An observer-based set-point controller for robot manipulators with flexible joints

We present a globally asymptotically stable controller for point-to-point regulation of robot manipulators with flexible joints that uses only position measurement on the motor side. Existing asymptotically stable schemes for the set point regulation problem without velocity measurement address only the rigid robot case. Furthermore, these solutions ensure only local stability provided some bounds on the dynamic part of the robot model are known. Also, they require the injection of high gains into the loop to enlarge the equilibrium domain of attraction. In contrast, our solution is global, applies for robots with flexible joints and assumes only that the gravity forces are known. The underlying rationale of the design is to ‘shape’ the potential energy of the closed loop system so that it has an absolute minimum at the desired equilibrium, and add the required dampingto achieve asymptotic stability. This is attained by adding a (linear) observer that converges to the position required to compensate the gravity forces and injects the damping, and a ‘spring-like’ effect between the observer and the robot that ‘pulls’ the robot to the desired target. This approach to observer-based controller design differs from the classical certainly equivalent approach and effectively exploits the dynamic properties of the physical system.     

A spreadsheet program to optimize kinetic parameters of organic matter from laboratory pyrolysis data

Transformation of sedimentary organic matter (OM) to hydrocarbons is best modeled by assuming the total reaction suite consists of parallel degradations of ‘i’ hypothetical components following the Arrhenius equation and first order kinetics. A kerogen can be defined by characterizing each constituent component by its activation energy (Ei) their initial potentials (Xios) and a single frequency factor (A). We present a user friendly Lotus 1-2-3 program to determine A, Ei and Xio distribution of OMs using a multiple linear regression utility and programmed macros. Rock Eval (RE) S₂ curves of three heating rates are required. Equally spaced time/temperature and peak height data for S₂ curves of ‘n’ temperature steps in increasing order of heating rates are the inputs for the program. The fraction of hydrocarbon generated (f) from 19 hypothetical components of Ei 30, 32,34…78 Kcal/mole for ‘n’ temperature/time steps, by using frequency factor (A) value and assuming Xios=1, are calculated and set up in a ‘n×19’ matrix (matrix M). The fraction of total hydrocarbon generated (f) at ‘n’ temperature steps, obtained from the observed peak heights, are set up in a ‘n×1’ matrix (matrix L). Matrix M is suitably reduced by the program to ‘n×k’ matrix (matrix N) where ‘k’ is a variable, facilitating matrix inversion. Regressing matrix N against matrix L by the program, gives the Xios for ‘k’ Ei components along with a standard error (ERR) of Y estimates and R². Xios and A are then optimized iteratively by varying A values and selecting the solution associated with the lowest ERR value. Results of applying the program on data sets of two widely different types of samples from Indian basins are shown. They match the results obtained from the more sophisticated proprietary software.     

H∞-state-feedback control of linear systems with small state delay

In this note, a H_∞ controller for systems with state delay is presented. For a prechosen γ the controller is obtained by solving Riccati partial differential inequalities (RPDIs). For small time delays, an asymptotic approximation of the controller is achieved by expanding the solution in powers of the delay. It is shown that a higher-order accuracy controller improves the performance. An example is brought where the results of the new method provide a satisfactory solution for a delay length comparable with the system ‘bandwidth’. The performance of the system under the zero-order accuracy controller, which corresponds to systems without delay, is studied. Explicit formula for the guaranteed performance level is obtained for the delay lengths that preserve the internal stability of the system.     

A spectral characterization of H-optimal feedback performance and its efficient computation

We study the spectral properties of a ‘Toeplitz⁺ Hankel’ operator which arises in the context of the mixed-sensitivity H^∞-optimization problem and whose largest eigenvalue characterizes the optimal achievable performance ε₀. The existence of such an operator was first shown by Verma and Jonckheere [26], who also'noted the potential numerical advantage of computing eo through its eigenvalue characterization rather than through the ε-iteration. Here, we investigate this operator in detail, with the objective of efficiency computing its spectrum. We define an ‘adjoint’ linear-quadratic problem that involves the same ‘Toeplitz⁺ Hankel’ operator, as shown by Jonckheere and Silverman [13–16]. Consequently, a finite polynomial algorithm allows ε₀ to be characterized as simply as the largest root of a polynomial. Finally, a computationally more attractive state space algorithm emerges from the H^t8/LQ relationship. This algorithm yields a very good accuracy evaluation of the performance ε₀ by solving just one algebraic Riccati equation. Thorough exploitation of this algorithm results in a drastic computation reduction with respect to the standard e-iteration.     

The parallel projection operators of a nonlinear feedback system

This paper defines and studies a pair of nonlinear parallel projection operators associated with a nonlinear feedback system. These operators have been seen to play an important role in the robustness and design of linear systems especially in the theory of the gap metric, the use of weighted gaps in control system design and Glover-McFarlane loop-shaping. We show that the input-output L₂-stability of a feedback system amounts to a ‘coordinatization’ of the input and output spaces, which is also equivalent to the existence of a pair of nonlinear parallel projection operators onto the graph of the plant and the inverse graph of the controller respectively. These projections are shown to have equal norms whenever one of the feedback elements is linear. A bound on this norm is given in the case of passive systems with unity negative feedback.     

Some counterexamples in robust stability theory

This note deals with the problem of determining if a linear system whose characteristics polynomial depends multilinearly on n independent uncertain real parameters Δ_i, I = 1,…,n, is robustly stable. It is shown by example that a polynomial in n variables may have a unique real root, and that this observation disposes of several natural conjectures in robust stability theory. In particular, we show that, in a certain sense, there are no ‘edge’ or ‘m-dimensional face’ Kharitonov-like theorems for the general multilinear case. The result holds even when restricted to that subset of multilinear functions which can be written in the form f(Δ₁,…, Δ_n) = det(I + diag(Δ₁,…,Δ_n)M) for some complex matrix M.     

The order of any stabilizing regulator is sufficient a priori information for adaptive stabilization

Let the unknown linear system , be given, together with the a priori information that for a known, nonnegative integer l, there is a (nonadaptive) regulator of order l which stabilizes the system. It is shown that this suffices as a priori information for an adaptive stabilizing controller. An example of such an algorithm is given. This yields a continuous regulator, which does not utilize probing signals. It is based on a dense search through parameter space, and does not utilize high-gain properties, as opposed to the ‘universal regulators’ proposed before [3–6]. In the absence of information of such an l, it is shown how to modify the algorithm to search over the regulator structures, i.e. the controller's dimension.     

Optimal diagonal scaling for infinity-norm optimization

A solution is presented for the previously unsolved diagonally scaled multivariable infinity-norm optimization problem of minimizing D(s)(A(s) + Ψ(s) X(s))D⁻¹(s)_∞ over the set of stable minimum-phase diagonal D(s) and stable X(s). This problem is of central importance in the synthesis of feedback control laws for robust stability and insensitivity in the presence of ‘structured’ plant uncertainty. The result facilitates the design of feedback controllers which optimize the ‘excess stability margin’ [3] (or, equivalently, the ‘structured singular value μ’ [4]) of diagonally perturbed feedback systems.     

Design of Digital State <Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript>-Controllers of the MIMO Systems with Prescribed Precision

For the linear MIMO plants subject to polyharmonic perturbations, a method of designing digital state controllers was developed with regard for the requirements on limitedness of the control actions in the stable state as well as on limitedness of the controlled variables. A notion of radius of the stable state of a closed-loop system was introduced, and the problem of using the full state vector to design a discrete controller providing the desired radius was formulated. Its necessary and sufficient solvability conditions were obtained using the procedure of H _∞-optimization by selecting the weight matrices of the discrete Lur’e-Riccati equation.__________Translated from Avtomatika i Telemekhanika, No. 8, 2005, pp. 46–51.Original Russian Text Copyright © 2005 by Chestnov.     

On the boundedness of the set of stabilizing controllers

This paper shows that the set of rational, strictly proper, robustly stabilizing controllers for single‐input single‐output linear‐time invariant plants will form a bounded (can even be empty) set in the controller parameter space if and only if the order of the stabilizing controller cannot be reduced any further; if the set of proper stabilizing controllers of order r is not empty and the set of strictly proper controllers of order r is bounded, then r is the minimal order of stabilization. The paper also extends this result to characterize the set of controllers that guarantee some pre‐specified performance specifications. In particular, it is shown here that the minimal order of a controller that guarantees specified performance is l iff (1) there is a controller of order l guaranteeing the specified performance and (2) the set of strictly proper, robustly stabilizing controllers of order l and guaranteeing the performance is bounded. Moreover, if the order of the controller is increased, the set of higher‐order controllers which satisfies the specified performance will necessarily be unbounded. This characterization is provided for performance specifications, such as gain margin and robust stability, which require a one‐parameter family of real polynomials to be Hurwitz, where the parameter is in a closed interval. Other performance specifications, such as phase margin and ℋ︁_∞ norm, can be reduced to the problem of determining a set of stabilizing controllers that renders a family of complex polynomials Hurwitz. The characterization of the set of controllers for the stabilization of complex polynomials is provided and is used to show the boundedness properties for the set of controllers that guarantee a given phase margin or an upper bound on the ℋ︁_∞ norm. Copyright © 2007 John Wiley & Sons, Ltd.     

Hurwitz testing sets for parallel polytopes of polynomials

In considering robustness of linear systems with uncertain paramenters, one is lead to consider simultaneous stability of families of polynomials. Efficient Hurwitz stability tests for polytopes of polynomials have earlier been developed using evaluations on the imaginary axis. This paper gives a stability criterion for parallel polytopes in terms of Hurwitz stability of a number of corners and edges. The ‘testing set’ of edges and corners depends entirely on the edge directions of the polytope, hence the results are particularly applicable in simultaneous analysis of several polytopes with equal edge directions.It follows as a consequence, that Kharitonov's four polynomial test for independent coefficient uncertainties is replaced by a test of 2q polynomials, when the stability region is a sector Ω = { e^iv | > 0, rπ/q < | v | ≤ π } and r/q is a rational number.     

Estimation of model quality

This paper provides an introduction to recent work on the problem of quantifying errors in the estimation of models for dynamic systems. This is a very large field. We therefore concentrate on approaches that have been motivated by the need for reliable models for control system design. This will involve a discussion of efforts that go under the titles of ‘estimation in ’, ‘worst-case estimation’, ‘estimation in ℓ₁’ and ‘stochastic embedding of undermodelling’. A central theme of this survey is to examine these new methods with reference to the classic bias/variance tradeoff in model structure selection.     

On bounds for the existence of a bistable controller

By we shall denote the set of polynomials nonzero on the closed disk {z: |z|r}. The problem to be studied is given as follows.Avoidance Problem. Given polynomials A and B, nonnegative integers n, m, and positive numbers r₁, r₂, r₃, determine if there exist , with degrees n and m, respectively, such that AP+BQ is in .It is shown that the Avoidance Problem for r₁=r₂=r₃=1 is equivalent to the well-known Bistable Stabilization Problem for a large class of control systems.Also, a general class of optimization problems for solving the Avoidance Problem is introduced. The class of optimization problems includes problems with sets more general than disks . In addition, an algorithm is proposed for solving these optimization problems, and numerical experience is reported.As an application, bistable controllers for the plant are shown to exist for values of β as low as 0.0246, thus improving on previously reported results by Blondel et al. for this difficult case.Furthermore, it is shown how in some cases it is possible to find optimal controllers by using a combination of numerical and algebraic techniques, and a variant of the algorithm already introduced. As an example, a low order controller is found for for the case when .This method could be applied to other problems where the plant depends on a parameter.     

A numerical method for deadbeat control of generalized state-space systems

In this paper we give a new numerical method for constructing a rank m correction BF to an n × n matrix A, such that the generalized eigenvalues of λE−(A+BF) are all at λ = 0. In the control literature, this problem is known as ‘deadbeat control’ of a generalized state-space system Ex_i+1 = Ax_i + Bu_i, whereby the matrix F is the ‘feedback matrix’ to be constructed.     

Certainty equivalence implies detectability

It is shown that any stabilizing, certainty equivalence control used within an adaptive control system, causes the familiar interconnection of a controlled process and associated output estimator to be detectable through the estimator’s output error e_p, for every frozen value of the index or parameter vector p upon which both the estimator and controller dynamics depend. The fact that certainty equivalence implies detectability has been known for some time – this has been shown to be so whenever the process model is linear and the controller and estimator models are also linear for every frozen value of p. In this paper, use is made of recently introduced concepts of input-to-state stability and detectability for nonlinear systems to prove that the same implication is valid in a more general, nonlinear setting.     

Robust control design for best mean performance over an uncertain-parameters box

This paper deals with robust, polytopic, probabilistic H_∞ analysis and state-feedback synthesis of linear systems and focuses on the performance distribution over the uncertainty region (rather than on the performance bound). The proposed approach allows different disturbance attenuation levels (DALs) at the vertices of the uncertainty polytope. It is shown that the mean disturbance attenuation level (DAL) over an uncertain parameters-box is the average of the DALs at the vertices, if each parameter has an independent, symmetrical, centered probability density function. In such a (most common) case, the mean DAL over the uncertain parameters-box can be optimized by minimizing the sum of the DALs at the vertices. The standard deviation of the DAL over the uncertain parameters-box is also addressed, and a method to minimize this standard deviation is shown. A new robust H_∞ state-feedback synthesis theorem is given; it is based on a recent, most efficient analysis method and applies the proposed multiple-vertex-DALs approach. A state-feedback design example utilizing the latter theorem shows that a control design which minimizes the sum of the vertex-DALs leads to a better actual closed-loop performance than a similar design which minimizes only the bound of the DAL over the uncertainty polytope. The comparison is based on the statistics of a population of closed-loop ‘point-wise’ H_∞-norms created by a Monte-Carlo mechanism.     

Robust control of a class of uncertain nonlinear systems

This paper considers the robust control of a class of nonlinear systems with real time-varying parameter uncertainty. Interest is focused on the design of linear dynamic output feedback control and two problems are addressed. The first one is the robust stabilization and the other is the problem of robust performance in an H_∞ sense. A technique is proposed for designing stabilizing controllers for both problems by converting them into ‘scaled’ H_∞ control problems which do not involve parameter uncertainty.     

Continuity properties of LQG optimal controllers

It is shown that the LQG optimal controller is a continuous function of the plant. The result is proved for a class of plants which contains the class of strictly proper finite-dimensional plants. The topology employed is the one generated by convergence of the closed-loop transfer functions in an induced _∞ sense. This topology is slightly stronger than the usual ₂ gap metric convergence on transfer functions.     

An ethnographic, action-based approach to human experience in virtual environments

This paper addresses a sensitive issue, of presence experienced by people interacting with a virtual environment (VE). Understanding ‘presence’, both theoretically and empirically, is important for designers interested in building effective computer-mediated environments for learning and work activities. The concept of presence has been treated mostly as a state of mind, to be investigated through ‘objective’ and ‘subjective’ measurement devices. The authors propose to add a different approach, which can address presence as an action-based process. This approach considers presence as the ongoing result of the actions performed in an environment and the local and cultural resources deployed by actors. In this sense, ‘presence’ can be captured by monitoring the sequence of participants’ actions and the aspects of the environment that are involved in this process; discourse/interaction analysis represents a fitting method for this goal. Sequences of interaction with a virtual library are used to illustrate some core aspects of an ethnographic, action-based approach to presence, such as the action possibilities envisaged by participants, the configuration of the virtual objects, the norms that regulate the interaction, the resources that are imported in the VE. These aspects are a necessary step to understand users’ presence in the VE and to plan consequent interventions to ameliorate the design of the interface.