Proportional Controls

P Controls

Proportional Controls

• In a block diagram, this is summing the output to the input
• This is controlled using a $k_p$ or some proportional gain which controls our “flow”
• As you increase $k_p$ you can make the DC gain as close as possible to 1 (This is the servo performance or why we use servo motors)
• Mathematically, $P_{out}=k_{p}e\left(t\right)+p_0$
• Increasing $k_p$ decreases $\zeta$, making the transient response more oscillatory

PD Controls

Proportional Derivative Controls

• A PD controller adds a derivative term to the P controller, which reacts to the rate of change of the error
• This is like implementing a virtual spring and damper
• Mathematically, $u\left(t\right)=K_{p}e\left(t\right)+K_d\frac{d}{dt}e\left(t\right)$ where we still have our proportional gain, but now have a derivative gain connected to the rate of change of the error
  • In LaPlace… $F\left(s\right)=G_K\left(s\right)E\left(s\right)$
• We use this when we need faster responses and to reduce overshoot
• Designing a PD controller means finding $k_p$ and $k_d$ which we can do by putting our transfer function into standard 2nd Order form and when doing a comparison of coefficients

Integral Control

• Based on the total history of the error
• $U\left(s\right)=\frac{k_I}{s}$ where the s in the denominator is the term that makes this a PI controller instead of a proportional controller
• Infinite gain at DC (when t = infinity or when s = 0)
• Best performance in steady state error

PI vs P Controller Example

• The goal is to describe the steady state behaviours for two inputs
• First find $G_P$ from whatever system we are analysizing
• Note that $G_C$ is $K_p=k_p+\frac{k_i}{s}$ for a PI controller
• Use the closed-loop TF equations which include our desired inputs to get transfer functions for our desired inputs
• Now we can construct the transfer functions for our P and PI systems by $G_C{G}_P$ and determine the type of system that they are
• For each of our input transfer functions, we want to apply a unit step input and find the error transfer function for each one

PD vs PI

• PD does not change the closed loop system order or the system type
• I-control and I-action increase the closed loop order by 1 which is why its used for steady state error

PID Controls

Soo Jeon Wisdom

PID gain tuning is more of an art than a science but these are the general guidelines

• PD gains are for transient response
  • Here, you start with a PD control with a small D gain, increase P until the oscillation is too big. The, increase the D gain and continue
  • Stop when you achieve a satisfactory transient response (settling time and overshoot)
• Integral gain is related to steady state error
  • I gain removes the steady state error and suppresses the disturbance effect by increasing the system order
• Requires lots of manual tuning of parameters
• May not be feasible to satisfy desired transient plants