Ultimate State. MEM 355 Performance Enhancement of Dynamical Systems

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1 Ultimate State MEM 355 Performance Enhancement of Dnamical Sstems Harr G. Kwatn Department of Mechanical Engineering & Mechanics Drexel Universit

2 Outline Design Criteria two step process Ultimate state error (this lecture) Transient response (following lectures) Ultimate Error Response to Command ~ Unit Feedback Sstems Tables ~ Franklin p. 35, Nise p. 383 Ultimate Error Response to Command ~ General Case Example Error Response to Disturbance

3 Control Sstem Performance Command tracking Match output to changing command (minimize/eliminate error) Shape transient response (overshoot, damping) Disturbance rejection Minimize/eliminate output error while sstem is subjected to disturbance Shape transient response (overshoot, damping) Stabilization Insure adequate degree of stabilit (deca rate, damping, pole locations) Insure stabilit robustness (gain & phase margins) Handling qualities From perspective of operator (pilot, driver) maneuverabilit, stabilit Reliabilit, Safet emerging theor

4 Unit Feedback Sstems e, error - G s c G ( p s ) p c G Y + G G s G s G s Y s E s + G s Y s open loop transfer function output response transfer function (closed loop) error response transfer function (closed loop) L, then plug in to get ( ) et Es t e or, much easier, use Final Value Theorem: e lim se s

5 Error Response to Polnomials Test inputs: polnomial in t, step, ramp, parabola,... step: u t s e lim s + G s s + lim G ramp: tu t s e lim s + G s lim s sg parab: tut s e( ) lim s G s s Position constant: k Velocit constant: k Acceleration constant: k lim s p v limg lim sg s a G s li m s G

6 Transfer Function Tpe A transfer function G s of the form n s G K where s is not a factor of d s n s k sd s or is said to be of tpe k. Note that: 0 k 0 c 0 k 0 kp limg kv lim sg c 0 k k k 0 k 0, ka lim s G c 0 k k 3

7 Ultimate Error Table ~ Unit Feedback Sstems Tpe 0 Input + k u(t) 0 0 p tu(t) 0 k v t u(t)/ k a

8 Example Consider a unit feedback, tpe sstem with G s s s s+ Error responses for standard step, ramp, parabola e t Output responses for standard step, ramp, parabola ,, 5, Step response t Ramp response t Parabola response t

9 General Case Gs () - H The closed loop input response transfer function is G G + GH The error response transfer function is (recall e ) e G s G s e ( ) lim sg Y e + GH G + GH

10 Ultimate Error Table ~ General Case Tpe 0 Input limg u(t) e 0 0 tu(t) lim sg e 0 t u(t)/ lim sge

11 Example: Power Sstem Load Tracking Tpical dail load pattern Load Each power plant that participates in load tracking receives a command of similar shape t

12 Example, cont d Area Controller k Plant Required generation - G k Plant Total generation k n Plant n Area Controller k, i,, n participation factors i k + k + + k n Required Generation - Gc Aggregate Power Plant Total Generation

13 Example: cont d - Aggregate Plant 0. 5 s + s 00s + controller plant Simple power plant model. e Error response to tpical Load command Step Response Note: On a 3000 MW sstem we have an error of ~ 5 MW t t

14 Example: cont d Area Controller Aggregate Plant Required Generation - Gc s + s 00s + Total Generation Outer Loop Compensator: Tpe : Tpe : G G c c s + s+ s + s+ s s e Original response Tpe Tpe t Error responses with and without outer loop

15 Ultimate Error Due to Disturbance d - G () s G () s H G Transfer Function d e: Ged + GG H e lim sg s D s ed

16 Example: Step Disturbance ( ) lim e sg s D s To reduce error: step disturbance: D s / s e lim + GG H s increase tpe or gain of reduce gain of G ed sg GH lim + limg G ( tpicall not possible) H

17 Example G, H s s ( + 5) ( a) : P 000 G s + ( b) : PI 000 s Response to command ( + ) Response to disturbance - G () s H s s 5 a Ge, G de s + 5s+ 000 s + 5s+ 000 s( s+ 5) e ( ) lim s 0, e lim 0 d s s 0 s + 5s+ 000 s s s + 5s+ 000 s 000 d G () s First confirm that closed loop is stable!! ( + 5) s s s, G 3 de 3 b Ge s + 5s + 000s+ 000 s + 5s + 000s+ 000 s ( s+ 5) s e ( ) lim s 0, e li 0 3 d ms 0 s 3 s + 5s + 000s+ 000 s s + 5s + 000s+ 000 s

18 Summar Concept of ultimate error, Formulae for unit feedback sstems in response to commands, Introduction of error constants, The concept of sstem tpe and its role in command tracking, Formulae for general feedback sstems in response to commands and disturbances.

Steady State Errors. Recall the closed-loop transfer function of the system, is

Steady State Errors. Recall the closed-loop transfer function of the system, is Steady State Errors Outline What is steady-state error? Steady-state error in unity feedback systems Type Number Steady-state error in non-unity feedback systems Steady-state error due to disturbance inputs