Improved Robustness for Dynamic Inversion Based Nonlinear Flight Control Laws

Copyright te IFAC Automatic Control in Aerospace. Palo Alto. Califomia. USA . 1994

IMPROVED ROBUSTNESS FOR DYNAMIC INVERSION BASED NONLINEAR FLIGHT CONTROL LAWS S.A.SNELL* and P.W.STOUT* * University of California at Davis. Department of Mechanical and Aeronautical Engineering. Davis. CA 95616. U.SA A bstract : Using a simple analysis it is shown that modifications to the desired, linear, closed-loop dynamics of a non linear dynamic inversion control law can improve the performance in the presence of static uncertainties . The idea is validated by designing and simulating a control law for post stall flight of a high performance aircraft with substantial uncertainties in the aerodynamic database . Keywords : Nonlinear control systems ; aerospace control ; robustness ;

1. INTRODUCTION

First-order dynamic inversion is applied to a system in which the inputs, u, act directly on the first derivative of the controlled variable, x, as follows:

S ne 11 et al (1992) presented the design of a nonlinear control law for post-stall flight control of a high performance X 31 type aircraft with thrust vectoring control (TVC). The design was accomplished using nonlinear dynamic inversion, which has been discussed in references by Morton et al (1987), Hauser et al (1988) and Menon et al (1987), among others . These references emphasize using the technique to construct maneuver generators to permit exact tracking of specific trajectories given exact knowledge of the aircraft model. Snell et al (1992) demonstrated that dynamic inversion could also be used to design on board control laws which yield accurate control of the angle-of-attack, a, the sideslip angle ~, and the rate of the velocity bank angle, Il . The control law used a two-timescale approach. An outer loop control law incorporating a first-order inversion uses the body-axis, angular

it ;:: f(x) + g(x)u

(1)

Dynamic inversion sets u according to equation (2) which yields the desired it exactly.

(2) However, the desired dynamics are only given if the inversion is exact and this can occur only with exact knowledge of f(x) and g(x). Snell (1992) carried out a preliminary investigation of the robustness of dynamic inversion by perturbing the total aerodynamic coefficients according to the schedules: CLactual (a) =(0.8 - 0.5 a ) CLnominal (a) CDactual (a) =(1.2 + 0.5 a ) CDnominal (a) CYactual (a) = (1 .2 + 0.5 a) CYnominal (a) Clactual (a) = (0.8 - 0.5 a ) Cl nominal (a) Cmactual (a) = (0.8 - 0.5 a ) Cmnominal (a) Cnactual (a) = (0.8 - 0.5 a ) Cnnominal (a)

rates, p, q and r to control A., pand~, while a set of three inner loops was used to accurately produce the commanded p, q and r by carrying out another firstorder inversion relating p q and r to the control surface deflections. The inner loops also incorporated a means of apportioning TVC and conventional aerodynamic surface deflections in such a way that control usage was minimized. Bugajski and Enns (1992) and Buffington et al (1993) also present dynamic inversion control. They employ an alternative control apportioning strategy based on "daisy-chaining", whereby TVC is not used until the aerodynamic surfaces reach their deflection limits. While Snell et al (1992) demonstrated that the performance of the dynamic inversion conlrOl system was very promising, tlle simulations presented assumed that the aircraft could be mode led exactly. This is not the case in practice so here we investigate how the closed-loop aircraft would perform in the presence of perturbations to the databa.<;e.

(3)

where a is the angle of attack, measured in radians. It was also noted that perturbations of the term g(x) in equation (1) may have a significant effect on performance. Examination of aircraft equations of motion shows that g(x) is directly related to the derivatives Cm dc, Clda, Cldr, Cnda and Cndr. This paper examines the effect of static perturbations of the three important terms, Cmdc, Clda and Cndr. These terms are reduced to 50% of their nominal values. Static perturbations are those which are simply functions of the states x alone and are not dependent on past history. If the perturbations have explicit time dependence, then they are tenned dynamic perturbations.

261

_~ (x )f(x ) _ ~g'(XO) (M(xo}-~g(xo)f(xo» g 0 0 l+~g(xo) (12)

2. ROBUSlNESS AND PERFORMANCE OF A FIRST-ORDER INVERSION CONTROLLER Consider the nominal, first-order system in equation (1). Now say it is desired to track a command Xc such that the closed loop response is given by:

The linear terms dominate, at least in a neighborhood of xo because the basic control given by equation (2) has at least approximately cancelled the major nonlinearities of the dynamics, leaving only the small residual nonlinearities in equation (8). To ensure stability of the linearized dynamics we require al <0. First consider the system with ~g(xo)=O. The requirement for stability of the linear dynamics is simply that ~f~f for all M in the set of possible perturbations. When ~g~ the situation is more complicated. The following criteria should be met to ensure the stability of the linear dynamics in (11):

(4)

Then using (2) :

(5)

u = g(x)-l (-f(x) +wC
However, now assume that the actual dynamics are given by the following : x = f(x) + M(x) + (1 +

~g(x»

g(x) u

(6)

Then the input u given by (5) above will produce the closed loop dynamics: x = f(x) + M(x) + ( I + ~g(x) )(wc(xc-x)-f(x»

Wc >0

(7)

l-~g

>0

nominal stability

(13)

~g to

(14)

for all

be considered

which simplifies to: x = (M(x) -

(15)

~g(x)f(x»

+ (1 +~g(x) ) wc(xc-x)

(8)

In essence (14) means that g(x) cannot change sign, which is a reasonable constraint. (15) requires that the gain Wc be sufficiently large that it exceeds any destabilizing uncertainties in M and ~g . Thus, to provide good tracking and robustness to the static perturbations M and ~g it is desirable to make Wc as large as practicable.

For zero steady-state errors we require: x = 0 if and only if X = Xc

(9)

But if M(x)-~g(x)f(x)~ then when x=x c, x~. This means that the system will operate with steady state errors which will have to be compensated by an xdependent offset of the control input. Since the perturbations ~f(x) and ~g(x) are not known, the controller, which is designed for the nominal system, must be made sufficiently robust that it can provide acceptable performance with any perturbations likely to be encountered. The equilibrium manifold for this system can be

4. PROPORTIONAL + INTEGRAL FEEDBACK

It is a fact that integral feedback can be applied to linear systems to eliminate steady-state errors. Dynamic inversion can also benefit from this idea. For example our desired nominal feedback should yield x=~es but because of modelling inaccuracies it instead gives something else . One strategy is to

found by solving equation (8) for x=O to give X in terms of Xc:

measure x and adjust the control input to give

o = (M(x)-~g(x)f(x» + (1 +~g(x»

X=Xdes . Such an approach can be problematic

wc(xc-x)

(10)

because the transfer function from Xdes to X is not

It is desirable that the solution x be close to Xc to ensure good tracking . In the nominal case where ~f=M=O, then (10) yields x=x c . In general, let x=xo(x c) be the set of solutions of (10).

strictly proper. However the difference ~es-x can be integrated and fed back to the input to give a well behaved system.

Taking a Taylor series expansion of x about x=xo and retaining only the linear terms in x.

Say Xdes=Wc(xcx) and let us denote integration with respect to time by lID. Then our integral feedback would be:

(11) where: al = -(1+~g(xO»wc

oD «Wc xcx}-x) . = Wc 0D

(Xc-x) -Ox

(16)

o above is a constant gain chosen to give desirable

+~f(xO) -~g'(XO)f(xO)

properties (usually O«w c). Once again the control

262

input u is based on the nominal system equation (1) and the actual system is assumed to be represented by equations (6). The proportional + integral (P+I) input u is:

It can be seen that these conditions are similar to those in equations (13)-(15) for the proportional only system . (28) is required for nominal stability with M =~g =0. (29) ensures that ~g(x) does not cause a change of sign in g(x). (30) requires sufficiently high gain to guarantee stability in the presence of destabilizing uncertainties . The PI controller therefore has the same stability as the proportional only controller with the added benefit that the steady state errors are zero regardless of the perturbations M and ~g. The disadvantage is that the system is now second order instead of first. Thus while the above two conditions may guarantee stability of the closed loop linearized system they do not guarantee desirable pole locations per se. This because a PI structure introduces a phase lag into the control loops. If wc =11 the damping ratio will be nominally be I. However. if g(x) is reduced to 50% of nominal then the damping ratio will drop to 0.7. It is preferrable to separate 11 and Wc by a factor of two or more to maintain real poles even when g(x) is only 50% of the nominal. which is quite a severe reduction. In the simulations discussed below 11=0.1 Wc was used . This introduces a phase Jag of only 6 degrees at the nominal crossover. Wc. and even if gain in g(x) is reduced by 50% the phase lag is only about 12 deg . So far only the linearization of the perturbed nonlinear system with nominal feedback has been studied to determine the stability of the system in a neighborhood of the equilibrium. This may not be representative outside this neighborhood . An example of where problems can occur is when the command Xc is changed very rapidly to a new value. This means that the new equilibrium point at the new value of Xc is far removed from the present state x so that x no longer falls within the neighborhood of attraction of xo. Then the nonlinear terms a2(xxO)2. a3(x-xo)3 ...• which were neglected from the Taylor series expansion (11) might. in fact. predominate over the linear term al (x-xo). Then x may then diverge catastrophically from the desired trajectory. This problem can be countered by incorporating a low pass pre-filter on the command. which would eliminate discontinuities in Xc caused by step commands. The bandwidth on the prefilter would be set to ensure that xo can never differ from x by more than a safe amount so that the local stability of the control law given by the linear analysis would suffice.

11 u = g(x)-I (-f(x)+wc(xc-x) +wc D (xc-x) -11x) (17) This input produces the closed loop dynamics:

x= (l+~g) (wc(xc-x)+wc

g

(xc-x)-11x)

+M-~gf (18)

which can be expanded to a two state equations:

y = xc-x

(20)

Selling x=y=O yields the equilibrium manifold of this system : x

= Xc

(21)

«(l+~g(xc»~hc +~g(xc)f(xc)

(l+~g(xc»wc11

y=

»

-M(x c

=YO (22)

Note that [21] gives us zero steady state error in x despite the presence of uncenainties. M and ~g . The steady state value of y is not of interest as it is simply internal to the controller. Linearizing about x=x c and y=YO yields: (23)

y= - I

(x - xc) + 0 (y - YO)

(24)

where bl and b2 are:

(25) (26)

Where no argument is shown f(x) and g(x) are evaluated at x = Xc. The characteristic equation of such a linear system is: (27) From the Routh criterion. stability of the linear system is gauranteed if b2>0 and bl <0 for all allowable M and ~g. This leads to the following sufficient conditions for stability: Wc and 11 are both positive

(28)

1+~g

(29)

>0

5. SIMULA nONS An aggressive maneuver was simulated using the nominal X-31 type aircraft model as discussed in Snell et al (1992). The maneuver was also simulated with the following model perturbations incorporated. Cmdc = 0.5 CIDctc nominal

263

(31)

Clda Cndr

performance robustness according to the criterion in equation (15). The improvements might be more substantial if we were required to use smaller wc.

= 0.5 Clda nominal = 0.5 Cndr nominal

Simulations were also carried out both with and :-vithout integral terms in the controller loops. When mtegrators were included in the inner loops, the form of the desired rates on p, q and r is as follows: . 10 x = lO(xc -x) -x + - (xc -x) s

6. CONCLUSIONS Dynamic inversion is a straight forward method for designing feedback control laws for non linear systems. The performance in the presence of static modeling errors can be enhanced by using integral terms within the loops to remove steady state errors. Care should be exercised in the choice of gain on the ~ntegral feedback to ensure that the phase lag mtroduced by the term does not cause phase margin problems. Further work needs to be done to assess the effects of dynamic perturbations, such as actuator lags and structural modes, on the high bandwidth control laws, where the introduction of phase lags might be critical.

(32)

This is equivalent to selling n= 1 rad/s =0.1Wc in equation (16), which causes an additional phase lag at crossover 10 rad/s of only 6 deg compared with the P-only control. The outer loops use wc= 2 rad/s and n = 0.4 rad/s. Figures 1-3 are respectively plots of alae, J..L'llc and ~ for both dynamic inversion systems without modeling penurbations. The responeses for the control Jaws without (a-figures) are almost identical to those with integral terms (b-figures). Both systems show satisfactory tracking of a with peaks of 28 deg. The Il tracking is also good, while ~ peaks at about 0.5 deg for both systems. The control deflections for the system with the nominal model, without integrators are shown in figures 4a and 5a and those for the system with integra.tors .in ~gures 4b and 5b. Again the responses are qUIte Similar and all are well below saturation levels. Figures 6-10 show the corresponding plots when the control effectiveness terms g(x) are reduced to 50% nominal in accordance with equations (31). Figures 6a and 6b show that a-tracking is still good with dynamic inversion but now it can be seen that the inclusion of the integrators reduces the difference between a and its command . The peak a is now 30 deg . However, both systems demonstrate good a command following, despite the model errors. Figure 7 shows that tracking of Il is almost unchanged from the unperturbed model. Thus, roll control remains accurate in the presence of the current choice of model perturbations. Figure 8 shows that both dynamic inversion controllers retain accurate control of ~ with peaks of 2 deg for the proportional-only control and about 1.3 deg for the PI control. However, the integral terms do appear to produce a more oscillatory response in ~, indicating a reduced damping effect as suggested in the previous section. The control deflections for the system with model perturbations, both without and with integrators are shown in figures 9 and 10. Notice that the lateral deflections of the PI controller, shown in figure 9b, although safely below the deflection limits, remain larger for longer than the P-only deflections. These appear to concide with the oscillations in ~ discussed in the previous paragraph. The longitudinal control deflections are shown in figures lOa and b and are almost identical . Overall, the improvement shown in the plots produced by the inclusion of the integrators is small. This is probably because the nominal crossover, Wc is 10 rad/s which provides a good deal of

7. ACKNOWLEDGEMENTS This research was supported in part by a University of California, Junior Faculty Research Grant.

8. REFERENCES Buffington, J.M., Adams, R.1 ., and Bancta, S.S ., (1993) "Robust, Nonlinear, High Angle-of-Attack Control Design for a Supermaneuverable Vehicle", Proc. AIAA Guidance, Navigation and Control ConJ, Monterey CA. Bugajski, DJ., and Enns, D.F., (1992) "Nonlinear Control Law with Application to High Angle-ofAttack Flight", 1. Guidance, Control and Dynamics, Vol. 15, No. 3, pp 761-767. Hauser, 1., Sastry, S ., and Meyer, G ., (1988) "Nonlinear Controller Design for Flight Control Systems" , University of Califonia, Berkley, Electronics Research Laboratory, Memorandum UCBIERL M88/76 Menon, P.K.A., Badgett, M.E., Walker, R.A., and Duke, E .L., (1987) "Nonlinear Flight Test Trajectory Controls for Aircraft", 1. Guidance, Control and Dynamics, Vol. 10, No. I, pp67-72. Morton, B.G., Elgersma, M.R. , Harvey, e.A., and Hines, G., (1987) "Nonlinear Flying Qualities Parameters Based on Dynamic Inversion ", AFWAL-TR-87-3079. Snell, S.A., Enns, D.F. and Garrard, W.L., (1992) "Non linear Inversion Flight Control for a Supermaneuverable Aircraft", 1. Guidance, Control and Dynamics, Vol. 15, No. 3. Snell, S.A., (1992) "Preliminary Assesment of the Robustness of Dynamic Inversion Based Flight Control Laws", Proc. AIAA Guidance, Navigation and Control Con!, Hilton Head Se.

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