Closed-loop Schemes for Position and Sway Control of a Gantry Crane System

Transcription

1 AHMAD ALHASSAN et al: CLOSED-LOOP SCHEMES FOR POSIION AND SWAY CONROL OF A GANRY Closed-loop Schemes for Position and Sway Control of a Gantry Crane System Ahmad Alhassan, Kumeresan A. Danapalasingam*, Muhammad Shehu, Auwalu M. Abdullahi, Auwal Shehu Department of Control and Mechatronics Engineering Faculty of Electrical Engineering, Universiti eknologi Malaysia Johor, Malaysia *Corresponding Author / kumeresan@fke.utm.my, EL: Abstract his paper presents the investigation into the performance of Lyapunov pole placement (), linear quadratic regulator () and proportional-integral-derivative () control schemes for payload sway control and trolley position tracking of a gantry crane system. A D gantry crane system is considered. he nonlinear model of the system is derived using the Lagrangian energy equation and then linearized using aylor s series expansion. o investigate the performances of the designed controllers, a unit step input as a reference perturbation is applied to the controllers. MALAB simulation results of the responses are analysed in time domain. he response time specifications of the trolley position, level of payload sway reduction, and robustness to parameter variation and uncertainties are used to assess the performances of the controllers. Keywords-closed-loop; gantry crane; Lagrange; linearization; ; Lyapunov;, simulation, aylor series I. INRODUCION Cranes are the most widely used tools to transport various types of goods efficiently and reliably from one point to another []. Gantry crane, tower crane and boom crane are the three major types of cranes used today []. Due to its cost effectiveness and ease of operation, gantry crane system (GCS) is the most preferred crane system in the industries, shipping yards, mining sites, power plants, warehouses etc [3][4]. However, GCS are prone to vibration and deflection of the payload during operation and or in the presence of external disturbance (obstacle). his lead to inaccurate positioning of the load, delay in task completion and even a damage to the system or the operating environment [],[5]. Interestingly, to improve the system throughput, guarantee safety of the environment and minimize maintenance cost due to system failure, many researchers have engaged in developing the mathematical model of the system for precise dynamic analysis and effective control []. he dynamic behaviour of the non linear GCS using varying system parameters; trolley mass, payload mass and cable length was presented in [],[3][6]. It was observed that, payload oscillation and trolley displacement are highly dependent on those parameters. In order to improve the performance of the GCS, many control strategies were presented. A simple state feedback controller (SFB) using Ackerman s formula was presented in [8]. he main issue with SFB is that the states of the system must be measurable and the gains depend on the accuracy of the model. Optimal controller was proposed using weight summation approach in [7]. Due to its simplicity of design and implementation, is applicable to many industrial applications. Adaptive controller was also designed using corrective control parameter in the presence of uncertainties in [8][9]. Sliding mode control (SMC) was also developed by assuming constant cable length in [],[]. his scheme was also improved by incorporating real time analysis using variable cable length in []. SMC retains the stability of the system and it is insensitive to modelling errors. However, it leads to dissipation of energy or even burn out of the system (chattering) [3]. More so, Fuzzy logic controller was also developed to stabilized the responses of the GCS in [4],[5]. Intelligent control offers ease of execution and efficient control due to its ability to treat inaccurate model. However, significant parameter variation affects its performance [3]. In addition, input shaping (IS) technique was proposed for vibration control of flexible manipulator in [6]. As an open loop control, IS it s simple to design and cost effective as it does not require feedback control or additional sensors. It only requires estimated natural frequency and damping ratio of the system [7]. Conversely, small disturbance or variation in the system parameters significantly affects its performance. Linear quadratic regulator () was also proposed in [8] for balancing and control of an inverted pendulum. modern control uses the ideas of weighting matrices to achieved optimal conditions for the states and the control input. As a closed loop optimal controller, is effective and robust to uncertainties [9][]. In this paper, an investigation into the performance of Lyapunov pole placement (), linear quadratic regulator () and proportional-integral-derivative () control schemes for payload sway control and trolley position tracking of a gantry crane system is presented. A D gantry crane system is considered. he non linear model of the system is derived using the Lagrangian energy equation and then linearized using aylor s series expansion. o investigate the performances of the designed controllers, a unit step input as a reference perturbation is applied to the controllers. MALAB simulation results of the responses are analysed in time domain. he response time specifications of the trolley position, level of payload sway reduction and robustness to parameter variation are used to assess the performances of the controllers. he robustness of the controllers is assessed by changing the payload mass, cable length and a sine wave input disturbance. DOI.53/IJSSS.a ISSN: x online, print

2 AHMAD ALHASSAN et al: CLOSED-LOOP SCHEMES FOR POSIION AND SWAY CONROL OF A GANRY he paper is organised as follows. In section II, description of a gantry crane system is discussed. Section III presents the mathematical modelling of the system. In section IV, the derived nonlinear model is linearized using aylor s series expansion and presented in state space form. Section V describes the proposed control schemes. he implementation and discussion of the proposed controllers is discussed in section VI. Conclusion is finally presented in section VII. II. DESCRIPION OF A GANRY CRANE SYSEM he gantry crane system considered in this work is shown in Fig.. he trolley slides along the horizontal jib. he jib is supported by a pair of legs. A payload is suspended via a suspension cable to a trolley. he schematic diagram of GCS is shown in Fig.. he applied force, F, causes the trolley of mass, m to move a distance, x. his motion leads to a deflection angle, of the payload. he trolley carries the load of mass, m via am of hoisting cable of length, l. he values of the system parameters used for this study are tabulated in able I[4] III. MODELLLING OF NONLINEAR GANRY CRANE his section presents the nonlinear mathematical modeling of a nonlinear gantry crane using the Lagrangian energy equations. o simplify the model derivation, the following assumptions were adopted: (i) the force, F is considered as the input to the system (ii) the mass of the hoisting cable is neglected (iii) external disturbances are neglected (iv) the effect of friction on the trolley is also neglected. he Lagrange s equations are given as [4] d L L Qi, i,,, n dt q qi i () L P () where P and are respectively the total potential and kinetic energy, n is total number of independent generalized coordinate and Q i is non conservative generalized forces. he kinetic energy of the trolley given as: mx (3) he kinetic energy of the payload is given as m v (4) he velocity analysis of fig. using cosine rule gives ABLE I. Fig.. Gantry crane system (GCS) Fig.. Schematic diagram of GCS Parameters rolley mass (m) Payload mass (m) Cable length (l) DESCRIPION OF PARAMEERS Value (unit). Kg.5 Kg.5 m Acceleration due to gravity (g) 9.8 m/s v x l xl cos (5) m( x l xl cos) (6) herefore, () becomes L mx l xl cos mx mglcos (7) herefore, the non-linear model of gantry crane system can be summarized as: ( ) (8) m m x mlcos ml sin F m l m lxcos m glsin (9) DOI.53/IJSSS.a ISSN: x online, print

3 AHMAD ALHASSAN et al: CLOSED-LOOP SCHEMES FOR POSIION AND SWAY CONROL OF A GANRY IV. LINEARIZAION OF HE NON LINEAR MODEL his section discusses the linearization process of the derived non linear model. Assuming a very small deflection angle of the payload ( << o ). he nonlinear terms; sine and cosine can be linearized about an operating point using ailor s series expansion ' ( x x ) '' f x f xx x f x f x! n ( x x ) n f x n! hus, for a small deflection angle of the payload, () ; ; f sin ; f cos () Hence, the derived equation (8) and (9) of the non-linear model can be approximately linearized as: ( m m ) x m l F () ml mlx mgl (3) Also, the desired responses; payload oscillation and the trolley displacement, can be re-arranged as: mg x F (4) m m ( m m ) g F ml ml (5) V. CONROL DESIGN In this section, three control strategies were designed based on Lyapunov pole placement, and schemes. he crane system is presented in the general form for statevector equation as follows []: x Ax bu (6) y Cx Du (7) where A represents the system state matrix, b is the output matrix, C represents the output matrix and y is the system output. By using the linear model of equation (4) and (5), the state-vector equation matrices A, b and C can be presented as: A m g m g( m m ) lm, m b lm C (8) A pre-requisite to designing a control strategy is to investigate the system's stability. In this case, the system has two poles located at the origin and two conjugate imaginary poles (,, +5j, -5j). For this reason, the response of GCS is undesirable and behaves as an undamped oscillator. he system controllability was investigated by determining the rank of the controllability matrix G c as G C b Ab A b 3 Ab C G 96.5 (9) hus, the controllability matrix is nonsingular; full rank and hence, the system is controllable. A. Lyapunov pole placement scheme In this section, pole placement control based on Lyapunov approach is presented. By assuming a measurable state vector, x [ x x ], a control law u kx can be implemented to the system. he Lyapunov function is given as A F bk () [ ] () k k k k k3 k4 he poles of the closed loop system can be selected along the negative s- plane arbitrarily. In this case, fourth order conjugate poles are chosen as.5.5 j and j. he matrix F of () can be formed from the assigned Eigen values in a block diagonal form as F, k [ ] () By substituting for A,b, k, and F in () yields DOI.53/IJSSS.a ISSN: x online, print

4 AHMAD ALHASSAN et al: CLOSED-LOOP SCHEMES FOR POSIION AND SWAY CONROL OF A GANRY DOI.53/IJSSS.a ISSN: x online, print By solving for in (3) and substituting the value in () yields the controller gains as [ ] k (4) he general block diagram of a state feedback control is shown in Fig. 3. o obtain zero steady state error, a gain is added to the reference signal for the position tracking as.5 ( ) k in C A bk b (5) B. Linear Quadratic Regulator () control is a common approach employed in the control of hub angle and position of crane system []. he structure of is given in Fig. 3. he design of control requires a linear state-vector model. Hence, linearized state space model of equation (8) was utilized. he method involves obtaining a control law U=-Kx that derives the system state to the origin (i.e. to zero) at the same time minimizing the performance index function, J with minimal control effort given in [3] as ( ( ) ( ) ( ) ( )) J xt Qxt U t RU t dt (6) where Q is a symmetrical positive semi-definite matrix called state penalty matrix and R is the positive definite symmetrical matrix known as control action penalty. For single input system, R reduces to a single number. hus, J represents weighted energy cost of the state and control. o design, the penalty weighting matrices Q and R are selected such that; If the elements of Q are relatively large compared to that of R, then heavy penalty is applied to the deviations of the state x from the origin in comparison to the deviations of the control action from zero. (3) On the other hand, selecting elements of Q to be relatively small compared to the elements of R will result in costly control action and the system state x will not return or converge fast to the origin. he control law U=-Kx that minimizes the performance index function J is called Kalman s gain. For a LI system with cost function J, the optimal regulator is always a linear control law. For the closed-loop system, the system takes the following form [] K R B P (7) () ( ) () x t A BK x t (8) ( ( ) ( ) J x Qx Kx R Kx dt (9) he matrix P is obtained by solving the algebraic Riccati equation given as A P P A Q P B R B P (3) he closed-loop form of equation (8) is always stable if matrix P is positive definite. he state penalty matrix Q and the control effort penalty matrix R were selected as.75 Q, /4 R (3) Using MALAB command; lqr(a,b,q,r), the control gain was calculated as

5 AHMAD ALHASSAN et al: CLOSED-LOOP SCHEMES FOR POSIION AND SWAY CONROL OF A GANRY k [ ], k. (3) C. Proportional-Integral-Derivative control () In this section, control is presented. he general representation of is given as d uc() t Kpe() t Ki e() t dtkd e() t (33) dt where u(t) is the control signal, K i, K p and K d are respectively the integral, proportional and derivative gains. e(t) is the undesired error calculated by taking the difference between the actual signal and the output response. can be tuned manually or automatically to meet the desired response based on the three available features. he proportional gain is responsible for the system response. However, faster response leads to steady state error. he integral gain takes care of the steady error. he derivative feature reduces overshoot. hus, if those features are not tuned properly, it may affect the closed loop stability of the system. he block diagram of a control scheme is shown in Fig. 4. he signal is applied to the gantry crane. he resulting responses will be feedback for comparison with the reference input. he controller is tuned to make this error zero. i VI. IMPLEMENAION AND RESUL In this section, implementation and discussion of the results is presented. o study the dynamics of the proposed controllers, a unit step input is applied to the system. his is sufficient to make the GCS moves and then stop at the desired position based on the given parameters (exact) of able I. and were implemented based on the obtained control gains whereas the was tuned by trial and error. For a better performance of the, double is utilized. One for position tracking and the other for sway control as shown in Fig. 5. he performance of the, and control for the position tracking and payload sway control are respectively shown in Fig ime response specifications and level of sway reduction were used to assess the control performances. For the sway control, mean absolute error (MAE) of the payload sway is utilized. Small MAE means less sway and hence, the better the performance of the controller. It can be observed that gives a better sway reduction as compares to and whereas provide better position tracking as compared to and in terms of settling time (S) and rise time (R) and steady state (SS) error. / controller Signal K i + Crane System Output response Fig. 3. Block diagram of / control K.4. Fig. 5. A simulink block for double controller + - et () Kp.() et t i K et () ut () Crane System Output Response rolley position (m) et () Kd t Fig. 4. Block diagram of controller ime (s) Fig. 6. rolley position the exact parameters. DOI.53/IJSSS.a ISSN: x online, print

6 AHMAD ALHASSAN et al: CLOSED-LOOP SCHEMES FOR POSIION AND SWAY CONROL OF A GANRY o investigate the robustness of the designed controllers, the payload mass was increased to.kg. Later, the cable length was also changed to.m. he results were obtained based on those values separately as shown in Fig 8-. his acts as a parameter variation procedure. It can also be noticed that the variation of the parameters affects the performance of LLP significantly as compared to and. o further test the external disturbance rejection capability of the controllers, a continuous sine wave disturbance (SWD) of.5 peak to peak magnitudes is introduced to the system. his makes the controller unstable as shown in Fig. -3. However, the performances of demonstrated that the controller effectively rejects external disturbance and variation of the system parameters while was affected slightly. able II summarized the position tracking and sway control performance of the designed controllers. Payload oscillation (rad) Payload oscillation (rad) Payload position (m) ime (s) Fig. 9. Payload sway for.kg trolley mass PP ime (s) Fig. 7. Payload sway for the exact parameters ime (s) Fig.. rolley position for.m cable length. rolley position (m).5 Payload oscillation (rad) ime (s) Fig. 8. rolley position for.kg trolley mass ime (s) Fig.. Payload sway for.m cable length. DOI.53/IJSSS.a ISSN: x online, print

7 AHMAD ALHASSAN et al: CLOSED-LOOP SCHEMES FOR POSIION AND SWAY CONROL OF A GANRY Payload oscillation (rad) rolley position (m) ABLE II. LLP ime (s) Fig.. rolley position for a sine wave disturbance ime (s) Fig. 3. Payload sway for a sine wave disturbance. LEVEL OF SWAY AND RESPONSE SPECIFICAIONS Controller R (s) S (s) SS error MAE Exact kg m SWD Exact kg m SWD Exact kg m SWD VII. CONCLUSION In conclusion, this paper investigates the performance of LLP, and controllers for trolley position tracking and payload sway suppression. he effectiveness of the designed controllers have been assessed in terms of trolley position tracking, level of payload sway reduction, and robustness to parameter variation and uncertainties. Without an external disturbance, acceptable performances have been achieved with all the controllers. A comparative analysis of the results has shown that double and provides precise position tracking with fast response whereas LLP gives better sway reduction. However, best disturbance rejection was achieved using double compared to both and LLP. ACKNOWLEDGMEN his work was supported by Universiti eknologi Malaysia (UM), the Fundamental Research Grant Scheme (R.J F73) from the Ministry of Higher Education Malaysia and the esciencefund (R.J S) from the Ministry of Science, echnology and Innovation Malaysia. REFERENCES [] N. Đ. Zrni, V. M. Ga, and S. M. Bo, Dynamic responses of a gantry crane system due to a moving body considered as moving oscillator, Arch. Civ. Mech. Eng., pp. 9, 4. [] A. Masoud, Dynamics and Control of Cranes : A Review, Vib. Control, vol. 9, pp , 3. [3] V. S. Renuka and A.. Mathew, Precise Modelling of a Gantry Crane System Including Friction, 3D Angular Swing and Hoisting Cable Flexibility, Int. J. heor. Appl. Res. Mech. Eng., vol., pp. 9 5, 3. [4] H. Izzuan, Z. Mohamed, J. J. Jamian, A. Faiz, and Z. Abidin, Dynamic Behaviour of a Nonlinear Gantry Crane System, Procedia echnol., vol., no. Iceei, pp , 3. [5] J. Yoon, S. Nation, W. Singhose, and J. E. Vaughan, Control of Crane Payloads hat Bounce During Hoisting, IEEE rans. Control Syst. echnol., vol., no. 3, pp , 4. [6] A. B. Alhassan, B. B. Muhammad, K. A. Danapalasingam, and Y. Sam, Optimal Analysis and Control of D Nonlinear Gantry Crane System, IEEE Int. Conf. Smart Sensors Appl., pp. 3 35, 5. [7] H. I. Jaafar and M. F. Sulaima, Optimal Controller Parameters for Nonlinear Gantry Crane System via MOPSO echnique, IEEE Conf. Sustain. Util. Dev. Eng. echnol., pp. 86 9, 3. [8] C. S. eo, K. K. an, S. Y. Lim, S. Huang, and E. B. ay, Dynamic modeling and adaptive control of a H-type gantry stage, Mechatronics, vol. 7, pp , 7. [9] N. Sun, Y. Fang, and H. Chen, Adaptive antiswing control for cranes in the presence of rail length constraints and uncertainties, 5. [] G. Bartolini, A. Pisano, and E. Usai, Second-order sliding-mode control of container cranes, Automatica, vol. 38, no., pp ,. [] Q. H. Ngo and K. Hong, Sliding-Mode Antisway Control of an Offshore Container Crane, vol. 7, no., pp. 9,. [] L. A. uan, S. Moon, W. G. Lee, and S. Lee, Adaptive sliding mode control of overhead cranes with varying cable length, J. Mech. Sci. echnol., vol. 7, no. 3, pp , 3. [3] C. ai and K. Andrew, Review of Control and Sensor System of Flexible Manipulator, J. Intell. Robot Syst., pp. 87 3, 4. [4] J. Jalani, Robust Fuzzy Logic Controller for an Intelligent Gantry Crane System, First Int. Conf. Ind. Inf. Syst. ICIIS, Sri Lanka, pp , August 6. DOI.53/IJSSS.a ISSN: x online, print

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