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Model Uncertainty and Robust Control
K. J. Åström
Department of Automatic Control
Lund University, Lund, Sweden
Email: kja@control.lth.se, Fax: +46 46 13 81 18
1 Introduction
A key reason for using feedback is to reduce the effects of uncertainty which may
appear in different forms as disturbances or as other imperfections in the models used
to design the feedback law. Model uncertainty and robustness have been a central theme
in the development of the ﬁeld of automatic control. This paper gives an elementary
presentation of the key results.
A central problem in the early development of automatic control was to construct
feedback ampliﬁers whose properties remain constant in spite of variations in supply
voltage and componentvariations. This problem was the key for the telephoneindustry
that emerged in the 1920s. The problem was solved by [9]. We quote from his paper:
“ .. by building an ampliﬁer whose gain is deliberately made say 40 deci
bels higher than necessary (10 000 fold excess on energy basis) and then
feeding the output back on the input in such a way as to throw away excess
gain, it has been found possible to effect extraordinarily improvement in
constancy of ampliﬁcation and freedom from nonlinearity.”
Blacks inventionhad a tremendous impact and it inspired much theoretical work. This
was required both for understanding and for development of design method. A novel
approach to stability was developed in [36], fundamental limitations were explored
by [10] who also developed methods for designing feedback ampliﬁers, see [11]. A
systematic approach to design controllers that were robust to gain variations were also
developed by Bode.
The work on feedback ampliﬁers became a central part of the theory of servomech
anisms that appeared in the 1940s, see [22], [27]. Systems were then described using
63
64 Model Uncertainty and Robust Control
transfer functions or frequency responses. It was very natural to capture uncertainty
in terms of deviations of the frequency responses. A number of measures such as am
plitude and phase margins and maximum sensitivities were also introduced to describe
robustness. Design tools such as the Bode diagram introduced to design feedback am
pliﬁers also found good use in design of servomechanisms. Bode’s work on robust
design was generalized to deal with arbitrary variations in the process by Horowitz
[24]. The design techniques used were largely graphical.
The statespace theory that appeared in the 1960s represented signiﬁcant paradigm
shift. Systems were now described using differentialequations. There was a very vigor
ous development that gave new insight, new concepts [32], [30] and new design meth
ods. Control design problems were formulated as optimization problems which gave
effective design methods, see [7], [8], and [40]. Control of linear systems with Gaus
sian disturbances and quadratic criteria, the LQG problem, was particularly attractive
because it admitted analytical solutions, see [8], [29], [28], and [31]. The design com
putations were also improved because it was possible to capitalize on advances in nu
merical linear algebra and efﬁcient software. The controller obtained from LQG theory
also had a very interesting structure. It was a composition of a Kalman ﬁlter and a state
feedback.
The statespace theory became the predominant approach, see [5]. Safonov and
Athans [42] showed that the phase margin is at least and the amplitude margin is
inﬁnite for an LQG problem where all state variables are measured. This result does
unfortunately not hold for output feedback as was demonstrated in [15]. There were
attempts to recover the robustness of state feedback using special design techniques
called loop transfer recovery. The central issue is howeverthat it is not straightforward
to capture model uncertainty in a state variable setting. There was also criticism of the
statespace theory, see [25].
The paper [49] represented a paradigm shift which brought robustness to the fore
front. It started a new development that led to the so called theory. The idea was
to develop systematic design methods that were guaranteed to give stable closed loop
systems for systems with model uncertainty. The original work was based on frequency
responses and interpolationtheory which led to compensators of high order. The sem
inal paper [17] showed how the problem could be solved using statespace methods.
Game theory is another approach to theory. The game is to ﬁnd a controller in the
presence of an adversary that changes the process, see [6]. The theory is now well
described in books, see [16],[21] and [52].
Major advances in robust design was made in the book [35] where the control
problem was regarded as a loop shaping problem. This gave effective design methods
and it also reestablished the links with classical control. This line of research has been
continued by [48] who has obtained deﬁnite results relating modeling errors and robust
control. To do this he also hadto invent a novel metricfor systems, see [46]. This work
brings even closer to the classical results.
In this chapter we will try to present the essence of the development in the simple
setting of singleinput singleoutput systems. We start with a presentation of some as
pects of classical control theory in Section 2. Robustness issues for statespace theory
Model Uncertainty and Robust Control 65
−
+
R2
R1
V2
V1
V
Fig. 1: Schematic diagram of a feedback ampliﬁer.
is reviewed brieﬂy in Section 3 where we also present an example that illustrates that a
blind use of statespace theory can lead to closed loop systems with very poor robust
ness properties. This can be overcome by analyzing the robustness and modifying the
design criteria. In Section 4 we discuss fundamental limitations on performance due
to time delays and right half plane poles and zeros. This does not appear naturally in
statespace theory where the major requirements are observability and controllability.
In Section 5 we present some key results from loop shaping. To do this we have
to discuss the important problem of determining if two systems are close from the point
of view of feedback. We also have to introduce better stability concepts and ways to
characterize model uncertainty. We end the section with a discussion of Vinnicombe’s
theory which gives much insight, necessary conditions and very nice ties to classical
control theory.
2 Classical Control Theory
The ﬁelds of automatic control emerged in the mid 1940s when it was realized that it
was a common framework for problems associated with feedback control from a wide
range of ﬁelds such as telecommunication, industrial processes, vehicles, power sys
tems etc. An essential component came from telecommunicationswhere a key problem
had been to design accurate reliable ampliﬁers from components with variable proper
ties.
The Feedback Ampliﬁer
A schematic diagram of the ampliﬁer is shown in Figure 1. Let the raw gain of the
ampliﬁer be . The feedback ampliﬁer has very remarkable properties as can be seen
from its inputoutput relation. It follows from Figure 1 that
(1)
66 Model Uncertainty and Robust Control
e

Fig. 2: Block diagram of a feedback system.
Notice that the gain is essentially given by the ratio . If the raw ampli
ﬁer gain is large the gain is virtually independent of . Assume for example that
and that .A change of the variation gives only a gain
variation of . Feedback thus has the amazing property of reducing the effects of
uncertainty drastically. The linearity is also increased signiﬁcantly.
The risk for instability is a drawback of feedback. Nyquist developed a theory
for analyzing stability of feedback ampliﬁers, see [36]. Systematic methods to design
feedback systems were developed in [10] and further elaborated in [11]. These ideas
formed one of the foundations of automatic control.
In today’s terminology,we could say that Black used feedback to design an ampli
ﬁer that was robust to variations in the gain of the process. As a sideeffect, he also
obtained a closedloop system that was extremely linear.
Generalization
The ideas of feedback are applicable to a wide range of systems. This is illustrated in
Figure 2 which shows a basic feedback loop consisting of a process and a controller.
The purpose of the system is to make the process variable follow the set point in
spite the disturbances and that act on the system. The properties of the closed loop
system should also be insensitive to variations in the process. There are two types of
disturbances. The load disturbance drives the system away from its desired state and
the measurement noise which corrupts the information about the system obtained
from the sensors.
The system in Figure 2 has three inputs , and , and four interesting signals ,
,and . There are thus 12 relations that are of potential interest. Assume that the
process and the controller are linear timeinvariant systems that are characterized by
their transfer functions and respectively. The relations between the signals are
Model Uncertainty and Robust Control 67
given by the transfer functions:
Here denotes the transfer function from signal to signal . Notice that there are
only four independent transfer functions.
(2)
The transfer functions and have special names, is called the
sensitivity function, and is called the complementary sensitivity function. Notice
that both and depend only on the loop transfer function . The sensitivity
functions are are related by
(3)
This explains the name complementary sensitivity function. These functions have in
teresting properties as is discussed in the following.
Stability and Stability Margins
Many properties of the system in Figure 2 can be derived from the loop transfer function
.
The stability of the system can be investigated by Nyquist’s stability criterion which
says that the closed loop system is stable if
(4)
where is the argument variation when traverses a contour that encloses the
right half plane (RHP) and is the number of poles of in the right half plane.
Stability is normally investigated by analyzing the Nyquist curve, see Figure 3.
68 Model Uncertainty and Robust Control
Fig. 3: Nyquist curve with phase and amplitude margins.
To achieve stability the Nyquist curvemust be sufﬁciently far away from the critical
point . The distance from the critical point can also be used as a measure of the
degree of stability. In this way the notions of amplitude margin and phase margin
, see Figure 3. An amplitude margin implies that the gain can be increased with
a factor less than without makingthe system unstable. Similarly for a system with
a phase margin it is possible to increase the phase shift in the loop by a quantity
less than without making the system unstable.
2.1 Small Process Variations
The effects of small variations in the process will now be investigated. The signal
transmission of the closed loop system from set point to output is given by the
complementary sensitivity function
We have
(5)
The function tells how the closed loop properties are inﬂuenced by small variations
in the process. The maximum sensitivities
(6)
are also used as robustness measures. The variable can be interpreted as the
shortest distance between the Nyquist curve and the critical point , see Figure 3.
classically the maximum complementary sensitivity is denoted by .
Model Uncertainty and Robust Control 69
−2 −1.5 −1 −0.5 0 0.5 1
−2
−1.5
−1
−0.5
0
0.5
1
Im
Re
Fig. 4: Nyquist curve with circle for constant sensitivity . Disturbances
with frequencies outside the circle are attenuated and frequencies inside the circle are
ampliﬁed by the feedback.
2.2 A Property of the Sensitivity Function
The sensitivity function also has other physical interpretations. Consider the system in
Figure 2. If there is no feedback the Laplace transform of the output is . The output
under closed loop controlis given by
It thus follows that
(7)
The sensitivity function thus tells how the disturbances are inﬂuenced by feedback.
Disturbances with frequencies such that is less than one are reduced by an
amount equal to the distance to the critical point and disturbances with frequencies
such that is larger than one are ampliﬁed by the feedback. This is illustrated in
Figure 4.
2.3 Large Process Variations
Equation 5 givesthe sensitivity for small perturbationsof the process. It is also possible
to get expressions for large variations in the process. To see how much the process
can change without making the closed loop system unstable we will use the Nyquist
70 Model Uncertainty and Robust Control
diagram. Consider a point on the Nyquist curve in Figure 4. The distance to the critical
point is . If the process changes by , the point changes by . The system
will remain stable as long as
and the number of right hand poles of does not change. This implies that the
perturbations must have the property that does not have any poles in the right half
plane.
The admissible variation is process dynamics is thus given by
(8)
which can also be expressed as
(9)
A crude estimate of the largest admissible variation in the process is thus given by the
largest value of the complementary sensitivity. It follows from the Figure 4 that
large variations in are permitted for the frequencies where either is large or small.
The smallest admissible variations are for frequencies where is large.
A similar estimate based on the maximum sensitivity is that
(10)
Bode’s Integrals
It follows from Equations (5) and (8) that it would be highly desirable to make the
sensitivity functions and as small as possible. This is unfortunately not possible
because it follows from Equation (3) that . There are also other constraints
on the sensitivities. It was shown in [11] that
(11)
where are the right half plane poles of and are the right half plane zeros of
. These equations imply that the sensitivities can be made small at one frequency
only at the expense of increasing the sensitivity at other frequencies. This phenomena
is sometimes called the water bed effect. It also follows from the equations that the
presence of poles in the right half plane increase the sensitivity and that zeros in the
right half plane increase the complementary sensitivity. A fast RHP pole gives higher
sensitivity than a slow pole, and a slow RHP zero gives higher sensitivity than a fast
zero.
Model Uncertainty and Robust Control 71
2.4 Bode’s Relations
The amplitude and the phase curves are also related. It is not possible to achieve high
phase advance without using high gains and it is not possible to obtain transfer func
tions that decrease rapidly without having large phase lags. These facts are expressed
analytically by some relations derived in [10].
Consider a transfer function with no poles or zeros in the right half plane.
Introduce
(12)
a logarithmic frequency scale ,, and the functions
Assume that goesto zero as goes to inﬁnity,then
(13)
an approximate version is that
(14)
This means that if the slope of the magnitude curve is constant the phase
is . This relation appears in practically all elementary courses in feedback control.
Bode’s relations imposes fundamental limitations on the performance that can be
achieved. A simple observation is that even if it is desirable that the loop gain decreases
rapidly at the crossover frequency, it is not possible to have a steeper slope than 2
without violating stability constraints.
An interesting problem is if the limitations imposed by Bode’s relations can be
avoided by using nonlinear systems. The Clegg integrator [13] is a nonlinear system
where the magnitude curve has the slope 1 and the phase lag is only .
2.5 Bode’s Ideal Loop Transfer Function
In his work on design of feedback ampliﬁers Bode suggested an ideal shape of the loop
transfer function. He proposed that the loop transfer function shouldhave the form
(15)
72 Model Uncertainty and Robust Control
Fig. 5: Nyquist curve for Bode’s ideal loop transfer function.
The Nyquist curve for this loop transfer function is simply a straight line through the
origin with , see Figure 5. Bode called (15) the ideal cutoff char
acteristic. In the terminology of automatic control we will call it Bode’s ideal loop
transfer function.
One reason why Bode made the particular choice of given by Equation (15)
is that it gives a closedloop system that is insensitive to gain changes. Changes in
the process gain will change the crossover frequency but the phase margin is
for all values of the gain. The amplitude margin is inﬁnite. The slopes
,and correspond to phase margins of ,and . Bode’s
idea to use loop shaping to design controller that are insensitive to gain variations were
later generalized by [24] to systems that are insensitive to other variations of the plant,
culminating in the QFT method, see [26].
The transfer function given by Equation (15) is an irrational transfer function for
noninteger . It can be approximated arbitrarily close by rational frequency functions.
Bode also suggested that it was sufﬁcient to approximate over a frequency range
around the desired crossover frequency . Assume for example that the gain of the
process varies between and and that it is desired to have a loop transfer
function that is close to (15) in the frequency range . It follows from (15)
that
With and a gain ratio of 100 we get a frequency ratio of about 16 and with
we get a frequency ratio of 32. To avoid having too large a frequencyrange it
is thus useful to have as small as possible. There is, however,a compromise because
the phase margin decreases with decreasing and the system becomes unstable for
.
Model Uncertainty and Robust Control 73
2.6 Fractional Systems
It follows from Equation (15) that the loop transfer function is not a rational function.
We illustrate this with an example.
Example.
Consider a process with the transfer function
(16)
Assume that we would like to have a closed loop system that is insensitive to gain
variations with a phase margin of . Bode’s ideal loop transfer function that gives
this phase margin is
(17)
Since we ﬁnd that the controller transfer function is
(18)
To implement a controller the transfer function is approximated with a rational function.
This can be done in many ways. One possibility is the following
(19)
where the gain is chosen to equal the gain of for . Notice that the
controller is composed of sections of equal length having slopes 0, +1 and 1 in the
Bode diagram. Figure 6 shows the Bode diagram for the loop transfer function. The
ﬁgure shows that the phase margin will be close to with a tolerance of less than
for a gain variation of 3 orders of magnitude. With a tolerance of we can even
allow a gain variation of 4 orders of magnitude. The range of gains can be extended
by making the controller more complex. Even if the closed loop system has the same
phase margin when the gain changes the response speed will change with the gain.
The example shows thatrobustness is obtained by increasing controller complexity.
The range of gain variation that the system can tolerate can be increased by increasing
the complexity of the controller.
Fractional systems did not receive much attention after Bodes work. In the 1990s
there was however a resurgence in the interest of fractional systems, see e.g. [53, 54, 55,
56, 57]. Oustaloup coined the acronyme CRONE from the french Commande Robuste
d’Ordre Non Entier (Robust Control of Fractional Order) for his controller.
74 Model Uncertainty and Robust Control
100
50
0
50
100
102 101 100101102
150
140
130
120
Fig. 6: Bode diagram of the loop transfer function obtained by approximatingthe frac
tional controller with a rational transfer function.
yc
u
Σ
Model e
y
−1
Process
ysp
Feedforward
Σ
Controller
uff
Fig. 7: Block diagram of a system with two degrees of freedom.
Two Degree of Freedom Systems (2DOF)
The system in Figure 2 is a system with error feedback because the controller acts on the
error which is the difference between the set point and the output. There are
signiﬁcant advantagesin having control systems with other conﬁgurations. An example
of such a system is shown in Figure 7. In this system the set point is fed through a
model before it is compared with the process output. There is also a feedforward link
which essentially is a combination of the model and the inverse process model or an
approximate inverse. Ideally the feedforward signal generates a signal which when
applied to the process produces the ideal output in response to set point changes. The
feedback controller which acts on the error will only make some corrections if there are
deviations from the ideal behavior.
A system with error feedback only is called a system with one degree of freedom.
The system in Figure 7 is called a system with two degrees of freedom (2DOF) because
the signal paths from the set point to the control signal is different from the signal path
Model Uncertainty and Robust Control 75
from the output to the control signal. This terminology was introduced by [24] who
analyzed these systems carefully.
A very nice property of systems with two degrees of freedom is that the problem
of set point response can be separated fromthe problems of robustness and disturbance
rejection. Referring to Figure 2 we will ﬁrst design a feedback by compromising be
tween disturbance attenuation and robustness. When this is done we will then design a
model and a feedforward which gives the desired response to the setpoint.
There are many variations of systems with two degrees of freedom, the following
quote from [24] is still valid
“Some structures have been presented as fundamentally different from the
others. It has been suggested that they have virtues not possessed by oth
ers, and have been given special names ... all 2DOF conﬁgurations have
basically the same properties and potentials ....”
Quantitative Feedback Theory (QFT)
Bode’s technique of dealing with gain variations were both elegant and effective. A
limitation of Bode’s work was that it was limited to gain variations only. A very nice
generalization of Bode’s work was done by Horowitz who extended it to arbitrary vari
ations of a process transfer function. He characterized model uncertainty by sets of
amplitudes and phase for each frequency called templates. Horowitz also developed a
graphical design techniques to design feedback systems that were robust to these types
of disturbances. He used a system conﬁguration with two degrees of freedom to deal
with set point responses. Horowitz design techniquecalled quantitative feedback theory
(QFT) is described in several books, see [24] and [26]. It has been applied successfully
to a wide range of problems.
Summary
In this section we have reviewed classical control theory with a focus on model un
certainty and robustness. It is worthwhile to note that model uncertainty was a key
motivation for introducing feedbackand that classical control theory had very effective
ways of dealing with uncertainty both qualitatively and quantitatively. Process uncer
tainty could be described very easily as a variation in the process transfer function with
the caveat that the disturbances do not change the number of right half plane poles of
the system. The theory has given important concepts and tools such as the transfer func
tion, Nyquist’s stability theory, the Nyquist curve, Bode diagrams, Bode’s integrals and
Bode’s ideal loop transfer function. Robustness measures such as amplitude and phase
margins and the maximum sensitivities were also introduced. Bode’s ideal loop trans
fer function is probably the ﬁrst design method that addressed robustness explicitly.
Horowitz quantitative feedback theory is a continuation of this idea.
76 Model Uncertainty and Robust Control
3 StateSpace Theory
The statespace theory represented a paradigm shift which led to many useful system
concepts and new methods for analysis and design. The systems was represented by
differential equations instead of transfer functions. For linear systems the standard
model used was
(20)
where is the input, the output and is the state. The uncertainty is represented by
the disturbances and and by variations in the elements of the matrices , and .
The disturbances and were typically described as stochastic processes, see [20] and
[4].
The control problem was formulated as to minimize the criterion
(21)
Since the equations are linear with stochastic disturbances and thecriterion is quadratic
the problem was called the linear quadratic Gaussian control problem (LQG). The so
lution to the control problem is given by
(22)
This control law has a very nice interpretation as feedback from the error
which is the difference between the ideal states and the estimated states .The
estimated states are given by the Kalman ﬁlter. Controllability and observability are
key conditions for solving the problem.
There are many other design methods based on the statespace formulation which
gives controllers with the structure (22) for example pole placement. They differ from
the LQG method in the sense that other techniques are used to obtain the matrices
and .
In Figure 8 we show a block diagram of the controller obtained from LQG theory.
In the ﬁgure we have also used a system conﬁguration with two degrees of freedom.
The system has a very attractive structure. The observer or the Kalman ﬁlter delivers
an estimate of the state based on a model of the system and the input and output sig
nals of the system. Notice that the state may also have components that represent the
disturbances. There is a feedback from the deviations of the estimated state from its
desired value . Set point following is obtained by the usual two degree of freedom
conﬁguration.
Model Uncertainty and Robust Control 77
∑ ∑
xm
uff
ˆ
x Observer
LProcess
−
ufb y
uc
Model and
Feedforward
Generator
Fig. 8: Block diagram of a system state feedback having two degrees of freedom.
Robustness
In the model (20) it is natural to describe model uncertainties as variations in the ele
ments of the matrices , and . This is, however, a very restricted class of perturba
tions which does not cover neglected dynamics or small time delays such uncertainties
are easier to describe in the frequency domain. The LQG theory was also criticized
heavily by classic control theorist because it did not take robustness into account, see
[25].
Very strong robustness properties could be established when all states were mea
sured. In this case the system has a phase margin of and inﬁnite amplitude margin,
see [42], which indicates a very good robustness.
The nice robustness properties of systems with state feedback do not hold for sys
tems with output feedback. Nice counterexamples were given in the paper [15]. For
systems with output feedback it was attempted to recover the robustness of full state
feedback making very fast observers. This approach led to a design technique called
loop transfer recovery, see [18].
The only formal requirements on the system to be controlled in statespace theory
is that the system is observable and controllable. There are no consideration of right
half plane poles and zeros or time delays. Because of this it is necessary to investigate
the robustness of the design and to make appropriate modiﬁcations to achieve good
robustness. We use an example to illustrate what happensif this is not done.
Example: A fast system with a low bandwidth
Consider a system that is described by
The system is controllable if . We will assume that . The system is of
second order and one state variable is measured directly. The system can be controlled
78 Model Uncertainty and Robust Control
with an observer of ﬁrstdegree. The closed loop system is then of order 3. We assume
that a state feedback and an observe is designed so that the closed loop system is
(23)
The transfer function of the system is
(24)
To obtain a fast closedloop system we choose and . straightforward
calculations show that the controller has the transfer function
(25)
with ,and . The loop transfer function is
Theprocesspoleat is almost canceled by the controller zero at
. The Bode diagram of the loop transfer function is shown in Figure 9.
The loop transfer function has a low frequency asymptote that intersects the line
at , i.e. at the slow unstable zero. The magnitude then
becomes close to one and it remains so until the the break point at ,i.e.
the controller pole. The phase is also close to over that frequency range which
means that the stability margin is very poor. The crossover frequency is and the
phase margin is . The maximum sensitivities are and
which also shows that the system is extremely sensitive. The slope of the magnitude
curve at crossover is also very small which is another indication of the poor robustness
of the system.
The example illustrates clearly the danger of using a design method in a routine
manner. It also shows that it is not sufﬁcient to check controllability and observabil
ity. For this particular problem there are severe limitations caused by the right half
plane zero. Trying to make designs which violate these limitations by makinga closed
loop system that is too fast we obtain a closedloop system that has very poor stability
margins even if the closedloop poles are quite well damped. Also notice that even if
the gain crossover frequency is rad/s the sensitivity becomes larger than one for
, which is close to the right half plane zero. Feedback is thus not effective
for disturbances having higher frequencies than , because disturbances will be
ampliﬁed by the feedback. The example shows that it is important to be aware of the
limitations when designing control systems. It is of course possible to obtain sensible
control designs, for example by choosing smaller values of .
Model Uncertainty and Robust Control 79
Frequency (rad/sec)
Phase (deg); Magnitude (dB)
Bode Diagrams
−5
0
5
10
15
20
25
10−2 10−1 100101102103104
−250
−200
−150
−100
−50
Fig. 9: Bode diagram for the loop transfer function of a system with a slow unstable
zero at , where the speciﬁcations are and .
Summary
statespace theory is an elegant way to approacha control problem. A nice aspect is that
it naturally deals with multivariable systems. The theory has given important concepts
such as observability and reachability, It has also given several design methods, such
as linear quadratic control, loop transfer recovery and pole placement. It has also in
troduced powerful computationalmethods based on numerical linear algebra. A severe
drawback is that robustnessis not dealt with properly. This means that it is possible to
formulate control problems which give solutions that have very poor robustness proper
ties. It is easy to avoid the difﬁculties when we are aware of them, simply by evaluating
the robustness of a design and to reduce the requirements until a suitable compromise
is reached.
80 Model Uncertainty and Robust Control
4 Fundamental Limitations
It is very useful to determine the performancethat can be achieved without sacriﬁcing
robustness. Such estimates will be provided in this section. In particular we will con
sider limitations that arise from poles and zeros in the right half plane and time delays.
The results are based on [2] and [1].
Consider a system with the transfer function . Factor the transfer function as
(26)
where is the minimum phase part and is the nonminimum phase part. Let
denote the uncertainty in the process transfer function. It is assumed that the
factorization is normalized so that and the sign is chosen so that
has negative phase. The achievable bandwidth is characterized by the gain crossover
frequency .
4.1 The Crossover Frequency Inequality
We will now derive an inequality for the gain crossover frequency. The loop transfer
function is . Requiring that the phase margin is we get.
(27)
Assume that the controller is chosen so that the loop transfer function is equal to
Bode’s ideal loop transfer function given by Equation (15), then
(28)
where is the slope of the loop transfer function at the crossover frequency. Equa
tion (28) is also a good approximation for other controllers because the amplitude curve
is typically close to a straight line at the crossoverfrequency. The parameter in (28)
is then the slope at the crossover frequency. It follows from Bode’s relations (13)
that the phase is . It follows from Equations (27) and (28) that the crossover
frequency satisﬁes the inequality
(29)
where
(30)
This crossover frequency inequality gives the limitations imposed by nonminimum
phase factors. A straightforward method to determine the crossover frequencies that
can be obtained is to plot the left hand side of Equation (29) and determine when the
inequality holds. The following example gives a simple rule of thumb.
Model Uncertainty and Robust Control 81
Example: A Simple Rule of Thumb
To see the implications of (29) we will make some reasonable design choices. With a
phase margin of (), and a slope of we get and
Equation (29) becomes
(31)
This gives the simple rule that the phase lag of the minimum phase componentsshould
be less than at the gain crossover frequency.
4.2 A Zero in the Right Half Plane
We will now discusslimitations imposedby right half plane zeros. Wewill ﬁrst consider
systems with only one zero in the right half plane. The nonminimum phase part of the
plant transfer function then becomes
(32)
Notice that should be chosen to have unit gain and negativephase. We have
and (29) gives the following upper bound on the crossover frequency.
(33)
and the simple ruleof thumb (31) we get .
A right half plane zero gives an upper bound to the achievable bandwidth. The
bandwidth decreases with decreasing frequency of the zero. It is thus more difﬁcult to
control systems with slow zeros.
4.3 Time Delays
The transfer function for such systems has an essential singularity at inﬁnity. The non
minimum phase part of the transfer functionof the process is
(34)
We have and the crossover frequency inequality, Equation (29)
becomes
(35)
The simple rule of thumb (31) gives .
Time delays thus give an upper bound on the achievable bandwidth.
82 Model Uncertainty and Robust Control
4.4 A Pole in the Right Half Plane
Consider a system with one pole in the right half plane. The nonminimum phase part
of the transfer function is thus
(36)
where . Notice that the transfer function is normalized so that has unit gain
and negative phase. We have
and the crossover frequency inequality] (29) gives
(37)
The simple rule of thumb (31) gives .
Unstable poles give a lower bound on the crossover frequency. For systems with
right half plane poles the bandwidth must thus be sufﬁciently large.
4.5 Poles and Zeros in the Right Half Plane
Consider a system with
(38)
For we have
The right hand side has its maximum for and the inequality (29) becomes
(39)
The simple rule of thumb (31) gives . Table 1 gives the phase margin as a
function of the ratio for and . The phasemargin that can
be achieved for a given ratio is
(40)
When the unstable zero is faster than the unstable pole, i.e. , the ratio
thus must be sufﬁciently large in order to have the desired phase margin. The largest
gain crossover frequency is the geometric mean of the unstable pole and zero.
Model Uncertainty and Robust Control 83
22.24 3.86 5 5.83 8.68 10 20
6.0 0 30 38.6 45 60 64.8 84.6
Table 1: Achievable phase margin for and different zeropole ratios .
Example: The X29
Considerable design effort has been devoted to the design of the ﬂight control system
for the X29 aircraft, see [12] and [41]. One of the design criteria was that the phase
margin should be greater than for all ﬂight conditions. At one ﬂight condition the
model has the following nonminimumphase component
Since , it follows from Equation (40) that the achievable phase margins for
and are and . It is interesting to
note that many design methods were used in a futile attempt to reach the design goal.
A simple calculation of the type given in this section would have given much insight.
Example: Klein’s Unridable Bicycle
An interesting bicycle with rear wheel steering which is impossible to ride was designed
by Professor Klein in Illinois, see [33]. The theory presented in this paper is well suited
to explain why it is impossible to ride this bicycle. The transfer function from steering
angle to tilt angle is given by
where is the total mass of the bicycle and the rider, the moment of inertia for tilt
with respect to the contact line of the wheels and the ground, the height of the center
of mass from the ground, the vertical distance from the center of mass to the contact
point of the front wheel, the forward velocity, and the acceleration of gravity. The
system has a RHP pole at , caused by the pendulum effect. Because
of the rear wheel steering the system also has a RHP zero at . Typical
values kg, m, , and m/s, give
rad/s and rad/s. The ratio of the zero and the pole is
thus , with the inequality (29) shows that the phase margin can
be at most .
The reason why the bicycle is impossible to ride is thus that the system has a right
half plane pole and a right half plane zero that are too close together. Klein has veriﬁed
this experimentally by making a bicycle where the ratio is larger. This bicycle is
indeed possible to ride.
84 Model Uncertainty and Robust Control
So far we have only discussed the case . When the unstable zero is slower
than the unstable pole the crossoverfrequency inequality (29) cannot be satisﬁed unless
and .
4.6 A Pole in the Right Half Plane and Time Delay
Consider a system with one pole in the right half plane and a time delay . The non
minimum phase part of the transfer functionis thus
(41)
The crossover frequency condition (29) gives
(42)
The system cannot be stabilized if .If the left hand side has its smallest
value for . Introducing this value of into (42) we get
The simple rule of thumb with to and gives
(43)
Example: Pole balancing
To illustrate the results we can consider balancing of an inverted pendulum. A pendu
lum of length has a right half plane pole . Assuming that the neural lag of a
human is s. The inequality (43) gives , hence .The
calculation thus indicate that a human with a lag of 0.07 s should be able to balance
a pendulum whose length is 0.5 m. To balance a pendulum whose length is 0.1 m the
time delay must be less than 0.03s. Pendulum balancing has also been done using video
cameras as angle sensors. The limited video rate imposes strong limitations on what can
be achieved. With a videorate of 20 Hz it follows from (43) that the shortest pendulum
that can be balanced with and is m.
4.7 Other Criteria
The phase margin is a crude indicator of the stability margin. By carrying out detailed
designs the results can be reﬁned. This is done in [1] which gives results for designs
with and .
Model Uncertainty and Robust Control 85
A RHP zero
for ,
for ,
A RHP pole
for ,
for ,
A time delay
for ,
for ,
A RHP polezero pair with
for ,
for ,
A RHP pole and a time delay
for ,
for ,
A time delay or a zero in the righthalf plane gives an upper boundof the bandwidth that
can be achieved. The bound decreases when the zero decreases and the time delay
increases. A pole in the right half plane gives a lower bound on the bandwidth. The
bandwidth increases with increasing . For a pole zero pair there is a upper bound on
the polezero ratio.
5 Loop Shaping
A consequence of the introduction of statespace theory was that interest shifted from
robustness to optimization. New developments that started by the developmentof
control by George Zames in the 1980s gave a strong revival of robustness, see [50]. This
led to a very vigorous developmentthat has given new insight and new design methods.
These results will be discussed in this Section. To keep the presentation simple we
will only deal with systems having one input and one output, but techniques as well as
results can be generalized to systems with many inputs and many outputs. For more
extensive treatmentswe refer to [16], [21], [52] [23] and [48].
86 Model Uncertainty and Robust Control
Fig. 10: Block diagram of a simple feedback system.
Problem Formulation
If a system structure with two degrees of freedom is used the problems of setpoint
response can can be dealt with separately and we can therefore focus on robustness
and attenuation of disturbances. The set point will not be considered and Figure 2
can be simpliﬁed to the system shown in Figure 10. The system has two inputs the
measurement noise and the load disturbance . The problem we will consider is to
design a controller with the properties.
Insensitive to changes in the process properties.
Ability to reduce the effects of the load disturbance .
Does not inject too much measurement noise into the system.
Stability
Stability is a primary robustness requirement. There is one problem with the stability
concept introduced in Section 2. Because the stability is based on the loop transfer
function only there may be cancellations of poles and zeros in the process and the
controller. This does not pose any problems if the cancelled factors are stable. The
results will however be strongly misleading if the canceled factors are unstable because
there will be internal signals in the system that will diverge. We illustrate this with an
example.
Model Uncertainty and Robust Control 87
Example: Pole Zero Cancellation
Consider the system in Figure 10 with
The loop transfer function is and the system thus appears stable. Notice
however that the transfer function from disturbance to output is.
A load disturbance will thus make the output diverge and it does not make sense to call
the system stable.
The problem illustrated in Example 5 is well known. Classically it is resolved by
formally introducing the rule that cancellation of unstable poles are not permissible.
This can also be encapsulated in an algebra for manipulating systems which does not
permit divisionby factors having rootsin the right half plane, see [38] and [39].
Another way is to introduce a stability concept that takes care of the problem di
rectly. It follows from the analysis in Section 2 that the closed loop system is completely
characterized by the transfer functions given by Equation (2). Based on this we can say
that a system is stable if all these transfer functions (2) are stable. This is sometimes
called internal stability. The transfer functions (2) can be conveniently combined in the
matrix
(44)
Notice that this transfer function (44) representsthe signal transmission from the distur
bances and to the signals and in the block diagram in Figure 2. Let the transfer
functions of the process and the controllerbe represented as
The matrix (44) can then be represented as
(45)
88 Model Uncertainty and Robust Control
and the stability criterion is that the equation
(46)
should have all its roots in the left half plane. This is also called internal stability. We
will simply say that the system is stable.
Example: Pole Zero Cancellation
Applying the result to the problemin Example 5 we ﬁnd that
The characteristic polynomial is
which clearly has a root in the right half plane.
How to Compare two Systems
A fundamental problem when discussing robustness is to determine when two systems
are close. This seemingly innocent problem is not as simple as it may appear. For
feedback control it would be natural to claim that two systems are close if they have
similar behavior under a given feedback, see [45] and [43]. The fact that two systems
have similar open loop characteristics does not mean that they will behave similarly
under feedback.
Example: Similar Open Loop Different Closed Loop
Systems with the transfer functions
have very similar open loop responses for large values of . This is illustrated in Fig
ure 11 which shows the step responses of for . The differences between the
step responses is barely noticeable in the ﬁgure. The closed loop systems obtained with
unit feedback have the transfer functions
The closed loop systems are very different because the system is unstable.
Model Uncertainty and Robust Control 89
0 1 2 3 4 5 6 7 8
0
200
400
600
800
1000
Fig. 11: Step responses for systems with the transfer functions
and .
Example: Different Open Loop Similar Closed Loop
Systems with the transfer functions
have very different open loop properties because one system is unstable and the other
is stable. The closed loop systems obtained with unit feedback are however
which are very close.
There are many examples of this in the literature of adaptive control where the
importance of considering the closed loop properties of a model has been recognized
for a long time, see e.g. [3]. The examples given above show that the naive way
of comparing two systems by analyzing their responses to a given input signal is not
appropriate for feedback control. The difﬁculty is that it does not work when one or
both systems are unstable as in Example 5 and 5.
One approach is to compare the outputs when the inputs are restricted to the class
of inputs that give bounded outputs. This approach was introduced in [51] and [19]
using the notion of gap metric. Another approach was introduced in [44] and [45]. To
describe this approach we assume that the process is described by the rational transfer
function
where and are polynomials. Introduce a stable polynomial whose
degree is not smaller than the degrees of and . The transfer function
90 Model Uncertainty and Robust Control
can then be written as
(47)
Vidyasagar proposed to compare two systems by comparing the stable rational transfer
functions and . This is called the graph metric. A difﬁculty was that the graph
metric was difﬁcult to compute.
Coprime Factorization
The polynomial in (47) can be chosen in many different ways. We will now discuss
a convenient choice.
We start by introducing a suitable concept. Two rational functions and are
called coprime if there exist rational functions and which satisfy the equation
The condition for coprimeness is essentially that and do nothave any com
mon factors. The functions and will now be chosen so that
(48)
where we have used the notation . A factorization (47) of where
and satisfy (48) is called a normalized coprime factorization of . Such a factoriza
tion the polynomials and in (47) do not have common factors.
5.1 Vinnicombe’s Metric
A very nice solution to the problem of comparing two systems that is appropriate for
feedback was givenby Vinnicombe, see [46] and [48]. Consider two systems with the
normalized coprime factorizations
To compare the systems it must be required that
(49)
where is the Nyquist contour. In the polynomial representation this condition implies
that
(50)
Model Uncertainty and Robust Control 91
Fig. 12: Geometric interpretation of the Vinnicombe metric.
If the winding number constraint is satisﬁed the distance between the systems is deﬁned
as
(51)
We have . If the winding number condition is not satisﬁed the distance
is deﬁned as . Vinnicombe showed that is a metric and he called it the gap
metric.
Geometric Interpretation
Vinnicombe’s metric is easy to compute and it also has a very nice geometric interpre
tation. The expression
can be interpreted graphically as follows. Let and be two complex numbers. The
Riemann sphere is located above the complex plane. It has diameter 1 and its south pole
is at the origin of the complex plane. Points in the complex plane are projected onto the
sphere by a line through the pointand the north pole, see Figure 12. Let and be
the projections of and on the Riemann sphere. The number is then the shortest
chordal distance between the points and , see Figure 12.
92 Model Uncertainty and Robust Control
Fig. 13: Block diagram of a process with coprime factor uncertainty and a controller.
Coprime Factor Uncertainty
Classical sensitivity results such as (8) were obtained based on additive perturbations.
The system was perturbed to where is a stable transfer function. These
types of perturbations are not well suited to deal with feedback systems as is illustrated
by Example 5. A more sophisticated way to describe perturbations are required for
this. The development of the metrics for systems gave good insight into what should be
done. Uncertainty will be described in terms of the normalized coprime factorization
of a system. Consider a system describedby
(52)
where and is a normalized coprime factorization of and the perturbations
and are stable proper transfer functions. Figure 13 shows a block diagram of the
closed loop system with a the perturbedplant.
We will now investigate how large the perturbations can be without violating the
stability condition. For the system in Figure 13 we have
and
The system can thus be represented with the block diagram in Figure 14. We can then
Model Uncertainty and Robust Control 93
,
Fig. 14: Simpliﬁcation of the block diagram in Figure 13.
invoke the small gain theorem and conclude that the perturbed system will be stable if
the loop gain is less than one, see [14]. Hence
(53)
This condition can be simpliﬁed if we use the fact that and is a normalized coprime
factorization. This gives
where denotes the system matrix
(54)
Introducing
(55)
we thus ﬁnd that the closed loop system is stable for all normalized coprime perturba
tions and such that
(56)
94 Model Uncertainty and Robust Control
This equation is a natural generalization of Equation (8) in classical control theory.
Notice that since the systems have one input and one output we have
(57)
and Equation (55) can thus be written
(58)
Loop Shaping
The goal of is to design control systems that are insensitive to model uncertainty.
It follows from Equations (55) and (56) that this can be accomplished by ﬁnding a
controller that gives a stable closed loop system and minimizes the norm of the
transfer function given by Equation eq:gmatrix. It follows from Equation (56)
that such a design permits the largest deviation of the normalized coprime deviations.
It is interesting to observe that the transfer function also describes the signal
transmission from the disturbances and to and in Figure 2. A robust controller
obtained in this way will also attenuate the disturbances very well.
A statespace solution to the control problem was given in [17]. A loop shaping
design procedure was developed in [34] and [35].
Frequency Weighting
In the design procedure presented in [35] it is also possible to introduce a frequency
weighting as a design parameter. The control problem for the process
is then solve giving the controller . The controller for the process is then .
In this way it is possible to obtain controller that have high gain at speciﬁed frequency
ranges and high frequency roll off.
Generalized Stability Margin
A generalization of the classical stability margin was also introduced in [34]. For a
closed loop system consisting of the process and the controller such that the closed
loop system we deﬁne the generalized stability margin as
if is stable
otherwise (59)
Model Uncertainty and Robust Control 95
Notice that the generalized stability margin takes values between 0 and 1. The margin
is 0 if the system is unstable. A value close to one indicates a good margin of stability.
Reasonable practical valuesof the margin are in the range of to .
The loop shaping in [34] gives a controller that maximizes the stability margin
giving
(60)
Vinnicombe’s Theorems
A number of interesting theorems that relate model uncertainty to robustness have been
derived in [47] and [48]. These results, which can be seen as the natural conclusion of
the work that began in [50], give very nice relations between robust control and model
uncertainty. Vinnicombe has proven the following results.
Proposition 1
Consider a nominal processes and a controller and a parameter . Then the con
troller stabilizes all plants such that , if and only if .
Proposition 2
Given a nominal process , a perturbed process and a number .
Then is stable for all compensators , such that if and only if
.
The ﬁrst proposition tells that a controller designed for process with a general
ized stability margin greater than will stabilize all processes in a environment
of provided that .
Proofs of these theorems are given in [48]. Vinnicombe has actually given sharper
results which only requires the inequalities to hold pointwise for each frequency.
Connections to the Classical Control Theory
The loop shaping cannot be directly related to the classical robustness criteria.
The classical robustness criteria such as and depend only on the loop transfer
function . But the generalized stability margin and the generalized
sensitivity depend on both and . The generalized stability margin will
therefore change if the process transfer function is multiplied by a constant and the
controller transfer function is divided by the same number. One reason for this is that
the criterion (57) implicitly assumes that the disturbances and have equal weight.
This is a reasonable assumption if sufﬁcient information about the disturbances are
96 Model Uncertainty and Robust Control
available but very often we do not have this information. One possibility to formulate
the design problem in this case is to choose the most favorable disturbance relation.
This can be done by introducing a weighting of the disturbances. Let be the
weighted process let be the weighted controller .Wehave
The loop transfer function is invariant to . The generalized sensitivity becomes
A straightforward calculation shows that is minimized for the weight
see [37]. This weight we get the following expression for the weightedsensitivity
(61)
Notice that the weighted only depends on the loop transfer function .
Using the weighted sensitivity function we thus obtain an interesting connection
between loop shaping and classical robustness theory. The number deﬁned by
Equation (55) and Equation (58) is a natural generalization of the maxima ,of
the sensitivity function and the complementary sensitivity. It is useful to introduce a
combined sensitivity by requiring that both and should at most be equal
to . The combined sensitivity implies that the Nyquist curve of the loop transfer
function is outside a circle with diameter on
The following inequalities are shown
in [37] where also sharper inequalitiesare presented.
The generalized stability margin is also a natural generalization of the classical
stability margin . There are howeversome scale changes. The normal stability mar
gin takes values between 1 and while the generalized stability margin takes values
between zero and one. To get compatibility the classical stability margin should be
redeﬁned as the distance between the critical point and the intersection of the Nyquist
curve with the negative real axis. Hence
Model Uncertainty and Robust Control 97
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