![]() ![]() We divide A 1 and A 2 by -19.3258 to form the matrix: To zero A 3,4, we apply (A 3 − (−1.017)) Įigenvectors associated with the eigenvalue (λ 3 = −0.6742) are We start by dividing 39.3258 to A 1 and obtain the matrix: Interchange A 2 and A 3(A 3↔A 2) to obtain the matrix:Įigenvectors associated with the eigenvalue (λ 2 = −20) are To zero the value at A 2,1, we apply (A 2 − 20A 1) → A 2 and form the matrix: Then, apply (A 1 − 0.5A 2) → A 1, and the equivalent homogeneous system of equations isĮigenvectors associated with the eigenvalue (λ1 = 0) are Next, we perform (A 1 − (−0.5)A 3) → A 1 and have the matrix: To zero the A 2,3, we apply (A 3 − (−0.3333)A 2) → A 2 and obtain To zero A 4, we subtracting A 3 to A 4 and get the matrix: A 1 → A 2, producing the following matrix:Īt A 2 divide each element by 30, and have the matrix:.To zero the A 2,1, we perform (A 2 − 20) We start with dividing the first row A1 by the A 1,1 and get We have four set of homogeneous system of equations for each eigenvalues and using Gaussian elimination, we solve for the eigenvector. The eigenvalues are defined asįor each of the eigenvalues, plug back in to the (λI − A) and solve the associated system of equation to determine the eigenvectors. We solve the eigenvalues using det (λI − A) = 0 as derived from Ax = λx. ![]() Initially, we calculate the eigenvalues and eigenvectors of the system matrix. In this section, we formed the transformation matrix, P, from the vector quantities expressed in the system matrix. ![]() We used partial fraction and signal flow graphs from the transfer function or by similarity transformation of the state space model. There are two ways to decoupled a system in state space. This allow easier analysis hence each differential equation is solved independently of the other equations. Decoupling a System in State Spaceĭiagonal system matrix define the state space equation as a function of one state variable. The paper implement some analytical solution in MATLAB for ease of computation and derivation of numerical values in the discussion. In the succeeding sections, the process of decoupling a system in state space is discussed in details. The state space model of the system in Figure 1 is given by The values of each elements are R 1, R2, R 3 = 10 Ω, L 1, L 2 = 0.5 H and C 1, C 2 = 50 mF. The output y(t) is equal to the voltage across the resistor, R2. The given system consist of a voltage input, u(t), resistors, R 1, R2 and R 3, inductors, L 1 and L 2, and capacitors, C 1 and C 2. Also, the signal flow graph for the decoupled system is presented. In this paper, we highlighted the derivation of the transformation matrix, P, and the decoupled system equations. ![]() Similarity transformation projects a new system, input, output, and feedforward matrix of the system for a new basis vector. In this paper, we explored and derived the equations for the decoupled representation of the system using similarity transformation. These systems are called similar system.įigure 1: Electrical circuit of the given system The various forms of the state space equations yield from the transfer function, drawing signal flow graphs, and deriving equations from signal flow graphs. The system can be represented with different state variables even though the transfer function relating the output to the input remains the same. ![]()
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