A performance comparison of robust adaptive controllers: linear systems

We consider robust adaptive control designs for relative degree one, minimum phase linear systems of known high frequency gain. The designs are based on the dead-zone and projection modifications, and we compare their performance w.r.t. a worst case transient cost functional with a penalty on the ^∞ norm of the output, control and control derivative. We establish two qualitative results. If a bound on the ^∞ norm of the disturbance is known and the known a priori bound on the uncertainty level is sufficiently conservative, then it is shown that a dead-zone controller outperforms a projection controller. The complementary result shows that the projection controller is superior to the dead-zone controller when the a priori information on the disturbance level is sufficiently conservative.     

Distributed \begin{document}$H_{2}/H_\infty$\end{document} Filter Design for Discrete-Time Switched Systems

This paper addresses an infinite horizon distributed H2/H∞ filtering for discrete-time systems under conditions of bounded power and white stochastic signals. The filter algorithm is designed by computing a pair of gains namely the estimator and the coupling. Herein, we implement a filter to estimate unknown parameters such that the closed-loop multi-sensor accomplishes the desired performances of the proposed H2 and H∞ schemes over a finite horizon. A switched strategy is implemented to switch between the states once the operation conditions have changed due to disturbances. It is shown that the stability of the overall filtering-error system with H2/H∞ performance can be established if a piecewise-quadratic Lyapunov function is properly constructed. A simulation example is given to show the effectiveness of the proposed approach.     

The Random Members of a Π 1 0 ${\Pi }_{1}^{0}$ Class

Theory of Computing Systems - We examine several notions of randomness for elements in a given ${\Pi }_{1}^{0}$ class $\mathcal {P}$ . Such an effectively closed subset $\mathcal {P}$ of 2 ω...     

Branching Time Logics \mathcal {BTL}^{\mathrm {U,S}}_{\mathrm {N},\mathrm {N}^{-1}}(\mathcal {Z})_{\alpha } with Operations Until and Since Based on Bundles of Integer Numbers, Logical Consecutions, Deciding Algorithms

This paper is intended as an attempt to describe logical consequence in branching time logics. We study temporal branching time logics $\mathcal {BTL}^{\mathrm {U,S}}_{\mathrm {N},\mathrm {N}^{-1}}(\mathcal {Z})_{\alpha }$ which use the standard operations Until and Next and dual operations Since and Previous (LTL, as standard, uses only Until and Next). Temporal logics $\mathcal {BTL}^{\mathrm {U,S}}_{\mathrm {N},\mathrm {N}^{-1}}(\mathcal {Z})_{\alpha }$ are generated by semantics based on Kripke/Hinttikka structures with linear frames of integer numbers $\mathcal {Z}$ with a single node (glued zeros). For $\mathcal {BTL}^{\mathrm {U,S}}_{\mathrm {N},\mathrm {N}^{-1}}(\mathcal {Z})_{\alpha }$ , the permissible branching of the node is limited by α (where 1≤α≤ω). We prove that any logic $\mathcal {BTL}^{\mathrm {U,S}}_{\mathrm {N},\mathrm {N}^{-1}}(\mathcal {Z})_{\alpha }$ is decidable w.r.t. admissible consecutions (inference rules), i.e. we find an algorithm recognizing consecutions admissible in $\mathcal {BTL}^{\mathrm {U,S}}_{\mathrm {N},\mathrm {N}^{-1}}(\mathcal {Z})_{\alpha }$ . As a consequence, it implies that $\mathcal {BTL}^{\mathrm {U,S}}_{\mathrm {N},\mathrm {N}^{-1}}(\mathcal {Z})_{\alpha }$ itself is decidable and solves the satisfiability problem.     

Rounds in Combinatorial Search

A set system $\mathcal{H} \subseteq2^{[m]}$ is said to be separating if for every pair of distinct elements x,y∈[m] there exists a set $H\in\mathcal{H}$ such that H contains exactly one of them. The search complexity of a separating system $\mathcal{H} \subseteq 2^{[m]}$ is the minimum number of questions of type “x∈H?” (where $H \in\mathcal{H}$ ) needed in the worst case to determine a hidden element x∈[m]. If we receive the answer before asking a new question then we speak of the adaptive complexity, denoted by $\mathrm{c} (\mathcal{H})$ ; if the questions are all fixed beforehand then we speak of the non-adaptive complexity, denoted by $\mathrm{c}_{na} (\mathcal{H})$ . If we are allowed to ask the questions in at most k rounds then we speak of the k-round complexity of $\mathcal{H}$ , denoted by $\mathrm{c}_{k} (\mathcal{H})$ . It is clear that $|\mathcal{H}| \geq\mathrm{c}_{na} (\mathcal{H}) = \mathrm{c}_{1} (\mathcal{H}) \geq\mathrm{c}_{2} (\mathcal{H}) \geq\cdots\geq\mathrm{c}_{m} (\mathcal{H}) = \mathrm{c} (\mathcal{H})$ . A group of problems raised by G.O.H. Katona is to characterize those separating systems for which some of these inequalities are tight. In this paper we are discussing set systems $\mathcal{H}$ with the property $|\mathcal{H}| = \mathrm{c}_{k} (\mathcal{H}) $ for any k≥3. We give a necessary condition for this property by proving a theorem about traces of hypergraphs which also has its own interest.     

The Calculus of Relations as a Foundation for Mathematics

A variable-free, equational logic $\mathcal{L}^\timesA variable-free, equational logic based on the calculus of relations (a theory of binary relations developed by De Morgan, Peirce, and Schr?der during the period 1864–1895) is shown to provide an adequate framework for the development of all of mathematics. The expressive and deductive powers of are equivalent to those of a system of first-order logic with just three variables. Therefore, three-variable first-order logic also provides an adequate framework for mathematics. Finally, it is shown that a variant of may be viewed as a subsystem of sentential logic. Hence, there are subsystems of sentential logic that are adequate to the task of formalizing mathematics. This paper is an expanded version of a talk given by the author at the Special Session on Automated Reasoning in Mathematics and Logic, held March 8–10, 2002, at the Georgia Institute of Technology, during the Joint Southeastern Section MAA/Southeast Regional AMS Meeting. The session was organized by Johan G. F. Belinfante.     

Performance Comparison of Controllers with Fault-Dependent Control Allocation for UAVs

This paper combines fault-dependent control allocation with three different control schemes to obtain fault tolerance in the longitudinal control of unmanned aerial vehicles. The paper shows that fault-dependent control allocation is able to accommodate actuator faults that would otherwise be critical and it makes a performance assessment for the different control algorithms: an \(\mathcal {L}_{1}\) adaptive backstepping controller; a robust sliding mode controller; and a standard PID controller. The actuator faults considered are the partial to total loss of the elevator, which is a critical component for the safe operation of unmanned aerial vehicles. During nominal operation, only the main actuator, namely the elevator, is active for pitch control. In the event of a partial or total loss of the elevator, fault-dependent control allocation is used to redistribute control to available healthy actuators. Using simulations of a Cessna 182 aircraft model, controller performance and robustness are evaluated by metrics that assess control accuracy and energy use. System uncertainties are investigated over an envelope of pertinent variation, showing that sliding mode and \(\mathcal {L}_{1}\) adaptive backstepping provide robustness, where PID control falls short. Additionally, a key finding is that the fault-dependent control allocation is instrumental when handling actuator faults.     

Decentralized adaptive fuzzy sliding mode control for reconfigurable modular manipulators

A stable decentralized adaptive fuzzy sliding mode control scheme is proposed for reconfigurable modular manipulators to satisfy the concept of modular software. For the development of the decentralized control, the dynamics of reconfigurable modular manipulators is represented as a set of interconnected subsystems. A first‐order Takagi–Sugeno fuzzy logic system is introduced to approximate the unknown dynamics of subsystem by using adaptive algorithm. The effect of interconnection term and fuzzy approximation error is removed by employing an adaptive sliding mode controller. All adaptive algorithms in the subsystem controller are derived from the sense of Lyapunov stability analysis, so that resulting closed‐loop system is stable and the trajectory tracking performance is guaranteed. The simulation results are presented to show the effectiveness of the proposed decentralized control scheme. Copyright © 2009 John Wiley & Sons, Ltd.     

Delayed theory combination vs. Nelson-Oppen for satisfiability modulo theories: a comparative analysis

Most state-of-the-art approaches for Satisfiability Modulo Theories $(SMT(\mathcal{T}))$ rely on the integration between a SAT solver and a decision procedure for sets of literals in the background theory $\mathcal{T} (\mathcal{T}{\text {-}}solver)$ . Often $\mathcal{T}$ is the combination $\mathcal{T}_1 \cup \mathcal{T}_2$ of two (or more) simpler theories $(SMT(\mathcal{T}_1 \cup \mathcal{T}_2))$ , s.t. the specific ${\mathcal{T}_i}{\text {-}}solvers$ must be combined. Up to a few years ago, the standard approach to $SMT(\mathcal{T}_1 \cup \mathcal{T}_2)$ was to integrate the SAT solver with one combined $\mathcal{T}_1 \cup \mathcal{T}_2{\text {-}}solver$ , obtained from two distinct ${\mathcal{T}_i}{\text {-}}solvers$ by means of evolutions of Nelson and Oppen’s (NO) combination procedure, in which the ${\mathcal{T}_i}{\text {-}}solvers$ deduce and exchange interface equalities. Nowadays many state-of-the-art SMT solvers use evolutions of a more recent $SMT(\mathcal{T}_1 \cup \mathcal{T}_2)$ procedure called Delayed Theory Combination (DTC), in which each ${\mathcal{T}_i}{\text {-}}solver$ interacts directly and only with the SAT solver, in such a way that part or all of the (possibly very expensive) reasoning effort on interface equalities is delegated to the SAT solver itself. In this paper we present a comparative analysis of DTC vs. NO for $SMT(\mathcal{T}_1 \cup \mathcal{T}_2)$ . On the one hand, we explain the advantages of DTC in exploiting the power of modern SAT solvers to reduce the search. On the other hand, we show that the extra amount of Boolean search required to the SAT solver can be controlled. In fact, we prove two novel theoretical results, for both convex and non-convex theories and for different deduction capabilities of the ${\mathcal{T}_i}{\text {-}}solvers$ , which relate the amount of extra Boolean search required to the SAT solver by DTC with the number of deductions and case-splits required to the ${\mathcal{T}_i}{\text {-}}solvers$ by NO in order to perform the same tasks: (i) under the same hypotheses of deduction capabilities of the ${\mathcal{T}_i}{\text {-}}solvers$ required by NO, DTC causes no extra Boolean search; (ii) using ${\mathcal{T}_i}{\text {-}}solvers$ with limited or no deduction capabilities, the extra Boolean search required can be reduced down to a negligible amount by controlling the quality of the $\mathcal{T}$ -conflict sets returned by the ${\mathcal{T}_i}{\text {-}}solvers$ .     

An adaptive integral sliding mode FTC scheme for dissimilar redundant actuation system of civil aircraft

ABSTRACT

This paper proposed a new adaptive integral sliding mode FTC scheme to deal with the actuator faults and failure. The scheme combines integral sliding mode control, control allocation scheme and adaptive strategy. The unknown actuator faults are handled by adaptive modulation gain of nonlinear ISMC law. To cope with complete failure, control allocation scheme is integrated with the baseline controller to provide tolerance. The proposed strategy relies on the estimate of actuator effectiveness. Therefore, an adaptive sliding mode observer based fault reconstruction scheme is proposed in this paper. The proposed scheme is implemented on dissimilar redundant actuation system driven by hydraulic and electro-hydraulic actuators. In nominal and faulty conditions, both actuators are contributing to achieving the desired control surface deflection. However, when the actuator failure occurs, the control signals are reallocated to the redundant actuator. The problem of dynamics mismatch is addressed using fractional order controller designed in an inner loop. The comparison with the existing literature is also conducted in the simulation to validate the dominant performance.     

Recursive filtering adaptive fault‐tolerant tracking control for uncertain switched multivariable nonlinear systems

This article synthesizes a recursive filtering adaptive fault‐tolerant tracking control method for uncertain switched multivariable nonlinear systems. The multivariable nonlinear systems under consideration have both matched and mismatched uncertainties, which satisfy the semiglobal Lipschitz condition. A piecewise constant adaptive law generates adaptive parameters by solving the error dynamics with the neglection of unknowns, and the recursive least squares is employed to minimize the residual error by categorizing the total uncertainty estimates into matched and mismatched components. A filtering control law is designed to compensate the actuator faults and nonlinear uncertainties such that a good tracking performance is delivered with guaranteed robustness. The matched component is canceled directly by adopting their opposite in the control signal, whereas a dynamic inversion of the system is performed to eliminate the effect of the mismatched component on the output. By exploiting the average dwell time principle, the error bounds are derived for the states and control inputs compared with the virtual reference system which defines the best performance that can be achieved by the closed‐loop system. Both numerical and practical examples are provided to illustrate the effectiveness of the proposed switching recursive filtering adaptive fault‐tolerant tracking control architecture, comparisons with model reference adaptive control are also carried out.     

Filtering adaptive neural network controller for multivariable nonlinear systems with mismatched uncertainties

This paper synthesizes a filtering adaptive neural network controller for multivariable nonlinear systems with mismatched uncertainties. The multivariable nonlinear systems under consideration have both matched and mismatched uncertainties, which satisfy the semiglobal Lipschitz condition. The nonlinear uncertainties are approximated by a Gaussian radial basis function (GRBF)‐based neural network incorporated with a piecewise constant adaptive law, where the adaptive law will generate adaptive parameters by solving the error dynamics between the real system and the state predictor with the neglection of unknowns. The combination of GRBF‐based neural network and piecewise constant adaptive law relaxes hardware limitations (CPU). A filtering control law is designed to handle the nonlinear uncertainties and deliver a good tracking performance with guaranteed robustness. The matched uncertainties are cancelled directly by adopting their opposite in the control signal, whereas a dynamic inversion of the system is required to eliminate the effect of the mismatched uncertainties on the output. Since the virtual reference system defines the best performance that can be achieved by the closed‐loop system, the uniform performance bounds are derived for the states and control signals via comparison. To validate the theoretical findings, comparisons between the model reference adaptive control method and the proposed filtering adaptive neural network control architecture with the implementation of different sampling time are carried out.     

On “Exploiting” Node-Heterogeneous Clusters Optimally

It is proved that “FIFO” worksharing protocols provide asymptotically optimal solutions to two problems related to sharing large collections of independent tasks in a heterogeneous network of workstations (HNOW) . In the , one seeks to accomplish as much work as possible on during a prespecified fixed period of L time units. In the , one seeks to complete W units of work by “renting” for as short a time as necessary. The worksharing protocols we study are crafted within an architectural model that characterizes via parameters that measure ’s workstations’ computational and communicational powers. All valid protocols are self-scheduling, in the sense that they determine completely both an amount of work to allocate to each of ’s workstations and a schedule for all related interworkstation communications. The schedules provide either a value for W given L, or a value for L given W, hence solve both of the motivating problems. A protocol observes a FIFO regimen if it has ’s workstations finish their assigned work, and return their results, in the same order in which they are supplied with their workloads. The proven optimality of FIFO protocols resides in the fact that they accomplish at least as much work as any other protocol during all sufficiently long worksharing episodes, and that they complete sufficiently large given collections of tasks at least as fast as any other protocol. Simulation experiments illustrate that the superiority of FIFO protocols is often observed during worksharing episodes of only a few minutes’ duration. A portion of this research was presented at the 15th ACM Symp. on Parallelism in Algorithms and Architectures (2003).     

Cardinal interpolation with periodic polysplines on strips

Abstract  We obtain a multivariate extension of a classical result of Schoenberg on cardinal spline interpolation. Specifically, we prove the existence of a unique function in , polyharmonic of order p on each strip , , and periodic in its last n variables, whose restriction to the parallel hyperplanes , , coincides with a prescribed sequence of n-variate periodic data functions satisfying a growth condition in . The constructive proof is based on separation of variables and on Micchelli’s theory of univariate cardinal -splines. Keywords: cardinal -splines, polyharmonic functions, multivariable interpolation Mathematics Subject Classification (2000): 41A05, 41A15, 41A63     

Development and Application of Controller for Transition Flight of Tail-Sitter UAV

There is renewed interest in tail-sitter airplanes on account of their vertical takeoff and landing capability as well as their efficient horizontal flight capabilities. The transition from a vertical near-hover mode to a horizontal cruise mode is a critical component of the tail-sitter flight profile. In practice, this transition is often achieved by a stall-and-tumble maneuver, which is somewhat risky and therefore not desirable, so alternative maneuvering strategies along controlled trajectories are sought. Accordingly, this paper presents the synthesis and application of a transition controller to a tail-sitter UAV for the first time. For practical reasons, linear controllers are designed using the PID technique and linked by gain scheduling. The limits of the PID controller are complemented by a so-called $\mathcal{L}_{1}$ adaptive controller that considers the coupling effect, reduces the effort for appropriate gain selection, and improves the tracking performance at different points during operation. Each transition trajectory is controlled by the flight velocity and path angle using dynamic inversion. The transition control law is tested on a tail-sitter UAV, an 18-kg vehicle that has a 2-m wingspan with an aspect ratio of 4.71 and is powered by a 100-cm³ gasoline engine driving an aft-mounted ducted fan. This paper describes not only the synthesis and the onboard implementation of the control law but also the flight testing of the fixed-wing UAV in hover, transition, and cruise modes.     

Robust formation control of multiagent systems on the Lie group SE(3)

This paper addresses the robust formation control problem of multiple rigid bodies whose kinematics and dynamics evolve on the Lie group SE(3). First, it is assumed that all followers have access to the state information of a virtual leader. Then, a novel adaptive super‐twisting sliding mode control with an intrinsic proportional‐integral‐derivative sliding surface is proposed for the formation control problem of multiagent system using a virtual structure (VS) approach. The advantages of this control scheme are twofold: elimination of the chattering phenomenon without affecting the control performance and no requirement of prior knowledge about the upper bound of uncertainty/disturbance due to adaptive‐tuning law. Since the VS method is suffering from the disadvantages of centralized control, in the second step, considering a network as an undirected connected graph, we assume that only a few agents have access to the state information of the leader. Afterward, using the gradient of modified error function, a distributed adaptive velocity‐free consensus‐based formation control law is proposed where reduced‐order observers are introduced to remove the requirements of velocity measurements. Furthermore, to relax the requirement that all agents have access to the states of the leader, a distributed finite‐time super‐twisting sliding mode estimator is proposed to obtain an accurate estimation of the leader's states in a finite time for each agent. In both steps, the proposed control schemes are directly developed on the Lie group SE(3) to avoid singularity and ambiguities associated with the attitude representations. Numerical simulation results illustrated the effectiveness of the proposed control schemes.