Plant Transfer Function Stability Analysis

To determine the condition for the stability of a closed-loop system with a given plant transfer function in a unity feedback configuration, we will use the Routh-Hurwitz stability criterion. This method helps us analyze the characteristic equation of the system to find the range of parameters for which the system remains stable.

Closed-Loop System Transfer Function Derivation

The given plant transfer function is:

\(G\left( {\rm{s}} \right) = \left( {{{\rm{K}}_{\rm{p}}} + \frac{{{{\rm{K}}_{\rm{I}}}}}{{\rm{s}}}} \right)\frac{1}{{{\rm{s}}\left( {{\rm{s}} + 2} \right)}}\)

First, simplify the expression for \(G(s)\):

\(G\left( {\rm{s}} \right) = \left( {\frac{{{{\rm{K}}_{\rm{p}}}{\rm{s}} + {{\rm{K}}_{\rm{I}}}}}{{\rm{s}}}} \right)\frac{1}{{{\rm{s}}\left( {{\rm{s}} + 2} \right)}}\)

\(G\left( {\rm{s}} \right) = \frac{{{{\rm{K}}_{\rm{p}}}{\rm{s}} + {{\rm{K}}_{\rm{I}}}}}{{{{\rm{s}}^2}\left( {{\rm{s}} + 2} \right)}}\)

\(G\left( {\rm{s}} \right) = \frac{{{{\rm{K}}_{\rm{p}}}{\rm{s}} + {{\rm{K}}_{\rm{I}}}}}{{{{\rm{s}}^3} + 2{{\rm{s}}^2}}}\)

For a unity feedback configuration, the closed-loop transfer function \(T(s)\) is given by:

\(T\left( {\rm{s}} \right) = \frac{{G\left( {\rm{s}} \right)}}{{1 + G\left( {\rm{s}} \right)H\left( {\rm{s}} \right)}}\)

Since it's a unity feedback system, \(H(s) = 1\).

The characteristic equation of the closed-loop system is \(1 + G(s)H(s) = 0\).

\(1 + \frac{{{{\rm{K}}_{\rm{p}}}{\rm{s}} + {{\rm{K}}_{\rm{I}}}}}{{{{\rm{s}}^3} + 2{{\rm{s}}^2}}} = 0\)

To find the characteristic equation, we clear the denominator:

\({{\rm{s}}^3} + 2{{\rm{s}}^2} + {{\rm{K}}_{\rm{p}}}{\rm{s}} + {{\rm{K}}_{\rm{I}}} = 0\)

Routh-Hurwitz Criterion Application

Now, we will apply the Routh-Hurwitz criterion to this characteristic equation to determine the stability conditions. The Routh array is constructed from the coefficients of the characteristic equation \(a_n s^n + \dots + a_1 s + a_0 = 0\).

For our equation, \({{\rm{s}}^3} + 2{{\rm{s}}^2} + {{\rm{K}}_{\rm{p}}}{\rm{s}} + {{\rm{K}}_{\rm{I}}} = 0\), the coefficients are:

• \(a_3 = 1\)
• \(a_2 = 2\)
• \(a_1 = K_P\)
• \(a_0 = K_I\)

The Routh array is formed as follows:

Now, we calculate the elements of the \(s^1\) and \(s^0\) rows:

For \(b_1\):

\(b_1 = \frac{{\left( {a_2} \right)\left( {a_1} \right) - \left( {a_3} \right)\left( {a_0} \right)}}{{a_2}}\)

\(b_1 = \frac{{\left( 2 \right)\left( {{{\rm{K}}_{\rm{P}}}} \right) - \left( 1 \right)\left( {{{\rm{K}}_{\rm{I}}}} \right)}}{2}\)

\(b_1 = \frac{{2{{\rm{K}}_{\rm{P}}} - {{\rm{K}}_{\rm{I}}}}}{2}\)

For \(c_1\):

\(c_1 = \frac{{\left( {b_1} \right)\left( {{{\rm{K}}_{\rm{I}}}} \right) - \left( 2 \right)\left( 0 \right)}}{{b_1}}\)

\(c_1 = \frac{{\left( {\frac{{2{{\rm{K}}_{\rm{P}}} - {{\rm{K}}_{\rm{I}}}}}{2}} \right)\left( {{{\rm{K}}_{\rm{I}}}} \right) - 0}}{{\frac{{2{{\rm{K}}_{\rm{P}}} - {{\rm{K}}_{\rm{I}}}}}{2}}}\)

\(c_1 = {{\rm{K}}_{\rm{I}}}\)

The completed Routh array is:

Stability Conditions for the System

For the closed-loop system to be stable, all the elements in the first column of the Routh array must have the same sign. Since the first two elements (1 and 2) are positive, all subsequent elements in the first column must also be positive.

From the \(s^1\) row, we require:

\(\frac{{2{{\rm{K}}_{\rm{P}}} - {{\rm{K}}_{\rm{I}}}}}{2} > 0\)

This implies:

\(2{{\rm{K}}_{\rm{P}}} - {{\rm{K}}_{\rm{I}}} > 0\)

\(2{{\rm{K}}_{\rm{P}}} > {{\rm{K}}_{\rm{I}}}\)

\({{\rm{K}}_{\rm{P}}} > \frac{{{{\rm{K}}_{\rm{I}}}}}{2}\)

From the \(s^0\) row, we require:

\({{\rm{K}}_{\rm{I}}} > 0\)

Combining these two conditions, for the system to be stable, we must have:

\({{\rm{K}}_{\rm{P}}} > \frac{{{{\rm{K}}_{\rm{I}}}}}{2}\) and \({{\rm{K}}_{\rm{I}}} > 0\)

These two conditions can be compactly written as:

\({{\rm{K}}_{\rm{P}}} > \frac{{{{\rm{K}}_{\rm{I}}}}}{2} > 0\)

This is the required condition for the stability of the closed-loop system.