Linear Time-invariant Single-input Single-output Systems Research Articles

Pole Identification using discrete Laguerre expansion and variable projection

We propose a novel algorithm for identifying the poles of transfer functions describing SISO-LTI (single input single output, linear time invariant) systems. Our Identification method works in the frequency domain and consists of two parts. In the first part, we extend a discrete Laguerre expansion based method with an automatic parameter selection scheme. This allows us to find an initial estimate of the poles of SISO-LTI transfer functions without the need for human intuition. Then, in the second part, we propose a novel optimization problem to improve our initial estimates. The proposed optimization aims to reduce the least squared error of a parameterized model, which can be interpreted as an orthogonal projection of the system's frequency response onto a subspace spanned by Generalized Orthogonal Rational Basis functions (GOBFs). We solve the corresponding nonlinear optimization task using gradient based methods, where we can analytically calculate the gradient of the error functional. Through robust numerical experiments, we investigate the behavior of the developed methods and show that they work even in scenarios, when the transfer function has a high number of poles.

CSD Based DR Approach for the Solution of Differential Equation of LTI SISO System of Causal Type

Dimensional reduction (DR) approach is one of the way by which the problem of higher dimensional space can reduces to lower one. The main advantage of this DR approach is that the space-time interval (Δs – Δt); where Δs represents the computation space while Δt represents the computation time; requirement for it being relatively short. Within this paper the consecutive single derivative (CSD) based DR approach is consider for the solution of differential equation of linear time invariant (LTI) single input single output (SISO) system of causal type on the matrix ground. Besides this CSD based DR solution, the behavioural operator of the system is also discussed which represents the identity of a given system relating the input-output variables of desired.

An Analytical Approach to Tuning of Delay-Based Controllers for LTI-SISO Systems

This paper explores the deliberate introduction of delays in feedback loops and their positive impact on linear time-invariant, single-input, single-output (LTI-SISO) systems. The delay, considered as a design parameter, is intentionally induced using a class of delay-based controllers, aiming at improving closed-loop system performance by deploying a pole-placement technique to attain the fastest velocity of response to control commands. The technique, based on an algebraic geometric analysis, is able to extract exact tuning formulas for the controller gains and the delay.

Multi-objective iterative learning control using convex optimization

This paper presents a multi-objective iterative learning control (ILC) design approach that realizes an optimal trade-off between robust convergence, converged tracking performance, convergence speed, and input constraints. Linear time-invariant single-input single-output systems which are represented by both parametric and nonparametric models are considered. The noncausal filter Q(q) and learning function L(q) are simultaneously optimized by solving a convex optimization problem. The proposed method is applied to a non-minimal phase system and compared with a model-inversion based ILC design. Using the developed ILC design the underlying trade-off between tracking performance and convergence speed is thoroughly/quantitatively analyzed.

Creating two disjoint stability intervals along the delay axis via controller design: a class of LTI SISO systems

Frequency domain technique is adapted here together with an optimization scheme to design controllers for a class of linear time-invariant single-input single-output systems (SISO) with a loop delay, such that the closed loop system can be made stable in two distinct pre-selected delay intervals $$[0,h_{1})$$ and $$(h_{2},h_{3})$$ , where $$h_{1}<h_{2}<h_{3}$$ . This stability-certifying approach is demonstrated over case studies, including an experiment.

Some Open Problems on Simultaneous Stabilization of Linear Systems

Simultaneous stabilization of linear systems is a fundamental issue in the system and control theory, and is of theoretical and practical significance. In this paper, the authors review the recent research progress and the state-of-art results on simultaneous stabilization of single-input single-output linear time-invariant systems. Especially, the authors list the ever best results on the parameters involved in the well known “French Champagne Problem” and “Belgian Chocolate Problem” from the point of view of mathematical theoretical analysis and numerical calculation. And the authors observed that Boston claimed the lower bound of δ can be enlarged to 0.976461 in 2012 is not accurate. The authors hope it will inspire further study on simultaneous stabilization of several linear systems.

Singular perturbation margin and generalised gain margin for linear time-invariant systems

In this paper, we propose a singular perturbation margin (SPM) and a generalised gain margin (GGM) as stability metrics for single input–single output (SISO) linear time-invariant (LTI) systems from the view of singular perturbations and regular perturbations, which have bijective correspondences with the classical phase margin (PM) and the gain margin (GM), respectively. Both of the numerical and analytical time-domain SPM and GGM assessment methods are provided, and relationships between the singular perturbation parameter, PM of the perturbed system, PM and SPM of the nominal system, and the (monotonically increasing) phase of the fast system are also revealed. These results make it possible to assess the PM of the nominal system in the time domain for SISO LTI systems using the SPM with a standardised testing system called ‘PM-gauge,’ as demonstrated by examples. The concepts of SPM and GGM can be used as metrics of stability margins for linear time-varying systems and nonlinear systems.

A Modeling-Free Inversion-Based Iterative Feedforward Control for Precision Output Tracking of Linear Time-Invariant Systems

In this paper, we propose a modeling-free inversion-based iterative feedforward control (MIIFC) approach for high-speed output tracking of single-input single-output linear time-invariant systems. The recently developed inversion-based iterative learning control (IIC) techniques provide a straightforward manner to quantify and account for the effect of dynamics uncertainty on iterative learning control performance, thereby arriving at rapid convergence of the iterative control input. However, dynamics model and thereby the modeling process are still needed, and the model quality directly limits the performance of the IIC techniques. The main contribution of this paper is the development of the MIIFC algorithm to eliminate the dynamics modeling process, and significantly improve the tracking performance. The disturbance (measurement noise) effect on the tracking precision is addressed in the convergence analysis of the MIIFC algorithm. The allowable disturbance/noise level to guarantee the convergence is quantified in frequency domain, and the noise level can be estimated through the noise spectrum measured before the whole operation. The MIIFC technique is demonstrated by applying it to the output tracking of a piezotube scanner on an atomic force microscope. The experimental results showed that precision output tracking of a frequency-rich desired trajectory with power spectrum similar to a band-limited white noise can be achieved.

Formula omitted] control of fractional linear systems

In this paper, the standard H∞ control problem for continuous-time fractional linear time-invariant single-input–single-output systems is solved. The adopted approach consists of extending to the fractional case the procedure followed within the classical solution for the integer case. According to the classical route, we first consider the generalization to fractional systems of the standard Youla parameterization of all the stabilizing controllers. The H∞ optimal controller is then found among the class of stabilizing controllers, recasting the control problem into a model-matching one. The results obtained naturally extend well-established results to a fractional setting, including both the commensurate and noncommensurate case, thus providing a framework where H∞ design can be recast along the well-known and fruitful lines of the integer case. A worked-through example is discussed in detail to illustrate the design procedure.

An adaptive method for consistent estimation of real-valued non-minimum phase zeros in stable LTI systems

An adaptive algorithm, consisting of a recursive estimator for a finite impulse response model having two non-zero lags only, and an adaptive input are presented. The model is parametrized in terms of the first impulse response coefficient and the model zero. For linear time-invariant single-input single-output systems with real rational transfer functions possessing at least one real-valued non-minimum phase zero of multiplicity one, it is shown that the model zero converges to such a zero of the true system. In the case of multiple non-minimum phase zeros, the algorithm can be tailored to converge to a particular zero. The result is shown to hold for systems and noise spectra of arbitrary degree. The algorithm requires prior knowledge of the sign of the high frequency gain of the system as well as an interval to which the non-minimum phase zero of interest belongs.

Uncertainty and Disturbance Estimator–Based Control for Uncertain LTI-SISO Systems With State Delays

In this paper, a robust control strategy based on the uncertainty and disturbance estimator (UDE) is proposed for uncertain Linear Time Invariant-Single Input Single Output (LTI-SISO) systems with state delays. The knowledge of the bounds of uncertainties and disturbances is not needed during the design process although it is required for the stability analysis. Both the cases with known and unknown delays are considered. In the case of unknown delays, the terms involving the delays are treated as additional disturbances to the system. The robust stability of the closed-loop system is analyzed in detail, and a stability condition is proposed. Simulations are given to demonstrate the excellent tracking and disturbance rejection capabilities of the UDE-based control strategy.

Frequency Response Estimation for Mechanical Systems Using Closed-Loop Step Response Data

In this paper, a simple scheme to estimate frequency transfer functions of linear time-invariant single-input single-output systems based on closed-loop step response data is investigated. The method uses only a single set of step response data of plants under closed-loop operating conditions. The input and output of a plant are filtered by a simple bandpass filter, and the frequency transfer function of the plant is obtained with the Discrete Fourier Transforms of the filtered input and output signals. The method is applied to a resonant mechanical system and its validity is illustrated through simulations and experiments.

RST Controller Design by Convex Optimization Using Frequency-Domain Data

AbstractA new robust fixed-order RST controller design approach by convex optimization is proposed in this paper. Only frequency-domain data are considered in the approach for linear time-invariant single-input single-output systems. It is shown that performance specifications given as the infinity norm of the weighted sensitivity functions can be represented as convex constraints in the Nyquist diagram. One of the advantage of the proposed method is that it can deal directly with multi-model uncertainty. Moreover, a solution to a real industrial problem is given showing the effectiveness of the method.

Stabilization of a Class of Even Order Linear Time-Invariant Systems

This paper deals with the problem of stabilization of a class of linear time-invariant (LTI) even order controllable, single-input single-output (SISO) systems. The systems considered are open-loop unstable and in particular not stabilizable by static time-invariant output feedback. We propose a set of easily computable necessary and sufficient conditions in terms of scalar inequalities for the asymptotic stabilization of a class of n-dimensional even order LTI SISO systems using composite output feedback.

Fixed-order [formula omitted] controller design for nonparametric models by convex optimization

A new approach for robust fixed-order H ∞ controller design by convex optimization is proposed. Linear time-invariant single-input single-output systems represented by nonparametric models in the frequency domain are considered. It is shown that the H ∞ robust performance condition can be represented by a set of linear or convex constraints with respect to the parameters of a linearly parameterized controller in the Nyquist diagram. Systems with time-delay and with multimodel and frequency-domain uncertainty can be considered straightforwardly in the proposed approach. The proposed method is compared with the standard H ∞ control problem by a simulation example on an unstable system.

Calculation of All H∞ Robust Stabilizing Gains for SISO LTI Systems

Robust stabilization of continuous time single-input single-output (SISO) linear time invariant (LTI) systems with multiplicative uncertainties is considered in this paper. In particular, it has been shown that all gains that robustly stabilize a given uncertain SISO LTI system can be found by utilizing a generalization of the Nyquist theorem. The method proposed involves calculation of roots of two real polynomials and does not require any search or gridding over a parameter, and as a result offers computational advantages over existing methods in literature.

Performance enhancement of a class of nonlinear systems by disturbance observers

A large class of single-input single-output (SISO) nonlinear systems is considered. The nonlinearity in a system of this class has the property that its output can be decomposed as the summation of the outputs of a stable SISO linear time-invariant system and a bounded function of time. For the class of systems under consideration, disturbance observers are designed to estimate the effects of nonlinearities and to cancel them subsequently. The designed disturbance observers are thus able to make the nonlinear systems behave linearly and to enhance their performance. Examples are given to show disturbance observers can suppress the effect of dead-zone nonlinearity and limit cycles caused by backlash.

Eine Charakterisierung allgemeiner LTI-Systeme anhand der Approximierbarkeit durch FIR-Systeme

The main idea of the theory of discrete-time single-input single-output linear time-invariant systems (LTI-system) is that every such system has an input-output map that can be represented by the familiar convolution. Such a system is called system of convolution type. It was recently discovered by Boyd and Sandberg, that not all LTI-systems are systems of convolution type. Since not all LTI-systems are of convolution type it would be interesting to characterize the systems of convolution type. In this paper such a characterization is given with the help of the approximation behaviour of the system with respect to FIR-systems. A necessary and sufficient condition is given, such that the system is a system of convolution type. Some applications for the filter design are also described.

Cyclic higher-order statistics: Input/output relations for discrete- and continuous-time MIMO linear almost-periodically time-variant systems

A contribution to the theory of higher-order cyclostationarity very recently introduced to generalize the second-order cyclostationarity theory is given. Input/output relations in terms of cyclic higher-order statistics for multi-input multi-output linear almost-periodically time-variant systems that are excited by cyclostationary inputs are derived. Both continuous- and discrete-time systems are considered. For a single-input single-output linear time-invariant system, the Wiener system identification formula based on second-order cyclic spectra is generalized to higher-order statistics. The problem of reconstructing cyclic higher-order statistics of a continuous-time-series from its samples is addressed and a sufficient condition on the sampling rate to prevent aliasing is stated. Examples of application of the theory are included.

Design of stable controllers attaining low H/sup infinity / weighted sensitivity

Strong stabilization and sensitivity reduction for a linear time-invariant single-input single-output system are considered. Stable controllers are practically desirable in the presence of possible actuator or sensor failures. It is verified precisely that the optimal controller of H/sup infinity / weighted sensitivity minimization is almost always unstable when the plant has more than two zeros in the right-half plane. An algorithm for attaining the low sensitivity property for stable controllers is presented. It is based on Nevanlinna-Pick interpolation. >