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Let d and n be positive integers, and E/F be a separable field extension of degree m = n + d n . We show that if | F | > 2 , then there exists a point P ∈ P n (E) which does not lie on any degree d hypersurface defined over F. In other words, the m Galois conjugates of P impose independent conditions on the m-dimensional F-vector space of degree d forms in x 0 , x 1 , ... , x n . As an application, we determine the maximal dimensions of linear systems L 1 and L 2 of hypersurfaces in P n over a finite field F, where every F-member of L 1 is reducible and every F-member of L 2 is irreducible. [ABSTRACT FROM AUTHOR]

The traditional multi-period soliton method takes too long to solve and the solution results are not satisfactory. To address the above problems, this study proposes a multi-period soliton method for the discrete Riccati equation. In this method, the multi-period soliton solution index of the discrete Riccati equation is firstly selected, and the multi-period soliton solution of the discrete Riccati equation is determined according to the index. The correlation matrix is solved and a positive semi-definite matrix is established. Next, the lower bound and eigenvalues of the matrix are solved using the matrix inequality property inequality and the perturbation parameter. The boundary estimation and eigenvalue properties of the reference matrix and the solution matrix are obtained. Another positive semi-definite matrix is defined. Finally, the soliton solution is constructed to complete the multi-period soliton solution of the discrete Riccati equation. The simulation results showed that the traditional method took more than 5 minutes, while the multi-period soliton solution of the discrete Riccati equation designed in the research only took 2.5 minutes. The solution time of the proposed method was shorter than the traditional method. The optimal solution could be obtained. It showed that the multi-period soliton discrete Riccati equation solution could deepen the understanding for multi-period soliton and nonlinear fluctuation phenomenon, providing important mathematical models and tools for practical application and basic research. These instructions provide guidelines for preparing papers. The multi-periodic soliton method for the discrete Riccati equation proposed in the study can effectively improve the solution time and efficiency, and has certain applications in controller optimization and other aspects. [ABSTRACT FROM AUTHOR]

A novel and unified design approach on higher-dimensional switching chaos generators is derived in this paper. The whole n-dimensional linear space is divided into two parts by a closed hyper-polyhedron. Two higher-dimensional linear systems with the simplest structures as switching chaos generators are designed successfully to generate chaos. State matrix of the first linear system is Hurwitz stable. State matrix of the second linear system is not Hurwitz stable. Chaotic dynamical behaviors take place due to switching two systems. The switching trajectories go through the boundary of the closed hyper-polyhedron endlessly. Moreover, the size of the hyper-polyhedron can determine and control the amplitude of the chaotic signals. Specific numerical examples on four-dimensional, five-dimensional and six-dimensional switching chaos generators are employed, respectively, to illustrate the effectiveness of the novel and advanced approach presented in this paper. The proposed approach can also be applied to designing other switching chaos generators with the higher dimension beyond six. [ABSTRACT FROM AUTHOR]

The subject of KCC theory is a second‐order ordinary differential equation, it is sometimes difficult to convert the high dimensional system into an equivalent second‐order system because of the analytical requirements of KCC theory. By means of the Euler‐Lagrange extension of a flow on a Riemannian manifold, this paper gives five geometric invariants of some three‐dimensional systems with great convenience, and focus on the analysis of two of them. The results show that the hyperbolic equilibria corresponding to the seven standard forms of three‐dimensional linear systems are Jacobi unstable. This is completely different from what we got before in two‐dimensional systems, where Jacobi stable and Jacobi unstable correspond to focus and node, respectively. All equilibria of classical Lü chaotic system and Yang‐Chen chaotic system are Jacobi unstable. Meanwhile, in three‐dimensional linear case, the torsion tensors at any point of the trajectory are identically equal to zero, but the two nonlinear systems have nonzero torsion tensors components. [ABSTRACT FROM AUTHOR]

This work investigates the less conservative stability conditions for linear systems with a time-varying delay. At first, augmented Lyapunov–Krasovskii functionals(LKFs) are constructed with state vectors that have not been utilized in the existing works, and an augmented zero equality that can be derived according to the augmented vector is proposed. By utilizing them, a stability condition is proposed in the form of a linear matrix inequality. And, by using novel delay-dependent LKFs and the introduced ones, improved results are obtained than the previous result. The addition of the delay-dependent LKFs increases the number of decision variables in the results. Therefore, any vectors of integral inequalities utilized in the proposed criterion are appropriately adjusted to reduce computational complexity. To check the excellence and validity of the proposed results, several numerical examples are applied. • The improved stability conditions for the linear system with a time-varying delay are proposed. • The proposed Lyapunov-Krasovskii functionals(LKFs) have more extended augmented vectors to obtain less conservative results. And LKFs based on an integral inequality (Lee et al., 2023) is utilized. • The novel LKFs which are the delay-dependent forms are proposed. The superiority of the results is checked according to the presence or absence of the forms. • The computational complexity is reduced by adjusting any vector of the integral inequality utilized in the proposed criteria. [ABSTRACT FROM AUTHOR]

New approaches to transformations of linear continuous-time systems to their positive asymptotically stable canonical controllable (observable) forms are proposed. It is shown that, if the system matrix is nonsingular, then the desired transformation matrix can be chosen in block diagonal form. Procedures for the computation of the transformation matrices are proposed and illustrated with simple numerical examples.

In this study, we analyse the local identifiability of parameters in linear systems whose state matrix is given by a graph Laplacian. Graph Laplacian is a matrix given by a graph and the weights of its edges, where the weights represent the parameters in those systems. There are cases in which parameter estimation has to be conducted with a single trajectory data. In this case, detecting whether the parameter is locally unidentifiable (non-locally identifiable (LI)) from a single-state trajectory a priori is important. This is because we need conditions to avoid estimating non-LI parameters. Therefore, we address a problem to find the condition which implies that the parameter is non-LI from any single-state trajectory. Then, we obtain a sufficient condition for the parameter to be non-LI from any single-state trajectory. The condition is given based on the number of vertices and edges of the graph and the number of distinct eigenvalues of the graph Laplacian. This paper also presents an example that satisfies the condition.

In this paper, the global stabilization problem of control systems with input saturation and multiple input delays is studied, and a new method is proposed to design nonlinear stabilization control laws. First, based on Luenberger's canonical decomposition, the multiple-input delay system is transformed into a series of linear time-delay systems with single inputs and input saturation. However, for the converted system, each subsystem is coupled to the others. Therefore, the idea of recursion is adopted to construct a special state transformation with time delay for each subsystem and convert it into a linear system with time delay for both state variables and input variables. For the conversion system, a nonlinear controller with cascade saturation control is designed, and the controller includes some free parameters. The control performance of the controller is improved by adjusting the free parameters online. At the same time, a less conservative stability condition is established to ensure the dynamic performance of the closed-loop system. Finally, the effectiveness and superiority of the proposed method are verified by numerical simulation and practical applications in a spacecraft rendezvous system. [ABSTRACT FROM AUTHOR]

This work investigates the stability conditions for linear systems with time-varying delays via an augmented Lyapunov–Krasovskii functional (LKF). Two types of augmented LKFs with cross terms in integrals are suggested to improve the stability conditions for interval time-varying linear systems. In this work, the compositions of the LKFs are considered to enhance the feasible region of the stability criterion for linear systems. Mathematical tools such as Wirtinger-based integral inequality (WBII), zero equalities, reciprocally convex approach, and Finsler's lemma are utilized to solve the problem of stability criteria. Two sufficient conditions are derived to guarantee the asymptotic stability of the systems using linear matrix inequality (LMI). First, asymptotic stability criteria are induced by constructing the new augmented LKFs in Theorem 1. Then, simplified LKFs in Corollary 1 are proposed to show the effectiveness of Theorem 1. Second, asymmetric LKFs are shown to reduce the conservatism and the number of decision variables in Theorem 2. Finally, the advantages of the proposed criteria are verified by comparing maximum delay bounds in four examples. Four numerical examples show that the proposed Theorems 1 and 2 obtain less conservative results than existing outcomes. Particularly, Example 2 shows that the asymmetric LKF methods of Theorem 2 can provide larger delay bounds and fewer decision variables than Theorem 1 in some specific systems. [ABSTRACT FROM AUTHOR]

We consider interval systems of linear equations and inequalities with a restriction to some a priori given set. We focus on a characterization of strong solvability, that is, solvability for each realization of interval values, and we compare this with an existence of a strong solution defined analogously. The motivation comes from the area of interval-valued optimization problems, where strong solvability means guaranteed feasibility of any realization of the problem. Strong solvability with strict inequalities implies the robust Slater condition, which ensures that standard optimality conditions can be used. We apply the issues particularly in two optimization classes, convex quadratic programming with quadratic constraints and posynomial geometric programming. For the former, we also utilize the presented result to improve a characterization of the worst case optimal value. Eventually, we state several open problems that emerged while deriving the results. [ABSTRACT FROM AUTHOR]

Summary: This paper studies the problem of sensor fault estimation for linear systems in the presence of unknown inputs in both input and output channels. Relying on matrix equation theory, the unknown inputs are first removed from the output channel. Second, an augmented descriptor system is constructed by treating both system state and sensor fault as (pseudo)‐state variables. Third, the augmented descriptor system is transformed into a regular one, followed by the elimination of the unknown input from the state equation. Finally, a reduced‐order prescribed‐time sensor fault estimator is obtained by using periodic delayed observers. This matrix equation based approach is then complemented by an alternative one, which relies on successive transformations of state, input and outputs. This approach sheds light on the adopted solvability conditions throughout the paper and on the role of different directions in the output space in handling unknown inputs, sensor faults and the core state estimation problem. Finally, two numerical examples are presented to illustrate the effectiveness of the proposed methodologies. [ABSTRACT FROM AUTHOR]

A new approach to the transformations of the matrices of the fractional linear systems with desired eigenvalues is proposed. Conditions for the existence of the solution to the transformation problem of the linear system to its asymptotically stable controllable and observable canonical forms with desired eigenvalues are given and illustrated by numerical examples of fractional linear systems. [ABSTRACT FROM AUTHOR]

Studying a variety through a linear system of homogenous forms with a base locus is a method that has many advantages not only for understanding important geometric properties but also the ability to accurately evaluate many invariants of the variety. This paper aims to study linear systems of cubic forms through 5 points in the projective plane with many configurations. Many interesting and profound properties of surfaces have been discovered through their corresponding linear systems. These results not only enrich correspondence theory in algebraic geometry and computational algebra but also promise to provide methods and tools for applications in the fields of information technology and other technologies.

In this study, we propose a novel backward shift algorithm for pole estimation of linear time‐invariant (LTI) systems. By making adjustments, we address the question: How should the frequencies of input signals be chosen? We select the frequencies based on algebraic accuracy in the numerical integral and use the Gaussian–Legendre formula in this process. After poles are located, the discrete Fourier transform (DFT) and the new backward shift algorithm (new BSA) along with the least squares (LS) method are used to identify the systems. Two examples show that the proposed method for pole estimation performs better and the corresponding identification algorithm is practical. [ABSTRACT FROM AUTHOR]

This paper develops a class of algorithms for general linear systems and eigenvalue problems. These algorithms apply fast randomized dimension reduction ("sketching") to accelerate standard subspace projection methods, such as GMRES and Rayleigh-Ritz. This modification makes it possible to incorporate nontraditional bases for the approximation subspace that are easier to construct. When the basis is numerically full rank, the new algorithms have accuracy similar to classic methods but run faster and may use less storage. For model problems, numerical experiments show large advantages over the optimized MATLAB routines, including a 70x speedup over gmres and a 10x speedup over eigs. [ABSTRACT FROM AUTHOR]

The research topic of this work is at the junction of the theory of Lyapunov exponents and oscillation theory. In this paper, we study the spectra (that is, the sets of different values on nonzero solutions) of the exponents of oscillation of signs (strict and nonstrict), zeros, roots, and hyperroots of linear homogeneous differential equations with coefficients continuous on the positive semiaxis. In the first part of the paper, we build a third-order linear differential equation with the following property: the spectra of all upper and lower strong and weak exponents of oscillation of strict and nonstrict signs, zeros, roots, and hyperrots contain a countable set of different essential values, both metrically and topologically. Moreover, all these values are implemented on the same sequence of solutions of the constructed equation, that is, for each solution from this sequence, all of the oscillation exponents coincide with each other. In the construction of the indicated equation and in the proof of the required results, we used analytical methods of the qualitative theory of differential equations and methods from the theory of perturbations of solutions to linear differential equations, in particular, the author's technique for controlling the fundamental system of solutions of such equations in one special case. In the second part of the paper, the existence of a third-order linear differential equation with continuum spectra of the oscillation exponents is established, wherein the spectra of all oscillation exponents fill the same segment of the number axis with predetermined arbitrary positive incommensurable ends. It turned out that for each solution of the constructed differential equation, all of the oscillation exponents coincide with each other. The results are theoretical in nature; they expand our understanding of the possible spectra of oscillation exponents of linear homogeneous differential equations. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the uniform exponential stability of the system dx(t)dt=Ax(t)−ρBx(t),(ρ>0),$$ \frac{dx(t)}{dt}= Ax(t)-\rho Bx(t),\kern3.0235pt \left(\rho >0\right), $$ where the unbounded operator A$$ A $$ is the infinitesimal generator of a linear C0−$$ {C}_0- $$semigroup of contractions S(t)$$ S(t) $$ in a Hilbert space X$$ X $$ and B$$ B $$ is a Desch–Schappacher operator. Then we give sufficient conditions for exponential stability of the above system. The obtained stability result is then applied to prove the uniform exponential stabilization of some bilinear partial differential equations. [ABSTRACT FROM AUTHOR]

This paper investigates a class of sampled-data control systems in which at least one of the discrete-time zeros remains on the unit circle regardless of the sampling period. This type of zero is referred to as an “uniformly marginally stable” (UMS) zero. Utilizing this novel concept, we present necessary conditions as well as sufficient conditions for the continuous-time plant whose sampled-data model has an UMS zero. The theoretical finding of this paper newly highlights the significance of the symmetry regarding the zeros and poles of the continuous-time plant, as it plays a vital role in enforcing a discrete-time zero of the sampled-data system to be UMS, for both discretization zeros and intrinsic zeros. We develop the theory for sampled-data systems employing a conventional sampler and a zero-order hold together with a single-input single-output continuous-time plant, and then observe that certain results can be extended to the generalized holds. As an application, we construct a discrete-time stable inversion-based control in the sampled-data setting and establish a condition for continuous-time plants that allow for this type of control, in which absence of such an UMS zero is crucial.

The asymptotic stability of the convex linear combination of continuous-time and discretetime linear systems is considered. Using the Gershgorin theorem it is shown that the convex linear combination of the linear asymptotically stable continuous-time and discretetime linear systems is also asymptotically stable. It is shown that the above thesis is also valid (even simpler) for positive linear systems.

By exploiting the concept of row partitioning, we propose an efficient variant of the greedy block Kaczmarz algorithm for solving consistent large linear systems. The number of blocks is determined a priori through numerical experiments. The new algorithm works with a reduced linear system, which dramatically diminishes the computational overhead per iteration. The theoretical result validates that this method converges to the unique least-norm solution of the linear system. The effectiveness of the proposed algorithm is also justified by comparing it with some block Kaczmarz algorithms in extensive numerical experiments.

We study the continuous and discontinuous planar piecewise differential systems separated by a straight line and formed by an arbitrary linear system and a class of quadratic center. We show that when these piecewise differential systems are continuous, they can have at most one limit cycle. However, when the piecewise differential systems are discontinuous, we show that they can have at most two limit cycles, and that there exist such systems with two limit cycles. Therefore, in particular, we have solved the extension of the 16th Hilbert problem to this class of differential systems.

Object tracking technology based on image processing has made great progress recently. Based on the track-by-detection framework, the tracking algorithms are often combined with deep neural networks to perform online target tracking. However, existing motion models assume linearity and are sensitive to sudden changes in trajectories due to occlusion, overlap, or other detection issues. In this paper, we modified the existing object tracking algorithms and introduced a strong tracking filter (STF) module to the track-by-detection framework for solving the sudden change problem of the target. The STF module is designed to have a strong ability to track sudden changes by orthogonalizing the residual sequence. When the trajectory of the target is stable, the STF module returns to the inactive state, behaving similarly to tracking algorithms that follow conventional linear models. Experimental results on a public pedestrian tracking dataset show that the proposed method improves tracking performance on various metrics, including the ability to rematch missed trajectories. Moreover, compared with existing algorithms, the proposed method exhibits strong stability under noisy conditions. [ABSTRACT FROM AUTHOR]

In response to the influence of motor interference, damping, friction, and other uncertain factors on the operation of electric power steering systems under extreme working conditions, this study proposes a control strategy for electric power steering systems based on an active disturbance rejection algorithm. In ADRC, the fastest tracking differentiator is used to arrange the transition process for the target signal, and the extended state observer compensates for the total disturbance in the system. Phase compensation has been performed on the monitoring torque by using the torque differentiation method. The Simulink/Carsim simulation results show that ADRC has significantly improved anti-disturbance performance compared to PID and fuzzy PID. When using ADRC, the tracking accuracy of the assisted current is enhanced by 45.8%–75.8%, and the current adjustment time is reduced by 35.6%–61.7%. After phase compensation, the monitoring torque overshoot is reduced by 83.3%. Therefore, the proposed control strategy improves EPS's robustness and steering feel. [ABSTRACT FROM AUTHOR]

We introduce a mixed precision algorithm for computing sparse matrix-vector products and use it to accelerate the solution of sparse linear systems by iterative methods. Our approach is based on the idea of adapting the precision of each matrix element to their magnitude: we split the elements into buckets and use progressively lower precisions for the buckets of progressively smaller elements. We carry out a rounding error analysis of this algorithm that provides us with an explicit rule to decide which element goes into which bucket and allows us to rigorously control the accuracy of the algorithm. We implement the algorithm on a multicore computer and obtain significant speedups (up to a factor 7\times) with respect to uniform precision algorithms, without loss of accuracy, on a range of sparse matrices from real-life applications. We showcase the effectiveness of our algorithm by plugging it into various Krylov solvers for sparse linear systems and observe that the convergence of the solution is essentially unaffected by the use of adaptive precision. [ABSTRACT FROM AUTHOR]

GMRES-based iterative refinement in three precisions (GMRES-IR3), proposed by Carson and Higham in 2018, uses a low precision LU factorization to accelerate the solution of a linear system without compromising numerical stability or robustness. GMRES-IR3 solves the update equation of iterative refinement using GMRES preconditioned by the LU factors, where all operations within GMRES are carried out in the working precision u, except for the matrix--vector products and the application of the preconditioner, which require the use of extra precision u2. The use of extra precision can be expensive, and is especially unattractive if it is not available in hardware; for this reason, existing implementations have not used extra precision, despite the absence of an error analysis for this approach. In this article, we propose to relax the requirements on the precisions used within GMRES, allowing the use of arbitrary precisions up for applying the preconditioned matrix--vector product and ug for the rest of the operations. We obtain the fiveprecision GMRES-based iterative refinement (GMRES-IR5) algorithm which has the potential to solve relatively badly conditioned problems in less time and memory than GMRES-IR3. We develop a rounding error analysis that generalizes that of GMRES-IR3, obtaining conditions under which the forward and backward errors converge to their limiting values. Our analysis makes use of a new result on the backward stability of MGS-GMRES in two precisions. On hardware where three or more arithmetics are available, which is becoming very common, the number of possible combinations of precisions in GMRES-IR5 is extremely large. We provide an analysis of our theoretical results that identifies a relatively small subset of relevant combinations. By choosing from within this subset one can achieve different levels of tradeoff between cost and robustness, which allows for a finer choice of precisions depending on the problem difficulty and the available hardware. We carry out numerical experiments on random dense and SuiteSparse matrices to validate our theoretical analysis and discuss the complexity of GMRES-IR5. [ABSTRACT FROM AUTHOR]

The level of agricultural development in European Union countries is characterized by great diversity. This is due to differences in natural conditions, the type of agricultural production, agrarian fragmentation, and the level of economic development. The concept of a circular economy is the latest vision of changing the current economic systems, the assumptions of which constitute an alternative to the linear model of resource use. The implementation of the principles of a circular economy aims to create a system that will contribute to the implementation of sustainable development. This could be a strategy to support agriculture in the absence of agricultural land and water resources. This research aimed to identify and assess the spatial diversification of agricultural production-economic conditions and their links with the circular economy at the level of EU countries. The basis for grouping countries was synthetic measures obtained in the areas of agriculture and the circular economy. The analyses were performed for 2012 and 2020. The obtained results indicate the existence of significant spatial dependencies in the development of agriculture and the circular economy. Countries with a higher level of agricultural development were also higher in the ranking of the advancement of the implementation of the circular economy concept. [ABSTRACT FROM AUTHOR]

In order to solve the coordinated operation of voltage and frequency of microgrids and achieve effective distribution of output power, we propose the application of fractional-order nonlinear equations in coordination. This method designs a distributed impulse coordinated control strategy, to achieve the coordinated operation of the system. The distributed coordinated control structure and mathematical model are established, and the distributed two-level coordinated control strategy of the microgrid system is adopted. Aiming at the secondary control problem of microgrids, a distributed coordinated control protocol is designed. The results showed that after adding the distributed second-level coordination control of frequency at 3 s, the output active power of the four distributed power sources in the microgrid model is maintained to an evenly divided state after about 1 s. The output voltage and frequency utilizing the microgrid’s decentralized power supply quickly reach the ideal reference value, within the allowable error range, the output of the system can achieve coordinated control, and the active power can be distributed proportionally.

Abstract Object tracking technology based on image processing has made great progress recently. Based on the track-by-detection framework, the tracking algorithms are often combined with deep neural networks to perform online target tracking. However, existing motion models assume linearity and are sensitive to sudden changes in trajectories due to occlusion, overlap, or other detection issues. In this paper, we modified the existing object tracking algorithms and introduced a strong tracking filter (STF) module to the track-by-detection framework for solving the sudden change problem of the target. The STF module is designed to have a strong ability to track sudden changes by orthogonalizing the residual sequence. When the trajectory of the target is stable, the STF module returns to the inactive state, behaving similarly to tracking algorithms that follow conventional linear models. Experimental results on a public pedestrian tracking dataset show that the proposed method improves tracking performance on various metrics, including the ability to rematch missed trajectories. Moreover, compared with existing algorithms, the proposed method exhibits strong stability under noisy conditions.

In this paper, the global stabilization problem of control systems with input saturation and multiple input delays is studied, and a new method is proposed to design nonlinear stabilization control laws. First, based on Luenberger’s canonical decomposition, the multiple-input delay system is transformed into a series of linear time-delay systems with single inputs and input saturation. However, for the converted system, each subsystem is coupled to the others. Therefore, the idea of recursion is adopted to construct a special state transformation with time delay for each subsystem and convert it into a linear system with time delay for both state variables and input variables. For the conversion system, a nonlinear controller with cascade saturation control is designed, and the controller includes some free parameters. The control performance of the controller is improved by adjusting the free parameters online. At the same time, a less conservative stability condition is established to ensure the dynamic performance of the closed-loop system. Finally, the effectiveness and superiority of the proposed method are verified by numerical simulation and practical applications in a spacecraft rendezvous system.

This work investigates the stability conditions for linear systems with time-varying delays via an augmented Lyapunov–Krasovskii functional (LKF). Two types of augmented LKFs with cross terms in integrals are suggested to improve the stability conditions for interval time-varying linear systems. In this work, the compositions of the LKFs are considered to enhance the feasible region of the stability criterion for linear systems. Mathematical tools such as Wirtinger-based integral inequality (WBII), zero equalities, reciprocally convex approach, and Finsler’s lemma are utilized to solve the problem of stability criteria. Two sufficient conditions are derived to guarantee the asymptotic stability of the systems using linear matrix inequality (LMI). First, asymptotic stability criteria are induced by constructing the new augmented LKFs in Theorem 1. Then, simplified LKFs in Corollary 1 are proposed to show the effectiveness of Theorem 1. Second, asymmetric LKFs are shown to reduce the conservatism and the number of decision variables in Theorem 2. Finally, the advantages of the proposed criteria are verified by comparing maximum delay bounds in four examples. Four numerical examples show that the proposed Theorems 1 and 2 obtain less conservative results than existing outcomes. Particularly, Example 2 shows that the asymmetric LKF methods of Theorem 2 can provide larger delay bounds and fewer decision variables than Theorem 1 in some specific systems.

Abstract This paper proposes moving horizon estimation (MHE) to estimate the state variables of autonomous vehicle linear systems under measurement noises. To solve the MHE optimization problem, quadratic programming is employed. The steering angle, yaw angle, and global position constraints of an autonomous vehicle are considered in the estimation design. According to the simulation results, it can be observed that although the longer MHE step can give better results compared to the shorter MHE step, the difference in the MHE step only slightly affects the estimated results. However, the longer MHE step can increase the computational time. Additionally, the proposed MHE scheme is compared to the Kalman filter (KF) estimator. Based on the obtained results, the KF gives a better estimation than the MHE, but this notion must be verified for other case studies. Keywords: autonomous vehicle; Kalman filter; linear system; MHE; quadratic programming. Abstrak Paper ini mengusulkan moving horizon estimation (MHE) untuk mengestimasi variabel keadaan sistem linier kendaraan otonom karena pengaruh noise pengukuran. Untuk menyelesaikan masalah optimasi MHE, digunakan pemrograman kuadratik. Kendala sudut kemudi, sudut yaw dan posisi global dari kendaraan otonom dipertimbangkan dalam desain estimasi. Dari hasil simulasi dapat diketahui bahwa meskipun langkah MHE yang lebih panjang dapat memberikan hasil yang lebih baik dibandingkan dengan langkah MHE yang lebih pendek, perbedaan langkah MHE hanya sedikit mempengaruhi hasil estimasi. Namun, langkah MHE yang semakin panjang dapat meningkatkan waktu komputasi. Selain itu, skema MHE yang diusulkan dibandingkan dengan estimator Kalman filter (KF). Berdasarkan hasil yang diperoleh, KF memberikan estimasi yang lebih baik daripada MHE, tetapi gagasan ini harus diverifikasi untuk studi kasus lainnya. Kata Kunci: kendaraan otonom; Kalman filter; sistem linier; MHE; pemrograman kuadratik. 2020MSC: 62P35, 65D19

Inverting land cover reflectance or derived indices from low-spatial-resolution images to refine the spatial resolution of this data is cost-effective for land surface monitoring applications that face technical or budget limitations. Based on the linear spectral mixing model, many approaches have successfully unmixed coarse mixed pixels using high-spatial-resolution land cover maps in the past decades. However, in some cases, the solutions of linear systems composed of several mixed pixels may not be acquired due to the underdetermined problem. This study presents the causes of this problem and proposes an iterative approximation strategy to address it. An elastic-window-based algorithm was developed, where the initial size of the window was calculated based on the land cover of the mixed pixel. Mixed pixels of neighborhoods with similar land covers were then selected to form the unmixing linear system, which was examined through a simulation test to ensure it was not underdetermined. Otherwise, the window would expand to include more adjacent pixels. This process was repeated until a successful solution was obtained. A statistical analysis of sixty land cover maps from around the globe shows that the underdetermined problem exists at a low level but becomes more serious with an increase in mixed scale. The results demonstrate that the proposed algorithm effectively prevents the underdetermined problem for mixed pixels of different scales and can be integrated into the coarse NDVI downscaling procedure to refine spatial resolution. This study provides a reference for estimating underdetermined mixed pixels and benefits applications that require dealing with the inversion of land cover values directly. [ABSTRACT FROM AUTHOR]

A new approach to the transformations of the matrices of linear continuous-time systems to their canonical forms with desired eigenvalues is proposed. Conditions for the existence of solutions to the problems were given and illustrated by simple numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, we propose an event‐triggered adaptive integral sliding mode control scheme for a class of linear systems with external disturbance. In this method, the controller is designed using a triggered‐state‐dependent integral sliding mode, which can ensure the robustness and avoid the shortcomings of traditional sliding mode arrival stage. The triggering mechanism utilizes a time‐varying trigger threshold instead of a traditional fixed threshold, which not only realizes the dynamic update of the control law, reduces the overhead of network communication but also ensures that the system trajectory enters the bounded area. The lower bound of inter event time guarantees the avoidance of the Zeno phenomenon. Next, in order to reduce the impact of high‐frequency chattering of the control signal effectively and allow the gain of the discontinuous control term to be adjusted automatically according to the rate of change of the disturbance, a dual‐layer nested adaptive gain scheme based on equivalent control is considered. This scheme does not require a priori boundedness of the disturbance and its rate of change. Through Lyapunov stability analysis, the event‐triggered adaptive sliding mode controller can make the system trajectory converge in finite time and ensure the robustness of the control. Finally, the simulation results verify the effectiveness and simplicity of this method. [ABSTRACT FROM AUTHOR]

The eigenvalues assignment problems for descriptor linear systems with state and its derivative feedbacks are considered herein. Necessary and sufficient conditions for the existence of solutions to the problems are established. The Euler and Tustin approximations of the continuous-time systems are analyzed. Procedures for computation of the feedbacks are given and illustrated by numerical examples. [ABSTRACT FROM AUTHOR]

A Kalman reduced form is obtained for linear systems over Hermite rings. This reduced form gives information of the set of assignable polynomials to a given linear system. [ABSTRACT FROM AUTHOR]

The conjugate gradient (CG) method is a classic Krylov subspace method for solving symmetric positive definite linear systems. We analyze an analogous semi-conjugate gradient (SCG) method, a special case of the existing semi-conjugate direction (SCD) methods, for unsymmetric positive definite linear systems. Unlike CG, SCG requires the solution of a lower triangular linear system to produce each semi-conjugate direction. We prove that SCG is theoretically equivalent to the full orthogonalization method (FOM), which is based on the Arnoldi process and converges in a finite number of steps. Because SCG's triangular system increases in size each iteration, Dai and Yuan [Study on semi-conjugate direction methods for non-symmetric systems, Int. J. Numer. Meth. Eng. 60(8) (2004), pp. 1383–1399] proposed a sliding window implementation (SWI) to improve efficiency. We show that the directions produced are still locally semi-conjugate. A counter-example illustrates that SWI is different from the direct incomplete orthogonalization method (DIOM), which is FOM with a sliding window. Numerical experiments from the convection-diffusion equation and other applications show that SCG is robust and that the sliding window implementation SWI allows SCG to solve large systems efficiently. [ABSTRACT FROM AUTHOR]

The asymptotic stability of the convex linear combination of continuous-time and discretetime linear systems is considered. Using the Gershgorin theorem it is shown that the convex linear combination of the linear asymptotically stable continuous-time and discretetime linear systems is also asymptotically stable. It is shown that the above thesis is also valid (even simpler) for positive linear systems. [ABSTRACT FROM AUTHOR]