Longitudinal Motion Control of Air-Breathing

Hypersonic Vehicles

Based on Time-Varying Models∗

Barıs Fidan†

National ICT Australia Ltd. and Australian National University, Canberra, ACT 2601, Australia

Matthew Kuipers ‡, Petros A. Ioannou§

University of Southern California, Los Angeles, CA 90089, USA

Maj Mirmirani¶

California State University, Los Angeles, CA 90032, USA

The non-standard dynamic characteristics of air-breathing hypersonic flight vehiclestogether with the aerodynamic effects of hypersonic flight make the flight control of suchsystems highly challenging. Moreover the wide range of speed during operation and thelack of a broad flight dynamics database add significant plant parameter variations anduncertainties to the problem of controlling air-breathing hypersonic flight. In this paper,we present a general approach to this challenging problem based on linear time-varyingplant models. This approach lets employment of some recent techniques developed foradaptive and non-adaptive control of systems with fast varying parameters via exploitationof a priori information about parameter variations. We discuss application of this approachto a generic analytical hypersonic flight vehicle model that is widely used in the literatureas well as a detailed generic hypersonic vehicle concept model developed at the CaliforniaState University - Los Angeles. The paper is concluded with some remarks on validationsand extensions of the presented approach.

Nomenclature

AD : diffuser exit area/inlet area ratio

c (ft) : mean aerodynamic chord

D (lbf) : drag

h (ft) : vehicle altitude

Iyy (slug.ft2): vehicle y-axis inertia per unit width

L (lbf) : lift

m (slug) : vehicle mass

M : vehicle flight Mach Number

Myy (lb.ft) : pitching moment

q (rad/s) : pitch rate

r (rt) : distance from the Earth’s center

RE (ft) : radius of the Earth

S (ft2) : reference area

T (lbf) : thrust

T0 (◦R) : temperature across the combustor

ux (ft/s) : speed along the vehicle x-axis

V (ft/s) : vehicle velocity

α (rad) : angle of attack

γ (rad) : flight path angle (γ = θ − α)

δe (rad) : pitch control surface deflection

δt : throttle setting

η : generalized elastic coordinate

θ (rad) : pitch angle

µ (ft3/s2): gravitational constant

∗The work of B. Fidan is supported by National ICT Australia, which is funded by the Australian Government’s Departmentof Communications, Information Technology and the Arts and the Australian Research Council through the Backing Australia’sAbility Initiative. The work of M. Kuipers, M. Mirmirani, and P.A. Ioannou is supported in part by AFOSR grant numberF49620-01-1-0489 and in part by NASA grant number NCC4-158.

†Researcher, Systems Engineering and Complex Systems Program, Member AIAA.‡Graduate student, Electrical Engineering Department,Student Member AIAA.§Professor, Electrical Engineering Department, Member AIAA.¶Professor, Mechanical Engineering Department, Senior Member AIAA.

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14th AIAA/AHI Space Planes and Hypersonic Systems and Technologies Conference AIAA 2006-8074

Copyright © 2006 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

I. Introduction

Air-breathing hypersonic flight vehicle (AHFV) research is critical to the development of the next genera-tion of aviation systems. Feasible and affordable atmospheric hypersonic flight using air-breathing propulsionsystems will expand the boundaries of aeronautics, and will provide for the development of new technologiesfor space access, high speed civilian transportation, and military application1–6 . A number of extensiveinitiatives around the world including NASA’s Space Launch Initiative (SLI) and the Second GenerationReusable Launch Vehicle (RLV) Program require research on all aspects of hypersonics including vehicle sys-tem design, propulsion system design and experimental capabilities. Development of advanced technologiesto enable cost effective vehicle systems for use in space launching, orbiting and maneuvering will translateeasily to improvements in military interceptors, tactical and strategic reconnaissance, as well as in high-speedand orbital transport activities.

Hypersonic flight technologies have been studied by NASA and a number of other research units aroundthe world for more than 60 years, since the evolvement of the hydrocarbon-fueled conventional ramjet(CRJ) engine concept1,2, 4 . In the late 1940s, researchers began to study the feasibility of developing ascramjet engine; in the early ’60s, the scramjet technological obstacles were identified. In 1964, a flight-weight, regeneratively cooled scramjet research engine was placed on the X-15 rocket-powered researchplane, which flew several times and reached a record-making speed of Mach 6.72. In 1965, the U.S. AirForce (IFTV Program) demonstrated acceleration from a boosted speed of 5400 ft/s to 6000 ft/s usingfour hydrogen-fueled scramjet modules placed around a vehicle’s central body. In the 1970s, focus wasshifted on the rectangular airframe-integrated engine configuration which was deemed to be superior to theaxisymmetric configuration. In 1986, the U.S. Air Force and NASA initiated the National Aerospace Plane(NASP) program which made great strides in the airframe-integrated engine concept for a single-stage-to-orbit (SSTO) full scale experimental hypersonic aircraft. The NASP program led to extensive studies on thedynamic characteristics of airbreathing hypersonic vehicles and uncovered challenges in modelling and controlarising from the airframe-integrated engine configuration. In 1996, the Hyper-X program was initialed andfocused on the development and flight testing of small scale (X-43A, X-43B, X-43C, and X-43D) and onefull-scale demonstrator vehicles. The X-43A is “smart scaled” from a 200-foot operational concept. It hasbeen used in three scramjet-powered and un-powered flight tests, two successful ones at Mach 7 and Mach106,7 .

Subsequent to the successful X-43 flights, NASA Aeronautics Directorate and the Air Force have turnedtheir focus to foundation hypersonic research and development of predictive capabilities in support of the nextgeneration of highly reliable RLVs (HRRLV). Two recent models developed in this light for the longitudinaldynamics of a generic AHFV are presented in8,9 . The model in8 represents the longitudinal dynamicsof a full scale vehicle developed based on basic principles to give flight control engineers a fundamentalunderstanding of the physics of such vehicles. The model in9 , namely CSULA-GHV, on the other hand,is a more detailed one and is developed following a more complex procedure involving initial model sizingbased on mission requirements, hypersonic flow dynamic analysis, CFD analysis, elasticity and aerodynamicinteraction considerations, etc. Several control designs have been developed based on both of these modelsas well10–12 .

However, after over four decades of research in hypersonics, the state of knowledge in this field stillincludes both “known unknowns” and “unknown unknowns”. This is primarily due to the following:1,2

• Unique design of these vehicles, i.e. the tightly integrated airframe engine configuration.

• The aero-thermodynamic effects of hypersonic speeds, which are not adequately understood.

• Wide range of speeds and fast changing flight conditions.

• Lack of adequate flight and ground test data for both the vehicle, the engine, and vehicle systems

These effects hinder accurate characterization of the vehicle for design, modeling and control. Theanalyst and designer must deal with significant uncertainties. Furthermore, the unique configuration ofthe vehicle produces strong aerodynamic-propulsion-aeroelastic couplings which researches have not yetcompletely quantified.

In this paper, after providing a brief review of the literature on above uncertainty, coupling and variationphenomena and their affects on modeling and control of air-breathing hypersonic flight vehicle (AHFV) inSection II, we focus on design of a supervisor scheme to handle the system variations on-line.

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The conventional supervision methodology in aerospace applications is gain scheduling13,14 . In spiteof the wide use of this methodology, there is a lack of tests to guarantee global stability of gain scheduledsystems in general. Furthermore to apply gain scheduling effectively, plant parameter variations are requiredto be slow and precise quantification of the system dynamics at sufficiently many flight conditions is needed.Taking into account the anticipated fast variations in the air-breathing hypersonic flight (AHF) system andthe lack of broad flight data, designing a gain scheduling scheme for AHFVs seems currently to be extremelyhard if possible.

As an attempt to fill in some of the gaps mentioned above, we propose an adaptive flight supervisordesign approach based on time-varying (TV) models of the longitudinal AHFV motion. Leaving hybriddesigns combining this approach with gain scheduling based on the available data as a future research topic,we focus on TV modeling of the longitudinal AHFV motion and control designs based on such TV models.We derive a class of TV AHF dynamic models in Section III assuming that some a priori information aboutthe system parameters is available for exploitation.

In Section IV, we consider the longitudinal model for a concept vehicle developed at California StateUniversity, Los Angeles - CSULA-GHV9,15 as a case study and develop a linear time-varying (LTV) modelfor it. The CSULA-GHV has an integrated airframe-propulsion system configuration and resembles an actualtest vehicle. The vehicle’s aerodynamic data has been developed by both flow theoretic models and using anintegrated aero-propulsion CFD model. The couplings between the aerodynamics and the propulsion systemare quantified and are incorporated into the model.

In Sections III and IV, the a priori information about the time-dependent system parameters are assumedto be provided by the guidance scheme of the complete flight and the available flight condition databases.The exact values of the time-dependent system parameters are not expected to be fully known since thedatabases are not complete, the guided and the actual trajectories will be different, and there exist unmodeleddisturbances and uncertainties. Hence on-line parameter estimation and adaptation is required in the controldesign. In Section V, we present a control design approach for the LTV models presented in Section III. Theapproach is based on some recent works in the literature on adaptive control of fast linear TV systems16–19 ,where the control designs allow exploitation of a priori information about variations and guarantee stabilityand admissible transient performance under mild conditions. Application of this control design approachto the case studies described in Sections III and IV is briefly discussed as well in Section V. The paper isconcluded with some remarks on validations and future extensions of the proposed approach in Section VI.

II. Air-Breathing Hypersonic Flight (AHF) Dynamics

In this paper, we consider generic hypersonic aircraft configurations with airframe-integrated scramjetengines similar to X-30 or X-43A6,7, 20–22 or the more recent concept models in8,9 . For these configurations,the primary lift generating surface is the body itself due to inefficiency of using a thin wing. Beside theaerothermodynamic effects of the hypersonic speed, the strong interactions between the elastic airframe,the propulsion system, and the structural dynamics make the explicit characterization of flight dynamics ofAHFVs challenging3,20,21 .

The major dynamic interactions are illustrated in Figure 1, which is drawn from1 . The dynamic charac-teristics of the hypersonic vehicle vary more significantly over the flight envelope than other aircraft due to itsextreme range of operating conditions and rapid change of mass distribution. Moreover, many aerodynamicand propulsion characteristics still remain uncertain and are hard to predict due to lack of sufficiently manyflight tests and inadequacy of the ground test facilities. Unpredictable aerodynamic and thermodynamic be-haviors due to hypersonic speed and aerodynamics/ propulsion/structural dynamics interactions constituteanother uncertainty source1,21,23,24 .

Although a number of studies on air-breathing hypersonic flight (AHF) control consider all the challengesabove21,22,24 , most of the works have ignored the coupling effects, have assumed parameter variations anduncertainties to be small, and have considered only some of the aerothermodynamic characteristics of thehypersonic flight. Even in these cases, significant nonlinearities had to be introduced making them distinctlydifferent from the conventional flight control problems25,26 .

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StructuralDynamics

Inlet/NozzleConditions

EngineDynamics

Thrust/Lift

AHFDynamics

Bending

EngineControl

FlightControl

Flexible Modes

AHFVAttitude&Loc.;Aerodynamic

Forces&Torques

ThermodynamicEffects

Ther

m.

Mea

s.

Figure 1. Dynamic interactions for AHFVs.

III. Time-Varying Models for the AHF Dynamics

In the literature, flight dynamics models are often formulated using flight condition-specific system pa-rameters. We encounter the same formulation in all the main AHF control studies in the open literature2 .An example is the linear AHF dynamic model

x = (A + ∆A)x + (B + ∆B)u + (E + ∆E)d (1)

where x, u, d denote the system state, control input, and external disturbance vector, respectively; A,B, Eare the corresponding (nominal) system matrices that have fixed values for any fixed flight condition; and∆A, ∆B, ∆E represent the uncertainties in A,B,E that are due to the reasons mentioned in Section II.

Noting the flight condition dependence in the linear model (1), if we lump the flight conditions of theAHFV at each time t in a vector p(t) then (1) can be more precisely written in the linear parameter-varying(LPV) form as

x = (A(p) + ∆A(p))x + (B(p) + ∆B(p))u + (E(p) + ∆E(p))d (2)

at a given flight condition p, where for a fixed p, A(p), B(p), E(p) are constant. The uncertainties ∆A(p),∆B(p), ∆E(p) are not necessarily constant for a fixed p, yet their behavior, e.g., the domains of their possiblevalues may be different for different values of p.

Since p is a function of time t, we can go one step further and rewrite (2) as

x = (At(t) + ∆At(t))x + (Bt(t) + ∆Bt(t))u + (Et(t) + ∆Et(t))d (3)

where the notation ·t(t) is equivalent to ·(p(t)). A similar procedure can be followed for the cases where theflight condition specific model is nonlinear as opposed to (1). The resultant TV model in this case will bea nonlinear one and the time-dependent system parameters will be some function description parametersinstead of the system matrix entries in (3).

Above we assume that a nominal or desired time-trajectory for the flight condition vector p(t) is available,i.e.,

p(t) = p∗(t) + p(t) (4)

where p∗(t) is known at time t for all t ≥ 0 and p(t) is some mild deviation of the actual flight conditionvector p(t) from its nominal value p∗(t). This a priori information is assumed to be provided by the guidancescheme of the complete flight and the available flight condition databases.

Next, as a case study, we consider the analytical aeropropulsive/aeroelastic hypersonic-vehicle modelderived for the X-30 configuration in22,27 and later elaborated in certain aspects in24,28 . This modelincorporates most of the issues summarized in Figure 1. In22,27 the corresponding AHFV longitudinalequations of motion were derived and expressed in the state space form as

˙x = Ax + Bu (5)

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where

x = [∆h, ∆ux, ∆α, ∆θ, ∆q, ∆η, ∆η]T

u =[∆δe, ∆AD, ∆T0

]T

represent small perturbations about the equilibrium values x0(p), u0(p) for a certain flight condition vectorp of the state and control vectors

x = [h, ux, α, θ, q, η, η]T

u =[δe, AD, T0

]T

and A and B are some flight condition (p) specific system matrices whose detailed analytic expressions canbe found in22,27 (see page 1 for the nomenclature). Noting that Ax0 + Bu0 = 0, (5) can also be written as

x = Ax + Bu (6)

Based on the analysis in22,24,28 , it further follows that the 7 × 7 matrix A and the 7 × 3 matrix B are ofthe form

A =

0 0 a13 a14 0 0 0a21 a22 a23 0 a25 a26 a27

a31 a32 a33 0 a35 a36 a37

0 0 0 0 1 0 0a51 a52 a53 0 a55 a56 a57

0 0 0 0 0 0 1a71 a72 a73 0 a75 a76 a77

, B =

0 0 0b21 b22 b23

b31 0 00 0 0

b51 b52 b53

0 0 00 0 0

The entries of the system matrices above can be thought as nonlinear functions of vehicle geometry andmass properties, atmospheric conditions, structural vibration mode shapes, and Mach number, i.e. eachnonzero entry aij ≡ aij(p) or bij ≡ bij(p) depends on the characteristic vehicle constants and the flightcondition vector p in (4). Via reordering the state vector entries we further obtain

˙x = A(p)x + B(p)u (7)

and˙x = At(t)x + Bt(t)u (8)

where ·t(t) ≡ ·(p(t)) and

x = [q, ux, α, η, h, η, θ]T

A =

a55 a52 a53 a57 a51 a56 0a25 a22 a23 a27 a21 a26 0a35 a32 a33 a37 a31 a36 0a75 a72 a73 a77 a71 a76 00 0 a13 0 0 0 a14

0 0 0 1 0 0 01 0 0 0 0 0 0

, B =

b51 b52 b53

b21 b22 b23

b31 0 00 0 00 0 00 0 00 0 0

Considering the essential states that need to track certain set-value trajectories to be ux, α, h, we mayadd the following output equation to (8):

y = Cx (9)where y = [ux, α, h]T and

C =

0 1 0 0 0 0 00 0 1 0 0 0 00 0 0 0 1 0 0

In the next section we consider the recently developed vehicle concept CSULA-GHV9,15,29 , which hasbeen briefly described in Section I and we derive equations of the longitudinal motion of CSULA-GHV inthe LPV and LTV forms.

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IV. Time-Varying Modelling of the CSULA-GHV

The AHFV model9,15,29 CSULA-GHV has been developed similar to the design and development of aconventional aircraft model. Initial sizing of the aircraft was derived from a set of mission requirements.Then hypersonic flow physics was used for determining the basic geometry and producing estimates of theaerodynamic properties at trim and the thrust produced by the scramjet engine. A CFD model constructedin FLUENT generated a complete set of aerodynamic and aero-propulsion data for different angles of attack,Mach numbers, and control settings, i.e. throttle and elevon settings. The flow has been assumed to be aninviscid ideal gas (thermally perfect gas) with changes in specific heat calculated using the kinetic theory. Toaccount for the changes in oblique shock angle due to changes in flight conditions, the GHV has a variableinlet design in that the cowl moves in a fashion that captures the shock-waves at the optimum point. Thisensures that the mass flow rate of air through the engine is maximized and no shock train is generated insidethe scramjet. The model captures the so-called pluming effect at the exhaust nozzle thereby accuratelypredicting lift, drag and the installed thrust.

Using the above methodology, the nonlinear rigid-body longitudinal equations of motion for the CSULA-GHV, which include the gravitation and centripetal acceleration effects for the non-rotating round Earth30,31 ,are developed. The corresponding state-space equations are governed by the following set of differential equa-tions (see page 1 for the nomenclature):

V =T cos α−D

m− µ sin γ

r2(10)

γ =L + T sin α

mV− (µ− V 2r) cos γ

V r2(11)

h = V sin γ (12)α = q − γ (13)

q =Myy

Iyy(14)

where L = 12ρV 2SCL, D = 1

2ρV 2SCD, T = 12ρV 2SCT , Myy = 1

2ρV 2ScCM , r = h + RE . The details ofthe non-dimensional coefficients CL, CD, CT , and CM , as well as the atmospheric model, are given in theAppendix. Note that each non-dimensional coefficient is a function of M , α, δt, and δe. Their dependence onboth aerodynamic and propulsion conditions is a consequence of the aero-propulsion coupling characteristicof the CSULA-GHV.

Next, we develop LPV and LTV state-space models for the state and control input vectors

x = [V, γ, h, α, q]T

u = [δt, δe]T

We assume that the states are available for measurement, i.e., we have the system output y = x. Also,fV (x, u), fγ(x, u), fh(x, u), fα(x, u), and fq(x, u) denote the right hand sides of (10)–(14), respectively.Therefore, the state equations can be re-written in compact notation as:

x = [fV (x, u), fγ(x, u), fh(x, u), fα(x, u), fq(x, u)]T (15)

There are a number of approaches to developing an LPV model based on (15)32 . The methodologyundertaken in this paper is the widely used Jacobian linearization approach based on the first order Taylor-series expansion of (15) about a trim point. Note that each such trim point can be parameterized by a (flightcondition) vector p which can be formed by a collection of the entries of the trim state and control inputvectors xtrim, utrim that uniquely define this particular trim point. Using this approach, we obtain

x = A(p)x + B(p)u (16)

where

A(p) =

∂V fV ∂γfV ∂hfV ∂αfV ∂qfV

∂V fγ ∂γfγ ∂hfγ ∂αfγ ∂qfγ

∂V fh ∂γfh ∂hfh ∂αfh ∂qfh

∂V fα ∂γfα ∂hfα ∂αfα ∂qfα

∂V fq ∂γfq ∂hfq ∂αfq ∂qfq

∣∣∣∣∣∣∣∣∣∣∣p(t)

, B(p) =

∂δtfV ∂δefV

∂δtfγ ∂δefγ

∂δtfh ∂δefh

∂δtfα ∂δefα

∂δtfq ∂δefq

∣∣∣∣∣∣∣∣∣∣∣p(t)

(17)

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where we denote the partial derivative operator as ∂y4= ∂

∂y . Deriving the entries of the matrices in 17)analytically, we see that these matrices can be expressed in the form

A(p) =

a11(p) a12(p) a13(p) a14(p) 0a21(p) a22(p) a23(p) a24(p) 0a31(p) a32(p) 0 0 0−a21(p) 0 −a23(p) −a24(p) 1a51(p) 0 a53(p) a54(p) 0

, B(p) =

b11(p) b12(p)b21(p) b22(p)

0 00 0

b51(p) b52(p)

(18)

where the explicit expressions for the entries aij and bij are given in the Appendix.Using the equality θ = α+γ and applying a linear system transformation similar to the one in Section III,

(16) can be rewritten in the form˙x = A(p)x + B(p)u (19)

and˙x = At(t)x + Bt(t)u (20)

where ·t(t) ≡ ·(p(t)) and

x = [x5, x2, x1, x3, x2 + x4]T = [q, γ, V, h, θ]T

A(p) =

0 −a54(p) a51(p) a53(p) a54(p)0 a22(p)− a24(p) a21(p) a23(p) a24(p)0 a12(p)− a14(p) a11(p) a13(p) a14(p)0 a32(p) a31(p) 0 01 a22(p) 0 0 0

, B(p) =

b51(p) b52(p)b21(p) b22(p)b11(p) b12(p)

0 0b21(p) b22(p)

Note that the actual flight condition vector p(t), as discussed in Section III, will not be fully known. Itis only the nominal flight condition p∗(t) that is provided by the guidance system. Hence for any controllerimplementation the available system matrices are A∗t (t) , A(p∗(t)),B∗

t (t) , B(p∗(t)) or equivalently A∗t (t) ,A(p∗(t)),B∗

t (t) , B(p∗(t)).In our work, we assume that p∗(t) is generated using the following procedure:

1. A desired velocity and altitude trajectory is generated and, in turn, determines the correspondingvelocity and altitude components of p∗(t).

2. With the V (t) and h(t) components of p∗(t) specified, the α(t), δt(t), and δe(t) components are gener-ated via a trim-table generated off-line and stored in the guidance system’s database.

3. q(t) and γ(t) components of p∗(t) are set to 0.

The first item in the above procedure indicates that the essential states that need to track certain set-valuetrajectories are V , h. Hence we may add the following output equation to (20):

y = Cx (21)

where y = [V, h]T and

C =

[0 0 1 0 00 0 0 1 0

]

In the following section, we present a TV multivariable control scheme that is applicable to the LPV andLTV models derived in this section as well as the ones presented in Section III. As a factor in determiningthe adaptive control approach to be focused on, it is worth to note that at certain trim conditions, thelinear model (19),(21) is non-minimum phase. For example, for the trimmed cruise condition with M = 10,h = 100, 000 ft, α = 0.77019◦, δt = 0.088264 , δe = 0.54211◦, the linear model has an unstable transmissionzero at 8.3175 rad/s. This fact motivates us to use a pole placement control (PPC) approach rather than amodel reference control (MRC) approach in the control design.

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V. A Time-Varying Control Scheme

TV and parameter-varying methods have already been used in designing controllers for various aerospacesystems33–35 . These control designs, however, are in general based on the assumption that the variationsare slow in some sense and they do not exploit the a priori information about the variation structures. Amethod to exploit the a priori information about system parameter in designing controllers for TV systemsis presented in19 . This method is used to design effective adaptive controllers for LTV systems in16–19 . Aparticular aerospace application of these adaptive control schemes can be seen in36 .

In this paper, we use the pole-placement control (PPC) methodology developed in17 for a class of mul-tivariable LTV systems. The methodology can be used to design non-adaptive controllers for plants whosesystem models (or system parameters) are fully known as well as to design adaptive controller for systemmodels with partially known parameters. In this section, we first describe this methodology and present itsguaranteed performance properties, and then discuss its application to the models presented in Sections IIIand IV.

V.A. Pole Placement Control (PPC) of Time-Varying Plants with Known Parameters

Consider the multivariable LTV system

x = At(t)x + Bt(t)u (22)y = Ct(t)x

where the input u, the state x, and the output y satisfy x ∈ <n, u, y ∈ <m for some positive integers n, m.Assume that the following hold:

Assumption 1 The entries of At(t),Bt(t),Ct(t) are bounded functions of time t and n-times continuouslydifferentiable with bounded derivatives.

Assumption 2 For any t ≥ 0, Bt(t) has full column rank and Ct(t) has full row rank.

Assumption 3 [At(t), Bt(t), Ct(t)] is strongly controllable and strongly observable, i.e.∣∣det(Qc(t)QT

c (t))∣∣ ≥

c and∣∣det(Qo(t)QT

o (t))∣∣ ≥ c, ∀t > 0 for some constant c > 0, where the TV controllability matrix Qc(t) and

the TV observability matrix Qo(t) are defined as17

Qc , [H0,H1, . . . ,Hn] ; H0 = B(t), Hk+1 = A(t)Hk − Hk

Qo , [O0, O1, . . . , On]T ; O0 = CT (t), Ok+1 = AT (t)Ok + Ok

Assumption 4 [At(t), Bt(t), Ct(t)] is index-invariant, i.e.17 the linearly independent columns of the TVcontrollability matrix Qc(t1) and the linearly independent rows of the TV observability matrix Qo(t1)forsome t1 > 0 are actually the linearly independent columns and the linearly independent rows of the respectivematrices for all t ≥ 0.

In17 , a PPC scheme is presented for LTV systems of the form (22) that satisfy Assumptions 1–4, for thecontrol objective (Multivariable TV PPC Objective) described below. Before stating this objective, we needto give the definitions of some notions related to such LTV systems.

Consider an LTV system of the form (22) satisfying Assumptions 1–4. The controllability indices nri ,i = 1, . . . , m of this system are the number of linearly independent columns associated with Bi(t) found inthe matrix Qc(t), where Bi denotes the ith column of B and

Qc =[B1, . . . , A

nr1−1B1, . . . , Bm, . . . , Anrm−1Bm

]

is obtained by searching linearly independent columns of the TV controllability matrix Qc in the order fromleft to right17 . The overall controllability index of the system is defined as nr , maxi nri . Similarly, theobservability indices nvi , i = 1, . . . , m of the system are defined to be the number of linearly independentrows associated with Ci(t) found in the matrix Qo(t), where Ci denotes the ith row of B and

Qo =[CT

1 , . . . , (C1Anv1−1)T , . . . , CT

m, . . . , (CmAnvm−1)T]T

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is obtained by searching linearly independent rows of the TV observability matrix Qo in the order from topto bottom. The overall observability index of the system is defined as nv , maxi nvi

.The LTV system, by satisfying Assumptions 1–4, is guaranteed to admit a so-called left polynomial

differential operator matrix (PDOM) fractional description

y = D−1p (s, t)Np(s, t)[u] (23)

where Dp(s, t), Np(s, t) are left PDOM s, i.e. polynomial matrices of the form

Dp(s, t) = Dpn(t)sn + · · ·+ Dp1(t)s + Dp0(t)Np(s, t) = Npn(t)sn + · · ·+ Np1(t)s + Np0(t),

s denotes the differential operator ddt (.), and Dpi, Npi ∈ <m×m (i = 0, 1, . . . , n) are matrices with entries

that are bounded functions of time and n-times continuously differentiable with bounded derivatives. Theentries of Dpi, Npi can be derived from the system matrices A,B, C in (22). The details of this derivationprocedure can be found in16,17 .

The left polynomial integral operator matrix (PIOM) of the system is the operator P−1(s, t) that mapsthe input u to the zero-state response of the vector differential equation P (s, t)[y] = u where P (s, t) is theleft PDOM with the row-degree coefficient matrix being the identity matrix16,17 .

Multivariable TV PPC Objective: Given an LTV system of the form (22) that satisfy Assumptions1–4, determine a control input such that:

(i) The closed-loop plant is internally stable.

(ii) The output vector y tracks a reference signal vector r which satisfies Λ(s)[r] = 0 for some a priorispecified TI PDOM Λ(s) of degree nλ.

(iii) The closed-loop left PIOM of the system matches an a priori selected exponentially stable (e.s.) PIOMA∗−1(s, t) of degree nr + nv + nλ − 2, i.e. a PIOM A∗−1(s, t) such that the system A∗(s, t) is an e.s.PDOM of degree nr + nv + nλ− 2, where nr and nv denote, respectively, the controllability index andthe observability index of the system.

In17 , a two-step PPC procedure is proposed to meet the Multivariable TV PPC Objective: In the firststep, polynomial equation techniques are used to determine the controller parameters. The PPC law isproposed to be in the form

u = Q(s, t)P−1(s, t)[r − y]− U(s, t)V −1(s)[y] (24)

where U(s, t) and V (s) are PDOMs of degree nλ − 1; P (s, t),Q(s, t) are PDOMs of degree nr + nλ − 2;P (s, t),V (s) have identity-matrix leading coefficients; V (s) is e.s; and the coefficients of P (s, t),Q(s, t),U(s, t)are smooth and bounded. The PDOMs U(s, t) and V (s) are chosen such that V −1(s)Np(s, t) is stronglycontrollable and the equation

Dp(s, t)V (s) + Np(s, t)U(s, t) = X(s, t)Λ(s) (25)

is satisfied for some PDOM X(s, t) of degree nv − 1 with smooth and bounded coefficients. The controllerPDOMs P (s, t),Q(s, t) are given by

P (s, t) = V (s)P (s, t) (26)Q(s, t) = Q(s, t)− U(s, t)P (s, t)

where Q(s, t) is a PDOM of degree nr + nλ − 2, P (s, t) is a PDOM of degree nr − 1 with identity-matrixleading coefficient, and the coefficients of P (s, t),Q(s, t) are selected to satisfy the matching equation

Dp(s, t)V (s)P (s, t) + Np(s, t)Q(s, t) = A∗(s, t) (27)

In the second step of the PPC design, the controller described by the left PDOM fractional equation (24)is realized in the state space form. Some illustrative derivations of such realizations can be found in16,17 .

The following theorem summarizes the properties of the PPC design above:

Theorem 1 17 Consider a multivariable LTV system with state-space description (22) whose parameters areknown functions of time and the corresponding left PDOM fractional description (23). Let Assumptions 1–4hold. Then there exist PDOMs P (s, t),Q(s, t),U(s, t),V (s) satisfying the matching equations (25),(27), andthe PPC design above, i.e. the control law (24)–(27), guarantees that the Multivariable TV PPC Objectiveis achieved and the tracking error r − y converges to zero exponentially fast.

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V.B. Adaptive PPC of Time-Varying Plants with Partially Known Parameters

The design presented in Section V.A assumes that the system parameters of the LTV plant (22) are fullyknown. In this subsection, we consider the case where the system parameters are only partially known andreview the adaptive PPC (APPC) scheme proposed for such plants in17 . For the APPC design procedure,mainly for re-parametrization of the system parameter in an appropriate form for certain parameter identi-fication algorithms, we need another type of system description equivalent to the state-space model (22) orthe left PDOM fractional formulation (23), which is described by the following lemma:

Lemma 1 17 Consider a multivariable LTV system with left PDOM fractional description (23) with observ-ability indices nvi

(i = 1, . . . ,m). Let Dpr denote the row-degree coefficient matrix of Dp(s, t). For any n×nmatrix pair Fp = diag(Fpi

),Qp = diag(Qpi) where each Fpi

is an nvi×nvi

stable matrix and each (Qpi, Fpi

isan observable pair, there exist Θp1(t),Θp2(t) ∈ <n×m with bounded and differentiable entries having boundedderivatives such that

ωp1 = Fpωp1 + Θp1(t)uωp2 = Fpωp2 + Θp2(t)y (28)

Dpry = Qp(ωp1 + ωp2)

where ωp1 , ωp2 ∈ <n, represents the multivariable LTV system.

For the APPC design in17 , in addition to Assumptions 1–4, the multivariable LTV plant described byone of the three equivalent descriptions (22),(23),(28) is assumed to satisfy the following as well:

Assumption 5 The observability indices nvi (i = 1, . . . , m),the controllability index nr, and the row-degreecoefficient of the left plant PDOM Dp(s, t) are known.

Assumption 6 In the plant description (28) the parameter matrices Θp1(t),Θp2(t) ∈ <n×m can be decom-posed as

Θpi(t) = Πi(t)Θpi(t), i = 1, 2 (29)

where Πi(t) ∈ <n×n are matrices of known, smooth bounded functions with bounded derivatives. Furthermorethe following hold for θpi(t) = vec(Θpi(t)), i.e. the column vector of dimension nm obtained by concatenatingthe columns of Θpi(t):

(i) θpi(t) ∈ C, ∀t ≥ 0, i = 1, 2 for some closed, bounded, convex set C that is known a priori.

(ii) There exist constants c, µ > 0 such that, for all t0, T ≥ 0

∫ t0+T

t0

|θpi(t)|dt ≤ c + µT,

∫ t0+T

t0

|θpi(t)|2dt ≤ c + µT, i = 1, 2

Note that the decomposition (29) helps to exploit the a priori information about the system parametersin the controller design. Within the TV system parameter matrix Θpi(t), it allows the known or structuredcomponent Πpi(t) to be arbitrarily fast varying while requiring only the unstructured component Θpi(t) tobe slowly varying.

The APPC scheme proposed in17 for a multivariable LTV plant with unknown parameters that satisfiesAssumptions 1–6 can be summarized by the following algorithm: At each time instant t ≥ 0;

1. Use a parameter identification algorithm with projection to obtain the estimates θp1(t), θp2(t) ofθp1(t), θp2(t), respectively.

2. Using θp1(t), θp2(t) and (29), obtain the estimates ˆΘpi(t) and Θpi(t) = Πi(t) ˆΘpi(t) (i = 1, 2).

3. Using Θpi(t) instead of Θpi(t) in (28), obtain the left PDOM estimates Np(s, t), Dp(s, t) in the equiv-alent left PDOM fraction description (23).

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4. Substituting Np(s, t), Dp(s, t) in place of Np(s, t), Dp(s, t) in the matching equations (25),(27), obtainthe controller PDOMs P (s, t),Q(s, t) to apply the control law (24).a

5. Implement (24) in the state-space form.

It is established in17 that the above APPC algorithm guarantees the internal stability of the closed-loopsystem, i.e. the boundedness of all the closed-loop signals. It is also demonstrated there that the algorithmhas an acceptable tracking performance.

V.C. Adaptive and Non-Adaptive PPC of Time-Varying AHF Models

The TV models for AHF dynamics derived in Sections III,IV, particularly the TV models for the two casestudies described by (8),(9) and (20),(21) are of the form (22), and hence the non-adaptive and adaptivePPC schemes presented in Sections V.A and V.B can be applied to these models provided that the necessaryplant assumptions are met.

The control design for these models will be straightforward applications of the procedures which is brieflydescribed in Sections V.A and V.B and whose details can be found in the corresponding references16,17 ,once they are described in the left PDOM fractional form (24) and, for APPC, in the state-space parametricform (28). Hence the essential step for PPC design for the AHF dynamics models (8),(9) and (20),(21) isfinding their equivalent left PDOM fractional and state-space parametric descriptions.

VI. Concluding Remarks

In this paper, we have presented a general approach to the AHF control problems based on LTV plantmodels. The proposed approach permits employment of some recent techniques developed for adaptivecontrol of systems with fast varying parameters via exploitation of a priori information about parametervariations. Application of the approach to a recently developed detailed generic hypersonic vehicle conceptmodel, CSULA-GHV, is discussed in detail.

The ongoing relevant studies include validation of the approach for the CSULA-GHV model via detailedsimulations. A future research direction is extension of the approach to certain classes of nonlinear TVmodels as well as cases with measurement noise and disturbances.

Appendix A: Non-Dimensional Coefficients and the Atmospheric Model for theCSULA-GHV

The non-dimensional coefficients for the CSULA-GHV are formulated as follows via fitting the aerody-namic data obtained for the CFD model of the vehicle to a set of second order polynomials:

CL = aTLX

CD = aTDX

CT = aTT X

CM = aTMX

where

X =[1 α δt δe M αδt αδe αM δtM δeM α2 M2

]T

and the vectors

ai for i = L, D, T, M

are the coefficients of the corresponding polynomial. The above equations are valid for α ∈ [−0.0873, 0.0873] (rad),M ∈ [8, 12], δt ∈ [0, 0.3], δe ∈ [−0.3491, 0.3491] (rad).

aThis step is slightly different and in a more basic form in the original design description in17 .

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The atmospheric model is given as

ρ = .00238e−h/24000

a = 8.99× 10−9h2 − 9.16× 10−4h + 996M = V/a

Appendix B: System Matrix Entries for the CSULA-GHV

The explicit expressions for the entries of the system matrices in (18) for a given flight condition ( ortrim point) vector p are as follows:

a11(p) a12(p) a13(p) a14(p) 0a21(p) a22(p) a23(p) a24(p) 0a31(p) a32(p) 0 0 0−a21(p) 0 −a23(p) −a24(p) 1a51(p) 0 a53(p) a54(p) 0

=

∂V fV ∂γfV ∂hfV ∂αfV 0∂V fγ ∂γfγ ∂hfγ ∂αfγ 0∂V fh ∂γfh 0 0 0−∂V fγ 0 −∂hfγ −∂αfγ 1∂V fq 0 ∂hfq ∂αfq 0

∣∣∣∣∣∣∣∣∣∣∣p

b11(p) b12(p)b21(p) b22(p)

0 00 0

b51(p) b52(p)

=

∂δtfV ∂δe

fV

∂δtfγ ∂δe

fγ

0 00 0

∂δtfq ∂δefq

∣∣∣∣∣∣∣∣∣∣∣p

∂V fV =∂V T cosα− ∂V D

m, ∂γfV =

−µ cos α

r2, ∂hfV =

∂hT cosα− ∂hD

m+

2µ sin γ

r3

∂αfV =∂αT cos α− T sin α− ∂αD

m

∂V fγ =(∂V L + ∂V T sin α)

mV− (L + T sinα)

mV 2+

2V 2r cos γ + (µ− V 2r) cos γ

V 2r2

∂γfγ = −∂γfα =µ− V 2r

V r2sin γ

∂hfγ =∂hL + ∂hT sin α

mV+

V 2r cos γ + 2(µ− V 2r) cos γ

V r3, ∂αfγ =

∂αL + ∂αT sinα + T cosα

mV

∂V fh = sin γ, ∂γfh = V cos γ, ∂V fq =∂V Myy

Iyy, ∂γfq =

∂γMyy

Iyy= 0, ∂hfq =

∂hMyy

Iyy

∂αfq =∂αMyy

Iyy, ∂qfq =

∂qMyy

Iyy= 0, ∂δtfV =

∂δtT cosα− ∂δtD

m, ∂δtfγ =

∂δtL + ∂δtT sin α

mV

∂δtfq =∂δtMyy

Iyy, ∂δefV =

∂δeT cos α− ∂δeD

m, ∂δefγ =

∂δeL + ∂δeT sin α

mV, ∂δefq =

∂δeMyy

Iyy

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