Debraj Chakraborty

An Architecture for FACTS Controllers to Deal With Bandwidth-Constrained Communication

Bandwidth constraints could have an adverse impact on the flexible AC transmission system (FACTS) controllers relying on signals communicated from distant locations. An observer-driven system copy (OSC) architecture is adopted here to deal with low data rates caused by the limited bandwidth availability. The basic idea is to use the knowledge of the nominal system to approximate its actual behavior during the intervals when data from the remote phasor measurement units (PMUs) are not available. This is corrected whenever the most recent states are obtained from the reduced-order observer at the PMU location. The closed-loop behavior deteriorates as the operating condition drifts away from the nominal. Nonetheless, significantly better response is achieved under limited bandwidth availability compared to the conventional approach of communicating the measured outputs. The deterioration is quantified in terms of the difference between the nominal and the offnominal condition.

Partial Pole Placement with Controller Optimization

An arbitrary subset (n - m) of the (n) closed loop eigenvalues of an nth order continuous time single input linear time invariant system is to be placed using full state feedback, at pre-specified locations in the complex plane. The remaining m closed loop eigenvalues can be placed anywhere inside a pre-defined region in the complex plane. This region constraint on the unspecified poles is translated into a linear matrix inequality constraint on the feedback gains through a convex inner approximation of the polynomial stability region. The closed loop locations for these m eigenvalues are optimized to obtain a minimum norm feedback gain vector. This reduces the controller effort leading to less expensive actuators required to be installed in the control system. The proposed algorithm is illustrated on a linearized model of a 4-machine, 2-area power system example.

Optimal control during feedback failure

The problem of controlling a perturbed open loop system so as to keep its performance errors within bounds is considered. The objective is to maximise the time during which performance errors remain below a prescribed ceiling, while the controlled systems parameters are within a specified neighbourhood of their nominal values. It is shown that there is an optimal open loop controller that achieves this objective. Conditions under which the optimal controller generates a bang-bang control input signal are characterised. In general, it is shown that the performance of the optimal controller can always be approximated by a bang-bang signal.

Damping Control in Power Systems Under Constrained Communication Bandwidth: A Predictor Corrector Strategy

Damping electromechanical oscillations in power systems using feedback signals from remote sensors is likely to be affected by occasional low bandwidth availability due to increasing use of shared communication in future. In this paper, a predictor corrector (PC) strategy is applied to deal with situations of low-feedback data rate (bandwidth), where conventional feedback (CF) would suffer. Knowledge of nominal system dynamics is used to approximate (predict) the actual system behavior during intervals when data from remote sensors are not available. Recent samples of the states from a reduced observer at the remote location are used to periodically reset (correct) the nominal dynamics. The closed-loop performance deteriorates as the actual operating condition drifts away from the nominal dynamics. Nonetheless, significantly better performance compared to CF is obtained under low-bandwidth situations. The analytical criterion for closed-loop stability of the overall system is validated through a simulation study. It is demonstrated that even for reasonably low data rates the closed-loop stability is usually ensured for a typical power system application confirming the effectiveness of this approach. The deterioration in performance is also quantified in terms of the difference between the nominal and off-nominal dynamics.

Computation of Time Optimal Feedback Control Using Groebner Basis

The synthesis of time optimal feedback control of a single input continuous time linear time invariant (LTI) system is considered. If the control input u(t) is constrained to obey |u(t)| ≤ 1, then it is known that the optimal input switches between the extreme values ±1, according to some “switching surfaces” in the state space. It is shown that for systems with non-zero, distinct and rational eigenvalues, these switching surfaces are semi-algebraic sets and a method to compute them using Groebner basis, is proposed. In the process the null-controllable region for such systems is characterized and computed. Numerical simulations illustrate the proposed computational methods.

Robust optimal control: low-error operation for the longest time

The problem of maximising the time during which an open-loop system can operate without exceeding a specified error bound is considered for linear systems that are subject to uncertainties about their parameters and their initial conditions, and whose operation is hampered by disturbance signals. The objective is to characterise an optimal input signal that keeps performance errors within specified bounds for the longest time. It is shown that such an input signal exists, and that it can be approximated by a bang-bang input signal without significantly affecting performance.

Partial pole placement with minimum norm controller

The problem of placing an arbitrary subset (m) of the (n) closed loop eigenvalues of a nth order continuous time single input linear time invariant(LTI) system, using full state feedback, is considered. The required locations of the remaining (n − m) closed loop eigenvalues are not precisely specified. However, they are required to be placed anywhere inside a pre-defined region in the complex plane. The resulting non-uniqueness is utilized to minimize the controller effort through optimization of the feedback gain vector norm. Using a variant of the boundary crossing theorem, the region constraint on the unspecified (n−m) poles is translated into a quadratic constraint on the characteristic polynomial coefficients. The resulting quadratically constrained quadratic program can be approximated by a quadratic program with linear constraints. The proposed theory is demonstrated for power oscillation damping controller design, where the eigenvalues corresponding to poorly damped electro-mechanical modes are critical for performance and hence are specified precisely by the designer, whereas the remaining eigenvalues are non-critical and need not be specified precisely. Acceptable closed loop pole placement is achieved for this example along with a 51% reduction in controller norm.

Preserving System Performance During Feedback Failure

The problem of maintaining acceptable performance of a perturbed control system under conditions of feedback failure is considered. The objective is to maximize the time during which performance remains within desirable bounds without feedback, given that the parameters of the controlled system are within a specified neighborhood of their nominal values. It is shown that there is an optimal open-loop controller that achieves this objective. The performance of this controller can be approximated by a bang-bang controller.

Singular LQ Control, Optimal PD Controller and Inadmissible Initial Conditions

We consider a classical control problem: the infinite horizon singular LQ problem, i.e., some inputs are unpenalized in the quadratic performance index. In this case, it is known that the slow dynamics is constrained to be in a proper subspace of the state-space, with the optimal input for the slow dynamics implementable by feedback. In this technical note we show that both the fast dynamics and the slow dynamics can be implemented by a feedback controller. Moreover, we show that the feedback controller cannot be a static feedback controller but can be PD, i.e., proportional+differentiate exactly once, in the state. We show that the closed loop system is a singular descriptor state space system and we also characterize the conditions on the system/performance index for existence of inadmissible initial conditions, i.e., initial conditions that cause impulsive solutions. There are no inadmissible initial conditions in the controlled system if and only if in the strictly proper transfer matrix from the unpenalized inputs to the penalized states, there exists at least one maximal minor of relative degree equal to the number of unpenalized inputs. In addition to the above, we prove solvability of the infinite horizon singular LQ problem under milder assumptions than in the literature.

Computation of feedback control for time optimal state transfer using Groebner basis

Abstract Computation of time optimal feedback control law for a controllable linear time invariant system with bounded inputs is considered. Unlike a recent paper by the authors, the target final state is not limited to the origin of state-space, but is allowed to be in the set of constrained controllable states. Switching surfaces are formulated as semi-algebraic sets using Groebner basis based elimination theory. Using these semi-algebraic sets, a nested switching logic is synthesized to generate the time optimal feedback control. However, the optimal control law enforces an unavoidable limit cycle in the time-optimal trajectory for most non-origin target points. The time-period of this limit-cycle is dependent on the target position. This dependence is algebraically characterized and a method to compute the time-period of the limit-cycle is provided. As a natural extension, the set of constrained controllable states is also computed.