Solving time-invariant differential matrix Riccati equations using GPGPU computing

Differential matrix Riccati equations (DMREs) enable to model many physical systems appearing in different branches of science, in some cases, involving very large problem sizes. In this paper, we propose an adaptive algorithm for time-invariant DMREs that uses a piecewise-linearized approach based on the Padé approximation of the matrix exponential. The algorithm designed is based upon intensive use of matrix products and linear system solutions so we can seize the large computational capability that modern graphics processing units (GPUs) have on these types of operations using CUBLAS and CULATOOLS libraries (general purpose GPU), which are efficient implementations of BLAS and LAPACK libraries, respectively, for NVIDIA \(\copyright \) GPUs. A thorough analysis showed that some parts of the algorithm proposed can be carried out in parallel, thus allowing to leverage the two GPUs available in many current compute nodes. Besides, our algorithm can be used by any interested researcher through a friendly MATLAB \(\copyright \) interface.     

Optimal robot‐environment interaction using inverse differential Riccati equation

An optimal robot‐environment interaction is designed by transforming an environment model into an optimal control problem. In the optimal control, the inverse differential Riccati equation is introduced as a fixed‐end‐point closed‐loop optimal control over a specific time interval. Then, the environment model, including interaction force, is formulated in a state equation, and the optimal trajectory is determined by minimizing a cost function. Position control is proposed, and the stability of the closed‐loop system is investigated using the Lyapunov direct method. Finally, theoretical developments are verified through numerical simulation.     

Time-varying Riccati differential equations with known analyticsolutions

This note presents several examples of large-scale time-varying stiff as well as nonstiff Riccati differential equations (RDEs) with known analytic solutions. These examples are useful for testing the accuracy and efficiency of algorithms for solving such equations. Analytic expressions of the eigenvalues of the solutions of the RDEs are also found. Eigenvectors of some of the solutions are given     

The asymptotic behavior of constant-coefficient Riccati differential equations

A simple and self-contained proof is given of a general theorem on the convergence of a constant coefficient Riccati differential equation to a unique limiting value. In particular our result, which includes (strictly) previous results, does not require any analysis of the algebraic Riccati equation.     

Speeding up solving of differential matrix Riccati equations using GPGPU computing and MATLAB

In this work, we developed a parallel algorithm to speed up the resolution of differential matrix Riccati equations using a backward differentiation formula algorithm based on a fixed‐point method. The role and use of differential matrix Riccati equations is especially important in several applications such as optimal control, filtering, and estimation. In some cases, the problem could be large, and it is interesting to speed it up as much as possible. Recently, modern graphic processing units (GPUs) have been used as a way to improve performance. In this paper, we used an approach based on general‐purpose computing on graphics processing units. We used NVIDIA © GPUs with unified architecture. To do this, a special version of basic linear algebra subprograms for GPUs, called CUBLAS, and a package (three different packages were studied) to solve linear systems using GPUs have been used. Moreover, we developed a MATLAB © toolkit to use our implementation from MATLAB in such a way that if the user has a graphic card, the performance of the implementation is improved. If the user does not have such a card, the algorithm can also be run using the machine CPU. Experimental results on a NVIDIA Quadro FX 5800 are shown. Copyright © 2011 John Wiley & Sons, Ltd.     

Riccati equations for second order spatially invariant partial differential systems

Recently, the class of spatially invariant systems was introduced with motivating examples of partial differential equations on an infinite domain. For these it was shown that by taking Fourier transforms, one obtains infinitely many finite-dimensional systems with a scalar parameter. The idea is that, for the LQR controller design for these systems, one can solve the parameterized LQR-Riccati equation pointwise. While for simple first order systems like the heat equations this approach works, for second order systems like wave or beam equations it is easy to construct examples for which this approach fails. Here we give a correct formulation for second order partial differential systems including wave and beam type equations.     

Constructing Riccati differential equations with known analyticsolutions for numerical experiments

Three examples of symmetric and nonsymmetric Riccati differential equations (RDEs) whose analytic solutions are known and whose sizes are variable and can be large are presented. The eigenvalues and eigenvectors of the Riccati solutions in some of the examples are also given. The numerical examples have proved to be useful for testing the accuracy and efficiency of algorithms for solving large-scale RDEs     

SDP-based approximation of stabilising solutions for periodic matrix Riccati differential equations

Numerically finding stabilising feedback control laws for linear systems of periodic differential equations is a nontrivial task with no known reliable solutions. The most successful method requires solving matrix differential Riccati equations with periodic coefficients. All previously proposed techniques for solving such equations involve numerical integration of unstable differential equations and consequently fail whenever the period is too large or the coefficients vary too much. Here, a new method for numerical computation of stabilising solutions for matrix differential Riccati equations with periodic coefficients is proposed. Our approach does not involve numerical solution of any differential equations. The approximation for a stabilising solution is found in the form of a trigonometric polynomial, matrix coefficients of which are found solving a specially constructed finite-dimensional semidefinite programming (SDP) problem. This problem is obtained using maximality property of the stabilising solution of the Riccati equation for the associated Riccati inequality and sampling technique. Our previously published numerical comparisons with other methods shows that for a class of problems only this technique provides a working solution. Asymptotic convergence of the computed approximations to the stabilising solution is proved below under the assumption that certain combinations of the key parameters are sufficiently large. Although the rate of convergence is not analysed, it appeared to be exponential in our numerical studies.     

A new fundamental solution for differential Riccati equations arising in control

The matrix differential Riccati equation (DRE) is ubiquitous in control and systems theory. The presence of the quadratic term implies that a simple linear-systems fundamental solution does not exist. Of course it is well-known that the Bernoulli substitution may be applied to obtain a linear system of doubled size. Here, however, tools from max-plus analysis and semiconvex duality are brought to bear on the DRE. We consider the DRE as a finite-dimensional solution to a deterministic linear/quadratic control problem. Taking the semiconvex dual of the associated semigroup, one obtains the solution operator as a max-plus integral operator with quadratic kernel. The kernel is equivalently represented as a matrix. Using the semigroup property of the dual operator, one obtains a matrix operation whereby the kernel matrix propagates as a semigroup. The propagation forward is through some simple matrix operations. This time-indexed family of matrices forms a new fundamental solution for the DRE. Solution for any initial condition is obtained by a few matrix operations on the fundamental solution and the initial condition. In analogy with standard-algebra linear systems, the fundamental solution can be viewed as an exponential form over a certain idempotent semiring. This fundamental solution has a particularly nice control interpretation, and might lead to improved DRE solution speeds.     

Optimal design and noise consideration of micromachined vibratingrate gyroscope with modulated integrative differential optical sensing

A novel design of a micromachined vibrating rate gyroscope is presented. The rate gyroscope consists of a suspended proof mass which is attached by indium bumps to a CMOS chip. The proof mass is excited to vibration by electrostatic force. The displacements due to rate are sensed optically, using CMOS-integrated photodiodes and analog electronics. System considerations, including the mechanical behaviour, optical sensing, electronics, and noise sources of the rate gyroscope, are discussed. An expression for the noise equivalent rate (NER) of the system is obtained in order to derive an optimal design approach for the rate gyroscope. Optimal design and simulations of a case study of a rate gyroscope are presented. The device shows the ability of sensing 1 deg/h even at moderate quality factors of the order of 5000 and low-excitation voltages of 2.25 V     

Superposition laws for solutions of differential matrix Riccati equations arising in control theory

Representation formulas are given for the general solution of theN times Nmatrix Riccati equationdot{W} = A + WB + CW + WDWusingnknown solutions, withn = 1, ..., 5(n-representations). The 5- representation is a superposition formula, in that it expresses the general solution explicitly as a function of five particular solutions and N²arbitrary constants (N geq 2), using no further information. The representation formulas can be used in numerical calculations. The 4- and 5- representations are specially useful when a solutionW(t)has a singularity for some finitet = t_{0}. They also clarify the properties of the solution space: the matrix elements ofW(t)are meromorphic functions ofthaving simple poles as the only possible singularities. The relation between the representation formulas and previously known results is discussed.     

Optimal control design for time-varying catalytic reactors: a Riccati equation-based approach

The linear quadratic (LQ) optimal control problem is studied for a partial differential equation model of a time-varying catalytic reactor. First, the dynamical properties of the linearised model are studied. Next, an LQ-control feedback is computed by using the corresponding operator Riccati differential equation, whose solution can be obtained via a related matrix Riccati partial differential equation. Finally, the designed controller is applied to the non-linear reactor system and tested numerically.     

Optimal design for 2D wave equations

In this paper We consider a problem of optimal design in 2D for the wave equation with Dirichlet boundary conditions. We introduce a finite element discrete version of this problem in which the domains under consideration are polygons defined on the numerical mesh. We prove that, as the mesh size tends to zero, any limit, in the sense of the complementary-Hausdorff convergence, of discrete optimal shapes is an optimal domain for the continuous optimal design problem. We work in the functional and geometric setting introduced by V. ?veràk in which the domains under consideration are assumed to have an a priori limited number of holes. We present in detail a numerical algorithm and show the efficiency of the method through various numerical experiments.     

Cyclic animation using partial differential equations

This work presents an efficient and fast method for achieving cyclic animation using partial differential equations (PDEs). The boundary-value nature associated with elliptic PDEs offers a fast analytic solution technique for setting up a framework for this type of animation. The surface of a given character is thus created from a set of pre-determined curves, which are used as boundary conditions so that a number of PDEs can be solved. Two different approaches to cyclic animation are presented here. The first of these approaches consists of attaching the set of curves to a skeletal system, which is responsible for holding the animation for cyclic motions through a set mathematical expressions. The second approach exploits the spine associated with the analytic solution of the PDE as a driving mechanism to achieve cyclic animation. The spine is also manipulated mathematically. In the interest of illustrating both approaches, the first one has been implemented within a framework related to cyclic motions inherent to human-like characters. Spine-based animation is illustrated by modelling the undulatory movement observed in fish when swimming. The proposed method is fast and accurate. Additionally, the animation can be either used in the PDE-based surface representation of the model or transferred to the original mesh model by means of a point to point map. Thus, the user is offered with the choice of using either of these two animation representations of the same object, the selection depends on the computing resources such as storage and memory capacity associated with each particular application.     

Optimal experiment design for regression polynomial models identification

In this paper the problem of optimal experimental design for parameter identification of static non-linear blocks is addressed. Non-linearities are assumed to be polynomial and represented according to the Vandermonde base. The optimality problem is formulated in a set membership context and the cost functions to be minimized are the worst case parameter uncertainties. Closed form optimal input sequences are derived when the input u is allowed to vary on a given interval [ u a, u b ]. Since optimal input sequences are, in general, not invariant to base changes, results and criteria for representing polymomials with different bases, still preserving the optimal set of input levels derived from the Vandermonde parameterization, are introduced as well. Finally numerical results are reported showing the effectiveness of using optimal input sequences especially when identifying some block described dynamic models that include in their structure static non-linearities (such as Hammerstein and LPV models). In such cases the improvement achieved in the confidence of the estimates can add up to a factor of several hundreds with respect to the case of random generated inputs.