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Journal articles on the topic 'Controllability gramian'

Author: Grafiati

Published: 4 June 2021

Last updated: 15 February 2022

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1

Himpe, Christian, and Mario Ohlberger. "Cross-Gramian-Based Combined State and Parameter Reduction for Large-Scale Control Systems." Mathematical Problems in Engineering 2014 (2014): 1–13. http://dx.doi.org/10.1155/2014/843869.

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This work introduces the empirical cross-gramian for multiple-input-multiple-output systems. The cross-gramian is a tool for reducing the state space of control systems, by conjoining controllability and observability information into a single matrix and does not require balancing. Its empirical gramian variant extends the applicability of the cross-gramian to nonlinear systems. Furthermore, for parametrized systems, the empirical gramians can also be utilized for sensitivity analysis or parameter identification and thus for parameter reduction. This work also introduces the empirical joint gramian, which is derived from the empirical cross-gramian. The joint gramian allows not only a reduction of the parameter space but also the combined state and parameter space reduction, which is tested on a linear and a nonlinear control system. Controllability- and observability-based combined reduction methods are also presented, which are benchmarked against the joint gramian.

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2

Dusser, Xavier, and Rabah Rabah. "On exponential stabilizability of linear neutral systems." Mathematical Problems in Engineering 7, no. 1 (2001): 67–86. http://dx.doi.org/10.1155/s1024123x01001533.

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In this paper, we deal with linear neutral functional differential systems. Using an extended state space and an extended control operator, we transform the initial neutral system in an infinite dimensional linear system. We give a sufficient condition for admissibility of the control operatorB, conditions under which operatorBcan be acceptable in order to work with controllability and stabilizability. Necessary and sufficient conditions for exact controllability are provided; in terms of a gramian of controllabilityN(μ). Assuming admissibility and exact controllability, a feedback control law is defined from the inverse of the operatorN(μ)in order to stabilize exponentially the closed loop system. In this case, the semigroup generated by the closed loop system has an arbitrary decay rate.

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3

Klickstein, Isaac, and Francesco Sorrentino. "The controllability Gramian of lattice graphs." Automatica 114 (April 2020): 108833. http://dx.doi.org/10.1016/j.automatica.2020.108833.

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4

Zhang, Lifei, Konstantin Avenirovich Neusypin, and Maria Sergeevna Selezneva. "A New Method for Determining the Degree of Controllability of State Variables for the LQR Problem Using the Duality Theorem." Applied Sciences 10, no. 15 (July 29, 2020): 5234. http://dx.doi.org/10.3390/app10155234.

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The control performance of a dynamic system can be checked by the degree of controllability. In this work, we present a new method for determining the degree of observability of state variables for the linear quadratic optimal estimation (LQE) problem. We carried out the calculation of the degree of controllability for the linear quadratic optimal control (LQR) problem using a duality theorem. Compared with the traditional measures of controllability such as determinant, trace, and maximal eigenvalue of the inverse controllability Gramian, the proposed degree of controllability was developed for each state variable and takes into account both the controllability Gramian and the cost function. The new method is convenient to apply to LQR problem. In the numerical simulation, we determined the influence of the model parameters on the degree of controllability. Besides that, we analyzed the degree of controllability, which gives an insight into the relationship between the system model design and the control performance.

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Choo, Younseok, and Dongmin Kim. "SISO Continuous-System Reduction via Impulse Response Gramian by Iterative Formulae." Journal of Dynamic Systems, Measurement, and Control 128, no. 2 (May 22, 2005): 391–93. http://dx.doi.org/10.1115/1.2194075.

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This paper concerns the order reduction of single-input single-output (SISO) linear time-invariant continuous systems based on the impulse response Gramian. From the recursive relationship among the Gramians, a new formula is derived for computing the system matrix in controllability canonical form. The result is applied to the model reduction problem. Reduced models obtained approximate the reduced-order Gramians while preserving some initial time moments and Markov parameters of the original system.

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6

Alves da Silveira, Otávio Augusto, Jun Sérgio Ono Fonseca, and Ilmar Ferreira Santos. "Actuator topology design using the controllability Gramian." Structural and Multidisciplinary Optimization 51, no. 1 (July 16, 2014): 145–57. http://dx.doi.org/10.1007/s00158-014-1121-z.

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7

Roy, Sandip, and Mengran Xue. "Controllability-Gramian submatrices for a network consensus model." Systems & Control Letters 136 (February 2020): 104575. http://dx.doi.org/10.1016/j.sysconle.2019.104575.

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8

Benner, Peter, and Christian Himpe. "Cross-Gramian-based dominant subspaces." Advances in Computational Mathematics 45, no. 5-6 (November 19, 2019): 2533–53. http://dx.doi.org/10.1007/s10444-019-09724-7.

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AbstractA standard approach for model reduction of linear input-output systems is balanced truncation, which is based on the controllability and observability properties of the underlying system. The related dominant subspaces projection model reduction method similarly utilizes these system properties, yet instead of balancing, the associated subspaces are directly conjoined. In this work, we extend the dominant subspace approach by computation via the cross Gramian for linear systems, and describe an a-priori error indicator for this method. Furthermore, efficient computation is discussed alongside numerical examples illustrating these findings.

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9

Chang, Wen-Jer. "Model-Based Fuzzy Controller Design With Common Observability Gramian Assignment." Journal of Dynamic Systems, Measurement, and Control 123, no. 1 (August 31, 1998): 113–16. http://dx.doi.org/10.1115/1.1344241.

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Almost all robust properties of linear systems can be related directly to properties of the controllability or observability Gramian. A model-based fuzzy controller for a class of uncertain nonlinear systems will be developed in this paper to achieve a common observability Gramian. Design of a model-based fuzzy controller begins by assigning a certain common observability Gramian. The nonlinear systems considered in this paper are represented by Takagi-Sugeno fuzzy models. An application to the inverted pendulum system will be given to demonstrate the effects of the present design method.

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10

Wang, Yanqing, and Zhiyong Yu. "On the partial controllability of SDEs and the exact controllability of FBSDES." ESAIM: Control, Optimisation and Calculus of Variations 26 (2020): 68. http://dx.doi.org/10.1051/cocv/2019052.

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A notion of partial controllability (also can be called directional controllability or output controllability) is proposed for linear controlled (forward) stochastic differential equations (SDEs), which characterizes the ability of the state to reach some given random hyperplane. It generalizes the classical notion of exact controllability. For time-invariant system, checkable rank conditions ensuring SDEs’ partial controllability are provided. With some special setting, the partial controllability for SDEs is proved to be equivalent to the exact controllability for linear controlled forward-backward stochastic differential equations (FBSDEs). Moreover, we obtain some equivalent conclusions to partial controllability for SDEs or exact controllability for FBSDEs, including the validity of observability inequalities for the adjoint equations, the solvability of some optimal control problems, the solvability of norm optimal control problems, and the non-singularity of a random version of Gramian matrix.

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11

Jneid, Maher, and Muath Awadalla. "On the Controllability of Conformable Fractional Deterministic Control Systems in Finite Dimensional Spaces." International Journal of Mathematics and Mathematical Sciences 2020 (March 10, 2020): 1–7. http://dx.doi.org/10.1155/2020/9026973.

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In this paper, we establish a set of convenient conditions of controllability for semilinear fractional finite dimensional control systems involving conformable fractional derivative. Indeed, sufficient conditions of controllability for a semilinear conformable fractional system are presented, assuming that the corresponding linear systems are controllable. The present method is based on conformable fractional exponential matrix, Gramian matrix, and the iterative technique. Two illustrated examples are carried out to establish the facility and efficiency of this technique.

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12

Chen, Pengyu, Xuping Zhang, and Yongxiang Li. "Existence and approximate controllability of fractional evolution equations with nonlocal conditions via resolvent operators." Fractional Calculus and Applied Analysis 23, no. 1 (February 25, 2020): 268–91. http://dx.doi.org/10.1515/fca-2020-0011.

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AbstractIn this article, we are concerned with the existence of mild solutions as well as approximate controllability for a class of fractional evolution equations with nonlocal conditions in Banach spaces. Sufficient conditions of existence of mild solutions and approximate controllability for the desired problem are presented by introducing a new Green’s function and constructing a control function involving Gramian controllability operator. The discussions are based on Schauder’s fixed point theorem as well as the theory of α-order solution operator and α-order resolvent operator. An example is given to illustrate the feasibility of our theoretical results.

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13

Younus, A., I. Javaid, and A. Zehra. "On controllability and observability of fractional continuous-time linear systems with regular pencils." Bulletin of the Polish Academy of Sciences Technical Sciences 65, no. 3 (June 27, 2017): 297–304. http://dx.doi.org/10.1515/bpasts-2017-0033.

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AbstractIn this paper necessary and sufficient conditions of controllability and observability for solutions of the state equations of fractional continuous time linear systems with regular pencils are proposed. The derivations of the conditions are based on the construction of Gramian matrices.

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14

Liu, Zeyu, and John Wagner. "Nonlinear Model Reduction for Dynamic and Automotive System Descriptions." Journal of Dynamic Systems, Measurement, and Control 124, no. 4 (December 1, 2002): 637–47. http://dx.doi.org/10.1115/1.1515327.

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The mathematical modeling of dynamic systems is an important task in the design, analysis, and implementation of advanced control systems. Although most vehicle control algorithms tend to use model-free calibration architectures, a need exists to migrate to model-based control algorithms which may offer greater operating performance. However, in many instances, the analytical descriptions are too complex for real-time powertrain and chassis model-based control algorithms. Thus, model reduction strategies may be applied to transform the original model into a simplified lower-order form while preserving the dynamic characteristics of the original high-order system. In this paper, an empirical gramian balanced nonlinear model reduction strategy is examined. The controllability gramian represents the energy needed to transport the system between states, while the observability gramian denotes the output energy transmitted. These gramians are then balanced and select system dynamics truncated. For comparison purposes, a Taylor Series linearization will also be introduced to linearize the original nonlinear system about an equilibrium operating point, and then a balanced realization linear reduction strategy applied to reduce the linearized model. To demonstrate the functionality of each model reduction strategy, a vehicle suspension system and exhaust gas recirculation valve are investigated, and respective transient performances are compared.

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15

Chang, Wen-Jer, Chein-Chung Sun, and Chyun-Chau Fuh. "DISCRETE OUTPUT FUZZY CONTROLLER DESIGN FOR ACHIEVING COMMON CONTROLLABILITY GRAMIAN." Asian Journal of Control 2, no. 4 (October 22, 2008): 284–89. http://dx.doi.org/10.1111/j.1934-6093.2000.tb00034.x.

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16

Yu, Zhiyong. "Controllability Gramian and Kalman rank condition for mean-field control systems." ESAIM: Control, Optimisation and Calculus of Variations 27 (2021): 30. http://dx.doi.org/10.1051/cocv/2021031.

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This paper is concerned with the exact controllability of linear mean-field stochastic systems with deterministic coefficients. With the help of the theory of mean-field backward stochastic differential equations (MF-BSDEs, for short) and some delicate analysis, we obtain a mean-field version of the Gramian matrix criterion for the general time-variant case, and a mean-field version of the Kalman rank condition for the special time-invariant case.

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17

Das, Sanjukta, Dwijendra N. Pandey, and N. Sukavanam. "Existence of Solution and Approximate Controllability for Neutral Differential Equation with State Dependent Delay." International Journal of Partial Differential Equations 2014 (October 2, 2014): 1–12. http://dx.doi.org/10.1155/2014/787092.

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This paper is divided in two parts. In the first part we study a second order neutral partial differential equation with state dependent delay and noninstantaneous impulses. The conditions for existence and uniqueness of the mild solution are investigated via Hausdorff measure of noncompactness and Darbo Sadovskii fixed point theorem. Thus we remove the need to assume the compactness assumption on the associated family of operators. The conditions for approximate controllability are investigated for the neutral second order system with respect to the approximate controllability of the corresponding linear system in a Hilbert space. A simple range condition is used to prove approximate controllability. Thereby, we remove the need to assume the invertibility of a controllability operator used by authors in (Balachandran and Park, 2003), which fails to exist in infinite dimensional spaces if the associated semigroup is compact. Our approach also removes the need to check the invertibility of the controllability Gramian operator and associated limit condition used by the authors in (Dauer and Mahmudov, 2002), which are practically difficult to verify and apply. Examples are provided to illustrate the presented theory.

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18

Sen. "On the Approximated Reachability of a Class of Time-Varying Nonlinear Dynamic Systems Based on Their Linearized Behavior about the Equilibria: Applications to Epidemic Models." Entropy 21, no. 11 (October 26, 2019): 1045. http://dx.doi.org/10.3390/e21111045.

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This paper formulates the properties of point reachability and approximate point reachability of either a targeted state or output values in a general dynamic system which possess a linear time-varying dynamics with respect to a given reference nominal one and, eventually, an unknown structured nonlinear dynamics. Such a dynamics is upper-bounded by a function of the state and input. The results are obtained for the case when the time-invariant nominal dynamics is perfectly known while its time-varying deviations together with the nonlinear dynamics are not precisely known and also for the case when only the nonlinear dynamics is not precisely known. Either the controllability gramian of the nominal linearized system with constant linear parameterization or that of the current linearized system (which includes the time-varying linear dynamics) are assumed to be non-singular. Also, some further results are obtained for the case when the control input is eventually saturated and for the case when the controllability gramians of the linear parts are singular. Examples of the derived theoretical results for some epidemic models are also discussed.

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19

Gupta, Rikin, Wei Zhao, and Rakesh K. Kapania. "Controllability Gramian as Control Design Objective in Aircraft Structural Design Optimization." AIAA Journal 58, no. 7 (July 2020): 3199–220. http://dx.doi.org/10.2514/1.j059102.

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20

Zhao, Shiyu, and Fabio Pasqualetti. "Networks with diagonal controllability Gramian: Analysis, graphical conditions, and design algorithms." Automatica 102 (April 2019): 10–18. http://dx.doi.org/10.1016/j.automatica.2018.12.038.

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21

Vadivoo, B. Sundara, R. Raja, Jinde Cao, G. Rajchakit, and Aly R. Seadawy. "Controllability criteria of fractional differential dynamical systems with non-instantaneous impulses." IMA Journal of Mathematical Control and Information 37, no. 3 (October 15, 2019): 777–93. http://dx.doi.org/10.1093/imamci/dnz025.

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Abstract This manuscript prospects the controllability criteria of non-instantaneous impulsive Volterra type fractional differential systems. By enroling an appropriate Gramian matrix that is often defined by the Mittag-Leffler function and with the assistance of Laplace transform, the necessary and sufficiency conditions for the controllability of non-instantaneous impulsive Volterra-type fractional differential equations are derived by using algebraic approach and Cayley–Hamilton theorem. An important feature present in our paper is that we have taken non-instantaneous impulses into the fractional order dynamical system and studied the controllability analysis, since this do not exist in the available source of literature. Inclusively, we have provided two illustrative examples with the existence of non-instantaneous impulse into the fractional dynamical system. So this demonstrates the validity and efficacy of our obtained criteria of the main section.

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22

Le, Vandai, Xinran Li, Yong Li, Yijia Cao, and Caoquyen Le. "Optimal placement of TCSC using controllability Gramian to damp power system oscillations." International Transactions on Electrical Energy Systems 26, no. 7 (November 5, 2015): 1493–510. http://dx.doi.org/10.1002/etep.2159.

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23

Casadei, Giacomo, Carlos Canudas-de-Wit, and Sandro Zampieri. "Model Reduction Based Approximation of the Output Controllability Gramian in Large-Scale Networks." IEEE Transactions on Control of Network Systems 7, no. 4 (December 2020): 1778–88. http://dx.doi.org/10.1109/tcns.2020.3000694.

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24

BITTANTI, SERGIO, and PAOLO BOLZERN. "Discrete-time linear periodic systems: gramian and modal criteria for reachability and controllability†." International Journal of Control 41, no. 4 (April 1985): 909–28. http://dx.doi.org/10.1080/0020718508961171.

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25

Walker, Douglas G., Jeffrey L. Stein, and A. Galip Ulsoy. "An Input-Output Criterion for Linear Model Deduction." Journal of Dynamic Systems, Measurement, and Control 122, no. 3 (August 6, 1996): 507–13. http://dx.doi.org/10.1115/1.1286819.

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Model order deduction algorithms have been developed in an effort to automate the production of accurate, minimal complexity models of dynamic systems in order to aid in the design of these systems. Previous algorithms, MODA and Extended MODA, deduce models independent of system inputs and outputs. FD-MODA uses frequency response methods to deduce models of a single input-output pair. In this paper, an input-output criterion based on controllability and observability is combined with the frequency based criterion used by MODA. The new model deduction algorithm, IO-MODA, compares the ratio of the adjacent diagonal values of the system gramian to a user specified threshold. The gramian is computed from a balanced realization of the system. IO-MODA generates an accurate multiple-input multiple-output model of minimum order with physically meaningful states. This model is called a proper MIMO model. An example problem is used to demonstrate this new model deduction algorithm. [S0022-0434(00)02103-1]

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Kashima, Kenji, and Misaki Imai. "Controllability Gramian of Nonlinear Gaussian Process State Space Models with Application to Model Sparsification." IFAC-PapersOnLine 53, no. 2 (2020): 443–48. http://dx.doi.org/10.1016/j.ifacol.2020.12.215.

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Khelassi, Ahmed, Didier Theilliol, and Philippe Weber. "Reconfigurability analysis for reliable fault-tolerant control design." International Journal of Applied Mathematics and Computer Science 21, no. 3 (September 1, 2011): 431–39. http://dx.doi.org/10.2478/v10006-011-0032-z.

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Reconfigurability analysis for reliable fault-tolerant control design In this paper the integration of reliability evaluation in reconfigurability analysis of a fault-tolerant control system is considered. The aim of this work is to contribute to reliable fault-tolerant control design. The admissibility of control reconfigurability is analyzed with respect to reliability requirements. This analysis shows the relationship between reliability and control reconfigurability defined generally through Gramian controllability. An admissible solution for reconfigurability is proposed according to reliability evaluation based on energy consumption under degraded functional conditions. The proposed study is illustrated with a flight control application.

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Dilip, A. Sanand Amita. "The Controllability Gramian, the Hadamard Product, and the Optimal Actuator/Leader and Sensor Selection Problem." IEEE Control Systems Letters 3, no. 4 (October 2019): 883–88. http://dx.doi.org/10.1109/lcsys.2019.2919278.

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Gusev, Mikhail I. "THE LIMITS OF APPLICABILITY OF THE LINEARIZATION METHOD IN CALCULATING SMALL–TIME REACHABLE SETS." Ural Mathematical Journal 6, no. 1 (July 29, 2020): 71. http://dx.doi.org/10.15826/umj.2020.1.006.

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The reachable sets of nonlinear systems are usually quite complicated. They, as a rule, are non-convex and arranged to have rather complex behavior. In this paper, the asymptotic behavior of reachable sets of nonlinear control-affine systems on small time intervals is studied. We assume that the initial state of the system is fixed, and the control is bounded in the \(\mathbb{L}_2\)-norm. The subject of the study is the applicability of the linearization method for a sufficiently small length of the time interval. We provide sufficient conditions under which the reachable set of a nonlinear system is convex and asymptotically equal to the reachable set of a linearized system. The concept of asymptotic equality is defined in terms of the Banach-Mazur metric in the space of sets. The conditions depend on the behavior of the controllability Gramian of the linearized system – the smallest eigenvalue of the Gramian should not tend to zero too quickly when the length of the time interval tends to zero. The indicated asymptotic behavior occurs for a reasonably wide class of second-order nonlinear control systems but can be violated for systems of higher dimension. The results of numerical simulation illustrate the theoretical conclusions of the paper.

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Osipov, I. O. "On the convexity of the reachable set with respect to a part of coordinates at small time intervals." Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki 31, no. 2 (June 2021): 210–25. http://dx.doi.org/10.35634/vm210204.

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We investigate the convexity of the reachable sets for some of the coordinates of nonlinear systems with integral constraints on the control at small time intervals. We have proved sufficient convexity conditions in the form of constraints on the asymptotics of the eigenvalues of the Gramian of the controllability of a linearized system for some of the coordinates. There are two nonlinear third-order systems under study as examples. The system linearized along a trajectory generated by zero control is uncontrollable, and the system in the other example is completely controllable. We investigate the sufficient conditions for convexity of projection of reachable sets. Numerical modeling has been carried out, demonstrating the non-convexity of some projections even for small time intervals.

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Van Dai, Le, Doan Duc Tung, Tran Le Thang Dong, and Le Cao Quyen. "Improving Power System Stability with Gramian Matrix-Based Optimal Setting of a Single Series FACTS Device: Feasibility Study in Vietnamese Power System." Complexity 2017 (2017): 1–21. http://dx.doi.org/10.1155/2017/3014510.

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The Vietnamese power system has experienced instabilities due to the effect of increase in peak load demand or contingency grid faults; hence, using flexible alternating-current transmission systems (FACTS) devices is a best choice for improving the stability margins. Among the FACTS devices, the thyristor-controlled series capacitor (TCSC) is a series connected FACTS device widely used in power systems. However, in practice, its influence and ability depend on setting. For solving the problem, this paper proposes a relevant method for optimal setting of a single TCSC for the purpose of damping the power system oscillations. This proposed method is developed from the combination between the energy method and Hankel-norm approximation approach based on the controllability Gramian matrix considering the Lyapunov equation to search for a number of feasible locations on the small-signal stability analysis. The transient stability analysis is used to compare and determine appropriate settings through various simulation cases. The effectiveness of the proposed method is confirmed by the simulation results based on the power system simulation engineering (PSS/E) and MATLAB programs. The obtained results show that the proposed method can apply to immediately solve the difficulties encountering in the Vietnamese power system.

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Zhao, Shiyu, and Fabio Pasqualetti. "Discrete-Time Dynamical Networks with Diagonal Controllability Gramian * *This material is based upon work supported in part by ONR award N00014-14-1-0816 and NSF award ECCS 1462530." IFAC-PapersOnLine 50, no. 1 (July 2017): 8297–302. http://dx.doi.org/10.1016/j.ifacol.2017.08.1407.

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De la Sen, Manuel, Asier Ibeas, and Raul Nistal. "About Partial Reachability Issues in an SEIR Epidemic Model and Related Infectious Disease Tracking in Finite Time under Vaccination and Treatment Controls." Discrete Dynamics in Nature and Society 2021 (February 26, 2021): 1–21. http://dx.doi.org/10.1155/2021/5556897.

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This paper studies some basic properties of an SEIR (Susceptible-Exposed-Infectious-Recovered) epidemic model subject to vaccination and treatment controls. Firstly, the basic stability, boundedness, and nonnegativity of the state trajectory solution are investigated. Then, the problem of partial state reachability from a certain state value to a targeted one in finite time is focused on since it turns out that epidemic models are, because of their nature, neither (state) controllable from a given state to the origin nor reachable from a given initial condition. The particular formal statement of the partial reachability is focused on as a problem of output-reachability by defining a measurable output or lower dimension than that of the state. A special case of interest is that when the output is defined as the infectious subpopulation to be step-to-step tracked under suitable amounts being compatible with the required constraints. As a result, and provided that the output-controllability Gramian is nonsingular on a certain time interval of interest, a feedback control effort might be designed so that a prescribed value of the output can be approximately tracked. A linearization approximation is performed to simplify and facilitate the above task which is based on a point-to-point linearization of the solution trajectory. To this end, an “ad hoc” sampled approximate output trajectory is defined as control objective to be targeted through a point-wise calculated Jacobian matrix. A supervised appropriate restatement of the targeted suited sampled output values is redefined, if necessary, to make the initial proposed sampled trajectory compatible with the various needed constraints on nonnegativity and control boundedness. The design can be optionally performed under constant or adaptive sampling rates. Finally, some numerical examples are given to test the theoretical aspects and the design efficiency of the model.

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Bueno, D. D., C. R. Marqui, V. Lopes Jr., M. J. Brennan, and Daniel J. Inman. "Structural Damage Identification and Location Using Grammian Matrices." Shock and Vibration 19, no. 3 (2012): 287–99. http://dx.doi.org/10.1155/2012/158723.

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In this paper, an approach using observability and controllability grammian matrices is proposed to determine if structural damage has occurred together with an estimate of its location. The theory is outlined and simulations are carried out on a simple structure to demonstrate the method. Experimental tests were also carried out to demonstrate the validity of the approach using real signals. The dynamic properties of the structure are identified using the eigensystem realization algorithm (ERA) and a reduced order state-space model of the system subsequently constructed. Either the observability or controllability grammians can then be used depending on the number of sensors available. It is shown that these are sensitive to both the degree and location of the damage and offer promise for structural health monitoring applications.

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Guangqian, Xing, and Peter M. Bainum. "Actuator Placement Using Degree of Controllability for Discrete-Time Systems." Journal of Dynamic Systems, Measurement, and Control 114, no. 3 (September 1, 1992): 508–16. http://dx.doi.org/10.1115/1.2897376.

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In this paper some definitions of degree of controllability (observability), which is based on the scalar measure of the controllability (observability) Grammian matrix, are presented for discrete-time systems. The emphasis here is in showing the physical and geometrical meanings of the degree of controllability (observability) and the general properties of degree of controllability (observability). The concepts of degree of controllability are applied to the actuator placement problem for the shallow spherical sheel control system. The LQG transient responses for several typical systems with different degrees of controllability are also shown.

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Balachandran, Krishnan, and Jayakumar Kokila. "On the controllability of fractional dynamical systems." International Journal of Applied Mathematics and Computer Science 22, no. 3 (September 1, 2012): 523–31. http://dx.doi.org/10.2478/v10006-012-0039-0.

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Abstract This paper is concerned with the controllability of linear and nonlinear fractional dynamical systems in finite dimensional spaces. Sufficient conditions for controllability are obtained using Schauder’s fixed point theorem and the controllability Grammian matrix which is defined by the Mittag-Leffler matrix function. Examples are given to illustrate the effectiveness of the theory.

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Guendouzi, Toufik, and Iqbal Hamada. "Global relative controllability of fractional stochastic dynamical systems with distributed delays in control." Boletim da Sociedade Paranaense de Matemática 32, no. 2 (September 11, 2014): 55. http://dx.doi.org/10.5269/bspm.v32i2.20583.

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This paper is concerned with the global relative controllability of linear and nonlinear fractional stochastic dynamical systems with distributed delays in control for finite dimensional spaces. Sufficient conditions for controllability results are obtained using the Banach fixed point theorem and the controllability Grammian matrix which is dened by the Mittag-Leffler matrix function. An example is provided to illustrate the theory.

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Ikeh, Onyekachukwu Henry Ikeh, and Chukwunenye Ukwu. "Fundamental Results on Determining Matrices for a Certain Class of Hereditary Systems." Journal of Mathematics and Statistics Studies 2, no. 1 (June 12, 2021): 62–87. http://dx.doi.org/10.32996/jmss.2021.2.1.7.

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Three major tools are required to investigate the controllability of control systems, namely, determining matrices, index of control systems and controllability Grammian. Determining matrices are the preferred choice for autonomous control systems due to the fact that they are devoid of integral operators in their computations. This article developed the structure of certain parameter-ordered determining matrices of generic double time-delay linear autonomous functional differential control systems, with a view to obtaining the controllability matrix associated with the rank condition for Euclidean controllability of the system. Expressions for the relevant determining matrices were formulated and it was established that the determining matrices for double time-delay linear autonomous functional differential control systems do not exist if one of the time-delays is not an integer multiple of the other paving the way for the investigation of the Euclidean controllability of generic double time-delay control systems.

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Yasuda, Kazunori, and Robert E. Skelton. "Assigning controllability and observability Gramians in feedback control." Journal of Guidance, Control, and Dynamics 14, no. 5 (September 1991): 878–85. http://dx.doi.org/10.2514/3.20727.

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YASUDA, Kazunori, Nobuyuki IMAI, and Kazumasa HIRAI. "The Class of Controllability Gramians Assignable by State Feedback." Transactions of the Society of Instrument and Control Engineers 24, no. 8 (1988): 802–8. http://dx.doi.org/10.9746/sicetr1965.24.802.

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Wu, Q. H., and M. Mansour. "Schur-Cohn, Nour Eldin-Markov Matrices and the Controllability Gramians." IFAC Proceedings Volumes 26, no. 2 (July 1993): 783–86. http://dx.doi.org/10.1016/s1474-6670(17)49236-6.

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Al-Iedani, Intessar, and Zoran Gajic. "Recursive Reduced-Order Algorithm for Singularly Perturbed Cross Grammian Algebraic Sylvester Equation." Mathematical Problems in Engineering 2016 (2016): 1–9. http://dx.doi.org/10.1155/2016/2452746.

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A new recursive algorithm is developed for solving the algebraic Sylvester equation that defines the cross Grammian of singularly perturbed linear systems. The cross Grammian matrix provides aggregate information about controllability and observability of a linear system. The solution is obtained in terms of reduced-order algebraic Sylvester equations that correspond to slow and fast subsystems of a singularly perturbed system. The rate of convergence of the proposed algorithm isOε, whereεis a small singular perturbation parameter that indicates separation of slow and fast state variables. Several real physical system examples are solved to demonstrate efficiency of the proposed algorithm.

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Peters, Diane L., Panos Y. Papalambros, and A. Galip Ulsoy. "Relationship between coupling and the controllability Grammian in co-design problems." Mechatronics 29 (August 2015): 36–45. http://dx.doi.org/10.1016/j.mechatronics.2015.05.002.

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Wang, Yumei. "Eccentricity-Induced Seismic Behavior of Curved Bridges Based on Controllability." Symmetry 12, no. 10 (October 3, 2020): 1633. http://dx.doi.org/10.3390/sym12101633.

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The difficulty in curved bridge design lies in the eccentricity. Eccentricities break the regularity and make it difficult to resist horizontal loads. However, relatively stable and robust performance can still be achieved through properly aligned eccentricity. This paper used the controllability-related concepts, the controllability Grammians and Hankel singular values (HSVs), to study the impact of eccentricities on the seismic performance of curved bridges. An analytical model was expressed by second order differential equations with rigid deck assumption. Six eccentricity cases: three different radii (resulting in different center of mass (CM)), three different bearing arrangements (resulting in different center of stiffness (CS)), and variable earthquake directions (resulting in different moment arms) were strategized for research. Analyses showed that effects of eccentricities (offsets of CS from CM) can be extensively interpreted by controllability indices. Proper eccentricity may “reach” and thus “control” the responses better and decrease the coupling effects, counteract the unfavorable excitation effects, and make the bridge less sensitive to excitation changes. In this sense, regularity or stability could be somewhat re-established through design. Time history analyses confirmed the results.

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PRADHAN, S., V. J. MODI, and M. S. BHAT. "Relationship between the controllability grammian and closed-loop eigenvalues: the single input case." International Journal of Systems Science 24, no. 6 (June 1993): 1133–44. http://dx.doi.org/10.1080/00207729308949548.

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Ko, Byeongsik, Benson H. Tongue, and Andrew Packard. "A Method for Determining the Optimal Location of a Distributed Sensor/Actuator." Shock and Vibration 1, no. 4 (1994): 357–74. http://dx.doi.org/10.1155/1994/387283.

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The optimal location problem of distributed sensor/actuator for observation and control of a flexible structure is investigated. Using a property of controllability and observability grammian matrices, this approach employs a nonlinear optimization technique to determine the optimal placement of a distributed sensor/actuator. The effect of unimportant modes that do not strongly affect the structural behavior of a system is minimized and the effect of important modes is maximized. The final objective function is expressed as the combinational form of two different objective functions. This technique is applied to several types of beam support conditions and the corresponding optimal locations are determined.

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Ukpai, Ukpai I., and Suhada Jayasuriya. "Using Controller Reduction Techniques for Efficient PID Controller Synthesis." Journal of Dynamic Systems, Measurement, and Control 126, no. 3 (September 1, 2004): 692–96. http://dx.doi.org/10.1115/1.1790541.

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Balanced reduction relies on the internal balancing of the controllability and observability grammians to eliminate weakly observable and controllable modes. Weighting functions are often used to improve this procedure. Several benefits exist in implementing low-order rather than high-order controllers. It is, therefore, imperative to reduce any errors that result from using the infinite grammians in the controller reduction process. By fixing the structure of the high-order controller and choosing an appropriate frequency interval for the balanced reduction a system that assumes a second-order controller is obtained, forming the basis for an efficient approximation/optimization towards obtaining a robust PID controller. This paper establishes a procedure for choosing the weighting functions.

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Chang, Shih-Chuan. "A Contrastive Study of Grammar Translation Method and Communicative Approach in Teaching English Grammar." English Language Teaching 4, no. 2 (May 31, 2011): 13. http://dx.doi.org/10.5539/elt.v4n2p13.

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The Grammar Translation Method and the Communicative Approach have both played important roles in grammar teaching. Which is better, the Grammar Translation Method or the Communicative Approach? This paper aims to compare the controllability and feasibility of these two approaches and find out which one is more suitable for grammar teaching in Taiwan. Two classes were selected and taught by the Grammar Translation Method and the Communicative Approach respectively. The college admission test showed that they share a similar level of the overall English proficiency before the intervention. The pre-test demonstrated that there wasn’t any distinction between the two classes in their grammatical competence. The post-test embodied that there was significant difference in their grammatical competence between the two classes. The scores of the students in the Experimental Class were higher than that in the Control Class. The result showed that grammar teaching in the framework of the Grammar Translation Method is better than the Communicative Approach. Nevertheless, the Communicative Approach emphasizes fluency and the Grammar Translation Method is concerned with accuracy. Fluency and accuracy are the target for English learning. So the best way to improve the situation is to combine both methods in teaching English Grammar.

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Çela, Arben, Silviu-Iulian Niculescu, René Natowicz, and Abdellatif Reama. "Controllability and Observability Gramians as Information Metrics for Optimal Design of Networked Control Systems." Mechanical Engineering 140, no. 12 (December 1, 2018): S8—S15. http://dx.doi.org/10.1115/1.2018-dec-8.

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This paper addresses the problem of optimal control and scheduling of Networked Control Systems over limited bandwidth deterministic networks using some insight on the interplay between the control and information theory. The motivation is related to the necessity of choice a communication sequence which maximize control signal impact on the plant behavior. The solution is obtained by decomposing the overall problem in a twofold one. The first level problem aims obtaining the periodic off-line or static scheduling function of control signals based on system properties, communication constrains, periodicity of scheduling sequence, performance criteria and maximization of the degree of reachability/observability of the periodic system. A Mixed Integer Quadratic Programming (MIQP) problem is formulated and solved obtaining a periodic and stable NCS. The solution of the second level problem is based on the structure of the static scheduling function obtained from the first level solution and the dependance of the degree of reachability/observability on the current state of the system.

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Biryukov, Dmitriy S., Natalia A. Dudarenko, Leonid A. Mironovskiy, and Anatoliy V. Ushakov. "Stochastic Gramians in the Problem of Controllability Quantitative Measuring for Continuous and Discrete Plants." IFAC-PapersOnLine 48, no. 11 (2015): 942–45. http://dx.doi.org/10.1016/j.ifacol.2015.09.312.

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