CN106094525A - A kind of TSM control device based on fractional calculus and control method - Google Patents

Abstract

The invention discloses a kind of TSM control device based on fractional calculus, its functional form is；Wherein μ ∈ (0,1], α is the normal number more than 0, β=q/p, wherein p and q all be meet 0 ＜ q/p ＜ 1 positive odd number.The control method using this TSM control device is: for the controlled device of fractional order TSM control, obtain its state equation；Then the state equation of controlled device is brought into described fractional order TSM control function, obtains the TSM control function of this controlled device, utilize this function that system is controlled.Carrying out system control by this controller, its response time is short, strong interference immunity, and has less steady-state error.

Description

A kind of TSM control device based on fractional calculus and control method

Technical field

The invention belongs to automatic control technology field, be specifically related to a kind of TSM control based on fractional calculus Device, has further related to use the control method of this controller.

Background technology

The order of fractional calculus (FOC) can be arbitrary, and it has expanded the descriptive power of integer rank calculus.Pass The integer rank calculus of system is fractional calculus special case under specific circumstances, and the exploitation for fractional calculus is at present The focus of research.

Sliding formwork controls (sliding mode control, SMC) and is also Sliding mode variable structure control, is to carry the 1950's A kind of effective robust stabili out, is used in nonlinear Control aspect more, and its basic thought is according to system institute's phase The dynamic characteristic hoped carrys out the sliding-mode surface of design system, has switching control rule functional, makes the state trajectory of system from initial shape State arrives the predefined sliding-mode surface with desired dynamic characteristic.Make system mode from sliding-mode surface by sliding mode control Extroversion sliding-mode surface motion.System once arrives sliding-mode surface, and guarantee system is arrived system balancing point along sliding-mode surface by control action.

Compared with controlling with sliding formwork, TSM control overcome its closed loop response for the uncertainty of inner parameter and The tender subject of external disturbance, it has the convergence property in system finite time, have at steady state higher accurately Degree, close to advantages such as equilibrium point everywhere convergent speed substantially quickenings.But it is poor for the robustness of disturbance, it is impossible to meet part Very high degree of precision demand for control.

Summary of the invention

It is an object of the present invention to provide a kind of TSM control device based on fractional calculus, solve tradition Sliding formwork controls and TSM control is for the insensitive problem of disturbance.

It is a further object to provide the method using controller noted above to carry out system control.

The technical solution adopted in the present invention is, a kind of TSM control device based on fractional calculus, its function Form is；

$s={{\mathrm{\αx}}_1}^{\mathrm{\β}}+t_0{D}_t^{\mathrm{\μ}-1}{x}_2---\left(4\right)$

Wherein μ ∈ (0,1], α is the normal number more than 0, β=q/p, wherein p and q all be meet 0 ＜ q/p ＜ 1 the strangest Number.

The feature of controller noted above also resides in:

In above-mentioned function, α=100, β=0.6,0.7≤μ≤0.9.

Above-mentioned formula (4) is obtained by following methods:

Step 1: selecting R-L to define fractional calculus, the fractional order differential operator definitions on its α rank is:

$D_t^{-\mathrm{\α}}{}_{0}f\left(t\right)=\frac{1}{\mathrm{\Γ}\left(\mathrm{\α}\right)}{\∫}_0^t{(t-\mathrm{\τ})}^{\mathrm{\α}-1}f\left(\mathrm{\τ}\right)d\mathrm{\τ}---\left(1\right)$

Wherein, the gamma function that Euler proposes is defined as:

$\mathrm{\Γ}\left(z\right)={\∫}_0^{\mathrm{\∞}}{e}^{-t}{t}^{z-1}dt---\left(2\right)$

Step 2: fractional calculus shape is such asAnd function can amass, and obtains

$D_t^{-\mathrm{\α}}{}_{\mathrm{\α}}\left(D_t^{\mathrm{\α}}{}_{\mathrm{\α}}f\left(t\right)\right)=f\left(t\right)-\underset{j=1}{\overset{k}{\Σ}}{\[D_t^{\mathrm{\α}-j}{}_{\mathrm{\α}}f\left(t\right)\]}_{t=a}\frac{{(t-a)}^{\mathrm{\α}-j}}{\mathrm{\Γ}(\mathrm{\μ}-j+1)}---\left(3\right)$

Step 3: the fractional calculus form of step 2 is combined with integer rank TSM control, obtains fractional order TSM control function:

$s={{\mathrm{\αx}}_1}^{\mathrm{\β}}+t_0{D}_t^{\mathrm{\μ}-1}{x}_2---\left(4\right)$

Wherein μ ∈ (0,1], α is the normal number more than 0, β=q/p, wherein p and q all be meet 0 ＜ q/p ＜ 1 the strangest Number.

This controller is all suitable in power system, mechanical system and chaos system, such as buck circuit, Boost circuit, buck-boost circuit, inverter circuit and rectification circuit.

Another technical scheme of the present invention is, TSM control method based on fractional calculus, uses above-mentioned TSM control device, comprises the following steps:

Step 1, for the controlled device of fractional order TSM control, obtains its state equation；

Step 2, brings the state equation of controlled device described fractional order TSM control function into, obtains this controlled right The TSM control function of elephant, utilizes this function to be controlled system.

The invention has the beneficial effects as follows, TSM control device based on fractional calculus proposed by the invention and biography System synovial membrane controls to compare with TSM control, and the method being controlled by this controller is simple, and response time is short, robustness By force, and there is less steady-state error.

Accompanying drawing explanation

Fig. 1 is fractional order TSM control structure chart of the present invention；

Fig. 2 is embodiment of the present invention BUCK circuit diagram；

Fig. 3 is the embodiment of the present invention output voltage response when changing μ；

Fig. 4 is that embodiment of the present invention R is reduced to output voltage response during 2 Ω by 8 Ω；

Output voltage response when being increased to 8 Ω by 2 Ω that Fig. 5 is embodiment of the present invention R；

Fig. 6 is the embodiment of the present invention output voltage response when input voltage change Spline smoothing.

Detailed description of the invention

The present invention is described in further detail with detailed description of the invention below in conjunction with the accompanying drawings, but the present invention is not limited to These embodiments.

The thought of the TSM control device based on fractional calculus of the present invention is: introduced by fractional calculus whole Number rank TSM control.The method has fast convergence rate, degree of accuracy high.

This controller is as it is shown in figure 1, its method for designing is:

Step 1: selecting Riemman-Liouville (R-L) definition fractional calculus, the fractional order differential on its α rank is calculated Sub-definite is:

$D_t^{-\mathrm{\α}}{}_{0}f\left(t\right)=\frac{1}{\mathrm{\Γ}\left(\mathrm{\α}\right)}{\∫}_0^t{(t-\mathrm{\τ})}^{\mathrm{\α}-1}f\left(\mathrm{\τ}\right)d\mathrm{\τ}---\left(1\right)$

Wherein, the gamma function that Euler proposes is defined as:

$\mathrm{\Γ}\left(z\right)={\∫}_0^{\mathrm{\∞}}{e}^{-t}{t}^{z-1}dt---\left(2\right)$

Step 2: when fractional calculus shape such asAnd function can amass, can there is following shape Formula:

$D_t^{-\mathrm{\α}}{}_{\mathrm{\α}}\left(D_t^{\mathrm{\α}}{}_{\mathrm{\α}}f\left(t\right)\right)=f\left(t\right)-\underset{j=1}{\overset{k}{\Σ}}{\[D_t^{\mathrm{\α}-j}{}_{\mathrm{\α}}f\left(t\right)\]}_{t=a}\frac{{(t-a)}^{\mathrm{\α}-j}}{\mathrm{\Γ}(\mathrm{\μ}-j+1)}---\left(3\right)$

Step 3: above-mentioned fractional calculus form is combined with integer rank TSM control, obtains fractional order terminal Sliding mode controller function shape is:

$s={{\mathrm{\αx}}_1}^{\mathrm{\β}}+t_0{D}_t^{\mathrm{\μ}-1}{x}_2---\left(4\right)$

Wherein μ ∈ (0,1], α is the normal number more than 0, β=q/p, wherein p and q all be meet 0 ＜ q/p ＜ 1 the strangest Number.

In the case, actual contrast finds preferred version, and its parameter is α=100, β=0.6,0.7≤μ≤0.9.

The method using above-mentioned TSM control device based on fractional calculus to carry out Circuits System control, specifically presses Implement according to following steps:

Step 1, for the controlled device of fractional order TSM control, draws the pass of its output error and error change System, and then draw state equation.

Step 2: being brought into by controlled device state equation in above-mentioned fractional order TSM control device function, it is controlled right to obtain The concrete control function of elephant, utilizes this control function to be controlled system.

Below as a example by BUCK circuit as shown in Figure 1, the method for the present invention is described in detail further.

Obtain the state equation of BUCK circuit according to following methods, specifically comprise the following steps that

(1), have when the switch s is closed:

$\left[\begin{array}{c}\frac{{\mathrm{di}}_L\left(t\right)}{dt}\\ \frac{{\mathrm{dv}}_0\left(t\right)}{dt}\end{array}\right]=\left[\begin{array}{cc}0& -\frac{1}{L}\\ \frac{1}{C}& -\frac{1}{CR}\end{array}\right]\left[\begin{array}{c}{i}_L\left(t\right)\\ v_0\left(t\right)\end{array}\right]+\left[\begin{array}{c}\frac{1}{L}\\ 0\end{array}\right]v_{in}\left(t\right)---\left(5\right)$

Have when switching S and disconnecting:

$\left[\begin{array}{c}\frac{{\mathrm{di}}_L\left(t\right)}{dt}\\ \frac{{\mathrm{dv}}_0\left(t\right)}{dt}\end{array}\right]=\left[\begin{array}{cc}0& -\frac{1}{L}\\ \frac{1}{C}& -\frac{1}{CR}\end{array}\right]\left[\begin{array}{c}{i}_L\left(t\right)\\ v_0\left(t\right)\end{array}\right]+\left[\begin{array}{c}0\\ 0\end{array}\right]v_{in}\left(t\right)---\left(6\right)$

(2), (5) are combined with (6):

$\left[\begin{array}{c}\frac{{\mathrm{di}}_L\left(t\right)}{dt}\\ \frac{{\mathrm{dv}}_0\left(t\right)}{dt}\end{array}\right]=\left[\begin{array}{cc}0& -\frac{1}{L}\\ \frac{1}{C}& -\frac{1}{CR}\end{array}\right]\left[\begin{array}{c}{i}_L\left(t\right)\\ v_0\left(t\right)\end{array}\right]+\left[\begin{array}{c}\frac{1}{L}\\ 0\end{array}\right]{\mathrm{uv}}_{in}\left(t\right)---\left(7\right)$

Wherein u controls for input, and u is 1 when the switch is closed, and u is 0 when the switches are opened.

(3), definition output voltage error is x₁, it is:

x₁=v₀-V_ref (8)

Wherein V_refIt is the reference value of output voltage, the time is differentiated by formula (8), x can be obtained₂, output voltage is by mistake The rate of change of difference:

$x_2={\stackrel{\·}{x}}_1={\stackrel{\·}{v}}_0---\left(9\right)$

(4), now x₂Dynamic characteristic can be expressed as:

${\stackrel{\·}{x}}_2=\frac{1}{LC}({\mathrm{uV}}_{in}-V_{ref}-x_1)-\frac{x_2}{RC}---\left(10\right)$

(5), finally, the output voltage error of BUCK circuit and the state equation model of rate of change thereof are:

$\left[\begin{array}{c}{\stackrel{\·}{x}}_1\\ {\stackrel{\·}{x}}_2\end{array}\right]=\left[\begin{array}{cc}0& 1\\ -\frac{1}{LC}& -\frac{1}{RC}\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]+\left[\begin{array}{c}0\\ {\mathrm{uV}}_{in}-V_{ref}\end{array}\right]\frac{1}{LC}---\left(11\right)$

Above state equation is brought in fractional order TSM control functional expression (4), obtains its concrete control function:

$u=\[\frac{x_2}{RC}-D^{1-\mathrm{\μ}}\left({\mathrm{\α\βx}}_1^{\mathrm{\β}-1}{x}_2\right)\]\frac{LC}{V_{in}}+\frac{(V_{ref}+x_1)}{V_{in}}-Ksign\left(s\right)-Ks---\left(12\right)$

Wherein 0 ＜ β ＜ 1, k ＞ 0,0 ＜ μ ＜ 1, end-state variable x₁, x₂At Finite-time convergence to zero.

In order to verify the effectiveness of the inventive method, this control method and traditional control method are carried out emulation respectively real Testing, BUCK circuit parameter is as shown in table 1.

Table 1 buck circuit parameter

Describe	Parameter	Value
			Input voltage	V<sub>in</sub>	10V
Electric capacity	C	1000μF
			Inductance	L	1mH
Minimum load impedance	R<sub>min</sub>	2Ω
			Maximum load impedance	R<sub>max</sub>	8Ω
Desired output voltage	V<sub>0</sub>	5V

TSM control for different rank is analyzed, and exponent number is as shown in table 2:

Project	α	β	μ
				Sliding formwork controls	100	-	-
TSM control	100	0.6	-
				Fractional order TSM control 1	100	0.6	0.9
Fractional order TSM control 2	100	0.6	0.8
				Fractional order TSM control 3	100	0.6	0.7

Carry out emulation experiment according to the controller in table 2, its result as it is shown on figure 3, it appeared that when reduce exponent number time defeated Go out response to accelerate rapidly, but along with the reduction of order, it may appear that overshoot.Therefore, the two compromise is processed during actual selection.

As seen from Figure 4, method proposed by the invention, fractional order sliding formwork control response time be considerably shorter than other two Kind, when 0.15 second, load R is reduced (8 Ω are reduced to 2 Ω), now find the output voltage response of method proposed by the invention When load reduces suddenly, the anti-interference for disturbance is also better than other two kinds controls.As seen from Figure 5, by load resistance Increasing to 8 Ω from 2 Ω, the fractional order TSM control that the present invention proposes is still better than other two kinds controls.It will be appreciated from fig. 6 that When input voltage generation Spline smoothing, method proposed by the invention has less steady compared with other two kinds of methods do not improved State error.

Except BUCK circuit, the TSM control device based on fractional calculus of the present invention can be also used for other and is System, such as boost circuit, buck-boost circuit, inverter circuit, rectification circuit etc..Such as, for boost circuit, by its state Equation can be written as:

$\left[\begin{array}{c}\frac{{\mathrm{di}}_d}{dt}\\ \frac{{\mathrm{du}}_0}{dt}\end{array}\right]=\left[\begin{array}{cc}0& -\frac{1-d}{L}\\ \frac{1-d}{C_0}& -\frac{1}{{\mathrm{RC}}_0}\end{array}\right]\left[\begin{array}{c}{i}_d\\ u_0\end{array}\right]+\left[\begin{array}{c}\frac{1}{L}\\ 0\end{array}\right]u_d---\left(13\right)$

Bring formula (4) into, utilize the function obtained that system is controlled；The most such as, for buck-boost circuit, will Its state equation can be written as:

$\frac{d}{dt}\left[\begin{array}{c}i\\ v\end{array}\right]=\left[\begin{array}{cc}0& -\frac{1}{L}\\ \frac{1}{C}& -\frac{1}{RC}\end{array}\right]\left[\begin{array}{c}i\\ v\end{array}\right]+\left[\begin{array}{c}\frac{1}{L}(v+V_s)\\ -\frac{i}{C}\end{array}\right]u---\left(14\right)$

Bring formula (4) into, utilize the function obtained that system is controlled；The method such as inverter circuit, rectification circuit order Consistent with preceding method, here is omitted.

The inventive method has high degree of accuracy and convergence rate, and the disturbance for load has response faster. The method is all suitable in power system, mechanical system and chaos system.

Above description of the present invention is section Example, but the invention is not limited in above-mentioned detailed description of the invention. Above-mentioned detailed description of the invention is schematic, is not restrictive.The material of every employing present invention and method, do not taking off In the case of present inventive concept and scope of the claimed protection, all concrete expansions all belong to protection scope of the present invention it In.

Claims (6)

1. a TSM control device based on fractional calculus, it is characterised in that its functional form is；

$s={\mathrm{\αx}}_1^{\mathrm{\β}}+t_0{D}_t^{\mathrm{\μ}-1}{x}_2---\left(4\right)$

Wherein μ ∈ (0,1], α is the normal number more than 0, β=q/p, wherein p and q all be meet 0 ＜ q/p ＜ 1 positive odd number.

TSM control device based on fractional calculus the most according to claim 1, it is characterised in that described function In, α=100, β=0.6,0.7≤μ≤0.9.

TSM control device based on fractional calculus the most according to claim 1, it is characterised in that described formula (4) obtained by following methods:

Step 1: selecting R-L to define fractional calculus, the fractional order differential operator definitions on its α rank is:

$D_t^{-\mathrm{\α}}{}_{0}f\left(t\right)=\frac{1}{\mathrm{\Γ}\left(\mathrm{\α}\right)}{\∫}_0^t{(t-\mathrm{\τ})}^{\mathrm{\α}-1}f\left(\mathrm{\τ}\right)d\mathrm{\τ}---\left(1\right)$

Wherein, the gamma function that Euler proposes is defined as:

$\mathrm{\Γ}\left(z\right)={\∫}_0^{}{e}^{-t}{t}^{z-1}dt---\left(2\right)$

Step 2: fractional calculus shape is such asAnd function can amass, and obtains

$D_t^{-\mathrm{\α}}{}_a\left(D_t^{\mathrm{\α}}{}_{a}f\right(t\left)\right)=f\left(t\right)-\underset{j=1}{\overset{k}{\Σ}}{\[D_t^{\mathrm{\α}-j}{}_{a}f\left(t\right)\]}_{t=a}\frac{{(t-a)}^{\mathrm{\α}-j}}{\mathrm{\Γ}(\mathrm{\μ}-j+1)}---\left(3\right)$

Step 3: the fractional calculus form of step 2 is combined with integer rank TSM control, obtains fractional order terminal Sliding formwork control function:

$s={{\mathrm{\αx}}_1}^{\mathrm{\β}}+t_0{D}_t^{\mathrm{\μ}-1}{x}_2---\left(4\right)$

Wherein μ ∈ (0,1], α is the normal number more than 0, β=q/p, wherein p and q all be meet 0 ＜ q/p ＜ 1 positive odd number.

TSM control device based on fractional calculus the most according to claim 1, it is characterised in that described control Device is for buck circuit, boost circuit, buck-boost circuit, inverter circuit and rectification circuit.

5. a TSM control method based on fractional calculus, it is characterised in that use as claimed in claim 1 TSM control device based on fractional calculus, comprises the following steps:

Step 1, for the controlled device of fractional order TSM control, obtains its state equation；

Step 2, brings the state equation of controlled device described fractional order TSM control function into, obtains this controlled device TSM control function, utilizes this function to be controlled system.

TSM control method based on fractional calculus the most according to claim 5, it is characterised in that step 2 Described controlled device is buck circuit, boost circuit, buck-boost circuit, inverter circuit and rectification circuit.