Feedback Path Fuzzy-PID-like Incremental Servo Controller

This controller is mainly utilized to deal with the disturbances from the external load in early learning stages. The controller receives the error between the desired joint angles and the actual ones. It generates a control action, which is combined with the action from the feedforward controller to form the net torque (control action) applied to the joints of the robot.

Conventional PID controllers’ output is proportional to an error, the time derivative of the error and the integral o f the error. The controller employs a proportional control action to reduce the settling time and the rise time o f the plant response, a derivative control action to reduce the overshoot and the oscillations o f the plant response during transient conditions, and an integral control action to eliminate the steady state error during steady state conditions. This controller is easy to implement and sufficient tuning rules are available to cover a wide range o f plant specifications. For example,

the well known Ziegler-Nichols [Ziegler and Nichols, 1942] tuning method can be applied to estimate the controller gains based on the transient response characteristics of a given system. Moreover, the available PID tuning heuristics are easy to understand and implement for simple practical control problems. This controller is more effective for linear plants than for nonlinear plants, due to its linear control policy. As explained before, FLC have been used successfully in nonlinear control applications. They generally provide nonlinear transfer elements for nonlinear control. The majority of FPID applications belong to the direct-action FPID type where the direct-action FPID is placed within the feedback control loop to compute the control actions through fuzzy inference. Several direct-action FPID structures have been reported using one, two or three inputs (error, rate o f change o f error and integral o f error) [Mann et. al., 1999]. In all of these direct-action FPID controllers, the derivative and integral functions are performed quantitatively outside the FLC. They do not employ a FLS as a function approximator to perform a fuzzy integral or fuzzy derivative function. In these controllers, the FLS performs the nonlinear amplifications associated with the three PID control actions. For this work a new Fuzzy-PID controller [Shankir, 2001] is adapted with extended rules; this controller functionally performs fuzzy derivative and fuzzy integral functions, so that no calculations are required outside the FLC. The suggested fuzzy-PID-like incremental controller employs only two inputs (present and previous errors), so that the design procedure is simpler. Each element of the fuzzy- PID-like incremental controller can approximate the corresponding control function with separate nonlinear gain using five fuzzy set partitions (NL, NS, ZE, PS, and PL) for both input and output universes o f discourses. The input universe o f discourse of

triangular membership functions with 50% overlap to allow continuous approximation of input signals as shown in figure (4.2). The left most and the right most membership functions of the input universe o f discourse are saturated to unity membership value in the domain less than -2L and more than +2L respectively, where L is the distance between two consecutive membership functions centres. The output universe of discourse is uniformly partitioned using fuzzy sets defined by symmetrical triangular membership functions with 50% overlap as shown in figure (4.3). The left most and the right most membership functions o f the output universe o f discourse are both limited to the output minimum and maximum range o f operation in the domain less than -2L, and more than +2L, respectively. These minimum and maximum ranges in addition to controller gains are related to the maximum permissible servo torque applied to the robot joints. L, represents the distance between two consecutive output membership functions centres where, i is replaced by P, I, or D according to the proportional, integral, or derivative control element respectively.

NS ZE PS PL

NL

+L +2L

-2L L 0

a //P, jA, or /jD

ZE PS

NS PL

NL

P, I, or D

m ^n Value - 2 L p j 5 or D "L p,!, or D 0 + L p 5i5 or D + 2 L p j or D MCIX Value

Figure (4.3). Output membership functions o f fuzzy controller.

The proportional, derivative and incremental part of the integral control actions of a fuzzy-PID-like incremental controller are mainly functions o f the two present and past error variables, e r r(k t) and e r r ( k t - t ) , or their normalized variables, e(kt) and

e ( k t - t ) . Consequently,

U piD(kt) = f P(e (k t),e (k t-t)) + f D(e (k t),e (k t-t))

(4.1) + t / 7 (& /-/) + f j(e (k t},e (k t-ty )

where the three functions/ p if D, and / 7 are the proportional, derivative and incremental integral functions to be implemented using the fuzzy logic controller and

Uj (kt-t) is the past output o f the integral controller element. It was proved in [Wang and

Mendel, 1992] that fuzzy logic systems are universal approximators. Therefore, the three functions in equation (4.1) can be approximated using three two-input Fuzzy Control Elements (FCEs). Consequently, the outputs o f the three FCEs are summed

(4.4). In the following sections, the design o f the operation rules and implementation of the three functions in equation (4.1) in the form o f three fuzzy control elements are explained. Proportional FCE > f Integral FCE UD Derivative FCE

Figure (4.4). Structure o f the fuzzy servo controller.