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In this paper, we study an adaptive method to select online the pole value for a Laguerre scheme in Model Predictive Control (MPC) that yields high performance. It has been observed that, while sti...

The existence of hysteresis phenomenon in piezoelectric actuators of nanopositioners adversely affects their performance, e.g. image distortion in Atomic Force Microscopy. A usual approach to circumnavigate hysteresis nonlinearity is feedforward compensation where the performance depends extensively on the accuracy of the hysteresis model. To achieve accurate modeling of hysteresis in nanopositioners driven by piezoelectric stacks, we used a dual-stack differential driving configuration. Comparing hysteresis in single-stack piezoelectric actuators with dual-stack piezoelectric actuators in differential driving configuration, we observed a more symmetric behavior for the hysteresis in dual-stack differential driving actuators. Then, we modeled the differential driving configuration by utilizing coupled electromechanical equations with hysteresis models applied to them. In particular, Duhem and Prandtl-Ishlinskii (P–I) methods were used for hysteresis modeling. Based on the models and experimental data, we observed that the maximum value of the Duhem modeling error reduced from 9.63% for the nondifferential configuration to 1.85% for the differential configuration. For the P–I method, the maximum modeling error decreased from 7.46% to 2.77%. This observation shows that the dual-actuated differential driving configuration improves hysteresis modeling accuracy. Therefore, this configuration is a suitable choice for the applications where accuracy is of prime importance.

International audience; The paper deals with the characterization of piezoelectric ceramic type sensors integrated in a compact high-speed 2-DOF nano-positioner to measure large signal stage displacement. Such transducers are very attractive because they provide measurements in a wide frequency bandwidth with low noise, they do not require bulky electronic modules and are cost effective compared to traditional sensors. The transducers are installed in series with piezoelectric actuators and thick ceramic insulators are incorporated between them to reduce the cross-talk. The sensor provides accurate position measurements as long as the nano-positioner displacement is within the sensing bandwidth. This is verified by large amplitude sinusoidal excitations of the nano-positioner along both x and y axes within the sensor bandwidth. The sensor can also measure the displacement in response to a single tone sinusoid at frequencies well above the dominant imaginary zeros of the interaction force, which is beyond the sensor bandwidth. Experimental measurements are compared with those obtained using a LASER interferometer sensor. The results allow obtaining precise calibration factors of the piezoelectric ceramic transducers with respect to the input frequencies and accurate displacement/voltage transfer characteristics for closed loop control purposes.

International audience; We address design, implementation, and characterization of semiconductor strain gauges as stage displacement sensors with small footprint for piezo-driven nanopositioners in differential actuation mode. The strain gauges are not collocated with the stage of the nanopositioner. They are attached to piezo stack actuators in both longitudinal and transverse directions. We address two differential configurations for the displacement sensing. The first configuration uses all strain gauges and provides three output signals. Two of them are proportional to the length variations of the piezo actuators, which are considered as conventional non-differential sensing method. The last output in the first configuration is the first proposed differential sensor. In the second configuration, we propose an alternative differential sensor using only the longitudinal strain gauges and a simpler readout circuit. The results indicate that in the differential actuation mode, a differential sensing strategy provides much better accuracy than the conventional non-differential schemes. Using a laser interferometer as an independent collocated sensor for stage displacement, we obtained constant calibration factors both proposed differential sensors and characterized their sensing accuracies.

It is well known that some practical difficulties are involved in the implementation of stabilising model predictive control for time-varying systems. In order to address the difficulty of computational load, this paper extends the orthonormal function method for model predictive control to linear time-varying systems. We provide sufficient conditions for a sub-optimal model predictive controller to be stabilising for a time-varying system. It is also shown that the orthonormal parametrisation method enables us to reduce the number of decision variables significantly and with a satisfactory performance. In addition, it is shown that orthonormality and, the called for, long prediction horizons are not necessary for stability. Examples are provided, illustrating the effectiveness of the method for linear time-varying systems.

We address design, implementation, and characterization of semiconductor strain gauges as stage displacement sensors with small footprint for piezo-driven nanopositioners in differential actuation mode. The results indicate that in this actuation mode, a differential sensing strategy provides much better accuracy than the conventional non-differential schemes. The proposed differential sensor has a large sensitivity of $4.4\frac{V}{{\mu {\text{m}}}},$ 3σ-resolution of 2.4 nm, and bandwidth of 4 kHz.

We study a novel piezo-driven nanopositioning mechanism in the horizontal plane. For each horizontal axis, we employ two externally leverage mechanisms with flexure hinges to provide bilateral displacement of the output stage as well as amplified displacement with respect to strokes of two piezo stack actuator. The bilateral amplified motion is achieved by differential actuation of the PZT stack pair of each axis. We also designed a housing structure for the nanopositioner with holes and trenches for safe and neat wiring. It also provides proper installation on the optical table and allows incorporation of auxiliary parts to hold and and align capacitive displacement sensors for precise measurements. We designed wedge mechanisms that together with the housing and the nanopositioner structure allow proper alignment of PZT stacks during installation as well as preloading them. Experiments were carried out to identify the ranges of displacements for the output stage as well as the inputs of the leverage mechanism, using Laser-Doppler-Vibrometry and capacitive sensors. We also developed a simple rigid-link-ideal-hinge kinematic model for the leverage mechanism, which was consistent with the experimental results under no external load conditions. However, due to the external loads and elasticity, large deviations exist between the experimental results and the predicted values by the model. The discrepancy revealed a non-reciprocal property of the individual leverage mechanism and the need to employ more accurate flexure hinge models. Experiments show that the proposed nanopositioner amplifies the input stroke of the PZT stacks by a factor around 12 in the differential actuation mode. Compared to the conventional non-differential actuation modes, the differential one provides almost twice stroke for the output stage as well as more linear input–output characteristics. In addition, the proposed structure considerably filters out the off-axis input displacements of the PZT actuators and provides very small parasitic displacements at the output stage. Both channels exhibit almost identical dynamic responses in time and frequency domains, indicating highly symmetric fabrications of the nanopositioner and auxiliary parts for the installation of actuators and sensors.

This paper presents a Laguerre parametrization approach to employ Model Predictive Control (MPC) for the trajectory tracking problem of a non-holonomic mobile robot with input and state constraints. A time-varying error model is obtained for the trajectory tracking of the mobile robot. Then, a Laguerre based MPC (LMPC) for time-varying systems is designed and tuned to ensure asymptotic stability of the system. The proposed algorithm considers input and states, including velocity and acceleration, constraints to provide stability. It is shown that the proposed method is able to reduce the computation times. In order to confirm the effectiveness of the proposed method, extensive simulations results are provided.

It has been shown that Laguerre functions successfully reduce the number of decision variables in model predictive control of linear time invariant (LTI) systems. In this paper we extend the use of Laguerre functions to deal with linear time-varying (LTV) systems in the model predictive control framework. It is shown that Laguerre functions enable us to reduce the number of decision variables significantly and with a satisfactory performance. An example illustrating the unconstrained and constrained cases is provided.