An Integrated Approach to Plant Optimization and Controller Design

Copyright © IF AC Robust Control Design, Budapest, Hungary, 1997

AN INTEGRATED APPROACH TO PLANT OPTIMIZATON AND CONTROLLER DESIGN Erik Berglund * Isaac Kaminer

**,1,2

* Defence Research Establishment (FOA), 17290 Stockholm,

Sweden, Email: [email protected] ** Department of Aeronautical and Astronautical Engineering, Naval Postgraduate School, Monterey, CA 93943, USA. Email:

kaminer@aa. nps. navy. mil

Abstract: This paper addresses the problem of integrating the design of an aircraft's control surfaces with that of the corresponding feedback controller. The key idea is to rewrite the aircraft control requirements as linear matrix inequalities (LMIs) and to optimize a linear cost function associated with the aircraft control surface sizes, while meeting the LMI constraints. A numerical solution has been developed and applied to integrated plant/controller design of an F-4 aircraft at five different flight conditions. Keywords: Control, Gain Scheduling, Linear Matrix Inequalities, Optimization

1. INTRODUCTION

ities (LMIs) . The authors propose a numerical solution to this problem which results in reduced aircraft surface sizes and a feedback controller that meets performance specifications at a single flight condition. In this paper the ideas presented by Niewoehner and Kaminer are extended to include multiple flight conditions.

Control systems of modern aircraft have become increasingly complex. That applies especially to fighter aircraft with relaxed stability and multiple control surfaces acting about each axis. The control system, control surfaces included, must satisfy all the performance requirements (stability, disturbance rejection, maneuverability etc.), while not causing excessive drag, weight or signature. It is thus important to minimize the size of certain aircraft control surfaces, subject to the constraint that a suitable controller can still be found that satisfies flight performance requirements. Previous work in this area is reported by Niewoehner and Kaminer (1994, 1996), where the authors have reduced the problem of integrated aircraft / controller design to a constrained optimization problem, where the cost is a function of the aircraft control surface sizes and the constraint set is described by Linear Matrix Inequal-

This paper is organized as follows. Section 2 formulates the integrated aircraft/controller synthesis problem as a constrained optimization problem (ACO). In Sestion 3 the numerical to the ACO proposed in Section 2 is applied to the longitudinal dynamics of an F-4 fighter aircraft. The optimization parameters are chosen to be the sizes of the moving horizontal tail and of the spoiler. The paper ends with conclusions.

2. PROBLEM FORMULATION The problem to be addressed in this paper is to minimize the size of certain aircraft parameters (typically the size of the control surfaces). Since reduction of the control surface sizes may cause the aircraft to become uncontrollable, it is critical

1 This work was supported in part by the Directly Funded Research Program at the Naval Postgraduate School. 2 The second author's visit to FOA was supported by FOA travel grant.

213

to include the feedback controller design in the optimization problem. Since control requirements for several flight conditions are considered, a solution of the ACO problem leads to a new single set of reduced control surface sizes and a multiple set of feedback controllers that meet performance requirements for all given flight conditions. In this paper the feedback performance requirements are reduced to constraining the Hoo norm of a certain feedback transfer matrix. The Hoo controller synthesis problem is formulated in Section 2.1. The aircraft / controller optimization problem is treated in Section 2.2. The numerical solution to the ACO is outlined in Section 2.3.

Let

:i;

9 =

{

z y

= Ax + B1 W + B 2 u = Cx+Du = x,

(1)

where x E Rn, w E Rm, u E Rq and z E RP . Assume that (A, B 2 ) is stabilizable, and that D has full column rank. Then there exists a feedback controller e such that IITzw (9 ,e )1100 < , if and only if there exist matrices Y = Y' E nnxn : Y > and W E qxn , such that Boyd et al. (1994); Khargonekar and Rotea (1991)

°

n

R(W, Y,-y) :=

AY+YA T +B2W+WTB[ Bl (CY+DWfhj Bi -J 0 [ (CY + DW)h 0 -J

2.1 Hoo Synthesis

< o. (2)

z

w

9

u

C

t--Y

If such matrices W and Y exist, then one such controller e is the constant gain matrix K given by K = Wy-1.

r--

2.2 Plant Parametrization In this paper we consider feedback performance requirements given for several aircraft flight conditions. At each flight condition the aircraft nonlinear six degree of freedom equations of motion were linearized to obtain linear models of the form

Fig. 1. Feedback interconnection of plant 9 and controller

e.

Consider Figure 1. Let 9 denote the generalized linear aircraft model. It may consist of the linear aircraft model plus the weights usually appended to this model as a part of the design process. Such weights may include gains, integrators, low pass filters, notches , etc. The exogenous inputs w represent commands and disturbances acting on the aircraft. The vector z may include outputs whose size should be kept small in the presence of inputs w . Input u denotes aircraft control inputs such as elevators, ailerons, rudder and thrust, and output y denotes the signals available from the sensor suite.

:i;

The aircraft models are parameterized using the vector of plant parameters ( = [(1, (2, ... , (k (j > 0, Vj = 1, ... , k. Associated with ( is a linear cost function J = cT (, where c E k, Cj > 0, Vj = 1, ... , k, is the vector of weights on each of the plant parameters. Furthermore, the plant matrices Ai, Bt and B~, i = 1, ... , l, can be expressed as a linear combination of the elements of (

V,

n

w

the maximum singular value of

k

Tzw(jw) .

Ai(()

= Ab + L

A;(j

j=l

The Hoo synthesis problem is to find a feedback controller which will stabilize the generalized aircraft model 9 and will make the infinity norm of the transfer matrix T zw from inputs w to outputs z less than a given number, > 0:

e

IITzw(9

(3)

where i = 1, ... , l, 1 is the number of flight conditions , x E Rn is the state-vector representing perturbations about the trim condition, w E Rq is the external input, u E Rm is the control input, and z E RP is the output to be controlled.

11 Tzw(9 ,e) 1100:= sup{a(Tzw(jw))},

a denotes

= Aix + Biw + B~u

z=Cix +Diu,

Suppose the closed loop system in Figure 1 is stable. Let T zw (9 ,C ) denote the closed-loop transfer matrix from w to z. Then the infinity norm of Tzw(9 ,e ) is defined as the supremum over all frequencies of its largest singular value:

where

9 admit the realization:

k

Bi (() = Bio + L

BL(j

j=1 k

B~(()

,e )1100 < ,.

= B~o + L j=l

214

B~j(j.

In general, C i and Di matrices could also depend on (, but that case is not considered here.

equations, see e.g. McCormick (1995) . It is, however, usually possible to analyze the longitudinal direction separately and to use a system of four non-linear differential equations for that direction. Normally the aircraft is flown at a trim condition, i.e. at equilibrium. When only small pert urbations around trim are of interest, the non-linear equations are linearized at the trim condition. This procedure results in a system of four linear differential equations describing the longitudinal dynamics of the aircraft.

For each of the flight conditions a steady state

Hoo controller can be found by solving (2) for W i and y i, where the plant matrices now depend on (. The ACO problem can thus be formulated as follows. For a given vector c E R} , Cj > 0, Vj = 1, ... , k , let J = cT ( and define the set <)>

yi

= {Wi E R m x n , y i E R n x n , ( : yi

= yiT , (j

> 0,

> O, R(Wi , yi , () < O}

Then the ACO problem is minJ s.t.(yi , W i , () E

<}>

2.3 Numerical Solution Following the methodology described in Niewoehner and Kaminer (1994) the optimization problem in W i , y i and ( can be solved using a D - K type iteration. Observe that for a fixed (, R is affine in W i and y i. Similarly, for fixed yi and W i , R is affine in ( . This suggests the following approach (1) Fix ( and find W i and y i > 0, i = 1,2, ... , 1 such that R(Wi , yi, (0) < 0 (2) For W i and yi obtained in step 1, find ( that minimizes J = cT ( S.t. R(Wi, y i , () < 0 and (j > 0 (3) Go to step 1 until exit criterion is satisfied.

In this paper the equations are formulated in a body-fixed coordinate system with the x-axis sticking out of the nose of the aircraft and the zaxis pointing down. The four states are the perturbations in relative velocity in the x-direction ujU (where U is the trim velocity) , the angle of attack a , the pitch rate q, and the climb angle () . The control inputs are elevator and spoiler deflection , oeand Os respectively, and thrust, Ot. The external inputs are commanded values of angle of attack , i.e. are! and flight path angle, i.e. I re!. The equations of motion are formulated using dimensional aerodynamic derivatives McCormick (1995).

3.1 F-4 Dynamic Model F-4 longitudinal dynamics can be described by four linear differential equations formulated in a body fixed coordinate system

(4) where

Step 1 requires a solution of I LMIs for the matrices W i and y i. Step 2, on the other hand involves a solution of a single LMI for the plant parameter ( . This approach to solving ACO problem does not have guaranteed conversion properties, see Goh et al. (1994). Therefore, it must be executed for multiple initial conditions.

uo U -0Zo, [ oo -Mo, 0

00] 00 10 0 1

UXU X Ot 0 -gCOSBO] UZu ZOt U + Zq -gsinBo 0 [ UMu MOt Mq o 0 1 0

3. LONGITUDINAL CONTROL OF F-4 The pilot controls the aircraft by using the stick to command the aerodynamic control surfaces and the throttle to change the thrust. The flight control system makes the pilot's task of flying easier by, among other things , stabilizing the aircraft through feedback control.

i B In

The purpose of this example is to find a controller for the longitudinal direction, that fulfills performance requirements given as the ability to track commands in angle of attack (a) and flight path angle (r), and t o reject gusts.

X

-

-

i B 2n

-

-

0 1]

Xs Zo Zs 0 Mo 0 0

[o

= [ujU, a , q, ()] T w = [are! "

0 0

re!]

T

u = [Oe, Os, Ot]T The plant parameters to be optimized are the size of the moving horizontal tail and the size of t he

The equations describing the dynamics of an aircraft consist of six coupled non-linear differential 215

spoiler. These are represented by the two coefficients Za and Zs, which give the vertical acceleration per radian of elevator and spoiler deflection, respectively. Hence, the vector of normalized plant z" , Z"nominat Z. IT. parameters is ( = [Zon.oYninal

Table 2. Aircraft Data Flight Condition Altitude (ft) Mach V(m / s) m(kg) Iy(kg . m~) QTrim(deg)

It is important to point out that changing the size of the spoiler affects only Xs and Zs. However, changing the size of the horizontal tail affects the dynamics of the aircraft drastically. The effect on the aerodynamic derivatives can be expressed following Roskam (1971)

= ZQwb + Za I Zq = Zqwb + UZa I

= Zo,wb + UZa y

12m

= Mqwb + I y U Za

Mo

= Mowb + I y U Za ,

(3) 35,000 0.9 267 17660 8370 2.6

(4) 35,000 1.2 356 17660 8370 1.6

To accommodate the steady state tracking requirement, the plant is augmented with the integrals of 0:' and ,. The augmented system has six states, x = [u/U,O:',q,O,O:'/s,,/slT and, therefore, admits the following representation

lm

M Q = M Qwb + yZo Mq

(2) 35,000 0.6 178 17660 8370 9.4

• Gust disturbances with a magnitude of 1.5 m/s in the z-direction should not yield elevator deflection greater than 20 degrees, spoiler deflection greater than 40 degrees or errors in angle of attack greater than 1.5 degrees. • A flight path angle of 1 degree must be generated using elevator and spoiler deflection of less than 20 and 40 degrees, respectively.

ZQ

Zo

(1) 15,000 0.9 290 17660 8370 0.5

12m

2

Xa

where the subscript wb denotes the contribution of the wings/body section of the aircraft, U is the trimmed velocity, I is the distance from the center of gravity to the elevator, m is the mass and Iy is the moment of inertia in the pitch direction. The data for the F -4 fighter are given in tables 1 and 2. The data were taken from Schmidt and have been adjusted according to Roskam (1971) to obtain the wing/body values.

= (A~o + LA~j(j)xa + (B~lO j=l

2

2

+ LB~lj(j)W + (B;20 + LB;2j(j)U j=l Z

j+l

= C~Xa + D~u,

where

Table 1. Dimensional Derivatives Flight Condition Xu(s '") Xo(m / s~)

Zu(s '") Zowb(m/s~)

Zo,wb(m/s) Z-'Lwb(m/s) M u (m '" S ) Mowds .~ ) Mo,wb(S ,") Mqwb (S '" ) X6nominal (m / s~) Zonominal (m / s~) Monominal(S .~ ) Xsnominal(m / s~ )

Zsnominal(m/ s~ )

(1) -0.0215 -1.81 -1.170 -303.6 0.199 -1.02 -0.0013 8.00 0.163 -0.373 0.00 -32.6 -25.00 -0 .280 2.80

(2) -0.0176 -7.39 -0 .337 -43 .04 0.078 -0.297 -0.00003 2.97 0.057 -0.109 -0 .003 -6.39 -4.90 -0.280 2.80

(3) -0.0123 -2.26 -0.571 -129.8 0.097 -0.475 -0.0009 3.52 0.071 -0 .182 0.00 -15 .13 -H.40 -0.280 2.80

(4) -0 .0136 -5 .04 -0.747 -231.0 0.168 -0.701 0.0007 -8 .33 0.122 -0.336 -0.003 -27.57 -20 .70 -0 .280 2.80

AiaO --

(5) -0.0072 -9.14 -0.515 -196.9 0.181 -0.344 0.0008 -12.94 0.130 -0.236 0.00 -21.54 -16 .00 -0.280 2.80

00 00 Ab 00 00 1 0000 0 o -1 o 1 0 0

00 00 00 00 0000 0000

AiI Aial --

00 00

A~2

=0

3.2 Design Requirements • The feedback system must be internally stable. • Step response - There should be no steady state error in the response to a step input in angle of attack, 0:', or flight path angle,

B~IO'Q i BalO

B~l1 =

-

o o

,=

0-0:' .

216

0

0

0

0'1

0

0

(5) 45,000 1.5 443 17660 8370 2.6

i B a20

-

-

3.3 Results

00 1 000 000 000 000 000

LMI Toolbox Gahinet et al. (1995) was used to implement the numerical algorithm proposed in Section 2.3 for the following values of input parameters

eT = [lI lT, B~l

(J"(

BZ2

Bi a21 -

= 5jUi, 8emax = 20 degrees, (Ja

Bia22 --

0 0 0 0 0 0

0 0 0 0 0 0

c5 smax = 40 degrees c5tmax

i -Ca

0 0 0

0 0 0 1

1 c5 emax

0 Da i-

00 0 00 0 00 0

0 0 0

0 1 c5 smax

0

0

0 0 0

0 0 0

= 0.05g.

Since the proposed numerical algorithm does not have guaranteed convergence properties, see Goh et al. (1994) , it was executed for multiple initial values of control surface parameters. With fixed initial value of the spoiler, i.e. (2initial = 1, the initial value of the elevator, (linitial , was varied between 0.1 and 1.5. The procedure was repeated with the initial value of the elevator fixed at (linitial = 0.5 and (2initial varying between 0.6 and 1.5. Selected optimization histories are shown in Figures 2 and 3. It can be seen that the algorithm converges to the solution (opt = [0.06,0.49f in all cases.

00 0 0 a max 0 0 00 w'" 0 0 0 00 0 w"( 0

= 1 degree,

0

0 1 c5tmax

0 0 0

30

Fig. 2. Optimization history of (1 for a variety of initial conditions Besides the optimal values of the spoiler and elevator control surface sizes, the AeO problem produced five controllers, one for each of the flight conditions considered in this example. Table 3 shows the eigenvalues and the damping for the closed loop system at one of the flight conditions. Similar values for closed loop eigenvalues and damping ratios were obtained for other flight conditions. The designed controllers resulted in feedback systems with rather slow dynamics. This can be corrected by adjusting the weights in the C~ matrix for the ith flight condition. The damping, which for the five flight conditions varies between 0.54 and 0.88 is , however, satisfactory.

The scaling of the input and output to get IITzw(G , K)lI oo < 1 shows up through (J", and (J"( in the Ba1 matrix and through the weights in the Ca and Da matrices. 217

References S. Boyd, L. El Ghaoui, E. Fero, and V. Balakrishan. Linear Matrix Inequalities in Systems and Control Theory, volume 15 of Studies in Applied Mathematics. Society for Industrial and Applied Mathematics , Philadelphia, PA, 1994. P. Gahinet, A. Nemirovskii, E. Laub, and M. Chilali. LMI Control Toolbox. Math Works Partner Series, 1.0 edition, May 1995. K.C. Goh, L. Turan, M.G : Safonov, G.P. Papavassilopoulos, and J .H. Ly. Biaffine matrix inequality properties and computational methods. In Proceedings of the American Control Conference, June 1994. P. P. Khargonekar and M. A. Rotea. Mixed H2/ Hoo control: A convex optimization approach. IEEE Trans. on Automatic Control, 36 (7):824-837, July 1991. B.W. McCormick. Aerodynamics, Aeronautics and Flight Mechanics. John Wiley & Sons, 1995. R.J. Niewoehner and I.I Kaminer. Design of an autoland controller for carrier based f-14 aircraft using hoo output feedback sythesis. In Procedings of American Control Conference, pages 2501-2505, Baltimore, MD , 1994. R.J . Niewoehner and I.I Kaminer. Integrated aircraft-controller design using linear matrix inequalities. Journal of Guidance, Control, and Dynamics, 19(2) :445-452 , 1996. J. Roskam. Methods for Estimating Stability and Control Derivatives of Conventional Subsonic Airplanes. Roskam Aviation And Engineering Corp, 1971.

Fig. 3. Optimization history of (2 for a variety of initial conditions Table 3. Closed loop eigenvalues and damping, flight condition 1 Eigenvalue -0.2343 + 0.1751i -0.2343 - 0.1751i -2.1488 -2.6423 -5.2115 + 4.9987i -5.2115 - 4 .9987i

Damping 0.8010 0.8010 1.0000 1.0000 0.7217 0.7217

Freq. (rad/ sec) 0.2924 0.2924 2.1488 2.6423 7.2213 7.2213

4. CONCLUSIONS This paper developed a technique for integrating the design of aircraft plant parameters with the design of a corresponding feedback controller for multiple aircraft flight conditions. The key idea was to formulate this problem as a constrained optimization problem, where the cost is a function of aircraft parameters to be minimized and the constraint set representing certain feedback performance requirements is expressed using LMIs. This technique was applied to the integrated design of control surfaces and feedback controllers for an F-4 aircraft at five flight conditions. Performance requirements for the feedback controllers were reduced to an Hoo norm constraint on the certain feedback transfer matrix. As a result the algorithm produced five feedback controllers, one for each flight condition, that meet performance requirements and a single optimized value for the size of the moving horizontal tail and of the spoiler. This set of controllers can be used to construct a gain scheduled controller, that works for the non-linear aircraft dynamics at all flight conditions. The gain scheduling must be done in a way that recovers the performance of the linear system at specified flight conditions. The linearized feedback system dynamics should be as independent of the gain scheduling parameters as possible. A continuation of the work presented in this paper is to find a way to interpolate between the different controllers such that this can be achieved. 218