RBF Neural Network Sliding Mode Control for Passification of Nonlinear Time-Varying Delay Systems with Application to Offshore Cranes

This paper is devoted to studying the passivity-based sliding mode control for nonlinear systems and its application to dock cranes through an adaptive neural network approach, where the system suffers from time-varying delay, external disturbance and unknown nonlinearity. First, relying on the generalized Lagrange formula, the mathematical model for the crane system is established. Second, by virtue of an integral-type sliding surface function and the equivalent control theory, a sliding mode dynamic system can be obtained with a satisfactory dynamic property. Third, based on the RBF neural network approach, an adaptive control law is designed to ensure the finite-time existence of sliding motion in the face of unknown nonlinearity. Fourth, feasible easy-checking linear matrix inequality conditions are developed to analyze passification performance of the resulting sliding motion. Finally, a simulation study is provided to confirm the validity of the proposed method.


Introduction
In industrial applications and the physical world, time delays are very common, such as their existence in chemical reactor systems [1], automotive powertrain systems [2], mechanical systems [3], population dynamics [4] and so on. The physical phenomena that produce time delays are caused by the transmission of information, energy, or different masses in long distance. The stability analysis of physical systems with time-delay is usually divided into two categories: the delay-independent criteria and the delay-dependent criteria; The latter condition includes information on the degree of system delay, which results in simpler and more conservative delay-independent stability criteria, particularly for small delays, than the delay-dependent stability criteria. Since last century, many efforts have been devoted to time-delay systems. For physical systems with constant time-delay, the stability issue for neural networks of neutral-type through a new Lyapunov functional was proposed in [5]; By proposing distributed dynamic controllers, the consensus of heterogeneous linear multi-agents with arbitrarily large constant communication delays was addressed in [6]. For physical systems with time-varying delay, the robust stability for time-varying structural and uncertain systems was studied by designing a novel delaydependent stability criterion in [7]; In [8], the problems of stability analysis and stabilization for Takagi-Sugeno fuzzy systems in a discrete-time domain was investigated by employing a fuzzy Lyapunov-Krasovskii functional. For physical systems with distributed time-delay, the mode-dependent state feedback H ∞ robust control design for uncertain distributed delay systems with Markov switching parameters was investigated in [9]; An improved distributed delay-dependent criterion for stability analysis and stabilization of uncertain

Model Establishing and Problem Statement
Crane, as a modern transfer equipment, has been widely used in modern factories, installation sites, container yards, indoor or outdoor warehouse loading and unloading, and transportation. The type offshore crane can be checked in [29], its equivalent physical model is simplified as shown in Figure 1, in which the load is attached to the cart by a cable; m is the load mass [kg], M is the cart mass [kg], L is the cable length [m], F(t) is the control input generated by the electric motor and θ is the load swing angle at the moment of movement. In real applications, the swing of loads is an important factor that disturbs the crane efficiency. In order to ensure that the load movement is smooth and steady, necessary efforts should be taken to deal with this issue. In general, the problem of fast transportation and positioning of heavy loads can be summarized as follows: Under the input F(t), the cart moves from position A to position B in the fastest time t s allowing the swing angle |θ(t s )| < ∆, in which ∆ is the minimum allowable swing angle for the load in the process of movement. To deal with this problem, let us establish the system mathematical model in a generalized coordinate; see Figure 2, in which x 1 is the position of the cart, x 2 is the cable length, α is the swing angle, F 1 is the pull imposed on the cart and F 2 is the lift for the load. D is the friction damping coefficient between the cart and the horizontal track, and η is the damping coefficient when the load swings. Therefore, the positions for the cart and the load in the generalized coordinate can be denoted by Correspondingly, the velocity components for the cart and the load are obtained as On the other hand, it is known that the kinetic energy T of the overall system is denoted by Then, according to the generalized Lagrange formula that in which q k is the generalized coordinate, n is the degree of freedom and F k is the generalized force. Therefore, (4) yields Further, the overall system model is obtained as follows: Regarding the dynamic model (6), linearization of the model in state-space is necessary for the purpose of stability analysis and control design. Considering that the length of cable is unchanged, that is x 2 = L = const., thanẋ 2 =ẍ 2 = 0. Considering that the swing angle of the crane is quite small in the equilibrium point α = 0 during the real movement process, so we can linearize the model with sin α ≈ α, cos α ≈ 0 andα 2 sin α ≈ 0. In addition, it is deemed that the damping coefficient is quite small and taken as η = 0, then the simplified model is taken as Now, define z = [z 1 z 2 z 3 z 4 ] T , where z 1 = x 1 , z 2 =ẋ 1 , z 3 = α and z 4 =α. Letting x 1 and α be the output variables, then the system (7) described in the state-space can be presented as: where u = F and Generally, let us consider a more complex environment in which the physical system suffers from unknown nonlinearity, external disturbance and time-varying state delays. Then, the description of state-space system will be denoted as in which z(t) ∈ R n is the state variable, u(t) ∈ R m is the control variable and y(t) ∈ R q is the controlled output. The system matrices A, A τ , B, C, B w and D w are with appropriate dimensions, and B is full column rank. The system unknown nonlinearity is f (z(t)) and w(t) ∈ L 2 [0, +∞) denotes the norm-bounded external disturbance. The system state delay is represented as τ(t), satisfying in which τ 1 and τ 2 are known constants.
The following definition and lemma are useful in the following analysis.

Definition 1. ([30,31])
The nonlinear system (9) is said to have a passification performance index γ, if the following two conditions are satisfied: (1) With w(t) = 0, the solution z(t) of the system (9) is internally exponentially stable, i.e, (2) For nonzero w(t) ∈ L 2 [0, +∞), the following inequality is satisfied under zero initial condition, in which γ is a positive constant and for all t ≥ 0.

Lemma 1. ([32])
For a given matrix satisfying 0 < P ∈ R n×n , and a differentiable vector function ζ(t) has appropriate dimensions. Then, it holds

Main Results
This section will propose an SMC strategy so as to ensure the closed-loop system has a passification performance with an exponential stability property. The steps here include: design of sliding surface, passification analysis and RBF neural network sliding mode controller design.

Sliding Motion Design
For the system (9), let us define the following integral-type switching hyperplane function where G and K are both real matrices to be designed. Particularly, it is required that GB is nonsingular.
According to the dynamics of the system (9), the solution z(t) follows Combining (14) with (15), it is obtained that According to the SMC theory [14], it holds that both s(z, t) = 0 andṡ(z, t) = 0 when the sliding surface s(z, t) = 0 is reached. Therefore, using the conditionṡ(z, t) = 0, one can derive an equivalent control variable: By substituting (7) into the system (9), one can obtain the following sliding mode dynamic (SMD) systeṁ (14) is proposed, which contains an integral term with time delay. The advantage of such a sliding surface is that, since the design of the memory controller is more complicated in practice, it will be more convenient in the subsequent controller design.

Remark 1. An integral sliding surface function of the form
In the sequel, an SMC law will be presented to ensure the finite-time existence of sliding motion on the sliding surface s(z, t) = 0. However, due to f (z(t)) being an unknown function, how to compensate the effect of f (z(t)) in the whole phase should be considered first. In the following, a radial basis function θ T ξ(z) is applied to estimate the unknown function f (·), in which θ ∈ R l×m , ξ(q) ∈ R l is a vector-valued function and q ∈ R p is the input vector of the RBF neural network. The structure of three-layer RBF network is presented in Figure 3.
where m j ∈ R p and σ j ∈ R represent the center and width of above Gaussian function, respectively. If the integer l is chosen to be as large as possible, then there exists a θ * ∈ R l×m such that in which δ * (q) ≤ δ, and δ is a known constant.
In the following,θ(t) is used to denote the estimation of θ * . The corresponding error is denoted byθ(t) =θ(t) − θ * . Theorem 1. Given the nonlinear system (9), we define its switching surface function in (14). Then, the SMC law designed below could ensure the state trajectories driven onto the proposed sliding surface s(z, t) = 0 in finite time, in which sgn(·) is the symbolic function, ρ is chosen such that ρ − δ ≥ ε > 0, and˙θ(t) = Λξ(q)s T (z, t) with Λ is a known matrix parameter.
Proof. Choose G = B T and the Lyapunov function candidate below: Then, it obtains for (16) thaṫ In view of˙θ(t) =˙θ(t) and the property that tr{AB} = tr{BA}, it is seen in (21) that Therefore, by substituting (19) into (21) and in view of s(z, t) ≤ |s(z, t)|. One can read from (21) and (22) Thus, the finite-time reachability condition is satisfied. This covers the proof.
Proof. First, let us check the exponential stability of the SMD system (18) for the case w(t) = 0. Now, consider the Lyapunov functional candidate below: in which Then, calculating the derivative of V(t) along the trajectories of the SMD system (18) yieldṡ in which it is seen that Particularly, it is easily obtained from the SMD system (18) that the following equation holds in whichΘ Applying (24), one can readV(t) + αV(t) ≤ η T (t)Θη(t) < 0 for η(t) = 0. Integrating both sides of this inequality from 0 to t. Then one derives that Recalling (25), we know there is a constant λ 1 that meets 0 < λ 1 ≤ λ min (X) so as to Further, denoting λ 2 = λ max (X), λ 3 = λ max (Q 1 ) and λ 4 = λ max (Q 2 ). Then, it is easily seen that Thus, one can obtain the following result from (31)-(33) that . At this moment, it is seen that the SMD system (18) is exponentially stable.
Next, let us check the passivity performance, denoted by Under zero-initial condition, it is seen that Since Θ < 0, it derives from (36) thaṫ Now, integrating (37) from both sides in the region 0 to t results in which means In (39), letting t → +∞, then it obtains 2 t 0 w T (s)y(s)ds ≥ −γ t 0 w T (s)w(s)ds. Hence, the passification performance is guaranteed. This completes the proof.

Computation of Gain Matrix
In Theorem 2, an exponential stability criterion for the SMD system (18) with a decay rate α is established. However, our attention is to design a feasible control gain matrix K for exponential stability and passification performance of closed-loop system. Therefore, a theorem is further proposed as follows.

Remark 2.
In Theorems 2 and 3, both delay-dependent and delay-derivative-dependent exponential stability criteria for the SMD system (18) are established, which bring the following advantages: (1) It is seen that these conditions are not only fit for constant time-delay systems, but also available for the time-varying case; (2) The LMI conditions are easily changed into the asymptotic stability criteria with the decay rate α = 0; (3) The condition is supposed to be less conservative compared with some recent results since a set of free weighting matrices are introduced.
As we know, to have a fast convergence of system state trajectories requires a relatively large value of decay rate. Thus, it is desired to allow a maximal α to ensure the exponential stability based on the above Theorem 3. To this end, we propose a Corollary below. Corollary 1. For given positive scalars α > 0, γ > 0 and i (i = 1, 2, 3), if it finds matrices X > 0,Q i > 0 (i = 1, 2), K j > 0 (j = 1, 2, 3), and proper matrices Υ and Y satisfy the following conditions min β s.t. where Then, the SMD system (18) satisfies the properties in Definition 1 with exponential stability and passification performance γ.
Overall, the above Figure 4 shows how the proposed controller is implemented with necessary steps for the considered system. It should be noted that the controller gain matrix K in Figure 4 is computed by LMI (40) off-line in advance, then we initialize corresponding parameters to implement the controller.

Simulation Study
In this part, an example with a simulation study is provided for the crane system. Letting M = 200, m = 100, L = 4, D = 0.1 and g = 9.81. Therefore, we can consider the following system parameters for the system (9): Assuming that the system subjects to the nonlinearity by f (z(t)) = 0.1sin(z 1 (t)). The system state-delay is chosen as τ(t) = 0.2 such that τ 1 = 0.2 and τ 2 = 0 can be easily obtained. Our aims here are that: (1) To compute the controller gain matrix K so as to ensure the SMD system (18) is exponentially stable and satisfies a passivity performance index γ; (2) To design an adaptive SMC law given in (19) so as to ensure finite-time existence of desired sliding motion. For computation, define G = B T , select scalars i = 1(i = 1, 2, 3), the decay rate α = 0.1 and γ = 2.5. According to Theorem 3, and by solving the condition (40), one can obtain feasible solutions as follows: Based on the above solutions, the controller gain matrix can be computed as: In view of the simulation purpose, the initial condition is set as ϕ(s) = [5 − 5 1 − 1] T and ψ(s) = [0 0 0 0] T , s ∈ [−0.2, 0]. The unknown uncertainty and external disturbance are provided as f (z(t)) = 0.1 sin(z 1 (t)) and w(t) = [1/(t 2 + 1) 1/(t 2 + 1)], respectively. In the RBF neural network, the width σ j = 1, the initial weightθ(0) = 0, Λ = 0.1, and the center point m j = [−1 − 0.5 0 0.5 1] T . In addition, the switching signal sgn(s(z, t)) is changed by s(z, t)/( s(z, t) + 0.01) and ρ = 0.01. Then, we have the simulation results as presented in Figures 5-7. Figure 5 plots that the original system is unstable without control; Figure 6 depicts that the system achieves a stability property by the proposed control algorithm; Figure 7 gives the SMC input, which shows a satisfactory system performance is achieved.   Seen from Theorem 3, the index γ is sensitive to the system time-delay. To reflect the relationship between γ and τ 1 , the following Table 1 shows the maximum allowable values for τ 1 when different values for γ are given in Theorem 3, in which it is seen that large γ allows bigger time-delay. Otherwise, to understand how the time-delay may affect γ, a trajectory is plotted in Figure 8 to show the minimum allowable γ for given τ 1 , and the result also reveals that large τ 1 demands bigger γ.

Conclusions
The problem of SMC for passification of nonlinear time-delay systems via an adaptive neural network approach has been tackled in this paper. First, a mathematical model for the crane has been established based on the generalized Lagrange formula. Second, by proposing an integral-type sliding surface, on which the SMD with desirable dynamic property has been derived. Third, based on the RBF neural network approach, an adaptive SMC law has been synthesized to ensure the sliding motion in finite-time. Fourth, in order to check the passivity performance of the SMD system, feasible easy-checking LMI conditions have been developed. Finally, a numerical study with simulation has been put forward to verify the correctness of the proposed method.