Fast algorithms for solving H∞-norm minimization problems

We propose an efficient computational approach to minimize the H ∞-norm of a transfer-function matrix depending affinely on a set of free parameters. The minimization problem, formulated as a semi-infinite convex programming problem, is solved via a relaxation approach over a finite set of frequency values. In this way, a significant speed up is achieved by avoiding the solution of high order LMIs resulting by equivalently formulating the minimization problem as a high dimensional semidefinite programming problem. Numerical results illustrate the superiority of proposed approach over LMIs based techniques in solving zero order H∞-norm approximation problems

Distributed Model Predictive Control with Asymmetric Adaptive Terminal Sets for the Regulation of Large-scale Systems

In this paper, a novel distributed model predictive control (MPC) scheme with asymmetric adaptive terminal sets is developed for the regulation of large-scale systems with a distributed structure. Similar to typical MPC schemes, a structured Lyapunov matrix and a distributed terminal controller, respecting the distributed structure of the system, are computed offline. However, in this scheme, a distributed positively invariant terminal set is computed online and updated at each time instant taking into consideration the current state of the system. In particular, we consider ellipsoidal terminal sets as they are easy to compute for large-scale systems. The size and the center of these terminal sets, together with the predicted state and input trajectories, are considered as decision variables in the online phase. Determining the terminal set center online is found to be useful specifically in the presence of asymmetric constraints. Finally, a relaxation of the resulting online optimal control problem is provided. The efficacy of the proposed scheme is illustrated in simulation by comparing it to a recent distributed MPC scheme with adaptive terminal sets

A semidefinite relaxation procedure for fault-tolerant observer design

A fault-tolerant observer design methodology is proposed. The aim is to guarantee a minimum level of closed-loop performance under all possible sensor fault combinations while optimizing performance under the nominal, fault-free condition. A novel approach is proposed to tackle the combinatorial nature of the problem, which is computationally intractable even for a moderate number of sensors, by recasting the problem as a robust performance problem, where the uncertainty set is composed of all combinations of a set of binary variables. A procedure based on an elimination lemma and an extension of a semidefinite relaxation procedure for binary variables is then used to derive sufficient conditions (necessary and sufficient in the case of one binary variable) for the solution of the problem which significantly reduces the number of matrix inequalities needed to solve the problem. The procedure is illustrated by considering a fault-tolerant observer switching scheme in which the observer outputs track the actual sensor fault condition. A numerical example from an electric power application is presented to illustrate the effectiveness of the design

Voltage stabilization in DC microgrids: an approach based on line-independent plug-and-play controllers

We consider the problem of stabilizing voltages in DC microGrids (mGs) given by the interconnection of Distributed Generation Units (DGUs), power lines and loads. We propose a decentralized control architecture where the primary controller of each DGU can be designed in a Plug-and-Play (PnP) fashion, allowing the seamless addition of new DGUs. Differently from several other approaches to primary control, local design is independent of the parameters of power lines. Moreover, differently from the PnP control scheme in [1], the plug-in of a DGU does not require to update controllers of neighboring DGUs. Local control design is cast into a Linear Matrix Inequality (LMI) problem that, if unfeasible, allows one to deny plug-in requests that might be dangerous for mG stability. The proof of closed-loop stability of voltages exploits structured Lyapunov functions, the LaSalle invariance theorem and properties of graph Laplacians. Theoretical results are backed up by simulations in PSCAD

Improving Efficiency and Scalability of Sum of Squares Optimization: Recent Advances and Limitations

It is well-known that any sum of squares (SOS) program can be cast as a semidefinite program (SDP) of a particular structure and that therein lies the computational bottleneck for SOS programs, as the SDPs generated by this procedure are large and costly to solve when the polynomials involved in the SOS programs have a large number of variables and degree. In this paper, we review SOS optimization techniques and present two new methods for improving their computational efficiency. The first method leverages the sparsity of the underlying SDP to obtain computational speed-ups. Further improvements can be obtained if the coefficients of the polynomials that describe the problem have a particular sparsity pattern, called chordal sparsity. The second method bypasses semidefinite programming altogether and relies instead on solving a sequence of more tractable convex programs, namely linear and second order cone programs. This opens up the question as to how well one can approximate the cone of SOS polynomials by second order representable cones. In the last part of the paper, we present some recent negative results related to this question.Comment: Tutorial for CDC 201

Robust nonlinear control of vectored thrust aircraft

An interdisciplinary program in robust control for nonlinear systems with applications to a variety of engineering problems is outlined. Major emphasis will be placed on flight control, with both experimental and analytical studies. This program builds on recent new results in control theory for stability, stabilization, robust stability, robust performance, synthesis, and model reduction in a unified framework using Linear Fractional Transformations (LFT's), Linear Matrix Inequalities (LMI's), and the structured singular value micron. Most of these new advances have been accomplished by the Caltech controls group independently or in collaboration with researchers in other institutions. These recent results offer a new and remarkably unified framework for all aspects of robust control, but what is particularly important for this program is that they also have important implications for system identification and control of nonlinear systems. This combines well with Caltech's expertise in nonlinear control theory, both in geometric methods and methods for systems with constraints and saturations

Robust Constrained Model Predictive Control using Linear Matrix Inequalities

The primary disadvantage of current design techniques for model predictive control (MPC) is their inability to deal explicitly with plant model uncertainty. In this paper, we present a new approach for robust MPC synthesis which allows explicit incorporation of the description of plant uncertainty in the problem formulation. The uncertainty is expressed both in the time domain and the frequency domain. The goal is to design, at each time step, a state-feedback control law which minimizes a "worst-case" infinite horizon objective function, subject to constraints on the control input and plant output. Using standard techniques, the problem of minimizing an upper bound on the "worst-case" objective function, subject to input and output constraints, is reduced to a convex optimization involving linear matrix inequalities (LMIs). It is shown that the feasible receding horizon state-feedback control design robustly stabilizes the set of uncertain plants under consideration. Several extensions, such as application to systems with time-delays and problems involving constant set-point tracking, trajectory tracking and disturbance rejection, which follow naturally from our formulation, are discussed. The controller design procedure is illustrated with two examples. Finally, conclusions are presented

Issues in the design of switched linear systems : a benchmark study

In this paper we present a tutorial overview of some of the issues that arise in the design of switched linear control systems. Particular emphasis is given to issues relating to stability and control system realisation. A benchmark regulation problem is then presented. This problem is most naturally solved by means of a switched control design. The challenge to the community is to design a control system that meets the required performance specifications and permits the application of rigorous analysis techniques. A simple design solution is presented and the limitations of currently available analysis techniques are illustrated with reference to this example

Robust stability of time-varying uncertain systems with rational dependence on the uncertainty

Robust stability of time-varying uncertain systems is a key problem in automatic control. This note considers the case of linear systems with rational dependence on an uncertain time-varying vector constrained in a polytope, which is typically addressed in the literature by using the linear fractional representation (LFR). A novel sufficient condition for robust stability is derived in terms of a linear matrix inequality (LMI) feasibility test by exploiting homogeneous polynomial Lyapunov functions, the square matrix representation and an extended version of Polya's theorem which considers structured matrix polynomials. It is shown that this condition is also necessary for second-order systems, and that this condition is less conservative than existing LMI conditions based on the LFR for any order. © 2010 IEEE.published_or_final_versio