Application of the second method of Lyapunov to the stability of certain position control systems containing a back-e.m.f. non-linearity

In the linearized analysis of control systems containing a d.c. field-excited motor the armature current is assumed constant. The effect of armature back-e.m.f. is to introduce a two-variable (field current, motor speed) non-linearity into the system equation. The second method of Lyapunov is used in the paper to investigate the effect of this non-linearity on the stability of certain position control systems. The system equations studied are of second-, third- and fourth-order, for various types of stabilization and lag configurations. A summary of the Lyapunov method is given. The Lyapunov functions V employed are simple quadratic functions, and asymptotic stability results are determined from negative semi-definite derivatives V using the Barba¿in limit point argument. A choice method for V enables asymptotic stability in the large to be established for a number of second- and third-order systems conditional on the Routh-Hurwitz criteria for the linearized systems; for fourth-order cases an amplitude-dependent condition is introduced in addition. Analogue-computer results indicate that the stability of the fourth-order cases is not amplitude-dependent. The weakness is due to the simple form of Lyapunov function employed. The results establish the validity of a linearized stability analysis for certain systems containing this type of non-linearity and imply it for others.