### State space, Transfer function, state response adjoint, inverse output, input.

Abstract

To
analyse the dynamics of a physical process, it is necessary to obtain its
mathematical description or representation. There are many ways by which this
mathematical representation or model can be obtained. For physical systems, we
usually apply the laws of physics to develop the required model. For instance,
we can use the Kirchhoff’s laws to derive the models for electrical systems and
Newtons laws to obtain the models for mechanical systems. However, this paper
presents a matrix method for converting the state space to transfer function of
a system using theoretical methods such as KVL, KCL to obtain the state
representation then apply the MATLAB to simulate.

*Keywords: State space, Transfer function, state response adjoint, inverse output, input.*

#
1.introduction

A control system
consists of subsystems and processes (or plants) assembled for obtaining a
desired output with desired performance, given a specified input. For example,
consider an elevator, when the 4

^{th}floor button is pressed on the 1^{st}floor, the elevator rises to fourth floor with a speed and floor levelling accuracy designed for passenger comfort. The push of the 4^{th}floor button is an input that represents our desired output. [2]
State-Space Models. In the time-domain, the
state-space model of an nth order SISO dynamical system with an input u (t),
output y (t), and internal states x1 (t); x2 (t); to
(t) consists of n first-order deferential
equations, called state equations, and an output equation which are usually
expressed in vector forms.[1].The state space representation provides the
potential for minimizing the data communication requirements for a given
algorithm without increasing computational complexity. Other advantages of the
state space implementation over direct implementation include decreased
sensitivity to parameter variations and improved performance when finite
arithmetic is used. [1,2]

In practice,
engineering problems are difficult to solve. Most often, numerical methods are
used as analytical solutions to such problems may be non-existent. Numerical
methods in themselves are usually iterative in nature requiring several
intermediate steps to arrive at a solution. An RLC circuit (also known as a
resonant circuit or a tuned circuit) is an electrical circuit consisting of a
resistor (R), an inductor (L), and a capacitor (C), connected in series or in
parallel. An RLC circuit is called a second-order circuit as any voltage or
current in the circuit can be described by a second-order differential equation
for circuit analysis. One very useful characterization of a linear RLC circuit
is given by its Transfer Function, which is (more or less) the frequency domain
equivalent of the time domain input-output relation. These methods do not use
any knowledge of the interior structure of the plant, and as we have seen
allows only limited control of the closed-loop behaviour when feedback control
is used. [4]

A system output is
defined to be any system variable of interest. A description of a physical
system in terms of a set of state variables does not necessarily include all
the variables of direct engineering interest. An important property of the
linear state equation description is that all system variables may be
represented by a linear combination of the state variables xi and the system
inputs. The state-space representation can be thought of as a partial reduction
of the equation list to a set of simultaneous differential equations rather
than to a single higher order differential equation. Although the state
variables of a system are not unique, and definition of many non-physical
variables is possible, we will work with physical variables, specifically the
energy storage variables of a system. There are two independent energy storages
in RLC circuit, the capacitor which stores energy in an electric field and the
inductor which stores energy in a magnetic field. The state variables are the
energy storage variables of these two elements, V and I. The energy storage
elements of a system are what make the system dynamic. The flow of energy into
or out of a storage element occurs at a finite rate and is described by a
differential equation relating the derivative of the energy storage variable (a
state variable) to the other power variable of the element [4,5]

.

#
2.Problem statement

In practice,
engineering problems are difficult to solve. Most often, numerical methods are
used as analytical solutions to such problems may be non-existent. Numerical
methods in themselves are usually iterative in nature requiring several
intermediate steps to arrive at a solution. This means much time will be needed
and more energy expended to solve a problem. In this project conversion of
state space derivatives to transfer function is required.

#
2.2 Possible solution

To lessen these
difficulties MATLAB needs to be used as this will also yield accurate results.
Theoretical calculations are still going to be employed as this will help
compare the results from the theoretical and MATLAB solutions.

**NOTE: CODES PRESENTED IN THIS PAPER ARE SCREEN SHOTS, I ATTACHED THE COMPLET CODES IN THE DISCS AS REQUESTED**

#
3. Theoretical calculation and
derivations

#
3.1 Question 1

Given the state space equations:

Take the Laplace transform of the given state space

Equation (7) shows the transfer function when the state space
equation.

Given the state space equation data:

Input
Matrix

Therefore, the
determinant of
is
calculated (note: the calculation is extremely long, so I omitted the step to
determine the det(
)

#
3.1.1 Question 1 MATLAB solution

# 3.1.1.1 Procedure

# Define the given state space equation coefficient equation.

In this case the coefficient of the state space is:

A, B, C and D

- Using the built-in commands of MATLAB, the following function is used

*[n, d] = ss2tf (A, B, C, D, in)*

The transfer function of a system is defined
as the ratio of output to input.

From the built-in
command:

- n = numerator value
- d = nominator value
- ss2tf = convert from state space to transfer function
- A, B, C, D state space matrix
- In =input (assumed to be 1)

#
6. Conclusion

To conclude using state
space method I can easily find the response and stability of an RCL circuit as
given in question 2,3 and this could also make it easier to determine the
transfer function from the state space equations using theoretical method as derived
in question 1. With the help of MATLAB, the analysis of an RLC circuit becomes
too simpler. However, this project was a success as I managed to use
theoretical calculations to obtain the transfer function, this transfer
function is comparable to the transfer function I obtained from MATLAB. For
future projects based on state space to transfer function always start a state
equation reduction with the elemental equation of an energy storage element.
From question 2,3 it is seen that I end up with many state equations as there
are independent energy storages in the electrical networks, having this
equation rearrange the energy storage elemental equation to place the
derivative of the state variable on the left side by itself then proceed to
eliminate all power variables except for input variable and the state
variables.

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