Presentation Topic 1: Feedback Control. Copyright 1998 DLMattern
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1 Presentation Topic 1: Feedback Control
2 Outline Feedback Terminology Purpose of Feedback Limitations of Feedback Linear Control Design Techniques Nonlinear Control Design Techniques Rapid Prototyping Environments Summary
3 Feedback: Terminology r + e u P(s) K(s) Plant Model, inclues G(s), L(s), d, n G(s) actuators Plant sensors d + y n + z L(s) r = reference input e = error signal u = plant input y = plant output d = system disturbance n = measurement noise z = measured output signal P(s) = Reference Signal Prefilter/Conditioning L(s) = multiplicative unstructured uncertainty K(s) = control system G(s) = design plant S(s) = Sensitivity Function = (I+GK) -1 T(s) = Complementary Sensitivity Function = 1-S(s) = (I+GK) -1 GK
4 Purpose of Feedback Provide better performance tighter regulation and tracking better disturbance rejection increased range of stable operation Provide higher reliability and repeatability remove human operators allowing multivariable control Provide lower cost operation less wear and tear leads to less maintenance tighter control of operation reduces the factor of safety required for other components in the system allowing a synergetic reduction of component requirements
5 Limitations of Feedback Feedback can make a stable system unstable There are physical limitation to what you can accomplish because of actuator rate and range limits, sensor time and spatial resolution, controllability, observability and the linear control design tools. The feedback controller is the result of one step in a design process. The other steps must be tested and validated for the entire process to be successful. Control design begins with understanding the system physics. Modeling and control go together.
6 Limitations of Feedback Limits of Performance, plot of loop gain G(s)K(s) mag (db) High gain at low frequencies provides robustness to disturbance and steady state regulation. Because S+T=1, typically the best trade-off is to have a -20 db/decade slope of the loop gain magnitude plot at the crossover frequency. 0 db freq (rad/sec) High frequency rolloff required because of model uncertainties.
7 Limitations of Feedback Limits of Performance, S+T=1 mag (db) e/r=s(s) freq (rad/sec) mag (db) y/r=t(s) Good control (small error) is possible at those frequencies where the uncertainty is smallest. As uncertainty increases at higher frequencies, closed loop performance must fall off. freq (rad/sec)
8 Linear Control Design Techniques Algebraic Matrix Ricatti Equation LQ - Linear Quadratic LQG - LQ Gaussian (noise terms - Kalman Filter) H 2 - Performance Index is the system H 2 norm. These methods still have their place in specific applications, (Kalman Filter for example), but generally other techniques have replaced them on multivariable problems in order to incorporate uncertainty models and to build in both stability and performance robustness.
9 Linear Control Design Techniques Small Gain Theory: Norms, Singular Values sufficient condition, bounds for stability for MIMO systems QFT - Quantitative Feedback Theory (Freq. Domain) LQG/LTR LQG/Loop Transfer Recovery» only provides robustness properies at plant input OR output H-infinity-minimize infinity norm (worst case response)» appropriate frequency weights are required to obtain stability robustness (weighted S, T frequency functions) Mixed Sensitivity Problem, include input sensitivy function to weight actuation requirements. µ-synthesis - robust stability AND performance» using the structured singular value» An approximate solution approach: D-K iteration
10 Why use a linear design method? It is the only method that offers guarantees, even if they are only valid locally. techniques to analyze the robustness to variations, other than brute force simulation. the best multivariable design tools. Even if you plan to use another control design method, you should start with a linear design as part of the analysis of the local linear system. You will understand the physics of the system better after this design.
11 Linear Control Design: How much effort? A linear design is only 15% of the total design. You want an automated method that yields stability robustness; is repeatable; and allows the pieces to be assembled together (scheduled) for the over all design. Large transient operation is typically limit operation. transient cycle deck linear point models & analysis mode selection & logic design operating point linear control designs tie together point designs & handle nonlinearities evaluate discrete controller combined with nonlinearities
12 Linear Control Design Tools If you have the µ Analysis Toolbox from the MathWorks and are familiar with these tools, then use them. The Robust Controls Toolbox from the MathWorks is suffice for robust designs using H-infinity and appropriate frequency weights. Engines typically do not have a large number of inputs, so if you choose to use single loop design methods, make sure you do multivariable robustness testing and not just look at phase and gain margins.
13 Linear Control Design Example: Introduction Example: linear F100 engine model. 2 inputs: fuel flow, nozzle area 2 outputs, low rotor speed, engine pressure ratio. 7 states, low speed rotor, high speed rotor, 3 temperatures, engine pressure ratio. Key: You have to factor in fuel accel/decel schedule into actuator design bandwidth otherwise you ll spend all you time on the limits. Re-visit this topic when we cover control design.
14 Linear Control Designs: Additional Techniques Controller Order Reduction High order control designs result from the incorporation of system uncertainty models. Robustness of the reduced order controller must be checked so that the robustness built-in to the original design is maintained in the reduced order design. This may require some iteration using standard model order reduction techniques, (see the frequency weighted model order reduction technique of Dale Enns, Model Reduction for Control System Design, Ph.D. Thesis, Stanford Univ., June 1984). Also, the resulting controller order size must fit into any preselected controller size for gain scheduling, if used, otherwise you have to worry about controller tracking for controllers with different number of state variables.
15 Nonlinear Control Design Techniques In addition to the Linear Design Method Limits» operational limits» actuator rate and range limits Integrator Windup Protection Gain schedule linear controllers of the entire flight envelope In place of the Linear Design Methods Fuzzy Logic» becomes difficult to use as the system size inceases because of you have to manage all the variable relationships. Nonlinear Optimization» Parameter Optimization, Neural Networks, Genetic Alogirthms
16 Limits: Operational Limits Accel/Decel Fuel Schedule: rich/lean flame blowout, Compressor Surge, Max. Turbine Temperature rotor + acceleration Rich Blowout Maximum Turbine Temperature Compressor Surge Use these limits to scale the fuel actuator in the linear design at an operating point. Linear design is symmetric (+/-) % of design Lean Blowout rotor speed Lines of Constant Fuel Flow
17 Limits: Operational Limits Typical Engine Accel/Decel Fuel Schedule Fuel Flow Divided By Compressor Pressure ( WF/PS3 ) Lines of Constant Power Steady State Line Accel Step Deceleration Limit 20% 40% 60% 80% 100% 120% Corrected Rotor Speed (% of Design) Acceleration Limit Line Symmetric bounds on WF/PS3 are reflected back onto the fuel flow actuator for the linear design.
18 Limits: Compressor Operation Fan Compressor Map Surge Line Fan Average Pressure Ratio 1.0 Surge Margin increase in nozzle area accel decel Fan InletCorrected Mass Flow Rate (% of Design Mass Flow Rate) decrease in nozzle area Controller Surge Limit Line 100% Steady State Operating Line Lines of Constant Corrected Percent Rotor Speed
19 Limits: Actuator limits Rate and Range Limits of Actuators Typically the small signal hydraulic actuator rate and range limits are more than sufficient for the linear control design and the constraints used in the linear design will be those imposed by the operational limits. Problems with actuator are more typically due to the servoloop nonlinearities like:» deadband which causes the control actuator value to hunt.» stiction which can be corrected using a small dither. More electric designs may have to revisit voltage and current requirements to avoid any power availability issues.
20 Limits: Actuator limits Using this information in the control design Operational Limits» Reflect back to actuator limits in linear design Actuator Limits Accel/Decel - Fuel Flow Actuator Bounds Blow-out & Rotor Overspeed - Linear System Scaling Stall Margin - combination fuel, vanes, nozzle area» Rate limits - linear actuator bandwidth contraint» Range limits - considered in linear system scaling.
21 Integrator Windup Protection Limits define the maximum system performance actuator limits bound control effectiveness operational limits (ex. stall margin, temperatures) bound the safe operating range Linear stability analysis fails during limit operation. the loop gain effectively changes when a limit occurs the loop phase effectively changes due to the delay cuased by a controller integral unwinding.
22 Integrator Windup Protection Multivariable Windup Protection Structure Controller r + e u P(s) D c B c 1/sI C c Known Limit u L G(s) z A c x c Λ x c =(A c -ΛLC c )x c +(B c -ΛLD c ) e y +ΛLu L L is a diagonal matrix of logic indicators. A one in element (j,j), indicates actuator #j is on a limit. Design of L needs to track limit without causing instability to the closed loop system.
23 Gain Schedule Linear Controllers Making all the controllers the same size simplifies the scheduling of the A c, B c, C c, D c, matrices e D c B c 1/sI C c u1 The controller outputs should match to allow for bumpless transfer. x c control #1 A c If you don t force the controller outputs to match, then you ll need to blend them during transistion from 1 to 2 and from 2 to 1. u = α u1 + ( 1-α )u2, 0< α <1 e D c B c control #2 1/sI A c x c C c u2 Need to check the controller robustness properties near the transistion regions.
24 Nonlinear Example: Neural Networks For Model Based Diagnostics Example Application: nonlinear observer to detect and accommodate fuel metering valve sensor fault. For dual redundant FADEC s, the observer signal provides a 3rd signal allowing a voting scheme to be used. FADEC PLANT PLA ALT XM P2, T2 outer loops servo controller WFMA Actuator Servos WF, CVG Engine MMVFB_est MMVFB Observer WFMA MMVFB XNH
25 Neural Networks can be used for diagnostics Two part observer: dynamic and steady state correction WFMA fuel servo milli-amp demand Speed Error MMVFB Servo Actuator Dynamic Model First Order Model With Deadband and Limited Integral Scaling MMVFB_est Rotor Steady State MMVFB Tracking Based on Measured Variable (XNH) Speed XNH XNH_est Neural Net #2 WF_est Neural Net #1 Functional Approximation * Functional Approximation * NN#2 actually is based on corrected speed and correct fuel and therefore requires P2, T Neural Networks handle the nonlinearites of full operating range.
26 Neural Networks can be used for diagnostics Neural Network used for nonlinear curve fitting MMVFB (inches) WF vs MMVFB WF (pph) 1.8 x 10 XNHc vs WFc XNHc (corrected) WFc (corrected) curves valid for 1260 steady state operating points mmvfb Observer Transient Response corrected fuel flow PLA Slam alt=0, xmn=0, dtamb=0 mmvfb_est mmvfb time (seconds)
27 Control Design is a Process The process should be refined and optimized like any other process. modeling model validation model analysis & mode selection operating point linear control designs tie together point designs & handle nonlinearities evaluate discrete controller combined with nonlinearities hardware testing hardware Implementation, realtime simulation & testing
28 Control Design in a Process The MathWorks Rapid Prototyping Tools Make the process repeatable, efficient, and and build reusable components. Simulink model validation model analysis & mode selection Control Design Toolboxes Simulink Nonlinear Models discrete controller testing on nonlinear models hardware testing Realtime WorkShop Realtime Testing
29 Summary Feedback improve your overall product. Feedback won t correct poor mechanical and fluid designs. It will only extract the maximum capability out of the system. Linear control design is contracted to small perturbation responses around an operating point. While stability guarantees are available, this is only for the small perturbation response. Nonlinear design techniques are best suited to the outer loop. The key to feedback control design is the understand of the system physics, thus modeling is an integral part of control design process. Control Design is a process. This process can be refined and optimized like any other process. Rapid prototyping tools will assist in this optimization process.
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