ACTIVE VIBRATION CONTROL OF A FLEXIBLE MANIPULATOR USING MODEL PREDICTIVE CONTROL AND KALMAN OPTIMAL FILTERING

Transcription

1 ACTIVE VIBRATION CONTROL OF A FLEXIBLE MANIPULATOR USING MODEL PREDICTIVE CONTROL AND KALMAN OPTIMAL FILTERING Mohammed BAKHTI (a), Badr BOUOULID IDRISSI (b) Ecole Nationale Supérieure d Arts et Métiers - Meknes, Marjane II, B.P Beni Mhamed, Meknes, 50000, Morocco (a) mdbakhti@yahoo.fr (b) bbououlid@yahoo.fr Abstract : This article presents an application of the multivariable Model-based Predictive Controller associated to a Kalman filter to damp the mechanical vibrations of a flexible one-link manipulator using state variables feedback. The flexible manipulator, seen as a single input multiple outputs system is modelled using the Lagrange equations. The state space equations are expressed in order to simplify the controller and the filter implementation. The performance of the proposed control scheme is evaluated relatively to its vibration damping ability. Keywords: Flexible manipulator; Active vibration control; Discrete-time MPC; Kalman filter. 1. Introduction The active damping of flexible manipulators is subject to a large number of research papers due to its high potential for industrial applications. For small amplitude vibration on very flexible structures, active approaches lead to lightweight and high performance control systems [1]. Used as sensors or actuators, piezoelectric materials have been well-studied [1], with [2] the first to suggest this idea. Bailey and Hubbard [3] used distributed-parameter control theory and a piezoceramic actuator to actively control vibration on a cantilever beam actively. Other researchers [4,5] studied the effect of the actuators on the host structures for vibration control through modal shape analysis. A variable structure adaptive controller developed by [6] to control contact forces on a cantilever beam used only the output force as feedback, resulting in undesirable chattering. Artificial neural networks (ANN) for identification and state feedback control of flexible structures have been implemented with good preliminary results [7]. Robust control focuses on the ability to have good control performance and stability in the presence of uncertainty in the system model as well as its exogenous inputs, including disturbances and noise. The H1 controller compensates for some of these uncertainties in active vibration control [8]. Recently, [9] developed a robust rejection method using a Kalman filter to estimate the system states under persistent excitation. A large amount of research has focused on the optimization of sensors/actuators numbers and location, an example being [10]. Wang et al. [11] have very recently introduced Predictive Control for vibration suppression on a motor driven flexible beam, with very good simulation and practical results. These initial results were used also to diminish tip vibration on flexible beams [12]. The Kalman filter is generally used as an optimal state observer for systems which cannot be modelled accurately using only a deterministic model. It estimates the state variables given the presence of modelling uncertainties or the measurement noise [13 15]. The gain of the filter is calculated adaptively to minimize the variance of the estimation mean square error (MSE) [16]. The Kalman filter has also been widely used in the development of structural system identification strategies [17, 18, and 19] for both linear and nonlinear dynamical systems. The extended Kalman filter is the best known variant to deal with non linear systems that are corrupted by non- Gaussian noises [20]. This filter is based upon the principle of linearizing the state and the measurement models by using Taylor series expansions [21,22]. However, it provides first-order approximations to optimal nonlinear parameter estimation, which may include large errors. As alternatives, the second extended Kalman filter (SEKF) and extended Kalman smoother filter are proposed [23]. Compared with the widely used EKF, the SEKF provides second-order approximation of process and measurement errors for both Gaussian and non- Gaussian distributions [24]. There are many alternate ways of writing the Kalman equations or algorithms [25]. The structures may appear different, but they are mathematically equivalent. ISSN : Vol. 5 No.01 January

2 The one-link flexible manipulator, considered as a Bernoulli cantilever beam rigidly attached at one end to the shaft of an electric servo-motor, is modelled in section 2. The modelling is based on the Euler-Lagrange formulation and the proper modes of vibration contributions. The proper modes of vibration are restricted to the first one that has the most significant effect. In section 3, the formulation of the State Feedback Model-based Predictive Controller is presented. The predictions of the outputs are detailed, the cost function is expressed and the control signal is derived. Section 4 will deal with the Kalman filter implementation to minimize the effect of noises that are corrupting the dynamic and measurement equations. Finally, results of simulation are illustrated in section 5, and conclusions are given in section Modeling of the Flexible One-Link manipulator In this section, a model is developed for the case of a manipulator with one flexible link constrained to act on a horizontal plane, and which is rigidly attached at one end to the shaft of an electric servomotor. Once the equation of movement is derived, a state space formulation is adopted to accommodate the simulation and the controller/observer formulation. Figure.2.1. Configuration of the flexible one-link manipulator The total displacement of the manipulator is considered to be the sum of rigid body rotation plus the flexible motion, as follows:,.,.. (1) Figure.2.2. The flexible one- link manipulator displacement The relative displacement is truncated to the first vibration mode that has the most significant contribution to the global behaviour of the system.,. (2) 2.1. Lagrange's equations of the flexible beam The behaviour of the link relative to the system can be analyzed using beam theory. Assuming that the link is a homogeneous slender beam of constant cross-section, the equations of motion can be found using the Bernoulli-Euler theory. The flexible modes are calculated using the finite element method. The elementary matrices of mass end stiffness are given by: (3) / (4) And the analysis leads to an equation like: 0 (5) ISSN : Vol. 5 No.01 January

3 The proper modes of vibration are deduced using the eigen vectors of the matrix:, and the pulsations of vibration are given by its respective eigen values. The magnitude of the velocity of any point of the link can be obtained by:,.,.. (6) The kinetic energy of the manipulator can be written as the sum of rotational and translational components as: ,.... (7) Where ρ is the density of the beam's material, S and I respectively its cross-sectional area and moment of inertia, and is the motor fixture inertia. The potential energy is stored in the manipulator as strain energy in the flexible link, and it can be expressed in terms of the mode shapes and modal coordinates as follows: 1 2,. (8) E is the Young's modulus of the beam. The joint angle and the modal coordinate can be grouped to form a vector of generalized coordinates, defined as: T (9) Similarly, the torque T applied at the joint and the modal forces can be grouped to form a vector of generalized forces. Since there are no modal forces being applied to the system, the vector of generalized forces can be written as: 0 (10) The work done by non-conservative external forces can then be written in terms of the generalized coordinates and forces as: W Q T qtθ (11) Since the kinetic and potential energy are already expressed in terms of the vector of generalized coordinates, the Lagrange's equations can be expressed: 1 2 (12) Substituting the expressions for kinetic and potential energy and performing the required operations, one obtains the following matrix equation, which is a set of 2 ordinary differential equations, which model the dynamic behaviour of the system: 0 0. (13). : 0 Where u is the control voltage sent to the servomotor amplifier. The system matrices are given by:.. (14) , (15) The effects of actuator friction and the link structural damping can be included in the model via a viscous damping matrix given by: 0 (16) 0 2 Where is the clamped-free natural pulsation of the link, is the corresponding element of the mass matrix, and is the modal damping coefficient. ISSN : Vol. 5 No.01 January

4 This yields to the following modified dynamic model: (17) 2.3. State Space equations formulation For modelling and control purposes it is convenient to write the model of the system in state-space form, as follows: (18) Where: (19) Matrices and are defined in terms of the stiffness, mass and damping matrices, and, respectively, and the force vector. And 0 (20) (21) The outputs of the process are selected from the state variables using an appropriate output matrix C. (22) The output matrix is defined based on the measured outputs of the process, and it s going to be discussed in the simulation section Discrete and Augmented State Space Model with Embedded Integrator To be more suitable for the implementation of the Kalman filter and the MPC controller, the state space model is discretized assuming a sampling time. The discrete state space model is given by: (23) Where: (24) And: 1 44 (25) When the input to the process is the increments of the control signal instead of the control signal, the state needs to be augmented with an embedded integrator. Using Eq. (23), we may write: (26) Using the increments of both the state vector and the control signal, we have: (27) Now the input to the process modelled by Eq. (27) is, and to connect to the output, an augmented state variable vector is proposed: (28) The increment of the output vector is: (29) Rewriting Eq. (27) using the augmented state variable vector defined by Eq. (28) and using Eq. (29) leads to: (30) In Eq. (30), is the number of state variables, and is the number of outputs. The output vector is given by: (31) ISSN : Vol. 5 No.01 January

5 The matrices given below, define the augmented model, which will be used in the design of the Model Predictive Controller Model Predictive Controller design 3..1 The approach of model predictive control The model predictive controller computes the trajectory of the future manipulated variable increments to optimize the future of the plant output. Obviously, the prediction of the future output will be based on a process model. When using a state-space model of the process, the current information required to predict the future of the output is the state variable. The future errors are predicted, over a prediction horizon, as the difference between the values of the output and the values of a desired set-point trajectory. The errors are calculated using future control signal increments. Both the errors and the control signal increments will define the cost function that reflects the control objective. The standard least square formulation is the most commonly used cost function in model predictive control, and it s generally affected by introducing weight matrices and. (33) The cost formulation expresses the objective of minimizing the errors between the predicted output and the setpoint signal, and, also, reflects the consideration given to the size of when the objective function is minimized Prediction of the Output Variables The process model we are going to use for prediction is an augmented state-space model with embedded integrator. This model must show the dependence of the output on the current state variable and the current/future input increments. The future control trajectory is denoted by:, 1, 1 (34) We denote the predicted state variables as: 1, 2, (35) Where is the predicted state variable at: with given current plant information. Based on the augmented state-space model, the future state variables are calculated sequentially using the set of future control parameters: (36) From the predicted state variables, the predicted output variables are formulated in terms of current state variable information and the future control signal increments, where 1,2, (37) Eq. (36) and Eq. (37) are collected in the compact matrix form as: Φ (38) Where: is the current state variable. (32) ISSN : Vol. 5 No.01 January

6 And (39) 0 Φ Calculation of the optimal control signal Using Eq. (33) and Eq. (37), we have: Φ Φ 2 Φ Φ Φ (41) (40) We can, then, calculate the first derivative of the cost function as: 2Φ 2Φ Φ (42) And the optimal control signal vector is calculated with the necessary condition of the minimum : 0 (43) So: Φ Φ Φ (44) 3.4. Receding Horizon and Model Predictive Control Gain Matrices Every sample instant, we calculate the optimal control moves, but only the first sample of this vector, i.e., is implemented for the controlled process. When the next sample period arrives, a new measurement is considered to form the state vector 1 for calculation of the new sequence of control signal. This procedure is repeated in real time to give the receding horizon control law. So, the control signal increment is given by: (45) Now, considering the set-point vector as: (46) we will define a matrix so as: (47) Thus, the matrix is given by: (48) Finally, the control signal can be expressed using the state feedback control framework: (49) Where: and: Φ Φ Φ (50) Φ Φ Φ (51) Matrices G and H are Model Predictive Control Gain Matrices, and they are constant matrices for a timeinvariant system Desired reference trajectory generation Facing instantaneous step change in the desired set-point profile, a less aggressive set-point trajectory:, that would cause the process output to reach in a smooth manner, can be generated using an exponential trajectory: ISSN : Vol. 5 No.01 January

7 (52) Where: is the most recent measured process output, and is a tunable parameter that allows closed loop performance control The Block Diagram of Model Predictive Control Below, we give the block diagram for the Model Predictive Control using a state feedback scheme. The state is observed using a Kalman filter that will be detailed in the next section. Figure.3.1. The control scheme based on Model Predictive Control and Kalman filter [26] 4. Observer Design and Kalman Filter In this section a Kalman filter type observer is synthesized using the linear model of the flexible beam developed in section 2. The Kalman filter estimates the state-vector based upon statistical (rather than deterministic) description of the measured outputs and plant state. It s based on the state space model and on a recursive least square estimation algorithm, and it estimates the state of a dynamic and disturbed system using measurements that are disturbed as well. The noise affecting the process may arise due to modelling errors such as neglecting nonlinear or higherfrequency dynamics, and it s assumed to be a white noise. Also, the measurements are corrupted by a white noise which is assumed to be uncorrelated with the process noise. We suppose that the state equation is modified including a noise vector : (53) Where is discrete zero-mean white noise vector with covariance : 0 if (54) if The measured outputs are selected from the state variables by an appropriate observation matrix, and they are affected by a white noise vector. (55) The measurement noise caused by respective sensors has the covariance matrix defined by: 0 if (56) if The Kalman filter algorithm consists of two groups of equations [25]. First, a prediction of the state vector is made using the dynamic model supposed undisturbed: 1 (57) Then, the error covariance is recursively estimated by: (58) The estimation of the state vector is corrected by the available new measurement vector : (59) Where, the gain k, updated so as to minimize the error covariance, is given by: (60) Finally, the error covariance is updated for the next algorithm iteration by: ISSN : Vol. 5 No.01 January

8 (61) 5. Simulation and results Filtering and control simulations are conducted on the flexible beam modelled in section II. In this section, we are going to present the results obtained by the Model Predictive Controller and the results of estimations made by the Kalman filter. The Kalman filter is feeded with the noisy control signal and measurements. First, we will show the result of the estimation based on a noisy measure of the joint angle only. Then, we will supply also the fist derivative of the tip vibration to the filter. Results of both cases, obtained in open-loop tests, are going to be compared. As sensors, we use a high precision potentiometer to measure the motor or joint angular position, and an accelerometer, bounded next to the tip of the beam, that measures the total acceleration including the rigid body acceleration and the vibration acceleration :. For the Model Predictive Control, first the performances are illustrated when the controller use predictions on the angular position only. Then the results are compared to the case when both the angular position and the tip vibration are predicted. Also, are going to be discussed the performances when the predictions are made on the angular velocity, instead of the angular position, and the tip vibration. The characteristics of the beam and the DC motor, needed for the numeric simulation, are shown in Table V.1 and Table V.2. Table V.1: Beam Properties Density ρ = 2700 Kg/m 3 Length L = 1 m Young s modulus E = Pa Cross-section area S = m² The quadratic moment I = m 4 Table V.2: DC motor Properties Motor-Fixture Inertia J m = Kg m² Friction Coefficient B m = Nm/rad/s Motor Constant k m = Nm/V 5.1. State estimation using Kalman filter To asses the ability of the filter to give a good estimation of the state, based on noisy measurements, we are going to corrupt the measures with a zero-mean Gaussian white noise with a variance of The dynamic of the process is affected by the same noise signal. Also the state-vector initial condition is set to: (62) The Kalman filter algorithm starts with a zero initial value for the state-vector. Figures V.1 to V.4 shows the estimations of the state variables vector made by the Kalman filter. FigureV.1. Angular position estimation using the Kalman filter ISSN : Vol. 5 No.01 January

9 FigureV.2. Tip vibration estimation using the Kalman filter FigureV.3. Angular velocity estimation using the Kalman filter FigureV.4. Tip vibration velocity estimation using the Kalman filter The Kalman filter provides a very good estimation of the state vector when both the angular position and the tip vibration velocity are measured. Also, the noise effect on the measurements is greatly diminished. Figures V.5 and V.6 shows the noisy measurements of and that are supplied to the Kalman filter. FigureV.5. Noisy angular position used to estimate the state vector ISSN : Vol. 5 No.01 January

10 5.2. Model Predictive Controller simulation FigureV.6. Noisy tip vibration velocity used to estimate the state vector The system has one input which is the motor voltage as manipulated variable and two outputs which are the joint angle and the tip vibration. Note that tip vibration, 1. 1, where is the beam length. So the output matrices are given by Eq. (63), Eq. (64) and Eq. (65) respectively when predictions are made on the angular position only, the angular position and the tip vibration, and the angular velocity and the tip vibration (63) (64) (65) The motor is rotated and controlled to an angular set-point of using the model predictive controller. During the rotation, the control signal is adapted until the zero vibration set-point is achieved. The simulation is conducted with the prediction horizon 2000, which is equivalent to a time window of 2s, given a sample time of 1ms. The control horizon, or number of moves for the control signal is set to 2. The weighting matrix, that allows different penalties to be placed on the predicted errors, is given by: 1 0 (66) 0 10 The weight on the control signal increment is set to: 1 (67) The scalar values that serves as the primary tuning parameter, known also as move suppression coefficient are well discussed in [27]. For the case of predictions made on the angular velocity, the set-point is given by:., 4 (68) 0, 4 The set point defined by Eq. (68) makes the angular position to reach rad (90 ) over 4s. Figures V.7 to V.9 shows the simulation results when the predictions used by the controller are made on the angular position. ISSN : Vol. 5 No.01 January

11 FigureV.7. Angular position simulation using predictions on θ FigureV.8. Tip vibration simulation using predictions on θ FigureV.9. Control signal simulation using predictions on θ The simulation results show the benefits of the prediction on the tip vibration. The rigid body motion is slower, but the control signal provides a very good vibration damping. When the predictions are made on the angular velocity instead of, the results are more satisfying. Figures V.10 to V.13 shows the simulation results when the predictions used by the controller are made on the angular velocity. FigureV.10. Angular position simulation using predictions on θ ISSN : Vol. 5 No.01 January

12 FigureV.11. Angular velocity simulation using predictions on θ FigureV.12. Tip vibration simulation using predictions on θ FigureV.13. Control signal simulation using predictions on θ 6. Conclusions The control strategy based on combining the MPC controller and the Kalman filter provides a very good technique to suppress vibration on the flexible arm. The simulation results demonstrate the efficiency of the Kalman filter to suppress the effect of noise that corrupt either the process dynamic model or the outputs measurements. The advantage of measuring the tip vibration velocity was clearly proven by simulation results. The MPC controller has proven its ability to diminish considerably the vibration amplitudes and then to provide a good active damping. It s convenient to note that, by using a predictive controller in a practical context, adjustments can be made on the prediction of beam vibration taking into account the model nonlinearities. The use of prediction on the joint angle velocity instead of its position is very useful for the vibration diminishing and the practical feasibility of the control signal. REFERENCES [1] Sunar M, Rao SS. Recent advances in sensing and control of flexible structures via piezoelectric materials Technology. Appl Mech Rev 1999;52(10):1 16. [2] Plump JM, Hubbard Jr JE. Modeling of an active constrained layer damper. proc. In: 12th intl congress on acoustics, Toronto, Canada, 1986; #D4-1. [3] Bailey T, Hubbard Jr JE. Distributed piezoelectric polymer active vibration control of a cantilever beam. J Guidance Control Dynam 1985;8(5): [4] Baz A, Poh S. Performance of an active control system with piezoelectric actuators. J Vib Control 1988;126(2): ISSN : Vol. 5 No.01 January

13 [5] Chonan S, Jiang ZW, Sakuma S. Force control of a miniature gripper driven by piezoceramic bimorph cells. J Adv Automat Technol 1994;6: [6] Yim W, Singh SN. Variable structure adaptive control of a cantilever beam using piezoelectric actuator. J Vib Control 2000;6: [7] Lin CL, Lee GP, Liu VT. Identification and control of a benchmark flexible structure using piezoelectric actuators and sensors. J Vib Control 2003;9(12): [8] Wang S, Yeh H, Roschke PN. Robust control for structural systems with parametric and unstructured uncertainties. J Vib Control 2001;7: [9] Zheng LA. A robust disturbance rejection method for uncertain flexible mechanical vibrating systems under persistent excitation. J Vib Control 2004;10(3): [10] Yam LH, Yan YJ. Optimal design of number and locations of actuators in active vibration control of a space truss. Smart Mater Struct 2002;11: [11] Wang R, Hassan M, Dubay R. Predictive control of flexible structures. In: Flexible automation & intelligent manufacturing conference, Toronto, Ontario, p [12] Bravo R. Vibration control of flexible structure using smart materials. PhD dissertation. McMaster University, Canada, [13] A. Gelb, Applied Optimal Estimation, MIT Press, Cambridge, MA, [14] R. Brown, P. Hwang, Introduction to Random Signals and Applied Kalman Filtering, Wiley, New York, [15] Y. Bar-Shalom, X.R. Li, T. Kirubarajan, Estimation with Applications to Tracking and Navigation, Springer, Berlin, [16] Dah-Jing Jwo, Ta-Shun Cho. A practical note on evaluating Kalman filter performance optimality and degradation Original Research Article Applied Mathematics and Computation, Volume 193, Issue 2, 1 November 2007, Pages [17] Yun CB, Shinozuka M. Identification of nonlinear structural dynamic systems. Journal of Structural Mechanics 1980;8(2): [18] Hoshiya M, Sato E. Structural identification by extended Kalman filter. ASCE Journal of Engineering Mechanics 1984;112(12). [19] Hoshiya M, Sutoh A. Kalman filter-finite element method in identification. ASCE Journal of Engineering Mechanics 1993;119(2): [20] R. Tipireddy, H.A. Nasrellah, C.S. Manohar. A Kalman filter based strategy for linear structural system identification based on multiple static and dynamic test data. Probabilistic Engineering Mechanics, Volume 24, Issue 1, January 2009, Pages [21] W. Li, H. Leung, Y. Shou, Space-time registration of Radar and ESM using unscented Kalman Filter, IEEE Trans. Aerosp. Electron. Syst. 40 (3) (2004) [22] D. Fred, Nonlinear filters: beyond the Kalman filter, IEEE Aerosp. Electron. Syst. Mag. 20 (8) (2005) [23] Haitao Zhang, Yujiao Zhao. The performance comparison and analysis of extended Kalman filters for GPS/DR navigation. Optik - International Journal for Light and Electron Optics, Volume 122, Issue 9, May 2011, Pages [24] K. Xiong, C.W. Chan, H.Y. Zhang, Detection of satellite attitude sensor faults using the UKF, IEEE Trans. Aerosp. Electron. Syst. 43 (2) (2007) [25] Dan Simon. Optimal State Estimation. New York: John Wiley & Sons, [26] Liuping Wang. Model Predictive Control System Design and Implementation Using MATLAB. Advances in Industrial Control. Springer, [27] Shridhar R, Cooper DJ. A Tuning Strategy for Unconstrained Multivariable Model Predictive Control. Ind. Eng. Chem. Res. 1998;37, ISSN : Vol. 5 No.01 January