Iterative algorithm and curve fitting technique for tuning with time delay system PID controller

1 Introduction

Time-Delay Systems (TDS) are stumbled upon in different areas, together with engineering, biology, and economics [2, 13]. In a system, a time delay is a cause of unsteadiness and fluctuations [1]. Two kinds of time delay systems are there: retarded and neutral [16]. In TDS, where time-delays stay alive between the applications of input to the system and their resulting outcome, can be signified by delay disparity equations (DDEs) [3]. Systems with delays signify a class inside infinite size mostly applied for the modeling and the study of transport and broadcast phenomena [7]. Time-delays in control loops generally mortify system presentation and make difficult the study and plan of feedback controllers [14].

Due to a number of functions of communication networks in biology and population dynamics, the steadiness of time-delay system is a setback of recurrent interest [19]. In common, stability analysis of time-delay systems can be categorized into two kinds. One is the delay-dependent stability analysis which contains the data on the size of the delay, and one more is the delay-independent stability analysis [15, 21]. The delay independent stabilization offers a controller which can calm down the system irrespective of the size of the delay [18]. The delay dependent stabilizing controller is distressed with the size of the delay and generally provides the upper bound of the delay [20] alternatively.

The presentation and steadiness of time-delay systems are manipulated by dead time [8]. Time delay all the time stays alive in the measurement loop or control loop, so it is further hard to control this type of process [5]. In order to improve the control presentation,a few novel control technologies, like predictive control, the neural type of a synthetic neutral delay in a control loop [16]. The PID controller is applied in a broad range of problems like automotive, instrumentation, motor drives etc. [6]. This controller offers comment, it has the capability to eradicate steady-state offsets via integral action, and it can await the future through an imitative action [12]. Its structural effortlessness and ability to work out various practical control problems of the system [10]. For most of the systems if the PID parameters are adjusted accurately, PID controller offers robust and dependable presentation [6]. A number of delay-independent adequate conditions for the asymptotic stability of neutral delay differential systems have been employed [17].

For the tracking and stabilization of the loops plan system, the traditional PID compensator was applied [9]. Conventional PID controllers are not fit for nonlinear systems and higher-ordered and time-delayed systems [11]. The presentations of the PID controllers extremely depend on the précised information of the system time delay [7]. The stabilizing PID controller can diminish the computation time and it stays away from the time-consuming constancy checking [4].

Time Delay System control is the apply of fractional calculus and the system being modeled in a traditional way or as a partial one. The PID controller acts a significant task for tuning the Time Delay System. In the document, a hybrid method suggested to refrain the factional order time delay system with PID controller. The curve fitting process employed to fit the time delay system reaction in the hybrid method. After that, to acquire the best polynomial coefficient of the curve fitting function, an iterative algorithm is applied.

Contribution of the paper:

This paper presents an iterative algorithm and curve fitting technique based PID controller tuning method.

The complete explanation of the suggested hybrid method is shown in Section 3. Before that, the current research works are explained in Section 2. The results and explanation expressed in Section 4. In Section 5 ends the paper.

2 Recent research works: A brief review

Many associated works are previously accessible in literature which based PID controller design for time delay fractional order system. A few of them are assessed here.

For a random (including unstable) linear time delay system, a technique to calculate the complete set of stabilizing PID controller parameters has been offered by Norbert Hohenbichler [22]. To hold the infinite number of constancy boundaries in the plane for a rigid proportional gain kp was the main involvement. It was illustrated that for hold backed open loops, the steady area in the plane contains convex polygons. A fact was commenced concerning neutral loops. For definite systems and certain kp, the precise steady area in the plane could be explained by the limit of a sequence of polygons with an endless number of vertices. This series might be fairly accurate by convex polygons. Besides, they explained an essential condition for kp-intervals potentially containing a steady area in the plane. As a result, after gridding kp in these intervals, the set of stabilizing controller parameters could be computed.

Taonian Liang et al. [23] have suggested a partial order PI ^λ D ^μ controller of robust constancy area for interval plant with time delay. With time delay, the difficulty of computing the robust constancy area for interval plant has examined by them. By the lower and upper bounds, the partial order interval quasi-polynomial was rotted into numerous vertex characteristic quasi-polynomials. To describe the constancy boundaries of each vertex characteristic quasi-polynomial in the space of controller parameters, the D-decomposition method was applied. By crossing the constancy area of each quasi-polynomial, the constancy area of interval characteristic quasi-polynomial was decided. The parameters of their suggested controller were attained by choosing the control parameters from the constancy area. The robust constancy was checked by means of the value set accompanied by the zero elimination principle. For interval plant, their suggested algorithm was helpful in examining and scheming the robust PI ^λ D ^μ controller.

A particle swarm optimizated I-PD controller for Second Order Time Delayed System has been suggested by Suji Prasad et al. [24]. Optimization was based on the presentation indices like settling time, rise time, peak overshoot, ISE (integral square error) and IAE (integral absolute error). PID controllers and its variants are most favored even if there are major improvements in the control systems in industrialprocesses. The needed control output may fail if the parameter of controller was not correctly planned. They have confirmed that their reproduction results with optimized I-PD controller to be specified improved presentations compared with Ziegler Nichols and Arvanitis tuning.

Rama Reddy et al. [25] have explained a PID controller for time delay systems. Their suggested technique fixed the constancy areas of PID and a novel PID with series leading correction (SLC) for networked control systems with time delay. The latest PID controller has a tuning parameter ‘β’. They have received that relation among the parameters of the system. The consequence of plant parameters on constancy area of PID controllers and SLC-PID controllers in first-order and second-order systems with time delay are moreover specified. At last, an open-loop zero was included into the plant-unstable second order system with time delay so that the constancy areas of PID and SLC-PID controllers get successfully widened.

Ying Luo et al. [26] have suggested a part having for selecting two possible or attainable conditions, and a FOPI/IOPID controller synthesis were applied for all the stable FOPTD systems. The entire possible area of two specifications can be attained and pictured in the plane by means of their suggested plan. All mixtures of two specifications can be confirmed by the controller design with those areas as the previous information. Particularly, it was attractive to compare the regions of these two possible regions for the IOPID and FOPI controllers. A simulation design demonstrates that their suggested plan has achieved and their presentation of the designed FOPI controller compared with the optimized integer order PI controller and the IOPID controller designed.

Shi et al. [26] have recommended to discover the proportional-integral-derivative (PID) controller with balanced concert in terms of transient response, actuator preservation and robustness. In the minor level problem, the transient presentation was optimized so that the balanced controller could be planned with negligible controller output deviation in the higher level problem; where the necessity on transient presentation was lighten up to a pre-specified level. The toughness of the system was assured by constraints on the maximum sensitivity in both problems. By a single parameter, the trade-off among transient performance and actuator preservation was controlled equally for diverse process dynamics, which distinguishes the uniformness of lessening in transient performance. By selecting dissimilar values of this parameter, tuning rules were offered for first order plus time delay (FOPTD) processes for set point next and load commotion rejection, correspondingly. Covering the entire plant family set, the competence of these tuning rules was shown by examples.

Kiyong Kim et al. [28] have suggested an indirect technique for self-tuning of the proportional, integral, and derivative (PID) controller gains. A few of the recent voltage regulator systems were applying PID control for stabilization. Based on specified excitation system parameters, numerous PID tuning strategies were accounted. As in common, these parameters were not accessible during commissioning, particularly the machine time constants, this need of information causes a substantial time delay and price of fuel usage for charging the automatic voltage regulator (AVR). Using well-developed algorithms, the excitation system parameters were automatically recognized and the PID gains were computed to diminish the charging time and cost. To recognize the system parameters, Recursive least-square (RLS) with linearization via feedback was suggested. With numerous generator sets, the presentation of the suggested technique was assessed. Commissioning was achieved very rapidly with outstanding presentation results with self-tuned PID gains.

With mode-dependent time-variant delays the problem of stochastic stability for a class of semi-Markovian systems has been explored by Fanbiao Li et al. [29]. By Lyapunov function approach, together with a piecewise analysis method, a satisfactory condition was suggested to assurance the stochastic stability of the underlying systems. In order to demonstrate the efficiency and benefits of the suggested techniques two examples were specified.

An effective Lyapunov functional has been proposed by Yajun Li et al. [30] to explore the H infinity filtering problems for a class of neural networks with time delay. By joining with some inequality technique or free weighting matrix approach, the delay dependent conditions have been suggested such that the filtering error system was worldwide asymptotically steady with guaranteed H infinity presentation. To demonstrate the efficiency and low conservatism of the proposed filter two examples and simulations have been offered.

Kyun-Sang Park et al. [31] inspected the exponential stability of singularly perturbed discrete systems (SPDSs) with time-delay through the Lyapunov’s direct method. A composite Lyapunov function was suggested to demonstrate that the SPDS with time-delay was exponentially constant with the decay rate γ. In addition, the exponential stability of the nonlinear SPDS with time delay was as well regarded based on the linear SPDS result.

With multiple state delays for a class of discrete-time descriptor Takagi-Sugeno (T-S) fuzzy systems, the problems of D-stability and nonfragile control has been distressed by Fanbiao Li et al. [32]. In order to make certain that all the poles of the descriptor T-S fuzzy system are placed inside a disk enclosed in the unit circle, a D-stability criterion were suggested. Additionally, an adequate condition was offered such that the closed-loop system is regular, causal, and D-stable, regardless of parameter uncertainties and multiple state delays. The consequent solvability conditions for the needed fuzzy-rule-dependent nonfragile controllers were in addition launched.

Enhanced presentations of the closed-loop can be accomplished by applying such sorts of controllers in Time Delay System. The PID controller is employed due to the plainness of operation, plan and low cost in various industrial applications. However, for making constancy system, the tuning process of PID controller is one of the complexes. A lot of tuning techniques PID gains have been improved however the Ziegler Nichols method is one of the traditional PID gain tuning method. Other than, the Ziegler- Nichols technique is still comprehensively employed for determining the parameters of PID controllers. Moreover, erection of a mathematical tuning model plan turns into sometimes expensive and these tuning rules frequently never give correct values because of modeling faults. The linear programming technique is successfully for tuning the PID gains. However, it is an iterative comment tuning; hence, the computational time and the difficulty of the tuning are far above the ground. Therefore, the linear programming technique is appropriate for exact case of tuning that is linear purpose function. To adjust the Time Delay System with PID controller, iterative algorithm and curve fitting method is suggested in the document.

3 PID controller for time delay system

Time Delay System control is the use of fractional calculus and the system being modeled in a classical way or as a fractional one. The control theory based controller techniques are used for controlling the fractional order system. The control theory based controller can improve performance as well as successfully stabilize unstable Time Delay Systems. The block diagram of fractional order system with PID controller is given in Fig. 1. Where, r e and y are the input, error and output of the system. Then, G _c  (s) is denoted as PID controller and e ^-θs, G _p  (s) is the fractional order system.

The fractional order time delay equation of the system G _s  (s) is the multiplication of e ^-θs and G _p  (s) which is described as follows, G s ( s ) = e - θ s * G p ( s )(1) where, G p ( s ) = K ( Ts + 1 ), then the Equation (1) is rewritten as follows, G s ( s ) = K * e - θ s ( Ts + 1 )(2) where, T is the time constant, θ is the time delay and K is represented as the steady state gain of the system. For tuning the time delay system, the PID controller is used. The combined system output equation Y (s) is described as following them, Y ( s ) = E ( s ) * K * e - θ s ( K p s + K i + K d s 2 ( Ts 2 + s ) )(3)

The Equation (3) is the combined output of the system. At initially, the E (s) depends on the system input (reference). During the time of process, the E (s) depends on the system feedback that is the system error multiply with the system. For maintaining the system stability, the iterative and curve fitting technique is utilized. The detailed description of the iterative and the curve fitting technique is described in the following section.

The proposed hybrid technique was implemented in MATLAB working platform (version 7.10.0). Then, the performance of the proposed technique was analyzed with time delay system which is described in Equation (2). The implementation parameters of the proposed technique is described in the following Table 1,

Then, the performance of the system analyzed after applying PID controller, Ziegler-Nichols tuning method and the proposed technique. The performance of the system with PID controller, Ziegler-Nichols method and the proposed technique are described in Figs. 3–5 respectively. In Fig. 5, PID controller without tuning system has large overshoot as 58% and high settling time as 850 sec. Figure 5, the Ziegler-Nichols based tuning system has overshoot as 8% and high settling time as 800 sec. But when using the proposed tuning method, the system has fast response and less settling time as 65 sec. The comparison performance of this technique is described in Fig. 6. The comparison is differentiated the fractional order system tuning performance of proposed method to other methods.

Then, the performance of the fraction order system analyzed at different time delay (θ = 20, 40, 60, 80 and 100) instants. The bode diagram of the proposed system is evaluated subsequent setting the described time delay values. The performance of the bode diagram are illustrated in Fig. 7a–e respectively. Then, the step response tuning performance of the system is analyzed by different tuning methods such as proposed iterative method, Ziegler-Nichols method and PID controller. The comparison performance of the system step response tuning is described in Fig. 8 at time delay θ = 20. The comparison performance of the system step response tuning is described in Fig. 9 at time delay θ = 40. The comparison performance of the system step response tuning is described in Fig. 10 at time delay θ = 60. The comparison performance of the system step response tuning is described in Fig. 11 at time delay θ = 80. The comparison performance of the system step response tuning is described in Fig. 12 at time delay θ = 100.

When time delay occur, the bode diagram shows that the system output is oscillated. Therefore, the rise time, overshoot, settling time and steady state error of the system gets increased. But after tuning by the proposed method, the oscillation and the instability of the system were decreased. Moreover, the proposed tuning method based system has reached the stability margin in all time delay cases. Then, the tuning performance of the proposed method compared with PID controller and Ziegler-Nichols method. The comparison analysis shows that, the proposed iterative and curve fitting technique get better for tuning the system response when compared to Ziegler-Nichols method and PID controller. In addition, the fast response and the settling time of the proposed tuning based system are maintained in all the time delay instants.

5 Conclusion

The paper a hybrid technique was proposed for tuning the Time Delay System with PID controller. In the hybrid technique, curve fitting technique was utilized to derive the time delay system response in the form of polynomial equation. Then, the best polynomial coefficients were selected by iterative algorithm and the system response was tuned. The tuning performance of proposed technique is examined by different delay time such as θ= 20 sec, 40 sec, 60 sec, 80 sec and 100 sec respectively. Then, the response of proposed hybrid tuning technique was compared with classical PID controller, Ziegler–Nichols PID tuning method. The comparative analysis shows that, the proposed hybrid technique tuning the Time Delay System better than those other tuning methods. When compared to classical tuning methods, the proposed hybrid technique has less rise time, overshoot, settling time and steady state error. The proposed method reduce the error 0.0011, 0.0170, and −2.7591e-05 by tuning proportional, integral and derivative. So, the proposed hybrid technique tuning the Time Delay System with PID controller perfectly.

In future work, the performance of the proposed tuning method will be evaluated with other Time Delay System. Moreover instead of polynomial curve fitting model, exponential curve fitting model will be used.