Suboptimal variable structure control of uncertain nonlinear slowly varying systems

ABSTRACT In this paper, a new robust suboptimal controller is designed to stabilize a class of uncertain nonlinear time-varying systems with slowly varying parameters. In the design procedure of the proposed controller, first a suboptimal control law is designed for the nominal system based on considering a given cost function and an appropriate Slowly Varying Control Lyapunov Function (SVCLF). After that, a robustifying term is added to the nominal controller in order to vanish the effects of model uncertainties and/or external disturbances in a finite time. For this purpose, a special sliding surface, which is a combination of terminal and integral sliding surfaces, is used. This surface has the advantages of both of terminal and integral surfaces. Due to the structure of this surface, the actual trajectories track the desired one in the finite time. The other innovation of the proposed approach is accessing a chattering-free Controller. Finally, in order to confirm the applicability of the proposed controller and verify the theoretical results, it is applied on a practical benchmark system (a time-varying inertia pendulum). Computer simulations show the efficiency of the proposed controller.


Introduction
The time-varying systems are an important class of dynamical systems, which have various applications in physics and control engineering. Since the time-varying systems include time-varying parameters, the control design methods reported in the literature for these systems are more complicated in comparison with the timeinvariant ones.
One of the major categories of the time-varying systems is systems with slowly varying parameters (called slowly varying systems) (Khalil, 2002). The slowly varying systems have various applications such as air vehicles cranes, underwater vehicles, satellites and so on (Binazadeh & Shafiei, 2013a;Feintuch, 2012;Liu, Haraguchi, & Hu, 2009;Shafiei & Yazdanpanah, 2010). Using the control algorithms of time-invariant systems to control the slowly varying systems may cause instability and bad performance (Feintuch, 2012). On the other hand, the given methods in the literature for the stabilization of the time-varying systems lead to complicated and conservative control laws for slowly varying systems (Feintuch, 2012).
Despite the advantages of the SVCLF method, it has not considered the external disturbances and/or model uncertainties which are the integral parts in modelling of the practical systems. Therefore, the design of the robust controller is necessary to guarantee the asymptotical stability of the closed-loop slowly varying systems in the presence of model uncertainties and/or external disturbances. Binazadeh and Shafiei (2014) proposed a robust stabilizing controller for uncertain nonlinear slowly vary-ing systems by adding an additional term to the SVCLF controller. This additional term was designed based on the Lyapunov redesign method.
Variable structure controllers (like sliding mode controllers) are powerful robust control methods which have effective performance in practical engineering problems (Hu, Wang, Gao, & Stergioulas, 2012;Hu, Wang, & Gao, 2011). The basis of this control method is the design of appropriate sliding surfaces (to achieve the control objectives) and switching control laws so that the system trajectories reach to the desired sliding surface in finite time and stay on it for all future times (Boum, Djidjio Keubeng, & Bitjoka, 2017;Chenarani & Binazadeh, 2017;Su, Liu, Shi, & Song, 2018). Eksin, Tokat, Güzelkaya, and Söylemez (2003) proposed a time-varying sliding surfaces in order to minimize the settling time for second-order systems. Optimal linear sliding surfaces for underactuated nonlinear systems were studied by Nikkhah, Ashrafiuon, and Muske (2006). Designing a sliding surface which results in the optimality of the closed-loop system was investigated by Janardhanan and Kariwala (2008). The optimal sliding mode with fuzzy approach was studied by Li, Wang, Wu, Lam, and Gao (2018). This technique was also studied for the control of nonlinear vehicle active suspension system (Chen, Wang, Yao, & Kim, 2017) and spacecraft position and attitude manoeuvres (Pukdeboon & Kumam, 2015).
In this paper, a variable structure controller is designed for a class of uncertain nonlinear slowly varying systems with a new approach. The main advantage of the proposed robust method is in the elimination of the effects of the model uncertainties and/or external disturbances, in a finite time. This control law is designed in two phases. First, the nominal part of the proposed controller is designed for the nominal system (by ignoring the model uncertainties and/or external disturbances) based on the SVCLF method with an optimal approach related to a given cost function. Then, the robustifying term, which is the main contribution of this paper, is added to the nominal controller to overcome the model uncertainties and/or external disturbances. The additional control term is a variable structure controller with a special sliding surface. This surface is created by the combination of two surfaces included terminal and integral ones. Due to the integral structure of this surface, the actual trajectories track the desired one in the finite time. The other innovation is accessing a specific type of chattering-free second-order sliding mode controller by the combination of the integral and terminal structures in the sliding surface. In this regard, a theorem is given which guarantees vanishing the effect of model uncertainties and/or external disturbances in a finite time and also the asymptotic stability of the uncertain closed-loop system. Finally, this proposed controller is applied to the time-varying inertia pendulum system and its efficiency is affirmed by simulation results.

Problem definition
Consider the following time-varying system: where x ∈ R n is the state vector, u ∈ R is the control input, θ(t) ∈ ⊂ R p is the vector of slowly varying parameters and the unknown nonlinear function d(x, t) is due to the external disturbances and/or model uncertainties. Moreover, f (0, θ(t)) = 0, ∀t ≥ 0. The task is to design a robust control law as follows: where u 1 (t) is the nominal term and u 2 (t) is the robustifying term of the control law. The nominal model of the system (1) can be described as follows: Furthermore, the cost function J is chosen as follows: where l(x(t), θ(t)) is a positive-definite function which has influences on the transient responses of state variables and may be chosen by the designer after some trial and errors. According to Binazadeh and Shafiei (2013b), if the changes of θ(t) be slow enough such that ||θ(t)|| satisfies the following conditions (where α ∈ (0, 1) is selected such that the infimum is made positive) then the following suboptimal controller guarantees the asymptotic stability of the closed-loop system (3).
Moreover, l(x, θ(t)) is given in the cost function (4) and effects on the characteristics of the transient responses, γ ( x ) is a class K function and V(x, θ(t)) : D × → R is an SVCLF that satisfies the following conditions: where α 1 (.) and α 2 (.) are class K functions. The closedloop system with the controller (6) is as follows: The time derivative ofV(x, θ(t)) along the system's trajectories of the above closed-loop system with considering the controller (6) iṡ In the region where b(x, θ(t)) = 0, according to (9) and considering (5), one haṡ In the region where b(x, θ(t)) = 0, according to (9), one hasV is equal to −a. Thus considering the relation (5) in this case one haṡ ThusV(x, θ(t)) ≤ −αγ ( x ) in both cases (i.e. for b = 0 and b = 0) and therefore, the closed-loop system (8) is asymptotically stable. The nominal control law (6) is a suboptimal controller with respect to the cost function (4) and the following related Hamilton-Jacobi-Bellman equation: Remark 1: It is worth noting that θ(t) is belonging to the known region for the considered problem and its sup estimation is related to physical aspects of the problem. Indeed, what is evaluated in (5) is the admissible upper bound of sup ||θ(t)|| that depends on the selected SVCLF. If one can find the SVCLF that is independent of θ(t), there is not any limiting bound on ||θ(t)|| (Binazadeh & Shafiei, 2013b).

Remark 2:
In order to decrease the value of the cost function as far as possible, it is better to put some free parameters in the chosen SVCLF. Then, some acceptable ranges are obtained for these free parameters due to satisfying the conditions (7.a), (7.b) and (7.c). Finally, using a computer program, the best parameters that lead to the less cost function are obtained. In this way, the nominal control law (6) is a very good suboptimal controller and the nominal system response is very close to an optimal solution which is resulted from the related time-varying HJB equation (Binazadeh & Shafiei, 2015).

Design of the robustifying term of the control law
In this section, the additional state feedback term (i.e. u 2 (t)) will be designed such that the whole controller (i.e. u(t) = u 1 (t) + u 2 (t)) guarantees the asymptotic stability of the uncertain closed-loop slowly varying system (1) and leads to vanishing the effect of model uncertainties and/or external disturbances in a finite time. Consider the following integral sliding surface: where E = E 1 . . . E n is a row vector of the design parameter which is chosen such that Eh(x, θ(t)) = 0.
Moreover, x nom (t) is the state vector which is resulted from the nominal closed-loop system (3) with the nominal controller (6) and x(t) is the actual one related to system (1). The difference between x(t) and x nom (t) occurs due to external disturbances and/or model uncertainties. The main characteristics of the surface (10) are that sliding motion on this surface (i.e. s(t) = 0) leads to the nullification of the difference between the actual states of the uncertain system and the desired states of the nominal closed-loop system. This results in performance recovery in the uncertain system and forces the uncertain system to follow the nominal closed-loop system. Using an appropriate control action, this difference can be nullified and the actual trajectory tracks the required one (Castaños & Fridman, 2006). The time derivative of s(t) is as follows: Using Equations (1), (2) and (3), the following relations are obtained aṡ In the conventional sliding mode, the switching control law u 2 (t) is designed such that the reaching law (i.e. s(t)ṡ(t) < −ρ|s(t)| where ρ > 0) is satisfied where its satisfaction guarantees that the system trajectories reach to the surface in a finite time and stay on it for all future times (Khalil, 2002). This law results in appearing the sign function in the control law and therefore leads to chattering phenomena. Now consider the following sliding surface: where ψ > 0 is the switching gain and the odd numbers n and m are selected such that they satisfy the following condition: The advantages of choosing a fractional order sliding surface (13) are the combination of the structure of terminal and integral surfaces which leads to accessing a specific type of chattering-free second-order sliding mode controller which nullify the difference of x(t) and x nom (t)in a finite time. More discussion in this regard is given in the proof of the following theorem.
Theorem 1: Consider the uncertain nonlinear slowly varying system (1). The control law u(t) = u 1 (t) + u 2 (t) (where u 1 (t) and u 2 (t) are proposed in (6) and (15)) guarantees vanishing the effect of model uncertainties and/or external disturbances in a finite time and also the asymptotic stability of the closed-loop system (1) in the presence of model uncertainties and/or external disturbances.
where β(x) is the design parameter function which is defined in the proof.
Integrating (28) According to (29), s(t) will be decreasing and at t = t s , s(t s ) is zero where Since n and m are odd numbers, n−m is an even number and for all s(t) = 0, the term s(t) (n−m)/n is positive. Consequently, it is proved that the integral sliding surface s(t) converges to zero in a finite time. Therefore, the difference between x(t) and x nom (t) is nullified and the actual trajectory tracks the required one in the finite time. In the other words, the effect of external disturbances and/or model uncertainties vanishes in the finite time and since x nom (t) has the desirable behaviour (the asymptotical stable behaviour, Binazadeh & Shafiei, 2013b), therefore the control law u(t) = u 1 (t) + u 2 (t) guarantees the asymptotic stability of the uncertain closed-loop slowly -varying system (1).

Remark 3:
The controller (15) is a variable structure controller (because of existence the sign function), however, main advantages for the proposed robust control law in comparison with the existing literature are that the sign function is appeared inside the integral, thus the controller u 2 (t) is smooth and the resulted robust controller u(t) = u 1 (t) + u 2 (t) is chattering free. On the other hand, the proposed controller leads to the closed-loop actual uncertain system tracks the nominal one (or the desired one) in the finite time. Consequently, the optimality approach in the design of the nominal control law (refer to Remark 2) leads to the suboptimal design of the robust control law, indirectly.

Design example (the time-varying inertia pendulum)
In order to show the applicability of the proposed controller, it is applied to a famous benchmark system: the nonlinear inertia pendulum. The nonlinear inertia pendulum that is used in this paper is time-varying one. It consists of a plate, a beam and a travelling mass (which moves along the beam) which is shown in Figure 1.

Time-varying inertia pendulum modelling
The following mathematical equation describes the motions of the time-varying inertia pendulum: (I 0 + m m y 2 (t))λ(t) + (c v + 2m m y(t)ẏ(t))λ(t) where and m p , m b , m m , c v and τ (t) are the plate mass, the beam mass, the travelling mass, the viscous damping coefficient and the applied torque to the pendulum, respectively (and the gravity acceleration is considered as g 0 = 9.8). y(t) is the slowly varying parameter which shows the distance of the travelling mass from the point O ( Figure 1) and it slowly changes in the range of 0.1 ≤ y(t) ≤ 1. The other parameters are available in Table 1. The goal is designing τ (t) in a way that λ(t) regulates to π. Thus choosing the state variables as x 1 = λ − π(rad), x 2 =λ(rad/s) and the control input u(t) = τ (t), the state-space equations can be written as follows: where ξ 1 , ξ 2 and ξ 3 are as follows: Suppose ξ 2 has uncertainties due to the unknown parameter c v and inaccurate measurement ofẏ(t). Therefore, ξ 2 can be written as follows: whereξ 2 is the known nominal part. Considering 0 ≤ c v ≤ 1 and the maximum error 0.5(rad/s)in the estimation ofẏ, one has |δ| ≤ 1.
Considering θ(t) = y(t) as the slowly varying parameter, Equation (33) has the similar structure of (1) and it can be rewritten as follows: where d(x, θ(t)) = −δx 2 . Therefore, one has |d| ≤ |x 2 |. The goal is designing a robust stabilizing controller for the system (36) in the set of D = {x ∈ R 2 : |x 1 | ≤ π&|x 2 | ≤ 1} by considering the following given cost function:

Controller design for the nonlinear time-varying inertia pendulum
In this section, the robust control law is designed for the time-varying inertia pendulum according to the proposed method. First, assume the nominal form of Equation (36) (i.e. consider d = 0) and consider the following parametric Lyapunov function as where P i s are free parameters. Since P should be a positive-definite matrix, the parameters P 1 , P 2 and P 1 P 2 − (P 3 ) 2 should be positive. The proposed function is independent of the slowly varying parameter (refer to Remark 1) and therefore, condition (9) is satisfied. Moreover, according to Remark 2, some acceptable ranges may be obtained for the free parameters P i s by satisfying the conditions on (7a), (7b) and (7c). In this way that, by defining Then, in the points where b(x, θ(t)) = 0 or x 2 = −P 2 x 1 / P 3 , one has a(x, θ(t))| x 2 =− P 2 P 3 Since the matrix P is positive definite, the coefficient P 1 P 2 − (P 3 ) 2 is positive. In the above equation, it is evident that if P 2 be also a positive value, the coefficient of (−x 2 1 ) is positive. Therefore, there is a positive constant ε, such that a(x, θ(t))| x 2 =−P 3 x 1 /P 2 ≤ −εx 2 1 . Now, if the class K function γ is chosen as γ ( x ) = λ 0 (x 2 1 + x 2 2 ), λ 0 > 0, then in the points where b(x, θ(t)) = 0, it can be rewritten as γ ( x )| x 2 =−P 3 x 1 /P 2 = λ 0 1 + P 3 P 2 2 x 2 1 . Therefore, by satisfying the inequality −εx 2 1 ≤ −λ 0 (1 + (P 2 / P 3 ) 2 )x 2 1 that leads to determine the admissible range of λ 0 (λ 0 ≤ ε/(1 + (P 2 /P 3 ) 2 )), condition (8) is also satisfied. Now, using simple computer programming, the best possible free parameter values are obtained to achieve the minimum feasible cost value as P 1 = 11.9, P 2 = 0.3, P 3 = 1. This value is substituted in a(x, θ(t)) and   b (x, θ(t)) and then, the nominal control law (i.e. u 1 (t)) is calculated according to (6).
After that, consider the system in the presence of model uncertainties and/or external disturbances (d(x, θ(t)) = 0). In a way that, the integral sliding surface (10) is designed, the design parameter E = E 1 E 2 = 0.5 0.5 is chosen. Then, it is combined with the terminal sliding surface as in (13) with the switching gain ψ = 0.005 and the parameters n = 9, m = 5. Considering the upper bound (24) as |E(ḣd + hḋ)| ≤ 4.088 and choosing the design parameter β(x) = 4.2, the robust term (u 2 (t)) is also designed.

Computer simulation
The computer simulation results are shown in Figures  2-4. As it is seen in Figure 2, by applying the proposed controller u(t) = u 1 (t) + u 2 (t), the state variables converge to zero from their initial conditions and consequently λ(t) is regulate to π; however, the nominal controller (i.e. u 1 (t)) cannot tolerate the effects of model uncertainties and/or the external disturbances (the dotted lines). Also, some advantages of the proposed method like the desired characteristics of the transient responses are seen as well. Moreover, in Figure 3, the effective elimination of the chattering in the proposed control input is shown. The cost function value is also shown in Figure 4. Therefore, as seen from the simulation results, adding the robust term succeed in achieving the robust and desired performance for the uncertain closed-loop system.

Conclusion
In this paper, a new chattering-free stabilizing control law was presented for uncertain nonlinear slowly varying systems. The proposed controller had a suboptimal feature in addition to its robust performance in presence of model uncertainties and/or external disturbances. The design of the proposed controller had two steps. The first step was the design of the nominal part of the controller which was designed based on the SVCLF method for the nominal system by considering the given cost function. The second step was the design of a robustifying term to eliminate the effects of the model uncertainties and/or external disturbances such that the time response of the actual system converged to the nominal system in a finite time. For this purpose, a special sliding surface was provided by the combination of terminal and integral ones and a theorem was given and proved. At the end of the paper, the proposed controller was applied to a timevarying inertia pendulum system and its efficiency was confirmed. Further extension may be based on output feedback instead of state feedback and extended the proposed method for multi-input nonlinear slowly varying systems.

Disclosure statement
No potential conflict of interest was reported by the authors.