Feedthrough Term Research Articles

On Strictly Positive Real Improper Transfer Matrices

In this article we consider linear systems having real rational transfer matrices that may be improper. We investigate the properties of strictly positive real and weakly strictly positive real systems and their connections. Our first contribution is to establish a necessary and sufficient condition for a rational and possibly improper transfer matrix to be strictly positive real. Beside stability, this condition only involves the properties of the transfer matrix in the imaginary axis and a geometric relation between the feedthrough term and the residue at infinity of the transfer matrix. Our condition connects some of the properties of the system with those of its proper part. This leads to our second contribution: integrate in the improper setting all the several conditions derived in the literature for the proper case that discriminate weakly strictly positive real and strictly positive real systems.

Nonlinear Sampled-data Systems—A lifting framework

This short note gives a new framework for dealing with nonlinear sampled-data systems. We introduce a new idea of lifting, which is well known for linear systems, but not successfully generalized to nonlinear systems. This paper introduces a new lifting technique for nonlinear, time-invariant systems, which are different from the linear counterpart as developed in Bamieh et al. (1991); Yamamoto (1994) etc. The main difficulty is that the direct feedthrough term effective in the linear case cannot be generalized to the nonlinear case. Instead, we will further lift the state trajectory, and obtain an equivalent time-invariant discrete-time system with function-space input and output spaces. The basic framework, as well as the closed-loop equation with discrete-time controller, is given. As an application of this framework, we give a representation for the Koopman operator derived from the given original nonlinear system.

Optimal Control of Port-Hamiltonian Descriptor Systems with Minimal Energy Supply

We consider the singular optimal control problem of minimizing the energy supply of linear dissipative port-Hamiltonian descriptor systems. We study the reachability properties of the system and prove that optimal states exhibit a turnpike behavior with respect to the conservative subspace. Further, we derive a input-state turnpike towards a subspace for optimal control of port-Hamiltonian ordinary differential equations with a feed-through term and a turnpike property for the corresponding adjoint states towards zero. In an appendix we characterize the class of dissipative Hamiltonian matrices and pencils.

Earthquake input and state estimation for buildings using absolute floor accelerations

AbstractAfter earthquakes, structural response such as interstory drift is critical for accurate structural assessment for buildings. Typically, direct integration of absolute floor accelerations does not yield reliable floor displacements due to the long‐period drifts caused by noise, a widely acknowledged challenge. In this case, model‐based estimation strategies can be employed, which often require the ground input for better accuracy. However, in many cases the ground input may not be available for lack of instrumentation or even be unmeasurable due to soil‐structure interaction, hence needs to be estimated. Earthquake input estimation in this case is particularly challenging due to the lack of direct feedthrough term, leading to low observability of system input. As a result, input estimation is sensitive to modeling error, measurement noise, and incomplete measurements. To address this challenge, a hybrid strategy is proposed to estimate earthquake input, states, and acceleration response at unmeasured floors using limited absolute floor acceleration measurements. First, the earthquake input is estimated through a maximum a posteriori (MAP) estimation method, and then the estimated input is combined with Kalman filter to further estimate states and unmeasured responses. A comprehensive assessment was performed through a series of numerical and experimental tests including a comparative study with a popular online model‐based method. While the online method demonstrated certain sensitivity to modeling error and measurement noise due to weak observability, the proposed strategy showed robustness and accuracy under realistic and challenging conditions. Further verification is also performed using a real‐world building structure that experienced earthquake events.

A Setting Method of a Feedthrough Term for Pneumatic Isolation Tables Considering Difference between Bandwidth of Pressure Fluctuation and Mechanical Resonance

A Setting Method of a Feedthrough Term for Pneumatic Isolation Tables Considering Difference between Bandwidth of Pressure Fluctuation and Mechanical Resonance

Design of Robust Trackers and Unknown Nonlinear Perturbation Estimators for a Class of Nonlinear Systems: HTRDNA Algorithm for Tracker Optimization

A robust linear quadratic analog tracker (LQAT) consisting of proportional-integral-derivative (PID) controller, sliding mode control (SMC), and perturbation estimator is proposed for a class of nonlinear systems with unknown nonlinear perturbation and direct feed-through term. Since the derivative type (D-type) controller is very sensitive to the state varying, a new D-type controller design algorithm is developed to avoid an unreasonable large value of the controller gain. Moreover, the boundary of D-type controller is discussed. To cope with the unknown perturbation effect, SMC is utilized. Based on the fast response of SMC controlled systems, the proposed perturbation estimator can estimate unknown nonlinear perturbation and improve the tracking performance. Furthermore, in order to tune the PID controller gains in the designed tracker, the nonlinear perturbation is eliminated by the SMC-based perturbation estimator first, then a hybrid Taguchi real coded DNA (HTRDNA) algorithm is newly proposed for the PID controller optimization. Compared with traditional DNA, a new HTRDNA is developed to improve the convergence performance and effectiveness. Numerical simulations are given to demonstrate the performance of the proposed method.

Analysis and Computation of the $\mathcal {H}_2$ Norm of Delay Differential Algebraic Equations

We consider a class of dynamical systems described by linear delay differential algebraic equations (DDAEs) called strangeness free, which is broader than the class commonly studied within the control theory field. Two problems arise in the study of the $\mathcal {H}_2$ norm of DDAEs: the first one is that it may be infinite even if the system is stable or has no seemingly feedthrough term, and the second one is the computation. In this paper, both problems are addressed. We provide a necessary and sufficient condition for the finiteness of the $\mathcal {H}_2$ norm, which is based on controllability and observability properties of the delay-difference part of the system, and we present a formula for computing the $\mathcal {H}_2$ norm whenever it is finite, which is obtained by means of a neutral-type system whose transfer matrix is equivalent to the transfer matrix of DDAEs.

Lossless Systems Storage Function: New Results and Numerically Stable and Non-Iterative Computational Methods

In this paper, we formulate and prove new results in the context of storage functions for lossless systems: we use these results to propose new algorithms to compute the storage function. The computation of the storage function for the lossless case is not possible using conventional algebraic Riccati equation-based algorithms, though the storage function itself is well-defined. This is because a certain “regularity condition” on the feedthrough term in the i/s/o representation of the lossless system does not hold. We formulate new results about the storage function matrix for the lossless case and use them to propose non-iterative and stable algorithms to compute the storage function directly from different representations of the given system, namely, a kernel representation, transfer function, and the i/s/o representation of the system. Across the methods, for randomly generated transfer functions, we compare: 1) the computational effort (in flops); 2) the computation time using numerical experiments; and 3) the computational error.

An Input-Constrained NARMAX Model-Based Adaptive Tracker with Fault Tolerance for Unknown Systems with an Input-Output Direct Feed-Through Term

An Input-Constrained NARMAX Model-Based Adaptive Tracker with Fault Tolerance for Unknown Systems with an Input-Output Direct Feed-Through Term

Fractional order implementation of Integral Resonant Control – A nanopositioning application

By exploiting the co-located sensor-actuator arrangement in typical flexure-based piezoelectric stack actuated nanopositioners, the polezero interlacing exhibited by their axial frequency response can be transformed to a zero-pole interlacing by adding a constant feed-through term. The Integral Resonant Control (IRC) utilizes this unique property to add substantial damping to the dominant resonant mode by the use of a simple integrator implemented in closed loop. IRC used in conjunction with an integral tracking scheme, effectively reduces positioning errors introduced by modelling inaccuracies or parameter uncertainties. Over the past few years, successful application of the IRC control technique to nanopositioning systems has demonstrated performance robustness, easy tunability and versatility. The main drawback has been the relatively small positioning bandwidth achievable. This paper proposes a fractional order implementation of the classical integral tracking scheme employed in tandem with the IRC scheme to deliver damping and tracking. The fractional order integrator introduces an additional design parameter which allows desired pole-placement, resulting in superior closed loop bandwidth. Simulations and experimental results are presented to validate the theory. A 250% improvement in the achievable positioning bandwidth is observed with proposed fractional order scheme.

Stabilization of linear systems with direct feedthrough term in the presence of output saturation

This paper investigates the stabilization problem for controllable and observable linear systems with saturated outputs, in which direct feedthrough term is involved. The necessity of considering the direct feedthrough term is clarified. We present a constructive design strategy to achieve globally asymptotic stability of the systems by output feedback. The controller is continuous during the interval we drive a specific output component out of saturation in order to handle the direct feedthrough term. Moreover, the parameters to be designed for the controller have close relationship with the transfer function and direct feedthrough matrix. Numerical simulations are included to illustrate the effectiveness of theoretical results.

A Gauss–Seidel type solver for the fast computation of input-constrained control systems

An important class of nonlinear control systems can be represented as the feedback interconnection of two parts: a linear time-invariant system and a block of decentralized nonlinearities. When the linear time-invariant part has a nontrivial feedthrough term or the nonlinearity has a feedback gain, control computation involves the online implementation of a multivariable algebraic loop which must be resolved at each sampling instant. The requirements for such online computation may result in several implementation issues, especially in real-time and embedded control applications. This paper considers the implementation of such algebraic loops arising from several input-constrained systems. The proposed solution algorithm is globally convergent for a very large class of feedthrough or feedback gains and shows promise for real-time and embedded control applications where a fast but approximate solution is of the essence.

Optimal integral force feedback for active vibration control

This paper proposes an improvement to Integral Force Feedback (IFF), which is a popular method for active vibration control of structures and mechanical systems. Benefits of IFF include robustness, guaranteed stability and simplicity. However, the maximum damping performance is dependent on the stiffness of the system; hence, some systems cannot be adequately controlled. In this paper, an improvement to the classical force feedback control scheme is proposed. The improved method achieves arbitrary damping for any mechanical system by introducing a feed-through term. The proposed improvement is experimentally demonstrated by actively damping an objective lens assembly for a high-speed confocal microscope.

Optimal integral force feedback and structured PI tracking control: Application for objective lens positioner

This paper describes a new vibration damping technique based on Integral Force Feedback (IFF). Classical IFF utilizes a force sensor and integral controller to damp the resonance modes of a mechanical system. However, the maximum modal damping depends on the frequency difference between the system’s poles and zeros. If the frequency difference is small, the achievable modal damping may be severely limited. The proposed technique allows an arbitrary damping ratio to be achieved by introducing an additional feed-through term to the control system. This results in an extra degree of freedom that allows the position of the zeros to be modified and the maximum modal damping to be increased. The second contribution of this paper is a structured PI tracking controller that is parameterized to cancel the additional pole introduced by integral force feedback. The parameterized controller has only one tuning parameter and does not suffer from reduced phase margin. The proposed techniques are demonstrated on a piezoelectric objective lens positioner. The results show exceptional tracking and damping performance while maintaining insensitivity to changes in resonance frequency. The maximum bandwidth achievable with a commercial PID controller is 26.1Hz. In contrast, with the proposed damping and tracking controller, the bandwidth is increased to 255Hz.

An Analytical Approach to Integral Resonant Control of Second-Order Systems

Systems with colocated sensor-actuator pairs exhibit the interesting property of pole-zero interlacing. Integral resonance control (IRC) exploits this property by changing the pole-zero interlacing to zero-pole interlacing. The unique phase response of this class of systems enables a simple integral feedback controller to add substantial damping. Over the past few years, IRC has proven to be extremely versatile and has been applied to a wide variety of systems whose dominating dynamics of interest can be accurately modeled by second-order transfer functions. To date, a manual approach has been employed to determine the parameters of the IRC scheme, namely the feed-through term and the integral gain. In this paper, the relationship between the feed-through term, integral gain, and achievable damping is derived analytically for undamped/lightly damped second-order systems. The relationship between damping controller and an outer servo loop is also derived. These results add to the current understanding of colocated systems and automate the design of IRC controllers with a specified damping and tracking bandwidth. The presented results are applied to design and implement a damping and tracking controller for a piezoelectric nanopositioning stage.

On the feedforward control problem for discretized port-Hamiltonian systems

The boundary feedforward control problem for a class of distributed-parameter port-Hamiltonian systems in one spatial dimension is addressed. The considered hyperbolic systems of two conservation laws (with dissipation) are discretized in the spatial coordinate using an energy-based, structure preserving discretization scheme. The resulting finite-dimensional approximate state representation has a feedthrough term which allows to directly express the differential equation for the inverse dynamics. The inverse system needs to be solved in order to determine the control inputs for given desired output trajectories. For non-collocated pairs of boundary in- and outputs the magnitude of dissipation determines whether the inverse discretized models are stable or not. In the unstable case, the problem at hand can be attacked with classical approaches for the dynamic inversion of non-minimum phase systems.

Optimal Integral Force Feedback and Structured PI Tracking Control: Application for High Speed Confocal Microscopy.

In this paper, an improvement to Integral Force Feedback (IFF) damping control is proposed. The main limitation of Integral Force Feedback is that the maximum modal damping depends on the system's parameters. Hence, for some system achievable damping is insignificant. The proposed improvement allows any arbitrary damping ratio to be achieved for a system by introducing a new feed-through term in the system. To achieve displacement tracking, one technique is to enclose the system in an integral feedback loop. However, the bandwidth is limited due to the effects of an additional pole in the damping loop. The proposed Structured PI controller is parameterised so that it contains a zero that cancel the additional pole. Experiment was conducted on a commercial objective lens positioner. The results show an exceptional tracking and damping performance and the system's insensitivity to changes in resonance frequency. The maximum bandwidth achievable with a commercial PID controller is 26.1 Hz. In contrast, with the proposed method, the bandwidth is increased to 255 Hz.

A modified NARMAX model-based self-tuner with fault tolerance for unknown nonlinear stochastic hybrid systems with an input–output direct feed-through term

A modified nonlinear autoregressive moving average with exogenous inputs (NARMAX) model-based state-space self-tuner with fault tolerance is proposed in this paper for the unknown nonlinear stochastic hybrid system with a direct transmission matrix from input to output. Through the off-line observer/Kalman filter identification method, one has a good initial guess of modified NARMAX model to reduce the on-line system identification process time. Then, based on the modified NARMAX-based system identification, a corresponding adaptive digital control scheme is presented for the unknown continuous-time nonlinear system, with an input–output direct transmission term, which also has measurement and system noises and inaccessible system states. Besides, an effective state space self-turner with fault tolerance scheme is presented for the unknown multivariable stochastic system. A quantitative criterion is suggested by comparing the innovation process error estimated by the Kalman filter estimation algorithm, so that a weighting matrix resetting technique by adjusting and resetting the covariance matrices of parameter estimate obtained by the Kalman filter estimation algorithm is utilized to achieve the parameter estimation for faulty system recovery. Consequently, the proposed method can effectively cope with partially abrupt and/or gradual system faults and input failures by the fault detection.

A novel linear quadratic observer and tracker for the linear sampled data regular system with a direct feedthrough term: a digital redesign approach

This paper develops a novel linear quadratic observer and tracker for the linear sampled data regular system with a direct transmission term from input to output. At the beginning, we use some existing techniques to decompose the singular system into an equivalent regular system which has a direct transmission term from input to output. Then, based on this equivalent regular system, a high-gain optimal linear quadratic analogue observer and tracker is proposed, so that it can effectively induce a high-quality performance on the trajectory-tracking design and the state estimation. Besides, the prediction-based digital redesign method is utilized to obtain a relatively low-gain and implementable digital tracker and observer from the theoretically well-designed high-gain analogue observer and tracker for the linear sampled data regular system with a direct transmission term from input to output. Several illustrated examples are given to demonstrate the performance and effectiveness of the proposed method. 15

Convergence characteristics of proportional-type iterative learning control in the sense of Lebesgue- p norm

This article studies convergence characteristics of first- and second-order P-type (proportional-type) iterative learning control laws for a class of partially known linear time-invariant systems with a direct feed-through term. In the study, the tracking error is measured in the sense of Lebesgue-p norm, based on which the concepts of the Qp factor and the convergence speed are specified. For the first-order updating law, the monotone convergence is achieved by means of the generalised Young inequality of convolution integral. Moreover, for the second-order updating rule, the convergence is also verified and its speed is compared with the first-order law. Through analysis, it is quantitatively noted that both the system dynamics and the learning gains affect the convergence. It is also observed that the iterative learning process with the second-order law can be Qp-faster, Qp-equivalent or Qp-slower than the system with the first-order rule, depending on the selected learning gains. Numerical simulation manifests the validity and the effectiveness.