1. Introduction

2. Background And System Modelling

2.1. The Hysteresis Model

The Preisach operator stands out among various hysteresis models due to its capability to accurately represent complex hysteresis curves, including multi-loop and asymmetric hysteresis curves, and it is constructed by the weighted superposition of infinity basic relay operators. Typically, the thermostat relay operators [21] are chosen as the fundamental components for constructing the Preisach operator as shown in Figure 1.

Thermostat relay operator: We first consider the Preisach plane as

$$T_0=\left\{(\beta ,\alpha )\in |\beta \ge {\beta }_0,\alpha \le {\alpha }_0,\alpha \ge \beta \right\},$$

which is a right triangle area and is consisted of a vertex coordinate (${\beta }_0,{\alpha }_0$) and a part of line $\alpha =\beta$. For a visual representation, we present the geometric interpretation of the Preisach plane $T_0$ in Figure 2. For any given point $(\beta ,\alpha )$ on the Preisach plane $T_0$, there is a corresponding thermostat relay operator

$$\begin{array}{cc}\hfill {\gamma }_{\alpha \beta }^{*}& \left(v\left(t\right),{\tau }_0(\beta ,\alpha )\right)\hfill \\ \hfill & =\left\{\begin{array}{ccc}+1,\hfill & \mathrm{if}\hfill & v\left(t\right)>\alpha \hfill \\ -1,\hfill & \mathrm{if}\hfill & v\left(t\right)<\beta \hfill \\ \mathrm{remain}\mathrm{unchanged},\hfill & \mathrm{if}\hfill & v\left(t\right)\in [\beta ,\alpha ],\hfill \end{array}\right.\hfill \end{array}$$

(1)

where $v\left(t\right)\in [0,t_m]$ is the input of the thermostat relay operator with continuity and piecewise monotonicity and ${\tau }_0(\beta ,\alpha )$ represents the initial value of the thermostat relay operator ${\gamma }_{\alpha \beta }^{*}(v\left(0\right),·)$. For example, ${\tau }_0(\beta ,\alpha )=1$ while $\forall (\beta ,\alpha )\in T_0$ and $v\left(0\right)>{\alpha }_0$.

Preisach operator: The Preisach operator is constructed by the weighted superposition of infinity thermostat relay operators on the Preisach plane $T_0$, which is presented as follows

$$\begin{array}{cc}\hfill u\left(t\right)& =\mathcal{H}(v\left(t\right),{\tau }_0(\beta ,\alpha ))\hfill \\ \hfill & =\int {\int }_{T_0}\mu (\beta ,\alpha ){\gamma }_{\alpha \beta }^{*}\left(v\left(t\right),{\tau }_0(\beta ,\alpha )\right)d\beta d\alpha ,\hfill \end{array}$$

(2)

where the weighting function $\mu (\beta ,\alpha )$ also referred to the density function, and according to the definition of the Preisach operator, all points $(\beta ,\alpha )\in T_0$ have the corresponding density function $\mu (\beta ,\alpha )\ne 0$ and when $(\beta ,\alpha )\notin T_0$, the density function $\mu (\beta ,\alpha )=0$ as shown in Figure 2.

Memory curve: The memory effects of the Preisach operator can be captured by the memory curve in the Preisach plane $T_0$ (as illustrated in [21]). When the Preisach input increases monotonically, the output of the thermostat relay operator above the $\alpha$ threshold switches to $+1$ and forms an upward-shifting curve. Similarly, when the Preisach input decreases monotonically, the output of the thermostat relay operator below the $\beta$ threshold switches to $-1$ and forms a leftward-shifting curve. Then, in the Preisach plane $T_0$, a piecewise monotone input signal $v\left(t\right)$ can create the memory curve $\mathsf{\Phi }(\beta ,v(t\left)\right)$ as shown in Figure 2, where the memory curve divides the plane $T_0$ into two parts:

$$\begin{array}{cc}\hfill S_{+}\left(t\right)& =\left\{(\beta ,\alpha )\in T_0|{\gamma }_{\alpha \beta }^{*}(v\left(t\right),·)=+1\right\},\hfill \\ \hfill S_{-}\left(t\right)& =\left\{(\beta ,\alpha )\in T_0|{\gamma }_{\alpha \beta }^{*}(v\left(t\right),·)=-1\right\},\hfill \end{array}$$

(3)

and we can rewrite the integral (2) as

(4)

which is an essential form to analyze the output range of the Preisach operator and to prove the piecewise monotonicity of the adaptive Preisach operator.

Following the description of the Preisach operator, we proceed to introduce the considered plant model and formulate the adaptive control problem.

3. Relative Degree Conditions and Stability of Zero Dynamics Subsystem

4. Adaptive Inverse Control Scheme For Relative-Degree-One Case with Preisach Hysteresis

4.1. System Parameterization

According to Lemma 1, the relative degree of the approximation system (6) is one when it satisfies $CB\ne 0$, which leads to the formulation of the tracking control dynamics subsystem that can be expressed as follows

$$\begin{array}{cc}\hfill \dot{y}\left(t\right)& =CAx\left(t\right)+C{\mathcal{W}}^{*}S\left(x\left(t\right)\right)+CBu\left(t\right),\hfill \\ \hfill u\left(t\right)& =\mathcal{H}(v\left(t\right),{\tau }_0(\beta ,\alpha )),\hfill \end{array}$$

(12)

where the system parameters $A,B,C,{\mathcal{W}}^{*}$ are all unknown. For the tracking control study, the following basic Assumption is needed.

Assumption 2.

Assuming that the sign of the control gain $CB$ in (12) is known and positive [31].

This assumption guarantees the design procedure of the control scheme is free from any unknown control direction problems.

For ease of adaptive control scheme design, the system (12) needs to be reparameterized. We introduce some new parameters to transform the system into a more suitable form for adaptive control scheme design. Let ${\vartheta }_1^{*}={[CA,C{\mathcal{W}}^{*}]}^T$ represent a parameter vector, ${\varpi }_1\left(t\right)={[x\left(t\right),S^T\left(x\left(t\right)\right)]}^T$ denote the state vector, and ${\mu }^{*}(\beta ,\alpha )=CB\mu (\beta ,\alpha )$ represent the modified density function, and then the system (12) can be expressed as follows

(13)

Assumption 3.

The modified density function ${\mu }^{*}(\beta ,\alpha )$ defined on a finite right triangle plane $T_0$ takes values between two known nonnegative bounded values ${\mu }_a(\beta ,\alpha )$ and ${\mu }_b(\beta ,\alpha )$ for $\forall (\beta ,\alpha )\in T_0$, which means that ${\mu }_a(\beta ,\alpha )\le {\mu }^{*}(\beta ,\alpha )\le {\mu }_b(\beta ,\alpha )$.

Assumption A3 will be used later on in a projection design to equip the adaptive estimate $\widehat{\mu }(\beta ,\alpha ,t)$ of ${\mu }^{*}(\beta ,\alpha )$ with nonnegativity and boundedness properties.

4.2. Implicit Controller Equation

To compensate for the input hysteresis nonlinearity $\mathcal{H}(v\left(t\right),{\tau }_0)$ and to construct a tracking error system with asymptotic convergence property, we develop an adaptive Preisach inverse implicit controller as follows

$$\begin{array}{cc}\hfill \int {\int }_{T_0}& \widehat{\mu }(\beta ,\alpha ,t){\gamma }_{\alpha \beta }^{*}\left(v\left(t\right),{\tau }_0\right)d\beta d\alpha \hfill \\ \hfill & =-\iota (y\left(t\right)-y_m\left(t\right))-{\vartheta }_1^T\left(t\right){\varpi }_1\left(t\right)+{\dot{y}}_m\left(t\right),\hfill \end{array}$$

(14)

where $\widehat{\mu }(\beta ,\alpha ,·)$ and ${\vartheta }_1(·)$ are the estimates of ${\mu }^{*}(\beta ,\alpha )$ and ${\vartheta }_1^{*}$ respectively, and $\iota$ is a positive constant. The implicit controller equation (14) consists of an adaptive Preisach operator with $\widehat{\mu }(\beta ,\alpha ,t)$ as the adaptive estimate of ${\mu }^{*}(\beta ,\alpha )$ on the left sides and the desired output of the adaptive Preisach operator on the right sides. Then, we define

$$\begin{array}{cc}\hfill & \widehat{\mathcal{H}}(v\left(t\right),{\tau }_0)=\int {\int }_{T_0}\widehat{\mu }(\beta ,\alpha ,t){\gamma }_{\alpha \beta }^{*}\left(v\left(t\right),{\tau }_0\right)d\beta d\alpha ,\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill & u_d\left(t\right)=-\iota (y\left(t\right)-y_m\left(t\right))-{\vartheta }_1^T\left(t\right){\varpi }_1\left(t\right)+{\dot{y}}_m\left(t\right),\hfill \end{array}$$

(16)

and the implicit controller equation (14) is rewritten as

$$\widehat{\mathcal{H}}(v\left(t\right),{\tau }_0)=u_d\left(t\right).$$

(17)

The next task is to solve the implicit controller equation (17), so that we can computed the control input $v\left(t\right)$ in real time. This is essentially equivalent to constructing the inverse function $v\left(t\right)={\widehat{\mathcal{H}}}^{-1}(u_d\left(t\right),{\tau }_0)$, and we will next propose an inverse iterative algorithm to solve it.

A closest match algorithm for solving the implicit controller equation (17) and its convergence proof: Given that the desired output of the adaptive Preisach operator $u_d\left(t\right)$ has a continuous, piecewise monotone behavior over the defined time interval $[0,t_E]$, where the partition is

$$0=t_0<t_1<\cdots <t_{N-1}<t_N=t_E,$$

(18)

for a positive integer $N\ge 1$, and during each sub-interval $(t_i,t_{i+1}]$, $i=0,1,2,\cdots ,N-1$, $u_d\left(t\right)$ is monotone. Then, the implicit controller equation (17) will be solved on each sub-interval $(t_i,t_{i+1}]$. It will be shown on the analysis of Remark 4 that the adaptive estimate density function $\widehat{\mu }(\beta ,\alpha ,t)$ changes slowly with time. In this sense, we can assume that $\widehat{\mu }(\beta ,\alpha ,t)=\widehat{\mu }(\beta ,\alpha ,t_i)$ during each sub-intervals $(t_i,t_{i+1}]$, $i=0,1,2,\cdots ,N-1$. With this in mind, the adaptive Preisach operator $\widehat{\mathcal{H}}(v\left(t\right),{\tau }_0)$ can be expressed as

$$\begin{array}{cc}\hfill \widehat{\mathcal{H}}(v\left(t\right),{\tau }_0)=& \int {\int }_{T_0}\widehat{\mu }(\beta ,\alpha ,t_i){\gamma }_{\alpha \beta }^{*}\left(v\left(t\right),{\tau }_0\right)d\beta d\alpha ,\hfill \\ \hfill & t\in (t_i,t_{i+1}],i=0,1,2,\cdots ,N-1.\hfill \end{array}$$

(19)

With a projection design latter, $\widehat{\mu }(\beta ,\alpha ,t)$ is ensured boundedness as ${\mu }_a(\beta ,\alpha )\le \widehat{\mu }(\beta ,\alpha ,t)\le {\mu }_b(\beta ,\alpha )$ and nonnegativity for $\forall t>0$ and $\forall (\beta ,\alpha )\in T_0$. Then, from the third equality of (4), it is not hard to prove that the adaptive Preisach operator $\widehat{\mathcal{H}}(v\left(t\right),{\tau }_0)$ has monotonicity on each sub-interval $(t_i,t_{i+1}]$, and the output range of $\widehat{\mathcal{H}}(v\left(t\right),{\tau }_0)$ can be obtained from the following equation during $(t_i,t_{i+1}]$:

$$H_{i,min}=-\int {\int }_{T_0}\widehat{\mu }(·)d\beta d\alpha ,H_{i,max}=\int {\int }_{T_0}\widehat{\mu }(·)d\beta d\alpha .$$

For the implicit controller equation (17) to have a solution, the following constructed saturation condition is necessary:

$$H_{i,min}\le u_d\left(t\right)\le H_{i,max}\mathrm{for}\forall t\in (t_i,t_{i+1}],$$

(20)

where $i=0,1,2,\cdots ,N-1$. The limitation of the output range (20) stems from the fact that the Preisach operator $\mathcal{H}(v\left(t\right),{\tau }_0)$ is a saturated hysteresis model and the saturation occurs when the control input $v\left(t\right)$ above the upper threshold ${\alpha }_0$ or below the lower threshold ${\beta }_0$.

Suppose that the condition (20) is satisfied. There are two discretization steps involved, the discretization of the time interval $[0,t_E]$ has been described in (18) and the discretization range $R=[v_{\min },v_{\max }]$ of the adaptive Preisach operator (15) input $v\left(t\right)$ is uniformly divided into L segments as $V_L=\left\{{\overline{v}}_j,j=1,2,\cdots ,L+1\right\}$ where ${\overline{v}}_j=v_{\min }+(j-1){\Delta }_v$, ${\Delta }_v=(v_{\max }-v_{\min })/L$, and L is called the discretization level. The result of discretizing the input range R is that the Preisach plane $T_0$ is divided into cells. Considering the plane $T_0$ with discretization degree L, and within each discretization cell, assuming the density function $\widehat{\mu }(\beta ,\alpha ,t)$ on (15) be nonnegative and remains constant. The inversion problem is, given the desired instaneous value of $u_d\left(t\right)$, and the memory curve $\mathsf{\Phi }(\beta ,v\left(t_d\right))$ generated by the previous input, to find the corresponding input signal $v^{*}\left(t\right)$ such that the equality $u_d\left(t\right)=\widehat{\mathcal{H}}(v^{*}\left(t\right),{\tau }_0)$ is satisfied, which can be calculated by the following algorithm [22]:

The algorithm is based on the piecewise monotonicity property of the adaptive Preisach operator $\widehat{\mathcal{H}}(v\left(t\right),{\tau }_0)$, and it is not hard to see that the algorithm obtains the solution $v_L^{*}\left(t\right)$ in at most L times. The convergence of the above iterative algorithm is given below.

Proposition 1.

Under Assumption 1-3, suppose that condition (20) is satisfied. Then, the iterative algorithm can find a solution $v_L^{*}\left(t\right)={\overline{v}}_j\in V_L$ such that

$$|\widehat{\mathcal{H}}(v_L^{*}\left(t\right),{\tau }_0)-u_d\left(t\right)|=\underset{{\overline{v}}_j\in V_L}{min}|\widehat{\mathcal{H}}({\overline{v}}_j,{\tau }_0)-u_d\left(t\right)|.$$

(21)

Besides, as the discretization degree goes to infinity, we can find the exact solution of the inverse problem, i.e., ${lim}_{L\to \infty }{v}_L^{*}\left(t\right)=v^{*}\left(t\right)$.

Proof. 

Our task is to prove the piecewise monotonicity and Lipschitz continuity of the adaptive Preisach operator (15), based on this property we can follow the arguments as Proposition 5.1 in [22] to prove Proposition 1.

As previously demonstrated, it has been established that the adaptive Preisach operator (15) can be represented in the form of (19) during the sub-intervals $(t_i,t_{i+1}]$ with the adaptive density function $\widehat{\mu }(\beta ,\alpha ,t_i)$ is nonnegative. Then, from the third equality of (4), the following inequality holds

$$\left(\widehat{\mathcal{H}}(v\left(t_2\right),{\tau }_0)-\widehat{\mathcal{H}}(v\left(t_1\right),{\tau }_0)\right)\left(v\left(t_2\right)-v\left(t_1\right)\right)\ge 0,$$

(22)

for $\forall t_1,t_2\in (t_i,t_{i+1}]$. Hence, the adaptive Preisach operator (15) is piecewise monotone on $[0,t_E]$ for the continuous, piecewise monotone control input signal $v\left(t\right)$ on $(0,t_E]$. Besides, with the projection design later, the adaptive density function $\widehat{\mu }(\beta ,\alpha ,t_i)$ is guaranteed to be nonnegative and bounded for $\forall t_i\ge 0$, and $\forall (\beta ,\alpha )\in T_0$. Then, based on the piecewise expression (19), we can obtain the following Lipschitz continuity property:

$$∥\widehat{\mathcal{H}}(v\left({\mathcal{T}}_2\right),{\tau }_0)-\widehat{\mathcal{H}}(v\left({\mathcal{T}}_1\right),{\tau }_0)∥\le K_L∥v\left({\mathcal{T}}_2\right)-v\left({\mathcal{T}}_1\right)∥,$$

(23)

for $\forall {\mathcal{T}}_1,{\mathcal{T}}_2\in [0,t_E]$, where $K_L$ is a Lipschitz constant.

Based on the piecewise monotonicity (22) and Lipschitz continuity (23) of the adaptive Preisach operator (15), we can follow the arguments as Proposition 5.1 in [22] to prove Proposition 1. □

Up to now, we have provided an iterative algorithm that the control input signal $v\left(t\right)$ in the implicit controller equation $\left(17\right)$ can be computed iteratively, and finally, the convergence of this iterative algorithm is proved. Next, we will analyze the performances of the adaptive control scheme.

5. Simulation Study

6. Conclusion

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