Linear Quadratic Regulator Procedure and Symmetric Root Locus Relationship Analysis
Abstract
The present paper is focused on linear quadratic regulator (LQR) synthesis. It is evident that the solution of the problem leads to finding a solution of symmetric root locus (SRL) problem. It is proven and analytical evidences is also presented that by using SRL synthesis method and further application of LQR method guarantees stability, minimum phase and gain margins of the closed loop system. The presented mathematical relationships are further proven with experimental investigation in MATLAB programming environment
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Introduction
The scope of the present paper is LQR synthesis in the state space. According to the state feedback control chosen, the synthesis procedure is related to determination of the state feedback matrix elements in a manner that a compromise between achieving the desired settling time and the control effort needed. In the presence of immeasurable states, observers are typically used for state vector estimation. The prerequisites needed for the synthesis are the system to be completely controllable and observable although advanced state variable design techniques can handle situations wherein the system is only stabilizable and detectable. In the present paper it is assumed that the system is completely controllable and observable and all the state variables are measurable.
The procedure for LQR synthesis requires the feedback matrix to be chosen in order to satisfy the condition for minimum value of the cost function presented as integral performance estimation. The closed loop is considered optimal as it guarantees the minimum possible value of the cost function. The synthesis procedure allows the operator to choose the weights so that the compromise between the state cost and the control cost is acceptable. The major disadvantage of this technique is that the exact poles position and the control effort value are initially unknown.
The present paper aims to investigate the relationship between the linear quadratic regulator procedure and the symmetric root locus of the closed loop system.
For the provision of this analysis few main aspects should be investigated in details and further proven:
1. Find analytic solution of the LQR synthesis based on the Pontriagin’s minimum principle.
2. Show that LQR synthesis can be solved by SRL.
3. Use SLR procedure to prove that when LQR synthesis is applied the closed loop system will always be stable and minimum phase.
Conclusion
The paper presents analytical evidence that LQR synthesis can be considered as SRL problem.
In the presence of evidence for completely controllable and observable (stabilizable and detectable) open loop system the state feedback synthesis is invariant in accordance with open loop system stability.
LQR can be also taken into consideration as a method for state feedback matrix generation with guarantees for stability, minimum phase and stability margins of the closed loop system.
This type of analysis, using graphic interpretation (ex. Fig.2) allows direct identification of the damping ratio and corresponding natural frequencies n and respectively further calculation of the weight coefficient 2n m n for any point positioned on the loci.
The results delivered give further option for investigation of possible relationship and application of the pole placement synthesis and LQR synthesis based on SRL in the frequency domain.