Lecture 3 Basic Feedback

Transcription

1 Lecture 3 Basic Feeback Simple control esign an analsis Linear moel with feeback control Use simple moel esign control valiate Simple P loop with an integrator Velocit estimation Time scale Cascae control loops Control Engineering 3-

2 Feeback Stabilit State Space x t Ax + Cx Bu Close-loop namics x t A K x + B K Simple feeback control u K ) A K B K ( A BKC BK Stabilit is escribe b the close-loop poles { λ j} eig( AK ) Control Engineering 3-2

3 Close-loop eigenvalues F6 Longituinal Moel Example A C 57.3 [ ] B Roots poles eigenvalues eig( A BKC Can be plotte for ifferent gains K Root locus plot ) % Take A from % the F6 example >> eig(a) ans i i.97 % Close-loop poles >> K -.2; >> eig(a-b*k*c) ans i i Control Engineering 3-3

4 Close-loop poles Transfer function poles tells ou everthing about stabilit Moel-base analsis for a simple feeback example: u H ( s) u K( ) H ( s) K L( s) + H ( s) K If H(s) is a rational transfer function Then L(s) also is a rational transfer function Stabilit is etermine b the poles of L(s) Same results as for the state space analsis Control Engineering 3-4

5 Control of a st orer sstem Simplest namics, st orer sstem Simple feeback works just fine Static output feeback is sometimes calle P control P proportional The name P is use in process inustries an in servosstems, less in flight control Close-loop namics are ver well unerstoo Can be use as a esign template for more complex sstems, cascae loops Control Engineering 3-5

6 Control of a st orer sstem First orer sstem, integrator namics x t x + bu x A B C b P Control u k ( ) Close loop namics x t kb ( ) Eigenvaluepole λ kb Control Engineering 3-6

7 P control - example Integrator plant: & bu + w (t) -u(t) P controller: u k (t) P( ) Example: Utilization control in a vieo server Vieo stream i processing time c[i], perio p[i] CPU utilization: U[i]c[i]/p[i] u(t) (t) (t) -(t) CPU amission rate server utilization completion rate t U new Control Engineering 3-7 t U one

8 P control Close-loop namics & + bkp bkp + w Stea-state (s ) Step response: T ( t) () e t / T + Frequenc response (banwith2π/t) ( t) ˆ ( iω) e ( t) ˆ( iω) e iωt iωt + ˆ( iω) SS bk P /( bkp ) w SS + bk P w bk P s + bk SS ( t / T e ) s + bk -5 ˆ ( iω) + wˆ ( iω) /( bkp ) - (iω) ω /( bkp ) Control Engineering 3-8 P + (t) P w 2 4

9 Control an Error Peaking Fast poles are not necessaril goo This might mean large peak resonse Example: P control of an integrator h * - close-loop impulse response u k h * - control impulse response P h( t) bk exp P ( bk t) P fast response slow response Engineering esign is a series of traeoffs Control Engineering 3-9

10 I control th orer (feethrough) sstem u + w, Introuce integrator into control Example: flow through a valve in-flow u(t) u& v, v k ( Close-loop namics I k I s + k I ) + s s + k I w valve out-flow Valves: (t) Mechanical: flui or gas Electrical: power Computing: tasks Comm: packets Control Engineering 3-

11 P an I control P control of an integrator u k P( & bu + w ) w - b k p I control of a th orer sstem. Basicall, the same feeback loop w u& k I u + ( w, ) k I - Control Engineering 3-

12 First orer estimation - ifferentiator Differentiating filter Velocit estimation: v xˆ Observer L( t ˆ xˆ ˆ) Moel: xˆ vˆ t x v t x L( ˆ) Velocit estimation filter ˆ s L v + s x ŷ - L vˆ Control Engineering 3-2

13 First orer estimation example Input Signal INPUT SIGNAL Feeback gain L ESTIMATED DERIVATIVE v ˆ Output Signal s + L v s Control Engineering TIME 3-3

14 Simplifie esign an analsis Simple esign moel Often th or st or 2 n orer moel Will consier tpical examples Use cascae loops Approximate moel, robustness Analze using a more etaile linear moel Valiate through simulation, Control Simple Design Moel Control Detaile Simulation Moel Control Engineering 3-4 Valiation an verification Design an analsis

15 Example F6 longituinal moel, constant velocit Assume that the velocit is maintaine b regulating thrust V & α 2. α +. 9q 25. & θ q q&. 82α 8. q 8. δ e -3 δ e θ α V x 57.3θ z u δ e δ e x t [ 57.3 ] x C x + u A B >> eig(a) ans Control Engineering 3-5

16 F6 Attitue Control Simulate step response At slow time scale, the integrator namics are ominant Approximate b a simple integrator moel θ 2δ e t STEP RESPONSE TIME (s) Control Engineering 3-6

17 F6 Attitue Control P control esign for an integrator Simple moel θ k P.5 T P /(2k ) δ ( e k P θ θ ).4 Time responses for the simple moel an for the etaile moel STEP RESPONSE θ t 2δ Detaile moel e θ δ ( e k P θ θ ) TIME (s) x& θ Ax Cx + Bδ e Control Engineering 3-7

18 Time scale The same plot at ifferent scales Banwith /Timescale Simple 2 n orer moel example: H(s) /(+s+s 2 ) Time scales:.4 Fast Intermeiate Slow H /s 2 H /(+s+s 2 ) H Control Engineering 3-8

19 Time Scale an Frequenc Response Time scales: Frequenc response for the example: H(s) /(+s+s 2 ) Magnitue (B) Slow Boe Diagram 5 Banwith/Timescale The banwith is limite b moel uncertaint: Lectures 9- Phase (eg) Intermeiate Fast Control Engineering Frequenc (ra/sec ) 3-9

20 Feeback loop time scale Slow feeback loop I control Plant as a feethrough Fast feeback loop P control, plant as an integrator PD control, plant as a ouble integrator Control Engineering 3-2

21 Cascae loop esign Inner loop has faster time scale than outer loop In the outer loop time scale, consier the inner loop as a th or st orer sstem that follows its setpoint input outer loop setpoint Plant output - Outer Loop Control inner loop setpoint - Inner Loop Control inner loop Control Engineering 3-2

22 Servomotor Spee Control Example Power amp Motor sensor control voltage u The control goal is to track a velocit setpoint Mechanical time constant T J is ominant. Use simple moel controller spee v - setpoint v s G T J u moel Jv& + bv ci LI& + RI u v Transfer function G ( + TJ s)( + TI s) Control Engineering 3-22 T J.sec, TI G u.2 sec

23 Servomotor Example, cont Design P control u k v v ) for the simple moel v u V OPEN LOOP STEP RESPONSE ( s G T J k T V loop k P G / T J CLOSED LOOP STEP RESPONSE Simplifie moel Detaile moel Control Engineering 3-23

24 Servomotor Position Control Power amp Motor sensor control voltage u spee control - spee v setpoint v position control - position x setpoint x Cascae with the spee control loop The control goal is to track the position setpoint Spee loop integrator iels the ominant namics Use simple moel of the plant (inner loop) x t s Control Engineering 3-24 v u

25 Servomotor Position, cont v s Design P control v k ) for the simple moel P ( STEP RESPONSE OF POSTION WITH OUTER LOOP OPEN k T P loop 2 k P CLOSED LOOP STEP RESPONSE Simplifie moel Detaile moel Control Engineering 3-25

26 Electric Motor Servo Broal available proucts 2-4 cascae loops epening on the sensor harware, motor harware, the application, an the require control functions Lecture 6 Control Engineering 3-26

27 Aircraft Cascae Loops In practice, multivarable control esign might be one for attitue control Otherwise, aircraft is represente as a chain of several integrators Control Engineering 3-27

28 Basic cascae loops in aircraft FMS/MMS Guiance an Autopilot Wapoint Altitue, coorinates Translational position Commane airspee Translational velocit Flight Control θ x α V Flight Actuators Actuators Attitue comman Angular position; Attitue Angular rate comman Angular rate Elevator position Actuator servos Control Engineering 3-28

29 Aircraft Cascae Loops Embee servo avionics Actuators, hz banwith Flight Control box Angular rate, 2Hz banwith Angular position,.5hz banwith Autopilot/Guiance Translational velocit, 5 sec Translational position, 3 sec FMS - Flight Management Sstem Wapoint, - sec Control Engineering 3-29

30 Cascae Loop Example Descent/Abort Guiance Dale Enns, Honewell, 989/997 X-38 - Space Station Lifeboat Control Engineering 3-3