Saturday, October 17, 2009

Kalman filter

The Kalman filter is an efficient recursive filter that estimates the state of a linear dynamic system from a series of noisy measurements. It is used in a wide range of engineering applications from radar to computer vision, and is an important topic in control theory and control systems engineering. Together with the linear-quadratic regulator (LQR), the Kalman filter solves the linear-quadratic-Gaussian control problem (LQG). The Kalman filter, the linear-quadratic regulator and the linear-quadratic-Gaussian controller are solutions to what probably are the most fundamental problems in control theory.

Example applications

An example application would be providing accurate, continuously updated information about the position and velocity of an object given only a sequence of observations about its position, each of which includes some error. For example, in a radar application where one is interested in tracking a target, information about the location, speed, and acceleration of the target is measured at each time instant with a great deal of corruption by noise. The Kalman filter exploits the trusted model of the dynamics of the target, which describes the kind of movement possible by the target, to remove the effects of the noise and get a good estimate of the location of the target at the present time (filtering), at a future time (prediction), or at a time in the past (interpolation or smoothing).
Alternatively, consider an old slow car that is known to go from 0 to 60 miles per hour (mph) in no less than 10 seconds. The speedometer on this car however shows very noisy measurements that vary wildly within a 40 mph window around the actual speed of the car. From stop – which is measured with certainty because the wheels are not turning – the driver of the car pushes its gas pedal as far as possible. Five seconds later, the speedometer reads 70 mph. The driver concludes that the slow car cannot be traveling that quickly and uses information about the known speedometer noise to conclude that the car is likely traveling at 30 mph instead. Similarly, a Kalman filter uses information about noise and system dynamics to reduce uncertainty from noisy measurements.

Underlying dynamic system model

Kalman filters are based on linear dynamical systems discretized in the time domain. They are modelled on a Markov chain built on linear operators perturbed by Gaussian noise. The state of the system is represented as a vector of real numbers. At each discrete time increment, a linear operator is applied to the state to generate the new state, with some noise mixed in, and optionally some information from the controls on the system if they are known. Then, another linear operator mixed with more noise generates the visible outputs from the hidden state. The Kalman filter may be regarded as analogous to the hidden Markov model, with the key difference that the hidden state variables take values in a continuous space (as opposed to a discrete state space as in the hidden Markov model). Additionally, the hidden Markov model can represent an arbitrary distribution for the next value of the state variables, in contrast to the Gaussian noise model that is used for the Kalman filter. There is a strong duality between the equations of the Kalman Filter and those of the hidden Markov model. A review of this and other models is given in Roweis and Ghahramani (1999).[1]
In order to use the Kalman filter to estimate the internal state of a process given only a sequence of noisy observations, one must model the process in accordance with the framework of the Kalman filter. This means specifying the following matrices: Fk, the state-transition model; Hk, the observation model; Qk, the covariance of the process noise; Rk, the observation noise; and sometimes Bk, the control-input model for each time-step, k, as described below.
Model underlying the Kalman filter. Squares represent matrices. Ellipses represent multivariate normal distributions (with the mean and covariance matrix enclosed). Unenclosed values are vectors.
The Kalman filter model assumes the true state at time k is evolved from the state at (k − 1) according to
\textbf{x}_{k} = \textbf{F}_{k} \textbf{x}_{k-1} + \textbf{B}_{k} \textbf{u}_{k} + \textbf{w}_{k}
where
  • Fk is the state transition model which is applied to the previous state xk−1;
  • Bk is the control-input model which is applied to the control vector uk;
  • wk is the process noise which is assumed to be drawn from a zero mean multivariate normal distribution with covariance Qk.
\textbf{w}_{k} \sim N(0, \textbf{Q}_k)
At time k an observation (or measurement) zk of the true state xk is made according to
\textbf{z}_{k} = \textbf{H}_{k} \textbf{x}_{k} + \textbf{v}_{k}
where Hk is the observation model which maps the true state space into the observed space and vk is the observation noise which is assumed to be zero mean Gaussian white noise with covariance Rk.
\textbf{v}_{k} \sim N(0, \textbf{R}_k)
The initial state, and the noise vectors at each step {x0, w1, ..., wk, v1 ... vk} are all assumed to be mutually independent.
Many real dynamical systems do not exactly fit this model. In fact, unmodelled dynamics can seriously degrade the filter performance, even when it was supposed to work with unknown stochastic signals as inputs. The reason for this is that the effect of unmodelled dynamics depends on the input, and, therefore, can bring the estimation algorithm to unstability (to diverge). On the other hand, independent white noise signals will not make the algorithm diverge. The problem of separating between measurement noise and unmodelled dynamics is a difficult one and is treated in control theory under the framework of robust control.

The Kalman filter

The Kalman filter is a recursive estimator. This means that only the estimated state from the previous time step and the current measurement are needed to compute the estimate for the current state. In contrast to batch estimation techniques, no history of observations and/or estimates is required. In what follows, the notation \hat{\textbf{x}}_{n|m} represents the estimate of \textbf{x} at time n given observations up to, and including time m.
The state of the filter is represented by two variables:
  • \hat{\textbf{x}}_{k|k}, the a posteriori state estimate at time k given observations up to and including at time k;
  • \textbf{P}_{k|k}, the a posteriori error covariance matrix (a measure of the estimated accuracy of the state estimate).
The Kalman filter has two distinct phases: Predict and Update. The predict phase uses the state estimate from the previous timestep to produce an estimate of the state at the current timestep. This predicted state estimate is also known as the a priori state estimate because, although it is an estimate of the state at the current timestep, it does not include observation information from the current timestep. In the update phase, the current a priori prediction is combined with current observation information to refine the state estimate. This improved estimate is termed the a posteriori state estimate.

Predict

Predicted (a priori) state \hat{\textbf{x}}_{k|k-1} = \textbf{F}_{k}\hat{\textbf{x}}_{k-1|k-1} + \textbf{B}_{k-1} \textbf{u}_{k-1}
Predicted (a priori) estimate covariance
 \textbf{P}_{k|k-1} = \textbf{F}_{k} \textbf{P}_{k-1|k-1} \textbf{F}_{k}^{\text{T}} + \textbf{Q}_{k-1}

Update

Innovation or measurement residual \tilde{\textbf{y}}_k = \textbf{z}_k - \textbf{H}_k\hat{\textbf{x}}_{k|k-1}
Innovation (or residual) covariance \textbf{S}_k = \textbf{H}_k \textbf{P}_{k|k-1} \textbf{H}_k^\text{T} + \textbf{R}_k
Optimal Kalman gain \textbf{K}_k = \textbf{P}_{k|k-1}\textbf{H}_k^\text{T}\textbf{S}_k^{-1}
Updated (a posteriori) state estimate \hat{\textbf{x}}_{k|k} = \hat{\textbf{x}}_{k|k-1} + \textbf{K}_k\tilde{\textbf{y}}_k
Updated (a posteriori) estimate covariance \textbf{P}_{k|k} = (I - \textbf{K}_k \textbf{H}_k) \textbf{P}_{k|k-1}
The formula for the updated estimate covariance above is only valid for the optimal Kalman gain. Usage of other gain values require a more complex formula found in the derivations section.

Invariants

If the model is accurate, and the values for \hat{\textbf{x}}_{0|0} and \textbf{P}_{0|0} accurately reflect the distribution of the initial state values, then the following invariants are preserved: all estimates have mean error zero
  • \textrm{E}[\textbf{x}_k - \hat{\textbf{x}}_{k|k}] = \textrm{E}[\textbf{x}_k - \hat{\textbf{x}}_{k|k-1}] = 0
  • \textrm{E}[\tilde{\textbf{y}}_k] = 0
where E[ξ] is the expected value of ξ, and covariance matrices accurately reflect the covariance of estimates
  • \textbf{P}_{k|k} = \textrm{cov}(\textbf{x}_k - \hat{\textbf{x}}_{k|k})
  • \textbf{P}_{k|k-1} = \textrm{cov}(\textbf{x}_k - \hat{\textbf{x}}_{k|k-1})
  • \textbf{S}_{k} = \textrm{cov}(\tilde{\textbf{y}}_k)

Examples

Consider a truck on perfectly frictionless, infinitely long straight rails. Initially the truck is stationary at position 0, but it is buffeted this way and that by random acceleration. We measure the position of the truck every Δt seconds, but these measurements are imprecise; we want to maintain a model of where the truck is and what its velocity is. We show here how we derive the model from which we create our Kalman filter.
There are no controls on the truck, so we ignore Bk and uk. Since F, H, R and Q are constant, their time indices are dropped.
The position and velocity of the truck is described by the linear state space
\textbf{x}_{k} = \begin{bmatrix} x \\ \dot{x} \end{bmatrix}
where \dot{x} is the velocity, that is, the derivative of position with respect to time.
We assume that between the (k − 1)th and kth timestep the truck undergoes a constant acceleration of ak that is normally distributed, with mean 0 and standard deviation σa. From Newton's laws of motion we conclude that
\textbf{x}_{k} = \textbf{F} \textbf{x}_{k-1} + \textbf{B}a_{k}
where
\textbf{F} = \begin{bmatrix} 1 & \Delta t \\ 0 & 1 \end{bmatrix}
and
\textbf{B} = \begin{bmatrix} \frac{\Delta t^{2}}{2} \\ \Delta t \end{bmatrix}
We find that
\textbf{Q} = \textrm{cov}(\textbf{B}a) = \textrm{E}[(\textbf{B}a)(\textbf{B}a)^{\text{T}}] = \textbf{B} \textrm{E}[a^2] \textbf{B}^{\text{T}} = \textbf{B}[\sigma_a^2]\textbf{B}^{\text{T}} = \sigma_a^2 \textbf{B}\textbf{B}^{\text{T}} (since σa is a scalar).
At each time step, a noisy measurement of the true position of the truck is made. Let us suppose the measurement noise vk is also normally distributed, with mean 0 and standard deviation σz.
\textbf{z}_{k} = \textbf{H x}_{k} + \textbf{v}_{k}
where
\textbf{H} = \begin{bmatrix} 1 & 0 \end{bmatrix}
and
\textbf{R} = \textrm{E}[\textbf{v}_k \textbf{v}_k^{\text{T}}] = \begin{bmatrix} \sigma_z^2 \end{bmatrix}
We know the initial starting state of the truck with perfect precision, so we initialize
\hat{\textbf{x}}_{0|0} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}
and to tell the filter that we know the exact position, we give it a zero covariance matrix:
\textbf{P}_{0|0} = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}
If the initial position and velocity are not known perfectly the covariance matrix should be initialized with a suitably large number, say L, on its diagonal.
\textbf{P}_{0|0} = \begin{bmatrix} L & 0 \\ 0 & L \end{bmatrix}
The filter will then prefer the information from the first measurements over the information already in the model.

Derivations

Deriving the posterior estimate covariance matrix

Starting with our invariant on the error covariance Pk|k as above
\textbf{P}_{k|k} = \textrm{cov}(\textbf{x}_{k} - \hat{\textbf{x}}_{k|k})
substitute in the definition of \hat{\textbf{x}}_{k|k}
\textbf{P}_{k|k} = \textrm{cov}(\textbf{x}_{k} - (\hat{\textbf{x}}_{k|k-1} + \textbf{K}_k\tilde{\textbf{y}}_{k}))
and substitute \tilde{\textbf{y}}_k
\textbf{P}_{k|k} = \textrm{cov}(\textbf{x}_{k} - (\hat{\textbf{x}}_{k|k-1} + \textbf{K}_k(\textbf{z}_k - \textbf{H}_k\hat{\textbf{x}}_{k|k-1})))
and \textbf{z}_{k}
\textbf{P}_{k|k} = \textrm{cov}(\textbf{x}_{k} - (\hat{\textbf{x}}_{k|k-1} + \textbf{K}_k(\textbf{H}_k\textbf{x}_k + \textbf{v}_k - \textbf{H}_k\hat{\textbf{x}}_{k|k-1})))
and by collecting the error vectors we get
\textbf{P}_{k|k} = \textrm{cov}((I - \textbf{K}_k \textbf{H}_{k})(\textbf{x}_k - \hat{\textbf{x}}_{k|k-1}) - \textbf{K}_k \textbf{v}_k )
Since the measurement error vk is uncorrelated with the other terms, this becomes
\textbf{P}_{k|k} = \textrm{cov}((I - \textbf{K}_k \textbf{H}_{k})(\textbf{x}_k - \hat{\textbf{x}}_{k|k-1})) + \textrm{cov}(\textbf{K}_k \textbf{v}_k )
by the properties of vector covariance this becomes
\textbf{P}_{k|k} = (I - \textbf{K}_k \textbf{H}_{k})\textrm{cov}(\textbf{x}_k - \hat{\textbf{x}}_{k|k-1})(I - \textbf{K}_k \textbf{H}_{k})^{\text{T}} + \textbf{K}_k\textrm{cov}(\textbf{v}_k )\textbf{K}_k^{\text{T}}
which, using our invariant on Pk|k-1 and the definition of Rk becomes
\textbf{P}_{k|k} = (I - \textbf{K}_k \textbf{H}_{k}) \textbf{P}_{k|k-1} (I - \textbf{K}_k \textbf{H}_{k})^\text{T} + \textbf{K}_k \textbf{R}_k \textbf{K}_k^\text{T}
This formula (sometimes known as the "Joseph form" of the covariance update equation) is valid no matter what the value of Kk. It turns out that if Kk is the optimal Kalman gain, this can be simplified further as shown below.

Kalman gain derivation

The Kalman filter is a minimum mean-square error estimator. The error in the posterior state estimation is
\textbf{x}_{k} - \hat{\textbf{x}}_{k|k}
We seek to minimize the expected value of the square of the magnitude of this vector, \textrm{E}[|\textbf{x}_{k} - \hat{\textbf{x}}_{k|k}|^2]. This is equivalent to minimizing the trace of the posterior estimate covariance matrix \textbf{P}_{k|k} . By expanding out the terms in the equation above and collecting, we get:
\textbf{P}_{k|k} = \textbf{P}_{k|k-1} - \textbf{K}_k \textbf{H}_k \textbf{P}_{k|k-1} - \textbf{P}_{k|k-1} \textbf{H}_k^\text{T} \textbf{K}_k^\text{T} + \textbf{K}_k (\textbf{H}_k \textbf{P}_{k|k-1} \textbf{H}_k^\text{T} + \textbf{R}_k) \textbf{K}_k^\text{T}

= \textbf{P}_{k|k-1} - \textbf{K}_k \textbf{H}_k \textbf{P}_{k|k-1} - \textbf{P}_{k|k-1} \textbf{H}_k^\text{T} \textbf{K}_k^\text{T} + \textbf{K}_k \textbf{S}_k\textbf{K}_k^\text{T}
The trace is minimized when the matrix derivative is zero:
\frac{\partial \; \textrm{tr}(\textbf{P}_{k|k})}{\partial \;\textbf{K}_k} = -2 (\textbf{H}_k \textbf{P}_{k|k-1})^\text{T} + 2 \textbf{K}_k \textbf{S}_k = 0
Solving this for Kk yields the Kalman gain:
\textbf{K}_k \textbf{S}_k = (\textbf{H}_k \textbf{P}_{k|k-1})^\text{T} = \textbf{P}_{k|k-1} \textbf{H}_k^\text{T}
\textbf{K}_{k} = \textbf{P}_{k|k-1} \textbf{H}_k^\text{T} \textbf{S}_k^{-1}
This gain, which is known as the optimal Kalman gain, is the one that yields MMSE estimates when used.

Simplification of the posterior error covariance formula

The formula used to calculate the posterior error covariance can be simplified when the Kalman gain equals the optimal value derived above. Multiplying both sides of our Kalman gain formula on the right by SkKkT, it follows that
\textbf{K}_k \textbf{S}_k \textbf{K}_k^T = \textbf{P}_{k|k-1} \textbf{H}_k^T \textbf{K}_k^T
Referring back to our expanded formula for the posterior error covariance,
\textbf{P}_{k|k} = \textbf{P}_{k|k-1} - \textbf{K}_k \textbf{H}_k \textbf{P}_{k|k-1} - \textbf{P}_{k|k-1} \textbf{H}_k^T \textbf{K}_k^T + \textbf{K}_k \textbf{S}_k \textbf{K}_k^T
we find the last two terms cancel out, giving
\textbf{P}_{k|k} = \textbf{P}_{k|k-1} - \textbf{K}_k \textbf{H}_k \textbf{P}_{k|k-1} = (I - \textbf{K}_{k} \textbf{H}_{k}) \textbf{P}_{k|k-1}.
This formula is computationally cheaper and thus nearly always used in practice, but is only correct for the optimal gain. If arithmetic precision is unusually low causing problems with numerical stability, or if a non-optimal Kalman gain is deliberately used, this simplification cannot be applied; the posterior error covariance formula as derived above must be used.

Relationship to recursive Bayesian estimation

The true state is assumed to be an unobserved Markov process, and the measurements are the observed states of a hidden Markov model.
Hidden Markov model
Because of the Markov assumption, the true state is conditionally independent of all earlier states given the immediately previous state.
p(\textbf{x}_k|\textbf{x}_0,\dots,\textbf{x}_{k-1}) = p(\textbf{x}_k|\textbf{x}_{k-1})
Similarly the measurement at the k-th timestep is dependent only upon the current state and is conditionally independent of all other states given the current state.
p(\textbf{z}_k|\textbf{x}_0,\dots,\textbf{x}_{k}) = p(\textbf{z}_k|\textbf{x}_{k} )
Using these assumptions the probability distribution over all states of the hidden Markov model can be written simply as:
p(\textbf{x}_0,\dots,\textbf{x}_k,\textbf{z}_1,\dots,\textbf{z}_k) = p(\textbf{x}_0)\prod_{i=1}^k p(\textbf{z}_i|\textbf{x}_i)p(\textbf{x}_i|\textbf{x}_{i-1})
However, when the Kalman filter is used to estimate the state x, the probability distribution of interest is that associated with the current states conditioned on the measurements up to the current timestep. (This is achieved by marginalizing out the previous states and dividing by the probability of the measurement set.)
This leads to the predict and update steps of the Kalman filter written probabilistically. The probability distribution associated with the predicted state is the sum (integral) of the products of the probability distribution associated with the transition from the (k - 1)-th timestep to the k-th and the probability distribution associated with the previous state, over all possible x_{k_-1}.
p(\textbf{x}_k|\textbf{Z}_{k-1}) = \int p(\textbf{x}_k | \textbf{x}_{k-1}) p(\textbf{x}_{k-1} | \textbf{Z}_{k-1} ) \, d\textbf{x}_{k-1}
The measurement set up to time t is
\textbf{Z}_{t} = \left \{ \textbf{z}_{1},\dots,\textbf{z}_{t} \right \}
The probability distribution of the update is proportional to the product of the measurement likelihood and the predicted state.
p(\textbf{x}_k|\textbf{Z}_{k}) = \frac{p(\textbf{z}_k|\textbf{x}_k) p(\textbf{x}_k|\textbf{Z}_{k-1})}{p(\textbf{z}_k|\textbf{Z}_{k-1})}
The denominator
p(\textbf{z}_k|\textbf{Z}_{k-1}) = \int p(\textbf{z}_k|\textbf{x}_k) p(\textbf{x}_k|\textbf{Z}_{k-1}) d\textbf{x}_k
is a normalization term.
The remaining probability density functions are
p(\textbf{x}_k | \textbf{x}_{k-1}) = N(\textbf{F}_k\textbf{x}_{k-1}, \textbf{Q}_k)
p(\textbf{z}_k|\textbf{x}_k) = N(\textbf{H}_{k}\textbf{x}_k, \textbf{R}_k)
p(\textbf{x}_{k-1}|\textbf{Z}_{k-1}) = N(\hat{\textbf{x}}_{k-1},\textbf{P}_{k-1} )
Note that the PDF at the previous timestep is inductively assumed to be the estimated state and covariance. This is justified because, as an optimal estimator, the Kalman filter makes best use of the measurements, therefore the PDF for \mathbf{x}_k given the measurements \mathbf{Z}_k is the Kalman filter estimate.

Information filter

In the information filter, or inverse covariance filter, the estimated covariance and estimated state are replaced by the information matrix and information vector respectively. These are defined as:
\textbf{Y}_{k|k} = \textbf{P}_{k|k}^{-1}
\hat{\textbf{y}}_{k|k} = \textbf{P}_{k|k}^{-1}\hat{\textbf{x}}_{k|k}
Similarly the predicted covariance and state have equivalent information forms, defined as:
\textbf{Y}_{k|k-1} = \textbf{P}_{k|k-1}^{-1}
\hat{\textbf{y}}_{k|k-1} = \textbf{P}_{k|k-1}^{-1}\hat{\textbf{x}}_{k|k-1}
as have the measurement covariance and measurement vector, which are defined as:
\textbf{I}_{k} = \textbf{H}_{k}^{\text{T}} \textbf{R}_{k}^{-1} \textbf{H}_{k}
\textbf{i}_{k} = \textbf{H}_{k}^{\text{T}} \textbf{R}_{k}^{-1} \textbf{z}_{k}
The information update now becomes a trivial sum.
\textbf{Y}_{k|k} = \textbf{Y}_{k|k-1} + \textbf{I}_{k}
\hat{\textbf{y}}_{k|k} = \hat{\textbf{y}}_{k|k-1} + \textbf{i}_{k}
The main advantage of the information filter is that N measurements can be filtered at each timestep simply by summing their information matrices and vectors.
\textbf{Y}_{k|k} = \textbf{Y}_{k|k-1} + \sum_{j=1}^N \textbf{I}_{k,j}
\hat{\textbf{y}}_{k|k} = \hat{\textbf{y}}_{k|k-1} + \sum_{j=1}^N \textbf{i}_{k,j}
To predict the information filter the information matrix and vector can be converted back to their state space equivalents, or alternatively the information space prediction can be used.
\textbf{M}_{k} = [\textbf{F}_{k}^{-1}]^{\text{T}} \textbf{Y}_{k-1|k-1} \textbf{F}_{k}^{-1}
\textbf{C}_{k} = \textbf{M}_{k} [\textbf{M}_{k}+\textbf{Q}_{k}^{-1}]^{-1}
\textbf{L}_{k} = I - \textbf{C}_{k}
\textbf{Y}_{k|k-1} = \textbf{L}_{k} \textbf{M}_{k} \textbf{L}_{k}^{\text{T}} + \textbf{C}_{k} \textbf{Q}_{k}^{-1} \textbf{C}_{k}^{\text{T}}
\hat{\textbf{y}}_{k|k-1} = \textbf{L}_{k} [\textbf{F}_{k}^{-1}]^{\text{T}}\hat{\textbf{y}}_{k-1|k-1}
Note that if F and Q are time invariant these values can be cached. Note also that F and Q need to be invertible.

Fixed-lag smoother

The optimal fixed-lag smoother provides the optimal estimate of \hat{\textbf{x}}_{k - N | k} for a given fixed-lag N using the measurements from \textbf{z}_{1} to \textbf{z}_{k}. It can be derived using the previous theory via an augmented state, and the main equation of the filter is the following:
\begin{bmatrix} \hat{\textbf{x}}_{t|t} \\ \hat{\textbf{x}}_{t-1|t} \\ \vdots \\ \hat{\textbf{x}}_{t-N+1|t} \\ \end{bmatrix} = \begin{bmatrix} I \\ 0 \\ \vdots \\ 0 \\ \end{bmatrix} \hat{\textbf{x}}_{t|t-1} + \begin{bmatrix} 0 & \ldots & 0 \\ I & 0 & \vdots \\ \vdots & \ddots & \vdots \\ 0 & \ldots & I \\ \end{bmatrix} \begin{bmatrix} \hat{\textbf{x}}_{t-1|t-1} \\ \hat{\textbf{x}}_{t-2|t-1} \\ \vdots \\ \hat{\textbf{x}}_{t-N|t-1} \\ \end{bmatrix} + \begin{bmatrix} K^{(1)} \\ K^{(2)} \\ \vdots \\ K^{(N)} \\ \end{bmatrix} y_{t|t-1}
where:
1) \hat{\textbf{x}}_{t|t-1} is estimated via a standard Kalman filter;
2) y_{t|t-1} = z(t) - \hat{\textbf{x}}_{t|t-1} is the innovation produced considering the estimate of the standard Kalman filter;
3) the various \hat{\textbf{x}}_{t-i|t} with i = 0,\ldots,N are new variables, i.e. they do not appear in the standard Kalman filter;
4) the gains are computed via the following scheme:
K^{(i)} = P^{(i)} H^{T} \left[ H P H^{T} + R \right]^{-1}
and
P^{(i)} = P \left[ \left[ F - K H \right]^{T} \right]^{i}
where P and K are the prediction error covariance and the gains of the standard Kalman filter.
Note that if we define the estimation error covariance
P_{i} := E \left[ \left( \textbf{x}_{t-i} - \hat{\textbf{x}}_{t-i|t} \right)^{*} \left( \textbf{x}_{t-i} - \hat{\textbf{x}}_{t-i|t} \right) | z_{1} \ldots z_{t} \right]
then we have that the improvement on the estimation of \textbf{x}_{t-i} is given by:
P - P_{i} = \sum_{j = 0}^{i} \left[ P^{(j)} H^{T} \left[ H P H^{T} + R \right]^{-1} H \left( P^{(i)} \right)^{T} \right]

Fixed-interval filters

The optimal fixed-interval smoother provides the optimal estimate of \hat{\textbf{x}}_{k | n} (k \leq n) using the measurements from a fixed interval \textbf{z}_{1} to \textbf{z}_{n}. This is also called Kalman Smoothing.
There exists an efficient two-pass algorithm, Rauch-Tung-Striebel Algorithm, for achieving this. The main equations of the smoother is the following (assuming \textbf{B}_{k} = \textbf{0}):
  • forward pass: regular Kalman filter algorithm
  • backward pass: \hat{\textbf{x}}_{k|n} = \tilde{F}_k \hat{\textbf{x}}_{k+1|n} + \tilde{K}_k \hat{\textbf{x}}_{k+1|k} , where
    • \tilde{\textbf{F}}_k = \textbf{F}_k^{-1} (\textbf{I} - \textbf{Q}_k \textbf{P}_{k+1|k}^{-1})
    • \tilde{\textbf{K}}_k = \textbf{F}_k^{-1} \textbf{Q}_k \textbf{P}_{k+1|k}^{-1}

Non-linear filters

The basic Kalman filter is limited to a linear assumption. However, most non-trivial systems are non-linear. The non-linearity can be associated either with the process model or with the observation model or with both.

Extended Kalman filter

Main article: Extended Kalman filter
In the extended Kalman filter, (EKF) the state transition and observation models need not be linear functions of the state but may instead be (differentiable) functions.
\textbf{x}_{k} = f(\textbf{x}_{k-1}, \textbf{u}_{k}) + \textbf{w}_{k}
\textbf{z}_{k} = h(\textbf{x}_{k}) + \textbf{v}_{k}
The function f can be used to compute the predicted state from the previous estimate and similarly the function h can be used to compute the predicted measurement from the predicted state. However, f and h cannot be applied to the covariance directly. Instead a matrix of partial derivatives (the Jacobian) is computed.
At each timestep the Jacobian is evaluated with current predicted states. These matrices can be used in the Kalman filter equations. This process essentially linearizes the non-linear function around the current estimate.

Unscented Kalman filter

When the state transition and observation models – that is, the predict and update functions f and h (see above) – are highly non-linear, the extended Kalman filter can give particularly poor performance.[2] This is because the mean and covariance are propagated through linearization of the underlying non-linear model. The unscented Kalman filter (UKF) [2] uses a deterministic sampling technique known as the unscented transform to pick a minimal set of sample points (called sigma points) around the mean. These sigma points are then propagated through the non-linear functions, from which the mean and covariance of the estimate are then recovered. The result is a filter which more accurately captures the true mean and covariance. (This can be verified using Monte Carlo sampling or through a Taylor series expansion of the posterior statistics.) In addition, this technique removes the requirement to explicitly calculate Jacobians, which for complex functions can be a difficult task in itself (i.e., requiring complicated derivatives if done analytically or being computationally costly if done numerically).
Predict
As with the EKF, the UKF prediction can be used independently from the UKF update, in combination with a linear (or indeed EKF) update, or vice versa.
The estimated state and covariance are augmented with the mean and covariance of the process noise.
\textbf{x}_{k-1|k-1}^{a} = [ \hat{\textbf{x}}_{k-1|k-1}^{T} \quad E[\textbf{w}_{k}^{T}] \ ]^{T}
\textbf{P}_{k-1|k-1}^{a} = \begin{bmatrix} & \textbf{P}_{k-1|k-1} & & 0 & \\ & 0 & &\textbf{Q}_{k} & \end{bmatrix}
A set of 2L+1 sigma points is derived from the augmented state and covariance where L is the dimension of the augmented state.
\chi_{k-1|k-1}^{0} = \textbf{x}_{k-1|k-1}^{a}
\chi_{k-1|k-1}^{i} =\textbf{x}_{k-1|k-1}^{a} + \left ( \sqrt{ (L + \lambda) \textbf{P}_{k-1|k-1}^{a} } \right )_{i} i = 1..L \,\!
\chi_{k-1|k-1}^{i} = \textbf{x}_{k-1|k-1}^{a} - \left ( \sqrt{ (L + \lambda) \textbf{P}_{k-1|k-1}^{a} } \right )_{i-L} i = L+1,\dots{}2L \,\!
where
\left ( \sqrt{ (L + \lambda) \textbf{P}_{k-1|k-1}^{a} } \right )_{i}
is the ith column of the matrix square root of
(L + \lambda) \textbf{P}_{k-1|k-1}^{a}
using the definition: square root A of matrix B satisfies
B \equiv A A^T.
The matrix square root should be calculated using numerically efficient and stable methods such as the Cholesky decomposition.
The sigma points are propagated through the transition function f.
\chi_{k|k-1}^{i} = f(\chi_{k-1|k-1}^{i}) \quad i = 0..2L
The weighted sigma points are recombined to produce the predicted state and covariance.
\hat{\textbf{x}}_{k|k-1} = \sum_{i=0}^{2L} W_{s}^{i} \chi_{k|k-1}^{i}
\textbf{P}_{k|k-1} = \sum_{i=0}^{2L} W_{c}^{i}\ [\chi_{k|k-1}^{i} - \hat{\textbf{x}}_{k|k-1}] [\chi_{k|k-1}^{i} - \hat{\textbf{x}}_{k|k-1}]^{T}
where the weights for the state and covariance are given by:
W_{s}^{0} = \frac{\lambda}{L+\lambda}
W_{c}^{0} = \frac{\lambda}{L+\lambda} + (1 - \alpha^2 + \beta)
W_{s}^{i} = W_{c}^{i} = \frac{1}{2(L+\lambda)}
\lambda = \alpha^2 (L+\kappa) - L \,\!
Typical values for α, β, and κ are 10 − 3, 2 and 0 respectively. (These values should suffice for most purposes.)[citation needed]
Update
The predicted state and covariance are augmented as before, except now with the mean and covariance of the measurement noise.
\textbf{x}_{k|k-1}^{a} = [ \hat{\textbf{x}}_{k|k-1}^{T} \quad E[\textbf{v}_{k}^{T}] \ ]^{T}
\textbf{P}_{k|k-1}^{a} = \begin{bmatrix} & \textbf{P}_{k|k-1} & & 0 & \\ & 0 & &\textbf{R}_{k} & \end{bmatrix}
As before, a set of 2L + 1 sigma points is derived from the augmented state and covariance where L is the dimension of the augmented state.
\chi_{k|k-1}^{0} = \textbf{x}_{k|k-1}^{a}
\chi_{k|k-1}^{i} =\textbf{x}_{k|k-1}^{a} + \left ( \sqrt{ (L + \lambda) \textbf{P}_{k|k-1}^{a} } \right )_{i} i = 1..L \,\!
\chi_{k|k-1}^{i} = \textbf{x}_{k|k-1}^{a} - \left ( \sqrt{ (L + \lambda) \textbf{P}_{k|k-1}^{a} } \right )_{i-L} i = L+1,\dots{}2L \,\!
Alternatively if the UKF prediction has been used the sigma points themselves can be augmented along the following lines
\chi_{k|k-1} := [ \chi_{k|k-1}^T \quad E[\textbf{v}_{k}^{T}] \ ]^{T} \pm \sqrt{ (L + \lambda) \textbf{R}_{k}^{a} }
where
\textbf{R}_{k}^{a} = \begin{bmatrix} & 0 & & 0 & \\ & 0 & &\textbf{R}_{k} & \end{bmatrix}
The sigma points are projected through the observation function h.
\gamma_{k}^{i} = h(\chi_{k|k-1}^{i}) \quad i = 0..2L
The weighted sigma points are recombined to produce the predicted measurement and predicted measurement covariance.
\hat{\textbf{z}}_{k} = \sum_{i=0}^{2L} W_{s}^{i} \gamma_{k}^{i}
\textbf{P}_{z_{k}z_{k}} = \sum_{i=0}^{2L} W_{c}^{i}\ [\gamma_{k}^{i} - \hat{\textbf{z}}_{k}] [\gamma_{k}^{i} - \hat{\textbf{z}}_{k}]^{T}
The state-measurement cross-covariance matrix,
\textbf{P}_{x_{k}z_{k}} = \sum_{i=0}^{2L} W_{c}^{i}\ [\chi_{k|k-1}^{i} - \hat{\textbf{x}}_{k|k-1}] [\gamma_{k}^{i} - \hat{\textbf{z}}_{k}]^{T}
is used to compute the UKF Kalman gain.
K_{k} = \textbf{P}_{x_{k}z_{k}} \textbf{P}_{z_{k}z_{k}}^{-1}
As with the Kalman filter, the updated state is the predicted state plus the innovation weighted by the Kalman gain,
\hat{\textbf{x}}_{k|k} = \hat{\textbf{x}}_{k|k-1} + K_{k}( \textbf{z}_{k} - \hat{\textbf{z}}_{k} )
And the updated covariance is the predicted covariance, minus the predicted measurement covariance, weighted by the Kalman gain.
\textbf{P}_{k|k} = \textbf{P}_{k|k-1} - K_{k} \textbf{P}_{z_{k}z_{k}} K_{k}^{T}

Kalman–Bucy filter

The Kalman–Bucy filter is a continuous time version of the Kalman filter.[3][4]
It is based on the state space model
\frac{d}{dt}\mathbf{x}(t) = \mathbf{F}(t)\mathbf{x}(t) + \mathbf{w}(t)
\mathbf{z}(t) = \mathbf{H}(t) \mathbf{x}(t) + \mathbf{v}(t)
where the covariances of the noise terms \mathbf{w}(t) and \mathbf{v}(t) are given by \mathbf{Q}(t) and \mathbf{R}(t), respectively.
The filter consists of two differential equations, one for the state estimate and one for the covariance:
\frac{d}{dt}\hat{\mathbf{x}}(t) = \mathbf{F}(t)\hat{\mathbf{x}}(t) + \mathbf{K}(t) (\mathbf{z}(t)-\mathbf{H}(t)\hat{\mathbf{x}}(t))
\frac{d}{dt}\mathbf{P}(t) = \mathbf{F}(t)\mathbf{P}(t) + \mathbf{P}(t)\mathbf{F}^{T}(t) + \mathbf{Q}(t) - \mathbf{K}(t)\mathbf{R}(t)\mathbf{K}^{T}(t)
where the Kalman gain is given by
\mathbf{K}(t)=\mathbf{P}(t)\mathbf{H}^{T}(t)\mathbf{R}^{-1}(t)
Note that in this expression for \mathbf{K}(t) the covariance of the observation noise \mathbf{R}(t) represents at the same time the covariance of the prediction error (or innovation) \tilde{\mathbf{y}}(t)=\mathbf{z}(t)-\mathbf{H}(t)\hat{\mathbf{x}}(t); these covariances are equal only in the case of continuous time.[5]
The distinction between the prediction and update steps of discrete-time Kalman filtering does not exist in continuous time.
The second differential equation, for the covariance, is an example of a Riccati equation.

Naming and historical development

The filter is named after Rudolf E. Kalman, though Thorvald Nicolai Thiele[6][7] and Peter Swerling developed a similar algorithm earlier. Stanley F. Schmidt is generally credited with developing the first implementation of a Kalman filter. It was during a visit of Kalman to the NASA Ames Research Center that he saw the applicability of his ideas to the problem of trajectory estimation for the Apollo program, leading to its incorporation in the Apollo navigation computer. This Kalman filter was first described and partially developed in technical papers by Swerling (1958), Kalman (1960) and Kalman and Bucy (1961).
Kalman filters have been vital in the implemention of the navigation systems of U.S. Navy nuclear ballistic missile submarines; and in the guidance and navigation sustems of cruise missiles such as the U.S. Navy's Tomahawk missile; the U.S. Air Force's Air Launched Cruise Missile; It is also used in the guidance and navigation systems of the NASA Space Shuttle and the attitude control and navigation systems of the International Space Station.
This digital filter is sometimes called the Stratonovich–Kalman–Bucy filter because it is a special case of a more general, non-linear filter developed somewhat earlier by the Soviet mathematician Ruslan L. Stratonovich.[8][9] In fact, some of the equations of the special case linear filter appeared in these papers by Stratonovich that were published before summer 1960, when Kalman met with Stratonovich during a conference in Moscow.
In control theory, the Kalman filter is most commonly referred to as linear quadratic estimation (LQE).
A wide variety of Kalman filters have now been developed, from Kalman's original formulation, now called the simple Kalman filter, the Kalman-Bucy filter, Schmidt's extended filter, the information filter, and a variety of square-root filters that were developed by Bierman, Thornton and many others. Perhaps the most commonly used type of very simple Kalman filter is the phase-locked loop, which is now ubiquitous in radios, especially frequency modulation (FM) radios, television sets, satellite communications receivers, outer space communications systems, and nearly any other electronic communications equipment.

Applications

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SCADA

SCADA stands for supervisory control and data acquisition. It generally refers to an industrial control system: a computer system monitoring and controlling a process. The process can be industrial, infrastructure or facility based as described below:
  • Industrial processes include those of manufacturing, production, power generation, fabrication, and refining, and may run in continuous, batch, repetitive, or discrete modes.
  • Infrastructure processes may be public or private, and include water treatment and distribution, wastewater collection and treatment, oil and gas pipelines, electrical power transmission and distribution, civil defense siren systems, and large communication systems.
  • Facility processes occur both in public facilities and private ones, including buildings, airports, ships, and space stations. They monitor and control HVAC, access, and energy consumption
  •  

Common system components

A SCADA System usually consists of the following subsystems:

Supervision vs. control

There is, in several industries, considerable confusion over the differences between SCADA systems and Distributed control systems (DCS). Generally speaking, a SCADA system usually refers to a system that coordinates, but does not control processes in real time. The discussion on real-time control is muddied somewhat by newer telecommunications technology, enabling reliable, low latency, high speed communications over wide areas. Most differences between SCADA and DCS are culturally determined and can usually be ignored. As communication infrastructures with higher capacity become available, the difference between SCADA and DCS will fade.

Systems concepts

The term SCADA usually refers to centralized systems which monitor and control entire sites, or complexes of systems spread out over large areas (anything between an industrial plant and a country). Most control actions are performed automatically by Remote Terminal Units ("RTUs") or by Programmable Logic Controllers ("PLCs"). Host control functions are usually restricted to basic overriding or supervisory level intervention. For example, a PLC may control the flow of cooling water through part of an industrial process, but the SCADA system may allow operators to change the set points for the flow,and enable alarm conditions, such as loss of flow and high temperature, to be displayed and recorded. The feedback control loop passes through the RTU or PLC, while the SCADA system monitors the overall performance of the loop.
SCADA schematic overview-s.svg
Data acquisition begins at the RTU or PLC level and includes meter readings and equipment status reports that are communicated to SCADA as required. Data is then compiled and formatted in such a way that a control room operator using the HMI can make supervisory decisions to adjust or override normal RTU (PLC) controls. Data may also be fed to a Historian, often built on a commodity Database Management System, to allow trending and other analytical auditing.
SCADA systems typically implement a distributed database, commonly referred to as a tag database, which contains data elements called tags or points. A point represents a single input or output value monitored or controlled by the system. Points can be either "hard" or "soft". A hard point represents an actual input or output within the system, while a soft point results from logic and math operations applied to other points. (Most implementations conceptually remove the distinction by making every property a "soft" point expression, which may, in the simplest case, equal a single hard point.) Points are normally stored as value-timestamp pairs: a value, and the timestamp when it was recorded or calculated. A series of value-timestamp pairs gives the history of that point. It's also common to store additional metadata with tags, such as the path to a field device or PLC register, design time comments, and alarm information.

Human Machine Interface

Typical Basic SCADA Animations [1]
A Human-Machine Interface or HMI is the apparatus which presents process data to a human operator, and through which the human operator controls the process.
An HMI is usually linked to the SCADA system's databases and software programs, to provide trending, diagnostic data, and management information such as scheduled maintenance procedures, logistic information, detailed schematics for a particular sensor or machine, and expert-system troubleshooting guides.
The HMI system usually presents the information to the operating personnel graphically, in the form of a mimic diagram. This means that the operator can see a schematic representation of the plant being controlled. For example, a picture of a pump connected to a pipe can show the operator that the pump is running and how much fluid it is pumping through the pipe at the moment. The operator can then switch the pump off. The HMI software will show the flow rate of the fluid in the pipe decrease in real time. Mimic diagrams may consist of line graphics and schematic symbols to represent process elements, or may consist of digital photographs of the process equipment overlain with animated symbols.
The HMI package for the SCADA system typically includes a drawing program that the operators or system maintenance personnel use to change the way these points are represented in the interface. These representations can be as simple as an on-screen traffic light, which represents the state of an actual traffic light in the field, or as complex as a multi-projector display representing the position of all of the elevators in a skyscraper or all of the trains on a railway.
An important part of most SCADA implementations are alarms. An alarm is a digital status point that has either the value NORMAL or ALARM. Alarms can be created in such a way that when their requirements are met, they are activated. An example of an alarm is the "fuel tank empty" light in a car. The SCADA operator's attention is drawn to the part of the system requiring attention by the alarm. Emails and text messages are often sent along with an alarm activation alerting managers along with the SCADA operator.

Hardware solutions

SCADA solutions often have Distributed Control System (DCS) components. Use of "smart" RTUs or PLCs, which are capable of autonomously executing simple logic processes without involving the master computer, is increasing. A functional block programming language, IEC 61131-3 (Ladder Logic), is frequently used to create programs which run on these RTUs and PLCs. Unlike a procedural language such as the C programming language or FORTRAN, IEC 61131-3 has minimal training requirements by virtue of resembling historic physical control arrays. This allows SCADA system engineers to perform both the design and implementation of a program to be executed on an RTU or PLC. A Programmable automation controller (PAC) is a compact controller that combines the features and capabilities of a PC-based control system with that of a typical PLC. PACs are deployed in SCADA systems to provide RTU and PLC functions. In many electrical substation SCADA applications, "distributed RTUs" use information processors or station computers to communicate with protective relays, PACS, and other devices for I/O, and communicate with the SCADA master in lieu of a traditional RTU.
Since about 1998, virtually all major PLC manufacturers have offered integrated HMI/SCADA systems, many of them using open and non-proprietary communications protocols. Numerous specialized third-party HMI/SCADA packages, offering built-in compatibility with most major PLCs, have also entered the market, allowing mechanical engineers, electrical engineers and technicians to configure HMIs themselves, without the need for a custom-made program written by a software developer.

Remote Terminal Unit (RTU)

The RTU connects to physical equipment. Typically, an RTU converts the electrical signals from the equipment to digital values such as the open/closed status from a switch or a valve, or measurements such as pressure, flow, voltage or current. By converting and sending these electrical signals out to equipment the RTU can control equipment, such as opening or closing a switch or a valve, or setting the speed of a pump.

Supervisory Station

The term "Supervisory Station" refers to the servers and software responsible for communicating with the field equipment (RTUs, PLCs, etc), and then to the HMI software running on workstations in the control room, or elsewhere. In smaller SCADA systems, the master station may be composed of a single PC. In larger SCADA systems, the master station may include multiple servers, distributed software applications, and disaster recovery sites. To increase the integrity of the system the multiple servers will often be configured in a dual-redundant or hot-standby formation providing continuous control and monitoring in the event of a server failure.
Initially, more "open" platforms such as Linux were not as widely used due to the highly dynamic development environment and because a SCADA customer that was able to afford the field hardware and devices to be controlled could usually also purchase UNIX or OpenVMS licenses. Today, all major operating systems are used for both master station servers and HMI workstations.

Operational philosophy

For some installations, the costs that would result from the control system failing are extremely high. Possibly even lives could be lost. Hardware for some SCADA systems is ruggedized to withstand temperature, vibration, and voltage extremes, but in most critical installations reliability is enhanced by having redundant hardware and communications channels, up to the point of having multiple fully equipped control centres. A failing part can be quickly identified and its functionality automatically taken over by backup hardware. A failed part can often be replaced without interrupting the process. The reliability of such systems can be calculated statistically and is stated as the mean time to failure, which is a variant of mean time between failures. The calculated mean time to failure of such high reliability systems can be on the order of centuries.

Communication infrastructure and methods

SCADA systems have traditionally used combinations of radio and direct serial or modem connections to meet communication requirements, although Ethernet and IP over SONET / SDH is also frequently used at large sites such as railways and power stations. The remote management or monitoring function of a SCADA system is often referred to as telemetry.
This has also come under threat with some customers wanting SCADA data to travel over their pre-established corporate networks or to share the network with other applications. The legacy of the early low-bandwidth protocols remains, though. SCADA protocols are designed to be very compact and many are designed to send information to the master station only when the master station polls the RTU. Typical legacy SCADA protocols include Modbus RTU, RP-570, Profibus and Conitel. These communication protocols are all SCADA-vendor specific but are widely adopted and used. Standard protocols are IEC 60870-5-101 or 104, IEC 61850 and DNP3. These communication protocols are standardized and recognized by all major SCADA vendors. Many of these protocols now contain extensions to operate over TCP/IP. It is good security engineering practice to avoid connecting SCADA systems to the Internet so the attack surface is reduced.
RTUs and other automatic controller devices were being developed before the advent of industry wide standards for interoperability. The result is that developers and their management created a multitude of control protocols. Among the larger vendors, there was also the incentive to create their own protocol to "lock in" their customer base. A list of automation protocols is being compiled here.
Recently, OLE for Process Control (OPC) has become a widely accepted solution for intercommunicating different hardware and software, allowing communication even between devices originally not intended to be part of an industrial network.

SCADA Architectures

SCADA systems have evolved through 3 generations as follows:

First Generation: "Monolithic"

In the first generation computing was done by Mainframe systems. Networks didn’t exist at the time SCADA was developed. Thus SCADA systems were independent systems with no connectivity to other systems. Wide Area Networks were later designed by RTU vendors to communicate with the RTU. The communication protocols used were often proprietary at that time. The first generation SCADA System was redundant since a back-up mainframe system was connected at the bus level and was used in the event of failure of the main mainframe system.

Second Generation: "Distributed"

The processing was distributed across multiple stations which were connected through LAN and they shared information in real time. Each station was responsible for a particular task thus making the size and cost of each station less than the one used in First Generation. The network protocols used were still mostly proprietary, which led to significant security problems for any SCADA system that received attention from a hacker. Since the protocols were proprietary, very few people beyond the developers and hackers knew enough to determine how secure a SCADA installation was. Since both parties had vested interests in keeping security issues quiet, the security of a SCADA installation was often badly overestimated, if it was considered at all.

Third Generation: "Networked"

These are the current generation SCADA systems which use open system architecture rather than a vendor controlled proprietary environment. The SCADA system utilizes open standard and protocols thus distributing functionality across a WAN rather than a LAN. It is easier to connect third party peripheral devices like printers, disk drives, tape drives due to the use of open architecture. WAN protocols such as Internet Protocol (IP) are used for communication between the master station and communications equipment. Due to the usage of standard protocols and the fact that many networked SCADA systems are accessible from the internet, the systems are potentially vulnerable to remote cyber attacks. On the other hand, the usage of standard protocols and security techniques means that standard security improvements are applicable to the SCADA systems, assuming they receive timely maintenance and updates.

Trends in SCADA

There is a trend for PLC and HMI/SCADA software to be more "mix-and-match". In the mid 1990s, the typical DAQ I/O manufacturer supplied equipment that communicated using proprietary protocols over a suitable-distance carrier like RS-485. End users who invested in a particular vendor's hardware solution often found themselves restricted to a limited choice of equipment when requirements changed (e.g. system expansions or performance improvement). To mitigate such problems, open communication protocols such as IEC870-5-101/104, DNP3 serial, and DNP3 LAN/WAN became increasingly popular among SCADA equipment manufacturers and solution providers alike. Open architecture SCADA systems enabled users to mix-and-match products from different vendors to develop solutions that were better than those that could be achieved when restricted to a single vendor's product offering.
Towards the late 1990s, the shift towards open communications continued with individual I/O manufacturers as well, who adopted open message structures such as Modbus RTU and Modbus ASCII (originally both developed by Modicon) over RS-485. By 2000, most I/O makers offered completely open interfacing such as Modbus TCP over Ethernet and IP.
The North American Electric Reliability Corporation (NERC) has specified that electrical system data should be time-tagged to the nearest millisecond. Electrical system SCADA systems provide this Sequence of events recorder function, using Radio clocks to synchronize the RTU or distributed RTU clocks.
SCADA systems are coming in line with standard networking technologies. Ethernet and TCP/IP based protocols are replacing the older proprietary standards. Although certain characteristics of frame-based network communication technology (determinism, synchronization, protocol selection, environment suitability) have restricted the adoption of Ethernet in a few specialized applications, the vast majority of markets have accepted Ethernet networks for HMI/SCADA.
With the emergence of software as a service in the broader software industry, a few vendors have begun offering application specific SCADA systems hosted on remote platforms over the Internet. This removes the need to install and commission systems at the end-user's facility and takes advantage of security features already available in Internet technology, VPNs and SSL. Some concerns include security,[2] Internet connection reliability, and latency.
SCADA systems are becoming increasingly ubiquitous. Thin clients, web portals, and web based products are gaining popularity with most major vendors. The increased convenience of end users viewing their processes remotely introduces security considerations. While these considerations are already considered solved in other sectors of internet services, not all entities responsible for deploying SCADA systems have understood the changes in accessibility and threat scope implicit in connecting a system to the internet.

Security issues

The move from proprietary technologies to more standardized and open solutions together with the increased number of connections between SCADA systems and office networks and the Internet has made them more vulnerable to attacks - see references. Consequently, the security of SCADA-based systems has come into question as they are increasingly seen as extremely vulnerable to cyberwarfare/cyberterrorism attacks.[3][4]
In particular, security researchers are concerned about:
  • the lack of concern about security and authentication in the design, deployment and operation of existing SCADA networks
  • the mistaken belief that SCADA systems have the benefit of security through obscurity through the use of specialized protocols and proprietary interfaces
  • the mistaken belief that SCADA networks are secure because they are purportedly physically secured
  • the mistaken belief that SCADA networks are secure because they are supposedly disconnected from the Internet
SCADA systems are used to control and monitor physical processes, examples of which are transmission of electricity, transportation of gas and oil in pipelines, water distribution, traffic lights, and other systems used as the basis of modern society. The security of these SCADA systems is important because compromise or destruction of these systems would impact multiple areas of society far removed from the original compromise. For example, a blackout caused by a compromised electrical SCADA system would cause financial losses to all the customers that received electricity from that source. How security will affect legacy SCADA and new deployments remains to be seen.
There are two distinct threats to a modern SCADA system. First is the threat of unauthorized access to the control software, whether it be human access or changes induced intentionally or accidentally by virus infections and other software threats residing on the control host machine. Second is the threat of packet access to the network segments hosting SCADA devices. In many cases, there is rudimentary or no security on the actual packet control protocol, so anyone who can send packets to the SCADA device can control it. In many cases SCADA users assume that a VPN is sufficient protection and are unaware that physical access to SCADA-related network jacks and switches provides the ability to totally bypass all security on the control software and fully control those SCADA networks. These kinds of physical access attacks bypass firewall and VPN security and are best addressed by endpoint-to-endpoint authentication and authorization such as are commonly provided in the non-SCADA world by in-device SSL or other cryptographic techniques.
Many vendors of SCADA and control products have begun to address these risks in a basic sense by developing lines of specialized industrial firewall and VPN solutions for TCP/IP-based SCADA networks. Additionally, application whitelisting solutions are being implemented because of their ability to prevent malware and unauthorized application changes without the performance impacts of traditional antivirus scans[citation needed]. Also, the ISA Security Compliance Institute (ISCI) is emerging to formalize SCADA security testing starting as soon as 2009. ISCI is conceptually similar to private testing and certification that has been performed by vendors since 2007. Eventually, standards being defined by ISA99 WG4 will supersede the initial industry consortia efforts, but probably not before 2011 .
The increased interest in SCADA vulnerabilities has resulted in vulnerability researchers discovering vulnerabilities in commercial SCADA software and more general offensive SCADA techniques presented to the general security community. In electric and gas utility SCADA systems, the vulnerability of the large installed base of wired and wireless serial communications links is addressed in some cases by applying bump-in-the-wire devices that employ authentication and Advanced Encryption Standard encryption rather than replacing all existing nodes.
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Flexible Alternating Current Transmission System

A Flexible Alternating Current Transmission System (FACTS) is a system composed of static equipment used for the AC transmission of electrical energy. It is meant to enhance controllability and increase power transfer capability of the network. It is generally a power electronics-based device.
FACTS is defined by the IEEE as "a power electronic based system and other static equipment that provide control of one or more AC transmission system parameters to enhance controllability and increase power transfer capability."

FACTS technology

In series compensation, the FACTS is connected in series with the power system. It works as a controllable voltage source.
Series inductance occurs in long transmission lines, and when a large current flow causes a large voltage drop. To compensate, series capacitors are connected.

Shunt compensation

In shunt compensation, power system is connected in shunt with the FACTS. It works as a controllable current source.
Shunt compensation is of two types:
Shunt capacitive compensation
This method is used improve the power factor. Whenever an inductive load is connected to the transmission line, power factor lags because of lagging load current. To compensate, a shunt capacitor is connected which draws current leading the source voltage. The net result is improvement in power factor.
Shunt inductive compensation
This method is used either when charging the transmission line, or, when there is very low load at the receiving end. Due to very low, or no load – very low current flows through the transmission line. Shunt capacitance in the transmission line causes voltage amplification (Ferranti Effect). The receiving end voltage may become double the sending end voltage (generally in case of very long transmission lines). To compensate, shunt inductors are connected across the transmission line.
Transmission on a no-loss line.
Series compensation.
Shunt compensation.

FACTS theory

In the case of a no-loss line, voltage magnitude at receiving end is the same as voltage magnitude at sending end: Vs = Vr=V. Transmission results in a phase lag δ that depends on line reactance X.
\underline{V_s}=V \cos(\frac{\delta}{2}) +jV \sin(\frac{\delta}{2})
\underline{V_r}=V \cos(\frac{\delta}{2})-jV \sin(\frac{\delta}{2})
\underline{I}=\frac{\underline{V_s}-\underline{V_r}}{jX}=\frac{2V\sin\frac{\delta}{2}}{X}
As it is a no-loss line, active power P is the same at any point of the line:
P_s=P_r=P=V \cos(\frac{\delta}{2}) . \frac{2V\sin\frac{\delta}{2}}{X}=\frac{V^2}{X}\sin \delta
Reactive power at sending end is the opposite of reactive power at receiving end: Q_s=-Q_r=Q=V \sin(\frac{\delta}{2}) . \frac{2V\sin\frac{\delta}{2}}{X}=\frac{V^2}{X}(1-\cos \delta)
As δ is very small, active power mainly depends on δ whereas reactive power mainly depends on voltage magnitude.

Series compensation

FACTS for series compensation modify line impedance: X is decreased so as to increase the transmittable active power. However, more reactive power must be provided.
P=\frac{V^2}{X-Xc}\sin \delta
Q=\frac{V^2}{X-Xc}(1-\cos \delta)

Shunt compensation

Reactive current is injected into the line to maintain voltage magnitude. Transmittable active power is increased but more reactive power is to be provided.
P=\frac{2V^2}{X}\sin (\frac{\delta}{2})
Q=\frac{2V^2}{X}(1-\cos(\frac{\delta}{2}))

Examples of FACTS for series compensation

Examples of FACTS for series compensation (schematic)
  • Static Synchronous Series Compensator (SSSC)
  • Thyristor Controlled Series Capacitor (TCSC): a series capacitor bank is shunted by a thyristor-controlled reactor
  • Thyristor Controlled Series Reactor (TCSR): a series reactor bank is shunted by a thyristor-controlled reactor
  • Thyristor Switched Series Capacitor (TSSC): a series capacitor bank is shunted by a thyristor-switched reactor
  • Thyristor Switched Series Reactor (TSSR): a series reactor bank is shunted by a thyristor-switched reactor

Examples of FACTS for shunt compensation

Examples of FACTS for shunt compensation (schematic)
  • Static Synchronous Compensator (STATCOM) previously known as a Static Condenser (STATCON)
  • Static VAR compensator (SVC). Most common SVCs are:
    • Thyristor Controlled Reactor (TCR): reactor is connected in series with a bidirectional thyristor valve. The thyristor valve is phase-controlled. Equivalent reactance is varied continuously.
    • Thyristor Switched Reactor (TSR): Same as TCR but thyristor is either in zero- or full- conduction. Equivalent reactance is varied in stepwise manner.
    • Thyristor Switched Capacitor (TSC): capacitor is connected in series with a bidirectional thyristor valve. Thyristor is either in zero- or full- conduction. Equivalent reactance is varied in stepwise manner.
    • Mechanically Switched Capacitor (MSC): capacitor is switched by circuit-breaker. It aims at compensating steady state reactive power. It is switched only a few times a day.


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