Rachel K. Kalaimani

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.

Fast modes in the set of minimal dissipation trajectories

Abstract In this paper, we study the set of trajectories satisfying both a given LTI system’s laws and also laws of the corresponding ‘adjoint’ system: in other words, trajectories in the intersection of the system’s behavior and that of the adjoint system. This intersection has important system theoretic significance: for example, it is known that the trajectories in this intersection are the ones with minimal ‘dissipation’. Underlying the notion of adjoint is that of a power supply: it is with respect to this supply rate that the trajectories in the intersection are known to be ‘stationary’. In this paper, we deal with half-line solutions to the differential equations governing both the system and its adjoint. Analysis of half-line solutions plays a central role for example in initial value problems and in well-posedness studies of an interconnection. We interpret the set of half-line trajectories allowed by a system and its adjoint as an interconnection of these two systems, and thus address issues about well-posedness/ill-posedness of the interconnection. We formulate necessary and sufficient conditions for this intersection to be autonomous. For the case of an ill-posed interconnection and resulting autonomous system, we derive conditions for existence of initial conditions that lead to impulsive solutions in the states of the system. We link our conditions with the strongly reachable and weakly unobservable subspaces of a state space system. We show that absence of impulsive initial conditions is equivalent to the well-known subspace iteration algorithms for these subspaces converging in one step.

On the interaction between personal comfort systems and centralized HVAC systems in office buildings

ABSTRACT Most modern HVAC systems in office buildings are unable to meet diverse comfort requirements of the occupants and are not energy efficient. We propose to mitigate both issues by using personal comfort systems (PCS). Specifically, we address the question, ‘How should an existing HVAC system modify its operation to benefit from the deployment of PCSs?’ For example, energy use could be reduced during periods of sparse occupancy by choosing appropriate thermal set points, with each PCS providing the additional offset in thermal comfort required by each occupant. We present the design of a PCS-aware HVAC control strategy based on Model Predictive Control (MPC) that employs a bi-linear thermal model. We use extensive simulations to compare the energy use and comfort offered by our PCS-aware HVAC system with that of a state-of-the-art MPC-based central HVAC system. We study different room layouts and scenarios with full or partial deployment of PCSs. Numerical evaluations show that our system yields significant savings in energy use in both summer and winter, compared both with a state-of-the-art system that does not deploy PCSs and with a similar system that deploys PCSs, but is not aware of them.

Multiple time-scale model predictive control for thermal comfort in buildings

Intelligent control of heating, ventilation, and air conditioning (HVAC) systems in commercial buildings have been extensively studied in the literature. Although prior work has shown the benefits of using Model Predictive Control (MPC), existing work falls short either by relying on linear HVAC models or using MPC assuming control actions at a single (hourly) time scale, although more frequent control is feasible for some HVAC elements. Our main contribution is the use of a bi-linear thermal model and the careful modeling of the multiple time-scales inherent in the operation of an HVAC system, which permits the design of a multiple time-scale MPC control. We find that employing a multiple time-scale MPC results in significantly better comfort in comparison to a single time-scale MPC, typically without an increase in power consumption. Moreover, there exist cases where there is a significant reduction in power consumption (40%) for the two time-scale MPC in comparison to the single time-scale, with no decrease in comfort.

Impulse Controllability: From Descriptor Systems to Higher Order DAEs

Impulsive solutions in LTI dynamical systems have received ample attention, but primarily for descriptor systems, i.e., first order Differential Algebraic Equations (DAEs). This paper focuses on the impulsive behavior of higher order dynamical systems and analyzes the causes of impulses in the context of interconnection of one or more dynamical systems. We extend the definition of impulse-controllability to the higher order case. Amongst the various nonequivalent notions of impulse-controllability for first order systems available in the literature, which mostly rely on the input/output structure of the system, our definition, based on a so-called state-map obtained directly from the system equations, generalizes many key first order results to the higher order case. In particular, we show that our higher-order-extension of the definition of impulse controllability generalizes the equivalence between impulse controllability and the ability to eliminate impulses in the closed loop by interconnecting with a suitable controller. This requires an extension of the definition of regularity of interconnection from behaviors involving only smooth trajectories to behaviors on the positive half line involving impulsive-smooth trajectories.

Nearest singular descriptor system having impulsive initial-conditions

This paper considers a distance problem: given a singular first order system, what is the distance to a nearest singular descriptor system that has impulsive initial conditions. The link between impulsive initial conditions and zeros at infinity is well-known. This paper provides bounds on the minimum perturbation required for a pair of matrices E and A such that the perturbed matrix pencil has one or more zeros at infinity. We provide closed form solutions for the minimum value for rank one perturbations.

Decentralized controllability of multi-inverter microgrids

Network of Voltage Source Inverters (VSI) to form a microgrid has played a central role in providing reliable power supply. It is very useful to control this network of VSI using decentralized controllers rather than centralized controllers due to less dependence of decentralized controllers on communication with other systems. A difficulty frequently encountered when studying and designing decentralized controllers is the controllability of the overall network using controllers with the constraint of being decentralized, i.e. the constraint that each controller can access only local sensors and only local actuators. This paper proves that the droop controller and its class of decentralized controllers indeed allows controllability of the system for generically all system parameter values. While this has been shown for specific parameter values using numerical methods, the graph theoretic approach followed in this paper helps in proving this generically for all system parameter values. We apply a recent result on arbitrary pole-placement by a controller with structural constraints to the case of a VSI connected to the grid, and to the case of two VSIs to show that indeed pole-placement can be achieved using decentralized controllers.