Salgado, Goodwin,
Graebe, Prentice Hall 2000
Chapter 9
Chapter 9
Frequency Domain Design Limitations
Goodwin,
Graebe, Salgado, Prentice Hall 2000
Chapter 9
The purpose of this chapter is to develop frequency domain constraints and to explore their interpretations. The results to be presented here have a long and rich history beginning with the seminal work of Bode published in his 1945 book on network synthesis. The results give an alternative view of the fundamental time domain limitations presented in Chapter 8.
Goodwin,
Graebe, Salgado, Prentice Hall 2000
Chapter 9
Some History
H. Bode, Network Analysis and Feedback Amplifier Design, Van Nostrand, 1945. (Work done 1938, 1939). (Based on
course at Bell Telephone Laboratories on Feedback Amplifier Design for Long Distance Telephone systems.) Typical results:
(1) For Stable Min. phase Transfer Functions gain can be
computed from phase and vice versa, e.g.
1 = 0 dA log coth u du du 2
(2) Log sensitivity trade-off (sometimes called Water bed effect)
log S dw = 0
Goodwin,
Graebe, Salgado, Prentice Hall 2000
Chapter 9
For historical interest we include (as the next few slides) the first few pages of the book written by Bode
Goodwin,
Graebe, Salgado, Prentice Hall 2000
Chapter 9
Goodwin,
Graebe, Salgado, Prentice Hall 2000
Chapter 9
Goodwin,
Graebe, Salgado, Prentice Hall 2000
Chapter 9
Goodwin,
Graebe, Salgado, Prentice Hall 2000
Chapter 9
Goodwin,
Graebe, Salgado, Prentice Hall 2000
Chapter 9
Goodwin,
Graebe, Salgado, Prentice Hall 2000
Chapter 9
Theme
In Advanced Control, understanding what can and cannot be done (and why) is often more important than producing a specific design.
Goodwin,
Graebe, Salgado, Prentice Hall 2000
Chapter 9
The constraints presented here mirror constraints that apply in many other areas, e.g.
x x
Second Law of Thermodynamics Cramer Rao Inequality of Estimation
Goodwin,
Graebe, Salgado, Prentice Hall 2000
Chapter 9
Design Constraints in Engineering
Examples: First and Second Laws of Thermodynamics
(a) It can rule out silly ideas:
For example, consider the following Perpetual Motion Machine?
Fan Motor Turbine Generator
(Ruled out by fundamental principle of conservation of energy)
Goodwin,
Graebe, Salgado, Prentice Hall 2000
Chapter 9
(b) They can also quickly identify fundamentally hard problems: For example, if faced with the following problem: Design a coal-fired generating plant with 80% efficiency This can be shown to be impractical using fundamental laws, e.g., using 80% efficiency and ideas of entropy we see that the required temperature is unrealistic, e.g.
T Ta > 80% T > 5 Ta T > 1227oC! T
Goodwin,
Graebe, Salgado, Prentice Hall 2000
Chapter 9
Examples from other fields
Estimation
x
Cramer Rao inequality
2 E ( )
{
}
log p ( y; ) 2 E
1
Communications
x
Probability of error can be made arbitrarily small provided
RC
S C = B log 2 1 + bits / sec N
Goodwin,
Graebe, Salgado, Prentice Hall 2000
Chapter 9
One of the best known results is that for a stable minimum phase system, the phase is uniquely determined by the magnitude and vice versa. The exact formula is given on the next slide.
Goodwin,
Graebe, Salgado, Prentice Hall 2000
Chapter 9
Weighting Function
d log | H ( j0eu ) | (0 ) = 1 log coth u 2 du
Thus, the slope of the magnitude curve in the vicinity of 0, say c, determines the phase (0):
c log coth u du = c 2 = c (0 ) 2 2 2
Goodwin,
Graebe, Salgado, Prentice Hall 2000
Chapter 9
Here we study the fundamental design constraints that apply to feedback systems of the type illustrated below
+ C G
The constraints we develop apply to frequency domain integrals of the sensitivity function and complementary sensitivity function.
Goodwin,
Graebe, Salgado, Prentice Hall 2000
Chapter 9
Conceptual Background
Before delving into technical details, we first review the conceptual nature of the results to be presented. The simplest (and perhaps best known) result is that, for an open loop stable plant, the integral of the logarithm of the closed loop sensitivity is zero; i.e.
0 ln | S0( jw) | dw = 0
Now we know that the logarithm function has the property that it is negative if |S0| < 1 and it is positive if |S0| > 1.
Goodwin,
Graebe, Salgado, Prentice Hall 2000
Chapter 9
Graphical interpretation of the area formula
Typical Nyquist plot for a stable rational transfer function, L, of relative degree two:
Note that S(j )-1 = 1 + L(j) is the vector from the -1 point to the point on the Nyquist plot corresponding to the frequency . It is clear from the plot that frequencies where |S(jw)| < 1 are balanced by frequencies where |S(jw)| > 1.
Goodwin,
Graebe, Salgado, Prentice Hall 2000
Chapter 9
The above result implies that set of frequencies over which sensitivity reduction occurs (i.e. where |S0| < 1) must be matched by a set of frequencies over which sensitivity magnification occurs (i.e. where |S0| > 1). This has been given a nice (cartoon like) interpretation as thinking of sensitivity as a pile of dirt. If we remove dirt from one set of frequencies, then it piles up at other frequencies. This is conceptually illustrated on the next slide.
Goodwin,
Graebe, Salgado, Prentice Hall 2000
Chapter 9
Physical Interpretation
log S
1
w
Sensitivity dirt
Goodwin,
Graebe, Salgado, Prentice Hall 2000
Chapter 9
We will show in the sequel that the above sensitivity trade-off is actually made more difficult if there are either (or both)
x x
Right half plane open loop zeros Right half plane open loop poles
Notice that these results hold irrespective of how the control system is designed; i.e. they are fundamental constraints that apply to all feedback solutions.
Goodwin,
Graebe, Salgado, Prentice Hall 2000
Chapter 9
Indeed, it will turn out that to avoid large frequency domain sensitivity peaks it is necessary to limit the range of sensitivity reduction to be: (i) less than any right half plane open loop zero (ii) greater than any right half plane open loop pole.
Goodwin,
Graebe, Salgado, Prentice Hall 2000
Chapter 9
This begs the question - What happens if there is a right half plane open loop zero having smaller magnitude than a right half plan open loop pole? Clearly the requirements specified on the previous slide are then mutually incompatible. The consequence is that large sensitivity peaks are unavoidable and, as a result, poor feedback performance is inevitable. An example precisely illustrating this conclusion will be presented later.
Goodwin,
Graebe, Salgado, Prentice Hall 2000
Chapter 9
We begin with Bodes integral constraint on sensitivity. This is a formal statement of the result discussed above at a conceptual level; namely
0 ln | S0( jw) | dw = 0
Goodwin,
Graebe, Salgado, Prentice Hall 2000
Chapter 9
Bodes Integral Constraints on Sensitivity
Consider a one d.o.f. stable control loop with open loop transfer function
Go (s)C(s) = es Hol (s) 0
where H0l(s) is a rational transfer function of relative degree nr > 0 and define
= lim sHol (s)
s
Assume that H0l(s) has no open loop poles in the open RHP. Then the nominal sensitivity function satisfies:
0
ln |So (j)|d =
0 2
for > 0 for = 0
Goodwin,
Graebe, Salgado, Prentice Hall 2000
Chapter 9
Graphically, the above statement can be appreciated in Figure 9.1 In Figure 9.1, the area A1 ( |S0(jw)| < 1) must be equal to the area A2 ( |S0(jw)| > 1), to ensure that the integral in equation (9.2.3) is equal to zero.
Sensitivity frequency response
4 2
ln |So(j)|
A
0 2 4 6
2
A1
Figure 9.1: Graphical interpretation of the Bode integral
4 6 8 10 12
0
2
Frequency [rad/s]
Goodwin,
Graebe, Salgado, Prentice Hall 2000
Chapter 9
Proof
We will not give a formal proof of this result. Suffice to say it is an elementary consequence of the well known Cauchy integral theorem of complex variable theory - see the book for details.
Goodwin,
Graebe, Salgado, Prentice Hall 2000
Chapter 9
The extension to open loop unstable systems is as follows: Consider a feedback control look with open loop transfer function as in Lemma 9.1, and having unstable poles located at p1, , pN, pure time delay , and relative degree nr 1. Then, the nominal sensitivity satisfies:
N 0 0
ln |So (j)| d =
i=1
R {pi }
N
for
nr > 1 for nr = 1
where = lims sH0l(s)
ln |So (j)| d = + R {pi } 2 i=1
Goodwin,
Graebe, Salgado, Prentice Hall 2000
Chapter 9
Observations
We see from the above results that with open loop RHP poles, the integral of log sensitivity is required to be greater than zero (previously) it had to be zero. This makes sensitivity minimization more difficult.
Goodwin,
Graebe, Salgado, Prentice Hall 2000
Chapter 9
Design Interpretations
Several factors such as undermodeling, sensor noise, plant bandwidth, etc., lead to the need of setting a limit on the bandwidth of the closed loop. Typically where < 1/2 and k > 0. Corollary: Suppose that L is a rational function of relative degree two or more and satisfying the bandwidth restriction. Then
3c . log S ( j ) d c 2k
L( j ) c
1+ k
,
c ,
Goodwin,
Graebe, Salgado, Prentice Hall 2000
Chapter 9
The above result shows that the area of the tail of the Bode sensitivity integral over the infinite frequency range [c, ) is limited.
Goodwin,
Graebe, Salgado, Prentice Hall 2000
Chapter 9
Implication: Peak in the sensitivity frequency response before c. Suppose S satisfies the reduction spec.
S ( j ) < 1, 1 < c ,
1+ k
and translating
L ( j ) c
,
c
on the shape of |S(jw)| yields
Goodwin,
Graebe, Salgado, Prentice Hall 2000
Chapter 9
Now, using the bounds (2) and (1) in the Bode sensitivity integral, it is easy to show that
1 1 3c . sup log S ( j ) p + 1 log c 1 2k (1, c ) pzs
Then, the larger the area of sensitivity reduction (i.e., small and/or 1 close to c ) will necessarily result in a large peak in the range(1, c). Hence, the Bode sensitivity integral imposes a design trade-off when natural bandwidth constraints are assumed for the closed-loop system.
Goodwin,
Graebe, Salgado, Prentice Hall 2000
Chapter 9
Example
The inequality
1 1 3c . sup log S ( j ) p + 1 log c 1 2k (1, c ) pzs
can be used to derive a lower bound on the closed-loop bandwidth in terms of the sum of open-loop unstable poles. Let
1 = k1 c .
Goodwin,
Graebe, Salgado, Prentice Hall 2000
Chapter 9
Imposing the condition that the RHS above be yields the following lower bound on the bandwidth, which we take as b B( S m ) p,
pz s
where
B(S m ) =
. (1 k1 )(S m + 1.5 / k + k1 log )
Goodwin,
Graebe, Salgado, Prentice Hall 2000
Chapter 9
b B( S m )
pZ s
p
The factor B(Sm) is plotted in the figure below as a function of the desired peak sensitivity Sm, for = 0.45, k = 1, k1 = 0.7 and = 0.5.
Goodwin,
Graebe, Salgado, Prentice Hall 2000
Chapter 9
For example, an open-loop unstable system having relative degree two requires a bandwidth of at least 6.5 times the sum of its ORHP poles if it is desired that S be smaller than 1/2 over 70% of the closed-loop bandwidth while keeping the lower bound on the peak sensitivity smaller than Sm = 2 .
Goodwin,
Graebe, Salgado, Prentice Hall 2000
Chapter 9
We next turn to dual results that hold for complementary sensitivity reduction.
Goodwin,
Graebe, Salgado, Prentice Hall 2000
Chapter 9
Integral Constraints on Complementary Sensitivity
Consider a one d.o.f. stable control loop with open loop transfer function
Go (s)C(s) = es Hol (s) 0
where H0l(s) is a rational transfer function of relative degree nr > 1 satisfying
Hol (0)1 = 0
Furthermore, assume that H0l(s) has no open loop zeros in the open RHP.
Goodwin,
Graebe, Salgado, Prentice Hall 2000
Chapter 9
Then the nominal complementary sensitivity function satisfies:
0
1 ln |To (j)|d = 2
2 k
v
where kv is the velocity constant of the open loop transfer function satisfying:
1 = lim dT (s) kv s 0 ds 1 = lim s 0 sH 0l ( s)
Goodwin,
Graebe, Salgado, Prentice Hall 2000
Chapter 9
As for sensitivity reduction, the trade-off described above becomes harder in the presence of open loop RHP singularities. For the case of complementary sensitivity reduction, it is the open loop RHP zeros that influence the result. The formal result is stated on the next slide.
Goodwin,
Graebe, Salgado, Prentice Hall 2000
Chapter 9
Assume that H0l(s) has open loop zeros in the open RHP, located at c1, c2, , cM, then
1 0 w2 ln | T0( jw) | dw =
+ 1 ci 2kv
M i =1
Goodwin,
Graebe, Salgado, Prentice Hall 2000
Chapter 9
We next turn to a set of frequency domain integrals which are closely related to the Bode type integrals presented above but which allow us to consider RHP open loop poles and zeros simultaneously. These integrals are usually called Poisson type integral constraints.
Goodwin,
Graebe, Salgado, Prentice Hall 2000
Chapter 9
Poisson Integral Constraint on Sensitivity
A trick we will use here is to express a complex function f(s) as the product of functions which are non-minimum phase and analytic in the RHP, times the following Blaschke products (or times the inverses of these products).
M
Bz (s) =
k=1
s ck s + c k
Bp (s) =
s pi s + p i i=1
N
We use the above idea to prove the results below.
Goodwin,
Graebe, presented Salgado, Prentice Hall 2000
Chapter 9
Poisson Integral for S0(jw)
Consider a feedback control loop with open loop RHP zeros located at c1, c2, , cM, where ck = k + jk and open loop unstable poles located at p1, p2, .., pN. Then the nominal sensitivity satisfies
k ln |So (j)| 2 d = ln |Bp (ck )| k + (k )2
for k = 1, 2, . . . M
To illustrate this formula, consider the requirement that the sensitivity be reduced to below for all frequencies up to wl. (See the next slide).
Goodwin,
Graebe, Salgado, Prentice Hall 2000
Chapter 9
Figure 9.2: Design specification for |S0(jw)|
Sensitivity frequency response 2
1.5
Magnitude
1
|So(j)|
0.5
0 1 10
l
10
0
10 Frequency [rad/s]
1
10
2
Notional closed loop bandwidth
Goodwin,
Graebe, Salgado, Prentice Hall 2000
Chapter 9
In the sequel we will need to use the integral of the term W (ck , ) = 2 k 2
k + (k )
W (ck , )d =
and we define
2
2
and
1
W (ck , )d = (ck , 2 ) (ck , 1 )
c k c k =
(ck , c ) = 2 arctan
c
(*)
(ck , ) = 2 lim arctan
Goodwin,
Graebe, Salgado, Prentice Hall 2000
Chapter 9
Discussion
(i) Consider the plot of sensitivity versus frequency shown on the previous slide. Say we were to require the closed loop bandwidth to be greater than the magnitude of a right half plane (real) zero. In terms of the notation used in the figure, this would imply wl > k. We can then show using the Poisson formula that there is necessarily a very large sensitivity peak occurring beyond wl. To estimate this peak, assume that wl = 2k and take O in Figure 6.2 as 0.3. Then, without considering the effect of any possible open loop unstable pole, the sensitivity peak will be bounded below as follows:
Goodwin,
Graebe, Salgado, Prentice Hall 2000
Chapter 9
ln Smax
1 [|(ln 0.3)(ck , 2ck )|] > (ck , 2ck )
Then, using equation (*) we have that (ck, 2ck) = 2 arctan(2) = 2.21, leading to Smax > 17.7. That is we have a VERY large sensitivity peak. Note that this in turn implies that the complementary sensitivity peak will be bounded below by Smax - 1 = 16.7.
Goodwin,
Graebe, Salgado, Prentice Hall 2000
Chapter 9
(ii) The observation in (i) is consistent with the analysis carried out in Chapter 8. In both cases the conclusion is that the closed loop bandwidth should not exceed the magnitude of the smallest RHP open loop zero. The penalty for not following this guideline is that a very large sensitivity peak will occur, leading to fragile loops (non robust) and large undershoots and overshoots.
Goodwin,
Graebe, Salgado, Prentice Hall 2000
Chapter 9
(iii) In the presence of unstable open loop poles, the problem is compounded through the presence of the factor |ln|Bp(ck)||. This factor grows without bound when one RHP zero approaches an unstable open loop pole.
Goodwin,
Graebe, Salgado, Prentice Hall 2000
Chapter 9
Constraints on both |S0| and |T0|
Actually constraints can be imposed on both |S0| and |T0|. For example, say that we require |S0(jw)| < |T0(jw)| < for for w < wl w > wh
This is illustrated on the next slide.
Goodwin,
Graebe, Salgado, Prentice Hall 2000
Chapter 9
Figure 9.3: Design specifications for |S0(jw)| and |T0(jw)|
Sensitivities frequency responses 2
1.5
Magnitudes
|To(j)|
1
|S (j)|
o
0.5
0 1 10
l
10
0
Frequency [rad/s]
101 h
10
2
Goodwin,
Graebe, Salgado, Prentice Hall 2000
Chapter 9
We can then use the Poisson sensitivity integral to bound the peak sensitivity. The result is:
ln Smax > 1 [| ln |Bp (ck )|| + |(ln )(ck , l )| (ck , h ) (ck , l ) ( (ck , h )) ln(1 + )]
Goodwin,
Graebe, Salgado, Prentice Hall 2000
Chapter 9
We next turn to the dual Poisson integral constraints that hold for the complementary sensitivity function. The formal result is stated on the next slide.
Goodwin,
Graebe, Salgado, Prentice Hall 2000
Chapter 9
Poisson Integral Constraint on Complementary Sensitivity
Poisson integral for T0(jw). Consider a feedback control loop with delay 0, and having open loop unstable poles located at p1, p2, , pN, where pi = i + ji and open loop zeros in the open RHP, located at c1, c2, , cM. Then,
ln |To (j)|
i d = ln |Bz (pi )| + i 2 i + (i )2
for i = 1, 2, . . . N
Goodwin,
Graebe, Salgado, Prentice Hall 2000
Chapter 9
Say that we require that |T0| < for w > wh. Then it follows from the above result that the peak value of the complementary sensitivity will be bounded from below as follows:
ln Tmax 1 [| ln |Bz (i )|| + i + | ln |( (i , h )] > (i , h )
Goodwin,
Graebe, Salgado, Prentice Hall 2000
Chapter 9
Discussion
(i) We see that the lower bound on the complementary sensitivity peak is larger for systems with pure delays, and the influence of a delay increases for unstable poles which are far away from the imaginary axis, i.e. large i. (ii) The peak, Tmax, grows unbounded when a RHP zero approaches an unstable pole, since then |ln|Bz(pi)|| grows unbounded.
Goodwin,
Graebe, Salgado, Prentice Hall 2000
Chapter 9
(iii) Say that we ask that the closed loop bandwidth be much smaller than the magnitude of a right half plane (real) pole. In terms of the notation used above, we would then have wh << i. Under these conditions, (pi, wh) will be very small, leading to a very large complementary sensitivity peak. This is an unacceptable result. Thus we conclude that the closed loop bandwidth should be greater than any RHP open loop poles.
Goodwin,
Graebe, Salgado, Prentice Hall 2000
Chapter 9
We note that this is consistent with the results based on time domain analysis presented in Chapter 8. There it was shown that, when the closed loop bandwidth is not greater than an open loop RHP pole, large time domain overshoot will occur.
Goodwin,
Graebe, Salgado, Prentice Hall 2000
Chapter 9
Example of Design Trade-offs
We will illustrate the application of the above ideas by considering the inverted pendulum problem. Such systems exist in many universities and are used to illustrate control principles. The key idea is to balance a rod on top of a moving cart (similar to balancing a broom on ones hand). A photograph of a real inverted pendulum system (at the University of Newcastle) is shown on the next slide.
Goodwin,
Graebe, Salgado, Prentice Hall 2000
Chapter 9
Example of an Inverted Pendulum
Goodwin,
Graebe, Salgado, Prentice Hall 2000
Chapter 9
We will consider the following control problem: Sensors: We measure the position of the cart (but NOT the angle of the pendulum) Actuators: We can apply forces to the cart. Goal: We want to position the cart at some location plus have the pendulum balancing vertically.
Goodwin,
Graebe, Salgado, Prentice Hall 2000
Chapter 9
Inverted pendulum without angle measurement
A schematic diagram of the system is shown below:
(t)
m l f (t)
M
y(t)
Figure 9.4: Inverted pendulum
Goodwin,
Graebe, Salgado, Prentice Hall 2000
Chapter 9
The model for this system was discussed in Chapter 3. A linearized model for the system has the following transfer function linking the cart position (Y) to the force applied to the cart (F).
(s 10)(s + 10) Y (s) =2 2 (s F (s) s 20)(s + 20)
Goodwin,
Graebe, Salgado, Prentice Hall 2000
Chapter 9
We note that this system has
x x
an open loop RHP zero at 10 an open loop RHP pole at 20
We observe that the RHP pole has greater magnitude than the RHP zero. The reader is reminded of the comments made in the following two slides which appeared in the earlier part of this chapter.
Goodwin,
Graebe, Salgado, Prentice Hall 2000
Chapter 9
Indeed, it will turn out that to avoid large frequency domain sensitivity peaks it is necessary to limit the range of sensitivity reduction to be: (i) less than any right half plane open loop zero (ii) greater than any right half plane open loop pole.
Goodwin,
Graebe, Salgado, Prentice Hall 2000
Chapter 9
This begs the question - What happens if there is a right half plane open loop zero having smaller magnitude than a right half...