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

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 𝐻_∞ 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 𝐻_∞ 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 𝐻_∞ 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: 𝑥̇=𝑓(𝑥)+Δ𝑓(𝑥,𝑡)+𝑔(𝑥)+Δ𝑔(𝑡)⋅𝑢+𝑤, (1) where Δ𝑓(𝑥,𝑡) and Δ𝑔(𝑡) are unknown system dynamics, which are bounded in 𝑥 and 𝑡; 𝑥=[𝑥₁,𝑥₂,…,𝑥_𝑛]^𝑇∈𝑅^𝑛×1 is the state vector, 𝑢=[𝑢₁,𝑢₂,…,𝑢_𝑚]^𝑇∈𝑅^𝑚×1 is the nonlinear input vector, and 𝑤∈𝑅^𝑛×1 denotes unknown and bounded disturbance. Furthermore, nonlinear functions 𝑓(𝑥) and 𝑔(𝑥) are Lipschitz in 𝑥.

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 𝑖th rule of the affine T-S fuzzy model is in the following form: Plant rule 𝑖: IF 𝑧₁(𝑡) is 𝑀_𝑖1 and ⋯ and 𝑧_𝑝(𝑡) is 𝑀_𝑖𝑝THEN𝑥̇=𝐴⋅𝑥+Δ𝐴(𝑡)⋅𝑥+[𝐵+Δ𝐵(𝑡)]⋅𝑢+𝑐_𝑖for𝑖=1,2,…,𝐿. (2)

In each rule, 𝑧₁(𝑡), 𝑧₂(𝑡), … and 𝑧_𝑝(𝑡) are the 𝑝 premise variables, which can be state variables or functions of state variables, 𝑀_𝑖𝑗 is the fuzzy set corresponding to the 𝑗th premise variable, 𝐴∈𝑅^𝑛×𝑛 is the system matrix, and 𝐵∈𝑅^𝑛×𝑚 denotes the control input matrix. Moreover, 𝑐_𝑖∈𝑅^𝑚×1 is the 𝑖th bias vector, Δ𝐴(𝑡)∈𝑅^𝑛×𝑛is the system uncertainty, and Δ𝐵(𝑡)∈𝑅^𝑛×𝑚 denotes the control input uncertainty.

Defining 𝜇_𝑖𝑗(⋅) as the membership function corresponding to fuzzy set 𝑀_𝑖𝑗, we have that 𝜇_𝑖𝑗(𝑧_𝑗(𝑡)) is the grade of membership of 𝑧_𝑗(𝑡) in 𝑀_𝑖𝑗. Using the sum-product composition, the firing strength of the 𝑖th fuzzy rule is represented as 𝜛_𝑖=𝜛_𝑖(𝑧)≡∏^𝑝_𝑗=1𝜇_𝑖𝑗(𝑧_𝑗(𝑡)) with 𝑧≡[𝑧₁(𝑡),𝑧₁(𝑡),…,𝑧_𝑝(𝑡)]^𝑇.

By defining ℎ_𝑖(𝑧)=𝜛_𝑖/∑^𝐿_𝑗=1𝜛_𝑗 as the normalized firing strength of the 𝑖th rule, hence ∑^𝐿_𝑖=1ℎ_𝑖(𝑧)=1, the overall fuzzy system model is then inferred as the weighted average of the consequent parts: 𝑥̇=𝐴𝑥+Δ𝐴(𝑡)𝑥+[𝐵+Δ𝐵(𝑡)]⋅𝑢+^𝐿_𝑖=1ℎ_𝑖(𝑧)⋅𝑐_𝑖. (3)

The proposed control scheme is of a two-level switching structure where the control input is composed of three parts, 𝑢_𝑆, 𝑢_𝐻, and 𝑢_𝑓, defined as follows: 𝑢=1−𝐼^∗⋅𝑢_𝑆+𝐼^∗⋅𝑢_𝐻+𝑢_𝑓=1−𝐼^∗⋅𝐾_𝑆⋅𝑒−𝐼^∗⋅𝐾_𝐻⋅𝑥−^𝐿_𝑖=1ℎ_𝑖(𝑧)⋅𝑐_𝑖, (4) where 𝐼^∗∈{0,1} is a switching function to be defined in Section 3. In (4), the first term, 𝑢_𝑆=𝐾_𝑆⋅𝑒, is a servo controller located in the inner loop responsible for accurate tracking, where 𝑒=𝑥_𝑟−𝑥 is the tracking error with 𝑥_𝑟 denoting the reference state trajectory. The second term, 𝑢_𝐻=−𝐾_𝐻⋅𝑥, is an 𝐻_∞ robust controller in the outer loop to ensure system stability. And 𝑢_𝑓=−∑^𝐿_𝑖=1ℎ_𝑖(𝑧)⋅𝑐_𝑖 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 𝑒_mod as 𝑒_mod≡𝑓(𝑥)+Δ𝑓(𝑥,𝑡)+𝑔(𝑥)+Δ𝑔(𝑡)⋅𝑢+𝑤−𝑦, (5) where 𝑦=𝐴⋅𝑥+Δ𝐴(𝑡)⋅𝑥+[𝐵+Δ𝐵(𝑡)]⋅[𝑢+∑^𝐿_𝑖=1ℎ_𝑖(𝑧)⋅𝑐_𝑖]. Hence the closed-loop system, formed by applying (4) to (1), can be expressed concisely as follows: ̇(𝑡)[(𝑡⋅𝑥=𝐴⋅𝑥+Δ𝐴⋅𝑥+𝐵+Δ𝐵)]1−𝐼^∗⋅𝑢_𝑆+𝐼^∗⋅𝑢_𝐻+𝑢_𝑓+^𝐿_𝑖=1ℎ_𝑖(𝑧)⋅𝑐_𝑖+𝑒_mod=𝐴⋅𝑥+Δ𝐴(𝑡)⋅𝑥+[𝐵+Δ𝐵(𝑡)]⋅1−𝐼^∗⋅𝑢_𝑆+𝐼^∗⋅𝑢_𝐻+𝑒_mod. (6)

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 𝑥_𝑟(𝑡). That is, If‖𝑒‖=𝑥_𝑟−𝑥≤𝜀_𝐸then𝐼^∗=0,otherwise𝐼^∗=1. (7) In the condition, the threshold 𝜀_𝐸 is a user-defined positive number. The value of it, for instance, may be designed as 0.1×max_𝑡(‖𝑥_𝑟(𝑡)‖).

Figure 1 

The proposed two-level switching control scheme.

The closed-loop system dynamics when ‖𝑒‖>𝜀_𝐸 is formed by assigning 𝐼^∗=1 in (6), as follows: 𝑥̇=[𝐴+Δ𝐴(𝑡)]⋅𝑥−[𝐵+Δ𝐵(𝑡)]⋅𝐾_𝐻⋅𝑥+𝑒_mod. (8)

If uncertainties in the plant dynamic matrices, Δ𝐴(𝑡) and Δ𝐵(𝑡), are bounded, we may introduce a time-varying matrix, 𝐹(𝑡) with 0≤‖𝐹(𝑡)‖≤1, and constant matrices, 𝐷, 𝐸₁, and 𝐸₂, such that Δ𝐴(𝑡)Δ𝐵(𝑡)𝐾_𝐻⋅𝑥=𝐷⋅𝐹(𝑡)⋅𝐸₁𝐸₂𝐾_𝐻+𝛿(𝑡)0,⋅𝑥 (9) with 𝛿(𝑡) being a bounded function in 𝑥: ‖𝛿(𝑡)‖≤𝑎⋅‖𝑥‖,where𝑎isapositiveconstant. (10) Using (9), the closed-loop system dynamics, (8), can then be written as 𝑥̇=𝐴−𝐵⋅𝐾_𝐻⋅𝑥+𝐷⋅𝐹(𝑡)⋅𝐸₁−𝐸₂⋅𝐾_𝐻⋅𝑥+𝑒_mod, (11) where 𝑒_mod=𝑒_mod+𝛿(𝑡).

When the system is under acceptable tracking, that is, ‖𝑒‖≤𝜀_𝐸, only the servo controller is in charge. The closed-loop system dynamics is then formed by assigning 𝐼^∗=0 in (6), as follows: 𝑥̇=[𝐴+Δ𝐴(𝑡)]⋅𝑥+[𝐵+Δ𝐵(𝑡)]⋅𝐾_𝑆⋅𝑒+𝑒_mod. (12)

3.1. Design of the Outer-Level 𝐻_∞ Stabilization Controller

The 𝐻_∞ stabilization performance of 𝑢_𝐻 is defined as follows: ∫^𝑡𝑓₀𝑥(𝑡)^𝑇⋅𝑄⋅𝑥(𝑡)⋅𝑑𝑡𝐸_mod≤𝜌², (13) where 𝐸_mod=^𝑡𝑓₀𝑒^𝑇_mod⋅𝑒_mod⋅𝑑𝑡, (14)𝑡_𝑓 is terminal time of control, 𝑄 is a positive definite weighting matrix, and 𝜌 denotes prescribed attenuation level with 𝜌² being the attenuation disturbance level. From the energy viewpoint, (13) confines the effect of 𝑒_mod on state, 𝑥(𝑡), to be attenuated below a desired level. If initial conditions are also considered, the 𝐻_∞ performance in (13) can be modified as follows: ^𝑡𝑓₀𝑥^𝑇⋅𝑄⋅𝑥𝑑𝑡≤𝑥^𝑇(0)⋅𝑃⋅𝑥(0)+𝜌²⋅𝐸_mod, (15) where 𝑄 and 𝑃 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 𝑢_𝐻=−𝐾_𝐻⋅𝑥, such that the 𝐻_∞ performance (15) is guaranteed to stabilize the closed-loop system (11).

Theorem 1. Assuming that the modeling error is bounded such that ‖𝑒_mod‖≤𝑒_𝑈, with 𝑒_𝑈 being a positive constant, the 𝐻_∞ control performance, defined in (15) is guaranteed for the closed-loop system (11) via the stabilizing control law, 𝑢_𝐻=−𝐾_𝐻⋅𝑥, and the feed-forward fuzzy compensator 𝑢_𝑓=−∑^𝐿_𝑖=1ℎ_𝑖(𝑧)⋅𝑐_𝑖, if there exist constant positive values 𝑣, 𝜌, positive-definite matrix 𝑃, and matrix 𝐾_𝐻, such that the following linear matrix inequality is satisfied 𝜙𝐸Φ≡¹⋅𝑊−𝐸₂⋅𝑌^𝑇𝐸₁⋅𝑊−𝐸₂⋅𝑌−𝑣²⋅𝐼<0, (16) where 𝑊≡𝑃⁻¹,𝑌≡𝐾_𝐻⋅𝑊,𝜙≡(𝐴⋅𝑊−𝐵⋅𝑌)^𝑇1+𝐴⋅𝑊−𝐵⋅𝑌+𝜌²⋅𝐼+𝑣²⋅𝐷⋅𝐷^𝑇. (17)(18)

The proof of Theorem 1 requires the following lemma.

Lemma 2 (see [31, 39]). Given constant matrices 𝐷 and 𝐸 and a symmetric constant matrix 𝑆 of appropriate dimensions, the following inequality holds: 𝑆+𝐷⋅𝐹(𝑡)⋅𝐸+𝐸^𝑇⋅𝐹^𝑇(𝑡)⋅𝐷^𝑇<0, (19) if and only if for some 𝑣>0𝑆+𝑣⁻¹⋅𝐸^𝑇⋅⋅𝑣𝑣⋅𝐷𝑅00𝐼⁻¹⋅𝐸𝑣⋅𝐷^𝑇<0, (20) where 𝐹(𝑡) satisfies 𝐹(𝑡)^𝑇⋅𝐹(𝑡)≤𝑅.

Proof of Theorem 1. Considering a Lyapunov function candidate composed of the Lyapunov function: 𝑉(𝑡)=𝑥^𝑇(𝑡)⋅𝑃⋅𝑥(𝑡), (21) its time derivative, 𝑉̇, can be obtained as 𝑉̇(𝑥(𝑡))=𝑥̇^𝑇⋅𝑃⋅𝑥+𝑥^𝑇⋅𝑃⋅𝑥̇=𝑥^𝑇⋅𝐴−𝐵⋅𝐾_𝐻^𝑇⋅𝑃+𝑃⋅𝐴−𝐵⋅𝐾_𝐻⋅𝑥+𝑥^𝑇⋅𝐷⋅𝐹(𝑡)⋅𝐸₁−𝐸₂⋅𝐾_𝐻^𝑇(𝑡)⋅𝐸⋅𝑃+𝑃⋅𝐷⋅𝐹₁−𝐸₂⋅𝐾_𝐻⋅𝑥+𝑒^𝑇_mod⋅𝑃⋅𝑥+𝑥^𝑇⋅𝑃⋅𝑒_mod. (22)
By Lemma 2, we have 𝑉̇≤𝑥^𝑇⋅𝐴−𝐵⋅𝐾_𝐻^𝑇⋅𝑃+𝑃⋅𝐴−𝐵⋅𝐾_𝐻+1𝜌²⋅𝑃^𝑇+𝑣⋅𝑃₋₁⋅𝐸₁−𝐸₂⋅𝐾_𝐻^𝑇⋅𝑣𝑣⋅𝑃⋅𝐷₋₁⋅𝐸₁−𝐸₂⋅𝐾_𝐻𝑣⋅𝐷^𝑇⋅𝑃⋅𝑥+𝜌²⋅𝑒^𝑇_mod⋅𝑒_mod=−𝑥^𝑇⋅𝑄⋅𝑥+𝜌²⋅𝑒^𝑇_mod⋅𝑒_mod, (23) where 𝑄≡−𝐴−𝐵⋅𝐾_𝐻^𝑇⋅𝑃+𝑃⋅𝐴−𝐵⋅𝐾_𝐻+1𝜌²⋅𝑃^𝑇+𝑣⋅𝑃₋₁⋅𝐸₁−𝐸₂⋅𝐾_𝐻^𝑇⋅𝑣𝑣⋅𝑃⋅𝐷₋₁⋅𝐸₁−𝐸₂⋅𝐾_𝐻𝑣⋅𝐷^𝑇.⋅𝑃 (24)
According to (16) and (24), we have 𝑊^𝑇⋅𝑄⋅𝑊=−(𝐴⋅𝑊−𝐵⋅𝑌)^𝑇1+𝐴⋅𝑊−𝐵⋅𝑌+𝜌²+𝑣⋅𝐼₋₁⋅𝐸₁⋅𝑊−𝐸₂⋅𝑌^𝑇⋅𝑣𝑣⋅𝐷₋₁⋅𝐸₁⋅𝑊−𝐸₂⋅𝑌𝑣⋅𝐷^𝑇. (25) From (18) and (25), we have 𝜙+𝑣⁻²⋅𝐸₁⋅𝑊−𝐸₂⋅𝑌^𝑇⋅𝐸₁⋅𝑊−𝐸₂⋅𝑌<0. (26) Equation (26) can be represented in the standard LMI form: 𝜙𝐸₁⋅𝑊−𝐸₂⋅𝑌^𝑇𝐸₁⋅𝑊−𝐸₂⋅𝑌−𝑣²⋅𝐼<0. (27) If (16) holds, then 𝑄>0. Equation (23) can be rewritten as 𝑉̇≤−𝑥^𝑇⋅𝑄⋅𝑥+𝜌²⋅𝑒^𝑇_mod⋅𝑒_mod≤−𝜆_min(𝑄)⋅‖𝑥‖²+𝜌²⋅𝑒_mod²≤−𝜆_min(𝑄)⋅‖𝑥‖²+𝜌²⋅𝑒²_𝑈, (28) where the property ‖𝑒_mod‖≤𝑒_𝑈 is applied.
Whenever ‖𝑥‖>(𝜌⋅𝑒_𝑈)/𝜆_min(𝑄), we have that 𝑉̇<0. 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, 𝑢_𝐻=−𝐾_𝐻⋅𝑥, 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, 𝐽, is defined as a weighted sum of the indices 1𝐽=𝑡_𝑓⋅^𝑡𝑓₀‖𝑢(𝑡𝜔)‖⋅𝑑𝑡+¹𝑡_𝑓⋅^𝑡𝑓₀‖𝑒(𝑡)‖⋅𝑑𝑡, (29) where 𝜔₁ is a weighting factor, which is defined according to practical trade-offs between desired tracking performance and physical constrains.

The inner-level controller, 𝑢_𝑆=𝐾_𝑆⋅𝑒, is designed by searching for the gain matrix 𝐾_𝑆 such that the overall performance index, 𝐽, 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 𝑁+1 vertices in 𝑁 dimensions for an optimization problem with 𝑁 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 𝑁-dimensional problem we may start the algorithm with 𝑁+1 simplexes with (𝑁+1)² randomly generated parameter sets for the vertices, and, after several steps, collect the 𝑁+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.

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 𝑀 = 6.78 kg, the payload is of mass 𝑚 = 1.5 kg, and the string is of length 𝑙 = 0.5 m. Furthermore, 𝑥₁ is the cart position, 𝜃 is the swing angle, 𝑢 is the control signal applied to the cart, and 𝑥_𝑟=[1,0,0,0]^𝑇 is the reference input. The position of payload, 𝑦, can be calculated from the relation: 𝑦=𝑥₁+𝑙⋅sin(𝜃). Besides, we assume that the viscous friction coefficient between the cart and the rail is 𝐷₁, and the wind resistance coefficient between the air and the string is 𝐷₂.

Lagrange analysis of the simplified two-dimensional crane system gives the dynamic equation 𝑥̈₁=𝑢+𝑚⋅𝑙⋅𝜃̇²⋅sin(𝜃)+𝑚⋅𝑔⋅sin(𝜃)⋅cos(𝜃)−𝐷₁⋅𝑥̇₁+𝐷₂⋅𝜃̇⋅cos(𝜃)𝑀+𝑚−𝑚⋅cos²(𝜃)⁻¹𝜃̈=𝑚⋅cos(𝜃)⋅𝑢+𝑚²⋅𝑙⋅𝜃̇²(𝜃)(𝜃)+()(𝜃)×𝑚⋅sin⋅cos𝑀+𝑚⋅𝑚⋅𝑔⋅sin₂⋅𝑙⋅cos²(𝜃)−𝑚⋅𝑙⋅(𝑀+𝑚)⁻¹+−𝐷₁⋅𝑥̇₁⋅𝑚⋅cos(𝜃)+𝐷₂⋅)⋅𝑙(𝑀+𝑚−𝑚⋅cos²(𝜃⋅𝜃̇×𝑚₂)⋅𝑙⋅cos²(𝜃)−𝑚⋅𝑙⋅(𝑀+𝑚)⁻¹+𝑤₄, (30) where 𝑔 is the gravitational acceleration and 𝑤₄ 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 𝑥̇=𝐴𝑥+Δ𝐴(𝑡)𝑥+[𝐵+Δ𝐵(𝑡)]⋅𝑢+²_𝑖=1ℎ_𝑖(𝑧)⋅𝑐_𝑖, (31) where 𝑥=[𝑥₁,𝑥₂,𝑥₃,𝑥₄]^𝑇=[𝑥₁,𝑥̇₁,𝜃,𝜃̇]^𝑇 is the state vector. And the matrices are 𝐷𝐴=01000−¹𝑀𝑚⋅𝑔𝑀𝐷₂𝑀0𝐷0001₁()−()𝑙⋅𝑀𝑀+𝑚⋅𝑔()𝑙⋅𝑀𝑀+𝑚⋅𝐷₂𝑚²(),01⋅𝑙−𝑚⋅𝑙⋅𝑀+𝑚𝐵=𝑀0−1(),𝑙⋅𝑀 (32) with 𝐷₁=588, 𝐷₂=0.01, 𝑔=9.81, 𝑧=𝑚⋅𝑙⋅sin(𝑥₃)⋅𝑥²₄, ℎ₁=0.5(1+𝑧), ℎ₂=0.5(1−𝑧), 𝑐₁=1, and 𝑐₂=−1. And Δ𝐴(𝑡)Δ𝐵(𝑡)=𝐷⋅𝐹(𝑡)⋅𝐸₁𝐸₂, where 𝐹(𝑡)=sin(𝑡), 𝐷=[0,−0.01,0,0.01]^𝑇, 𝐸₁=[2,0,0,0], and 𝐸₂=0.02.

By selecting 𝑣=3 and 𝜌=18, we are able to obtain 𝑃=,90.366718.834713.06809.078318.834714.55880.37787.193813.06800.377851.44260.66809.07837.19380.66803.5618 (33) and 𝐾_𝐻=[1250,315.7,−1766.5,129.5] using the standard LMI techniques. The optimal servo control gains are found to be 𝐾_𝑆=[1140.47,230.47,69.97,315.22] 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.

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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 𝑢(𝑡); and (d) fuzzy rule map.

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 𝑤=[0,0,0,𝑤₄]^𝑇 with 𝑤₄=𝜋3,4.5≤𝑡≤6.5,0,otherwise (34) 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≤𝑡<4.5. However, while the proposed approach remains stable and exhibits accurate tracking after 𝑡≥4.5, the controller of AFCM cannot effectively compensate the applied disturbance 𝑤₄, shown in Figure 3(b), and eventually goes unstable. Note also that the control signal 𝑢(𝑡) generated by the proposed controller is much smoother and less violent than that of AFCM, further justifying it as a more efficient control strategy.

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Figure 3 

Comparative simulation for the proposed control strategy and the AFCM. (a) Position of payload, 𝑦(𝑡); (b) control input, 𝑢(𝑡), and the external disturbance, 𝑤4(𝑡).

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 𝜃. 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.

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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.

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 𝑥₂ is 𝑀₁₁, then𝑥̇=𝐴⋅𝑥+Δ𝐴(𝑡)⋅𝑥+[𝐵+Δ𝐵(𝑡)]⋅𝑢+𝑐₁. (35)(ii) Plant rule 2: If 𝑥₂ is 𝑀₂₁, then𝑥̇=𝐴⋅𝑥+Δ𝐴(𝑡)⋅𝑥+[𝐵+Δ𝐵(𝑡)]⋅𝑢+𝑐₂. (36)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, 𝑀₁₁ and 𝑀₂₁, are shown in Figure 5, with ,00.𝐴=0100−23.936300000012.1681000𝐵=−0.2950.1475 (37)

Figure 5 

The antecedent membership functions, 𝑀11 and 𝑀21, of the fuzzy control law 𝑢𝑓.

Furthermore, 00𝐷=−0.10.01,𝐸₁=,𝐸2000₂=0.02with𝐹(𝑡)∈[−1,1]. (38) These are used to define Δ𝐴(𝑡) and Δ𝐵(𝑡) according to (9). Interesting enough, if we draw the magnitude of ∑^𝐿_𝑖=1ℎ_𝑖(𝑧)⋅𝑐_𝑖 versus 𝑥₄, 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].

Figure 6 

The magnitude of −𝑢𝑓 as a function of 𝑥4 (cart velocity).

Next, by selecting 𝑣=1 and 𝜌=0.54, we are able to obtain 𝑃=,1.70700.3014−0.1362−0.26310.30140.0869−0.0188−0.0444−0.1362−0.01880.02980.0269−0.2631−0.04440.02690.0658 (39)𝑐₁=−7.772, 𝑐₂=4.0561, and 𝐾_𝐻=[26.23,8.02,23.44,14.92] by the standard LMI techniques.

Figure 7 shows the performance of the outer-level stabilizing controller, 𝑢_𝐻. 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 𝐻_∞ robustness against significant disturbances, in spite of variations in the plant dynamics.

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Figure 7 

Performance of 𝑢𝐻 in the face of impact during manipulation. The values of string length and payload weight, [length, weight], are: Case 1: [0.5, 0.531], Case 2: [0.55, 1.041], and Case 3: [0.6, 1.484], respectively.

Next, considering that string length and payload weight dominate system dynamics, we implemented the servo control law, 𝑢_𝑆=𝐾_𝑆⋅𝑒, as a fuzzy controller composed of four fuzzy rules: Servo control rule 𝑖𝑗 If string length is 𝐴_𝑖 and payload weight is 𝐵_𝑗, then𝑢_𝑆=𝐾_𝑆𝑖𝑗⋅𝑒. (40)That is, both string length and payload weight are fuzzified with two membership functions, 𝐴₁, 𝐴₂, 𝐵₁,and 𝐵₂, respectively. The corresponding membership grades of these four fuzzy sets are shown in Figure 8.

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Figure 8 

The antecedent membership functions of the fuzzy sets, 𝐴1, 𝐴2, 𝐵1, and 𝐵2.

Furthermore, by assigning 𝜔₁=10 in the definition of the overall performance index 𝐽, defined in (29), the Nelder-Mead simplex method was applied to search for the best gains 𝐾_𝑆𝑖𝑗 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: 𝐾_𝑆11=[56,44,34,23]forlength,weight=[],𝐾0.5,0.531_𝑆21=[62,40,53,39]forlength,weight=[],𝐾0.6,0.531_𝑆12=[53,48,42,24]forlength,weight=[],𝐾0.5,1.484_𝑆22=[56,41,46,38]forlength,weight=[0.6,1.484]. (41)

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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.

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.

Figure 10 

The simplex convergence history of the overall performance index, 𝐽, in the four rules.

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 𝑦=𝑥₁+𝑙⋅sin𝜃 to move in this way, the trajectory of 𝜃 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: 𝐾_𝑆 = [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.

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Figure 11 

Experimental results of the crane control system. Case 1: [length, weight, distance] = [0.6, 1.484, 1.0], Case 2: [length, weight, distance] = [0.6, 1.041, 0.8], Case 3: [length, weight, distance] = [0.5, 0.531, 0.6].

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 𝐻_∞ 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 𝐻_∞ 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.