Robust PID controller design for nonlinear processes using JITL technique
Introduction
For most chemical processes, the first-principle models are usually unavailable because of the lacking of physicochemical knowledge. An attractive alternative for controller design is to rely on the models extracted from process input and output measurements. These models generally have varying degrees of accuracy. If the plant/model mismatch is not taken into account in controller design, the resulting control performance may become poor or even unstable in the presence of significant modeling error. This design problem has motivated the researchers to pursue various robust controller design methods in the last two decades (Morari and Zafiriou, 1989, Malan et al., 2004). The objective of robust controller design is to ensure closed-loop stability and maintain control performance not only for the nominal model but also for a set of possible process models that captures the actual process dynamics. For linear system, various robust control problems had been tackled by using the transfer function model approaches (Morari and Zafiriou, 1989, Packard and Doyle, 1993). Normally, a given set of process models is represented by a nominal model together with a suitable uncertainty description equation to account for the modeling error between the nominal model and actual process dynamics. The associated design issue of estimating the uncertainty models also attracted much research investigations (Gustafsson and Makila, 2001, Boling et al., 2004).
Some robustness analysis results developed for linear systems have also been applied to the nonlinear systems. For example, Doyle et al. (1989) proposed a robust controller design method for a nonlinear CSTR for which a first-principle model is assumed to be available. By assuming that the process/model mismatch is entirely due to the nonlinearities of the process, bounds on the conic sectors that describe the process nonlinearities were developed and used in the standard structure for robust stability analysis. However, the identification of the conic bounds is not trivial and the resulting robust stability analysis tends to give conservative result, not to mention that first-principle models are generally not available for many chemical processes. To lessen the modeling requirement, Knapp and Budman (2000) developed a robust PID controller design methodology for nonlinear processes using an empirical state affine model developed from the available process data, which can be readily transformed into a suitable form for the robust stability analysis. Although their result provides an attractive alternative to the Doyle's approach, the construction of the state affine models is rather tedious and computational demanding. According to Knapp and Budman (2000), in order to obtain a state affine model, a NARMA model is initially constructed from the available process input and output data. Subsequently, an algorithm developed by Diaz and Desrochers (1988) is employed to find the parameters for a truncated Volterra model based on the NARMA model identified previously. Once the Volterra kernels are obtained, a generalized Hankel matrix can be developed to find a state affine model (Sontag, 1979). Obviously, the modeling efforts required to identify a state affine model are extensive and thus hampers the application of robust controller design method developed based on such a model.
To circumvent the aforementioned drawbacks by using the conic sectors and state affine models in robust controller designs for nonlinear systems, a robust PID controller design methodology using the just-in-time learning (JITL) technique (Cybenko, 1996, Atkeson et al., 1997, Bontempi et al., 2001) is developed in this paper. It is noted that robust PID controller design methods have been developed using different techniques in the literature, for example linear matrix inequalities (LMIs) approach in the multiple-model modeling framework (Ge et al., 2002, Toscano, 2007) and numerical optimization approach by solving the maximization of the shortest distance from the Nyquist curve of the open-loop transfer function to the critical point (Toscano, 2005). However, these methods require a priori process knowledge or ad-hoc procedure to address the associated design issue of partition of the operating space in the multiple-model approach and determine the worst-case model (or optimal model for that matter) in the operating space of interest in order to obtain the best viable PID design in the numerical optimization approach. In contrast, the proposed design method is a one-shot design approach in the sense that the PID parameters are obtained directly from a set of process data characterizing the process dynamics of interest. In this respect, the proposed method is more advantageous than the previous model-based robust PID design methods because the required a priori process information may not be readily available in practical applications, leading to the tedious trial and error design procedure. In the proposed method, it is assumed that the process nonlinearity is the only source of the model uncertainty. A composite model consisting of a nominal ARX model and the JITL, where the former is used to capture linear process dynamics and the latter is applied to approximate the inevitable modeling error caused by the process nonlinearity. The state space realization of this composite model is then reformulated as an uncertain system, by which the robust stability condition of this uncertain system under PID control is developed. Literature examples are used to illustrate the proposed method and a comparison with the previous result is made.
Section snippets
Modeling methodology
In this paper, a composite model consisting of a nominal ARX model and JITL models is used to describe the nonlinear process in the operating range of interest, where the former can be identified by using the process input and output data around a nominal operating condition and the latter is used to capture the modeling error caused by the process nonlinearity, i.e. the difference between the predicted output of nominal ARX model and actual process output. Suppose that an input sequence
Robust stability analysis
Since PID controller is widely used in the process industries, it is considered in the ensuing robust stability analysis. To do so, PID controller is represented by the following state space equation:where is a state variable vector of PID controller, is the tracking error, i.e. the difference between the set-point and process output, and other model parameters are given bywhere , , and are
Examples
Example 1 Consider a CSTR described by the following equations (Doyle et al., 1989):where and are the dimensionless concentration and temperature of the reactor, and is the cooling temperature chosen as manipulated variable while is the controlled variable. The process has one stable steady state when , , , and . The following operating space and is considered in this
Conclusion
A new methodology for robust PID controller design of nonlinear processes is developed. In the proposed method, a composite model is first constructed to model the process dynamics for the operating space of interest. This composite model consists of a nominal ARX model to capture linear process dynamics and the JITL to approximate the modeling error caused by process nonlinearity. The state space realizations of this composite model and PID controller are then reformulated as an uncertain
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