Identification of Reduced-Oder Dynamic Models of Gas Turbines CSC Student Seminars (Spring/Summer, 2006) Identification of Reduced-Oder Dynamic Models of Gas Turbines PhD Student: Xuewu Dai Supervisor: Tim Breikin and Hong Wang

Introduction 1. Introduction 2. Reduced-order Model 3. Long-term Prediction 4. Dynamic Gradient Descent 5. Nonlinear Least-Squares Optimization 6. Future Works

1. Introduction Modlling of Gas Turbines Fault Detection Condition Monitoring

Aims Reducing Computational Complexity: Real time Improving Prediction Accuracy: Long-term prediction Robustness

2. Reduced Order Thermodynamic models: 1. High order : 26th 2. Non-linear Linearisation Our ARX models : 1. Reduced order: 1st, 2nd … 2. Linear:

3. Long-term Prediction Model Model b. Long-term Prediction Model a. One-step Ahead Prediction Model

Model Equations One-step ahead prediction 2. Long-term prediction

Challenges Computational Burden How many iterations need to identify the parameters? Dependency of Prediction Errors (Non-Gaussian Noise) MSE=9.1318 Autocorrelation of prediction errors

4. Dynamic Gradient Descent Objective Function Global Gradient and local gradient

Dynamic Gradient Descent

Results 1: deepest direction

BFGS direction

5. Nonlinear Least-squares Optimization (Gauss-Newton)

Search direction, step size and initial value Deepest descent: inverse global gradient Nonlinear Least Squares: Gauss-Newton Step size: fixed, adjustable, line search Initial value: Blind guess: [0.5 0.5 0.5 0.5] LSE: [1.2805 -0.29191 0.10582 0.15903]

Result 3 Gauss-Newton

Prediction of 1st Order Model

Comparison of 1st Order Model Methods MSE a b Iterations LSE 23.49449 0.987395 0.032551 1 ANFIS 22.2925 N/A 200 GD 11.0163 0.9809 0.0376 Exhausted Search 9.131926 0.977774 0.043542 10000 DGD1* 9.131816 0.977764 0.043568 1000 DGD2* 9.131786 0.777774 0.043544 101 DGD3* 9.131785 0.977776 0.043543 98 DGD1: Deepest descent direction and adjusting step size DGD2: BFGS direction and adjusting step size DGD3: Gauss-Newton and line search

High Order Model initial (by LSE) : [1.2805 -0.29191 0.10582 0.15903] final: [1.8604 -0.8641 0.07045 -0.007475]

6. Future Works Initial value Problem: Robustness Problem: ??? Applying such learning algorithm to Neural Networks Model structure selection by autocorrelation of prediction errors NARMX models

CSC Student Seminars (Spring/Summer, 2006) Thanks

Appendix

Initial value problem manual setting of initial value [0.5 0.5 0.5 0.5] [1.8604 -0.8641 0.07045 -0.007475] Final MSE=8.60188 setting initial value by LSE [1.2805 -0.29191 0.10582 0.15903] [1.8604 -0.8641 0.07045 -0.007475] Final MSE=3.313612

appendix