Time-invariant Control Systems Research Articles

Stability analysis of hybrid integrator-gain systems: A frequency-domain approach

The hybrid integrator-gain system (HIGS) has been introduced recently with the aim to overcome fundamental limitations of linear time-invariant (LTI) control systems. To support the analysis and design of HIGS-based controllers, in this paper a novel frequency-domain condition for stability analysis of the feedback interconnection of an LTI system and HIGS is presented. Compared to existing frequency-domain stability conditions such as the one extending the circle-criterion, the condition presented in this paper exploits explicit knowledge regarding HIGS’ switching strategy, thereby potentially providing a significantly less conservative condition. In particular, the novel condition in this paper guarantees the existence of a quadratic Lyapunov function that does not need to be positive definite within the full state space. The proposed condition can be verified graphically in a manner that is reminiscent of the classical Popov plot, as will be illustrated in an experimental case-study.

Controllability and Observability of Linear Time-Invariant Control System on Superspace

Controllability and Observability of Linear Time-Invariant Control System on Superspace

Representations and Regularity of Vector-Valued Right-Shift Invariant Operators Between Half-Line Bessel Potential Spaces

Representation and boundedness properties of linear, right-shift invariant operators on half-line Bessel potential spaces (also known as fractional-order Sobolev spaces) as operator-valued multiplication operators in terms of the Laplace transform are considered. These objects are closely related to the input–output operators of linear, time-invariant control systems. Characterisations of when such operators map continuously between certain interpolation spaces and/or Bessel potential spaces are provided, including characterisations in terms of boundedness and integrability properties of the symbol, also known as the transfer function in this setting. The paper considers the Hilbert space case, and the theory is illustrated by a range of examples.

A combined method for stability analysis of linear time invariant control systems based on Hermite‐Fujiwara matrix and Cholesky decomposition

AbstractIn this paper, we have developed an integrative method for checking the stability of linear time‐invariant (LTI) systems as well as nonlinear continuous‐time ones. In our method, we first apply the iterative Faddeev–Leverrier algorithm to obtain the characteristic polynomial of the LTI system. Subsequently, the associated Hermite‐Fujiwara matrix will be evaluated by a particularly efficient technique for the calculation of the Bézoutian matrices. The positive‐definiteness of the Hermite‐Fujiwara form, as the stability criterion, is next tested by performing the Cholesky decomposition. Our method is extended to assess the local stability of nonlinear continuous‐time systems with the help of the Jacobian matrix concept. The proposed method is demonstrated to approximately be 2.2 times faster than the classical Hurwitz algorithm in average, at least for matrices with dimensions less than 40, according to a performed central processing unit (CPU) time analysis. For the sake of illustration, four numerical examples are given, including dynamical models for a real‐world hydrolysis reactor and a well‐mixed bioreactor.

Topological equivalence of linear time-varying control systems

Abstract In this paper, we mainly studied the topological equivalence of linear time-varying (LTV) control system $\dot{x}\left ( t\right )=A(t)x(t)+B(t)u(t)$ defined on an interval $I \subset \mathbb{R}^{+}$. After giving a new definition of the topological equivalence, we investigated the local equivalence of LTV control systems under two new hypotheses. These hypotheses were made by the local behavior of Krylov indices (which turned out to be controllability indices for the linear time-invariant (LTI) control systems). It was found out that Krylov indices play an important role in the classification problem of LTV control systems. Compared with our former work on the topological equivalence of LTI control systems, new methods and techniques were taken to deal with new difficulties occurred for LTV control systems.

Contraction analysis of Volterra integral equations via realization theory and frequency-domain methods

<p style='text-indent:20px;'>Realization theory for linear input-output operators and frequency-domain methods for the solvability of Riccati operator equations are used for the contraction analysis of a class of nonlinear Volterra integral equations in some Hilbert space. The key idea is to consider, similar to the Volterra equation, a time-invariant control system generated by an abstract ODE in some weighted Sobolev space which has the same stability properties as the Volterra equation.</p>

Stealthy attacks and attack-resilient interval observers

Industrial control systems have been frequent targets of cyber attacks during the last decade. Adversaries can hinder the safe operation of these systems by tampering with their sensors and actuators, while ensuring that the monitoring systems are not able to detect such attacks in time. This paper presents methods to design and overcome stealthy attacks on linear time-invariant control systems that estimate their state using an interval observer, in the presence of unknown but bounded noise and perturbations. We analyze scenarios in which a malicious agent compromises the sensors and/or the actuators of the system with additive attack signals to steer the state estimate outside of the bounds provided by the interval observer. We first show that maximally disruptive attack sequences that remain undetected by a linear monitor can be computed recursively via linear programming. We then design an attack-resilient interval observer for the system’s state, identifying sufficient conditions on the sensor data for such an observer to be realizable. We propose a computational method to determine optimal observer gains using semi-definite programming and compute bounds for the unknown attack signal as well. In numerical simulations, we illustrate and compare the ability of such interval observers to still provide accurate estimates when under attack.

Distinguishability criteria of conformable hybrid linear systems

Abstract We relate this article to the emerging idea of distinguishability of conformable linear hybrid time-invariant control systems. To obtain the necessary and sufficient conditions of α \alpha -distinguishability for fractional cases, we develop the Leibnitz rule for conformable derivatives. Furthermore, with the help of a study of Laplace techniques, a more simple criterion of α \alpha -distinguishability for the fractional linear system is developed.

Decoupling fault detection vectors for fault detection in the TS fuzzy system

ABSTRACT The parity space method can be applied to detect the fault of the time-invariant control system simply and effectively. However, the parity space method is greatly limited in the time-variant system. Therefore, in this paper, a TS fuzzy system is used to describe this kind of system. The system parameters are obtained based on TS fuzzy rules, and the decoupled fault detection vectors for different channels are designed. Using this vector, the residual representing fault information is extracted directly. Ideally, the fault detection of different channels is decoupled. However, sometimes the fully decoupled detection vector cannot be designed, then a suboptimal decoupled fault detection vector is proposed. Simulation and verification are carried out on a steady-state system and a TS fuzzy system.

An Application of the Z-Transform in Extracting the Density Regulation from the Periodic Output of an Age-Structured Population Model

The application of the Z-transform, a manipulation tool from the discrete signal processing (DSP) toolbox, on an ecological model was motivated by the mathematical similarities between an age-structured fish population model with a non linear density regulation and a linear time invariant (LTI) control system. Both models include a switching mechanism in regulating stock/signal throughput in accordance with a given density limitation/set value and both models can be expressed in terms of a negative feedback loop difference equations (Getz & Haight,1989; °Astr¨om & Murray, 2008). In the fish model, the switching mechanism is a density regulated stock-recruitment (SR) function which models the strategies implemented by the population in keeping the vulnerable egg-larvaejuvenile densities within an environmental limitation thereof (Subbey et al, 2014). A switching mechanism is also present in control engineering, for example, in the mechanism associated with cruise control in cars which keeps traveling speed close to a chosen set value midst varying weather and road conditions (Antsaklis and Gao, 2005). In both cases, the choosing of the control action and the tuning of its parameters requires careful consideration to avoid failures such as incorrectly timed switching actions in a control plant (see Kuphaldt (2019)) and errors in estimating total allowable catch (TAC) in the fishing industry (see Borlestean et al (2015), Skagen et al (2013) and Taboadai and R. Anadn (2016)). The Z-transform has proven itself useful in tuning LTI controlmodels for a desired control action (see Orfanidis, (2010) and Smith, (1999)) and it is on this account that its application was extended to the ecological model in pursuit of a more efficient way of estimating SR parameters to simulate an already existing output. It was however found that it could not be used for parameter tuning but rather for the extraction of the SR component hidden in the output together with components resulting from the age structure itself. Such an extraction can greatly assist in the mathematical identification of the SR, reducing the complexity of its choosing as there are many different types used in the fishing industry such as the classic Beverton-Holt model, the Ricker model and Shepherd model (Myers, 2001; Iles, 1994; Shepherd, 1982). It can also be used to monitor changes in the SR over time which can indicate the presence of strategy evolution (Apaloo et al, 2009; Br¨annstr¨om et al, 2013). In 1998 Schoombie and Getz investigated the latter by subjecting the Shepherd SR to strategy optimization with regards to a parameter associated with population interventions in regulating recruitment throughput and it is because of this versatility that the Shepherd SR is chosen for the intended extraction. In true control style, Simulink, a graphic environment for designing control simulations, is used to visualize the production of the output as well as the extraction of the SR from it. This paper showcases the versatility of the Z transform and the possibilities and unexpected finds when applied to similar systems designed to regulate signals or, in this case, recruitment densities.

Reset Control of Parallel MISO Systems

The proportional–integral plus Clegg integrator (PI + CI) controller is a hybrid extension of the proportional–integral (PI) controller that is able to overcome fundamental limitations of the linear and time-invariant control systems, potentially obtaining faster responses without increasing overshooting. This work focused on the analysis and design of PI + CI controllers and reset controllers in general, for the case of parallel multiple-input single-output (MISO) systems, extending previous design methods developed for the single-input single-output (SISO) case. Several design strategies were developed: one for first-order MISO plants achieving a flat response with a finite settling time, and for second-order MISO plants obtaining a fast response with a reduced overshoot and settling time in comparison with non-hybrid strategies. Several case studies were also developed to illustrate the potential of the proposed methods.

On the geometry of the set of control matrices of linear systems with prescribed controllability indices

Given a square matrix, F, the geometry of the set of control matrices G such that the linear time-invariant controllable system (F,G) has prescribed controllability or, equivalently, Brunovsky indices is investigated. This set is proved to be a differentiable manifold, its dimension is computed and a local coordinate system and a local parametrization are provided.

Frequency-domain stability conditions for asynchronously sampled decentralized LTI systems

This paper deals with the exponential stability analysis of decentralized, sampled-data, Linear Time Invariant (LTI) control systems with asynchronous sensors and actuators. We consider the case where each controller in the decentralized setting has its own sampling and actuation frequency, which translates to asynchrony between sensors and actuators. Additionally, asynchrony may be induced by delays between the sampling instants and actuation update instants as relevant in a networked context. The decentralized, asynchronous LTI system is represented as the feedback interconnection of a continuous-time LTI system operator and an operator that captures the effects of asynchrony induced by sampling and delay. By characterizing the properties of the operators using small-gain type Integral Quadratic Constraints (IQC), we provide criteria for exponential stability of the asynchronous, decentralized LTI state-space models. The approach provided in this paper considers two scenarios, namely the ‘large-delay’ case and the ‘small-delay’ case where the delays are larger and smaller than the sampling interval, respectively. The effectiveness of the proposed results is corroborated by a numerical example.

Matrix eigenvalue spectrum assignment for linear control systems by static output feedback

For a linear time-invariant control system defined by a linear differential equation of the n-th order with a multidimensional state, input and output, we set and study the problem of arbitrary matrix eigenvalue spectrum assignment by linear static output feedback. We obtain controllability-like rank conditions that are necessary and sufficient for the problem of arbitrary matrix eigenvalue spectrum assignment and are sufficient for the problem of arbitrary eigenvalue spectrum assignment by linear static output feedback. It is proved that, in particular cases, when the system has block scalar matrix coefficients, these conditions can be relaxed. The obtained results generalize the known results for the corresponding problem by static state feedback and by static output feedback in the case of the one-dimensional equation. Examples are presented to illustrate the results.

On the observability of conformable linear time-invariant control systems

<p style='text-indent:20px;'>In this paper, we analyze the concept of observability in the case of conformable time-invariant linear control systems. Also, we study the Gramian observability matrix of the conformable linear system, its rank criteria, null space, and some other conditions. We also discuss some properties of conformable Laplace transform.</p>

On Assignment of the Upper Bohl Exponent for Linear Time-Invariant Control Systems in a Hilbert Space by State Feedback

We consider a linear continuous-time control system with time-invariant linear bounded operator coefficients in a Hilbert space. The controller in the system has the form of linear state feedback with a time-varying linear bounded gain operator function. We study the problem of arbitrary assignment for the upper Bohl exponent by state feedback control. We prove that if the open-loop system is exactly controllable then one can shift the upper Bohl exponent of the closed-loop system by any pregiven number with respect to the upper Bohl exponent of the free system. This implies arbitrary assignability of the upper Bohl exponent by linear state feedback. Finally, an illustrative example is presented.

ON EQUIVALENCE OF LINEAR CONTROL SYSTEMS AND ITS USAGE TO VERIFICATION OF THE ADEQUACY OF DIFFERENT MODELS FOR A REAL DYNAMIC PROCESS

A problem of description of algebraic invariants for a linear control system that determine its structure is considered. With the help of these invariants, the equivalence problem of two linear time-invariant control systems with respect to actions of some linear groups on the spaces of inputs, outputs, and states of these systems is solved. The invariants are used to establish the necessary equivalence conditions for two nonlinear systems of differential equations generalizing the well-known Hopfield neural network model. Finally, these conditions are applied to establish the adequacy of two neural network models designed to describe the behavior of a real dynamic process given by two different sets of time series.

A new Solution of the Fractional-order Singular Linear Time-Invariant System

A new solution, the distributional solution of the fractional-order singular linear time-invariant control system (FOSLTIS) is studied in the paper. On the basis of proposing the definition of the Caputo fractional derivatives of Dirac function δ(t), the Laplace transform was obtained, then was derived. With regard to FOSLTIS, the system was decomposed into fast subsystem and slow subsystem by restricted equivalent transformation. Using Laplace transformation and taking into account, distributional solution for the fast subsystem was obtained. Two examples illustrated the correctness of the distributional solution of the FOSLTIS.

Discrete Time Pontryagin Maximum Principle Under State-Action-Frequency Constraints

We establish a Pontryagin maximum principle for discrete time optimal control problems under the following three types of constraints: a) constraints on the states pointwise in time, b) constraints on the control actions pointwise in time, and c) constraints on the frequency spectrum of the optimal control trajectories. While the first two types of constraints are already included in the existing versions of the Pontryagin maximum principle, it turns out that the third type of constraints cannot be recast in any of the standard forms of the existing results for the original control system. We provide two different proofs of our Pontryagin maximum principle in this article, and include several special cases fine-tuned to control-affine nonlinear and linear system models. In particular, for minimization of quadratic cost functions and linear time invariant control systems, we provide tight conditions under which the optimal controls under frequency constraints are either normal or abnormal.

АДАПТИВНОЕ ОЦЕНИВАНИЕ И УПРАВЛЕНИЕ ДЛЯ СЛОЖНЫХ СИСТЕМ:ПРОШЛЫЕ ДОСТИЖЕНИЯ И НЕРЕШЕННЫЕ ЗАДАЧИ

Engineering systems’ adaptability to the unknown and unexpectedly changing equip-ment condition and environmental data is provided artificially by equipping the main loop of control with the adaptor loop. The paper characterizes the current status of re-search and practical studies in the field. As shown for estimation and control systems, the adaptor’s task is to ensure the efficient and conforming solution of the following three problems. Problem 1 is the quickest change point detection in the acting model relative to the previous model. Problem 2 is the unbiased and adequately quick identification of the acting model. Problem 3 is the computationally efficient modifying the feedback in the main loop of control. The paper characterizes the current status of research and prac-tical studies in the field over many publications and some author’s developments. Among the latter, there is the active principle of adaptation for model identification in the class of linear time-invariant control systems.