Design and Modeling of Anti Wind Up PID Controllers

Abstract

In this chapter several anti windup control strategies for SISO and MIMO systems are proposed to diminish or eliminate the unwanted effects produced by this phenomena, when it occurs in PI or PID controllers. Windup is a phenomena found in PI and PID controllers due to the increase in the integral action when the input of the system is saturated according to the actuator limits. As it is known, the actuators have physical limits, for this reason, the input of the controller must be saturated in order to avoid damages. When a PI or PID controller saturates, the integral part of the controller increases its magnitude producing performance deterioration or even instability. In this chapter several anti windup controllers are proposed to eliminate the effects yielded by this phenomena. The first part of the chapter is devoted to explain classical anti windup architectures implemented in SISO and MIMO systems. Then in the second part of the chapter, the development of an anti windup controller for SISO systems is shown based on the approximation of the saturation model. The derivation of PID SISO (single input single output) anti windup controllers for continuous and discrete time systems is implemented adding an anti windup compensator in the feedback loop, so the unwanted effects are eliminated and the system performance is improved. Some illustrative examples are shown to test and compare the performance of the proposed techniques. In the third part of this chapter, the derivation of a suitable anti windup PID control architecture is shown for MIMO (multiple input multiple output) continuous and discrete time systems. These strategies consist in finding the controller parameters by static output feedback (SOF) solving the necessary linear matrix inequalities (LMI’s) by an appropriate anti windup control scheme. In order to obtain the control gains and parameters, the saturation is modeled with describing functions for the continuous time case and a suitable model to deal with this nonlinearity in the discrete time case. Finally a discussion and conclusions sections are shown in this chapter to analyze the advantages and other characteristics of the proposed control algorithms explained in this work.

Appendices

Appendix 1

In this appendix the internal model PID controller, explained in Sect. 3 the gain and time constants are found with the following equations.

Define:

$$D(s) = ((\lambda s + 1)^{r} - P_{1A} (s))/s$$

(83)

and

$$K_{p} = p_{1m} (0)$$

(84)

Then the following gain and time constants are obtained using (15) with the following equations of the function f(s) and its derivatives (Lee et al. 1998):

$$f(0) = \frac{1}{{K_{p} D(0)}}$$

(85)

where D(0) is

$$D(0) = r\lambda - \dot{P}_{1A} (0)$$

(86)

the derivatives of D(0), $\dot{D}(0)$ and $\ddot{D}(0)$ are shown in (Lee et al. 1998). Then the derivative of f(0) is given by:

$$\begin{aligned} \dot{f}(0) & = \left( {\frac{{K\left( {\alpha + \beta {\kern 1pt} {\varDelta }_{\phi } } \right)a_{1} }}{{a_{0} - \alpha - \beta {\kern 1pt} {\varDelta }_{\phi } }} - \frac{{K\left( {\alpha + \beta {\kern 1pt} {\varDelta }_{\phi } } \right)a_{0} a_{1} }}{{\left( {a_{0} - \alpha - \beta {\varDelta }_{\phi } } \right)^{2} }} - \frac{{K\left( {\alpha + \beta {\kern 1pt} {\varDelta }_{\phi } } \right)a_{0} \tau }}{{a_{0} - \alpha - \beta {\kern 1pt} {\varDelta }_{\phi } }}} \right)\left( {r\lambda + \theta } \right)^{ - 1} K_{p}^{ - 2} \\ & \quad + \frac{{1/2{\kern 1pt} r\left( {r - 1} \right)\lambda^{2} - 1/2{\kern 1pt} \theta^{2} }}{{K_{p} {\kern 1pt} \left( {r\lambda + \theta } \right)^{2} }} \\ \end{aligned}$$

(87)

$$\ddot{f}(0)=\dot{f}(0)\left ( \left ( \frac{\ddot{p}_{1m}(0)D(0)+2\dot{p}_{1m}(0)\dot{D}(0)+K_{p}\ddot{D}(0)}{\dot{p}_{1m}(0)D(0)+ K_{p}\dot{D}(0)} \right ) + 2\dot{f}(0)/f(0) \right )$$

(88)

Appendix 2

In this appendix the internal model PID controller, explained in Sect. 4 the gain and time constant are found and shown in the following equations. Consider the following representation in Taylor series of the digital PID controller (31) based on the analog controller design shown in (Lee et al. 1998)

$$G_{c} (s) = \frac{f(z)}{z - 1} = \frac{1}{z - 1}(f(1) + f^{\prime}(1)(z - 1) + \frac{{f^{\prime\prime}(1)}}{2}(z - 1)^{2} + \cdots )$$

(89)

Due to $G_{c} (s) = \frac{f(z)}{z - 1}$ the following equation can be considered:

$$D(z) = \frac{{(z - \alpha ) - P_{\gamma A}^{*} (1 - \alpha )z}}{z - 1}$$

(90)

because of (30) can be represented by:

$$G_{c} (z) = \frac{{(1 - \alpha )zP_{\gamma M}^{* - 1} }}{{(z - \alpha ) - P_{\gamma A}^{*} (1 - \alpha )z}}$$

(91)

The design procedure of the discrete time SISO controller is similar to the continuous time SISO case, (Lee et al. 1998) where (90) can be represented by:

$$D(z) = \frac{N(z)}{z - 1}$$

(92)

where

$$N(z) = (z - \alpha ) - P_{\gamma A}^{*} (1 - \alpha )z$$

(93)

Then by the Taylor series expansion of D(z) the following equation is obtained:

$$D(z) = \frac{1}{z - 1}(N(1) + N^{\prime}(1)(z - 1) + \frac{{N^{\prime\prime}(1)}}{2}(z - 1)^{2} + \frac{{N^{\prime\prime\prime}(1)}}{6}(z - 1)^{3} + \cdots )$$

(94)

Considering that N(1) = 0, (94) becomes in:

$$D(z) = N^{\prime}(1) + \frac{{N^{\prime\prime}(1)}}{2}(z - 1) + \frac{{N^{\prime\prime\prime}(1)}}{6}(z - 1)^{2} + \cdots$$

(95)

Expanding D(z) in Taylor series expansion as an only term, the following result is obtained:

$$D(z) = D(1) + D^{\prime}(1)(z - 1) + \frac{{D^{\prime\prime}(1)}}{2}(z - 1)^{2} + \cdots$$

(96)

Then associating the similar terms of (95) and (96) the following values for D(1) and its derivatives are obtained:

$$\begin{aligned} D(1) & = N^{\prime}(1) \\ D^{\prime}(1) & = N^{\prime\prime}(1)/2 \\ D^{\prime\prime}(1) & = N^{\prime\prime\prime}(1)/3 \\ \end{aligned}$$

(97)

the values of D(1) and its derivatives can be found by:

$$D(1) = 1 + (N - 1)(1 - \alpha )$$

(98)

$$D^{\prime}(1) = ( - N(N - 1)(1 - \alpha ))/2$$

(99)

$$D^{\prime\prime}(1) = ((N + 1)N(N - 1)(1 - \alpha ))/3$$

(100)

Then the values for f(1) and its derivatives are found by (Lee et al. 1998):

$$f(1)=\frac{1}{(p^{*}_{\gamma M}(1)/(1 - \alpha))D(1)}$$

(101)

$$f'(1)=-\frac{(p'^{*}_{\gamma M}(1)/(1 - \alpha))D(1) + (p^{*}_{\gamma M}(1)/(1 - \alpha))D'(1)}{((p^{*}_{\gamma M}(1)/(1 - \alpha))D(1))^2}$$

(102)

$$f''(1)= f'(1) \left( \frac{(p''^{*}_{\gamma M}(1)/(1 - \alpha))D(1)+2(p'^{*}_{\gamma M}(1)/(1 - \alpha))D'(1) + (p^{*}_{\gamma M}(1)/(1 - \alpha))D''(1)}{(p'^{*}_{\gamma M}(1)/(1 - \alpha))D(1) + (p^{*}_{\gamma M}(1)/(1 - \alpha))D'(1)} \right) + 2f'^{2}(1)/f(1)$$

(103)

With f(1) and its respective derivatives, the parameters of the digital PID controllers can be found using (32).