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Unformatted text preview: EE2010 Systems and Control (Chapter 8) 81 Performance refers to how well the system responds to inputs. It is usually given in terms of time domain specifications, such as rise time, settling time, steadystate error, etc. Control Performance 5 10 15 5 10 15 20 25 Step Response Time (sec) Angular Velocity K = 50, J = 5, b = 5 K = 40, J = 5, b = 2 K = 50, J = 5, b = 2 Motor performance without control: Different transient behaviors with different parameters and input value. Open Loop Performance b Js K s V s V s G in ) ( ) ( ) ( EE2010 Systems and Control (Chapter 8) 82 Suppose Closed loop performance depends on K c : K c = 1, K c = 0.2 s + 1, ) 2 5 ( 50 ) ( s s s G G ( s ) + ( t ) ref ( t ) 50 2 5 50 ) ( 2 1 s s s G cl K c 50 12 5 50 10 ) ( 2 2 s s s s G cl c c c c cl K s s K s G K s G K s G 50 2 5 50 ) ( 1 ) ( ) ( 2 5 10 15 20 25 30 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Step Response Time (sec) Amplitude ) ( 2 s G cl ) ( 1 s G cl EE2010 Systems and Control (Chapter 8) 83 The performance depends on gains, time constants, damping ratios, natural frequencies and poles DC gains affect the steadystate response Time constants, damping ratios and natural frequencies affect the transient response. These are, roughly speaking, factors in time domain Position of poles also affect the transient response but this is a factor in frequency domain. Above all, poles determine the stability of the closed loop system We are interested in 2 kinds of performances: steadystate and transient These 2 are closely inter related. What does performance depend on? EE2010 Systems and Control (Chapter 8) 84 Consider the following feedback system The error function is, DC gain is defined as the gain at zero frequency . Hence, static/steady state/DC gain is given as G (0) or G cl (0), e.g., closed loop DC gain is If we want y ss = r , we require G cl (0) = 1. Hence for any step input tracking without steady state error , we require G cl (0) = 1 SteadyState Error of Feedback Control Systems G ( s ) + K ( s ) ) ( ) ( 1 ) ( ) ( ) ( ) ( ) ( K G K G R Y G cl Y ( s ) R ( s ) E ( s ) ) ( ) ( ) ( 1 1 ) ( s R s K s G s E EE2010 Systems and Control (Chapter 8) 85 We will consider steadystate errors in stable systems for general polynomial inputs, r ( t ) = r t ( n − 1) Taking L.T., , where C = 1 for step or ramp; C = 2 for parabolic n = 1 : step input n = 2 : ramp input n = 3 : parabolic input Using Final Value Theorem: n s r C s R ) ( r r t t r t 2 t t r 3 2 ) ( s r s R 2 ) ( s r s R s r s R ) ( ) ( ) ( lim ) ( ) ( 1 1 lim ) ( lim 1 1 s K s G s s r C s r C s K s G s s sE e n n s n s s ss EE2010 Systems and Control (Chapter 8) 86 ) ( 1 ) ( ) ( s P s s K s G m Suppose , where P ( s ) does not contain any integrator...
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 Spring '11
 TanKayChen

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