On the Evaluation of Cost Functionals, with Particular Emphasis on Time-delay Systems

“…For linear systems without time delays, the exact value of complex integral J can be computed by efficient parametric algorithms as described in [1,2,3,5,13]. Each of these algorithms involves only simple algebraic operations on the coefficients of the numerator and denominator polynomials of the rational transfer functions E{s) and F(s).…”

Section: J= E(s)e(-s)ds E(s)f(-s)dsmentioning

confidence: 99%

“…It follows from (13) and (14) that Y(s) satisfies the relation (15) where I* denotes the kxk identity matrix. Since the matrix K(s) satisfies the relation …”

Section: Ris)xis)=vis)mentioning

confidence: 99%

“…stable transfer function polynomial matrix stable transfer function adjoint of polynomial matrix G(s) identity matrix of dimension k complex integral defined in (3) complex integral defined in (4) positive integer kxk scalar matrix positive integers coefficients of a z-polynomial P(s,z) z-polynomials defined in (12) coefficients of a z-polynomial Q(s,z) z-polynomials defined in (12) polynomial matrix defined in (13) coefficients of a z-polynomial R(s,z) the least common multiple of A(s,z) and C(s,z) partitions of the polynomial matrix R(s) polynomial matrix z-polynomial defined in (9) z-polynomial defined in (10) …”

mentioning

confidence: 99%

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Numerical computation of cross‐covariance functionals for linear systems with multiple time delays

Hwang

Tsay

Hwang

1998

Journal of the Chinese Institute of Engineers

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The problem of evaluating a cross-covariance functional of the form 1 / 2πi i -i F(s)G(-s)ds arises, e.g., in parametric optimization of linear systems and in optimal linear filtering. In this paper, an algorithm is presented for computing cross-covariance functionals for linear systems with multiple commensurate time delays. The algorithm is based on converting a cross-covariance functional into an equivalent auto-covariance one. Then by utilizing an existing partial-fraction-expansion scheme for a class of two-dimensional rational transfer functions, the evaluation of a cross-covariance functional is reduced to the determination of symbolic inversion of a polynomial matrix and the spectral factorization of a finite-degree real-coefficient polynomial. It is shown that the algorithm presented here can be extended to systems having non-commensurate time delays by either introducing fictitious delays or adjusting time delays with rational approximations such that the resulting delays are commensurate. In order to illustrate the effectiveness of the algorithm, several examples are provided.

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“…For the most proposed PID design methods of using optimization approach, the integral of the square error (ISE) subject to a step change in set-point or load is used as the closed-loop control performance index because of its mathematical tractability. In general, an Downloaded by [FU Berlin] at 05:56 14 October 2014 ISE performance index can be evaluated efficiently (Astrom, 1970;Babister, 1957a,b;Babister, 1958;Walton and Gorecki, 1984;Gorecki and Korytowski, 1993) from the closed-loop transfer function and it is differentiable so that a computationally efficient gradient-based optimization technique can be applied to search for the optimal PID controller parameters. However, the ISE criterion puts more weight on large errors and as a consequence results in higher controller gains, a relatively oscillatory step response and the loss of robustness.…”

Section: Introductionmentioning

confidence: 99%

Designing Pid Controllers With a Minimum Iae Criterion by a Differential Evolution Algorithm

Cheng

Hwang

1998

Chemical Engineering Communications

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Although the integral of absolute error (IAE) has long been used as a performana: measure for controller evaluation, a systematic approach is still lacking for the design of optinlal PID controller with the minimum IAE performance criterion. This may be ascribed to the nonsmoothness of the IAE objective function and the lack of closed-form expression for the IAE performance index. In this paper, we propose the use of a differential evolution algorithm (DEA) to design a minimum-IAE PID controller lo overcome the nondifferentiability of the IAE objective function. To achieve also a specified least relative stability margin and least damping ratio, the D-partition technique is used to identify the allowable domain in the tuning parameter spnce. Three examples are provided to illustrate the effectiveness of the DEA.

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On the Evaluation of Cost Functionals, with Particular Emphasis on Time-delay Systems

Cited by 19 publications

References 0 publications

Direct method for TDS stability analysis

Direct method for TDS stability analysis

Numerical computation of cross‐covariance functionals for linear systems with multiple time delays

Designing Pid Controllers With a Minimum Iae Criterion by a Differential Evolution Algorithm

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