Nini ni Tahlilizi Nafasi ya Hali?

Ni nini State Space Analysis?

Maana ya State Space Analysis

Tathmini nyanja ya hali ya mikakati ya kudhibiti ni njia ya kutathmini mikakati madogo na magumu kutumia seti ya viwango kwa kuwasifu utendaji wao kwa muda.

Maelezo ya Nyanja ya Hali

Hivi sasa tujenge maelezo ya nyanja ya hali ya mfumo ambayo ni wa kihesabu na haiingie muda.

Tuangalie mfumo wa vingineko vingi na matumizi mengi ambayo ana vingineko r na matumizi m.

Kwenye, r = u1, u2, u3 ……….. ur.

Na m = y1, y2 ……….. ym.

Sasa tunapata n viwango vya hali kusaidia kusaidia kuelezea mfumo uliyotolewa kwa hiyo n = x1, x2, ……….. xn.

Pia tunaelezea vekta za vingineko na matumizi kama,

Vekta ya vingineko transposha,

Kwenye, T ni transposha ya matrix.

Vekta ya matumizi transposha,

Kwenye, T ni transposha ya matrix.

Vekta ya hali transposha,

Kwenye, T ni transposha ya matrix.

Viwango hivi vinajungwa na seti ya maelezo ambayo yameandikwa chini na yanayojulikana kama maelezo ya nyanja ya hali

Uelezaji wa Modeli ya Hali kwa Kutumia Function ya Kupitisha

Gawanya : Ina maana ya mchakato wa kupata modeli ya hali kutoka kwa function ya kupitisha iliyotolewa. Sasa tunaweza kugawanya function ya kupitisha kwa mitaala gawa mbalimbali:

• Gawanya moja kwa moja,

• Gawanya ya kipaza au series,

• Gawanya parallel.

Katika zote mizizi ya gawanya tulizopewa hapo juu tuwanapo badilisha function ya kupitisha kwa maelezo ya differential ambayo inatafsiriwa kama maelezo ya dynamic. Baada ya kubadilisha kwa maelezo ya differential tutapata inverse Laplace transform ya maelezo hili halisi kisha kulingana na aina ya gawanya tunaweza kurudia modeli. Tunaweza kurudia yoyote aina ya function ya kupitisha kwa modeli ya hali. Tunapewa aina mbalimbali za modeli kama vile modeli ya umeme, modeli ya mifano ya mekani ya mifano.

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).