Microsecond State Monitoring of Nonlinear Time-Varying Dynamic Systems" (2017)

Home , Microsecond

Civil, Construction and Environmental Engineering Civil, Construction and Environmental Engineering Conference Presentations and Proceedings

2017 Microsecond State Monitoring of Nonlinear Time- Varying Dynamic Systems Jacob Dodson Air Force Research Laboratory

Bryan Joyce University of Dayton Research Institute

Applied Research Associates Inc.

Simon Laflamme Iowa State University, [email protected]

Janet Wolfson Air Force Research Laboratory

Follow this and additional works at: https://lib.dr.iastate.edu/ccee_conf Part of the Civil Engineering Commons, Dynamics and Dynamical Systems Commons, Structural Engineering Commons, and the VLSI and Circuits, Embedded and Hardware Systems Commons

Recommended Citation Dodson, Jacob; Joyce, Bryan; Applied Research Associates Inc.; Laflamme, Simon; and Wolfson, Janet, "Microsecond State Monitoring of Nonlinear Time-Varying Dynamic Systems" (2017). Civil, Construction and Environmental Engineering Conference Presentations and Proceedings. 67. https://lib.dr.iastate.edu/ccee_conf/67

This Conference Proceeding is brought to you for free and open access by the Civil, Construction and Environmental Engineering at Iowa State University Digital Repository. It has been accepted for inclusion in Civil, Construction and Environmental Engineering Conference Presentations and Proceedings by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. Microsecond State Monitoring of Nonlinear Time-Varying Dynamic Systems

Abstract Reliable operation of next generation high-speed complex structures (e.g. hypersonic air vehicles, space structures, and weapons) relies on the development of microsecond structural health monitoring (μSHM) systems. High amplitude impacts may damage or alter the structure, and therefore change the underlying system configuration and the dynamic response of these systems. While state-of-the-art structural health monitoring (SHM) systems can measure structures which change on the order of seconds to minutes, there are no real-time methods for detection and characterization of damage in the microsecond timescales.

This paper presents preliminary analysis addressing the need for microsecond detection of state and parameter changes. A background of current SHM methods is presented, and the need for high rate, adaptive state estimators is illustrated. Example observers are tested on simulations of a two-degree of freedom system with a nonlinear, time-varying stiffness coupling the two masses. These results illustrate some of the challenges facing high speed damage detection.

Keywords Dynamic systems, Structural health monitoring, Damage, Weapons, Simulation, Space frame structures, Engineering simulation, Vehicles, Dynamic response, Stiffness

Disciplines Civil Engineering | Dynamics and Dynamical Systems | Structural Engineering | VLSI and Circuits, Embedded and Hardware Systems

Comments This proceeding is published as Dodson, Jacob, Bryan Joyce, Jonathan Hong, Simon Laflamme, and Janet Wolfson. "Microsecond State Monitoring of Nonlinear Time-Varying Dynamic Systems." In ASME 2017 Conference on Smart Materials, Adaptive Structures and Intelligent Systems, pp. V002T05A013-V002T05A013. American Society of Mechanical Engineers, 2017. DOI: 10.1115/ SMASIS2017-3999. Posted with permission.

Rights Works produced by employees of the U.S. Government as part of their official duties are not copyrighted within the U.S. The onc tent of this document is not copyrighted.

This conference proceeding is available at Iowa State University Digital Repository: https://lib.dr.iastate.edu/ccee_conf/67 Proceedings of the ASME 2017 Conference on Smart Materials, Adaptive Structures and Intelligent Systems SMASIS2017 September 18-20, 2017, Snowbird, UT, USA

SMASIS2017-3999

MICROSECOND STATE MONITORING OF NONLINEAR TIME-VARYING DYNAMIC SYSTEMS

Jacob Dodson Bryan Joyce Air Force Research Laboratory University of Dayton Research Institute Eglin AFB, FL, USA Eglin AFB, FL, USA

Jonathan Hong Simon Laflamme Janet Wolfson Applied Research Associates Inc. Iowa State University Air Force Research Laboratory Niceville, FL 32578, USA Ames, Iowa, USA Eglin AFB, FL, USA

ABSTRACT remaining useful life, and allow for mitigating the risk of Reliable operation of next generation high-speed complex destructive circumstances [3]. Current SHM systems can structures (e.g. hypersonic air vehicles, space structures, and measure and process slowly varying structures on the order of weapons) relies on the development of microsecond structural seconds to minutes. While system state estimation and control health monitoring (μSHM) systems. High amplitude impacts techniques are capable of operating on a microsecond scale may damage or alter the structure, and therefore change the [4], there are no existing, real-time methods that can detect underlying system configuration and the dynamic response of and characterize damage in complex, time-varying structures these systems. While state-of-the-art structural health in microseconds timescales. The development of microsecond monitoring (SHM) systems can measure structures which SHM (μSHM) methods on the timescales of 10 μs to 10 ms change on the order of seconds to minutes, there are no real- will allow for the structural integrity of a system to be time methods for detection and characterization of damage in monitored on timescales where speed of damage detection is the microsecond timescales. critical. This paper presents preliminary analysis addressing the Several challenges arise when detecting and predicting need for microsecond detection of state and parameter damage at these timescales. As summarized in a recent changes. A background of current SHM methods is presented, newsletter by the authors [5], the feasibility of microsecond and the need for high rate, adaptive state estimators is prognostics must account for high rate physics of failure, illustrated. Example observers are tested on simulations of a complexity of the rate-dependent structural dynamics, high two-degree of freedom system with a nonlinear, time-varying rate real-time processing requirements, identifying optimal stiffness coupling the two masses. These results illustrate quantity and placement of sensors [6], and uncertainties some of the challenges facing high speed damage detection. throughout the system (material, structural, history/lifecycle, etc.). This paper will outline some of the high rate, state-of-the- INTRODUCTION art state estimation and SHM techniques currently in use. A comparative study of observers for induction motors is given Hypervelocity air vehicles, space structures, and weapon to demonstrate the feasibility of high rate damage detection systems are subject to extremely high speed impacts and highlight some of the capabilities and limitations seen in (>4 km/s), causing damage to propagate through the structures high rate observers. A Luenberger observer and a Kalman in microseconds [1, 2]. Here, damage refers to a change in the filter are developed that use output feedback to adapt the structure’s configuration, material failure, or change in the observer parameters. These observers are simulated on a two system boundary conditions. High amplitude impacts may degree of freedom, time-varying system, and performance cause damage that alters the underlying dynamic response of limitations are demonstrated. the structure, resulting in loss of system functionality or other severe consequences. Structural health monitoring (SHM) and prediction systems with integrated sensor networks can identify changes in the operation of a system, predict

1 Copyright © 2017 ASME This work was authored in part by a U.S. Government employee in the scope of his/her employment. ASME disclaims all interest in the U.S. Government’s contribution. Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 02/20/2018 Terms of Use: http://www.asme.org/about-asme/terms-of-use CURRENT STATE-OF-THE-ART applications of state estimation for fault detection and prognosis, data-based methods generally only provide a This section will discuss some of the current methods sequential measure of fault based on pattern recognition and used to detect system damage and provide examples of high classification. Alternatively, they require precise examples speed estimators for induction motors. and extensive training over available data sets to associate data features to types and measures of fault. Statistical methods, such as the least squares estimator Microsecond Structural Health Monitoring (LSE), the maximum likelihood estimator (MLE), and particle filters, provide probabilistic predictions based on known There is a need for adaptive, model-based approaches parameters. They bypass the need for linearized dynamic which incorporate relevant, nonlinear, rate-dependent equations allowing global convergence of estimations. A phenomena and methods of handling uncertainties of the general limitation of statistical methods is their reliance on system dynamics into a fast-running microsecond state available data sets for training and/or extraction of probability estimator. distribution functions. Statistical methods can identify faults The authors [7-10] and collaborators [11, 12] have through a probabilistic measure, and they may be used to previously conducted preliminary work on health monitoring conduct prognosis by evaluating the probability of faults, but methods for detecting time-varying damage at the require knowledge of probability distribution functions. microsecond timescale (10 µs to 10 ms) and have illustrated Model-based observers use a numerical representation of areas which need further technical development. One the system to be identified. Model-based methods include the empirical-based damage detection method used milliseconds Kalman filter (KF), sliding mode observer (SMO), Luenburger of time data, but the damage was stationary in the system and Observers (LO), and variations of these methods for nonlinear existed before the dynamics of the structure illustrated damage systems (e.g., Extended Kalman Filter). Additionally adaptive [8]. Other damage detection methods examine structures that observers (AO) are variations of traditional model-based or are damaged during an impact event (therefore creating time- data-based observers and continuously change the gains and varying systems). In one work, a model was used with strain system parameters with a pre-determined feedback rule. energy methods to detect local stiffness changes in a plate, but These methods have attracted much attention because they the algorithm required large computational resources [7]. produce accurate state estimations when it is not possible or Another technique uses electromechanical impedance (EMI) practical to have sensors to characterize every state [16]. on the microsecond time scale to detect damage [11, 12]. Furthermore, the nominal models required for control and However this EMI method can only detect nearby defects in estimation purposes are readily available [17]. Model-based the structure because the high-frequency vibration used for methods have the advantage of providing precise measures of excitation is heavily damped in typical structures. damage due to the availability of models, therefore enabling There is a clear need for a model-based data fusion condition assessment and system prognosis. However, they method to incorporate local and global measurements in order require knowledge of the physical model, which may add to implement microsecond monitoring and prognosis on a full significant burden on computational time. Nevertheless, these structure. methods are able to provide measures of parameter changes following a fault, which is desirable in damage estimation.

Nonlinear State Observers

Observers are used in the parameter estimation and feedback control communities for estimating the internal states and parameters of a dynamic system using measurements of the system and any applied forces. A block diagram of an observer used for damage detection is shown in Figure 1. The observer takes measurements of the system’s inputs and responses, computes estimates of internal system states and parameters, and feeds this information to an algorithm to determine the presence and severity of structural damage. There are three main categories of observers for nonlinear systems: data-based, statistical, and model-based methods [13, 14]. Data-based methods, such as fuzzy logic and neural network estimators, are methods that process information Figure 1. Block diagram of an observer used for without knowledge of a system’s dynamics. These methods damage detection. are particularly useful for handling highly complex systems. The performance of data-based methods is dependent on the quality of data mining and interpretation algorithms, and these methods are limited by the lengthy computational time required to achieve an appropriate estimate [15]. In

Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 02/20/2018 Terms of Use: http://www.asme.org/about-asme/terms-of-use Adaptive Observers of LO and SMO were essentially identical and were much simpler than the complex calculations of the EKF. Using a Adaptive observers (AO) are a set of methods that are an 150 MHz processor, one update took the LO and SMO about extension of the traditional observers by adding feedback rules 5 µs to achieve while the EKF took about 100 µs. to continuously update the parameters of the observer based Table 1 summarizes the main results from these two on measurements of the system’s inputs and outputs, see studies. Although these numbers are based on the Figure 2 [18, 19]. This quality makes them ideal for handling programmer’s proficiency, it serves as a good example for the uncertainty in state estimation [20]. In particular, adaptive feasibility of high rate estimation and health monitoring of observers have been proposed to estimate the un-measurable complex systems using observers. states for different classes of nonlinear systems [21]. The nature of AOs gives them a unique advantage of asymptotic Table 1. Summary of comparative study of observers for system performance without unnecessary dependence on induction motors. models and in the presence of uncertainties through their ability to adapt to unexpected changes in system conditions. Observer Type Computational Time Ref.

Luenberger 5 µs for one update using [24] Observer (LO) 150 MHz processor Sliding Mode 5 µs for one update using [24] Observer (SMO) 150 MHz processor Adaptive Sliding 19 µs computational time. [23] Observer (ASO) 86 µs computational time Extended Kalman and 100 µs for one update [23, 24] Filter (EKF) using 150 MHz processor.

MODEL-BASED OBSERVERS

The use of adaptive, model-based observers for parameter estimation will be demonstrated on an example system. Two Figure 2. Block diagram of an adaptive observer used for adaptive observers will be examined: one based on a damage detection Luenberger observer and the other based on a Kalman filter. Both estimators will incorporate time-varying system dynamics. Comparative Study of Observers for Induction

Motors

This section compares observers for applications Description of a Mechanical System restricted to induction motors. The induction motor’s highly

nonlinear and uncertain dynamics make it a useful baseline for Consider a second-order, degree of freedom, time- comparing observers applicable to other time-varying dynamic varying dynamic system representing a mechanical structure ( 𝑛𝑛) ( ) system [22]. with displacement vector , input vector , and ( ) In [23], a comparative study of an adaptive sliding measurements vector of the form 𝑞𝑞 𝑡𝑡 𝑢𝑢 𝑡𝑡 observer (ASO) and an extended Kalman filter (EKF) was ( ) ( ) (𝑦𝑦) 𝑡𝑡( ) ( ) ( ) ( ) ( ) conducted for a sensor-less motor drive. The dynamic + + = , (1) performance of the estimators was tested by accelerating the 𝑀𝑀 𝑡𝑡 𝑞𝑞̈ 𝑡𝑡 𝑆𝑆 𝑡𝑡 𝑞𝑞̇ 𝑡𝑡 𝐾𝐾 𝑡𝑡 𝑞𝑞 𝑡𝑡 𝐻𝐻 𝑡𝑡 𝑢𝑢 𝑡𝑡 motor from 478 rpm to 1260 rpm. The results showed the ( ) = ( ) ( ) + ( ) ( ) + ( ), (2) ASO had a computation time of 19 µs, was easy to implement, and produced an accurate estimation. The results of the EKF 𝑦𝑦 𝑡𝑡 𝑃𝑃 𝑡𝑡 𝑞𝑞 𝑡𝑡 𝑁𝑁 𝑡𝑡 𝑞𝑞̇ 𝑡𝑡 𝑤𝑤 𝑡𝑡 showed a computation time of 86 µs, was complicated to where ( ) is the velocity vector, ( ) is the acceleration implement, and produced a more accurate estimation than the vector, ( ), ( ), and ( ) are the mass, damping and adaptive sliding observer. This study exhibits the tradeoff stiffness𝑞𝑞̇ matrices𝑡𝑡 respectively, ( ) 𝑞𝑞̈is 𝑡𝑡the input matrix, ( ) between accuracy and computation time. and (𝑀𝑀) 𝑡𝑡are𝑆𝑆 the𝑡𝑡 output𝐾𝐾 𝑡𝑡matrices associated with the In [24], another comparative study was conducted on a measurement of the displacement𝐻𝐻 𝑡𝑡 and velocity respectively𝑃𝑃 𝑡𝑡 , Luenberger observer (LO), sliding mode observer (SMO), and and the𝑁𝑁 𝑡𝑡overdot denotes time derivatives. The term ( ) the EKF. All observers delivered high accuracy estimations at represents measurement noise. Many of the state estimation high motor speeds. The LO and SMO results were more methods in the literature are designed for systems of 𝑤𝑤first𝑡𝑡- robust to parameter variations than the EKF. The EKF’s order equations. Noting this, define the state vector ( ) as performance proved to be immune to external noise while the

LO and SMO’s performance was excellent only in the ( ) 𝑥𝑥 𝑡𝑡 presence of small external noise. The calculation complexity ( ) = . ( ) (3) 𝑞𝑞 𝑡𝑡 𝑥𝑥 𝑡𝑡 � � 𝑞𝑞̇ 𝑡𝑡

Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 02/20/2018 Terms of Use: http://www.asme.org/about-asme/terms-of-use

The second-order system above can also be written as the Kalman Filter first-order (state-space) system Assume the process noise ( ) and measurement noise ( ) = ( ) ( ) + ( ) ( ) + ( ) ( ), (4) ( ) are white, Gaussian, zero mean, and independent of each other. From these assumptions,𝑣𝑣 the𝑡𝑡 Kalman filter equations 𝑥𝑥̇ 𝑡𝑡 𝐴𝐴 𝑡𝑡 𝑥𝑥 𝑡𝑡 𝐵𝐵 𝑡𝑡 𝑢𝑢 𝑡𝑡 𝐺𝐺 𝑡𝑡 𝑣𝑣 𝑡𝑡 𝑤𝑤can𝑡𝑡 be written as ( ) = ( ) ( ) + ( ), (5) ( ) ( ) 0 ( ) = 𝑦𝑦 𝑡𝑡 𝐶𝐶 𝑡𝑡 𝑥𝑥 𝑡𝑡 𝑤𝑤 𝑡𝑡 ( ) ( ) ( ) ( ) ( ) ( ) ( ) where ( ) is time derivative of the state vector ( ), ( ) is 𝑥𝑥̇ 𝑡𝑡 𝐴𝐴 𝑡𝑡 𝑥𝑥 𝑡𝑡 � � � ( ) ( ) ( ) � � � the state matrix, ( ) is the input matrix, and ( ) is the 𝑥𝑥� ̇ 𝑡𝑡 +𝐹𝐹 𝑡𝑡 𝐶𝐶 𝑡𝑡 ( )𝐴𝐴̂+𝑡𝑡 − 𝐹𝐹 𝑡𝑡 𝐶𝐶̂ 𝑡𝑡, 𝑥𝑥� 𝑡𝑡 (10) 𝑥𝑥̇ 𝑡𝑡 𝑥𝑥 𝑡𝑡 𝐴𝐴 𝑡𝑡 ( ) ( ) ( ) measurement matrix. These state-space matrices are defined 𝐵𝐵 𝑡𝑡 𝐺𝐺 𝑡𝑡 𝑣𝑣 𝑡𝑡 in terms of the matrices𝐵𝐵 𝑡𝑡 describing the physical system𝐶𝐶 𝑡𝑡 as � � 𝑢𝑢 𝑡𝑡 � � 𝐵𝐵� 𝑡𝑡 𝐹𝐹 𝑡𝑡 𝑤𝑤 𝑡𝑡 ( ) 0 where is the time-varying observer gain. The optimal ( ) = , ( ), ( ) ( ) ( ) ( ) (6) observer gain, is given by 𝐹𝐹 𝑡𝑡 𝐼𝐼 𝑜𝑜𝑜𝑜𝑜𝑜 𝐴𝐴 𝑡𝑡 � −1 −1 � 𝐹𝐹 𝑡𝑡 −𝑀𝑀 𝑡𝑡 𝐾𝐾 𝑡𝑡 −𝑀𝑀 𝑡𝑡 𝑆𝑆 𝑡𝑡 ( ) = ( ) ( ) ( ) ( ) , (11) 0 𝑜𝑜𝑜𝑜𝑜𝑜 𝑇𝑇 −1 ( ) = , ( ) ( ) (7) where ( ) 𝐹𝐹is the𝑡𝑡 covarianceΠ 𝑡𝑡 𝐶𝐶̂ 𝑡𝑡matrix𝐶𝐶̂ 𝑡𝑡 𝑊𝑊of 𝑡𝑡the measurement noise ( ) and ( ) is the covariance of the state error ( ) 𝐵𝐵 𝑡𝑡 � −1 � 𝑀𝑀 𝑡𝑡 𝐻𝐻 𝑡𝑡 ( ). The𝑊𝑊 𝑡𝑡 state error covariance is the solution to the time- ( ) = [ ( ) ( )], (8) dependent𝑤𝑤 𝑡𝑡 RicattiΠ equation𝑡𝑡 𝑥𝑥 𝑡𝑡 − 𝑥𝑥� 𝑡𝑡 𝐶𝐶 𝑡𝑡 𝑃𝑃 𝑡𝑡 𝑁𝑁 𝑡𝑡 ( ) = ( ) ( ) + ( ) ( ) ( ) ( ) ( ) ( ) ( ), (12) where 0 is a × matrix of zeros and is a × identity 𝑇𝑇 𝑇𝑇 −1 ̇ ̂ ̂ ̂ ̂ matrix. The term ( ) ( ) is added to equation (4) to whereΠ 𝑡𝑡 𝐴𝐴(𝑡𝑡) Πis𝑡𝑡 theΠ time𝑡𝑡 𝐴𝐴 derivative𝑡𝑡 − Π 𝑡𝑡 𝐶𝐶 of𝑡𝑡 𝑊𝑊( 𝑡𝑡). Given𝐶𝐶 𝑡𝑡 Π 𝑡𝑡enough represent un-modeled𝑛𝑛 𝑛𝑛 dynamics or unmeasured𝐼𝐼 𝑛𝑛input𝑛𝑛 sources. convergence time, this estimator can compensate for The term ( ) is this process𝐺𝐺 𝑡𝑡 𝑣𝑣 𝑡𝑡 noise, and ( ) is a noise matrix. disturbancesΠ̇ 𝑡𝑡 or inaccuracies in the initial Πstate𝑡𝑡 estimate. This system and the state matrices will be used in the following 𝑣𝑣sections𝑡𝑡 for defining the Luenberger𝐺𝐺 𝑡𝑡 observer and Kalman filter. Parameter Adaptation Rule

The matrices in the Luenberger observer and Kalman Luenberger Observer filter, ( ), ( ), and ( ), can be updated using measurements from the system to calculate estimates of the The Luenberger observer (LO) is a common estimator system parameters.𝐴𝐴̂ 𝑡𝑡 𝐵𝐵� 𝑡𝑡 Consider𝐶𝐶 ̂the𝑡𝑡 equation (1) written as used for state estimation of linear, deterministic systems. The Luenberger observer finds an estimate of the state vector, ( ) = [ ( )] [ ( ) ( ) ( ) ( ) ( ) ( )], (13) ( ) , through the equations −1 𝑞𝑞̈ 𝑡𝑡 𝑀𝑀 𝑡𝑡 𝐻𝐻 𝑡𝑡 𝑢𝑢 𝑡𝑡 − 𝑆𝑆 𝑡𝑡 𝑞𝑞̇ 𝑡𝑡 − 𝐾𝐾 𝑡𝑡 𝑞𝑞 𝑡𝑡 𝑥𝑥� 𝑡𝑡 ( ) ( ) 0 ( ) ( ) which can be expressed as = + ( ), (9) ( ) ( ) ( ) ( ) ( ) ( ) 𝑥𝑥̇ 𝑡𝑡 𝐴𝐴 𝑡𝑡 𝑥𝑥 𝑡𝑡 𝐵𝐵 𝑡𝑡 � � � � � � � � 𝑢𝑢 𝑡𝑡 ( ) = ( )[ ( )] , (14) 𝑥𝑥�̇ 𝑡𝑡 𝐿𝐿𝐿𝐿 𝑡𝑡 𝐴𝐴̂ 𝑡𝑡 − 𝐿𝐿𝐶𝐶̂ 𝑡𝑡 𝑥𝑥� 𝑡𝑡 𝐵𝐵� 𝑡𝑡 T where L is a constant observer gain and the hat denotes the where (t) is a parameter𝑞𝑞̈ 𝑡𝑡 Ψvector𝑡𝑡 Θ composed𝑡𝑡 of the stiffness vector or matrix is the estimate used in the model. These and damping parameters and ( ) is a matrix formed from the estimates may be adapted over time using an additional state vectorΘ and the input forces. Next an adaptive back- adaption law. The first row of equation (9) is the original propagation rule for updatingΨ 𝑡𝑡 (t) is calculated using system dynamics, while the second row describes the measurements of the velocity vector ( ) and the estimated dynamics of the state estimate. Here the process noise velocity vector ( ), namely Θ� ( ) ( ) and measurement noise ( ) are ignored. The 𝑞𝑞̇ 𝑡𝑡 Luenberger observer can have fast convergence rates when � T 𝑞𝑞̇( 𝑡𝑡) = ( ) [ ( ) ( )], (15) 𝐺𝐺given𝑡𝑡 𝑣𝑣 an𝑡𝑡 accurate system model and𝑤𝑤 appropriate𝑡𝑡 selection of

L, but the observer is not robust in the presence of noise. The Θ�̇ 𝑡𝑡 −Γ �Ψ� 𝑡𝑡 � 𝑞𝑞̇ 𝑡𝑡 − 𝑞𝑞̇� 𝑡𝑡 fixed observer gain can be calculated from the desired where Γ is a pre-defined learning rate matrix [25, 26]. The eigenvalues of the matrix [ ( ) ( )] at some point in time, performance of this rule will depend on the value of the such as when the 𝐿𝐿system is in its initial, undamaged learning rate matrix. ̂ ̂ configuration. 𝐴𝐴 𝑡𝑡 − 𝐿𝐿𝐶𝐶 𝑡𝑡

Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 02/20/2018 Terms of Use: http://www.asme.org/about-asme/terms-of-use SIMULATION STUDY capable to converge to the new, correct stiffness in approximately 16.5 ms, and a 2% change in stiffness A preliminary study is used to evaluate the estimators’ (indicative of damage to the system) is detectable in performance and illustrate opportunities and limitations of approximately 12 ms. The Kalman filter is not able to high rate estimation. The previously mentioned Luenberger converge to the new, correct stiffness in the simulation time. observer and Kalman filter are evaluated on simulations from However, it reaches 80% of the stiffness change in about a time-varying system. The simulation consists of a two 51.5 ms in the presence of process and measurement noise, degrees-of-freedom (DOF) system shown in Figure 3. The and a 2% change in stiffness is detectable in approximately simulations use 1 kg for both masses (m1 and m2), damping 25 ms. A summary of the high rate estimation simulation -1 values of 0.1 N s m for both dampers (s1 and s2), and a results are shown in Table 2. simulation rate of 50 kHz. Both stiffness values (k1 and k2) are initially set to 400 N m-1. The stiffness of the spring between the masses (k2) suddenly decreases at the moment an external force is applied to the second mass. This stiffness change simulates damage to the linkage between the two masses.

Figure 3. Two degree of freedom dynamic system with variable stiffness k2.

For the two degree of freedom system in Figure 3, the equations of motion can be written as Figure 4. Simulation and estimation results for the two degree ( ) 0 of freedom system with changing stiffness. The top plot shows ( ) ( ) ( ) ( ) 𝑇𝑇 the impulse force ( ) on the second mass, the middle plot ( ) 0 1 1 = ⎧ −𝑞𝑞 (𝑡𝑡) 0 ⎫ 𝑘𝑘 , (16) shows the velocity of the two masses ( and ), and the ( ) 0 −1 ⎡ 2 1 1 2 ⎤ ⎡ 2⎤ 2 1 ⎪ 1 𝑞𝑞 (𝑡𝑡) − 𝑞𝑞 (𝑡𝑡) 𝑞𝑞 (𝑡𝑡) − 𝑞𝑞 (𝑡𝑡) ⎪ 𝑘𝑘 bottom plot shows 𝑢𝑢the real and estimated values for the 𝑞𝑞̈ 𝑡𝑡 𝑚𝑚 ⎢ 1 ⎥ ⎢ 1 ⎥ 1 2 � 2 � � 2� ⎢ −𝑞𝑞̇ ( 𝑡𝑡) ( ) ⎥ ⎢𝑠𝑠1 ⎥ stiffness of the connecting spring ( ). 𝑞𝑞̇ 𝑞𝑞̇ 𝑞𝑞̈ 𝑡𝑡 ⎨ 𝑚𝑚 2 1 1 2 ⎬ 2 ⎢𝑞𝑞̇ 𝑡𝑡 − 𝑞𝑞̇ 𝑡𝑡 𝑞𝑞̇ 𝑡𝑡 − 𝑞𝑞̇ 𝑡𝑡 ⎥ ⎢𝑠𝑠 ⎥ ⎪ 1 2 ⎪ ⎣ ⎦ 2 ⎩ (⎣ ) 𝑢𝑢 𝑡𝑡 𝑢𝑢 𝑡𝑡 ⎦ ⎭ 𝑘𝑘 = { ( )} ( ) . (17) 1 𝑇𝑇 Table 2. Summary of simulation results from two degree of 𝑞𝑞̈ 𝑡𝑡 � 2 � Ψ 𝑡𝑡 Θ freedom, high rate estimation. This forms the matrix 𝑞𝑞̈ (𝑡𝑡 ) used in the adaption rule given in equation (15). For the Luenberger observer, a rough Luenberger Kalman

optimization is performedΨ 𝑡𝑡 on the both the constant observer Observer Filter gain and learning matrix gain to produce the best Final value 380 N/m 384 N/m estimate. Because the Kalman filter already includes rules for Final error 0% 20% optimizing𝐿𝐿 the observer gain, Γ( ), only the learning matrix is tuned for optimal parameter𝑜𝑜𝑜𝑜𝑜𝑜 estimation. Convergence time 16.5 ms - Figure 4 shows a representative𝐹𝐹 simulation𝑡𝑡 of the system Time for 2% value change 12 ms 25 ms experiencing an impulsive force of amplitude 50 N over a Robust with noise No – unstable Yes duration of 0.2 ms. During this simulation, the initial stiffness value is reduced by 5% (to 380 N m-1) at the onset of 2 the force input to simulate damage. Process and measurement𝑘𝑘 noises are included in the Kalman filter implementation as The performance of the estimators is sensitive to tuning of they are needed to calculate the gain in equation (11), and observer and learning rate gains, input amplitudes, and the rate without noise the calculation becomes numerically unstable. of change of system parameters. Figure 5 plots estimates of The Luenberger is found to become unstable when noise is the stiffness k2 from both observers for input amplitudes of added to the system and therefore the case of Luenburger with 50 N and 100 N. Doubling the input amplitude decreases the noise is excluded from this paper. The Luenberger estimator is estimation time of the Luenberger observer and worsens the

Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 02/20/2018 Terms of Use: http://www.asme.org/about-asme/terms-of-use convergence rate of the Kalman filter. The rate of parameter that will need to be addressed in estimators for more complex change can also affect results. Figure 6 plots the estimations systems. Characteristics of the input and the rate of change of during different durations for the stiffness change. The parameters produce different qualities of estimation. Adaption original input force of 50 N is used here. Both observers are laws that can account for this broader array of inputs and better at tracking a slow stiffness change than a fast one. parameter changes are needed for more robust SHM at high speeds. In addition, noise affects the estimation results. The high rate Luenberger observer became unstable in the 405 Real k presence of large noise in the system. New methods are 2 400 Kalman Filter [50 N] needed which account for system and measurement noise 395 Luenberger Observer [50 N] while balancing the convergence time. The Kalman filter was Kalman Filter [100 N]

(N/m) able to give a stiffness estimate in the presence of noise, but 2 390 Luenberger Observer [100 N] with our selected learning rate matrix, the parameter estimates 385 were slow to converge to the final estimate. 380 Stiffness k 375

370 CONCLUSIONS 0 50 100 150 200 Time (ms) This paper discussed the motivation for developing structural health monitoring (SHM) systems that can detect Figure 5. The effect of pulse amplitude on observer and characterize damage in timescales on the order of tens or performance. The amplitude of the input force is noted in the hundreds of microseconds. Such systems are needed in a legend. range of aerospace, automotive, and military applications. High rate SHM methods will have to handle system 405 complexities, system uncertainties, and real-time processing (a) Real k [20 µs] 2 requirements. A case study with observers for induction Luenberger Observer [20 µs] 400 Real k [10 ms] motors and simulation examples presented here showed the 2 Luenberger Observer [10 ms] feasibility of damage detection using adaptive observers on a Real k [20 ms] 395 2 linear system in the millisecond timescales. The estimation (N/m) 2 Luenberger Observer k [20 ms] 2 and detection of damage in the high rate loading with Real k [40 ms] 390 2 changing nonlinear dynamics and large noise will require Luenberger Observer [40 ms] implementing advanced observer design methods.

Stiffness k 385 Nonetheless, these simulations have shown that the approach of developing a microsecond state detection method is 380 possible.

10 20 30 40 50 60 70 80 Time (ms) ACKNOWLEDGEMENTS 405 (b) Real k [20 µs] 2 Kalman Filter [20 µs] The material is based upon work supported by the Air 400 Real k [10 ms] 2 Force Office of Scientific Research, Program Office Dr. Jamie Kalman Filter [10 ms] Tiley, under award number FA9550-17RWCOR503. Any Real k [20 ms] 395 2 (N/m)

2 opinions, findings, and conclusions or recommendations Kalman Filter k [20 ms] 2 Real k [40 ms] expressed in this material are those of the authors and do not 390 2 Kalman Filter [40 ms] necessarily reflect the views of the United States Air Force.

Stiffness k 385 Distribution A. Approved for public release; distribution unlimited (96TW-2017-0180). 380

10 20 30 40 50 60 70 80

Time (ms) REFERENCES Figure 6. The role of the rate of change of stiffness on observer performance. (a) Luenberger observer results. (b) [1] Stein, C., Roybal, R., Tlomak, P., and Wilson, W., 2000, Kalman filter results. The times indicated in the legend are the "A review of hypervelocity debris testing at the Air Force duration of the 5% stiffness change. Research Laboratory," Space Debris, 2(4), pp. 331-356. [2] Hallion, R. P., Bedke, C. M., and Schanz, M. V., 2016, "Hypersonic weapons and US national security, a 21st The implementation of the Kalman filter and Luenberger century breakthrough," Mitchell Institute for Aerospace observer state estimates on this time-varying mechanical Studies, Air Force Association. system illustrate a few issues with high rate state estimation

Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 02/20/2018 Terms of Use: http://www.asme.org/about-asme/terms-of-use [3] Farrar, C. R., and Worden, K., 2007, "An introduction to [19] Sastry, S., and Bodson, M., 2011, Adaptive Control: structural health monitoring," Philosophical Transactions Stability, Convergence and Robustness, Courier of the Royal Society A: Mathematical Physical and Corporation. Engineering Sciences, 365(1851), pp. 303-315. [20] Yingjuan, Y., and Pengzhang, X., 2014, "Design of a [4] Hong, J., Laflamme, S., and Dodson, J., 2016, "Survey of nonlinear adaptive observer for a class of Lipschitz observers for high-rate state estimation," Journal of systems," 33rd Chinese Control Conference (CCC), pp. Dynamic Systems, Measurement, and Control. 2240-2243. (Submitted, under review). [21] Khayati, K., and Zhu, J., 2013, "Adaptive observer for a [5] Lowe, R., Dodson, J., and Foley, J., 2014, "Microsecond large class of nonlinear systems with exponential prognostics and health monitoring," IEEE Reliability convergence of parameter estimation," International Society Newsletter, 60. Conference on Control, Decision and Information [6] Dodson, J. C., 2010, "Optimal sensor placement for state Technologies (CoDIT), pp. 100-105. estimation of linear distributed parameter systems," [22] Daum, F. E., 2005, "Nonlinear filters: beyond the Kalman Masters Project, Applied Mathematics, Virginia filter," IEEE Aerospace and Electronic Systems Polytechnic Institute and State University. Magazine, 20(8), pp. 57-69. [7] Dodson, J., Inman, D. J., and Foley, J. R., 2009, [23] Xu, Z., Rahman, F., and Xu, D., "Comparative study of "Microsecond structural health montoring in impact an adaptive sliding observer and an EKF for speed sensor- loaded structures," Proceedings in SPIESan Diego, CA. less DTC IPM synchronous motor drives," Proc. Power [8] Dodson, J. C., Foley, J. R., and Inman, D. J., 2010, Electronics Specialists Conference, pp. 2586-2592. "Structural health monitoring of an impulsively loaded [24] Zhang, Y., Zhao, Z., Lu, T., Yuan, L., Xu, W., and Zhu, structure using wave-propagation based instantanous J., "A comparative study of Luenberger observer, sliding baseline," Proceedings of SPIESan Diego, CA. mode observer and extended Kalman filter for sensorless [9] Hong, J., Laflamme, S., and Dodson, J., "Variable input vector control of induction motor drives," Proc. Power spce observer for structural health monitoring of high-rate Electronics Specialists Conference, pp. 2586-2592. system," Proc. Review of Progress in Quantitative [25] Åström, K. J., and Witternmark, B., 1994, Adaptive Nondestructive Evaluation (QNDE). Control, Addison-Wesley Longman Publishing Co., Inc., [10] Hong, J., Laflamme, S., Cao, L., and Dodson, J., Boston, MA. "Variable input observer for structural health monitoring [26] Laflamme, S., Slotine, J., and Connor, J., 2012, "Self- of high-rate systems," Proc. AIP Conference Proceedings, organizing input space for control of structures," Smart p. 07003. Materials and Structures, 21(11), p. 115015. [11] Kettle, R., Dick, A., Dodson, J., Foley, J., and Anton, S. R., 2015, "Real-time detection in highly dynamic systems," IMAC XXXIV A conference and exposition on structural dynamicsOrlando, Fl. [12] Dodson, J., Kettle, R., and Anton, S. R., 2016, "Microsecond state detection and prognosis using high- rate electromechanical impedance," IEEE Prognostics and Health Management Conference, PHM '16. [13] Kye-Lyong, K., Jang-Mok, K., Keun-Bae, H., and Kyung-Hoon, K., 2004, "Sensorless control of PMSM in high speed range with iterative sliding mode observer," IEEE Appied Power Electronics Confernece and Exposition, APEC '04, Nineteenth Annual. [14] Xu, J., Kum, K., Xie, L., and Loh, A., 2012, "Fault detection and isolation of nonlinear systems: An unknown input observer approach with sum-of-squares techniques," Journal of Dynamic Systems, Measurement, and Control, 134(4), pp. 1-7. [15] Guyon, I., 1991, "Neural networks and applications tutorial," Physics Reports, 207(3), pp. 215-259. [16] Besancon, G., 2007, "An overview on observer tools for nonlinear systems," Nonlinear Observers and Applications, Springer, pp. 1-33. [17] Zarei, J., and Shokri, E., 2014, "Robust sensor fault detection based on nonlinear unknown input observer," Measurement, 48, pp. 355-367. [18] Ioannou, P. A., and Kosmatopoulos, E. B., 1999, "Adaptive control," Wiley Encyclopedia of Elecrical and Electronics Engineering, John Wiley & Sons, Inc.

Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 02/20/2018 Terms of Use: http://www.asme.org/about-asme/terms-of-use