Continuous Linear Systems Research Articles

Control and Filtering Problems for Linear Time-Varying Systems Based on Ellipsoidal Reachable Sets

The paper is devoted to reachable sets of linear time-varying continuous systems under uncertain initial states and disturbances with a bounded uncertainty measure. The uncertainty measure is the sum of a quadratic form of the initial state and the integral over the finite-time interval from a quadratic form of the disturbance. It is shown that the reachable set of the system under this assumption is an evolving ellipsoid with a matrix being a solution to the linear matrix differential equation. This result is used to synthesize the optimal observer providing the minimal ellipsoidal set as the estimate of the system state, as well as optimal controllers steering the system state into a final target ellipsoidal set or keeping the entire system trajectory in a prescribed ellipsoidal tube under all admissible initial states and disturbances. The relationship between the optimal ellipsoidal observer and the Kalman filter are established. Numerical modeling with the Mathieu equation for parametric vibrations of a linear oscillator illustrates the results.

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.

Robust Anomaly Detection Based on a Dynamical Observer for Continuous Linear Roesser Systems

Monitoring of industrial systems for anomalies such as faults and cyber-attacks as unknown and extremely undesirable inputs in the presence of other inputs (like disturbances) is an important issue for ensuring the safety and the reliability of their operation. In this study, a robust anomaly detection filter is proposed for continuous linear Roesser systems using dynamic observer framework. Sufficient conditions for the existence of the observer and its sensitivity to anomaly as well as its robustness to disturbances are addressed via linear matrix inequalities (LMIs). The mentioned sensitivity and robustness are based on the H_ and H∞ performance indices, respectively. Finally, the performance of the proposed observer is demonstrated through a numerical example.

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.

Вибiркова неперервнiсть з iмовiрнiстюодиниця для оцiнки iмпульсноїперехiдної функцiї

The problem of estimation of a stochastic linear system has been a matter of active research for the last years. One of the simplest models considers a ‘black box’ with some input and a certain output. The input may be single or multiple and there is the same choice for the output. This generates a great amount of models that can be considered. The sphere of applications of these models is very extensive, ranging from signal processing and automatic control to econometrics (errors-in-variables models). In this paper a time-invariant continuous linear system is considered with a real-valued impulse response function. We assume that impulse function is square-integrable. Input signal is supposed to be Gaussian stationary stochastic process with known spectral density. A sample input–output crosscorrelogram is taken as an estimator of the response function. The conditions on sample continuousness with probability one for impulse response function are investigated.

SYNTHESIS OF THE MATHEMATICAL MODEL OF THE AUTOMATIC CONTROL SYSTEM OF THE UNMANNED AIRCRAFT COURSE

Unmanned aerial vehicles are by far the most promising military and civilian systems. There is a tendency to increase the efforts of a number of leading countries in the development of unmanned aerial vehicles and their complexes. The mathematical model of any system reflects in one way or another its real properties, including the existing limitations. It has been found that one of the most favorable and efficient methods for constructing mathematical models of automatic control systems is to develop them using transfer functions. In order to solve this problem, the article deals with the composition of the control system of a drone. A mathematical model consisting of the joint design of the unmanned aerial vehicle and its automatic control system has been synthesized. The description of the proposed mathematical model of the system is based on the representation of a linear continuous system by the difference equations obtained using the Tustin relation. The mathematical model proposed in the article can be used for the study of typical aircraft whose course management system is built according to the considered structure. The practical significance of the obtained results is the possibility of applying the developed mathematical model to study the dynamics of the change of state and to set up the system of automatic control of the course of the unmanned aerial vehicle through computer simulation. Prospects for further research in this area are computer simulation of an unmanned aerial vehicle control system and estimation of the accuracy of the mathematical model developed.

Characteristic Modeling Approach for High-Order Linear Dynamical Systems

This article presents a full mathematical proof of the characteristic modeling approach for high-order linear dynamical systems. It explores the nature of the characteristic model in rigorous mathematical forms, also showing why and how the high-order dynamics can be compressed into the lower-order characteristic model. The relationships between high-order linear continuous dynamical systems, discrete-time characteristic model coefficients, and sampling-time intervals are investigated.

Bifurcations from a center at infinity in 3D piecewise linear systems with two zones

We consider continuous piecewise linear systems in R3 with two zones under the assumption of having a linear center in the invariant manifold of the point at infinity. A specific conic projection is introduced, so that it is possible to analyze in a convenient way the dynamics near such a center at infinity in two qualitative different situations.In the semi-homogeneous case, such a center is associated to the existence of a continuum of invariant semi-cones sharing the vertex at the origin; perturbing the configuration it is possible to detect the bifurcation of a limit cycle at infinity leading to the bifurcation of isolated invariant semi-cones.For the non-homogeneous case, the cycles at infinity does not imply invariant semi-cones. However, the non-generic case when the center at infinity is associated to the existence of invariant cylinders for one of the involved vector fields, becomes rather interesting. It is possible then, by perturbing the other vector field, to get the bifurcation of a big limit cycle from infinity without destroying the center.

A common Lyapunov function for a differential inclusion with matrices having small commutators

Let A1 and A2 be asymptotically stable matrices. We consider switched continuous time linear systems described by the differential inclusion ẋ(t)={y(t):y(t)=Ax(t),A∈{A1,A2}}. The paper demonstrates that a common quadratic Lyapunov function exists for such systems, provided the commutator [A1,A2]=A1A2−A2A1 has a sufficiently small norm. Our results generalize the ones of Narendra and Balakrishnan (1994).

Adaptive iterative learning control for unknown linear time-varying continuous systems

This paper presents a novel model reference adaptive iterative learning control (ILC) for unknown continuous-time linear time-varying systems. The unknown time-varying parameters of the system are neither required to vary slowly nor to have known bounds. The system is not required to be minimum-phase, stable, controllable or observable. The input of the system is determined by a differentiator-free control law. The used reference model is time-invariant and first order and thus choosing its parameters is easily possible, even though, the system under control is high order and time variant. Almost all of the components of the system initial condition can be iteration variant. By introducing a novel kind of Lyapunov function the convergence of the proposed adaptive ILC (AILC) and achieving asymptotic tracking are proved. Also, by rigorous mathematical analysis and with the help of some mathematical key techniques such as Bellman-Gronwall lemma, it is shown that all signals and quantities in the closed-loop system are bounded in the sense of at least one norm. Finally, the effectiveness of the proposed method is verified by two simulation examples.

Applications of Linear Systems in Contemporary Urban Design

One of the fundamental disputes in the contemporary urban design is principal of shaping urban systems. The main focus of disputes and research is the clash of urban interior (and its multiplication – residential quarter) systems with freestanding building urbanism. The issue is also discussed from perspective of neighbourly bonds, contact with nature and access to transport. The concept of linear systems was supposed to be an alternative solution to the dispute. The father of the idea of linear cities was Arturo Soria y Mata. The first drafts about the concept appeared in 1882. Its priorities were based on transport accessibility and green zones. In the Polish urban design, the original concepts of linear systems were created by Oskar Hansen (Linear Continuous System) and Włodzimierz Gruszczyński (Ribbon City of Conjugate Transportation) at the turn of 60’s into 70’s. The theory of Hansen predicts creation of four stripes North-South across Poland. The theory was based on demographical and geographical research. To certain extend, Hansen managed to achieve some aspects of the theory. The examples can be seen in the housing estate Przyczółek Grochowski in Warszawa and in the housing estate Lubelska Spółdzielnia Mieszkaniowa in Lublin. The article tackles upon the strengths and weaknesses of linear systems. Their main assets are: transport accessibility, contact with greenery and ability to counteract the urban sprawl. However, the basic weaknesses are schematism and dullness of buildings and the problem of residential cell divided into two parts by long transport belt. Another discussed aspect is comparison between urban interior systems and linear systems. Transforming the structure of linear systems includes three processes – reduction of the scale of systems, naturalisation of the central transport belt and individualisation of the structure of building range. One of the methods opposing the dullness is integration of various types of buildings – “hybridisation.”Besides the enumerated above transformation processes, I think that it is crucial to break the schematism of the linear systems. The parallel belts of residential areas should be replaced by networks. Such networks would leave the enormous fields to untamed and natural space.

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.

Preview control for a class of linear stochastic systems with multiplicative noise

ABSTRACT In this paper, the preview control problem for a class of linear continuous time stochastic systems with multiplicative noise is studied based on the augmented error system method. First, a deterministic assistant system is introduced, and the original system is translated to the assistant system. Then, the integrator is employed to ensure the output of the closed-loop system tracking the reference signal accurately. Second, the augmented error system, which includes integrator vector, control vector and reference signal, is constructed based on the system after translation. As a result, the tracking problem is transformed into the optimal control problem of the augmented error system, and the optimal control input is obtained by the dynamic programming method. This control input is regarded as the preview controller of the original system. For a linear stochastic system with multiplicative noise, the difficulty being unable to construct an augmented error system by the derivation method is solved in this paper. And, the existence and uniqueness solution of the Riccati equation corresponding to the stochastic augmented error system is discussed. The numerical simulations show that the preview controller designed in this paper is very effective.

Extended adaptive optimal control of linear systems with unknown dynamics using adaptive dynamic programming

AbstractThe extended infinite horizon optimal control problem of continuous time linear systems with unknown dynamics is investigated in this paper. This optimal control problem can be solved using the corresponding extended algebraic Riccati equation. A new policy iteration algorithm is proposed to approximate the solution of the extended algebraic Riccati equation when the system dynamics are known. The convergence of the proposed algorithm is proved. Based on the proposed policy iteration algorithm, an online adaptive dynamic programming (ADP) algorithm is developed to find the solution to the extended infinite horizon optimal control problem of unknown continuous time linear systems. The convergence of the online ADP algorithm is analyzed. Finally, two simulation examples are given to demonstrate the effectiveness of the developed online ADP algorithm.

Positivity of continuous time irregular linear descriptor systems

Positivity of continuous time irregular linear descriptor systems

Stabilization of a Class of Hybrid Systems by Switching Controllers with Input Constraints

In this paper, a hybrid control strategy using appropriate switching between a set of linear controllers is presented for stabilization of a class of hybrid systems with input constraints. The input constraints under consideration are novel reverse polytopic constraints avoiding a set of specified input values. The design procedure of the switching controllers is performed by solving a set of linear matrix inequalities for the full state feedback case and by solving a set of bilinear matrix inequalities for the output feedback case. Under the designed controllers, the closed-loop system is shown to be globally uniformly pre-asymptotically stable if a specific set of matrix inequalities ensuring the existence of a common Lyapunov function are satisfied. To avoid the Zeno behavior, a tunable parameter related to the controller gains is introduced and assessed. It is shown that the proposed switching control is superior to continuous state feedback control with input constraints. In addition, besides continuous linear systems, the proposed control strategy is applicable to hybrid or impulsive systems. Numerical examples are included to illustrate the proposed results.

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.

State estimators and observers for continuous and discrete linear systems. Part 2. Integral observers for exact state reconstruction

In the paper, the exact state observers will be presented. The state estimators and observers can be used in technical processes for many purposes like the fault detection and diagnosis, the implementation of the state controllers, and soft reconstruction of inaccessible for measurements variables of the system. As the standard, for continuous systems the differential estimators of Kalman filter or Luenberger type observer are commonly used. However, if the initial conditions of the real state are unknown, both estimators guarantee only an asymptotic quality of the real state tracking. The paper presents another type of the state observers, which for continuous system have the structure given by two integral operators. Based on measurements of the system input and output signals on some predefined finite time interval T, they can reconstruct the initial state exactly. In on-line version, the exact state reconstruction is performed continuously for every t, based on special procedure executed within two moving windows of width T, on sliding time interval [t-T, t].

Performance boundary mapping for continuous and discrete time linear systems

Mapping of performance specification into parameter space for LTI systems is discussed from the perspective of the Lyapunov equation. After setting the main results on stability, corresponding formulations for a variety of performance criteria (e.g. settling time, damping and frequency-based criteria) are discussed. Thereby, the original performance mapping problem is transformed to a stability problem with modified system matrices that include free control and/or design parameters. The proposed performance mapping approach is demonstrated on two benchmark case studies of practical interest.

New results on restricted static output feedback controller design with regional pole placement

This study addresses the design of static output feedback (SOF) controller for continuous time linear systems. New sufficient conditions are derived for designing SOF controllers. The development involves suitable decomposition of Lyapunov matrices and deriving linear matrix inequality (LMI) criterion that ensures H ∞ performance. LMI criteria for pole placement in the LMI region are also derived. The proposed criteria can be used for restricted SOF controller design effectively. Several numerical examples are presented to demonstrate the effectiveness of the proposed results.