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Hybrid explicit model predictive control of a nonlinear process approximated with a piecewise affine model
Journal of Process Control 20 (2010) 832–839
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Journal of Process Control journal homepage: www.elsevier.com/locate/jprocont
Hybrid explicit model predictive control of a nonlinear process approximated with a piecewise affine model Boˇstjan Pregelj ∗ , Samo Gerkˇsiˇc Joˇzef Stefan Institute, Ljubljana, Slovenia
a r t i c l e
i n f o
Article history: Received 5 March 2010 Received in revised form 13 April 2010 Accepted 4 May 2010
Keywords: Predictive control Model-based control Hybrid systems Nonlinear control Tracking Pressure control Tuning regulators
a b s t r a c t The recently developed methods of explicit (multi-parametric) model predictive control (e-MPC) for hybrid systems provide an interesting opportunity for solving a class of nonlinear control problems. With this approach, the nonlinear process is approximated by a piecewise affine (PWA) hybrid model containing a set of local linear dynamics. Compared to linear-model-based MPC, a performance improvement is expected with the reduction of the plant-to-model mismatch; however at a cost of controller computation complexity. In order to reduce the computational load, so that desired horizon lengths may be used, we present an efficient sub-optimal solution. The feasibility of the approach for the application was evaluated in an experimental case study, where an output feedback, offset-free-tracking hybrid eMPC controller was considered as a replacement for a PID-controller-based scheme for the control of the pressure in a wire-annealing machine. © 2010 Elsevier Ltd. All rights reserved.
1. Introduction Following the appearance of model predictive control (MPC) methods for hybrid systems, composed of continuous dynamic components and logical discontinuous components [1–3], and linear-model-based explicit (e-) MPC methods [4–6], explicit variants of MPC methods for hybrid systems have appeared [7–15]. Indeed, several software toolboxes have been created [16–18]. Typically, such hybrid e-MPC methods rely on multi-parametric, mixed-integer, linear or quadratic programming solvers (mpMILP/MIQP) [19–21], whereas the related on-line, hybrid MPC methods employ conventional MILP or MIQP solvers. With the explicit (multi-parametric) formulation of the control problem, the optimization problem may be solved in advance, off-line. This allows a very simple implementation of the on-line controller in the form of a table look-up, avoiding the need for on-line optimization. Therefore, an implementation is possible in industry-standard programmable logic controllers, embedded controllers, and even on FPGA chips [22]. Due to the parametric explosion of the off-line mp-MILP/MIQP computation with the problem’s dimensions, this approach is practically feasible for MPC problems of relatively small sizes. Therefore, it is not interesting in the conventional application niche of MPC, but rather for low-
∗ Corresponding author. E-mail address: [email protected] (B. Pregelj). 0959-1524/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.jprocont.2010.05.002
level control applications, such as advanced PID replacement and small-scale multivariate processes. The theoretical papers on hybrid e-MPC listed above mostly focus on the state-feedback problem and related stability, feasibility and computational efficiency issues. However, in most practical applications output feedback tracking controllers with steady-state offset elimination are required [23–27]. A hybrid state estimator is required for the output feedback [28–31]. In addition, the controller-estimator interplay must not be underestimated, as it is well known that, despite the favourable properties of both, the joint control system may exhibit arbitrarily poor stability margins and robustness [32]. The developed hybrid MPC methods provide an opportunity to solve a class of practical nonlinear control problems by approximating the nonlinear system with a piecewise affine (PWA) hybrid model [33]. In [34], such an approach was recently studied, with the focus on the stability of the hybrid control system. It may be possible to pursue a proof of the stability of the control system, also considering the actual nonlinear nature of the plant, by applying a robust hybrid MPC approach [11], taking into account the approximation error of the PWA model. An engineering approach similar to ours has recently been applied to automotive turbocharged-engine control [39]; however, it does not address the issues of switching among the dynamics and complexity reduction. This paper presents an experimental case-study evaluation of an output feedback, offset-free-tracking, 2-norm hybrid finite horizon e-MPC controller used for the control of pressure in the vac-
B. Pregelj, S. Gerkˇsiˇc / Journal of Process Control 20 (2010) 832–839
uum chamber of a wire-annealing machine, a high-order nonlinear system approximated with a hybrid PWA model with low-order dynamics. The plant is non-square with two inputs and a single output. For the output feedback a switching Kalman filter (KF) is used; the logical variables can be determined directly from the control signals. Particular attention is paid to the implementation issues that are important in low-level control: efficient disturbance rejection, and robustness to modelling error [35,36]. Particularly with hybrid models, predictable behaviour during switching among the dynamics is required, as discontinuities in the model may lead to formally correct results that are not useful in practice. An important consideration is that due to the off-line computational complexity and the resulting number of regions, the hybrid e-MPC controller is implementable only with very short horizon lengths, practically resulting in performance inferior to PID control. As a solution to this problem we propose a sub-optimal variant of the hybrid e-MPC controller, with which we are able to achieve the desired horizon lengths while maintaining a reasonable computational load and without significant practical performance deterioration. The following sections contain: a brief overview of hybrid e-MPC; the tracking and steady-state offset removal issues; a description of the plant; the simulation model of the plant, and the PWA hybrid model; the control tuning and results with the original and the simplified hybrid e-MPC controller; and the conclusions. 2. Hybrid e-MPC Consider a hybrid system described by a PWA model:
x(k + 1) = Ai x(k) + Bi u(k) + f i
if
x(k) u(k)
∈ Pi,
i = {1, . . . , s} (1)
where Ai , Bi , fi are hybrid-system matrices or vectors, k is the discrete-time index, u ∈ m is the input, x ∈ n is the state, and i s P is the bounded compact partition of the state + input space i=1 in n+m , and s is the number of local models. The system is subject to input and state constraints defined by the following inequality: Ex(k) + Lu(k) ≤ Mc
(2)
The cost function of the 2-norm state-cost constrained finitetime optimal control (CFTOC) problem is defined as: T QN xk+N + J(UN ; xk ) ≡ xk+N
N−1
T xk+i Qxk+i + uTk+i Ru uk+i
(3)
i=0 T
where N is the prediction horizon, UN ≡ [uT1 , . . . , uTN ] is the optimization vector, Q is the state cost, QN is the terminal state cost, and Ru is the control cost. The CFTOC problem is the minimization of the cost (3) subject to the prediction model based on (1) and (2), xk+N ∈ Xf , and x0 = x(0), where Xf is the terminal feasible set. The solution UN∗ is sought in the form of a multi-parametric function of xk . It is shown in [8] that the solution is a PWA-on-polyhedra statefeedback law: uk ∗ (x(k)) = Fki x(k) + Gki
i Nk
if
x(k) ∈ Rki ,
i = {1, . . . , Nk }
(4)
where Rk is a partition of the set Xk of feasible states x(k) and i=1 its closure is of the form: R¯ ki ≡ {x : xT (k)Lki (j)x(k) + Mki (j)x(k) ≤ Nki (j), k = 0, . . . , N − 1
j = 1, . . . , nik }, (5)
The regions of the partition have affine or quadratic boundaries. uk * may be non-unique or discontinuous (on certain types of region
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boundaries). The value function Jk * is a piecewise quadratic function and may be discontinuous. In a practical computation of the control law, a partition is computed for each switching sequence of active dynamics (and binary inputs if present) through the prediction horizon. If several partitions include feasible solutions for a given x(0), the one with the lowest value function is selected. 3. Tracking controller and offset removal The first step towards a practical tracking controller is offset-free output reference yr tracking. Extensions to the output cost formulation and output reference tracking are described in [5,14,27]. Reference tracking may be implemented by tracking-velocity-form model augmentation [5,17,27] or by a direct extension of the cost function [16]. Integral action is required for the removal of the steady-state offset with asymptotically non-zero disturbances [23–26]. It may be achieved either by tracking-error integration [18,37,25] or by a disturbance estimation [36,25,26]. The estimation approach is preferred, as the tracking-error integration approach is prone to integrator wind-up in the case of unreachable targets, and tends to increase the overshoots in the nominal performance in the absence of disturbances. Furthermore, there is a choice of using a scheme with a target calculator (TC) or a “joint” scheme with an integrated controller [36]; while there was no decisive performance difference in our case study, the latter was selected for the sake of the off-line computational complexity. An integrating disturbance estimation state d is added at the output of the affine dynamics using a disturbance augmentation:
x(k + 1) d(k + 1)
y(k)
= =
Ai
0
0
I
[ Ci
1]
x(k) d(k)
x(k) d(k)
+
+ [ Di
Bi
u(k) + f i
0
, 0
(6)
]u(k) + g i
where Ci , Di , gi are hybrid-system matrices or vectors. A Kalman filter (KF) is used for the state estimation of the linear component of the dynamics, by assuming that a stochastic variable w(k) with covariance QK acts on the augmented state [xT (k) dT (k)], and that there is measurement noise v(k) with covariance RK at the output. The switching was performed by changing the model gains and offset in the output equation; the currently active dynamic was determined from the control signal u(k). Notice that the well-known output disturbance model may be achieved with proper tuning. The control problem is ill-conditioned for a plant with more inputs than outputs. With the joint scheme, this issue may be solved by using an input reference value ur with an appropriate (small) penalty Ru in the cost function for the surplus control input(s). In e-MPC the use of fixed reference values urf via a coordinate transformation un = u − urf is beneficial in order not to increase the computational demand. If the implementation does not allow u signal references [17], this may be achieved by using a state reference and penalty for the past control signal state u(k − 1) within the tracking augmented state vector. 4. Pressure control in the wire annealer The case-study control problem is related to the vacuum subsystem of a novel wire-annealing machine from PlasmaIt GmbH. In the annealer, the processed metal wire is heated using magnetofocused plasma in an appropriate inert-gas atmosphere. The controller maintains the specified pressure p (process output y) in the vacuum chamber of the annealer that may vary depend-
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Fig. 1. Vacuum subsystem of the annealer.
Fig. 2. Static characteristics: (left) p [mbar] as a function of u1 when u2 = 30%; (right) p [mbar] as a function of u2 , when u1 = 15 Hz.
ing on the type of wire and gas. The construction of the vacuum subsystem ensures that a certain desired pressure profile along the vacuum chamber is maintained to prevent any undesired leakage. Several vacuum pumps are connected to the different chambers that are separated by seals. The pressure p (process output y) is controlled roughly by adjusting the frequency converters of the pumps u1 connected to the chambers at the wire exit (right-hand side of Fig. 1). Additionally, a valve u2 bypassing the seal before the main chamber is used for rapid regulation, with an approximately five times faster response, but with a limited action range. The controller must be able to rapidly suppress fast-acting disturbances that appear during the plant’s operation, such as momentary sealing changes, the ignition of plasma, etc. It must be able to operate over a wide range of operating points, affected by the pressure setpoint, the wire diameter, the machine temperature during start-up, etc. It must also be able to efficiently suppress the measurement noise. Because of the properties of the actuators, the amplitude and rate constraints must be taken into account: 0 < u1 < 50 [s−1 ], 0 < u2 < 100 [%], −5 < u1 < 5 [s−2 ], −50 < u2 < 50 [%/s]. The measurement range is 0 < p < 133 [mbar]; however, the pumps at the wire input always maintain it below 22 mbar. Regarding the spare
degree of freedom, which is available when the narrow u2 constraints are inactive, it is reasonable to keep u2 in the centre of its linear range when possible, so that the controller can react effectively to disturbances in any direction. Both actuators exhibit static, nonlinear characteristics. Approximations of the characteristics, based on measurement data from a limited number of operating points, are shown in Fig. 2. From the experimental results with a linear-model-based e-MPC [36] it was evident that the control performance varies considerably with the operating point; the experiments indicated that the dependence on u1 was the most evident, whereas u2 would mostly remain within the linear region with proper controller tuning. In addition, the static characteristic of p as a function of u2 is dependent on u1 .
5. Process model The process dynamics exhibits high-order nature due to the sealings, dividing the vacuum chamber into many sub-chambers; this makes the complete physical model very complex and impractical for control purposes. Therefore a discrete-time state-space model with second-order dynamics for each input with a sampling
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Table 2 Piecewise affine dynamics gains and offsets.
Table 1 Examined operation points—gains and offsets. OP
u1 [Hz]
u2 [%]
u1 gain
u2 gain
PWA dynamic (i)
Output (gain) matrix Ci
Offset (gi )
1 (low extreme) 2 (high extreme) 3 (intermediate)
15 10 12.5
30 30 30
−0.3203 −1.0010 −0.7007
−1.0057 −2.4136 −1.7096
1 2 3
[0 −1.0010 0 −2.4136] [0 −0.7007 0 −1.7096] [0 −0.3203 0 −1.0057]
4.2408 −1.2497 −8.5920
time Ts = 0.2 s was identified in the nominal operating point:
⎡
0.7762 −0.0600 0.1769 0.9937 ⎢ A=⎣ 0 0 0 0
⎡
⎤
⎤
0.1769 0 ⎢ 0.0184 0 ⎥ , B=⎣ 0 0.1362 ⎦ 0 0.0155
C= 0
−0.2376
each PWA dynamic, the gains for the u1 and u2 model branches and the output offset are assigned in the output equation:
0 0 0 0 ⎥ , 0.4209 −0.4622 ⎦ 0.1362 0.9473
0 −5.769 ,
D= 0 0
y = Ci x + Du + gi
(7)
Within the limited time for experimentation, two more operating points (OPs) were examined; the respective gains are displayed in Table 1. There are considerable changes in the local gains while the changes in the dynamics are less expressed, and therefore not included in the models. 5.1. Simulation model A simple nonlinear simulation model was built for the purpose of evaluating the controller in a simulation prior to any experimental application. The model is composed of invariant linear dynamics and static nonlinear functions at its inputs ulin . The static nonlinearity at ulin1 is a polynomial approximation of the static characteristic y = f(u1 ),1 including an appropriate offset. The static nonlinearity at ulin2 is a two-dimensional PWA look-up table, defining u2 with respect to u1 according to Table 1, and including the offset.2 The static characteristic y = f(u2 ) is disregarded, except for additional limits set to include only the linear region: 15 < u2 < 45 [%]; not much effective range can be gained outside this region in practice. The measurement noise is also included.
(9)
where Ci and gi , which are coefficients for a particular region, are given in Table 2. In the upper chart of Fig. 3, the steady-state characteristics p = f(u1 , u2 ) of the PWA model with the initial assumption of switching with regard to u1 are shown; the u1 boundaries are at 11.25 Hz and 13.75 Hz. However, some undesired control performance may appear around the discontinuous boundaries. For example, the controller may refuse to follow the reference signal across the region boundary if the cost of the performance at the boundary (with the tracking offset) is lower. Therefore, continuous boundaries are desired. Different solutions to this issue may be possible. The lower chart of Fig. 3 shows the adopted solution with the switching lines moved to plane intersections, while the number and the parameters of the dynamics are unaffected. The corresponding plane boundaries of the plane pairs (1, 2) and (2, 3) are: u2(1,2) =
(C2 (2) − C1 (2)) u1 + (g2 − g1 ) (C1 (4) − C2 (4))
u2(2,3) =
(C3 (2) − C2 (2)) u1 + (g3 − g2 ) (C2 (4) − C3 (4))
(10)
Notice that a continuous transition of the model states is also required. Otherwise, bumps may be observed at the boundaries, as is illustrated by the open-loop PWA model simulation in Fig. 4 (case b), where the switching is implemented at the B matrix instead of C. 6. Hybrid controller tuning
5.2. PWA hybrid model
6.1. E-MPC tuning
The aim of the hybrid modelling was to extend the model of the original linear e-MPC controller with the known information regarding the operating-point-dependent changes of the gains and offset in Table 1. All the dynamics are based on a unity-gain linear model:
For practical purposes, computation times of no more than a few minutes are considered useful; therefore, only short finite horizon lengths can be used. Local linear analysis (LLA) [35,36] is used, because tuning via simulation requires long computations and because the effects of the tuning parameters are not always obvious in the time domain. Root-locus diagrams of the controller poles and singular values diagrams are the most valuable. The tuning is first made for the unconstrained region with the nominal model from the intermediate OP and verified with models of other OPs. After the e-MPC controller is calculated, the constrained regions and the sequences with varying dynamics may be analysed. The following set of parameters was selected with the simulation model: N = 6, Nu = 2, Rdu = diag([0.1 0.05]), Ru = diag([10−6 0.02]), where Nu is the control horizon. The major concern with MPC tuning in our case study is the robustness to the modelling error;
⎡
0.7762 −0.0600 0.1769 0.9937 ⎢ A=⎣ 0 0 0 0
⎡
⎤
0.0600 0 ⎢ 0.0063 0 ⎥ , B=⎣ 0 0.4622 ⎦ 0 0.0527
⎤
0 0 0 0 ⎥ , 0.4209 −0.4622 ⎦ 0.1362 0.9473
C= 0
−1
0
−1 ,
D= 0
0
(8)
1
u1 = kn ulin1 n
n
8
7
6
5
4
3
2
1
0
kn
9.7617 × 10−11
−2.3282 × 10−8
2.3355 × 10−6
−1.2713 × 10−4
4.0165 × 10−3
−7.1880 × 10−2
0.6160
−0.5150
2.9540 × 10−4
2
u2 = (2.4 ulin2 − 3 × 2.4) × 1.6 + 3, u2 = (1.7 ulin2 − 3 × 1.7) × 1.4 + 3, u2 = (1.0 ulin2 − 3 × 1.0) × 1.0 + 3,
for ulin1 < 9.7 for 9.7 < ulin1 < 12.5 . for 12.5 < ulin1
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Fig. 3. Steady-state static characteristics p = f(u1 , u2 ) of the PWA model: (above) discontinuous with switching from u1 ; (below) continuous with switching on the plane intersection lines.
tuning on the robust side of the performance-vs-robustness compromise is preferred. In principle, the use of the hybrid controller should reduce the modelling error compared to the linear-modelbased controller [36] and therefore allow less conservative tuning. Unfortunately, this is not the case with the short horizon lengths that are practically achievable with the hybrid e-MPC (the desired values determined using LLA being N = 43, Nu = 3). Notice that relatively fast sampling is required for the efficient suppression of disturbances. An infinite horizon approach would likely allow better unconstrained performance with short constraints horizons; however, this would still pose a practical problem for a constrained performance. 6.2. Kalman filter tuning KF tuning is also based on the LLA of the closed-loop system. Primarily, the local linear dynamics in the three OPs are studied for
the unconstrained regions. In addition to the nominal dynamics, the effect of plant-to-model mismatch for a selected set of models is always examined. The root-locus diagrams and the frequency characteristics of the sensitivity functions are observed. Useful results with a reasonably fast feedback response and good robustness to the modelling error are obtained with the MPC standard output step disturbance (OSD) model, with the KF parameters: Q K = diag([00001]),
RK = 10−3
although the bandwidth and the robustness are not as good as for linear e-MPC with longer horizons [36]. Fig. 5 shows a sample rootlocus diagram. It is evident that underdamped complex pairs of poles or slow poles on the real axis are present in the case of a plant-to-model mismatch. The feedback efficiency of the OSD model is well known to be limited. A considerably faster nominal feedback performance was achieved by using a closed-loop estimation for the whole state vector and the input disturbance model, and tuning using covariance estimation techniques, and by using a pole-placement observer design. Unfortunately, increasing the bandwidth contradicts the requirement for robustness to the modelling error, resulting in control that tends to oscillate when away from the nominal conditions. Using LLA we were able to achieve small, but in practice rather negligible, improvements to the robustness at approximately the same response speed using several different approaches: by manually adjusting other diagonal elements of QK , by combining the covariance estimation techniques with manual tuning of the OSD model, and by using the Loop Transfer Recovery technique [32]. 6.3. Experimental results
Fig. 4. Open-loop simulation of PWA model variants: (above) output signal y for (a) discontinuous static characteristic, (b) continuous static characteristic with gain switching in Bi , (c) continuous static characteristic with gain switching in Ci ; (below) input signals u1 , u2 .
The controller was experimentally tested on the wire-annealing machine at PlasmaIt GmbH, Lebring, AT. Due to the high level of measurement noise the control costs were retuned, Rdu = diag([1 0.5]), Ru = diag([10−5 0.2]), with both the linear and hybrid controllers. Fig. 6 shows the tracking of a “staircase” yr signal across the relevant operating range of the process, including all three OPs (PWA
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Fig. 5. Root-locus diagram of the closed-loop system with the (e-)MPC controller, the KF, and the process model from the intermediate OP3, for the unconstrained region. The red + signs show the locations of poles with a set of candidate true models, including models from OP1 and OP2, and the same models with an additional sample of delay. The result, assuming the same set of true models, is valid both for the linear e-MPC controller and the hybrid one, when the latter is in OP3 throughout the prediction horizon; however, less plant-to-model mismatch is expected with the hybrid controller, as it switches to another dynamic before reaching OP1 or OP2.
dynamics), with the hybrid-model-based e-MPC with N = 6, Nu = 2 (grey) and with the linear-model-based e-MPC with N = 27, Nu = 2 (black). The tracking is offset-free with both controllers. The effects of the plant-to-model mismatch can be observed in the linear eMPC response (black): overshoot is present at higher yr values, where the process gain is higher, while the response gets sluggish at lower yr values. These effects are reduced with the hybrid e-MPC controller; however, the short horizon length results in an impaired robustness to modelling mismatch, and in irregularities related to the u2 saturation (particularly at times of 170 s and 250 s).
Fig. 6. Plant experiment: staircase yr signal tracking across the operating range using linear (left axis, black line) and hybrid-model-based e-MPC (right axis, grey). Top to bottom: yr (dotted) and y; u1 ; u2 ; the active PWA dynamic i used by the controller and switching KF.
In Fig. 7 the sample local controller performance is shown around the intermediate OP, at 4.3 mbar. A sequence of step changes of yr and disturbances at u1 , u2 , and y in both directions is made in 20 s intervals. Due to the operation being near steady state, the saturations of u2 do not occur.
Fig. 7. Sequence of step changes of yr and disturbances at u1 , u2 , and y using hybrid-model-based e-MPC at intermediate OP 4.3 mbar. Signals as in Fig. 6.
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Table 3 Comparison of the eMPC controller complexity. eMPC controller
Linear
Predictive hor., N Control hor., Nu Switches, Nsw Partitions, sNsw +1 Regions total Off-line comp. time
27 2 0 1 300 13 s
a
Hyb. (orig.) 6 2 5 729 200,000 2100 sa
Hyb. (simpl.)
Hyb. (simpl.)
6 2 2 27 5500 210 s
6 2 0 3 260 9s
Hyb. (simpl.) 43 3 2 27 31,000 2500 s
Hyb. (simpl.) 43 3 0 3 2300 120 s
On-line calculation is too slow for real-time application.
7. Simplified hybrid e-MPC As mentioned earlier, the control law is in practice determined on-line by choosing the switching sequence with the lowest value function and requires storage of one controller partition per each possible switching sequence. The main problem prohibiting the practical use of hybrid e-MPC with longer horizons is the rapidly increasing number of possible switching sequences, with the number of local models and the horizon length. However, the number of sequences can be efficiently limited by a sub-optimal strategy allowing switching among the dynamics at specified “switching samples” within the predictive horizon only. The switching strategy is in principle the same as with the original hybrid e-MPC, except that many of the candidate switching sequences are discarded. Conceptually this is very similar to control move blocking. With appropriate placement of the switching samples, this suboptimality does not cause a significant performance deterioration in practice. An important consideration is to relax the guardlines of the current dynamics at the remaining samples (where the dynamic is constrained to remain fixed) beyond the original boundaries of
this dynamics; otherwise the controller struggles to achieve the guardline transition exactly during switching samples, resulting in erratic performance. The simplest case, when switching is only allowed at the start of the horizon, results in a single partition for each local dynamic. The controller does not anticipate the coming changes in dynamics; therefore, sub-optimal performance is observed during the rapid changes of the operating point. The performance is very similar to the multiple-model-based e-MPC [38]. In theory the simplification influences the performance when the operating point moves to another local model, while the performance within each local dynamics is unaffected. However, like with the control move blocking strategy, the sub-optimality stands in theory, but proves effective in practice and this way widens the application area of hybrid e-MPC. In Fig. 8, the described simplified approach is compared to the original hybrid e-MPC; one can see that the control performance of both is very similar. Because of the considerable variation in the operating conditions during the experiments, the variants of the controller are compared in the simulation.
Fig. 8. Simulation: staircase yr signal tracking across the operating range using hybrid e-MPC (left axis, black line) and corrected (relaxed) simplified hybrid e-MPC (right axis, grey); both using Nu = 2, N = 6. Signals as in Fig. 6.
Fig. 9. Simulation: staircase yr signal tracking across the operating range using the optimally tuned linear (Nu = 2, N = 27) e-MPC (left axis, black line) and the simplified hybrid (Nu = 3, N = 43) e-MPC (right axis, grey). Signals as in Fig. 6.
B. Pregelj, S. Gerkˇsiˇc / Journal of Process Control 20 (2010) 832–839
By allowing more switching samples, a performance even closer to the non-simplified hybrid MPC can be achieved, while enabling use of the desired horizon lengths, determined using LLA, that are computationally infeasible with the original hybrid controller. The finite horizon implementation is therefore not perceived as a limitation; rather it represents additional flexibility for tuning, although this does come at the cost of tuning complication. As discussed earlier, slightly less conservative tuning can be used (optimized using LLA and monitoring the root-locus diagrams): Nu = 3, N = 43. Again the OSD model is used for feedback tuning. In Fig. 9 the simplified hybrid e-MPC with switching samples at the indices 0, 3, 6 is compared to the best tuned linear e-MPC (Nu = 2, N = 27). The simplified hybrid e-MPC achieves a more uniform response without the side effects seen in Fig. 8. By reducing the number of switching samples from all steps in horizon N to Nsw , the number of generated partitions decreases from sN to sNsw +1 . The impact of the simplification on the controller complexity is presented in Table 3 displaying the data from the controllers used in this paper. 8. Conclusions In our case study we were investigating the viability of offsetfree-tracking hybrid e-MPC controller for solving a range of practical, smoothly nonlinear, low-level control problems. Our experience indicates that due to the off-line computational load and the rapidly increasing number of partitions and regions, the original e-MPC controller is applicable only with very short horizon lengths that result in severely impaired feedback performance and robustness. However, the discussed sub-optimal simplification, which restricts controller switching among the local dynamics, allows the computation of controller partitions with the desired horizon lengths while retaining an acceptable computational load without a perceivable performance sacrifice due to the sub-optimality. Due to the decreased plant-to-model mismatch, a less conservative tuning may be used and improved performance is achieved in comparison to linear-model-based e-MPC, so that the feedback response is accelerated in low-gain operating points and overshoots are reduced in high-gain operating points, while maintaining the required robustness to the modelling error. Acknowledgements The authors are grateful for the support of PlasmaIt GmbH, Lebring AT with regard to the plant modelling. The financial support of the EC (CONNECT, COOP-CT-2006-031638) and the Slovenian Research Agency (P2-0001) is gratefully acknowledged. References [1] A. Bemporad, M. Morari, Control of systems integrating logic, dynamics, and constraints, Automatica 35 (3) (1999) 407–427. ´ Model predictive control of constrained piecewise affine [2] D.Q. Mayne, S. Rakovic, discrete-time systems, International Journal of Robust and Nonlinear Control 13 (3–4) (2003) 261–279. ´ Optimal control of constrained piecewise affine dis[3] D.Q. Mayne, S. Rakovic, crete time systems, Computational Optimization and Applications 25 (2003) 167–191. [4] E.N. Pistikopoulos, V. Dua, N.A. Bozinis, A. Bemporad, M. Morari, On-line optimization via off-line parametric optimization tools, Computers and Chemical Engineering 24 (2000) 183–188. [5] A. Bemporad, M. Morari, V. Dua, E.N. Pistikopoulos, The explicit linear quadratic regulator for constrained systems, Automatica 38 (1) (2002) 3–20. [6] J. Spjøtvold, P. Tøndel, T.A. Johansen, Continuous selection and unique polyhedral representation of solutions to convex parametric quadratic programs, Journal of Optimization Theory and Applications 134 (2007) 177–189. [7] V. Sakizlis, V. Dua, J.D. Perkins, E.N. Pistikopoulos, The explicit control law for hybrid systems via parametric programming, in: 2002 Proc. American Control Conference, vol. 1, Anchorage, 2002, pp. 674–679.
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Contents lists available at ScienceDirect
Journal of Process Control journal homepage: www.elsevier.com/locate/jprocont
Hybrid explicit model predictive control of a nonlinear process approximated with a piecewise affine model Boˇstjan Pregelj ∗ , Samo Gerkˇsiˇc Joˇzef Stefan Institute, Ljubljana, Slovenia
a r t i c l e
i n f o
Article history: Received 5 March 2010 Received in revised form 13 April 2010 Accepted 4 May 2010
Keywords: Predictive control Model-based control Hybrid systems Nonlinear control Tracking Pressure control Tuning regulators
a b s t r a c t The recently developed methods of explicit (multi-parametric) model predictive control (e-MPC) for hybrid systems provide an interesting opportunity for solving a class of nonlinear control problems. With this approach, the nonlinear process is approximated by a piecewise affine (PWA) hybrid model containing a set of local linear dynamics. Compared to linear-model-based MPC, a performance improvement is expected with the reduction of the plant-to-model mismatch; however at a cost of controller computation complexity. In order to reduce the computational load, so that desired horizon lengths may be used, we present an efficient sub-optimal solution. The feasibility of the approach for the application was evaluated in an experimental case study, where an output feedback, offset-free-tracking hybrid eMPC controller was considered as a replacement for a PID-controller-based scheme for the control of the pressure in a wire-annealing machine. © 2010 Elsevier Ltd. All rights reserved.
1. Introduction Following the appearance of model predictive control (MPC) methods for hybrid systems, composed of continuous dynamic components and logical discontinuous components [1–3], and linear-model-based explicit (e-) MPC methods [4–6], explicit variants of MPC methods for hybrid systems have appeared [7–15]. Indeed, several software toolboxes have been created [16–18]. Typically, such hybrid e-MPC methods rely on multi-parametric, mixed-integer, linear or quadratic programming solvers (mpMILP/MIQP) [19–21], whereas the related on-line, hybrid MPC methods employ conventional MILP or MIQP solvers. With the explicit (multi-parametric) formulation of the control problem, the optimization problem may be solved in advance, off-line. This allows a very simple implementation of the on-line controller in the form of a table look-up, avoiding the need for on-line optimization. Therefore, an implementation is possible in industry-standard programmable logic controllers, embedded controllers, and even on FPGA chips [22]. Due to the parametric explosion of the off-line mp-MILP/MIQP computation with the problem’s dimensions, this approach is practically feasible for MPC problems of relatively small sizes. Therefore, it is not interesting in the conventional application niche of MPC, but rather for low-
∗ Corresponding author. E-mail address: [email protected] (B. Pregelj). 0959-1524/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.jprocont.2010.05.002
level control applications, such as advanced PID replacement and small-scale multivariate processes. The theoretical papers on hybrid e-MPC listed above mostly focus on the state-feedback problem and related stability, feasibility and computational efficiency issues. However, in most practical applications output feedback tracking controllers with steady-state offset elimination are required [23–27]. A hybrid state estimator is required for the output feedback [28–31]. In addition, the controller-estimator interplay must not be underestimated, as it is well known that, despite the favourable properties of both, the joint control system may exhibit arbitrarily poor stability margins and robustness [32]. The developed hybrid MPC methods provide an opportunity to solve a class of practical nonlinear control problems by approximating the nonlinear system with a piecewise affine (PWA) hybrid model [33]. In [34], such an approach was recently studied, with the focus on the stability of the hybrid control system. It may be possible to pursue a proof of the stability of the control system, also considering the actual nonlinear nature of the plant, by applying a robust hybrid MPC approach [11], taking into account the approximation error of the PWA model. An engineering approach similar to ours has recently been applied to automotive turbocharged-engine control [39]; however, it does not address the issues of switching among the dynamics and complexity reduction. This paper presents an experimental case-study evaluation of an output feedback, offset-free-tracking, 2-norm hybrid finite horizon e-MPC controller used for the control of pressure in the vac-
B. Pregelj, S. Gerkˇsiˇc / Journal of Process Control 20 (2010) 832–839
uum chamber of a wire-annealing machine, a high-order nonlinear system approximated with a hybrid PWA model with low-order dynamics. The plant is non-square with two inputs and a single output. For the output feedback a switching Kalman filter (KF) is used; the logical variables can be determined directly from the control signals. Particular attention is paid to the implementation issues that are important in low-level control: efficient disturbance rejection, and robustness to modelling error [35,36]. Particularly with hybrid models, predictable behaviour during switching among the dynamics is required, as discontinuities in the model may lead to formally correct results that are not useful in practice. An important consideration is that due to the off-line computational complexity and the resulting number of regions, the hybrid e-MPC controller is implementable only with very short horizon lengths, practically resulting in performance inferior to PID control. As a solution to this problem we propose a sub-optimal variant of the hybrid e-MPC controller, with which we are able to achieve the desired horizon lengths while maintaining a reasonable computational load and without significant practical performance deterioration. The following sections contain: a brief overview of hybrid e-MPC; the tracking and steady-state offset removal issues; a description of the plant; the simulation model of the plant, and the PWA hybrid model; the control tuning and results with the original and the simplified hybrid e-MPC controller; and the conclusions. 2. Hybrid e-MPC Consider a hybrid system described by a PWA model:
x(k + 1) = Ai x(k) + Bi u(k) + f i
if
x(k) u(k)
∈ Pi,
i = {1, . . . , s} (1)
where Ai , Bi , fi are hybrid-system matrices or vectors, k is the discrete-time index, u ∈ m is the input, x ∈ n is the state, and i s P is the bounded compact partition of the state + input space i=1 in n+m , and s is the number of local models. The system is subject to input and state constraints defined by the following inequality: Ex(k) + Lu(k) ≤ Mc
(2)
The cost function of the 2-norm state-cost constrained finitetime optimal control (CFTOC) problem is defined as: T QN xk+N + J(UN ; xk ) ≡ xk+N
N−1
T xk+i Qxk+i + uTk+i Ru uk+i
(3)
i=0 T
where N is the prediction horizon, UN ≡ [uT1 , . . . , uTN ] is the optimization vector, Q is the state cost, QN is the terminal state cost, and Ru is the control cost. The CFTOC problem is the minimization of the cost (3) subject to the prediction model based on (1) and (2), xk+N ∈ Xf , and x0 = x(0), where Xf is the terminal feasible set. The solution UN∗ is sought in the form of a multi-parametric function of xk . It is shown in [8] that the solution is a PWA-on-polyhedra statefeedback law: uk ∗ (x(k)) = Fki x(k) + Gki
i Nk
if
x(k) ∈ Rki ,
i = {1, . . . , Nk }
(4)
where Rk is a partition of the set Xk of feasible states x(k) and i=1 its closure is of the form: R¯ ki ≡ {x : xT (k)Lki (j)x(k) + Mki (j)x(k) ≤ Nki (j), k = 0, . . . , N − 1
j = 1, . . . , nik }, (5)
The regions of the partition have affine or quadratic boundaries. uk * may be non-unique or discontinuous (on certain types of region
833
boundaries). The value function Jk * is a piecewise quadratic function and may be discontinuous. In a practical computation of the control law, a partition is computed for each switching sequence of active dynamics (and binary inputs if present) through the prediction horizon. If several partitions include feasible solutions for a given x(0), the one with the lowest value function is selected. 3. Tracking controller and offset removal The first step towards a practical tracking controller is offset-free output reference yr tracking. Extensions to the output cost formulation and output reference tracking are described in [5,14,27]. Reference tracking may be implemented by tracking-velocity-form model augmentation [5,17,27] or by a direct extension of the cost function [16]. Integral action is required for the removal of the steady-state offset with asymptotically non-zero disturbances [23–26]. It may be achieved either by tracking-error integration [18,37,25] or by a disturbance estimation [36,25,26]. The estimation approach is preferred, as the tracking-error integration approach is prone to integrator wind-up in the case of unreachable targets, and tends to increase the overshoots in the nominal performance in the absence of disturbances. Furthermore, there is a choice of using a scheme with a target calculator (TC) or a “joint” scheme with an integrated controller [36]; while there was no decisive performance difference in our case study, the latter was selected for the sake of the off-line computational complexity. An integrating disturbance estimation state d is added at the output of the affine dynamics using a disturbance augmentation:
x(k + 1) d(k + 1)
y(k)
= =
Ai
0
0
I
[ Ci
1]
x(k) d(k)
x(k) d(k)
+
+ [ Di
Bi
u(k) + f i
0
, 0
(6)
]u(k) + g i
where Ci , Di , gi are hybrid-system matrices or vectors. A Kalman filter (KF) is used for the state estimation of the linear component of the dynamics, by assuming that a stochastic variable w(k) with covariance QK acts on the augmented state [xT (k) dT (k)], and that there is measurement noise v(k) with covariance RK at the output. The switching was performed by changing the model gains and offset in the output equation; the currently active dynamic was determined from the control signal u(k). Notice that the well-known output disturbance model may be achieved with proper tuning. The control problem is ill-conditioned for a plant with more inputs than outputs. With the joint scheme, this issue may be solved by using an input reference value ur with an appropriate (small) penalty Ru in the cost function for the surplus control input(s). In e-MPC the use of fixed reference values urf via a coordinate transformation un = u − urf is beneficial in order not to increase the computational demand. If the implementation does not allow u signal references [17], this may be achieved by using a state reference and penalty for the past control signal state u(k − 1) within the tracking augmented state vector. 4. Pressure control in the wire annealer The case-study control problem is related to the vacuum subsystem of a novel wire-annealing machine from PlasmaIt GmbH. In the annealer, the processed metal wire is heated using magnetofocused plasma in an appropriate inert-gas atmosphere. The controller maintains the specified pressure p (process output y) in the vacuum chamber of the annealer that may vary depend-
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Fig. 1. Vacuum subsystem of the annealer.
Fig. 2. Static characteristics: (left) p [mbar] as a function of u1 when u2 = 30%; (right) p [mbar] as a function of u2 , when u1 = 15 Hz.
ing on the type of wire and gas. The construction of the vacuum subsystem ensures that a certain desired pressure profile along the vacuum chamber is maintained to prevent any undesired leakage. Several vacuum pumps are connected to the different chambers that are separated by seals. The pressure p (process output y) is controlled roughly by adjusting the frequency converters of the pumps u1 connected to the chambers at the wire exit (right-hand side of Fig. 1). Additionally, a valve u2 bypassing the seal before the main chamber is used for rapid regulation, with an approximately five times faster response, but with a limited action range. The controller must be able to rapidly suppress fast-acting disturbances that appear during the plant’s operation, such as momentary sealing changes, the ignition of plasma, etc. It must be able to operate over a wide range of operating points, affected by the pressure setpoint, the wire diameter, the machine temperature during start-up, etc. It must also be able to efficiently suppress the measurement noise. Because of the properties of the actuators, the amplitude and rate constraints must be taken into account: 0 < u1 < 50 [s−1 ], 0 < u2 < 100 [%], −5 < u1 < 5 [s−2 ], −50 < u2 < 50 [%/s]. The measurement range is 0 < p < 133 [mbar]; however, the pumps at the wire input always maintain it below 22 mbar. Regarding the spare
degree of freedom, which is available when the narrow u2 constraints are inactive, it is reasonable to keep u2 in the centre of its linear range when possible, so that the controller can react effectively to disturbances in any direction. Both actuators exhibit static, nonlinear characteristics. Approximations of the characteristics, based on measurement data from a limited number of operating points, are shown in Fig. 2. From the experimental results with a linear-model-based e-MPC [36] it was evident that the control performance varies considerably with the operating point; the experiments indicated that the dependence on u1 was the most evident, whereas u2 would mostly remain within the linear region with proper controller tuning. In addition, the static characteristic of p as a function of u2 is dependent on u1 .
5. Process model The process dynamics exhibits high-order nature due to the sealings, dividing the vacuum chamber into many sub-chambers; this makes the complete physical model very complex and impractical for control purposes. Therefore a discrete-time state-space model with second-order dynamics for each input with a sampling
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Table 2 Piecewise affine dynamics gains and offsets.
Table 1 Examined operation points—gains and offsets. OP
u1 [Hz]
u2 [%]
u1 gain
u2 gain
PWA dynamic (i)
Output (gain) matrix Ci
Offset (gi )
1 (low extreme) 2 (high extreme) 3 (intermediate)
15 10 12.5
30 30 30
−0.3203 −1.0010 −0.7007
−1.0057 −2.4136 −1.7096
1 2 3
[0 −1.0010 0 −2.4136] [0 −0.7007 0 −1.7096] [0 −0.3203 0 −1.0057]
4.2408 −1.2497 −8.5920
time Ts = 0.2 s was identified in the nominal operating point:
⎡
0.7762 −0.0600 0.1769 0.9937 ⎢ A=⎣ 0 0 0 0
⎡
⎤
⎤
0.1769 0 ⎢ 0.0184 0 ⎥ , B=⎣ 0 0.1362 ⎦ 0 0.0155
C= 0
−0.2376
each PWA dynamic, the gains for the u1 and u2 model branches and the output offset are assigned in the output equation:
0 0 0 0 ⎥ , 0.4209 −0.4622 ⎦ 0.1362 0.9473
0 −5.769 ,
D= 0 0
y = Ci x + Du + gi
(7)
Within the limited time for experimentation, two more operating points (OPs) were examined; the respective gains are displayed in Table 1. There are considerable changes in the local gains while the changes in the dynamics are less expressed, and therefore not included in the models. 5.1. Simulation model A simple nonlinear simulation model was built for the purpose of evaluating the controller in a simulation prior to any experimental application. The model is composed of invariant linear dynamics and static nonlinear functions at its inputs ulin . The static nonlinearity at ulin1 is a polynomial approximation of the static characteristic y = f(u1 ),1 including an appropriate offset. The static nonlinearity at ulin2 is a two-dimensional PWA look-up table, defining u2 with respect to u1 according to Table 1, and including the offset.2 The static characteristic y = f(u2 ) is disregarded, except for additional limits set to include only the linear region: 15 < u2 < 45 [%]; not much effective range can be gained outside this region in practice. The measurement noise is also included.
(9)
where Ci and gi , which are coefficients for a particular region, are given in Table 2. In the upper chart of Fig. 3, the steady-state characteristics p = f(u1 , u2 ) of the PWA model with the initial assumption of switching with regard to u1 are shown; the u1 boundaries are at 11.25 Hz and 13.75 Hz. However, some undesired control performance may appear around the discontinuous boundaries. For example, the controller may refuse to follow the reference signal across the region boundary if the cost of the performance at the boundary (with the tracking offset) is lower. Therefore, continuous boundaries are desired. Different solutions to this issue may be possible. The lower chart of Fig. 3 shows the adopted solution with the switching lines moved to plane intersections, while the number and the parameters of the dynamics are unaffected. The corresponding plane boundaries of the plane pairs (1, 2) and (2, 3) are: u2(1,2) =
(C2 (2) − C1 (2)) u1 + (g2 − g1 ) (C1 (4) − C2 (4))
u2(2,3) =
(C3 (2) − C2 (2)) u1 + (g3 − g2 ) (C2 (4) − C3 (4))
(10)
Notice that a continuous transition of the model states is also required. Otherwise, bumps may be observed at the boundaries, as is illustrated by the open-loop PWA model simulation in Fig. 4 (case b), where the switching is implemented at the B matrix instead of C. 6. Hybrid controller tuning
5.2. PWA hybrid model
6.1. E-MPC tuning
The aim of the hybrid modelling was to extend the model of the original linear e-MPC controller with the known information regarding the operating-point-dependent changes of the gains and offset in Table 1. All the dynamics are based on a unity-gain linear model:
For practical purposes, computation times of no more than a few minutes are considered useful; therefore, only short finite horizon lengths can be used. Local linear analysis (LLA) [35,36] is used, because tuning via simulation requires long computations and because the effects of the tuning parameters are not always obvious in the time domain. Root-locus diagrams of the controller poles and singular values diagrams are the most valuable. The tuning is first made for the unconstrained region with the nominal model from the intermediate OP and verified with models of other OPs. After the e-MPC controller is calculated, the constrained regions and the sequences with varying dynamics may be analysed. The following set of parameters was selected with the simulation model: N = 6, Nu = 2, Rdu = diag([0.1 0.05]), Ru = diag([10−6 0.02]), where Nu is the control horizon. The major concern with MPC tuning in our case study is the robustness to the modelling error;
⎡
0.7762 −0.0600 0.1769 0.9937 ⎢ A=⎣ 0 0 0 0
⎡
⎤
0.0600 0 ⎢ 0.0063 0 ⎥ , B=⎣ 0 0.4622 ⎦ 0 0.0527
⎤
0 0 0 0 ⎥ , 0.4209 −0.4622 ⎦ 0.1362 0.9473
C= 0
−1
0
−1 ,
D= 0
0
(8)
1
u1 = kn ulin1 n
n
8
7
6
5
4
3
2
1
0
kn
9.7617 × 10−11
−2.3282 × 10−8
2.3355 × 10−6
−1.2713 × 10−4
4.0165 × 10−3
−7.1880 × 10−2
0.6160
−0.5150
2.9540 × 10−4
2
u2 = (2.4 ulin2 − 3 × 2.4) × 1.6 + 3, u2 = (1.7 ulin2 − 3 × 1.7) × 1.4 + 3, u2 = (1.0 ulin2 − 3 × 1.0) × 1.0 + 3,
for ulin1 < 9.7 for 9.7 < ulin1 < 12.5 . for 12.5 < ulin1
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B. Pregelj, S. Gerkˇsiˇc / Journal of Process Control 20 (2010) 832–839
Fig. 3. Steady-state static characteristics p = f(u1 , u2 ) of the PWA model: (above) discontinuous with switching from u1 ; (below) continuous with switching on the plane intersection lines.
tuning on the robust side of the performance-vs-robustness compromise is preferred. In principle, the use of the hybrid controller should reduce the modelling error compared to the linear-modelbased controller [36] and therefore allow less conservative tuning. Unfortunately, this is not the case with the short horizon lengths that are practically achievable with the hybrid e-MPC (the desired values determined using LLA being N = 43, Nu = 3). Notice that relatively fast sampling is required for the efficient suppression of disturbances. An infinite horizon approach would likely allow better unconstrained performance with short constraints horizons; however, this would still pose a practical problem for a constrained performance. 6.2. Kalman filter tuning KF tuning is also based on the LLA of the closed-loop system. Primarily, the local linear dynamics in the three OPs are studied for
the unconstrained regions. In addition to the nominal dynamics, the effect of plant-to-model mismatch for a selected set of models is always examined. The root-locus diagrams and the frequency characteristics of the sensitivity functions are observed. Useful results with a reasonably fast feedback response and good robustness to the modelling error are obtained with the MPC standard output step disturbance (OSD) model, with the KF parameters: Q K = diag([00001]),
RK = 10−3
although the bandwidth and the robustness are not as good as for linear e-MPC with longer horizons [36]. Fig. 5 shows a sample rootlocus diagram. It is evident that underdamped complex pairs of poles or slow poles on the real axis are present in the case of a plant-to-model mismatch. The feedback efficiency of the OSD model is well known to be limited. A considerably faster nominal feedback performance was achieved by using a closed-loop estimation for the whole state vector and the input disturbance model, and tuning using covariance estimation techniques, and by using a pole-placement observer design. Unfortunately, increasing the bandwidth contradicts the requirement for robustness to the modelling error, resulting in control that tends to oscillate when away from the nominal conditions. Using LLA we were able to achieve small, but in practice rather negligible, improvements to the robustness at approximately the same response speed using several different approaches: by manually adjusting other diagonal elements of QK , by combining the covariance estimation techniques with manual tuning of the OSD model, and by using the Loop Transfer Recovery technique [32]. 6.3. Experimental results
Fig. 4. Open-loop simulation of PWA model variants: (above) output signal y for (a) discontinuous static characteristic, (b) continuous static characteristic with gain switching in Bi , (c) continuous static characteristic with gain switching in Ci ; (below) input signals u1 , u2 .
The controller was experimentally tested on the wire-annealing machine at PlasmaIt GmbH, Lebring, AT. Due to the high level of measurement noise the control costs were retuned, Rdu = diag([1 0.5]), Ru = diag([10−5 0.2]), with both the linear and hybrid controllers. Fig. 6 shows the tracking of a “staircase” yr signal across the relevant operating range of the process, including all three OPs (PWA
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Fig. 5. Root-locus diagram of the closed-loop system with the (e-)MPC controller, the KF, and the process model from the intermediate OP3, for the unconstrained region. The red + signs show the locations of poles with a set of candidate true models, including models from OP1 and OP2, and the same models with an additional sample of delay. The result, assuming the same set of true models, is valid both for the linear e-MPC controller and the hybrid one, when the latter is in OP3 throughout the prediction horizon; however, less plant-to-model mismatch is expected with the hybrid controller, as it switches to another dynamic before reaching OP1 or OP2.
dynamics), with the hybrid-model-based e-MPC with N = 6, Nu = 2 (grey) and with the linear-model-based e-MPC with N = 27, Nu = 2 (black). The tracking is offset-free with both controllers. The effects of the plant-to-model mismatch can be observed in the linear eMPC response (black): overshoot is present at higher yr values, where the process gain is higher, while the response gets sluggish at lower yr values. These effects are reduced with the hybrid e-MPC controller; however, the short horizon length results in an impaired robustness to modelling mismatch, and in irregularities related to the u2 saturation (particularly at times of 170 s and 250 s).
Fig. 6. Plant experiment: staircase yr signal tracking across the operating range using linear (left axis, black line) and hybrid-model-based e-MPC (right axis, grey). Top to bottom: yr (dotted) and y; u1 ; u2 ; the active PWA dynamic i used by the controller and switching KF.
In Fig. 7 the sample local controller performance is shown around the intermediate OP, at 4.3 mbar. A sequence of step changes of yr and disturbances at u1 , u2 , and y in both directions is made in 20 s intervals. Due to the operation being near steady state, the saturations of u2 do not occur.
Fig. 7. Sequence of step changes of yr and disturbances at u1 , u2 , and y using hybrid-model-based e-MPC at intermediate OP 4.3 mbar. Signals as in Fig. 6.
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Table 3 Comparison of the eMPC controller complexity. eMPC controller
Linear
Predictive hor., N Control hor., Nu Switches, Nsw Partitions, sNsw +1 Regions total Off-line comp. time
27 2 0 1 300 13 s
a
Hyb. (orig.) 6 2 5 729 200,000 2100 sa
Hyb. (simpl.)
Hyb. (simpl.)
6 2 2 27 5500 210 s
6 2 0 3 260 9s
Hyb. (simpl.) 43 3 2 27 31,000 2500 s
Hyb. (simpl.) 43 3 0 3 2300 120 s
On-line calculation is too slow for real-time application.
7. Simplified hybrid e-MPC As mentioned earlier, the control law is in practice determined on-line by choosing the switching sequence with the lowest value function and requires storage of one controller partition per each possible switching sequence. The main problem prohibiting the practical use of hybrid e-MPC with longer horizons is the rapidly increasing number of possible switching sequences, with the number of local models and the horizon length. However, the number of sequences can be efficiently limited by a sub-optimal strategy allowing switching among the dynamics at specified “switching samples” within the predictive horizon only. The switching strategy is in principle the same as with the original hybrid e-MPC, except that many of the candidate switching sequences are discarded. Conceptually this is very similar to control move blocking. With appropriate placement of the switching samples, this suboptimality does not cause a significant performance deterioration in practice. An important consideration is to relax the guardlines of the current dynamics at the remaining samples (where the dynamic is constrained to remain fixed) beyond the original boundaries of
this dynamics; otherwise the controller struggles to achieve the guardline transition exactly during switching samples, resulting in erratic performance. The simplest case, when switching is only allowed at the start of the horizon, results in a single partition for each local dynamic. The controller does not anticipate the coming changes in dynamics; therefore, sub-optimal performance is observed during the rapid changes of the operating point. The performance is very similar to the multiple-model-based e-MPC [38]. In theory the simplification influences the performance when the operating point moves to another local model, while the performance within each local dynamics is unaffected. However, like with the control move blocking strategy, the sub-optimality stands in theory, but proves effective in practice and this way widens the application area of hybrid e-MPC. In Fig. 8, the described simplified approach is compared to the original hybrid e-MPC; one can see that the control performance of both is very similar. Because of the considerable variation in the operating conditions during the experiments, the variants of the controller are compared in the simulation.
Fig. 8. Simulation: staircase yr signal tracking across the operating range using hybrid e-MPC (left axis, black line) and corrected (relaxed) simplified hybrid e-MPC (right axis, grey); both using Nu = 2, N = 6. Signals as in Fig. 6.
Fig. 9. Simulation: staircase yr signal tracking across the operating range using the optimally tuned linear (Nu = 2, N = 27) e-MPC (left axis, black line) and the simplified hybrid (Nu = 3, N = 43) e-MPC (right axis, grey). Signals as in Fig. 6.
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By allowing more switching samples, a performance even closer to the non-simplified hybrid MPC can be achieved, while enabling use of the desired horizon lengths, determined using LLA, that are computationally infeasible with the original hybrid controller. The finite horizon implementation is therefore not perceived as a limitation; rather it represents additional flexibility for tuning, although this does come at the cost of tuning complication. As discussed earlier, slightly less conservative tuning can be used (optimized using LLA and monitoring the root-locus diagrams): Nu = 3, N = 43. Again the OSD model is used for feedback tuning. In Fig. 9 the simplified hybrid e-MPC with switching samples at the indices 0, 3, 6 is compared to the best tuned linear e-MPC (Nu = 2, N = 27). The simplified hybrid e-MPC achieves a more uniform response without the side effects seen in Fig. 8. By reducing the number of switching samples from all steps in horizon N to Nsw , the number of generated partitions decreases from sN to sNsw +1 . The impact of the simplification on the controller complexity is presented in Table 3 displaying the data from the controllers used in this paper. 8. Conclusions In our case study we were investigating the viability of offsetfree-tracking hybrid e-MPC controller for solving a range of practical, smoothly nonlinear, low-level control problems. Our experience indicates that due to the off-line computational load and the rapidly increasing number of partitions and regions, the original e-MPC controller is applicable only with very short horizon lengths that result in severely impaired feedback performance and robustness. However, the discussed sub-optimal simplification, which restricts controller switching among the local dynamics, allows the computation of controller partitions with the desired horizon lengths while retaining an acceptable computational load without a perceivable performance sacrifice due to the sub-optimality. Due to the decreased plant-to-model mismatch, a less conservative tuning may be used and improved performance is achieved in comparison to linear-model-based e-MPC, so that the feedback response is accelerated in low-gain operating points and overshoots are reduced in high-gain operating points, while maintaining the required robustness to the modelling error. Acknowledgements The authors are grateful for the support of PlasmaIt GmbH, Lebring AT with regard to the plant modelling. The financial support of the EC (CONNECT, COOP-CT-2006-031638) and the Slovenian Research Agency (P2-0001) is gratefully acknowledged. References [1] A. Bemporad, M. Morari, Control of systems integrating logic, dynamics, and constraints, Automatica 35 (3) (1999) 407–427. ´ Model predictive control of constrained piecewise affine [2] D.Q. Mayne, S. Rakovic, discrete-time systems, International Journal of Robust and Nonlinear Control 13 (3–4) (2003) 261–279. ´ Optimal control of constrained piecewise affine dis[3] D.Q. Mayne, S. Rakovic, crete time systems, Computational Optimization and Applications 25 (2003) 167–191. [4] E.N. Pistikopoulos, V. Dua, N.A. Bozinis, A. Bemporad, M. Morari, On-line optimization via off-line parametric optimization tools, Computers and Chemical Engineering 24 (2000) 183–188. [5] A. Bemporad, M. Morari, V. Dua, E.N. Pistikopoulos, The explicit linear quadratic regulator for constrained systems, Automatica 38 (1) (2002) 3–20. [6] J. Spjøtvold, P. Tøndel, T.A. Johansen, Continuous selection and unique polyhedral representation of solutions to convex parametric quadratic programs, Journal of Optimization Theory and Applications 134 (2007) 177–189. [7] V. Sakizlis, V. Dua, J.D. Perkins, E.N. Pistikopoulos, The explicit control law for hybrid systems via parametric programming, in: 2002 Proc. American Control Conference, vol. 1, Anchorage, 2002, pp. 674–679.
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