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Lectures 12&13: Persistent Excitation for Off-line and On-line Parameter Estimation. Dr Martin Brown Room: E1k Email: [email protected] Telephone: 0161 306 4672 http://www.eee.manchester.ac.uk/intranet/pg/coursematerial/. Outline 13&14. Persistent excitation and identifiability

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Lectures 12&13: Persistent Excitation for Off-line and On-line Parameter Estimation

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Lectures 12&13: Persistent Excitation for Off-line and On-line Parameter Estimation

Dr Martin Brown

Room: E1k

Email: [email protected]

Telephone: 0161 306 4672

http://www.eee.manchester.ac.uk/intranet/pg/coursematerial/


Outline 13&14

  • Persistent excitation and identifiability

  • Structure of XTX

  • Role of signal magnitude

  • Role of signal correlation

  • Types of system identification signals for experimental design)

  • On-line estimation and persistent excitation

  • On-line persistent excitation

  • Time-varying parameters

  • Exponential Recursive Least Squares (RLS)


Resources 13&14

  • Core reading

  • Ljung chapter 13

  • On-line notes, chapter 5

  • Norton, Chapter 8


Central Question: Experimental Design

  • An important part of system identification is experimental design

  • Experimental design is involved with answering the question of how experiments should be constructed to to maximise the information collected with the minimum amount of effort/cost

  • For system identification, this corresponds to how the input/control signal injected into the plant should be chosen to best identify the parameters

  • N.B. This is relative to the model structure (i.e. different model structures will have different optimal model designs).


What is Persistent Excitation

  • Persistent excitation refers to the design of a signal, u(t), that produces estimation data D={X,y} which is rich enough to satisfactorily identify the parameters

  • The parameter accuracy/covariance is determined by:

  • Ideally, and E(xi2)>>sy2

  • The variance/covariance can be made smaller (better) by:

  • Reducing the measurement error variance (hard)

  • Collecting more data (but this often costs money)

  • Make the signals larger (but there are physical limits)

  • Make the signals independent (difficult for dynamics)


Review: XTX Matrix

  • The variance/covariance matrix, XTX, (and its inverse) is central in many system identification/parameter estimation tasks

  • Consider a model


Identifying Parameters

  • For a set of measured exemplars D={X,y}, there are several (related) concepts that determine how well the parameters can be estimated (off-line, in batch mode)

  • 1 , i.e. how well can the parameters be identified or equivalently, what is the region of uncertainty about the estimated values q.

  • Is (XTX) non-singular? i.e. can the normal equations be solved uniquely

  • Are the parameter estimates significantly non-zero?

  • All of these are related and influenced/determined by how the input data X is generated/collected.

^


Example: Signal Magnitude & Noise

  • Consider feeding steps of magnitude 0.01, 0.1 and 1 into the first order, electrical circuit with

  • The magnitude of the signals strongly influences the identifiability of the parameters. Typically, each signal should be of similar magnitude and high in relation to the measurement noise.


Example: Signals Interactions

  • Consider collecting data from a model of the form:

  • Each input is ui(t) = sin(0.5t), 20 samples:

  • Note that X = [u1u2] is singular

  • Now consider u1(t) = sin(0.5t), u2(t) = cos(0.5t), E(u1u2)0, E(ui2)=c

  • The input signals are ~orthogonal

  • This is difficult with feedback …


Good and Bad Covariance Matrices

  • Ideal structure of (XTX)-1 is

  • which means that:

  • Each parameter has the same variance, and the estimates are uncorrelated. In addition, if E(xi2)>>sy2, the parameter variances are small.

  • Each parameter can be identified to the same accuracy

  • For modelling and control, we want to feed an input signal in produces a matrix with these properties.


Well determined l = 12.8

l = 0.52

poorly determined

How to Measure Goodness?

  • There are several ways to assess/compare how good a particular signal is:

    • Cond(XTX) = lmax/lmin

  • This measures the ratio of the maximum signal to the minimum signal correlations

  • Smaller Cond(XTX) is better

  • Cond(I) = 1

  • Choose u to minu Cond(XTX)

  • Insensitive to the signal

  • magnitude, just measures the

  • degree of correlation


u(t)

y(t)

Signal Correlation and Dynamics

  • So far, we have just discussed choosing input signals that are uncorrelated/orthogonal

  • However, dynamics/feedback introduce correlation between individual signals (i.e. between u(t) and u(t-1) and y(t-1) and y(t)):

  • E(y(t-1)u(t))  0

  • This is because y(t) is related to u(t), especially when they change slowly

  • A stable plant will track (correlate

  • with) the input signal

  • Condition will be worse


Example 1: Impulse/Step Signal

u(t)

u(t)

u(t)

  • Any linear system is completely identified by its impulse (or step) response – because convolution can be used to calculate the output.

  • However, as shown in Slide 8, there are several aspects that may make this identification difficult

  • Magnitude of the step signal (relative to the noise) & impulse

  • Length of the transient period, relative to the steady state

  • Generation of the impulse/step signal which may be infeasible due to control magnitude and/or actuator dynamics limits

  • High correlation between u(t) and u(t-k), steady state adds little

  • Note that if the plant model is non-linear, an impulse/step only collects information at one operating point, so if the aim is to reject non-linear components, step/impulse trains of different amplitudes must be used

t

t

t


Example 2: Sinusoidal Signal

  • While a sinusoid may look to be a rich enough signal to identify linear models

  • It can be used to identify the gain margin and phase advance for one particular frequency

  • However, can only be used when the maximum control delay is 1, because

  • u(t) = q1u(t-1) + q2u(t-2)

  • Similar for the output feedback delay as well (because in the steady state, the output is also sinusoidal).


Example 3: Random Signal

  • A random signal is persistently exciting for a linear model of any order

  • It involves a range of amplitudes and so can be used for non-linear terms as well. However,

  • It is a bit of a “scatter gun” approach

  • It can be wasteful when the model structure is reasonably well-known

  • There may be limits on the actuator dynamics

  • Difficult to use on-line, where the control action is “smooth”


On-Line Parameter Estimation

  • So far, it has been assumed that the parameter estimation is being performed off-line

    • Collect a fixed size data set

    • Estimate the parameters

    • Issues of parameter identifiability are related to a fixed data set

  • On-line parameter estimation is more complex

    • Typically a plant is controlled to a set-point for a long period of time

    • The recursive calculation is often re-set after fixed intervals (re-set floating point errors)

    • Sometimes need to track time-varying parameters


Time Varying Parameters

  • One reason for considering on-line/recursive parameter estimation is to model systems where the linear parameters vary slowly with time

  • Common parameter changes are step or slow drifts

  • The aim is to treat the systems as slowly changing, and the model must be kept “plastic enough” to respond to changes in the parameters

  • Note that, strictly speaking, this is now a non-linear system where the dynamics of the parameters are much slower than the dynamics of the system’s states.


Long Term Convergence & Plasticity

  • Using either the normal equations or the equivalent on-line, recursive version, when the amount of data increases, the parameter estimates tend to the true values and the effect of a new datum is close to zero.

  • To model parametric drifts, the parameter estimates must include a term that makes the model more dependent on recent large residuals

  • This can be achieved by defining a modified performance function where the residuals are weighted by a time decay factor


Exponential RLS

  • Form the new input vector x(t+1) using the new data

  • Form e(t+1) from the model using

  • Form P(t+1) using

  • Update the least squares estimate

  • Proceed with next time step


Example: Exponential RLS

  • Consider the first order electrical circuit example

  • Here a and k are functions of time and both linearly vary between 1 and 2 during the length of the simulation

  • Input signal is sinusoidal and noise N(0,0.01) is added

  • There is a balance between noise filtering and model/parameter plasticity


Parameter Convergence & Persistent Excitation

  • While this algorithm is relatively simple, it has two important, related aspects that must be considered

  • What is the value of l?

  • What form of persistently exciting input is needed?

  • When l is 1, this is just standard RLS estimation.

  • When l<0.9, the model is extremely adaptive and the parameters will not generally converge when the measurement noise is significant

  • As the model becomes more plastic, the input signal must be sufficiently persistently exciting over every significant time window to stop random parameter drift/premature convergence


Summary 13&14

  • The engineer’s aim is to minimise the amount of data collected to identify the parameters sufficiently accurately

  • Signal magnitude should be as large as possible to improve the signal/noise ratio and to minimize the parameter covariances. However, the signal should not to large enough to violate any system constraints or to make the unknown system significantly non-linear

  • Signal type & frequency must be smooth enough not to exceed any dynamic constraints, however the dynamics must excite any potential dynamics.

  • When parameter estimation is on-line, this imposes additional constraints as the signals must be sufficiently exciting for each time period

  • Exponential-forgetting can be used to track time-varying parameters, but previous comments must hold


Laboratory 13&14

  • 1. Prove Slide 14 relationship for a sin function – what are q1 and q2

  • 2. Measure the Cond(XTX) and the parameter estimates for:

    • Step

    • Sin

    • Random

  • for the electrical simulation. Try varying the magnitudes of the step signal as well.

  • 3. Implement the exponential RLS for the electrical simulation for time-varying parameters on Slide 20. Try changing the input/control signals and compare the responses.


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