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

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Feedback scheme Controller design PID Performances and requirements

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.

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Feedback scheme Controller design PID Performances and requirements

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

Wyr(s) = y(s) r(s) = P(s)C(s) 1 + P(s)C(s) ,

(complementary sensitivity transfer function)

(1) Wer(s) = e(s) r(s) = 1 1 + P(s)C(s) ,

(sensitivity transfer function)

(2)

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Feedback scheme Controller design PID Performances and requirements

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)−1x0 + (sI − A)−1Bu(s) (3) L[y(t)](s) := y(s) = C(sI − A)−1x0 +

• C(sI − A)−1B + D
• u(s).

(4) Definition (Transfer function) Assuming x0 = 0, then FDT(s) := y(s) u(s) = C(sI − A)−1B + D = Γm

i=0(s − zi)

Γn

i=0(s − pi),

(5) 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) pi of the FDT has non-positive real part (pi ∈ C−

0 ) then if

u(t) = E cos(ωt + θ) it holds lim

t→∞ y(t) = ρ(ω)E cos(ωt + θ + ϕ(ω)),

(6) where ρ(ω) := |FDT(jω)| and ϕ(ω) := FDT(jω).

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Feedback scheme Controller design PID Performances and requirements

Performances: Steady state requirements

Consider the factorization Π(s) = µΠ ΠN(s) sgΠΠD(s), ΠD(0) = ΠN(0) = 1, gΠ ∈ Z, µΠ ∈ R, (7)

• f a generic transfer function Π(s), yielding the Laplace transform of the error

as e(s) = Wer(s)r(s) = sgP +gCPD(s)CD(s) sgP +gCPD(s)CD(s) + µP µCPN(s)CN(s)r(s). (8) Then, if limt→+∞ e(t) = e∞ exists and is finite, it holds lim

t→+∞ e(t) = lim s→0 s Wer(s)r(s).

(9)

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Feedback scheme Controller design PID

Performances: Steady state requirements (cont’d)

When r(s) = R/sq then e∞ = lim

s→0 sWer(s) R

sq , lim

s→0

sgP +gC−q+1PD(s)CD(s) sgP +gCPD(s)CD(s) + µP µCPN(s)CN(s)R, (10) =        if gP + gC − q + 1 ≥ 1, lims→0

1 sq−1+µP µC R

if gP + gC − q + 1 = 0, ∞ if gP + gC − q + 1 ≤ −1, (11) 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) = Rω2/(s2 + ω2)?

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Feedback scheme Controller design PID

Performances: Steady state requirements (cont’d)

Asymptotic properties (steady state) Selection of the controller C(s) =

µC sgC CN (s) CD(s), with CN(0) = CD(0) = 1:

gain µC gC number of poles in zero to satisfy asymptotic requirements on e∞.

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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 Wyr(s) is asymptotically stable, or equivalently that den(Wyr(s)) = sgP +gCPD(s)CD(s) + µP µCPN(s)CN(s) (12) is Hurwitz. A possible approach consists to iteratively increase the controller numerator and denominator degree j as CN(s) = 1 +

j

• i=1

bisi, CD(s) = 1 +

j

• i=1

aisi and retrieve the coefficients ai and bi (also µC and gC if necessary and not fixed in the previous step) such that (12) is Hurwitz (Routh-Hurwitz criteria could be considered).

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Feedback scheme Controller design PID

Standard regulator: PID

A well-established standard controller is the PID (Proportional-Integral-Derivative) controller whose transfer function is PID(s) = kp + ki s + kds s/N + 1 = s2(kd + kp/N) + s(kp + ki/N) + ki s(s/N + 1) . (13) Other possible combinations/configurations are P(s) = kp, (14) PI(s) = skp + ki s , (15) PD(s) = s(kp/N + kd) + kp s/N + 1 . (16) The controller parameters have to be selected in order to satisfy the asymptotic performances and closed loop stability.

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