Neural network based predictive control with optimized search space for dynamic tracking of a piezo-actuated nano stage

Abstract

This paper presents an intelligent modified predictive control approach with squeezed search space, for tracking control of piezo-actuated nano stage. The model obtained from the gray box neural network is first dynamically linearized to avoid calculation of inverse hysteresis model. The optimum control values of the previous control cycle are used to construct a squeezed search space, which reduces the computation burden and improves the tracking control performance. The effectiveness of the proposed scheme is verified theoretically by deriving a convergence analysis and by experimental results. The results show that the proposed approach significantly improves the dynamic tracking performance for high-frequency reference signals than existing results in the literature.

Keywords

Nano-positioning piezo actuator hysteresis intelligent control

1. Introduction

Piezo-actuated (PZT) nano systems are being increasingly used in industry for nanopositioning (Al-Nadawi et al., 2021b) and tracking (Al-Nadawi et al., 2021a; Chowdhury et al., 2022) applications. But the critical challenge, in achieving the nano scale accuracy for these applications, is intrinsic hysteresis nonlinearity which reduces its tracking accuracy. Therefore significant research has been done on hysteresis modeling and control in recent past (Devasia et al., 2007).

Hysteresis models presented in literature can be classified into physics-based models and phenomenology-based models. Physics-based models have clear physical meaning but they usually have complex mathematical structure and high computational cost, for example, Preisach model (Nguyen et al., 2018), Prandtl-Ishlinskii (PI) model (Li et al., 2020), Maxwell model (Madsen et al., 2021). The problem with these models is their rate-independent nature that is, they can describe the hysteresis behavior for a fixed input frequency signal only. On the other hand, phenomenology-based models for example, Duhem model (Wang and Chen, 2017), Buce Wen (Wang et al., 2015) etc. have simple mathematical structure which makes them very popular in hysteresis modeling but their parameter identification is a complicated problem.

To control the effects of hysteresis, one of the predominant approach is using inverse hysteresis model as feedforward compensator (Li et al., 2019; Wang et al., 2014; Yang et al., 2021). Although the open loop feedforward compensation method shows reasonable control performance, the system’s robustness against external disturbances and modeling uncertainties still needs to be ensured. Therefore, the combination of feedback control loop with feedforward compensation is often presented in the literature Mao et al. (2018). Zhang and Yan (2021) presented a robust nano-positioning control scheme for systems with sensor delays. Recently some neural network based modeling methods are becoming popular due to their model identification properties. Elman neural network–based hysteresis model is presented in Zhao et al. (2020) and a dynamic hysteretic operator is used to handle the multivalued problem of hysteresis. To avoid computing inverse hysteresis model a neural network based adaptive controller scheme is presented in Zhao et al. (2021). Although the conventional neural network based modeling and control methods are proven to be effective but training time is too long which increases computation burden. Gray box neural network based model presented in Ahmed and Yan (2021) can describe the behavior hysteresis behavior of piezo-actuated nano stage with high precision and low computation burden. Due to its high performance and constraints handling abilities, Model Predictive Control (MPC) is widely used in industrial applications (Gong et al., 2016; Zhou et al., 2021). An inverse ferromagnetic material hysteresis compensation with MPC based feedback control is proposed in Cao et al. (2013) to control the piezo-actuated system.

Finding an inverse hysteresis model is a complicated task, particularly for high dynamic applications such as tracking. Therefore, Cheng et al. (2015) and Liu et al. (2015) presented MPC based tracking control scheme where inverse hysteresis model is no longer needed. In Cheng et al. (2015) Nonlinear AutoRegressive Moving Average with eXogenous input (NARMAX) based hysteresis model was proposed using neural networks and nonlinear MPC (NMPC) method was used to implement the tracking control, hence avoiding the calculation of inverse hysteresis model. A dynamically linearized neural network model along with MPC controller is presented in Liu et al. (2015). But in optimum control based schemes, the control law is obtained by solving a complex optimization problem during each control cycle, which slows down the control action, making it less suitable for high frequency tracking references.

This paper proposes a modified predictive control approach with squeezed search space supporting high dynamic tracking performance. To approximate the behavior of piezo-actuated nano stage, a gray box neural network based displacement predictor is proposed. The displacement predictor is linearized in each control cycle using dynamic linearization method to avoid inverse model for the compensation of nonlinearities such as hysteresis and creep. The modified predictive controller utilizes the optimal solution values of the previous control cycle to construct a new optimized search space during each control cycle. This can solve the slow convergence problem of the predictive controller. In addition to that, convergence analysis of modified predictive control scheme is presented, which guarantees a finite time convergence as well as the upper bound of the time of convergence under mild assumptions. Finally, to verify the effectiveness of the proposed scheme, experiments are conducted on the piezo-actuated nano stage.

2. System description and problem formulation

The high performance tracking control of piezo-driven nano systems, is a challenging task due to inherent nonlinearities such as rate-dependent hysteresis and creep. Experimental setup to excite the piezo-actuated X-Y nano stage is shown in Figure 1. The high bandwidth voltage amplifier is used to drive the piezo-actuated nano stage. The displacement of the nano stage is measured using MicroE systems Mercury II 6000 series linear encoder, with 1.2 nm resolution and maximum speed of 61 mm/s. NI-6259 data acquisition card is used to extract the displacement data from the linear encoder at the sampling rate of 20 kHz. The control signals are generated by Matlab Simulink/xPc Target. NI-6259 data acquisition card communicates between the host computer and the high bandwidth voltage amplifier. Figure 2 shows the response of the piezo-actuated nano stage against the 1 Hz triangular input signal with decreasing amplitude. It can be seen that nano-stage shows strong nonlinear hysteresis behavior. The static hysteresis behavior of the nano-stage increases with the increase in the amplitude of the input signals. Thus to implement tracking control, it is necessary to develop a model to describe the behavior of nano stage.

Figure 1.

Experimental setup to collect input/output data from piezo-actuated nano stage.

Figure 2.

Hysteresis response of piezo-actuated nano stage using triangular input signal.

2.1. Neural network based model of piezo-actuated nano stage

The overall neural network model of the piezo-actuated nano stage is shown in Figure 3. The gray box neural network is used to model the quasi-static hysteresis behavior 3(a). The dynamic behavior of the piezo-actuated nano stage is described by NARMAX-based dynamic sub-model 3(b). For the gray box neural network-based static sub-model the displacement $d_{H} (k)$ is given as (Ahmed and Yan, 2021):

\begin{array}{l} d_{H} (k) = d_{o} + Δ d_{o} T (1 - γ_{o} e^{\frac{- | u (k) - V_{shift} (k) |}{V_{c o}}}) u (k) \\ + h_{s} f [u (k)], \end{array}

(1)

d_{H} (k) = ϕ_{1} + ϕ_{2},

(2)

where $d_{o}$ is the initial condition, $T$ is the sampling time, $h_{s} \in ℜ$ is hysteresis mapping coefficient, $V_{shift} \in u (k)$ is the value of input voltage at last zero value of input derivative, $Δ d_{o}, γ_{o}, h_{s}$ , and $V_{co}$ are the unknown parameters, $ϕ_{1} = d_{o} + h_{s} f [u (k)]$ $ϕ_{2} = Δ d_{o} T (1 - γ_{o} e^{\frac{- | u (k) - V_{shift} (k) |}{V_{co}}}) u (k)$ and $f [u (k)]$ are given as (Ahmed and Yan, 2021):

f [u (k)] = \frac{u (k) - u (k - 1)}{T}

(3)

Figure 3.

Neural network based model of piezo-actuated nano stage: (a) static sub model of piezo-actuated nano stage (Ahmed and Yan, 2021) and (b) dynamic sub-model of piezo-actuated nano stage.

To describe the dynamics of the system (Figure 3), NARMAX based dynamic sub-model, is defined as:

y_{D} (k) = \sum_{j = 1}^{m} v_{j}^{o} σ (\sum_{i = 1}^{l} v_{ij}^{h} ψ_{i} (t_{k})),

(4)

where $l$ and $m$ are the number of neurons in input and hidden layers respectively, such that $l = n_{y} + n_{u} + 1$ . $v^{0}$ and $v^{h}$ are the weights of output and hidden layer, $n_{y}$ and $n_{u}$ respectively are the lags of output and input and $ψ = [y (k - 1), y (k - 2) \dots y (k - n_{y}), u (k), u (k - 1) \dots u (k - n_{u})]$ . Levenberg-Marquardt (LM) method can be used to train the neural network. $σ$ is the tangent sigmoid activation function $σ (x) = (e^{2 x} - 1) / (e^{2 x} + 1)$ . The overall output $y (k)$ of nano stage is given as:

y (k) = d_{H} (k) + y_{D} (k) .

(5)

2.2. Dynamic linearization of neural network based model

Dynamical linearization of the model gives the precise approximation of the behavior of the piezo-actuated nano stage, since it is linearized specifically for the current time sample (Cheng et al., 2015). Taylor series expansion of feedforward neural network presented in (5), at time $k_{p}$ , is given by

\begin{matrix} y (k) - y (k_{p}) = a_{1} [y (k - 1) - y (k_{p} - 1)] \\ + a_{2} [y (k - 2) - y (k_{p} - 2)] \dots \\ a_{n_{y}} [y (k - n_{y}) - y (k_{p} - n_{y})] \\ + b_{0} [(u (k) - u (k_{p})) (ϕ_{2} (k) - ϕ_{2} (k_{p}))] \dots \\ b_{n_{u}} [u (k - n_{u}) - u (k_{p} - n_{u})] + ϕ_{1} (k), \end{matrix}

(6)

\begin{matrix} y (k) = a_{1} y (k - 1) + a_{2} y (k - 2) \dots \\ a_{n_{y}} y (k - n_{y}) + b_{0} u (k) ϕ_{2} (k) \dots \\ b_{n_{u}} u (k - n_{u}) + ϕ_{1} (k) + ζ (k_{p}), \end{matrix}

(7)

where, $a_{i} \in ℜ$ , for $i = [1, 2 . . ., n_{y}]$ and $b_{j} \in ℜ$ , for $j = [1, 2 . . ., n_{u}]$ , are the linearization parameters, $ζ (k_{p})$ , is the constant bias term which represents the nonlinearities and disturbances in the system.

At each control cycle linearized dynamic model of piezo-actuated nano stage is obtained, which can be used to construct the displacement predictor for the modified predictive controller.

2.3. Problem formulation

Consider the displacement function $y (k)$ in (7) for tracking control of piezo-actuated nano stage. The predictive controller solves a complex fitness function $J$ , in given search space, to obtain control law.

J = \sum_{k = 1}^{N} {(r (k) - \hat{y} (k))}^{2} + \sum_{k = 1}^{N} Δ u (k)^{2},

(8)

where $r (k)$ and $\hat{y} (k)$ are the reference and augmented outputs and $Δ u (k)$ is the change in input. Search space of fitness function is defined by constraints and the upper and lower limits of the decision variables. Since for hysteresis based system such as piezo-actuated nano stage, $y (k) \propto u (k)$ , the search space is formed by constraints on input $u_{\max}$ and $u_{\min}$ (Figure 4(a)).

Figure 4.

Formulation of search space in: (a) conventional predictive controller, (b) modified predictive controller, and (c) search space formation.

Finding optimum solution in large search space can slow down the optimization process which can deteriorate the tracking control performance. One of the solution to this problem is to reduce the number of decision variables (Zimmer et al., 2015), but this can lead to compromise the tracking performance.

Therefore, modified predictive controller with squeezed search space is proposed in this work Figure 5. The main idea is to construct the squeezed search space in real time which can reduce the computation burden and improve the tracking performance. The search space in conventional predictive control is formed by maximum displacement that can be obtained by the actuator that is, $y_{\max}$ and $y_{\min}$ , and the saturation input voltage $u_{\max}$ and $u_{\min}$ . Assuming the bounded disturbance $δ \in D$ and high precision of neural network based output model, the predictive horizon values, from previous control cycle, are used to predict the optima of fitness function for current control cycle. The predicted optima values are then used to construct a squeezed search space (Figure 4(b)). The search space in proposed modified predictive controller is formed by adjusting the upper and lower limits of decision variables for current control cycle that is, $u_{UB}$ and $u_{LB}$ . The modified predictive control formulation in following section will describe the necessary conditions, and method to construct the squeezed search space.

Figure 5.

Schematic diagram of modified predictive control.

3. Formulation of modified predictive control

This section presents the displacement predictor, for modified predictive controller, based on dynamically linearized model. The displacement predictor is then used with modified predictive control formulation to implement tracking control.

3.1. Displacement predictor for predictive control scheme

Since during one control cycle, bias term $ζ (k_{p})$ , remains constant thus it can be eliminated by employing adjacent difference of the predicted displacement (7). Predicted displacement at $j^{th}$ time step can be found as follows:

\begin{matrix} \hat{y} (k + j) = \hat{y} (k + j - 1) (1 - a_{1}) + \hat{y} (k + j - 1) (a_{2} - a_{1}) \dots \\ - \hat{y} (k + j - n_{y} - 1) a_{n_{y}} + Δ u (k + j) ϕ_{2} (k + j) b_{o} \\ + Δ u (k + j - 1) b_{1} \dots u (k + j - n_{u}) b_{n_{u}} . \end{matrix}

(9)

Let’s define, $\hat{Y} (k) = {[\begin{matrix} \hat{y} (k + 1) & \dots & \hat{y} (k + N_{y}) \end{matrix}]}^{T}$ , $Y (k) = {[\begin{matrix} y (k) & y (k - 1) & \dots & y (k - n_{y}) \end{matrix}]}^{T}$ , $Δ U^{'} (k) = {[\begin{matrix} Δ u (k) & Δ u (k - 1) & \dots & Δ u (k - n_{u} + 1) \end{matrix}]}^{T}$ , $Δ U (k) = {[\begin{matrix} Δ u (k + 1) & Δ u (k + 2) & \dots & Δ u (k + N_{y}) \end{matrix}]}^{T}$ ,

where $y (k)$ and $\hat{y} (k)$ are the measured and augmented values of output, and $Δ u (k) = u (k) - Δ u (k - 1)$ . For the prediction horizon, $N_{y}$ , the predicted displacement is expressed as:

\hat{Y} (k) = G Δ U (k) + H Δ U^{'} (k) + SY (k),

(10)

where $G^{'} \in ℜ^{N_{y} \times N_{y}}, H^{'} \in ℜ^{N_{y} \times n_{u}}, S^{'} \in ℜ^{N_{y} \times (n_{y} + 1)}$ , and $K \in ℜ^{N_{y} \times N_{y}}$ are the matrices of dynamic linearization from Liu et al. (2015), such that, $G = K^{- 1} G^{'}, H = K^{- 1} H^{'}$ , and $S = K^{- 1} S^{'}$ . At time $k + 1$ , the general form of $\hat{Y} (k)$ is given as:

y (k + 1) = F (y (k), U (k), δ (k)),

(11)

where $Δ U (k) \in ℜ^{N_{y} \times 1},$ $Δ U^{'} (k) \in ℜ^{(n_{y} + 1) \times 1}$ , $Y (k) \in ℜ^{n_{y} \times 1}$ , $y (k) \in Y, u (k) \in U, δ (k) \in D$ , and $D \in ℜ^{N_{y} \times 1}$ is the set of bounded disturbance. Since displacement predictor is derived from dynamic linearization model, the prediction horizon $N_{y}$ should not be too large because the accuracy of linearized model decrease as we move away from current operating point.

Based on predicted displacement in (10) the objective function of conventional predictive controller for the horizon $N_{y}$ is as follows:

\begin{matrix} J_{N_{y}} (Y, Δ U, R) = [R (k) - \hat{Y} (k)] [R (k) - \hat{Y} (k)]^{T} \\ + ρ Δ U (k) Δ U^{T} (k), \end{matrix}

(12)

J_{N_{y}} (Y, Δ U, R) = J_{r} (Y) + J_{Π (r)} (Δ U),

(13)

where $J_{r} (Y) = [R (k) - \hat{Y} (k)] [R (k) - \hat{Y} (k)]^{T}$ and $J_{Π (r)} (Δ U) = ρ Δ U (k) Δ U^{T} (k)$ . $R (k) \in ℜ^{(N_{y} \times 1)}$ is the matrix representing the desired reference trajectory and $ρ$ is the term that limits the rate of change of input hence avoiding lower gain margin problem, $J_{r} (Y)$ and $J_{Π (r)} (U)$ are the Euclidean distances from $\hat{y}$ to $r$ and from $u$ to $Π (r)$ respectively and $Π (r)$ is the set of input $u (k) \in U (k)$ which can keep system at convergence $r$ once system has already achieved it that is, $Π (r) \dot{=} {u \in U : \exists y \in r | F (y, u, δ) \in r, \forall δ \in D}$ . If there are no constraints on $Δ U (k)$ the control law can be determined from quadratic problem presented in (12) as:

\begin{matrix} Δ U (k) = {(G^{T} G + ρ I)}^{- 1} G^{T} \\ (R (k) - H Δ U^{'} (k) - S Y^{'} (k)) . \end{matrix}

(14)

Since the first value of matrix $Δ U (k)$ is used as control increment, thus control signal $u (k + 1)$ is given by,

u (k + 1) = u (k) + Δ u (k + 1) .

(15)

3.2. Modified predictive controller

Predictive controller solve fitness function, in given search space, to obtain control law in every control cycle. Search space is defined by constraints and the upper and lower limits of the decision variables.

Modified predictive controller use the predicted horizon of previous control cycle to estimate the possible optima. This information is then used to construct a new squeezed search space (Figure 4(b)), thus improving the tracking performance.

First the tolerance factor $σ_{m} \in ℜ^{2 \times 1}$ is defined. $σ_{m}$ indicate the maximum allowable deviation of augmented output $\hat{y} (k)$ from the measured output value $y (k)$ . If the augmented output is within the tolerance range of measured output, the upper and lower bounds of decision variables can be redefined to construct the new search space. From (10),

\hat{Y} (k - 1) = {\hat{Y}}_{Δ U} (k - 1) + {\hat{Y}}_{Δ U^{'}} (k - 1) + Y_{y_{m}} (k - 1),

(16)

where ${\hat{Y}}_{Δ U} (k - 1) = G Δ U (k - 1)$ , ${\hat{Y}}_{Δ U^{'}} (k - 1) = H Δ U^{'} (k - 1)$ , and $Y_{y_{m}} (k - 1) = S Y^{'} (k - 1)$ .

The augmented output $\hat{y} (k)$ can be found as:

\hat{y} (k) = C {\hat{Y}}_{Δ U} (k) + C {\hat{Y}}_{Δ U^{'}} (k) + C Y_{y_{m}} (k),

(17)

where $C \in ℜ^{1 \times N_{y}}$ is defined as $C = [\begin{matrix} 1 & 0 & 0 & \dots & 0 \end{matrix}]$ , and from tolerance term $σ_{m} = {[\begin{matrix} σ_{LB} & σ_{UB} \end{matrix}]}^{T}$ we get,

\hat{y} (k) σ_{LB} \leq y_{m} (k) \leq σ_{UB} \hat{y} (k),

(18)

where $y_{m} (k)$ is the measured value of output from previous control cycle. If the criterion in (18) is fulfilled, then search space can be updated as follows. Form (16),

C {\hat{Y}}_{Δ U} (k - 1) = \hat{y} (k) - C {\hat{Y}}_{Δ U^{'}} (k - 1) - C {\hat{Y}}_{Y_{m}} (k - 1),

(19)

Form (18) the upper and lower bounds of search space at time $k$ can be found as:

[\begin{matrix} Δ u_{LB} (k) \\ Δ u_{UB} (k) \end{matrix}] = \frac{1}{b_{0}} [\begin{matrix} \hat{y} (k) μ_{LB} \\ \hat{y} (k) μ_{UB} \end{matrix}] - \frac{1}{b_{0}} [\begin{matrix} C {\hat{Y}}_{Δ U} (k - 1) \\ C Y_{y_{m}} (k - 1) \end{matrix}],

(20)

where $Δ u_{LB} (k) \in Δ U_{LB}, Δ u_{UB} (k) \in Δ U_{UB}$ such that $Δ U_{LB}, Δ U_{LB} \in ℜ^{N_{y} \times N_{y}}$ are the upper and lower bounds input search space and $μ = {[\begin{matrix} μ_{LB} & μ_{UB} \end{matrix}]}^{T}$ defines the upper and lower limits of search space. If the criterion in (18) does not meets, then search space can be updated as:

[\begin{matrix} Δ u_{LB} (k) \\ Δ u_{UB} (k) \end{matrix}] = [\begin{matrix} Δ u_{max} \\ Δ u_{min} \end{matrix}],

(21)

where $Δ u_{min}$ and $Δ u_{max}$ are the minimum and maximum values of $Δ u$ depending on dynamics of system and constraints on $Δ u$ . For modified predictive controller the value of $μ$ and $σ_{m}$ depends on the accuracy of augmented output, (7). If the augmented output is closer to the measured one, smaller values of $μ$ and $σ_{m}$ may be selected. Choosing higher values of $μ$ and $σ_{m}$ will increase the size of search space which will increase the computation burden to find optimal solution.

3.3. Modified predictive controller with integral error compensation

The predictive controller generates the control actions similar to classical proportional controller thus modeling uncertainties may introduce a steady state error. Therefore an integral compensation term is proposed to avoid steady state error (Liu et al., 2015). The final control law with integral compensation term $u_{e} (k)$ is as follows:

u (k) = \frac{K_{y}}{(1 + K_{u} z^{- 1}) (z - 1)} (\frac{K_{r}}{K_{y}} r (k) - y (k) + u_{e} (k)),

(22)

where $z$ is the shift operator, and,

\begin{matrix} K_{r} = K_{g} {[\begin{matrix} z & z^{2} & \dots & z^{N_{y}} \end{matrix}]}^{T} \\ K_{u} = K_{g} H {[\begin{matrix} 1 & z^{- 1} & \dots & z^{- (n_{u} + 1)} \end{matrix}]}^{T} \\ K_{y} = K_{g} S {[\begin{matrix} 1 & z^{- 1} & \dots & z^{n_{y}} \end{matrix}]}^{T} \end{matrix}

(23)

$K_{g}$ is the first row of ${(G^{T} G + ρ I)}^{- 1} G^{T}$ and the error compensation term $u_{e} (k)$ , in (22), is defined as:

u_{e} (k + 1) = u_{e} (k) + K_{e} (r (k + 1) - y (k + 1),

(24)

where $K_{e}$ is the gain of integral action.

The objective function for the modified predictive control, $J_{N_{y}}^{m}$ , can defined as:

\begin{matrix} J_{N_{y}}^{m} (Y, U, R) = [R (k) - \hat{Y} (k)] [R (k) \\ {- \hat{Y} (k)]}^{T} + ρ U (k) U^{T} (k), \end{matrix}

(25)

J_{N_{y}}^{m} (Y, U, R) = J_{r}^{m} (Y) + J_{Π (r)}^{m} (U),

(26)

where $J_{r}^{m}$ and $J_{Π (r)}^{m}$ are analogs to $J_{r}$ and $J_{Π (r)}$ of conventional predictive controller in (13).

4. Convergence analysis of modified predictive controller

This section investigates the necessary conditions for the convergence and upper bound of convergence time for the modified predictive control formulation.

Displacement predictor from (11) is given as:

y (k + 1) = F (y (k), U (k), δ (k)),

(27)

with optimal control,

\begin{matrix} J_{N_{y}} (Y, R, U) = [R (k) - Y (k)] [R (k) - Y (k {)]}^{T} + \\ [Π_{N} (u) - U (k)] {[Π_{N} (u) - U (k)]}^{T}, \end{matrix}

(28)

J_{N_{y}} (Y, R, U, δ) = J_{r} (Y) + J_{Π (r)} (U) .

(29)

The finite time convergence to a target set $r$ can be guaranteed, if the target set $r = [R, Δ U_{UB}, Δ U_{UB}]$ is chosen to be disturbance $γ -$ control invariant set, for open loop system, and its associated input set $Π (r)$ is also consider in cost function. Before we proceed further let’s first define disturbance $γ -$ control invariant set $(D γ - CIS)$ and N-step disturbance controllable set $C^{N} (r, U)$ .

Disturbance $γ -$ control invariant set is defined as, for a given invariance $γ = (0, 1]$ , the target set $r \in \hat{Y}$ is disturbance $γ -$ control invariant set $(D γ - CIS)$ , if there exists an associated input $u \in U$ such that $F (y, U, δ) \in γ r$ for all $y \in r$ and $δ \in D$ .

N-step disturbance controllable set Given an admissible control input set $u (k) \subset U (k)$ , the one-step disturbance controllable set $C (r, u)$ , for desired convergence $r$ , exists such that

C (r, u) \dot{=} {y \in Y : \exists u \in U | F (y, u, δ) \in r \forall δ \in D} .

(30)

In other words $C (r, u)$ is a set of all $y \in Y$ for which there exists a control input $u \in U$ , such that the system will converge to $r$ in one step regardless of disturbance (bounded) effect that is, $F (y, u, δ) \in r .$ Similarly, an N-step disturbance controllable set can be defined as:

C^{N} (r, u) \dot{=} C [C^{N - 1} (r, u), u] .

(31)

If $r \in Y^{o}$ is the disturbance closed invariant set with $γ < 1$ and $Π (r)$ as input set, then $r \subseteq C (r, Π (r))$ , where $Y^{o}$ is the boundary set of $Y$ . In the following it is shown that if the desired convergence is a disturbance closed invariant set with invariance $γ = (0, 1]$ and its associated input $Π (r)$ is considered in cost functions (13) and (26) of the predictive controller, then a finite time convergence to $r$ can be ensured and modified predictive controller can ensure faster convergence.

Lemma 1. For the predictive controller defined in (11) and cost functions in (29), the closed loop system $y (k + 1) = F (y (k), U (k + 1), δ (k))$ satisfies,

\begin{matrix} J_{N_{y}}^{m} (y (k + 1), r) - J_{N_{y}}^{m} (y (k), r) \leq \\ J_{N_{y}} (y (k + 1), r) - J_{N_{y}} (y (k), r) \\ \leq - J_{r} (y (k)) - J_{Π (r)} (u (k)), \end{matrix}

(32)

for all $k > 0$ , $lim_{k \to \infty} J_{r} (y (k)) = 0$ , and $lim_{k \to \infty} J_{Π (r)} (u (k)) = 0$ where $y = y (0) \in C^{N} (r, U)$ , $\forall$ $δ \in D$ .

Proof. For the time $k$ , let $y (k) \in C^{N} (r, U)$ , the optimal cost form (13) is given by:

J_{N_{y}} (y, r) = min \sum_{i = 0}^{N_{y}} J_{r} (y_{o} (i; y) + J_{Π (r)} (u_{o} (i; y))),

(33)

where $y_{o} (i; y)$ and $u_{o} (i; y)$ for $i = {1 \dots N_{y}}$ are optimal output and input trajectories. Suppose that $y^{+} = y_{o} (1; y)$ is the successor optimal output value, such that $y^{+} \in {y_{o} (1; y, δ)}$ , and ${y_{o} (N_{y}; y, δ)} \subset r$ . Then ${\hat{u}}_{o} \dot{=} {u_{o} (1; y), \dots u_{o} (N_{y}; y), \hat{u}}$ can be a possible solution at time $k + 1$ to problem in (11) where $\hat{u} \in Π (r)$ and ${\hat{y}}_{o} = F (y_{o} (N_{y}; y), \hat{u}, δ) \subset r$ such that $r$ is the disturbance closed invariant.

Consider control trajectory ${\hat{u}}^{m}$ for optimization problem (26), where ${\hat{u}}^{m} \dot{=} {{u^{m}}_{o} (1; y), \dots u_{o}^{m} (N_{y}; y)}$ and $Δ U_{LB} \leq Δ U \leq Δ U_{UB}$ . Assuming (18) holds, we get:

J_{N_{y}}^{m} (y^{+}, r; {\hat{u}}_{o}) \leq J_{N_{y}} (y^{+}, r; {\hat{u}}_{o}) .

(34)

Since at any time $k$ cost function take into account the worst case scenario, the feasible cost function is written as:

\begin{matrix} J_{N_{y}} (y^{+}, r, \hat{u}) = J_{N_{y}}^{o} (y^{+}, r) + J_{r} (\hat{y}) \\ + J_{Π (r)} (\hat{u}) - J_{r} (y) + J_{Π (r)} ({\hat{u}}_{o} (0; y)), \end{matrix}

(35)

where $J_{r} (\hat{y}) = 0$ and $J_{Π (r)} (\hat{u}) = 0$ , and from optimality theory,

J_{N_{y}} (y^{+}, r) \leq J_{N_{y}} (y^{+}, r; {\hat{u}}_{o}),

(36)

thus,

J_{N_{y}} (y^{+}, r) - J_{N_{y}} (y, r) \leq - J_{r} (y) - J_{Π (r)} (u_{o} (0; y)),

(37)

from (34),

\begin{matrix} J_{N_{y}}^{m} (y^{+}, r) - J_{N_{y}}^{m} (y, r) \leq J_{N_{y}} (y^{+}, r) - J_{N_{y}} (y, r) \\ \leq - J_{r} (y) - J_{Π (r)} (u_{o} (0; y)) . \end{matrix}

(38)

This proves that for any time $k > 0$ , the property in (32) is satisfied by optimal cost function, and $lim_{k \to \infty} J_{r} (\hat{y}) = 0$ and $lim_{k \to \infty} J_{Π (r)} (\hat{u}) = 0$ , provided $J_{N_{y}}^{o} (y^{+}, r)$ is positive.

The following lemma illustrates the role of $Π (r)$ in MPC formulation (29).

Lemma 2. Consider predictive control formulation in (11), with optimal cost (29) and $u_{o}$ is the initial condition, if $y$ is N-step disturbance controllable that is, $y \in C^{N} (r, Π (r))$ then, $F (y (k), u_{o}, δ (k)) \in r$ for all bounded disturbances $δ \in D$ .

Proof. Let $y (0) = y \in C (r, Π (r))$ such that there is a control input $u_{o} \in Π (r)$ then,

y (1) = F (y (0), u_{o}, δ (0)) \in r, \forall δ (0) \in D .

(39)

Provided that target set $r$ is disturbance $γ$ invariance set, and there exists a control input $u (k) \in Π (r)$ for which $y (k) \in γ r \subseteq r$ $\forall k = i :$ where $i = 1, 2 \dots N_{y - 1}$ , then we get the cost:

\begin{array}{l} J_{N_{y}} (y, r, u) = J_{r} (y (0)) + J_{Π (r)} (u_{0}) + \min \sum_{i = 0}^{N} [J_{r} (y (i)) + J_{Π (r)} (u (i))] \\ = J_{r} (y (0)) . \end{array}

(40)

Lemma 2 shows that once the system has achieved convergence $r$ , the control action bounded by the set $Π (r)$ does not add positive cost to the cost function of modified predictive control defined in (29).

The following theorem will establish the finite time convergence and upper bound for the time of convergence.

Theorem 1. For the predictive control formulation presented in (29), if $y = y (0) \in C^{N} (r, U)$ , then the convergence $r$ is reached in finite time by the system $y (k + 1) = F (y (k), Δ u (k + 1), δ (k))$ and the convergence $r$ is reached in upper bound of $k_{o}$ time steps, such that:

k_{o} = \frac{J_{N_{y}}^{o} (y, r)}{{min}_{y \in \partial C (r, Π (r))} d_{r} (y)} + 1 .

(41)

Proof. We take contradiction approach to prove the following theorem. Consider system output $y (0) \in C^{N} (r, U) / C (r, Π (r))$ and scalar value $p$ such that

p > \frac{J_{N_{y}}^{o} (y, r)}{{min}_{y \in \partial C (r, Π (r))} d_{r} (y)} = k_{o} - 1

(42)

and assume that,

y (k) \notin C (r, Π (r)),

(43)

that is, $- J_{r} (y (k)) \leq - min_{y \in \partial C (r, Π (r))} d_{r} (y) < 0$ for $k = 1, 2, \dots p$ , then from lemma 1:

\begin{matrix} J_{N_{y}} (y (k + 1), r) - J_{N_{y}} (y (k), r) \leq \\ - min_{y \in \partial C (r, Π (r))} J_{r} (y) < 0, \end{matrix}

(44)

and

\begin{matrix} J_{N_{y}}^{m} (y (k + 1), r) - J_{N_{y}}^{m} (y (k), r) \leq J_{N_{y}} (y (k + 1), r) \\ - J_{N_{y}} (y (k), r) . \end{matrix}

(45)

Summing up the inequality terms from $k = 0$ to $p$ ,

J_{N_{y}}^{m} (y (p), r) - J_{N_{y}}^{m} (y (k), r) \leq - p min_{y \in \partial C (r, Π (r))} J_{r} (y) .

(46)

This shows that:

J_{N_{y}}^{m} (y (p), r) \leq - p min_{y \in \partial C (r, Π (r))} J_{r} (y) + J_{N_{y}}^{m} (y (0), r) < 0,

(47)

which is a contradiction to the assumption made in (43) thus, it is necessary that $y (k) \in C (r, Π (r))$ for any $k \leq k_{o} - 1$ also from lemma 2 it is known that $y (k) \in r$ for any $k \leq k_{o}$ this concludes the proof.

5. Results and discussion

In order to verify the performance of proposed control scheme, experiments were conducted on piezo-actuated nano stage using setup shown in Figure 6. The control signal generated by Matlab Simulink/xPc Target, in the target PC, is fed to high bandwidth voltage amplifier, magnification factor of 20 and high precision with ripple less than 0.1 mV. NI-6259 data acquisition interface, with 16 bit resolution, is used as interface between voltage amplifier and target PC. The output of voltage amplifier was given to piezo-actuated nano positioning stage. To measure the output displacement of nano positioning stage, MicroE systems Mercury II 6000 series linear encoder, with high speed and high resolution, is used at a sampling rate of 10 kHz. To avoid the external disturbances, the experimental setup was mounted on an air flotation platform.

Figure 6.

Schematic diagram of experimental setup.

5.1. Verification of neural network dynamic model

First the static sub-module (Figure 3) was identified on an input frequency as low as 0.5 Hz. A low voltage DC signal is applied to estimate $d_{o}$ from (1), where it shows negligible hysteresis. An input voltage of 5 VDC was applied to piezo-actuated stage and displacement was measured. To train the static sub-model (Figure 3(a)), we need input/output data of the piezo-actuated nano stage at low frequency where nano-stage does not show any dynamic behavior. The input/output data of the piezo-actuated nano stage was recorded using the linear encoder at a frequency of 0.5 Hz. The parameters $h_{s}$ , $Δ d_{o}, γ_{o}$ , and $V_{co}$ from (1) are then found by training static submodel using LM training method.

The nano stage was excited using an input voltage of amplitude $0 - 20 V$ and frequency of 1–450 Hz to obtain the dataset for the training of the dynamical model. The model from (5) was then trained using the LM training method. After trying different combinations of the model parameters, best results were obtained on $n_{y} = 2$ and $n_{u} = 2$ with five hidden layers (Figure 3).

To validate the model performance, sinusoid input signal with constant frequency and varying amplitude was employed. Figure 7 shows the comparison between experimental results and predicted output of the model. The performance of the proposed neural network dynamic model with respect to mean squared error (MSE), root mean squared error (RMSE), and mean absolute error (MAE) metrics is presented in Table 1. The error of the predicted model with measured value is well within the tolerance range. This shows that the proposed model can approximate the behavior of the piezo-actuated nano stage with high precision.

Figure 7.

Comparison between predicted and actual displacement of piezo-actuated nano stage: (a) 1 Hz and (b) 300 Hz.

Table 1.

Performance analysis of neural network based dynamical model of piezo-actuated nano stage.

Error type	1 Hz input (nm)	300 Hz input (nm)
MSE	0.8410	6.1677
RMSE	0.9171	2.4835
MAE	0.8284	2.2474

5.2. Verification of modified predictive control scheme

This section presents the validation of modified model predictive control (MMPC) proposed in this article and its comparison with the previously presented state of the art MPC based method (Liu et al., 2015). From here on the method presented in Liu et al. (2015) will be referred to as MPC method in this article. For fair comparison both MMPC and MPC schemes are initialized with similar parameters.

For both MPC and MMPC methods, $ρ = 30$ from (12) and (26) respectively, whereas $N_{y} = 7$ , and sampling interval is $0.0001 s$ . The error tolerance factor $σ_{m}$ from (18) and search space range $μ$ from (20), in MMPC, can be chosen based on the accuracy of the model. Here we consider a control scheme to track various frequencies, so we choose both the parameters based on the error in model identification, thus $σ_{m} = \pm 1.35 μ m$ and $μ = \pm 1.5 μ m$ .

To evaluate the performance of the proposed control scheme, high-frequency reference tracking is performed and compared with MPC based state-of-the-art method presented in Liu et al. (2015) (Figures 8 and 9). Sinusoid references with frequencies 10, 100, 200, and 300 Hz are selected as input references. The results show that the performance of the proposed control scheme is slightly better than the conventional predictive controller for lower frequency tracking references. But as the frequency of input reference signal increases, the proposed modified predictive controller with squeezed search space outperformed the conventional predictive controller significantly, with lower tracking error. This shows that the modified predictive controller with squeezed search space has faster convergence properties and can precisely track high-frequency references. The maximum tracking error at 300 Hz is between $0.036 μ m$ and $- 0.17 μ m$ for the MMPC method, whereas $0.204 μ m$ and $- 0.22 μ m$ , for MPC scheme (Figure 8(b)) which is under acceptable range.

Figure 8.

Tracking performance of the predictive controller under reference of: (a) 10 Hz signal and (b) 100 Hz signal.

Figure 9.

Tracking performance of the predictive controller under reference of: (a) 200 Hz signal and (b) 300 Hz signal.

To further validate the performance of proposed modified predictive controller, we used non-periodic signals. A special non-periodic reference (48) was introduced in the experiment. Where $f_{\max}$ is the maximum frequency. Figure 10 shows non-periodic references with $f_{\max} = 100 Hz$ .

\begin{matrix} w = \frac{30}{7} \sin [\frac{2 π f_{max}}{20} (t - 0.1) + π] \\ + \frac{25}{7} \sin [\frac{2 π f_{max}}{5} (t - 0.1) + \frac{1}{2} π] \\ + \frac{5}{7} \sin [\frac{2 π f_{max}}{2} (t - 0.1) + \frac{1}{5} π] + 5 . \end{matrix}

(48)

Figure 10.

Performance of the proposed control scheme under mixed frequency references.

The performance of proposed scheme was satisfactory under mixed frequency signals. The maximum error range (Figure 10) was between $- 0.147 μ m$ to $0.192 μ m$ .

To compare the performance of proposed modified predictive control scheme with other existing state of the art predictive controller based schemes such as, the inversion-based model predictive control scheme (Cao et al., 2013) and neural network based nonlinear model predictive control scheme presented in Cheng et al. (2015), the experiments were conducted on sinusoid references of different frequencies. Table 2 shows the results of comparison between proposed scheme and existing techniques. The proposed method shows significant improvement than existing solution both in lower frequencies as well as higher frequencies reference signal.

Table 2.

Comparison between proposed and existing control schemes.

Reference
Frequency (Hz)	Proposed scheme error $(nm)$	Inversion-based model predictive control (Cao et al., 2013) error $(nm)$	Nonlinear model predictive control (Cheng et al., 2015) error $(nm)$
f = 1	2.3	6.2	7.6
f = 10	3.1	7.3	7.9
f = 50	3.7	9.4	18
f = 100	4.9	16	33
f = 150	8.6	27	49

The control performance of proposed scheme is further verified using triangular reference and compared with the one presented in Liu et al. (2015). Figures 11 and 12 shows the performance of both control schemes at triangular references of 100 and 200 Hz respectively. It is evident that proposed MMPC method out performed control scheme presented in Liu et al. (2015). The maximum error observed was $0.92 μ m$ and $0.7 μ m$ for MPC and MMPC respectively for 200 Hz frequency reference.

Figure 11.

Performance of the proposed control scheme under the triangular reference signal of 100 Hz.

Figure 12.

Performance of the proposed control scheme under the triangular reference signal of 200 Hz.

6. Conclusions

A modified predictive controller with squeezed search space is proposed in this paper which can track fast moving reference signals without relying on inverse plant model to compensate inherent nonlinearities (such as hysteresis and creep in case of piezo-actuated nano stage). First neural network based model of piezo-actuated nano stage was constructed and linearized using dynamic linearization technique. The linearized model was then used to implement the modified predictive controller proposed in the paper. The modified predictive controller can squeeze the search space of the optimum controller by using the optimum solutions of previous control cycle, hence improving the controller speed. An integral action was then introduced to reduce the possible steady state error. The effectiveness of proposed modified predictive controller was validated theocratically by convergence analysis of the proposed control scheme.

To further validate the performance of the proposed scheme, experiments were conducted on the piezo-actuated nano stage. The results show that the proposed scheme outperformed the predictive control based state of the art methods presented in the literature, especially in the case of high-frequency reference tracking. The tracking error improvement of about 68 $%$ and 82 $%$ was observed from the schemes presented in Cao et al. (2013) and Cheng et al. (2015), whereas up to 22% improvement from Liu et al. (2015) was observed.

Footnotes

Appendix

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported in part by the Major Basic Research Program of the Natural Science Foundation of Shandong Province under Grant ZR2019ZD08, and in part by the National Natural Science Foundation of China under Grant 51775319.

ORCID iD

Peng Yan

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