# Frequency domain analysis

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1 Automatic Control 2 Frequency domain analysis Prof. Alberto Bemporad University of Trento Academic year Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year / 37

2 Frequency response Frequency response Definition The frequency response of a linear dynamical system with transfer function G(s) is the complex function G(jω) of the real angular frequency ω 0 u(t) =Ū sin(ωt) G(s) y(t) Theorem If G(s) is asymptotically stable (poles with negative real part), for u(t) = Ū sin(ωt) in steady-state conditions lim t y(t) y ss (t) = 0, where y ss (t) = Ū G(jω) sin(ωt + G(jω)) The frequency response G(jω) of a system allows us to analyze the response of the system to sinusoidal excitations at different frequencies ω Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year / 37

3 Frequency response Frequency response Proof Set Ū = 1 for simplicity. The input u(t) = sin ωt 1I(t) has Laplace transform U(s) = ω s 2 + ω 2 As we are interested in the steady-state response, we can neglect the natural response y l (t) of the output. In fact, as G(s) is asymptotically stable, y l (t) 0 for t The forced response of the output y(t) has Laplace transform ω Y f (s) = G(s) s 2 + ω = G(s)ω 2 (s jω)(s + jω) Let s compute the partial fraction decomposition of Y f (s) Y f (s) = ωg(jω) 2jω(s jω) + ωg( jω) n R i + 2jω(s + jω) s p }{{} i=1 i }{{} Y ss (s) = steady-state response transient forced response 1 1 ω R i =residue of G(s) s 2 +ω 2 in s = p i. We are assuming here that G(s) has distinct poles Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year / 37

4 Frequency response Frequency response Proof (cont d) As G(s) is asymptotically stable, the inverse Laplace transform n 1 R n i = R i e pit tends to zero asymptotically s p i=1 i i=1 The remaining steady-state output response is therefore G(jω) y ss (t) = 1 2j(s jω) + G( jω) 2j(s + jω) = j 2 G(jω)ejωt + j 2 G( jω)e jωt = j j 2 G(jω)ejωt + 2 G(jω)ejωt j = 2R 2 G(jω)ejωt = Im G(jω) e j G(jω)+jωt ) }{{} = G(jω) sin(ωt + G(jω)) Re( j(a + jb)) = Re( ja j 2 b) = Re(b ja) = b We can repeat the proof for Ū 1 and get the general result. Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year / 37

5 Frequency response Example u(t) = 5 sin 10t G(s) = 10 ( s) 2 y(t) The poles are p 1 = p 2 = 10, the system is asymptotically stable The steady-state response is where For ω = 10 rad/s, Finally, we get G(jω) = y ss (t) = 5 G(jω) sin(ωt + G(jω)) 10 ( jω) = jω 0.01ω 2 G(10j) = j 1 = 5 j = 5j y ss (t) = 5 5 sin(10t π 2 ) = 25 sin(10t π 2 ) Wednesday, April 28, 2010 Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year / 37

6 Frequency response Example (cont d) y ss (t) y(t) u(t) time (s) u(t) = 5 sin(10t) y ss (t) = 25 sin(10t π 2 ) Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year / 37

7 Frequency response Frequency response of discrete-time systems u(k) =Ū sin(kθ) G(z) y(k) A similar result holds for discrete-time systems: Theorem If G(z) is asymptotically stable (poles inside the unit circle), for u(k) = Ū sin(kθ) in steady-state conditions lim k y(k) y ss (k) = 0, where y ss (k) = Ū G(e jθ ) sin(kθ + G(e jθ )) Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year / 37

8 Bode plot Bode plot Ū sin(ωt) G(s) Ū G(jω) sin(ωt + G(jω)) The Bode plot is a graph of the module G(jω) and phase G(jω) of a transfer function G(s), evaluated in s = jω The Bode plot shows the system s frequency response as a function of ω, for all ω > 0 Hendrik Wade Bode ( ) Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year / 37

9 Bode plot Bode plot Bode plot of G(s) Bode magnitude plot Magnitude (db) G(s) = ( s)( s s 2 ) Bode phase plot Phase (deg) Frequency (rad/sec) Example: frequency response of a telephone, approx ,400 Hz, good enough for speech transmission The frequency ω axis is in logarithmic scale The module G(jω) is expressed as decibel (db) G(jω) db = 20 log 10 G(jω) Example: the DC-gain G(0) = 1, G(j0) db = 20 log 10 1 = 0 MATLAB»bode(G)»evalfr(G,w) Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year / 37

10 Bode Bode form To study the frequency response of the system is useful to rewrite the transfer function G(s) in Bode form K is the Bode gain G(s) = K s h Π i (1 + sτ i ) Π j (1 + st j ) Π i 1 + 2ζ i s + 1 s 2 ω ni ω 2 ni Π j 1 + 2ζ j ω nj s + 1 s ω 2 2 nj h is the type of the system, that is the number of poles in s = 0 T j (for real T j > 0) is said a time constant of the system ζ j is a damping ratio of the system, 1 < ζ j < 1 ω nj is a natural frequency of the system Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year / 37

11 Bode Bode magnitude plot K Π i (1 + jωτ i ) G(jω) db = 20 log 10 (jω) h Π j (1 + jωt j ) Recall the following properties of logarithms: Thus we get Π i 1 + 2jζ i ω ω ni Π j 1 + 2jζ jω ω nj ω2 ω 2 ni ω2 ω 2 nj log αβ = log α + log β, log α β = log α log β, log αβ = β log α G(jω) db = 20 log 10 K h 20 log 10 ω + 20 log jωτ i 20 log jωt j i + 20 log jζ i ω ω ni i j ω2 ω 2 ni j 20 log jζ jω ω nj ω2 ω 2 nj Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year / 37

12 Bode Bode magnitude plot We can restrict our attention to four basic components only: G(jω) db = 20 log 10 K h 20 log }{{} 10 ω }{{} #1 # log jωτ i i + 20 log jζ i ω ω ni i j ω2 ω 2 20 log jωt j }{{} ni #3 20 log jζ jω j ω2 ω nj ω 2 nj }{{} #4 Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year / 37

13 Bode Bode magnitude plot Basic component #1 20 log 10 K jg(j!)jdb log 10 jkj 20 Basic component #2 20 log 10 ω 20 log 10 ω ! jg(j!)jdb ! Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year / 37

14 Bode Bode magnitude plot Basic component #3 jg(j!)jdb log jωt G(jω) 1 for ω 1 T G(jω) ωt for ω 1 T Basic component #4 20 log jζ ω ω2 ω n G(jω) 1 for ω ω n G(jω) ω2 ω 2 for ω ω n n ω 2 n ! T -20 jg(j!)jdb db/ decade 6 ³= !n 1000! db/ decade ³ 0 ³ crescente For ζ < 1 2 the peak of the response is obtained at the resonant frequency ω peak = ω n 1 2ζ 2 Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year / 37

15 Bode Example Draw the Bode plot of the transfer function db/ decade jg(j!)j db G(s) = 1000(1 + 10s) s 2 (1 + s) db/ decade -60 db/ decade 0! For ω 0.1: G(jω) 1000 ω 2 For ω = 0.1: 20 log ω 2 = 20 log = 100 For 0.1 < ω < 1: effect of zero s = 0.1, increase by 20 db/decade For ω > 1: effect of double pole s = 1, decrease by 40 db/decade Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year / 37

16 Bode Bode phase plot G(jω) = K Π i (1 + jωτ i ) (jω) h Π j (1 + jωt j ) Π i 1 + 2jζ i ω ω ni Π j 1 + 2jζ jω ω nj Because of the following properties of exponentials we get ω2 ω 2 ni ω2 ω 2 nj (ρe jθ ) = θ (αβ) = (ρ α e jθ α ρβ e jθ β ) = (ρα ρ β e j(θ α+θ β ) ) = θ α + θ β = α + β α β = ρ αe jθ α ρ β e jθ β = ρα ρ β e j(θ α θ β ) = θ α θ β = α β G(jω) = K (jω) h + (1 + jωτ i ) (1 + jωt j ) i jζ i ω ω ni i j ω2 ω 2 ni j 1 + 2jζ jω ω nj ω2 ω 2 nj Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year / 37

17 Bode Bode phase plot We can again restrict our attention to four basic components only: G(jω) = K (jω) }{{}}{{} #1 #2 + (1 + jωτ i ) i jζ i ω ω ni i j (1 + jωt j ) }{{} ω2 ω 2 ni # jζ jω j ω nj ω2 ω 2 nj } {{ } #4 Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year / 37

18 Bode Bode phase plot Basic component #1 ¼=2 \ G(j!) K = 0 for K > 0 π for K < 0 0 K> ! -¼=2 Basic component #2 -¼ K<0 ¼=2 \ G(j!) (jω) h = hπ 2 0 h= ! -¼=2 h=1 -¼ h=2 Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year / 37

19 Bode Bode phase plot Basic component #3 (1 + jωt) = atan(ωt) G(jω) 0 for ω 1 T G(jω) π 2 for ω 1 T, T > 0 (1 jωt) = atan(ωt) ¼=2 \ G(j!) ¼=4 T> ¼= T ! -¼=2 T<0 Basic component # jζ ω ω2 ω n ω 2 n G(jω) 0 for ω ω n For ζ 0, G(jω) = π 2 for ω = ω n, G(jω) ω2 ω 2 = π for ω ω n n ¼ \ G(j!) ³ 0 ¼=2 ³ crescente !n 1000! Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year / 37

20 Bode Example (cont d) -¼=2 jg(j!)j db G(s) = 1000(1 + 10s) s 2 (1 + s) 2-3¼=4 -¼ -5¼=4! For ω 0.1: G(jω) π -3¼= For 0.1 < ω < 1: effect of zero in s = 0.1, add π 2 Per ω > 1: effect of double pole in s = 1, subtract 2 π 2 = π Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year / 37

21 Bode Example 100 G(s) = 10(1 + s) s 2 (1 10s)( s) Fase (deg) Modulo (db) Frequenza (rad/s) For ω 0.1: G(jω) 10 ω (slope=-40 db/dec), G(jω) π 2 10 For ω = 0.1: 20 log 10 ω = 60 db 2 For 0.1 < ω < 1: effect of unstable pole s = 0.1, decrease module by 20 db/dec, increase phase by π 2 For 1 < ω < 10: effect of zero s = 1, +20 db/dec module, + π 2 phase For ω > 10: effect of pole s = 10, -20 db/dec module, - π 2 phase Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year / 37

22 Bode Summary table for drawing asymptotic Bode plots magnitude phase stable real pole T > 0 20 db/dec π/2 unstable real pole T < 0 20 db/dec π/2 stable real zero τ > db/dec π/2 unstable real zero τ < db/dec π/2 pair of stable complex poles ζ > 0 40 db/dec π pair of unstable complex poles ζ < 0 40 db/dec π pair of stable complex zeros ζ > db/dec π pair of unstable complex zeros ζ < db/dec π Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year / 37

23 Bode More on damped oscillatory modes Let s = a ± jb be complex poles, b 0 Since (s (a jb))(s (a + jb)) = (s a) 2 + b 2 = s 2 2as + (a 2 + b 2 ) = (a 2 + b 2 )(1 2a a 2 +b s a 2 +b s 2 ), we get 2 a ω n = a 2 + b 2 = a ± jb, ζ = = cos (a ± jb) a2 + b2 Note that ζ > 0 if and only if a < 0 Vice versa, a = ζω n, and b = ω n 1 ζ 2 The natural response is Me ζω nt sin(ω n 1 ζ2 t + φ) where M, φ depend on the initial conditions The frequency ω d = ω n 1 ζ2 is called damped natural frequency Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year / 37

24 Bode Zero/pole/gain form and Bode form Sometimes the transfer function G(s) is given in zero/pole/gain form (ZPK) G(s) = K s h Π i (s z i ) Π j (s p j ) Π i s 2 + 2ζ i ω ni s + ω 2 ni Π j s 2 + 2ζ j ω nj s + ω 2 nj The relations between Bode form and ZPK form are z i = 1 τ i, p j = 1 T i, K = K Π i( z i )Π i ω 2 ni Π j ( p j )Π i ω 2 ni MATLAB»zpk(G) Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year / 37

25 Nyquist plot Nyquist (or polar) plot Let G(s) be a transfer function of a linear time-invariant dynamical system The Nyquist plot (or polar plot) is the graph in polar coordinates of G(jω) for ω [0, + ) in the complex plane G(jω) = ρ(ω)e jφ(ω), where ρ(ω) = G(jω) and φ(ω) = G(jω) The Nyquist plot is one of the classical methods used in stability analysis of linear systems The Nyquist plot combines the Bode magnitude & phase plots in a single plot Harry Nyquist ( ) Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year / 37

26 Nyquist plot Nyquist plot To draw a Nyquist plot of a transfer function G(s), we can give some hints: Imaginary Axis db 2 db 6 db 10 db 20 db 0 db 2 db Nyquist plot of G(s) 4 db 20 db 10 db 6 db Real Axis G(s) = 1 ( s)( s s 2 ) For ω = 0, the Nyquist plot equals the DC gain G(0) If G(s) is strictly proper, lim ω G(jω) = 0 In this case the angle of arrival equals (n z n+ z n p + n+ p ) π 2, where n+[ ] is the # z[p] of zeros [poles] with positive [negative] real part G( jω) = G(jω) A system without any zero with positive real part is called minimum phase Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year / 37

27 Nyquist criterion Nyquist criterion Consider a transfer function G(s) = N(s)/D(s) under unit static feedback u(t) = (y(t) r(t)) As y(t) = G(s)( y(t) + r(t)), the closed-loop transfer function from r(t) to y(t) is W(s) = G(s) 1 + G(s) = N(s) D(s) + N(s) Nyquist stability criterion The number N W of closed-loop unstable poles of W(s) is equal to the number N R of clock-wise rotations of the Nyquist plot around s = 1 + j0 plus the number N G of unstable poles of G(s). [N W = N R + N G ] Corollary: simplified Nyquist criterion For open-loop asympt. stable systems G(s) (N G = 0), the closed-loop system W(s) is asymptotically stable if and only if the Nyquist plot G(jω) does not encircle clock-wise the critical point 1 + j0. [N W = N R ] Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year / 37

28 Nyquist criterion Nyquist stability criterion Proof: Follows by the Argument principle: The polar diagram of 1 + G(s) has a number of clock-wise rotations around the origin equal to the number of zeros of 1 + G(s) (=roots of N(s) + G(s)) minus the number of poles of 1 + G(s) (=roots of D(s)) of 1 + G(s) The poles of W(s) are the zeros of 1 + G(s) Note: the number of counter-clockwise encirclements counts as a negative number of clockwise encirclements Use of Nyquist criterion for open-loop stable systems: 1 draw the Nyquist plot of G(s) 2 count the number of clockwise rotations around 1 + j0 3 if 1 + j0 is not encircled, the closed-loop system W(s) = G(s)/(1 + G(s)) is stable The Nyquist criterion is limited to SISO systems Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year / 37

29 Nyquist criterion Example Consider the transfer function G(s) = 10 (s + 1) = 10 under static output 2 s 2 + 2s + 1 feedback u(t) = (y(t) r(t)) Nyquist Diagram 8 6 Imaginary Axis The closed-loop transfer function is W(s) = G(s) 1 + G(s) Real Axis MATLAB»G=tf(10,[1 2 1])»nyquist(G) The Nyquist plot of G(s) does not encircle 1 + j0 By Nyquist criterion, W(s) is asymptotically stable Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year / 37

30 Nyquist criterion Example Consider the transfer function G(s) = u(t) = 2(y(t) r(t)) 10 under unit output feedback s 3 + 3s 2 + 2s + 1 Nyquist Diagram Imaginary Axis Real Axis MATLAB»G=tf(10,[ ]); K=2»nyquist(K*G)»L=feedback(G,K)»pole(L) i i The closed-loop transfer function is W(s) = 2G(s) 1 + 2G(s) The Nyquist plot of 2G(s) encircles 1 + j0 twice By Nyquist criterion, W(s) is unstable Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year / 37

31 Nyquist criterion Dealing with imaginary poles The Nyquist plot is generated by a closed curve, the Nyquist contour, rotating clock-wise from 0 j to 0 + j and back to 0 j along a semi-circle of radius, avoiding singularities of G(jω) We distinguish three types of curves, depending on the number of poles on the imaginary axis: Im(ω) Im(ω) jω 0 Im(ω) ω 0 + x ω0 j0 x + Re(ω) 0 x + Re(ω) 0 x + Re(ω) jω 0 x ω + 0 ω 0 no poles on imaginary axis pole(s) in s = 0 pole(s) in s = ±jω (counted as stable) (counted as stable) Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year / 37

32 Nyquist criterion Stability analysis of static output feedback Under static output feedback u(t) = K(y(t) r(t)), the closed-loop transfer function from r(t) to y(t) is W(s) = KG(s) 1 + KG(s) The number of encirclements of 1 + j0 of KG(s) is equal to the number of encirclements of 1 K + j0 of G(s) To analyze closed-loop stability for different values of K is enough to draw Nyquist plot of G(s) and move the point 1 K + j0 on the real axis G(s) = 60 (s 2 + 2s + 20)(s + 1) I0=no unstable closed-loop poles I2 =two unstable closed-loop poles I1=an unstable closed-loop pole Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year / 37

33 Phase and gain margins Phase margin Assume G(s) is open-loop asymptotically stable Let ω c be the gain crossover frequency, that is G(jω c ) = 1 To avoid encircling the point 1 + j0, we want the phase G(jω c ) as far away as possible from π The phase margin is the quantity M p = G(jω c ) ( π) If M p > 0, unit negative feedback control is asymptotically stabilizing For robustness of stability we would like a large positive phase margin IG(jω) 1+j0 M p RG(jω) G(jω c ) Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year / 37

34 Phase and gain margins Gain margin Assume G(s) is open-loop asymptotically stable Let ω π be the phase crossover frequency such that G(jω π ) = π To avoid encircling the point 1 + j0, we want the point G(jω π ) as far as possible away from 1 + j0. The gain margin is the inverse of G(jω π ), expressed in db M g = 20 log 10 1 G(jω π ) = G(jω π) db For robustness of stability we like to have a large gain margin IG(jω) 1+j0 G(jω π ) 1 M g RG(jω) Sometimes the gain margin is defined as M g = 1 G(jω π ) and therefore G(jω π ) = 1 M g + j0 Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year / 37

35 Phase and gain margins Stability analysis using phase and gain margins G(s) = 8 s 3 + 4s 2 + 4s + 2 The phase and gain margins help assessing the degree of robustness of a closed-loop system against uncertainties in the magnitude and phase of the process model They can be also applied to analyze the stability of dynamic output feedback laws u(t) = C(s)(y(t) r(t)), by looking at the Bode plots of the loop function C(jω)G(jω) (see next lecture on loop shaping ) Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year / 37

36 Phase and gain margins Stability analysis using phase and gain margins However in some cases phase and gain margins are not good indicators, see the following example IG(jω) 1+j0 RG(jω) M p The system is characterized by a large phase margin, but the polar plot is very close to the point ( 1, 0) The system is not very robust to model uncertainties changing G(s) from its nominal value In conclusion, phase and gain margins give good indications on how the loop Thursday, April 29, 2010 function should be modified, but the complete Bode and Nyquist plots must be checked to conclude about closed-loop stability Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year / 37

37 Phase and gain margins English-Italian Vocabulary frequency response angular frequency steady-state Bode plot Bode form Bode gain system s type time constant damping ratio natural frequency zero/pole/gain form minimum phase system Nyquist plot crossover frequency risposta in frequenza pulsazione regime permanente diagramma di Bode forma di Bode guadagno di Bode tipo del sistema costante di tempo fattore di smorzamento frequenza naturale forma poli/zeri sistema a fase minima diagramma di Nyquist frequenza di attraversamento Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year / 37

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