arXiv subjects
Publications and source records attributed to Anders Rantzer.
It is proved that network realizability of controllers can be enforced without conservatism using convex constraints on the closed loop transfer function. Once a network realizable closed loop transfer matrix has been found, a corresponding controller can be implemented using a network structured version of Internal Model Control.
We address a class of systems for which the solution to an H-infinity optimal control problem can be given on a very simple closed form. In fact, both the control law and optimal performance value are explicitly given. The class of systems include models for large-scale systems such as temperature dynamics in buildings, buffer networks and transportations systems. Furthermore, the structure of the control law is suitable for distributed control of such large-scale systems, which is illustrated through examples.
The problem of low-rank approximation with convex constraints, which appears in data analysis, system identification, model order reduction, low-order controller design and low-complexity modelling is considered. Given a matrix, the objective is to find a low-rank approximation that meets rank and convex constraints, while minimizing the distance to the matrix in the squared Frobenius norm. In many situations, this non-convex problem is convexified by nuclear norm regularization. However, we will see that the approximations obtained by this method may be far from optimal. In this paper, we propose an alternative convex relaxation that uses the convex envelope of the squared Frobenius norm and the rank constraint. With this approach, easily verifiable conditions are obtained under which the solutions to the convex relaxation and the original non-convex problem coincide. An SDP representation of the convex envelope is derived, which allows us to apply this approach to several known problems. Our example on optimal low-rank Hankel approximation/model reduction illustrates that the proposed convex relaxation performs consistently better than nuclear norm regularization and may outperform balanced truncation.
Sufficient and necessary conditions for the stability of positive feedback interconnections of negative imaginary systems are derived via an integral quadratic constraint (IQC) approach. The IQC framework accommodates distributed-parameter systems with irrational transfer function representations, while generalising existing results in the literature and allowing exploitation of flexibility at zero and infinite frequencies to reduce conservatism in the analysis. The main results manifest the important property that the negative imaginariness of systems gives rise to a certain form of IQCs on positive frequencies that are bounded away from zero and infinity. Two additional sets of IQCs on the DC and instantaneous gains of the systems are shown to be sufficient and necessary for closed-loop stability along a homotopy of systems.
We develop a simple routine unifying the analysis of several important recently-developed stochastic optimization methods including SAGA, Finito, and stochastic dual coordinate ascent (SDCA). First, we show an intrinsic connection between stochastic optimization methods and dynamic jump systems, and propose a general jump system model for stochastic optimization methods. Our proposed model recovers SAGA, SDCA, Finito, and SAG as special cases. Then we combine jump system theory with several simple quadratic inequalities to derive sufficient conditions for convergence rate certifications of the proposed jump system model under various assumptions (with or without individual convexity, etc). The derived conditions are linear matrix inequalities (LMIs) whose sizes roughly scale with the size of the training set. We make use of the symmetry in the stochastic optimization methods and reduce these LMIs to some equivalent small LMIs whose sizes are at most 3 by 3. We solve these small LMIs to provide analytical proofs of new convergence rates for SAGA, Finito and SDCA (with or without individual convexity). We also explain why our proposed LMI fails in analyzing SAG. We reveal a key difference between SAG and other methods, and briefly discuss how to extend our LMI analysis for SAG. An advantage of our approach is that the proposed analysis can be automated for a large class of stochastic methods under various assumptions (with or without individual convexity, etc).
A multi-variable PI (proportional integrating) controller is proved to be optimal for an important class of control problems where performance is specified in terms of frequency weighted H-infinity norms. The problem class includes networked systems with a subsystem in each node and control action along each edge. For such systems, the optimal PI controller is decentralized in the sense that control action along a given network edge is entirely determined by states at nodes connected by that edge.
We consider a class of monotone systems in which the control signal multiplies the state. Among other applications, such bilinear systems can be used to model the evolutionary dynamics of HIV in the presence of combination drug therapy. For this class of systems, we formulate an infinite horizon optimal control problem, prove that the optimal control signal is constant over time, and show that it can be computed by solving a finite-dimensional non-smooth convex optimization problem. We provide an explicit expression for the subdifferential set of the objective function and use a subgradient algorithm to design the optimal controller. We further extend our results to characterize the optimal robust controller for systems with uncertain dynamics and show that computing the robust controller is no harder than computing the nominal controller. We illustrate our results with an example motivated by combination drug therapy.
Controllers with a diagonal-plus-low-rank structure constitute a scalable class of controllers for multi-agent systems. Previous research has shown that diagonal-plus-low-rank control laws appear as the optimal solution to a class of multi-agent H2 coordination problems, which arise in the control of wind farms. In this paper we show that this result extends to the case where the information exchange between agents is subject to limitations. We also show that the computational effort required to obtain the optimal controller is independent of the number of agents and provide analytical expressions that quantify the usefulness of information exchange.
We address H-infinity structured static state feedback and give a simple form for an optimal control law applicable to linear time invariant systems with symmetric and Hurwitz state matrix. More specifically, the control law as well as the minimal value of the norm can be expressed in the matrices of the system's state space representation, given separate cost on state and control input. Thus, the control law is transparent, easy to synthesize and scalable. Furthermore, if the plant possess a compatible sparsity pattern it is also distributed. Examples of such sparsity patterns are included. Furthermore, we give an extension of the optimal control law that incorporate coordination among subsystems. We demonstrate by a numerical example that the derived optimal controller is equal in performance to an optimal controller derived by the riccati equation approach.
Closed-loop positivity of feedback interconnections of positive monotone nonlinear systems is investigated. It is shown that an instantaneous gain condition on the open-loop systems which implies feedback well-posedness also guarantees feedback positivity. Furthermore, the notion of integral linear constraints (ILC) is utilised as a tool to characterise uncertainty in positive feedback systems. Robustness analysis of positive linear time-varying and nonlinear feedback systems is studied using ILC, paralleling the well-known results based on integral quadratic constraints.
Distributed algorithms for solving coupled semidefinite programs (SDPs) commonly require many iterations to converge. They also put high computational demand on the computational agents. In this paper we show that in case the coupled problem has an inherent tree structure, it is possible to devise an efficient distributed algorithm for solving such problems. This algorithm can potentially enjoy the same efficiency as centralized solvers that exploit sparsity. The proposed algorithm relies on predictor-corrector primal-dual interior-point methods, where we use a message-passing algorithm to compute the search directions distributedly. Message-passing here is closely related to dynamic programming over trees. This allows us to compute the exact search directions in a finite number of steps. Furthermore this number can be computed a priori and only depends on the coupling structure of the problem. We use the proposed algorithm for analyzing robustness of large-scale uncertain systems distributedly. We test the performance of this algorithm using numerical examples.
We study dynamical transportation networks in a framework that includes extensions of the classical Cell Transmission Model to arbitrary network topologies. The dynamics are modeled as systems of ordinary differential equations describing the traffic flow among a finite number of cells interpreted as links of a directed network. Flows between contiguous cells, in particular at junctions, are determined by merging and splitting rules within constraints imposed by the cells' demand and supply functions as well as by the drivers' turning preferences, while inflows at on-ramps are modeled as exogenous and possibly time-varying. First, we analyze stability properties of dynamical transportation networks. We associate to the dynamics a state-dependent dual graph whose connectivity depends on the signs of the derivatives of the inter-cell flows with respect to the densities. Sufficient conditions for the stability of equilibria and periodic solutions are then provided in terms of the connectivity of such dual graph. Then, we consider synthesis of control policies that use a combination of turning preferences, speed limits, and ramp metering, in order to optimize convex objectives. We first show that, in the general case, the optimal control synthesis problem can be cast as a convex optimization problem, and that the equilibrium of the controlled network is in free-flow. If the control policies are restricted to speed limits and ramp metering, then the resulting synthesis problem is still convex for networks where every node is either a merge or a diverge junction, and where the dynamics is monotone. These results apply both to the optimal selection of equilibria and periodic solutions, as well as to finite-horizon network trajectory optimization. Finally, we illustrate our findings through simulations on a road network inspired by the freeway system in southern Los Angeles.
As distributed systems increase in size, the need for scalable algorithms becomes more and more important. We argue that in the context of system identification, an essential building block of any scalable algorithm is the ability to estimate local dynamics within a large interconnected system. We show that in what we term the "full interconnection measurement" setting, this task is easily solved using existing system identification methods. We also propose a promising heuristic for the "hidden interconnection measurement" case, in which contributions to local measurements from both local and global dynamics need to be separated. Inspired by the machine learning literature, and in particular by convex approaches to rank minimization and matrix decomposition, we exploit the fact that the transfer function of the local dynamics is low-order, but full-rank, while the transfer function of the global dynamics is high-order, but low-rank, to formulate this separation task as a nuclear norm minimization.
A system is called positive if the set of non-negative states is left invariant by the dynamics. Stability analysis and controller optimization are greatly simplified for such systems. For example, linear Lyapunov functions and storage functions can be used instead of quadratic ones. This paper shows how such methods can be used for synthesis of distributed controllers. It also shows that stability and performance of such control systems can be verified with a complexity that scales linearly with the number of interconnections. Several results regarding scalable synthesis and verfication are derived, including a new stronger version of the Kalman-Yakubovich-Popov lemma for positive systems. Some main results are stated for frequency domain models using the notion of positively dominated system. The analysis is illustrated with applications to transportation networks, vehicle formations and power systems.
Large-scale interconnected uncertain systems commonly have large state and uncertainty dimensions. Aside from the heavy computational cost of solving centralized robust stability analysis techniques, privacy requirements in the network can also introduce further issues. In this paper, we utilize IQC analysis for analyzing large-scale interconnected uncertain systems and we evade these issues by describing a decomposition scheme that is based on the interconnection structure of the system. This scheme is based on the so-called chordal decomposition and does not add any conservativeness to the analysis approach. The decomposed problem can be solved using distributed computational algorithms without the need for a centralized computational unit. We further discuss the merits of the proposed analysis approach using a numerical experiment.
In this paper, we consider robust stability analysis of large-scale sparsely interconnected uncertain systems. By modeling the interconnections among the subsystems with integral quadratic constraints, we show that robust stability analysis of such systems can be performed by solving a set of sparse linear matrix inequalities. We also show that a sparse formulation of the analysis problem is equivalent to the classical formulation of the robustness analysis problem and hence does not introduce any additional conservativeness. The sparse formulation of the analysis problem allows us to apply methods that rely on efficient sparse factorization techniques, and our numerical results illustrate the effectiveness of this approach compared to methods that are based on the standard formulation of the analysis problem.
In distributed model predictive control (DMPC), where a centralized optimization problem is solved in distributed fashion using dual decomposition, it is important to keep the number of iterations in the solution algorithm, i.e. the amount of communication between subsystems, as small as possible. At the same time, the number of iterations must be enough to give a feasible solution to the optimization problem and to guarantee stability of the closed loop system. In this paper, a stopping condition to the distributed optimization algorithm that guarantees these properties, is presented. The stopping condition is based on two theoretical contributions. First, since the optimization problem is solved using dual decomposition, standard techniques to prove stability in model predictive control (MPC), i.e. with a terminal cost and a terminal constraint set that involve all state variables, do not apply. For the case without a terminal cost or a terminal constraint set, we present a new method to quantify the control horizon needed to ensure stability and a prespecified performance. Second, the stopping condition is based on a novel adaptive constraint tightening approach. Using this adaptive constraint tightening approach, we guarantee that a primal feasible solution to the optimization problem is found and that closed loop stability and performance is obtained. Numerical examples show that the number of iterations needed to guarantee feasibility of the optimization problem, stability and a prespecified performance of the closed-loop system can be reduced significantly using the proposed stopping condition.
A distributed model predictive control (DMPC) approach based on distributed optimization is applied to the power reference tracking problem of a hydro power valley (HPV) system. The applied optimization algorithm is based on accelerated gradient methods and achieves a convergence rate of O(1/k^2), where k is the iteration number. Major challenges in the control of the HPV include a nonlinear and large-scale model, nonsmoothness in the power-production functions, and a globally coupled cost function that prevents distributed schemes to be applied directly. We propose a linearization and approximation approach that accommodates the proposed the DMPC framework and provides very similar performance compared to a centralized solution in simulations. The provided numerical studies also suggest that for the sparsely interconnected system at hand, the distributed algorithm we propose is faster than a centralized state-of-the-art solver such as CPLEX.