Stabilization of Linear Time-Invariant Systems by State-Derivative
Feedback
KONSTADINOS H. KIRITSIS
Hellenic Air Force Academy,
Department of Aeronautical Sciences,
Division of Automatic Control,
Dekelia Air Base, PC 13671,
Acharnes, Attikis, Tatoi,
GREECE
Abstract: - In this paper is studied the stabilization problem by state-derivative feedback for linear time-
invariant continuous-time systems. In particular, explicit necessary and sufficient conditions are established for
the stability of a closed-loop system, obtained by state-derivative feedback from the given linear time-invariant
continuous-time system. Furthermore a procedure is given for the computation of stabilizing state-derivative
feedback. Our approach is based on properties of real and polynomial matrices.
Key-Words: - stabilization, state-derivative feedback, linear time-invariant continuous-time systems.
Received: October 15, 2022. Revised: January 17, 2023. Accepted: February 20, 2023. Published: March 24, 2023.
1 Introduction
What are the conditions under which the closed-loop
system obtained by state-derivative feedback from a
given linear time-invariant continuous-time system
is stable? This simple question is known as
stabilization of linear time-invariant continuous-
time systems by state-derivative feedback. In [1] are
established sufficient conditions for the solution of
the stabilization problem by state-derivative
feedback for linear time-invariant continuous-time
systems. In particular is proven that if the given
linear time-invariant continuous-time system is
either controllable or uncontrollable with stable
uncontrollable poles and all its controllable poles
are nonzero then the closed-loop system obtained by
state- derivative feedback from a given linear time-
invariant continuous-time system is stable. In [2],
see also [3], is proven that if the given linear time-
invariant continuous-time system has at least one
zero pole then the closed-loop system obtained by
state- derivative feedback from a given linear time-
invariant continuous-time system has also at least
one zero pole; therefore the stabilization problem by
state-derivative feedback has no solution. The state-
derivative feedback design methods have been
extensively studied over the last twenty years. The
motivation for the study of these methods comes
from some practical applications such as controlled
vibration suppression of mechanical systems, for
more complete references we refer the reader to [1-
5] and references given therein.
To the best of our knowledge the stabilization
problem by state-derivative state feedback for linear
time-invariant continuous-time systems in its full
generality, is still an open problem. This motivates
the present study. In this paper, are established
explicit necessary and sufficient conditions for the
solution of the stabilization problem by state-
derivative feedback for linear time-invariant
continuous-time systems. In particular it is proved
that the sufficient conditions of [1] for the solution
of stabilization problem by state-derivative feedback
for linear time-invariant continuous-time systems
are also necessary. Furthermore a procedure is given
for the computation of stabilizing state- derivative
feedback.
2 Problem Formulation
Consider a linear time-invariant continuous-time
system described by the following state-space
equations
=Ax(t) + Bu(t) (1)
where A and B are real matrices of size (n x n) and
(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 (m x 1). Consider the control law
u(t) = Dv(t) (2)
where D is a real matrix of size (m x n) and v(t) is
the reference input vector of size (m x 1). By
applying the state-derivative feedback (2) to the
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system (1), the state-space equations of closed–loop
system are
[I -= Ax(t) + Bv(t) (3)
Let R be the field of real numbers. Also let R[s] be
the ring of polynomials with coefficients in. R. The
stabilization problem by state-derivative feedback
considered in this paper can be stated as follows:
Does there exists a state-derivative state feedback
(2) such that
(4)
where is a monic, strictly Hurwitz polynomial
over R[s] of degree
n
(i.e., all roots of have
negative real parts . If so, give conditions for
existence and a procedure for the computation of
matrix D. It is pointed out that relationship (4)
ensures that the closed-loop system (3) is a stable
regular state-space system [6].
3 Basic Concepts and Preliminary
Results
This section contains lemmas which are needed to
prove the main results of this paper and some basic
notions from linear control theory that are used
throughout the paper. Let C be the field of complex
numbers, also let C+ be the set of complex numbers
λ with Re(λ) . A matrix whose elements are
polynomials over R[s] termed polynomial 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)] = μ (5)
where μ is a finite nonzero real number; therefore
every unimodular polynomial matrix has a
polynomial inverse. Every polynomial matrix W(
s
)
of size (p x m) with rank[W(
s
)]=r, can be
expressed as [7]
U1(
s
)W(
s
)U2(
s
)=M(
s
) (6)
The polynomial matrices U1(s) and U2(s) are
unimodular and the matrix M(s) is given by
(7)
The non-singular polynomial matrix of
size (
r
x
r)
in (7) is given by
= diag[a1(
s
)
,
a2(
s), ...,
ar(
s
)] (8)
The nonzero polynomials ai(s) for i=1,2,...,r are
termed invariant polynomials of W(s) and have the
following property
ai(
s
) divides ai+1(
s
), for
i = 1,2,..., r-1
(9)
The relationship (6) is called Smith-McMillan form
of W(s) over R[s]. Since the matrices U1(
s
) and
U2(
s
) are unimodular and the polynomial matrix
given by (8) is non-singular, from (6) and
(7) it follows that
rank[W(
s
)] =
rank
=
r
(10)
Let A(s), be a polynomial matrix over R[s] if there
are polynomial matrices P(s) and Q(s) of
appropriate dimensions such that
A(s) = P(s) Q(s) (11)
Then the polynomial matrix P(s) over R[s] termed
the left divisor of A(s) [7]. Let A(s) and B(s), be
polynomial matrices over R[s] if
A(s) = D(s) M(s) (12)
B(s) = D(s) N(s) (13)
for polynomial matrices M(s), N(s) and D(s) over
R[s], then D(s) termed the common left divisor of
polynomial matrices A(s) and B(s) [7]. 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 [8]. Let
A and B be real matrices of size (n x n) and (
n
x
m)
respectively. Then there always exists a unimodular
matrix U(s) over R[s] such that
[, B] = [V(s), 0] U(s) (14)
The non-singular polynomial matrix V(s) of size
(n x n) is a greatest common right divisor of the
polynomial matrices and B [8]. Since the
polynomial matrix U(s) is unimodular from (14) it
follows that
rank[,B] = rank[V(s),0]=rank[V(
s
)] =
n
(15)
Definition 1: The nonzero polynomial
c
(
s
) over
R[s] is said to be strictly Hurwitz if and only if
c
(
s
) , C+.
Definition 2: Let V(
s
) be a non-singular matrix
over R[s] of size (
n
x
n)
. Also let
c
i(
s
) for
i
=
1,2,
n
be the invariant polynomials of polynomial
matrix V(
s
). The polynomial matrix V(
s
) is said to
be strictly Hurwitz if and only if the polynomials
c
i(s) are strictly Hurwitz for every
n
, or
alternatively if and only if det[V(s)] is a strictly
Hurwitz polynomial.
Definition 3: The matrix A over R matrices of size
(n x n), is said to be Hurwitz stable if and only if all
eigenvalues of the matrix A have negative real parts
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or alternatively if and only if the characteristic
polynomial of matrix A is a strictly Hurwitz
polynomial.
Definition 4: Let A and B be matrices over R
matrices of size (n x
n)
and (n x m), respectively.
Then the pair (A, B) is said to be stabilizable if and
only if there exists a real matrix K of size (m x n),
such that the matrix [AK] is Hurwitz stable [9].
The following Lemma is taken from [10].
Lemma 1: Let A and B be matrices over R of size
(n x n) and (
n
x
m)
, respectively. The pair (A, B) is
stabilizable if and only if the following condition
holds:
= C+
Lemma 2: Let V(s) be a non-singular polynomial
matrix over R[s], of size (n x n)
.
Also let
c
i(s) for
i=1,2, n
be the invariant polynomials of the
polynomial matrix V(s). The polynomial matrix
V(s) is strictly Hurwitz if and only if the following
condition holds
(a) rank[V(
s
)] =
n
, C+
Proof: Let V(
s
) be a non-singular and strictly
Hurwitz polynomial matrix of size (n x n) with
invariant polynomials
ci
(
s
) for
n
From
Definition 2 it follows that the polynomials
c
i(s)
are strictly Hurwitz for every
n
and
therefore from Definition 1 it follows that
ci
(
s
) ) C+ ,
(16)
we define the polynomial matrix
= diag [
c
1(
s
)
, c
2(
s
)
, c
n(
s
)] (17)
From (16) and (17) it follows that
]=
rank
[
diag
[
c
1(
s
)
,c
2(
s
)
n(
s
)]}=
n
C+ (18)
The Smith-McMillan form of polynomial matrix
V(
s
) over R[s] is given by
K(s) V(s) L(s) = (19)
where K(s) and L(s) are unimodular matrices.
Since the matrices K(s), L(s) are unimodular,
from (10), (17), (18) and (19) it follows that
rank[V(
s
)]=
n
, C+ (20)
This is condition (a) of the Lemma. To prove
sufficiency, we assume that condition (a) holds.
Using (10) from (17) and (19) we have that
rank[=]=
=
rank
{
diag
[
c
1(
s
)
,c
2(
s
)
n(
s
)]}=
n
(21)
Since by assumption condition (a) holds we have
that
rank[V(
s
)]=
n
, C+ (22)
Relationships (21) and (22) imply
]=
rank
{
diag
[
c
1(
s
)
,c
2(
s
)
cn(
s
)]}=
=
n
C+ (23)
From (23) it follows that
ci
(
s
) ) C+ ,
i
(24)
Relationship (24) and Definition 1 imply that
polynomials ci(s) are strictly Hurwitz for every
i=1,2,…,n, and therefore according to Definition 2
the non-singular polynomial matrix V(
s
) over R[s],
is strictly Hurwitz. This completes the proof.
Lemma 3: Let A and B be matrices over R
matrices of size (n x n) and (n x m), respectively and
B not zero. Further let V(s) be a greatest common
left divisor of polynomial matrices [Is-A] and B of
size (n x n). The pair (A, B) is stabilizable if and
only if the following condition holds:
(a) The polynomial matrix V(s) is strictly Hurwitz.
Proof: Let the pair (A, B) is stabilizable. Then
from Lemma 1 it follows that
= C+ (25)
Since by assumption the polynomial matrix V(s) is
the greatest common left divisor of polynomial
matrices [Is-A] and B, from (14) it follows that
there exists a unimodular matrix U(s) such that
[, B] = [V(s, 0] U(s) (26)
Since the polynomial matrix U(
s
) is unimodular
from (15) and (26) it follows that
rank[,B] = rank[V(s, 0]=rank[V(
s
)] =
n
(27)
From relationships (25) and (27) it follows that
rank[V(
s
)]=
n
, C+ (28)
Relationship (28) and Lemma 2 imply that the
polynomial matrix V(s) is strictly Hurwitz. This is
condition (a) of the Lemma. To prove sufficiency,
we assume that the polynomial matrix V(s) is
strictly Hurwitz. Then from Lemma 2 it follows that
rank[V(
s
)]=
n
, C+ (29)
Since the polynomial matrix V(s) is the greatest
common left divisor of polynomial matrices [Is-A]
and B, from (29) and (27) it follows that
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= C+ (30)
Lemma 1 and (30) imply that the pair (A, B) is
stabilizable. This completes the proof.
Lemma 4: Let A and B be matrices over R
matrices of size (n x n) and (n x m), respectively and
B not zero. Further let V(s) be a greatest common
left divisor of polynomial matrices [Is-A] and B of
size (n x n). Further let D be a matrix over R of size
(m x n) such that det[ Then the
following condition holds:
(a) The polynomial matrix V(s) is a
left divisor of the matrix ].
Proof: Since by assumption det[ the
matrix is non-singular and therefore the
matrix ] can be rewritten as
follows
] =
=[s – A] =
= Is–A, B]
(31)
Since by assumption the polynomial matrix V(s) is
the greatest common left divisor of polynomial
matrices [Is-A] and B, from (12) and (13) 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. Using (32) and (33) and
after simple algebraic manipulations, the
relationship (31) can be rewritten as
] =
V(s)[X(s) Y(s)(Ds] (34)
Using (11) from (34) it follows that the matrix
V(s) is a left divisor of the polynomial
matrix ]. This is condition (a) of
the Lemma and the proof is complete.
Lemma 5: Let A be a Hurwitz stable matrix over
R of size (n x n). Then the following condition
holds:
(a) The matrix A is non-singular.
Proof: Let A be a Hurwitz stable matrix over R of
size (n x n). The characteristic polynomial of
matrix A is given by [11]
(35)
From Definition 3 it follows that is a strictly
Hurwitz polynomial over R[s] of degree
n.
Let
i
for i=1,2,…,n, be the roots of . Then
c(
i
) = 0
(36)
Since is a strictly Hurwitz polynomial of
degree
n
from Definition 1 and (36) it follows that
Re(
i
) 0 ,
(37)
Since the polynomial
in (35) is the
characteristic polynomial of the matrix A, the
complex numbers
i
for i=1,2,…,n, are the
eigenvalues of the matrix A. From (37) it follows
that
i
0 ,
(38)
From (38) it follows that all eigenvalues of the
matrix A are nonzero and therefore the matrix A is
non-singular [11]. This is condition (a) of the
Lemma and the proof is complete.
The following Lemma is partially based on the
main results of [1] and [12]
Lemma 6: Let A and B be matrices over R
matrices of size (n x n) and (n x m), respectively
with A being non-singular and B not zero. Further
let D be a matrix over R of size (m x n) such that
det[ .
Then there exists real matrices F of appropriate size
and D given by
D = F[A BF]-1
such that the matrix is Hurwitz stable
if and only if the following condition holds:
(a) The pair (A, B) is stabilizable.
Proof: Let there exists real matrices F and D of
appropriate dimensions with D given by
D = F[A BF]-1 (39)
such that the matrix is Hurwitz stable.
Since by assumption the matrix A is non-singular,
from (39) we have that
]-1 A=
= [ ]-1A=
= A-1A = ] (40)
Since by assumption the matrix is
Hurwitz stable, from (40) it follows that the matrix
] is Hurwitz stable. Hurwitz stability of the
matrix and Definition 4 imply that the
pair (A, B) is stabilizable. This is condition (a) of
the Lemma. To prove sufficiency we assume that
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the pair (A, B) is stabilizable. Stabilizability of the
pair (A, B) and Definition 4 imply the existence of
matrix F of appropriate size such that the
] is Hurwitz stable. Hurwitz stability
of the matrix and Lemma 5 imply that the
matrix ] is non-singular and therefore
invertible. Invertibility of the real matrix
implies the possibility of calculation of the matrix D
given by (39).Taking in the mind the above, from
relationship (40) we conclude that the real matrix
(41)
with D given by (39) is Hurwitz stable. This
completes the proof.
The following Lemma was first published by
Wonham in [9] and can be also found in any
standard text of linear control theory.
Lemma 7. Let A and B be matrices over R
matrices of size (n x n) and (n x m), respectively.
Then the pair (A, B) is controllable if and only if for
every monic polynomial over R[s] of degree
n
there exists a matrix F over R of size m x n, such
that the matrix [A+BF] has characteristic
polynomial .
The standard decomposition of uncontrollable
systems given in the following Lemma was first
published by Kalman in [13] and can be also
found in any standard text of linear control
theory.
The following Lemma is taken from [7].
Lemma 8: Let A and B be matrices over R
matrices of size (n x n) and (n x m), respectively
Further let the pair (A, B) is uncontrollable and B
not zero. Then there exists a non-singular matrix
T such that
AT =
B =
The pair is controllable and the
eigenvalues of the matrix are the uncontrollable
eigenvalues of the pair (A, B).
Lemma 9: Let A and B be matrices over R matrices
of size (n x n), (n x m), respectively and B not zero.
Further let
AT=
B =
with controllable. The pair (A, B) is
stabilizable if and only if the following condition
holds:
(a) The matrix is Hurwitz stable or
alternatively all uncontrollable eigenvalues of the
pair (A, B) are stable (i.e., the eigenvalues of the
matrix have negative real parts).
Proof: From the statement of the Lemma we have
that
A =
B =
(42)
with controllable. If the pair (A, B) is
stabilizable, then from Definition 4 it follows that
there exists a matrix F such that the matrix [A +BF]
is Hurwitz stable. Using (42) we have that
A BF =
F =
=
+
FT} (43)
Let
FT = [, ] (44)
Substituting (44) to (43) and after simple algebraic
manipulations we have that
A +BF =
(45)
From (45) it follows that the matrices
[A + BF],
(46)
are similar; therefore Hurwitz stability of [A+BF]
implies Hurwitz stability of . Since the matrix
is Hurwitz stable, from Lemma 8 and
Definition 3 it follows that all uncontrollable
eigenvalues of the pair (A, B) are stable. This is
condition (a) of the Lemma. To prove sufficiency
we assume that condition (a) holds. Controllability
of the pair and Lemma 7 imply the
existence of matrix of appropriate size such that
the matrix
det] φ(s) (47)
where φ(s) is an arbitrary monic, strictly Hurwitz
polynomial over R[s] of appropriate degree. The
matrix can be calculated using known methods
for the solution of pole assignment problem by state
feedback [7]. Let
F= [, 0] (48)
Substituting (48) to (45) we have that
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A + BF =
(49)
Using (47), from (49) it follows that Hurwitz
stability of implies Hurwitz stability of
[A+BF]. Hurwitz stability of [A+BF] and Definition
4 imply stabilizability of the pair (A, B). This
completes the proof.
4 Problem Solution
The following theorem is the main result of this
paper and gives explicit necessary and sufficient
conditions for the solution over R of the
stabilization problem by state-derivative feedback
for linear time-invariant continuous-time systems
with state-space equations given by (1).
Theorem 1. The stabilization problem by state-
derivative feedback for linear time-invariant
continuous-time systems with state-space equations
given by (1) has solution over R if and only if the
following condition holds:
(a) The matrix A is non-singular.
(b) The pair (A, B) is stabilizable.
Proof: Let system (1) is stabilizable by state-
derivative feedback. Then from (4) we have that
(50)
where is a monic, strictly Hurwitz polynomial
over R[s] of degree
n.
Relationship (50) and
Definition 3 imply that the matrix ] is
Hurwitz stable. Hurwitz stability of the matrix
and Lemma 5 imply non-singularity
of and therefore non-singularity of
matrix A. This is condition (a) of the Theorem. Let
V(s) be a greatest common left divisor of
polynomial matrices [Is-A] and B of size (n x n).
Then from Lemma 4 it follows that the polynomial
matrix V(s) is a left divisor of the
polynomial matrix ] that is
] = V(s) X(s) (51)
where X(s) is a matrix over R[s] of appropriate size.
From (51) we have that
det] =
det[V(s)] det[X(s)] (52)
From relationship (50) and (52) it follows that
det[V(s)] divides (53)
Since by assumption is a monic, strictly
Hurwitz polynomial over R[s] of degree n, from
(53) it follows that det[V(s)] is a strictly Hurwitz
polynomial over R[s]; therefore by Definition 2 the
polynomial matrix V(s) is strictly Hurwitz. Since
V(s) is strictly Hurwitz, from Lemma 3 it follows
that the pair (A, B) is stabilizable. This is condition
(b) of the Theorem.
To prove sufficiency, we assume that conditions (a)
and (b) hold. Stabilizability of the pair (A, B) imply
that the pair (A, B) is either controllable or
uncontrollable with stable uncontrollable
eigenvalues (i.e. all uncontrollable eigenvalues have
negative real parts).
If the pair (A, B) is controllable, then from
Lemma 7 it follows that there exists a matrix F of
appropriate size over R such that
det[ A BF] = A BF] = χ(s) (54)
where χ be an arbitrary monic, strictly Hurwitz
polynomial over R[s] of degree
n.
The matrix F can
be calculated using known methods for the solution
of pole assignment problem by state feedback [7].
If the pair (A, B) is uncontrollable with stable
uncontrollable eigenvalues, then from Lemma 8 and
Lemma 9 it follows that there exists a matrix T
such that
AT =
B =
(55)
The pair is controllable and the matrix
is Hurwitz stable. Controllability of the pair
and Lemma 7 imply the existence of a
matrix over R of appropriate dimensions such
that
det] φ(s) (56)
where φ(s) is an arbitrary monic, strictly Hurwitz
polynomial over R[s] of appropriate degree. The
matrix can be calculated using known methods
for the solution of pole assignment by state feedback
[7]. According to (49) the matrix [A+BF] with F
given by
F= [, 0] (57)
is Hurwitz stable. Conditions (a) and (b) and
Lemma 6 imply the existence of the matrix D given
by
D = F[A BF]-1 (58)
such that the matrix is Hurwitz stable
that is
(59)
where is a monic, strictly Hurwitz polynomial
over R[s] of degree
n.
From (59) and (4) it follows
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that the closed-loop system (3) is a stable regular
state-space system. This completes the proof.
The sufficiency part of the proof of Theorem 1
provides a construction of the matrix D of state-
derivative feedback which stabilizes the system (1).
The major steps of this construction are given
below.
Construction
Given: A, B
Find: D
Step 1: Check conditions (a) and (b) of Theorem
1. If these conditions are satisfied go to Step 2. If
conditions (a) and (b) are not satisfied go to Step 4.
Step 2: Stabilizability of the pair (A, B) implies
that the pair (A, B) is either controllable or
uncontrollable with stable uncontrollable
eigenvalues. If the pair (A, B) is controllable, then
from Lemma 7 it follows that there exists a matrix
F over R such that
det[ A BF] = A BF] = χ(s)
where be an arbitrary monic and strictly
Hurwitz polynomial over R[s] of degree
n.
The
matrix F can be calculated using known methods
for the solution of pole assignment problem by state
feedback [7].
If the pair (A,B) is uncontrollable with stable
uncontrollable eigenvalues then from Lemma 8 and
Lemma 9 it follows that there exists a matrix T
such that
AT =
B =
The pair is controllable and the matrix
is Hurwitz stable. Controllability of the pair
and Lemma 7 imply the existence of a
matrix over R of appropriate dimensions such
that
det] φ(s)
where φ(s) is an arbitrary monic, strictly Hurwitz
polynomial over R[s] of appropriate degree. The
matrix can be calculated using known methods
for the solution of pole assignment by state feedback
[7]. According to (49) the matrix [A+BF] with F
given by
F= [, 0]
is Hurwitz stable.
Step 3: Put
D = F[A BF]-1
Step 4: The stabilization problem by state-
derivative state feedback has no solution.
5 Conclusions
In this paper the stabilization problem by state-
derivative feedback for linear time-invariant
continuous-time systems is studied and completely
solved. The proof of the main results of this paper is
constructive and furnishes a procedure for the
computation of stabilizing state-derivative feedback.
As far as we know the stabilization problem by
state-derivative feedback for linear time-invariant
continuous-time systems in its full generality, is still
an open problem. This clearly demonstrates the
originality of the contribution of the main results of
this paper with respect to existing results.
References:
[1] T.H.S. Abdelaziz and M. Valasek, Direct
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WSEAS TRANSACTIONS on SYSTEMS and CONTROL
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