Systems & Control Letters 61 (2012) 292–297

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Systems & Control Letters

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Pole placement by parametric output feedbackUlrich Konigorski ∗Institute of Automatic Control, Control Engineering and Mechatronics Lab., Technische Universität Darmstadt, Landgraf-Georg-Strasse 4, 64283 Darmstadt, Germany

a r t i c l e i n f o

Article history:Received 29 September 2011Received in revised form18 November 2011Accepted 21 November 2011Available online 3 January 2012

Keywords:Pole placementOutput feedbackEigenstructure assignment

a b s t r a c t

This note presents a new analytical solution to the problem of pole placement via constant outputfeedback under the condition m + p ≥ n, where n, m, and p are the number of states, inputs andoutputs, respectively. The approach is based upon parametric eigenstructure assignment of linear time-invariantmultivariable systems in combinationwith a special explicit formulation of the pole assignmentequations. Thus, the resulting analytical solution explicitly offers all remainingmp−n degrees of freedombeyond eigenvalue assignment which can be used for additional design goals such as response shaping,minimizing the norm of the feedback matrix, and robust control, respectively.

© 2011 Elsevier B.V. All rights reserved.

1. Introduction

Besides optimal control the pole placement approach is one ofthe most popular design methods in linear control theory. Whilein the case of complete state feedback the problem of finding aconstant feedback matrix which assigns an arbitrary selected setof self-conjugate complex numbers as spectrum of the closed-loop system is completely solved (see e.g. [1]) to deal with staticoutput feedback is much harder. This is mainly due to the factthat in multi-input–multi-output (MIMO) systems pole placementis a nonlinear problem and demands solving a set of nonlinearalgebraic equations in the unknown gain parameters whosesolution may not exist in the case of output feedback. However,for controllable systems and state feedback these equations alwayshave a solution [1] and in MIMO systems this solution is even notunique. In this case, there are additional degrees-of-freedom (dof)beyond pole placement which can be used for further design goalssuch as eigenvector or eigenstructure assignment.

Based on the results in [2] on eigenstructure assignment inthe case of state feedback several solutions to the problem ofpole placement by static output feedback have been reportedduring the last three decades [3–10] and just recently another newapproach has been presented in [11,12]. Most of them rely on thefundamental result of [13,14], which is also known as Kimura’scondition, that for the generic system all closed-loop poles can beassigned almost arbitrary ifm+p ≥ n+1where n,m, p denote thesystem order and the number of inputs and outputs, respectively.

∗ Tel.: +49 6151163014; fax: +49 6151166114.E-mail address: ukonigorski@iat.tu-darmstadt.de.

0167-6911/$ – see front matter© 2011 Elsevier B.V. All rights reserved.doi:10.1016/j.sysconle.2011.11.015

On the other hand in [15] it was shown that a necessary andsufficient condition for arbitrary pole assignment for the genericsystem is mp ≥ n if complex feedback gains are allowed andfinally in [16,17] Wang presented the important result that forreal static output feedback mp > n is sufficient for generic poleassignability. However, up to now there is no closed-form solutionto the problem of finding a real feedback under Wang’s conditionwhich is certainly due to the fact that in general pole placementvia static output feedback is N P -hard [18]. Moreover, if mp > none is interested in a solution which encompasses all remainingmp − n dof beyond pole placement. Meanwhile, under Kimura’scondition there exist several design techniques (see e.g. [6,7,10])which offer such a parametric solution and in [11] the authorspresented a new noniterative approach to pole placement basedon eigenstructure assignment which improves Kimura’s sufficientcondition to m + p ≥ n while in [12] this result was extended toeven encompass some cases for whichm+ p < n < mp. A generaloverview on static output feedback which covers several differentdesign techniques can be found in [19].

In this note, we seize the suggestions from [10,11] and by com-bining them with the results from [20] we are able to developa straightforward noniterative procedure for pole placement byparametric output feedback. In Section 2, after statement of theproblem the fundamental properties of eigenstructure assignmentare shortly reviewed and for m = p = 2 an analytical expressionfor the direct solution of the pole assignment equation is devel-oped. Based on these preliminary results a closed-form parametricsolution to the problemof pole placement by constant output feed-back under the conditionm+p ≥ n is presented in Section 3whileSection 4 gives a numerical example before themain results of thisnote are summarized in Section 5.

U. Konigorski / Systems & Control Letters 61 (2012) 292–297 293

2. Problem statement and preliminaries

In this section, the problem of pole placement by constantoutput feedback is stated and some fundamental results oneigenstructure assignment as well as a formulation of the problembased on an explicit expression of the closed loop characteristicpolynomial are shortly reviewed.

Consider the completely controllable and observable lineartime-invariant multivariable system

x = Ax + Bu, y = Cx (1)

where x ∈ Rn, u ∈ Rm, y ∈ Rp. The real constant matricesA, B, and C are of appropriate dimensions and it is assumed thatrank(B) = m ≥ 2 and rank(C) = p ≥ 2. If the real constant outputfeedback

u = Ky (2)

is applied to (1) the closed-loop state equation becomes

x = (A + BKC)x = Acx. (3)

The problem of pole placement or eigenvalue assignment byconstant output feedback then means to find the real matrix K in(2) such that the spectrum σ {Ac} of Ac coincides with the given setΛ = {λ1, λ2, . . . , λn} of self-conjugate complex values.

2.1. Eigenstructure assignment

In the context of eigenstructure assignment by constant outputfeedback the set Λ = {Λ1, Λ2} of closed-loop eigenvalues isusually divided into two self-conjugate sets of arbitrarily selecteddistinct complex numbers Λ1 = {λ1, λ2, . . . , λr} and Λ2 =

{λr+1, λr+2, . . . , λn}. Then in a first step the set Λ1 is associatedwith the spectrum of Ac and the closed-loop eigenvectors vi via

Acvi = (A + BKC)vi = λivi, i = 1, . . . , r (4)

which can also be written as

[A − λiI, B]

viKCvi

= 0, i = 1, . . . , r. (5)

If (A, B) is completely controllable rank[A − λiI, B] = n, ∀λi ∈

C [21] and thus the right nullspace

ker{[A − λiI, B]} = ker{Si} =

NiMi

(6)

is of dimension m. Then the closed-loop eigenvectors vi and theinput directions

hi = KCvi (7)

can be parameterized by nonzero parameter vectors qi ∈ Cm (seee.g. [2,10,11])

vi = Niqi (8)hi = Miqi. (9)

Remark 1. Since the columns of ker{Si} can arbitrarily be scaledby any nonzero scalar the eigenvectors vi and input directions hi in(8), (9) are only determined except for their length and so are theparameter vectors qi. Thus, each parameter vector only provides(m − 1) dof to assign the corresponding eigenvector vi within them-dimensional subspace of Cn spanned by the columns of Ni [2].Moreover, it directly follows from elementary matrix theory thatfor a self-conjugate complex pair λi1 = λ∗

i2 in the set Λ1 we alsohave vi1 = v∗

i2,Ni1 = N∗

i2 and this implies qi1 = q∗

i2. Therefore, theparameter vectors qi associated with the setΛ1 are not completelyfree but must also constitute a self-conjugate set. However, thisdoes not reduce the available dof provided by the set qi, i =

1, . . . , r since a complex qi offers m − 1 complex and 2(m − 1)real dof, respectively.

Nowwe come back to (8), (9) and substitute them into (7) to getthe homogeneous equation

(Mi − KCNi)qi = 0 (10)

which has a nonzero solution qi = 0 iff

det(Mi − KCNi) = 0. (11)

Obviously, (11) can be used to assign λi as closed-loop pole [22]while the parameter vector qi associated with λi via (10) thenexplicitly offers (m − 1) additional dof beyond eigenvalueassignment. Thus, to assign the r numbers in Λ1 as closed-loopeigenvalues K must solve the linear equation

K(CVr) = Qr (12)

where the qi = 0, i = 1, . . . , r are considered as free parametersand

Vr = [N1q1, . . . ,Nrqr ] (13)

Qr = [M1q1, . . . ,Mrqr ]. (14)

Obviously, a solution of (12) exists for almost any choice of the setΛ1 and qi = 0, i = 1, . . . , r if the condition rank(CVr) = r holdswhich in turn implies r ≤ p. For r = p, the usual choice in the lit-erature on eigenstructure assignment, the solution K = Qr(CVr)

−1

of (12) explicitly exhibits all mp dof provided by K ∈ Rm×p in theshape of the p eigenvalues from Λ1 and the p corresponding pa-rameter vectors qi = 0, i = 1, . . . , p. Thus, to assign the remainingn−p eigenvalues inΛ2 the parameter vectors qi = 0, i = 1, . . . , pare not arbitrary butmust undergo some restrictions. In the follow-ing this can be seen if all investigations carried out so far with righteigenvectors vi, input directions hi and (right) parameter vectors qiare accomplishedwith left eigenvectorsw′

j , output directions l′

j andcorresponding (left) parameter vectors z ′

j for the eigenvalues inΛ2where the prime denotes transpose. To this end instead of (4) westart with the relation

w′

jAc = w′

j(A + BKC) = λjw′

j, j = r + 1, . . . , n (15)

or

[w′

j, w′

jBK ]

A − λjI

C

= 0′, j = r + 1, . . . , n (16)

where

rankA − λjI

C

= rank(Tj) = n, ∀λj ∈ C

if (A, C) is an observable pair [21]. Therefore, the p-dimensionalleft nullspace of Tj can be calculated from

[Dj, Ej] · Tj = 0 (17)

and the closed-loop left eigenvectors w′

j and output directions

l′j = w′

jBK (18)

are parameterized by nonzero left parameter vectors z ′

j ∈ Cp

w′

j = z ′

jDj (19)

l′j = z ′

jEj. (20)

Finally, from (18)–(20) we get the dual version of (10)

z ′

j (Ej − DjBK) = 0′ (21)

with the corresponding necessary condition

det(Ej − DjBK) = 0 (22)

and

(W ′

n−rB)K = Z ′

n−r (23)

294 U. Konigorski / Systems & Control Letters 61 (2012) 292–297

with

Wn−r = [D′

r+1zr+1, . . . ,D′

nzn] (24)

Zn−r = [E ′

r+1zr+1, . . . , E ′

nzn] (25)

as dual version of (12). Combining (10) and (21) results in a bilinearequation with respect to qi, i = 1, . . . , r and zj, j = r + 1, . . . , n(see [10,11]). Other approaches to eigenstructure assignmentby output feedback solve (10), (21) via two coupled Sylvesterequations [4,5,8] or a bilinear generalized Sylvester equation [9].

In the sequel it is shown that under the condition m + p ≥ n aclosed form parametric solution to the pole assignment problemcan be obtained without referring to the solution of a bilinearmatrix equation. To that purpose (10), (21) are combined withthe direct evaluation of the pole-assignment Eq. (11). By meansof exterior algebra the following section summarizes the resultsin [20] on the general formulation of the closed-loop characteristicequation for the special case K ∈ R2×2.

2.2. Direct solution of the pole assignment equation

As already discussed in the previous section for λi to be a rootof the closed-loop characteristic polynomial Pc(λ) = det(λI −

A − BKC) the constant output feedback matrix K must solve oneof the Eqs. (11) and (22), respectively. In [20] the expansion ofPc(λ) has been discussed in great detail and it turns out that forarbitrary m and p the pole-assignment equations are multilinearin the coefficients of K and therefore an analytical solution is notobvious. However, in what follows we concentrate on the specialcase K ∈ R2×2,Mi ∈ R2×2, and CNi ∈ R2×2 where an analyticalsolution of (11) can still be obtained. To that purpose the evaluationof the determinant in (11) by means of the exterior product of itscorresponding column vectors is used and we write

K = [k1, k2] =

k11 k12k21 k22

and

Mi = [µi1, µ

i2] = {µi

jl}

CNi = [ηi1, η

i2] = {ηi

jl}

respectively. Then taking into account the special properties ofthe exterior product x ∧ y of two vectors x, y (e.g. distributivity,associativity, etc.) and after some elementary calculations theevaluation of

det(Mi − KCNi) = (µi1 − Kηi

1) ∧ (µi2 − Kηi

2) = 0

finally results in

det(Mi − KCNi) = det(Mi) − δ′

i k = 0 (26)

with

δ′

i =[ηi

11, ηi12]Πi, [η

i21, η

i22]Πi, − det(CNi)

k′

= [k11, k21, k12, k22, det(K)] (27)

Πi =

µi

22 −µi12

−µi21 µi

11

.

With (10), (21) and (26) we are now able do derive the main resultof this note.

3. Parametric pole assignment

To avoid a bilinear relation between the right and left parametervectors qi and z ′

j inwhat followswe consider a consecutive solutionof (10), (21) or (12), (23), respectively. In a first stepwe assign the reigenvalues inΛ1 via solving (12) while at the same time retainingenough dof to subsequently solve the remaining n− r Eqs. (23) for

Λ2. To that purpose we choose r < p and nonzero column vectors[q1, . . . , qr ] such that (CVr) in (12) has full rank r . Then K can besolved from (12)

K = Qr(CVr)+

+ K1U ′

1 = K0 + K1U ′

1 (28)

where K0 = Qr(CVr)+ while (CVr)

+= [(CVr)

′(CVr)]−1(CVr)

′

denotes the Moore–Penrose pseudo-inverse of CVr and U ′

1 is a(p − r) × p orthonormal matrix whose rows form a basis for theleft kernel of (CVr), i.e.,

U ′

1(CVr) = C1Vr = 0. (29)

By assumption, rank(C) = p and thus rank(U ′C) = rank(C) = pwhere U = [U1,U⊥

1 ] is a full rank p × p orthonormal matrix withU ′

1U⊥

1 = 0. Therefore, all rows of C are linear independent and thedownsized output matrix C1 = U ′

1C always has full rank p − r . Asshown in the Appendix the output feedback K given by (28) assignsthe r numbers in Λ1 as closed-loop eigenvalues of

Ac = A + BK0C + BK1U ′

1C = A1 + BK1C1 (30)

for arbitrary qi = 0, i = 1, . . . , r and K1 ∈ Rm×(p−r) andrenders the pair (A1, C1) unobservable for λ ∈ Λ1. However,since all unobservable eigenvalues of (A1, C1) cannot be changedby the constant output feedback K1, if some eigenvalues of A1 notbelonging to Λ1 are unobservable, they also remain fixed in thesubsequent design steps. Applying the coordinate transformationx = [Vr , R]xwith R′Vr = 0, R′R = In−r to (A1, C1) results in

A1 =

V+

r A1Vr V+

r A1RR′A1Vr R′A1R

=

diag{Λ1} V+

r A1R0 R′A1R

C1 =

C1Vr C1R

=

0 C1R

since according to (A.2) the columns of Vr are eigenvectors of A1corresponding to the r unobservable eigenvalues in Λ1. i.e. A1Vr =

Vr diag{Λ1} and C1Vr = 0. Then a straightforward investigation ofthe PBH eigenvector test (A.2) for (A1, C1) reveals that λ ∈ Λ1 isunobservable, if there exist a vector v2 = 0 of dimension n − rwhich simultaneously solves the two equations

C1Rv2 = U ′

1CRv2 = 0 (31)R′A1R − λIn−r

v2 = 0. (32)

Obviously, from (31) we get

Rv2 = C⊥β + Vrζ

v2 = R′C⊥β

with ζ , β = 0 arbitrary vectors of appropriate dimensions.Substituting these two latter expressions for Rv2 and v2 in (32)finally results in the condition

R′Hoβ = 0, β = 0 (33)

with Ho = (A − λI)C⊥. Since C⊥∈ Rn×(n−p), R′

∈ R(n−r)×n andr < p (33) can only be fulfilled if rank(R′Ho) < n − p or in otherwords any eigenvalue λ ∈ Λ1 of A1 is observable if

rank(R′Ho) = n − p (34a)

which can equivalently be written

Im(Ho) ∩ Im(Vr) = {0} (34b)

(see also Theorem 2.3 and its proof in [23]). But since Vr (andthus R) depend on the choice of the eigenvalues in Λ1 and theparameter vectors q1, . . . , qr either (34) is never true, which canbe seen as singular case, or only violated on a hypersurface of theparameter space Po = {Λ1, q1, . . . , qr}. Hence, we may assumethat generically rank(R′Ho) = n− p and thus λ ∈ Λ1 is observablefor almost all parameter values in the set Po.

U. Konigorski / Systems & Control Letters 61 (2012) 292–297 295

Therefore K1 can subsequently be used to solve (23) for(A1, B, C1) whereas only the r eigenvalues Λ1 of A1 are unobserv-able and thus remain unchanged. So we choose ρ self-conjugatenumbersΛρ = {λr+1, . . . , λr+ρ} from the setΛ2 and nonzero rowvectors [z1, . . . , zρ] such that (W ′

ρB) in (23) has rank ρ which im-plies ρ ≤ m. Then K1 can be solved from (23)

K1 = (W ′

ρB)+Z ′

ρ + U2K3 = K2 + U2K3 (35)

where K2 = (W ′ρB)

+Z ′ρ while (W ′

ρB)+

= (W ′ρB)

′[(W ′

ρB)(W′ρB)

′]−1

denotes the Moore–Penrose pseudo-inverse of W ′ρB and U2 is an

m × (m − ρ) orthonormal matrix whose columns form a basis forthe right kernel ofW ′

ρB, i.e.,

(W ′

ρB)U2 = W ′

ρB2 = 0. (36)

Obviously, U2 and thus K3 and the downsized input matrix B2 =

BU2 only exist if ρ < m and by a similar argumentation as beforefor C1 we always have rank(B2) = m−ρ. Then it can be shown (seeAppendix) that K1 assigns the ρ numbers from Λρ as closed-loopeigenvalues of

Ac = A1 + BK2C1 + BU2K3C1 = A2 + B2K3C1 (37)

for arbitrary self-conjugate z ′

j = 0′, j = 1, . . . , ρ and K3 ∈

R(m−ρ)×(p−r) and renders the pair (A2, B2) uncontrollable for λ ∈

Λρ . Therefore, if r + ρ < n in the last step K3 can be used tosolve (11) for (A2, B2, C1) and the ℓ = n − r − ρ remainingself-conjugate values from the set Λ2 without changing the r + ρpreassigned eigenvalues of A2. But to assure the controllability ofthe ℓ remaining eigenvalues of A2, with W ′

ρS = 0, S ′S = In−ρ andH ′

c = B⊥(A−λI) a similar reasoning as before for the observabilityof λ ∈ Λ1 results in the condition

rank(H ′

cS) = n − m (38a)

or equivalently

Im(Hc) ∩ Im(Wρ) = {0} (38b)

for λ ∈ {Λ1, Λρ} which should generically be fulfilled. Then tosolve (11) we must bear in mind that in general for arbitrary m ≥

2, p ≥ 2, 0 ≤ r < p, 0 ≤ ρ ≤ m the resulting ℓ equations

det(Mi − K3C1Ni) = 0, i = 1, . . . , ℓ (39)

are multilinear in the elements of K3 and will only have a solutionif the number ℓ of equations does not exceed the number (m −

ρ)(p − r) of unknowns, i.e.,

(m − ρ)(p − r) ≥ n − r − ρ. (40)

Hence, in the followingwe concentrate on two special cases wherea solution to (39) is not required or an analytical solution to (39)can be obtained by means of (26).

3.1. r = p − 1(m + p ≥ n + 1)

For r = p − 1 (40) yields Kimura’s condition m + p ≥ n + 1with ρ arbitrary. Thus, to avoid solving (39)we set ρ = n−r whichresults in ℓ = 0, ρ ≤ m and the parametric solution to the poleassignment problem is given by (28) and (35).

3.2. r = p − 2 (m + p ≥ n)

For r = p−2 the evaluation of (40) givesm+p ≥ n+(ρ−m+2)and sinceρ ≤ m the choiceρ = m andρ = m−1 results inm+p ≥

n + 2 and m + p ≥ n + 1, respectively, so that both conditionsare already covered by Kimura’s condition (see Section 3.1). On theother hand, for 0 ≤ ρ ≤ m−2 only the choice ρ = m−2 results inan easily accessible analytical solution to (39) under the condition

m + p ≥ n. In this case K3 ∈ R2×2, ℓ = n − m − p + 4 andespecially form+p = nwe thus have ℓ = 4 so that the solution of(39) turns to the problem discussed in Section 2.2. In what followswe therefore address the solution of this problem in more detail.

To this end for (A2, B2, C1), K3 ∈ R2×2 and the 4 remaining self-conjugate values {λn−3, . . . , λn} from the set Λ2 we refer to (26)and set ∆′

= [δ1, . . . , δ4], β ′= [det(M1), . . . , det(M4)], which

results ink3 = ∆+β + f α = γ + f α (41)where ∆+

= ∆′[∆∆′

]−1 denotes the right inverse of the 4 × 5

matrix ∆ if rank(∆) = 4 and f is a one-dimensional basis for theright nullspace of ∆, i.e., 1f = 0. At first glance there seems tobe an infinite set of solutions parameterized by the scalar α butto comply with (27) α must solve a quadratic equation. To seethis according to the notation introduced in Section 2.2 we setγ = [γ11, γ21, γ12, γ22, γ5]

′ and f = [f11, f21, f12, f22, f5]′. Thenwith respect to (27) and γ1 = [γ11, γ21]

′, γ2 = [γ12, γ22]′, f1 =

[f11, f21]′, f2 = [f12, f22]′ (41) can be split into the matrix equationK3 = [γ1, γ2] + α[f1, f2] = Γ + αF (42)and the scalar equationdet(K3) = γ5 + αf5 (43)where in turn the left hand side of (43) is replaced by the exteriorproductdet(K3) = (γ1 + αf1) ∧ (γ2 + αf2)

= (γ1 ∧ γ2) + (γ1 ∧ αf2) + (αf1 ∧ γ2) + (αf1 ∧ αf2)= det(Γ ) + α{det([γ1, f2]) + det([f1, γ2])} + α2 det(F)

to finally result in the quadratic equation0 = {det(Γ ) − γ5} + α{det([γ1, f2]) + det([f1, γ2]) − f5}

+ α2 det(F)

0 = p0 + αp1 + α2p2. (44)Obviously, only for a real solution α of (44) we get a real k3 in

(41) and a real K3 from (42), respectively. With this K3 the finalsolution to the pole assignment problem is then given by (28) and(35).

3.3. Main results

Before proceeding with a numerical example the main resultsof this note are summarized in the following propositions.

Proposition 1. For m ≥ 2, p ≥ 2 let m + p = n and Λ1 =

{λ1, . . . , λp−2}, Λ2 = {λp−1, . . . , λn−4}, Λ3 = {λn−3, . . . , λn} beself-conjugate sets of distinct complex numbers and qi = 0, i =

1, . . . , p − 2, z ′

j = 0′, j = p − 1, . . . , n − 4 corresponding self-conjugate sets of right and left parameter vectors, respectively. If

(I) rank(CVp−2) = p − 2, Vp−2 ∈ Rn×(p−2)

(II) rank(W ′

m−2B) = m − 2, W ′

m−2 ∈ R(m−2)×n

(III) rank(∆) = 4, ∆ ∈ R4×5

(IV) (44) has a real solution α

then there exists a real output feedback matrix K = K0 + K2U ′

1 +

U2K3U ′

1 according to (28), (35), (42) that assigns to (A + BKC) theclosed-loop spectrum Λ = {Λ1, Λ2, Λ3}.

In Proposition 1 conditions (III), (IV) are obsolete if instead ofm + p = n the stronger condition m + p ≥ n + 1 (Kimura’scondition) holds.

Proposition 2. For m ≥ 2, p ≥ 2 let m + p ≥ n + 1 andΛ1 = {λ1, . . . , λp−1}, Λ2 = {λp, . . . , λn} be two self-conjugate setsof distinct complex numbers and qi = 0, i = 1, . . . , p − 1, z ′

j =

0′, j = p, . . . , n corresponding self-conjugate sets of right and left

296 U. Konigorski / Systems & Control Letters 61 (2012) 292–297

parameter vectors, respectively. If for ρ = n + 1 − p

(I) rank(CVp−1) = p − 1, Vp−1 ∈ Rn×(p−1)

(II) rank(W ′

ρB) = ρ, W ′

ρ ∈ Rρ×n

then there exists a real output feedback matrix K = K0 + K2U ′

1 +

U2K3U ′

1 according to (28), (35) with K3 arbitrary that assigns to(A + BKC) the closed-loop spectrum Λ = {Λ1, Λ2}.

Remark 2. As aforementioned the case m + p ≥ n + 1 is notonly covered by Kimura’s condition (Section 3.1) but also by theapproach presented in Section 3.2 for m − 1 ≤ ρ ≤ m. Therefore,to gain greater flexibility in the separation of the spectrum intoself-conjugate subsets one can freely choose between the twoapproaches. For instance, if n = 6, p = 4,m = 4 the separation ofthe spectrum according to Section 3.1 yields three eigenvalues inΛ1 and Λ2, respectively. Thus, two closed-loop eigenvalues mustbe real. However, the approach according to Section 3.2 results intwo subsets with r = p − 2 = 2 and ρ = m = 4 so that alln = 6 closed-loop eigenvalues can be complex and K is given by(28), (35).

Remark 3. Obviously, according to the derivations in Section 3 itis generically always possible to place at leastm + p − 1 orm + pclosed loop poles, respectively. Therefore, as in [11] the approachpresented in this note can also be used for partial pole placementalthough this might be of less practical importance.

Remark 4. Of course, instead of (A, B, C) the dual system (A′, C ′,B′) can be used in all preceding calculations to interchange the roleof p and m as appropriate.

A simple calculation shows that form+p ≥ n (28), (35) provideall remainingmp−n dof beyond eigenvalue assignment. The r self-conjugate right parameter vectors qi = 0, i = 1, . . . , r in (28)offer r(m− 1) dof while ρ(p− r − 1) dof are provided by the self-conjugate set z ′

j = 0′, j = 1, . . . , ρ of left parameter vectors. InProposition 1 r = p−2, ρ = m−2 and therefore the left parametervectors z ′

j = 0′ are of dimension 2 and each of them offers 1dof whereas in Proposition 2 r = p − 1 and the left parametervectors z ′

j = 0′ are of dimension 1, i.e., nonzero scalars and offerno additional dof. But if in Proposition 2 m + p = n + 1 + d withd > 0 then there are d additional real dof provided by K3 ∈ Rd×1

in (35).For m + p = n we also have mp > n if m ≥ 3 and/or

p ≥ 3. Therefore, in this case Wang’s condition for the genericsolvability of the pole assignment problem over the reals is alwaysfulfilled and the remaining mp − n dof covered by the left andright parameter vectors qi = 0, i = 1, . . . , p − 2, z ′

j = 0′, j =

p − 1, . . . , n − 4 can be used to generically assure one or tworeal solutions of (44). This assertion has been verified by hundredsof numerical test runs with randomly assigned matrices A, B andC of order up to 20 and with different closed-loop eigenvaluesand combinations of m and p, respectively. All these runs gave anumerically efficient and reliable solution. However, in some rarecases at first a self-conjugate complex solution for the feedbackmatrix K was obtained but finally after some additional runs withthe same A, B, C and closed-loop eigenvalues but randomly chosenparameter vectors in all cases one or two real solutions have beenobtained.

However, if m = p = 2, m + p = mp = n = 4 the sets Λ1 andΛ2 in Proposition 1 are both empty and conditions (I) and (II) canbe omitted. Thus there are no dof beyond eigenvalue assignmentand the solution is solely given by K3 in (42) which depends onthe solutions α of (44) and thus might be complex. Indeed, thecondition mp ≥ n is in general only necessary and sufficient forthe generic solvability of the pole assignment problem if complexsolutions are allowed [24,25]. Especially in the case m = p =

2, n = 4 it can be shown [26] that the problem is not genericallysolvable over the reals.

As shown in [12] even form+p < n < mp there are some caseswhich allow for a direct solution of the pole assignment problem.Of course, this interesting result is not covered by the approachpresented in this note. However, for r = p − 3, ρ = m − 2the evaluation of (40) gives m + p ≥ n − 1 and especially form + p = n − 1(A2, B2, C1) in (37) reduces to a system of McMillandegree n = 6 with m = 2 inputs and p = 3 outputs, respectively.For this systemwemust assign the remaining ℓ = 6 self-conjugateeigenvalues {λn−5, . . . , λn} from the set Λ2 by solving (39) forK3 ∈ R2×3.

Due to the results in [24,25] for this kind of system thereexists (generically) at least one real solution to the pole placementproblem. However, instead of (44) in this case the expansion of(39) results in a system of 3 quadratic equations in 3 unknowns(see e.g. [20]) which in general cannot be solved analytically. Butobviously the results in [12] suggest that the solution of thesespecial quadratic equations deserves further investigations.

4. Numerical example

In this section anumerical example from the literature is used toillustrate the application of Proposition 1. The systemdata (A, B, C)originate from [11] and describe a system with n = 5,m = 3, p =

2. Since r = p − 2 = 0 the set Λ1 is empty and condition (I) inProposition 1 can be omitted. So we can directly evaluate (35) forA1 = A, C1 = C and the ρ = m − 2 = 1 real eigenvalue fromthe desired closed-loop spectrum Λ = {Λ2, Λ3} = {−0.5, −3 ±

2i, −2 ± 2i}. For that purpose we must choose one nonzero realleft parameter vector associated with the eigenvalue λ1 = −0.5from the set Λ2. With z ′

1 = [1, 1] the evaluation of (44) gives tworeal solutions for α and thus we readily get from (35), (42) twocorresponding real feedback matrices

Kα1 =

20.2391 18.1572−7.5363 −6.966545.0720 39.1186

,

Kα2 =

−788.530 775.014−83.736 82.257

−4003.256 3933.974

which assign both the spectrum Λ to the closed-loop system A +

BKα1C and A + BKα2C , respectively. Obviously, ∥Kα2∥ ≫ ∥Kα1∥

and even the norm of Kα1 is quite large and should be reduced. Tothis end the single dof offered by z ′

1 can be used, e.g., the choicez ′

1 = [1, 0.36] leads to the two solutions

Kα1 =

3.0150 2.3515−1.8156 −3.65622.8693 −3.0012

,

Kα2 =

−560.317 547.490−65.655 64.091

−2839.943 2774.271

with much smaller entries of both feedback matrices.

This example shows the advantage over the approach in [11]where form+p = nnot allmp dof are accessible by the designer. Infact, there is always one dof missing and therefore in this examplethe technique in [11] yields exactly one real solution which onlydepends on the separation of the desired closed-loop spectrum.However, as shown above there are in general two real solutionswhich in addition depend on the remaining dof offered by z ′

1.

5. Conclusion

In this paper a closed form solution to the problem of poleplacement by constant output feedback has been presented.Under the condition m + p ≥ n beside eigenvalue assignmentthe approach explicitly offers all remaining mp − n degrees offreedom provided by two self-conjugate sets of right and leftparameter vectors, qi and z ′

j and possibly a real matrix K3. For

U. Konigorski / Systems & Control Letters 61 (2012) 292–297 297

instance these dof can be used to minimize the norm of theresulting feedback matrix K as shown by a numerical example.Furthermore the maximization of the complex stability radiusor the minimization of the closed-loop eigenvalue sensitivity asproposed in [20,27], respectively can be considered. Finally, theless restrictive condition m + p + 1 = n ≤ mp deserves somefurther investigation.

Appendix

From (8), (13), (14) we have vi = Niqi = Vrϵi and Qrϵi = Miqiwhere ϵi denotes the ith unit vector. Then with (28), (29) and (6)for Ac in (30) we get

(λiI − Ac)vi = (λiI − A − BQr(CVr)+C − BK1U ′

1C)vi

= (λiI − A)vi − BQr(CVr)+(CVr)ϵi − BK1U ′

1(CVr)ϵi

= (λiI − A)Niqi − BMiqi = 0, ∀qi, i = 1, . . . , r.

This shows that for any choice of self-conjugate right parametervectors qi = 0 and arbitrary real K1 in (28) λi ∈ Λ1 and vi = Niqiare eigenvalues and corresponding right eigenvectors of A1 = A +

BQr(CVr)+C = A + BK0C , i.e.,

(λiI − A1)vi = 0, i = 1, . . . , r. (A.1)

Moreover, the two Eqs. (A.1) and (29) directly correspond to thePopov–Belevitch–Hautus (PBH) eigenvector test for observabil-ity [21], i.e.,

A1vi = λivi, C1vi = 0, i = 1, . . . , r (A.2)

thus (A1, C1) is an unobservable pair for the r eigenvalues in Λ1.Now following an analogue chain of proof as before by

application of (35), (36), (17) and with w′

j = z ′

jDj = ϵ′

jW′ρ and

ϵ′

j Z′ρ = z ′

jEj according to (19), (24), (25) we get for Ac in (37)

w′

j(λjI − Ac)

= ϵ′

jW′

ρ(λjI − A1 − B(W ′

ρB)+Z ′

ρC1 − BU2K3C1)

= z ′

jDj(λjI − A1) − z ′

jEjC1

= −z ′

j [Dj, Ej]Tj = 0′, ∀z ′

j , j = 1, . . . , ρ.

Therefore λj ∈ Λ2 and w′

j = z ′

jDj are eigenvalues and accompany-ing left eigenvectors of A2 = A1 + B(W ′

ρB)+Z ′

ρC1 = A1 + BK2C1 forany self-conjugate set of left parameter vectors z ′

j = 0′ and arbi-trary real K3 in (35), i.e.,

w′

j(λjI − A2) = 0′, j = 1, . . . , ρ. (A.3)

Finally, the combination of (A.3) and (36) gives the PHB eigenvectortest for controllability

w′

jA2 = λjw′

j, w′

jB2 = 0′, j = 1, . . . , ρ (A.4)

hence (A2, B2) is an uncontrollable pair for the ρ selected eigenval-ues from Λ2.

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