Controller performance evaluation

The main issue when choosing the controller to be used is to decide between the delity of the controller and its complexity: in many cases one has to be satised with a non-optimal solution.

The following controller-types are supported by MPT:

• Innite Time Optimal Control. This type covers the maximum controllable set of states and guarantees the asymptotic stability and constraint satisfaction. Accordingly the complexity may be prohibitive and also computation of this controller may take forever. Output tracking is not allowed for this type of controller.

• Finite Time Optimal Control. It is a restriction of the previous type, i.e., a nite time optimal controller using limited length of prediction horizon on which the complexity of the controller highly depends.

• Minimum Time Control. This class of controllers is for the cases when you prefer faster run to the cheaper one. This class covers all controllable states and guarantees asymptotic stability and constraint satisfaction (similarly to the rst type). In case of this controller the prediction horizon is innity and the dynamic has to have the origin as equilibrium point.

Output weights are also not supported for minimum time solutions, i.e., output regulation is not possible with this type of controller.

• Low Complexity Controller. In this case the prediction horizon is one. It behaves just like the previous type but is not supported for systems with discrete inputs.

The eect of the design parameters on the controller performance was investigated by simulation experiments using the vaporizer model described in section 5.1. As it can be seen below, the careful choice of these parameters is of high importance from the perspective of controller design. It is very important to build as much knowledge into the controller description as much we can.

Evaluation method The operation, structure and performance of the various controllers that have been investigated are demonstrated on two kinds of gures, i.e., a pair of gures belongs to each investigated case.

The rst type of gures is a picture-triplet (for example see Fig. 5.5) that contains the following sub-gures:

1. Evolution of states. It shows the states as a function of discrete time steps (sampling in-stances). The states are the temperature of the water (x1) and of the wall (x2) accordingly.

The initial state is[270; 265].

2. Evolution of outputs. It shows the output of the system as a function of steps.

3. Evolution of control moves. It shows how the controller acts in each step. In our case it shows the number of heating elements that are switched on.

The second type of gures (for example see Fig. 5.8) shows the regions in the domain of possible states of the state-space (a rectangle). Note that the number of dierent local controllers generated by the optimization algorithm is equal to the number of regions.

To make the dierent controllers comparable the same initial state has been chosen ([270; 265]) to start the control process from, i.e., the system was not considered being in a steady state. The controllers designed with dierent tuning parameters have been examined and compared based on the speed of convergence, the complexity of the controller (via the number of regions dened) and the output's deviance from the reference value.

Norms For systems with discrete inputs solving the CFTOC problem withl = 2 is very slow.

Moreover for the case of the vaporizer even when prediction horizon is 5 the output of the resulting closed-loop system does not converge to the reference point.

For the vaporizer system it is practically irrelevant what linear norm (l= 1orl=∞) is chosen.

In Fig. 5.3 the output of the controller designed with 1-norm and with innity-norm (sampling time isτ= 60, prediction horizon isN = 4) can be seen. In eect the two controllers work identically.

However in the followings the usage of innity-norm is supposed because the highest temper-ature matters the most between the water's and the wall's tempertemper-ature.

Sampling time Choosing the value of sampling time has great eect on the controller. The smaller sampling time is chosen, the approximation of the CT-model becomes more precise. On the other hand as the time-resolution increases, the complexity of the controller will increase as well. In addition, choosing small sampling times, the controller for the vaporizer cannot drive the system to the desired state. In the case of the investigated model it is good idea to chooseτ= 60 seconds which can be explained by the physics of the system. Due to the large mass hold-ups in the original system of industrial size, the time constants of the system are in the region of hours so sampling it with frequency of 1 minute is precise enough and still does not lead to too complex controller.

If the sampling time is less than about τ = 20 sec, the resulting controller cannot drive the system to the desired state thus it cannot track the reference output. In Fig. 5.4 an example can be seen for how the controller runs having relatively high prediction horizon (N = 5) and the system was discretised with sampling time of τ = 20sec. In this case the controller does not work since the heating elements are all switched o and thus the system simmers down.

Using τ = 30 sec as sampling time with prediction horizon N = 5 the controller will work acceptable as seen in Fig. 5.5. However this controller has 321 polyhedral regions over the set of states that may be too complex solution in demanding environments.

Using τ = 60 sec as sampling time with prediction horizonN = 4 leads to very good results (See Fig. 5.7). The number of regions in the controller is 258.

Usingτ= 60sec as sampling time with prediction horizonN= 3is enough to get good results (See Fig. 5.6). This nal controller has only 41 regions that is signicantly less than in the previous cases. That means choosingτ = 60seconds as sampling time with prediction horizon ofN = 3 is a good choice from engineering viewpoint as well.

Length of prediction horizon The unit of the prediction horizon is 1 sampling instance and its length has great eect on the controller complexity: the longer the prediction horizon is, the more precise tracking the controller can obtain. The higher prediction horizon results in higher number of polytopic regions of the controller (see Fig. 5.2). A compromise has to be found between complexity and accuracy. The table below gives a short comparison on how the complexity increases as longer prediction horizon is chosen (forl=∞):

N 2 3 4 5

#polytopes forτ= 30 6 11 73 321

#polytopes forτ= 60 6 41 258 902

#polytopes forτ = 120 20 186 754 1924

(a) Number of polytopes for 1-norm (b) Number of polytopes for∞-norm

Figure 5.2: Number of polytopes (Z) of the controller in function of sampling time (τ) and length of horizon (N)

To repeat the ndings from the previous paragraph, prediction horizonN = 3is a good choice for the compromise between complexity and accuracy.

5.3 Summary

In this chapter a PWA state-space model of an industrial vaporizer with discrete-valued inputs has been developed for controller design purposes. The third-party free MPT-toolbox has been used for designing hybrid controllers by using dierent tuning parameters.

By investigating the eect of tuning parameters, it can be ascertained that the prediction horizon and the sampling time are the two most important parameters. To get shorter calculation times and to stabilize the closed-loop system it is highly encouraged to choose the longest sampling time and the shortest prediction horizon possible. The best controller-parameters have been chosen by performing preliminary simulation experiments. It was found to be irrelevant whether 1-norm or innity-norm is chosen.

0 20 40 60 80

(a) Operation of the controller for 1-norm

0 20 40 60 80

(b) Operation of the controller for∞-norm

Figure 5.3: Operation of the controller forτ= 60,N = 4for dierent norms

0 20 40 60 80

0 20 40 60 80

Controller partition with 41 regions.

(a) Partition of controller of Fig. 5.6

260 280 300 320 340

Controller partition with 258 regions.

(b) Partition of controller of Fig. 5.7

Figure 5.8: Partitions of the investigated controllers

270.5 271 271.5 272 272.5 273 271

271.5 272 272.5 273

x₁ x2

Closed−Loop Trajectories

(a) Operation of controller of Fig. 5.6, started from several dierent initial states

262 264 266 268 270 272 274 276 278 280 282 264

266 268 270 272 274 276 278 280 282 284

x₁ x2

Closed−Loop Trajectories

(b) Operation of controller of Fig. 5.7, started from several dierent initial states

Figure 5.9: Trajectories of the controlled system

Chapter 6

Conclusions