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Controlling Inverted Pendulum Using Performance-Oriented PDC Method

Kamran Vafaee*, Behdad Geranmehr

School of Mechanical Engineering, Iran University of Science and Technology (IUST), Tehran, IRAN

Abstract: In this paper a performance-oriented Parallel Distributed Compensation (PDC) controller for Stabilizing the Linear Single Inverted Pendulum system is presented as a classical challenging benchmark problem in control engineering. The main idea of the original PDC method is to partition the dynamics of a nonlinear system into a number of linear subsystems, design a number of state feedback gains for each linear subsystem, and finally generate the overall state feedback gain by fuzzy blending of such gains. In Performance-Oriented PDC algorithm the state feedback gains are not considered constant through the linear subsystems, rather, based on some prescribed performance criteria, several feedback gains are associated to every subsystem, and the final gain for every subsystem is obtained by fuzzy blending of such gains. The model-based design methodology for mechatronic systems is a key factor for innovative and operative excellence in the design process. It was shown through this method of designing and simulation studies that application of the Performance-Oriented PDC method to such a challenging problem is robust against the uncertainties, while the control effort is still kept at an acceptable level.

Keywords: Fuzzy Takagi–Sugeno model, Parallel Distributed Compensation, model-based mechatronic design, Inverted Pendulum

Received November 20, 2013; Revised March 15, 2014; Accepted March 18, 2014

Citation: Kamran Vafaee, Behdad Geranmehr, Controlling Inverted Pendulum Using Performance-Oriented PDC Method. Journal of Automation and Control, 2 (2). 39-44. January 2014. doi:10.12691/automation-2-2-1

1. Introduction

Fuzzy logic is known as a powerful and flexible tool for modeling and control of nonlinear systems with uncertainties, especially in systems that are structurally difficult to model due to their inherent natural nonlinearities and other modeling complexities [21].

Model based fuzzy control, with the possibility of guaranteeing the closed loop stability, is an attractive method for control of nonlinear systems. In recent years, many studies are devoted to the stability analysis of continuous time or discrete time model based fuzzy control systems [1,2,3,4]. Among such methods, the method of Takagi–Sugeno [1] has found many applications in modelling of complex nonlinear systems [5,6]. Since T–S fuzzy model is a universal approximator, it can be used to approximate any complex nonlinear system, and can also produce any complex control signal. The concept of sector nonlinearity [7] provided means for exact approximation of nonlinear systems by fuzzy blending of a few locally linear subsystems.

Parallel Distributed Compensation (PDC) is a model-based T–S fuzzy controller whose rule antecedents are equivalent to those of the fuzzy model of the plant [3]. It is a generalization of the state feedback controller to the case of nonlinear systems, using the Takagi–Sugeno fuzzy model [2]. This method is based on partitioning of a nonlinear system dynamics into a number of linear subsystems, for which state feedback gains are designed and blended in a fuzzy sense. Using this technique, the control design procedure will be conceptually simple and natural. The unique advantage of PDC technique is that many of conventional linear control design methods based on both classical and modern theory can be employed in designing the nonlinear T–S fuzzy controllers [22]. One important advantage of using such a method for control design is that the closed-loop stability analysis, using the Lyapunov method, becomes easier to apply. Various stability conditions are proposed for such systems, [3,8], where the existence of a common solution to a set of Lyapunov equations is shown to be sufficient for guaranteeing the closed-loop stability. Some relaxed conditions are also proposed in [9,10,11,12].

Takagi–Sugeno model and parallel distributed compensation are applied to many applications successfully [13,15]. Linear matrix inequality (LMI) technique offers a numerically tractable way for designing a PDC controller, with various design objectives such as stability [2,10,11,16], control [17], H2 control [18] and pole-placement technique [19,20] and others [3].

A Linear Single Inverted Pendulum is comprised of a vertical pendulum pivoted in the end, over a cart moving linearly. The cart can be forced to translate linearly using an electrical motor. The objective is to retain the pendulum around its upper unstable equilibrium position. Despite the simplicity of the structure, some characteristics such as strong nonlinearity, inherent instability, and the under-actuation of the system, made it a good motivation for researchers to adopt it as a benchmark problem for verification of the effectiveness of new control strategies. Various recent works are reported for control and stabilization of such systems which have been used as benchmarks for control of linear unstable systems, and highly nonlinear dynamical systems [24,25].

This paper, proposes the application of an extension to the conventional PDC method - Performance-Oriented parallel distributed compensation – (proposed in [23]) to the challenging control engineering problem, Linear Single Inverted Pendulum. Unlike the method in [23], it is not used the pole placement method to obtain stabilizing feedback gains, instead LQR control scheme is applied.

This research is organized as follows: the mechatronic design, the Takagi–Sugeno fuzzy model and the PDC design method are shortly reviewed in Section 2 and 3. In section 4 the Performance-Oriented parallel distributed compensation method is presented. Section 5 describes the mathematical modeling of a Linear Single Inverted Pendulum system, which has been manufactured. In section 6, the controller designing procedure is followed. And finally Obtained results from simulating the controlled system, are presented in section 7.

2. Mechatronic Design

Mechatronics can be considered to be an integrative methodology utilizing the technologies of mechanical engineering, electrical engineering, control engineering and computer engineering in order to provide enhanced products, processes and systems [28,29].

Figure 1. a) Model based design and b) Mechatronic design process

A model-based design methodology for mechatronic systems is a key factor for innovative and operative excellence in the design process. Here we present an approach for using hierarchical models in the design process of a mechatronic system, called Linear Single Inverted Pendulum system which is a multi domain system. Demanding mechatronic solutions include an optimized result spanning various domains. The utilization and proper combination of solution principles from different mechatronic domains allow an extended variety and quality of principal solutions. Using traditional design methods this often leads to several iteration steps concerning the whole design process. In the case of a mechatronic system these iterations are especially time consuming as they include intense communication across domain boundaries.

Figure 2. Hierarchical models in the conceptual design

In this study, CATIA is used for modeling and Assembly, ANSYS is used for rigid body dynamics, finite element vibration and strength analyses, and The integration of the design process is achieved with a MATLAB/Simscape (SimMechanics, SimElectronics and SimDriveline) for mechatronic modeling of cart-pole and control simulation (Figure 2 and Figure 3).

Figure 3. Conceptual design

3. Fuzzy-PDC Control

The history of the PDC method begins with the work of Sugeno and Kang [3]; the name PDC; however, was first used by Wang et al.[4]. The PDC design methodology was improved and the associated stability issues were studied by Tanaka and Sugeno [5]. Wang et al. [4] verified the effectiveness of the PDC by simulating the stabilization of a moving cart inverted pendulum, where the objective of the control was to stabilize the pendulum, without any specific control on the position of the cart.

The fuzzy-parallel distributed compensation (fuzzy-PDC) design approach consists of three steps [2]. The first step is to use the main feature of the Takagi-Sugeno fuzzy model to express the local dynamics of each fuzzy implication rule by a linear model. The overall fuzzy model of the system is achieved by fuzzy "blending" of the linear models. The second step is implementation of the so-called parallel distributed compensation (PDC) scheme on the fuzzy models. And the last step is to check, weather the stability conditions are satisfied by using the linear matrix inequality (LMI) analysis.

Suppose the original nonlinear system satisfies the sector nonlinearity condition [3] i.e., the values of nonlinear terms in the state-space equation remain within a sector of hyper-planes passing through the origin. The fuzzy rule associated with the i th linear subsystem, can then be defined as:

i th rule: If is , and is ; . . ., and is then

[picture] | (1)

where, is the state vector, is the input vector, ,,, Here, are some nonlinear functions of the state variables obtained from the original nonlinear equation, and are the degree of membership of in a fuzzy set . Whenever there is no ambiguity, the time argument in is dropped. The overall output, using the fuzzy blend of the linear subsystems, will then be as follows:

[picture] | (2)

And

[picture] | (3)

Where

[picture] | (4)

And

[picture] | (5)

It is also true, for all t, that

[picture] | (6)

Using the Takagi–Sugeno fuzzy model, a fuzzy combination of the stabilizing state feedback gains, , i=1,2,…,r, associated with every linear subsystem is used as the overall state feedback controller. The general structure of the controller is then as

If is , and is ; . . ., and is then

[picture] | (7)

The Takagi–Sugeno model and the Parallel Distributed Compensation have the same number of fuzzy rules and use the same membership functions.

4. Performance-Oriented PDC

In the proposed modified PDC, unlike the conventional PDC, state feedback gains, associated with every linear subsystem, are not assumed fixed. Instead, based on some pre- specified performance criteria, several feedback gains are designed and used for every subsystem. The overall gain associated with each of the subsystems, is then determined by a fuzzy blending of such gains, so that a better closed-loop performance can be achieved [23]. The dynamics of the resulting closed-loop control system can be analyzed as follows. Consider the following Takagi–Sugeno model of the plant:

[picture] | (8)

The following structure is proposed for the fuzzy controller rules:

ith rule: If is , and is ; . . ., and is , J(t) is ; . .., and J(t) is then:

[picture] | (9)

where is the number of gain coefficients in the i th subsystem, is the relevant membership degree for , is the n th state feedback gain associated with the i th subsystem, and is the n th membership function for J(t), defined in the i th rule. Here, J(t) is a term depicting a selected performance index; for instance, if one wants to limit the magnitude of the control signal, u(t), then. For such a selection, the resulting block diagram of the fuzzy blended parallel distributed controller is as shown in Fig 3, where the overall control input generated by the PDC controller is in the form of where, , the overall feedback gain for the i th subsystem, is in the form of

[picture] | (10)

[picture] | (11)

Lemma 1. The fuzzy control system (6), with the control strategy (8), is globally, Asymptotically, stable, if there exists a common positive definite matrix, P, such that

[picture] | (12)

[picture] | (13)

where i < j, , and. The proof is presented step by step in [23].

Figure 4. Block diagram of input-bounded acontroller

5. Single Inverted Pendulum Modeling

A Linear Single Inverted Pendulum is comprised of a vertical pendulum pivoted in the end, over a cart moving linearly. The cart can be forced to translate linearly using an electrical motor. The objective is to retain the pendulum around its upper unstable equilibrium position. Despite the simplicity of the structure, some characteristics such as strong nonlinearity, inherent instability, and the under-actuation of the system, made it a good motivation for researchers to adopt it as a benchmark problem for verification of the effectiveness of new control strategies. Similar behavior of a Linear Single Inverted Pendulum to a number of well-known commercially available systems, e.g., Segway self-balancing human transporter, further motivates such studies.

In order to derive the mathematical model of the Linear Single Inverted Pendulum, some simplifying assumptions are made, e.g., all frictions and the gearbox backlash are neglected. The derived equations are as follows:

Figure 5. Model for a Linear Single Inverted Pendulum

[picture] | (14)

[picture] | (15)

6. Controller Design

In order to overcome the nonlinearities, and the, usually unknown, friction and mechanical backlash of the gearbox conventional heuristic fuzzy control methods can be applied. However, in this research, a Performance-Oriented Parallel Distributed Compensation (PDC) scheme is employed. One of the most important characteristics of the PDC method is its model-based nature, which provides means for a closed-loop stability analysis. Unlike the traditional fuzzy approach in which the control membership functions are determined through a tedious trial and error procedure, the PDC membership functions are derived by a systematic approach using the mathematical model of the system.

In order for the sector nonlinearity condition satisfied, the nonlinear expressions, , and for , are expressed as

[picture] | (16)

[picture] | (17)

[picture] | (18)

[picture] | (19)

where,,,,,,,. Hence, the membership functions for z are obtained as

[picture] | (20)

[picture] | (21)

[picture] | (22)

[picture] | (23)

[picture] | (24)

[picture] | (25)

[picture] | (26)

[picture] | (27)

Now, the following stabilizing feedback gains are chosen using the LQR method, so that and produce large magnitude inputs for subsystems 1 and 2, respectively, andand induce low magnitude inputs for those subsystems, respectively.

7. Experimental Results

In this section, the performance of the proposed approach is evaluated for the Linear Single Inverted Pendulum problems. In this experiment, simulation results of proposed performance-oriented PDC controller are given for different initial conditions.

Figure 6. performance- oriented PDC controller for a Linear Single Inverted Pendulum

Figure 7. Error comparison of between PDC controller and the Performance-Oriented Controller for Inverted Pendulum system, a) with, b) with and , c) with and

8. Conclusions

In this paper, the Performance-oriented PDC control scheme has been applied to the stabilization control of Linear Single Inverted Pendulum System. The Performance-oriented PDC control scheme realized stabilization of the Inverted Pendulum System with good performance. In comparison with PDC method, it was shown that the Performance-Oriented method is more acceptable. Also It was shown through simulation studies that application of the Performance-oriented PDC method to such a challenging problem is robust against the uncertainties, while the control effort is still kept at an acceptable level.

The framework used in this paper can also be applied to generate nonlinear controllers for other uncertain systems. This plant is designed through a mechatronic process. This way of model-base design leaded the plant to fulfill the control requirements as well as other requirements.

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