Robust chattering-free optimal sliding-mode control using recurrent neural networks: An H ∞ -based approach

Abstract

In this paper, a robust and chattering-free sliding-mode control strategy using recurrent neural networks (RNNs) and H_∞ approach for a class of nonlinear systems with uncertainties is proposed. The dynamic and algebraic models of the RNN are extracted based on the nominal model of the system and formulation of a quadratic programming problem. For tuning the parameters of the sliding surface, the performance index and the switching coefficient, a robust approach based on the H_∞ method is developed. To this end, the control law is divided into two parts: (1) the main term, which includes the feedback error and (2) other terms, which include the network states, the reference input and its derivatives and the effects of the uncertainties. The feedback error gain is tuned by solving a linear matrix inequality. The neural optimizer determines the sliding-mode control law without being directly affected by the uncertainties. By applying the proposed method to the continuous-stirred reactor tank and the inverted pendulum problems, the performance of the proposed controller has been evaluated in terms of the tracking accuracy, elimination of the chattering, robustness against the uncertainties and feasibility of the control signals. Moreover, the results are compared with the conventional and twisting sliding-mode control methods.

Keywords

Sliding-mode control recurrent neural network optimisation projection theorem

Introduction

Achieving performance goals such as transient response and good accuracy in the steady state, feasibility of the solution and robustness in the face of uncertainties and nonlinearities are essential in nonlinear control systems. Robust control methods can be appropriate strategies for the realization of system regulation or tracking in the presence of model mismatches and disturbances in the system dynamics.

Methods such as H_∞ control approaches are perfect tools for tuning control parameters in linear systems with uncertainties. In this approach, the state or output feedback gains are tuned based on the infinite norm of the closed-loop transfer function and the relation between the robust stability and solving the linear matrix inequalities (LMIs). However, development of the H_∞-based approaches for nonlinear systems is a difficult task. Variable structure methods with high accuracy and good performance are appropriate approaches for the linear and nonlinear systems, especially in the case of model uncertainties and nonlinearities. Sliding-mode control (SMC) method is an example of such methods, where the definition of a stable sliding surface causes the system states to move along the sliding surface and remain on it from any initial condition.

Generally, the SMC law consists of an equivalent control law to achieve nominal stability of the closed-loop system and a switching control law to provide its robustness. However, there are major issues regarding the design of this control method including chattering phenomenon (Feng et al., 2014), optimal sliding surface (Das and Mahanta, 2014) and robustness of the closed-loop system (Moradi and Majd, 2015; Yang et al., 2016). Adverse fluctuations around the sliding surface, technically known as the chattering, are one of the major challenges in the SMCs. The chattering is caused by the switching nature of the control law due to the non-idealities in the switching term such as time delay, high-frequency effects of the parasites and the unmolded dynamics, which results in dramatic decline in the control system performance and damage to the actuators (Incremona et al., 2017). Numerous strategies have been developed in recent years to remove or reduce the effects of chattering.

The boundary layer is the simplest approach, where the approximation of the switching function with the saturation operator causes the system trajectories to converge to the area around the sliding surface (Saghafinia et al., 2014). In addition, hyperbolic tangent function (Aghababa and Akbari, 2012) and fuzzy logic (Li et al., 2017; Niknam et al., 2014) have been employed in order to approximate the switching function. The fundamental problem with these methods is the lack of good accuracy of the closed-loop performance. By using an integrator between the controller and the main system, the chattering can be eliminated (Mobayen, 2014; Wang et al., 2017). In this approach, a switching is included in the derivation of the control signal that removes the chattering in the control signal. Nevertheless, sensitivity to the measurement noise and the initial conditions, integration drift and need to estimate the derivatives of the uncertainties are some of the problems with this SMC method. The high-order SMCs have been considered by many researchers during the last decade. The premise in these approaches is forming a sliding manifold as a function of the sliding surface and its derivatives such that the switching function is implicitly or explicitly excluded from the control law. As a result, the chattering is significantly reduced or completely eliminated. The most significant high-order SMCs are twisting and super twisting (Benbouhenni et al., 2018; Evangelista et al., 2013; Shtessel et al., 2017), sub-optimal (Ferrara and Incremona, 2015; Liu and Li, 2014), methods based on the reaching law modification (Mondal and Mahanta, 2012) and the method based on the high-order derivatives of the sliding surface (Ding et al., 2015; Bartolini et al., 2016; Shtessel et al., 2014a). In these methods, the sign of the sliding surface and its derivative is used. This provides more degrees of freedom to the control law. However, this method can only reduce, and not eliminate, the chattering. In the modified reaching law, the sliding surface and its sign are simultaneously used for the convergence of the states towards the sliding surface. The high-order derivatives of the sliding surface can also be used to remove the switching term directly from the control signal. Therefore, it can be expected that the chattering is reduced or eliminated. Major challenges in the design of the high-order SMCs include sensitivity to the measurement noise and initial conditions, which increase the design complexity, inability to remove chattering completely and differentiator problems. Further issues regarding the high-order SMCs are finite-time convergence (Behnamgol and Vali, 2015; Song et al., 2017) and chattering analysis (Ren et al., 2016; Shtessel et al., 2014b).

Conventionally, the optimization problem in the SMCs is defined as finding an optimum sliding manifold corresponding to an integral performance index. The index is defined based on the tracking, regulation and minimization of the control costs. In the majority of these methods, the switching is designed separately from the optimized term. Particularly, the main goal of these methods is dealing with the uncertainties (Tang et al., 2008). Therefore, optimality is merely taken into account in the sliding-surface parameters or in the equivalent control law. This means that the sliding-mode law is not optimal. Since the impacts of uncertainties are included in the sliding-mode gain, the optimality of the control law tends to diminish. On the other hand, it is possible to increase the signal amplitude, especially in the transient conditions. To address this problem, a number of studies have employed a saturation operator in designing the control law (Ding and Zheng, 2016; Huang et al., 2016; Li et al., 2016). In general, the saturation function causes a performance decline in the closed-loop control system and requires a systematic method to guide the optimal control law within its permissible range. In general, in the optimal SMC methods, the constrained optimization problem is not developed and only a tuning of the sliding-surface parameters is considered.

In this paper, a constrained optimal and chattering-free SMC method using recurrent neural networks (RNNs) and H_∞ robust method is developed. In the proposed approach, the optimality is realized using the definition of a performance index based on the derivative of the sliding surface and minimization of the control cost. Moreover, the optimization problem, which is solved using the RNN, considers the physical limitations of the actuator. Accordingly, the switching is explicitly removed from the control law, yielding a chattering-free SMC. In addition, in order to obtain robustness against uncertainties in the system parameters, the coefficients of the sliding surface and the performance index are determined using the H_∞ approach, which is based on the infinite norm of the closed-loop transfer function. Furthermore, the convergence analysis of the RNN is performed using the Lyapunov stability theory.

The advantages of the proposed control method can be summarized as follows:

Explicit elimination of the switching function in the SMC law and hence, complete removal of the chattering;

Access to the optimal control law by defining a quadratic performance index and solving it using the RNN;

Considering the constraints on the control signals and improving their applicability in the practical systems;

Making use of a neural optimizer with a simple structure and solid convergence analysis;

Automatic offline adjustment of the control parameters using the H_∞ algorithm without any need for real-time calculations;

High-rate convergence of solving the optimization problem and the ability to develop it for the real-time applications;

No need for input matrix inversion and hence, avoiding the singularity problem in the control law;

No need for any adaptation to the dynamic model of the system;

Applicability of the proposed method to linear and nonlinear systems;

Ability to deal with the structural and non-structural uncertainties.

This paper is organized as follows. In Section 2, the dynamic model of the system is presented along with the assumptions, which are necessary to design the control law. The constrained SMC problem is presented in Section 3. In Section 4, the dynamic model of the RNN for solving the constrained quadratic programming (QP) problem is introduced and the convergence analysis is provided. In Section 5, the problem of H_∞ control is described for robustness of the closed-loop system in the presence of uncertainties. Section 6 shows simulating example and discussions. Conclusions are drawn in Section 7.

Problem statement

Consider the following nonlinear single-input single-output (SISO) dynamic system

\begin{matrix} \overset{\cdot}{x} (t) = f (x, t) + B u (t) + d (t) \\ y (t) = Cx (t) \end{matrix}

(1)

where $x \in R^{n}$ , $u \in R$ and $y \in R$ denote state vector, input and output of the system, respectively, $f \in R^{n}$ and $d \in R^{n}$ denote the nonlinear part and the model uncertainties of the system, respectively, and $B \in R^{n}$ and $C^{T} \in R^{n}$ are the input and output matrices of the system, respectively.

Assumption 1: The dynamic model of the system in (1) can be rewritten as a linear time-invariant system including the uncertainties as follows

\begin{matrix} \overset{\cdot}{x} (t) = Ax (t) + B u (t) + \bar{d} (t) \\ y (t) = Cx (t) \end{matrix}

(2)

where $\bar{d} \in R^{n}$ denotes the vector containing model uncertainties, external disturbances and nonlinear part of system (1).

Assumption 2: The upper bound of $d (t)$ is known or can be estimated; that is, $‖ d (t) ‖ \leq Γ$ . It should be noticed that such assumption is necessary to ensure robust stability of the closed-loop system.

The purpose of the proposed control scheme is to design the sliding-mode control law without any chattering. Moreover, the control signal should be optimal considering the system constraints. In addition, the closed-loop system should be robust against uncertainties in the system parameters. The design of this controller is performed in the following steps: (1) defining a suitable performance index based on the derivatives of the sliding surface and constructing a constraint QP problem; (2) formulating the dynamic and algebraic models of the RNN for solving the constraint optimization problem and extracting the control law for the normal operation of the system; (3) using the $H_{\infty}$ approach for offline self-tuning parameters of the controller to obtain robustness of the closed-loop system in the presence of uncertainties.

Introducing constrained optimal sliding-mode controller

Let the relative degree of the system in (2) be m. Then

CB = CAB = \dots = C A^{m - 1} B

(3)

The sliding surface is defined in the integral–derivative form as follows

s (t) = e^{(m)} (t) + λ_{1} e^{(m - 1)} (t) + \dots + λ_{m + 1} \int_{0}^{t} e (t) dt

(4)

where $e (t) = y (t) - y_{d} (t)$ is the tracking error, in which $y_{d} (t)$ is the reference signal and $e^{(m)} (t)$ is the mth time-derivative of $e (t)$ . In order to formulate an optimal sliding-mode controller, consider the following primal performance index

\begin{matrix} min_{u} \frac{1}{2} q {\overset{\cdot}{s}}^{2} (t) + \frac{1}{2} p u^{2} (t) \\ u_{min} \leq u (t) \leq u_{max} \end{matrix}

(5)

where $q > 0$ and $p > 0$ play important roles in the stability of the closed-loop system. Moreover, $u_{min}$ and $u_{max}$ denote the lower and the upper bounds of the control signal, respectively.

Remark 1: Based on the constraint optimization problem in (5), the goal of the proposed method, in addition to the feasibility of the solution and the minimum control cost, is to obtain asymptotic convergence of the derivative of the sliding surface to zero.

Remark 2: Since the main goal in the control system is the tracking problem, q must be greater than p. That is to say, the smaller the value of p, the less important the minimum cost problem as compared to the tracking problem.

Differentiating the sliding surface in (4) yields

\overset{\cdot}{s} (t) = e^{(m + 1)} (t) + λ_{1} e^{(m)} (t) + \dots + λ_{m + 1} e (t) .

(6)

Based on the relative degree of the system (m) and the nominal dynamic model in (1), it gives

\begin{matrix} \overset{\cdot}{s} (t) = & C A^{m + 1} x (t) + C A^{m} B u (t) - y_{d}^{(m + 1)} (t) \\ + λ_{1} e^{(m)} (t) + \dots + λ_{m + 1} e (t) \end{matrix}

(7)

which can be rewritten as

\overset{\cdot}{s} (t) = ax (t) + bu (t) + g (t)

(8)

where

\begin{matrix} a = C (A^{m + 1} + λ_{1} A^{m} + \dots + λ_{m + 1} I_{n \times n}) \\ b = C A^{m} B \\ g (t) = - (y_{d}^{(m + 1)} (t) + λ_{1} y_{d}^{(m)} (t) + . . . + λ_{m + 1} y_{d} (t)) . \end{matrix}

Substituting (8) in (5) yields

\begin{matrix} min_{u} \frac{1}{2} q {(ax (t) + bu (t) + g (t))}^{2} + \frac{1}{2} p u^{2} (t) . \\ u_{min} \leq u (t) \leq u_{max} \end{matrix}

(9)

Simplifying (9) gives

\begin{matrix} min_{u} \frac{1}{2} (q b^{2} + p) u^{2} (t) + q (ax (t) + g (t)) u (t) + \frac{1}{2} q {(ax (t) + g (t))}^{2} . \\ u_{min} \leq u (t) \leq u_{max} \end{matrix}

(10)

The third term in (10) is independent of the control signal, which is the optimization variable. Hence, this term can be ignored in the optimization problem. Therefore, the constraint optimization problem can be stated as follows

\begin{matrix} min_{u} \frac{1}{2} (q b^{2} + p) u^{2} (t) + q (ax (t) + g (t)) u (t) \\ u_{min} \leq u (t) \leq u_{max} \end{matrix}

(11)

To guarantee stability of the closed-loop system, the states of the system must move toward the sliding surface from any initial condition and remain on it. This means that the reaching law $\overset{\cdot}{s} (t) \leq - η sgn (s (t)), η > 0$ must always be satisfied. Hence, based on (8), the reaching law can be rewritten as

bu (t) \leq - ax (t) - g (t) - η sgn (s (t))

(12)

By defining $ϖ : = q b^{2} + p$ and $h (t) : = q (ax (t) + g (t))$ , the performance index can be reformulated as follows

\begin{matrix} min_{u} \frac{1}{2} ϖ u^{2} (t) + h (t) u (t) \\ u_{min} \leq u (t) \leq u_{max} \\ bu (t) \leq - ax (t) - g (t) - η sgn (s (t)) \end{matrix}

(13)

Equation (13) represents a constrained QP problem with static and dynamic constraints that will be solved using RNNs in this paper. This provides good tracking performance of the closed-loop system in addition to the minimum control cost and satisfying the control constraints and the reaching law (i.e. the fundamental condition of the SMC).

It should be noted that the optimization problem in (13) denotes a QP with linear and nonlinear constraints and hence, in general, it is a nonlinear optimization problem.

Recurrent neural network for solving QP problem

In order to solve the constrained optimization problem in (13), the dynamic model of the RNN that is based on the variational inequality (VI) and the projection theorems, will be used. First, the primary optimization problem will be converted to a dual problem. Then, based on the Karush-Kuhn-Tucker (KKT) optimal conditions (Kinderlehrer and Stampacchia, 1980), the dynamic and algebraic model of the RNN will be derived (Liu and Wang, 2006). To formulate the control law, some preliminaries must be introduced first.

Suppose $U (x)$ is a continuous vector-valued function from $R^{n}$ to $R^{m}$ that denotes the gradient of the cost function and $X = {x \in R^{n} | x_{i min} \leq x_{i} \leq x_{i max}}$ indicates the constrained space. The point $x^{*}$ is the optimal solution of the QP problem in (13) if it is a solution of the following inequality (Kinderlehrer and Stampacchia, 1980)

U (x^{*})^{T} (x - x^{*}) \geq 0, \forall x \in X

(14)

which is known as the VI. If the control signal is the optimization variable, one can write

U (u (t)) = \frac{\partial (\frac{1}{2} ϖ u^{2} (t) + h (t) u (t))}{\partial u (t)} = ϖ u (t) + h (t)

Based on the fixed-point theorem, the optimal solution of VI in (14) corresponds to the optimal solution of the following projection problem (Kinderlehrer and Stampacchia, 1980)

\Pr_{X} {x - U (x)} = x

(15)

where

\Pr_{X} {x} = \arg min_{v \in X} ‖ x - v ‖

(16)

where v denotes a point in the constrained space X, which has the nearest possible distance to vector x and $\Pr_{X} {\cdot}$ is a projection operator that maps vector x into a feasible space of the optimization problem. According to (16), the main idea in the projection problem is to find a vector in the constrained space that has the minimum distance from the optimal point.

First, the QP Problem in (13) is converted to the following dual problem (Liu and Wang, 2006)

\begin{matrix} max_{u (t)} - \frac{1}{2} ϖ u^{2} (t) + κ_{1}^{T} ν (t) - κ_{2}^{T} (t) χ (t) \\ s . t . ϖ u (t) + h (t) - G^{T} ν (t) + G^{T} χ (t) = 0 \end{matrix}

(17)

where $ν, χ \in R^{2}$ are the decision variables and $κ_{1}$ , $κ_{2}$ and $G$ are as follows

\begin{matrix} κ_{1} = {[u_{min} - \infty]}^{T} \\ κ_{2} (t) = {[u_{max} - ax (t) - g (t) - η sgn (s (t))]}^{T} \\ G = {[1 b]}^{T} . \end{matrix}

Defining a new variable $z : = ν - χ$ , the constraint in (17) becomes

ϖ u (t) + h (t) - G^{T} z (t) = 0

(18)

According to the constrained space in (13), we have

{(G u)}_{i} = {\begin{matrix} {(κ_{1})}_{i} if {(G u)}_{i} < {(κ_{1})}_{i} \\ {(κ_{2})}_{i} if {(G u)}_{i} > {(κ_{2})}_{i} \\ {(G u)}_{i} if {(κ_{1})}_{i} \leq {(G u)}_{i} \leq {(κ_{2})}_{i} \end{matrix}

(19)

where ${(κ_{1})}_{i}$ denotes the ith component of vector $κ_{1}$ , and so forth. On the other hand, based on the projection theorem in (15) and by solving the optimization problem, the following relation holds

G u (t) = \Pr_{X} {G u (t) - z (t)} .

(20)

Based on (16) and inevitability of $ϖ$ , we can get

u (t) = ϖ^{- 1} (G^{T} z (t) - h (t)) .

(21)

Since the optimal solution of the problem in (17) is equivalent to the optimal solution of the projection theorem in (20), the dynamic model of the RNN can be considered as follows

ε \frac{d z}{dt} = \Pr_{X} {G u (t) - z (t)} - G u (t)

(22)

where $ε > 0$ is the learning rate that affects the convergence of the neural network. Equation (22) represents the state equations of the projection RNN (PRNN). Moreover, (21) represents the network output, which is the optimal solution of the optimization problem. As it can be seen in (21), the PRNN only optimizes the control signal and not the state variables by adapting a proper search path for the optimization variable to achieve the minimum tracking error and the minimum control effort.

Remark 3: Based on (21) and (22), if the optimization problem is constrained, the control law would be directly available as a function of the PRNN states, the main system states and the reference input. This condition is of prime importance in the control systems especially for analyzing the closed-loop stability.

Remark 4: The order of the dynamic model of the PRNN is equals to the number of constraints considered for the optimization problem. This means that the structural complexity of the RNN in (22) in directly related on the number of constraints.

The block diagram of the PRNN including its dynamic model and the output equation is shown in Figure 1. As this figure shows, the activation functions are piecewise linear. Moreover, when the number of constraints increases, the network can perform parallel processing. The simplicity of the network structure along with the parallel processing and the piecewise-linear form of the activation functions provides a fast solver for the constraint optimization problem in hand. This makes it very suitable for real-time applications.

Remark 5: Based on the second-order constrained optimization problem in (17), the PRNN encounters two types of constraints: (1) the static constraint representing the limitations on the control signal and (2) the dynamic constraint representing the reaching law of the sliding condition.

Based on the proposed approach, the control law is formulated using the PRNN by solving the QP problem in (13). It should be noted that there is no explicit switching function in the proposed control signal in (21). This is similar to the dynamic sliding-mode and high-order sliding-mode control methods. The switching function exists only in the reaching law as a dynamic constraint. In other words, the switching function only affects the bounds of the optimization problem. In fact, the PRNN finds an optimal solution for the control signal in the feasible control space and avoids the critical bounds, which create chattering.

In the following, convergence analysis of the PRNN using the Lyapunov stability theory will be presented. First, the following lemmas should be considered.

Lemma 1: For the projection operator Pr_X in the closed-convex space X and $α, β \in X$ , the following relation is true

{(\Pr_{X} (α) - β)}^{T} (α - \Pr_{X} (α)) \geq 0

(23)

Proof: See Kinderlehrer and Stampacchia (1980).

Lemma 2: The following inequality holds for the optimal point $x^{*}$ and the optimal state $z^{*} :$

{(δ - u^{*})}^{T} G^{T} z^{*} \geq 0, \forall δ \in X

(24)

Proof: Based on (16), the gradient of the cost function is equal to $ϖ u (t) + h (t)$ . According to the output equation of the PRNN in (21), we have $G^{T} z^{*} = ϖ u (t) + h (t)$ . Accordingly, relation (24) is a VI problem.

Theorem 1: The dynamic model of the PRNN in (22) asymptotically converges to its equilibrium point.

Proof: Consider the following Lyapunov function

v (t) = \frac{1}{2} {(z - z^{*})}^{T} (z - z^{*}) + \frac{1}{ε} ϖ^{- 1} {(z - z^{*})}^{T} G^{T} G (z - z^{*})

(25)

The time derivative of (25) is

\overset{\cdot}{v} (t) = {(z - z^{*})}^{T} \overset{\cdot}{z} + ϖ^{- 1} G^{T} G {(z - z^{*})}^{T} \overset{\cdot}{z}

(26)

Using the dynamic model of the PRNN in (22) yields

\begin{matrix} \overset{\cdot}{v} (t) = & \frac{1}{ε} {(z - z^{*})}^{T} (\Pr (G u - z) - G u) \\ + \frac{1}{ε} ϖ^{- 1} G^{T} G {(z - z^{*})}^{T} (\Pr (G u - z) - G u) . \end{matrix}

(27)

By defining $α : = G u - z$ and $β : = G u$ , (27) can be rewritten as

\overset{\cdot}{v} (t) = \frac{1}{ε} (z - z^{*})^{T} (\Pr (α) - β) + \frac{1}{ε} ϖ^{- 1} G^{T} G (z - z^{*})^{T} (\Pr (α) - β)

(28)

Based on Lemma 1 for the optimal point $u^{*}$ , we have

{(\Pr_{X} (α) - G u^{*})}^{T} (α - \Pr_{X} (α)) \geq 0

(29)

Using Lemma 2, (29) can be reformulated as

{(\Pr_{X} (α) - G u^{*})}^{T} (z^{*} + α - \Pr_{X} (α)) \geq 0

(30)

which can be rewritten as

{(\Pr_{X} (α) - G u^{*} + G u - G u)}^{T} (z - z^{*} + \Pr_{X} (α) - G u) \leq 0

(31)

Moreover, based on (21)

u - u^{*} = ϖ^{- 1} G^{T} (z - z^{*})

(32)

Therefore, (31) can be represented as follows

\begin{matrix} {(\Pr_{X} (α) - G u)}^{T} (z - z^{*}) + {(\Pr_{X} (α) - G u)}^{T} G^{T} G (u - u^{*}) \leq \\ - ‖ (\Pr_{X} (α) - G u) ‖^{2} - ϖ^{- 1} ‖ z - z^{*} ‖^{2} \leq 0 \end{matrix}

(33)

The relation in (33) shows that the time derivative of the Lyapunov function is negative. Therefore, the PRNN in (22) is asymptotic convergent to its equilibrium point.

Equation (33) shows stability of the PRNN. The network, which includes the dynamics and output equations, is derived for minimization of the QP problem in (13). The QP problem has been obtained by reformulating the optimization problem in (5) for the control signal (u) as the optimization variable. Therefore, minimizing the QP is equal to minimizing the primary problem in (5). Moreover, equation (33) has been obtained using some lemmas about the projection theorem and variational inequality and only concludes that the time derivative of the Lyapunov function in (28) is negative.

It should be noted that, the PRNN-based SMC in (21) has been derived based on the nominal model of the system. In order to insure robustness of the closed-loop system against uncertainties in the system parameters, the control parameters such as the weighting coefficients of the performance index (q and p) and the parameters of the sliding surface (λ_i) will be obtained using the H_∞ approach, which will be described in the next section.

Parameter fine tuning using H_∞ control strategy

An important goal in the SMC approaches is to encounter model uncertainties and external disturbance. In this way, ensuring robustness of the closed-loop system is essential. For this purpose, the $H_{\infty}$ method, which is based on maintaining the stability in the worst case of the uncertainties, is utilized in this paper. This conservative but powerful method is capable of determining the control parameters as the solution of an LMI problem.

Consider the dynamic model in (2). By defining the tracking error as $\bar{e} = x - x_{d}$ , it yields

\begin{matrix} \bar{\overset{\cdot}{e}} (t) & = Ax (t) + B u (t) + \bar{d} (t) - {\overset{\cdot}{x}}_{d} (t) \\ = A \bar{e} (t) + B_{1} u (t) + B_{2} w (t) \end{matrix}

(34)

where

\begin{matrix} B_{1} = B \\ B_{2} = {[\begin{matrix} I_{n \times n} & A_{n \times n} & - I_{n \times n} \end{matrix}]}_{n \times 3 n} \\ w (t) = {[\begin{matrix} \bar{d} (t) & x_{d} (t) & {\overset{\cdot}{x}}_{d} (t) \end{matrix}]}^{T} . \end{matrix}

Considering $ζ$ and $y$ as the regulated and measured outputs, respectively, it gives

\begin{matrix} \bar{\overset{\cdot}{e}} (t) = A \bar{e} (t) + B_{1} u (t) + B_{2} w (t), \\ ζ = C_{1} \bar{e} (t), \\ y = C_{2} \bar{e} (t) \end{matrix}

(35)

where

C_{1} = I_{n \times n}, C_{2} = C .

The control law in (21) can be written as

u (t) = ϖ^{- 1} G^{T} z (t) - q ϖ^{- 1} ax (t) - q ϖ^{- 1} g (t)

(36)

Equation (36) can be decomposed into an error feedback and a feedback including the neural network states and the reference inputs. In other words

u (t) = k_{1} \bar{e} (t) + k_{2} μ (t)

(37)

where

\begin{matrix} k_{1} = - q ϖ^{- 1} a \\ k_{2} = ϖ^{- 1} [\begin{matrix} G^{T} & - q a & - q \end{matrix}] \\ μ (t) = {[\begin{matrix} z (t) & x_{d} (t) & g (t) \end{matrix}]}^{T} . \end{matrix}

The aim of the H_∞ strategy is to determine the error feedback gain $k_{1}$ in such a way that the robust stability of the closed-loop system is maintained in the worst uncertainty condition. To this end, consider the control law as

u (t) = u_{f} (t) + k_{2} μ (t)

(38)

where $u_{f} (t) = k_{1} \bar{e} (t)$ . In this way, the dynamic in (35) can be reformulated as

\begin{matrix} \bar{\overset{\cdot}{e}} (t) = A \bar{e} (t) + B_{1} u_{f} (t) + \bar{B} σ (t) \\ ζ = C_{1} \bar{e} (t) \\ y = C_{2} \bar{e} (t) \end{matrix}

(39)

where

\begin{matrix} \bar{B} = [\begin{matrix} B_{2} & B_{1} k_{2} \end{matrix}] \\ σ (t) = {[\begin{matrix} w (t) & μ (t) \end{matrix}]}^{T} . \end{matrix}

The control law $u_{f} (t)$ is H_∞ robust if the following conditions hold for a given scalar $γ > 0$ (Iwasaki and Skelton, 1994):

Matrix $A + B_{1} k_{1}$ is asymptotic stable.

The closed-loop transfer function between $ζ$ and $σ$ (i.e. $T_{ζ σ}$ ) satisfies $‖ T_{ζ σ} ‖_{\infty} < γ$ .

The block diagram of the proposed PRNN-based optimal SMC using the H_∞ strategy is shown in Figure 2. As this figure shows, the control parameters have been determined offline using the H_∞ algorithm. Then, using the data received from the sliding surface and the control constraints, the SMC law is applied to the system based on the dynamic model of the RNN.

Lemma 3: By considering $u_{f} (t) = k_{1} \bar{e} (t)$ as the feedback control law, the closed-loop system can be described as follows

\begin{matrix} \bar{\overset{\cdot}{e}} (t) = A_{cl} x (t) + B_{cl} σ (t), \\ ζ (t) = C_{cl} x (t) + D_{cl} σ (t) \end{matrix}

(40)

where

[\begin{matrix} A_{cl} & B_{cl} \\ C_{cl} & D_{cl} \end{matrix}] = [\begin{matrix} A & \bar{B} \\ C_{1} & 0 \end{matrix}] + [\begin{matrix} B_{1} \\ 0 \end{matrix}] k_{1} [\begin{matrix} C_{2} & 0 \end{matrix}] .

The control law $u_{f} (t) = k_{1} \bar{e} (t)$ is an H_∞ controller if for any $R > 0$ there exist $Φ > 0$ such that

\begin{matrix} A_{cl} Φ + {Φ A}_{cl}^{T} + ({Φ C}_{cl}^{T} + B_{cl} D_{cl}^{T}) R^{- 1} {({Φ C}_{cl}^{T} + B_{cl} D_{cl}^{T})}^{T} \\ + B_{cl} B_{cl}^{T} < 0 \\ R = γ I - D_{cl} {D_{cl}}^{T} > 0 \end{matrix}

(41)

Proof: See (Zhou and Khargonekar, 1988).

Lemma 4: The inequality in (41) is equivalent to the following LMI

[\begin{matrix} A_{cl} Φ + {Φ A}_{cl}^{T} + B_{cl} B_{cl}^{T} & {Φ C}_{cl}^{T} + B_{cl} D_{cl}^{T} \\ C_{cl} Φ + D_{cl} B_{cl}^{T} & D_{cl} D_{cl}^{T} - γ I \end{matrix}] < 0 .

(42)

Proof: By defining the following matrices

\begin{matrix} M_{11} : = A_{cl} Φ + {Φ A}_{cl}^{T} + B_{cl} B_{cl}^{T} \\ M_{12} : = ({Φ C}_{cl}^{T} + B_{cl} D_{cl}^{T}) \\ M_{22} : = D_{cl} D_{cl}^{T} - γ I \end{matrix}

the inequalities in (41) can be rewritten as

{\begin{matrix} M_{11} - M_{12} M_{22}^{- 1} M_{12}^{- 1} < 0 \\ M_{22} < 0 \end{matrix} .

(43)

By defining $M : = [\begin{matrix} M_{11} & M_{12} \\ M_{12}^{T} & M_{22} \end{matrix}]$ and based on the Schur complement, (43) shows that matrix $M$ is negative definite. Therefore, the inequality in (42) is negative.

Lemma 5: The LMI in (42) is equivalent to

[\begin{matrix} A_{cl} Φ + {Φ A}_{cl}^{T} & {Φ C}_{cl}^{T} & B_{cl} \\ C_{cl} Φ & - I & D_{cl} \\ B_{cl}^{T} & D_{cl}^{T} & - γ I \end{matrix}] < 0

(44)

Proof: See Iwasaki and Skelton (1994).

Lemma 6: The LMI in (44) is equivalent to the following inequality

B_{n} k_{1} C_{n} + {(B_{n} k_{1} C_{n})}^{T} + Λ < 0

(45)

where

\begin{matrix} B_{n} = [\begin{matrix} B_{1} \\ 0 \\ 0 \end{matrix}] \\ C_{n} = [\begin{matrix} C_{2} Φ & 0 & 0 \end{matrix}] \\ Λ = [\begin{matrix} A Φ + Φ A^{T} & {Φ C}_{1}^{T} & \bar{B} \\ C_{1} Φ & - I & 0 \\ {\bar{B}}^{T} & 0 & - γ I \end{matrix}] . \end{matrix}

Proof: Substituting the closed-loop matrices in (40) in (42) yields

[\begin{matrix} (A + B_{1} k_{1} C_{2}) Φ + Φ {(A + B_{1} k_{1} C_{2})}^{T} & {Φ C}_{1}^{T} & \bar{B} \\ C_{1} Φ & - I & 0 \\ {\bar{B}}^{T} & 0 & - γ I \end{matrix}] < 0 .

(46)

Simplifying (46) gives

\begin{matrix} [\begin{matrix} A Φ + Φ A^{T} & {Φ C}_{1}^{T} & \bar{B} \\ C_{1} Φ & - I & 0 \\ {\bar{B}}^{T} & 0 & - γ I \end{matrix}] + \\ [\begin{matrix} (B_{1} k_{1} C_{2}) Φ + Φ {(B_{1} k_{1} C_{2})}^{T} & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}] < 0 . \end{matrix}

(47)

The second part of (47) can be rewritten as

\begin{matrix} [\begin{matrix} (B_{1} k_{1} C_{2}) Φ + Φ {(B_{1} k_{1} C_{2})}^{T} & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}] = \\ [\begin{matrix} B_{1} \\ 0 \\ 0 \end{matrix}] k_{1} [\begin{matrix} C_{2} Φ & 0 & 0 \end{matrix}] + [\begin{matrix} {Φ C}_{2}^{T} \\ 0 \\ 0 \end{matrix}] k_{1} [\begin{matrix} B_{1}^{T} & 0 & 0 \end{matrix}] < 0 . \end{matrix}

(48)

Therefore

\begin{matrix} [\begin{matrix} A_{cl} Φ + {Φ A}_{cl}^{T} & {Φ C}_{cl}^{T} & B_{cl} \\ C_{cl} Φ & - I & D_{cl} \\ B_{cl}^{T} & D_{cl}^{T} & - γ I \end{matrix}] = \\ [\begin{matrix} A Φ + Φ A^{T} & {Φ C}_{1}^{T} & \bar{B} \\ C_{1} Φ & - I & 0 \\ {\bar{B}}^{T} & 0 & - γ I \end{matrix}] + [\begin{matrix} B_{1} \\ 0 \\ 0 \end{matrix}] k_{1} [\begin{matrix} C_{2} Φ & 0 & 0 \end{matrix}] \\ + [\begin{matrix} {Φ C}_{2}^{T} \\ 0 \\ 0 \end{matrix}] k_{1} [\begin{matrix} B_{1}^{T} & 0 & 0 \end{matrix}] < 0 \end{matrix}

(49)

which is the same as (45).

For satisfying (45), it is sufficient that there exists matrix $Φ > 0$ and scalar $ϕ_{k} > 0$ such that the following inequality holds (Zhou and Khargonekar, 1988)

[\begin{matrix} A Φ + Φ^{T} A + B_{2}^{T} B_{2} + B_{1} ϕ_{k} B_{1}^{T} & {Φ C}_{1}^{T} \\ C_{1} Φ & - γ I \end{matrix}] < 0

(50)

Theorem 2: The closed-loop system in (38) with the feedback control law $u_{f} (t) = k_{1} \bar{e} (t)$ is robust stable if there exist $Φ > 0$ and scalar $ϕ_{k} > 0$ such that the inequality (50) holds. Then, the H_∞ feedback gains can be obtained as follows

k_{1} = {({(B_{n}^{R})}^{T} B_{n}^{R})}^{- 1} {(B_{n}^{R})}^{T} Ψ {(C_{n}^{L})}^{T} {(C_{n}^{L} {(C_{n}^{L})}^{T})}^{- 1}

(51)

where $B_{n} \in R^{n \times m}$ and $C_{n} \in R^{k \times n}$ are given in (46), $r_{b}$ and $r_{c}$ denote their ranks, respectively, $(B_{n}^{L}, B_{n}^{R})$ and $(C_{n}^{L}, C_{n}^{R})$ denote the full rank factors of $B_{n}$ and $C_{n}$ such that $B_{n} = B_{n}^{L} B_{n}^{R}$ and $C_{n} = C_{n}^{L} C_{n}^{R}$ , respectively and

\begin{matrix} Ψ : = & - H^{- 1} {(B_{n}^{L})}^{T} N {(C_{n}^{R})}^{T} {(C_{n}^{R} N {(C_{n}^{R})}^{T})}^{- 1} \\ + H^{- 1} S^{1 / 2} Θ {(C_{n}^{R} N {(C_{n}^{R})}^{T})}^{- 1 / 2} \end{matrix}

in which

\begin{matrix} N : = {(B_{n}^{L} H^{- 1} {(B_{n}^{L})}^{T} - Λ)}^{- 1} \\ S : = H - {(B_{n}^{L})}^{T} [N - N {(C_{n}^{R})}^{T} {((C_{n}^{R}) N {(C_{n}^{R})}^{T})}^{- 1} (C_{n}^{R}) N] B_{n}^{L} \\ H : = ρ I > 0 \\ Θ = S^{- 1 / 2} \hat{Θ} {(C_{n}^{R} N (C_{n}^{R}))}^{1 / 2}, ‖ Θ ‖_{2} < 1 \\ \hat{Θ} = H Ψ + {(B_{n}^{L})}^{T} N {(C_{n}^{R})}^{T} {(C_{n}^{R} N {(C_{n}^{R})}^{T})}^{- 1} \end{matrix}

Proof: Based on Lemma 3, $u_{f} (t) = k_{1} \bar{e} (t)$ in (38) is an H_∞ controller if there exist $Φ > 0$ such that inequality (42) holds. Moreover, based on Lemma 6, (45) is equivalent to (41).

Using the right and left factors of matrices $B_{n}$ and $C_{n}$ , (45) can be rewritten as

B_{n}^{L} B_{n}^{R} k_{1} C_{n}^{L} C_{n}^{R} + {(B_{n}^{L} B_{n}^{R} k_{1} C_{n}^{L} C_{n}^{R})}^{T} + Λ < 0 .

(52)

By defining $Ψ : = B_{n}^{R} k_{1} C_{n}^{L}$ we have

B_{n}^{L} {Ψ C}_{n}^{R} + {(B_{n}^{L} {Ψ C}_{n}^{R})}^{T} + Λ < 0 .

(53)

Suppose there exists $H = ρ I > 0$ for sufficiently small $ρ > 0$ such that

B_{n}^{L} {Ψ C}_{n}^{R} + {(B_{n}^{L} {Ψ C}_{n}^{R})}^{T} + Λ + {(C_{n}^{R})}^{T} Ψ^{T} {HC}_{n}^{R} < 0

(54)

Or equivalently

\begin{matrix} {({Ψ C}_{n}^{R} + H^{- 1} {(B_{n}^{L})}^{T})}^{T} H ({Ψ C}_{n}^{R} + H^{- 1} {(B_{n}^{L})}^{T}) \\ < B_{n}^{L} H^{- 1} {(B_{n}^{L})}^{T} - Λ \end{matrix}

(55)

which can be simplified as

H > ({H Ψ C}_{n}^{R} + {(B_{n}^{L})}^{T}) (B_{n}^{L} H^{- 1} {(B_{n}^{L})}^{T} - Λ) {({H Ψ C}_{n}^{R} + {(B_{n}^{L})}^{T})}^{T} .

(56)

Manipulating (56) gives (Iwasaki and Skelton, 1994)

\hat{Θ} (C_{n}^{R} N (C_{n}^{R})) {\hat{Θ}}^{T} < S

(57)

where

\begin{matrix} N = {(B_{n}^{L} H^{- 1} {(B_{n}^{L})}^{T} - Λ)}^{- 1} \\ \hat{Θ} = H Ψ + {(B_{n}^{L})}^{T} N {(C_{n}^{R})}^{T} {(C_{n}^{R} N {(C_{n}^{R})}^{T})}^{- 1} \\ S = H - {(B_{n}^{L})}^{T} [N - N {(C_{n}^{R})}^{T} {((C_{n}^{R}) N {(C_{n}^{R})}^{T})}^{- 1} (C_{n}^{R}) N] B_{n}^{L} . \end{matrix}

(58)

Inequality (57) is equivalent to

Θ = S^{- 1 / 2} \hat{Θ} {(C_{n}^{R} N (C_{n}^{R}))}^{1 / 2} .

(59)

Using (59), the solution of (57) can be given as follows (Iwasaki and Skelton, 1994)

\begin{matrix} Ψ = & - H^{- 1} {(B_{n}^{L})}^{T} N {(C_{n}^{R})}^{T} {(C_{n}^{R} N {(C_{n}^{R})}^{T})}^{- 1} \\ + H^{- 1} S^{1 / 2} Θ {(C_{n}^{R} N {(C_{n}^{R})}^{T})}^{- 1 / 2} . \end{matrix}

(60)

Therefore

Ψ = B_{n}^{R} k_{1} C_{n}^{L}

(61)

where

k_{1} = {({(B_{n}^{R})}^{T} B_{n}^{R})}^{- 1} {(B_{n}^{R})}^{T} Ψ {(C_{n}^{L})}^{T} {(C_{n}^{L} {(C_{n}^{L})}^{T})}^{- 1} .

(62)

Hence, the robust feedback control law, which can be determined explicitly using (61), is a solution of the LMI problem in (45).

The procedure for finding the control parameters using the $H_{\infty}$ strategy can be summarized as follows

Compute matrix $Φ$ and scalar $ϕ_{k}$ by solving LMI $[\begin{matrix} A Φ + Φ^{T} A + B_{2}^{T} B_{2} + B_{1} ϕ_{k} B_{1}^{T} & {Φ C}_{1}^{T} \\ C_{1} Φ & - γ I \end{matrix}] < 0$ .

Select $ρ > 0$ sufficiently small for $H = ρ I$ .

Select an arbitrary positive $Θ$ such that $‖ Θ ‖ < 1$ .

Compute matrices $B_{n}$ and $C_{n}$ in (45) and their left and right factors.

Compute matrices $N$ and $S$ in (57) and $Ψ$ in (59).

Compute $k_{2}$ using $ϕ_{k}$ .

Compute the feedback gain $k_{1}$ in (61).

Compute the control parameters, i.e. q, p, and $λ_{i} (i = 1, . . ., m + 1)$ based on (37).

Simulating examples

Example 1: In order to show performance of the proposed SMC method, it is applied to the continuous stirred tank reactor (CSTR) system, whose state-space model can be expressed as follows (Chen and Peng, 1999)

\begin{matrix} {\overset{\cdot}{x}}_{1} = - x_{1} + D_{a} (1 - x_{1}) \exp (\frac{x_{2}}{1 + x_{2} / φ_{a}}), \\ {\overset{\cdot}{x}}_{2} = - (1 + δ) x_{2} + B_{a} D_{a} (1 - x_{1}) \exp (\frac{x_{2}}{1 + x_{2} / φ_{a}}) + δ u \end{matrix}

(63)

where $x_{1}$ , $x_{2}$ and $u$ denote the reactant concentration, the reactor temperature and the cooling jacket temperature, respectively. The parameters $B_{a}$ , $D_{a}$ , $φ_{a}$ and $δ$ denote the heat of reaction, the Damokhler number, the activated energy and the heat transfer coefficient, respectively. The numerical values of the parameters are $B_{a} = 8$ , $D_{a} = 0.072$ , $φ_{a} = 20$ and $δ = 0.3$ . Equation (63) is dimensionless in order to simplify the control system analysis. A schematic view of the CSTR system is shown in Figure 3. In this system, liquids enter the tank while stirred rapidly. The output of this system (i.e. the product) is a liquid with appropriate concentration and temperature.

Based on the proposed control structure, the dynamics of CSTR is decomposed into linear and nonlinear parts as follows

\begin{matrix} [\begin{matrix} {\overset{\cdot}{x}}_{1} \\ {\overset{\cdot}{x}}_{2} \end{matrix}] = & [\begin{matrix} - 1 & 0 \\ 0 & - (1 + δ) \end{matrix}] [\begin{matrix} x_{1} \\ x_{2} \end{matrix}] + [\begin{matrix} 0 \\ δ \end{matrix}] u \\ + [\begin{matrix} D_{a} (1 - x_{1}) \exp (\frac{x_{2}}{1 + x_{2} / φ_{a}}) \\ B_{a} D_{a} (1 - x_{1}) \exp (\frac{x_{2}}{1 + x_{2} / φ_{a}}) \end{matrix}] . \end{matrix}

(64)

By considering the reactant temperature as the output, the relative degree of the system is equal to zero. Therefore, the sliding surface is defined as follows

s (t) = e (t) + λ \int_{0}^{t} e (t) dt

(65)

where $e = x_{2} - x_{d 2}$ denotes the tracking error of the reactor temperature. Using the H_∞ approach the control parameters can be found as

q = 2.5, p = 0.1, λ = 50, η = 1, ε = 10^{- 4}

It should be noted that the nonlinear part of (64) is considered as the uncertainty. Moreover, since the variations of $x_{1}$ is small ( $0 < x_{1} \leq 0.5$ ), by considering the variations of $x_{2}$ as $0 < x_{2} \leq 4$ , the upper bound of uncertainty is equal to 3.

The sampling time is 0.2 sec. The limitations on the control signal is $0 \leq u \leq 18$ . The reference input (i.e. the desired reactant temperature) is a step signal with the amplitude equal to 1.5 for $15 < t < 30 \sec .$ and equal to 2 for other times. The response of the closed-loop system in Figure 4 indicates very accurate input tracking as well as good transient response with small overshoots. Moreover, Figure 5 shows that the control signal is free of chattering and within its predefined limits, which is very important in practical applications. In other words, the proposed optimal SMC approach guarantees feasibility of the control action as well. Figure 6 shows that the sliding condition, which is a dynamic constraint in the optimization problem, is satisfied by the PRNN. Satisfying this constraint ensures stability of the closed-loop system. The reactant concentration of the CSTR system (Figure 7) implies that the variations of $x_{1}$ are small and bounded. Variations of the nonlinear parts of the system (as the uncertainty terms in equation (64)) are shown in Figure 8, which indicates that the maximum range of variations in uncertainties is less than 3, which is considered for the robustness of the controller.

Based on the results, it can be observed that no chattering exists in state variables as well as in the control signal. Moreover, the control method is capable of effectively realizing the performance goals by avoiding singularity in the control law and keeping its magnitude in the predefined range. As mentioned in the previous sections, the simple structure of the PRNN makes it suitable for real time applications. This issue can be seen in Figure 9, where the computational time required to obtain the control signal corresponding to each sample, is much less than the sampling time. Figure 10 shows that the proposed approach is also able to track a sinusoidal input with fast converging time.

Hence, the advantages of the proposed method may be summarized as follows: (1) removing the chattering, (2) optimality of the control signal, (3) feasibility of the solution, (4) offline fine tuning of the control parameters for robustness against uncertainties in the system parameters, (5) avoiding singularities, (6) online optimization, (7) simple structure of the neural optimizer, (8) powerful convergence analysis.

Next, performance of the proposed algorithm will be compared with the classic SMC (Slotine and Li, 1991) and the second-order SMC known as the twisting algorithm (Evangelista et al., 2013). Considering the sliding surface in (65). Then, the control law for the classic SMC is equal to

u (t) = {(CB)}^{- 1} (- CAx - {\overset{\cdot}{x}}_{2 d} - λ e (t) - η sgn (s (t)))

(66)

In the twisting algorithm, a switching term based on the time derivative of the sliding surface is added to the control law

u (t) = {(CB)}^{- 1} (- CAx - {\overset{\cdot}{x}}_{2 d} - λ e (t) - η_{1} sgn (s (t)) - η_{2} sgn (\overset{\cdot}{s} (t)))

(67)

where $η_{2} > η_{1} > 0$ . Figures 10 and 11 show the response of the classic SMC. As Figure 11 shows, small switching gains do not provide satisfactory responses. On the other hand, increasing the switching gains (e.g. $η = 3$ and $λ = 5$ ) provides good tracking but at the cost of large overshoots and incurring chattering to the control signal (Figure 12).

Figures 13 and 14 show performance of the twisting algorithm. Increasing the switching gains (e.g., $(η_{1} η_{2}) = (1 1.4)$ ) yields relatively good tracking. However, the chattering still exists in the control action. Moreover, the input signal has exceeded the permissible range because this is not an optimal control method and there is no systematic way to incorporate the input constraint to the controller design. A typical comparison between the proposed approach, the conventional and the second order SMCs has been shown in Table. According to this table, the proposed algorithm has much less overshoot and less steady-state error as compared with other approaches. Moreover, three methods have almost the same rise time and settling time.

Table 1.

Compression between the control methods

Method/property	Overshoot %	Rise time (sec.)	Settling time (sec.)	Steady state error	Considering input signal constraint
PRNN-based SMC	4	1	4.5	0.005	Yes
Conventional SMC	25	1	4	−0.003<~<0.007	Yes
Second order SMC	15	0.8	4	−0.001<~<0.006	No

Next, for more evaluations, it is supposed that an uncertainty has been occurred in the parameter $β$ (0.3 to 0.5) at time 7 seconds and a step external disturbance has been applied to the output at time 23 second. As shown in Figure 15, the proposed method is more robust than the other controllers. Moreover, according to Figure 16, there is no chattering in the control signal obtained by the proposed method and control feasibility has been realized as compared with other controllers.

Example 2: In this example, performance of the proposed approach is evaluated by applying it to inverted pendulum including the actuator dynamics. The equations of this system can be described as follows (Campbell et al., 2008)

\begin{matrix} (M + m) \overset{\cdot\cdot}{x} (t) - ml \sin θ (t) {\overset{\cdot}{θ}}^{2} (t) + ml \cos θ (t) \overset{\cdot\cdot}{θ} (t) = F (t) + F_{fric} (t), \\ ml \cos θ (t) \overset{\cdot\cdot}{x} (t) - mgl \sin θ (t) + \frac{4}{3} ml \overset{\cdot\cdot}{θ} (t) = 0 . \end{matrix}

(68)

where M, m, l and g are mass of the cart, mass and length of the pendulum and gravity acceleration constant, respectively, x and $θ$ are position of the cart and pendulum angle, respectively, $F (t)$ is the applied force to the cart, and $F_{fric} (t)$ is the friction force that is modeled as

F_{fric} (t) = {\begin{matrix} F_{static} & if \overset{\cdot}{x} (t) = 0 \\ - μ_{c} F_{N} sign (\overset{\cdot}{x} (t)) - υ \overset{\cdot}{x} (t) & if \overset{\cdot}{x} (t) \neq 0 \end{matrix}

(69)

in which $μ_{c}$ and $υ$ are the coefficients of the Coulomb and viscous frictions, respectively, $F_{N} = (m + M) g$ is the magnitude of the normal force, and

F_{static} (t) = {\begin{matrix} - F_{0} & if | F_{0} | < μ_{s} F_{N} \\ - μ_{s} F_{N} sign (F_{0}) & if | F_{0} | \geq μ_{s} F_{N} \end{matrix}

(70)

where $μ_{s}$ is coefficient of the static friction and $F_{0}$ is the applied force before the cart begins to move. The dynamic model of the actuator, which is an armature-controlled dc servomotor, can be expressed in the following form (Wai and Muthusamy, 2014)

\begin{matrix} τ_{e} = K_{T} i_{a} \\ τ_{e} = J_{m} {\overset{\cdot\cdot}{θ}}_{m} + B_{m} {\overset{\cdot}{θ}}_{m} + τ_{m} \\ v_{t} = R_{a} i_{a} + L_{a} {\overset{\cdot}{i}}_{a} + K_{E} {\overset{\cdot}{θ}}_{m} \end{matrix}

(71)

where $τ_{e}$ is the electromagnetic torque, $K_{T}$ is the motor torque constants, $i_{a}$ is the armature current, $J_{m}$ is the moment of inertia, $B_{m}$ is the torsional damping coefficient, $θ_{m}, {\overset{\cdot}{θ}}_{m} and {\overset{\cdot\cdot}{θ}}_{m}$ denote the motor shaft position, velocity and acceleration, respectively, $τ_{m}$ is the load torque, $v_{t}$ is the armature input voltage, $R_{a}$ is the armature resistance, $L_{a}$ is the armature inductance, and $K_{E}$ is the back electromotive force coefficient. Moreover, the relationship between the pendulum angle ( $θ$ ) and the motor-shaft position ( $θ_{m}$ ) can be represented as follows

G_{r} = \frac{θ_{m}}{θ} = \frac{τ}{τ_{m}}

(72)

where $G_{r}$ is the gear ratio, $τ$ is the control torque developed at the joint side and $θ$ is the pendulum angle. The applied force to the pendulum (F) can be reformulated as follows

F = \frac{τ}{r} = \frac{G_{r}}{r} τ_{m} = \frac{G_{r}}{r} (K_{T} i_{a} - J_{m} G_{r} \overset{\cdot\cdot}{θ} - B_{m} G_{r} \overset{\cdot}{θ}) .

(73)

where r is the wheel radius. By considering the state vector as $x = {[\begin{matrix} x & \overset{\cdot}{x} & θ & \overset{\cdot}{θ} & i_{a} \end{matrix}]}^{T}$ and the input signal as $u (t) = v (t)$ , the dynamic equations of the inverted pendulum can be decomposed into the linear and nonlinear parts as

\begin{matrix} \overset{\cdot}{x} (t) = Ax (t) + B u (t) + d (x, t) \\ y (t) = Cx (t) \end{matrix}

(74)

where

\begin{matrix} A = [\begin{matrix} 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & a_{23} & a_{24} & a_{25} \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & a_{43} & a_{44} & a_{45} \\ 0 & 0 & 0 & - K_{E} G_{r} / L_{a} & - R_{a} / L_{a} \end{matrix}], B = [\begin{matrix} 0 \\ 0 \\ 0 \\ 0 \\ \frac{1}{L_{a}} \end{matrix}], \\ C^{T} = [\begin{matrix} 0 \\ 0 \\ 1 \\ 0 \\ 0 \end{matrix}], d (x, t) = [\begin{matrix} 0 \\ f_{1} \\ 0 \\ f_{2} \\ 0 \end{matrix}] \end{matrix}

in which

\begin{matrix} a_{23} = \frac{- 3 g ({G_{r}}^{2} J_{m} + mlr)}{4 r (M + m) - 3 {G_{r}}^{2} J_{m} - 3 lmr}, a_{24} = \frac{- 4 B_{m} {G_{r}}^{2}}{4 r (M + m) - 3 {G_{r}}^{2} J_{m} - 3 lmr} \\ a_{25} = \frac{4 G_{r} K_{T}}{4 r (M + m) - 3 {G_{r}}^{2} J_{m} - 3 lmr}, a_{43} = \frac{gr (M + m)}{4 r (M + m) - 3 {G_{r}}^{2} J_{m} - 3 lmr} \\ a_{44} = \frac{B_{m} G_{r} r^{2}}{4 r (M + m) - 3 {G_{r}}^{2} J_{m} - 3 lmr}, a_{45} = \frac{- K_{T} G_{r}}{4 r (M + m) - 3 {G_{r}}^{2} J_{m} - 3 lmr} \\ f_{1} = \frac{4 F_{fric} r + 4 G_{r} K_{T} x_{5} - 4 B_{m} {G_{r}}^{2} x_{4} - 3 {G_{r}}^{2} J_{m} g \sin x_{3} + 4 Lmr x_{4} \sin {x_{3}}^{2} - 3 lgmr \cos x_{3} \sin x_{3}}{4 Mr + 4 mr - 3 {G_{r}}^{2} J_{m} \cos x_{3} - 3 lmr \cos {x_{3}}^{2}} \\ f_{2} = \frac{- 3 (F_{fric} r \cos x_{3} + G_{r} K_{T} x_{5} \cos x_{3} - Mgr \sin x_{3} - gmr \sin x_{3} - B_{m} {G_{r}}^{2} x_{4} \cos x_{3} + lmr x_{4} \cos x_{3} \sin {x_{3}}^{2})}{4 Mr + 4 mr - 3 {G_{r}}^{2} J_{m} \cos x_{3} - 3 lmr \cos {x_{3}}^{2}} \end{matrix}

The numeral values of the parameters have been selected as (Campbell et al., 2008)

\begin{matrix} M = 0.815 Kg, m = 0.21 Kg, l = 0.305 m, \\ g = 9.81 m / \sec^{2}, \\ μ_{s} = 0.08328, μ_{c} = 0.04287, υ = 2.3156 . \end{matrix}

The initial states of the inverted pendulum and the constraints on the input signal are considered as $[θ (0) \overset{\cdot}{θ} (0)] = [π / 6 0]$ and $- 48 \leq u (t) \leq 48$ , respectively. The results of the proposed approach and the twisting algorithm are presented in Figures 17 and 18. As shown in Figure 17, the output of the PRNN-based SMC has no overshoot in the transient response. On the other hand, the twisting algorithm has faster response. However, according to Figure 18, the twisting algorithm has severely violated the limits of the control signals while the proposed control method is able to properly consider the control constraints. If the control constraint is applied to the twisting algorithm, it degrades performance of the closed-loop system (Figures 19 and 20).

Conclusion

In this paper, an optimal and robust SMC method using RNNs based on the projection theorem and the H_∞ strategy was designed. The cost function includes the derivative of the sliding surface and the control cost. The constraints were the control signal limitations and the reaching law of the SMC. The convergence of the neural optimizer was analyzed using the Lyapunov stability theory. Adjusting the control parameters in the presence of model uncertainties has been performed by solving an LMI problem and an appropriate analysis method. Convergence speed of the RNN was considerably high, which could make it a viable tool for solving online optimization problems. The proposed method was compared with the classic and twisting SMCs, which suffer from chattering, inappropriate transient responses, and violating the input constraints. It was shown that the proposed algorithm is able to remove chattering from the control signals, avoids the singularity and encounter the parametric and non-parametric uncertainties in the model. Moreover, it is able to consider the actuator limits and has an offline methodology to adjust the control parameters. Simple structure of the neural optimizer and its convergence analysis are some other benefits of the proposed algorithm.

Footnotes

Appendix

Declaration of conflicting interests

The author(s) declared no potential conflict of interests with respect to the research, authorship and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Mohammad Farrokhi

References

Aghababa

Akbari

(2012) A chattering-free robust adaptive sliding mode controller for synchronization of two different chaotic systems with unknown uncertainties and external disturbances. Applied Mathematics and Computation 218(9): 5757–5768.

Bartolini

Levant

Pisano

Usai

(2016) Adaptive second-order sliding mode control with uncertainty compensation. International Journal of Control 89(9): 1747–1758.

Behnamgol

Vali

(2015) Terminal sliding mode control for nonlinear systems with both matched and unmatched uncertainties. Iranian Journal of Electrical and Electronic Engineering 11(2): 109–117.

Benbouhenni

Boudjema

Belaidi

(2018) Neuro-second order sliding mode control of a DFIG supplied by a two-level NSVM inverter for wind turbine system. Iranian Journal of Electrical and Electronic Engineering 14(4): 362–373.

Chen

Peng

(1999) Intelligent process control using neural fuzzy techniques. Journal of Process Control 9(6): 493–503.

Campbell

Crawford

Morris

(2008) Friction and the inverted pendulum stabilization problem. Journal of Dynamic Systems, Measurement, and Control 130(5): 1–7.

Das

Mahanta

(2014) Optimal second order sliding mode control for nonlinear uncertain systems. ISA Transactions 53(4): 1191–1198.

Ding

Zheng

(2016) Nonsingular terminal sliding mode control of nonlinear second-order systems with input saturation. International Journal of Robust and Nonlinear Control 26(9): 1857–1872.

Ding

Wang

Zheng

(2015) Second-order sliding mode control for nonlinear uncertain systems bounded by positive functions. IEEE Transactions on Industrial Electronics 62(9): 5899–5909.

10.

Evangelista

Puleston

Valenciaga

Fridman

(2013) Lyapunov-designed super-twisting sliding mode control for wind energy conversion optimization. IEEE Transactions on Industrial Electronics 60(2): 538–545.

11.

Feng

(2014) Chattering free full-order sliding-mode control. Automatica 50(4): 1310–1314.

12.

Ferrara

Incremona

(2015) Design of an integral suboptimal second-order sliding mode controller for the robust motion control of robot manipulators. IEEE Transactions on Control Systems Technology 23(6): 2316–2325.

13.

Huang

Zhang

(2016) Adaptive reaching law based sliding mode control for electromagnetic formation flight with input saturation. Journal of the Franklin Institute 353(11): 2398–2417.

14.

Incremona

Rubagotti

Ferrara (2017) A sliding mode control of constrained nonlinear systems. IEEE Transactions on Automatic Control 62(6): 2965–2972.

15.

Iwasaki

Skelton

(1994) All controllers for the general H∞ control problem: LMI existence conditions and state space formulas. Automatica 30(8): 1307–1317.

16.

Kinderlehrer

Stampacchia

(1980) An Introduction to Variational Inequalities and Their Applications. Philadelphia: Society for Industrial and Applications, Mathematics.

17.

Wang

Shi

(2016) Output-feedback based sliding mode control for fuzzy systems with actuator saturation. IEEE Transactions on Fuzzy Systems 24(6): 1282–1293.

18.

Wang

Lam

Gao

(2017) Optimal guaranteed cost sliding mode control of interval type-2 fuzzy time-delay systems. IEEE Transactions on Fuzzy Systems 26(1): 246–257.

19.

Liu

Li (2014) Suboptimal integral sliding mode controller design for a class of affine systems. Journal of Optimization Theory and Applications 161(3): 877–904.

20.

Liu

Wang

(2006) A simplified dual neural network for quadratic programming with its KWTA application. IEEE Transactions on Neural Networks 17(6): 1500–1510.

21.

Mobayen

(2015) An adaptive chattering-free PID sliding mode control based on dynamic sliding manifolds for a class of uncertain nonlinear systems. Nonlinear Dynamics 82(1–2): 53–60.

22.

Mondal

Mahanta

(2012) Adaptive second-order sliding mode controller for a twin rotor multi-input-multi-output system. IET Control Theory and Applications 6(14): 2157–2167.

23.

Moradi

Johari Majd

(2015) A dissipative integral sliding mode control redesign method for uncertain nonlinear switched systems. Iranian Journal of Electrical & Electronic Engineering 11(4): 310–318.

24.

Niknam

Khooban

Kavousifard

Soltanpour

(2014) An optimal type II fuzzy sliding mode control design for a class of nonlinear systems. Nonlinear Dynamics 75(1–2): 73–83.

25.

Ren

Xie

Zhang

, et al. (2016) Chattering analysis for discrete sliding mode control of distributed control systems. Journal of Systems Engineering and Electronics 27(5): 1096–1107.

26.

Suryawanshi

Shendge

Phadke

(2016) A boundary layer sliding mode control design for chatter reduction using uncertainty and disturbance estimator. International Journal of Dynamics and Control 4(4): 456–465.

27.

Saghafinia

Ping

Uddin

M.N

(2014) Fuzzy sliding mode control based on boundary layer theory for chattering-free and robust induction motor drive. International Journal of Advanced Manufacturing Technology 71(1–4): 57–68.

28.

Shtessel

Moreno

Fridman

(2017) Twisting sliding mode control with adaptation: Lyapunov design, methodology and application. Automatica 75(1): 229–235.

29.

Shtessel

Edwards

Fridman

Levant (2014a) Second-order sliding mode controllers and differentiators. In: Sliding Mode Control and Observation. Basel: Birkhäuser, pp. 143–182.

30.

Shtessel

Edwards

Fridman

Levant (2014b) Analysis of sliding mode controllers in the frequency domain. In: Sliding Mode Control and Observation. Basel: Birkhäuser, pp. 183–211.

31.

Song

Niu

Zou

(2016) Finite-time sliding mode control synthesis under explicit output constraint. Automatica 65(3): 111–114.

32.

Song

Niu

Zou

(2017) Finite-time stabilization via sliding mode control. IEEE Transactions on Automatic Control 62(3): 1478–1483.

33.

Slotine

JJE

(2008) Applied Nonlinear Control. New Jersey: Prentice Hall.

34.

Tang

Dong

Gao

(2008) Optimal sliding mode control for nonlinear systems with time-delay. Nonlinear Analysis: Hybrid Systems 2(3): 891–899.

35.

Wang

Bao

(2017) Second-order dynamic sliding-mode control for nonminimum Phase under actuated hypersonic vehicles. IEEE Transactions on Industrial Electronics 64(2): 3105–3112.

36.

Wai

Muthusamy

(2014) Design of fuzzy-neural-network-inherited backstepping control for robot manipulator including actuator dynamics. IEEE Transactions on Fuzzy Systems 22(4): 709–722.

37.

Yang

Pan

Liu

(2016) Global robust sliding mode control for time-delay systems with mismatched uncertainties. Circuits, Systems, and Signal Processing 35(8): 3015–3026.

38.

Zhou

Khargonekar

(1988) An algebraic Riccati equation approach to H_∞ optimization. Systems and Control Letters 11(2): 85–91.

Robust chattering-free optimal sliding-mode control using recurrent neural networks: An H ∞ -based approach

Abstract

Keywords

Introduction

Problem statement

Introducing constrained optimal sliding-mode controller

Recurrent neural network for solving QP problem

Parameter fine tuning using H∞ control strategy

Simulating examples

Conclusion

Footnotes

Appendix

Declaration of conflicting interests

Funding

ORCID iD

References

Parameter fine tuning using H_∞ control strategy