Bài giảng Electric circuit theory - Chapter IV: Feedback control system characteristics - Nguyễn Công Phương

Nguyễn Công Phương CONTROL SYSTEM DESIGN Feedback Control System Characteristics Contents I. Introduction II. Mathematical Models of Systems III. State Variable Models IV. Feedback Control System Characteristics V. The Performance of Feedback Control Systems VI. The Stability of Linear Feedback Systems VII. The Root Locus Method VIII.Frequency Response Methods IX. Stability in the Frequency Domain X. The Design of Feedback Control Systems XI. The Design of State Variable Feedback Systems XII. Robust Control Systems XIII.Digital Control Systems sites.google.com/site/ncpdhbkhn 2 Feedback Control System Characteristics 1. Introduction 2. Error Signal Analysis 3. Sensitivity of Control Systems to Parameter Variations 4. Disturbance Signals in a Feedback Control System 5. Control of the Transient Response 6. Steady – State Error 7. The Cost of Feedback 8. Control System Characteristics Using Control Design Software sites.google.com/site/ncpdhbkhn 3 Introduction (1) • An open – loop system operates without feedback & directly generates the output in response to an input signal. • It is highly sensitive to disturbances & to changes in parameters of the process. sites.google.com/site/ncpdhbkhn 4 Process Desired output response OutputActuatorController Open – loop control system (without feedback) Disturbance Introduction (2) • A closed – loop system uses a measurement of the input signal & a comparison with the desired output to generate an error signal that is used by the controller to adjust the actuator. • Advantages: – Decreased sensitivity of the system to variations in the parameters of the process. – Improved rejection of the disturbances. – Improved measurement noise attenuation – Improved reduction of the steady – state error of the system. – Easy control & adjustment of the transient response of the system. sites.google.com/site/ncpdhbkhn 5 Process Desired output response Actual outputActuatorController Closed – loop control system with external disturbances & measurement noise Sensor (–) Error Measurement output Feedback Measurement noise Disturbance Feedback Control System Characteristics 1. Introduction 2. Error Signal Analysis 3. Sensitivity of Control Systems to Parameter Variations 4. Disturbance Signals in a Feedback Control System 5. Control of the Transient Response 6. Steady – State Error 7. The Cost of Feedback 8. Control System Characteristics Using Control Design Software sites.google.com/site/ncpdhbkhn 6 sites.google.com/site/ncpdhbkhn 7 Error Signal Analysis (1) ( )Y s( )R s ( )N s ( )G s ( ) ( )dT s ( )cG s ( )H s ( )aE s Controller Process Sensor ( ) ( ) ( )E s R s Y s  ( ) ( ) ( ) ( )( )( ) ( ) ( ) ( ) 1 ( ) ( ) 1 ( ) ( ) 1 ( ) ( ) c c d c c c G s G s G s G sG sY s R s T s N s G s G s G s G s G s G s      ( ) ( )1 ( )( ) ( ) ( ) ( ) 1 ( ) ( ) 1 ( ) ( ) 1 ( ) ( ) c d c c c G s G sG sE s R s T s N s G s G s G s G s G s G s       ( ) ( ) ( )cL s G s G s 1 ( ) ( )( ) ( ) ( ) ( ) 1 ( ) 1 ( ) 1 ( )d G s L sE s R s T s N s L s L s L s       sites.google.com/site/ncpdhbkhn 8 Error Signal Analysis (2) ( )Y s( )R s ( )N s ( )G s ( ) ( )dT s ( )cG s ( )H s ( )aE s Controller Process Sensor ( ) ( ) ( )cL s G s G s ( ) 1 ( )F s L s  1 1( ) ( ) 1 ( )S s F s L s   ( )( ) 1 ( ) L sC s L s   ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )dE s S s R s S s G s T s C s N s   ( ) ( ) 1S s C s  Feedback Control System Characteristics 1. Introduction 2. Error Signal Analysis 3. Sensitivity of Control Systems to Parameter Variations 4. Disturbance Signals in a Feedback Control System 5. Control of the Transient Response 6. Steady – State Error 7. The Cost of Feedback 8. Control System Characteristics Using Control Design Software sites.google.com/site/ncpdhbkhn 9 ` sites.google.com/site/ncpdhbkhn 10 Sensitivity of Control Systems to Parameter Variations (1) ( )Y s( )R s ( )N s ( )G s ( ) ( )dT s ( )cG s ( )H s ( )aE s Controller Process Sensor ( ) ( ) ( ) ( )( )( ) ( ) ( ) ( ) 1 ( ) ( ) 1 ( ) ( ) 1 ( ) ( ) c c d c c c G s G s G s G sG sY s R s T s N s G s G s G s G s G s G s      ( ) ( ) 1, ( ) 0, ( ) 0c dG s G s T s N s  ( ) ( )Y s R s  sites.google.com/site/ncpdhbkhn 11 Sensitivity of Control Systems to Parameter Variations (2) ( ) ( )1 ( )( ) ( ) ( ) ( ) 1 ( ) ( ) 1 ( ) ( ) 1 ( ) ( ) c d c c c G s G sG sE s R s T s N s G s G s G s G s G s G s      ( ) ( ) ( ), ( ) 0, ( ) 0dG s G s G s T s N s     1( ) ( ) ( ) 1 ( )[ ( ) ( )]c E s E s R s G s G s G s        ( ) ( )( ) ( )[1 ( ) ( ) ( ) ( )][1 ( ) ( )] c c c c G s G sE s R s G s G s G s G s G s G s          ( ) ( ) ( ) ( )c cG s G s G s G s 2 ( ) ( )( ) ( ) [1 ( )] cG s G sE s R s L s      1 ( ) ( )L s L s  1 ( )( ) ( )( ) ( ) G sE s R s L s G s     sites.google.com/site/ncpdhbkhn 12 Sensitivity of Control Systems to Parameter Variations (3) ( )( ) ( ) Y sT s R s  ( ) / ( ) ( ) / ( ) T s T sS G s G s   ( )Y s( )R s ( )N s ( )G s ( ) ( )dT s ( )cG s ( )H s ( )aE s Controller Process Sensor ( ) / ( ) ln ( ) / ( ) ln T s T s T G s G s G     System sensitivity is the ratio of the change in the system transfer function to the change of a process transfer function (or parameter) for a small incremental change. sites.google.com/site/ncpdhbkhn 13 Sensitivity of Control Systems to Parameter Variations (4) ( ) ( )( ) 1 ( ) ( ) c c G s G sT s G s G s   ( )Y s( )R s ( )N s ( )G s ( ) ( )dT s ( )cG s ( )H s ( )aE s Controller Process Sensor ln ln TS G   2(1 ) 1 . . 1 ( ) ( ) 1 c T c G c c c G G G GT G GS G GG T G G s G s G G        sites.google.com/site/ncpdhbkhn 14 Sensitivity of Control Systems to Parameter Variations (5)Ex. out a inv K v  aT K  ( 1) . . 1 a T a a a K a a a K K KTS K T K K       aK outvinv  aK out vinv 1 1 a a K GT K G    1 1 1 1 1 ( 1) a a a a a a K KTG KT K K K              1 ( 1) a a KG K     outvinv sites.google.com/site/ncpdhbkhn 15 Sensitivity of Control Systems to Parameter Variations (6)Ex. a a T T G K G KS S S 1 1 a a K GT K G    1 ( 1) a a KG K     outvinv 2 1 1(1 ) . . 1 1 G T G T G GGS GG T G G G        1 ( 1)1 11 1 ( 1) a a a a K K K K          2 1 [1 ( 1)] 1 . . 1 ( 1) 1 ( 1) a a G a a a K aa a a a K K K KGS KK G K K K              1 ( 1) 1 1 . 1 1 ( 1) 1a T a K a a a KS K K K            sites.google.com/site/ncpdhbkhn 16 Sensitivity of Control Systems to Parameter Variations (7)Ex. 1 a T KS  aK outvinv  aK out vinv 1 ( 1) a a KG K     outvinv 1 1a T K a S K    System sensitivity is the ratio of the change in the system transfer function to the change of a process transfer function (or parameter) for a small incremental change. Feedback Control System Characteristics 1. Introduction 2. Error Signal Analysis 3. Sensitivity of Control Systems to Parameter Variations 4. Disturbance Signals in a Feedback Control System 5. Control of the Transient Response 6. Steady – State Error 7. The Cost of Feedback 8. Control System Characteristics Using Control Design Software sites.google.com/site/ncpdhbkhn 17 sites.google.com/site/ncpdhbkhn 18 Disturbance Signals in a Feedbak Control System (1) 1 ( ) ( )( ) ( ) ( ) ( ) 1 ( ) 1 ( ) 1 ( )d G s L sE s R s T s N s L s L s L s      1( ) ( ) 0, ( ) 1 ( )R s N s S s L s    ( )( ) ( ) ( ) ( ) ( ) 1 ( ) d d G sE s T s S s G s T s L s      sites.google.com/site/ncpdhbkhn 19 Disturbance Signals in a Feedbak Control System (2) ( )mT s ( )dT s ( ) ( )LT s ( )aI s( )aV s 1 aR ( )s1 Js b bK ( ) Motor back electromotive force mK ( )E s 1 1( ) ( ) ( )1 1 /1 d d b m a b m a Js bE s T s T s Js b K K RK K R Js b      ( ) /dT s D s 0 0 1lim ( ) lim ( ) lim . /t s s b m a DE t sE s s Js b K K R s        0 ( )/b m a D b K K R     Ex. sites.google.com/site/ncpdhbkhn 20 Disturbance Signals in a Feedbak Control System (3)Ex. ( )mT s ( )dT s ( ) ( )LT s ( )aI s( )aV s 1 aR ( )s1 Js b bK ( ) Motor back electromotive force mK ( )E s lim ( ) /t b m a DE t b K K R   ( )mT s ( )dT s ( ) ( )LT s ( )R s ( )s1 Js b bK ( ) m a K Ra K tK ( )aE s Amplifier ( ) Tachometer ( )tV s 1 ( ) ( ) 11 . d a m b t a a Js bE s T s K K KK R Js b K        lim ( ) a t a m t RE t D K K K   sites.google.com/site/ncpdhbkhn 21 Disturbance Signals in a Feedbak Control System (4) ( )Y s( )R s ( )N s ( )G s ( ) ( )dT s ( )cG s ( )H s ( )aE s Controller Process Sensor ( ) ( )1 ( )( ) ( ) ( ) ( ) 1 ( ) ( ) 1 ( ) ( ) 1 ( ) ( ) c d c c c G s G sG sE s R s T s N s G s G s G s G s G s G s      Feedback Control System Characteristics 1. Introduction 2. Error Signal Analysis 3. Sensitivity of Control Systems to Parameter Variations 4. Disturbance Signals in a Feedback Control System 5. Control of the Transient Response 6. Steady – State Error 7. The Cost of Feedback 8. Control System Characteristics Using Control Design Software sites.google.com/site/ncpdhbkhn 22 sites.google.com/site/ncpdhbkhn 23 Control of the Transient Response (1) ( ) ( ) [( ) ] m a a b m Ks V s s Js b R K K     ( )mT s ( )dT s ( ) ( )LT s 1 s ( )s( )aV s m a K R Armature ( )s1 Js b Load bK ( ) Back electromotive force( )( ) ( ) [( ) ] m a a b m KsG s V s Js b R K K      1 1 11 m a b m a a b m K R b K K K R J s s R b K K     2( )a k EV s s  1 2 1 ( ) . 1 K k E s s s      1/ 1 2( ) (1 )tt K k E e     sites.google.com/site/ncpdhbkhn 24 Control of the Transient Response (2) ( )mT s ( )dT s ( ) ( )LT s 1 s ( )s( )aV s m a K R Armature ( )s1 Js b Load bK ( ) Back electromotive force 1/ 1 2( ) (1 )tt K k E e    1 1, a m a b m a b m R J KK R b K K R b K K     ( )mT s ( )dT s ( ) ( )LT s 2( ) k ER s s  ( )s1 Js b bK ( ) m a K Ra K tK ( )aE s Amplifier ( ) Tachometer ( )tV s ( )( ) ( ) 1 ( ) a a t K G ss R s K K G s    1 1 11 a a t K K s K K K    1 2 1 1 1 ( ) 1 exp 1 a a t a t K K k E K K K t t K K K              12 1 1 exp a t t K K Kk E t K           sites.google.com/site/ncpdhbkhn 25 Control of the Transient Response (3) 0 2 4 6 8 10 12 14 16 18 20 0 0.2 0.4 0.6 0.8 1 Time (s) Sp e e d With feedback Without feedback Feedback Control System Characteristics 1. Introduction 2. Error Signal Analysis 3. Sensitivity of Control Systems to Parameter Variations 4. Disturbance Signals in a Feedback Control System 5. Control of the Transient Response 6. Steady – State Error 7. The Cost of Feedback 8. Control System Characteristics Using Control Design Software sites.google.com/site/ncpdhbkhn 26 sites.google.com/site/ncpdhbkhn 27 Steady – State Error (1) ( ) ( ) ( ) [1 ( )] ( )open loopE s R s Y s G s R s     1( )R s s  1 ( )( )o G sE s s   0 0 1 ( )( ) lim ( ) lim ( ) lim 1 (0)o o o t s s G s e e t sE s s G s         1( ) ( ) 1 ( ) ( )closed loop c E s R s G s G s   1( )R s s  1 1( ) . 1 ( ) ( )c c E s G s G s s    0 0 1 1 1( ) lim ( ) lim ( ) lim . 1 ( ) ( ) 1 (0) (0)c c ct s s c c e e t sE s s G s G s s G G         sites.google.com/site/ncpdhbkhn 28 Steady – State Error (2) Open – loop Closed – loop ( ) 1 (0)oe G   1( ) 1 (0) (0)c c e G G    (0)G K ( ) 1oe K    (0) 1; (0)cG G K  1( ) 1c e K     1 ( ) 0oK e    100 ( ) 1/101cK e    0.1 ( ) 0.1o K e K       ( ) 0.1 0.1| ( ) | 1 oe r t     1 10.1 ( ) 101 91c K e K        ( ) 0.0011| ( ) | ce r t    Feedback Control System Characteristics 1. Introduction 2. Error Signal Analysis 3. Sensitivity of Control Systems to Parameter Variations 4. Disturbance Signals in a Feedback Control System 5. Control of the Transient Response 6. Steady – State Error 7. The Cost of Feedback 8. Control System Characteristics Using Control Design Software sites.google.com/site/ncpdhbkhn 29 The Cost of Feedback • Feedback control is very good, but ... • An increased number of components & complexity: – Requires sensor(s) – The sensor is often the most expensive component in a control system – The sensor introduces noise & inaccuracies into the system • The loss of gain: • The possibility of instability. sites.google.com/site/ncpdhbkhn 30 ( ) ( )( ) ( ) 1 ( ) ( ) c c c G s G sG s G s G s G s   Feedback Control System Characteristics 1. Introduction 2. Error Signal Analysis 3. Sensitivity of Control Systems to Parameter Variations 4. Disturbance Signals in a Feedback Control System 5. Control of the Transient Response 6. Steady – State Error 7. The Cost of Feedback 8. Control System Characteristics Using Control Design Software sites.google.com/site/ncpdhbkhn 31 sites.google.com/site/ncpdhbkhn 32 Control System Characteristics Using Control Design Software ( )dT s ( )( )R s 11s K Controller ( )Y s1 ( 1)s s  Load ( ) Ex.