This article was published in an Elsevier journal. The attached copyis furnished to the author for non-commercial research and

education use, including for instruction at the author’s institution,sharing with colleagues and providing to institution administration.

Other uses, including reproduction and distribution, or selling orlicensing copies, or posting to personal, institutional or third party

websites are prohibited.

In most cases authors are permitted to post their version of thearticle (e.g. in Word or Tex form) to their personal website orinstitutional repository. Authors requiring further information

regarding Elsevier’s archiving and manuscript policies areencouraged to visit:

http://www.elsevier.com/copyright

Author's personal copy

Automatica 44 (2008) 1 – 14www.elsevier.com/locate/automatica

Simultaneous performance achievement via compensator blending�

Franco Blanchinia,∗, Patrizio Colanerib, Felice Andrea Pellegrinoc

aDipartimento di Matematica e Informatica, Università degli Studi di Udine, Via delle Scienze 208, 33100 Udine, ItalybDipartimento di Elettronica e Informazione, Piazza Leonardo da Vinci 32, 20133 Milano, Italy

cDipartimento di Elettrotecnica, Elettronica e Informatica, Università degli Studi di Trieste, Via Valerio 10, 34127 Trieste, Italy

Received 20 March 2006; received in revised form 8 March 2007; accepted 16 April 2007Available online 4 September 2007

Abstract

In this paper we consider the problem of designing a state-feedback controller that simultaneously achieves different optimality criteria definedon different input–output pairs. Precisely, if r “optimal” target transfer functions are given (as the result of local “optimal” controllers), it isshown that (under mild assumptions) there exists a unique controller capable of replicating these transfer functions in the closed-loop system,so simultaneously achieving the performances inherited by the chosen local transfer functions. An explicit and constructive procedure (we referto such procedure as “compensator blending”) is provided. The possibility of designing a stable blending compensator or the generalization todynamic local controllers or time varying systems are also discussed. We finally consider the dual version of the problem, precisely, we showhow to achieve simultaneous optimality by filter blending.� 2007 Elsevier Ltd. All rights reserved.

Keywords: Optimal control; Mixed-objective control; Constrained control; Robustness

1. Introduction and motivation

It is well known that addressing different (often conflict-ing) objectives is a major issue in control design. A typicalexample is the mixed control problem, in which the designeraims at simultaneously satisfying two (or more) performancerequirements. Celebrated is the so-called mixed H2–H∞ con-trol problem, that has received so much attention in the recentliterature (Khargonekar & Rotea, 1991; Khargonekar, Rotea, &Sivashankar, 1993; Rotea, 1990; Rotea & Khargonekar, 1991;Saberi, Stoorvogel, & Sannuti, 2000; Sznaier, Rotstein, Bu, &Sideris, 2000).

In the present paper we aim at assessing the potentiality givenby increasing the complexity (order) of the compensator in or-der to achieve mixed requirements without any sort of approx-imations. The core of the paper is the design of a controller

� A preliminary version of this paper was submitted to ROCOND, Toulose,July 2006. This paper was not presented at any IFAC meeting. This paperwas recommended for publication in revised form by Associate Editor BenM. Chen under the direction of Editor Ian Petersen.

∗ Corresponding author.E-mail address: blanchini@uniud.it (F. Blanchini).

0005-1098/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.doi:10.1016/j.automatica.2007.04.010

capable of coping simultaneously cope with different optimal-ity criteria defined on different input–output pairs. This prob-lem was previously faced in Khargonekar et al. (1993), Roteaand Khargonekar (1991), and Rotea (1990) where the follow-ing result was established: If r “optimal” target transfer func-tions are given (as the result of local “optimal” controllers),(under mild assumptions) there exists a unique state-feedbackcontroller (of suitable dimensions) capable to replicate thesetransfer functions in the closed-loop system, so simultaneouslyachieving the performances inherited by the chosen local trans-fer functions.

Here we reconsider those results and we provide a new in-sight along the same line, both for the state-feedback case andthe output-feedback case. In the former the stability of theclosed-loop system is guaranteed by mild stabilizability andrank assumptions, whereas in the latter case additional assump-tions made on the zero dynamics of the open-loop system haveto be added in order to get a stable closed-loop system.

The main result of the paper is based on a very easy pro-cedure based on simple linear algebra tools, mostly borrowedfrom Blanchini and Pellegrino (2003, 2006), which lead to asingle compensator achieving simultaneously the performanceof several given compensators. We refer to such procedure as

Author's personal copy

2 F. Blanchini et al. / Automatica 44 (2008) 1 – 14

“compensator blending”. A fundamental fact is that such ablending procedure does not depend on the criteria which areadopted in the determination of the compensators. These can bestatic (or dynamic) state feedback linear compensators derivedby means of classical criteria such as LQ control (Kwakernaak& Sivan, 1972) or H∞-control (Doyle, Glover, Khargonekar, &Francis, 1989). However, they may result as well from morerecent design techniques such as constrained synthesis (Hu &Lin, 2001; Saberi et al., 2000), robust control (Sanchez-Peña &(Sznaier, 1998; Zhou, Doyle, & Glower, 1996), peak-minimi-zation design (Abedor, Nagpal, & Poolla, 1996; Dahleh &Pearson, 1987, 1988), model matching (Chu & Mehrmann,2000; Chu & Van Dooren, 2006; Kucera, 1991) or disturbancedecoupling (Chen, Mareels, Zheng, & Zhang, 2000; Chu &Mehrmann, 2001; Willems & Commault, 1981).

The paper also provides some aside results that are interest-ing per se. For instance, it is shown that the design of a sta-ble blending compensator (strong blending design) is possibleonly if a suitable defined system is minimum phase. We finallyconsider the dual version of the problem, precisely, we showhow to achieve simultaneous optimality by filter blending. Themain contributions of this paper are summarized:

• We provide explicit and simple formulas for blendingcompensators. These formulas are simpler than those pro-vided in previous work (Khargonekar et al., 1993; Rotea& Khargonekar, 1991) which were of a recursive nature.

• We consider the problem of strong compensator blending,precisely we show how to assign the poles of the resultingcompensator (while preserving the closed-loop poles whichare the union of those assigned by the original compensators,in the state-feedback case).

• We relate the achievable compensator poles and extra closed-loop poles (in the output-feedback case) with the systemtransmission zeros.

• We discuss some specific cases such as the blending of rel-atively optimal compensators and the optimal robust design.

• We consider the dual problem of filter blending.• We provide experimental results in the control of a cart-

pendulum system.

2. Control blending

Consider the continuous-time system

x(t) = Ax(t) + Bu(t) +r∑

j=1Ejwj (t),

y1(t) = C1x(t) + D1u(t),

y2(t) = C2x(t) + D2u(t),

...

yr (t) = Crx(t) + Dru(t),

(1)

where x(t) ∈ Rn, u(t) ∈ Rm, yi(t) ∈ Rpi , wi ∈ Rqi and A,B, Ci , Di Ei are matrices of appropriate dimensions. In thissection we consider the feedback scheme in Fig. 1, where the

u x

wr

y1

yr

y2w1w2

Plant

Control

Fig. 1. Feedback scheme.

plant is given in (1) and the controller takes on the form

z(t) = Fz(t) + Gx(t),

u(t) = Hz(t) + Kx(t). (2)

The closed-loop system with inputs wi , i = 1, 2, . . . , r andoutputs yi , i=1, 2, . . . , r , is therefore described by the compactexpression:

(3)

Throughout this section we make the following assumptions:

Assumption 1. (A, B) is stabilizable.

Assumption 2. The composed n × s matrix,

E = [E1E2 . . . Er ],where s

.= ∑rj=1qj , has full column rank.

The latter assumption is a restriction on which we will com-ment later on. Note that this avoids the presence of two matricesEj = Ek , j �= k.

Given system (1) we assume that

u(t) = Kix(t), i = 1, 2, . . . , r (4)

are assigned stabilizing compensators and define

Ai = A + BKi, Ci = Ci + DiKi .

The corresponding i–i desired transfer function is given by

Hi(s) = Ci(sI − Ai)−1Ei .

The following theorem holds, see also Khargonekar et al.(1993).

Theorem 2.1. There exists a single dynamic compensator ofform (2) of order � = n(r − 1) such that

• the closed-loop system is stable with spectrum (formed byn × r elements)

�(Acl) =⋃i

�(Ai);

Author's personal copy

F. Blanchini et al. / Automatica 44 (2008) 1–14 3

• for all i, the transfer functions Hi(s) from the ith input to theith output coincide with the ones given by the ith gain Ki , i.e.

Hi(s) = Hi(s).

The importance of the previous theorem whose proof is givenlater, is explained by the fact that if Ki , i = 1, 2, . . . , r , are thestatic compensators which optimize some performance criteria,then there exists a single (dynamic) compensator which simul-taneously achieves the same performances. The compensatormay be found following the next procedure.

Procedure 2.1.

(1) Let Ki be assigned and let Ai = A + BKi .(2) Take matrices Ei of dimension n×(n−qi) such that [Ei Ei]

is square and invertible (note that this is possible becauseeach Ei has full column rank in view of Assumption 2).

(3) Fix matrices Zi of dimension � × (n − qi) (remind that� = n(r − 1)) such that the matrix[

E1 E2 . . . Er E1 E2 . . . Er

0 0 . . . 0 Z1 Z2 . . . Zr

](5)

is invertible.(4) Define

Zi.= [0 Zi][Ei Ei]−1. (6)

(5) Define

Vi.= ZiAi .

(6) Form the square nr-dimensional matrix T as

T =[

I I . . . I

Z1 Z2 . . . Zr

]. (7)

(7) Compute the compensator matrices F, G, H, K as[K H

G F

]=[K1 K2 . . . Kr

V1 V2 . . . Vr

]

×[

I I . . . I

Z1 Z2 . . . Zr

]−1

. (8)

The procedure is quite simple to implement. The only miss-ing point concerns Eq. (8). We fix it by showing that T is indeedan invertible matrix. Indeed we have

[I I . . . I

Z1 Z2 . . . Zr

]︸ ︷︷ ︸

T

⎡⎢⎢⎢⎣

[E1E1] 0 . . . 0

0 [E2E2] . . . 0

: : . . . :0 0 . . . [ErEr ]

⎤⎥⎥⎥⎦

=[E1 E1 E2 E2 . . . Er Er

0 Z1 0 Z2 . . . 0 Zr

]. (9)

The last matrix is achieved by a column permutation of (5),which is invertible by construction. Since also the block-diagonal matrix in the leftmost term is invertible, T is invertibleas well.

Remark 2.1. Note that there is no assumption on the matricesCi and Di . Therefore, the construction is valid no matter howwe choose them. In particular, the state itself might be chosenas performance output. In particular, we might have yi(t)=x(t)

for all i and this means that for any input wi(t) (assuming zerothe others) we can achieve the same (desired) closed loop statetrajectory.

Proof of Theorem 2.1. The proof of the theorem is based onexpression (3) of the closed-loop plant. By applying the statetransformation

[x

z

]=[

I I . . . I

Z1 Z2 . . . Zr

]� = T �,

where �(t) ∈ Rnr , the next equivalent representation isachieved

This implies that the transfer function from the ith input tothe ith output is indeed Hi(s) as claimed. We omit here themathematical details since they can be derived as a special casefrom those reported in Section 4. �

We remind that in some cases the optimal compensator isnot static but dynamic. For instance the L1 optimal control(Dahleh & Pearson, 1987) can be dynamic (Diaz-Bobillo &Dahleh, 1992) (although static suboptimal ones can be com-puted Abedor et al., 1996). The mentioned relatively optimalcontrol (Blanchini & Pellegrino, 2003, 2006) provides a dy-namic compensator. Dynamic compensators can be blendedas well since, as it is known, any dynamic compensator canbe seen as a static compensator for a properly augmentedplant.

Remark 2.2. One problem of the procedure is the order of thecompensator n(r −1) which can be very high (although knowna priori). Reducing the order is not generically possible sincewe are requiring to match r different transfer functions that, ingeneral, may have different poles. This means that nr differ-ent poles have to be allocated and a full order compensator isrequired. Introducing appropriate requirements on the transferfunctions to be matched brings to a different problem that isnot considered in the present work.

Author's personal copy

4 F. Blanchini et al. / Automatica 44 (2008) 1 – 14

3. Extensions and applications

3.1. Strong blending design

Pursuing an idea similar to Blanchini and Pellegrino (2003),we can exploit the degrees of freedom in the construction ofthe compensator in order to achieve our goal by means of a sta-ble compensator. This goal is quite important since “blending”introduces an additional dynamics which can have undesirableconsequences in the system. In particular an unstable compen-sator can have fatal consequences in plants with output or inputsaturation. We first notice that for our purpose the fundamentalconditions are (6) and (8). In particular from these two equa-tions we need the conditions

ZiEi = 0 ∀i, (10)

G + FZi = ZiAi ∀i. (11)

Now we investigate the possibility of achieving Eq. (8) andconditions (10), (11) by fixing F as an arbitrary stable matrix.The unknowns are matrix G ∈ R�×n and matrices Zi ∈ R�×n

where �=n(r − 1) is the compensator order. Letting again s =∑i qi it turns out that the number of unknowns and equations

are

nun = r × n × � + � × n, neq = r × n × � + � × s.

Therefore, system (10) and (11) is generically solvable underAssumption 2 which implies s�n. Once the solution is found,if the matrix T is invertible, we can determine the matrices Hand K as

[K H ] = [K1 K2 . . . Kr ]

[I I . . . I

Z1 Z2 . . . Zr

]−1

︸ ︷︷ ︸T −1

.

(12)

In this way we achieve again the key equation (8), but nowwith a fixed stable F. However, there is a restriction we musttake into account. Indeed consider the system

(13)

and let us apply again to this system the inverse transformationT −1. It turns out that this system is equivalent to the followingone:

(14)

This last expression implies that the zeros of (14) are eigenval-ues of F. On the other hand, the zeros of system (14) coincide

with those of (13) which are fixed by the choices of the nom-inal compensators Ki , i = 1, 2, . . . , r . As a first conclusion, ifthe system has unstable transmission zeros the strong blend-ing design is not possible. Moreover, notice that if matrix E issquare, then the eigenvalues of F coincide with the invariantzeros of (14) so that in this case F turns out to be stable if andonly if such a system is minimum phase.1 We can summarizethese facts in the following statement.

Proposition 1. (i) The strong blending design is not possibleif system (13) has unstable invariant zeros.

(ii) The strong blending design is generically possible if sys-tem (13) is minimum phase.

(iii) If E is square then the strong blending design is possibleif and only if system (13) is minimum phase.

3.2. The discrete-time case: blending relatively optimalcontrollers

The discrete time case can be handled easily with only techni-cal changes in the equations. There is a point worth of attention.Precisely we would like to make a connection with a previouswork on the relatively optimal control proposed in Blanchiniand Pellegrino (2003, 2006). A relatively optimal control is adynamic controller which provides the optimal trajectory fora specific initial condition and is stabilizing for all the others.For the sake of brevity, assume that the problem is that of opti-mizing the transient from a nominal initial condition x(1)=E1(i.e. optimizing the impulse response with input matrix E1, forinstance using time domain criteria such as Dahleh & Pear-son, 1987, 1988) and simultaneously optimizing the transferfunction from another signal entering through a different inputmatrix E2 to a certain output y2. If we assume, in line withAssumption 2, that

[E1 E2] = [x(1) E2]has full column rank, then we can adapt the blending procedureto achieve both relative optimality and an optimal w2 to y2transfer function.

3.3. Blending robust and optimal controllers

Consider the system

x = (A + E2�H2)x + (B + E2�G2)u + E1w1,

z1 = H1x + G1u,

where E2, H2 and G2 are known structural matrices and � is anuncertain perturbation matrix with bounded norm, say one. Itis well known that the system can be put in the equivalent form

x = Ax + Bu + E1w1 + E2w2,

z1 = H1x + G1u,

1 We use “minimum phase” to mean a system with stable zeros.

Author's personal copy

F. Blanchini et al. / Automatica 44 (2008) 1–14 5

z2 = H2x + G2u,

w2 = �z2.

Therefore, the blending setting can be clearly adopted to solvethe so-called robust H2 problem. This problem consists in find-ing a state-feedback controller that minimizes the H2 normfrom w1 to z1 when � = 0 and ensures stability for all pertur-bations � with ‖�‖ < 1. For ease of computations we assumefurther that the pair (A, B) is stabilizable, the pairs (A, Hi),i = 1, 2 are detectable and

G′iGi = I, G′

iHi = 0, i = 1, 2.

Then take K1 = −B ′P where P �0 is the unique stabilizingsolution of the Riccati equation

A′P + PA − PBB′P + H ′1H1 = 0.

Such a feedback gain minimizes the H2 norm of the nominalclosed-loop system for � = 0. Moreover, assume that thereexists the positive semidefinite and stabilizing solution of theRiccati equation

A′� + �A − �BB′� + �E′2E2� + H ′

2H2 = 0,

and take the control gain K2 = −B ′�. Such a gain ensuresthat the closed-loop system matrix A+BK2 +E2�H2 is stablefor any bounded � with norm less than one. By applying thestate-space transformation worked out in the control blendingsection, namely

� = T −1[x

z

], T =

[I I

Z1 Z2

].

A simple computation shows that the closed-loop system isdescribed by

� =[A + BK1 0

E2�H2 A2 + BK2 + E2�H2

]� +

[E1

0

]w1,

z1 = [H1 + G1K1 H1 + G1K1]�.

Therefore, the closed-loop is stable for each ‖�‖ < 1. Noticethat the single K = K1 minimizing the H2 norm for � = 0,in general does not yield a robustly stable closed-loop system.Conversely, a single K = K2 ensuring robust stability doesnot possess any optimality property for the nominal system. Itwould be interesting to compare this procedure with the so-called post optimization procedure introduced in Chapter 6 ofColaneri, Geromel, and Locatelli (1997). This will be the sub-ject of further investigation.

3.4. Input output decoupling and model matching

The proposed investigation can be successfully combined withthe disturbance decoupling problem (Willems & Commault,1981) (see also Chen et al., 2000; Chu & Mehrmann, 2001 andthe references therein for recent developments). Basically, thedisturbance decoupling problem is a special case in which onedesired transfer function is zero.

Also the model matching problem (Kucera, 1991) can besuccessfully combined by means of the proposed technique.The classical model matching problem (a generalization of theformer) consists in determining K and R such that, if we takeu = Kx + Ru, the resulting transfer function from u to theoutput is

C(sI − A − BK)−1BR = H (s),

with H (s) assigned. Several conditions for this problem to besolvable are known (see for instance Chu & Mehrmann, 2000;Chu & Van Dooren, 2006 and the included references). Underappropriate assumptions on BR, we can recover the transferfunction C(sI − A − BK)−1BR, among others. The details areomitted for brevity.

3.5. Time-varying systems

The material presented so far makes reference to the controlof linear time-invariant systems. However, the idea underlyingthe blending property can be transferred without major prob-lems to linear time-varying systems. Basically, it can be shownthat we can arrive to a time-varying compensator of the form(for short, the explicit dependence on the time is avoided)[K H

G F

]=[

K1 K2 . . . Kr

LZ1A1 LZ2A2 . . . LZr Ar

]T −1, (15)

where the time-varying bounded state transformation T has thesame form as in Eq. (7) (it is differentiable and with differen-tiable inverse in continuous-time). The linear operators LZi

,i = 1, . . . , r are such that LZi

Ai = ZiAi + Zi in continuous-time and LZi

Ai = Zi(t + 1)Ai(t) in discrete-time.

4. Output blending

In this section we extend the results obtained so far to thecase when the full state variables are not available for feedbackand we can only rely on some measurements described by thevector

�(t) = Nx(t), (16)

where � ∈ Rp. Hence, the controller takes the form

z(t) = Fz(t) + G�(t),

u(t) = Hz(t) + K�(t), (17)

and the closed-loop system with inputs wi , i = 1, 2, . . . , r andoutputs yi , i = 1, 2, . . . , r , is therefore described by

(18)

Author's personal copy

6 F. Blanchini et al. / Automatica 44 (2008) 1 – 14

In this case the relevant assumption for controller blending isthe following:

Assumption 3. The composed p × s matrix,

EN = [NE1 NE2 . . . NEr ],where s

.= ∑rj=1qj , has full column rank.

Given system (1) and a candidate control of form (17) defineas Hi(s) the transfer function from the ith input wi to the ithoutput yi . Hence, with wj = 0, j �= i we have

yi(s) = Hi(s)wi(s).

Now, assume that

u(t) = Kix(t), i = 1, 2, . . . , r (19)

are assigned stabilizing compensators and define

Ai = A + BKi, Ci = Ci + DiKi .

The corresponding i–i desired transfer function is given by

Hi(s) = Ci(sI − Ai)−1Ei .

The following theorem holds.

Theorem 4.1. There exists a single dynamic compensator ofform (17) of order � = nr − p such that

• the spectrum of the closed-loop system (formed by n × r

elements) satisfies⋃i

�(Ai) ⊆ �(Acl);

• for all i, the transfer function Hi(s) from the ith input to theith output coincide with the one given by the ith gain Ki , i.e.

Hi(s) = Hi(s).

The compensator may be found as follows.

Procedure 4.1.

(1) Let Ki be assigned and let Ai = A + BKi .(2) Take matrices Ei of dimension n×(n−qi) such that [Ei Ei]

is square and invertible (note that this is possible becauseeach Ei has full column rank in view of Assumption 3).

(3) Fix matrices Zi of dimension � × (n − qi) (remind that� = nr − p) such that the matrix[

NE1 NE2 . . . NEr NE1 NE2 . . . NEr

0 0 . . . 0 Z1 Z2 . . . Zr

](20)

is invertible.

(4) Define the � × n matrices as follows:

Zi.= [0 Zi][Ei Ei]−1. (21)

(5) Define

Vi.= ZiAi .

(6) Form the invertible nr × nr-dimensional matrix � as

� =[

N N . . . N

Z1 Z2 . . . Zr

]. (22)

(7) Compute the compensator matrices F, G, H, K as

[K H

G F

]=[K1 K2 . . . Kr

V1 V2 . . . Vr

] [N N . . . N

Z1 Z2 . . . Zr

]−1.

(23)

Again, the missing point concerns Eq. (23). We fix it soonby showing that � is indeed an invertible matrix.

Proof of Theorem 4.1. In order to prove the main theorem, weneed to check the invertibility of matrix � appearing in (22).Consider the following expression:

[N N . . . N

Z1 Z2 . . . Zr

]︸ ︷︷ ︸

�

⎡⎢⎢⎢⎣

[E1 E1] 0 . . . 0

0 [E2 E2] . . . 0

: : . . . :0 0 . . . [Er Er ]

⎤⎥⎥⎥⎦

=[NE1 NE1 NE2 NE2 . . . NEr NEr

0 Z1 0 Z2 . . . 0 Zr

], (24)

and notice that the last matrix is achieved by a column permu-tation of (20), which is invertible by construction, hence � isinvertible. The closed loop system is represented by (18). Con-sider the (tall) matrix

T.=[

I I . . . I

Z1 Z2 . . . Zr

].

The closed-loop system matrix Acl satisfies the equation

AclT =[A + BKN BH

GN F

]T

= T

⎡⎢⎢⎢⎣

A1 0 . . . 0

0 A2 . . . 0

: : . . . :0 0 . . . Ar

⎤⎥⎥⎥⎦

︸ ︷︷ ︸.=Acl

, (25)

which means that the new matrix Acl is the restriction of Aclwith respect to the basis induced by T. Therefore the first as-sertion of the theorem is proved.

Author's personal copy

F. Blanchini et al. / Automatica 44 (2008) 1–14 7

Consider new matrices Ecl and Ccl as follows:

[I I . . . I

Z1 Z2 . . . Zr

]︸ ︷︷ ︸

T

⎡⎢⎢⎢⎢⎣

E1 0 . . . 0

0 E2 . . . 0...

... . . ....

0 0 . . . Er

⎤⎥⎥⎥⎥⎦

︸ ︷︷ ︸Ecl=new input matrix

=[E1 E2 . . . Er

0 0 . . . 0

]︸ ︷︷ ︸

Ecl=old input matrix

,

Ccl = CclT

=⎡⎢⎣

C1 + D1KN D1H

......

Cr + DrKN DrH

⎤⎥⎦[ I I . . . I

Z1 Z2 . . . Zr

]

=

⎡⎢⎢⎢⎢⎣

C1 + D1K1 C1 + D1K2 . . . C1 + D1Kr

C2 + D2K1 C2 + D2K2 . . . C2 + D2Kr

...... . . .

...

Cr + DrK1 Cr + DrK2 . . . Cr + DrKr

⎤⎥⎥⎥⎥⎦

︸ ︷︷ ︸Ccl

,

where we exploited Eq. (23). From the formulas AclT =T Acl, Ccl = CclT and T Ecl = Ecl it follows Ccl(sI −Acl)

−1Ecl = CclT (sI − Acl)−1Ecl = Ccl(sI − Acl)

−1T Ecl =Ccl(sI −Acl)

−1Ecl, so that, in view of the structure of Acl, Ccland Ecl, the proof is concluded. �

Remark 4.1. Notice that the first conclusion of Theorem 4.1,namely

⋃i�(Ai) ⊆ �(Acl), does not guarantee the stability of

the closed-loop system. In order to ensure stability, additionalassumptions (basically minimum-phase requirements) have tobe made as discussed in the final part of this section.

For the simpler exposition, we investigate first the possibilityof finding a stable compensator, while internal stability will bediscussed later. Consider the system of equations

FZi + GN = ZiAi, ZiEi = 0 ∀i. (26)

If we fix a stable matrix F, the unknowns are the elements ofmatrices G ∈ R�×n and Zi ∈ R�×n. Letting again s =∑

iqi itturns out that the number of unknowns and equations are

nun = r × n × � + � × p, neq = r × n × � + � × s.

Therefore, system (26) is generically solvable under Assump-tion 3 which implies s�p. Once the solution is found, we canachieve a stable compensator. However, consider the transfor-mation

�−1 =[N ′

1 N ′2 · · · N ′

r

Z′1 Z′

2 · · · Z′r

]′. (27)

It follows that F = ∑ri=1ZiAiZi , with

∑ri=1ZiZi = I (here

we used Eq. (25)). It turns out that an important restrictionfor strong blending design has to be taken into account (thenatural extension on that imposed on the zeros of (13) and (14))according to Proposition 1. Precisely, let

(28)

By applying the inverse transformation �, it turns out that sys-tem (28) is equivalent to the following one

(29)

where � denotes a non important entry. Hence, the zeros of N

are also eigenvalues of F, the two sets coinciding when EN isinvertible. Therefore, the following proposition holds.

Proposition 2. (i) The controller matrix is not stable if system(28) has unstable invariant zeros.

(ii) The controller matrix can be generically chosen as astability matrix if system (28) is minimum phase.

(iii) If EN is square then the controller matrix is stable ifand only if system (28) is minimum phase.

Finally, we consider the crucial problem of internal stabilityand basically we show that, again, some minimum phase typeof conditions are required. The eigenvalues of the closed-loopsystem are those of matrix Acl. These includes all those of ma-trices Ai = A + BKi , which are stable by construction. How-ever there are additional n − p eigenvalues of a matrix, say W,that can be unstable. Indeed, it is easy to see that there exists a(nr − p) × (n − p) full column rank matrix Y such that

T =[

I I · · · I 0

Z1 Z2 · · · Zr Y

]is invertible. Recalling the definition of �−1 in (27), it turnsout that (the easy check is left to the reader)

AclT =[A + BKN BH

GN F

]T

= T

⎡⎢⎢⎢⎢⎢⎢⎣

A1 0 · · · 0 BK1Z1Y

0 A2 · · · 0 BK2Z2Y

......

. . ....

...

0 0 · · · Ar BKr ZrY

0 0 · · · 0 W

⎤⎥⎥⎥⎥⎥⎥⎦ ,

YW =(

r∑i=1

ZiAZi

)Y . (30)

Author's personal copy

8 F. Blanchini et al. / Automatica 44 (2008) 1 – 14

Thus the eigenvalues of W are a subset of those of the matrixbetween brackets. This last formula points out the relationshipsbetween W and the zeros of a properly defined system. Indeed,consider the system

(31)

and, again, let us apply to this system the inverse transformation�. It turns out that this system is equivalent to the followingone:

(32)

where � denotes a non important entry. When EN is square, theeigenvalues of

∑ri=1ZiAZi coincide with the invariant zeros

of system (31), so that, being W a restriction of∑r

i=1ZiAZi

(see (30)), W (and the closed-loop system) is stable if such asystem is minimum phase. In the simple case r = 1, matrix Ycan be chosen to be the identity (of dimension n − p), so thatthe extra matrix W is given by the formula W = F − Z1BH .

Remark 4.2. Note that we start from the state feedback com-pensators and we wish to recover the same transfer functions.Therefore, some minimum-phase assumptions are crucial, seeDoyle and Stein (1981), even to recover a single closed-loopfunction as in the example below.

Example 4.1. Let r = 1, p = 2, n = 3, and

A =⎡⎢⎣

0 1 0

0 0 1

−1 2 5

⎤⎥⎦ , B =

⎡⎢⎣

0

0

1

⎤⎥⎦ ,

E =⎡⎢⎣

1

0

0

⎤⎥⎦ , N =

[1 4

9 0

0 2 3

],

and the stabilizing control matrix K1 = [−1 − 5 − 10]. For �= 3

2� it results:

F = 2� − 113

2 − 3�, W = −2

3.

Hence, strong stabilizability is possible since W < 0 and F canbe chosen negative as well. Notice that W does not depend onthe free parameters � and and equals the unique invariantzero of the triple (A, E, N).

Now, for the same choice of A, B, E and K1, take

N =[

1 − 329 0

0 −2 3

].

For �= − 32� it results:

F = 2� − 1213

2 + 3�, W = 2

3.

Hence, the controller matrix can be chosen stable (F negative),but stabilizability is not possible. Indeed the closed-loop systemis internally unstable for any choice of � and since the uniqueinvariant zero of (A, E, N) is W = 2

3 .

5. Filter blending

In this section we consider the dual problem of filtering.Given some measured variables y, we aim at finding a filteryielding appropriate estimates zi of the performance variableszi in order to achieve the so-called blending property. Thisproperty is characterized by the fact that all the transfer func-tions from the input noises wi to the estimation errors zi − zi

must equal the transfer functions associated with target localfilters worked out for the single pairs (wi, zi − zi). To put theproblem in the right setting, consider the system

x(t) = Ax(t) + Bu(t) +r∑

j=1

Ejwj (t), (33)

y(t) = Cx(t) +r∑

j=1

Djwj (t), (34)

zi(t) = Hix(t) +r∑

j=1

Mijwj (t), i = 1, 2, . . . , r , (35)

where x(t) ∈ Rn is the state, u(t) ∈ Rm is the input, wj(t) ∈Rqj are the input noises, y(t) ∈ Rp is the measured variable,and zi(t) ∈ Rpi the performance variables.

For each input output pair (wi, zi), under the assumption ofdetectability of the pair (A, C), it is possible to work out astable local filter in observer form

˙x(i)(t) = Ax(i)(t) + Bu + Li(Cx(i)(t) − y(t)), (36)

zi (t) = Hix(i)(t). (37)

Matrix Li is the local output injection matrix which can bedesigned in order to achieve the two following objectives:

(i) Stability of the local filter, i.e. Ai = A + LiC is Hurwitz.(ii) Some desired norm constraint on the input output map Ti (s)

from wi to zi − zi , where

Ti (s) : =Hi(sI − Ai)−1(Ei + LiDi) + Mii . (38)

We want to find a (unique) filter (see Fig. 2) such that eachtransfer function from wk to zk − zk is exactly equal to Tk(s).This problem is referred to as filter blending problem and theassociated filter will be named as blending filter. To work outthe result, we introduce the dual of Assumptions 1 and 2:

(A1) The pair (A, C) is detectable.

Author's personal copy

F. Blanchini et al. / Automatica 44 (2008) 1–14 9

PLANT FILTER

w1

w2

wr

z1ˆ

z2ˆ

zrˆ

z1 z2 zr

y

u

Fig. 2. Filter structure.

(A2) The performance v × n matrix

H = [H ′1 H ′

2 · · · H ′r ]′,

where v : =∑ipi , has full row rank.

Again, notice that assumption (A2) prevents the possibilitythat some output matrices Hi are equal or lined. Now, defineA ∈ Rnr×nr , C ∈ R(p+n(r−1))×nr and � ∈ Rn(r−1)×n(r−1) as

A =[A 0

0 0

], C =

[C 0

0 I

], � =

[L11 L12

L21 L22

]. (39)

Thanks to these definitions of the matrices of the augmentedsystem and filter gain, we can introduce the blending filterstructure as follows:

˙x(t) = Ax(t) + L11(Cx(t) − y(t)) + L12 (t) + Bu(t), (40)

(t) = L21(Cx(t) − y(t)) + L22 (t), (41)

zi (t) = Hix(t), i = 1, 2, . . . , r , (42)

where (t) ∈ Rn(r−1).

Remark 5.1. We stress the following point to avoid trivialities.The purpose of this mechanism is provide a single estimate x ofx. Variables zi are only introduced to express the filter perfor-mances. For instance, we could decide to “clean” different com-ponents of x from the noise introduced by inputs of differentnature (see the example) by setting Hj =[0 . . . 0 1 0 . . . 0]with 1 in the proper position. The problem of estimating allvariables zi by means of zi can be obviously solved by imple-menting all the optimal filters.

The input output map from wi to zi − zi can be written in acompact way as (the computations are left to the reader):

Ti(s) = [Hi 0](sI − A − �C)−1[Ei + L11Di

L21Di

]+ Mii .

The problem is to find � in such a way that, for each i =1, 2, . . . , r

Ti(s) = Ti (s).

The following result provides a closed-form solution to theproblem:

Theorem 5.1. Let the local output injection matrices Li , i =1, 2, . . . , r be given and consider the stable transfer functionsTi (s), i = 1, 2, . . . , r . Under assumptions (A1) and (A2), thereexists a n(r − 1)-dimensional blending filter of form (40)–(42)such that:

(i) The spectrum of the blending filter is given by the union ofthe spectra of the matrices A + LiC.

(ii) For each i = 1, 2, . . . , r , it results Ti (s) = Ti(s).

Proof. The proof is constructive. However, it is only sketchedbecause it can be derived by dualizing the rationale behind theproof of Theorem 2.1 provided in Procedure 2.1. Thanks tothe assumption on the rank of H , there exist r matrices Qi ofdimensions n × n(r − 1) such that matrix

R =[

I I · · · I

Q′1 Q′

2 · · · Q′r

]′

is invertible. The filter matrices are then given by

� = R−1[

L1 L2 · · · Lr

(A1Q1)′ (A2Q2)

′ · · · (ArQr)′

]′. (43)

We now miss to show that this definition of the filter matricesbrings to the desired input output matching. This conclusioncan be drawn by noticing that

R

([A 0

0 0

]+ �

[C 0

0 I

])= diag{Aj , j = 1, . . . , r}R,

R

[Ei + L11Di

L21Di

]= col{Ei + LjDi, j = 1, . . . , r},

i = 1, . . . , r ,

[Hi 0] = [0 · · · Hi︸︷︷︸ith block column

0 · · · 0]R,

i = 1, 2, . . . , r. �

The blending filter is stable, but stability is not a priori guar-anteed under structural variations of the measurement devices.In other words, we would like to guarantee the stability of ma-trix L22 appearing in the filter equations. To this aim, one canget a closer insight into formula (43), by inverting matrix R.Letting S = col{Qi − Q1, i = 2, . . . , r} it follows:

SL22 = diag{Ai , i = 2, . . . , r}S + col{Ai − A1, i = 2, . . . , r}Q1.

If a stable matrix L22 is fixed, the filter blending problem withstability requires finding matrices Qi such that

(Qi − Q1)L22 = AiQi − A1Q1, i > 1, HiQi = 0 ∀i

Letting � = n(r − 1) and v =∑ipi , the number of unknowns

is �nr and the number of equations is �n(r − 1) + �v. Thanksto Assumption (A2) we have v�n, so that the equations aregenerically solvable. Once found matrices Qi , and assuming

Author's personal copy

10 F. Blanchini et al. / Automatica 44 (2008) 1 – 14

sku

θ

d

Fig. 3. A scheme of the cart-pole system.

that the transformation R is invertible, one can compute theblending filter matrices from (43).

Remark 5.2. The observations made for the blending controldesign can be cast in the realm of filtering design by a simpledualization of the relevant results. This includes the possibilityof building a blending filter from target filters which are notin observer form, the extension to time-varying systems and todiscrete-time systems (relatively optimal filter design).

6. Experiments and simulations

Consider the cart-pole system whose scheme is reportedin Fig. 3. The system state vector is x = [� � s s]′. Thecontinuous–time model, linearized around a stable equilibriumpoint has the following state and input matrices

A =

⎡⎢⎢⎢⎣

0 1 0 0

−19.62 −0.125 0 −9.886

0 0 0 1

0 0 0 −4.943

⎤⎥⎥⎥⎦ , B =

⎡⎢⎢⎢⎣

0

11.53

0

5.767

⎤⎥⎥⎥⎦ .

Part 1: In this part we tackle the control problem. To thisend, the model we consider is

x(t) = Ax(t) + Bu(t) + E1w1(t) + E2w2(t),

y1(t) = C1x(t) + D1u(t),

y2(t) = C2x(t) + D2u(t), (44)

with D1 = D2 = 0 and

E1 = [0 0 1 0]′, E2 = [0 1 0 0]′,C1 = [0 0 1 0], C2 = [1 0 0 0].For this system we are interested in solving two different prob-lems, namely an optimal step response problem and a distur-bance attenuation problem for the pendulum oscillation givendisturbances d acting as indicated in the figure.

In the step response problem, the primary interest is the cartposition to be driven to zero, consequently (assuming 0 thearrival state) we take into account an initial condition of theform [0 0 �s 0]′. This justifies the choice of E1 =[0 0 1 0]′.Since the cart position is the main concern the natural outputis y1(t)= s(t) and this justifies the choice C1 =[0 0 1 0] andD1 = 0. We consider a standard LQ regulator with high costcorresponding to the cart position:

J =∫ ∞

0(�2x′(t)Qx(t) + u2(t)) dt ,

with Q = diag{0.1, 0, 1, 0} and � = 17.32. The correspond-ing optimal compensator turns out to be u = K1x with K1 =[−3.9799 0.73366 −17.321 −3.7031], leading to the closed-loop eigenvalues {−7.8451±7.3929i and −1.1352±3.9466i}.

For the disturbance attenuation problem we should choose adifferent objective. We consider the H∞-type problem of min-imizing the peak of the frequency response from input d to theangle: then we choose E2 = [0 1 0 0]′, C2 = [1 0 0 0] andD2 =0. By solving the attenuation problem we obtain the feed-back gain K2=[−8.0415 −0.65903 −1.6897 −1.9322], withclosed-loop eigenvalues {−17.805, −0.8338, −2.5859 ±2.4881i}.

In Fig. 4 (left) it is reported the step response of the w1–y1transfer function for the two state feedback gains K1 and K2.It is apparent that the controller K1 (solid line) drives the cartto a neighborhood of the origin basically two times faster thanthe controller K2 (dotted line) hence it outperforms controllerK2 with respect to the step response objective (in the rightside of the same figure, the angle of the pole during the steptransient is reported). The situation is reversed as long as thedisturbance rejection problem is taken into account: in Fig. 5(right) it is reported the impulse response for the transfer func-tion w2–y2 for the two controllers. It is clear that in this casecontroller K2 behaves much better than K1, that results in apoor damping of the oscillations. By means of our techniquewe can simultaneously achieve the two patterns. According toProcedure 2.1, we computed the following (stable) compensator

F =

⎡⎢⎢⎢⎣

11.4022 7.1768 −31.1887 −10.4908

0.858881 −17.7704 −17.3809 5.63393

13.0299 5.41171 −26.5652 −9.69492

−18.8215 10.2619 30.1316 3.92007

⎤⎥⎥⎥⎦ ,

H = [−1.86191 −0.242811 3.52709 1.2161] ,

K = [12.3277 −0.659 −17.3205 −0.0155166] .

As shown in Fig. 6, the two input–output behaviors of thestatic controllers K1 and K2 are simultaneously achieved bythe dynamic blending controller: precisely, the step response is(close to) that of K1 and the impulse response is (close to) thatof K2. Movies of the experimental results are available at theURL: http://control.units.it/pellegrino.

Part 2: In this part we present, as an example of applicationof filter blending, the problem of estimating velocities basedon position and angle measurements for the described equip-ment. We present simulation results only, because the currentexperimental setup is not suitable for measuring the linear andthe angular speed exactly. Consider the system

x(t) = Ax(t) + E1w1(t) + E2w2(t),

y(t) = Cx(t) + D1w1(t) + D2w2(t),

z1(t) = H1x(t),

z2(t) = H2x(t), (45)

Author's personal copy

F. Blanchini et al. / Automatica 44 (2008) 1–14 11

0 2 4 6 8−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

cart

positio

n [m

]

time [s]

0 2 4 6 8−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

pole

angle

[ra

d]

time [s]

Fig. 4. Left: step response for the transfer function 1–1 with K1 (solid line) and K2 (dotted line). Right: the behavior of the angle of the pole during thesame transient.

0 2 4 6 8-0.2

-0.15

-0.1

-0.05

0

0.05

cart

positio

n [m

]

time [s]

0 2 4 6 8-0.3

-0.2

-0.1

0

0.1

0.2

pole

angle

[ra

d]

time [s]

Fig. 5. Right: impulse response for the transfer function 2–2 with K1 (solid line) and K2 (dotted line). Left: the behavior of the cart position during the sametransient.

0 2 4 6 8-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

cart

positio

n [m

]

time [s]

0 2 4 6 8-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

pole

angle

[ra

d]

time [s]

Fig. 6. Step response (solid line) and impulse response (dotted line) for the compensator obtained by blending K1 and K2.

Author's personal copy

12 F. Blanchini et al. / Automatica 44 (2008) 1 – 14

with A as in the first part and

E1 =

⎡⎢⎢⎢⎣

0 0

0 0.01

0 0

1 0

⎤⎥⎥⎥⎦ , E2 =

⎡⎢⎢⎢⎣

0 0

0 1

0 0

0.01 0

⎤⎥⎥⎥⎦ , C =

[1 0 0 0

0 0 1 0

],

D1 = D2 =[

0.01 0

0 1

], H1 = [0 1 0 0 ] ,

H2 = [0 0 0 1] .

Matrix C reflects the fact that we have two measurements (an-gular displacement of the pole and the position of the cart). Asfor H1 and H2, they express the need of estimating the twovelocities. Matrices D1 and D2 (which are equal) explain thefact that the measurement on the angular position is much moreaccurate than that on the position of the cart. Finally, matri-ces E1 and E2 take into account the model mismatch in thetwo dynamic equations. In the first filtering problem we findan estimate z1 of z1 minimizing the H2 norm between w1 andthe estimation error z1 − z1. The solution is a filter in observerform with gain L1. Notice that being A − E1D

−11 C unstable,

the trivial solution L1 =−E1D−11 that would give a zero norm

is not feasible. In the second problem we find the estimate z2of z2 that bounds the H∞ norm of the transfer function fromw2 to the estimation error. Again, being A − E2D

−12 C unsta-

ble, the trivial solution L2 = −E12D−12 that would give a zero

norm is not feasible. As a value of the attenuation level we havechosen � = 0.08. The resulting gains are

L1 =

⎡⎢⎢⎢⎣

−0.0222 0

0.0003 −1

0.3915 −0.0007

−0.9955 0

⎤⎥⎥⎥⎦ ,

L2 =

⎡⎢⎢⎢⎣

−0.152242 0.0016

−116.2057 0.0021

15.4195 −0.0017

21.3152 −0.0127

⎤⎥⎥⎥⎦ .

With these gains the target transfer functions T1(s) and T2(s)

in (38) are fixed. In order to compare the results, we denote asT12(s) the transfer function T1(s) with L1 replaced by L2 andwith T21(s) the transfer function T2(s) with L2 replaced byL1. The filter assumes the form (40)–(42) with suitable �11,�21, �12, �22. In Fig. 7 it is plotted the response of the targettransfer function T1(s) in (38) to an impulse at the first channelof w1, and compared with the response of T12(s). As apparentthe gain L1 ensures a smooth response to zero error whereasL2 would bring to an oscillatory behavior. In order to illustratethe characteristics of the target function T2(s), we have takenthe second input of w2 and drawn in Fig. 8 the associatedBode plots due to L2 and L1. Again, we can see the betterperformance of L2 in terms of attenuation level with respectto L1. Finally, we have taken the complete blending filter.In Fig. 9 it is plotted the response of the errors z1 − z1 and

0 0.5 1 1.5 2 2.5 3 3.5 4-1.5

-1

-0.5

0

0.5

1

1.5

Impulse Response

Time (sec)

Am

plit

ude

Fig. 7. Response to an impulse at the first component of w1 of T1(s) (smooth)and T12(s) (oscillatory).

-150

-100

-50

0

Magnitude (

dB

)

10-4 10-3 10-2 10-1 100 101 102 103

-90

0

90

180

Phase (

deg)

Bode Diagram

Frequency (rad/sec)

Fig. 8. Bode plots of the second component of T2(s) (down) and T21(s) (up).

z2 − z2 due to an impulse at the first component of w1.In Fig. 10 the Bode plots are drawn of the transfer functionsfrom w2 to z2 − z2 and z1 − z1.

7. Conclusions

In this paper we have introduced a new design control tech-nique, called control blending. Roughly speaking, this tech-nique consists in the construction of dynamic state-space com-pensator that simultaneously achieves different (and possiblyconflicting) target objectives that are defined off-line throughdifferent controllers. To this aim, a simple procedure has beenintroduced which requires only matrix manipulations and linearprogramming tools. The blending procedure does not depend

Author's personal copy

F. Blanchini et al. / Automatica 44 (2008) 1–14 13

0 0.5 1 1.5 2 2.5 3 3.5 4-1.5

-1

-0.5

0

0.5

1

Impulse Response

Time (sec)

Am

plit

ude

Fig. 9. Response to an impulse at the first component of w1 of z1 − z1(down) and z2 − z2 (up).

-150

-100

-50

0

Magnitude (

dB

)

10-4 10-3 10-2 10-1 100 101 102 103

-90

0

90

180

270

360

Phase (

deg)

Bode Diagram

Frequency (rad/sec)

Fig. 10. Bode plots of the transfer functions from the second component ofw2 to z2 − z2 (down) and z1 − z1 (up).

on the criteria which are adopted in the determination of thecompensators, which can be independently designed. Aside im-portant subproblems have been also investigated, for instancethe possibility of coming up with a stable blending compensatorand the extension to discrete-time and/or time-varying systems.

The dual counterpart of filter blending has been also devised.At first glance, putting together a blending compensator anda blending filter does not seem to bring to a blending outputcompensator, since the well-known separation principle doesnot directly hold in this case. Therefore, further work is neededto investigate the output feedback case, and the critical issue ofimposing the internal stability constraints (Kucera, 1991;Youla,Jabr, & Bongiorno, 1976) in the blending procedure.

Acknowledgments

The authors thank ProfessorYasumasa Fujisaki for suggestingimportant references. Helpful comments from Professor MarioRotea were sincerely appreciated.

References

Abedor, J., Nagpal, K., & Poolla, K. (1996). A linear matrix inequalityapproach to peak-to-peak gain minimization. Linear matrix inequalitiesin control theory and applications. International Journal on Robust andNonlinear Control, 6(9–10), 899–927.

Blanchini, F., & Pellegrino, F. A. (2003). Relatively optimal control and itslinear implementation. IEEE Transactions on Automatic Control, 48(12),2151–2162.

Blanchini, F., & Pellegrino, F. A. (2006). Relatively optimal controlwith characteristic polynomial assignment and output feedback. IEEETransactions on Automatic Control, 51(2), 183–191.

Chen, B. M., Mareels, M. Y., Zheng, Y., & Zhang, C. (2000). Solutions todisturbance decoupling problem with constant measurement feedback forlinear systems. Automatica, 36(11), 1717–1724.

Colaneri, P., Geromel, J. C., & Locatelli, A. (1997). Control theory anddesign—an RH 2 and RH∞ viewpoint. New York: Academic Press.

Chu, D., & Mehrmann, V. (2000). Disturbance decoupling for descriptorsystems by state feedback. SIAM Journal on Control Optimization, 38(6),1830–1858.

Chu, D., & Mehrmann, V. (2001). Disturbance decoupling for linear time-invariant systems: a matrix pencil approach. IEEE Transactions onAutomatic Control, 46(5), 802–808.

Chu, D., & Van Dooren, P. (2006). A novel numerical method forexact model matching problem with stability. Automatica, 42(10),1621–1834.

Dahleh, M. A., & Pearson, J. B. (1987). l1-Optimal feedback controllers forMIMO discrete-time systems. IEEE Transactions on Automatic Control,32(4), 314–322.

Dahleh, M. A., & Pearson, J. B. (1988). Minimization of a regulated responseto a fixed input. IEEE Transactions on Automatic Control, AC33(10),924–930.

Diaz-Bobillo, I. J., & Dahleh, M. A. (1992). State feedback l1-optimalcontrollers can be dynamic. Systems Control Letters, 19(2), 87–93.

Doyle, J. C., Glover, K., Khargonekar, P. P., & Francis, B. A. (1989).State-space solutions to standard H2 and H∞ control problems. IEEETransactions on Automatic Control, 34(8), 831–847.

Doyle, J. C., & Stein, G. (1981). Multivariable feedback design: Conceptsfor a classical/modern synthesis. IEEE Transactions on Automatic Control,26, 4–16.

Hu, T., & Lin, Z. (2001). Control of systems with actuator saturation. Boston,MA: Birkhauser.

Khargonekar, P. P., & Rotea, M. A. (1991). Mixed H2/H∞ control: A convexoptimization approach. IEEE Transactions on Automatic Control, 36(7),824–837.

Khargonekar, P. P., Rotea, M. A., & Sivashankar, N. (1993). Exact andapproximate solutions to a class of multiobjective controller synthesisproblems. American control Conference (Vol. 2, pp. 1602–1606), SanFrancisco, CA.

Kucera, V. (1991). Analysis and design of discrete linear control systems,Prentice-Hall international series in systems and control engineering.Englewood Cliffs, NJ: Prentice-Hall International.

Kwakernaak, H., & Sivan, R. (1972). Linear optimal control systems. NewYork–London–Sydney: Wiley-Interscience.

Rotea, M. A. (1990). Multiple objective and robust control for linear systems.Ph.D. Dissertation, University of Minnesota.

Rotea, M. A., & Khargonekar, P. P. (1991). H2-optimal control with an H∞constraint: The state-feedback case. Automatica, 27(2), 307–316.

Author's personal copy

14 F. Blanchini et al. / Automatica 44 (2008) 1 – 14

Saberi, A., Stoorvogel, A., & Sannuti, P. (2000). Control of linear systemswith regulation and input constraints. London: Springer.

Sanchez-Peña, & Sznaier, M. (1998). Robust systems theory and Applications.New York: Wiley.

Sznaier, M., Rotstein, H., Bu, J., & Sideris, A. (2000). An exact solution tocontinuous-time mixed H2–H∞ control problems. IEEE Transactions onAutomatic Control, 45(11), 2095–2101.

Willems, J. C., & Commault, J. C. (1981). Disturbance decoupling bymeasurement feedback with stability or pole placement. SIAM Journal onControl Optimizations, 19(4), 490–504.

Youla, D. C., Jabr, H. A., & Bongiorno, J. J. (1976). Modern Wiener-Hopfdesign of optimal controllers II. The multivariable case. IEEE Transactionson Automatic Control, AC-21(3), 319–338.

Zhou, K., Doyle, J. C., & Glower, K. (1996). Robust and optimal control.Englewood Cliff, NJ: Prentice-Hall.

Franco Blanchini was born in Legnano (MI)on 29 December 1959. He is Full Professor atthe Engineering Faculty of the University ofUdine where he teaches Dynamic System The-ory and Automatic Control. He is a Memberof the Department of Mathematics and Com-puter Science of the same university and he isDirector of the System Dynamics Laboratory.He has been Associate Editor of Automaticafrom 1996 to 2006. In 1997 he was memberof the Program Committee of the 36th IEEEConference on Decision and Control, S. Diego,

California. In 1999 he was member of the Program Committee of the 38thIEEE Conference on Decision and Control, Phoenix, Arizona. In 2001 hewas member of the Program Committee of the 40th IEEE Conference onDecision and Control, Orlando, Florida, Arizona. In 2003 he was Memberof the Program Committee of the 42th IEEE Conference on Decision andControl, Maui, Hawaii. He was Chairman of the 2002 IFAC Workshop onRobust Control, Cascais, Portugal. He has been Program Vice-Chairman forthe conference Joint CDC-ECC, Seville, Spain, December 2005. He is therecipient of 2001 ASME Oil and Gas Application Committee Best PaperAward as a co-author of the article “Experimental Evaluation of a High-GainControl for Compressor Surge Instability”. He is the recipient of the 2002IFAC prize survey paper award as author or the article “Set Invariance inControl—A Survey”, Automatica, November, 1999. He is recipient of theaward Automatica Certificate of Outstanding Service.

Patrizio Colaneri was born in Palmoli, Italy on12 October 1956. He received the Doctor’s de-gree (Laurea) in Electrical Engineering in 1981from the Politecnico di Milano, Italy, and thePh.D. degree (Dottorato di Ricerca) in Auto-matic Control in 1987 from the Ministero dellaPubblica Istruzione of Italy. From 1982 to 1984,he worked in industry on simulation and con-trol of electrical power plants. From 1984 to1992, he was with the Centro di Teoria dei Sis-temi of the Italian National Research Council(CNR). He spent a period of research at the

Systems Research Center of the University of Maryland and held a visitingposition at the Johannes Kepler University in Linz since 2000. He is currentlyProfessor of Automatica at the Faculty of Engineering of the Politecnico diMilano. Dr. Colaneri was a YAP (Young Author Prize) finalist at the 1990IFAC World Congress, Tallin, USSR. He is the Chair the IFAC TechnicalCommittee on Control Design, a Member of the IFAC Technical Committeeon Robust Control, a Senior Member of the IEEE, a Member of the Council ofEUCA (European Union Control Association) and a Member of the editorialboard of International Journal of Applied and Computational Mathematics.He was a member of the International Program Committee of the 1999Conference of Decision and Control. Dr. Colaneri has been serving for sixyears as Associate Editor of Automatica. His main interests are in the area ofperiodic systems and control, robust filtering and control, digital and multiratecontrol, switching control. On these subjects, he has authored or co-authoredabout 170 papers, two books (in Italian) and the book “Control SystemsDesign: an RH-2 and RH-infinity viewpoint”, published by Academic Pressin 1997.

Felice Andrea Pellegrino was born inConegliano (Italy) in 1974. He received theLaurea degree Magna cum Laude in Engineer-ing from the University of Udine, Italy, in Juneof 2000 and the Ph.D. degree from the sameuniversity in May of 2005, discussing a thesison Constrained and Optimal Control. From2001 to 2003 he was Research Fellow at Inter-national School for Advanced Studies (Trieste,Italy). From 2005 to 2006 he was ResearchFellow at DIMI (Dipartimento di Matematica eInformatica, University of Udine). At present

he is Research Assistant at DEEI (Dipartimento di Elettrotecnica, Elettronicae Informatica, University of Trieste). His research interests include ControlTheory, Computer Vision and Pattern Recognition.