State observer
It is typically computer-implemented, and provides the basis of many practical applications.
In most practical cases, the physical state of the system cannot be determined by direct observation.
Instead, indirect effects of the internal state are observed by way of the system outputs.
Linear, delayed, sliding mode, high gain, Tau, homogeneity-based, extended and cubic observers are among several observer structures used for state estimation of linear and nonlinear systems.
The state of a linear, time-invariant discrete-time system is assumed to satisfy where, at time
The observer model of the physical system is then typically derived from the above equations.
Additional terms may be included in order to ensure that, on receiving successive measured values of the plant's inputs and outputs, the model's state converges to that of the plant.
Note that the variables of a state observer are commonly denoted by a "hat":
The Luenberger observer for this discrete-time system is therefore asymptotically stable when the matrix
The observer equations then become: or, more simply, Due to the separation principle we know that we can choose
are chosen to make the continuous-time error dynamics converge to zero asymptotically (i.e., when
is high, the linear Luenberger observer converges to the system states very quickly.
However, high observer gain leads to a peaking phenomenon in which initial estimator error can be prohibitively large (i.e., impractical or unsafe to use).
[1] As a consequence, nonlinear high-gain observer methods are available that converge quickly without the peaking phenomenon.
Sliding mode observers also have attractive noise resilience properties that are similar to a Kalman filter.
Ciccarella, Dalla Mora, and Germani[9] obtained more advanced and general results, removing the need for a nonlinear transform and proving global asymptotic convergence of the estimated state to the true state using only simple assumptions on regularity.
[13] The sliding mode observer uses non-linear high-gain feedback to drive estimated states to a hypersurface where there is no difference between the estimated output and the measured output.
The non-linear gain used in the observer is typically implemented with a scaled switching function, like the signum (i.e., sgn) of the estimated – measured output error.
Hence, due to this high-gain feedback, the vector field of the observer has a crease in it so that observer trajectories slide along a curve where the estimated output matches the measured output exactly.
Hence, some sliding mode observers have attractive properties similar to the Kalman filter but with simpler implementation.
[2][3] As suggested by Drakunov,[14] a sliding mode observer can also be designed for a class of non-linear systems.
In practice, it switches (chatters) with high frequency with slow component being equal to the equivalent value.
Applying appropriate lowpass filter to get rid of the high frequency component on can obtain the value of the equivalent control, which contains more information about the state of the estimated system.
The algorithm is simple to implement and does not contain any risky operations like differentiation.
[4] The idea of multiple models was previously applied to obtain information in adaptive control.
observers are combined into one to obtain single state vector estimation where
These factors are changed to provide the estimation in the second layer and to improve the observation process.
Some transformation yields to linear regression problem This formula gives possibility to estimate
These bounds are very important in practical applications,[19][20] as they make possible to know at each time the precision of the estimation.
is properly selected, using, for example, positive systems properties:[21] one for the upper bound
