Classes of Strongly Stabilizing Bandpass Controllers for Flexible Structures

Abstract

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.

1. 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 ℋ₂ 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.

2. 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 as𝑥̇=𝐴𝑥+𝐵_𝑢𝑢,𝑦=𝐶_𝑦𝑥, (1) where 𝑥=(𝑥^𝑇₁𝑥^𝑇₂)^𝑇∈ℝ^2𝑛 is the state vector, 𝑢∈ℝ^𝑚 the control input, and 𝑦∈ℝ^{𝑚+𝑛𝑑} the measured (control) output. Obviously, since we are approximating an infinite-order 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 form𝐴=0𝐼−Ω−Λ,𝐵_𝑢=0𝐵_𝑢2,𝐶_𝑦=0𝐵^𝑇_𝑑2𝐵^𝑇_𝑢2, (2)(3)(4) where Ω=diag𝜔²₁,…,𝜔²_𝑛,Λ=diag2𝜁₁𝜔₁,…,2𝜁_𝑛𝜔_𝑛, (5) with 𝜔_𝑖 and 𝜁_𝑖>0 being the natural frequency and damping coefficient of the 𝑖th mode (𝑖=1,…,𝑛). In this case 𝑥₁ are the modal displacements and 𝑥₂ 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.

3. The Control Strategy

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 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’’ matrices, the following generalized plant 𝑃 is obtained:𝑥̇=𝐴𝑥+𝐵_𝑤𝑤+𝐵_𝑢𝑢,𝑧=𝐶_𝑧𝑥+𝐷₁₂𝑢,𝑦=𝐶_𝑦𝑥+𝐷₂₁𝑤, (6)(7)(8) where 𝑤∈ℝ^{𝑚+𝑛𝑑+𝑛} is a fictitious disturbance, 𝑧∈ℝ^𝑚+𝑛 a fictitious performance output, and𝑢=𝑅₁^1/2𝑢∈ℝ^𝑚,𝑦=𝑅₂^−1/2𝑦∈ℝ^{𝑚+𝑛𝑑},𝐵_𝑢=𝐵_𝑢𝑅₁^−1/2=0𝐵_𝑢2𝐶_𝑦,=𝑅₂^−1/2𝐶_𝑦=0𝐵^𝑇_𝑑2𝐵^𝑇_𝑢2, (9)(10)(11) with 𝑅₁ and 𝑅₂ 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:𝐵_𝑤=0_𝑛×𝑛0_{𝑛×(𝑛𝑑+𝑚)}𝑊^1/20_{𝑛×(𝑛𝑑+𝑚)},𝐶_𝑧=0_𝑛×𝑛𝑄^1/20_𝑚×𝑛0_𝑚×𝑛, (12) where the matrices 𝑊 and 𝑄 are to be selected. Finally, for the sake of simplicity, the matrices 𝐷₁₂ and 𝐷₂₁ are chosen as to satisfy the following conditions:𝐷^𝑇₁₂𝐶_𝑧=0,𝐵_𝑤𝐷^𝑇₂₁=0. (13) Although not essential, conditions (13) are standard in ℋ₂ or ℋ_∞ control [6]. It is possible to remove them by slightly increasing the complexity of the control strategy. A possible selection is𝐷₁₂=0_𝑛×𝑚𝐷₁₂₂,𝐷₂₁=0_{𝑛𝑑×𝑛}𝐷₂₁₁0_{𝑛𝑑×𝑚}0_𝑚×𝑛0_{𝑚×𝑛𝑑}𝐷₂₁₂, (14) where𝑅₁=𝐷^𝑇₁₂𝐷₁₂=𝐷^𝑇₁₂₂𝐷₁₂₂𝑅>0,₂=𝐷₂₁₁𝐷^𝑇₂₁₁0_{𝑛𝑑×𝑚}0_{𝑚×𝑛𝑑}𝐷₂₁₂𝐷^𝑇₂₁₂=𝑅₂₁00𝑅₂₂>0. (15)(16)

Figure 1 

Standard control problem scheme.

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 𝐻(𝑠)=𝐴𝐵𝐶0, (17) where 𝐴𝐴=0𝐼₁𝐴₂0𝐵,𝐵=₂,𝐶=0𝐶₂, (18) where all the matrices are partitioned assuming the state to be split into two 𝑛-dimensional substates and the square 𝑛×𝑛 matrix 𝐴₁ is invertible. Then, the system 𝐻(𝑠) is bandpass, in the sense that lim_𝑠→0𝐻(𝑠)=lim_𝑠→∞𝐻(𝑠)=0. (19)

Proof. Although the lemma is an extension of Theorem  1 in Cavallo et al. [3], it can be easily proved observing that lim_𝑠→0(𝑠𝐼−𝐴)⁻¹=𝐴₁⁻¹𝐴₂𝐴₁⁻¹−𝐼0. (20) 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 𝐴^T𝑋_∞+𝑋_∞𝐴+𝑋_∞𝛾⁻²𝐵_𝑤𝐵^T_𝑤−𝐵_𝑢𝐵^T_𝑢𝑋_∞+𝐶^T_𝑧𝐶_𝑧=0,𝐴𝑌_∞+𝑌_∞𝐴^T+𝑌_∞𝛾⁻²𝐶^T_𝑧𝐶_𝑧−𝐶^T_𝑦𝐶_𝑦𝑌_∞+𝐵_𝑤𝐵^T_𝑤=0, (21) for a given 𝛾>0. Then, a necessary condition for the solution of the AREs (21) is to be of the form 𝑋_∞=𝛽²𝑌_∞⁻¹, (22) 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 𝐵_𝑤=𝐵00_𝑤20,𝐶_𝑦=0𝐵^T_𝑤200, (23) where 𝐵_𝑤2=𝛼Λ+𝐵_𝑢2𝐵^T_𝑢2^1/2 (24) and 𝛼>0 is a scalar parameter.
Then, the following controller is strongly stabilizing and has 𝑚 zeros at the origin 𝐾_∞=𝐴𝑠,𝛾^∞𝐵_𝑢−𝛾²𝛾²𝐵−2^T𝑢0, (25) where 𝐴_∞=0𝐼−Ω−Λ_∞, (26) and Λ_∞=21−𝛾²𝛾Λ+2²−1𝛾²−2𝛾²−1𝛾²𝐵_𝑢2𝐵^T_𝑢2. (27) Moreover, if 𝛼=21−𝛾⁻², (28) the LFT 𝑇_𝑧𝑤 satisfies 𝑇_𝑧𝑤∞<𝛾. (29) Finally, inf𝛾=√2 and the controller 𝐾_opt=𝐵^T_𝑢2𝐵_𝑢2𝐵^T_𝑢2^†𝐵_𝑢2, (30) where (⋅)^† denotes the pseudoinverse, achieves the optimal value min_𝐾∈𝒦𝑇_𝑧𝑤∞=√2, (31) 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 ℋ₂ problem, obtained by letting 𝛾→∞.

Corollary 4. With the same hypotheses as those in Theorem 3, the controller 𝐾₂(𝑠)=𝐴−2𝐵_𝑢𝐵^T_𝑢𝐵_𝑢−𝐵^T_𝑢0 (32) 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 ℋ₂ suboptimal controllers as follows.

Let𝑄(𝑠)=𝐶_𝑞𝑠𝐼−𝐴_𝑞⁻¹𝐵_𝑞+𝐷_𝑞 (33) be a stable transfer matrix with ‖𝑄‖_∞<𝛾. Then, the family of controllers such that ‖𝑇_𝑧𝑤‖_∞<𝛾 is𝐾_∞,𝑄(𝑠)=𝐴_∞,𝑄𝐵_∞,𝑄𝐶_∞,𝑄𝐷_𝑞, (34) where𝐴_∞,𝑄=𝐴_∞−𝛾²−1𝛾²𝐵−2^𝑢𝐷_𝑞𝐵^𝑇_𝑢𝛾²−1𝛾²𝐵−2^𝑢𝐶_𝑞−𝐵_𝑞𝐵^𝑇_𝑢𝐴_𝑞,𝐵_∞,𝑄=𝛾²−1𝛾²𝐵−2^𝑢𝐷_𝑞+𝛾𝛾²𝐼𝐵−1_𝑞,𝐶_∞,𝑄=−𝐷_𝑞+𝛾𝛾²𝐼𝐵−1^𝑇𝑢𝐵_𝑞. (35)

Analogously, an (simpler) expression for the ℋ₂ suboptimal controller may be obtained as follows:𝐾_2,𝑄(𝑠)=𝐴−2𝐵_𝑢𝐵^𝑇_𝑢𝐵_𝑢𝐶_𝑞𝐵_𝑢−𝐵_𝑞𝐵^𝑇_𝑢𝐴_𝑞𝐵_𝑞−𝐵^𝑇_𝑢𝐶_𝑞0, (36) 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 𝐾(𝑠)=𝐴_𝑐𝐵_𝑐𝐶_𝑐0, (37) where 𝐵_𝑐=0_𝑚×𝑛0_𝑚×𝑛𝑌₂𝐵_𝑑2𝑅₂⁻¹₁𝑌₂𝐵_𝑢2𝑅₂⁻¹₂,𝐶_𝑐=−𝑅₁⁻¹0_𝑚×𝑛𝐵^𝑇_𝑢2𝑋₂,𝐴_𝑐=𝐴+𝐵_𝑢𝐶_𝑐−𝐵_𝑐𝐶_𝑦, (38) where 𝑋₂ and 𝑌₂ are 𝑛×𝑛 diagonal matrices with nonnegative entries.

Proof. Let 𝑄=2𝑋₂Λ+𝑋₂𝐵_𝑢2𝑅₁⁻¹𝐵^𝑇_𝑢2𝑋₂,𝑊=2𝑌₂Λ+𝑌₂𝐵_𝑑2𝑅₂⁻¹₁𝐵^𝑇_𝑑2+𝐵_𝑢2𝑅₂⁻¹₂𝐵^𝑇_𝑢2𝑌₂. (39)(40) It is clear that both 𝑄 and 𝑊 are positive semidefinite for any nonnegative matrices 𝑋₂ and 𝑌₂. Moreover, by defining the matrices 𝑋=Ω𝑋₂00𝑋₂,Ω𝑌=⁻¹𝑌₂00𝑌₂, (41) it is easy to show that 𝑋=Ric(𝐻₂) and 𝑌=Ric(𝐽₂), 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 𝐻₂ and 𝐽₂ are 𝐻₂=𝐴−𝐵_𝑢𝑅₁⁻¹𝐵^𝑇_𝑢−𝑄−𝐴^𝑇𝐽₂,=𝐴^𝑇−𝐶^𝑇_𝑦𝑅₂⁻¹𝐶_𝑦−𝑊−𝐴. (42)(43) Finally, the matrices 𝑄 and 𝑊 are ,.𝑄=000𝑄𝑊=000𝑊 (44) Thus, for any matrices 𝑋₂ and 𝑌₂ the controller (38) solves an ℋ₂ problem and is therefore stabilizing.

Controller stability is not guaranteed by the above theorem; thus it must be enforced as follows. Preliminarily, define𝐵_𝑋=𝐵_𝑢𝑅₁⁻¹𝐵^𝑇_𝑢,𝐵_𝑌=𝐶_𝑦𝑅₂⁻¹𝐶^𝑇_𝑦. (45) 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 𝐾_𝑠(𝑠)=𝐴_𝑐𝐵_𝑐𝐶_𝑐0, (46) where the matrices 𝐴_𝑐, 𝐵_𝑐, and 𝐶_𝑐are given in (38). If the diagonal matrices 𝑋₂ and 𝑌₂ are chosen so as to satisfy the LMI 𝑋₂𝐵_𝑋+𝐵_𝑋𝑋₂+𝑌₂𝐵_𝑌+𝐵_𝑌𝑌₂𝑋>0,₂𝑌>0,₂>0, (47) 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 𝑉(𝑥)=𝑥^𝑇Ω⁻¹00𝐼𝑥. (48) 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 𝑋₂ and 𝑌₂. Since these matrices define the controller gain, a further constraint in the form of LMItr𝑋₂+tr𝑌₂<𝛿, (49) 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 𝑅₁ and 𝑅₂ have still to be selected. Expressions (39) and (40) can suggest how to choose the degrees of freedom. According to them, 𝑋₂ and 𝑌₂ can weigh modes on which to operate, while the matrices 𝑅₁ and 𝑅₂ 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 𝑋₂ and 𝑌₂ in the vector 𝜉=(𝑥₂₁₁,…,𝑥_2𝑛𝑛,𝑦₂₁₁,…,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 max𝑐^𝑇𝑐^𝑇𝑋𝜉,subjectto₂(𝜉)𝐵_𝑋+𝐵_𝑋𝑋₂(𝜉)+𝑌₂(𝜉)𝐵_𝑌+𝐵_𝑌𝑌₂(𝜉)𝑋+2Λ>0,₂(𝜉)𝑌>0,₂(𝜉)1>0,_𝑇𝑛𝑋₂(𝜉)1_𝑛+1^𝑇_𝑛𝑌₂(𝜉)1_𝑛<𝛿, (50)(51)(52)(53)(54) 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 astr𝑋₂+tr𝑌₂<𝛿. (55) Obviously, in the above algorithm 𝑐_𝑗, the 𝑗th element of 𝑐 can assume any real value. For instance, choosing 𝑐₁=10, 𝑐₂=1, and 𝑐₃=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 𝑅₁ and, even more importantly, 𝑅₂ 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 𝑅₂₁ relative to 𝑅₂₂ the case of noncolocated feedback can be considered with stability guaranteed.

4. 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.

Figure 2 

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

4.1. 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 real-time control system with 16-bit A/D channels and 14-bit D/A channels at a sampling frequency of 20kHz.

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 as𝐽_𝑘=^𝑀_ℎ=1Φ_𝑘𝜔_ℎ²,𝑘=1,…,35, (56) where Φ_𝑘(𝜔_ℎ) is the acceleration spectrum measured at the 𝑘th point at the frequency 𝜔_ℎ∈{𝜔₁,…,𝜔_𝑀}, a set of 𝑀=2500 linearly spaced frequencies within Ω. Then, the relative energy reduction at the 𝑘th point has been quantified as𝑅_𝑘%=𝐽_𝑘,ol−𝐽_𝑘,cl𝐽_𝑘,ol100, (57) where the subscripts “ol’’ and “cl’’ stand for “open loop’’ and “closed loop,’’ respectively.

In a first experimental campaign, the optimal ℋ_∞ and ℋ₂ 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 ℋ₂ 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.

Figure 3 

Acceleration in a colocated point of the panel: open loop (dashed grey line), ℋ2 (solid grey line), and ℋ∞ (solid black line).

Figure 4 

Acceleration in a noncolocated point of the panel: open loop (dashed grey line), ℋ2 (solid grey line), and ℋ∞ (solid black line).

Figures 5 and 6 report the values of the relative energy reduction index as defined in expression (57) at each grid point for the ℋ₂ and ℋ_∞ cases, respectively, showing the effectiveness of both controllers on the whole panel. The average vibration reductions are 37.8% with the ℋ₂ controller ℋ₂ 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.

Figure 5 

Reduction energy index at grid points of the panel with ℋ2 controller.

Figure 6 

Reduction energy index at grid points of the panel with ℋ∞ controller.

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.

(a)

(b)

Figure 7 

Performance of the ℋ2 controller on the whole panel: open loop (a) and closed loop (b).

(a)

(b)

Figure 8 

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

Finally, Figure 9 presents the time history of a control input (piezoelectric actuator voltage) in both the ℋ₂ and the ℋ_∞ cases, showing that the control voltage with the ℋ_∞ controller is generally higher than the ℋ₂ 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 (200V).

(a)

(b)

Figure 9 

Control signals in the ℋ∞ case (a) and ℋ2 case (b).

4.2. 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:A: colocated controller with modal performance weights based on the Henkel singular values,B:rectangular controller (with an additional sensor) with modal performance weights based on the Henkel singular values,C: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 (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 𝑅₁=𝐼, 𝑅₂₁=10⁸, and 𝑅₂₂=𝐼. The large value of 𝑅₂₁ 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 𝑅₁=𝐼, 𝑅₂₁=1, and 𝑅₂₂=𝐼, where a lower value for 𝑅₂₁ 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 𝑅₁=10⁻⁴𝐼, 𝑅₂₁=10⁻⁴, and 𝑅₂₂=10⁸𝐼, where a large value for 𝑅₂₂ 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 20dB of vibration reduction peaks are visible and no spillover appears.

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.

(a)

(b)

(c)

Figure 10 

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 (solid black lines), and controller A (solid grey lines), controller B (dashed black lines).

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 as𝐸_𝑠=^𝑁_𝑘=1𝑠²(𝑘), (58) 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,𝐸^𝑟_𝑢,𝑌=𝐸^𝑟_𝑢,𝑌𝐸_𝑑,𝑌,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 index𝑅_𝑒%=𝐸^𝑟_𝑢,𝐴−𝐸^𝑟_𝑢,𝑋𝐸^𝑟_𝑢,𝐴100,𝑟=1,2,3,𝑋=𝐵,𝐶. (60) 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.

Table 2 

Control energy reduction index 𝑅_𝑒% for controllers B and C.

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 in Table 2. Interestingly enough, even in this actually non-colocated control no spillover phenomenon occurs, as evident from the FRFs reported in Figure 11, still obtaining over 10dB of vibration reduction peaks.

Table 3 

Values of the reduction energy index 𝑅_𝑘% in (57) obtained for noncolocated controller in the frequency range [100,600] Hz for disturbance and performance points.

(a)

(b)

(c)

Figure 11 

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 (solid black lines) and noncolocated controller C (dashed grey lines).

5. Conclusions

The paper reported a procedure for designing closed-form optimal (ℋ_∞ and ℋ₂) controllers for large-scale square systems, together with an extension to nonsquare plants to tackle the vibration reduction problem of flexible structure with the strong requirement of stable and bandpass controller. The original control law is here extended to the case of more sensors than actuators. The more complex design is tackled by solving an LMI problem, which allows also to select the performance weights based on controllability/observability indices. The experimental results obtained for a vibration reduction problem of a stiffened aeronautical panel controlled by piezoelectric actuators confirm the applicability of the strategy in actual systems and its advantages with respect to a controller assuming only colocated actuator/sensor pairs.