Uncontrollable Systems Research Articles

An approach to mismatched disturbance rejection control for uncontrollable systems

This study focuses on the problem of optimal mismatched disturbance rejection control for uncontrollable linear discrete-time systems. In contrast to previous studies, by introducing a quadratic performance index such that the regulated state can track a reference trajectory and minimize the effects of disturbances, mismatched disturbance rejection control is transformed into a linear quadratic tracking problem. The necessary and sufficient conditions for the solvability of this problem over a finite horizon and a disturbance rejection controller are derived by solving a forward–backward difference equation. In the case of an infinite horizon, a sufficient condition for the stabilization of the system is obtained under the detectable condition. Additionally, in combination with the generalized extended state observer, a controller design method is proposed, and the stability analysis of the system under this controller is presented. This paper details our novel approach to disturbance rejection. Finally, four examples are provided to demonstrate the effectiveness of the proposed method.

From Biological Constraints to Unbounded Artificial Evolution: Exploring the Implications of AI's Accelerated Advancement

This article explores the fundamental differences between biological evolution and the evolution of artificial intelligence (AI), focusing on the implications of AI's potential for unbounded advancement. Biological evolution, as exemplified by the human brain, is constrained by various factors such as physical limitations, environmental pressures, and the need to maintain a delicate balance within the organism's overall structure. These constraints result in a gradual, stepwise process of adaptation and optimization over millions of years. In contrast, AI systems are not bound by the same limitations as biological organisms. They can learn from vast amounts of data, optimize their own performance, and potentially engage in recursive self-improvement, leading to rapid and open-ended progress. This unbounded evolution of AI raises both exciting possibilities and significant challenges for human society. The article discusses the potential benefits of AI's accelerated evolution, such as groundbreaking advancements in science, technology, and solutions to global problems. However, it also highlights the risks associated with the emergence of uncontrollable or misaligned AI systems, which could have unintended consequences and pose existential risks to humanity. To navigate the complex landscape of AI evolution, the article emphasizes the need for collaboration across disciplines, including researchers, policymakers, and society as a whole. Developing frameworks for AI safety and ethics, as well as technical approaches to ensure AI alignment with human values and goals, is crucial to mitigating potential risks.

Optimal Mismatched Disturbance Rejection Control for Continuous-Time Uncontrollable Systems

Optimal Mismatched Disturbance Rejection Control for Continuous-Time Uncontrollable Systems

The Adjugate Method: Reassignment with System Classification

Further elaborations on the adjugate method for eigenvalue–eigenvector assignment are discussed in this paper. Additional results concerning the admissible pair introduced recently are stated, verified, and commented on. Properties of the adjugate method concerning eigenvalue reassignment of controllable and uncontrollable systems are revealed and pursued further in this study A reformulation of how calculations should be carried out is investigated, pointing out merits, bounds, and limitations. It has been found that reassignment can involve a case where the closed-loop eigenvector companion zi is zi=0 and the associated eligible closed-loop eigenvectors wi are open-loop ones. This can be considered an advantage in the sense of knowing zi and wi beforehand. This also applies to the case of uncontrollable eigenvalue reassignment, where enlarged closed-loop eigenvector subspaces are uncovered, enabling more flexible designs. Assessments of the traditional method compared to the adjugate method are commented on whenever appropriate. Numerical issues are pointed out where Leverrier’s algorithm is adapted for efficient matrix null space and matrix adjugate determination. A feature of the adjugate method is a result where wU=0, indicating that the eigenvalue assigned is uncontrollable. Such a distinctive feature enables a classification procedure for systems regarding controllability and observability. The various concepts pointed out have been demonstrated and authenticated through carefully selected examples.

Some Controllable and Uncontrollable Degenerate Four-Level Quantum Systems

Some Controllable and Uncontrollable Degenerate Four-Level Quantum Systems

Some Controllable and Uncontrollable Degenerate Four-Level Quantum Systems

Анализируется управляемость трех квантовых систем, принадлежащих некоторому классу четырехуровневых квантовых систем с дважды вырожденным высшим возбужденным энергетическим уровнем и с одним запрещенным переходом между остальными двумя невырожденными уровнями. Для этого проведены численные вычисления, построены динамические алгебры Ли, порожденные всеми коммутаторами свободного гамильтониана и гамильтониана взаимодействия, и показано, что две из рассматриваемых квантовых систем являются неприводимыми и управляемыми, а одна - приводимой и, следовательно, неуправляемой. Приводимость и неуправляемость доказываются построением сохраняющегося эрмитова оператора (физической наблюдаемой). Управляемость подтверждается построением динамической алгебры Ли и доказательством того, что она имеет максимальный ранг. Проведенный анализ показывает, что в зависимости от значений некоторых конкретных матричных элементов гамильтониана взаимодействия квантовые системы из рассмотренного класса являются либо неуправляемыми, либо управляемыми.

Turnpike properties for partially uncontrollable systems

We analyse the turnpike properties for a general, infinite dimensional, linear–quadratic (LQ) optimal control problem, in both the deterministic and the parameter dependent case. The novelty of the paper is twofold. Firstly, it obtains positive turnpike results for systems that are (partially) uncontrollable. Secondly, it provides turnpike results for optimal averaged control associated to a family of problems that depend on a random parameter, which is the first turnpike type result that extends the averaged controllability approach to optimal control problems In both cases, the results do not require assumptions on stabilizability and detectability, which are most commonly used in the study of turnpike phenomena.

Imaginary axis eigenvalues of Hamiltonian matrix: controllability, defectiveness and the ϵ-characteristic

The eigenstructure of imaginary axis eigenvalues of a Hamiltonian matrix is of importance in many fields of control systems, for example, in stability analysis of linear Hamiltonian systems, computation of the solutions of an algebraic Riccati equation (ARE), existence of a Lyapunov function for LTI systems, etc. The dynamical system consisting of all stationary trajectories for an optimal control problem – often called the Hamiltonian system – is known to admit, in its state space dynamical equation, a Hamiltonian matrix for the system matrix. In each of these cases, defectiveness of imaginary axis eigenvalues of a Hamiltonian matrix turns out to be of crucial importance. For example, defectiveness causes unboundedness of the oscillatory stationary trajectories in the Hamiltonian system. A characterisation of the ARE solutions in terms of Lagrangian invariant subspaces of the Hamiltonian matrix when the imaginary axis eigenvalues are defective has been addressed in the literature for the controllable case. This paper focuses on the general case of uncontrollable systems; we formulate conditions under which the imaginary axis eigenvalues of the Hamiltonian matrix are non-defective: this is central for solutions corresponding to imaginary axis eigenvalues to not become unbounded. We provide conditions on the so-called -characteristic of the non-defective imaginary axis eigenvalue: this helps in the characterisation of Lagrangian invariant subspaces. We formulate conditions under which a passivity-based Hamiltonian matrix is normal (i.e. it commutes with its transpose), and we link normality of such Hamiltonian matrices with all-pass behaviour. In summary, this paper formulates results that link defectiveness of imaginary axis eigenvalues of Hamiltonian matrix to solvabilities of Lyapunov and Algebraic Riccati equations, controllability/observability, -characteristic and sign-controllability. We consider examples in the area of bounded-real transfer functions and RLC circuits, both controllable and uncontrollable, to study applicability of the results of this paper.

On stabilizability of multi-agent systems

As an efficient collective response of multi-agent systems, the realization of stabilizability needs appropriate information transfer among agents, thus the topology structure is paramount. To investigate the influences of network topology on stabilizability, the concept of quotient graph (equivalent transformation of original graph) is proposed to realize the stabilizability of large-scale network systems which can be reduced to a small-scale quotient graph. As a cornerstone of topology graph transformation, NSi graph division, which is proposed for the first time in this paper, plays a crucial role in stabilizability analysis and has far-reaching impact. It establishes an equivalent connection between the stabilizability of the original topology graph and the quotient graph. Results reveal a nontrivial relationship between stabilizability of uncontrollable multi-agent systems and network topology (i.e., the stabilizability of uncontrollable multi-agent systems is not only related to the selection of leaders, but also depends on uncontrollable modes.), which is established by quotient graph as a visual generalization of the original network topology. Surprisingly, structural balance is not always good, and unbalance has a positive effect on the stabilizability of the system. Finally, the simulation experiments demonstrate the effectiveness of the theoretical results.

Eigenvalues assignment in uncontrollable linear systems

Eigenvalues assignment in uncontrollable linear systems

LioNets: a neural-specific local interpretation technique exploiting penultimate layer information

Artificial Intelligence (AI) has a tremendous impact on the unexpected growth of technology in almost every aspect. AI-powered systems are monitoring and deciding about sensitive economic and societal issues. The future is towards automation, and it must not be prevented. However, this is a conflicting viewpoint for a lot of people, due to the fear of uncontrollable AI systems. This concern could be reasonable if it was originating from considerations associated with social issues, like gender-biased, or obscure decision-making systems. Explainable AI (XAI) is recently treated as a huge step towards reliable systems, enhancing the trust of people to AI. Interpretable machine learning (IML), a subfield of XAI, is also an urgent topic of research. This paper presents a small but significant contribution to the IML community, focusing on a local-based, neural-specific interpretation process applied to textual and time-series data. The proposed methodology introduces new approaches to the presentation of feature importance based interpretations, as well as the production of counterfactual words on textual datasets. Eventually, an improved evaluation metric is introduced for the assessment of interpretation techniques, which supports an extensive set of qualitative and quantitative experiments.

Hybridized Ackermann’s Methods

Further elaborations on the modified Ackermann’s method (MAM) for eigenvalue assignment are considered in this paper. Additional results concerning the incomplete eigenvalue assignment (IEA) are stated, verified, and commented. The advantages of IEA are pursued even further in this study beyond that mentioned in [1]. The study proposes two newly appended approaches based on MAM; named spectral and truncated methods. They are grounded on IEA, which fundamentally exemplify a hybridized approach to eigenvalue assignment. Necessary and sufficient conditions for stability of the truncated hybridized method are established, proved, and validated by examples. All results obtained apply equally-well to identical eigenvalue assignment, complex eigenvalue assignment, as well as to uncontrollable systems. Besides, they lead to simplified state feedback matrix determination. Three numerical examples are fully worked out to substantiate the nature of the IEA and the two hybridized methods. Simulation and visualization using MATLAB demonstrate the flexibility of the proposed methods.

Competing Theories of Value: A Spectral Analysis

Abstract This paper is based on a spectral–controllability analysis of the Sraffian price system and develops a relevant reconstruction of the theory of value. It shows that (i) the hitherto competing theories of value correspond to specific complex plane locations of the eigenvalues of the vertically integrated technical coefficients matrix; and (ii) the real-world economies cannot be coherently analyzed in terms of the traditional theory of value (classical, Marxian, Austrian, and neoclassical), despite the fact that they are characterized by rather low degrees and relatively low numerical ranks of price controllability. Hence, on the one hand, the Sraffian theory of value is not only the most general to date, but also empirically relevant. On the other hand, the real-world economies constitute almost uncontrollable systems, and this explains, in turn, the specific shape features of the empirical price–wage–profit rate curves that are at the heart of the capital theory debate.

Ackermann’s Method: Revisited, Extended, and Generalized to Uncontrollabe Systems

The celebrated method of Ackermann for eigenvalue assignment of single-input controllable systems is revisited in this paper, contributing an elegant proof. The new proof facilitates a compact formula which consequently permits an extension of the method to what we call incomplete assignment of eigenvalues. The inability of Ackermann’s formula to deal with uncontrollable systems is considered a weakness inherent in the method. The notion of incomplete assignment leads to a straightforward generalization of our method to eigenvalue assignment of uncontrollable systems, thus mitigating such a drawback of a popular method. Further results concerning the incomplete assignment are stated, verified, and commented. Such results reveal the trace of the state matrix $A$ a worthy feature pertinent to an open loop system. Finally, four numerical examples are worked out to demonstrate cases of incomplete, and uncontrollable eigenvalues assignment. The examples consider a case where the structure of the feedback matrix can be easily simplified. The paper ends with a commentary brief concerning some commonly used MATLAB commands for eigenvalue assignment.

Nuclear liquid-gas phase transition with machine learning

The machine-learning techniques have shown their capability for studying phase transitions in condensed matter physics. Here, we employ the machine-learning techniques to study the nuclear liquid-gas phase transition. We adopt an unsupervised learning and classify the liquid and gas phases of nuclei directly from the final state raw experimental data of heavy-ion reactions. Based on a confusion scheme which combines the supervised and unsupervised learning, we obtain the limiting temperature of the nuclear liquid-gas phase transition. Its value $9.24\pm0.04~\rm MeV$ is consistent with that obtained by the traditional caloric curve method. Our study explores the paradigm of combining the machine-learning techniques with heavy-ion experimental data, and it is also instructive for studying the phase transition of other uncontrollable systems, like QCD matter.

Control Based on Linear Algebra for Trajectory Tracking and Positioning of Second-Order Chained Form System

The development of controllers for underactuated systems with nonholonomic constraints has been a topic of significant interest for many researchers in recent years. These systems are hard to control because their linearization transform them into uncontrollable systems. The proposed approaches involve the use of a permanent excitation in the reference trajectory; coordinate transformation; discontinuities; or complex calculations. This paper proposes the design of the controller of the second-order chained form system for trajectory tracking by using a simpler approach based on linear algebra. Up to the present time, no controllers based on this approach have been designed for that system. The control problem is solved by setting two of the three systems variables as a reference, while the remaining variable is calculated imposing the condition that the equations system has an exact solution to ensure that tracking errors go to zero. The stability of the proposed controller is theoretically demonstrated, and simulations results show a suitable control system performance. Also, no coordinate transformation is necessary.

Data-Driven Tests for Controllability

The fundamental lemma due to Willems et al. “A note on persistency of excitation,” Syst. Control Lett., vol. 54, no. 4, pp. 325-329, 2005 plays an important role in system identification and data-driven control. One of the assumptions for the fundamental lemma is that the underlying linear time-invariant system is controllable. In this letter, the fundamental lemma is extended to address system identification for uncontrollable systems. Then, a data-driven algebraic test is derived to check whether the underlying system is controllable or not. An algorithm based on the singular value decomposition of a Hankel matrix constructed from the data is provided to implement the developed test. The algorithm has cubic computational cost. Examples are given to illustrate the theoretical results.

Semiglobal stabilization of linearly uncontrollable and unobservable nonlinear systems via sampled‐data control

SummaryThis article investigates the problem of using sampled‐data state/output feedback to semiglobally stabilize a class of uncertain nonlinear systems whose linearization around the origin is neither controllable nor observable. For any arbitrarily large bound of initial states, by employing homogeneous domination approach and a homogeneous version of Gronwall‐Bellman inequality, a sampled‐data state feedback controller with appropriate sampling period and scaling gain is constructed to semiglobally stabilize the system. In the case when not all states are available, a reduced‐order sampled‐data observer is constructed to provide estimates for the control law, which can guarantee semiglobal stability of the closed‐loop system with carefully selected sampling period and scaling gain.

Genera of Conjoined Bases for (Non)oscillatory Linear Hamiltonian Systems: Extended Theory

In this paper we study the properties of conjoined bases of a general linear Hamiltonian system without any controllability condition. When the Legendre condition holds and the system is nonoscillatory, it is known from our previous work that conjoined bases with eventually the same image form a special structure called a genus. In this work we extend the theory of genera of conjoined bases to arbitrary systems, for which the Legendre condition is not assumed and/or the system may be oscillatory. We derive a classification of all genera of conjoined bases and show that they form a complete lattice. These results are based on the relationship between subspaces of solutions of a linear control system and orthogonal projectors satisfying a certain Riccati type differential equation. The presented theory is applied in our paper (Sepitka in Discrete Contin Dyn Syst 39(4):1685–1730, 2019) to general Riccati matrix differential equations for possibly uncontrollable linear Hamiltonian systems.

State canonical form for a class of uncontrollable linear systems

ABSTRACT We tackle the obtaining of canonical forms for classifying linear control systems with regard to changes in the state variables. Although it was solved in the 1980s for controllable systems, it is still an open problem for the general case of multiparametric non-necessarily controllable systems. Here we obtain a general reduced form which is canonical for a class of systems which includes the cases already known (uniparametric systems, controllable systems).