Development of the adaptive control system described in this chapter relied on a computer simulation of the problem environment. Such a computer simulation was described in Chapter 3 where the numerical integration of the governing equations of motion was discussed. The same computer simulation is used here but the system parameters can now be altered. For the reader’s benefit, Table 9.1 lists both the system parameters and state variables associated with the cart-pole system.

Table 9.1 System parameters and state variables for the cart-pole system

System parameters:
m	=	cart mass
l	=	pole length
mc	=	mass of cart;
mp	=	mass of pole;
l	=	length of pole;
μc	=	coefficient of friction of cart on track;
μp	=	coefficient of friction of pole on cart.
State variables:
x	=	cart position;
	=	cart velocity;
θ	=	angle of the pole with respect to the vertical;
	=	angular velocity of the pole.

The model equations are the basis of a mathematical model for the cart-pole system used to evaluate the effectiveness of potential fuzzy control strategies in Chapter 7. In the current chapter, this model is used for much more. The computer model is used by the analysis element to determine the magnitude of changes to the system parameters and by the learning element to investigate new and improved control strategies.

In the current effort, the control problem is made considerably more difficult by considering a system in which an unmeasured parameter of the system can change with time; a time-varying control problem. The parameter can change by a large amount. For instance the cart mass can increase by as much as 500 percent. These changes significantly alter the response of the cart-pole system to a given force stimulus. If the system parameters change too rapidly, we will not have time to determine the extent of the changes and update our control strategy accordingly.

9.3 Alternative to Additional Rules

There are three main tasks that have to be accomplished by an adaptive, autonomous control system: (1) it must effectively manipulate the problem environment driving it from an arbitrary state toward a desired state, (2) it must recognize when the problem environment has changed in a manner that requires an adjustment to the control strategy (the rules and membership functions) being employed to accomplish the task of manipulating the problem environment, and (3) it must alter the control strategy in response to the new problem environment. A three-tiered computer software architecture has been developed to accomplish these three tasks. Together, GAs and fuzzy controllers include all of the capabilities necessary to produce powerful, efficient, and robust adaptive control systems and therefore form the core of the computer software architecture. To perform efficiently, such control systems require a control element to manipulate the problem environment, an analysis element to recognize changes in the problem environment, and an adaptive element to adjust to the changes in the problem environment. The remainder of this chapter describes the use of a GA in the development of a process control system that includes each of the three elements.

Figure 9.3 shows the adaptive process control system. The top two elements of Figure 9.3 are the elements of a traditional feedback control loop. The control element receives information from sensors in the problem environment concerning the status of the condition variables, i.e., x, , θ, and . It then computes a desirable state for a set of action variables, i.e., force to be added to the cart (F). Changes in the action variable forces the problem environment toward the setpoint (x = = θ = = 0). This is the basic approach adopted for the design of virtually any closed loop control system. However, this basic loop includes no mechanism for adaptive control.

Figure 9.3  The computer software architecture shown above allows for effective, robust, adaptive process control systems.

The adaptive capabilities of the system shown in Figure 9.3 are due to two new elements: (1) the analysis element and (2) the learning element. In general, the analysis element must recognize when a change in the problem environment has occurred. A “change,” as it is used here, consists of any alteration of a system parameter that affects the response characteristics of the cart-pole system. Further, the change must occur to a parameter that is not included directly in the rule set. The analysis element uses information concerning the condition and action variables over some finite time period to recognize changes in the environment and to compute the new parameters appropriate to these changes.

Control Element

The control element receives feedback from the cart-pole system, and based on the current state of x, , θ , and, must prescribe appropriate values of F. Any of a number of closed-loop controllers could be used for this element. However, since we have a perfectly good fuzzy controller for the cart-pole system, we use it here for the control element. The details of the cart-pole fuzzy controller are found in Chapters 3 and 7.