solving the controller design problemstanford.edu/~boyd/lcdbook/out8.pdf · solving the controller...
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Chapter 15
Solving the Controller DesignProblem
In this chapter we describe methods for forming and solving �nite�dimensionalapproximations to the controller design problem� A method based on theparametrization described in chapter � yields an inner approximation of the re�gion of achievable speci�cations in performance space� For some problems� anouter approximation of this region can be found by considering a dual problem�By forming both approximations� the controller design problem can be solved toan arbitrary� and guaranteed� accuracy�
In chapter � we argued that many approaches to controller design could be describedin terms of a family of design speci�cations that is parametrized by a performance
vector a � RL�
H satis�es Dhard� ���H� � a�� � � � � �L�H� � aL� ������
Some of these speci�cations are unachievable� the designer must choose amongthe speci�cations that are achievable� In terms of the performance vectors� thedesigner must choose an a � A� where A denotes the set of performance vectors thatcorrespond to achievable speci�cations of the form ������� We noted in chapter �that the actual controller design problem can take several speci�c forms� e�g�� aconstrained optimization problem with weightedsum or weightedmax objective�or a simple feasibility problem�
In chapters ��� we found that in many controller design problems� the hardconstraint Dhard is convex �or even a�ne� and the functionals ��� � � � � �L are convex�we refer to these as convex controller design problems� We refer to a controllerdesign problem in which one or more of the functionals is quasiconvex but notconvex as a quasiconvex controller design problem� These controller design problemscan be considered convex �or quasiconvex� optimization problems over H� since H
351
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352 CHAPTER 15 SOLVING THE CONTROLLER DESIGN PROBLEM
has in�nite dimension� the algorithms described in the previous chapter cannot bedirectly applied�
15.1 Ritz Approximations
The Ritz method for solving in�nitedimensional optimization problems consists ofsolving the problem over larger and larger �nitedimensional subsets� For the controller design problem� the Ritz approximation method is determined by a sequenceof nz � nw transfer matrices
R�� R�� R�� � � � � H� ���� �
We let
HN��
���R� �
X��i�N
xiRi
������ xi � R� � � i � N
���
denote the �nitedimensional a�ne subset of H that is determined by R� and thenext N transfer matrices in the sequence� The Nth Ritz approximation to thefamily of design speci�cations ������ is then
H satis�es Dhard� ���H� � a�� � � � � �L�H� � aL� H � HN � ������
The Ritz approximation yields a convex �or quasiconvex� controller design problem�if the original controller problem is convex �or quasiconvex�� since it is the originalproblem with the a�ne speci�cation H � HN adjoined�
The Nth Ritz approximation to the controller design problem can be considereda �nitedimensional optimization problem� so the algorithms described in chapter ��can be applied� With each x � RN we associate the transfer matrix
HN �x��� R� �
X��i�N
xiRi� ������
with each functional �i we associate the function ��N�i � RN � R given by
��N�i �x�
�� �i�HN �x��� ������
and we de�ne
D�N� �� fx j HN �x� satis�es Dhardg � ������
Since the mapping from x � RN into H given by ������ is a�ne� the functions
��N�i given by ������ are convex �or quasiconvex� if the functionals �i are� similarly
the subsets D�N� � RN are convex �or a�ne� if the hard constraint Dhard is� In
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15.1 RITZ APPROXIMATIONS 353
section ���� we showed how to compute subgradients of the functions ��N�i � given
subgradients of the functionals �i�
Let AN denote the set of performance vectors that correspond to achievablespeci�cations for the Nth Ritz approximation ������� Then we have
A� � � � � � AN � � � � � A�
i�e�� the Ritz approximations yield inner or conservative approximations of theregion of achievable speci�cations in performance space�
If the sequence ���� � is chosen well� and the family of speci�cations ������ iswell behaved� then the approximations AN should in some sense converge to A asN � �� There are many conditions known that guarantee this convergence� seethe Notes and References at the end of this chapter�
We note that the speci�cation Hslice of chapter �� corresponds to the N � Ritz approximation�
R� � H�c�� R� � H�a� �H�c�� R� � H�b� �H�c��
15.1.1 A Specific Ritz Approximation Method
A speci�c method for forming Ritz approximations is based on the parametrization of closedloop transfer matrices achievable by stabilizing controllers �see section � ����
Hstable � fT� � T�QT� j Q stableg � �����
We choose a sequence of stable nu � ny transfer matrices Q�� Q�� � � � and form
R� � T�� Rk � T�QkT�� k � �� � � � � ������
as our Ritz sequence� Then we have HN � Hstable� i�e�� we have automaticallytaken care of the speci�cation Hstable�
To each x � RN there corresponds the controllerKN �x� that achieves the closedloop transfer matrix HN �x� � HN � in the Nth Ritz approximation� we search overa set of controllers that is parametrized by x � RN � in the same way that the familyof PID controllers is parametrized by the vector of gains� which is in R�� But theparametrization KN �x� has a very special property� it preserves the geometry of the
underlying controller design problem� If a design speci�cation or functional is closedloop convex or a�ne� so is the resulting constraint on or function of x � RN � Thisis not true of more general parametrizations of controllers� e�g�� the PID controllers�The controller architecture that corresponds to the parametrizationKN �x� is shownin �gure �����
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354 CHAPTER 15 SOLVING THE CONTROLLER DESIGN PROBLEM
Knom
Q
KN �x�
u y
v e
Q�
QN
x�
xN
r
�
�
r
�
�
r
�
��
�
�
�
�
�
��
q
q
q
Figure ���� The Ritz approximation ��������� corresponds to aparametrized controller KN �x that consists of two parts� a nominal con�troller Knom� and a stable transfer matrix Q that is a linear combination ofthe �xed transfer matrices Q�� � � � � QN � See also section ��� and �gure ����
15.2 An Example with an Analytic Solution
In this section we demonstrate the Ritz method on a problem that has an analyticsolution� This allows us to see how closely the solutions of the approximations agreewith the exact� known solution�
15.2.1 The Problem and Solution
The example we will study is the standard plant from section ��� We consider theRMS actuator e�ort and RMS regulation functionals described in sections ��� ��and ����� �
RMS�yp��� �rms yp�H� �
�kH��Wsensork
�� � kH��Wprock
��
����RMS�u�
�� �rms u�H� �
�kH��Wsensork
�� � kH��Wprock
��
�����
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15.3 AN EXAMPLE WITH NO ANALYTIC SOLUTION 355
We consider the speci�c problem�
min�rms yp�H� � ���
�rms u�H�� ������
The solution� ��rms u � ������ can be found by solving an LQG problem withweights determined by the algorithm given in section �����
15.2.2 Four Ritz Approximations
We will demonstrate four di�erent Ritz approximations� by considering two di�erentparametrizations �i�e�� T�� T�� and T� in ������ and two di�erent sequences of stableQ�s�
The parametrizations are given by the formulas in section �� using the twoestimatedstatefeedback controllers K�a� and K�d� from section �� �see the Notesand References for more details�� The sequences of stable transfer matrices weconsider are
Qi �
�
s� �
i
� �Qi �
�
s� �
i
� i � �� � � � �������
We will denote the four resulting Ritz approximations as
�K�a�� Q�� �K�a�� �Q�� �K�d�� Q�� �K�d�� �Q�� �������
The resulting �nitedimensional Ritz approximations of the problem ������ turnout to have a simple form� both the objective and the constraint function are convex quadratic �with linear and constant terms� in x� These problems were solvedexactly using a special algorithm for such problems� see the Notes and Referencesat the end of this chapter� The performance of these four approximations is plotted in �gure ��� along with a dotted line that shows the exact optimum� ������Figure ���� shows the same data on a more detailed scale�
15.3 An Example with no Analytic Solution
We now consider a simple modi�cation to the problem ������ considered in theprevious section� we add a constraint on the overshoot of the step response fromthe reference input r to the plant output yp� i�e��
min�rms yp�H� � ����os�H��� � ���
�rms u�H�� ����� �
Unlike ������� no analytic solution to ����� � is known� For comparison� the optimaldesign for the problem ������ has a step response overshoot of �����
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356 CHAPTER 15 SOLVING THE CONTROLLER DESIGN PROBLEM
RMS�u�
N
����K�a�� Q
����
�K�a�� Q
���
�K�d�� Q���
�K�d�� Q
��Iexact
� � � � � �� �� �� �� �� ���
����
����
����
����
���
����
����
����
����
���
Figure ���� The optimum value of the �nite�dimensional inner approxi�mation of the optimization problem ����� versus the number of terms Nfor the four di�erent Ritz approximations ������� The dotted line showsthe exact solution� RMS�u � �������
RMS�u�
N
����K�a�� Q
��I �K�a�� Q��I �K�d�� Q
����
�K�d�� Q
��Iexact
� � � � � �� �� �� �� �� �������
����
�����
�����
�����
�����
�����
Figure ���� Figure ���� is re�plotted to show the convergence of the �nite�dimensional inner approximations to the exact solution� RMS�u � �������
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15.3 AN EXAMPLE WITH NO ANALYTIC SOLUTION 357
The same four Ritz approximations ������� were formed for the problem ����� ��and the ellipsoid algorithm was used to solve them� The performance of the approximations is plotted in �gure ����� which the reader should compare to �gure ��� �The minimum objective values for the Ritz approximations appear to be convergingto ������ whereas without the step response overshoot speci�cation� the minimumobjective is ������ We can interpret the di�erence between these two numbers asthe cost of reducing the step response overshoot from ���� to ����
RMS�u�
N
����K�a�� Q
����K�a�� Q
����K�d�� Q
����K�d�� Q
��Iwithout step spec�
� � � � � �� �� �� �� �� ���
����
����
����
����
���
����
����
����
����
���
Figure ���� The optimum value of the �nite�dimensional inner approxi�mation of the optimization problem ������ versus the number of terms Nfor the four di�erent Ritz approximations ������� No analytic solution tothis problem is known� The dotted line shows the optimum value of RMS�uwithout the step response overshoot speci�cation�
15.3.1 Ellipsoid Algorithm Performance
The �nitedimensional optimization problems produced by the Ritz approximationsof ����� � are much more substantial than any of the numerical example problemswe encountered in chapter ��� which were limited to two variables �so we could plotthe progress of the algorithms�� It is therefore worthwhile to brie�y describe howthe ellipsoid algorithms performed on the N � � �K�a�� Q� Ritz approximation� asan example of a demanding numerical optimization problem�
The basic ellipsoid algorithm was initialized with A� � ����I� so the initialellipsoid was a sphere with radius ��� All iterates were well inside this initialellipsoid� Moreover� increasing the radius had no e�ect on the �nal solution �and�indeed� only a small e�ect on the total computation time��
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358 CHAPTER 15 SOLVING THE CONTROLLER DESIGN PROBLEM
The algorithm took �� iterations to �nd a feasible point� and � �� iterations forthe maximum relative error to fall below ����� The maximum and actual relativeerrors versus iteration number are shown in �gure ����� The relative constraintviolation and the normalized objective function value versus iteration number areshown in �gure ����� From these �gures it can be seen that the ellipsoid algorithm produces designs that are within a few percent of optimal within about ����iterations�
�error
k
���
maximumrelative error
���actual
relative error
� ���� ���� ���� ���� �������
�
��
���
Figure ���� The ellipsoid algorithm maximum and actual relative errorsversus iteration number� k� for the solution of the N � �� �K�a�� Q Ritzapproximation of ������� After �� iterations a feasible point has been found�and after ���� iterations the objective has been computed to a maximumrelative error of ����� Note the similarity to �gure ������ which shows asimilar plot for a much simpler� two variable� problem�
For comparison� we solved the same problem using the ellipsoid algorithm withdeepcuts for both objective and constraint cuts� with the same initial ellipsoid�A few more iterations were required to �nd a feasible point ����� and somewhatfewer iterations were required to �nd the optimum to within ���� � � ��� Itsperformance is shown in �gure ���� The objective and constraint function valuesversus iteration number for the deepcut ellipsoid algorithm were similar to the basicellipsoid algorithm�
The number of iterations required to �nd the optimum to within a guaranteedmaximum relative error of ���� for each N �for the �K�a�� Q� Ritz approximation�is shown in �gure ���� for both the regular and deepcut ellipsoid algorithms� �ForN � � the step response overshoot constraint was infeasible in the �K�a�� Q� Ritzapproximation��
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15.3 AN EXAMPLE WITH NO ANALYTIC SOLUTION 359
�violation
k
�����rms u � ��rms u���
rms u
�����rms yp � ������������os � �������
� ���� ���� ���� ���� ����
��
��
��
�
�
�
�
�
��
Figure ���� The ellipsoid algorithm constraint and objective functionalsversus iteration number� k� for the solution of the N � �� �K�a�� Q Ritzapproximation of ������� For the constraints� the percentage violation isplotted� For the objective� �rms u� the percentage di�erence between thecurrent and �nal objective value� ��rms u� is plotted� It is hard to distinguishthe plots for the constraints� but the important point here is the �steady�stochastic� nature of the convergence� Note that within ���� iterations�designs were obtained with objective values and contraints within a fewpercent of optimal and feasible� repsectively�
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360 CHAPTER 15 SOLVING THE CONTROLLER DESIGN PROBLEM
�error
k
���
maximumrelative error
���actual
relative error
� ��� ���� ���� ���� ���� �������
�
��
���
Figure ���� The deep�cut ellipsoid algorithm maximum and actual rel�ative errors versus iteration number� k� for the solution of the N � ���K�a�� Q Ritz approximation of ������� After �� iterations a feasible pointhas been found� and after ���� iterations the objective has been computedto a maximum relative error of ����� Note the similarity to �gure �����
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15.3 AN EXAMPLE WITH NO ANALYTIC SOLUTION 361
iterations
N
��Rregular ellipsoid
��Ideep�cut ellipsoid
� � � � � �� �� �� �� �� ���
���
����
����
����
����
����
����
����
����
����
Figure ���� The number of ellipsoid iterations required to compute upperbounds of RMS�u to within ����� versus the number of terms N � is shown
for the Ritz approximation �K�a�� Q� The upper curve shows the iterationsfor the regular ellipsoid algorithm� The lower curve shows the iterationsfor the deep�cut ellipsoid algorithm� using deep�cuts for both objective andconstraint cuts�
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362 CHAPTER 15 SOLVING THE CONTROLLER DESIGN PROBLEM
15.4 An Outer Approximation via Duality
Figure ���� suggests that the minimum value of the problem ����� � is close to������ since several di�erent Ritz approximations appear to be converging to thisvalue� at least over the small range of N plotted� The value ����� could reasonablybe accepted on this basis alone�
To further strengthen the plausibility of this conclusion� we could appeal to someconvergence theorem �one is given in the Notes and References�� But even knowingthat each of the four curves shown in �gure ���� converges to the exact value ofthe problem ����� �� we can only assert with certainty that the optimum value liesbetween ����� �the true optimum of ������� and ������ �the lowest objective valuecomputed with a Ritz approximation��
This is a problem of stopping criterion� which we discussed in chapter �� in thecontext of optimization algorithms� accepting ����� as the optimum value of ����� �corresponds to a �quite reasonable� heuristic stopping criterion� Just as in chapter ��� however� a stopping criterion that is based on a known lower bound may beworth the extra computation involved�
In this section we describe a method for computing lower bounds on the valueof ����� �� by forming an appropriate dual problem� Unlike the stopping criteria forthe algorithms in chapter ��� which involve little or no additional computation� thelower bound computations that we will describe require the solution of an auxiliaryminimum H� norm problem�
The dual function introduced in section ���� produces lower bounds on the solution of ����� �� but is not useful here since we cannot exactly evaluate the dualfunction� except by using the same approximations that we use to approximatelysolve ����� �� To form a dual problem that we can solve requires some manipulationof the problem ����� � and a generalization of the dual function described in section ���� � The generalization is easily described informally� but a careful treatmentis beyond the scope of this book� see the Notes and References at the end of thischapter�
We replace the RMS actuator e�ort objective and the RMS regulation constraintin ����� � with the corresponding variance objective and constraint� i�e�� we squarethe objective and constraint functionals �rms u and �rms yp� Instead of consideringthe step response overshoot constraint in ����� � as a single functional inequality�we will view it as a family of constraints on the step response� one constraint foreach t � �� that requires the step response at time t not exceed ����
min�rms yp�H�� � ����
�step�t�H� � ���� t � �
�rms u�H�� �������
where �step�t is the a�ne functional that evaluates the step response of the ��� entryat time t�
�step�t�H� � s���t��
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15.4 AN OUTER APPROXIMATION VIA DUALITY 363
In this transformed optimization problem� the objective is quadratic and the constraints consist of one quadratic constraint along with a family of a�ne constraintsthat is parametrized by t � R��
This suggests the following generalized dual functional for the problem ��������
���u� �y� �s�
�� minH � H
�u�rms u�H�� � �y�rms yp�H�� �
Z �
�
�s�t��step�t�H� dt
� �������
where the �weights� now consist of the positive numbers �u and �y� along withthe function �s � R� � R�� So in equation ������� we have replaced the weightedsum of �a �nite number of� constraint functionals with a weighted integral over thefamily of constraint functionals that appears in ��������
It is easily established that �� is a convex functional of ��u� �y� �s� and thatwhenever �y � � and �s�t� � � for all t � �� we have
���� �y� �s�� �����y �
Z �
�
����s�t� dt � �pri
where �pri is the optimum value of ����� �� Thus� computing ���� �y� �s� yields alower bound on the optimum of ����� ��
The convex duality principle �equations ��������� � of section ���� suggests thatwe actually have
�pri � max�y � �� �s�t� � �
���� �y� �s�� �����y �
Z �
�
����s�t� dt
� �������
which in fact is true� So the optimization problem on the righthand side of �������can be considered a dual of ����� ��
We can compute ���� �y� �s�� provided we have a statespace realization withimpulse response �s�t�� The objective in ������� is an LQG objective� with theaddition of the integral term� which is an a�ne functional of H� By completingthe square� it can be recast as an H�optimal controller problem� and solved by �anextension of� the method described in section � � � Since we can �nd a minimizerH��� for the problem �������� we can evaluate a subgradient for ���
�sg��u� �y� �s� � ��u�rms u�H���
� � �y�rms yp�H���
� �
Z �
�
�s�t��step�t�H�
�� dt
�c�f� section ��������By applying a Ritz approximation to the in�nitedimensional optimization prob
lem �������� we obtain lower bounds on the righthand� and hence lefthand sidesof ��������
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364 CHAPTER 15 SOLVING THE CONTROLLER DESIGN PROBLEM
To demonstrate this� we use two Ritz sequences for �s given by the transferfunctions
�� � �� �i�s� �
s�
i
� ��i�s� �
�
s� �
i
� i � �� � � � � �������
We shall denote these two Ritz approximations �� ��� The solution of the �nitedimensional inner approximation of the dual problem ������� is shown in �gure ����for various values of N � Each curve gives lower bounds on the solution of ����� ��
RMS�u�
N
��R�
��I �
��Iwithout step spec�
� � � � � �� �� �� �� �� ������
�����
����
�����
����
�����
����
Figure ��� The optimum value of the �nite�dimensional inner approxi�mation of the dual problem ������ versus the number of termsN for the twodi�erent Ritz approximations ������� The dotted line shows the optimumvalue of RMS�u without the step response overshoot speci�cation�
The upper bounds from �gure ���� and lower bounds from �gure ���� are showntogether in �gure ������ The exact solution of ����� � is known to lie between thedashed lines� this band is shown in �gure ����� on a larger scale for clarity�
The best lower bound on the value of ����� � is ������ corresponding to theN � � f��g Ritz approximation of �������� The best upper bound on the valueof ����� � is ������� corresponding to the N � � �K�a�� Q� Ritz approximationof ����� �� Thus we can state with certainty that
���� � min�rms yp�H� � ����os�H��� � ���
�rms u�H� � ������
We now know that ����� is within ������ �i�e�� ��� of the minimum value of theproblem ����� �� the plots in �gure ���� only strongly hint that this is so�
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15.4 AN OUTER APPROXIMATION VIA DUALITY 365
RMS�u�
N
����K�a�� Q
����K�a�� Q
����K�d�� Q
����K�d�� Q
��I�
��I ���I
without step spec�
� � � � � �� �� �� �� �� ���
����
����
����
����
���
����
����
����
����
���
Figure ���� The curves from �gures ���� and ���� are shown� The solu�tion of each �nite�dimensional inner approximation of ������ gives an upperbound of the exact solution� The solution of each �nite�dimensional innerapproximation of the dual problem ������ gives a lower bound of the exactsolution� The exact solution is therefore known to lie between the dashedlines�
RMS�u�
N
�
�K�a�� Q���
�K�a�� Q
���
�K�d�� Q
����K�d�� Q
��I�
��I �
�� �� �� � �� �� �������
�����
����
�����
�����
����
�����
Figure ����� Figure ����� is shown in greater detail� The exact solutionof ������ is known to lie inside the shaded band�
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366 CHAPTER 15 SOLVING THE CONTROLLER DESIGN PROBLEM
We should warn the reader that we do not know how to form a solvable dualproblem for the most general convex controller design problem� see the Notes andReferences�
15.5 Some Tradeoff Curves
In the previous three sections we studied two speci�c optimization problems� andthe performance of several approximate solution methods� In this section we consider some related twoparameter families of design speci�cations and the sameapproximate solution methods� concentrating on the e�ect of the approximationson the computed region of achievable speci�cations in performance space�
15.5.1 Tradeoff for Example with Analytic Solution
We consider the family of design speci�cations given by
�rms yp�H� � �� �rms u�H� � �� ������
for the same example that we have been studying�Figure ���� shows the tradeo� curves for the �K�a�� Q� Ritz approximations
of ������� for three values of N � The regions above these curves are thus AN � Ais the region above the solid curve� which is the exact tradeo� curve� This �guremakes clear the nomenclature �inner approximation��
15.5.2 Tradeoff for Example with no Analytic Solution
We now consider the family of design speci�cations given by
D����� � �rms yp�H� � �� �rms u�H� � �� �os�H��� � ���� �������
Inner and outer approximations for the tradeo� curve for ������� can be computed using Ritz approximations to the primal and dual problems� For example� anN � � �K�d�� Q� Ritz approximation to ������� shows that the speci�cations in thetop right region in �gure ����� are achievable� On the other hand� an N � � � Ritzapproximation to the dual of ������� shows that the speci�cations in the bottomleft region in �gure ����� are unachievable� Therefore the exact tradeo� curve isknown to lie in the shaded region in �gure ������
The inner and outer approximations shown in �gure ����� can be improved bysolving larger �nitedimensional approximations to ������� and its dual� as shownin �gure ������
Figure ����� shows the tradeo� curve for ������� for comparison� The gapbetween this curve and the shaded region shows the cost of the additional stepresponse speci�cation� �The reader should compare these curves with those of�gure ����� in section ����� which shows the cost of the additional speci�cationRMS� �yp� � ���� on the family of design speci�cations ��������
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15.5 SOME TRADEOFF CURVES 367
RMS�yp�
RMS�u�
���N � �
���N � �
���N � ��
���exact
� ���� ��� ���� ��� ���� ����
����
���
����
���
����
���
Figure ����� The exact tradeo� between RMS�yp and RMS�u for theproblem ������ can be computed using LQG theory� The tradeo� curves
computed using three �K�a�� Q Ritz inner approximations are also shown�
RMS�yp�
RMS�u�
achievable
unachievable
� ���� ��� ���� ��� �����
����
���
����
���
����
Figure ����� With the speci�cation �os�H�� � ���� speci�cations on�rms yp and �rms u in the upper shaded region are shown to be achievable
by solving an N � � �K�d�� Q �nite�dimensional approximation of ������Speci�cations in the lower shaded region are shown to be unachievable bysolving an N � � � �nite�dimensional approximation of the dual of ������
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368 CHAPTER 15 SOLVING THE CONTROLLER DESIGN PROBLEM
RMS�yp�
RMS�u�
����K�d�� Q� N � �
����� N � �
� ���� ��� ���� ��� �����
����
���
����
���
����
Figure ����� From �gure ����� we can conclude that the exact tradeo�curve lies in the shaded region�
RMS�yp�
RMS�u�
����K�d�� Q� N � ��
����
�� N � ��
� ���� ��� ���� ��� �����
����
���
����
���
����
Figure ����� With larger �nite�dimensional approximations to ����� andits dual� the bounds on the achievable limit of performance are substantiallyimproved� The dashed curve is the boundary of the region of achievableperformance without the step response overshoot constraint�
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NOTES AND REFERENCES 369
Notes and References
Ritz Approximations
The term comes from Rayliegh�Ritz approximations to in�nite�dimensional eigenvalueproblems� see� e�g�� Courant and Hilbert �CH��� p����� The topic is treated in� e�g��section ��� of Daniel �Dan���� A very clear discussion of Ritz methods� and their conver�gence properties� appears in sections and � of the paper by Levitin and Polyak �LP����
Proof that the Example Ritz Approximations Converge
It is often possible to prove that a Ritz approximation �works�� i�e�� that AN � A �insome sense as N � �� As an example� we give a complete proof that for the four Ritzapproximations ������ of the controller design problem with objectives RMS actuatore�ort� RMS regulation� and step response overshoot� we have
�N
AN � A� ������
This means that every achievable speci�cation that is not on the boundary �i�e�� not Paretooptimal can be achieved by a Ritz approximation with large enough N �
We �rst express the problem in terms of the parameter Q in the free parameter represen�tation of Dstable�
���Q�� �rms yp�T� � T�QT� � k G� �G�Qk��
���Q�� �rms u�T� � T�QT� � k G� �G�Qk��
���Q�� �os��� ���T� � T�QT��� � ��T � k G� �G�Qkpk step � ��
where the Gi depend on submatrices of T�� and the Gi depend on the appropriate sub�matrices of T� and T�� �These transfer matrices incorporate the constant power spectraldensities of the sensor and process noises� and combine the T� and T� parts since Q isscalar� These transfer matrices are stable and rational� We also have G��� � �� since P�has a pole at s � ��
We have
A � fa j �i�Q � ai� i � �� �� �� for some Q � H�g �
AN �
�a
����� �i�Q � ai� i � �� �� �� for some Q �
NXi��
xiQi
�H� is the Hilbert space of Laplace transforms of square integrable functions from R� intoR�
We now observe that ��� ��� and �� are continuous functionals on H�� in fact� there is anM �� such that
���i�Q� �i� Q�� �MkQ� Qk�� i � �� �� �� ������
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370 CHAPTER 15 SOLVING THE CONTROLLER DESIGN PROBLEM
This follows from the inequalities�����Q� ��� Q�� � kG�k�kQ� Qk�������Q� ��� Q�� � kG�k�kQ� Qk�������Q� ��� Q�� � kG��sk�kQ� Qk��
�The �rst two are obvious� the last uses the general fact that kABkpk step � kA�sk�kBk��which follows from the Cauchy�Schwarz inequality�
We now observe that the sequence �s � ��i� i � �� �� � � �� where � � �� is complete� i�e��has dense span in H�� In fact� if this sequence is orthonormalized we have the Laplacetransforms of the Laguerre functions on R��
We can now prove ������� Suppose a � A and � � �� Since a � A� there is a Q� � H�
such that �i�Q� � ai� i � �� �� �� Using completeness of the Qi�s� �nd N and x� � RN
such that
kQ� �Q�Nk� � ��M�
where
Q�N��
NXi��
x�iQi�
By ������� �i�Q�
N � ai � �� i � �� �� �� This proves �������
The Example Ritz ApproximationsWe used the state�space parametrization in section ���� with
P std� �s � C�sI � A��B�
where
A �
���� � �� � �� � �
�� B �
����
�� C �
�� �� ��
��
The controllers K�a� and K�b� are estimated�state�feedback controllers with
L�a�est �
��������
�������������
�� K
�a�sfb �
��������� ������� �������
��
L�d�est �
��������
���������������
�� K
�d�sfb �
������� ������� �������
��
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NOTES AND REFERENCES 371
Problem with Analytic SolutionWith the Ritz approximations� the problem has a single convex quadratic constraint� anda convex quadratic objective� which are found by solving appropriate Lyapunov equations�The resulting optimization problems are solved using a standard method that is describedin� e�g�� Golub and Van Loan �GL��� p�������� Of course an ellipsoid or cutting�planealgorithm could also be used�
Dual Outer Approximation for Linear Controller DesignGeneral duality in in�nite�dimensional optimization problems is treated in the book byRockafellar �Roc���� which also has a complete reference list of other sources covering thismaterial in detail� See also the book by Anderson and Nash �AN��� and the paper byReiland �Rei���
As far as we know� the idea of forming �nite�dimensional outer approximations to a convexlinear controller design problem� by Ritz approximation of an appropriate dual problem�is new�
The method can be applied when the functionals are quadratic �e�g�� weighted H� normsof submatrices of H� or involve envelope constraints on time domain responses� We do notknow how to form a solvable dual problem of a general convex controller design problem�
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372 CHAPTER 15 SOLVING THE CONTROLLER DESIGN PROBLEM
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Chapter 16
Discussion and Conclusions
We summarize the main points that we have tried to make� and we discuss someapplications and extensions of the methods described in this book� as well as somehistory of the main ideas�
16.1 The Main Points
An explicit framework� A sensible formulation of the controller design problemis possible only by considering simultaneously all of the closedloop transferfunctions of interest� i�e�� the closedloop transfer matrix H� which shouldinclude every closedloop transfer function necessary to evaluate a candidatedesign�
Convexity of many speci�cations� The set of transfer matrices that meet adesign speci�cation often has simple geometry�a�ne or convex� In manyother cases it is possible to form convex inner �conservative� approximations�
E�ectiveness of convex optimization� Many controller design problems canbe cast as convex optimization problems� and therefore can be �e�ciently�solved�
Numerical methods for performance limits� The methods described in thisbook can be used both to design controllers �via the primal problem� and to�nd the limits of performance �via the dual problem��
16.2 Control Engineering Revisited
In this section we return to the broader topic of control engineering� Some of themajor tasks of control engineering are shown in �gure �����
373
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374 CHAPTER 16 DISCUSSION AND CONCLUSIONS
The system to be controlled� along with its sensors and actuators� is modeledas the plant P �
Vague goals for the behavior of the closedloop system are formulated as a setof design speci�cations �chapters ������
If the plant is LTI and the speci�cations are closedloop convex� the resultingfeasibility problem can be solved �chapters �������
If the speci�cations are achievable� the designer will check that the design issatisfactory� perhaps by extensive simulation with a detailed �probably nonlinear� model of the system�
System
Goals
Plant
Specs�
Feas�Prob�
OK�yes yes
nono
Figure ���� A partial �owchart of the control engineer�s tasks�
One design will involve many iterations of the steps shown in �gure ����� Wenow discuss some possible design iterations�
Modifying the Specifications
The speci�cations are weakened if they are infeasible� and possibly tightened if theyare feasible� as shown in �gure ��� � This iteration may take the form of a searchover Pareto optimal designs �chapter ���
System
Goals
Plant
Specs�
Feas�Prob�
OK�yes yes
nono
Figure ���� Based on the outcome of the feasibility problem� the designermay decide to modify �e�g�� tighten or weaken some of the speci�cations�
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16.2 CONTROL ENGINEERING REVISITED 375
Modifying the Control Configuration
Based on the outcome of the feasibility problem� the designer may modify thechoice and placement of the sensors and actuators� as shown in �gure ����� If thespeci�cations are feasible� the designer might remove actuators and sensors to seeif the speci�cations are still feasible� if the speci�cations are infeasible� the designermay add or relocate actuators and sensors until the speci�cations become achievable�The value of knowing that a given set of design speci�cations cannot be achievedwith a given con�guration should be clear�
System
Goals
Plant
Specs�
Feas�Prob�
OK�yes yes
nono
Figure ���� Based on the outcome of the feasibility problem� the designermay decide to add or remove sensors or actuators�
These iterations can take a form that is analogous to the iteration describedabove� in which the speci�cations are modi�ed� We consider a �xed set of speci�cations� and a family �which is usually �nite� of candidate control con�gurations�Figure ���� shows fourteen possible control con�gurations� each of which consists ofsome selection among the two potential actuators A� and A� and the three sensorsS�� S�� and S�� �These are the con�gurations that use at least one sensor� and one�but not both� actuators� A� and A� might represent two candidate motors for asystem that can only accommodate one�� These control con�gurations are partiallyordered by inclusion� for example� A�S� consists of deleting the sensor S� from thecon�guration A�S�S��
These di�erent control con�gurations correspond to di�erent plants� and therefore di�erent feasibility problems� some of which may be feasible� and others infeasible� One possible outcome is shown in �gure ����� nine of the con�gurationsresult in the speci�cations being feasible� and �ve of the con�gurations result inthe speci�cations being infeasible� In the iteration described above� the designercould choose among the achievable speci�cations� here� the designer can chooseamong the control con�gurations that result in the design speci�cation being feasible� Continuing the analogy� we might say that A�S�S� is a Pareto optimal controlcon�guration� on the boundary between feasibility and infeasibility�
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376 CHAPTER 16 DISCUSSION AND CONCLUSIONS
A�S�S�S�
A�S�S� A�S�S� A�S�S�
A�S� A�S� A�S�
A�S�S�S�
A�S�S� A�S�S� A�S�S�
A�S� A�S� A�S�
Figure ���� The possible actuator and sensor con�gurations� with thepartial ordering induced by achievability of a set of speci�cations�
A�S�S�S�
A�S�S� A�S�S� A�S�S�
A�S� A�S� A�S�
A�S�S�S�
A�S�S� A�S�S� A�S�S�
A�S� A�S� A�S�
D achievable
D unachievable
Figure ���� The actuator and sensor con�gurations that can meet thespeci�cation D�
Modifying the Plant Model and Specifications
After choosing an achievable set of design speci�cations� the design is veri�ed� doesthe controller� designed on the basis of the LTI model P and the design speci�cations D� achieve the original goals when connected in the real closedloop system If the answer is no� the plant P and design speci�cations D have failed to accurately represent the original system and goals� and must be modi�ed� as shown in�gure �����
Perhaps some unstated goals were not included in the design speci�cations� Forexample� if some critical signal is too big in the closedloop system� it should beadded to the regulated variables signal� and suitable speci�cations added to D� toconstrain the its size�
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16.3 SOME HISTORY OF THE MAIN IDEAS 377
System
Goals
Plant
Specs�
Feas�Prob�
OK�yes yes
nono
Figure ���� If the outcome of the feasibility problem is inconsistent withdesigner�s criteria then the plant and speci�cations must be modi�ed tocapture the designer�s intent�
As a speci�c example� the controller designed in the rise time versus undershoottradeo� example in section � ��� would probably be unsatisfactory� since our designspeci�cations did not constrain actuator e�ort� This unsatisfactory aspect of thedesign would not be apparent from the speci�cations�indeed� our design cannotbe greatly improved in terms of rise time or undershoot� The excessive actuatore�ort would become apparent during design veri�cation� however� The solution� ofcourse� is to add an appropriate speci�cation that limits actuator e�ort�
An achievable design might also be unsatisfactory because the LTI plant P isnot a su�ciently good model of the system to be controlled� Constraining varioussignals to be smaller may improve the accuracy with which the system can bemodeled by an LTI P � adding appropriate robustness speci�cations �chapter ���may also help�
16.3 Some History of the Main Ideas
16.3.1 Truxal’s Closed-Loop Design Method
The idea of �rst designing the closedloop system and then determining the controller required to achieve this closedloop system is at least forty years old� Anexplicit presentation of a such a method appears in Truxal�s ���� Ph�D� thesis !Tru��"� and chapter � of Truxal�s ���� book� Automatic Feedback Control
System Synthesis� in which we �nd !Tru��� p���"�
Guillemin in ��� proposed that the synthesis of feedback control systems take the form � � �
�� The closedloop transfer function is determined from the speci�cations�
� The corresponding openloop transfer function is found�
�� The appropriate compensation networks are synthesized�
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378 CHAPTER 16 DISCUSSION AND CONCLUSIONS
Truxal cites his Ph�D� thesis and a ���� article by Aaron !Aar��"�On the di�erence between classical controller synthesis and the method he pro
poses� he states �!Tru��� p�������"��
The word synthesis rigorously implies a logical procedure for the transition from speci�cations to system� In pure synthesis� the designer is ableto take the speci�cations and in a straightforward path proceed to the�nal system� In this sense� neither the conventional methods of servodesign nor the root locus method is pure synthesis� for in each case thedesigner attempts to modify and to build up the openloop system untilhe has reached a point where the system� after the loop is closed� willbe satisfactory�
� � � !The closedloop design" approach to the synthesis of closedloop systems represents a complete change in basic thinking� No longer is thedesigner working inside the loop and trying to splice things up so thatthe overall system will do the job required� On the contrary� he is nowsaying� �I have a certain job that has to be done� I will force the systemto do it��
So Truxal views his closedloop controller design method as a �more logical synthesispattern� �p �� than classical methods� He does not extensively justify this view�except to point out the simple relation between the classical error constants and theclosedloop transfer function �p ���� �In chapter � we saw that the classical errorconstants are a�ne functionals of the closedloop transfer matrix��
The closedloop design method is described in the books !NGK��"� !RF���ch�"� !Hor� x����"� and !FPW��� x���" �see also the Notes and References fromchapter on the interpolation conditions��
16.3.2 Fegley’s Linear and Quadratic Programming Approach
The observation that some controller design problems can be solved by numericaloptimization that involves closedloop transfer functions is made in a series of papersstarting in ���� by Fegley and colleagues� In !Feg�" and !FH�"� Fegley applieslinear programming to the closedloop controller design approach� incorporatingsuch speci�cations as asymptotic tracking of a speci�c command signal and anovershoot limit� This method is extended to use quadratic programming in !PF�BF�"� In !CF�" and !MF��"� speci�cations on RMS values of signals are included�A summary of most of the results of Fegley and his colleagues appears in !FBB��"�which includes examples such as a minimum variance design with a step responseenvelope constraint� This paper has the summary�
Linear and quadratic programming are applicable � � � to the design ofcontrol systems� The use of linear and quadratic programming frequently represents the easiest approach to an optimal solution and often
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16.3 SOME HISTORY OF THE MAIN IDEAS 379
makes it possible to impose constraints that could not be imposed inother methods of solution�
So� several important ideas in this book appear in this series of papers by Fegleyand colleagues� designing the closed�loop system directly� noting the restrictionsplaced by the plant on the achievable closed�loop system� expressing performancespeci�cations as closed�loop convex constraints� and using numerical optimizationto solve problems that do not have an analytical solution �see the quote above��
Several other important ideas� however� do not appear in this series of papers�Convexity is never mentioned as the property of the problems that makes e�ectivesolution possible� linear and quadratic programming are treated as useful toolswhich they apply to the controller design problem� The casual reader mightconclude that an extension of the method to inde�nite �nonconvex� quadratic pro�gramming is straightforward� and might allow the designer to incorporate someother useful speci�cations� This is not the case� numerically solving nonconvexQP�s is vastly more di�cult than solving convex QP�s �see section ���� ��
Another important idea that does not appear in the early literature on theclosed�loop design method is that it can potentially search over all possible LTIcontrollers� whereas a classical design method �or indeed� a modern state�spacemethod� searches over a restricted �but often adequate� set of LTI controllers� Fi�nally� this early form of the closed�loop design method is restricted to the designof one closed�loop transfer function� for example� from command input to systemoutput�
16.3.3 Q-Parameter Design
The closed�loop design method was �rst extended to MAMS control systems �i�e�� byconsidering closed�loop transfer matrices instead of a particular transfer function��in a series of papers by Desoer and Chen �DC��a� DC��b� CD��b� CD��� andGustafson and Desoer �GD��� DG��b� DG��a� GD���� These papers emphasizethe design of controllers� and not the determination that a set of design speci�cationscannot be achieved by any controller�
In his ��� Ph� D� thesis� Salcudean �Sal��� uses the parametrization of achiev�able closed�loop transfer matrices described in chapter � to formulate the controllerdesign problem as a constrained convex optimization problem� He describes many ofthe closed�loop convex speci�cations we encountered in chapters �� �� and discussesthe importance of convexity� See also the article by Polak and Salcudean �PS����
The authors of this book and colleagues have developed a program called qdes�which is described in the article �BBB���� The program accepts input writtenin a control speci�cation language that allows the user to describe a discrete�timecontroller design problem in terms of many of the closed�loop convex speci�cationspresented in this book� A simple method is used to approximate the controllerdesign problem as a �nite�dimensional linear or quadratic programming problem�
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380 CHAPTER 16 DISCUSSION AND CONCLUSIONS
which is then solved� The simple organization and approximations made in qdesmake it practical only for small problems� The paper by Oakley and Barratt �OB��describes the use of qdes to design a controller for a �exible mechanical structure�
16.3.4 FIR Filter Design via Convex Optimization
A relevant parallel development took place in the area of digital signal processing�In about ���� several researchers observed that many �nite impulse response �FIR��lter design problems could be cast as linear programs� see� for example� the arti�cles �CRR��� Rab�� or the books by Oppenheim and Schaefer �OS� x���� andRabiner and Gold �RG�� ch���� In �RG�� x����� we even �nd designs subject toboth time and frequency domain speci�cations�
Quite often one would like to impose simultaneous restrictions on boththe time and frequency response of the �lter� For example� in the designof lowpass �lters� one would often like to limit the step response over�shoot or ripple� at the same time maintaining some reasonable controlover the frequency response of the �lter� Since the step response is alinear function of the impulse response coe�cients� a linear program iscapable of setting up constraints of the type discussed above�
A recent article on this topic is �OKU����Like the early work on the closed�loop design method� convexity is not recognized
as the property of the FIR �lter design problem that allows e�cient solution� Nor isit noted that the method actually computes the global optimum� i�e�� if the methodfails to design an FIR �lter that meets some set of convex speci�cations� then thespeci�cations cannot be achieved by any FIR �lter �of that order��
16.4 Some Extensions
16.4.1 Discrete-Time Plants
Essentially all of the material in this book applies to single�rate discrete�time plantsand controllers� provided the obvious changes are made �e�g�� rede�ning stability tomean no poles on or outside the unit disk�� For a discrete�time development� thereis a natural choice of stable transfer matrices that can be used to form the �analogof the� Ritz sequence � ���� described in section �� �
Qijk�z� � Eijz��k���� � i � nu� � j � ny� k � � �� � � � �
�Eij is the matrix with a unit i� j entry� and all other entries zero�� which correspondsto a delay of k� time steps� from the jth input of Q to its ith output� Thus in theRitz approximation� the entries of the transfer matrix Q are polynomials in z���i�e�� FIR �lters� This approach is taken in the program qdes �BBB����
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16.4 SOME EXTENSIONS 381
Many of the results can be extended to multi�rate plant and controllers� i�e��a plant in which di�erent sensor signals are sampled at di�erent rates� or di�er�ent actuator signals are updated at di�erent rates� A parametrization of stabi�lizing multi�rate controllers has recently been developed by Meyer �Mey��� thisparametrization uses a transfer matrix Q�z� that ranges over all stable transfermatrices that satisfy some additional convex constraints�
16.4.2 Nonlinear Plants
There are several heuristic methods for designing a nonlinear controller for a non�linear plant� based on the design of an LTI controller for an LTI plant �or a familyof LTI controllers for a family of LTI plants�� see the Notes and References for chap�ter �� In the Notes and References for chapter �� we saw a method of designing anonlinear controller for a plant that has saturating actuators� These methods oftenwork well in practice� but do not qualify as extensions of the methods and ideasdescribed in this book� since they do not consider all possible closed�loop systemsthat can be achieved� In a few cases� however� stronger results have been obtained�
In �DL���� Desoer and Liu have shown that for stable nonlinear plants� there isa parametrization of stabilizing controllers that is similar to the one described insection ������ provided a technical condition on P holds �incremental stability��
For unstable nonlinear plants� however� only partial results have been obtained�In �DL��� and �AD���� it is shown how a family of stabilizing controllers can beobtained by �rst �nding one stabilizing controller� and then applying the resultsof Desoer and Liu mentioned above� But even in the case of an LTI plant andcontroller� this two�step compensation approach can fail to yield all controllersthat stabilize the plant� This approach is discussed further in the articles �DL��a�DL��b� DL����
In a series of papers� Hammer has investigated an extension of the stable factor�ization theory �see the Notes and References for chapter �� to nonlinear systems�see �Ham��� and the references therein� Stable factorizations of nonlinear systemsare also discussed in Verma �Ver����
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382 CHAPTER 16 DISCUSSION AND CONCLUSIONS
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Notation and Symbols
Basic Notation
Notation Meaning
f � � � g Delimiters for sets� and for statement grouping inalgorithms in chapter ��
� � � � � Delimiters for expressions�
f � X � Y A function from the set X into the set Y �
� The empty set�
� Conjunction of predicates� and�
k � k A norm� see page ��� A particular norm is indicatedwith a mnemonic subscript�
���x� The subdi�erential of the functional � at the point x�see page ����
�� Equals by de�nition�
� Equals to �rst order�
� Approximately equal to �used in vague discussions��
�
� The inequality holds to �rst order�
�X A �rst order change in X�
argmin A minimizer of the argument� See page ���
C The complex numbers�
Cn The vector space of n�component complex vectors�
Cm�n The vector space of m� n complex matrices�
EX The expected value of the random variable X�
�z� The imaginary part of a complex number z�
383
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384 NOTATION AND SYMBOLS
inf The in�mum of a function or set� The readerunfamiliar with the notation inf can substitute minwithout ill e�ect�
j A square root of � �lim sup The asymptotic supremum of a function� see page ���
Prob�Z� The probability of the event Z�
�z� The real part of a complex number z�
R The real numbers�
R� The nonnegative real numbers�
Rn The vector space of n�component real vectors�
Rm�n The vector space of m� n real matrices�
�i�M� The ith singular value of a matrix M � the square rootof the ith largest eigenvalue of M�M �
�max�M� The maximum singular value of a matrix M � thesquare root of the largest eigenvalue of M�M �
sup The supremum of a function or set� The readerunfamiliar with the notation sup can substitute maxwithout ill e�ect�
TrM The trace of a matrix M � the sum of its entries on thediagonal�
M � � The n� n complex matrix M is positive semide�nite�i�e�� z�Mz � � for all z � Cn�
M � � The n� n complex matrix M is positive de�nite� i�e��z�Mz � � for all nonzero z � Cn�
� � � The n�component real�valued vector � hasnonnegative entries� i�e�� � � Rn
��
MT The transpose of a matrix or transfer matrix M �
M� The complex conjugate transpose of a matrix ortransfer matrix M �
M��� A symmetric square root of a matrix M �M� � ��i�e�� M���M��� �M �
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NOTATION AND SYMBOLS 385
Global Symbols
Symbol Meaning Page
� A function from the space of transfer matrices to realnumbers� i�e�� a functional on H� A particularfunction is indicated with a mnemonic subscript�
��
� The restriction of a functional � to a�nite�dimensional domain� i�e�� a function from R
n toR� A particular function is indicated with amnemonic subscript�
���
D A design speci�cation� a predicate or boolean functionon H� A particular design speci�cation is indicatedwith a mnemonic subscript�
��
H The closed�loop transfer matrix from w to z� ��
Hab The closed�loop transfer matrix from the signal b tothe signal a�
��
H The set of all nz � nw transfer matrices� A particularsubset of H �i�e�� a design speci�cation� is indicatedwith a mnemonic subscript�
��
K The transfer matrix of the controller� ��
L The classical loop gain� L � P�K� ��
nw The number of exogenous inputs� i�e�� the size of w� ��
nu The number of actuator inputs� i�e�� the size of u� ��
nz The number of regulated variables� i�e�� the size of z� ��
ny The number of sensed outputs� i�e�� the size of y� ��
P The transfer matrix of the plant� �
P� The transfer matrix of a classical plant� which isusually one part of the plant model P �
��
S The classical sensitivity transfer function or matrix� ��� �
T The classical I�O transfer function or matrix� ��� �
w Exogenous input signal vector� ��
u Actuator input signal vector� ��
z Regulated output signal vector� ��
y Sensed output signal vector� ��
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386 NOTATION AND SYMBOLS
Other Symbols
Symbol Meaning Page
� A feedback perturbation� ��
� A set of feedback perturbations� ��
kxk The Euclidean norm of a vector x � Rn or x � Cn�i�e��
px�x�
��
kuk� The L� norm of the signal u� �
kuk� The L� norm of the signal u� ��
kuk� The peak magnitude norm of the signal u� ��
kukaa The average�absolute value norm of the signal u� ��
kukrms The RMS norm of the signal u� ��
kukss� The steady�state peak magnitude norm of the signal u� ��
kHk� The H� norm of the transfer function H� ��� �
kHk� The H� norm of the transfer function H� �� �
kHk��a The a�shifted H� norm of the transfer function H� ��
kHkhankel The Hankel norm of the transfer function H� ��
kHkpk step The peak of the step response of the transfer functionH�
��
kHkpk gn The peak gain of the transfer function H� ���
kHkrms�w The RMS response of the transfer function H whendriven by the stochastic signal w�
��
kHkrms gn The RMS gain of the transfer function H� equal to itsH� norm�
��� �
kHkwc A worst case norm of the transfer function H� ��
A The region of achievable speci�cations in performancespace�
��
AP � Bw� Bu�
Cz� Cy�Dzw�
Dzu�Dyw�Dyu
The matrices in a state�space representation of theplant�
��
CF�u� The crest factor of a signal u� �
Dz��� The dead�zone function� ���
I��H� The �entropy of the transfer function H� �
nproc A process noise� often actuator�referred� ��
nsensor A sensor noise� ��
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NOTATION AND SYMBOLS 387
Fu�a� The amplitude distribution function of the signal u� ��
h�t� The impulse response of the transfer function H attime t�
��
H�a�� H�b��
H�c�� H�d�
The closed�loop transfer matrices from w to z achievedby the four controllers K�a�� K�b�� K�c�� K�d� in ourstandard example system�
��
K�a�� K�b��
K�c�� K�d�
The four controllers in our standard example system� ��
P A perturbed plant set� �
P std� The transfer function of our standard example
classical plant��
p� q Auxiliary inputs and outputs used in the perturbationfeedback form�
��
s Used for both complex frequency� s � � � j� and thestep response of a transfer function or matrix�although not usually in the same equation��
Sat��� The saturation function� ���
sgn��� The sign function� ��
T�� T�� T� Stable transfer matrices used in the free parameterrepresentation of achievable closed�loop transfermatrices�
��
� A submatrix or entry of H not relevant to the currentdiscussion�
��
�� The minimum value of the function �� �
x� A minimizing argument of the function �� i�e���� � ��x���
�
Tv�f� The total variation of the function f � ��
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388 NOTATION AND SYMBOLS
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List of Acronyms
Acronym Meaning Page
�DOF One Degree�Of�Freedom ��
��DOF Two Degree�Of�Freedom ��
ARE Algebraic Riccati Equation ��
FIR Finite Impulse Response ���
I�O Input�Output ��
LQG Linear Quadratic Gaussian ���
LQR Linear Quadratic Regulator ���
LTI Linear Time�Invariant ��
MAMS Multiple�Actuator� Multiple�Sensor ��
MIMO Multiple�Input� Multiple�Output �
PID Proportional plus Integral plus Derivative �
QP Quadratic Program ���
RMS Root�Mean�Square ��
SASS Single�Actuator� Single�Sensor ��
SISO Single�Input� Single�Output ��
389
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390 LIST OF ACRONYMS
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Index
��relaxed problem� ���� ���kHk�� ��� ���� ���kHk�� ��� ���� ���� ���kHkhankel� ���� ���� ���kHkL� gn
� ���
kHkL� gn� ��
kHkpk gn� �� ���� ���� ��kHkpk step� �� ���� ��kHkrms�w� �� ���kHkrms gn� ��� ���kHkwc� �� ��kuk�� ��� ��kuk�� ��� ��kuk�� �� ��kukaa� �� �kukitae� ��kukrms� �� ����DOF control system� �� ���� �����DOF control system� ��
MAMS� ���
AAbsolute tolerance� ���Achievable design speci�cations� ��� �Actuator� �
boolean� �e ort� ��� �� ��� ���� ��� �� ����
�� ��e ort allocation� ���integrated� ��multiple� ��placement� �referred process noise� �� ��RMS limit� ��� �� ���saturation� ���� ���� ���� ���� ���selection� �� �signal� �
Adaptive control� ��� �� ��Adjoint system� ���
A�necombination� ���de�nition for functional� ���de�nition for set� ���
Amplitude distribution� �� ��Approximation
�nite�dimensional inner� ���nite�dimensional outer� ���inner� ��� ���� ���� ���outer� ��� ���
ARE� ��� ��� ���� ��� �� ���� ���argmin� ���a�shifted H� norm� ���Assumptions� ��� ��
plant� ��Asymptotic
decoupling� ��disturbance rejection� ��tracking� ��� ��
Autocorrelation� vector signal� ��
Average�absolutegain� ��� ���norm� �vector norm� �
Average poweradmittance load� ��resistive load� �
BBandwidth
generalized� ���limit for robustness� ���regulation� ���signal� ���� ���tracking� ���
Bilinear dependence� ��� �Bit complexity� ���Black box model� �Block diagram� �
405
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406 INDEX
Bode plot� ��� ���Bode�s log sensitivity� ���
MAMS system� ���Book
outline� ��purpose� ��
Booleanactuator� �algebra of subsets� ��sensor� �
Branch�and�bound� ��
CCACSD� ��� ��Cancellation
unstable pole�zero� ��Chebychev norm� ��Circle theorem� ��Classical
controller design� � ��� ��� ���� ��optimization paradigm� regulator� ��� ��
Closed�loopcontroller design� ��� ��� ��� ����
��convex speci�cation� ��design� �realizable transfer matrices� ��stability� ��� ��stability� stable plant� �system� ��transfer matrices achieved by stabi�
lizing controllers� ��� ��transfer matrix� ��
Commanddecoupling� ��following� ���input� ��interaction� ��signal� ��
Comparing norms� ��� ��Comparison sensitivity� ���Complementary sensitivity transfer func�
tion� ��Complexity
bit� ���controller� �information based� ���quadratic programs� ���
Computing norms� ���Concave� ���
Conservative design� ���Constrained optimization� ���� ���� ���
��Continuous�time signal� �� ��Control con�guration� �� �Control engineering� �Controllability
Gramian� ���� ��Control law�
irrelevant speci�cation� ��speci�cation�
Controller� ��DOF� �� ���� �� ���� �����DOF� ��adaptive� ��� �� ��assumptions� ��complexity� �� ��decentralized� �diagnostic output� ��digital� ��discrete�time� �� ���estimated�state feedback� ���� ���gain�scheduled� ��� ��H��optimal� ��H� bound� ���implementation� � �� ��LQG�optimal� � ��LQR�optimal� �minimum entropy� ���nonlinear� ��� �� ���� ���� ���observer�based� ���open�loop stable� ���order� ��� ��parameters� ��parametrization� ��PD� � ��� ��PID� � ��� ��rapid prototyping� ��state�feedback� �state�space� ��structure� ��
Controller design� �analytic solution� �challenges� ��classical� � ��� ��� ��convex problem� ��empirical� feasibility problem� �� ��goals� ��modern�� open vs� closed�loop� ��
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INDEX 407
problem� �� ��� ��software� ��state�space� trial and error� ��
Control processor� �� �� ��DSP chips� ��power� ��
Control system� ���DOF� �� ���� �����DOF� ����DOF MAMS� ��� ���architecture� �classical regulator� ��� ��continuous�time� �speci�cations� ��� �testing� �
Control theoryclassical� geometric� �optimal� ��� �state�space�
Convexanalysis� ���combination� ���controller design problem� ��de�nition for functional� ���de�nition for set� ���duality� ���functional� ���inner approximation� ��� ���� ����
���optimization� ���outer approximation� ��set� ���supporting hyperplane for set� ���
Convexitymidpoint rule� ���multicriterion optimization� ���Pareto optimal� ���performance space� ���
Convex optimization� ��� �����relaxed problem� ���� ���complexity� ��cutting�plane algorithm� ���descent methods� ���duality� ���ellipsoid algorithm� ���linear program� ���� ��� ���� ����
���� ���� ��quadratic program� ��� ��� ���stopping criterion� ��
subgradient algorithm� ���upper and lower bounds� ��
Convolution� ��Crest factor� ��Cutting�plane� ���
algorithm� ���algorithm initialization� ���deep�cut� ���
DDamping ratio� ���dB� ��Dead�zone nonlinearity� ���� ���Decentralized controller� �Decibel� ��Decoupling
asymptotic� ��exact� ���
Deep�cut� ���ellipsoid algorithm� ��
Descentdirection� ��� ���method� ���
Describing function� ���� ���Design
closed�loop� �conservative� ���FIR �lters� ���Pareto optimal� � ���� ��� ���procedure� ��trial and error� ��
Design speci�cation� ��� �achievable� ��� �as sets� ��boolean� ��comparing� ��consistent� �families of� �feasible� �hardness� �inconsistent� �infeasible� �nonconvex� ��ordering� ��qualitative� ��quantitative� �robust stability� ���time domain� ���total ordering� �
Diagnostic output� ��� �
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408 INDEX
Di erential sensitivity� � ��versus robust stability� ���� ���� ���
Digitalcontroller� ��signal processing� ��� ���
Directional derivative� ��Discrete�time
plant� ���signal� �� ��
Disturbance� �� �modeling� ��stochastic� ���� ���unknown�but�bounded� ���
Dualfunctional� ��� ���� ���� ���functional via LQG� ���objective functional� ��optimization problems� ���problem� ��� ���
Duality� ��� ���� ���� ���convex optimization� ���for LQG weight selection� ���gap� ���in�nite�dimensional� ��
EEllipsoid� ���� ���Ellipsoid algorithm� ���
deep�cut� ��initialization� ���
Empirical model� �Entropy� ���� ���� ���
de�nition� ���interpretation� ���state�space computation� ���
Envelope speci�cation� �� ���Error
model reference� ���nulling paradigm� ��signal� ��� ��� ��tracking� ��
Estimated�state feedback� ���� ��� ���controller� ���
Estimator gain� ���Exact decoupling� ���Exogenous input� �
FFactorization
polynomial transfer matrix� ���spectral� ��� ���
stable transfer matrix� ���Failure modes� � �� ���False intuition� ��� ���� ��� ��Feasibility problem� �� ���� �
families of� �RMS speci�cation� ���� ��
Feasibility tolerance� ���Feasible design speci�cations� �Feedback
estimated�state� ���� ��� ���linearization� ��� �output� �paradigm� ��perturbation� ���plant perturbations� ���random� ���state� �well�posed� ��� ���
Fictitious inputs� �Finite�dimensional approximation
inner� ��outer� ���
Finite element model� ��FIR� ���
�lter design� ���Fractional perturbation� ���Frequency domain weights� ��� ���Functional
a�ne� ���bandwidth� ���convex� ���de�nition� �dual� ��� ���� ���equality speci�cation� ���� ��general response�time� �inequality speci�cation� �� ���level curve� ��linear� ���maximum envelope violation� �objective� �of a submatrix� ���overshoot� ��quasiconcave� ���quasiconvex� ���� ��� ���� ���relative overshoot� ��relative undershoot� ��response�time� �rise time� �settling time� �subgradient of� ���sub�level set� ���
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INDEX 409
undershoot� ��weighted�max� ��� ���� ���� ���weighted�sum� �� ���� ���� ���
GGain� ��� ��� ���
average�absolute� ��� ���cascade connection� ��feedback connection� ���generalized margin� ���� ���margin� ��� ��� ���� ���peak� ���scaling� ��scheduling� �small gain theorem� ���variation� ���
Generalizedbandwidth� ���gain margin� ���response time� ���stability� ��� ���
Geometric control theory� �Global
optimization� ��� ��� ��positioning system� �� ��
Goal programming� ��Gramian
controllability� ���� ��observability� ���� ���� ��
G�stability� ��
HH� norm� ��
subgradient� ���H��optimal
controller� ��H� norm� ��� ���
bound and Lyapunov function� ��shifted� ���subgradient� ���
Half�space� ���Hamiltonian matrix� ���� �� ���Hankel norm� ���� ���� ���Hardness� �History
closed�loop design� �feedback control� ��� ���stability� ���
Hunting� ���
II�O speci�cation� ��� ��Identi�cation� �� ��� ��� ��Impulse response� ��� ��� ���� ��� ���Inequality speci�cation� ���Infeasible design speci�cation� �Information based complexity� ���Inner approximation� ��� ���� ���� ���
de�nition� ���nite�dimensional� ��
Input� �actuator� �exogenous� ��ctitious� �signal� �
Input�outputspeci�cation� ��� ��transfer function� ��
Integratedactuators� ��sensors� �� ��
Interaction of commands� ��Interactive design� ��Internal model principle� ���Internal stability� ��
free parameter representation� ��with stable plant� �
Interpolation conditions� �� ���� ������
Intuitioncorrect� ���� ���� ��� ���false� ��� ���� ��� ��
ITAE norm� ��
JJerk norm� ��� ���
LLaguerre functions� ��Laplace transform� ��Level curve� ��Limit of performance� ��� ��� ���� ����
��Linear
fractional form� ��� �functional� ���program� ���� ��� ���� ���� ����
���� ��time�invariant system� ��transformation� ��
Linearity consequence� ���� ���
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410 INDEX
Linearly ordered speci�cations� ���� ���Logarithmic sensitivity� ��
MAMS system� ���Loop gain� ��LQG� ��� ��� ��� ���
multicriterion optimization� ���optimal controller� � ��
LQG weightsselection� ���� ���trial and error� ��tweaking� ���� ��via dual function� ��
LQR� � �LTI� �� ��Lumped system� ��Lyapunov equation� ���� ���� ��� ���
��Lyapunov function� ��� ��
MMAMS system� ��
logarithmic sensitivity� ���Margin� ���Matrix
Hamiltonian� ���� �� ���Schur form� ���� �weight� ��� ��
Maximum absolute step� ���Maximum value method for weights� ��M �circle constraint� ���Memoryless nonlinearity� ���Midpoint rule� ���Minimax� ��Minimizer� ���
half�space where it is not� ���Model� �� ��� ��� ��
black box� �detailed� ���empirical� �LTI approximation� �physical� �� ��� ��reference� ���
Modi�ed controller paradigm� �for a stable plant� ���observer�based controller� ���
Multicriterion optimization� �convexity� ���LQG� ���� ���
Multipleactuator� ��sensor� ��
Multi�rate plants� ��� ���
NNeglected
nonlinearities� ���plant dynamics� ���� ��� ���
Nepers� ��Nested family of speci�cations� ���� ���
���Noise� �
white� ��� ���� ���� ���� �� ��Nominal plant� ���Nominal value method for weights� ��Nonconvex
convex approximation for speci�ca�tion� ��
design speci�cation� ��speci�cation� ��� ��
Noninferior speci�cation� Nonlinear
controller� ��� �� ���� ���� ���plant� ��� �� ���� ���� ���� ���
Nonlinearitydead�zone� ���describing function� ���memoryless� ���neglected� ���
Non�minimum phase� ���Nonparametric plant perturbations� ���Norm� ��
kHk�� ��� ���� ���� ��kHk�� ��� ���� ���� ���� ���kHkhankel� ���� ���� ���kHkL� gn
� ���
kHkL� gn� ��
kHkpk gn� �� ���� ���� ��kHkpk step� �� ���� ��kHkrms�w� �� ���kHkrms gn� ��� ���kHkwc� �� ��kuk�� ��� ��kuk�� ��� ��amusing� ��average�absolute value� �Chebychev� ��comparing� ��comparison of� ��computation of� ���� ��de�nition� ��� ��false intuition� ��
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INDEX 411
frequency domain weight� ��gains� ��� ��Hankel norm� ���� ���� ���inequality speci�cation� ���� ��ITAE� ��jerk� ��� ���maximum absolute step� ���MIMO� ���multivariable weight� ��� ��peak� �peak acceleration� ��peak gain� �� ���� ���� ��� ����
��peak�step� �� ��RMS� �RMS gain� ��� ���RMS noise response� �� ���scaling� ��� ��seminorm� ��shifted H�� ���signal� ��SISO� �snap� crackle and pop� ��stochastic signals� time domain weights� ��total absolute area� ��total energy� ��using an average response� ��using a particular response� ��using worst case response� ��vector average�absolute value� �vector peak� ��vector RMS� ��vector signal� ��vector total area� ��vector total energy� ��weight� ��� ��worst case response� �� ���
Notation� ��� ���Nyquist plot� ���
OObjective functional� �
dual� ��weighted�max� ��weighted�sum� �� ���
ObservabilityGramian� ���� ���� ��
Observer�based controller� ���� ���Observer gain� ���
Open�loopcontroller design� ��equivalent system� ���
Operating condition� ��Optimal
control� ��� �controller paradigm� �Pareto� � ���� ���� ���� ��� ���
���� ��Optimization
branch�and�bound� ��classical paradigm� complexity� ��constrained� ���convex� ���cutting�plane algorithm� ���descent method� ���� ��ellipsoid algorithm� ���feasibility problem� ���global� ��� ��linear program� ���� ��� ���� ����
���� ���� ��multicriterion� �� �� ���quadratic program� ��� ��� ���unconstrained� ���
Order of controller� ��� ��Outer approximation� ��� ���
de�nition� ���nite�dimensional� ���
Output� ��diagnostic� ��referred perturbation� ���referred sensitivity matrix� ��regulated� ��sensor� ��
Overshoot� ��� ��� ��
PP� ���Parametrization
stabilizing controllers in state�space����
Pareto optimal� � ���� ���� ��� ������� ��� ���� ��
convexity� ���interactive search� ��
Parseval�s theorem� ��� ���PD controller� � ��� ��Peak
acceleration norm� ��gain from step response� �
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412 INDEX
gain norm� �� ���� ���� ��� ���signal norm� �step� �� ��subgradient of norm� ��vector signal norm� ��
Performancelimit� ��� ��� ���� ��loop� ���speci�cation� �� ��
Performance space� �achievable region� ���convexity� ���
Perturbation� fractional� ���nonlinear� ���� ���output�referred� ���plant gain� ���plant phase� ���singular� ��step response� ��time�varying� ���� ���
Perturbed plant set� ���Phase
angle� ��margin� ���unwrapping� ��variation� ���
Physicalmodel� �units� ��
PID controller� ��� ��Plant� ��
assumptions� ��� ��discrete�time� ���failure modes� ���gain variation� ���large perturbation� ���multi�rate� ���neglected dynamics� ���� ��� ���nominal� ���nonlinear� ��� �� ���� ���� ���� ���parameters� ���perturbation feedback form� ���perturbation set� ���phase variation� ���pole variation� ���� ���� ���relative uncertainty� ��� ���� ���small perturbation� ��standard example� ��� ���� ��� ����
��� ���� ���� ���� ��� �������� ��� ��
state�space� ��� ���strictly proper� ��time�varying� ���� ���transfer function uncertainty� ���typical perturbations� ���
Poleplant pole variation� ���� ���zero cancellation� ��
Polynomial matrix factorization� ���Power spectral density� Predicate� ��Primal�dual problems� ���Process noise
actuator�referred� �Programmable logic controllers� �� ��
Qqdes� ��Q�parametrization� ���� ���� ��� ���
���Quadratic program� ��� ��� ���Quantization error� �Quasiconcave� ���Quasiconvex functional� ���� ��� ����
���relative overshoot� undershoot� �
Quasigradient� ���settling time� ���
RRealizability� ��Real�valued signal� �Reference signal� ��� ��Regulated output� ��Regulation� �� ��� ��� ��
bandwidth� ���RMS limit� ���
Regulatorclassical� ��� ��
Relativedegree� ��overshoot� ��tolerance� ��
Resource consumption� ��Response time� �
functional� �generalized� ���
Riccati equation� ��� ���� �� ���� ���Rise time� �
unstable zero� ���
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INDEX 413
Ritzapproximation� ���� ��� ���
RMSgain norm� ��� ���noise response� �� ���signal norm� �speci�cation� ���� ���vector signal norm� ��
Robustness� �bandwidth limit� ���de�nition for speci�cation� ���speci�cation� � ���� ���
Robust performance� ���� ���� ���Robust stability� ���� ���
classical Nyquist interpretation� ���norm�bound speci�cation for� ���versus di erential sensitivity� ���� ����
���via Lyapunov function� ��
Root locus� ��� ��� �� ��Root�mean�square value� �
SSaturation� ��� ���� ���� ���� ���� ���
actuator� ���� ���� ���Saturator� ���Scaling� ��� ��Schur form� ���� �Seminorm� ��Sensible speci�cation� ��� ���� �� ���
��Sensitivity
comparison� ���complementary transfer function� ��logarithmic� ��� ���matrix� ��output�referred matrix� ���step response� ��� ���transfer function� ��� ���versus robustness� ���� ���� ���
Sensor� �boolean� �dynamics� ���integrated� �� ��many high quality� ��multiple� ��noise� �� ��output� ��placement� �selection� �� �
Seta�ne� ���boolean algebra of subsets� ��convex� ���representing design speci�cation� ��supporting hyperplane� ���
Settling time� �� ��quasigradient� ���
Shifted H� norm� ���� ���Side information� ��� �� ��� ��� ��Signal
amplitude distribution of� �average�absolute value norm� �average�absolute value vector norm�
�bandwidth� ���� ���comparing norms of� ��continuous� �continuous�time� ��crest factor� ��discrete�time� �� ��ITAE norm� ��norm of� ��peak norm� �peak vector norm� ��real�valued� �RMS norm� �size of� ��stochastic norm of� total area� ��total energy� ��transient� �� ��� ��unknown�but�bounded� �� ���vector norm of� ��vector RMS norm� ��vector total area� ��vector total energy� ��
Simulation� �� ��Singing� ��Singular
perturbation� ��� ���structured value� ��value� ���� ���� ���� ���value plot� ���
Sizeof a linear system� ��of a signal� ��
Slew rate� ��� ���� ��limit� �
Small gainmethod� ���� ���� ���
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414 INDEX
method state�space connection� ��theorem� ��� ���
Snap� crackle and pop norms� ��Software
controller design� ��Speci�cation
actuator e ort� ��� ���� ��� ���circle theorem� ��closed�loop convex� ��� ��controller order� ��di erential sensitivity� ��disturbance rejection� ��functional equality� ���functional inequality� �� �� ���input�output� ��� ��jerk� ���linear ordering� ���� ���M �circle constraint� ���model reference� ���nested family� ���� �� ���nonconvex� ��� ��� ��� ���noninferior� norm�bound� ��on a submatrix� ���open�loop stability of controller� ���overshoot� ��� ��Pareto optimal� PD controller� ��peak tracking error� ���� ���performance� ��realizability� ��regulation� ��� ��� ��relative overshoot� ��rise time� �RMS limit� ��� ���� ���� ��robustness� ���� ���sensible� ��� ���� �� ��� ��settling time� �slew�rate limit� ���stability� ���step response� ���� ��step response envelope� �time domain� ���undershoot� ��unstated� ��well�posed�
Spectral factorization� ��� ��� ���Stability� ��
augmentation� �closed�loop� ��� ��controller� ���
degree� ���� ���� ���� ���� ���for a stable plant� ���free parameter representation� ��generalized� ��� ���historical� ���internal� ��modi�ed controller paradigm� �robust� ���speci�cation� ���transfer function� ��� ��� ��transfer matrix� ���� ��transfer matrix factorization� ���using interpolation conditions� ��
���via Lyapunov function� ��via small gain theorem� ��
Stabletransfer matrix� ��
Standard example plant� ��� ���� ������� ��� ���� ���� ���� ������� ���� ��� ��
tradeo curve� ��� ���� ���State�estimator gain� ��State�feedback� �
gain� ���� ��State�space
computing norms� ���� ��controller� ��parametrization of stabilizing con�
trollers� ���plant� ��small gain method connection� ��
Step response� �and peak gain� ��envelope constraint� �interaction� ��overshoot� ��� ��� �overshoot subgradient� ���perturbation� ��relative overshoot� ��sensitivity� ��settling time� ��slew�rate limit� ���speci�cation� ���� ��total variation� ��undershoot� ��
Stochastic signal norm� Stopping criterion� ���� ���
absolute tolerance� ���convex optimization� ��relative tolerance� ��
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INDEX 415
Strictly proper� ��Structured singular value� ��Subdi erential
de�nition� ���Subgradient� ���
algorithm� ���computing� ���de�nition� ���H� norm� ���H� norm� ���in�nite�dimensional� ���peak gain� ��quasigradient� ���step response overshoot� ���tools for computing� ���worst case norm� ���
Sub�level set� ���Supporting hyperplane� ���System
failure� �identi�cation� �� ��� ��� ��linear time�invariant� ��lumped� ��
TTechnology� �Theorem of alternatives� ���Time domain
design speci�cation� ���weight� ��
Tolerance� ���feasibility� ���
Totalenergy norm� ��fuel norm� ��variation� ��
Trackingasymptotic� ��bandwidth� ���error� �� ���� ��peak error� ���� ���RMS error� ���
Tradeo curve� ��� � ���� ���� ���interactive search� ��norm selection� ���rise time versus undershoot� ���RMS regulated variables� ��� ����
���standard example plant� ��� ���� ���surface� � ���
tangent to surface� ��tracking bandwidth versus error� ���
Transducer� ��Transfer function� ��
Chebychev norm� ��G�stable� ��maximum magnitude� ��peak gain� �RMS response to white noise� ��sensitivity� ���shifted� ���stability� ��� ��� ��uncertainty� ���unstable� ��
Transfer matrix� ��closed�loop� ��factorization� ���singular value plot� ���singular values� ���size of� ��stability� ���� ��� ��weight� ��
Transient� �� ��� ��Trial and error
controller design� ��LQG weight selection� ��
Tuning� �rules� � ��
Tv�f�� ��Tweaking� ��
UUnconstrained optimization� ���Undershoot� ��
unstable zero� ���Unknown�but�bounded
model� �signal� ���� ���
Unstablepole�zero cancellation rule� ��� ���transfer function� ��zero and rise time� ���zero and undershoot� ���
VVector signal
autocorrelation� ��
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416 INDEX
WWcontr� ���Wobs� ���Weight
a smart way to tweak� ���constant matrix� ��diagonal matrix� ��for norm� ��frequency domain� ��� ���H�� ���informal adjustment� ��L� norm� ���matrix� ��� ��� ��maximum value method� ��multivariable� ��nominal value method� �� ��selection for LQG controller� ����
���time domain� ��transfer matrix� ��tweaking� ��tweaking for LQG� ���� ��vector� �
Weighted�max functional� ��� ���� �������
Weighted�sum functional� �� ���� �������
Well�posedfeedback� ��� ���problem�
White noise� ��� ���� ���� ���� �� ��Worst case
norm� ��response norm� �� ��subgradient for norm� ���
ZZero
unstable� ���