System Analysis

Objective

Systems analysis is a problem solving technique that decomposes a system into its component pieces for the purpose of the studying how well those component parts work and interact to accomplish their purpose.
So in order to do so we need to first represent the system is a way that makes this analysis easier. Here I will try to do the following,

• Understand the relationship between the two representations - Transfer function and the State Space.
• Discuss the tools in MATLAB to convert between these representations.
• To work with both SISO as well as MIMO systems and discuss its stability criteria.
• We use Bode plot, Polar plot and Root locus for achieving the same.
• Getting familiar with MATLAB’s sisotool which makes the compensator design process much easier.

Transfer Function

This is a representation of the relation between the input and output of a linear time-invariant system (LTI) with zero initial conditions and zero-point equilibrium (This representation is in terms of spatial or temporal frequency - $s$ ).

Thus, for continuous-time input signal $x(t)$ and output $y(t)$, the transfer function $H(s)$ is the linear mapping of the Laplace transform of the input, $X(s) = \mathcal{L}\left\{x(t)\right\}$, to the Laplace transform of the output $Y(s) = \mathcal{L}\left\{y(t)\right\}$.

$$H(s) = \frac{Y(s)}{X(s)} = \frac{ \mathcal{L}\left\{y(t)\right\} }{ \mathcal{L}\left\{x(t)\right\} }$$

You may wonder why a LTI system, this is because LTI systems have this really useful property - the eigenfunctions of all LTI are complex exponentials.

What this means is that if we give a input $A e^{st}$ for some complex amplitude $A$ and complex frequency $s$, the output will be some complex constant times the input, say $B e^{st}$ for some new complex amplitude $B$. Then this ratio $\frac{B}{A}$ is the transfer function at frequency s.

What we can interpret from this is that if the input to the system is a sinusoid (sum of complex exponentials with complex-conjugate frequencies), then the output of the system will also be a sinusoid, perhaps with a different amplitude and a different phase, but always with the same frequency upon reaching steady-state.

LTI systems cannot produce frequency components that are not in the input.

LTI system theory is good at describing many important systems. Most LTI systems are considered “easy” to analyze, at least compared to the time-varying and/or nonlinear case. Any system that can be modeled as a linear homogeneous differential equation with constant coefficients is an LTI system.

So we can use this to transform the governing differential equation into an algebraic equation which is often easier to analyze.

Further, for a LTI system its impulse response can be used to fully describe the system’s output to any input. The transfer function is the Laplace transform of the impulse response.

We can continue discussing the Transfer function but for now lets see what limitations it faces when it comes to having to analyze MIMO systems.

State-space representation

In reality when we need to control a realistic system it usually has many sensors and actuators. Now since all of them are related with a single system their performances are coupled.

What this means is that since for $m$ number of actuators and $n$ number of sensors, one will have $mn$ number of transfer functions and one has to design as many controllers for the system!

With the advent of computers and computer based digital control, one can design control systems for the MIMO systems directly in the time domain.

This is achieved by replaces an $n$th order differential equation with $n$ first order differential equations - which are represented as a matrix .

Also unlike the frequency domain approach, the use of the state-space representation is not limited to systems with linear components and zero initial conditions.

The most general state-space representation of a linear system with $p$ inputs, $q$ outputs and $n$ state variables is written in the following form
$$\dot{\mathbf{x}}(t) = A(t) \mathbf{x}(t) + B(t) \mathbf{u}(t)$$
$$\mathbf{y}(t) = C(t) \mathbf{x}(t) + D(t) \mathbf{u}(t)$$
where:
$\mathbf{x}(\cdot)$ is called the “state vector”, $\mathbf{x}(t) \in \mathbb{R}^n;$
$\mathbf{y}(\cdot)$ is called the “output vector”, $\mathbf{y}(t) \in \mathbb{R}^q;$
$\mathbf{u}(\cdot)$ is called the “input (or control) vector”, $\mathbf{u}(t) \in \mathbb{R}^p;$
$A(\cdot)$ is the “state (or system) matrix”, $\operatorname{dim}[A(\cdot)] = n \times n,$
$B(\cdot)$ is the “input matrix”, $\operatorname{dim}[B(\cdot)] = n \times p,$
$C(\cdot)$ is the “output matrix”, $\operatorname{dim}[C(\cdot)] = q \times n,$
$D(\cdot)$ is the “feedthrough (or feedforward) matrix”
$\dot{\mathbf{x}}(t) := \frac{\operatorname{d}}{\operatorname{d}t} \mathbf{x}(t).$

Note that in cases where the system model does not have a direct feedthrough, $D(\cdot)$ is the zero matrix.

Answers to questions

Question 1

Given the zeros and poles of a transfer function, construct the equivalent state space representation.

As you see, we used zpk command which constructed the TF (Transfer Function) Object -$G$ from the zeros and poles.

Then we used ssdata to extract the State Space model from $G$.

Question 2

Given a State Space representation of a system, find its transfer function.

We can use tf command to get the transfer function.

Question 3

Given a system in its state space representation, find its Transmission Zeros.

So with the tzero command we got both the transmission zeros.

Question 4

Given a matrix of transfer functions (MIMO system), find its state space equivalent representation.

So, we use the same command as before and MATLAB does all the work.

Question 5

We can construct the various Matrices which uniquely define the State Space representation of this system.

The equivalent Transfer function using ss2tf (State-space to transfer function conversion).

The poles and zeros are as follows,

Question 6

The closed loop plant transfer function will be,

Now its State Space representation will be,

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