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.
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
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,
B
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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),
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
We consider the interconnected system (A, Bd) defined by x
Ax
Bdu 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
A11 A21
A12 A22 n 1 and n2
Bd b11 0 0 b22 with n=n1+n2. System (A, Bd) consists of the interconnected ni-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
1 a11 A21 0 system (A, Bd ) : D(s) s
. (9) (10)
As the following lemma indicates, D(s) expresses the conditions for the controllability of (A, Bd):
Lemma 2.1:
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]
In (8) G is an (n-2)x2 (constant) matrix, consisting of columns n1and 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) (
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 U (s)
P
TS (s) ˆ Gm 1 (s) where P, T, S(s), G$ m , and δ(s) are as previously explained.
2.3
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)+Bdu(t) In the operator domain, this equation corresponds to the equation this, in its turn, reduces to the equations: and (sI - A) x(s) = Bd u(s)
D(s) x(s) = 0 (sE - Am ) x(s) = u(s) x(s) = U(s) ξ(s)
M(s) ξ(s) = u(s) M(s) = (sE - Am) U(s).
M (D) ξ (t) = u (t)
X (t) = U (D) ξ (t)
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: 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:
The matrix M(s) appearing in (14) is termed Characteristic Matrix of the
interconnected system (A, Bd)
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):
are equivalents
x(t) = U(D) ξ(t) (in the relations (16), (17), the symbol D denotes the differential operator d/dt). (11) (12a) (12b) (13) (14) (15) (16a) (16b) (17)
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. Then, there exists a static LLSVF of the form u=Kdx, so that the resulting closedloop system is stable.
Proof: For the proof we consider the equivalent system {M(s), I2} 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:
K d = . . . . 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 m 1 (s) = -(s-an1-a1,n1)[11]-a(s)+a1,n [21] -(s-αn1 -a1,n1)[12]-α1(s)+a1,n [22] = -(s-βn2 -a2,n1 )[21]-β1(s)+a2,n1[ ] -(s-βn2 -a2,n )[22]-β(s)+a 2,n1[12] = M11(s)
M 21(s)
M12 (s) M 22 (s) (18) (19) 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 matrix G$ m 1 (s) . Then the matrix in (19) is equivalent to the following matrix: Μd (s)=
M11 (s)+M 21(s)
M21 (s)
M 12(s)+M 22(s)
M 22 (s) (20) h(s)=detMd'(s)={M11(s)+M21(s)}M22(s)-{M12(s)+M22(s)}M21(s)= ={M11(s)+M21(s)}{-β(s)- (s-βn2 -a 2,n ) [22]+a2,n1[12]}-{Μ12(s)+M22(s)}M21(s)=
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-βn2 -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
(21)
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 αn1.
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. 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 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
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 blockdiagonal Cd=Ld-1Kd corresponding, thus, to the case where the measurements are also decentralized. 4
The controllable system (A, Bd) is: 1 4 3 5 2 0 0 2 1 1 2 3 0 2 4 1 1 1 0 0 0 0 1 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: T1 2
1 The intercontrollability matrix D(s), of system (A, Bd ) is:
D(s)
1 1.667 1
1 0.571 1.667 0.571 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:
Fˆ= 1.268 0.418
-1.114 -1.268
U(s)= -1 -s+1.268 -1.667 1.114 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.
ρ(s)=s2+3.25s+2.60 Polynomial (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
BdKd . The matrix of the feedback parameters is:
Kd = -25 0 -28.22 0 0 -25 0 -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
L= 10 20 30 50 C= the output matrix C is If we suppose, as corollary 3.1 diagonal L the corresponding matrix Cd is block-diagonal, where the measurements are indeed decentralized.
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