Implementation Analysis of State Space Modelling and Control of Nonlinear Process using PID algorithm in MATLAB and PROTEUS Environment

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1 Applied Mechanics and Materials Online: ISSN: , Vol. 573, pp doi: / Trans Tech Publications, Switzerland Implementation Analysis of State Space Modelling and Control of Nonlinear Process using PID algorithm in MATLAB and PROTEUS Environment G.Prabhakar 1, a*, P. Nedumal Pugazhenthi 2, b, Dr.S.Selvaperumal 3, c 1 Assistant Professor, 2, 3 Associate Professor, Department of EEE, Syed Ammal Engineering College, Ramanathapuram, Tamilnadu, India a gprabhakar2488@gmail.com, b neduaupci@rediffmail.com, c perumalvnr@gmail.com Keywords: State space model, Non-linear systems, PID controller, Cruise control, Simulink, Proteus Abstract. The scope of this paper is to look beyond linear solutions and discuss briefly about the recent developments in nonlinear system control & controller tuning methods. An adaptive cruise control model is taken as case study to illustrate the practicality of implementation in the simulation environment in terms of modelling and control of non-linear process. Effort is made to analyse the nonlinear system control in MATLAB and PROTEUS environment. Introduction Many & modern real-time systems are inherently nonlinear in nature. Nonlinear systems can have multiple equilibrium points and limit cycles. In general, most of the non linear system equations are linearised around a stable operating point and analysis is carried out. In this work, the first order non linear adaptive cruise control system is taken into consideration and its model is expressed by the state space representation. Also the model is controlled by PI & PID controller with different tuning algorithms [12], [22], [23] in simulation environment. Non Linear Design Analysis The differential geometric approach [1] is the transformation of a system into a linear one with the help of feedback and coordinates. The notion of zero dynamics is used to achieve the local asymptotic stability, asymptotic tracking, disturbance decoupling and model matching. Inputto-state stability (ISS) [3] is used to obtain a desirable stability condition with respect to actuator errors by redesigning the feedback & it provides a necessary and sufficiency test in terms of ISS Lyapunov function. A recursive control design procedure which has an upper triangular structure for non linear systems is called as forwarding technique. In this design, parameters are carefully selected to obtain the feedback interconnection between the systems which satisfies the small gain conditions. Lyapunov approaches [5-7] are practically very difficult due to the hard construction of exact cross term. Backstepping [6], [8] has a lower triangular structure with different recursive design. A misconception is that the interlaced designs [5], [6] apply also to special structures (half upper and half lower structures). Sliding mode control [9] can be taken as a recursive design procedure similar to backstepping. Singular perturbations considering the neglected high frequency phenomena as a separate fast time scale which is achieved by treating a change in the dynamic order of a system of differential equations as a parameter perturbation [2]. Stable inversion/output regulation [10] The Byrnes-Isidori [1] regulator generalizes internal model principle to nonlinear systems that can be applied to track any trajectory generated by a given ecosystem, if one can solve the associated Partial Differential Equations. The stable inversion technique [10] trades the requirement of solving these general PDES for a specific trajectory. Both tools can deal with the unstable zero dynamics that cannot be dealt with by the conventional inversion technique. Local model (LM) networks by Johansen and Foss in [13], [14] as a means of decomposing non-linear auto-regressive moving average with exogenous inputs (NARMAX) models into an insightful structure for system identification and control. Murray-Smith [15], [16] presented further All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of Trans Tech Publications, (# , Pennsylvania State University, University Park, USA-19/09/16,18:26:01)

2 298 Advancements in Automation and Control Technologies reports on LM network, which presented this approach as one of the standard techniques to combine linear models and ANN to characterise the non-linearity. Analysis of the PID Controller A PID controller may be considered as an extreme form of a phase lead-lag compensator. It consists of one pole at the origin and another pole at infinity. Similarly, it relates, the PI and the PD controllers, can also be regarded respectively as extreme models of phase-lag and phase-lead compensators. A standard PID controller is also named as the three-term controller. The transfer function is generally written in the parallel form given by (1) = + + (1) Where k p is the proportional gain, k d the derivative gain, k i the integral gain, T i the integral time constant and, T d the derivative time constant. The three-term functionalities are highlighted by the following. 1) The proportional term provides an overall control action proportional to the error signal through the all-pass gain factor. 2) Steady-state errors are reduced by the integral term through low-frequency compensation by an integrator. 3) The transient response is improved by the derivative term through high-frequency compensation by a differentiator. Selection of Controller Tuning Based on Model The process model is first-order lag plus delay (FOLPD) The FOPDT model is widely used in process control, and various tuning methods based on the model are proposed in the literature [20], [21]. An important factor of the model is the normalized delay (ratio of the time delay and the time constant τ m /T m ), which shows how large the real delay of the system is [12], [18-20], [22]. In this paper, FOPDT models whose τ m /T m is less than equal to 1 are taken into consideration to attain a quarter decay ratios for the nominal closed loop control system. The tuning formulas considered seem quite out of date. However, we believe that these methods are representative, and they are generally bases of comparison for more recent tuning methods. We find out that there are lots of PID design and tuning methods reported in the papers, for example refer [21-23] the performance of these methods can be evaluated via the proposed method. Due to space limit and our emphasis, we compare only few tuning rules such as Ziegler and Nichols (1942), Astrom and Hagglund (1995), Chien et al. (1952) servo, Murrill (1967) min. ISE, St. Clair (1997) Case Study This paper focuses on cruise control which has a nonlinear model and for better control the following considerations are made Design Consideration. A cruise control system needs to accelerate to the desired speed in a short time without overshooting the speed of the car. Also, it needs to maintain the speed with little deviation, when the car is driving up or down a steep hill. Physical Model. This case study begins with model construction, starting from basic physics concept. An important consideration is given to incorporate wind resistance and include as a system disturbance, the onset of a hill. The Newton s law of motion states that given a vehicle mass m, air resistance B and engine supplied force cu(t), where c is proportionality constant and 0 u(t) 1 represents the engine throttle, F tot (t) = mv(t) = cu(t), t 0. Air resistance, proportional to the velocity squared times constant B, produces drag on the vehicle. Additionally when the vehicle encounters a hill of angle θ, gravity creates a second counter force termed as mg sin (θ), where g is

3 Applied Mechanics and Materials Vol the gravitational constant of 9.8 m/s 2. A small angle is assumed in this case, so the following is valid. Sinθ θ. The equation of motion is now Fig.1 System Model = (2) Vehicle velocity is a nonlinear differential equation in terms of v(t) because of the air resistance term v 2 (t). Observations from Nonlinear Differential Equation. Although the nonlinear differential equation is not in a form suitable for design of an adaptive cruise control, some interesting observations that can be made. This understanding also helps with the linearising process. Observation 1. When the vehicle is on the horizontal floor (θ = 0) and at the highest throttle value of 1.0, the reduction of nonlinear differential equation is = The highest velocity will be reached as t becomes large. At highest velocity the acceleration must be zero, so 0 h The equation (2) further simplifies to = = Observation 2. When climbing a hill at full throttle the vehicle stalls for some critical angle θ s. The stall represents the meaning of vehicle velocity is zero and the acceleration is also zero. From the full nonlinear differential equation = If we solve for θ s as sin 1 [c/(mg)]. This discussion assumes that the vehicle stays in a fixed gear. While driving a car we probably downshift to the lowest gear to avoid stalling. Observation 3. The solution to the nonlinear differential equation at highest throttle and θ = 0. Defining the constant =, Consider =, = (3) = Where, = Or =. Differentiating with respect to t, we get = Substitute in equation 4 (4) = (5) Hence the state space car model equation is = (6)

4 300 Advancements in Automation and Control Technologies The absolute solution to this simplified form is v(t) = v max tanh(t ). This final information provides you the velocity profile versus time with the accelerator held to the floor. Design Specifications. For analysis, we have considered the following system with design specifications such as Mass (m)-1250 kg, Resistance (B) Nsec/m, proportionality constant(c) N Tuning PID Controllers The result of a PID controller is given by = [+ + ] (7) Where = proportional gain, = integral time, = derivative time As a mathematical model of the plant can be derived, we can able to apply various design methods for determining parameters of the controller that will meet the transient and steady-state specifications of the closed-loop system. The selection process of controller parameters to meet given performance specifications is known as controller tuning. Ziegler and Nichols (1942) proposed rules for tuning PID controllers (for determining values of, & ) based on the transient response characteristics of a given plant. Ziegler-Nichols Open Loop Response Method. This idea is also known as reaction curve (openloop) method. The conceptual idea of open loop testing is to begin with a steady-state process by making a step change to the final control element and record the results of the process output. Zeigler-Nichols transient response method will work on any system that has an open-loop step response which is an essentially critically damped or over damped character like that shown in Figure 2. Information produced by the open loop test is the open-loop gain K m, the loop apparent dead time, and the loop time constant, T m. The nonlinear state model is given a step response and from the open loop step response, the model is approximated as a FOPDT. = 1+ Fig. 2 Open Loop Response Curve By using this method, k m = 1, T m = 8 & 8 are calculated. Comparison of Tuning Rules Based on ISE, IAE & ITAE. The measured values k m, T m & are applied to the above mentioned tuning rules to find out, which is the versatile rule for our nonlinear model to achieve the stable conditions and the corresponding ISE, IAE & ITAE values are tabulated below

5 Applied Mechanics and Materials Vol Table 1. Integral Error Analysis-PI Table 2. Integral Error Analysis-PID Simulink Modeling & Results The Simulation results provided in fig 4(B) and 4(C) is based on Murrill (1967) proposed tuning algorithm, since it shows less error than other tuning methods. (A) (B) (C) Fig. 4 A) System Model [ = ] B) Servo Response C) Regulatory Response Proteus Real time Modeling & Results (A) (B) Fig. 5 A) Before Controller Action [Set RPM=100, Attained RPM=82 B) After Controller Action [Set RPM=100, Attained RPM=100]

6 302 Advancements in Automation and Control Technologies The real time implementation analysis of non linear model of cruise control prototype using DC motor is simulated in Proteus virtual system modelling. The Murrill (1967) proposed PID tuning algorithm with values k p =1.36, T i = 0.1, T d = 3.04 can be used to control a DC Motor with Encoder. The disc attached with encoder generates a pulse signal depending upon the rotation of shaft. These pulses are read by a Microcontroller (PIC18F458), and the Motor Voltage is controlled by the PWM pulses which are regulated with the help of PID controller algorithm, and thus the resulting RPM. Conclusion and Future Scope The nonlinear cruise control system considered in the case study is modeled using state space representation and it is approximated using open loop Z-N tuning methods as FOPDT. Various PI and PID tuning rules were applied to verify the performance measures like ISE, IAE, and ITAE etc., which are available in the literature. This analysis is simulated in MATLAB and PROTEUS environment. The authors have an idea of building a real time model and analyze the performance of the nonlinear system in the future. The performance analysis can also be tested by applying nonlinear PID in future. References [1] Isidori, A. (1995), Nonlinear Control System (3rd edition), Springer. [2] Kokotović; P. Khalil, H. & O Reilly, J. (1986), Singular Perturbation Methods in Control Analysis and Design, Academic Press Inc... [3] Sontag, E. (1990), Further facts about input to state stabilization, IEEE Transactions on Automatic Control, Vol. 35, pp [4] Sontag, E. (2005), Input to state stability Basic concepts and results, Springer Lecture Notes in Mathematics, Springer. [5] Sepulchre, R., Janković, M. & Kokotović, P. (1997). Constructive Nonlinear Control, Springer, pp [6] Sepulchre, R.; Janković M. & Kokotović, P. (1997), Integrator forwarding a new recursive nonlinear robust design, Automatica, Vol. 393, pp [7] Mazenc, F. & Praly, L. (1996), Adding integrations, saturated controls, and stabilization for feed-forward systems, IEEE Transactions on Automatic Control, Vol.41, pp [8] Krstić M., Kanellakopoulos, L. & Kokotović, P. (1995). Nonlinear and Adaptive Control Design, John Wiley & Sons. [9] Utkin, V. (1992), Sliding modes in control optimization, Springer-Verlag. [10] Devasia, D.; Chen, D. & Paden, B. (1996), Nonlinear inversion based output tracking, IEEE Transactions on Automatic Control, Vol. 41, pp [11] Benchmark, Guangyu Liu and Yanxin Zhang, On Nonlinear Control Perspectives of a Challenging, the University of Auckland New Zealand [12] Ruiyao Gao, Aidan O'Dwyer, Eugene Coyle, A non-linear PID controller for CSTR using local model networks, Proceedings of the IEEE 4th World Congress on Intelligent Control and Automation (WCICA 2002), Shanghai, China, June. [13] Johansen T.A. and Foss B.A., A NARMAX model representation for adaptive control based on local models, Modelling, Identification, and Control, Vol.13, No.1, 1992, pp [14] Johansen, T.A and Foss B.A., Constructing NARMAX models using ARMAX models, International Journal of Control, Vol.58, 1993, pp [15] Murray-Smith R, Local Model networks and local learning, in Fuzzy Duisburg, 94, pp. p , Feb.1994 [16] Murray-Smith R. and Johansen T. A., Multiple Model approaches to Modelling and Control, Taylor and Francis, 1997.

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