Reconfigurable Controller Design in Descriptor Systems Obtained from Second Order Dynamic Systems via State Feedback Eigenstructure Assignment

this paper proposes a new method for designing a reconfigurable controller for descriptor systems which is obtained from the second order dynamics of a robot control situation. Using mathematical tools the equations of robot control situation have been translated to descriptor systems, thus, after this change the controller has been designed for control performances. The proposed reconfigurable controller can recover the nominal closed-loop performances after fault occurrence in the system. The dynamics of descriptor systems contain infinite and finite elements, so a complete response of descriptor systems can be represented by an eigenstructure which involves finite and infinite elements. In this paper eigenstructure assignment is used to design a reconfigurable controller in general such that the reconfigured system can recover the complete response of a nominal system as much as possible. Finally, an example represents the effectiveness of the new method.


Introduction
Reconfigurable control system is a system that is capable of dealing with large variations in the system being controlled by means of adjusting or modifying the nominal control law.The stability and performance of the original closed-loop system is maintained as much as possible using the reconfigurable control systems.The reconfigurable control law must provide stability and excellent performance under conditions of failure and damage, as well as during normal operation [1].In recent years, reconfigurable control has attracted much research attention and many new approaches have been proposed.In [2], the linear quadratic regulator method has been considered.In addition [2], pseudo inverse method [3], feedback linearization [4] , the Lyapunov method [5], sliding mode control [6] are all considered, in [10] a reconfigurable controller design via output feedback in the case of post order fault is represented.[11] considers a modified approach that guarantees the stability of the closed-loop system by using an appropriate Lyapunov equation -all examples of approaches for designing reconfigurable control.
The eigenstructure assignment is one of the most powerful tools for control systems.According to the fact that the response of the system can be stated based on eigenvalues and corresponding eigenvectors, in this paper the eigenstructure assignment method is considered.
We know that the second order dynamic system is one of the most important systems in the dynamics of robot manipulators [12].It is well known that these systems can be translated as descriptor systems by changes in variables.Designing a reconfigurable controller in these systems is important because fault occurrence in these systems may disturb the appropriate performances.Designing a reconfigurable controller in a descriptor system and high order dynamics system has been considered in some research.In [7], designing a reconfigurable controller in a second order dynamic system via p-d feedback eigenstructure assignment is considered.By changing variables in the second order system, this system is translated to a descriptor system.Based on finite eigenvalues and corresponding eigenvectors, the reconfigurable system has been designed.However, we know that the complete response of the descriptor system [8] consists of finite and infinite eigenstructures.So this method does not give any information about infinite eigenstructures.
The major contribution of this paper is to develop using eigenstructure assignment in second order dynamic systems in general case.In the proposed method there isn't any constraint on eigenvalues and eigenvectors of system, while in the previous work [7] eigenvalues are distinct.The proposed method can be used for descriptor systems which are important in network and other systems.According to the fact that the complete response of descriptor systems involves finite eigenstructure and infinite eigenstructure, the proposed method considers both finite eigenstructure and infinite eigenstructure.In this paper, a new method is suggested for the design of a reconfigurable controller in descriptor systems using finite and infinite eigenstructure assignment.Based on parametric eigenstructure assignment by state feedback in the descriptor system [9], the reconfigurable controller is designed.This paper is organized as follows: Section 2 describes the parametric approach.In section3 the main problem is represented.Section 4 demonstrates the effectiveness of the proposed method.

Parametric Eigenstructure Assignment
Consider the second order dynamic system Where  are state and input vectors respectively.Suppose that Consider the following transformation By applying the above transformation the following descriptor system is obtained Where Suppose that due to the fault occurrence in the system, the dynamics of the nominal system (3) is changed.The new model of the faulty system is described as follows A E B are real matrices with appropriate dimension and , ,  and the faulty system is controllable.Our objective is to design a new state feedback ( ) such that the performances of the nominal system are recovered.The closed-loop system is given by ( ) In order to solve this problem, first we describe the parametric eigenstructure assignment by state feedback which is presented in [9].
, ,..., , where is the set of finite eigenvalues of the closed-loop system ( , ) . The eigenvalue f i  has the algebraic multiplicity i m and a geometric multiplicity i q .
The Jordan canonical form f J of ( , ) . According to the above definition the following equation is satisfied [9] 1 2 ...
... m m m m m    according to definition of eigenvalues and eigenvector in descriptor system, the following equations are satisfied  are eigenvalue and corresponding eigenvector of faulty system.Based on [9], (9) can be written as [ , ,..., ] By applying singular value decomposition to the matrix Where * ( ) ( ) .The parametric form of eigenstructure assignment is  are parameter vectors and ( ) is the infinite eigenvalue of ( , ) According to definition of eigenstructure assignment, we have Note that the algebraic and geometric multiplicities are Based on the results of [9], the parametric infinite eigenvector is 1, 2,..., are obtained from the following equations 0 ( ) Similar to (10) , we have By combining (10) and (20) the following equation is obtained In order to calculate the matrix f K , the following constraints must be satisfied Where f T  is the left eigenvector matrix of the closed- loop system associated with the finite closed-loop eigenvalue s  .

Main problem
Based on [8], the closed-loop response of descriptor systems (1) using state feedback and in terms of eigenstructure assignment can be stated as follows Where f T and f T  are left eigenvector matrices of the closed-loop system associated with the finite closed-loop eigenvalues f i  and left eigenvector matrices of the closed-loop system associated with the finite closed-loop eigenvalue f s  respectively, such that The complete response of closed-loop system can be rewritten as follows inf ( ) Clearly, this response consists of two parts, the finite part f fin x that related to finite eigenstructure and the infinite part inf f x related to infinite eigenstructures.
According to the above discussions, similarly the closedloop response of the nominal system contains two parts inf , fin x x .Based on (24) and (26), it is clear that the response of the system is related to its eigenstructure, so in order to recover nominal performance, the state feedback controller f K must be designed such that the behaviour of the faulty system with this controller is closed to nominal system much as possible.The reconfiguration objective can be translated to the following equations  h h  and then set to zero we have In order to describe the method, consider 2 k  so the fin J is stated as follows According to (30) Taking the above gradient and then by simplifying the results, we have , ij ij h h can be calculated from the above equations.In the special case of distinct finite eigenvalues ( 1, 1, 2,..., ) The parametric eigenstructure of distinct eigenvalues is computed as By substituting (35) in (28) and then take gradient the solution is obtained as Where v is eigenvector of faulty system in distinct case.
One of the most important specifications that should be recovered by the reconfigured system is steady state response of the nominal systems.The following theorem states how a feedforward matrix should be designed to minimize the difference between the after fault and prefault steady state response Theorem1: For the faulty system (5), consider the control law ( ) ( ) Where is feedforward matrix.The steady state response of the nominal system to step input can be recovered if R is selected as ) Proof: The closed-loop systems of nominal and faulty systems are given by (

Ex t A BK x t Br t u t Kx t r t y Cx t
By applying Laplace transformation to (40) and (41), we have In the above equation suppose that ( , ) E A BK  and ( , ) The steady state output of the nominal closed-loop system to a unit step input is given by In addition, for the faulty system So for recovering of the steady state response of the nominal system we have to design L such that min min ( ) ( ) The solution is described in (38).

Conclusion
An eigenstructure assignment-based method is suggested to design a reconfigurable controller for descriptor systems via state feedback.The controller can be reconfigured to compensate for the effect of change in the system dynamics.This work deals with the control reconfiguration in the general case where the complete response of systems includes the infinite and finite eigenstructure.Finally, an example demonstrates the effectiveness of the proposed method via simulation.