Aircraft Stability & Control

Transcription

1 Aircraft Stability & Control Textbook Automatic control of Aircraft and missiles 2 nd Edition by John H Blakelock References Aircraft Dynamics and Automatic Control - McRuler & Ashkenas Aerodynamics, Aeronautics and Flight Mechanics - McCormick Automatic Flight Control Systems - Donald McLean

2 Overviews of Aircraft Flight Control Designs 1

3 Aircraft flight and the control designs An aircraft flies in various steady state flight modes, and the pilot-actuated maneuvers are performed only during change of flight modes Takeoff maneuver Climb Level off maneuver Cruise Cruise to turn maneuver Taxi Steady state flight modes: Coordinated Turn Flare and touch down Decent Cruise to decent maneuver Cruise Turn to cruise maneuver In any steady flight, all accelerations (translational and rotational alike) of the aircraft are supposed to be zero This condition is maintained by various automatic control designs which are collectively called as the autopilots In other word, autopilots of an aircraft are automatic control systems designed to hold the vehicle in various steady state flight modes There are many different autopilots in an aircraft, with each autopilot being designed to perform a particular function There are also control designs to improve the stability of an aircraft These control designs of an A/C are called the stability argumentation systems (SAS) 2

4 Overviews of autopilot designs Design strategies of aircraft autopilots and SAS: Provide the aircraft with an acceptable level of stability Provide the aircraft with the specified command-response relationships Enable the aircraft to maintain the designated flight modes, suppress the effects of disturbances and variations of aircraft component Provide adequate maneuverability for steering the aircraft between flight modes Modify or, if possible, eliminate vehicle cross-coupling --- By vehicle cross-coupling we mean interactions between the longitudinal and the lateral dynamics of an aircraft In general, this coupling destabilizes the aircraft, and occurs when the vehicle excites a fast rolling motion Classifications on autopilots and SAS: Normally, the six degrees-of-freedom of an aircraft are separated into two uncorrelated groups of motion: the longitudinal motions and the lateral motions As a result, autopilots for the longitudinal motions, the longitudinal autopilots, and autopilots for the lateral motions, the lateral autopilots, are designed separately Basic features of autopilot designs: Feedback control strategies are fully employed Classical control methodologies are still the main design tool Multi-loop feedback structures are common place 3

5 A Reviews of Classical Control Theory and Aircraft Flight Mechanics Review of a closed-loop control design Closed-loop control design - Why using it? Closed-loop control design - A brief overview Root-locus based Classical Control Analysis Multi-loop feedback control design Reviews of A/C flight mechanics A/C motions and the controls Basic features of flight mechanics B Longitudinal Flight Control Systems Basic longitudinal autopilots Displacement autopilot Pitch orientation control system Integrated longitudinal control systems Glide slope coupler and Automatic flare control Flight Path Stabilization System Course Contents C Lateral Flight Control Systems Basic lateral controllers Dutch roll damper Lateral coordination controller Intermediate lateral autopilots Yaw orientation autopilot Heading autopilot Roll orientation autopilot Integrated lateral control systems Automatic turn maneuver Automatic lateral beam guidance D Further Discussions Effects of high roll rate Aircraft stability in the presence of inertial cross-coupling Stability argumentation of A/C with inertial cross-coupling Appendix A: A/C navigational aids Appendix B: Lateral controllability of a coordinated A/C 4

6 Reviews of Classical Control Theory and Aircraft Flight Mechanics 5

7 Review of a closed-loop control design Closed-loop control design - Why using it? Reason 1: The first goal of control design is to have a nicely stable system When the plant is barely stable or unstable, only feedback design can improve stability of the system 1 Consider a plant y( u( --- y: Output u: input s For the step input, u( t ) 1, t 0, or u( 1 / s, the open-loop response will be y( ; hence, 01 y( t) 10( e t 1), a divergent solution s( s 01 ) s 01 s To obtain a convergent output, let u r ay --- r: Command, a > 0 Re r( ay( --- Then, y (, which implies that y( r s s + a 01 1 s + ( a 01 ) ( ) --- For the same step input, this closed-loop system will respond as follows: y( s( s + a 01 ) a 01 s s + a 01 y t a t ( ) e a ( ( ) ) A convergent solution is obtained provided that a > Open-loop system response y(t) u(t) r(t) Closed-loop system response

8 Reason 2: It is vital that the system output follows the given command, and does so without regard to plant dynamics variation, and to the presence of disturbance This robust command following goal can be achieved only through feedback design, Consider a plant y( h( u( --- h(: Rational function, eq h( b ( s + a) For a stable plant, we may want the output y(t) to track some slow varying function f(t) --- We often represent the slowness of f(t) with a step function, f ( t ) c, and we want output y(t) to track f(t) asymptotically, namely we seek lim t y( t) f ( t) c --- It is impractical to seek y( t) f ( t) for all t and for any f(t), since it would require h( 1, s, a dynamic transfer function never exists --- We refer lim t y( t ) c to the steady state following of a slow command lim t y( t ) c can be achieved in open-loop sense as follows: --- Let u( t ) c, t 0 u( c / s y( h( c / s --- From the final value theorem: lim t y( t) sy( ch, h h( s s Then, an open-loop command following is achieved if we choose 1 / h 0 Drawbacks of an open-loop command following design --- Plant dynamics often vary due to change in environment or due to wear of its part( --- The model we used for design, y( h( u(, may fail to represent the true plant, which we shall put it as y( h ( u( --- Then, the open-loop design, u( t ) c, 1/ h0 will only result in lim s sy( c h / h, h h( s, and most often, c / h0 c 0 0 7

9 Closed-loop command following design : --- Let u( t ) k( c y( t)) u( k( c / s y( ) --- For the nominal design, we will have y( h( k( c / s y( ), which implies that kh( kh y( c ; hence, lim lim s( 1+ k h( ) t y( t ) s sy( 0 0 kh c For the true plant, we will have y( h ( k( c / s y( ), which then implies that kh( kh y( c ; hence, lim lim s( 1+ kh( ) t y( t ) s sy( 0 0 c 1 + kh0 We shall compare the closed-loop design with the open-loop design, --- Let h( 1 ( s + 01 ) and h( 1 ( s ); hence, h 0 10 and h For the open-loop command following design, namely u( t ) 01 c, t 0: AThe nominal design says that we will have lim t y( t) c B But to the true plant, it will result in lim t y( t ) 01 c h 0 0 5c A 50% error will occur! --- For the closed-loop command following design, namely u( t ) k( c y( t)) : 10k AThe nominal design says that lim t y( t ) k c c for k k B And to the true plant, it will result in lim t y( t ) k c c + 5 Tracking error < 2%! Even smaller error is possible with a larger k 8

10 Closed-loop control design - A brief overview Stability argumentation with a feedback controller - A formal treatment Plant equation: y ( s s ) ( ) b ( u ( s ) --- y: Output, u: Input, a(,b(: Polynomials a The open-loop system stability depends on the roots of a( If the roots of a( is not acceptable, we can use feedback controller: v( u( ( r( y( ) --- r: Closed-loop command, d(,v(: Polynomials d( so that the closed-loop system becomes: v( u b( s s y ( s ) b( ) v( ) a( d( + b( v( r( r - y d( a( Now, the stability issue depends on the roots of a( d( + b( v(, which can be adjusted with appropriately chosen polynomials, d( and v( --- If we let d ( s ) 1 and v( K P, then we have u( KP[ r( y( ], which mean u( t) KP[ r( t) y( t)], namely a proportional control --- If we let d ( s ) 1 and v( sk D, then we have u( KDs[ r( y( ], which mean u( t) KD [ r& ( t) y& ( t)], namely a differentiator control --- We can also let d ( s ) 1 and v ( s ) sk D + K P to form a PD controller --- An integrator control would mean d( s and v( K I In general, however, an integrator feedback will destabilize the system We will go back to this later 9

11 Closed-loop command following design It is shown that open-loop command following is not practical Even with a closed-loop design, steady-state following of slow command is the only practical goal to achieve/ Steady-state following of slow command in closed-loop sense, lim t y( t ) r( t), will be b( v( true provided that lim 0 1, which can be achieved if we let s a( d( + b( v( d( s then a( d( sa( which vanishes at s 0 K --- The design, d( s and v( K I, will correspond to u( P [ r( y( ], s which means u( t) KI [ r( τ) y( τ)] dτ, namely an integrator feedback control Because that an integrator feedback will destabilize the system, we may want to replace the design with a proportional control, d ( s ) 1, v( K P and K P >> 1 Then, b( v( K b( ) lim 0 lim P s s s 0 1, K p >> 1 a( d( + b( v( a( + K Pb( In theory, we can never achieve lim t y( t ) r( t) with a proportional control But a large K P will drive the command following error small enough to be ignored On the other hand, a differentiator control, d ( s ) 1 and v( sk D, will result in b( v( K lim lim Dsb( s 0 s 0 0 which means a total loss of a( d( + b( v( a( + K Psb( output at steady-state, a disaster in command following design 10

12 Command following and closed-loop stability at the presence of plant dynamics variation, and of the process disturbance Often, y ( s s ) ( ) b ( u ( s ) does not model the true plant A more appropriate model will be a b( y( + Γ( u( + γ( where Γ ( and γ ( s ) are frequency domain envelope a( functions representing, respectively, the uncertain modeling error of the plant and the unknown, and most likely stochastic, process disturbance When the feedback design is used, the closed-loop system will become: [ H( + Γ( ] 1 y( r( + γ( 1+ [ H( + Γ( ] 1+ [ H ( + Γ( ] where H ( b( / a( and G ( v( / d ( For command following design, we want to set G ( H ( 1 so that [ H( + Γ( ] 1+ [ H ( + Γ( ] s0 >> 1 1, and 0 1+ [ H ( + Γ( ] s 0 s 0 t 0 y( t) sy( r lim r( t), r( r / s s 0 t hence, lim With an integrator control, G ( KI / s, hence s )H( --- For a proportional control, G ( KP; hence, a high gain design, K P >> 1, is needed in order that the condition, G ( H ( 1, can be met s0 >> s0 11

13 The above result applies only if the closed-loop system is stable Stability of the system now depends on the roots of the denominator polynomial Φ ( a( d( + b( v( + a( v( Γ( We may write Φ( a( d( ( 1+ H ( + Γ( ) a( d( { 1+ H ( ( 1+ Ψ( )} where H ( b( a( is the transfer function of the plant, G ( v( d( that of the controller, and Ψ ( Γ( H ( the fractional error of the model Then, the closed-loop system will be stable provided that the polar plot of the function H( jω) jω) ( 1+ Ψ( jω) ) satisfies the Nyquist stability criteria --- The polar plot of H( jω) jω) ( 1+ Ψ( jω) ) must not encircle the point (-1,0) --- Since Ψ ( jω) is unknown, we can only make the polar plot of H ( jω) jω) --- In order to ensure closed-loop system stability, we embed gain margin (GM) and phase margin (PM) into the design to account for differences in magnitude and phase of ( jω) jω) H( jω) jω) 1+ Ψ( jω) H and ( ) ~ ( ) Π jω -1 Need GM > Π (jω) - Π (jω) jν Need PM > Π (jω) - Π (jω) Π( jω) Π (jω) Π (jω) Π (H((1+Ψ () Π (H( -1 υ ~ ( ) Π jω Π( jω) Π (jω) Unit circle jν Π (jω) υ 12

14 Root-locus based Classical Control Analysis Root locus shows the roots of Φ( a( d( + b( v( as the controller gain varies To systematize the analysis, we can rewrite Φ( into Φ( ~ ~~ a ( + K b ( : --- We can consider ~ ~ a( s ) and b( s ) as the denominator and the numerator of a generalized ~ loop transfer function, and K the generalized gain of the design --- For P-control: Φ( a( + K b( ~ ~ ~ a( a(, b( b(, K KP P --- For D-control: Φ( a( + K sb( ~ ~ ~ a( a(, b( s b(, K KD P --- For I-control: Φ( sa( + KPb( ~ ~ ~ a( sa(, b( b(, K K I ~ Then, we will examine the roots of Φ( as K changes from 0 to infinity Some fundamentals about root locus analysis: Φ( ~ ~ a ( s ) at K 0 and Φ( ~ ~ b( s ) as K ; hence, all root locus will start from the roots of ~ ~ a( s ) and end at the roots of b( s ) LHP If ~ ~ p3 jω a( s ) has n roots and b( s ) has m roots, then there will be Open-loop poles Open-loop zeros RHP p n m locus, called the asymptote(, that end at infinity For p 3, portion of the asymptote will fall into the RHP, indicating unstable closed-loop design for large feedback gain p 3 is common for physical systems; hence, using feedback for command following purpose often destabilizes the system Good command following calls for high gain design, which is often Asymptote unacceptable as closed-loop system stability is concerned 13

15 Some design guidelines about root locus analysis: A leftward, or inward, movement of the locus is stabilizing On the other hand, a rightward, or outward, movement of the locus is destabilizing Normally, zero( attracts the locus while pole( repeals them Outward jω ~ --- In general, adding real and LHP zeros into b( s ) Oscillation pulls the Rightward locus inward, thereby stabilizing the design Convergence ~ --- But adding poles into b( s ) pushes the locus outward and Leftward Convergence rightward, thereby destabilizing the design Adding zero( into ~ Inward b( s ) also increases m hence decreases p If Oscillation p can be reduced to the extent that p 2, all locus will remain ~ in LHP for all K Adding poles into the system, however, does just the opposite ~ In general, differentiator feedback adds zero( into b( s ), hence is good for stability Conversely, integrator feedback adds pole( into ~ a( s ), hence is bad for stability PD feedback, K P zk D -z One zero added, p2 jω RHP Differentiator feedback jω LHP Locus remains entirely in LHP Enhance stability in this region One zero added, p2 RHP Basic design LHP Proportional feedback, p3 jω RHP Integrator feedback LHP Locus move to RHP faster than with p3 One pole added, p4 In reality, differentiation of measured signal will amplify the noise and is rarely used jω RHP 14

16 Multi-loop feedback Control Design Often, engineering systems are designed for command following operations However, the use of closed-loop command-following design often destabilizes the system Need to enhance system stability before doing closed-loop command-following design The whole design will consist of two procedures: (a) Improve (or say enhance) stability of the system (b) Apply closed-loop command following design on the stabilized system Both procedures will involve feedback control synthesis; the two-step feedback control design forms a two-loop feedback structure An illustrative example: --- Open-loop plant: y ( h( u( A direct closed-loop command following design: u( K[ r( y( ], K > The closed-loop root locus will rush into RHP (right figure), indicating a small working range for K A two-step command following design: Step 1 Enhancement of system stability A PD feedback stabilization is applied: u( K1( s + γ )[ u( y( ] --- γ : The pre-selected controller zero --- u ( : An intermediate command The resulting closed-loop root locus (figure right) indicates that the closed-loop stability is improved with the increasing of K 1 u - u K 1 (s+γ) Open-loop zeros Open-loop poles h( Stability enhancement with PD feedback u(k 1 (s+γ)[u(-y(] --- One zero added y -γ jω jω 15

17 The following procedure will allow us to determine a desired closed-loop design, and compute the corresponding value of K 1 1) Select, from the locus, desired locations for the dominant jω closed-loop poles --- Say, the red triangles in the plot 2) Plot vectors that connect this location to each and every open-loop poles and zeros on the locus, including the 3 controller zero --- The green and the purple vectors d d 1 p p 3) Measure the lengths of these vectors; denote the lengths 2 i of those vectors that connect the open-loop poles as d p, d z 1 d z j and the lengths of those vectors connect to zeros as d z -γ d 2 4) Let K 1 denotes the value of K 1 such that this design is p achieved Then, we can compute K 1 as follows: d 4 p K1 ( d p d p L ) /( dz d z L) Resulting closed-loop equation of the stabilized system: The desired closedloop pole locations K1h( y( u( 1+ K1( s +γ ) h( The remaining poles of the closed-loop system (the blue triangles in the plot) can be obtained from the above closed-loop equation Remark: The use of u( K1( s + γ )[ u( y( ] will require data for y& (t) If this data is not measurable, differentiation on the measured signal of y (t) is needed In practice, differentiation of the measured signal often amplify the noise level 16

18 Step 2 Command following design with the stabilized system Command following control law: u( K2[ r( y( ] u r K By building the control law on u (, a two-loop feedback structure is formed u K 1 (s+γ) h( y --- The stability enhancement portion of this structure is called the inner-loop design, while its command-following portion is termed the outer-loop design Procedure of the outer-loop design: 1) Perform outer-loop design only after the inner-loop design is done In this case, after K1 is computed and all of the closed-loop poles u K of the inner-loop design are determined r K 1 h( 2) Redraw the block diagram, by replacing the K1 (s+γ)h( y inner-loop with its closed-loop design --- Note that the closed-loop system of the inner-loop design will be used as the openloop system for the outer-loop design 3) Perform outer-loop analysis accordingly Select the final closed-loop design and Closed-loop poles of the inner-loop design jω compute the appropriate value for K 2 from the outer-loop locus, by using the same procedure we did in the inner-loop design 4) Re-do Step 1, if no satisfactory outer-loop design can be found Remark: Other control structure with more than two feedback loops may also exist Open-loop zeros of the system Desired closed-loop poles of the outer-loop design -γ 17