Time-invariant Feedback Systems Research Articles

The Determination of Periodic Modes in Relay Systems Using the State Space Approach†

ABSTRACT A method for determining the exact mode of oscillation of a time-invariant feedback system containing relays is presented. This method is based on tho state space approach in the real time domain, and results in a set of transcendental algebraic equations which can easily be solved by digital computation. General state transition and control matrices for plants with n poles and m zeros are derived. For complicated plant characteristics, the switching equations can be obtained easily by simple superposition of the switching equations of ‘elementary plants’. The general expression is given in discrete form, and may be reduced to a continuous form when no impulse control is present, otherwise a combined form is used. Switching equations for elementary plants up to third order are listed for reference, and illustrative examples are given with results compared by analogue simulation.

On the input-output stability of time-varying nonlinear feedback systems Part one: Conditions derived using concepts of loop gain, conicity, and positivity

The object of this paper is to outline a stability theory for input-output problems using functional methods. More particularly, the aim is to derive open loop conditions for the boundedness and continuity of feedback systems, without, at the beginning, placing restrictions on linearity or time invariance. It will be recalled that, in the special case of a linear time invariant feedback system, stability can be assessed using Nyquist's criterion; roughly speaking, stability depends on the mounts by which signals are amplified and delayed in flowing around the loop. An attempt is made here to show that similar considerations govern the behavior of feedback systems in general-that stability of nonlinear time-varying feedback systems can often be assessed from certain gross features of input-output behavior, which are related to amplification and delay. This paper is divided into two parts: Part I contains general theorems, free of restrictions on linearity or time invariance; Part II, which will appear in a later issue, contains applications to a loop with one nonlinear element. There are three main results in Part I, which follow the introduction of concepts of gain, conicity, positivity, and strong positivity: THEOREM 1: If the open loop gain is less than one, then the closed loop is bounded. THEOREM 2: If the open loop can be factored into two, suitably proportioned, conic relations, then the closed loop is bounded. THEOREM 3: If the open loop can be factored into two positive relations, one of which is strongly positive and has finite gain, then the closed loop is bounded. Results analogous to Theorems I-3, but with boundedness replaced by continuity, are also obtained.