Constrained Inner-Loop Control of a Hypersonic Glider Using ...
2013 American Control Conference (ACC) Washington, DC, USA, June 17-19, 2013
Constrained Inner-Loop Control of a Hypersonic Glider Using Extended Command Governor Morgan Baldwin
Ilya Kolmanovsky
Abstract— The paper considers inner-loop control of a hypersonic glider. To maintain vehicle flight, pointwise-in-time constraints on vehicle state and control variables need to be enforced. To enforce constraints, an Extended Command Governor (ECG) is augmented to the nominal LQ-PI controller. Simulation results demonstrate the capability of the ECG to enforce the constraints while minimizing the degradation in the vehicle responsiveness.
I. I NTRODUCTION Hypersonic vehicles are an area of great interest for the United States. Several experimental vehicles have been launched; DARPA’s Falcon Project being an example. Two test vehicles were launched under the Falcon Project; both flights ended unsuccessfully. It was determined that the main technical challenges are the assumptions from known flight regimes are inadequate and the advanced thermal modeling and ground testing are not successful at predicting reality [15]. Conventional control techniques appropriate for subsonic and supersonic flight are not necessarily applicable for hypersonic flight. A hypersonic vehicle requires a control system that can maintain stability and active damping during flight, be robust to uncertainties, and cater to structural effects [6]. Current research projects into hypersonic vehicles can benefit from an advanced guidance, navigation, and control system with capabilities of real time optimization. The capability of determining a guidance and control solution onboard the vehicle would be an enabling technology to achieve the DoD and NASA’s hypersonic vehicle goals. Due to the uncertainties of hypersonic flight, predictive and adaptive control for hypersonic vehicles has been a major area of research. Predictive control was researched in Refs. [16], [4], and [5]. In Refs. [18] and [17], a Model Predictive Control (MPC) approach was used. In a companion paper [1], we implement an MPC approach for guidance of the hypersonic vehicle. Adaptive control has also been actively researched for control of a hypersonic vehicle [14], [7], [8]. This paper presents an inner-loop control scheme for the longitudinal dynamics of a hypersonic glider vehicle with constraint enforcement capability. For the inner-loop, angle-of-attack command tracking is achieved using a Linear This work is supported in part by ASEE through the Air Force Summer Faculty Fellow Program (SFFP) and by the National Science Foundation Award Number 1130160. Ilya Kolmanovsky is with the Department of Aerospace Engineering, The University of Michigan, Ann Arbor, Michigan. [email protected] Morgan Baldwin is with the Space Vehicles Directorate, Air Force Research Laboratory, Albuquerque, New Mexico.
[email protected] 978-1-4799-0176-0/$31.00 ©2013 AACC
Quadratic Proportional plus Integral (LQ-PI) controller. An Extended Command Governor (ECG) is augmented to the nominal closed-loop system to handle the constraints. The ECG is an advanced reference type governor algorithm that manipulates the command to enforce constraints. Reference governors have been an active area of research since the 1990s. They are add-on schemes used to enforce constraints in stable closed-loop systems. In the late 1990s, the theory was further developed to include command governors and, more recently, to also include extended command governors, both of which give a larger constrained domain of attraction and have a potential for a faster response than the reference governor. See Ref. [13] and references therein. In Ref. [19], a reference governor was implemented to avoid the occurrence of saturation of the control system and prevent windup. It was augmented to an existing adaptive controller, which was designed to track reference signals. In Ref. [10], a reference governor was used to enforce constraints on elevator deflection for conventional UAV gliders. In this work, the ECG ensures that the implemented control will keep the vehicle in a safe operating region; its added benefit is an increase in the domain of attraction for the set of initial conditions, and the potential for improving system responsiveness. II. L ONGITUDINAL E QUATIONS OF M OTION FOR A F LEXIBLE H YPERSONIC G LIDER The flexible aircraft model used is from Ref. [3]. The longitudinal equations of motion for a flexible aircraft are as follows T cos α − D − g sin γ, (1) V˙ t = m T sin α + L g Vt α˙ = − + ( − ) cos γ + Q, (2) mVt Vt r M , (3) Q˙ = Iyy ˙ = Q, Θ (4) ˙h = Vt sin γ, (5) η¨i
=
−2ξωi η˙ i − ωi2 ηi + Ni , i = 1, 2, . . . , n,
(6)
where Vt is the true airspeed, T is the thrust, D is the drag, L is the lift, α is the angle-of-attack, γ is the flight path angle, M is the pitching moment, Q is the pitch rate, h is the altitude, and r is the radius of the vehicle. The flexible dynamics of the vehicle are modeled in Eqn. 6, where ηi is the ith modal coordinate of the flexible dynamics, ξ is the damping ratio, ωi is the natural frequency, and Ni is the ith
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generalized force. The modal method and its derivation is more completely described in Ref. [3]. This model is useful for preliminary control analysis and provides insight in the interactions between the flexible vehicle dynamics and the implemented control. The thrust is set to zero in the gliding phase. III. C ONSTRAINTS Given the model in the previous section, constraints were added to reflect requirements relevant to maintaining vehicle flight. The angle-of-attack, pitch rate, elevator deflection, elevator deflection rate, and flexible deflections were constrained. The elevator deflection is used to control the vehicle by changing the pressure distribution on the vehicle and thereby altering the lift and drag forces affecting the vehicle. The rate of elevator deflection is constrained to reflect the actuator physical limits and to ensure safe control of the vehicle by preventing control maneuvers that can lead to unstable flight. The angle-of-attack is constrained so the vehicle stays within its flight envelope during hypersonic flight. If large angle-of-attack values are allowed, the vehicle may transition to uncontrollable flight. These constraints are represented by αmin ≤ α ≤ αmax , (7) δemin ≤ δe ≤ δemax ,
(8)
∆δemin ≤ ∆δe ≤ ∆δemax .
(9)
are the modal coordinates and their time derivatives (η1 ,η˙ 1 , η2 , η˙ 2 , η3 , η˙ 3 ). The control input for the system is the elevator deflection, δe . A. LQ-PI controller The LQ-PI controller is designed using the short period subsystem extracted from the continuous-time linearized system model. This subsystem includes the angle-of-attack, pitch rate, the modal coordinates, and their derivatives. The controller is designed for set-point tracking of angle-of-attack commands. To treat the case when the elastic states are not measured or estimated, a second LQ-PI controller was designed based on a second order subsystem with states being the angle-of-attack and pitch rate. In this case, the LQ-PI controller uses feedback on the integral of the angle of attack tracking error, angle-of-attack and pitch rate. B. Extended Command Governor Overview An ECG is implemented to enforce state and control constraints by modifying the set-point commands to the closed-loop system, as shown in Fig. 1. In Fig. 1, x ˆ denotes the state estimate, y is the system output constrained by specifying the set inclusion conditions y(t) ∈ Y for all t, w is the disturbance/uncertainty, v is the ECG output, and r is the reference command. In this paper, w(t) = 0 as the characterization and treatment of uncertainty is left to future publications.
The pitch rate is constrained as follows Qmin ≤ Q(t) ≤ Qmax .
(10)
Due to high speeds of the vehicle in the hypersonic flight regime, structural dynamics, including aeroelastic effects, play a significant role; constraints must be imposed to ensure acceptable elastic deflections. Specifically, at specified distances back from the nose of the vehicle, the contributions of the three modal frequencies are constrained to be less than a specific value, EDmin ≤
3 X
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(11)
i=1
where ED is the total elastic deflection of the vehicle and Φi is the ith mode shape at the specified location on the vehicle, xloc,i . This constraint enables the designed controller to keep the vehicle’s structure safe during the gliding portion of flight. The selected locations are at 1/10, 3/10, 5/10, 7/10 and 9/10 of vehicle length from the nose. IV. LQ-PI C ONTROLLER WITH E XTENDED C OMMAND G OVERNOR Given the model and constraints proposed above, a nominal LQ-PI controller is augmented with an extended command governor to handle the constraints. The equations of motion are linearized about a trim point assuming zero thrust for the vehicle. This yields an 11 dimensional system with the first five states given above (velocity V , angle-of-attack α, pitch rate q, altitude h and pitch attitude θ) and the last six
Fig. 1: Extended command governor applied to a closed-loop system. ˇ ∞ , of states and parameterized The ECG uses the set, O inputs such that the constraints are satisfied for all time. The ECG output is defined by v(t) = ρ(t) + C¯ x ¯(t),
(12)
where ρ(t) and x ¯(t) are solutions to the following optimization problem minimize ||ρ(t) − r(t)||2Q + ||¯ x(t)||2S , ˇ∞ . subject to (ρ(t), x ¯(t), x(t)) ∈ O
(13)
The x ¯ ∈ Rn¯ , n ¯ ≥ 0, and ρ are states of a stable auxiliary dynamic system which evolve over the semi-infinite prediction horizon according to ¯x(t + k), k ≥ 0 x ¯(t + k + 1) = A¯ (14) ρ(t + k) = ρ(t). Compared to reference governors and conventional command governors (the case n ¯ = 0 above), the ECG yields a larger
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domain of recoverable initial states and faster response. It is also able to react to mitigate constraint violation even if there are no changes in r(t). ˇ ∞ is a finitely determined inner approximation to The set O the set of all (ρ(t), x ¯(t), x(t)) that do not induce subsequent constraint violation when the input sequence v(t + k) is determined by the fictitious dynamics per (12) and (14). In the case when Y is polyhedral, computational procedures ˇ ∞ , see [13]. Without the exist that lead to polyhedral O fictitious states, i.e., when n ¯ = 0, the ECG becomes a simple command governor [9]. The optimization problem can be solved online using conventional quadratic programming techniques. The iterative procedures can be avoided by using explicit multi-parametric quadratic programming [2], leading to ρ and x ¯ being given as a piecewise-affine function of the state x(t) and reference r. Various choices of A¯ and C¯ can be made. The shift sequences used in Ref. [9] are generated by the fictitious dynamics with, 0 Im 0 0 ··· 0 0 Im 0 · · · 0 Im · · · (15) A¯ = 0 0 0 0 0 0 · · · .. .. .. .. .. . . . . . C¯ = Im 0 0 0 · · · ,
pitch rate, −δe,min = δe,max = 0.349 rad for the elevator deflection (20 deg), −∆δe,min = ∆δe,max = 0.349 rad/sec for the elevator deflection rate, and −EDmin = EDmax = 1.0 for the elastic deflections. To illustrate how the imposition of constraints progressively affects the response, we considered a subset of constraints and a full set of constraints. The subset of constraints contained just the limits on the elevator deflection and elastic deflections (8), (11). The full set of constraints contained all of the constraints (7)-(11) V. S IMULATION R ESULTS
The simulation results are based on vehicle model linearized at a dynamic pressure of 1500 psi and an altitude of 95000 ft with zero thrust. We first show the closed-loop responses for the case of the nominal LQ controller that includes elastic states in feedback. With a pitch command of r = ±0.29, very close to the imposed angle-of-attack limit, constraints on angle-of-attack, pitch rate, elevator deflection, elevator deflection rate, and elastic deflections are all violated in transients, see Fig. 2. Note that the elastic deflections are largest near the nose of the vehicle, near the tail of the vehicle, and near the middle of the vehicle, with smaller deflections elsewhere. Enforcing the constraints on elastic deflections is thus important to ensure that the vehicle does not break apart during the flight. The response with ECG designed for n ¯ = 5, α = 0.0 to enforce constraints on anglewhere Im is an m × m identity matrix. Another approach of-attack, pitch rate, elevator deflection rates of change, and uses Laguerre sequences [12]. These sequences possess elastic mode deflections is shown in Fig. 3. The imposed orthogonality properties and are generated by the fictitious constraints are enforced and the response of the system with dynamics with ECG is slowed down. The LQ-PI controller does not use elastic states for feedback and the response of the system is αIm βIm −αβIm α2 βIm · · · 0 similar for the cases when the LQ-PI controller is designed αIm βIm −αβIm · · · with elastic states for feedback. 0 0 αI βI · · · ¯ m m A= (16) 0 Different design choices can lead to differences in the 0 0 αI · · · m .. .. .. .. .. speed of response of the closed-loop system. Fig. 4 shows . . . . . ¯ = 1, α = 0, which essentially is the case p the case of n C¯ = β Im −αIm α2 Im −α3 Im · · · (−α)N −1 Im , of the conventional command governor. While in all cases (¯ n = 1, 5, 10) the system is protected against the constraint where β = 1 − α2 and 0 ≤ α ≤ 1 is a selectable parameter violation, there is a considerable speed up in responses as that corresponds to the time-constant of the fictitious dynamn ¯ , i.e., the dimensionality of the auxiliary system increases. ics. Note that with the choice of α = 0, (16) coincides with Non-zero α helps to speed up the response further. With the shift register considered in Ref. [9]. n ¯ = 10, α = 0.4 as shown in Fig. 5 versus n ¯ = 5, C. Extended Command Governor Design α = 0.0 as shown in Fig. 3, the response is significantly For this work, the constraints are on angle-of-attack, more aggressive, and more constraints become active as the pitch rate, elevator deflection, elevator deflection rate and ECG has considerable flexibility to improve the performance elastic deflection. The discrete-time model of the closed-loop under constraints, see Fig. 5. The computational complexity ¯. system necessary to implement ECG is based on the LQ-PI of the ECG increases, however, with the increase in n controller (one of two designs depending on whether elastic VI. T OWARDS O N -B OARD I MPLEMENTATION modes are used for feedback), the short period subsystem The preceding sections demonstrated the potential of the (with elastic modes), and a sampling period of 0.01 sec. ECG to handle constraints. The development of ECG for This model has the following form on-board implementation require several additional steps. x(t + 1) = Ax(t) + Bv(t) (17) The ECG based on partial state information and reduced y(t) = Cx(t) + Dv(t) ∈ Y. order model (without the elastic dynamics) can be developed The limits were as follows: −αmin = αmax = 0.3 rad for using the reduced order reference governor techniques [11]. the angle of attack; −Qmin = Qmax = 0.2 rad/sec for the Both observer errors and contributions of the omitted states 5578
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Fig. 2: The response without the ECG for (a) angle-of-attack, (b) pitch rate, (c) elevator deflection, (d) elastic deflection, (e) elevator deflection time rate of change.
Fig. 3: The response with the ECG formulated for n ¯ = 5, α = 0, LQ-PI controller that does not use elastic mode measurements/estimates for feedback and full set of the constraints: (a) angle-of-attack, (b) pitch rate, (c) elevator deflection, (d) elastic deflection, (e) elevator deflection time rate of change.
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to the constrained outputs can be handled by imposing an additional constraint on δv(t) = v(t) − v(t − 1). The uncertainties in forces and moments in hypersonic flight can be handled by representing these uncertainties as additive set-bounded disturbances. The existing extended command governor theory can handle these uncertainties by empoying disturbance invariant sets. Finally, if explicit implementation of ECG is used (in the form of Piecewise Affine controller that represents a multi-parametric solution to the QP (13)), the ECGs designed for different trim points during the flight can be merged into a single Piecewise Affine explicit solution applicable in the full flight range. If the model is periodically re-identified or updated on-board during the flight, ˇ ∞ with O ˇ ∞ computed the condition (ρ(t), x ¯(t), x(t)) ∈ O off-line, will need to be replaced by the conditions on the predicted response of the system, y(t + k|t) ∈ Y , for k = 0, · · · , nc , where nc is the constraint horizon. We leave these developments as next steps for future publications. VII. C ONCLUDING R EMARKS The paper has proposed an augmentation of an innerloop LQ-PI controller with an Extended Command Governor (ECG) to handle constraints faced during hypersonic vehicle flight. The advantages of ECG, which is based on higher order auxilliary dynamics and Laguerre’s sequence generation have been demonstrated for the actuator rate limited case. Future research will address steps which facilitate the on-board implementation of ECG design. These include implementation based on reduced order models and based on models with uncertainties in forces and moments during the hypersonic flight. VIII. ACKNOWLEDGMENTS The authors thank Michael Bolender for providing a copy of the model of the glider and guidance on the modeling and control aspects. The ECG code used in this paper has been developed at the University of Michigan by Uros Kalabic.
[8] Fiorentini, L., Serrani, A., Bolender, M., and Doman, D., “Nonlinear robust adaptive control of flexible air-breathing hypersonic vehicles,” Journal of Guidance, Control, and Dynamics, Vol. 32, No. 2, March - April 2009. [9] Gilbert, E.G., and Ong, C.-J. “Constrained linear systems with hard constraints and disturbances: An extended command governor with large domain of attraction”, Automatica, vol. 47, pp. 334-340, 2011. [10] Kahveci, N.E., and Kolmanovsky, I.V., “Constrained control of UAVs using adaptive anti-windup compensation and reference governors,” Proceedings of SAE 2009 AeroTech Congress and Exhibition, Seattle, WA, November, Paper Number 2009-01-3097. [11] Kalabic, U., Gilbert, E.G., and Kolmanovsky, I., “Reduced order reference governor,” Proc. 51st IEEE Conf. Decision and Control, Hawaii, USA, 2012. [12] Kalabic, U., Kolmanovsky, I., Buckland, J.H., and Gilbert, E.G., “Reference and extended command governors for control of turbocharged gasoline engines based on linear models,” Proc. of 2011 IEEE MultiConference on Systems and Control, Denver, Co., pp. 319–325, 2011. [13] Kolmanovsky, I.V., Kalabic, U., and Gilbert, E.G., “Developments in constrained control using reference governors,” Proceedings of the 2012 IFAC Conference on Nonlinear Model Predictive Control (NMPC), Noordwijkerhout, the Netherlands, August 23 - 27, 2012, pp. 282–290. [14] Lei, Y., Cao, C., Cliff, E., Hovakimyan, N., and Kurdila, A., “Design of an L1 adaptive controller for air-breathing hypersonic vehicle model in the presence of unmodeled dynamics,” AIAA Guidance, Navigation and Control Conference and Exhibit, Hilton Head, S.C., August 2007. [15] Malik, T., “Superfast military aircraft lost in test flight,” http://www.space.com/12607-darpa-launches-hypersonic-glidermach-20-test-flight.html, 11 August 2011. [16] Soloway, D., Rodriguez, A., Dickeson, J., Cifdaloz, O., Benavides, J., Sridharan, S., Kelkar, A., and Vogel, J., “Constraint enforcement for scramjet-powered hypersonic vehicles with significant aero-elasticpropulsion interactions,” 2009 American Control Conference, St. Louis, MO, June 2009. [17] Tao, X., Hua, C., Li, N., and Li, S., “Robust Model Predictive Controller design for a hypersonic flight vehicle,” Proceedings of 2012 International Conference on Modelling, Identification and Control, Wuhan, China June 2012. [18] Weiwei, Q., Zhiqiang, Z., Li, Z., and Gang, L., “Robust Model Predictive Control for hypersonic vehicle based on LPV,” Proceedings of the 2010 IEEE International Conference on Information and Automation, Harbin, China, June 2010. [19] Zinnecker, A., Serrani, A., Bolender, M.A., and Doman, D.B, “Combined reference governor and anti-windup design for constrained hypersonic vehicles models,” AIAA Guidance, Navigation, and Control Conference, Chicago, IL, 10 - 13 August 2009.
R EFERENCES [1] Baldwin, M. and Kolmanovsky, I., “Hypersonic glider guidance using Model Predictive Control,”Proceedings of 2013 American Control Conference, Washington, DC, June 2013. [2] Bemporad, A., Morari, M., Dua, V., and Pistikopoulos, “The explicit solution of Model Predictive Control via multiparametric quadratic programming,” Automatica, Vol. 38, No. 3, 2002. [3] Bolender, M.A., “An overview on dynamics and controls modelling of hypersonic vehicles,” Proceedings of 2009 American Controls Conference, St. Louis, MO, 10-12 June 2009. [4] Chen, H., Ning, L., Shaoyuan, L., “Switching multi-model predictive control for hypersonic vehicle,” Proceedings of 2011 8th Asian Control Conference (ASCC), Kaohsiung, May 2011. [5] Cheng, L., Jiang, C., Chen, M., and Pu, M., “Online-SVR-based GPC control for airframe/engine integrated near-space hypersonic vehicle,” Proceedings of 2011 8th Asian Control Conference (ASCC), Kaohsiung, May 2011 [6] Coleman, C.C., and Faruqi, F.A., “On stability and control of hypersonic vehicles,” Defence Science and Technology Organisation, Edinburgh, Australia, DSTO-TR-2358, November 2009. [7] Creagh, M.A., Beasley, P., and Kearney, M., “Adaptive control for a hypersonic glider using parameter feedback from system identification,” AIAA Guidance, Navigation, and Control Conference, Portland, OR, August 2011.
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Constrained Inner-Loop Control of a Hypersonic Glider Using Extended Command Governor Morgan Baldwin
Ilya Kolmanovsky
Abstract— The paper considers inner-loop control of a hypersonic glider. To maintain vehicle flight, pointwise-in-time constraints on vehicle state and control variables need to be enforced. To enforce constraints, an Extended Command Governor (ECG) is augmented to the nominal LQ-PI controller. Simulation results demonstrate the capability of the ECG to enforce the constraints while minimizing the degradation in the vehicle responsiveness.
I. I NTRODUCTION Hypersonic vehicles are an area of great interest for the United States. Several experimental vehicles have been launched; DARPA’s Falcon Project being an example. Two test vehicles were launched under the Falcon Project; both flights ended unsuccessfully. It was determined that the main technical challenges are the assumptions from known flight regimes are inadequate and the advanced thermal modeling and ground testing are not successful at predicting reality [15]. Conventional control techniques appropriate for subsonic and supersonic flight are not necessarily applicable for hypersonic flight. A hypersonic vehicle requires a control system that can maintain stability and active damping during flight, be robust to uncertainties, and cater to structural effects [6]. Current research projects into hypersonic vehicles can benefit from an advanced guidance, navigation, and control system with capabilities of real time optimization. The capability of determining a guidance and control solution onboard the vehicle would be an enabling technology to achieve the DoD and NASA’s hypersonic vehicle goals. Due to the uncertainties of hypersonic flight, predictive and adaptive control for hypersonic vehicles has been a major area of research. Predictive control was researched in Refs. [16], [4], and [5]. In Refs. [18] and [17], a Model Predictive Control (MPC) approach was used. In a companion paper [1], we implement an MPC approach for guidance of the hypersonic vehicle. Adaptive control has also been actively researched for control of a hypersonic vehicle [14], [7], [8]. This paper presents an inner-loop control scheme for the longitudinal dynamics of a hypersonic glider vehicle with constraint enforcement capability. For the inner-loop, angle-of-attack command tracking is achieved using a Linear This work is supported in part by ASEE through the Air Force Summer Faculty Fellow Program (SFFP) and by the National Science Foundation Award Number 1130160. Ilya Kolmanovsky is with the Department of Aerospace Engineering, The University of Michigan, Ann Arbor, Michigan. [email protected] Morgan Baldwin is with the Space Vehicles Directorate, Air Force Research Laboratory, Albuquerque, New Mexico.
[email protected] 978-1-4799-0176-0/$31.00 ©2013 AACC
Quadratic Proportional plus Integral (LQ-PI) controller. An Extended Command Governor (ECG) is augmented to the nominal closed-loop system to handle the constraints. The ECG is an advanced reference type governor algorithm that manipulates the command to enforce constraints. Reference governors have been an active area of research since the 1990s. They are add-on schemes used to enforce constraints in stable closed-loop systems. In the late 1990s, the theory was further developed to include command governors and, more recently, to also include extended command governors, both of which give a larger constrained domain of attraction and have a potential for a faster response than the reference governor. See Ref. [13] and references therein. In Ref. [19], a reference governor was implemented to avoid the occurrence of saturation of the control system and prevent windup. It was augmented to an existing adaptive controller, which was designed to track reference signals. In Ref. [10], a reference governor was used to enforce constraints on elevator deflection for conventional UAV gliders. In this work, the ECG ensures that the implemented control will keep the vehicle in a safe operating region; its added benefit is an increase in the domain of attraction for the set of initial conditions, and the potential for improving system responsiveness. II. L ONGITUDINAL E QUATIONS OF M OTION FOR A F LEXIBLE H YPERSONIC G LIDER The flexible aircraft model used is from Ref. [3]. The longitudinal equations of motion for a flexible aircraft are as follows T cos α − D − g sin γ, (1) V˙ t = m T sin α + L g Vt α˙ = − + ( − ) cos γ + Q, (2) mVt Vt r M , (3) Q˙ = Iyy ˙ = Q, Θ (4) ˙h = Vt sin γ, (5) η¨i
=
−2ξωi η˙ i − ωi2 ηi + Ni , i = 1, 2, . . . , n,
(6)
where Vt is the true airspeed, T is the thrust, D is the drag, L is the lift, α is the angle-of-attack, γ is the flight path angle, M is the pitching moment, Q is the pitch rate, h is the altitude, and r is the radius of the vehicle. The flexible dynamics of the vehicle are modeled in Eqn. 6, where ηi is the ith modal coordinate of the flexible dynamics, ξ is the damping ratio, ωi is the natural frequency, and Ni is the ith
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generalized force. The modal method and its derivation is more completely described in Ref. [3]. This model is useful for preliminary control analysis and provides insight in the interactions between the flexible vehicle dynamics and the implemented control. The thrust is set to zero in the gliding phase. III. C ONSTRAINTS Given the model in the previous section, constraints were added to reflect requirements relevant to maintaining vehicle flight. The angle-of-attack, pitch rate, elevator deflection, elevator deflection rate, and flexible deflections were constrained. The elevator deflection is used to control the vehicle by changing the pressure distribution on the vehicle and thereby altering the lift and drag forces affecting the vehicle. The rate of elevator deflection is constrained to reflect the actuator physical limits and to ensure safe control of the vehicle by preventing control maneuvers that can lead to unstable flight. The angle-of-attack is constrained so the vehicle stays within its flight envelope during hypersonic flight. If large angle-of-attack values are allowed, the vehicle may transition to uncontrollable flight. These constraints are represented by αmin ≤ α ≤ αmax , (7) δemin ≤ δe ≤ δemax ,
(8)
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(9)
are the modal coordinates and their time derivatives (η1 ,η˙ 1 , η2 , η˙ 2 , η3 , η˙ 3 ). The control input for the system is the elevator deflection, δe . A. LQ-PI controller The LQ-PI controller is designed using the short period subsystem extracted from the continuous-time linearized system model. This subsystem includes the angle-of-attack, pitch rate, the modal coordinates, and their derivatives. The controller is designed for set-point tracking of angle-of-attack commands. To treat the case when the elastic states are not measured or estimated, a second LQ-PI controller was designed based on a second order subsystem with states being the angle-of-attack and pitch rate. In this case, the LQ-PI controller uses feedback on the integral of the angle of attack tracking error, angle-of-attack and pitch rate. B. Extended Command Governor Overview An ECG is implemented to enforce state and control constraints by modifying the set-point commands to the closed-loop system, as shown in Fig. 1. In Fig. 1, x ˆ denotes the state estimate, y is the system output constrained by specifying the set inclusion conditions y(t) ∈ Y for all t, w is the disturbance/uncertainty, v is the ECG output, and r is the reference command. In this paper, w(t) = 0 as the characterization and treatment of uncertainty is left to future publications.
The pitch rate is constrained as follows Qmin ≤ Q(t) ≤ Qmax .
(10)
Due to high speeds of the vehicle in the hypersonic flight regime, structural dynamics, including aeroelastic effects, play a significant role; constraints must be imposed to ensure acceptable elastic deflections. Specifically, at specified distances back from the nose of the vehicle, the contributions of the three modal frequencies are constrained to be less than a specific value, EDmin ≤
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where ED is the total elastic deflection of the vehicle and Φi is the ith mode shape at the specified location on the vehicle, xloc,i . This constraint enables the designed controller to keep the vehicle’s structure safe during the gliding portion of flight. The selected locations are at 1/10, 3/10, 5/10, 7/10 and 9/10 of vehicle length from the nose. IV. LQ-PI C ONTROLLER WITH E XTENDED C OMMAND G OVERNOR Given the model and constraints proposed above, a nominal LQ-PI controller is augmented with an extended command governor to handle the constraints. The equations of motion are linearized about a trim point assuming zero thrust for the vehicle. This yields an 11 dimensional system with the first five states given above (velocity V , angle-of-attack α, pitch rate q, altitude h and pitch attitude θ) and the last six
Fig. 1: Extended command governor applied to a closed-loop system. ˇ ∞ , of states and parameterized The ECG uses the set, O inputs such that the constraints are satisfied for all time. The ECG output is defined by v(t) = ρ(t) + C¯ x ¯(t),
(12)
where ρ(t) and x ¯(t) are solutions to the following optimization problem minimize ||ρ(t) − r(t)||2Q + ||¯ x(t)||2S , ˇ∞ . subject to (ρ(t), x ¯(t), x(t)) ∈ O
(13)
The x ¯ ∈ Rn¯ , n ¯ ≥ 0, and ρ are states of a stable auxiliary dynamic system which evolve over the semi-infinite prediction horizon according to ¯x(t + k), k ≥ 0 x ¯(t + k + 1) = A¯ (14) ρ(t + k) = ρ(t). Compared to reference governors and conventional command governors (the case n ¯ = 0 above), the ECG yields a larger
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domain of recoverable initial states and faster response. It is also able to react to mitigate constraint violation even if there are no changes in r(t). ˇ ∞ is a finitely determined inner approximation to The set O the set of all (ρ(t), x ¯(t), x(t)) that do not induce subsequent constraint violation when the input sequence v(t + k) is determined by the fictitious dynamics per (12) and (14). In the case when Y is polyhedral, computational procedures ˇ ∞ , see [13]. Without the exist that lead to polyhedral O fictitious states, i.e., when n ¯ = 0, the ECG becomes a simple command governor [9]. The optimization problem can be solved online using conventional quadratic programming techniques. The iterative procedures can be avoided by using explicit multi-parametric quadratic programming [2], leading to ρ and x ¯ being given as a piecewise-affine function of the state x(t) and reference r. Various choices of A¯ and C¯ can be made. The shift sequences used in Ref. [9] are generated by the fictitious dynamics with, 0 Im 0 0 ··· 0 0 Im 0 · · · 0 Im · · · (15) A¯ = 0 0 0 0 0 0 · · · .. .. .. .. .. . . . . . C¯ = Im 0 0 0 · · · ,
pitch rate, −δe,min = δe,max = 0.349 rad for the elevator deflection (20 deg), −∆δe,min = ∆δe,max = 0.349 rad/sec for the elevator deflection rate, and −EDmin = EDmax = 1.0 for the elastic deflections. To illustrate how the imposition of constraints progressively affects the response, we considered a subset of constraints and a full set of constraints. The subset of constraints contained just the limits on the elevator deflection and elastic deflections (8), (11). The full set of constraints contained all of the constraints (7)-(11) V. S IMULATION R ESULTS
The simulation results are based on vehicle model linearized at a dynamic pressure of 1500 psi and an altitude of 95000 ft with zero thrust. We first show the closed-loop responses for the case of the nominal LQ controller that includes elastic states in feedback. With a pitch command of r = ±0.29, very close to the imposed angle-of-attack limit, constraints on angle-of-attack, pitch rate, elevator deflection, elevator deflection rate, and elastic deflections are all violated in transients, see Fig. 2. Note that the elastic deflections are largest near the nose of the vehicle, near the tail of the vehicle, and near the middle of the vehicle, with smaller deflections elsewhere. Enforcing the constraints on elastic deflections is thus important to ensure that the vehicle does not break apart during the flight. The response with ECG designed for n ¯ = 5, α = 0.0 to enforce constraints on anglewhere Im is an m × m identity matrix. Another approach of-attack, pitch rate, elevator deflection rates of change, and uses Laguerre sequences [12]. These sequences possess elastic mode deflections is shown in Fig. 3. The imposed orthogonality properties and are generated by the fictitious constraints are enforced and the response of the system with dynamics with ECG is slowed down. The LQ-PI controller does not use elastic states for feedback and the response of the system is αIm βIm −αβIm α2 βIm · · · 0 similar for the cases when the LQ-PI controller is designed αIm βIm −αβIm · · · with elastic states for feedback. 0 0 αI βI · · · ¯ m m A= (16) 0 Different design choices can lead to differences in the 0 0 αI · · · m .. .. .. .. .. speed of response of the closed-loop system. Fig. 4 shows . . . . . ¯ = 1, α = 0, which essentially is the case p the case of n C¯ = β Im −αIm α2 Im −α3 Im · · · (−α)N −1 Im , of the conventional command governor. While in all cases (¯ n = 1, 5, 10) the system is protected against the constraint where β = 1 − α2 and 0 ≤ α ≤ 1 is a selectable parameter violation, there is a considerable speed up in responses as that corresponds to the time-constant of the fictitious dynamn ¯ , i.e., the dimensionality of the auxiliary system increases. ics. Note that with the choice of α = 0, (16) coincides with Non-zero α helps to speed up the response further. With the shift register considered in Ref. [9]. n ¯ = 10, α = 0.4 as shown in Fig. 5 versus n ¯ = 5, C. Extended Command Governor Design α = 0.0 as shown in Fig. 3, the response is significantly For this work, the constraints are on angle-of-attack, more aggressive, and more constraints become active as the pitch rate, elevator deflection, elevator deflection rate and ECG has considerable flexibility to improve the performance elastic deflection. The discrete-time model of the closed-loop under constraints, see Fig. 5. The computational complexity ¯. system necessary to implement ECG is based on the LQ-PI of the ECG increases, however, with the increase in n controller (one of two designs depending on whether elastic VI. T OWARDS O N -B OARD I MPLEMENTATION modes are used for feedback), the short period subsystem The preceding sections demonstrated the potential of the (with elastic modes), and a sampling period of 0.01 sec. ECG to handle constraints. The development of ECG for This model has the following form on-board implementation require several additional steps. x(t + 1) = Ax(t) + Bv(t) (17) The ECG based on partial state information and reduced y(t) = Cx(t) + Dv(t) ∈ Y. order model (without the elastic dynamics) can be developed The limits were as follows: −αmin = αmax = 0.3 rad for using the reduced order reference governor techniques [11]. the angle of attack; −Qmin = Qmax = 0.2 rad/sec for the Both observer errors and contributions of the omitted states 5578
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Fig. 3: The response with the ECG formulated for n ¯ = 5, α = 0, LQ-PI controller that does not use elastic mode measurements/estimates for feedback and full set of the constraints: (a) angle-of-attack, (b) pitch rate, (c) elevator deflection, (d) elastic deflection, (e) elevator deflection time rate of change.
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to the constrained outputs can be handled by imposing an additional constraint on δv(t) = v(t) − v(t − 1). The uncertainties in forces and moments in hypersonic flight can be handled by representing these uncertainties as additive set-bounded disturbances. The existing extended command governor theory can handle these uncertainties by empoying disturbance invariant sets. Finally, if explicit implementation of ECG is used (in the form of Piecewise Affine controller that represents a multi-parametric solution to the QP (13)), the ECGs designed for different trim points during the flight can be merged into a single Piecewise Affine explicit solution applicable in the full flight range. If the model is periodically re-identified or updated on-board during the flight, ˇ ∞ with O ˇ ∞ computed the condition (ρ(t), x ¯(t), x(t)) ∈ O off-line, will need to be replaced by the conditions on the predicted response of the system, y(t + k|t) ∈ Y , for k = 0, · · · , nc , where nc is the constraint horizon. We leave these developments as next steps for future publications. VII. C ONCLUDING R EMARKS The paper has proposed an augmentation of an innerloop LQ-PI controller with an Extended Command Governor (ECG) to handle constraints faced during hypersonic vehicle flight. The advantages of ECG, which is based on higher order auxilliary dynamics and Laguerre’s sequence generation have been demonstrated for the actuator rate limited case. Future research will address steps which facilitate the on-board implementation of ECG design. These include implementation based on reduced order models and based on models with uncertainties in forces and moments during the hypersonic flight. VIII. ACKNOWLEDGMENTS The authors thank Michael Bolender for providing a copy of the model of the glider and guidance on the modeling and control aspects. The ECG code used in this paper has been developed at the University of Michigan by Uros Kalabic.
[8] Fiorentini, L., Serrani, A., Bolender, M., and Doman, D., “Nonlinear robust adaptive control of flexible air-breathing hypersonic vehicles,” Journal of Guidance, Control, and Dynamics, Vol. 32, No. 2, March - April 2009. [9] Gilbert, E.G., and Ong, C.-J. “Constrained linear systems with hard constraints and disturbances: An extended command governor with large domain of attraction”, Automatica, vol. 47, pp. 334-340, 2011. [10] Kahveci, N.E., and Kolmanovsky, I.V., “Constrained control of UAVs using adaptive anti-windup compensation and reference governors,” Proceedings of SAE 2009 AeroTech Congress and Exhibition, Seattle, WA, November, Paper Number 2009-01-3097. [11] Kalabic, U., Gilbert, E.G., and Kolmanovsky, I., “Reduced order reference governor,” Proc. 51st IEEE Conf. Decision and Control, Hawaii, USA, 2012. [12] Kalabic, U., Kolmanovsky, I., Buckland, J.H., and Gilbert, E.G., “Reference and extended command governors for control of turbocharged gasoline engines based on linear models,” Proc. of 2011 IEEE MultiConference on Systems and Control, Denver, Co., pp. 319–325, 2011. [13] Kolmanovsky, I.V., Kalabic, U., and Gilbert, E.G., “Developments in constrained control using reference governors,” Proceedings of the 2012 IFAC Conference on Nonlinear Model Predictive Control (NMPC), Noordwijkerhout, the Netherlands, August 23 - 27, 2012, pp. 282–290. [14] Lei, Y., Cao, C., Cliff, E., Hovakimyan, N., and Kurdila, A., “Design of an L1 adaptive controller for air-breathing hypersonic vehicle model in the presence of unmodeled dynamics,” AIAA Guidance, Navigation and Control Conference and Exhibit, Hilton Head, S.C., August 2007. [15] Malik, T., “Superfast military aircraft lost in test flight,” http://www.space.com/12607-darpa-launches-hypersonic-glidermach-20-test-flight.html, 11 August 2011. [16] Soloway, D., Rodriguez, A., Dickeson, J., Cifdaloz, O., Benavides, J., Sridharan, S., Kelkar, A., and Vogel, J., “Constraint enforcement for scramjet-powered hypersonic vehicles with significant aero-elasticpropulsion interactions,” 2009 American Control Conference, St. Louis, MO, June 2009. [17] Tao, X., Hua, C., Li, N., and Li, S., “Robust Model Predictive Controller design for a hypersonic flight vehicle,” Proceedings of 2012 International Conference on Modelling, Identification and Control, Wuhan, China June 2012. [18] Weiwei, Q., Zhiqiang, Z., Li, Z., and Gang, L., “Robust Model Predictive Control for hypersonic vehicle based on LPV,” Proceedings of the 2010 IEEE International Conference on Information and Automation, Harbin, China, June 2010. [19] Zinnecker, A., Serrani, A., Bolender, M.A., and Doman, D.B, “Combined reference governor and anti-windup design for constrained hypersonic vehicles models,” AIAA Guidance, Navigation, and Control Conference, Chicago, IL, 10 - 13 August 2009.
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