Nonlinear control synthesis by convex optimization

A stability criterion for nonlinear systems, recently derived by the third author, can be viewed as a dual to Lyapunov's second theorem. The criterion is stated in terms of a function which can be interpreted as the stationary density of a substance that is generated all over the state-space and flows along the system trajectories toward the equilibrium. The new criterion has a remarkable convexity property, which in this note is used for controller synthesis via convex optimization. Recent numerical methods for verification of positivity of multivariate polynomials based on sum of squares decompositions are used

Output-Feedback Control of Nonlinear Systems using Control Contraction Metrics and Convex Optimization

Control contraction metrics (CCMs) are a new approach to nonlinear control design based on contraction theory. The resulting design problems are expressed as pointwise linear matrix inequalities and are and well-suited to solution via convex optimization. In this paper, we extend the theory on CCMs by showing that a pair of "dual" observer and controller problems can be solved using pointwise linear matrix inequalities, and that when a solution exists a separation principle holds. That is, a stabilizing output-feedback controller can be found. The procedure is demonstrated using a benchmark problem of nonlinear control: the Moore-Greitzer jet engine compressor model.Comment: Conference submissio

Continuity argument revisited: geometry of root clustering via symmetric products

We study the spaces of polynomials stratified into the sets of polynomial with fixed number of roots inside certain semialgebraic region $\Omega$, on its border, and at the complement to its closure. Presented approach is a generalisation, unification and development of several classical approaches to stability problems in control theory: root clustering ($D$-stability) developed by R.E. Kalman, B.R. Barmish, S. Gutman et al., $D$-decomposition(Yu.I. Neimark, B.T. Polyak, E.N. Gryazina) and universal parameter space method(A. Fam, J. Meditch, J.Ackermann). Our approach is based on the interpretation of correspondence between roots and coefficients of a polynomial as a symmetric product morphism. We describe the topology of strata up to homotopy equivalence and, for many important cases, up to homeomorphism. Adjacencies between strata are also described. Moreover, we provide an explanation for the special position of classical stability problems: Hurwitz stability, Schur stability, hyperbolicity.Comment: 45 pages, 4 figure

H-Infinity Optimal Interconnections

In this paper, a general ℋ∞ problem for continuous time, linear time invariant systems is formulated and solved in a behavioral framework. This general formulation, which includes standard ℋ∞ optimization as a special case, provides added freedom in the design of sub-optimal compensators, and can in fact be viewed as a means of designing optimal systems. In particular, the formulation presented allows for singular interconnections, which naturally occur when interconnecting first principles models

ℋ∞ optimization with spatial constraints

A generalized ℋ∞ synthesis problem where non-euclidian spatial norms on the disturbances and output error are used is posed and solved. The solution takes the form of a linear matrix inequality. Some problems which fall into this class are presented. In particular, solutions are presented to two problems: a variant of ℋ∞ synthesis where norm constraints on each component of the disturbance can be imposed, and synthesis for a certain class of robust performance problems

Optimal pricing control in distribution networks with time-varying supply and demand

This paper studies the problem of optimal flow control in dynamic inventory systems. A dynamic optimal distribution problem, including time-varying supply and demand, capacity constraints on the transportation lines, and convex flow cost functions of Legendre-type, is formalized and solved. The time-varying optimal flow is characterized in terms of the time-varying dual variables of a corresponding network optimization problem. A dynamic feedback controller is proposed that regulates the flows asymptotically to the optimal flows and achieves in addition a balancing of all storage levels.Comment: Submitted to 21st International Symposium on Mathematical Theory of Networks and Systems (MTNS) in December 201

Convex Duality Approach to Robust Stabilization of Uncertain Plants

In this thesis we are study the problem of designing the controllers that are robust with respect to the parametric uncertainty. In Part I "The Rank-One Problem" we consider the class of systems with restriction that the structure of uncertainty is limited to a vector. In Chapter " Canonical Parametrization of the Dual Problem in Robust Optimization: Non-Rational Case" we extend the class of allowed systems. The main result is the canonical parametrization of all destabilizing uncertainties in the dual problem. The corresponding result in the rational case was previously stated in terms of unstable zero-pole cancellations. For non-rational systems the situation with common zeros is more complicated. The nominal factors can contain a singular component and cannot be treated by unstable cancellations. We have shown that in the general case the common zeros of the plant factors are naturally replaced by a scalar function with the positive winding number. To illustrate the duality principle, the result is applied to a system with delay. By dual parametrization we can easily calculate the optimal uncertainty bound and the optimal controller. Since the optimal controller is not robustly stabilizing in the strong sense,as it is only a limit of suboptimal robustly stabilizing controllers,we have to regularize the limiting controller. In Chapter "Regularization of the Limiting Optimal Controller in Robust Stabilization" we present a method of obtaining the suboptimal controller of lower order that provides the stability margin as close to the optimal one as we wish. The method is illustrated with some scalar examples. In Chapter "Robust Control via Linear Programming" we propose the numerical algorithm for the optimal robust control synthesis. The algorithm proposed is a sequence of the standard linear programming problems of growing dimensions which approximate the initial problem. In the special case, when the uncertainty parameter is real-valued, it is shown that the initial problem can be considered as finite-dimensional in the space of variables. In Part II "Convex Duality: Matrix Case" we generalize the results to the system with matrix uncertainties. We obtain a canonical factorization of a plant with unstructured uncertainty in terms of an unitary matrix function with finite winding number and an outer matrix function. We introduce a metric in the space of factorization and discuss connection with nu-gap metric