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Distributed Control Design for Spatially Interconnected Systems
Abstract
This paper deals with analysis, synthesis, and implementation of distributed controllers, designed for spatially interconnected systems. We develop a state space framework for posing problems of this type, and focus on systems whose model is spatially discrete. In this paper, analysis and synthesis results are developed for this class of systems using the l<sub>2</sub>-induced norm as the performance criterion. The results are stated in terms of linear matrix inequalities and are thus readily amenable to computation. A special implementation of the resulting controllers is presented, which is particularly attractive for distributed operation of the controller. Several examples are provided to further illustrate the application of the results.
... Ces systèmes ont été popularisés par, e.g., [5,1,6] et peuvent être particulièrement adaptés à des dispositifs ou phénomènes que l'on rencontre dans de nombreux problèmes pratiques d'ingénierie. La plupart des résultats se concentrent sur les modèles 2D linéaires discrets, continus ou hybrides, qui possèdent plusieurs applications telles que le contrôle d'apprentissage itératif (CAI) [7,8], très utile en robotique, le filtrage numérique, le contrôle des modèles répétitifs [9], l'étude de pelotons de véhicules [10], les systèmes inter-connectés [11,12,13,14], etc. Les deux modèles à espace d'état les plus populaires sont le modèle de Fornasini-Marchesini [15] et le modèle de Roesser [16], qui ont été à l'origine développés pour le filtrage d'images numériques. ...
... Une possibilité offerte par les systèmes nD est de fournir une approximation de certaines EDP en les échantillonnant le long d'une dimension (souvent la dimension spatiale) tout en préservant l'autre dimension (souvent la dimension temporelle), ce qui amène à certains modèles nD hybrides discrets-continus. C'est ce qui fût proposé dans, e.g., [12,60,61], où l'EDP approximée prend la forme de sous modèles inter-connectés, ce qui donne lieu à de nombreuses analyses, et une approche qui utilise les outils et les techniques originellement développées pour les modèles nD hybrides, ou, de façon plus spécifique, les processus répétitifs où les schémas de Contrôle d'Apprentissage Itératif (CAI) [61]. La même stratégie fut utilisée dans [62]. ...
... La ressemblance de (B.6) avec les systèmes linéaires usuels vont permettre son étude par des outils utilisés pour les systèmes 1D linéaires classiques. Quelques problèmes seront reliés à la question de savoir si les matrices A(s), B(s), C(s), et D(s) sont bornées ou non pour s ∈ Iω, ou, autrement dit, si ces matrices peuvent être associées à des opérateurs bornés agissant sur les signaux x t et u t , i.e., sur L 2 ω (R) [12,64]. Si c'est le cas, le modèle (B.7) peut être vu comme un modèle 1D comme dans l'analyse ci-dessous. ...
Cette thèse présente les résultats de travaux sur lLes différentes notions de stabilité utilisées dans la littérature des systèmes dynamiques multidimensionnels. Plus précisément, dans le cadre des modèles 2D de Roesser et de Fornasini-Marchesini, nous analysons les notions de stabilité au sens de Lyapunov, stabilité asymptotique, stabilité(s) exponentielle(s) et stabilité structurelle, ainsi que les relations entre ces différentes propriétés. Le premier chapitre de ce mémoire effectue un certain nombre de rappels concernant les définitions de stabilité et les liens qui existent entre celles-ci, dans le but d'établir un cadre solide en vue d'étendre ces notions du cas 1D au cas 2D. Une fois ces rappels établis, nous présentons les modèles 2D que nous étudions. Le deuxième chapitre dresse la liste des définitions de stabilité utilisées pour les modèles 2D de Roesser et de Fornasini-Marchesini et établit les liens entre ces différentes définitions. Au cours du troisième chapitre, nous proposons une condition nécessaire et suffisante de stabilité asymptotique pour une certaine classe de modèles de Fornasini-Marchesini 2D discrets linéaires. Le quatrième et dernier chapitre propose une étude détaillée d'un modèle 1D non-linéaire qui possède la particularité rare d'être à la fois attractif et instable, et nous généralisons ce modèle particulier au cas 2D afin d'établir quelles propriétés se conservent ou non lorsque l'on passe du cas 1D au cas 2D.
... The work [10], [14] characterized the structural properties of optimal controllers for SIS. Related research has followed since then: many have focused on conceiving tractable design strategies for optimal distributed control of SIS e.g., [15], [16], [17], [18], [19]; others on extensions to more general classes of systems, including generalizations to multiple dimensions and heterogeneous systems [20], time-varying SIS [21], and systems with spatially decaying operators [22], [23], [24]. However, only a few have analyzed the Kalman filtering problem for particular examples of SIS (e.g., [25], [26], [27], [28]). ...
... Spatial Localization of the Kalman gain L: The branch points of (19) are |z ∞ | = ∞ and ...
We consider the centralized optimal estimation problem in spatially distributed systems. We use the setting of spatially invariant systems as an idealization for which concrete and detailed results are given. Such estimators are known to have a degree of spatial localization in the sense that the estimator gains decay in space, with the spatial decay rates serving as a proxy for how far measurements need to be shared in an optimal distributed estimator. In particular, we examine the dependence of spatial decay rates on problem specifications such as system dynamics, measurement and process noise variances, as well as their spatial autocorrelations. We propose non-dimensional parameters that characterize the decay rates as a function of problem specifications. In particular, we find an interesting matching condition between the characteristic lengthscale of the dynamics and the measurement noise correlation lengthscale for which the optimal centralized estimator is completely decentralized. A new technique - termed the branch point locus - is introduced to quantify spatial decay rates in terms of analyticity regions in the complex spatial frequency plane. Our results are illustrated through two case studies of systems with dynamics modeled by diffusion and the Swift-Hohenberg equation, respectively.
... The design and architecture of a PID controller-based system for setting reactive power of energy storage to regulate voltage in a grid involves several key components and considerations [10]. The system consists of various components, including energy storage systems, grid voltage monitoring sensors, a PID controller, control algorithms, communication interfaces, system optimization and tuning, and safety and protection mechanisms. ...
This article proposes a PID controller-based approach to optimize voltage regulation in smart grids by leveraging the reactive power capabilities of energy storage systems. The research focuses on designing a robust control framework that utilizes the PID control technique to maintain grid voltage within desired limits. The article presents the implementation details, performance evaluation, and analysis of the proposed solution, demonstrating its effectiveness in voltage regulation and its potential for improving grid stability and reliability. The findings contribute to the development of advanced control strategies for smart grids, enabling more efficient energy management and enhanced system performance.
... On the other hand, data-driven approaches are flexible, adaptable, and can handle complex dynamics, but they may be difficult to interpret (Narayanan et al., 2022). In the industry, the most common systems used to control equipment, units, and processes -whether locally or remotely -are supervisory control and data acquisition (SCADA) (Aydogmus, 2009) and distributed control systems (DCS) (D'Andrea and Dullerud, 2003). These systems are mainly based on local controllers such as model predictive control (MPC) (Fernandez-Camacho and Bordons-Alba, 1995) and proportional-integral-derivative controllers (PID) (Visioli, 2006). ...
Possessing efficient supervisory control systems is crucial for maintaining the desired operational performance of complex industrial processes. Several challenges face the developers of these systems, such as requiring accurate physical models, dealing with the variability and uncertainty of process operating conditions and coordinating between local controllers to reach desired global performance. This paper proposes an intelligent supervisory control approach based on causal reinforcement learning (CRL) to effectively manipulate the controllers' setpoints of the process in a way that optimizes its key performance indicators (KPIs), thereby improving the energy efficiency of the process. The approach adopts deep reinforcement learning (DRL) to develop an efficient control policy through interaction with a process simulation. The DRL training history is then exploited using interpretable machine learning and process mining to build a discrete event system (DES) model, in the form of a state-event graph. The DES model identifies causal relationships between events and provides interpretability to the control policy developed by the DRL method. The DES discovered is exploited as a Markov decision process to apply the Q-learning algorithm as a CRL supervisor. The supervisor incorporates causal knowledge into its training process, thus improving the DRL control policy developed and identifying the event paths that optimize the process's KPIs. The proposed approach is validated using two heat recovery systems in a pulp & paper mill. It successfully achieves a control policy that reduces energy consumption by up to 15.6% for the first system and 5.02% for the second, compared to the expert's baseline methods.
... LSNSs contain a great many of subsystems, which have various dynamics [7]. Although the subsystems interact with each other in a simple way, the overall plant can exhibit extremely complex behaviors [8]. However, in practical engineering applications, the complexity of LSNSs makes it difficult to avoid the existence of model errors between the ideal model and the actual system, which may deteriorate the system performance. ...
This paper considers the robust H∞ performance problem of continuous-time uncertain large-scale networked systems (LSNSs). The systems consist of numerous arbitrarily connected subsystems, each of which has different dynamics. Currently it is computationally difficult to manage systems with the existing lumped analysis method; therefore, exploiting the structural properties of the systems, sufficient conditions are derived for robust H∞ performance. Based on these results, an analysis condition that depends only on the parameters of each single subsystem is further obtained. Some numerical simulations are proposed to verify the validity and superiority of the developed conditions.
... Although the subsystems can be analyzed and controlled 5 individually, complex and unpredictable behaviors might appear in the analysis of the whole interconnected system. Even in cases where the interactions among the subsystems are considered simple, the interconnections can lead to undesirable responses [2]. ...
This paper addresses the distributed control of delayed interconnected nonlinear systems with time-varying delays in both the local subsystems’ dynamics and the physical interconnections among the subsystems. The Takagi-Sugeno fuzzy model with nonlinear consequent parts (N-TS), which is capable to provide
less complex representations than standard T-S fuzzy models, is considered to efficiently deal with this class of complex systems. Then, based on Lyapunov-Krasovskii stability arguments, a synthesis condition is proposed to design a distributed control law such that the origin of the closed-loop interconnected system is locally asymptotically stable together with a guaranteed set of admissible
initial conditions for which the validity of the N-TS fuzzy model is ensured. Moreover, a quasi-convex optimization procedure is formulated to enlarge the set of admissible initial conditions. The effectiveness of the proposed synthesis condition is validated in two numerical examples, including an interconnected power network with seven generators.
... Structural symmetry can be extended to linear dynamical systems [8], in which case most controllers inherit the symmetry properties. When V is a computationally efficient transformation, such as the Fast Fourier Transformation (FFT) for circulant matrices [12], structural symmetries can also be exploited to speed up controller computations, which is particularly useful for large-scale and high-speed systems [10,11,22]. Structural symmetries are frequently encountered in cross-directional (CD) systems [19], y(s) = Rg(s)u(s) + d(s), where s ∈ C is the Laplace variable, g : C → C the scalar actuator dynamics, u : C → C nbu are the control inputs, y : C → C nby the outputs and d : C → C nby the disturbances. ...
Structural symmetries of linear dynamical systems can be exploited for decoupling the dynamics and reducing the computational complexity of the controller implementation. However, in practical applications, inexact structural symmetries undermine the ability to decouple the system, resulting in the loss of any potential complexity reduction. To address this, we propose substituting an approximation with exact structural symmetries for the original system model, thereby introducing an approximation error. We focus on internal model controllers for cross-directional systems encountered in large-scale and high-speed control problems of synchrotrons or the process industry and characterise the stability, performance, and robustness properties of the resulting closed loop. While existing approaches replace the original system model with one that minimises the Frobenius norm of the approximation error, we show that this can lead to instability or poor performance. Instead, we propose approximations that are obtained from semidefinite programming problems. We show that our proposed approximations can yield stable systems even when the Frobenius norm approximation does not. The paper concludes with numerical examples and a case study of a synchrotron light source with inexact structural symmetries. Exploiting structural symmetries in large-scale and high-speed systems enables faster sampling times and the use of more advanced control techniques, even when the symmetries are approximate.
... The well-developed linear time-invariant (LTI) system allows the use of the dynamic frequency domain for the design and analysis of control systems. The linear shiftinvariant (LSI) control systems [34], [6] give a similar advantage by exploiting the LSI property in the spatial frequency domain. Here the symmetric circulant system model is considered for modeling spatial parameter systems. ...
The paper manufacturing process is a large-scale process in which the control performance is required to be optimal in the machine direction (MD) and cross direction (CD). This work focuses on CD control, which is more complicated than MD control. Uncertainties in the process arise from different sources, so there is a requirement for better modeling techniques. The control of paper profile properties, namely basis weight, moisture, and caliper, is done with the help of actuators that are distributed spatially. Basis weight is taken as the profile parameter in this work. The work aims to improve robust performance and simplify the control design problem. Here, μ synthesis methods are used for controller design, as this method provides guaranteed robust performance. To simplify the design problem two-dimensional modeling is explained explicitly. The conventional methods of μ analysis are extended to spatial coordinates. The performance of the μ synthesis designed method is compared with the open loop shaping method. Control performance is judged in terms of sensitivity, stability margin, 2σ performance, and control effort. The effectiveness of the developed μ synthesis designed method is demonstrated using the industrial example from the pulp and paper industry.
... To consider the noncausality along the spatial dimension, a descriptor Roesser model would be more appropriate (see [8] and the references therein). Note that considering both directions of the same dimension as independent, as we here do, is not new since, in [22], a model is considered with a forward shift operator along one dimension and a backward shift operator along the same dimension. However, the relation between both operators is never taken into account. ...
This article provides new conditions for testing the structural stability of 3D Roesser models. The models can be discrete, continuous, or mixed discrete/continuous. The conditions consist in a few tests on the eigenvalues of matrices and one test on an auxiliary 2D model. The latter test is based upon a hierarchy of linear matrix inequalities relaxations. The global test for structural stability is necessary and sufficient for a large enough value of the hierarchy level.
... In real-world systems, the size of networks and the number of connections between their components may be unknown and time-varying, which poses new challenges for stability analysis and control design of such networks. A promising approach to this problem is to over-approximate the network by an infinite network and to perform the stability analysis and control design for this infinite over-approximation [43,108,47]. This approach has received significant attention during the last two decades. ...
Input-to-state stability (ISS) allows estimating the impact of inputs and initial conditions on both the intermediate values and the asymptotic bound on the solutions. ISS has unified the input-output and Lyapunov stability theories and is a crucial property in the stability theory of control systems as well as for many applications whose dynamics depend on parameters, unknown perturbations, or other inputs. In this habilitation thesis, we provide a broad picture of infinite-dimensional input-to-state stability theory.
... The main focus has been on the network connectivity aspect of designing a feedback control system. More specifically, in [4][5][6][7][8][9][10][11][12][13] and the references therein, LQR control designs with predetermined structure of network topologies were studied. In order to reduce the number of communication links in a distributed multi-agents system, the proposed solutions in [10,[14][15][16][17] took into account sparse LQR control design models which simultaneously minimize LQR cost as well as sparsity level of the network topology by considering the sparsity condition on the topology as a regularization term or as a constraint in optimization problems. ...
This paper considers a LQR optimal control design problem for distributed control systems with multi-agents. To control large-scale distributed systems such as smart-grid and multi-agent robotic systems over wireless communication networks, it is desired to design a feedback controller by considering various constraints on communication such as limited power, limited energy, or limited communication bandwidth, etc. In this paper, we focus on the reduction of communication energy in an LQR optimal control design problem on wireless communication networks. By considering the characteristic of wireless communication, i.e., Radio Frequency (RF) signal can spread in all directions in a broadcast way, we formulate a low-rank LQR optimal control model to reduce the communication energy in the distributed feedback control system. To solve the problem, we propose an Alternating Direction Method of Multipliers (ADMM) based algorithm. Through various numerical experiments, we demonstrate that a feedback controller designed using low-rank structure can outperform the previous work on sparse LQR optimal control design, which focuses on reducing the number of communication links in a network, in terms of energy consumption, system stability margin against noise and error in communication.
... Recent developments in robotics and sensing have created significant interest among researchers to deploy multiple robots to operate cooperatively towards achieving a common goal. Many works have developed techniques to tackle realworld problems using multi-robot systems (MRS), like conducting surveys or automating warehouses [1,2], [3]. The major developments in MRS for enabling multiple robots to behave cooperatively have been based on interactions within a single team, i.e., a robot interacts with other robots in its group to achieve a given objective [4,5]. ...
In this paper, we consider the problem of protecting a high-value area from being breached by sheep agents by crafting motions for dog robots. We use control barrier functions to pose constraints on the dogs' velocities that induce repulsions in the sheep relative to the high-value area. This paper extends the results developed in our prior work on the same topic in three ways. Firstly, we implement and validate our previously developed centralized herding algorithm on many robots. We show herding of up to five sheep agents using three dog robots. Secondly, as an extension to the centralized approach, we develop two distributed herding algorithms, one favoring feasibility while the other favoring optimality. In the first algorithm, we allocate a unique sheep to a unique dog, making that dog responsible for herding its allocated sheep away from the protected zone. We provide feasibility proof for this approach, along with numerical simulations. In the second algorithm, we develop an iterative distributed reformulation of the centralized algorithm, which inherits the optimality (i.e. budget efficiency) from the centralized approach. Lastly, we conduct real-world experiments of these distributed algorithms and demonstrate herding of up to five sheep agents using five dog robots.
... Designing optimal feedback controllers for distributed and interconnected systems has been studied in various research [1,2,[6][7][8][9][10][11] and the references therein. Among them, we are interested in the optimal control design problem with sparse network constraints considered in [1,2,10] in order to reduce the number of communication links in large-scale distributed control systems. ...
In this paper, we consider an LQR design problem for distributed control systems. For large-scale distributed systems, finding a solution might be computationally demanding due to communications among agents. To this aim, we deal with LQR minimization problem with a regularization for sparse feedback matrix, which can lead to achieve the reduction of the communication links in the distributed control systems. For this work, we introduce a simple but efficient iterative algorithms - Iterative Shrinkage Thresholding Algorithm (ISTA) and Iterative Sparse Projection Algorithm (ISPA). They can give us a trade-off solution between LQR cost and sparsity level on feedback matrix. Moreover, in order to improve the speed of the proposed algorithms, we design deep neural network models based on the proposed iterative algorithms. Numerical experiments demonstrate that our algorithms can outperform the previous methods using the Alternating Direction Method of Multiplier (ADMM) [1] and the Gradient Support Pursuit (GraSP) [2], and their deep neural network models can improve the performance of the proposed algorithms in convergence speed.
... Also large-scale MIMO 1 systems are not protected from the increasing requirements. These and the increasing complexity of the systems enhance the system order of the dynamic large-scale MIMO systems [3], [4]. The use of such complex and large-scale models in simulation is very cumbersome and sometimes impossible. ...
... Since stability and stabilization of large-scale networks has many meaningful applications [11,25,26], another recent popular topic has become infinite networks [5,35,6,41]. The main focus was the infinite networks of finite-dimensional linear control systems with linear interconnections. ...
We prove a small-gain sufficient condition for (global) finite-time input-to-state stability (FTISS) of infinite networks. The network under consideration is composed of a countable set of finite-dimensional subsystems of ordinary differential equations, each of which is interconnected with a finite number of its “neighbors” only and is affected by some external disturbances. We assume that each node (subsystem) of our network is finite-time input-to-state stable (FTISS) with respect to its finite-dimensional inputs produced by this finite set of the neighbors and with respect to the corresponding external disturbance. As an application we obtain a new theorem on decentralized finite-time input-to-state stabilization with respect to external disturbances for infinite networks composed of a countable set of strict-feedback form systems of ordinary differential equations. For this we combine our small-gain theorem proposed in the current work with the controllers design developed by S. Pavlichkov and C. K. Pang (NOLCOS-2016) for the gain assignment of the strict-feedback form systems in the case of finite networks. The current results address the finite-time input-to-state stability and decentralized finite-time input-to-state stabilization and redesign the technique proposed in recent work S. Dashkovskiy and S. Pavlichkov, Stability conditions for infinite networks of nonlinear systems and their application for stabilization, Automatica. – 2020. – 112. – 108643, in which the case of $\ell_{\infty}$-ISS of infinite networks was investigated. The current paper extends and generalizes its conference predecessor to the case of finite-time ISS stability and decentralized stabilization in presence of external disturbance inputs and with respect to these disturbance inputs. In the special case when all these external disturbances are zeroes (i.e. are abscent), we just obtain finite-time stability and finite-time decentralized stabilization of infinite networks accordingly.
... In this section we consider a toy example to confirm our results. The toy example shares the same B, Q, R, S matrices as the counterexample defined in (6), whereas A is set as a stable matrix A = e −ρ I. Clearly the above system satisfies both Assumption 2 and 3. Figure 3 plots the heatmap of the absolute values of the entries of K, visually demonstrating that entries of K is spatially decaying, which forms a sharp contrast with the counter example (figure 1) in Section II. Figure 4 demonstrates that K ij indeed decays exponentially with respect to dist(i, j), and that the decay rate is faster when ρ is larger. This phenomenon can be explained by Remark 2, which concludes that the decay rate is faster if ρ is larger. ...
This paper studies network LQR problems with system matrices being spatially-exponential decaying (SED) between nodes in the network. The major objective is to study whether the optimal controller also enjoys a SED structure, which is an appealing property for ensuring the optimality of decentralized control over the network. We start with studying the open-loop asymptotically stable system and show that the optimal LQR state feedback gain $K$ is `quasi'-SED in this setting, i.e. $\|[K]_{ij}\|\sim O\left(e^{-\frac{c}{\mathrm{poly}\ln(N)}}\mathrm{dist}(i,j)\right)$. The decaying rate $c$ depends on the decaying rate and norms of system matrices and the open-loop exponential stability constants. Then the result is further generalized to unstable systems under a SED stabilizability assumption. Building upon the `quasi'-SED result on $K$, we give an upper-bound on the performance of $\kappa$-truncated local controllers, suggesting that distributed controllers can achieve near-optimal performance for SED systems. We develop these results via studying the structure of another type of controller, disturbance response control, which has been studied and used in recent online control literature; thus as a side result, we also prove the `quasi'-SED property of the optimal disturbance response control, which serves as a contribution on its own merit.
This paper designs a decentralized sliding mode preview repetitive control (SMPRC) for interconnected nonlinear systems. First, an augmented error system, which is composed of the derivative equation of the available future reference dynamics, the output of the modified repetitive controller and the tracking error dynamics, is constructed. Next, an integral switching surface is designed to verify the robust asymptotic stability of the considered system subject to nonlinearity. Then, the preview repetitive-control (PRC) law design problem is transformed into the stability problem of the augmented system considering the nominal conditions. Thereafter, a sliding mode control (SMC) law cooperates with the PRC law with the purpose to ensure the robustness of the interconnected systems in the presence of nonlinearity. Further, the required stability conditions are derived on the basis of a Lyapunov analysis, linear matrix inequalities (LMI) and the singular-value-decomposition (SVD) technique. Finally, the numerical simulation results demonstrate the effectiveness of the proposed method.
In this paper, we consider the problem of protecting a high-value area from being breached by sheep agents by crafting motions for dog robots. We use control barrier functions to pose constraints on the dogs’ velocities that induce repulsions in the sheep relative to the high-value area. This paper extends the results developed in our prior work on the same topic in three ways. Firstly, we implement and validate our previously developed centralized herding algorithm on many robots. We show herding of up to five sheep agents using three dog robots. Secondly, as an extension to the centralized approach, we develop two distributed herding algorithms, one favoring feasibility while the other favoring optimality. In the first algorithm, we allocate a unique sheep to a unique dog, making that dog responsible for herding its allocated sheep away from the protected zone. We provide feasibility proof for this approach, along with numerical simulations. In the second algorithm, we develop an iterative distributed reformulation of the centralized algorithm, which inherits the optimality (i.e. budget efficiency) from the centralized approach. Lastly, we conduct real-world experiments of these distributed algorithms and demonstrate herding of up to five sheep agents using five dog robots. Videos of these results are available at https://bit.ly/3bZq0dB.
The synthesis problem of the interconnection of homogeneous Linear Time-Invariant (LTI) systems such that the frequency-response satisfies magnitude constraints is investigated. To this end, the usual synthesis approach of traditional filters, viewed as the interconnection of integrators
$\frac{1}{s}$
, is revisited based on the Linear Fractional Transformation (LFT) representation and the
$\lbrace x,y,z\rbrace$
-dissipative characterization. This approach, based on convex optimization, consists of two steps: the magnitude synthesis and the spectral factorization steps. When the systems are modeled by a lossless dissipative transfer function
$T(s)$
, it is demonstrated that each step can be extended. However, a factorization error appears when considering a general dissipative
$T(s)$
, preventing the direct extension of the two-step approach. It is then revealed how to overcome this issue by coupling both steps, generalizing thereby the usual synthesis approach. Finally, the interest of this work is illustrated through two applications: the design of
$LC$
-bandpass filters and the weighted
$\mathcal {H}_{\infty }$
-control of interconnected homogeneous systems.
The robust stability problem of spatially interconnected systems with signal saturation among the many composed subsystems is considered. The system structure is usually sparse, and each subsystem has different dynamics. Firstly, a robust stability condition is established based on integral quadratic constraint (IQC) theory, which makes full use of the sparseness of the subsystem connection topology. Secondly, a decoupling robust condition that only depends on the subsystem parameters is proved. Finally, it is shown through numerical simulations that the obtained conditions are computationally valid in analyzing spatially interconnected systems with signal saturation constraints among the subsystems.
Internet of Things (IoT) designers frequently must determine whether action-oriented decisions should be made by edge computers or whether they should be made only by central servers combining input from all edge computers. An important example of this design problem occurs in fire protection IoT where individual edge computers attached to sensors might be empowered to make decisions (have decision rights) about how to manage the fire. Alternately, decision rights could be held exclusively by a central server isolated from the fire, because the designer is concerned damage to edge computers could cause them to act unreliably. This research models this allocation of decision rights to identify the relative influence of various decision factors. We first model the allocation of decision rights under the following assumptions (1) the central server cannot make an error the edge computer cannot make, (2) the central server cannot update the edge computer with its information in a timely manner, and (3) the central server cannot reverse an action initiated by the edge computer to explore the factors impacting decision rights conferral. We then relax each of these three assumptions. We show how relaxing each assumption radically changes the factors impacting decision rights conferral. We also show that allowing the central server to update information on the edge computer or reverse the edge computer's decision making can result in overall lower system performance. We then perform a series of numerical experiments to understand how changing various parameters affect the problem. We show for the general real-world scenario, the key factor influencing the decision is the ability of the edge computer to detect false alarms. We also show magnitude of loss and ratio of real to false incidents have a linear and logarithmic relationship to the reliability of the edge computer.
Microgrids are rapidly evolving as cyber-physical systems to have interoperability among the connected sources. These cyber-physical microgrids (CPM) are highly dependent on communication networks for realizing power control strategies, making them highly vulnerable to cyber intrusions. Cyber-attacks are usually intended to meddle with data availability, integrity, and confidentiality. This paper first summarizes mathematical models of commonly occurring cyber-attacks, which include denial of service, false data injection, replay, and covert attacks. Further, a brief discussion about their impacts on distributed controlled AC and DC CPM is detailed. As consensus among the sources is an essential requirement in distributed CPM, the technical and physical challenges arising in achieving the consensus due to different attacks in AC and DC microgrids are analyzed using simulation case studies. Viewing the importance of security assessment and enhancement, state-of-the-art cyber defense strategies for AC and DC CPM are discussed. For better understanding, few defense methods are simulated and validated on CPM. Finally, this paper identifies a few challenges in security and presents some basic resilient strategies along with future research scope in this field.
Over the past decades, robotics has shown great potential to impact the built environment, from automation to differentiation and efficient construction. However, construction processes are highly complex and require tackling a multitude of problems, from safety and robustness to ease of control and interactivity. For this reason, the field of construction robotics is still evolving, requiring finding solutions for new challenges every day. The present review analyzes the role of robotics in construction and architecture over time and highlights current trends in shifting from pure automation toward collaborative and adaptive processes that have the potential to fully integrate robotics into a rigid and challenging industry, such as construction.
Expected final online publication date for the Annual Review of Control, Robotics, and Autonomous Systems, Volume 14 is May 2023. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates.
In this paper, we consider two classes of spatially interconnected systems (including ladder circuits) modelled as mixed discrete-continuous linear 2D systems. Within the algebraic analysis approach to linear systems theory, we prove that these systems can always be transformed into equivalent implicit Roesser models. Moreover we show that ladder circuits can also be transformed into equivalent implicit Fornasini-Marchesini models. An advantage of our results compared to previous ones obtained through the zero coprime system equivalence approach is that the dimension of the state vector of the equivalent Roesser (or Fornasini-Marchesini) models is significantly smaller.
This comprehensive text provides an excellent introduction to the state of the art in the identification of network-connected systems. It covers models and methods in detail, includes a case study showing how many of these methods are applied in adaptive optics and addresses open research questions. Specific models covered include generic modelling for MIMO LTI systems, signal flow models of dynamic networks and models of networks of local LTI systems. A variety of different identification methods are discussed, including identification of signal flow dynamics networks, subspace-like identification of multi-dimensional systems and subspace identification of local systems in an NDS. Researchers working in system identification and/or networked systems will appreciate the comprehensive overview provided, and the emphasis on algorithm design will interest those wishing to test the theory on real-life applications. This is the ideal text for researchers and graduate students interested in system identification for networked systems.
The continuous- and discrete-time H∞ control problems are solved via elementary manipulations on linear matrix inequalities (LMI). Two interesting new features emerge through this approach: solvability conditions valid for both regular and singular problems, and an LMI-based parametrization of all H∞-suboptimal controllers, including reduced-order controllers.
The solvability conditions involve Riccati inequalities rather than the usual indefinite Riccati equations. Alternatively, these conditions can be expressed as a system of three LMIs. Efficient convex optimization techniques are available to solve this system. Moreover, its solutions parametrize the set of H∞ controllers and bear important connections with the controller order and the closed-loop Lyapunov functions.
Thanks to such connections, the LMI-based characterization of H∞ controllers opens new perspectives for the refinement of H∞ design. Applications to cancellation-free design and controller order reduction are discussed and illustrated by examples.
The purpose of this paper is to show how the important problems of linear system theory can be solved concisely for a particular class of linear systems, namely block circulant systems, by exploiting the algebraic structure. This type of system arises in lumped approximations to linear partial differential equations. The computation of the transition matrix, the variation of constants formula, observability, controllability, pole allocation, realization theory, stability and quadratic optimal control are discussed. In principle, all questions which are solved here could also be solved by standard methods; the present paper clearly exposes the structure of the solution, and thus permits various savings in computational effort.
We present a mathematical model for the dynamics of an array of
capacitively actuated microcantilevers. We propose a system where the
current measured at each cantilever is used as the sensing signal of the
cantilever state through an observer. We show that such an array is a
spatially invariant system with distributed control and sensing. For the
common case of periodically excited cantilevers, we show that the
underlying dynamics are those of a periodic system described by a
Mathieu equation. We exploit the spatial invariance of the problem to
design an optimal distributed observer, where the temporal periodicity
is handled using the lifting technique
It has long been known that aircraft flying in close formation can
achieve an overall efficiency much greater than is possible for each
aircraft flying alone. When coupled with possible near-term advances in
solar-electric power storage and aeronautical propulsion systems, it is
theoretically possible to create a formation flight system that is
capable of truly infinite endurance. This paper discusses a research
effort currently underway that addresses the design and implementation
of such a system. A brief overview of the system concept is given, and
some of the particular problems inherent are more fully discussed. In
particular, the difficulties involved in the autonomous control of such
systems are presented and discussed
The purpose of this paper is to design and test a vehicle control
system in order to achieve full vehicle automation in the longitudinal
direction for several modes of operation, where the infrastructure
manages the vehicle following. These modes include autonomous vehicles,
cooperative vehicle following and platooning. The vehicle control system
consists of a supervisory controller that processes the inputs from the
driver, the infrastructure, other vehicles and the on-board sensors and
sends the appropriate commands to the brake and throttle controllers. In
addition, the controller makes decisions about normal, emergency and
transition operations. Simulation results of some of the basic vehicle
following manoeuvres are used to verify the claimed performance of the
designed controllers
We consider distributed parameter systems where the underlying
dynamics are spatially invariant, and where the controls and
measurements are spatially distributed. These systems arise in many
applications such as the control of vehicular platoons, flow control,
microelectromechanical systems (MEMS), smart structures, and systems
described by partial differential equations with constant coefficients
and distributed controls and measurements. For fully actuated
distributed control problems involving quadratic criteria such as linear
quadratic regulator (LQR), H<sub>2</sub> and H<sub>∞</sub>,
optimal controllers can be obtained by solving a parameterized family of
standard finite-dimensional problems. We show that optimal controllers
have an inherent degree of decentralization, and this provides a
practical distributed controller architecture. We also prove a general
result that applies to partially distributed control and a variety of
performance criteria, stating that optimal controllers inherit the
spatial invariance structure of the plant. Connections of this work to
that on systems over rings, and systems with dynamical symmetries are
discussed
An important class of linear time-varying systems consists of
plants where the state-space matrices are fixed functions of some
time-varying physical parameters θ. Small gain techniques can be
applied to such systems to derive robust time-invariant controllers.
Yet, this approach is often overly conservative when the parameters
θ undergo large variations during system operation. In general,
higher performance can be achieved by control laws that incorporate
available measurements of θ and therefore “adjust” to
the current plant dynamics. This paper discusses extensions of
H<sub>∞</sub> synthesis techniques to allow for controller
dependence on time-varying but measured parameters. When this dependence
is linear fractional, the existence of such gain-scheduled H<sub>∞
</sub> controllers is fully characterized in terms of linear matrix
inequalities. The underlying synthesis problem is therefore a convex
program for which efficient optimization techniques are available. The
formalism and derivation techniques developed here apply to both the
continuous- and discrete-time problems. Existence conditions for robust
time-invariant controllers are recovered as a special case, and
extensions to gain-scheduling in the face of parametric uncertainty are
discussed. In particular, simple heuristics are proposed to compute such
controllers
Simple state-space formulas are derived for all controllers
solving the following standard H <sub>∞</sub> problem: For
a given number γ>0, find all controllers such that the H
<sub>∞</sub> norm of the closed-loop transfer function is
(strictly) less than γ. It is known that a controller exists if
and only if the unique stabilizing solutions to two algebraic Riccati
equations are positive definite and the spectral radius of their product
is less than γ<sup>2</sup>. Under these conditions, a
parameterization of all controllers solving the problem is given as a
linear fractional transformation (LFT) on a contractive, stable, free
parameter. The state dimension of the coefficient matrix for the LFT,
constructed using the two Riccati solutions, equals that of the plant
and has a separation structure reminiscent of classical LQG (i.e. H
<sub>2</sub>) theory. This paper is intended to be of tutorial
value, so a standard H <sub>2</sub> solution is developed in
parallel
A linear finite-dimensional plant with rational state-space parameter dependence is controlled using a parameter-dependent controller. The parameters are known to take on values in a unit ball, and are known in real time. The goal of control is to stabilize the parameter-dependent closed-loop system, and provide disturbance/error attenuation as measured in induced l2 norms. The approach taken uses the optimally scaled small-gain theorem, and solves the control synthesis problem by reformulating the existence conditions into a finite-dimensional convex optimization.
This paper describes a new MATLAB-based toolbox for control design via linear matrix inequality (LMI) techniques. After a brief review of LMIs and of some of their applications to control, the toolbox contents and capabilities are presented.
The purpose of this note is to present a new elementary proof for the multivariable K-Y-P lemma. A minimum of linear algebra and finite dimensional convexity theory is used.
A generalized analysis methodology for assessing the performance and cost of distributed satellite systems is presented. The framework is completely generalizable and can be applied to any satellite mission, including communications, sensing or navigation. The primary enabler for this work is that all current satellite applications involve the collection and dissemination of information. Satellite systems can then be treated as modular information processing networks. This generalization allows the adoption of well-developed mathematics for information network flow. The resulting analysis methodology involves the definition of quantitative metrics for the cost, capability, performance and adaptability of satellite systems. The capabilities of a satellite system are shown to be characterized by four quality of service parameters that relate to the isolation, rate and integrity of information being transferred between origin-destination pairs. The Cost per Function metric is identified as a valuable tool for design and comparative analysis, allowing the impacts of system architecture, deployment strategy, schedule slip, market demographics and technical risk to be assessed. As a case study to demonstrate the metrics utility, the comparative analysis of several proposed Ka-band communication systems is presented.
We present a state space approach to controlling systems with a highly structured interconnection topology. It is shown that by capturing these systems as fractional transformations on temporal and spatial operators, many standard results in control – such as the bounded real lemma, H-infinity optimization, and robustness analysis – can be generalized accordingly. The state space formulation yields conditions that can be expressed as linear matrix inequalities.
In this paper we apply, recently developed, distributed control techniques to the control of an adaptive secondary mirror. The control technique implemented uses the l2 induced norm as a performance criterion. A finite difference model for the mirror is developed using shift operators. We determine the necessary constants from the physical properties of the mirror, from the typical dimensions of such a system, and from our performance requirements for operating the telescope in the near infrared region.The closed loop system is simulated by modelling the deformable mirror in Abaqus, a commercial finite element software, and by implementing the controller in Matlab.
This paper investigates the control problems associated with autonomous vehicles in formation flight. An inviscid flow model of formation flight explains the increase in efficiency, and describes the effects that the craft in the formation have on each other. Decentralized controllers are investigated for a formation of five aircraft. The formation consists of a single line, with each plane flying one wingspan behind the plane to its left, with its left wingtip aligned with the right wingtip of the leading plane. The controllers are derived using a linear model of the system dynamics, and evaluated in a linear simulation and in a simulation incorporating a vortex-lattice aerodynamics routine. (Author)
1 Introduction.- 1.1 Motivation.- 1.2 Systems theory concepts in finite dimensions.- 1.3 Aims of this book.- 2 Semigroup Theory.- 2.1 Strongly continuous semigroups.- 2.2 Contraction and dual semigroups.- 2.3 Riesz-spectral operators.- 2.4 Delay equations.- 2.5 Invariant subspaces.- 2.6 Exercises.- 2.7 Notes and references.- 3 The Cauchy Problem.- 3.1 The abstract Cauchy problem.- 3.2 Perturbations and composite systems.- 3.3 Boundary control systems.- 3.4 Exercises.- 3.5 Notes and references.- 4 Inputs and Outputs.- 4.1 Controllability and observability.- 4.2 Tests for approximate controllability and observability.- 4.3 Input-output maps.- 4.4 Exercises.- 4.5 Notes and references.- 5 Stability, Stabilizability, and Detectability.- 5.1 Exponential stability.- 5.2 Exponential stabilizability and detectability.- 5.3 Compensator design.- 5.4 Exercises.- 5.5 Notes and references.- 6 Linear Quadratic Optimal Control.- 6.1 The problem on a finite-time interval.- 6.2 The problem on the infinite-time interval.- 6.3 Exercises.- 6.4 Notes and references.- 7 Frequency-Domain Descriptions.- 7.1 The Callier-Desoer class of scalar transfer functions.- 7.2 The multivariable extension.- 7.3 State-space interpretations.- 7.4 Exercises.- 7.5 Notes and references.- 8 Hankel Operators and the Nehari Problem.- 8.1 Frequency-domain formulation.- 8.2 Hankel operators in the time domain.- 8.3The Nehari extension problem for state linear systems.- 8.4 Exercises.- 8.5 Notes and references.- 9 Robust Finite-Dimensional Controller Synthesis.- 9.1 Closed-loop stability and coprime factorizations.- 9.2 Robust stabilization of uncertain systems.- 9.3 Robust stabilization under additive uncertainty.- 9.4 Robust stabilization under normalized left-coprime-factor uncertainty.- 9.5 Robustness in the presence of small delays.- 9.6 Exercises.- 9.7 Notes and references.- A. Mathematical Background.- A.1 Complex analysis.- A.2 Normed linear spaces.- A.2.1 General theory.- A.2.2 Hilbert spaces.- A.3 Operators on normed linear spaces.- A.3.1 General theory.- A.3.2 Operators on Hilbert spaces.- A.4 Spectral theory.- A.4.1 General spectral theory.- A.4.2 Spectral theory for compact normal operators.- A.5 Integration and differentiation theory.- A.5.1 Integration theory.- A.5.2 Differentiation theory.- A.6 Frequency-domain spaces.- A.6.1 Laplace and Fourier transforms.- A.6.2 Frequency-domain spaces.- A.6.3 The Hardy spaces.- A.7 Algebraic concepts.- A.7.1 General definitions.- A.7.2 Coprime factorizations over principal ideal domains.- A.7.3 Coprime factorizations over commutative integral domains.- References.- Notation.
A general theory for the optimal regulation of linear systems, comprised of infinite identical objects, is developed. The bilateral z transform generating function is used to eliminate the object indices, which leads to a Riccati equation in the z domain. The theory is applied to the string of vehicles problem where the number of vehicles is infinite. The proper structure for a typical vehicle controller and numerical results for the amount of feedback a typical vehicle requires from other vehicles are obtained.
The authors consider four control-related problems, all of which
involve reformulation into linear matrix inequalities (LMIs). The
problems are: structured singular value (μ) upper bound synthesis for
constant matrix problems; robust-state-feedback problem with quadratic
stability criteria for uncertain systems; optimal, constant, block
diagonal, similarity scaling for full information and state feedback
H <sub>∞</sub> problem; and a partial theory for optimal
performance in systems which depend on several independent variables
The 2-Riccati H <sup>∞</sup> controller formulas and
their derivations are simplified via various loop-shifting
transformations that are naturally expressed in terms of a degree-one
polynomial system matrix closely related to the Luenberger descriptor
form of a system. The technique enables one, without loss of generality,
to restrict attention to a simple case. Matrix fraction descriptions for
the algebraic Riccati equation solutions afford another change of
variables which brings the 2-Riccati H <sup>∞</sup>
controller formulas into a cleaner, more symmetric descriptor form, with
the important practical advantage that it eliminates the numerical
difficulties that can occur in cases where one or both of the Riccati
solutions blowup
A tutorial introduction to the complex structured singular value (μ) is presented, with an emphasis on the mathematical aspects of μ. The μ-based methods discussed here have been useful for analysing the performance and robustness properties of linear feedback systems. Several tests for robust stability and performance with computable bounds for transfer functions and their state space realizations are compared, and a simple synthesis problem is studied. Uncertain systems are represented using Linear Fractional Transformations (LFTs) which naturally unify the frequency-domain and state space methods.
We present a class of spatially interconnected systems with boundary conditions that have close links with their spatially invariant extensions. In particular well-posedness, stability and performance of the extension imply the same characteristics for the actual, finite extent system. In turn existing synthesis methods for control of spatially invariant systems can be extended to this class. The relation between the two kinds of system is proved using ideas based on the 'Method of Images' of partial differential equations theory and uses symmetry properties of the interconnection as a key tool.
Applied new structured distributed control methods to an experimental apparatus representing aerial vehicles flying in close formation for the purpose of induced drag reduction. Four wings with roll and lateral degrees of freedom are set in a wind tunnel in a half-vee formation. Upon system identification via grey box modelling, the performance of three types of control were compared, by applying the control to a model of the experimental apparatus and to a simulation of a larger array of wings. The hypothesis states that the centralized controller will result in the smallest closed-loop gains, but that the distributed controller will perform substantially better than the decentralized controller. Experimental results support this hypothesis, so that the distributed, scheme can provide much of the performance of the centralized approach, while presenting a much more tractable synthesis problem.
In this paper, we consider the analysis and synthesis problem for linear time-invariant distributed discrete time systems. These systems occur in many practical applications such as formation flight, control of systems described by partial differential equations, and control of automated vehicles moving in a platoon. Computationally attractive linear matrix inequality conditions are obtained for ensuring stability and induced L<sub>2</sub> performance. An example is given to motivate the class of systems we consider.
This paper considers the analysis and synthesis of a distributed parameter dependent controller designed for a parameter-varying spatially interconnected system. It is assumed that the distributed linear parameter varying system depends on spatial varying parameters in linear fractional manner. The controller takes a similar structure as the system. Since the dependence is in linear fractional form, the. controller synthesis condition is characterized in terms of linear matrix inequalities, which are convex optimization problems for which efficient computation techniques are readily available. This distributed control approach is demonstrated on a flexible beam problem with nonuniform cross-section geometry.
Considers one of the fundamental issues in design and analysis of
sampled multidimensional systems-that of uncertainty modeling and robust
stability analysis. The paper extends methods of structured uncertainty
analysis (μ-analysis) towards spatially distributed system with
dynamical and spatial coordinates. The main contribution with respect to
earlier work in this area is in clarification of stability issues for
multidimensional systems with noncausal coordinates. Here stability is
understood in a broad sense and includes decay (localization) of system
response along noncausal spatial coordinates. The presented framework
allows to address such practically important issues as robustness of
dynamical stability and spatial localization of multidimensional
closed-loop feedback system response and boundary effects in a unified
way
There have been efforts previously to use multidimensional system
representations for the control of spatially distributed systems. By
expressing spatially distributed systems as multidimensional system
models, it has been shown that semidefinite programming techniques can
be used for control design and analysis. This formulation can in fact be
interpreted as the natural generalization of linear fractional
transformation based robust control tools to spatially distributed
systems. While most models of physical systems are continuous time
models, discrete models are required for control implementation. In this
paper, we address the issue of constructing discrete time models from
continuous time models for spatially distributed systems
Considers control design for distributed systems, where the
controller is to adopt and preserve the distributed spatial structure of
the system. Specifically systems that are inhomogeneous with respect to
the spatial variable, and may have imposed boundary conditions, are
considered. Operator theoretic tools for working with these systems are
developed, and lead to convex conditions for analysis and synthesis with
respect to the l<sub>2</sub>-induced norm
A technique for the simplification of spatially distributed
systems is presented. This technique relies on linear matrix inequality
(LMI) based reduction results originally developed for the
simplification of uncertain and multi-dimensional systems. The original
results are applicable to systems that can be modelled by linear
fractional transformations (LFTs) on structured operator sets whose
elements are assumed to be temporal variables. In the paper, the
original LFT results are extended to systems written as LFTs on spatial
variables as well as on temporal variables. The main technical
difference in the derivation of the new reduction results is a
relaxation of standard causality requirements (or equivalently stability
requirements), which in turn leads to a relaxation of the constraints to
the relevant LMI solutions
In this paper we describe a Matlab Toolbox for modeling, analysis,
and control design for multidimensional (MD) systems. This allows one to
perform: MD system representations from first principles models; MD
systems model reduction; MD systems manipulation, such as addition,
multiplication, block transformations, etc.; spatially distributed
controller design for MD systems; performance analysis of MD systems;
and MD systems simulation
The problem of distributed control by an array of sensors and
actuators is discussed. A method is presented to implement control laws
with regard to global objectives, but only requiring the communication
of each unit with the adjacent elements of the array. The method is
based on local estimates of the distant measurements which are
recursively updated by communicating with their neighbors. The
implications of this approach on the question of global stability is
analyzed, leading to guidelines on the communication requirements of the
array
In this paper the analytical and software tools developed by the
author for the control of spatially distributed and interconnected
systems are applied to a numerical example: the control of a vibrating,
elastic cable. Tractable solutions and practical controller
implementations are obtained by considering systems with distributed
actuation and sensing capabilities
Considers control design for distributed systems, where the
controller is to adopt and preserve the spatial structure of the system.
Specifically systems that vary spatially are considered; namely those
that are not shift invariant with respect to spatial variables. Operator
theoretic tools for working with these systems are developed, and lead
to convex conditions for analysis and synthesis with respect to the
L<sub>2</sub>-induced norm
In this paper preliminary results in the use of linear matrix
inequalities for the decentralized control of distributed parameter
systems is presented. The class of systems being considered are those
that can be expressed as multidimensional systems. It is shown that
linear matrix inequalities can be used to provide tractable solutions to
this problem; the conditions are in general conservative, but are
computationally attractive and lead to controllers which have a
decentralized structure
A formation flight testbed highlights the intricate interplay between technology and control theory for interconnected systems. Successfully controlling airplanes that fly in formation will entail a complete control design cycle: first principles modeling to understand the underlying physical phenomena, system identification to fit the experimental data to a physically motivated model, model-based control design, control system implementation, and finally, assessing the performance of the controller.
In this paper, the matrix version of Parrott's theorem is extended
to encompass nondefinite scalings. This extension can be used to relax
the causality requirements for many linear matrix inequality based
synthesis problems
The structured singular-value function (μ) is defined with
respect to a given uncertainty set. This function is continuous if the
uncertainties are allowed to be complex. However, if some uncertainties
are required to be real, then it can be discontinuous. It is shown that
μ is always upper semi-continuous, and conditions are derived under
which it is also lower semi-continuous. With these results, the
real-parameter robustness problem is reexamined. A related (although not
equivalent) problem is formulated, which is always continuous, and the
relationship between the new problem and the original real-μm problem
is made explicit. A numerical example and results obtained via this
related problem are presented
An important class of linear time-varying systems consists of plants where the state-space matrices are fixed functions of some time-varying physical parameters `. Small Gain techniques can be applied to such systems to derive robust time-invariant controllers. Yet, this approach is often overly conservative when the parameters ` undergo large variations during system operation. In general, much higher performance can be achieved by control laws that incorporate available measurements of ` and therefore "adjust" to the current plant dynamics. This paper discusses extensions of H1 synthesis techniques to allow for controller dependence on time-varying but measured parameters. When this dependence is linear fractional, the existence of such gain-scheduled H1 controllers is fully characterized in terms of linear matrix inequalities (LMIs). The underlying synthesis problem is therefore a convex program for which efficient optimization techniques are available. The formalism and derivatio...
A distributed gain-scheduling control design for a flexible beam problem
- F Wu
- E Yildizoglu
F. Wu and E. Yildizoglu, "A distributed gain-scheduling control design
for a flexible beam problem," in Proc. Amer. Control Conf., 2002, pp.
3174-3179.
Software for modeling, analysis, and control design for multidimensional systems Extension of Parrott's theorem to nondefinite scalings
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