Experimental evaluations of proportional–integral–derivative type fuzzy controllers with parameter adaptive methods for an active magnetic bearing system

Abstract: This paper deals with the experimental control of a rotating active magnetic bearing (AMB) system using proportional–integral–derivative type fuzzy controllers (PIDFCs) with parameter adaptive methods. Three kinds of parameter adaptive method, including fuzzy tuner, function tuner and relative rate observer, have been proposed in the literature for tuning the coefficients of PIDFCs. However, only a simulation comparison between these methods for control of a second‐order linear system with varying parameters and time delay has been done. In general, theoretical models need to be confirmed and modified through experimental results. This paper provides experimental verification by applying PIDFCs with self‐tuning algorithms for control of a highly nonlinear AMB system. It is shown that the steady‐state error of the AMB system using the function tuner method is lower and the first resonant frequency of the AMB system using the relative rate observer method is higher than the other two methods, and the proportional–integral–derivative controller is quite unstable. The experimental results also show that all of the tuning methods can support a high rotation frequency of the AMB system. In practice, there are only a few differences between the three kinds of parameter adaptive methods.     

Graphical tuning method of FOPID controllers for fractional order uncertain system achieving robust ‐stability

This paper focuses on the graphical tuning method of fractional order proportional integral derivative (FOPID) controllers for fractional order uncertain system achieving robust ‐stability. Firstly, general result is presented to check the robust ‐stability of the linear fractional order interval polynomial. Then some alternative algorithms and results are proposed to reduce the computational effort of the general result. Secondly, a general graphical tuning method together with some computational efficient algorithms are proposed to determine the complete set of FOPID controllers that provides ‐stability for interval fractional order plant. These methods will combine the results for fractional order parametric robust control with the method of FOPID ‐stabilization for a fixed plant. At last, two important extensions will be given to the proposed graphical tuning methods: determine the ‐stabilizing region for fractional order systems with two kinds of more general and complex uncertainty structures: multi‐linear interval uncertainty and mixed‐type uncertainties. Numerical examples are followed to illustrate the effectiveness of the method. Copyright © 2015 John Wiley & Sons, Ltd.     

Design of a fuzzy-PID controller for a nonlinear hydraulic turbine governing system by using a novel gravitational search algorithm based on Cauchy mutation and mass weighting

Hydraulic turbine governing system (HTGS) is a complicated nonlinear system, which regulates frequency and power of hydropower generating unit. In previous study, control model of HTGS is always overly simplified and the elastic water hammer model is seldom considered. In this paper, a nonlinear HTGS model with elastic water hammer effect has been studied and a fuzzy-PID controller is designed to improve control quality of this system. In order to optimize the fuzzy-PID controller, a novel gravitational search algorithm based on Cauchy mutation and mass weighing (GSA-CW) has been proposed with two improvements: a weighting strategy is designed to accelerate the convergence by assigning weights to agents in mass calculation; a combined mutation strategy based on Cauchy and Gaussian distribution is proposed to balance the exploration and exploitation ability of the proposed algorithm. At first, the searching ability of the GSA-CW has been verified on a set of 13 complex benchmark functions by statistical analysis. And then, the GSA-CW is applied to optimize the fuzzy-PID controller, while different meta-heuristics and different PID controllers are employed for comparison. Experimental results indicate that the fuzzy-PID controller optimized by the GSA-CW is more effective to improve the control quality of the nonlinear HTGS.     

Robust FOPID controller design for fractional‐order delay systems using positive stability region analysis

In this paper, a robust fractional‐order PID (FOPID) controller design method for fractional‐order delay systems is proposed based on positive stability region (PSR) analysis. Firstly, the PSR is presented to improve the existing stability region (SR) in D‐decomposition method. Then, the optimal fractional orders λ and μ of FOPID controller are achieved at the biggest three‐dimensional PSR, which means the best robustness. Given the optimal λ and μ, the other FOPID controller parameters k_p, k_i, k_d can be solved under the control specifications, including gain crossover frequency, phase margin, and an extended flat phase constraint. In addition, the steps of the proposed robust FOPID controller design process are listed at length, and an example is given to illustrate the corresponding steps. At last, the control performances of the obtained robust FOPID controller are compared with some other controllers (PID and FOPI). The simulation results illustrate the superior robustness as well as the transient performance of the proposed control algorithm.     

A computational method for solving two‐dimensional nonlinear variable‐order fractional optimal control problems

In this paper, a new class of two‐dimensional nonlinear variable‐order fractional optimal control problems (V‐OFOCPs) is introduced where the variable‐order fractional derivative is defined in the Caputo type. The general procedure for solving theses systems is expanding the state variable and the control variable based on the Legendre cardinal functions in the matrix form. Hence, we derive their operational matrix of derivative (OMD) and operational matrix of variable‐order fractional derivative (OMV‐OFD). More significantly, some properties of these basis functions are proved to be exploited in our approach. Using these achieved results, we simply expand the matrix form of the nonlinear performance index in terms of the Legendre cardinal functions and subsequently convert it to an algebraic equation. We emphasize that it is a valuable advantage of applying cardinal functions in approximation theory. Then, we implement the OMD and the OMV‐OFD of the Legendre cardinal functions to transform the variable‐order fractional dynamical system to a system of algebraic equations. Next, the method of constrained extremum is applied to adjoin the constraint equations including the given dynamical system and the initial‐boundary conditions to the performance index by a set of undetermined Lagrange multipliers. Finally, the necessary conditions of the optimality are derived as a system of nonlinear algebraic equations including the unknown coefficients of the state variable, the control variable and the Lagrange multipliers. The applicability and efficiency of the proposed approach are investigated through the various types of test problems.     

The velocity control of an electro‐hydraulic displacement‐controlled system using adaptive fuzzy controller with self‐tuning fuzzy sliding mode compensation

Hydraulic servo control systems have been used widely in industry. Within the realm of hydraulic control systems, conventional hydraulic valve‐controlled systems have higher response and lower energy efficiency, whereas hydraulic displacement‐controlled servo systems have higher energy efficiency. This paper aims to investigate the velocity control performance of an electro‐hydraulic displacement‐controlled system (EHDCS), where the controlled hydraulic cylinder is altered by a variable displacement axial piston pump to achieve velocity control. For that, a novel adaptive fuzzy controller with self‐tuning fuzzy sliding‐mode compensation (AFC‐STFSMC) is proposed for velocity control in EHDCS. The AFC‐STFSMC approach combining adaptive fuzzy control and the self‐tuning fuzzy sliding‐mode control scheme, has the advantages of the capability of automatically adjusting the fuzzy rules and of reducing the fuzzy rules. The proposed AFC‐STFSMC scheme can design the sliding‐mode controller with no requirement on the system dynamic model, and it can be free of chattering, thereby providing stable tracking control performance and robustness against uncertainties. Moreover, the stability of the proposed scheme via the Lyapunov method is proven. Therefore, the velocity control of EHDCS controlled by AFC‐STFSMC is implemented and verified experimentally in different velocity targets and loading conditions. The experimental results show that the proposed AFC‐STFSMC method can achieve good velocity control performance and robustness in EHDCS with regard to parameter variations and external disturbance. Copyright © 2011 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

An H∞non‐fragile observer‐based adaptive sliding mode controller design for uncertain fractional‐order nonlinear systems with time delay and input nonlinearity

In this paper the problem of non‐fragile adaptive sliding mode observer design is addressed for a class of nonlinear fractional‐order time‐delay systems with uncertainties, external disturbance, exogenous noise, and input nonlinearity. An H_∞ observer‐based adaptive sliding mode control considering the non‐fragility of the observer is proposed for this system. The sufficient asymptotic stability conditions are derived in the form of linear matrix inequalities. It is proven that the sliding surface is reachable in finite time. An illustrative example is provided which corroborates the effectiveness of the theoretical results.     

Optimal fractional order PIλDμ controller for stabilization of cart‐inverted pendulum system: Experimental results

The cart‐inverted pendulum is a non‐minimum phase system having right half s‐plane pole and zero in close vicinity to each other. Linear time invariant (LTI) classical controllers cannot achieve satisfactory loop robustness for such systems. Therefore, in the present work the fractional order PI^λD^μ (FOPID) controller is addressed for robust stabilization of the system, since fractional order controller design allows more degrees of freedom compared to its integer order counterparts by virtue of its two parameters λ and μ. The controller parameters are tuned by three evolutionary optimization techniques. In order to select the controller parameters optimally, a novel non‐linear fitness function using integral time square error (ITSE), settling‐time, and rise time is proposed here. The control algorithm is implemented successfully in real‐time. Moreover, stability analysis of the system compensated with a fractional order controller is presented using Riemann surface. Robustness of the physical cart‐inverted pendulum system towards multiplicative gain variations and plant parameter variations is verified. In this regard, it is shown that the fractional order controller provides satisfactory robust performance in both simulation and real‐time system.     

Guaranteed‐cost fuzzy controller design for a self‐sustaining bicycle with practical constraints

This study presents a guaranteed‐cost fuzzy controller for a self‐sustaining bicycle. First, the nonlinear dynamics of the bicycle are exactly transformed into a T‐S fuzzy system with model uncertainty. The guaranteed‐cost fuzzy controller is then designed for the transformed T‐S fuzzy system. For practical considerations, the input/state constraints are also satisfied in the design. The main contribution of this study is the guaranteed‐cost control design for a T‐S fuzzy system with model uncertainty and input/state constraints. Finally, simulation results show the validity of the proposed controller design method. Copyright © 2011 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society