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Dynamic inversion: an evolving methodology for flightcontrol designDALE ENNS a , DAN BUGAJSKI a , RUSS HENDRICK a & GUNTER STEIN aa Honeywell Inc, MN-15-2321, SRC, 1625 Zaarthan Avenue, St. Louis Park, MN 55416, U.S.A.Published online: 12 Mar 2007.

To cite this article: DALE ENNS , DAN BUGAJSKI , RUSS HENDRICK & GUNTER STEIN (1994) Dynamic inversion: an evolvingmethodology for flight control design, International Journal of Control, 59:1, 71-91

To link to this article: http://dx.doi.org/10.1080/00207179408923070

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INT. J. CONTROL, 1994, YOLo 59, No.1, 71-91

Dynamic inversion: an evolving methodology for flight controldesign

DALE ENNSt* DAN BUGAJSKIt RUSS HENDRICKt andGUNTER STEINt§

This paper describes nonlinear dynamic inversion as an alternative designmethod for flight controls. The method is illustrated with super-manoeuvringcontrol laws for the F-18 high angle-of-attack research vehicle.

1. IntroductionFlight control systems have evolved dramatically over the past few decades.

They started as limited authority analogue systems, intended to provide a bit ofstability augmentation for otherwise well-behaved airframes. They have evolvedto full-authority digital systems, critical to stability and full envelope perform­ance for otherwise unflyable airframes.

Control laws and design methods for these systems have evolved dramaticallyas well. They have progressed from very simple fixed-form feedback structures(e.g. pitch rate to elevator) with gains tuned by control engineers in flight, tocomplex multivariable feedback laws, designed with modern multivariable toolsthat optimally trade off command responses, disturbance responses, and robust­ness characteristics of the final closed-loop airframe/controller combination.

Today's prevailing paradigm for flight control design is based on thedivide-and-conquer approach common to many complicated engineering tasks.An airframe is first partitioned into many separate operating regimes (flightconditions). For each of these regimes, a linearized dynamic model approxi­mates the airframe well, and the tools of linear control theory can be used todesign individual compensators to satisfy closed-loop specifications. Next, theindividual compensators are 'stitched together' with gain schedules to cover thefull flight envelope. The resulting scheduled control law is then verified withextensive nonlinear simulations and with carefully executed flight tests beforethe design is finalized. While this overall design process is intended to avoidcostly surprises at the last moment, recent experiences with the YF-22, theJAS-39 and other new airplanes demonstrate that we are not always successful.

Over the last decade or so, control researchers have begun to apply analternate methodology to flight control design (Meyer et al . 1984, Lane andStengel 1988, Bugajski and Enns 1992, Bugajski et al. 1992). This alternative isvariously called dynamic inversion (Morton 1987 et al ., Elgersma 1988) orfeedback linearization (Brockett 1978, Hunt and Su 1981, Isidori 1985). We

Received in final revised form 2 November 1992.t Honeywell Inc., MN-15-2321, SRC, 1625 Zaarthan Avenue, St. Louis Park, MN 55416,

U.S.A.:j: Also Adjunct Professor, Aerospace Engineering and Mechanics Department, University

of Minnesota, Minneapolis, Minnesota, U.S.A.§ Also Adjunct Professor, Electrical Engineering and Computer Science, Massachusetts

Institute of Technology, Cambridge, Massachusetts, U.S.A.

0020-7179/94 $10.00 © 1994 Taylor & Francis Ltd

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72 D. Enns et al.

believe that this methodology will eventually replace the divide-and-conquerapproach as the prevailing paradigm. With dynamic inversion, the initial step ofdividing the design problem into separate operating regimes is bypassed.Instead, a nonlinear control law is fashioned which globally reduces thedynamics of selected controlled variables (CVs) to integrators. A closed loopsystem is then designed to make the CVs exhibit specified command responseswhile satisfying the usual disturbance response and robustness requirements forthe overall system and the various physical limitations of the aircraft's controleffectors.

The chief advantage of this emerging alternative is that it avoids thegain-scheduling step of the current method. This step is time consuming, costlyto iterate, and still relies substantially on engineering art. The new alternativealso offers greater generality for re-use across different airframes, greaterflexibility for handling changing models as an airframe evolves during its designcycle, and greater power to address non-standard flight regimes such assupermanoeuvres. These advantages, and some of their costs, will becomeevident in our discussion of the methodology below. We begin in § 2 with areview of the basic principles behind dynamic inversion. We then discuss itscentral step for flight control, namely the choice of specific controlled variables,in § 3. Some implementation issues are described in § 4, a serious designexample for the F-18 HARV follows in § 5, and concluding comments are madein § 6.

2. Basic principles of dynamic inversion

2.1. Aircraft equations of motionRigid body dynamics of aircraft are described globally (over the full-flight

envelope) by a set of 12 nonlinear differential equations, two each for sixdegrees of freedom. We will summarize these equations as follows;

dx-=F(x,u)dt

Y = H(x)

(1)

(2)

where the symbols, F(.,.) and H(.), denote nonlinear functions known to usreasonably accurately as a mix of analytic expressions and tabular data (see forexample, Etkin 1972, McRuer et al . 1973).

The symbol, x, denotes the usual state vector which comprises the followingcomponents:

(a) three rotation rates about body axes (p, q, r),

(b) three attitudes, measured with respect to the airstream, (rr, (3, /1),

(c) three velocity components, described by total velocity, flight path angle,and heading angle, (V, y, X), and

(d) three inertial position coordinates, (X, Y, H).

The symbol, u, denotes the positions of all control effectors. This includesthe usual elevator, aileron, and rudder surface deflections, but it also includesany additional surfaces, such as canards or leading edge devices, forebody

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Dynamic inversion 73

(4)

controls, and any thrust modulation and vectoring capabilities available on theairframe.

Finally, the symbol y denotes selected CVs, variables to be controlled. Thesevariables are discussed at length in § 3.

2.2. Alternative formsBy means of appropriate changes of variables, it is often possible to rewrite

equations (1)-(2) in the following alternate form

dy/dt = Yi

dYi/d t = Yz

(3)

dYrn/dt = /y(x, u)

dz/dt = !z(x, u)

where (y, Yi, yz, ... , Yrn, Z)T = T(x) is a transformed version of the originalstate vector.

To obtain this alternative form formally, we can time-differentiate y(x(t» in(2) and define its derivative expression to be a new function, Yi(X). Next, wetime-differentiate Yi(X(t» and define its derivative expression to be yz(x), thenwe time-differentiate yz(x(t» and define Y3(X), etc. We stop when the controlvariables, u, appear explicitly in the next time-derivative. Then, as a final step,we must complete the state transformation by selecting a function, z(x), whichfills out (Yi(X), yz(x), ... , Ym(x), Z(x»T = T(x) and makes it invertible for allx.

This formal process does not always work for arbitrary nonlinear systems.However, sufficient conditions under which the process is rigorous are known,as are conditions under which the last derivative expression, !r(x, u), turns outto be invertible in u (Hunt et al. 1981, Isidori 1985). Fortunately, for rigid bodyaircraft dynamics, the process always succeeds, and indeed it does so withm = o. This happens because (1)-(2) have CVs for the three pitch, roll and yawaxes, each including a body rate, and the control effectors primarily producetorques in these same three axes. As a result, we get a single three-elementequation

~ = [aH]F(X, u)dt ax T

Furthermore, the right-hand side of this equation turns out, in most cases, to belinear in u, thus yielding the form

~ = f(x) + g(x)u (5)dt

with g(x) invertible for all values of x.

2.3. Globally linearizing controllersGiven (5), it is easy to fashion control laws which reduce the controlled

variables' dynamics to linear ones. For example, the control law

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74 D. Enns et al.

u = g(X)-1 [-[(x) + v]

yields simple integrations for y

~=vdt

(6)

(7)

and it also provides new control variables, v, which will be used later to makeyet) behave as desired. Of course, while these new controls are three-dimen­sional, they are, in general, produced by more than three physical controleffectors. For such cases, g(X)-l in (6) should be interpreted as a (non-unique)right inverse. Potential versions of this inverse are discussed further in § 4.

2.4. Zero dynamicsAt this point, astute readers will have raised at least two major concerns

about the concept.

(1) How good must airframe models be to realize (6) adequately?

(2) What happens to the remaining variables, z , in the new state vectorwhen y is controlled but not z?

The first question deals with robustness of the control law and is addressed indetail in § 3. The second question deals with so-called zero dynamics (Isidori1985) or complementary dynamics (Elgersma 1986) of (1)-(2) and must beaddressed by ensuring that these dynamics are stable and well-behaved.

Conceptually, zero dynamics are nothing more than the remaining motionspermitted by equation (1) when the CVs in (2) are constrained to be constant orprescribed. That is, they are the solutions of

dx = F (x, u) with constraints H (x) = c (8)dt

If F(.,.) and H(.) were linear functions, i.e. F(x, u) = Fx + Gu andH(x) = Hx , then these constrained solutions would be determined by the zerosof system (1)-(2). Specifically, with v = 0 they would satisfy

x( t) = Lajxj exp (Zit) (9)

where a., i = 1, 2, ... , p, are arbitrary constants, and (Zi, Xi)' i = 1, 2, ... , p,are zero/zero-direction pairs, defined by a generalized eigenvalue problem(Rosenbrock 1970).

[

Z .[ - F0= I

H(10)

Note that these constrained solutions are not observable in the outputs, i.e.Hx =0, and they are stable and well-behaved whenever all zeros of the CVs arelocated in the left half-plane and have reasonable damping ratios.

Zero dynamics are simply nonlinear generalizations of these same ideas, andCVs must likewise be chosen to make them stable and well-behaved. We willuse the linear interpretation in § 3 to verify that our selected CVs do indeedhave such properties.

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Dynamic inversion 75

3. Controlled variable selectionsIt is already evident that CVs playa central role in the dynamic inversion

concept. We must actually be concerned with two aspects of CVs. The first isthe choice of the variables themselves, and the second is the closed-loopdynamic characteristic we elect to impress upon them by means of an (as yetundetermined) control law for v. Both aspects influence the quality of finaldesigns. They determine how we satisfy handling quality specifications, how weattenuate disturbance responses, how we get good zero dynamics, and how weget favourable performance/robustness trade-offs for the overall closed loopsystem. We treat each of these considerations in turn.

3.1. Handling quality specificationsOne of the main jobs of a flight control system is to produce good responses

for pilot commands. The characteristics of 'good responses' are well known fromyears of studies, piloted experiments, and flight experience, and are documentedin existing military specifications (MIL-STD-1797A 1987). They usually take theform of equivalent linear system models for the primary commanded variables,with model parameters constrained to fall into specified ranges. The followingtable summarizes CV selections and associated closed-loop dynamic character­istics which provide effective ways to satisfy these specifications.

These recommendations work well for most conventional flight regimes andpiloting tasks. They may need modifications, however, for other situations, suchas high-a and very low speed flight, and for other command modes, such asdirect lift and side-force.

For example, the pitch axis variable, q + nz!Vco , is motivated by thehistorical C*-criterion for conventional flight. It has one free parameter, thecrossover velocity Vco , which must be adjusted to match the separate equivalentsystem requirements for q- and n,-responses in current military specifications.Moreover, its closed-loop response time constant must be selected to satisfyspecific Level 1 flying quality numbers identified in the specifications for theappropriate vehicle class. (The numbers in Table 1 are for fighters.) In order tomake this variable work properly in high-a flight, it is desirable to replace n,with K a (this avoids the destabilizing effect of lift-curve slope reversal) and toadd pitch/Iateral-directional decoupling terms, (g cos (I/» cos «(;I) - Vp)/U. Wewill also add an airspeed term later, in order to fine-tune the zero dynamics.

Similarly, the roll axis variable, p + ar, which represents the stability axisroll rate (roll about the velocity vector), would be replaced by a full nonlinearversion, Ps =cos (a)p + sin(a)r, and the lateral-directional variable, r­ap - gsin(l/»cos «(;I)/V + k{3, which enforces conventional co-ordinated turns

Axis

Pitch

Roll

Yaw

Variable (Yi)

q + -Jv;

p + a:r

r - a:p-gsin(4))cos(II)/V + kf3

Closed-loop response

First order withw~ 5rads- J , r~ 0·2s

Same as pitch

Same as pitch

Table 1. Recommended controlled variables.

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76 D. Enns et al.

with zero sideslip, would include the full nonlinear version of the stability axisyaw rate.

3.2. Disturbance attenuationIn addition to providing good command responses, flight controllers must, of

course, also attenuate undesirable external disturbances. These are causedprimarily by atmospheric turbulence and gusts, and their responses must besuppressed to the point where ride-quality is acceptable for the crew, residualmotions are acceptable for the payload and, in some cases, loads are acceptablefor the structure.

Fortunately, the CV selections in Table 1, when controlled to their recom­mended bandwidth (w), do a generally good gust-suppression job. Issues usuallyarise only in the pitch axis and can be easily addressed by changing Veo (to placemore emphasis on acceleration) and/or by increasing the bandwidth beyond thehandling-quality derived number.

Of course, when these changes are made, the command responses of theclosed-loop system also change. This effect must then be corrected withcommand pre-filters to restore the original responses. We will see shortly thatthere are also other reasons to change the properties of the CV feedback loops.These reasons also call for pre-filters to restore nominal command responses, sothe presence of pre-filters is inevitable.

3.3. Fine-tuned zero dynamicsAs discussed above, one aspect of the dynamic inversion concept is the

presence of hidden zero dynamics. These dynamics are implicitly defined by ourselected CVs, and we must examine them separately to make certain that theyare stable and well-behaved.

While it is far from trivial to establish properties of zero dynamics globally(for an example, see Morton and Enns 1991), local properties obtained fromlinearizations are often enough to identify potential problems. To illustrate this,Table 2 examines the pitch axis CV from Table 1, using a linearized pitch axismodel. The model has typical short period and phugoid modes, with thephugoid stable but lightly damped. The CV has one stable zero in the shortperiod frequency range and two more zeros in the phugoid range. Unfortun­ately, one of the latter is slightly unstable. As a result, dynamic inversioncontrollers based on this CV would destabilize the phugoid motions.

Although the instability would not be severe (time to double ""150 s), it canbe alleviated entirely by adding a small airspeed term to the CV. This option isillustrated as CV' in Table 2. Of course, this modification also requires that weinclude an airspeed trim term (in addition to the usual av; trim term) in orderto command equilibrium flight. This additional trim term can consist oflow-passed actual airspeed, or it ' can be avoided entirely by high-passingairspeed in CV'.

Similar analyses for the roll and lateral-directional CVs in Table 1 show nolocal zero dynamics problems, and simulations studies described later in § 5provide high confidence that none exist globally.

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Dynamic inversion 77

State-space model

V a q g ddV/dt -8·29OOE - 03 -2·9680E -t- 01 9·8570E - 02 -3·2170E -t- 01 -3·5080E - 02da/dt -1·63OOE - 04 -1·l300E +. 00 9·8840E - 01 3-6380E - 06 -1·3650E - 03dq/dt -7·5080E - 05 -4·9820E +. 00 -5·0880E - 01 -9·6620E - 07 -8·6590E - 02dg/dt 1·6300E - 04 1·l3ooE +. 00 1·1580E - 02 -3·6380E - 06 1.3650E - 03

Poles

Real Imaginary Magnitude Damping

-3·3222E - 03 ±6·5146E - 02 6·5231E - 02 5·0930E - 02-8·2022E - 01 ±2·1966E +. 00 2·3448E +. 00 3·4981E - 01

CV

2·7I60E - 04 1·7384E +. 00 1·oo36E +. 00 O·OooOE - 00 O·OOooE - 00

Real Imaginary Magnitude Damping

Zeros { 4·6836E - 03 O.OOooE - 01 4·6836E - 03 -I·OOOOE -l- 00ofCV -1'0525E - 02 O·ooOOE - 01 1·0525E - 02 1·0000E +. 00

-2·7058E +. 00 O.ooooE - 01 2·7058E +. 00 1·0000E +. 00

CV'

O·OOOOE - 00 1·7384E -t- 00 l-0036E -t- 00 O·OOOOE - 00 O'OOOOE - 00

Real Imaginary Magnitude Damping

Zeros { -3·6455E - 03 5·6852E - 02 5·6969E - 02 6·3991E - 02ofCV' -3·6455E - 03 -5·6852E - 02 5·6969E - 02 6·3991E - 02

-2·7045E +. 00 O·OOOOE - 01 2·7045E +. 00 1·0000E +. 00

Table 2. Local properties of pitch zero dynamics F-18 HARV, Mach = 0,65, H = 16000 ft.

3.4. Robustness propertiesOf course, good nominal responses and well-behaved zero dynamics are not

enough. These qualities must also be robust with respect to various modellingerrors inherent in aircraft systems. Unfortunately, robustness properties ofdynamic inversion have received too little attention in the literature so far (somereferences are Spong and Vidyasagar 1987, Kravaris 1987, Akhrif and Blanken­ship 1988). We summarize some of the known results below. Much strongerresults will be needed to encourage broader use and acceptance of themethodology.

The major dynamic inversion robustness issues are exhibited by replacing thenominal aircraft model in (5) with a perturbed model, i.e.

~ = (f +. Df) -t- (g -t- Dg)udt

(11)

and also replacing the ideal control effector position in (6) with a perturbedvalue obtained by passing the ideal position through actuator dynamics, flexiblestructural elements, and other high-frequency uncertainties, i.e.

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78 D. Enns et aI.

u = (I + 6)g(x)-1(-f(x) + v) (12)

where 6(s) is an arbitrary stable dynamic perturbation, small for low frequencysignals, but increasing in size to unity and beyond as frequency increases. Ashort derivation shows that the resulting dynamic model for y is then given by

with

~ = (of - Df) + (I + D)vdt

(13)

(14)

These equations replace the integrators in (7) as a new dynamic model for ourCYs. Note that there are two major uncertainty terms. The first term,(of - Df), is a direct disturbance input to the integrators, while the secondterm, (I + D)v, is a multiplicative perturbation on the control inputs of theintegrators. Both terms are correlated through their common perturbationoperator D, and all functions in this operator, f, g, Of and Sg ; remaindependent on the state vector, x = r-1(y, z).

3.4.1. The linear case. Note that the new model is still almost linear. Only theperturbation terms are not linear. If we ignore this fact for the moment, thereare well-established design methods available to construct robust controllers.Perhaps the simplest of these are loop-shaping methods which satisfy norm­based robustness constraints on the CY-feedback loops (Doyle and Stein 1981).We have two basic constraints.

(1) Sufficient conditions for robust stability with respect to the multiplicativeterm alone

0max[(I + K(s)P(sn- J K(s)P(s)] < 1 for all s = jw (15)omax[Dm(s)]

(2) Sufficient condition for robust stability with respect to the direct disturb­ance term alone

0min[(I + K(s)P(s»] > omax[Da(s)P(s)] for all s = jw (16)

In these expressions, P(s) = lis is the nominal plant, K(s) is the feedbackcompensator for the CY-loop, and omaxl.] and Omin['] denote the largest andsmallest singular values, respectively, of their matrix arguments.

The symbols Dm(s) and Da(s) in (15)-(16) are slightly modified versions ofthe multiplicative and direct disturbance terms. These modifications arise whenwe re-write the direct disturbances as follows.

(of - Df) = (6[ - D)f(x)

= (6[ - D)(Fyyy + Fy,z)

= (6[ - D)(M(s)y + N(s)v) (17)

Here, of(x) is represented by a multiplicative dynamic error on f(x) (i.e.Of = 6jf), and M(s) and N(s) are transfer matrices obtained from the zerodynamics. The latter can be derived by substituting the linear version of (6) intothe linear version of (1)-(2), expressed in (y, z) coordinates. This gives a

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Dynamic inversion 79

state-space representation for z , with transfer matrices, M(s) = fly andN(s) = flv

dz -I ( G-1 ) G G-1- = (Fzz - GzG y Fyz)z + Fzy - G, y Fyy y + z y vdt

f = FyZz + FyyY (18)

The y-dependent part of expression (17) is now defined as DaY and thev-dependent part is lumped together with D to form D m , i.e.

D; = (""r - D)M(s) and Dm = D(I - N(s)) + ""rN(s) (19)

Note that conditions (15)-(16) constrain two frequency extremes of the CVfeedback loop. High-frequency constraints are imposed by the multiplicativeterm, which requires that the loop be rolled off before the magnitude of Dm(s)exceeds unity. Low-frequency constraints are imposed by the direct disturbanceterm, which calls for sufficient low-frequency gain to overpower any destabiliz­ing effects of Da(s). Combining these two extremes gives us a familiar loop-shaping requirements plot illustrated in Fig. 1. .

Whenever the above loop-shape constraints are widely separated, simple CVcompensators suffice. A typical example is the 5 rad S-I proportional-plus­integral design in Fig. 2, which readily satisfies Fig. 1. It also includesfeedforwards around the integrator (pre-filters) to restore first-order commandresponses.

In more challenging situations, when the frequency-domain constraints aretight, we can no longer ignore the fact that the direct disturbance andmultiplicative perturbations occur together and in a correlated way. Any of thesophisticated modern multivariable design tools could then be brought to bear toexecute the design. Arguably the most powerful of these is the It-synthesismethod (Stein and Doyle 1991, Packard et al. 1993).

Magnitude of II D II Frequency DomainConstraints

0.1.01 0.1 1.0 10.

log frequency 0.1

lJl.

1.0

~i I

UpJlOfBoun(\for (\+KP)·lKP,

V-r-, ~.......<, <,

.Lower :rn_~ for (\~I~>.JI

]'§,~

~100•

V.-/

10.

1.0

.01 0.1 1.0 10. 100.

Figure 1. Loop-shape requirements for F-l8 HARV model in Table 2.

CVandpes)

cv

Figure 2. A simple CV control law.

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80 D. Enns et al.

(21)

An informative special correlated case arises if we set of = 0 and keep onlythe perturbation, D, in (13). A short derivation shows that the following singlecondition then assures stability robustness

Gmax[(I + KP)-l KP] < 1 for all s = jw (20)GmaxlD(I - N + K- 1M)]

Now compare (15) and (20), using (19) for Dm and Ae= O. The right-hand sideis modified by the term K- 1M. This term makes the effective multiplicativeperturbation larger or smaller, depending upon details of the plant. Its effect iseasy to see for scalar plants. For stable scalar plants, K- 1M will be negative,reducing the effective perturbation and permitting larger bandwidth in theCY-Ioop. On the other hand, for unstable scalar plants, K-1M will be positive,increasing the effective perturbation and reducing the allowed bandwidth. Thishappens because the total 'inner loop' bandwidth (the bandwidth used for bothfeedback linearization (6) and CY control) has a maximum imposed by D. Inthe unstable case, a part of this maximum is already used by (6) to reduce theplant to an integrator, and only the remaining part is available to control theCY. (Indeed, for sufficiently large open-loop instabilities, the remaining partmay not exist, indicating that (6) itself is unstable in the presence of D.)

3.4.2. The nonlinear case. Of course, no matter which design tools or conditionswe choose, a critical assumption is still linearity. When this assumption isremoved, the formal design options are much less powerful. This happensbecause formal norm-based robustness conditions for general nonlinear perturb­ations are blunt instruments. Consider the following generalizations of (15) and(16), derived from the small gain theorem (Desoer and Yidyasagar 1975)

(1) Sufficient condition for robust stability with respect to the nonlinearmultiplicative term alone

11(1 + KP)-l KPII < _1_IIDml1

(2) Sufficient condition for robust stability with respect to the nonlineardirect disturbance term alone

11(1 + KP)-l PII < _1_ (22)IIDal1

. Here the symbol 11·11 denotes an induced operator norm. This norm correspondsto the largest input/output gain of the operator taken over all signals. If theoperator is linear, the norm is equal to the largest magnitude of its frequencyresponse taken over all frequencies. Thus, the left-hand side of (21) is nothingmore than the M-peak of the CY-loop's closed loop transfer function.

On the other hand, the right-hand side of (21) is more complicated. Toexamine this term, consider an 'optimistic' version of Dm , obtained by setting Ogto zero and ignoring the'N v' term in the nonlinear analogue of (17). From (14),the perturbation is then given by

Dm = g(x)Ag(X)-1 (23)

and it is not difficult to show that the norm of this operator satisfies thefollowing inequalities

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Dynamic inversion 81

sup {amax[g(X)] }II~II"; IIDmll.,; {~uPx amax[g(X»)}II~11 (24)x amin[g(X») inf, amin[g(X»)

The left-hand side follows from the linear robustness literature: simply treatg(x) as a constant matrix with x fixed to produce the worst case conditionnumber, K(x) = amax/amin' The right-hand side follows by letting x vary strate­gically in relation to the dynamics of ~: first, pass signals through g(X)-l with xchosen to maximize the gain, then pass the resulting signals through a ~ whichdelays them and rotates them into the worst direction of g(x), and finally passthose delayed signals through g(x) with x chosen (after the delay) to maximizethe gain of g(x). (These arguments show that the inequalities are, in fact, tightfor the two extremes.) The norm II~II on both sides of (24) is equal to theM-peak of the (linear) actuator/flex perturbation in the control channels of theaircraft. This norm is typically much greater than unity.

The bottom line of all this is that (19) can be satisfied only if we make theM-peak of the CV-Ioop much smaller than unity. This requires very small loopgains, i.e. IK(s)p(s)1 « 1 for all s = jw. Unfortunately, such gains cannot satisfy(20) which requires large loop gains to make the M-peak of (1 + KP)-l P smallenough. Thus, the formal robustness guarantees of the nonlinear theory cur­rently do not help us in design.

There are several avenues of research which promise to alleviate the currentlimitations of the nonlinear theory. One approach is to reduce conservatism in(19)-(20) with weighting functions (Safonov 1980). For example, (19) can bereplaced by

11(1 + KP)-I KPwl1 < 1 (25)IIW-1Dmll

where W(s) is any stable and stably invertible linear operator. This condition isstill sufficient for robust stability with respect to D m . Now let W(s) be chosencleverly, such that the input/output gain of W- 1Dm is nearly unity for all signals,not large for some signals and small for others. Then (25) would reduce to amuch less conservative condition, similar to (15), e.g.

11(1 + KP)-l KPWII < 1

¢> amax[(I + K(s)p(s»-l K(s)P(s») < 1 for all s = jw (26)amax[W(s»)

Similar weighted versions are also possible for (20).In the absence of such improved tests, we are left for now with the linear

design methods described earlier, and we are obliged to verify robustnessproperties after the fact via overall system linearizations and analyses and vianonlinear simulations. Some results using these interim methods are discussed in§ 5.

4. Implementation issuesThe final control laws obtained from dynamic inversion are summarized in

Fig. 3. They consist of a nonlinear block which performs the feedbacklinearization and a CV controller block which implements the CV-compensator.Some key aspects of this implementation include on-board models, full state

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82 D. Enns et al.

feedback, and aircraft-unique surface allocation and limiting logic. We deal withthese three topics briefly below.

4.1. On-board modelsAs shown in Fig. 3, dynamic inversion controllers require aircraft models

stored on-board to implement the functions g(X)-1 and f(x). In the past, thememory needed for these functions and the computing time needed to evaluatethem at inner loop update rates have been prohibitive. Modern flight computertechnology has all but eliminated these prohibitions.

For the F-18 HARV control design used in § 5, for example, the completecontrol law requires approximately 1200 floating point memory locations, ofwhich 900 are dedicated to the aircraft model. This covers a limited flightenvelope (Mach < 0·7, 15,000 < H < 45,000 feet). Supersonic flight regimes willrequire additional memory.

The current control law software for the HARV is still in developmentalform, so timing estimates are somewhat premature. However, the code runs inreal-time on a SPARC processor in NASA's fixed-base piloted simulator. It hasexecution times consistent with the HARV's 80 Hz inner loop update rate. Ofthe 12·5 ms available, approximately 9·5 are due to the CV-compensator,feedback linearizer, formation of the CV and pre-filter. This equates approxi­mately to 2500 indexed floating point multiply-add operations.

4.2. Full-state measurementsAnother obstacle from the past is the availability of measurements for the

complete state vector. Most modern aircraft now carry a full complement ofsensors, from inner loop rate gyros to complete navigators providing inertialorientation, velocity and position. Moreover, with multi-channel redundancy andother fault-tolerant configurations, these measurements are increasingly availablein flight-critical form. The only persisting sensing issues revolve around air­stream-relative measurements, i.e. true airspeed, angle-of-attack and angle-of­sideslip. Currently, observer-based blends of inertial measurements and air-dataprovide the preferred solution for these signals.

CV·Compcnsatot FeedbackLinearizer

Figure 3. Full dynamic inversion control laws.

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Dynamic inversion 83

4.3. Surface allocation and limitingAs noted earlier, many aircraft have more than three surfaces available to

produce the desired dy/dt in (5). There are many ways to build g(x)-I to utilizethis redundancy, and no one way is necessarily best for every aircraft and everymission. The approach we describe here and, utilize in § 5, is one historicaloption. It is the so-called daisy chain (Bugajski et al. 1992). This approach isillustrated in Fig. 4. It designates primary effectors for each of the three aircraftaxes, and treats other controls as auxiliary effectors to be used only when theprimaries saturate. Generally, the primaries will be the standard elevator,aileron and rudder aero-surfaces, or combinations of other surfaces slavedtogether to provide equivalent moments. The auxiliaries are whatever else isavailable on the aircraft, such as thrust vectoring vanes on the F-18 HARV.

The logic for switching from primaries to auxiliaries is illustrated in Fig. 4.Basically, the primaries are always asked to produce the commanded moments.However, their predicted output, subject to rate limits and saturations, issubtracted from the command and the difference is passed on to the auxiliarieswhenever the primaries are not powerful enough. In this way, primary effectorsare active full-time, while auxiliaries are only active when they are needed.

Of course, it can also happen that even the primaries and auxiliaries togetherare too weak to generate the desired moments. In this case, there is noalternative but to lower the commanded dy/dt in the affected axes. It is thenalso necessary to include anti-windup logic on the corresponding integrators inthe CV-control law. As another difficulty, the inverse, gp(x)-I, in the primarychannel of Fig, 3 can become poorly conditioned for some x. It is thennecessary to use a modified inverse, such as (gp(x) + e1)-I. These 'fixes' areincorporated in the super-manoeuvring simulations shown in the next section.

5. An F-18 HARV design example5.1. The F-18 high angle-of-attack research vehicle

The National Aeronautics and Space Administration (NASA) is conductingan on-going high angle-of-attack flight research programme. This programmeuses an F-18 testbed aircraft, selected in part because the production version ofthis aircraft has exceptional angle-of-attack capability (> 55°), recovers well fromspins, has excellent engine performance at high angle-of-attack, and has digitalflight controls. An aircraft was obtained by NASA from the US Navy andmodified in the following ways to become the HARV.

RateLimits Deflection Limits

Auxiliarynand

Primary

1-----1=t= f---

Figure 4. 'Daisy chain' logic for redundant control effectors.

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84 D. Enns et al.

(a) First, the testbed was instrumented: two elementry systems, five videocameras, two still cameras, and various other data-taking devices.

(b) Second, the two engine nozzles were replaced with an asymmetricarrangement of six thrust vectoring vanes; three per engine. Thrustvectoring is provided by deflecting the vanes into the engine exhaustplume.

(c) Third, the research flight control system (RFCS) was installed. Thiscontrol system architecture retains the production F-18's flight controlsystem, but adds another processor (Ada programmable) to store andexecute the research flight control laws. The two systems communicatethrough dual port RAM, and any failure in the research control lawscauses a reversion to the production control laws. This architecturefacilitates the development and flight testing of various flight controllaws.

A photograph of the final modified vehicle is shown in Fig. 5.For our purposes, the aircraft has 13 control effectors: ten aerodynamic

surfaces and three axes of thrust vectoring. These effectors are slaved togetherin various ':\'ays to provide the following effective control inputs.

(a) Pich axis: two horizontal stabilators slaved together for primary (aero­dynamic) control. Thrust vectoring for auxiliary control.

(b) Lateral/directional axes: two slaved aileron surfaces, twin slaved verticaltails, and two differentially deflected stabilators for primary control.Roll and yaw thrust vectoring for auxiliary control.

Note that since the F-18 is a twin engine aircraft, some roll vectoring canalso be obtained. Pairs of leading and trailing edge flaps follow the productionF-18 control law.

A complete set of sensors is available, with air-mass measurements providedby a blend of inertial and air-data sensors, as previously discussed.

The dynamic inversion design methodology described in § 2 to § 4 has been

FigureS. The F-18 HARV.

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Dynamic inversion 85

used to design control laws for this research vehicle. These control laws arecurrently being evaluated on piloted simulators at NASA's Dryden FlightResearch Facility. They are scheduled for flight evaluation in 1994. Some linearanalyses conducted to support these designs and a typical supermanoeuvresimulation to demonstrate their global performance are described below.

5.2. Linear analysesAs discussed earlier, the mathematics supporting dynamic inversion do not

yet provide effective tools to assure stability and performance robustness fornonlinear perturbations. In the absence of such tools, we rely upon linearanalyses of the overall system, plant and controller in closed loop, linearizedabout various operating points. This subsection summarizes some of theseanalyses.

Analyses were performed at a variety of flight conditions whose angles-of­attack ranged from 2° to 80° and whose dynamic pressures ranged from 34 to390 psf. Lateral/directional and longitudinal axes were analysed separatelydespite some known aerodynamic asymmetries above £l' =40. The daisy-chainmethod of allocating control effectiveness (§ 4) was not employed for the linearanalyses. Instead, multiple control effectors (aero surfaces and thrust vectoring)were employed in a manner which minimizes control deflections (Snell 1991).This change does not affect the stability and robustness results which follow.The closed loop systems for each axis included the control laws, rigid bodyaircraft dynamics, 30 rad s"! first-order actuators, and 50 ms time delays thatapproximate the delays observed in the HARV's real-time simulator.

Results of the linear analyses are summarized in Table 3. This table includesfour parameters for both the pitch and lateral/directional axes. The parametersare

(1) minimum single loop-at-a-time phase margins across flight conditions forloops broken at inputs and outputs of the controller,

(2) typical crossover frequencies,

(3) robust stability margins for unstructured uncertainty at the aircraft'sactuator inputs, and

(4) robust stability margins for diagonally structured uncertainty at theaircraft's sensor outputs.

The first two parameters are obtained from classical frequency response

Minimum phase Typical crossover Peak amax at Peak I' at sensorAxes margin frequency actuator inputs outputs

Pitch =55' =5 rad s" <2·0 < 2·0 in shortperiod range

< 15 in phgoidrange

Lateral/ =48' =6·5 rad s"! <3·0 <3·0directional

Table 3. Linear analysis results.

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86 D. Enns et al.

analyses. The third is obtained from the peak magnitude of the maximumsingular value, 0max[(I + KP)-l KP], across frequency (analogous to (15», andthe fourth is obtained from the peak magnitude of the structured singular valueJ1[ PK(I + PK)-l] across frequency (Doyle 1982). Peak magnitudes of 2 for 0max

or J1 indicate that normalized modelling errors of size 1/2, or 50%, can betolerated without instability, while peak magnitudes of 3 indicate that 33%modelling errors can be tolerated without instability.

In addition to these parameters, we also examined nominal closed loop polesover the flight envelope. For pitch, the short period poles are well damped, butmany of the high-a flight conditions have unstable phugoids (resulting fromunstable zero dynamics in need of fine-tuning (§ 3». These phugoid sensitivitiesare also evident in Table 3. For lateral directional, all dutch roll and othercomplex poles are stable and well damped (~> 0·4).

The results in Table 3 and the analyses of closed loop poles verify that thecontrol laws have generally good small signal feedback properties.

5.3. A simulated supermanoeuvreA high angle-of-attack, high angular rate, Herbst manoeuvre (Well et al.

1982, Herbst 1980) was simulated to demonstrate the performance of thedynamic inversion control laws over wider operating regimes. Since the controllaws are based on CVs for piloted operation (as described in § 3) the manoeuvrewas executed with an additional outer loop commanding the normal pilot inputs(i.e. longitudinal stick, lateral stick, rudder pedals, and power lever). Detaileddescriptions of the manoeuvre and the various aircraft responses are givenbelow. They demonstrate that the control laws handle the aircraft well, evenwhen the aircraft is pushed to very high angles-of-attack and angular rates andto the limits of available control authority.

5.3.1. The manoeuvre. The intent of a Herbst manoeuvre is to reverse aircraftvelocity in minimum time. The manoeuvre begins in straight-and-level flight andproceeds into a full-afterburner climb with longitudinal stick full aft. During theclimb, the aircraft develops very large angles of attack and reduces airspeeddramatically. The climb eventually stops and is followed by a rapid descent backto the aircraft's initial altitude and speed. However, near the top of the climb atvery low airspeed, lateral stick and rudder pedal inputs are applied to reversethe heading and body attitude by 1800

, so the descent occurs back along nearlythe same path as the climb. This returns the aircraft approximately to the samepoint in space where the manoeuvre started but heading 'the other way'.

5.3.2. Detailed aircraft responses. Selected response variables of the F-18HARV executing such a supermanoeuvre are summarized in Fig. 6. The basictrajectory is shown in Fig. 6(e), where altitude and east position are plottedversus north position. At the beginning of the manoeuvre, the aircraft is flyingnorth, and at the end of the manoeuvre it is flying south. The entire trajectorystays within 1300 ft of the north-south vertical plane. Note how the downwardand upward portions of the trajectory are nearly coincident. The flight pathangle reaches a maximum of 860 near the top of the manoeuvre and a minimumof -570 during the dive as shown in Fig. 6(d). The heading is reversed by 1800

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Dynamic inversion 87

as shown in Fig. 6(/), near the top of the manoeuvre when the velocity is lessthan 100 fts- I .

Angle-of-attack, shown in Fig. 6(b), reaches 92° about 8 s after the man­oeuvre is initiated. The fuselage has pitched nose up through 150° during theinitial 8 s. At 6·1 s, the nose is straight up. After this, the vertical tail pointsdown and the aircraft rolls to wings-level and simultaneously pitches to nearnose-level attitude. This can be seen in Fig. 6(m), where the aircraft's Eulerangles for pitch, roll and yaw are shown (by convention, the pitch Euler angle isbetween -90° and 90°). During the lateral transient to reverse the body attitudeand velocity direction, the angle-of-attack decreases to a minimum of 7° andthen increases to the value for maximum lift coefficient for the pullout from thedive. The corresponding pitch rate, shown in Fig. 6(c), is between -30 deg s"and 30 deg s" throughout the manoeuvre.

Normal acceleration, shown in Fig. 6(g), is less than 4·7 g for the initialpull-up, is zero at the top of the manoeuvre when velocity in Fig. 6(a), anddynamic pressure, in Fig. 6(1), are less than 20 fts- I and 0·3 psf, respectively.Acceleration then increases to 1·5 g during the final pull-out from the steepdive. Throughout the manoeuvre the lateral acceleration, shown in Fig. 6(g), isless than 0·11 g despite the large sideslip, shown in Fig. 6(i), which occurs whenthe airspeed is negligible. The 0·11 g lateral acceleration is a direct result of yawthrust vectoring, shown in Fig. 6(p). Sideslip angle is less than 2° when dynamicpressure is larger than 50 psf.

Lateral stick and rudder pedal inputs are first issued 11 s after the man­oeuvre is initiated. These inputs lead to the transients in roll rate and yaw rateshown in Fig. 6(j). These rates have magnitudes of about 10 deg s"! whenangle-of-attack is between 60° to 70° and increase in magnitude to 65 deg s"? forroll rate and 28 deg s"! for yaw rate during the rapid decrease in angle-of-attackfrom 60° to 7°. These are considered to be high angular rates for this low-speed,high angle-of-attack condition.

The lateral-directional inputs roll the aircraft 140° about the velocity vectorduring the period when velocity is less than 100 ft S-I. This roll angle (about thevelocity vector) is shown in Fig. 6(k). The heading overshoots 180° and it isnecessary to roll to -30° to return to the north-south plane in which themanoeuvre originated. The aircraft Euler angles for roll and yaw are shown inFig. 6(m), where it can be seen that the body attitude has been reversed by 180°in yaw and is close to a normal wings level attitude at the end of themanoeuvre.

Horizontal tail and pitch thrust vectoring are shown in Fig. 6(h). Thehorizontal tail and pitch vectoring reach position limits during the initial pull-upand stay on the limits for about 2 s. A demand for nose-down moment thenoccurs which puts both effectors on the positive rate limit. Nose up is thenrequired and the controls are rate limited in the opposite direction. For theremainder of the manoeuvre these controls are neither rate or position limited.Note that the daisy chain does not call for pitch thrust vectoring until thehorizontal tail limits in the initial pull-up. Likewise, when dynamic pressure hasincreased to 50 psf and (l' is about 42° during the pullout from the dive, thehorizontal tail is sufficiently effective and no pitch thrust vectoring is employed.

The rudder control surface and the yaw thrust vectoring are shown in Fig.6(p). They are zero until 11 s after the manoeuvre is initiated, when the reversal

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88 D. Enns et al.

v"

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"

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"'\ /

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\ --../V

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\;II

" 20 20 " so 0

a

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/ -," 20 0 " sc 0

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Figure 6. Herbst manoeuvre traces.

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Dynamic inversion 89

~.-r, t:..

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90 D. Enns et al.

is commanded. At this time, the angle-of-attack is 600 and the rudder is not very. effective. Thus, the daisy chain calls for yaw thrust vectoring. Since the daisychain gives priority to pitch vectoring, the yaw thrust vectoring limits briefly forabout 1 s, in the nose right direction and then for about 2 s in the nose leftdirection. Subsequently, the yaw thrust vectoring limits on both sides once morebefore dynamic pressure increases and angle-of-attack decreases enough for therudder to become sufficiently effective again. During the low velocity period,the rudder changes from positive to negative position limits, but at the end ofthe manoeuvre the rudder is well behaved while completing the turn, and yawthrust vectoring is not employed.

There are two aerodynamic controls for roll, namely the aileron and therolling (or differential) tail. Since there are two engines, there is also a thrustvectoring roll control based on differential pitch thrust vectoring. The aileronand differential tail control surfaces are shown in Fig. 6(n), and the roll thrustvectoring angle is shown in Fig. 6(0). The rolling tail is only used to the extentthat it does not interfere with tail deflections needed for pitch control. When thereversal is commanded near the top of the manoeuvre, the aerodynamic surfacesare not sufficiently effective and the daisy chain calls for roll thrust vectoring.Since roll is last in priority (with pitch first and yaw second), the desireddynamics for roll cannot be satisfied for most of the 10 s following the reversalcommand. At the end of the manoeuvre, the thrust vectoring is no longeremployed because aerodynamic controls for roll are sufficiently effective.

The various traces in Fig. 6 demonstrate that the control laws handle theaircraft well throughout the manoeuvre. This is true even at very high angles-of­attack and high angular rates, and also when commands exceed the total controlauthority available on the aircraft.

6. ConclusionsWe have tried to demonstrate in this paper that dynamic inversion offers a

potentially powerful alternative design methodology for flight control. Dynamicinversion avoids gain schedules. Instead, it uses on-board dynamic models andfull-state feedback to globally linearize dynamics of selected controlled variables.Simple controllers can then be designed to regulate these variables withdesirable closed loop dynamics. The variables themselves and their closed loopdynamics must be selected to meet handling quality specifications, to satisfy gustresponse requirements, to exhibit well-behaved zero dynamics, and to achievegood robustness properties for the overall system.

Each of these aspects was discussed in the paper and illustrated with a designfor the F-I8 HARV. While these discussions showed that the method's currentstatus is already adequate for serious designs, the method will benefit substan­tially from additional research developments, particularly in areas of nonlinearzero dynamics and nonlinear robustness.

Because of its general control law structure, dynamic inversion is well suitedfor control law re-use across different airframes, for easy control law updates asairframe models change, and for non-standard flight applications such assupermanoeuvring. This power is possible because modern flight computerhardware and instrumentation make on-board models and full-state feedback nolonger prohibitive.

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Dynamic inversion 91

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