CN109764539A - Dynamic system and control model for load group of electric water heater - Google Patents
Dynamic system and control model for load group of electric water heater Download PDFInfo
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Abstract
The invention belongs to the field of power distribution network control of an electric power system, and particularly relates to a dynamic system and a control model of a load group of an electric water heater. The linear state equation constructed in the invention is suitable for modeling of various electric water heaters, the sliding mode control strategy is used for controlling the load group of the electric water heaters in real time, the design of the control rate is related to the number of the electric water heater groups, the use condition of hot water and the like, the robustness for wind power consumption is stronger, a corresponding mathematical model is established according to the physical characteristics of the electric water heater groups by using a linear differential equation, a discrete sliding mode is adopted to set a temperature value, and the load of the electric water heater groups is controlled to achieve the aim of wind power consumption.
Description
Technical field
The invention belongs to system for distribution network of power control fields, more particularly, to a kind of electric heater load group dynamical system
And Controlling model.
Background technique
As environmental pressure incrementally increases, wind power generation construction cost constantly declines, cause wind-powered electricity generation construction and grid-connected upsurge
It is constantly surging, currently, China's wind-powered electricity generation total installation of generating capacity is more than the U.S., leap to the first in the world.However wind power generation output power has
There are natural strong randomness and high uncertainty in traffic, the whole nation abandonment of the first half of the year in 2018 electricity reaches 18,200,000,000 kilowatt hours, mentions
High wind electricity digestion is horizontal, reduces abandonment electricity, be substantially improved Operation of Electric Systems economic well-being of workers and staff and environment emission reduction benefit must
By road.
Hot water demand is the basic living demand of ordinarily resident, and resident produces hot water using electric heater, is had pollution-free
The advantages that discharge, at low cost, high-efficient, using flexible, simple to install, large-scale popularization is obtained.Resident's electric heater is one
The typical hot energy storage load of kind, the hot water that electricity consumption generates have hot storage, the consumption to fluctuation wind-powered electricity generation may be implemented, with
Home intelligent power terminal is popularized, such that family's electric heater dissolves wind-powered electricity generation.
However, existing electric heater temperature control mode lacks optimization, wind power fluctuation cannot be responded, is realized to wind
Effective consumption of electricity, therefore, electric heater optimization operation control strategy require study.
Summary of the invention
The problem to be solved in the present invention is to provide a kind of electric heater load group dynamical system and Controlling models, thus to wind
The renewable energy such as energy are effectively dissolved.
In order to achieve the above object, the technical scheme adopted by the invention is that: a kind of electric heater load group dynamical system,
Including following mathematical model,
For indicate electric water heater heating process and with the Thermokinetic equation formula of air heat exchanging process:
The dynamic equation of electric heater group:
The switch function equation of electric heater:
Temperature equation formula is arranged in environment temperature and water heater:
The general power equation of electric heater group:
Temperature change equation is set, it is assumed that the adjustment of electric heater i temperature is instantaneous to be realized:
Tsp,i=Tsp0,i+ΔTsp(t),|ΔTsp(t)|≤ΔTsp,max (6)
Wherein, i represents electric heater load i；ThIt is water tank water temperature；T is environment temperature；TspIt is setting temperature；Tsp0
It is best setting temperature；T is the time；QhThe heat that water in heating water tank needs；β is unit conversion parameter；AwIt is electric heater
Capacity；αhIt is heat transference efficiency；S is switch function；P is electric heater rated power；R is effective thermal resistance；C is cold water injection
Amount；Δ db is temperature bandwidth, εtIt is time delay；PTCLIt is electric heater group's general power；η is the efficiency of electric heater.
A kind of electric heater load group Controlling model, by the way that the mathematical model in (1)~(6) is quasi- using approximate continuous value
It closes and obtains.
Further, electric heater is included at least from response model, electric heater group Temperature Control Model, electric heater group
Finite dimension state-space model and the control of electric heater group's Discrete-time Sliding Mode.
Further, electric heater from response model include following equation,
Assuming that switching switch is in temperature range [TL,TH] between carry out, if Xon(t, T) and Xoff(t, T) respectively indicates electric hot water
Device group unlatching/closing quantity in time t and temperature T, and Xon(t, T) corresponds to t ∈ [TL,Tmax]；Xoff(t, T) corresponds to t ∈
[Tmin,TH], wherein definition:
Wherein,Represent the change rate of electric heater temperature；
The dynamic equation of electric heater group in (2) is substituted into equation defined in (7), obtains following equation:
Wherein, αonAnd αoffRespectively change rate of the electric heater temperature when opening and closing；
Assuming that environment temperature is constant, and ignores the variation of initial temperature set-point, parameter is indicated with average value, obtains electric heating
Rate of temperature change equation when hydrophone group opens/closes:
If step-length is dT, obtains electric heater group and opens/close amount change equation:
Gained equation in (8) is substituted into gained equation in formula (10), obtains the partial differential equation of the system:
Electric heater group's gross energy is the energy consumption of open state to time integral, and equation is as follows:
Further, partial differential equation described in (11) indicate two first-order linear processes, and equation is as follows:
Further, when temperature is minimum and highest, load changing rate goes to zero, and can get following equation:
αonXon(t, T)=αoffXoff(t,TL)=0
(13)。
Further, electric heater group Temperature Control Model includes following equation,
Electric heater group opens/closes that amount change equation is as follows, is the difference of adjacent time variable quantity:
The functional equation that electric heater group is opened/closed between amount change and set temperature is as follows:
Equation in (12) is converted by equation in (16) are as follows:
Further, the electric heater group finite dimension state-space model includes following equation,
The state space equation of amount change is opened/closed for describing electric heater group:
Wherein, xj(t) the water heater quantity that period j is opened is indicated, Δ T indicates step-length, and subscript M indicates TLBefore switching to ON
One period, N indicate that ON switches to the OFF previous period, and P indicates that ON switches to the OFF latter period, and Q is indicated to THThe period of approach；
The water heater total energy consumption equation of unlatching, obtains multiplied by net power:
Y (t)==Cx (t) (20)
Wherein, x (t)=[x1(t), x2(t) ..., xQ(t)]T,
Y (t)=PT(t), C=[P/ η ..., P/ η | N, 0 ... 0],
A is Q × Q state matrix:
B is the matrix of Q × Q:
Further, it enablesIf F (t, Tsp) it is control variable u (t), state equation turns to mark
Quasi- form:
X (t)=Ax (t)+Bu (t)
Y (t)=Cx (t) (2)
Above-mentioned state equation is write as to the form of transmission function:
Discretization is carried out again, at this point, A and B is the matrix of 2X2.
Further, the electric heater group Discrete-time Sliding Mode Controlling model includes following equation,
Define the switching band of an encirclement diverter surface:
SΔ={ x ∈ Rn|-Δ < s (x)=cx <+Δ }
The reaching condition of continuous system is generalized to discrete system, the condition of reaching is,
[s (k+1)-s (k)] s (k) < 0 (26)
Liapunov function is chosen,
Meet condition,
Δ V (k)=s2(k+1)-s2(k) 0 <, s (k) ≠ 0
(4)
According to Lyapunov theorem of stability, s (k)=0 is the balanced surface of Globally asymptotic, i.e. arbitrary initial position
State can all be intended to diverter surface s (k), take the reaching condition to be,
s2(k+1) < s2(k)
(29)
When the sampling time, T was becoming tight infinitely small, the presence of discrete sliding mode and reaches performance condition and is,
[s (k+1)-s (k)] sgn (s (k)) < 0
[s (k+1)+s (k)] sgn (s (k)) > 0 (30)
In discrete system, exponentially approaching rule is,
Wherein ε > 0, q > 0,1-qT > 0, T is the sampling period；
Exponentially approaching rule meets,
S (k+1)-s (k)=- T ε sgn (s (k)) -- qTs (k)=- qT | s (k) |-T ε | s (k) | < 0 (32)
Meanwhile when sampling time T approach it is infinitely small when, 2-qT be much larger than 0, therefore
S (k+1)+s (k)=[(2-qT) s (k)-T ε sgn (s (k))] sgn (s (k))=(2-qT) | s (k) |-T ε | s (k) |
> 0 (33)
S (k+1)=Cx (k+1)=CAx (k)+CBu (k) is substituted into Reaching Law and obtained by discrete sliding mode face s (k)=Cx (k),
(1-qT) s (k)-T ε sgn (s (k))=CAx (k)+GBu (k) (5)
Assuming that sliding moding structure controlled condition CB ≠ 0 establishment, discrete sliding mode control rate are,
U (k)=- (CB)-1[CAx(k)-(1-qT)s(k)+Tεsgn(s(k))] (35)。
Through the above technical solutions, the beneficial effects of the present invention are:
Electric heater load group dynamical system proposed by the present invention and its Controlling model, using linear differential equation according to electricity
The physical characteristic of water heater group establishes corresponding mathematical model, temperature value is arranged using discrete sliding mode, control electric heater group is negative
Lotus dissolves the target of wind-powered electricity generation to reach.By theory analysis and sample calculation analysis, show electric heater load proposed by the present invention
Group control strategy can effectively dispatch water heater group, and wind power is followed to fluctuate, and achieve the purpose that dissolve wind-powered electricity generation.
Detailed description of the invention
In order to more clearly explain the embodiment of the invention or the technical proposal in the existing technology, to embodiment or will show below
There is attached drawing needed in technical description to be briefly described, it should be apparent that, the accompanying drawings in the following description is only this
The some embodiments recorded in invention, for those of ordinary skill in the art, without creative efforts,
It is also possible to obtain other drawings based on these drawings.
Fig. 1 is 24 hours hot water usage amounts；
Fig. 2 is water heater group handoff procedure；
Fig. 3 be temperature set points it is constant when electric heater group open/close amount change；
Fig. 4 be temperature set points change when electric heater group open/close amount change；
Fig. 5 is the state-space model of finite difference discrete system；
Fig. 6 is 24 hours energy consumptions of fixed temperature setting value lower water-heater；
Fig. 7 is 24 hours wind-powered electricity generation data；
Fig. 8 is wind electricity digestion error tracking frequency figure.
Specific embodiment
Following will be combined with the drawings in the embodiments of the present invention, and technical solution in the embodiment of the present invention carries out clear, complete
Site preparation description, it is clear that described embodiments are only a part of the embodiments of the present invention, instead of all the embodiments.
A kind of electric heater load group dynamical system, using electric heater as research object, it is assumed that water heater water storage is constant, then
Cold water injection rate be equal to hot water usage amount, when electric heater open, water temperature rise, when electric heater close, water temperature drop, including
Following mathematical model,
For indicate electric water heater heating process and with the Thermokinetic equation formula of air heat exchanging process:
The dynamic equation of electric heater group:
The switch function equation of electric heater:
Assuming that Tmax,iAnd Tmin,iWith setting temperature TspTemperature equation formula is arranged in correlation, environment temperature and water heater:
The general power equation of electric heater group:
Temperature change equation is set, it is assumed that the adjustment of electric heater i temperature is instantaneous to be realized:
Tsp,i=Tsp0,i+ΔTsp(t),|ΔTsp(t)|≤ΔTsp,max (6)
Wherein, i represents electric heater load i；ThIt is water tank water temperature；T is environment temperature；TspIt is setting temperature；Tsp0
It is best setting temperature；T is the time；QhThe heat that water in heating water tank needs；β is unit conversion parameter；AwIt is electric heater
Capacity；αhIt is heat transference efficiency；S is switch function；P is electric heater rated power；R is effective thermal resistance；C is cold water injection
Amount, value are as shown in Figure 1；Δ db is temperature bandwidth；εtIt is time delay (simulation step length)；PTCLIt is electric heater group's total work
Rate；η is the efficiency of electric heater.
Formula (1)-formula (6) represents single-input single-output (SISO) dynamical system comprising a large amount of water heaters, Δ Tsp(t) and
PTCLIt (t) is respectively to output and input variable, above system criteria of right and wrong control form cannot be directly used to control.
A kind of electric heater load group Controlling model uses approximate continuous value by the mathematical model in above-mentioned (1)~(6)
Fitting obtains, and establishes the electric heater team control simulation for meeting comfort level, is maintained at electric heater temperature in comfort standard, has
Body, electric heater load group Controlling model includes electric heater from response model, electric heater group Temperature Control Model, electric heating
Hydrophone group's finite dimension state-space model and the control of electric heater group's Discrete-time Sliding Mode.
(1) assume switching switch in temperature range [TL,TH] between carry out, switch switch state, electric hot water as shown in Fig. 2
Device is opened, and water temperature rises, and arrives temperature upper limit, switching；Vice versa, electric heater from response model include following equation
Formula,
If Xon(t, T) and Xoff(t, T) respectively indicates electric heater group unlatching/closing quantity in time t and temperature T,
And Xon(t, T) corresponds to t ∈ [TL,Tmax]；Xoff(t, T) corresponds to t ∈ [Tmin,TH], wherein definition:
Wherein,Represent the change rate of electric heater temperature；
The dynamic equation of electric heater group in (2) is substituted into equation defined in (7), obtains following equation:
Wherein, αonAnd αoffRespectively change rate of the electric heater temperature when opening and closing；
Assuming that environment temperature is constant, and ignores the variation of initial temperature set-point, parameter is indicated with average value, obtains electric heating
Rate of temperature change equation when hydrophone group opens/closes:
If step-length is dT, electric heater group opens/closes the difference that amount change is single step variable quantity, such as the institute of attached drawing 3 and 4
Show, obtain electric heater group and open/close amount change equation:
Gained equation in (8) is substituted into gained equation in formula (10), obtains the partial differential equation of the system:
(11) partial differential equation described in indicate two first-order linear processes, and equation is as follows:
When temperature is minimum and highest, load changing rate goes to zero, and can get following equation:
αonXon(t, T)=αoffXoff(t,TL)=0 (13)
Electric heater group's gross energy is the energy consumption of open state to time integral, and equation is as follows:
When formula (11)-formula (14) research desired temperature is fixed, the dynamic response of water-heater system.(2) electric heater
Group's Temperature Control Model, studies the influence that the temperature set points of variation respond electric heater group's system, including following equation,
Electric heater group opens/closes that amount change equation is as follows, is the difference of adjacent time variable quantity:
The functional equation that electric heater group is opened/closed between amount change and set temperature is as follows:
Equation in (12) is converted by equation in (16) are as follows:
Assuming that electric heater group is within the scope of given temperature, and [TL,TH] sufficiently wide, load changing rate 0 meets formula
(13)。
By equation (16) and (17) and formula (13) and formula (14), continuous passive system is indicated, in this model, control
Input is the change rate of set temperature.
(3) the electric heater group finite dimension state-space model includes following equation, is asked using finite difference method
Actual temperature range is separated into segment by partial differential equation, ifUsing backward-difference method, otherwise, selection
Forward-difference method.When set point temperatures variation is slower, backward-difference method indicates the electric heater of off state, forward-difference method
Indicate the electric heater of open state.
The state space equation of amount change is opened/closed for describing electric heater group:
Wherein, xj (t) indicates the water heater quantity that period j is opened, and Δ T indicates step-length, subscript M, N, P and Q such as attached drawing 5
Shown, open state is divided into N sections, and Q sections of subscript M indicate T in totalLThe ON previous period is switched to, it is previous that N indicates that ON switches to OFF
Period, P indicate that ON switches to the OFF latter period, and Q is indicated to THThe period of approach；
The water heater total energy consumption equation of unlatching, obtains multiplied by net power:
Y (t)=Cx (t) (20)
Its
In, x (t)=[x1(t),x2(t),…,xQ(t)]T, y (t)=PT(t), C=[P/ η ..., P/ η | N, 0 ... 0], A
It is Q × Q state matrix:
B is the matrix of Q × Q:
Formula (20) is off-rating equation, more troublesome when calculating, and is enabledIf F (t,
Tsp) it is control variable u (t), state equation turns to canonical form:
Y (t)=Cx (t) (7)
Above-mentioned state equation is write as to the form of transmission function:
s2X (s)=Ax (s)+Bu (s)
Discretization is carried out again, at this point, A and B is the matrix of 2X2.
(4) electric heater group Discrete-time Sliding Mode Controlling model includes following equation,
Define the switching band of an encirclement diverter surface:
SΔ={ x ∈ Rn|-Δ < s (x)=cx <+Δ }
(25)
The reaching condition of continuous system is generalized to discrete system, the condition of reaching is,
[s (k+1)-s (k)] s (k) < 0 (26)
Liapunov function is chosen,
Meet condition,
Δ V (k)=s2(k+1)-s2(k) 0 <, s (k) ≠ 0
(9)
According to Lyapunov theorem of stability, s (k)=0 is the balanced surface of Globally asymptotic, i.e. arbitrary initial position
State can all be intended to diverter surface s (k), take the reaching condition to be,
s2(k+1) < s2(k)
(29)
When the sampling time, T was becoming tight infinitely small, the presence of discrete sliding mode and reaches performance condition and is,
In discrete system, exponentially approaching rule is,
Wherein ε > 0, q > 0,1-qT > 0, T is the sampling period；
Exponentially approaching rule meets,
S (k+1)-s (k)=- T ε sgn (s (k)) -- qTs (k)=- qT | s (k) |-T ε | s (k) | < 0 (32)
Meanwhile when sampling time T approach it is infinitely small when, 2-qT be much larger than 0, therefore
S (k+1)+s (k)=[(2-qT) s (k)-T ε sgn (s (k))] sgn (s (k))=(2-qT) | s (k) |-T ε | s (k) |
> 0 (33)
S (k+1)=Cx (k+1)=CAx (k)+CBu (k) is substituted into Reaching Law and obtained by discrete sliding mode face s (k)=Cx (k),
(1-qT) s (k)-T ε sgn (s (k))=CAx (k)+GBu (k) (10)
Assuming that sliding moding structure controlled condition CB ≠ 0 establishment, discrete sliding mode control rate are,
U (k)=- (CB)-1[CAx(k)-(1-qT)s(k)+Tεsgn(s(k))] (35)。
In the present invention, the load group formed using the resident's electric heater of certain residential block 1350 is research object, initially
42.8% water heater is opened, remaining closing, and hot water is in comfort standard in all electric heaters, to study set temperature pair
The desired temperature of electric heater is promoted 0.1 DEG C with 0.1 DEG C/h by the influence of single power of electric water heater and general power,
It maintains 4 hours, is then restored to initial set value at a same speed.
(1) without control electric heater group energy consumes when
The temperature set points of electric heater group are constant, and the energy consumption of electric heater group is related to 24 hours amount of hot water, and Fig. 1 is
24 hours amount of hot water of normalized, attached drawing 6 are uncontrolled electric heater group energy consumption curve, and 24:00 reaches at night
Peak value.Fig. 7 provides 24 hours wind power outputs.Comparative drawings figs 6 and attached drawing 7 as it can be seen that if only adjust Generation Side realize power generation and
Balancing the load is more difficult.
(2) electric heater group energy consumes when having control
Reach supply and demand Real-time Balancing by adjusting the desired temperature of electric heater group in real time to respond wind power.From attached
Fig. 7 as it can be seen that wind-powered electricity generation in 20:00~24:00 period, electric heater group wind power output very little instead when energy consumption is high, and in real time
Fluctuation is larger.Herein, it is assumed that Generation Side includes using wind-powered electricity generation as the renewable energy of representative and conventional power generation unit.
Attached drawing 8 is the frequency distribution of output power under sliding formwork control (electric heater load) error, from attached drawing 8 as it can be seen that
The period of more than half is in no error following state.
It should be understood by those skilled in the art that the present invention is not limited to the above embodiments, above-described embodiment and explanation
It is merely illustrated the principles of the invention described in book, without departing from the spirit and scope of the present invention, the present invention also has
Various changes and modifications, these changes and improvements all fall within the protetion scope of the claimed invention.The claimed scope of the invention
It is defined by the appending claims and its equivalent thereof.
Claims (10)
1. a kind of electric heater load group dynamical system, which is characterized in that including following mathematical model,
For indicate electric water heater heating process and with the Thermokinetic equation formula of air heat exchanging process:
The dynamic equation of electric heater group:
The switch function equation of electric heater:
Temperature equation formula is arranged in environment temperature and water heater:
The general power equation of electric heater group:
Temperature change equation is set, it is assumed that the adjustment of electric heater i temperature is instantaneous to be realized:
TSp, i=TSp0, i+ΔTsp(t), | Δ Tsp(t)|≤ΔTSp, max (6)
Wherein, i represents electric heater load i；ThIt is water tank water temperature；T is environment temperature；TspIt is setting temperature；Tsp0It is most
Good setting temperature；T is the time；QhThe heat that water in heating water tank needs；β is unit conversion parameter；AwIt is that electric heater holds
Amount；αhIt is heat transference efficiency；S is switch function；P is electric heater rated power；R is effective thermal resistance；C is cold water injection
Amount；Δ db is temperature bandwidth, εtIt is time delay；PTCLIt is electric heater group's general power；η is the efficiency of electric heater.
2. a kind of electric heater load group Controlling model, which is characterized in that by by the mathematics in (1) in claim 1~(6)
Model is obtained using the fitting of approximate continuous value.
3. electric heater load group Controlling model according to claim 2, which is characterized in that include at least electric heater certainly
Response model, electric heater group Temperature Control Model, electric heater group's finite dimension state-space model and electric heater group are discrete
Time sliding formwork control.
4. electric heater load group Controlling model according to claim 3, which is characterized in that the electric heater responds certainly
Model includes following equation,
Assuming that switching switch is in temperature range [TL,TH] between carry out, if Xon(t, T) and Xoff(t, T) respectively indicates electric heater group
Unlatching/closing quantity in time t and temperature T, and Xon(t, T) corresponds to t ∈ [TL,Tmax]；Xoff(t, T) corresponds to t ∈ [Tmin,
TH], wherein definition:
Wherein,Represent the change rate of electric heater temperature；
The dynamic equation of electric heater group in (2) is substituted into equation defined in (7), obtains following equation:
Wherein, αonAnd αoffRespectively change rate of the electric heater temperature when opening and closing；
Assuming that environment temperature is constant, and ignores the variation of initial temperature set-point, parameter is indicated with average value, obtains electric heater
Rate of temperature change equation when group's unlatching/closing:
If step-length is dT, obtains electric heater group and opens/close amount change equation:
Gained equation in (8) is substituted into gained equation in formula (10), obtains the partial differential equation of the system:
Electric heater group's gross energy is the energy consumption of open state to time integral, and equation is as follows:
5. electric heater load group Controlling model according to claim 4, which is characterized in that (11) partial differential side described in
Journey indicates two first-order linear processes, and equation is as follows:
6. electric heater load group Controlling model according to claim 5, which is characterized in that when temperature is minimum and highest
Load changing rate goes to zero, and can get following equation:
αonXon(t, T)=αoffXoff(t,TL)=0 (13).
7. electric heater load group Controlling model according to claim 5, which is characterized in that the control of electric heater group's temperature
Model includes following equation,
Electric heater group opens/closes that amount change equation is as follows, is the difference of adjacent time variable quantity:
The functional equation that electric heater group is opened/closed between amount change and set temperature is as follows:
Equation in (12) is converted by equation in (16) are as follows:
8. according to the described in any item electric heater load group Controlling models of claim 4~7, which is characterized in that the electric heating
Hydrophone group's finite dimension state-space model includes following equation,
The state space equation of amount change is opened/closed for describing electric heater group:
Wherein, xj(t) the water heater quantity that period j is opened is indicated, Δ T indicates step-length, and subscript M indicates TLSwitch to ON it is previous when
Section, N indicate that ON switches to the OFF previous period, and P indicates that ON switches to the OFF latter period, and Q is indicated to THThe period of approach；
The water heater total energy consumption equation of unlatching, obtains multiplied by net power:
Y (t)=Cx (t)
(20)
Wherein, x (t)=[x1(t),x2(t),…,xQ(t)]T,
Y (t)=PT(t), C=[P/ η ..., P/ η | N, 0 ... 0],
A is Q × Q state matrix:
B is the matrix of Q × Q:
9. electric heater load group Controlling model according to claim 8, which is characterized in that enableIf F (t, Tsp) it is control variable u (t), state equation turns to canonical form:
Y (t)=Cx (t) (2)
Above-mentioned state equation is write as to the form of transmission function:
Discretization is carried out again, at this point, A and B is the matrix of 2X2.
10. electric heater load group Controlling model according to claim 9, which is characterized in that the electric heater group from
Dissipating time sliding formwork control model includes following equation,
Define the switching band of an encirclement diverter surface:
SΔ={ x ∈ Rn|-Δ < s (x)=cx <+Δ }
The reaching condition of continuous system is generalized to discrete system, the condition of reaching is,
[s (k+1)-s (k)] s (k) < 0 (26)
Liapunov function is chosen,
Meet condition,
Δ V (k)=s2(k+1)-s2(k) 0 <, s (k) ≠ 0
(4)
According to Lyapunov theorem of stability, s (k)=0 is the balanced surface of Globally asymptotic, i.e. the shape of arbitrary initial position
State can all be intended to diverter surface s (k), take the reaching condition to be,
s2(k+1) < s2(k)
(29)
When the sampling time, T was becoming tight infinitely small, the presence of discrete sliding mode and reaches performance condition and is,
[s (k+1)-s (k)] sgn (s (k)) < 0
[s (k+1)+s (k)] sgn (s (k)) > 0 (30)
In discrete system, exponentially approaching rule is,
Wherein ε > 0, q > 0,1-qT > 0, T is the sampling period；
Exponentially approaching rule meets,
S (k+1)-s (k)=- T ε sgn (s (k)) -- qTs (k)=- qT | s (k) |-T ε | s (k) | < 0
(32)
Meanwhile when sampling time T approach it is infinitely small when, 2-qT be much larger than 0, therefore
S (k+1)+s (k)=[(2-qT) s (k)-T ε sgn (s (k))] sgn (s (k))=(2-qT) | s (k) |-T ε | s (k) | > 0
(33)
S (k+1)=Cx (k+1)=CAx (k)+CBu (k) is substituted into Reaching Law and obtained by discrete sliding mode face s (k)=Cx (k),
(1-qT) s (k)-T ε sgn (s (k))=CAx (k)+GBu (k) (5)
Assuming that sliding moding structure controlled condition CB ≠ 0 establishment, discrete sliding mode control rate are,
U (k)=- (CB)-1[CAx(k)-(1-qT)s(k)+Tεsgn(s(k))] (35)。
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