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2.2.4 The Design of the LQR Controller
With Advanced Numerical Methods, the feedback controller can be designed using several different approaches, such as solving the LQR problem, using constrained feedback stabilization, or implementing the pole assignment approach. This section illustrates the LQR design. For the experiment,  is a symmetrical matrix constructed from the output matrix of the system and  is an identity matrix multiplied by a scalar coefficient  . Evaluation of how different choices of the coefficient  affect the performance of the closed-loop system completes the process.
These are the weighting matrices needed for the LQR design.
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The LQR gains with control weight  range from  (cheap control) to  (minimum energy control). The underlying Riccati equation is solved using the Newton method.
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This displays the poles of the closed-loop system with varying  .
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Make sure LogLinearListPlot is available.
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Increasing improves the stability and the damping factor of the closed-loop system. However, changes in the damping factor are not significant when is either very low or very high.
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Varying also changes the norm of the feedback gain matrix. The LQR gains are high when is small.
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These are the closed-loop system output responses for various feedback matrices. For convenience, the original open-loop system is represented as a closed-loop system with a dummy (zero) feedback.
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In the following plots, the closed-loop responses are represented by colored graphs and the open-loop responses are represented by thick black graphs.
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These are the output responses of the original and closed-loop systems. From the spectrum, the designer can choose a feedback that brings about suitable values of the drum pressure and the drum liquid level.
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