123, v2'0

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Lectures 1213 Persistent Excitation for
Off-line and On-line Parameter Estimation

• Dr Martin Brown
• Room E1k
• Email martin.brown_at_manchester.ac.uk
• Telephone [phone]
• http//www.eee.manchester.ac.uk/intranet/pg/course
  material/

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Outline 1314

• 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)

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Resources 1314

• Core reading
• Ljung chapter 13
• On-line notes, chapter 5
• Norton, Chapter 8

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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).

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What is Persistent Excitation

• Persistent excitation refers to the design of a
  signal, u(t), that produces estimation data
  DX,y which is rich enough to satisfactorily
  identify the parameters
• The parameter accuracy/covariance is determined
  by
• Ideally, and E(xi2)gtgtsy2
• 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)

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Review XTX Matrix

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

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Identifying Parameters

• For a set of measured exemplars DX,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.

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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.

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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 u1 u2 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

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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)gtgtsy2, 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.

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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

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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

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Example 1 Impulse/Step Signal
u(t)
u(t)
u(t)
t
t
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

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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).

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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

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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

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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 systems states.

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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

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Exponential RLS

• Form the new input vector x(t1) using the new
  data
• Form e(t1) from the model using
• Form P(t1) using
• Update the least squares estimate
• Proceed with next time step

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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

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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 llt0.9, 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

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Summary 1314

• The engineers 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

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Laboratory 1314

• 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.