Switched Two-Level H ∞ and Robust Fuzzy Learning Control of an Overhead Crane

Overhead cranes are typical dynamic systems which can be modeled as a combination of a nominal linear part and a highly nonlinear part. For such kind of systems, we propose a control scheme that deals with each part separately, yet ensures global Lyapunov stability. The former part is readily controllable by theH ∞ PDC techniques, and the latter part is compensated by fuzzy mixture of affine constants, leaving the remaining unmodeled dynamics or modeling error under robust learning control using the Nelder-Mead simplex algorithm. Comparison with the adaptive fuzzy control method is given via simulation studies, and the validity of the proposed control scheme is demonstrated by experiments on a prototype crane system.


Introduction
Overhead cranes are used in workshops or harbors to transport massive goods within short distance.The manipulation of overhead cranes is affected by the existence of unavoidable disturbances, such as friction, winds, unbalanced load, and accidental collision.Besides, change of payloads and string length can result in tremendous variations in system dynamics.Due to these inherent problems, most of the overhead cranes are still operated by skilled labors.An automatic crane system should be able to accurately carry payloads to the desired position as fast as possible without swing.Many works have been focused on automatic control of the overhead crane in the literature.For instance, Park et al. and Singhose et al. [1,2] investigated the input shaping control of the crane systems.[3][4][5] used the variable structure control with sliding modes to control the overhead crane.Moreno et al. [6] used neural networks to tune the parameters of state feedback control law to improve the performance of an overhead crane.Lee and Cho [7] proposed an antiswing fuzzy controller to enhance a servo controller that was used for positioning.Moreover, Nalley and Trabia [8] adopted fuzzy control for both positioning control and swing damping.Moreover, a standard discrete-time fuzzy model [9][10][11][12][13] and continuous-time fuzzy controller [14] have been proposed in the literature.While the controllers of [15][16][17][18][19][20] are based on the so-called Single Input Rule Modules (SIRMs) and [21] focused on the construction of a reducedorder model to approximate the original system.
In the above researches, [1,2] lack robustness consideration for external disturbances and plant uncertainty, while stability is not guaranteed in [6][7][8].Successful implementation of these schemes might depend on unreliable and hard-to-obtain consequent parts (linguistic value), such as the schemes of [3,14] and the dynamic importance degree defined in [15], respectively.
In this paper, we model the nonlinear plant as a combination of a continuous-time linear nominal model and fuzzily blended supplemental affine terms.These terms are added mainly to account for dominant friction effects and residual nonlinear dynamics.The model not only simplifies subsequent control design but also enhances system robustness, because assumptions on the plant dynamics are significantly reduced.The nominal model allows linear control techniques, specifically, the  ∞ linear control technique [22,23], to be applied to the nonlinear plant.
To further alleviate the requirement for accurate fuzzy modeling of the plant, a two-level  ∞ robust nonlinear control scheme is proposed.The inner-level controller is responsible for accurate servo control, while the outer-level controller compensates for unmodeled system dynamics and bounded disturbances.Besides, each part of the proposed control laws can be independently designed satisfying its own specification.This incremental design procedure avoids solving the problem at one time and allows each part to be designed with different guidelines.Also, global stability of the closed-loop system is ensured against bounded disturbances with guaranteed disturbance attenuation level.
A particular switching controller is proposed in [32] for nonlinear systems with unknown parameters based on a fuzzy logic approach.The major difference between our proposed scheme and the controller of [32] is that the switching of our scheme is between the inner-loop and the outer-loop controllers, while the controller of [32] is switched constantly between many (which is 8 in the simulation example) linear controllers.Furthermore, the fuzzy terms in our controller are dedicated for the compensation of highly nonlinear effects that deviate from the nominal linear dynamics.Nevertheless, in [32], a fuzzy plant model is required for the construction of the switching plant model, which is then used for the modelbased design of the switching controller.The switching Takagi-Sugeno fuzzy control proposed in [33] also requires the plant to be accurately represented by a fuzzy system.
As the closed-loop stability is ensured by the outer-level controller, we are able to optimize the inner-level controller by the Nelder-Mead simplex algorithm [34] based on actual closed-loop control performance, rather than deriving from the plant model.The optimization algorithm converges faster than particle swarm optimization (PSO) [35], which is adequate for online applications.This scheme, which incorporates online trials, can be applied to many applications such as self-guided robot and evolvable systems.Furthermore, considering that the swing dynamics depend on both string length and load mass, fuzzy rules are created to interpolate control gains obtained from trial experiments [36][37][38].
In the following sections, this paper is divided into four parts.Section 2 describes the plant model and the problem, Section 3 proposes the two-level control scheme, and Section 4 evaluates the effectiveness of the proposed scheme using both simulation comparison with a recently proposed strategy in the literature and experimental studies.Finally, Section 5 concludes the results.

Problem Formulation
The plant under consideration is assumed to be a disturbed nonlinear system which is affine in the input and contains uncertain dynamics: where Δ(, ) and Δ() are unknown system dynamics, which are bounded in  and ;  = [ Next, we approximate the nonlinear system as a nominal linear system augmented with Takagi-Sugeno type fuzzy blending of affine terms.Note that these affine terms, which are usually dominated by friction in many mechatronic systems, are added to the control-input term, rather than being added directly.This form closely reflects the practical effects of friction on system dynamics.Specifically, the th rule of the affine T-S fuzzy model is in the following form: Plant rule : for  = 1, 2, . . ., . (2) In each rule,  1 (),  2 (), . . .and   () are the  premise variables, which can be state variables or functions of state variables,   is the fuzzy set corresponding to the th premise variable,  ∈  × is the system matrix, and  ∈  × denotes the control input matrix.Moreover,   ∈  ×1 is the th bias vector, Δ() ∈  × is the system uncertainty, and Δ() ∈  × denotes the control input uncertainty.
Defining   (⋅) as the membership function corresponding to fuzzy set   , we have that   (  ()) is the grade of membership of   () in   .Using the sum-product composition, the firing strength of the th fuzzy rule is represented as   =   () ≡ ∏  =1   (  ()) with  ≡ [ 1 (),  1 (), . . .,   ()]  .By defining ℎ  () =   / ∑  =1   as the normalized firing strength of the th rule, hence ∑  =1 ℎ  () = 1, the overall fuzzy system model is then inferred as the weighted average of the consequent parts: The proposed control scheme is of a two-level switching structure where the control input is composed of three parts,   ,   , and   , defined as follows: where  * ∈ {0, 1} is a switching function to be defined in Section 3. In (4), the first term,   =   ⋅ , is a servo controller located in the inner loop responsible for accurate tracking, where  =   −  is the tracking error with   denoting the reference state trajectory.The second term,   = −  ⋅ , is an  ∞ robust controller in the outer loop to ensure system stability.And   = − ∑  =1 ℎ  () ⋅   is a fuzzycombination term that compensates for nonlinear dynamics, such as friction and other effects that deviate from nominal linear dynamics.
Next, let us define the modeling error  mod as where Hence the closed-loop system, formed by applying ( 4) to (1), can be expressed concisely as follows:

The Proposed Two-Level Control Scheme
As shown in Figure 1, the overall control scheme is composed of an outer-level stabilizing controller and an inner-level servo controller.Each of the controllers is designed according to a switching condition defined by the deviation of tracking errors from a prescribed reference vector   ().That is, In the condition, the threshold   is a user-defined positive number.The value of it, for instance, may be designed as 0.1× max  (‖  ()‖).

Plant Fuzzy compensator
The proposed two-level switching control scheme.
When the system is under acceptable tracking, that is, ‖‖ ≤   , only the servo controller is in charge.The closedloop system dynamics is then formed by assigning  * = 0 in (6), as follows:

Design of the Outer-Level 𝐻 ∞ Stabilization Controller.
The  ∞ stabilization performance of   is defined as follows: where is terminal time of control,  is a positive definite weighting matrix, and  denotes prescribed attenuation level with  2 being the attenuation disturbance level.From the energy viewpoint, (13) confines the effect of  mod on state, (), to be attenuated below a desired level.If initial conditions are also considered, the  ∞ performance in (13) can be modified as follows: where  and  are symmetric and positive definite weighting matrices.The design of the stabilizing controller in the outer level corresponds to find a linear controller in the form of   = −  ⋅ , such that the  ∞ performance ( 15) is guaranteed to stabilize the closed-loop system (11).
Theorem 1. Assuming that the modeling error is bounded such that ‖ mod ‖ ≤   , with   being a positive constant, the  ∞ control performance, defined in (15) is guaranteed for the closed-loop system (11) via the stabilizing control law,   = −  ⋅ , and the feed-forward fuzzy compensator   = − ∑  =1 ℎ  () ⋅   , if there exist constant positive values V, , positive-definite matrix , and matrix   , such that the following linear matrix inequality is satisfied where The proof of Theorem 1 requires the following lemma.
Lemma 2 (see [31,39]).Given constant matrices  and  and a symmetric constant matrix  of appropriate dimensions, the following inequality holds: if and only if for some where () satisfies ()  ⋅ () ≤ .
Proof of Theorem 1. Considering a Lyapunov function candidate composed of the Lyapunov function: its time derivative, V, can be obtained as By Lemma 2, we have where According to ( 16) and ( 24), we have From ( 18) and ( 25), we have Equation ( 26) can be represented in the standard LMI form: If ( 16) holds, then  > 0. Equation ( 23) can be rewritten as where the property ‖ mod ‖ ≤   is applied.Whenever ‖‖ > ( ⋅   )/√ min (), we have that V < 0. It is clear that if (16) is satisfied, then the system (11) is UUB stable.This completes the proof.

Design of the Inner-Level Tracking Controller.
Once the outer-level stabilization controller,   = −  ⋅ , has been designed, we are able to put the system undergoing safe trials.Taking tracking performance together with control effort into consideration, the overall performance index, , is defined as a weighted sum of the indices where  1 is a weighting factor, which is defined according to practical trade-offs between desired tracking performance and physical constrains.
The inner-level controller,   =   ⋅ , is designed by searching for the gain matrix   such that the overall performance index, , is minimized.We propose to use the Nelder-Mead simplex method [34] to guide the minimization procedure in this paper.The method deals with nonlinear optimization problems without derivative information, which normally requires fewer steps to find a solution close to global optimum, when proper initial values are given, in comparison with the more powerful DIRECT (DIviding RECTangle) algorithm or evolutionary computation techniques.
The Nelder-Mead simplex method uses the concept of a simplex, which has  + 1 vertices in  dimensions for an optimization problem with  design parameters.In each step of the algorithm, one of the four possible operations is conducted: reflection, expansion, contraction, and shrink.As the method is sensitive to initial guess, for an -dimensional problem we may start the algorithm with  + 1 simplexes with ( + 1) 2 randomly generated parameter sets for the vertices, and, after several steps, collect the  + 1 best solutions of the simplexes to form a simplex for final convergence.With this strategy, we have more initial guesses to avoid being trapped at local minimum.Details are presented in the subsequent case study.

Case Study
In order to verify performance of the proposed control scheme, case studies of simulations and experiments are conducted.In the simulations, a comparison with the adaptive fuzzy control method (AFCM) of [40] is made.In experimental studies, a two-dimensional prototype crane system is used.

Simulation Study.
The crane system under control is composed of a motor-driven cart running along a horizontal rail, a payload, and a string carrying the payload, which is attached to a joint on the cart.We assume that the cart and the load can move only in the vertical plane.In the following study, the cart is of mass  = 6.78 kg, the payload is of mass  = 1.5 kg, and the string is of length  = 0.5 m.Furthermore,  1 is the cart position,  is the swing angle,  is the control signal applied to the cart, and   = [1, 0, 0, 0]  is the reference input.The position of payload, , can be calculated from the relation:  =  1 +⋅sin().Besides, we assume that the viscous friction coefficient between the cart and the rail is  1 , and the wind resistance coefficient between the air and the string is  2 .
Lagrange analysis of the simplified two-dimensional crane system gives the dynamic equation where  is the gravitational acceleration and  4 represents the external disturbance.
By selecting V = 3 and  = (2) Controller Design of [40].For comparison purpose, the adaptive fuzzy controller of [40], abbreviated as AFCM, is implemented.Design parameters of the AFCM include membership functions of the antecedents in the fuzzy rules, values of the consequent forces, and the fuzzy rule map.Detailed values obtained by the procedures described in [40] are shown in Figure 2.
In the fuzzy rules, each of the universe of discourse of the variables is divided into 6 linguistic values as {NB, NS, ZO, PS, PM, PB}, which represent Negative Big, Negative Small, Zero, Positive Small, and Positive Big, respectively.
(3) Performance Comparison.In order to compare relative performance of the two approaches, a significant disturbance of  = [0, 0, 0,  4 ]  with is applied to the crane model.
From the time history of the payload position of these two approaches, shown in Figure 3(a), it is clear that both can successfully demonstrate stable tracking during 0 ≤  < 4.5.However, while the proposed approach remains stable and exhibits accurate tracking after  ≥ 4.5, the controller of AFCM cannot effectively compensate the applied disturbance  4 , shown in Figure 3(b), and eventually goes unstable.Note also that the control signal () generated by the proposed controller is much smoother and less violent than that of AFCM, further justifying it as a more efficient control strategy.

Experimental Study.
A prototype crane system, shown in Figure 4(a), is built to test the proposed control strategy.As shown in the pictures of Figures 4(b) and 4(c), an encoder with resolution of 2000 pulse/rev is installed in the hanging joint to measure the swing angle .To investigate robustness of the control system, the string length can vary between 0.5 to 0.6 m, and the payload weight has three choices: 0.531, 1.041, and 1.484 kg.
The system is firstly identified using the parallel genetic algorithms [41] as T-S type fuzzy combination of the following two rules.(i) Plant rule 1: (ii) Plant rule 2: In the identification, a set of commands are designed to perform various maneuvers satisfying persistent excitation requirements for system identification.The identified two antecedent membership functions of these two rules,  11 and  21 , are shown in Figure 5, with These are used to define Δ() and Δ() according to (9).
Interesting enough, if we draw the magnitude of ∑  =1 ℎ  () ⋅   versus  4 , the velocity of the cart, we are able to obtain the relationship of Figure 6, which shows the behavior similar to a combination of Coulomb friction with Stribeck effects [42].
Next, by selecting V = 1 and  = 0. Figure 7 shows the performance of the outer-level stabilizing controller,   .In this figure, three cases were recorded where impacts were applied to the payload at 1.72, 1.45, and 0.38 sec, respectively.The string length and payload weight, [length, weight], of these cases were [0.5, 0.531], [0.55, 1.041], and [0.6, 1.484], respectively.According to these experimental results, the stabilizing controller applied at the outer level exhibits  ∞ robustness against significant disturbances, in spite of variations in the plant dynamics.
Next, considering that string length and payload weight dominate system dynamics, we implemented the servo control law,   =   ⋅ , as a fuzzy controller composed of four fuzzy rules: That is, both string length and payload weight are fuzzified with two membership functions,  1 ,  2 ,  1 , and  2 , respectively.The corresponding membership grades of these four fuzzy sets are shown in Figure 8. Furthermore, by assigning  1 = 10 in the definition of the overall performance index , defined in (29), the Nelder-Mead simplex method was applied to search for the best gains   in the four rules.The learning history of gains is depicted in Figure 9.Note that only the gains corresponding to [length, weight] = [0.5, 0.531] underwent 116 steps; all the other gains were initiated with the gains of fuzzy rule 1, hence less than 30 steps were required.According to considerations Figure 9: The learning history of gains guided by the simplex method showing convergence of the gain parameters.In each of the 3 dimensional cases, the method starts with 4 simplexes and a new simplex, is formed for the following steps by collecting the 4 best solutions so far at the 30th step.corresponds to 1 to 3 steps in Figure 9, since only improved step is regarded as an effective iteration.
In the experiments of automatic repetitive trials to find the optimal gains, reference state trajectories were designed such that the payload moves smoothly forward without swinging back.In fulfilling the requirement, the reference trajectories should be a function of the nature frequency that, in turn, depends on both the string length and the load weight.Specifically, for the payload position  =  1 +  ⋅ sin  to move in this way, the trajectory of  should contain integer multiple of a full nature-frequency cycle.
Finally, experiments were conducted to justify the control performance.Three experiments were designed:  40.5352, 49.2539, 38.4648].The performance of the crane control system is demonstrated in Figure 11.According to the experimental results, the proposed control strategy can guide the payload smoothly forward without swinging back in a reasonable period of time.

Conclusions
By the antiswing control approach, a two-level control scheme is proposed for crane systems.The plant is modeled as a combination of a nominal linear system and a T-S fuzzy blending of affine terms.This type of dynamic model significantly simplifies the subsequent analysis and control designs, because assumptions on the plant dynamics can be significantly reduced.The proposed control scheme can also be applied to other nonlinear plants, such as ships, mobile robots, and aircrafts, but is not applicable for systems with considerable time delay, which is the issue to be addressed in our future investigation.
In the scheme, the outer-level control law serves as an  ∞ robust controller, which is responsible for closedloop stability in the face of disturbances and plant dynamic variations.Optimal gains of the inner-loop servo control law are obtained using the Nelder-Mead simplex algorithm in a learning control manner.Close observation of the obtained fuzzy model reveals that the fuzzy compensator mainly counteracts the effects of friction.The dynamics of Coulomb friction, viscous friction, and Stribeck effects are distinguishable as functions of relative velocity.
A simulation study shows superior performance of the proposed control strategy in compensating significant disturbances.Experimental results of a prototype two-dimensional crane control system also demonstrate smooth manipulation of the payload with  ∞ robust stability.The control strategy can be extended to full dimensional crane systems and is within our plans of future research.

Figure 2 :
Figure 2: Linguistic terms, membership functions, and rule table of the fuzzy control rules for AFCM.(a) Definition of membership functions of position error; (b) definition of membership functions of swing angle; (c) consequent part membership function of control input (); and (d) fuzzy rule map.

Figure 4 :
Figure 4: Pictures of the experimental crane system.(a) A whole view of the system.The image was generated by overlapping five snapshots taken during operation.(b) An upper view of the cart showing a servo motor to drive the cart and a bearing-supported shaft to hang the payload.(c) Close view of an encoder, also shown in (b), which is attached to the shaft for swing angle measurement.

Figure 5 :Figure 6 :
Figure 5: The antecedent membership functions,  11 and  21 , of the fuzzy control law   .

Figure 10 :
Figure 10: The simplex convergence history of the overall performance index, , in the four rules.