Nejat Olgac

An exact method for the stability analysis of time-delayed linear time-invariant (LTI) systems

A general class of linear time invariant systems with time delay is studied. Recently, they attracted considerable interest in the systems and control community. The complexity arises due to the exponential type transcendental terms in their characteristic equation. The transcendentality brings infinitely many characteristic roots, which are cumbersome to elaborate as evident from the literature. A number of methodologies have been suggested with limited ability to assess the stability in the parametric domain of time delay. This study offers an exact, structured and robust methodology to bring a closure to the question at hand. Ultimately we present a unique explicit analytical expression in terms of the system parameters which not only reveals the stability regions (pockets) in the domain of time delay, but it also declares the number of unstable characteristic roots at any given pocket. The method starts with the determination of all possible purely imaginary (resonant) characteristic roots for any positive time delay. To achieve this a simplifying substitution is used for the transcendental terms in the characteristic equation. It is proven that the number of such resonant roots for a given dynamics is finite. Each one of these roots is created by infinitely many time delays, which are periodically distributed. An interesting property is also claimed next, that the root crossing directions at these locations are invariant with respect to the delay and dependent only on the crossing frequency. These two unique findings facilitate a simple and practical stability method, which is the highlight of the work.

Robust output tracking control of nonlinear MIMO systems via sliding mode technique

The robust output tracking control problem of general nonlinear multi-input multi-output (MIMO) systems is discussed. The robustness against parameter uncertainties and unknown disturbances is considered. A second order sliding mode control (SMC) technique is used to establish the desired tracking. Input/output (I/O) linearization, relative degree, minimum phase and matching condition concepts are reviewed. Some earlier SMC strategies which are restricted to the systems in canonical form are extended to a much broader class of nonlinear dynamics. It is also shown that for unperturbed dynamics, the sliding phase of the SMC applications have a direct correspondence to the I/O linearization operations. Interesting parametric flexibilities emanate within the formation of the second order SMC, designating the “s dynamics” and the “error dynamics” segments as frequency domain filters. However, a critical impasse is posed in the off-line selections of the design parameters. A set of example cases is presented for a spacecraft attitude control problem. These examples manifest that the proposed control strategy is tunable to a desired response despite the disturbances and uncertainties.

Sliding mode control with perturbation estimation (SMCPE): a new approach

Abstract Sliding mode control (SMC) of a general class of nonlinear control systems is considered in this work. The conventional SMC technique requires knowledge of the upperbounds of disturbances and modelling uncertainties to assure robustness. However, this may not be easy to obtain. As a remedy, an estimation process for these dynamic perturbations is employed jointly with the SMC technique. This new methodology, sliding mode control with perturbation estimation (SMCPE), offers a robust feedback control with much lower gains than its conventional counterparts against slowly varying perturbations. This resolves one of the problematic issues which has caused concern over the years of development of SMC applications. An interesting perspective of selecting the cut-off frequency for s dynamics is presented with a novel upperbound argument. Much desirable tracking fidelity is arrived through SMCPE in the computer simulation studies for a two-link manipulator. A companion approach to SMCPE, the discrete equ...

Implementation of sliding mode control with perturbation estimation (SMCPE)

Experimental verification of a recently developed algorithm, sliding mode control with perturbation estimation (SMCPE), is performed, a two-axes planar SCARA type robot is used as the test platform. The controller is a PC-based microprocessor with transducer and actuator interfaces. The objective of trajectory tracking is achieved by directly controlling the joint torques, despite the modeling deficiencies and unknown disturbances. Two major practical issues are considered. One of them is the measurement noise and the other is the hard/software limitations on the control loop closure speed. Both of these issues affect the parametric selections with the SMCPE algorithm. A sample test result is presented, to compare the performance of SMCPE with the classical SMC.

Active Vibration Control of Distributed Systems Using Delayed Resonator With Acceleration Feedback

The Delayed Resonator (DR) and Dual Frequency Fixed Delayed Resonator (DFFDR) are newly introduced control techniques for active vibration absorption. Both methods propose a delayed position feedback within the absorber section of the structure to impart ideal resonance features to the absorber. When installed on an oscillating primary body, they form notch filters at their resonance frequencies attenuating the response of the primary structure. The DR absorber is shown to be real-time tunable to time varying disturbance frequencies. In this article, a number of new issues are considered. First, the basic theory is modified for acceleration feedback instead of position, which was originally proposed for the DR methodology. Second, the new absorption methods are implemented on distributed parameter structures which are under high frequency excitation (around 1 KHz). Stability of the combined structure is studied on a reduced order multi-degree-of-freedom primary structure together with the DR absorber. Experimental tests are conducted on a steel beam to verify the analytical findings. Piezoelectric actuators are used both to generate harmonic disturbances and to implement the control. The correspondence observed between the theoretical and experimental results is encouraging. The efficiency of the DR and DFFDR absorption techniques is demonstrated.

Sliding Mode Control With Sliding Perturbation Observer

This work introduces a new robust motion control algorithm using partial state feedback for a class of nonlinear systems in the presence of modelling uncertainties and external disturbances. The effects of these uncertainties are combined into a single quantity called perturbation. The major contribution of this work comes as the development and design of a robust observer for the state and the perturbation which is integrated into a Variable Structure Controller (VSC) structure. The proposed observer combines the procedures of Sliding Observers (Slotine et al, 1987) with the idea of Perturbation Estimation (Elmali and Olgac, 1992). The result is what is called Sliding Perturbation Observer (SPO). The VSC follows the philosophy of Sliding Mode Control (SMC) (Slotine and Sastry, 1983). This combination of controller/observer gives rise to the new routine called Sliding Mode Control with Sliding Perturbation Observer (SMCSPO). The stability analysis shows how the algorithm parameters are scheduled in order to assure the sliding modes of both controller and observer. A simplified form of the general design procedure is also presented in order to ease the practical applications of SMCSPO. Simulations are presented for a two-link manipulator to verify the proposed approach. Experimental validation of the methodology is also performed on a PUMA 560 robot. A superior control performance is obtained over some full state feedback techniques such as SMC and Computed Torque Method.

Brief A practical method for analyzing the stability of neutral type LTI-time delayed systems

A new paradigm is presented for assessing the stability posture of a general class of linear time invariant-neutral time delayed systems (LTI-NTDS). The ensuing method, which we name the direct method (DM), offers several unique features: It returns the number of unstable characteristic roots of the system in an explicit and non-sequentially evaluated function of time delay, @t. Consequently, the direct method creates exclusively all possible stability intervals of @t. Furthermore, it is shown that this method inherently verifies a widely accepted necessary condition for the @t-stabilizability of a LTI-NTDS. In the core of the DM lie a root clustering paradigm and the strength of Rekasius transformation in mapping a transcendental characteristic equation into an equivalent rational polynomial. In addition, we also demonstrate by an example that DM can tackle systems with unstable starting posture for @t=0, only to stabilize for higher values of delay, which is rather unique in the literature.

A unique methodology for the stability robustness of multiple time delay systems

Abstract The stability robustness is considered for linear time invariant (LTI) systems with rationally independent multiple time delays against delay uncertainties. The problem is known to be notoriously complex, primarily because the systems are infinite dimensional due to delays. Multiplicity of the delays in this study complicates the analysis even further. And “rationally independent” feature of the delays makes the problem prohibitively challenging as opposed to the TDS with commensurate time delays (where time delays are rationally related). A unique framework is described for this broadly studied problem and the enabling propositions are proven. We show that this procedure analytically reveals all possible stability regions exclusively in the space of the delays. As an added strength, it does not require the delay-free system under consideration to be stable. Our methodology offers a resolution to this question, which has been studied from variety of directions in the past four decades. None of these respectable investigations can, however, deliver an exact and exhaustive robustness declaration. From this stand point the new method has a unique contribution.

Extended Kronecker Summation for Cluster Treatment of LTI Systems with Multiple Delays

A new procedure is presented for determining the kernel and the offspring hypersurfaces for general linear time invariant (LTI) dynamics with multiple delays. These hypersurfaces, as they have very recently been introduced in a concept paper [R. Sipahi and N. Olgac, Automatica, 41 (2005), pp. 1413-1422], form the basis of the overriding paradigm which is called the cluster treatment of characteristic roots (CTCR). In fact, these two sets of hypersurfaces exhaustively represent the locations in the domain of the delays where the system possesses at least one pair of imaginary characteristic roots. To determine the kernel and offspring we use the extraordinary features of the “extended Kronecker summation” operation in this paper. The end result is that the infinite-dimensional problem reduces to a finite-dimensional one (and preferably into an eigenvalue problem). Following the procedure described in this paper, we are able to shorten the computational time considerably in determining these hypersurfaces. We demonstrate these concepts via some example case studies. One of the examples treats a 3-delay system. For this case another interesting perspective, called the “building block,” is also utilized to display the kernel in three-dimensional space in the domain of “spectral delays.”

A new perspective and analysis for regenerative machine tool chatter

Abstract The paper contains a practical perspective on regenerative machine tool chatter. Chatter is a well known phenomenon, occurrence of which is undesired in manufacturing. Aggressive machining conditions, in the sense of removing more metal rapidly, usually cause chatter. In most cases, these conditions can be determined a priori to the operation. A chatter stability study and its reasoning based on root locus plot analysis of time delayed systems is presented as a new and practical perspective in the field. At the junction of root locus and chatter concepts an area of particular interest to the authors arises: a new method for active vibration suppression, the Delayed Resonator. It is an active vibration absorber tuning of which is achieved utilizing a simple time delayed feedback. The cross linking between the Delayed Resonator study and the subject matter, machine tool chatter, is exciting to share. This is the primary motivation in pursuing this study. One of the highlights of the work appears at the phenomenon called Dual Frequency Delayed Resonator. This feature has been conjectured in the literature using the well known “stability lobes”, but never discussed with detail.