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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.

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

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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