CN117588313A - Aeroengine finite time control method based on event trigger control - Google Patents

Aeroengine finite time control method based on event trigger control Download PDF

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CN117588313A
CN117588313A CN202311680226.6A CN202311680226A CN117588313A CN 117588313 A CN117588313 A CN 117588313A CN 202311680226 A CN202311680226 A CN 202311680226A CN 117588313 A CN117588313 A CN 117588313A
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event
control
time
trigger
state
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赵旭东
葛佰恩
许宁
李鹏
刘硕硕
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Dalian University of Technology
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    • G—PHYSICS
    • G05—CONTROLLING; REGULATING
    • G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
    • G05B13/00—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion
    • G05B13/02—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric
    • G05B13/04—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric involving the use of models or simulators
    • G05B13/042—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric involving the use of models or simulators in which a parameter or coefficient is automatically adjusted to optimise the performance
    • F—MECHANICAL ENGINEERING; LIGHTING; HEATING; WEAPONS; BLASTING
    • F02—COMBUSTION ENGINES; HOT-GAS OR COMBUSTION-PRODUCT ENGINE PLANTS
    • F02C—GAS-TURBINE PLANTS; AIR INTAKES FOR JET-PROPULSION PLANTS; CONTROLLING FUEL SUPPLY IN AIR-BREATHING JET-PROPULSION PLANTS
    • F02C9/00—Controlling gas-turbine plants; Controlling fuel supply in air- breathing jet-propulsion plants
    • F—MECHANICAL ENGINEERING; LIGHTING; HEATING; WEAPONS; BLASTING
    • F05—INDEXING SCHEMES RELATING TO ENGINES OR PUMPS IN VARIOUS SUBCLASSES OF CLASSES F01-F04
    • F05D—INDEXING SCHEME FOR ASPECTS RELATING TO NON-POSITIVE-DISPLACEMENT MACHINES OR ENGINES, GAS-TURBINES OR JET-PROPULSION PLANTS
    • F05D2270/00—Control
    • F05D2270/01—Purpose of the control system
    • F05D2270/02—Purpose of the control system to control rotational speed (n)
    • F—MECHANICAL ENGINEERING; LIGHTING; HEATING; WEAPONS; BLASTING
    • F05—INDEXING SCHEMES RELATING TO ENGINES OR PUMPS IN VARIOUS SUBCLASSES OF CLASSES F01-F04
    • F05D—INDEXING SCHEME FOR ASPECTS RELATING TO NON-POSITIVE-DISPLACEMENT MACHINES OR ENGINES, GAS-TURBINES OR JET-PROPULSION PLANTS
    • F05D2270/00—Control
    • F05D2270/70—Type of control algorithm

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  • Feedback Control In General (AREA)

Abstract

The invention belongs to the field of transition state control of aero-engines, and provides an aero-engine finite time control method based on event trigger control. Firstly, establishing a mathematical model of an aeroengine; secondly, designing a limited time H based on an event trigger mechanism ∞ A controller to control the problem; then, H in a limited time range under the designed DETM is verified ∞ Tracking performance; and finally, applying the designed DETM to an engine model to carry out tracking simulation. The invention can ensure the closed loop system to have a finite time H ∞ In addition, in the event triggering control scheme, the update frequency of the controller can be greatly reduced while the control performance is ensured, and the event triggering times are reduced, so that the consumption of system resources is reduced, the uncertainty and external disturbance can be adapted, and the robustness and the stability of the control system are improved.

Description

Aeroengine finite time control method based on event trigger control
Technical Field
The invention belongs to the field of transition state control of aero-engines, and provides an aero-engine finite time H based on event trigger control ∞ A control method.
Background
Because of the strong nonlinearity of aeroengines and lack of theoretical support, some conventional control methods are applicable to linear models but not to aero-engine control. H ∞ Control is a modern control theory aimed at designing a robust controller that enables the system to maintain its stability and performance in the presence of uncertainties and disturbances. H ∞ Control has been applied in many fields, however, many studies have considered H over an infinite time range ∞ Control, limited time H ∞ Control is a new control strategy developed in recent years, aimed at achieving robust stability and performance requirements of the system within a limited time.
And conventional H ∞ Controlled differently, limited time H ∞ Control takes into account control problems of the system over a limited time, and thus has great applicability to some systems that require a fast response. Considering H of aeroengine in limited time range ∞ The performance has important significance and the finite time H ∞ The controller can ensure that the system converges in a limited time and has better stability and performance requirements in the limited time.
Time triggered control is a common control scheme that performs control operations based on predetermined time intervals. In time-triggered control, the controller performs control operations at predetermined time intervals, while the sensor and the actuator sample and perform at fixed time intervals, the main advantage of time-triggered control is easy implementation and analysis. Since the information transfer is determined by a given time interval, the control strategy of the time triggered control can be relatively easily constructed and implemented. However, time triggered control also has some drawbacks. Since the sampling rate and the number of transmissions are manually determined, excessive resource consumption may result. Furthermore, in some cases, time-triggered control may not be able to accommodate changes or anomalies in the system.
To overcome the drawbacks of time triggered control, a new control strategy is proposed, namely event triggered control. The event triggering mechanism designs an event triggering condition to screen data, and the controller only executes control operation when the system state changes, so that the consumption of resources is greatly reduced.
Based on the above discussion, the present invention has studied H in a limited time range of event triggering of an aeroengine model based on switching linear variable parameters ∞ A tracking control method.
Disclosure of Invention
Aiming at the problems existing in the prior art, the invention designs a group of dynamic event trigger controllers and a state-dependent switching signal, is applied to an aeroengine control system, and ensures that the system has H in a given time period ∞ In the event trigger control scheme, only state information at sampling time is detected, discrete state information is utilized to design a state-dependent switching signal, and finally the designed DETM is applied to transition state control of an aeroengine, so that the trigger times are greatly reduced while the tracking effect of a control method is basically ensured, system resources are saved, and finally the validity of the designed DETM is verified through simulation.
In order to achieve the above purpose, the invention adopts the following technical scheme:
h in limited time range based on event triggering ∞ Firstly, establishing a mathematical model of an aeroengine; secondly, designing a limited time H based on an event trigger mechanism ∞ A controller to control the problem; then, H in a limited time range under the designed DETM is verified ∞ Tracking performance; and finally, applying the designed DETM to an engine model to carry out tracking simulation. The method specifically comprises the following steps:
first, establishing a mathematical model of the aero-engine.
(1.1) defining related variables and related parameters.
Definition Z + ＝{1,2,...,},R＝(-∞,+∞),R + = [0, + -infinity), the expression of the real space R n The Euclidean norm on, the symmetric block in the symmetric matrix is denoted as O, where n represents the positive integerThe number represents an n-dimensional vector. If A is a matrix, A>0 means that a is a symmetrical positive definite matrix. Lambda (lambda) max (A)，λ min (A) The eigenvalues representing a have the largest (smallest) real part.
(1.2) establishing a switched aero-engine model with external disturbances, as shown in equation (1):
wherein the system state x (t) ∈R n Representing low pressure turbine rotor speed; the control input being denoted as u (t) ∈R m ；J(t)∈R n Representing an external disturbance input; sigma (t) R + ρ is the switching signal, where ρ= {1,2,3,..n } is the index set; omega is an exogenous parameter; a is that σ(t) And B σ(t) Coefficient matrices representing system status and control inputs respectively,representing the derivative of the system state x (t).
(1.3) in order to track the switched aeroengine model established in step (1.2), establishing a parameter independent reference system:
wherein reference state x r (t)∈R n Reference input R (t) ∈R n ，And->A r ∈R n×n Is a constant matrix, where n represents a positive integer, represents an n-dimensional matrix, +.>Representing reference state x r (t) differentiation.
The following tracking errors are defined:
e(t)＝x(t k )-x(D k,r ),t∈[D k,r ,D k,r+1 ]
wherein t is k Is an event-triggered transient, D k,r =0, 1,2,..Up-sampling time, wherein->Representation … …; t is t k+1 Representing the next event triggered transient. When->At this time, in the time periodUpper e (t) =0. From the system (1) shown in formula (1) and the system (2) shown in formula (2), the following augmentation system is derived:
wherein, a disturbance estimation matrix representing the external disturbance J (t) and r (t), is +.>Representing reference state x r (t) and a system state x (t).
(1.4) the following assumptions and theorem are put forward for the augmentation system (3) obtained in step (1.3):
assuming a given constant k, the disturbance is assumedIs time-varying and satisfies constraintsWhere c > 0 is a constant.
Theorem 1, given positive definite matrix G, normal numbers a, b, k and a < b, switching signal ζ, ifThen it can be deduced
Then the system (3) is bounded (FTB) for a limited time range for (a, b, c, k, G) and the switching signal ζ.
Theorem 2. If there is a positive definite matrix G, normal numbers a, b, k and a < b, switching signal ζ, when the initial condition is zero, if the pair (a, b, c, k, G) of the system (3) is bounded in a finite time range, and the following inequality is satisfied
Where α > 0 represents the degree of attenuation of the disturbance. Then the system (3) has a finite time range H for (a, b, c, k, G) and ζ ∞ Performance.
Second step, based on event trigger mechanism finite time H ∞ Controller design to control problems.
(2.1) continuous event triggering mechanism needs to continuously detect the signal, which leads to higher cost, the invention adopts a discrete mode to sample the signal data, and samples the signal data with a fixed period M, the event triggering condition is only detected at the sampling moments, and the event triggering transient can be expressed as:
t k ＝i k M,i k ∈N,k∈N， (5)
D k,r ＝(i k +r)M,r＝0,1,2,...
wherein t is k Indicating the trigger time; n represents a positive integer; d (D) k,r ＝(i k +r) M, r=0, 1, 2..represents the sampling moment, x (D k,r ) Representation D k,r System state at time. According to the prior literature study, the switching condition needs hardware facility detection, which causes the phenomena of cost addition and frequent switching, and the patent designs that only discrete state information x (D k,r ) State switching law of (2), switching being only at discrete instants D k,r The occurrence of frequent switching phenomenon can be avoided.
Defining a Dynamic Event Trigger Mechanism (DETM) as:
wherein y (t) =e T r (t)e r (t)-δ(t)e T (t) e (t), wherein e r (t)＝x(t k )-x r (D k,r ) Is the sampling error, { t k * -represents a set of event trigger occurrences, k e {0,1,2,., }, δ (t) represents an event trigger threshold condition, ψ > 0 is a trigger parameter; t is t k Is event-triggered transient, t k+1 The next event triggering transient time is represented, in the event triggering process, the triggering time of the event is changed, and whether the triggering condition is met or not is judged through delta (t), wherein the following triggering condition is delta (t):
where τ > 0, ε > 0 are constants, and the magnitude of δ (t) varies with system tracking performance.
If the tracking error e (t) ε, the trigger threshold δ (t) will increase untilResulting inThe trigger time for the next transmission becomes long. If ||e (t) || > ε, the trigger threshold δ (t) will decrease until +.>The next transmission moment will happen immediately, and the dynamic event triggering mechanism designed by the patent adopts discrete state information, so that on one hand, the invention does not need additional hardware equipment to carry out continuous detection and calculation, and the cost can be reduced. On the other hand, the event condition is triggered only at the sampling instant, so +.>This means that the minimum trigger interval time is M. Thus, the Zeno phenomenon of the event trigger mechanism can be avoided.
(2.2) construction System feedback controller
u(t)＝K i (ω)x r (t k ) (7)
Wherein t is k K e {0,1,2,., } represents the number of triggers, K i (ω) represents the controller gain, under the controller (7), the system (3) shown in equation (3) is:
under the dynamic event triggering mechanism DETM (6), the Zeno phenomenon of the system (9) shown in the formula (9) can be eliminated.
Third step, prove the second step event triggering mechanism has finite time H ∞ The controller system (9) controlling the problem is relative to (a) in the least hysteresis switching law 1 ,a 2 ,d,T f G) is progressively stable, demonstrating the deduction as follows.
(3.1) assume that there is a positive definite matrix G > 0 and four positive numbers a 1 ,a 2 ,d,T f Satisfy a 1 ＜a 2 Assuming positive numbers η, γ>0,λ i 0.gtoreq.2n2n2n matrix function P i (omega) > 0, let arbitraryThe following inequality constraint is satisfied:
P i (ω)-P j (ω)≥0，ω∈S ij (11)
wherein the method comprises the steps of
(3.2) first define the Liapunov function asThen according to V σ (t) derivative along the system (9) with respect to t and dynamic trigger mechanism DETM (6), always present +.>It is possible to obtain that when t is not less than 0,
the equation (10) can be obtained by:
through some operations, the product with t more than or equal to 0 can be finally obtained
Wherein the total operating time of the p-th subsystem during the interval s, T is denoted as T p (s,t)。
Then combining inequality (12) and inequality (15), and finally derivingUnder the established controller (7) and the designed DETM (6), the system (9) is relative to (a) 1 ,a 2 ,d,T f G) is system global asymptotically stable.
Fourth, verify H in a limited time range under the designed DETM ∞ Performance is tracked.
(4.1) from the conclusion in the third step, the system (9) is relative to (a 1 ,a 2 ,d,T f G) is globally asymptotically stable, the following demonstrates that the system (8) has a finite time range of H under the action of the dynamic event trigger mechanism (6) and the state dependent switching law (5) ∞ Performance.
Assume that for any oneThe inequalities (11), (12) of 3.1 of the third step and the following inequalities are all true:
wherein H= (I) n×n ,-I n×n )。
First define the Lyapunov function asThe conclusion (10) is established from (13). From the conclusion of the third stepAnd the system (8) is asymptotically stable in a limited time range, satisfies inequality (4), when t is more than or equal to 0,
wherein,using the procedure in 3.1, deducing that when t is not less than 0,due to->Further deriving->By setting t=t f Can obtain the result of the constraint condition (4) being established, the system (8) has H in a limited time range under the action of the dynamic event trigger mechanism (6) and the state dependent switching law (5) ∞ Performance.
(4.2) finite time H ∞ Control problem system controller gain matrix design.
The system controller gain is derived from the following theorem. Assuming that there is a constant matrix G > 0, four positive numbers a 1 ,a 2 ,d,T f Satisfy a 1 ＜a 2 T of (2) f And a matrix function u (ω) is present, assuming a constant λ is present i Not less than 0, eta, gamma > 0, m x n matrix function W i (omega) to any oneThe following inequality is satisfied:
M i (ω)-M i (ω)≤0,ω∈S ij (1)
under the action of the dynamic event trigger mechanism (6) and the state-dependent switching law (5), the system (9) is bounded in a limited time range, and the system control gain is obtained as
And fifthly, applying the designed DETM to a JT9D engine model to carry out tracking simulation.
The aim is to control the speed n of the low-pressure turbine of a JT9D aeroengine l Tracking a preset reference track, and system state x 2 Tracking a preset reference system state track. The state output of the engine model is selected as the fan speed increase ΔN relative to the steady state value f And an increase ΔN in low pressure turbine speed relative to a steady state value c The engine model reference input is selected as the increase in fuel flow delta N relative to the steady state value F . Considering a JT9D type aeroengine model, then designing a controller gain matrix, according to the conclusion in the second step, the magnitude of a trigger threshold delta (t) changes along with the change of the tracking performance of the system, the smaller the upper limit of the trigger threshold is, the more the trigger times are caused, the larger the lower limit of the trigger threshold is, the fewer the trigger times are caused, and the proper upper and lower limits of the trigger threshold are selected to ensure the transmission frequency, and the parameter of event triggering is selected to be 0.5. Further, selecting iota=0.01, then DETM (6) can be determined, assuming the reference input asAnd (5) performing tracking performance simulation.
The beneficial effects of the invention are as follows:
the invention designs a group of dynamic event trigger controllers (DETM) and a state-dependent switching signal, which can ensure that a closed-loop system has a finite time H ∞ In addition, in the event triggering control scheme, the update frequency of the controller can be greatly reduced while the control performance is ensured, and the event triggering times are reduced, so that the consumption of system resources is reduced, the uncertainty and external disturbance can be adapted, and the robustness and the stability of the control system are improved. In addition, the invention takes JT9D engine model as an example, and the effectiveness of the method is verified.
Drawings
FIG. 1 is a Simulink model diagram of a JT9D engine;
FIG. 2 is a schematic illustration of a JT9D dynamic gas turbine simulation;
FIG. 3 is a schematic diagram of a variation of flight Mach number;
FIG. 4 is x 2 And x r2 Tracking a simulation schematic;
FIG. 5 is N c And N r2 Tracking a simulation schematic;
FIG. 6 is a schematic diagram of a change curve of a switching signal;
FIG. 7 is a diagram of a dynamic event trigger mechanism.
In the figure: 1 fan, 2 low-pressure compressor, 3 high-pressure compressor, 4 combustion chamber, 5 high-pressure turbine, 6 low-pressure turbine, 7 tail pipe.
Detailed Description
The invention is further illustrated below with reference to specific examples.
H in a limited time range of event triggering of an aeroengine model based on a switched linear variable parameter ∞ Tracking control, comprising the following steps:
firstly, establishing an aeroengine mathematical model based on switching linear variable parameters
1.1 define the relevant variables and the relevant tuning parameters.
Definition Z + ＝{1,2,...,},R＝(-∞,+∞),R + = [0, + -infinity), the expression of the real space R n The euclidean norm above, and the symmetric blocks in the symmetric matrix are denoted as onium, where n represents a positive integer and represents an n-dimensional vector. If A is a matrix, A>0 means that a is a symmetrical positive definite matrix. Lambda (lambda) max (A)，λ min (A) The eigenvalues representing a have the largest (smallest) real part.
1.2 building a switched aeroengine model with external disturbances:
wherein the system state x (t) ∈R n Representing low pressure turbine rotor speed; the control input being denoted as u (t) ∈R m ；J(t)∈R n Representing an external disturbance input; sigma (t) R + ρ is the switching signal, where ρ= {1,2,3,..n } is the index set; omega is an exogenous parameter; a is that σ(t) And B σ(t) The matrix of coefficients is represented and,representing the derivative of the system state x (t).
1.3 to track the model built in 1.2, build a system of independent:
wherein reference state x r (t)∈R n Input R (t) ∈R n ，And->And A is r ∈R n×n Is a constant matrix. Combining the systems (1) and (2), deriving an augmentation system as follows:
wherein the method comprises the steps of A disturbance estimation matrix representing the external disturbance J (t) and r (t), is +.>Representing reference state x r (t) and a system state x (t).
The following event trigger errors are defined:
e(t)＝x(t k )-x(D k,r ),t∈[D k,r ,D k,r+1 ]
wherein t is k Is an event-triggered transient, D k,r =0, 1,2,..Up-sampling time, wherein->Representation … …; t is t k+1 Representing the next event triggered transient. When->At this time, in the time periodUpper e (t) =0.
1.4 the following assumptions and theorem are proposed for the augmentation system (3) obtained in 1.3:
assuming a given constant k, the disturbance is assumedIs time-varying and satisfies constraintsWhere c > 0 is a constant.
Theorem 1, given positive definite matrix G, normal numbers a, b, k and a < b, switching signal ζ, ifThen it can be deduced
Then the system (3) is bounded (FTB) for a limited time range for (a, b, c, k, G) and the switching signal ζ.
Theorem 2. If there is a positive definite matrix G, normal numbers a, b, k and a < b, switching signal ζ, when the initial condition is zero, if the pair (a, b, c, k, G) of the system (3) is bounded in a finite time range, and the following inequality is satisfied
Where α > 0 represents the degree of attenuation of the disturbance. Then the system (3) has a finite time range H for (a, b, c, k, G) and ζ ∞ Performance.
Second step, based on event trigger mechanism finite time H ∞ Controller design to control problems.
(2.1) continuous event triggering mechanism needs to continuously detect signals, which can lead to higher cost, the patent adopts a discrete mode to sample signal data, samples with a fixed period M, event triggering conditions are detected only at the sampling moments, and event triggering transient can be expressed as:
t k ＝i k M,i k ∈N,k∈N， (5)
D k,r ＝(i k +r)M,r＝0,1,2,...
wherein t is k Indicating the trigger time; n represents a positive integer; d (D) k,r ＝(i k +r) M, r=0, 1, 2..represents the sampling moment, x (D k,r ) Representation D k,r System state at time. According to the previous research, the switching condition needs to be continuously detected by hardware facilities, which can lead to the phenomena of cost addition and frequent switching, and the patent adopts the discrete state information x (D k,r ) Switching is only at discrete instants D k,r The occurrence of frequent switching phenomenon can be avoided.
The Dynamic Event Trigger Mechanism (DETM) is:
wherein y (t) =e T r (t)e r (t)-δ(t)e T (t) e (t), … …, wherein e r (t)＝x(t k )-x r (D k,r ) Is the sampling error, { t k * -represents a set of event trigger occurrences, k e {0,1,2,., }, δ (t) represents an event trigger threshold condition, ψ > 0 is a trigger parameter; t is t k Is event-triggered transient, t k+1 Indicating the next event triggering transient time, wherein the triggering time of the event is not determined in the event triggering process, and judging delta (t) can judge whether the triggering condition is met or not, wherein the triggering condition is delta (t) below
Where τ > 0, ε > 0 are constants, and the magnitude of δ (t) varies with system tracking performance.
If the tracking error e (t) ε, the trigger threshold δ (t) will increase untilResulting in a longer trigger time for the next transmission. If ||e (t) || > ε, the trigger threshold δ (t) will decrease until +.>The next transmission moment will happen immediately, and the designed dynamic event trigger mechanism adopts discrete state information, so that on one hand, the invention does not need additional hardware equipment to carry out continuous detection and calculation, thereby reducing the cost. On the other hand, the event condition is triggered only at the sampling instant, so +.>This means that the minimum trigger interval time is M, and the Zeno phenomenon of the event trigger mechanism can be avoided.
(2.2) construction System feedback controller
u(t)＝K i (ω)x r (t k ) (7)
Wherein t is k K e {0,1,2,., } represents the number of triggers, K i (ω) represents the controller gain, under the controller (7), the system (3) shown in equation (3) is:
under the dynamic event triggering mechanism DETM (6), the Zeno phenomenon of the system (9) shown in the formula (9) can be eliminated.
Third step, prove the second step event triggering mechanism has finite time H ∞ The controller system (9) controlling the problem is relative to (a) in the least hysteresis switching law 1 ,a 2 ,d,T f G) is progressively stable.
(3.1) assume that there is a positive definite matrix G > 0 and four positive numbers a 1 ,a 2 ,d,T f Satisfy a 1 ＜a 2 Assuming positive numbers η, γ>0,λ i 0.gtoreq.2n2n2n matrix function P i (omega) > 0, let arbitraryThe following inequality constraint is satisfied:
P i (ω)-P j (ω)≥0，ω∈S ij (11)
wherein the method comprises the steps of
(3.2) first define the Liapunov function asThen according to V σ (t) derivative with respect to t along the system (9) and dynamic trigger mechanism DETM (6), whenWhen t=tk+1, the state information of the reference system is sampled and updated. Deducing->Thus, for all t.gtoreq.0 there is always +.>It is possible to obtain that when t is not less than 0,
it can be obtained with the condition (10):
through some operations, the product with t more than or equal to 0 can be finally obtained
Wherein the total operating time of the p-th subsystem during interval s, T is denoted as T p (s,t)。
Then combining inequality (12) and inequality (15), and finally derivingUnder the established controller (7) and the designed DETM (6), the system (9) is relative to (a) 1 ,a 2 ,d,T f G) is system global asymptotically stable.
Fourth step, H in a limited time range ∞ Controller design to control problems.
4.1 from the conclusion in the third step, the system (9) is relative to (a 1 ,a 2 ,d,T f G) is globally asymptotically stable, the following demonstrates that the system (8) has a finite time range of H under the action of the dynamic event trigger mechanism (6) and the state dependent switching law (5) ∞ Performance.
If to arbitraryThe inequalities (11), (12) and the following inequalities hold in 3.1 of the third step:
wherein H= (I) n×n ,-I n×n )。
First define the Lyapunov function asThe conclusion (10) is established from (13). From the conclusion of the third step, the system (8) is asymptotically stable for a limited time range, satisfying the inequality (4), when t is not less than 0,
wherein the method comprises the steps ofUsing the procedure in 3.1, deducing that when t is not less than 0,
due toFurther deriving->By setting t=t f Can obtain the result of the constraint condition (4) being established, the system (8) has H in a limited time range under the action of the dynamic event trigger mechanism (6) and the state dependent switching law (5) ∞ Performance.
4.2 finite time H ∞ Control problem system controller gain matrix design.
The system controller gain matrix is derived from the following theorem. Assume that there are four positive constants a for a positive definite matrix G > 0 1 ,a 2 ,d,T f Satisfy a 1 ＜a 2 T of (2) f And a positive definite matrix function u (ω), assuming that λ is present i Not less than 0, eta, gamma > 0, m x n matrix function W i (omega) to any oneAnd the following inequality is satisfied:
M i (ω)-M i (ω)≤0,ω∈S ij (17)
then the system (9) is asymptotically stable over a finite time range under the action of the dynamic event trigger mechanism (6) and the state dependent switching law (5), the system control gain being
Fifthly, simulating application on an actual engine model, selecting a JT9D engine model, and verifying whether the target rotating speed and the system state of the system accurately track a preset reference track under a designed dynamic trigger mechanism.
5.1 selecting an increment of fan speed ΔN f And a relative increase in low pressure turbine rotor speed from steady state value ΔN c As the output result of the model, the relative increment delta N of the fuel flow and the steady state value F Is the input of the model.
Firstly, selecting a JT9D engine, wherein a Simulink model is shown in a figure 1, and the structure of the JT9D engine model is schematically shown, in the figure, 1 is a fan, 2 is a low-pressure compressor, 3 is a high-pressure compressor, 4 is a combustion chamber, 5 is a high-pressure turbine, 6 is a low-pressure turbine, 7 is a tail nozzle, the leftmost side is a fan, the low-pressure compressor and the high-pressure compressor are close to the fan, the middle part is the combustion chamber, then the right side is the high-pressure turbine and the low-pressure turbine in sequence, and finally the tail partIs a tail pipe. Consider the aero-engine model input/output as x (t) = (Δn) f ,ΔN c ) T ,u(t)＝ΔN F J (t) is an external disturbance. The parameter of control is flight Mach number. Suppose ω ε [0.1,0.775 ]]，Ρ 1 ＝[0.1,0.55]，Ρ 2 ＝[0.45,0.775]，Ρ＝Ρ 1 ∪P 2 . The variation of the flight mach number is shown in figure 2. Using MATLAB toolbox to find:
5.2 Next, consider reference model (2), where x r (t)＝(ΔN f ,ΔN c ) T Andthe DETM control scheme of the event triggering mechanism designed by the invention is applied to JT9D aeroengines. The aim is to control the speed n of the low-pressure turbine of a JT9D aeroengine l To track the preset reference trajectory shown in fig. 5. Selecting lambda 1 ＝0.01,λ 2 =0.2, η=45, γ=0.42, u (ω) =0.5I, and the controller gain matrix is set to
K 1 ＝(-0.000784,0.002321)+ω(0.001127,0.002256),
K 2 ＝(-0.0007952,-0.001202)+ω(0.0001217,0.0001219)
5.3 according to the conclusion in 2.1, the magnitude of the trigger threshold δ (t) varies with the variation of the tracking performance of the system. If the error between the tracking output and the reference exceeds a threshold, then an event is triggered, epsilon=0.01 is selected and the trigger constant is selected to be 0.5. Further, if iota=0.01 is selected, thenTo determine DETM (6). The external disturbance is J (t) = (e -2t ,e -2t ) T ，r(t)＝(0.01sin(0.01t),0.01sin(0.01t)) T . For simulation results, the initial state of a given model isThe tracking simulation of the system state of the model under the preconditions described above is shown in fig. 4 and 5. From the simulation, it can be concluded that the open loop system is limited in time under the action of the designed dynamic event trigger controller, and the system eventually tends to be stable, and the designed dynamic event trigger controller and state-dependent switching law can ensure that the closed loop system has limited time H ∞ Performance, it can be seen that the dynamic event triggering mechanism greatly reduces the update frequency of the controller, proving the effectiveness of the scheme.
The invention designs a group of dynamic event trigger controllers and a state-dependent switching signal, is applied to an aeroengine control system, and ensures that the system has H in a given time period ∞ The method has the advantages that the performance is improved, the resource loss caused by time trigger control is avoided, in the event trigger control method, only state information at sampling time is needed to be detected, discrete state information is utilized to design a state-dependent switching signal, when the DETM method is applied, the calculation burden is remarkably reduced, the dynamic trigger mechanism is benefited, frequent trigger events are not needed when a strategy is used, finally, the designed event trigger mechanism is applied to aeroengine transition state control, and the effectiveness of the designed DETM is verified through simulation.
The examples described above represent only embodiments of the invention and are not to be understood as limiting the scope of the invention, it being understood that variations and modifications can be made by those skilled in the art without departing from the spirit of the invention, which are all within the scope of the invention.

Claims (3)

1. An aeroengine finite time control method based on event trigger control, which is characterized in thatFirstly, establishing a mathematical model of the aero-engine; secondly, designing a limited time H based on an event trigger mechanism ∞ A controller to control the problem; then, H in a limited time range under the designed DETM is verified ∞ Tracking performance; and finally, applying the designed DETM to an engine model to carry out tracking simulation.
2. The method for controlling the limited time of an aeroengine based on event triggering control according to claim 1, comprising the following steps:
firstly, establishing a mathematical model of the aero-engine;
(1.1) defining related variables and related parameters;
definition Z + ＝{1,2,...,},R＝(-∞,+∞),R + = [0, + -infinity), the expression of the real space R n The euclidean norm above, the symmetric block in the symmetric matrix is denoted ∈, where n represents a positive integer, representing an n-dimensional vector;
(1.2) establishing a switched aero-engine model with external disturbances, as shown in equation (1):
wherein the system state x (t) ∈R n Representing low pressure turbine rotor speed; the control input being denoted as u (t) ∈R m ；J(t)∈R n Representing an external disturbance input; sigma (t) R + ρ is the switching signal, where ρ= {1,2,3,..n } is the index set; omega is an exogenous parameter; a is that σ(t) And B σ(t) Coefficient matrices representing system status and control inputs respectively,a derivative representing the system state x (t);
(1.3) in order to track the switched aeroengine model established in step (1.2), establishing a parameter independent reference system:
wherein reference state x r (t)∈R n Reference input R (t) ∈R n ，And->A r ∈R n×n Is a constant matrix, where n represents a positive integer, represents an n-dimensional matrix, +.>Representing reference state x r (t) differentiating;
the following tracking errors are defined:
e(t)＝x(t k )-x(D k,r ),t∈[D k,r ,D k,r+1 ]
wherein t is k Is an event-triggered transient, D k,r =0, 1,2,..Up-sampling time, wherein->Representation … …; t is t k+1 Representing a next event triggered transient; when->At this time, in the time periodUpper e (t) =0;
from the system (1) shown in formula (1) and the system (2) shown in formula (2), the following augmentation system is derived:
wherein, a disturbance estimation matrix representing the external disturbance J (t) and r (t), is +.>Representing reference state x r (t) a reference state estimation matrix consisting of system states x (t);
(1.4) the following assumptions and theorem are put forward for the augmentation system (3) obtained in step (1.3):
assuming a given constant k, the disturbance is assumedIs time-dependent and satisfies the constraint +.>Wherein c > 0 is a constant;
theorem 1, given positive definite matrix G, normal numbers a, b, k and a < b, switching signal ζ, ifIt can be deduced that:
then the system (3) is bounded (FTB) for a finite time range for (a, b, c, k, G) and the switching signal ζ;
theorem 2. If there is a positive definite matrix G, normal numbers a, b, k and a < b, switching signal ζ, when the initial condition is zero, if the pair (a, b, c, k, G) of the system (3) is bounded within a finite time range, and the following inequality is satisfied:
wherein α > 0 represents the attenuation degree of the disturbance; then the system (3) has a finite time range H for (a, b, c, k, G) and ζ ∞ Performance;
second step, based on event trigger mechanism finite time H ∞ A controller design to control the problem;
(2.1) continuous event triggering mechanism requires continuous detection of the signal, sampling of the signal data in a discrete manner, sampling with a fixed period M, event triggering condition being detected only at these sampling instants, event triggering transient can be expressed as:
t k ＝i k M,i k ∈N,k∈N，(5)
D k,r ＝(i k +r)M,r＝0,1,2,...
wherein t is k Indicating the trigger time; n represents a positive integer; d (D) k,r ＝(i k +r) M, r=0, 1, 2..represents the sampling moment, x (D k,r ) Representation D k,r System state at time;
design uses only discrete state information x (D k,r ) State switching law of (2), switching being only at discrete instants D k,r The occurrence of the position can avoid the frequent switching phenomenon;
defining a dynamic event trigger mechanism DETM as follows:
wherein y (t) =e T r (t)e r (t)-δ(t)e T (t) e (t), wherein e r (t)＝x(t k )-x r (D k,r ) Is the sampling error, { t k * The event touch occurrence times are representedA set of numbers, k e {0,1,2, }, δ (t) represents an event trigger threshold condition, ψ > 0 is a trigger parameter; t is t k Is event-triggered transient, t k+1 The next event triggering transient time is represented, in the event triggering process, the triggering time of the event is changed, and whether the triggering condition is met or not is judged through delta (t), wherein the following triggering condition is delta (t):
wherein, tau > 0 and epsilon > 0 are constants, and delta (t) changes along with the change of the tracking performance of the system;
if the tracking error e (t) ε, the trigger threshold δ (t) will increase untilThe triggering time of the next transmission becomes long; if ||e (t) || > ε, the trigger threshold δ (t) will decrease until +.>The next transmission moment will happen immediately, the dynamic event triggering mechanism designed adopts discrete state information, on one hand, no extra hardware equipment is needed to carry out continuous detection and calculation, which can reduce the cost, on the other hand, the event condition is triggered only at the sampling moment, so->The minimum trigger interval time is M; the Zeno phenomenon of an event triggering mechanism can be avoided;
(2.2) construction System feedback controller
u(t)＝K i (ω)x r (t k )(7)
Wherein t is k K e {0,1,2,., } represents the number of triggers, K i (ω) represents the controller gain, under the controller (7), the system (3) shown in equation (3) is expressed as:
under a dynamic event triggering mechanism DETM (6), the Zeno phenomenon of the controller system (9) shown in the formula (9) can be eliminated;
third, under the established feedback controller (7) and designed DETM (6), the second step event triggering mechanism has a finite time H ∞ A controller system (9) controlling the problem, which is system global asymptotically stable on a minimum hysteresis switching law;
fourth, verify H in a limited time range under the designed DETM ∞ Tracking performance;
under the action of a dynamic event trigger mechanism (6) and a state-dependent switching law (5), a controller system (9) is bounded within a limited time range, and can obtain a system control gain;
and fifthly, applying the designed DETM to an engine model to carry out tracking simulation.
3. The method for controlling the finite time of the aeroengine based on the event triggering control according to claim 2, wherein the tracking simulation of the fifth step comprises the following specific steps:
the aim is to control the speed n of the low-pressure turbine of an aeroengine l Tracking a preset reference track, and system state x 2 Tracking a preset reference system state track; the state output of the engine model is selected as the fan speed increase ΔN relative to the steady state value f And an increase ΔN in low pressure turbine speed relative to a steady state value c The engine model reference input is selected as the increase in fuel flow delta N relative to the steady state value F The method comprises the steps of carrying out a first treatment on the surface of the Then, designing a controller gain matrix, and according to the conclusion in the second step, the magnitude of the trigger threshold delta (t) is changed along with the change of the tracking performance of the systemThe smaller the upper limit of the trigger threshold value is, the more the trigger times are caused, the larger the lower limit of the trigger threshold value is, the fewer the trigger times are caused, and the proper upper and lower limits of the trigger threshold value are selected; finally, the tracking performance simulation is performed with reference to the input.
CN202311680226.6A 2023-12-08 2023-12-08 Aeroengine finite time control method based on event trigger control Pending CN117588313A (en)

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