# PID controller tunning: CHR and CC methods

This post’s subject is the description of other methods to tune a PID controller. CHR (Chien, Hrone e Reswick) and CC (Cohen e Coon).

In case you don’t know what is PID, access the link below before continuing.

### The CHR method

All tuning methods for PID controller consist in calculate values of proportional ($K_{p}$), derivative ($K_{p}$) and integral ($K_{i}$) gains. The CHR method was developed in 1952, as an alternative to solve problems with step response. Despite gains being smaller than the ones obtained by Ziegler-Nichols, the system has more stability when adopts the CHR method. Comparison of stability between Ziegler-Nichols (Z&N) and CHR (lines below Z & N). Limite de estabilidade = Stability limit, Sobrevalor = Overshoot. Source: UnB.

Defines tuning laws for setpoint change and regulation resistant to disturbances.

#### Performance criteria

The following tables show gain calculations to obtain the fastest response as possible with 0 or 20% of overshoot for both problems. T_{u} is delay time, T_{g} is time constant and K_{s} is the gain factor. Source: Beckhoff Information System.

#### Gain calculation table for problems with regulation with robustness

$T_{N}$ and $T_{V}$ are integral and derivative times. Reminding that integral $Ki$ and derivative $Kd$ gains are calculated with the equations below.

$Ki=\frac{Kp}{T_{N}}$

$Kd=Kp\cdot T_{V}$

### Cohen-Coon (CC) method

Developed in 1953, this method is also used for step response and for a system with longer dead time, considering $0,6<\frac{\theta }{\tau }<4,5$. However, robustness with be bad if $\frac{\theta }{\tau }\leq 2$. Symbols used by many sources to represent dead time \theta and time constant \tau. Source: Responde Aí. Table for calculation of gain and times with CC method. Source: Control de Procesos.

The method Ziegler-Nichols and Cohen-Coon were created to obtain a quarter amplitude damping. In case you want to avoid oscillation, the gain $K_{c}$ must be smaller than the calculated. 