Uncontrollable Systems Research Articles

On the Optimal Control of Passive or Nonexpansive Systems

The positive-real and bounded-real lemmas solve two important linear-quadratic optimal control problems for passive and nonexpansive systems, respectively. The lemmas assume controllability, yet a passive or nonexpansive system can be uncontrollable. In this paper, we solve these optimal control problems without making any assumptions. In particular, we show how to extract the greatest possible amount of energy from a passive but not necessarily a controllable system (e.g., a passive electric circuit) using state feedback. A complete characterization of the set of solutions to the linear matrix inequalities in the positive-real and bounded-real lemmas is also obtained. In addition, we obtain necessary and sufficient conditions for a system to be nonexpansive that augment the bounded-real condition with new conditions relevant to uncontrollable systems.

Singular Sturmian separation theorems for nonoscillatory symplectic difference systems

ABSTRACTIn this paper, we derive new singular Sturmian separation theorems for nonoscillatory symplectic difference systems on unbounded intervals. The novelty of the presented theory resides in two aspects. We introduce the multiplicity of a focal point at infinity for conjoined bases, which we incorporate into our new singular Sturmian separation theorems. At the same time we do not impose any controllability assumption on the symplectic system. The presented results naturally extend and complete the known Sturmian separation theorems on bounded intervals by J. V. Elyseeva [Comparative index for solutions of symplectic difference systems, Differential Equations 45(3) (2009), pp. 445–459, translated from Differencial'nyje Uravnenija 45 (2009), no. 3, 431–444], as well as the singular Sturmian separation theorems for eventually controllable symplectic systems on unbounded intervals by O. Došlý and J. Elyseeva [Singular comparison theorems for discrete symplectic systems, J. Difference Equ. Appl. 20(8) (2014), pp. 1268–1288]. Our approach is based on developing the theory of comparative index on unbounded intervals and on the recent theory of recessive and dominant solutions at infinity for possibly uncontrollable symplectic systems by the authors [P. Šepitka and R. Šimon Hilscher, Recessive solutions for nonoscillatory discrete symplectic systems, Linear Algebra Appl. 469 (2015), pp. 243–275; P. Šepitka and R. Šimon Hilscher, Dominant and recessive solutions at infinity and genera of conjoined bases for discrete symplectic systems, J. Difference Equ. Appl. 23(4) (2017), pp. 657–698]. Some of our results, including the notion of the multiplicity of a focal point at infinity, are new even for an eventually controllable symplectic difference system.

On the internal signature and minimal electric network realizations of reciprocal behaviors

In a recent paper, it was shown that (i) any reciprocal system with a proper transfer function possesses a signature-symmetric realization in which each state has either even or odd parity; and (ii) any reciprocal and passive behavior can be realized as the driving-point behavior of an electric network comprising resistors, inductors, capacitors and transformers. These results extended classical results to include uncontrollable systems. In this paper, we establish new lower bounds on the number of states with even parity (capacitors) and odd parity (inductors) for reciprocal systems that need not be controllable.

A theory of passive linear systems with no assumptions

We present two linked theorems on passivity: the passive behavior theorem, parts 1 and 2. Part 1 provides necessary and sufficient conditions for a general linear system, described by a set of high order differential equations, to be passive. Part 2 extends the positive-real lemma to include uncontrollable and unobservable state-space systems.

On discrete algebraic Riccati equations: A rank characterization of solutions

We give a rank characterization of the solution set of discrete algebraic Riccati equations (DAREs) involving real matrices. Our characterization involves the use of invariant subspaces of the system matrix. We observe that a DARE can be considered as a specific type of a non-symmetric continuous-time algebraic Riccati equation. We show how to extract the low rank solutions from the full rank solution of a homogeneous DARE. We also study the eigen-structure of various feedback/closed loop matrices associated with solutions of a DARE. We show as a consequence of our observations that a solution X of a DARE for controllable systems and certain specific types of uncontrollable systems satisfies the inequality Xmin≤X≤Xmax. We characterize conditions under which the solution set of DARE is bounded/unbounded. We use the method of invariant subspaces to identify some invariant sets of a difference Riccati equation.

Controllability of networked higher-dimensional systems with one-dimensional communication.

In this paper, the state controllability of networked higher-dimensional linear time-invariant dynamical systems is considered, where communications are performed through one-dimensional connections. The influences on the controllability of such a networked system are investigated, which come from a combination of network topology, node-system dynamics, external control inputs and inner interactions. Particularly, necessary and sufficient conditions are presented for the controllability of the network with a general topology, as well as for some special settings such as cycles and chains, which show that the observability of the node system is necessary in general and the controllability of the node system is necessary for chains but not necessary for cycles. Moreover, two examples are constructed to illustrate that uncontrollable node systems can be assembled to a controllable networked system, while controllable node systems may lead to uncontrollable systems even for the cycle topology.This article is part of the themed issue 'Horizons of cybernetical physics'.

Global Stabilization via Sampled-Data Output Feedback for a Class of Linearly Uncontrollable and Unobservable Systems

This note considers the problem of global stabilization using sampled-data output feedback for a class of nonlinear systems in the $p$ - normal form which have uncontrollable and unobservable linearizations around the origin. A new sampled-data observer, which is featured with a special feedforward propagation structure, is constructed to estimate the unmeasurable states. The output feedback control law is obtained by discretizing the continuous-time homogeneous control law. Based on the trajectory estimation of the homogeneous system, a sampling period is determined to guarantee global asymptotic stability of the closed-loop system.

Local uncontrollability for affine control systems with jumps

ABSTRACTThis paper investigates affine control systems with jumps for which the ideal If(g1, …, gm) generated by the drift vector field f in the Lie algebra L(f, g1, …, gm) can be imbedded as a kernel of a linear first-order partial differential equation. It will lead us to uncontrollable affine control systems with jumps for which the corresponding reachable sets are included in explicitly described differentiable manifolds.

On controllability of homogeneous and inhomogeneous discrete-time multi-input bilinear systems in dimension two

ABSTRACTIn this paper, the controllability problem of two-dimensional discrete-time multi-input bilinear systems is completely solved. The homogeneous and the inhomogeneous cases are studied separately and necessary and sufficient conditions for controllability are established by using a linear algebraic method, which are easy to apply. Moreover, for the uncontrollable systems, near-controllability is considered and similar necessary and sufficient conditions are also obtained. Finally, examples are provided to demonstrate the results of this paper.

Yet another characterization of solutions of the Algebraic Riccati Equation

This paper deals with a characterization of the solution set of algebraic Riccati equation (ARE) (over reals) for both controllable and uncontrollable systems. We characterize all solutions using simple linear algebraic arguments. It turns out that solutions of ARE of maximal rank have lower rank solutions encoded within it. We demonstrate how these lower rank solutions are encoded within the full rank solution and how one can retrieve the lower rank solutions from the maximal rank solution. We characterize situations where there are no full rank solutions to the ARE. We also characterize situations when the number of solutions to the ARE is finite, when they are infinite and when they are bounded. We also explore the poset structure on the solution set of ARE, which in some specific cases turns out to be a lattice which is isomorphic to lattice of invariant subspaces of a certain matrix. We provide several examples that bring out the essence of these results.

Simplified Methods for Eigenvalue Assignment

A state feedback method of reduced order for eigenvalue assignment is developed in this paper. It offers immediate assignment of m eigenvalues, with freedom to assign the remaining n-m eigenvalues. The method also enjoys a systematic one-step application in the case where the system has a square submatrix. Further simplification is also possible in certain cases. The method is shown to be applicable to uncontrollable systems, offering the simplest control law when having maximum uncontrollable eigenvalues.

Explicit Determination of State Feedback Matrices

Methods which calculate state feedback matrices explicitly for uncontrollable systems are considered in this paper. They are based on the well-known method of the entire eigenstructure assignment. The use of a particular similarity transformation exposes certain intrinsic properties of the closed loop w-eigenvectors together with their companion z-vectors. The methods are extended further to deal with multi-input control systems. Existence of eigenvectors solution is established. A differentiation property of the z-vectors is proved for the repeated eigenvalues assignment case. Two examples are worked out in detail.

Uncontrollable dissipative systems: observability and embeddability

The theory of dissipativity is well developed for controllable systems. A more appropriate definition of dissipativity in the context of uncontrollable systems is in terms of the existence of a storage function, namely a function such that, along every system trajectory, its rate of change at each time instant is at most the power supplied to the system at that time. However, even when the supplied power is expressible in terms of just the external variables, the dissipativity property for uncontrollable systems crucially hinges on whether or not the storage function depends on variables unobservable/hidden from the external variables: this paper investigates the key aspects of both cases, and also proposes another intuitive definition of dissipativity. These three definitions are compared: we show that drawbacks of one definition are addressed by another.Dealing first with observable storage functions, under the conditions that no two uncontrollable poles add to zero and that dissipativity is strict as frequency tends to infinity, we prove that the dissipativities of a system and its controllable part are equivalent. We use the behavioural approach for formalising key notions: a system behaviour is the set of all system trajectories. We prove that storage functions have to be unobservable for ‘lossless’ uncontrollable systems. It is known, however, that unobservable storage functions result in certain ‘fallacious’ examples of lossless systems. We propose an intuitive definition of dissipativity: a system/behaviour is called dissipative if it can be embedded in a controllable dissipative superbehaviour. We prove embeddability results and use them to resolve the fallacy in the example termed ‘lossless’ due to unobservable storage functions. We next show that, quite unreasonably, the embeddability definition admits behaviours that are both strictly dissipative and strictly antidissipative. Drawbacks of the embeddability definition in the context of RLC circuits are finally related to the inability to realise/synthesise the special one-port electrical network, called the nullator, using only passive electrical components.

A Class of Uncontrollable Diffusively Coupled Multiagent Systems with Multichain Topologies

We construct systematically a class of uncontrollable diffusively coupled multiagent systems with a single leader and multichain topologies. For studying the controllability of diffusively coupled multiagent systems, such identified uncontrollable systems serve as counterexamples that prove the need to modify the existing sufficient condition using graph partitioning characterization. The uncontrollability of the constructed multichain structures can be preserved when the structures are further augmented to get better connected. The paper also provides an algorithm to obtain the minimal leader-invariant relaxed equitable partition for the graph associated with any diffusively coupled multiagent system guided by a single leader.

The ROSA26-iPSC Mouse: A Conditional, Inducible, and Exchangeable Resource for Studying Cellular (De)Differentiation

Control of cellular (de)differentiation in a temporal, cell-specific, and exchangeable manner is of paramount importance in the field of reprogramming. Here, we have generated and characterized a mouse strain that allows iPSC generation through the Cre/loxP conditional and doxycycline/rtTA-controlled inducible expression of the OSKM reprogramming factors entirely from within the ROSA26 locus. After reprogramming, these factors can be replaced by genes of interest-for example, to enhance lineage-directed differentiation-with the use of a trap-coupled RMCE reaction. We show that, similar toESCs, Dox-controlled expression of the cardiac transcriptional regulator Mesp1 together with Wnt inhibition enhances the generation of functional cardiomyocytes upon invitro differentiation of such RMCE-retargeted iPSCs. This ROSA26-iPSC mouse model is therefore an excellent tool for studying both cellular reprogramming and lineage-directed differentiation factors from the same locus and will greatly facilitate the identification and ease of functional characterization of the genetic/epigenetic determinants involved in these complex processes.

Ferromagnetism in Hydrogenated Graphene Nanopore Arrays

Theoretically, the so-called zigzag edge of graphenes provides localized electrons due to the presence of flat energy bands near the Fermi level. Spin interaction makes the localized spins strongly polarized, yielding ferromagnetism. However, in most experimental studies, ferromagnetism has been observed in uncontrollable and complicated carbon-based systems. Here, we fabricate graphenes with honeycomblike arrays of hexagonal nanopores, which have a large ensemble of hydrogen-terminated and low-defect pore edges that are prepared by a nonlithographic method (nanoporous alumina templates). We observe large-magnitude ferromagnetism derived from electron spins localizing at the zigzag nanopore edges. This promises to be a realization of graphene magnets and novel spintronic devices.

Uncontrollable dissipative dynamical systems

AbstractThe classical definition of dissipativity for Linear Time Invariant differential (LTI) systems requires non-negativity of the net power absorbed over all system trajectories of compact support. This definition forces one to deal with only controllable systems. The more general definition uses the existence of an internal storage function as the required condition for dissipativity. While this definition allows considering uncontrollable systems also, the existential aspect of the definition raises questions about what systems admit a storage function. In this paper we provide conditions on an LTI system under which a storage function exists. Further, we also prove that this storage function is a state function, meaning the storage of energy requires no more variables than those that have to satisfy constraints for impulse-free concatenability of system trajectories. While such results have been proved for controllable systems or for a special class of uncontrollable systems, we use techniques from indefinite linear algebra to strengthen past results.The proofs of the main results above requires new results in Algebraic Riccati Equation (ARE) solvability in the context of uncontrollable state space systems: these ARE solvability results are also stated and proved in this paper.

On Uncontrollable Discrete-Time Bilinear Systems Which are “Nearly” Controllable

In this note, for a class of uncontrollable discrete-time bilinear systems, it is shown that the controllable region “nearly” covers the whole space while the uncontrollable region is only a hypersurface. As a result, for almost any initial state and any terminal state of the system, the former can be transferred to the latter. In addition, the two-dimensional controllability counterexamples in are generalized to arbitrary finite-dimensional cases.

On pole structure assignment in linear systems

The problem of pole structure assignment (PSA) by state feedback in implicit, linear and uncontrollable systems is discussed in the article. It is shown that the problem is solvable if the system is regularisable. Then necessary and sufficient conditions for characteristic polynomial assignment are established. In the case of PSA (invariant polynomials assignment) just necessary conditions have been obtained. But it turns out that these conditions are also sufficient in some special cases. This happens, for example, when the system does not possess any non-proper controllability indexes. A possible application of the achieved results to modelling a constrained movement of a robot arm is outlined, too.

Uncontrollable quantum systems: A classification scheme based on Lie subalgebras

It is well known that a finite level quantum system is controllable if and only if the Lie algebra of its generators has full rank. When the rank of the Lie algebra is not full, there is a rich mathematical and physical structure to the subalgebra that to date has been analyzed only in special cases. We show that uncontrollable systems can be classified into reducible and irreducible ones. The irreducible class is the more subtle and can be related to a notion of generalized entanglement. We give a general prescription for revealing irreducible uncontrollable systems: the fundamental representation of $\text{su}(N)$, where $N$ is the number of levels, must remain irreducible in the subalgebra of $\text{su}(N)$. We illustrate the concepts with a variety of physical examples.