Systematic design of weighting matrices for the mixed sensitivity problem M
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Systematic design of weighting matrices for the mixed sensitivity problem M

G Ortega FR Rubio Departamento Ingenier a de Sistemas y Automa tica Escuela Superior de Ingenieros Universidad de Sevilla 41092Sevilla Spain Received 4December 2002 received in revised form pril 2003 accepted 2 May 2003 Abstract systematic design

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Systematic design of weighting matrices for the mixed sensitivity problem M

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Systematic design of weighting matrices for the mixed sensitivity problem M.G. Ortega , F.R. Rubio Departamento Ingenier a de Sistemas y Automa tica, Escuela Superior de Ingenieros, Universidad de Sevilla, 41092Sevilla, Spain Received 4December 2002$ received in revised form % &pril 2003$ accepted %2 May 2003 Abstract & systematic design of weighting matrices for the mixed sensitivity problem is presented in this paper. Once a nominal model has been chosen, an initial design of the weighting matrices based on the multiplicative output uncertainty is proposed. The final

weighting matrices *which permit that an appropriate closed loop behavior is achieved+are obtained by tuning just one parameter for each output of the system. & multivariable control of two temperatures of a pilot plant *which constitutes a typical example of an industrial process+ is included as an application where the validity of the proposed methodology has been tested. 2003 -lsevier .td. &ll rights reserved. Keywords: control$ Mixed sensitivity problem$ Weighting matrix$ 1rocess control 1. Introduction The control systems community has been paying increasingattentiontothe

controltheorythroughout the past decade because of the robustness characteristics supplied by its controllers. This control theory is based on a more realistic hypotheses with respect to the restrictionsimposedonthedifferentsignalswhichappear in a control scheme. 3n this case, no special statistical distribution is imposed, and it is only supposed that the signal energy are bounded. This article deals with the design of appropriate weighting functions for the mixed sensitivity problem, which takes place in the frame of the synthesis of controller. Many articles have been published during

the last few years in which a particular control applica5 tionhasbeensolvedbymeansofthisapproach*see 6%–%08 as representative examples+. /owever, despite the importance of the design of the weighting functions, no methodology appears in these papers regarding the chosen functions. Furthermore, most of them confirm that these functions are tuned through a trial and error process. On the other hand, there are some publications where the main aim is to expose some considerations about the design of these functions *see, for example, 6%%–%98 +. /owever, it is necessary to have a wide

knowledge of control theory to understand them, and above all to use them. There are many industrial applications where the engineers in charge to control them do not possess an advanced knowledge about control theory. This is the reason why most of these applications are controlled by means of 13D control laws which regulate S3SO process loops. 3n these cases, it may be better to apply advanced control laws that improve the behavior of the controlled system. :evertheless, being able to tune these con5 trollers in an easy and intuitive mode, through no more

thanoneortwoparameterswhichwererepresentativeof theclosedloopsystemperformance,couldbeinteresting. 3n this paper, a simplified design methodology of the weighting functions is proposed. Thereby, after an initial adjustment based on the estimated uncertainty, the system performance will basically depend on the value of adimensional parameters ;%, ... , no. out5 puts+. These parameters have been normalized so that they should adopt positive values for a usual behavior. =alues equal to zero are proposed for initial selection. 09@95%@24A03AB 5 see front matter 2003 -lsevier .td. &ll rights

reserved. doiC%0.%0%DAS09@95%@24*03+0003@50 Eournal of 1rocess Fontrol %4*2004+ 99–99 www.elsevier.comAlocateAjprocont * Forresponding author. Tel.C G34595@4459H3@D$ faxC G34595@445 9H@40. E-mail addresses: *M.G. Ortega+, rubioIesi. *F.R. Rubio+.
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&s these parameter values increase, faster *although more oscillatory+ responses are obtained. Therefore, the final selection of these parameter values would be a compromised solution between quality and speed of the response. The remainder of this article is organized as followsC the mixed sensitivity

problem in the frame of the control is presented in the first part. This is followed by an exposition of how each weighting matrix has to be calculated and a summary of the design methodology. Finally, a case study is exposed in order to check the validity of the proposed methodology. 2. The mixed sensitivity problem The feedback controller design problem can be for5 mulated as an optimization problem, which can be posed under the general configuration shown in Fig. % 3n this figure, + is the generalized plant, + is the controller, are the control signals, the measured

variables, the exogenous signals and are the so5 called error variables. The optimal control problem with this con5 figuration consists of computing a controller such that the ratio between the energy of the error vector and the energy of the exogenous signals vector is mini5 mized. This optimal problem is not solved yet, but a solution exists for the suboptimal problem *see 6%98 for the continuous time domain case and 6208 in case of discrete time domain+. Thereby, the value of the energy ratio is decreased as much as possible by means of an iteration process. This is the synthesis

process which has been implemented in various well5known software packages such as 62%8 or 6228 & configuration for building up the generalized plant is the mixed sensitivity problem *see, for example, 6%H8 +, which is exposed in Fig. 2 . 3n this case, the expression of the resulting closed loop transfer function + is as followsC where + is the output sensitivity transfer matrix and + is the output complementary sensitivity transfer matrixC )s Ks )s Ks )s Ks The terms + and + constitute their respective weighting matrices, which allow the range of frequencies of main importance for the

corresponding closed5loop transfer matrix to be specified. &s it is known, shaping + is desirable for tracking problems, noise attenua5 tion and for robust stability with respect to multi5 plicative output uncertainties. On the other hand, since + relates the error signals with references and dis5 turbances 6238 , shaping the sensitivity function will allow the performance *in terms of command tracking and disturbance attenuation+ of the system to be con5 trolled. Therefore, since the controller is obtained from the generalized plant, the synthesis problem with this con5 figuration

is reduced to the design of a nominal model + and some appropriate weighting matrices which will impose the control specifications. Once these designs have been carried out, the generalized plant can be built up, and consequently the controller can be computed by means of a synthesis algorithm imple5 mented in a computer. 3. Design of the weighting matrices &ccording to what has been stated in the introduc5 tion, this paper supplies some design rules for the weighting matrices which automate the synthesis pro5 cess of controllers through the mixed sensitivity problem. Steps to be

followed for an appropriate design of each term of the generalized plant are exposed in the following section. Fig. %. General formulation of the control problem. Fig. 2. mixed sensitivity configuration. 90 M.). ,rtega, -... .u/io01ournal o2 Process 3ontrol 14 420045 89–98
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8.1. Selection o2 t9e nominal plant and estimating uncertainties The synthesis of a robust controller is based on the knowledge of the system uncertainty. 3n this paper, the unstructured multiplicative output description of the uncertainty is used, where the choice of the nominal model is the

first step to estimate it. & low order model is proposed as the nominal plant. This selection yields a simplified nominal system, which benefits the calculus of the controller. /owever, an inconvenience to take into account is that the resulting uncertainty region may be widespread. Once the nominal model + is chosen, the multi5 plicative output uncertainty can be estimated as followsC )s )s ... where stands for the different nonnominal systems at each operating point where the controller is needed to work properly. Thereby, this calculus must be repeated as many times

as the number of nonnominal models are being considered. Jy definition, the multi5 plicative uncertainty means the percentage of ignorance of the plant at each frequency. Ksually, this percentage increases as the frequency does, and there always exists a frequency value that starting from this one, the system ignorance is complete, that is, the value of the multi5 plicative uncertainty is greater than one. 8.2. Design o2 t9e weig9ting matrix Since the class of uncertainty employed corresponds to the multiplicative output one, its associated robust stability condition is given by the

following expression 6238 where + is the output complementary sensitivity function, and +and + matrices are such that they normalize the estimated uncertainty, that isC 3f the value of + is set to the identity and +is recalled as +, the resulting robust stability condi5 tion has the following expressionC sup 3t can be seen how the term + is the same that appears in the mixed sensitivity models where the complementary sensitivity function is involved. There5 fore, to diminish the infinity norm in the mixed sensi5 tivity problem implies making the infinity norm of + small, and so,

making the system robust against this uncertainty class. The proposed design of + consists of a square diagonal matrix with all its diagonal elements with the same transfer functionC diag such that the matrix dimension is equal to the number of system outputs. The transfer function diag + must be stable, minimum phase and with module greater than the maximum singular value of the uncertainty pre5 viously calculated for each nonnominal model and fre5 quency, that is, Tdiag !; Moreover, taking into account that + must weight to the complementary sensitivity function, it is desirable that the

module of diag + has a high value in order to impose that + has a small gain at high fre5 quencies. :otice that since + is diagonal, its singular values for each frequency are equal to the module of the transfer function diag +, that is, diag ... Fonsequently, this selection of + ensures the closed loop robustness respect to any multiplicative error in the setC Tdiag Tdiag no and obviously, all calculated uncertainties are elements of this set, that is, 2 8.8. Design o2 t9e weig9ting matrix The weighting matrix + will be employed to impose performance conditions to the system.

&s it is known, the sensitivity function is especially interesting from the control point of view as it has properties that characterize both the quality and the speed of the closed loop temporal response *see, for example, 6%H8 +. Jearing in mind that the matrix + must weight the sensitivity function, its design is proposed to be a square diagonal matrix of transfer functionsC diag %% ... ii ... qq where the matrix dimension is again the number of the systemLs outputs. -ach diagonal element ii + must be M.). ,rtega, -... .u/io01ournal o2 Process 3ontrol 14 420045 89–98 9%
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designed bearing in mind that its inverse should shape to ii as an upper bound, where ii + is the th element of the diagonal of +. Thereby, these ele5 ments are proposed with the following expressionC ii :otice that since the computed controller tries to decouple the outputs, the resulting matrix function will be almost diagonal. :ext it will be shown how to design each one of these parametersC is the function gain at high frequency. &s it is well known, a characteristic value for the max5 imum peak of the magnitude of ii + is about 2 if an acceptable system behavior is required. Thereby,since

ii mustimposeanupper bound on ii , it is desirable that the gain of ii at high frequency, that is % = , is about 2. Fonsequently, an appropriate value for each should be approximately about 0.@. is the function gain at low frequency. This value supposes an upper bound on the allowed steady state error. From the control point of view, it would be desirable to get null steady state errors, which implies adopting values for equal to zero. /owever, this selection would give numeric problems in the synthesis algo5 rithm because the augmented plant would have null values on the imaginary axis. &n

appro5 priate small value for these parameter may be between %0 and %0 depending on the application. is the crossover frequency of the function. These values indicate the minimum bandwidth of the transfer functions which are weighted. &s an initial value a decade below the crossover fre5 quency of the function diag + previously designed is proposed. This frequency will be called in the following as . &ccording to the proposed design for +, this frequency must be lower than the one at which the maximum singular value of the uncertainty gets the unity value. This initial selection yields slow

responses and rarely oscillatory ones. Different experiments showed the convenience of varying the value of in a logarithmic way. 3n this sense, it is pro5 posed to express this value as the following expressionC %0 where the parameter is employed to vary the value of once the value of has been obtained. The initial value of is obtained for equal to zero, while a value of this parameter equal to one shows that is equal to Therefore, the final selection of this frequency is determined by an adimensional parameter where the value must be higher as the desired response speed

increases. 4. Design methodology Once it has been shown how to design each weighting matrix, in following section the steps to follow to syn5 thesize the final controller is summarized. %. Fhoose a nominal model with a low order. 2. -stimate the multiplicative output uncertainty of the system with respect to the chosen nominal model. 3. Design the weighting matrix + as exposed in Section 3.2 4. Design an initial weighting matrix +as exposed in Section 3.3 with ;0 and with standard values of both and @. Juildup theaugmentedplant +andsynthesize an initial controller +. D. Design + again by

adjusting the values of until obtaining the desired behavior of the tem5 poral response. Finally, it is convenient to remember the importance of the systemscaling as a previous stage to the controller design. Jesides avoiding some numeric problems *for example, decreasing the condition number of the sys5 tem+, in the case of multivariable systems, scaling per5 mits that all signals are comparable in magnitude, and particularly the errors at the different channels. Jearing in mind that the control attenuates the energy of the error vector , all its components will have the same relative

weight if the system is scaled. 5. A case study 3n this section the validity of the proposed design methodology of controllers is tested in a pilot plant, which constitutes a typical example of industrial pro5 cess. 5.1. System description The pilot plant consists of industrial components, such as a water tank, a resistor to heat the water, a heat exchanger, pumps and different pneumatic valves. & diagram and a photograph of the plant which shows its main elements as well as the localization of the various 92 M.). ,rtega, -... .u/io01ournal o2 Process 3ontrol 14 420045 89–98

instruments are presented in Fig. 3 . The plant is used nowadays as a test bed of control strategies *see 624,2@8 which can be implemented on an industrial SF&D& which is connected to it. The aim of this application is to control some tem5 peratures, as is usual in any industrial process. 3n this case, they are the tank temperature * TT + and the out5 put exchange temperature * TT + *see Fig. 3b +. The con5 trol variables are the heat supplied to the tank water by a resistor * +, and the water flow in the recirculated circuit * -T +, which is regulated by the aperture of the valve

. & heat exchanger reduces the temperature of the recirculation water driven by the pump, using a constant flow of cold water *temperature TT +$ the other valves are closed. This constitutes a M3MO system with two inputs and two outputs. & robust controller is needed to keep a good perfor5 mance of the temperature values due to the different system behaviors depending on the operating point. The heat exchanger is the main cause of these different behaviors, where the efficiency may change drastically from one operating point to another. This inserts a high nonlinearity

into the system, that will be dealt like an unstructured uncertainty. 5.2. 3ontroller design Steps to synthesize the controller using a mixed sensitivity model are related in this section. Since the system is multivariable, a scaling of the plant will be made as a previous step to the design of the weighting matrices. 5.2.1. ,/taining a linear model .inear models of the system were obtained by identi5 fication at several operating points by supplying a pseudo random binary sequence to both actuators, i.e. the water flow -T and the resistor . 3n this plant the working space is

bounded due to physic conditions. This working space is shown in Fig. 4 , where the minimum and maximum values of the allowed control variables for a proper behavior of the plant can be observed. Taking into account this working space, the nominal operating point * NP + has been chosen with values of ;D0O and ;90O. This selection yields equilibrium temperatures of TT and TT equal to 3D.H and @D.3 F respectively. ¬her four points are Fig. 3. 1ilot plant Fig. 4. Working space of the control variables. M.). ,rtega, -... .u/io01ournal o2 Process 3ontrol 14 420045 89–98 93
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in Fig. 4 , which correspond to different working points around the nominal one. These points will be used for the estimation of the uncertainty with respect to the nominal model. -xperimental results at the different operating points were given to an identification algorithm * 62D8 + using a multivariable &RP model with a sample time of one second. The transfer functions obtained in the case of the nominal operating point were the followingC TT TT %% %2 2% 22 -T whereC %% D@%% D@%@ 999D% %2 000%H09H 00000%HD33 0002H@DH 999D% 2% %09H9 %030D 999@D 22 000D9H99 0000039%92

0002990D 999@D 5.2.2. Scaling t9e plants and selection o2 t9e nominal model To carry out the scaling, it is necessary to determine the expected magnitudes of the maximum changes of control signals over each input and the maximum variations allowed of the outputs. 3n this application the following values have been chosen as maximum variationsC TT max 3TT max 3-T max max %0 Taking into account these values, the system can be scaled by means of the following expression * 6%H8 +C )z where the two scaling matrices and can be built as followsC TT max TT max @0 0@ -T max max 20 0%0 Once the systems

are scaled, the nominal model is chosen as the model identified at the nominal operating point but removing the delays which appear in it. This is carried out in order to reduce the controller order. 5.2.8. Estimating uncertainties The output multiplicative uncertainty is estimated by means of the expressionC where is the nominal model and stand for each nonnominal identified model. Maximum singular values of these estimated uncertainties *calculated using the transformation with ;% s+ are shown in Fig. @ . 3t can be observed how there is a model *dashed line+ where its maximum

singular value of the uncer5 tainty is greater than one at low frequency. This means that there is no knowledge at all of the system for this operating point * in the working space of Fig. 4 +with respect to the chosen nominal model. Taking into account that this working point is close to limit of the allowed region, it will not be considered for the con5 troller design. Therefore, it will be expected that the controller does not work properly at this operating point. 5.2.4. Initial design o2 t9e weig9ting matrices The continuous time domain is used to design the weighting filters, which

may be discretized by a bilinear transformation in case a discrete synthesis algorithm is applied. 3n this application, the weighting matrix +has been designed as followsC diag D3%0 %00 Themoduleofitsdiagonalelementsisplottedin Fig.@ where it can be seen how it satisfies the constraints imposed in Section 3.2 The initial + is designed as a square diagonal matrixwhereeachofitsdiagonalelementsareasfollowsC ii %0 %0 The parameters of these functions have been selec5 ted as the proposed values, that is, ;0.@ and ;%0 . 3n addition, a value of equal to 0.0%@ radAs approximately is obtained

from the design of + carried out previously. Finally, values of and *associated to the response speed of TT and TT respectively+ equal to zero are selected for the initial controller. &s will be exposed afterwards, these values must be adjusted to obtain an appropriate performance. 94 M.). ,rtega, -... .u/io01ournal o2 Process 3ontrol 14 420045 89–98
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5.2.5. Building up t9e generalized plant and o/taining t9e controller 3n this case, a continuous time synthesis algo5 rithm * 6%98 + has been employed to obtain the controller. This was carried out by converting the nominal model

to the continuous time domain by a bilinear inverse transformation as a previous step of building up the generalized plant. The synthesized controller is then converted to the discrete time domain again by a bilinear transformation. The synthesis process in this case is summarized in the following schemeC Finally, at this point it is important to remember that the controller has been synthesized from a scaled model. To obtain the controller to be implemented in the real application it will be necessary to carry out a scaling inverse process. Thereby, the real controller has to be calculated

from the following expressionC Kz where and matrices are the same ones used in the scaling process of the plant. 5.8. Experimental results The step responses of the output temperatures will be plotted in order to evaluate the performance of the controllers. & step of 2.@ F is supplied as reference for TT while one of @ F is supplied for TT . These mag5 nitudes coincide with the maximum variation allowed for each output in the plant scaling. Fig. D shows the results obtained with the initial selection for +. 3t can be noted that, as expected, the system response is too slow and without any

overshoot. :ext, both and were increased until values were equal to 0.@. The system response in this case is shown in Fig. H . The evolution of TT is still too slow while the one of TT starts to be oscillatory. This means that the value of could be increased again while it would be convenient to decrease the value of Finally, in Fig. 9 are presented the step response of the system corresponding to a controller computed with and equal to %.3 and 0.2, respectively. 3n this case, both responses are faster than the ones obtained with Fig. @. + matrix as an upper bound of the singular values of the

multiplicative output uncertainty. M.). ,rtega, -... .u/io01ournal o2 Process 3ontrol 14 420045 89–98 9@
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Fig. D. -xperimental results with ;0. Fig. H. -xperimental results with ;0.@. 9D M.). ,rtega, -... .u/io01ournal o2 Process 3ontrol 14 420045 89–98
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the initial controller, with a reasonable rise time and overshoot. 3n addition, it can be seen how a change in a temperature reference hardly affects the other output. 6. Conclusions & design methodology of weighting functions has been proposed for the mixed sensitivity problem. This method allows the

application of this control tech5 nique without a wide theoretical knowledge of advanced control, although a large amount of information of the real system is recommended. Despite the simplicity of the method, which is practically reduced to an easy tuning of a parameter for each system output, the controller designed is able to cope with changing oper5 ating conditions and disturbances acting on the systems. The validity and characteristics of the controllers have been tested experimentally with a real plant. Acknowledgements The authors wish to acknowledge F3FQT for funding this work under

grants D135200%524245F0250% and D1320005%2%95F0450%. They also acknowledge the many helpful suggestions by reviewers and editors that have been incorporated into the manuscript. References 6%8 =. &thans, S. &garwal, Designing of a robust controller for a supersonic aircraft using approach, Fontrol -ng. 1ractice D *%994+ %0@%–%0D%. 628 .opez, M.E., Rubio, F.R. / Multivariable design methodology for a ship, inC 1roc. of 3rd -uropean Fontrol Fonference, Rome, 3taly, %99@, pp. D0H–D%2. 638 R. Janavar, 1. Dominic, &n AB) Fontroller for a flexible manipulator, 3--- Trans. on Fontrol Systems

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