Direct controller order reduction by identification in closed loop

�A preliminary version of this paper has been presented to theIFAC-SYSID Symposium, June 2000, S. Barbara, USA. This paperwas recommended for publication in revised form by AssoicateEditor Brett Ninness under the direction of Editor TorstenSoK derstroK m.

*Corresponding author. Tel.: #033-4-7682-6391; fax: #033-4-7682-6382.E-mail addresses: [email protected] (I. DoreH Landau),

alireza.karimi@ep#.ch (A. Karimi), [email protected](A. Constantinescu).

�Alireza Karimi is currently with the Automatic Institute of SwissFederal Institute of Technology at Lausanne (EPFL).

Automatica 37 (2001) 1689}1702

Direct controller order reduction by identi"cation in closed loop�

Ioan DoreH Landau��*, Alireza Karimi��, Aurelian Constantinescu��Laboratoire d'Automatique de Grenoble, ENSIEG, BP 46, 38402 Saint Martin d 'He% res, France

�Sharif University of Technology, Electrical Engineering Dept., Tehran 11365-9363, Iran�Laboratoire d'Automatique de Grenoble, ENSIEG, BP 46, 38402 Saint Martin d'He% res, France

Received 7 February 2000; revised 16 October 2000; received in "nal form 17 April 2001

Abstract

The paper addresses the problem of directly estimating the parameters of a reduced order digital controller using a closed looptype identi"cation algorithm. The algorithm minimizes the closed loop plant input error between the nominal closed loop systemand the closed loop system using the reduced order controller. It is assumed that a plant model (if necessary validated in closedloop with the nominal controller) is available. One of the original features of this approach is that it can use either simulated or realdata. The frequency bias distribution of the parameter estimates shows that the reduced order controller maintains the criticalperformance of the nominal closed loop system. A theoretical analysis is provided. Validation tests are proposed. Experimentalresults, obtained on an active suspension, illustrate the performance of the proposed algorithms. � 2001 Published by ElsevierScience Ltd.

Keywords: Controller reduction; Closed loop identi"cation; Active suspension

Notations

Consider the system shown in Fig. 1 where the plantmodel transfer function is given by

G(z��)"z��B(z��)

A(z��)(1)

and the nominal controller by

K(z��)"R(z��)

S(z��). (2)

The following sensitivity functions are de"ned:

� output sensitivity function:

S��(z��)"

1

1#KG"

A(z��)S(z��)

P(z��),

� input sensitivity function:

S��(z��)"!

K

1#KG"!

A(z��)R(z��)

P(z��),

� output sensitivity function with respect to an inputdisturbance:

S��(z��)"

G

1#KG"

z��B(z��)S(z��)

P(z��),

� complementary sensitivity function:

S��(z��)"

KG

1#KG"

z��B(z��)R(z��)

P(z��),

where

P(z��)"A(z��)S(z��)#z��B(z��)R(z��). (3)

The system of Fig. 1 will be denoted as the `true closedloop systema. Throughout the paper we will consider

0005-1098/01/$ - see front matter � 2001 Published by Elsevier Science Ltd.PII: S 0 0 0 5 - 1 0 9 8 ( 0 1 ) 0 0 1 2 7 - 3

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Fig. 1. The closed loop system

feedback systems which will use either an estimation ofG (denoted GK ) or a reduced order estimation of K(denoted KK ). The corresponding sensitivity functionswill be denoted as follows:

� S��*Sensitivity function of the true closed loop

system (K, G).� SK

��*Sensitivity function of the nominal simulated

closed loop system (nominal controllerK# estimatedplant model GK ).

� SKK��*Sensitivity function of the simulated closed loop

system using a reduced order controller (reduced con-troller KK # estimated plant model GK ).

Similar notations are used for P(z��): PK (z��) when usingK and GK , PKK (z��) when using KK and GK .

1. Introduction

Controller design frequently results in high order con-trollers. On one hand this may be the consequence of thecomplexity of the model used for design, on the otherhand robust controller design often results in complexhigh order controllers even if the design model is ofreasonable size. See for example Landau, Rey, Karimi,Voda, and Franco (1995).Controller reduction is a very important issue in many

control applications either because the size of the con-troller is limited by hardware and computation time orbecause simpler controllers are easier to implement andto understand. There are a number of approaches andmethods for obtaining reduced order controllers. See forexample Zhou and Doyle (1998); Anderson (1993) andAnderson and Liu (1989).What is most important is to remember that controller

reduction should aim to preserve the required closedloop properties as far as possible. Direct simpli"cation ofthe controller using standard techniques (pole-zero can-cellation within a certain radius, balanced reduction)without taking into account the closed loop behaviourgenerally yields unsatisfactory results.There are basically two approaches for controller

reduction:

(1) Indirect approach:

� Obtain a reduced order model which will capture theessential characteristics of the nominal model in thecritical frequency regions for design.

� Design a new controller using the new low-ordermodel.

(2) Direct approach:

� Obtain an approximate reduced order controllerwhich will preserve the nominal closed loop properties.

The indirect approach is subject to a number of criti-cisms. First of all, use of a reduced order model does notnecessarily guarantee that the resulting controller will beof a su$ciently low order (design speci"cations usuallybecome more complex when using model approxima-tion). Secondly, the errors caused by model approxima-tion will spread to the subsequent design steps, seeAnderson and Liu (1994) and Cordons, Bendotti,Falin-ower, and Gevers (1999).The direct approach to controller reduction seems

more appropriate because the approximation is carriedout in the "nal step of controller design and the resultscan be easily understood. It should be noted that thecontroller resulting from an indirect reduction procedurecan be further reduced, if necessary, by application in thelast step of a direct reduction approach.Identi"cation in closed loop o!ers an e$cient meth-

odology both for model order reduction and directcontroller order reduction, see (Bendotti, Cordons,Falinower, & Gevers, 1998).In this paper we will focus on the use of closed loop

identi"cation techniques for direct controller order re-duction. One of the basic block diagrams for reducedorder controller identi"cation is shown in Fig. 2 (inputmatching scheme). The upper part represents thesimulated nominal closed loop system. It is made up ofthe nominal designed controller (denoted by K) and thebest identi"ed plant model (i.e. which results in theclosest behaviour of the true closed loop system and ofthe nominal simulated one and denoted by GK ).The lower part is made up of the estimated reduced

order controller (denoted by KK ) in feedback connectionwith the plant model used for simulation of the nominalsystem. The parametric adaptation algorithm will tryto "nd the best reduced order controller of a givenorder which will minimize the closed loop input errorexpressed as the di!erence between the input to theplant model generated in the nominal simulated closedloop and the input to the plant model generated by theclosed loop using the reduced order controller (i.e. whichwill minimize the discrepancy between the two closedloops).However, another objective for controller order reduc-

tion can be to minimize the closed loop error between theplant output generated in the nominal simulated closedloop and the plant output generated by the closed loopusing the reduced order controller. As will be shown, thisis possible using either the scheme of Fig. 2 but "ltering

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Fig. 2. Closed loop identi"cation of reduced order controllers usingsimulated data (input matching). Fig. 3. Closed loop identi"cation of the reduced order controllers using

real data (input matching).

the external excitation r through GK or by addingthe external excitation to the input of the plant insteadof to the input of the controller (in both cases theclosed loop error, in the absence of disturbances,will re#ect the di!erence between the two plant modeloutputs).Identi"cation of a reduced order controller is also

possible using real data as shown in Fig. 3 (the upper partrepresents the true closed loop system). Note that theadjustable closed loop predictor (lower part) may usea plant model identi"ed in closed loop with the nominalcontroller or more precisely the plant model availablewhich yields the best results when a `closed loopa modelvalidation test is used, see Landau and Karimi (1997) andLandau, Lozano, and M'Saad (1997). In short, this plantmodel will minimize the discrepancy between the nom-inal true closed loop system and the simulated closedloop system using the nominal controller.The method for direct controller reduction using real

data may be related to iterative feedback tuning (IFT) inHjalmarsson, Gevers, Gunnarsson, and Lequin (1998)and Hjalmarsson, Gunnarsson, and Gevers (1994). Themain di!erences are: (1) the IFT requires several experi-ments; (2) the IFT does not use an estimated model of theplant to tune a reduced order controller.The paper is organized as follows. In Section 2 some

basic relationships considered in controller reductionwill be reviewed. In Section 3 the recursive algorithms fordirect identi"cation of the reduced order controllers willbe presented and analyzed. Validation of the estimatedreduced order controller will be discussed in Section 4.The practical aspects of the methodology will be sum-marized in Section 5. Experimental results concerningthe identi"cation of reduced order controllers for activesuspension will be given in Section 6. Finally we will havesome concluding remarks.

2. Direct controller reduction=some basic facts

Several criteria for controller reduction have been pro-posed by Anderson (1993) and Zhou and Doyle (1998).Their objective is the minimization of the discrepancybetween the nominal closed loop and the loop using thereduced order controller. Let us consider the block dia-gram of Fig. 2. Consider the `input matchinga objectivewhich aims to preserve performance with respect to thee!ect of the output disturbance upon the plant input byminimization of the closed loop error between the twoplant inputs u and u( . This is equivalent to minimizing thefollowing norm:

��SK��

!SKK��

��"��K

1#KGK!

KK1#KK GK �� (4)

where SK��is the input sensitivity function of the nominal

simulated closed loop and SKK��is the input sensitivity

function when using the reduced order controller. There-fore the optimal reduced order controller will be givenby

KK H"argmin�K

��SK��

!SKK��

��

"argmin�K

��SK��(K!KK )SKK

��. (5)

If we now consider preservation of performance in track-ing, the reduced order controller should minimize thefollowing norm:

��SK��

!SKK��

��"��KGK

1#KGK!

KK GK1#KK GK ��. (6)

In the same way as when preserving performancefor output disturbance rejection, the reduced order

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controller should minimize:

��SK��

!SKK��

��"��1

1#KGK!

1

1#KK GK ��. (7)

Fortunately, these two norms are equal and the reducedorder controller can be obtained by the followingexpression:

KK H"argmin�K

��SK��

!SKK��

��

"argmin�K

��SK��(K!KK )SKK

��. (8)

Eqs. (5) and (8) show that a weighted norm of K!KKshould be minimized. Several methods for solving thisproblem have been proposed in the literature. The modelreduction using weighted balanced realization in Enns(1984) solves an approximation of the problem, i.e. itneither minimizes the in"nity norm of the weighted errornor gives the error bounds. Another approach proposedby Anderson (1986) minimizes the Hankel norm of theweighted error as well as giving the error bounds.In this paper we present an approach for minimization

of the 2-norms considered in (5) and (8) by direct identi-"cation of a reduced controller in closed loop. The identi-"cation algorithm will minimize a 2-norm of a closedloop prediction error. We will show that the modelingerror between the nominal and reduced order controlleris exactly weighted with the "lters of Eq. (5) or (8).Unstable parts and "xed parts of the controller, whichshould be preserved in the reduced order controller, canbe easily treated. It is assumed that a stabilizing control-ler of a given complexity exists and that the algorithmwill search the parameters of this controller. Once sucha reduced order controller is obtained, it should bevalidated and validation tests are proposed. The Vinni-combe distance between the two closed loop transferfunctions (the nominal closed loop and the closed loopusing the reduced order controller) provides useful in-formation on the quality of the approximation. What iscertainly very interesting and introduces new features forthe proposed procedure is that our presentation showsthat the reduced order controller can be identi"ed usingreal data.

3. Algorithms for direct closed loop identi5cationof reduced order controllers

The parametric adaptation algorithms which will beused to identify the parameters of a reduced order con-troller are very similar to the `closed loop output errora(CLOE) algorithms used for plant model identi"cation inclosed loop. To a large extent, the problem of identi"ca-tion of reduced order controllers can be considered to bethe dual of the reduced order plant model identi"cation

in closed loop (Landau & Karimi, 1997a,b; Landau et al.,1997; Landau & Karimi, 2000). We will often refer toLandau and Karimi (1997a, b) and Landau et al. (1997)for details of the algorithms and proofs.

3.1. Algorithms

Consider the upper part of Fig. 2. The nominalsimulated closed loop is formed by the estimated plantmodel and the nominal controller. The estimated plantmodel is de"ned by the transfer operator (q��-unit delayoperator):

GK (q��)"q��BK (q��)

AK (q��), (9)

where

BK (q��)"bK�q��#2#bK

q�"q��BK H(q��), (10)

AK (q��)"1#a(�q��#2#a(

�q��

"1#q��AK H(q��). (11)

The plant model is operated in closed loop with a digitalcontroller:

K"

R(q��)

S(q��), (12)

where

R(q��)"r�#r

�q��#2#r

�q�� , (13)

S(q��)"1#s�q��#2#s

q��

"1#q��SH(q��), (14)

u(t) is the plant input, y(t) is the plant output and r(t) isthe external excitation signal ("ltered if necessary).The output of the nominal controller is given by

u(t#1)"!SH(q��)u(t)#R(q��)c(t#1)

"��(t), (15)

where

c(t#1)"r(t#1)!y(t#1), (16)

y(t#1)"!AK Hy(t)#BK Hu(t!d), (17)

��(t)"[!u(t),2,!u(t!n�#1),

c(t#1),2, c(t!n�

#1)], (18)

��"[s�,2, s

�, r

�,2, r

�]. (19)

To implement and analyse the algorithm, we need, re-spectively, the a priori (based on �K (t)) and the a posteriori(based on �K (t#1)) predicted outputs of the estimatedreduced order controller (of orders n

�Kand n

�K) which are


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Fig. 4. Closed loop output matching (CLOM): (a) CLIM algorithmwith "ltered external excitation (CLOM1); (b) CLIM algorithm withexternal excitation applied to the plant input (CLOM2).

given by (see the lower part of Fig. 2):

a priori:

u( �(t#1)"u( (t#1��K (t))

"!SK H(t, q��) u( (t)#RK (t, q��) c( (t#1)

"�K �(t)�(t), (20)

a posteriori:

u( (t#1)"�K �(t#1)�(t), (21)

where

�K �(t)"[s�((t),2, s(

�K(t), r(

�(t),2,r(

�K(t)], (22)

��(t)"[!u( (t),2,!u( (t!n�K#1),

c( (t#1),2, c( (t!n�K

#1)], (23)

c( (t#1)"r(t#1)!y( (t#1)

"r(t#1)#AK Hy( (t)!BK Hu( (t!d). (24)

3.1.1. Closed loop input matching algorithm (CLIM)The closed loop input error is given by

a priori:

��(t#1)"u(t#1)!u( �(t#1), (25)

a posteriori:

��(t#1)"u(t#1)!u( (t#1) (26)

and the parameter adaptation algorithm will be given by

�K (t#1)"�K (t)#F(t)�(t)��(t#1), (27)

F��(t#1)"��(t)F��(t)#�

�(t)�(t)��(t), (28)

0(��(t))1; 0)�

�(t)(2; F(0)'0,

��(t#1)"

��(t#1)

1#��(t)F(t)�(t)

"

u(t#1)!u( �(t#1)

1#��(t)F(t)�(t). (29)

As we can see from (29), the a posteriori closed loop inputerror �

��(t#1) can be expressed in terms of the a priori

(measurable) closed loop input error ��(t#1). Therefore

the right hand side of (27) will depend only on measur-able quantities at t#1. For more details see Landauet al. (1997).Speci"c algorithms will be obtained by an appropriate

choice of observation vector �(t) as follows:

� CLIM: �(t)"�(t),

� F-CLIM: �(t)"AK (q��)

PK (q��)�(t),

where

PK (q��)"AK (q��)S(q��)#q��BK (q��)R(q��). (30)

3.1.2. Closed loop output matching algorithm (CLOM)The objective is to create a closed loop error which

re#ects the di!erence between the output y(t) of thenominal closed loop simulated system and the output y( (t)of the simulated closed loop system using the reducedorder controller. Two solutions are proposed:

(1) Filtering the external excitation (CLOM1)Consider the upper part of the Fig. 2. The output of the

plant model in the absence of disturbances is given by

y(t)"KGK

1#KGKr(t). (31)

Now, if instead of directly using r(t) we apply to thesystem r(t) "ltered through GK we obtain (see Fig. 4a):

u� (t)"KGK

1#KGKr(t) (32)

and therefore, in the absence of disturbances, the twotransfer operators are the same. The situation is similarfor the feedback system using the lower order controllerKK . Therefore, if we use the CLIM algorithms but "lter


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"rst r(t) by GK , the algorithm will minimize the closedloop output error instead of the closed loop inputerror.

(2) Adding the external excitation to the plant input(CLOM2)This is illustrated in Fig. 4b. By changing the point

of application of the external excitation, the relationbetween r(t) and u(t) is given by (32), i.e. in the absenceof disturbances, the new u(t) will be equal to y(t).For e!ective implementation of the algorithm, theonly changes occur in Eqs. (15) and (18) where c(t) isreplaced by

x(t)"GK (r(t)!u(t)). (33)

Note also that for real time experiments (as well as insimulation), the order of the blocks in the upper part ofthe scheme given in Fig. 4b can be interchanged.

3.1.3. Imposing constraints on the reduced ordercontrollersWithout di$culty, "xed parts ("lters) can be forced in

the reduced order controllers (like integrators, opening ofthe loop at 0.5f

, "xed "lters, etc.). For this, the nominal

controller is factorized as K"K�K� and the reduced

order controller is factorized as KK "K�KK � where

K�corresponds to the "xed part of the controller that we

would like to be contained in the reduced order control-ler. Then a new input to the controller KK � will be de"nedas c( �"K

�c( and c is replaced by c( � in the observation

vector �.

3.2. Stability analysis

Using the results of the analysis given in Landau andKarimi (1997a, b) and Landau et al. (1997) and by dualityarguments (interchanging B and A with R and S, respec-tively), it can be shown that if the estimated controller hasthe same structure as the nominal controller, then theclosed loop error goes to zero and the signals remainbounded, provided that a su$cient strict passivity condi-tion of the form

H�(z��)"H(z��)!�2; max

�

��(t))�(2 (34)

is a strictly positive real transfer function is satis"ed,where

H"�AK /PK for CLIM,

1 for F-CLIM.(35)

Strictly positive real condition also requires that forCLIM (as well as for F-CLIM) AK be an asymptoticallystable polynomial. Note that for F-CLIM the passivity

condition is always satis"ed since the "lters which haveto be used in the algorithm contain known quantities.The same passivity conditions are valid for the CLOMalgorithm.The passivity conditions (34) and (35) also guarantee

the asymptotic unbiasedness of the estimates in the pres-ence of measurement noise (noise and external excitationare assumed to be independent).In the case of estimation of a reduced order controller,

the following assumptions are made:

� There is a reduced order controller characterized byunknown polynomials SK (of order n

�K) and RK (of order

n�K) which stabilizes the closed loop system.

� r(t) is norm bounded.� The output of the nominal controller (15) can beexpressed as

u(t#1)"!SK H(q��) u(t)#RK (q��) c(t#1)

#�(t#1), (36)

where �(t) is a norm bounded signal.

Eq. (36) can be interpreted as a decomposition of thenominal controller into two parallel blocks: one is thereduced order controller and the other is the neglectedpart generating �(t). The boundedness of �(t) requires theneglected part to be stable. The practical consequence ofthis assumption is that any unstable parts of the nominalcontroller should remain in the reduced order controller.This can be imposed, for example, as a "xed part in thereduced order controller.Based on the results of Landau and Karimi (1997b,

pp. 1505}1506) and assuming that r(t) is norm bounded,we can show that all the signals are norm bounded underthe passivity conditions (34) and (35). Therefore, whenthe estimated controller does not have the same structureas the nominal controller, the above conditions ensurethe boundedness of the closed loop input error andclosed loop output error, respectively. For more detailssee Landau and Karimi (1997a, b).Under the above assumptions a stable closed loop

system will be obtained if the data length is long enoughto allow convergence to the stabilizing controller (if thereis not enough data, recycling of the data is possible).

3.3. Bias analysis

In the case of controller order reduction, by de"nitionthe estimated reduced order controller does not have thesame structure as the nominal controller. Therefore anasymptotic bias will occur which can be characterizedin the frequency domain.We will now examine the bias for the various algo-

rithms when using simulated and real time data.


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3.3.1. Use of simulated data(1) Closed loop input matching algorithmThe output of the nominal and reduced order control-

ler can be expressed as

u(t)"K

1#KGKr(t), (37)

u( (t)"KK

1#KK GKr(t). (38)

The closed loop input error will be given by

��(t)"u(t)!u( (t)"�

K

1#KGK!

KK1#KK GK � r(t) (40)

"!(SK��

!SKK��) r(t)"SK

��(K!KK )SKK

��r(t). (41)

Using Parseval's relation, the asymptotic bias distribu-tion of the parameter estimates in the frequency domainwill be given by

�K H"argmin� �

�

��SK

��K!KK ��SKK

��

�() d

"argmin� �

�

��SK

��!SKK

��

�() d, (42)

where �K H is the vector of estimated parameters of thereduced order controller and �

�() is the spectral density

of the excitation signal.

(2) Closed loop output matching algorithmsIn this case the output of the nominal and reduced

order controller can be expressed as

u(t)"KGK

1#KGKr(t), (43)

u( (t)"KK GK

1#KK GKr(t). (44)

The closed loop output error will be given by

��(t)"u(t)!u( (t)

"�KGK

1#KGK!

KK GK1#KK GK � r(t) (45)

"(SK��

!SKK��) r(t)"SK

��(K!KK )SKK

��r(t). (46)

Using Parseval's relation, the asymptotic bias distribu-tion of the parameter estimates in the frequency domainwill be given by


�

��SK

��K!KK ��SKK

��

�() d

"argmin� �

�

��SK

��!SKK

��

�() d, (47)

where �K H is the vector of estimated parameters of thereduced order controller and �

�is the spectral density of

the excitation signal.Expressions (42) and (47) clearly show that:

� The two norm expression of either Eq. (8) for CLIM(the di!erence between SK

��and SKK

��) or Eq. (5) for

CLOM (the di!erence between SK��and SKK

��) are mini-

mized when r(t) is a white noise signal (since the spec-tral density of a white noise is a constant).

� The frequency distribution of the bias is weighted bythe spectrum of the sensitivity functions of the nominalsimulated system and the spectrum of the estimatedsensitivity functions (given by the plant model and theestimated reduced order controller).

� The di!erence between K and KK is minimized in thecritical frequency regions for control where themodulus of the sensitivity functions is large.

� The frequency distribution of the bias can be tuned bythe choice of r(t).

3.3.2. Use of the real data(1) Closed loop input matching algorithmThe same algorithm applies when the real data are

used (Fig. 3). In this case the expression of the closed looperror will be given by

��(t)"�

K

1#KG!

KK1#KK GK � r(t)#

1

1#KGv�(t) (48)

"!(S��

!SKK��)r(t)#S

��v�(t), (49)

where

v�(t)"v(t)!Kp(t). (50)

In Eq. (50) v(t) and p(t) represent the input and outputdisturbances (noise), respectively. The asymptotic biasdistribution of the parameter estimates in the frequencydomain will be given by


�

��S

��!SKK

��

�()#�S

��

�()�d

"argmin� �

�

��S

��(K!KK )SKK

��#S

��(GK !G)

�SKK��

�� ()#�S

��

�()�d. (51)

(2) Closed loop output matching algorithmsFor CLOM1, the controller output of the nominal and

reduced order controller can be expressed as:

u(t)"KGK

1#KGr(t)#

1

1#KGv�(t), (52)

u( (t)"KK GK

1#KK GKr(t). (53)


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The closed loop output error will be given by

��(t)"u(t)!u( (t)"�

KGK1#KG

!

KK GK1#KK GK �r(t)

#

1

1#KGv�(t) (54)

"S��(K!KK )SKK

��r(t)

#S��(GK !G)SKK

��r(t)#S

��v�(t). (55)

Using Parseval's relation again, the asymptotic biasdistribution will be given by


�

��S

��(K!KK )SKK

��#S

��(GK !G)SKK

��

��()#�S

��

�()�d. (56)

For CLOM2 the output of the nominal controller isgiven by Eq. (52) in which GK is replaced by G. The outputof the reduced order controller is given by Eq. (53). Thiswill "nally yields to

�K H"argmin�K ��

�

��S

��!SKK

��

�()#�S

��

�()�d

"argmin�K ��

�

��SKK

��(GK!GK KK )S

��

�()

# �S��

��()�d

"argmin�K ��

�

��SKK

��(G!GK )S

��!SKK

��(K!KK )S

��

��()#�S

��

�()�d. (57)

Expressions (51), (56) and (57) show that:

� The noise does not a!ect the asymptotic bias distribu-tion of the controller parameters estimate.

� The feedback system using the reduced order control-ler approximates the real closed loop system instead ofthe nominal simulated system for CLIM and CLOM2.

� The frequency distribution of the bias can be tuned bythe choice of r(t).

Eq. (57) (as well as Eqs. (51) and (56)) shows that the realdata allows to take into account (in the good direction)the modelling error which always exists between theplant model used for controller reduction and the trueplant model. Consider that G!GK is small in some fre-quency regions (normally at low frequencies), thereforethe term SKK

��(G!GK )S

��can be ignored and the minimiz-

ation of the integral leads to the weighted minimizationof �K!KK � at low frequencies. Now consider that �G�;1in some frequency regions (normally at high frequencies),therefore the second term can be ignored and the minim-ization of the integral leads to the minimization of themagnitude of the input sensitivity function �SK K

�� at the

frequencies where the modeling errorG!GK is large, thusimproving the robustness. For an example which illus-trates this property see Karimi and Landau (2000).

4. Validation of the estimated reduced order controller

Once a reduced order controller has been estimated, itmust be validated before it is applied on the real system.

4.1. Simulated data

In this case we assume that the nominal controllerstabilizes both the real plant and the plant model used inthe reduction procedure. In actual fact controller reduc-tion takes place with respect to the available plant modelwhich is assumed to correspond to the real plant model.Also any uncertainties were taken into account when thenominal controller was designed (as in all model-basedcontroller reduction techniques). The resulting reducedorder controller should stabilize the nominal model andyield sensitivity functions which are close to nominalones in the critical frequency regions for performance androbustness.In many applications, the output sensitivity function

and the input sensitivity function are important. There-fore, in addition to testing closed loop stability whenusing a reduced order controller, it is necessary to checkthe closeness of the nominal and reduced order sensitiv-ity functions in the frequency domain since they arerelevant both for performance and robustness.The Vinnicombe distance (�-gap) between the sensitiv-

ity functions obtained with the nominal and the reducedorder controller, proposed in Vinnicombe (1993), usesa single number to make a "rst evaluation and classi"ca-tion of the results for various reduced order controllers.Then, a visual comparison of the frequency character-istics of the various sensitivity functions will allow us todecide whether the results obtained are satisfactory.Another important aspect is tolerance with respect to

the uncertainties between the true plant model and thenominal plant model used for design and controllerreduction. If we assume that this robustness issue wastaken into account when the nominal controller wasdesigned, we have to try to preserve this property for thecase of the reduced order controller. Comparison of thesensitivity functions already provides valuable informa-tion, but more complete results can be obtained using theVinnicombe stability test in Vinnicombe (1993) and Zhouand Doyle (1998). A controller K

�which stabilizes the

plant model G�will also stabilize all the plant models

G�satisfying the condition

�(G�,G

�))b(K

�,G

�), (58)

where � (G�,G

�) is the �-gap between the two plant

models and b(K�,G

�) is the generalized stability margin


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de"ned as (Vinnicombe, 1993; Zhou & Doyle, 1998):

b(K�,G

�)"�

��¹(K�,G

�)��

�if (K

�,G

�) is stable,

0 otherwise,(59)

where

¹(K�,G

�)"�

S��

S��

!S��

S�� (60)

and S��are computed for (K

�,G

�).

Therefore, in order to preserve the robustness of thereduced order controller with respect to the plant modeluncertainties, we must check that b(K

�,G

�) is close to

b(K,G�), where K is the nominal controller and K

�is

a reduced order controller.

4.2. Real time data

In this case there are two options. The "rst one is tovalidate the reduced order controller obtained with realtime data using the validation methods for `simulateddataa (see above). The second option is to take advantageof the available real time data. We will focus on this issuebelow.With reference to Fig. 3, the objective is to test to what

extent the estimated closed loop with a reduced ordercontroller is close to the real closed loop system formedby the plant and the nominal controller.Basic information is provided by the variance of the

residual closed loop error. Moreover, as long as thecomplexity of the two closed loop systems is not toodi!erent, the values of the cross-correlations between theresidual closed loop error and the output of estimatedcontroller generated in closed loop with the plant modelare good indicators. This is similar to the statistical testsused in closed loop identi"cation (see Landau & Karimi,1997b; Landau et al., 1997).The other alternative for validation using the real data

is to compute the Vinnicombe gap between the identi"edtransfer function of the true closed loop system (betweenr and u) which uses the nominal controller and thecomputed transfer function of the closed loop formed bythe reduced order controller and the estimated plantmodel.

5. Practical aspects

The controller reduction problem can be approachedin two ways. The "rst is to assume that the plant model isgiven and that the nominal controller has been designedallowing for both performance speci"cation and sup-posed uncertainty of the model. This is the basic situationconsidered in all classical reduction techniques, whichresults in the context of this paper in performing reduc-tion using the simulated data.

However, if access to the real-system is possible andthe nominal controller can be implemented, then thereare two possibilities:

� improve the quality of the model by identi"cationin closed loop (see Landau & Karimi, 1997b; Landauet al., 1997);

� use the real data for controller reduction.

Once a plant model has been selected, the procedurefor controller reduction is applied using real data orsimulated data (with a preference for the real data). Thereduction procedure can scan all the orders for the poly-nomials R and S below the nominal ones using a singleset of data. Those o!ering good approximation of theclosed loop behaviour are selected.

6. Identi5cation of reduced order controllers=experimental results

The procedure for obtaining reduced order controllersby identi"cation in closed loop will be illustrated for thecase of control of an active suspension. The active sus-pension is shown in Fig. 5 and the corresponding blockdiagram in Fig. 6. The controller via the power ampli"erwill act on the piston in order to reduce the residualacceleration. The frequency spectrum of the vibrationswhich have to be reduced is limited by an upper fre-quency lower than 200 Hz. The system is controlled bya PC via an I/O board. Sampling frequency is 800 Hz.The primary acceleration was generated using a

shaker. Its input is a signal given by the computer. Theprimary path model, between the excitation of the shakerand the residual acceleration, has several vibrationmodes: the "rst is at 31.47 Hz with a damping factor0.093. The goal is to compute a controller which minim-izes the residual acceleration around the "rst vibrationmode and to try to distribute the ampli"cation of thedisturbances over the high frequency range.The identi"ed plant model, between piston position

and the residual acceleration (the secondary path), hasthe following complexity: n

"11, n

�"12, d"2. The

system contains a double di!erentiator. Fig. 7 illustratesthe frequency characteristics of two models for thesecondary path:

(1) The open loop identi"ed model, used to design thenominal controller. This model was identi"ed inopen loop using a PRBS generated by a 9-bit shiftregister, with a clock frequency of fs/4 (data length is¸"2048 samples).

(2) The closed loop identi"ed model, used to identify thereduced order controllers. This model gives betterresults in terms of closed loop validation and wasidenti"ed in closed loop using the F-CLOE method


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Fig. 5. The active suspension system.

Fig. 6. Block diagram of the active suspension system.

Fig. 7. The frequency characteristics of the secondary path models(input: piston displacement, output: residual acceleration).

(see Landau & Karimi, 1997a), the nominal control-ler and the same excitation signal as in open loopidenti"cation.

The results for controller reduction are given for themodel identi"ed in closed loop. The model has six vibra-tion modes. Five of the six vibration modes have a very

low damping ((0.085). Note that an important attenu-ation on S

��is required at the frequency of the "rst

vibration mode.The nominal controller was designed using the `pole

placement with sensitivity function shaping by convexoptimizationa method (see Langer & Landau, 1999).A pair of dominant poles was "xed at the frequency of the"rst vibration mode, with a damping �"0.8. In addition,a "xed partH

�"1#q�� was introduced in the control-

ler (R"H�R�) to ensure opening of the loop at 0.5f

. The

resulting nominal controller satisfying the residualacceleration speci"cations was obtained. Its complexityis given by the orders of polynomial R and S:n�

"27, n�"28. Note that if standard pole placement is

used, by solving the Bezout equation, the result will bea controller with the orders: n

�"12, n

�"13 (consider-

ing the "xed part H�).

First, the CLIM direct identi"cation method for a re-duced order controller was used, based on the simulateddata. The external input was a PRBS generated by a10-bit shift register, with a clock frequency of f

/2

(data length is ¸"4096 samples). A variable forgettingfactor with �

�"0.95 and �

�"0.9 was used (�

�(t)"

��(t!1)#1!�

�).

The frequency characteristics of the sensitivity functions(S

��,S

��) and of the various controllers for the nominal

controller (K) with n

�"27, n

�"28, and for three re-

duced order controllers:K�with n

�"19, n

�"20,K

�with

n�

"12, n�"13 and K

with n

�"9, n

�"10 are shown

in Figs. 8 and 9, respectively (a "xed part H�

"1#q��

was imposed in the reduced order controllers).Note that the reduced controller K

�corresponds to

the complexity of the pole placement controller withthe "xed part H

�, (i.e. as a side result, the approach

proposed here seeks out an optimal con"guration forthe closed loop poles in order to achieve the stipulatedperformance). Complexity of controller K

is lower

than that corresponding to pole placement. It is also


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Table 1Comparison of the various controllers (controller reduction using CLIM algorithm on the simulated data)

Controller K

K�

K�

K

n�

"27 n�

"19 n�

"12 n�

"9n�"28 n

�"20 n

�"13 n

�"10

1 �(K,K

�) 0 0.1810 0.5049 0.5180

2 �(S��, S�

��) 0 0.1487 0.4388 0.4503

3 �(S��, S�

��) 0 0.0928 0.1206 0.1233

4 b(K�,G) 0.0800 0.0786 0.0685 0.0810

5 �(CL(K),CL(K

�)) 0.1296 0.2461 0.5435 0.5522

6 C.L. error variance 0.0023 0.0083 0.0399 0.0398

Fig. 8. Output sensitivity for the active suspension (controller reduction using CLIM algorithm on the simulated data).

important to see the values of the various �-gap. Theseresults are summarized in the Table 1 (the rows 1}3). Thelast 2 rows in Table 1 summarize real time results. Weobserve that the generalized stability margins b(K

�,G)

computed with the nominal model for the various re-duced order controllers are close to the stability marginobtained with the nominal controller. Row 5 gives the�-gap between the input/output transfer function corre-sponding to the input sensitivity function S

��of the true

closed loop system formed by the nominal designed con-troller with the real plant (obtained by identi"cation) andthe input/output transfer function of the simulated closedloop system (SKK

��) formed by the various controllers

(including the nominal one and the reduced onesobtained using simulated data) in feedback connectionwith the plant model. This quantity is denoted by �(CL(K

), CL(K�)). This is a good criterion for valida-

tion of the reduced order controller. Row 6 gives thevariance of the residual closed loop input error betweenthe true system and the simulated one. We observe that

the �-gap and the closed loop error variance have a co-herent evolution.To illustrate the performance of the resulting controllers

(K,K

�,K

�andK

) on the real system, the spectral density

of the residual acceleration in open and in closed loop isshown in Fig. 10. The primary source of vibration (shaker'sexcitation) is a PRBS. The characteristics obtained in closedloop operation with the four controllers are compared withthe characteristic of the open loop operation (the interestingfrequency range is 0}0.25f

(200 Hz)). We note that the

performance of the reduced order controllers is similar tothat of the nominal controller and that they all achievea signi"cant reduction of the vibrations around the "rstvibration mode of the plant model.The results obtained using the real data (Landau,

Karimi, & Constantinescu, 2001) show that the perfor-mance of the reduced order controllers is very similar tothe case with the simulated data. This can be accountedfor by the quality of the identi"edmodel used for control-ler reduction.


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Fig. 9. Input sensitivity for the active suspension (controller reduction using CLIM algorithm on the simulated data).

Fig. 10. Spectral density of the residual acceleration in open and closed loops (controller reduction using CLIM algorithm on simulated data).

Good results have been obtained also with the CLOMalgorithms (Landau, Karimi, & Constantinescu, 2001)using simulated and real data. Comparison of the resultsobtained with CLIM and CLOM algorithms shows thatthe �-gap between the nominal and the reduced orderoutput sensitivity function �(S

��,S�

��) for CLOM algo-

rithms is slightly smaller, while the �-gap between thenominal and reduced order input sensitivity � (S

��,S�

��)

for CLOM algorithms is larger. This is in agreementwith the bias analysis, since the CLIM algorithm gives

preference to minimizing the error between the inputsensitivity functions while the CLOM algorithm givespreference to minimizing the error between the outputsensitivity functions.

7. Conclusions

Algorithms for direct closed loop identi"cation ofreduced order digital controllers have been proposed,


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analysed and experimentally tested. These algorithms arethe dual of closed loop output error identi"cation algo-rithms. The algorithms try to minimize either the errorbetween inputs generated in closed loop operation by thenominal controller and the estimated reduced order con-troller, or the closed loop output error between the twoloops.The possibility of using real data for estimation of

reduced order controllers makes it possible to take intoaccount to a certain extent in the reduction procedure,the discrepancies between the true plant and the plantmodel used for design and controller reduction.While the speci"c algorithms proposed have a recur-

sive form, it is also possible to develop iterative algo-rithms along the lines used for closed loop identi"cation(see Van Donkelaar & Van den Hof, 2000). Unfortunate-ly this will raise similar theoretical problems.Validation tests related to the �-gap and Vinnicombe

stability test have been proposed. Use of the results givenin Vinnicombe, (1993, 1996), Zhou and Doyle (1998)and Anderson and Gevers (1998), for reduced order con-troller validation deserve further investigation. Futurework should consider the extension to the multivariablecase.

Acknowledgements

The authors would like to thank one of the anonymousreviewers for his suggestions concerning the interpreta-tion of the validation tests.

References

Anderson, B., & Gevers, M. (1998). Fundamental problems in adaptivecontrol. In D. Normand-Cyrot (Ed.), Perspectives in control.London, UK: Springer.

Anderson, B. D. O. (1986). Weighted Hankel norm approximation:calculation of bounds. Systems and Control Letters, 7(4), 247}255.

Anderson, B. D. O. (1993). Controller design: moving from theory topractice. IEEE Control Magazine, 13, 16}25.

Anderson, B. D. O., & Liu, Y. (1989). Controller reduction: conceptsand approaches. IEEE Transactions on Automatic Control, 34(8),802}812.

Bendotti, P., Cordons, B., Falinower, C., & Gevers, M. (1998). Controloriented low order modeling of a complex PWR plant: a compari-son between open loop and closed loop methods. In IEEE-CDC1998, FAO 7-2. FL. USA: Tampa.

Cordons, B., Bendotti, P., Falinower, C., & Gevers, M. (1999). A com-parison between model reduction and controller reduction: applica-tion to a PWR nuclear plant. In IEEE-CDC, Vol. ix, USA(pp. 4625}4630).

Enns, D. (1984). Model reduction with balanced realization: an errorbound and a frequency weighted generalization. In 23rd CDC, LasVegas, Nevada, USA (pp. 127}132).

Hjalmarsson, H., Gevers, M., Gunnarsson, S., & Lequin, O. (1998).Iterative feedback tuning: theory and application. IEEE ControlSystems Magazine, 18, 26}41.

Hjalmarsson, H., Gunnarsson, S., & Gevers, M. (1994). A convergentiterative restricted complexity control design scheme. In 33rdIEEE-CDC.

Karimi, A., & Landau, I. D. 2000. Controller order reduction by directclosed loop identi"cation (output matching). In 3rd Rocond IFAC,Prague.

Landau, I. D., & Karimi, A. (1997a). An output error recursive algo-rithm for unbiased identi"cation in closed loop. Automatica, 33(5),933}938.

Landau, I. D., &Karimi, A. (1997b). Recursive algorithms for identi"ca-tion in closed loop*a uni"ed approach and evaluation. Automatica,33(8), 1499}1523.

Landau, I. D., & Karimi, A. (2000). Model estimation and controllerreduction: Dual closed loop identi"cation problems. In 39th IEEE-CDC, Sydney.

Landau, I. D., & Karimi, A., Constantinescu, A. (2001).Direct controllerorder reduction by identixcation in closed-loop*application to activesuspension control. Technical report, Laboratoire d'Automatique deGrenoble, ENSIEG, BP. 46, 38402 St. Martin d'Heres.

Landau, I. D., Lozano, R., & M'Saad, M. (1997). Adaptive control.London: Springer.

Landau, I. D., Rey, D., Karimi, A., Voda, A., & Franco, A. (1995).A #exible transmission system as a benchmark for robust digitalcontrol. European Journal of Control, 1(2), 77}96.

Langer, J., & Landau, I. (1999). Combined pole placement/sensitivityfunction shaping method using convex optimization criteria. Auto-matica, 35, 1111}1120.

Van Donkelaar, E. T., & Van den Hof, P. M. J. (2000). Analysisof closed-loop identi"cation with tailor-made parameterization.European Journal of Control, 6(1), 54}62.

Vinnicombe, G. (1993). Frequency domain uncertainty and the graphtopology. IEEE Transactions on Automatic Control, 38, 1571}1583.

Vinnicombe, G. (1996). The robustness of feedback systems withbounded complexity. IEEE Transaction on Automatic Control, 41(6),1571}1583.

Zhou, K., & Doyle, J. C. (1998). Essentials of robust control. NY:Prentice-Hall.

Ioan DoreH Landau is Research Director atC.N.R.S. (National Centre for Scienti"c Re-search) and works at Laboratoire d'Auto-matique de Grenoble (CNRS/INPG) of theInstitutNationalPolytechniquedeGrenoble.His research interests encompass theory

and applications in system identi"cation,adaptive control, robust digital controland nonlinear systems. He has authoredand co-authored over 200 papers on thesesubjects. He is the author and co-author ofseveral books including: Adaptive Control~

The Model Reference Approach (Dekker 1979), System Identixcation andControl Design (Prentice Hall 1990) and Adaptive Control (Springer-Verlag 1997). He delivered a number of Plenary Talks at InternationalConferences including American Control Conference in Seattle in 1995and the European Control Conference in Bruxelles in 1997.Dr. Landau received the Rufus Oldenburger Medal 2000

from the American Society of Medical Engineering for his pioneeringcontributions in adaptive control and system identi"cation. He wasa R. Severance Springer Professor at University of California.Berkeley, Dept. of Mechanical Engineering in 1992. He receivedthe Price Monpetit from the French Academy of Science in 1991,the `Best Review Paper Award (1981}1984)a for his paper onadaptive control published in ASME Journal of Dynamical SystemsMeasurement and Control, the C.N.R.S. Silver Medal in 1982and the Great Gold Medal at the Invention Exhibition, Vienna in1968 for his patent on the variable frequency control of asyn-chronous motors. He is an IEEE-CSS `Distinguished Lecturera for2001.


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Dr. Landau was the "rst President of the European Union ControlAssociation (EUCA) from 1991 to 1993 and he is Editor in Chief of theEuropean Journal of Control (a publication of the European UnionControl Association).

Alireza karimi was born in Mashad, Iran,in 1964. He received the B.Sc. (1987) andM.Sc. (1990) degrees in electrical engineer-ing both from Amir Kabir University(Tehran Polytechnic). From 1990 to 1993he was in charge of Research and Develop-ment Department in Iran-Switch andsapta Companies. Then he joined the Au-tomatic Control Laboratory of Grenobleand received the Ph.D. degree in automaticcontrol from National Polytechnic Insti-tute of Grenoble, France in 1997. He was

an Assistant Professor at Electrical Engineering Department of SharifUniversity of Technology, Tehran, Iran. He is currently ResearchAssociate at the Institut d'Automatique of Suisse Federal Institute ofTechnology at Lausanne (EPFL). His research interest include closed-loop identi"cation, adaptive and robust control.

Aurelian Constantinescu was born inBucarest, Romania, in 1974. He receivedan aeronautical engineer degree from thePolytechnical University of Bucarest in1992. He received the master's degree incontrol in 1993 from the National Poly-technical Institute of Grenoble, France. Heis currently preparing a Ph.D. Thesis at theLaboratoire d'Automatique de Grenoble inFrance. His current research interests in-clude robust, adaptive and active control.


Direct controller order reduction by identification in closed loop

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