A conformal mapping based fractional order approach for sub-optimal tuning of PID controllers with guaranteed dominant pole placement

A novel conformal mapping based fractional order (FO) methodology is developed in this paper for tuning existing classical (Integer Order) Proportional Integral Derivative (PID) controllers especially for sluggish and oscillatory second order systems. The conventional pole placement tuning via Linear Quadratic Regulator (LQR) method is extended for open loop oscillatory systems as well. The locations of the open loop zeros of a fractional order PID (FOPID or PI^λD^μ) controller have been approximated in this paper vis-à-vis a LQR tuned conventional integer order PID controller, to achieve equivalent integer order PID control system. This approach eases the implementation of analog/digital realization of a FOPID controller with its integer order counterpart along with the advantages of fractional order controller preserved. It is shown here in the paper that decrease in the integro-differential operators of the FOPID/PI^λD^μ controller pushes the open loop zeros of the equivalent PID controller towards greater damping regions which gives a trajectory of the controller zeros and dominant closed loop poles. This trajectory is termed as “M-curve”. This phenomena is used to design a two-stage tuning algorithm which reduces the existing PID controller’s effort in a significant manner compared to that with a single stage LQR based pole placement method at a desired closed loop damping and frequency.     

Real-coded Genetic Algorithm for system identification and tuning of a modified Model Reference Adaptive Controller for a hybrid tank system

Modeling and controlling of level process is one of the most common problems in the process industry. As the level process is nonlinear, Model Reference Adaptive Control (MRAC) strategy is employed in this paper. To design an MRAC with equally good transient and steady state performance is a challenging task. The main objective of this paper is to design an MRAC with very good steady-state and transient performance for a nonlinear process such as the hybrid tank process. A modification to the MRAC scheme is proposed in this study. Real-coded Genetic Algorithm (RGA) is used to tune off-line the controller parameters. Three different versions of MRAC and also a Proportional Integral Derivative (PID) controller are employed, and their performances are compared by using MATLAB. Input–output data of a coupled tank setup of the hybrid tank process are obtained by using Lab VIEW and a system identification procedure is carried out. The accuracy of the resultant model is further improved by parameter tuning using RGA. The simulation results shows that the proposed controller gives better transient performance than the well-designed PID controller or the MRAC does; while giving equally good steady-state performance. It is concluded that the proposed controllers can be used to achieve very good transient and steady state performance during the control of any nonlinear process.     

Auto-Tuning and Fractional Order Controller Implementation on Hardware in the Loop System

The paper deals with the design of an auto-tuning fractional order proportional-integrative-derivative controllers and its implementation on hardware in the loop simulator for the real-time control of unknown plants. The proposed procedure can be applied to systems with delay and order greater than one, once specifications on cross-over frequency and phase margin are given. The auto-tuning procedure consists of two phases: the first one dedicated to the identification of the process at the desired cross-over frequency and the second one to determine all the parameters of the fractional order proportional-integrative-derivative controllers. The obtained controller ensures an iso-damping response of the plant. Experimental results are given to confirm the effectiveness of the proposed approach and show that the requirements are totally met for the system to be controlled.     

Digital PID controller design for multivariable analogue systems with computational input-delay

This paper presents a new methodology to design MIMO digitalPID controllers for multivariable analogue systems with computationalinput time-delay. The preliminarily designed analogue PID controlleris refined using a newly developed state-feedback and state-feedforwardLQR approach. The optimally designed closed-loop system withthe refined MIMO analogue PID controller has pre-assigned closed-loopeigenvalues. A prediction-based digital redesign technique isdeveloped to discretize the cascaded MIMO analogue PID controller,such that the states of the digitally redesigned closed-loopsampled-data system with the MIMO digital PID controller areclose to those of the analogously designed closed-loop systemwith the refined MIMO analogue PID controller. The aforementioneddigital redesign technique is further modified based on thepredictive control method to cope with MIMO analogue systemswith input delay.     

Applications of an IMC based PID Controller tuning strategy in atmospheric and vacuum distillation units

Applications of internal model control (IMC) based single loop controller tuning in atmospheric and vacuum distillation units were investigated. The robust IMC-PID controller not only inherits the virtues that the IMC controller has, but also has a simple and general structure such as that of a PID controller. Tuning and optimization of controllers becomes more convenient using the IMC-PID controller. It can also become easier to achieve in a distributed control system (DCS) via control module configuration. In order to make it easier to apply in industrial processes, the modeling problem of the industrial process should be resolved. In this paper, a convenient closed-loop system identification strategy based on new Luus-Jaakola (NLJ) algorithm was presented, meanwhile, the principle of IMC-PID was interpreted. A software package was developed, capable of collecting actual data on-line, obtaining the process model and optimizing the parameters of the controllers. It was applied in an atmospheric and vacuum distillation unit of a refinery to tune the PID parameters of all controllers. The application results demonstrate the validity of the proposed method.     

Fractional order adaptive high-gain controllers for a class of linear systems

In this paper we show that a fractional adaptive controller based on high gain output feedback can always be found to stabilize any given linear, time-invariant, minimum phase, siso systems of relative degree one. We generalize the stability theorem of integer order controllers to the fractional order case, and we introduce a new tuning parameter for the performance behaviour of the controlled plant. A simulation example is given to illustrate the effectiveness of the proposed algorithm.     

LQR based improved discrete PID controller design via optimum selection of weighting matrices using fractional order integral performance index

The continuous and discrete time Linear Quadratic Regulator (LQR) theory has been used in this paper for the design of optimal analog and discrete PID controllers respectively. The PID controller gains are formulated as the optimal state-feedback gains, corresponding to the standard quadratic cost function involving the state variables and the controller effort. A real coded Genetic Algorithm (GA) has been used next to optimally find out the weighting matrices, associated with the respective optimal state-feedback regulator design while minimizing another time domain integral performance index, comprising of a weighted sum of Integral of Time multiplied Squared Error (ITSE) and the controller effort. The proposed methodology is extended for a new kind of fractional order (FO) integral performance indices. The impact of fractional order (as any arbitrary real order) cost function on the LQR tuned PID control loops is highlighted in the present work, along with the achievable cost of control. Guidelines for the choice of integral order of the performance index are given depending on the characteristics of the process, to be controlled.     

Inventory control of supply chains: Mitigating the bullwhip effect by centralized and decentralized Internal Model Control approaches

In this paper, a two-degrees-of-freedom Internal Model Control structure is incorporated in production inventory control for a supply chain system. This scheme presents an intuitive and simple parametrization of controllers, where inventory target tracking and disturbance (demand) rejection in the inventory level problems are treated separately. Moreover, considering that the lead times are known, this scheme presents a perfect compensation of the delay making the stabilization problem easier to handle. This control structure is formulated for a serial supply chain in two ways (by using a centralized and a decentralized control approach). The behavior of these inventory control strategies is analyzed in the entire supply chain. Analytical tuning rules for bullwhip effect avoidance are developed for both strategies. The results of controller evaluations demonstrate that centralized control approach enhances the behavior with respect to the inventory target tracking, demand rejection and bullwhip effect in the supply chain systems.     

Hardware Implementation of Fuzzy PID Controllers

For traditional hardware implementation of fuzzy PID controllers, it is large at computation and bad in real-time performance, so, a kind of PID control algorithm, whose gain parameters could be tuned by their fuzzy system, was selected as studying example for a novel idea of hardware implementation. In this paper, authors presented hardware network of memory address mapping to implement fuzzy PID control algorithm, and designed the corresponding hardware system. The idea actually realizes fusion of hardware and intelligent algorithm. The implementation effectively simplified hardware circuits, the whole controller is very simple without CPU. Meanwhile, it is very easy to use, only connecting the sensor/transducer, the driver and the actuator is OK. The controller is very rapid in response, it need only two A/D conversion periods for outputting a required control signal. So the implementation could meet real-time performance effectively.     

Steady-state,self-oscillating and chaotic behavior of a PID controlled nonlinear servomechanism by using Bogdanov–Takens and Andronov–Poincaré–Hopf bifurcations

This paper analyzes a controlled servomechanism with feedback and a cubic nonlinearity by means of the Bogdanov–Takens and Andronov–Poincaré–Hopf bifurcations, from which steady-state, self-oscillating and chaotic behaviors will be investigated using the center manifold theorem. The system controller is formed by a Proportional plus Integral plus Derivative action (PID) that allows to stabilize and drive to a prescribed set point a body connected to the shaft of a DC motor. The Bogdanov–Takens bifurcation is analyzed through the second Lyapunov stability method and the harmonic-balance method, whereas the first Lyapunov value is used for the Andronov–Poincaré–Hopf bifurcation. On the basis of the results deduced from the bifurcation analysis, we show a procedure to select the parameters of the PID controller so that an arbitrary steady-state position of the servomechanism can be reached even in presence of noise. We also show how chaotic behavior can be obtained by applying a harmonical external torque to the device in self-oscillating regime. The advantage of achieving chaotic behavior is that it can be used so that the system reaches a set point inside a strange attractor with a small control effort. The analytical calculations have been verified through detailed numerical simulations.     

Simple, robust, selftuning controller

This paper proposes a robust method for automatic tuning of parameters of a discrete PID controller. The tuning rules for SISO and MIMO systems are based on automatic determination of critical gain and critical frequency from the estimated model parameters. The plant model can be expressed by a transfer function in continuous and/or discrete form or by differential and/or difference equation. A simple control law using Takahashi discrete form is proposed. Simulations results prove that it is easy to use being able to handle minimum and nonminimum phase plant as well.     

Robust control of a spatial robot using fuzzy sliding modes

In order to improve the performance of the sliding mode controller, fuzzy logic sliding mode controller is proposed in this study. The control gain of the conventional sliding mode controller is tuned by a fuzzy logic rule base and, also dynamic sliding surfaces are obtained by changing their slopes using the error states of the system in another fuzzy logic algorithm. These controllers are then combined in order to enhance the performance. Afterwards, proposed controllers were used in trajectory control of a three degrees of freedom spatial robot, which is subjected to noise and parameter variations. Finally, the controllers introduced are compared with a PID controller which is commonly used for control of robotic manipulators in industry. The results indicate the superior performance of the proposed controller.     

Outcome of special vibration controller techniques linked to a cracked beam

This paper presents a comparison between three different controller methods added to a cracked beam under the action of a harmonic excitation. Those three controllers are Positive Position Feedback (PPF), Integral Resonant Control (IRC) and Nonlinear Integral Positive Position Feedback (NIPPF) which be added to the measured system. The multiple scales method (MSM) is applied for getting the approximate solution on behalf of measured design. This method is effective to solve the major equations of measured system. Stability and effect of different coefficients of the system are demonstrated. The approximate solution response is established via numerical simulation outcome. NIPPF controller is the best one gives better results compared to the other two controllers in decreasing the high amplitude of the system. Comparison between mathematical solution and numerical simulation are considered. Relationship of formerly available papers is considered.     

Joint swing-up and stabilization of the Reaction Wheel Pendulum using Discontinuous Integral algorithm

In this paper, a third-order discontinuous integral controller is designed to jointly swing-up and robustly stabilize in finite-time the upright position of the Reaction Wheel Pendulum (RWP) system, despite some uncertainties and perturbations. To show this, a numerical estimation of the domain of attraction of the controller is used. The control algorithm produces a continuous control signal, thus reducing the usual chattering effect of the sliding-mode controllers. The theoretical results of the paper are verified by simulations and experiments in a laboratory setup.     

Observer‐based dissipative control for networked control systems: A switched system approach

This article studies the problem of observer‐based dissipative control problem for wireless networked control systems (NCSs). The packet loss and time delay in the network are modeled by a set of switches, using that a discrete‐time switched system is formulated. First, results for the exponential dissipativity of discrete‐time switched system with time‐varying delays are proposed by using the average dwell time approach and multiple Lyapunov–Krasovskii function. Then, the results are extended to drive the controller design for considered wireless NCS. The attention is focused on designing an observer‐based state feedback controller which ensures that, for all network‐induced delay and packet loss, the resulting error system is exponentially stable and strictly dissipative. The sufficient conditions for existence of controllers are formulated in the form of linear matrix inequalities (LMIs), which can be easily solved using some standard numerical packages. Both observer and controller gains can be obtained by the solutions of set of LMIs. Finally, numerical examples are provided to illustrate the applicability and effectiveness of the proposed method. © 2014 Wiley Periodicals, Inc. Complexity 21: 297–308, 2015     

A new hybrid algorithm based on optimal fuzzy controller in multimachine power system

In this article, a new methodology based on fuzzy proportional‐integral‐derivative (PID) controller is proposed to damp low frequency oscillation in multimachine power system where the parameters of proposed controller are optimized offline automatically by hybrid genetic algorithm (GA) and particle swarm optimization (PSO) techniques. This newly proposed method is more efficient because it cope with oscillations and different operating points. In this strategy, the controller is tuned online from the knowledge base and fuzzy interference. In the proposed method, for achieving the desired level of robust performance exact tuning of rule base and membership functions (MF) are very important. The motivation for using the GA and PSO as a hybrid method are to reduce fuzzy effort and take large parametric uncertainties in to account. This newly developed control strategy mixed the advantage of GA and PSO techniques to optimally tune the rule base and MF parameters of fuzzy controller that leads to a flexible controller with simple structure while is easy to implement. The proposed method is tested on three machine nine buses and 16 machine power systems with different operating conditions in present of disturbance and nonlinearity. The effectiveness of proposed controller is compared with robust PSS that tune using PSO and the fuzzy controller which is optimized rule base by GA through figure of demerit and integral of the time multiplied absolute value of the error performance indices. The results evaluation shows that the proposed method achieves good robust performance for a wide range of load change in the presents of disturbance and system nonlinearities and is superior to the other controllers. © 2014 Wiley Periodicals, Inc. Complexity 21: 78–93, 2015     

Delayed state feedback and chaos control for time-periodic systems via a symbolic approach

This paper presents a symbolic method for a delayed state feedback controller (DSFC) design for linear time-periodic delay (LTPD) systems that are open loop unstable and its extension to incorporate regulation and tracking of nonlinear time-periodic delay (NTPD) systems exhibiting chaos. By using shifted Chebyshev polynomials, the closed loop monodromy matrix of the LTPD system (or the linearized error dynamics of the NTPD system) is obtained symbolically in terms of controller parameters. The symbolic closed loop monodromy matrix, which is a finite dimensional approximation of an infinite dimensional operator, is used in conjunction with the Routh–Hurwitz criterion to design a DSFC to asymptotically stabilize the unstable dynamic system. Two controllers designs are presented. The first design is a constant gain DSFC and the second one is a periodic gain DSFC. The periodic gain DSFC has a larger region of stability in the parameter space than the constant gain DSFC. The asymptotic stability of the LTPD system obtained by the proposed method is illustrated by asymptotically stabilizing an open loop unstable delayed Mathieu equation. Control of a chaotic nonlinear system to any desired periodic orbit is achieved by rendering asymptotic stability to the error dynamics system. To accommodate large initial conditions, an open loop controller is also designed. This open loop controller is used first to control the error trajectories close to zero states and then the DSFC is switched on to achieve asymptotic stability of error states and consequently tracking of the original system states. The methodology is illustrated by two examples.     

Synthesis of damping controllers for the solution of completely regular differential-algebraic delay systems

For linear autonomous completely regular differential-algebraic delay systems, we develop methods for synthesizing state feedback controllers that simultaneously provide solution damping and finite spectrum assignment in the closed-loop system. We prove necessary and sufficient conditions for the existence of such controllers defining finite- and infinite-time controls depending on the system parameters (complete 0-controllability and 0-controllability criteria, respectively). The proofs are of constructive character and allow one to synthesize the corresponding controller for every system. An illustrative example is given.     

Predictor-based controls: The implementation problem

The dynamic controller design problem for systems with delay in the state and control variables is studied. The main attention is paid to the practical implementation of the controllers. We show that the closed-loop system remains exponentially stable if the integral terms are approximated by finite Riemann type sums.