Last updated: August 31, 2019
Topic: ArtBooks
Sample donated:

Proportional Integral Derivative (PID) controller is one of the most common control loop algorithms used, as more than 95 $\%$ of all the regulatory controller utilize PID feedback system. Based on the heuristic approach feedback controllers allow for the response to be sufficiently close to the desired value and are therefore ideal for most practical control problems.A control feedback system was first established in the industry by James Watt’s “conical pendulum” governor. This control system was operating with only a proportional gain ($K_p$) resulting in wider oscillations of the Process Variable for bigger values of the $K_p$ or higher load. The history of PID controllers in the form that we know them today started on 1868 when James Clerk Maxwell explored the mathematical basis of controlling systems stability in his famous paper “On Governors”.  In the same year, a pendulum-and-hydro-stat control was introduced in the newly invented Whitehead torpedo being one of the first PID control systems to be used in the industry.  The last step of defining the PID controllers came from Nicolas Minorsky, who presented a formal control law using theoretical analysis. Since then PID controllers are spreading in the industry becoming until today the most used control loop in the industry.egin{figure}Hcenteringincludegraphicswidth=linewidth,height=6cm{pip.png} itle{Figure 1: Step function}end{figure} A step function as seen above, can be used to measure control system performance requirements representing the set point. Then by measuring the process variable response  over time and using waveform characteristics (represented in figure 1 above) such as, rise time, overshoot and steady-state error the response is evaluated.Feedback control allows the correction of errors in a system based on the desired and actual output. One of the most widely used control algorithms is the Proportional Integral Derivative controller, thanks to its effective performance in a variety of operating conditions and its simplicity. A PID consists of 3 coefficients the values of which can be varied in order to achieve the optimal response.The PID controller can be represented by the following equation where, u(t) is the control signal and $e( au)$ the error, which is the difference between the desired result and the actual response of the system.egin{equation}u(t)=K_P(t)+K_I int_{t}^{0}e( au)d au+K_Dfrac{de}{dt}end{equation}The control signal is comprised of 3 terms, proportional, integral and derivative, with control parameters $k_p$, $k_i$ and $k_d$ respectively.The above equation can also be expressed in the Laplace domain, where the roles of each term  is better represented:egin{equation}G_R(s)=frac{U(s)}{E(s)}=K_P(1+frac{1}{T_Is}+frac{T_Ds}{1+frac{T_D}{N} imes s})end{equation}In its Laplace form it easier to manipulate and relate to practical results, especially in higher order problems.egin{figure}Hcenteringincludegraphicswidth=linewidth,height=5cm{Block.png} itle{Figure 2: PID controller block diagram}end{figure}For the proportional component, the error magnitude is multiplied by the proportional gain $K_p$  resulting in the proportional response. Therefore the greater the percentage difference of the output signal from the desired output is, the greater the proportional response will be.Integral:egin{equation} K_P imes e(t) end{equation}The integral component sums the error term over time, multiplying it by the integral coefficient $K_i$ and then integrating.Even a minor deviation from the desired value will result in a large integral component over time. The steady-state errors, the final difference between the process variable and set point, will be driven to zero through the integral component since the integral response will rise until the error is zero.Proportional:egin{equation} K_I imes int_{t_D}^{t} e(t) end{equation}The derivative response is proportional to the rate of change of the process variable. Hence the control system is able to minimize the effect of a disturbance and the time it takes to correct one, through the use of a projection of the process variable value. This allows higher Integral and Proportional coefficients to be used without the system becoming unstable.However since any changes in the error term affect it, the component is highly sensitive to noise in the sensor feedback and can easily render the system unstable if it is set too large.Derivative:egin{equation} K_D imes frac{d}{dt} e(t) end{equation}section{Block Diagram} The effects of a corresponding increase at each individual component of the PID are represented below in Table 1. The term rise time refers to the time taken for the helicopter to rise from 10\% to 90\% of its steady value. Overshoot is when the helicopter goes past the required elevation, desired response signal. Settling time is the time period for the helicopter to remain within a certain margin from the required elevation with no further deviations. Steady State Error (SSE) is the difference between the desired result and the actual response of the system, $u(t)-e(t)$. The term stability refers to the oscillatory behavior of the system.egin{table}ht!centeringcaption{Increasing the independent variable values of a PID control}label{tab:table1}egin{tabular}{|c|c|c|c|c|c|}hlineParameter & Rise time & Overshoot & Settling Time & Steady State Error & Stability \hline$K_P$ & Decrease & Increase & Slight Change & Decrease & Degrade \hline$K_I$ & Decrease & Increase & Increase & Gradually Eliminate & Degrade \hline$K_D$ & Slight Change & Decrease & Decrease & No effect in theory & Improve if $K_D$ is small\hlineend{tabular}end{table}section{Qunaser Helicopter PID control practical results}In this part of the laboratory, the values of $K_P$, $K_D$, $K_I$ where tested individually, by adjusting their values and observing the visual result on the signal. The overall results, satisfy our theoretical background.egin{figure}Hcenteringincludegraphicswidth=1linewidth,height=8cm{kP_-0_2__0__+0_2__+0_4_NEW.jpg} itle{Figure 3: Test values for $K_{P}$, a:0.3, b:0.5, c:0.7, d:0.9}end{figure}When increasing the value of $K_{P}$ we can see that the overall Rise time decreases, the Overshoot increases and there is no particular change Settling time. However, having a high $K_{P}$ results to instabilities, as can be seen from Figure 3: d ($K_{P}$=$K_{Pi}+0.4)$. Furthermore, the benefits than can be obtained by rising $K_{P}$ can be achieved by adjusting $K_{I}$ to a higher value too. Therefore, increasing $K_{p}$ does not ensures us with a gradual and smooth result in steady state. As a result, the final value is decreased compared to the initial one, Figure 3: b ($K_{Pi}=0.5$), to $K_{Pf}=0.27$, to avoid unwanted oscillations.egin{figure}Hcenteringincludegraphicswidth=linewidth,height=9cm{___-0_05__0__0_10__0_20_NEW.jpg} itle{Figure 4: Test values for $K_{I}$, I:0.03, II:0.08, III:0.18, IV:0.28}end{figure}Adjusting $K_{I}$ variable to a higher value, as can be seen from the above Figure, the Rise time and the Steady State Error decreases, while the Overshoot and the Settling time increase. Moreover, the change of the value $K_{I}$, as it is demonstrated in Figure 4, doesn’t affect the stability greatly, like $K_{P}$. Consequently, the value of $K_{I}$ has been raised to $K_{I}=0.23$ from an initial value of $K_{Ii}=0.08$egin{figure}Hcenteringincludegraphicswidth=1linewidth,height=9cm{Kd_-0_14__0__+0_06__+0_14_NEW.png} itle{Figure 5: Test values for $K_{D}$, A:0.20, B:0.34, C:0.40, D:0.48}end{figure}Altering the value of $K_{D}$ in this the overshoot and the Settling Time.  Raising $K_{D}$, gives smaller overshoot and shorter Settling Time with no visual effect on Rise Time or Steady State Error. As a result, the final value of $K_{D}$ is 0.40, +0.06 from the initial value.section{Discussion} In order to achieve the desired response some tuning had to be undertaken, adjusting the proportional, differential and integral gains. For that purpose a trial and error method was followed. A $0.2$ , 0.07 and  0.01, step was used for each of $K_P$,$K_D$ and $K_I$ respectively, hence finding the optimum value for each component and balance between them.The progress can be seen in Figures 3 to 6, as each component is adjusted the rise time, overshoot and settling time are stabilized and the signal deviation from the ideal response is minimized. As seen in Figure 6 the final values were $+0.15$ for $K_I$, $-0.23$ for $K_P$ and $+0.06$ for $K_D$.  egin{figure}Hcenteringincludegraphicswidth=0.8linewidth,height=7cm{+0_15_-0_23_+0_06.jpg}centering itle{Figure 6 Final Results : $K_I +0.15, K_P -0.23, K_D +0.06$}end{figure} section{References} egin{thebibliography}{2} ibitem{1} $Desborough Honeywell. “PID Control.” Caltech.edu, 2000, pp. 301–322., \www.cds.caltech.edu/~murray/books/AM08/pdf/am06-pid_16Sep06.pdf.$ibitem{2}$”PID for Dummies.” PID for Dummies – Control Solutions, www.csimn.com/CSI_pages/PIDforDummies.html.$ibitem{3}”PID Theory Explained.” PID Theory Explained – National Instruments, www.ni.com/white-paper/3782/en/.ibitem{4}Smuts, Jacques. “PID Controllers Explained.” Blog.opticontrols, 7 Mar. 2011, blog.opticontrols.com/archives/344.ibitem{5}Smuts, Jacques F. Process Control for Practitioners: How to Tune PID Controllers and Optimize Control Loops. OptiControls, 2011.ibitem{6}$”PID Controller.” Wikipedia, Wikimedia Foundation, 26 Jan. 2018, en.wikipedia.org/wiki/PID_controller.$end{thebibliography}end{document}