Controller design by symbolic regression
Introduction
Controllers are usually defined for a formulaic plant model (i.e., represented by differential equations) and are analytic. When plants are linear or can be approximated by linear models, linear controllers of standard form (e.g., lead, lag, or state feedback) can be considered, with the controller parameters determined according to a design method, such as root locus, pole placement, or frequency response [1]. For nonlinear plants, nonlinear controllers may be considered and designed using a method such as feedback linearization, Lyapunov’s direct method, sliding mode, or back-stepping [2], [3], [4]. The noted feature of analytic controllers is their legible form which makes them transparent in how their output is computed in relation to the controller inputs (see A for the intended definition of controller transparency).
Analytic control, however, has its limitations: (1) it requires an analytic plant model, (2) it needs expertise, especially for nonlinear controller design, and (3) its form is confined to established controller forms (e.g., state feedback). These limitations have motivated the application of empirical design methods, whereby both the controller structure and its parameters are derived through learning. Noted examples of such empirical methods are fuzzy control [5] and neuro-control [6], [7], [8], [9]. The advantage of these controllers is that they can be developed without much control design expertise. Their disadvantage is their “black-box” form which offers little insight about their structure. The objective of this work is to rectify the black-box characteristic of empirically derived controllers, by deriving through learning empirical controllers that are in analytic form.
To produce legible controllers, we use symbolic regression (SR) [10], [11], [12] to derive controllers in analytic form (see B for a general overview of SR). Briefly, SR represents the time-series associated with loop variables such as the plant states as symbols and integrates them as blocks to form candidate equations that produce as output a time series close to the target output [13]. As such, these equations are transparent in their structure by informing how their output relates to their inputs. If these equations qualify as controllers, their transparency provides insight into their form, offering the possibility of discovering new controller forms. For instance, an equation found by SR (shown later in Table 1) defines the controller output u as , where denotes the velocity. Despite its empirical basis, the controller is transparent in how its output is computed. The objective of this paper is to make controller design by symbolic regression possible.
Ideally, the SR search, free of restrictions in form/structure, can be conducted by genetic programming (GP) for equations that tested as controllers generate closed-loop outputs close to a target output. However, SR is computationally demanding, requiring anywhere from millions to billions of evaluations. If the candidate controllers were to be evaluated for their performance in closed-loop, as traditionally conducted in empirical controller design, then each evaluation would require a time-consuming closed-loop simulation. But it is unfeasible to conduct so many simulation-based evaluations of candidate controllers during a genetic search, rendering closed-loop evaluation of candidate controllers impractical. Because of this computational impediment, the use in controls of evolutionary and/or genetic algorithms has been confined to parameter optimization [14], [15] or search among a limited number of structural components [16], [17], [18], leaving unexplored the true benefit of SR in unrestricted search of controller forms.
An alternative to time-consuming closed-loop evaluation of candidate controllers by simulation is algebraic evaluation, which can be much more rapidly obtained. Such algebraic evaluation would be possible if the desired plant input were available to use as target for the candidate controller output. But, even with the availability of the desired plant input, and derivation of the candidate controllers by SR, it remains to be determined, as a fundamental hypothesis, whether such candidate controllers designed in open-loop would function as controllers in closed-loop. The paper examines this hypothesis by (1) offering a method of inverse solution to provide the plant input for a desired plant output, (2) deriving candidate controllers by SR, and (3) evaluating the performance of these candidate controllers in closed-loop. The above strategy is illustrated in Fig. 1.
As to the inverse solution, the first contribution of this work (termed “Input Estimation” in Fig. 1 and elsewhere in the paper), it is trivial for a linear plant since it can be computed using the discrete-time transfer operator of the plant as a difference equation. It incorporates the target output in the auto-regressive part so as to compute the plant input from the moving average part. For a nonlinear plant, however, the solution requires a second phase, which also relies on the linearized model of the plant. In the first phase, as for the linear plant, the approximate input to the linearized model of the plant is obtained with the desired output as the target. In the second phase, this input is adapted in a method akin to iterative learning control (ILC) [19], [20] toward an input that renders the target output by the nonlinear plant.
Once the desired/target plant input becomes available, the potential controller can be constructed by SR to yield as its output the estimated plant input. SR uses the space of plant states and inputs as time-series to search for this potential controller (see Fig. 1). For SR, we use Epigenetic Linear Genetic Programming (ELGP) [21], [22], [23], [24] which is developed to yield concise models. Equations that yield as output the target plant input (representing the inverse solution to the desired plant output) may qualify as controllers and will be tested via simulation in closed-loop.
We present the results from application of the proposed SR-Based Controller Design (SRBCD) method to three nonlinear plants. The results not only affirm the fundamental hypothesis of the study by confirming the efficacy of controllers generated by the SRBCD method, they provide valuable insight about the choice of variables to be included in SR to render an effective controller form for closed-loop operation. These results are, therefore, significant in not only validating the proposed method but also affirming the possibility of controller design in open-loop, whereby controllers can be designed with the objective of producing as output the desired plant input. Even though the proposed method of open-loop controller design, together with its input estimation, is examined and validated with controllers derived by SR, the open-loop design strategy, in and of itself, ought to be considered the main contribution of the present work. The justification for this claim is rooted in the paper’s finding that controllers can be designed in open-loop to target the desired plant input. Such a strategy may be implemented, independent of SR, with any design strategy, including linear regression.
The paper is organized as follows. The input estimation method, which is an integral part of the proposed open-loop controller design strategy, is discussed in Section 2. A brief description of ELGP, which is an efficient method of search developed by the authors for creating concise dynamic models [23], [24], is presented in Section 3. Section 4 presents the results from the application of the proposed method to three nonlinear plants, selected for the various challenges they pose to controller design. Some of the controllers presented in this section manifest the uniqueness of controller forms derived by SR, potentially unattainable by traditional design techniques. A necessary step to open-loop design, is stability analysis. An example of such analysis is provided in Section 5 for one of the studied cases via loop gain describing functions. Discussion of the results, future work, and conclusion are discussed in Sections 6 Discussion, 7 Future work, 8 Conclusion, respectively.
Section snippets
Plant input estimation
Estimation of plant input for a desired plant output, commonly referred to as inverse modeling [25], is generally hindered by non-minimum phase characteristics and nonlinearities [26]. We avoid here the construction of an inverse model [27] and focus instead on estimating the input (i.e., the inverse solution) alone, independent of the inverse model that would generate it. To this end, we rely on the discrete-time model of the plant as a difference equation.
Consider the transfer operator of the
Symbolic regression by ELGP
In traditional applications of SR to dynamic modeling, analytical models are developed according to building blocks that comprise the state variables, inputs, constants (coefficients and exponents), and algebraic functions and operators. Symbolic regression generally uses genetic programming (GP) to search for the nonlinear differential equations that can fit the observations [10], [28], [29], [30]. Classical GP [13] represents the equations by tree structures, although linear representations
Study platforms
A salient feature of the SRBCD method is its capacity to present novel and exotic controller forms. As such, the potential value of SRBCD is in application to nonlinear plants, since it is unlikely to render controllers of superior performance to the linear controllers already available for linear plants. As such, we present here only the results from the SRBCD’s application to nonlinear plants. Several controllers are developed for each of the three plants. Two of the derived controllers are
Stability analysis
For illustration purposes, stability analysis is performed on the inverted pendulum for which several controllers are available for comparison. The input/output (IO) sinusoidal-input describing functions (SIDFs) (see D for further explanation) were obtained via simulation for the closed-loop systems comprising the controllers of the inverted pendulum in Table 4. To ensure steady state response, the last twelve cycles of each output were used to compute their Fourier transforms. The magnitude of
Discussion
The two components of SRBCD are input estimation/adaptation and controller construction (see Appendix C for more details). Some of the issues observed during the implementation of these components are briefly discussed below.
Future work
The proposed SRBCD method is in its early stages of development, so its results, although promising, cannot yet address all the issues of concern in a practical control method. Our thoughts about addressing some of these issues are expressed below.
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Robustness analysis of the controllers. The controllers produced by SRBCD are tested on plants having the same parameter values as during input estimation and controller construction. It behooves us, therefore, to evaluate the robustness of these
Conclusion
A novel method of empirical controller design is introduced with the potential to produce unique controller forms by symbolic regression (SR). To make SR applicable in controller design, time-consuming closed-loop evaluation of controller candidates is replaced by algebraic evaluation. For this purpose, the desired plant input is estimated and used as target for derivation of candidate controllers, testing, in effect, the following hypothesis: Given the availability of the plant input that
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
The development of ELGP was partially supported by the NSF-sponsored IGERT: Offshore Wind Energy Engineering, Environmental Science, and Policy (Grant No. 1068864).
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