This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.

# EstimatorGains

 EstimatorGains gives the estimator gain matrix for the StateSpaceModel object ss, such that the poles of the estimator are .
• The state-space model ss can be given as StateSpaceModel, where a, b, c, and d represent the state, input, output, and transmission matrices in either a continuous-time or a discrete-time system:
 continuous-time system discrete-time system
• If ss is observable, the eigenvalues of will be , where is the computed estimator gain matrix.
• The observer dynamics are given by:
 continuous-time system discrete-time system
• In the case of a square nonsingular matrix , the state vector can be computed as .
• The estimator gains are computed as the state feedback gains of the dual system.
Compute estimator gains for a continuous-time system:
A discrete-time system:
Compute estimator gains for a continuous-time system:
 Out[1]=

A discrete-time system:
 Out[1]=

 Scope   (4)
A set of gains for a SISO system:
Verify the solution:
An observer gain matrix for a two-output system:
Compute the estimator gains with poles specified as the roots of a polynomial:
Determine estimator gains symbolically:
 Options   (3)
The Ackermann method is used by default for systems with exact values:
For inexact systems with multiple output channels, the Ackermann method selects the output channel for which the observability matrix has the smallest condition number:
The second output has the lower condition number:
The gains associated with the first output:
The KNVD method uses all available outputs to estimate the states:
 Applications   (2)
Construct an observer for a continuous-time system:
Simulate the system with input Sin[t] and from a random initial condition:
Compare each state and its estimate:
Construct an observer for a zero-input sampled-data system:
Compute the actual and estimated states for initial states and initial observer states :
Examine the state tracking achieved by the observer:
The error dynamics:
The observer dynamics:
StateOutputEstimator assembles an observer that estimates both the states and outputs:
The estimator gains are the conjugate transpose of the state feedback gains of the dual system:
Or vice versa:
The KNVD method does not handle exact systems when the number of measurements is less than the number of states:
Evaluate numerically:
The KNVD method cannot handle cases in which the multiplicity of the poles exceeds the number of measurements:
Use the Ackermann method:
The KNVD method can give different sets of gains on different computer systems:
The observer's eigenvalues are the same:
The system must be observable:
It is not observable:
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