Model Predictive Control Part 11 docx

Model Predictive Trajectory Control for High-Speed Rack Feeders 193 Using this simple discretisation method, the computational effort for the MPC-algorithm can be kept acceptable. By the way, no significant improvement could be obtained for the given system with the Heun discretisation method because of the small sampling time t s = 3 ms. Only in the case of large sampling times, e.g. t s > 20 ms, the increased computational effort caused by a sophisticated time discretisation method is advantageous. Then, the smaller dis- cretisation error allows for less time integration steps for a specified prediction horizon, i.e. a smaller number M. As a result, the smaller number of time steps can overcompensate the larger effort necessary for a single time step. The ideal input u d (t) can be obtained in continous time as function of the output variable y K (t) = c T y x y (t) =  1 1 2 κ 2 ( 3 − κ ) 0 0  x y (t) , (43) and a certain number of its time derivatives. For this purpose the corresponding transfer function of the system under consideration is employed Y K ( s ) U d ( s ) = c T y  sI − A y  −1 b y =  b 0 + b 1 · s + b 2 · s 2  N ( s ) . (44) Obviously, the numerator of the control transfer function contains a second degree polynomial in s, leading to two transfer zeros. This shows that the considered output y K (t) represents a non-flat output variable that makes computing of the feedforward term more difficult. A pos- sible way for calculating the desired input variable is given by a modification of the numerator of the control transfer function by introducing a polynomial ansatz for the feedforward action according to U d ( s ) =  k V0 + k V1 · s + . . . + k V4 · s 4  Y Kd ( s ) . (45) For its realisation the desired trajectory y Kd (t) as well as the first four time derivatives are available from a trajectory planning module. The feedforward gains can be computed from a comparison of the corresponding coefficients in the numerator as well as the denominator polynomials of Y K ( s ) Y Kd ( s ) =  b 0 + . . . + b 2 · s 2  k V0 + . . . + k V4 · s 4  N ( s ) = b V0  k Vj  + b V1  k Vj  · s + . . . + b V6  k Vj  · s 6 a 0 + a 1 · s + . . . + s 4 (46) according to a i = b Vi  k Vj  , i = 0, . . . , n = 4 . (47) This leads to parameter-dependent feedforward gains k Vj = k Vj (κ). It is obvious that due the higher numerator degree in the modified control transfer function a remaining dynamics must be accepted. Lastly, the desired input variable in the time domain is represented by u d (t) = u d  ˙ y Kd (t), ¨ y Kd (t), y Kd (t), y (4) Kd (t), κ  . (48) To obtain the desired system states as function of the output trajectory the output equation 0 5 10 15 0 0.2 0.4 0.6 0.8 t in s y K in m 0 5 10 15 −2 −1 0 1 2 t in s y Kpd in m/s 0 5 10 15 0 0.2 0.4 0.6 0.8 t in s x K in m 0 5 10 15 −1.5 −1 −0.5 0 0.5 1 1.5 t in s x Kpd in m/s y Kd y K x Kd x K Fig. 4. Desired trajectories for the cage motion: desired and actual position in horizontal direction (upper left corner), desired and actual position in vertical direction (upper right corner), actual velocity in horizontal direction (lower left corner) and actual velocity in vertical direction (lower right corner). and its first three time derivatives are considered. Including the equations of motion (12) yields the following set of equations y Kd (t) = y S (t) + 1 2 κ 2 ( 3 − κ ) · v 1 (t), (49) ˙ y Kd (t) = ˙ y S (t) + 1 2 κ 2 ( 3 − κ ) · ˙ v 1 (t), (50) ¨ y Kd (t) = ¨ y S (t) + 1 2 κ 2 ( 3 − κ ) · ¨ v 1 (t) = ¨ y K ( v 1 (t), ˙ y S (t), ˙ v 1 (t), u d (t), κ ) , (51) y Kd (t) = y K ( v 1 (t), ˙ y S (t), ˙ v 1 (t), u d (t), ˙ u d (t), κ ) . (52) Solving equation (49) to (52) for the system states results in the desired state vector x d (t) =     y Sd ( y Kd (t), ˙ y Kd (t), ¨ y Kd (t), y Kd (t), u d (t), ˙ u d (t), κ ) v 1d ( ˙ y Kd (t), ¨ y Kd (t), y Kd (t), u d (t), ˙ u d (t), κ ) ˙ y Sd ( ˙ y Kd (t), ¨ y Kd (t), y Kd (t), u d (t), ˙ u d (t), κ ) ˙ v 1d ( ˙ y Kd (t), ¨ y Kd (t), y Kd (t), u d (t), ˙ u d (t), κ )     . (53) This equation still contains the inverse dynamics u d (t) and its time derivative ˙ u d . Substituting u d for equation (48) and ˙ u d (t) for the time derivative of (48), which can be calculated analyti- Model Predictive Control194 cally, finally leads to x d (t) =         y Sd  y Kd (t), ˙ y Kd (t), ¨ y Kd (t), y Kd (t), y (4) Kd (t), y (5) Kd (t), κ  v 1d  y Kd (t), ˙ y Kd (t), ¨ y Kd (t), y Kd (t), y (4) Kd (t), y (5) Kd (t), κ  ˙ y Sd  y Kd (t), ˙ y Kd (t), ¨ y Kd (t), y Kd (t), y (4) Kd (t), y (5) Kd (t), κ  ˙ v 1d  y Kd (t), ˙ y Kd (t), ¨ y Kd (t), y Kd (t), y (4) Kd (t), y (5) Kd (t), κ          . (54) 0 5 10 15 −4 −2 0 2 4 6 8 x 10 −3 t in s e y in m Fig. 5. Tracking error e y ( t ) for the cage motion in horizontal direction. 0 5 10 15 −4 −3 −2 −1 0 1 2 3 4 5 x 10 −3 t in s e x in m Fig. 6. Tracking error e x ( t ) for the cage motion in vertical direction. 5. Experimental validation on the test rig The benefits and the efficiency of the proposed control measures shall be pointed out by exper- imental results obtained from the test set-up available at the Chair of Mechatronics, University of Rostock. For this purpose, a synchronous four times continuously differentiable desired trajectory is considered for the position of the cage in both x- and y-direction. The desired trajectory is given by polynomial functions that comply with specified kinematic constraints, which is achieved by taking advantage of time scaling techniques. The desired trajectory shown in Figure 4 comprises a sequence of reciprocating motions with maximum velocities of 2 m/s in horizontal direction and 1.5 m/s in vertical direction. The resulting tracking errors e y ( t ) = y Kd ( t ) − y K ( t ) (55) and e x ( t ) = x Kd ( t ) − x K ( t ) (56) are depicted in Figure 5 and Figure 6. As can be seen, the maximum position error in y- direction during the movements is about 6 mm and the steady-state position error is smaller than 0.2 mm, whereas the maximum position error in x -direction is approx. 4 mm. Figure 7 0 5 10 15 −0.015 −0.01 −0.005 0 0.005 0.01 0.015 t in s v 1 in m v 1d v 1 Fig. 7. Comparison of the desired values v 1d ( t ) and the actual values v 1 ( t ) for the bending deflection. shows the comparison of the bending deflection measured by strain gauges attached to the flexible beam with desired values. During the acceleration as well as the deceleration inter- vals, physically unavoidable bending deflections could be noticed. The achieved benefit is given by the fact the remaining oscillatons are negligible when the rack feeder arrives at its target position. This underlines both the high model accuracy and the quality of the active damping of the first bending mode. Figure 8 depicts the disturbance rejection properties due to an external excitation by hand. At the beginning, the control structure is deactivated, and the excited bending oscillations decay only due to the very weak material damping. After approx. 2.8 seconds, the control structure is activated and, hence, the first bending mode is actively damped. The remaining oscillations are characterised by higher bending modes that decay with material damping. In future work, the number of Ritz ansatz functions shall be Model Predictive Trajectory Control for High-Speed Rack Feeders 195 cally, finally leads to x d (t) =         y Sd  y Kd (t), ˙ y Kd (t), ¨ y Kd (t), y Kd (t), y (4) Kd (t), y (5) Kd (t), κ  v 1d  y Kd (t), ˙ y Kd (t), ¨ y Kd (t), y Kd (t), y (4) Kd (t), y (5) Kd (t), κ  ˙ y Sd  y Kd (t), ˙ y Kd (t), ¨ y Kd (t), y Kd (t), y (4) Kd (t), y (5) Kd (t), κ  ˙ v 1d  y Kd (t), ˙ y Kd (t), ¨ y Kd (t), y Kd (t), y (4) Kd (t), y (5) Kd (t), κ          . (54) 0 5 10 15 −4 −2 0 2 4 6 8 x 10 −3 t in s e y in m Fig. 5. Tracking error e y ( t ) for the cage motion in horizontal direction. 0 5 10 15 −4 −3 −2 −1 0 1 2 3 4 5 x 10 −3 t in s e x in m Fig. 6. Tracking error e x ( t ) for the cage motion in vertical direction. 5. Experimental validation on the test rig The benefits and the efficiency of the proposed control measures shall be pointed out by exper- imental results obtained from the test set-up available at the Chair of Mechatronics, University of Rostock. For this purpose, a synchronous four times continuously differentiable desired trajectory is considered for the position of the cage in both x- and y-direction. The desired trajectory is given by polynomial functions that comply with specified kinematic constraints, which is achieved by taking advantage of time scaling techniques. The desired trajectory shown in Figure 4 comprises a sequence of reciprocating motions with maximum velocities of 2 m/s in horizontal direction and 1.5 m/s in vertical direction. The resulting tracking errors e y ( t ) = y Kd ( t ) − y K ( t ) (55) and e x ( t ) = x Kd ( t ) − x K ( t ) (56) are depicted in Figure 5 and Figure 6. As can be seen, the maximum position error in y- direction during the movements is about 6 mm and the steady-state position error is smaller than 0.2 mm, whereas the maximum position error in x -direction is approx. 4 mm. Figure 7 0 5 10 15 −0.015 −0.01 −0.005 0 0.005 0.01 0.015 t in s v 1 in m v 1d v 1 Fig. 7. Comparison of the desired values v 1d ( t ) and the actual values v 1 ( t ) for the bending deflection. shows the comparison of the bending deflection measured by strain gauges attached to the flexible beam with desired values. During the acceleration as well as the deceleration inter- vals, physically unavoidable bending deflections could be noticed. The achieved benefit is given by the fact the remaining oscillatons are negligible when the rack feeder arrives at its target position. This underlines both the high model accuracy and the quality of the active damping of the first bending mode. Figure 8 depicts the disturbance rejection properties due to an external excitation by hand. At the beginning, the control structure is deactivated, and the excited bending oscillations decay only due to the very weak material damping. After approx. 2.8 seconds, the control structure is activated and, hence, the first bending mode is actively damped. The remaining oscillations are characterised by higher bending modes that decay with material damping. In future work, the number of Ritz ansatz functions shall be Model Predictive Control196 0 1 2 3 4 5 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 t in s v 1 in m Control activated Manual excitation Fig. 8. Transient response after a manual excitation of the bending deflection: at first without feedback control, after approx. 2.8 seconds with active control. increased to include the higher bending modes as well in the active damping. The correspond- ing elastic coordinates and their time derivatives can be determined by observer techniques. 6. Conclusions In this paper, a gain-scheduled fast model predictive control strategy for high-speed rack feed- ers is presented. The control design is based on a control-oriented elastic multibody system. The suggested control algorithm aims at reducing the future tracking error at the end of the prediction horizon. Beneath an active oscillation damping of the first bending mode, an accu- rate trajectory tracking for the cage position in x- and y-direction is achieved. Experimental results from a prototypic test set-up point out the benefits of the proposed control structure. Experimental results show maximum tracking errors of approx. 6 mm in transient phases, whereas the steady-state tracking error is approx. 0.2 mm. Future work will address an active oscillation damping of higher bending modes as well as an additional gain-scheduling with respect to the varying payload. 7. References Aschemann, H. & Ritzke, J. (2009). Adaptive aktive Schwingungsdämpfung und Trajektorien- folgeregelung für hochdynamische Regalbediengeräte (in German), Schwingungen in Antrieben, Vorträge der 6. VDI-Fachtagung in Leonberg, Germany. (in German). Aschemann, H. & Ritzke, J. (2010). Gain-scheduled tracking control for high-speed rack feed- ers, Proc. of the first joint international conference on multibody system dynamics (IMSD), 2010, Lappeenranta, Finland . Bachmayer, M., Rudolph, J. & Ulbrich, H. (2008). Flatness based feed forward control for a horizontally moving beam with a point mass, European Conference on Structural Con- trol, St. Petersburg pp. 74–81. Fliess, M., Levine, J., Martin, P. & Rouchon, P. (1995). Flatness and defect of nonlinear systems: Introductory theory and examples, Int. J. Control 61: 1327–1361. Jung, S. & Wen, J. (2004). Nonlinear model predictive control for the swing-up of a rotary in- verted pendulum, ASME J. of Dynamic Systems, Measurement and Control 126(3): 666– 673. Kostin, G. V. & Saurin, V. V. (2006). The Optimization of the Motion of an Elastic Rod by the Method of Integro-Differential Relations, Journal of computer and Systems Sciences International, Vol. 45, Pleiades Publishing, Inc., pp. 217–225. Lizarralde, F., Wen, J. & Hsu, L. (1999). A new model predictive control strategy for affine nonlinear control systems, Proc of the American Control Conference (ACC ’99), San Diego pp. 4263 – 4267. M. Bachmayer, J. R. & Ulbrich, H. (2008). Acceleration of linearly actuated elastic robots avoid- ing residual vibrations, Proceedings of the 9th International Conference on Motion and Vibration Control, Munich, Germany. Magni, L. & Scattolini, R. (2004). Model predictive control of continuous-time nonlin- ear systems with piecewise constant control, IEEE Transactions on automatic control 49(6): 900–906. Schindele, D. & Aschemann, H. (2008). Nonlinear model predictive control of a high-speed lin- ear axis driven by pneumatic muscles, Proc. of the American Control Conference (ACC), 2008, Seattle, USA pp. 3017–3022. Shabana, A. A. (2005). Dynamics of multibody systems, Cambridge University Press, Cambridge. Staudecker, M., Schlacher, K. & Hansl, R. (2008). Passivity based control and time optimal tra- jectory planning of a single mast stacker crane, Proc. of the 17th IFAC World Congress, Seoul, Korea pp. 875–880. Wang, Y. & Boyd, S. (2010). Fast model predictive control using online optimization, IEEE Transactions on control systems technology 18(2): 267–278. Weidemann, D., Scherm, N. & Heimann, B. (2004). Discrete-time control by nonlinear online optimization on multiple shrinking horizons for underactuated manipulators, Pro- ceedings of the 15th CISM-IFToMM Symposium on Robot Design, Dynamics and Control, Montreal . Model Predictive Trajectory Control for High-Speed Rack Feeders 197 0 1 2 3 4 5 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 t in s v 1 in m Control activated Manual excitation Fig. 8. Transient response after a manual excitation of the bending deflection: at first without feedback control, after approx. 2.8 seconds with active control. increased to include the higher bending modes as well in the active damping. The correspond- ing elastic coordinates and their time derivatives can be determined by observer techniques. 6. Conclusions In this paper, a gain-scheduled fast model predictive control strategy for high-speed rack feed- ers is presented. The control design is based on a control-oriented elastic multibody system. The suggested control algorithm aims at reducing the future tracking error at the end of the prediction horizon. Beneath an active oscillation damping of the first bending mode, an accu- rate trajectory tracking for the cage position in x- and y-direction is achieved. Experimental results from a prototypic test set-up point out the benefits of the proposed control structure. Experimental results show maximum tracking errors of approx. 6 mm in transient phases, whereas the steady-state tracking error is approx. 0.2 mm. Future work will address an active oscillation damping of higher bending modes as well as an additional gain-scheduling with respect to the varying payload. 7. References Aschemann, H. & Ritzke, J. (2009). Adaptive aktive Schwingungsdämpfung und Trajektorien- folgeregelung für hochdynamische Regalbediengeräte (in German), Schwingungen in Antrieben, Vorträge der 6. VDI-Fachtagung in Leonberg, Germany. (in German). Aschemann, H. & Ritzke, J. (2010). Gain-scheduled tracking control for high-speed rack feed- ers, Proc. of the first joint international conference on multibody system dynamics (IMSD), 2010, Lappeenranta, Finland . Bachmayer, M., Rudolph, J. & Ulbrich, H. (2008). Flatness based feed forward control for a horizontally moving beam with a point mass, European Conference on Structural Con- trol, St. Petersburg pp. 74–81. Fliess, M., Levine, J., Martin, P. & Rouchon, P. (1995). Flatness and defect of nonlinear systems: Introductory theory and examples, Int. J. Control 61: 1327–1361. Jung, S. & Wen, J. (2004). Nonlinear model predictive control for the swing-up of a rotary in- verted pendulum, ASME J. of Dynamic Systems, Measurement and Control 126(3): 666– 673. Kostin, G. V. & Saurin, V. V. (2006). The Optimization of the Motion of an Elastic Rod by the Method of Integro-Differential Relations, Journal of computer and Systems Sciences International, Vol. 45, Pleiades Publishing, Inc., pp. 217–225. Lizarralde, F., Wen, J. & Hsu, L. (1999). A new model predictive control strategy for affine nonlinear control systems, Proc of the American Control Conference (ACC ’99), San Diego pp. 4263 – 4267. M. Bachmayer, J. R. & Ulbrich, H. (2008). Acceleration of linearly actuated elastic robots avoid- ing residual vibrations, Proceedings of the 9th International Conference on Motion and Vibration Control, Munich, Germany. Magni, L. & Scattolini, R. (2004). Model predictive control of continuous-time nonlin- ear systems with piecewise constant control, IEEE Transactions on automatic control 49(6): 900–906. Schindele, D. & Aschemann, H. (2008). Nonlinear model predictive control of a high-speed lin- ear axis driven by pneumatic muscles, Proc. of the American Control Conference (ACC), 2008, Seattle, USA pp. 3017–3022. Shabana, A. A. (2005). Dynamics of multibody systems, Cambridge University Press, Cambridge. Staudecker, M., Schlacher, K. & Hansl, R. (2008). Passivity based control and time optimal tra- jectory planning of a single mast stacker crane, Proc. of the 17th IFAC World Congress, Seoul, Korea pp. 875–880. Wang, Y. & Boyd, S. (2010). Fast model predictive control using online optimization, IEEE Transactions on control systems technology 18(2): 267–278. Weidemann, D., Scherm, N. & Heimann, B. (2004). Discrete-time control by nonlinear online optimization on multiple shrinking horizons for underactuated manipulators, Pro- ceedings of the 15th CISM-IFToMM Symposium on Robot Design, Dynamics and Control, Montreal . Model Predictive Control198 Plasma stabilization system design on the base of model predictive control 199 Plasma stabilization system design on the base of model predictive control Evgeny Veremey and Margarita Sotnikova 0 Plasma stabilization system design on the base of model predictive control Evgeny Veremey and Margarita Sotnikova Saint-Petersburg State University, Faculty of Applied Mathematics and Control Processes Russia 1. Introduction Tokamaks, as future nuclear power plants, currently present exceptionally significant re- search area. The basic problems are electromagnetic control of the plasma current, shape and position. High-performance plasma control in a modern tokamak is the complex prob- lem (Belyakov et al., 1999). This is mainly connected with the design requirements imposed on magnetic control system and power supply physical constraints. Besides that, plasma is an extremely complicated dynamical object from the modeling point of view and usually con- trol system design is based on simplified linear system, representing plasma dynamics in the vicinity of the operating point (Ovsyannikov et al., 2005). This chapter is focused on the con- trol systems design on the base of Model Predictive Control (MPC) (Camacho & Bordons, 1999; Morari et al., 1994). Such systems provide high-performance control in the case when accurate mathematical model of the plant to be controlled is unknown. In addition, these systems allow to take into account constraints, imposed both on the controlled and manip- ulated variables (Maciejowski, 2002). Furthermore, MPC algorithms can base on both linear and nonlinear mathematical models of the plant. So MPC control scheme is quite suitable for plasma stabilization problems. In this chapter two different approaches to the plasma stabilization system design on the base of model predictive control are considered. First of them is based on the traditional MPC scheme. The most significant drawback of this variant is that it does not guarantee stability of the closed-loop control circuit. In order to eliminate this problem, a new control algorithm is proposed. This algorithm allows to stabilize control plant in neighborhood of the plasma equilibrium position. Proposed approach is based on the ideas of MPC and modal paramet- ric optimization. Within the suggested framework linear closed-loop system eigenvalues are placed in the specific desired areas on the complex plane for each sample instant. Such areas are located inside the unit circle and reflect specific requirements and constraints imposed on closed-loop system stability and oscillations. It is well known that the MPC algorithms are very time-consuming, since they require the repeated on-line solution of the optimization problem at each sampling instant. In order to re- duce computational load, algorithms parameters tuning are performed and a special method is proposed in the case of modal parametric optimization based MPC algorithms. 9 Model Predictive Control200 The working capacity and effectiveness of the MPC algorithms is demonstrated by the exam- ple of ITER-FEAT plasma vertical stabilization problem. The comparison of the approaches is done. 2. Control Problem Formulation 2.1 Mathematical model of the plasma vertical stabilization process in ITER-FEAT tokamak The dynamics of plasma control process can be commonly described by the system of ordinary differential equations (Misenov, 2000; Ovsyannikov et al., 2006) dΨ dt + RI = V, (1) where Ψ is the poloidal flux vector, R is a diagonal resistance matrix, I is a vector of active and passive currents, V is a vector of voltages applied to coils. The vector Ψ is given by nonlinear relation Ψ = Ψ (I, I p ), (2) where I p is the plasma current. The vector of output variables is given by y = y (I, I p ). (3) Linearizing equations (1)–(3) in the vicinity of the operating point, we obtain a linear model of the process in the state space form. In particular, the linear model describing plasma vertical control in ITER-FEAT tokamak is presented below. ITER-FEAT tokamak (Gribov et al., 2000) has a separate fast feedback loop for plasma vertical stabilization. The Vertical Stabilization (VS) converter is applied in this loop. Its voltage is evaluated in the feedback controller, which uses the vertical velocity of plasma current cen- troid as an input. So the linear model can be written as follows ˙ x = A x + bu, y = c x + du, (4) where x ∈ E 58 is a state space vector, u ∈ E 1 is the voltage of the VS converter, y ∈ E 1 is the vertical velocity of the plasma current centroid. Since the order of this linear model is very high, an order reduction is desirable to simplify the controller synthesis problem. The standard Matlab function schmr was used to perform model reduction from 58th to 3rd order. As a result, we obtain a transfer function of the reduced SISO model (from input u to output y) P (s) = 1.732 · 10 −6 (s − 121.1)(s + 158.2)(s + 9.641) (s + 29.21)(s + 8.348)(s − 12.21) . (5) This transfer function has poles which dominate the dynamics of the initial plant. The un- stable pole corresponds to vertical instability. It is natural to assume that two other poles are determined by the virtual circuit dynamic related to the most significant elements in the tokamak vessel construction. The quality of the model reduction can be illustrated by the comparison of the Bode diagram for both initial and reduced models. Fig. 1 shows the Bode diagrams for initial and reduced 3 rd order models on the left and for initial and reduced 2 nd order model on the right. It is easy to see that the curves for initial model and reduced 3 rd order model are actually indistinguishable, contrary to the 2 nd order model. −120 −110 −100 −90 −80 −70 Magnitude (dB) 10 0 10 2 10 4 −5 0 5 10 15 20 Phase (deg) Bode Diagram Frequency (rad/sec) −120 −110 −100 −90 −80 −70 Magnitude (dB) 10 0 10 2 10 4 −5 0 5 10 15 20 Phase (deg) Bode Diagram Frequency (rad/sec) Fig. 1. Bode diagrams for initial (solid lines) and reduced (dotted lines) models. In addition to plant model (5), we must take into account the following limits that are imposed on the power supply system V VS max = 0.6kV, I VS max = 20.7kA, (6) where V VS max is the maximum voltage, I VS max is the maximum current in the VS converter. So, the linear model (5) together with constraints (6) is considered in the following as the basis for controller synthesis. 2.2 Optimal control problem formulation The desired controller must stabilize vertical velocity of the plasma current centroid. One of the approaches to control synthesis is based on the optimal control theory. In this framework, plasma vertical stabilization problem can be stated as follows. One needs to find a feedback control algorithm u = u (t, y) that provides a minimum of the quadratic cost functional J = J(u) =  ∞ 0 (y 2 (t) + λu 2 (t))dt, (7) subject to plant model (5) and constraints (6), and guarantees closed-loop stability. Here λ is a constant multiplier setting the trade-off between controller’s performance and control energy costs. Specifically, in order to find an optimal controller, LQG-synthesis can be performed. Such a controller has high stabilization performance in the unconstrained case. However, it is per- haps not the best choice in the presence of constraints. Contrary to this, the MPC synthesis allows to take into account constraints. Its basic scheme implies on-line optimization of the cost functional (7) over a finite horizon subject to plant model (5) and imposed constraints (6). Plasma stabilization system design on the base of model predictive control 201 The working capacity and effectiveness of the MPC algorithms is demonstrated by the exam- ple of ITER-FEAT plasma vertical stabilization problem. The comparison of the approaches is done. 2. Control Problem Formulation 2.1 Mathematical model of the plasma vertical stabilization process in ITER-FEAT tokamak The dynamics of plasma control process can be commonly described by the system of ordinary differential equations (Misenov, 2000; Ovsyannikov et al., 2006) dΨ dt + RI = V, (1) where Ψ is the poloidal flux vector, R is a diagonal resistance matrix, I is a vector of active and passive currents, V is a vector of voltages applied to coils. The vector Ψ is given by nonlinear relation Ψ = Ψ (I, I p ), (2) where I p is the plasma current. The vector of output variables is given by y = y (I, I p ). (3) Linearizing equations (1)–(3) in the vicinity of the operating point, we obtain a linear model of the process in the state space form. In particular, the linear model describing plasma vertical control in ITER-FEAT tokamak is presented below. ITER-FEAT tokamak (Gribov et al., 2000) has a separate fast feedback loop for plasma vertical stabilization. The Vertical Stabilization (VS) converter is applied in this loop. Its voltage is evaluated in the feedback controller, which uses the vertical velocity of plasma current cen- troid as an input. So the linear model can be written as follows ˙ x = A x + bu, y = c x + du, (4) where x ∈ E 58 is a state space vector, u ∈ E 1 is the voltage of the VS converter, y ∈ E 1 is the vertical velocity of the plasma current centroid. Since the order of this linear model is very high, an order reduction is desirable to simplify the controller synthesis problem. The standard Matlab function schmr was used to perform model reduction from 58th to 3rd order. As a result, we obtain a transfer function of the reduced SISO model (from input u to output y) P (s) = 1.732 · 10 −6 (s − 121.1)(s + 158.2)(s + 9.641) ( s + 29.21)(s + 8.348)(s − 12.21) . (5) This transfer function has poles which dominate the dynamics of the initial plant. The un- stable pole corresponds to vertical instability. It is natural to assume that two other poles are determined by the virtual circuit dynamic related to the most significant elements in the tokamak vessel construction. The quality of the model reduction can be illustrated by the comparison of the Bode diagram for both initial and reduced models. Fig. 1 shows the Bode diagrams for initial and reduced 3 rd order models on the left and for initial and reduced 2 nd order model on the right. It is easy to see that the curves for initial model and reduced 3 rd order model are actually indistinguishable, contrary to the 2 nd order model. −120 −110 −100 −90 −80 −70 Magnitude (dB) 10 0 10 2 10 4 −5 0 5 10 15 20 Phase (deg) Bode Diagram Frequency (rad/sec) −120 −110 −100 −90 −80 −70 Magnitude (dB) 10 0 10 2 10 4 −5 0 5 10 15 20 Phase (deg) Bode Diagram Frequency (rad/sec) Fig. 1. Bode diagrams for initial (solid lines) and reduced (dotted lines) models. In addition to plant model (5), we must take into account the following limits that are imposed on the power supply system V VS max = 0.6kV, I VS max = 20.7kA, (6) where V VS max is the maximum voltage, I VS max is the maximum current in the VS converter. So, the linear model (5) together with constraints (6) is considered in the following as the basis for controller synthesis. 2.2 Optimal control problem formulation The desired controller must stabilize vertical velocity of the plasma current centroid. One of the approaches to control synthesis is based on the optimal control theory. In this framework, plasma vertical stabilization problem can be stated as follows. One needs to find a feedback control algorithm u = u (t, y) that provides a minimum of the quadratic cost functional J = J(u) =  ∞ 0 (y 2 (t) + λu 2 (t))dt, (7) subject to plant model (5) and constraints (6), and guarantees closed-loop stability. Here λ is a constant multiplier setting the trade-off between controller’s performance and control energy costs. Specifically, in order to find an optimal controller, LQG-synthesis can be performed. Such a controller has high stabilization performance in the unconstrained case. However, it is per- haps not the best choice in the presence of constraints. Contrary to this, the MPC synthesis allows to take into account constraints. Its basic scheme implies on-line optimization of the cost functional (7) over a finite horizon subject to plant model (5) and imposed constraints (6). Model Predictive Control202 3. Model Predictive Control Algorithms 3.1 MPC Basic Principles Suppose we have a mathematical model, which approximately describes control process dy- namics ˙ ˜ x (τ) = f(τ, ˜ x(τ), ˜ u(τ)), ˜ x | τ=t = x (t). (8) Here ˜ x (τ) ∈ E n is a state vector, ˜ u(τ) ∈ E m is a control vector, τ ∈ [t, ∞), x(t) is the actual state of the plant at the instant t or its estimation based on measurement output. This model is used to predict future outputs of the process given the programmed control ˜ u (τ) over a finite time interval τ ∈ [t, t + T p ]. Such a model is called prediction model and the parameter T p is named prediction horizon. Integrating system (8) we obtain ˜ x(τ) = ˜ x (τ, x(t), ˜ u(τ))—predicted process evolution over time interval τ ∈ [t, t + T p ]. The programmed control ˜ u (τ) is chosen in order to minimize quadratic cost functional over the prediction horizon J = J(x(t), ˜ u(·), T p ) =  t+T p t (( ˜ x − r x )  R( ˜ x )( ˜ x − r x ) + ( ˜ u − r u )  Q( ˜ x )( ˜ u − r u ))dτ, (9) where R ( ˜ x ) , Q ( ˜ x ) are positive definite symmetric weight matrices, r x , r u are state and con- trol input reference signals. In addition, the programmed control ˜ u (τ) should satisfy all of the constraints imposed on the state and control variables. Therefore, the programmed control ˜ u (τ) over prediction horizon is chosen to provide minimum of the following optimization problem J (x ( t ) , ˜ u ( · ) , T p ) → min ˜ u ( · ) ∈ Ω u , (10) where Ω u is the admissible set given by Ω u =  ˜ u (·) ∈ K 0 n [t, t + T p ] : ˜ u(τ) ∈ U, ˜ x(τ, x(t ), ˜ u(τ)) ∈ X, ∀τ ∈ [t, t + T p ]  . (11) Here, K 0 n [t, t + T p ] is the set of piecewise continuous vector functions over the interval [t, t + T p ], U ⊂ E m is the set of feasible input values, X ⊂ E n is the set of feasible state values. Denote by ˜ u ∗ (τ) the solution of the optimization problem (10), (11). In order to implement feedback loop, the obtained optimal programmed control ˜ u ∗ (τ) is used as the input only on the time interval [t, t + δ], where δ << T p . So, only a small part of ˜ u ∗ (τ) is implemented. At time t + δ the whole procedure—prediction and optimization—is repeated again to find new optimal programmed control over time interval [t + δ, t + δ + T p ]. Summarizing, the basic MPC scheme works as follows: 1. Obtain the state estimation ˆ x on the base of measurements y. 2. Solve the optimization problem (10), (11) subject to prediction model (8) with initial conditions ˜ x | τ=t = ˆ x (t) and cost functional (9). 3. Implement obtained optimal control ˜ u ∗ (τ) over time interval [t, t + δ]. 4. Repeat the whole procedure 1–3 at time t + δ. From the previous discussion, the most significant MPC features can be noted: • Both linear and nonlinear model of the plant can be used as a prediction model. • MPC allows taking into account constraints imposed both on the input and output vari- ables. • MPC is the feedback control with the discrete entering of the measurement information at each sampling instant 0, δ, 2δ, . . • MPC control algorithms imply the repeated (at each sampling instant with interval δ) on-line solution of the optimization problems. It is especially important from the real- time implementation point of view, because fast calculations are needed. 3.2 MPC real-time implementation In order for real-time implementation, piece-wise constant functions are used as a pro- grammed control over the prediction horizon. That is, the programmed control ˜ u (τ) is pre- sented by the sequence { ˜ u k , ˜ u k+1 , , ˜ u k+P−1 }, where ˜ u i ∈ E m is the control input at the time interval [ iδ, (i + 1)δ ] , δ is the sampling interval. Note that, P is a number of sampling intervals over the prediction horizon, that is T p = Pδ. Likewise, general MPC formulation presented above consider nonlinear prediction model in the discrete form ˜ x i+1 = f ( ˜ x i , ˜ u i ), i = k + j, j = 0, 1, 2, , ˜ x k = x k , ˜y i = C ˜ x i . (12) Here ˜y i ∈ E r is the vector of output variables, x k ∈ E n is the actual state of the plant at time instant k or its estimation on the base of measurement output. We shall say that the sequence of vectors { ˜y k+1 , ˜y k+2 , , ˜y k+P } represents the prediction of future plant behavior. Similar to the cost functional (9), consider also its discrete analog given by J k = J k ( ¯y, ¯ u) = ∑ P j =1  ( ˜y k+j − r y k +j ) T R k+j ( ˜y k+j − r y k +j ) + ( ˜ u k+j−1 − r u k +j−1 ) T Q k+j ( ˜ u k+j−1 − r u k +j−1 )  , (13) where R k+j and Q k+j are the weight matrices as in the functional (9), r y i and r u i are the output and input reference signals, ¯y =  ˜y k+1 ˜y k+2 ˜y k+P  T ∈ E rP , ¯ u =  ˜ u k ˜ u k+1 ˜ u k+P−1  T ∈ E mP are the auxiliary vectors. The optimization problem (10), (11) can now be stated as follows J k (x k , ˜ u k , ˜ u k+1 , ˜ u k+P−1 ) → min { ˜ u k , ˜ u k+1 , , ˜ u k+P−1 }∈Ω∈E mP , (14) where Ω =  ¯ u ∈ E mP : ˜ u k+j−1 ∈ U, ˜ x k+j ∈ X, j = 1, 2, , P  is the admissible set. Generally, the function J (x k , ˜ u k , ˜ u k+1 , ˜ u k+P−1 ) is a nonlinear function of mP variables and Ω is a non-convex set. Therefore, the optimization task (14) is a nonlinear programming prob- lem. Now real-time MPC algorithm can be presented as follows: 1. Obtain the state estimation ˆ x k based on measurements y k using the observer. 2. Solve the nonlinear programming problem (14) subject to prediction model (12) with initial conditions ˜ x k = ˆ x k and cost functional (13). It should be noted, that the value of the function J k (x k , ˜ u k , ˜ u k+1 , ˜ u k+P−1 ) is obtained by numerically integrating the pre- diction model (12) and then substituting the predicted behavior ¯ x ∈ E nP into the cost function (13) given the programmed control { ˜ u k , ˜ u k+1 , , ˜ u k+P−1 } over the prediction horizon and initial conditions ˆ x k . [...]... feedback control law Note that the optimization problem (19) can be solved analytically for the unconstrained case The result is the linear controller ˜ ˜ uk = Kxk , (20) which converges to the LQR-optimal one as P is increased This convergence is obvious, because the discrete LQR controller minimizes the functional (13) with infinity prediction horizon for linear model (15) 4 Model Predictive Control. .. The computational consumption also depends on the prediction model used So, one needs to use as simple models as possible But the prediction model should adequately reflect the dynamics of the plant considered The simplest case is using the linear prediction model 3.3 Linear MPC In this particular case, MPC is based on the linear prediction model These algorithms are computationally efficient which is...Plasma stabilization system design on the base of model predictive control 203 • MPC is the feedback control with the discrete entering of the measurement information at each sampling instant 0, δ, 2δ, • MPC control algorithms imply the repeated (at each sampling instant with interval δ) on-line solution of the optimization problems... prediction model is given by ˜ ˜ ˜ x i +1 = f ( x i , u i ), ˜ yi = C xi ˜ i = k + j, j = 0, 1, 2, , ˜ xk = xk , (22) 206 Model Predictive Control Here xk ∈ En is the actual state of the plant at time instant k or its estimation on the base of measurement output x Let desired object dynamics is presented by the given vector sequences {rk } and {ru }, k = k 0,1,2, The linear mathematical model of the... = where νi { 11 , γ12 , γ21 , γ22 , , γd1 , γd2 , γd0 } The function f is such that f (·) : (−∞, +∞) → (0, 1) and its inverse function exists in the whole region of the definition; the function ψ(ξ ) is a real function from the variable ξ ∈ (0, r ], which takes the values in the interval [0, π ] and ψ(r ) = 0 Plasma stabilization system design on the base of model predictive control 211 Proof Similar... propositions of the theorem are evident 212 Model Predictive Control Now let us show how introced areas C∆1 and C∆2 are related to the standart areas on the complex plane, which are commonly used in the analysis and synthesis of the continuos time systems Primarily, it may be noticed that the eigenvalues of the continues linear model and the discrete linear model are connected by the following rule (Hendricks... mathematical model of the plant to be controlled is described by the following system of difference equations ˆ ˆ ˆ ˆ x k +1 = F ( x k , u k , ϕ k ), ˆ yk = Cxk ˆ (21) ˆ ˆ Here yk ∈ Es is the vector of output variables, xk ∈ En is the state space vector, uk ∈ Em is the ˆ ˆ vector of controls, ϕk ∈ El is the vector of external disturbances Equations (21) can be used as a basis for nonlinear prediction model. .. obtain the system of (nd + 1) nonlinear equations with r unknown components of vector h in the form L(h) = γ (29) It is evident that the controller (24) has a full structure if and only if the system of equations (33) has a solution for any vector γ 208 Model Predictive Control It can be easy shown that if the parameter vector h consists of the coefficients of numerator and denominator polynomials of matrix... substituting the predicted behavior x ∈ EnP into the cost ˜ ˜ ˜ function (13) given the programmed control {u k , uk+1 , , uk+ P−1 } over the prediction ˆ horizon and initial conditions xk 204 Model Predictive Control ˜ k ˜k ˜k 3 Let {u ∗ , u∗+1 , , u∗+ P−1 } be the solution of the problem (14) Implement only the first ˜k component u∗ of the obtained optimal sequence over time interval [kδ, (k + 1)δ]... computational load can be proposed: 1 Using the control horizon The positive integer number M < P is called the control horizon if the following condition hold: ˜ ˜ ˜ uk+ M−1 = uk+ M = = uk+ P−1 Thus, the number of independent variables is decreased from mP to mM This approach allows to essentially reduce the optimization problem order However, if the control horizon M is too small, the closed-loop . and Control, Montreal . Model Predictive Control1 98 Plasma stabilization system design on the base of model predictive control 199 Plasma stabilization system design on the base of model predictive. subject to plant model (5) and imposed constraints (6). Model Predictive Control2 02 3. Model Predictive Control Algorithms 3.1 MPC Basic Principles Suppose we have a mathematical model, which approximately. on the base of model predictive control 203 3. Model Predictive Control Algorithms 3.1 MPC Basic Principles Suppose we have a mathematical model, which approximately describes control process