System Identification Overview & Session Goals GWADW 2015 Controls Session A: Identification Dennis Coyne

G1500673-v1 What is System Identification (SID)?

• SID is the process of developing or improving a mathematical representation of a using experimental • or grey box (Physical insight must be used to develop the underlying model structure) • and domain • Model reduction • Plant, or plant + controller • Open loop, closed loop • design • SID for structural dynamics • Modal identification (the focus of this talk) • Structural-model parameter identification (motivation given in this talk) • Control-model identification

G1500673-v1 What does SID offer?

• Does adaptive control negate the need for SID? • No; Adaptive techniques are essentially recursive identification algorithms applied to a specific model structure • Does robust control design render SID obsolete? • No; Robust control design requires a description of the model/plant uncertainty that SID can provide.

G1500673-v1 SID Process

• Basic steps: • Build a physics-based model • Design the experiment • Collect the data • Select, filter, de-trend the data, as appropriate • Fit the model • Validate the model (compare prediction to data) • Iterate until converged on a validated model • Software tools: • Matlab Sys ID Toolbox (Ljung) • Matlab Frequency-Domain Identification (Kollar) • Matlab System/Observer/Controller Identification Toolbox (SOCIT) • …

G1500673-v1 SID for structural dynamics

• Modal parameter identification • the focus of this talk • Structural-model parameter identification • motivation given in this talk • Control-model identification

G1500673-v1 Structural Modeling & SID

SIGNAL FILTERED FFTs IMPULSE REALIZATION STRUCTURES SENSORS DISTURBANCES CONDITIONING DATA RESPONSE (STATE SPACE)

REALIZATION ACTIVE CONTROL MODELS/SYNTHESIS

STRUCTURAL MODIFICATION, PHYSICAL MODEL REPAIR, MODELS MONITORING, ... (e.g. FINITE ELEMENT MODEL (FEM) (M, D, K, modal representation, Nonlinearities)

Derived from a lecture on STRUCTURAL SYSTEM IDENTIFICATION USING THEORY AND WAVELETS VIBRATION ISOLATION DESIGN K. C. PARK Department of Aerospace Engineering Sciences and CONTROL DESIGN Center for Aerospace Structures, University of Colorado DAMAGE DETECTION, ...

G1500673-v1 Continuous-Time, Finite Dimensional, Linear Dynamic System Model

G1500673-v1 Continuous-Time State-Space Model

G1500673-v1 Discrete-Time, State-Space, Observer Model

• Discrete-time: • Since the state vector is 푥 푘 + 1 = 퐴 푥 푘 + 퐵 푢(푘) (generally) not accessible, a 푦 푘 = 퐶 푥 푘 + 퐷 푢(푘) state (observer) is added: k = 0, 1, 2, … 푥 푘 + 1 = 퐴 푥 푘 + 퐵 푣(푘) where 푦 푘 = 퐶 푥 푘 + 퐷 푢(푘) 퐴 = 푒퐴푐 ∆푡 where ∆푡 퐴 = 퐴 + 퐺퐶 퐴푐 휏 퐵 = 푒 휕휏 퐵푐 퐵 = [퐵 + 퐺퐷 − 퐺] 0 푢(푘) A , B are the continuous versions 푣 푘 = c c 푦(푘)

G1500673-v1 System Realization Theory

• Construct a model with controllability & observability • Seek a minimum realization (smallest state space dimensions) • Time domain models for modal parameter identification are based on the matrix, which yields Markov parameters (i.e. pulse response) 푥 푘 + 1 = 퐴 푥 푘 + 퐵 푢(푘) 푦 푘 = 퐶 푥 푘 + 퐷 푢(푘) Let 푢푖 0 = 1 푖 = 1, 2, … , 푟 and 푢푖 푘 = 0 푘 = 1, 2, … Pulse response matrix Y (m x r): 푘−1 푌0 = 퐷; 푌1 = 퐶퐵; 푌2 = 퐶퐴퐵; … ; 푌푘 = 퐶퐴 퐵 where Y are the Markov parameters

• A realization is the computation of the triplet [A, B, C] (D = Y0)

G1500673-v1 System Realization Theory

• The triplet [A, B, C] can be transformed into modal parameters by computing the eigenvectors, 휑 = {휑1, 휑2, …, 휑푛}, with the corresponding eigenvalues, 훾 = {훾1, 훾2, …, 훾푛}, of matrix A • Transform to realization 훾 휑−1퐵 퐶휑

Diagonal Matrix 훾 contains the Shapes at the modal damping rates and damped sensor locations natural frequencies Re & Im parts of 훾푐 = ln 훾 /∆푡

Initial Modal Amplitudes

G1500673-v1 PULSE-RESPONSE FUNCTIONS Eigensystem Realization (MARKOV PARAMETERS)

Algorithm (ERA) HANKEL MATRIX, H(0) • The Hankel matrix is formed from the Markov parameters LEFT SINGULAR VECTORS SINGULAR VALUES RIGHT SINGULAR VECTORS 푌푘 푌푘+1 … 푌푘+훽−1 푌 퐻 푘 − 1 = 푘+1 푌 … … 푘+2 TIME-SHIFTED HANKEL MATRIX, H(1) 푌푘+훼−1 … 푌푘+훼+훽−2 • ERA is a fit to the pulse response measurements OUTPUT MATRIX STATE MATRIX INPUT MATRIX • Variant ERA/DC • with Data Correlations EIGENSOLUTION • Fit to the output auto-correlations and cross-correlations over a finite number of lag values NATURAL FREQUENCIES MODE SHAPES MODAL AMPLITUDES • Juang, J. et. al., “An eigensystem AND MODAL DAMPING realization algorithm using data correlations (ERA/DC) for modal parameter identification”, and Advanced Technology, REDUCED MODEL v4,n1,1988. VALADATION (RECONSTRUCTION & COMPARISON WITH DATA) G1500673-v1 Other SID Techniques

• Ultimately we seek a recursive, real-time parameter identification of our MIMO systems (at their operating states) • Many techniques are available; Potential candidates include: • Generalized Least Squares and Maximum Likelihood (e.g. the Prediction Error Method) are computationally simple • Observer/ ldentification (OKID) -- time domain based, can be extended to identification of closed loop effective controller/observer combination (Observer Controller ldentification, OCID) • State-Space (SSFD) identification techniques

G1500673-v1 How and what to measure – modal testing

• Frequency Response Function (FRF) is the Fourier transform of an impulse or pulse response sequence (Markov parameters) • Typically the FRFs are computed before application of a SID technique • However many SID tools can analyze the free or forced response time histories directly

G1500673-v1 SID Session Goals

• Compare techniques • Share experiences • Exchange tools and ideas • Define a “road map” for application to our systems and further research • Explore the state of the art, new tools … • Define suitable problems/applications • Define upgrades & next generation system targets (early stages of testing and commissioning) • A new GW working group? • Host a wiki page? • Meet virtually on some cadence?

G1500673-v1