=Paper=
{{Paper
|id=Vol-1152/paper38
|storemode=property
|title=Stabilization of Interconnected Systems with Decentralized State and/or Output Feedback
|pdfUrl=https://ceur-ws.org/Vol-1152/paper38.pdf
|volume=Vol-1152
|dblpUrl=https://dblp.org/rec/conf/haicta/Parisses11
}}
==Stabilization of Interconnected Systems with Decentralized State and/or Output Feedback==
https://ceur-ws.org/Vol-1152/paper38.pdf
Stabilization of Interconnected Systems with
Decentralized State and/or Output Feedback
Parisses Constantinos
Department of Electrical Engineering, Technology Educational Institute of Western
Macedonia, Greece, e-mail: kpar@teikoz.gr
Abstract. This paper investigates the problem of the stabilization of a system
(A, Bd) consisting of two interconnected subsystems, with decentralized state
or output feedback. Following the initial definition of the global system and of
its two subsystems in the state-space, and based on the intercontrollability
matrix D(s) of system (A, Bd) and on the kernel U(s) of D(s), an equivalent
system {M(s), I2} defined in the operator domain by an appropriate polynomial
matrix description (PMD) is determined. The interconnected system can then
stabilized with a suitable local feedback, based on which, a decentralized
output feedback can be determined as well.
Keywords: control systems, modeling and simulation interconnected systems,
decentralized stabilization, local output feedback.
1 Introduction
Decentralized control has been a control of choice for large-scale systems (consist
of many interconnected subsystems) for over four decades. It is computationally
efficient to formulate control laws that use only locally available subsystem states or
outputs. Such an approach is also economical; since it is easy to implement and can
significantly reduce costly communication overhead. Also, when exchange of state
information among the subsystems is prohibited, decentralized structure becomes an
essential design constraint. Necessary and sufficient conditions, as well as methods
and algorithms have been proposed in these four decades, to find decentralized
feedback controllers which stabilize the overall system (see (Ikeda,1980), (Sandell,
1978), (Siljak, 1978 ), (Wang, 1973) and the references therein). In recent years, the
problems of decentralized robust stabilization for interconnected uncertain linear
systems have been studied by many researchers. Different design approaches have
been proposed, such as the Riccati approach (Ge, 1996), (Ugrinovsskii , 1998), the
LMI (Linear Matrix Inequality) approach (Liu, 2004), (Souza, 1999), a combination
of genetic algorithms and gradient-based optimization (Labibi, 2003), (Patton, 1994).
It is the main purpose of this paper to present the stabilization problem of an
interconnected (global) system with decentralized state or output feedback. The
interconnected (global) system (A, Bd), consists of two local scalar subsystems,
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431
under the very general assumptions of the global and the local controllability. It is
noted that only the case of two interconnected subsystems is examined, since only
then the global system will have no decentralized fixed modes, when local static
state-vector feedbacks are applied ((Aderson, 1979), (Caloyiannis, 1982), (Davison,
1983), (Fessas, 1982,1987,1988) and (Wolovich, 1974). Additionally, we assume,
without loss of generality (Davison, 1983) that both input channels of system are
scalar.
Following the initial definition of the system in the state-space, an equivalent
system - defined in the operator domain- is first determined. This is presented in the
next section, together with some known results concerning (a) the intercontrollability
matrix D(s) of (A, Bd) (b) the kernel U(s) of D(s), (c) the equivalent system
{M(s), I2} in the operator domain, and (d) the stabilization of the interconnected
system with linear, local, state-vector feedback (LLSVF), introducing linear
programming methods for computing them (Parisses, 1998). In case the values of
these feedbacks are considered to be large for practical implementation, an algorithm
for designing “optimal” decentralized control can be applied (Parisses, 2006). In
section 3, the main result on the stabilizing local output feedbacks, and on a method
to design a suitable output matrix C, is presented. As a corollary, the decentralized
version of all theses is given. To demonstrate this illustrative example is given, in
section 4.
2 Preliminaries
2.1 Form of Matrices A and Bd
We consider the interconnected system (A, Bd) defined by
x Ax Bd u
Ax
(1)
where x is the n-dimensional state of (A, Bd), u is the 2-dimensional input vector, A
is the nxn system matrix, and Bd is its nx2 input matrix. Matrices A and Bd admits
the following partitioning:
A A n b11 0
11 12 1 and B
A d (2)
A A n 0 b 22
21 22 2
with n=n1+n2. System (A, Bd) consists of the interconnected n i-dimensional
subsystems (Aii, bii) -i=1,2- of local state vectors x1and x2, with x=[x1' x2']', u1
and u2 being, respectively, the scalar inputs of these subsystems, with u=[u1 u2]'. We
further assume that the global system (A, Bd), as well as its two subsystems (Aii, bii)
-i=1,2- are controllable. In that case, and in order to have some analytical results,
subsystems (Aii, bii) are supposed to be in their companion controllable form
(Kailath, 1980)
432
0 1 . 0 0
0 0 . . .
Aii (nixni) and bii = (nix1) (3)
0 0 . 1 0
aii 1
where aii' denotes the last row elements of Aii. When (Aii, bii) are in the above form,
the nixnj sub matrices Aij assume no particular form; for them we use the notation
Aijo
Aij = (i j) (4)
aij'
where aij' denotes the last row elements of Aij, and Aijo the others. It is obvious
that when the various sub matrices of (A, Bd) are in the above form, system
(A,
A Bd ) is called Canonical Interconnected Form (Fessas, 1982) and is as:
0 1 A120
0
. . 0
1 .
a11 a12 1
A Bd (5)
0
A 210 0 1
0 .
. .
1 1
a21 a22
Finally, with the elements of rows n 1 and n1+n2(=n) of A, we form matrix Am:
a11 a12
Am (6)
a 21 a 22
2.2 The intercontrollability matrix D(s) and its kernel
The following (n-2)xn polynomial matrix is the intercontrollability matrix of
system (A,
A Bd ) :
433
s 1
. .
. . A120
s 1 (7)
D(s)
s 1
. .
A 210 . .
s 1
As the following lemma indicates, D(s) expresses the conditions for the
controllability of (A, Bd):
Lemma 2.1: (Caloyiannis, 1982) System (A, Bd) is controllable if and only if rank
D(s) = n-2 for all complex numbers s. (Caloyiannis, 1982).
Thus the matrix D(s) of a controllable system is a full-rank matrix. Its kernel U(s)
is an nx2 polynomial matrix of rank 2, such that D(s) U(s) = 0. The analytical
determination of U(s) is as follows: P is the matrix representing the column
permutations of matrix D(s), which brings it to the form of the matrix pencil:
D(s) = D(s) P = [s In-2 - F | G] (8)
In (8) G is an (n-2)x2 (constant) matrix, consisting of columns n 1and n1+n2=n of
D(s), F is an (n-2)x(n-2) constant matrix, and In-2 is the unity matrix of order n-2.
Since D(s) is a full rank matrix, the pair (F,G) is controllable, and can be brought to
its Multivariable Controllable Form (MCF) ((Kailath, 1980), (Wolovich, 1974))
$ ) by a similarity transformation T; let d1, d2 be the controllability indices of
(F$ , G
(F,G), S(s) be the associated structure operator, let δ(s) be the characteristic
(polynomial) matrix of F, and (in case rank[G]=2) let G $ m be the 2x2 matrix
consisting of rows d1 and d1+d2=n-2 of G. The precise form of U(s) is the content of
the following lemma:
Lemma 2.2 Let D(s) be the intercontrollability matrix of (A, Bd) as in (7), and
suppose that rank[G]=2, for G as in (8). Then the kernel U(s) of D(s) is equal to
TS ( s)
U ( s) P (9)
Gˆ
1
m ( s)
where P, T, S(s), G
$ m , and δ(s) are as previously explained.
2.3 An equivalent system defined by a PMD
Consider the interconnected system (A, Bd), with A and Bd as in (5). In that case,
the corresponding differential equation in the state space is:
x(t)=Ax(t)+Bd u(t)
x(t)=Ax(t) (10)
In the operator domain, this equation corresponds to the equation
434
(sI - A) x(s) = Bd u(s) (11)
this, in its turn, reduces to the equations:
D(s) x(s) = 0 (12a)
and
(sE - Am ) x(s) = u(s) (12b)
In these equations, D(s) is as in (7), E is a 2xn (constant) matrix, of the form:
E = diag{e1' e2'}, the ni-dimensional vector ei being equal to : ei = [0...0 1]' -for
i=1,2-, and Am is the matrix defined in (6). From (12a) it follows that x(s) must
satisfy the relation:
x(s) = U(s) ξ(s) (13)
where U(s) is the kernel of D(s), and ξ(s) is any two-dimensional vector. It follows
that ξ(s) must satisfy the equation:
M(s) ξ(s) = u(s) (14)
The matrix M(s) appearing in (14) is termed Characteristic Matrix of the
interconnected system (A, Bd) (Fessas, 1982) and is defined by the relation:
M(s) = (sE - Am) U(s). (15)
The three systems defined respectively (i) in the state space by the pair of matrices
(A, Bd), (ii) in the operator domain by {sI-A, Bd}, and (iii) by the polynomial matrix
description (PMD):
M (D) ξ (t) = u (t) (16a)
X (t) = U (D) ξ (t) (16b)
are equivalents (Chen, 1984), (Fessas, 1987), (Kailath, 1980). It is noted that in (16)
ξ(t) is the pseudo state vector of the system, and is related to the state vector x(t) of
(A, Bd), by the relation
x(t) = U(D) ξ(t) (17)
(in the relations (16), (17), the symbol D denotes the differential operator d/dt).
2.4 Stabilizability with local state-vector feedback
We present analytically Theorem 2.1, on the stabilizability of the interconnected
system (A, Bd) with LLSVF, as well as a result, which is needed in the proof of it.
Lemma 2.3 : Let h(s) be a polynomial of the form: h(s)=r(s)p(s)+q(s), for which
the following assumptions hold: (i) The polynomials r(s), p(s), q(s) are monic (ii)
r(s) is arbitrary, (iii) degree r(s)p(s) > degree q(s) (iv) p(s) is a stable polynomial.
Then, the arbitrary polynomial r(s) can be chosen so, that h(s) is stable (Seser, 1978).
Theorem 2.1: Consider the interconnected system (A, Bd) as in (1), and suppose
that the global system (A, Bd), and the local ones (Aii, bii) -i=1,2- are controllable.
435
Then, there exists a static LLSVF of the form u=Kdx, so that the resulting closed-
loop system is stable.
Proof: For the proof we consider the equivalent system {M(s), I 2} and examine
the stability of the polynomial matrix Md(s)=(sE-Am-Kd)U(s). We assume that the
feedback matrix Kd has the form:
α1 . . α n1 0 . . 0 α 0
Kd = = (18)
0 . . 0 β1 . . β n2 0 β
where αi (i=1,...,n1), and βj (j=1,...,n2) are some unknown, real numbers. We shall
deal with the case where rank [G]=2, which is the usual one for the matrix G. Then
the matrix Md(s) takes the form:
TS ( s)
Md(s)=(sΕ-Am-Kd)U(s)=(sΕ-Am-Kd) P =
Gˆ
1
m ( s)
-(s-a n1 -a1,n1
1, )[11]-a(s)+a1,n [ 21] -(s-
--(s α n1 -a1,n1 )[12]-α1 (s)+a1,n
-(s-α 1 [ 22]
= =
-( β n2 -a 2,n1 )[21]-β1 (s)+aa 22,n1[ ] -(s-β
-(s-β - -β n2 -a 2,n
2, )[ 22]-β(s)+a 2,n1[12]
= M 11 ( s) M 12 ( s) (19)
M 21 ( s) M 22 ( s )
where α(s)=[α1+a1,1....αn1-1+a1,n1-1 a1,n1+1....a1,n-1]TS1(s)
α1(s)=[α1+a1,1....αn1-1+a1,n1-1 a1,n1+1....a1,n-1]TS2(s)
β(s)=[a2,1....a2,n1-1 β1+a2,n1+1....βn2-1+a2,n-1]TS2(s)
β1(s)=[a2,1....a2,n1-1 β1+a2,n1+1....βn2-1+a2,n-1]TS1(s)
are scalar polynomials, not monic,
TS1(s) = T [ 1 s ... sd1-1 0 ... 0 ]'
TS2(s) = T [ 0 ... 0 1 s ... sd2-1 ]'
(i.e., TS(s)=[TS1(s) TS2(s)] ,and [ij] -for i,j=1,2- are the entries of the polynomial
$m 1
matrix G ((ss) . Then the matrix in (19) is equivalent to the following matrix:
M11 ((s)+M 21 (s) M 12(s)+M 22(s)
Μ d (s)= (20)
M 21 (s) M 22 (s)
h(s)=detMd'(s)={M11(s)+M21(s)}M22(s)-{M12(s)+M22(s)}M21(s)=
={M11(s)+M21(s)}{-β(s)- (s-β n2 -a 2,n ) [22]+a2,n1[12]}-{Μ12(s)+M22(s)}M21(s)=
436
=-[22]{M11(s)+M21(s)} (s-β n2 -a 2,n ) -{M12(s)+M22(s)}M21(s)+{M11(s)+M21(s)}
{-β(s)+a2,n1[12]}=r(s)p(s)+q(s) (21)
The determinant of this matrix is actually a monic polynomial of degree n, by
identifying r(s) as the polynomial -[22][M11(s)+M21(s)], which is of degree (n-1),
arbitrary and monic, p(s) as the polynomial (s-β n2 -a 2,n ) , which is stable by choice of
βn2, and q(s) as the polynomial -
{M12(s)+M22(s)}M21(s)+{M11(s)+M21(s)}{-β(s)+a2,n1[12]}, of degree (n-1). Then,
according to lemma, the arbitrary polynomial r(s) can be chosen so that the
polynomial h(s) is stable. Q.E.D.
This proof is completed with an iterative method (Parisses, 1998), in order to
compute the feedback coefficients. The central idea is to compute the feedback
parameters by solving a linear programming problem (Luenberger, 1984)
corresponding to choosing positive the coefficients of the polynomials that should be
stable. A set of such polynomials (with positive coefficients) is generated. They are
then examined whether they are stable or not.
ALGORITHM
Step1 Choose the feedback parameter βn2+α2,n<0 so that a stable p(s) results.
Step2 Write the polynomial r(s) in the following form:
r(s) =sn-1+kρ(s)=sn-1+k(sn-2+k1sn-3+...+kn-2).
By viewing the degrees of the polynomials α(s) and β(s), it is seen that k-the
leading coefficient of the polynomial ρ(s)- contains only the parameters βn2 and
αn1. It follows that by giving a value to k, we can also compute α n1.
Step3 Form n-2 inequalities with the n-2 unknown feedback parameters, by setting
positive the coefficients ki of the polynomial ρ(s) (ki>0, for i=1,n-2).
Step4 Solve the linear programming problem, by putting an objective function with
unity weighting coefficients, and find all feedback parameters αi and βj.
Step5 Evaluate the polynomial ρ(s), and check if it is stable. If it is not, go back to
Step 1, and select another βn2.
Step6 Evaluate the polynomial r(s), and check if it is stable. If it is not, go back to
Step 2, and select another k.
Step7 Evaluate the polynomial h(s), and check if it is stable. If it is not, go back to
Step 1, and select another βn2.
Step8 The feedback matrix Kd can be evaluated from steps 1, 2, and 4.
END OF THE ALGORITHM
3 Main Result
Theorem 3.1 Consider the interconnected system (A, Bd) as in (1), under the usual
assumptions of the global and the local controllability. Then, this system can
437
stabilized with the feedback u=Ly, where the output feedback matrix L is arbitrary,
and the output matrix C is: C=L-1Kd, matrix Kd being the feedback stabilizing matrix.
Proof: Since system (A, Bd) satisfies the assumptions of the global and the local
controllability, there exists a local feedback stabilizing matrix Kd, such that A+BdKd
is stable. According to lemma 2.1 of (Fessas, 1994), system (A, Bd, C) can stabilized
with the output feedback u=Ly, when the output matrix C is given by the relation
C=L-1Kd.
Remark 3.1 It is remarked that, while the 2x2 output matrix L is arbitrary, it is the
2xn matrix C that takes care of the stabilization. As an extreme case consider L=I2
(the unity matrix); it is follows that the output matrix C is identical to the stabilizing
local state feedback matrix Kd.
Corollary 3.1 We consider matrix L as a diagonal Ld matrix (corresponding to the
control with local feedbacks). It follows that the output matrix C is also block-
diagonal Cd=Ld-1Kd corresponding, thus, to the case where the measurements are also
decentralized.
4 An illustrative example
The controllable system (A, Bd) is:
1 2 1 0 1 0
4 0 1 2 1 0
Bd =
3 2 2 1 0 1
5 0 3 4 0 1
The system is unstable, since the eigenvalues of A are: {0.315 2.732j, 3.185
1.511j}, and it is asked to be stabilized by the d-control u=Kdx. Subsystems Aii -
i=1,2- are transformed into their companion forms, by the transformation matrices:
2 1 5 1
T1 T2
5 1 1 1
while matrix , as in (5), is:
0 1 1.143
3 0.571
0
8 1 2.714 0.143
1 1.667
1 0 1
9 3.333 11 6
The intercontrollability matrix D(s), of system (A,
A Bd ) is:
s 1 1.143 0.571
D(s)
1 1.667 s 1
It follows that system (F, G) is given by:
438
0 1.143 1 0.571
F= G=
1 0 1.667 1
The controllability indices d1 and d2 of (F, G) are: d1=1, d2=1. Obviously, rank
[G]=2. The canonical form of matrix F is:
1.268 0.418
F̂=
-1.114 -1.268
The kernel U(s), of D(s), is:
-1 0.571
-s+1.268
+1.268 0.418
U(s)=
-1.667 -1
1.114 -s-1.268
At this point begins the search for stable polynomials, by applying simultaneously
the linear programming method, as described by the algorithm. We use the same
notation as in the text, and give the final results: βn2=-8.0, αn1=-28.22.
Polynomial ρ(s)=s2+3.25s+2.60 (roots of ρ(s):-1.84, -1.41). Polynomial
r(s)=s3+25.00s2+81.35s+ 65.00 (roots of r(s) : -21.33, -2.40, -1.27), and finally h(s)
= s4+29.22s3+122.44s2+1108.14s+1624.01. The roots of this polynomial are the
numbers {-26.01, -0.75 6.09j, -1.66}, which are the eigenvalues of the closed-loop
system, i.e., of system A Bd K d . The matrix of the feedback parameters is:
-25 -28.22
2 0 0
Kd =
0 0 -25 -8
It is remarked that the above values of Kd are in the transformed system of
coordinates (used to apply the method based on the equivalent system defined by a
PMD). For a given matrix
10 30
L=
20 50
the output matrix C is
12.5 14.11
1 --7.5 -2.4
C= .
-5 -5.644 2.5 0.8
If we suppose, as corollary 3.1 diagonal L
10 0
Ld =
0 50
the corresponding matrix Cd is block-diagonal, where the measurements are indeed
decentralized.
-2.5 -2.822 0 0
Cd = .
0 0 -0.5 -0.16
439
5 Conclusion
In this paper we considered the stabilization of a global system (A, Bd), resulting
from the interconnection of subsystems (Aii, bii) -i=1,2-, with decentralized state
and/or output feedback control. We studied initially the problem of the stabilization
of (A, Bd) with linear, static feedback of the local state-vectors, under the weak
conditions of the global and the local controllability. Although the problem was
defined in the state-space, it was transformed into the frequency domain and studied
therein. The existence of a local, feedback stabilizing matrix was formally proven
and it is completed by a numerical procedure -based on linear programming
methods- for the numerical computation of the feedback parameters (Kd). It is
supposed the output feedback matrix L is arbitrary, and one wishes to determine the
appropriate output matrix C which ‘realizes’ the decentralized feedback u=Kdx, by
the matrix C=L-1Kd . That corresponds to what (Zheng , 1989 ) refers to as ‘the
designer’s possibility to choose the output matrix C’.
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