Short introduction to standard regulator Daniele Carnevale - PowerPoint PPT Presentation

Feedback scheme Controller design PID Short introduction to standard regulator Daniele Carnevale Dipartimento di Ing. Civile ed Ing. Informatica (DICII), University of Rome “Tor Vergata” Corso di Fondamenti di Automatica e Controlli Automatici, A.A. 2014-2015 1 / 9

Feedback scheme Controller design Performances and requirements PID The standard control scheme The standard control scheme r reference (desired) signal y is the controlled output of the plant P u is the control variable (plant input) generated by the controller C e is difference between the reference r and plant response y r e u y C P + − Figure : The feedback control scheme. 2 / 9

Feedback scheme Controller design Performances and requirements PID The closed loop transfer function Via the Laplace transform on the plant and the signals, it is possible to evaluate two important transfer functions of the feedback scheme in Fig. 1 such as W yr ( s ) = y ( s ) P ( s ) C ( s ) r ( s ) = (1) 1 + P ( s ) C ( s ) , (complementary sensitivity transfer function) W er ( s ) = e ( s ) 1 r ( s ) = 1 + P ( s ) C ( s ) , (2) (sensitivity transfer function) 3 / 9

Feedback scheme Controller design Performances and requirements PID Laplace transform: the transfer function Let the state space data of the plant be ( A, B, C, D ) , then L [ x ( t )]( s ) := x ( s ) = ( sI − A ) − 1 x 0 + ( sI − A ) − 1 Bu ( s ) (3) L [ y ( t )]( s ) := y ( s ) = C ( sI − A ) − 1 x 0 + C ( sI − A ) − 1 B + D � � u ( s ) . (4) Definition (Transfer function) Assuming x 0 = 0 , then u ( s ) = C ( sI − A ) − 1 B + D = Γ m FDT ( s ) := y ( s ) i =0 ( s − z i ) i =0 ( s − p i ) , (5) Γ n is the system transfer function and is a proper rational function (ratio of polynomials of s where m ≤ n ≤ n ). Furthermore, if every pole (root of the denominator) p i of the FDT has non-positive real part ( p i ∈ C − 0 ) then if u ( t ) = E cos( ωt + θ ) it holds t →∞ y ( t ) = ρ ( ω ) E cos( ωt + θ + ϕ ( ω )) , lim (6) where ρ ( ω ) := | FDT ( jω ) | and ϕ ( ω ) := FDT ( jω ) . 4 / 9

Feedback scheme Controller design Performances and requirements PID Performances: Steady state requirements Consider the factorization Π N ( s ) Π( s ) = µ Π s g Π Π D ( s ) , Π D (0) = Π N (0) = 1 , g Π ∈ Z , µ Π ∈ R , (7) of a generic transfer function Π( s ) , yielding the Laplace transform of the error as s g P + g C P D ( s ) C D ( s ) e ( s ) = W er ( s ) r ( s ) = s g P + g C P D ( s ) C D ( s ) + µ P µ C P N ( s ) C N ( s ) r ( s ) . (8) Then, if lim t → + ∞ e ( t ) = e ∞ exists and is finite, it holds t → + ∞ e ( t ) = lim lim s → 0 s W er ( s ) r ( s ) . (9) 5 / 9

Feedback scheme Controller design PID Performances: Steady state requirements (cont’d) When r ( s ) = R/s q then s → 0 sW er ( s ) R e ∞ = lim s q , s g P + g C − q +1 P D ( s ) C D ( s ) lim s g P + g C P D ( s ) C D ( s ) + µ P µ C P N ( s ) C N ( s ) R, (10) s → 0  0 if g P + g C − q + 1 ≥ 1 ,    = (11) 1 lim s → 0 s q − 1 + µ P µ C R if g P + g C − q + 1 = 0 ,   ∞ if g P + g C − q + 1 ≤ − 1 ,  i.e. the error can be rendered zero or smaller (modulus) than certain desired value (or percentage with respect to R ). Does the same reasoning hold when considering sinusoidal inputs, i.e. with r ( s ) = Rω 2 / ( s 2 + ω 2 ) ? 6 / 9

Feedback scheme Controller design PID Performances: Steady state requirements (cont’d) Asymptotic properties (steady state) C N ( s ) µ C Selection of the controller C ( s ) = C D ( s ) , with C N (0) = C D (0) = 1 : s gC gain µ C g C number of poles in zero to satisfy asymptotic requirements on e ∞ . 7 / 9

Feedback scheme Controller design PID Stability: closed loop asymptotic stability Closed loop asymptotic stability Selection of the other controller parameters has to be such that the closed loop transfer function W yr ( s ) is asymptotically stable, or equivalently that den ( W yr ( s )) = s g P + g C P D ( s ) C D ( s ) + µ P µ C P N ( s ) C N ( s ) (12) is Hurwitz . A possible approach consists to iteratively increase the controller numerator and denominator degree j as j j � b i s i , � a i s i C N ( s ) = 1 + C D ( s ) = 1 + i =1 i =1 and retrieve the coefficients a i and b i (also µ C and g C if necessary and not fixed in the previous step) such that (12) is Hurwitz (Routh-Hurwitz criteria could be considered). 8 / 9

Feedback scheme Controller design PID Standard regulator: PID A well-established standard controller is the PID (Proportional-Integral-Derivative) controller whose transfer function is s/N + 1 = s 2 ( k d + k p /N ) + s ( k p + k i /N ) + k i PID ( s ) = k p + k i k d s s + . (13) s ( s/N + 1) Other possible combinations/configurations are P ( s ) = k p , (14) PI ( s ) = sk p + k i , (15) s PD ( s ) = s ( k p /N + k d ) + k p . (16) s/N + 1 The controller parameters have to be selected in order to satisfy the asymptotic performances and closed loop stability. 9 / 9