Loop gain

Abstract: 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.

Abstract: A new class of fast-converging timing recovery methods for synchronous digital data receivers is investigated. Starting with a worst-case timing offset, convergence with random binary data will typically occur within 10-20 symbols. The input signal is sampled at the baud rate; these samples are then processed to derive a suitable control signal to adjust the timing phase. A general method is outlined to obtain near-minimum-variance estimates of the timing offset with respect to a given steady-state sampling criterion. Although we make certain independence assumptions between successive samples and postulate ideal decisions to obtain convenient analytical results, our simulations with a decision-directed reference and baud-to-baud adjustments yield very similar results. Convergence is exponential, and for small loop gains the residual jitter is proportional and convergence time is inversely proportional to the loop gain. The proposed algorithms are simple and economic to implement. They apply to binary or multilevel PAM signals as well as to partial response signals.

Abstract: A necessary and sufficient condition, expressed simply as the dc loop gain (i.e., the loop gain at zero frequency) being less than unity, is given in this note to guarantee the internal stability of a feedback interconnection of linear time-invariant (LTI) multiple-input multiple-output systems with negative imaginary frequency response. Systems with negative imaginary frequency response arise, for example, when considering transfer functions from force actuators to colocated position sensors, and are commonly important in, for example, lightly damped structures. The key result presented here has similar application to the small-gain theorem, which refers to the stability of feedback interconnections of contractive gain systems, and the passivity theorem, which refers to the stability of feedback interconnections of positive real (or passive) systems. A complete state-space characterization of systems with negative imaginary frequency response is also given in this note and also an example that demonstrates the application of the key result is provided.

Abstract: The grid-connected inverter with an LCL filter has the ability of attenuating the high-frequency current harmonics. However, the current distortion caused by harmonics in the grid voltage is difficult to be eliminated. Increasing the loop gain can reduce the current distortion, but this approach is compromised by the system stability requirement. Without increasing the loop gain, applying feedforward of the grid voltage can suppress the effect of grid voltage harmonics. This paper proposes the feedforward function of the grid voltage for the grid-connected inverter with an LCL filter. Specifically, the proposed feedforward function involves proportional, derivative, and second derivative of the grid voltage, and can be simplified according to the dominant harmonics in the grid voltage. The proposed feedforward scheme can effectively suppress the current distortion arising from the grid voltage harmonics, and the steady-state error of the injected current can be substantially reduced even if a conventional proportional and integral regulator is applied. A 6-kW experimental prototype has been tested to verify the effectiveness of the proposed feedforward scheme.