Methods, apparatus and systems for real time identification and control of modes of oscillation

A method for real time identification of and compensation for a mode of oscillation of a system, the method comprising the steps of:

• a) sensing the system to generate a signal f(t);
• b) determining values of modal functions a(t) and b(t) of the mode of oscillation of the signal f(t), based on a pair of integrals .intg..PSI..sub.a f(t)dt and .intg..PSI..sub.b f(t)dt, respectively, .PSI..sub.a and .PSI..sub.b being predetermined wavelet functions that are orthogonal and localized at an estimated frequency .omega. of the mode of oscillation, the pair of integrals having limits of integration based on a time t and an estimated period T=2p/.omega. of the mode of oscillation, the estimated frequency .omega. being predetermined initially;
• c) determining an updated value of the estimated frequency .omega., based on the modal functions a(t) and b(t);
• d) updating the wavelet functions .PSI..sub.a and .PSI..sub.b, based on the updated value of the estimated frequency .omega.;
• e) generating a control signal to control the oscillation in the system, based on the values of the modal functions a(t) and b(t), the wavelet functions .PSI..sub.a and .PSI..sub.b, and the updated value of the estimated frequency .omega.;
• and, wherein the system includes a compressor having a plenum receiving a fluid,
• f) bleeding the fluid from the plenum, based on the control signal.

A method for real time identification of and compensation for a mode of oscillation of a system, the method comprising the steps of:

• a) sensing the system to generate a signal f(t);
• b) determining values of modal functions a(t) and b(t) of the mode of oscillation of the signal f(t), based on a pair of integrals .intg..PSI..sub.a f(t)dt and .intg..PSI..sub.b f(t)dt, respectively, .PSI..sub.a and .PSI..sub.b being predetermined wavelet functions that are orthogonal and localized at an estimated frequency .omega. of the mode of oscillation, the pair of integrals having limits of integration based on a time t and an estimated period T=2p/.omega. of the mode of oscillation, the estimated frequency .omega. being predetermined initially;
• c) determining an updated value of the estimated frequency .omega., based on the modal functions a(t) and b(t);
• d) updating the wavelet functions .PHI..sub.a and .PSI..sub.d, based on the updated value of the estimated frequency .omega.;
• and
• e) replacing the modal functions a(t) and b(t) at an end of the estimated period T with average values A(T) and B(T) accumulated over the estimated period T of the mode of oscillation.

A method for real time identification of the modes of oscillation of a system, the method comprising the steps of:

• a) initializing modal output signals y(t) for L modes of oscillation, L being a predetermined integer;
• b) sensing the system and generating a signal indicative of the L modes of oscillation;
• c) combining the sensed signal and the modal output signals y(t) of the L modes of oscillation multiplied by respective predetermined weighting functions, to generate L modal input signals f(t);
• d) determining modal functions a(t) and b(t) for each of the L modes of oscillation of the sensed signal, based on L pairs of integrals .intg..PSI..sub.a f(t)dt and .intg..PSI..sub.b f(t)dt, respectively, .PSI..sub.a and .PSI..sub.b being predetermined wavelet functions that are orthogonal and localized at an estimated frequency .omega. of each of the L modes of oscillation, the pair of integrals having limits of integration based on a time t and an estimated period T=2p/.omega. of the mode of oscillation, the estimated frequencies .omega. being predetermined initially for each of the L modes of oscillation;
• e) determining updated values of the estimated frequencies .omega. for the L modes of oscillation, based on the modal functions a(t) and b(t);
• f) updating the wavelet functions .PSI..sub.a and .PSI..sub.b for the L modes of oscillation, based on respective updated values of the estimated frequencies .omega.;
• g) generating a control signal to control the oscillation in the system, based on the values of the modal functions a(t) and b(t), the wavelet functions .PSI..sub.a and .PSI..sub.b and the updated values of the estimated frequencies .omega.;
• and, wherein the system includes a combustor,
• h) supplying fuel to the combustor, based on the control signal.

A method for real time identification of the modes of oscillation of a system, the method comprising the steps of:

• a) initializing modal output signals y(t) for L modes of oscillation, L being a predetermined integer;
• b) sensing the system and generating a signal indicative of the L modes of oscillation;
• c) combining the sensed signal and the modal output signals Y(t) of the L modes of oscillation multiplied by respective predetermined weighting functions, to generate L modal input signals f(t);
• d) determining modal functions a(t) and b(t) for each of the L modes of oscillation of the sensed signal, based on L pairs of integrals .intg..PSI..sub.a f(t)dt and .intg..PSI..sub.b f(t)dt, respectively, .PSI..sub.a and .PSI..sub.b being predetermined wavelet functions that are orthogonal and localized at an estimated frequency .omega. of each of the L modes of oscillation, the pair of integrals having limits of integration based on a time t and an estimated period T=2p/.omega. of the mode of oscillation, the estimated frequencies .omega. being predetermined initially for each of the L modes of oscillation;
• e) determining updated values of the estimated frequencies .omega. for the L modes of oscillation, based on the modal functions a(t) and b(t);
• f) updating the wavelet functions .PSI..sub.a and .PSI..sub.b for the L modes of oscillation, based on respective updated values of the estimated frequencies .omega.;
• g) generating a control signal to control the oscillation in the system, based on the values of the modal functions a(t) and b(t), the wavelet functions .PSI..sub.a and .PSI..sub.b and the updated values of the estimated frequencies .omega.;
• and, wherein the system includes a compressor having a plenum receiving a fluid,
• h) bleeding the fluid from the plenum, based on the control signal.

A method for real time identification of the modes of oscillation of a system, the method comprising the steps of:

• a) initializing modal output signals y(t) for L modes of oscillation, L being a predetermined integer;
• b) sensing the system and generating a signal indicative of the L modes of oscillation;
• c) combining the sensed signal and the modal output signals y(t) of the L modes of oscillation multiplied by respective predetermined weighting functions, to generate L modal input signals f(t);
• d) determining modal functions a(t) and b(t) for each of the L modes of oscillation of the sensed signal, based on L pairs of integrals .intg..PSI..sub.a f(t)dt and .intg..PSI..sub.b f(t)dt, respectively, .PSI..sub.a and .PSI..sub.b being predetermined wavelet functions that are orthogonal and localized at an estimated frequency .omega. of each of the L modes of oscillation, the pair of integrals having limits of integration based on a time t and an estimated period T=2p/.omega. of the mode of oscillation, the estimated frequencies .omega. being predetermined initially for each of the L modes of oscillation;
• e) determining updated values of the estimated frequencies .omega. for the L modes of oscillation, based on the modal functions a(t) and b(t);
• f) updating the wavelet functions .PSI..sub.a and .PSI..sub.b for the L modes of oscillation, based on respective updated values of the estimated frequencies .omega.;
• and
• g) replacing the modal functions a(t) and b(t) at an end of an estimated period T for each of the L modes of oscillation with respective average values A(T) and B(T) accumulated over a respective estimated period T.

A system receiving a signal indicative of a mode of oscillation and a fluid flow, the system comprising:

• an observer including a processor, coupled to receive the signal, determining values of modal functions a(t) and b(t) of the mode of oscillation of the signal f(t), based on a pair of integrals .intg..PSI..sub.a f(t)dt and .intg..PSI..sub.b f(t)dt, respectively, .PSI..sub.a and .PSI..sub.b being predetermined wavelet functions that are orthogonal, localized at an estimated frequency .omega. of each of the L modes of oscillation, the pair of integrals having limits of integration based on a time t and an estimated period T=2p/.omega. of the mode of oscillation, the estimated frequencies .omega. being predetermined initially for each of the L modes of oscillation, the observer determining updated values of the estimated frequencies .omega. for the L modes of oscillation, based on the modal functions a(t) and b(t), and the observer updating the wavelet functions .PSI..sub.a and .PSI..sub.b for the L modes of oscillation, based on respective updated values of the estimated frequencies .omega.;
• a first controller coupled to receive the values of the modal functions a(t) and b(t) and the frequency .omega. determined by the observer, the first controller generating a first control signal based on the modal functions a(t) and b(t), the frequency .omega. determined by the observer, a gain G and a phase shift f, and outputting a first control signal;
• a combustor generating the mode of oscillation based on the fluid flow;
• a first sensor arranged in proximity to the combustor, sensing the mode of oscillation to generate the signal indicative of the mode of oscillation;
• and
• a first actuator coupled to receive the first control signal from the first controller, controlling the fluid flow into the combustor, based on the first control signal.