An analysis of separable least squares data driven local coordinates for maximum likelihood estimation of linear systems

In this paper we study a novel parametrization for state-space systems, namely separable least squares data driven local coordinates (slsDDLC). The parametrization by slsDDLC has recently been successfully applied to maximum likelihood estimation of linear dynamic systems. In a simulation study, the use of slsDDLC has led to numerical advantages in comparison to the use of more conventional parametrizations, including data driven local coordinates (DDLC). However, an analysis of properties of slsDDLC, which are relevant to identification, has not been performed up to now. In this paper, we provide insights into the geometry and topology of the slsDDLC construction and show a number of results which are important for actual identification, in particular for maximum likelihood estimation. We also prove that the separable least squares methodology is indeed guaranteed to be applicable to maximum likelihood estimation of linear dynamic systems in typical situations.     

An analysis of the parametrization by data driven local coordinates for multivariable linear systems

In this paper, we study a novel parametrization for state-space systems, namely data driven local coordinates (DDLC) which have recently been introduced and applied. Even though DDLC has meanwhile become the default parametrization used in the system identification toolbox of the software package MATLAB, an analysis of properties of DDLC, which are relevant to identification, has not been performed up to now. In this paper, we provide insights into the geometry and topology of the DDLC construction and show a number of results which are important for actual identification such as maximum likelihood-type estimation.     

Constrained state-space system identification with application to structural dynamics

Constrained identification of state-space models representing structural dynamic systems is addressed. Based on physical insight, transfer function constraints are formulated in terms of the state-space parametrization. A simple example shows that a method tailored for this application, which utilizes the non-uniqueness of a state-space model, outperforms the classic sequential quadratic programming method in terms of robustness and convergence properties. The method is also successfully applied to real experimental data of a plane frame structure.     

Every stabilizing dead-time controller has an observer-predictor-based structure

This paper considers the stabilization problem for linear time-invariant systems with a single delay h in the feedback loop. Two parametrizations of all stabilizing controllers are derived in terms of a state-space realization of the rational part of the plant. These parametrizations have simple structures and clear interpretations: the first is in the form of a generalized dead-time compensator (Smith predictor) and the second is in the observer-predictor form. Applications of the proposed parametrization to the H² optimization and the robust stabilization problems are discussed.     

Min-max control of constrained uncertain discrete-time linear systems

For discrete-time uncertain linear systems with constraints on inputs and states, we develop an approach to determine state feedback controllers based on a min-max control formulation. Robustness is achieved against additive norm-bounded input disturbances and/or polyhedral parametric uncertainties in the state-space matrices. We show that the finite-horizon robust optimal control law is a continuous piecewise affine function of the state vector and can be calculated by solving a sequence of multiparametric linear programs. When the optimal control law is implemented in a receding horizon scheme, only a piecewise affine function needs to be evaluated on line at each time step. The technique computes the robust optimal feedback controller for a rather general class of systems with modest computational effort without needing to resort to gridding of the state-space.     

State-space realizations of linear differential-algebraic-equationsystems with control-dependent state space

This note addresses the derivation of state-space realizations for the feedback control of linear, high-index differential-algebraic-equation systems that are not controllable at infinity. In particular, a class of systems is considered for which the underlying algebraic constraints involve the control inputs, and thus a state-space realization cannot be derived independently of the feedback controller. The proposed methodology involves the design of a dynamic state feedback compensator such that the underlying algebraic constraints in the resulting modified system are independent of the new inputs. A state-space realization of the feedback-modified system is then derived that can be used as the basis for controller synthesis     

On unique representations of certain dynamical systems produced by continuous-time recurrent neural networks

This article extends previous mathematical studies on elucidating the redundancy for describing functions by feedforward neural networks (FNNs) to the elucidation of redundancy for describing dynamical systems (DSs) by continuous-time recurrent neural networks (RNNs). In order to approximate a DS on R(n) using an RNN with n visible units, an n-dimensional affine neural dynamical system (A-NDS) can be used as the DS actually produced by the above RNN under an affine map from its visible state-space R(n) to its hidden state-space. Therefore, we consider the problem of clarifying the redundancy for describing A-NDSs by RNNs and affine maps. We clarify to what extent a pair of an RNN and an affine map is uniquely determined by its corresponding A-NDS and also give a nonredundant sufficient search set for the DS approximation problem based on A-NDS.     

State-space realizations for output feedback control of linear differential-algebraic-equation systems

This work addresses the derivation of state-space realizations for the output feedback control of linear, high-index differential-algebraic-equation systems that are not controllable at infinity and for which the control inputs appear explicitly in the underlying algebraic constraints. The constrained state space of such systems depends on the control inputs, and thus, a state-space realization cannot be derived independently of the controller design. Motivated by this, initially a dynamic output feedback compensator is designed that yields a modified system for which the algebraic constraints are independent of the new control inputs. For this feedback modified system, a state-space realization is then derived which can be used for output feedback controller synthesis.     

RNGSSELIB: Program library for random number generation, SSE2 realization

The library RNGSSELIB for random number generators (RNGs) based upon the SSE2 command set is presented. The library contains realization of a number of modern and most reliable generators. Usage of SSE2 command set allows to substantially improve performance of the generators. Three new RNG realizations are also constructed. We present detailed analysis of the speed depending on compiler usage and associated optimization level, as well as results of extensive statistical testing for all generators using available test packages. Fast SSE implementations produce exactly the same output sequence as the original algorithms.

Program summary

Program title: RNGSSELIBCatalogue identifier: AEIT_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEIT_v1_0.htmlProgram obtainable from: CPC Program Library, Queen?s University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 4177No. of bytes in distributed program, including test data, etc.: 21 228Distribution format: tar.gzProgramming language: C.Computer: PC.Operating system: UNIX, Windows.RAM: 1 MbytesClassification: 4.13.Nature of problem: Any calculation requiring uniform pseudorandom number generator, in particular, Monte Carlo calculations.Solution method: The library contains realization of a number of modern and reliable generators: mt19937, mrg32k3a and lfsr113. Also new realizations for the method based on parallel evolution of an ensemble of dynamical systems are constructed: GM19, GM31 and GM61. The library contains both usual realizations and realizations based on SSE command set. Usage of SSE commands allows the performance of all generators to be substantially improved.Restrictions: For SSE realizations of the generators, Intel or AMD CPU supporting SSE2 command set is required. In order to use the realization lfsr113sse, CPU must support SSE4 command set.Running time: Running time is of the order of 20 sec for generating 10⁹ pseudorandom numbers with a PC based on Intel Core i7-940 CPU. Running time is analysed in detail in Section 5 of the paper.     

Robustness analysis for linear dynamical systems with linearlycorrelated parametric uncertainties

The authors propose an approach for robust pole location analysis of linear dynamical systems with parametric uncertainties. Linear control systems with characteristic polynomials whose coefficients are affine in a vector of uncertain physical parameters are considered. A design region in complex plane for system pole placement and a nominal parameter vector generating a characteristic polynomial with roots in that region are given. The proposed method allows the computation of maximal domains bounded by linear inequalities and centered at the nominal point in system parameter space, preserving system poles in the given region. The solution of this problem is shown to also solve the problem of testing robot location of a given polytope of polynomials in parameter space. It is proved that for stability problems for continuous-time systems with independent perturbations on polynomial coefficients, this method generates the four extreme Kharitonov polynomials     

Identification and data-driven model reduction of state-space representations of lossless and dissipative systems from noise-free data

We illustrate procedures to identify a state-space representation of a lossless or dissipative system from a given noise-free trajectory; important special cases are passive systems and bounded-real systems. Computing a rank-revealing factorization of a Gramian-like matrix constructed from the data, a state sequence can be obtained; the state-space equations are then computed by solving a system of linear equations. This idea is also applied to perform model reduction by obtaining a balanced realization directly from data and truncating it to obtain a reduced-order model.     

eHDECAY: An implementation of the Higgs effective Lagrangian into HDECAY

We present eHDECAY, a modified version of the program HDECAY which includes the full list of leading bosonic operators of the Higgs effective Lagrangian with a linear or non-linear realization of the electroweak symmetry and implements two benchmark composite Higgs models.     

Minimum phase robustness for uncertain state-space systems

The minimum phase robustness of an uncertain state-space system with affine parametric uncertainties in the state-space matrices is studied. A tolerable margin in terms of the structured singular value is given for uncertain parameters to guarantee the minimum phase property of the system. Based on the linear fractional transformation methodology, the matrix sizes involved in the computation of structured singular value are reduced significantly to improve computational burden. The approach can be applied to the proper or strictly proper linear uncertain systems.     

On affine state-space neural networks for system identification: Global stability conditions and complexity management

A nonlinear black box structure represented by an affine three-layered state-space neural network with exogenous inputs is considered for nonlinear systems identification. Global stability conditions are derived based on Lyapunov's stability theory and on the contraction mapping theorem. The problem of structural complexity is also addressed by defining an upper bound for the number of hidden layer neurons expressed as the cardinality of the dominant singular values of the oblique subspace projection of data driven Hankel matrices. Rank degeneracy stemming from nonlinearities, colored noise or finite data sets are dealt with either in terms of the sensitivity of singular values or by comparing the relevance of including additional coordinates in a given realization. Two examples are provided for illustrating the feasibility of derived global stability condition and the proposed approach for delaying with the complexity problem.     

Conjugation and H ∞ control of discrete-time systems

We establish conjugation notion in discrete-time systems, first introduced into the H ^∞ control theory of continuous-time systems by Kimura (1989). In discrete-time systems, conjugation is a very elementary operation on rational transfer functions that replaces some of their poles by their reflections with respect to the unit circle. With the aid of J-lossless conjugation conjugation by a J.lossless system), it is shown that the parametrization of sub-optimal solutions of H ^∞ model-matching problems is reduced to a Lyapunov-type equation. The parametrization of all solutions is given in an extremely simple way. It is further proved that the J-lossless conjugation of the H ^∞ model-matching problem is a natural state-space representation of classical interpolation in discrete-time systems.     

On a generalization of the Youla–Ku era parametrization. Part I: the fractional ideal approach to SISO systems

In this paper, we show how to use the theory of fractional ideals in order to study the fractional representation approach to analysis and synthesis problems for SISO systems. Within this mathematical framework, we give necessary and sufficient conditions so that a plant is internally/strongly/bistably stabilizable or admits a (weak) coprime factorization. Moreover, we show how to generalize the Youla–Ku era parametrization of the stabilizing controllers to any stabilizable plant which does not necessarily admit a coprime factorization. This parametrization is generally affine in two free parameters.     

Robust end-point optimizing feedback for nonlinear dynamic processes

The paper suggests two novel approaches to the synthesis of robust end-point optimizing feedback for nonlinear dynamic processes. Classically, end-point optimization is performed only for the nominal process model using optimal control methods, and the question of performance robustness to disturbances and model-plant mismatch remains unaddressed. The present contribution addresses the end-point optimization problem for nonlinear affine systems with fixed final time through robust optimal feedback methods. In the first approach, a nonlinear state feedback is derived that robustly optimizes the final process state. This solution is obtained through series expansion of the Hamilton-Jacobi-Bellman PDE with an active opponent disturbance. As reliable measurements or estimates of all states may not always be available, the second approach also robustly optimizes the process end-point, but uses output rather than state information. This direct use of measurement information is preferred since the choice of a state estimator for robust state feedback is non-trivial even when the observability issue is addressed. A linear time-variant output corrector is obtained by feedback parametrization and numerical optimization of a nonlinear H _∞ cost functional. A number of possible variations and alternatives to both approaches are also discussed. As model-plant mismatch is particularly common with chemical batch processes, the suitability of the robust optimizing feedback is demonstrated on a semi-batch reactor simulation example, where robustness to several realistic mismatches is investigated and the results are compared against those for the optimal open-loop policy and the optimal feedback designed for the nominal model.