Frequency Design

Control Theory

Frequency domain design

Germn Andrs Ramos, PhD.


Design based on the system frequency response (bode plot, nyquist plot) System robustness can be defined using the gain and phase margins Nyquist stability criterion can be applied also to establish gain and phase margins System error constants can be read directly from the low frequency asimptote in the magnitude plot System speed response is proportional to the crossover frequency


In continuous time The bode plots are easily plotted and interpreted (by hand) Corrections to the frequency response can be easily made using the controller poles and zeros Gs(j!) is a rational function of j! then plot bode plots is simple In discrete time Computer tools Gz(ej!) is an irational function of !

Frequency domain design Principio del argumento

F(s)

C1 closed curve in the s-plane

F(s) rational function of s (no pole nor zero on C1) Z, P numbers of zeros and poles of F(s) encircled by C1 C2 maping of C1 by F(s) into the F-plane

C2 will encircle the origin of the F-plane (Z-P) times in the same direction as C1

From [1]


Nyquist stability criterion

(s) =1+Gl (s)Closed-loop characteristic equation:

Loop transfer function: Gl (s)

Go(s) =Gl (s)1+Gl (s)

Closed-loop transfer function:

Finn Haugen, TechTeach: Discrete-time signals and systems 61

ymSP ymL(s)

ymSP uHc(z) Hu(z) Hs(z)

ymy

ymSP uHc(z) Hp(z)

ym

Hp(z)

L(z)

Making compact

Making compact

Figure 31: Converting an extracted part of the detailed block diagram in Figure 30into a compact block diagram. L(z) is the loop transfer function.

where nL(z) and dL(z) are the numerator and denominator polynomials ofL(z), respectively. The stability of the feedback system is determined by thestability of T (z).

11.4.2 Pole placement based stability analysis

The characteristic polynomial of the tracking transfer function (290) is

c(z) = dL(z) + nL(z) (291)

Hence, the stability of the control system is determined by the placement ofthe roots of (291) in the complex plane.

Example 11.2 Pole based stability analysis of feedback system

Assume given a control system where a P controller controls an integratingprocess. The controller transfer function is

Hc(z) = Kp (292)

Gl (z)Gl (s)

G(s)C(s)R(s) Y (s)

Gl (s)


Nyquist plot

Gl (s)

From [5]


Nyquist plot

Gl (s)

Gl (s) =s+1

s2 (s 2) F(s) =1+Gl (s)

From [1]


Nyquist plot polar plot Gl (s)

3.5 Frequency Response

1

Ultimate point

a G(i) Re

G(i) Im

Figure 3.8 The Nyquist plot of a transfer function G(i ).

AB

1

L(s)

Figure 3.9 Block diagram of a simple feedback system.

numerically it is not easy to determine how the roots are influenced bythe properties of the controller. It is for example not easy to see how tomodify the controller if the closed loop system is stable. We have also de-fined stability as a binary property, a system is either stable or unstable.In practice it is useful to be able to talk about degrees of stability. Allof these issues are addressed by Nyquists stability criterion. This resulthas a strong intuitive component which we will discuss first. There is alsosome beautiful mathematics associated with it that will be discussed in aseparate section.

Consider the feedback system in Figure 3.9. Let the transfer functionsof the process and the controller be P(s) and C(s) respectively. Introducethe loop transfer function

L(s) = P(s)C(s) (3.24)

To get insight into the problem of stability we will start by investigating

93

Polar plot From [6]


Nyquist plot


system outside the unit circle, and let arg [1 + L(z)] be the angularchange of the vector [1 + L(z)] as z have followed the contour once inpositive direction of circulation. Then, the number poles PCL of theclosed loop system outside the unit circle, is

PCL =arg [1 + L(z)]

360+ POL (306)

If PCL = 0, the closed loop system is asymptotically stable.

Let us take a closer look at the terms on the right side of (306): POL are thenumber of the roots of dL(z), and there should not be any problem calculatingthat number. What about determining the angular change of the vector1 + L(z)? Figure 34 shows how the vector (or complex number) 1 + L(z)appears in a Nyquist diagram for a typical plot of L(z). A Nyquist diagram issimply a Cartesian diagram of the complex plane in which L is plotted.1 + L(z) is the vector from the point (1, 0j), which is denoted the criticalpoint, to the Nyquist curve of L(z).

Re L(z)

Im L(z)

1

1 + L(z)

The critical point

Decreasing ZPositive Z

Negative Z

0

Nyquist curve of L(z)

Curve Bis mappedto origo

Curve A1is mappedto here


Figure 34: Typical Nyquist curve of L(z). The vector 1 + L(z) is drawn.

More about the Nyquist curve of L(z) Let us take a more detailed lookat the Nyquist curve of L as z follows the contour in the z-plane, see Figure33. In practice, the denominator polynomial of L(z) has higher order than thenumerator polynomial. This implies that L(z) is mapped to the origin of the

Gl (s)

Gl (s)

Gl (s)

Gl (s)

Distance of the Nyquist curve to the critical point (-1,0):

1+Gl (s)From [7]


Nyquist stability criterion Theorem:

Go(s) is stable if and only if the Nyquist plot of Gl(s) does not pass through critical point (-1,0) and the n u m b e r o f c o u n t e r c l o c k w i s e encirclements of (-1,0) equals the number of open right-half-plane poles of Gl(s)



PCL =arg [1 + L(z)]

360+ POL (306)



Re L(z)

Im L(z)

1

1 + L(z)

The critical point


Negative Z

0







Gl (s)

Gl (s)

Gl (s)

Gl (s)

From [7]


Nyquist stability criterion Corollary:

If Gl(s) has no open right-half-plane poles, then Go(s) is stable if and only if the Nyquist plot of Gl(s) does not encircle nor pass through critical point (-1,0)



PCL =arg [1 + L(z)]

360+ POL (306)



Re L(z)

Im L(z)

1

1 + L(z)

The critical point


Negative Z

0







Gl (s)

Gl (s)

Gl (s)

Gl (s)

From [7]

Frequency domain design Design criteria Gain and phase margins:

From [1]


From [6]


From [1]

Frequency domain design Gain and phase margins (OL) transient response (CL)

Mp = e

1 2 Mr =1

2 1 2 < 0.707

1 0.707

#

$%

&%

Go(s) = 2

s2 + 2s+ 2For a second order continuous time system:

overshoot

Resonant peak From [5]

Go( j)

c

cutoff frequency

bandwidth

Frequency domain design Gain and phase margins (OL) transient response (CL)

Mp = e

1 2 Mr =1

2 1 2 < 0.707

1 0.707

#

$%

&%

Go(s) = 2

s2 + 2s+ 2For a second order continuous time system:

overshoot

Resonant peak From [5] From [1]


Gain and phase margins (OL) transient response (CL)

Go(s) = 2

s2 + 2s+ 2

For a second order continuous time system:

Gain margin > 10 dB, Phase margin > 45 Mr 1.3 Gain margin > 12 dB, Phase margin > 60 Mr 1.0

Frequency domain design Gain and phase margins (OL) transient response (CL) For the second order continuous time system: Phase margin (G(s)) and Mp (Go(s)) are directly related: Which is usually approximated by:

m = tan12

4 4 +1 2 2( )12

"

#

$$$

%

&

'''

G(s) = 2

s2 + 2s

m =100

Go(s) =G(s)1+G(s)closed-loop

Frequency domain design Gain and phase margins (OL) transient response (CL) Discrete proportional feedback of Open loop phase margin and closed-loop damping ratio as the feedback gain was varied

m =100

G(s) = 1s2 + s

From [6]

Frequency domain design Gain and phase margins (OL) transient response (CL) The resonant peak (CL) increases as the frequency response (OL) appoaches the point -1

Larger resonance small stability margins time domain frequency domain CL CL

Frequency domain design Gain and phase margins (OL) transient response (CL) Sensitivity to parameter variation: is in general reduced increasing the open loop gain Disturbance rejection can be improved using hign loop gain, provided that this gain is not in the direct path between the disturbance input and system output Control effort: constraints on the control action need to be considered. High loop gain can cause system saturation


Low frequency gains (OL) steady state error (CL) Position error constant: Velocity error constant:

Kp = lims0 C(s)G(s)

Kv = lims0 sC(z)G(z)


Given the closed loop transfer function: Position error constant:

- For a step reference: Obtaining

E(s) = 11+G(s)C(s) R(s)

R(s) = 1sE(z) = 11+G(s)C(s)

1s

ess = lims0 sE(s) = lims0 s1

1+G(s)C(s)1s = lims0

11+G(s)C(s)

ep =1

1+Kpwhere Kp = lims0 C(s)G(s)


Given the closed loop transfer function: Position error constant:

- For a step reference: Obtaining

E(s) = 11+G(s)C(s) R(s)

R(s) = 1s2

E(z) = 11+G(s)C(s)1s2

ess = lims0 sE(s) = lims0 s1

1+G(s)C(s)1s2 = lims0

1s+ sG(s)C(s)

where ev =1Kv

Kv = lims0 sC(z)G(z)


Syteady state error vs system type

System Type \ Error

constant

Kp Kv Ka

0 K 0 0 I K 0 II K

System Type \ Error

ep ev ea

0 k I 0 k II 0 0 k


Syteady state error in bode plot

From [1]


Specifications of control systems


Lag phase network

C(s) = 1+ aT1s1+T1s

0 < a


Design procedure: Compute the error constant from steady state error specification Plot bode plot gain and phase margins Determine the frequency at which the bode plot has the required phase margin plus 6. This is the new gain crossover frequency !g' Measure the attenuation needed to bring the gain plot down to !g' Compute T1 from:

= 20 log(a)

1aT1

=g!

10


Lead phase network

C2 (s) =1+ bT2s1+T2s

b >1

b = 1+ sinm1 sinmFrom [1]

Frequency domain design Design procedure: Compute the error constant from steady state error specification Plot bode plot gain margin, phase margin 1, gain crossover frequency !g and phase crossover frequency !p Calculate Compute Draw a horizontal line with gain The intersection with the bode plot will be the new gain crossover frequency !g' Measure the resulting phase margin

= required 1 +

b = 1+ sin1 sin10 log(b)


Ejemplo: Velocity error ep 60 Gain margin > 12dB

G(s) = 1s(s+ 2)


Sensitivity function:


ymSP ymL(s)

ymSP uHc(z) Hu(z) Hs(z)

ymy

ymSP uHc(z) Hp(z)

ym

Hp(z)

L(z)

Making compact

Making compact

Figure 31: Converting an extracted part of the detailed block diagram in Figure 30into a compact block diagram. L(z) is the loop transfer function.

where nL(z) and dL(z) are the numerator and denominator polynomials ofL(z), respectively. The stability of the feedback system is determined by thestability of T (z).

11.4.2 Pole placement based stability analysis

The characteristic polynomial of the tracking transfer function (290) is

c(z) = dL(z) + nL(z) (291)

Hence, the stability of the control system is determined by the placement ofthe roots of (291) in the complex plane.

Example 11.2 Pole based stability analysis of feedback system

Assume given a control system where a P controller controls an integratingprocess. The controller transfer function is

Hc(z) = Kp (292)

Gl (z)Gl (s)

G(s)C(s)R(s) Y (s)E(s)

E(s) = 11+C(s)G(s) R(s) =1

1+Gl (s)R(s)

S(s) = 11+Gl (s)

The error is given by

The sensitivity function is defined as

5.4 Disturbance Attenuation

102 101 100 101102

101

100

101

!S(i

)!

Figure 5.9 Gain curve of the sensitivity function for PI control (k = 0.8, ki = 0.4)of process with the transfer function P(s) = (s + 1)4. The sensitivity crossoverfrequency is indicated by + and the maximum sensitivity by o.

0 10 20 30 40 50 60 70 80 90 10030

20

10

0

10

20

30

t

y

Figure 5.10 Outputs of process with control (full line) and without control (dashedline).

is an important variable which gives the largest amplification of the dis-turbances. The maximum occurs at the frequency ms.

A quick overview of how disturbances are influenced by feedback isobtained from the gain curve of the Bode plot of the sensitivity function.An example is given in Figure 5.9. The figure shows that the sensitivitycrossover frequency is 0.32 and that the maximum sensitivity 2.1 occurs atms = 0.56. Feedback will thus reduce disturbances with frequencies lessthan 0.32 rad/s, but it will amplify disturbances with higher frequencies.The largest amplification is 2.1.

If a record of the disturbance is available and a controller has beendesigned the output obtained under closed loop with the same disturbancecan be visualized by sending the recorded output through a filter with thetransfer function S(s). Figure 5.10 shows the output of the system withand without control.

189


Sensitivity function:

S( j) = 11+Gl ( j)

In the frequency domain

S = S(s) =max S( j)

The following can be defined

S

sc ms From [2]

Frequency domain design Sensitivity function:

For a larger S the Nyquist plot comes closer to the point of instability


1+Gl (s) =1S(s)

Minimum distance:

1S(s)

Chapter 5. Feedback Fundamentals

1/Ms ms

sc

1

Figure 5.11 Nyquist curve of loop transfer function showing graphical interpre-tation of maximum sensitivity. The sensitivity crossover frequency sc and the fre-quency ms where the sensitivity has its largest value are indicated in the figure.All points inside the dashed circle have sensitivities greater than 1.

The sensitivity function can be written as

S(s) = 11+ P(s)C(s) =

11+ L(s) . (5.8)

Since it only depends on the loop transfer function it can be visualizedgraphically in the Nyquist plot of the loop transfer function. This is illus-trated in Figure 5.11. The complex number 1+ L(i ) can be representedas the vector from the point 1 to the point L(i ) on the Nyquist curve.The sensitivity is thus less than one for all points outside a circle withradius 1 and center at 1. Disturbances of these frequencies are attenu-ated by the feedback. If a control system has been designed based on agiven model it is straight forward to estimated the potential disturbancereduction simply by recording a typical output and filtering it through thesensitivity function.

Slow Load DisturbancesLoad disturbances typically have low frequencies. To estimate their effectson the process variable it is then natural to approximate the transferfunction from load disturbances to process output for small s, i.e.

Gxd(s) = P(s)1+ P(s)C(s) " c0 + c1s+ c2s2 + (5.9)

190

S

1+Gl (s)

All points inside the dashed circle have sensitivities greater than 1

From [2]

ImGl (s)

ReGl (s)

Frequency domain design Sensitivity function:

Vector gain margin (VGM): VGM = SS 1

For a larger Sthe Nyquist plot comes closer to the point of instability

1+Gl (s)


1+Gl (s) =1S(s)

Minimum distance:

1S(s)

From [6]

Chapter 5. Feedback Fundamentals

1

1+ L

CP

Figure 5.12 Nyquist curve of a nominal loop transfer function and its uncertaintycaused by process variations P.

A technical condition, namely that the perturbation P is a stable trans-fer function, must also be required. If this does not hold the encirclementcondition required by Nyquists stability condition is not satisfied. Alsonotice that the condition (5.10) is conservative because it follows fromFigure 5.12 that the critical perturbation is in the direction towards thecritical point 1. Larger perturbations can be permitted in the other di-rections.

This formula (5.10) is one of the reasons why feedback systems workso well in practice. The mathematical models used to design control sys-tem are often strongly simplified. There may be model errors and theproperties of a process may change during operation. Equation (5.10) im-plies that the closed loop system will at least be stable for substantialvariations in the process dynamics.

It follows from (5.10) that the variations can be large for those fre-quencies where T is small and that smaller variations are allowed forfrequencies where T is large. A conservative estimate of permissible pro-cess variations that will not cause instability is given by

"P(i )""P(i )"

Frequency Design

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