The general form of the recursive estimation algorithm is as follows:
$$\widehat{\theta}\left(t\right)=\widehat{\theta}\left(t-1\right)+K\left(t\right)\left(y\left(t\right)-\widehat{y}\left(t\right)\right)$$
$$\widehat{\theta}\left(t\right)$$ is the parameter estimate at time t. y(t) is the observed output at time t and $$\widehat{y}\left(t\right)$$ is the prediction of y(t) based on observations up to time t-1. The gain, K(t), determines how much the current prediction error $$y\left(t\right)-\widehat{y}\left(t\right)$$ affects the update of the parameter estimate. The estimation algorithms minimize the prediction-error term $$y\left(t\right)-\widehat{y}\left(t\right)$$.
The gain has the following form:
$$K\left(t\right)=Q\left(t\right)\psi \left(t\right)$$
The recursive algorithms supported by the System Identification Toolbox™ product differ based on different approaches for choosing the form of Q(t) and computing $$\psi \left(t\right)$$, where $$\psi \left(t\right)$$ represents the gradient of the predicted model output $$\widehat{y}\left(t|\theta \right)$$ with respect to the parameters $$\theta $$.
The simplest way to visualize the role of the gradient $$\psi \left(t\right)$$ of the parameters, is to consider models with a linear-regression form:
$$y\left(t\right)={\psi}^{T}\left(t\right){\theta}_{0}\left(t\right)+e\left(t\right)$$
In this equation, $$\psi \left(t\right)$$ is the regression vector that is computed based on previous values of measured inputs and outputs. $${\theta}_{0}\left(t\right)$$ represents the true parameters. e(t) is the noise source (innovations), which is assumed to be white noise. The specific form of $$\psi \left(t\right)$$ depends on the structure of the polynomial model.
For linear regression equations, the predicted output is given by the following equation:
$$\widehat{y}\left(t\right)={\psi}^{T}\left(t\right)\widehat{\theta}\left(t-1\right)$$
For models that do not have the linear regression form, it is not possible to compute exactly the predicted output and the gradient $$\psi \left(t\right)$$ for the current parameter estimate $$\widehat{\theta}\left(t-1\right)$$. To learn how you can compute approximation for $$\psi \left(t\right)$$ and $$\widehat{\theta}\left(t-1\right)$$ for general model structures, see the section on recursive prediction-error methods in [1].
The System Identification Toolbox software provides the following recursive estimation algorithms for online estimation:
The forgetting factor and Kalman Filter formulations are more computationally intensive than gradient and unnormalized gradient methods. However, they typically have better convergence properties.
The following set of equations summarizes the forgetting factor adaptation algorithm:
$$\widehat{\theta}\left(t\right)=\widehat{\theta}\left(t-1\right)+K\left(t\right)\left(y\left(t\right)-\widehat{y}\left(t\right)\right)$$
$$\widehat{y}\left(t\right)={\psi}^{T}\left(t\right)\widehat{\theta}\left(t-1\right)$$
$$K\left(t\right)=Q\left(t\right)\psi \left(t\right)$$
$$Q\left(t\right)=\frac{P\left(t-1\right)}{\lambda +{\psi}^{T}\left(t\right)P\left(t-1\right)\psi \left(t\right)}$$
$$P\left(t\right)={\scriptscriptstyle \frac{1}{\lambda}}\left(P\left(t-1\right)-\frac{P\left(t-1\right)\psi \left(t\right)\psi {\left(t\right)}^{T}P\left(t-1\right)}{\lambda +\psi {\left(t\right)}^{T}P\left(t-1\right)\psi \left(t\right)}\right)$$
The software ensures P(t) is a positive-definite
matrix by using a square-root algorithm to update it [2].
The software computes P
assuming that the residuals
(difference between estimated and measured outputs) are white noise,
and the variance of these residuals is 1. R_{2}/2
* P
is
approximately equal to the covariance matrix of the estimated parameters,
where R_{2} is the true variance
of the residuals.
Q(t) is obtained by minimizing the following function at time t:
$$\sum}_{k=1}^{t}{\lambda}^{t-k}{(y(k)-\widehat{y}(k))}^{2$$
See section 11.2 in [1] for details.
This approach discounts old measurements exponentially such that an observation that is $$\tau $$ samples old carries a weight that is equal to $${\lambda}^{\tau}$$ times the weight of the most recent observation. $$\tau ={\scriptscriptstyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$1-\lambda $}\right.}$$ represents the memory horizon of this algorithm. Measurements older than $$\tau ={\scriptscriptstyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$1-\lambda $}\right.}$$ typically carry a weight that is less than about 0.3.
$$\lambda $$ is called the forgetting factor
and typically has a positive value between 0.98
and 0.995
.
Set $$\lambda =1$$ to
estimate time-invariant (constant) parameters. Set $$\lambda <1$$ to
estimate time-varying parameters.
Note: The forgetting factor algorithm for $$\lambda $$ = 1 is equivalent to the Kalman filter algorithm with R_{1}=0 and R_{2}=1. For more information about the Kalman filter algorithm, see Kalman Filter. |
The following set of equations summarizes the Kalman filter adaptation algorithm:
$$\widehat{\theta}\left(t\right)=\widehat{\theta}\left(t-1\right)+K\left(t\right)\left(y\left(t\right)-\widehat{y}\left(t\right)\right)$$
$$\widehat{y}\left(t\right)={\psi}^{T}\left(t\right)\widehat{\theta}\left(t-1\right)$$
$$K\left(t\right)=Q\left(t\right)\psi \left(t\right)$$
$$Q\left(t\right)=\frac{P\left(t-1\right)}{{R}_{2}+{\psi}^{T}\left(t\right)P\left(t-1\right)\psi \left(t\right)}$$
$$P\left(t\right)=P\left(t-1\right)+{R}_{1}-\frac{P\left(t-1\right)\psi \left(t\right)\psi {\left(t\right)}^{T}P\left(t-1\right)}{{R}_{2}+\psi {\left(t\right)}^{T}P\left(t-1\right)\psi \left(t\right)}$$
The software ensures P(t) is a positive-definite
matrix by using a square-root algorithm to update it [2].
The software computes P
assuming that the residuals
(difference between estimated and measured outputs) are white noise,
and the variance of these residuals is 1. R_{2}* P
is
the covariance matrix of the estimated parameters, and R_{1} /R_{2} is
the covariance matrix of the parameter changes. Where, R_{1} is
the covariance matrix of parameter changes that you specify.
This formulation assumes the linear-regression form of the model:
$$y\left(t\right)={\psi}^{T}\left(t\right){\theta}_{0}\left(t\right)+e\left(t\right)$$
Q(t) is computed using a Kalman filter.
This formulation also assumes that the true parameters $${\theta}_{0}\left(t\right)$$ are described by a random walk:
$${\theta}_{0}\left(t\right)={\theta}_{0}\left(t-1\right)+w\left(t\right)$$
w(t) is Gaussian white noise with the following covariance matrix, or drift matrix R_{1}:
$$Ew\left(t\right){w}^{T}\left(t\right)={R}_{1}$$
R_{2} is the variance of the innovations e(t) in the following equation:
$$y\left(t\right)={\psi}^{T}\left(t\right){\theta}_{0}\left(t\right)+e\left(t\right)$$
The Kalman filter algorithm is entirely specified by the sequence of data y(t), the gradient $$\psi \left(t\right)$$, R_{1}, R_{2}, and the initial conditions $$\theta \left(t=0\right)$$ (initial guess of the parameters) and $$P\left(t=0\right)$$ (covariance matrix that indicates parameters errors).
Note: It is assumed that R_{1} and P(t = 0) matrices are scaled such that R_{2} = 1. This scaling does not affect the parameter estimates. |
In the linear regression case, the gradient methods are also known as the least mean squares (LMS) methods.
The following set of equations summarizes the unnormalized gradient and normalized gradient adaptation algorithm:
$$\widehat{\theta}\left(t\right)=\widehat{\theta}\left(t-1\right)+K\left(t\right)\left(y\left(t\right)-\widehat{y}\left(t\right)\right)$$
$$\widehat{y}\left(t\right)={\psi}^{T}\left(t\right)\widehat{\theta}\left(t-1\right)$$
$$K\left(t\right)=Q\left(t\right)\psi \left(t\right)$$
In the unnormalized gradient approach, Q(t) is given by:
$$Q(t)=\gamma $$
In the normalized gradient approach, Q(t) is given by:
$$Q\left(t\right)=\frac{\gamma}{{\left|\psi \left(t\right)\right|}^{2}+Bias}$$
The normalized gradient algorithm scales the adaptation gain, γ, at each step by the square of the two-norm of the gradient vector. If the gradient is close to zero, this can cause jumps in the estimated parameters. To prevent these jumps, a bias term is introduced in the scaling factor.
These choices of Q(t) for the gradient algorithms update the parameters in the negative gradient direction, where the gradient is computed with respect to the parameters. See pg. 372 in [1] for details.
[1] Ljung, L. System Identification: Theory for the User. Upper Saddle River, NJ: Prentice-Hall PTR, 1999.
[2] Carlson, N.A. "Fast triangular formulation of the square root filter." AIAA Journal, Vol. 11, Number 9, 1973, pp. 1259-1265.
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