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

Dr Martin Brown Room: E1k Telephone: EE-M /7: EF L12&13

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**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) EE-M /7: EF L12&13

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**Resources 13&14 Core reading Ljung chapter 13 On-line notes, chapter 5**

Norton, Chapter 8 EE-M /7: EF L12&13

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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). EE-M /7: EF L12&13

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**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) EE-M /7: EF L12&13

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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 EE-M /7: EF L12&13

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**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) , 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. ^ EE-M /7: EF L12&13

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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. EE-M /7: EF L12&13

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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 … EE-M /7: EF L12&13

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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)>>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. EE-M /7: EF L12&13

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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 Well determined l = 12.8 l = 0.52 poorly determined EE-M /7: EF L12&13

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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 u(t) y(t) EE-M /7: EF L12&13

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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 EE-M /7: EF L12&13

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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). EE-M /7: EF L12&13

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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” EE-M /7: EF L12&13

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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 EE-M /7: EF L12&13

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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 system’s states. EE-M /7: EF L12&13

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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 EE-M /7: EF L12&13

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**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 EE-M /7: EF L12&13

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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 EE-M /7: EF L12&13

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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 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 EE-M /7: EF L12&13

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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 EE-M /7: EF L12&13

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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. EE-M /7: EF L12&13

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