Mai Da Dacewa na Iya Kafa?

Me kadan Wani Abin da Yawanci?

Takardun Wani Abin da Yawanci

Wani abin da yawanci na dandamai ita ce hanyar bayyana masu yawan da keke da kuma mafi karfi game da wata matsayin cikakken bayanin tarihin su.

Muhimman Tushen Wani Abin da Yawanci

A zama tushen wani abin da yawanci na dandamai na tushen da suka gudana da tarihi da ya fi inganta.

Za a iya duba masu shirya da kuma masu fadada rike da r inputs da m outputs.

Idan, r = u1, u2, u3 ……….. ur.

Da m = y1, y2 ……….. ym.

Daga baya za mu iya samun n state variables don in bayyana dandamai na tarihi, saboda haka n = x1, x2, ……….. xn.

Kuma za mu iya takarda input da output vectors kamar yadda ake bayyana a nan,

Transpose of input vectors,

Idan, T ita ce transpose of the matrix.

Transpose of output vectors,

Idan, T ita ce transpose of the matrix.

Transpose of state vectors,

Idan, T ita ce transpose of the matrix.

Wasu waɗannan variables suna da take da wasu tushen da ake rubuta a nan kuma suna canza a kan wani abin da yawanci.

Bayanin Model na State Ta Hanyar Transfer Function

Decomposition : Ana ƙirƙira wannan a matsayin yanayin samun model na state daga transfer function. Idan kuna iya ƙirƙira transfer function ta hanyar uku hanyoyi:

• Decomposition na ɗaya,

• Decomposition na series ko cascade,

• Decomposition na parallel.

A cikin duk waɗannan hanyoyi na ƙirƙiri, a gaba muna ƙirƙira transfer function ta zuwa tushen differential equations, wanda ake kira tushen dynamic equations. Ba da ƙirƙirar zuwa tushen differential equations, muna ƙirƙira inverse Laplace transform ta, sannan ba da ƙirƙirar zuwa tushen decomposition, muna iya ƙirƙira model. Muna iya ƙirƙira wani abu a model na state. Akwai wasu hanyoyi na model kamar electrical model, mechanical model, da sauransu.

Expression of Transfer Matrix in terms of A, B, C and D. We define transfer matrix as the Laplace transform of output to the Laplace transform of input.On writing the state equations again and taking the Laplace transform of both the state equation (assuming initial conditions equal to zero) we have

We can write the equation as

Where, I is an identity matrix

Now substituting the value of X(s) in the equation Y(s) and putting D = 0 (means is a null matrix) we have

Inverse of matrix can substitute by adj of matrix divided by the determinant of the matrix, now on rewriting the expression we have of

|sI-A| is also known as characteristic equation when equated to zero.

Concept of Eigen Values and Eigen Vectors

The roots of characteristic equation that we have described above are known as eigen values or eigen values of matrix A.Now there are some properties related to eigen values and these properties are written below-

• Any square matrix A and its transpose At have the same eigen values.

• Sum of eigen values of any matrix A is equal to the trace of the matrix A.

• Product of the eigen values of any matrix A is equal to the determinant of the matrix A.

• If we multiply a scalar quantity to matrix A then the eigen values are also get multiplied by the same value of scalar.

• If we inverse the given matrix A then its eigen values are also get inverses.

• If all the elements of the matrix are real then the eigen values corresponding to that matrix are either real or exists in complex conjugate pair.

Now there exists one eigen vector corresponding to one Eigen value, if it satisfy the following condition (ek × I – A)Pk = 0. Where, k = 1, 2, 3, ……..n.

State Transition Matrix and Zero State Response

We are here interested in deriving the expressions for the state transition matrix and zero state response. Again taking the state equations that we have derived above and taking their Laplace transformation we have,

Now on rewriting the above equation we have

Let [sI-A] -1 = θ(s) and taking the inverse Laplace of the above equation we have

The expression θ(t) is known as state transition matrix.

L-1.θ(t)BU(s) = zero state response.

Now let us discuss some of the properties of the state transition matrix.

• If we substitute t = 0 in the above equation then we will get 1. Mathematically we can write θ(0) =1.

• If we substitute t = -t in the θ(t) then we will get inverse of θ(t). Mathematically we can write θ(-t) = [θ(t)]-1.

• We also another important property [θ(t)]n = θ(nt).