Synthesis of Output Feedback Static Regulators by Using the Gradient Flows: LQR and LS-Matching Eigenvalue Assignment

Synthesis of regulators based on the output measured variable is considered more difficult than synthesis based on the state variables and synthesis of dynamic regulators. This problem has not been solved in its general form so far. In this paper, we consider the case when neither the necessary (mp ≥ n) nor sufficient (mp > n) conditions for the pole placement synthesis at the output are fulfilled, where m is the number of inputs, p the number of outputs, and n the order of the system. At the same time, two approaches to solving this problem are being investigated: LQR and the pole placement approximation using gradient flows. In the first case, Lyapunov algebraic equations are solved and gradient equations are integrated. In this case, an extremum (local or global) is always reached. The second approach uses gradient flows on group Lie GL(n, R) Ѕ R mЅp , where a least squares pole assignment (mp < n) is performed with the choice of a part of the desired spectrum of a closed system. Here, optimization is performed before the modes enter a small neighborhood of the desired modes. To obtain gradient equations in both approaches, gradients of objective functions tr( M ) from the matrix argument are derived. The properties of the function tr( M ) and the rules for finding its gradient from the matrix argument are considered in detail. The obtained gradient equations are verified using a practical example: a 4th-order object with weakly damped dynamics with a large range of parameter uncertainty, where the task is to synthesize a robust static regulator of the minimum (second) order. This problem has been solved by both methods with the fulfillment of the set technical requirements.