Classes of Strongly Stabilizing Bandpass Controllers for Flexible Structures

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.


Introduction
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 [1] may lead to control problems, even to the instability of the closed-loop system, in the most unfavorable situation.A second problem is that, in order to base the design technique for the controller on a meaningful model of the structure, many structural modes must be included in the model, especially for structures with high modal density, so that control problems for flexible structures are intrinsically large-scale problems.Only in the simpler case of low modal density nonoptimal control strategies (e.g., sliding manifold-based control [2] may be used).Finally, stability of the controller (in order to reduce sensitivity to disturbances) and limited bandwidth, for example, to avoid spillover, due to interaction of unmodelled dynamics with control action, must be enforced.From a practical point of view, vibrations are often measured by using accelerometers, which are prone to offsets which can even saturate actuators if not suitably filtered out in the feedback loop.Thus, the need for a filtering action both at high and at low frequency arises.Hence, a limited-bandwidth strongly stabilizing controller is needed.Further discussions on this topic can be found in [3,4].Furthermore, in [5] a broader list of references on the topic is presented and discussed.
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 H 2 approach, adopts an LMI (linear matrix inequality) technique to compute the control parameters to allow the user to select the design weights based on various design requirements or by taking into account modal controllability/observability indices.The paper reports experimental results of broadband vibration reduction for a stiffened aeronautical panel controlled with piezoelectric actuators.
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.

The Model
In this paper a flexible structure with m control inputs and m + n d measured (control) outputs is considered.The structure is assumed to be infinite-dimensional, linear, and time invariant.Since in general n d ≥ 0, the system is not necessarily square, that is, it may have more outputs than inputs.This assumption is quite reasonable in flexible systems control applications, where actuators are heavy and expensive, while sensors (accelerometers) are light, small, and relatively cheap.
As extensively discussed in [5], the mathematical model of a flexible structure can be written as where the control input, and y ∈ R m+nd the measured (control) output.Obviously, since we are approximating an infiniteorder system, the number of retained modes n is generally very high (with systems of order of hundreds).By resorting to the "modal coordinates" the matrices of the state space model assume the form where with ω i and ζ i > 0 being the natural frequency and damping coefficient of the ith mode (i = 1, . . ., n).In this case x 1 are the modal displacements and x 2 the modal velocities.Moreover, assuming that the velocity measurement is available, the output matrix C y has the form (4).By following the gray-box approach proposed by Cavallo et al. [5], the model above can be accurately identified through a measurement campaign carried out on the structure.The gray-box approach assumes that the structure of the system to identify is exactly the one in ( 2)-( 4), and this assumption is of paramount importance in the design of the controller, as it will be illustrated in Section 3.

The Control Strategy
In order to design a stabilizing controller for the flexible system it is convenient to resort to the standard TITO (twoinput-two-output) control problem framework depicted in Figure 1.In this figure, P is the system to control, with measured output y, control input u, generalized disturbance w, and performance output z.Augmenting the flexible system with suitable fictitious "disturbance" and "performance" Advances in Acoustics and Vibration 3 matrices, the following generalized plant P is obtained: where w ∈ R m+nd+n is a fictitious disturbance, z ∈ R m+n a fictitious performance output, and with R 1 and R 2 being the weighting matrices to be selected.The matrices B w and C z are 2n×(m+n d +n) and (m+n)×2n real matrices, respectively, with the following structure: where the matrices W and Q are to be selected.Finally, for the sake of simplicity, the matrices D 12 and D 21 are chosen as to satisfy the following conditions: Although not essential, conditions (13) are standard in H 2 or H ∞ control [6].It is possible to remove them by slightly increasing the complexity of the control strategy.A possible selection is where Finally, let T zw denote the lower LFT (linear fractional transformation) T zw = F l (P, K) [6].
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.

Lemma 1. Let H(s) be a linear time invariant (LTI) 2n-(McMillan) degree system with minimal state space representation
where where all the matrices are partitioned assuming the state to be split into two n-dimensional substates and the square n × n matrix A 1 is invertible.Then, the system H(s) is bandpass, in the sense that Proof.Although the lemma is an extension of Theorem 1 in Cavallo et al. [3], it can be easily proved observing that Thus, direct computation shows that the static gain is zero.The high-frequency behavior is trivial.
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 (18).Thus, the lemma is useful to generate different families of bandpass controllers.
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 [7].Moreover, in the case of flexible structures the high-bandwidth requirement conflicts with the need to avoid spillover effects.
A first possible family of controllers is obtained in the case of square plants, that is, when n d = 0. Indeed, in this case it is possible to compute the explicit solution of a stable stabilizing controller satisfying an H ∞ index.Preliminarily, the following lemma is needed.Lemma 2. Consider the system (6)-( 8), assume, n d = 0, and let X ∞ and Y ∞ be (2n × 2n) positive semidefinite matrices satisfying for a given γ > 0.Then, a necessary condition for the solution of the AREs (21) is to be of the form where β < γ is a real scalar, that is, X ∞ and Y ∞ are block diagonal.
Although rather technical, Lemma 2 is useful to characterize the closed-form solution to an H ∞ control problem, as stated by the following theorem.Theorem 3. Consider the system (6)-( 8), with matrices given by (2), (10), and (11), assume, n d = 0, let the pair (A, B u ) be stabilizable and the pair (A, C y ) detectable, and let where and α > 0 is a scalar parameter.Then, the following controller is strongly stabilizing and has m zeros at the origin where and Moreover, if the LFT T zw satisfies Finally, inf γ = √ 2 and the controller where (•) † denotes the pseudoinverse, achieves the optimal value where K is the set of stable, stabilizing controllers with m zeros at the origin s = 0.
The proofs of Lemma 2 and Theorem 3 can be found in Cavallo et al. [5].They are based on the classic solution to the H ∞ problem involving two coupled Riccati equations [6], while the lower bound for γ is obtained by resorting to the theory of singularly perturbed systems.It is worth noticing that the above theorem yields as a byproduct the closed-form solution of the H 2 problem, obtained by letting γ → ∞.

Corollary 4. With the same hypotheses as those in Theorem 3, the controller
minimizes the norm T zw 2 .
Proof.It is sufficient to let γ → ∞ in the controller in Theorem 3.
Finally, by resorting to the equations for the parameterization of all the stabilizing controllers [8], it is possible to give a general expression for the whole family of H ∞ optimal controllers and H 2 suboptimal controllers as follows. Let be a stable transfer matrix with Q ∞ < γ.Then, the family of controllers such that T zw ∞ < γ is where Analogously, an (simpler) expression for the H 2 suboptimal controller may be obtained as follows: where, it has been, obviously, assumed that D q = 0.In the general case of n d / = 0 the above formulation does not hold any longer, and the design of the controller has to be redefined.The main reason is that the Riccati equations to be solved for the H ∞ problem are strongly coupled.The choice of a stabilizing controller with the structure in Lemma 1 can be carried out by exploiting the following theorem.
Theorem 5. Any system of the form (6)-( 8) with matrices given by (2), (10), and (11) is closed-loop stabilized by a controller with state space representation where Advances in Acoustics and Vibration 5 where X 2 and Y 2 are n × n diagonal matrices with nonnegative entries. Proof.Let It is clear that both Q and W are positive semidefinite for any nonnegative matrices X 2 and Y 2 .Moreover, by defining the matrices it is easy to show that X = Ric(H 2 ) and Y = Ric(J 2 ), where, for a given Hamiltonian matrix Z, Ric(Z) denotes the stabilizing solution of the Riccati equation associated to the Hamiltonian matrix [6], and the Hamiltonian matrices H 2 and J 2 are Finally, the matrices Q and W are Thus, for any matrices X 2 and Y 2 the controller (38) solves an H 2 problem and is therefore stabilizing.
Controller stability is not guaranteed by the above theorem; thus it must be enforced as follows.Preliminarily, define From the above formulation it is easy to deduce the following theorem.
Theorem 6.Consider the system K s (s) with minimal state space representation where the matrices A c , B c , and C c are given in (38).If the diagonal matrices X 2 and Y 2 are chosen so as to satisfy the LMI then the system K s (s) strongly stabilizes the system (6)-( 8) and is bandpass.
Proof.The theorem can be easily proved by considering the Lyapunov function The rest of the theorem follows from Theorem 5 and Lemma 1.
The above theorem shows that a parametrization of stable, stabilizing, and bandpass controller is obtained, with 2n free parameters, namely, the diagonal entries of the matrices X 2 and Y 2 .Since these matrices define the controller gain, a further constraint in the form of LMI tr(X 2 ) + tr(Y 2 ) < δ, ( where tr(X) is the trace of the matrix X and δ is an upper bound on the controller gain depending on the problem, can be imposed.Moreover, also the matrices R 1 and R 2 have still to be selected.Expressions (39) and (40) can suggest how to choose the degrees of freedom.According to them, X 2 and Y 2 can weigh modes on which to operate, while the matrices R 1 and R 2 define the authority of the control.
The above considerations can be summarized in a procedure for controller design, namely, (1) preliminarily, collect the diagonal elements of X 2 and Y 2 in the vector ξ = (x 211 , . . ., x 2nn , y 211 , . . ., y 2nn ) T , define a vector of performance weights c ∈ R n , whose jth element is the weight of the jth mode, (2) solve the optimization problem max c T c T ξ, subject to (50) where δ > 0 and 1 n is the n-dimensional column vector with all elements equal to 1; (3) compute the controller according to (37)-(38).
In the above procedure, the scalar δ > 0 is chosen to limit the controller gain since (54) can be written as tr(X 2 ) + tr(Y 2 ) < δ. (55) Obviously, in the above algorithm c j , the jth element of c can assume any real value.For instance, choosing c 1 = 10, Advances in Acoustics and Vibration c 2 = 1, and c 3 = 0.1 would pose different penalties on different modes, thus focussing the control action more on mode one than on mode two and on mode three.Cavallo et al. [9] showed that the optimal choice of the input scaling matrices is performed based on the controllability Gramian of the open-loop and closed-loop systems.However, it is clear that the efficiency of the control action depends on both the controllability and the observability Gramians.As an example, a well-known strategy for model order reduction [10] consists of the computation of a balanced realization and a reduction based on neglecting the modes associated with the Hankel singular values, that is, square roots of the eigenvalues of the product of the controllability Gramian and the observability Gramian, with the lowest magnitude.This kind of consideration leads to the idea of selecting the modal performance weights in (50) based on the Hankel singular values.Another aspect of the presented control strategy to highlight is the selection of the matrices R 1 and, even more importantly, R 2 in ( 15) and ( 16), respectively.The former is responsible for the authority of the control.The latter is responsible for the relative weights of colocated measurements with respect to noncolocated ones.By varying the magnitude of R 21 relative to R 22 the case of noncolocated feedback can be considered with stability guaranteed.

Experimental Results
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 2).The panel is stiffened by two bulkheads and three orthogonal stiffeners riveted to the rear of the panel itself.In order to reproduce the conditions usually adopted in the structural testing phase, the panel has been suspended by a couple of soft springs to simulate free boundary conditions.In the experiments presented in this section piezoelectric ceramic actuators have been used both for producing the primary disturbance field and for generating the control forces on the structure aimed at counteracting the effects of the primary field.

Square Case.
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 Ω = [100, 600] Hz and a model with 148 modes has been identified as proposed by Cavallo et al. [11].The experiments were performed by using a dSPACE rapid prototyping realtime control system with 16-bit A/D channels and 14-bit D/A channels at a sampling frequency of 20 kHz.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 B u in (6).
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 Ω has been estimated for the kth of the five used sensors as where Φ k (ω h ) is the acceleration spectrum measured at the kth point at the frequency ω h ∈ {ω 1 , . . ., ω M }, a set of M = 2500 linearly spaced frequencies within Ω.Then, the relative energy reduction at the kth point has been quantified as where the subscripts "ol" and "cl" stand for "open loop" and "closed loop," respectively.In a first experimental campaign, the optimal H ∞ and H 2 controllers are tested.One of the known drawbacks in the application of optimal feedback control to vibration reduction problems is that the control action may focus only on the performance objectives, reducing vibrations only at the controlled points, while increasing vibration levels at other points of the structure.In order to check that the controller actually has a beneficial effect on the whole structure a matrix of 35 points evenly spaced on the panel has been considered, as shown in Figure 2.
A band-limited white noise in the range [100, 600] Hz has been chosen as primary field, to excite all the system natural modes in the frequency range Ω.The noise has been applied to the structure on all four input channels so that the disturbance has components both in the range of B u and off range.Figures 3 and 4 present results for both H 2 and H ∞ cases.The H ∞ controller has been designed fixing γ = 1.5 in (25).The figures show the acceleration spectrum measured in a control point and in a performance test point.Obviously, performance is better at the control points than at the test points, where a vibration reduction is still apparent, anyway.Figures 5 and 6 report the values of the relative energy reduction index as defined in expression (57) at each grid point for the H 2 and H ∞ cases, respectively, showing the effectiveness of both controllers on the whole panel.The average vibration reductions are 37.8% with the H 2 controller H 2 and 55.1% with the H ∞ controller almost uniformly on the whole panel, despite the small number of control actuators, the high modal density, and the broadband nature of the disturbance.
Figures 7 and 8 further clarify the effectiveness of the proposed controllers reporting the values of the J k energy index in all the grid points with the same color scale in order to compare open loop with closed loop.
Finally, Figure 9 presents the time history of a control input (piezoelectric actuator voltage) in both the H 2 and the H ∞ cases, showing that the control voltage with the H ∞ controller is generally higher than the H 2 case (this explains why the H ∞ controller behaves better).Nevertheless, in both cases the input voltage is far below the saturation limits of the actuators (200 V).

Rectangular Case.
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 only through the fourth piezo actuator not used as control input, thus the disturbance is totally out of the range of the input matrix B u .Although this makes the two sets of experiments not comparable, this choice allows better highlighting of the feature of the controller when used with additional sensors located near the entry point of the primary field.Since the controller computation depends on the choice of modal performance weights and on the number of feedback sensors, the guidelines described at the end of Section 3 have been followed.As a consequence, three controllers have been computed and compared: 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 (50) according to the Henkel singular values.However, controller A practically uses, as output accelerations, only the measurements provided by the three colocated sensors, since weighting matrices are selected as R 1 = I, R 21 = 10 8 , and R 22 = I.The large value of R 21 makes the information of the fourth sensor meaningless for the control action.Controller B takes into account the problem of vibration reduction when information on the location of the application point of the disturbance force is available.Of course, in such a case, the optimal solution would be to locate an actuator/sensor pair in this entry point.However, assume that mounting the actuator in such a location is not possible for some reason, while placement of the sensor is still allowed.Under this assumption, controller B uses the information provided by the fourth sensor by selecting the weighting matrices R 1 = I, R 21 = 1, and R 22 = I, where a lower value for R 21 has been set.The controller C has been considered to assess the behaviour of the active vibration control system when the application at hand does not allow to place actuators and sensors in colocated pairs.In particular, it practically uses only the signal of the fourth sensor placed in the disturbance entry point, by selecting the weighting matrices R 1 = 10 −4 I, R 21 = 10 −4 , and R 22 = 10 8 I, where a large value for R 22 has been set.The rectangular noncolocated controller C has no information on what happens to the structure in the control points.The performance obtained with the controllers above are evaluated both in terms of vibration reduction quantified according to the index in (57) and by comparing the FRFs from disturbance input point to a control point, the disturbance point, and the performance point.In the first experiment controller A, used as reference, has been compared to controller B. Table 1 shows that the reduction energy index for controller B is significantly higher in the disturbance and performance points.Moreover, the effects of controller B, compared with controller A and open loop, on the entire frequency range Ω are shown in Figure 10, where over 20 dB of vibration reduction peaks are visible and no spillover appears.The controllers have been also compared in terms of required control energy according to the index defined below.First, the energy E s of a generic discrete-time signal s(k) has been evaluated as where N is a fixed number of samples equal for all experiments.Taking into account that the primary field has been randomly generated, the energy E r u,Y of the rth control signal in the experiment Y (Y denotes the control label as listed above) has been normalized with respect to the actual energy of the disturbance E d,Y in the same experiment, that is, The results are summarized in Table 2, which shows that controller B requires less control energy than controller A despite its higher performance.It is apparent that the information from the additional sensor improves the control action.
In a subsequent experiment controller C has been tested.Although in the design phase the weighting matrices have been selected positive definite according to (15) and ( 16), in the real experiment, only the fourth sensor placed in the disturbance point has been physically used, in order to demonstrate the possibility to use the proposed controller as a true non-colocated one.The performance in terms of reduction energy index is reported in Table 3.The reduction energy index values are lower than in the previous cases, but the same applies to the required control energy indicated

Figure 2 :
Figure 2: Front view of the panel with piezoelectric actuators and grid performance point.

Figure 3 :Figure 4 :
Figure 3: Acceleration in a colocated point of the panel: open loop (dashed grey line), H 2 (solid grey line), and H ∞ (solid black line).

Figure 5 :Figure 6 :
Figure 5: Reduction energy index at grid points of the panel with H 2 controller.

Figure 7 :
Figure 7: Performance of the H 2 controller on the whole panel: open loop (a) and closed loop (b).

Figure 8 :
Figure 8: Performance of the H ∞ controller on the whole panel: open loop (a) and closed loop (b).

Figure 9 :
Figure 9: Control signals in the H ∞ case (a) and H 2 case (b).
, r = 1, 2, 3, Y = A, B, C. (59)The control energy required by controllers B and C has been then compared to the one required by controller A, by considering the control energy reduction index

Table 1 :
Values of the reduction energy index R k% in (57) obtained in the frequency range [100, 600] Hz for disturbance and performance points.

Table 2 :
Control energy reduction index R e% for controllers B and C.