Document
URC97051
AIRCRAFT PITCH CONTROL WITH FIXED ORDER LQ COMPENSATORS A. Homaifar$ James Green* CR. Ashokkumar; NASA Center of Research Excellence The North Carolina A & T State University Greensboro, NC 27411 A B S T R A C T This paper considers a given set of fixed order compensators for aircraft pitch control problem. By augment- ing compensator variables to the original state equations of the aircraft, a new dynamic model is considered to seek a LQ controller. While the fixed order compensators can achieve a set of desired poles in a specified region, LQ formulation provides the inherent robustness properties. The time response for ride quality is significantly improved with a set of dynamic compensators.
1. Introduction: While designing a feedback control, ride and handling qualities are major performance objectives in aircraft control problems. Such objectives are normally achieved by closed closed loop pole assignment [I]. Preserving these closed loop poles ( within the desired regions ) in the presence of perturbations is another requirement [2]. LQ problems have inherent stability margins to tolerate unstructured uncertainties. LQ design techniques with regional pole constraints have been studied extensively in the literature see [3], and its references ].
Similar approach, but with dynamic compensators, hwe been investigated for automotive applications [4].
The compensators given in [5] for aircraft control problem are considered in LQ problem setting. The objective of this approach is to improve aircraft ride quality defined in [1].
II. Prol}lem Formulation: An aircraft model in pitch plane [ with normal acceleration ( 712 ), pitch rate ( q ) and elevator deflection ( 6e ) as state variables and command input ( tiC ) as control variable], is given by [2;: (1)
‘=[’~ ‘: ~’Hj’j+hJ
A z(t) b It is well known that the control law
u(t) = –R–lb’ l%(t) + r(t) (2)
minimizes the performance index
.1 = ‘{zrQz + ?;R?J}(it (3)
J
o and satisfies the algebraic riccati equation A’P– PbR-lb’P+PA+Q=O (4) Selection of weighting matrices to achieve a controller in equation 2 for exact pole assignment has been extensively investigated in reference [3]. Suppose, we choose a set of dynamic compensators given in [s] for the control law structure ~2] ( see Figure-1 ), then the state equations for the compensators are: * [Undergraduate Student, Dept. of Electrical Engineering.
t post Doctoral R=earcl, .4ss0 ciate.
I Associate prof~sor, Dept of Electrical Engineering.
t- Red Figure 2: Regional Constraints for Aircraft II. Simulation Results: For F-4 aircraft model at Mach= 1..5, Altitude =35,000ft, the system dynamic matrices are given by: 1’78.9 –0.5162 26.96 A = –0.6896 –1.225 –30.38 0 –14 o [ –17.5.6 b = –14
[1
The matrices ~ and b for the state vector ~(t) = [z(t), ZI(t), z2(t)]’ arc 0 0 —T1 o 0 — TS ~=[;:” 1 –175.6 o $= –14 where, o Ii A1=[l OO] AZ = [ –0.6896 (–1,225 + r.. ) –30.38 ] At this flight condition, the short period damping ( <.P ) and frequency ( W,P ) requirements are: 0.35< <.p <1.3 (12) and (13) 3.29< w~p <11.8 In complex plane, these constraints impose regional pole constraints shown in Figure 2.
Table- 1 Design Variables Wsv <s, 0.4789 Q. R 4.50’78 0.5316 Q, R 6.44.58 Desired [3.29, 11.8] [0.35, 1.3] FIIs) n: - F L+
(f
Z2
F21S} Figure 1: Control Law Structure [Ref 2] with Filters F’1 (s) = ~ and F’z(s) = ~ = —Tlzl + 7-12 (5) . .
—T3Z2 ~ T2~ (6) %2-Y= From the aircraft dynamical equations 1, substituting for ~, we have ~~ = azlrzz + (azz + r~)q + aztfi, — T3Z2 (7) For the new state vector z(t),
z(t) = [z(t), Zl(t), z2(f)]’
the state space equations become, i(t) = m(t) + k(t) (8) where, (211 alz 013 0 0 LZ21 (L22 Q3 o 0 0 0 –14 o 0 bl o 6=14 o
II
It can be verified that for these dynamic compensators, the system in equation 8 is completely controllable.
Thus the control law m(t) = –R–16’FT(t) + ?-(t) (9) minimizes the performance index 3= ‘{Z’QZ + 12’Rti}dt (10) / .0 and satisfies the algebraic riccati equation ——— (11) ii’~-PbR-li~+~~+~=O With the above formulations, we shall now present the closed loop eigenvalues for various values of the design parameters. The design parameters for J are obviously the weighting matrices Q and R. However, note that the performance index ~ is significantly influenced by the other design parameters T1, T2, and r3, in addition to Q and ~. The next, section presents the simulation results.
. . . . . . . . . . . . . . . . . .,,.,’..
/ with Compensators = 2.5 T1 TQ = 75 T3 = 4.0 -1 -., .,.
-2 + without compensators -3 -4
o 0.5 1
1.5 2 2.5 3 Figure 3: Time response pelts due to step input 4 4 The weighting matrices Q = 1 and R. = 10 as well as the weighting matrices ~ = 1 and ~ = 10 provide 3 3 the acceptable closed loop poles [ see Table 1 ].
I-Iowever, what needs to be observed is the time response plots ( due to step input ) shown in Figure 3. We observe that the normal acceleration at the sensor location is nonminimal. Moreover, the peak accelerations are significantly reduced with dynamic compensators ( about 50% ).
Acknowledgements: This work is partially supported by grant from the NASA Center of Research of Excellence at NC A&T’ State University under grant # NAGW-2924. The authors wish to thank the NASA-CORE administration.
References: 1. “Flying Qualities of Piloted Airplanes,” MIL-F-8785B(ASG), Aug 7, 1969 2. S.N. Franklin and J. Ackerman, “Robust Flight Control:A Design Example,” Journal of Guidance, Control, and Dynamics, v 4, n 6, 1981, pp 597.
Y. Ochi and K. Kanai, “Pole Placement in Optimal Regulator by Continuous Pole-S hefting,” Journal 3.
of Guidance, Control, and Dynamics, v 18, n 6, 1995, pp 1253-1258 4. H. Peng and M. Tomizuka, “Preview Control for Vehicle Lateral Guidance in Highway Automation: Journal of Dynamic systems, Measurement and Control, v 115, 1993, pp 679-686 5. CR. Ashokkumar, ‘LRobust Optimal Compensators with Tight Control Philosophy: Rep. GCD/CRA/2, NASA Center of Research Excellence, North Carolina A & T State U-niversity, Greensboro, NC