Continuous-time Systems Research Articles

Synchronizations control of fractional-order multidimension-valued memristive neural networks with delays

In this paper, systems of the fractional-order multidimension-valued memristive neural networks (FMVMNNs) with leakage delay and time-varying delays are studied in continuous-time and discrete-time cases. Comparative studies on synchronizations control between discrete-time FMVMNNs and the continuous-time system are investigated directly rather than through the decomposition method. By employing Lyapunov theory and fractional derivative inequalities, sufficient criteria about the quasi-synchronization and global ultimate Mittag-Leffler lag quasi-synchronization are obtained in multidimension-valued linear matrix inequality form, while succinct conclusions of asymptotical synchronization and global Mittag-Leffler synchronization are derived in vector form. The obtained criteria have many advantages in a less computation, lower conservatism, more generality and higher flexibility. Finally, four examples are given to demonstrate the effectiveness and availability of the derived conclusions.

Observer-based SMC design for stochastic systems with Levy noise

Abstract This work addresses the problem of sliding mode control (SMC) design for a continuous-time non-linear stochastic system with Levy-type noise. A state observer model is constructed to estimate the unavailable state information. Furthermore, Levy-type noise is considered to analyse small perturbations and to characterize the appearance of large samples that will occur in the system. Lyapunov stability and SMC theory are used to provide some sufficient conditions that ensure the stochastic stability of the error system and reachability of the predefined sliding surface. Finally, an example is given to demonstrate the feasibility of the proposed approach.

Input–output finite-time stabilisation for periodic piecewise polynomial systems with nonlinear actuator faults: an observer-based approach

This paper is concerned with the input–output finite-time (IO-FT) stabilisation problem for a class of continuous-time periodic piecewise polynomial systems (PPPSs) with immeasurable states and external disturbances via state estimation-based robust reliable controller. Firstly, to reflect the actuality, a parameter uncertainty that exhibits the random nature and the actuator faults with nonlinear character are considered in the addressed PPPSs and the control scheme, respectively. In detail, the randomness of uncertain parameters is portrayed by the stochastic variable and it is presumed to pursue the Bernoulli distributions. Secondly, to estimate the immeasurable states of PPPSs, a periodic piecewise polynomial observer is designed depending on the output of PPPSs. The main intent of this article is to devise a state estimation-based robust reliable controller to ascertain the IO-FT stabilisation of the PPPSs. Moreover, by bridging the Lyapunov stability theory, linear matrix inequality technique and IO-FT stability theory, the required IO-FT stabilisation conditions in the frame of linear matrix inequality are procured for the system under consideration. Eventually, simulation results of the addressed PPPSs are shown in line with the proposed analytical findings, revealing the competence, inherent capability and utility of the devised control protocol.

Optimal robust control of nonlinear systems: inter-sample optimisation in sampled-data control

A methodology is presented for the design and implementation of robust controllers that optimise inter-sample performance for a broad range of nonlinear systems operated within a sampled-data environment. The methodology applies to nonlinear continuous-time systems described by state equations, and it allows for modelling uncertainties and constraints on maximal control effort. It is shown that there exist optimal robust state-feedback controllers that minimise inter-sample tracking errors for such systems, as long as an appropriate controllability requirement is met. A relatively simple design and implementation procedure for such controllers is described.

Input-to-state stabilization of networked impulsive systems under communication constraints: A refined Lyapunov functional

In this article, the protocol-based stabilization issue is considered for a class of continuous-time networked impulsive systems with external noises. For the purpose of saving communication cost, the Round-Robin communication protocol is implemented to dispatch the data transmission. Under this kind of communication protocol, the transmission order of sensor nodes is predefined. And, only the selected sensor nodes are able to get the privileges to visit the communication network at sampling instants. In addition, because of the network-induced transmission delays, the addressed system becomes an impulsive time-delayed one. For the protocol-induced system, sufficient conditions are obtained based on a refined Lyapunov functional to ensure the so-called input-to-state stability. Moreover, a suitable algorithm is developed to calculate the corresponding controller gains. In the end, two simulation examples are proposed to validate the effectiveness of our designed control strategy.

Robust Recursive Least-Squares Wiener Filter for Linear Continuous-Time Uncertain Stochastic Systems

For linear continuous-time systems with uncertainties in the system and observation matrices, an original robust RLS Wiener filter is designed in this study. The robust RLS Wiener filter does not assume norm-bounded uncertainty for the system and observation matrices, in contrast to the robust Kalman filter. In the design of the robust RLS Wiener filter, the degraded signal, affected by the uncertainties in the system and observation matrices, is modeled by an autoregressive (AR) model. The system and observation matrices for the degraded signal are formulated from the relationship between the AR model of the degraded signal and the state-space model. Estimation formulas for the system and observation matrices are proposed in Section 2. The robust filtering problem is introduced based on the minimization of the mean-square value of the filtering errors for the system states. The robust filtering estimate is given as an integral transformation of the degraded observations using the impulse response function. The integral equation that an optimal impulse response function satisfies is given in Section 3. Theorem 1 presents a robust RLS Wiener filtering algorithm starting from this integral equation. The proposed robust RLS Wiener filter outperforms the existing robust Kalman filter regarding estimate accuracy, as shown by a numerical simulation example.

Performance analysis of digitally controlled nonlinear systems considering time delay issues

In this paper, a comprehensive investigation into discretization, effective sample time selection considering delays in the system, and time and frequency domain analysis of a DC-DC buck converter, which plays a vital role in photovoltaic (PV) systems, is conducted to enhance the understanding of their dynamic behavior, optimize control algorithms, improve system efficiency, and ensure reliable power conversion in photovoltaic applications. To effectively address the non-linear behavior and enhance digital control of a buck converter by selecting the best sample time, several approaches can be employed. These include accurate modeling and identification of non-linear elements, development of advanced control algorithms that account for non-linearities, implementation of adaptive control techniques, and utilization of feedback mechanisms to compensate for deviations from linearity. By considering and mitigating the non-linear behavior, digital control systems can achieve improved accuracy, stability, and transient behavior in regulating the buck converter's output waveforms (voltage or current). The results of the study demonstrated that the trapezoidal integration method which is also known as bilinear approximation, or Tustin's approach outperformed other commonly used discretization methods, such as first-order hold (FOH), zero-order hold (ZOH), impulse response matching (impulse invariant), and matched pole-zero (MPZ) technique, in dual-domain (both time and frequency) analysis. The key finding highlighting the superiority of the bilinear approximation was its ability to achieve the closest match in the frequency domain bridging the continuous-time and discrete systems. This finding emphasizes the significance of the bilinear approach in preserving the frequency characteristics of the original continuous-time system during discretization. By employing this method, the discrete system closely approximated the behavior of its continuous-time counterpart, ensuring accurate frequency-domain representation.

Unknown input observer for Takagi-Sugeno implicit models with unmeasurable premise variables

<p>Recent years have seen a great deal of interest in implicit nonlinear systems, which are used in many different engineering applications. This study is dedicated to presenting a new method of fuzzy unknown inputs observer design to estimate simultaneously both non-measurable states and unknown inputs of continuous-time nonlinear implicit systems defined by Takagi-Sugeno (T-S) models with unmeasurable premise variables. The suggested observer is based on the singular value decomposition approach and rewritten the continuous-time T-S implicit models into an augmented fuzzy system, which gathers the unknown inputs and the state vector. The exponential convergence condition of the observer is established by using the Lyapunov theory and linear matrix inequalities are solved to determine the gains of the observer. Finally, the effectiveness of the suggested method is then assessed using a numerical application. It demonstrates that the estimated variables and the unknown input converge to the real variables accurately and quickly (less than 0.5 s).</p>

Linear-Like Policy Iteration Based Optimal Control for Continuous-Time Nonlinear Systems

We propose a novel strategy to construct optimal controllers for continuous-time nonlinear systems by means of linear-like techniques, provided that the optimal value function is differentiable and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">quadratic-like</i> . This assumption covers a wide range of cases and holds locally around an equilibrium under mild assumptions. The proposed strategy does not require solving the Hamilton-Jacobi-Bellman equation, that is a nonlinear partial differential equation, which is known to be hard or impossible to solve. Instead, the Hamilton-Jacobi-Bellman equation is replaced with an easy-solvable state-dependent Lyapunov matrix equation. We exploit a linear-like factorization of the underlying nonlinear system and a policy-iteration algorithm to yield a linear-like policy-iteration for nonlinear systems. The proposed control strategy solves optimal nonlinear control problems in an asymptotically exact, yet still linear-like manner. We prove optimality of the resulting solution and illustrate the results via four examples.

H∞ Control for Observer-Based Non-Negative Scaled Edge Consensus of Continuous-Time and Discrete-Time Networked Systems

H_∞ Control for Observer-Based Non-Negative Scaled Edge Consensus of Continuous-Time and Discrete-Time Networked Systems

Serial digital QAM-modem based on complex band-pass IIR filters with LF-prototypes of Bessel

The article is devoted to the development and implementation of algorithms for the single-channel sequential quadrature amplitude modulation and demodulation based on digital complex band-pass IIR filters with LF-prototypes of Bessel. The use of complex IIR filters of Bessel allows to obtain orthogonal carrier signals which are close in shape to symmetric signals with practical non-overlapping spectra. The procedure for calculating the 4th-order digital low-pass Bessel filter using the generalized bilinear transformation method, which is used in the theory of digital signal processing to convert a continuous-time system into a discrete-time system, is considered. The variants of implementing the digital complex bandpass filters with a sequential structure using the complex delay method are considered, which allows to obtain the block diagrams of complex bandpass filters without additional calculations and to realize the restructuring of the central frequency without changing the frequency response shape by changing only two coefficients in complex delays. The procedure for obtaining of the expression of the impulse response of digital complex band-pass IIR filters of Bessel is considered. With the help of circuit modeling, the amplitude-frequency response and impulse response of such filter are determined. If the IIR filters are used in the modulator, the pulse characteristics of which are infinite and asymmetric, then it is necessary to use digital complex FIR filters consistent with the truncated pulse characteristic in the demodulator. But as an exception, the Bessel digital IIR filter has a phase-frequency response similar in shape closely to the linear one and, accordingly, a pulse response close to the symmetric one. In this case, the same recursive complex Bessel filters, similar to the filter in the modulator, can be used as matched filters in the demodulator. Therefore, the serial block diagram of the modem contains the same complex band-pass IIR filters with LF-prototypes of Bessel. The choice of the central frequency of the channel digital complex bandpass IIR filter is carried out by setting two coefficients. The results of circuit modeling of the data transmission system with the quadrature amplitude modulation (QAM-16) in the Microcap environment are presented, confirming the operability of the proposed option.

Analysis of the pointwise completeness and the pointwise degeneracy of the standard and fractional descriptor linear systems and electrical circuits

The Drazin inverse of matrices is applied to analysis of the pointwise completeness and the pointwise degeneracy of the descriptor standard and fractional linear continuous-time and discrete-time systems. It is shown that: 1) The descriptor linear continuous-time system is pointwise complete if and only if the initial and final states belong to the same subspace. 2) The descriptor linear discrete-time system is not pointwise complete if its system matrix is singular. 3) System obtained by discretization of continuous-time system is always not pointwise complete. 4) The descriptor linear continuous-time system is not pointwise degenerated in any nonzero direction for all nonzero initial conditions. 5) The descriptor fractional system is pointwise complete if the matrix defined by (36) is invertible. 6) The descriptor fractional system is pointwise degenerated if and only if the condition (41) is satisfied. Considerations are illustrated by examples of descriptor linear electrical circuits.

General solution of two-dimensional singular fractional linear continuous-time system using the conformable derivative and Sumudu transform

The effectiveness of this paper lies in presenting a new solution for the singular fractional two-dimensional linear continuous-time systems using the conformable derivative and Sumudu transform. The proposed technique combines the new advantageous features of conformal derivative and double-delta-Kronecker, which efficiently handles singularities and Sumudu transform, and provides an efficient solution for 2D singular Fornasini–Marchesini fractional models. Applying these approaches, we then derive new explicit expressions for the fundamental matrices of the considered model. The applicability and usefulness of our proposed methods are validated and evaluated by numerical simulations in order to show the accuracy of the obtained results.

Nonlinear Continuous-Time System $$\mathcal {H}_\infty $$ Control Based on Dynamic Quantization and Event-triggered Mechanism

Nonlinear Continuous-Time System $$\mathcal {H}_\infty $$ Control Based on Dynamic Quantization and Event-triggered Mechanism

Adaptive optimal output regulation of unknown linear continuous-time systems by dynamic output feedback and value iteration

This paper proposes a value iteration based learning algorithm to solve the optimal output regulation problem for linear continuous-time systems, which aims at achieving disturbance rejection and asymptotic tracking simultaneously. Notably, the state is unmeasurable and the system dynamics is completely unknown, which greatly increases the challenge of solving the output regulation problem. Firstly, we present a dynamic output feedback controller design method by combining the internal model, setting a virtual exo-system, and constructing the augmented internal state without using any knowledge of system. Then, by establishing a novel iterative learning equation which requires no repeated finite window integral operations, an adaptive dynamic programming based learning algorithm with employing value-iteration scheme is proposed to estimate the optimal feedback control gain, which may lead to a reduction of the computational load. The analysis on solvability and convergence shows that the estimated control gain converges to the optimal control gain. Finally, a physical experiment on control of an unmanned quadrotor illustrates the effectiveness of the proposed algorithm.

Polynomial methods to construct inputs for uniformly ensemble reachable linear systems

This paper is concerned with linear parameter-dependent systems and considers the notion of uniform ensemble reachability. The focus of this work is on constructive methods to compute suitable parameter-independent open-loop inputs for such systems. In contrast to necessary and sufficient conditions for ensemble reachability, computational methods have to distinguish between continuous-time and discrete-time systems. Based on recently derived sufficient conditions and techniques from polynomial approximation, we present two methods for discrete-time single-input linear systems. Moreover, we illustrate that one method can also be applied to certain continuous-time single-input systems.

Formalizing developmental phenomena as continuous-time systems: Relations between mathematics and language development.

We demonstrate how developmental theories may be instantiated as statistical models, using hierarchical continuous-time dynamic systems. This approach offers a flexible specification and an often more direct link between theory and model parameters than common modeling frameworks. We address developmental theories of the relation between the academic competencies of mathematics and language, using data from the online learning system Mindsteps. We use ability estimates from 160,164 observation occasions, across N = 4623 3rd to 9th grade students and five ability domains. Model development is step-by-step from simple to complex, with ramifications for theory and modeling discussed at each step.

Adaptive optimal control of continuous-time nonlinear affine systems via hybrid iteration

In this paper, a novel successive approximation framework, named hybrid iteration (HI), is proposed to fill up the performance gap between two well-known dynamic programming algorithms, namely policy iteration (PI) and value iteration (VI). Using HI, an approximated optimal control policy can be learned without prior knowledge of an initial admissible control policy required by PI. Additionally, the HI algorithm converges to the optimal solution much faster than VI, and thus requires tremendously less number of learning iterations and CPU-time, compared to VI. Initially, we develop a model-based HI algorithm, and then extend it to a data-driven HI algorithm which learns the optimal control policy without any information of the physics of the system. Simulation results demonstrate the efficacy of the proposed HI algorithm.

Generalized Necessary and Sufficient Robust Boundedness Results for Feedback Systems

Classical sufficient conditions for ensuring the robust stability of a dynamical system in feedback with a nonlinearity include passivity, small gain, circle, and conicity theorems. We present a generalized version of these results for arbitrary semi-inner product spaces. Our result is purely algebraic, and holds even when the conventional discrete or continuous-time causal dynamical systems are replaced by general nonlinear relations, where there need not exist a notion of time. Our result clarifies when the sufficient conditions for robust stability are also necessary, and explains why stronger assumptions such as linearity and time-invariance are typically needed to prove necessity in the conventional dynamical systems setting.

A “fundamental lemma” for continuous-time systems, with applications to data-driven simulation

A “fundamental lemma” for continuous-time systems, with applications to data-driven simulation