Control and hysteresis reduction in prestressed curved unimorph actuators using model predictive control

Abstract

Curved piezoceramic unimorph actuators exhibit strong nonlinearities due to their special architecture that enables large motion amplification. Among these nonlinearities, hysteresis is the most problematic as it makes it difficult to predict the displacement of the actuator for a given input voltage. Therefore, it has been difficult to use these actuators in precision displacement control applications. In order to overcome this difficulty, this research is focused on the development of an effective reference-tracking displacement control algorithm for such actuators. For this purpose, two linear (proportional–integral and internal model) and two nonlinear (sliding mode and model predictive sliding mode) controllers are designed and implemented. These controllers are applied to the curved piezoceramic unimorph actuator to control the displacement of the actuator for multiple sinusoidal voltage inputs at various frequencies. Experimental results are obtained, and their performance is compared both qualitatively and quantitatively. As a part of the model-based controller design, a new actuator model is also developed based on the mechanical second-order equation with an additional phase lag term to describe the hysteretic effect.

Keywords

Prestressed curved actuator hysteresis model predictive control sliding mode control

Introduction

Piezoelectric devices have been actively studied by many researchers, and their applications are expanding in many fields. These devices are lightweight, compact, relatively inexpensive, and generating very high forces. However, they typically exhibit very small mechanical displacements, and this constitutes one of the main disadvantages of the piezoelectric devices (Lawver, 2001).

For applications requiring large displacements, several stroke amplification actuators and mechanisms have been developed, including C-blocks (Moskalik and Brei, 1996), Moonies (Xu et al., 1991), RAINBOW (Haertling, 1994), leveraged stack (Prechtl and Hall, 1997), telescopic actuator (Brei et al., 2001), and piezoelectric hybrid actuator using hydraulic displacement amplification mechanism (Yoon et al., 2006). The objective of this research is real-time control of a prestressed curved unimorph actuator. These actuators are produced under the trade name THUNDER^® (THin layer UNimorph ferroelectric Driver and sEnsoR) (Lawver, 2001). The THUNDER actuator is widely used in many applications including precision positioning (Homer and Taleghani, 1999), airfoil shape control (Pinkerton and Moses, 1997), autonomous swimming vehicles (Borgen et al., 2003), active noise control (Jayachandran et al., 1999), and active aperture antennas (Fukashi et al, 2004; Granger et al., 2000; Yoon et al., 2000). This type of curved actuator can be fabricated by attaching a thin piezoceramic to a stainless steel substrate using a high-temperature polyimide adhesive, which is cyclically heated and cooled beyond the activation temperature of the adhesive (Capozzoli et al., 1999). During the cooling process, an internal stress is induced in the piezoceramic material due to the mismatch of the thermal expansion coefficients of the ceramic and metal combination, resulting in a prestressed curved actuator that can produce relatively large displacements. The mechanical output displacement is as high as 30 times the thickness of the device itself. Another example is an actuator called LIghtweight Piezo-composite Curved Actuator (LIPCA) (Yoon et al., 2002). In this actuator, the stainless steel substrate is replaced by a lightweight fiber–reinforced plastic to reduce the weight of the actuator. Thus, LIPCA can be fabricated to be lighter than THUNDER by up to 40%.

A number of models for the THUNDER actuator have been developed since the introduction of the device. These models can be categorized into three different types: static deformation, deformation by applied field, and hysteresis. The first two models, static deformation and deformation by applied field, are usually formulated by the finite element method (Chung and Kim, 2005; Taleghani, 2000; Usher et al., 2004) and other methods such as the Rayleigh–Ritz model (Aimmanee and Hyer, 2004), general shell theory (Washington and Song, 2000; Wieman et al., 2001), partial differential equation models with linear thermoelastic relations (Capozzoli et al., 1999), and Newtonian principles (Ball et al., 2003). The hysteretic effect of the curved actuator has been modeled using several theories such as free energy–based relationship (Ball et al., 2003), Preisach model (Zhou et al., 2003), and the ferroelastic switching model (Ball et al., 2007; Ball and Smith, 2004).

From the perspective of precision positioning, hysteresis introduces inaccuracy to the positioning system as the output displacement depends on the direction of the input signal change as well as the applied voltage. Therefore, it is desirable to reduce the hysteretic effect so that the piezoelectric device can exhibit a linear relationship between the input voltage and the output displacement. For this purpose, various control methodologies have been applied to different types of piezoelectric devices. Among them, feedforward control techniques based on the inverse model of the hysteresis have been prevalent (Ahrens et al., 2007; Bashash and Jalili, 2007; Croft et al., 2001; Iyer et al., 2005). This concept has been further investigated by integrating feedback controllers to increase the control accuracy for the systems with unknown disturbance, model uncertainty, and sampling effect. Controllers of this type include high-gain feedback control (Leang and Devasia, 2002), lead-lag feedback control (Song et al., 2005), and sliding mode control (SMC) and neural networks (Yu et al., 2005). In addition, feedback controllers without any feedforward hysteresis compensation have been designed for reducing hysteresis using fuzzy logic (Song and Washington, 1999) and H-infinity controls (Pare and How, 1998). However, applications of these controllers have been limited only to quasi-static (<0.05 Hz) position control problems. The applicability of many of these algorithms in real time is also lacking.

The main objective of this research is to reduce the hysteretic effect of the THUNDER actuator for a position control applications in real time using model-based control. Model-based controllers are ideal for these applications as they have the ability to structurally incorporate (or compensate) for nonlinearities and modeling uncertainties. In this study, the following four control methodologies are developed, experimentally evaluated, and compared: proportional–integral (PI) control, internal model control (IMC), SMC, and model predictive sliding mode control (MPSMC). In addition, for the design of the two model-based controls, a simple actuator model with hysteresis is proposed.

Actuator description

THUNDER actuators are thin, lightweight, composite unimorphs consisting of five different layers: a top aluminum conductor, a middle piezoceramic (lead zirconate titanate (PZT)) transducer, and a bottom stainless steel substrate with two additional adhesive layers for bonding. Among many different sizes and shapes of THUNDER actuators that are commercially available, the one used in this study is shown in Figure 1.

For the fabrication of the THUNDER actuators, those five layers are placed in a sandwich configuration, as shown in Figure 2. Then, the entire assembly is placed in an autoclave to be heated and pressurized so that the different layers are bonded together. When it is cooled down, the mismatch in thermal expansion coefficients causes the metal and ceramic layers to contract at different rates putting the ceramic in compression at room temperature. The result is a curved actuator with the prestressed PZT layer.

THUNDER actuators have a couple of advantages over other types of piezoelectric actuators. First, they produce considerably large displacements that are about 30 times more than conventional piezoceramic-based actuators. Although the amount of force that an actuator can generate tends to decrease as the actuator displacement increases, THUNDER actuators have force profiles that do not decrease significantly despite their large displacements. In addition, whereas bare PZT materials are brittle, THUNDER actuators are rugged due to the protective stainless steel substrate and the aluminum top cover. This property allows them to be used in harsh environments.

In spite of these advantages, they also have some drawbacks. Like any other piezoceramic material–based devices, THUNDER actuators exhibit significant nonlinearities such as hysteresis. Because of these effects, an accurate prediction of the actuator displacement is difficult to achieve for any given input voltage. This can apply a significant limitation to precision positioning applications. Moreover, the hysteretic effect depends on the frequency of the input voltage. When the frequency of the input voltage varies, the hysteresis curve changes its shape accordingly. As shown in Figure 3, the hysteresis curve for the case when the input frequency is 10 Hz shows two crossings at extremums, while the one for the case of 1 Hz does not.

The nonlinear behaviors are more pronounced for the THUNDER actuators than typical piezoceramic devices due to their nonconventional manufacturing process (Washington and Song, 2000). First, THUNDER actuators employ a softer PZT material than conventional piezoceramic actuators, which causes more hysteresis. Second, the initially curved shape introduces a geometrical nonlinearity in actuation. Finally, two adhesive layers used to bond the PZT and the metal layers together induce an additional nonlinearity.

THUNDER actuator modeling

Most of the currently existing models for THUNDER actuators are too complicated to be applied to a controller design. Thus, a simple model with appropriate accuracy suitable for the controller design will be developed in this section. This model is then applied to the design of two model-based controllers in this research.

The relationship between the displacement of the THUNDER actuator and the input voltage can be expressed as a simple phase lag in the time domain. Tsai and Chen (2003) developed a simple model called “variable delay and gain model” for the hysteresis of a piezoceramic actuator. In the following subsections, this model is discussed in detail and one of the main features of this model is applied to a new actuator model for two model-based displacement control algorithms in this study: IMC and the MPSMC.

Variable delay and gain hysteresis model

The basic concept of the variable time delay and gain model is as follows: If the input voltage is a sinusoid and the output displacement of the actuator has no phase lag, the voltage versus displacement plot will be a straight line with a certain slope. However, if the output has some phase lag, the plot will deviate from a straight line resembling a typical hysteresis curve shown in Figure 4. From this curve, it can be observed that (1) the phase lag is 0 when the input voltage is 0, (2) then it increases until the input voltage reaches half of the maximum, and (3) finally the phase lag decreases to 0 as the input voltage approaches to its maximum. The phase lag during the voltage reversal follows the similar pattern: (3) → (4) → (1). In other words, the output response has zero phase lag at points 1 and 3 and the maximum phase lag occurs at points 2 and 4.

When the maximum voltage of the sinusoidal input changes, so does the slope of the corresponding hysteresis curve, as shown in Figure 5. In this figure, it can be observed that the slope of the curve increases as the maximum input voltage increases.

Thus, a proportional relationship can be established between the actuator displacement and the maximum input voltage. With the variable time delay, this relationship is incorporated in the following hysteresis model.

If v denotes the input voltage, the output displacement of the actuator, y, can be written as

y (t) = a (V_{max}) v (t - τ (t, ω))

(1)

where

τ (t, ω) = τ_{max} | sin (ω t) | : variable time delay

(2)

and

a (V_{max}) = a_{0} (1 + b V_{max}) : variable gain

(3)

In equation (2), the variable time delay τ is shown to be a function of time $t$ and frequency $ω$ of the input voltage with τ_max to be the maximum time delay during one cycle. In equation (3), the variable gain a is described as a linear function of the maximum input voltage V_max with two parameters, a₀ and b, that determine the slope of the hysteresis curve. Equation (1) in the time domain can be Laplace transformed into the s-domain as

Y (s) = a (V_{max}) e^{- τ s} V (s)

(4)

where $e^{- τ s}$ is the time delay term in the s-domain. To use this model in the controller design, the three model parameters $τ_{max}$ , $a_{0}$ , and b need to be determined by matching the predicted response with experimental data. Note that τ in equation (4) is now a parameter, which is chosen semi-empirically with the knowledge of ω and V_max.

Second-order model with phase lag for THUNDER actuator

Among the two features in the above hysteresis model, the variable time delay term is incorporated in a typical mechanical second-order model to describe the dynamic behavior of the THUNDER actuator. As the actuator studied in this research is set up in a cantilever configuration, the baseline system can be assumed to be a single-degree-of-freedom mass–spring–damper system. Thus, the equation of motion can be written as

\ddot{y} + 2 ζ ω_{n} \overset{\cdot}{y} + ω_{n}^{2} y = a ω_{n}^{2} v

(5)

where y is the actuator tip displacement, v is the input voltage, ω_n is the natural frequency, and ζ is the damping ratio of the structure. A proportional gain a is also included in the equation to relate the voltage input to the actuator displacement. The transfer function of this structure can be written as

\frac{Y}{V} (s) = \frac{a ω_{n}^{2}}{s^{2} + 2 ζ ω_{n} s + ω_{n}^{2}}

(6)

The hysteresis of the actuator is modeled only as a variable time delay whose Laplace transform is

D (s) = e^{- τ s}

(7)

where τ is the delayed time. By multiplying the time delay term to the second-order model, a simple system model can be obtained for the THUNDER actuator

T (s) = \frac{Y}{V} (s) \cdot D (s) = \frac{a ω_{n}^{2}}{s^{2} + 2 ζ ω_{n} s + ω_{n}^{2}} \cdot e^{- τ s}

(8)

This model has four parameters to be determined: the natural frequency ω_n, damping ratio ζ, variable gain a, and the variable time delay τ. The first two parameters from the mechanical second-order model can be determined from a voltage step response of the actuator, and the time delay function can be obtained from a hysteresis curve: ω_n = 37 Hz, ζ = 0.03. The gain a will be calculated for a certain maximum input voltage and fixed to it until the maximum voltage is changed: a = 2.0425 × 10⁻⁴, τ = 0.01 (τ is assumed to be a fixed value for the purpose of controller design). Although this model based on the variable time delay is relatively simple and not appropriate to describe the hysteretic behavior of THUNDER actuators in detail, this state-space–based crude model is good for designing the MPSMC and implementing it in real time. This model in equation (8) will be used in the design of an IMC, and a simplified model based only on the variable time delay without the second-order dynamics will be used in the MPSMC design.

Experimental setup

In order to measure the tip displacement of the THUNDER actuator for an input voltage, a simple experimental setup is prepared, as shown in Figure 6.

The dimension of the actuator is 99 × 25 × 0.25 mm³ and the range of applicable voltage is $- 300 \leq V \leq 600$ . In the setup, the actuator is clamped with a vise in a cantilever configuration so that the tip displacement is measured by a capacitance displacement sensor. For the data acquisition and control, dSPACE system was used with Simulink. As the piezoceramic actuator requires a high-voltage input, the control signal from the dSPACE controller board is amplified by a power amplifier, as shown in Figure 6. Each controller developed in the next section is implemented in MATLAB/Simulink and downloaded into the dSPACE after compilation.

Displacement controls with hysteresis reduction

Among many possible control methodologies, four approaches are selected and studied here: PI control, IMC, SMC, and MPSMC. These are chosen based on their different degrees of linearity, complexity, and control capability. In the following subsections, these controllers are designed and their performances are experimentally evaluated.

PI control

Proportional–integral–derivative (PID) control is the simplest yet very effective linear control algorithm that is widely applied in many applications. Since the hysteresis of the THUNDER actuator shows up as a cyclic lag in the time domain, the control effort is focused on reducing the phase lag. It is known that derivative control does not improve the response of a system with dominant phase lag and even destabilizes the system as can be proved by the Nyquist stability criterion (Li et al., 2006). For this reason, the derivative term is not included in the controller among the three terms in PID control, resulting in a PI controller for the current application. The structure of the PI-controlled THUNDER actuator is shown in Figure 7. In designing the PI controller for the actuator, proper values for the PI gains need to be determined for different input frequencies. The best set of gains for each case is obtained manually by trial and error.

Performance of the designed PI controller is experimentally evaluated for different input signals. Figure 8 compares time domain tracking performance of the PI controller at 0.1 and 2 Hz. As shown in Figure 9(a), the PI controller can eliminate the phase lag almost completely when the input voltage is at 0.1 Hz showing an almost straight line in the hysteresis plot. However, as the input frequency is increased above 0.1 Hz, the performance of the PI controller deteriorates showing an increasing gap in the hysteresis curve. These results are shown in Figures 9(b) and 10.

Although PI control successfully reduces hysteresis of the THUNDER actuator for low-frequency input signals, its performance deteriorates significantly as the frequency increases. Also, the fact that different control gains are required whenever the input signal changes is one of the major drawbacks of the PI control. In summary, the linear PI control is too simple to control such an actuator that has a complicated nonlinear dynamics.

IMC

IMC employs the plant model as a part of the controller (Garcia and Morari, 1982). Compared with PI control, IMC has several advantages. First, the closed-loop stability is assured simply by choosing a stable controller if an exact plant model is available. Moreover, controller design is relatively straightforward, and the closed-loop performance characteristics such as the time constant, rising, and settling time are directly related to the controller parameters. This fact allows an online tuning of the controller to be very convenient. The IMC structure is a variation of the classical feedback structure with the process model employed explicitly as an internal part of the controller (Brosilow and Joseph, 2002). A general structure of the IMC controlled system is shown in Figure 11.

From a previous section, a model for the THUNDER actuator was derived as

T (s) = \frac{Y}{V} (s) \cdot D (s) = \frac{a ω_{n}^{2}}{s^{2} + 2 ζ ω_{n} s + ω_{n}^{2}} \cdot e^{- τ s}

(9)

Since IMC is a linear controller requiring a linear system model, however, the exponential function in the model needs to be linearized. For this purpose, (0, 1) Padé approximation is applied to convert the exponential function into a rational function such that

e^{- τ s} \approx \frac{1}{τ s + 1} = \tilde{D} (s)

(10)

Then, the actuator model becomes linearized as

\tilde{T} (s) = \frac{Y}{V} (s) \cdot \tilde{D} (s) = \frac{a ω_{n}^{2}}{s^{2} + 2 ζ ω_{n} s + ω_{n}^{2}} \cdot \frac{1}{τ s + 1}

(11)

With this model, an IMC can be designed. By taking the reciprocal of the plant model, the controller transfer function can be written as

C (s) = \frac{s^{2} + 2 ζ ω_{n} s + ω_{n}^{2}}{a ω_{n}^{2}} \cdot \frac{τ s + 1}{{(λ s + 1)}^{p}}

(12)

where the term (λs+ 1)^p in the denominator is introduced to make the transfer function proper, that is, the order of the denominator is not lower than that of the numerator. Therefore, p needs to be determined by the order of the numerator in the controller transfer function. In this case, p = 3 is utilized. The resulting IMC controller is

C (s) = \frac{s^{2} + 2 ζ ω_{n} s + ω_{n}^{2}}{a ω_{n}^{2}} \cdot \frac{τ s + 1}{{(λ s + 1)}^{3}}

(13)

The time constant $λ$ in the new term depends on the modeling errors and the allowable noise amplification by the controller. In this research, this constant is tuned manually for the best input tracking performance.

Performance of the IMC controller is experimentally evaluated for the same four input signals that were used for the PI control. As shown in Figures 12 and 13, the IMC controller does not show any improvement over PI control in the hysteresis reduction.

Ideally, the IMC will show the perfect track performance with the exact model of the plant, which cannot be obtained in real-life situations. There are difficulties in achieving a certain accuracy of the model since good approximated models can make the IMC perform highly effective, while it can render the controlled system unstable with some discrepancies between the plant model and the actual plant dynamics. The main limitation of the PI and IMC stems from the fact that they are basically linear control methodologies while the plant is nonlinear. Thus, it is a natural procedure to consider nonlinear control approaches for the actuator with nonlinear dynamics.

SMC

SMC is one of the most frequently applied nonlinear control methods. SMC alters the dynamics of a nonlinear system by applying a high-frequency switching control so that the state trajectory follows a predefined discontinuity surface called the sliding surface in the state space (Utkin et al., 1999). In this way, the motion of the system does not rely on its original dynamics but rather depends on the dynamics of the sliding surface. The main advantage of SMC lies in its robustness and insensitivity to noise. Once the system reaches the sliding surface, it will have prescribed dynamics associated with desirable performance characteristics, and the system response will be insensitive to disturbances or uncertainties in the system parameters. However, in order for an ideal sliding mode to occur, the discontinuous control input should be able to switch at a very high, theoretically infinite frequency while most of the real actuators cannot. This often causes a so-called chattering problem and a special technique such as a boundary layer or a sliding mode observer needs to be applied to eliminate this phenomenon (Young et al., 1999). The chattering with the ideal SMC could be observed apparently from Figure 14(a) to (c), showing the control input of the SMC, time domain tracking, and hysteresis curves, respectively.

The design process of SMC consists of two steps. First, a sliding surface is selected to have dynamics associated with desirable performance characteristics. A discontinuous control input is then designed to force the current system state to approach to the sliding surface and move along it.

For the current reference-tracking problem with reduced hysteresis, the sliding surface s is defined as a function of the error e between the reference r and the system state x such that

s = \overset{\cdot}{e} + α e, where e = x - r and \overset{\cdot}{e} = \frac{de}{dt}

(14)

The constant α represents the slope of the sliding surface in the phase plane. If a discontinuous control input, $u = - M sgn (s)$ , with a large constant M is applied, the chattering phenomenon is inevitable due to the switching limit of any real actuator. Among several methods for preventing chattering, the boundary layer method is applied here. The boundary layer smoothes the discontinuity of the SMC by inserting a continuous function between the sliding surfaces (Slotine and Sastry, 1983). The chattering phenomenon could be avoided with this, while the dynamics of the state trajectory is not the exact one expected but an approximation. Also, to minimize the motion of the actuator, the control magnitude M is reduced by adding the reference state r to the discontinuous control input. Thus, the following control input u is applied to the THUNDER actuator

u = r + u_{d}

(15)

where

u_{d} = {\begin{matrix} - M sgn (s), | s | > ε_{b} \\ - \frac{M}{ε_{b}} s, | s | \leq ε_{b} \end{matrix}

(16)

In equation (16), ε_b represents the thickness of the boundary layer. The boundary layer thickness ε_b is directly related to the precision for the ideal sliding mode (Slotine, 1984). Thus, the value ε_b is set first based on the precision of the control system we desire. Next, the control law is determined outside the boundary layer (M is adjusted) satisfying the convergence condition of the sliding mode. The actuator driving limit should be considered also in order to prevent the controller saturation. Parameters used are as follows: ε_b = 1600, M = 3.

However, when this method is utilized, there will be a non-zero steady-state error in the response due to the additional reference input r in equation (15). Therefore, in order to eliminate the steady-state error, an integral control is applied with the SMC. The overall SMC system utilized in this research is shown in Figure 15.

An integral control is a type of state feedback control designs (Friedland, 2005) whose performance characteristics are similar to those of the PI controller. Let us assume that the SMC block is a dynamic system in the state-space form defined as follows

\overset{\cdot}{x} = A_{s} x + B_{s} u_{s}

(17)

y = x

(18)

Next, a feedback control law is designed

u_{s} = K_{P} x + K_{I} x_{I} where {\overset{\cdot}{x}}_{I} = y - r

(19)

Equations (17) to (19) could be integrated to create an augmented state-space system

(\begin{matrix} \overset{\cdot}{x} \\ {\overset{\cdot}{x}}_{I} \end{matrix}) = (\begin{matrix} A & 0 \\ I & 0 \end{matrix}) (\begin{matrix} x \\ x_{I} \end{matrix}) + (\begin{matrix} B \\ 0 \end{matrix}) u_{s} + (\begin{matrix} 0 \\ - I \end{matrix}) r

(20)

y = (\begin{matrix} I & 0 \end{matrix}) (\begin{matrix} x \\ x_{I} \end{matrix}) where u_{s} = (\begin{matrix} K_{P} & K_{I} \end{matrix}) (\begin{matrix} x \\ x_{I} \end{matrix})

(21)

Therefore, the integral control problem becomes a traditional state feedback problem, and a state feedback u should be determined to obtain expected controller performances. The integral control problem is described in Figure 15.

With this modified SMC, experiments are conducted for the previous four reference inputs and the results are shown in Figures 16 and 17. Although an ideal SMC is expected to eliminate the hysteresis completely, with the two additional control schemes introduced to eliminate the chattering phenomenon, the system performance was deteriorated and no significant improvement is observed in the controller performance.

MPSMC

By combining the model predictive control and the SMC, a new control scheme called MPSMC was formulated (Neelakantan, 2005). As described in the previous section, the SMC suffers from the chattering problem. Moreover, when SMC is implemented in the discrete time domain, control input saturation may occur because the sliding mode is enforced at the very next sampling instant. This causes the controller to operate in a suboptimal manner and may even destabilize the system. In order to resolve these issues, model predictive control (Rossiter, 2003) is applied to modify the way the sliding mode surface is reached by the state trajectory.

A key feature of MPSMC is the fact that the system state converges to the sliding surface in an optimized fashion in this method. Consider the following discrete-time state-space equation with a disturbance vector h_k

x_{k + 1} = A x_{k} + B u_{k} + h_{k}, y_{k} = C x_{k}

(22)

where A, B, and C represent the system, input, and output matrices, respectively, and x_k and u_k are the system and input vectors, respectively. A new state error vector e_k is introduced by subtracting a desired reference vector r_k from the state vector x_k as in equation (23) for reference-tracking problems. The system equations can then be rewritten with respect to e as equation (24)

e_{k} = x_{k} - r_{k}

(23)

e_{k + 1} = A e_{k} + B u_{k} + d_{k}

(24)

where $d_{k} = A r_{k} + h_{k} - r_{k + 1}$ .

Note that the additional term d_k encompasses disturbance, nonlinearity, hysteresis, model uncertainties, and effects of the reference input. In the following derivation, it is more convenient to augment the system state by adding the control input u_k_–1 at the precedent instant to the state vector. Thus, the augmented state vector z_k becomes

z_{k} = [\begin{matrix} e_{k} \\ u_{k - 1} \end{matrix}]

(25)

With the new state vector, the state equation can be rewritten as

[\begin{matrix} e_{k + 1} \\ u_{k} \end{matrix}] = [\begin{matrix} A & B \\ 0 & I \end{matrix}] [\begin{matrix} e_{k} \\ u_{k - 1} \end{matrix}] + [\begin{matrix} B \\ I \end{matrix}] Δ u_{k} + [\begin{matrix} I \\ 0 \end{matrix}] d_{k}

(26)

or simply

z_{k + 1} = Γ z_{k} + Φ Δ u_{k} + Θ d_{k}

(27)

where

Γ = [\begin{matrix} A & B \\ 0 & I \end{matrix}], Φ = [\begin{matrix} B \\ I \end{matrix}], and Θ = [\begin{matrix} I \\ 0 \end{matrix}]

In the new state equation, the difference in the control signal Δu, defined as $Δ u_{k} = u_{k} - u_{k - 1}$ , is used as the control input. With the new state vector, the desired sliding surface s_k can be defined as

s_{k} = G_{s} z_{k}

(28)

where G_s determines the dynamic characteristics of the sliding mode. By substituting equation (27) into equation (28), the sliding mode equation can be obtained. Then, by combining the sliding mode equations at the (k+ 1)th through the (k+N)th sampling instants, a prediction equation for the sliding mode can be derived as follows

s_{(k + 1 : k + N)} = \hat{Γ} z_{k} + \hat{Φ} Δ u_{(k : k + N - 1)} + \hat{Θ} d_{(k : k + N - 1)}

(29)

where

\begin{array}{l} s_{(k + 1 : k + N)} = {[\begin{matrix} s_{k + 1} & \dots & s_{k + N} \end{matrix}]}^{T}, \\ Δ u_{(k : k + N - 1)} = {[\begin{matrix} Δ u_{k} & \dots & Δ u_{k + N - 1} \end{matrix}]}^{T}, \\ a n d_{(k : k + N - 1)} = {[\begin{matrix} d_{k} & \dots & d_{k + N - 1} \end{matrix}]}^{T} \end{array}

(30)

and the matrices $\hat{Γ}$ , $\hat{Φ}$ , and $\hat{Θ}$ are defined as

\begin{matrix} \hat{Γ} = [\begin{matrix} G_{s} Γ \\ G_{s} Γ^{2} \\ ⋮ \\ G_{s} Γ^{N} \end{matrix}], \hat{Φ} = [\begin{matrix} G_{s} Φ & 0 & \dots & 0 \\ G_{s} Γ Φ & G_{s} Φ & \dots & 0 \\ ⋮ & ⋮ & ⋱ & 0 \\ G_{s} Γ^{N - 1} Φ & G_{s} Γ^{N - 2} Φ & \dots & G_{s} Φ \end{matrix}], \\ and \hat{Θ} = [\begin{matrix} G_{s} Θ & 0 & \dots & 0 \\ G_{s} Γ Θ & G_{s} Θ & \dots & 0 \\ ⋮ & ⋮ & ⋱ & 0 \\ G_{s} Γ^{N - 1} Θ & G_{s} Γ^{N - 2} Θ & \dots & G_{s} Θ \end{matrix}] \end{matrix}

(31)

Note that $s_{(k + 1 : k + N)}$ , $Δ u_{(k : k + N - 1)}$ , and $d_{(k : k + N - 1)}$ are arrays of the sliding mode, control input difference, and disturbance vectors up to N future time instants, respectively. In order to obtain an optimal vector for the control input difference, a cost function J is defined as

J = ‖ s_{(:)} ‖^{2} + λ_{m} ‖ Δ u_{(:)} ‖^{2}

(32)

where λ_m is a weighting factor. Minimization of the cost function leads to the optimal solution for the array of the control difference vectors as follows

Δ u_{(:)} = - {({\hat{Φ}}^{T} \hat{Φ} + λ_{m} I)}^{- 1} {\hat{Φ}}^{T} [\hat{Γ} z_{k} + \hat{Θ} d_{(:)}]

(33)

Finally, the first vector at the kth time instant is extracted from the array of the control difference vectors in equation (33) by multiplying it by a row vector as follows

Δ u_{k} = [\begin{matrix} I & 0 & \dots & 0 \end{matrix}] Δ u_{(:)}

(34)

Note that in equation (34), $d_{(:)}$ represents the future prediction of the disturbance that is generally unknown. A simple way of overcoming this difficulty is to obtain the previous value of the disturbance and assume that the disturbance remains the same for the next N time instants, which is basically an extended zero-order estimate. In this case, $d_{(:)}$ can be replaced by its estimate such that

d_{(:)}^{est} = {[\begin{matrix} d_{k - 1} & d_{k - 1} & \dots & d_{k - 1} \end{matrix}]}^{T}

(35)

where $d_{k - 1} = e_{k} - A e_{k - 1} - B u_{k - 1}$ . A block diagram for MPSMC system is shown in Figure 18.

This control methodology has been verified with a two-stage actuation system (Neelakantan, 2005) in a previous study. In the study, MPSMC exhibited superior performances when step inputs are applied to the system. However, its performance for sinusoidal inputs has not been examined until now.

While a third-order model shown in equation (11) was employed in the design of IMC, a simpler first-order model based only on the time delay is utilized to design the MPSMC controller. This is because the velocity signal introduced by the second-order mechanical model in equation (6) is prone to have amplified noise due to the derivative of the displacement signal. Thus, the model used for the MPSMC is

\tilde{p} (s) = \frac{1}{τ_{\max} s + 1}

(36)

where the time constant τ_max corresponds to the maximum time delay in the given voltage input. Note that this model is a (0, 1) Padé approximation for the variable time delay. In addition, the sliding surface and the size of the receding horizon are selected to be $G_{s} = [\begin{matrix} 1 & 0 \end{matrix}]$ and $N = 5$ , respectively, and the weighting factor λ_m in the cost function is adjusted to get the best result for each different voltage input. Control performance of MPSMC for the four different input signals is shown in Figures 19 to 21. In these figures, the performance improvement of MPSMC is clearly identified over the other three control methods. Note that the hysteresis reduction could be observed up to 7 Hz with MPSMC.

Hysteresis curves of all four controllers for the input voltage at 2 Hz are compared in Figure 22. It can be clearly seen that MPSMC exhibits the best hysteresis reduction performance followed by SMC, indicating that nonlinear controllers are superior to linear ones in hysteresis reduction for high input frequencies.

Additionally, it could be observed that the chattering phenomenon is eliminated with the employment of the MPSMC. Figure 23(a) to (c) shows the control input of the MPSMC, time domain tracking, and hysteresis curves, respectively.

For a quantitative comparison of the controller performance, the enclosed area of the hysteresis curve is chosen as a performance metric. This performance metric is calculated for the four controllers against the four different input signals, and the results are summarized in Table 1. Note that each quantity is normalized by dividing it by the area of the open-loop curve for the same frequency input. This table clearly shows the superiority of MPSMC in the hysteresis reduction, especially at higher frequencies.

Table 1.

Enclosed area of hysteresis curves.

Frequency (Hz)	Open-loop curve	PI	IMC	SMC	MPSMC
0.1	1.000	0.0484	0.1292	0.0492	0.0527
0.5	1.000	0.2215	0.2763	0.2348	0.0829
1	1.000	0.3960	0.7645	0.3446	0.1680
2	1.000	0.8689	1.2738	0.7994	0.2942
3	1.000	Enlarged	Enlarged	Enlarged	0.3227
5	1.000	Enlarged	Enlarged	Enlarged	0.3509
7	1.000	Enlarged	Enlarged	Enlarged	0.3754

PI: proportional–integral; IMC: internal model control; SMC: sliding mode control; MPSMC: model predictive sliding mode control.

Conclusion

This research has studied and compared reference-tracking performances of four different control methods for a curved piezoceramic unimorph actuator, which has a strong nonlinear behavior called hysteresis. Two of the controllers are linear controls (PI and IMC) and the other two are nonlinear controls (SMC and MPSMC). The experimental results show that all the controllers can effectively reduce the hysteretic effect when the input frequency is relatively low (less than 1 Hz). However, with the input frequency of 1 Hz and over, the linear controllers are outperformed by the nonlinear controllers with MPSMC showing the most hysteresis reduction up to 7 Hz. The MPSMC, which combines the robust disturbance rejection and uncertainty minimization capabilities of SMC and the optimal tracking performance of model predictive control, is validated to be a very effective method for the reference-tracking displacement control applications. Possible applications include the actuation of bio-inspired autonomous vehicles, positioning control for aerospace, automotive, and biomedical robotics devices, and precise position control for valves, lids, switches, and so on.

Future works would focus on a novel control method with less computational burden, since it prevents the MPSMC from working properly at higher frequencies. Applying an inverse model of the actuator, developed offline, could be an alternative approach. It can reduce nonlinear effect of the actuator, and so the load of the controller could be decreased. Finally, the transient behavior and the Raleigh loops of the actuator could be considered for validating the new control technique.

Footnotes

Funding

This research was supported by the National Science Foundation.

References

Ahrens

Tan

Khalil

(2007) Multirate sampled-data output feedback control of smart material actuated systems. In: Proceedings of American control conference, New York, 9–13 July.

Aimmanee

Hyer

(2004) Analysis of the manufactured shape of rectangular THUNDER^®-type actuators. Smart Materials and Structures 13: 1389–1406.

Ball

Smith

(2004) A stress-dependent hysteresis model for PZT-based transducers. Proceedings of SPIE Conference on Smart Structures and Materials 5383: 23–30.

Ball

Smith

Ounaies

(2003) A dynamic hysteresis model for THUNDER^® transducers. Proceedings of SPIE Conference on Smart Structures and Materials 5049: 100–111.

Ball

Smith

Kim

. (2007) A stress-dependent hysteresis model for ferroelectric materials. Journal of Intelligent Material Systems and Structures 18: 69–88.

Bashash

Jalili

(2007) Intelligence rules of hysteresis in the feedforward trajectory control of piezoelectrically-driven nanostagers. Journal of Micromechanics and Microengineering 17: 342–349.

Borgen

Washington

Kinzel

(2003) Design and evolution of a piezoelectrically actuated miniature swimming vehicle. IEEE-ASME Transactions on Mechatronics 8(1): 66–76.

Brei

Berner

Alexander

(2001) Modeling and study of the quasi-static behavior of piezoceramic telescopic actuation architectures. Journal of Intelligent Material Systems and Structures 12(5): 303–313.

Brosilow

Joseph

(2002) Techniques of Model-Based Control. Upper Saddle River, NJ: Prentice Hall PTR.

10.

Capozzoli

Gopalakrishnan

Hogan

. (1999) Modeling aspects concerning THUNDER actuators. Proceedings of SPIE Conference on Smart Structures and Materials 3667: 719–727.

11.

Chung

Kim

(2005) Investigation of actuating displacement performance of curved actuator by large-scale computation. Smart Materials and Structures 14(4): 615–623.

12.

Croft

Shed

Devasia

(2001) Creep, hysteresis, and vibration compensation for piezoactuators: atomic force microscopy application. ASME Journal of Dynamic Systems, Measurement and Control 123: 35–43.

13.

Friedland

(2005) Control System Design: An Introduction to State-Space Methods. Mineola, NY: Dover Publications.

14.

Fukashi

Washington

Yoon

. (2004) Efficient shape control of distributed reflectors with discrete piezoelectric actuators. Journal of Intelligent Material Systems and Structures 15: 3–15.

15.

Garcia

Morari

(1982) Internal model control. 1. A unifying review and some new results. Industrial & Engineering Chemistry Process Design and Development 21: 308–323.

16.

Granger

Washington

Kwak

(2000) Modeling and control of a singly curved active aperture antenna using curved piezoceramic actuators. Journal of Intelligent Material Systems and Structures 11: 225–233.

17.

Haertling

(1994) Rainbow ceramics—a new type of ultra-high-displacement actuator. American Ceramic Society Bulletin 73(1): 93–96.

18.

Homer

Taleghani

(1999) Single Axis Piezoceramic Gimbal. Hampton, VA: NASA Langley Research Center.

19.

Iyer

Tan

Krishnaprasad

(2005) Approximate inversion of the Preisach hysteresis operator with application to control of smart actuators. IEEE Transactions on Automatic Control 50(6): 798–810.

20.

Jayachandran

Meyer

Westervelt

. (1999) Piezoelectrically driven speakers for active aircraft interior noise suppression. Applied Acoustics 57: 263–277.

21.

Lawver

(2001) Thunder^®: a new frontier in research for smart materials. Smart Materials Bulletin 2001(8): 5–9.

22.

Leang

Devasia

(2002) Hysteresis, creep, and vibration compensation for piezoactuators: feedback and feedforward control. In: The 2nd IFAC conference on mechatronic systems, Berkeley, CA, December 9–11, pp. 283–289. New York: IFAC.

23.

Ang

Chong

GCY

(2006) PID control system analysis and design. IEEE Control Systems Magazine 26(1): 32–41.

24.

Moskalik

Brei

(1996) Force-deflection behavior of individual unimorph piezoceramic C-block actuators. Proceedings of ASME Aerospace Division Conference 52: 679–687.

25.

Neelakantan

(2005) Modeling, design, testing and control of a two-stage actuation mechanism using piezoelectric actuators for automotive applications. PhD Dissertation, The Ohio State University, Columbus, OH.

26.

Pare

How

(1998) Robust H-infinity controller design for systems with hysteresis nonlinearities. Proceedings of IEEE Conference on Decision and Control 4: 4057–4062.

27.

Pinkerton

Moses

(1997) A Feasibility Study to Control Airfoil Shape Using THUNDER. Hampton, VA: NASA Langley Research Center.

28.

Prechtl

Hall

(1997) Design of a high efficiency discrete servo-flap actuator for helicopter rotor control. Proceedings of SPIE Conference on Smart Structures and Integrated Systems 3041: 158–182.

29.

Rossiter

(2003) Model-Based Predictive Control: A Practical Approach. Boca Raton, FL: CRC Press.

30.

Slotine

(1984) Sliding controller design for non-linear systems. International Journal of Control 40(2): 421–434.

31.

Slotine

Sastry

(1983) Tracking control of non-linear systems using sliding surfaces, with application to robot manipulators. International Journal of Control 38(2): 465–492.

32.

Song

Zhao

Zhou

. (2005) Tracking control of a piezoceramic actuator with hysteresis compensation using inverse Preisach model. IEEE-ASME Transactions on Mechatronics 10(2): 198–209.

33.

Song

Washington

(1999) Thunder actuator modeling and control with classical and fuzzy control algorithm. Proceedings of SPIE Conference on Smart Structures and Materials 3668: 866–877.

34.

Taleghani

(2000) Validation of high displacement piezoelectric actuator finite element models. Proceedings of SPIE Conference on Smart Structures and Materials 4073: 37–45.

35.

Tsai

Chen

(2003) Robust tracking control of a piezoactuator using a new approximate hysteresis model. ASME Journal of Dynamic Systems, Measurement and Control 23: 96–102.

36.

Usher

Sim

Ashford

. (2004) Modeling and applications of new piezoelectric actuator technologies. Proceedings of SPIE Conference on Smart Structures and Materials 5383: 31–38.

37.

Utkin

Guldner

Shi

(1999) Sliding Mode Control in Electro-Mechanical Systems. Philadelphia, PA: Taylor & Francis.

38.

Washington

Song

(2000) Mechatronic design and control of singly and doubly curved composite mesoscale actuator systems. IEEE-ASME Transactions on Mechatronics 5(1): 49–57.

39.

Wieman

Smith

Kackley

. (2001) Displacement models for THUNDER actuators having general loads and boundary conditions. Proceedings of SPIE Conference on Smart Structures and Materials 4326: 253–263.

40.

Dogan

Tressler

. (1991) Ceramic-metal composite actuator. Proceedings of IEEE Ultrasonics Symposium 2: 923–928.

41.

Yoon

Washington

Theunissen

(2000) Analysis and design of doubly curved piezoelectric strip-actuated aperture antennas. IEEE Transactions on Antennas and Propagation 48(5): 755–763.

42.

Yoon

Washington

Eyabi

. (2006) A millimeter-stroke piezoelectric hybrid actuator using hydraulic displacement amplification mechanism. IEEE International Symposium on Industrial Electronics 4: 2809–2813.

43.

Yoon

Shin

Park

. (2002) Design and manufacture of a lightweight piezo-composite curved actuator. Smart Materials and Structures 11: 163–168.

44.

Young

Utkin

Ozguner

(1999) A control engineer’s guide to sliding mode control. IEEE Transactions on Control Systems Technology 7(3): 328–342.

45.

Alici

Shirinzadeh

. (2005) Sliding mode control of a piezoelectric actuator with neural network compensating rate-dependent hysteresis. Proceedings of IEEE Conference on Robotics and Automation 3641–3646.

46.

Zhou

Zhao

Song

. (2003) Preisach modeling of hysteresis and tracking control of a Thunder actuator system. Proceedings of SPIE Conference on Smart Structures and Materials 5049: 112–125.