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Scalar Reference Governor for Constrained Maneuver and Shape Control of Nonlinear Multibody Aircraft
11th IFAC Symposium on Nonlinear Control Systems 11th IFAC Symposium on Nonlinear Control Systems Vienna, Austria, Sept. 4-6, 2019 11th IFAC Symposium on Nonlinear Nonlinear Control Systems Systems 11th IFAC Symposium on Control Vienna, Austria, Sept. 4-6, 2019 Available online at www.sciencedirect.com 11th IFAC Symposium on Nonlinear Control Systems Vienna, Austria, Sept. 2019 Vienna, Austria, Sept. 4-6, 4-6, 2019 Vienna, Austria, Sept. 4-6, 2019
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IFAC PapersOnLine 52-16 (2019) 819–824
Scalar Scalar Reference Reference Governor Governor for for Constrained Constrained Scalar Reference Governor for Constrained Maneuver and Shape Control Nonlinear Scalar Reference Governor for of Constrained Maneuver and Shape Control of Maneuver and Shape Control of Nonlinear Nonlinear Multibody Aircraft Maneuver and Shape Control Multibody Aircraftof Nonlinear Multibody Multibody Aircraft Aircraft Ian O’Rourke ∗∗ Ilya Kolmanovsky ∗∗ Emanuele Garone ∗∗ ∗∗ Ian Ian Ian Ian
O’Rourke ∗∗ Ilya Kolmanovsky ∗∗ ∗Emanuele Garone ∗∗ O’Rourke Ilya Ilya Anouck Kolmanovsky Emanuele Garone Garone ∗∗ Girard O’Rourke Kolmanovsky Girard∗ ∗∗∗Emanuele O’Rourke ∗ Ilya Anouck Kolmanovsky Emanuele Garone ∗∗ Anouck Girard Girard Anouck ∗ Anouck Girard ∗ ∗ University of Michigan, Ann Arbor, MI 48109 USA University of Michigan, Ann Arbor, MI 48109 USA ∗ ∗ University of Michigan, Ann Arbor, (e-mail: {ianor, ilya, anouck}@umich.edu) Michigan, Arbor, MI MI 48109 48109 USA USA (e-mail: of {ianor, ilya, Ann anouck}@umich.edu) ∗∗∗University of Michigan, Ann Arbor, MI 48109 USA (e-mail: {ianor, ilya, anouck}@umich.edu) Universit´ e Libre de Bruxelles, Bruxelles, Belgium ∗∗University (e-mail: e{ianor, ilya, anouck}@umich.edu) Universit´ Libre de Bruxelles, Bruxelles, Belgium ∗∗ (e-mail:(e-mail: ilya, anouck}@umich.edu) ∗∗ Universit´ e{ianor, Libre de de Bruxelles, Bruxelles, Belgium Belgium [email protected]) Universit´ e Libre Bruxelles, Bruxelles, (e-mail: [email protected]) ∗∗ Universit´ e Libre de Bruxelles, Bruxelles, Belgium (e-mail: [email protected]) (e-mail: [email protected]) (e-mail: [email protected]) Abstract: Abstract: A A control control system system for for constrained constrained control control of of aa nonlinear nonlinear multibody multibody aircraft aircraft is is Abstract: A control system for constrained control of aa governor nonlinear multibody aircraft is described which exploits a combination of a scalar reference and a Linear Quadratic Abstract: A control system for constrained control of nonlinear multibody aircraft is described which exploits a combination of a scalar reference governor and a Linear Quadratic Abstract: A control system for constrained control of a nonlinear multibody aircraft is described which exploits a combination of a scalar reference governor and a Linear Quadratic Integral (LQ-I) controller for maneuver load alleviation, trajectory control and shape control. described which exploits a combination of a scalar reference governor and a Linear Quadratic Integral (LQ-I) controller for maneuver of load alleviation, trajectory control and shape control. described which exploits a combination a scalar reference governor and a Linear Quadratic Integral (LQ-I) controller for maneuver load alleviation, trajectory control and shape control. A nonlinear multibody aircraft model for longitudinal flight is used to develop and validate the Integral (LQ-I) controller for maneuver alleviation, trajectory control andand shape control. A nonlinear multibody aircraft model forload longitudinal flight is used to develop validate the Integral (LQ-I) controller for The maneuver alleviation, trajectory control and shape control. A nonlinear nonlinear multibody aircraft model forload longitudinal flight is designed used to to develop and validate the control system performance. scalar reference governor is based on the linearized A multibody aircraft model for longitudinal flight is used develop and validate the control system performance. The scalar reference governor is designed based on the linearized A nonlinear multibody aircraft model forreference longitudinal flight is designed used are to develop and validate the control system performance. The scalar governor is based on the linearized model and ensures that the imposed state and control constraints satisfied. The mismatch control system performance. The scalar reference governor is designed based on the linearized model and ensures that the imposed state and control constraints are satisfied. The mismatch control system performance. The scalar reference governor is designed based on the linearized model and ensures that the imposed state and control constraints are satisfied. The mismatch between the linearized model used by the reference governor for prediction and the nonlinear model and thatmodel the imposed and control constraints are satisfied. between theensures linearized used bystate the reference governor for prediction andThe the mismatch nonlinear model and ensures thatmodel theaircraft imposed and control constraints are satisfied. The mismatch between the linearized model used is bystate the reference reference governor for prediction prediction and theThe nonlinear dynamics of the multibody accounted for in the reference governor design. paper between the linearized used by the governor for and the nonlinear dynamics of the multibody aircraft is accounted for in the reference governor design. The paper between the linearized model used by the reference governor for prediction and the nonlinear dynamics of of the the multibody aircraft is accounted accounted forapproach, in the the reference reference governor design. The paper demonstrates the feasibility of the adopted control including constraint enforcement, dynamics multibody aircraft is for in governor design. The paper demonstrates the feasibility of the adopted control approach, including constraint enforcement, dynamics of the multibody aircraft is accounted forapproach, in maneuvers. the reference governor design. The paper demonstrates the feasibility of the adopted control including constraint enforcement, based on the nonlinear model simulations of pitching demonstrates the feasibility of the adopted control approach, including constraint enforcement, based on the nonlinear model pitching maneuvers. demonstrates the feasibility of simulations the adoptedof approach, including constraint enforcement, based model simulations of pitching maneuvers. based on on the the nonlinear nonlinear model simulations ofcontrol pitching maneuvers. © 2019,onIFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. based the nonlinear model simulations of pitching maneuvers. Keywords: Keywords: Flight Flight Control, Control, Multibody Multibody Aircraft, Aircraft, Maneuver Maneuver Load Load Alleviation, Alleviation, Shape Shape Control, Control, Keywords: Flight Control, Multibody Aircraft, Maneuver Load Alleviation, Shape Constrained Control, Reference Governors, Predictive Control Keywords: Flight Control, Multibody Aircraft, Maneuver Load Alleviation, Shape Control, Control, Constrained Control, Reference Governors, Predictive Control Keywords: Control, Multibody Aircraft, Maneuver Load Alleviation, Shape Control, Constrained Control, Reference Governors, Predictive Control ConstrainedFlight Control, Reference Governors, Predictive Control Constrained Control, Reference Governors, Predictive Control 1. INTRODUCTION INTRODUCTION (MPC) to to gust gust load load alleviation alleviation with with constraint constraint handling handling 1. (MPC) 1. (MPC) to gust load alleviation with constraint handling for flexible and very flexible aircraft have been considered 1. INTRODUCTION INTRODUCTION (MPC) to gust load alleviation with constraint handling for flexible and very aircraft have been considered 1. INTRODUCTION (MPC) to gust load flexible alleviation with constraint handling for flexible and very flexible aircraft have been considered in Kopf et al. (2018), Haghighat et al. (2012) and Simpson for flexible and very flexible aircraft have been considered In this paper a maneuver and shape control problem for in Kopf et al. (2018), Haghighat et al. (2012) and Simpson In this paper a maneuver and shape control problem for for flexible and very flexible aircraft have been considered in Kopf et al. (2018), Haghighat et al. (2012) and et al. (2014). The control system for very flexible aircraft In this paper a maneuver and shape control problem for in Kopf et al. (2018), Haghighat etfor al. very (2012) and Simpson Simpson a multibody aircraft is considered. The aircraft is formed In this paper a maneuver and shape control problem for et al. (2014). The control system flexible aircraft a multibody aircraft is considered. The aircraft is formed in Kopf etinal.Gonzalez (2018), Haghighat etfor al. (2012) and Simpson et al. (2014). The control system very flexible aircraft In this paper a maneuver and shape control problem for proposed et al. (2016) uses two control loops, afrom multibody aircraft is considered. The aircraft is formed et al. (2014). The control system for very flexible aircraft three rigid wing sections connected with torsional a multibody aircraft is considered. The aircraft is formed proposed in Gonzalez et al. (2016) uses two control loops, from three rigid wing connected with et al. (2014). The control system for very flexible aircraft proposed in Gonzalez et al. (2016) uses two control loops, a multibody aircraft is sections considered. The aircraft istorsional formed an inner-loop to hold the aircraft’s shape to a desired from three rigid wing sections connected with torsional proposed in Gonzalez et al. (2016) uses two control loops, springs. This model has been proposed by Gibson et al. from three rigid wing sections connected with torsional an inner-loop to hold aircraft’s to a springs. This model has been proposed by Gibson et al. proposed inan Gonzalez etthe al. (2016) usesshape two control loops, an inner-loop to hold the aircraft’s shape to aa desired desired from three rigid wing sections connected with torsional shape, and outer-loop to track desired flight commands. springs. This model has been proposed by Gibson et al. an inner-loop to hold the aircraft’s shape to desired (2011) to capture the dynamics and control challenges of springs. This model has been proposed by Gibson et al. shape, and an outer-loop to track desired flight commands. (2011) to capture the dynamics and control challenges of an inner-loop to hold the aircraft’s shape to a desired shape, and an outer-loop to track desired flight commands. springs. This model has been proposed by Gibson et al. (2011) to capture the dynamics and control challenges of shape, and an outer-loop to track desired flight commands. very flexible aircraft. (2011) to capture the dynamics and control challenges of Gibson et al. (2011) take a different approach by introvery flexible aircraft. shape, and an outer-loop to atrack desired flight commands. (2011) to capture the dynamics and control challenges of Gibson et al. (2011) take different approach by introvery flexible aircraft. very flexible aircraft. Gibson et al. (2011) take aastate different approach introducing a low order, seven nonlinear modelby for the Gibson et al. (2011) take different approach byfor introVery flexible aircraft exploit long, thin, high aspect ratio very flexible aircraft. ducing a low order, seven state nonlinear model the Very flexible aircraft exploit long, thin, high aspect ratio Gibson et al. (2011) take a different approach by introducing a low order, seven state nonlinear model for the longitudinal flight of a very flexible aircraft based on Very flexible aircraft exploit long, thin, high aspect ratio ducing a low order, seven state nonlinear model for wings and low lowaircraft structural weight inthin, orderhigh to improve improve the longitudinal flight of a very flexible aircraft based the Very flexible exploit long,in aspect ratio on wings and structural weight order to the ducing a low order,of seven state nonlinear model for the longitudinal flight a very flexible aircraft based on Very flexible aircraft exploit long, thin, high aspect ratio the rigid multibody approximation rather than flexible wings and low structural weight in order to improve the longitudinal flight of a very flexible aircraft based on aircraft’s aerodynamic efficiency and maximize its lift-towings andaerodynamic low structural weight and in order to improve the the rigid multibody approximation rather than flexible aircraft’s efficiency maximize its lift-tolongitudinal flight ofapproximation a very flexible aircraft based on rigid multibody rather than flexible wings andaerodynamic low structural weightaircraft in order to improve the the beam approximation. This low order model is then used aircraft’s efficiency and maximize its lift-tothe rigid multibody approximation rather than flexible drag ratio. This enables such to maintain long aircraft’s aerodynamic efficiency and maximize its lift-tobeam approximation. This low order model is then used drag ratio. This enables such aircraft to maintain long the rigid multibody approximation rather than flexible beam approximation. This low order model is then used aircraft’s aerodynamic efficiency and maximize its lift-toby Gibson et al. (2011) to design and compare a linear drag enables such aircraft to long beam approximation. This order is then used loiter times This at high high altitudes and potentially serve as as drag ratio. ratio. This enables such and aircraft to maintain maintain long by Gibson et al. (2011) to low design andmodel compare a linear loiter times at altitudes potentially serve aaa by beam approximation. This order model is then used et to design and compare aa linear drag ratio. This enables such and aircraft to maintain long controller and an(2011) adaptive Linear Quadratic Gaussian loiter times at high altitudes potentially serve as by Gibson Gibson et al. al.an (2011) to low design and compare linear replacement for communication satellites. loiter times at high altitudes and potentially serve as a controller and adaptive Linear Quadratic Gaussian replacement for communication satellites. by Gibson et al. (2011) to design and compare a linear and an Linear Quadratic Gaussian loiter times at altitudes and potentially serve as a controller (LQG) Loop Transfer Recovery (LTR) based controller. controller. replacement for communication satellites. controller and Transfer an adaptive adaptive Linear Quadratic Gaussian replacement for high communication satellites. /// Loop Recovery (LTR) based The combination of long long span span and and low structural structural stiffness stiffness (LQG) controller and Transfer antheadaptive Linear Quadratic Gaussian (LQG) Loop Recovery (LTR) based controller. replacement for communication satellites. In Gibson (2014) effects of turbulence are also studied; (LQG) / Loop Transfer Recovery (LTR) based controller. The combination of low In Gibson (2014) the effects of turbulence are also studied; The combination of long span and low structural stiffness leads the wing to exhibit large amounts of flexibility. To (LQG) / Loop Transfer Recovery (LTR) based controller. The combination of long span and low structural stiffness In Gibson (2014) the effects of turbulence are also studied; however, constraints have not been addressed. In Gibson (2014) the effects of turbulence are also studied; leads the wing to exhibit large amounts of flexibility. To however, constraints have not been addressed. The longdeformation span low structural stiffness leads the wing exhibit large amounts of flexibility. To avoid significant wing and high structural In Gibsonconstraints (2014) the have effectsnot of turbulence are also studied; leadscombination the wing to to of exhibit largeand amounts flexibility. To however, been addressed. avoid significant wing deformation and of high structural however, constraints have not been addressed. leads the wing to exhibit large amounts of flexibility. To In this paper, paper, we consider consider the application of aa scalar scalar avoid significant wing deformation high loads during the flight flight and maximize and the efficiency, efficiency, active In however, constraints have notthe been addressed. of avoid during significant wing and deformation and high structural structural this we application loads the maximize the active In we consider the application of aa(2014); scalar avoid significant wing deformation and high structural reference governor Kolmanovsky loads during the flight and maximize the efficiency, active In this this paper, paper, wedescribed consider in the application et ofal. scalar load alleviation and shape control become necessary. The loads during the flight and maximize the efficiency, active reference governor described in Kolmanovsky et al. (2014); load alleviation shape become necessary. The In this et paper, wedescribed consider application scalar reference governor in Kolmanovsky et loads during theand flight and control maximize the efficiency, active Garone al. (2017) (2017) and Linear Linear Quadratic Integral (LQ-I) load alleviation and shape control become necessary. The reference governor described inthe Kolmanovsky etofal. al.a(2014); (2014); control problem is challenging due to the need to account load alleviation and shape control become necessary. The Garone et al. and Quadratic Integral (LQ-I) control problem is challenging due to the need to account reference governor described in Kolmanovsky et al. (2014); al. (2017) Linear Quadratic Integral (LQ-I) load alleviation and shape control become necessary. The Garone controller to maneuver load alleviation, trajectory control control problem is due to to account Garone et et to al. maneuver (2017) and andload Linear Quadratic Integral control (LQ-I) for nonlinear dynamics when large control problem is challenging challenging due deformations to the the need need toare account alleviation, trajectory for nonlinear dynamics when the the deformations are large controller Garone et to al. (2017)based andload Linear Quadratic Integral (LQ-I) controller maneuver alleviation, trajectory control control problem is challenging due deformations to the need toare account and shape control on the nonlinear multibody airfor nonlinear dynamics when the large controller to maneuver load alleviation, trajectory control and for aeroelastic effects. for nonlinear dynamics when the deformations are large and shape control based on the nonlinear multibody airand for aeroelastic effects. controller to maneuver load alleviation, trajectory control and shape control based on the nonlinear multibody airfor nonlinear dynamics when the deformations are large craft model in Gibson et al. (2011). The LQ-I controller and for aeroelastic effects. and shape control based on the nonlinear multibody airand for aeroelastic effects. craft model in Gibson et al. (2011). The LQ-I controller Several control problems problems for flexible flexible and and very very flexible flexible craft and control on nonlinear multibody airmodel in al. (2011). The LQ-I controller and for aeroelastic effects. for tracks set-points inbased the et angle of attack, attack, pitch angle, flight craftshape model in Gibson Gibson et al. the (2011). The LQ-I controller Several control tracks set-points in the angle of pitch angle, flight Several control problems for flexible and very flexible aircraft have been studied in the existing literature and craft model in Gibson et al. (2011). The LQ-I controller Several control problems for flexible and very flexible tracks in angle of pitch flight speed, and dihedral dihedral angle. The scalar reference reference governor tracks set-points set-points in the the angle of attack, attack, pitch angle, angle, flight aircraft have been studied in the existing and and angle. The scalar governor Several control problems for flexible and literature veryIn flexible aircraft have been studied in existing literature and several control approaches have been developed. partictracks set-points in the angle ofdihedral attack, pitch and angle, flight aircraft control have been studiedhave in the the existing literature and speed, speed, and dihedral angle. The scalar reference governor enforces the constraints on the angle on posispeed, and dihedral angle. The scalar reference governor several approaches been developed. In particthe constraints on the dihedral angle and on posiaircraft have been studied in the existing literature and enforces several control approaches have been developed. In particular, a trajectory control problem for very flexible aircraft speed, and dihedral angle. The scalar reference governor several control approaches have been developed. In particenforces the constraints on the dihedral angle and on positions of outer ailerons, center elevator, and outer elevators. enforces the constraints on the dihedral angle and on posiular, a trajectory control problem for very flexible aircraft tions of outer ailerons, center elevator, and outer elevators. several control approaches have been developed. Inaircraft particular, aa trajectory control problem for very has been considered in Shearer Shearer and Cesnik (2008) based tions enforces thereference constraints on theelevator, dihedraland and on posiular, trajectory control problem for Cesnik very flexible flexible aircraft of ailerons, center outer elevators. This scalar governor exploits aangle linear prediction tions scalar of outer outer ailerons,governor center elevator, and outer elevators. has been considered in and (2008) based This reference exploits a linear prediction ular, a trajectory control problem for very flexible aircraft has in (2008) based on highconsidered order dynamic dynamic modeland in Cesnik UM/NAST. An ex- This tions ofand, outer center elevator, and outer elevators. has aabeen been considered in Shearer Shearer and Cesnik (2008)An based reference exploits aa linear prediction model inailerons, order to togovernor accommodate larger maneuvers, it This scalar scalar reference governor exploitslarger linear prediction on high order model in UM/NAST. exmodel and, in order accommodate maneuvers, it has considered in Shearer and Cesnik (2008) based on aabeen high order dynamic model in UM/NAST. An extension in the presence of gusts is addressed in Dillsaver This scalar reference governor exploits a linear prediction on high order dynamic model in UM/NAST. An exmodel and, in order to accommodate larger maneuvers, it is augmented with extra states to account for the mismatch model and, in order to accommodate larger maneuvers, it tension in the presence of gusts is addressed in Dillsaver augmented extra states to account for the mismatch on a high order dynamic model inaddressed UM/NAST. An ex- is tension in presence of ismodel in et al. The applications of predictive control model and, inwith order to accommodate larger it tension in the the presence of gusts gusts addressed in Dillsaver Dillsaver with extra states for the et al. (2013b). (2013b). The applications ofis model predictive control is is augmented augmented with extra states to to account account for maneuvers, the mismatch mismatch tension in the presence of gusts is addressed in Dillsaver et et al. al. (2013b). (2013b). The The applications applications of of model model predictive predictive control control is augmented with extra states to account for the mismatch et al. (2013b). The applications ofFederation model predictive control 2405-8963 © 2019, IFAC (International of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.
Copyright © 2019 IFAC 1469 Copyright © under 2019 IFAC 1469Control. Peer review responsibility of International Federation of Automatic Copyright © 1469 Copyright © 2019 2019 IFAC IFAC 1469 10.1016/j.ifacol.2019.12.064 Copyright © 2019 IFAC 1469
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in the prediction of constrained outputs between the linear model and the nonlinear system. As compared to MPC, the scalar reference governor has smaller computational footprint as noted by Kolmanovsky et al. (2014); Garone et al. (2017). Simulation results will be presented on the nonlinear multibody aircraft model which demonstrate the capability of the LQ-I controller and scalar reference governor combination to accomplish the control objectives.
Tail
Sec. 1 δa o
2. LONGITUDINAL FLIGHT MODEL The model from Gibson et al. (2011) is for the longitudinal flight dynamics of aircraft formed out of three rigid wing sections connected by torsional springs. The model represents aircraft flexibility through the dihedral angle η, shown in Fig. 1. The hinges each have torsional stiffness and damping coefficients κk and κc , respectively, to emulate the structural parameters of a flexible wing.
yˆ1 zˆ1
where
∗ , c1 =3Iyy
s2 ∗ ∗ − 2Iyy + m∗ , c2 =2Izz 6 s d1 = m∗ V˙ sin(α)V cos(α)α˙ cos(η), 2 2s cos(η) sin(η)η˙ 2 , − V sin(α) sin(η)η˙ − 3 2 ∗ ∗ ∗s d2 =(Iyy − Izz − m ) sin(η) cos(η)q 2 , 12 s − m∗ cos(η)V cos(α)q, 2 s2 s2 ∗ d3 =Ixx + cos2 (η) , + m∗ 4 6 and all constants are provided in Table 1. Parameter Sectional Mass Sectional Inertia
Sectional Span Wing Chord Tail Span Tail Chord Tail Boom Length Sectional Wing Reference Area Sectional Tail Reference Area Lift Coefficients Moment Coefficients
The resulting model has six states: the velocity V , the angle of attack α, the pitch angle θ, the pitch rate q, the dihedral angle η, and the dihedral angle derivative η. ˙ The state vector is given by
The equations of motion are given by
Parasitic Drag Coefficient Induced Drag Coefficient Dihedral Joint Damping Dihedral Joint Stiffness Air Density at 40,000 ft Gravitational Acceleration
(1)
The control inputs are the deflections of the center aileron δac , the outer ailerons δao , the center elevator δec , the outer elevators δeo , and the thrust T . A top-down aircraft view in Fig. 2 shows the locations of the control surfaces and the rectangular planforms of the wing and tail. Input saturation is also added, as compared to Gibson et al. (2011), with control surfaces saturating at ±35◦ .
(3)
Table 1. Constants for the aircraft model
Fig. 1. Front view of aircraft with coordinate frames for each section and the overall body frame B, as in Gibson et al. (2011)
As compared to Gibson et al. (2011), the altitude state, h, is removed and constant air density is assumed.
(2)
˙ M − 2c2 sin(η) cos(η)ηq , 2 c1 + c2 sin η H − κc η˙ − κk η + d1 − d2 η¨ = , d3 q˙ =
3
T
Sec. 3 δa o
V˙ = (T cos α − D)/m − g sin γ, α˙ = −(T sin α + L)/(mV ) + q + g cos(γ)/V, θ˙ = q,
2
˙ . X = [V α θ q η η]
Sec. 2 δa c
Fig. 2. Top-down view of aircraft configuration
yˆ3 zˆ η
yˆB zˆB yˆ2 zˆ
δ eo
Nose
Dillsaver et al. (2013a) considered the use of scalar reference governors and extended command governors to constrain root curvature, and thus bending moment, of very flexible aircraft during simulated maneuvers on UM/NAST model; however, in this paper the model, the constraints, and the control objectives, which include shape control, and the results are different. The paper is organized as follows. The nonlinear multibody aircraft model is described in Section 2. The LQ-I controller design is presented in Section 3. The constraints and the reference governor formulation are the subject of Section 4. The results of the closed-loop simulations are reported in Section 5. Robustness assessment is made in Section 6. Concluding remarks are made in Section 7.
δ ec
δeo
Symbol m∗ ∗ Ixx ∗ Iyy ∗ Izz s cw st ct lb ∗ Sw St∗ CL α CL δ CM 0 CMδ C D0 κD κc κk ρ g
Quantity 9.32 slugs 280 slugs . ft2 18.63 slugs . ft2 167.7 slugs . ft2 80 ft 8 ft 20 ft 2 ft 36 ft 640 ft2 40 ft2 2π 2 0.025 −0.25 0.007 0.07 141400 lbf/s 4900 lbf 5.8572 × 10−4 slug/ft2 32.2 ft/s2
To obtain lift, drag, and moment for each of the sections, the sectional velocity, Vi , is calculated from the bodyframe velocities ui , vi , and wi , as (4) Vi = u2i + vi2 + wi2 , and is used to find the sectional dynamic pressure Qi as 1 (5) Qi = ρVi2 . 2
The sectional angle of attack αi and sideslip angle βi are computed as
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wi vi , βi = arcsin . (6) ui Vi The sectional body-frame velocities ui , vi , and wi , are determined from s u2 = V cos α + q sin η, 3 v2 = 0, (7) s w2 = V sin α + η˙ cos η, 3 for the center section, and from s u3 = u1 = V cos α − q sin η, 6 s v3 = −v1 = (V sin α + η˙ cos η) sin η, (8) 3 s s w3 = w1 = (V sin α + η˙ cos η) cos η − η˙ , 3 2 for the outer two sections. The wing sectional lift and drag are computed as ∗ L∗w,i = Qi CLw,i Sw , CLwi = CLα αi + CLδ δa,i , (9) ∗ ∗ Dw,i = Qi CDw,i Sw , CDwi = CD0 + κD CL2 w,i ,
3. LQ-I CONTROLLER DESIGN
αi = arctan
and tail sectional lift and drag are computed as ∗ , CLti = CLα (αi + δe,i ), L∗t,i = Qi CLt,i Sw ∗ ∗ , = Qi CDt,i Sw Dt,i
CDti = CD0 + κD CL2 t,i .
(10)
The above expressions can be combined into a single force matrix Pi∗ as ∗ ∗ + Dt,i ) −(Dw,i −Di∗ . 0 0 (11) = Pi∗ = ∗ ∗ ∗ −(Lw,i + Lt,i ) −Li
To combine these sectional lift and drag parameters, rotation matrices are used to convert the lift and drag from the wind frame to the local section body frame with cos α cos β − cos α sin β − sin α sin β cos β 1 , (12) Rb/w (α, β) = sin α cos β − sin α sin β cos β and then to rotate the outer sections into the center sections’ body frame with 1 0 0 T R2/1 = 0 cos η sin η , R2/3 = R2/1 . (13) 0 − sin η cos η This then allows the total lift and drag to be written as −D 0 =Rb/w (α, β)T R2/1 Rb/w (α1 , β1 )P1∗ + (14) −L ∗ ∗ Rb/w (α2 , β2 )P2 + R2/3 Rb/w (α3 , β3 )P3 .
The sectional pitching moment, M∗i , is found using ∗ , CMi = CM0 + CMδ δa,i M∗i = Qi CMi cw Sw (15) ∗ and combined with the forces from the wing, Dw,i , and tail, L∗t,i , as M=
3 i=1
∗ (M∗i + lw,i Dw,i − lt L∗t,i ).
821
The LQ-I controller is designed to track the set-points in the angle of attack α, pitch angle θ, flight speed V and dihedral angle η using the thrust T , the outer ailerons δao (operated symmetrically), the center elevator δec , and the center aileron δac as control inputs. Let T
U = [T δao δec δac ] ,
T
y = [α θ V η] .
(19)
The control design exploits a linearized model about a steady state condition corresponding to V = 30 ft/s, η = 5◦ , h = 40, 000 ft, α = 8◦ and θ = 13◦ (i.e., γ = 0◦ ). The model has the form, ˜˙ = AX ˜ + BU ˜ , y˜ = C X, ˜ X (20) 0 ˜ 0 0 ˜ where X = X − X , U = U − U and y˜ = y − y are deviations of the state, control and tracking output, y, from the respective trim points. The eigenvalues of A are λ1 = −0.0207, λ2,3 = −0.308 ± 1.4917i, (21) λ4 = −6.6181, λ5,6 = −2.5697 ± 6.7663i. The LQ-I design is based on applying LQR theory to the extended system, d y˜ − r y˜ − r 0 C 0 ˜˙ = + U, ˜ ˜ 0 A B X X dt where r designates the vector of the set-points for y˜. The state and control weighting matrices are defined by Q = diag([1000 1000 100 100 0.1 (22) 0.1 0.1 0.1 0.1 0.1]), R = diag([0.01 0.01 0.03 0.04]). (23) The LQ-I controller has the form, ˜ z˙ = y˜ − r, ˜ (t) = Ki z + Kp X, (24) U where K = [Ki , Kp ] is the LQR gain.
4. SCALAR REFERENCE GOVERNOR Let
z z , (25) = 0 ˜ X −X X denote the state of the closed-loop system (20), (24). The design of the scalar reference governor is based on the discrete-time model of the closed-loop system, (26) Xcl (t + 1) = Acl Xcl (t) + Bcl r, obtained assuming the update period for r of Ts = 0.01 s. Xcl =
The scalar reference governor modifies the set-point r when this set-point has a potential to induce future constraint violation. See Figure 3 and Garone et al. (2017); Kolmanovsky et al. (2014). Xcl (t)
r(t)
(16)
Reference v(t) LQ-I ClosedGovernor Loop
yc (t)
Fig. 3. Reference governor and LQ-I block diagram
Finally, the moment at the hinge, H, is calculated as ∗ ∗ + Pt,1 ) − m∗ g cos η cos θ. (17) H = −(Pw,1 z z The flight path angle γ and climb rate h˙ is given by (18) γ = θ − α, h˙ = V sin γ.
Thus in (26), r is replaced by v, where v is updated according to v(t) = v(t − 1) + κ(t) r(t) − v(t − 1) , (27) and where κ(t), 0 ≤ κ(t) ≤ 1, is a scalar parameter.
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If κ(t) = 1, then v(t) = r(t) and the scalar reference governor is inactive. If κ(t) = 0, then v(t) = v(t − 1) and r(t) is isolated from the system, assuring command safety. Consequently, the value of κ(t) is maximized subject to the constraint 0 ≤ κ(t) ≤ 1 and that the updated command (27), if it were applied for all future time instants, would not cause the constraint violation. To make the last constraint explicit, suppose the constraints for the model (26) can be written in the form, HCcl Xcl ≤ h, h > 0. (28) Then, (28) imposed over the future prediction horizon leads to conditions, HCcl Xcl (t + k) ≤ h, (29) for k = 0, 1, · · · . Assuming v(t + k) = v(t) and given (27), (29) becomes Hv,k (r−v(t−1))κ(t) ≤ h−Hx,k Xcl (t)−Hv,k v(t−1), (30) where Hx,k = Ccl Akcl , Hv,k = Ccl (I − Acl )−1 (I − Akcl )Bcl . (31) In the implementation, the constraints (30) are imposed up to a finite prediction horizon, k ∗ , i.e., only for k = 0, 1, · · · , k ∗ and an extra constraint is imposed as Hv,∞ (r − v(t − 1))κ(t) ≤ h(1 − ) − Hv,∞ v(t − 1), (32)
where Hv,∞ = Ccl (I − Acl )−1 Bcl and > 0 is small. With (32) imposed, under standard assumptions, if k ∗ is sufficiently large, then all constraints (30) for k > k ∗ are redundant and need not be explicitly imposed. It is also possible that some of the constraints for k ≤ k ∗ are redundant and can be a priori eliminated resulting in a smaller subset of constraints to be considered. The reader is referred to Garone et al. (2017); Kolmanovsky et al. (2014) for details. In either case, maximizing the scalar parameter κ(t) subject to the above affine constraints is an explicitly and easily solvable optimization problem. While recursive feasibility is guaranteed in absence of model mismatch, no feasible value for κ(t) may exist if model mismatch is present; in such a case an infeasibility handling mechanism is used with κ(t) set to zero. 5. SIMULATION CASE STUDIES In this section, simulation results are reported. In all cases, the aircraft starts from steady-state at the trim point and the commands are applied 2 sec after the simulation starts. Constraints are imposed on the dihedral angle, center and outer aileron position, and center aileron position, 4.5◦ ≤ η ≤ 5.5◦ , 1.5◦ ≤ δec ≤ 7.5◦ , (33) 25◦ ≤ δao ≤ 32◦ , 17◦ ≤ δac ≤ 23◦ , to emulate structural and control deflection limitations. These constraints are reformulated as in (29). The chosen value of k ∗ = 1000 provides a look-ahead horizon of 10 sec.
While the linearized models are used in the design of LQ-I controller and the scalar reference governor, the subsequent closed-loop simulations are performed on the nonlinear model of the multibody aircraft described in Section 2. The maneuvers correspond to changes in the flight path angle and are implemented through commands in the pitch angle θr and angle of attack αr (γr = θr − αr ). The commands for the aircraft velocity and the dihedral
angle are to maintain trim values of 30 ft/s and 5◦ , respectively. 5.1 Descent with γ = 5◦ We first consider a maneuver in which the flight path angle γr is decreased to −5◦ by setting θr to 3◦ and αr to 8◦ . The closed-loop response without the reference governor is shown in Fig. 4. While the desired flight path angle is achieved, nearly every constraint is violated. The chosen weights in the LQ-I controller design provide fast response and good tracking, but the dihedral angle constraint is violated and initial deflections for all the control surfaces exceed their bounds.
Fig. 4. Descent command without reference governor violates dihedral (top) and control deflection (center) constraints, but reaches desired flight path angle γr = −5◦ in about 20 sec (bottom). Now, the same flight path angle decreased maneuver is commanded to the closed-loop system with the reference governor. See Fig. 5. The constraints are met and the commanded flight path angle is achieved. The reference governor slowly decreases the applied pitch angle and angle of attack commands to prevent constraint violation. Note that the commanded flight path angle is reconstructed as the difference between the pitch angle and angle of attack commands that the reference governor outputs. The active constraints are both the control surface deflections and dihedral angle, with the outer and center ailerons being the most actively constrained. The desired flight path angle is reached in about 30 sec, as opposed to approximately 20 sec for the prior unconstrained maneuver. 5.2 Climb with γ = 10◦ Next a climb maneuver with a larger flight path angle command of 10◦ is considered. The closed-loop response with the reference governor is shown in Fig. 6. While the dihedral angle satisfies the constraints, there is a brief exceedance (by 0.66◦ ) of the limit by the center aileron position. This exceedance is attributed to larger mismatch
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0 η η(t) δa0 δao (t) g(t) = − HCcl Xcl (t) − 0o δec (t) δec δac (t) δa0c
823
,
(35)
where η 0 = 5◦ , δa0o = 28.6◦ , δe0c = 4.5◦ , δa0c = 19.8◦ are the values of dihedral angle, outer aileron position, center elevator position, and center aileron position at trim at which the linearized model is obtained and Xcl (t) is determined from (25). The reference governor with the mismatch compensation provides the results in Fig. 7, with only a minor exceedance in the outer aileron position of 0.1◦ .
Fig. 5. Descent with reference governor keeps dihedral (top) and control deflection (center) constraints, and still reaches desired flight path angle γ = −5◦ in about 30 sec (bottom). between the linear model used by the reference governor and the actual nonlinear system for larger maneuvers.
Fig. 7. The response with the reference governor enhanced with mismatch compensation meets dihedral constraints (top) but violates outer aileron deflection by less than 0.1◦ (center) for the flight path angle command of 10◦ . 5.3 Climb with γr = 10◦ and trim point with η 0 = 23◦
Fig. 6. Climb meets dihedral constraints (top) but violates center aileron deflection by 0.66◦ (center) for 10◦ flight path angle command. To accommodate the mismatch between the linear model and the nonlinear system, the approach described in Garone et al. (2017); Kolmanovsky et al. (2014) is used. In this approach, the constraint (29) over the prediction horizon, is replaced by the constraint, HCcl Xcl (t + k) + g(t) ≤ h,
(34)
where g(t) is set to the mismatch at the current time instant t between the constrained outputs and their estimates based on the linearized model, i.e.,
If the nominal trim point is modified to η 0 = 23◦ , the open-loop model (20) becomes unstable, with open-loop eigenvalues about the trim condition given by λ1 = 0.0120, λ2,3 = 0.3523 ± 1.0421i, (36) λ4 = −6.9121, λ5,6 = −2.3515 ± 1.0232i Fig. 8 shows simulated closed-loop responses and the constraints for the same Q and R weights but with LQI controller and reference governor reconfigured for the linearized model and updated constraints at the new dihedral angle. Without the reference governor the closed loop response is unstable as the control inputs saturate. 6. CONTROLLER ROBUSTNESS If we augment the system with a simple turbulence model, such that normal zero-mean disturbances with standard deviation of 1 ft/s are symmetrically added to ui and vi for each section from (7) and (8), as done by Gibson (2014), the reference governor is still able to enforce the constraints with minor exceedances. See Fig. 9. The flight
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7. CONCLUSION The combination of the scalar reference governor and LQI controller has been shown to be effective for control of longitudinal flight of a nonlinear multibody aircraft based on the model in Gibson et al. (2011). ACKNOWLEDGEMENTS The authors thank Dr. Travis Gibson for providing the multibody aircraft model described in Gibson et al. (2011) and assisting them in its use that has greatly facilitated this work. REFERENCES
Fig. 8. The response with the reference governor enhanced with mismatch compensation meets dihedral constraints (top), control deflection constraints (center) and tracks the flight path angle command of 10◦ from a trim condition with higher dihedral angle of 23◦ . path angle command is set to a larger value of 15◦ , but with dihedral angle command at 23◦ as before. In simulations the center aileron is constrained against its limit, limiting flight path angle that can be achieved.
Fig. 9. Reference governor with turbulence can constrain the average values of the disturbances, not overall trajectory, due to the linear model in prediction. Robustness tests were also performed in simulations that used the original controller for the aircraft with 10% of the original dihedral angle stiffness. In response to 10◦ flight path angle command and 5◦ dihedral angle command, the reference governor was still able to account for the model mismatch and enforce the constraints between the model with minor constraint violations under 1◦ in center aileron deflection during the maneuver.
Dillsaver, M.J., Kalabic, U.V., Kolmanovsky, I.V., and Cesnik, C.E.S. (2013a). Constrained control of very flexible aircraft using reference and extended command governors. In Proceedings of American Control Conference, 1608–1613. Dillsaver, M., Cesnik, C.E., and Kolmanovsky, I. (2013b). Trajectory control of very flexible aircraft with gust disturbance. In Proceedings of AIAA Atmospheric Flight Mechanics (AFM) Conference. Garone, E., Cairano, S.D., and Kolmanovsky, I. (2017). Reference and command governors for systems with constraints: A survey on theory and applications. Automatica, 75, 306 – 328. Gibson, T., Annaswamy, A., and Lavretsky, E. (2011). Modeling for control of very flexible aircraft. In Proceedings of AIAA Guidance, Navigation, and Control Conference. Gibson, T.E. (2014). Closed-loop Reference Model adaptive control : with application to very flexible aircraft. Ph.D. thesis, Massachusetts Institute of Technology. Gonzalez, P.J., Silvestre, F.J., Paglione, P., K¨ othe, A., Pang, Z.Y., and Cesnik, C.E. (2016). Linear control of highly flexible aircraft based on loop separation. In Proceedings of AIAA Atmospheric Flight Mechanics Conference. Haghighat, S., T. Liu, H.H., and RA Martins, J.R. (2012). Model-predictive gust load alleviation controller for a highly flexible aircraft. Journal of Guidance, Control, and Dynamics, 35(6), 1751–1766. Kolmanovsky, I., Garone, E., and Cairano, S.D. (2014). Reference and command governors: A tutorial on their theory and automotive applications. In Proceedings of 2014 American Control Conference, 226–241. Kopf, M., Bullinger, E., Giesseler, H.G., Adden, S., and Findeisen, R. (2018). Model predictive control for aircraft load alleviation: Opportunities and challenges. In Proceedings of American Control Conference, 2417– 2424. Shearer, C.M. and Cesnik, C.E. (2008). Trajectory control for very flexible aircraft. AIAA Journal of Guidance, Control, and Dynamics, 31(2), 340–357. Simpson, R.J., Palacios, R., Hesse, H., and Goulart, P. (2014). Predictive control for alleviation of gust loads on very flexible aircraft. In Proceedings of 55th AIAA/ASME/ASCE/AHS/SC Structures, Structural Dynamics, and Materials Conference, AIAA SciTech Forum.
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Scalar Scalar Reference Reference Governor Governor for for Constrained Constrained Scalar Reference Governor for Constrained Maneuver and Shape Control Nonlinear Scalar Reference Governor for of Constrained Maneuver and Shape Control of Maneuver and Shape Control of Nonlinear Nonlinear Multibody Aircraft Maneuver and Shape Control Multibody Aircraftof Nonlinear Multibody Multibody Aircraft Aircraft Ian O’Rourke ∗∗ Ilya Kolmanovsky ∗∗ Emanuele Garone ∗∗ ∗∗ Ian Ian Ian Ian
O’Rourke ∗∗ Ilya Kolmanovsky ∗∗ ∗Emanuele Garone ∗∗ O’Rourke Ilya Ilya Anouck Kolmanovsky Emanuele Garone Garone ∗∗ Girard O’Rourke Kolmanovsky Girard∗ ∗∗∗Emanuele O’Rourke ∗ Ilya Anouck Kolmanovsky Emanuele Garone ∗∗ Anouck Girard Girard Anouck ∗ Anouck Girard ∗ ∗ University of Michigan, Ann Arbor, MI 48109 USA University of Michigan, Ann Arbor, MI 48109 USA ∗ ∗ University of Michigan, Ann Arbor, (e-mail: {ianor, ilya, anouck}@umich.edu) Michigan, Arbor, MI MI 48109 48109 USA USA (e-mail: of {ianor, ilya, Ann anouck}@umich.edu) ∗∗∗University of Michigan, Ann Arbor, MI 48109 USA (e-mail: {ianor, ilya, anouck}@umich.edu) Universit´ e Libre de Bruxelles, Bruxelles, Belgium ∗∗University (e-mail: e{ianor, ilya, anouck}@umich.edu) Universit´ Libre de Bruxelles, Bruxelles, Belgium ∗∗ (e-mail:(e-mail: ilya, anouck}@umich.edu) ∗∗ Universit´ e{ianor, Libre de de Bruxelles, Bruxelles, Belgium Belgium [email protected]) Universit´ e Libre Bruxelles, Bruxelles, (e-mail: [email protected]) ∗∗ Universit´ e Libre de Bruxelles, Bruxelles, Belgium (e-mail: [email protected]) (e-mail: [email protected]) (e-mail: [email protected]) Abstract: Abstract: A A control control system system for for constrained constrained control control of of aa nonlinear nonlinear multibody multibody aircraft aircraft is is Abstract: A control system for constrained control of aa governor nonlinear multibody aircraft is described which exploits a combination of a scalar reference and a Linear Quadratic Abstract: A control system for constrained control of nonlinear multibody aircraft is described which exploits a combination of a scalar reference governor and a Linear Quadratic Abstract: A control system for constrained control of a nonlinear multibody aircraft is described which exploits a combination of a scalar reference governor and a Linear Quadratic Integral (LQ-I) controller for maneuver load alleviation, trajectory control and shape control. described which exploits a combination of a scalar reference governor and a Linear Quadratic Integral (LQ-I) controller for maneuver of load alleviation, trajectory control and shape control. described which exploits a combination a scalar reference governor and a Linear Quadratic Integral (LQ-I) controller for maneuver load alleviation, trajectory control and shape control. A nonlinear multibody aircraft model for longitudinal flight is used to develop and validate the Integral (LQ-I) controller for maneuver alleviation, trajectory control andand shape control. A nonlinear multibody aircraft model forload longitudinal flight is used to develop validate the Integral (LQ-I) controller for The maneuver alleviation, trajectory control and shape control. A nonlinear nonlinear multibody aircraft model forload longitudinal flight is designed used to to develop and validate the control system performance. scalar reference governor is based on the linearized A multibody aircraft model for longitudinal flight is used develop and validate the control system performance. The scalar reference governor is designed based on the linearized A nonlinear multibody aircraft model forreference longitudinal flight is designed used are to develop and validate the control system performance. The scalar governor is based on the linearized model and ensures that the imposed state and control constraints satisfied. The mismatch control system performance. The scalar reference governor is designed based on the linearized model and ensures that the imposed state and control constraints are satisfied. The mismatch control system performance. The scalar reference governor is designed based on the linearized model and ensures that the imposed state and control constraints are satisfied. The mismatch between the linearized model used by the reference governor for prediction and the nonlinear model and thatmodel the imposed and control constraints are satisfied. between theensures linearized used bystate the reference governor for prediction andThe the mismatch nonlinear model and ensures thatmodel theaircraft imposed and control constraints are satisfied. The mismatch between the linearized model used is bystate the reference reference governor for prediction prediction and theThe nonlinear dynamics of the multibody accounted for in the reference governor design. paper between the linearized used by the governor for and the nonlinear dynamics of the multibody aircraft is accounted for in the reference governor design. The paper between the linearized model used by the reference governor for prediction and the nonlinear dynamics of of the the multibody aircraft is accounted accounted forapproach, in the the reference reference governor design. The paper demonstrates the feasibility of the adopted control including constraint enforcement, dynamics multibody aircraft is for in governor design. The paper demonstrates the feasibility of the adopted control approach, including constraint enforcement, dynamics of the multibody aircraft is accounted forapproach, in maneuvers. the reference governor design. The paper demonstrates the feasibility of the adopted control including constraint enforcement, based on the nonlinear model simulations of pitching demonstrates the feasibility of the adopted control approach, including constraint enforcement, based on the nonlinear model pitching maneuvers. demonstrates the feasibility of simulations the adoptedof approach, including constraint enforcement, based model simulations of pitching maneuvers. based on on the the nonlinear nonlinear model simulations ofcontrol pitching maneuvers. © 2019,onIFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. based the nonlinear model simulations of pitching maneuvers. Keywords: Keywords: Flight Flight Control, Control, Multibody Multibody Aircraft, Aircraft, Maneuver Maneuver Load Load Alleviation, Alleviation, Shape Shape Control, Control, Keywords: Flight Control, Multibody Aircraft, Maneuver Load Alleviation, Shape Constrained Control, Reference Governors, Predictive Control Keywords: Flight Control, Multibody Aircraft, Maneuver Load Alleviation, Shape Control, Control, Constrained Control, Reference Governors, Predictive Control Keywords: Control, Multibody Aircraft, Maneuver Load Alleviation, Shape Control, Constrained Control, Reference Governors, Predictive Control ConstrainedFlight Control, Reference Governors, Predictive Control Constrained Control, Reference Governors, Predictive Control 1. INTRODUCTION INTRODUCTION (MPC) to to gust gust load load alleviation alleviation with with constraint constraint handling handling 1. (MPC) 1. (MPC) to gust load alleviation with constraint handling for flexible and very flexible aircraft have been considered 1. INTRODUCTION INTRODUCTION (MPC) to gust load alleviation with constraint handling for flexible and very aircraft have been considered 1. INTRODUCTION (MPC) to gust load flexible alleviation with constraint handling for flexible and very flexible aircraft have been considered in Kopf et al. (2018), Haghighat et al. (2012) and Simpson for flexible and very flexible aircraft have been considered In this paper a maneuver and shape control problem for in Kopf et al. (2018), Haghighat et al. (2012) and Simpson In this paper a maneuver and shape control problem for for flexible and very flexible aircraft have been considered in Kopf et al. (2018), Haghighat et al. (2012) and et al. (2014). The control system for very flexible aircraft In this paper a maneuver and shape control problem for in Kopf et al. (2018), Haghighat etfor al. very (2012) and Simpson Simpson a multibody aircraft is considered. The aircraft is formed In this paper a maneuver and shape control problem for et al. (2014). The control system flexible aircraft a multibody aircraft is considered. The aircraft is formed in Kopf etinal.Gonzalez (2018), Haghighat etfor al. (2012) and Simpson et al. (2014). The control system very flexible aircraft In this paper a maneuver and shape control problem for proposed et al. (2016) uses two control loops, afrom multibody aircraft is considered. The aircraft is formed et al. (2014). The control system for very flexible aircraft three rigid wing sections connected with torsional a multibody aircraft is considered. The aircraft is formed proposed in Gonzalez et al. (2016) uses two control loops, from three rigid wing connected with et al. (2014). The control system for very flexible aircraft proposed in Gonzalez et al. (2016) uses two control loops, a multibody aircraft is sections considered. The aircraft istorsional formed an inner-loop to hold the aircraft’s shape to a desired from three rigid wing sections connected with torsional proposed in Gonzalez et al. (2016) uses two control loops, springs. This model has been proposed by Gibson et al. from three rigid wing sections connected with torsional an inner-loop to hold aircraft’s to a springs. This model has been proposed by Gibson et al. proposed inan Gonzalez etthe al. (2016) usesshape two control loops, an inner-loop to hold the aircraft’s shape to aa desired desired from three rigid wing sections connected with torsional shape, and outer-loop to track desired flight commands. springs. This model has been proposed by Gibson et al. an inner-loop to hold the aircraft’s shape to desired (2011) to capture the dynamics and control challenges of springs. This model has been proposed by Gibson et al. shape, and an outer-loop to track desired flight commands. (2011) to capture the dynamics and control challenges of an inner-loop to hold the aircraft’s shape to a desired shape, and an outer-loop to track desired flight commands. springs. This model has been proposed by Gibson et al. (2011) to capture the dynamics and control challenges of shape, and an outer-loop to track desired flight commands. very flexible aircraft. (2011) to capture the dynamics and control challenges of Gibson et al. (2011) take a different approach by introvery flexible aircraft. shape, and an outer-loop to atrack desired flight commands. (2011) to capture the dynamics and control challenges of Gibson et al. (2011) take different approach by introvery flexible aircraft. very flexible aircraft. Gibson et al. (2011) take aastate different approach introducing a low order, seven nonlinear modelby for the Gibson et al. (2011) take different approach byfor introVery flexible aircraft exploit long, thin, high aspect ratio very flexible aircraft. ducing a low order, seven state nonlinear model the Very flexible aircraft exploit long, thin, high aspect ratio Gibson et al. (2011) take a different approach by introducing a low order, seven state nonlinear model for the longitudinal flight of a very flexible aircraft based on Very flexible aircraft exploit long, thin, high aspect ratio ducing a low order, seven state nonlinear model for wings and low lowaircraft structural weight inthin, orderhigh to improve improve the longitudinal flight of a very flexible aircraft based the Very flexible exploit long,in aspect ratio on wings and structural weight order to the ducing a low order,of seven state nonlinear model for the longitudinal flight a very flexible aircraft based on Very flexible aircraft exploit long, thin, high aspect ratio the rigid multibody approximation rather than flexible wings and low structural weight in order to improve the longitudinal flight of a very flexible aircraft based on aircraft’s aerodynamic efficiency and maximize its lift-towings andaerodynamic low structural weight and in order to improve the the rigid multibody approximation rather than flexible aircraft’s efficiency maximize its lift-tolongitudinal flight ofapproximation a very flexible aircraft based on rigid multibody rather than flexible wings andaerodynamic low structural weightaircraft in order to improve the the beam approximation. This low order model is then used aircraft’s efficiency and maximize its lift-tothe rigid multibody approximation rather than flexible drag ratio. This enables such to maintain long aircraft’s aerodynamic efficiency and maximize its lift-tobeam approximation. This low order model is then used drag ratio. This enables such aircraft to maintain long the rigid multibody approximation rather than flexible beam approximation. This low order model is then used aircraft’s aerodynamic efficiency and maximize its lift-toby Gibson et al. (2011) to design and compare a linear drag enables such aircraft to long beam approximation. This order is then used loiter times This at high high altitudes and potentially serve as as drag ratio. ratio. This enables such and aircraft to maintain maintain long by Gibson et al. (2011) to low design andmodel compare a linear loiter times at altitudes potentially serve aaa by beam approximation. This order model is then used et to design and compare aa linear drag ratio. This enables such and aircraft to maintain long controller and an(2011) adaptive Linear Quadratic Gaussian loiter times at high altitudes potentially serve as by Gibson Gibson et al. al.an (2011) to low design and compare linear replacement for communication satellites. loiter times at high altitudes and potentially serve as a controller and adaptive Linear Quadratic Gaussian replacement for communication satellites. by Gibson et al. (2011) to design and compare a linear and an Linear Quadratic Gaussian loiter times at altitudes and potentially serve as a controller (LQG) Loop Transfer Recovery (LTR) based controller. controller. replacement for communication satellites. controller and Transfer an adaptive adaptive Linear Quadratic Gaussian replacement for high communication satellites. /// Loop Recovery (LTR) based The combination of long long span span and and low structural structural stiffness stiffness (LQG) controller and Transfer antheadaptive Linear Quadratic Gaussian (LQG) Loop Recovery (LTR) based controller. replacement for communication satellites. In Gibson (2014) effects of turbulence are also studied; (LQG) / Loop Transfer Recovery (LTR) based controller. The combination of low In Gibson (2014) the effects of turbulence are also studied; The combination of long span and low structural stiffness leads the wing to exhibit large amounts of flexibility. To (LQG) / Loop Transfer Recovery (LTR) based controller. The combination of long span and low structural stiffness In Gibson (2014) the effects of turbulence are also studied; however, constraints have not been addressed. In Gibson (2014) the effects of turbulence are also studied; leads the wing to exhibit large amounts of flexibility. To however, constraints have not been addressed. The longdeformation span low structural stiffness leads the wing exhibit large amounts of flexibility. To avoid significant wing and high structural In Gibsonconstraints (2014) the have effectsnot of turbulence are also studied; leadscombination the wing to to of exhibit largeand amounts flexibility. To however, been addressed. avoid significant wing deformation and of high structural however, constraints have not been addressed. leads the wing to exhibit large amounts of flexibility. To In this paper, paper, we consider consider the application of aa scalar scalar avoid significant wing deformation high loads during the flight flight and maximize and the efficiency, efficiency, active In however, constraints have notthe been addressed. of avoid during significant wing and deformation and high structural structural this we application loads the maximize the active In we consider the application of aa(2014); scalar avoid significant wing deformation and high structural reference governor Kolmanovsky loads during the flight and maximize the efficiency, active In this this paper, paper, wedescribed consider in the application et ofal. scalar load alleviation and shape control become necessary. The loads during the flight and maximize the efficiency, active reference governor described in Kolmanovsky et al. (2014); load alleviation shape become necessary. The In this et paper, wedescribed consider application scalar reference governor in Kolmanovsky et loads during theand flight and control maximize the efficiency, active Garone al. (2017) (2017) and Linear Linear Quadratic Integral (LQ-I) load alleviation and shape control become necessary. The reference governor described inthe Kolmanovsky etofal. al.a(2014); (2014); control problem is challenging due to the need to account load alleviation and shape control become necessary. The Garone et al. and Quadratic Integral (LQ-I) control problem is challenging due to the need to account reference governor described in Kolmanovsky et al. (2014); al. (2017) Linear Quadratic Integral (LQ-I) load alleviation and shape control become necessary. The Garone controller to maneuver load alleviation, trajectory control control problem is due to to account Garone et et to al. maneuver (2017) and andload Linear Quadratic Integral control (LQ-I) for nonlinear dynamics when large control problem is challenging challenging due deformations to the the need need toare account alleviation, trajectory for nonlinear dynamics when the the deformations are large controller Garone et to al. (2017)based andload Linear Quadratic Integral (LQ-I) controller maneuver alleviation, trajectory control control problem is challenging due deformations to the need toare account and shape control on the nonlinear multibody airfor nonlinear dynamics when the large controller to maneuver load alleviation, trajectory control and for aeroelastic effects. for nonlinear dynamics when the deformations are large and shape control based on the nonlinear multibody airand for aeroelastic effects. controller to maneuver load alleviation, trajectory control and shape control based on the nonlinear multibody airfor nonlinear dynamics when the deformations are large craft model in Gibson et al. (2011). The LQ-I controller and for aeroelastic effects. and shape control based on the nonlinear multibody airand for aeroelastic effects. craft model in Gibson et al. (2011). The LQ-I controller Several control problems problems for flexible flexible and and very very flexible flexible craft and control on nonlinear multibody airmodel in al. (2011). The LQ-I controller and for aeroelastic effects. for tracks set-points inbased the et angle of attack, attack, pitch angle, flight craftshape model in Gibson Gibson et al. the (2011). The LQ-I controller Several control tracks set-points in the angle of pitch angle, flight Several control problems for flexible and very flexible aircraft have been studied in the existing literature and craft model in Gibson et al. (2011). The LQ-I controller Several control problems for flexible and very flexible tracks in angle of pitch flight speed, and dihedral dihedral angle. The scalar reference reference governor tracks set-points set-points in the the angle of attack, attack, pitch angle, angle, flight aircraft have been studied in the existing and and angle. The scalar governor Several control problems for flexible and literature veryIn flexible aircraft have been studied in existing literature and several control approaches have been developed. partictracks set-points in the angle ofdihedral attack, pitch and angle, flight aircraft control have been studiedhave in the the existing literature and speed, speed, and dihedral angle. The scalar reference governor enforces the constraints on the angle on posispeed, and dihedral angle. The scalar reference governor several approaches been developed. In particthe constraints on the dihedral angle and on posiaircraft have been studied in the existing literature and enforces several control approaches have been developed. In particular, a trajectory control problem for very flexible aircraft speed, and dihedral angle. The scalar reference governor several control approaches have been developed. In particenforces the constraints on the dihedral angle and on positions of outer ailerons, center elevator, and outer elevators. enforces the constraints on the dihedral angle and on posiular, a trajectory control problem for very flexible aircraft tions of outer ailerons, center elevator, and outer elevators. several control approaches have been developed. Inaircraft particular, aa trajectory control problem for very has been considered in Shearer Shearer and Cesnik (2008) based tions enforces thereference constraints on theelevator, dihedraland and on posiular, trajectory control problem for Cesnik very flexible flexible aircraft of ailerons, center outer elevators. This scalar governor exploits aangle linear prediction tions scalar of outer outer ailerons,governor center elevator, and outer elevators. has been considered in and (2008) based This reference exploits a linear prediction ular, a trajectory control problem for very flexible aircraft has in (2008) based on highconsidered order dynamic dynamic modeland in Cesnik UM/NAST. An ex- This tions ofand, outer center elevator, and outer elevators. has aabeen been considered in Shearer Shearer and Cesnik (2008)An based reference exploits aa linear prediction model inailerons, order to togovernor accommodate larger maneuvers, it This scalar scalar reference governor exploitslarger linear prediction on high order model in UM/NAST. exmodel and, in order accommodate maneuvers, it has considered in Shearer and Cesnik (2008) based on aabeen high order dynamic model in UM/NAST. An extension in the presence of gusts is addressed in Dillsaver This scalar reference governor exploits a linear prediction on high order dynamic model in UM/NAST. An exmodel and, in order to accommodate larger maneuvers, it is augmented with extra states to account for the mismatch model and, in order to accommodate larger maneuvers, it tension in the presence of gusts is addressed in Dillsaver augmented extra states to account for the mismatch on a high order dynamic model inaddressed UM/NAST. An ex- is tension in presence of ismodel in et al. The applications of predictive control model and, inwith order to accommodate larger it tension in the the presence of gusts gusts addressed in Dillsaver Dillsaver with extra states for the et al. (2013b). (2013b). The applications ofis model predictive control is is augmented augmented with extra states to to account account for maneuvers, the mismatch mismatch tension in the presence of gusts is addressed in Dillsaver et et al. al. (2013b). (2013b). The The applications applications of of model model predictive predictive control control is augmented with extra states to account for the mismatch et al. (2013b). The applications ofFederation model predictive control 2405-8963 © 2019, IFAC (International of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.
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in the prediction of constrained outputs between the linear model and the nonlinear system. As compared to MPC, the scalar reference governor has smaller computational footprint as noted by Kolmanovsky et al. (2014); Garone et al. (2017). Simulation results will be presented on the nonlinear multibody aircraft model which demonstrate the capability of the LQ-I controller and scalar reference governor combination to accomplish the control objectives.
Tail
Sec. 1 δa o
2. LONGITUDINAL FLIGHT MODEL The model from Gibson et al. (2011) is for the longitudinal flight dynamics of aircraft formed out of three rigid wing sections connected by torsional springs. The model represents aircraft flexibility through the dihedral angle η, shown in Fig. 1. The hinges each have torsional stiffness and damping coefficients κk and κc , respectively, to emulate the structural parameters of a flexible wing.
yˆ1 zˆ1
where
∗ , c1 =3Iyy
s2 ∗ ∗ − 2Iyy + m∗ , c2 =2Izz 6 s d1 = m∗ V˙ sin(α)V cos(α)α˙ cos(η), 2 2s cos(η) sin(η)η˙ 2 , − V sin(α) sin(η)η˙ − 3 2 ∗ ∗ ∗s d2 =(Iyy − Izz − m ) sin(η) cos(η)q 2 , 12 s − m∗ cos(η)V cos(α)q, 2 s2 s2 ∗ d3 =Ixx + cos2 (η) , + m∗ 4 6 and all constants are provided in Table 1. Parameter Sectional Mass Sectional Inertia
Sectional Span Wing Chord Tail Span Tail Chord Tail Boom Length Sectional Wing Reference Area Sectional Tail Reference Area Lift Coefficients Moment Coefficients
The resulting model has six states: the velocity V , the angle of attack α, the pitch angle θ, the pitch rate q, the dihedral angle η, and the dihedral angle derivative η. ˙ The state vector is given by
The equations of motion are given by
Parasitic Drag Coefficient Induced Drag Coefficient Dihedral Joint Damping Dihedral Joint Stiffness Air Density at 40,000 ft Gravitational Acceleration
(1)
The control inputs are the deflections of the center aileron δac , the outer ailerons δao , the center elevator δec , the outer elevators δeo , and the thrust T . A top-down aircraft view in Fig. 2 shows the locations of the control surfaces and the rectangular planforms of the wing and tail. Input saturation is also added, as compared to Gibson et al. (2011), with control surfaces saturating at ±35◦ .
(3)
Table 1. Constants for the aircraft model
Fig. 1. Front view of aircraft with coordinate frames for each section and the overall body frame B, as in Gibson et al. (2011)
As compared to Gibson et al. (2011), the altitude state, h, is removed and constant air density is assumed.
(2)
˙ M − 2c2 sin(η) cos(η)ηq , 2 c1 + c2 sin η H − κc η˙ − κk η + d1 − d2 η¨ = , d3 q˙ =
3
T
Sec. 3 δa o
V˙ = (T cos α − D)/m − g sin γ, α˙ = −(T sin α + L)/(mV ) + q + g cos(γ)/V, θ˙ = q,
2
˙ . X = [V α θ q η η]
Sec. 2 δa c
Fig. 2. Top-down view of aircraft configuration
yˆ3 zˆ η
yˆB zˆB yˆ2 zˆ
δ eo
Nose
Dillsaver et al. (2013a) considered the use of scalar reference governors and extended command governors to constrain root curvature, and thus bending moment, of very flexible aircraft during simulated maneuvers on UM/NAST model; however, in this paper the model, the constraints, and the control objectives, which include shape control, and the results are different. The paper is organized as follows. The nonlinear multibody aircraft model is described in Section 2. The LQ-I controller design is presented in Section 3. The constraints and the reference governor formulation are the subject of Section 4. The results of the closed-loop simulations are reported in Section 5. Robustness assessment is made in Section 6. Concluding remarks are made in Section 7.
δ ec
δeo
Symbol m∗ ∗ Ixx ∗ Iyy ∗ Izz s cw st ct lb ∗ Sw St∗ CL α CL δ CM 0 CMδ C D0 κD κc κk ρ g
Quantity 9.32 slugs 280 slugs . ft2 18.63 slugs . ft2 167.7 slugs . ft2 80 ft 8 ft 20 ft 2 ft 36 ft 640 ft2 40 ft2 2π 2 0.025 −0.25 0.007 0.07 141400 lbf/s 4900 lbf 5.8572 × 10−4 slug/ft2 32.2 ft/s2
To obtain lift, drag, and moment for each of the sections, the sectional velocity, Vi , is calculated from the bodyframe velocities ui , vi , and wi , as (4) Vi = u2i + vi2 + wi2 , and is used to find the sectional dynamic pressure Qi as 1 (5) Qi = ρVi2 . 2
The sectional angle of attack αi and sideslip angle βi are computed as
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wi vi , βi = arcsin . (6) ui Vi The sectional body-frame velocities ui , vi , and wi , are determined from s u2 = V cos α + q sin η, 3 v2 = 0, (7) s w2 = V sin α + η˙ cos η, 3 for the center section, and from s u3 = u1 = V cos α − q sin η, 6 s v3 = −v1 = (V sin α + η˙ cos η) sin η, (8) 3 s s w3 = w1 = (V sin α + η˙ cos η) cos η − η˙ , 3 2 for the outer two sections. The wing sectional lift and drag are computed as ∗ L∗w,i = Qi CLw,i Sw , CLwi = CLα αi + CLδ δa,i , (9) ∗ ∗ Dw,i = Qi CDw,i Sw , CDwi = CD0 + κD CL2 w,i ,
3. LQ-I CONTROLLER DESIGN
αi = arctan
and tail sectional lift and drag are computed as ∗ , CLti = CLα (αi + δe,i ), L∗t,i = Qi CLt,i Sw ∗ ∗ , = Qi CDt,i Sw Dt,i
CDti = CD0 + κD CL2 t,i .
(10)
The above expressions can be combined into a single force matrix Pi∗ as ∗ ∗ + Dt,i ) −(Dw,i −Di∗ . 0 0 (11) = Pi∗ = ∗ ∗ ∗ −(Lw,i + Lt,i ) −Li
To combine these sectional lift and drag parameters, rotation matrices are used to convert the lift and drag from the wind frame to the local section body frame with cos α cos β − cos α sin β − sin α sin β cos β 1 , (12) Rb/w (α, β) = sin α cos β − sin α sin β cos β and then to rotate the outer sections into the center sections’ body frame with 1 0 0 T R2/1 = 0 cos η sin η , R2/3 = R2/1 . (13) 0 − sin η cos η This then allows the total lift and drag to be written as −D 0 =Rb/w (α, β)T R2/1 Rb/w (α1 , β1 )P1∗ + (14) −L ∗ ∗ Rb/w (α2 , β2 )P2 + R2/3 Rb/w (α3 , β3 )P3 .
The sectional pitching moment, M∗i , is found using ∗ , CMi = CM0 + CMδ δa,i M∗i = Qi CMi cw Sw (15) ∗ and combined with the forces from the wing, Dw,i , and tail, L∗t,i , as M=
3 i=1
∗ (M∗i + lw,i Dw,i − lt L∗t,i ).
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The LQ-I controller is designed to track the set-points in the angle of attack α, pitch angle θ, flight speed V and dihedral angle η using the thrust T , the outer ailerons δao (operated symmetrically), the center elevator δec , and the center aileron δac as control inputs. Let T
U = [T δao δec δac ] ,
T
y = [α θ V η] .
(19)
The control design exploits a linearized model about a steady state condition corresponding to V = 30 ft/s, η = 5◦ , h = 40, 000 ft, α = 8◦ and θ = 13◦ (i.e., γ = 0◦ ). The model has the form, ˜˙ = AX ˜ + BU ˜ , y˜ = C X, ˜ X (20) 0 ˜ 0 0 ˜ where X = X − X , U = U − U and y˜ = y − y are deviations of the state, control and tracking output, y, from the respective trim points. The eigenvalues of A are λ1 = −0.0207, λ2,3 = −0.308 ± 1.4917i, (21) λ4 = −6.6181, λ5,6 = −2.5697 ± 6.7663i. The LQ-I design is based on applying LQR theory to the extended system, d y˜ − r y˜ − r 0 C 0 ˜˙ = + U, ˜ ˜ 0 A B X X dt where r designates the vector of the set-points for y˜. The state and control weighting matrices are defined by Q = diag([1000 1000 100 100 0.1 (22) 0.1 0.1 0.1 0.1 0.1]), R = diag([0.01 0.01 0.03 0.04]). (23) The LQ-I controller has the form, ˜ z˙ = y˜ − r, ˜ (t) = Ki z + Kp X, (24) U where K = [Ki , Kp ] is the LQR gain.
4. SCALAR REFERENCE GOVERNOR Let
z z , (25) = 0 ˜ X −X X denote the state of the closed-loop system (20), (24). The design of the scalar reference governor is based on the discrete-time model of the closed-loop system, (26) Xcl (t + 1) = Acl Xcl (t) + Bcl r, obtained assuming the update period for r of Ts = 0.01 s. Xcl =
The scalar reference governor modifies the set-point r when this set-point has a potential to induce future constraint violation. See Figure 3 and Garone et al. (2017); Kolmanovsky et al. (2014). Xcl (t)
r(t)
(16)
Reference v(t) LQ-I ClosedGovernor Loop
yc (t)
Fig. 3. Reference governor and LQ-I block diagram
Finally, the moment at the hinge, H, is calculated as ∗ ∗ + Pt,1 ) − m∗ g cos η cos θ. (17) H = −(Pw,1 z z The flight path angle γ and climb rate h˙ is given by (18) γ = θ − α, h˙ = V sin γ.
Thus in (26), r is replaced by v, where v is updated according to v(t) = v(t − 1) + κ(t) r(t) − v(t − 1) , (27) and where κ(t), 0 ≤ κ(t) ≤ 1, is a scalar parameter.
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If κ(t) = 1, then v(t) = r(t) and the scalar reference governor is inactive. If κ(t) = 0, then v(t) = v(t − 1) and r(t) is isolated from the system, assuring command safety. Consequently, the value of κ(t) is maximized subject to the constraint 0 ≤ κ(t) ≤ 1 and that the updated command (27), if it were applied for all future time instants, would not cause the constraint violation. To make the last constraint explicit, suppose the constraints for the model (26) can be written in the form, HCcl Xcl ≤ h, h > 0. (28) Then, (28) imposed over the future prediction horizon leads to conditions, HCcl Xcl (t + k) ≤ h, (29) for k = 0, 1, · · · . Assuming v(t + k) = v(t) and given (27), (29) becomes Hv,k (r−v(t−1))κ(t) ≤ h−Hx,k Xcl (t)−Hv,k v(t−1), (30) where Hx,k = Ccl Akcl , Hv,k = Ccl (I − Acl )−1 (I − Akcl )Bcl . (31) In the implementation, the constraints (30) are imposed up to a finite prediction horizon, k ∗ , i.e., only for k = 0, 1, · · · , k ∗ and an extra constraint is imposed as Hv,∞ (r − v(t − 1))κ(t) ≤ h(1 − ) − Hv,∞ v(t − 1), (32)
where Hv,∞ = Ccl (I − Acl )−1 Bcl and > 0 is small. With (32) imposed, under standard assumptions, if k ∗ is sufficiently large, then all constraints (30) for k > k ∗ are redundant and need not be explicitly imposed. It is also possible that some of the constraints for k ≤ k ∗ are redundant and can be a priori eliminated resulting in a smaller subset of constraints to be considered. The reader is referred to Garone et al. (2017); Kolmanovsky et al. (2014) for details. In either case, maximizing the scalar parameter κ(t) subject to the above affine constraints is an explicitly and easily solvable optimization problem. While recursive feasibility is guaranteed in absence of model mismatch, no feasible value for κ(t) may exist if model mismatch is present; in such a case an infeasibility handling mechanism is used with κ(t) set to zero. 5. SIMULATION CASE STUDIES In this section, simulation results are reported. In all cases, the aircraft starts from steady-state at the trim point and the commands are applied 2 sec after the simulation starts. Constraints are imposed on the dihedral angle, center and outer aileron position, and center aileron position, 4.5◦ ≤ η ≤ 5.5◦ , 1.5◦ ≤ δec ≤ 7.5◦ , (33) 25◦ ≤ δao ≤ 32◦ , 17◦ ≤ δac ≤ 23◦ , to emulate structural and control deflection limitations. These constraints are reformulated as in (29). The chosen value of k ∗ = 1000 provides a look-ahead horizon of 10 sec.
While the linearized models are used in the design of LQ-I controller and the scalar reference governor, the subsequent closed-loop simulations are performed on the nonlinear model of the multibody aircraft described in Section 2. The maneuvers correspond to changes in the flight path angle and are implemented through commands in the pitch angle θr and angle of attack αr (γr = θr − αr ). The commands for the aircraft velocity and the dihedral
angle are to maintain trim values of 30 ft/s and 5◦ , respectively. 5.1 Descent with γ = 5◦ We first consider a maneuver in which the flight path angle γr is decreased to −5◦ by setting θr to 3◦ and αr to 8◦ . The closed-loop response without the reference governor is shown in Fig. 4. While the desired flight path angle is achieved, nearly every constraint is violated. The chosen weights in the LQ-I controller design provide fast response and good tracking, but the dihedral angle constraint is violated and initial deflections for all the control surfaces exceed their bounds.
Fig. 4. Descent command without reference governor violates dihedral (top) and control deflection (center) constraints, but reaches desired flight path angle γr = −5◦ in about 20 sec (bottom). Now, the same flight path angle decreased maneuver is commanded to the closed-loop system with the reference governor. See Fig. 5. The constraints are met and the commanded flight path angle is achieved. The reference governor slowly decreases the applied pitch angle and angle of attack commands to prevent constraint violation. Note that the commanded flight path angle is reconstructed as the difference between the pitch angle and angle of attack commands that the reference governor outputs. The active constraints are both the control surface deflections and dihedral angle, with the outer and center ailerons being the most actively constrained. The desired flight path angle is reached in about 30 sec, as opposed to approximately 20 sec for the prior unconstrained maneuver. 5.2 Climb with γ = 10◦ Next a climb maneuver with a larger flight path angle command of 10◦ is considered. The closed-loop response with the reference governor is shown in Fig. 6. While the dihedral angle satisfies the constraints, there is a brief exceedance (by 0.66◦ ) of the limit by the center aileron position. This exceedance is attributed to larger mismatch
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0 η η(t) δa0 δao (t) g(t) = − HCcl Xcl (t) − 0o δec (t) δec δac (t) δa0c
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,
(35)
where η 0 = 5◦ , δa0o = 28.6◦ , δe0c = 4.5◦ , δa0c = 19.8◦ are the values of dihedral angle, outer aileron position, center elevator position, and center aileron position at trim at which the linearized model is obtained and Xcl (t) is determined from (25). The reference governor with the mismatch compensation provides the results in Fig. 7, with only a minor exceedance in the outer aileron position of 0.1◦ .
Fig. 5. Descent with reference governor keeps dihedral (top) and control deflection (center) constraints, and still reaches desired flight path angle γ = −5◦ in about 30 sec (bottom). between the linear model used by the reference governor and the actual nonlinear system for larger maneuvers.
Fig. 7. The response with the reference governor enhanced with mismatch compensation meets dihedral constraints (top) but violates outer aileron deflection by less than 0.1◦ (center) for the flight path angle command of 10◦ . 5.3 Climb with γr = 10◦ and trim point with η 0 = 23◦
Fig. 6. Climb meets dihedral constraints (top) but violates center aileron deflection by 0.66◦ (center) for 10◦ flight path angle command. To accommodate the mismatch between the linear model and the nonlinear system, the approach described in Garone et al. (2017); Kolmanovsky et al. (2014) is used. In this approach, the constraint (29) over the prediction horizon, is replaced by the constraint, HCcl Xcl (t + k) + g(t) ≤ h,
(34)
where g(t) is set to the mismatch at the current time instant t between the constrained outputs and their estimates based on the linearized model, i.e.,
If the nominal trim point is modified to η 0 = 23◦ , the open-loop model (20) becomes unstable, with open-loop eigenvalues about the trim condition given by λ1 = 0.0120, λ2,3 = 0.3523 ± 1.0421i, (36) λ4 = −6.9121, λ5,6 = −2.3515 ± 1.0232i Fig. 8 shows simulated closed-loop responses and the constraints for the same Q and R weights but with LQI controller and reference governor reconfigured for the linearized model and updated constraints at the new dihedral angle. Without the reference governor the closed loop response is unstable as the control inputs saturate. 6. CONTROLLER ROBUSTNESS If we augment the system with a simple turbulence model, such that normal zero-mean disturbances with standard deviation of 1 ft/s are symmetrically added to ui and vi for each section from (7) and (8), as done by Gibson (2014), the reference governor is still able to enforce the constraints with minor exceedances. See Fig. 9. The flight
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7. CONCLUSION The combination of the scalar reference governor and LQI controller has been shown to be effective for control of longitudinal flight of a nonlinear multibody aircraft based on the model in Gibson et al. (2011). ACKNOWLEDGEMENTS The authors thank Dr. Travis Gibson for providing the multibody aircraft model described in Gibson et al. (2011) and assisting them in its use that has greatly facilitated this work. REFERENCES
Fig. 8. The response with the reference governor enhanced with mismatch compensation meets dihedral constraints (top), control deflection constraints (center) and tracks the flight path angle command of 10◦ from a trim condition with higher dihedral angle of 23◦ . path angle command is set to a larger value of 15◦ , but with dihedral angle command at 23◦ as before. In simulations the center aileron is constrained against its limit, limiting flight path angle that can be achieved.
Fig. 9. Reference governor with turbulence can constrain the average values of the disturbances, not overall trajectory, due to the linear model in prediction. Robustness tests were also performed in simulations that used the original controller for the aircraft with 10% of the original dihedral angle stiffness. In response to 10◦ flight path angle command and 5◦ dihedral angle command, the reference governor was still able to account for the model mismatch and enforce the constraints between the model with minor constraint violations under 1◦ in center aileron deflection during the maneuver.
Dillsaver, M.J., Kalabic, U.V., Kolmanovsky, I.V., and Cesnik, C.E.S. (2013a). Constrained control of very flexible aircraft using reference and extended command governors. In Proceedings of American Control Conference, 1608–1613. Dillsaver, M., Cesnik, C.E., and Kolmanovsky, I. (2013b). Trajectory control of very flexible aircraft with gust disturbance. In Proceedings of AIAA Atmospheric Flight Mechanics (AFM) Conference. Garone, E., Cairano, S.D., and Kolmanovsky, I. (2017). Reference and command governors for systems with constraints: A survey on theory and applications. Automatica, 75, 306 – 328. Gibson, T., Annaswamy, A., and Lavretsky, E. (2011). Modeling for control of very flexible aircraft. In Proceedings of AIAA Guidance, Navigation, and Control Conference. Gibson, T.E. (2014). Closed-loop Reference Model adaptive control : with application to very flexible aircraft. Ph.D. thesis, Massachusetts Institute of Technology. Gonzalez, P.J., Silvestre, F.J., Paglione, P., K¨ othe, A., Pang, Z.Y., and Cesnik, C.E. (2016). Linear control of highly flexible aircraft based on loop separation. In Proceedings of AIAA Atmospheric Flight Mechanics Conference. Haghighat, S., T. Liu, H.H., and RA Martins, J.R. (2012). Model-predictive gust load alleviation controller for a highly flexible aircraft. Journal of Guidance, Control, and Dynamics, 35(6), 1751–1766. Kolmanovsky, I., Garone, E., and Cairano, S.D. (2014). Reference and command governors: A tutorial on their theory and automotive applications. In Proceedings of 2014 American Control Conference, 226–241. Kopf, M., Bullinger, E., Giesseler, H.G., Adden, S., and Findeisen, R. (2018). Model predictive control for aircraft load alleviation: Opportunities and challenges. In Proceedings of American Control Conference, 2417– 2424. Shearer, C.M. and Cesnik, C.E. (2008). Trajectory control for very flexible aircraft. AIAA Journal of Guidance, Control, and Dynamics, 31(2), 340–357. Simpson, R.J., Palacios, R., Hesse, H., and Goulart, P. (2014). Predictive control for alleviation of gust loads on very flexible aircraft. In Proceedings of 55th AIAA/ASME/ASCE/AHS/SC Structures, Structural Dynamics, and Materials Conference, AIAA SciTech Forum.
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