Bode Integral Research Articles

Bode Integral Limitation For Irrational Systems

Bode integrals of sensitivity and sensitivity-like functions along with complementary sensitivity and complementary sensitivity-like functions are conventionally used for describing performance limitations of a feedback control system. In this paper, we investigate the Bode integral and evaluate what happens when a fractional order Proportional-Integral-Derivative (PID) controller is used in a feedback control system. We extend our analysis to when fractal PID controllers are applied to irrational systems. We split this into two cases: when the sequence of infinitely many right half plane open-loop poles doesn't have any limit points and when it does have a limit point. In both cases, we prove that the structure of the Bode Integral is analogous to the classical version under certain conditions of convergence. We also provide a sufficient condition for the controller to lower the Bode sensitivity integral.

H∞ Optimal Performance Design of an Unstable Plant under Bode Integral Constraint

This paper proposed the H∞ state feedback and H∞ output feedback design methods for unstable plants, which improved the original H∞ state feedback and H∞ output feedback. For the H∞ state feedback design of unstable plants, it presents the complete robustness constraint which is based on solving Riccati equation and Bode integral. For the H∞ output feedback design of unstable plants, the medium‐frequency band should be considered in particular. Besides, this paper presents the method to select weight function or coefficients in the H∞ design, which employs Bode integral to optimize the H∞ design. It takes a magnetic levitation system as an example. The simulation results demonstrate that the optimal performance of perturbation suppression is obtained with the design of robustness constraint. The presented method is of benefit to the general H∞ design.

Convergence of fundamental limitations in feedback communication, estimation, and feedback control over Gaussian channels

In this paper, we establish the connections of the fundamental limitations in feedback communication, estimation, and feedback control over Gaussian channels, from a unifying perspective for information, estimation, and control. The optimal feedback communication system over a Gaussian necessarily employs the Kalman filter (KF) algorithm, and hence can be transformed into an estimation system and a feedback control system over the same channel. This follows that the information rate of the communication system is alternatively given by the decay rate of the Cramer-Rao bound (CRB) of the estimation system and by the Bode integral (BI) of the control system. Furthermore, the optimal tradeoff between the channel input power and information rate in feedback communication is alternatively characterized by the optimal tradeoff between the (causal) one-step prediction mean-square error (MSE) and (anti-causal) smoothing MSE (of an appropriate form) in estimation, and by the optimal tradeoff between the regulated output variance with causal feedback and the disturbance rejection measure (BI or degree of anti-causality) in feedback control. All these optimal tradeoffs have an interpretation as the tradeoff between causality and anti-causality. Utilizing and motivated by these relations, we provide several new results regarding the feedback codes and information theoretic characterization of KF. Finally, the extension of the finite-horizon results to infinite horizon is briefly discussed under specific dimension assumptions (the asymptotic feedback capacity problem is left open in this paper).

MOBIUS AND DELTA TRANSFORMS IN THE UNIFICATION OF CONTINUOUS-DISCRETE SPACES

It is well-known that in control theory the stability region of continuous- time system is laid in the left half plane of complex space, while that of discrete-time system is dwelled inside a unit circle. The former fact might be shown by exploiting the Laplace transform and the later by utilizing the corresponding zeta transform. In this paper we revealed the connectivity of both regions by employing M¨obius transform. We also used the same transform to derive continuous/discrete-time counterpart of several existing results, including Bode integral and Poisson-Jensen formula. We then demonstrated their unification property by using delta transform. Some numerical examples were provided to verify our results.

The Design of QFT Robust Compensators with Magnitude and Phase Specifications

The frequency response is an important tool for practical and efficient design of control systems. Control techniques based on frequency response are of special interest to dealing with important subjects such as the bandwidth and the cost of feedback. Furthermore, these techniques are easily adapted to deal with the uncertainty of the process to control. Quantitative feedback theory (QFT) is an engineering design technique of uncertain feedback systems that uses frequency domain specifications. This paper analyzes the phase specifications problem in frequency domain using QFT. This type of specification is not commonly taken into account due to the fundamental limitations of the linear control given by Bode's integral. An algorithm is proposed aimed at achieving prespecified closed-loop transfer function phase and magnitude variations, taking into account the plant uncertainty. A two-degrees-of-freedom feedback control structure is used and a new type of boundary is defined to satisfy these objectives. As the control effort heavily depends on a good estimation of these boundaries, the proposed algorithm allows avoiding overdesign.

Robust Inversion-Based 2-DOF Control Design for Output Tracking: Piezoelectric-Actuator Example

In this paper, a novel robust inversion-based 2-DOF control approach for output tracking is proposed. Inversion-based feedforward control techniques have been successfully implemented in various applications. Usually, to account for adverse effects such as dynamics variations and disturbances, the inverse feedforward control is applied by augmenting it with a feedback control. However, such effects have not been directly addressed in existing system-inversion methods, and the integration of the feedback control with the inversion-based feedforward control is performed in an ad hoc manner, which may not lead to an optimal complement of the inversion-based feedforward control with the feedback control. The contribution of this paper is the development of the following: 1) a robust system-inversion approach to directly account for and then minimize the dynamics-uncertainty effect when finding the inversion-based feedforward controller, and 2) a systematic integration (of such a feedforward controller) with a robust feedback controller. The proposed robust inversion method achieves a guaranteed tracking performance of the feedforward control for bounded dynamics uncertainties. Then, the quantified bound of the feedforward control tracking error is utilized in designing an H infin robust feedback controller to complement the feedforward control. Based on the concept of Bode's integral, it is shown that the feedback bandwidth can be improved from that obtained by using feedback alone. We illustrated the proposed approach by implementing it in experiments on a piezotube actuator of an atomic force microscope for precision positioning.

Nonlinear extension of Bode's integral based on an information-theoretic interpretation

For linear systems, we show that the constraint known as Bode's sensitivity integral has an information-theoretic interpretation in terms of the difference in the entropy rates between the input and output of the systems. For nonlinear system, we use this interpretation to show that, if the open loop system is globally exponentially stable and has fading memory, this difference is zero. For nonlinear systems that are not stable, we investigate the method to calculate this difference using new results on the inner/outer factorization of nonlinear systems.

Logarithmic integrals and system dynamics: an analogue of Bode's sensitivity integral for continuous-time, time-varying systems

A new time-domain interpretation of Bode's integral is presented. This allows for a generalization to the class of time-varying systems which possess an exponential dichotomy. It is shown that the sensitivity function is constrained, on average, by the spectral values in the dichotomy spectrum of the antistable component of the open-loop dynamics.

Tradeoffs in linear time-varying systems: an analogue of Bode's sensitivity integral

A new time-domain interpretation of Bode's integral is presented. This allows for a generalization to the class of time-varying systems which possess an exponential dichotomy. It is shown that the sensitivity function is constrained, on average, by the Lyapunov exponents of the open-loop system which take the place of the magnitude of the open-loop poles.

Minimization of the Bode sensitivity integral

In this paper we present two new results on Bode-type integrals. Firstly, we obtain, for a given scalar or multivariable continuous-time plant, the infimum of the Bode sensitivity integral which can be obtained with any stabilizing controller. The result involves the unstable plant poles and, perhaps surprisingly, a subset of the plant nonminimum phase zeros. Secondly, we obtain an apparently new expression for the Bode integral for the complementary sensitivity for a stable discrete-time scalar system.

Integral Relations for Disturbance Isolation

Introduction P ASSIVE and active dynamic systems of high order are employed for the purpose of disturbance isolation. Such systems’ theory and designmethods could beneŽ t from using the design theory of active electrical networks. The feasibility and the available performanceof such systemscan be evaluatedusingBode integrals. This enables the system engineer to resolve the design tradeoffs without actually designing the system and the subsystems.A Bode integral is applied to estimate the performanceof a disturbance isolation system. Consider the system in Fig. 1a of two bodies connected with an active strut,2 which is a linearmotor. A force disturbancesource F1 is applied to the body M1 (capital letters designate Laplace transforms). The force F3 is applied via the massless active strut to the body M3 . To increase the disturbance isolation, the force division ratio KF D F3/ F1 should be made small. The strut mobility (in some literature called mechanical impedance) Z2 is the ratio of the difference in the velocities at the ends of the strut to the force (because we neglect the strut’s mass, the force is same at the both ends of the strut). Feedback is employed in the active strut to increase jZ2j in order to reduce jKF j. For the purpose of analysiswe use the following electromechanical analogy: power to power, voltage to velocity, current to force, capacitance to mass, and inductance to the inverse of the stiffness coefŽ cient. The electrical equivalentcircuit for the system is shown in Fig. 1b. The current division ratio I3/ I1 is equivalent to KF . The electrical impedance Z2 is the equivalent of the strut mobility. The current division ratio, i.e., the force division ratio, is

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.

Imaginary part of antenna's admittance from its real part using Bode's integrals

The imaginary part of an antenna input admittance is calculated from its real part using Bode's integrals. Since the real part is typically a smoother function of the frequency than the imaginary part, the procedure presented here requires computation at a smaller number of frequency points, thus saves time, and is ideal for systems whose input conductance exhibits sharp peaks. A numerical procedure to evaluate the singular Bode's integral is also presented. Numerical examples using a wire antenna are used to illustrate the advantages of this approach compared to calculations involving a densely scanned frequency range. The noise stability and robustness of the algorithm are demonstrated through the successful prediction of the susceptance and the resonant frequencies of the antenna in the presence of random noise in the conductance.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Directional sensitivity tradeoffs in multivariable feedback systems

In this paper tradeoffs in multivariable sensitivity reduction are developed for a new Directional Sensitivity Function (DSF). The directions considered belong to direction modules of the nonminimum phase zeros (NMP) and unstable poles of the transfer matrices. The well known Bode Integral is extended for the DSF whenever the direction vector coincides with the direction module of an unstable pole. Similarly the tradeoff known as the waterbed effect is extended whenever the direction vector coincides with the direction module of a NMP zero. These results intuitively exhibit the effect of spatial positions of poles and zeros of multivariable systems on sensitivity.

Bode's integral revisited

According to quantitative feedback design theory, the synthesis of a closed-loop transfer function requires the evaluation of a Bode integral. Series expansion techniques and an extension of the Gauss-Chebyshev numerical quadrature exist to compute this integral. A survey and comparison of these methods is presented. Some examples are introduced to show their utility.