Invariant Zeros Research Articles

Modification of Infinite and Unstable Invariant Zeros in Linear Systems Using Stability-Preserving State Feedback

Modification of Infinite and Unstable Invariant Zeros in Linear Systems Using Stability-Preserving State Feedback

A way to identify whether a DFT gap is from right reasons or error cancellations: The case of copper chalcogenides.

Gap opening remains elusive in copper chalcogenides (Cu2X, X = S, Se, and Te), not least because Hubbard + U, hybrid functional, and GW methods have also failed. In this work, we elucidate that their failure originates from a severe underestimation of the 4s-3d orbital splitting of the Cu atom, which leads to a band-order inversion in the presence of an anionic crystal field. As a result, the Fermi energy is pinned due to symmetry, yielding an invariant zero gap. Utilizing the hybrid pseudopotentials to correct the underestimation on the atomic side opens up gaps of experimental magnitude in Cu2X, suggesting their predominantly electronic nature. Our work not only clarifies the debate about the Cu2X gap but also provides a way to identify which of the different methods really captures the physical essence and which is the result of error cancellation.

Interpolation with strongly F-positive real matrix and application to strong stabilization

Necessary and sufficient condition for the newly introduced problem to find a strongly F-positive real rational interpolant for a two-sided interpolation problem are given, for some matrix F to-be-found, as well as the set of all solutions. The strong F-positive realness of the interpolant H implies that H is bi-stable and bi-proper. Since such an interpolant is needed in the strong stabilization problem, the interpolation result is used to develop a control design algorithm, so that a stable stabilizing controller is constructed using the matrices F and H. It is shown (also by examples) that the algorithm works with large-scale plants possessing both real and complex invariant zeros in ℜ[s]≥0. The design algorithm is applied to the sugar mild process, and it is shown that the obtained controller is more suitable for practical implementation, in respect to the literature. By a simple analogy, it is shown that a minimum-phase stabilizing controller can be found by the same interpolation algorithm.

A graph-oriented approach to address generically flat outputs in structured LTI discrete-time systems

This paper addresses difference flatness for structured LTI discrete-time systems. Two forms of necessary and sufficient conditions for an output to be a structural flat output are given. First, a preliminary result algebraically defines a flat output in terms of invariant zeros regardless whether an LTI system is structured or not. Next, the conditions are expressed in terms of graphical conditions to define a structural flat output. Checking for the graphical conditions calls for algorithms that have polynomial-time complexity and that are commonly used for digraphs. The tractability of the conditions is illustrated on several examples.

Co-design of linear low-and-high gain feedback and high gain observer for suppression of effects of peaking on semi-global stabilization

This paper presents a co-design of linear low-and-high gain state feedback and linear high gain observer to suppress the adverse effects of peaking and thus to enlarge the class of nonlinear systems that can be semi-globally asymptotically stabilized by output feedback. It is known that a globally asymptotically stable nonlinear system cascaded by a stabilizable linear system through its output can be semi-globally asymptotically stabilized by a combination of linear low gain and high gain feedback of the state of the linear subsystem as long as the linear subsystem is right invertible and has all its invariant zeros located on the closed left-half plane. Such low-and-high state feedback can be implemented by an observer. To retain a domain of attraction arbitrarily close to that under a given state feedback law, a high gain observer is employed to achieve arbitrarily fast decay to zero of the observation errors of all states and the effect of the peaking phenomenon associated with the high observer gain is overcome by saturating the control input outside the desired domain of attraction under the state feedback law. Existing high gain observer designs thus require the linear subsystem to be observable. The proposed co-design does not resort to making all state observation errors to decay to zero arbitrarily fast and thus allows the linear subsystem to be detectable but not observable. Moreover, the proposed design does not resort to saturating the control input and results in linear feedback laws that are parameterized in a single parameter and easy to tune.

Data-Driven Meets Geometric Control: Zero Dynamics, Subspace Stabilization, and Malicious Attacks

Studying structural properties of linear dynamical systems through invariant subspaces is one of the key contributions of the geometric approach to system theory. In general, a model of the dynamics is required in order to compute the invariant subspaces of interest. In this letter we overcome this limitation by finding direct data-driven formulas for some of the foundational tools of the geometric approach. We use our results to (i) find a feedback gain that confines the system state within a subspace, (ii) compute the invariant zeros of the unknown system, and (iii) design attacks that remain undetectable.

Trajectory tracking in networks of linear systems

This paper proposes a control method for trajectory tracking in interconnected linear systems. The physical couplings among all subsystems are considered by a modification of the reference signals of the individual subsystems. Feedforward controllers are designed for the modified isolated systems. This way of solution leads to a set of differential–algebraic equations whose solution depends on the invariant zeros of the subsystems. The design of the controllers needs only a single communication step for the information exchange between neighboring subsystems. The main results of the paper are a new representation of the internal dynamics as differential–algebraic equations that lead to a relation between the tracked flat output and the original subsystem output and the proof that the proposed networked controllers ensure trajectory tracking.

Reduction of SISO H-infinity output feedback control problem

We consider the linear matrix inequality (LMI) problem of H∞ output feedback control problem for a generalized plant whose control input, measured output, disturbance input, and controlled output are scalar. We provide an explicit form of the optimal value. This form is the unification of some results in the literature of H∞ performance limitation analysis. To obtain the form of the optimal value, we focus on the non-uniqueness of perpendicular matrices, which appear in the LMI problem. We use the null vectors of invariant zeros associated with the dynamical system for the expression of the perpendicular matrices. This expression enables us to reduce and simplify the LMI problem. Our approach uses some well-known fundamental tools, e.g., the Schur complement, Lyapunov equation, Sylvester equation, and matrix completion. We use these techniques for the simplification of the LMI problem. Also, we investigate the structure of dual feasible solutions and reduce the size of the dual. This reduction is called a facial reduction in the literature of convex optimization.

Characterization of the dual problem of linear matrix inequality for H-infinity output feedback control problem via facial reduction

This paper deals with the minimization of H_infty output feedback control. This minimization can be formulated as a linear matrix inequality (LMI) problem via a result of Iwasaki and Skelton 1994. The strict feasibility of the dual problem of such an LMI problem is a valuable property to guarantee the existence of an optimal solution of the LMI problem. If this property fails, then the LMI problem may not have any optimal solutions. Even if one can compute parameters of controllers from a computed solution of the LMI problem, then the computed H_infty norm may be very sensitive to a small change of parameters in the controller. In other words, the non-strict feasibility of the dual tells us that the considered design problem may be poorly formulated. We reveal that the strict feasibility of the dual is closely related to invariant zeros of the given generalized plant. The facial reduction is useful in analyzing the relationship. The facial reduction is an iterative algorithm to convert a non-strictly feasible problem into a strictly feasible one. We also show that facial reduction spends only one iteration for so-called regular H_infty output feedback control. In particular, we can obtain a strictly feasible problem by using null vectors associated with some invariant zeros. This reduction is more straightforward than the direct application of facial reduction.

On Unknown Input Observers of Linear Systems: Asymptotic Unknown Input Decoupling Approach

This note presents a new unknown input observer (UIO) with adjustable performances for linear systems. It focuses on state estimation in the presence of zeros with slow dynamics. It is known that if the system has stable invariant zeros and satisfies the decoupling condition then a UIO exists. However, the invariant zeros constrain the pole placement and consequently the state estimation error convergence rate, especially, when the zeros dynamics are slow. In the proposed UIO, the exact decoupling condition is relaxed by the introduction of the asymptotic decoupling notion which allows to act on the invariant zeros and then enhance the convergence rate of the state estimation error. Numerical examples are provided to illustrate the performances of the UIO.

Exact and Causal Inversion of Nonminimum-Phase Systems: A Squaring-Down Approach for Overactuated Systems

Nonminimum-phase zeros pose challenges for controller design, i.e., in inversion-based control approaches where inverting these zeros may result in “unstable” poles. The aim of this paper is to exploit the additional freedom in overactuated systems to eliminate these zeros, facilitating subsequent inversion. In particular, an approach for causal and exact inversion of systems with nonminimum-phase behavior is presented. In an either static or dynamic squaring-down step prior to inversion, the approach exploits the fact that nonsquare systems typically have no invariant zeros. The proposed approach is successfully demonstrated in experiments on an overactuated motion system. The method enables exact inversion for nonsquare systems without requiring preview or preactuation.

Linear Hybrid Systems With Periodic Jumps: A Notion of Strong Observability and Strong Detectability

In this paper, two important structural properties, i.e., strong observability and strong detectability , are introduced for linear hybrid systems with periodic jumps. These properties are characterized in terms of geometric and algebraic conditions over the system matrices. The concepts and characterizations of the weakly unobservable subspace and the hybrid invariant zeros are also introduced, respectively. An algorithm to compute the weakly unobservable subspace is provided . In addition, it is shown that there exists a close relationship between the hybrid invariant zeros and the properties of strong observability and strong detectability. Some examples illustrate the proposed properties.

Asymptotic Unknown Input Decoupling Observer for Discrete-Time LTI Systems

This letter addresses a new observer structure and its corresponding design for linear time invariant discrete-time systems affected by unknown inputs (UIs). We show that the performances of standard unknown input observer (UIO) can be enhanced, especially, in the presence of stable invariant zeros with slow dynamics (detectable systems). Under mild condition, the proposed UIO design avoids the drawback of the invariant zeros, at least in a time interval, and allows to arbitrary place the eigenvalues during that interval. The main idea is to relax the exact UI decoupling condition for an asymptotic decoupling condition. The convergence analysis is analyzed using the Lyapunov theory and the asymptotic convergence conditions of the state estimation error are expressed in linear matrix inequality formalism.

A Further Result on Semi-global Stabilization of Minimum-Phase Input–Output Linearizable Nonlinear Systems by Linear Partial State Feedback

It has been established that a globally asymptotically stable nonlinear system cascaded by a controllable linear system through its output can be semi-globally asymptotically stabilized by linear feedback of the state of the linear subsystem as long as the linear system is right invertible and has all its invariant zeros located on the closed left-half plane. In particular, a globally asymptotically stable nonlinear system cascaded by a chain of integrator through the state of any one integrator or the control input is semi-globally asymptotically stabilizable by linear feedback of the states of the integrators. In this note, we show that the restriction on the interconnection between the linear and nonlinear subsystems can be further relaxed. For example, a globally asymptotically stable nonlinear system cascaded by a chain of integrator can still be semi-globally stabilized even when the cascade is through the states of two consecutive integrators, with the input regarded as an additional state following the last integrator.

Inversion of hyperoutput time-delay systems

An inversion problem for LTI hyperoutput time-delay system is considered. For such systems canonical form with isolated zero dynamics is obtained, system invariant zeros and their relation to spectral observability of zero dynamics subsystem are investigated. Using this results, inversion algorithm is suggested for time-delay systems.

MIMO tracking control of LTI systems: A geometric approach

This paper addresses the tracking problem of constant references for multiple-input multiple-output linear time-invariant systems. We provide necessary and sufficient conditions for the solvability of the problem under the minimal set of assumptions that guarantee the well-posedness of every control problem requiring stability. Our approach allows to solve tracking problems for systems which are possibly nonright-invertible, simply (not asymptotically) stabilizable, and possibly with invariant zeros at the origin. Our methodology is constructive and results in the design of a stabilizing feedback matrix and a feedforward signal which solve the problem.

Deadbeat unknown-input state estimation and input reconstruction for linear discrete-time systems

This paper considers discrete-time input reconstruction and state estimation assuming that the system has no invariant zeros, without assuming that the initial condition is known, and without assuming that at least one Markov parameter has full column rank. Algorithms based on the generalized inverse of a block-Toeplitz matrix are given for unknown-input state estimation and simultaneous input reconstruction and state estimation. In both cases, the unknown input is an arbitrary signal. Both algorithms are deadbeat, which means that exact input reconstruction and state estimation are achieved in a finite number of steps.

Equivariant minimal surfaces in $\mathbb{CH}^2$ and their Higgs bundles

This paper gives a construction for all minimal immersions $f$ of the Poincar\'{e} disc into the complex hyperbolic plane $\mathbb{CH}^2$ which are equivariant with respect to an irreducible representation $\rho$ of a hyperbolic surface group into $PU(2,1)$. We exploit the fact that each such immersion is a twisted conformal harmonic map and therefore has a corresponding Higgs bundle. We identify the structure of these Higgs bundles and show how each is determined by properties of the map, including the induced metric and a holomorphic cubic differential on the surface. We show that the moduli space of pairs $(\rho,f)$ is a disjoint union of finitely many complex manifolds, whose structure we fully describe. The holomorphic (or anti-holomorphic) maps provide multiple components of this union, as do the non-holomorphic maps. Each of the latter components has the same dimension as the representation variety for $PU(2,1)$, and is indexed by the number of complex and anti-complex points of the immersion. These numbers determine the Toledo invariant and the Euler number of the normal bundle of the immersion. We show that there is an open set of quasi-Fuchsian representations of Toledo invariant zero for which the minimal surface is unique and Lagrangian.

Reduction of H∞ state feedback control problems for the MIMO servo systems

AbstractWe deal with H∞ state feedback control problem for the multi‐input‐multi‐output (MIMO) servo system and discuss the advantages of the facial reduction (FR) to the resulting linear matrix inequality (LMI) problems. In fact, as far as our usual setting, the dual of the LMI problem is not strictly feasible because the generalized plant has always stable invariant zeros. Thus FR is available to such LMI problems, and we can reduce and simplify the original LMI problem to a smaller‐size LMI problem. As a result, we observe that the numerical performance of the SDP solvers is improved. Also, as a by‐product, we obtain the best performance index of the reduced LMI problem with a closed‐form expression. This helps the H∞ performance limitation analysis. Another contribution is to reveal that the resulting LMI problem obtained from H∞ control problem has a finite optimal value, but no optimal solutions under an additional assumption. This is also confirmed in the numerical experiment of this paper. FR also plays an essential role in this analysis.

Unknown input observer with stability: A structural analysis approach in bond graph

This paper presents a solution for the state and unknown input estimator of linear MIMO systems with a structural approach based on Bond Graph methodology.A systematic procedure is initially proposed at the analysis stage by studying the finite and infinite structures of the modeled system from a structural approach, similar to the case study for the classical input-output decoupling problem and the disturbance rejection problem. Afterwards, the observer that can be represented by a Bond Graph model is directly implemented from the original model at the synthesis stage. The monovariable case as well as the multivariable case are treated and the stability property of the classical error equations (state and disturbance) of the observer is studied, based on the stability property of the invariant zeros of the model. The observer is tested by an illustrative example that considers a real bar system used for different cases.