GB2404999A

GB2404999A - Adaptive sensor model

Info

Publication number: GB2404999A
Application number: GB0318794A
Authority: GB
Inventors: Roger Dixon; Andrew W Pike
Original assignee: Alstom SA
Current assignee: Alstom SA
Priority date: 2003-08-09
Filing date: 2003-08-09
Publication date: 2005-02-16
Anticipated expiration: 2023-08-09
Also published as: GB2404999B; GB0318794D0

Abstract

Output values of a physical system are predicted from a set of measured inputs of the system using an adaptive model. At each time when a prediction is made, the model is re-initialized to an initial, off-line model 1 and is then refined to incorporate on-line data using a predetermined number of recent sets of measured inputs and output 2, 3. The model thus always remains "tethered" to the initial, off-line model and if operating conditions remain steady, the model does not become too specific to those operating conditions.

Description

TITLE

Adaptive sensor model

DESCRIPTION

Technical field

The invention relates to the field of modelling physical systems. One example is the modelling of gas turbine engines but the method is in principle applicable to any physical system where a set of measured inputs of the system can be used to predict an output of the system.

Background of the invention

It is often desirable to operate a model of a plant or system in real time to predict the output(s) of that system based on a set of measured inputs. The predicted output(s) can then be used to control the system in order to obtain the desired outputs.

In the case where the model was created and fitted to data off-line, it is usually beneficial to permit the model to be refined based on on-line measurements of the system output, in order for a generic model to be adapted to an individual system or to the particular operating conditions of the system.

Adaptive algorithms are known, which can refine a model of a system based on the measured inputs and outputs of the system while on-line. Such algorithms typically use methods based on recursive least squares to refine the model and it is known to tailor the way in which such models are refined using "forget factors" to determine how much effect the new data has on the model, and covariance matrices to incorporate estimated parameter uncertainty information into the adaptive parameter updates.

A significant problem of known real-time adaptive algorithms is that they may over- adapt the model to the current data. The longer the system runs at a steady operating point, the more data about that operating point accumulates and the more closely the model will be tuned to behaviour at that operating point. The consequence of over A080GO5

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adaptation is a model that gives accurate predictions of response around the current operating point but predicts badly when the operating point changes: i.e. the generality of the model is lost.

The invention The invention provides a method of predicting output values y of a physical system from a set of measured inputs rf of the system, wherein at each time t when a prediction is made, the method comprises the following steps: (a) storing the set of measured inputs rj(ti) and the corresponding measured output y(tJ at each of a predetermined number n of times to In earlier than i; (b) initializing a model of the system, which generates a predicted output 9(t) from the set of measurements rj(t), to a predetermined initial model; (c) using each of the predetermined number n of sets of measured inputs rj(ti) and output y(td to adapt the model; and (d) using the adapted model to predict the output value y(t) of the system at time t from the set of measured inputs refit) of the system at time I. The invention thus differs from the prior art in that, when each prediction is made, the method starts with the initial (e.g. off-line) model and then revises the model using only the most recent n sets of online data. The model always remains "tethered" to the initial, generic model, while also taking into account recent data. If operating conditions remain steady, the model does not become too specific to those operating conditions. The number n can be chosen for any given system to achieve the desired balance between contributions from the initial model and from on-line measurements.

The invention allows an initial model to be developed that is applicable to a whole group of systems, such as an engine fleet, whilst also being capable of adapting itself to be directly applicable to the individual engine on which it is installed.

The invention is applicable to the modelling of almost any physical system but specific examples of the predicted output value 9(t) in the context of a turbine engine A080005 might be the compressor outlet pressure (units Pa) or the first stage turbine outlet temperature (units deg.K).

Preferably, the times t...tn are the most recent n times preceding time t at which measurements were made, so that the most current data are used to refine the model.

However, there may be applications in which it is desirable to use a set of sampling times spread more widely over the lifetime of the system in order to base the revised model on a broader range of experience. In that situation, it is likely that the sampling times will not be evenly spread but clustered more closely in the recent past.

Preferably, step (c) of the method is applied using the n sets of measurements r'(tJ end y(t) in chronological order.

In a preferred method according to the invention, for each of the n sets of measurements of the inputs and the output, step (c) comprises the following steps: (cl) applying the model to the set of inputs rj(t) to generate a predicted output y(ti) at time ti, (c2) calculating an error eats by finding the difference between the predicted output y(ti) and the measured output yet) at time I, and (c3) using the error e(tJ to revise a set of parameters b' of the model.

Step (c3) of the method may use a recursive estimation algorithm to revise the parameters b' of the model. This is preferably a Recursive Least Squares (RLS) algorithm but other established recursive parameter estimation methods could be applied, for example Recursive Maximum Likelihood (RML), Recursive Instrumental Variables (RIV) etc. The model may further include an error covariance matrix P. which is used in step (c3) to revise the parameters of the model, in which case the method further comprises a step (c4) of using the measured input values rj(ti) to revise the error covariance matrix P. It is within this error covariance matrix that the information about all the previous data and the model's fit to that data is encoded.

A080005 The invention further provides a control apparatus for a physical system, the apparatus including means for predicting output values of the system from measured inputs of the system in accordance with a method as previously defined.

The drawing Figure l is a flowchart showing the cycle of steps that is followed each time a prediction is made in accordance with the invention.

Description of a preferred embodiment

Initially a fixed parameter model is fitted off-line to input/output data from an item of plant. The nature of these inputs and outputs will depend on the individual plant.

For example, the plant may be a gas turbine engine. One model for a gas turbine engine may predict the compressor outlet pressure as a function of the following measured inputs: generated power, turbine exit pressure, compressor outlet temperature and shaft speed. Another model may predict the first stage turbine outlet temperature as a function of the following measured inputs: compressor outlet pressure, water injection flow and engine speed. In principle, similar models can be used to predict any variable of interest such as pressure temperature, speed or power, based on other measurements from around the engine; and of course such models are

not limited to the field of engines.

After development of the initial, off-line model, the model algorithm is run in software as part of the control system for the plant. At each sample time of the control system, the model uses the measured plant inputs to predict the plant output.

In addition, the model is updated at each sample time based on the measured output data received from the plant. That is, it is tuned to be a better representation of the inputloutput relationship.

Figure l shows a cycle of steps that is followed once for each sample interval. In Step l, the model is re-initialized to the off-line model. In particular, the model parameters and an error covariance matrix are initialized to the values obtained from A080005 the off-line fitting. It is important to note that this is done at every sample interval, whereby the adapted model always remains tethered to the initial model and the online data do not accumulate disproportionately.

Next, in Step 2, a window of n previous samples of input/output data is updated by adding the most recent set of values and deleting the oldest set of values.

In Step 3, the model adaptation takes place, using the recursive least squares (RLS) method. Alternatively, another established recursive parameter estimation methods could be applied, for example Recursive Maximum Likelihood (RML), Recursive Instrumental Variables (RIV) etc. The recursion begins with the oldest set of sampled input values and sweeps through the n samples in the window of historical data in sequence from old to new, updating the model parameters and the error covariance matrix each time to refine the model.

After n recursions of the adaptive algorithm, the final parameters of the adapted model are used in Step 4 to predict the current output based on the measured current inputs.

Step 5 represents a pause until the next sample interval, when the model is re- initialized and the cycle begins again from Step l.

These steps are described in more detail below in relation to the particular model implementing the RLS parameter estimation method.

Model Structure The structure of one particular model used for gas turbine engine sensor to sensor modelling is of the form: A00005 - 6 y(t) = be (I) +b'l (t)rl (I) + b2l (t)r2 (I) + .. + bNRI (t)rNR (I) + bl2 (t)rl (I) + b22 (t)r2 (I) + .. + bNR2 (t)rNR (I) (l) + bl3 (t)rl (I) + b23 (t)r2 (t) + À. + bNR3 (t)rNR (I) Some of the coefficients bit) in (l) may be set to zero according to the module parameter 'order', discussed below. However, every model contains the term be (I) . Equation (l) can be written as follows: y(t) = [be (I) tell (t) À-- bNR3 (tail = * Aft) (2) rNR (t) Nomenclature t - Index of current sample interval t* - Index of one of the sample intervals in the history window (t-NH < t* ≤ I) y(t) Current measured value of response variable (output) {scalar} r(t) - Current measured value of model regressors (inputs) {vector NR* l 3 y(t) -Current modelpredicted output {scalar} B(t) - Current model parameters {vector NTH* l} NTH - number of model parameters (bo, b',, À. bNR3) NR - number of model regressors (rat, r2, . . . rNR) NH - number of samples in the data memory (also referred to as n) 00 - initial model parameter values {vector NTH* l} PO - initial model parameter confidence {matrix NTH*NTH} A080005 7 - RLS forget factor (0<≤1), default value 1 order - model structure identifier, {Binary vector 1 *3NR} 1 indicates term included, 0 indicates term not included.

For example, -vith reference to equation (1) above, if NR = 4 and order = [1 1 1 1 1 0 0 1 1 0 1 1] then y(f) = bo (t) + tell (t)rl (t) + b21 (t)r2 (t) + b31 (t)r3 (t) + b41 (t)r4 (t) + bl2 (t)rl (t) + b42 (t)r42 (t) + bl3 (t)rl3 (t) + b33 (t)r33 (t) + b43 (t)r4 (t) Algorithm Step 1) RLS initialisation: 1.1) Sett*=t-NH 1.2) Set B(t*)= Oo 1.3) Set P(t*) = Po Step 2) Update data memoy by adding most recent measurements r(t), y(t) and discarding oldest measurements y(t-NH), r(t- NH) I y(t-NH+l) r(t-NH+l) 1: : data moves up one row I in the delay regiser a I y(t-2) r(t-2) each sample update I I y(t-l) r(t-1) l y(t) r(t) Step 3) RLS recursion: 3.1) Update recursion time index: A080005 - a t* = t*+l 3.2) Compute adaptation gain ci(t *) = P(t * -l)a, (I*) + al (t*)P(t -Ha) (I) where a, (I*) represents the reduced terms vector a, (t), defined in equation (2) above. In the reduced terms vector, the elements for which the corresponding element in the binary vector order are equal to O have been removed.

Il(t*) is an {NTH* 1} vector 3.3) Compute model error (I*) = y (I*) -wT(t*)(t *-l) 3.4) Update model parameters 8(t*) = 8(t * -l) + P(t*) s(t*) 3.5) Update -the error covariance matrix ( ) Pit -l) _ ( )a7 (I)) (t*)P(t *-Al l + a) (t*)P(t -ba, (I) 3.6) IF (I* < t) Return to Step 3.1 ELSE go to Step 4.

Step 4) Update and output model prediction y(t) = (t)a, (I) Step S) Wait for next sample interval before returning to Step 1.

A080005 - 9 -

Claims

1. A method of predicting output values y of a physical system from a set of measured inputs rj of the system, wherein at each time t when a prediction is made, the method comprises the following steps: (a) storing the set of measured inputs rj(tJ and the corresponding measured output y(t) at each of a predetermined number n of times to... In earlier than I, (b) initializing a model of the system, which generates a predicted output y(t) from the set of measurements rj(t), to a predetermined initial model, (c) using each of the predetermined number n of sets of measured inputs rj(tJ and output y(tJ to adapt the model, and (d) using the adapted model to predict the output value y(t) of the system at time t from the set of measured inputs rift) of the system at time I.

2. A method according to claim l, wherein step (c) uses the n sets of measurements rj(tJ end y(tJ in chronological order.

3. A method according to claim l or claim 2, wherein the times t...tn are the most recent n times preceding time t at which measurements were made.

4. A method according to any preceding claim, wherein, for each of the n sets of measurements of the inputs and the output, step (c) comprises the following steps: (cl) applying the model to the set of inputs rj(tJ to generate a predicted output y(ti) at time ti; (c2) calculating an error e(tJ by finding the difference between the predicted output y(tJ and the measured output y(tJ at time ti, and (c3) using the error e(tJ to revise a set of parameters b' ofthe model.

5. A method according to claim 4, wherein step (c3) uses a recursive estimation algorithm to revise the parameters be of the model.

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6. A method according to claim 5, wherein step (c3) uses a recursive least squares algorithm to revise the parameters hi of the model.

7. A method according to claim 5 or claim 6, wherein the model further includes an error covariance matrix P. which is used in step (c3) to revise the parameters of the model, and wherein the method further comprises a step (c4) of using the measured input values rj(ti) to revise the error covariance matrix P.

8. A method according to claim 7, wherein in step (c4) the error covariance matrix P is revised using the measurements from time ti according to the following formula: P(t.) =-Spat)_ (ti,))(ti)Q) (ti)P(ti ') L -' + er(ti)P(tj,)(tj) ] where: P(ti) is the revised error covariance matrix based on the measured input values up to time ti; P(ti i) is the previous error covariance matrix based on the measured input values up to time ti i; Aft; ) is a vector representing the set of measured input values rj(ti) at time t and powers of those input values {rj(t)}n that are used in the model; and is a scalar "forget factor" in the range 0 < < 1.

9. A method according to claim 8, wherein: in step (c2) the scalar error eat) is calculated in accordance with the following formula: (tj) = y(t, )- (t' , )a'(t' ) where is a vector representing the set of model parameters bk; and in step (c3) the parameters of the model are revised in accordance with the following formula: A080005 - 11 B(t, ) = 49(t' l) + u(t, ) e(t)) whereat is a vector calculated from the following formula: + a, (t')P(t, )a'(t, )

10. A method according to any preceding claim, wherein the physical system is an engine.

A method according to claim 10, wherein the physical system is a gas turbine engine.

12. A method of predicting an output value of a physical system from a set of measured inputs of the system substantially as described herein.

13. Control apparatus for a physical system, the apparatus including means for predicting output values of the system from measured inputs of the system in accordance with a method as defined in any preceding claim.

A080005