Time Domain Tolerance Research Articles

Computationally efficient large-change statistical analysis of linear electronic circuits

A new method is presented for efficient statistical analysis of linear electronic circuits, when small and large parameter tolerances are given. The statistically generated value of the parameter is considered as a faulted value, as it deviates from the nominal thus enabling the application of a simulation method which uses a new approach of concurrent fault simulation. This method adds new elements to the circuit, representing individual parameter increments, while keeping the topology of the original one. The equations for the original and several perturbed circuits are formulated and solved simultaneously. In this way, redundant computations are avoided in both the equation formulation and equation solving phases, which shorten the simulation time. A statistical frequency and time domain tolerance simulator of linear circuits was developed on the basis of this method with effective user-friendly interface. The method is especially suited for yield sensitivity to some selected circuit parameters estimation. Here simulation results of several benchmark circuits are presented. Efficiency analysis is also included.

Frequency-based nonlinear controller design for regulating systems subjected to time-domain constraints

Presented is a nonlinear controller design methodology for a class of linear regulating systems subjected to quantitative time-domain constraints. The design objective is to satisfy an output time-domain tolerance given an actuator saturation constraint despite an external step disturbance. The goal is to increase the allowable magnitude of the external disturbance beyond that achievable via linear control subject to the time-domain specifications. The controller design process is comprised of two phases. In the first phase, a linear controller is designed that balances the trade-off between output regulation and required actuation. To realize the linear design, the time-domain performance specifications are mapped into amplitude and phase constraints which are in turn imposed on the frequency response of the linear open-loop transfer function. In the second phase, the linear controller is then augmented with an odd nonlinearity. The coefficient for the nonlinear term is designed such that the gain and phase distortions (in the sense of describing functions) meet the frequency-domain constraints. The describing function calculation is automated by a recursive Volterra Series relationship. The nonlinear controller design methodology is experimentally verified on the idle speed control of a Ford 4.6L V-8 fuel injected engine. Copyright © 2000 John Wiley & Sons, Ltd.

Frequency-based controller design for a class of nonlinear systems

Presented in this paper is a frequency-based controller design methodology for a class of nonlinear systems that can be characterized by a class of Hammerstein models. The controller design is directed toward the disturbance rejection problem where the output performance specification appears as a time domain tolerance. The controller design process is executed in three stages. First, the system nonlinearities are quasilinearized via describing function techniques. Second, the design methodology enforces a time domain tolerance ∣y(t)∣⩽α via frequency domain constraints on the quasilinearized closed-loop system. Finally, the controller is designed via classical loop shaping of the quasilinearized system on the open-loop gain-phase plane. The design process is illustrated on the idle speed control of a Ford 4.6L-2 valve V-8 fuel injected engine. Copyright © 1999 John Wiley & Sons, Ltd.

NARMAX modelling and robust control of internal combustion engines

Detailed in this paper is a SISO non-linear modelling and robust controller design methodology experimentally verified on an internal combustion engine. The methodology begins with the identification of a NARMAX model that captures the non-linear dynamics relating the input to the output of a system. This model is converted to a describing function representation for the purpose of robust feedback controller design. The ideology for the describing function recovery is developed in the form of an algorithm which can be extended to other NARMAX model structures not considered here. The controller design is executed in the frequency domain where the output performance specification is |y(t)|≤β∀t>0 and the actuator saturation constraint is |u(t)|≤K∀t>0. For the engine idle speed control application of this study, a SISO NARMAX model of the engine is developed between the by-pass idle air valve (BPAV) and engine speed. The performance objective for the controller design is the time domain tolerance of |Δ rpm| ≤ 100 rpm on idle speed perturbations despite a non-measurable 20 N m external torque disturbance. The controller is validated through numerical simulations as well as experimental verification.

Frequency Based Multivariable Feedback Controller Design

Presented in this paper is a multivariable feedback controller design methodology for regulating systems required to satisfy hard time domain tolerances despite a vector of external step disturbances. In addition to satisfying time domain tolerances, this design methodology seeks to minimize the bandwidth of each feedback controller so that the cost of feedback is minimized. The two key contributions of this work are (i) the rigorous justification of enforcing time domain tolerances via frequeucy domain amplitude inequalities, and (ii) the design of a fully populated, multivariable controller via sequential loop closure. This multivariable controller design methodology will be demonstrated on the control of an distillation column.

Novel methods for circuit worst-case tolerance analysis

The method, presented by E. R. Hansen (see SIAM J. Numer. Anal., vol. 29, no. 5, p. 1493-1503, 1992) for solving interval linear equations, is improved to meet the requirements of circuit worst-case tolerance analysis. Based on the solutions of interval linear equations, an analytic algorithm and an accurate algorithm are constructed. It is shown that the performances of the two algorithms are complementary: the analytic algorithm runs faster than the accurate algorithm, but has more interval expansion error. Techniques for AC tolerance analysis, time-domain tolerance analysis and DC tolerance analysis are discussed respectively. Some examples and comparisons are given.

Selecting the Performance Weights for the μ and H∞ Synthesis Methods for SISO Regulating Systems

Presented in this paper is a method in which the performance weights W1(s) and W2(s) of the robust performance problem and the H∞ optimization problem can be chosen to directly enforce hard time domain constraints. The class of systems addressed are single-input-single-output (SISO) regulating systems required to maintain the output within a prespecified time domain tolerance despite (i) plant uncertainty, (ii) an external step disturbance, and (iii) actuator output saturation. The performance weights are extracted from the feedback configuration and facilitate performance maximization.

Optimal loop-shaping for systems with large parameter uncertainty via linear programming

This paper describes an optimization method for designing feedback systems subject to large parameter uncertainty. Following the design philosophy of the Quantitative Feedback Theory (Horowitz and Sidi 1972 HOROWITZ , I. M. , and SIDI , M. , 1972 , Synthesis of feedback systems with large plant ignorance for prescribed time-domain tolerances . International Journal of Control , 16 , 287 – 309 ; 1978, Optimum synthesis of non-minimum phase systems with plant uncertainty. International Journal of Control, 27, 361–386 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar], 1978) the objective is to minimize the magnitude of the open loop L(jω) at high frequencies subject to: (a) low and intermediate-frequency bounds capturing the closed-loop robust-performance objectives; (b) a universal-frequency bound on L(jω) which limits the effects of disturbances; and (c) realization constraints on L(s) in the form of Bode's integral. This last relation is discretized at a number of frequencies and defines, together with (a) and (b), the overall set of linear constraints in a resulting linear programming optimization problem.

Synthesis of controllers for uncertain multivariable plants for prescribed time-domain tolerances†

Abstract The paper considers the synthesis of controllers for the feedback control of multi-variable plants with large parameter ignorance. Here, the output response of each loop is required to satisfy some a priori specified time domain tolerance bounds irrespective of the parameter variations of the plant and the interaction carried through other loops. By an interplay between the Nyquist array and Bode array, it becomes possible to obtain a design that not only satisfies the required specifications but also offers some advantages over other design techniques for the same class of problems.

New time-domain sensitivity formulas and their applications

This paper proposes two time-domain tolerance measures: the sum of squared sensitivities of output sample values (SS), and the sum of squared errors of output sample values (SE), in order to investigate the parasitic effect on the impulse responses of linear time-invariant networks. The measure SS is expressed in a Hermitian form with 1) a matrix to be determined uniquely from a transfer function and time parameters, and 2) a vector whose elements are pole and zero sensitivities. In the case of cascade networks, the dependence of SS on an active element contained in a second-order stage can be expressed in terms of a pole sensitivity of this stage. This leads to computational savings. As an application, we present a method of finding the pole sensitivity which provides the lower limit of SS. This method is applicable to various network structures although only one case is discussed here. The measure SE is expressed in a quadratic form, with 1) a matrix to be computed using the formula for SS, and 2) a vector whose elements are relative errors of network parameters. We apply SE to the problem of the optimum tolerance assignment which produces the cheapest network.

Quantitative synthesis of multiple-loop feedback systems with large uncertainty†

Abstract This work extends single input-output linear time invariant minimum-phase ‘ quantitative feedback synthesis ’ to two new complex plant structures with internal sensing points. One is the triangular structure. The second consists of parallel branches, each with cascaded sections. Due to uncertainty, the plant parameters are elements of given sets. The system response must satisfy specified time or frequency domain tolerances. The basic problem is how to divide the feedback burden among the available loops so as to minimize the net rms effect at the plant input, of the various sensor noise sources. Frequency-response formulations are presented which provide a clear understanding of the trade-off among the feedback loops. One vital feature is ‘ free uncertainty ’, wherein a loop optimized to cope with uncertainty U1 , may in fact for some frequency ranges, handle uncertainty U≫U1 . A second is ‘ bandwidth propagation ’ wherein the loops take turns in dominating the design over the frequency range. T...

Synthesis of sampled feedback systems for prescribed time domain tolerance

This paper presents a design procedure for single-loop sampled feedback systems in which the plant transfer function has prescribed bounds on its parameters. The plant is imbedded in a ‘ two-degree-of-freedom system’, which is to be designed such that the system time response to a deterministic input lies within specified boundaries. The design procedure is based on frequency concepts. The time-domain tolerances are translated into equivalent frequency response tolerances. The latter lead to bounds on the loop-transmission L(jv) [v =tang (π/WS)[, in the form of continuous curves on the Nichols chart. However, due to inherent theoretical difficulties existing in sampled feedback systems, it might happen that no L(jv) can be found so that the design specifications of the problem are achieved. The design procedure is transparent enough to guide the designer in which direction to change the specifications of the problem so that the next trial design will be successful.

Feedback synthesis with plant ignorance, nonminimum-phase, and time-domain tolerances

This paper presents a design procedure for single-loop feedback systems in which the plant is nonminimum-phase, with ignorance in its parameters. The feedback system is to be designed such that the time response to a deterministic input lies within specified boundaries. Subject to the above, the design should be such as to minimize the effect of sensor noise at the input to the plant. The time domain tolerances are translated into equivalent frequency response tolerances. The latter lead to bounds on the loop-transmission L( jw), in the form of continuous curves on the Nichols chart. However due to the nonminimum-phase nature of the plant, it may happen that no L( jw) can be found so that the design specifications of the problem are achieved, unless they are changed accordingly.

Synthesis of Feedback Systems with Large Plant Ignorance for Prescribed Time Domain Tolerances

There is given a minimum-phase plant transfer function, with prescribed bounds on its parameter values. The plant is imbedded in a two-degree-of-freedom feedback system, which is to be designed such that the system time response to a deterministic input lies within specified boundaries. Subject to the above, the design should be such as to minimize the effect of sensor white noise at the input to the plant. This report presents a design procedure for this purpose, based on frequency response concepts. The time-domain tolerances are translated into equivalent frequency response tolerances. The latter lead to bounds on the loop transmission function L(jω), in the form of continuous curves on the Nichols chart. Properties of L(jω) which satisfy these bounds with minimum effect of sensor white noise, are derived. The design procedure is quite transparent, providing the designer with the insight to make necessary trade-offs, at every step in the design process. The same design philosophy may be used to attenuate the effect of disturbances on plants with parameter ignorance.

Synthesis of feedback systems with large plant ignorance for prescribed time-domain tolerances†

There is given a minimum-phase plant transfer function, with prescribed bounds on its parameter values. The plant is imbedded in a two-degree-of-freedom feedback system, which is to be designed such that the system time response to a deterministic input lies within specified boundaries. Subject to the above, the design should be such as to minimize the effect of sensor white noise at the input to the plant. This report presents a design procedure for this purpose, based on frequency response concepts. The time-domain tolerances are translated into equivalent frequency response tolerances. The latter lead to bounds on the loop-transmission function L(jω), in the form of continuous curves on the Nichols chart. Properties of L(jω) which satisfy these bounds with minimum effect of sensor white noise are derived. The design procedure is quite transparent, providing the designer with the insight to make necessary tradeoffs, at every step in the design process. The same design philosophy may be used to attenuate the effect of disturbances on plants with parameter ignorance.