Tahalilin Kukaci na Nizalin Tattalin Arzikin IEE-Business

Idan na bani kana game da fahimta kan state space analysis of control system, yana da kyau a tattauna wannan yanayin ya dace bayan iyakoki da cewa akwai farkon bayanin control system da teori mai tsawo ta control system.

1. Teori mai tsawo ta control system yana da shirya ne a hankali domin wata domain, amma teori mai tsawo ta control system yana da shirya ne a hankali domin lokaci domain.

2. A cikin teori mai tsawo ta control system muna da systems linear da time invariant single input single output (SISO) kawai, amma idan an yi amfani da teori mai tsawo ta control system za a iya yi amfani da systems non linear da time variant multiple inputs multiple outputs (MIMO) kuma.

3. A cikin teori mai tsawo ta control system, zai iya yi analysis stability da lokacin response analysis da wasu hanyoyi grafik da kuma analytical method da gaskiya.

Yanzu state space analysis of control system yana da shirya ne a hankali domin teori mai tsawo wanda yake da shirya zuwa dukkan irin systems kamar systems single input single output, multiple inputs and multiple outputs systems, linear and non linear systems, time varying and time invariant systems. Za mu duba wasu abubuwan da suka fi sani don state space analysis of modern theory of control systems.

1. State a State Space Analysis : Wannan yana nufin set masu yawan variables wadanda ba a tabbas ba, inda ake magance a lokacin t = t0 da kuma magance input a lokacin t ≥ t0 za a bayar da cikakken bayanin rayuwarsa a lokacin t ≥ t0.

2. State Variables a State Space analysis : Wannan yana nufin set masu yawan variables wadanda ba a tabbas ba wanda ke taimaka muna da cikakken bayanin state dynamic system. State variables suna cika da x1(t), x2(t)……..Xn(t).

3. State Vector : Idan an bukatar n state variables don bayyana cikakken bayanin system, sa, za a iya cewa waɗannan n state variables suna da n components of a vector x(t). Wannan vector yana nufin state vector.

4. State Space : Wannan yana nufin n dimensional space wanda yake da x1 axis, x2 axis ………xn axis.

State Space Equations

Za a rubuta state space equations don system wanda yake linear da time invariant.
Za a duba multiple inputs and multiple outputs system wanda yake da r inputs da m outputs.
Inda, r = u1, u2, u3 ……….. ur.
Kuma m = y1, y2 ……….. ym.
Yanzu muna da n state variables don bayyana system, saboda haka n = x1, x2, ……….. xn.
Amsa muna define input and output vectors as,
Transpose of input vectors,

Inda, T ita ce transpose of the matrix.

Transpose of output vectors,

Inda, T ita ce transpose of the matrix.
Transpose of state vectors,

Inda, T ita ce transpose of the matrix.
Waɗannan variables suna da shiga cewa set of equations wanda ake rubuta a nan kuma suna nufin state space equations

Representation of State Model using Transfer Function

Decomposition : Wannan yana nufin process of obtaining the state model from the given transfer function. Yanzu muna iya decompose the transfer function using three different ways:

1. Direct decomposition,

2. Cascade or series decomposition,

3. Parallel decomposition.

A cikin waɗannan hanyoyi na decomposition, muna iya convert the given transfer function into the differential equations wanda ake cewa dynamic equations. Bayan ake convert into differential equations muna iya take inverse Laplace transform of the above equation then corresponding to the type of decomposition we can create model. Muna iya represent any type of transfer function in state model. Muna da various types of model like electrical model, mechanical model etc.

Expression of Transfer Matrix in terms of A, B, C and D. Muna 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

Muna iya rubuta the equation as

Inda, I ita ce 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| ita ce 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-

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

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

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

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

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

6. 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