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# PI-based first-order controller

## Introduction

The aim of this post is to explain and demonstrate how to calculate a simple PI controller for a first-order system. Let’s assume that the system is given by its transfer function G. The closed loop system is given by the following diagram:

## Close loop transfer function

The transfer function of the closed-loop system is given by:

$$\frac{y}{y_c} = \frac {CG}{1+CG}$$

as $$G$$ is assumed to be a first-order system, its equation is given by:

$$G=\frac{b}{z-a} = \frac{b} { z-e^{-\Delta/\tau} }$$

where $$\Delta$$ is the sampling time, and $$\tau$$ the time constant of the open loop system.

$$C$$ is the PI controller, its equation is given by:

$$C=K . \frac{z-a}{z-1}$$

The transfer function of the closed loop system is now given by:

$$\frac{y}{y_c} = \frac { K \frac{z-a}{z-1} \frac{b}{z-a} }{1+ K \frac{z-a}{z-1} \frac{b}{z-a}}$$

Previous equation can be simplified:

$$\frac{y}{y_c} = \frac { K \frac{b}{z-1} }{1+ K \frac{b}{z-1} }$$

The new transfer function is given by:

$$\frac{y}{y_c} = \frac { Kb }{ z -1 + Kb }$$

## Static gain of the closed loop system

Let's consider the response of the closed loop system when the input is a unity step ( $$\frac{z}{z-1}$$ ):

$$y(z) = \frac { zKb }{ (z -1 + Kb)(z-1) }$$

According to the final value theorem for Z-transforms, the static gain of the system is given by :

$$\lim\limits_{z \to 1} (z-1).y(z) = \lim\limits_{z \to 1} \frac { zKb(z-1) }{ (z -1 + Kb)(z-1) } = \lim\limits_{z \to 1} \frac { zKb }{ (z -1 + Kb) } = 1$$

The static gain of the system is equal to 1, the static error will be equal to zero.

## Time constant of the closed loop system

The closed loop system is also a first-order :

$$\frac{y}{y_c} = \frac { Kb }{ z -1 + Kb } = \frac { Kb }{ z - e^{-\Delta/\tau_c} }$$

where $$\Delta$$ is the sampling time, and $$\tau_c$$ the time constant of the closed loop system. Note that $$\tau_c$$ can be less than $$\tau$$ (the time constant of the open loop system). If $$\tau_c<\tau$$ the system is more reactive (but is also more energy consuming). In practice, $$\tau_c=\tau/2$$ is a good compromise. From the previous equation :

$$1-Kb = e^{-\Delta/\tau_c}$$

and:

$$K= \frac {1-e^{-\Delta/\tau_c}} {b}$$

To get the same response time in closed and open loop, the previous equation becomes :

$$K= \frac {1-a} {b}$$

## Stability

The system is stable if all the poles are located inside the unity circle. Here, as the system is a first-order, there is one pole : $$1-Kb$$. The system is stable if:

$$-1 < 1-Kb < 1$$

The previous equation can be rewritten as:

$$0 < Kb < 2$$

and:

$$0 < K < \dfrac{2}{b}$$

Note that if $$K$$ is calculated from $$\tau_c$$ ( $$K = \dfrac {1-e^{-\Delta / \tau_c}}{b}$$ ), the term $$e^{-\Delta / \tau_c}$$ is included in the range [0,1[ and the system is necessarely stable because it leads to $$0 < Kb < 1$$.

## Discrete-time function

The transfert function of the controller expressed in the discrete-time domain is given by:

$$\frac{u(z)}{\epsilon(z)} = K . \frac{z-a}{z-1}$$

The previous can be rewritten as:

$$(z-1)u(z) = K(z-a)\epsilon(z)$$ $$z.u(z) - u(z) = K.z.\epsilon(z) -K.a.\epsilon(z)$$ $$z.u(z) = K.z.\epsilon(z) - K.a.\epsilon(z) + u(z)$$

Let's expressed the previous equation in the discrete-time domain:

$$u_{n+1}= K\epsilon_{n+1} - K.a.\epsilon_n + u_{n}$$

or :

$$u_{n}= K\epsilon_{n} - K.a.\epsilon_{n-1} + u_{n-1}$$