Plant Friendly Input Design: Convex Relaxation and Quality

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 56, NO. 6, JUNE 2011 1467

the �� norms are higher in general as expected, but for �� thenorm is actually lower than what is achieved when using default mode,which is the result of the nonconvexity of the problem combined withrandomized starting points.

VIII. CONCLUSION

A new algorithm for low order �� controller synthesis has beenproposed. The new algorithm has been evaluated and the results hasbeen compared with HIFOO. The conclusion is that the proposed algo-rithm has comparable performance and speed, but HIFOO has an edgefor higher order systems. For lower order systems (�10 states), e.g.,AC6, the proposed algorithm is much faster than HIFOO. When tryingto synthesize controllers for some high order systems (�20 states), itresulted in too big matrices for the proposed method to handle, whileaccording to [21], HIFOO is able to find controllers for AC10 (55 states),which is one of these systems. How to handle systems with higher di-mensions is something we are going to look into when developing theproposed method further. It would also be interesting to investigate ifthe calculation of the initial point can be done in a better way.

REFERENCES

[1] J. Doyle, “Guaranteed margins for LQG regulators,” IEEE Trans.Autom. Control, vol. AC-23, no. 4, pp. 756–757, Aug. 1978.

[2] G. Zames, “Feedback and optimal sensitivity: Model reference trans-formations, multiplicative seminorms, and approximate inverses,”IEEE Trans. Autom. Control, vol. AC-26, no. 2, pp. 301–320, Apr.1981.

[3] J. Doyle, K. Glover, P. Khargonekar, and B. Francis, “State-space solu-tions to standard� and� control problems,” IEEE Trans. Autom.Control, vol. AC-34, no. 8, pp. 831–847, Aug. 1989.

[4] A. Helmersson, “Methods for Robust Gain Scheduling,” Ph.D. disser-tation, Linköping Univ., Linköping, Sweden, Dec. 1995.

[5] D. Ankelhed, A. Helmersson, and A. Hansson, “A primal-dual methodfor low order � controller synthesis,” in Proc. IEEE Conf. DecisionControl, Shanghai, China, Dec. 2009, pp. 6674–6674.

[6] M. J. Todd, K. C. Toh, and R. H. Tütüncü, “On the Nesterov-Todddirection in semidefinite programming,” SIAM J. Optim., vol. 8, no.3, pp. 769–796, 1998.

[7] P. Gahinet and P. Apkarian, “A linear matrix inequality approach to� control,” Int. J. Robust Nonlin. Control, vol. 4, no. 4, pp. 421–448,1994.

[8] A. Helmersson, “On Polynomial Coefficients and Rank Constraints,”Dept. Autom. Control, Linköping Univ., Linköping, Sweden, Tech.Rep. LiTH-ISY-R-2878, 2009 [Online]. Available: http://www.con-trol.isy.liu.se/publications/doc?id=2119

[9] H. Lütkepohl, Handbook of Matrices. New York: Wiley, 1996.[10] J. Nocedal and S. Wright, Numerical Optimization, 2nd ed. New

York: Springer, 2006.[11] Y. Zhang, “On extending some primal-dual interior-point algorithms

from linear programming to semidefinite programming,” SIAM J.Optim., vol. 8, no. 2, pp. 365–386, 1998.

[12] M. J. Todd, “A study of search directions in primal-dual interior-pointmethods for semidefinite programming,” Optim. Methods Software,vol. 11, no. 1–4, pp. 1–46, 1999.

[13] F. Alizadeh, J. Haeberly, and M. Overton, “Primal-dual interior-pointmethods for semidefinite programming: Convergence rates, stabilityand numerical results,” SIAM J. Optim., vol. 8, no. 3, pp. 746–768,1998.

[14] K. Toh, M. Todd, and R. Tütüncü, “SDPT3 a Matlab software packagefor semidefinite programming, version 1.3,” Optim. Methods Software,vol. 11, no. 1–4, pp. 545–581, 1999.

[15] M. Fazel, H. Hindi, and S. Boyd, “A rank minimization heuristic withapplication to minimum order system approximation,” in Proc. Amer.Control Conf., Jun. 2001, pp. 4734–4739.

[16] S. Mehrotra, “On the implementation of a primal-dual interior pointmethod,” SIAM J. Optim., vol. 2, no. 4, pp. 575–601, 1992.

[17] F. Leibfritz, �� : COnstrained Matrix Optimization ProblemLibrary 2006 [Online]. Available: http://www.complib.de

[18] F. Leibfritz, “�� : COnstraint Matrix Optimization ProblemLibrary—A Collection of Test Examples for Nonlinear SemidefinitePrograms, Control System Design and Related Problems,” Dept. Math.,Univ. Trier, Trier, Germany, Tech. Rep., 2004.

[19] S. Gumussoy, D. Henrion, M. Millstone, and M. Overton, “Multiob-jective robust control with HIFOO 2.0,” in Proc. IFAC Symp. RobustControl Design, Haifa, Israel, Jun. 2009, pp. 144–149.

[20] D. Ankelhed, A. Helmersson, and A. Hansson, Additional Nu-merical Results for the Quasi-Newton Interior Point Method forLow Order H-Infinity Controller Synthesis Dept. Autom. Control,Linköping Univ., Linköping, Sweden, Tech. Rep. LiTH-ISY-R-2964,2010 [Online]. Available: http://www.control.isy.liu.se/publica-tions/doc?id=2313

[21] S. Gumussoy and M. Overton, “Fixed-order � controller designvia HIFOO, a specialized nonsmooth optimization package,” in Proc.Amer. Control Conf., 2008, pp. 2750–2754.

Plant Friendly Input Design:Convex Relaxation and Quality

Sridharakumar Narasimhan and Raghunathan Rengaswamy

Abstract—A common practice in a system identification exercise is to per-turb the system of interest and use the resulting data to build a model. Theproblem of interest in this contribution is to synthesize an input signal thatis maximally informative for generating good quality models while being“plant friendly,” i.e., least hostile to plant operation. In this contribution,limits on input move sizes are the plant friendly specifications. The resultingoptimization problem is nonlinear and nonconvex. Hence, the original plantfriendly input design problem is relaxed which results in a convex optimiza-tion problem. We formulate a SemiDefinite Programme using the theory ofgeneralized Tchebysheff inequalities to derive tight bounds on the qualityof relaxation. Simulations show that the relaxation results in more plantfriendly input signals.

Index Terms—Convex optimization, input design, semidefinite program-ming (SDP), system identification, Tchebycheff inequalities.

I. INTRODUCTION

Design of optimal experiments or perturbations that are maximallyinformative for purposes of parameter estimation has been well studiedin the statistics and systems engineering literature [1]–[3]. While theearly focus was largely on identification in open loop, the problemof control relevant input design has recently gained attention [4]–[6]with closed loop performance specifications posed in the frequency do-main. One common feature of some of the classical and latest resultsis that input design can be reformulated as convex optimization prob-lems which can be solved efficiently to determine globally optimal so-lutions [7].

The problem of designing “plant friendly” inputs that are least hos-tile to operating conditions has received attention from the process en-gineering community [6], [8]–[11]. Common considerations in plant

Manuscript received July 24, 2009; revised February 26, 2010 and August 24,2010; accepted February 01, 2011. Date of publication March 24, 2011; date ofcurrent version June 08, 2011. Recommended by Associate Editor F. Dabbene.

S. Narasimhan was with the Department of Chemical Engineering, ClarksonUniversity, Potsdam, NY 13699-5705 USA. He is now with the Indian Instituteof Technology, Chennai 600036, India (e-mail: [email protected]).

R. Rengaswamy is with the Department of Chemical Engineering, TexasTech. University, Lubbock, TX, 79409-3121 USA (e-mail: [email protected]).

Digital Object Identifier 10.1109/TAC.2011.2132290

0018-9286/$26.00 © 2011 IEEE

1468 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 56, NO. 6, JUNE 2011

friendly input design are to minimize input move sizes (to reduce wearand tear on valves and actuators), input and output amplitudes or ener-gies and experiment time or perform a closed loop test [9]. This con-tribution addresses one of the plant friendly requirements, viz., to keepinput move sizes small.

Literature on input design is largely devoted frequency domaindesign problems while plant friendliness constraints are specified inthe time domain. There has been a suggestion to solve a time domainD-optimal input design problem with an explicit move size constraint[10], though no further progress has been reported. Another possibility,though suboptimal, is to solve the frequency domain input designproblem for the optimal spectrum and then impose the plant friend-liness constraints when realizing the input in the time domain [12].Other approaches include keeping track of the number of changes forbinary signals [8]. Constrained optimization formulations that allowsimultaneous specification of time and frequency domain properties ofmulti-sine signals have been presented [13], [14]. The disadvantagesinclude computational difficulties arising from non-convexities, po-tentially large size of the optimization problem (since a large numberof amplitudes and phases are to be determined) and restriction to aparticular class of signals.

In this contribution, we pose a plant friendly D-optimal input designproblem, but solve a relaxed problem instead by constraining the sumof squares of the move sizes. When the input length is large, this isequivalent to constraining the variance of the input move size. This hasthe following advantages:

• The relaxed problem is shown to be a convex problem. In thefrequency domain, the relaxed constraint can be interpreted as aweighted constraint on the input.

• The plant friendliness constraint is affine in the power spectrumand hence this constraint can be readily appended to other designformulations, e.g., the finite dimensional spectrum parameteriza-tion [5], periodic input design [6] etc.

• The optimal solution of the relaxed problem provides a non-trivialupper bound on the solution of the original plant friendly D-op-timal design problem. This allows the user to study the trade-offsbetween input power, parameter estimation accuracy and plantfriendliness, (albiet in a relaxed manner) [11].

• The quality of this relaxation is determined in a novel, prob-abilistic manner using the theory of Tchebycheff inequalities.Standard Tchebycheff inequality formulations are shown to beapplicable only for short input sequences. Using the theory ofduality for general convex optimization problems, we formulatean appropriate SemiDefinite Programme (SDP) for longer inputsequences.

II. PROBLEM FORMULATION

We consider the following SISO system with input �� , output �� [3]:

��

��

��

�� (1)

where �� is the backward shift operator, ��

��

� ,��

��

� , ��

��

� , ��

��

� ,�� is a discrete time Gaussian White Noise sequence with zero meanand unit variance. We make the following assumptions:

• �� and �� have no zeroes on the closed unit disk.• �� and �� are co-prime.• There are no pole-zero cancellations.• The true system is in the model set.• The input has unit power.• The experiment time is large.Let � � �� , � � ��

�.Then � � ��

� is the overall vector of parameters to be esti-

mated. Suppose an identification experiment is performed with input�� resulting in the output �� . Let � be an unbi-ased, asymptotically efficient estimate of �. The quality of � can bedescribed by its covariance, which is simply ��

, where � is theFisher information matrix. For the above dynamic system, it has beenshown that � can be partitioned as [3]

� ��

��

(2)

where � is related to the �� parameters in � and dependenton the input. �� is related to the noise parameters and importantly, isindependent of the input. Hence, in an input design problem, it is perti-nent and sufficient to consider � . A classical input design problem isthe well-known D optimal experiment design. The plant-friendly inputdesign problem that we wish to solve is the following:

Problem 1 (Plant Friendly D-Optimal):

��

� ��

�

�

��

��

where is a user specified upper bound on the input move size. How-ever, the above problem is not known to be convex and solution of thesame is not trivial. Rather than solve the above, we relax the above con-straint and solve the following constrained problem instead:

Problem 2 (Relaxed Plant Friendly D-Optimal):

��

� ��

�

�

��

�

��

�

��

� � �

where � is a design parameter. Since is large, ��

��

� can be interpreted as the variance of themove size. A major advantage of the above formulation is that it canbe reformulated as a convex optimization problem by randomization[2], [15].

There are considerable advantages in reformulating the input designproblem in the frequency domain where the decision variable is thepower spectrum. The input is assumed to be a stationary process withpower spectrum (two-sided) �� defined on �� . �� is theauto-correlation of�� at lag � and forms a Fourier transform pair with��. Corresponding to the two-sided power spectrum, we define anequivalent one-sided power spectrum�� on �� . The relationshipbetween the two is as follows [4]: given �� defined on �� ,�� is defined such that:

�

��

��

�

�

��

�� (3)

where �� is a �� function on �� .We define an average information matrix related to � which can be

expressed as a function of the input spectrum in the following form byapplication of Parseval’s theorem [3]:

� � ��

�

� �

��

��

� ��

(4)

where each �� is a constant matrix and � is given by:

� �

�

�

��

��

�

�� (5)

Using Parseval’s theorem, the left hand side of the relaxed plantfriendliness constraint can be expressed in the frequency domain as

��

�

� �

�

��

��

�

�

�� (6)


and hence the above problem can be reposed in the frequency domainas follows:

Problem 3 (Frequency Domain Relaxed Plant Friendly D-Optimal):

��

� ��

� ��

��

��

��

Theorem 1: Problem 3 is a convex optimization problem (in thesense that the objective function is convex and the feasible set isconvex).

Proof: This follows from the fact that the set of feasible infor-mation matrices �� is convex [3] and � �� is a convexfunction on the set of positive semi-definite matrices [15]. The plantfriendliness constraint is clearly a convex constraint. Hence, the fea-sible set is convex and hence the optimization problem is convex.

III. SEMIDEFINITE PROGRAMMING FORMULATION

While Problem 3 is a convex optimization problem, the optimizationvariable �� is infinite dimensional. The theory of Tchebycheff sys-tems [1] and its applications to input design for dynamic system iden-tification was comprehensively studied by [3]. This idea was revisitedrecently [4] to solve a control relevant input design problem and by thecurrent authors [11] who solved a plant friendly input design problemin a multi-objective setting. We follow this partial correlation approachand reformulate Problem 3 as an SDP using the theory of Tchebycheffsystems. The resulting problem can be solved using software such asCVX, a freely available tool for specifying and solving standard convexprograms [16], [17].

Define �� , �� and �� as follows:

��

��

�

�

�� (7)

Let � � �� . It has been shown that� �� constitute a Tchebycheff system [3], [4]. Let � ��

� and let �� denote the conical hull of all feasible �. Inview of the above definitions

��

��

��

��

� �

��

��

�� (8)

where �� can be expressed in terms of the coefficients of �� and the original ��

�. Hence, any feasible information matrix can berepresented as a point in a moment space induced by a Tchebycheffsystem. The condition that a point belong to a moment space inducedby the Tchebycheff system of interest is known to be a Linear MatrixInequality (LMI):

Theorem 2: [1], [4] A point � � � belongs to�� iff the followingmatrix is positive semi-definite, i.e.

��

��

�� . . . � ��

. . . �. . . �

. . . �. . .

�� . . . � ��

� �� (9)

Problems involving minimization of � �� can be re-cast as an SDP [18]. However, since the CVX library includes suchfunctions, we do not attempt to rewrite the same as an LMI. The inputpower and plant friendliness constraints are now enforced and shownto be affine.

Since �� , it is a trigonometric polynomialof degree �� in �� and we denote it by ��

��

� � . Hence the constraint �� can be replaced by a linearequality as follows:

��

�

�

��

��

��

��

�� (10)

Likewise, using standard trigonometric identities, we replace the plantfriendliness constraint by a linear inequality

�� (11)

where ��

�� can be evaluated from �� . Problem 3 can becompletely parameterized in terms of �� and can be written asfollows:

Problem 4 (SDP):

��

��

��

��

��

��

Problem 4 is a standard convex optimization problem and can besolved using CVX [16], [17] for the optimal �� . Since the feasible setis the convex hull of all single frequency designs (a subset of �),straightforward application of Caratheodory’s theorem [1, pg 40]shows that corresponding to any continuous spectrum, there exists adiscrete spectrum with no more than � � � frequencies in the supportof �� , i.e., ��

�� , � � �� , � � �� .Hence a natural, but not the only choice for the optimal input is amulti-sine: ��

�� , where �� , �� ,

if �� and �� and �� chosen arbitrarily for all

other �. One technique to determine these frequencies �� and theassociated weights �� by solving convex optimization problems hasbeen proposed [4].

However, under certain conditions, it is possible to obtain a morecompact representation using the theory of Tchebcyheff systems [1],[3], [11]. Determining conditions which guarantee minimal designs isrelated to a similar problem in statistical experiment design [1], [3].

The �� can also be interpreted in terms of the partial auto-correlationsequence of the input, �� , � � �� . The input power constraintand the plant friendliness constraint specify �� . If the optimal�� lies on the boundary of��, there exists a unique design�� thatinduces �� [1] and hence the auto-correlation sequence is completelyspecified. On the other hand, if the optimal�� lies in the strict interior of��, then the auto-correlation sequence �� is only partially specifiedfor � � , where � �� and can be expressed asa linear function of the optimal ��. Further elements of the sequenceare not uniquely determined [5].

IV. TCHEBYCHEFF INEQUALITIES

Given a a random variable! with mean" and variance#�,$��!�"� � % � #�&%�. This bound also known as Tchebycheff’s in-equality is computed over all classes of distributions and known to betight [1]. Given a random vector ! � � and some knowledge ofthe moments of ! , the multivariate generalization seeks to determinetight bounds for $��! � ' , where ' � � [1]. Recent activity inthis area has focussed on solving this problem using semidefinite pro-gramming [19]. The question that we seek to answer is this: Given the�� that solve Probelm 4 or equivalently, the input auto-correlation se-quence (partial or complete), and the corresponding input sequence,( � �� , what is the probability that �� %?

A. Short Sequence Lengths

Assume that the plant friendly input design Problem 4 has beensolved and the auto-correlation sequence is only partially specified:


��

��. Defining � � �� , with � �

� �, the second order moments of � are thus known

��

��

�� ...

. . ....

��

�

� (12)

Define � � � �� which can berepresented by the following inequalities:

� ��

� � �� ...

. . ....

...� � � � � ��

��

� �� ...

. . ....

...� � � � ��

�

��...

��

...

...

� (13)

With knowledge of the second order moments of � , we wish to findtight upper and lower bounds on �� with no restriction onthe underlying distributions. Defining � � �� , we wish to solve

� ��

�

�� (14)

��

�

��

��

�

�� (15)

where � varies over all admissible probability (positive) measures. Wewill first discuss the optimization problem in (14). � is a convex set.Assuming � � �, the optimal value is related to the squared distanceof ��, to the set � under the norm induced by �� [19], i.e.

� �

�

��

� ��

�

� �� (16)

Remark 4.1: The above bound is only asymptotically achievable,i.e., there exist a sequence of random variables�� suchthat �� and �� and �� .

It has been shown that the dual of the problem in (15) is the followingSDP [7, pg 376] with strong duality holding true:

��

� � ��

� ��

� ��

��

� � ��

(17)

where � is the �th row of � and �,� , � are the optimization variables.Remark 4.2: In [7], the objective function and the quadratic form

have a linear and constant term. However, in this case, it can be safelyneglected as they are optimally 0 when the set is symmetric and whenthe mean is unspecified [1].

B. Long Sequence Lengths

If the auto-correlation sequence is completely specified, the SDP in(17) can be solved for any � . However, in other cases, the auto-cor-relation sequence is only partially specified. As before define � �� , with � � � �, and hence the moment matrix�� is only partially known. We are interested in solving the op-timization problem in (15) and (14), with � only partially known. Asbefore, when � � �, the supremum of the objective function in (14) is1, which is however, only asymptotically achievable. We now focus onthe problem in (15). It should be noted that the unknown or unspecified

�� cannot be arbitrarily chosen. Indeed, they must be chosen so that� ispositive definite (the well known moment completion problem) and thespectrum�� should induce��. The �� can be expressed as an affinefunction of known ��

� and unknown �� as follows:

��

�� ......

��

� �� (18)

where �� is a known constant vector and � is a constant matrix ofappropriate dimension. The ��

�� belong to a higher

dimensional moment space and hence have to satisfy the LMI in (9)

�� (19)

where �� are constant matrices of appropriate dimen-sion. In addition, define a � �� matrix � as follows:

� � ��

��

��

��

��

(20)

where Toeplitz[] is a Toepltiz matrix consisting of the indicated ele-ments on the diagonals. Hence, the above problem is rewritten as

��

��

�

��

�� (21a)

�� (21b)

�� (21c)

�� (21d)

�� (21e)

� � �� (21f)

As shown in the Appendix, the dual of the above problem (again withstrong duality holding true) is the following SDP which can be solvedefficiently:

��

��

� ��

� �

� ��

� ��

��

��

��

��

��

��

��

�� (22)

V. EXAMPLES

Example 1: Consider the following example:

� � �� !�

where !� is a white noise sequence with variance 1. We first solveProblem 4 to obtain optimal values of �� . As explained previously, theupper bound on �� is 1, which is achievable, but only asymp-totically. By solving the SDPs in (17) and (22), we get a lower bound onthe probability that the plant friendly constraint is satisfied in short and


TABLE ILOWER BOUNDS ON ��

long input sequences respectively. These bounds are calculated explic-itly for different plant-friendly input designs, with different values of �and � and presented in Table I. As � is reduced, the lower bound on theprobability that � � � gets larger and hence, it is worthwhile to con-sider the plant-friendly input design. For large � , the lower bound onthis probability is 0. While this may be discouraging, it should be notedthat a smaller value of � reduces the probability that the move size con-straint is violated. E.g., when � � � and � � ��, the probability that�� is at least 0.12. However, when � � ��, the probabilitythat �� is significantly higher, viz., 0.54. Although tight,the probability estimates are pessimistic in the context of input design.This is demonstrated through numerical simulations with inputs of thefollowing form: �� .When � � �, the maximum and minimum input move sizes were1.9508 and 1.2334, in a simulated experiment where � and � werechosen randomly and the experiment repeated 1000 times. When � ��, the corresponding figures were 1.2565 and 0.8482, thus demon-strating the effect of incorporating a plant friendliness constraint ex-plicitly. The trade-off is that the quality of the model estimated de-teriorates: when � � �, �� and when � � ��,�� .

VI. CONCLUSION

The bounds on the quality of the relaxed problem obtained by solvingTchebycheff inequalities, though tight are often pessimistic as the dis-crete distributions that achieve this bound may not find application inpractice [20]. In the context of input design, we may impose additionalconstraints to obtain more realistic estimates of the quality of relax-ation, e.g., the input is known to be periodic of known period (sum ofsines), the input is shaped by an FIR filter etc. It has been argued thatin identification of process plants, output constraints (on absolute mag-nitude, deviation, rate of change etc.) are more important than inputconstraints. The above methodology can be extended, in principle tooutput constraints. Other possible extensions could include partial orimperfect knowledge of the system under test.

APPENDIX

Definition 6.1: [21] Let � be a vector space, � be a normed space,� a convex subset of � and � be the positive cone in � . Assume that� contains an interior point. Let � be a convex real-valued functionalon � and � be a convex mapping from � into � and � be an affinemap of � into a finite dimensional normed space � , i.e., �� . Denote �� to mean that �� . Denote ��

and � � to be the dual spaces of � and � respectively, i.e., the space ofcontinuous linear functionals on � and � respectively. Denote �� to represent the value of the linear functional �� evaluated at� � � . Denote �� to be dual cone of � . Define the followingconvex optimization problem (in primal form) as follows:

�� (23)

If the infimum is finite, denote it by ��.

Definition 6.2: [21] Consider the convex optimization problemdefined above. The Lagrangian and dual functionals are definedas follows: �� ,��

�� . The dual optimization problem is:�� , subject to �� . If the maximum is finite, denote itby ��.

Theorem 3: [21, pg 236] Consider the convex optimizationproblem as defined above. Assume �� is finite and � � � isan interior point of � � �� and thereexists �� such that �� is an interior point of � and�� . Then �� , �� and �� such that�� and if a feasible �� is a solution of the primalproblem, it is a minimizer of the Lagrangian with ��

��

and �� . In addition, the optimal values of the primal anddual cost functions are the same, i.e., �� .

Consider the optimization problem in (21). Let be the space offinite signed measures which is a vector space. Let � � � � �� . The set of admissible (positive) probability measures with mea-

sure equal to 1 forms a convex subset of and the feasible set in� is convex. The normed vector space � is �� , where�� is the space of symmetric matrices of order �� ,� is �, where is total number of linear equalities in (21). To eachequality in the (21a) and (21b) we associate a dual variable �� , !� " �� . To each equality in (21c) we associate a dual variable ��,! � �� . To the LMI (21d) we associate positive semidef-inite symmetric matrix �� and to (21e) a scalar �. The positive conesin the dual spaces of � and �� are the positive orthant and theset of symmetric positive semi-definite matrices respectively. From theRiesz-Markov representation theorem for positive linear functionals[22, pg 87], the dual space of of is the space of scalar valued contin-uous functionals of compact support on � , ��

��. The non-neg-

ativity condition in the dual space of is # � �, # � ��

��. Wedecompose � as � � $ � , collect like terms and rewrite theLagrangian as follows:

� ��

��

�� # � �%

�

� ��

�� # � �%

��

��

&��' ��

��

��

�� (��

��

��

��

� ��&�

� ��!� "�� (�� )�� (24)


Recall that � � �� as �� vary. Since, they appear lin-early, the infimum is �� unless the coefficients of these quantities(the bracketed quantities) in (24) are identically 0. The dual problem isthen simply the maximum of � subject to � � � and �� being positivesemidefinite. After some algebraic manipulations, the dual problem canbe simplified to

� ��

��

��

��

��

��

��

� ��

��

��

Collecting �� as follows:

� �

��

......

......

��

�

The first of the two inequality constraints is equivalent to requiring thatthe following matrix be positive semidefinite

� �

� ��

From the definition of �, the quadratic form is non-negative whenever�� . This condition can be replaced by the following LMIs usingthe S-procedure [7]:

� �

� ��

� ��

��

Putting them together, we arrive at (22), which is the dual to (21). Forstrong duality to hold, we need the constraint qualification to be satis-fied, i.e., there is an point � in the strict interior of the positive cone, �and �� . Whenever �� is in the interior of��, one can alwaysfind a valid spectrum, appropriately chosen �� , anda valid probability measure � such that the LMI is strictly satisfied andthe linear equalities are satisfied.

REFERENCES

[1] S. Karlin and W. J. Studden, Tchebycheff Systems: With Applicationsin Analysis and Statistics. New York: Interscience, 1966.

[2] R. Mehra, “Optimal input signals for parameter estimation in dynamicsystems—Survey and new results,” IEEE Trans. Autom. Control, vol.AC-19, no. 6, pp. 753–768, Dec. 1974.

[3] M. Zarrop, “A Chebyshev system approach to optimal input design,”IEEE Trans. Autom. Control, vol. AC-24, no. 5, pp. 687–698, Oct.1979.

[4] R. Hildebrand and M. Gevers, “Identification for control: Optimal inputdesign with respect to a worst-case �-gap cost function,” SIAM J. Con-trol Optim., vol. 41, no. 5, pp. 1586–1608, 2003.

[5] H. Jansson and H. Hjalmarsson, “Input design via LMIs admitting fre-quency-wise model specifications in confidence regions,” IEEE Trans.Autom. Control, vol. 50, no. 10, pp. 1534–1549, Oct. 2005.

[6] B. Cooley, J. Lee, and S. Boyd, “Control-relevant experiment design:A plant-friendly, LMI-based approach,” in Proc. Amer. Control Conf.,Philadelphia, PA, 1998, pp. 1240–1244.

[7] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge,U.K.: Cambridge Univ. Press, 2004.

[8] R. Parker, D. Heemstra, F. Doyle, III, R. Pearson, and B. Ogunnaike,“The identification of nonlinear models for process control using tai-lored “plant-friendly” input sequences,” J. Process Control, vol. 11, pp.237–250, 2001.

[9] D. Rivera, H. Lee, M. Braun, and H. Mittelmann, “Plant friendlysystem identification: A challenge for the process industries,” in Proc.SYSID’03, Rotterdam, The Netherlands, 2003, pp. 917–922.

[10] Q. Zheng, “A Volterra Series Approach to Nonlinear Process Con-trol and Control-Relevant Identification,” Ph.D. dissertation, Univ. ofMaryland, College Park, 1995.

[11] S. Narasimhan and R. Rengaswamy, “Multi-objective optimal inputdesign for plant friendly identification,” in Proc. Amer. Control Conf.,Seattle, WA, 2008, pp. 1304–1309.

[12] K. Godfrey, Perturbation Signals for System Identification. Engle-wood Cliffs, NJ: Prentice-Hall, 1993.

[13] D. Rivera, H. Lee, H. Mittelmann, and M. Braun, “High purity distil-lation: Using plant-friendly multisine signals to identify a strongly in-teractive process,” IEEE Control Syst. Mag., vol. 27, no. 5, pp. 72–89,2007.

[14] D. Rivera, H. Lee, H. Mittelmann, and M. Braun, “Constrained multi-sine input signals for plant-friendly identification of chemical processsystems,” J. Process Control, vol. 19, no. 4, pp. 623–635, 2009.

[15] V. V. Fedorov, Theory of Optimal Experiments. New York: Aca-demic Press, 1972.

[16] M. Grant and S. Boyd, CVX: Matlab Software for Disciplined ConvexProgramming, Version 1.21 [Online]. Available: http://cvxr.com/cvxJan. 2011

[17] M. Grant and S. Boyd, “Graph implementations for nonsmooth convexprograms,” in Recent Advances in Learning and Control (Tribute to M.Vidyasagar), V. Blondel, S. Boyd, and H. Kimura, Eds. New York:Springer Verlag, pp. 95–110.

[18] S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Linear MatrixInequalities in System and Control Theory. Singapore: SIAM, 1994.

[19] D. Bertsimas and I. Popescu, “Optimal inequalities in probabilitytheory: A convex optimization approach,” SIAM J. Optim., vol. 15, pp.780–804, 2005.

[20] I. Popescu, “A semidefinite programming approach to optimal momentbounds for convex classes of distributions,” Math. Oper. Res., vol. 30,no. 3, pp. 632–657, 2005.

[21] D. Leunberger, Optimization by Vector Space Methods. New York:Wiley, 1969.

[22] S. Kantorovitz, Introduction to Modern Analysis. Oxford, U.K.: Ox-ford Univ. Press, 2003.

Plant Friendly Input Design: Convex Relaxation and Quality

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