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1560 JOURNAL OF COMPUTERS, VOL. 8, NO. 6, JUNE 2013 MPC Controller Performance Evaluation and Tuning of Single Inverted Pendulum Device Chao Cheng Department of Automation, Beijing University of Chemical Technology, Beijing, China Email: beryle117@163.com Zhong Zhao 1 , Haixia Li Department of Automation, Beijing University of Chemical Technology, Beijing, China Email: zhaozhong@mail.buct.edu.cn Abstract—Inverted pendulum is a non-linear, multivariable and unstable device, a model predictive control (MPC) performance evaluation and tuning method for inverted pendulum device is proposed. MPC was designed to control the inverted pendulum device, and the minimum variance covariance constrained control (MVC 3 ) was applied to evaluate the performance of the MPC controller and tune its parameters. The application results to a single inverted pendulum device have verified the feasibility and effectiveness of the proposed method. Index Terms—Inverted pendulum, Model Predict Control, Minimum Variance Covariance Constrained Control, Performance evaluation, Controller-tuning I. INTRODUCTION Inverted pendulum is a non-linear, strongly coupled, multivariable and unstable system. Because it can effectively reflect a lot of key control problems, such as the stabilization, robustness, tracking performance, many control theories and control methods can be verified with the inverted pendulum experiment. Google Technology LTD [1] designed its LQR controller. D. Chatterjee et al. [2] described the swing-up and stabilization with a restricted cart track length and restricted control force using generalized energy control methods. M. Bugeja [3] presented a swing-up and stabilizing controller on inverted pendulum non-linear model. S.Y. Zhang [4] and Y. Fan et al. [5] designed the fuzzy controllers for inverted pendulum. L.X. Deng [6] designed a controller based on back stepping for inverted pendulum. Model Predictive Controllers (MPC) was proposed by J. Richalet et al. in 1978[7]. It is a model-based optimal control strategy [8]. Its ability to incorporate meaningful limits on manipulative as well as control variables has allowed the industry to move away from traditional regulation-type control and focus on the economics of operating point selection [9]. Model predictive control has been widely applied to process control [10]. On the 1, Corresponding author, zhaozhong@mail.buct.edu.cn; other hand, it is noted that less effort has been made on the performance monitoring of MPC applications, while the performance monitoring of conventional controllers has been well studied such as in Harris (1989) [11], Harris, Boudreau, and Macgregor (1996) [12], Huang, Shah, and Kwok (1997) [13], Huang and Shah (1999) [14], Jelali (2005) [15], Srinivasan, Rengaswamy, and Miller (2005) [16] [17], Xu, Lee, and Huang (2006) [18], Salsbury (2007) [19] and Bauer and Craig (2008) [20]. The Minimum Variance Covariance Constrained Control (MVC 3 ) principle was proposed by R.E. Skelton et al. [21] as the solution of linear feedback control problem. For multivariable systems, D.J. Chmielewski* et al. [22] solved it with LQR method. In this work, the model predictive control (MPC) performance evaluation and tuning system has been developed by extended MVC 3 and applied to a single inverted pendulum device. The application results have verified the feasibility and effectiveness of the developed system. II. THE MATHEMATIC MODEL OF LINEAR SINGLE INVERTED PENDULUM The linear single inverted pendulum can be described as a system composed of a cart and a homogeneous rod without air resistance and all kinds of frictions, as is illustrated in Fig. 1. Figure 1. Linear single inverted pendulum model Where, M , m , x , F , l ,θ ,denote cart weight, rod weight, cart level displacement, force on cart, the length from the © 2013 ACADEMY PUBLISHER doi:10.4304/jcp.8.6.1560-1570
JOURNAL OF COMPUTERS, VOL. 8, NO. 6, JUNE 2013 1561 axis of the rod angle to the center of rod mass, the angle of the rod from the vertical upward direction respectively. A. State Space Model of Linear Single Inverted Pendulum The method of this work is based on linear constant state space model as: x = Ax + Bu . (1) y = Cx + Du The mathematical model of single inverted pendulum can be obtained by mechanism analysis [1], shown in (2), where, u , is cart angular velocity. x andθ , are the same as shown in Fig 1. ⎛x ⎞ ⎛0 1 0 0⎞⎛x ⎞ ⎛0⎞ ⎜ x ⎟ ⎜ 0 0 0 0 ⎟⎜ x ⎟ ⎜ 1 ⎟ ⎜ ⎟ = ⎜ ⎟⎜ ⎟+ ⎜ ⎟u ⎜ θ ⎟ ⎜0 0 0 1⎟⎜θ ⎟ ⎜0⎟ ⎜ θ ⎟ ⎜ ⎟ 0 0 29.4 0 ⎜θ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝3⎠ ⎛x ⎞ ⎜ x 1 0 0 0 x ⎟ ⎛ ⎞ ⎛ ⎞ ⎛0⎞ y = ⎜ ⎟ ⎜ = + u θ ⎟ ⎜ 0 0 1 0 ⎟⎜θ ⎟ ⎜ 0 ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎜ θ ⎟ ⎝ ⎠ B. Controllability & Observability Analysis of Single Inverted Pendulum The controllability and observability of a system is prerequisite for analysis and controller design. Here, n− Uc = B AB A 1 B and Controllability matrix ( ) n observability matrix Uo ( C CA CA −1 ) T (2) = were obtained and then rank criterion was employed to analysis its controllability and observability. It has been proved that the system as (2) has both controllability and observability. C. Stability Analysis of Single Inverted Pendulum The extended MVC 3 performance evolution and MPC tuning system is based on the stabilization system. Hence, Stability analysis is necessary. The poles of the inverted pendulum as (2) are ( 5.4222 − 5.4222 0 0) , where positive real root appears. It shows that the system as (2) is instable and this requires a stabilizer before designing a MPC controller for the generalized controlled system [23]. III. MPC PERFORMANCE EVOLUTION AND TUNING METHODE BASED ON EXTENDED MVC 3 A. Introductions of LMI and Lemmas • About LMI: Assume a linear matrix inequality (LMI) can be stated as: F( x) = F0 + x1F1+⋅⋅⋅+ xmFm < 0 .Where, variable x constitutes a convex set, LMI can be solved using the method of convex optimization problem [24]. 1) Feasible solution of LMI: If there exists x makes, F( x ) < 0 , established, then the LMI is feasible [24].This can be expressed using the following formulation: min. t . s.. tF( x) < tI 2) Minimization problem of LMI: The problem can be stated as a optimality problem that minimize the largest eigenvalue, λ , of the matrix Gx ( ) under inequality constraint H( x ) < 0 [24]: st .. Another expression is as: T min λ G < λI . < 0 ( x) H ( x) T min c x , st .. F < 0 ( x) where, c x is object function. • Lemmas: Consider a linear time-invariant state-space system: x( k + 1) = Ax( k) + Bu( k) + Ew( k) , (3) y( k) = Cx( k) where, x (k) , u ( k ) , y( k ) are state variable, manipulative variable and control output variable respectively, A , B , C and E are process model matrix, w ( k) denotes stationary, Gaussian noise with zero mean and covariance as ∑ w . State feedback controller can be expressed as: u( k) = Kx ( k) . (4) Then, the closed-loop system can be written as: x( k + 1) = ( A+ BK) x( k) + Ew ( k) . (5) Lemma1: LMI of MVC 3 [22]: For system (5), ∃ stabilizing K and ∑x ≥ 0 s.t. ∑ , = 1... p yi i T T x w x T ∑ y = ϕ i iC∑ xC ϕi 2 ∑ y < y , 1 i i i = … p ( A+ BK) ∑ ( A+ BK) + E∑ E = ∑ If and only if ∃ W, X > 0and Yi , = 1... ps.t. ⎡ T X − E∑ E AX BW⎤ w + ⎢ ⎥> 0 T ⎢⎣( AX + BW ) X ⎥⎦ ⎡ Yi ϕiCX⎤ ⎢ ( ) T 0 T ⎥ > ⎢⎣ CX ϕi X ⎥⎦ © 2013 ACADEMY PUBLISHER
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Journal of Computers ISSN 1796-203X
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Corn Moisture Measurement using a C
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Call for Papers and Special Issues
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(Contents Continued from Back Cover
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