Continuous Time Model Predictive Control for a Magnetic Bearing System

Transcription

1 PIERS ONLINE, VOL. 3, NO. 2, Continuous Time Model Predictive Control for a Magnetic Bearing System Jianming Huang College of Automation, Chongqing University, Chongqing, China Liuping Wang and Yang Huang School of Electrical and Computer Engineering, RMIT University, Melbourne, Australia Abstract The nature of active magnetic bearings has many advantages over the conventional bearing, as its operation is energy efficient and potentially leads to cleaner and noise-free environment. However, the successful operation of an active magnetic bearing system requires a complex real-time control system, because of its unstable characteristics, as well as its nature of being a multi-input and multi-output system. This paper presents design and implementation of a continuous time model predictive control algorithm (CMPC) to an active magnetic bearing system (AMB). In this application, the plant continuous time model is identified from experimental data using prediction error method. The control performance of this algorithm is studied using an experimental AMB laboratory system. A host-target development environment of real-time digital control system with hardware in the loop (HIL) is implemented and demonstrated by controlling a nonlinear, open-loop unstable, and multivariable magnetic levitation device. DOI:.2529/PIERS INTRODUCTION This paper proposes a continuous time model predictive control scheme for an active magnetic bearing system. The AMB, an alternative to the conventional mechanical bearings, is an emerging technology with potential for application to a variety of products such as turbomolecular vacum pumps, flywheel energy storage systems and other high speed rotating machinery. Active magnetic bearing (AMB) systems have been widely applied due to their unique advantages such as: noncontact, the lubricant-free operation, high rotational speed, and the controllability of the bearing characteristics [ 3]. Firstly, the AMB rotor system always requires feedback control for stable levitation of the rotor. Therefore, the control system plays a central role in achieving high performance of the system. Secondly, the magnetic bearing system is strongly nonlinear due to electromagnetic forces, state and control constraints, and electrical characteristics of the power-supply actuator unit [4]. To apply active control to the AMB system, it is necessary to construct a high performance, realtime feedback controller to stabilize and control the suspended rotor. The control design problem is usually quite complicated. This has received much attention in recent years [5]. In this paper, we explore the model predictive control (MPC) algorithms in the application to AMB systems. Model predictive control has many advantages. It can handle constraints explicitly, can handle multivariable problems in a natural way and is model based design in the sense that it uses an explicit internal model to generate predictions of future plant behaviour [6]. However, most of the MPC techniques have focused on discrete time systems. The corresponding continuous time counterpart has received relatively little attention in the development. The continuous-time model predictive control scheme using orthonormal functions (Laguerre functions in our case) developed in [7] is computationally effective and easy to tune to achieve desirable closed-loop performance. These advantages are demonstrated in this application. Due to fast dynamics of the AMB systems, the application of real-time control is always a challenging task. With the development of today s PC/DSP-based technology, advanced control methods are more often used in applied AMB control algorithms. In our application, the hosttarget real-time environment with the combination of real time workshop (RTW) and xpc target is implemented to develop and control an experimental AMB laboratory system [8, 9]. 2. SYSTEM DESCRIPTION 2.. MBC 5 Magnetic Bearing System Figure shows the configuration of the MBC 5 magnetic bearing system. The MBC 5 magnetic bearing system includes a stainless steel shaft or rotor, four magnetic bearing actuators with eight

2 horseshoe electromagnets to levitate the rotor. Hall effect displacement sensors are placed just outside the electromagnets at the end of the rotor. The electromagnets are driven by four current amplifiers which can be actuated by four internal or external controllers. The system has four-degree of freedom with two degrees of freedom at each end of the rotor. These two degrees of freedom are translation (x and x 2 ) in the horizontal direction perpendicular to the z axis and translation (y and y 2 ) in the vertical direction. There are four on-board analogue controllers provided for the MBC 5 magnetic bearing which levitate the bearing when connected in the feedback. On the PIERS ONLINE, VOL. 3, NO. 2, Figure : Configuration of MBC 5 magnetic bearing system. front panel there are four switches for disconnecting each of the controllers so that any one or all of them can be replaced by the external controller []. The radial dynamics of the magnetic bearing system constitute a 4 input-4 output MIMO plant with motion in the x-z and y-z planes. The coupling of the motion in the two planes for the open loop plant is due to gyroscopic cross coupling and cross coupling effects from the bearings. In this paper, we assume that these effects be small. Therefore it is reasonable to control the entire system via two subsystems in different plane independently. In this paper, two axes in y-z plane are controlled by on-board controllers and two axes in x-z plane are controlled by a 2 input-2 output continuous time model predictive controller as proposed in this paper Host-target Real-time Control System Architecture The hardware and software tools used in the development, implementation and testing of the digital control algorithm are discussed in this section. The overall host-target real-time control system architecture is shown in Figure 2, which is similar to the structure discussed in []. Host Computer Software Windows MATLAB/SIMULINK Real Time Workshop Target Computer Software Real Time Operating System xpc Target Kernel (Loaded from a boot disk) xpc Target Watcom C Ethenet Ethenet PCI-624E Hardware in the Loop Figure 2: Host-target real-time control system architecture. The host-target real-time environment is implemented using MathWorks tools. MATLAB is the basic engine with add-on components called toolboxes. SIMULINK is a MATLAB add-on that provides a graphical user interface for model development and system simulation. The RTW toolbox is capable of generating real time code for SIMULINK models. The xpc Target toolbox allows access to input/output data directly from a compatible data acquisition card and generates, compiles, and creates real-time executable code for SIMULINK models without the user having to write some low-level code. The real-time code is downloaded to the target computer to control a system in a HIL environment. The Watcom C compiler is used for generating C code. The target computer is booted using an xpc boot floppy that loads the xpc target real-time kernel. Subsequently, the generated executable real-time code is downloaded to the target PC via the Ethernet. The I/O board and the PCI-624E card by National Instruments provide the interface between the target computer and the HIL system.

3 PIERS ONLINE, VOL. 3, NO. 2, Implementation Experimental setup using host-target real time control system architecture to control the MBC 5 magnetic bearing system is shown in Figure 3. SIMULINK block diagram for xpc target to control the plant is shown in Figure 4. PCI-624E and PCI-624E are I/O Target blocks for National Instruments PCI-624E I/O board with A/D channels and D/A channels. The plant outputs are sampled via analog input (A/D) channels and the control sig- Host Magnetic Bearing nals are generated via analog output (D/A) channels to control the plant. The MATLAB Function block receives the control changes and the plant outputs from which the observed states are generated. The MATLAB Function block implements the CMPC algorithm. The Figure 3: Experimental setup. MATLAB Function2 block and the MATLAB Function3 block generate the reference inputs. The signal generator block and the signal generator block generate the disturbance inputs. signal signal2 r MATLAB Function2 r2 MATLAB Function3 Memory r r2 x_hat cmpc x_hat observer control udot MATLAB Function udot out MATLAB Function Signal Generator Signal Generator 2 PCI-624E National Instr. Analog Input PCI-624E Add Add 2 PCI-624E National Instr. Analog Output PCI-624E Figure 4: Simulink block diagram for magnetic bearing control system. 3. CONTINUOUS TIME MODEL PREDICTIVE CONTROL This section provides a brief discussion of the continuous time model predictive control [7] used in this paper. The approach uses orthonormal functions to describe the trajectory of the control variable, and a multivariable state space model is used in the design. With respect to a set of real functions l i (t), i =, 2,..., that is orthogonal and complete over the interval (, ), it is known that an arbitrary function f(t) is expressed in terms of a series expansion as f(t) = ξ i l i (t) () i= where ξ i, i =, 2,..., are the coefficients.

4 PIERS ONLINE, VOL. 3, NO. 2, Specifically, we let L(t) = [l (t)l 2 (t)... l N (t)] T, where l i (t) s are Laguerre functions defined as l (t) = 2p e pt l 2 (t) = 2p( 2pt + )e pt. =. l i (t) = i (2p)i 2p[( ) (i )! ti i (i )(2p)i 2 + ( ) (i 2)! i (i )(i 2)(2p)i 3 +( ) t i ]e pt 2!(i 3)! where the parameter p is positive and often called scaling factor for Laguerre functions and L() = 2p [ ] T. Then we can describe the derivative of the control signal using Laguerre functions based series expansion as u(t) = t i 2 N ξ i l i (t) = L(t) T η (2) i= where N is the number of terms used and η = [ξ ξ 2 ξ N ] T is the vector of coefficients. Suppose that the plant to be controlled is an r input - q output multivariable system having a state space model } Ẋ m (t) = A m X m (t) + B m u(t) (3) y(t) = C m X m (t) where X m (t) is the state vector of dimension n. Let us now define an auxiliary variable Z(t) = Ẋm(t) (4) We then write an augmented state space description of the system (3) as } Ẋ(t) = AX(t) + B u(t) (5) y(t) = CX(t) where X(t) = [ [ ] Z(t) T y(t) T] T Am, A =, B = [ B C m m T ] T, C = [ I], I is the q q unit matrix. Note that the augmented state space description (5) takes the first derivative of the control signal as its input and its output remains the same. Suppose that in a multi-input and multi-output setting, a set of future setpoint r(t i + τ) = [r (t i + τ) r 2 (t i + τ) r q (t i + τ)], τ T p are available, where T p is the prediction horizon. The objective of model predictive control is to find the control law that will drive the predicted plant output y(t i + τ) as close as possible, in a least squares sense, to the future trajectory of the setpoint r(t i + τ). The cost function is taken as J = [r(t i + τ) y(t i + τ)] T Q[r(t i + τ) y(t i + τ)]dτ + u(τ) T R u(τ)dτ (6) where Q and R are symmetric matrices with Q > and R. Let the ith ( i r) set of Laguerre functions be L i (t) T = [ l i (t) li 2 (t) li N i (t) ], and define the convolution integral corresponding to the ith input I int (τ) i = τ e A(τ γ) B i L i (γ) T dγ (7) To minimize J, in a least squares sense, the derivative of the optimal control for the unconstrained problem with finite horizon prediction is derived as L () T... L 2 () T... u(t i ) =. Π {Ψ r(t i ) Ψ 2 X(t i )} (8)... L r () T

5 PIERS ONLINE, VOL. 3, NO. 2, where Π = φ(τ)qφ(τ) T dτ + R; Ψ = φ(τ) T = ( C [ I int (τ) I int (τ) 2 I int (τ) r] ) T. To integrate the u(t), we get u(t) = φ(τ)qdτ; Ψ 2 = t φ(τ)qce Aτ dτ; u(τ)dτ (9) The prediction of the future plant behaviour is built on the availability of the state variable at time t i. The state variable is estimated by the observer equation that is needed for implementing the CMPC ˆX(t) = A ˆX(t) + B u(t) + J ob (y(t) C ˆX(t)) () where ˆX(t) is the estimate of X(t) and Job is the observer gain. 4. EXPERIMENTAL RESULTS 4.. System Identification The CMPC algorithm is a model based approach. The plant model has to be obtained first for the purpose of control design. Here the plant model is estimated using experimental data. Two sets of test signals that have white noise characteristics are applied to two reference inputs. The experimental data of inputs and outputs of the plant is shown in Figure 5. The MATLAB function pem in System Identification Toolbox is used to identify the plant model in discrete-time transfer function form. The identified model is converted into the continuous-time transfer function matrix as follows: [ ] y (s) = y 2 (s) [ 43.98(s 424)(s+42.)(s 299.2) (s 45.7)(s 252.9)(s+45.4)(s+359.4) 9.8(s+4)(s+83.8)(s 92.44) (s 45.7)(s 252.9)(s+45.4)(s+359.4) 2.38(s 68.23)(s+27.2)(s+3739) (s 45.7)(s 252.9)(s+45.4)(s+359.4) 55.72(s 3762)(s+387.3)(s 37.2) (s 45.7)(s 252.9)(s+45.4)(s+359.4) ] [ ] u (s) u 2 (s) Figure 6 gives the fit of measured outputs and -step ahead predicted model outputs, 98.% and 94.67% respectively, and the prediction errors for axis x and axis x 2. ().5 Output of x (v).5 Measured and Predicted Output of x (v).5 Measured and Predicted Input of x (v).2.2 Prediction Error of x (v).6 Prediction Error of x2 (v) Figure 5: Experimental data of the plant. Figure 6: Validation result of the plant Simulation The CMPC controller with an observer based on the identified plant model is designed. We select number of terms used in the Laguerre model to capture the control signal as N = N 2 = 2. The prediction horizon is set to be 4 ms. Two tuning parameters for the closed-loop response speed are selected to be p = p 2 = 25. The observer poles are selected close to the region of 3 max(p, p 2 ). The observer is designed using the pole assignment technique by putting the observer poles at [ ]. The simulated results are shown in Figures 7.

6 PIERS ONLINE, VOL. 3, NO. 2, Reference of x (v).. Reference of x (v) Output of x (v) 5 x 3 5 x 3 Output of x (v) Input of x (v).5.5 Input of x (v) Figure 7: Staircase response of axis x. Figure 8: Staircase response of axis x 2. Disturbance of x (v) Disturbance of x (v) Output of x (v) Output of x (v) Input of x (v) Input of x (v) Figure 9: Disturbance response of axis x Figure : Disturbance response of axis x Reference of x (v) Output of x (v) Input of x (v) Figure : Staircase response of axis x.. Reference of x (v).. Output of x (v)..2. Input of x (v) Figure 2: Staircase response of axis x Real-time Control The CMPC algorithm is used to control the MBC 5 magnetic bearing system via the host-target real-time control system architecture shown in Figure 4. The design parameters are the same as the ones in Section 4.2. Sampling time is.2 ms. Experimental results are illustrated in Figures 4. Figure shows the system output of axis x (voltage equivalent of measured rotor translation in axis x direction). Figure 2 shows the system output of axis x 2 (voltage equivalent of measured rotor translation in axis x 2 direction). Each of the system outputs tracks the corresponding staircase setpoint signal. The interaction is very small for the system output of zero-setpoint axis. Figures 3 and 4 show the disturbance responses when the square disturbances were applied

7 PIERS ONLINE, VOL. 3, NO. 2, to one of the plant inputs respectively. The system outputs converge to the equilibrium points rapidly. Disturbance of x (v) Disturbance of x (v) Output of x (v) Output of x (v) Input of x (v) Input of x (v) Figure 3: Disturbance response of axis x. Figure 4: Disturbance response of axis x CONCLUSIONS This paper has designed and implemented a continuous time model predictive control system on an active magnetic bearing system, which is unstable and in the framework of a multivariable system. The predictive control was designed using Laguerre functions to describe the future control trajectory. As a result, the closed-loop performance of the predictive control system can be tuned through the number of terms of the Laguerre functions in conjunction with the scaling factor. In the implementation of predictive control system, a host-target environment is used where the embedded MATLAB function blocks linking with the SIMULINK models provide a successful development platform for this application. REFERENCES. Schweitzer, G., G. H. Bleuler, and A. Traxler, Active Magnetic Bearings: Basic, Properties and Applications of Active Magnetic Bearings, VDF Hochschulverlag, Mohd-Mokhtar, R., L. P. Wang, L. J. Qin, and T. Barry, Continuous time system identification of magnetic bearing systems using frequency response data, in Proceedings of 5th Asian Control Conference, Melbourne, Australia, , July Srinivasan, S. and Y. M. Cho, Modeling and system identification of active magnetic bearing system, in Proceedings of the 4th IEEE conference on Control Application, New York, USA, , September Ahn, H. J., S. W. Lee, S. B. Lee, and D. C. Han, Frequency domain control-relevant identification of MIMO AMB rigid rotor, Automatica, Vol. 39, No. 2, Grega, W. and A. Pilat, Comparison of linear control methods for an AMB system, International Journal of Applied Mathematics and Computer Science, Vol. 5, No. 2, Maciejowski, J. M., Predictive Control with Constraints, Pearson education limited, Wang, L. P., Continuous time model predictive control using orthonormal functions, International Journal of Control, Vol. 74, No. 6, Real Time Workshop User s Guide, MathWorks, Inc., MA, xpc Target User s Guide, MathWorks, Inc., MA, MBC 5 Magnetic Bearing System Operating Instructions, Magnetic Moments, LLC, 24.. Shiakolas, P. S. and D. Piyabongkarn, Development of a real-time digital control system with a hardware-in-the-loop magnetic levitation device for reinforcement of controls education, IEEE Transactions on Education, Vol. 46, No.,