Showing papers by "Hassan K. Khalil published in 2013"

A Nonlinear High-Gain Observer for Systems With Measurement Noise in a Feedback Control Framework

Abstract: We address the problem of state estimation for a class of nonlinear systems with measurement noise in the context of feedback control. It is well-known that high-gain observers are robust against model uncertainty and disturbances, but sensitive to measurement noise when implemented in a feedback loop. This work presents the benefits of a nonlinear-gain structure in the innovation process of the high-gain observer, in order to overcome the tradeoff between fast state reconstruction and measurement noise attenuation. The goal is to generate a larger observer gain during the transient response than in the steady-state response. Thus, by reducing the observer gain after achieving satisfactory state estimates, the effect of noise on the steady-state performance is reduced. Moreover, the nonlinear-gain observer presented in this paper is shown to surpass the system performance achieved when using comparable linear-gain observers. The proof argues boundedness and ultimate boundedness of the closed-loop system under the proposed output feedback.

Control of Systems With Hysteresis Via Servocompensation and Its Application to Nanopositioning

Abstract: Partly motivated by nanopositioning applications in scanning probe microscopy systems, we consider the problem of tracking periodic signals for a class of systems consisting of linear dynamics preceded by a hysteresis operator, where uncertainties exist in both the dynamics and the hysteresis. A robustified servocompensator is proposed, in combination with an approximate hysteresis inverse, to achieve high-precision tracking. The servocompensator accommodates the internal model of the reference signal and a finite number of harmonic terms. Using a Prandtl-Ishlinskii (PI) operator for modeling hysteresis, we show that the closed-loop system admits a unique and asymptotically stable periodic solution, which justifies treating the inversion error as an exogenous periodic disturbance. Consequently, the asymptotic tracking error can be made arbitrarily small as the servocompensator accommodates a sufficient number of harmonic terms. The analysis is further extended to the case where the hysteresis is modeled by a modified PI operator. Experiments on a commercial nanopositioner show that, with the proposed method, tracking can be achieved for a 200-Hz reference signal with 0.52% mean error and 1.5% peak error, for a travel range of 40 μm. The performance of the proposed method in tracking both sinusoidal and sawtooth signals does not fall off with increasing frequency as fast as the proportional-integral controller and the iterative learning controller, both adopted in this paper for comparison purposes. Further, the proposed controller shows excellent robustness to loading conditions.

Control of flexible joint manipulators using only motor position feedback: A separation principle approach

Abstract: We consider the problem of output feedback tracking of flexible joint manipulators where the link angle is the output to be controlled and the motor angle is the measured output. We allow the use of any globally stabilizing full state feedback control. We show recovery of the stability and trajectory performance of the state feedback by using a high gain observer-based output feedback control. Finally, we demonstrate the effectiveness of the proposed scheme in the single link case, where the zero dynamics are not asymptotically stable.

Application of dynamic inversion with extended high-gain observers to inverted pendulum on a cart

Abstract: In this paper we utilize dynamic inversion together with high-gain observers to stabilize an inverted pendulum on a cart. Dynamic inversion is used to invert the nonlinear map involving the control input and the high-gain observers are used to estimate the states and terms related to acceleration variables. Through numerical simulations and experiments, it is shown that it is possible to stabilize the inverted pendulum on a cart and recover the performance of state feedback control.

Closed-loop analysis for systems with fast linear dynamics preceded by hysteresis

Abstract: Piezoelectric actuators are commonly modelled by a hysteresis operator preceding fast, stable linear dynamics. This motivates our work to analyze systems with these characteristics when a popular control architecture involving both hysteresis inversion and feedback is adopted. In particular, we are interested in the frequency-scaling behavior of the tracking error for such systems, which is of practical interest but has received little attention in the literature. The hysteresis nonlinearity in our analysis is represented by piecewise linear segments, which is applicable to many hysteresis operators. To fix ideas, we consider a proportional integral controller for the feedback component, as well as the case where a constant-gain feedforward component is added to the feedback term. This work is a continuation of our previous work where we only examined the system behavior for a given hysteresis segment. Here we use singular perturbation techniques to separate the slow variables of the controller from the fast variables of the plant dynamics, and derive the solution of the closed-loop system and the tracking error at the steady state under a sinusoidal reference. The analysis incorporates the effect of uncertainty in the hysteresis model, and offers insight into how the tracking error scales with the reference frequency. The analysis is confirmed with experimental and simulation results for the control of a piezo-actuated nanopositioner.

Semi-global output feedback stabilization of a class of non-minimum phase nonlinear systems

Abstract: We solve the problem of semi-global output feedback stabilization of a class of non-minimum phase nonlinear systems using a full order Extended Kalman Filter-Extended High Gain Observer proposed in [1]. We allow for any globally stabilizing full state feedback control scheme to be used as long as it satisfies a particular ISS condition. The proposed output feedback control system strategy is constructive and simple. Finally, we provide an example that illustrates the procedure and efficacy of the control design.

Design and analysis of a sliding mode controller for systems with hysteresis

Abstract: A sliding mode controller is proposed for a class of systems comprising a hysteresis operator preceding an kth-order linear plant without zero dynamics. The hysteresis operator is modeled with piecewise linear characteristics with uncertainties, and a nominal inverse operator is included to mitigate the hysteresis effect. A bound on the inversion error is used in the design of the sliding mode controller. The stability of the closed-loop system is established, and singular perturbation is exploited to analyze the system behavior within the boundary layer. In particular, analytical insight is gained on the frequency-scaling behavior of the tracking error under a periodic reference. Simulation and experimental results on a piezoelectric actuator-based nanopositioner are presented to illustrate the design and analysis, where the hysteresis nonlinearity is represented by a Prandtal-Ishlinskii operator.

Inversion-free stabilization and regulation of systems with hysteresis using integral action

Abstract: In this paper, we present analysis on the stabilization and regulation of the tracking error for an n-dimensional dynamic system with zero dynamics, which is preceded by a Prandtl-Ishlinskii hysteresis operator. A general controller structure is considered; however, we assume that an integral action is present. We treat this problem from the perspective of switched systems, where the state of the hysteresis operator defines the switching surfaces. The common Lyapunov function theorem is utilized together with an LMI condition to show that, under suitable conditions, the tracking error of the system goes to zero exponentially fast when a constant reference is considered. A key feature of this LMI condition is that it does not require the hysteresis effect to be small, meaning that hysteresis inversion is not required. We use this condition together with a periodicity assumption to prove that a servocompensator-based controller can stabilize a system with hysteresis without using hysteresis inversion. Finally, we conduct experiments using a servocompensator-based controller, where we verify the stability of the system and achieve a mean tracking error of 0.5% at 200 Hz using a sinusoidal reference.

Self-excited limit cycles in an integral-controlled system with backlash

Abstract: The stability of systems with hysteresis, driven by developments in smart material applications, has been an important topic of research over the past two decades. Most results provide sufficient conditions for boundedness of the system states, but do not further investigate the steady state solutions. In this paper, we present an example of a system with hysteresis that possesses self-excited limit cycles. In particular, we consider an integral-controlled system with backlash (also known as play operator). A Newton-Raphson algorithm is formulated to calculate the limit cycles in the system. We then prove that the amplitude and period of these limit cycles have linear relationships to parameters within the system. These results are then confirmed in simulation, where we demonstrate our ability to predict and modify the properties of the limit cycles.

Inversion-freeStabilizationandRegulationofSystemswith HysteresisviaIntegralAction 1

Abstract: In this paper, we present conditions for the stabilization and regulation of the tracking error for an n-dimensional minimum-phase system preceded by a Prandtl-Ishlinskii hysteresis operator. A general controller structure is considered; however, we assume that an integral action is present. The common Lyapunov function theorem is utilized together with a Linear Matrix Inequality (LMI) condition to show that, under suitable conditions, the tracking error of the system goes to zero exponentially fast when a constant reference is considered. A key feature of this LMI condition is that it does not require the hysteresis effect to be small, meaning that hysteresis inversion is not required. We use this condition together with a periodicity assumption to prove that a servocompensator-based controller can stabilize the system without using hysteresis inversion. Additionally, we draw parallels between our LMI condition and passivity-based results achieved in the literature. We then verify our LMI results in simulation, where we show that the LMI condition can accurately predict the stability margins of a system with hysteresis. Finally, we conduct experiments using a servocompensator-based controller, where we verify the stability of the system and achieve a mean tracking error of 0.5% for a 200 Hz sinusoidal reference.