An improved compact-form antisaturation model-free adaptive control algorithm for a class of nonlinear systems with time delays

Abstract

To solve the time-delay problem and actuator saturation problem of nonlinear plants in industrial processes, an improved compact-form antisaturation model-free adaptive control (ICF-AS-MFAC) method is proposed in this work. The ICF-AS-MFAC scheme is based on the concept of the pseudo partial derivative (PPD) and adopts equivalent dynamic linearization technology. Then, a tracking differentiator is used to predict the future output of a time-delay system to effectively control the system. Additionally, the concept of the saturation parameter is proposed, and the ICF-AS-MFAC controller is designed to ensure that the control system will not exhibit actuator saturation. The proposed algorithm is more flexible, has faster output responses for time-delay systems, and solves the problem of actuator saturation. The convergence and stability of the proposed method are rigorously proven mathematically. The effectiveness of the proposed method is verified by numerical simulations, and the applicability of the proposed method is verified by a series of experimental results based on double tanks.

Introduction

Computer technology has rapidly developed since the 1950s, and advanced control theory has greatly advanced in recent years. Advanced control theory can generally be divided into two control methods: data-based or data-driven control (DDC) and model-based control.^1,2 Because most industrial processes are complex (nonlinear, high-order, etc.), it is difficult to obtain a precise kinetic model^3,4; with the rapid development of computing speed, the DDC method, a control and optimization method based on collected data, is playing an increasingly important role in the modern industrial field. At present, the DDC method has been widely used in industrial fields.

The main DDC methods include the iterative method,^5–7 the virtual reference feedback tuning (VRFT) method, ⁸ adaptive dynamic programming,^9–13 the lazy learning (LL) control method, ³ and the model-free adaptive control (MFAC) method. ¹⁴ Among them, the MFAC method has the advantages of good robustness and low computation. MFAC has been studied in recent years by some scholars and is widely used in industry, such as in autopilot excavators ¹⁵ and four-rotor aircraft. ¹⁶ In addition, the MFAC method has been used to address sliding mode control, ¹⁷ formation control, ¹⁸ predictive control, ¹⁹ iterative learning control, ²⁰ and other problems in nonlinear systems.^1,21

However, most modern industrial systems have time delays, such as permanent-magnet synchronous-motor-position servo systems ²² and cement calciner outlet temperature systems. ²³ The problem of time delay often leads to deterioration of control system performance and even disruption of system stability, increasing the difficulty of controller design for the controlled object. The MFAC method cannot produce an ideal control effect for time-delay systems, and ideal control performance is difficult to achieve. At the same time, some control systems cannot reach the expected value quickly, which leads to a large number of errors in the control process of the MFAC algorithm. This will lead to actuator saturation, which will cause a large overshoot and impact the control effect.

In order to achieve ideal control effect for nonlinear systems with large time delay and effectively solve actuator saturation problem, this article proposes an improved compact-form antisaturation model-free adaptive control (ICF-AS-MFAC) method. By introducing the concept of saturation parameter, Smith prediction technology, and tracking differentiator, the proposed algorithm can effectively handle actuator saturation and time delay problems. The contributions of this article are as follows:

1. The article introduces a compact form dynamic linearization (CFDL) method and dynamically linearizes the studied nonlinear system. Based on the dynamic linearization model, this paper proposes the concept of saturation parameter, which is introduced into the controller criterion function to enable the algorithm to effectively handle the problem of actuator saturation.

2. Based on the introduction of antisaturation, the article introduces the control of differential term into the controller, adopts Smith prediction technology and tracking differentiator, and proposes ICF-AS-MFAC algorithm, which makes the proposed control method can effectively solve the time-delay problem and actuator saturation problem of nonlinear systems. Through rigorous Mathematical analysis, the stability and convergence of ICF-AS-MFAC method are verified.

3. Conduct a series of numerical simulations and physical dual tank experiments. The ICF-AS-MFAC algorithm was applied to the simulation system and its applicability was verified through several simulation experimental results.

The outline of this article is as follows. The next section introduces the ICF-MFAC control strategy and proves the algorithm convergence. Numerical simulations and semiphysical experiments are presented in the sections “Numerical simulations” and “Double-tank experiments.” The conclusions are given in the last section.

ICF-AS-MFAC algorithm

Dynamic linearization of a nonlinear system

Consider a single-input and single-output (SISO) system that is nonlinear and time-delayed. The system can be expressed as follows.

$$y(i+1)=\mathrm{\Theta }(Y_i,U_{i-\tau },i),$$ (1)

where $\mathrm{\Theta }(\cdot ):R^{n_y+n_u+2}\mapsto R$ is an unknown nonlinear function and $\tau$ is the delay time constant. $Y_i=\{y(i),y(i-1),\dots ,y(i-n_y)\}$ , $U_{i-\tau }=\{u_{cfdl}(i-\tau ),u_{cfdl}(i-\tau -1),\dots u_{cfdl}(i-\tau -n_u)\}$ , $u_{cfdl}(i)$ and $y(i)$ represent measurements of the input and output (I/O) at an instant i, $n_y$ and $n_u$ are the system orders which is unknown.

Assumption 1

The system (1) input is measurable and controllable; which means, for a series of uniformly bounded expected output signals $y^{*}$ , there must be a series of corresponding bounded input signals, so that the system can make the system output signal equal to the expected value under the effect of changing the input signal.

Assumption 2

The system partial derivatives described in (1) for outputs are continuous.

Assumption 3

System (1) satisfies the general Lipschitz ²⁴ criteria, which means that for any time i and $u_{cfdl}(i-\tau )-u_{cfdl}(i-\tau -1)\ne 0$ :

$$|\mathrm{\Theta }(Y_{i_1},U_{i_1-\tau },i_1)-\mathrm{\Theta }(Y_{i_2},U_{i_2-\tau },i_2)|\le \mathrm{\Gamma }|u_{cfdl}(i_1)-u_{cfdl}(i_2)|,$$ (2)

where $\mathrm{\Gamma }$ is a constant.

Theorem 1

According to reference, ²⁵ under Assumptions 1, 2 and 3, there must be a time-varying parameter $\phi (i)\in R$ called the pseudo partial derivative (PPD) such that the nonlinear system (1) can be converted to the following equivalent CFDL data model:

$$y(i+\tau +1)=y(i+\tau )+\phi (i)\mathrm{\Delta }{u}_{cfdl}(i),$$ (3)

where $\mathrm{\Delta }{u}_{cfdl}(i)=u_{cfdl}(i)-u_{cfdl}(i-1)$ .

Proof

See Appendix A.

Remark 1: The dynamic linearization method proposed in this article has essential differences from other linearization methods such as Feedback linearization, ²⁶ Taylor linearization, ²⁷ piecewise linearization, ²⁸ etc. Firstly, the generation of PPD is not dependent on the structure and parameters of the original system, but only on the I/O data generated by the system. Secondly, the dynamic linearization data model provides a direct mapping relationship between the control input signal increment and the system output signal increment, which is a linearization method aimed at control system design. At the same time, all possible complex behavior characteristics in the original dynamic nonlinear system, such as nonlinearity, time-varying parameters or time-varying structures, are compressed into PPD, and the numerical behavior of PPD is simple and easy to estimate.

Control system design

Controller design

To ensure that the system achieves desaturation and can complete the control task well in the case of actuator saturation, this paper introduces a parameter called the saturation parameter, $\mathrm{\Xi }(i)$ , to implement the controller antisaturation function, and the definition of $\mathrm{\Xi }(i)$ is given below.

$$\{\begin{array}{c}\mathrm{\Xi }(k)=\vartheta (u_m(i)-u_{cfdl}(i-1)),\\ ifsign(e^{*}(i))\ne sign(y(i))and{u}_{cfdl}(i-1)<u_m(i).\\ \mathrm{\Xi }(i)=1,\\ else.\end{array}$$ (4)

where $\vartheta >0$ is a weight parameter and $u_m$ is the maximum saturated input upper limit. It is worth noting that $\mathrm{\Xi }(i)$ refers to a way in which the controller handles the phenomenon of actuator saturation after it occurs. That is, when there is no saturation, the controller remains unchanged, and when it is saturated, the controller is modified to $\vartheta (u_m(i)-u_{cfdl}(i-1))$ .

Considering the tracking error, the control input cost function is constructed as follows.

$$\begin{array}{rl}J(u_{cfdl}(i))& =|e(i)\mathrm{\Xi }(i)-\mathrm{\Delta }y(i+1){|}^2+\lambda |u_{cfdl}(i)-u_{cfdl}(i-1){|}^2\\ & =|e(i)\mathrm{\Xi }(i)-y(i+\tau +1)-y(i+\tau ){|}^2+\lambda |u_{cfdl}(i)-u_{cfdl}(i-1){|}^2\end{array}$$ (5)

Consider Equation (4), and take the derivative of $J(u_{cfdl}(i))$ with respect to $u_{cfdl}(i)$ . This yields Equation (6).

$$\begin{array}{rl}{\displaystyle \frac{dJ(u_{cfdl}(i))}{d(u_{cfdl}(i))}=}& -2\phi ((y^{*}(i+\tau )-y(i)-\tau y^{\mathrm{\prime }}(i))\mathrm{\Xi }(i)\\ & -\phi (i)\mathrm{\Delta }{u}_{cfdl}(i))+2\lambda (u_{cfdl}(i)-u_{cfdl}(i-1)).\\ & \end{array}$$ (6)

For the optimality condition $dJ(u_{cfdl}(i))/d{u}_{cfdl}(i)=0$ and introduce parameters ${\rho }_1$ and ${\rho }_2$ to obtain the control algorithm:

$$u_{cfdl}(i)=u_{cfdl}(i-1)+{\displaystyle \frac{\phi (i)}{\lambda +\phi {(i)}^2}\cdot \mathrm{\Xi }(i)\cdot ({\rho }_1(y^{*}(i+\tau +1)-y(i))+{\rho }_2\tau y^{\mathrm{\prime }}(i))}$$ (7)

where ${\rho }_1,{\rho }_2\in (0,1)$ are step factors which have impacts on the smoothness of system operation. In addition, since $\phi (i)$ and $y^{\mathrm{\prime }}(i)$ in Equation (7) is unknown, their value need to be estimated.

Differential estimation

The structure of the ICF-AS-MFAC method is shown in Figure 1. The delay system transfer function can be expressed by $G_p(s)e^{-\tau s}$ , $\tau$ can be obtained by the relevant change method in general industrial processes.^29,30 In the feedback part, a leading link is added to predict the future output. When the system are slowly time-varying, the Smith predictor ³¹ can be applied by using $\tau s+1$ instead of $G_p(s)e^{-\tau s}$ . For large delay systems, the Equation (3) can be converted into Equation (8).

$$y(i+\tau +1)=y(i+\tau )+\phi (i)\mathrm{\Delta }{u}_{cfdl}(i)=y(i)+\tau y^{\mathrm{\prime }}(i)+\phi (i)\mathrm{\Delta }{u}_{cfdl}(i).$$ (8)

Figure 1.

Schematic diagram of the ICF-AS-MFAC system.

From Equation (8), the differential value of the output signal should be obtained. However, the traditional method of obtaining the differential will amplify the noise of the system. Therefore, this article uses a second-order differentiator to obtain the differential. The second-order tracking differentiator can track the differential signal well without amplifying noise, and the obtained differential signal is smoother. ³² The second-order tracking differentiator is described below.

$$\begin{array}{rl}& x_1(i+1)=x_1(i)+h{x}_2(i),\\ & x_2(i+1)=x_2(i)+h\mathrm{fst}(x_1(i)-y(i),x_2(i),r,h_0).\end{array}$$ (9)

The nonlinear function $\mathrm{fst}(\cdot )$ is defined below.

$$\begin{array}{rl}\mathrm{fst}(x_1(i)-y(k),x_2(i),r,h_0)& =\mathrm{fst}(A,h_0)\\ & =\{\begin{array}{cc}\begin{array}{c}sign(A)\\ A/h_0\end{array}& \begin{array}{c}|A|>h_0,\\ |A|\le h_0,h_0>0\end{array}\end{array}\end{array}$$ (10)

$$A=x_1(i)-y(i)+{\displaystyle \frac{x_2(i)|x_2(i)|}{2r}}$$ (11)

where $x_1(i)$ and $x_2(i)$ are the two input signals of the differentiator, h is the sampling time, $x_1(i)$ tracks $y(i)$ and $x_2(i)$ tracks the $y^{\mathrm{\prime }}(i)$ , r is a tracking parameter, and $h_0$ is a filtering parameter. $y(i)$ and $y^{\mathrm{\prime }}(i)$ can be well tracked by using Equation (11).

Remark 2: The delay of the system addressed in this article is assumed to be known. If the exact time delay is unknown, there are two situations. One is that the time delay varies with time, and the time delay $\tau (i)$ at this time can be replaced by the time delay from the previous time, that is, $\tau (i)$ = $\tau (i-1)$ . On the other hand, if the system delay is fixed and unknown, $\tau$ can be processed by taking the mean of the lower and upper limits of the delay time $[{\tau }_{min},{\tau }_{max}]$ .

Pseudo partial derivative estimation

The PPD true value exists, and Theorem 1 proves the existence of $\phi (i)$ . However, it cannot be obtained, so an estimated value $\hat{\phi }(i)$ is needed to replace the true value. In this study, a projection algorithm is used to obtain $\hat{\phi }(i)$ . First, the following cost function for estimating $\hat{\phi }(i)$ is defined:

$$\begin{array}{rl}J(\phi (i))=& |u_{cfdl}(i)-u_{cfdl}(i-\tau -1)-\phi (i)\mathrm{\Delta }y(i-1){|}^2\\ & +\mu |\phi (i)-\hat{\phi }(i-1){|}^2.\end{array}$$ (12)

where $\mu$ is a weight factor used to limit PPD changes. The optimal condition $dJ(\phi (i))/d\phi (i)=0$ is taken, through matrix inverse lemma, ³³ the following equation can be obtained:

$$\begin{array}{rl}& \hat{\phi }(i)=\hat{\phi }(i-\tau -1)+{\displaystyle \frac{\eta \mathrm{\Delta }{u}_{cfdl}(i-\tau -1)}{\mu +\mathrm{\Delta }{u}_{cfdl}(i-\tau -1)}\cdot }\\ & (\mathrm{\Delta }y(i)-\hat{\phi }(i-\tau -1)\mathrm{\Delta }{u}_{cfdl}(i-\tau -1))\end{array}$$ (13)

where $\mathrm{\Delta }y(i+1)=y(i+1)-y(i)$ and $\eta \in (0,1]$ is an added step-size constant. To adapt the PPD estimation algorithm to the actual industrial system and better track the time-varying parameters, the reset algorithm is defined as follows.

$$\hat{\phi }(i)=\hat{\phi }(1)$$ (14)

if $|\hat{\phi }(i)|<1$ , $|\mathrm{\Delta }x(i-1)|\le \epsilon$ or $sign(\hat{\phi }(i))=sign(\hat{\phi }(1))$ , and $\epsilon$ is a sufficiently small positive number.

As of now, the ICF-AS-MFAC controller has been designed and completed. The controller is summarized in Equation (15) :

$$\{\begin{array}{c}y(i)=x_1(i),y^{\mathrm{\prime }}(i)=x_2(i),\\ x_1(i+1)=x_1(i)+h{x}_2(i),\\ x_2(i+1)=x_2(i)+h\mathrm{fst}(x_1(i)-y(i),x_2(i),r,h_0),\\ A=x_1(i)-y(i)+{\displaystyle \frac{x_2(i)|x_2(i)|}{2r},}\\ \hat{\phi }(i)=\hat{\phi }(i-\tau -1)+{\displaystyle \frac{\eta \mathrm{\Delta }{u}_{cfdl}(i-\tau -1)}{\mu +\mathrm{\Delta }{u}_{cfdl}(i-\tau -1)}\cdot }\\ (\mathrm{\Delta }y(i)-\hat{\phi }(i-\tau -1)\mathrm{\Delta }{u}_{cfdl}(i-\tau -1)),\\ \hat{\phi }(i)=\hat{\phi }(1),\\ if|\hat{\phi }(i)|<1or|\mathrm{\Delta }{u}_{cfdl}(i-1)|\le \epsilon \\ orsign(\hat{\phi }(i))=sign(\hat{\phi }(1)),\\ \{\begin{array}{c}\mathrm{\Xi }(k)=\vartheta (u_m(i)-u_{cfdl}(i-1)),\\ ifsign(e(i))\ne sign(y(i))and{u}_{cfdl}(i-1)<u_m(i).\\ \mathrm{\Xi }(i)=1,\\ else.\end{array}\\ u_{cfdl}(i)=u_{cfdl}(i-1)+{\displaystyle \frac{\phi (i)}{\lambda +\phi {(i)}^2}\cdot \mathrm{\Xi }(i)\cdot }\\ ({\rho }_1(y^{*}(i+\tau +1)-y(i))+{\rho }_2\tau y^{\mathrm{\prime }}(i)).\end{array}$$ (15)

The steps of the ICF-AS-MFAC algorithm can be summarized as follows:

• Step 1: Differential estimation using a second-order differential tracker,

• Step 2: Estimate PPD based on input and output data,

• Step 3: Determine whether actuator saturation occurs based on the output of the controller,

• Step 4: Calculate the input at the next moment based on the saturation of the controller actuator, as well as the differential and output data

• Step 5: Transfer the calculated input at the next moment to the controlled object, obtain the output value at the next moment, and repeat Step 1.

With the designed controller, the stability of the ICF-ASMFAC method is considered next.

Stability analysis and other discussions

Stability analysis

Assumption 4. For sufficiently large i and $\mathrm{\Delta }x(i)\ne 0$ , the PPD of system (1) satisfies $\phi (i)\ge 0$ , and $\phi (i)=0$ only at a finite time i.

Theorem 2. Under the conditions of Assumptions 1–4, if ${\rho }_1+2({\rho }_1+{\rho }_2)\tau$ < $min\left\{4\sqrt{\lambda }/{\phi }_{max}(i),2\sqrt{\lambda }\right\}$ and $y^{*}(i)=\mathrm{const}=y^{*}$ , the application of the ICF-AS-MFAC scheme to a nonlinear system with the later delay meets the following two conditions:

1. The tracking output error of the system is convergent, that is, $\underset{k\to \mathrm{\infty }}{lim}|y^{*}-y(i+\tau +1)|=0$ .

2. After selecting appropriate values for $\eta ,\mu ,{\rho }_1,{\rho }_2$ , the system is closed-loop stable, which means that $y(i+\tau +1)$ and $u_{cfdl}(i)$ are bounded.

Proof

See Appendix B.

Discussion

The ICF-AS-MFAC scheme has been designed, and the pseudocode of the ICF-AS-MFAC algorithm is presented in Table 1.

Table 1.

Pseudocode of the ICF-AS-MFAC algorithm.

ICF-AS-MFAC

Evaluate the delay time $\tau$ of the plant; While i < max_execution time do Calculate the difference $y^{*}(i+\tau +1)-y(i)$ ; Calculate the differential of $y(i)$ according to Equations (9)-(11); Calculate the ith time input ucfdl(k) according to Equation (9); If ucfdl(i)> um(i) Perform anti-saturation treatment; End if Input $u_{cfdl}(i)$ into the Equation (1) to obtain the output $y(i+1+\tau )$ ; Calculate $\mathrm{\Delta }y(i+1)$ ; Estimate the $\hat{\phi }(i+1)$ according to Equation (13); Store $\hat{\phi }(i+1)$ , $u_{cfdl}(i)$ and $y(i+1+\tau )$ ; End

The flow chart of the controller is shown in Figure 2.

Figure 2.

ICF-AS-MFAC flow chart.

Remark 3: Equation (7) shows that when ${\rho }_2$ =0 and $\mathrm{\Xi }(i)$  = 1, the system reduces to the traditional CF-MFAC controller; that is:

$$u_{cfdl}(i)=u_{cfdl}(i-1)+{\displaystyle \frac{\phi (i)}{\lambda +\phi {(i)}^2}({\rho }_1(y^{*}(i+1)-y(i)))}$$ (16)

It is obvious that the traditional compact-form model-free adaptive control (CF-MFAC) controller is a special state of the improved method proposed in this article. Therefore, the ICF-AS-MFAC method is more versatile and flexible than the traditional CF-MFAC scheme.

Remark 4: There are three forms of model-free adaptive control (MFAC), namely, compact form (CF), partial form (PF), and full form (FF). ²⁵ For complex nonlinear systems, the improved method proposed in this paper can also be applied to PF-MFAC and FF-MFAC.

Remark 5: Compared with other DDC methods such as neural network-based control methods, ³⁴ VRFT, ⁸ IFT ³⁵ etc., ICF-AS-MFAC has more advantages. Firstly, ICF-AS-MFAC does not require external test signals or training processes, which is necessary for some DDC methods such as neural network-based control methods. Secondly, ICF-AS-MFAC does not require offline data adjustment, while algorithms such as VRFT require this step. Thirdly, there are strict mathematical proofs for the stability and convergence of ICF-AS-MFAC, which are not found in most other DDC algorithms such as PID.

Remark 6: The ICF-AS-MFAC algorithm proposed in this article only uses online data of the controlled object, without offline data of the controlled object. In fact, offline data of controlled objects can also be used reasonably, as described in relevant literature such as reference ³⁴ and. ³⁶ The application of offline data can be added according to the actual situation to increase the control effect of the algorithm.

Remark 7: At present, some papers have studied the delay and actuator saturation issues of MFAC. For example, Reference ³⁷ proposed an IOC-MFAC method for actuator saturation, but this method only limits output saturation and does not further desaturate. Reference ³⁸ proposed a method to resist actuator saturation in MFAC, but this method did not consider the delay problem of the system.

Numerical simulations

This section describes numerical simulations that use MATLAB 2020b to verify the effectiveness of the ICF-ASMFAC algorithm, include the time-delay system simulation and actuator saturation simulation.

Time-delay system simulation

The following third-order inertia delay objects are used in the simulations.

$$G(s)={\displaystyle \frac{5e^{-60s}}{{(5s+1)}^3}}$$ (17)

The simulation methods are the VRFT, CF-MFAC and ICF-AS-MFAC methods. The CF-MFAC and ICF-AS-MFAC schemes are optimized with the trial-and-error method, with a sampling time of 1 s and a simulation time of 8000 s. The VRFT algorithm is shown in Equation (18).

$$u(i)=u(i-1)+{\sigma }_{1}e(i)+{\sigma }_{2}e(i-1)+\dots +{\sigma }_I(i-I+1)$$ (18)

where $\mathrm{{\rm Y}}=[{\sigma }_1,{\sigma }_2\dots {\sigma }_I]$ . The VRFT toolbox in MATLAB 2020 was used to tune the controller parameter $\mathrm{{\rm Y}}$ , $M=z^{-\tau }$ , and the weight function $W(z)=1$ . ³⁹ To choose an appropriate I, six experiments were carried out with different $I$ (I= 3, 4, and 5). For the selected length I, two different sets of 500 pairs of I/O data were collected to calculate the corresponding parameters. According to the experimental results, the parameter vector $\mathrm{{\rm Y}}=[14.2501,1.5520,2.5417,-1.0011,8.3254]$ was adopted.

The expected signal is a pulse signal with a period of 2000s, a 50% pulse width and a single amplitude. The simulation parameters are shown in Table 2, and Figures 3 and 4 show the simulation results. The simulation results demonstrate that all three methods can complete control tasks; however, in the case of small overshoot, the ICF-AS-MFAC scheme has the fastest response and better control effects.

Table 2.

Simulation parameters.

Algorithm	Parameters
ICF-AS-MFAC	η = 0.001, ρ1 = 0.14, ρ2 = -1, λ = 2, μ = 1 $\phi (1)=0.1$
CF-MFAC	η = 0.001, ρ1 = 0.14, ρ2 = -1, λ = 2, μ = 1

Figure 3.

The tracking performance of the three algorithms for simulation 1.

Figure 4.

The inputs of the three algorithms for simulation 1.

In the ICF-AS-MFAC controller, ${\rho }_2$ is a key parameter that plays an important role in the control process, and therefore requires in-depth research. To verify the influence of ${\rho }_2$ , the system uses the same pulse signal and takes different ${\rho }_2$ values in the simulation. The values of ${\rho }_2$ are 0, −1, and −2. The simulation results are shown in Figure 5. In this range, when the value of ${\rho }_2$ is large, the overshoot of the system notably increases. In addition, the system response speed increases, although this change is not as obvious.

Figure 5.

The tracking performance of systems with different ${\rho }_2$ .

Actuator saturation simulation

To test the anti-actuator saturation effect of ICF-AS-MFAC, a second-order system is designed for testing:

$$G(s)={\displaystyle \frac{4}{s^2+3s+5}.}$$ (19)

The CF-MFAC and ICF-AS-MFAC proposed in this paper are selected for simulation. The simulation time of the system is 200 s, and the expected value is set to 100. If the output of the system cannot meet the expected value in a timely manner, resulting in error accumulation, the system will experience actuator saturation phenomenon. The greater the difference between the system and the expected value, the longer it takes to achieve the expected value, and the higher the degree of actuator saturation. Based on this phenomenon, to simulate different levels actuator saturation, three forces are applied at 50–55 s, 110–140 s and 160–165 s to cause error accumulation. The main parameters are set as follows: $\rho =0.3$ , $\eta =1$ , $\mu =10$ , $\epsilon =0.001$ . set $\vartheta =3$ , ρ₁= 0.3, ρ₂= 0, $u_m(i)=150$ for the ICF-AS-MFAC algorithm. The simulation results are shown in Figures 6 and 7.

Figure 6.

The tracking performance of the two algorithms for simulation 2.

Figure 7.

The inputs of the two algorithmsn for simulation 2.

As shown in Figures 6 and 7, both control algorithms can quickly reach the expected value. However, when the system encounters external forces and the output value cannot reach the expected value within a certain period of time, the traditional CF-MFAC will have large error accumulation, leading to a serious actuator saturation problem. It can be seen that under the influence of actuator saturation, the input and output values of CF-MFAC have a sudden change, which causes great damage to the controlled object in the actual system. However, ICF-AS-MFAC can rapidly desaturate without incurring the problem of the traditional CF-MFAC that there is a sharp increase in the output due to actuator saturation.

Time-delay and actuator saturation simulation

In order to observe the control effect of the proposed algorithm on systems with both time delay and actuator saturation, equation (17) is still selected as the controlled object for simulation experiments. The expected values of the system are set as follows:

$$y(i)=\{\begin{array}{c}1.3\\ 0.2\\ 1.5\\ 1\end{array}\begin{array}{c}i\le 500\\ 500<i\le 1000\\ 1000<i\le 1500\\ 1500<i\le 3000\end{array}$$

In order to cause actuator saturation, a force was applied between 2000 and 2100 s. The simulation methods are the VRFT, CF-MFAC and ICF-AS-MFAC methods, the simulation parameters are shown in Table 2, and the simulation time of the system is 200 s. The simulation results are shown in Figures 8 and 9.

Figure 8.

The inputs of the three algorithms for simulation 3.

Figure 9.

The tracking performance of the three algorithms for simulation 3.

The experimental results show that compared to CF-MFAC and VRFT algorithms, ICF-AS-MFAC has better control performance in environments with actuator saturation and time delay. The proposed algorithm can effectively handle time delay problems and achieve fast controller dropout.

Double-tank experiments

This section demonstrates the effectiveness of the algorithm through double slot experiments. Scenario 1 demonstrates that the algorithm can effectively complete the level control task of a dual tank, while Scenario 2 demonstrates that even after experiencing external disturbances, the algorithm can still quickly complete the level control task of a dual tank and has good robustness.

Scenario 1

Figure 10(a) and (b) shows the double-tank system equipment. Specifically, water tanks 1–2 are stacked without direct connections. In addition, the pump flow can be regulated by the electric water pump. The maximum flow rate of the water pump is 6000 rmp.

Figure 10.

Double-tank liquid level control system. (a) Schematic. (b) Equipment.

The control of the liquid level is a complex unstable nonlinear system. The input is the speed change of the pump, and the output is the second water tank liquid level. The incoming water enters the tank 1 and then flows out of the tank 2. Compared with a single tank, the response of the controlled quantity is one step behind in time due to the addition of the second tank; that is, there is a volume delay, which makes the process difficult to control.

It is worth noting that only the liquid level of water tank 2 is sent to the host computer. In this case, the control process shows delays and high nonlinearity. The task is to maintain the liquid level in tank 2 at 300 ml, as shown in Figure 11.

Figure 11.

Schematic diagram of the water tank level change. (a) The liquid level is 50 ml. (b) The liquid level is 200 ml. (c) The liquid level is 300 ml.

Four DDC algorithms were used in these experiments, namely, the conventional PID algorithm, VRFT algorithm, CF-MFAC algorithm and ICF-MFAC algorithm. The simulation time was 400 s.

The PID controller is described in Equation (20), and the PID parameters were set as follows: $K_p=30$ , $K_i=0.8$ , and $K_d=0$ . The CF-MFAC parameters were set as follows: η= 0.001, ρ= 1, λ= 2.4, and μ= 1. The ICF-AS-MFAC parameters were set as follows: η= 0.001, ρ₁= 1, ρ₂= -20, λ= 2.4, $u_m(i)=6000$ , $\vartheta =0.3$ , and μ= 1.

$$u(i)=K_p\left[e(i)+\sum _{j=0}^i{\displaystyle \frac{e(j)}{K_I}+K_d(e(i)-e(i-1))}\right]$$ (20)

According to the experimental results, the parameter vector ϒ= [7.5214, −6.4454, 0.4585, −2.1011, 1.4854] for VRFT algorithm was adopted.

Figures 12, 13, and 14 describe the performance of the four DDC algorithms. Based on the comparison results in Figures 12, 13, and 14, the following conclusions can be obtained:

1. The convergence effect of PID is good, and it converges to the expected level after approximately 230 s. However, the PID input fluctuates violently during the convergence process, which may damage the controller.

2. Due to its offline tuning scheme, the VRFT controller cannot quickly converge to the desired level. Therefore, the corresponding output converges to 300 ml at approximately 250 s. In addition, compared with the other three methods, the parameter adjustment procedure of the VRFT algorithm is time consuming because many experiments are needed to obtain the required parameters.

3. Both the CF-MFAC and ICF-AS-MFAC schemes can control the water tank level. However, under the control of the CF-MFAC scheme, the level of the dual-capacity water tank becomes stable after approximately 220 s, while the ICF-AS-MFAC algorithm reaches a stable level after approximately 200 s. At the same time, there is a certain delay between the input and output of the water tank, while the traditional MFAC lacks a mechanism to address the system delay. In contrast, the proposed ICF-AS-MFAC method can effectively address the system delay. Therefore, the control effect of the ICF-AS-MFAC algorithm is significantly better than that of the CF-MFAC algorithm in the double water tank and other time-delay systems.

Figure 12.

Change in liquid level of lower water tank.

Figure 13.

Change in water pump speed.

Figure 14.

Change in liquid level of upper water tank.

Scenario 2

The controlled object and selected control algorithm in Scenario 2 are no different from Scenario 1, and the selection of parameters is also completely consistent. It is worth noting that at the 350th second, 50 ml of water will be injected into the upper water tank to cause disturbance and observe the control effects of the four algorithms under disturbance conditions. The control results of the four algorithms are shown in Figures 15, 16, and 17, and the following conclusions can be drawn.

Figure 15.

Change in liquid level of lower water tank after adding water.

Figure 16.

Change in pump speed after adding water.

Figure 17.

Change in liquid level of upper water tank after adding water.

Compared to the original algorithm, the robustness of ICF-AS-MFAC has also been improved to a certain extent. In the experiment, after being disturbed, ICF-AS-MFAC can quickly restore normal control and achieve the expected water level. The proposed algorithm has good resistance to disturbance and robustness.

In summary, for large time-delay systems similar to double tank systems, the ICF-AS-MFAC scheme proposed in this paper has better control performance than the other three control methods, and there is no serious actuator saturation problem. It should be noted that the ICF-AS-MFAC method is easy to implement, and the entire control process does not use an accurate mathematical model of the controlled object, only uses its input and output data to complete the control task. The smaller the vibration of the control input signal, the less wear on the actuator valve. In the actual production process, there are generally various time delays. In the control process, choosing ICF-AS-MFAC with better time delay processing effect and faster control speed is clearly a better choice than the original MFAC and PID.

Conclusion

cIn this article, an ICF-AS-MFAC method is proposed for a class of time-delay discrete-time nonlinear systems based on a new dynamic linearization technique. This method uses only the plant's online I/O data to directly design the controller. The ICF-AS-MFAC controller has a better control effect for nonlinear time-delay systems and can effectively deal with the actuator saturation problem in the control process. The bounded-input/bounded-output (BIBO) stability and monotonic convergence of the ICF-AS-MFAC method are supported by rigorous mathematical analyses. The results of the MATLAB simulations and double-tank experiments show the effectiveness of the ICF-AS-MFAC scheme.

Author biographies

Lipu Wu received MS degrees from North China University of Technology. He is currently working toward the PhD degree with the School of Electrical and Control Engineering, North China University of Technology. His research interests include control and cooperative adaptive cruise control system.

Zhen Li obtained a bachelor's degree in engineering from Changchun University of Technology in 2019 and is currently pursuing a master's degree in engineering from Northern University of Technology. His research interest is data-driven control and model free adaptive control.

Shida Liu received the bachelor's degree from Inner Mongolia University, Hohhot, China, in 2011, and the PhD degree from Beijing Jiaotong University in 2017. He completed the post-doctoral degree in Beihang University in 2019. Until now, he is an associate professor working in the North China University of Technology. His research interests are the data-driven control, model free adaptive control, learning control, and intelligent transportation systems.

Zhijun Li received the bachelor's degree from Northeastern University, and the Ph.D. degree from University of Science and Technology Beiling. He is a professor working in North China University of Technology. His research interests include Control and decision-making, system control.

Dehui Sun received the bachelor's degree from North China Electric Power University (Bao Ding), and the PhD degree from North China Electric Power University. He is an professor working in North China University of Technology. His research interests include control, autonomous vehicles, and big data analytics.

Appendices

Appendix A

$$\begin{array}{rl}y(i+1)& =\mathrm{\Theta }(Y_i,U_{i-\tau },i)\\ & =\mathrm{\Theta }(y(i),y(i-1),\dots ,y(i-n_y),u_{cfdl}(i-\tau ),u_{cfdl}(i-\tau -1),\dots ,u_{cfdl}(i-\tau -n_u)),\end{array}$$ \rm (A1)

$$\begin{array}{rl}\mathrm{\Delta }y(i+1)& =\mathrm{\Theta }(y(i),y(i-1),\dots ,y(i-n_y),u_{cfdl}(i-\tau ),u_{cfdl}(i-\tau -1),\dots ,u_{cfdl}(i-\tau -n_u))\\ & -(y(i),y(i-1),\dots ,y(i-n_y),u_{cfdl}(i-\tau -1),u_{cfdl}(i-\tau -1),\dots ,u_{cfdl}(i-\tau -n_u)\\ & +(y(i),y(i-1),\dots ,y(i-n_y),u_{cfdl}(i-\tau -1),u_{cfdl}(i-\tau -1),\dots ,u_{cfdl}(i-\tau -n_u)\\ & -(y(i-1),\dots ,y(i-n_y-1),u_{cfdl}(i-\tau -1),\dots ,u_{cfdl}(i-\tau -n_u-1))\end{array}$$ \rm (A2)

Let:

$$\begin{array}{rl}\psi (i)=& (y(i),y(i-1),\dots ,y(i-n_y),u_{cfdl}(i-\tau -1),u_{cfdl}(i-\tau -1),\dots ,u_{cfdl}(i-\tau -n_u))\\ & -(y(i-1),\dots ,y(i-n_y-1),u_{cfdl}(i-\tau -1),\dots ,u_{cfdl}(i-\tau -n_u-1))\end{array}$$ \rm (A3)

According to Theorem 3 and Cauchy differential Mean value theorem, equation (1.1) can be written as follows:

$$\mathrm{\Delta }y(i+1)={\displaystyle \frac{\mathrm{\partial }\mathrm{\Theta }}{\mathrm{\partial }{u}_{cfdl}(i)}\mathrm{\Delta }{u}_{cfdl}(i)+\psi (i)}$$ \rm (A4)

where $\mathrm{\partial }\mathrm{\Theta }/\mathrm{\partial }{u}_{cfdl}(i)$ represents the value of $\mathrm{\Theta }$ (…) about the Partial derivative of the $(n_y+2)\mathrm{th}$ variable at a point between

$$(y(i),y(i-1),\dots ,y(i-n_y),u_{cfdl}(i-\tau -1),u_{cfdl}(i-\tau -1),\dots ,u_{cfdl}(i-\tau -n_u)$$

and

$$(y(i),y(i-1),\dots ,y(i-n_y),u_{cfdl}(i-\tau ),u_{cfdl}(i-\tau -1),\dots ,u_{cfdl}(i-\tau -n_u)$$

For each fixed time i, consider the following equation with variable $\eta (i)$ :

$$\psi (i)=\eta (i)\mathrm{\Delta }{u}_{cfdl}(i)$$

Let $\phi (i)=\eta (i)+\mathrm{\partial }\mathrm{\Theta }/\mathrm{\partial }{u}_{cfdl}(i)$ , then equation (3) can be written as $y(i+\tau +1)=y(i+\tau )+\phi (i)\mathrm{\Delta }{u}_{cfdl}(i)$ Assumption 1 is proof.

Appendix B

In this part, Theorem 2 is proved. First, we need to prove the boundedness of the PPD.

If $\hat{\phi }(i)\le \epsilon$ , $|\mathrm{\Delta }{u}_{cfdl}(i-1)|\le \epsilon$ or $\mathrm{sign}(\hat{\phi }(i))\ne \mathrm{sign}(\hat{\phi }(1))$ , then $\hat{\phi }(i)$ is obviously bounded. $\tilde{\phi }(i)=\hat{\phi }(i)-\phi (i)$ can be defined as the estimation error of the PPD. Subtract $\phi (i)$ from both sides of equation (13) to obtain:

$$\begin{array}{rl}\tilde{\phi }(i)=& \left[1-{\displaystyle \frac{\eta {|\mathrm{\Delta }{u}_{cfdl}(i-1)|}^{{}^2}}{\mu +{|\mathrm{\Delta }{u}_{cfdl}(i-1)|}^{{}^2}}}\right]\tilde{\phi }(i-\tau -1)\\ & +\phi (i-\tau -1)-\phi (i)\end{array}$$ \rm (B1)

Taking the absolute value on both sides of equation (B1):

$$|\tilde{\phi }(i)|=\left|\left[1-{\displaystyle \frac{\eta {|\mathrm{\Delta }{u}_{cfdl}(i-1)|}^{{}^2}}{\mu +{|\mathrm{\Delta }{u}_{cfdl}(i-1)|}^{{}^2}}}\right]\right||\tilde{\phi }(i-\tau -1)|+|\phi (i-\tau -1)-\phi (i)|$$ \rm (B2)

It is worth noting that for the variable $|\mathrm{\Delta }{u}_{cfdl}(i-1){|}^{{}^2}$ , the function $\eta |\mathrm{\Delta }{u}_{cfdl}(i-1){|}^{{}^2}/(\mu +|\mathrm{\Delta }{u}_{cfdl}(i-1){|}^{{}^2})$ increases monotonically, and its minimum is $\eta {\epsilon }^2/(\mu +{\epsilon }^2)$ . Then, if $0\le \eta \le 1$ and $\mu \ge 0$ , there must exist $d_1$ , such that:

$$0\le \left|\left[1-{\displaystyle \frac{\eta {|\mathrm{\Delta }{u}_{cfdl}(i-1)|}^{{}^2}}{\mu +{|\mathrm{\Delta }{u}_{cfdl}(i-1)|}^{{}^2}}}\right]\right|\le 1-{\displaystyle \frac{\eta {\epsilon }^2}{\mu +{\epsilon }^2}=d_1<1}$$ \rm (B3)

According to Assumption 1, $\phi (i)$ is bounded, and $|\phi (i)|\le \mathrm{\Gamma }$ , $|\phi (i-1)-\phi (i)|\le 2\mathrm{\Gamma }$ and $\mathrm{\Gamma }$ are constant. Using (B3), the following inequalities are obtained:

$$|\tilde{\phi }(i)|\le d_1|\tilde{\phi }(i-1)|+2\mathrm{\Gamma }\le d_1^2|\tilde{\phi }(i-2)|+2d_1\mathrm{\Gamma }+2\mathrm{\Gamma }\le \dots \le d_1^{i-1}|\tilde{\phi }(1)|+{\displaystyle \frac{2\mathrm{\Gamma }(1-d_1^{i-1})}{1-d_1},}$$ \rm (B4)

which means $\tilde{\phi }(i)$ is bounded, and since $\phi (k)$ has the upper bound ${\phi }_{max}(i)$ , $\tilde{\phi }(i)$ is bounded.

If the actuator is in an integral saturation state, $e(i)\ge 0$ , $\mathrm{\Xi }(i)\le 0$ , It can be seen from the reset equation:

$${\displaystyle \frac{\rho \hat{\phi }(i)}{\lambda +{|\hat{\phi }(i)|}^2}>0}$$ \rm (B5)

Then, in this case, the input is monotonically reduced and $u_{cfdl}(i)$ can apparently be restored below $u_m(k)$ . When $u_{cfdl}(i)<u_m(i)$ , the system ends the integral saturation status.

When the system is not in the integral state, $\mathrm{\Xi }(i)=1$ . The following equation can be obtained from Equations (3) and (7):

$$\begin{array}{c}e(i+1)=y^{*}(i+\tau +1)-y(i+\tau +1)\\ =y^{*}(i+\tau +1)-y(i+\tau )-\phi (i)\mathrm{\Delta }{u}_{cfdl}(i)\\ =e(i)-\phi (i)\psi (i)({\rho }_1(y^{*}(i+\tau +1)-y(i)-\tau y^{\mathrm{\prime }}(i))+({\rho }_2+{\rho }_1)\tau y^{\mathrm{\prime }}(i))\\ =(1-\phi (i)\psi (i)({\rho }_1+({\rho }_2+{\rho }_1)\tau ))e(i)+({\rho }_2+{\rho }_1)\tau \phi (i)\psi (i)e(i-1)\end{array},$$ \rm (B6)

where $\psi (i)=\phi (i)/(\lambda +\phi {(i)}^2)$ .

Since $\phi (i)>0$ , the disposal of the PPD can be chosen as $\hat{\phi }(0)=0$ . Based on Equation (14), for any time i, $\hat{\phi }(k)>0$ , $\psi (i)>0$ . If ${\xi }_1=\phi (i)\psi (i)({\rho }_1+({\rho }_2+{\rho }_1)\tau )$ , ${\xi }_2=({\rho }_2+{\rho }_1)\tau \phi (i)\psi (i)$ , then ${\xi }_1>{\xi }_2\ge 0$ . The absolute value of Equation (B6) is:

$$\begin{array}{rl}|e(i+1)|& =|(1-{\xi }_1(i))e(i)+{\xi }_2(i)e(i-1)|\\ & \le |1-{\xi }_1(i)||e(i)|+{\xi }_2(i)|e(i-1)|\\ & \le \chi (i)max\{|e(i)|,|e(i-1)|\},\end{array}$$ \rm (B7)

where $\chi (i)=|1-{\xi }_1(i)|+{\xi }_2(i)$ .

The following inequality can be obtained from ${\xi }_1>{\xi }_2\ge 0$ and ${\rho }_1+2({\rho }_1+{\rho }_2)\tau <4\sqrt{\lambda }/{\phi }_{max}(i)$ .

$$0<{\xi }_1(i)+{\xi }_2(i)=({\rho }_1+2({\rho }_1+{\rho }_2)\tau )\phi (i)\psi (i)\le ({\rho }_1+2({\rho }_1+{\rho }_2)\tau )\phi (i{)}_{max}{\displaystyle \frac{\phi (i)}{2\sqrt{\lambda }\phi (i)}<2,}$$ \rm (B8)

$$0\le {\xi }_2(i)<({\xi }_1(i)+{\xi }_2(i))<1.$$ \rm (B9)

According to Equation (B9), $0<1-{\xi }_2(i)\le 1$ . Moreover, according to Equation (B9):

$$-1\le {\xi }_2(i)-1<1-{\xi }_1(i)<1-{\xi }_2(i)\le 1$$ \rm (B10)

That is, $\chi (i)=|1-{\xi }_1(i)|+{\xi }_2(i)<1$ . Let the constant $v=max\{\chi (i)\}$ , k = 0, 1, 2,…; then, $\chi (i)\le v<1$ . According to Equation (13):

$$|e(i+1)|\le vmax\{|e(i)|,|e(i-1)|\},$$ \rm (B11)

$$|e(i)|\le vmax\{|e(i-1)|,|e(i-2)|\},$$ \rm (B12)

$$|e(i-1)|\le vmax\{|e(i-2)|,|e(i-3)|\}.$$ \rm (B13)

Equations (B12) and (B13) can be combined to obtain:

$$\begin{array}{rl}& max\{|e(i)|,|e(i-1)|\}\le \\ & vmax\{|e(i)|,|e(i-2)|,|e(i-3)|\}.\end{array}$$ \rm (B14)

Because $v<1$ , $v|e(i-1)|\le |e(i-1)|$ ; then, Equation (B14) can be simplified as:

$$\begin{array}{rl}& max\{|e(i)|,|e(i-1)|\}\le \\ & vmax\{|e(i-2)|,|e(i-3)|\}.\end{array}$$ \rm (B15)

Equation (B15) can be substituted into Equation (B11) to obtain:

$$\begin{array}{rl}& |e(i+1)|\le \\ & vmax\{|e(i)|,|e(i-1)|\}\le \\ & v^{2}max\{|e(i-2)|,|e(i-3)|\}\le \\ & \{\begin{array}{c}{v}^{(i/2)+1}max\{|e(0)|,|e(-1)|\},i\mathrm{is}\mathrm{an}\mathrm{even}\mathrm{number}\\ v^{(i+1)/2}max\{|e(0)|,|e(-1)|\},i\mathrm{is}\mathrm{an}\mathrm{odd}\mathrm{number},\end{array}\end{array}$$ \rm (B16)

Since $0\le d<1$ , Equation (B16) suggests that:

$$\underset{i\to \mathrm{\infty }}{lim}|y*-y(i+\tau +1)|=\underset{i\to \mathrm{\infty }}{lim}|e(i+1)|=0.$$ \rm (B17)

The conclusion of Theorem 2 a) is proved. Equation (B17) shows that $e(i)$ is bounded, and because the expected signal $y^{*}$ is bounded, the output $y(i+\tau +1)$ of the system is bounded.

Next, the article proves that the input signal $u(k)$ is bounded.

According to Equation (B11), there is a constant N, that is, the upper bound of equation $\phi (i)/(\lambda +\phi {(i)}^2)$ , such that:

$$|\mathrm{\Delta }{u}_{cfdl}(i)|<N|{\rho }_1(y^{*}(i+\tau +1)-y(i))+{\rho }_2\tau y^{\mathrm{\prime }}(i)|.$$ \rm (B18)

Then, the boundedness of the control input can be deduced from the following inequality:

$$|u_{cfdl}(i)|\le |\mathrm{\Delta }{u}_{cfdl}(i)+\mathrm{\Delta }{u}_{cfdl}(i-1)+\dots +\mathrm{\Delta }{u}_{cfdl}(2)|+u_{cfdl}(1).$$ \rm (B19)

The conclusion of Theorem 2 b) is proved. Thus, Theorem 2 is proved.