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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

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

�� 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

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 ��

KN �x�

��

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

��

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� �

� �Qi �

s� �

� 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 ��

RMS�u�

��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�

��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 � ��

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�

��K�a�� 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��

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

��

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��

�violation

��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�

�error

��

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 ��

iterations

��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�

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��

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��

�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 �

��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 �

��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��

� �

�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 ��

To demonstrate this� we use two Ritz sequences for �s given by the transferfunctions

�� i�s� �

� ��i�s� �

s� �

� 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�

��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�

15.4 AN OUTER APPROXIMATION VIA DUALITY 365

RMS�u�

��K�a�� 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�

�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�

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 ��

15.5 SOME TRADEOFF CURVES 367

RMS�yp�

RMS�u�

��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 ��

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�

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

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 �

�� i�Q � ai� i � �� for some Q �

NXi��

�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 � ��

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�

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�

��

�� 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 �

��

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�

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 ��

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

Specs�

Feas�Prob�

OK�yes yes

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

Specs�

Feas�Prob�

OK�yes yes

Figure �� Based on the outcome of the feasibility problem� the designermay decide to modify �e�g�� tighten or weaken some of the speci�cations�

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

Specs�

Feas�Prob�

OK�yes yes

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�

A�S�S�S�

A�S�S� A�S�S� A�S�S�

A�S� A�S� A�S�

A�S�S�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�

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�

16.3 SOME HISTORY OF THE MAIN IDEAS 377

System

Specs�

Feas�Prob�

OK�yes yes

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�

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

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�

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��

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��

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�

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 �

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� ��

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��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� ��

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 � ��

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 ��

390 LIST OF ACRONYMS

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��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� �

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� ��

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� ��

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� ��

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� ��

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� ��

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� �

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� ��

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� ��

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� ��

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� ��

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� ��

unstable� ��

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