [2] S. G. Kim and M. J. Tahk, “Output-redefinition based on robust zero dynamics,” in Proc. AIAA Guidance, Navigation, and Control Conf., 98-4493, Boston, MA, Aug. 1998, pp. 1–8. [3] J. Kim and J. Jang, “Nonlinear model inversion control for bank-to-turn missile,” in Proc. AIAA Guidance, Navigation, and Control Conf., 95-3318, Baltimore, MD, Aug. 1995, pp. 1308–1315. [4] M. B. McFarland and C. N. D’Souza, “Missile flight control with dynamic inversion and structured singular value synthesis,” in Proc. AIAA Guidance, Navigation, and Control Conf., Scottsdale, AZ, Aug. 1994, pp. 544–550. [5] C. Schumacher and P. P. Khargonekar, “Missile autopilot designs using control with gain scheduling and nonlinear dynamic inversion,” J. Guid. Control. Dyn., vol. 21, no. 2, pp. 234–243, 1998. [6] R. A. Hull, D. Schumacher, and Z. Qu, “Design and evaluation of robust nonlinear missile autopilots from a performance perspective,” in Proc. American Control Conf., Seattle, WA, June 1995, pp. 189–193. [7] R. A. Hull and Z. Qu, “Dynamic robust recursive control design and its application to a nonlinear missile autopilot,” in Proc. American Control Conf., Albuquerque, NM, June 1997, pp. 833–837.
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[8] J. Huang, C. F. Lin, J. R. Cloutier, J. H. Evers, and C. D’Souza, “Robust feedback linearization approach to autopilot design,” in Proc. IEEE Conf. Control Applications, Dayton, OH, 1992, pp. 220–225. [9] D. Chwa and J. Y. Choi, “New parametric affine modeling and control for skid-to-turn missiles,” IEEE Trans. Contr. Syst. Technol., vol. 9, pp. 335–347, Mar. 2001. [10] J. H. Oh and I. J. Ha, “Missile autopilot design via functional inversion and time-scaled transformation,” IEEE Trans. Aerosp. Electron. Syst. , vol. 33, pp. 64–76, Jan. 1997. [11] J. I. Lee and I. J. Ha, “Autopilot design for highly maneuvering STT missiles via singular perturbation-like techniques,” IEEE Trans. Contr. Syst. Technol., vol. 7, pp. 466–477, July 1999. [12] M. Krstic, I. Kanellakopoulos, and P. Kokotovic, Nonlinear and Adaptive Control Design. New York: Wiley, 1995. [13] “Autopilot Design for Missile Systems with Reduced Static Stability,” Agency for Defense Development, Taejon, Korea, Tech. Rep. TECD-416-020 423, May 2002. [14] R. A. Hess and S. A. Snell, “Flight control system design with rate saturating actuators,” J. Guid. Control Dynam., vol. 20, no. 1, pp. 90–96, 1997. [15] S. A. Snell and R. A. Hess, “Robust, decoupled flight control design with rate-saturating actuators,” J. Guid. Control Dynam., vol. 21, no. 3, pp. 361–367, 1998.
IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 12, NO. 4, JULY 2004
Compensation of Actuator Dynamics in Nonlinear Missile Control Dongkyoung Chwa, Jin Young Choi, and Jin H. Seo
Abstract—This brief presents an approach to the compensation of the actuator dynamics in nonlinear missile control. In general, actuator dynamics are not considered in nonlinear missile control systems. Here, we analyzed the influences of actuator dynamics on the nonlinear missile controller and found that the second-order actuator dynamics showed risks of potentially destabilizing the overall system. Thus, to accommodate for the influence of the actuator dynamics, a compensator is proposed where the information of the nonlinear control input and the actual fin deflection is incorporated. We also conducted performance and stability analysis for the proposed control system, including actuator dynamics. We were able to confirm the validity of the proposed approach based on simulation results for the first- and second-order actuator dynamics. Index Terms—Actuator dynamics, compensation, nonlinear missile control.
Fig. 1.
Missile control system including actuator dynamics.
The nonlinear missile controller in this brief adopts the method proposed by the authors in [9]. Analysis on the influence of the actuator dynamics shows that the reduced parametric-affine missile model is not valid when the actuator dynamics are not fast enough. Thus, to accommodate for the influence of the actuator, a compensator using a backstepping technique in [12] is designed. The simulation results show that the proposed compensator can significantly improve the performance compared to the approach without compensation.
I. INTRODUCTION
S
EVERAL methods on autopilot designs have been studied to deal with the nonminimum phase problems in missiles with highly nonlinear characteristics [1]–[11]. The output-redefinition method for the acceleration control of tail-controlled skid-to-turn (STT) missiles was carried out in [1] and [2]. Also, the model inversion control using time-scaling separation and the robust nonlinear control were applied to the missile-autopilot system in [3]–[5] and [6]–[8], respectively. This brief proposed a nonlinear control approach [9] using a partial-linearization method [10] and a singular perturbation-like method [11]. Most approaches of nonlinear autopilot design assume that actuator dynamics are fast enough to be negligible. However, actuators show limited performance in real situations and, accordingly, the control performance would be severely degraded if the actuator dynamics are neglected. In this brief, we analyze the influence of actuator dynamics on the parametric-affine missile model proposed in [9] and design a controller considering actuator dynamics. The overall structure of the control-loop system in this brief consists of nonlinear controller, actuator compensator, actuator, is pitch and missile dynamics as shown in Fig. 1. Here, is pitch acceleration output; acceleration command and is control input from nonlinear controller and is control input from actuator compensator; and and are angle and angular rate of actual fin deflection. Manuscript received January 14, 2003; revised September 15, 2003. Manuscript received in final form October 31, 2003. Recommended by Associate Editor J. M. Buffington. This work was supported by the Agency for Defense Development, Automatic Control Research Center, Seoul National University, and Brain Korea 21 Project. The authors are with the School of Electrical Engineering and Computer Science, Seoul National University, Kwanak, Seoul 151-742, Korea (e-mail: [email protected]; [email protected]; [email protected]). Digital Object Identifier 10.1109/TCST.2004.825046
II. NONLINEAR CONTROLLER WITHOUT ACTUATOR COMPENSATION The controller is designed in accordance with the approach in [9] for the pitch missile dynamics where the aerodynamic coefficients in [13] are used. As discussed in [9], we assume that , and are missile mass , and the moments of inertia constants; the missile is symmetric in the plane; angular ; and linear rates of roll and pitch dynamics are zero velocities in and -directions are constants . Then, the pitch dynamics can be given by
(2.1)
where and are linear and angular velocities in -direction, is dynamic pressure, is aerodynamic is angle of attack, and aerodynamic reference area, and are decomposed coefficients and in [13] into where and are constants. Here, we introduce the new aerodynamic function as independent of
(2.2) where namics in (2.1) can be expressed as
1063-6536/04$20.00 © 2004 IEEE
. Since -dy-
(2.3)
IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 12, NO. 4, JULY 2004
and becomes
holds from
in assumptions, (2.1)
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theoretical analysis. Actuator dynamics in Fig. 1 are considered for the following two cases: the first-order low-pass filter (3.1a)
(2.4) and the second-order low-pass filter In the following, we design the controller without considering . To make the above system the actuator dynamics, i.e., almost linear, the control law is designed to satisfy
(3.1b)
A. Analysis of Actuator Influence (2.5) Then, (2.4) becomes (2.6)
It is assumed that the performance of the controller designed in the previous section may be degraded if the actuator dynamics in (3.1a) or (3.1b) are included. We will now discuss the validity of the reduction of -dynamics to (2.7) depending on the type of actuator dynamics. 1) First-Order Actuator Dynamics: When the first-order actuator dynamics are used, the actual fin deflection becomes
Since has a physically large value, angular rate can be assumed to converge to its steady-state value quickly via singular perturbation technique [11]
(3.2) unlike (2.5). Then, -dynamics in (2.3) becomes
(2.7) Then, the reduced missile model (2.8) becomes minimum phase from to system output , which makes it possible to design the feedback linearizing controller. Differentiation of the output in (2.8) yields and the choice yields . When of is generated by
(3.3) which can be expressed in state space form as
(3.4)
(2.9) the transfer function from the acceleration command to the acceleration output becomes the second-order linear system
Here, the eigenvalues of istic polynomial
become the roots of the character-
(3.5) (2.10) and are design parameters, and Here, and can be adjusted to obtain the desired response. The result of the nonlinear controller in (2.5) is effective when the response of the actuator dynamics is sufficiently fast, which is not necessarily so in actual situations. Thus, further analysis of the control system including actuator dynamics is required.
The positive guarantees the stability and the sufficiently large leads to the quick stabilization of -dynamics, which implies that singular perturbation method remains valid. 2) Second-Order Actuator Dynamics: When the secondorder actuator dynamics are used, -dynamics along with the actual control input become
III. INFLUENCE OF ACTUATOR DYNAMICS ON CONTROL PERFORMANCE In this section, the influence of actuator dynamics on control performance is analyzed by numerical simulations as well as
(3.6)
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This can be expressed in state–space form as
(3.7) and the eigenvalues of
are the roots of (3.8)
Even though the roots of (3.5) always have the negative real parts, the roots of (3.8) can have positive real parts for small . In this case, -dynamics do not converge to the steady-state value, which implies that singular perturbation method is no longer valid.
Fig. 2. Acceleration (solid: output, dotted: command) and fin deflection of the nonlinear controller in (2.5) for first-order actuator in (3.1a).
B. Numerical Verification of Actuator Influence Next, we will attempt to verify the above analysis through numerical experiments. The tracking performance is evaluated with respect to square wave acceleration commands for m/s. 1) First-Order Actuator Dynamics: The actuator model is and the design parameters of selected as (3.1a) with , and the control law in (2.9) are (rad/s). As the roots of (3.5) are -dynamics converge to its steady state within a short time as shown in Fig. 2, where the desired response can be obtained for the given acceleration commands. 2) Second-Order Actuator Dynamics: The actuator model , and is selected as (3.1b) with (rad/s). The roots of (3.8) change with as in Fig. 3. Two of the roots among the three remain in unstable less than (rad/s), in which case -dynamics region for cannot converge to its steady-state value. Thus, in this case, should be greater than (rad/s) to ensure the stable -dynamics, but this is not a realistic step. The roots of (3.8) , two of which have positive become real parts. As shown in Fig. 4, the control input as well as the system output shows chattering phenomenon. Of course, is greater than (rad/s), the results similar to Fig. 2 if could be obtained. As a result, it is necessary to modify the control law in (2.5) to accommodate for the influence of the actuator dynamics. IV. COMPENSATION OF ACTUATOR DYNAMICS IN NONLINEAR MISSILE CONTROL
Fig. 3. Root locus of (3.8) with respect to !
.
Fig. 4. Acceleration (solid: output, dotted: command) and fin deflection of the nonlinear controller in (2.5) for second-order actuator in (3.1b).
A. Compensator Design for the Second-Order Actuator A compensator is developed to reduce the difference between the nonlinear control input and the actual fin-deflection angle and this compensator can maintain the validity of the non-
linear control law in Section II. The structure of the compensation scheme is described in Fig. 1 in Section I. Here, and are assumed to be available.
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in the actuator dynamics (3.1b) is changed into
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as
(4.1) where is a control input from the actuator compensator. As the desired performance of the nonlinear controller can be maintained when becomes , the error is defined as
Fig. 5. Modified nonlinear control system with filtering without actuator dynamics.
or, in state–space form
(4.2) Taking the time derivative, we have (4.3) By backstepping procedure [12], the virtual input is designed , where is a positive constant. Then, we as which corresponds to the case of the firsthave order actuator dynamics in (3.1a) for the constant set point . Following the backstepping procedure, the variable:
(4.12) where the eigenvalues of
(4.13)
(4.4) is further introduced and its time derivative becomes
(4.5)
are the roots of
Here, the roots of (4.13) always have negative real parts. Thus, by increasing , the -dynamics converge to the steady-state value sufficiently fast. Now, instead of (2.8), we have
Now, we choose the Lyapunov function as
(4.14) (4.6) or
and we also take its time derivative to obtain
(4.15) (4.7) Choosing the control input to be (4.8)
where asymptotically converges to zero. Also, using (2.9), we can to becomes the show that the transfer function from second-order linear system
(4.7) becomes (4.16)
(4.9) which can satisfy the control objective by appropriately choosing . Substituting the control input in (4.8) into the actuator dynamics in (4.1), -dynamics can be given by
(4.10) . Then, using in (2.5), we can show that where -dynamics along with -dynamics become
which asymptotically becomes (2.10). Thus, the compensator in (4.8) can guarantee the expected control performance. The implementation of the control law in (4.8), however, requires the information of and , which may incur large control energy and also excite the unmodeled dynamics. Thus, we modify the compensator to reduce the high frequency components in and . When the actuator dynamics are neglected, the control input in (2.5) can be modified as as Fig. 5. Here, is filtered through the second-order low-pass filter
(4.17) where , and (rad/s). The previous section shows that can when (rad/s) holds. Thus, be used instead of to obtain
(4.11)
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Fig. 6. Modified backstepping compensation for actuator dynamics.
by substituting spectively) into input as
and and
(or and , rein (4.8), we can obtain a compensator
(4.18) and are used instead where of (4.2) and (4.4), respectively. Next, we will attempt to validate the compensation structure in Fig. 6. Substituting the compensator input in (4.18) into the actuator dynamics in (4.1), -dynamics become
Fig. 7. Acceleration (solid: output, dotted: command) and fin deflection of the compensator in (4.18) for actuator in (4.1).
(4.19) where
as in (4.10). Then, -dynamics become Fig. 8. Second-order actuator dynamics with amplitude and rate saturation.
(4.20) or, along with (4.17) and (4.19) in state–space form
(4.21) where
and the eigenvalues of
B. Numerical Simulation for the Compensator Performance The performance of the compensator in (4.18) was evaluated through simulations. The design parameters in (4.17) and (4.18) (rad/s) and . Simulations were selected as are performed by using several types of actuators. Firstly, the actuator was given by (4.1) with , and (rad/s). The tracking performance of the compensator in (4.18) was deemed satisfactory as shown in Fig. 7, whereas the controller in (2.5) without compenless than (rad/s), as shown sator showed chattering for in Fig. 4. We also used the actuator dynamics of the first-order ], third-order [(3.1a) cascaded with (4.1)], [(3.1a) with and fourth-order [(4.1) cascaded with (4.1)] and we could see good performance as in Fig. 7, which are omitted here. Additional simulation was conducted for the case where the actuator had amplitude and rate saturation in control fin deflection as in [14] and [15], where the high-performance fighter aircraft and a rate and amplitude limited second-order actuator was employed. Instead of (4.1), the actuator can be re-expressed like Fig. 8 as
are the roots of (4.23)
(4.22) where As shown in the previous section, every root of (4.22) has a negative part for (rad/s). This shows that the compensation structure in Fig. 6 with the control input (4.18) can guarantee the stability of the overall missile control system.
(4.24)
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Fig. 9. Acceleration (solid: output, dotted: command) and fin deflection of the nonlinear controller in (2.5) without compensator for actuator in (4.23) with _ = 150 deg/s.
j j
and
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Fig. 10. Acceleration (solid: output, dotted: command) and fin deflection = of the compensator in (4.25) for actuator in (4.23) with _ 150 deg/s.
j j
and are rate and amplitude saturation limits. Since and were used in the compensator instead of , (4.18) becomes
(4.25) (rad/s) and The parameters are shosen as deg/s. As shown in Fig. 9, the controller (2.5) without actuator compensator becomes more severely degraded than in Fig. 4 due to the rate saturation of the actuator. The compensator in (4.25), however, shows good performance in the rate saturated situation, as shown in Fig. 10. For accel, the amplitude saturation with eration command deg arises as in Figs. 11 and 12 for nonlinear controller (2.5) without and with actuator compensator in (4.25), respectively. The proposed compensator was found to be able to retain good tracking performance even with rate and amplitude saturation. From the above simulations, we could verify that the proposed method is valid.
Fig. 11. Acceleration (solid: output, dotted: command) and fin deflection of the nonlinear controller in (2.5) without compensator for actuator in (4.23) with = 15 deg.
V. CONCLUSION In this brief, the influence of actuator dynamics on nonlinear missile control system has been analyzed. We found that the first-order actuator dynamics does not influence the approach of the reduction of -dynamics via singular perturbation, but the second-order actuator dynamics may potentially destabilize the -dynamics, which means that the singular perturbation approach is no longer valid. To reduce the influence of actuator dynamics, we have developed a compensator using the backstepping technique. The proposed approach has been verified through simulations as well as analytic methods. Simulations showed that amplitude and rate saturations greatly influence the stability in the form of chattering. Thus, further analysis of saturations on the stability could be a subject for future research.
Fig. 12. Acceleration (solid: output, dotted: command) and fin deflection of = 15 deg. the compensator in (4.25) for actuator in (4.23) with
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