In control theory, a state observer, state estimator, or Luenberger observer is a system that provides an estimate of the internal state of a given real system, from measurements of the input and output of the real system. It is typically computer-implemented, and provides the basis of many practical applications.

Knowing the system state is necessary to solve many control theory problems; for example, stabilizing a system using state feedback. 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. A simple example is that of vehicles in a tunnel: the rates and velocities at which vehicles enter and leave the tunnel can be observed directly, but the exact state inside the tunnel can only be estimated. If a system is observable, it is possible to fully reconstruct the system state from its output measurements using the state observer.

Typical observer model

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. A linear observer structure is described in the following sections.

Discrete-time case

The state of a linear, time-invariant discrete-time system is assumed to satisfy

is its outputs. These equations simply say that the plant's current outputs and its future state are both determined solely by its current states and the current inputs. (Although these equations are expressed in terms of discrete time steps, very similar equations hold for continuous systems). If this system is observable, can be used to steer the state of the state observer.

The observer model of the physical system; this is then added to the equations for the state of the observer to produce a so-called Luenberger observer to distinguish them from the variables of the equations satisfied by the physical system.

has all the eigenvalues inside the unit circle.

.

The observer equations then become:

or, more simply,

Due to the separation principle independently without harm to the overall stability of the systems. As a rule of thumb.

Continuous-time case

is a Hurwitz matrix).

For a continuous-time linear system

, the observer looks similar to discrete-time case described above:

1. .

satisfies the equation

1. .

is observable, i.e. observability.

Peaking and other observer methods

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). As a consequence, nonlinear high-gain observer methods are available that converge quickly without the peaking phenomenon. For example, sliding mode control can be used to design an observer that brings one estimated state's error to zero in finite time even in the presence of measurement error; the other states have error that behaves similarly to the error in a Luenberger observer after peaking has subsided. Sliding mode observers also have attractive noise resilience properties that are similar to a Kalman filter. Another approach is to apply multi observer, that significantly improves transients and reduces observer overshoot. Multi-observer can be adapted to every system where high-gain observer is applicable.

State observers for nonlinear systems

High gain, sliding mode and extended observers are the most common observers for nonlinear systems. To illustrate the application of sliding mode observers for nonlinear systems, first consider the no-input non-linear system:

given by

There are several non-approximate approaches for designing an observer. The two observers given below also apply to the case when the system has an input. That is,

Linearizable error dynamics

One suggestion by Krener and Isidori and Krener and Respondek can be applied in a situation when there exists a linearizing transformation (i.e., a diffeomorphism, like the one used in feedback linearization such that in new variables the system equations read

The Luenberger observer is then designed as

1. .

satisfies the same equation as in classical linear case.

1. .

As shown by Gauthier, Hammouri, and Othman and Hammouri and Kinnaert, such that the system can be transformed into the form

then the observer is designed as

1. ,

is a time-varying observer gain.

Ciccarella, Dalla Mora, and Germani 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.

Switched observers

As discussed for the linear case above, the peaking phenomenon present in Luenberger observers justifies the use of switched observers. A switched observer encompasses a relay or binary switch that acts upon detecting minute changes in the measured output. Some common types of switched observers include the sliding mode observer, nonlinear extended state observer, fixed time observer, switched high gain observer and uniting observer. 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. So, if the system is observable from its output, the observer states will all be driven to the actual system states. Additionally, by using the sign of the error to drive the sliding mode observer, the observer trajectories become insensitive to many forms of noise. Hence, some sliding mode observers have attractive properties similar to the Kalman filter but with simpler implementation.

As suggested by Drakunov, a sliding mode observer and has the form

where:

• vector extends the scalar signum function dimensions. That is,

  1. .

  and its repeated Lie derivatives. In particular,

  1. is the ith Lie derivative trajectories of the non-linear system). In the special case where the system has no input or has a relative degree of n derivatives. Because the inverse of the Jacobian linearization is guaranteed to be a local diffeomorphism.

  The diagonal matrix of gains is such that

  1. and suitably large to ensure reachability of the sliding mode.

  is such that

  1. here is the normal signum function denotes an "equivalent value operator" of a discontinuous function in sliding mode.

should be replaced by equivalent values (see equivalent control in the theory of sliding modess). 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 observer described above uses this method several times to obtain the state of the nonlinear system ideally in finite time.

. In particular,

and so

So:

1. sliding mode in finite time.

2. sliding mode in finite time.

3. th sliding mode in finite time.

is a diffeomorphism (i.e., that its Jacobian linearization is invertible) asserts that convergence of the estimated output implies convergence of the estimated state. That is, the requirement is an observability condition.

In the case of the sliding mode observer for the system with the input, additional conditions are needed for the observation error to be independent of the input. For example, that

does not depend on time. The observer is then

Multi-observer

Multi-observer extends the high-gain observer structure from single to multi observer, with many models working simultaneously. This has two layers: the first consists of multiple high-gain observers with different estimation states, and the second determines the importance weights of the first layer observers. The algorithm is simple to implement and does not contain any risky operations like differentiation. 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

are weight factors. These factors are changed to provide the estimation in the second layer and to improve the observation process.

Let assume that

and

.

Some transformation yields to linear regression problem

from observer error

1. .

defined as

is some time constant coefficients. The mapping error is specified as

.

Bounding observers

Bounding or interval observers constitute a class of observers that provide two estimations of the state simultaneously: one of the estimations provides an upper bound on the real value of the state, whereas the second one provides a lower bound. The real value of the state is then known to be always within these two estimations.

These bounds are very important in practical applications, as they make possible to know at each time the precision of the estimation.

is properly selected, using, for example, positive systems properties:, in the absence of noise and uncertainty

See also

• Moving horizon estimation

• Kalman filter

• Extended Kalman filter

• Positive systems

References

1. In-line references

   General references

External links

• Kalman Filter Explained Simply, Step-by-Step Tutorial of the Kalman Filter with Equations