IMC-PID controller design based on loop shaping via LMI approach
Graphical abstract
Introduction
PID controllers have been utilized for many purposes due to their simple structure, acceptable performance, and satisfactory robustness (Åström and Hägglund, 1995). The literature shows a variety of approaches to tuning PID controllers. A few examples are the internal model control (IMC) method (Skogestad, 2003, Grimholt and Skogestad, 2012, Shamsuzzoha, 2016, Kumar and Sree, 2016, Tan et al., 2003, Vu and Lee, 2013, Vuppu et al., 2015, Jin and Liu, 2014a, Jin and Liu, 2014b), the loop shaping method (Grassi et al., 2001, Hara et al., 2006, Saeki, 2009), and methods based on linear matrix inequality (LMI) (Huang and Huang, 2004; Wang et al., 2007; Wu et al., 2011; Boyd et al., 2016; Souza et al., 2016). The most interesting approaches to IMC are Skogestad IMC (SIMC), which was proposed by Skogestad (2003), and improved SIMC (Grimholt and Skogestad, 2012). To enhance disturbance rejection, Shamsuzzoha (2016) developed a closed-loop tuning method for the IMC-PID controller using the set-point step test. In a recent paper by Kumar and Sree (2016), a set-point filter was designed to enhance the servo response of the controller. A set-point filter was employed in many studies (Tan et al., 2003, Vu and Lee, 2013, Vuppu et al., 2015) to improve the servo response of the closed-loop system. IMC-PID tuning rules based on model matching approach and closed-loop shaping was presented by Jin and Liu, 2014a, Jin and Liu, 2014b.
The method proposed by Grassi et al. (2001) drew upon the principles of loop shaping and integrated system identification and PID controller tuning. Hara et al. (2006) introduced a method for designing PID controllers with the purpose of fulfilling multiple frequency domain inequalities in the open-loop transfer function in finite and semi-infinite frequency ranges. In Saeki (2009), a method was proposed for tuning multi-loop PID controllers based on loop shaping.
Huang and Huang (2004) designed the multi-loop or decentralized PID controller based on generalized covariance constraints and propounded an iterative LMI approach to solve the problem. Wang et al. (2007) determined the parameter ranges for multi-loop PID controllers. In so doing, they developed an efficient computational design by recasting the problem under study as a quasi-LMI problem. Wu et al. (2011) developed algorithms on the basis of LMIs to design multivariable PID controllers for discrete-time systems. Boyd et al. (2016) formulated MIMO PID controller design as an optimization problem that entails nonconvex quadratic matrix inequalities and introduced a simple method in order to replace the nonconvex matrix inequalities with an LMI constraint. Souza et al. (2016) obtained LMI conditions for the design of PID controllers and used a Lyapunov–Krasovskii functional to substantiate closed-loop stability in the presence of time-varying delays, network-caused delays, and packet dropouts.
This paper formulates IMC-PID controller design as an optimization problem via combining loop shaping and LMI approaches. In this problem, obtaining desired bandwidth and phase margin are considered as the main objectives of the optimization, and the control objectives such as disturbance rejection, noise attenuation, and robustness to variations in the gain of the process are regarded as the corresponding constraints. Moreover, stability conditions are formulated as LMI constraints using the Routh–Hurwitz criterion. This optimization problem can be stated in the form of a generalized eigenvalue problem (GEVP).
The remainder of the paper is organized as follows: Section 2 outlines the problem statement. IMC-PID controller design is introduced in Section 3. The proposed IMC-PID tuning rules and their relationship to the LMI approach are discussed in Section 4. Simulation results are given in Section 5. Finally, the conclusion is reported in Section 6.
Section snippets
Problem statement
In some controllers, a linear relationship exists between the controller and its variables. Hence, the structure of the controller can be considered C(s) = W(s)X, where X is a vector of control variables and W(s) is a vector of functions in s. The vectors X and W(s) are specified according to the type of the controller. For example, a PID controller has a transfer function like (1)Design parameters in PID controller are KP, KI, and KD. Thus, the vectors W(s) and X are considered
IMC-PID controller design
The standard IMC structure is shown in Fig. 4, where P(s) is the process to be controlled, M(s) is the process model, and Q(s) is the IMC controller. The standard IMC structure can be reduced to the equivalent classic feedback system shown in Fig. 1, where C(s) is in the form of a PID controller. Utilizing the IMC design approach (Wang et al., 2001), the process model is decomposed into invertible M−(s) and non-invertible M+(s) portions, where M+(s) typically includes time delay and RHP zeros
First-order plus time delay (FOPTD)
The process transfer function of the FOPTD system is stated as follows:If first-order Pade approximation is used for approximating the time delay term (i.e., e−Ls = (1 − 0.5Ls)/(1 + 0.5Ls)), the IMC controller is obtained as follows:Therefore, the feedback controller C(s) is obtained as follows:which can be rewritten as a PID controller:Hence, we have
Simulation results and discussions
This section presents the results of simulations performed on several types of processes with time delay. The proposed method is compared, by means of four examples, to a few similar approaches recently introduced. In order to measure the performance and robustness of the control system, the integral square error (ISE), overshoot (OS), total variation (TV) and phase margin are considered as the quantitative indices. To make the comparison fair, the same maximum magnitude of sensitivity function
Conclusions
Using open-loop shaping approach, the IMC-PID controller was tuned so that desired frequency domain specifications such as bandwidth and phase margin could be met. The control objectives which were satisfied by open-loop shaping were recast as LMIs, which were then easily solved on the MATLAB software. Also, the stability conditions were formulated as an LMI constraint. The proposed method and some other similar approaches in the recent literature were applied to time-delay processes. In order
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