Pole Assignment by Proportional-Plus-Derivative State Feedback for
Multivariable Linear Time-Invariant Systems
KONSTADINOS H. KIRITSIS
Hellenic Air Force Academy,
Department of Aeronautical Sciences,
Division of Automatic Control,
Air Base of Dekelia, ΤΓΑ 1010,
Dekelia, Athens, GREECE
Abstract: - In this paper the pole assignment problem by proportional-plus-derivative state feedback for
multivariable linear-time invariant systems is studied. In particular, explicit necessary and sufficient conditions
are established for a given polynomial with real coefficients to be characteristic polynomial of closed-loop
system obtained by proportional-plus-derivative state feedback from the given multivariable linear time-
invariant system. A procedure is given for the calculation of proportional-plus-derivative state feedback which
places the poles of closed-loop system at desired locations. Our approach is based on properties of polynomial
matrices.
Key-Words: - pole assignment, proportional-plus-derivative state feedback, linear time-invariant systems.
Received: June 11, 2021. Revised: April 15, 2022. Accepted: May 13, 2022. Published: June 16, 2022.
1 Introduction
The problem of pole assignment by proportional-
plus-derivative output feedback or equivalently by
incomplete proportional-plus-derivative state
feedback for multivariable linear time-invariant
systems was introduced in [1]. In particular in [1] a
method was given for placing up to max(2m, 2p),
poles of closed-loop system, where m and p are the
numbers of inputs and outputs respectively of
closed-loop system. A method for assigning up to
max(2m+p-1, 2p+m-1) poles of closed-loop system
by proportional-plus-derivative output feedback(or
equivalently by incomplete proportional-plus
derivative state feedback) is presented in [2]. In [3]
was introduced a new class of multivariable output
feedback controllers consisting of proportional plus
multiple derivative terms. It is shown that all poles
of closed-loop system, can be placed at desired
positions, provided a sufficient number of derivative
terms. The pole placement equations for the
proportional-plus-derivative output feedback
compensator are derived in [4]. In [5] is proven that
controllability is sufficient condition for the solution
of pole assignment problem by full proportional-
plus-derivative feedback for single input single
output linear time-invariant systems.
In this paper, are established explicit necessary and
sufficient conditions for the solution of pole
assignment problem by proportional-plus- derivative
state feedback for multivariable linear time-
invariant systems. Furthermore a procedure is given
for the computation of proportional-plus- derivative
state feedback which assigns the poles of closed-
loop system to any desired positions.
Problem Formulation
Consider a multivariable linear time-invariant
system described by the following state-space
equations
=Mx(t) + Nu(t) (1)
where M and N are real matrices of size n x n ,
n x m, respectively, x(t) is the state vector of
dimensions n x 1 and u(t) is the vector of inputs of
dimensions mx1. Without any loss of generality we
assume that
rank[N]=m (2)
Let T be a non-singular matrix of size n x n such
that
TN =
(3)
where is the identity matrix of size m x m. Using
the following similarity transformation
x(t) = z(t) (4)
and the relationship (3), the state–space equations of
system (1) can be rewritten as follows
= Az(t) + Bu(t) (5)
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The real matrices A and B of appropriate
dimensions are given by
A=TM , B =
(6)
Consider the control law
u(t) = –Fz(t) + Dv(t) (7)
where F and D are real matrices of size , m x n and
m x n respectively and v(t) is the reference input
vector of size mx1. By applying the proportional-
plus-derivative state feedback (7) to the system (5)
the state-space equations of closed–loop system are
[I-= (A-BF) z(t) + Bv(t) (8)
Let R be the field of real numbers. Also let R[s] be
the ring of polynomials with coefficients in R. Let
c(s) be a given arbitrary monic polynomial over
R[s] of degree
n
. Further let μ is finite nonzero real
number. The pole assignment problem considered in
this paper can be stated as follows: Does there exist
a proportional-plus-derivative state feedback (7)
such that
(9)
if so, give conditions for existence and a procedure
for the computation of matrices F and D.
3 Basic concepts and preliminary
results
Let us first introduce some notations that are used
throughout the paper. Let D(s) be a non-singular
polynomial matrix over R[s] of dimensions m x m,
write degci for the degree of column i of D(s). If
degciD(s)≥ degcjD(s), i< j (10)
the polynomial matrix D(s) is said to be column
degree ordered. Denote Dh the highest column
degree coefficient matrix of polynomial matrix D(s).
The polynomial matrix D(s) is said to be column
reduced if the real matrix Dh is non-singular. The
matrix D(s) is said to be column monic if its highest
column degree coefficient matrix is the identity
matrix. A polynomial matrix U(s) over R[s] of
dimensions k x k is said to be unimodular if and
only if
det[U(s)] = λ (11)
where λ is finite nonzero real number; therefore
every unimodular polynomial matrix has
polynomial inverse. Let D(s) be a non-singular
polynomial matrix over R[s], then there exist
unimodular matrices U(s) and V(s) over R[s] such
that
D(s)=U(s) diag [a1(s), a2(s), ….am(s)]V(s) (12)
where the polynomials ai(s) for i=1,2, .., m are
termed invariant polynomials of D(s) and have the
following property
ai(s) divides ai+1(s), for i=1,2,….m-1
(13)
Furthermore we have that
1,2,…….m
(14)
where do(s) =1 by definition and di(s) is the monic
greatest common divisor of all minors of order i in
D(s), for i=1,2,….., m. Let A(s), be a polynomial
matrix over R[s] if there are polynomial matrices
P(s) and Q(s) such that
A(s) = P(s) Q(s) (15)
Then, the polynomial matrix P(s) over R[s] is
termed left divisor of A(s). Let A(s) and B(s), be
polynomial matrices over R[s] if
A(s) = D(s) M(s) (16)
B(s) = D(s) N(s) (17)
for polynomial matrices M(s), N(s) and D(s) over
R[s], then D(s) is termed common left divisor of
polynomial matrices A(s) and B(s).
A greatest common left divisor of two polynomial
matrices A(s) and B(s) is a common left divisor
which is a right multiple of every common left
divisor. Let V(s) be a greatest common left divisor
of two polynomial matrices A(s) and B(s), then
there is a unimodular matrix U(s) over R[s], such
that
[A(s), B(s)] = [V(s, 0] U(s) (18)
The system (5) is controllable if and only if
rank [Is – A, B] = n (19)
for all complex numbers s.
The material on polynomial matrices and their
properties presented in this section was obtained
primarily from references [6], [7], [8] and [9].
Definiton 1. Relatively right prime polynomials
matrices D(s) and N(s) of dimensions m x m and
n x m respectively with D(s) to be column reduced
and column degree ordered such that
(20)
are said to form a standard right matrix fraction
description of system (5). The column degrees of
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the matrix D(s) are the controllability indices of
system (5).
The following Lemmas are needed to prove the
main theorem of this paper.
Lemma 1. Let A(s) and B(s) be matrices over R[s]
of size m x k and m x p respectively. The following
are equivalent:
(a) The polynomial matrices A(s) and B(s) are
relatively left prime over R[s].
(b) The greatest common left divisor of
polynomial matrices A(s) and B(s) over
R[s] is unimodular matrix.
(c) rank[A(s), B(s)] = m, for all complex
numbers s.
Proof. See [7, p. 538, Theorem 2.4].
Lemma 2. Let the matrices D(s) and N(s) of
dimensions m x m and n x m over R[s] be a
standard matrix fraction description of controllable
system (5). Then
(a) The rows of polynomial matrix N(s) form a
basis for the linear space over R of all
polynomial vectors v(s) of size 1 x m such
that v is strictly proper.
Proof. See [8, pp. 73-74, Theorem 2.17].
The following lemma was first proved in [9,
pp.184-185].
Lemma 3. Let C(s) be a column monic
polynomial matrix over R[s] of size m x m.
Suppose that degcpC(s< degcqC(s) for some p and
q. Then C(s) can be transformed by unimodular
transformations to a column reduced polynomial
matrix (s) with column degrees given by
degci(s) = degciC(s) for i ≠ p,q
degci(s) = degcpC(s) + 1 for i = p
degci(s) = degcqC(s) - 1 for i ≠q
Proof. Add s times row p to row q in C(s). This
leaves the degree of each column but p and q
unchanged. It also leaves unchanged the degrees of
the elements in position ( i, p), i ≠ q and ( i, q), i ≠
q; places a monic polynomial of degree degcpC(s)+1
in position (p, q) and does not increase the degree
of the element in position (q, q). Let β= degcqC(s).
Further let α be the coefficient of sβ in the element
in position (q, q). Put d = degcpC(s) – 1. Substract
αsd times column p from column q. This reduces the
degree of column q below degcqC(s). The resulting
matrix (s) then satisfies the conditions of the
lemma. This completes the proof of the Lemma.
The following lemma was first proved in [10].
Lemma 4. Let (5) be a controllable system. Further
let D(s), N(s) be a standard right matrix fraction
description of system (5). Then for every m x n real
matrix F the polynomial matrices [Is – A+BF] and
[D(s)+FN(s)] have the same non-unit invariant
polynomials.
Proof. Let D(s) and N(s) are relatively right prime
polynomial matrices over R[s] of respective
dimensions m x m and n x m respectively such that
(21)
We have that
(22)
We add BFNto both sides of the above identity
and rearrange to get
B=N
Since and B are relatively left prime over
R[s] by controllability of (5) and since
(24)
it follows that and B are relatively
left prime over R[s]. On the other hand D(s) and
N(s) are relatively right prime over R[s] and
Hence and are relatively right
prime over R[s]. It follows that the matrices
and must share the
same non-unit invariant polynomials. This
completes the proof of the Lemma.
Lemma 5. Let (5) be a controllable system. Further
let D be an arbitrary matrix over R of size m x n.
Then the following condition holds:
(a) rank [s – A , B] = n for all complex
numbers s.
Proof. Suppose that the system (5) is controllable.
From (19) it follows
rank [Is – A , B] = n (26)
we rewrite the polynomial matrix [s – A ,
B] as
[s – A , B ]= [Is – A , B]
Since the following polynomial matrix over R[s]
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is unimodular, condition (a) of the Lemma 5 follows
from (26) and (27) and the proof is complete.
Lemma 6. Let (5) be a controllable system. Further
let D be a matrix over R of size m x n such that
det[ Then the following condition
holds:
(a) The pair A, B] is
controllable.
Proof. We rewrite the polynomial matrix [
s – A , B] as
[s – A , B] = – A,
B] (28)
Since by assumption the matrix [ is non-
singular, from (28) and Lemma 5 it follows that
A, B] = n (29)
for all complex numbers s .
Condition (a) of the Lemma 6 follows from (29) and
(19) and the proof is complete.
Lemma 7. Let D(s) and N(s) be a standard right
matrix fraction description of a controllable system
(5) and F is a matrix over R of size m x n. Further,
let c1(s), c2(s), …., cr(s) be the non-unit invariant
polynomials of polynomial matrix [D(s)+FN(s) of
size m x m, r Then
(a) det[Is – A+BF[] =
(s)
Proof. Let (s) for i=1,2,…,r be the non-unit
invariant polynomials of the matrix det[D(s)+FN(s)]
By Lemma 4, the polynomials c1(s), c2(s), …., cr(s)
are the non-unit invariant polynomials of [Is –
A+BF] and remaining invariant polynomials are
cr+1(s),=c2(s)=….=cn(s)=1. The characteristic
polynomial of the matrix [Is – A+BF] is given by
[8, p. 47]
det[Is – A+BF] =
(s) (30)
condition (a) of the Lemma 7 follows from (30) and
the proof is complete.
4 Problem Solution
The following theorem is the main result of this
paper and gives explicit necessary and sufficient
conditions for the solution of the pole assignment
problem by proportional-plus-derivative state
feedback for multivariable linear time-invariant
systems.
Theorem 1. The pole assignment problem by
proportional-plus-derivative state feedback for
multivariable linear time-invariant systems has
solution over R if and only if the following
condition holds:
(a) The open-loop system (5) is controllable.
Proof. Let c(s) be an arbitrary monic polynomial
over R[s] of degree n. Suppose that the pole
assignment problem by proportional-plus-derivative
state feedback has a solution over R. From (9) it
follows that
(31)
where μ is finite nonzero real number Let V(s) be
the greatest common left divisor of polynomial
matrices [ Is-A] and B. Then, from (16) and (17) it
follows that
[Is – A] = V(s) X(s) (32)
B = V(s) Y(s) (33)
for polynomial matrices X(s) and Y(s) over R[s] of
appropriate dimensions. We rewrite the polynomial
matrix [s – A + BF ] as
[s–A+BF]=[Is–A, B]
(34)
Using (32), (33) and (34) and after simple algebraic
manipulations, the relationship (31) can be rewritten
as
det[V(s)]det[[X(s), Y(s)]
] = (35)
From relationship (35) it follows that
det[V(s)] divides (36)
Since by assumption is an arbitrary monic
polynomial over R[s] of degree n, relationship (36)
is satisfied if and only if
det[V(s)] = λ (37)
where λ is finite nonzero real number. From (37)
and (11) it follows that the polynomial matrix V(s)
is unimodular; therefore by Lemma 1 the
polynomial matrices [ Is-A] and B are relatively left
prime over R[s] or equivalently
rank [Is – A , B] = n (38)
for all complex numbers s.
From (38) and (19) it follows that the pair (A, B) is
controllable. This is condition (a).
To prove sufficiency, we assume that condition (a)
holds. We form the matrix
D = [0, (-2)Im] (39)
From (6) and (39) it follows that the matrix
[ = diag[In-m , -Im] (40)
is non-singular. Let A1 and B1 be real matrices of
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appropriate dimensions given by
A1 = (41)
B1 = (42)
By Lemma 6 the pair (A1, B1) with [ given
by (39), is controllable. Let D1(s) and N1(s) be
relatively prime polynomial matrices over R[s],
with D1(s) column reduced and column degree
ordered, which satisfy
(Is – A1)-1B1 = N1(s)
(s) (43)
Let
12m
and Dh be the ordered list of
column degrees and the highest column degree
coefficient matrix of polynomial matrix D1(s)
respectively. Further, let c(s) be an arbitrary monic
polynomial over R[s] of degree n. We form the
polynomial matrix C(s) of size m x m
C(s) = diag[c(s), 1, …, 1] (44)
whose invariant polynomials are c(s), 1, …, 1. Then,
Lemma 3 can be applied several times if necessary
in order to make the polynomial matrix C(s) column
reduced with column degrees equal to those of
D1(s), without changing its invariant polynomials.
We call the resulting matrix C1(s). Let Ch be the
highest column degree coefficient matrix of
polynomial matrix C1(s) [10]. We form the
polynomial matrix C2(s)
C2(s) =
C1(s) (45)
From (45) it follows that the polynomial matrices
C2(s) and D1(s) have the same highest column
degree coefficient matrix [8, p.123], [10]; therefore
degci(s) - D1(s)] ≤
i
-1
i
= 1,2, .....,
m
(46)
from (46) it follows directly that the following
rational matrix
(s) - D1(s)]
(s) (47)
is strictly proper [8, p.39]. From (47) and Lemma 2
it follows directly that the equation
D1(s) + F N1(s) = C2(s) (48)
or equivalently the equation
F N1(s) = C2(s) - D1(s) (49)
has a solution for F over R. From (44), (45) and
(48) it follows that the invariant polynomials of the
polynomial matrix D1(s) + F N1(s) of size m x m are
(s) = c(s), c2(s)=….=cm(s)=1 (50)
Since by Lemma 4 the polynomial matrices [Is –
A1+ B1F] and [D(s)+FN(s)] have the same non-unit
invariant polynomials, from (48) it follows that the
invariant polynomials of the matrix [Is – A1 + B1F]
of size n x n are
(s) = c(s), c2(s)=….=cn(s)=1 (51)
Then by Lemma 7
det[Is – A1 + B1F] =
(s) = c(s) (52)
using (41) and (42) the relationship (52) can be
rewritten as
A + BF] =c(s) (53)
since according to (40) the matrix [ is non-
singular, the relationship (53) after simple algebraic
manipulations, can be rewritten as
A + BF] =
=det]
= c(s) (54)
From (54) it follows that
(55)
where is nonzero real number given by
= 1/( det) (56)
from (55) it follows that the polynomial []
over R[s], is the characteristic polynomial of
closed-loop system (8). This completes the proof.
Corollary 1. For every multivariable linear time-
invariant controllable system (5) there exists a
proportional-plus-derivative state feedback (7) such
that the closed-loop system is stable.
Proof. Let c(s) be a monic polynomial over R[s] of
degree n whose roots lie in the open left half
complex plane. From Theorem 1 it follows that
there exists matrices F and D over R of appropriate
dimensions such that
(57)
where is nonzero real number given by
= 1/( det) (58)
From (57) it follows that the polynomial []
over R[s], is the characteristic polynomial of
closed-loop system (8). Since by assumption the
roots of polynomial c(s) over R[s] of degree n, lie
in the open left half complex plane, the closed-loop
system (8) is stable. This completes the proof.
The sufficiency part of the proof of Theorem 1
suggests a simple procedure to compute the matrices
F and D of proportional-plus-derivative state
feedback (7) which assigns the poles of closed-loop
system (8) to desired positions.
Given: A, B and
Find: F and D
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Step 1: Form the polynomial matrix
[Is – A, B]
and check whether has full row rank for all
complex numbers s. If not, the open-loop system
(5) is uncontrollable; therefore the pole
assignment problem by proportional-plus-
derivative state feedback is impossible.
Step 2: Set
D = [0, (-2)Im]
Step 3:Calculate the matrices
[ = diag[In-m , -Im]
A1 =
B1 =
Step 4: Calculate relatively prime polynomial
matrices D1(s) and N1(s) over R[s] [10], with D1(s)
column reduced and column degree ordered, which
satisfy
(Is – A1)-1B1 = N1(s)
(s)
Step 5: Read out
1, 2m
, the column
degrees of polynomial matrix D1(s).
Step 6: Form the matrix C(s) of size m x m
C(s) = diag[, 1, …, 1]
Step 7: Apply Lemma 3 several times if
necessary in order to make the polynomial matrix
C(s) column reduced with column degrees
1, 2,
m.
Call the resulting matrix C1(s).
Step 8: Set
C2(s) =
C1(s)
where Dh and Ch are the highest column degree
coefficient matrices of polynomial matrices D1(s)
and C1(s) respectively
Step 9: Calculate the solution for F over R of the
equation
D1(s) + F N1(s) = C2(s)
In [10] has been proposed a computationally
efficient method for the calculation of solution
for F over R of the above equation. In particular
the proposed method in [10] reduces the solution
of the equation D1(s) + F N1(s) = C2(s) to that of
solving a system of linear equations of the form
FK=L, where the real matrices K and L of
appropriate dimensions, are constructed from
polynomial matrices N1(s), D1(s) and C2(s) over
R[s]. For more details the reader is referred to
[10].
Remark. The pole assignment problem by
proportional state feedback for multivariable linear
time invariant systems has been completely solved
in [11] and [12](according to [13]). As far as we know
the pole assignment problem by proportional-plus-
derivative state feedback for multivariable linear
time invariant systems is an open problem yet. The
main theorem of this paper gives explicit necessary
and sufficient conditions for the solution of the pole
assignment problem by proportional-plus-derivative
state feedback for multivariable linear time invariant
systems. This clearly demonstrates the originality of
the contribution of the main theorem of this paper
with respect to existing results.
5 Conclusions
In this paper is proven that the pole assignment
problem by proportional-plus-derivative state
feedback for multivariable linear time invariant
systems has solution over the field of real numbers
if and only if the given open-loop system is
controllable. Furthermore is proven that every
multivariable linear time-invariant controllable
system is stabilizable by proportional-plus -
derivative state feedback. The proof of the main
result of this paper is constructive and furnishes a
procedure for the computation of proportional-plus-
derivative state feedback which assigns the poles of
closed-loop system to any desired positions.
References:
[1] E. Seraji and M. Tarokh, Design of
proportional-plus-derivative output feedback
for pole assignment, Proceedings IEE,
Vol.124, No.8, 1977, pp.729-732.
[2] H. Seraji, Pole placement in multivariable
systems using proportional plus derivative
output feedback, International Journal of
Control, Vol.31, No.1, 1980, pp.195-207.
[3] M. Tarokh and H. Seraj, Proportional –plus-
multiple derivative output feedback: A new
multivariable controller for pole placement,
International Journal of Control, Vol.25,
No.2, 1977, pp.293-302.
[4] .T. Rajagopalan, Pole assignment with output-
feedback, Automatica, Vol.20, No.1,1984,
pp.127-128.
[5] T.H.S. Abdelaziz, Pole placement for single
input linear system by proportional-derivative
state feedback, ASME J. Dynamic Systems,
Meas. & Control, Vol. 137, No. 4, 2015, pp.
041015-1-041015-10.
[6] W.A. Wolowich, Linear Multivarible systems,
Springer Verlag, Berlin, New York, 1974
[7] P. J. Antsaklis and A. Michel, Linear Systems
Birkhauser, Boston, 2006.
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[8] V.Kucera, Analysis and Design of Linear
Control Systems, Prentice Hall, London,
1991.
[9] H.H.Rosenbrock, State-space and
Multivariable Systems, Nelson, London,
1970.
[10] V. Kucera, Assigning the invariant factors by
feedback Kybernetika, Vol. 2, No. 17, 1981,
pp.118–127.
[11] W. M. Wonham, On Pole assignment in multi-
Input controllable linear systems. IEEE Trans.
Automat. Control, Vol. 12, No.6, 1967, pp. 660-
665.
[12] V.I. Zubov, Theory of Optimal Control,
Sudostroenie, Leningrad 1966. (in Russian).
[13] G.A. Leonov and M.M. Shumafov, Stabilization
of controllable linear systems, Nonlinear dynamics
and systems theory, Vol. 10, No. 3, 2010, pp.235-
268.
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