Longitudinal Flight Control Systems 9
Basic Longitudinal Autopilots (I) Attitude Control System First idea: A Displacement Autopilot Typical block diagram: Vertical e g e δ Elevator δ A/C θ ref Amplifier e θ(t) Gyro Servo Dynamics Basic features: A direct command following design The simplest from of autopilot Designed to hold the A/C in straight and level flight by maintaining θ ( t) θref The A/C has to be trimmed for straight and level flight before this autopilot is engaged Working principle: Whenever θ( t) θref e g 0 e δ 0 A restoring δ e appears Mathematically, if we represent the A/C dynamics as θ( s) H ( δ, ) δe( s) e θ, the elevator servo as δ e ( s) H voeg ( s), and the amplifier as a gain K a, then we will have For KaH(, ) H e vo θ( s) δ θ θref ( s) eg ( s) θref ( s + KaH( δ, ) H ) e θ vo + KaH( δ, ) H e θ vo θ ref ( s) θ* s lim e g( t) seg ( s) s 0 <<, if KaH(, ) H vo >> e θ s 0 t δ 0
Design considerations: Selection of K a must provide adequate stability margin for errors in H ( δ e, θ ) and H vo Root locus analysis is often used to examine the closed-loop design Design example (a) Conventional transport: Simplified longitudinal dynamics: Simplified servo dynamics: H vo H( δ e, θ ) s( s 5 s + 5 Line of 06 damping ratio Dominant mode damping ratio decreases when K a increases ( s + ) + 8s + 4) CL poles at K a 064 when the dominant mode damping ratio still > 06-4 - -0-8 -6-4 - 0 e g 5 δ θ e ( s + ) ref K a θ s + 5 s( s + 8s + 4) 4 K a max 596
Difficulties of the system: (a) Little maneuvering capability is provided with θ ref being the command Pitching maneuver is often conducted by commanding the pitching rate (b) The design works only if the open-loop system is adequately stable Example design for problem (b) Modern jet aircraft: Simplified longitudinal dynamics: H ( δ e, θ ) 9( s + 006) s( s + 0805s + 5) Dominant pole damping ratio of the A/C: 0 5 ( For the A/C in : 77 Line of 06 damping ratio ) Dominant mode damping ratio decreases rapidly as K a increases ) No acceptable design can be found -4 - -0-8 -6-4 - 0 e g 0 δ θ e 9( s + 006) ref K a θ s + 0 s( s + 0805s + 5) 0 & ) K a max 4565 System goes unstable at K 565 (In, K 96) a max 4 a max 5
Circumvention for system stability Attitude Autopilot with a rate feedback inner loop Typical block diagram: Basic features: Vertical e θ Amplifier δ Elevator A/C θ ref Gyro e g e a Servo δ e Dynamics e Rate Gyro θ rg A rate gyro which senses the pitching rate is used to feedback θ & A multi-loop feedback structure is employed The differentiator inner loop will increase the damping of the A/C (short period) dynamics Example design for the jet aircraft in Block diagram representation of the system: e g e δ θ a e δ 0 e -9(s+006) θ ref Ka θ s + 0 s +0805s+5 s inner-loop e rg K b K a : outer-loop feedback gain K b : inner-loop feedback gain Note that the rate gyro senses the pitching rate θ &, and the system output is pitch angle θ
Inner-loop root locus Stability increases with increasing K b e a e δ -0 δ e -9(s+006) θ s+0 s +0805s+5 K b Solutions with K b Solutions with K b - -0-8 -6-4 - 0 CL poles of the inner-loop locus become the OL poles of the outer-loop locus Outer-loop root locus The outer-loop locus will retain the open-loop zero The pole at s 0 reappears in the outer-loop locus with K b, K a max > 49 with K b, K a max < 44 e g e θ a ref Ka inner-loop s θ θ - -0-8 -6-4 - 0 4
Discussions: A differentiator feedback inner-loop greatly enhances the system stability In, the dominant pole damping ratio (ζ ) of the CL system is < 0 5 for any K a In this design, ζ of the CL system will be > 0 6 for reasonable values of K a and K b The larger the value of K b, the larger the value of ζ will be for the same K a Ka 05 Ka 0 95 Ka Kb 0 ζ 0 5 ζ 0 6 56 Kb ζ 0 76 ζ 0 604 4 K ζ 0 940 ζ 0 96 9 b max 4 No inner-loop( ) 4 With inner-loop( ) 4 With inner-loop( ) If we fix the desired ζ for the CL system, then the larger the value of K b will mean a larger K a max, hence a larger gain margin for the design However, it may not always be desirable to use a large K b Overly increase in ζ will drag down the response speed of the CL system Often, rate data measurement is noisy Consequently, increase the value of K b will mean amplification of the noise signal Remaining difficulties of the system: With θ ref being the command, the problem of little maneuvering capability remains Pitching maneuver is often conducted by commanding the pitching rate Remedy of the problem: Replace θ ref with a pitch rate command, ie θ & ref The resulting design: A Pitch Orientation Control System 5
An attitude autopilot with maneuverability Pitch Orientation Control System Typical block diagram: θ Integrating Amplifier Elevator ref Gyro e e e Servo g δ δ a e e g ( θ & θ& ) dτ t 0 ref Rate Gyro A/C Dynamics Basic features and the design considerations: Provide the A/C with a stable longitudinal dynamics and a responsive pitch rate maneuver The responsive pitch rate maneuver is achieved by replacing θ ref with θ & ref A rate gyro feedback inner-loop remains, thereby maintaining the two-loop feedback structure An integrator (through the integrating gyro) is used to nullify the steady state output error The current feedback structure is the same as that of the Attitude autopilot e g e δ θ a p e ref Ka s s + p e δ H δe,θ θ -b(s+z) s +a s+a θ inner-loop K b Like previous designs, we will decide the inner-loop gain K b and the outer-loop gain K a The system can be analyzed through the same procedure as that performed previously 6
Special stability issue on attitude control designs Both the Attitude Autopilot and the Pitch Orientation Control System are basic to all longitudinal flight controllers Being the basic flight control systems, they must maintain A/C stability in all working ranges In general, this working range will cover both low AOA and high AOA conditions, where widely different longitudinal dynamics may appear at different AOA ranges For instance, high AOA pitch-up phenomena for high tail fighter aircraft 0 dc M / dc L > 0 C M 0 4 5 High AOA pitch up -0 Due to loss of tail lift U -0 4 5-0 High AOA 0 0 06 09 seperation A single feedback design must ensure CL stability in both working ranges Control design must proceed with two paths: (a) Low AOA case & (b) High AOA case Aircraft with pitch up at high AOA ranges: At low AOA: At high AOA: H H 5( s + 04) s + 09s + 8 9( s + 0) ( s + )( s 9) ( δ, &,low) Stable, without pitch up e θ ( δ, &,high ) Unstable, with pitch up e θ C L 7
Inner-loop root locus The samek b is applied to the both cases Solutions with K b 0 Solutions with K b 06 The low AOA case -0-8 -6-4 - 9 6 - e a e δ -0 δ e -b(s+z) θ s+0 s +a s+a -6-9 K b 9 The high AOA case 6 Increase in K b will move the unstable pole leftward, but will harm the others Often, we seek a tradeoff design -0-8 -6-4 - - -6-9 8
Outer-loop root locus The high AOA case 9 The selected values of K b and K a must be suitable for both cases Locus with K b 0 Locus with K b 06 Sloution with K b 06 and K a -0-8 -6-4 - e g θ e a ref s Ka inner-loop θ 6 - -6-9 9 6 Tune up K b will boost K amax, hence the gain margin -0-8 -6-4 - The low AOA case - -6-9 9
A simulation of the closed-loop system response: θ 8 ref (t) 4 θ (t) 0 0 deg 5 0 5 α(t) A/C is neutrally stable at this AOA 0 deg 5 4 deg 0-4 δ e (t) 0 5 0 5 t, sec 0 The closed-loop stability is maintained for both the low AOA and the high AOA regimes The pitch rate follows the command alright, though with a time delay Indicating maneuverability with control stick steeling Further comments on attitude control designs Though a pitch rate maneuver is emphasized in the design of the Pitch orientation control system, it is true that a pitch angle command may be more desirable in certain instances In these cases, an Attitude autopilot will be the basic attitude control design used Often, an attitude autopilot does not work along without some sort of velocity control Next 40