IIT Madras is a public technical and research university located in Chennai, Tamil Nadu. Founded in 1959, it is recognised as an Institute of National Importance.

IIT Madras has been ranked as the top engineering institute in India for four years in a row by the National Institutional Ranking Framework of the MHRD

It currently offers undergraduate, postgraduate and research degrees across 16 disciplines in Engineering, Sciences, Humanities and Management. About 596 faculty belonging to science and engineering departments and centres of the Institute are engaged in teaching, research and industrial consultancy.

IIT Madras

In this paper, we consider the controllability of a networked system where each node in the network has higher order linear time-invariant (LTI) dynamics. We employ a quantitative measure for controllability based on average controllability. We relate this metric to the network topology and the dynamics of individual subsystems that constitute each node of the networked system. Using this, we show that, under certain assumptions, the average controllability increases with increased interactions across subsystems in the network. Next, we consider the problem of identifying an appropriate network topology when there are constraints on the number of links that exist in the network. This problem is formulated as a set function optimization problem. We show that for our problem, this set function is a monotone increasing supermodular function. Since maximization of such a function with cardinality constraints is a hard problem, we implement a greedy heuristic to obtain a sub-optimal solution. © 2019 IEEE.

Department of Electrical Engineering

World University root node

Proceedings of the IEEE Conference on Decision and Control

Optimizing Average Controllability of Networked Systems

We address the problem of stabilizing a system, with a certain desired dynamic performance, using control signal sent over a communication channel having a limited capacity. A desired dynamic performance of the closed-loop system is obtained by placing its poles in a specific region of the open left-half of the complex plane. Denoting such a region by D, this procedure is also called D-stabilization. We first analyze a single-input linear time-invariant (LTI) system with state-feedback control over a channel subjected to additive noise. Using tools from H2 control theory, we pose the above stabilization problem as an optimization problem involving linear matrix inequalities (LMIs). Using this, we derive a sufficient condition for D-stabilization of the system subject to the channel capacity constraint. Further, we extend our analysis to a multi-input LTI system with state-feedback over a multiinput multi-output (MIMO) channel subjected to additive noise. The channel consists of multiple single-input single-output (SISO) subchannels connected in parallel. The total capacity of the MIMO channel is fixed, while the individual subchannel capacities can be freely allocated. We provide a necessary and sufficient condition for stabilization and a sufficient condition for D-stabilization of this system subject to the channel capacity constraints. We also propose a method to design the channel and an optimal controller.

D-stability under Channel Capacity Constraints

We address the problem of stabilizing a set of discrete-time systems over a communication network. The network consists of capacity-constrained discrete-time Additive White Gaussian Noise (AWGN) channels. We consider the case when the number of channels is limited and propose a dynamic scheduling scheme that, at a given time, determines which subset of systems get access to the channel for feedback control. This problem is addressed by considering two separate problems - scheduling systems over noiseless channels and stabilizing a system over an AWGN channel. The scheduling problem is addressed in the switched system framework by making use of a min-type Lyapunov function. We provide a sufficient condition in the form of Linear Matrix Inequalities (LMIs) to schedule a subset of systems while achieving stability of all systems. We also explicitly determine the scheduling scheme. Next, we provide a novel LMI-based necessary and sufficient condition for stabilization of a discrete-time system over a discrete-time AWGN channel. Finally, we appropriately combine the two results to obtain an LMI-based sufficient condition for the join scheduling-stabilization problem. © 2019 IEEE.

Scheduling for Stabilization over Capacity-Constrained Channels

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. © 2018, © 2018 Informa UK Limited, trading as Taylor & Francis Group.

Fulltext

Advances in Building Energy Research

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

This paper deals with finding a 'least interaction' controller that generically achieves pole placement, given a actuator-sensor interaction possibility for the controller. The structure of the plant and the controller are modeled as an undirected and bipartite graph. Assuming that the plant/controller structures are specified, we find a minimum controller substructure within the specified controller structure such that the controller substructure allows generic eigenvalue assignability. The minimum is in the sense that the bipartite graph consisting of the proposed controller has the minimum number of edges. The complexity of a brute-force algorithm to identify a minimum controller substructure would be exponential and hence we propose two formulations for solving this problem, using recent results about equivalence of generic pole-assignability and covering of plant edges using cycles. The first uses multi-commodity flow networks to include all plant edges in some cycle using the least number of controller edges. We show that an integer solution to this formulation gives a minimum controller substructure for arbitrary pole placement problem. Since the problem of finding a feasible integer flow in multicommodity networks is NP-complete, there is ample reason that identifying a minimum controller substructure is NP-hard. The second formulation uses the framework of travelling salesman with profits (TSP with profits) to cover all vertices of the bipartite graph by cycles using the least number of controller edges. The TSP-with-profits problem too belongs to the class of NP-hard problems. We show that our formulation is equivalent to the so-called Generalized Travelling Salesman Problem (GTSP) thus allowing branch-cut algorithms developed for GTSP problems. © 2016 EUCA.

2016 European Control Conference, ECC 2016

Minimum controller substructure for generic arbitrary pole placement: Multicommodity flow and TSP based formulations

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. © 2016 Elsevier B.V.

Systems and Control Letters

Fast modes in the set of minimal dissipation trajectories

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. © 2016 Copyright held by the owner/author(s).

Proceedings of the 7th International Conference on Future Energy Systems Poster Sessions, e-Energy 2016

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

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. © 1963-2012 IEEE.

IEEE Transactions on Automatic Control

Impulse Controllability: From Descriptor Systems to Higher Order DAEs

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. © 2014 IEEE.

Nearest singular descriptor system having impulsive initial-conditions

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. © 2013 IEEE.

Proceedings of the IEEE International Conference on Industrial Technology

Decentralized controllability of multi-inverter microgrids

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., $˚m proportional+˚m differentiate exactly once$ , 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. © 2013 IEEE.

Department Of Chemistry

Singular LQ control, optimal PD controller and inadmissible initial conditions

In this paper we address the generic pole placement problem for a system represented by differential algebraic equations. The genericity aspect is relevant when dealing with large dynamical systems where the plant equations are sparse. We capture the sparsity structure of the plant equations into an edge weighted and undirected bipartite graph. We propose an algorithm that furnishes a 'minimal' controller structure for achieving generic arbitrary pole placement: minimality in the sense of the sparsity within controller equations. More precisely, we introduce a procedure to come up with a set of controller equations such that, in addition to generically achieving arbitrary pole placement, the bipartite graph constructed for this controller has the minimum number of edges amongst all controllers that generically achieve arbitrary pole placement. The algorithm we propose involves finding a minimum number of paths that cover a given set of vertices corresponding to plant equations. We introduce an integer that captures the extent of MIMO features inside the plant equations, since this turns out to crucially decide the minimum number of required edges. This paper's minimal controller structure problem and the proposed solution turn out to also solve the problem of generically completing a given rectangular polynomial matrix into a unimodular matrix using the minimum number of nonzero entries. © 2013 EUCA.

2013 European Control Conference, ECC 2013

Minimal controller structure for generic pole placement

In this paper we assume dynamical systems are represented by linear differential-algebraic equations (DAEs) of order possibly higher than one. We consider a structured system of DAEs for both the to-be-controlled plant and the controller. We model the structure of the plant and the controller as an undirected and <a href="https://www.sciencedirect.com/topics/mathematics/bipartite-graph">bipartite graph</a> and formulate <a href="https://www.sciencedirect.com/topics/mathematics/necessary-and-sufficient-condition">necessary and sufficient conditions</a> on this graph for the structured controller to generically achieve arbitrary pole placement. A special case of this problem also gives new equivalent conditions for structural controllability of a plant. Use of results in matching theory, and in particular, ‘admissibility’ of edges and ‘elementary bipartite graphs’, make the problem and the solution very intuitive. Further, our approach requires standard graph algorithms to check the required conditions for generic arbitrary pole placement, thus helping in easily obtaining running time estimates for checking this. When applied to the state space case, for which the literature has running time estimates, our algorithm is faster for sparse state space systems and comparable for general state space systems.The solution to the above problem also provides a necessary and sufficient condition for the following matrix completion problem. Given a structured rectangular <a href="https://www.sciencedirect.com/topics/mathematics/polynomial-matrix">polynomial matrix</a>, when can it be completed to a unimodular matrix such that the additional rows that are added during the completion process are constrained to have zeros at certain locations.

Linear Algebra and its Applications

Generic pole assignability, structurally constrained controllers and unimodular completion

We study Hamiltonian systems, namely, systems comprising of trajectories which are `stationary' with respect to a quadratic performance index: they play a central role in many optimal control problems. A typical assumption in the literature is that of `regularity': the resulting first-order dynamical system is a regular state space system and not a singular descriptor system. In this paper we show that the first order representation of a Hamiltonian is a singular descriptor system if and only if the interconnection of a related MIMO system G(s) with its dual (i.e. its adjoint) is ill-posed. We address the possibility of existence of inadmissible initial conditions, i.e. initial conditions that give rise to impulsive solutions. We characterize conditions on G(s) under which the corresponding singular Hamiltonian system has inadmissible initial conditions. Under suitable simplifying assumptions, which amount to studying an extreme case of ill-posedness, our main result states that there exist no inadmissible initial conditions if and only if the skew-symmetric part of the first moment about s = ∞ of the transfer matrix G(s) is nonsingular; a condition we show that is opposite to that for G(s) to be an all-pass filter. As a corollary, ill-posed interconnection of a square MIMO system with odd number of inputs (in particular, SISO systems) with its adjoint always contains in inadmissible initial conditions.

2013 European Control Conference

Stationary trajectories, singular Hamiltonian systems and ill-posed interconnection

In this paper we consider the LQR control problem with no penalty on the input; this is addressed in the literature as the singular LQR control problem. We show that here the optimal controller is no longer a static controller but a PD controller. We also show that the closed loop system, i.e. the controlled system is a singular state space system. Singular system brings in the concern of existence of inadmissible initial conditions, i.e. initial conditions for which the solution is impulsive. Our main result is that there are no inadmissible initial conditions in the controlled system if and only if states which have relative degree one with respect to the input are penalised. Though the Algebraic Riccati equation is not defined for the singular case, we use the notion of storage function in dissipative systems theory to obtain the optimal cost function explicitly in terms of the initial conditions. We use this to prove that the initial conditions for which states of the autonomous (i.e. closed loop), singular system immediately jump to 0 have optimal cost 0. Our result that the optimal controller is a PD controller underlines a key intuitive statement for the dual problem, namely the Kalman-Bucy filter when measurements are noiseless: the minimum variance estimator differentiates the noiseless measurements. The MIMO case is not dealt in this paper due to space constraints and since it involves controllability indices and Forney indices in the result statements and proofs. © 2011 IEEE.