6.1 Full-Order State Estimation
The state estimation problem is described briefly in the following only for the continuous-time systems Eq. (6.1). Adaptation to the discrete-time systems is straightforward. The state-space system is assumed to be observable.
Given the matrices A, B, C, and D and the input and output vectors u(t) and y(t) of the continuous-time system,
where is , is (), and is (), the state estimation problem is one of finding an order estimator of , satisfying
such that the error approaches zero as . Figure 6.1 presents a feedback design scheme that uses an estimator in block diagram form. As seen in the diagram, the estimator uses the control input and the actual output measurement to produce the estimate of the system states. When the estimator computes values of all states, the problem is called the full-order state estimation problem to distinguish it from the reduced-order state estimation problem, where only values of some states are estimated while the remaining ones are estimated directly from the input and output vectors (see Section 6.2). To obtain from Eq. (6.2), one finds the matrix such that the matrix is stable. This can be done using either the pole assignment algorithms or solving the associated Sylvester-observer equation. The function EstimatorGains in Control System Professional (Section 9.2) uses the pole assignment approach to solve the full-order state estimation problem.
Figure 6.1. The feedback design scheme that uses the state estimator.
New numerical methods introduced in Advanced Numerical Methods to solve the pole assignment problem are available for the function EstimatorGains.
Make sure that the package is available.
Load the collection of test examples.
This is a simplified magnetic tape control problem.
Here are the system poles.
This finds the estimator gains for the system by replacing the unstable pole with the pole at while preserving the remaining stable poles.