Adaptive Finite-Time Prescribed Performance Control of Nonlinear Power Systems with Symmetry Full-State Constraints

Abstract

Power system control is commonly based on linear controllers, where linear controllers are designed using a linearized model of the system at a specific operating point. However, when the system’s operating point is changed, the dynamic characteristics of the system shift significantly. At this point, linear controllers often fail to meet system stability requirements. Furthermore, the range of state variables in the power system is limited by the objective conditions. In addition, the power system has high-precision constraints on the deviation of the load frequency and so on. Therefore, it is worth designing a finite-time controller that satisfies the prescribed performance and full-state constraints based on the nonlinear model of the power systems. Firstly, the prescribed performance is incorporated into the barrier Lyapunov function to ensure that the tracking error is within the desired accuracy. Then, the tracking strategy is designed based on backstepping and incorporating a first-order filter to ensure that the controlled system’s signals and tracking errors remain bounded in finite time. Finally, two simulations are given to illustrate the effectiveness of the proposed control scheme, confirming that all states keep within the predefined range.

1. Introduction

The power system is widely recognized as a typical complex nonlinear system; therefore, it is crucial to consider the nonlinearity of the power system for practical purposes. Many effective control methods have been developed for nonlinear systems. These include the robust control technique [1,2], the sliding mode control technique [2,3], the fuzzy control technique [4,5], neural network control [6,7], adaptive control techniques [8,9], and backstepping control methods [10,11,12]. In controller design, these control methods have been widely used. Considering the various objective conditions and subjective requirements in practical power system operation, it is necessary to constrain all system state variables. For example, the chamber constrains the electrohydraulic system cylinder position [13]. The most commonly used approach is the barrier Lyapunov function (BLF) [14,15,16,17,18,19]. These take into account symmetry constraints or asymmetry constraints. A switching barrier Lyapunov function technique has been developed in [20]. The controller designs with their characteristics for full-state constraints have been proposed in [21,22,23]. It can be observed that the above results only achieve the state constraints. This is not enough for the control requirements of the power grid. For example, power systems have precise requirements for the time and range of frequency variations that meters must be able to withstand.

Prescribed performance control (PPC) is a viable solution to this problem. In [24], the visualization of a bounded output constraint is defined as a specific performance characteristic, and a new controller design is proposed. In [25], a control scheme with a specified performance function was proposed to improve the transient adjustment performance of active suspension in case of actuator failure. In [26], a control strategy based on the fuzzy adaptive event-triggering performance of an observer was proposed. However, if the power system becomes unstable, it must be quickly restored to achieve new transient stability to avoid destabilization or collapse. Therefore, the need for the power system to achieve stability for a limited period must also be considered.

Finite-time control compensates for the drawback of the convergence time approaching infinity. Finite time control can ensure that the system state quickly converges to the equilibrium point to provide a faster convergence speed. Some results [27,28,29,30,31,32] for the finite-time control problems of nonlinear systems are reported. In [33], a finite-time control strategy for ship course tracking is proposed when the marine environment interferes. A disturbance compensator with finite-time convergence was proposed in [34]. In [35], finite-time quantized feedback control was considered for the first time and a novel finite-time adaptive neural output feedback control method was proposed. Most of them only achieve finite-time tracking control for systems without dead zones. Dead zone characteristics are common in various mechanical and electrical components, especially in actuators, such as the dead-zone problem in the frequency control system of the thermal steam turbine. It is impossible to overlook the detrimental effects of the dead zone. In [36], the modeling of the drive system of a type of permanent magnet synchronous motor with dead-zone input nonlinearity was studied. In [37], the system subject to the unknown input of the dead zone was studied, and an observer-based adaptive tracking controller has been developed. Other results can be found in [38,39,40]. Therefore, constructing an improved finite-time control method that ensures the tracking error stays within a prescribed limit and keeps all the signals constrained is a problem worth solving.

Motivated by the above considerations, the finite time tracking control problem of nonlinear systems with prescribed performance and a dead zone is studied in this paper. It enables the system to achieve a prescribed performance in a finite time while ensuring that all states are constrained to a specific range. The main contributions of this study are: (1) A barrier function with prescribed performance is designed. It can guarantee that all state constraints are not violated while the tracking error is kept within a specified prescribed range. (2) A Finite-time adaptive tracking controller is proposed to ensure that the output of a non-linear system with a dead zone can track a given reference signal for a finite time and ensure that all variables in the control system are bounded.

The structure of the rest of this paper is as follows. Section 2 presents the research questions and background information. Section 3 then presents the main results of this study. Section 4 presents some simulation results to validate the developed controller. Finally, a summary is given in Section 5.

2. Problem Formulation and Basic Knowledge

2.1. Problem Formulation

Consider the following class of nonlinear mathematical models in power systems.

$$\left\{\begin{array}{c}{\dot{x}}_i=f_i\left({\overline{x}}_i\right)+g_i\left({\overline{x}}_i\right)x_{i+1},i=1,2,\cdots ,n-1\hfill \\ {\dot{x}}_n=f_n\left({\overline{x}}_n\right)+g_n\left({\overline{x}}_n\right)D\left(u\left(t\right)\right)\hfill \\ y=x_1\hfill \end{array}\right.$$

(1)

where ${\overline{x}}_i={[x_1,x_2,\cdots ,x_i]}^T,i=1,2,\cdots ,n$ is the state vector, $f_i\left({\overline{x}}_i\right)={\theta }_i^T{\psi }_i\left({\overline{x}}_i\right)$, with ${\theta }_i\in R^m$ is the unknown constant vector, ${\psi }_i\left(\overline{x_i}\right)$ is the known smooth nonlinear function vector, $g_i\left({\overline{x}}_i\right),i=1,2,\cdots ,n$ is the known smooth nonlinear function. $y\in R$ is the system output. $D\left(u\right(t\left)\right)$ is the dead-zone input. In this study, the focus is on symmetric constraints on states, i.e., all states are restricted to the symmetric predefined compact sets, i.e., $\left|x_i\right|<k_{ci}$, where $k_{ci},i=1,\cdots ,n$ are predefined positive constants.

The following is a description of the dead-zone input.

$$\begin{array}{c}\hfill D\left(u\left(t\right)\right)=\left\{\begin{array}{cc}{m}_r(u\left(t\right)-b_r),& u\left(t\right)\ge b_r\\ 0,& -b_l<u\left(t\right)<b_r\\ m_l(u\left(t\right)+b_l),& u\left(t\right)\le -b_l\end{array}\right.\end{array}$$

(2)

By defining $m_r=m_l=m>0,b_r,b_l>0$, (2) can be rewritten in the following form:

$$D\left(u\right(t\left)\right)=mu\left(t\right)+d\left(t\right),$$

where

$$d\left(t\right)=\left\{\begin{array}{cc}-m{b}_r,& u\left(t\right)\ge b_r\\ -mu\left(t\right),& -b_l<u\left(t\right)<b_r\\ m{b}_l,& u\left(t\right)\le -b_l\end{array}\right.$$

and $d\left(t\right)\le max\{m{b}_r,m{b}_l\}=\overline{d}$.

The control objective is to develop an adaptive finite-time control scheme for system (1), such that the system output y tracks a given reference signal $y_r$ for a finite time according to a prescribed error constraint and does not violate any state constraint.

2.2. Basic Knowledge

The following definitions and lemmas support the design work of this paper.

Definition 1

([41]). For the equilibrium $\zeta =0$ of the nonlinear system $\dot{\zeta }=f\left(\zeta \right)$, if there exists a scalar $\epsilon >0$ and a settling time $T(\epsilon ,{\zeta }_0)<\infty$ such that $∥\zeta \left(t\right)∥<\epsilon$, for all $t\ge t_0+T$, the system is practical finite-time stable (PFS).

Lemma 1

([42,43]). For any real variables z, ζ, and any positive constants $\mu ,\vartheta$ and ι, the following inequation holds:

$${\left|z\right|}^{\mu }{\left|\zeta \right|}^{\vartheta }\le \frac{\mu }{\mu +\vartheta }\iota {\left|z\right|}^{\mu +\vartheta }+\frac{\vartheta }{\mu +\vartheta }{\iota }^{\frac{-\mu }{\vartheta }}{\left|\zeta \right|}^{\mu +\vartheta }$$

Lemma 2

([42]). The following inequality holds with $z_j\in R,j=1,2,\cdots ,m,0<p\le 1$:

$${\left(\sum _{j=1}^m\left|z_j\right|\right)}^p\le {\sum _{j=1}^m\left|z_j\right|}^p\le m^{1-p}{\left(\sum _{j=1}^m\left|z_j\right|\right)}^p$$

Lemma 3

([44]). For the system, $\dot{\zeta }=f\left(\zeta \right)$. If there exists a continuous positive-definite function $V\left(\zeta \right)$, scalars ${\lambda }_1>0$, ${\lambda }_2>0$, $0<\beta <1$, and $\rho >0$ such that

$$\dot{V}\left(\zeta \right)\le -{\lambda }_{1}V\left(\zeta \right)-{\lambda }_2{V}^{\beta }\left(\zeta \right)+\rho ,t\ge 0$$

then the system $\dot{\zeta }=f\left(\zeta \right)$ is PFS.

Assumption 1

([14,41]). It is assumed that the reference trajectory $y_r\left(t\right)$ and its derivatives of the jth order satisfy $\left|y_r^{\left(j\right)}\left(t\right)\right|\le B_j$, $j=0,1,\cdots n$, where $B_0,B_1,\cdots ,B_n$ are positive constants.

Assumption 2

([5,14]). The sign of $g_i\left({\overline{x}}_i\right),i=1,2,\cdots ,n$ satisfy $0<g_{i0}\le \left|g_i\left({\overline{x}}_i\right)\right|<g_{i1}$, where $g_{i0}$ and $g_{i1}$ are positive constants.

Remark 1.

Since $g_i\left({\overline{x}}_i\right),i=1,2,\cdots ,n$ are a known smooth function, and being away from zero is a controllable condition for system (1), Assumption 2 is reasonable. From Assumption 2, it can be obtained that either $0<g_{i0}\le g_i\left({\overline{x}}_i\right)\le g_{i1}$ or $-g_{i1}\le g_i\left({\overline{x}}_i\right)\le -g_{i0}<0$, i.e., $g_i\left({\overline{x}}_i\right)$ are strictly positive or negative. Without loss of generality, it can be further assumed that $g_i\left({\overline{x}}_i\right)$ is strictly positive. When $g_i\left({\overline{x}}_i\right)$ are negative, the stability of the control scheme remains unchanged. In addition, the constants $g_{i0}$ and $g_{i1}$ in Assumption 2 are used for analytical purposes only. It is not necessary to know their true values for the design of the controller.

3. Controller Design

This section describes the entire design process. First, the notation ${\psi }_i={\psi }_i\left({\overline{x}}_i\right)$ and $g_i=g_i\left({\overline{x}}_i\right)$ is introduced to simplify the design. To ensure that the output does not violate its constraints, a BLF is defined on the set ${\mathsf{\Omega }}_{z_i}=\left\{z_i:\left|z_i\right|<k_{bi},i=2,3,\cdots n\right\}$, where $k_{bi}>0$ is a design constant. The following design process details the definition of BLF and $z_i$.

Step 1: Define the tracking error as $z_1=x_1-y_r$, which has

$${\dot{z}}_1={\dot{x}}_1-{\dot{y}}_r={\theta }_1^T{\psi }_1+g_1{x}_2-{\dot{y}}_r$$

(3)

Define the following transformation with the set ${\mathsf{\Omega }}_{z_1}=\left\{z_1:\left|z_1\right|<\rho \right\}$

$$v_1=tan\left(\frac{\pi z_1}{2\rho }\right)$$

(4)

where $\rho$ is the PPF. In this paper, the exponential function $\rho =(k_{b1}-{\rho }_{\infty })e^{-kt}+{\rho }_{\infty },({\rho }_{\infty }>0,k>0)$ is chosen as PPF, with $k_{b1}=k_{c1}-B_0>{\rho }_{\infty }$.

From (3) and (4), the following is obtained

$${\dot{v}}_1={\phi }_1\left({\theta }_1^T{\psi }_1+g_1{x}_2-{\dot{y}}_r-{\rho }^{-1}\dot{\rho }{z}_1\right)$$

(5)

where ${\phi }_1=\frac{\pi (1+v_1^2)}{2\rho }$.

Remark 2.

It follows from (4) that $z_1=\frac{2}{\pi }\rho arctan{v}_1$. Noting that the nonlinear function $\frac{2}{\pi }arctan{v}_1$ has the following properties:

(1) 

$-1<\frac{2}{\pi }arctan{v}_1<1$;

(2) 

${\displaystyle \underset{v_1\to -\infty }{lim}}\frac{2}{\pi }arctan{v}_1=-1$;

(3) 

${\displaystyle \underset{v_1\to +\infty }{lim}}\frac{2}{\pi }arctan{v}_1=1$.

Then, there is $-\rho <z_1<\rho$, $t>0$.

The virtual control signal ${\alpha }_1$ is designed as follows

$${\alpha }_1=\frac{1}{g_1}\left({\dot{y}}_r+{\rho }^{-1}\dot{\rho }{z}_1-\frac{s_1{v}_1}{{\phi }_1}-\frac{k_1{v}_1^{2\sigma -1}}{{\phi }_1}-g_1^2{v}_1{\phi }_1-\frac{v_1{\phi }_1{\widehat{\delta }}_1{∥{\psi }_1∥}^2}{2b_1^2}\right)$$

(6)

where $s_1,k_1,b_1>0,0<\sigma <1$ are design constants.

To avoid the “complexity explosion” that results from the repetition of differentiating ${\alpha }_1$. A first-order filter with a time constant ${\beta }_1$ is applied to the virtual control signal ${\alpha }_1$.

$${\beta }_1{\dot{\overline{\alpha }}}_1+{\overline{\alpha }}_1={\alpha }_1$$

(7)

The output error ${\xi }_1$ of this filter and the tracking error $z_2$ are defined as follows

$${\xi }_1={\overline{\alpha }}_1-{\alpha }_1$$

(8)

$$z_2=x_2-{\overline{\alpha }}_1$$

(9)

Remark 3.

With the above definitions, it is obtained that ${\dot{\alpha }}_1=\frac{\partial {\alpha }_1}{\partial x_1}{\dot{x}}_1+\frac{\partial {\alpha }_1}{\partial {\dot{y}}_r}{\ddot{y}}_r+\frac{\partial {\alpha }_1}{\partial \rho }\dot{\rho }+\frac{\partial {\alpha }_1}{\partial \dot{\rho }}\ddot{\rho }+\frac{\partial {\alpha }_1}{\partial z_1}{\dot{z}}_1+\frac{\partial {\alpha }_1}{\partial v_1}{\dot{v}}_1+\frac{\partial {\alpha }_1}{\partial {\phi }_1}{\dot{\phi }}_1+\frac{\partial {\alpha }_1}{\partial {\widehat{\delta }}_1}{\dot{\widehat{\delta }}}_1=h_1(z_1,z_2,{\xi }_1,\rho ,{\widehat{\delta }}_1,y_r,{\dot{y}}_r,{\ddot{y}}_r)$. All variables in the continuous function $h_1$ come from compact sets. Then, according to the boundedness theorem of continuous functions on compact sets, there is $\left|h_1\right|\le H_1$ with $H_1>0$ as a constant.

Define the Lyapunov function as follows

$$V_1=\frac{1}{2}{v_1}^2+\frac{1}{2}{{\tilde{\delta }}^2}_1+\frac{1}{2}{\xi }_1^2$$

(10)

where ${\delta }_1={∥{\theta }_1∥}^2$, ${\tilde{\delta }}_1={\widehat{\delta }}_1-{\delta }_1$, and ${\widehat{\delta }}_1$ is the estimation of ${\delta }_1$.

It follows from (5) and (10) that

$${\dot{V}}_1=v_1{\phi }_1\left({\theta }_1^T{\psi }_1+g_1{x}_2-{\dot{y}}_r-{\rho }^{-1}\dot{\rho }{z}_1\right)+{\tilde{\delta }}_1{\dot{\widehat{\delta }}}_1+{\xi }_1{\dot{\xi }}_1$$

(11)

Based on (8) and (9), it gives $x_2=z_2+{\xi }_1+{\alpha }_1$. Then, the following holds

$${\dot{V}}_1=v_1{\phi }_1\left({\theta }_1^T{\psi }_1+g_1{z}_2+g_1{\xi }_1+g_1{\alpha }_1-{\dot{y}}_r-{\rho }^{-1}\dot{\rho }{z}_1\right)+{\tilde{\delta }}_1{\dot{\widehat{\delta }}}_1+{\xi }_1{\dot{\xi }}_1$$

(12)

Design the following adaptive law

$${\dot{\widehat{\delta }}}_1=\frac{{\left(v_1{\phi }_1\right)}^2{∥{\psi }_1∥}^2}{2b_1^2}-2{\mu }_1{\widehat{\delta }}_1$$

(13)

where ${\mu }_1>0$ is a design constant.

Using Young’s inequality, it holds that

$$v_1{\phi }_1{\theta }_1^T{\psi }_1\le \frac{{\left(v_1{\phi }_1\right)}^2{\delta }_1{∥{\psi }_1∥}^2}{2b_1^2}+\frac{b_1^2}{2}$$

(14)

$$v_1{\phi }_1{g}_1{z}_2\le \frac{1}{2}{\left(v_1{\phi }_1{g}_1\right)}^2+\frac{1}{2}{z}_2^2$$

(15)

$$v_1{\phi }_1{g}_1{\xi }_1\le \frac{1}{2}{\left(v_1{\phi }_1{g}_1\right)}^2+\frac{1}{2}{\xi }_1^2$$

(16)

$$-2{\mu }_1{\tilde{\delta }}_1{\widehat{\delta }}_1\le {\mu }_1\left({{\delta }_1}^2-{{\tilde{\delta }}_1}^2\right)$$

(17)

Let $z=1,\zeta =\frac{1}{2}{{\tilde{\delta }}_1}^2,\mu =1-\sigma ,\vartheta =\sigma$ and $\iota ={\sigma }^{\frac{\sigma }{1-\sigma }}$, then Lemma 1 gives

$${\left(\frac{1}{2}{{\tilde{\delta }}_1}^2\right)}^{\sigma }\le \left(1-\sigma \right)\iota +\frac{1}{2}{{\tilde{\delta }}_1}^2$$

(18)

From (7) and (8), it follows that

$${\xi }_1{\dot{\xi }}_1={\xi }_1\left(-\frac{1}{{\beta }_1}{\xi }_1-{\dot{\alpha }}_1\right)\le -\left(\frac{1}{{\beta }_1}-\frac{1}{2}\right){\xi }_1^2+\frac{H_1^2}{2}$$

(19)

Given $z=1,\zeta =\frac{1}{2}{{\xi }_1}^2,\mu =1-\sigma ,\vartheta =\sigma$ and $\iota ={\sigma }^{\frac{\sigma }{1-\sigma }}$, then Lemma 1 gives

$${\left(\frac{1}{2}{{\xi }_1}^2\right)}^{\sigma }\le \left(1-\sigma \right)\iota +\frac{1}{2}{{\xi }_1}^2$$

(20)

Substituting (6), (13)–(20) into (12), gives

$$\begin{array}{cc}\hfill {\dot{V}}_1\le & -s_1{v}_1^2-{\mu }_1\left(\frac{1}{2}{{\tilde{\delta }}_1}^2\right)-\left(\frac{1}{{\beta }_1}-1\right)\left(\frac{1}{2}{{\xi }_1}^2\right)-k_1{v}_1^{2\sigma }-{\mu }_1{\left(\frac{1}{2}{{\tilde{\delta }}_1}^2\right)}^{\sigma }-\left(\frac{1}{{\beta }_1}-1\right){\left(\frac{1}{2}{{\xi }_1}^2\right)}^{\sigma }\hfill \\ \hfill & +\left(\frac{1}{{\beta }_1}-1+{\mu }_1\right)\left(1-\sigma \right)\iota +\frac{b_1^2}{2}+\frac{H_1^2}{2}+{\mu }_1{{\delta }_1}^2+\frac{1}{2}{z}_2^2\hfill \\ \hfill \le & -{\sigma }_{1,1}{V}_1-{\sigma }_{1,2}{V}_1^{\sigma }+{\sigma }_{1,3}+\frac{1}{2}{z}_2^2\hfill \end{array}$$

(21)

where ${\sigma }_{1,1}=min\left\{2s_1,{\mu }_1,\left(\frac{1}{{\beta }_1}-1\right)\right\}>0$, ${\sigma }_{1,2}=min\left\{2^{\sigma }{k}_1,{\mu }_1,\left(\frac{1}{{\beta }_1}-1\right)\right\}>0$,

${\sigma }_{1,3}=\frac{b_1^2}{2}+\frac{H_1^2}{2}+\left(\frac{1}{{\beta }_1}-1+{\mu }_1\right)\left(1-\sigma \right)\iota +{\mu }_1{{\delta }_1}^2$.

Step i (i = 2, …, n − 1): The structure of the virtual control signal ${\alpha }_i$ is as follows

$${\alpha }_i=\frac{1}{g_i}\left({\dot{\overline{\alpha }}}_{i-1}-\frac{s_i{v}_i}{{\phi }_i}-\frac{k_i{v}_i^{2\sigma -1}}{{\phi }_i}-g_i^2{v}_i{\phi }_i-\frac{v_i{\phi }_i{\widehat{\delta }}_i{∥{\psi }_i∥}^2}{2b_i^2}-\frac{z_i^2}{2v_i{\phi }_i}\right)$$

(22)

where $s_i,k_i,b_i>0$ are design constants.

Similarly to step 1, the ${\alpha }_i$ is used as an input to the first-order filter to obtain ${\overline{\alpha }}_i$.

$${\beta }_i{\dot{\overline{\alpha }}}_i+{\overline{\alpha }}_i={\alpha }_i,{\overline{\alpha }}_i\left(0\right)={\alpha }_i\left(0\right)$$

(23)

The output error ${\xi }_i$ of this filter and the tracking error $z_{i+1}$ are defined as follows

$${\xi }_i={\overline{\alpha }}_i-{\alpha }_i$$

(24)

$$z_{i+1}=x_{i+1}-{\overline{\alpha }}_i$$

(25)

Define the following transformation

$$v_i=tan\left(\frac{\pi z_i}{2k_{bi}}\right)$$

(26)

From (9) and (24)–(26), it follows that

$${\dot{v}}_i={\phi }_i\left({\theta }_i^T{\psi }_i+g_i{z}_{i+1}+g_i{\xi }_i+g_i{\alpha }_i-{\dot{\overline{\alpha }}}_{i-1}\right)$$

(27)

where ${\phi }_i=\frac{\pi (1+v_i^2)}{2k_{bi}}$.

Define the Lyapunov function

$$V_i=\frac{1}{2}{v_i}^2+\frac{1}{2}{\tilde{\delta }}_i^2+\frac{1}{2}{\xi }_i^2$$

(28)

where ${\delta }_i={∥{\theta }_i∥}^2$, ${\tilde{\delta }}_i={\widehat{\delta }}_i-{\delta }_i$, and ${\widehat{\delta }}_i$ is the estimation of ${\delta }_i$.

Based on (27) and (28), it gives

$$\begin{array}{c}\hfill {\dot{V}}_i=v_i{\phi }_i\left({\theta }_i^T{\psi }_i+g_i{z}_{i+1}+g_i{\xi }_i+g_i{\alpha }_i-{\dot{\overline{\alpha }}}_{i-1}\right)+{\tilde{\delta }}_i{\dot{\widehat{\delta }}}_i+{\xi }_i{\dot{\xi }}_i\end{array}$$

(29)

Design the following adaptive law

$${\dot{\widehat{\delta }}}_i=\frac{{\left(v_i{\phi }_i\right)}^2{∥{\psi }_i∥}^2}{2b_i^2}-2{\mu }_i{\widehat{\delta }}_i$$

(30)

where ${\mu }_i>0$ is a design constant.

Using Young’s inequality yields

$$v_i{\phi }_i{\theta }_i^T{\psi }_i\le \frac{{\left(v_i{\phi }_i\right)}^2{\delta }_i{∥{\psi }_i∥}^2}{2b_i^2}+\frac{b_i^2}{2}$$

(31)

$$v_i{\phi }_i{g}_i{z}_{i+1}\le \frac{1}{2}{\left(v_i{\phi }_i{g}_i\right)}^2+\frac{1}{2}{z}_{i+1}^2$$

(32)

$$v_i{\phi }_i{g}_i{\xi }_i\le \frac{1}{2}{\left(v_i{\phi }_i{g}_i\right)}^2+\frac{1}{2}{\xi }_i^2$$

(33)

$$-2{\mu }_i{\tilde{\delta }}_i{\widehat{\delta }}_i\le {\mu }_i\left({{\delta }_i}^2-{{\tilde{\delta }}_i}^2\right)$$

(34)

For $z=1,\zeta =\frac{1}{2}{{\tilde{\delta }}_i}^2,\mu =1-\sigma ,\vartheta =\sigma$ and $\iota ={\sigma }^{\frac{\sigma }{1-\sigma }}$, then Lemma 1 gives

$${\left(\frac{1}{2}{\tilde{\delta }}_i^2\right)}^{\sigma }\le \left(1-\sigma \right)\iota +\frac{1}{2}{\tilde{\delta }}_i^2$$

(35)

Similarly to step 1, define $\left|{\dot{\alpha }}_i\right|\le H_i,H_i>0$. From (23) and (24), it follows that

$${\xi }_i{\dot{\xi }}_i={\xi }_i\left(-\frac{1}{{\beta }_i}{\xi }_i-{\dot{\alpha }}_i\right)\le -\left(\frac{1}{{\beta }_i}-\frac{1}{2}\right){\xi }_i^2+\frac{H_i^2}{2}$$

(36)

With $z=1,\zeta =\frac{1}{2}{{\xi }_i}^2,\mu =1-\sigma ,\vartheta =\sigma$ and $\iota ={\sigma }^{\frac{\sigma }{1-\sigma }}$, the following holds by Lemma 1

$${\left(\frac{1}{2}{{\xi }_i}^2\right)}^{\sigma }\le \left(1-\sigma \right)\iota +\frac{1}{2}{{\xi }_i}^2$$

(37)

Substituting (29)–(37) into (28) gives

$$\begin{array}{cc}\hfill {\dot{V}}_i\le & -s_i{v}_i^2-{\mu }_i\left(\frac{1}{2}{\tilde{\delta }}_i^2\right)-\left(\frac{1}{{\beta }_i}-1\right)\left(\frac{1}{2}{{\xi }_i}^2\right)-k_i{v}_i^{2\sigma }-{\mu }_i{\left(\frac{1}{2}{\tilde{\delta }}_i^2\right)}^{\sigma }-\left(\frac{1}{{\beta }_i}-1\right){\left(\frac{1}{2}{{\xi }_i}^2\right)}^{\sigma }\hfill \\ \hfill & +\left(\frac{1}{{\beta }_i}-1+{\mu }_i\right)\left(1-\sigma \right)\iota +\frac{b_i^2}{2}+\frac{H_i^2}{2}+{\mu }_i{{\delta }_i}^2+\frac{1}{2}{z}_{i+1}^2-\frac{1}{2}{z}_i^2\hfill \\ \hfill \le & -{\sigma }_{i,1}{V}_i-{\sigma }_{i,2}{V}_i^{\sigma }+{\sigma }_{i,3}+\frac{1}{2}{z}_{i+1}^2-\frac{1}{2}{z}_i^2\hfill \end{array}$$

(38)

where ${\sigma }_{i,1}=min\left\{2s_i,{\mu }_i,\left(\frac{1}{{\beta }_i}-1\right)\right\}>0$, ${\sigma }_{i,2}=min\left\{2^{\sigma }{k}_i,{\mu }_i,\left(\frac{1}{{\beta }_i}-1\right)\right\}>0$, ${\sigma }_{i,3}=\frac{b_i^2}{2}+\frac{H_i^2}{2}+\left(\frac{1}{{\beta }_i}-1+{\mu }_i\right)\left(1-\sigma \right)\iota +{\mu }_i{{\delta }_i}^2$.

Step n: It follows from (25) that

$${\dot{z}}_n={\dot{x}}_n-{\dot{\overline{\alpha }}}_{n-1}={\theta }_n^T{\psi }_n+g_{n}mu\left(t\right)+g_{n}d\left(t\right)-{\dot{\overline{\alpha }}}_{n-1}$$

(39)

Define the following transformation

$$v_n=tan\left(\frac{\pi z_n}{2k_{bn}}\right)$$

(40)

Based on (39) and (40), we obtain that

$${\dot{v}}_n={\phi }_n\left({\theta }_n^T{\psi }_n+g_{n}mu\left(t\right)+g_{n}d\left(t\right)-{\dot{\overline{\alpha }}}_{n-1}\right)$$

(41)

where ${\phi }_n=\frac{\pi (1+v_n^2)}{2k_{bn}}$.

Define the Lyapunov function

$$V_n=\frac{1}{2}{v_n}^2+\frac{1}{2}{{\tilde{\delta }}^2}_n$$

(42)

where ${\delta }_n={∥{\theta }_n∥}^2$, ${\tilde{\delta }}_n={\widehat{\delta }}_n-{\delta }_n$, and ${\widehat{\delta }}_n$ is the estimation of ${\delta }_n$.

From (41) and (42), it follows that

$${\dot{V}}_n=v_n{\phi }_n\left({\theta }_n^T{\psi }_n+g_{n}mu\left(t\right)+g_{n}d\left(t\right)-{\dot{\overline{\alpha }}}_{n-1}\right)+{\tilde{\delta }}_n{\dot{\widehat{\delta }}}_n$$

(43)

Design the following control signal and adaptive law as

$$u\left(t\right)=\frac{1}{m{g}_n}\left({\dot{\overline{\alpha }}}_{n-1}-\frac{s_n{v}_n}{{\phi }_n}-\frac{k_n{v}_n^{2\sigma -1}}{{\phi }_n}-\frac{1}{2}{g}_n^2{v}_n{\phi }_n-\frac{v_n{\phi }_n{\widehat{\delta }}_n{∥{\psi }_n∥}^2}{2b_n^2}-\frac{z_n^2}{2v_n{\phi }_n}\right)$$

(44)

$${\dot{\widehat{\delta }}}_n=\frac{{\left(v_n{\phi }_n\right)}^2{∥{\psi }_n∥}^2}{2b_n^2}-2{\mu }_n{\widehat{\delta }}_n$$

(45)

where $s_n,k_n,{\mu }_n,b_n>0$ are design constants.

According to Young’s inequality, there are

$$v_n{\phi }_n{\theta }_n^T{\psi }_n\le \frac{{\left(v_n{\phi }_n\right)}^2{\delta }_n{∥{\psi }_n∥}^2}{2b_n^2}+\frac{b_n^2}{2}$$

(46)

$$v_n{\phi }_n{g}_{n}d\left(t\right)\le \frac{1}{2}{\left(v_n{\phi }_n{g}_n\right)}^2+\frac{{\overline{d}}^2}{2}$$

(47)

$$-2{\mu }_n{\tilde{\delta }}_n{\widehat{\delta }}_n\le {\mu }_n\left({{\delta }_n}^2-{\tilde{\delta }}_n^2\right)$$

(48)

Let $z=1,\zeta =\frac{1}{2}{{\tilde{\delta }}_n}^2,\mu =1-\sigma ,\vartheta =\sigma$ and $\iota ={\sigma }^{\frac{\sigma }{1-\sigma }}$; then, Lemma 1 gives

$${\left(\frac{1}{2}{\tilde{\delta }}_n^2\right)}^{\sigma }\le \left(1-\sigma \right)\iota +\frac{1}{2}{\tilde{\delta }}_n^2$$

(49)

Substituting (44)–(49) into (43) yields

$$\begin{array}{cc}\hfill {\dot{V}}_n\le & -s_n{v}_n^2-{\mu }_n\left(\frac{1}{2}{\tilde{\delta }}_n^2\right)-k_n{v}_n^{2\sigma }-{\mu }_n{\left(\frac{1}{2}{\tilde{\delta }}_n^2\right)}^{\sigma }+{\mu }_n\left(1-\sigma \right)\iota +{\mu }_n{{\delta }_n}^2+\frac{b_n^2}{2}+\frac{{\overline{d}}^2}{2}-\frac{1}{2}{z}_n^2\hfill \\ \hfill \le & -{\sigma }_{n,1}{V}_n-{\sigma }_{n,2}{V}_n^{\sigma }+{\sigma }_{n,3}-\frac{1}{2}{z}_n^2\hfill \end{array}$$

(50)

where ${\sigma }_{n,1}=min\left\{2s_n,{\mu }_n\right\},{\sigma }_{n,2}=min\left\{2^{\sigma }{k}_n,{\mu }_n\right\},{\sigma }_{n,3}=\frac{b_n^2}{2}+\frac{{\overline{d}}^2}{2}+{\mu }_n\left(1-\sigma \right)\iota +{\mu }_n{{\delta }_n}^2$.

Theorem 1.

Consider the system (1) with Assumptions 1 and 2. With the virtual control signals (6), (22), the actual control law (44), and the adaptive laws (13), (30), and (45), then the system output tracks the given reference signal in finite time while the tracking error remains within the predetermined range and all the states of the closed-loop system are bounded.

Proof. 

Define

$$V=\sum _{j=1}^n{V}_j$$

Then, there is

$$\dot{V}=\sum _{j=1}^n{\dot{V}}_j$$

(51)

Substituting (21), (38) and (50) into (51) yields

$$\dot{V}\le -\sum _{j=1}^n{\sigma }_{j,1}V-\sum _{j=1}^n{\sigma }_{j,2}{V}_j^{\sigma }+\sum _{j=1}^n{\sigma }_{j,3}$$

According to Lemma 2, it holds that

$$\dot{V}\le -{\sigma }_{1}V-{\sigma }_2{V}^{\sigma }+{\sigma }_3$$

where ${\sigma }_1=min\left\{{\sigma }_{j,1},j=1,2,\cdots ,n\right\}$, ${\sigma }_2=min\left\{{\sigma }_{j,2},j=1,\right.$$\left.2,\cdots ,n\right\}$, and ${\sigma }_3={\displaystyle \sum _{j=1}^n}{\sigma }_{j,3}$, which means that the closed-loop system is PFS by Lemma 3. The end of the proof. □

Remark 4.

According to the above proof, it is obvious that $\left|x_1\right|\le \left|z_1\right|+\left|y_r\right|<\rho +B_0$ and $\left|x_i\right|\le \left|z_i\right|+\left|{\overline{\alpha }}_{i-1}\right|<k_{bi}+T_i$, $i=2,3,\cdots ,n$. Define $k_{c_1}=\rho \left(0\right)+B_0$ and $k_{ci}=k_{bi}+T_i$, $i=2,3,\cdots ,n$, then all the states are constrained.

To express the control scheme more clearly, the design procedure of the algorithm is shown in Figure 1.

4. Numerical Simulations

In this section. the designed controller is compared to two cases to demonstrate its advantages. Case 1, Case 2, and Case 3 represent the control method of this paper proposed, the general BLF control, and the free constraint control, respectively.

Example 1.

Consider the following single-ink rigid robot system:

$$J\ddot{\theta }=-(0.5mgl+Mgl)sin\left(\theta \right)+u$$

where $M=3kg$ is the mass of the rigid link, $J=M{l}^2+\frac{1}{3}m{l}^2$ is the moment of inertia, the angle of joint rotation is $\theta \in [0,\frac{\pi }{2}]$, $m=1.5kg$ is the load, the gravity coefficient, and the length of the robot link is $l=0.5m$. The system can be expressed as the following if set $x_1=\theta ,x_2=\dot{\theta }$, and the dead-zone input are combined.

$$\left\{\begin{array}{cc}& {\dot{x}}_1=x_2\hfill \\ & {\dot{x}}_2=\frac{1}{J}\left[-(0.5mgl+Mgl)sin{x}_1+D\left(u\left(t\right)\right)\right]\hfill \\ & y=x_1\hfill \end{array}\right..$$

In the simulation, the states are constrained in $\left|x_1\right|<k_{c1}=0.5$ and $\left|x_2\right|<k_{c2}=1$. The given tracking signal is $y_r=0.3cos\left(0.5t\right)$. The given performance function is ${\rho }_1=0.199e^{-6t}+0.001$. That is, the steady-state error is allowed not to exceed 0.001. The initial states are set to $x_1\left(0\right)=0.4$, $x_2\left(0\right)=-0.3$, ${\widehat{\delta }}_1=0.005$, ${\widehat{\delta }}_2=3.8$, and the input parameters of the dead zone are $m_r=m_l=0.8,b_r=0.4,b_l=0.6$. The parameters are set as $k_1=1,k_2=1,\sigma =0.5$, $s_1=1,s_2=0.01$, $b_1=1,b_2=1$, ${\mu }_1=20,{\mu }_2=10$. Figure 2, Figure 3, Figure 4, Figure 5, Figure 6 and Figure 7 show the results of the simulation.

From Figure 2, Figure 3, Figure 4 and Figure 5, it is seen that using the control method designed in this paper, y can fast track the reference signal $y_r$ in finite-time, and the state $x_2$ constraints are not violated. Moreover, Figure 6 shows the response of the adaptive law. Figure 7 shows the trajectory of $D\left(u\right(t\left)\right)$. To further clarify the comparison, some statistics are given in the following.

From

Table 1. The system output y and the reference signal $y_r$.

t (s)	0.0017	0.3007	0.7548	0.9963	1.0709	3.5123	6.0010	7.1269	8.5197	10
$y_r$	0.3000	0.2966	0.2789	0.26353	0.2580	−0.0553	−0.2970	−0.2737	−0.1312	0.0851
Case 1	0.3994	0.3024	0.2789	0.26354	0.2580	−0.0553	−0.2970	−0.2737	0.1312	0.0851
Case 2	0.3996	0.3835	0.3384	0.3014	0.2880	−0.0543	−0.2939	−0.27067	−0.1292	0.07902
Case 3	0.3994	0.2853	0.26839	0.2535	0.2481	−0.05730	−0.2885	−0.2658	−0.1273	0.0808

, using the presented control method, the signal is tracked at $t=1.0709$ and the tracking remains stable. The two other methods do not track the reference signal at this time. As can be seen from

Table 2. The tracking error $z_1$ and the prescribed performance function $\rho$.

t (s)	0.0017	0.3007	0.7548	0.9963	1.0709	3.5123	6.0010	7.1269	8.5197	10
$\rho$	0.1980	0.0337	0.0031	0.0015	0.0013	0.0010	0.0010	0.0010	0.00100	0.0010
Case 1	0.0994	0.0058	3.19 × 10−5	4.72 × 10−6	3.09 × 10−6	3.72 × 10−8	1.35 × 10−7	1.33 × 10−7	1.13 × 10−7	−1.00 × 10−7
Case 2	0.0996	0.0869	0.05958	0.0379	0.0300	−0.0020	0.0032	0.0031	0.0020	−0.0061
Case 3	0.0994	−0.0113	−0.0105	−0.0101	−0.0099	0.0009	0.0085	0.0079	0.0038	−0.0043

, the errors in Case 1 remain within the prescribed range, while the errors in Case 2 and Case 3 exceed the preset range at some moments. Moreover, the mean and RMS of the error are shown in

Table 3. The mean and RMS of the tracking $z_1$.

	Case 1	Case 2	Case 3
Mean	4.478 × 10−4	2.33 × 10−3	2.564 × 10−3
RMS	3.481 × 10−3	1.088 × 10−2	7.310 × 10−3

.

According to

Table 4. The system state $x_2$.

t (s)	0.0188	0.0600	0.3007	0.9963	1.0709	3.5123	6.0010	7.1269	8.5197	10
Case 1	−0.6996	−0.5270	−0.0883	−0.07175	−0.0766	−0.14746	−0.0216	0.0615	0.1350	0.1439
Case 2	−0.0389	−0.0431	−0.0697	−0.1741	−0.1863	−0.1403	−0.0210	0.0612	0.1331	0.1421
Case 3	−1.0043	−1.0641	−0.0114	−0.0697	−0.0744	−0.1430	−0.0203	0.0597	0.1309	0.1394

below, the state $x_2$ remains within the constraint range $k_{c2}=1$ in Case 1 and Case 2, while the constraint is violated at some moments in Case 3.

Example 2.

The second-order strict feedback nonlinear system is as follows.

$$\left\{\begin{array}{cc}& {\dot{x}}_1=0.1x_1^2+x_2\hfill \\ & {\dot{x}}_2=0.1x_1{x}_2-0.2x_1+[3cos\left(x_1{x}_2\right)+6.1]D\left(u\left(t\right)\right)\hfill \\ & y=x_1\hfill \end{array}\right.$$

where the states are constrained in $\left|x_1\right|<k_{c1}=0.5$ and $\left|x_2\right|<k_{c2}=0.6$. The initial states are set to $x_1\left(0\right)=0.4$, $x_2\left(0\right)=-0.3$, ${\widehat{\delta }}_1=0.005$, ${\widehat{\delta }}_2=0.01$. The parameters are set as $k_1=20,k_2=15,\sigma =0.5$, $s_1=15,s_2=5$, $b_1=40,b_2=40$, ${\mu }_1=80,{\mu }_2=60$. The rest is the same as in Example 1. Figure 8, Figure 9, Figure 10, Figure 11, Figure 12 and Figure 13 show the simulation results.

From Figure 8, Figure 9, Figure 10 and Figure 11, it can be seen that using the control method designed in this paper, y can fast track the reference signal $y_r$ in finite-time, and the state $x_2$ constraints are not violated. Figure 12 shows the response of the adaptive law. Furthermore, some statistics are given in the following.

From

Table 5. The system output y and the reference signal $y_r$.

t (s)	0.0010	0.3073	0.7389	0.9026	1.0217	3.6101	5.2730	7.1419	8.8566	10
$y_r$	0.3000	0.2965	0.27978	0.2700	0.2617	−0.0696	−0.2625	−0.2728	−0.0841	0.0851
Case 1	0.3997	0.3100	0.2798	0.2700	0.2617	−0.0696	−0.2625	−0.2728	−0.0841	0.0851
Case 2	0.3998	0.3591	0.2854	0.2679	0.2597	−0.0716	−0.2646	−0.2748	−0.0862	0.0832
Case 3	0.3997	0.2949	0.2752	0.2656	0.2576	−0.0644	−0.2630	−0.2772	−0.0891	0.0804

,

Table 6. The tracking error $z_1$ and the prescribed performance function $\rho$.

t (s)	0.0010	0.3073	0.7389	0.9026	1.0217	3.6101	5.2730	7.1419	8.8566	10
$\rho$	0.1988	0.0325	0.0034	0.0019	0.0014	0.0010	0.0010	0.0010	0.0010	0.0010
Case 1	0.09974	0.0135	4.25 × 10−5	−3.30 × 10−8	−9.87 × 10−9	−5.58 × 10−9	−5.63 10−9	−5.41 × 10−9	−8.14 × 10−9	−5.01 × 10−9
Case 2	0.0998	0.0626	0.0056	−0.0020	−0.0020	−0.0020	−0.0020	−0.0020	−0.0021	−0.0019
Case 3	0.1000	−0.0016	−0.0046	−0.0044	−0.0041	0.0053	−0.0004	−0.0045	−0.0051	−0.0046

and

Table 7. The mean and RMS of the tracking $z_1$.

	Case 1	Case 2	Case 3
Mean	6.883 × 10−4	−3.015 × 10−4	−7.916 × 104
RMS	5.984 × 10−3	1.105 × 10−2	6.259 × 10−3

. The signal is tracked at t = 1.0217 and the tracking remains stable in Case 1. The other two methods do not track the reference signal at this time. As can be seen from Table 2, the errors in Case 1 remain within the prescribed range, while the errors in Case 2 and Case 3 exceed the preset range at some moments. Furthermore, the mean and RMS of the error are shown in Table 7.

According to

Table 8. The system state $x_2$.

t (s)	0.0010	0.0276	0.5010	0.0737	0.1834	3.4979	6.5700	7.5069	8.9619	10
Case 1	−0.3123	−0.3866	−0.0686	−0.4004	−0.2869	−0.1479	0.0126	0.0802	0.1455	0.14319
Case 2	−0.2145	−0.1283	−0.1802	−0.1361	−0.1523	−0.1479	0.0125	0.0800	0.1455	0.1431
Case 3	−0.3447	−0.7690	−0.0489	−0.6542	−0.1983	−0.1489	0.0123	0.0795	0.1452	0.1459

below, the state $x_2$ remains within the constraint range $k_{c2}=0.6$ in Case 1 and Case 2, while the constraint is violated at some moments in Case 3.

Moreover, the effect of three different groups of control parameters on the performance of the control system is also investigated. They are, respectively, Group 1, which is $k_1=20,k_2=15$, $s_1=15,s_2=5$, Group 2, which is $k_1=60,k_2=60$, $s_1=2,s_2=2$, and Group 3, which is $k_1=0.01,k_2=0.01$, $s_1=0.2,s_2=0.2$. Figure 13, Figure 14 and Figure 15 show the results under different parameters.

5. Conclusions

The power system satisfies the prescribed performance and full state constraints. This approach ensures that in a finite time, the state variables in the power system reach and remain within the boundary constraints and that the error of the power system from the load frequency, etc., reaches and keeps within a predefined range in a finite time. The future investigation will include considering asymmetric constraints on states and time-varying constraint control. The proposed method will be applied to the study of multi-machine power systems, considering the interconnection between multiple motors; moreover, it will be simulated using a simulation system that more closely resembles the real system.