Dynamics Of Constrained Systems Research Articles

Projection continuation for minimal coordinate set formulation and singularity detection of redundantly constrained system dynamics

AbstractThe formulation of (possibly redundantly) constrained system dynamics using coordinate projection onto a subspace locally tangent to the constraint manifold is revisited using the QR factorization of the constraint Jacobian matrix, using column pivoting to identify a suitable subspace, possibly detect any singular configurations that may arise, and extract it. The evolution of the QR factorization is integrated along with that of the constraint Jacobian matrix as the solution evolves, generalizing to redundant constraints a recently proposed true continuation algorithm that tracks the evolution of the subspace of independent coordinates. The resulting subspace does not visibly affect the quality of the solution, as it is merely a recombination of that resulting from the blind application of the QR factorization but avoids the artificial algorithmic irregularities or discontinuities in the generalized velocities that could otherwise result from arbitrary reparameterizations of the coordinate set, and identifies and discriminates any further possible motions that arise at singular configurations. The characteristics of the proposed subspace evolution approach are exemplified by solving simple problems with incremental levels of redundancy and singularity orders.

A projection continuation approach for minimal coordinate set constrained dynamics

The formulation of constrained system dynamics using coordinate projection onto a subspace locally tangent to the constraint manifold is revisited using the QR factorization of the constraint Jacobian matrix to extract a suitable subspace and integrating the evolution of the QR factorization along with that of the constraint Jacobian matrix, as the solution evolves. A true continuation algorithm is thus proposed to track the evolution of the subspace of independent coordinates. It does not visibly affect the quality of the solution but avoids the artificial algorithmic irregularities or discontinuities in the generalized velocities that could otherwise result from arbitrary reparameterizations of the coordinate set. The characteristics of the proposed subspace evolution approach are exemplified by solving simple single- and multi-degree-of-freedom problems.

Hierarchical clustering of constrained dynamic systems using robust positively invariant sets

A systematic Agglomerative Hierarchical Clustering (AHC) approach based on minimal Robust Positively Invariant (mRPI) sets is presented for decomposing linear constrained dynamic systems. Specifically, a novel distance metric is proposed where the distance between neighboring subsystems is computed based on the volume of the mRPI sets. Through this distance metric, the proposed AHC approach simultaneously accounts for subsystem dynamics, the constraints on states and inputs, and disturbances created by the dynamic coupling between subsystems. A combination of simple and complex numerical examples is used to demonstrate the key features of the approach including the sensitivity of the proposed AHC to system structure, parameters, and constraints. Finally, numerical results show the benefit of using the proposed AHC when determining potential system decompositions for robust decentralized Model Predictive Control (MPC).

Mode separability-based state estimation for uncertain constrained dynamic systems

This paper addresses the state estimation problem for dynamic systems subject to uncertain constraints, i.e., all possible constraints are described by a finite set, and only one constraint is satisfied at each moment. For this typical hybrid system estimation problem with coupled discrete constraint modes and continuous states, we design a mode separability-based state estimation (MSSE) framework. Based on the measurement model over a window, the hypothesis testing is performed to detect whether the mode changes firstly. Then the maximum-likelihood criterion is used to estimate the mode change sequence once detecting the change. Next, to consider the possible impact of the decided modes on the state estimation, a metric of mode separability is proposed to evaluate the separability of the recognized modes, and two different state estimation methods are introduced. Specifically, if the recognized modes are separable from the others, the state estimates are obtained by a recursive mode-based constraint Kalman filter (MCKF) which is proved that the estimation error is bounded in mean square. Otherwise, the estimation results output the fused state estimates (FSE) of the inseparable modes. Finally, simulation results of road-constrained vehicle tracking are provided to demonstrate the effectiveness of the proposed framework.

Protein-Ligand Complexes as Constrained Dynamical Systems.

This study focuses on how the low-frequency end of the vibrational spectrum related to the functional motions changes as a protein binds to a small ligand(s). Our recently proposed residue-specific (RESPEC) elastic network model provides a natural laboratory for this aim due to its systematic mixed coarse-graining approach and parametrization. Current analysis on a large data set of protein-ligand complexes reveals a universal curve enclosing the frequency distributions, which bears the features of previous computational and experimental studies. We mostly observe positive frequency shifts in the collective modes of the protein upon ligand binding. This observation, conforming to the Rayleigh-Courant-Fisher theorem, points to a constraining effect imposed by ligands on protein dynamics, which may be accompanied by a negative vibrational entropy difference. Positive frequency shifts in the global modes can thus be linked to the harmonic well getting steeper, because of interactions with the ligand(s).

Linear port-Hamiltonian descriptor systems

The modeling framework of port-Hamiltonian systems is systematically extended to linear constrained dynamical systems (descriptor systems, differential-algebraic equations) of arbitrary index and with time-varying constraints. A new algebraically and geometrically defined system structure is derived. It is shown that this structure is invariant under equivalence transformations, and that it is adequate also for the modeling of high-index descriptor systems. The regularization procedure for descriptor systems to make them suitable for simulation and control is modified to preserve the port-Hamiltonian form. The relevance of the new structure is demonstrated with several examples.

Constrained dynamical systems, robust model reference adaptive control, and unreliable reference signals

ABSTRACTIn this paper, we address the robust control design problem for nonlinear dynamical systems tracking unreliable reference signals. Specifically, we present robust model reference adaptive control laws that guarantee uniform ultimate boundedness of the trajectory tracking error for nonlinear plants that are affected by matched, unmatched, and parametric uncertainties, and are subject to constraints on the state space and the measured output. These control laws guarantee satisfactory results even in case the reference trajectory or the reference output signal do not verify the given constraints and hence, may draw the plant's trajectory or measured output outside their constraint sets. A numerical example involving the attitude control of a spacecraft illustrates the feasibility of the theoretical results presented.

A Method for Solving Large-Scale Multiloop Constrained Dynamical Systems Using Structural Decomposition

A structural decomposition method based on symbol operation for solving differential algebraic equations (DAEs) is developed. Constrained dynamical systems are represented in terms of DAEs. State-space methods are universal for solving DAEs in general forms, but for complex systems with multiple degrees-of-freedom, these methods will become difficult and time consuming because they involve detecting Jacobian singularities and reselecting the state variables. Therefore, we adopted a strategy of dividing and conquering. A large-scale system with multiple degrees-of-freedom can be divided into several subsystems based on the topology. Next, the problem of selecting all of the state variables from the whole system can be transformed into selecting one or several from each subsystem successively. At the same time, Jacobian singularities can also be easily detected in each subsystem. To decompose the original dynamical system completely, as the algebraic constraint equations are underdetermined, we proposed a principle of minimum variable reference degree to achieve the bipartite matching. Subsequently, the subsystems are determined by aggregating the strongly connected components in the algebraic constraint equations. After that determination, the free variables remain; therefore, a merging algorithm is proposed to allocate these variables into each subsystem optimally. Several examples are given to show that the proposed method is not only easy to implement but also efficient.

Constrained evolution algebras and dynamical systems of a bisexual population

Consider a bisexual population such that the set of females can be partitioned into finitely many different types indexed by {1,2,…,n} and, similarly, that the male types are indexed by {1,2,…,ν}. Recently an evolution algebra of bisexual population was introduced by identifying the coefficients of inheritance of a bisexual population as the structure constants of the algebra. In this paper we study constrained evolution algebra of bisexual population in which type “1” of females and males have preference. For such algebras sets of idempotent and absolute nilpotent elements are known. We consider two particular cases of this algebra, giving more constraints on the structural constants of the algebra. By the first our constraint we obtain an n+ν-dimensional algebra with a matrix of structural constants containing only 0 and 1. In the second case we consider n=ν=2 but with general constraints. In both cases we study dynamical systems generated by the quadratic evolution operators of corresponding constrained algebras. We find all fixed points, limit points and some 2-periodic points of the dynamical systems. Moreover we study several properties of the constrained algebras connecting them to the dynamical systems. We give some biological interpretation of our results.

Synchronization of uncertain constrained hyperchaotic systems and chaos-based secure communications via a novel decomposed nonlinear stochastic estimator

In this paper, a secure chaotic scheme for communications in noisy public channel is proposed. This scheme is based on the concept of carrier encryption in addition to the typical data encryption techniques. At the transmitter end, the unified chaotic system with adaptive parameter and a hyperchaotic Rossler system with uncertain parameters are coupled, constrained and used as a new hyperchaotic system which generates waveforms that are different from those of any known chaotic oscillator. After modulating one of the outputs of the system with the encrypted data signal, the outputs of the system are encrypted using a set of pre-defined encryption rules and transmitted to the receiving end through a noisy public communications channel. At the receiving end, the received outputs are decrypted and the transmitted data are retrieved by reconstructing the constrained hyperchaotic signals using the novel discrete-time iterative decomposed uncertain constrained extended Kalman filter (IDUCEKF). The proposed state estimator, besides being used to handle the estimation problem of uncertain constrained nonlinear dynamical systems, reduces the required processing time and gives good numerical performance. Simulation results are firstly presented to illustrate the applicability of the IDUCEKF in synchronizing the states of the constrained hyperchaotic system. Then, the proposed secure communication scheme is applied to transmit images, and the quality of the transmission process is assessed. The obtained results show the effectiveness of the proposed approach.

Contributions to the Method of Coordinate Splitting for Solving Differential Algebraic Equations

The Coordinate Splitting method is a numerical technique which was developed to aid in solving constrained multibody dynamic systems modelled as differential algebraic equations (DAEs). This paper presents a discussion about the implementation of this technique, analyzing what happens to the structure of the DAEs and how the quantity called the annihilation matrix is utilized to compute a solution. Also, an explicit formula is described for computing the Jacobian of the DAE formulation of the Coordinate Splitting method when using QR-factorization to determine the annihilation matrix. Finally, an illustrative example is used to demonstrate that the Coordinate Splitting method can be successfully applied to systems involving a singular inertia matrix.

Potentials based optimization with embedded Markov chain for stochastic constrained system

In this paper, a RBF neural network based on-line optimization algorithm with performance potentials analysis method is presented for a class of stochastic constrained dynamic systems. The control signals of the considered systems are constrained to a range according to a subset of the whole state space. With the conception of an embedded Markov chain, an optimization approach on the basis of potentials is presented for a stochastic constrained system, where the optimization criterion is the long-time average performance. With this approach, the computation burden has been reduced because it only requires one to compute the control strategy on the states concerned, which are a subset of the whole state space. Furthermore, with the characteristic of approximation performance of RBF neural network, the potentials and the transition probability matrix are estimated conveniently by a sample path compared with the statistic approach or the method by solving the Poisson equation. The effectiveness of the optimization approach has been shown by the simulation results, finally.

Robust Design of Linear Control Laws for Constrained Nonlinear Dynamic Systems

AbstractIn this paper we present techniques to solve robust optimal control problems for nonlinear dynamic systems in a conservative approximation. Here, we assume that the nonlinear dynamic system is affected by a time-varying uncertainty whose L-infinity norm is known to be bounded. By employing specialized explicit upper estimates for the nonlinear terms in the dynamics we propose a strategy to design a linear control law which guarantees that given constraints on the states and controls are robustly satisfied when running the system in closed-loop mode. Finally, the mathematical techniques are illustrated by applying them to a tutorial example.

Direct eigenanalysis of constrained system dynamics

This work addresses the direct eigenanalysis of the problem that results from the linearization of the dynamics of constrained systems about arbitrary, problem-dependent steady solutions. The procedure is outlined; the expected properties of the eigensolution are illustrated; implementation issues, related to the application of the procedure to an existing multi-body formulation, are addressed. The soundness of the proposed approach is illustrated by a simple, yet exhaustive example, and by the analysis of a relatively detailed model of a flexible helicopter rotor blade.

Fractional action-like variational problems in holonomic, non-holonomic and semi-holonomic constrained and dissipative dynamical systems

We communicate through this work the fractional calculus of variations and its corresponding Euler–Lagrange equations in 1D constrained holonomic, non-holonomic, and semi-holonomic dissipative dynamical system. The extension of the laws obtained to the 2D space state is done. Some interesting consequences are revealed.

A Network Coding Approach to Secret Key Distribution

We consider the problem of secret key distribution in a sensor network with multiple scattered sensor nodes and a mobile device that can be used to bootstrap the network. Our main contribution is a set of secure protocols that rely on simple network coding operations to provide a robust and low-complexity solution for sharing secret keys among sensor nodes, including pairwise keys, cluster keys, key revocation, and mobile node authentication. Despite its role as a key enabler for this approach, the mobile node only has access to an encrypted version of the keys, providing information-theoretic security with respect to attacks focused on the mobile node. Our results include performance evaluation in terms of security metrics and a detailed analysis of resource utilization. The basic scheme was implemented and tested in a real-life sensor network testbed. We deem this class of network coding protocols to be particularly well suited for highly constrained dynamic systems such as wireless sensor networks.

Some results on optimal control with unilateral state constraints

In this paper, we study the problem of quadratic optimal control with state variables unilateral constraints, for linear time-invariant systems. The necessary conditions are formulated as a linear invariant system with complementary slackness conditions. Some structural properties of this system are examined. Then it is shown that the problem can benefit from the higher order Moreau’s sweeping process, that is, a specific distributional differential inclusion, and from ten Dam’s geometric theory [A.A. ten Dam, K.F. Dwarshuis, J.C. Willems, The contact problem for linear continuous-time dynamical systems: A geometric approach, IEEE Trans. Automat. Control 42 (4) (1997) 458–472; A.A. ten Dam, Unilaterally Constrained Dynamical Systems, Ph.D. Thesis, Rijsuniversiteit Groningen, NL, available at http://irs.ub.rug.nl/ppn/159407869, 1997] for partitioning of the admissible domain boundary (in particular for the case of multivariable systems). In fact, the first step may be also seen as follows: does the higher order Moreau’s sweeping process (developed in Acary et al. [V. Acary, B. Brogliato, D. Goeleven, Higher order Moreau’s sweeping process: Mathematical formulation and numerical simulation, Math. Programm. A 113 (2008) 133–217]) correspond to the necessary conditions of some optimal control problem with an extended integral action? The knowledge of the qualitative behaviour of optimal trajectories at junction times is improved with the approach, which also paves the way towards efficient time-stepping numerical algorithms to solve the optimal control boundary value problem.

Parameter-Free Second Order Numerical Scheme for Constrained Multibody Dynamical Systems

This paper describes a second-order numerical scheme for constrained multibody dynamics that is simple to implement and does not require the selection of parameters for constraint satisfaction. The method is a predictor-corrector type of method, and both the predictor and the corrector steps require solution of a symmetric positive-definite linear system with no iteration of nonlinear equations. The basic philosophy of the method is to satisfy the constraint equations to the same order as the discretization of the ordinary differential equations.

Bookshelf [Cold, small, hot, and queued

The following books are reviewed in this issue: 1) Analytical Mechanics: A Comprehensive Treatise on the Dynamics of Constrained Systems; For Engineers, Physicists and Mathematicians (Papastavridis, J.G.; 2002); 2) Mechanics of Micromechanical Systems (Lobontiu, N. and Garcia, E.; 2004); 3) Plasma Physics and Controlled Nuclear Fusion (Miyamoto, K.; 2005); and 4) Feedback Control of Computing Systems (Hellerstein, J.L. et al.; 2004).

Haptic Display of Constrained Dynamic Systems via Admittance Displays

In the Cobotic Hand Controller, we have introduced an admittance display that can render very high impedances (up to its own structural stiffness). This is due to its use of infinitely variable transmissions. While admittance displays typically excel at rendering high impedances, the incorporation of infinitely variable transmissions in the Cobotic Hand Controller allows the stable display of a wide dynamic range, including low impedances. The existence of a display that excels at rendering high-impedance constraints, but has high-fidelity control of low impedances tangent to those constraints, has led us to describe an admittance control architecture not often examined in the haptics community. In this paper, we develop a comprehensive approach that enables rendering of rigid motion constraints while simultaneously preserving the physical integrity of the intended inertial dynamics tangent to those constraints. This is in contrast to conventional impedance-control algorithms that focus primarily on rendering reaction forces along contact normals with constraints. We present this algorithm here, which is general to all admittance displays, and report on its implementation with the Cobotic Hand Controller. We offer examples of rigid bodies and linkages subject to holonomic and/or nonholonomic constraints