5.3 Constrained Feedback Stabilization
In many applications, it may not be enough to stabilize the system. The user should be able to place the eigenvalues in a certain stability region according to prescribed design constraints, such as the lower bound for minimal damping ratio, minimal damping factor (decay rate), settling time, minimal (undamped) natural frequency, and so on. This gives rise to the constrained feedback stabilization problem that is concerned with placing the eigenvalues in a userspecified region of the complex plane using feedback. The four regions available in Advanced Numerical Methods are: DampingFactorRegion[], SettlingTimeRegion[, ], DampingRatioRegion[], and NaturalFrequencyRegion[]. These regions for the continuoustime system are defined in the following discussion. For the discretetime system, the transformation ( is the sampling period of the system) maps a continuoustime stability region to the corresponding discretetime stability region.
Consider a continuoustime system with a pair of closedloop stable eigenvalues specified by . The four userspecified regions, which could contain such paired eigenvalues, are defined in the following discussion.
DampingFactorRegion[] designates the region that consists of all such that as shown in Figure 5.1. The parameter is called the damping factor of the statespace system.
Figure 5.1. The minimal damping factor of the continuoustime closedloop system: the poles lie in the region .
SettlingTimeRegion[, ] denotes the region that consists of all such that as shown in Figure 5.2. The parameter is called the settling time of the statespace system and the parameter species that the system response should settle within % of its final value. By default, if the parameter is not specified, the system response is assumed to settle within 5% of its final value with and . Obviously, the same region on the complex plane can be defined using either DampingFactorRegion or SettlingTimeRegion.
Figure 5.2. The maximal settling time of the continuoustime closedloop system: the poles lie in the region .
DampingRatioRegion[] designates the region that consists of all such that and as shown in Figure 5.3. The parameter is called the damping ratio of the statespace system.
Figure 5.3. The minimum damping ratio of the continuoustime closedloop system: the poles lie in the region .
NaturalFrequencyRegion[] specifies the region that consists of all such that and as shown in Figure 5.4. The parameter is called the (undamped) natural frequency of the statespace system.
Figure 5.4. The minimum (undamped) natural frequency of the continuoustime closedloop system: the poles lie in the region .
As mentioned, the defined stability regions for the continuoustime systems have their discretetime counterparts. In general, the discretetime stability region is the image of the complex transformation applied to the corresponding continuoustime stability region. For example, the continuoustime stability region depicted in Figure 5.2 is transformed to the following discretetime stability region.
Figure 5.5. The maximal settling time of the discretetime closedloop system: the poles lie in the region .
The closedloop system performance specifications.
The function StateFeedbackGains can be used to solve a constrained feedback stabilization problem if the list of closedloop poles is replaced by a list of the form or, equivalently, by an expression of the form , signifying the intersection of the , , and so on.
Constrained feedback stabilization.
All pole assignment methods work equally well for constrained feedback stabilization problems. Additionally, if the only constraint is either DampingFactorRegion[] or SettlingTimeRegion[, ], then the method called Lyapunov feedback stabilization method, described in Section 5.4, can be used as an alternative.
Make sure the application is loaded.
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Load the collection of test examples.
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This is a model of a drum boiler.
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These are the openloop poles, arranged in ascending order of their real parts.
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The feedback gain matrix guarantees that the closedloop system has both the damping factor less than and the damping ratio greater than .
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Make sure the function MultipleListPlot is available.
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This displays the closedloop poles of the drum boiler model on the complex plane. Both design specifications are met.
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