Continuous Linear Time-invariant Systems Research Articles

Guaranteed Robust Performance of $\mathcal {H}_{\infty }$ Filters With Sparse and Low Precision Sensing

The performance of estimation algorithms depends on the available sensors and their precisions. While higher precisions of sensors provide better estimation accuracy, it may lead to unnecessarily expensive designs and higher operational costs due to higher costs of high-precision sensing modalities. Also, higher precision can cause interference for other sensors in the environment, such as RADARS, resulting in degradation in the overall system's performance. This paper presents a framework for co-designing a sparse sensing network with the least precise sensors and the filter that guarantees the prescribed <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {H}_{\infty }$</tex-math></inline-formula> estimation accuracy. Convex optimization problems for minimizing sensor precisions are formulated for continuous and discrete-time linear time-invariant systems, with and without model uncertainties. Different heuristics for determining a sparse sensor set with the least precise sensors are discussed, and their performances are compared using numerical simulations. The application of the proposed framework is demonstrated using numerical examples.

Identification of Linear Dynamic Systems in State Sate

System identification has been growing in an engineering community over the last 60 years, and an intensive research has been published in this field. This study presents an identification method of a linear time-invariant continuous dynamic system directly from time series data. The considered system is represented in a state space form. All states are assumed to be measured. The proposed identification algorithm is demonstrated and investigated, with noise-free data and noisy data, on a MATLAB simulation environment. The results confirm that the simple procedures of the proposed algorithm give an effective and successful estimation of the system parameter matrices.

An effective controller design strategy for higher order systems

The recent methods that guarantee the stability of the reduced-order models are Mihailov stability, improved Pade approximation and truncation, improved generalized pole clustering, moment matching and salp swarm optimization. Further, these methods could overcome the limitations such as non-uniqueness, pole clustering, gain adjustment and difficulty to maintain the dominant roots in the lower order system for non-minimum higher order plants. This research emphasizes the design of the proposed compensating algorithm by using an additional open loop zero for the stable reduced-order models of large-scale single-input single-output linear time-invariant continuous time systems. The results of the proposed algorithm are compared with the existing compensating methods and the design is validated and illustrated numerically.

Impulsive time series modeling with application to luteinizing hormone data.

This work considers the estimation of impulsive time series pertaining to biomedical systems and, in particular, to endocrine ones. We assume a signal model in the form of the output of a continuous linear time-invariant system driven by a sequence of instantaneous impulses, which concept is utilized here, in particular, for modeling of the male reproductive hormone axis. An estimation method to identify the impulsive sequence and the continuous system dynamics from sampled measurements of the output is proposed. Hinging on thorough mathematical analysis, the method improves upon a previously developed least-squares algorithm by resolving the trade-off between model fit and input sparsity, thus removing the need for manual tuning of user-defined estimation algorithm parameters. Experiments with synthetic data and Markov chain Monte-Carlo estimation demonstrate the viability of the proposed method, but also indicate that measurement noise renders the estimation problem ill-posed, as multiple estimates along a curve in the parameter space yield similar fits to data. The method is furthermore applied to clinical luteinizing hormone data collected from healthy males and, for comparability, one female, with similar results. Comparison between the estimated and theoretical elimination rates, as well as simulation of the estimated models, demonstrate the efficacy of the method. The sensitivity of the impulse distribution to the estimated elimination rates is investigated on a subject-specific data subset, revealing that the input sequence and elimination rate estimates can be interdependent. The dose-dependent effect of a selective gonadotropin releasing hormone receptor antagonist on the frequency and weights of the estimated impulses is also analyzed; a significant impact of the medication on the impulse weights is confirmed. To demonstrate the feasibility of the estimation approach for other hormones with pulsatile secretion, the modeling of cortisol data sets collected from three female adolescents was performed.

Practical Prescribed-Time Sampled-Data Control of Linear Systems With Applications to the Air-Bearing Testbed

This article studies the practical prescribed time sampled-data control problem for input-constrained linear time invariant continuous time systems. By using the solution to a parametric Lyapunov equation, a bounded linear time-varying (LTV) sampled-data controller is designed to solve such a problem. In addition, stability analysis and performance analysis of the closed-loop sampled-data control system are presented by using the methodology of scalarization. Finally, the developed LTV sampled-data state controller is utilized to the whole physical simulation system of air-bearing testbed and experimental results verify the effectiveness of the proposed approach.

Order reduction of linear time invariant large-scale system by preserving the impact of dominant poles in the reduced model

ABSTRACT This paper introduces a new reduction method for linear time-invariant continuous and discrete-time systems. The method is created considering only the effect of the dominance pole to get the reduced-order model. The denominator coefficient of reduced order model (ROM) is obtained by conserving the dominant poles of a higher order system while the numerator polynomial of the reduced model is found by comparing the original system transfer function with the transfer function of desired reduced order model. Five standard numeric examples are examined from literature considering single input single output (SISO) and discrete systems. To prove the validity, accuracy and simplicity of the proposed method, the results obtained are compared with other recent established and well-known techniques like Truncation and Routh Stability methods.

On σ-entropy Analysis of Linear Stochastic Systems in State Space

In this paper the problem of random disturbance attenuation capabilities in linear continuous systems is studied. It is supposed that the system operates under random disturbances with bounded σ-entropy level. σ-entropy norm indicates a performance index of the continuous system on the set of the random signals with bounded σ-entropy. This paper presents a time-domain solution to the calculation of σ-entropy norm of the continuous linear time-invariant system. σ-entropy norm is defined after solving coupled matrix equations: one algebraic Riccati equation, one nonlinear equation over log determinant function, and two Lyapunov equations.

Second order discrete time-varying and time-invariant linear continuous systems and Kalman type conditions

&lt;p style='text-indent:20px;'&gt;The paper deals with the controllability and observability of second order discrete linear time varying and linear time-invariant continuous systems in matrix form. To this case, we generalize the classical conditions for linear systems of the first order, without reducing them to systems of the first order. Within the framework of Kalman-type criteria, we investigate these concepts for second-order linear systems with discrete / continuous time; we define the initial values and input functions uniquely if and only if the observability and controllability matrices have full rank, respectively. Also a conceptual partner of controllability, that is, reachability of second order discrete time-varying systems is formulated and a necessary and sufficient condition for complete reachability is derived. Also the transfer function of the second order continuous-time linear state-space system is constructed. We have given numerical examples to illustrate the feasibility and effectiveness of the theoretical results obtained.&lt;/p&gt;

A class of optimal control problems governed by singular systems via Balakrishnan’s method

Abstract The purpose of this paper is to discuss, by the use of the Balakrishnan’s epsilon method, a class of optimal control problems governed by continuous linear time invariant singular systems which have only a finite dynamic mode. The linear differential algebraic equation is handled using the epsilon technique to obtain a sequence of the calculus of variations problems. A convergence theorem is given to obtain approximate and, in the limit, an optimal solution of this class of optimal control problem by the use of the necessary optimality conditions of Euler–Lagrange type. A correspondence has been also shown between this penalty function and duality for this class of optimal control problems considered. As an application, an example of optimal linear quadratic problem is also given.

Lower Order Approximation of Bounded-Parameter Uncertain Systems using Amplitude Matching and Whale Optimization Algorithm with Constriction Factor

Lower order modelling of uncertain bounded systems is presented in this paper using amplitude matching and Whale Optimization Algorithm (WOA) with Constriction Factor (CF). The amplitude matching technique ensures to retain the stability and dynamics of the original higher-order systems. The proposed optimization algorithm for minimum Integral Square Error (ISE) approximates the model so that the responses match closely. The results are validated by comparing the ISE values of existing models. The presented technique is applicable for both Linear Time Invariant Continuous Systems (LTICS) and Linear Time Invariant Discrete Systems (LTIDS) without any need for tedious calculation of domain transformation.

Interval set‐membership estimation for continuous linear systems

SummaryThis article proposes a mixed interval set‐membership estimation (ISME) method for continuous linear time‐invariant (LTI) systems by combining the positive system theory and the set theory. The proposed ISME method gives a new mixed interval‐set estimation framework for continuous LTI systems, whose benefit consists in that it has potential to achieve a balance of computational complexity and robust state estimation conservatism with respect to the interval observer (IO) and the set‐valued observer (SVO) for continuous LTI systems. Particularly, the proposed ISME method first uses a coordinate transformation such that the original system is transformed into an equivalent system. Second, the equivalent system is partitioned into two subsystems, where the first subsystem has a Meztler and Hurwitz subsystem matrix and then an IO is designed for the first subsystem based on the positive system theory. Because it is not guaranteed that the second subsystem also has a Meztler and Hurwitz subsystem matrix, a zonotopic SVO is further designed for the second subsystem based on the set theory. Consequently, an integration of the two steps above provides the whole SE results for the original system. At the end of this article, an example is used to illustrate the effectiveness of the proposed ISME method.

On Topological Properties of the Set of Stabilizing Feedback Gains

This article presents a fairly complete account on various topological and metrical aspects of feedback stabilization for single-input single-output (SISO) continuous and discrete time linear time-invariant (LTI) systems. In particular, we prove that the set of stabilizing output feedback gains for a SISO system with n states have at most [n/2] connected components. Furthermore, our analysis yields an algorithm for determining intervals of stabilizing gains for general continuous and discrete LTI systems; the proposed algorithm also computes the number of unstable roots in each unstable interval.

Singularity Excitations and Initial Value Problem in Continuous LTI Systems

A modern challenge in electrical engineering education is to keep the math at a sufficient level, with a goal to find an optimal balance between calculus competence and operative skills needed for real-life technical applications. It is not uncommon that some gaps emerge during this quest, which makes it difficult for undergraduates to entirely understand topics related on prior knowledge. This paper aims to draw attention to several important moments concerning total response of Continuous Linear Time Invariant systems, which are superficially or incorrectly explained in many textbooks, and to offer logical arrangement which can be easily understood and accepted by students. The base of discussion relies on classical calculus background, particularly on Picard’s theorem on existence and uniqueness. This theorem is rarely mentioned in signals and systems textbooks. However, mathematical models of many types of signals don’t satisfy the condition for continuity, which can easily produce difficulties in the learning process. It is shown that some reported disagreements and issues related to initial conditions, can be easily cleared out by using the smooth transition from classical calculus to mathematics used in system theory. It is also shown that the classical method is always the primary tool, even for determining the impulse response, while the impulse response is unnecessary or sufficient to determine the total system response, regardless of whether the convolution integral is used.

Some Remarks on the Regular Splitting of Quasi-Polynomials with Two Delays. Characterization of Double Roots in Degenerate Cases

This paper addresses the classification of multiple critical roots of dynamical continuous linear time-invariant systems including two constant delays in their mathematical representation. By considering the associated Weierstrass polynomial and its algebraic properties, we investigate the splitting behavior of such critical roots when the delays are subject to small variations. Some degenerate cases are also considered. The effectiveness of the proposed approach is illustrated through several numerical examples.

Sampling Strategies for Data-Driven Inference of Input–Output System Properties

Due to their relevance in controller design, we consider the problem of determining the $\mathcal{L}^2$-gain, passivity properties and conic relations of an input-output system. While, in practice, the input-output relation is often undisclosed, input-output data tuples can be sampled by performing (numerical) experiments. Hence, we present sampling strategies for discrete time and continuous time linear time-invariant systems to iteratively determine the $\mathcal{L}^2$-gain, the shortage of passivity and the cone with minimal radius that the input-output relation is confined to. These sampling strategies are based on gradient dynamical systems and saddle point flows to solve the reformulated optimization problems, where the gradients can be evaluated from only input-output data samples. This leads us to evolution equations, whose convergence properties are then discussed in continuous time and discrete time.

Reliable control of cyber-physical systems under sensor and actuator attacks: An identifier-critic based integral sliding-mode control approach

This study focuses on reliable control problems for cyber-physical systems (CPSs), described by linear continuous time-invariant systems, under a class of time-varying state-dependent sensor and actuator attacks. According to the state-dependent property of sensor attacks, the original reliable control problem is converted into a problem to stabilise a completely unknown time-varying system with the actuator attacks. Firstly, a dynamic neural network (NN) identifier with an indicator-function term is developed to model the unknown system dynamics. Due to the indicator term, the input distribution matrix of the identifier is guaranteed to be of full column rank. It provides a technical condition for designing an effective identifier-based integral sliding-mode (I-SM) attack compensator. It is shown that, under the proposed adaptive law, the identification errors and NN weight estimation errors are uniformly ultimately bounded. Furthermore, an identifier-critic based I-SM controller, which consists of an ADP-based controller and an adaptive I-SM attack compensator, is online learned using real-time control input and compromised sensor data, such that the unknown dynamics with the actuator attacks is stable with a nearly optimal performance. Meanwhile, it is shown that, the developed control approach ensures that the states in CPSs under sensor and actuator attacks converge to a compact set around zero. Finally, the effectiveness of the proposed one is verified by two illustrative examples.

Non-Existence of Convolution Sum System Representations

Convolution sum system representations are commonly used in signal processing. It is known that the convolution sum, treated as the limit of its partial sums, can be divergent for certain continuous signals and stable linear time-invariant (LTI) systems, even when the convergence of the partial sums is treated in a distributional setting. In this paper, we ask a far more general question: is it at all possible to define a generalized convolution sum with natural properties that works for all absolutely integrable continuous signals that vanish at infinity and all stable LTI systems? We prove that the answer is “no.” Further, for certain subspaces, we give a sufficient and necessary condition for uniform convergence. Finally, we discuss the implications of our results on the effectiveness of window functions in the convolution sum.

Linear Differential Equations and Related Continuous LTI Systems

In this paper, we consider the problem, usual in analog signal processing, to find a continuous linear time-invariant system related to a linear differential equation $$P(D)x = Q(D)f$$ , i.e. a system $$\mathscr {L}$$ such that for every input signal f yields an output $$\mathscr {L}(f)$$ which verifies $$P(D)\mathscr {L}(f ) = Q(D)f$$ . We give a systematic theoretical analysis of the existence and uniqueness of such systems (both causal and non-causal ones) defined on $${L^{p}}$$ functions and $${\mathscr {D'}_{L^{p}}}$$ distributions (input spaces which include signals with not necessarily left-bounded support), for every p. More precisely, by finding all their possible impulse responses, we characterise all these systems apart two pathologies arising when $$p = \infty $$ . Finally, we give necessary and sufficient conditions on P, Q for causality and stability of the systems. As an application, we consider the problem of finding the inverse of a causal continuous linear time-invariant system, defined on $${L^{p}}$$ , related to a simple differential equation. We also show a digital simulation of this inverse system.

Robust performance for uncertain systems via Lyapunov functions with higher order terms

It is proposed in this paper new linear matrix inequality (LMI) conditions to certify stability and robust performance in terms of the H∞ guaranteed cost for uncertain linear time-invariant continuous and discrete-time systems. The proposed conditions are based on the existence of a Lyapunov function that considers higher order state derivatives for continuous-time systems and higher order state shifts for discrete-time systems. Moreover, the Lyapunov candidate function depends on the uncertain system matrices and on the polynomial structure imposed to the candidate Lyapunov matrices. Benchmark examples from the literature illustrate the performance, in terms of scalar decision variables and LMI rows, of the proposed method when compared to other approaches.

Regulation of Linear Systems Using Event-Based Detection Sensors

In this paper, we investigate the problem of regulating a continuous-time linear time-invariant (LTI) system to a desired point using discrete event measurements from signal change detection sensors with a logarithmic event generation threshold (trigger), which includes the recently developed neuromorphic vision sensors. Existing control algorithms typically only process periodic measurements, and thus, a new class of algorithms needs to be developed that can efficiently process the sensors’ asynchronous discrete events for control tasks. Thus, we present a novel control design procedure that regulates the hybrid system, consisting of the continuous LTI system and a discrete-event signal change observation model, to a desired set point. Moreover, we provide the set of thresholds (sufficient conditions) for the given system to fulfill the regulation task. The effectiveness of our approach is illustrated on an unstable system.