Performance bounds for optimal feedback control in networks

Many important complex networks, including critical infrastructure and emerging industrial automation systems, are becoming increasingly intricate webs of interacting feedback control loops. A fundamental concern is to quantify the control properties and performance limitations of the network as a function of its dynamical structure and control architecture. We study performance bounds for networks in terms of optimal feedback control costs. We provide a set of complementary bounds as a function of the system dynamics and actuator structure. For unstable network dynamics, we characterize a tradeoff between feedback control performance and the number of control inputs, in particular showing that optimal cost can increase exponentially with the size of the network. We also derive a bound on the performance of the worst-case actuator subset for stable networks, providing insight into dynamics properties that affect the potential efficacy of actuator selection. We illustrate our results with numerical experiments that analyze performance in regular and random networks

Minimal Actuator Placement with Optimal Control Constraints

We introduce the problem of minimal actuator placement in a linear control system so that a bound on the minimum control effort for a given state transfer is satisfied while controllability is ensured. We first show that this is an NP-hard problem following the recent work of Olshevsky. Next, we prove that this problem has a supermodular structure. Afterwards, we provide an efficient algorithm that approximates up to a multiplicative factor of O(logn), where n is the size of the multi-agent network, any optimal actuator set that meets the specified energy criterion. Moreover, we show that this is the best approximation factor one can achieve in polynomial-time for the worst case. Finally, we test this algorithm over large Erdos-Renyi random networks to further demonstrate its efficiency.Comment: This version includes all the omitted proofs from the one to appear in the American Control Conference (ACC) 2015 proceeding

On Submodularity and Controllability in Complex Dynamical Networks

Controllability and observability have long been recognized as fundamental structural properties of dynamical systems, but have recently seen renewed interest in the context of large, complex networks of dynamical systems. A basic problem is sensor and actuator placement: choose a subset from a finite set of possible placements to optimize some real-valued controllability and observability metrics of the network. Surprisingly little is known about the structure of such combinatorial optimization problems. In this paper, we show that several important classes of metrics based on the controllability and observability Gramians have a strong structural property that allows for either efficient global optimization or an approximation guarantee by using a simple greedy heuristic for their maximization. In particular, the mapping from possible placements to several scalar functions of the associated Gramian is either a modular or submodular set function. The results are illustrated on randomly generated systems and on a problem of power electronic actuator placement in a model of the European power grid.Comment: Original arXiv version of IEEE Transactions on Control of Network Systems paper (Volume 3, Issue 1), with a addendum (located in the ancillary documents) that explains an error in a proof of the original paper and provides a counterexample to the corresponding resul

Combined structures-controls optimization of lattice trusses

The role that distributed parameter model can play in CSI is demonstrated, in particular in combined structures controls optimization problems of importance in preliminary design. Closed form solutions can be obtained for performance criteria such as rms attitude error, making possible analytical solutions of the optimization problem. This is in contrast to the need for numerical computer solution involving the inversion of large matrices in traditional finite element model (FEM) use. Another advantage of the analytic solution is that it can provide much needed insight into phenomena that can otherwise be obscured or difficult to discern from numerical computer results. As a compromise in level of complexity between a toy lab model and a real space structure, the lattice truss used in the EPS (Earth Pointing Satellite) was chosen. The optimization problem chosen is a generic one: of minimizing the structure mass subject to a specified stability margin and to a specified upper bond on the rms attitude error, using a co-located controller and sensors. Standard FEM treating each bar as a truss element is used, while the continuum model is anisotropic Timoshenko beam model. Performance criteria are derived for each model, except that for the distributed parameter model, explicit closed form solutions was obtained. Numerical results obtained by the two model show complete agreement

Techniques for Optimum Design of Actively Controlled Structures Including Topological Considerations

The design and performance of complex engineering systems often depends on several conflicting objectives which, in many cases, cannot be represented as a single measure of performance. This thesis presents a multi-objective formulation for a comprehensive treatment of the structural and topological considerations in the design of actively controlled structures. The dissertation addresses three main problems. The first problem deals with optimum placement of actuators in actively controlled structures. The purpose of control is to reduce the vibrations when the structure is subjected to a disturbance. In order to mitigate the structural vibrations as quickly as possible, it is necessary to place the actuators at locations such that their ability to control the vibrations is maximized. Since the actuator locations are discrete (0-1) variables, a genetic algorithm based approach is used to solve the resulting optimization problem. The second problem this dissertation addresses is the multi-objective design of actively controlled structures. Structural weight, controller performance index and energy dissipated by the actuators are considered as the objective functions. It is assumed that a hierarchical structure exist between the actuator placement and structural-control design objective functions with the actuator placement problem considered being more important. The resulting multi-objective optimization problem is solved using Stackelberg game and cooperative game theory approaches. The exchange of information between different levels of the multi-level problem is done by constructing the rational reaction set of follower solution using design of experiments and response surface methods. The third problem addressed in this dissertation is the optimization of structural topology in the context of structural/control system design. Despite the recognition that an optimization of topology can significantly improve structural performance, most of the work in design of actively controlled structures has been done with structures of a known topology. The combined topology and sizing optimization of actively controlled structures is also considered in this thesis. The approach presented involves the determination of optimum topology followed by a sizing and control system optimization of the optimum topology. Using two numerical examples, it is shown that a simultaneous consideration of topological, control and structural aspects yields solutions that outperform designs when topological considerations are neglected

Optimal sensor/actuator placement and switching schemes for control of flexible structures

The vibration control problem for flexible structures is examined within the context of overall controller performance and power reduction. First, the issue of optimal sensor and actuator placement is considered along with its associated control robustness aspects. Then the option of alternately activating subsets of the available devices is investigated. Such option is considered in order to better address the effects of spatiotemporally varying disturbances acting on a flexible structure while reducing the overall energy consumption. Towards the solution to the problem of optimal device placement, three different approaches are proposed. First, a computationally efficient scheme for the simultaneous placement of multiple devices is presented. The second approach proposes a strategy for the optimal placement of sensors and collocated sensor/actuator pairs, taking into account the influence of the spatial distribution of disturbances. The third approach provides a solution to the actuator location problem by incorporating considerations with respect to preferred spatial regions within the flexible structure. Then the second problem named above is considered. Activating a subset of the available and optimally placed actuators and sensors in a flexible structure provides enhanced performance with reduced energy consumption. Such approach of switching on and off different actuating devices, depending on their local-in-time authority, results in a hybrid system. Therefore the proposed work draws on existing results on hybrid systems and includes an additional degree of freedom, whereby both the actuating devices and the control signals allocated to them are switched in and out. To enable this switching an activation strategy, which insures also that stability-under-switching is guaranteed, is required. Three different strategies are considered for such actuators allocation: first a cost-to-go index is considered, then a cost function based on the mechanical energy of the flexible structure and finally a performance index based on the maximum deviation of the transverse displacement. A flexible aluminum plate was chosen to validate and test the proposed approaches. The set up utilized four pairs of collocated piezoceramic patches that serve to provide sensing and actuating capabilities. Extensive numerical simulations were performed for both the placement strategies and the switching policies proposed, in order to predict the behavior of the flexible plate and provide the optimal actuator and sensor locations that were to be affixed on the flexible structure. Finally, to complete the validation process a sequence of experimental tests were performed. The objective of these tests was to compare the performance of the proposed hybrid control system to traditional non switched control schemes. In order to provide a repeatable perturbation, four of the piezoceramic patches were allocated to simulate a spatiotemporally varying disturbance, while the remaining four patches were used as sensors and controlling actuators. The experimental results showed a significant performance improvement for the switched controller over the traditional controller. Moreover the switched controller exhibited improved robustness towards spatiotemporally varying disturbances while the traditional controller showed a significant loss of controller performance. The improvement achieved in vibration control problems could be extended to a wider range of applications. In particular, although this study was concentrated on a rectangular thin plate, the proposed strategies can be applied to emph{any} structure and more generally to any plant whose dynamics can be represented by a second order linear system. For example, by removing the restriction of spatially fixed actuators and sensors, the proposed theory can be applied to the problem of unmanned vehicles control

Controller Switching Policy in Flexible Plates Using PZT Actuators Subject to Spatiotemporal Variations of Disturbances

The primary goal of this thesis is to evaluate vibration control of an all-clamped plate having an unknown disturbance. The vibration control is implemented using a piezoelectric actuator placed at an optimal location. The first part of this thesis considers a robust actuator placement with respect to varying spatial distributions of disturbances. The treatment here, is different from performance-based LQR approaches, since it is based on minimizing the effect of the disturbance distributions. The second part of this thesis addresses a more general case where the plate is under an unknown disturbance. An unknown disturbance is also characterized by the case where the disturbance signal moves randomly over the entire spatial domain. An optimal switching controller algorithm is developed, based on LQR performance, which switches between piezoelectric actuators employed for the vibration control of the plate. A single actuator is selected from the various actuator locations during each time interval, which leads to performance enhancement