Fuzzy PID Control

As explored by Cerecero-Natale et al. (2012), the past three decades have been prolific in the field of fuzzy logic controllers; not only has a great deal of work gone into these types of controller, but their applications have broadened and come to include PID fuzzy gain scheduling. Classical continuous gain scheduling is the notion of switching between PID control parameters, in an effort to more accurately control a system with a non-linear response. In essence, this establishes a non-linear controller through the splicing of a number of linear controllers (Naus, 2009).

Traditional gain scheduling implementation as described by Naus, (2009) consists of four main steps. The first step involves deriving a number of linear approximations from a non- linear system at constant operating points, characterised by measureable and changeable variables. From there, the second step calls for the construction of multiple controllers that satisfy performance and stability requirements for the previously acquired set of models at their respective operating points. Step three then implements continuous scheduling of the gain coefficients of the PID controller, corresponding to the previously found linear controllers, based on the current operating point of the system. The final step is simply performance assessment; the system is tested for local and global performance and robustness analytically. Criteria for good system behaviour may be established by the users’ desired control outcome.

It should be noted that Naus, (2009) has identified a ‘chattering behaviour’ which may occur when implementing this type of control incorrectly. This refers to observed jerky performance as the controller switches between control regions, which may be a result of a large jump in coefficient size. Naus, (2009) states that this can be avoided through linearization of the schedule; additionally, increasing the number of regions will provide smoother control region transitions.

The difficulty with this type of control arises when deciding on how to schedule the varying gains and to what degree. One approach is fuzzy logic.

A paper published by Cerecero-Natale et al. (2012), has evaluated the use of fuzzy logic for gain scheduling on a PI controller for position control of a mechatronic system. Implementation and robustness of this type of control was assessed through both simulation and experimentation. A Two Input Two Output (TITO), 5 linguistic term (fig. 2 - 81) fuzzy controller was used where error and error derivative serve as inputs and proportional gain ( ) and integral gain ( ) as outputs. Cerecero-Natale et al. (2012) defined the fuzzy rule set as per Figure 2 - 82, comprising of 25 AND logic operator rules.

Figure 2 - 82. Fuzzy rule set. (Cerecero-Natale et al., 2012).

For convenience, error and error derivative were normalised by Cerecero-Natale et al. (2012) as follows:

= −

−

̇ = ̇ − ̇

̇ − ̇

Figure 2 - 83. Error and error derivative equations. (Cerecero-Natale et al., 2012).

The membership function plot (fig. 2 - 81) and fuzzy rule set (fig. 2 - 82) result in non- linear control surfaces for outputs (fig. 2 - 84) and output (fig 2 - 84) after applying a centroid defuzzifier (fig 2 - 80).

Figure 2 - 84. Resulting control surfaces. (Cerecero-Natale et al., 2012).

Figure 2 - 85. Fuzzy PI control set-point response. (Cerecero-Natale et al., 2012).

Figure 2 - 86. Proportional and integral gain responses. (Cerecero-Natale et al., 2012).

Once all parameters needed for the proposed fuzzy logic PI controller were established, Cerecero-Natale et al. (2012) used a first order mathematical system for comparing the simulated behaviour of a traditional PI controller and the proposed controller (fig. 2 - 87). The Ziegler-Nichols method was used to determine fixed controller gains and .

Figure 2 - 87. FLC PI vs. Classical ZM PI set-point response. (Cerecero-Natale et al., 2012).

Although this paper claims to have implemented this system on a light-weight, 8 bit PIC 18F4550 microcontroller (Cerecero-Natale et al., 2012), no significant amount of experimental results are provided. Cerecero-Natale et al. (2012) concluded that the main point of response control improvement associated with the fuzzy logic gain scheduling PI controller is its ability to reach the desired set point in less time (fig. 2 - 87), while reliably avoiding overshoot. It is recommended that a Gaussian membership such as Figure 2 - 81 is used, as this will provide smoother control in comparison to triangular memberships (fig. 2 - 78).

In a similar paper produced by Zhao et al. (1992), fuzzy logic gain scheduling of PID controllers are described in a broader sense. Zhao et al. (1992) recommend using a discrete-time equivalent expression for the PID algorithm, referred to as the standard

form in the PID section 2.7.1 (fig. 2 - 62). For simulation, this paper has decided to use a 7- term linguistic membership function, with error and error derivative as inputs, yielding two outputs, and . is calculated with reference to the derivative time constant (fig. 2 - 88).

= =

( )= ( )

Figure 2 - 88. Integral term calculation. (Zhao et al., 1992).

The fuzzy rule set comprises of 49 rules, similarly defined by Cerecero-Natale et al. (2012), producing similar membership function output plots. Inputs and outputs are also normalised in a similar fashion. Although full PID is used in this fuzzy gain scheduling scheme by Zhao et al. (1992), in this case simulation does not seem to suggest a significant improvement over traditional, Ziegler-Nichols tuned PID control (fig. 2 - 89) when dealing with a 3rd _{order process.}

Figure 2 - 89. Controller 3rd order response. (Zhao et al., 1992).

Zhao et al. (1992) briefly touch on the concept of a ‘hybrid controller’. It suggests that in some situations it may be viable to use dynamic fuzzy controlled PID in tandem with traditional fixed gain PID. For large set point changes, fuzzy PID is employed to utilise its quick transient response, as error reduces and the system begins to settle, the controller is then switched to fixed gain PID. Zhao et al. (1992) states that sometimes set PID parameters may be excellent for good set point response, but lack in appropriate performance for load disturbance rejection.