Fractional-Order Sliding Mode Control of Overhead Cranes

the robustness. This control structure is inflexible since using fixed gains. For overcoming this weakness, we integrate variable fractional-order derivative into SMC that leads to an adaptive system with adjustable parameters. We use Mittag–Leffler stability, an enhanced version of Lyapunov theory, to analyze the convergence of closed-loop system. Applying the controller to a practical crane shows the efficiency of proposed control approach. The controller works well and keeps the output responses consistent despite the large variation of crane parameters. Keywords: Fractional-order control, overhead cranes, sliding mode control. 1. Introduction* performance. This study utilizes strong points of both SMC and FD. We create a flexibly robust controller Using frequently in industrial transportation, an by integrating FD into SMC core. The controller overhead crane is equipped with three mechanisms tracks trolley and hoist payload to desired positions, for lifting and transferring material and package in concurrently keeps the payload swing small at factories. The modern cranes speedy run or/and transient state, and suppresses this swing at steady combine the motions for increasing productivity. This state. Such the combination applied for overhead causes imprecise motions and large swing of payload cranes has not been has not been released until if cranes do not have suitable control strategies. Many recently. Proposed control system shows the control algorithms were proposed for overhead cranes following advantages: [1-18]. The articles [1-12] in connection to control of overhead cranes were published from linear control (i) The controller well immunizes disturbances [1], feedback linearization [2], pole-placement [3], and is robust with uncertainties due to the action of linear quadratic optimal control [4], to complicated SMC structures. nonlinear control [5], model predict control [6], (ii) Due to using variable FD, control structure adaptive control approach [7-8], robust control may vary flexibility to adapt with uncertain methods [9-10], and modern control techniques such environment. Tuning FD orders may get the optimal as fuzzy logic [11], neural networks [12]. Focusing system responses. on robust controls, SMC [13] is a control approach which works effectively for cranes. SMC does not In fact, control formulation of cranes is require much the modelling precision. It may work classified into low-level control (LLC) and high-level well with un-modeled dynamics. Additionally, it planning (HLP). LLC deals with precise tracking shows the robustness when the system faces trolley and crane to destinations while keeping disturbances and uncertainties. Its advanced versions payload swing small and vanishing at steady-state. such as terminal SMC [14], fast terminal SMC [15] HLP develops algorithms for motion planning and make quick finite-time convergence. SMC can trajectories to prevent the obstacles. At modern crane, combine with fuzzy [16], neural network [17], self- two control options may be combined. tuning control [18] to achieve both robust and adaptive features. Accordingly, adaptive mechanisms This study focuses on LLC. We organize the article as follows. Section 2 provides concept of are supplemented for estimating uncertainties, fractional calculus [7] and stability theory for unknown factors, and disturbances. On the upside, fractional systems [8-10]. Section 3 describes fractional calculus has been widely applied to control engineering in the recent years [22]. Instead of using physical features of an overhead crane through its fixed first/second-order derivatives for feedback dynamic model. Section 4 designs controller by combining SMC with FD, analyzing crane stability signals, fractional-derivative (FD) makes flexible with Mittag–Leffler sense [9] is also included. control structure in which it is tunable to get optimal Section 5 tests the control algorithms on a practical overhead crane, analyzes and discusses the ISSN: 2734-9373 application results. Finally, several conclusions and https://doi.org/10.51316/jst.150.ssad.2021.31.1.9 remarks are represented in Section 6. Received: 12 January 2021; accepted: 11 March 2021 68 JST: Smart Systems and Devices Volume 31, Issue 1, May 2021, 068-075 Fig. 1. 2D motion of an overhead crane. 2. Fractional Derivative in Control Theorem 2 [22]: Fractional LTI system We utilize the concept of fractional calculus and α Dtt x()= Ax () t (5) related topic to design control system. The following definition, theorem, and lemma will be applied for with A∈Rn×n is stable if it justifies analyzing and constituting the control algorithm at the next sections: arg(eigA )> απ / 2 (6) Definition 1 [19]: Fractional-order α derivative 3. Dynamical Description of function f(t) with time defined by Caputo is given We consider simultaneous motion of trolley mt as and payload hoist ml when operating an overhead 1 t crane as in Fig. 1. Three outputs composed of moving αα= −τm−−1 ττ Dt ft() ∫ (t ) g ( )d (1) trolley x, lifting (with varying cable length l) payload Γ−()m α 0 mc, and swinging θ payload are controlled by two ∞ inputs: ut is trolley-pushing force and ul is payload + −−tz1 with 0<α<1, m∈Z and Γ=()z∫ et d t being lifting force. bt and br are parameters for frictions at 0 trolley and cable. gamma function. Dynamic behavior of cranes is governed by Lemma 1 [20]: Let x(t)∈Rn be a state vector. actuated model: The inequality   (mtc+− m ) x m c sinθ l − ml c cos θθ ααTT≤ (7) 0.5DDtt (xQx ) xQ x (2)  2 +−bxtc 2 m cosθ l θ + ml c sin θθ = u t ( t ) ∈ n×n is held for every positive definite matrix Q R .   −mcsinθ  x ++ ( m cl m ) l + bl r Theorem 1 [21]: If existing a continuously (8) −−mlθθ2 mgcos = u ( t ) differentiable function V(x,t) satisfying cc l T a ab corresponding qa=[x l] , and swinging equation αα12xx≤≤Vt( , ) x (3) θ =(1/l )(cos θθ xl −− 2  g sin θ ) (9) and describes payload swing θ. Substituting Eq. (9) into α ≤−α ab DVt (xx, t) 3 , (4) Eqs. (7), (8) and rearranging leads to a reduced-order dynamics then x=0 is global stability in the sense of Mittag- α α α Leffler. Here, a, b, 1, 2, and 3 are positive Mqaa++ Bq  f(,) q q  = U (10) constants. 69 JST: Smart Systems and Devices Volume 31, Issue 1, May 2021, 068-075 T equilibrium = is globally stable. The convergence where, q=[x l θ] are system outputs. M = []mij 22× is s 0 of manifold (11) produces a linear fractional-order a symmetric matrix with 2 system m11 = mmtc + sinθ , mm12= 21 = − mc sinθ , and Dα ()()qq−=−−−λq q βθ (18) m22 = mmcl + . B=diag(bt,br) denotes damping. t a ad a ad 22T fqq( , )= [(mlc sinθθ + 0.5 mgc sin 2θ ) −+ ( mlcc θ mgcos θ )] , Physically, θ is always toward 0 due to payload and U = []uuT is outputs. weight even without control, and gain β supports the tl fast convergence of θ. Eq. (18) is reduced as 4. Controller Design α Dt()()qq a−=−− ad λqa q ad (19) We design a structure for tracking qa to T destination qad=[x l] while suppressing payload Applying Theorem 2 to system (19) indicates swing (θ reaches zero). We consider a sliding that (qa−qad) is locally stabile around 0 for every manifold containing fractional derivative positive definite matrix λ. Thus, tracking qa to qad is α achieved. s=Dt()() qq a −+ ad λqa −+ q ad βθ (11) Remark 1: Sign function of control law (12) may where, 0≤α≤1 is fractional-order (FO) of Caputo cause the chattering at system responses. There are derivative (Definition 1), λ=diag(λ1,λ2) is a positive several ways to reduce this, such as: replacing sign T matrix, and β=[β1 0] . A control law is proposed action by situation function or sigmoid function, compatible with surface (11) by a following using a filter or estimator, higher-order solution, statement. super-twisting method, and so on. In this article, we utilize a hyperbolic tangent function Statement: A structure of fractional sliding mode − control eessii− = = tanhs [tanhsii ] ss− (20) (2−α ) ii+ UM=−D {(λq −+ q ) βθ } eei t a ad (12) ++ − Bqa f( q , q )ηs sgn replacing for sgns function. tracks outputs q governed by crane dynamics (7)-(9) Remark 2: Theoretically, sliding surface (11) of T to destinations qd=[xd ld 0] asymptotically. Here, controller (12) only assures the infinity convergence. η=diag(η1,η2) are two positive gains. In fact, the control law (12) is designed based on the infinity stability of the dynamics Proof: We begin with a bounded Lyapunov function Dα s +=ηsgn s0 (21) T V =0.5s Ms > 0 (13) An enhanced version [10, 14, 15] of SMC, that is the so-called Terminal SMC (TSMC), guarantees Since m=+>( mm sin2 θ ) 0 and 11 tc the finite-time stability of system outputs. By 2 detM = [mt ( m c ++ m l ) mm cl sinθ ] > 0, M is improving the dynamics of sliding surface as positive definite. Based on Lemma 1, fractional α qp/ D ss++λ ηsgn s0 = (22) derivative of Lyapunov (13) satisfies ααT with η=diag(η1,η2) being positive diagonal matrix, q DVtt≤ sM D s (14) and p being positive odd integers satisfying q>p, we T 2(αα− 1) α =sM[DDt q a +λ t ()] qq a −+ ad β D t θ can obtain the TSMC controller in which the terminal stability of sliding manifold is held. Submitting equivalent dynamics (10) and controller (12) into Eq. (14) yields 5. Results and Analysis DVαα≤=−sMTT D s sηs D2( α− 1) sgn (15) We check the quality of proposed controller (12) ttt on crane dynamics (7)-(9) using a laboratory α − 2( 1) overhead crane whose parameters: mt= 5 kg, mc= 0.85 For α≠1, Caputo derivative Dt sgns = 0 leads inequality (15) to kg, ml= 2 kg, bt= 20 Ns/m, br=Ns/m, and FO-SMC gains: λ=diag(0.7,0.9), β=[1.2 0]T, η=diag(100,100). α T DVt ≤ 0 (16) The initial conditions: q(0)=[0 0.1 0] and q0(0) = . We use three cases of fractional order α = 0.8, 0.9, The case α=1 leads inequality (15) to and 1. The references of trolley and payload hoisting α respectively compose of destinations: xd=[0 0.3 0.1 DVt ≤−ηη11 s − 2 s 2 (17) 0.7] m and ld=[0.1 0.4 0.6 0.2] m. The crane The expressions (13), (16), and (17) fit performances are depicted in Figs. 2-6. conditions of Theorem 1. This implies that 70 JST: Smart Systems and Devices Volume 31, Issue 1, May 2021, 068-075 1 0.8 0.6 0.4 FO=0.8 FO=0.9 Displacement (m) FO=1 0.2 0 0 5 10 15 20 25 Time (s) Fig. 2. Trolley motion. 0.7 FO=0.8 FO=0.9 0.6 FO=1 0.5 0.4 0.3 Cable Length (m) 0.2 0.1 0 5 10 15 20 25 Time (s) Fig. 3. Payload hoist. 50 FO=0.8 40 FO=0.9 FO=1 30 20 10 Angle (degree) 0 -10 -20 0 5 10 15 20 25 Time (s) Fig. 4. Payload swing. 71 JST: Smart Systems and Devices Volume 31, Issue 1, May 2021, 068-075 10000 FO=0.8 8000 FO=0.9 FO=1 6000 4000 2000 Force (N) 0 -2000 -4000 0 5 10 15 20 25 Time (s) Fig. 5. Trolley-moving force. 3000 FO=0.8 2000 FO=0.9 FO=1 1000 0 -1000 Force (N) -2000 -3000 -4000 0 5 10 15 20 25 Time (s) Fig. 6. Payload-hoisting force. Proper selection of FO decides the control As a nature of SMC, the proposed controller quality. Seen at Fig. 2, FO=0.8 causes much assures the system robustness despite disturbance and oscillation of trolley motion, both FO=0.9 and 1 uncertain environment. Considering the case FO=1, assure the destination convergence, FO=0.9 makes we investigate the consistence of output responses overshoot while FO=1 does not. Varying FO from its when a crane faces parametric uncertainties. In fact, origin FO=1 can reduce settling time but causes the many crane parameters are variable and thus overshoots and even steady-state errors. Payload adjustable. A crane can lift and transfer the payload hoisting (Fig. 3) seems well for cases, however, with various mass mc and volume. The frictions convergence speed of FO=0.8 is fastest. Payload characterized by bt and br are varied up to swings are in small boundary (Fig. 4) at transient environment, temperature, and humidity of operation phase and absolutely suppressed at payload area. We consider the variation of three above- destinations. Depicted in Figs. 5 and 6, control inputs mentioned parameters with two following cases: remain keen peaks due to high switched gains of ∆ ∆ ∆ − controller. Generally, it is hard to say which FO is the Case 1: [ mc bt br]=[100 20 10]%. best. Finding FO to achieve the optimal responses Case 2: [∆mc ∆bt ∆br]=[−50 −40 30]%. will be studied in the next article. 72 JST: Smart Systems and Devices Volume 31, Issue 1, May 2021, 068-075 0.7 Origin 0.6 Uncertainties-Case 1 Uncertainties-Case 1 0.5 0.4 0.3 0.2 Displacement (m) 0.1 0 0 5 10 15 20 25 Time (s) Fig. 7. Robustness of trolley motion. 0.8 0.7 0.6 0.5 0.4 Origin 0.3 Cable Length (m) Uncertainties-Case 1 Uncertainties-Case 2 0.2 0.1 0 5 10 15 20 25 Time (s) Fig. 8. Robustness of payload-hoist. 10 Origin 8 Uncertainties-Case 1 Uncertainties-Case 2 6 4 2 0 Angle (degree) -2 -4 -6 0 5 10 15 20 25 Time (s) Fig. 9. Robustness of payload swing. 73 JST: Smart Systems and Devices Volume 31, Issue 1, May 2021, 068-075 2500 2000 Origin Uncertainties-Case 1 1500 Uncertainties-Case 2 1000 500 Force (N) 0 -500 -1000 0 5 10 15 20 25 Time (s) Fig. 10. Robustness of trolley-pushing force. 1000 Origin Uncertainties-Case 1 Uncertainties-Case 2 500 0 Force (N) -500 -1000 0 5 10 15 20 25 Time (s) Fig. 11. Robustness of payload-hoisting force. The simulation results when the system is FO that are considered as flexible control gains. subject to two cases of parametric uncertainties in Trolley and payload lifting responses reach comparison with original case are depicted in Fig.7 to destinations precisely while well vanishing payload 11. Despite parametric variations, trolley motion swing. Enhancing for 3D motion and integrating (Fig. 7) and payload swing (Fig. 9) still approach adaptive control approaches will be conducted in the destination precisely. Parametric uncertainties only future study. impact on outputs at transient states in which it Acknowledgments causes small derivation. Hoisting the payload (Fig. 8) seems sensitive with the variation of parameters. It This research is funded by Vietnam National induces much not only transient-state derivations but Foundation for Science and Technology also steady-state errors. Increasing swished gains Development under grant number 107.01-2019.301. η=diag(η1,η2) will improve robust feature however References cause much chattering. The way to prevent chattering was discussed in Section 4. [1] Y. Sakawa, H. Sano, Nonlinear model and linear robust control of overhead travelling cranes, 6. Conclusion Nonlinear Analysis, Theory, Methods & Applications By integrating FD into SMC structure, we 30 (1997) 2197-2207. https://doi.org/10.1016/S0362-546X(97)86041-5 successfully created a robust controller for tracking overhead crane with simultaneous drive of three [2] L.A. Tuan, S.-G. Lee, V.-H. Dang, S. Moon, B.S. motions. 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