CN103605288B - A kind of based on the 1/4 cycle repetitive controller attracting rule - Google Patents

Abstract

The invention discloses a kind of based on the 1/4 cycle repetitive controller attracting rule.The present invention include given link,Periodic feedback link,Signal conversion module and subtract/addition ring.Given link producesThe reference signal of periodic symmetry；StructurePeriodic feedback link；According to attracting rule method, structureSignal conversion module, its output signal is for the correction of repetitive controller；Then the output signal input as controlled device of repetitive controller is calculated.Concrete attitude conirol work can be carried out according to characterizing system convergence performance indications, and provides the sign monotone decreasing region of tracking error convergence process, absolute attractable layer and steady-state error band border.Closed loop system can be completely eliminatedPeriodic symmetry interference signal, realizes closed loop system and accurately controls.The present invention provides a kind of Time domain design 1/4 cycle repetitive controller that can effectively suppress Symmtry Signals interference, raising control accuracy and greatly reduce EMS memory occupation.

Description

1/4 periodic repetitive controller based on attraction law

Technical Field

The invention relates to a repetitive control technology, in particular to a quarter-cycle repetitive controller suitable for an inverter and also suitable for a periodic operation process in industrial occasions.

Background

A repetitive controller is designed according to an internal model principle, and a polynomial generated by a periodic signal needs to be placed in a system closed loop to form a periodic delay feedback link. No matter what the waveform of the input signal is, the controller outputs the input signal to accumulate cycle by cycle as long as the waveform appears repeatedly in the fundamental wave cycle, thereby playing the role of completely inhibiting the periodic interference. Current repetitive control techniques focus on frequency domain design based on an internal model principle. To achieve complete tracking/suppression of external periodic signals along the entire cycle, the internal model principle requires an implementation mechanism that includes the same periodic signal inside the system. The implementation mode is two types: one is that the pure time-lag link is positioned in the front channel; the other is in the feedback path. The feedback system with pure time-delay link can produce any periodic signal, the period of the signal is determined by the lag time constant, the formed closed loop system is infinite dimensional, and it has infinite poles on the imaginary axis. For a system with zero relative order (object transfer function is true), the closed-loop system can be exponentially stabilized by repetitive control, but for a strictly true system, the closed-loop system cannot be exponentially stabilized. Thus, to realize a controlled system to track any periodic signal (track any high frequency component), very strict requirements are required on the system structure. Tracking of the high frequency component of the reference input is usually abandoned and only the steady state error requirement on the low frequency band is guaranteed. One way is to set a low-pass filter before the pure time lag link to cut off the high frequency components above the shearing frequency.

When a discrete time-lag internal model is used, the closed-loop system is of finite dimension. In general, a stable minimum phase system may employ a feed-forward compensation design with zero-pole cancellation, and a stable non-minimum phase system may employ zero-phase compensation. The repetitive controller is designed by a Zero Phase Error Tracking Control (ZPETC) method, the stability of the repetitive controller is easy to judge and only depends on the selection of the gain of the repetitive controller. The zero phase compensation can cancel the phase shift introduced by the unstable zero, but cannot make the gain 1 after cancellation. The incomplete cancellation of the gain affects the tracking performance of the high frequency component, and further measures are needed to eliminate the effect.

Discrete repetitive control requires constructing an arbitrary periodic signal internal model with a period of N. The generation mechanism of the symmetric signal with the period of N is

$x\left(t\right)=\frac{q^{-N}}{1-q^{-N}}{x}_0\left(t\right)---\left(1\right)$

Here, q is^-1For one-step post-shift operators, i.e. for the signal f (t), there is q^-1f (t) = f (t-1); x (t) is a periodic signal generated by the mechanism, x₀(t) is the initial value of the signal, i.e. the value of the first period of the periodic signal x (t). We call 1-q^-NTo generate a polynomial. The periodic signal generation mechanism represented by formula (1) is shown in fig. 1. According to the generation mode of the periodic signal, a periodic feedback link can be constructed

$u\left(t\right)=\frac{q^{-N}}{1-q^{-N}}v\left(t\right)---\left(2\right)$

Here, v (t), u (t) are input and output signals of the controller, respectively, and u (t) is also called a control signal. The repetitive controller comprises a periodic feedback link and an e/v signal transformation link, wherein the e/v signal transformation link can be written as v (t) = f (e (t)). The periodic feedback loop represented by equation (2) is shown in fig. 2. Thus, the formula (2) can be written again

u(t)=u(t-N)+v(e(t-N)) (3)

As can be seen from equation (3), when the controller is implemented repeatedly, the data of the whole period of u (t-N) and e (t-N) needs to be stored.

Of interest is a finite order approximation of the time-lag internal model or a finite order internal model. For example, finite dimensional approximation of continuous internal models, quasi-feed forward methods (PFFs) model band-limited interference with finite order polynomials, comb filters are used as discrete time-lag internal models. In a simpler case, only the inner sine model is constructed for the tracking/suppression problem of the sine signal. Reducing the memory requirements of the controller is an important issue to be solved for real-time control. The odd harmonic repetitive controller designed by Ramon et al effectively utilizes the half-cycle symmetry of signals to derive a generator of half-cycle symmetric signals in the frequency domain. The use of such a generator results in a reduction of memory footprint by half. Because only frequency domain factors are considered, and signal symmetry is often depicted in a time domain, more complex symmetric signals cannot be effectively processed, and the technical defect that the memory occupation is still large exists.

Disclosure of Invention

In order to overcome the defects that the existing full-period repetitive controller and half-period repetitive controller can not effectively process more complicated symmetric signals and the memory occupation amount is large, the invention provides an 1/4 period repetitive controller based on an attraction law method, so that more complicated symmetric signals can be effectively processed, and the memory occupation space is obviously reduced.

The technical scheme adopted by the invention is as follows:

an 1/4 periodic repetitive controller based on an inverter sets a reference signal r (t) with 1/4 periodic symmetry, and has the following two symmetric conditions

$p1.r\left(t\right)=\left\{\begin{array}{c}\&PlusMinus;r(t-N/4)+r_0\left(t\right),0<\mathrm{mod}(t,N)\&le;N/4\\ \&PlusMinus;r(t-N/4)+r_0\left(t\right),N/4<\mathrm{mod}(t,N)\&le;N/2\\ \&PlusMinus;r(t-N/4)+r_0\left(t\right),N/2<\mathrm{mod}(t,N)\&le;3N/4\\ \&PlusMinus;r(t-N/4)+r_0\left(t\right),3N/4<\mathrm{mod}(t,N)<\mathrm{N\; or}\mathrm{mod}(t,N)=0\end{array}\right.---\left(4\right)$

And t is more than or equal to N/4; or

$p2.r\left(t\right)=\left\{\begin{array}{c}\&PlusMinus;r(t-t^{\&prime;})+r_0\left(t\right),0<\mathrm{mod}(t,N)\&le;N/4\\ \&PlusMinus;r(t-t^{\&prime;})+r_0\left(t\right),N/4<\mathrm{mod}(t,N)\&le;N/2\\ \&PlusMinus;r(t-t^{\&prime;})+r_0\left(t\right),N/2<\mathrm{mod}(t,N)\&le;3N/4\\ \&PlusMinus;r(t-t^{\&prime;})+r_0\left(t\right),3N/4<\mathrm{mod}(t,N)<\mathrm{N\; or}\mathrm{mod}(t,N)=0\end{array}\right.---\left(5\right)$

And t > N/4, wherein,

$t^{\&prime;}=\left\{\begin{array}{cc}2(t\mathrm{mod}N/4)& t\mathrm{mod}N\&NotEqual;0\\ N/2& t\mathrm{mod}N/4=0\end{array}\right.$

r₀(t) is the initial value of the signal, i.e. the value of the first 1/4 cycles of the periodic signal, during the interval t>A constant value of zero is taken at N/4. Here, ,

the current time is judged to be at a specific position in a period according to specific conditions, and then whether +/-is +/-or +/-is judged according to 1/4 period symmetry characteristics.

Setting 1/4 a periodic feedback loop, the general structure of the repetitive controller can be expressed as: in the case of the situation p1,

$u\left(t\right)=\left\{\begin{array}{c}\&PlusMinus;u(t-N/4)+v\left(t\right),0<\mathrm{mod}(t,N)\&le;N/4\\ \&PlusMinus;u(t-N/4)+v\left(t\right),N/4<\mathrm{mod}(t,N)\&le;N/2\\ \&PlusMinus;u(t-N/4)+v\left(t\right),N/2<\mathrm{mod}(t,N)\&le;3N/4\\ \&PlusMinus;u(t-N/4)+v\left(t\right),3N/4<\mathrm{mod}(t,N)<\mathrm{N\; or}\mathrm{mod}(t,N)=0\end{array}\right.---\left(6\right)$

and t is more than or equal to N/4; in the case of the situation p2,

$u\left(t\right)=\left\{\begin{array}{c}\&PlusMinus;u(t-t^{\&prime;})+v\left(t\right),0<\mathrm{mod}(t,N)\&le;N/4\\ \&PlusMinus;u(t-t^{\&prime;})+v\left(t\right),N/4<\mathrm{mod}(t,N)\&le;N/2\\ \&PlusMinus;u(t-t^{\&prime;})+v\left(t\right),N/2<\mathrm{mod}(t,N)\&le;3N/4\\ \&PlusMinus;u(t-t^{\&prime;})+v\left(t\right),3N/4<\mathrm{mod}(t,N)<\mathrm{N\; or}\mathrm{mod}(t,N)=0\end{array}\right.---\left(7\right)$

and t is more than N/4, wherein u (t) is an output signal of the repetitive controller, and v (t) is an input signal of the periodic feedback link, and is obtained by an e/v signal conversion link. Aiming at a specific model of the system, a conversion mode of the e/v signal can be obtained according to an attraction law method.

Furthermore, the reference signal is a sine signal, and when t is more than or equal to N/4, the 1/4 periodic symmetry characteristic is

$r\left(t\right)=\left\{\begin{array}{c}-r(t-N/4)+r_0\left(t\right),0<\mathrm{mod}(t,N)\&le;N/4\\ r(t-N/4)+r_0\left(t\right),N/4<\mathrm{mod}(t,N)\&le;N/2\\ -r(t-N/4)+r_0\left(t\right),N/2<\mathrm{mod}(t,N)\&le;3N/4\\ r(t-N/4)+r_0\left(t\right),3N/4<\mathrm{mod}(t,N)<\mathrm{N\; or}\mathrm{mod}(t,N)=0\end{array}\right.---\left(8\right)$

According to the symmetrical characteristic of the sine signal, when t is more than or equal to N/4, the periodic feedback link of the repetitive controller is

$u\left(t\right)=\left\{\begin{array}{c}-u(t-N/4)+v\left(t\right),0<\mathrm{mod}(t,N)\&le;N/4\\ u(t-N/4)+v\left(t\right),N/4<\mathrm{mod}(t,N)\&le;N/2\\ -u(t-N/4)+v\left(t\right),N/2<\mathrm{mod}(t,N)\&le;3N/4\\ u(t-N/4)+v\left(t\right),3N/4<\mathrm{mod}(t,N)<\mathrm{N\; or}\mathrm{mod}(t,N)=0\end{array}\right.---\left(9\right)$

The following continuous suction law was constructed

$\stackrel{\&CenterDot;}{e}\left(t\right)=-\mathrm{\&rho;e}\left(t\right)-\mathrm{\&epsiv;}\sqrt{1-{\mathrm{\&delta;}}^{\left|e\left(t\right)\right|}}\mathrm{sgne}\left(t\right)---\left(10\right)$

Wherein ρ >0, >0,0< < 1; e (t) is a tracking error signal. Equation (10) is the finite time law of attraction, with a time of arrival of

$t_1<\frac{\ln (1-\sqrt{1-{\mathrm{\&delta;}}^{e\left(0\right)}})-\ln (1+\sqrt{1-{\mathrm{\&delta;}}^{e\left(0\right)}})}{\mathrm{\&epsiv;}\ln \mathrm{\&delta;}}---\left(11\right)$

The discretization form of the continuous suction law (10) is

$e(t+1)=(1-\mathrm{\&rho;})e\left(t\right)-\mathrm{\&epsiv;}\sqrt{1-{\mathrm{\&delta;}}^{\left|e\right|\left(t\right)|}}\mathrm{sgne}\left(t\right)---\left(12\right)$

Wherein 0< ρ <1, >0,0< < 1. To improve the system's immunity to periodic disturbances, the discrete attraction law (12) may be modified to

$e(t+1)=(1-\mathrm{\&rho;})e\left(t\right)-\mathrm{\&epsiv;}\sqrt{1-{\mathrm{\&delta;}}^{\left|e\left(t\right)\right|}}\mathrm{sgne}\left(t\right)+d^{*}(t+1)-d(t+1)---\left(13\right)$

Wherein, for the case p1,in the case of the situation p2,d (t +1) is equivalent disturbance, d^*(t +1) is a compensation signal for equivalent disturbance; w (t +1) is a system disturbance term.

The inverter dynamics model may be expressed in the form (and other periodic operation processes expressed by this model apply as well):

y(t+1)+a₁y(t)+a₂y(t-1)=b₁u(t)+b₂u(t-1)+w(t+1) (14)

wherein y (t) represents output at time t, u (t) represents a control quantity at time t, and w (t +1) is a disturbance signal at time t + 1. Accordingly, a specific expression of the e/v transformation link can be given as follows:

in the case of the situation p1,

$v\left(t\right)=\begin{array}{c}\left\{\begin{array}{cc}\frac{1}{b_1}[-(1-\mathrm{\&rho;})e\left(t\right)+\mathrm{\&epsiv;}\sqrt{1-{\mathrm{\&delta;}}^{\left|e\left(t\right)\right|}}\mathrm{sgne}\left(t\right)-d^{*}(t+1)& 0\&le;\mathrm{mod}(t,N)<N/4\\ +z\left(t\right)+z(t-\frac{N}{4})-e(t-\frac{N}{4}+1)]& \\ \frac{1}{b_1}[-(1-\mathrm{\&rho;})e\left(t\right)+\mathrm{\&epsiv;}\sqrt{t-{\mathrm{\&delta;}}^{\left|e\left(t\right)\right|}}\mathrm{sgne}\left(t\right)-d^{*}(t+1)& N/4\&le;\mathrm{mod}(t,N)<N/2\\ +z\left(t\right)-z(t-\frac{N}{4})+e(t-\frac{N}{4}+1)],& \\ \frac{1}{b_1}[-(1-\mathrm{\&rho;})e\left(t\right)+\mathrm{\&epsiv;}\sqrt{1-{\mathrm{\&delta;}}^{\left|e\left(t\right)\right|}}\mathrm{sgne}\left(t\right)-d^{*}(t+1)& N/2\&le;\mathrm{mod}(t,N)<3N/4\\ +z\left(t\right)+z(t-\frac{N}{4})-e(t-\frac{N}{4}+1)]& \\ \frac{1}{b_1}[-(1-\mathrm{\&rho;})e\left(t\right)+\mathrm{\&epsiv;}\sqrt{1-{\mathrm{\&delta;}}^{\left|e\left(t\right)\right|}}\mathrm{sgne}\left(t\right)-d^{*}(t+1)& 3N/4\&le;\mathrm{mod}(t,N)<\mathrm{N\; or}\mathrm{mod}(t,N)=0\\ +z\left(t\right)-z(t-\frac{N}{4})+e(t-\frac{N}{4}+1)]& \end{array}\right.---\left(15\right)\end{array}$

in the case of the situation p2,

$v\left(t\right)=\begin{array}{c}\left\{\begin{array}{cc}\frac{1}{b_1}[-(1-\mathrm{\&rho;})e\left(t\right)+\mathrm{\&epsiv;}\sqrt{1-{\mathrm{\&delta;}}^{\left|e\left(t\right)\right|}}\mathrm{sgne}\left(t\right)-d^{*}(t+1)& 0\&le;\mathrm{mod}(t,N)<N/4\\ +z\left(t\right)+z(t-t^{\&prime;})-e(t-t^{\&prime;}+1)]& \\ \frac{1}{b_1}[-(1-\mathrm{\&rho;})e\left(t\right)+\mathrm{\&epsiv;}\sqrt{t-{\mathrm{\&delta;}}^{\left|e\left(t\right)\right|}}\mathrm{sgne}\left(t\right)-d^{*}(t+1)& N/4\&le;\mathrm{mod}(t,N)<N/2\\ +z\left(t\right)-z(t-t^{\&prime;})+e(t-t^{\&prime;}+1)],& \\ \frac{1}{b_1}[-(1-\mathrm{\&rho;})e\left(t\right)+\mathrm{\&epsiv;}\sqrt{1-{\mathrm{\&delta;}}^{\left|e\left(t\right)\right|}}\mathrm{sgne}\left(t\right)-d^{*}(t+1)& N/2\&le;\mathrm{mod}(t,N)<3N/4\\ +z\left(t\right)+z(t-t^{\&prime;})-e(t-t^{\&prime;}+1)]& \\ \frac{1}{b_1}[-(1-\mathrm{\&rho;})e\left(t\right)+\mathrm{\&epsiv;}\sqrt{1-{\mathrm{\&delta;}}^{\left|e\left(t\right)\right|}}\mathrm{sgne}\left(t\right)-d^{*}(t+1)& 3N/4\&le;\mathrm{mod}(t,N)<\mathrm{N\; or}\mathrm{mod}(t,N)=0\\ +z\left(t\right)-z(t-t^{\&prime;})+e(t-t^{\&prime;}+1)]& \end{array}\right.---\left(16\right)\end{array}$

the reference signal for the inverter is a sinusoidal signal

$v\left(t\right)=\begin{array}{c}\left\{\begin{array}{cc}\frac{1}{b_1}[-(1-\mathrm{\&rho;})e\left(t\right)+\mathrm{\&epsiv;}\sqrt{1-{\mathrm{\&delta;}}^{\left|e\left(t\right)\right|}}\mathrm{sgne}\left(t\right)-d^{*}(t+1)& 0\&le;\mathrm{mod}(t,N)<N/4\\ +z\left(t\right)+z(t-\frac{N}{4})-e(t-\frac{N}{4}+1)]& \\ \frac{1}{b_1}[-(1-\mathrm{\&rho;})e\left(t\right)+\mathrm{\&epsiv;}\sqrt{t-{\mathrm{\&delta;}}^{\left|e\left(t\right)\right|}}\mathrm{sgne}\left(t\right)-d^{*}(t+1)& N/4\&le;\mathrm{mod}(t,N)<N/2\\ +z\left(t\right)-z(t-\frac{N}{4})+e(t-\frac{N}{4}+1)],& \\ \frac{1}{b_1}[-(1-\mathrm{\&rho;})e\left(t\right)+\mathrm{\&epsiv;}\sqrt{1-{\mathrm{\&delta;}}^{\left|e\left(t\right)\right|}}\mathrm{sgne}\left(t\right)-d^{*}(t+1)& N/2\&le;\mathrm{mod}(t,N)<3N/4\\ +z\left(t\right)+z(t-\frac{N}{4})-e(t-\frac{N}{4}+1)]& \\ \frac{1}{b_1}[-(1-\mathrm{\&rho;})e\left(t\right)+\mathrm{\&epsiv;}\sqrt{1-{\mathrm{\&delta;}}^{\left|e\left(t\right)\right|}}\mathrm{sgne}\left(t\right)-d^{*}(t+1)& 3N/4\&le;\mathrm{mod}(t,N)<\mathrm{N\; or}\mathrm{mod}(t,N)=0\\ +z\left(t\right)-z(t-\frac{N}{4})+e(t-\frac{N}{4}+1)]& \end{array}\right.---\left(17\right)\end{array}$

Wherein z (t) = r (t +1) + a₁y(t)+a₂y(t-1)-b₂u(t-1)，e(t)=r(t)-y(t)。

After the repetitive controller design is complete, the performance of the 1/4 cycle repetitive controller can be measured by the following indicators characterizing the convergence process, which are monotonically decreasing zone boundaries Δ_MDRAbsolute attraction layer boundary Δ_AALSteady state error band boundary Δ_SSEThe concrete expression is

1) Monotonous decreasing area (delta)_MDR)

Δ_MDR=max{Δ_MDR1,Δ_MDR2} (18)

In the formula,. DELTA._MDR1，Δ_MDR2Is positive and real, satisfies

$\left\{\begin{array}{c}\mathrm{\&rho;}{\mathrm{\&Delta;}}_{\mathrm{MDR}1}+\mathrm{\&epsiv;}\sqrt{1-{\mathrm{\&delta;}}^{{\mathrm{\&Delta;}}_{\mathrm{MDR}1}}}-\mathrm{\&Delta;}=0\\ (1-\mathrm{\&rho;}){\mathrm{\&Delta;}}_{\mathrm{MDR}2}-\mathrm{\&epsiv;}\sqrt{1-{\mathrm{\&delta;}}^{{\mathrm{\&Delta;}}_{\mathrm{MDR}2}}}-\mathrm{\&Delta;}=0\end{array}---\left(19\right)\right.$

2) Absolute attraction layer (. DELTA.)_AAL)

Δ_AAL=max{Δ_AAL1,Δ_AAL2} (20)

In the formula,. DELTA._AAL1，Δ_AAL2Is positive and real, and can be determined by

$\begin{array}{c}\mathrm{\&rho;}{\mathrm{\&Delta;}}_{\mathrm{AAL}1}+\mathrm{\&epsiv;}\sqrt{1-{\mathrm{\&delta;}}^{{\mathrm{\&Delta;}}_{\mathrm{AAL}1}}}-\mathrm{\&Delta;}=0\\ (2-\mathrm{\&rho;}){\mathrm{\&Delta;}}_{\mathrm{AAL}2}-\mathrm{\&epsiv;}\sqrt{1-{\mathrm{\&delta;}}^{{\mathrm{\&Delta;}}_{\mathrm{AAL}2}}}-\mathrm{\&Delta;}=0\end{array}---\left(21\right)$

3) Steady state error band (Δ)_SSE)

Δ_SSECan be based on_AALDetermining:

a. when 0 is present<Δ_AAL<x₁Time of flight

Δ_SSE=Δ_AAL(22)

b. When x is₁≤Δ_AAL<x₂Time of flight

${\mathrm{\&Delta;}}_{\mathrm{SSE}}=-(1-\mathrm{\&rho;})x_1+\mathrm{\&epsiv;}\sqrt{1-{\mathrm{\&delta;}}^{x_1}}+\mathrm{\&Delta;}---\left(23\right)$

c. When delta_AAL≥x₂Time of flight

Δ_SSE=Δ_AAL(24)

Wherein x is₁Is an equation

$(1-\mathrm{\&rho;})+\frac{\mathrm{\&epsiv;}{\mathrm{\&delta;}}^x\ln \mathrm{\&delta;}}{2\sqrt{1-{\mathrm{\&delta;}}^x}}=0---\left(25\right)$

Root of Zhengguo, x₂Is an equation

$(1-\mathrm{\&rho;})x-\mathrm{\&epsiv;}\sqrt{1-{\mathrm{\&delta;}}^x}+(1-\mathrm{\&rho;})x_1-\mathrm{\&epsiv;}\sqrt{1-{\mathrm{\&delta;}}^{x_1}}=0---\left(26\right)$

The root of Zhengguo.

1/4 the adjustable parameters of the periodic repetitive controller include rho, and the parameter setting can be performed according to the above-mentioned indexes characterizing the convergence performance of the system.

The attraction law method provided by the invention is also suitable for the whole-period repetitive control. When t is more than or equal to N, the reference signal has the symmetrical characteristic of

r(t)=r(t-N)+r₀(t) (27)

Wherein r is₀(t) is the initial value of the signal, i.e. the value of the first period of the periodic signal, during the interval t>A value of zero is constantly taken above N; the controller has the expression of

$\begin{array}{c}u\left(t\right)=u(t-N)+\frac{1}{b_1}[-(1-\mathrm{\&rho;})e\left(t\right)+\mathrm{\&epsiv;}\sqrt{1-{\mathrm{\&delta;}}^{\left|e\left(t\right)\right|}}\mathrm{sgne}\left(t\right)\\ -d^{*}(t+1)+z\left(t\right)-z(t-N)+e(t-N+1)]\end{array}---\left(28\right)$

Wherein z (t) = r (t +1) + a₁y(t)+a₂y(t-1)-b₂u (t-1); the boundary condition is the same as the 1/4 cycle repetitive control boundary condition, and is not described herein.

The attraction law method provided by the invention is also suitable for feedback control. When t is more than or equal to 1, the reference signal satisfies

r(t)=r(t-1)+r₀(t) (29)

Wherein r is₀(t) is the initial value of the signal, i.e. the value of the first period of the periodic signal, during the interval t>1, taking a zero value constantly; the controller is

$\begin{array}{c}u\left(t\right)=u(t-1)+\frac{1}{b_1}[-(1-\mathrm{\&rho;})e\left(t\right)+\mathrm{\&epsiv;}\sqrt{1-{\mathrm{\&delta;}}^{\left|e\left(t\right)\right|}}\mathrm{sgne}\left(t\right)\\ -d^{*}(t+1)+z\left(t\right)-z(t-1)+e\left(t\right)]\end{array}---\left(30\right)$

Wherein z (t) = r (t +1) + a₁y(t)+a₂y(t-1)-b₂u(t-1)。

The technical idea of the invention is that the current design of the repetitive controller method mostly focuses on frequency domain design, while the signal symmetry characteristic is shown in the time domain. Therefore, the time domain design method is more direct and unique in designing the repetitive controller, and the designed controller is simpler and more visual.

The invention provides a time domain design of 1/4 periodic repetitive controller for a reference signal with 1/4 periodic symmetry. 1/4 the cycle repeat controller can further reduce the occupied space of the memory, and the memory requirement is only one fourth of the whole cycle repeat controller; the design method provided designs the repetitive controller according to different symmetrical characteristics of signals in different time intervals, and corrects the control signal every 1/4 cycles to realize complete suppression of interference signals with the same cycle characteristics. And compared with a full-period repetitive controller, the response time is faster, and the method is beneficial to accelerating disturbance elimination.

The invention can effectively process more complex symmetrical reference signals, greatly reduce the occupied space of the memory, and has the advantages of quick convergence performance, accelerated interference suppression and high control precision.

Drawings

Fig. 1 is a block diagram of a periodic signal generator.

Fig. 2 is a block diagram of a periodic feedback link.

FIG. 3 is a block diagram of 1/4 periodic signal generators, wherein FIG. 3a is for the case of p1 and FIG. 3b is for the case of p 2.

FIG. 4 is a diagram illustrating various types of reference signals according to the present invention, wherein FIG. 4a is a diagram illustrating various types of reference signals with 1/4 periodic symmetry, for a total of 60 cases; fig. 4b is a diagram equivalent to one-half symmetrical characteristic reference signal types, totaling 4 types.

Fig. 5 is a block diagram of 1/4 periodic feedback links provided by the present invention, wherein fig. 5a is for the case of p1 and fig. 5b is for the case of p 2.

Fig. 6 is a block diagram of an 1/4 cycle repetitive control system provided by the present invention, wherein fig. 6a is for the case of p1, and fig. 6b is for the case of p 2.

Fig. 7 is a block diagram of an inverter control system according to an embodiment of the present invention.

Fig. 8 is a schematic block diagram of an inverter according to an embodiment of the present invention.

Fig. 9 shows an error signal when feedback control is used in the embodiment of the present invention.

FIG. 10 is a diagram illustrating an error signal when the full-period repetitive control is adopted in the embodiment of the present invention.

FIG. 11 is a diagram illustrating control signals generated when the full-period repetitive control is adopted in the embodiment of the present invention.

Fig. 12 shows an error signal when 1/4 cycle repetitive control is adopted in the embodiment of the invention.

Fig. 13 is a control signal generated when 1/4 cycle repetitive control is employed in an embodiment of the present invention.

Detailed Description

The following further describes embodiments of the present invention with reference to the accompanying drawings.

Referring to fig. 3 to 8, the controller is repeated at 1/4 cycles based on the attraction law method.

First step, determining 1/4 a periodic reference signal

The generation of the periodic reference signal with 1/4 periodic symmetry may employ the following mechanisms, which may be represented as: for case p1

$r\left(t\right)=\left\{\begin{array}{c}\frac{1}{1\&PlusMinus;q^{-N/4}}{r}_0\left(t\right),0<\mathrm{mod}(t,N)\&le;N/4\\ \frac{1}{1\&PlusMinus;q^{-N/4}}{r}_0\left(t\right),N/4<(t,N)\&le;N/2\\ \frac{1}{1\&PlusMinus;q^{-N/4}}{r}_0\left(t\right),N/2<\mathrm{mod}(t,N)\&le;3N/4\\ \frac{1}{1\&PlusMinus;q^{-N/4}}{r}_0\left(t\right),3N/4<\mathrm{mod}(t,N)<\mathrm{N\; or}\mathrm{mod}(t,N)=0\end{array}\right.---\left(1\right)$

And t is more than or equal to N/4;

for case p2

$r\left(t\right)=\left\{\begin{array}{c}\frac{1}{1\&PlusMinus;q^{-t^{\&prime;}}}{r}_0\left(t\right),0<\mathrm{mod}(t,N)\&le;N/4\\ \frac{1}{1\&PlusMinus;q^{-t^{\&prime;}}}{r}_0\left(t\right),N/4<(t,N)\&le;N/2\\ \frac{1}{1\&PlusMinus;q^{-t^{\&prime;}}}{r}_0\left(t\right),N/2<\mathrm{mod}(t,N)\&le;3N/4\\ \frac{1}{1\&PlusMinus;q^{-t^{\&prime;}}}{r}_0\left(t\right),3N/4<\mathrm{mod}(t,N)<\mathrm{N\; or}\mathrm{mod}(t,N)=0\end{array}\right.---\left(2\right)$

And t > N/4.

1/4 periodic reference signal generation mechanism3, respectively. The 1/4 periodic reference signal generated under this mechanism is shown in FIG. 4, where r₀(t) is the initial value of the signal, i.e. the value of the first 1/4 cycles of the periodic signal, during the interval t>The value of N/4 is constantly zero; the quarter-period symmetry of the reference signal has 64 cases, of which there are 60 cases with quarter-symmetry properties, as shown in fig. 4 a; there are 4 that can be equated to satisfying half the symmetry property, as shown in fig. 4 b.

For a sinusoidal reference signal, when t ≧ N/4, the specific generation mechanism of 1/4 periodic symmetry can be expressed as

$r\left(t\right)=\left\{\begin{array}{c}\frac{1}{1-q^{-N/4}}{r}_0\left(t\right),0<\mathrm{mod}(t,N)\&le;N/4\\ \frac{1}{1+q^{-N/4}}{r}_0\left(t\right),N/4<(t,N)\&le;N/2\\ \frac{1}{1-q^{-N/4}}{r}_0\left(t\right),N/2<\mathrm{mod}(t,N)\&le;3N/4\\ \frac{1}{1+q^{-N/4}}{r}_0\left(t\right),3N/4<\mathrm{mod}(t,N)<\mathrm{N\; or}\mathrm{mod}(t,N)=0\end{array}\right.---\left(3\right)$

Design 1/4 periodic repetitive controller

With this generation mechanism, the 1/4 cycle repetitive controller is designed to correspond to the general form: for case p1

$u\left(t\right)=\left\{\begin{array}{c}\frac{1}{1\&PlusMinus;q^{-N/4}}v\left(t\right),0<\mathrm{mod}(t,N)\&le;N/4\\ \frac{1}{1\&PlusMinus;q^{-N/4}}v\left(t\right),N/4<(t,N)\&le;N/2\\ \frac{1}{1\&PlusMinus;q^{-N/4}}v\left(t\right),N/2<\mathrm{mod}(t,N)\&le;3N/4\\ \frac{1}{1\&PlusMinus;q^{-N/4}}v\left(t\right),3N/4<\mathrm{mod}(t,N)<\mathrm{N\; or}\mathrm{mod}(t,N)=0\end{array}\right.---\left(4\right)$

And t is more than or equal to N/4; for case p2

$u\left(t\right)=\left\{\begin{array}{c}\frac{1}{1\&PlusMinus;q^{-t^{\&prime;}}}v\left(t\right),0<\mathrm{mod}(t,N)\&le;N/4\\ \frac{1}{1\&PlusMinus;q^{-t^{\&prime;}}}v\left(t\right),N/4<(t,N)\&le;N/2\\ \frac{1}{1\&PlusMinus;q^{-t^{\&prime;}}}v\left(t\right),N/2<\mathrm{mod}(t,N)\&le;3N/4\\ \frac{1}{1\&PlusMinus;q^{-t^{\&prime;}}}v\left(t\right),3N/4<\mathrm{mod}(t,N)<\mathrm{N\; or}\mathrm{mod}(t,N)=0\end{array}\right.---\left(5\right)$

And t > N/4. The periodic feedback loop is shown in fig. 5.

Aiming at the sinusoidal reference signal, when t is more than or equal to N/4, the feedback link of the 1/4 periodic repetitive controller is

$u\left(t\right)=\left\{\begin{array}{c}\frac{1}{1-q^{-N/4}}v\left(t\right),0<\mathrm{mod}(t,N)\&le;N/4\\ \frac{1}{1+q^{-N/4}}v\left(t\right),N/4<(t,N)\&le;N/2\\ \frac{1}{1-q^{-N/4}}v\left(t\right),N/2<\mathrm{mod}(t,N)\&le;3N/4\\ \frac{1}{1+q^{-N/4}}v\left(t\right),3N/4<\mathrm{mod}(t,N)<\mathrm{N\; or}\mathrm{mod}(t,N)=0\end{array}\right.---\left(6\right)$

Thirdly, determining a controlled object model

To give v (t) in the controller, take the following inverter control system model as an example:

y(t+1)+a₁y(t)+a₂y(t-1)=b₁u(t)+b₂u(t-1)+w(t+1) (7)

wherein the parameter a₁, a₂, b₁, b₂Can be obtained through mechanism modeling or experimental modeling; perturbation signal w (t) has 1/4 periodic symmetry; given a reference signal r (t), an output signal y (t), a tracking error e (t) = r (t) -y (t).

Fourthly, constructing a tracking error attraction law equation of the discrete system

$e(t+1)=(1-\mathrm{\&rho;})e\left(t\right)-\mathrm{\&epsiv;}\sqrt{1-{\mathrm{\&delta;}}^{\left|e\left(t\right)\right|}}\mathrm{sgne}\left(t\right)+d^{*}(t+1)-d(t+1)---\left(8\right)$

Wherein for the case of p1For the case of p2The current time is needed to be judged to be at a specific position in a period, and then whether +/-is +/-or +/-is judged according to 1/4 periodic symmetry characteristics.

And fifthly, giving a specific v (t) expression of the e/v conversion link: for case p1

$v\left(t\right)=\begin{array}{c}\left\{\begin{array}{cc}\frac{1}{b_1}[-(1-\mathrm{\&rho;})e\left(t\right)+\mathrm{\&epsiv;}\sqrt{1-{\mathrm{\&delta;}}^{\left|e\left(t\right)\right|}}\mathrm{sgne}\left(t\right)-d^{*}(t+1)& 0\&le;\mathrm{mod}(t,N)<N/4\\ +z\left(t\right)+z(t-\frac{N}{4})-e(t-\frac{N}{4}+1)]& \\ \frac{1}{b_1}[-(1-\mathrm{\&rho;})e\left(t\right)+\mathrm{\&epsiv;}\sqrt{t-{\mathrm{\&delta;}}^{\left|e\left(t\right)\right|}}\mathrm{sgne}\left(t\right)-d^{*}(t+1)& N/4\&le;\mathrm{mod}(t,N)<N/2\\ +z\left(t\right)-z(t-\frac{N}{4})+e(t-\frac{N}{4}+1)],& \\ \frac{1}{b_1}[-(1-\mathrm{\&rho;})e\left(t\right)+\mathrm{\&epsiv;}\sqrt{1-{\mathrm{\&delta;}}^{\left|e\left(t\right)\right|}}\mathrm{sgne}\left(t\right)-d^{*}(t+1)& N/2\&le;\mathrm{mod}(t,N)<3N/4\\ +z\left(t\right)+z(t-\frac{N}{4})-e(t-\frac{N}{4}+1)]& \\ \frac{1}{b_1}[-(1-\mathrm{\&rho;})e\left(t\right)+\mathrm{\&epsiv;}\sqrt{1-{\mathrm{\&delta;}}^{\left|e\left(t\right)\right|}}\mathrm{sgne}\left(t\right)-d^{*}(t+1)& 3N/4\&le;\mathrm{mod}(t,N)<\mathrm{N\; or}\mathrm{mod}(t,N)=0\\ +z\left(t\right)-z(t-\frac{N}{4})+e(t-\frac{N}{4}+1)]& \end{array}\right.---\left(9\right)\end{array}$

For case p2

$v\left(t\right)=\begin{array}{c}\left\{\begin{array}{cc}\frac{1}{b_1}[-(1-\mathrm{\&rho;})e\left(t\right)+\mathrm{\&epsiv;}\sqrt{1-{\mathrm{\&delta;}}^{\left|e\left(t\right)\right|}}\mathrm{sgne}\left(t\right)-d^{*}(t+1)& 0\&le;\mathrm{mod}(t,N)<N/4\\ +z\left(t\right)+z(t-t^{\&prime;})-e(t-t^{\&prime;}+1)]& \\ \frac{1}{b_1}[-(1-\mathrm{\&rho;})e\left(t\right)+\mathrm{\&epsiv;}\sqrt{t-{\mathrm{\&delta;}}^{\left|e\left(t\right)\right|}}\mathrm{sgne}\left(t\right)-d^{*}(t+1)& N/4\&le;\mathrm{mod}(t,N)<N/2\\ +z\left(t\right)-z(t-t^{\&prime;})+e(t-t^{\&prime;}+1)],& \\ \frac{1}{b_1}[-(1-\mathrm{\&rho;})e\left(t\right)+\mathrm{\&epsiv;}\sqrt{1-{\mathrm{\&delta;}}^{\left|e\left(t\right)\right|}}\mathrm{sgne}\left(t\right)-d^{*}(t+1)& N/2\&le;\mathrm{mod}(t,N)<3N/4\\ +z\left(t\right)+z(t-t^{\&prime;})-e(t-t^{\&prime;}+1)]& \\ \frac{1}{b_1}[-(1-\mathrm{\&rho;})e\left(t\right)+\mathrm{\&epsiv;}\sqrt{1-{\mathrm{\&delta;}}^{\left|e\left(t\right)\right|}}\mathrm{sgne}\left(t\right)-d^{*}(t+1)& 3N/4\&le;\mathrm{mod}(t,N)<\mathrm{N\; or}\mathrm{mod}(t,N)=0\\ +z\left(t\right)-z(t-t^{\&prime;})+e(t-t^{\&prime;}+1)]& \end{array}\right.---\left(10\right)\end{array}$

The reference signal for the inverter is a sinusoidal signal

$v\left(t\right)=\begin{array}{c}\left\{\begin{array}{cc}\frac{1}{b_1}[-(1-\mathrm{\&rho;})e\left(t\right)+\mathrm{\&epsiv;}\sqrt{1-{\mathrm{\&delta;}}^{\left|e\left(t\right)\right|}}\mathrm{sgne}\left(t\right)-d^{*}(t+1)& 0\&le;\mathrm{mod}(t,N)<N/4\\ +z\left(t\right)+z(t-\frac{N}{4})-e(t-\frac{N}{4}+1)]& \\ \frac{1}{b_1}[-(1-\mathrm{\&rho;})e\left(t\right)+\mathrm{\&epsiv;}\sqrt{t-{\mathrm{\&delta;}}^{\left|e\left(t\right)\right|}}\mathrm{sgne}\left(t\right)-d^{*}(t+1)& N/4\&le;\mathrm{mod}(t,N)<N/2\\ +z\left(t\right)-z(t-\frac{N}{4})+e(t-\frac{N}{4}+1)],& \\ \frac{1}{b_1}[-(1-\mathrm{\&rho;})e\left(t\right)+\mathrm{\&epsiv;}\sqrt{1-{\mathrm{\&delta;}}^{\left|e\left(t\right)\right|}}\mathrm{sgne}\left(t\right)-d^{*}(t+1)& N/2\&le;\mathrm{mod}(t,N)<3N/4\\ +z\left(t\right)+z(t-\frac{N}{4})-e(t-\frac{N}{4}+1)]& \\ \frac{1}{b_1}[-(1-\mathrm{\&rho;})e\left(t\right)+\mathrm{\&epsiv;}\sqrt{1-{\mathrm{\&delta;}}^{\left|e\left(t\right)\right|}}\mathrm{sgne}\left(t\right)-d^{*}(t+1)& 3N/4\&le;\mathrm{mod}(t,N)<\mathrm{N\; or}\mathrm{mod}(t,N)=0\\ +z\left(t\right)-z(t-\frac{N}{4})+e(t-\frac{N}{4}+1)]& \end{array}\right.---\left(11\right)\end{array}$

Wherein z (t) = r (t +1) + a₁y(t)+a₂y(t-1)-b₂u (t-1), tracking error e (t) = r (t) -y (t).

A block diagram of a control system employing the repetitive controller is shown in fig. 6. Comprising a function f (e (t)) converted from an error signal e (t) to a control signal v (t), 1/4 periodically repeating the controller module 101 and the controlled system module 102.

The sixth step, the controller parameter setting

After the repetitive controller design is completed, the controller parameters thereof need to be set. The adjustable parameters comprise rho, and the specific parameter setting work can be carried out according to the following indexes representing the system convergence. In order to characterize the tracking error convergence process, the invention introduces concepts of a monotone decreasing area, an absolute attraction layer and a steady-state error band, which are specifically defined as follows:

monotonous decreasing region delta_MDR

$\left\{\begin{array}{c}0<e(t+1)<\left(t\right),e\left(t\right)>{\mathrm{\&Delta;}}_{\mathrm{MDR}}\\ e\left(t\right)<e(t+1)<0,e\left(t\right)<-{\mathrm{\&Delta;}}_{\mathrm{MDR}}\end{array}\right.---\left(12\right)$

Absolute attraction layer Δ_AAL

$\left|e\left(t\right)\right|>{\mathrm{\&Delta;}}_{\mathrm{AAL}}\&DoubleRightArrow;\left|e(t+1)\right|<\left|e\left(t\right)\right|---\left(13\right)$

Steady state error band Δ_SSE

$\left|e\right|\left(t\right)|\&le;{\mathrm{\&Delta;}}_{\mathrm{SSE}}\&DoubleRightArrow;|e(t+1)|\&le;{\mathrm{\&Delta;}}_{\mathrm{SSE}}---\left(14\right)$

1) Monotonous decreasing area (delta)_MDR)

Δ_MDR=max{Δ_MDR1,Δ_MDR2} (15)

In the formula,. DELTA._MDR1，Δ_MDR2Is positive and real, and satisfies

$\begin{array}{c}\mathrm{\&rho;}{\mathrm{\&Delta;}}_{\mathrm{MDR}1}+\mathrm{\&epsiv;}\sqrt{1-{\mathrm{\&delta;}}^{{\mathrm{\&Delta;}}_{\mathrm{MDR}1}}}-\mathrm{\&Delta;}=0\\ (1-\mathrm{\&rho;}){\mathrm{\&Delta;}}_{\mathrm{MDR}2}-\mathrm{\&epsiv;}\sqrt{1-{\mathrm{\&delta;}}^{{\mathrm{\&Delta;}}_{\mathrm{MDR}2}}}-\mathrm{\&Delta;}=0\end{array}---\left(16\right)$

2) Absolute attraction layer (. DELTA.)_AAL)

Δ_AAL=max{Δ_AAL1,Δ_AAL2} (17)

In the formula,. DELTA._AAL1，Δ_AAL2Is a positive real number, can be determined by the following formula,

$\begin{array}{c}\mathrm{\&rho;}{\mathrm{\&Delta;}}_{\mathrm{AAL}1}+\mathrm{\&epsiv;}\sqrt{1-{\mathrm{\&delta;}}^{{\mathrm{\&Delta;}}_{\mathrm{AAL}1}}}-\mathrm{\&Delta;}=0\\ (2-\mathrm{\&rho;}){\mathrm{\&Delta;}}_{\mathrm{AAL}2}-\mathrm{\&epsiv;}\sqrt{1-{\mathrm{\&delta;}}^{{\mathrm{\&Delta;}}_{\mathrm{AAL}2}}}-\mathrm{\&Delta;}=0\end{array}---\left(18\right)$

3) steady state error band (Δ)_SSE)

Δ_SSECan be based on_AALTo be determined.

a. When 0 is present<Δ_AAL<x₁Time of flight

Δ_SSE=Δ_AAL(19)

b. When x is₁≤Δ_AAL<x₂Time of flight

${\mathrm{\&Delta;}}_{\mathrm{SSE}}=-(1-\mathrm{\&rho;})x_1+\mathrm{\&epsiv;}\sqrt{1-{\mathrm{\&delta;}}^{x_1}}+\mathrm{\&Delta;}---\left(20\right)$

c. When delta_AAL≥x₂Time of flight

Δ_SSE=Δ_AAL(21)

Wherein x is₁Is an equation

$(1-\mathrm{\&rho;})+\frac{\mathrm{\&epsiv;}{\mathrm{\&delta;}}^x\ln \mathrm{\&delta;}}{2\sqrt{1-{\mathrm{\&delta;}}^x}}=0---\left(22\right)$

Root of Zhengguo, x₂Is an equation

$(1-\mathrm{\&rho;})x-\mathrm{\&epsiv;}\sqrt{1-{\mathrm{\&delta;}}^x}+(1-\mathrm{\&rho;})x_1-\mathrm{\&delta;}\sqrt{1-{\mathrm{\&delta;}}^{x_1}}=0---\left(23\right)$

The root of Zhengguo.

The 1/4 cycle repetitive controller design described above is described as follows:

1) introduction of d into the law of attraction^*(t +1) reflects the suppression measure for the disturbance signal of a given periodic pattern, d^*And (t +1) is a compensation value of d (t +1) for compensating the aperiodic disturbance.

A direct method for determining the compensation value is d^*(t+1)=d(t)。

Here, a method of determining the compensation value when the d (t) bound is known is provided. Let the upper and lower bounds of the equivalent disturbance d (t) be d_u、d_lThen d (t +1) satisfies the inequality

d_l≤d(t)≤d_u

Note the book $\stackrel{\&OverBar;}{d}=\frac{d_u+d_1}{2},\mathrm{\&Delta;}=\frac{d_u-d_l}{2},$ Then the process of the first step is carried out,

$|d\left(t\right)-\stackrel{\&OverBar;}{d}|\&le;\mathrm{\&Delta;}$

is convenient to use

$d^{*}(t+1)=\stackrel{\&OverBar;}{d}=\frac{d_u+d_l}{2}$

2) In the formulas (8), (9), (10) and (11), e (t), y (t-1), y (t-t ' +1), y (t-t '), y (t-t ' -1) can be obtained through measurement, and u (t-1), u (t-t '), u (t-t ' -1) are stored values of the control signals and can be read from the memory.

3) For feedback control, its reference signal symmetry characteristic is r (t) = r (t-1). Therefore, the 1/4 periodic repetitive controller proposed in the present invention is also suitable for the constant regulation problem, where the equivalent disturbance is d (t) = w (t-1).

4) The control is repeated in a whole period, and the reference signal symmetry characteristic is r (t) = r (t-N). The equivalent disturbance is d (t) = w (t) — w (t-N), and the invention is also applicable to the whole period repetition control.

5) The method proposed by the invention is also applicable to five-parameter models (inverter models usually employ five-parameter models). The five-parameter model is as follows

y(t+1)+a₁y(t)+a₂y(t-1)=b₁u(t)+b₂u(t-1)+c₁+w(t+1) (24)

In the formula, c₁Is a constant value. GetIt is substituted into the formula (29),

$y(t+1)=-a_{1}y\left(t\right)-a_{2}y(t-1)b_1{u}_c\left(t\right)+b_2{u}_2(t-1)+c_1+b_1\stackrel{\&OverBar;}{u}+b_2\stackrel{\&OverBar;}{u}+d(t+1)---\left(25\right)$

get

$\stackrel{\&OverBar;}{u}=-\frac{c_1}{b_1+b_2}---\left(26\right)$

Namely, it is

$u\left(t\right)=u_c\left(t\right)-\frac{c_1}{b_1+b_2}---\left(27\right)$

That is, formula (25) can be changed to

y(t+1)=-a₁y(t)-a₂y(t-1)+b₁u_c(t)+b₂u_c(t-1)+d(t+1) (28)

The five-parameter model can be converted into a four-parameter model by equation (28). In this way, iterative controller design with respect to a five-parameter model may be performed with reference to a four-parameter model.

6) The 1/4 cycle repeat controller is given for a second order system (7), and the design results of a higher order system can be given in the same way.

Examples

The reference signal is a sinusoidal signal, and according to the embodiment shown in fig. 3a, 1/4 period symmetry characteristic of the sinusoidal signal is formula (3), and according to the symmetry characteristic of the signal, the repetitive controller can be designed as formula (6).

The present embodiment controls the inverter output waveform. The adopted inverter system comprises a given signal part, an 1/4 period repetition controller, a PWM modulation part, an inverter main circuit (comprising a later-stage LC filter circuit and the like) and a detection circuit. Wherein, the given signal, the repetitive controller and the PWM module are all realized by the DSP, and the rest parts are realized by the hardware circuit. The whole system is given the desired signal to be output by the DSP, which is equivalent to the input signal of the whole closed loop system. The high-low pulse signals which are subjected to PWM modulation and can drive a switching tube of a bridge circuit of the inverter are changed into high-low pulse signals, the output of the inverter is reduced into sine signals through an LC filter circuit, the sine signals are sampled by a detection circuit and returned to a DSP, and the input signals are corrected after 1/4-period repetitive control action, so that the waveform tracking control of the inverter is realized.

The following provides the design process and implementation results of the inverter 1/4 periodic repetitive controller, and compares the results with the effects of conventional feedback control and full-period repetitive control.

First, a system mathematical model is established. The inverter control system used in the present embodiment is as shown in fig. 7. The system comprises an e/v conversion link f (e (t)), a period feedback link, a period signal PWM modulation module 201, an inverter main circuit 202 and a detection circuit 203. The first two constitute 1/4 cycle repetitive controller 301, and the last three constitute inverter system 302.

Modeling is carried out by taking the main circuit of the inverter, the later-stage LC filter circuit and the sampling circuit as objects to obtain the following second-order difference equation model

y(t+1)+a₁y(t)+a₂y(t-1)=b₁u(t)+b₂u(t-1)+w(t+1) (29)

Wherein y (t) represents the output voltage of the inverter at the time t, u (t) represents the control quantity of the inverter at the time t, and w (t +1) represents the uncertain characteristic of the system and consists of external interference, measurement noise and unmodeled characteristic. Parameter a in the model₁,a₂,b₁,b₂Is obtained by mechanism modeling, and the specific value is

a₁=-1.3414,a₂=0.7254,b₁=0.1867,b₂=0.1769

Feedback control: formula (6), taking d (t +1) = w (t +1) -w (t),

$e(t+1)=(1-\mathrm{\&rho;})e\left(t\right)-\mathrm{\&epsiv;}\sqrt{1-{\mathrm{\&delta;}}^{\left|e\left(t\right)\right|}}\mathrm{sgne}\left(t\right)+d^{*}(t+1)-w(t+1)w\left(t\right)---\left(30\right)$

1/4 periodic repetitive control: given a reference signal r (t), the system tracking error dynamic equation is

$e(t+1)=(1-\mathrm{\&rho;})e\left(t\right)-\mathrm{\&epsiv;}\sqrt{1-{\mathrm{\&delta;}}^{\left|e\left(t\right)\right|}}\mathrm{sgne}\left(t\right)+d^{*}(t+1)-d(t+1)---\left(8\right)$

1/4 the e/v signal conversion link in the periodic repetitive controller is designed as follows

$v\left(t\right)=\begin{array}{c}\left\{\begin{array}{cc}\frac{1}{b_1}[-(1-\mathrm{\&rho;})e\left(t\right)+\mathrm{\&epsiv;}\sqrt{1-{\mathrm{\&delta;}}^{\left|e\left(t\right)\right|}}\mathrm{sgne}\left(t\right)-d^{*}(t+1)& 0\&le;\mathrm{mod}(t,N)<N/4\\ +z\left(t\right)+z(t-t^{\&prime;})-e(t-t^{\&prime;}+1)]& \\ \frac{1}{b_1}[-(1-\mathrm{\&rho;})e\left(t\right)+\mathrm{\&epsiv;}\sqrt{t-{\mathrm{\&delta;}}^{\left|e\left(t\right)\right|}}\mathrm{sgne}\left(t\right)-d^{*}(t+1)& N/4\&le;\mathrm{mod}(t,N)<N/2\\ +z\left(t\right)-z(t-t^{\&prime;})+e(t-t^{\&prime;}+1)],& \\ \frac{1}{b_1}[-(1-\mathrm{\&rho;})e\left(t\right)+\mathrm{\&epsiv;}\sqrt{1-{\mathrm{\&delta;}}^{\left|e\left(t\right)\right|}}\mathrm{sgne}\left(t\right)-d^{*}(t+1)& N/2\&le;\mathrm{mod}(t,N)<3N/4\\ +z\left(t\right)+z(t-t^{\&prime;})-e(t-t^{\&prime;}+1)]& \\ \frac{1}{b_1}[-(1-\mathrm{\&rho;})e\left(t\right)+\mathrm{\&epsiv;}\sqrt{1-{\mathrm{\&delta;}}^{\left|e\left(t\right)\right|}}\mathrm{sgne}\left(t\right)-d^{*}(t+1)& 3N/4\&le;\mathrm{mod}(t,N)<\mathrm{N\; or}\mathrm{mod}(t,N)=0\\ +z\left(t\right)-z(t-t^{\&prime;})+e(t-t^{\&prime;}+1)]& \end{array}\right.---\left(11\right)\end{array}$

In an embodiment, the reference signal r (k) =19.8sin (2 pi fkT) of the inverter_s) In volts (V), signal frequency f =50Hz, sampling time T_sIn equation (15) =0.0001s, ρ =0.3, =0.01, = 0.5. Set the disturbance signal as

$w\left(k\right)=0.1[\mathrm{rand}\left(1\right)-0.5]+\underset{i=1}{\overset{3}{\mathrm{\&Sigma;}}}{c}_i\sin (2\mathrm{\&pi;}50*(2i-1)\frac{k{T}_s}{N}),c_i=1/(2i-1)---\left(31\right)$

The antecedent is a random disturbance signal, and the consequent is used for simulating an odd harmonic disturbance signal of the inverter. In this case, the upper and lower bounds of d (t +1) are equal in value and opposite in sign. Therefore, can be taken d^*(t +1) =0, error e (t) will converge to radius Δ_SSEIn the neighborhood of the origin of (c).

In practice, the upper and lower bounds of d (t +1) will be closer to zero and the error will converge to a smaller neighborhood of the origin.

And (3) simulating the system parameters and the control parameters, checking 1/4 the implementation result of the periodic repetitive control strategy on the inverter system, and comparing the implementation result with the implementation results of feedback control and full-period repetitive control:

1) the error signal under the action of the feedback controller is shown in fig. 9. It can be seen that although the error signal converges, the error signal has a large amplitude and exhibits a distinct periodic characteristic, since the periodic harmonic disturbance signal of the control is not eliminated, it is known that the feedback controller does not have the periodic error suppression capability.

2) The error signal under the action of the full-period repetitive controller is shown in fig. 10, and the control signal is shown in fig. 11. Estimates of the controller parameters ρ =0.3, =0.01, =0.5 and Δ (Δ =0.05 after removing the odd harmonic disturbance) can give a threshold value characterizing the convergence performance of the system: monotonous decreasing region delta_MDR=0.7756V, absolute attraction layer Δ_AAL=0.2797V, steady state error band Δ_SSE= 0.2994V. The boundary values in FIG. 10 are monotonically decreasing regions (Δ) from large to small_MDR) Steady state error band (delta)_SSE) And absolute attraction layer (. DELTA.)_AAL). As shown in fig. 10, the error convergence is faster in the case of disturbance by using the full-period repetitive control, the effect of the error signal is the same as that by using the feedback control in the first full period, and the error signal quickly converges into the characterization system convergence boundary after 0.02 second.

3) The error signal under the action of the controller repeated at 1/4 cycles is shown in fig. 12, and the control signal is shown in fig. 13. Reference deviceEstimates of p =0.3, =0.01, =0.5 and Δ for the cycle-repeated controller parameters (Δ =0.05 after removing odd harmonic disturbances) can give a threshold value characterizing the convergence performance of the system: monotonous decreasing region delta_MDR=0.7756V, absolute attraction layer Δ_AAL=0.2797V, steady state error band Δ_SSE= 0.2994V. The boundary values in FIG. 12 are monotonically decreasing regions (Δ) from large to small_MDR) Steady state error band (delta)_SSE) And absolute attraction layer (. DELTA.)_AAL). As shown in fig. 12, the error converges more rapidly under the influence of disturbance with 1/4 cycle repeat control than with full cycle repeat control, the control effect is the same in the first quarter cycle error and with feedback control, and the error signal converges rapidly within the characterized system convergence bounds after 0.005 seconds. The odd harmonic disturbing signals are eliminated, leaving only small error fluctuations caused by the random disturbing signals. Therefore, the 1/4 periodic repetitive controller can effectively and quickly eliminate periodic harmonic disturbance signals and greatly reduce the occupied memory space.

Claims (3)

1. An 1/4 cycle repetitive controller based on attraction law, which is suitable for an inverter, and is characterized in that:

(1) given a reference signal r (t), the reference signal has 1/4 periodic symmetry

$\begin{array}{cc}p1.& r\left(t\right)=\left\{\begin{array}{cc}\&PlusMinus;r(t-N/4)+r_0\left(t\right),& 0<\mathrm{mod}(t,N)\&le;N/4\\ \&PlusMinus;r(t-N/4)+r_0\left(t\right),& N/4<\mathrm{mod}(t,N)\&le;N/2\\ \&PlusMinus;r(t-N/4)+r_0\left(t\right),& N/2<\mathrm{mod}(t,N)\&le;3N/4\\ \&PlusMinus;r(t-N/4)+r_0\left(t\right),& 3N/4<\mathrm{mod}(t,N)<Nor\mathrm{mod}(t,N)=0\end{array}\right.\end{array}---\left(1\right)$

And t is more than or equal to N/4;

$\begin{array}{cc}p2.& r\left(t\right)=\left\{\begin{array}{cc}\&PlusMinus;r(t-t^{\&prime;})+r_0\left(t\right),& 0<\mathrm{mod}(t,N)\&le;N/4\\ \&PlusMinus;r(t-t^{\&prime;})+r_0\left(t\right),& N/4<\mathrm{mod}(t,N)\&le;N/2\\ \&PlusMinus;r(t-t^{\&prime;})+r_0\left(t\right),& N/2<\mathrm{mod}(t,N)\&le;3N/4\\ \&PlusMinus;r(t-t^{\&prime;})+r_0\left(t\right),& 3N/4<\mathrm{mod}(t,N)<Nor\mathrm{mod}(t,N)=0\end{array}\right.\end{array}---\left(2\right)$

and t > N/4; wherein,

$t^{\&prime;}=\left\{\begin{array}{cc}2(t\mathrm{mod}N/4)& t\mathrm{mod}N/4\&NotEqual;0\\ N/2& t\mathrm{mod}N/4=0\end{array}\right.$

n is a reference signal period; r is₀(t) is the initial value of the signal, i.e. the value of the first 1/4 cycles of the periodic signal, during the interval t>The value of N/4 is constantly zero; here, the specific position in a cycle at the current moment needs to be judged according to specific conditions, and then whether "+" -or "-" is taken according to 1/4 cycle symmetry characteristics is judged;

the reference signal of the inverter is a sine signal, and when t is more than or equal to N/4, the 1/4 periodic symmetry characteristic is

$r\left(t\right)=\left\{\begin{array}{cc}-r(t-N/4)+r_0\left(t\right),& 0<\mathrm{mod}(t,N)\&le;N/4\\ r(t-N/4)+r_0\left(t\right),& N/4<\mathrm{mod}(t,N)\&le;N/2\\ -r(t-N/4)+r_0\left(t\right),& N/2<\mathrm{mod}(t,N)\&le;3N/4\\ r(t-N/4)+r_0\left(t\right),& 3N/4<\mathrm{mod}(t,N)<Nor\mathrm{mod}(t,N)=0\end{array}\right.---\left(3\right)$

(2) Law of structural continuous suction

$\stackrel{\&CenterDot;}{e}\left(t\right)=-\mathrm{\&rho;}e\left(t\right)-\mathrm{\&epsiv;}\sqrt{1-{\mathrm{\&delta;}}^{\left|e\left(t\right)\right|}}\mathrm{sgn}e\left(t\right)---\left(4\right)$

Where e (t) ═ r (t) -y (t) denotes a tracking error, ρ >0, >0,0< < 1; equation (4) is the finite time law of attraction, with an arrival time of

$t_1<\frac{ln(1-\sqrt{1-{\mathrm{\&delta;}}^{e\left(0\right)}})-ln(1+\sqrt{1-{\mathrm{\&delta;}}^{e\left(0\right)}})}{\mathrm{\&epsiv;}ln\mathrm{\&delta;}}---\left(5\right)$

The discretization form of the attraction law (4) is

$e(t+1)=(1-\mathrm{\&rho;})e\left(t\right)-\mathrm{\&epsiv;}\sqrt{1-{\mathrm{\&delta;}}^{\left|e\left(t\right)\right|}}\mathrm{sgn}e\left(t\right)---\left(6\right)$

Wherein 0< ρ <1, >0,0< < 1;

(3) adding interference suppression term and establishing discrete attraction law

$e(t+1)=(1-\mathrm{\&rho;})e\left(t\right)-\mathrm{\&epsiv;}\sqrt{1-{\mathrm{\&delta;}}^{\left|e\left(t\right)\right|}}\mathrm{sgn}e\left(t\right)+d^{*}(t+1)-d(t+1)---\left(7\right)$

Wherein the equivalent disturbancew (t +1) is a system disturbance term, d^*(t +1) is compensation of the equivalent disturbance, and is used for inhibiting the influence of the equivalent disturbance d (t + 1);

(4) 1/4 periodic repetitive controller based on reference signal periodic symmetry

(a)1/4 periodic feedback loop of the periodic repetitive controller: for case p1

$u\left(t\right)=\left\{\begin{array}{cc}\&PlusMinus;u(t-N/4)+v\left(t\right),& 0<\mathrm{mod}(t,N)\&le;N/4\\ \&PlusMinus;u(t-N/4)+v\left(t\right),& N/4<\mathrm{mod}(t,N)\&le;N/2\\ \&PlusMinus;u(t-N/4)+v\left(t\right),& N/2<\mathrm{mod}(t,N)\&le;3N/4\\ \&PlusMinus;u(t-N/4)+v\left(t\right),& 3N/4<\mathrm{mod}(t,N)<Nor\mathrm{mod}(t,N)=0\end{array}\right.---\left(8\right)$

And t is more than or equal to N/4; for case p2

$u\left(t\right)=\left\{\begin{array}{cc}\&PlusMinus;u(t-t^{\&prime;})+v\left(t\right),& 0<\mathrm{mod}(t,N)\&le;N/4\\ \&PlusMinus;u(t-t^{\&prime;})+v\left(t\right),& N/4<\mathrm{mod}(t,N)\&le;N/2\\ \&PlusMinus;u(t-t^{\&prime;})+v\left(t\right),& N/2<\mathrm{mod}(t,N)\&le;3N/4\\ \&PlusMinus;u(t-t^{\&prime;})+v\left(t\right),& 3N/4<\mathrm{mod}(t,N)<Nor\mathrm{mod}(t,N)=0\end{array}\right.---\left(9\right)$

And t > N/4; wherein u (t) is an output signal of the repetitive controller, v (t) is an input signal of the periodic feedback link, and the input signal is obtained through an e/v signal conversion link; aiming at a system concrete model, a conversion mode of an e/v signal can be obtained according to an attraction law method;

(b) the periodic feedback link of the repetitive controller for the inverter is

$u\left(t\right)=\left\{\begin{array}{cc}-u(t-N/4)+v\left(t\right),& 0<\mathrm{mod}(t,N)\&le;N/4\\ u(t-N/4)+v\left(t\right),& N/4<\mathrm{mod}(t,N)\&le;N/2\\ -u(t-N/4)+v\left(t\right),& N/2<\mathrm{mod}(t,N)\&le;3N/4\\ u(t-N/4)+v\left(t\right),& 3N/4<\mathrm{mod}(t,N)<Nor\mathrm{mod}(t,N)=0\end{array}\right.---\left(10\right)$

And t is more than or equal to N/4

(5) The specific v (t) expression of the e/v transformation link is given as follows: for case p1

$v\left(t\right)=\left\{\begin{array}{cc}\begin{array}{c}\frac{1}{b_1}\&lsqb;-(1-\mathrm{\&rho;})e\left(t\right)+\mathrm{\&epsiv;}\sqrt{1-{\mathrm{\&delta;}}^{\left|e\left(t\right)\right|}}\mathrm{sgn}e\left(t\right)-d^{*}(t+1)\\ +z\left(t\right)+z(t-\frac{N}{4})-e(t-\frac{N}{4}+1)\&rsqb;\end{array}& 0\&le;\mathrm{mod}(t,N)<N/4\\ \begin{array}{c}\frac{1}{b_1}\&lsqb;-(1-\mathrm{\&rho;})e\left(t\right)+\mathrm{\&epsiv;}\sqrt{1-{\mathrm{\&delta;}}^{\left|e\left(t\right)\right|}}\mathrm{sgn}e\left(t\right)-d^{*}(t+1)\\ +z\left(t\right)-z(t-\frac{N}{4})+e(t-\frac{N}{4}+1)\&rsqb;,\end{array}& N/4\&le;\mathrm{mod}(t,N)<N/2\\ \begin{array}{c}\frac{1}{b_1}\&lsqb;-(1-\mathrm{\&rho;})e\left(t\right)+\mathrm{\&epsiv;}\sqrt{1-{\mathrm{\&delta;}}^{\left|e\left(t\right)\right|}}\mathrm{sgn}e\left(t\right)-d^{*}(t+1)\\ +z\left(t\right)+z(t-\frac{N}{4})-e(t-\frac{N}{4}+1)\&rsqb;\end{array}& N/2\&le;\mathrm{mod}(t,N)<3N/4\\ \begin{array}{c}\frac{1}{b_1}\&lsqb;-(1-\mathrm{\&rho;})e\left(t\right)+\mathrm{\&epsiv;}\sqrt{1-{\mathrm{\&delta;}}^{\left|e\left(t\right)\right|}}\mathrm{sgn}e\left(t\right)-d^{*}(t+1)\\ +z\left(t\right)-z(t-\frac{N}{4})+e(t-\frac{N}{4}+1)\&rsqb;\end{array}& 3N/4\&le;\mathrm{mod}(t,N)<Nor\mathrm{mod}(t,N)=0\end{array}\right.---\left(11\right)$

And t is more than or equal to N/4

For case p2

$v\left(t\right)=\left\{\begin{array}{cc}\begin{array}{c}\frac{1}{b_1}\&lsqb;-(1-\mathrm{\&rho;})e\left(t\right)+\mathrm{\&epsiv;}\sqrt{1-{\mathrm{\&delta;}}^{\left|e\left(t\right)\right|}}\mathrm{sgn}e\left(t\right)-d^{*}(t+1)\\ +z\left(t\right)+z(t-t^{\&prime;})-e(t-t^{\&prime;}+1)\&rsqb;\end{array}& 0\&le;\mathrm{mod}(t,N)<N/4\\ \begin{array}{c}\frac{1}{b_1}\&lsqb;-(1-\mathrm{\&rho;})e\left(t\right)+\mathrm{\&epsiv;}\sqrt{1-{\mathrm{\&delta;}}^{\left|e\left(t\right)\right|}}\mathrm{sgn}e\left(t\right)-d^{*}(t+1)\\ +z\left(t\right)-z(t-t^{\&prime;})+e(t-t^{\&prime;}+1)\&rsqb;,\end{array}& N/4\&le;\mathrm{mod}(t,N)<N/2\\ \begin{array}{c}\frac{1}{b_1}\&lsqb;-(1-\mathrm{\&rho;})e\left(t\right)+\mathrm{\&epsiv;}\sqrt{1-{\mathrm{\&delta;}}^{\left|e\left(t\right)\right|}}\mathrm{sgn}e\left(t\right)-d^{*}(t+1)\\ +z\left(t\right)+z(t-t^{\&prime;})-e(t-t^{\&prime;}+1)\&rsqb;\end{array}& N/2\&le;\mathrm{mod}(t,N)<3N/4\\ \begin{array}{c}\frac{1}{b_1}\&lsqb;-(1-\mathrm{\&rho;})e\left(t\right)+\mathrm{\&epsiv;}\sqrt{1-{\mathrm{\&delta;}}^{\left|e\left(t\right)\right|}}\mathrm{sgn}e\left(t\right)-d^{*}(t+1)\\ +z\left(t\right)-z(t-t^{\&prime;})+e(t-t^{\&prime;}+1)\&rsqb;\end{array}& 3N/4\&le;\mathrm{mod}(t,N)<Nor\mathrm{mod}(t,N)=0\end{array}\right.---\left(12\right)$

And t > N/4

The reference signal for the inverter is a sinusoidal signal

$v\left(t\right)=\left\{\begin{array}{cc}\begin{array}{c}\frac{1}{b_1}\&lsqb;-(1-\mathrm{\&rho;})e\left(t\right)+\mathrm{\&epsiv;}\sqrt{1-{\mathrm{\&delta;}}^{\left|e\left(t\right)\right|}}\mathrm{sgn}e\left(t\right)-d^{*}(t+1)\\ +z\left(t\right)+z(t-\frac{N}{4})-e(t-\frac{N}{4}+1)\&rsqb;\end{array}& 0\&le;\mathrm{mod}(t,N)<N/4\\ \begin{array}{c}\frac{1}{b_1}\&lsqb;-(1-\mathrm{\&rho;})e\left(t\right)+\mathrm{\&epsiv;}\sqrt{1-{\mathrm{\&delta;}}^{\left|e\left(t\right)\right|}}\mathrm{sgn}e\left(t\right)-d^{*}(t+1)\\ +z\left(t\right)-z(t-\frac{N}{4})+e(t-\frac{N}{4}+1)\&rsqb;,\end{array}& N/4\&le;\mathrm{mod}(t,N)<N/2\\ \begin{array}{c}\frac{1}{b_1}\&lsqb;-(1-\mathrm{\&rho;})e\left(t\right)+\mathrm{\&epsiv;}\sqrt{1-{\mathrm{\&delta;}}^{\left|e\left(t\right)\right|}}\mathrm{sgn}e\left(t\right)-d^{*}(t+1)\\ +z\left(t\right)+z(t-\frac{N}{4})-e(t-\frac{N}{4}+1)\&rsqb;\end{array}& N/2\&le;\mathrm{mod}(t,N)<3N/4\\ \begin{array}{c}\frac{1}{b_1}\&lsqb;-(1-\mathrm{\&rho;})e\left(t\right)+\mathrm{\&epsiv;}\sqrt{1-{\mathrm{\&delta;}}^{\left|e\left(t\right)\right|}}\mathrm{sgn}e\left(t\right)-d^{*}(t+1)\\ +z\left(t\right)-z(t-\frac{N}{4})+e(t-\frac{N}{4}+1)\&rsqb;\end{array}& 3N/4\&le;\mathrm{mod}(t,N)<Nor\mathrm{mod}(t,N)=0\end{array}\right.---\left(13\right)$

And t is more than or equal to N/4; wherein z (t) ═ r (t +1) + a₁y(t)+a₂y(t-1)-b₂u (t-1), where z (t) is given for the inverter dynamics model:

y(t+1)+a₁y(t)+a₂y(t-1)＝b₁u(t)+b₂u(t-1)+w(t+1) (14)

y (t) represents the output at time t, and u (t) represents the controlled variable at time t; a is₁,a₂,b₁,b₂Is a system parameter; w (t +1) represents a system disturbance term.

2. The controller of claim 1, wherein: the performance of the controller is measured by the following indicators characterizing the convergence process, including the monotone decreasing area, the absolute attraction layer and the steady state error band, which are specifically defined as follows:

monotonous decreasing region delta_MDR

$\left\{\begin{array}{cc}0<e(t+1)<e\left(t\right),& e\left(t\right)>{\mathrm{\&Delta;}}_{MDR}\\ e\left(t\right)<e(t+1)<0,& e\left(t\right)<-{\mathrm{\&Delta;}}_{MDR}\end{array}\right.---\left(15\right)$

Absolute attraction layer Δ_AAL

$\left|e\left(t\right)\right|>{\mathrm{\&Delta;}}_{AAL}\&DoubleRightArrow;\left|e(t+1)\right|<\left|e\left(t\right)\right|---\left(16\right)$

Steady state error band Δ_SSE

$\left|e\left(t\right)\right|\&le;{\mathrm{\&Delta;}}_{SSE}\&DoubleRightArrow;\left|e(t+1)\right|\&le;{\mathrm{\&Delta;}}_{SSE}---\left(17\right)$

Determining 1/4 cycle weights from the discrete attraction lawMonotonous decreasing area delta of complex controller_MDRAbsolute attraction layer Δ_AALSteady state error band Δ_SSEThe specific expression is as follows:

(1) monotone decreasing area

Δ_MDR＝max{Δ_MDR1,Δ_MDR2} (18)

In the formula,. DELTA._MDR1，Δ_MDR2Is positive and real, and satisfies

$\left\{\begin{array}{c}{\mathrm{\&rho;\&Delta;}}_{MDR1}+\mathrm{\&epsiv;}\sqrt{1-{\mathrm{\&delta;}}^{{\mathrm{\&Delta;}}_{MDR1}}}-\mathrm{\&Delta;}=0\\ (1-\mathrm{\&rho;}){\mathrm{\&Delta;}}_{MDR2}-\mathrm{\&epsiv;}\sqrt{1-{\mathrm{\&delta;}}^{{\mathrm{\&Delta;}}_{MDR2}}}-\mathrm{\&Delta;}=0\end{array}\right.---\left(19\right)$

(2) Absolute attraction layer

Δ_AAL＝max{Δ_AAL1,Δ_AAL2} (20)

In the formula,. DELTA._AAL1，Δ_AAL2Is positive and real, and can be determined by

$\left\{\begin{array}{c}{\mathrm{\&rho;\&Delta;}}_{AAL1}+\mathrm{\&epsiv;}\sqrt{1-{\mathrm{\&delta;}}^{{\mathrm{\&Delta;}}_{AAL1}}}-\mathrm{\&Delta;}=0\\ (2-\mathrm{\&rho;}){\mathrm{\&Delta;}}_{AAL2}-\mathrm{\&epsiv;}\sqrt{1-{\mathrm{\&delta;}}^{{\mathrm{\&Delta;}}_{AAL2}}}-\mathrm{\&Delta;}=0\end{array}\right.---\left(21\right)$

(3) Steady state error band

Δ_SSECan be based on_AALTo determine

a. When 0 is present<Δ_AAL<x₁Time of flight

Δ_SSE＝Δ_AAL(22)

b. When x is₁≤Δ_AAL<x₂Time of flight

${\mathrm{\&Delta;}}_{SSE}=-(1-\mathrm{\&rho;})x_1+\mathrm{\&epsiv;}\sqrt{1-{\mathrm{\&delta;}}^{x_1}}+\mathrm{\&Delta;}---\left(23\right)$

c. When delta_AAL≥x₂Time of flight

Δ_SSE＝Δ_AAL(24)

Wherein，x₁Is an equation

$(1-\mathrm{\&rho;})+\frac{{\mathrm{\&epsiv;\&delta;}}^{x}ln\mathrm{\&delta;}}{2\sqrt{1-{\mathrm{\&delta;}}^x}}=0---\left(25\right)$

Root of Zhengguo, x₂Is an equation

$(1-\mathrm{\&rho;})x-\mathrm{\&epsiv;}\sqrt{1-{\mathrm{\&delta;}}^x}+(1-\mathrm{\&rho;})x_1-\mathrm{\&delta;}\sqrt{1-{\mathrm{\&delta;}}^{x_1}}=0---\left(26\right)$

Root of Zhengguo;

whereind_lLower bound value of equivalent disturbance d (t) at time t, d_uIs the upper bound of the equivalent disturbance d (t) at the time t.

3. The controller of claim 2, wherein: the adjustable parameters of the controller include ρ,; parameter setting can be performed according to indexes representing a convergence process.