This paper proposes different design strategies of robust controllers for high-order plants. The design is tailored on the structure of the equations resulting from modeling flexible structures by using modal coordinates. Moreover, the control laws have some characteristics which make them specially suited for active vibration reduction, such as strong stabilization property and bandpass frequency shape. The approach is also targeted the case of more sensors than actuators, which is very frequent in practical applications. Indeed, actuators are often rather heavy and bulky, while small and light sensors may be placed more freely. In such cases, sensors can be usefully placed in the locations where the primary force fields act on the structure, so as to provide the controller with a direct information on the disturbance effects in terms of structural vibrations. Eventually, this approach may lead to uncolocated control strategies. The design problem is here solved by resorting to a Linear Matrix Inequality technique, which allows also to select the performance weights based on different design requirements, for example, a suitable bandpass frequency shape. Experimental results are presented for a vibration reduction problem of a stiffened aeronautical panel controlled by piezoelectric actuators.

Reduction of vibrations in flexible systems has been considered for years a challenging problem, due to some specific aspects. First, as the model of the structure is described by partial differential equations that can be treated as an infinite set of ordinary differential equations, any finite-order controller must rely on a limited (truncated) description of the plant. Thus, neglected dynamics actions must be taken into account by robustness properties of the controller, otherwise, spillover effects [

In this paper the selection of a stable stabilizing controller with bandpass frequency shape is discussed. First, optimal controllers are discussed, with analytical expression in closed form that is useful when high-order models may result into increased computational burden. The main limitation of this approach is that it can deal only with square plants, that is, as many inputs as outputs must be present. Next, a further technique, able to deal with rectangular plants (i.e., with more outputs than inputs), is proposed. The motivation is provided by the requirements of many applications, where the rejection of a disturbance acting on the structure is desired in locations where the installation of actuator/sensor couples is prohibited. Therefore, the possibility to exploit the vibration measurement in those locations appears the only viable option. The second control strategy proposed in this paper is specifically designed to usefully exploit the knowledge of the disturbance entry point on the structure to place additional sensors able to provide the controller with the necessary information on the vibration status caused by the disturbance. The resulting controller thus has more inputs than outputs, but it is still guaranteed to be strongly stabilizing and with bandpass frequency response. Moreover, selection of free control parameters in order to improve effectiveness of the control, by emphasizing the action of the most controllable and observable modes, is discussed. To achieve such an objective, the control design procedure, still based on an

A final consideration is in order in this section. It is clear that the placement of the actuators and sensors is of paramount importance in the whole response of the flexible structure to external disturbances. Thus, an efficient control system should in general address also the problem of the optimal allocation of actuators and sensors, for example, based on maximizing vibrational energy reduction. However, in practical situations the position of sensors and actuators is often constrained by different considerations (geometry, weight limitations, safety, and cost); thus the control designer has very limited degrees of freedom in deciding sensors and actuators position and even number. For this reason, in this paper the location of sensors and actuators is assumed to be given.

In this paper a flexible structure with

As extensively discussed in [

In order to design a stabilizing controller for the flexible system it is convenient to resort to the standard TITO (two-input-two-output) control problem framework depicted in Figure

Standard control problem scheme.

Finally, let

Before computing the controller, a discussion on the required properties of the controller is in order. As stated in the introduction, stable bandpass controllers are preferred options in the control of flexible systems, due to their ability to filter out accelerometer biases. The following lemma is useful for characterizing the state space representation of a general class of bandpass systems.

Let

Although the lemma is an extension of Theorem 1 in Cavallo et al. [

The above lemma has a key role in our context. Indeed, it guarantees that a controller with a bandpass frequency shape, as required in the introduction, can be imposed by satisfying the rather mild conditions (

The last issue to address in order to design a useful controller for flexible structures is stability of the controller itself. It is well known that inserting unstable elements into the control loop increases sensitivity to disturbances and calls for high-bandwidth controller, thus also increasing the effect of measurement noise [

A first possible family of controllers is obtained in the case of square plants, that is, when

Consider the system (

Although rather technical, Lemma

Consider the system (

Then, the following controller is strongly stabilizing and has

The proofs of Lemma

With the same hypotheses as those in Theorem

It is sufficient to let

Finally, by resorting to the equations for the parameterization of all the stabilizing controllers [

Let

Analogously, an (simpler) expression for the

In the general case of

Any system of the form (

Let

Controller stability is not guaranteed by the above theorem; thus it must be enforced as follows. Preliminarily, define

Consider the system

The theorem can be easily proved by considering the Lyapunov function

The above theorem shows that a parametrization of stable, stabilizing, and bandpass controller is obtained, with

The above considerations can be summarized in a procedure for controller design, namely,

preliminarily, collect the diagonal elements of

solve the optimization problem

compute the controller according to (

In the above procedure, the scalar

Control procedures described in the previous section have been experimentally tested on a very complex flexible structure. In particular, the selected test article is a fuselage skin panel of a BOEING 717 (depicted in Figure

Front view of the panel with piezoelectric actuators and grid performance point.

In detail, the control inputs are produced using three of the four piezoelectric patches bonded on the panel surface. The output acceleration is measured by means of the three colocated accelerometers placed on the rear part of the panel behind the piezoelectric actuators. The structure has a high modal density in the considered frequency range

A fourth piezo is used only as a disturbance input so as to introduce a disturbance contribution not belonging to the range of the input matrix

The effectiveness of the control strategy has been evaluated comparing the frequency response functions (FRFs) from disturbance input to acceleration outputs and in terms of an energy-based index defined as follows. First, the energy of the measured acceleration in the given frequency range

In a first experimental campaign, the optimal

A band-limited white noise in the range

Acceleration in a colocated point of the panel: open loop (

Acceleration in a noncolocated point of the panel: open loop (

Figures

Reduction energy index at grid points of the panel with

Reduction energy index at grid points of the panel with

Figures

Performance of the

Performance of the

Finally, Figure

Control signals in the

The next campaign of experiments addresses the case of more measurements than controls available. In this case, a fifth accelerometer is used to measure the acceleration in a performance point far from both control and disturbance inputs. Moreover, the noise has been applied to the structure

colocated controller with modal performance weights based on the Henkel singular values,

rectangular controller (with an additional sensor) with modal performance weights based on the Henkel singular values,

noncolocated controller with modal performance weights based on the Henkel singular values.

All these controllers use the same three piezoelectric actuators as control inputs and the four accelerometers as measured outputs and are designed by selecting the modal performance weights in (

The performance obtained with the controllers above are evaluated both in terms of vibration reduction quantified according to the index in (

Values of the reduction energy index

Disturbance pt. | Performance pt. | |
---|---|---|

30.74 | 30.58 | |

39.16 | 39.50 |

FRFs from disturbance input to acceleration output in a control point (a), in the disturbance point (b), and in the performance point (c): open loop (

The controllers have been also compared in terms of required control energy according to the index defined below. First, the energy

Control energy reduction index

Controller B | Controller C | |
---|---|---|

Channel 1 | 11.18 | 56.35 |

Channel 2 | 12.83 | 64.59 |

Channel 3 | 10.14 | 75.57 |

In a subsequent experiment controller C has been tested. Although in the design phase the weighting matrices have been selected positive definite according to (

Values of the reduction energy index

Disturbance pt. | Performance pt. | |
---|---|---|

27.44 | 18.25 |

FRFs from disturbance input to acceleration output in a control point (a), in the disturbance point (b), and in the performance point (c): open loop (

The paper reported a procedure for designing closed-form optimal (

_{2}strongly stabilizing bandpass controllers