Computation of robustly stabilizing PID controllers for interval systems
© The Author(s). 2016
Received: 13 April 2016
Accepted: 12 May 2016
Published: 20 May 2016
The paper is focused on the computation of all possible robustly stabilizing Proportional-Integral-Derivative (PID) controllers for plants with interval uncertainty. The main idea of the proposed method is based on Tan’s (et al.) technique for calculation of (nominally) stabilizing PI and PID controllers or robustly stabilizing PI controllers by means of plotting the stability boundary locus in either P-I plane or P-I-D space. Refinement of the existing method by consideration of 16 segment plants instead of 16 Kharitonov plants provides an elegant and efficient tool for finding all robustly stabilizing PID controllers for an interval system. The validity and relatively effortless application of presented theoretical concepts are demonstrated through a computation and simulation example in which the uncertain mathematical model of an experimental oblique wing aircraft is robustly stabilized.
The Proportional-Integral-Derivative (PID) control algorithms and their simplifications (P, I, PD and especially PI) comprise the great majority of contemporary industrial control applications. It has been reported that they represent over 90 % of all practically applied controllers in process control (Åström and Hägglund 1995; O‘Dwyer 2003). Thus, despite the existence of many more sophisticated control design methods and modern approaches (see e.g. (Selma and Chouraqui 2013) for the example of neuro-fuzzy control, (Ahmed et al. 2014) for the static synchronous series compensator based damping control, (Ibraheem et al. 2014) for automatic generation control, or (Shang 2016) for the stochastic consensus problems for multi-agent systems over Markovian switching networks with time-varying delays and topology uncertainties), the effective tuning of PI and PID controllers is still very topical because it can bring significant saving on energy as well as expenses. Evidently, the systematic research on the application of the PI(D) controllers under various conditions of uncertainty contributes to this mosaic.
Obviously, the stability is the first and most critical requirement of all control applications. However, the real-life control circumstances differ from the ideal nominal ones and so the uncertainty of the mathematical models has to be frequently taken into considerations. The attention of many researchers has been focused on the investigation of robust stability for systems with parametric uncertainty—see e.g. (Barmish 1994; Bhattacharyya et al. 1995, 2009; Matušů and Prokop 2011). Typical problem of practical PI(D) controller design is to ensure, that the calculated controller will guarantee stability not only for one assumed nominal controlled system but also for the whole family of systems described by a model with parametric uncertainty. Such closed-loop control system is called as “robustly stable” and the controller itself is then robustly stabilizing one.
An array of techniques for calculation of (nominally) stabilizing PI and PID controllers have been already published, such as rules presented in (Söylemez et al. 2003), the Tan’s method described in (Tan and Kaya 2003; Tan et al. 2006) or the Kronecker summation method from (Fang et al. 2009). Furthermore, these methods have been also extended for robust stabilization of interval plants by their combination with the sixteen plant theorem (Barmish et al. 1992; Barmish 1994). Nevertheless, this extension works only for PI but not for PID controllers.
The main aim of this paper is to present a method for computation of all possible robustly stabilizing PID controllers for interval plants and to demonstrate its serviceability by robust stabilization of an oblique wing aircraft model. More specifically, the goal is to refine the elegant and effective Tan’s method (Tan and Kaya 2003; Tan et al. 2006) by the ideas from (Ho et al. 1998, 2001), i.e. to use 16 segment plants instead of 16 Kharitonov plants, and to make the existing method applicable for computation of robustly stabilizing PID controllers. Previously, the computation of all (nominally) stabilizing PI or PID controllers, robustly stabilizing PI controllers and consequent choice of the specific controller with desired performance on the basis of the desired model method (formerly known as dynamics inversion method) (Vítečková 2000) is shown in (Matušů 2011). Then, the application of Kronecker summation method (Fang et al. 2009) to robust stabilization of a chemical reactor or robust stabilization of a third order nonlinear electronic model is given in (Matušů et al. 2011) or (Matušů et al. 2010a), respectively. The robust stabilization of the same nonlinear electronic plant using the Tan’s method (Tan and Kaya 2003; Tan et al. 2006) is presented e.g. in (Matušů et al. 2010b).
The paper is organized as follows. In “Computation of (nominal) stability regions for PI controllers” section, a graphical method for computation of (nominally) stabilizing PI controllers is recalled. “Computation of (nominal) stability regions for PID controllers” section has the same purpose but for PID controllers. Next, the computation of robustly stabilizing PI controllers for interval plants is presented in “Robust stabilization using PI controllers” section. The key “Robust stabilization using PID controllers” section extends the existing Tan’s (et al.) method, combines it with the segment plants concept and makes it applicable for calculation of robustly stabilizing PID controllers. Further, the extensive “Illustrative example: Robust stabilization of oblique wing aircraft” section confirms the obtained results by means of the simulation example with an experimental oblique wing aircraft model. And finally, “Conclusion” section offers some conclusion remarks.
Computation of (nominal) stability regions for PI controllers
First, the fundamentals related to computation of (nominal) stability regions for PI controllers are going to be summarized.
The obtained intervals could be helpful for the proper frequency scaling.
Computation of (nominal) stability regions for PID controllers
The principal idea for obtaining the relevant stability regions is to fix one controller parameter to a certain value and calculate the stability boundary locus using two remaining parameters analogously to the procedure presented in the previous “Computation of (nominal) stability regions for PI controllers” section.
Note that the last two terms in (9) depend on derivative constant k D . From the viewpoint of practical computation, k D is considered to be chosen and the corresponding set of boundary parameters k P , k I is consequently calculated while this process is repeated for several selected values of k D . Thus, the final stability regions are successively plotted through the “(k P , k I ) sections” in the (k P , k I , k D ) space.
Obviously, the final stability regions are given by the “(k P , k D ) sections” in the (k P , k I , k D ) space.
However, the stability region in the (k I , k D ) plane for a fixed k P can be acquired using the stability region in the (k P , k I ) plane and (k P , k D ) plane together as it has been presented in (Tan et al. 2006). In accordance with a linear programming based approach from (Ho et al. 1997), the stability region in the (k I , k D ) plane under fixed k P is a convex polygon which can be sometimes advantageous for easier plotting.
Robust stabilization using PI controllers
Consequently, the robust stabilization of an interval plant directly follows from the simultaneous stabilization of all 16 fixed Kharitonov plants. Hence, the final area of stability for original interval plant is given by the intersection of all 16 related partial areas obtained individually using the techniques from the “Computation of (nominal) stability regions for PI controllers” section.
Robust stabilization using PID controllers
Unfortunately, the sixteen plant theorem is not applicable for robust stabilization of interval systems by PID controllers as it is not valid anymore (Pujara and Roy 2001). However, the suitable method based on the generalized Kharitonov theorem and linear programming techniques has been presented e.g. in (Ho et al. 1998, 2001). This paper adopts the idea of Kharitonov segments used in the generalized Kharitonov theorem (Chapellat and Bhattacharyya 1989) and similar thirty-two edge theorem (Barmish 1994; Chapellat and Bhattacharyya 1989) and combines it with the stability boundary locus technique (Tan and Kaya 2003; Tan et al. 2006).
The family of interval systems (17) is stabilized by a fixed PID controller if and only if each of sixteen segment plants related to the interval family is stabilized by the same PID controller (Ho et al. 2001).
The computation of robustly stabilizing PID controllers can be performed as follows: First, a certain value of controller parameter k D is chosen and fixed (alternatively, also the parameter k I or k P can be fixed according to “Computation of (nominal) stability regions for PID controllers” section, but the fixed k D is supposed here). Then, the stability boundary for one of segment plants (18) is calculated for several sampled values of λ ∊ 〈0, 1〉 using the Eqs. (8) and (9). The intersection of the obtained areas in (k P , k I ) plane gives the stability boundary locus for this specific segment plant. The calculations are repeated for all the remaining segment plants and the robust stability region for the original interval plant and chosen value of k D is determined by the intersection of areas for all 16 segment plants. From the practical viewpoint, the curves for all sampled λ ∊ 〈0, 1〉 and all 16 segment plants can be plotted in one figure and intersection can be found at a time. Anyway, the whole process should be repeated for the other selected values of k D and the very final robust stability region can be visualized by the simultaneous plotting of the “(k P , k I ) sections” into one graph in (k P , k I , k D ) space.
Illustrative example: Robust stabilization of oblique wing aircraft
This Section is intended to practically demonstrate the theoretical results from the previous parts by means of the illustrative example.
In this part, all robustly stabilizing PID controllers are going to be found for the same oblique wing aircraft model (20).
The same process can be analogously repeated for the remaining 15 segment plants and then the intersection of all 16 partial intersections would lead to the stability boundary locus for the original interval family (20) under the assumption of k D = 1.
The main aim of paper has been to present the improved method for computation of stabilizing controllers with the conventional structure on the basis of plotting the stability boundary locus in either P-I plane or P-I-D space. Now, thanks to the combination of the original method with stabilization of so-called segment plants, the modified technique can be conveniently used for determination of all possible robustly stabilizing PID controllers for interval plants. In the illustrative example, the model of an experimental oblique wing aircraft is considered as a controlled object. Two final robust stability regions have been computed and visualized, one for PI and the other for PID controller, and selected representatives from stable or even intentionally unstable areas have been chosen and used for supporting control simulations.
RM did the literature review, designed the extension of the existing method, constructed and performed the simulations and wrote the manuscript. RP contributed to the basic ideas behind the paper. Both authors read and approved the final manuscript.
This work was supported by the European Regional Development Fund under the project CEBIA-Tech Instrumentation No. CZ.1.05/2.1.00/19.0376. This assistance is very gratefully acknowledged.
A preliminary version of this paper was presented at the 8th IFAC Symposium on Robust Control Design, Bratislava, Slovak Republic, 2015 (Matušů and Prokop 2015).
The authors declare that they have no competing interests.
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