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Convexity of Optimal Linear Controller Design
"... Abstract — We develop a general class of stochastic optimal control problems for which the problem of designing optimal linear feedback gains is convex. The class of problems includes arbitrary time varying linear systems and costs that are mixtures of exponentiated quadratics. This allows us to mod ..."
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Abstract — We develop a general class of stochastic optimal control problems for which the problem of designing optimal linear feedback gains is convex. The class of problems includes arbitrary time varying linear systems and costs that are mixtures of exponentiated quadratics. This allows us to model problems with quadratic state costs and linear constraints on states and state transitions. Further, convexity in the feedback gains lets us impose arbitrary convex constraints or penalties on the feedback matrix: Thus we can model problems like distributed control (by imposing a sparsity structure on the feedback matrix) and variablestiffness control (by applying timevarying penalties to feedback gain matrices). We show that the convex optimization problem can be solved efficiently by using the structure of the matrices involved. Finally, we present an application of these ideas to a practical problem arising in distributed control of power systems. I.
Convex Control Design via Covariance Minimization
"... Abstract — We consider the problem of synthesizing optimal linear feedback policies subject to arbitrary convex constraints on the feedback matrix. This is known to be a hard problem in the usual formulations (H2,H∞,LQR) and previous works have focussed on characterizing classes of structural constr ..."
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Abstract — We consider the problem of synthesizing optimal linear feedback policies subject to arbitrary convex constraints on the feedback matrix. This is known to be a hard problem in the usual formulations (H2,H∞,LQR) and previous works have focussed on characterizing classes of structural constraints that allow efficient solution through convex optimization or dynamic programming techniques. In this paper, we propose a new control objective based on eigenvalues of the covariance matrix of trajectories of the system and show that this formulation makes the problem of computing optimal linear feedback matrices convex under arbitrary convex constraints on the feedback matrix. This allows us to solve problems in distributed control (sparsity in the feedback matrices), control with delays and variable impedance control. Although the control objective is nonstandard, we present theoretical and empirical evidence that it agrees well with standard notions of control. We numerically validate the our approach on problems arising in power systems and simple mechanical systems. I.
Convex Structured Controller Design
 SUBMITTED TO IEEE TRANSACTIONS ON NETWORKED CONTROL SYSTEMS
, 2013
"... We consider the problem of synthesizing optimal linear feedback policies subject to arbitrary convex constraints on the feedback matrix. This is known to be a hard problem in the usual formulations (H2,H∞,LQR) and previous works have focused on characterizing classes of structural constraints that a ..."
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We consider the problem of synthesizing optimal linear feedback policies subject to arbitrary convex constraints on the feedback matrix. This is known to be a hard problem in the usual formulations (H2,H∞,LQR) and previous works have focused on characterizing classes of structural constraints that allow efficient solution through convex optimization or dynamic programming techniques. In this paper, we propose a new control objective and show that this formulation makes the problem of computing optimal linear feedback matrices convex under arbitrary convex constraints on the feedback matrix. This allows us to solve problems in decentralized control (sparsity in the feedback matrices), control with delays and variable impedance control. Although the control objective is nonstandard, we present theoretical and empirical evidence that it agrees well with standard notions of control. We also present an extension to nonlinear control affine systems. We present numerical experiments validating our approach.
Automating Stochastic Control
, 2014
"... Stochastic Optimal Control is an elegant and general framework for specifying and solving control problems. However, a number of issues have impeded its adoption in practical situations. In this thesis, we describe algorithmic and theoretical developments that address some of these issues. In the fi ..."
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Stochastic Optimal Control is an elegant and general framework for specifying and solving control problems. However, a number of issues have impeded its adoption in practical situations. In this thesis, we describe algorithmic and theoretical developments that address some of these issues. In the first part of the thesis, we address the problem of designing cost functions for control tasks. For many tasks, the appropriate cost functions are difficult to specify and highlevel cost functions may not be amenable to numerical optimization. We adopt a datadriven approach to solving this problem and develop a convex optimization based algorithm for learning costs given demonstrations of desirable behavior. The next problem we tackle is modelling riskaversion. We develop a general theory of linearly solvable optimal control capable of modelling all these preferences in a computationally tractable manner. We then study the problem of optimizing parameterized control policies. The study presents the first convex formulation of control policy optimization for arbitrary dynamical systems. Using algorithms for stochastic convex optimization, this approach leads to algorithms that are guaranteed to find the optimal policy efficiently. We discuss particular applications including decentralized control and training neural networks. Finally,
Convex Structured Controller Design in Finite Horizon
, 2014
"... We consider the problem of synthesizing optimal linear feedback policies subject to arbitrary convex constraints on the feedback matrix. This is known to be a hard problem in the usual formulations (H2,H∞,LQR) and previous works have focused on characterizing classes of structural constraints that a ..."
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We consider the problem of synthesizing optimal linear feedback policies subject to arbitrary convex constraints on the feedback matrix. This is known to be a hard problem in the usual formulations (H2,H∞,LQR) and previous works have focused on characterizing classes of structural constraints that allow efficient solution through convex optimization or dynamic programming techniques. In this paper, we propose a new control objective for finite horizon discretetime problems and show that this formulation makes the problem of computing optimal linear feedback matrices convex under arbitrary convex constraints on the feedback matrix. This allows us to solve problems in decentralized control (sparsity in the feedback matrices), control with delays and variable impedance control. Although the control objective is nonstandard, we present theoretical and empirical evidence showing that it agrees well with standard notions of control. We show that the theoretical approach carries over to nonlinear systems, although the computational tractability of the extension is not investigated in this work. We present numerical experiments validating our approach.
Convex Risk Averse Control Design
"... Abstract — In this paper, we show that for arbitrary stochastic linear dynamical systems, the problem of optimizing parameters of a feedback control policy can be cast as a convex optimization problem when a riskaverse objective (similar to LEQG) is used. The only restriction is a condition relatin ..."
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Abstract — In this paper, we show that for arbitrary stochastic linear dynamical systems, the problem of optimizing parameters of a feedback control policy can be cast as a convex optimization problem when a riskaverse objective (similar to LEQG) is used. The only restriction is a condition relating the control cost, risk factor and noise in the system. The resulting approach allows us to synthesize riskaverse controllers efficiently for finite horizon problems. For the standard quadratic costs in infinite horizon, the resulting problems become degenerate if the uncontrolled system is unstable. As an alternative, we propose using a discountbased approach that ensures that costs do not blow up. We show that the discount factor can effectively be used as a homotopy parameter to gradually synthesize stabilizing controllers for unstable systems. We also propose extensions where nonquadratic costs can be used for controller synthesis, and in this case, as long as the costs are bounded, the optimization problems are wellposed and have nontrivial solutions. I.