A.A. Mohammad

Optimal control, LQR, root locus, Lyapunov equations, balanced realization

In this paper, a new optimal root locus technique is presented. It relies on the choice of the performance index in the linear quadratic regulator (LQR) problem. The choice of the states weighting matrix Q is made so that the solution of the resulting algebraic Riccati equation (ARE) is known in advance. This eliminates the need to solve the (2n × 2n) eigenvalue/eigenvector problem usually needed to solve the ARE. With this choice of Q, the solution of the ARE becomes a constant multiple of the observability Gramian. The control law is easily calculated in O(n2) ﬂops. This is a good improvement over the Chang-Letov method, which in general requires O(2n)3 ﬂops. The weighting matrix Q, the feedback control gain K, and the cost J are directly given as functions of the observability Gramian. Although the algorithm is given for any coordinate system, the development in balanced coordinates is particularly interesting. In this coordinate system, Q, K, and J are directly given as functions of the observability and controllability Gramians and the Hankel singular values of the system. Finally, it is shown that the optimal root locus can be obtained using ordinary root locus and develops efficient rules and algorithms.

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