Intelligent decoupled controller for mobile inverted pendulum real-time implementation

Abstract

In this work, an intelligent decoupled backstepping control system (IDBCS) is proposed for mobile inverted pendulums (MIPs) real-time control. This control system combined with adaptive output recurrent cerebellar model articulation controller (AORCMAC) and H^∞ control theory. The AORCMAC is designed to imitate an ideal backstepping controller, and the H^∞ controller is used to mitigate the effect of the approximation errors and outer disturbances. The decoupled method provides an easy way to achieve asymptotic stability control for a fourth-order nonlinear mobile inverted pendulum system. The concept of the decoupled approach is to decouple the whole system into two subsystems such that each subsystem has an individual control target. Then, the secondary subsystem provides information for the main subsystem, which generates a control action to make both subsystems move to their targets, respectively. In other words, it means that a fourth-order MIP system can be controlled well based on a second-order dynamic model. Moreover, all the adaptation laws of the IDBCS are obtained based on Lyapunov stability criterion, Taylor linearization technique and H^∞ control technique, so that the stability of the system can be guaranteed. Experiment results show that the MIP can stand stably when it moves toward a given position.

Keywords

Mobile inverted pendulum backstepping tracking control output recurrent cerebellar model articulation control decoupled intelligent controller

1 Introduction

A large number of literatures have proposed to study the inverted pendulum [1 –6] in the past several years. The rail-cart type inverted pendulum is the most usual type in control experiments. Recently, some studies about extensions of the rail-cart type inverted pendulum control system have been discussed. The most challenging task is to control a mobile inverted pendulum when the cart is not on a guide rail. Xu et al. [7] proposed a Takagi–Sugeno-type fuzzy logic controller (FLC) on a two-wheeled mobile robot (2WMR), which consists of two wheels in parallel and an inverse pendulum. Its objective of the 2WMR is to achieve position control of the wheels while keeping the pendulum around the upright position that is an unstable equilibrium. The omnidirectional rehabilitative training walker in [8] was controlled using an asymptotically stable controller that can guarantee the safety of user. Moreover, the proposed method can ensure the walker tracking on a training trajectory planned by a physical therapist. In [9], a Hankel-norm output feedback controller design for a class of T–S fuzzy stochastic systems was proposed. It was focused on the designing of full-order output feedback controller to guarantee the corresponding closed loop system to be mean-square asymptotically stable. Huang et al. [10] introduced the design and implementation of a two-wheel inverted pendulum (TWIP) system with a fuzzy control scheme and the system-on-a-programmable-chip (SoPC) technology. In [11], a new model transformation was analyzed and applied for a dynamic output feedback controller design of discrete-time T–S fuzzy systems with time varying delays. In [12], a novel model-free adaptive output recurrent cerebellar model articulation controller (AORCMAC) was utilized to control wheeled inverted pendulums that have a pendulum mounted on two coaxial wheels. In [13], an intelligent control system has been proposed for mobile wheeled inverted pendulum control. Simulation results were used to verify the effectiveness of the proposed robust control scheme.

Many researchers have argued that neural networks (NNs) are powerful building blocks for a wide class of complex nonlinear system control strategies when model information is absent or when a controlled plant is considered a “black box”. However, learning is slow as all weights are updated during each learning cycle. Therefore, the appropriateness of NNs is limited for problems requiring on-line learning. Cerebellar model articulation controller (CMAC) had been proposed in the works [14, 15]. In general, simple computation, fast learning property, good generalization capability, and easier hardware implementation are the major advantages of CMAC. The CMAC network can approximate a nonlinear function over a domain of interest to any desired accuracy. The advantages of using a CMAC over conventional NNs in many practical applications have recently been identified. The CMAC has been already validated that it can approximate a nonlinear function over a domain of interest to any desired accuracy. However, the main drawback of CMAC is that they are static networks. In other words, the application domain of CMAC will be limited to static mapping due to its static network structure[16, 17].

Backstepping control technique is a mighty and systematic design methodology for nonlinear systems [18 –21]. It offers a choice to accommodate the unmodelled nonlinear effects and parameter uncertainties. The backstepping design is to select recursively some appropriate functions of state variables as fictitious control inputs for lower dimension subsystems of the overall system. Each backstepping stage results in a new fictitious control design, expressed in terms of the fictitious control design from preceding design stages. The procedure terminates a feedback design for the true control input which achieves the original design objective by virtue of a final Lyapunov function, which is formed by summing the Lyapunov functions associated with each individual design stage. Thus the backstepping control approach is capable of keeping the robustness properties with respect to the uncertainties.

In this study, a mobile inverted pendulum (MIP) system is realized. The hardware of the MIP includes a chassis carrying a dc motor coupled to a gearbox for each wheels, the input/output (I/O) board connects the MIP system to a personal computer (PC), driver circuit for the motors, the necessary sensors and filter circuit needed to measure system states. Obviously, the MIP is a nonlinear system, the traditional control approaches are hard to control such a system very well because precise models are difficult to obtain. To deal with this disadvantage, an intelligent decoupled backstepping control system (IDBCS) has been proposed for MIPs.

Obviously, a MIP system is the extension of one-dimensional typical inverted pendulum. It is a fourth-order nonlinear system for cart position and pole angle control. Although there has been some research on the system analysis of such a control system, however, most of these studies are limited the pole-cart system to a second-order system whose controller is designed with respect to pole angle control only. Therefore, it is impossible to achieve a good control around the set point of cart position. In other words, the cart-subsystem is not under control if system model is unknown. It is well know that the design of a fourth-order system control will be a difficulty issue. In order to realize high-precision MIP system control, each dimension in MIP should be realized decoupling control. The usual decoupling control methods are state feedback control based on system dynamics, neural network inverse system decoupling control, fuzzy slide-mode decoupled control, linear decoupling control based on inverse system, etc [22, 23]. Although the proposed methods can realize dynamic decoupling for a multivariable and nonlinear coupled system well, their mathematical tool is too complex. With the decoupled control proposed in this paper, a lot of fourth-order system can be controlled without increasing system complexity but reducing the process of controller design. By the IDBCS proposed in this study, a fourth-order MIP system can be controlled based on second-order pole-subsystem combined cart-subsystem information. The ideas of the controller are as follows. First, decouple the whole system into two subsystems (i.e., pole-subsystem and cart-subsystem) such that each subsystem has a separate control target. Then, information from the cart-subsystem is feed-in the pole-subsystem, which generates a control force to make both subsystems move toward their targets, respectively.

The developed IDBCS system is comprised of an AORCMAC and a robust H^∞ controller. The AORCMAC is used to mimic an ideal backstepping control (IBC), and a robust H^∞ controller is designed to attenuate the effect of the residual approximation errors and external desired attenuation level. Similarity robust intelligent tracking control structure had been proposed by Peng [19]. However, the structure in [19] uses CMAC to copy an IBC. Moreover, the learning rates are constants in [19]. Here, adaptive output recurrent cerebellar model articulation controller (AORCMAC) architecture [12] is a modified version of the conventional CMAC network. In the AORCMAC control structure, a small number of receptive fields are used to convert the static CMAC into a dynamic controller. Since the proposed controller to capture the dynamics response of controlled system, the AORCMAC will achieve good control performance for a nonlinear system. In addition, all the adaptation laws of the IDBCS system are derived based on Lyapunov stability analysis, Taylor linearization technique and H^∞ control theory, so that the stability of the closed-loop system and H^∞ tracking performance can be guaranteed. Moreover, the learning rates of the AORCMAC are determined based on Lyapunov function to ensure the convergence of tracking errors.

The contribution of the paper is summarized as follows: (1) The IDBCS system is developed. The AORCMAC is used to mimic an IBC, and a robust H^∞ controller is designed to attenuate the effect of the residual approximation errors and external desired attenuation level. (2) The decoupled method provides an easy way to achieve asymptotic stability control for a fourth-order nonlinear mobile inverted pendulum system based on two second-order controllers. (3) In the AORCMAC control structure, a small number of receptive fields are used to convert the static CMAC into a dynamic controller. Therefore, The IDBCS system comprised of an AORCMAC and a robust H^∞ controller will achieve good control performance for a nonlinear system than the method in [19].

Finally, the proposed IDBCS system is applied to control a MIP system. The effectiveness of the proposed MIP system control is verified by several real-time experimental results.

2 Mobile inverted pendulum

The dynamic equation of the MIP is described as following [24]:

$\begin{matrix} \ddot{x} (t) = F (x, t) + G (x, t) u (t) + D (t) \\ \begin{matrix} = \frac{(a_{4} - a_{2}) M_{b} gl}{a_{1} a_{4} - a_{2} a_{3}} x (t) + \frac{- μ_{s} a_{22}}{a_{1} a_{4} - a_{2} a_{3}} \dot{x} (t) \\ + \frac{- a_{4} η τ_{t}}{a_{1} a_{4} - a_{2} a_{3}} u (t) + D (t) \end{matrix} \end{matrix}$ (1) where $\begin{matrix} (\begin{matrix} a_{1} & a_{2} \\ a_{3} & a_{4} \end{matrix}) \\ = (\begin{matrix} M_{b} l^{2} + I_{b} + η^{2} I_{M} & M_{b} Rl - η^{2} I_{M} \\ M_{b} Rl + M_{b} l^{2} + I_{b} & (M_{b} + M_{w}) R^{2} + M_{b} Rl + I_{W} \end{matrix}) \end{matrix}$

The parameters in (1) are defined in Table 1.

Table 1

Symble description

Symbol	Description	Unit
M _b	Mass of the body	Kg
M _w	Mass of the wheel	Kg
R	Radius of the wheel	m
g	Gravitational acceleration	m/sec²
l	Length between the axleof wheel and gravitationalcenter of the robot body	m
I _M	Rotational inertia of the motor axis	kg m²
I _b	Rotational inertia of the body	kg m²
I _w	Rotational inertia of the wheel	kg m²
η	Reduction ratio of gear
τ _t	Torque constant of the motor	N · m/A
μ _s	Viscous between the wheel axle including motor and gem	N · m/(rad/sec)

Moreover, x (t) is the state vector of the MIP’s angle, F (x, t) ∈ R is the dynamic function, G (x, t) ∈ R denotes the control gain for all x and t; u (t) ∈ R is the control input and D (t) ∈ R denotes the outer disturbance. The control object is to design a suitable control law for the system (1) so that the MIP’s angle x can track a desired reference trajectory vector x_d. In practical application, F (x, t) can not be exactly obtained in general, and the external disturbance D (t) is always unknown.

2.1 Angle and position decoupled controlfor MIPs

The MIP system shown in above can be represented by Equation (1). But for such nonlinear model is only a second-order system (for pole system only), the system dynamic representation is not in the pole-cart dynamic form (fourth-order system) exactly. In other words, the pendulum angle is the only considered state in system (1), therefore, the position of the MIP is uncontrolled by using the above proposed equation. However, the purpose of this study is to keep the pendulum at upright position when the MIP moves to a set point. Obviously, the above dynamic equation is no good enough to achieve this control purpose. In this study, an angle and position decoupled controller is used to overcome such a problem. The main idea of the decoupled controller proposed in this study can be structured as follows. First, the whole system is decoupled into two subsystems (pole-subsystem and cart-subsystem). Pole-subsystem contains error and change of error of pole angle as it inputs and cart-subsystem contains error and change of error of cart position as it inputs.

Because the main target is to stabilize pole-subsystem, it is reasonable to consider the information from cart-subsystem as secondary information and this secondary information must be reflected through a mechanism to the pole-subsystem. In other words, an intermediate value, which contains secondary information, is incorporated into primary subsystem. This modified reflects the fact that the subtask of cart-subsystem is embedded to the pole-subsystem through the intermediate value. Then, both primary and secondary targets can be controlled simultaneously. By doing this, the main control objective is to keep states of pole-subsystem move toward the equilibrium point and converge to zero degree stably and a subtask is to keep states of cart-subsystem moving toward the set position.

Then, the interpretation of the decoupled control is described in the follows. The configuration of the proposed decoupled controller is shown in Fig. 1. The modified angle signal Δx_θ is specified by a decouple machine following a position command input ${\bar{x}}_{dp}$ and position output of the MIP, x_p. Clearly, Δx_θ is an intermediate value which is used to move the MIP forward or backward to the set position. Obviously, Δx_θ will be zero when x_p is equal to ${\bar{x}}_{dp}$ . It means that the MIP moves to the set positionalready.

Fig.1

Decoupled controller.

By the above discussion, the angle value after the decouple machine is expressed as $x (t) = x (t) + Δ x_{θ} .$ (2)

In this study, $Δ x_{θ} = k ({\bar{x}}_{dp} - x_{p})$ where k is a positive constant.

2.2 Intelligent backstepping tracking control

The control law of an intelligent backstepping tracking control system is defined to take the following form: $u = u_{AORCMAC} + u_{R}$ (3) where u_AORCMAC and u_R are outputs of an AORCMAC and a H^∞ controller, respectively. Here, the AORCMAC is used to reproduce the IBC, and the compensated controller is designed to retrieve the residual approximation error.

2.2.1 Output recurrent cerebellar model articulation controller

Here, an output recurrent cerebellar model articulation controller (ORCMAC) is proposed. In general, it is composed of input space, association memory space, receptive-field space, weight memory space and output space.

For a given x = [x₁, x₂, ⋯ , x_n] ^T ∈ Rⁿ, each input state variable x_i must be quantized into discrete regions (called elements) according to given control space. The number of element, n_E, is termed as a resolution. Several elements can be accumulated as a block. ρ is the number of elements in a complete block. In this space, each block performs a receptive-field basis function. The Gaussian function is accepted here as the receptive-field basis function, which can expressed as $Φ_{ik} = e^{\frac{- (x_{i} - m_{ik})^{2}}{σ_{ik}^{2}}}, for k = 1, 2, \dots, n_{B}$ (4) where Φ_ik is the kth block of the ith input x_i with the mean m_ik and variance σ_ik and n_B is the number of blocks. An association memory space has n_A (n_A = n × n_B) constituents. In addition, the input of this block can be represented as $x_{ri} (t) = x_{i} (t) + r_{i} Y (t - T)$ (5) where r_i is the recurrent weight of the recurrent unit, Y (t - T) denotes the value of Y through delay time T. It is clear that the input of this block contains the memory terms, which store the past information of the network. This is the apparent difference between the proposed ORCMAC and the conventional CMAC.

Areas formed by blocks are called receptive-fields. The multidimensional receptive-field function is define as $B_{k} = \prod_{i = 1}^{n} Φ_{ik} (x_{ri}) = e^{\sum_{i = 1}^{n} \frac{- (x_{ri} - m_{ik})^{2}}{σ_{ik}^{2}}}, k = 1, 2, \dots, n_{R}$ (6) where B_k is associated with the kth receptive fieldand n_R is the number of receptive field. The multi-dimensional receptive-field function can be expres-sed in a vector form as Ψ (x, m, σ, r) = [B₁, ⋯ , B_k, ⋯, B_{n
_R}] ^T, where $m = [m_{1}^{T}, \dots, {m_{k}^{T}, \dots, m_{n_{R}}^{T}]}^{T}$ ∈R^{nn
_R}, $σ = {[σ_{1}^{T}, \dots σ_{k}^{T}, \dots, σ_{n_{R}}^{T}]}^{T} \in R^{{nn}_{R}}$ and $r = [r_{1}^{T}, \dots, r_{k}^{T}, \dots, r_{n_{R}}^{T}]^{T} \in R^{{nn}_{R}}$ .

Every location of the receptive-field space to aparticular adjustable value in the weight memoryspace can be expressed as w = [w₁, w₂, ⋯ , w_k, ⋯ , w_{n
_R}] ^T, where w_k denotes the connecting weight value of the output associated with the kth receptive-field.

The output of the ORCMAC is the algebraic sum of the activated weights in the weight memory, and is expressed as $Y = w^{T} Ψ (x, m, σ, r) = \sum_{k = 1}^{n_{R}} w_{k} B_{k}$ (7)

2.2.2 Intelligent backstepping tracking controller

The design of intelligent backstepping system for the uncertain nonlinear MIP system is described as follows:

Define the tracking error $e_{1} (t) = x_{d} (t) - x (t)$ (8)

Then the deductive of tracking error can be represented as ${\dot{e}}_{1} (t) = {\dot{x}}_{d} (t) - \dot{x} (t)$ (9)

The $\dot{x} (t)$ can be viewed as a virtual control in above equation. Define the following stabilizing function $α (t) = k_{1} e_{1} (t) + {\dot{x}}_{d} (t)$ (10) where k₁ is a positive constant. The first Lyapunov function is chosen as $V_{1} (t) = \frac{1}{2} e_{1}^{2} (t)$ (11)

Define $e_{2} (t) = α (t) - \dot{x} (t) = {\dot{e}}_{1} (t) + k_{1} e_{1} (t)$ (12)

If the dynamic system is known, an ideal backstepping control law can be obtained as $\begin{matrix} u_{IBC}^{*} = \frac{1}{G (x, t)} (k_{1} {\dot{e}}_{1} (t) + {\ddot{x}}_{d} (t) + e_{1} (t) \\ + k_{2} e_{2} (t) - F (x, t) - D (t)) \end{matrix}$ (13) where k₂ is a positive constant. The Lyapunov function is chosen as $V_{2} (t) = V_{1} (t) + \frac{1}{2} e_{2}^{2} (t)$ (14)

Theorem 1: The adaptive laws of IDBCS are chosen as $\dot{\hat{w}} = η_{1} e_{2} (t) G (x, t) {\hat{Ψ}}^{T}$ (15) $\dot{\hat{m}} = η_{2} e_{2} (t) G (x, t) L {\hat{w}}^{T}$ (16) $\dot{\hat{σ}} = η_{3} e_{2} (t) G (x, t) Q {\hat{w}}^{T}$ (17) $\dot{\hat{r}} = η_{4} e_{2} (t) G (x, t) S {\hat{w}}^{T}$ (18) where η₁, η₂, η₃ and η₄ are positive constants and ${L = [\frac{\partial B_{1}}{\partial m} \dots \dots \frac{\partial B_{n_{R}}}{\partial m}] |}_{m = \hat{m}} \in R^{{nn}_{R} \times n_{R}}$ , ${Q = [\frac{\partial B_{1}}{\partial σ} \dots \dots \frac{\partial B_{n_{R}}}{\partial σ}] |}_{σ = \hat{σ}} \in R^{{nn}_{R} \times n_{R}}$ , ${S = [\frac{\partial B_{1}}{\partial r} \dots \dots \frac{\partial B_{n_{R}}}{\partial r}] |}_{r = \hat{r}} \in R^{n \times n_{R}}$ .

The H^∞ controller is chosen as $u_{R} = \frac{(φ^{2} + 1)}{2 φ^{2}} e_{2} (t)$ (19) where φ is a positive constant.

Proof: Since the AORCMAC is utilized to estimate the IBC, so that u_AORCMAC can be written as follows $u_{AORCMAC} (x, m, σ, r) = w^{T} Ψ (x, m, σ, r)$ (20)

Assume there exists an optimal $u_{AORCMAC}^{*}$ to approach the ideal $u_{IBC}^{*}$ such that

$\begin{matrix} u_{IBC}^{*} = u_{AORCMAC} (x, w^{*}, m^{*}, σ^{*}, r^{*}) + ɛ \\ = w^{*^{T}} Ψ^{*} + ɛ \end{matrix}$ (21) where ɛ is a minimum approximation error, w^*, m^*, σ^*, Ψ^*, and r^* are optimal parameters of w, m, σ, Ψ, and r, respectively. However, the optimal $u_{AORCMAC}^{*}$ can not be obtained, so that the on-line estimation u_AORCMAC is used to approach the $u_{IBC}^{*}$ . From (20), the control law (3) can be rewritten as follows: $\begin{matrix} u = u_{AORCMAC} (x, \hat{w}, \hat{m}, \hat{σ}, \hat{r}) + u_{R} \\ = {\hat{w}}^{T} \hat{Ψ} + u_{R} \end{matrix}$ (22) where $\hat{w}, \hat{m}, \hat{σ}, \hat{Ψ}$ and $\hat{r}$ are some estimates of the optimal parameters w^*, m^*, σ^*, Ψ^* and r^*, respectively. Subtracting (22) from (21), an approximation error $\tilde{u}$ is defined as $\begin{matrix} \tilde{u} = u_{IBC}^{*} - u = w^{*^{T}} Ψ^{*} + ɛ - {\hat{w}}^{T} \hat{Ψ} - u_{R} \\ = {\tilde{w}}^{T} Ψ^{*} + {\hat{w}}^{T} \tilde{Ψ} + ɛ - u_{R} \end{matrix}$ (23) where $\tilde{w} = w^{*} - \hat{w}$ and $\tilde{Ψ} = Ψ^{*} - \hat{Ψ}$ . Moreover, the linearization technique is employed to transform the multidimensional receptive-field basis functions into partially linear form so that the expansion of $\tilde{Ψ}$ in Taylor series can be obtained as [19] $\begin{matrix} {\tilde{Ψ} = [\begin{matrix} {\tilde{B}}_{1} \\ ⋮ \\ {\tilde{B}}_{k} \\ ⋮ \\ {\tilde{B}}_{n_{R}} \end{matrix}] = [\begin{matrix} {(\frac{\partial B_{1}}{\partial m})}^{T} \\ ⋮ \\ {(\frac{\partial B_{k}}{\partial m})}^{T} \\ ⋮ \\ {(\frac{\partial B_{n_{R}}}{\partial m})}^{T} \end{matrix}] |}_{m = \hat{m}} \cdot (m^{*} - \hat{m}) \\ + {[\begin{matrix} {(\frac{\partial B_{1}}{\partial σ})}^{T} \\ ⋮ \\ {(\frac{\partial B_{k}}{\partial σ})}^{T} \\ ⋮ \\ {(\frac{\partial B_{n_{R}}}{\partial σ})}^{T} \end{matrix}] |}_{σ = \hat{σ}} \cdot (σ^{*} - \hat{σ}) + {[\begin{matrix} {(\frac{\partial B_{1}}{\partial r})}^{T} \\ ⋮ \\ {(\frac{\partial B_{k}}{\partial r})}^{T} \\ ⋮ \\ {(\frac{\partial B_{n_{R}}}{\partial r})}^{T} \end{matrix}] |}_{r = \hat{r}} \\ \cdot (r^{*} - \hat{r}) + O_{t} \\ \equiv L^{T} \tilde{m} + Q^{T} \tilde{σ} + S^{T} \tilde{r} + O_{t} \end{matrix}$ (24)

Where ${\tilde{B}}_{k} = B_{k}^{*} - {\hat{B}}_{k}; B_{k}^{*}$ is the optimal parameter of $B_{k}; {\hat{B}}_{k}$ is an estimate of $B_{k}^{*}; \tilde{m} = m^{*} - \hat{m}; \tilde{σ} = σ^{*} - \hat{σ}; O_{t} \in R^{n_{R}}$ is a vector of higher-order terms. Substituting (24) into (23), yields $\tilde{u} = {\tilde{w}}^{T} \hat{Ψ} + {\hat{w}}^{T} (L^{T} \tilde{m} + Q^{T} \tilde{σ} + S^{T} \tilde{r}) + ξ - u_{R}$ (25) where $ξ = \tilde{w} [L^{T} m + Q^{T} \tilde{σ} + S^{T} \tilde{r}] + w^{*} O_{t} + ɛ$ .

In order to develop a H^∞ controller, the derivative of e₂ (t) can be expressed as

$\begin{matrix} {\dot{e}}_{2} (t) & = & G (x, t) (u_{IBC}^{*} - u_{1}) - e_{1} (t) - k_{2} e_{2} (t) \\ = & G (x, t) \tilde{u} - e_{1} (t) - k_{2} e_{2} (t) \\ = & G (x, t) [\tilde{w} \hat{Ψ} + \hat{w} (L^{T} \tilde{m} + Q^{T} \tilde{σ} + S^{T} \tilde{r}) \\ + ξ - u_{R}] - e_{1} (t) - k_{2} e_{2} (t) \end{matrix}$ (26) where k₂ is a positive constant. Define the Lyapunov function as

$\begin{matrix} V_{3} (t) & = & V_{2} (t) + \frac{1}{2 η_{1}} {\tilde{w}}^{T} \tilde{w} + \frac{1}{2 η_{2}} {\tilde{m}}^{T} \tilde{m} \\ + \frac{1}{2 η_{3}} {\tilde{σ}}^{T} \tilde{σ} + \frac{1}{2 η_{4}} {\tilde{r}}^{T} \tilde{r} . \end{matrix}$ (27)

Taking the derivative of the Lyapunov function (27) and use (26), it is concluded that

$\begin{matrix} {\dot{V}}_{3} (t) = - E^{T} KE + {\tilde{w}}^{T} [e_{2} (t) G (x, t) \hat{Ψ} - \frac{1}{η_{1}} \dot{\hat{w}}] \\ + [e_{2} (t) G (x, t) {\hat{wL}}^{T} - \frac{1}{η_{2}} {\dot{\hat{m}}}^{T}] \tilde{m} \\ + [e_{2} (t) G (x, t) {\hat{wQ}}^{T} - \frac{1}{η_{3}} {\dot{\hat{σ}}}^{T}] \tilde{σ} \\ + [e_{2} (t) G (x, t) {\hat{wS}}^{T} - \frac{1}{η_{4}} {\dot{\hat{r}}}^{T}] \tilde{r} \\ + e_{2} (t) G (x, t) (ξ - u_{R}) \end{matrix}$ (28) where E = [e₁ (t) , e₂ (t)] ^T and K = diag (k₁, k₂). From (15–18), Equation (28) can be rewritten as

$\begin{matrix} {\dot{V}}_{3} (t) = - E^{T} KE + e_{2} (t) G (x, t) (ξ - u_{R}) \\ = - E^{T} KE + e_{2} (t) G (x, t) ξ - e_{2} (t) G (x, t) u_{R} \\ = - E^{T} KE + e_{2} (t) G (x, t) ξ \\ - \frac{e_{2}^{2} (t) G (x, t) (φ^{2} + 1)}{2 φ^{2}} \\ = - E^{T} KE - \frac{1}{2} G (x, t) e_{2}^{2} (t) \\ - \frac{1}{2} G (x, t) [\frac{e_{2} (t)}{φ} - φ ξ]^{2} + \frac{1}{2} G (x, t) φ^{2} ξ^{2} \\ \leq - \frac{1}{2} G (x, t) e_{2}^{2} (t) + \frac{1}{2} G (x, t) φ^{2} ξ^{2} \end{matrix}$ (29)

Assume ξ ∈ L₂ [0, T] , ∀ T ∈ [0, ∞). Integrating the above equation from t = 0 to t = T, yields

$\begin{matrix} - \frac{1}{2} \int_{0}^{\begin{matrix} T \end{matrix}} e_{2}^{2} (t) G (x, t) dt + \frac{1}{2} φ^{2} \int_{\begin{matrix} 0 \end{matrix}}^{\begin{matrix} T \end{matrix}} G (x, t) ξ^{2} (t) dt \\ \geq V_{3} (T) - V_{3} (0) \end{matrix}$ (30)

Since V₃ (T) ≥0, the above inequality implies the following inequality $\frac{1}{2} \int_{0}^{T} e_{2}^{2} (t) G (x, t) dt \leq V_{3} (0) + \frac{1}{2} φ^{2} \int_{0}^{T} G (x, t) ξ^{2} (t) dt$ (31)

Using (27), the above inequality is equivalent to the following

$\begin{matrix} \int_{0}^{T} e_{2}^{2} (t) G (x, t) dt \leq E^{T} (0) E (0) + \frac{1}{η_{1}} \tilde{w} (0) {\tilde{w}}^{T} (0) \\ + \frac{1}{η_{2}} {\tilde{m}}^{T} (0) \tilde{m} (0) + \frac{1}{η_{3}} {\tilde{σ}}^{T} (0) \tilde{σ} (0) \\ + \frac{1}{η_{4}} {\tilde{r}}^{T} (0) \tilde{r} (0) + φ^{2} \int_{0}^{T} G (x, t) ξ^{2} (t) dt \end{matrix}$ (32)

If the system starts with initial condition E (0) =0, $\tilde{w} (0) = 0$ , $\tilde{m} (0) = 0$ , $\tilde{σ} (0) = 0$ , $\tilde{r} (0) = 0$ , the H^∞ tracking performance in (32) can be rewritten as $sup_{ξ \in L_{2} [0, T]} \frac{∥ e_{2} ∥}{∥ ξ ∥} \leq φ$ (33) where ${∥ e_{2} ∥}^{2} = \int_{0}^{T} e_{2}^{2} (t) G (x, t) dt$ and ${∥ ξ_{2} ∥}^{2} = \int_{0}^{T} ξ^{2} (t) G (x, t) dt$ . The attenuation constant φ can be specified by the designer to achieve desired attenuation ratio between ∥e₂∥ and ∥ξ∥. If φ =∞, this is the case of minimum error tracking control without disturbance attenuation [25]. Then, the desired robust tracking performance in (33) can be achieved for a prescribed attenuation level φ. An IDBCS is shown in Fig. 2

Fig.2

Block diagram of IDBCS.

2.2.3 Convergence analyses

The constants η₁, η₂, η₃ and η₄ shown in of Equations (15–18) are arbitrary positive constants. Obviously, the learning laws of (15–18) call for a proper choice of the constants η₁, η₂, η₃ and η₄. For a small value, the adaptive speed is slow. On the other hand, the performances of the intelligent backstepping system may be not well if the constants are too large. Definitely, it is hard to choose suitable learning rates for the four adaptive laws in real time control by user.

To choose those constants effectively, the variable learning constants, which guarantee the convergence of the output error, are given in the following.

Lemma 1. Let η₁ be the learning-rate parameter for the AORCMAC weight and let I_w max be defined as I_w max = max_N∥ I_w (N) ∥, where ∂u/∂w = I_w (N) for discrete time N and || · || is the Euclidean norm. Then, the convergence of tracking error is guaranteed if η₁ is chosen as $0 < η_{1} < \frac{2}{(I_{w max})^{2}} = \frac{2}{n_{A}}$ (34)

Lemma 2. [12] Let η₂ and η₃ be the learning-rate parameters of the mean and variance of the Gaussian receptive-field basis function for the AORCMAC. Then the convergence of tracking error is guaranteed if η₂ and η₃ are chosen as $0 < η_{2}, η_{3} < \frac{η_{1}}{n_{B}} {(\frac{| σ_{ik} |_{min}}{2 | w_{k} |_{max}})}^{2}$ (35) where |w_k|_max = max_N|w_k (N) | and |σ_ik|_min = min_N|σ_ik (N) |; | · | is the absolute value.

Lemma 3. [12] Let η₄ be the learning-rate parameter for the AORCMAC recurrent weight and let I_r max be defined as I_r max = max_N∥ I_r (N) ∥, where ∂u/ ∂r = I_r (N). Then, the convergence of tracking error is guaranteed if η₄ is chosen as $0 < η_{4} < η_{1} {[\frac{| r_{i} |_{min}}{Y_{max}}]}^{2}$ (36) where Y_max = max_N|Y (N - 1) | and |r_i|_min = min_N|r_i (N) |; | · | is the absolute value.

3 Experiment results

3.1 Experimental system

The MIP in this study is composed of a chassis carrying a 12V dc motor with a gearbox for each wheels, the input/output (I/O) board connects the MIP system to a personal computer (PC), motor driver circuits, circuit for sensors and filter circuit used to measure pendulum states. Figure 3 shows the system hardware block diagram. Processor unit is the control center for signal processing and control algorithms. A servo control card installed in PC is the channel for feedback signals and command signals. Sensors and a filter circuit are used to acquire the signals from an inclinometer, gyro and two incremental encoders. The encoders mounted on each dc motor are used to gauge the angle and angular velocity of the wheels. The angle and angle rate of the pendulum are measured by an inclinometer and a gyro on the chassis, respectively. The noise signals combined with sensor signals are filtered by sensors and the filter circuit to actuate the MIP. The driver circuit has two H-bridge circuits. The Pulse-width Modulation (PWM) signals from PWM generator are sent to driver circuit delivering PWM power to the motors.

Fig.3

Hardware structure.

Figure 4 shows the configuration of the PC-based experimental system. A servo control card is installed in PC, which includes multi-channels of digital to analog (D/A) converter, analog to digital (A/D) converter, programmable input/output (PIO) and encoder interface circuits. The measured analog signals are converted to digital values using the A/D converter, which has 12-bit resolution. The proposed IDBCS is realized in the PC using the

Fig.4

PC-based experimental system.

“Borland C++ Builder (BCB)” language and its control interval is set at 1.5 ms. The whole system is driven using the control voltage from the PWM. The output of the DC-DC converter is a 0–12V PWM square wave (10 kHz) with a variable duty cycle.

Figure 5 presents the signal block diagram of the MIP. A gyro, inclinometer, and two encoders mounted on each dc motor measure all system states. The encoders are utilized to gauge the angle and angular velocity of the wheels. The angle and angle rate of the pendulum are measured by the inclinometer and gyro on the chassis, respectively.

Fig.5

Sensors signal block diagram of MIP.

3.2 Experimental results

The receptive-field basis functions are chosen as $Φ_{ik} = e^{- (x_{ri} - m_{ik})^{2} / σ_{ik}^{2}}$ . Moreover, the parameters of the MIP are given as follows: total weight of the MIP is about 7Kg(M_b is about 6Kg and M_w is about 1Kg), I_b = 0.0338 Kg · m², I_w = 0.0269 Kg · m², I_M = 0.00003 Kg · m², η = 24, τ_t = 0.12N · m/A, and μ_s = 0.005N · m/(rad · sec), lateral distance between wheel and the center of chassis is about 0 . 13 m and the height of the MIP from the chassis is 0.18 m. In this work, the intermediate value, Δx_θ, can be gotten as $({\bar{x}}_{dp} - x_{p}) / 2$ .

The proposed AORCMAC in this study is characterized by ρ = 4, n_E = 5, and n_B = n_R = 2 ×4. Initial conditions of the AORCMAC control system parameters are chosen as r₁ = 0.01, r₂ = 0.01, m_i1 = -3, m_i2 = -2, m_i3 = -1, m_i4 = -0.5, m_i5 = 0.5, m_i6 = 1, m_i7 = 2, m_i8 = 3 and σ_ik = 2 for all i and k. The adaptive laws of IDBCS are chosen as Equation (15–18). And its control interval is set at 1.5 ms. Moreover, according to Lemma 1, Lemma 2 and Lemma 3, the proposed variable learning-rates were selected as $η_{1} = 0.1 < \frac{2}{n_{A}} = \frac{2}{16} = 0.125$ , η₂ = η₃ = (η₁/n_B) · (1/ (|w_k|_max · (2/ |σ_ik|_min))) ², $η_{4} = η_{1} \cdot ({| r_{i} |}_{min}^{2} / Y_{max}^{2})$ , k₁ = 1, and k₂ = 1.

For comparison, several existing methods, including the adaptive output recurrent cerebellar model articulation controller (AORCMAC) in [12], the adaptive cerebellar model articulation controller (ACMAC) in [26], the Elman NN (ENN) in [27], and a robust intelligent backstepping tracking control system combined with CMAC and H^∞ control technique in [19] is used to control the MIP. The system parameters of the AORCMAC are same as those parameters of the above description. Figure 6 shows experimental results for the AORCMAC. The system parameters of the ACMAC are the same as those for the AROCMAC system except the recurrent units. Figure 7 shows the experimental results for the ACMAC. The adopted ENN had 2, 9, and 1 neurons at the input, hidden, and output layers, respectively. Figure 8 shows the experimental results obtained using the ENN. Figure 9 is the experimental results for the MIP system using IDBCS combined with CMAC and H^∞ for the MIP system. Figure 10 is the experimental results for the MIP system using the proposed method.

Fig.6

System response, AORCMAC for MIP.

Fig.7

System response, ACMAC for MIP control.

Fig.8

System response, ENN for MIP control.

Fig.9

System response, IDBCS combined with CMAC and H^∞ for the MIP system when φ = 0.1.

Fig.10

System response, IDBCS combined with AORCMAC and H^∞ for the MIP system when φ = 0.1.

The five experiments above use the same initial conditions. Then, at about 1000 sampling times, an external disturbance is added to the system by tapping the MIP. This external disturbance force given by tapping is about 1 N. Comparisons of experimental results for AORCMAC (Fig. 6), the conventional ACMAC (Fig. 7), the ENN (Fig. 8), the adopted robust intelligent backstepping tracking control with CMAC and H^∞ (Fig. 9) and, proposed IDBCS (Fig. 10), the IDBCS tracking error converges faster than that using the AORCMAC, ACMAC, ENN, and the adopted robust intelligent backstepping tracking control with CMAC and H^∞. MIP chattering is clearly reduced due to the output recurrent structure of the IDBCS. The proposed method is superior to AORCMAC, ACMAC, and ENN in capturing system dynamics and H^∞ controller attenuating the effect of the residual approximation errors. Moreover, comparisons of the experimental results in Figs. 9 and 10 indicate that tracking responses for the IDBCS (Fig. 10) converge faster than the method proposed by Peng (Fig. 9). This rapid tracking response convergence is due to capturing system dynamics and online adjustment of learning rates. Obviously, the IDBCS is better than the other four methods in capturing system dynamics and choosing a prescribed attenuation lever for MIP system.

Fig.11

Tracking response, experiment results of IDBCS when x_d = 20°, ${\bar{x}}_{dp} = 0 m$ : (a), (b) for φ = 0.1; (c), (d) for φ = 1.

The following cases, including the nonzero initial condition and the alteration in the attenuation constant, φ, are used to examine the adaptive and robust control performance of the proposed method. The experiment results are depicted in Fig. 11. Figure 11 shows the control response of the MIP system with x_d = 20°, ${\bar{x}}_{dp} = 0 m$ . The tracking responses of angle and position are plotted in Fig. 11(a), (b) for φ = 0.1; and Fig. 11(c), (d) for φ = 1.

Since the proposed AORCMAC has excellent properties including capture the dynamic response of controlled system, simple computation, fast learning and good generalization capability, and a robust H^∞ controller is designed to attenuate the effect of the residual approximation errors and external disturbances with desired attenuation level, the proposed method will achieve good control performance for the MIP system. Moreover, the backstepping control technique proposes a powerful ability to accommodate the unmodelled disturbance and uncertainty effects, Furthermore, the better tracking performance can be achieved as the attenuation constant φ is chosen smaller. The effectiveness of the proposed robust control scheme is verified.

4 Conclusions

In this study, an intelligent decoupled backstepping control system is proposed for MIPs. The IDBCS comprises an AORCMAC and a H^∞ controller. The AORCMAC is used to mimic an ideal backstepping control, and the H^∞ controller is designed to recover the residual approximation error and to achieve H^∞ tracking performance with desired attenuation level. To choose those learning rates of AORCMAC effectively, an analytical method based on Lyapunov function is utilized. In order to control the whole forth order system based on a second-order system. A decoupled structure has been presented. The main concept is to decouple the whole system into two subsystems. By proposing an intermediate value, the subtask of cart-subsystem is embedded to the pole-subsystem. Then, both primary and secondary targets can be controlled simultaneously. Finally, the proposed IDBCS system is applied to control the MIP system. The experiment results demonstrate the effectiveness of the proposed robust scheme for the MIP system.

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