Constant Energy Control by TimeVarying Gain for SteadyState

Constant Energy Control by Time-Varying Gain for Steady-State Oscillation of Thermoacoustic Engines to Estimate Critical Temperature Ratio • Rijke tube • thermoacoustic instability Yasuhide Kobayashi, Kazuaki Sakurai, Noboru Yamada Nagaoka University of Technology, JAPAN

Introduction • 　Attenuation of thermoacoustic instability (TAI) sound wave – to prevent mechanical damage – active noise control [1 -9] • Utilization of TAI cold output – less application of control eng. waste heat – not admitted electrical power input stack thermoacoustic refrigerator Aim: contribute thermoacoustic system design by applying control engineering

Introduction (cont. ) • Estimation of CTR (= TH /TC) – <CTR: Q values in freq. resp. [15][20] – >CTR: smallest temp. ratio 1/Q TH pressure TC estimation methods depend on temperature ratio range no unified method has been proposed temp. CTR ratio ≒CTR temp. ratio

GOAL 1. unified method by steady-state oscillation control – PI control … pressure amplitude → reference value – PI controller’s output … time-varying gain G(t) – CTR is empirically estimated with G(∞) 2. closed-loop stability analysis – simple 2 nd-order model – condition on PI gains – numerical simulation

Experimental apparatus • 　Standing-wave thermoacoustic engine – TC =27℃，TH is varied

Experimental apparatus (cont. ) • 　steady-state oscillation control system – pressure amplitude P → P* (resist / assist) – phase-delay control: u(t) = G p(t-τ) … TAI – G: const ⇒ P=0 (ANC), G(t) G=0 ⇒ P: const (TAI) – oscillation freq. is automatically determined KP = 0. 5, KI = 0. 1 estimation of pressure amplitude P^ ωF = 1 G

Experimental results • example of time response – TH / TC = 1. 0，P*=200 Pa – oscillation @ 60 Hz 0 ＾P gain(Pa) G Pressure (Pa), Gain p (Pa) P (Pa) Gain G p p (Pa) -50 400 Pa -100 G Time (s) 1 1. 1 Time (s) 1. 2 TH / T C 1. 3 CTR

Experimental results (cont. ) • 　example of time response (P*=100 Pa) TH / TC = 1. 0 < CTR TH / TC = 1. 46 > CTR stable TH / TC = 1. 0 t n e t it unstable le: b a t ns m r e int u s le b ta on i t a l l i osc e t s : y d a e t a st n o i llat i c s o - 2. 6 KI < KP

Closed-loop stability analysis nd order physical connection between model and exp. • • 2 no model • ANC and TAI are simulated by this model … simulink 　　　　 • not trivial if the amplitude is stabilized or not α<0

Closed-loop stability analysis (cont. ) • Assumption in amplitude estimation　　

Closed-loop stability analysis (cont. ) • PI controller　　　　

Closed-loop stability analysis (cont. ) • dynamics on amplitude 　　　　 independent on Y*

Closed-loop stability analysis (cont. ) • Stability condition 　　　　 　 ⇒　　 0 < KI < KP 2. 6 KI < KP

Numerical simulation • 　Matlab/Simulink • 　absolute value function • Y*=10, α = -0. 01

CONCLUSIONS • an empirical method for estimating the CTR – steady-state oscillation control – PI controller’s output = time-varying gain – a linear relationship between temp. ratio and gain • stability analysis based on a 2 nd order model – an inequality condition on PI gains and LPF – consistent with experiment and numerical simulation • Future work – quantitative difference between exp. and theory

Closed-loop stability analysis (cont. ) • Interpretation of constant energy control system – constant amplitude ⇔　constant energy