# 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 open-loop 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 Closed-loop function n G(s) d r! " e!! C(s) u G(s)!! y The closed-loop 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 closed-loop responses (i.e., similar to 2nd-order underdamped systems) 1 W(s) = W (jω) resonant M r 1 + 2ζ s + 1 peak ω n ω 2 s 2 n B 3 closed-loop 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: steady-state tracking We look at tracking of constant set-points, 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: steady-state 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 set-points with zero offset in steady-state (integral action!) Constraint on Bode gain K of L(s): need K sufficiently large to bound steady-state 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 closed-loop response due to a unit step the rise time t r is the time required to rise from to 1% of steady-state (1 9% for non-oscillating 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 steady-state, 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 second-order closed-loop 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 closed-loop 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 second-order closed-loop 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 closed-loop specifications include static specs system type h tracking error e(+ ) in steady-state 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 loop-shaping 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 steady-state 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 steady-state error r d = 1 since the type of G(s) is h g =, h c = max{, 1} = 1, and C 1 (s) = 1 s steady-state 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 closed-loop specs to open-loop specs Closed-loop specifications must be translated into specifications on the loop transfer function L(jω) = C(jω)G(jω) In particular closed-loop 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 closed-loop 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

18 phase [rad] magnitude [db] Lecture: Loop shaping Lead-lag 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 Lead-lag 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 Lead-lag 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 Lead-lag 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 Lead-lag 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

24 Loop-shaping example Example of loop shaping Bode Diagram 5 Open-loop 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) Closed-loop specifications 1 Track a ramp reference r(t) = t with finite steady-state error e d.2 2 Rise-time 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 Loop-shaping example Example of loop shaping Static performance 1 Track a ramp reference r(t) = t with finite steady-state 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 steady-state 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 Loop-shaping example Example of loop shaping Dynamic performance 2 Rise-time of unit step response t r.4 Since B 3 3/t r, we get a closed-loop 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 Loop-shaping 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

29 Loop-shaping 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 Loop-shaping example Example of loop shaping Validation of the controller 1 Track a ramp reference r(t) = t with finite steady-state 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 Loop-shaping example Example of loop shaping Validation of the controller 2 Rise-time 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 (=closed-loop 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 high-frequency 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 high-frequency 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 Loop-shaping techniques are most adequate for SISO (single-input single-output) systems They provide good insight in frequency domain properties of the closed loop (bandwidth, noise filtering, robustness to uncertainty, etc.) Most traditional single-loop 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 English-Italian 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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