Annraoi M. de Paor

Controllers of Ziegler-Nichols type for unstable process with time delay

Abstract Proportional (P), proportional plus integral (PI) and proportional plus integral plus derivative (PID) controllers are designed for a time-delayed process having latency L and a single right half-plane pole at s=λ, with λL < 1. Explicit designs are derived from the concept of phase margin, and an optimum stability approach for PI controllers is based on a parameter plane study. An optimum gain margin design is also given for P control.

A modified Smith predictor and controller for unstable processes with time delay

An integrated design procedure is developed for a modified Smith predictor and associated controller for linear time-delay systems having transfer functions of the form k 1, A exp (—sT)/B, where A and B are monic polynomials in s of degree n — l and n, respectively. A is Hurwitz and B has a single right-half-plane root at s = λ. For l=1,2,3, an augmented PI controller guarantees asymptotic stability for λT less than an l-dependent limit. The procedure for l = 3 is extended to l = 4 with the introduction of derivative action into the controller. Design arguments are on root locus topology, and on Nyquist analysis applied to an auxiliary system.

Application of Describing Function Analysis to a Model of Deep Brain Stimulation

Deep brain stimulation effectively alleviates motor symptoms of medically refractory Parkinsons disease, and also relieves many other treatment-resistant movement and affective disorders. Despite its relative success as a treatment option, the basis of its efficacy remains elusive. In Parkinsons disease, increased functional connectivity and oscillatory activity occur within the basal ganglia as a result of dopamine loss. A correlative relationship between pathological oscillatory activity and the motor symptoms of the disease, in particular bradykinesia, rigidity, and tremor, has been established. Suppression of the oscillations by either dopamine replacement or DBS also correlates with an improvement in motor symptoms. DBS parameters are currently chosen empirically using a “trial and error” approach, which can be time-consuming and costly. The work presented here amalgamates concepts from theories of neural network modeling with nonlinear control engineering to describe and analyze a model of synchronous neural activity and applied stimulation. A theoretical expression for the optimum stimulation parameters necessary to suppress oscillations is derived. The effect of changing stimulation parameters (amplitude and pulse duration) on induced oscillations is studied in the model. Increasing either stimulation pulse duration or amplitude enhanced the level of suppression. The predicted parameters were found to agree well with clinical measurements reported in the literature for individual patients. It is anticipated that the simplified model described may facilitate the development of protocols to aid optimum stimulation parameter choice on a patient by patient basis.

The Root Locus Method: Famous Curves, Control Designs and Non-Control Applications

Famous named curves generated as root loci produce optimum and pseudo-optimum stability designs for realistic systems. Non-minimum-phase control involves a principle of topological certainty. A roo...Famous named curves generated as root loci produce optimum and pseudo-optimum stability designs for realistic systems. Non-minimum-phase control involves a principle of topological certainty. A root locus with complex breakpoints is discussed. Cassini’s ovals in Brauer’s method of eigenvalue localisation are illustrated. A problem in dielectric theory, recast into an imaginary-parameter root locus, is solved via real-parameter theory. Continuation, translation and scaling are invoked. It is hoped to impart an appreciation of the versatility of root-locus-inspired thinking.

Analysis of Oscillatory Neural Activity in Series Network Models of Parkinson's Disease During Deep Brain Stimulation

Parkinsons disease is a progressive, neurodegenerative disorder, characterized by hallmark motor symptoms. It is associated with pathological, oscillatory neural activity in the basal ganglia. Deep brain stimulation (DBS) is often successfully used to treat medically refractive Parkinsons disease. However, the selection of stimulation parameters is based on qualitative assessment of the patient, which can result in a lengthy tuning period and a suboptimal choice of parameters. This study explores fourth-order, control theory-based models of oscillatory activity in the basal ganglia. Describing function analysis is applied to examine possible mechanisms for the generation of oscillations in interacting nuclei and to investigate the suppression of oscillations with high-frequency stimulation. The theoretical results for the suppression of the oscillatory activity obtained using both the fourth-order model, and a previously described second-order model, are optimized to fit clinically recorded local field potential data obtained from Parkinsonian patients with implanted DBS. Close agreement between the power of oscillations recorded for a range of stimulation amplitudes is observed (R2 = 0.69-0.99). The results suggest that the behavior of the system and the suppression of pathological neural oscillations with DBS is well described by the macroscopic models presented. The results also demonstrate that in this instance, a second-order model is sufficient to model the clinical data, without the need for added complexity. Describing the system behavior with computationally efficient models could aid in the identification of optimal stimulation parameters for patients in a clinical environment.

Concepts of Optimum Stability for Linear Feedback Systems

Based on undergraduate teaching experience in control theory, the concept of gain margin is revised, vector margin systematised and the rudiments of its calculation exposed, and the concept of phase margin is illustrated. Some two-parameter problems are tackled via the Routh array. Concepts of centroid design and generalised vector margin design are discussed and finally, an eigenvalue-based optimum stability idea suggested by root locus analysis is explored.Based on undergraduate teaching experience in control theory, the concept of gain margin is revised, vector margin systematised and the rudiments of its calculation exposed, and the concept of phase margin is illustrated. Some two-parameter problems are tackled via the Routh array. Concepts of centroid design and generalised vector margin design are discussed and finally, an eigenvalue-based optimum stability idea suggested by root locus analysis is explored.

Application of non-linear control theory to a model of deep brain stimulation

Deep brain stimulation (DBS) effectively alleviates the pathological neural activity associated with Parkinsons disease. Its exact mode of action is not entirely understood. This paper explores theoretically the optimum stimulation parameters necessary to quench oscillations in a neural-mass type model with second order dynamics. This model applies well established nonlinear control system theory to DBS. The analysis here determines the minimum criteria in terms of amplitude and pulse duration of stimulation, necessary to quench the unwanted oscillations in a closed loop system, and outlines the relationship between this model and the actual physiological system.

Analytic Root Locus and Lambert W Function in Control of a Process with Time Delay

Analytic Root Locus and Lambert W Function in Control of a Process with Time Delay Recently, the Lambert W function has arisen in the analysis of many systems including a restricted class of time-delay systems. An alternative approach to this analysis, based on the well-established root locus method, is shown here to contain the Lambert W function as a special case. As a purely illustrative example of the equivalence between the Lambert W function and analytic root locus a system comprising a Proportional controller with a time-delay process is analysed. Controller designs based on rightmost eigenvalue location and the dominant eigenvalue method are described.

PI control of first-order lag plus time-delay plants: root locus design for optimal stability

We present a procedure for the design of a PI controller for a general first-order lag plus time-delay plant. We derive two equations that allow a designer to calculate the PI controller parameters using only the plant parameters. The central idea of this design procedure is to use the root locus method to select controller parameters that put the system’s rightmost eigenvalue as far to the left as possible in the (σ, ω) plane. When the system is operating at this point, it is said to be operating at a point of optimum stability in the root locus sense.

Optimum Stability and Minimum Complexity as Desiderata in Feedback Control System Design

Abstract The deceptively simple problem of designing a single-loop, error-actuated feedback system is considered anew. Fundamental concepts only are invoked. Minimum controller complexity achieves arbitrary eigenvalue assignment. Optimum stability places the rightmost eigenvalue as deep in the left half plane as possible. Desirable side conditions confer static disturbance rejection and unity static gain between reference input and process output. Root loci show that any system designed to have all eigenvalues equal is optimally stable with respect to variation of any design parameter from its nominal value. In a cautionary vein, gain and phase margins are used to compare a design arrived at here with an overly complex one yielded by the H ∞ approach.