Design of Optimal PID Controller with <svg style="vertical-align:-0.18pt;width:11.225px;" id="M1" height="10.7" version="1.1" viewBox="0 0 11.225 10.7" width="11.225" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.022,-0,0,-.022,.062,10.412)"><path id="x1D73A" d="M451 389l-18 -7q-3 4 -19 20q-24 20 -84 20q-52 0 -90 -32q-37 -31 -37 -76q0 -24 16.5 -37.5t43.5 -13.5q8 0 31 4q28 5 43 5q47 0 47 -22q0 -37 -65 -37q-5 0 -28 3t-31 3q-39 0 -77 -31q-39 -30 -39 -78q0 -36 27.5 -55.5t73.5 -19.5q43 0 76.5 17.5t74.5 61.5l18 -8
q-92 -116 -224 -116q-71 0 -116.5 31t-45.5 77q0 48 29.5 86.5t78.5 55.5v3q-21 7 -35 26.5t-14 40.5q0 62 61 107q62 45 166 45q47 0 87 -21.5t50 -51.5z" /></g></svg>-Routh Stability for Different Processes

<table>Step response of the proposed method under various <svg height="7.9499998" id="M407" style="vertical-align:-0.1638pt" version="1.1" viewbox="0 0 7.0999999 7.9499998" width="7.0999999" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,7.675)"><path d="M387 375q0 -16 -17 -31t-29 -15q-9 0 -12 13q-13 74 -80 74q-36 0 -61 -24.5t-25 -56.5t24 -51.5t68 -19.5q33 0 47 2l2 -7l-32 -42q-20 2 -54 2q-45 0 -74 -21.5t-29 -60.5q0 -41 28 -65.5t73 -24.5q78 0 145 67l17 -23q-33 -47 -84.5 -75t-111.5 -28q-70 0 -114.5 35.5
t-44.5 92.5q0 46 38 78t95 45v2q-35 10 -54.5 33t-19.5 52q0 55 53 88.5t122 33.5q67 0 98.5 -23.5t31.5 -49.5z" id="x1D700"></path></g>
</svg>-stability (for plant.2).</table>

Mathematical Problems in Engineering

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Figure 5

Figure 5: Design of Optimal PID Controller with <svg style="vertical-align:-0.18pt;width:11.225px;" id="M1" height="10.7" version="1.1" viewBox="0 0 11.225 10.7" width="11.225" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.022,-0,0,-.022,.062,10.412)"><path id="x1D73A" d="M451 389l-18 -7q-3 4 -19 20q-24 20 -84 20q-52 0 -90 -32q-37 -31 -37 -76q0 -24 16.5 -37.5t43.5 -13.5q8 0 31 4q28 5 43 5q47 0 47 -22q0 -37 -65 -37q-5 0 -28 3t-31 3q-39 0 -77 -31q-39 -30 -39 -78q0 -36 27.5 -55.5t73.5 -19.5q43 0 76.5 17.5t74.5 61.5l18 -8
q-92 -116 -224 -116q-71 0 -116.5 31t-45.5 77q0 48 29.5 86.5t78.5 55.5v3q-21 7 -35 26.5t-14 40.5q0 62 61 107q62 45 166 45q47 0 87 -21.5t50 -51.5z" /></g></svg>-Routh Stability for Different Processes 

Mathematical Problems in Engineering

Design of Optimal PID Controller with -Routh Stability for Different Processes

Figure 5