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An alternating direction method of multipliers algorithm for symmetric model predictive control
Abstract
This article presents an alternating direction method of multipliers (ADMM) algorithm for solving large‐scale model predictive control (MPC) problems that are invariant under the symmetric‐group. Symmetry was used to find transformations of the inputs, states, and constraints of the MPC problem that decompose the dynamics and cost. We prove an important property of the symmetric decomposition for the symmetric‐group that allows us to efficiently transform between the original and decomposed symmetric domains. This allows us to solve different subproblems of a baseline ADMM algorithm in different domains where the computations are less expensive. This reduces the computational cost of each iteration from quadratic to linear in the number of repetitions in the system. In addition, we show that the memory complexity for our ADMM algorithm is also linear in number of repetitions in the system, rather than the typical quadratic complexity. We demonstrate our algorithm for two case studies; battery balancing and heating, ventilation, and air conditioning. In both case studies, the symmetric algorithm reduced the computation‐time from minutes to seconds and memory usage from tens of megabytes to tens or hundreds of kilobytes, allowing the previously nonviable MPCs to be implemented in real time on embedded computers with limited computational and memory resources.
... An efficient method has been proposed in [12], [13] where an equivalent simple affine function has been employed to remove the regions whose associated control laws attain a saturated value, leading to significant reduction in the storage requirement. In another proposed approach [14], [15], symmetries of the MPC problem have been computed as a mathematical problem in order to reduce inherent complexity of explicit MPC with no change on optimality of the original representation. Identification of the set of all controller symmetries plays an important role in efficiency of the method of eliminating the symmetric regions [14]. ...
... Identification of the set of all controller symmetries plays an important role in efficiency of the method of eliminating the symmetric regions [14]. To identify these controller symmetries, however, [15], [16] has shown the fact that the symmetries corresponding to control regions can be identified by using graph theory. For this objective, it is shown that the problem of finding controller symmetries is converted to graph automorphism problem which can easily be solved by the standard graph automorphism software packages [17], [19]. ...
... Ref. [34] exploited symmetries in interior-point algorithms involved in the solution of optimization problems. References [35], [36], [37] used symmetry to reduce the computational complexity required by the solution of model predictive control problems. While all these papers have exploited symmetry to reduce the computational complexity involved in the solution of problems of different nature, in this paper we recognize that sometimes dimensionality reductions are possible even in systems that do not possess symmetry. ...
... As an example of this, the reader may consider the distinction between 'orbit partitions' and 'equitable partitions' in graph theory [38], where the former is generated by symmetry, while the latter is not. References [36], [26], [32], [39], [40], [41] have focused on a decomposition based on the symmetries of the system and, if applicable, of the objective function, which requires calculation of the irreducible representations (IRR) of the symmetry group. The goal of this paper is to compare the symmetry approach with an alternative symmetry-independent approach, which we will show yields better and faster decompositions. ...
In this paper, we consider optimal control problems (OCPs) applied to large-scale linear dynamical systems with a large number of states and inputs. We attempt to reduce such problems into a set of independent OCPs of lower dimensions. Our decomposition is ‘exact’ in the sense that it preserves all the information about the original system and the objective function. Previous work in this area has focused on strategies that exploit symmetries of the underlying system and of the objective function. Here, instead, we implement the algebraic method of simultaneous block diagonalization of matrices (SBD), which we show provides advantages both in terms of the dimension of the subproblems that are obtained and of the computation time. We provide practical examples with networked systems that demonstrate the benefits of applying the SBD decomposition over the decomposition method based on group symmetries.
... Therefore, computational efficiency is crucial for solving MPC problems. There have been many improved works in this area, for example, the extended Newton Raphson algorithm [16], the gradient algorithm [17], and the ADMM algorithm (the common MPC problems: the Lasso MPC problem for time-varying systems [18], the MPCT problem [19], the MPC problem for systems with feedback gain [20], the BCMPC problem [21], and the symmetric MPC problem [22]). ...
In this paper, we consider optimal control problems with linear discrete state space model, which originate from a class of turbofan engines. The optimization problem associated with each moving horizon estimation (MHE) in classical model predictive control (MPC) is a qua-dratic programming (QP) problem, and the general QP algorithms does not exploit the structural features of the turbofan engine itself to improve the computational efficiency of the algorithm. In the framework of model predictive control, the turbofan engine model makes the rolling optimization subproblem exhibit a sparse structure. Based on this feature, the alternating direction method of multipliers (ADMM) is employed to solve each optimization sub-problem and design an improved MPC-ADMM algorithm for solving this class of optimal control problems. The simulation results are compared with the MPC-QP algorithm by numerical examples to show the effectiveness and superiority of the novel algorithm.
This paper is concerned with the application of model predictive control (MPC) to large-scale linear dynamical systems with linear inequality constraints. A decomposition is proposed of such problems into sets of independent MPCs of lower dimensions that preserves all information about the system, cost function, and constraints. Different from previous work, the constraints are incorporated in the decomposition procedure, which is attained by generalizing a previously developed technique to simultaneously block diagonalize a set of matrices. This approach is applied to practical examples involving large-scale systems with inequality constraints. It is shown that the computational complexity and the CPU time required to solve the transformed MPC problems are lower than those required by the solution of the original MPC problem.
Real-world systems in epidemiology, social sciences, power transportation, economics and engineering are often described as multilayer networks. Here we first define and compute the symmetries of multilayer networks, and then study the emergence of cluster synchronization in these networks. We distinguish between independent layer symmetries, which occur in one layer and are independent of the other layers, and dependent layer symmetries, which involve nodes in different layers. We study stability of the cluster synchronous solution by decoupling the problem into a number of independent blocks and assessing stability of each block through a Master Stability Function. We see that blocks associated with dependent layer symmetries have a different structure to the other blocks, which affects the stability of clusters associated with these symmetries. Finally, we validate the theory in a fully analog experiment in which seven electronic oscillators of three kinds are connected with two kinds of coupling. Complex systems in the real world are often characterized by connected patterns interacting between each other in multiple ways. Here, Della Rossa et al. describe a general method to determine symmetries in multilayer networks and then relate them to different synchronization modes that the networks can exhibit.
This paper presents the design and realization of a linear Model Predictive Controller (MPC) and state estimator for a multi-zone heat pump in the Modelica modeling language, in order to validate closed-loop performance prior to experimental testing. The vapor compression system uses a variable speed compressor and a set of expansion valves for control, and it is required to regulate zone temperatures to set-points without offset. Constraints are imposed on all control inputs and also the values of both measured and unmeasured system outputs. Because experimental testing is both expensive and time-consuming, we have developed a tool chain for software-in-the-loop validation that uses a Modelica model for the plant, integrated with a software representation of the MPC that is realized in a combination of Modelica and C that is suitable for real-time use. We show the results of closedloop tests of the controller with a nonlinear system model, which provide a partial validation of the controller and tool chain.
In this paper we derive a formulation for Model Predictive Control (MPC) of linear time-invariant systems based on H infinity loop-shaping. The design provides an optimized stability margin for problems that require state estimation. Input and output weights are designed in the frequency domain to satisfy steady-state and transient performance requirements, in lieu of conventional MPC plant model augmentations. The H infinity loop-shaping synthesis results in an observer-based state feedback structure. Using the linear state feedback law, an inverse optimal control problem is solved to design the MPC cost function, and the H infinity state estimator is used to initialize the prediction model at each time step. The MPC inherits the closed-loop performance and stability margin of the loopshaped design when constraints are inactive. We apply the methodology to a multi-zone heat pump system in simulation. The design rejects constant unmeasured disturbances and tracks constant references with zero steady-state error, has good transient performance, provides an excellent stability margin, and enforces input and output constraints.
We derive a power-optimizing output feedback controller for a multi-zone heat pump that (1) regulates individual zone temperatures, rejecting unknown heat load disturbances, (2) regulates condenser subcooling and (3) the compressor discharge temperature, and (4) minimizes electrical power consumption at steady-state operating conditions. The design is a cascade of a linear inner-loop and a nonlinear outer-loop. The inner-loop is designed for robust disturbance rejection using H-infinity loop-shaping methods. The outer-loop uses a model of compressor and fan power consumption and a gradient descent feedback to drive the system to its power-minimizing equilibrium for constant values of references and disturbances. The controller uses only temperature measurements for feedback; refrigerant pressure sensors, which are not present in many products for cost reasons, are not required. A proof of exponential stability is provided and preliminary experimental tests demonstrate satisfactory transient responses for a commercial multi-zone heat pump.
Real-time optimal control algorithms for fast, mechatronic systems need to be run on embedded hardware and they need to respect tight timing constraints. When using nonlinear models, the simulation and generation of sensitivities forms a computationally demanding part of any algorithm. Automatic code generation of Implicit Runge-Kutta (IRK) methods has been shown to reduce its CPU time significantly. However, a typical model also shows a lot of structure that can be exploited in a rather elegant and efficient way. The focus of this paper is on nonlinear models with linear subsystems. With the proposed model formulation, the new auto generated integrators can be considered a powerful generalization of other solvers, e.g. those that support quadrature variables. A speedup of up to 5 - 10 is shown in the integration time for two examples from the literature.
We investigate the infeasibility detection in the alternating direction method of multipliers (ADMM) when minimizing a convex quadratic objective subject to linear equalities and simple bounds. The ADMM formulation consists of alternating between an equality constrained quadratic program (QP) and a projection onto the bounds. We show that: (i) the sequence of iterates generated by ADMM diverges, (ii) the divergence is restricted to the component of the multipliers along the range space of the constraints and (iii) the primal iterates converge to a minimizer of the Euclidean distance between the subspace defined by equality constraints and the convex set defined by bounds. In addition, we derive the optimal value for the step size parameter in the ADMM algorithm that maximizes the rate of convergence of the primal iterates and dual iterates along the null space. In fact, the optimal step size parameter for the infeasible instances is identical to that for the feasible instances. The theoretical results allow us to specify a practical termination condition for infeasibility and the performance of such criterion is demonstrated in a model predictive control application.
We consider an approach for solving strictly convex quadratic programs (QPs) with general linear inequalities by the alternating direction method of multipliers (ADMM). In particular, we focus on the application of ADMM to the QPs of constrained Model Predictive Control (MPC). After introducing our ADMM iteration, we provide a proof of convergence closely related to the theory of maximal monotone operators. The proof relies on a general measure to monitor the rate of convergence and hence to characterize the optimal step size for the iterations. We show that the identified measure converges at a Q-linear rate while the iterates converge at a 2-step Q-linear rate. This result allows us to relax some of the existing assumptions in optimal step size selection, that currently limit the applicability to the QPs of MPC. The results are validated through a large public benchmark set of QPs of MPC for controlling a four tank process.
Motivated by recent interest in group-symmetry in semidefinite programming, we propose a numerical method for finding a finest simultaneous block-diagonalization of a finite number of matrices, or equivalently the irreducible decomposition of the generated matrix
${*}$
-algebra. The method is composed of numerical-linear algebraic computations such as eigenvalue computation, and automatically makes full use of the underlying algebraic structure, which is often an outcome of physical or geometrical symmetry, sparsity, and structural or numerical degeneracy in the given matrices. The main issues of the proposed approach are presented in this paper under some assumptions, while the companion paper gives an algorithm with full generality. Numerical examples of truss and frame designs are also presented.
Semidefinite programming (SDP) is one of the most active areas in mathematical programming, due to varied applications and
the availability of interior point algorithms. In this paper we propose a new pre-processing technique for SDP instances that
exhibit algebraic symmetry. We present computational results to show that the solution times of certain SDP instances may
be greatly reduced via the new approach.
KeywordsSemidefinite programming–Algebraic symmetry–Pre-processing–Interior point methods
We describe a methodology for modeling, analysis and distributed control design of a large vehicular formation whose information graph is a D -dimensional lattice. We derive asymptotic formulae for the closed-loop stability margin based on a partial differential equation (PDE) approximation of the formation. We show that the exponent in the scaling law for the stability margin is influenced by the structure of the information graph and by the control architecture (symmetric or asymmetric). For a given fixed number of vehicles, we show that the scaling law can be improved significantly by employing a higher dimensional information graph and/or by introducing small asymmetry (mistuning) in the nominally symmetric proportional control gains. We also provide a characterization of the error introduced by the PDE approximation.
This paper deals with exploiting symmetry for solving linear and integer
programming problems. Basic properties of linear representations of finite
groups can be used to reduce symmetric linear programming to solving linear
programs of lower dimension. Combining this approach with knowledge of the
geometry of feasible integer solutions yields an algorithm for solving highly
symmetric integer linear programs which only takes time which is linear in the
number of constraints and quadratic in the dimension.
We study dynamical systems that switch between two different vector fields depending on a discrete variable and with a delay. When the delay reaches a problem-dependent critical value so-called event collisions occur. This paper classifies and analyzes event collisions, a special type of discontinuity induced bifurcations, for periodic orbits. Our focus is on event collisions of symmetric periodic orbits in systems with full reflection symmetry, a symmetry that is prevalent in applications. We derive an implicit expression for the Poincare map near the colliding periodic orbit. The Poincar map is piecewise smooth, finite-dimensional, and changes the dimension of its image at the collision. In the second part of the paper we apply this general result to the class of unstable linear single-degree-of-freedom oscillators where we detect and continue numerically collisions of invariant tori. Moreover, we observe that attracting closed invariant polygons emerge at the torus collision. The research of J.S. and P.K. was partially supported by by EPSRC grant GR/R72020/01.
There has been substantial work studying consensus problems for which there is a single common final state, although there are many real-world complex networks for which the complete consensus may be undesirable. More recently, the concept of group consensus whereby subsets of nodes are chosen to reach a common final state distinct from others has been developed, but the methods tend to be independent of the underlying network topology. Here, an alternative type of group consensus is achieved for which nodes that are “symmetric” achieve a common final state. The dynamic behavior may be distinct between nodes that are not symmetric. We show how group consensus for heterogeneous linear agents can be achieved via a simple coupling protocol that exploits the topology of the network. We see that group consensus is possible on both stable and unstable trajectories. We observe and characterize the phenomenon of “isolated group consensus,” where one or more clusters may achieve group consensus while the other clusters do not.
Readers of this book will be shown how, with the adoption of ubiquituous sensing, extensive data-gathering and forecasting, and building-embedded advanced actuation, intelligent building systems with the ability to respond to occupant preferences in a safe and energy-efficient manner are becoming a reality. The articles collected present a holistic perspective on the state of the art and current research directions in building automation, advanced sensing and control, including:
• model-based and model-free control design for temperature control;
• smart lighting systems;
• smart sensors and actuators (such as smart thermostats, lighting fixtures and HVAC equipment with embedded intelligence); and
• energy management, including consideration of grid connectivity and distributed intelligence.
These articles are both educational for practitioners and graduate students interested in design and implementation, and foundational for researchers interested in understanding the state of the art and the challenges that must be overcome in realizing the potential benefits of smart building systems. This edited volume also includes case studies from implementation of these algorithms/sensing strategies in to-scale building systems. These demonstrate the benefits and pitfalls of using smart sensing and control for enhanced occupant comfort and energy efficiency.
Multi-evaporator vapor compression systems (ME-VCS) are inherently multi-input multi-output (MIMO) systems, often with complex, highly coupled dynamics. Thus, they require more sophisticated control schemes than traditional on-off logic, or decentralized proportional-integral controllers. Unfortunately, many MIMO control design techniques are not well suited for this problem since they require complex numerical computations that do not scale gracefully for the high-dimensional dynamics of ME-VCS systems. This paper exploits the observed similarity of the room dynamics to reduce the computational complexity of designing controllers. We use a linear matrix inequality based controller synthesis technique that exploits symmetry for designing controllers for large-scale ME-VCS systems. This controller synthesis technique was applied to an ME-VCS system with 50 rooms. Using tradition control design methods required 41 hours to synthesize a controller, while our technique designed an identical controller in less than 1 second.
Two of the most common pattern formation mechanisms are Turing-patterning in reaction-diffusion systems and lateral inhibition of neighboring cells. In this paper, we introduce a broad dynamical model of interconnected cells to study the emergence of patterns, with the above mentioned two mechanisms as special cases. This model comprises modules encapsulating the biochemical reactions in individual cells, and interconnections are captured by a weighted directed graph. Leveraging only the static input/output properties of the subsystems and the spectral properties of the adjacency matrix, we characterize the stability of the homogeneous fixed points as well as sufficient conditions for the emergence of spatially non-homogeneous patterns. To obtain these results, we rely on properties of the graphs (bipartiteness, equitable partitions) together with tools from monotone systems theory. As application example, we consider pattern formation in neural networks to illustrate the practical implications of our results. Our results do not restrict the number of cells or reactants, and do not assume symmetric connections between two connected cells.
This paper considers the control of a multievaporator vapor compression system (ME-VCS) where individual evaporators are permitted to turn ON or OFF. We present a model predictive controller (MPC) that can be easily reconfigured for different ON/OFF configurations of the system. In this approach, only the cost function of the constrained finite-time optimal control problem is updated depending on the system configuration. Exploiting the structure of the system dynamics, the cost function is modified by zeroing elements of the state, input, and terminal cost matrices. The advantage of this approach is that cost matrices for each configuration of the ME-VCS do not need to be stored or computed online. This reduces the effort required to tune and calibrate the controller and the amount of memory required to store the controller parameters in a microprocessor. The reconfigurable MPC is compared with a conventional approach in which individual model predictive controllers are independently designed for each ON/OFF configuration. The simulations show that the reconfigurable MPC method provides a similar closed-loop performance in terms of reference tracking and constraint satisfaction to the set of individual model predictive controllers. Further, we show that our controller requires substantially less memory than the alternative approaches. Experiments on a residential two-zone vapor compression system further validate the reconfigurable MPC method.
We study a flow network model for vehicular traffic that captures congestion effects at diverging junctions. Standard approaches which rely on monotonicity of the flow dynamics do not immediately apply to such first-in-first-out models. The network model nonetheless exhibits a mixed monotonicity property. Mixed monotonicity enables the original system to be embedded in a system of twice the dimension that is monotone and symmetric. The dynamics of the original system are recovered on a subspace of the embedding system, and we prove global asymptotic stability for a class of networks by considering convergence properties of the embedding system.
We exploit symmetries in the interconnection topology of a networked system to provide a dimensionality reduction in the certification of performance. The certification method exploits the dissipativity properties of the subsystems; thus the conservatism introduced by the reduction is minimal when the subsystems possess similar dissipativity characteristics. We combine this reduction with distributed optimization techniques to be able to analyze large interconnections efficiently.
We study cluster synchronization in networks with symmetries in the presence of small generic parametric mismatches of two different types: mismatches affecting the dynamics of the individual uncoupled systems and mismatches affecting the network couplings. We perform a stability analysis of the nearly synchronous cluster synchronization solution and reduce the stability problem to a low-dimensional form. We also show how under certain conditions the low dimensional analysis can be used to predict the overall synchronization error, i.e., how close the individual nearly synchronous trajectories are to each other.
This paper extends previous results on symmetry in strictly convex linear model predictive control to non-strictly convex and nonlinear model predictive control. We define symmetry for constrained systems, controllers, and model predictive control problems. We show that symmetric model predictive control problems produce symmetric controllers. We show that the previously established methods of memory reduction can be applied to non-strictly convex problems. We apply these memory reduction techniques to the battery balancing problem. Exploiting symmetry leads to an exponential memory reduction and simple, intuitive optimal controllers.
This paper studies symmetry in linear model predictive control (MPC). We define symmetry for model predictive control laws and for model predictive control problems. Properties of both MPC symmetries are studied by using a group theory formalism. We show how to efficiently compute MPC symmetries by transforming the search of MPC symmetry generators into a graph automorphism problem. MPC symmetries are then used to design model predictive control algorithms with reduced complexity. The effectiveness of the proposed approach is shown through a simple large-scale MPC problem whose explicit solution can only be found with the method presented in this paper.
In this paper we consider the problem of finding all the state-space and input-space transformations that preserve the parameters of a constrained linear system. Such transformations are called symmetries. For systems constrained by bounded polytopes the set of all symmetries is a finite group and requires techniques from discrete mathematics to find. We transform the problem of finding the symmetries of a constrained linear system into the problem of finding the symmetries of a vertex colored graph. The symmetries of a vertex colored graph can be found efficiently using graph automorphism software. We demonstrate our symmetry identification procedure on a quadcopter example.
We exploit symmetry to reduce the memory storage requirements of linear explicit model predictive controllers. In the first part of the paper we define controller symmetry. We describe how symmetry can be used to compress the explicit controller and discuss the implementation of the resulting compressed controller. In the second part we develop a method for computing the reduced-memory controller without first computing the full-memory controller by employing the concept of fundamental domain.
Every finite (permutation) group is the full symmetry group of a suitable linear program.
This paper studies the control of distributed storage networks with guarantees of constraints satisfaction and asymptotic stability. We consider two problems: network capacity maximization and network balancing. In the first part of the paper we describe the two problems, highlight their importance in a wide number of engineering applications, and compare them by analyzing the properties of their solutions. In the second part we present algorithms for solving both problems by using a convex one-step model predictive controller (MPC) which guarantees persistent state and flow constraints satisfaction. We present simple conditions which link the network topology, the MPC weights and the asymptotic stability of the closed-loop system. A numerical example illustrates the effectiveness of the proposed approach.
In this paper we evaluate the performance of seven proposed hardware topologies for balancing the cells in a battery pack. We consider four classes of battery balancing hardware; shunting, cell-to-stack, storage element, and dissipative hardware. We present models of these hardware topologies that capture the steady-state behavior of the balancing hardware dynamics. We evaluate the hardware topologies based on time required to balance the cells and energy dissipated during balancing. A linear programming based method for calculating the worst-case time to balance and energy dissipated during balance is provided. The number of linear programs required to compute both metrics grows exponentially with the number of cells. We show how to use symmetries to efficiently compute both metrics. Our approach scales well for large-scale battery packs and provides non intuitive solutions.
We consider the following problem: Given a set of m×n real (or complex) matrices A1,…,AN, find an m×m orthogonal (or unitary) matrix P and an n×n orthogonal (or unitary) matrix Q such that P*A1Q,…,P*ANQ are in a common block-diagonal form with possibly rectangular diagonal blocks. We call this the simultaneous singular value decomposition (simultaneous SVD). The name is motivated by the fact that the special case with N=1, where a single matrix is given, reduces to the ordinary SVD. With the aid of the theory of *-algebra and bimodule it is shown that a finest simultaneous SVD is uniquely determined. An algorithm is proposed for finding the finest simultaneous SVD on the basis of recent algorithms of Murota–Kanno–Kojima–Kojima and Maehara–Murota for simultaneous block-diagonalization of square matrices under orthogonal (or unitary) similarity.
This paper presents a new lumped-parameter model for describing the dynamics of vapor compression cycles. In particular, the dynamics associated with the two heat exchangers, i.e., the evaporator and the condenser, are modeled based on a moving-interface approach by which the position of the two-phase/single-phase interface inside the one-dimensional heat exchanger can be properly predicted. This interface information has never been included in previous lumped-parameter models developed for control design purpose, although it is essential in predicting the refrigerant superheat or subcool value. This model relates critical performance outputs, such as evaporating pressure, condensing pressure, and the refrigerant superheat, to actuating inputs including compressor speed, fan speed, and expansion valve opening. The dominating dynamic characteristics of the cycle around an operating point is studied based an the linearized model. From the resultant transfer function matrix, an interaction measure based on the Relative Cain Array reveals strong cross-couplings between various input-output pairs, and therefore indicates the inadequacy of independent SISO control techniques. In view of regulating multiple performance outputs in modem hear pumps and air-conditioning systems, this model is highly useful for design of multivariable feedback control.
Representations and characters: generalities on linear representations character theory subgroups, products, induced representation compact groups examples. Representations in characteristic zero: the group algebra induced representations Mackey's criterion examples of induced representations Artin's theorem a theorem of Brauer applications of Brauer's theorem rationality questions - examples. Introduction to Brauer theory: the groups RK(G), RX(G) and PK(G) the cde triangle theorems proofs modular characters application to Artin representations.
An integer linear program (ILP) is symmetric if its variables can be permuted without changing the structure of the problem.
Areas where symmetric ILPs arise range from applied settings (scheduling on identical machines), to combinatorics (code construction),
and to statistics (statistical designs construction). Relatively small symmetric ILPs are extremely difficult to solve using
branch-and-cut codes oblivious to the symmetry in the problem. This paper reviews techniques developed to take advantage of
the symmetry in an ILP during its solution. It also surveys related topics, such as symmetry detection, polyhedral studies
of symmetric ILPs, and enumeration of all non isomorphic optimal solutions.
In this paper, the passivity indices for both linear and nonlinear multi-agent systems with feedforward and feedback interconnections are derived. For linear systems, the passivity indices are explicitly characterized, while the passivity indices in the nonlinear case are characterized by a set of matrix inequalities. We also focus on symmetric interconnections and specialize the passivity indices results to this case. An illustrative example is also given.
This paper presents results related to the multiagent formation control problem. Symmetries in the system are exploited to simplify the stability analysis and control synthesis problem for symmetric systems. The type of symmetry considered is a discrete symmetry where individual agents are either identical or have dynamics that are diffeomorphically related. These results are applicable to both distributed as well as non-distributed coordination methods and are demonstrated with simulation results for systems taken from the literature.
In this paper we show how the symmetry present in many linear systems can be exploited to significantly reduce the computational effort required for controller synthesis. This approach may be applied when controller design specifications are expressible via semidefinite programming. In particular, when the overall system description is invariant under unitary coordinate transformations of the state space matrices, synthesis semidefinite programs can be decomposed into a collection of smaller semidefinite programs.
We show how to exploit symmetries of a graph to efficiently compute the fastest mixing Markov chain on the graph (i.e., find the transition probabilities on the edges to minimize the second-largest eigenvalue modulus of the transition probability matrix). Exploiting symmetry can lead to significant reduction in both the number of variables and the size of matrices in the corresponding semidefinite program, thus enable numerical solution of large-scale instances that are otherwise computationally infeasible. We obtain analytic or semi-analytic results for particular classes of graphs, such as edge-transitive and distance-transitive graphs. We describe two general approaches for symmetry exploitation, based on orbit theory and block-diagonalization, respectively. We also establish the connection between these two approaches. Comment: 39 pages, 15 figures
Symmetric explicit model predictive control
- Danielsonc Borrellif
Pattern formation in large‐scale networks with asymmetric connections
- Gyorgy A
Balancing of battery networks via constrained optimal control. Paper presented at: Proceedings of the American Control Conference Montreal Canada
- Danielsonc Borrellif Oliverd Andersond Kuangm Phillipst
Low-complexity anti-windup with projection
- Schwerdtnerp Bortoffs Danielsonc Di Cairanos
Sparse feedback synthesis via the alternating direction method of multipliers. Paper presented at: Proceedings of the American Control Conference Montreal Canada
- Jovanovićm Linf Fardadm
A PDE viewpoint on basic properties of coordination algorithms with symmetries
- Sarlettea Sepulchrer
Symmetric Constrained Optimal Control: Theory Algorithms and Applications
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