HEC Montréal, Canada, May 6  8, 2013
2013 Optimization Days
HEC Montréal, Canada, 6 — 8 May 2013
WA10 Systêmes et contrôle / Systems and Control
May 8, 2013 10:30 AM – 12:10 PM
Location: Nancy et MichelGaucher
Chaired by Lyne Woodward
5 Presentations

10:30 AM  10:55 AM
Ergodicity and ClassErgodicity of Balanced Asymmetric Stochastic Chains
Unconditional consensus is the property of a consensus algorithm for multiple agents, to produce consensus irrespective of the particular time or state at which the agent states are initialized. Under a weak condition, socalled balanced asymmetry, on the sequence (An) of stochastic matrices in the agents states update algorithm, it is shown that (i) the set of accumulation points of states as n grows large is finite, (ii) the asymptotic unconditional occurrence of single consensus or multiple consensuses is directly related to the property of absolute infinite flow of this sequence, as introduced by Touri and Nedic. The latter condition must be satisfied on each of the islands of the socalled unbounded interactions graph induced by (An), defined by Hendrickx et al. The property of balanced asymmetry is satisfied by many of the well known discrete time consensus models studied in the literature.

10:55 AM  11:20 AM
MultiUnit Optimization for Systems with Multiple Inputs  Application to Photovoltaic Arrays
Multiunit optimization is an extremum seeking control method in which the gradient is calculated based on differences between the outputs of each unit. Although this method is useful when the system consists of multiple units, convergence to the optimal point is proved provided units are identical, whereas a photovoltaic (PV) array consists of more than two nonidentical PV cells. Therefore, an optimization procedure based on the multiunit method is developed for three nonidentical units with two inputs. The proposed algorithm consists of sequential and adaptive correction to compensate for the differences between static curves of the objective functions related to each unit. The algorithm is tested on a PV array model and is compared to other common methods of maximum power point tracking (MPPT) such as perturb and observe, and incremental conductance algorithms.

11:20 AM  11:45 AM
MFG Systems with Recursive Estimation of Common Partially Observed Disturbances: Application to Large Scale Power Markets
Following (KMC CDC 2012), power markets are modelled as dynamic large population games where suppliers and consumers submit their bids in realtime. The agents are coupled in their dynamics and cost functions through the price process. Extending the model in (KMC CDC 2012), a common unpredictable Partially Observed Neutral Major Agent is added to the system which is game theoretically neutral and which represents common unpredictable disturbance factors (e.g. wind) and exogenous market factors (e.g. competing energy resource prices), etc.
In (KMC CDC 2012), the Mean Field Game (MFG) methodology is used to study the limit (i.e. infinite population) behaviour of large population market systems without a Major Agent; this results in a decentralized algorithm where agents submit their bids solely following the price signal and using statistical information on the dynamics of the entire population. When a Major Agent is absent, the system exhibits the standard counter intuitive property of MFG solutions that agents need not observe the behaviour (i.e. inputs and state trajectories, market price evolution, etc) of any other agent (individually or collectively) in order that simple decentralized control actions achieve a mass eNash equilibrium (with e vanishing as the population goes to infinity) and individual L2 stability.
The contribution of this paper is the extension of the MFG theory to cover the addition of a Neutral Major Agent to the power market problem. In general, the addition of a Major Agent in the MFG framework (see MYH, MYS, MPEC) makes the mean field stochastic and this gives rise to different Nash equilibria (see [MYH, MYHS, MNPEC]).
In the general situation of sporadic noisy observations of the mean field and the state of the Major Agent, the extended MFG theory (with estimation of the mean field and the Major Agent state) yields simple decentralized control laws which achieve a mass eNash equilibrium (with e vanishing as the population goes to infinity) and individual L2 stability. In this paper, this is carried out for the MFG formulation of the power market problem in order to fit the situation where sporadic noisy observations of the state of the Major Agent and of the market price are available for recursive mean field state estimation.

11:45 AM  12:10 PM
Distributed Estimation and Control for Large Population Stochastic MultiAgent Systems with Coupling in the Measurements
In this paper, we investigate a class of large population stochastic multiagent systems where the agents have linear stochastic dynamics and are coupled via their measurement equations. Using the state aggregation technique, we propose a distributed estimation and control algorithm that combines the Kalman filtering for state estimation and the linearquadraticGaussian (LQG) feedback controller. Moreover, the stability analysis in terms of exponential boundedness in the mean square is given for the proposed algorithm.

12:10 PM  12:35 PM
ConicSectorBased Control to Circumvent Passivity Violations
Strictly positive real controller design is widely used to ensure inputoutput stability via the Passivity Theorem, a special case of the Conic Sector Theorem. This presentation discusses use of the Conic Sector Theorem itself in stability analysis and optimal controller design for passivityviolated systems. Given an existing controller and a plant that has experienced a (partially unknown) passivity violation, a novel sector bound selection procedure is presented to assess inputoutput stability via the Conic Sector Theorem. Should inputoutput stability not be ensured, two original controller synthesis methods are designed to mimic the original controller. Both methods guarantee inputoutput stability by selecting controllers within appropriate conic sectors, and involve only the evaluation of readily solvable convex optimization problems constrained by linear matrix inequalities. A numerical simulation involving a flexible manipulator is provided as a proof of concept.