This worksheet illustrates how the MapleSim Control Design Toolbox can be used to design PID controllers using several methods. This worksheet is divided into 3 sections. In the first section, we will use the Pole Placement method to design a PI controller for a second-order system so that we can confine the closed-loop poles to a desired region. In the second section, we will use the Exact Pole Placement method to design a PID controller so that we can specify the exact location of the dominant poles. Finally, in the last section, we will use the Gain-Phase Margin method to design a PID controller for a fifth-order system.

The Pole Placement method is a unique feature of the MapleSim ControlDesign toolbox. It allows you to specify a desired region for the closed-loop poles based on damping ratio, $\mathrm{\ζ}$, and the natural frequency, $\mathrm{\ωn}$.  In this section, we will show how the Pole Placement method can be used to design a PI controller for an arbitrary second-order system.

We can create a second-order system using the DynamicSystems[NewSystem] command

$\mathrm{sys1} ≔ \mathrm{DynamicSystems}\left[\mathrm{NewSystem}\right]\left(\frac{s\+3}{s^2-3\cdot s\+5}\right)\:$

$\mathrm{DynamicSystems}\left[\mathrm{PrintSystem}\right]\left(\mathrm{sys1}\right)$

As mentioned above, the desired region for the closed-loop poles is defined by the damping ratio, $\mathrm{\ζ}$, and the natural frequency, $\mathrm{\ωn}$. For this example, the values of 0.7 and 5/2 have been chosen for the damping ratio and natural frequency, respectively.

$\mathrm{\ζ1}≔0.7$

$\mathrm{\ωn1}≔\frac{5}{2}$

Based on the values of $\mathrm{\ζ1}$ and $\mathrm{\ωn1}$, the command ControlDesign[FeasibleGain] provides you with the option to output a list of 3 possible gain values for the PI controller as sample designs for which all the closed-loop poles will lie in the desired region. Alternatively, the command gives you the option to plot a region of feasible gain values.

$\mathrm{solution1}≔\mathrm{ControlDesign}\left[\mathrm{FeasibleGains}\right]\left(\mathrm{sys1}\,\mathrm{\ζ1}\, \mathrm{\ωn1}\, \'\mathrm{output}\'\=\mathrm{samplepoints}\, \'\mathrm{condition}\' \=\mathrm{all}\, \mathrm{posgains}\=\mathrm{true}\, \mathrm{controller}\= \'\mathrm{PI}\'\right)$

$\mathrm{ControlDesign}\left[\mathrm{FeasibleGains}\right]\left(\mathrm{sys1}\,\mathrm{\ζ1}\, \mathrm{\ωn1}\, \'\mathrm{output}\' \=\mathrm{region}\, \'\mathrm{condition}\' \=\mathrm{all}\, \'\mathrm{posgains}\' \=\mathrm{false}\, \'\mathrm{controller}\' \= \'\mathrm{PI}\'\right)$

For this example, we will use the gain values for $\mathrm{kc}$ and $\mathrm{ki}$ as provided in (4)

$\mathrm{kp1}≔\mathrm{rhs}\left(\left[\mathrm{solution1}\right]\left[2\right]\left[1\right]\right)\;\mathrm{ki1}≔\mathrm{rhs}\left(\left[\mathrm{solution1}\right]\left[2\right]\left[2\right]\right)$

Using these gain values, the transfer function for the PI controller can be determined as:

$\mathrm{Gc1} ≔ \mathrm{DynamicSystems}\left[\mathrm{NewSystem}\right]\left(\mathrm{kp1}\+\frac{\mathrm{ki1}}{s}\right)\:$

$\mathrm{DynamicSystems}\left[\mathrm{PrintSystem}\right]\left(\mathrm{Gc1}\right)$

At this point, we can automatically obtain the transfer function for the closed-loop system using the command ControlDesign[PIDClosedLoop].

$\mathrm{ClosedLoopSys1} ≔ \mathrm{ControlDesign}\left[\mathrm{PIDClosedLoop}\right]\left(\mathrm{sys1}\,\mathrm{Gc1}\right)\:$

$\mathrm{DynamicSystems}\left[\mathrm{PrintSystem}\right]\left(\mathrm{ClosedLoopSys1}\right)$

As expected, the poles of the closed-loop reside in the region defined by $\mathrm{\ζ1}$ and $\mathrm{\ωn1}$.

The step response of the open and closed-loop system can be determined using the DynamicSystems[ResponsePlot] command. From the plots, we can see how the including of a PI controller was able to stabilize the system.

$\mathrm{p1}≔\mathrm{DynamicSystems}\left[\mathrm{ResponsePlot}\right]\left(\mathrm{ClosedLoopSys1}\,\mathrm{Step}\left(\right)\, \'\mathrm{duration}\' \=5\, \'\mathrm{numpoints}\' \=300\, \'\mathrm{color}\' \=\mathrm{blue}\, \'\mathrm{legend}\' \=''Closed\; Loop\; Response''\, \'\mathrm{labels}\' \=\left[''time''\,''Amplitude''\right]\right)\:$

$\mathrm{p2}≔\mathrm{DynamicSystems}\left[\mathrm{ResponsePlot}\right]\left(\mathrm{sys1}\,\mathrm{Step}\left(\right)\, \'\mathrm{duration}\' \=3\, \'\mathrm{numpoints}\' \=300\, \'\mathrm{color}\' \=\mathrm{red}\, \'\mathrm{legend}\' \= ''Open\; Loop\; Response''\, \'\mathrm{labels}\' \=\left[''time''\,''Amplitude''\right]\right)\:$

$\mathrm{plots}\left[\mathrm{dualaxisplot}\right]\left(\mathrm{p1}\,\mathrm{p2}\right)$

In this section, we will show how the Exact Pole Placement method is used to design a PID controller for a 2nd order system.

As in the previous section, the transfer function of the open loop system can be defined as follows:

$\mathrm{sys2} ≔ \mathrm{DynamicSystems}\left[\mathrm{NewSystem}\right]\left(\frac{3}{s^2\+3\cdot s\+5}\right)\:$

$\mathrm{DynamicSystems}\left[\mathrm{PrintSystem}\right]\left(\mathrm{sys2}\right)$

Let us assume that the desired closed-loop pole locations are:

$\mathrm{poles2}≔\left[-4\+2\cdot I\,-4-2\cdot I\,-\frac{5}{2}\right]$

Using the ControlDesign[DominantPole] command we can obtain the PID controller transfer function

$$\begin{gathered} \mathrm{Gc2}≔\mathrm{ControlDesign}\left[\mathrm{DominantPole}\right]\left(\mathrm{sys2}\,\mathrm{poles2}\, \'\mathrm{controller}\' \=\mathrm{PID}\, \'\mathrm{returntype}\' \= \mathrm{system}\right)\: \\ \mathrm{DynamicSystems}\left[\mathrm{PrintSystem}\right]\left(\mathrm{Gc2}\right)\; \end{gathered}$$

At this point, we can automatically obtain the transfer function for the closed-loop system using the command ControlDesign[PIDClosedLoop].

$$\begin{gathered} \mathrm{ClosedLoopSys2}≔\mathrm{ControlDesign}\left[\mathrm{PIDClosedLoop}\right]\left(\mathrm{sys2}\, \mathrm{Gc2}\right)\: \\ \mathrm{DynamicSystems}\left[\mathrm{PrintSystem}\right]\left(\mathrm{ClosedLoopSys2}\right) \end{gathered}$$

The step response of the open and closed-loop system is shown in the following plots. The closed-loop response in addition to tracking the input signal is much faster than then open-loop response.

$\mathrm{p3}≔\mathrm{DynamicSystems}\left[\mathrm{ResponsePlot}\right]\left(\mathrm{ClosedLoopSys2}\,\mathrm{Step}\left(\right)\,\mathrm{duration}\=7\,\mathrm{numpoints}\=300\,\mathrm{color}\=\mathrm{blue}\,\mathrm{legend}\=''Closed\; Loop\; Response''\,\mathrm{parameters}\=\left[a\=3\,b\=5\,c\=2\right]\right)\:$

$\mathrm{p4}≔\mathrm{DynamicSystems}\left[\mathrm{ResponsePlot}\right]\left(\mathrm{sys2}\,\mathrm{Step}\left(\right)\,\mathrm{duration}\=7\,\mathrm{numpoints}\=300\, \mathrm{color}\=\mathrm{red}\, \mathrm{legend} \= ''Open\; Loop\; Response''\,\mathrm{parameters}\=\left[a\=3\,b\=5\,c\=2\right]\right)\:$

$\mathrm{plots}\left[\mathrm{display}\right]\left(\left[\mathrm{p3}\,\mathrm{p4}\right]\, \mathrm{labels}\=\left[''time''\,''Amplitude''\right]\right)$

In this section, we will show how the Gain-Phase Margin method is used to design a PID controller for the following fifth-order system.

$\mathrm{sys3} ≔ \mathrm{NewSystem}\left(\frac{70\cdot \left(s\+1\right)}{{\left(s\+2\right)}^5}\right)\:$

$\mathrm{DynamicSystems}\left[\mathrm{PrintSystem}\right]\left(\mathrm{sys3}\right)$

The system gain margin (in decibels) and associated crossover frequency (in rad/sec) is:

$\mathrm{DynamicSystems}\left[\mathrm{GainMargin}\right]\left(\mathrm{sys3}\,\mathrm{decibels}\=\mathrm{true}\right)$

Similarly, the system phase margin (in degrees) and crossover frequency (in rad/sec) is:

$\mathrm{DynamicSystems}\left[\mathrm{PhaseMargin}\right]\left(\mathrm{sys3}\,\mathrm{radians}\=\mathrm{false}\right)$

Using the ControlDesign[GainPhaseMargin] command we can obtain the PID gain values so that the desired gain and phase margin for the closed-loop system is at least, $15 \mathrm{dB}$ and $60°$, respectively.

$$\begin{gathered} \mathrm{Gc3} ≔ \mathrm{ControlDesign}\left[\mathrm{GainPhaseMargin}\right]\left(\mathrm{sys3}\,15\,60\, \'\mathrm{controller}\' \=\mathrm{PID}\, \'\mathrm{decibels}\' \=\mathrm{true}\, \'\mathrm{radians}\' \=\mathrm{false}\, \'\mathrm{returntype}\' \= \mathrm{system}\right)\: \\ \mathrm{DynamicSystems}\left[\mathrm{PrintSystem}\right]\left(\mathrm{Gc3}\right)\; \end{gathered}$$

At this point, we can automatically obtain the transfer function for the closed-loop system using the command ControlDesign[PIDClosedLoop].

$\mathrm{ClosedLoopSys3} ≔ \mathrm{ControlDesign}\left[\mathrm{PIDClosedLoop}\right]\left(\mathrm{sys3}\, \mathrm{Gc3}\right)\:$

$\mathrm{DynamicSystems}\left[\mathrm{PrintSystem}\right]\left(\mathrm{ClosedLoopSys3}\right)$

The gain and phase margin and crossover frequencies for the loop transfer function are shown below. These values exceed the minimum requirements specified above.

$\mathrm{LoopTf}≔\mathrm{DynamicSystems}\left[\mathrm{TransferFunction}\right]\left(\mathrm{sys3}:-\mathrm{tf}\left[1\,1\right]\cdot \mathrm{Gc3}:-\mathrm{tf}\left[1\,1\right]\right)\:$

$\mathrm{DynamicSystems}\left[\mathrm{GainMargin}\right]\left(\mathrm{LoopTf}\, \mathrm{decibels}\=\mathrm{true}\right)$

$\mathrm{DynamicSystems}\left[\mathrm{PhaseMargin}\right]\left(\mathrm{LoopTf}\, \mathrm{radians}\=\mathrm{false}\right)$

The step response of the open and closed-loop system is shown in the following plots. Unlike the open-loop response which was more than 100% off in the steady state, the closed-loop response now tracks the input signal.

$\mathrm{p5}≔\mathrm{DynamicSystems}\left[\mathrm{ResponsePlot}\right]\left(\mathrm{ClosedLoopSys3}\,\mathrm{Step}\left(\right)\,\mathrm{duration}\=20\,\mathrm{numpoints}\=300\,\mathrm{color}\=\mathrm{blue}\,\mathrm{legend}\=''Closed\; Loop\; Response''\right)\:$

$\mathrm{p6}≔\mathrm{DynamicSystems}\left[\mathrm{ResponsePlot}\right]\left(\mathrm{sys3}\,\mathrm{Step}\left(\right)\,\mathrm{duration}\=20\,\mathrm{numpoints}\=300\, \mathrm{color}\=\mathrm{red}\, \mathrm{legend} \= ''Open\; Loop\; Response''\right)\:$

$\mathrm{plots}\left[\mathrm{display}\right]\left(\left[\mathrm{p5}\,\mathrm{p6}\right]\, \mathrm{labels}\=\left[''time''\,''Amplitude''\right]\right)$

The bode plots of the open-loop system and the loop transfer function are shown below.

$\mathrm{DynamicSystems}\left[\mathrm{BodePlot}\right]\left(\mathrm{sys3}\, \mathrm{color}\=''Red''\right)$

$\mathrm{DynamicSystems}\left[\mathrm{BodePlot}\right]\left(\mathrm{LoopTf}\,\mathrm{color}\=''Blue''\right)$

In this section, we will show how to tune the parameters of a PID controller using a single tuning parameter t, which is equivalent to the desired time constant of the closed-loop system.

$\mathrm{sys4} ≔ \mathrm{NewSystem}\left(\frac{5}{\left(s\+1\right)\cdot \left(3\cdot s\+1\right)\cdot \left(5\cdot s\+1\right)}\right)\:$

$\mathrm{DynamicSystems}\left[\mathrm{PrintSystem}\right]\left(\mathrm{sys4}\right)$

The desired closed-loop time constant t is 1.0 second (for a settling time of approximately $6\cdot \mathrm{\τ}\=6$ seconds).

  $\mathrm{\τ}≔1.0$

$\mathrm{infolevel}\left[\mathrm{ControlDesign}\right]≔3\:$

$$\begin{gathered} \mathrm{pidtf} ≔ \mathrm{ControlDesign}\left[\mathrm{PIDAuto}\right]\left(\mathrm{sys4}\, \mathrm{\τ}\, \'\mathrm{returntype}\' \= \mathrm{system}\right)\: \\ \mathrm{DynamicSystems}\left[\mathrm{PrintSystem}\right]\left(\mathrm{pidtf}\right) \end{gathered}$$

$\mathrm{ClosedLoopSys4} ≔ \mathrm{ControlDesign}\left[\mathrm{PIDClosedLoop}\right]\left(\mathrm{sys4}\, \mathrm{pidtf}\right)\:$

$\mathrm{DynamicSystems}\left[\mathrm{PrintSystem}\right]\left(\mathrm{ClosedLoopSys4}\right)$

$\mathrm{p7}≔\mathrm{DynamicSystems}\left[\mathrm{ResponsePlot}\right]\left(\mathrm{ClosedLoopSys4}\,\mathrm{Step}\left(\right)\,\mathrm{duration}\=30\,\mathrm{numpoints}\=300\,\mathrm{color}\=\mathrm{blue}\,\mathrm{legend}\=''Closed\; Loop\; Response''\right)\:$

$\mathrm{p8}≔\mathrm{DynamicSystems}\left[\mathrm{ResponsePlot}\right]\left(\mathrm{sys4}\,\mathrm{Step}\left(\right)\,\mathrm{duration}\=30\,\mathrm{numpoints}\=300\, \mathrm{color}\=\mathrm{red}\, \mathrm{legend} \= ''Open\; Loop\; Response''\right)\:$

$\mathrm{plots}\left[\mathrm{display}\right]\left(\left[\mathrm{p7}\,\mathrm{p8}\right]\, \mathrm{labels}\=\left[''time''\,''Amplitude''\right]\right)$

Verify the resulting closed-loop settling time

$t_{\mathrm{settling}} ≔ \mathrm{DynamicSystems}\left[\mathrm{StepProperties}\right]\left(\mathrm{ClosedLoopSys4}\right)\left[7\right]$