Proceedings of the 25th Chinese Control Conference 7-11 August, 2006, Harbin, Heilongjiang
Variable Structure Control Design of Process Plant Based on Sliding Mode Approach Hazim H. Tahir, Ali A. A. Al-Rawi Mechatronics Department, Control and Mechatronics Research Center, Electronics Systems and Communication Office, Ministry of Science and Technology, Baghdad, Iraq. E-mails:
[email protected] ,
[email protected] Abstract: This paper deals with basic concepts, mathematics, and design aspects of variable structure controller based on
sliding mode as a principle operation approach. After reviewing the operation of the plant, a variable –structure system (VSS) based scheme is formulated to control the outflow rate. The scheme is then designed and tested by Simulink Toolbox Ver. 3.0 of Matlab Ver. 6P5. Simulations are presented and have shown that VSS control is better and more versatile compared with the conventional approach using PID controller. Key Words: Variable Structure Control, Sliding Mode Control, Non-linear Control Design, Process Control.
1 Introduction 1.1
This is due to the fact that all the states of the system have to be accessible and have to be monitored by measuring sensors. Generally, this kind of design is used widely in military applications [17, 18].
A Brief Survey
Variable structure control (VSC) with sliding mode control was first proposed and developed in the early 1950’s in Russia by Emel'yanov and several co researchers[2,3,4,5]. In their pioneer works, the plant considered a linear second order system modelled in phase variable form. Since then, VSC has developed into a general design method that is examined for a wide spectrum of system types including Non-linear System, Multi–Input /Multi–Output systems (MIMO), Stochastic Systems, Discrete–Time Models, Large–Scale and Infinite – Dimensional Systems. In addition, the objective of VSC has been greatly extended from stabilisation to other control functions. The most distinguished feature of VSC is its ability to result in very robust control systems that possess good transient performance, fast response, insensitive to parameter variations and external disturbances (noises)[8, 11, 16, 17]. Sliding mode control design is an approach whose structure is intentionally changed with a discontinuous control which drives the phase trajectory to a stable hyperplane or manifold [12, 13, 14]. This design method is basically formulated from the time domain point of view which uses a Lyapunov function– based control law to ensure the closed–loop stability. Specifically, stability to the switching surface, requires selecting a generalised Lyapunov function which is positive definite and has a negative time derivative in the region of attraction [4, 5, 12]. Owing to all the mentioned features, there is a logical design of other types of control systems, but it has to be said that this design is very expensive to implement practically. ___________________________
1.2 The Aim The purpose of this paper is to present a new scheme, using the theory of variable–structure systems (VSSs) based on sliding mode control, to control a laboratory process control [6]. The process control consists of a reservoir tank that feeds water via a precisor–driven control valve and delay channel to a capacitance tank from the bottom of which the water flows through an orifice–type flow meter to a sump tank. From this it is returned via a pump to the reservoir. This arrangement is shown in figures (1- 2) respectively.
IEEE Catalog Number: 06EX1310
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The two “ capacitance–resistance “ time constants (1.1 and 0.21 minutes) are associated with the capacitance tank and delay channel respectively, while the “ transport lag “ (0.25 minutes) is associated with the delay channel. The transport lag element e – 0. 25s is easily represented as a fairly good first–order approximation by:
Fig. 5 shows the way of measuring the transfer function of the plant G(s).
The transducers and the controllers could easily be electronic, but in a system of this kind are more typically mixed electronic and pneumatic units. Two types of controllers were designed to monitor the capacitance tank outflow rate [controlled variable (c.v.)], compare this with a set point and drive the valve process input m(t) [manipulated variable (m.v.)] according to the difference. This paper documents a laboratory process control simulation, which has been conducted firstly by using PID controller and secondly by using VSC control based on sliding mode. All simulations were performed using Matlab package Ver. 6P5 plus Simulink toolbox Ver. 3.0 on a personal computer (PC), see figures (3 - 4) respectively. The results are presented for comparison and further studies.
3 PID Controller Design The controller transfer function, F (s), is:
Where: KP is the proportional gain of the controller, KP / τi is the integral gain of the controller, Kp τd is the differential gain of the controller. Empirical method of Ziegler & Nichols [1] was used and the following controller parameters found experimentally:
2 Plant Transfer Function A series of tests using a transfer function analyser and suitable elector– pneumatic transducers give open–loop transfer function of the plant, or system [6] (Time units, minutes):
KP = 1.2 τi = 0.96 min. τd = 0.3 min.
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Fig. 6 shows the system open–loop and the closed–loop responses of the plant that provides interesting comparison.
Where Ci are the coefficients in the switching function in (4.4) [5, 7, 8, 15], which define the characteristic equation of the sliding mode, (i.e. allowing the control to depend on the state X1 and its derivatives up to order (k – 1)), and one can get:
Hence necessary and sufficient conditions for the existence of a sliding mode are:
4 The Basic of VSC To explain the basic concept of VSC, one can take nth– order plant with derivative feedback by considering the general system [7]:
As for stability on the hyperplane, σ , it can be shown that under the conditions (4. 6) the sliding modes on the hyperplane σ ≈ 0 are stable if and only if all roots of the characteristic equation:
have negative real parts, with the possible exception of the root λ = Cn-1 – an .
With feedback control (depending on Xi):
5
The Design Problem
The design problem is to determine values of αi, βi and Ci such that the states would be brought from any initial position in the phase plane to the switching hyperplane σ = 0. Then, the state trajectory would slide along the switching hyperplane to the phase–plane origin [9, 10]. The phase trajectory along the switching line is known as the "Sliding Mode" of this controller [4, 5, 8]. Whenever the state leaves the switching hyperplane, the controller changes structure to force the point back to the switching hyperplane.
Where:
Which is switched on the hyperplane or manifold σ when S: σ ≈ 0,
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Second Case: if x3 is not accessible, new values of αi, βi, Ci and σ should be calculated as follow:
5.1 VSC of The Process Plant The transfer function of the plant (2.1) can be written as:
6 Simulation Results The Simulink implementation of the process plant and PID controller were straightforward and very easily done. The researchers found difficulties and problems in implementing the “VSC“ controller, especially “ψ“ the switching gains, (i.e. α and β), since this type of problem was tackled and tried first time by the researchers using Simulink Toolbox. After continuous trials they overcame the problem. They found the solutions and found another way of obtaining optimum tuning for the PID controller by using NCD toolbox [Non linear control Design]. Fig. 3 presents the process control with the conventional PID controller design and the NCD design together for comparison purposes. Fig. 4 shows the process control with VSC controller design. Comparative results, shown in fig. 7, which describes the output responses for step input, obtained using the three versions of controllers, namely: a- Conventional PID controller b- Non–linear Control Design (NCD Optimum Tuning). c- VSC controller based on sliding mode.
By using (4. 1) one can transform the system transfer function to companion canonical form as:
From (4.2) the following Control Law was used to enable fast hitting and minimal chattering in steady state:
The first term (proportional term) of (5.3) is standard control law of sliding mode controller, the second term ψ2 x2 and third term ψ3 x3 are included to enable an improvement in rejecting disturbance and faster hitting [4, 5, 12, 13] and the hyperplane or manifold σ was found using (4.4):
From (4.6a – 4.6d), (5.2), (5.3) & (5.4) one can calculate the values of αi, βi and Ci and they were found to be: First Case: It can be concluded that the proposed VSC controller yielded a better dynamic performance than the other two controllers, in terms of Rising Time, Settling Time, Maximum Overshoot and Steady–State Error. The results of the proposed VSC controller can be compared with other two controllers namely:
(i.e. the parameters αi, βi and Ci were initiated and tuned by trial and error method).
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Conventional PID and Non-linear Control Design NCDPID. Table (1) summarises the outcome results.
Figures (8 - 9) show the sliding mode of the VSC hyperplane in 3– Dimension and in 2-Dimension respectively (i.e. when all states are presents and when x3 is absent).
Fig. 10 shows the hyperplane σ tends to zero as soon as possible for the two cases.
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[7] S. P. Banks: “Control Systems Engineering “, Prentice / Hall International Ltd., 1986. [8] J. Y. Hung, W. Gao and J. C. Hung: “Variable Structure Control: A survey ", IEEE Transactions on Industrial Electronics, Vol. 40, No. 1, Feb. 1993. [9] K. S. Young, C. Cheng and C. Kwan: “A Unifying Design of Sliding Mode and Classical Controller “, IEEE Trans. Automat. Contr., Vol. 38, PP. 1422 – 1427, 1993. [10] S. H. Zak and S, Hui: “On Variable Structure Output Feedback Controllers for Uncertain Dynamic Systems”, IEEE Trans. Automat. Contr., Vol. 38, PP. 1509 – 1512, 1993. [11] V. I. UTKIN: “Sliding Mode Control Design Principles and Applications to Electric Drives “, IEEE Trans. Ind. Electr., Vol. 40, No. 1, PP. 23 – 36, Feb. 1993. [12] W. Gao, Y. Wang and A. Homaifa: “ Discrete – Time Variable Structure Control Systems ", IEEE Trans. Ind. Electr., Vol. 42, No .2, PP. 117 – 122, April 1995. [13] Y. B. Shtessel: “Non-linear Output Tracking in Conventional and Dynamic Sliding Manifolds “, IEEE Trans. Automat. Contr., Vol. 42, No. 9, PP. 1282 – 1286, Sept. 1997. [14] S. C. Chung and C. Lin:” A General Class of Sliding Surface for Sliding Mode Control “, IEEE Trans. Automat. Contr., Vol. 43, No. 1, PP. 115 – 119, Jan. 1998. [15] A. Bartoszewicz: “Discrete–Time Quasi– Sliding Mode Control Strategies “, IEEE Trans. Ind. Electronics. Vol. 45, No. 4, PP. 633 – 637, August 1998. [16] A. A. Ahmed, H. H. TAHIR and A. A. ALI: “Digital Simulation of D. C. Motor Position Control Based on the Variable – Structure System (VSS)", 1st Specialist Conference on Control, M.P.E., Nov. 1999, Baghdad, IRAQ. [17] A. A. Ali, H. H. Tahir and A. A. Ahmed: “Designing Digital Autopilot for SSM Based on Variable Structure Controller (VSC)", 3rd Scientific Conference for M.E.C., June 2000, Baghdad, IRAQ. [18] Jian Wang, Hendrik Van Brussel, and Jan Swevers:” Robust Perfect Tracking Control with Discrete Sliding Mode Controller”, Trans. of the ASME J. of Dynamic Systems, Measurement, and Control, 125:27–32, March 2003.
7 Conclusions The purpose of this paper is to present a new scheme, using the theory of variable–structure systems (VSSs) based on sliding mode control, to control a laboratory process control. After reviewing the operation of the plant, a variable –structure system (VSS) based scheme is formulated to control the outflow rate. Simulations are presented and are showed that VSS control is better and more versatile compared with the conventional approach using PID controller. Complete simulations on three versions of controllers were done namely: a- Conventional PID controller. b- Non–linear Control Design (NCD Optimum Tuning). c- VSC controller based on sliding mode. The conventional, Non–linear Control Design (NCD) and VSC controllers are presented for reference and comparison purposes. Any of the controllers may be approved by themselves without any lag in concept. It can be concluded that the proposed VSC controller yielded a better dynamic performance than the other two controllers, in terms of Rising Time, Settling Time, Maximum Overshoot and Steady–State Error, but it has to be said that this design is very expensive to implement practically. Generally, this kind of design is used widely in military applications.
Acknowledgments The authors wish to thank the Control and Mechatronics research centre for their technical support, especially their colleagues Dr. Samir I. Abbas, Dr. Saad A. R. Makki and Dr. Rami A. Mahir for their comments and suggestions.
REFERENCES [1] J. G. Ziegler and N. B. Nichols: “Optimum Settings for Automatic Controllers”, Trans. of ASME, Vol. 64, Ch. 4, PP. 759 – 768, 1942. [2] S. V. Emel’yanov: “Use of Non-linear Correcting Devices of Switch Type to Improve the Quality of Second Order Automatic Control System ", Automate. I Telemekh, Vol. 20, No 7, 1959, (In Russian). [3] E. A. Barbashin and E. I. Gerashchenko. :" On Introduction to Sliding Modes in Control Systems", Differential Equations, Vol. 1, No. 1, 1965. [4] U. ITKIS: “Control Systems of Variable Structure “, John Wiley, 1976. [5] V. I. UTKIN: “Sliding Modes and Their Applications in Variable Structure Systems ", MIR, Moscow, 1978. [6] J. A. Borrie, “Modern Control Systems: A manual of Design Methods ", Prentice / Hall International Ltd., 1986. 937
