## New Developments in Mathematical Control and Information for Fuzzy Systems

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Kao-Ting Hung, Zhi-Ren Tsai, Yau-Zen Chang, "Switched Two-Level and Robust Fuzzy Learning Control of an Overhead Crane", *Mathematical Problems in Engineering*, vol. 2013, Article ID 712615, 12 pages, 2013. https://doi.org/10.1155/2013/712615

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

**Academic Editor:**Peng Shi

#### Abstract

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 the 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.

#### 1. 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. [3–5] 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 [9–13] and continuous-time fuzzy controller [14] have been proposed in the literature. While the controllers of [15–20] are based on the so-called Single Input Rule Modules (SIRMs) and [21] focused on the construction of a reduced-order 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 [6–8]. 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.

In the closely related literature of [15–20, 24–31], fuzzy controllers are developed to simultaneously stabilize these fuzzy linear models using the parallel distributed control (PDC) scheme that satisfies the linear matrix inequality (LMI) relations. However, these control design strategies rely on accurate fuzzy modeling of the plant, which usually results in a large number of fuzzy rules and, hence, complex and conservative designs.

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 model-based 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 [36–38].

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.

#### 2. 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 and are unknown system dynamics, which are bounded in and ; is the state vector, is the nonlinear input vector, and denotes unknown and bounded disturbance. Furthermore, nonlinear functions and are Lipschitz in .

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 : IF is and and is

In each rule, , , 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, is the th bias vector, is the system uncertainty, and 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 with .

By defining as the normalized firing strength of the th rule, hence , 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 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 is a fuzzy-combination 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 as where . Hence the closed-loop system, formed by applying (4) to (1), can be expressed concisely as follows:

#### 3. 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 .

The closed-loop system dynamics when is formed by assigning in (6), as follows:

If uncertainties in the plant dynamic matrices, and , are bounded, we may introduce a time-varying matrix, with , and constant matrices, , , and , such that with being a bounded function in : Using (9), the closed-loop system dynamics, (8), can then be written as where .

When the system is under acceptable tracking, that is, , only the servo controller is in charge. The closed-loop system dynamics is then formed by assigning in (6), as follows:

##### 3.1. Design of the Outer-Level 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 being the attenuation disturbance level. From the energy viewpoint, (13) confines the effect of 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 , 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 , if there exist constant positive values , , 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, , 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 . Equation (23) can be rewritten as
where the property is applied.

Whenever , we have that . It is clear that if (16) is satisfied, then the system (11) is UUB stable. This completes the proof.

##### 3.2. 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 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 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 simplexes with randomly generated parameter sets for the vertices, and, after several steps, collect the 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.

#### 4. 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.

##### 4.1. 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, is the cart position, is the swing angle, is the control signal applied to the cart, and is the reference input. The position of payload, , can be calculated from the relation: . Besides, we assume that the viscous friction coefficient between the cart and the rail is , and the wind resistance coefficient between the air and the string is .

Lagrange analysis of the simplified two-dimensional crane system gives the dynamic equation where is the gravitational acceleration and represents the external disturbance.

* (1) Controller Design of the Proposed Control Strategy. *From (3), the overall fuzzy model of the overhead crane system (30) is inferred to be
where is the state vector. And the matrices are
with , , , , , , , and . And , where , , , and .

By selecting and , we are able to obtain and using the standard LMI techniques. The optimal servo control gains are found to be by the simplex method.

*(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.

**(a)**

**(b)**

**(c)**

**(d)**

In the fuzzy rules, each of the universe of discourse of the variables is divided into 6 linguistic values as , 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 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 . However, while the proposed approach remains stable and exhibits accurate tracking after , the controller of AFCM cannot effectively compensate the applied disturbance , 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.

**(a)**

**(b)**

##### 4.2. 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.

**(a)**

**(b)**

**(c)**

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: If is , (ii) Plant rule 2: If is , 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, and , are shown in Figure 5, with

Furthermore, These are used to define and according to (9). Interesting enough, if we draw the magnitude of versus , 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 and , we are able to obtain , , and by the standard LMI techniques.

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.

**(a)**

**(b)**

**(c)**

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: Servo control rule If string length is and payload weight is , That is, both string length and payload weight are fuzzified with two membership functions, , , ,and , respectively. The corresponding membership grades of these four fuzzy sets are shown in Figure 8.

**(a)**

**(b)**

Furthermore, by assigning 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 and procedures detailed in Remark 3, the gains are found to be of the following values:

**(a)**

**(b)**

**(c)**

**(d)**

*Remark 3. *Four fuzzy rules defined in (40), each of them contains a set of optimized control gains, are designed to compensate for the uncertainties in the weight of payload and string length. As shown in the learning history of Figure 10, the gains of rule 1 took 50 iterations and those of the rest of the rules took only 12 iterations. That is, only the first gain set requires complete search. This is because the performance of the Nelder-Mead simplex algorithm is sensitive to initial trial values, and the optimal control gains are close to each other. If the search for the other sets begin with the optimized first set, less iteration is required. Also, an iteration of Figure 10 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 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: Case 1: [length, weight, distance] = [0.6, 1.484, 1.0], Case 2: [length, weight, distance] = [0.6, 1.041, 0.8], and Case 3: [length, weight, distance] = [0.5, 0.531, 0.6]. The gains of Case 2 are interpolated from four fuzzy rules to be: = [58.7891, 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.

**(a)**

**(b)**

**(c)**

#### 5. 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 closed-loop 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.

#### Acknowledgments

The authors would like to express their sincere appreciation to the editor and all the reviewers for their helpful and constructive comments. Besides, the authors are enormously grateful for the supports from the National Science Council of Taiwan, under Grants nos. NSC 101-2221-E-182-006 and 100-2221-E-182-008.

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Copyright © 2013 Kao-Ting Hung et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.