Pole Assignment for Symmetric Quadratic Dynamical Systems: An

Algorithmic Method

Abstract: In this article an algorithmic method is proposed for the solution of the pole assignment problem which

is associated with a symmetric quadratic dynamical system, when it is completely controllable. The problem is

shown to be equivalent to two subproblems, one linear and the other multi-linear. Solutions of the linear problem

must be decomposable vectors, i.e. they must lie in an appropriate Grassmann variety. The proposed method

computes a reduced set of quadratic Plucker relations, with only three terms each, which describe completely

the specific Grassmann variety. Using these relations one can solve the multi-linear problem and consequently

calculate the feedback matrices which give a solution to the pole assignment problem. An illustrative example

of the proposed algorithmic procedure is given. The main advantage of our approach is that the complete set of

feedback solutions is obtained, over which further optimisation can be carried out, if desired. This is important

for problems with structural constraints (e.g. decentralization) or norm-constraints on the feedback gain-matrix.

Key-Words: Control Theory, Pole assignment, Quadratic matrix pencils, Grassmann variety, Plucker relations,

numerical algorithm

Received: May 29, 2023. Revised: May 26, 2024. Accepted: June 27, 2024. Published: July 29, 2024.

1 Introduction

The fundamental criterion of controllability is useful

in deciding whether a symmetric quadratic system is

state-controllable. This condition is intimately linked

to state-feedback control which allows the designer

to modify the behavior of the closed-loop dynamics

in order to relocate troublesome eigenvalues in

the complex plane, thus avoiding long transients,

resonance phenomena and potential instability under

model uncertainty. The paper provides a systematic

approach for designing state-feedback controllers of

symmetric quadratic dynamical systems, by decom-

posing the problem into two separate sub-problems,

one linear and the other multi-linear, [1], [2], [3], [4].

To avoid excessive computational complexity, care

has been taken to reduce algorithmic redundancy,

via the computation of a reduced set of quadratic

Plucker relations. The paper describes the two design

steps of the approach which are illustrated with a

numerical example. A background on aspects of

algebraic geometry, on which many of the results

are based, is also provided. The main advantage

of our method is that it can be used to parametrize

the complete set of state-feedback solutions in the

multivariable case which is useful when additional

optimization constraints are imposed, e.g. structural

constraints in the form of decentralization patterns, or

norm-constraints on the state-feedback matrix which

may address robust-stability issues under model

uncertainty. A further advantage of the proposed

method is that it can be extended in a relatively

straightforward manner to wider classes of models,

e.g. higher-order matrix models or implicit (descrip-

tor) systems described by a set of differential and

algebraic equations, [5], [6], [7], [8]. Several results

obtained in these two directions will be reported in

future publications.

2 Problem definition and

Methodology

The equation of motion for a matrix second-order sys-

tem, e.g. a structural system with viscus damping and

without externally applying forces is expressed as fol-

lows:

Mq00(t) + Dq0(t)+Kq(t) = 0 (1)

Here, M,D,Kare the n×nmass, damping and stiff-

ness matrices respectively. Also, q(t)is the displace-

ment vector, q0(t)the velocity vector and q00(t)the ac-

celeration vector. In most applications M,Dand K

are symmetric matrices. Furthermore, Mis typically

positive definite and D,Kare positive semi-definite

or positive definite.

Separation of variables q(t) = eλtcwhere cis

a constant vector gives us the quadratic eigenvalue

S. PANTAZOPOULOU, 1

M. TOMAS-RODRIGUEZ, 2

G. KALOGEROPOULOS, 2

G. HALIKIAS

1School of Science and Technology, City University, London

Northampton Square, London EC1V 0HB, UK

2Department of Mathematics, University of Athens

Panepistimiopolis, Athens 15784, GREECE

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problem:

P(λj)cj= 0, j = 1,2, . . . , 2n

in which P(λ)denotes the quadratic matrix pencil:

P(λ) = λ2M+λD +K. The eigenvalues and

eigenvectors of the pencil govern the free response

of the system. The model can be used to identify

poorly damped oscillations and resonance phenom-

ena, which we may be able to avoid by the appropriate

selection of the matrix parameters.

When an actuating force F(t)is applied, the model

described by the above equations becomes:

Mq00(t) + Dq0(t) + Kq(t) = F(t)(2)

Often the force can be selected by the designer, in

which case

F(t) = Bu(t)(3)

where u(t)is the m×1control input vector and Bis

the n×minput matrix.

One of the major concerns for the control engineer

is to ensure stability of the control system. Thus, the

behaviour of the system is usually modified by ap-

plying state-feedback control to relocate the trouble-

some eigenvalues in the complex plane. In the single-

input case, under the assumption of controllability, the

problem has a unique solution. This is no longer the

case in the multivariable case, in which typically an

infinite set of state-feedback matrices can be used to

assign the closed-loop eigenvalues at specified loca-

tions. Existing algorithms [9], [10], [11] do not in

general parameterize completely the solution set, or

give an implicit parametrization which is difficult to

apply in practice. In contrast, our approach gives a

complete solution to the problem, although intermedi-

ate steps of the algorithm may be computationally de-

manding, especially for high-dimensional problems,

if certain redundant relations are not removed. The

analysis in the present work involves quadratic sym-

metric models, which arise naturally in mechanical

vibration and electrical circuit systems, however the

techniques can be extended with minor modifications

to matrix higher order systems and implicit (descrip-

tor) systems. Deriving parametric solutions to a par-

ticular design objectives is important in order to select

specific solutions which optimize additional objec-

tives or constraints. In the present context, in addition

to placing the eigenvalues of the closed-loop system

to specific locations, a designer may need to consider

further issues, such as sparsity in the state-feedback

matrix corresponding to specific decentralization ob-

jectives, minimizing the norm of the state-feedback

matrix to improve the sensitivity properties of the de-

sign, or more general problems of eigen-structure as-

signment. If the system is (open-loop) unstable, we

aim, as a minimum requirement, to stabilize it. If the

system is (open-loop) stable, it is desirable to maintain

some degree of relative stability and robust-stability

margins, i.e. ensure that stability is maintained un-

der realistic uncertainty conditions. In general, the

notion of stabilization is connected to the problem of

relocation of troublesome eigenvalues of the system

in equation (2), also called pole placement. In the-

ory, eigenvalue relocation to arbitrary locations of the

complex plane can always be achieved if the system is

state-controllable, although in practice this is limited

by constraints on the amplitude or energy of the con-

trol signal, system bandwidth constraints, and robust

stability issues. For the model described in equations

(2) and (3), for any choice of a conjugate-symmetric

set of poles, suitable real m×nmatrices F1and F2

can be found such that under the state-feedback con-

trol law:

u(t) = −F1q(t)−F2q0(t)(4)

the corresponding closed-loop system described by

the equation

Mq00(t)+(D+BF2)q0(t)+(K+BF1)q(t) = 0 (5)

has the chosen set of poles. Equivalently, suppose that

the required closed-loop polynomial is:

f(s) = a0+a1s+. . . +a2n−1s2n−1+s2n(6)

This can be written more compactly in the form

f(s) = et

2n(s)˜a(7)

where

2n(s) = [1 s . . . s2n]and ˜aT= [a0a1. . . a2n−11]

We wish to compute the matrices F1and F2such that

the characteristic polynomial

˜ϕ(s) = det(s2M+s(D+BF2)+(K+BF1)) (8)

of the quadratic pencil

Pc:= s2M+s(D+B F2)+(K+B F1)(9)

associated with the closed-loop system (5) is equal to:

˜ϕ(s) = (det M)f(s)(10)

Define:

Q(s) := [s2M+sD +K|sB |B](11)

and

V="In

F1#(12)

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Then we can rewrite equation (9) as:

Pc(s) = Q(s)V(13)

Using compound matrices and applying the Binet-

Cauchy theorem we can write,

˜ϕ(s) = det(Pc(s)) ≡Cn(Pc(s)) (14)

=Cn(Q(s)V) = Cn(Q(s))Cn(V)(15)

We now define:

P:= P(M, D, K, B)∈R(2n+1)×`(16)

Since Cn(Q(s)) := et

2n(s)P, the columns of Pare

the coefficient vectors of the elements of Cn(Q(s)),

which are polynomials of maximum degree 2nand

g:= Cn(V) = Cn " In

F1#! ∈R`(17)

where `:= n+2m

n. Combining equations (6), (10)

and (15) we get

2n(s)P g =et

2n(s)(det M)˜a:= et

2n(s)a

which in turn implies that:

P g =a(18)

3 Techniques from algebraic

geometry and algorithmic solution

In this section we construct the quadratic Plucker rela-

tions of the problem which are algebraically indepen-

dent. In order to achieve this, a criterion based on the

correspondence between vectors of the Grassmann

variety and the lexicographical ordering is applied.

Let us assume that Ω(m, n)is the Grassmann Vari-

ety of the projective space Pk(F)with k=n

m−1

whose dimension is given by the equation

dimΩ(m, n) = m(n−m)(19)

It can be proved, [12], that the total number of Plucker

relations which define the Grassmann Variety is:

n

m−1 n

m+ 1(20)

However these are dependent, in the sense that some

of these relations are linear combinations of the oth-

ers. Hence, if all these relations are used as equality

constraints in the remaining part of the algorithm, this

would raise the complexity of the solution unneces-

sarily. The minimum number of independent equa-

tions neq that describe a variety, of dimension equal

to dimV, is

neq =dimP−dimV(21)

and so

neq =n

m−1−m(n−m)(22)

In this section, we propose an algorithm which re-

duces the full set of equations that describes the

Grassmann variety to an exact minimal set of inde-

pendent equations whose number is given in equa-

tion (22). To give an illustration of the reduction in

complexity that this entails, note that, for example if

m= 2 and n= 4, the full number of Plucker relations

according to the equation (20) is 16. This is reduced

to only 1equation which consists the reduced set (see

(22)).In case where m= 3 and n= 5 the total num-

ber of Plucker relations according to equation (20) is

50 equations while the reduced set from the equation

(22) has only 3, i.e. in this case 47 of the total num-

ber of equations are redundant. It should be clear that

the reduction in complexity is highly significant and

grows with the dimensionality of the problem, [12].

Before stating the Theorems on which the algo-

rithm is based, we introduce the following notation

and definitions, [12], [13], [14]. Let the binomial co-

efficient be

k:= n

m, n, m ∈N, n ≥m

Denote the set of the first nnatural numbers as:

Tn:= {κ∈N∗: 1 ≤κ≤n}

with N∗the set of naturals (excluding zero). Let

m:= {(a1, . . . , am) : aj∈ Tn,1≤j≤m}

be the set of sequences of length mwith elements aj,

1≤j≤m, not necessarily distinct. Also, let

m:= {(a1, . . . , am)∈ T n

m:aκ6=aλ,if κ6=λ}

be the set of sequences, where the elements of each

sequence are distinct. Denote by Dn

mall m-tuples <

a1, a2, . . . , am>such that (a1, . . . , am)∈˜

mand

aκ< aλif κ<λ. We can order the elements of the

set Dn

mas follows:

< a1, a2, . . . , am>≺< j1, j2, . . . , jm>

if and only if there exists κ∈ Tnsuch that aκ< jκ

and aλ=jλfor every λ < κ. The above relation is a

total ordering called lexicographical ordering. Using

the lexicographical ordering, we can now characterize

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the kelements of Dn

mby

a0=<1,2, . . . , m −1, m >

a1=<1,2, . . . , m −1, m + 1 >

ak−1=< n −m+ 1, . . . , n −1, n >

Note that a0≺a1≺. . . ≺ak−1. Next, we define the

map

δ:˜

m→N×Dn

such that:

(a1, a2, . . . , am)δ

−→ (λ, < a0

1, a0

2, . . . , a0

m>)

where λis the number of permutations needed to or-

der the elements of (a1, a2, . . . , am)in normal order-

ing < a0

1, a0

2, . . . , a0

m>.

If [x0, x1, . . . , xk−1]T∈ Pk−1(R), we define the

following maps: First,

φ:Dn

m→[x0, x1, . . . , xk−1]T

by aκ

−→ xκ,0≤κ≤k−1, where it is clear from the

above definition of φthat φ−1(xκ) = aκ. Secondly

we define: ˜

φ:N×Dn

m→R

(λ, aκ)˜

−→ (−1)λφ(aκ) = (−1)xκ

Finally, if we denote β:= (a1, a2, . . . , am)∈ T n

m, we

define the map

g:Tn

m→R

g(β) = (˜

φ◦δ)(β),if β∈˜

0,if β∈Tn

m−˜

We can now construct the Plucker relations: For every

group of m−1indices

< t1, t2, . . . , tm−1>∈˜

m−1

and from every group of m+ 1 indices

< p1, p2, . . . , pm+1 >∈˜

m+1

we define for κ= 1, . . . , m + 1,

βκ:= (t1, t2, . . . , tm−1, pκ)

and

γκ:= (p1, p2, . . . , pκ−1, pκ+1, . . . , pm+1),

for κ= 1, . . . , m + 1 and the corresponding Plucker

relation is as follows

m+1

κ=1

(−1)κg(β)g(γκ) = 0

Next we state the following theorems which extract

from the whole set of quadratic Plucker relations a

reduced set of relations which have simple form and

describe completely the Grassmann variety of the

corresponding projective space.

Theorem 1: Assume that

x= [x0, x1, . . . , xk−1]t∈Ω(m, n)

and

φ−1(xκ) =< a1, a2, . . . , am>∈Dn

with a2< n −m+ 1. Then, there exists a three-term

Plucker relation

σ(xκ, xκ1, xκ2, xκ3, xκ4, xκ5) = 0

where κ < κi, i = 1,2, . . . , 5.

Theorem 2: Assume that x=

[x0, x1, . . . , xk−1]t∈Ω(m, n). The full num-

ber of the coordinates xκhaving the property

φ−1(xκ) =< a1, a2, . . . , am>with a2< n −m+ 1

r:= n

m−(n−m)m−1

Corollary 1: For every x= [x0, x1, . . . , xk−1]t∈

Ω(m, n), there exists a set Sof three-term quadratic

Plucker relations, of the form

σi(xκi, xλi, xµi, xνi, xξi, xρi) = 0

where 1≤i≤rwith

κi≤min(λi, µi, νi, ξi, ρi),for 1≤i≤r,

and

κj< κj+1,1≤i≤r−1.

Theorem 3 [14]: The three-term quadratic

Plucker relations given by the set Sof Corollary 1

describe completely the Grassmann variety Ω(m, n)

of the projective space Pk−1(R).

Next, we propose the following algorithm which

computes the Reduced Set of Quadratic Plucker Re-

lations (RSQPR).

Algorithm RSQPR

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• Step 1: Read the dimensions n, m.

• Step 2: Compute k=n

m.

• Step 3: Repeat for k= 0,1, . . . , k −1.

a. Find the κ-th order multi-index

< a1, a2, . . . , am>.

b. If λ2< n −m+ 1 then

b1. Find indices j1, j2∈Tn− {a1, a2, . . . , am}

such that j1> a2and j2> a2.

b2. Define am+1 := j2,

βρ:= (a3, . . . , am, j1, aρ)and

γρ:= (a1, . . . , aρ−1, aρ+1, . . . , am+1).

b3. Type Plucker relation

X(−1)ρ·g(βρ)·g(γρ) = 0

• Step 4: End.

Finally, suppose we have computed the coordinates of

a decomposable vector as shown below

x= [x0, x1, . . . , xk−1]t∈Ω(m, n)

using the quadratic Plucker relations generated by the

above algorithm. Then, we would like to construct

H∈Rn×mwith the property:

Cm(H) = x(23)

The following Proposition is helpful for this purpose.

Proposition 1 [14]: Let

x= [x0, x1, . . . , xk−1]t∈Rk

be a decomposable vector and xp6= 0 for an index p,

1≤p≤k−1, such that

φ−1(xp) =< a1, a2, . . . , am>

Then, the elements hij of matrix H∈Rn×mwhich

satisfies equation (23) are given by

hij =g((a1, a2, . . . , aj−1, i, aj+1, . . . , am)) (24)

for 1≤i≤nand 1≤j≤m.

4 Numerical Example

The benefits of the proposed algorithmic procedure

are clarified in the following example. We consider

the system of the form (2), with

M= 10I3, K ="40 −40 0

−40 80 −40

0−40 80 #

and

D= 03×3, B ="1

The open-loop e-values are

{±3.6039i, ±2.4940i, ±0.8901i}

. Suppose we want to shift these to

{−1,−2,−3,−4,−5,−6}

using the state feedback law (4) where F1, F2∈

R1×3. In this case the problem consists of finding ma-

trices F1, F2such that the closed-loop system (5) has

a charactristic polynomial

φ(s) = (detM)f(s)

where

f(s) =

i=1

(s−λi) =

i=1

(s+i)

.After several computations, we arrive at a linear

system of the form (18) which is P g =a, the solution

of which is given by:

g0= 1, g1= 2932.5, g2=−4318.571, g3= 3345

g4=−5737.143, g6= 1447.5, g7=−2705.714

Variables g5, g8, g9can be chosen arbitrarily since

they do not appear in equations of the linear system.

Now we must solve a multinear subproblem, which

consists of the following three equations, the so-called

reduced set of quadratic Plucker relations:

g4g6−g3g7+g0g9= 0

−g5g6+g3g8−g1g9= 0

g5g7−g4g8+g2g9= 0

These equations are generated by the proposed algo-

rithm. Using a symbolic package, i.e. Mathematica,

and substituting the solutions of the linear system to

the above set of quadratic relations the following so-

lutions are obtained:

g5=−2378500, g8=−1683375, g9=−746100

Now, using equation (18) together with Proposi-

tion 1, we obtain:

F1= [g7,−g4, g2]

= [−2705.714 ,5737.143 ,−4318.571]

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and

F2= [g6,−g3, g1] = [1447.5,−3345 ,2932.5]

In order to avoid cumbersome calculations by

hand, a numerical software package can be used to

verify that the closed-loop system (5) has the follow-

ing poles:

λ1=−1.000000000047105

λ2=−1.999999999594935

λ3=−3.000000000111104

λ4=−3.999999999929748

λ5=−5.000000000132230

λ6=−6.000000000120886

Note that whatever the starting guess for the approx-

imation of the eigenvalues, the error |λi−˜

λi|, for

i= 1, . . . , 6, is always smaller than 10−9.

5 Conclusion

An efficient algorithmic framework has been pro-

posed for the solution of the pole-assignment problem

of symmetric quadratic systems. This involves the

computation of a reduced set of quadratic Plucker

relations describing completely the Grassmann

variety of the corresponding projective space. The

extraction of this reduced set has been achieved by

the use of a simple criterion based on the correspon-

dence between the coordinates of a decomposable

vector and lexicographical orderings. The minimum

number of the linear independent quadratic Plucker

relations which describe completely the Grassmann

variety is given in equation (22). Each equation of

this reduced set is homogeneous and contains only

three terms. We have to emphasize that the proposed

algorithm is free from numerical errors because it

does not contain any numerical calculation.This fact

has beneficial influence on the overall complexity of

the solution of the problem which is mainly due to

its non-linearity. The redundant Plucker relations can

be expressed as linear combinations of the remaining

minimal set, and hence can be completely ignored

in the subsequent steps of the algorithm, which

reduces the complexity of the solution significantly.

An advantage of the proposed method relative to

existing ones is that it generates the whole family of

feedback matrices with the specified pole-placement

properties; this family can be subsequently used for

further optimization purposes if additional design

objectives are desirable or further constraints are

imposed. Another advantage of the approach is that

it can be extended to more general classes of models,

including descriptor (implicit) systems and matrix

higher-order models (only quadratic matrix models

have been considered in this paper). Descriptor

systems are widely employed in systems and con-

trol engineering to model and simulate dynamical

systems with algebraic constraints, i.e. systems de-

scribed by mixed differential and algebraic equations.

They have a wide range of applications, such as in

mechanical multi-body systems, electrical circuits,

chemical (process) engineering, fluid dynamics and

many others. The methods can be extended also to

non-linear systems and can be used to solve a wide

range of related initial-value and boundary value

problems. The method can also be extended to the

solution of other problems of control theory hav-

ing a similar multi-linear nature, [15], [16], [17], [18].

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