Non-collocated Control Research Articles

Distributed Control of Truss Structures

Continuous element methods have frequently been used in the modal analysis of structures modeled as flexible beams, extensible strings and rods, but only rarely for purposes of control. In this paper, we define a strategy of active control using the exact transfer matrix of a two-cell planar truss. The closed-loop performance predictions of both collocated and non-collocated controllers are compared as a function of feedback controller parameters.

Effects of piezoelectric sensor/actuator debonding on vibration control of smart beams

Abstract In vibration control of smart structures, piezoelectric sensors/actuators are usually bonded to the surface of a host structure. Debonding may occur between the piezoelectric sensors/actuators and the host structure, and it may decrease the control efficiency of vibration suppression or even lead to an unstable closed loop control system. This paper investigates the effect of the debonding on vibration control of beams with piezoelectric sensors and actuators. Presented is a novel model, which takes into account both flexural and longitudinal displacements of the host beam and piezoelectric layers as well as the peel and shear strains of the adhesive layer. Both collocated and non-collocated control schemes are used to study the effects of sensor or actuator debonding on active vibration control of smart beams.

ACHIEVING MINIMUM PHASE TRANSFER FUNCTION FOR A NONCOLLOCATED SINGLE‐LINK FLEXIBLE MANIPULATOR

ABSTRACTA noncollocated system has the potential of providing more precise tracking, improved disturbance rejection and increased bandwidth at the sensor location, but is considerably more difficult to stabilize than a collocated system due to its nonminimum phase nature. For a flexible manipulator, the problem becomes even more complicated because the system is inherently infinite dimensional. In this paper, a single‐link flexible manipulator having a tip payload and a noncollocated actuator and sensor is investigated using a linear distributed parameter model. With the joint torque as the input and the joint angle plus a weighted value of tip deflection as the measured output, an exact transfer function involving transcendental functions is derived. Using the methods of infinite product expansion, the root locus, and the asymptotic property of the roots of the transcendental equation, a necessary and sufficient condition in terms of the weighting factor of tip deflection is obtained such that the transfer function does not have any open right‐half plane zeros. This condition depends neither on the physical properties of the link nor on the mass properties of the tip payload. In order to correct the misinterpretation which has occurred in some closely‐related works concerning the stability of the infinite‐dimensional zero‐dynamics, the equivalence of the zeros of the transfer function and the eigenvalues of zero‐dynamics is also verified. Numerical results for the closed‐loop performance of a single‐link flexible arm using collocated and noncollocated PD controllers are given to show the efficacy of the proposed minimum phase transfer function approach.

ON THE FEEDBACK CONTROL OF THE WAVE EQUATION

This paper addresses the problem of design of collocated and non-collocated controllers for a uniform bar without structural damping. The bar whose dynamics are described by the wave equation is required to perform a rest-to-rest maneuver. A time delay controller whose gains are determined using the root-locus technique is used to control the non-collocated system. The effect of sensor locations on the stability of the system is investigated when the actuator is located at the one end of the bar. The critical gains which correspond to a pair of poles entering the right-half of the s -plane and the optimal gains corresponding to locating the closed-loop poles at the left extreme of the root-locus for each vibration mode are determined. The gain which minimizes a quadratic cost, in the range of the critical gains, is selected as the optimum gain.

Nonlinear strategies for longitudinal control in the stabilization of an oscillating suspended cable

Thefeasibility of reducing vibrations in suspended cables by animposed longitudinal displacement of one support is investigatedthrough an analytical model. A noncollocated active control schemeis considered where modal amplitudes describing transverse oscillationsare used in the feedback control action at the boundary. A 2DOFnonlinear model of the cable is used to design state-feedbackcontroller. Linear and nonlinear velocity feedback and feedbackbilinearisation are shown to produce a nonlinear and bilinearactively-damped system, respectively. The controlled system performanceis analysed comparing the effectiveness of the different strategieson both control demand and response amplitudes in the stabilizationof the equilibrium position. Control spillover effects are commentedthrough an enlarged 8DOF modal description of the controlledcable oscillations.

Endpoint Control of a Two-Link Flexible Manipulator

This article deals with the endpoint control of a lightweight, two-link planar manipulator with flexible links. Three types of control methods are presented. The first method is based on the computed torque techniques where the actuators and the angular position sensors are collocated at the joints. The second method uses the deflections of the links and the joint angles to control the angle of a selected point along each link. This method is referred to as the noncollocated control. To add active damping to the flexible links, a third method that uses the noncollocated control combined with a delayed deflection has been introduced. Simulation studies have been conducted to compare the tracking performance of each method.

On the use of differential drives for overcoming transmission nonlinearities

Nonlinear friction can couple with the internal dynamics of geared reducers to exhibit the phase characteristics of noncollocation. In the setting of robot control this is significant, in so far as the additional phase-lag limits bandwidth. This paper suggests a solution based on the concept of differential actuation, with two actuators driving a single joint arbitrarily in speed and direction. The advantage of the approach is that the internal gears are always in constant motion, thereby avoiding low and zero velocities where friction is strongly nonlinear. The potential performance of a differential transmission is compared with that of a conventional gearbox. A scheme for solving the intrinsic redundancy of the device is presented and its feasibility demonstrated through two controllers: a position collocated controller and a force noncollocated controller. The performance of these controllers is compared with their counterparts implemented on a nondifferential gear train.

Control of Flexible Manipulator Systems

This paper presents an investigation into the development of open-loop and closed-loop control strategies for flexible manipulator systems. Shaped torque inputs, including Gaussian-shaped and low-pass (Butter-worth and elliptic) filtered input torque functions, are developed and used in an open-loop configuration and their performance studied in comparison to a bang-bang input torque through experimentation on a single-link flexible manipulator system. Closed-loop control strategies that use both collocated (hub angle and hub velocity) and non-collocated (end-point acceleration) feedback are then proposed. A collocated proportional and derivative (PD) control is first developed and its performance studied through experimentation. The collocated control is then extended to incorporate, additionally, non-collocated feedback through a proportional integral derivative (PID) configuration. The performance of the hybrid collocated and non-collocated control strategy thus developed is studied through experimentation. Experimental results verifying the performance of the developed control strategies are presented and discussed.

Effects of time delays on the stability of collocated and noncollocated point control of discrete dynamic structural systems

The effects of time delays on collocated as well as noncollocated point control of discrete dynamic systems have been examined. Controllers of proportional-integral-derivative (PID) type have been considered. Analytical estimates of time delays to maintain/obtain stability for small gains have been given. Several new results dealing with the effect of time delays on collocated and noncollocated control designs are obtained. It is shown that undamped structural systems cannot be stabilized with pure velocity (or integral) feedback without time delays while using a controller which is not collocated with the sensor when the mass matrix is diagonal. However, with the appropriate choice of time delays for certain classes of commonly occurring structural systems, stable noncollocated control can be achieved. Analytical results providing the upper bound on the controller's gain necessary for stability have been presented. The theoretical results obtained are illustrated and verified with numerical examples.

Design of reduced-order, H2 optimal controllers using a homotopy algorithm

The minimal dimension of a linear-quadratic-gaussian (LQG) compensator is almost always equal to the dimension of the design plant. This deficiency can lead to implementation problems when considering control design for high-order systems such as flexible structures and has led to the development of methodologies for the design of optimal (or near optimal) controllers whose dimension is less than that of the design plant. This paper presents a new (gradient-based) homotopy algorithm for the design of reduced-order, H2 optimal controllers. An important result is the development of an efficient methodology for computation of the cost functional hessian which is required by the algorithm. The optimal controller is represented by a parameter vector and various parametrizations of the optimal controller are considered to reduce the algorithm dimensionality. The algorithm has been implemented in MAT-LAB and the results are illustrated using a benchmark, non-collocated flexible structural control problem. It is ...

Time delayed control of classically damped structural systems

The effects of time delays on collocated as well as non-collocated point control of classically damped discrete dynamic systems have been examined. Controllers of PID type nave been considered. Analytical estimates of time delays to maintain/obtain stability for small gains have been given. Several new results dealing with the effect of time delays on collocated and non-collocated control designs are obtained. It is shown that undamped structural systems cannot be stabilized with pure velocity (or integral) feedback without time delays while using a controller that is not collocated with the sensor, when the mass matrix is diagonal. However, with the appropriate choice of time delays, for certain classes of commonly occurring structural systems, stable non-collocated control can be achieved. Analytical results providing the upper bound on the controller's gain necessary for stability have been presented. The theoretical results obtained are illustrated and verified with numerical examples.

Discrete Time Noncollocated Control of Flexible Mechanical Systems Using Time Delay

This paper studies discrete time noncollocated control of flexible mechanical systems. A sampled data system with one sensor and one actuator is formulated. Specific time delay is introduced in the feedback loop to eliminate the destabilizing effect of noncollocation of the sensor and actuator. A stability criterion that explicitly relates the closed-loop stability to the sampling frequency, time delay, and sensor and actuator locations, is derived. Based on the criterion, particular noncollocated controllers are designed for a simply supported beam. A sensitivity analysis yields a robust control condition in terms of the sampling frequency, system parameter uncertainties, and inherent delays. In addition, the relationship between frequency folding in data sampling and spill-over instability is discussed.

Robust H-infinity control synthesis method and its application to benchmark problems

This paper presents a robust H-infinity control synthesis method for structured parameter uncertainty. The robust H-infinity control design methodology is also incorporated with the so-called internal model principle for persistent-disturbance rejection. A noncollocated control problem of flexible space structures subject to parameter variations is used to illustrate the design methodology. It is shown that the proposed design method invariably makes use of nonminimum-phase compensation and that it achieves the desired asymptotic disturbance rejection by having a disturbance rejection 'dipole'.

Experiment of a noncollocated controller for wave cancellation

Wave-absorbing control for vibration suppression of flexible space structures is demonstrated experimentally. The experimental model is a hung flexible beam with a sensor and a noncollocated torque actuator. The wave-absorbing control employs partial differential equations as the mathematical model of structural dynamics, and the controller can be designed without any use of the modal expansion. Thus, the control is free from the crucial truncation effect. The structural vibration is expressed in terms of propagating disturbances in the frequency domain. The control forces are applied to eliminate outgoing waves at the boundaries of the structure. The controller is implemented in the form of a nonrecursive digital filter. Details of the controller implementation are discussed, including the digital filter as a compensator, filter truncation, and the hardware. Experimental results show satisfactory performance of the controller and good agreement with the performance predicted analytically. ARGE space structures (LSS) will be inevitably flexible in their structure, and their structural vibration will be easily excited. Therefore, control techniques of structural vibration are necessary to satisfy performance of LSS. Most of the studies on the control problem of LSS employ the dynamic model with a discretized description for flexible structures that are inherently distributed parameter systems. The discretization is usually employed to adopt the conventional state-space method. Methods free from the discretization are natural and suitable to the control of such distributed parameter systems as the LSS. This paper treats through the experiment the wave-absorbing control for vibration suppression of flexible structures. The wave-absorbing control employs partial differential equations as the mathematical model of structural dynamics to design the controller, and the discretization method is not employed in the analyses. The controller obtained is thus free from the crucial truncation effect. The equations are Laplace transformed with respect to time to describe the structural responses in terms of propagating disturbances in the frequency domain. Boundary conditions of the system are expressed as relations between incoming waves, outgoing waves, and external control forces at the boundaries. External control forces are applied to eliminate outgoing waves at the boundaries. Resulting closed-loop behavior of the system shows drastic improvement for disturbance attenuation in comparison with open-loop behavior. The concept of the traveling wave approach is applied to the control problem of LSS by von Flotow.13 He introduces the viewpoint that the elastic response of LSS may be aptly viewed in terms of the disturbance propagation characteristics of the structure. Active vibration isolation by cancelling traveling waves based on the same concept is studied in Refs. 4-8. The technique aims to isolate one part of a structure from disturbances excited elsewhere in the structure. Actuators are located at midpoints of structural members and cancel passing disturbances so that the disturbances do not disturb downstream portions. A generic description of the traveling wave control for structural networks is given in Ref. 9. Miller et al.

Modal analysis of controlled multilink systems with flexible links and joints

In this study, a systematic finite element transfer matrix scheme for the modal analysis of integrated multilink control systems with flexible links and joints is presented. This proposed scheme treats the control signals as parts of physical degrees of freedom of the system and then modifies the transfer matrices considering the coupling and noncoupling characteristics between the electric and mechanical interfaces. Since it is systematic, the effects of interaction between control and structure can be considered, and reference control signals can be directly input to the system without calculating equivalent static actuating torques. An example of a twolink/two-joint control system is given to illustrate its availability in sensitivity analysis of both structural and control design parameters by a systematic manner. The illustrated example also shows its capability in the application of both collocated and noncollocated control simulations.

A new controller design for a flexible one-link manipulator

A novel controller design for noncollocated flexible one-link manipulator arm tip position control based on variable structure sliding mode control is presented. Using the assumed-mode method, the plant model is derived. The discontinuous control law based on the variable structure system theory for the noncollocated manipulator tip position control is then designed. The position state variables are obtained directly from the inversion of the output submatrix multiplied by the sensor measurements. The velocity state variables are estimated through decoupled estimators-a separate first-order observer for each of the system's modes under consideration. Different sampling periods are used for the estimator and the controller. The performance of the controller is evaluated through a series of simulations, followed by an analysis of the designed control system. >

Modeling and Design Implications of Noncollocated Control in Flexible Systems

In this paper we investigate generic properties of structural modeling pertinent to structural control, with emphasis on noncollocated systems. Analysis is performed on a representative example of a pinned-free Euler-Bernoulli beam with distributed sensors. Analysis in the wave number plane highlights the crucial qualitative characteristics common to all structural systems. High sensitivity of the transfer function zeros to errors in model parameters and sensor locations is demonstrated. The existence of finite right half plane zeros in noncollocated systems, along with this high sensitivity, further complicates noncollocated controls design. A numerical method for accurate computation of the transfer function zeros is proposed. Wiener-Hopf factorization is used to compute equivalent delay time, which is important in controls design.