Experimental veriﬁcation of SMC with moving switching lines applied to hoisting crane vertical motion control

1. Experimental verification of SMC with moving switching lines applied to hoisting crane vertical motion control A. Nowacka-Leverton a,n , M. Micha"ek b , D. Pazderski b , A. Bartoszewicz a a Technical University of Ło´dz´, Institute of Automatic Control, Ło´dz´, Poland b Poznan´ University of Technology, Chair of Control and Systems Engineering, Poznan´, Poland a r t i c l e i n f o Article history: Received 10 November 2011 Received in revised form 9 April 2012 Accepted 13 May 2012 Available online 12 June 2012 Keywords: Switching line design Sliding mode control State constraints Hoisting crane a b s t r a c t In this paper we propose sliding mode control strategies for the point-to-point motion control of a hoisting crane. The strategies employ time-varying switching lines (characterized by a constant angle of inclination) which move either with a constant deceleration or a constant velocity to the origin of the error state space. An appropriate design of these switching lines results in non-oscillatory convergence of the regulation error in the closed-loop system. Parameters of the lines are selected optimally in the sense of two criteria, i.e. integral absolute error (IAE) and integral of the time multiplied by the absolute error (ITAE). Furthermore, the velocity and acceleration constraints are explicitly taken into account in the optimization process. Theoretical considerations are verified by experimental tests conducted on a laboratory scale hoisting crane. & 2012 ISA. Published by Elsevier Ltd. All rights reserved. 1. Introduction Position control of hoisting cranes and other cable-driven mechanisms have recently become an important research issue [5–7,13–15,17–19]. Hoisting cranes are widely present in the industry as well as in everyday life (for example lifts in high buildings). In many practical applications a reference set-point value of the crane payload should be achieved monotonically (without overshoots or oscillations) and as fast as possible, however subject to acceleration and velocity constraints. Position control of rope-suspended systems is also complicated by the fact that the suspension rope can exert only unidirectional force on the payload. Consequently, the rate of motion velocity change should not exceed the gravitational acceleration during the payload lowering in order to maintain positive tensions in cables and preserve forcing capability. Moreover, it is expected in practice that the properties mentioned above will be achieved in the presence of (bounded) model uncertainties or external disturbances. In this paper we present an application of the sliding mode technique [1,8–12,16,21] for a point-to-point motion control of a hoisting crane payload under parametric uncertainties of the system model. We consider two alternative control strategies, both of them employing time-varying sliding lines [2–4,20,22]. The lines are designed in such a way that the system representative point on the phase plane belongs to them already at the initial time. As a consequence, the reaching phase is eliminated leading to robustness of the closed-loop system with respect to model uncertainties and external disturbances from the very beginning of the control process. We propose and compare alternative line synthesis procedures using two quality criteria. Design procedures proposed in the paper ensure that the acceleration and velocity constraints imposed by a user are satisfied and the non-oscillatory, fast error convergence in the resultant control systems is obtained. At this stage it is worth to point out that dynamic properties of every control system operating in the sliding mode can be analysed with respect to two different time scales. The first one – called ‘‘fast motion’’ time scale – may be applied to analyse the system behaviour in any, even an infinitely small time period. With reference to this time scale the system velocity is a non-differentiable function of time and the system acceleration is undefined on any finite interval. This is because in any finite time period the control signal switches an infinite number of times between its two different values: the one generated when the switching variable is positive, and the other one obtained when this variable is negative. With reference to this time scale the system acceleration cannot be determined at any non-zero measure set, i.e. at any set of practical engineering importance. On the other hand, the second time scale – some- times called ‘‘slow motion’’ or macroscopic time scale – makes it possible to analyse the ‘‘average’’ system dynamics. With reference to this time scale the system velocity is a differentiable function of time and the system acceleration indeed exists. These Contents lists available at SciVerse ScienceDirect journal homepage: www.elsevier.com/locate/isatrans ISA Transactions 0019-0578/$ - see front matter & 2012 ISA. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.isatra.2012.05.003 n Corresponding author. E-mail address: aleksandra.nowacka-leverton@p.lodz.pl (A. Nowacka-Leverton). ISA Transactions 51 (2012) 682–693

2. remarks show that the system acceleration can only be practically meaningful when defined in the ‘‘average’’ sense. Therefore, whenever in this paper acceleration is considered, it is understood as the smooth, average signal (related to the notion of equivalent control) and not as its discontinuous counterpart. Indeed further in the paper, the system acceleration is defined as a signal of practical importance which can be measured with accelerometers or perceived for example by lift passengers. The paper consists of two parts. The first, theoretical part of the paper introduces the switching control law and its continuous fractional approximation which allows attenuating the chattering phenomenon. Stability of the closed-loop system for both control laws is analyzed using the Lyapunov method. Next, the optimal synthesis of the sliding lines is presented and the conditions necessary to satisfy the imposed constraints are explained. The second, experimental part of the paper is devoted to practical verification of the proposed algorithms on a laboratory-scale hoisting crane commercially available from Inteco Ltd. (www. inteco.com.pl). Finally, based on the obtained results qualitative comparison of the proposed solutions is presented. 2. System model In this paper we take into account a hoisting crane whose mechanical structure is illustrated in Fig. 1. From the d’Alembert principle, assuming for simplicity (similarly as in [6,7,13]) that the rope flexibility is negligibly small, we obtain the following three equations of motion Jm €qm þfmðqm, _qmÞ ¼ tmÀZtd, ð1Þ Jd €qd þf dðqd, _qdÞ ¼ tdÀtg, ð2Þ m€x þfgðx, _xÞþm Á g ¼ tg=r, ð3Þ where we have, respectively: qm,qd,x is an angular position of the hoisting motor, an angular position of the hoisting drum, and a linear position of the payload; tm,td,tg is motor driving torque, motor torque exerted on a drum shaft, and dynamic load torque (resulting from the motion effects and static influence of the payload mass); Jm,Jd is motor moment of inertia, and hoisting drum moment of inertia; m is payload mass; f m,f d,f g are functions representing unmodeled dynamics of the hoisting motor, the hoisting drum, and the payload mass, respectively; g is gravity acceleration; r is hoisting drum radius; Z is gear reduction ratio between the hoisting motor and the hoisting drum (Z ¼ _qd=_qm o1). Assuming that inductance of the armature winding is negli- gible the driving torque tm may be expressed as a function of armature winding voltage tm ¼ ki Á i % ki R ðuÀke Á _qmÞ, ð4Þ where i is an armature circuit current, u is an armature voltage (physically realized control input signal), ki,ke are machine con- stants, R is an armature winding resistance. Combining Eqs. (1)–(4), then using linear relations between qm, qd and x (related to each other by the gear ratio Z), and introducing the state variables x1 ¼ x and x2 ¼ _x, one can obtain the following hoisting crane dynamics model with the armature voltage as a control input _x1 ¼ x2, _x2 ¼ Fðx1,x2Þþ 1 a u, ð5Þ with Fðx1,x2Þ ¼ À R kia Zf d x1 r , x2 r h þf m x1 rZ , x2 rZ þZrf gðx1,x2ÞþZrmg À b a x2, ð6Þ where a ¼ R rki ðJdZþJmZÀ1 þmr2 ZÞ, b ¼ ke rZ : ð7Þ Motivated by possible practical difficulties in modeling and identification of crane dynamics we assume that: A1. parameter a40 is not known exactly, A2. function Fðx1,x2Þ describing model uncertainty of system (5) is bounded, i.e. for any x1 and x2 the following inequality holds 9Fðx1,x2Þ9om. In the next section, we propose a discontinuous sliding mode control for system (5) and its continuous approximation appropriate for practical applications. 3. Sliding mode controller In this paper, we denote by x1d and x2d the desired value of the payload position and its velocity. The system state error can be determined as follows e1 ¼ x1Àx1d, e2 ¼ x2Àx2d: ð8Þ We analyze point-to-point motion control problem for which x1d ¼ const: and xd2 ¼ 0. As a result velocity error e2 ¼ x2. Further- more, in order to design a sliding mode control we introduce a time-varying switching line. The line is described as follows s ¼ e2 þce1 þwðtÞ ¼ 0, ð9Þ where c40 is a constant parameter determining the line angle of inclination. We assume that function w : Rþ /R is continuous and determined as wðtÞ ¼ pðtÞ for tA½0,tf Þ, 0 for tA½tf ,1Þ, ( ð10Þ where tf 40 denotes the time instant when the line stops moving. Furthermore, p : Rþ /R is a differentiable function with bounded first derivative (compare [8]) defined in such a way that condition pðtf Þ ¼ 0 holds, which guarantees that w(t) is continuous at t ¼ tf . In general, the line motion is determined by the selection of function p(t). As c ¼ const:, the line considered in this paper moves to the origin of the error state space with a constant angle of inclination. Having reached the origin at time tf, it stops moving and remains fixed.Fig. 1. Mechanical structure of a hoisting crane. A. Nowacka-Leverton et al. / ISA Transactions 51 (2012) 682–693 683

3. In the paper we consider the robust control problem in the presence of state constraints. The problem may be defined as follows Problem 1. Find bounded control u ¼ uðe1,e2,tÞ for system (5) which guarantees that: R1. position and velocity errors, e1 and e2, converge to zero asymptotically, i.e. lim t-1 ½e1ðtÞ e2ðtÞŠT ¼ 0, ð11Þ R2. the payload does not exhibit oscillations during transient stage, i.e. 8tZ0 signðe1ðtÞÞ ¼ signðe1ð0ÞÞ and 8t40 signðe2ðtÞÞ ¼ const ., ð12Þ R3. payload velocity and acceleration are constrained during regulation process, i.e. 8tZ0 9x2ðtÞ9rVm ð13Þ and 8tZ0 9_x2ðtÞ9rAm, ð14Þ where Vm 40 and Am 40 determine the maximum admissible values of the velocity and the acceleration, respectively, R4. requirements R1, R2 and R3 are satisfied under assumptions A1 and A2. Note that the upper bounds Am and Vm can result directly from drive limits or can be chosen arbitrarily by a user treating them as additional design parameters. First we formulate the following proposition for the control law which ensures sliding motion on the time-varying line defined by (9). Proposition 1. The following sliding mode control law u ¼ À â½g sgnðsÞþce2 þ _wðtÞŠ, g40, ð15Þ with the sufficiently large coefficient g and â40 being an estimate of parameter a, applied to system (5) with uncertainty (6), ensures sliding motion on switching line (9). Proof. Calculating the time derivative of function V ¼ 1 2 s2 one obtains _V ¼ s½_e2 þc_e1 þ _wðtÞŠ ¼ sF þs 1 a uþsce2 þs _wðtÞ ¼ sFÀ â a g9s9þs 1À â a ½ce2 þ _wðtÞŠ ¼ sFÀ â a g9s9þsf a½cx2 þ _wðtÞŠ r9s9 9F9À â a g þ9s99f a99cx2 þ _wðtÞ9 r9s9 mÀ â a gþ9fa99cx2 þ _wðtÞ9 , ð16Þ where f a ¼ 1À â=a. Furthermore, if gZsup x2 ½Dðx2ÞŠ, ð17Þ where Dðx2Þ ¼ a â ½mþkþ9fa99cx2 þ _wðtÞ9Š, ð18Þ then _V rÀk9s9, ð19Þ where k40 is an arbitrarily small constant. Remark 1. Notice that in practical application for any positive value of â one can find big enough value of parameter g. Furthermore, as a result of (15), in the design procedure one should rather consider proper selection of product â Á g than selection of â and g separately. In practical applications, in order to attenuate chattering, function sgnðsÞ in Eq. (15) can be replaced with its continuous approximation given by s=ð9s9þnÞ, where n40 is a small positive design coefficient. Then, the following proposition can be proved. Proposition 2. The following approximation û ¼ À â g s 9s9þn þce2 þ _wðtÞ # ð20Þ of control signal (15) with â40 being an estimate of parameter a, applied to system (5) with uncertainty (6) ensures that variable s remains in the neighborhood of zero with the radius E ¼ n ^Dðx2Þ k 40, ð21Þ where ^Dðx2Þ ¼ mþ9fa99cx2 þ _wðtÞ9, ð22Þ and n40, k40 are design coefficients. Proof. Using, in relation (16), modified control signal (20) instead of (15) one gets _V ¼ s F þ 1 a û þcx2 þ _wðtÞ rfaðcx2 þ _wðtÞÞ9s9þ9s99F9À â a g s2 9s9þn r mÀ â a g 9s9 9s9þn ! 9s9þ9fa99cx2 þ _wðtÞ99s9: ð23Þ Assuming now that g satisfies (17) with Dðx2Þ given by (18) one obtains the following upper bound _V r n ^Dðx2Þ9s9 9s9þn À ks2 9s9þn ¼ n ^Dðx2ÞÀk9s9 9s9þn 9s9, ð24Þ where ^Dðx2Þ is given by (22). Then one can conclude that 9s941=k9n ^Dðx2Þ9 ) _V o0. The above considerations lead to the conclusion that modified control (20) does not bring the representative point of the system exactly onto the switching line but it forces the representative point to the neighborhood of the line, with radius (21). Note that E can be made arbitrarily small by the appropriate choice of coefficient n and parameter g ¼ gðk,ÁÞ. However, practical selection of n and g should be a result of a compromise between motion precision and chattering attenuation. 4. Switching line synthesis In this section we select a time-varying switching line which will lead to an optimal (in the sense of introduced criteria) solution of Problem 1, stated in the previous section. In order to design switching line (9), we assume, similarly as in [2], that function p is selected in the following form pðtÞ ¼ Ct2 þBtþA, ð25Þ where A, B and C are constant parameters. Note that for C a0 the line moves to the origin of the error state space with a constant deceleration and when C¼0, the line moves with a constant velocity. Having reached the origin, line (9) stops moving at the A. Nowacka-Leverton et al. / ISA Transactions 51 (2012) 682–693684

4. time instant tf and remains fixed. First of all, parameters of the line should be chosen in such a way that the system representative point belongs to the line at the initial time t¼0. In this way the insensitivity of the closed-loop system with respect to model uncertainty from the very beginning of the control action is guaranteed. In this paper it is assumed that at the initial time t¼0, e1ð0Þ ¼ e10 a0 and e2ð0Þ ¼ 0. Therefore, the following condition should be satisfied e2ð0Þþce1ð0ÞþA ¼ 0, which implies A ¼ Àce10: ð26Þ Moreover, the selected parameters of the switching line are supposed to ensure the minimum value of the following control quality criteria, either IAE criterion JIAE ¼ Z 1 0 9e1ðtÞ9 dt, ð27Þ or ITAE criterion JITAE ¼ Z 1 0 t9e1ðtÞ9 dt, ð28Þ subject to constraints of the payload velocity and the payload acceleration given by inequalities (13) and (14), respectively. Further in the paper we show how to choose the optimal, in the sense of both criteria, switching lines moving with a constant deceleration and a constant velocity. The procedure for finding the optimal switching line parameters begins with calculation of the regulation error and its derivative. For that purpose first we solve Eq. (9) with function w(t) determined by (25) for tA½0,tf Þ. This reflects the situation when the line moves. Taking into account initial conditions e10 a0 and e20 ¼ 0 and relation (26), we obtain e1ðtÞ ¼ À ðBcÀ2CÞt c2 À Ct2 c þ ÀBcþ2C c3 ðeÀct À1Þþe10, ð29Þ e2ðtÞ ¼ ÀBcþ2C c2 ðÀeÀct þ1ÞÀ 2Ct c : ð30Þ Now we calculate values of system error (29) and its derivative (30) for t ¼ tf e1ðtf Þ ¼ À ðBcÀ2CÞtf c2 À Ct2 f c þ ÀBcþ2C c3 ðeÀctf À1Þþe10, ð31Þ e2ðtf Þ ¼ ÀBcþ2C c2 ðÀeÀctf þ1ÞÀ 2Ctf c , ð32Þ which are initial conditions necessary to solve Eq. (9) for tA½tf ,1Þ, which represents the system dynamics when the line does not move. Then, after some calculations, we obtain the evolution of the error for tZtf e1ðtÞ ¼ ÀðBcÀ2CÞtf c2 þ ðÀBcþ2CÞ c3 ðeÀctf À1ÞÀ Ct2 f c þe10 # eÀct þctf , ð33Þ e2ðtÞ ¼ ðBcÀ2CÞtf c À ðÀBcþ2CÞ c2 ðeÀctf À1ÞþCt2 f Àce10 eÀct þctf : ð34Þ Notice that the regulation error described by (29) and (33) converges to zero monotonically. Then, criterion (27) can be expressed as JIAE ¼ Z 1 0 e1 dt

12. : ð35Þ Substituting (29) and (33) into (35) and calculating appropriate integrals, we obtain JIAE ¼ e10tf þ e10 c À Bt2 f 2c À Ct3 f 3c

22. : ð36Þ On the other hand, the ITAE criterion can be rewritten as JITAE ¼ Z 1 0 te1 dt

30. : ð37Þ Now, substituting (29) and (33) into (37) and calculating appropriate integrals, we obtain JITAE ¼ e10 c2 þ e10tf c À Bt2 f 2c2 þ e10t2 f 2 À Bt3 f 3c À Ct3 f 3c2 À Ct4 f 4c

40. : ð38Þ Since the switching line selection fully determines the system motion and its performance, switching line parameters A, B, C and c should be carefully chosen in accordance with the specified requirements. In order to select these parameters, further in the paper, we minimize (36) and (38) with constraints (13) and (14). 4.1. Constant deceleration switching line Now we carefully analyze the case when C a0, i.e. when for trtf the line moves with a constant deceleration to the origin of the error state space. Notice that for the time tZtf , switching line (9) is fixed and passes through the origin of the error state space. Taking into account relation (25), we conclude that Ct2 f þBtf þA ¼ 0: ð39Þ Furthermore, in order to avoid rapid input changes, the velocity of the introduced line should change smoothly. Thus, the following condition should also hold 2Ctf þB ¼ 0: ð40Þ Using (26), (39) and (40), we obtain tf ¼ 2 e10c B : ð41Þ In order to facilitate further minimization procedure, we define the following positive constant k ¼ e10c2 B : ð42Þ From (42) we obtain c ¼ ffiffiffiffiffiffiffi Bk e10 s : ð43Þ As it was stated above both criteria (36) and (38) will be minimized with two constraints, i.e. with system velocity constraint (13) and acceleration constraint (14). Calculating the maximum value of 9x2ðtÞ9, we get max t 9x2ðtÞ9 ¼ B c lnð1þ2kÞ 2k À1

48. : ð44Þ Then using relation (43) and taking into account condition 9x2ðtÞ9rVm, we obtain the following inequality 9B9r V2 mk 9e109 lnð1þ2kÞ 2k À1 2 : ð45Þ Now we consider the acceleration constraint given by inequality 9_x2ðtÞ9rAm. Similarly as in [2], it can be calculated that the maximum value of 9_x2ðtÞ9 is achieved for t¼0, and it is equal 9B9. Consequently, condition 9_x2ðtÞ9rAm is satisﬁed when the following relation holds 9B9rAm: ð46Þ Further in the paper we show the minimization procedure for both criteria introduced in this section (IAE and ITAE) with the velocity and acceleration constraints. A. Nowacka-Leverton et al. / ISA Transactions 51 (2012) 682–693 685

49. 4.1.1. Minimization of IAE Taking into account relations (40), (41) and (43) from (36), we get the following form of control quality criterion JIAE ¼ 9e109 3=2 ffiffiffiffiffiffiffi 9B9 q 1 ffiffiffi k p þ 2 3 ffiffiffi k p : ð47Þ Fig. 2 illustrates criterion (47) as a function of two variables k and 9B9. It can be seen from the figure that for any value of argument k, the criterion decreases with increasing value of 9B9. As mentioned before, criterion (47) will be minimized with two constraints, i.e. with the system velocity constraint and the system acceleration constraint which are satisfied when relations (45) and (46) hold. Because criterion (47) decreases with increasing value of 9B9, the minimization of criterion JIAE as a function of two variables ðk,9B9Þ with constraint (45) may be replaced by the minimization of the following single variable function JV IAEðkÞ ¼ e2 10 Vm lnð1þ2kÞ 2k À1

57. 2 3 þ 1 k : ð48Þ Closer analysis of this function shows that it reaches its minimum for numerically found argument k IAE vopt % 13:467. On the other hand, considering constraint (46), we get the following single variable function JA IAEðkÞ ¼ 9e109 3=2 ffiffiffiffiffiffiffi Am p 1 ffiffiffi k p þ 2 3 ffiffiffi k p : ð49Þ Criterion (49) achieves the minimum value for k IAE aopt ¼ 3 2. In order to find the minimum of the IAE criterion with both constraints satisfied at the same time, we consider three cases: JV IAEð3 2 ÞrJA IAEð3 2Þ, JV IAEðk IAE voptÞZJA IAEðk IAE voptÞ and JV IAEð3 2 Þ4JA IAEð3 2Þ4JV IAEðk IAE voptÞoJA IAEðk IAE voptÞ. This approach is justified by the fact that criterion (47) reaches its minimum for such value kopt that JIAEðkoptÞ ¼ min max k 40 fJV IAEðkÞ,JA IAEðkÞg : ð50Þ Thus, in the first considered case (when the acceleration constraint is dominant), we obtain that the optimal parameter kopt ¼ k IAE aopt ¼ 3 2. Then, parameter B can be obtained from B ¼ Am sgnðe0Þ: ð51Þ For the second case, the optimal value of k is equal to k IAE vopt % 13:467. Then, the optimal value of parameter B may be calculated from B ¼ V2 mk 9e109 lnð1þ2kÞ 2k À1 h i2 sgnðe0Þ, ð52Þ for k ¼ k IAE vopt. Finally, in the third case, i.e. when JIAE V ð3 2 Þ4 JIAE A ð3 2Þ4JIAE V ðk IAE voptÞoJIAE A ðk IAE voptÞ in order to find kopt, we solve equation f1ðkÞ ¼ 0 in the interval ð3 2,k IAE voptÞ where f1ðkÞ ¼ 9e109 3 2 1 ffiffiffi k p þ 2 3 ffiffiffi k p ffiffiffiffiffiffiffiffiffiffiffi 9e109 q Vm ffiffiffi k p lnð1þ2kÞ 2k À1

65. À ffiffiffiffiffiffiffi 1 Am s0 @ 1 A: ð53Þ Having calculated the optimal value of k, we can derive parameter B using either (51) or (52). Then, other switching line parameters A, C and c can be found from (26), (40) and (43), respectively. The line designed in this way moves with a constant deceleration and it stops moving at the time instant tf given by (41). That ends the first procedure of the switching line design based on the IAE minimization. In the next subsection we take into account the ITAE criterion. Fig. 2. Criterion JIAEðk,9B9Þ. A. Nowacka-Leverton et al. / ISA Transactions 51 (2012) 682–693686

66. 4.1.2. Minimization of ITAE Now we present the ITAE criterion as a function of ðk,BÞ. Substituting (39), (41) and (43) into (38), we get the following form of ITAE JITAE ¼ e2 10 39B9 3 k þ2þk : ð54Þ Notice that, similarly as before, considered criterion (54) reaches its smaller values as argument 9B9 increases which is shown in Fig. 3. Therefore, repeating the considerations presented above, in order to find minimum value of criterion (54) with constraints (45) and (46), we minimize the following single variable function JITAEðkÞ ¼ max k 40 fJV ITAEðkÞ,JA ITAEðkÞg, ð55Þ where JV ITAEðkÞ ¼ 9e109 3 3V2 m lnð1þ2kÞ 2k À1 2 3 k 2 þ 2 k þ1 , ð56Þ and JA ITAEðkÞ ¼ e2 10 39Am9 3 k þ2þk : ð57Þ Notice, that minimization of function (56) results in such a value of k which minimizes the ITAE criterion subject to velocity constraint (13). Analysis of this function leads to the conclusion that criterion (56) reaches its minimum for the numerically found value of k which equals k ITAE vopt % 6:263. On the other hand, when minimization of ITAE with acceleration constraint (14) is required, we minimize function (57) and we obtain that its minimum value is achieved for k ITAE aopt ¼ ffiffiffi 3 p . Since in this paper we consider both constraints (45) and (46) satisfied by the system at the same time, then in order to find optimal parameters of the switching line, we consider three cases: JV ITAEð ffiffiffi 3 p ÞrJA ITAEð ffiffiffi 3 p Þ, JV ITAEðk ITAE voptÞZJA ITAEðk ITAE voptÞ and JV ITAEð ffiffiffi 3 p Þ4 JA ITAEð ffiffiffi 3 p Þ4 JV ITAEðk ITAE voptÞoJA ITAEðk ITAE voptÞ. In the first one, we obtain that the optimal parameter kopt ¼ ffiffiffi 3 p . Then, the optimal parameter B can be calculated using (51). For the second case, optimal value kopt is equal to k ITAE vopt % 6:263, and as a consequence, parameter B is given by (52). In the last case the optimal value of k is the root of function f 2ðkÞ in the interval ð ffiffiffi 3 p ,k ITAE voptÞ where f2ðkÞ ¼ e2 10 3 3 k þ2þk 9e109 lnð1þ2kÞ 2k À1 2 V2 mk À 1 Am 8 : 9 = ; : ð58Þ Having calculated the optimal value of k, we can determine the optimal value of B, using one of expressions (51) or (52). The other switching line parameters A, C, c and tf may be found using formulas (26), (40), (43) and (41), respectively. It is worth pointing out that condition JV ITAEð ffiffiffi 3 p Þ4 JA ITAEð ffiffiffi 3 p Þ 4JV ITAEðk ITAE voptÞoJA ITAEðk ITAE voptÞ ensures that IAE and ITAE solutions are identical. This is a direct consequence of the fact that roots of functions f1ðkÞ and f2ðkÞ are the same for any fixed values of Am, Vm and e10 which can be seen from the following relation 9e109 lnð1þ2kÞ 2k À1 2 V2 mk À 1 Am ¼ ffiffiffiffiffiffiffiffiffiffiffi 9e109 q lnð1þ2kÞ 2k À1

74. Vm ffiffiffi k p À ffiffiffiffiffiffiffi 1 Am s 2 6 6 4 3 7 7 5 Â ffiffiffiffiffiffiffiffiffiffiffi 9e109 q lnð1þ2kÞ 2k À1

82. Vm ffiffiffi k p þ ffiffiffiffiffiffiffi 1 Am s 2 6 6 4 3 7 7 5: ð59Þ In order to illustrate cases which appear during the minimization procedure, Figs. 4–6 are presented. Since the general concept with Fig. 3. Criterion JITAEðk,9B9Þ. A. Nowacka-Leverton et al. / ISA Transactions 51 (2012) 682–693 687

83. the IAE minimization is similar to the method with the ITAE minimization, then in these figures subscript i might represent either IAE or ITAE. The next part of this section is dedicated to the design of the switching line which moves with a constant velocity to the origin of the error state space. Similarly to the previous considerations we will consider two ways of obtaining the line parameters, i.e. the IAE and ITAE minimization with the velocity and acceleration constraints. 4.2. Constant velocity switching line In this subsection we introduce the switching line which moves to the origin of the error state space with a constant velocity. It means that in definition (25) we assume C¼0. That leads to the following conclusions. Because at the time t ¼ tf the line passes through the origin of the space, then the following condition Btf þA ¼ 0 ð60Þ is satisfied. Using relation (26) which is also true now since the system representative point belongs to the line at the initial time t¼0, we obtain tf ¼ e10c B : ð61Þ Using Eqs. (29) and (33) for C¼0, after some calculations one concludes that velocity constraint (13) is satisfied if 9B9r V2 mk 9e109½1ÀexpðÀkÞŠ2 , ð62Þ and furthermore, acceleration constraint (14) holds when inequality (46) is true. Now we move on to the precise analysis of the minimization procedure. It means that we present two algorithms for finding optimal parameter k when criteria IAE and ITAE are minimized subject to constraints (46) and (62). 4.2.1. Minimization of IAE Under the considered assumption C¼0, criterion (36) can be expressed as follows JIAE ¼ e10tf þ e10 c À Bt2 f 2c

93. : ð63Þ Substituting (61) and (43) into (63), we obtain the following form of this criterion JIAE ¼ 9e109 3=2 ffiffiffiffiffiffiffi 9B9 q ffiffiffi k p 2 þ 1 ffiffiffi k p ! , ð64Þ as a function of ðk,BÞ. Notice that also this time, the criterion decreases with increasing value of 9B9, and the plot of criterion (64) is very similar to the one shown in Fig. 2. Minimization of (64) subject to constraints (62) and (46) might be replaced by minimization of an appropriate single variable function without any constraints. Thus, taking into account the velocity constraint expressed by (62), the criterion can be presented as JV IAEðkÞ ¼ e2 10ð1ÀexpðÀkÞÞ Vm 1 k þ 1 2 , ð65Þ and for acceleration constraint (46) we obtain JA IAEðkÞ ¼ 9e109 3=2 ffiffiffiffiffiffiffi Am p ffiffiffi k p 2 þ 1 ffiffiffi k p ! : ð66Þ Minimizing these functions separately, we conclude that JV IAEðkÞ reaches its minimum value when k tends to infinity, and that JA IAEðkÞ achieves its minimum value for k IAE opt ¼ 2. Finding minimum of function (64) with constraints (62) and (46), we consider two cases: JV IAEð2ÞrJA IAEð2Þ and JV IAEð2Þ4JA IAEð2Þ. In the first one, we obtain that parameter kopt ¼ 2 and consequently, B can be derived from (51). This is the case when the acceleration constraint is dominant. In the second case in order to find minimum value of k, we calculate numerically in the interval ð2,kzÞ the root of the following function f3ðkÞ ¼ 9e109 3=2 1 ffiffiffi k p þ ffiffiffi k p 2 ! ffiffiffiffiffiffiffiffiffiffiffi 9e109 q ½1ÀexpðÀkÞŠ Vm ffiffiffi k p À ffiffiffiffiffiffiffi 1 Am s2 4 3 5, ð67Þ where kz is given by the following formula [3] kz ¼ 9e109Am V2 m : ð68Þ Then, having calculated the optimal value of k, we can determine the optimal value of parameter B either from (51) or from the following relation B ¼ V2 mk 9e109½1ÀexpðÀkÞŠ2 sgnðe0Þ: ð69Þ Fig. 5. Minimization procedure — case 2: JV i ðkvoptÞZJA i ðkvoptÞ. Fig. 6. Minimization procedure — case 3: JV i ðkaopt Þ4JA i ðkaoptÞ4JV i ðkvoptÞoJA i ðkvoptÞ. Fig. 4. Minimization procedure — case 1: JV i ðkaopt ÞrJA i ðkaoptÞ. A. Nowacka-Leverton et al. / ISA Transactions 51 (2012) 682–693688

94. Next, we can calculate A and c using formulas (26) and (43), respectively. The line designed in this way stops moving at the time instant tf given by (61). 4.2.2. Minimization of ITAE Substituting C¼0, (43) and (61) into (38), we obtain the following two variable function JITAE ¼ e2 10 69B9 6 k þ3þk : ð70Þ The 3D plot of criterion (70) is similar to the one shown in Fig. 3. Notice that in this case we also can replace the minimization of this function with constraints (62) and (46) by the minimization of the following single variable functions without any constraints JV ITAEðkÞ ¼ 9e109 3 6V2 m 6 k 2 þ 3 k þ1 ½1ÀexpðÀkÞŠ2 , ð71Þ and JA ITAEðkÞ ¼ e2 10 6Am 6 k þ3þk : ð72Þ Repeating our earlier considerations, first we minimize these functions separately. Then, we get that JV ITAEðkÞ reaches its minimum value for k-1. On the other hand, criterion JA ITAEðkÞ achieves its minimum value for k ITAE opt ¼ ffiffiffi 6 p . Therefore, in order to solve the minimization task for criterion (55) where JV ITAEðkÞ and JA ITAEðkÞ are defined by (71) and (72), respectively, we consider two cases: JV ITAEð ffiffiffi 6 p ÞrJA ITAEð ffiffiffi 6 p Þ and JV ITAEð ffiffiffi 6 p Þ4JA ITAEð ffiffiffi 6 p Þ. In the first one, we obtain that parameter kopt ¼ ffiffiffi 6 p which leads to the conclusion that B is given by (51). In the second case the optimal value of k is a numerically found solution of equation f4ðkÞ ¼ 0 in the interval ð ffiffiffi 6 p ,kzÞ where kz is given by (68) and f4 is expressed as f4ðkÞ ¼ e2 10 6 6 k þ3þk 9e109½1ÀexpðÀkÞŠ2 V2 mk À 1 Am ( ) : ð73Þ Next, the optimal value of parameter B can be found from either (51) or (69). Having calculated the optimal values of k and B we determine other switching line parameters A, c and tf using formulas (26), (43) and (61), respectively. Notice that when condition JV ITAEð ffiffiffi 6 p Þ4JV ITAEð ffiffiffi 6 p Þ is satisfied, the optimal value of k is the same when either ITAE or IAE is minimized, which can be justified by the fact that for any fixed values of Vm, Am and e10 we obtain 9e109½1ÀexpðÀkÞŠ2 V2 mk À 1 Am ¼ ffiffiffiffiffiffiffiffiffiffiffi 9e109 q ½1ÀexpðÀkÞŠ Vm ffiffiffi k p À ffiffiffiffiffiffiffi 1 Am s2 4 3 5 Â ffiffiffiffiffiffiffiffiffiffiffi 9e109 q ½1ÀexpðÀkÞŠ Vm ffiffiffi k p þ ffiffiffiffiffiffiffi 1 Am s2 4 3 5: ð74Þ Considering the similarity between the presented minimization procedures for IAE and ITAE, we show the general idea of finding minimum value of the criteria in Figs. 7 and 8. In these figures i might stand either for IAE or for ITAE. In this way we proposed the sliding mode control algorithms which employ switching lines moving with a constant velocity ðC ¼ 0Þ and a constant deceleration ðC a0Þ. In the next section, the methods specified above will be applied to the position control of system (5). 5. Experimental results The sliding mode control strategy proposed above has been applied to a laboratory model of an industrial hoisting crane commercially available from Inteco Ltd. (www.inteco.pl. The mechanical structure of the hoisting subsystem is compatible with the model illustrated in Fig. 1. The maximum control voltage is limited to 24 V, i.e. 9u9rUm ¼ 24 V. For each experiment approximate control (20) has been used with â ¼ 3:0 V s2 =m, g ¼ 50=3 m=s2 and n ¼ 0:05 m=s. The algorithm has been imple- mented in discrete time with sampling frequency fs ¼ 100 Hz. The payload velocity and acceleration have been estimated by linear observer with dominant pole À15. Velocity and acceleration limits have been selected as follows Vm ¼ 0:1 m=s and Am ¼ 0:3 m=s2 . The desired position has been changed as xd1 ¼ 0:5 m for 0rto10 s, 0:1 m for tZ10 s, ( ð75Þ while initial payload position x1ð0Þ ¼ 0:1 m. Considering the initial position error and the constraints of the velocity and the acceleration, we calculate parameters of the switching line using the proposed methods. For the line moving with a constant velocity we obtain the same optimal value of k no matter which criterion (IAE or ITAE) is minimized. On the other hand, for the constant deceleration case, two different optimal values of k have been obtained. Consequently, three experiments (E1, E2 and E3) have been conducted. Experiment E1 concerns the line moving with a constant velocity. Then, the optimal value of k, calculated using the method presented in Section 4.2, is equal to 11.999 and minimization procedure is similar to the one, illustrated by Fig. 8. When we take into account the line with a constant deceleration, we get two different sets of optimal parameters (one for the IAE minimization — E2 and another one for the ITAE minimization — E3). From the IAE minimization, described in subsection 4.1.1, we obtain kopt % 8:177. This case corresponds to the one presented in Fig. 5 and parameter B can be calculated from (51). However, for the ITAE minimization the optimal value of parameter k is given as k ITAE vopt % 6:263, which means that parameter B should be calculated from (52). To Fig. 7. Minimization procedure — case 1: JV i ðkaopt ÞrJA i ðkaopt Þ. Fig. 8. Minimization procedure — case 2: JV i ðkaoptÞ4JA i ðkaoptÞ. A. Nowacka-Leverton et al. / ISA Transactions 51 (2012) 682–693 689

95. illustrate better the optimization procedure, we present two additional ﬁgures for experiment E3 (for the other experiments the illustrations would be quite similar). First in Fig. 9, two criteria JITAE A and JITAE V are presented, both as functions of single variable k. Then, from Fig. 10, the evolution of the ITAE criterion as a function of 9B9 for the optimal value of k may be seen. The ﬁgures show that for the obtained value kopt, the ITAE criterion reaches its minimum value for the greatest admissible 9B9. The optimal parameters required to conduct the experiments, calculated using relations given in Section 4, are collected in Table 1. Notice that signs of parameters C, B and A depend on the sign of the initial error e10. Results of the experiments are illustrated in Figs. 11–28. From Figs. 11–16 one can observe that regulation error converges to some neighborhood of zero. For the worst case recorded (Fig. 16) position error in the steady state does not exceed 1 mm. The payload velocity is presented in Figs. 17–19 from which it can be seen how the behavior of the closed-loop control system depends on the switching line design method. For the switching line moving with a constant acceleration, the payload velocity is smooth, however for the line moving with constant velocity, it reaches assumed maximum bound for longer time period. Fig. 9. Criterion JITAE as a function of k. Fig. 10. Criterion JITAE as a function of 9B9 for optimal value of k. Fig. 11. Position error (E1). Table 1 Optimal parameters for experiments E1–E3. Experiment 9C9 (m/s3 ) 9B9 (m/s2 ) 9A9 (m/s) c (1/s) tf (s) E1 0.0 0.30 1.20 3.00 4.00 E2 0.023 0.30 0.99 2.477 6.60 E3 0.020 0.25 0.87 2.167 6.93 Fig. 12. Position error (E2). A. Nowacka-Leverton et al. / ISA Transactions 51 (2012) 682–693690

96. The experimental results confirm that both the velocity and acceleration constraints are always satisfied (see Figs. 17–22). The acceleration estimate shows significant amount of high frequency signal (noise) which results from inaccuracy of measurement method. In practice ideal sliding is not achieved — variable s remains in the neighborhood of zero with non-zero radius Fig. 13. Position error (E3). Fig. 14. Logarithmic position error (E1). Fig. 15. Logarithmic position error (E2). Fig. 16. Logarithmic position error (E3). Fig. 17. Velocity error (E1). Fig. 18. Velocity error (E2). Fig. 19. Velocity error (E3). Fig. 20. Filtered acceleration (E1). Fig. 21. Filtered acceleration (E2). Fig. 22. Filtered acceleration (E3). A. Nowacka-Leverton et al. / ISA Transactions 51 (2012) 682–693 691

97. (Figs. 23–25) as a result of continuous approximation of switching function. It can be observed that for higher velocity, the radius of the neighborhood increases. It implies that in the considered case dominant part of uncertain dynamics depends on payload velocity. Moreover, it can be seen that due to gravity effects in the steady state after the lowering stage, variable s becomes smaller than after the lift stage. The control voltage presented in Figs. 26–28 is not saturated. The chattering phenomenon is not completely avoided and in particular it can appear for lower values of the payload velocity (at the end stage of deceleration) as a result of worse quality of the velocity estimation and also the increasing importance of Coulomb friction. Phase portraits depicted in Figs. 29–31 refer to the lift and lowering stage (the initial conditions are denoted by ‘Å’ and ‘’, respectively) and show that experimental results correspond quite well to the theoretical ones. Additionally, in order to make comparison of experimental results with the theoretical ones (assuming perfect sliding) more detailed and objective, settling time and respective values of Fig. 23. Sliding variable (E1). Fig. 24. Sliding variable (E2). Fig. 25. Sliding variable (E3). Fig. 26. Control voltage input (E1). Fig. 27. Control voltage input (E2). Fig. 28. Control voltage input (E3). Fig. 29. Phase portrait (grey—simulated, black—experimental results) (E1). Fig. 30. Phase portrait (grey—simulated, black—experimental results) (E2). A. Nowacka-Leverton et al. / ISA Transactions 51 (2012) 682–693692

98. integral criteria have been determined. For each experiment average values of considered criteria have been obtained taking into account lift and lowering stage. Settling time has been determined assuming 1% and 5% tolerance. The data is collected in Table 2 — in parenthesis theoretical values are given. 6. Conclusions In this paper experimental results on a point-to-point sliding mode control of a hoisting crane with time-varying switching lines are presented. The proposed SMC algorithm seems to be an effective method in practical applications. The optimal synthesis of the switching lines results in good dynamic performance (in the sense of IAE and ITAE criteria) and guarantees meeting velocity and acceleration constraints during the whole control process. The robustness of the system is ensured from the very beginning of the control action. Furthermore, monotonic error convergence to zero is guaranteed. The application of the proposed control scheme reveals its main benefits in the case when a crane model cannot be easily identified. Acknowledgments This work has been performed in the framework of a project ‘‘Optimal sliding mode control of time delay systems’’ financed by the National Science Centre of Poland – decision number DEC 2011/01/B/ST7/02582. References [1] Bartolini G, Fridman L, Pisano A, Usai E, editors. Modern sliding mode control theory. New perspectives and applications. Series LNCIS, vol. 375. Springer- Verlag; 2008. [2] Bartoszewicz A. Time-varying sliding modes for second-order systems. IEE Proceedings on Control Theory and Applications 1996;143:455–62. [3] Bartoszewicz A, Nowacka-Leverton A. Time-varying sliding modes for second and third order systems. Series LNCIS, vol. 382. Springer-Verlag; 2009. [4] Betin F, Pinchon D, Capolino G. A time-varying sliding surface for robust position control of a DC motor drive. IEEE Transactions on Industrial Electronics 2002;49:462–73. [5] Borgstrom PH, Jordan BL, Borgstrom BJ, Stealey MJ, Sukhatme GS, Batalin MA, Kaiser WJ. A cable-driven robot with self-calibration capabilities. IEEE Transactions on Robotics 2009;25:1005–15. [6] Chang C. Adaptive fuzzy controller of the overhead cranes with nonlinear disturbance. IEEE Transactions on Industrial Informatics 2007;3:164–72. [7] Chang C, Hsu K, Chiang K, Huang G. An enhanced adaptive sliding mode fuzzy control for positioning and anti-swing control of the overhead crane system. In: Proceedings of IEEE international conference on systems, man, and cybernetics; 2006. p. 992–7. [8] Choi HS, Park YH, Cho Y, Lee M. Global sliding-mode control: improved design for a brushless DC Motor. IEEE Control Systems Magazine 2001;21:27–35. [9] DeCarlo RS, Zak S, Mathews G. Variable structure control of nonlinear multivariable systems: a tutorial. Proceedings of the IEEE 1988;76:212–32. [10] Edwards C, Fossas Colet E, Fridman L, editors. Advances in variable structure and sliding mode control, Series LNCIS, vol. 334. Springer-Verlag; 2006. [11] Edwards C, Spurgeon SK. Sliding mode control: theory and applications. Taylor and Francis; 1998. [12] Hung JY, Gao W, Hung JC. Variable structure control: a survey. IEEE Transactions on Industrial Electronics 1993;40:2–22. [13] Neupert J, Arnold E, Schneider K, Sawodny O. Tracking and anti-sway control for boom cranes. Control Engineering Practice 2010;18:31–44. [14] Oh S, Agrawal SK. A control Lyapunov approach for feedback control of cable- suspended robots. In: IEEE international conference on robotics and automation; 2007. p. 4544–9. [15] Oh S, Ryu J, Agrawal SK. Dynamics and control of a helicopter carrying a payload using a cable-suspended robot. Journal of Mechanical Design 2006;128:1113–21. [16] Orowska-Kowalska T, Dybkowski M, Szabat K. Adaptive sliding-mode neuro- fuzzy control of the two-mass induction motor drive without mechanical sensors. IEEE Transactions on Industrial Electronics 2010;57:553–64. [17] Park M, Chwa D, Hong S. Adaptive fuzzy nonlinear anti-sway trajectory tracking control of uncertain overhead cranes with high-speed load hoisting motion. In: Proceedings of international conference on control, automation and systems; 2007. p. 2886-91. [18] Prommaneewat K, Roengruen P, Kongratana V. Anti-sway control for overhead crane, In: Proceedings of international conference on control, automation and systems; 2007. p. 1954–7. [19] Stergiopoulos J, Konstantopoulos G, Tzes A. Experimental verification of an adaptive input shaping scheme for hoisting cranes. In: 17th mediterranean conference on control and automation, MED ’09; 2009. p. 730–5. [20] Tokat S, Eksin I, Guzelkaya M. A new design method for sliding mode controllers using a linear time-varying sliding surface. IMechE 2002;216: 455–66. [21] Utkin V. Variable structure systems with sliding modes. IEEE Transactions on Automatic Control 1977;22:212–22. [22] Yagiz N, Hacioglu Y, Taskin Y. Fuzzy sliding-mode control of active suspen- sions. IEEE Transactions on Industrial Electronics 2008;55(11):3883–90. Table 2 Comparison of settling time for experiments E1–E3. Experiment Settling time (s) D ¼ 1% D ¼ 5% E1 5.16 (4.71) 4.36 (4.17) E2 6.79 (6.48) 5.74 (5.59) E3 6.95 (6.46) 5.73 (5.52) Fig. 31. Phase portrait (grey—simulated, black—experimental results) (E3). A. Nowacka-Leverton et al. / ISA Transactions 51 (2012) 682–693 693

Experimental veriﬁcation of SMC with moving switching lines applied to hoisting crane vertical motion control

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Experimental veriﬁcation of SMC with moving switching lines applied to hoisting crane vertical motion control