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Independent motion control of a tower crane through wireless sensor and actuator networks
ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
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Independent motion control of a tower crane through wireless sensor and actuator networks Fotis N. Koumboulis a,n, Nikolaos D. Kouvakas a, George L. Giannaris a,b, Demosthenes Vouyioukas b a b
Sterea Ellada Institute of Technology, Department of Automation Engineering, Psahna Evias 34400 Greece University of the Aegean, Department of Information & Communication Systems Engineering, 83200 Karlovasi, Samos, Greece
art ic l e i nf o
a b s t r a c t
Article history: Received 2 January 2015 Received in revised form 24 August 2015 Accepted 9 November 2015 This paper was recommended for publication by Y. Chen.
The problem of independent control of the performance variables of a tower crane through a wireless sensor and actuator network is investigated. The complete nonlinear mathematical model of the tower crane is developed. Based on appropriate data driven norms an accurate linear approximant of the system, including an upper bound of the communication delays, is derived. Using this linear approximant, a dynamic measurable output multi delay controller for independent control of the performance outputs of the system is proposed. The controller performs satisfactory despite the nonlinearities of the model and the communication delays of the wireless network. & 2015 ISA. Published by Elsevier Ltd. All rights reserved.
Keywords: Dynamic multi-delay controllers I/O decoupling Time delay Tower crane Wireless network
1. Introduction Cranes are used in several applications having significant economic impact. Classical uses are in construction (bridges, dams, buildings), transportation (loading and unloading cargo), industry (oil platforms, refineries), in nuclear power plants and in bio/ecological applications (see [1–3] and the references therein). Efficient crane maneuverability needs appropriate control, requiring accurate mathematical model of the crane (see f.e. [4–14]). For the case of tower cranes significant results have been presented in [1–4]. In [4] a simplified nonlinear model of a tower crane is being developed in the form of a spherical pendulum in a non-inertial frame. The model is then used to analyze the cases of linearly accelerating support and the support describing a circular path at constant speed. In [5], a simplified mathematical model of a tower crane is being developed. The model is then used to demonstrate the performance of event trigger controllers for wireless control of multiple 3d tower cranes. The influence of the network uncertainties such as time delays and packet dropouts are not taken into account in the controller design stage but they are used in the computational experiments of the closed loop system. n
Corresponding author. E-mail address: [email protected] (F.N. Koumboulis).
Similar issues are investigated in [6]. In [7], the mathematical model of a laboratory tower crane is being developed and a local nonlinear model predictive control is proposed for the performance output to follow a desired path. The dynamics of the trolley, the length and the angular displacement of the jib are neglected and the respective accelerations are considered as actuatable input to the system. For other categories of cranes (such as bridge cranes, overhead cranes, rotary cranes and boom cranes), several results have been published. In particular in [8] the Bluetooth protocol is used for short range wireless control of a bridge crane system. The controller is of the rule based type not using information from the system dynamics. The influence of network parameters is not investigated. In [9], a local fuzzy logic control scheme and a local LQG controller are being used to regulate the performance output of an 2d overhead crane made up of a platform carrying a cable holding the load. The results are validated using physical experiments. In [10] the mathematical model of the 2d overhead crane system is used to develop a local controller which combines a feedback linearization approach and a time delay control scheme. In [11] a simplified mathematical model of a rotary crane system is being used to develop a local neuro-controller for vibration control of the load involving the rotation about the vertical axis only. In [12], based upon a simplified mathematical model of a boom
http://dx.doi.org/10.1016/j.isatra.2015.11.011 0019-0578/& 2015 ISA. Published by Elsevier Ltd. All rights reserved.
Please cite this article as: Koumboulis FN, et al. Independent motion control of a tower crane through wireless sensor and actuator networks. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.11.011i
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2
crane, the I/O linearization approach is being used to locally control slewing and luffing motion. The efficiency of the proposed approach is demonstrated through physical experiments on an industrial harbor mobile crane. In [13] a simplified mathematical model of a rotary crane is being developed and an optimal local open loop control approach is being used for sway free, point to point motion of the load mass. The performance of the proposed approach is demonstrated through physical and computational experiments. In [14] the mathematical model of a crane system which is equipped with a flexible cable is being developed in the form of hybrid system represented by partial-ordinary differential equations. Based upon the mathematical model of the system, a local integral-barrier Lyapunov function based control scheme is proposed to suppress the undesirable vibrations. From the communication point of view, the controllers in [5,6,8] use wireless connections. The controllers proposed in [7,9,10–14] use wired connections. From the control point of view, in [5,6] event triggered schemes have been proposed. In [8,9] pure rule based and fuzzy approaches have been used. In [9,13] optimal approaches have been applied. In [11] an intelligent neural approach has being used. In [7] model predictive controllers are used. In [10,12] feedback linearization approaches are used. In [10] a delay is also included to improve the control performance. In [14] a nonlinear Lyapunov based controller is proposed. Here, we propose Input Output (I/O) Decoupling of the motion variables of a tower crane through a wireless sensor and actuator network. The complete nonlinear dynamic mathematical model of a tower crane is developed. The crane is considered to be equipped with three preinstalled approximate PID controllers. Using the nonlinear model, the linear approximant of the system is derived. Its accuracy is investigated via series of computational experiments. For the remote signal transmission among the sensors, the actuators and the controller, the ZigBee protocol is used. A synchronization algorithm, guaranteeing constant transmission delay, is proposed. Additionally, an algorithm towards signal reconstruction of the analogue continuous time signals from the digital transmitted signals is proposed. The contribution of the present paper is summarized to the following points: 1. A dynamic model of the system is developed. The model is more general, as compared to those in [4–7], in the sense that it includes variable cable length and all coupling terms in the system kinetics. 2. As compared to [5,6,8] the delays of the wireless communication are taken in to account in the controller design. The issue is treaded as follows: a) Due to the presence of the delays, an inverse dynamics controller compensating the nonlinearities of the system is not adequate. This is why a linear approximant, as well as a linear controller, performing satisfactory despite the nonlinearities of the crane is proposed. b) A synchronization algorithm is proposed. Using this algorithm, the uncertain communication delay becomes constant and known to the designer. The values of the delay are used to develop an appropriate realizable multi-delay dynamic controller providing I/O Decoupling for the performance outputs preserving the accuracy of the closed loop linear approximant. 2. Dynamics of a tower crane Cranes are worksite mechanisms used to lift and lower loads as well as to place them in the site. A tower crane (see Fig. 1) is a
modern form of balance crane consisting of three mechanical parts: an arm rotating around a vertical mast, a trolley moving along the arm and a cable drum with a load at its end. The arm rotates around the mast by an arm motor (actuator 1), the trolley moves along the arm by another motor (actuator 2) and the payload is lifted or lowered by a third motor (actuator 3) rotating the cable drum to gather or release the cable. The tower crane is a highly oscillatory system with linear and nonlinear dynamics. The nonlinear dynamics come mainly from the rotational motion inducing centripetal and Coriolis accelerations producing instability. In what follows, the mathematical model of the tower crane will be developed using the Euler–Lagrange approach. The tower crane is inherently unstable with respect to all motion variables. In such mechanical systems it is a common practice (see [15]) to use pre-installed three term controllers to regulate local performance variables. Such pre-installed local PID controllers are necessary for the safe ground operation. Here, 3 pre-installed approximate PID controllers are considered. The 1st PID stabilizes the velocity of the arm, the 2nd the position of the trolley along the arm and the 3rd the cable length. 2.1. 2.1 State space model of the crane
hDefine x ¼ x1 ⋯ ¼ q1 q2 q3 q4 q5 q_ 1
x16 q_ 2
T q_ 3
q_ 4
q_ 5
χ 1;1 χ 1;2 χ 2;1 χ 2;2 χ 3;1 χ 3;2
iT
h iT h iT T w ¼ w1 w2 w3 , y ¼ y1 y2 y3 and ψ ¼ ψ 1 ⋯ ψ 5 , where x is the state vector, y is the performance output vector, ψ is the measurable output vector, w is the vector of external commands of the pre-installed PID controllers (w1 ,w2 and w3 are the external commands for the rotational velocity of the crane, the position of the trolley and the cable length, respectively), q1 is the arm rotation angle with respect to an inertial frame, q2 is the distance of the trolley from the crane’s revolution axis, q3 and q4 are the cable’s angles and q5 is the cable length, χ i;j (i ¼ 1; 2; 3 and j ¼ 1; 2) are internal variables of the approximate PID controllers. The mathematical model of the tower crane is developed to be dx ¼ ½EðxÞ 1 f ðx; wÞ; y ¼ r ðxÞ; ψ ¼ Lx; EðxÞ A ℝ1616 ; f ðx; wÞ A ℝ161 ; dt r ðxÞ A ℝ31 ; L A ℝ516 ð1Þ The nonzero elements of EðxÞ and F ðx; wÞ are: e1;1 ðxÞ ¼ 1, e2;2 ðxÞ ¼ 1, e3;3 ðxÞ ¼ 1, e4;4 ðxÞ ¼ 1, e5;5 ðxÞ ¼ 1, 2 e6;6 ðxÞ ¼ J 1 þ J 2 þJ 5 þ lc m1 þ ðm2 þm5 Þx22 2sx4 m5 x2 x5 0:25 c2x3 2 2 þ 2cx3 c2x4 3Þm5 x5 , e6;7 ðxÞ ¼ cx4 sx3 m5 x5 , e7;10 ðxÞ ¼ sx4 m5 , e6;8 ðxÞ ¼ cx3 cx4 m5 x5 x2 sx4 x5 , e6;9 ðxÞ ¼ sx3 J 5 þ m5 sx3 x5 x5 sx4 x2 , e6;10 ðxÞ ¼ cx4 sx3 m5 x2 , e7;6 ðxÞ ¼ cx4 sx3 m5x5 , e7;7 ðxÞ ¼ m2 þ m5 , e7;9 ðxÞ ¼ cx4 m5 x5 ; e8;6 ðxÞ ¼ cx3 cx4 m5 x5 x2 sx4 x5 , e8;8 ðxÞ ¼ c2x4 m5 x25 þJ 5 ; e9;6 ðxÞ ¼ sx3 J 5 þ m5 sx3 x5 x5 sx4 x2 , e9;7 ðxÞ ¼ cx4 m5 x5 , e9;9 ðxÞ ¼ m5 x25 þ J 5 , e10;6 ðxÞ ¼ cx4 sx3 m5 x2 , e10;7 ðxÞ ¼ sx4 m5 , e10;10 ðxÞ ¼ m5 , e11;11 ðxÞ ¼ 1, e12;12 ðxÞ ¼ 1, e13;13 ðxÞ ¼ 1, e14;14 ðxÞ ¼ 1, e15;15 ðxÞ ¼ 1, e16;16 ðxÞ ¼ 1, f 1 ðx; wÞ ¼ x6 , f 2 ðx; wÞ ¼ x7 , f 3 ðx; wÞ ¼ x8 , f 4 ðx; w Þ ¼ x9 , f 6 ðx; wÞ ¼ 2m2 x2 x6 x7 þ 0:5m5 x5 x6 4sx4 x7 2x5 f 5 ðx; wÞ ¼ x10 ,
c2x4 s2x3 x8 þ c2x3 s2x4 x9 x5 x8 sx3 s2x4 x8 þ 4cx3 s2x4 x9 þ i c2x3 þ 2c2x3 c2x4 3 x6 þ 2cx3 s2x4 x8 4sx3 x9 x10 þ m5 x2 ð 2x6 x7 þ
cx4 x5 2x6 x9þ sx3 x28 þ x29 2cx3 x8 x10 þ2sx4 cx3 x5x8 x9 þ x6 þ sx3 2 x9 Þx10 Þ þ w1 γ 1 f D;1 þ f P;1 J 5 cx3 x8 x9 γ 1 x12 f D;1 þ γ 1 x11 þ x12 f I;1 x6 γ 1 f D;1 þ f P;1 , f 7 ðx; wÞ ¼ m2 x2 x26 þ m5 x2 x26 þ 2m5 x5 cx3 cx4 x6 x8 m5 x5 sx4 x26 þ 2sx3 x6 x9 þ x29 þ 2m5 cx4 sx3 x6 þ x9 x10 þ γ 2 f D;2 ðw2 x2 Þ γ 22 x14 f D;2 þ γ 2 x13 þ x14 Þf I;2 þ ðw2 x2 Þf P;2 , f 8 ðx; wÞ ¼ J 5 cx3 x6 x9 gm5 cx4 x5 sx3 2m5 c2x4 x5 x8 x10 þ m5 cx4 x25 2sx4 x8 x9 þ cx3 cx4 x6 sx3 x6 þ2x9
Please cite this article as: Koumboulis FN, et al. Independent motion control of a tower crane through wireless sensor and actuator networks. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.11.011i
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2m5 cx3 cx4 x5 x6 x7 sx4 x10 ,
f 9 ðx; wÞ ¼ m5 sx4 x5 2sx3 x6 x7 gcx3 J 5 cx3 x6 x8 m5 cx4 x5 x2 x26 þ m5 cx4 x25 c2x3 sx4 x26 2cx3 cx4 x6 x8 sx4 x28 2m5 x5 sx3 x6 þx9 x10 , f 10 ðx; wÞ ¼ m5sx4 x2 x26 þm 5 cx4 gcx3 2sx3 x6 x7 Þ þ m5 x5 x6 2sx3 x9 cx3 s2x4 x8 þm5 x5 c2x4 x28 þ x29 0:25m5 x5 2c2x3 c2x4 þ c2x4 3 x26 γ 23 x16 f D;3 þ γ 3 x15 þ x16 f I;3 þ ðw3 x5 Þ γ 3 f D;3 f 13 ðx; wÞ ¼ x14 ; þf P;3 Þf 11 ðx; wÞ ¼ x12 ; f 12 ðx; wÞ ¼ w1 x6 γ 1 x12 ; f 14 ðx; wÞ ¼ w2 x2 γ 2 x14 ; f 15 ðx; wÞ ¼ x16 ; f 16 ðx; wÞ ¼ w3 x5 γ 3 x16 The 1st performance output is the angular velocity of the payload (around the mast), i.e. y1 ¼ r 1 ðxÞ ¼ x6 þ sx3 x9 . The 2nd is the h radial position of the payload, i.e. y2 ¼ r 2 ðxÞ ¼ x22 2sx4 x2 x5 þ i0:5 and the 3rd is the vertical distance of the payc2x4 s2x3 þ s2x4 x25 load from the ground, i.e. y3 ¼ r 3 ðxÞ ¼ h0 cx3 cx4 x5 . So, r ðxÞ ¼ r 1 ðxÞ r 2 ðxÞ r 3 ðxÞ T . The 1st measurable variable is the position of the trolley, i.e. x2 , the 2nd is the cable length, i.e. x5 , the 3rd is the angular velocity of the arm, i.e. x6 , the 4th is the velocity of the trolley, i.e. x7 and the 5th is the rate of change of the cable’s length, i.e. x10 . Hence, the nonzero elements of L are l1;2 ¼ 1, l2;5 ¼ 1, l3;6 ¼ 1, l4;7 ¼ 1 and l5;10 ¼ 1. m1 , m2 and m5 are the arm, trolley and payload masses respectively, J 1 , J 2 and J 5 are the arm, trolley and payload moments of inertia respectively, g is the gravitational acceleration, lc is the distance of the center of mass of the arm from the revolution axis of the crane, h0 is the length of the mast and cα ¼ cos ðαÞ, sα ¼ sin ðαÞ. Each approximate PID controller regulates one part of the crane. Let f P;i , f I;i and f D;i (i ¼ 1; 2; 3) are the proportional, integral and derivative gains respectively of the three PID controllers and γ 1 , γ 2 and γ 3 are the three filter coefficients for the proper approximate implementation of the respective derivative terms. The parameters of the PID controllers are dedicated to achieve pole placement and asymptotic command following for the respective outputs. Additionally, the parameters of the 1st PID controller increase the properness of the respective closed loop transfer function. Hence, the PID parameters are derived to be 2 2 2 f P;1 ¼ J 1 þ lc m1 λ1;1 λ2;1 λ0;1 λ2;1 ; f P;2 ¼ m2 λ1;2 λ3;2 λ0;2 λ3;2 ; 2 1 2 f P;3 ¼ m5 λ1;3 λ3;3 λ0;3 λ3;3 ; f I;1 ¼ J 1 þ lc m1 λ0;1 λ2;1 ; 1
1
f I;2 ¼ m2 λ0;2 λ3;2 ; f I;3 ¼ m5 λ0;3 λ3;3 ; 3 2 f D;1 ¼ J 1 þ lc m1 λ0;1 λ1;1 λ2;1 λ2;1 ; γ 1 ¼ λ2;1 ; 3 f D;2 ¼ m2 λ0;2 þ λ3;2 λ2;2 λ3;2 λ1;2 λ3;2 ; γ 2 ¼ λ3;2 ; 3 f D;3 ¼ m5 λ0;3 þ λ3;3 λ2;3 λ3;3 λ1;3 λ3;3 ; γ 3 ¼ λ3;3 where the parameters λi;j (i ¼ 0; 1; 2; 3, j ¼ 1; 2; 3) are free to be chosen by the designer. They are the coefficients of the characteristic polynomials of each subsystem of the crane, i.e. P P p1 ðsÞ ¼ s3 þ 2j ¼ 0 λj;1 sj , pi ðsÞ ¼ s4 þ 3j ¼ 0 λj;i sj (i ¼ 2; 3). Clearly, λj;i are chosen such that pi ðsÞ (i ¼ 1; 2; 3) are stable. The developed here model of the crane is more general than those presented in [5,6], in the sense that it includes variable cable length, as well as all coupling terms in the system dynamics. 2.2. Linear approximant around constant velocity of the arm Here, a linear approximant (LA) of the nonlinear model (1) around a constant angular velocity ω of the arm of the crane will be produced. Let xi be the nominal points of xi for i ¼ 2; :::; 16, where x6 ¼ ω. The nominal trajectory of the angle of the arm is x1 ðt Þ ¼ ωt þ q1;0 , where q1;0 is the initial angle of the mast. Let xðt Þ, yðt Þ and wðt Þ be the nominal trajectories of the state, performance output and input vectors of the model (1). Clearly, x_ ðt Þ ¼ h iT ω 0115 T . Let wðt Þ ¼ ω x2 x5 , where x2 and x5 are the nominal values of the trolley position along the arm and the cable
3
length respectively. Thus, xðt Þ and yðt Þ are evaluated by x1 ðt Þ ¼ ωt þ q1;0 ; x2 ¼ x2 ; x3 ¼ 0; x5 ¼ x5 ; x6 ¼ ω; x7 ¼ x8 ¼ x9 ¼ x10 ¼ x11 ¼ x12 ¼ x14 ¼ x16 ¼ 0; x13 ¼ m5 x5 sx4 ðm2 þ m5 Þx2 ω2 =f I;2 γ 2 ; x15 ¼ m5 ω2 sx4 x2 x5 sx4 gcx4 =f I;3 γ 3 ; y1 ¼ ω; y2 ¼ x2 x5 sx4 ; y3 ¼ h0 x5 cx4 where the nominal value x4 is evaluated by the equation gsx4 þ ω2 cx4 x2 x5 sx4 ¼ 0. The deviations around the nominal values of the external commands and the performance, state and measurable variables are Δw ¼ w w, Δy ¼ y y, Δx ¼ x x and Δψ ¼ ψ ψ . The LA of (1) is d δx ¼ Aδx þ Bδw; δy ¼ C δx; δψ ¼ Lδx dt
ð2Þ
where δw ¼ Δw and δx, δy and δψ are the LA responses, approximating the deviations Δx, Δy and Δψ . LA operates around
an operating and
point oðt Þ ¼ oðt Þ where oðt Þ ¼ w; x; y A B o ð t Þ ¼ w; x; y . The LA system matrices are ¼ h i ∂ ∂x
∂ ∂w
½EðxÞ 1 F ðx; wÞjo ¼ o , C ¼ ∂Γ ðxÞ=∂xjo ¼ o . A and B can be
1 1 ~ The nonzero eleexpressed in the form A ¼ E~ A~ and B ¼ E~ B. ~ A~ and B~ are ments of E,
e~ 1;1 ¼ 1, e~ 2;2 ¼ 1, e~ 3;3 ¼ 1, e~ 4;4 ¼ 1, e~ 5;5 ¼ 1, e~ 6;6 ¼ J 1 þ J 2 þJ 5 þ 2 lc m1 þx5 sx4 x5 sx4 2x2 m5 þx2 e~ 6;8 ¼ x5 cx4 x2 x5 sx4 2 ðm2 þ m5 Þ, m5 , e~ 7;7 ¼ m2 þ m5 , e~ 7;9 ¼ x5 cx4 m5 , e~ 7;10 ¼ sx4 m5 , e~ 8;6 ¼ x5 cx4 2 ~ ~ x2 x5 sx4 Þm5 , e~ 8;8 ¼ J 5 þx2 5 cx4 m5 , e 9;7 ¼ x5 cx4 m5 e 9;9 ¼ J 5 þ 2 x5 m5 , e~ 10;7 ¼ sx4 m5 , e~ 10;10 ¼ m5 , e~ 11;11 ¼ e~ 12;12 ¼ e~ 13;13 ¼ e~ 14;14 ¼ e~ 15;15 ¼ e~ 16;16 ¼ 1, a~ 7;2 ¼ ω2 ðm2 þ m5 Þ γ 2 f D;2 f P;2 , a~ 9;2 ¼ x5 ω2 cx4 m5 , a~ 10;2 ¼ ω2 sx4 m5 , a~ 14;2 ¼ 1, a~ 8;3 ¼ x5 cx4 x5 ω2 cx4 g m5 , 2 2 a~ 7;4 ¼ x5 ω cx4 m5 , a~ 9;4 ¼ x5 ω x5 c2x4 þx2 sx4 gcx4 m5 , a~ 10;4 ¼ 2 ω cx 2x5 sx4 x2 gsx4 m5 , a~ 7;5 ¼ ω2 sx 4 m5 , a~ 9;5 ¼ ω2 cx4 4 2 a~ 10;5 ¼ ω2 s2x m5 2x5 sx4 x2 gsx4 m5 ,a~ 8;9 ¼ ω J 5 þ 2m5 x2 5 cx4 4 ~ ~ ~ γ 3 f D;3 f P;3 , a 16;5 ¼ 1, a 1;6 ¼ 1, a 6;6 ¼ γ 1 f D;1 f P;1 , a~ 7;6 ¼ 2ω x2ðm2 þ m5 Þ x5 sx4 m5 , a~ 9;6 ¼ 2x5 ωcx4 x5 sx4 x2 m5 , a~ 10;6 ¼ 2ωsx4 x5 sx4 x2 m5 ,a~ 12;6 ¼ 1, a~ 2;7 ¼ 1, a~ 6;7 ¼ 2x5 ωsx4 m5 2x2 ω ðm2 þ m5 Þ, a~ 8;7 ¼ 2x5 ωcx4 m5 , a~ 3;8 ¼ 1, a~ 7;8 ¼ 2x5 ωcx4 m5 , a~ 9;8 ¼ ω J 5 þ 2x2 c2 m5 , a~ 10;8 ¼ 2x5 ωcx4 sx4 m5 , a~ 4;9 ¼ 1, a~ 6;9 ¼ 5 x4 a~ 5;10 ¼ 1, a~ 6;10 ¼ 2ωsx4 x2 x5 sx4 m5 , 2x5 ωcx4 x2 x5 sx4 m5 , 2 a~ 8;10 ¼ x5 ωs2x4 m5 , a~ 6;11 ¼ γ 1 f I;1 , a~ 6;12 ¼ γ 1 f D;1 þ f I;1 , a~ 11;12 ¼ 1, a~ 12;12 ¼ γ 1 , a~ 7;13 ¼ γ 2 f I;2 , a~ 7;14 ¼ γ 22 f D;2 þ f I;2 , a~ 13;14 ¼ 1, a~ 14;14 ¼ γ 2 , a~ 10;15 ¼ γ 3 f I;3 , a~ 10;16 ¼ γ 23 f D;3 þ f I;3 , a~ 15;16 ¼ 1, a~ 16;16 ¼ γ 3 , b~ 6;1 ¼ γ 1 f D;1 þ f P;1 , b~ 7;2 ¼ γ 2 f D;2 þ f P;2 , b~ 10;3 ¼ γ 3 f D;3 þ f P;3 , b~ 12;1 ¼ 1, b~ 14;2 ¼ 1, b~ 16;3 ¼ 1 The nonzero elements of C are c1;6 ¼ 1, c2;2 ¼ 1, c2;4 ¼ x5 cx4 , c2;5 ¼ sx4 , c3;4 ¼ x5 sx4 and c3;5 ¼ cx4 . 2.3. Accuracy of the linear approximant Here, it will be investigated via series of computational experiments if the LA (2) of the tower crane performs similarly to the nonlinear model (1). At first, assume that the model (1) operates on certain operating conditions. In particular, let w1 , w2 and w3 be the nominal values of the external commands of the approximate PID controllers yielding the respective operating trajectories for the state variables, let x1 ðt Þ and xj for j ¼ 2; …; 16. In what follows, small variations of the model will be considered. To this end, the external commands are chosen to be of the form ^ i ρðt; τ1 ; τ 2 ; T 1 ; T 2 ; T 3 Þ; i ¼ 1; 2; 3; w ^ iA w ^ i min ; w ^ i max ; w i ðt Þ ¼ w i þ w
ρðt; τ1 ; τ2 ; T 1 ; T 2 ; T 3 Þ ¼ σ ðt; τ1 Þ σ ðt; τ2 Þ τ1 t
e
T1
τ1 t
τ1 t
T 21 ðT 2 T 3 Þ e T 2 T 22 ðT 1 T 3 Þ þ e T 3 T 23 ðT 1 T 2 Þ σ ðt; τ1 Þ ðT 1 T 2 ÞðT 1 T 3 ÞðT 2 T 3 Þ
Please cite this article as: Koumboulis FN, et al. Independent motion control of a tower crane through wireless sensor and actuator networks. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.11.011i
F.N. Koumboulis et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
4 τ2 t
τ2 t
τ2 t
T 21 ðT 2 T 3 Þ e T 2 T 22 ðT 1 T 3 Þ þ e T 3 T 23 ðT 1 T 2 Þ σ ðt; τ2 Þ; ðT 1 T 2 ÞðT 1 T 3 ÞðT 2 T 3 Þ ( 0 t o τi σ ðt; τi Þ ¼ ; τ ¼ 1; 2 1 t Z τi i þ
e
T1
with τ1 o τ2 and T 1 , T 2 , T 3 being positive different among themselves constants. The form of ρ is such that its 1st and 2nd derivatives with respect to time are smooth. Using these input changes, the response of the nonlinear model (1) regarding the performance outputs yi (i ¼ 1; 2; 3) and the measurable variables ψ k (k ¼ 1; …; 5) is computed. The responses of the LA and nonlinear model are compared using a Euclidian type norm the form [16] p φj ; δφj ¼ 100% ‖φj ðt Þ δφj ðt Þ φj ‖2 =‖φj ðt Þ φj ‖2 ð3Þ where φj is a signal representing the state, performance or mea i2 1=2 Rτ h surable variables, ‖φj ðt Þ‖2 ¼ 0 max φj ðt Þ dt and τmax A ℝ þ where τmax represents a time instance, greater than the relaxation time of the argument response. To evaluate the cost in (3), consider the test case presented in [17]: m1 ¼ 73:377½Kgr, m2 ¼ 2:492½Kgr, m5 ¼ 3:1415½Kgr, J 1 ¼ 2 13:609 ½Kgr m2 , J 2 ¼ 8:572½Kgr m 2,l c ¼ 0:4517½m, J 5 ¼ 0:01256 2 ½Kgr m , h0 ¼ 2:55½m, g ¼ 9:81 m=s The nominal and the parameters are: values controller ω ¼ π =20 rad=s , q1;0 ¼ 0 rad , x2 ¼ 0:5½m, x5 ¼ 0:5½m, λ0;1 ¼ 4766:52, λ1;1 ¼ 889:21, λ2;1 ¼ 53, λ0;2 ¼ 1712:2557, λ1;2 ¼ 1425:512, λ2;2 ¼ 371:78, λ3;2 ¼ 35:2, λ0;3 ¼ 122138:016, λ1;3 ¼ 30449:576, λ2;3 ¼ 2475:06, λ3;3 ¼ 83:1, w 1 ¼ π =6 rad=s , w2 ¼ 0:5½m, w3 ¼ 0:75½m. For the above values of the system parameters, the inner controller coefficients are computed to be: f p;1 ¼ 431:01, f i;1 ¼ 2570:35, f d;1 ¼ 8:13227, γ 1 ¼ 53, f p;2 ¼ 97:476, f i;2 ¼ 121:22, f d;2 ¼ 23:5511, γ 2 ¼ 35:2, f p;3 ¼ 1095:58, f i;3 ¼ 4617:42, f d;3 ¼ 80:3857, γ 3 ¼ 83:1 while the operating trajectories for the state variables and performance outputs are evaluated to be: ½m x3 ¼ 0 rad , x4 ¼ 0:00125918 x1 ðt Þ ¼ π t=20 rad , x2 ¼ 0:5 rad , x5 ¼ 0:5 ½m, x 6 ¼ π =20 rad=s , x 7 ¼ 0 m=s , x 8 ¼ 0 rad=s , x 9 ¼ 0 rad=s , x 10 ¼ 0 m=s , x11 ¼ 0½, x12 ¼ 0½, x13 ¼ 0½, x14 ¼ 0½, x15 ¼ 0½, x16 ¼ 0½, y 1 ¼ π =20 rad=s , y2 ¼ 0:5006½m, y3 ¼ 2:05016½m. To evaluate the accuracy of the LA through the cost criterion (3) for the measurable and performance variables of the plant, the
Fig. 1. Tower crane configuration.
external commands are to deviate from their nominal considered ^ i =wi o0:15 (i ¼ 1; 2; 3). Here, the level of values about 15%, i.e. w accuracy of the LA is considered to be the maximum of p for all performance and measurable variables. After computing the level of accuracy for all combinations in the aforementioned indicative deviation bounds of the external commands, it has been shown that a level of accuracy of 10.16% is derived, i.e. for all combina w ^ i =wi o0:15, the maximum of tions of the external commands in
p φj ; δφj for φj A y1 ; y2 ; y3 ; ψ 1 ; …; ψ 5 is less than or equal to 10.16%. Hence, the LA (3) is an accurate representation of the nonlinear model (1) and so a controller can be designed on the basis of it.
3. Wireless controller implementation Wireless control overcomes mobility limitations and enhances the system's flexibility especially for the case of controlling simultaneously multiple mechanisms. However, wireless control necessitates high fidelity equipment. For our purposes, neither data storage limitations, transmission rate nor long distances are crucial factors to consider. A major limitation to consider is the need for a telecommunication signal resisting to noise and other users' interference. This is an inherent situation to any industrial environment. 3.1. Measurable variables Since a wireless controller has to be designed, a transmission delay is expected between the sensors and the controller base. Furthermore, this transmission delay is in general time-varying. Hence, the vector of the system’s measurable outputs is ψ ðt Þ ¼ Lxðt τ1 ðt ÞÞ. The delay τ1 ðt Þ stands for the transmission/ reconstruction delay taking place between the sensors and the controller. Without loss of generality, this delay is assumed to be the same for all measurable variables. So, an appropriate transmission/ reconstruction algorithm will be proposed in order to eliminate delay variations. Elimination of the delay variations will clearly facilitate controller design and implementation. 3.2. Controller configuration With respect to the controller configuration, it is considered that the setup consists of two separate networks (see Fig. 3).
Fig. 2. Signal reconstruction error.
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Fig. 3. Block diagram of the closed loop system.
The 1st is dedicated to the transmission of the measurable outputs from the sensors to the controller, while the 2nd is dedicated to the transmission of the inner external commands to the preinstalled PIDs on the tower crane. Each network consists of a single transmitter and a single receiver. The transmitter and receiver of the tower crane are assumed to be located at the highest point of the mast while the 2nd transmitter/receiver system is located at the controller base at the ground. Since the inner PID external commands are transmitted to the local preinstalled controllers, a transmission delay is expected between the base and the crane. The transmission delay, denoted by τ2 ðt Þ, is time-varying and the nonlinear model in (1) takes on the form dxðt Þ ¼ ½Eðxðt ÞÞ 1 f ðxðt Þ; wðt τ2 ðt ÞÞÞ dt
section a signal transmission/synchronization approach will be proposed to synchronize the transmitted signals and eliminate delay variations. To this end, assume that a set of signals, let φi (i ¼ 1; …; ns ), needs to be transmitted. Furthermore, define a base clock period, let T s , upon which signal transmission changes take place. The synchronization algorithm is proposed to be: Synchronization Algorithm Step 1: Step 2: Step 3:
ð4Þ
Clearly, the signals of each network cannot be transmitted simultaneously and an appropriate synchronization algorithm is needed. This algorithm must satisfy the specification of equal delay for all signals.
Step 4: Step 5: Step 6:
3.3. Communication protocol With the aforementioned factors in mind, the Wi-Fi, the Bluetooth and the ZigBee protocols were recognized as suitable candidates. The connectivity of Bluetooth is practically limited to a several meters distance. Moreover, its activation latency, typically about three seconds, was considered rather large for the present application. ZigBee as compared to Wi-Fi is cheaper, easier to implement, more robust from a networking topology perspective and less power consuming. Further-on, the robustness of both systems' direct sequence spread spectrum (ds-ss) modulation is evaluated and compared, since it constitutes the most determining factor against interference. To account for ZigBee’s IEEE 802.15.4 Mac/Physical layer, a 31 chips Gold sequence is used. For the Wi-Fi case, the known 11-chip Barker pn-sequence ‘110110111000’ is used. At both cases, a zero-mean average white Gaussian noise is added using a uniform random number generator. According to our emulation test results, the comparison between ZigBee and Wi-Fi clearly demonstrated the effectiveness of the 802.15.4 modulation type, in terms of signal to noise ratio. 3.4. Elimination of the transmission delay variations As mentioned in the previous subsections, time varying delays appear during wireless communication between the base and the tower crane. Furthermore, due to the system’s configuration, the signals cannot be transmitted simultaneously. In the present
Step 7: Step 8:
At t ¼ t 0 sample and hold φi ðt 0 Þ for i ¼ 1; …; ns . Let i ¼ 1. For the time period t 0 þ T s ði 1Þ r t o t 0 þT s i the transmitter is triggered to transmit the value φi ðt 0 Þ. Within the above time period, triggering takes place multiple times at a sub period T s =r p where r p Z 2. The receiver holds the value φi ðt 0 Þ. Let i ¼ i þ 1. If ir ns go to Step 3 else go to Step 6. For the time period t 0 þ T s ns r t o t 0 þ T s ðns þ 1Þ the transmitter is triggered to transmit a “loop completion” signal to the receiver. Within the above time period, triggering takes place multiple times at a sub period T s =r p where r p Z 2. As soon as the receiver gets the “loop completion” signal releases as output φi ðt 0 Þ (i ¼ 1; …; ns ). Let t 0 ¼ t 0 þ T s ðns þ 1Þ. Go to Step 1.
Although in the above procedure multiple transmissions of the same signal take place, it guaranties that all transmitted signals are synchronized, while a constant delay transmission T s ðns þ 1Þ is achieved. 3.5. Signal reconstruction The final task in the signal transmission scheme lies in the reconstruction of the continuous time signal at the receiver, using discrete time signal data. To this end, there exist several well established procedures, e.g. “zero order hold” (ZOH) and “first order hold” (FOH). The main disadvantage of these procedures is that they generate discontinuous non-smooth signals that may produce significant errors if used for control purposes, especially if the controller includes derivative terms. To avoid these drawbacks, a “delayed interpolation” (DI) scheme will be proposed. The main
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idea of this approach is to use past data to reconstruct the signal at present time. The reconstruction is accomplished by evaluating appropriate interpolating functions, based on past data, through which the value of the signal at the present time will be generated. The proposed procedure will be applied using a 2nd order interpolation scheme. Let T be the sampling period. Three consecutive samples are available from a transmitted signal φðt Þ, let φðkT Þ, φðkT T Þ and φðkT 2T Þ. The 2nd order interpolating function passing through all data points is ϑðt Þ ¼ β2 ðkT Þt 2 þ β1 ðkT Þt þ β0 ðkT Þ. The coefficients β0 ðkT Þ, β1 ðkT Þ and β2 ðkT Þ are evaluated by 2 32 3 2 3 2 β2 ðkT Þ φðkT Þ ðkT Þ kT 1 6 76 7 6 7 2 6 ðkT T Þ ð5Þ kT T 1 7 4 54 β 1 ðkT Þ 5 ¼ 4 φðkT T Þ 5 2 β ð kT Þ φ ð kT 2T Þ 0 ðkT 2T Þ kT 2T 1 The system in (5) is solvable if and only if T a 0 which is obviously satisfied. For t A ðkT; kT þ T the reconstructed signal, let φr ðt Þ, becomes φr ðt Þ ¼ ϑðt 2T Þ. This scheme does not use extrapolation of past data thus providing accurate reconstruction and introduces an additional time delay to the reconstructed signals. Nevertheless, here this issue is overcome using a controller that incorporates the time delays. To demonstrate the efficiency of this scheme the test signal φðt Þ ¼ sin ð2t Þ sin ð3t Þ=ð2 þ cos ð4t ÞÞ is considered. Different sampling periods from 0:1½s to 0:5½s are used and the sampled signal is reconstructed using ZOH, FOH and the DI approach proposed here. Defining a percentile error norm of the form p φ; φr ¼ 100 %‖φðt Þ φr ðt Þ‖2 =‖φðt Þ‖2 it is observed that for all sampling periods examined, the DI scheme provides far more accurate reconstruction than the ZOH or FOH schemes. See Fig. 2 where two sampling periods have been considered.
4. Controller design In the present section a dynamic controller will be designed in order to control the performance outputs of the tower crane. In particular the design goal will be that of Input/Output (I/O) Decoupling (see [18]), where each performance output is controlled by only one external input. Using the results presented in the previous sections and considering that constant transmission/reconstruction delays exists between the controller and the actuators, the overall LA of the nonlinear model takes on the form d δxðt Þ ¼ Aδxðt Þ þ Bδwðt τ2 Þ; δψ ðt Þ ¼ Lδxðt τ1 Þ dt
ð6Þ
where τ1 and τ2 are the constant transmission/reconstruction delays between the sensors and the controller respectively and the controller and the actuators respectively. For the I/O Decoupling goal, the interest is focused on the forced behavior of the system, i.e. for zero initial and past conditions (xðt Þ ¼ 0,wðt Þ ¼ 0 for t r 0). The system in (6) can be described in the frequency domain as follows sδX ðsÞ ¼ AδX ðsÞ þz2 BδW ðsÞ ð7Þ
where δX ðsÞ ¼ L δxðt Þ , δW ðsÞ ¼ L δwðt Þ , δY ðsÞ ¼ L δyðt Þ and δΨ ðsÞ ¼ L δψ ðt Þ with L fg be the Laplace transform of the signal and z1 ¼ e sτ1 ,z2 ¼ e sτ2 . To derive I/O decoupling, the feedback is proposed to be of the dynamic multi delay type (see [18]), i.e.
δW ðsÞ ¼ K ðs; z1 ; z2 ÞδΨ ðsÞ þ Gðs; z1 ; z2 ÞΩðsÞ
ð8Þ
where ΩðsÞ is the 3 1 vector of external inputs. The elements of K ðs; z1 ; z2 Þ and Gðs; z1 ; z2 Þ are rational functions of s with coefficients rational functions of z1 ,z2 . The implementability of the controller requires that the elements of K ðs; z1 ; z2 Þ and Gðs; z1 ; z2 Þ must be realizable (see [18]). Substituting (8) to (7) the I/O decoupling problem is formally stated as follows (see [18]): Find K ðs; z1 ; z2 Þ and Gðs; z1 ; z2 Þ such that
z2 C ½sI 16 A z1 z2 BK ðs; z1 ; z2 ÞL 1 BGðs; z1 ; z2 Þ ¼ diag hi ðs; z1 ; z2 Þ i ¼ 1;:::;3
ð9Þ where hi ðs; z1 ; z2 Þ are appropriate different than zero rational functions of s with coefficients being functions of z1 and z2 and I n is the n-dimensional unitary matrix. Eq. (9) formulates the problem in a normal system form. For (9) to be well defined the precompensator Gðs; z1 ; z2 Þ must be invertible and the feedback to satisfy the inequality matrix K ðs; z1 ; z2 Þ is restricted h i det sI 16 A z1 z2 BK s; z1 ; z2 L ≢0. Let LB ðs; z1 ; z2 Þ ¼ z1 z2 LðsI 16 AÞ 1 B; H B ðs; z2 Þ ¼ z2 C ðsI 16 AÞ 1 B
ð10Þ
where LB ðs; z1 ; z2 Þ is the transfer matrix mapping the actuatable inputs to the measurable variables and H B ðs; z2 Þ is the transfer matrix mapping the actuatable inputs to the performance variables. Similarly to [18], it can be verified that the solution of the controller matrices solving the I/O decoupling problem are K ðs; z1 ; z2 Þ : arbitrary proper and realizable
ð11aÞ
Gðs; z1 ; z2 Þ ¼ I 5 K ðs; z1 ; z2 ÞLB ðs; z1 ; z2 Þ ½H B ðs; z2 Þ 1 diag hi ðs; z1 ; z2 Þ i ¼ 1;:::;3
ð11bÞ For Gðs; z1 ; z2 Þ to be invertible and realizable, it is necessary for hi ðs; z1 ; z2 Þ (i ¼ 1; …; 3) to be different than zero and sufficiently realizable. Thus, the index of realizability of hi ðs; z1 ; z2 Þ will be selected to be greater than or equal to the minus of the index of realizability of the i-th column of I 5 K ðs; z1 ; z2 ÞLB ðs; z1 ; 1 z2 Þg½H B ðs; z2 Þ . LB ðs; z1 ; z2 Þ is realizable. It can be proven that the index of realizability ½H B ðs; z2 Þ 1 is τ2 . Thus, the index of realizability of hi ðs; z1 ; z2 Þ (i ¼ 1; …; 3) must be greater than or equal to τ2 . The diagonal elements of the closed loop transfer matrix connecting the external commands to the respective performance outputs are selected to be of the form X5 α sj =∏7j ¼ 1 ζ i;j s þ 1 ð12Þ hi ðsÞ ¼ z2 1 þ j¼1 j where ζ i;j (i ¼ 1; 2; 3, j ¼ 1; …; 7) are positive numbers. The independent of the delays denominator guaranties asymptotic command following for the performance outputs of the system. The parameters αi (i ¼ 1; …; 5) are dedicated to canceling out the
possible unstable roots of I 5 K ðs; z1 ; z2 ÞLB ðs; z1 ; z2 Þ ½H B ðs; z2 Þ 1 by hi ðsÞ in formula (11b).
5. Performance of the closed loop system To demonstrate the performance of the control scheme proposed in Section 4, consider the data presented in Section 2. To simulate the influence of the wireless network, the True Time Simulator [19] will be used. With regard to the network para meters, minimum frame size is the data rate is 250 kbps , the 30 byte , the transmit power is 0 dBm , the receiver signal threshold is 85 dbm , the path loss exponent is 5:5½ , the ACK
Please cite this article as: Koumboulis FN, et al. Independent motion control of a tower crane through wireless sensor and actuator networks. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.11.011i
F.N. Koumboulis et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎ 4 ½s, the retry limit is 3½ and the packet size timeout is 8:64 U10 is 126 byte . Let T s ¼ 0:01½s and r p ¼ 2. In order to compute the I/O decoupling controller, the following steps are carried out:
Step 1. Using the data in Section 2 and the relations in (10) the transfer matrices LB ðs; z1 ; z2 Þ and H B ðs; z2 Þ are evaluated. Step 2. The feedback matrix K ðs; z1 ; z2 Þ is chosen to be static, thus guaranteeing properness and realizability. Using the 20
0:86779497 6B K ¼ 4@ 0:91902173 0:636785193
5.1. Simulation results The external commands are chosen to be unit step changes with amplitudes 0:1y1 , 0:1y2 and 0:15y3 , respectively To demonstrate the performance of the proposed control scheme when applied to the nonlinear model (1), Figs. 4–9 are presented. In Figs. 4–6 the closed loop responses of the performance outputs for both the linear (dotted line) and the nonlinear (continuous
0:710134519
0:097179456
0:0921621528
0:76829232
0:0477334395
0:0117649226
0:7668754
0:06596675
0:0109753486
metaheuristic algorithm in [16], the feedback matrix is derived to be The above selection increases the closed loop stability margin, as compared to the open loop system, to more than 50%. Step 3. The parameters of the closed loop transfer functions in (12) are selected to be
ζ i;1 ¼ 1:0; ζ i;2 ¼ 1:2; ζ i;3 ¼ 1:4; ζ i;4 ¼ 1:6; ζ i;5 ¼ 1:8; ζ i;6 ¼ 2:0; ζ i;7 ¼ 2:2 f or i ¼ 1; 2; 3 α1 ¼ 6:71012 10 18 ; α2 ¼ 0:0526666; α3 ¼ 3:53399 10 19 ; α4 ¼ 0:0000422826; α5 ¼ 2:83721 10 22 This selection cancels out the unstable roots of I 5 K ðs; z1 ; z2 ÞLB ðs; z1 ; z2 Þg½H B ðs; z2 Þ 1 thus guaranteeing a stable precompensator. Furthermore, the closed loop response for the performance outputs has zero overshoot and a rise time of about 14:2½s. Step 4. Using (11b), the precompensator is derived to be of the form: h i ^ ðsÞ =GðsÞ Gðs; z1 ; z2 Þ ¼ G~ ðsÞ þ z1 z2 G
(14)
^ ðsÞ are appropriate polynomial matrices and where G~ ðsÞ and G ^ ðsÞ, GðsÞ is a stable polynomial. The i; j elements of G~ ðsÞ and G ^ ~ denoted by G i;j ðsÞ and Gi;j ðsÞ respectively, as well as the polynomial GðsÞ are presented in Appendix.
Fig. 4. Nonlinear and linear closed loop response for y1 ðt Þ.
7
0:0885801612
13
7 0:0331280923 C A5
ð13Þ
0:0107663
line) closed loop systems are presented, while in Figs. 7–9 the actuator forces are presented. With respect to the performance outputs (Figs. 4–6), it can readily be observed that the closed loop responses follow accurately (being almost visually identical) the reference responses. With respect to the actuator forces (Figs. 7– 9), it is observed that they do not present significant fluctuations, especially for the mast rotation and cart position actuators. With respect to the cable length actuator (Fig. 9), fast varying changes are observed. Nevertheless, their amplitude is small thus not imposing implementation difficulties. All the state variables remain within acceptable limits. It is pointed that the accuracy range of the LA of the closed loop system as compared to the nonlinear one is preserved (Section 2).
5.2. Influence of the communication delays To investigate the influence of the delays to the controller performance and the necessity for the design of a delay dependent controller, consider the case where the controller δW ðsÞ ¼ K δΨ ðsÞ þ Gðs; 1; 1ÞΩðsÞ is used where K and Gðs; z1 ; z2 Þ are given in (11a), (11b) and (12). The nonlinear closed loop response for this case will be compared to the respective one using the delay dependent controller presented in the previous paragraph. To compare the responses consider the percentile error of the form (3) with τmax ¼ 30½s. It is observed that the percentile error for the first performance variable using the proposed controller in Section 4 as compared to the one using the delayless controller is 96.23% smaller. For the second performance variable the respective error is 91.45% smaller while for the third the respective error is 78.62% smaller. The proposed in the present paper delay dependent controller approach produces far more accurate results.
Fig. 5. Nonlinear and linear closed loop response for y2 ðt Þ.
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5.3. Accuracy of the reconstruction algorithm The final issue that needs to be addressed is whether the use of the DI scheme (for the analog signal reconstruction from the transmitted signals) is advantageous as compared to the use of classical reconstruction approaches such as the ZOH approach, especially after recalling that the DI scheme as used in the present paper increases the transmission delay by two cycles. With respect to the accuracy of the signal reconstruction, the issue has been examined in Section 3 where it has been observed that the reconstructed signal using the DI technique is far more accurate. With respect to the performance outputs it can be observed that using the ZOH approach the results are similar to those using the DI approach. The main advantage of the DI approach lies in the fact that it produces smooth responses for the state variables in the sense that the acceleration of the state variables x1 to x5 does not present high oscillatory behavior. Furthermore, the peak values in the ZOH case are significantly larger as compared to the respective ones in the DI approach. In the ZOH case, the variation ranges for x_ 6 ðt Þ, x_ 7 ðt Þ, x_ 8 ðt Þ, x_ 9 ðt Þ and x_ 10 ðt Þ are 12.91, 250.85, 282.7, 9713.9 and
Fig. 9. Nonlinear closed loop response for the cable actuator force.
426.88 times greater than those in the DI case. High variations in the accelerations of the system motion variables may lead to destructive conditions to the system and reveal flexible structure characteristics that should be included in the model (1), thus producing a more complex mathematical model to describe the system’s dynamics and consequently requiring the design of a much more complex controller. Similar conclusions can be drawn for the implementation of the actuator forces where an additional issue regarding the implementability of these forces arises. Indeed, in the ZOH case, the variation ranges for the actuator forces are 10.85, 47.76 and 426.37 times greater than in the DI case.
6. Conclusions
Fig. 6. Nonlinear and linear closed loop response for y3 ðt Þ.
The general dynamic model of the crane, in the sense that it includes variable cable length and all coupling kinetic terms, has been developed. An accurate LA of the nonlinear plant has been derived. An appropriate realizable multi-delay dynamic controller providing I/O decoupling, depending upon constant communication delays, has been designed. Constant communication delays and accurate signal transmission through digital wireless communication, has been achieved via developing a synchronization algorithm and a signal reconstruction algorithm. The LA accuracy of the open loop and the closed loop system have been preserved.
Appendix
Fig. 7. Nonlinear closed loop response for the arm actuator torque.
Fig. 8. Nonlinear closed loop response for the trolley actuator force.
GðsÞ ¼ s12 þ 26:18s11 þ297:574s10 þ1876:21s9 þ 7200:55s8 þ 17807:9s7 þ þ 29290:5s6 þ32544:7s5 þ 24459:6s4 þ 12232:3s3 þ 3894:36s2 þ 713:511s þ 57:2256 G~ 1;1 ðsÞ ¼ 2:954137 10 9 s11 þ 2:042398 10 7 s10 þ 8:492199 10 6 s9 þ 0:000313735s8 þ þ 0:0064772s7 þ 0:0803441s6 þ 0:63119s5 þ 3:40546s4 þ 13:5813s3 þ 40:8844s2 þ72:5845s þ57:2256 G~ 1;2 ðsÞ ¼ 1:9621839 10 9 s11 þ1:3565922 10 7 s10 þ 1:953201 10 6 s9 þ 0:0000162957s8 þ þ 0:000100922s7 þ0:000566182s6 þ 0:00258026s5 þ 0:0100925 s4 þ 0:0216889s3 þ 0:0197045s2 G~ 1;3 ðsÞ ¼ 2:46997 10 12 s11 þ1:7076622 10 10 s10 þ 2:4566418 10 8 s9 þ 2:037286 10 8 s8 þ 1:23955 10 7 s7 þ 6:21972 10 7 s6 þ 2:1231 10 6 s5 þ 6:243978 10 6 s4 þ 0:0000121s3 þ 0:000011s2 G~ 2;1 ðsÞ ¼ 8:5173 10 27 s10 1:26932276 10 9 s9 6:71001575 10 8 s8 3:18817448 10 6 s7 0:000119747s6 0:00200904s5 0:0196612s4 0:123502s3 0:490858s2 0:999666s 0:45873 G~ 2;2 ðsÞ ¼ 2:321466 10 24 s12 þ 3:459652 10 7 s11 þ 0:00001829s10 þ 0:00040261s9 þ 0:0053433s8 þ
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þ 0:0478925s7 þ 0:31226s6 þ 1:56763s5 þ 6:23987s4 þ 18:9939s3 þ 42:8764s2 þ 72:4277s þ 57:1536 G~ 2;3 ðsÞ ¼ 2:866 10 27 s12 þ 4:2715 10 10 s11 þ 2:25805 8 10 10 s þ 4:901 10 7 s9 þ 6:131 10 6 s8 þ þ 0:000052s7 þ 0:0003376s6 þ0:001725s5 þ 0:006981s4 þ 0:02204s3 þ0:0541s2 þ 0:0914s þ 0:0720572 G~ 3;1 ðsÞ ¼ 1:6793 10 32 s10 2:5027 10 15 s9 2:3098 10 13 s8 þ 1:85461 10 10 s7 þ3:58979 10 9 s6 þ þ 5:01846 10 7 s5 þ 9:9852 10 6 s4 þ 0:0000863:s3 þ 0:0003659s2 þ 0:000724942s þ 0:000577623 G~ 3;2 ðsÞ ¼ 6:7801 10 30 s12 þ 1:0104 10 12 s11 þ 9:3258 10 11 s10 1:5425 10 7 s9 3:2418 10 6 s8 0:0000313s7 0:00018s6 0:000788s5 0:00252s4 0:00461s3 0:003629s2 þ0:0001143s þ 0:000091 G~ 3;3 ðsÞ ¼ 5:38457 10 27 s12 8:02456 10 10 s11 7:40624 10 8 s10 3:619182 10 6 s9 0:00013667s8 0:00365607s7 0:0586131s6 0:531235s5 3:00164s4 12:675s3 40:0256s2 72:5846s 57:2256 ^ 1;1 ðsÞ ¼ 1:29512455 10 24 s10 1:9301077 10 7 s9 G 4:135049719 10 6 s8 0:000276455s7 0:00528582s6 0:0495661s5 0:288407s4 1:26221s3 4:094s2 7:51694s 5:95965 ^ 1;2 ðsÞ ¼ 7:79015 10 23 s11 0:0000116096s10 G 0:000140017s9 0:000272292s8 þ þ 0:0059854s7 þ 0:0708063s6 þ 0:543449s5 þ2:83979s4 þ 10:2277s3 þ30:5558s2 þ 57:6418s þ 49:5976 ^ 1;3 ðsÞ ¼ 1:08242957 10 24 s11 þ 1:613131 10 7 s10 þ G 5:01615769 10 6 s9 þ 0:000282125s8 þ þ 0:0068785s7 þ 0:084515s6 þ0:62817s5 þ 3:2259s4 þ 12:5316s3 þ 34:8916s2 þ 56:6865s þ 40:7004 ^ 2;1 ðsÞ ¼ 6:3615039 10 25 s10 þ 9:4804685 10 8 s9 þ G 2:039842 10 6 s8 þ 0:000136055s7 þ þ 0:00262104s6 þ 0:0250133s5 þ 0:149735s4 þ 0:671718s3 þ 2:18993s2 þ4:00426s þ3:15271 ^ 2;2 ðsÞ ¼ 9:921598 10 24 s11 1:4786031 10 6 s10 G 0:000147572s9 0:00282378s8 0:0286005s7 0:201121s6 1:12136s5 4:78983s4 15:5165s3 40:2824s2 67:2954s 52:5255 ^ 2;3 ðsÞ ¼ 4:290101 10 25 s11 þ 6:3934834 10 8 s10 þ G 2:7545552 10 6 s9 þ 0:000123548s8 þ þ 0:0039646s7 þ 0:058843s6 þ0:491759s5 þ 2:68787s4 þ 11:0146s3 þ33:1052s2 þ 57:5771s þ 43:8998 ^ 3;1 ðsÞ ¼ 8:79148355 10 25 s10 1:31018363 10 7 G 2:8182194 10 6 s8 0:000187911s7 0:00361802s6 0:0342682s5 0:20136s4 0:881882s3 2:8355s2 5:16315s 4:06666 ^ 3;2 ðsÞ ¼ 9:276856 10 24 s11 þ 1:382518 10 6 s10 þ G 0:000109911s9 þ0:00203677s8 þ þ 0:0203885s7 þ 0:142587s6 þ0:790958s5 þ 3:36443s4 þ 10:8714s3 þ28:1153s2 þ 46:79s þ 36:3947
9
^ 3;3 ðsÞ ¼ 1:3180293 10 25 s11 1:9642423 10 8 s10 G 1:8443968 10 6 s9 0:0000608249s8 0:0027467s7 0:0473408s6 0:425221s5 2:4029s4 10:1075s3 31:4407s2 56:2204s 43:8391
References [1] Singhose W, Lawrence J, Sorensen K, Dooroo K. Applications and educational uses of crane oscillation control. FME Trans 2006;34:175–83. [2] Parker GG, Smith AP, Hogan KP. Access to the upper forest canopy with a large tower crane. BioScience 1992;42:664–70. [3] Abdel-Rahman EM, Nayfeh AH, Masoud ZN. Dynamics and control of cranes: a review. J Vib Control 2003;9:863–908. [4] Ghigliazza R, Holmes P. On the dynamics of cranes, or spherical pendula with moving supports. Int J Non-Linear Mech 2002;37:1211–21. [5] Altaf F. Modeling and event-triggered control of multiple 3d tower cranes over WSNs. Stockholm, Sweden: Royal Institute of Technology (KTH); 2010 Master thesis. [6] Altaf F, Araújo J, Hernandez A, Sandberg H, Johansson KH. Wireless eventtriggered controller for a 3D tower crane lab process. Proc 19th Mediterr Conf Control Autom (Med 2011) 2011:994–1001. [7] Bock M, Kugi A. Real-time nonlinear model predictive path-following control of a laboratory tower crane. IEEE Trans Control Syst Technol 2014;22(4):1461– 73. [8] Capella JV, Bonastre A, Ors R. Industrial applications of wireless networks: a bridge crane distributed control system based on Bluetooth. IEEE Int Conf Ind Technol (ICIT 04) 2004:824–9. [9] Benhidjeb A, Gissinger G. Fuzzy control of an overhead crane performance comparison with classic control. Control Eng Pract 1995;3:1687–96. [10] Cheng C-C, Chen C-Y. Controller design for an overhead crane system with uncretainty. Control Eng Pract 1996;4:645–53. [11] Nakazono K, Ohnishi K, Kinjo H, Yamamoto T. Vibration control of load for rotary crane system using neural network with GA-based training. Artif Life Robot 2008;13:98–101. [12] Neupert J, Arnold E, Schneider K, Sawodny O. Tracking and anti-sway control for boom cranes. Control Eng Pract 2010;18:31–44. [13] Terashima K, Shen Y, Yano K i. Modeling and optimal control of a rotary crane using the straight transfer transformation method. Control Eng Pract 2007;15:1179–92. [14] He W, Zhang S, Ge SS. Adaptive control of a flexible crane system with the boundary output constraint. IEEE Trans Ind Electron 2014;61(8):4126–33. [15] Tsujita K, Tsuchiya K, Onat A. Decenrtalized autonomous control of a quadrupedal locomotion robot using oscillators. Artif Life Robot 2001;5(3):153–8. [16] Koumboulis F, Kouvakas N, Paraskevopoulos P. Linear approximant-based metaheuristic proportional- integral-derivative controller for a neutral time delay central heating system. Proc Inst Mech Eng, Part I: J Syst Control Eng 2009;223:605–18. [17] Pearce HM. The Design and Construction of an Intelligent Power Assist Jib Crane. Illinois, USA: Northwestern University; 1999 Master Thesis. [18] Koumboulis FN, Kouvakas ND. I/O decoupling with simultaneous Disturbance Rejection of general neutral time delay systems via a measurement output feedback dynamic controller. Proc 21St Mediterr Conf Control Autom (Med) 2013:890–5. [19] Cervin A, Henriksson D, Lincoln B, Eker J, Årzén KE. How does control timing affect performance? analysis and simulation of timing using jitterbug and truetime IEEE Control Syst Mag 2003;23:16–30.
Please cite this article as: Koumboulis FN, et al. Independent motion control of a tower crane through wireless sensor and actuator networks. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.11.011i
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Independent motion control of a tower crane through wireless sensor and actuator networks Fotis N. Koumboulis a,n, Nikolaos D. Kouvakas a, George L. Giannaris a,b, Demosthenes Vouyioukas b a b
Sterea Ellada Institute of Technology, Department of Automation Engineering, Psahna Evias 34400 Greece University of the Aegean, Department of Information & Communication Systems Engineering, 83200 Karlovasi, Samos, Greece
art ic l e i nf o
a b s t r a c t
Article history: Received 2 January 2015 Received in revised form 24 August 2015 Accepted 9 November 2015 This paper was recommended for publication by Y. Chen.
The problem of independent control of the performance variables of a tower crane through a wireless sensor and actuator network is investigated. The complete nonlinear mathematical model of the tower crane is developed. Based on appropriate data driven norms an accurate linear approximant of the system, including an upper bound of the communication delays, is derived. Using this linear approximant, a dynamic measurable output multi delay controller for independent control of the performance outputs of the system is proposed. The controller performs satisfactory despite the nonlinearities of the model and the communication delays of the wireless network. & 2015 ISA. Published by Elsevier Ltd. All rights reserved.
Keywords: Dynamic multi-delay controllers I/O decoupling Time delay Tower crane Wireless network
1. Introduction Cranes are used in several applications having significant economic impact. Classical uses are in construction (bridges, dams, buildings), transportation (loading and unloading cargo), industry (oil platforms, refineries), in nuclear power plants and in bio/ecological applications (see [1–3] and the references therein). Efficient crane maneuverability needs appropriate control, requiring accurate mathematical model of the crane (see f.e. [4–14]). For the case of tower cranes significant results have been presented in [1–4]. In [4] a simplified nonlinear model of a tower crane is being developed in the form of a spherical pendulum in a non-inertial frame. The model is then used to analyze the cases of linearly accelerating support and the support describing a circular path at constant speed. In [5], a simplified mathematical model of a tower crane is being developed. The model is then used to demonstrate the performance of event trigger controllers for wireless control of multiple 3d tower cranes. The influence of the network uncertainties such as time delays and packet dropouts are not taken into account in the controller design stage but they are used in the computational experiments of the closed loop system. n
Corresponding author. E-mail address: [email protected] (F.N. Koumboulis).
Similar issues are investigated in [6]. In [7], the mathematical model of a laboratory tower crane is being developed and a local nonlinear model predictive control is proposed for the performance output to follow a desired path. The dynamics of the trolley, the length and the angular displacement of the jib are neglected and the respective accelerations are considered as actuatable input to the system. For other categories of cranes (such as bridge cranes, overhead cranes, rotary cranes and boom cranes), several results have been published. In particular in [8] the Bluetooth protocol is used for short range wireless control of a bridge crane system. The controller is of the rule based type not using information from the system dynamics. The influence of network parameters is not investigated. In [9], a local fuzzy logic control scheme and a local LQG controller are being used to regulate the performance output of an 2d overhead crane made up of a platform carrying a cable holding the load. The results are validated using physical experiments. In [10] the mathematical model of the 2d overhead crane system is used to develop a local controller which combines a feedback linearization approach and a time delay control scheme. In [11] a simplified mathematical model of a rotary crane system is being used to develop a local neuro-controller for vibration control of the load involving the rotation about the vertical axis only. In [12], based upon a simplified mathematical model of a boom
http://dx.doi.org/10.1016/j.isatra.2015.11.011 0019-0578/& 2015 ISA. Published by Elsevier Ltd. All rights reserved.
Please cite this article as: Koumboulis FN, et al. Independent motion control of a tower crane through wireless sensor and actuator networks. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.11.011i
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crane, the I/O linearization approach is being used to locally control slewing and luffing motion. The efficiency of the proposed approach is demonstrated through physical experiments on an industrial harbor mobile crane. In [13] a simplified mathematical model of a rotary crane is being developed and an optimal local open loop control approach is being used for sway free, point to point motion of the load mass. The performance of the proposed approach is demonstrated through physical and computational experiments. In [14] the mathematical model of a crane system which is equipped with a flexible cable is being developed in the form of hybrid system represented by partial-ordinary differential equations. Based upon the mathematical model of the system, a local integral-barrier Lyapunov function based control scheme is proposed to suppress the undesirable vibrations. From the communication point of view, the controllers in [5,6,8] use wireless connections. The controllers proposed in [7,9,10–14] use wired connections. From the control point of view, in [5,6] event triggered schemes have been proposed. In [8,9] pure rule based and fuzzy approaches have been used. In [9,13] optimal approaches have been applied. In [11] an intelligent neural approach has being used. In [7] model predictive controllers are used. In [10,12] feedback linearization approaches are used. In [10] a delay is also included to improve the control performance. In [14] a nonlinear Lyapunov based controller is proposed. Here, we propose Input Output (I/O) Decoupling of the motion variables of a tower crane through a wireless sensor and actuator network. The complete nonlinear dynamic mathematical model of a tower crane is developed. The crane is considered to be equipped with three preinstalled approximate PID controllers. Using the nonlinear model, the linear approximant of the system is derived. Its accuracy is investigated via series of computational experiments. For the remote signal transmission among the sensors, the actuators and the controller, the ZigBee protocol is used. A synchronization algorithm, guaranteeing constant transmission delay, is proposed. Additionally, an algorithm towards signal reconstruction of the analogue continuous time signals from the digital transmitted signals is proposed. The contribution of the present paper is summarized to the following points: 1. A dynamic model of the system is developed. The model is more general, as compared to those in [4–7], in the sense that it includes variable cable length and all coupling terms in the system kinetics. 2. As compared to [5,6,8] the delays of the wireless communication are taken in to account in the controller design. The issue is treaded as follows: a) Due to the presence of the delays, an inverse dynamics controller compensating the nonlinearities of the system is not adequate. This is why a linear approximant, as well as a linear controller, performing satisfactory despite the nonlinearities of the crane is proposed. b) A synchronization algorithm is proposed. Using this algorithm, the uncertain communication delay becomes constant and known to the designer. The values of the delay are used to develop an appropriate realizable multi-delay dynamic controller providing I/O Decoupling for the performance outputs preserving the accuracy of the closed loop linear approximant. 2. Dynamics of a tower crane Cranes are worksite mechanisms used to lift and lower loads as well as to place them in the site. A tower crane (see Fig. 1) is a
modern form of balance crane consisting of three mechanical parts: an arm rotating around a vertical mast, a trolley moving along the arm and a cable drum with a load at its end. The arm rotates around the mast by an arm motor (actuator 1), the trolley moves along the arm by another motor (actuator 2) and the payload is lifted or lowered by a third motor (actuator 3) rotating the cable drum to gather or release the cable. The tower crane is a highly oscillatory system with linear and nonlinear dynamics. The nonlinear dynamics come mainly from the rotational motion inducing centripetal and Coriolis accelerations producing instability. In what follows, the mathematical model of the tower crane will be developed using the Euler–Lagrange approach. The tower crane is inherently unstable with respect to all motion variables. In such mechanical systems it is a common practice (see [15]) to use pre-installed three term controllers to regulate local performance variables. Such pre-installed local PID controllers are necessary for the safe ground operation. Here, 3 pre-installed approximate PID controllers are considered. The 1st PID stabilizes the velocity of the arm, the 2nd the position of the trolley along the arm and the 3rd the cable length. 2.1. 2.1 State space model of the crane
hDefine x ¼ x1 ⋯ ¼ q1 q2 q3 q4 q5 q_ 1
x16 q_ 2
T q_ 3
q_ 4
q_ 5
χ 1;1 χ 1;2 χ 2;1 χ 2;2 χ 3;1 χ 3;2
iT
h iT h iT T w ¼ w1 w2 w3 , y ¼ y1 y2 y3 and ψ ¼ ψ 1 ⋯ ψ 5 , where x is the state vector, y is the performance output vector, ψ is the measurable output vector, w is the vector of external commands of the pre-installed PID controllers (w1 ,w2 and w3 are the external commands for the rotational velocity of the crane, the position of the trolley and the cable length, respectively), q1 is the arm rotation angle with respect to an inertial frame, q2 is the distance of the trolley from the crane’s revolution axis, q3 and q4 are the cable’s angles and q5 is the cable length, χ i;j (i ¼ 1; 2; 3 and j ¼ 1; 2) are internal variables of the approximate PID controllers. The mathematical model of the tower crane is developed to be dx ¼ ½EðxÞ 1 f ðx; wÞ; y ¼ r ðxÞ; ψ ¼ Lx; EðxÞ A ℝ1616 ; f ðx; wÞ A ℝ161 ; dt r ðxÞ A ℝ31 ; L A ℝ516 ð1Þ The nonzero elements of EðxÞ and F ðx; wÞ are: e1;1 ðxÞ ¼ 1, e2;2 ðxÞ ¼ 1, e3;3 ðxÞ ¼ 1, e4;4 ðxÞ ¼ 1, e5;5 ðxÞ ¼ 1, 2 e6;6 ðxÞ ¼ J 1 þ J 2 þJ 5 þ lc m1 þ ðm2 þm5 Þx22 2sx4 m5 x2 x5 0:25 c2x3 2 2 þ 2cx3 c2x4 3Þm5 x5 , e6;7 ðxÞ ¼ cx4 sx3 m5 x5 , e7;10 ðxÞ ¼ sx4 m5 , e6;8 ðxÞ ¼ cx3 cx4 m5 x5 x2 sx4 x5 , e6;9 ðxÞ ¼ sx3 J 5 þ m5 sx3 x5 x5 sx4 x2 , e6;10 ðxÞ ¼ cx4 sx3 m5 x2 , e7;6 ðxÞ ¼ cx4 sx3 m5x5 , e7;7 ðxÞ ¼ m2 þ m5 , e7;9 ðxÞ ¼ cx4 m5 x5 ; e8;6 ðxÞ ¼ cx3 cx4 m5 x5 x2 sx4 x5 , e8;8 ðxÞ ¼ c2x4 m5 x25 þJ 5 ; e9;6 ðxÞ ¼ sx3 J 5 þ m5 sx3 x5 x5 sx4 x2 , e9;7 ðxÞ ¼ cx4 m5 x5 , e9;9 ðxÞ ¼ m5 x25 þ J 5 , e10;6 ðxÞ ¼ cx4 sx3 m5 x2 , e10;7 ðxÞ ¼ sx4 m5 , e10;10 ðxÞ ¼ m5 , e11;11 ðxÞ ¼ 1, e12;12 ðxÞ ¼ 1, e13;13 ðxÞ ¼ 1, e14;14 ðxÞ ¼ 1, e15;15 ðxÞ ¼ 1, e16;16 ðxÞ ¼ 1, f 1 ðx; wÞ ¼ x6 , f 2 ðx; wÞ ¼ x7 , f 3 ðx; wÞ ¼ x8 , f 4 ðx; w Þ ¼ x9 , f 6 ðx; wÞ ¼ 2m2 x2 x6 x7 þ 0:5m5 x5 x6 4sx4 x7 2x5 f 5 ðx; wÞ ¼ x10 ,
c2x4 s2x3 x8 þ c2x3 s2x4 x9 x5 x8 sx3 s2x4 x8 þ 4cx3 s2x4 x9 þ i c2x3 þ 2c2x3 c2x4 3 x6 þ 2cx3 s2x4 x8 4sx3 x9 x10 þ m5 x2 ð 2x6 x7 þ
cx4 x5 2x6 x9þ sx3 x28 þ x29 2cx3 x8 x10 þ2sx4 cx3 x5x8 x9 þ x6 þ sx3 2 x9 Þx10 Þ þ w1 γ 1 f D;1 þ f P;1 J 5 cx3 x8 x9 γ 1 x12 f D;1 þ γ 1 x11 þ x12 f I;1 x6 γ 1 f D;1 þ f P;1 , f 7 ðx; wÞ ¼ m2 x2 x26 þ m5 x2 x26 þ 2m5 x5 cx3 cx4 x6 x8 m5 x5 sx4 x26 þ 2sx3 x6 x9 þ x29 þ 2m5 cx4 sx3 x6 þ x9 x10 þ γ 2 f D;2 ðw2 x2 Þ γ 22 x14 f D;2 þ γ 2 x13 þ x14 Þf I;2 þ ðw2 x2 Þf P;2 , f 8 ðx; wÞ ¼ J 5 cx3 x6 x9 gm5 cx4 x5 sx3 2m5 c2x4 x5 x8 x10 þ m5 cx4 x25 2sx4 x8 x9 þ cx3 cx4 x6 sx3 x6 þ2x9
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2m5 cx3 cx4 x5 x6 x7 sx4 x10 ,
f 9 ðx; wÞ ¼ m5 sx4 x5 2sx3 x6 x7 gcx3 J 5 cx3 x6 x8 m5 cx4 x5 x2 x26 þ m5 cx4 x25 c2x3 sx4 x26 2cx3 cx4 x6 x8 sx4 x28 2m5 x5 sx3 x6 þx9 x10 , f 10 ðx; wÞ ¼ m5sx4 x2 x26 þm 5 cx4 gcx3 2sx3 x6 x7 Þ þ m5 x5 x6 2sx3 x9 cx3 s2x4 x8 þm5 x5 c2x4 x28 þ x29 0:25m5 x5 2c2x3 c2x4 þ c2x4 3 x26 γ 23 x16 f D;3 þ γ 3 x15 þ x16 f I;3 þ ðw3 x5 Þ γ 3 f D;3 f 13 ðx; wÞ ¼ x14 ; þf P;3 Þf 11 ðx; wÞ ¼ x12 ; f 12 ðx; wÞ ¼ w1 x6 γ 1 x12 ; f 14 ðx; wÞ ¼ w2 x2 γ 2 x14 ; f 15 ðx; wÞ ¼ x16 ; f 16 ðx; wÞ ¼ w3 x5 γ 3 x16 The 1st performance output is the angular velocity of the payload (around the mast), i.e. y1 ¼ r 1 ðxÞ ¼ x6 þ sx3 x9 . The 2nd is the h radial position of the payload, i.e. y2 ¼ r 2 ðxÞ ¼ x22 2sx4 x2 x5 þ i0:5 and the 3rd is the vertical distance of the payc2x4 s2x3 þ s2x4 x25 load from the ground, i.e. y3 ¼ r 3 ðxÞ ¼ h0 cx3 cx4 x5 . So, r ðxÞ ¼ r 1 ðxÞ r 2 ðxÞ r 3 ðxÞ T . The 1st measurable variable is the position of the trolley, i.e. x2 , the 2nd is the cable length, i.e. x5 , the 3rd is the angular velocity of the arm, i.e. x6 , the 4th is the velocity of the trolley, i.e. x7 and the 5th is the rate of change of the cable’s length, i.e. x10 . Hence, the nonzero elements of L are l1;2 ¼ 1, l2;5 ¼ 1, l3;6 ¼ 1, l4;7 ¼ 1 and l5;10 ¼ 1. m1 , m2 and m5 are the arm, trolley and payload masses respectively, J 1 , J 2 and J 5 are the arm, trolley and payload moments of inertia respectively, g is the gravitational acceleration, lc is the distance of the center of mass of the arm from the revolution axis of the crane, h0 is the length of the mast and cα ¼ cos ðαÞ, sα ¼ sin ðαÞ. Each approximate PID controller regulates one part of the crane. Let f P;i , f I;i and f D;i (i ¼ 1; 2; 3) are the proportional, integral and derivative gains respectively of the three PID controllers and γ 1 , γ 2 and γ 3 are the three filter coefficients for the proper approximate implementation of the respective derivative terms. The parameters of the PID controllers are dedicated to achieve pole placement and asymptotic command following for the respective outputs. Additionally, the parameters of the 1st PID controller increase the properness of the respective closed loop transfer function. Hence, the PID parameters are derived to be 2 2 2 f P;1 ¼ J 1 þ lc m1 λ1;1 λ2;1 λ0;1 λ2;1 ; f P;2 ¼ m2 λ1;2 λ3;2 λ0;2 λ3;2 ; 2 1 2 f P;3 ¼ m5 λ1;3 λ3;3 λ0;3 λ3;3 ; f I;1 ¼ J 1 þ lc m1 λ0;1 λ2;1 ; 1
1
f I;2 ¼ m2 λ0;2 λ3;2 ; f I;3 ¼ m5 λ0;3 λ3;3 ; 3 2 f D;1 ¼ J 1 þ lc m1 λ0;1 λ1;1 λ2;1 λ2;1 ; γ 1 ¼ λ2;1 ; 3 f D;2 ¼ m2 λ0;2 þ λ3;2 λ2;2 λ3;2 λ1;2 λ3;2 ; γ 2 ¼ λ3;2 ; 3 f D;3 ¼ m5 λ0;3 þ λ3;3 λ2;3 λ3;3 λ1;3 λ3;3 ; γ 3 ¼ λ3;3 where the parameters λi;j (i ¼ 0; 1; 2; 3, j ¼ 1; 2; 3) are free to be chosen by the designer. They are the coefficients of the characteristic polynomials of each subsystem of the crane, i.e. P P p1 ðsÞ ¼ s3 þ 2j ¼ 0 λj;1 sj , pi ðsÞ ¼ s4 þ 3j ¼ 0 λj;i sj (i ¼ 2; 3). Clearly, λj;i are chosen such that pi ðsÞ (i ¼ 1; 2; 3) are stable. The developed here model of the crane is more general than those presented in [5,6], in the sense that it includes variable cable length, as well as all coupling terms in the system dynamics. 2.2. Linear approximant around constant velocity of the arm Here, a linear approximant (LA) of the nonlinear model (1) around a constant angular velocity ω of the arm of the crane will be produced. Let xi be the nominal points of xi for i ¼ 2; :::; 16, where x6 ¼ ω. The nominal trajectory of the angle of the arm is x1 ðt Þ ¼ ωt þ q1;0 , where q1;0 is the initial angle of the mast. Let xðt Þ, yðt Þ and wðt Þ be the nominal trajectories of the state, performance output and input vectors of the model (1). Clearly, x_ ðt Þ ¼ h iT ω 0115 T . Let wðt Þ ¼ ω x2 x5 , where x2 and x5 are the nominal values of the trolley position along the arm and the cable
3
length respectively. Thus, xðt Þ and yðt Þ are evaluated by x1 ðt Þ ¼ ωt þ q1;0 ; x2 ¼ x2 ; x3 ¼ 0; x5 ¼ x5 ; x6 ¼ ω; x7 ¼ x8 ¼ x9 ¼ x10 ¼ x11 ¼ x12 ¼ x14 ¼ x16 ¼ 0; x13 ¼ m5 x5 sx4 ðm2 þ m5 Þx2 ω2 =f I;2 γ 2 ; x15 ¼ m5 ω2 sx4 x2 x5 sx4 gcx4 =f I;3 γ 3 ; y1 ¼ ω; y2 ¼ x2 x5 sx4 ; y3 ¼ h0 x5 cx4 where the nominal value x4 is evaluated by the equation gsx4 þ ω2 cx4 x2 x5 sx4 ¼ 0. The deviations around the nominal values of the external commands and the performance, state and measurable variables are Δw ¼ w w, Δy ¼ y y, Δx ¼ x x and Δψ ¼ ψ ψ . The LA of (1) is d δx ¼ Aδx þ Bδw; δy ¼ C δx; δψ ¼ Lδx dt
ð2Þ
where δw ¼ Δw and δx, δy and δψ are the LA responses, approximating the deviations Δx, Δy and Δψ . LA operates around
an operating and
point oðt Þ ¼ oðt Þ where oðt Þ ¼ w; x; y A B o ð t Þ ¼ w; x; y . The LA system matrices are ¼ h i ∂ ∂x
∂ ∂w
½EðxÞ 1 F ðx; wÞjo ¼ o , C ¼ ∂Γ ðxÞ=∂xjo ¼ o . A and B can be
1 1 ~ The nonzero eleexpressed in the form A ¼ E~ A~ and B ¼ E~ B. ~ A~ and B~ are ments of E,
e~ 1;1 ¼ 1, e~ 2;2 ¼ 1, e~ 3;3 ¼ 1, e~ 4;4 ¼ 1, e~ 5;5 ¼ 1, e~ 6;6 ¼ J 1 þ J 2 þJ 5 þ 2 lc m1 þx5 sx4 x5 sx4 2x2 m5 þx2 e~ 6;8 ¼ x5 cx4 x2 x5 sx4 2 ðm2 þ m5 Þ, m5 , e~ 7;7 ¼ m2 þ m5 , e~ 7;9 ¼ x5 cx4 m5 , e~ 7;10 ¼ sx4 m5 , e~ 8;6 ¼ x5 cx4 2 ~ ~ x2 x5 sx4 Þm5 , e~ 8;8 ¼ J 5 þx2 5 cx4 m5 , e 9;7 ¼ x5 cx4 m5 e 9;9 ¼ J 5 þ 2 x5 m5 , e~ 10;7 ¼ sx4 m5 , e~ 10;10 ¼ m5 , e~ 11;11 ¼ e~ 12;12 ¼ e~ 13;13 ¼ e~ 14;14 ¼ e~ 15;15 ¼ e~ 16;16 ¼ 1, a~ 7;2 ¼ ω2 ðm2 þ m5 Þ γ 2 f D;2 f P;2 , a~ 9;2 ¼ x5 ω2 cx4 m5 , a~ 10;2 ¼ ω2 sx4 m5 , a~ 14;2 ¼ 1, a~ 8;3 ¼ x5 cx4 x5 ω2 cx4 g m5 , 2 2 a~ 7;4 ¼ x5 ω cx4 m5 , a~ 9;4 ¼ x5 ω x5 c2x4 þx2 sx4 gcx4 m5 , a~ 10;4 ¼ 2 ω cx 2x5 sx4 x2 gsx4 m5 , a~ 7;5 ¼ ω2 sx 4 m5 , a~ 9;5 ¼ ω2 cx4 4 2 a~ 10;5 ¼ ω2 s2x m5 2x5 sx4 x2 gsx4 m5 ,a~ 8;9 ¼ ω J 5 þ 2m5 x2 5 cx4 4 ~ ~ ~ γ 3 f D;3 f P;3 , a 16;5 ¼ 1, a 1;6 ¼ 1, a 6;6 ¼ γ 1 f D;1 f P;1 , a~ 7;6 ¼ 2ω x2ðm2 þ m5 Þ x5 sx4 m5 , a~ 9;6 ¼ 2x5 ωcx4 x5 sx4 x2 m5 , a~ 10;6 ¼ 2ωsx4 x5 sx4 x2 m5 ,a~ 12;6 ¼ 1, a~ 2;7 ¼ 1, a~ 6;7 ¼ 2x5 ωsx4 m5 2x2 ω ðm2 þ m5 Þ, a~ 8;7 ¼ 2x5 ωcx4 m5 , a~ 3;8 ¼ 1, a~ 7;8 ¼ 2x5 ωcx4 m5 , a~ 9;8 ¼ ω J 5 þ 2x2 c2 m5 , a~ 10;8 ¼ 2x5 ωcx4 sx4 m5 , a~ 4;9 ¼ 1, a~ 6;9 ¼ 5 x4 a~ 5;10 ¼ 1, a~ 6;10 ¼ 2ωsx4 x2 x5 sx4 m5 , 2x5 ωcx4 x2 x5 sx4 m5 , 2 a~ 8;10 ¼ x5 ωs2x4 m5 , a~ 6;11 ¼ γ 1 f I;1 , a~ 6;12 ¼ γ 1 f D;1 þ f I;1 , a~ 11;12 ¼ 1, a~ 12;12 ¼ γ 1 , a~ 7;13 ¼ γ 2 f I;2 , a~ 7;14 ¼ γ 22 f D;2 þ f I;2 , a~ 13;14 ¼ 1, a~ 14;14 ¼ γ 2 , a~ 10;15 ¼ γ 3 f I;3 , a~ 10;16 ¼ γ 23 f D;3 þ f I;3 , a~ 15;16 ¼ 1, a~ 16;16 ¼ γ 3 , b~ 6;1 ¼ γ 1 f D;1 þ f P;1 , b~ 7;2 ¼ γ 2 f D;2 þ f P;2 , b~ 10;3 ¼ γ 3 f D;3 þ f P;3 , b~ 12;1 ¼ 1, b~ 14;2 ¼ 1, b~ 16;3 ¼ 1 The nonzero elements of C are c1;6 ¼ 1, c2;2 ¼ 1, c2;4 ¼ x5 cx4 , c2;5 ¼ sx4 , c3;4 ¼ x5 sx4 and c3;5 ¼ cx4 . 2.3. Accuracy of the linear approximant Here, it will be investigated via series of computational experiments if the LA (2) of the tower crane performs similarly to the nonlinear model (1). At first, assume that the model (1) operates on certain operating conditions. In particular, let w1 , w2 and w3 be the nominal values of the external commands of the approximate PID controllers yielding the respective operating trajectories for the state variables, let x1 ðt Þ and xj for j ¼ 2; …; 16. In what follows, small variations of the model will be considered. To this end, the external commands are chosen to be of the form ^ i ρðt; τ1 ; τ 2 ; T 1 ; T 2 ; T 3 Þ; i ¼ 1; 2; 3; w ^ iA w ^ i min ; w ^ i max ; w i ðt Þ ¼ w i þ w
ρðt; τ1 ; τ2 ; T 1 ; T 2 ; T 3 Þ ¼ σ ðt; τ1 Þ σ ðt; τ2 Þ τ1 t
e
T1
τ1 t
τ1 t
T 21 ðT 2 T 3 Þ e T 2 T 22 ðT 1 T 3 Þ þ e T 3 T 23 ðT 1 T 2 Þ σ ðt; τ1 Þ ðT 1 T 2 ÞðT 1 T 3 ÞðT 2 T 3 Þ
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4 τ2 t
τ2 t
τ2 t
T 21 ðT 2 T 3 Þ e T 2 T 22 ðT 1 T 3 Þ þ e T 3 T 23 ðT 1 T 2 Þ σ ðt; τ2 Þ; ðT 1 T 2 ÞðT 1 T 3 ÞðT 2 T 3 Þ ( 0 t o τi σ ðt; τi Þ ¼ ; τ ¼ 1; 2 1 t Z τi i þ
e
T1
with τ1 o τ2 and T 1 , T 2 , T 3 being positive different among themselves constants. The form of ρ is such that its 1st and 2nd derivatives with respect to time are smooth. Using these input changes, the response of the nonlinear model (1) regarding the performance outputs yi (i ¼ 1; 2; 3) and the measurable variables ψ k (k ¼ 1; …; 5) is computed. The responses of the LA and nonlinear model are compared using a Euclidian type norm the form [16] p φj ; δφj ¼ 100% ‖φj ðt Þ δφj ðt Þ φj ‖2 =‖φj ðt Þ φj ‖2 ð3Þ where φj is a signal representing the state, performance or mea i2 1=2 Rτ h surable variables, ‖φj ðt Þ‖2 ¼ 0 max φj ðt Þ dt and τmax A ℝ þ where τmax represents a time instance, greater than the relaxation time of the argument response. To evaluate the cost in (3), consider the test case presented in [17]: m1 ¼ 73:377½Kgr, m2 ¼ 2:492½Kgr, m5 ¼ 3:1415½Kgr, J 1 ¼ 2 13:609 ½Kgr m2 , J 2 ¼ 8:572½Kgr m 2,l c ¼ 0:4517½m, J 5 ¼ 0:01256 2 ½Kgr m , h0 ¼ 2:55½m, g ¼ 9:81 m=s The nominal and the parameters are: values controller ω ¼ π =20 rad=s , q1;0 ¼ 0 rad , x2 ¼ 0:5½m, x5 ¼ 0:5½m, λ0;1 ¼ 4766:52, λ1;1 ¼ 889:21, λ2;1 ¼ 53, λ0;2 ¼ 1712:2557, λ1;2 ¼ 1425:512, λ2;2 ¼ 371:78, λ3;2 ¼ 35:2, λ0;3 ¼ 122138:016, λ1;3 ¼ 30449:576, λ2;3 ¼ 2475:06, λ3;3 ¼ 83:1, w 1 ¼ π =6 rad=s , w2 ¼ 0:5½m, w3 ¼ 0:75½m. For the above values of the system parameters, the inner controller coefficients are computed to be: f p;1 ¼ 431:01, f i;1 ¼ 2570:35, f d;1 ¼ 8:13227, γ 1 ¼ 53, f p;2 ¼ 97:476, f i;2 ¼ 121:22, f d;2 ¼ 23:5511, γ 2 ¼ 35:2, f p;3 ¼ 1095:58, f i;3 ¼ 4617:42, f d;3 ¼ 80:3857, γ 3 ¼ 83:1 while the operating trajectories for the state variables and performance outputs are evaluated to be: ½m x3 ¼ 0 rad , x4 ¼ 0:00125918 x1 ðt Þ ¼ π t=20 rad , x2 ¼ 0:5 rad , x5 ¼ 0:5 ½m, x 6 ¼ π =20 rad=s , x 7 ¼ 0 m=s , x 8 ¼ 0 rad=s , x 9 ¼ 0 rad=s , x 10 ¼ 0 m=s , x11 ¼ 0½, x12 ¼ 0½, x13 ¼ 0½, x14 ¼ 0½, x15 ¼ 0½, x16 ¼ 0½, y 1 ¼ π =20 rad=s , y2 ¼ 0:5006½m, y3 ¼ 2:05016½m. To evaluate the accuracy of the LA through the cost criterion (3) for the measurable and performance variables of the plant, the
Fig. 1. Tower crane configuration.
external commands are to deviate from their nominal considered ^ i =wi o0:15 (i ¼ 1; 2; 3). Here, the level of values about 15%, i.e. w accuracy of the LA is considered to be the maximum of p for all performance and measurable variables. After computing the level of accuracy for all combinations in the aforementioned indicative deviation bounds of the external commands, it has been shown that a level of accuracy of 10.16% is derived, i.e. for all combina w ^ i =wi o0:15, the maximum of tions of the external commands in
p φj ; δφj for φj A y1 ; y2 ; y3 ; ψ 1 ; …; ψ 5 is less than or equal to 10.16%. Hence, the LA (3) is an accurate representation of the nonlinear model (1) and so a controller can be designed on the basis of it.
3. Wireless controller implementation Wireless control overcomes mobility limitations and enhances the system's flexibility especially for the case of controlling simultaneously multiple mechanisms. However, wireless control necessitates high fidelity equipment. For our purposes, neither data storage limitations, transmission rate nor long distances are crucial factors to consider. A major limitation to consider is the need for a telecommunication signal resisting to noise and other users' interference. This is an inherent situation to any industrial environment. 3.1. Measurable variables Since a wireless controller has to be designed, a transmission delay is expected between the sensors and the controller base. Furthermore, this transmission delay is in general time-varying. Hence, the vector of the system’s measurable outputs is ψ ðt Þ ¼ Lxðt τ1 ðt ÞÞ. The delay τ1 ðt Þ stands for the transmission/ reconstruction delay taking place between the sensors and the controller. Without loss of generality, this delay is assumed to be the same for all measurable variables. So, an appropriate transmission/ reconstruction algorithm will be proposed in order to eliminate delay variations. Elimination of the delay variations will clearly facilitate controller design and implementation. 3.2. Controller configuration With respect to the controller configuration, it is considered that the setup consists of two separate networks (see Fig. 3).
Fig. 2. Signal reconstruction error.
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Fig. 3. Block diagram of the closed loop system.
The 1st is dedicated to the transmission of the measurable outputs from the sensors to the controller, while the 2nd is dedicated to the transmission of the inner external commands to the preinstalled PIDs on the tower crane. Each network consists of a single transmitter and a single receiver. The transmitter and receiver of the tower crane are assumed to be located at the highest point of the mast while the 2nd transmitter/receiver system is located at the controller base at the ground. Since the inner PID external commands are transmitted to the local preinstalled controllers, a transmission delay is expected between the base and the crane. The transmission delay, denoted by τ2 ðt Þ, is time-varying and the nonlinear model in (1) takes on the form dxðt Þ ¼ ½Eðxðt ÞÞ 1 f ðxðt Þ; wðt τ2 ðt ÞÞÞ dt
section a signal transmission/synchronization approach will be proposed to synchronize the transmitted signals and eliminate delay variations. To this end, assume that a set of signals, let φi (i ¼ 1; …; ns ), needs to be transmitted. Furthermore, define a base clock period, let T s , upon which signal transmission changes take place. The synchronization algorithm is proposed to be: Synchronization Algorithm Step 1: Step 2: Step 3:
ð4Þ
Clearly, the signals of each network cannot be transmitted simultaneously and an appropriate synchronization algorithm is needed. This algorithm must satisfy the specification of equal delay for all signals.
Step 4: Step 5: Step 6:
3.3. Communication protocol With the aforementioned factors in mind, the Wi-Fi, the Bluetooth and the ZigBee protocols were recognized as suitable candidates. The connectivity of Bluetooth is practically limited to a several meters distance. Moreover, its activation latency, typically about three seconds, was considered rather large for the present application. ZigBee as compared to Wi-Fi is cheaper, easier to implement, more robust from a networking topology perspective and less power consuming. Further-on, the robustness of both systems' direct sequence spread spectrum (ds-ss) modulation is evaluated and compared, since it constitutes the most determining factor against interference. To account for ZigBee’s IEEE 802.15.4 Mac/Physical layer, a 31 chips Gold sequence is used. For the Wi-Fi case, the known 11-chip Barker pn-sequence ‘110110111000’ is used. At both cases, a zero-mean average white Gaussian noise is added using a uniform random number generator. According to our emulation test results, the comparison between ZigBee and Wi-Fi clearly demonstrated the effectiveness of the 802.15.4 modulation type, in terms of signal to noise ratio. 3.4. Elimination of the transmission delay variations As mentioned in the previous subsections, time varying delays appear during wireless communication between the base and the tower crane. Furthermore, due to the system’s configuration, the signals cannot be transmitted simultaneously. In the present
Step 7: Step 8:
At t ¼ t 0 sample and hold φi ðt 0 Þ for i ¼ 1; …; ns . Let i ¼ 1. For the time period t 0 þ T s ði 1Þ r t o t 0 þT s i the transmitter is triggered to transmit the value φi ðt 0 Þ. Within the above time period, triggering takes place multiple times at a sub period T s =r p where r p Z 2. The receiver holds the value φi ðt 0 Þ. Let i ¼ i þ 1. If ir ns go to Step 3 else go to Step 6. For the time period t 0 þ T s ns r t o t 0 þ T s ðns þ 1Þ the transmitter is triggered to transmit a “loop completion” signal to the receiver. Within the above time period, triggering takes place multiple times at a sub period T s =r p where r p Z 2. As soon as the receiver gets the “loop completion” signal releases as output φi ðt 0 Þ (i ¼ 1; …; ns ). Let t 0 ¼ t 0 þ T s ðns þ 1Þ. Go to Step 1.
Although in the above procedure multiple transmissions of the same signal take place, it guaranties that all transmitted signals are synchronized, while a constant delay transmission T s ðns þ 1Þ is achieved. 3.5. Signal reconstruction The final task in the signal transmission scheme lies in the reconstruction of the continuous time signal at the receiver, using discrete time signal data. To this end, there exist several well established procedures, e.g. “zero order hold” (ZOH) and “first order hold” (FOH). The main disadvantage of these procedures is that they generate discontinuous non-smooth signals that may produce significant errors if used for control purposes, especially if the controller includes derivative terms. To avoid these drawbacks, a “delayed interpolation” (DI) scheme will be proposed. The main
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idea of this approach is to use past data to reconstruct the signal at present time. The reconstruction is accomplished by evaluating appropriate interpolating functions, based on past data, through which the value of the signal at the present time will be generated. The proposed procedure will be applied using a 2nd order interpolation scheme. Let T be the sampling period. Three consecutive samples are available from a transmitted signal φðt Þ, let φðkT Þ, φðkT T Þ and φðkT 2T Þ. The 2nd order interpolating function passing through all data points is ϑðt Þ ¼ β2 ðkT Þt 2 þ β1 ðkT Þt þ β0 ðkT Þ. The coefficients β0 ðkT Þ, β1 ðkT Þ and β2 ðkT Þ are evaluated by 2 32 3 2 3 2 β2 ðkT Þ φðkT Þ ðkT Þ kT 1 6 76 7 6 7 2 6 ðkT T Þ ð5Þ kT T 1 7 4 54 β 1 ðkT Þ 5 ¼ 4 φðkT T Þ 5 2 β ð kT Þ φ ð kT 2T Þ 0 ðkT 2T Þ kT 2T 1 The system in (5) is solvable if and only if T a 0 which is obviously satisfied. For t A ðkT; kT þ T the reconstructed signal, let φr ðt Þ, becomes φr ðt Þ ¼ ϑðt 2T Þ. This scheme does not use extrapolation of past data thus providing accurate reconstruction and introduces an additional time delay to the reconstructed signals. Nevertheless, here this issue is overcome using a controller that incorporates the time delays. To demonstrate the efficiency of this scheme the test signal φðt Þ ¼ sin ð2t Þ sin ð3t Þ=ð2 þ cos ð4t ÞÞ is considered. Different sampling periods from 0:1½s to 0:5½s are used and the sampled signal is reconstructed using ZOH, FOH and the DI approach proposed here. Defining a percentile error norm of the form p φ; φr ¼ 100 %‖φðt Þ φr ðt Þ‖2 =‖φðt Þ‖2 it is observed that for all sampling periods examined, the DI scheme provides far more accurate reconstruction than the ZOH or FOH schemes. See Fig. 2 where two sampling periods have been considered.
4. Controller design In the present section a dynamic controller will be designed in order to control the performance outputs of the tower crane. In particular the design goal will be that of Input/Output (I/O) Decoupling (see [18]), where each performance output is controlled by only one external input. Using the results presented in the previous sections and considering that constant transmission/reconstruction delays exists between the controller and the actuators, the overall LA of the nonlinear model takes on the form d δxðt Þ ¼ Aδxðt Þ þ Bδwðt τ2 Þ; δψ ðt Þ ¼ Lδxðt τ1 Þ dt
ð6Þ
where τ1 and τ2 are the constant transmission/reconstruction delays between the sensors and the controller respectively and the controller and the actuators respectively. For the I/O Decoupling goal, the interest is focused on the forced behavior of the system, i.e. for zero initial and past conditions (xðt Þ ¼ 0,wðt Þ ¼ 0 for t r 0). The system in (6) can be described in the frequency domain as follows sδX ðsÞ ¼ AδX ðsÞ þz2 BδW ðsÞ ð7Þ
where δX ðsÞ ¼ L δxðt Þ , δW ðsÞ ¼ L δwðt Þ , δY ðsÞ ¼ L δyðt Þ and δΨ ðsÞ ¼ L δψ ðt Þ with L fg be the Laplace transform of the signal and z1 ¼ e sτ1 ,z2 ¼ e sτ2 . To derive I/O decoupling, the feedback is proposed to be of the dynamic multi delay type (see [18]), i.e.
δW ðsÞ ¼ K ðs; z1 ; z2 ÞδΨ ðsÞ þ Gðs; z1 ; z2 ÞΩðsÞ
ð8Þ
where ΩðsÞ is the 3 1 vector of external inputs. The elements of K ðs; z1 ; z2 Þ and Gðs; z1 ; z2 Þ are rational functions of s with coefficients rational functions of z1 ,z2 . The implementability of the controller requires that the elements of K ðs; z1 ; z2 Þ and Gðs; z1 ; z2 Þ must be realizable (see [18]). Substituting (8) to (7) the I/O decoupling problem is formally stated as follows (see [18]): Find K ðs; z1 ; z2 Þ and Gðs; z1 ; z2 Þ such that
z2 C ½sI 16 A z1 z2 BK ðs; z1 ; z2 ÞL 1 BGðs; z1 ; z2 Þ ¼ diag hi ðs; z1 ; z2 Þ i ¼ 1;:::;3
ð9Þ where hi ðs; z1 ; z2 Þ are appropriate different than zero rational functions of s with coefficients being functions of z1 and z2 and I n is the n-dimensional unitary matrix. Eq. (9) formulates the problem in a normal system form. For (9) to be well defined the precompensator Gðs; z1 ; z2 Þ must be invertible and the feedback to satisfy the inequality matrix K ðs; z1 ; z2 Þ is restricted h i det sI 16 A z1 z2 BK s; z1 ; z2 L ≢0. Let LB ðs; z1 ; z2 Þ ¼ z1 z2 LðsI 16 AÞ 1 B; H B ðs; z2 Þ ¼ z2 C ðsI 16 AÞ 1 B
ð10Þ
where LB ðs; z1 ; z2 Þ is the transfer matrix mapping the actuatable inputs to the measurable variables and H B ðs; z2 Þ is the transfer matrix mapping the actuatable inputs to the performance variables. Similarly to [18], it can be verified that the solution of the controller matrices solving the I/O decoupling problem are K ðs; z1 ; z2 Þ : arbitrary proper and realizable
ð11aÞ
Gðs; z1 ; z2 Þ ¼ I 5 K ðs; z1 ; z2 ÞLB ðs; z1 ; z2 Þ ½H B ðs; z2 Þ 1 diag hi ðs; z1 ; z2 Þ i ¼ 1;:::;3
ð11bÞ For Gðs; z1 ; z2 Þ to be invertible and realizable, it is necessary for hi ðs; z1 ; z2 Þ (i ¼ 1; …; 3) to be different than zero and sufficiently realizable. Thus, the index of realizability of hi ðs; z1 ; z2 Þ will be selected to be greater than or equal to the minus of the index of realizability of the i-th column of I 5 K ðs; z1 ; z2 ÞLB ðs; z1 ; 1 z2 Þg½H B ðs; z2 Þ . LB ðs; z1 ; z2 Þ is realizable. It can be proven that the index of realizability ½H B ðs; z2 Þ 1 is τ2 . Thus, the index of realizability of hi ðs; z1 ; z2 Þ (i ¼ 1; …; 3) must be greater than or equal to τ2 . The diagonal elements of the closed loop transfer matrix connecting the external commands to the respective performance outputs are selected to be of the form X5 α sj =∏7j ¼ 1 ζ i;j s þ 1 ð12Þ hi ðsÞ ¼ z2 1 þ j¼1 j where ζ i;j (i ¼ 1; 2; 3, j ¼ 1; …; 7) are positive numbers. The independent of the delays denominator guaranties asymptotic command following for the performance outputs of the system. The parameters αi (i ¼ 1; …; 5) are dedicated to canceling out the
possible unstable roots of I 5 K ðs; z1 ; z2 ÞLB ðs; z1 ; z2 Þ ½H B ðs; z2 Þ 1 by hi ðsÞ in formula (11b).
5. Performance of the closed loop system To demonstrate the performance of the control scheme proposed in Section 4, consider the data presented in Section 2. To simulate the influence of the wireless network, the True Time Simulator [19] will be used. With regard to the network para meters, minimum frame size is the data rate is 250 kbps , the 30 byte , the transmit power is 0 dBm , the receiver signal threshold is 85 dbm , the path loss exponent is 5:5½ , the ACK
Please cite this article as: Koumboulis FN, et al. Independent motion control of a tower crane through wireless sensor and actuator networks. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.11.011i
F.N. Koumboulis et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎ 4 ½s, the retry limit is 3½ and the packet size timeout is 8:64 U10 is 126 byte . Let T s ¼ 0:01½s and r p ¼ 2. In order to compute the I/O decoupling controller, the following steps are carried out:
Step 1. Using the data in Section 2 and the relations in (10) the transfer matrices LB ðs; z1 ; z2 Þ and H B ðs; z2 Þ are evaluated. Step 2. The feedback matrix K ðs; z1 ; z2 Þ is chosen to be static, thus guaranteeing properness and realizability. Using the 20
0:86779497 6B K ¼ 4@ 0:91902173 0:636785193
5.1. Simulation results The external commands are chosen to be unit step changes with amplitudes 0:1y1 , 0:1y2 and 0:15y3 , respectively To demonstrate the performance of the proposed control scheme when applied to the nonlinear model (1), Figs. 4–9 are presented. In Figs. 4–6 the closed loop responses of the performance outputs for both the linear (dotted line) and the nonlinear (continuous
0:710134519
0:097179456
0:0921621528
0:76829232
0:0477334395
0:0117649226
0:7668754
0:06596675
0:0109753486
metaheuristic algorithm in [16], the feedback matrix is derived to be The above selection increases the closed loop stability margin, as compared to the open loop system, to more than 50%. Step 3. The parameters of the closed loop transfer functions in (12) are selected to be
ζ i;1 ¼ 1:0; ζ i;2 ¼ 1:2; ζ i;3 ¼ 1:4; ζ i;4 ¼ 1:6; ζ i;5 ¼ 1:8; ζ i;6 ¼ 2:0; ζ i;7 ¼ 2:2 f or i ¼ 1; 2; 3 α1 ¼ 6:71012 10 18 ; α2 ¼ 0:0526666; α3 ¼ 3:53399 10 19 ; α4 ¼ 0:0000422826; α5 ¼ 2:83721 10 22 This selection cancels out the unstable roots of I 5 K ðs; z1 ; z2 ÞLB ðs; z1 ; z2 Þg½H B ðs; z2 Þ 1 thus guaranteeing a stable precompensator. Furthermore, the closed loop response for the performance outputs has zero overshoot and a rise time of about 14:2½s. Step 4. Using (11b), the precompensator is derived to be of the form: h i ^ ðsÞ =GðsÞ Gðs; z1 ; z2 Þ ¼ G~ ðsÞ þ z1 z2 G
(14)
^ ðsÞ are appropriate polynomial matrices and where G~ ðsÞ and G ^ ðsÞ, GðsÞ is a stable polynomial. The i; j elements of G~ ðsÞ and G ^ ~ denoted by G i;j ðsÞ and Gi;j ðsÞ respectively, as well as the polynomial GðsÞ are presented in Appendix.
Fig. 4. Nonlinear and linear closed loop response for y1 ðt Þ.
7
0:0885801612
13
7 0:0331280923 C A5
ð13Þ
0:0107663
line) closed loop systems are presented, while in Figs. 7–9 the actuator forces are presented. With respect to the performance outputs (Figs. 4–6), it can readily be observed that the closed loop responses follow accurately (being almost visually identical) the reference responses. With respect to the actuator forces (Figs. 7– 9), it is observed that they do not present significant fluctuations, especially for the mast rotation and cart position actuators. With respect to the cable length actuator (Fig. 9), fast varying changes are observed. Nevertheless, their amplitude is small thus not imposing implementation difficulties. All the state variables remain within acceptable limits. It is pointed that the accuracy range of the LA of the closed loop system as compared to the nonlinear one is preserved (Section 2).
5.2. Influence of the communication delays To investigate the influence of the delays to the controller performance and the necessity for the design of a delay dependent controller, consider the case where the controller δW ðsÞ ¼ K δΨ ðsÞ þ Gðs; 1; 1ÞΩðsÞ is used where K and Gðs; z1 ; z2 Þ are given in (11a), (11b) and (12). The nonlinear closed loop response for this case will be compared to the respective one using the delay dependent controller presented in the previous paragraph. To compare the responses consider the percentile error of the form (3) with τmax ¼ 30½s. It is observed that the percentile error for the first performance variable using the proposed controller in Section 4 as compared to the one using the delayless controller is 96.23% smaller. For the second performance variable the respective error is 91.45% smaller while for the third the respective error is 78.62% smaller. The proposed in the present paper delay dependent controller approach produces far more accurate results.
Fig. 5. Nonlinear and linear closed loop response for y2 ðt Þ.
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5.3. Accuracy of the reconstruction algorithm The final issue that needs to be addressed is whether the use of the DI scheme (for the analog signal reconstruction from the transmitted signals) is advantageous as compared to the use of classical reconstruction approaches such as the ZOH approach, especially after recalling that the DI scheme as used in the present paper increases the transmission delay by two cycles. With respect to the accuracy of the signal reconstruction, the issue has been examined in Section 3 where it has been observed that the reconstructed signal using the DI technique is far more accurate. With respect to the performance outputs it can be observed that using the ZOH approach the results are similar to those using the DI approach. The main advantage of the DI approach lies in the fact that it produces smooth responses for the state variables in the sense that the acceleration of the state variables x1 to x5 does not present high oscillatory behavior. Furthermore, the peak values in the ZOH case are significantly larger as compared to the respective ones in the DI approach. In the ZOH case, the variation ranges for x_ 6 ðt Þ, x_ 7 ðt Þ, x_ 8 ðt Þ, x_ 9 ðt Þ and x_ 10 ðt Þ are 12.91, 250.85, 282.7, 9713.9 and
Fig. 9. Nonlinear closed loop response for the cable actuator force.
426.88 times greater than those in the DI case. High variations in the accelerations of the system motion variables may lead to destructive conditions to the system and reveal flexible structure characteristics that should be included in the model (1), thus producing a more complex mathematical model to describe the system’s dynamics and consequently requiring the design of a much more complex controller. Similar conclusions can be drawn for the implementation of the actuator forces where an additional issue regarding the implementability of these forces arises. Indeed, in the ZOH case, the variation ranges for the actuator forces are 10.85, 47.76 and 426.37 times greater than in the DI case.
6. Conclusions
Fig. 6. Nonlinear and linear closed loop response for y3 ðt Þ.
The general dynamic model of the crane, in the sense that it includes variable cable length and all coupling kinetic terms, has been developed. An accurate LA of the nonlinear plant has been derived. An appropriate realizable multi-delay dynamic controller providing I/O decoupling, depending upon constant communication delays, has been designed. Constant communication delays and accurate signal transmission through digital wireless communication, has been achieved via developing a synchronization algorithm and a signal reconstruction algorithm. The LA accuracy of the open loop and the closed loop system have been preserved.
Appendix
Fig. 7. Nonlinear closed loop response for the arm actuator torque.
Fig. 8. Nonlinear closed loop response for the trolley actuator force.
GðsÞ ¼ s12 þ 26:18s11 þ297:574s10 þ1876:21s9 þ 7200:55s8 þ 17807:9s7 þ þ 29290:5s6 þ32544:7s5 þ 24459:6s4 þ 12232:3s3 þ 3894:36s2 þ 713:511s þ 57:2256 G~ 1;1 ðsÞ ¼ 2:954137 10 9 s11 þ 2:042398 10 7 s10 þ 8:492199 10 6 s9 þ 0:000313735s8 þ þ 0:0064772s7 þ 0:0803441s6 þ 0:63119s5 þ 3:40546s4 þ 13:5813s3 þ 40:8844s2 þ72:5845s þ57:2256 G~ 1;2 ðsÞ ¼ 1:9621839 10 9 s11 þ1:3565922 10 7 s10 þ 1:953201 10 6 s9 þ 0:0000162957s8 þ þ 0:000100922s7 þ0:000566182s6 þ 0:00258026s5 þ 0:0100925 s4 þ 0:0216889s3 þ 0:0197045s2 G~ 1;3 ðsÞ ¼ 2:46997 10 12 s11 þ1:7076622 10 10 s10 þ 2:4566418 10 8 s9 þ 2:037286 10 8 s8 þ 1:23955 10 7 s7 þ 6:21972 10 7 s6 þ 2:1231 10 6 s5 þ 6:243978 10 6 s4 þ 0:0000121s3 þ 0:000011s2 G~ 2;1 ðsÞ ¼ 8:5173 10 27 s10 1:26932276 10 9 s9 6:71001575 10 8 s8 3:18817448 10 6 s7 0:000119747s6 0:00200904s5 0:0196612s4 0:123502s3 0:490858s2 0:999666s 0:45873 G~ 2;2 ðsÞ ¼ 2:321466 10 24 s12 þ 3:459652 10 7 s11 þ 0:00001829s10 þ 0:00040261s9 þ 0:0053433s8 þ
Please cite this article as: Koumboulis FN, et al. Independent motion control of a tower crane through wireless sensor and actuator networks. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.11.011i
F.N. Koumboulis et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
þ 0:0478925s7 þ 0:31226s6 þ 1:56763s5 þ 6:23987s4 þ 18:9939s3 þ 42:8764s2 þ 72:4277s þ 57:1536 G~ 2;3 ðsÞ ¼ 2:866 10 27 s12 þ 4:2715 10 10 s11 þ 2:25805 8 10 10 s þ 4:901 10 7 s9 þ 6:131 10 6 s8 þ þ 0:000052s7 þ 0:0003376s6 þ0:001725s5 þ 0:006981s4 þ 0:02204s3 þ0:0541s2 þ 0:0914s þ 0:0720572 G~ 3;1 ðsÞ ¼ 1:6793 10 32 s10 2:5027 10 15 s9 2:3098 10 13 s8 þ 1:85461 10 10 s7 þ3:58979 10 9 s6 þ þ 5:01846 10 7 s5 þ 9:9852 10 6 s4 þ 0:0000863:s3 þ 0:0003659s2 þ 0:000724942s þ 0:000577623 G~ 3;2 ðsÞ ¼ 6:7801 10 30 s12 þ 1:0104 10 12 s11 þ 9:3258 10 11 s10 1:5425 10 7 s9 3:2418 10 6 s8 0:0000313s7 0:00018s6 0:000788s5 0:00252s4 0:00461s3 0:003629s2 þ0:0001143s þ 0:000091 G~ 3;3 ðsÞ ¼ 5:38457 10 27 s12 8:02456 10 10 s11 7:40624 10 8 s10 3:619182 10 6 s9 0:00013667s8 0:00365607s7 0:0586131s6 0:531235s5 3:00164s4 12:675s3 40:0256s2 72:5846s 57:2256 ^ 1;1 ðsÞ ¼ 1:29512455 10 24 s10 1:9301077 10 7 s9 G 4:135049719 10 6 s8 0:000276455s7 0:00528582s6 0:0495661s5 0:288407s4 1:26221s3 4:094s2 7:51694s 5:95965 ^ 1;2 ðsÞ ¼ 7:79015 10 23 s11 0:0000116096s10 G 0:000140017s9 0:000272292s8 þ þ 0:0059854s7 þ 0:0708063s6 þ 0:543449s5 þ2:83979s4 þ 10:2277s3 þ30:5558s2 þ 57:6418s þ 49:5976 ^ 1;3 ðsÞ ¼ 1:08242957 10 24 s11 þ 1:613131 10 7 s10 þ G 5:01615769 10 6 s9 þ 0:000282125s8 þ þ 0:0068785s7 þ 0:084515s6 þ0:62817s5 þ 3:2259s4 þ 12:5316s3 þ 34:8916s2 þ 56:6865s þ 40:7004 ^ 2;1 ðsÞ ¼ 6:3615039 10 25 s10 þ 9:4804685 10 8 s9 þ G 2:039842 10 6 s8 þ 0:000136055s7 þ þ 0:00262104s6 þ 0:0250133s5 þ 0:149735s4 þ 0:671718s3 þ 2:18993s2 þ4:00426s þ3:15271 ^ 2;2 ðsÞ ¼ 9:921598 10 24 s11 1:4786031 10 6 s10 G 0:000147572s9 0:00282378s8 0:0286005s7 0:201121s6 1:12136s5 4:78983s4 15:5165s3 40:2824s2 67:2954s 52:5255 ^ 2;3 ðsÞ ¼ 4:290101 10 25 s11 þ 6:3934834 10 8 s10 þ G 2:7545552 10 6 s9 þ 0:000123548s8 þ þ 0:0039646s7 þ 0:058843s6 þ0:491759s5 þ 2:68787s4 þ 11:0146s3 þ33:1052s2 þ 57:5771s þ 43:8998 ^ 3;1 ðsÞ ¼ 8:79148355 10 25 s10 1:31018363 10 7 G 2:8182194 10 6 s8 0:000187911s7 0:00361802s6 0:0342682s5 0:20136s4 0:881882s3 2:8355s2 5:16315s 4:06666 ^ 3;2 ðsÞ ¼ 9:276856 10 24 s11 þ 1:382518 10 6 s10 þ G 0:000109911s9 þ0:00203677s8 þ þ 0:0203885s7 þ 0:142587s6 þ0:790958s5 þ 3:36443s4 þ 10:8714s3 þ28:1153s2 þ 46:79s þ 36:3947
9
^ 3;3 ðsÞ ¼ 1:3180293 10 25 s11 1:9642423 10 8 s10 G 1:8443968 10 6 s9 0:0000608249s8 0:0027467s7 0:0473408s6 0:425221s5 2:4029s4 10:1075s3 31:4407s2 56:2204s 43:8391
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Please cite this article as: Koumboulis FN, et al. Independent motion control of a tower crane through wireless sensor and actuator networks. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.11.011i