CN114296355B - Self-adaptive event trigger control method containing dynamic anti-saturation compensator system - Google Patents
Self-adaptive event trigger control method containing dynamic anti-saturation compensator system Download PDFInfo
- Publication number
- CN114296355B CN114296355B CN202210000999.4A CN202210000999A CN114296355B CN 114296355 B CN114296355 B CN 114296355B CN 202210000999 A CN202210000999 A CN 202210000999A CN 114296355 B CN114296355 B CN 114296355B
- Authority
- CN
- China
- Prior art keywords
- saturation
- adaptive event
- dynamic
- self
- dynamic anti
- Prior art date
- Legal status (The legal status is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the status listed.)
- Active
Links
Images
Classifications
-
- Y—GENERAL TAGGING OF NEW TECHNOLOGICAL DEVELOPMENTS; GENERAL TAGGING OF CROSS-SECTIONAL TECHNOLOGIES SPANNING OVER SEVERAL SECTIONS OF THE IPC; TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
- Y02—TECHNOLOGIES OR APPLICATIONS FOR MITIGATION OR ADAPTATION AGAINST CLIMATE CHANGE
- Y02P—CLIMATE CHANGE MITIGATION TECHNOLOGIES IN THE PRODUCTION OR PROCESSING OF GOODS
- Y02P90/00—Enabling technologies with a potential contribution to greenhouse gas [GHG] emissions mitigation
- Y02P90/02—Total factory control, e.g. smart factories, flexible manufacturing systems [FMS] or integrated manufacturing systems [IMS]
Abstract
The invention relates to a self-adaptive event triggering control method of a system with a dynamic anti-saturation compensator, which comprises the steps of firstly establishing a system model with the dynamic anti-saturation compensator, then designing self-adaptive event triggering conditions, and updating signals received by a dynamic controller only when the self-adaptive event triggering conditions are met; then, a system model under the control of self-adaptive event triggering is established, and under the condition of a given dynamic anti-saturation compensator, self-adaptive event triggering conditions are designed to enable a saturation system to be asymptotically stable; in the case that the dynamic anti-saturation compensator is not given, the dynamic anti-saturation compensator and the self-adaptive event triggering condition are designed at the same time; finally, it is proved that the system does not generate Zeno phenomenon under the designed self-adaptive event trigger control. According to the method, a dynamic anti-saturation compensator and a self-adaptive event triggering mechanism are simultaneously introduced into the system, so that the bad influence of the saturation of an actuator on the system can be reduced and network communication resources can be saved while the asymptotic stability of the system is ensured.
Description
Technical Field
The invention belongs to the technical field of event trigger control design, and particularly relates to a self-adaptive event trigger control method comprising a dynamic anti-saturation compensator system.
Background
Saturation is one of the non-linear problems common in practical systems, and its output is often limited due to the physical condition limitation of the actuator itself, and if the actuator is not considered for saturation in designing the controller, the performance of the system may be reduced, and even the system may be unstable. At present, there are two main methods for processing the saturation of an actuator, one is a direct method, namely, the influence of the saturation of the actuator is directly considered when a controller is designed, so that a system works in a linear region of the actuator; the other is a two-step method, which first ignores the saturation nonlinearity to design the controller, and then adds an anti-saturation compensator to the system to reduce the adverse effect of the saturation nonlinearity on the saturation system. In addition, compared with a static anti-saturation compensator, the dynamic anti-saturation compensator has more coefficient matrixes, so that more degrees of freedom can be provided for the whole system, and better control performance is realized.
The event triggering mechanism has the advantage of saving network communication resources, and is attracting attention. Specifically, in event-triggered control, once a designed event-triggered condition is satisfied, a control task is executed. The self-adaptive event triggering mechanism comprises auxiliary dynamic variables meeting a certain differential equation, and can achieve the purpose of saving communication resources by adjusting the triggering threshold value on line, so that the self-adaptive event triggering mechanism is more effective than the traditional event triggering mechanism. Thus, the adaptive event triggering mechanism may be more flexible and more efficient in reducing network communication resources, thereby improving system performance.
The anti-saturation compensator is introduced into the saturation system based on the event triggering mechanism, so that network communication resources can be saved, and adverse effects of the saturation of the actuator on the system can be greatly reduced.
Disclosure of Invention
In order to overcome the defects of the prior art, the invention aims to provide a self-adaptive event triggering control method with a dynamic anti-saturation compensator system.
The technical scheme adopted by the invention for solving the technical problems is as follows:
an adaptive event triggering control method comprising a dynamic anti-saturation compensator system is characterized by comprising the following specific steps:
the method comprises the steps of firstly, establishing a system model containing a dynamic anti-saturation compensator, wherein the system model comprises a continuous system with an actuator for saturation, a dynamic controller and the dynamic anti-saturation compensator;
the second step, designing self-adaptive event triggering conditions is as follows:
e(t) T Φ 1 e(t)≤δ 1 (t)y p (t) T Φ 2 y p (t)+δ 1 (t)∈e -∈t (5)
wherein e (t) is a dynamic error, e (t) =y p (t k )-y p (t)，y p (t k ) Indicating the transmission time t k Data successfully transmitted to the dynamic controller, y p (t) is the output vector of the controlled object, Φ 1 、Φ 2 For the free weight matrix, e is a positive scalar, T represents the matrix transpose; delta 1 (t) is an adaptive event trigger threshold, and satisfies the condition: delta 1 (t)＝max{δ,η(t)}，δ∈(0,1]Represents the lower bound of the adaptive event trigger threshold, eta (t) represents the threshold function, and the initial value eta (0) >0 and η (t) satisfies formula (6):
wherein, the liquid crystal display device comprises a liquid crystal display device,
representing the first derivative of the threshold function eta (t), theta representing the adjustment parameter of the convergence rate of the threshold function, theta > 1;
thirdly, establishing a system model under the triggering control of the self-adaptive event;
step four, under the condition of a given dynamic anti-saturation compensator, designing a self-adaptive event triggering condition, and ensuring the asymptotic stability of the system; constructing an optimization problem, and maximizing an estimated attraction domain of the system by solving the optimization problem;
when delta < eta (t), delta 1 (t) =η (t), substituting formula (5) yields an adaptive event triggering condition of: e (t) T Φ 1 e(t)≤δ 1 (t)y p (t) T Φ 2 y p (t)+δ 1 (t)∈e -∈t ；
When delta is larger than or equal to eta (t), delta 1 (t) =δ, substituting formula (5) yields an adaptive event triggering condition of: e (t) T Φ 1 e(t)≤δy p (t) T Φ 2 y p (t)+δ∈e -∈t ；
Fifthly, under the condition that the dynamic anti-saturation compensator is not given, designing the anti-saturation compensator and the self-adaptive event triggering condition at the same time, and ensuring the asymptotic stability of the system; constructing an optimization problem, and maximizing an estimated attraction domain of the system by solving the optimization problem;
and step six, calculating the minimum event triggering interval, and proving that the system with the dynamic anti-saturation compensator cannot generate Zeno phenomenon under the self-adaptive event triggering control of the design.
In the first step, the continuous system with actuator saturation is:
wherein t represents the time, and the time,
is a state vector of a continuous system, n p 、/>
Respectively state vector x p Dimension and first derivative of (t),>
is the output of the dynamic controller, m is the dimension, A p 、B p 、C p Are coefficient matrices, sat (u (t))= [ sat (u)) = [ sat (u) 1 (t))，…，sat(u m (t))] T Representing a vector saturation function;
the dynamic controller is as follows:
wherein, the liquid crystal display device comprises a liquid crystal display device,
representing a state vector of the dynamic controller, n c 、/>
Respectively state vector x c Dimension and first derivative of (t),>
representing the output of a dynamic anti-saturation compensator, A c 、B c 、C c 、D c Are coefficient matrixes;
the dynamic anti-saturation compensator is:
wherein, the liquid crystal display device comprises a liquid crystal display device,
is the state vector of the dynamic anti-saturation compensator, n aw 、/>
Respectively state vector x aw The dimensions and first derivatives of (t); phi (t) =sat (u (t)) -u (t) represents a dead zone function, which is the input of the dynamic anti-saturation compensator;
in the third step, after the self-adaptive event triggering mechanism is introduced, the dynamic controller is as follows:
without considering the dynamic anti-saturation compensator, the system model is:
wherein x. cl (t) is x cl First derivative of (t), x cl (t) is a state vector of the system without consideration of the dynamic anti-saturation compensator; a is that cl 、B φcl 、B ecl 、C ucl 、D uecl 、C ycl Are coefficient matrixes;
considering the dynamic anti-saturation compensator, the system model is:
where x (t) is the state vector of the system when considering the dynamic anti-saturation compensator,
is the first derivative of x (t); A. b (B) φ 、B e 、C u 、D uφ 、D ue 、C y Are coefficient matrices.
In the fourth step, the optimization problem of the construction is:
wherein ρ is a positive scalar introduced to maximize the estimation system attraction domain, Q is a positive definite matrix, R is a diagonal positive definite matrix, N i I is an identity matrix for the ith row of the matrix N, K is a matrix introduced to maximize the estimation system attraction domain, α i Is the absolute value of the saturation bound of the i-th dimension control input.
In the fifth step, the optimization problem of the construction is:
wherein Y is 1 Is a sub-matrix of matrix Y, N1 is a sub-matrix of matrix N,
N1 i for matrix N i Is L is a matrix, K 11 Is a sub-matrix of matrix K.
Compared with the prior art, the invention has the beneficial effects that:
1. the invention adopts the self-adaptive event triggering mechanism and the dynamic anti-saturation compensator in the system at the same time, designs a new self-adaptive event triggering condition delta 1 (t) =max { δ, η (t) }, the existing threshold function of the adaptive event trigger condition contains δ only 1 (t), while the adaptive event trigger threshold delta is employed in the present invention 1 The form of (t) =max { δ, η (t) } is such that the adaptive event triggers a threshold δ 1 (t) there is a lower bound delta, which allows the system to save more communication resources.
2. For the problem of saturation of an actuator in a system, a general static anti-saturation compensator is convenient to design, and can improve the performance of the saturated system to a certain extent, but often cannot meet the higher requirement of the actual system on the performance.
3. The self-adaptive event triggering control is based on the output of the system, which can be widely applied to the situation that the system state can not be completely measured, and the self-adaptive event triggering mechanism adopted by the invention can only transmit information when the self-adaptive event triggering condition is met, thereby ensuring the asymptotic stability of the system and saving communication resources.
4. By designing the free weight matrix phi in the self-adaptive event triggering condition 1 ，Φ 2 And coefficient matrix A of dynamic anti-saturation compensator aw 、B aw 、C aw1 、C aw2 、D aw1 、D aw2 And the like, the system can realize asymptotic stability. In addition, compared with the related event trigger mechanism with constant parameters, the self-adaptive event trigger mechanism can save network communication resources to a greater extent, and the introduction of the anti-saturation compensator greatly reduces the bad influence of the saturation of the actuator on the systemAnd (5) sounding.
Drawings
FIG. 1 is a block diagram of the system architecture of the present invention;
FIG. 2 is a system input of the present invention including a dynamic anti-saturation compensator;
FIG. 3 is a system output of the present invention including a dynamic anti-saturation compensator;
FIG. 4 is an adaptive event triggering interval incorporating a dynamic anti-saturation compensator of the present invention;
FIG. 5 is a system input of the present invention including a static anti-saturation compensator;
FIG. 6 is a system output of the present invention including a static anti-saturation compensator;
FIG. 7 is an adaptive event triggering interval including a static anti-saturation compensator of the present invention;
FIG. 8 is a system state diagram of the present invention including a dynamic anti-saturation compensator;
FIG. 9 is a system state diagram of the invention including a static anti-saturation compensator;
FIG. 10 is a graph of adaptive event triggering condition parameters over time with a dynamic anti-saturation compensator;
FIG. 11 is a graph of adaptive event triggering condition parameters over time with a static anti-saturation compensator;
fig. 12 is an attraction domain of a system estimate with dynamic anti-saturation compensators, static anti-saturation compensators, and without anti-saturation compensators under adaptive event-triggered control.
Detailed Description
The following describes the technical scheme of the present invention in detail with reference to specific embodiments and drawings, but is not intended to limit the scope of protection of the present application.
The invention relates to a self-adaptive event triggering control method of a system with a dynamic anti-saturation compensator, which comprises the steps of firstly establishing a system model with the dynamic anti-saturation compensator, then designing self-adaptive event triggering conditions, and updating signals received by a dynamic controller only when the self-adaptive event triggering conditions are met; then, a system model under the control of self-adaptive event triggering is established, and under the condition of a given dynamic anti-saturation compensator, self-adaptive event triggering conditions are designed to enable a saturation system to be asymptotically stable; in the case that the dynamic anti-saturation compensator is not given, the dynamic anti-saturation compensator and the self-adaptive event triggering condition are designed at the same time; finally, the system is proved not to have Zeno phenomenon under the designed self-adaptive event trigger control; the method comprises the following specific steps:
firstly, establishing a system model containing a dynamic anti-saturation compensator:
1-1, establishing a continuous system with actuator saturation as shown in formula (1):
wherein t represents the time, and the time,
is a state vector of a continuous system, n p 、/>
Respectively state vector x p Dimension and first derivative of (t),>
is the output of the dynamic controller, m is the dimension, < >>
Is the p-dimensional output vector of the controlled object, A p 、B p 、C p Are coefficient matrices, sat (u (t))= [ sat (u)) = [ sat (u) 1 (t))，…，sat(u m (t))] T Representing a vector saturation function, T representing a matrix transpose; furthermore, the controlled object is considered to be controllably measurable;
each component in the vector saturation function satisfies equation (2):
wherein alpha is i ＞0,i＝1,...,m，u i (t) represents the system ith dimension control input, α i Is the absolute value of the saturation bound of the i-th dimensional control input;
1-2, establishing a dynamic controller of formula (3):
wherein, the liquid crystal display device comprises a liquid crystal display device,
representing a state vector of the dynamic controller, n c 、/>
Respectively state vector x c Dimension and first derivative of (t),>
representing the output of a dynamic anti-saturation compensator, A c 、B c 、C c 、D c Are coefficient matrixes;
1-3, building a dynamic anti-saturation compensator of the formula (4):
wherein, the liquid crystal display device comprises a liquid crystal display device,
is the state vector of the dynamic anti-saturation compensator, n aw 、/>
Respectively state vector x aw The dimensions and first derivatives of (t); phi (t) =sat (u (t)) -u (t) represents a dead zone function, which is the input of the dynamic anti-saturation compensator; a is that aw 、B aw 、C aw1 、C aw2 、D aw1 And D aw2 All are coefficient matrixes, and the system is asymptotically stabilized by designing the matrixes;
secondly, designing self-adaptive event triggering conditions:
in order to save network communication resources, the invention provides a self-adaptive event triggering mechanism, and only when the self-adaptive event triggering condition is met, the signal received by the dynamic controller is updated; definition of dynamic error e (t) =y p (t k )-y p (t)，y p (t k ) Indicating the transmission time t k The data successfully transmitted to the dynamic controller designs the self-adaptive event triggering conditions as follows:
e(t) T Φ 1 e(t)≤δ 1 (t)y p (t) T Φ 2 y p (t)+δ 1 (t)∈e -∈t (5)
wherein phi is 1 、
For the free weight matrix, e is a positive scalar; delta 1 And (t) is an adaptive event trigger threshold, can be adaptively adjusted according to the dynamic performance of the system, and meets the condition: delta 1 (t)＝max{δ,η(t)}，δ∈(0,1]Representing the lower bound of the adaptive event trigger threshold, η (t) represents the threshold function, initial value η (0) > 0 and η (t) satisfies equation (6):
wherein, the liquid crystal display device comprises a liquid crystal display device,
representing the first derivative of the threshold function eta (t), theta representing the adjustment parameter of the convergence rate of the threshold function eta (t), theta > 1;
thirdly, establishing a system model under the control of self-adaptive event triggering:
after the self-adaptive event triggering mechanism is introduced, the dynamic controller is as follows:
3-1, without considering the dynamic anti-saturation compensator, building a system model of formula (8):
wherein, the liquid crystal display device comprises a liquid crystal display device,
is x cl First derivative of (t), x cl (t) is a state vector of the system without consideration of the dynamic anti-saturation compensator; a is that cl 、B φcl 、B ecl 、C ucl 、D uecl 、C ycl Are coefficient matrixes; wherein (1)>
Then there is the following equation:
3-2, under the condition of considering the dynamic anti-saturation compensator, establishing a system model of the formula (9):
where x (t) is the state vector of the system when considering the dynamic anti-saturation compensator,
is the first derivative of x (t); A. b (B) φ 、B e 、C u 、D uφ 、D ue 、C y Are coefficient matrixes; wherein (1)>
Then there isThe following equation:
wherein, the liquid crystal display device comprises a liquid crystal display device,
step four, under the condition of a given dynamic anti-saturation compensator, designing a self-adaptive event triggering condition, and ensuring the asymptotic stability of the system in the formula (9); constructing an optimization problem, and maximizing an estimation system attraction domain by solving the optimization problem;
4-1, defining the lyapunov function:
wherein P is a positive definite matrix;
case 1: when delta < eta (t), delta 1 (t) =max { δ, η (t) } =η (t), δ will be 1 Substitution of (t) =η (t) into equation (5) yields an adaptive event trigger condition of: e (t) T Φ 1 e(t)≤η(t)y p (t) T Φ 2 y p (t)+η(t)∈e -∈t ；
The following inequality can be obtained from the sector condition, J being an arbitrary diagonal positive definite matrix;
φ(t) T J(φ(t)+u(t)+Gx(t))≤0 (11)
deriving the Lyapunov function of formula (10), and further transforming by combining formula (11) can obtain:
wherein the method comprises the steps of
To ensure asymptotically stable systems of formula (9), it is necessary to satisfy:
case 2: when delta is larger than or equal to eta (t), delta 1 (t) =max { δ, η (t) } =δ, δ will be δ 1 (t) =δ is substituted into equation (5), resulting in an adaptive event trigger condition of: e (t) T Φ 1 e(t)≤δy p (t) T Φ 2 y p (t)+δ∈e -∈t ；
To ensure asymptotically stable system of equation (9), the derivative of the Lyapunov function
The following should be satisfied:
that is, the requirement
Obviously, formula (12) is a sufficient condition for formula (13), so when formula (12) is satisfied, for case 2, asymptotic stability of the system can still be ensured;
for the left and right co-multiplier matrices diag { Q, R, I, I } of inequality (12), let Q=P -1 ,R＝J -1 ,N＝GQ T The following inequality can be obtained:
wherein if there is a free weight matrix Φ 1 、
Positive definite matrix
And diagonal positive matrix ++>
Satisfy inequality (14), then the system of formula (9) satisfies
The asymptotic stability of the system is realized;
making x (t) belong to a set according to sector conditions
G i Representing the ith row of matrix G, i.e. ellipse +.>
Attracting domain for the system to be estimated, +.>
Can also be expressed as +.>
Further transformations may yield formula (15):
wherein I is an identity matrix, q=p -1 ；N i Is the ith row of matrix N;
4-2, constructing an optimization problem of formula (16), and obtaining a system estimated attraction domain epsilon (P, 1) by solving the optimization problem;
wherein inf represents minimization, ρ is a positive scalar introduced to maximize the estimation system attraction domain, s.t. represents constraint; matrix array
A matrix introduced to maximize the estimation system attraction domain;
fifthly, under the condition that the dynamic anti-saturation compensator is not given, designing the anti-saturation compensator and the self-adaptive event triggering condition at the same time, and ensuring the asymptotic stability of the system; constructing an optimization problem, and maximizing an estimation system attraction domain by solving the optimization problem;
according to the matrix factorization theorem, if the following set of inequalities is satisfied:
rank represents the rank of the matrix;
then the positive definite matrix Q may be decomposed into:
order the
Multiplying the set of inequalities (17) by the matrix L and the matrix Y to obtain
Further, according to the projection theorem, the inequality (15) can be expressed as follows:
Ψ+F T ΛH+H T ΛF＜0 (19)
let n= [ N1N 2 ]]，
Substituting the decomposed matrix Q, i.e., equation (18), into inequality (15) can yield each matrix in equation (19):
wherein, the liquid crystal display device comprises a liquid crystal display device,
according to the projection theorem, equation (19) is equivalent to:
wherein W is F And W is H Is a matrix composed of the basis vectors of the null spaces of matrices F and H, respectively, and is therefore obtainable according to equation (20):
further, substituting the formula (20) and the formula (22) into the inequality group (21) can obtain the following formula:
further, matrix N in formula (15) i ＝[N1 i N2 i ]＝[N1 i N1 i Y 1 -1 Y 12 N1 i Y 1 -1 U 1 T ]The formula (15) can be expressed as follows:
by further transformation it is possible to obtain:
substitution of formula (18) into formula (23) yields:
further get alpha i 2 I-N1 i Y 1 -1 N1 i T And (3) not less than 0, and reusing Schur complement to obtain:
further, the constraint in the problem equation (16) is optimized
The constraint is written as:
ρK-Q -1 ≥0 (25)
order the
Thus formula (25) can be expressed as:
let ρK be 12 ＝V T ,ρK 21 ＝V,
Then equation (25) may be further expressed as ρK 11 -L -1 Not less than 0; thus, when the dynamic anti-saturation compensator is not given, the optimization problem of design formula (27) can achieve asymptotic stabilization of the system;
but the inequality rank (Y-L). Ltoreq.n in formula (27) aw Is nonlinear, which is detrimental to further calculations, thus n is considered aw ≥n p And n aw Two more widely used cases =0;
when n is aw ≥n p At the time, let Y 12 ＝L 12 ,Y 22 ＝L 22 Then
let U T U＝YL -1 Y-Y，W＝I+UY -1 U T ，N a ＝[N1 N1Y 1 -1 Y 12 ]，
The free weight matrix phi in the adaptive event triggering condition of the formula (5) can be solved by the formula (28) 1 、Φ 2 Then, obtaining a coefficient matrix of the dynamic anti-saturation compensator through solving a formula (29);
wherein, the liquid crystal display device comprises a liquid crystal display device,
solving the set of inequalities of equation (29) to obtain the coefficient matrices of the dynamic anti-saturation compensator of equation (4) as follows:
up to now at n aw ≥n p In the process, the design of a dynamic anti-saturation compensator and a self-adaptive event triggering condition is completed, and the asymptotic stability of the system is realized;
when n is aw At=0, the anti-saturation compensator is static, in which case y=l should be satisfied, so the following optimization problem is designed:
let q=y, u=w=0, Φ 1 ，Φ 2 The coefficient matrix of the anti-saturation compensator can be obtained through the method (30) and then the method (31);
wherein, the liquid crystal display device comprises a liquid crystal display device,
if positive definite matrix +.>
And positive scaling ρ satisfies equation (31), then the system of equation (9) may be asymptotically stable;
solving the inequality group of (31) to obtain the coefficient matrix of the anti-saturation compensator as
Up to now finish n aw The design of the anti-saturation compensator and the self-adaptive event triggering condition when the system is=0 realizes the asymptotic stabilization of the system;
step six, calculating the minimum event triggering interval, and proving that the system with the dynamic anti-saturation compensator cannot generate Zeno phenomenon under the self-adaptive event triggering control;
at time t e [ t ] k ,t k+1 ) In this, the dynamic error e (t) is derived to obtain:
let a= |λ max (A p )|，b(t k )＝|λ max (A p )|·‖y p (t k )‖+‖C p ‖·‖B p II, solving the formula (32):
wherein lambda is max (A p ) Representation matrix A p Is the largest feature root of (1);
and because of the adaptive event triggering condition e (t) T Φ 1 e(t)≤δ 1 (t)y p (t) T Φ 2 y p (t)+δ 1 (t)∈e -∈t Can be defined by e (t) T Φ 1 e(t)≤δy p (t) T Φ 2 y p (t)+δ∈e -∈t Is guaranteed, so that it is possible to obtain:
wherein lambda is 1 ＝λ min (Φ 1 ),λ 2 ＝λ max (Φ 2 )，
Is->
Lower bound of (2);
from equations (33) and (34), a minimum trigger event interval can thus be obtained as:
wherein t is k ,t k+1 Respectively representing the time when the k-th and k+1th event triggers occur;
from formula (35), T (T) k ,t k+1 ) > 0, indicating that the system will not appear Zeno under adaptive event-triggered control (5).
Examples
The system structure diagram of the self-adaptive event trigger control method with the dynamic anti-saturation compensator system is shown in figure 1, and the system comprises a dynamic controller, an executor, a controlled object, an event generator, the dynamic anti-saturation compensator and a zero-order retainer; the specific steps of the embodiment are as follows:
firstly, establishing a system model containing a dynamic anti-saturation compensator; the coefficient matrix of the continuous system model with actuator saturation is designed as follows:
assuming that the saturation limit alpha=0.3 of the actuator, on the basis of ensuring the stability of the system, the coefficient matrix of the dynamic controller is designed as follows:
C c ＝[0.0091 0.0438]，D c ＝1.5933；
secondly, designing a self-adaptive event triggering mechanism;
in order to save network communication resources, the invention provides an adaptive event trigger mechanism, and a signal received by a dynamic controller is updated only when an adaptive event trigger condition is met, and a dynamic error e (t) =y is defined p (t k )-y p (t)，y p (t k ) Indicating the transmission time t k The data successfully transmitted to the controller designs the self-adaptive event triggering conditions as follows:
e(t) T Φ 1 e(t)≤δ 1 (t)y p (t) T Φ 2 y p (t)+δ 1 (t)∈e -∈t (5)
wherein phi is 1 、
For the free weight matrix, e is a positive scalar; delta 1 And (t) is an adaptive event trigger threshold, can be adaptively adjusted according to the dynamic performance of the system, and meets the condition: delta 1 (t)＝max{δ,η(t)}，δ∈(0,1]Eta (t) represents a threshold function, the initial value eta (0) > 0 and eta (t) satisfies the formula (6):
wherein, the liquid crystal display device comprises a liquid crystal display device,
represents the first derivative of the threshold function η (t), θ > 1 and θ represents the adjustment parameter of the convergence rate of the threshold function η (t);
thirdly, establishing a system model under the triggering control of the self-adaptive event;
step four, under the condition of a given dynamic anti-saturation compensator, designing a self-adaptive event triggering condition, ensuring the asymptotic stability of the system in the formula (9), and maximizing the suction domain estimated by the system by solving an optimization problem;
fifthly, under the condition that the dynamic anti-saturation compensator is not given, designing the anti-saturation compensator and the self-adaptive event triggering condition at the same time, and ensuring the asymptotic stability of the system; maximizing the attraction domain of the system estimation by solving an optimization problem;
let θ=1.2, δ 2 =0.1, e=0.2, δ=0.99, η (0) =1; when n is aw ≥n p When solving the inequality group of the formula (29):
N 1 ＝[-0.2581 0.0917]，Φ 1 ＝71.8853，Φ 2 ＝10.8463，
N＝[-0.2581 0.0917 16.7638 -3.2489 0.3135 0.0780]
when n is aw When=0, solving the inequality group of the formula (31) results in:
Φ 1 ＝70.7461，Φ 2 ＝11.2614，N＝[-0.1275-0.0773-2.21130.1110]，
and step six, calculating the minimum event triggering interval, and proving that the system with the dynamic anti-saturation compensator cannot generate Zeno phenomenon under the designed self-adaptive event triggering control.
To verify the feasibility of the method, numerical simulation is carried out in a MATLAB environment, and simulation results are shown in FIGS. 2-12; FIGS. 2-4 depict system input, output and adaptive event trigger intervals with dynamic anti-saturation compensators, respectively, and FIGS. 5-7 depict system input, output and adaptive event trigger intervals with static anti-saturation compensators, respectively; it can be seen from the figure that the time for the input and output of the dynamic anti-saturation compensator to converge into the actuator linear region is significantly shorter than for a system with a static anti-saturation compensator, and that the Zeno phenomenon is avoided.
Table 1 statistics of the adjustment time and adaptive event trigger times for two anti-saturation compensators
| Anti-saturation compensator type | Adjusting the time | Adaptive event trigger times |
| Dynamic anti-saturation compensator | 2.93 seconds | 23 times |
| Static anti-saturation compensator | 5.68 seconds | 34 times |
Fig. 8 and 9 are system state diagrams including a dynamic anti-saturation compensator and a static anti-saturation compensator, respectively, and it can be seen from fig. 8 that the system state can converge to the origin, i.e., the asymptotic stability of the system is ensured, under the adaptive event triggering control proposed herein. In addition, the system state with the dynamic anti-saturation compensator of FIG. 4 has a smaller overshoot and a faster convergence speed than the system state with the static anti-saturation compensator of FIG. 5, which indicates that the dynamic anti-saturation compensator can improve system performance to a greater extent than the static anti-saturation compensator. Table 1 shows statistics of the adjustment time and the number of times of triggering of the adaptive event for two anti-saturation compensators, and the adjustment time and the number of times of triggering of the adaptive event for a system with a dynamic anti-saturation compensator are obviously less than those of a system with a static anti-saturation compensator, so that the system with the dynamic anti-saturation compensator has more advantages in the aspect of saving communication resources than the system with the static anti-saturation compensator.
Fig. 10 and 11 show the time-dependent changes in the parameters of the adaptive event triggering conditions for systems with dynamic anti-saturation compensators and with static anti-saturation compensators, respectively.
Fig. 12 depicts the attractive domain of a system with a dynamic anti-saturation compensator, a static anti-saturation compensator and no anti-saturation compensator under adaptive event-triggered control, the system with a dynamic anti-saturation compensator having a larger attractive domain than the system with a static anti-saturation compensator and no anti-saturation compensator, the larger the attractive domain is to illustrate the better system performance, therefore, under adaptive event-triggered control, the system with a dynamic anti-saturation compensator can both ensure system stability and effectively save communication resources.
TABLE 2 statistics of trigger times under different trigger mechanisms
As can be seen from Table 2, compared with the related event trigger mechanism with constant variable, the adaptive event trigger control provided by the invention has fewer trigger times and can save more communication resources.
The invention is applicable to the prior art where it is not described.
Claims (4)
1. An adaptive event triggering control method comprising a dynamic anti-saturation compensator system is characterized by comprising the following specific steps:
the method comprises the steps of firstly, establishing a system model containing a dynamic anti-saturation compensator, wherein the system model comprises a continuous system with an actuator for saturation, a dynamic controller and the dynamic anti-saturation compensator;
the second step, designing self-adaptive event triggering conditions is as follows:
e(t) T Φ 1 e(t)≤δ 1 (t)y p (t) T Φ 2 y p (t)+δ 1 (t)∈e -∈t (5)
wherein e (t) is a dynamic error, e (t) =y p (t k )-y p (t)，y p (t k ) Indicating the transmission time t k Data successfully transmitted to the dynamic controller, y p (t) is the output vector of the controlled object, Φ 1 、Φ 2 For the free weight matrix, e is a positive scalar, T represents the matrix transpose; delta 1 (t) is an adaptive event trigger threshold, and satisfies the condition: delta 1 (t)＝max{δ,η(t)}，δ∈(0,1]Representing the lower bound of the adaptive event trigger threshold, η (t) represents the threshold function, initial value η (0) > 0 and η (t) satisfies equation (6):
wherein, the liquid crystal display device comprises a liquid crystal display device,
representing the first derivative of the threshold function eta (t), theta representing the adjustment parameter of the convergence rate of the threshold function, theta > 1;
thirdly, establishing a system model under the triggering control of the self-adaptive event;
step four, under the condition of a given dynamic anti-saturation compensator, designing a self-adaptive event triggering condition, and ensuring the asymptotic stability of the system; constructing an optimization problem, and maximizing an attraction domain estimated by a system by solving the optimization problem;
when delta < eta (t), delta 1 (t) =η (t), substituting formula (5) yields an adaptive event triggering condition of: e (t) T Φ 1 e(t)≤η(t)y p (t) T Φ 2 y p (t)+η(t)∈e -∈t ；
When delta is larger than or equal to eta (t), delta 1 (t) =δ, substituting formula (5) yields an adaptive event triggering condition of: e (t) T Φ 1 e(t)≤δy p (t) T Φ 2 y p (t)+δ∈e -∈t ；
Fifthly, under the condition that the dynamic anti-saturation compensator is not given, designing the anti-saturation compensator and the self-adaptive event triggering condition at the same time, and ensuring the asymptotic stability of the system; constructing an optimization problem, and maximizing an estimation system attraction domain by solving the optimization problem;
and step six, calculating the minimum event triggering interval, and proving that the system with the dynamic anti-saturation compensator cannot generate Zeno phenomenon under the self-adaptive event triggering control of the design.
2. The method of adaptive event-triggered control with dynamic anti-saturation compensator system according to claim 1, wherein in a first step, the continuous system with actuator saturation is:
wherein t represents the time, and the time,
is a state vector of a continuous system, n p 、/>
Respectively state vector x p Dimension and first derivative of (t),>
is the output of the dynamic controller, m is the dimension, A p 、B p 、C p Are coefficient matrices, sat (u (t))= [ sat (u)) = [ sat (u) 1 (t))，…，sat(u m (t))] T Representing a vector saturation function;
the dynamic controller is as follows:
wherein, the liquid crystal display device comprises a liquid crystal display device,
representing a state vector of the dynamic controller, n c 、/>
Respectively state vector x c Dimension and first derivative of (t),>
representing the output of a dynamic anti-saturation compensator, A c 、B c 、C c 、D c Are coefficient matrixes;
the dynamic anti-saturation compensator is:
wherein, the liquid crystal display device comprises a liquid crystal display device,
is the state vector of the dynamic anti-saturation compensator, n aw 、/>
Respectively state vector x aw The dimensions and first derivatives of (t); phi (t) =sat (u (t)) -u (t) represents a dead zone function, which is the input of the dynamic anti-saturation compensator;
in the third step, after the self-adaptive event triggering mechanism is introduced, the dynamic controller is as follows:
without considering the dynamic anti-saturation compensator, the system model is:
wherein, the liquid crystal display device comprises a liquid crystal display device,
is x cl First derivative of (t), x cl (t) is a state vector of the system without consideration of the dynamic anti-saturation compensator; a is that cl 、B φcl 、B ecl 、C ucl 、D uecl 、C ycl Are coefficient matrixes;
considering the dynamic anti-saturation compensator, the system model is:
where x (t) is the state vector of the system when considering the dynamic anti-saturation compensator,
is the first derivative of x (t); A. b (B) φ 、B e 、C u 、D uφ 、D ue 、C y Are coefficient matrices.
3. The adaptive event-triggered control method comprising a dynamic anti-saturation compensator system according to claim 2, wherein in the fourth step, the optimization problem of the configuration is:
wherein ρ is a positive scalar introduced to maximize the estimation system attraction domain, Q is a positive definite matrix, R is a diagonal positive definite matrix, N i I is an identity matrix for the ith row of the matrix N, K is a matrix introduced to maximize the estimation system attraction domain, α i Is the absolute value of the saturation bound of the i-th dimension control input.
4. The adaptive event-triggered control method comprising a dynamic anti-saturation compensator system according to claim 2, wherein in the fifth step, the optimization problem of the configuration is:
wherein Y is 1 Is a sub-matrix of matrix Y, N1 is a sub-matrix of matrix N,
N1 i for matrix N i Is L is a matrix, K 11 For the sub-moment of matrix KAn array.
Priority Applications (1)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| CN202210000999.4A CN114296355B (en) | 2022-01-04 | 2022-01-04 | Self-adaptive event trigger control method containing dynamic anti-saturation compensator system |
Applications Claiming Priority (1)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| CN202210000999.4A CN114296355B (en) | 2022-01-04 | 2022-01-04 | Self-adaptive event trigger control method containing dynamic anti-saturation compensator system |
Publications (2)
| Publication Number | Publication Date |
|---|---|
| CN114296355A CN114296355A (en) | 2022-04-08 |
| CN114296355B true CN114296355B (en) | 2023-07-07 |
Family
ID=80975108
Family Applications (1)
| Application Number | Title | Priority Date | Filing Date |
|---|---|---|---|
| CN202210000999.4A Active CN114296355B (en) | 2022-01-04 | 2022-01-04 | Self-adaptive event trigger control method containing dynamic anti-saturation compensator system |
Country Status (1)
| Country | Link |
|---|---|
| CN (1) | CN114296355B (en) |
Citations (4)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| US5394322A (en) * | 1990-07-16 | 1995-02-28 | The Foxboro Company | Self-tuning controller that extracts process model characteristics |
| EP1950636A1 (en) * | 2007-01-23 | 2008-07-30 | General Electric Company | Multivariable controller design method for multiple input/output systems with multiple input/output constraints |
| CN103207568A (en) * | 2013-03-18 | 2013-07-17 | 哈尔滨工程大学 | Steering engine saturation resistant self-adaptive control method for ship courses |
| CN110456681A (en) * | 2019-07-01 | 2019-11-15 | 天津大学 | The output feedback controller of neutral stability saturation system based on event triggering |
-
2022
- 2022-01-04 CN CN202210000999.4A patent/CN114296355B/en active Active
Patent Citations (4)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| US5394322A (en) * | 1990-07-16 | 1995-02-28 | The Foxboro Company | Self-tuning controller that extracts process model characteristics |
| EP1950636A1 (en) * | 2007-01-23 | 2008-07-30 | General Electric Company | Multivariable controller design method for multiple input/output systems with multiple input/output constraints |
| CN103207568A (en) * | 2013-03-18 | 2013-07-17 | 哈尔滨工程大学 | Steering engine saturation resistant self-adaptive control method for ship courses |
| CN110456681A (en) * | 2019-07-01 | 2019-11-15 | 天津大学 | The output feedback controller of neutral stability saturation system based on event triggering |
Non-Patent Citations (1)
| Title |
|---|
| 基于扩张状态观测器的动态抗饱和补偿器设计方法;刘晨;董朝阳;王青;冉茂鹏;;控制与决策(第11期);全文 * |
Also Published As
| Publication number | Publication date |
|---|---|
| CN114296355A (en) | 2022-04-08 |
Similar Documents
| Publication | Publication Date | Title |
|---|---|---|
| Chen et al. | Adaptive synchronization of multiple uncertain coupled chaotic systems via sliding mode control | |
| Hu et al. | Special functions-based fixed-time estimation and stabilization for dynamic systems | |
| Hua et al. | Event-based dynamic output feedback adaptive fuzzy control for stochastic nonlinear systems | |
| Liu et al. | Adaptive tracking control for perturbed strict-feedback nonlinear systems based on optimized backstepping technique | |
| Zhu et al. | Adaptive synchronization of chaotic Cohen–Crossberg neural networks with mixed time delays | |
| Liu et al. | Adaptive fixed-time hierarchical sliding mode control for switched under-actuated systems with dead-zone constraints via event-triggered strategy | |
| He et al. | Quantized adaptive pinning control for fixed/preassigned-time cluster synchronization of multi-weighted complex networks with stochastic disturbances | |
| Yan et al. | $ H_ {\infty} $ weighted integral event-triggered synchronization of neural networks with mixed delays | |
| Wang et al. | Event-triggered adaptive containment control for heterogeneous stochastic nonlinear multiagent systems | |
| CN111221311B (en) | Complex network distributed pulse synchronization method and system based on parameter variational method | |
| Lü et al. | Robust adaptive estimation and tracking control for perturbed cyber-physical systems against denial of service | |
| CN114296355B (en) | Self-adaptive event trigger control method containing dynamic anti-saturation compensator system | |
| Chen et al. | Observer-based event-triggered consensus of leader-following linear multi-agent systems with input saturation and switching topologies | |
| Mi et al. | Fixed-time consensus tracking for multi-agent systems with a nonholomonic dynamics | |
| CN112131693B (en) | Lur' e network clustering synchronization method based on pulse containment adaptive control | |
| Chu et al. | Integrated event-triggered fault estimation and fault-tolerant control for discrete-time fuzzy systems with input quantization and incomplete measurements | |
| Rokhforoz et al. | Large-scale dynamic system optimization using dual decomposition method with approximate dynamic programming | |
| CN112731801B (en) | Symmetric dead zone nonlinear self-adaptive dynamic surface output feedback control method | |
| Zhou et al. | Nonperiodic intermittent control for pinning synchronization of directed dynamical networks with internal delay and hybrid coupling | |
| Feng et al. | Multiobjective H_2/H_∞ Control Design Subject to Multiplicative Input Dependent Noises | |
| Sui et al. | Finite-Time Adaptive Fuzzy Event-Triggered Consensus Control for High-Order MIMO Nonlinear MASs | |
| CN113225045B (en) | Sparse-facilitated affine projection adaptive filter with low computational complexity | |
| Ji et al. | Fixed-Time Synchronization for Different Dimensional Complex Network Systems with Unknown Parameters via Adaptive Control | |
| Wang et al. | Event-triggered H∞ control for networked TS fuzzy systems with time delay | |
| CN113777920B (en) | Fractional order chaotic synchronization control method based on RBF-NN and observer |
Legal Events
| Date | Code | Title | Description |
|---|---|---|---|
| PB01 | Publication | ||
| PB01 | Publication | ||
| SE01 | Entry into force of request for substantive examination | ||
| SE01 | Entry into force of request for substantive examination | ||
| GR01 | Patent grant | ||
| GR01 | Patent grant |