9.2 State Reconstruction
The prerequisite for pole assignment using state feedback is knowledge of the state variables. So far in this chapter we have assumed that this prerequisite was somehow met. Now we address the problem of how that can be done. In the trivial case of a square nonsingular matrix , the state vector can simply be computed from the input and output vectors as ; otherwise an attempt can be made to reconstruct the state vector by forming a device called an estimator (or observer), with which the approximation to the state vector can be obtained as
where is the estimator gain matrix. Figure 9.4 presents such a device in block diagram form (Brogan (1991)). As seen from the diagram, that the estimator is driven by the difference between the actual output measurement and the "expected" output , provided that the direct transmission term is taken into account.
Figure 9.4. Continuoustime estimator schematic.
Figure 9.5. Discretetime estimator schematic.
If the initial system is completely observable, it is possible to choose gains to be such that the eigenvalues of assume any desired location, thereby controlling the rate at which follows . This reduces the problem of finding the estimator gains to one of finding the (transposed) controller gains for the dual system:
Therefore, the function StateFeedbackGains can be employed for the purpose. Alternatively, one may use the function EstimatorGains, which performs the procedure in one step. EstimatorGains does not introduce any new options of its own, but accepts and passes on the options for StateFeedbackGains and DualSystem.
Estimator design.
The preceding diagram and equations refer to continuoustime systems. The estimator for a discretetime system is determined by an equation similar to Eq. (9.4):
the corresponding block diagram is presented in Figure 9.5. EstimatorGains handles both continuous and discretetime cases.
Another linearized statespace model of an inverted pendulum, reproduced after Gopal (1993), will be used in the rest of this section to illustrate obtaining the estimator gain matrix. The model assumes that the state and input vectors are, respectively,
where is the angular displacement of the pendulum, is the horizontal position of the cart, and is the external force applied to the wheels. The model further assumes that the only variable available for measurement is , making it, therefore, the only output variable:
Here is the statespace model.
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Because we know that all symbolic variables in the input system are realvalued, we may set the option ComplexVariables correspondingly to get a simpler result. This option will be picked up by the function DualSystem. At this point, we will not specify the desired poles for the estimator, but use a generic list of four symbolic values, {p1, p2, p3, p4}. Therefore, the expression will be obtained in its general form.
This finds the estimator for the system and somewhat simplifies the result.
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This is the result for a particular set of numerical values.
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