Swarm intelligence-based extremum seeking control
Highlights
► This paper propses an extremum seeking control scheme based on particle swarm optimization (PSO). ► The system states are steered to the optimal point by the controller, and the explicit form of the performance function is not needed. ► We also propose an reshuffle-then-insertion algorithm to reduce the regulator gain and osillation induced by population-based stochastic searching algorithms.
Introduction
Regulation and tracking of system states to optimal setpoints or trajectories are typical tasks in control engineering. However, these optimal setpoints are sometimes difficult to be chosen a priori, or vary with the environmental condition changes. Extremum seeking control (ESC) is a kind of adaptive control schemes that can search for the optimal setpoints online, based on the measurement of the performance output or its gradient. ESC can be regarded as an optimization problem, and many of the schemes used in ESC are transferred from optimization algorithms. However, some optimization algorithms cannot be incorporated into the ESC framework easily, for the reason that, practical issues, such as stability, noise, regulation time, control gain and oscillation limitation, will prevent the use of some optimization algorithms from ESC context. Thus, the study on suitable combination of ESC and optimization algorithms is of great interest both in academics and in engineering.
Unlike the traditional variational calculus-involved optimal control method, the explicit form of the performance function is not needed in ESC. Therefore, ESC is useful in the applications that the performance functions are difficult to model. After Krstic and Wang’s stability studies (Krstic & Wang, 2000), research on ESC has received significant attention in recent years. The recent ESC application examples include active flow control (Beaudoin, Cadot, Aider, & Wesfreid, 2006), bioreactor or chemical process control (Bastin et al., 2009, Hudon et al., 2008, Hudon et al., 2005), cascaded Raman optical amplifiers (Dower, Farrell, & Nesic, 2008), antilock braking system design (Zhang & Ordonez, 2007), thermoacoustic cooler (Li, Rotea, Chiu, Mongeau, & Paek, 2005), and fuel cell power plant (Zhong, Huo, Zhu, Cao, & Ren, 2008). There also have been considerable theoretical studies in ESC, such as stability studies on perturbation-based ESC (Krstic, 2000, Krstic and Wang, 2000), ESC for discrete-time systems (Joon-Young, Krstic, Ariyur, & Lee, 2002), PID tuning by ESC (Killingsworth & Krstic, 2006), ESC for nonlinear dynamic systems with parametric uncertainties (Guay & Zhang, 2003), and ESC for state-constrained nonlinear systems (DeHaan & Guay, 2005). The majority of ESC literature focused on two issues, the one is the searching for the optima, and the other is the regulation of the systems. The recent studies of Zhang and Ordonez, 2007, Zhang and Ordonez, 2009 presented a numerical optimization-based ESC (NOESC) framework that takes the advantage of numerical algorithms to find the optima online. However, these algorithms are unable to find the global optima if the assumption that the performance functions are convex and continuous does not hold. Furthermore, the NOESC is sensitive to measurement noise, due to the poor robustness of the numerical algorithms.
Particle swarm optimization (PSO) algorithm is a population-based stochastic optimization method which first devised by Kennedy and Eberhart (1995). PSO algorithm mimics the food-seeking behavior of birds or fishes. Due to its simplicity and effectiveness, PSO algorithm witnesses a considerable interest and is applied in many areas. The convergence of PSO algorithm is studied by deterministic method (Eberhart & Shi, 2001) or stochastic process theory (Jiang, Luo, & Yang, 2007). Clerc and Kennedy (2002) presented a convergence condition of PSO algorithm. Rapaic and Kanovic (2009) studied the time-varied parameter PSO algorithm and the selection of the parameters. Studies have shown that PSO algorithm is able to handle a wide range of problems, such as integer optimization (Laskari, Parsopoulos, & Vrahatis, 2002), multi-objective optimization (Dasheng, Tan, Goh, & Ho, 2007), and global optimization of multimodal functions (Liang, Qin, Suganthan, & Baskar, 2006). The recent application of PSO algorithm includes power systems (del Valle, Venayagamoorthy, Mohagheghi, Hernandez, & Harley, 2008), flights control (Duan, Ma, & Luo, 2008), and nuclear power plants (Meneses, Machado, & Schirru, 2009), to name a few. In control engineering, PSO algorithm is usually employed to identify the models (Panda, Mohanty, Majhi, & Sahoo, 2007), or to optimize the parameters of the controller offline (El-Zonkoly, 2006). PSO algorithm is usually regarded as an effective global optimization method. However, it is often used in offline optimization, and depends on the explicit form and the solvability of the performance functions. However, for some complex models, e.g. active flow control problems which described by Navier–Stokes equations, it is difficult to obtain the optimal parameters of the controllers by time-consuming numerical simulations.
In this paper, we extend the numerical optimization-based ESC (Zhang & Ordonez, 2007) by incorporating PSO algorithm into the extremum seeking framework. We also address the practicability issues of this scheme, and propose a reshuffle-then-insertion algorithm to reduce the control gain and oscillation. In the proposed scheme, a sequence converging to the global optima is generated by PSO with reshuffle-then-insertion algorithm. The sequence used as a guidance to regulate the state of the plant approach to the optimal set point. This paper is organized as follows. Section 2 gives a problem statement, where a PSO-based ESC (PSOESC) framework is introduced. We then review the standard PSO algorithm in Section 3. The details of the PSOESC scheme for linear time invariant (LTI) systems and feedback linearizable systems are presented in Section 4, where the reshuffle-then-insertion approach for improving the practicability of the PSOESC is also proposed. Section 5 presents the results of the numerical experiments. Finally, Section 6 concludes the paper.
Section snippets
Problem statement
In control practice, the problem of seeking for an optimal set point is encountered usually. In general, this problem can be represented as modelwhere is the state, is the input, is the performance output to be optimized, is a sufficiently smooth function on D, and is an unknown function. For simplicity, we assume in this paper. Without loss of generality, we consider the minimization of the performance function (2). Unlike optimal control,
Particle swarm optimization
PSO is a population-based random optimization algorithm that mimics the behavior of bird or fish swarm in searching food. In the swarm, each particle has a variable speed, moving toward the positions of its own best fitness achieved so far and the best fitness achieved so far by any of its neighbors.
Let is an n-dimensional search space. The size of the particle swarm is denoted as N. The position of the ith particle is represented as an n-dimensional vector , and its
PSO-based extremum seeking scheme
Similar to Zhang and Ordonez (2007), we discuss the PSO-based extremum seeking control scheme in the order of linear time-invariant systems (LTI), feedback linearizable systems, and input-output feedback linearizable systems.
Numerical experiments
The numerical experiments were carried out on a personal computer running on the Matlab/Simulink environment. The parameters of the standard PSO were chosen as w = 0.9, c1 = 0.12, and c2 = 0.012. For the purpose of simulation, performance functions were provided explicitly, while they would be unknown in real-world applications. As mentioned above, the convergence of the PSOESC scheme is described in mean square sense, and the convergence to the global optima is not guaranteed by the standard PSO.
Conclusion
This paper presents a PSO-based ESC scheme that is able to find the optimal set point online. The PSO algorithm produces a sequence converging to the global optima. The sequence serves as a guidance to regulate the state of the plant approaching to the optimal set point. We also propose a reshuffle-then-insertion algorithm that is able to reduce the control gain and oscillation, thus improving the practicability of PSOESC scheme.
The numerical experiments show the effectiveness of the scheme.
Acknowledgement
The authors would like to thank Dr. C. Zhang in Etch Engineering Technology for his comments and useful suggestions, especially the naming of the reshuffle-then-insertion algorithm.
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