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Reference.org
Ziegler–Nichols method
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<p>The Ziegler–Nichols tuning method is a <a href="/facts/Heuristic/BcBEkv9c">heuristic</a> method of tuning a <a href="/facts/PID_controller/7XcYk0CH">PID controller</a>. It was developed by <a href="/facts/John_G._Ziegler/EG7y8Sby">John G. Ziegler</a> and <a href="/facts/Nathaniel_B._Nichols/nPj1bNdI">Nathaniel B. Nichols</a>. It is performed by setting the <i>I</i> (integral) and <i>D</i> (derivative) gains to zero. The "P" (proportional) gain, K p {\displaystyle K_{p}} is then increased (from zero) until it reaches the ultimate gain K u {\displaystyle K_{u}} , at which the output of the control loop has stable and consistent oscillations. K u {\displaystyle K_{u}} and the oscillation period T u {\displaystyle T_{u}} are then used to set the P, I, and D gains depending on the type of controller used and behaviour desired: </p> Ziegler–Nichols method<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a><table><tbody><tr><th>Control Type</th><th> K p {\displaystyle K_{p}} </th><th> T i {\displaystyle T_{i}} </th><th> T d {\displaystyle T_{d}} </th><th> K i {\displaystyle K_{i}} </th><th> K d {\displaystyle K_{d}} </th></tr><tr><td>P</td><td> 0.5 K u {\displaystyle 0.5K_{u}} </td><td>–</td><td>–</td><td>–</td><td>–</td></tr><tr><td>PI</td><td> 0.45 K u {\displaystyle 0.45K_{u}} </td><td> 0.8 3 ¯ T u {\displaystyle 0.8{\overline {3}}T_{u}} </td><td>–</td><td> 0.54 K u / T u {\displaystyle 0.54K_{u}/T_{u}} </td><td>–</td></tr><tr><td>PD</td><td> 0.8 K u {\displaystyle 0.8K_{u}} </td><td>–</td><td> 0.125 T u {\displaystyle 0.125T_{u}} </td><td>–</td><td> 0.10 K u T u {\displaystyle 0.10K_{u}T_{u}} </td></tr><tr><td>classic PID</td><td> 0.6 K u {\displaystyle 0.6K_{u}} </td><td> 0.5 T u {\displaystyle 0.5T_{u}} </td><td> 0.125 T u {\displaystyle 0.125T_{u}} </td><td> 1.2 K u / T u {\displaystyle 1.2K_{u}/T_{u}} </td><td> 0.075 K u T u {\displaystyle 0.075K_{u}T_{u}} </td></tr><tr><td>Pessen Integral Rule</td><td> 0.7 K u {\displaystyle 0.7K_{u}} </td><td> 0.4 T u {\displaystyle 0.4T_{u}} </td><td> 0.15 T u {\displaystyle 0.15T_{u}} </td><td> 1.75 K u / T u {\displaystyle 1.75K_{u}/T_{u}} </td><td> 0.105 K u T u {\displaystyle 0.105K_{u}T_{u}} </td></tr><tr><td>some overshoot</td><td> 0.3 3 ¯ K u {\displaystyle 0.3{\overline {3}}K_{u}} </td><td> 0.50 T u {\displaystyle 0.50T_{u}} </td><td> 0.3 3 ¯ T u {\displaystyle 0.3{\overline {3}}T_{u}} </td><td> 0.6 6 ¯ K u / T u {\displaystyle 0.6{\overline {6}}K_{u}/T_{u}} </td><td> 0.1 1 ¯ K u T u {\displaystyle 0.1{\overline {1}}K_{u}T_{u}} </td></tr><tr><td>no overshoot</td><td> 0.20 K u {\displaystyle 0.20K_{u}} </td><td> 0.50 T u {\displaystyle 0.50T_{u}} </td><td> 0.3 3 ¯ T u {\displaystyle 0.3{\overline {3}}T_{u}} </td><td> 0.40 K u / T u {\displaystyle 0.40K_{u}/T_{u}} </td><td> 0.06 6 ¯ K u T u {\displaystyle 0.06{\overline {6}}K_{u}T_{u}} </td></tr></tbody></table> <p>The ultimate gain ( K u ) {\displaystyle (K_{u})} is defined as 1/M, where M = the amplitude ratio, K i = K p / T i {\displaystyle K_{i}=K_{p}/T_{i}} and K d = K p T d {\displaystyle K_{d}=K_{p}T_{d}} . </p><p>These 3 parameters are used to establish the correction u ( t ) {\displaystyle u(t)} from the error e ( t ) {\displaystyle e(t)} via the equation: </p> u ( t ) = K p ( e ( t ) + 1 T i ∫ 0 t e ( τ ) d τ + T d d e ( t ) d t ) {\displaystyle u(t)=K_{p}\left(e(t)+{\frac {1}{T_{i}}}\int _{0}^{t}e(\tau )\,d\tau +T_{d}{\frac {de(t)}{dt}}\right)} <p>which has the following transfer function relationship between error and controller output: </p> u ( s ) = K p ( 1 + 1 T i s + T d s ) e ( s ) = K p ( T d T i s 2 + T i s + 1 T i s ) e ( s ) {\displaystyle u(s)=K_{p}\left(1+{\frac {1}{T_{i}s}}+T_{d}s\right)e(s)=K_{p}\left({\frac {T_{d}T_{i}s^{2}+T_{i}s+1}{T_{i}s}}\right)e(s)}