## Abstract

Recently, a sensitivity-enhancement technique for system interrogation using linear controllers and eigenstructure assignment has been extended from linear to nonlinear systems. Nonlinearities have been accounted for by forming (higher dimensional) augmented systems that are designed for each trajectory of the nonlinear system, and are characterized by a specific forcing which ensures that the augmented systems follow that trajectory (when projected onto the original lower dimensional space). The use of system augmentation has several benefits beyond its ability to handle nonlinearities. For example, sensitivity can be increased compared with existing linear techniques through nonlinear feedback auxiliary signals (NFASs) because the constraint that the system is stable during its interrogation has to be applied only to the linearized closed-loop system, while the augmented linear system does not have that constraint. In this work, NFASs are designed for interrogating *linear* systems. System augmentation is used in a linear system because a nonlinear controller is employed to enhance sensitivity. In addition to the increased sensitivity, fewer controller actuator points and sensors are required compared with existing linear techniques due to the efficient use of added (augmented) equations. To demonstrate the approach, damage detection is considered as an application. Numerical simulations for linear mass–spring and mass–spring–damper systems are used to validate the approach and discuss the effects of noise.

## 1. Introduction

Recently, methods that use measured resonant frequencies for applications involving model updating in structural dynamics have received increased attention. These methods (herein referred to as frequency shift-based methods) have several advantages over other approaches such as those that use mode shapes, for instance. First, measuring mode shapes requires the use of many sensors, which is difficult to implement in practical structures. Second, mode shapes are more sensitive to noise and measurement errors than resonant frequencies (Dascotte 1990).

However, there are several drawbacks to using classical frequency shift-based methods. The first key drawback is the limited number of frequencies that can be extracted accurately, which leads to an underdetermined problem when solving for parameter variations (Stubbs & Osegueda 1990*a*,*b*). Several approaches have been suggested to handle this problem. Trivailo *et al*. (1997) proposed attaching an additional (‘twin’) structure to the tested structure to obtain additional frequency information. However, attaching these twin structures is difficult in practice. To overcome this issue, a virtual passive controller was proposed by Lew & Juang (2002). They obtained additional frequencies using output and feedback controllers instead of physically adding mass and stiffness elements to the interrogated structure.

The second key drawback of classical frequency shift-based model updating methods is that often the sensitivity of the resonant frequencies to parameter variations is low, which has been shown both numerically (Swamidas & Chen 1995) and experimentally (Adams *et al*. 1978). To overcome this drawback, Ray & Tian (1999) proposed to enhance the sensitivity of the resonant frequencies through feedback control. They used feedback control to increase the sensitivity of the poles of a system to changes in mass or stiffness. Their approach has been demonstrated on a cantilevered beam both numerically (Ray & Tian 1999) and experimentally (Ray *et al*. 2000). For fixed actuator locations and minimized control effort, Ray & Marini (2000) explored the optimality of the sensitivity of the closed-loop frequencies to mass or stiffness changes. Juang *et al*. (1989) explored the use of eigenstructure assignment techniques for the multi-input case. They assigned the closed-loop eigenvectors close to the open-loop eigenvectors to minimize the control gains (and hence the control effort). Koh & Ray (2004) enriched the frequency information and increased the sensitivity of the resonant frequencies using multiple independent closed-loop systems. Jiang *et al*. (2007) formulated an optimization algorithm for eigenstructure assignment in the multi-input case. Their algorithm minimizes the control effort while maximizing the sensitivity enhancement by optimally placing both the closed-loop frequencies and the eigenvectors of the system.

The third key drawback of existent frequency shift-based methods for model updating and system interrogation is their inability to handle nonlinearities. Nonlinearities, however, can be particularly beneficial for system interrogation. Recently, augmented linear systems have been proposed by the authors to handle discrete nonlinearities (D'Souza & Epureanu 2005, 2007, 2008, in press). These augmented (fictitious) systems are of higher dimension than the corresponding nonlinear (physical) system, and are designed to follow a single trajectory of the nonlinear system when projected onto the original (physical) space. The authors have used the augmentation together with the approaches that use mode-shape information (D'Souza & Epureanu 2005, 2007, 2008) and frequency shift-based approaches (D'Souza & Epureanu in press). When handling nonlinear systems using optimal feedback auxiliary signals, the authors discovered several benefits of system augmentation combined with optimal feedback auxiliary signals beyond handling nonlinear systems. For instance, nonlinear feedback auxiliary signals (NFASs) provide the ability to control the augmented (fictitious) degrees of freedom without changing the input actuation. Also, the system augmentation approach provides the ability to use unstable augmented (fictitious) systems to enhance sensitivity (while maintaining the stability of the physical linearized system). These features are key enablers for the approach proposed herein, where optimal NFASs are designed and used for identifying parameter variations in *linear* systems. This work is distinct from (and complements) other results of the authors where nonlinearity (with given functional form and characteristics) was already present in the interrogated system, and where that nonlinearity was tailored through feedback controllers (D'Souza & Epureanu in press).

In this work, NFASs are optimally designed and applied to *linear* systems by means of system augmentation. The nonlinear feedback signal design developed herein is a strong solution to a very challenging problem (Campbell & Nikoukhah 2004) in the vibration-based identification of changes in structural parameters (which is currently used in a wide variety of technologies). In particular, two areas, sensing and damage detection, focus closely on identifying parameter variations such as mass and stiffness by exploiting variations in resonant frequencies. For example, recent sensing techniques for chemical and biological detection as well as atomic force microscopes in tapping mode (Wolf & Gottlieb 2002) have used the vibration of microstructures such as microchannel resonators (Burg *et al*. 2007; Godin *et al*. 2007), microbeams (DeVoe 2001) and microcantilevers (Chen *et al*. 1995; Thundat *et al*. 1995; Ilic *et al*. 2001; Gupta *et al*. 2004; Braun *et al*. 2005). These sensing and detection methods based on frequency shifts (Salawu 1997) have recently become of increasing interest. They have been developed because frequency extraction can be done robustly for both micro- and large-scale applications. Embedding (nonlinear) controllers within the interrogated system also enables additional applications of NFASs such as vibration confinement (Tang & Wang 2004) and vibration suppression (Ray *et al*. 2000).

As an application for the approach herein, damage detection is discussed in §3. An overview of damage detection (and structural health monitoring applied to civil infrastructure) is provided by Brownjohn (2007). Currently, many existent damage detection approaches are model based (Friswell *et al*. 2001; Yin & Epureanu 2006; Friswell 2007). These approaches use a model of the healthy structure along with various features of the damaged structure (e.g. mode shapes, mode shape curvature, natural frequencies) to assess changes in structural properties. Here, the results focus on this application because a general powerful approach to interrogate the monitored system is essential to the success of damage detection.

The systems explored herein are linear mass–spring and mass–spring–damper ones. In addition to the benefits of using nonlinear control to enable the use of system augmentation, the effects of modelling generalized damping in the optimization algorithm are discussed. Also, a more realistic output feedback approach is presented, which allows for the eigenstructure assignment to be carried out with an incomplete set of measurements (where only a few displacements need to be measured). Additionally, the optimal feedback auxiliary signals have been adaptively designed to the expected level of parameter variation using a linearity constraint in the optimization algorithm. Numerical simulations are presented, and the effects of random noise are discussed.

## 2. Methodology

In this section, the procedure for using system augmentation and optimal NFASs for linear systems is explained. First, the implementation of system augmentation to linear systems is discussed. Next, a frequency shift-based method for identifying parameter variations is reviewed. Then, the eigenstructure assignment technique is detailed. Finally, the optimization algorithm that determines the parameters of the optimal NFASs and system augmentation is presented.

### (a) System augmentation for linear systems using NFASs

A detailed explanation of system augmentation for nonlinear systems can be found in the previous works of the authors (D'Souza & Epureanu 2005, 2007, 2008, in press). A key attribute of all augmented systems is that they have a higher dimension than their corresponding nonlinear system. Also, each augmented system is designed to follow a single trajectory of the nonlinear system when projected onto the original (physical) subspace.

As a simple example, consider a single degree-of-freedom linear mass–spring–damper system excited by a forcing *g*(*t*). The equation of motion of the linear system is given by(2.1)where *m*, *d*, *k* and *x* correspond to the mass, damping, stiffness and position, respectively, and *t* is the time. Figure 1*a* depicts this linear system with a nonlinear controller given by , where *K*_{CL} is the linear control gain and *K*_{CN} is the nonlinear (cubic) control gain. The nonlinear equation of motion of the closed-loop system is(2.2)Since the closed-loop system is nonlinear, the usual linear methods do not apply. However, system augmentation can be applied. Figure 1*b* depicts an augmented system that can be created for equation (2.2), with the following equations of motion:(2.3)where *m*_{a}, *k*_{c}, *k*_{a}, *y* and *h*(*t*) are the augmented mass, coupled stiffness, augmented stiffness, augmented variable and augmented forcing, respectively, and *k*′=*k*+*K*_{CL}. The constants *m*_{a}, *k*_{c} and *k*_{a} are chosen to provide sensitivity enhancement and their optimization is discussed in §2*a*,*c*,*d*. The augmented variable is chosen as the nonlinearity, i.e. *y*=*x*^{3}. The augmented forcing *h*(*t*) is calculated directly from the left-hand side of equation (2.3) since *m*_{a}, *k*_{c} and *k*_{a} are known (chosen), *x*(*t*) is measured and *y*(*t*) is calculated from its (known) dependence on *x*(*t*).

The augmented forcing *h*(*t*) constrains the higher dimensional (augmented) system to follow a single trajectory of the nonlinear system when projected onto the lower dimensional (physical) space. The use of this augmented forcing is also the reason why frequency extraction or modal analysis techniques that are used with augmented systems must be input–output approaches (as opposed to output-only approaches). For example, a multi-input multi-output approach called DSPI (Leuridan 1984) has been used successfully with augmented systems to extract both natural frequencies and mode shapes (D'Souza & Epureanu 2005, 2008).

Designing optimal NFASs by means of system augmentation has several advantages over standard linear feedback (LF) excitation. Consider a general multiple degree-of-freedom linear system that is actuated upon by a multi-input nonlinear controller. The equations of motion can be expressed as(2.4)where ** M**,

**and**

*D***are the linear mass, damping and stiffness matrices of the physical system, respectively;**

*K*

*N*_{AI},

*N*_{AD},

*N*_{AS}are the augmented matrices that contain parameters of the augmentation (such as

*m*

_{a}and

*k*

_{a});

*N*_{CS}is the coupled stiffness matrix (which can be used to maintain the symmetry of the augmented system matrices for the healthy structure); and

*K*_{CL}and

*K*_{CN}are the linear and nonlinear control gain matrices. In contrast to equation (2.4), a linear system with standard LF excitation would have the following equation of motion:(2.5)

There are several advantages of using a system characterized by equation (2.4) rather than equation (2.5). First, the additional inputs in equation (2.4) (all the augmented degrees of freedom ** y**) allow for additional controller configurations. This can significantly increase the amount of frequency information, which in turn helps identifying more types of parameters (which may vary simultaneously). Second, having displacement information and control over additional degrees of freedom (all the augmented degrees of freedom

**) helps significantly when performing eigenstructure assignment without full-state feedback. Also, the placement of**

*y*

*N*_{AI},

*N*_{AD},

*N*_{CS}and

*N*_{AS}requires no actuation energy because those matrices affect only the signals used to calculate the augmented forcing

**. Finally, the system that must be stable during interrogation is the physical linearized system, while the fictitious augmented system in equation (2.4) does not have to be stable. This can greatly increase the performance of the interrogation. Note that a physical linearized system is readily extractable from the augmented system by setting the nonlinear terms to zero and removing the augmented equations.**

*h*### (b) First-order frequency-shift based methods

In this work, parameter variations are identified using a first-order frequency shift-based method. This method has been used in the literature with sensitivity-enhancing feedback control (Koh & Ray 2004; Jiang *et al*. 2007) and optimal NFASs (D'Souza & Epureanu in press). A summary of this approach is presented next. In general, perturbations Δ** p** to parameters lead to changes Δ

**in the resonant frequencies, which can be expressed as(2.6)where**

*ω***is the first-order sensitivity matrix and**

*S***(Δ**

*N***) is a nonlinear function of the change Δ**

*p***in parameters. The first-order perturbation approach neglects the nonlinear term**

*p***. The entries of matrix**

*N***are expressed as , where**

*S**i*=1, …,

*q*and

*j*=1, …,

*r*, where

*r*represents the number of variable parameters

*p*

_{i}and

*q*represents the number of measurable frequencies

*ω*

_{j}. Thus, Δ

**is a vector of dimension**

*p**r*, while Δ

**is a vector of dimension**

*ω**q*. The first-order perturbation approximation is expected to be valid for small parameter variations. In this work, a linearity constraint is added to the optimization function to ensure the validity of equation (2.6) up to the desired/expected level of parameter variation. Assuming that the matrix

**has full rank, one may use the pseudo-inverse of**

*S***to solve equation (2.6) for the unknown vector of parameter variations as Δ**

*S***=**

*p*

*S*^{+}Δ

**.**

*ω*Designing optimal NFASs for linear systems has two goals. The first goal is to create multiple independent closed-loop systems to increase the amount of measured frequency information. This is particularly important when the number of variable parameters *r* is greater than the number of measured frequencies *q*. The independent closed-loop sensitivity matrices (denoted by *S*^{ci}) can be combined to form the overall sensitivity matrix expressed as , where *u* is the number of unique controller configurations used and the superscript T indicates a transpose. When ** S** is full rank and

*q*.

*u*>

*r*, then an overdetermined set of equations exist, and Δ

**can be obtained as Δ**

*p***=**

*p*

*S*^{+}Δ

**. The second goal is to increase the sensitivity of the closed-loop sensitivity matrices (denoted by**

*ω*

*S*^{ci}) compared with the open-loop sensitivity matrix.

### (c) Eigenstructure assignment via singular value decomposition

Eigenstructure assignment via singular value decomposition has been studied extensively for linear systems (Cunningham 1980; Shelley & Clark 2000*a*,*b*; Tang & Wang 2004; Wang & Wang 2007). The following is a brief overview of eigenstructure assignment that follows closely the work presented by Jiang *et al*. (2007), but it is extended for augmented systems and includes the option for the direct output feedback of Juang *et al*. (1989; as opposed to full-state feedback). Consider equation (2.4) transformed into state space. The augmented equations of motion in state-space form are(2.7)whereand where *M*_{A}, *D*_{A} and *K*_{A} are *N*_{o}×*N*_{o} augmented mass, damping and stiffness matrices, respectively; ** x** and

**are the**

*y**N*×1 coordinate vector and

*n*×1 augmented variable vector;

**and**

*g***are the**

*h**N*×1 physical forcing and

*n*×1 augmented forcing; and

**,**

*B***and**

*C***are the 2**

*H**N*

_{o}×

*c*controller input,

*c*×

*s*control and

*s*×2

*N*

_{o}sensor matrices, respectively. Here,

*c*is the number of actuators and

*s*is the number of sensors used. Note that

*N*

_{o}=

*N*+

*n*, and the control gain matrix is defined as

*K*_{C}=

**. Also, the bottom**

*CH**N*

_{o}rows of the controller input matrix

**are always zero. Equation (2.7) can be written as(2.8)whereThe eigenvalue problem for the augmented closed-loop system can be expressed as(2.9)where**

*B**ω*

_{cj}is the

*j*th closed-loop eigenvalue and

*ϕ*

_{j}is the corresponding eigenvector. Equation (2.9) can alternatively be written as(2.10)which means that must fall into the null space of . The symbol | indicates a partition in the matrix; for example, the matrix is composed of two matrices side by side.

Next, perform a singular value decomposition of to obtain(2.11)where *U*_{j} and *V*_{j} correspond to the left and right singular matrices, respectively, and *D*_{j} is a diagonal matrix that contains the singular values. The superscript asterisk indicates a complex conjugate. The right singular matrix can be partitioned into four sub-matrices(2.12)where =2*N*_{o}×2*N*_{o}, =2*N*_{o}×*c*, =*c*×2*N*_{o}, and =*c*×*c*.

To exploit the orthogonality property of the singular value decomposition, one can multiply equation (2.11) by to obtain(2.13)

Next, a coefficient vector *f*_{j} can be defined for the *j*th eigenvector in its admissible subspace. This subspace is the span of , as shown in equation (2.13). Hence, the assigned eigenvector can be expressed as , while . Next, gather all 2*N*_{o} eigen-solutions for all *j* to obtain *CH**Φ*^{a}=** W**, where ; ; and , . Then, invert

*HΦ*^{a}to solve for the control matrix as , where the superscript + indicates the pseudo-inverse. Finally, the control gain matrix

*K*_{C}can then be calculated as

*K*_{C}=

**.**

*CH*The number of frequencies that can be placed by this technique is limited for the case where full-state feedback is not employed (i.e. when *s*<2*N*_{o}). The number of assignable frequencies *q* was given by Juang *et al*. (1989) as 2*q*≤max(*c*, *s*).

### (d) Optimization algorithm for sensitivity enhancement

The relationship between eigenvectors and resonant frequencies (with sensitivity enhancement) on one side and control effort on the other is complex in the multiple degree-of-freedom case (Jiang *et al*. 2007). As a result, optimization algorithms are used to design *K*_{C} to place the frequencies and eigenvectors optimally by maximizing sensitivity while minimizing control effort. In this work, a linearity constraint is also included in the optimization algorithm to enforce the linearity needed when using equation (2.6). The optimization algorithm follows closely the algorithm presented previously (D'Souza & Epureanu in press; Jiang *et al*. 2007), which maximizes the sensitivity while minimizing the control effort.

The parameters optimized are the coefficient vectors *f*_{j}, the frequencies *ω*_{cj} of the *q* measurable frequencies and the *n* augmented masses (where one assumes that the augmented mass matrix is diagonal). Primarily, for computational reasons, the last two sets of these parameters are defined relative to their open-loop or nominal values. Hence, the optimization is applied to parameters *γ*_{j} and *Γ*_{i}, defined by and , where *ω*_{oj} are the open-loop eigenvalues of the system, *m*_{ai} are the nominal values for the augmented masses (values chosen by the user are typically close to the value of the mass at the degree of freedom the nonlinearity originates from), and *j*=1, 2, …, *q*; *i*=1, 2, …, *n*. Note that, in general, the augmented equations are not unique for a given nonlinear system; in fact, multiple augmentations of a single nonlinear system have been exploited for damage detection (D'Souza & Epureanu 2008). Also, note that the stiffness portion of the augmentation is already being optimized via the calculation of the rows in the gain matrix, which correspond to the augmented equations.

The assigned closed-loop eigenvectors can be expressed as for *j*=1, 2, …, *q*. The remainder of the eigenvectors (for *j*=*q*+1, *q*+2, …, *N*_{o}) have to be placed as close as possible to the open-loop eigenvectors to minimize the control effort. Hence, one searches for the coefficient vectors so that is as close as possible to . The coefficient vectors can be found by minimizing (and ). The coefficient vectors are obtained as . Finally, the closest achievable eigenvector is given by , where *j*=*q*+1, *q*+2, …, *N*_{o}.

The optimization cost function (which is maximized) includes the level of sensitivity enhancement. This level is defined as the sum of the (absolute values of the) element-by-element ratios of the closed-loop sensitivity matrices *S*^{ci} to the open-loop sensitivity matrix, divided by the number of elements. Also, the singular values of ** S** are maximized (particularly, the minimum singular value). This is important when noisy data are used, and in the cases where very few measurements are taken.

The control effort for the nonlinear feedback actuation is optimized by minimizing the absolute value of the maximum entry in the controller gain matrix *K*_{C}. The rows of the gain matrix that correspond to the augmented equations are not included in the controller effort because the augmented degrees of freedom do not require any physical actuation (but just signal processing). The relative importance of these components of the optimization is taken into account by weighing the sensitivity enhancement with a coefficient *c*_{1} and the control effort with a coefficient *c*_{2}.

The optimization is subject to stability constraints because the physical system cannot be unstable during its interrogation. Note that the stability of the physical linearized system is (related to but) independent of the stability of the augmented system. The important issue is the stability of the physical linearized system. This is ensured by a constraint in the optimization algorithm. Note that the instability in the augmented system is due to the augmentation and not the physics. Hence, it is possible for an augmented system to be unstable while the physical system is stable. The reason why the augmented system can be unstable while the response of the physical system remains stable is the specialized augmented forcing that limits the growth of the motion of the augmented system. Thus, the important issue is ensuring the stability of the linearized physical system, whereas the augmented linear system does not have to be stable. The linearized physical system must be stable both when it is healthy and after each damage scenario. Thus, the optimization is subject to the following constraints:where are the linearized closed-loop eigenvalues of the baseline system and are the linearized closed-loop eigenvalues of the system with changed parameters. These constraints must be satisfied for the maximum allowable (expected) level of parameter variations for all *r* scenarios of parameter variations.

Furthermore, in order for equation (2.6) to hold, a relationship close to linear must exist between Δ** ω** and Δ

**. A linearity constraint can be included into the optimization algorithm to enforce this relationship. In this work, cost was added proportional to . Here, is obtained for a small change in parameter**

*p**p*

_{j}, and is obtained for the maximum allowable damage level for parameter

*p*

_{j}.

An important aspect of any frequency-based method is ensuring that the sensitivity matrix used for parameter reconstruction is full rank. Hence, the optimization process can include maximizing the sensitivity matrices *S*^{ci} (which compose ** S**) while also maximizing the minimum singular value of

**. Maximizing the minimum singular value of**

*S***in the optimization process ensures that this overall sensitivity matrix is full rank (when the minimum singular value is large). Hence, the independence of the sensitivities of the closed-loop systems is ensured as closely as possible. However, given the limited measurements and control input actuators, it is possible that only a subset of the parameters is identifiable, and variations in the unidentifiable parameters have to be detected by other means.**

*S*## 3. Results

To demonstrate the proposed methods, numerical simulations were performed for the systems shown in figures 2 and 3. The system in figure 2 is a six degree-of-freedom linear mass–spring system. Considering the parameter identification as a damage detection problem, the damageable elements consist of six linear springs with stiffnesses denoted as *p*_{1}=*k*_{1g}, *p*_{2}=*k*_{12}, *p*_{3}=*k*_{23}, *p*_{4}=*k*_{34}, *p*_{5}=*k*_{45} and *p*_{6}=*k*_{56}. The controller input actuators were located at the first and sixth masses. The system in figure 3 is similar to the one shown in figure 2, with six dampers placed between the masses and ground. Augmented linear systems were created by generating matrices ** M**,

**,**

*D***,**

*K*

*K*_{CL},

*K*_{CN},

*N*_{AI},

*N*_{AD},

*N*_{AS}and

*N*_{CS}in equation (2.4). The non-zero entries of the

*K*_{CN}matrix consist of cubic nonlinearities that are generated using nonlinear feedback control. The matrix

*N*_{AI}consists of the augmented masses. The nominal values for the augmented masses are chosen to have the same values as the (physical) masses they are coupled to. The matrix

*N*_{CS}is constructed to create a symmetric augmentation for the nominal (healthy) system. The matrix

*N*_{AS}is a diagonal matrix that contains the terms

*k*

_{a}chosen as

*k*

_{a}=2

*k*

_{n}. The matrix

*N*_{AD}is a zero matrix. The matrices

**,**

*M***and**

*D***have been constructed for the physical uncontrolled system using the parameters given in figures 2 and 3. The entries in the matrices**

*K*

*K*_{CL},

*K*_{CN},

*N*_{AI},

*N*_{CS}and

*N*_{AS}are all then optimized. The constrained optimization problem was solved using the

`fmincon`function in Matlab (Coleman

*et al*. 1999). Note that the global optimum was not always found owing to the initial guesses used. Three controllers (NFAS1–3) were designed to maximize the sensitivity of the first three frequencies of the system to changes in the six (damageable) elements. The different combinations of feedback auxiliary signals are given by actuation locations as follows: NFAS1, [1, 6, 7, 8]; NFAS2, [1, 7, 8]; and NFAS3, [6, 7, 8]. Note that locations 7 and 8 correspond to the augmented degrees of freedom, which require no actual actuation, but only signal processing. Additionally, three different combinations of LF controllers were also used: LF1, [1, 6]; LF2, [1]; and LF3, [6].

### (a) Frequency extraction for unstable augmented system

In this section, an example of how the frequencies of an augmented system can be extracted using DSPI (Leuridan 1984), a time-based multi-input multi-output modal analysis technique, is demonstrated. In particular, the system explored was physically stable during its interrogation. However, the augmented system was unstable. An unstable augmented system can be physically stable because the augmented system is a fictitious system excited by a specific augmented forcing. In the case of the unstable augmented system, the augmented forcing actually stabilizes the response of the augmented system such that the response is bounded.

Table 1 shows the eigenvalues of a physically stable system characterized by an unstable augmented system. These eigenvalues are those of a six degree-of-freedom mass–spring system (shown in figure 2) with NFASs applied to the first and sixth masses. The first column of table 1 consists of the eigenvalues of the linearized system. There are six eigenvalues because there are six physical degrees of freedom. The second column of table 1 consists of the exact eigenvalues of the augmented system. There are eight eigenvalues for the augmented system: six for the physical degrees of freedom and two for the augmented degrees of freedom (one for each nonlinearity). The last column of table 1 consists of the eigenvalues extracted using DSPI, when the physical system is forced by harmonic excitation at all six physical degrees of freedom. Comparing the second and third columns of table 1, it is clear that DSPI can accurately extract the eigenvalues of an unstable augmented system. Figure 4 shows the response of the six physical degrees of freedom of the system to the harmonic excitation used.

### (b) LF versus optimal NFASs

In this section, a comparison of LF excitation and optimal NFASs is provided. In these results, the measurements collected are the positions of the six linear degrees of freedom (the velocities of the masses are not measured). The weighting factors were chosen as *c*_{1}=10^{3} and *c*_{2}=10^{−4}.

The design variables (*γ*_{1}, *γ*_{2}, *γ*_{3}, *f*_{1}, *f*_{2}, *f*_{3}, *Γ*_{1} and *Γ*_{2}) for the LF controller and the NFASs are shown in table 2. The optimal closed-loop augmented mass parameters are not listed in the LF section of the table since no augmentation is created for the case of LF applied to a linear system. It should be noted that the control effort is considerably (4–10 times) larger for the cases where LF is used.

The results in figure 5 show the increased performance when using NFASs and system augmentation versus using just LF excitation for the six damage scenarios. The *x*-axis in each plot corresponds to the damaged elements, while the *y*-axis in each plot corresponds to the percent damage. Each plot shows the exact value of the damage and the average predicted damage for LF and NFASs. One hundred separate calculations were performed for the case of ±0.25 (approx. 1% of the lowest frequency) random noise applied to the frequencies of the damaged system. Standard deviation error bars are plotted. Note that elements 4, 5 and 6 have smaller error bars when NFASs are used instead of standard LF. Also, elements 2 and 3 have slightly decreased performance, but this is a small issue compared with the significant improvement at elements 4, 5 and 6, and the fact that the control effort required when using the nonlinear feedback is 4–10 times smaller than that required by the LF.

### (c) Linearity and sensitivity

In this section, the relationships between the linearity constraint, the sensitivity to noise and the damage level are explored. Two sets of controllers were designed to maximize the sensitivity of frequencies. The first set of controllers 0.1 NFAS1–3 was designed to maximize the sensitivity of the first three frequencies to a 0.1 per cent change in parameters. The second set of controllers 5.0 NFAS1–3 was designed to maximize the sensitivity of the first three frequencies to a 5 per cent change in parameters. In each case, the positions of the six masses were used for feedback.

The relationship between sensitivity and linearity of each set of controllers is shown in figure 6. The plots show the changes in the first frequency for damages in one linear spring for the open-loop and each set of closed-loop systems (i.e. 0.1 NFAS1–3 and 5.0 NFAS1–3). The *x*-axis in each plot corresponds to the change in parameter *p*_{1}, while the *y*-axis in each plot corresponds to the change in the first frequency *ω*_{1}. Figure 6*a*,*b* corresponds to a parameter change of up to 0.1 per cent, while figure 6*c*,*d* corresponds to a parameter change of up to 5 per cent. Figure 6*a*,*b* shows the linear relationship between the change in frequency and the change in parameter required by equation (2.6). These results also show that the sensitivity of the controllers designed for a 0.1 per cent parameter change (0.1 NFAS1–3) is much larger than that designed for a 5 per cent parameter change (5.0 NFAS1–3). Figure 6*c*,*d* shows that the controllers designed for a 0.1 per cent parameter change do not have a linear relationship between the change in frequency and the change in parameter *p*_{1} over the 5 per cent range, while the linear relationship does exist for 5.0 NFAS1–3. Similar results can be obtained for the other two frequencies and five parameters, but are omitted here for the sake of brevity.

The results in figure 7 show the excellent performance of the nonlinear approach for two different cases. The first case consists of a 0.1 per cent damage using the 0.1 NFAS1–3 controllers. One hundred separate calculations were performed for the case of ±0.1 random noise in the measured frequencies. The second case consists of 5 per cent damage using the 5.0 NFAS1–3 controllers with ±1.0 random noise applied to the measured frequencies. The plots are laid out in a manner similar to that shown in figure 5. The *y*-axis consists of a normalized damage (the damage for each case is divided by the maximum damage) instead of a per cent damage. Standard deviation error bars are plotted for both noisy cases.

### (d) Modelling generalized damping

In this section, the effects of modelling damping in the optimization process are considered. The mass–spring–damper system explored is shown in figure 3. The form of damping included in the system is a generalized linear damping. The more challenging case of generalized damping is investigated instead of the common proportional damping. Note that the modes of a system with generalized damping are not the same as the ones for the undamped system. By contrast, the modes of a proportionally damped system are the same with or without proportional damping.

The results in figure 8 compare the performance of the approach for two different control optimizations. The first corresponds to a case where the generalized damping is not modelled in the optimization algorithm. The second corresponds to a case where the generalized damping is modelled in the optimization algorithm. In both cases, the only measured locations are the positions of the first and sixth degrees of freedom. In addition to these linear degrees of freedom, the positions of the two augmented degrees of freedom are also known (by their relation to the positions of the first and sixth degrees of freedom). Owing to the eigenvector assignment constraint 2*q*≤max(*c*, *s*), the maximum number of frequencies and mode shapes *q* that can be assigned when *s*=4 and *c*=4 is *q*=2. Note that, if a purely linear control were used, *s* and *c* would be equal to 2 and the maximum number of frequencies and mode shapes that could be placed would be 1, which would lead to poor damage detection performance.

For these results, the optimization algorithm included the maximization of the minimum singular value of the sensitivity matrix. However, given the limited measurements and control input actuators (sensors and actuators only at the first and sixth masses), only five parameters of the six were identifiable. This indicates that the changes in the two frequencies for the three control configurations could not create a matrix ** S** of full rank for all six parameters. Therefore, the parameter

*p*

_{6}was removed as a damageable parameter. One hundred separate calculations were performed for the case of ±0.12 random noise (which corresponds to approx. 2% noise in the lowest resonant frequency) in the frequencies of the damaged system. The plots are laid out in a manner similar to that shown in figure 5. Standard deviation error bars are plotted for both noisy cases.

## 4. Conclusions and discussion

A novel approach for improving the sensitivity enhancement for linear systems using optimal NFASs and system augmentation was presented. The NFASs apply linear and nonlinear feedback to the structure. The nonlinearity is handled with system augmentation, which consists of higher dimensional linear models for a trajectory of the physical nonlinear system.

The optimal NFASs have been adaptively designed to the level of parameter variation expected. In general, some feedback signals work better for larger parameter variations, when the needed level of sensitivity enhancement is low. However, those signals are not effective for cases where very small parameter variations are expected. There, a hyperenhancement of sensitivity is needed, and that can be accomplished only by distinct feedback signals. If one naively applies the hypersensitive signals to large parameter variations, failures occur in the interrogation because the assumption of linearity of the frequency shifts with respect to parameter variations is no longer accurate. The adaptive design of the feedback signals is done by a linearity constraint built into the optimization algorithm. This constraint ensures the linearity for any level of parameter variation below the expected values (and it also alleviates other limits on the frequencies of the augmented system).

The eigenstructure assignment used in the design of the feedback signals was done using only partial measurements (of an augmented model). This is an important advantage over other techniques because in many practical implementations only a few degrees of freedom can be measured.

The optimal NFASs have been applied to linear systems. Hence, the benefits of nonlinearities compared to traditional linear sensitivity-enhancing approaches can be clearly identified. These benefits include a reduced number of actuator points and a reduced number of sensor locations needed.

The NFASs and the system augmentation approach also have the benefit of allowing for physically stable systems to be characterized by unstable augmented systems. Although the augmented system is unstable, the response of this system is still bounded because the linearized physical system is actually stable. The response of the augmented system is bounded because the specified augmented forcing stabilizes the response of the system. Furthermore, the frequencies of these systems can be found using linear multi-input multi-output approaches such as DSPI.

The ability to place the augmented degrees of freedom arbitrarily without extra controller effort is an additional benefit of the proposed approach. This allows for additional controller configurations to diversify the frequency information. Moreover, the added ‘control’ helps when there are limited numbers of sensors and actuators. The eigenvector assignment procedure allows for the placement of *q* frequencies and modes, where 2*q* is less than or equal to the number of actuators or sensors.

The benefits of modelling generalized damping in the optimization algorithm were shown. A state-space formulation of the equations of motion is required for the optimization algorithm when generalized damping is present. However, the benefits for damage detection are significant. When damping is ignored in the optimization algorithm, the optimizer places the poles and modes optimally for the wrong system, which leads to a lower performance.

To demonstrate the use of NFASs and system augmentation for sensitivity enhancement, a damage detection problem has been studied for linear mass–spring and mass–spring–damper systems. The effects of the level of damage and measurement noise were also discussed.

## Acknowledgments

The authors wish to acknowledge the National Science Foundation (CAREER and Graduate Research Fellowship programmes) for the generous support of this work.

## Footnotes

- Received February 19, 2008.
- Accepted July 1, 2008.

- © 2008 The Royal Society