Partial Eigenvalue Assignment for High-Order Linear Systems in a Time Delayed System

Corresponding Author: Ehab A. El-Sayed Department of Mathematics, College of Science and Humanitarian Studies, Prince Sattam Bin Abdulaziz University, Al-Kharj, Saudi Arabia Email: janine.tiu@gmail.com Abstract: This paper introduces a solution to the partial eigenvalues assignment problem of high order linear systems with time delayed system in both single-input and multi-input cases using orthogonality relations between eigenvectors of the matrix polynomial. The solution requires the knowledge of only a few eigenvalues and with corresponding left eigenvectors of matrix polynomial. The numerical examples are done to illustrate the proposed method.

The system modeled by (1.1) can be controlled with the application of a forcing function Bu(t-τ), B∈R n×m , (m≤n) (B the control matrix) and u(t-τ) is a timedependent m×1 real vector and τ is a constant time delay. The matrix B has full column rank, that is, rank(B) = m in which case (1.1) is replaced by: With the choice of the controlling force: Such that, F 1 , F 2 ,…,F k ∈R n×m constant matrices, leads to the closed loop system: Differential Equation (1.6) leads, with the separation of variables v(t) = xe λt , to the problem of finding the eigenvalues of the modified matrix polynomial: The partial eigenvalue assignment problem for the time delayed system (1.6) is to find the control gain matrices F 1 , F 2 , …,F k such that: has the desired eigenvalues {µ 1 , µ 2 ,…,µ m , λ m+1 ,…,λ kn }. That it contains m numbers of new assigned desired eigenvalues {µ 1 , µ 2 ,…,µ m } without affecting the remaining eigenvalues {λ m+1 ,…,λ kn } of the Equation (1.1). Throughout this paper, notion (PEAP) is used instead of partial eigenvalue assignment problem.
PEAP arises from many practical situations such as electrical circuit simulation and acoustic system, vibration analysis of structural mechanical, fluid mechanics, finite element model updating in automobile industries and aerospace (Arévalo and Lötstedt, 1995;De Boor and Kreiss, 1986;Sand, 2002;Mackey et al., 2006). PEAP for high-order linear systems without time delay introduced in many paper (Wang and Zhang, 2014;Ramadan and El-Sayed, 2010). Prattand and Singh (2009) were solution PEAP for second order linear system with known time delay in single-input case. Also Singh et al. (2014) were solution the same problem but in Multi-input case and study its stability with known time delay. Wang and Zhang (2013) were introduced the direct method of PEAP for high order systems in single input case using receptance method with known time delay. In this paper, we proposed a method to solve PEAP for high order control systems in single input case and multi-input case with known time delay, using orthogonality relations between the eigenvectors of matrix polynomial. This solution requires only a partial knowledge of the eigenvalue and the corresponding eigenvector of the matrix polynomial which allows the partial assignment of desired eigenvalues with no spillover. The numerical examples illustrate that the proposed method.

Orthogonality Relations between the Eigenvectors of Matrix Polynomial
In this section, we introduce some Orthogonality relations which are play role important for solving our problem. If, in addition, A is real symmetric, then: The following theorems establish the orthogonality relations between the eigenvectors for the matrix polynomial using its connection with the standard eigenvalues problem.
Theorem 2 (Ramadan and El-Sayed, 2010) A scalar λ∈C is an eigenvalue of the matrix polynomial with the corresponding right eigenvector x and the left eigenvector y if and only if λ is an eigenvalue of the kn×kn matrix: With the corresponding right eigenvector x and left eigenvector ŷ such that:

Theorem 3 (Ramadan and El-Sayed, 2010) (Orthogonality of the Eigenvectors of the Matrix Polynomial)
Let λ 1 , λ 2 ,…,λ kn be the eigenvalues of the kn×kn matrix polynomial P(λ) = λ k M k +λ k-1 M k-1 +⋯+λM 1 +M 0 and let X and Y be respectively the right and left eigenvector matrices. Assume Then: And:

PEAP for High Order Control Systems in a Time Delayed System in Multi-Input Case
Write Equation (1.2) in the following form: Or in the matrix form: where, X = (x 1 , x 2 …,x kn )∈C n×kn and Λ = diag(λ 1 , λ 2 …,λ kn )∈C kn×kn where λ i are all distinct. Where λ 1 , …,λ kn are the eigenvalues of matrix polynomial: Let us partition the n×kn right eigenvector matrix X, the kn×n left eigenvector matrix Y H and kn×kn eigenvalues matrix Λ as follows: Given m complex numbers µ 1 , µ 2 ,…,µ m closed under complex conjugation, m<n and a matrix B∈R n×m with known time delay τ, we are required to find F 1 , F 2 ,…,F k ∈R n×m such that the modified matrix polynomial: and F k be defined by: Then for any choice of ϕ we have: In words, this theorem assures us that any choice of ϕ with the feedback matrices { } 1 k i i F = as in (3.6) guarantees that the last kn-m eigenpairs (Λ 2 , X 2 ) of the matrix polynomial P(λ) are also eigenpairs of the modified matrix polynomial P c (λ, τ).

Proof
Our goal is to prove that: Expanding the left hand side of (3.8) by substituting (3.6) in (3.8), we obtain: Since (X 2 , Λ 2 ) is an eigenpair of the open loop pencil (3.2) we have: From theorem 3. Thus: The theorem is then proved. In order to use theorem 4 to solve PEAP, we need to

∑∑
(3.14) Then: Wφ H = Γ H is matrix that will depend on the scaling chosen for the eigenvectors in Z. To obtain Z, we choosing the matrix Γ in the following form Γ = (γ 1 , γ 2 ,…,γ m ). Then the equation ( So, we computed the eigenvectors Z and hence we computed the matrix W from: We solve the m×m square linear system for ϕ: Finally, determine the control gain matrices F i , i = 1,2,…,k are obtained from (3.6).

PEAP for High Order Control Systems in a Time Delayed System in Single-Input Case
For single-input control application, the system (1.1) is modified by applying a controlling force bu(t-τ) as: where, b∈R n×1 , (b is the control vector) and control force u(t-τ), is defined as: Such that, f 1 , f 2 ,…,f k vectors, leads to the closed loop system: where, τ is the known time delay with the separation of variables v(t) = xe λt , to the problem of finding the eigenvalues of the modified matrix polynomial: PEAP with the time delayed system (4.4) is to find the control vectors f 1 , f 2 ,…,f k such that det(P c (λ i,τ )) = 0, for i = 1,2,…,kn, has the desired eigenvalues {µ 1 , µ 2 ,…,µ m , λ m+1 ,…,λ kn }. That it contains m numbers of newly assign need desired e igenvalues µ 1 , µ 2 ,…,µ m } without affecting the remaining eigenvalues {λ m+1 ,…,λ kn } of the open loop system (1.1).

Theorem 5
Let the feedback vectors { } 1 1 k i i f − = and f k be defined by: Then for any choice of ϕ the following relation holds:

Conclusion
We derived a solution to PEAP for high-order linear systems (1.4) in both multi-input and single-input case by using orthogonality relations between eigenvectors for matrix polynomial P(λ). The approach needs the knowledge of only a few eigenvalues and with corresponding left eigenvectors of matrix polynomial P(λ) without turning high order systems into first order form. When τ = 0 is a special case of our results see the reference (Ramadan and El-Sayed, 2010).