Decentralized robust control for multiconnected objects with structural uncertainty

We solve the problem of constructing a decentralized robust regulating system for a multiconnected control object that provides for compensating interconnections in local subsystems and also parametric and external bounded perturbations, compensating with precision δ if one does not measure the derivatives of the local subsystems output vector and in full if the derivatives are measured.     

Adaptive controller design and disturbance attenuation for SISO linear systems with noisy output measurements and partly measured disturbances

In this article, we present robust adaptive controller design for SISO linear systems with noisy output measurements and partly measured disturbances. Using the worst-case analysis approach, we formulate the robust adaptive control problem as a non-linear H ^∞-optimal control problem under imperfect state measurements and solve it using game theory. The design paradigm is the same as (Pan, Z. and Ba?ar, T., 1998, Adaptive Controller Design for Tracking and Disturbance Attenuation for SISO Linear Systems with Noisy Output Measurements, CSL report, Urbana, IL: University of Illinois at Urbana-Champaign) with the only difference being the treatment of the measured disturbances. The same result as (Pan and Ba?ar l998) is achieved. In addition, when the relative degrees from the measured disturbances to the output are no less than that from the control input, the controller designed achieves the zero disturbance attenuation level with respect to the measured disturbance inputs. The asymptotic tracking objective is achieved even if the measured disturbance is only uniformly bounded, without requiring it to be of finite energy. This strong robustness property is then illustrated by numerical examples.     

Weak admissibility does not imply admissibility for analytic semigroups

Two conjectures on admissible control operators by George Weiss are disproved in this paper. One conjecture says that an operator B defined on an infinite-dimensional Hilbert space U is an admissible control operator if for every element uU the vector Bu defines an admissible control operator. The other conjecture says that B is an admissible control operator if a certain resolvent estimate is satisfied. The examples given in this paper show that even for analytic semigroups the conjectures do not hold. In the last section we construct a semigroup example showing that the first estimate in the Hille–Yosida theorem is not sufficient to conclude boundedness of the semigroup.     

Reduced‐order design of high‐order sliding mode control system

To design an rth (r>2) order sliding mode control system, a sliding variable and its derivatives of up to (r ? 1) are in general required for the control implementation. This paper proposes a reduced‐order design algorithm using only the sliding variable and its derivatives of up to (r ? 2) as the extension of the second‐order asymptotic sliding mode control. For a linear time‐invariant continuous‐time system with disturbances, it is found that a high‐order sliding mode can be reached locally and asymptotically by a reduced‐order sliding mode control law if the sum of the system poles is less than the sum of the system zeros. The robust stability of the reduced‐order high‐order sliding mode control system, including the convergence to the high‐order sliding mode and the convergence to the origin is proved by two Lyapunov functions. Simulation results show the effectiveness of the proposed control algorithm. Copyright © 2010 John Wiley & Sons, Ltd.     

LMI‐based second‐order sliding set design using reduced order of derivatives

Sliding mode control design for systems with relative degree r requires a number r ? 1 of time‐derivatives of the system output, which usually leads to deterioration of the whole scheme; if the highest‐order derivative is spared, a better precision is ensured. This paper proposes a control algorithm that guarantees reaching a second‐order sliding manifold using only r ? 2 derivatives of the system output. This objective is achieved at the price of yielding finite‐time convergence while preserving the essential feature of insensitivity to matched disturbances. The results take full advantage of convex representations and linear matrix inequalities, whose formulation easily allows dealing with unmatched disturbances by convex optimization techniques already implemented in commercially available software. Simulation examples are included to show the effectiveness of the proposed approach. Copyright © 2014 John Wiley & Sons, Ltd.     

Flow regularity and optimality conditions with controls inL p

In this paper we study theC ^p regularity of the flow of a nonlinear nonautonomous control system with respect to control maps belonging toL ^p withp≥r. The results obtained are applied to get first- and second-order optimality conditions when the control space isL ^p . The problem which we consider is in the Mayer form and includes endpoint constraints. We present first-order necessary conditions for a wide class of control systems. Moreover, we show that the usual second-order sufficient conditions are effective only if the mapf that defines the control system is a polynomial of degree two in the control variable and the controls belong toL ².     

ADF95: Tool for automatic differentiation of a FORTRAN code designed for large numbers of independent variables

ADF95 is a tool to automatically calculate numerical first derivatives for any mathematical expression as a function of user defined independent variables. Accuracy of derivatives is achieved within machine precision. ADF95 may be applied to any FORTRAN 77/90/95 conforming code and requires minimal changes by the user. It provides a new derived data type that holds the value and derivatives and applies forward differencing by overloading all FORTRAN operators and intrinsic functions. An efficient indexing technique leads to a reduced memory usage and a substantially increased performance gain over other available tools with operator overloading. This gain is especially pronounced for sparse systems with large number of independent variables. A wide class of numerical simulations, e.g., those employing implicit solvers, can profit from ADF95.

Program summary

Title of program:ADF95Catalogue identifier: ADVIProgram summary URL:http://cpc.cs.qub.ac.uk/summaries/ADVIProgram obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandComputer for which the program is designed: all platforms with a FORTRAN 95 compilerProgramming language used:FORTRAN 95No. of lines in distributed program, including test data, etc.: 3103No. of bytes in distributed program, including test data, etc.: 9862Distribution format: tar.gzNature of problem: In many areas in the computational sciences first order partial derivatives for large and complex sets of equations are needed with machine precision accuracy. For example, any implicit or semi-implicit solver requires the computation of the Jacobian matrix, which contains the first derivatives with respect to the independent variables. ADF95 is a software module to facilitate the automatic computation of the first partial derivatives of any arbitrarily complex mathematical FORTRAN expression. The program exploits the sparsity inherited by many set of equations thereby enabling faster computations compared to alternate differentiation toolsSolution method: A class is constructed which applies the chain rule of differentiation to any FORTRAN expression, to compute the first derivatives by forward differencing. An efficient indexing technique leads to a reduced memory usage and a substantially increased performance gain when sparsity can be exploited. From a users point of view, only minimal changes to his/her original code are needed in order to compute the first derivatives of any expression in the codeRestrictions: Processor and memory hardware may restrict both the possible number of independent variables and the computation timeUnusual features:ADF95 can operate on user code that makes use of the array features introduced in FORTRAN 90. A convenient extraction subroutine for the Jacobian matrix is also providedRunning time: In many realistic cases, the evaluation of the first order derivatives of a mathematical expression is only six times slower compared to the evaluation of analytically derived and hard-coded expressions. The actual factor depends on the underlying set of equations for which derivatives are to be calculated, the number of independent variables, the sparsity and on the FORTRAN 95 compiler     

Global L2-gain design for a class of nonlinear systems

It is known that the so-called H_∞ control problem of a nonlinear system is locally solvable if the corresponding problem for the linearized system can be solved by linear feedback. In this paper we prove that this condition suffices to solve also a globalH_∞ control problem, for a fairly large class of nonlinear systems, if one is free to choose a state-dependent weight of the control input. Using a two-way (backward and forward) recursive induction argument, we simultaneously construct, starting from a solution of the Riccati algebraic equation, a global solution of the Hamilton–Jacobi–Isaacs partial differential equation arising in the nonlinear H_∞ control, as well as a state feedback control law that achieves global disturbance attenuation with internal stability for the nonlinear systems.     

Application of facial reduction to H ∞ state feedback control problem

One often encounters numerical difficulties in solving linear matrix inequality (LMI) problems obtained from H _∞ control problems. For semidefinite programming (SDP) relaxations for combinatorial problems, it is known that when either an SDP relaxation problem or its dual is not strongly feasible, one may encounter such numerical difficulties. We discuss necessary and sufficient conditions to be not strongly feasible for an LMI problem obtained from H _∞ state feedback control problems and its dual. Moreover, we interpret the conditions in terms of control theory. In this analysis, facial reduction, which was proposed by Borwein and Wolkowicz, plays an important role. We show that the dual of the LMI problem is not strongly feasible if and only if there exist invariant zeros in the closed left-half plane in the system, and present a remedy to remove the numerical difficulty with the null vectors associated with invariant zeros in the closed left-half plane. Numerical results show that the numerical stability is improved by applying it.     

Gain scheduling via linear fractional transformations

A linear finite-dimensional plant with rational state-space parameter dependence is controlled using a parameter-dependent controller. The parameters are known to take on values in a unit ball, and are known in real time. The goal of control is to stabilize the parameter-dependent closed-loop system, and provide disturbance/error attenuation as measured in induced l₂ norms. The approach taken uses the optimally scaled small-gain theorem, and solves the control synthesis problem by reformulating the existence conditions into a finite-dimensional convex optimization.     

Control of complex ecological-economic systems

The paper is devoted to developing methods for control of ecological-economic systems consisting of three hierarchically subordinate subjects of control. In describing the dynamics of a system state, equations in partial derivatives that are solved numerically according to a semi-implicit scheme of the finite-difference method are used. To achieve its main goal (maintenance of an ecological subsystem in stable state), the subject of control of the upper level applies different control methods. Methods of hierarchical control (motivation, enforcement, and persuasion) that differ in the direction of action (on the objective function or domain of admissible controls) are proposed; examples of their use are represented and a comparative analysis of the obtained results is made.     

Input–output approach to stability and -gain analysis of systems with time-varying delays

Stability and L₂ (l₂)-gain of linear (continuous-time and discrete-time) systems with uncertain bounded time-varying delays are analyzed under the assumption that the nominal delay values are not equal to zero. The delay derivatives (in the continuous-time) are not assumed to be less than q<1. An input–output approach is applied by introducing a new input–output model, which leads to effective frequency domain and time domain criteria. The new method significantly improves the existing results for delays with derivatives not greater than 1, which were treated in the past as fast-varying delays (without any constraints on the delay derivatives). New bounded real lemmas (BRLs) are derived for systems with state and objective vector delays and norm-bounded uncertainties. Numerical examples illustrate the efficiency of the new method.     

Generalized derivative and π-derivative for set-valued functions

In this paper we study the generalized derivative and the π-derivative for interval-valued functions. We show the connections between these derivatives. Some illustrative examples and applications to interval differential equations and fuzzy functions are presented.     

H ∞ sampled data control of systems with time-delays

Sampled-data H _∞ control of linear systems with constant state, control and measurement delays is considered. The sampling of the controlled input and of the measured output is not assumed to be uniform. The system is modelled as a continuous-time one, where the controlled input and the measurement output have piecewise-continuous delays. The input–output approach to stability and L ₂-gain analysis is applied to the resulting system. The discretised Lyapunov functional method is extended to the case of multiple delays, where the Lyapunov functional is complete in one of the delays (in the state) and is simple in the other delays (those in the input and in the output), which are constant. Solutions to the state-feedback and the output-feedback H _∞ control problems are derived in terms of linear matrix inequalities (LMIs).     

LMI-based H ∞-optimal control with transients

In this article, the worst-case norm of the regulated output over all exogenous signals and initial states as a performance measure of the system is characterised in terms of linear matrix inequalities (LMIs). Optimal time-invariant state- and output-feedback controllers are synthesised as minimising this performance measure. The essential role in this synthesis plays a weighting matrix reflecting the relative importance of the uncertainty in the initial state contrary to the uncertainty in the exogenous signal. H _∞-optimal control with transients is shown to be actually a trade-off between H _∞-control, being optimal under unknown exogenous disturbances and zero initial state, and γ-control, being optimal under zero exogenous signal and unknown initial conditions, if and only if the weighting matrix satisfies a fundamental inequality. If this inequality is met, the performance measure is achieved and the explicit formulae for the worst-case disturbance and initial state are provided. If this inequality fails, the performance measure coincides with the H _∞-norm and the trade-off gets broken.     

An inequality governing nonlinear H∞ control

This note gives necessary and sufficient conditions for solving a reasonable version of the nonlinear H^∞ control problem. The most objectionable hypothesis is elegant and holds in the linear case, but every possibly may not be forced for nonlinear systems. What we discover in distinction to Isidori and Astolfi (1992) and Ball et al. (1993) is that the key formula is not a (nonlinear) Riccati partial differential inequality, but a much more complicated inequality mixing partial derivatives and an approximation theoretic construction called the best approximation operator. This Chebeshev-Riccati inequality when specialized to the linear case gives the famous solution to the H^∞ control problem found in Doyle et al. (1989). While complicated the Chebeshev-Riccati inequality is (modulo a considerable number of hypotheses behind it) a solution to the nonlinear H^∞ control problem. It should serve as a rational basis for discovering new formulas and compromises. We follow the conventions of Ball et al. (1993) and this note adds directly to that paper.     

Reduced-order controllers for the H∞ control problem with unstable invariant zeros

This paper addresses the existence and design methods of reduced-order controllers for the H_∞ control problem with unstable invariant zeros in the state-space realization of the transfer function matrix from the control input to the controlled error or from the exogenous input to the observation output, where the realization is induced from a stabilizable and detectable realization of the generalized plant. This paper presents a new controller degree bound for the H_∞ control problem in terms of the minimal rank of the system matrix pencils of these two transfer function matrices in the unstable region. When the unstable invariant zero exists, this paper shows that reduced-order controllers with orders strictly less than that of the generalized plant exist if the H_∞ control problem is solvable. Moreover, this paper shows that the computational problem of finding the controllers with the new degree bound is convex by providing two linear matrix inequality-based design methods (algorithms) for constructing the reduced-order controllers. The results developed in this paper are valid both for the continuous- and discrete-time H_∞ control problems.     

Differential flatness of two one-forms in arbitrary number of variables

Given a differentially flat system of ODEs, flat outputs that depend only on original variables but not on their derivatives are called zero-flat outputs and systems possessing such outputs are called zero-flat. In this paper we present a theory of zero-flatness for a system of two one-forms in arbitrary number of variables (t,x¹,…,x^N). Our approach splits the task of finding zero-flat outputs into two parts. First part involves solving for distributions that satisfy a set of algebraic conditions. The second part involves finding an integrable distribution from the solution set of the first part. Typically this part involves solving PDEs. Our results are also applicable in determining if a control affine system in n states and n−2 controls has flat outputs that depend only on states. We illustrate our method by examples.     

Left time derivatives in mathematics, mechanics and control of motion

By the traditional representation accepted in mathematics, mechanics and theoretical physics, the time-derivatives are defined as the right derivatives and are used in this way in differential equations describing processes in nature and technology. The fact that even for infinitely smooth x(t), the right time-derivatives do not physically exist, due to the positive orientation of time, somehow escaped the attention of scientists. This led to misconceptions and omissions in mechanics, physics and engineering, with unexpected consequences in some cases. All measurements and experiments contain and use only left time-derivatives, thereby with time delays. All processes require some kind of transmittal of information (forces, actions) which takes time, so the expressions that define their evolution from a current state actually contain the left and delayed time derivatives, even if they are written with the exact right time-derivatives, according to the classical tradition. In this paper, the causal representations of physical processes by differential equations with the left time-derivatives on the right-hand side are considered for some basic problems in classical mechanics, physics and technology. The use of the left time-derivatives explicitly takes into account the causality of processes depending on the transmission of information and defines the motions subject to external forces that may depend on accelerations and higher order derivatives of velocities. Such forces are exhibited in Weber’s electro-dynamic law of attraction; they are produced by the Kirchhoff-Thomson adjoint fluid acceleration resistance acting on a body moving in a fluid, and they are also involved in the manual control of aircraft or spacecraft that depends on accelerations of the craft itself. The consistency condition is presented, and the existence of solutions for equations of motion driven by forces with higher order derivatives of velocity is proved. The inclusion of such forces in the autopilot design is proposed to assure the safety of the aircraft in case of a failure of its outboard velocity sensors. It is demonstrated that the classical form of the 2nd law of Newton is preserved with respect to the effective forces for which the parallelogram law of addition is valid. Then the Lagrange and Hamilton equations are extended to include the generalized forces with the left higher order derivatives, and a method for the solution of such equations with the left and delayed higher order derivatives is presented with the example of a physical pendulum. The results open new avenues in science and technology providing the basis for correct design in the projects sensitive to information transmittal.     

Improved DAE formulation for inverse dynamics simulation of cranes

Cranes are underactuated systems with less control inputs than degrees of freedom. Dynamics and control of such systems is a challenging task, and the existence of solution to the inverse dynamics simulation problem in which an r-degree-of-freedom system with m actuators, m<r, is subject to m specified motion task (servo-constraints) is conditioned upon the system is differentially flat (all the system states and control inputs can be algebraically expressed in terms of the outputs and their time derivatives up to a certain order). The outputs are often designed as specified in time load coordinates to model a rest-to-rest maneuver along a trajectory in the working space, from the initial load position to its desired destination. The flatness-based methodology results then in the required control inputs determined in terms of the fourth time derivatives of the imposed outputs, and the derivations are featured by substantial complexity. The DAE formulation motivated in this contribution offers a more convenient approach to the prediction of dynamics and control of cranes executing prescribed load motions, and only the second time derivatives of the specified outputs are involved. While most of the inverse simulation formulations, both flatness-based and DAE ones, are performed using independent state variables, the use of dependent coordinates and velocities may lead to substantial modeling simplifications and gains in computational efficiency. An improved DAE formulation of this type is presented in this paper.