Open Right-half Plane Research Articles

Stability Analysis of Distributed-Order Systems

The paper deals with stability of distributed order systems. A recently proposed method for stability analysis of a wide class of linear stationary systems is shown to be applicable to distributed order systems as well. In addition, we demonstrate that the well-known distributed order PID control algorithm should be used with utmost care, since the corresponding transfer functions introduce dipoles in the open right-half complex plane. The analysis is illustrated by a numerical example.

Wave analysis and control of two-dimensionally connected damped mass-spring systems (Analysis based on analytic properties of secondary constants)

This paper considers wave analysis and control of two-dimensionally connected damped mass-spring systems, focusing on the properties of the secondary constants as an analytic function of the Laplace transform variable s. Mass motion in the longitudinal direction is considered. The system can be viewed as a cascade connection of layers comprising the lateral direction elements. The dynamics can be described by a first order recurrence formula in the Laplace transform domain. We show that the characteristic polynomial of the coefficient matrix can be decomposed into second order polynomials, which reveals analyticity of the secondary constants in the open right-half plane, as well as the separation property of the propagation constants and the positive real property of the characteristic admittances. These properties justify the harmonic analysis in the wave analysis and guarantee the closed loop stability of the impedance matching controller. Numerical examples illustrate the derived results and show effectiveness for vibration control.

Some spectral properties of operator-valued positive-real functions

We consider operator-valued positive-real functions H and show that the intersections of the point, continuous and residual spectra of H ( s ) with the imaginary axis do not depend on s . In particular, if H is positive real and H ( z ) is invertible for some z in the open right-half plane, then H ( s ) is invertible for all s in the open right-half plane. Furthermore, we prove that the eigenspace of H ( s ) corresponding to an imaginary eigenvalue does not depend on s . It is also shown that the intersection of the numerical range of H ( s ) with the imaginary axis is independent of s . Finally, we prove that, under suitable assumptions, application of a “sufficiently positive-real” static output feedback to a positive-real transfer function leads to a strictly positive-real closed-loop system.

Analytical properties of secondary constants of uniform and uniformly varying mono-coupled periodic structures

This study considers the wave analysis and control of mono-coupled periodic mechanical systems, focusing on the properties of the secondary constants (propagation constants and characteristic impedances) as analytic functions of the Laplace transform variable s. In a general wave analysis process, we derive the secondary constants from the system dynamics, and evaluate the steady-state response for harmonic excitation (harmonic analysis). This corresponds to evaluating the transfer functions on a point on the imaginary axis s=jω. A correct answer is provided if the secondary constants are defined as analytic in the open right-half plane (RHP); however, the result is invalid if singularities exist there. This study addresses this issue, and reveals that the secondary constants can actually be defined as analytic in the open RHP and can satisfy several required properties. We firstly investigate the uniform case, and then apply the procedure to the non-uniform case. We derive a system condition to be applied to the wave analysis, and demonstrate that series connections of several uniformly varying systems satisfy this condition. A numerical example is presented to illustrate the results and the effectiveness of the impedance matching controller for vibration control.

Nyquist Interpretation of the Large Gain Theorem

This paper presents a proof of the Large Gain Theorem using the Nyquist Stability Criterion. The minimum gain constraint stipulated by the Large Gain Theorem guarantees the open-loop transfer function encircles the point (-1,0) exactly P times in the counterclockwise direction, where P is the number of open-loop open right-half plane poles. This guarantees asymptotic stability of the feedback system, even in the presence of an unstable open-loop transfer function. The Nyquist interpretation of the Large Gain Theorem is compared to Nyquist interpretations of the Large Gain and Passivity Theorems. Applications of the Large Gain Theorem are discussed and numerical examples illustrating the concept of minimum gain and the Large Gain Theorem are presented.

A New Look at the Target Feedback Loop Parameterization for $$\mathcal {H}_{\infty }$$ H ∞ /LTR Control

In this paper, a new approach on the target feedback loop (TFL) parameterization for the \(\mathcal {H}_{\infty }\)/loop transfer recovery (LTR) control is presented. The proposed procedure, which is based on a similarity transformation, allows to obtain a first-order system behavior through a particular shaping of the TFL sensitivity in terms of singular values. Furthermore, the proposed methodology makes possible to augment the design plant model with any stable real-finite pole, and not only a pole at the origin, as classical TFL parameterization procedures suggest. Thus, the proposed methodology should gather great attention by LTR-based control applications in which the use of integrators in the open-loop is undesirable or even prohibitive. Additional properties are derived if the design plant model does not have poles at the open right-half plane. Since \(\mathcal {H}_{\infty }\)/LTR control can be viewed as a generalization of well-known LQG/LTR control, the proposed methodology in this paper to shape the TFL can be properly extended to the LQG/LTR control.

Schur Algorithm in The Class $${\mathcal{SI}}$$ of J-contractive Functions Intertwining Solutions of Linear Differential Equations

In the PhD thesis of the second author under the supervision of the third author was defined the class SI of J-contractive functions, depending on a parameter and arising as transfer functions of overdetermined conservative 2D systems invariant in one direction. In this paper we extend and solve in the class SI, a number of problems originally set for the class SC of functions contractive in the open right-half plane, and unitary on the imaginary line with respect to some preassigned signature matrix J. The problems we consider include the Schur algorithm, the partial realization problem and the Nevanlinna-Pick interpolation problem. The arguments rely on a correspondence between elements in a given subclass of SI and elements in SC. Another important tool in the arguments is a new result pertaining to the classical tangential Schur algorithm.

Longitudinal Aircraft Dynamics and the Instantaneous Acceleration Center of Rotation

Nonminimum-phase zeros, that is, closed-right-half-plane (CRHP) zeros, affect both the open-and closed-loop behavior of continuous-time linear systems in undesirable ways [1]. For example, an asymptotically stable linear system with an odd number of positive zeros experiences initial undershoot to a step input (see “Initial Undershoot”). Moreover, under the rules of root locus, zeros in the open-right-half plane (ORHP) attract closed-loop poles, which limits the controller gain and thus the performance of the closed-loop system.

Dynamic consensus of linear multi-agent systems

This study concerns the consensus of a network of agents with general linear or linearised dynamics, whose communication topology contains a directed spanning tree. An observer-type consensus protocol based on the relative outputs of the neighbouring agents is adopted. The notion of consensus region is introduced, as a measure for the robustness of the protocol and as a basis for the protocol design. For neutrally stable agents, it is shown that there exists a protocol achieving consensus together with a consensus region that is the entire open right-half plane if and only if each agent is stabilisable and detectable. An algorithm is further presented for constructing such a protocol. For consensus with a prescribed convergence speed, a multi-step protocol design procedure is given, which yields an unbounded consensus region and at the same time maintains a favourable decoupling property. Finally, the consensus algorithms are extended to solve the formation control problems.

Stabilization of a Riderless Bicycle [Applications of Control

The bicycle provides transportation for leisure, recreation, and travel between home and work, throughout the world, in big cities as well as in small villages, supporting human mobility for more than a century. Modeling, analysis, and control of bicycle dynamics has been an attractive area of research in the last century as well as in recent years. Bicycle dynamics has attracted the attention of the automatic control research community because of its peculiar features, such as, for example, the fact that it depends strongly on the bicycle speed and that, under certain conditions, it exhibits both open right-half plane poles and zeros, making the design of feedback controllers for either balancing the bicycle in the upright position or moving it along a predefined path a challenging problem. In this article, the LPV nature of the bicycle dynamics is exploited to design a control system that automatically balances a riderless bicycle in the upright position. More precisely, the problem is formulated as the design of an LPV state-feedback controller that guarantees stability of this two-wheeled vehicle when the speed varies within a given range and its derivative is bounded.

Synthesis of Complete Orthonormal Fractional Basis Functions With Prescribed Poles

In this paper, fractional orthonormal basis functions that generalize the well-known fixed pole rational basis functions are synthesized. For a range of noninteger differentiation orders and under mild restrictions on the choice of the basis poles, the synthesized basis functions are shown to be complete in the space of functions which are analytic on the open right-half plane and square-integrable on the imaginary axis. This presents an extension of the completeness results for the fractional Laguerre and Kautz bases to fractional orthonormal bases with prescribed pole locations.

SYNTHESIS OF COMPLETE ORTHONORMAL FRACTIONAL BASIS FUNCTIONS WITH PRESCRIBED POLES

In this paper, fractional orthonormal basis functions which generalize the well-known fixed pole rational basis functions are synthesized. For a range of non-integer differentiation orders and under mild restrictions on the choice of the basis poles, the synthesized basis functions are shown to be complete in the space of functions which are analytic on the open right-half plane and square-integrable on the imaginary axis. This presents an extension of the completeness results for the fractional Laguerre and Kautz bases to fractional orthonormal bases with prescribed pole locations.

Adaptive Low-Gain Integral Control of Multivariable Well-Posed Linear Systems

The principle of low-gain integral control for finite-dimensional systems is well known. More recently, low-gain integral control results have been obtained for classes of infinite-dimensional systems. In this paper we show that integral control with a simple and natural adaptation of the integrator gain achieves tracking of constant reference signals for every exponentially stable, multivariable, well-posed, infinite-dimensional, linear system whose steady-state gain matrix has its spectrum in the open right-half plane. Our results considerably extend, improve, and simplify previous work by the authors [SIAM J. Control Optim. {35} (1997), pp. 78--116].

Complementarity forms of theorems of Lyapunov and Stein, and related results

The well-known Lyapunov's theorem in matrix theory / continuous dynamical systems asserts that a (complex) square matrix A is positive stable (i.e., all eigenvalues lie in the open right-half plane) if and only if there exists a positive definite matrix X such that AX+XA * is positive definite. In this paper, we prove a complementarity form of this theorem: A is positive stable if and only if for any Hermitian matrix Q, there exists a positive semidefinite matrix X such that AX+XA *+Q is positive semidefinite and X[AX+XA *+Q]=0 . By considering cone complementarity problems corresponding to linear transformations of the form I−S, we show that a (complex) matrix A has all eigenvalues in the open unit disk of the complex plane if and only if for every Hermitian matrix Q, there exists a positive semidefinite matrix X such that X−AXA *+Q is positive semidefinite and X[X−AXA *+Q]=0 . By specializing Q (to −I), we deduce the well known Stein's theorem in discrete linear dynamical systems: A has all eigenvalues in the open unit disk if and only if there exists a positive definite matrix X such that X−AXA * is positive definite.

ACHIEVING MINIMUM PHASE TRANSFER FUNCTION FOR A NONCOLLOCATED SINGLE‐LINK FLEXIBLE MANIPULATOR

ABSTRACTA noncollocated system has the potential of providing more precise tracking, improved disturbance rejection and increased bandwidth at the sensor location, but is considerably more difficult to stabilize than a collocated system due to its nonminimum phase nature. For a flexible manipulator, the problem becomes even more complicated because the system is inherently infinite dimensional. In this paper, a single‐link flexible manipulator having a tip payload and a noncollocated actuator and sensor is investigated using a linear distributed parameter model. With the joint torque as the input and the joint angle plus a weighted value of tip deflection as the measured output, an exact transfer function involving transcendental functions is derived. Using the methods of infinite product expansion, the root locus, and the asymptotic property of the roots of the transcendental equation, a necessary and sufficient condition in terms of the weighting factor of tip deflection is obtained such that the transfer function does not have any open right‐half plane zeros. This condition depends neither on the physical properties of the link nor on the mass properties of the tip payload. In order to correct the misinterpretation which has occurred in some closely‐related works concerning the stability of the infinite‐dimensional zero‐dynamics, the equivalence of the zeros of the transfer function and the eigenvalues of zero‐dynamics is also verified. Numerical results for the closed‐loop performance of a single‐link flexible arm using collocated and noncollocated PD controllers are given to show the efficacy of the proposed minimum phase transfer function approach.

Inner-outer factorization for non-square proper functions with infinite and finite j omega -axis zeros

The inner-outer factorization of a non-square proper function which has infinite and finite jomega -axis zeros and has no poles in the open right-half plane is studied in this paper via the concept of an infinite zero compensator. For a stabilizable and detectable realization of the above function with full normal column rank, the eigenstructures of the system matrix pencils of the function and its spectral density matrix are discussed based on Kronecker's theory, and a simple state-space solution to the inner-outer factorization is given via the solutions of the generalized eigenvalue problems in terms of the original realization of the function.

On a root distribution criterion for interval polynomials

H. Kokame and T. Mori (1991) and C.B. Soh (1990) derived conditions under which an interval polynomial has a given number of roots in the open left-half plane and the other roots in the open right-half plane. However, the one-shot-test approach using Sylvester's resultant matrices and Bezoutian matrices implies that the implemented conditions are only sufficient (not necessary) for an interval polynomial to have at least one root in the open left-half plane and open right-half plane. Alternative necessary and sufficient conditions, which only require the root locations of four polynomials to check the root distribution of an interval polynomial, are presented.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A root distribution criterion for interval polynomials

The problem of finding the conditions under which an interval polynomial has a given number of roots in the open left-half plane and the other roots in the open right-half plane, irrespective of the values of its coefficients, is considered. A simple criterion is provided to test interval polynomials for the root distribution invariance, viewed as an extension of Kharitonov's theorem. The goal is to provide an alternative theorem and then give an efficient means of checking the root distribution invariance. >

Quantization and overflow effects in digital implementations of linear dynamic controllers

The stability of a class of single-input/single-output (SISO) digital-feedback control systems is investigated. The systems considered contain a linear dynamic plant, a digital controller, and suitable D/A (digital-to-analog) and A/D converters. The design of such systems is usually accomplished by ignoring the nonlinear effects caused by quantization and overflow truncation. Simulations of specific examples establish the quantization in such systems can lead to loss of asymptotic stability of the origin. Obtained results prove that when quantization is taken into account (but overflow is neglected), one only has convergence to some (small) neighborhood about the origin. The results also prove that for initial conditions sufficiently far from the origin, overflow effects can lead to unbounded solutions. In particular, this is the case for observable systems with at least one eigenvalue in the open right-half plane (RHP). The case of systems with no eigenvalues in the open RHP, but with eigenvalues on the j omega -axis, is still unresolved. >

A stable state-space realization in the formulation of H&amp;lt;sup&amp;gt;∞&amp;lt;/sup&amp;gt;norm computation

In the two block Hinftyoptimization problem, usually we are given the state-space realizations of the proper rational matrices R_{1}(s) and R_{2}(s) whose poles are all the open right-half plane. Two problems are studied in the note. The first is the evaluation of \phi(s)R_{1}(s) at s = s_{k}, k = 1, 2, ..., n , where \phi(s) is an inner function whose zeros {s_{k}, k = 1, 2, ..., n } are the poles of R_{1}(s) . This evaluation is essential if Chang and Pearson's method is used for computing the optimal Hinftynorm. The problem is solved in state space via the solutions of Lyapunov equations. Neither polynomial matrix manipulations nor numerical pole-zero cancellations are involved in the evaluation. The second problem is to find a stable state-space realization of S(s) = U(s)R_{2}(s) where U(s) is an inner matrix. This problem arises in the spectral factorization of \gamma^{2} - R_{2}^{\ast}R_{2} . Doyle and Chu had a method for constructing stable S(s) based on a minimal realization of R_{2}(s) . An alternate method is proposed. The alternate method does not require a minimal realization of R_{2}(s) and only a Lyapunov equation is involved.