Automatic Control 2. Loop shaping. Prof. Alberto Bemporad. University of Trento. Academic year


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1 Automatic Control 2 Loop shaping Prof. Alberto Bemporad University of Trento Academic year Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year / 39
2 Feedback control problem Feedback control problem measurement noise n model uncertainty G(s) d process disturbance r reference! " e!! tracking error C(s) u input G(s)!! y output Objective: make the tracking error e(t) = r(t) y(t), despite the dynamics of the openloop system G(s) (slow, unstable, etc.) disturbance d(t) affecting the process model uncertainty G(s) (the system is not exactly as we modeled it) measurement noise n(t) Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year / 39
3 Feedback control problem Feedback control problem r reference! " measurement noise e!! tracking error n C(s) model uncertainty G(s) u input G(s) d!! process disturbance y output To achieve the objective, we want to design a feedback control law satisfying a number of requirements: stability in nominal conditions ( G(s) =, d(t) =, n(t) = ) stability in perturbed conditions static performances (tracking error for constant r(t)) dynamic performances (transients and frequency response) noise attenuation (be insensitive to measurement noise n(t)) Tuesday, May 4, 21 feasibility of the controller (strictly proper transfer function C(s)) Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year / 39
4 Feedback control problem Closedloop function n G(s) d r! " e!! C(s) u G(s)!! y The closedloop function is the transfer function from r to y W(s) = H yr (s) = C(s)G(s) 1 + C(s)G(s) We call loop function the transfer function Tuesday, May 4, 21 L(s) = C(s)G(s) For L(jω) 1 we get W(jω) 1, W(jω) For L(jω) 1 we get W(jω) Given G(s), we will choose C(s) to shape the loop function response L(jω) Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year / 39
5 Feedback control problem Sensitivity functions n G(s) d r! " e!! C(s) u G(s)!! y The sensitivity function S(s) is the function Tuesday, May 4, 21 S(s) = W(s)/W(s) G(s)/G(s) = W(s) G(s) G(s) W(s) = L(s) The sensitivity function S(s) is also the transfer function from d to y Y(s) D(s) = L(s) = S(s) The complementary sensitivity function T(s) is the transfer function from n to y T(s) = Y(s) N(s) = L(s) 1 + L(s) = W(s) Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year / 39
6 Performance specifications Performance specifications: frequency domain sensitivity transfer function from d to y 1 S(s) = 1 + C(s)G(s) complementary sensitivity transfer function from n to y T(s) = C(s)G(s) 1 + C(s)G(s) = W(s) We want to be immune (=small tracking errors) to both process disturbances and measurement noise both S and T small Problem: S(s) + T(s) = 1! How to make both them small? Solution: keep S small at low frequencies, and hence W 1 (=good tracking) keep T small at high frequencies (=good noise rejection) Magnitude (db) db good tracking C(jω)G(jω) 1 frequency (rad/s) Constraint on performance: tracking cannot be too good at high frequencies! C(jω)G(jω) 1 good noise rejection Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year / 39
7 Performance specifications Performance specifications: frequency domain We usually refer to regular closedloop responses (i.e., similar to 2ndorder underdamped systems) 1 W(s) = W (jω) resonant M r 1 + 2ζ s + 1 peak ω n ω 2 s 2 n B 3 closedloop bandwidth 3 db the resonant peak M r is the max value of W(jω) the bandwidth of the closed loop system is the range of frequencies at which W(jω) 2 ( W(jω) 2, in general) W() 1 note that = 2 log db 1 2 3dB For 2nd order systems W(s): Mr W 1 = 2ζ 1, ζ 2 BW 3 = ω n 1 2ζ ζ 2 + 4ζ 4 Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year / 39
8 Performance specifications Performance specifications: steadystate tracking We look at tracking of constant setpoints, ramps, etc., R(s) = 1 s k Write the loop function in Bode form L(s) = K s h L 1(s), with L 1 () = 1 Compute lim t + e(t) using final value theorem for h k 1: lim t + e(t) = lim s = lim s se(s) = lim s(r(s) W(s)R(s)) = lim s s s h k+1 s h + KL 1 (s) = L(s) if h k K if h =, k = 1 1 K if h = k 1, k > 1 1 s k 1 Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year / 39
9 Performance specifications Performance specifications: steadystate tracking lim e(t) = t + if h k K if h =, k = 1 1 K if h = k 1, k > 1 Constraint on type h of L(s) = C(s)G(s) = KL 1(s) s h : need h k 1 to track r(t) = t k 1 need h k to track t k 1 with zero asymptotic error special case: need h 1 to track constant setpoints with zero offset in steadystate (integral action!) Constraint on Bode gain K of L(s): need K sufficiently large to bound steadystate tracking error when h = k 1 Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year / 39
10 Performance specifications Performance specifications: time response ŝ y(t) r(t) rise time t r peak overshoot settling time t s r(t) = 1 for t Tuesday,, May 4, 21 r(t) = for t < time t We look at the shape of the transient closedloop response due to a unit step the rise time t r is the time required to rise from to 1% of steadystate (1 9% for nonoscillating systems) the settling time t s is the time to reach and stay within a specified tolerance band (usually 2% or 5%) the peak overshoot ŝ is the max relative deviation from steadystate, ŝ = max t y(t) 1 y(+ ) Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year / 39
11 Performance specifications Relations between frequency and time response Usually we refer to the step response of a secondorder closedloop system peak overshoot r(t) ŝ y(t) rise time tr settling time ts W(s) = ζ s + 1 ω n ω 2 s 2 n Tuesday, May 4, 21 time t In this case we have ŝ = e πζ 1 ζ 2, t r = 1 1 π tan 1 ω n 1 ζ 2 1 ζ 2 Good average formulas for closedloop systems W(s) are ζ, t s[5%] 3 ζω n t s[2%] 4 ζω n t r B W 3 3 ŝ.85 MW r 1 where t r [s], B W 3 [rad/s], ŝ [no unit], MW r [not in db] Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year / 39
12 Performance specifications Relations between frequency and time response Relations between frequency and time response for secondorder closedloop systems as a function of the damping factor ζ B 3 ω n M r ω n t r ζ ζ ζ B 3 t r ζ 1 + ŝ M r ζ ŝ ζ Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year / 39
13 Synthesis via loop shaping Controller synthesis via loop shaping Typical closedloop specifications include static specs system type h tracking error e(+ ) in steadystate for r(t) = t k Bode gain K and dynamic specs peak overshoot ŝ resonant peak M r rise time t r bandwidth B 3 Designing a regulator that meets all specs in one shot can be a hard task In loopshaping synthesis the controller C(s) is designed in a series of steps C(s) r! y " C 1 (s) C 2 (s) G(s) where C 1 (s) satisfies static specs, while C 2 (s) dynamic specs Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year / 39
14 Synthesis via loop shaping Synthesis of C 1 (s) (static performance) The general form of C 1 (s) is C 1 (s) = K c s h c where K c is the controller gain and h c is the type of the controller If the type of G(s) is h g and the desired type is h d, then h c = max{, h d h g } The gain K c is chosen by imposing the desired steadystate tracking error K c G() e d if r d = 1 K c G() e d if r d > Example: G(s) = 1 (s + 1) Track a step reference with zero steadystate error r d = 1 since the type of G(s) is h g =, h c = max{, 1} = 1, and C 1 (s) = 1 s steadystate error e d to unit ramp reference is not required, so K c is free Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year / 39
15 Synthesis via loop shaping Mapping closedloop specs to openloop specs Closedloop specifications must be translated into specifications on the loop transfer function L(jω) = C(jω)G(jω) In particular closedloop specs should be translated into a desired phase margin M L p and desired crossover frequency ωl c of L(jω) We have some approximate formulas to do that: M L p 2.3 MW r 1.25 where M r is in not expressed in db, and M p is expressed in rad, and ω L c [.5,.8]BW 3 The next step of loop shaping is to synthesize C 2 (s) so that L(jω) has the right phase margin M L p M pd and crossover frequency ω L c = ω cd Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year / 39
16 Synthesis via loop shaping Synthesis of C 2 (s) (dynamic performance) The general form of C 2 (s) is C 2 (s) = i (1 + τ is) i (1 + 2ζ i s ω ni j (1 + T js) j (1 + 2ζ js ω nj + s2 ω 2 ) ni + s2 ) ω 2 nj C 2 (s) must be designed to guarantee closedloop asymptotic stability C 2 (s) must be designed to satisfy the dynamic specifications We focus here on controllers containing only real poles/zeros Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year / 39
17 Leadlag networks magnitude [db] phase [rad] Lead network A lead network has transfer function C Lead (s) = 1 + τs 1 + ατs where τ >, < α < 1 C Lead (s) provides phase lead in the frequency range [ 1 τ, α τ ] The maximum phase lead is at ω max = 1 logarithmic scale) 1/τ 1/(ατ) φmax ωmax frequency [rad/s] 2 log 1 ( 1 α π/2 π/4 Tuesday, May 4, 21 τ (midpoint between 1 and 1 α τ ατ in ) Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year / 39
18 phase [rad] magnitude [db] Lecture: Loop shaping Leadlag networks Lead network G(jω) G(jω)C Lead (jω) 1 ατ 1 ω c ω! τ G(jω) (G(jω)C Lead (jω)) M p M p The main goal of the lead network is to increase the phase margin As a side effect, the loop gain is increased at high frequencies (=reduced complementary sensitivity) Tuesday, May 4, 21 Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year / 39
19 Leadlag networks Lead network Bode Diagram 3 25 Magnitude (db) Phase (deg) alpha=.5 alpha=.1 alpha=.15 6 alpha=.2 alpha=.3 alpha=.4 alpha=.5 3 alpha=.6 alpha=.7 alpha=.8 alpha= Frequency (rad/sec) C Lead (s) = 1 + s 1 + αs τ = 1 Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year / 39
20 Leadlag networks magnitude [db] phase [rad] Lag network A lag network has transfer function C Lag (s) = 1 + ατs 1 + τs where τ >, < α < 1 C Lag (s) provides phase lag in the frequency range [ 1 τ, α τ ] The maximum phase lag is at ω min = 1 logarithmic scale) 1/τ 1/(ατ) φmin ωmin frequency [rad/s] Tuesday, May 4, 21 τ (midpoint between 1 and 1 α τ 2 log 1 (α) π/4 π/2 ατ in Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year / 39
21 phase [rad] magnitude [db] Lecture: Loop shaping Leadlag networks Lag network G(jω) 1 τ 1 ατ G(j!)C Lag (j!) ω c ω c M p The main goal of the lag network is to decrease the loop gain at high frequencies (=reduce complementary sensitivity) To avoid decreasing the phase margin, set ω min at low frequencies, where the loop gain Tuesday, May L(jω) 4, 21 is still high, therefore avoiding ω c ω min Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year / 39
22 Leadlag networks Lag network Bode Diagram Magnitude (db) Phase (deg) alpha=.5 alpha=.1 alpha=.15 3 alpha=.2 alpha=.3 alpha=.4 alpha=.5 6 alpha=.6 alpha=.7 alpha=.8 alpha= Frequency (rad/sec) Note that C Lag (s) = 1 + αs 1 + s τ = ατjω 1 + τjω 1 + τjω = db 1 + ατjω db and that 1 + ατjω 1 + τjω = 1 + τjω 1 + ατjω Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year / 39
23 Leadlag networks magnitude [db] phase [rad] Leadlag network A leadlag network has transfer function C Lead Lag (s) = (1 + α 1τ 1 s) (1 + τ 1 s) where τ 1, τ 2 >, < α 1, α 2 < 1 (1 + τ 2 s) (1 + α 2 τ 2 s) The leadlag network C Lead Lag (s) provides the coupled effect of a lead and a lag network: increase the phase margin without increasing the closedloop bandwidth ω γ 1/τ1 1/(ατ1) 1/(ατ2) 1/τ2 φmax ωmin ωmax φmin frequency [rad/s] 2 log 1 (α) π/2 π/2 Tuesday, May 4, 21 A special form of leadlag networks most used in industrial practice are proportional integral derivative (PID) controllers (see more later...) PID temperature controller Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year / 39
24 Loopshaping example Example of loop shaping Bode Diagram 5 Openloop transfer function of the process Magnitude (db) 5 System: G Gain Margin (db): 13.6 At frequency (rad/sec): 2.83 Closed Loop Stable? Yes 1 G(s) = s(s + 2)(s + 4) = s (1 +.5s)(1 +.25s) Phase (deg) System: G Phase Margin (deg): 47 Delay Margin (sec):.77 At frequency (rad/sec): 1.7 Closed Loop Stable? Yes Frequency (rad/sec) Closedloop specifications 1 Track a ramp reference r(t) = t with finite steadystate error e d.2 2 Risetime of unit step response t r.4 s 3 Overshoot of unit step reference ŝ 25% Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year / 39
25 Loopshaping example Example of loop shaping Static performance 1 Track a ramp reference r(t) = t with finite steadystate error e d.2 Since G(s) is of type r g = 1, no need to add integrators C 1 (s) = K c = K c s h c s = K c Choose K c by looking at steadystate tracking error of unit ramp 1 K c 1.25 e d =.2 K c 4 Finally, set C 1 (s) = 4 Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year / 39
26 Loopshaping example Example of loop shaping Dynamic performance 2 Risetime of unit step response t r.4 Since B 3 3/t r, we get a closedloop bandwidth constraint B 3 3 = 3 = 7.5 rad/s t r.4 As the desired ω c [.5,.8]B 3, we get a target for the crossover frequency of the loop function L(jω) ω c 4.7 rad/s 2 Overshoot of unit step reference ŝ 25% As ŝ.85 M r 1 and M p 2.3 M r we get M r 1 or 1.25 M r = The resulting specification on the phase margin of L(jω) is M p.664 rad 38 deg Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year / 39
27 Loopshaping example Example of loop shaping Dynamic performance Let s examine the current loop gain L 1 (s) = C 1 (s)g(s) = s = jω c = j4.7 Bode diagram of L 1 (s) 4 s(s + 2)(s + 4) for 1 Phase (deg) Magnitude (db) System: GC1 Frequency (rad/sec): 4.7 Magnitude (db): 11.3 System: GC1 Frequency (rad/sec): 4.7 Phase (deg): 26 L 1 (j4.7) = 11.4 db L 1 (j4.7) 26 deg at ω = 4.7 rad/s we need to increase the gain by M = = 11.4 db and the phase by φ = 26 (18 38) = 64 deg Frequency (rad/sec) Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year / 39
28 Loopshaping example Example of loop shaping Lead network A suitable network is a lead network. As we need to gain φ = 64 deg, we choose a cascade of two identical lead networks C Lead (s) = 1 + τs 1 + ατs to gain 32 deg at ω c = 4.7 rad/s, we set α =.3 and 4.7 = 1 τ α, or τ =.39 s 25 Bode Diagram s C 2Lead (s) = s Magnitude (db) 15 9 System: CLead Frequency (rad/sec): 4.7 Magnitude (db): 1.4 Phase (deg) 6 3 System: CLead Frequency (rad/sec): 4.7 Phase (deg): Frequency (rad/sec) Note that at the desired crossover frequency ω c = 4.7 rad/s, C 2Lead (j4.7) = 1.4 db Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year / 39
29 Loopshaping example Example of loop shaping Resulting controller The gain difference = 1 db at ω c = 4.7 rad/s is tolerable If the gain difference was too large, we should also have designed and cascaded a lag network The resulting feedback controller is s C(s) = C 1 (s)c 2Lead (s) = s Bode Diagram Phase (deg) Magnitude (db) System: L Frequency (rad/sec): 5.54 Magnitude (db): 3 System: L Phase Margin (deg): 41.5 Delay Margin (sec):.168 At frequency (rad/sec): 4.32 Closed Loop Stable? Yes Bode plot of loop transfer function L(jω) = C(jω)G(jω) B 3 = 5.54 rad/s M p = 41.5 deg Frequency (rad/sec) Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year / 39
30 Loopshaping example Example of loop shaping Validation of the controller 1 Track a ramp reference r(t) = t with finite steadystate error e d closed loop ramp response Amplitude Input: In(1) Time (sec): 2 Amplitude: 2 System: W Time (sec): 2 Amplitude: 1.8 tracking error e(2) = =.2 MATLAB»W = L/(1+L);»t = :.1:2.5;»ramp=t;»lsim(W,ramp,t); Time (sec) Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year / 39
31 Loopshaping example Example of loop shaping Validation of the controller 2 Risetime of unit step response t r.4 3 Overshoot of unit step reference ŝ 25% 1.4 closed loop step response 1.2 System: W Time (sec):.674 Amplitude: 1.28 Amplitude System: W Time (sec):.449 Amplitude: 1 MATLAB»step = ones(1,length(t));»lsim(w,step,t); Time (sec) Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year / 39
32 PID controllers Proportional integral derivative (PID) controllers PID (proportional integrative derivative) controllers are the most used controllers in industrial automation since the 3s u(t) = K p e(t) + 1 t de(t) e(τ)dτ + T d T i dt where e(t) = r(t) y(t) is the tracking error Initially constructed by analog electronic components, today they are implemented digitally ad hoc digital devices just few lines of C code included in the control unit Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year / 39
33 PID controllers PID parameters r e 1 st i u y K p G(s) st d C(s) u(t) = K p e(t) + 1 t de(t) e(τ)dτ + T d T i dt K p is the controller gain, determining the aggressiveness (=closedloop bandwidth) of the controller T i is the reset time, determining the weight of the integral action T d is the derivative time, determining the phase lead of the controller Wednesday, May 5, 21 we call the controller P, PD, PI, or PID depending on the feedback terms included in the control law Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year / 39
34 PID controllers Derivative term The derivative term has transfer function st d, a high pass filter To avoid amplifying highfrequency noise (and to make the PID transfer function proper) st d gets replaced by st d st d 1 + s T d N No effect of the new pole s = N at low frequencies, but the highfrequency T d gain is limited to N (typically N = 3 2) The derivative term has the effect of predicting the future tracking error de(t) ê(t + T d ) = e(t) + T d (linear extrapolation) dt e(t) de(t) e(t)+td dt e(t+td) There are more advanced controllers that use a more refined prediction, based on the mathematical model of the process (model predictive control, MPC See more later...) t t+td Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year / 39
35 PID controllers Frequency response of PD controller {}} { N + 1 G PD (s) = K p 1 + st 1 + T d s d N 1 + s T = Kp d N N + 1 T d s } N {{ + 1 } N }{{} PD regulator α = 1 τ N+1 25 τ 2 Magnitude (db) Phase (deg) K p = 1 K d = 1 N = Frequency (rad/sec) The PD controller is equivalent to a lead network Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year / 39
36 PID controllers Frequency response of PI controller G PI (s) = K p st i = K p/t i (1 + T i s) s PI regulator 5 4 Magnitude (db) K p = 1 T i = 1 Phase (deg) Frequency (rad/sec) The PI controller introduces integral action The zero 1 T i compensates the decrease of phase margin of the integrator Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year / 39
37 PID controllers Implementation of PID controller r b! "! " e 1 st i C(s)!! u K p G(s) y " std 1+s Td N u(t) = K p br(t) y(t) + 1 st i (r(t) y(t)) Wednesday, May 5, 21 The proportional action only uses a fraction b 1 of the reference signal r. st d 1 + s T d N y(t) The reference signal r is not included in the derivative term (r(t) may have abrupt changes) Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year / 39
38 PID controllers Final remarks on loop shaping Loopshaping techniques are most adequate for SISO (singleinput singleoutput) systems They provide good insight in frequency domain properties of the closed loop (bandwidth, noise filtering, robustness to uncertainty, etc.) Most traditional singleloop industrial controllers are PID, and over 9% of PIDs are PI Curiosity: PID Temperature Control Retrofit KIT for Gaggia This PID controller kit is designed for retrofitting into the Gaggia Classic, Gaggia Coffee, and Gaggia Coffee Deluxe espresso machine. By adding a PID controller to the heater control circuit, brewing water temperature can be controlled to ±1 F accuracy. Thus, it will significantly improve the taste of your espresso. Users can also easily adjust the brew water temperature to suit their own tastes. Auber Instruments Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year / 39
39 PID controllers EnglishItalian Vocabulary loop shaping peak overshoot rise time settling time lead network lag network sintesi per tentativi sovraelongazione massima tempo di salita tempo di assestamento rete anticipatrice rete attenuatrice Translation is obvious otherwise. Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year / 39
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