# FREQUENCY-RESPONSE ANALYSIS

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1 ECE450/550: Feedback Control Systems. 8 FREQUENCY-RESPONSE ANALYSIS 8.: Motivation to study frequency-response methods Advantages and disadvantages to root-locus design approach: ADVANTAGES: Good indicator of transient response. Explicitly shows location of closed-loop poles. Tradeoffs are clear. DISADVANTAGES: Requires transfer function of plant be known. Difficult to infer all performance values. Hard to extract steady-state response (sinusoidal inputs). Frequency-response methods can be used to supplement root locus: Can infer performance and stability from same plot. Can use measured data when no model is available. Design process is independent of system order (# poles). Time delays handled correctly (e sτ ). Graphical techniques (analysis/synthesis) are quite simple. What is a frequency response? We want to know how a linear system responds to sinusoidal input, in steady state.

2 ECE450/ECE550, FREQUENCY-RESPONSE ANALYSIS 8 2 Consider system Y (s) = G(s)U(s) with input u(t) = u0 cos(ωt), so s U(s) = u 0 s 2 + ω 2. With zero initial conditions, s Y (s) = u 0 G(s) s 2 + ω 2. Do a partial-fraction expansion (assume distinct roots) Y (s) = α s a + α 2 s a α n s a n + α 0 s jω + α 0 s + jω y(t) = α e at + α 2 e a2t + +α n e a nt }{{} +α 0 e jωt + α0 e jωt. If stable, these decay to zero. y ss (t) = α 0 e jωt + α 0 e jωt. Let α0 = Ae jφ.then, y ss = Ae jφ e jωt + Ae jφ e jωt = A ( e j (ωt+φ) + e j (ωt+φ)) = 2A cos (ωt + φ). We find α 0 via standard partial-fraction-expansion means: α 0 = [ (s jω)y (s) s= jω [ u0 sg(s) = (s + jω) s= jω = u 0( jω)g( jω) (2 jω) Substituting into our prior result = u 0G( jω). 2 y ss = u 0 G( jω) cos (ωt + G( jω)).

3 ECE450/ECE550, FREQUENCY-RESPONSE ANALYSIS 8 3 Important LTI-system fact: If the input to an LTI system is a sinusoid, the steady-state output is a sinusoid of the same frequency but different amplitude and phase. FORESHADOWING: Transfer function at s = jω tells us response to a sinusoid...but also about stability as jω-axis is stability boundary! EXAMPLE: Suppose that we have a system with transfer function G(s) = s. Then, the system s frequency response is G( jω) = s = s= jω jω. The magnitude response is A( jω) = jω = jω = 2 = (3 + jω)(3 jω) The phase response is ( ) 2 φ(jω) = 3 + jω = (2) (3 + jω) = 0 tan (ω/3). θ ω 2. Now that we know the amplitude and phase response, we can find the amplitude gain and phase change caused by the system for any specific frequency. For example, if ω = 3rads, A( j3) = = 2 φ(j3) = tan (3/3) = π/4. 3 ω

4 ECE450/ECE550, FREQUENCY-RESPONSE ANALYSIS : Plotting a frequency response There are two common ways to plot a frequency response the magnitude and phase for all frequencies. EXAMPLE: U(s) R C Y (s) G(s) = + RCs Frequency response G( jω) = = = + jωrc + jω + ω 2 (let RC = ) tan (ω). We will need to separate magnitude and phase information from rational polynomials in jω. Magnitude = magnitude of numerator / magnitude of denominator R(num)2 + I(num) 2 R(den)2 + I(den) 2. Phase = phase of numerator phase of denominator ( ) ( ) I(num) I(den) tan tan. R(num) R(den) Plot method #: Polar plot in complex plane Evaluate G( jω) at each frequency for 0 ω<. Result will be a complex number at each frequency: a + jb or Ae jφ.

5 ECE450/ECE550, FREQUENCY-RESPONSE ANALYSIS 8 5 Plot each point on the complex plane at (a + jb) or Ae jφ for each frequency-response value. Result = polar plot. We will later call this a Nyquist plot. ω G( jω) The polar plot is parametric in ω, soitishardtoreadthe frequency-response for a specific frequency from the plot. We will see later that the polar plot will help us determine stability properties of the plant and closed-loop system. Plot method #2: Magnitude and phase plots We can replot the data by separating the plots for magnitude and phase making two plots versus frequency.

6 ECE450/ECE550, FREQUENCY-RESPONSE ANALYSIS 8 6 G( jω) Frequency, (rads/sec.) G( jω) Frequency, (rads/sec.) The above plots are in a natural scale, but usually a log-log plot is made This is called a Bode plot or Bode diagram. Reason for using a logarithmic scale Simplest way to display the frequency response of a rational-polynomial transfer function is to use a Bode Plot. Logarithmic G( jω) versus logarithmic ω, andlogarithmic versus ω. REASON: G( jω) ( ) ab log 0 = log cd 0 a + log 0 b log 0 c log 0 d. The polynomial factors that contribute to the transfer function can be split up and evaluated separately. (s + ) G(s) = (s/0 + ) G( jω) = ( jω + ) ( jω/0 + ) jω + G( jω) = jω/0 + ( ω ) 2. log 0 G( jω) =log 0 + ω2 log 0 + 0

7 ECE450/ECE550, FREQUENCY-RESPONSE ANALYSIS 8 7 Consider: For ω ωn, ( ) ω 2 log 0 +. ( ω log 0 + ) 2 log 0 () = 0. For ω ωn, ( ) ω 2 ( ) ω log 0 + log 0. KEY POINT: Two straight lines on a log-log plot; intersect at ω =. Typically plot 20 log0 G( jω) ; thatis,indb. 20dB Exact Approximation 0. 0 Atransferfunctionismadeupoffirst-orderzerosandpoles,complex zeros and poles, constant gains and delays. We will see how to make straight-line (magnitude- and phase-plot) approximations forallthese, and combine them to form the appropriate Bode diagram.

8 ECE450/ECE550, FREQUENCY-RESPONSE ANALYSIS : Bode magnitude diagrams (a) The log0 ( ) operator lets us break a transfer function up into pieces. If we know how to plot the Bode plot of each piece, then we simply add all the pieces together when we re done. Bode magnitude: Constant gain db = 20 log0 K. Not a function of frequency. Horizontal straight line. If K <, thennegative,else positive. db K > 0. 0 K < Bode magnitude: Zero or pole at origin For a zero at the origin, G(s) = s db = 20 log 0 G( jω) = 20 log 0 jω db. For a pole at the origin, G(s) = s db = 20 log 0 G( jω) = 20 log 0 jω db. 20 db 20 db 20 db 20 db 20 db per decade db per decade 0. 0

9 ECE450/ECE550, FREQUENCY-RESPONSE ANALYSIS 8 9 Both are straight lines, slope =±20 db per decade of frequency. Line intersects ω-axis at ω =. For an nth-order pole or zero at the origin, Still straight lines. Still intersect ω-axis at ω =. db =±20 log 0 ( jω) n =±20 log 0 =±20n log 0 ω. But, slope =±20n db per decade. Bode magnitude: Zero or pole on real axis, but not at origin For a zero on the real axis, (LHP or RHP), the standard Bode form is ( ) s G(s) = ±, which ensures unity dc-gain. If you start out with something like G(s) = (s + ), then factor as ( ) s G(s) = +. Draw the gain term ( )separatelyfromthezeroterm(s/ + ). In general, a LHP or RHP zero has standard Bode form ( ) s G(s) = ±

10 ECE450/ECE550, FREQUENCY-RESPONSE ANALYSIS 8 0 ( ) ω G( jω) =± + j ( ω 20 log 0 G( jω) =20 log 0 + ( ) ω 2 For ω ωn, 20 log log 0 = 0. ( ) ω 2 ( ω For ω ωn, 20 log log 0 Two straight lines on a log scale which intersect at ω = ωn. For a pole on the real axis, (LHP or RHP) standard Bode form is ) 2 ). G(s) = ( s ± ) ( ) ω 2 20 log 0 G( jω) = 20 log 0 +. This is the same except for a minus sign db 20 db per decade db 20 db per decade

11 ECE450/ECE550, FREQUENCY-RESPONSE ANALYSIS 8 8.4: Bode magnitude diagrams (b) Bode magnitude: Complex zero pair or complex pole pair For a complex-zero pair (LHP or RHP) standard Bode form is ( ) s 2 ( ) s ± 2ζ +, which has unity dc-gain. If you start out with something like s 2 ± 2ζ s + ω 2 n, which we have seen before as a standard form, the dc-gain is ω 2 n. Convert forms by factoring out ω 2 n [ ( s s 2 ± 2ζ s + ωn 2 = ω2 n ) 2 ± 2ζ ( ) ] s +. Complex zeros do not lend themselves very well to straight-line approximation. ( s 2 If ζ =, thenthisis ± ). Double real zero at ωn slope of 40 db/decade. For ζ =, therewillbeovershootorundershootatω ωn. For other values of ζ : 0 Dip amount for 0 < ζ < Dip frequency: ω d = 2ζ 2 Value of H( jω d ) is: 20 log 0 (2ζ ( ζ 2 )). Note: There is no dip unless 0 <ζ </ ζ Dip (db)

12 ECE450/ECE550, FREQUENCY-RESPONSE ANALYSIS 8 2 We write complex poles (LHP or RHP) as [ ( ) s 2 ( s G(s) = ± 2ζ ) + ]. The resonant peak frequency is ω r = 2ζ 2 Value of H( jω r ) is 20 log 0 (2ζ ( ζ 2 )). Same graph as for dip for complex-conjugate zeros. Note that there is no peak unless 0 <ζ </ For ω,magnitude 0dB. For ω,magnitudeslope= 40 db/decade. 40 db 20 db 0 db 20 db Bode Mag: Complex zeros ζ = Bode magnitude: Time delay 20 db 0 db 20 db Bode Mag: Complex poles ζ = ζ = db 0. 0 G(s) = e sτ... G( jω) =. 20 log0 = 0dB. Does not change magnitude response. EXAMPLE: Sketch the Bode magnitude plot for G(s) = 2000(s + 0.5) s(s + 0)(s + 50).

13 ECE450/ECE550, FREQUENCY-RESPONSE ANALYSIS 8 3 The first step is to convert the terms of the transfer function into Bode standard form. G(s) = G( jω) = (s + 0.5) s(s + 0)(s + 50) = 0 50 ( ) 2 jω jω ( jω 0 + )( jω 50 + ). ( s ) s ( s 0 + )( s 50 + ) We can see that the components of the transfer function are: DC gain of 20 log 0 2 6dB; Pole at origin; One real zero not at origin, and Two real poles not at origin

14 ECE450/ECE550, FREQUENCY-RESPONSE ANALYSIS : Bode phase diagrams (a) Bode diagrams consist of the magnitude plots we have seen so far, BUT, also phase plots. These are just as easy to draw. BUT, they differ depending on whether the dynamics are RHP or LHP. Finding the phase of a complex number Plot the location of the number as a vector in the complex plane. Use trigonometry to find the phase. For numbers with positive real part, ( ) I(#) (#) = tan. R(#) For numbers with negative real part, ( ) I(#) (#) = 80 tan. R(#) p 2 (p 2 ) If you are lucky enough to have the atan2(y, x) function, then I R p (p ) (#) = atan2(i(#), R(#)) for any complex number. Also note, ( ) ab cd = (a) + (b) (c) (d). Finding the phase of a complex function of ω This is the same as finding the phase of a complex number, if specific values of ω are substituted into the function.

15 ECE450/ECE550, FREQUENCY-RESPONSE ANALYSIS 8 5 Bode phase: Constant gain G(s) = K. { 0, K 0; (K ) = 80, K < 0.. Constant phase of 0 or 80. Bode phase: Zero or pole at origin Zero: G(s) = s,... G( jω) = jω = ω 90. Pole: G(s) =,... G( jω) = s jω = j ω = 90 ω. Constant phase of ±90. Bode phase: Real LHP zero or pole ( ) s Zero: G(s) = +. G( jω) = ( j ω ) + ( ) ω = tan. Pole: G(s) = ( ), s + ( G( jω) = () j ω ) + ( ) ω = tan per decade per decade

16 ECE450/ECE550, FREQUENCY-RESPONSE ANALYSIS 8 6 Bode phase: Real RHP zero or pole ( ) s Zero: G(s) =. G( jω) = ( j ω ) ( ) ω = 80 tan per decade 90 Pole: G(s) = ( ), s ( G( jω) = () j ω ) ( )) = (80 tan ωωn ( ) ω = 80 + tan per decade

17 ECE450/ECE550, FREQUENCY-RESPONSE ANALYSIS : Bode phase diagrams (b) Bode phase: Complex LHP zero pair or pole pair Complex LHP zeros cause phase to go from 0 to 80. Complex LHP poles cause phase to go from 80 to 0. Transition happens in about ±ζ decades, centered at ωn. Bode Phase: Complex LHP zeros 80 ζ = Bode Phase: Complex LHP poles 0 ζ = Bode phase: Complex RHP zero pair or pole pair 0. 0 Complex RHP zeros cause phase to go from 360 to 80. Complex RHP poles cause phase to go from 360 to Bode Phase: Complex RHP zeros Bode Phase: Complex RHP poles

18 ECE450/ECE550, FREQUENCY-RESPONSE ANALYSIS 8 8 Bode phase: Time delay G(s) = e sτ, G( jω) = e jωτ = ωτ 0.ω i ω i 0ω i G( jω) = ωτ in radians. = 56.3ωτ in degrees. Note: Line curve in log scale. EXAMPLE: Sketch the Bode phase plot for 2000(s + 0.5) G(s) = or G( jω) = s(s + 0)(s + 50) 2 ( jω/0.5 + ) jω ( jω/0 + )(jω/50 + ), where we converted to Bode standard form in a prior example. Constant: K =+2. Zerophasecontribution. Pole at origin: Phase contribution of 90. Two real LHP poles: Phase from 0 to 90,each. One real LHP zero: Phase from 0 to

19 ECE450/ECE550, FREQUENCY-RESPONSE ANALYSIS 8 9 EXAMPLE: Sketch the Bode magnitude and phase plots for G(s) = 200 (s + 3) s (s + 2)(s + 50). First, we convert to Bode standard form, which gives 200 (3) ( ) + s 3 G(s) = s (2)(50) ( )( ) + s 2 + s 50 ( ) 6 + jω 3 G( jω) = ( )( ). jω + jω + jω 2 50 Positive gain, one real LHP zero, one pole at origin, two real LHP poles

20 ECE450/ECE550, FREQUENCY-RESPONSE ANALYSIS : Some observations based on Bode plots Nonminimum-phase systems Asystemiscalledanonminimum-phaseifithaspole(s)orzero(s) in the RHP. Consider G (s) = 0 s + } s + 0 G 2 (s) = 0 s } s + 0 zero at pole at 0 zero at + pole at 0 } } minimum phase nonminimum phase The magnitude responses of these two systems are: jω + G ( jω) =0 jω + 0 = 0 ω2 + ω jω G 2 ( jω) =0 jω + 0 = 0 ω2 + ω which are the same! The phase responses are very different: G( jω), G2( jω) Bode-Magnitude Plot Frequency, (rads/sec.) (G( jω)), (G2( jω)) Minimum Phase, G Bode-Phase Plot Non-minimum Phase, G Frequency, (rads/sec.) Note that the change in phase of G is much smaller than change of phase in G 2.HenceG is minimum phase and G 2 is nonminimum-phase

21 ECE450/ECE550, FREQUENCY-RESPONSE ANALYSIS 8 2 Non-minimum phase usually associated with delay. G 2 (s) = G (s) s s + }{{} Delay s Note: is very similar to a first-order Padé approximation to a s + delay. It is the same when evaluated at s = jω. Consider using feedback to control a nonminimum-phase system. What do the root-locus plotting techniques tell us? Consequently, nonminimum-phase systems are harder to design controllers for; step response often tends to go the wrong way, at least initially. Steady-state errors from Bode magnitude plot Recall our discussion of steady-state errors to step/ramp/parabolic inputs versus system type (summarized on pg. 4 24) Consider a unity-feedback system. If the open-loop plant transfer function has N poles at s = 0 then the system is type N K p is error constant for type 0. K v is error constant for type. K a is error constant for type 2... For a unity-feedback system, K p = lim s 0 G(s). At low frequency, atype0systemwillhaveg(s) K p. We can read this off the Bode-magnitude plot directly!

22 ECE450/ECE550, FREQUENCY-RESPONSE ANALYSIS 8 22 Horizontal y-intercept at low frequency = K p. e ss = + K p for step input. Kv = lim s 0 sg(s), andisnonzeroforatypesystem. At low frequency, atypesystemwillhaveg(s) K v s. At low frequency, G( jω) K v.slopeof 20 db/decade. ω Use the above approximation to extend the low-frequency asymptote to ω =. Theasymptote(NOT THE ORIGINAL G( jω) ) evaluatedatω = is K v. e ss = K v for ramp input. Ka = lim s 0 s 2 G(s), andisnonzeroforatype2system. At low frequency, atype2systemwillhaveg(s) K a s 2. At low frequency, G( jω) K a.slopeof 40 db/decade. ω2 Again, use approximation to extend low-frequency asymptote to ω =. Theasymptoteevaluatedatω = is K a. Similar for higher-order systems. e ss = K a for parabolic input.

23 ECE450/ECE550, FREQUENCY-RESPONSE ANALYSIS Example 60 Example Magnitude Magnitude Frequency, (rads/sec.) EXAMPLE : Frequency, (rads/sec.) EXAMPLE 2: Horizontal as ω 0, soweknowthisistype0. Intercept = 6dB...K p = 6dB= 2 [linear units]. Slope = 20 db/decade as ω 0, soweknowthisistype. Extend slope at low frequency to ω =. Intercept = 20 db... K v = 20 db = 0 [linear units].

24 ECE450/ECE550, FREQUENCY-RESPONSE ANALYSIS : Stability revisited If we know the closed-loop transfer function of a system in rationalpolynomial form, we can use Routh to find stable ranges for K. Motivation: Whatifweonlyhaveopen-loopfrequency response? Asimpleexample Consider, for now, that we know the transfer-function of the system, and can plot the root-locus. EXAMPLE: I(s) r(t) K s(s + ) 2 y(t) K = 2 R(s) We see neutral stability at K = 2. ThesystemisstableforK < 2 and unstable for K > 2. Recall that a point is on the root locus if KG(s) = and G(s) = 80. If system is neutrally stable, jω-axis will have a point (points) where KG( jω) = and G( jω) = 80.

25 ECE450/ECE550, FREQUENCY-RESPONSE ANALYSIS 8 25 Consider the Bode plot of KG(s)... Aneutral-stabilityconditionfrom Bode plot is: KG( jω o ) = AND KG( jω o ) = 80 at the same frequency ω o K = 0. K = 2 K = In this case, increasing K instability KG( jω) < at KG( jω) = 80 =stability. In some cases, decreasing K instability KG( jω) > at KG( jω) = 80 =stability KEY POINT: We can find neutral stability point on Bode plot, but don t (yet) have a way of determining if the system is stable or not. Nyquist found a frequency-domain method to do so. Nyquist stability Poles of closed-loop transfer function in RHP the system is unstable. Nyquist found way to count closed-loop poles in RHP. If count is greater than zero, system is unstable. Idea: First, find a way to count closed-loop poles inside a contour. Second, make the contour equal to the RHP. Counting is related to complex functional mapping.

26 ECE450/ECE550, FREQUENCY-RESPONSE ANALYSIS : Interlude: Complex functional mapping Nyquist technique is a graphical method to determine system stability, regions of stability and MARGINS of stability. Involves graphing complex functions of s as a polar plot. EXAMPLE: Plotting f (x), arealfunctionofarealvariablex. f (x) This can be done. EXAMPLE: PlottingF(s), acomplexfunctionofacomplexvariables. F(s)? s x NO! This is wrong! Must draw mapping of points or lines from s-plane to F(s)-plane. jω s I(F) F(s 0 ) F(s) s 0 mapping σ R(F) EXAMPLE: F(s) = 2s +... map the four points: A, B, C, D D j A I(s) R(s) C j B

27 ECE450/ECE550, FREQUENCY-RESPONSE ANALYSIS 8 27 EXAMPLE: Mapasquarecontour(closedpath)byF(s) = D j A j I(s) F(D) s s + 2. F(A) F(B) R(s) C j B j F(C) FORESHADOWING: By drawing maps of a specific contour, using a mapping function related to the plant open-loop frequency-response, we will be able to determine closed-loop stability of systems. Mapping function: Poles of the function When we map a contour containing (encircling) poles and zeros of the mapping function, this map will give us information about how many poles and zeros are encircled by the contour. Practice drawing maps when we know poles and zeros. Evaluate G(s) s=so = G(s o ) = v e jα α = (zeros) (poles). EXAMPLE: I(s) F(c ) c R(s) α

28 ECE450/ECE550, FREQUENCY-RESPONSE ANALYSIS 8 28 In this example, there are no zeros or poles inside the contour. The phase α increases and decreases, but never undergoes a net change of 360 (does not encircle the origin). EXAMPLE: I(s) c 2 F(c 2 ) R(s) α One pole inside contour. Resulting map undergoes 360 net phase change. (Encircles the origin). EXAMPLE: I(s) c 3 F(c 3 ) R(s) In this example, there are two poles inside the contour, and the map encircles the origin twice.

29 ECE450/ECE550, FREQUENCY-RESPONSE ANALYSIS : Cauchy s theorem and Nyquist s rule These examples give heuristic evidence of the general rule: Cauchy s theorem Let F(s) be the ratio of two polynomials in s. Lettheclosed curve C in the s-plane be mapped into the complex plane through the mapping F(s). IfthecurveC does not pass through any zeros or poles of F(s) as it is traversed in the CW direction, the corresponding map in the F(s)-plane encircles the origin N = Z P times in the CW direction, where Z = # of zeros of F(s) in C, P = # of poles of F(s) in C. Consider the following feedback system: r(t) D(s) H(s) G(s) y(t) T (s) = D(s)G(s) + D(s)G(s)H(s). For closed-loop stability, no poles of T (s) I(s) in RHP. R No zeros of + D(s)G(s)H(s) in RHP. Let F(s) = + D(s)G(s)H(s). R(s) Count zeros in RHP using Cauchy theorem! (Contour=entire RHP). The Nyquist criterion simplifies Cauchy s criterion for feedback systems of the above form.

30 ECE450/ECE550, FREQUENCY-RESPONSE ANALYSIS 8 30 Cauchy: F(s) = + D(s)G(s)H(s). N = # of encirclements of origin. Nyquist: F(s) = D(s)G(s)H(s). N = # of encirclements of. I(s) Cauchy I(s) Nyquist R(s) R(s) Simple? YES!!! Think of Nyquist path as four parts: I(s) I. Origin. Sometimes a special case (later examples). II III II. + jω-axis. FREQUENCY-response of O.L. system! Just plot it as a polar plot. III. For physical systems=0. I IV R(s) IV. Complex conjugate of II. So, for most physical systems, the Nyquist plot, used to determine CLOSED-LOOP stability, is merely a polar plot of LOOP frequency response D( jω)g( jω)h( jω). We don t even need a mathematical model of the system. Measured data of G( jω) combined with our known D( jω) and H( jω) are enough to determine closed-loop stability.

31 ECE450/ECE550, FREQUENCY-RESPONSE ANALYSIS 8 3 THE TEST: N = # encirclements of point when F(s) = D(s)G(s)H(s). P = # poles of + F(s) in RHP= # of open-loop unstable poles. (assuming that H(s) is stable reasonable). Z = # of zeros of + F(s) in RHP= # of closed-loop unstable poles. Z = N + P The system is stable iff Z = 0. Be careful counting encirclements! Draw line from in any direction. Count # crossings of line and diagram. N = #CW crossings #CCW crossings. Changing the gain K of F(s) MAGNIFIES the entire plot. ENHANCED TEST: Loop transfer function is KD(s)G(s)H(s). N = # encirclements of /K point when F(s) = D(s)G(s)H(s). Rest of test is the same. Gives ranges of K for stability.

32 ECE450/ECE550, FREQUENCY-RESPONSE ANALYSIS : Nyquist test example EXAMPLE: D(s) = H(s) =. G(s) = or, G( jω) = I: At s = 0, G(s) = 5. II: At s = jω, G( jω) = 5 (s + ) 2 III: At s =, G(s) = 0. 5 ( jω + ) 2 5 ( + jω) 2. IV: At s = jω, G(s) = 5 ( jω) 2. I(s) ω R(G( jω)) I(G( jω)) R(s) No encirclements of, N = 0. No open-loop unstable poles P = 0. Z = N + P = 0. Closed-loopsystemisstable. No encirclements of /K for any K > 0. So, system is stable for any K > 0.

33 ECE450/ECE550, FREQUENCY-RESPONSE ANALYSIS 8 33 Confirm by checking Routh array. Routh array: a(s) = + KG(s) = s 2 + 2s + + 5K. s 2 + 5K s 2 s 0 + 5K Stable for any K > EXAMPLE: G(s) = (s + ) 2 (s + 0). I: G(0) = 50/0 = II: G( jω) = ( jω + ) 2 ( jω + 0). III: G( ) = 0. IV: G( jω) = G( jω). Note loop to left of origin. System is NOT stable for all K > 0. I(s) ω R(G( jω)) I(G( jω)) Zoom R(s)

34 ECE450/ECE550, FREQUENCY-RESPONSE ANALYSIS : Nyquist test example with pole on jω-axis EXAMPLE: Pole(s) at origin. G(s) = s(τs + ). WARNING! Wecannotblindlyfollowprocedure! Nyquist path goes through pole at zero! (Remember from Cauchy s theorem that the path cannot pass directly through a pole or zero.) Remember: We want to count closed-loop poles inside a box that encompasses the RHP. So, we use a slightly-modified Nyquist path. I(s) I(s) Zoom II III ρ 0 I R(s) θ R(s) IV The bump at the origin makes a detour around the offending pole. Bump defined by curve: s = lim ρ 0 ρe jθ, 0 θ 90. From above, G(s) s=ρe jθ = ρe jθ (τρe jθ + ), 0 θ 90 Consider magnitude as ρ 0 lim G(s) s=ρe ρ 0 jθ = ρ τρe jθ + ρ.

35 ECE450/ECE550, FREQUENCY-RESPONSE ANALYSIS 8 35 Consider phase as ρ 0 So, lim G(s) s=ρe jθ = θ (τρe jθ + ). ρ 0 lim G(ρe jθ ) = lim ρ 0 ρ 0 ρ θ + This is an arc of infinite radius, sweeping from 0 to 90 + (a little more than 90 because of contribution from (τs + ) term). WE CANNOT DRAW THIS TO SCALE! Z = N + P. N = # encirclements of. N = 0. P = # Loop transfer function poles inside MODIFIED contour. P = 0. Z = 0. Closed-loopsystemisstable. EXAMPLE: G(s) = Use modified Nyquist path again s 2 (s + ) I: Near origin G(s) s=ρe jθ = Magnitude: lim ρ 0 G(ρe jθ ) = ρ 2 e j2θ ( + ρe jθ ). ρ 2 + ρe jθ ρ 2. Phase: lim ρ 0 G(ρe jθ ) = 0 [2θ + ( + ρe jθ )] 2θ +.So, lim G(ρe jθ ) = lim ρ 0 ρ 0 ρ 2 2θ + 0 θ 90.

36 ECE450/ECE550, FREQUENCY-RESPONSE ANALYSIS 8 36 Infinite arc from 0 to 80 + (a little more than 80 because of + s term.) Z = N + P = = 2. Unstablefor K =. In fact, unstable for any K > 0! Matlab for above G(s) = num=[0 0 0 ]; den=[ 0 0]; nyquist(num,den); s 3 + s 2 + 0s + 0 axis([xmin xmax ymin ymax]); nyquist.m is available on course web site. It repairs the standard Matlab nyquist.m program, which doesn t work when poles are on imaginary axis. nyquist2.m is also available. It draws contours around poles on the imaginary axis in the opposite way to nyquist.m. Counting is different.

37 ECE450/ECE550, FREQUENCY-RESPONSE ANALYSIS : Stability (gain and phase) margins Alargefractionofsystemstobecontrolledarestableforsmall gain but become unstable if gain is increased beyond a certain point. The distance between the current (stable) system and an unstable system is called a stability margin. Can have a gain margin and a phase margin. GAIN MARGIN: Factor by which the gain is less than the neutral stability value. Gain margin measures How much can we increase the gain of the loop transfer function L(s) = D(s)G(s)H(s) and still have a stable system? Many Nyquist plots are like this one. Increasing loop gain magnifies the plot. GM GM =/(distance between origin and place where Nyquist map crosses real axis). PM If we increase gain, Nyquist map stretches and we may encircle. For a stable system, GM > (linear units) or GM > 0dB. PHASE MARGIN: Phase factor by which phase is greater than neutral stability value. Phase margin measures How much delay can we add to the loop transfer function and still have a stable system?

38 ECE450/ECE550, FREQUENCY-RESPONSE ANALYSIS 8 38 PM = Angle to rotate Nyquist plot to achieve neutral stability = intersection of Nyquist with circle of radius. If we increase open-loop delay, Nyquist map rotates and we may encircle. For a stable system, PM > 0. IRONY: This is usually easiest to check on Bode plot, even though derived on Nyquist plot! Define gain crossover as frequency where Bode magnitude is 0 db. Define phase crossover as frequency where Bode phase is 80. GM = /(Bode gain at phase-crossover frequency) if Bode gain is measured in linear units. GM = ( Bode gain at phase-crossover frequency) [db] if Bode gain measured in db. PM = Bode phase at gain-crossover ( 80 ). Magnitude Phase Frequency, (rads/sec.) Frequency, (rads/sec.) We can also determine stability as K changes. Instead of defining gain crossover where G( jω) =, usethefrequencywhere KG( jω) =.

39 ECE450/ECE550, FREQUENCY-RESPONSE ANALYSIS 8 39 You need to be careful using this test. It works if you apply it blindly and the system is minimum-phase. You need to think harder if the system is nonminimum-phase. Nyquist is the safest bet. PM and performance AbonusofcomputingPM from the open-loop frequency response graph is that it can help us predict closed-loop system performance. PM is related to damping. Consider open-loop 2nd-order system with unity feedback, G(s) = ω 2 n s(s + 2ζ ) T (s) = ω 2 n. s 2 + 2ζ + ωn 2 The relationship between PM and ζ is: (for this system) PM = tan 2ζ + 4ζ 4 2ζ 2 For PM 60, ζ PM 00,socanalsoinferM p from PM. Damping ratio ζ Damping ratio versus PM Phase margin M p, overshoot Overshoot fraction versus PM Phase margin

40 ECE450/ECE550, FREQUENCY-RESPONSE ANALYSIS : Preparing for control using frequency-response methods Bode s gain-phase relationship For any stable minimum-phase system (that is, one with no RHP zeros or poles), the phase of G( jω) is uniquely related to the magnitude of G( jω) Relationship: G( jω o ) = π M = ln G( jω) ( ) ω u = ln ω o ( dm du ) W (u) du dm du slope n of log-mag curve at ω = ω o W (u) = weighting function = ln(coth u /2) π 2 W (u) δ(u). Usingthisrelationship, G( jω) n 90 2 (in radians) if slope of Bode magnitudeplot is constant in the decadeneighborhood of ω So, if G( jω) 90 if n =. W (u) Normalized freq. u So, if G( jω) 80 if n = 2. KEY POINT: Want crossover G( jω) = at a slope of about for good PM. Wewillsoonseehowtodothis(design!).

41 ECE450/ECE550, FREQUENCY-RESPONSE ANALYSIS 8 4 Closed-loop frequency response Most of the notes in this section have used the open-loop frequency response to predict closed-loop behavior. How about closed-loop frequency response? T (s) = KD(s)G(s) + KD(s)G(s). General approximations are simple to make. If, KD( jω)g( jω) and KD( jω)g( jω) for ω ω c for ω ω c where ω c is the cutoff frequency where open-loop magnitude response crosses magnitude=. { T ( jω) = KD( jω)g( jω) + KD( jω)g( jω), ω ω c ; KD( jω)g( jω), ω ω c. Note: ωc ω bw 2ω c.

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