This paper presents a new approach to distributed controller design that exploits a partial-structure representation of linear time invariant systems to characterize the structure of a system. This partial-structure representation, called the dynamical structure function, characterizes the signal structure , or open-loop causal dependencies among manifest variables, capturing a significantly richer notion of structure than the sparsity pattern of the transfer function. The design technique sequentially constructs each link in an arbitrary controller signal structure, and the main result proves that the resulting controller is either stabilizing or no controller with the desired structure can stabilize the system.
In @cite_2 the authors show that if the structure of the transfer function matrices of the plant and the controller meets the quadratic invariance condition then the problem of synthesizing the optimal controller is convex. In @cite_1 the authors show that the quadratic invariance condition is necessary and sufficient for the problem of synthesizing the optimal controller to be convex. This method requires a decentralized stabilizing controller to initialize the convex optimization problem, so to complete the process, an algorithm to obtain such a controller is provided in @cite_5 .
@cite_5 @cite_1 @cite_2
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In this paper we deal with the problem of stabilizing linear, time-invariant plants using feedback control configurations that are subject to sparsity constraints. Recent results show that given a strongly stabilizable plant, the class of all stabilizing controllers that satisfy certain given sparsity constraints admits a convex representation via Zames's Q-parametrization. More precisely, if the pre-specified sparsity constraints imposed on the controller are quadratically invariant with respect to the plant, then such a convex representation is guaranteed to exist. The most useful feature of the aforementioned results is that the sparsity constraints on the controller can be recast as convex constraints on the Q-parameter, which makes this approach suitable for optimal controller design (in the ℋ2 sense) using numerical tools readily available from the classical, centralized optimal ℋ2 synthesis. All these procedures rely crucially on the fact that some stabilizing controller that verifies the imposed sparsity constraints is a priori known, while design procedures for such a controller to initialize the aforementioned optimization schemes are not yet available. This paper provides necessary and sufficient conditions for such a plant to be stabilizable with a controller having the given sparsity pattern. These conditions are formulated in terms of the existence of a doubly coprime factorization of the plant with additional sparsity constraints on certain factors. We show that the computation of such a factorization is equivalent to solving an exact model-matching problem. We also give the parametrization of the set of all decentralized stabilizing controllers by imposing additional constraints on the Youla parameter. These constraints are for the Youla parameter to lie in the set of all stable transfer function matrices belonging to a certain linear subspace. We consider the problem of constructing optimal decentralized controllers. We formulate this problem as one of minimizing the closed-loop norm of a feedback system subject to constraints on the controller structure. We define the notion of quadratic invariance of a constraint set with respect to a system, and show that if the constraint set has this property, then the constrained minimum-norm problem may be solved via convex programming. We also show that quadratic invariance is necessary and sufficient for the constraint set to be preserved under feedback. These results are developed in a very general framework, and are shown to hold in both continuous and discrete time, for both stable and unstable systems, and for any norm. This notion unifies many previous results identifying specific tractable decentralized control problems, and delineates the largest known class of convex problems in decentralized control. As an example, we show that optimal stabilizing controllers may be efficiently computed in the case where distributed controllers can communicate faster than their dynamics propagate. We also show that symmetric synthesis is included in this classification, and provide a test for sparsity constraints to be quadratically invariant, and thus amenable to convex synthesis. We consider multiple sensors randomly accessing a shared wireless medium to transmit measurements of their respective plants to a controller. To mitigate the packet collisions arising from simultaneously transmitting sensors, we appropriately design the sensor access rates. This is posed as an optimization problem, where the total transmit power of the sensors is minimized, and control performance for all control loops needs to be guaranteed. Control performance of each loop is abstracted as a desired expected decrease rate of a given Lyapunov function. By establishing an equivalent convex optimization problem, the optimal access rates are shown to decouple among sensors. Moreover, based on the Lagrange dual problem, we develop an easily implementable distributed procedure to find the optimal sensor access rates. This paper addresses the optimal control problem known as the Linear Quadratic Regulator in the case when the dynamics are unknown. We propose a multi-stage procedure, called Coarse-ID control, that estimates a model from a few experimental trials, estimates the error in that model with respect to the truth, and then designs a controller using both the model and uncertainty estimate. Our technique uses contemporary tools from random matrix theory to bound the error in the estimation procedure. We also employ a recently developed approach to control synthesis called System Level Synthesis that enables robust control design by solving a convex optimization problem. We provide end-to-end bounds on the relative error in control cost that are nearly optimal in the number of parameters and that highlight salient properties of the system to be controlled such as closed-loop sensitivity and optimal control magnitude. We show experimentally that the Coarse-ID approach enables efficient computation of a stabilizing controller in regimes where simple control schemes that do not take the model uncertainty into account fail to stabilize the true system.