A new model predictive control strategy for affine nonlinear control systems, Proc of the American Control Conference ACC ’99, San Diego pp.. A new model predictive control strategy for
Trang 1Model 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) =
2κ2(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=
b0+b1· s+b2· s2
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 · s4Y 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) =
b0+ .+b2· s2 k V0+ .+k V4 · s4
N(s)
= b V0
k Vj
+b V1
k Vj
· s+ .+b V6
k Vj
· s6
according to
a i=b Vi
k Vj
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), ¨yKd(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 0.2 0.4 0.6 0.8
t in s
y K
−2
−1 0 1 2
t in s
y Kpd
0 0.2 0.4 0.6 0.8
t in s
x K
−1.5
−1
−0.5 0 0.5 1 1.5
t in s
x Kpd
yKd
yK
xKd
xK
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) +12κ2(3− κ)· ˙v1(t), (50)
¨y Kd(t) = ¨y S(t) +12κ2(3− κ)· ¨v1(t) = ¨y K(v1(t), ˙yS(t), ˙v1(t), ud(t), κ), (51) .y Kd(t) = y K(v1(t), ˙yS(t), ˙v1(t), ud(t), ˙ud(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), ˙yKd(t), ¨yKd(t), y Kd(t), ud(t), ˙u d(t), κ)
v 1d(˙y Kd(t), ¨yKd(t), y Kd(t), ud(t), ˙u d(t), κ)
˙y Sd(˙y Kd(t), ¨yKd(t), y Kd(t), ud(t), ˙u d(t), κ)
˙v 1d(˙y Kd(t), ¨yKd(t), y Kd(t), ud(t), ˙u d(t), κ)
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
Trang 2analyti-cally, finally leads to
x d(t) =
y Sd y Kd(t), ˙yKd(t), ¨yKd(t), y Kd(t), y(4)Kd(t), y(5)Kd(t), κ
v 1d y Kd(t), ˙yKd(t), ¨yKd(t), y Kd(t), y(4)
Kd(t), y(5)Kd(t), κ
˙y Sd y Kd(t), ˙yKd(t), ¨yKd(t), y Kd(t), y(4)Kd(t), y(5)Kd(t), κ
˙v 1d y Kd(t), ˙yKd(t), ¨yKd(t), y Kd(t), y(4)Kd(t), y(5)Kd(t), κ
−4
−2 0 2 4 6
8x 10
−3
t in s
ey
Fig 5 Tracking error e y(t)for the cage motion in horizontal direction
−4
−3
−2
−1 0 1 2 3 4
5x 10−3
t in s
ex
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
and
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
−0.015
−0.01
−0.005 0 0.005 0.01 0.015
t in s
v1
v1d
v1
Fig 7 Comparison of the desired values v 1d(t)and the actual values v1(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
Trang 3Model Predictive Trajectory Control for High-Speed Rack Feeders 195
cally, finally leads to
x d(t) =
y Sd y Kd(t), ˙yKd(t), ¨yKd(t), y Kd(t), y(4)Kd(t), y(5)Kd(t), κ
v 1d y Kd(t), ˙yKd(t), ¨yKd(t), y Kd(t), y(4)
Kd(t), y(5)Kd(t), κ
˙y Sd y Kd(t), ˙yKd(t), ¨yKd(t), y Kd(t), y(4)Kd(t), y(5)Kd(t), κ
˙v 1d y Kd(t), ˙yKd(t), ¨yKd(t), y Kd(t), y(4)Kd(t), y(5)Kd(t), κ
−4
−2 0 2 4 6
8x 10
−3
t in s
ey
Fig 5 Tracking error e y(t)for the cage motion in horizontal direction
−4
−3
−2
−1 0 1 2 3 4
5x 10−3
t in s
e x
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
and
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
−0.015
−0.01
−0.005 0 0.005 0.01 0.015
t in s
v 1
v1d
v1
Fig 7 Comparison of the desired values v 1d(t)and the actual values v1(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
Trang 40 1 2 3 4 5
−0.03
−0.02
−0.01 0 0.01 0.02 0.03
t in s
v1
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, Proceedavoid-ings 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
Trang 5Model Predictive Trajectory Control for High-Speed Rack Feeders 197
−0.03
−0.02
−0.01 0 0.01 0.02 0.03
t in s
v1
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, Proceedavoid-ings 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
Trang 7Plasma 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 Concon-trol (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
Trang 8The 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
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
where I pis the plasma current The vector of output variables is given by
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=Ax+bu,
where x∈E58is a state space vector, u ∈ E1is the voltage of the VS converter, y ∈ E1is 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 3rd order models on the left and for initial and reduced 2nd
order model on the right It is easy to see that the curves for initial model and reduced 3rd
order model are actually indistinguishable, contrary to the 2ndorder model
−120
−110
−100
−90
−80
−70
10 0 10 2 10 4
−5 0 5 10 15 20
Bode Diagram
Frequency (rad/sec)
−120
−110
−100
−90
−80
−70
10 0 10 2 10 4
−5 0 5 10 15 20
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
where V VS
maxis the maximum voltage, I VS
maxis 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) =
∞
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)
Trang 9Plasma 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
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
where I pis the plasma current The vector of output variables is given by
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=Ax+bu,
where x∈E58is a state space vector, u ∈ E1is the voltage of the VS converter, y ∈ E1is 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 3rdorder models on the left and for initial and reduced 2nd
order model on the right It is easy to see that the curves for initial model and reduced 3rd
order model are actually indistinguishable, contrary to the 2ndorder model
−120
−110
−100
−90
−80
−70
10 0 10 2 10 4
−5 0 5 10 15 20
Bode Diagram
Frequency (rad/sec)
−120
−110
−100
−90
−80
−70
10 0 10 2 10 4
−5 0 5 10 15 20
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
where V VS
maxis the maximum voltage, I VS
maxis 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) =
∞
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)
Trang 103 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) ∈ Enis a state vector, ˜u(τ) ∈ Em 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−rx)R(˜x)( ˜x−rx) + (˜u−ru)Q(˜x)( ˜u−ru))dτ, (9)
where R(˜x) , Q(˜x) are positive definite symmetric weight matrices, rx, ruare state and
con-trol input reference signals In addition, the programmed concon-trol ˜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
where Ωuis the admissible set given by
Ωu=
˜u(·) ∈K0n[t, t+T p]: ˜u(τ)∈U, ˜x(τ, x(t), ˜u(τ))∈X, ∀τ ∈ [t, t+T p]
Here, K0
n[t, t+T p] is the set of piecewise continuous vector functions over the interval
[t, t+T p], U⊂Emis the set of feasible input values, X⊂Enis 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+δthe 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 ˆx on the base of measurements y.
2 Solve the optimization problem (10), (11) subject to prediction model (8) with initial
conditions ˜x| τ=t =ˆx(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 propro-grammed control ˜u(τ)is pre-sented by the sequence{˜uk, ˜uk+1, , ˜uk+P−1 }, where ˜ui ∈ Emis 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
˜xi+1=f(˜xi, ˜ui), i=k+j, j=0, 1, 2, , ˜xk=xk,
Here ˜yi ∈Eris the vector of output variables, xk ∈ Enis 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{˜yk+1, ˜yk+2, , ˜yk+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 (˜yk+j −ry k+j)TRk+j(˜yk+j −ry k+j)
+ (˜uk+j−1 −ru k+j−1)TQk+j(˜uk+j−1 −ru k+j−1) , (13)
where Rk+jand Qk+jare the weight matrices as in the functional (9), ry i and ru
i are the output and input reference signals,
¯y=
˜yk+1 ˜yk+2 ˜yk+P T ∈ErP,
¯u= ˜uk ˜uk+1 ˜uk+P−1 T ∈EmP
are the auxiliary vectors
The optimization problem (10), (11) can now be stated as follows
J k(xk, ˜uk, ˜uk+1, ˜uk+P−1)→ min
{ ˜u k, ˜uk+1, , ˜uk+P−1 }∈Ω∈E mP, (14) where Ω=
¯u∈EmP: ˜uk+j−1 ∈U, ˜xk+j ∈X, j=1, 2, , Pis the admissible set
Generally, the function J(xk, ˜uk, ˜uk+1, ˜uk+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 ˆxkbased on measurements ykusing the observer
2 Solve the nonlinear programming problem (14) subject to prediction model (12) with
initial conditions ˜xk = ˆxkand cost functional (13) It should be noted, that the value
of the function J k(xk, ˜uk, ˜uk+1, ˜uk+P−1)is obtained by numerically integrating the
pre-diction model (12) and then substituting the predicted behavior ¯x ∈ EnPinto the cost function (13) given the programmed control{˜uk, ˜uk+1, , ˜uk+P−1 }over the prediction
horizon and initial conditions ˆxk