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Pole Assignment and Decoupling of Descriptor Systems



M. Chaabane
 
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ABSTRACT

In this study, we develop a numerical algorithm for decoupling a multi-input-multi-output singular system into non-interacting subsystems using PDF (Proportional and Derivate Feedback) control laws. In this study, the bilinear transformation from continuous systems is considered. Necessary and sufficient conditions for a solution of the decoupling problem are established. When the system satisfies these conditions, the class of controllers which decouple the system is synthesized. This study presents a method for simultaneous decoupling and pole assignment of singular systems is presented. Allowing, the method is used, when systems are not statically decoupled. Finally, we give numerical examples in order to show the advantages and the simplicity of the presented approach.

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  How to cite this article:

M. Chaabane , 2005. Pole Assignment and Decoupling of Descriptor Systems. Journal of Applied Sciences, 5: 1836-1842.

DOI: 10.3923/jas.2005.1836.1842

URL: https://scialert.net/abstract/?doi=jas.2005.1836.1842

INTRODUCTION

It is well-known that within a multivariable control system, every input affects several (if not all) outputs resulting in a complicated input-output relationship. Decoupling control strategies have been developed for the transformation of coupled input-output systems to equivalent decoupled systems. In case that each input effects only one output, the MIMO plant can be greatly simplified into a number of SISO plants. Propelled by this idea, the decoupling controllers of linear multivariable systems retain a great deal of attention since the early work see Falb and Wolovich[1], also Howze and Pearson[2]. Since then, there have been additional important contributions on this line[3-5]. Work on decoupling in the design and synthesis of descriptor systems has been developed first by Christodoulou et al.[6,7]. Decoupling control strategies of descriptor systems have been further developed[8-10]. The study by Dai[5] not only emphasizes the decoupling of the closed-loop dynamics or statics, but it also ensures. In particulars decoupling descriptor system by state feedback and regular input transformation on the Matrix Fraction Descriptions (MFDs) in frequency domain, associated with a poles assignment, developed by Vafiadis and Karcanias[11] and the references therein. Recently, Duan and Zhang[12] have proposed the dynamical order assignment approach for linear descriptor systems via state derivate feedback. In this work, PDF (Proportional and Derivate Feedback) controllers were employed for simultaneous decoupling and pole assignment of descriptor systems, necessary and sufficient conditions for a solution have been established. Yet, PDF control is also used in standard systems[3]. The theory of descriptor systems has a wide variety of applications in the domains of robotics, aerodynamics, electrical networks, perturbed systems, population models in biology[8,13,14].

FORMULATION OF THE PROBLEM

In this study, an approach for the input-output decoupling of singular, generalized or descriptor systems of the form Image for - Pole Assignment and Decoupling of Descriptor Systems is presented. Here, Proportional and Derivate Feedback (PDF) control laws are used. We consider the bilinear transformation[15] for the decoupling control problem of continuous descriptor systems.

Consider the linear time-invariant multivariable continuous descriptor systems described by a general state space model such that:

Image for - Pole Assignment and Decoupling of Descriptor Systems
(1)

Where, x is the n-dimensional state vector , u is the m-dimensional input vector and y is the l-dimensional output vector. E, A, B and C are matrices of appropriate dimensions and noting that the E is a singular matrix.

For the existence of a solution to system (1), we assume that det Image for - Pole Assignment and Decoupling of Descriptor Systems ≠ 0 where p is the complex variable associated with the Laplace transformation, i.e., Image for - Pole Assignment and Decoupling of Descriptor Systems is assumed to be a regular pencil matrix.

Let the transfer function matrix of system (1) be defined as follows:

Image for - Pole Assignment and Decoupling of Descriptor Systems
(2)

Here, we assume that m = l, i.e., the system has an equal number of inputs and outputs. Then, it is called single-input-single-output decoupled if and only if H(p) is diagonal and nonsingular.

Many feedback laws have been used in the regular system case in order to achieve decoupling systems. Most commonly used is the static state feedback law. Also, the case with dynamic state feedback and/or dynamic precompensator is used[6]. Here, we use a PDF control law, that is:

Image for - Pole Assignment and Decoupling of Descriptor Systems
(3)

The problem of decoupling the descriptor system (1) by PDF controller is to choose the matrices F1, F2 and G so that H(p) can be nonsingular and diagonal. In the next section, we first develop control laws for decoupled descriptor systems and use the basic necessary and sufficient conditions for decoupling system (1). Second, a compact procedure for computing the parameters of PDF control law is developed. Finally, we deduce a method permitting simultaneous decoupling and pole assignment for descriptor system.

DEVELOPMENT OF THE CONTROL STRATEGY

Consider the bilinear transformation defined by:

Image for - Pole Assignment and Decoupling of Descriptor Systems
(4)

for Image for - Pole Assignment and Decoupling of Descriptor Systems

with h assumed to be a strictly positive real parameter.

Applying this transformation to system (1), we obtain the discrete system defined by:

Image for - Pole Assignment and Decoupling of Descriptor Systems
(5)

where:

Image for - Pole Assignment and Decoupling of Descriptor Systems
(6)

and

Image for - Pole Assignment and Decoupling of Descriptor Systems
(7)

In the following, system (5) will be said to the Discrete Bilinear Transform (DBT) of system (1).

It is seen that the parameter h/2 must be selected such that it does not match any of the eigenvalues of the matrix pencil (E, A) i.e., det Image for - Pole Assignment and Decoupling of Descriptor Systems

In the rest of the study, the parameter h is selected to satisfy det Image for - Pole Assignment and Decoupling of Descriptor Systems

In order to make this study self-contained we recall some useful results from the literature.

Proposition[14]: Consider the generalized transfer function matrix H(p) of the system (E, A, B, C) and let Image for - Pole Assignment and Decoupling of Descriptor Systems be the transfer function of its DBT (5), then :

Image for - Pole Assignment and Decoupling of Descriptor Systems
(8)

Theorem[1]: The system (A, B, C) is decoupled if and only if the transfer function is diagonal and not singular.

Let the matrix D:

Image for - Pole Assignment and Decoupling of Descriptor Systems
(9)

where: Ci is the ith row of the matrix C

Image for - Pole Assignment and Decoupling of Descriptor Systems

there exist a pair (F, G) decouple the system (A, B, C) if only if D is not singular.

A particular solution for the pair (F,G) is given by:

Image for - Pole Assignment and Decoupling of Descriptor Systems

and D is defined in (9) and A* is given by:

Image for - Pole Assignment and Decoupling of Descriptor Systems

In order to obtain H(p), the transfer function Image for - Pole Assignment and Decoupling of Descriptor Systems is determined using any procedure for standard linear systems (5).

Relation (8) directly implies that H(p) is decoupled if and only if Image for - Pole Assignment and Decoupling of Descriptor Systems is decoupled. This shows decoupling control problem for descriptor systems can be solved using classical decoupling control engineers.

In such a case, the problem of determining the necessary and sufficient conditions for the descriptor system (1) to be decoupled is reduced to determining the corresponding conditions for the DBT system (5).

Falb and Wolovich[1] have proposed a necessary and sufficient condition for the existence of a control law which decouples the DBT system (4)[16,17].

Image for - Pole Assignment and Decoupling of Descriptor Systems
(10)

where v(k) represents the new m-vector control and the constants matrices F and G are appropriate dimensions. The closed loop transfer function becomes:

Image for - Pole Assignment and Decoupling of Descriptor Systems
(11)

Using results of Falb and Wolovich[1] in DBT system (5), we obtain the following proposition:

Proposition: Let Image for - Pole Assignment and Decoupling of Descriptor Systems be the m x m matrix given by:

Image for - Pole Assignment and Decoupling of Descriptor Systems
(12)

where Ci is the ith row of the matrix C. The superscripts di, i=1, 2,.., m are defined by:

Image for - Pole Assignment and Decoupling of Descriptor Systems

Then, there is a pair of matrices (F, G) which decouples the descriptor system (1) if and only if B* is nonsingular, i.e.,

Image for - Pole Assignment and Decoupling of Descriptor Systems
(13)

Proof of proposition: From propositions and theorem it comes that there exists a pair (F, G) such that Image for - Pole Assignment and Decoupling of Descriptor Systems is diagonal and nonsingular if and only if (13) holds. In this case H(p) is also diagonal and non singular.

It should be noted that this result provides a handy tool to decouple descriptor systems since the control law can be determined using classical methods of standard systems. The problem of determining the control law for decoupled descriptor systems (1), can be reduced to constructing the matrices F and G that decouple the DBT system (5). Then, the control law which decouples system (1), provided that condition (13) holds, reduces to the following form:

Image for - Pole Assignment and Decoupling of Descriptor Systems
(14)

where the controller matrices F1 and F2 are related by the following relationship:

Image for - Pole Assignment and Decoupling of Descriptor Systems
(15)

and h is a scalar which satisfies:

Image for - Pole Assignment and Decoupling of Descriptor Systems
(16)

In (14), x is the state vector,Image for - Pole Assignment and Decoupling of Descriptor Systems is the derivative of the state and v is the new input.

The following proposition suggests a set of pairs (F, G) capable of decoupling descriptor systems.

Proposition: If condition (13) holds, then the PDF con4rol law (14) is given by:

where, Image for - Pole Assignment and Decoupling of Descriptor Systems

Image for - Pole Assignment and Decoupling of Descriptor Systems
(17)

Image for - Pole Assignment and Decoupling of Descriptor Systems
(18)

Image for - Pole Assignment and Decoupling of Descriptor Systems
(19)

and B* is defined in (12) and K is given by:

Image for - Pole Assignment and Decoupling of Descriptor Systems
(20)

Proof of proposition: According to Eq 8, it turnout that if Image for - Pole Assignment and Decoupling of Descriptor Systems is diagonal then H(p) is also diagonal.

Let us note that, the problem of decoupled DBT system is equivalent to the problem of decoupled descriptor systems.

The pairs (F, G) to decouple DBT system is established by Descusse[16], in theorem. The feedback law which decouples descriptor systems is given under the following PDF structure:

from the bilinear transformation (4), we can write:

Image for - Pole Assignment and Decoupling of Descriptor Systems

when replacing x(k) in expression (10), we obtain:

Image for - Pole Assignment and Decoupling of Descriptor Systems

When we apply a PDF control to system (1), then the transfer function of closed loop descriptor system becomes:

Image for - Pole Assignment and Decoupling of Descriptor Systems
(21)

From the above considerations, it is clear that the crucial step in evaluating PDF controllers is the computation of matrices F1, F2 and G. The following algorithm can be used for this purpose.

Algorithm: Evaluation of PDF controllers for decoupled descriptor systems

Step 1: Compute Image for - Pole Assignment and Decoupling of Descriptor Systems given by (6)-(7)
Step 2: Using relation (12), determine B*
Step 3: if det(B*) ≠ 0 then go to step 4 Else PDF controllers for decoupling system (1) do not exist.
Step 4: Using relations (17-19), compute F and G.
Step 5: Deduce F1 = F and F2=h/2 F.
Step 6: Determine the closed-loop system(Ef,Af,B G, C) from:

Image for - Pole Assignment and Decoupling of Descriptor Systems

PDF POLE SHIFTING

The main problem here is the determination of the pole assignment of decoupled descriptor systems.

Let us first note that, the transfer function of closed loop DBT system is defined by Descusse[16].

Image for - Pole Assignment and Decoupling of Descriptor Systems
(22)

From above, we note that the closed loop poles are located at the origin.

The pole assignment of the decoupled descriptor system is specified by the next corollary.

Corollary: If system (1) satisfies the condition ² Image for - Pole Assignment and Decoupling of Descriptor Systemsregular² then the transfer function of the decoupled descriptor system is given by the following:

Image for - Pole Assignment and Decoupling of Descriptor Systems
(23)

where, the polynomials Ni(p) and Di(p) given by:

Image for - Pole Assignment and Decoupling of Descriptor Systems

and the parameter λi are those defined in theorem, with i ∈[1, m].

Proof of corollary: The transfer function of the decoupled descriptor system can be written as:

Image for - Pole Assignment and Decoupling of Descriptor Systems
(24)

and since:

Image for - Pole Assignment and Decoupling of Descriptor Systems

then:

Image for - Pole Assignment and Decoupling of Descriptor Systems
(25)

Let

Image for - Pole Assignment and Decoupling of Descriptor Systems
(26)

When replacing Image for - Pole Assignment and Decoupling of Descriptor Systems in expression (26), we obtain:

Image for - Pole Assignment and Decoupling of Descriptor Systems
(27)

Factorizing, (27) becomes:

Image for - Pole Assignment and Decoupling of Descriptor Systems
(28)

Using relations (6) and (7) for Image for - Pole Assignment and Decoupling of Descriptor Systems expression (28) becomes:

Image for - Pole Assignment and Decoupling of Descriptor Systems

and with (22) is mind we get:

Image for - Pole Assignment and Decoupling of Descriptor Systems

Going back to the p-domain we get:

Image for - Pole Assignment and Decoupling of Descriptor Systems

Taking for the expression above in H(p) one can deduce easily that H(p) is given by (23).

It should be noted that this result provides a method for simultaneous decoupling and pole assignment to "-2/h" of descriptor systems.

To illustrate these theoretical results, let us consider two examples.

ILLUSTRATIVE EXAMPLES

Example 1: DAI,[8]: Let a continuous system {E, A, B, C} be descriptor and given by:

Image for - Pole Assignment and Decoupling of Descriptor Systems

where, n=3, m=2 and rank Image for - Pole Assignment and Decoupling of Descriptor Systems

there are two design ways :

Thus, by Dai,[8], there exists a proportional feedback:

Image for - Pole Assignment and Decoupling of Descriptor Systems

such that the closed-loop is statically decoupled.

In the next Image for - Pole Assignment and Decoupling of Descriptor Systems is chosen as: Image for - Pole Assignment and Decoupling of Descriptor Systems λi=1, i∈[1, 2], where, I2: identity matrix

System (1) may be decoupled via PDF controllers, here, h is chosen to be 2. For the above system, we find

d1= d2=0 and from (7), we obtain :Image for - Pole Assignment and Decoupling of Descriptor Systems

which is nonsingular and, therefore, the descriptor system can be decoupled. The matrices F and G are then:

Image for - Pole Assignment and Decoupling of Descriptor Systems

The transfer function of the closed-loop system is:

Image for - Pole Assignment and Decoupling of Descriptor Systems

Note that the resulting closed-loop system is descriptor:

Image for - Pole Assignment and Decoupling of Descriptor Systems

Where, h is chosen to be 4, the matrices F and G which decoupled systems are:

Image for - Pole Assignment and Decoupling of Descriptor Systems

then the transfer function of the closed-loop system is:

Image for - Pole Assignment and Decoupling of Descriptor Systems

we can easy verify corollary, the decoupled transfer function in two case of h.

Example 2: Consider the following descriptor system, described by: DAI,[8]

Image for - Pole Assignment and Decoupling of Descriptor Systems

This system cannot be statically decoupled because, DAI,[8]:

Image for - Pole Assignment and Decoupling of Descriptor Systems

For all value of h, the matrix Image for - Pole Assignment and Decoupling of Descriptor Systems is chosen as: Image for - Pole Assignment and Decoupling of Descriptor Systems λi = 1, i ∈ [1, 2]

Here, we choose h=2, we find d1 = d2 = 0 and from steps 3 and 4, we obtain:

Image for - Pole Assignment and Decoupling of Descriptor Systems

The transfer function of the closed loop system is:

Image for - Pole Assignment and Decoupling of Descriptor Systems

If h is chosen to be 3, the matrices F and G which decoupled systems are:

Image for - Pole Assignment and Decoupling of Descriptor Systems

then the closed loop decoupled system has the following transfer function :

Image for - Pole Assignment and Decoupling of Descriptor Systems

We have verified for other values of h the transfer function of the closed loop decoupled system can be written as:

Image for - Pole Assignment and Decoupling of Descriptor Systems

The pole of the decoupled system is assigned to "-2/h".

CONCLUSIONS

A strategy for decoupling a linear descriptor system using PDF control laws was proposed. Necessary and sufficient conditions for decoupling were given on the basis that the matrix B*defined in (11) is non-singular. The class of PDF controllers which decouple a descriptor system were shown to be in the form of (h/2 F, F). Also the method is used, when systems are not statically decoupled. The closed loop decoupled system was realized with an assigned pole at "2/h" A method for simultaneous decoupling and pole assignment was presented. It was verified that the proposed decoupling strategy can be easily applied to discrete-time descriptor systems.

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