This work focuses on the problem of automatic loop shaping in the context of robust control. More specifically in the framework given by Quantitative Feedback Theory (QFT), traditionally the search of an optimum design, a non convex and nonlinear optimization problem, is simplified by linearizing and/or convexifying the problem. In this work, the authors propose a suboptimal solution using a fixed structure in the compensator and evolutionary optimization. The main idea in relation to previous work consists of the study of the use of fractional compensators, which give singular properties to automatically shape the open loop gain function with a minimum set of parameters, which is crucial for the success of evolutionary algorithms. Additional heuristics are proposed in order to guide evolutionary process towards close to optimum solutions, focusing on local optima avoidance.

It is a well-known fact that there is no general procedure to exactly solve nonlinear nonconvex optimization problems when the solutions belong to continuous solution sets ([

Quantitative Feedback Theory is a robust frequency domain control design methodology which has been successfully applied in practical problems from different domains [

Optimal QFT loop computation is a nonlinear nonconvex optimization problem, for which there is not yet an optimization algorithm which computes a globally optimum solution in a reasonable time, in terms of interactive design purposes. It must be noticed, however, that the work by Nataraj and others on this subject, based on deterministic optimization procedures, combining branch and bound optimization and interval analysis techniques, is very promising (see e.g., [

Other typical approaches to solve this problem have tried to find approximate solutions in different ways. For instance, some authors have simplified the problem somehow, in order to obtain a different optimization problem for which there exists a closed solution or an optimization algorithm which does guarantee a global optimum in a shorter computation time. A trade-off between necessarily conservative simplification of the problem and computational solvability has to be chosen. This is the approach in, for instance, [

Evolutionary algorithms are computationally demanding, specially as the dimension of the search space increases. In this paper the authors study the use of evolutionary algorithms-based optimization, proposing the addition, with respect to previous work, of some heuristics, very much specific to the particular problem under consideration, which help to improve obtained solutions accuracy and computation time needed to obtain these solutions. In this sense, a good structure for the compensator, in terms of using a reduced set of parameters, but with a rich frequency domain behavior, is of crucial importance. This is the main heuristic proposed in this paper: to use evolutionary algorithms together with a flexible structure, able to get a close to optimum solution, but with a reduced number of parameters. In previous work, the compensator has been fixed to a rational structure, with a finite (but no necessarily small) number of zeros and poles. In this work, the main contribution is to introduce a fractional compensator that, with a minimum number of parameters, gives a flexible structure in the frequency domain regarding automatic loop shaping. In fact, it can be approximated by a rational compensator, but with a considerably large number of parameters. This dramatic reduction in the number of parameters has shown to be of capital importance for the success of evolutionary algorithms in the solution of the automatic loop shaping problem. Other applied heuristics have to do with including some features in the objective function that guide the evolutionary search towards close to optimum solutions, paying special attention to prevent the search from getting stacked in local minima, which is specially likely to happen in the problem under consideration.

In this work, following [

From here onwards, the structure of the paper stands as follows. In Section

The basic idea in QFT ([

Two degrees of freedom control system configuration.

Consider an uncertain plant, represented as the set of transfer functions

The design of the controller

The open lines horizontally crossing the Nichols plane from

The closed line around the point (0 dB,

The basic step in the design process,

Note that this criterion is defined irrespective of the particular structure used for

Since

For a given

The meaning of this optimum definition is minimizing the cost of feedback related with sensor noise amplification at high frequencies by minimizing

Nichols plot of

Bode plot of

Note the shape of

The used method for automatic QFT controller design consists of using a fixed structure in the compensator with a certain number

The main contribution of this work has to do with the first factor: to use a fractional structure in the compensator is proposed as a key idea to get flexible structures, able to yield close to optimum solutions, but with a reduced

The second contribution of this work has to do with the second factor. Some ad hoc heuristics, with features very much specific to the particular problem under consideration, have been developed in order to help evolutionary search to improve obtained solutions accuracy and computation time needed to obtain these solutions, specially in terms of local minima avoidance. These heuristics are presented in Section

In Section

TID controller [

The CRONE approach [

The second compensator uses CRONE 3 structure, consisting of the substitution of the (real) order

The third compensator is decoupled CRONE 3 ([

Fractional order Complex Terms (FCTs) [

Once the problem of a large number of parameters to be optimized has been solved by the use of fractional structures, the main problem is the fact that, due to the typically convex nature of the constraints in the solutions space, the evolutionary search can easily get stacked in local minima. The main goal for the heuristics design has been to help the evolutionary search to quickly avoid these local minima when they are detected.

The objective function used as the criterion for the natural selection of individuals during the evolutionary optimization process,

For open boundaries, let

For closed boundaries, let

Another important consideration, in order to avoid the evolutionary search getting stacked, is an

This section describes the optimization algorithm which has been implemented for controllers synthesis. It consists on the use of commercial evolutionary algorithm software (the

The evolutionary algorithm is, in particular, a multiple subpopulations evolutionary search algorithm. In this kind of evolutionary search, each subpopulation evolves in an isolated way for a few generations (as it happens in a single population evolutionary algorithm). After that, one or more individuals are exchanged between subpopulations. The way this process models species evolution is more similar to nature, compared to single population evolutionary algorithms, which helps to avoid local minima. The basic structure of this kind of algorithm is the following (adapted from [

This is the code which implements the objective function

Values

Values

Value

Weights are conceived to give more importance to a certain penalization compared to others, but have not been used for the moment; that is, all of them are equal.

As explained in Section

For each design frequency

Figure

Evolution of

Evolution of

PID-based loop design.

To illustrate the behavior of the proposed optimization method, the QFT Toolbox for MATLAB [

For comparison purposes, a classical PID (TID with

The result obtained with TID controller is shown in Figure

TID-based loop design.

CRONE 2 structure yields the loop shown in Figure

CRONE-2-based loop design.

In Figure

CRONE-3-based loop design.

In Figure

Decoupled CRONE-3-based loop design.

Finally, in Figure

FCT-based loop design.

Figure

CONTROLLER | |
---|---|

PID | 152 |

TID | 140 |

128 | |

CRONE 2 | 129.5 |

CRONE 3 | 126.8 |

decoupled CRONE 3 | 105.3 |

FCT | 94.2 |

An automatic QFT controller design procedure, based on evolutionary algorithms optimization on the parameters of a fixed structure, has been proposed. The key idea behind this proposal is the introduction of a structure with few parameters (a must in order to get good results from evolutionary optimization) but, at the same time, flexible enough, thanks to its fractional nature, to get results which are close to the optimum. Fractional structures have been proposed as ideal candidates. Additional heuristics, focused on guiding the evolutionary search to prevent it from getting stacked in local minima, have been proposed. These structures and heuristics have achieved very good results in terms of QFT classical optimization criterion.

This work was partially supported by the Spanish Government DPI2007-66455-C02-01 project, which is greatly appreciated by the authors.