Section 1
where j = 6.
The condition for complete controllability, from equation (A7) i n the appendix, i s s a t i s f i e d with the numerical data of the example.
The inequality i s The solution of equation (13), with use of the example data, resulted i n the following gains: k l = 0.939796 k2 = 0.0026195 k4 = 0.0391162 A similar set of gains can be computed f o r any s e t of desired roots hi of the characteristic equation (eq. (9)), and, i n a complete analysis, the roots may have t o be chosen selectively i n order t o insure reasonable gains.
Figure 1 presents the motions of the controlled airplane, f o r the partic- The ular set o f chosen roots, following an i n i t i a l sideslip angle of 10.go.
period of the oscillatory motion i s 2 seconds and the time t o damp t o one-half amplitude is 2 seconds. The motions of the f r e e airplane system are included i n figure 1 for comparison. The f r e e airplane has a period of about 3.5 sec- onds, and the time t o damp t o one-half amplitude i s about 64 seconds. Although the motions were computed for a large i n i t i a l sideslip angle, the aileron and rudder deflections are within probable limits f o r an airplane of the type considered.
LO ~~ ~ .
c CONCLUDING REMAKKS State-vector contrc theory has been applied t o a s t a b i l i t y problem of high-performance aircraft. Analytical expressions have been derived f o r feed- back gains that will produce any desired roots of the characteristic equation of the augmented system. The gains are given i n terms of the parameters of the system and the desired roots. A numerical example i l l u s t r a t e s the practica- bility of the method.
Langley Research Center, National Aeronautics and Space Administration, Langley Station, -ton, V a . , April 7, 1965.
1 1
APPENDIX
APPENDIX C0MPL;ETE CONTROLLABILITY A linear dynamical system is s a i d t o be completely controllable i f , a t any i n i t i a l time t, any i n i t i a l state x can be taken t o t h e o r i g i n i n a f i n i t e length of time by the application of a s u i t a b l e control function.
Consider t h e mathematical model of a l i n e a r dynamical system
where 9 i s an n-vector, the state of t h e system, and u’ i s an m-vector, t h e
control of the system. The rectangular matrices [ A ] and [ G ] are, i n gen- eral, functions of t i m e . However, f o r present purposes, they are assumed t o be constant.
A necessary and s u f f i c i e n t condition f o r t h e constant system (eq. ( A l ) ) t o be completely controllable i s Suppose t h a t the control vector 7 ? i s a s c a l a r s times a known vector 2, o r Define a new n-vector
$’= [GI2
(A4 1
so t h a t the dynamical system is represented by
e = [ A ] ” + 2 s
(A5 1
d t The condition f o r complete c o n t r o l l a b i l i t y of equation ( A 5 ) i s
rank@] E r a n k k [ A ] < [ A ] * < . . . , [ I AIn-%? = n
or, since [ i J
is an n x n matrix, det A more detailed and general discussion of complete controllability is presented i n reference 3 .
.
REFERJiNCES 1. Bass, R. W.; and Mendelson, P.: Aspects of General Control Theory.
AFOSR 2754, Aeronca Manufacturing Corp., Aug. 1962.
2 . Campbell, John P . ; and McKinney, Marion 0.: Summary of Methods for Calcu- lating Dynamic Lateral Stability and Response and for Estimating Lateral Stability Derivatives. NACA Rept. 1098, 1952. (Supersedes NACA TN 2 4 0 9 . ) 3. Kalman, R. E . : On the General Theory of Control Systems. Automatic and Remote Control, Vol. I , J. F. Coales, J. R. Ragazzini, and A. T . Fuller, eds., Butterworth, Inc., 1961, pp. 481-492.
4. Anon.: Flying Qualities of Piloted Airplanes. Mil. Specification ~1~-~-878 (ASG), Sept. 1, 1954.
I -I Controlled airplane 32 0 2 4 6 a 1 0 1 2 0 2 4 6 a 1 0 1 2 Time, t, sec Time, 1, sec Figure I . - Motion o f airplane and controls.
, Figure I.- Concluded NASA - Langley,