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Ind. Eng. Chem. Res. 1996, 35, 746-757
Nonlinear Control of a Fluidized Bed Reactor Using Approximate Feedback Linearization Thomas A. Kendi and Francis J. Doyle III* School of Chemical Engineering, Purdue University, West Lafayette, Indiana 47907-1283
Two approaches to approximate feedback linearization are applied to the nonlinear control of a fluidized bed reactor. The first approach, third order approximate input-state linearization, results in improved performance for setpoint tracking over a narrow region around the operating point. On the basis of an analysis of the assumptions of approximate input-state linearization, an approximate input-output linearization scheme is proposed. Implementation of a control scheme for the fluidized bed reactor based upon this approach resulted in excellent setpoint tracking over a 30 °C temperature range in the reactor. The issues of nonlinear model selection and computational issues are discussed. Finally, a novel internal model control based nonlinear anti-windup scheme is implemented to ensure that improved performance is obtained without violating hard actuator constraints. 1. Introduction
Given the following control affine model structure:
Over the past years several feedback linearization techniques, such as input-state linearization and inputoutput linearization, have received considerably attention in the literature [26]. These approaches, based upon the tools of differential geometry, decouple the nonlinear control problem into the design of an optimal linear controller and a nonlinear compensator. Two difficulties often encountered in the design of the nonlinear compensator are (i) restrictive applicability conditions and (ii) the computational complexity involved in the design for large problems. One approach to reducing these difficulties is to use approximate linearization techniques. In this paper, two approximate linearization approaches, approximate inputstate linearization [28] and approximate input-output linearization [11], will be applied to a fluidized bed reactor. Also, a new nonlinear anti-windup scheme is proposed and applied to the fluidized bed reactor. 1.1. Input-Output Linearization. Input-output linearization is a control synthesis technique which involves a change of coordinates, u ) ψ(x,v), to be constructed such that the mapping between the original system output, y, and the new input, v, is linear. One of the earliest formulations of input-output linearization, referred to as Volterra linearization, was proposed by Isidori and Ruberti [18]. Below, the basic structure and applicability conditions of input-output linearization for SISO systems will be summarized. The interested reader is referred to the original references for full details [17, 24, 39]. Input-output linearization can be applied to a system if and only if the following two conditions are upheld [26]: 1. The system must have a well-defined relative degree. 2. The system must be minimum phase. The relative degree of a system is the number of times that the output must be differentiated until the input appears explicitly. A more rigorous mathematical definition is as follows: * To whom correspondence should be addressed. E-mail:
[email protected].
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x˘ ) f(x) + g(x)u
(1)
y ) h(x)
(2)
The relative degree of the system described in eqs 1 and 2 is defined as the integer r which satisfies
LgLi-1 f h(x) ) 0
∀i
