Disturbance Rejection Problem with Stability by Static Output Feedback of Linear Continuous TIME System

Disturbance rejection problem with stability by static output feedback of linear-time invariant continuous time system is solvable if there is found a static output feedback control law, u(t) = Ky(t) (if possible), such that disturbance q(t) has no in°uence in controlled output z(t). So, it is needed the necessary and su±cient condition disturbance rejection problem is solvable. By using the de¯nition and characteristics of (A, B)-invariant subspace, and (C, A)-invariant subspace, then it will be ¯nd the necessary and su±cient condition disturbance rejection problem of that system will be solved if and only if maximal element of a set of (C, A)-invariant is an (A, B)-invariant subspace that internally stabilizable and externally stabilizable.


Introduction
In system theory, there are many system models. Once is a linear, continuous-time system described by:ẋ (t) = Ax(t) + Bu(t) where x ∈ R n is the state, u ∈ R m is the control input, y ∈ R h is the output. A ∈ R n is a matrix, B ∈ R n×m is a matrix, and C ∈ R h×n is a matrix. If that system has a disturbance, we can make the model of the system likes: where q ∈ R q is the disturbance input, and E ∈ R n×q is a matrix. When there is a disturbance in a system, it may cause deviation between the expecting result and the obtained output. Therefore, it needs a feedback control which becomes control rule which gives information from output deviation and entry that information as a new input, then all the deviations from the expecting result can be corrected. A solution of Disturbance Rejection Problem with stability by Static Input Feedback, u(t) = F x(t) has been recently proposed in [9]. When the state vector is not available for measurement, [3] are led to consider the Disturbance Rejection Problem by Static Output Feedback, u(t) = Ky(t) with the system is: where y ∈ R h is measured output z ∈ R z is the controlled output, and D ∈ R z×n is a matrix . For next, the system in 2 will be called (A, B, C, D, E) system. In this paper will be studied the solution of Disturbance Rejection with Stability by Output Static Feedback related to the system with the system class is: , and S * = minimal (C, A) invariant subspace containing Image (E).

Preliminaries
Consider (A, B, C, D, E) system, if we give Static Output Feedback, u(t) = Ky(t), where K ∈ R m×h is a matrix, then system in (1) becomes: The solution of the equation above by using linear differential equation if x(0) = x 0 is: The goal of the Disturbance Rejection Problem by Static Output Feedback is to determine, if possible, a static output feedback control law, u(t) = Ky(t), such that the transfer matrix from disturbance q to controlled output z is null, or:   The set of all (C, A)-invariant subspaces containing Im (E)which is defined by ξ(C, A; Im(E) has a minimal element: From the above definition we can prove this theorem: Gathering definition and characteristic of (A, B) invariant and (C, A)invariant, we can make a subspace V which is (A, B) invariant and (C, A)-invariant become a new form of invariant subspace that will be explained in this following theorem:

Internally Stabilizable and Externally Stabilizable
An invariant subspace of a matrix A ∈ R n always has linear transformation. From that, we can classification an invariant subspace becoming internally stabilizable or externally stabilizable, which will be explained in these following theorem and definition: is a matrix, and subspace J ∈ R n is an A-invariant, then there exist linear transformation T such that with A ′ 11 ∈ R h×h is a matrix and dimension of J = h.
We can use this following theorem to look for determinant of A' in (3). .

Consider now the following class of subspaces
where (A + BKC) n−1 is composition function as n-1 times.
Definition 2.8 [9] Disturbance rejection problem at (A, B, C, D, E) system with stability by static output feedback can be solved if there exist a matrix K ∈ R m×h such that A + BKC|ImE ⊂ Ker(D) and σ(A+BKC) ⊂ C g where σ(A+BKC) represents the set of (A + BKC)'s eigen value, and C g = {s; s ∈ C, Re(λ s ) < 0} Class of System: Self Hidden In this paper, we want to establish the geometric condition which defined the class of system. The class is satisfying condition that is explained in this following definition: Definition 2.9 [3] S ∈ ξ(C, A; Im(E) is said to be self hidden with respect to Im(E) if S ⊂ S * + Ker(C)

Discussion
The preceding Theorem establishes necessary and sufficient condition for existence of static output feedback which solves Disturbance Rejection with Stability by Output Static Feedback with the system class is γ * ⊂ Ker(C) + S * .
The first theorem that we have to prove is showing that there is a subspace which become (A, B)-invariant and also (C, A) invariant likes in this following theorem

From Definition 2.1 and Theorem 2.4, A + BKC|ImE is (A, B) and (C, A)-invariant.
Before we prove the last theorem, we need to find necessary and sufficient condition of disturbance rejection problem which not contain a system class which is included in this following theorem: 1. By using Definition 2.8 there exists a matrix K ∈ R mxh such that A + BKC|ImE ⊂ Ker(D) and σ(A + BKC) ⊂ C g .
We take V = A+BKC|ImE . For arbitrary p ∈ Im(E), there exists z ∈ R q such that p = E(z).
Next, let v ∈ V . Because V is a subspace and (A + BKC) is linear transformation, there exists 0 ∈ V , so v can be wrote as: From (4), (5), and (6) we get then A + BKC|Im(E) ⊂ Ker (D), or disturbance rejection problem at(A, B, C, D, E) system with stability by static output feedback can be solved.
Suppose Ψ (C, A; Im (E)) represent the set of (C, A)-invariant subspace containing Im (E) which are self hidden or we can wrote as: Ψ(C, A; Im (E)) = {S; (A+GC)S⊂ S, S ⊃ Im (E) and S ⊂ Ker (C) + S * } Based on Theorem 2.3 we find that (C, A)-invariant is closed under subspace addition. We also can prove that (Ψ(C, A; Im (E)) is closed under subspace addition Suppose Γ(C, A; Im(E)) represent the set of (C, A)-invariant subspace containing Im (E) and contained in γ * or it is can be wrote as: Γ(C, A; Im (E))= {S: S ∈ ξ(C, A; Im(E) and S ⊂ γ * } Based on above sentences we find that (C, A) invariant is closed under subspace addition.
Based on Definition 2.2, we find that set of all (C, A)-invariant subspaces containing Im (E) has a minimal element. In this section, we also can prove that Γ(C, A; Im (E)) also has a maximal element if γ * ⊂ Ker(C) + S * which is: The last theorem we need to prove is necessary and sufficient condition of disturbance rejection problem at (A, B, C, D, E) system with the system class is γ * ⊂ Ker(C) + S * with stability by static output feedback can be solved which is mentioned in following theorem: Theorem 3.3 Disturbance rejection problem at (A, B, C, D, E) system with the system class is γ * ⊂ Ker(C) + S * with stability by static output feedback can be solved if and only if S * is an (A, B)-invariant subspace that internally stabilizable and externally stabilizable.
Because S * is an (A, B)-invariant and (C, A)-invariant then based on Theorem 2.4 S * is an (A+BKC)-invariant, which is implies based on Definition 2.1 (A+BKC)S * ⊂ S * .

Conclusion
Disturbance rejection problem at (A, B, C, D, E) system with the system class is γ * ⊂ Ker(D) + S * with stability by static output feedback can be solved if and only if S * is an (A, B)-invariant subspace that internally stabilizable and externally stabilizable.