CN108983616A - A kind of nonuniform sampling systems by output feedback control method and controller based on switching principle - Google Patents
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Abstract
The present invention relates to intelligent control algorithms, more particularly to a kind of nonuniform sampling systems by output feedback control method and controller based on switching principle, in nonuniform sampling system, the non-homogeneous refreshing of input signal, output signal are uniform samplings within a frame period.The present invention handles refresh time uncertain problem using a kind of switching system method, it is higher than the strategy of sensor sample frequency using the frequency that non-homogeneous retainer reads data, regard refresh interval as time delay variable, system dynamic change under different delay size cases is portrayed with different sub-systems model, to which variable when index is decomposed into multiple permanent items, the switching law between different submodels is converted by the variation rule of time delay, it is finally that nonuniform sampling is system converting for a kind of discrete time switched systems for containing several limited subsystems.The present invention solves the design challenges of system controller in the case that nonuniform sampling system refresh time interval is not known with time-varying.
Description
Technical Field
The invention relates to an intelligent control algorithm, in particular to a non-uniform sampling system output feedback control method and a controller based on a switching principle.
Background
With the development of modern industry, advanced computer technology, control technology and communication technology are adopted, and in the 3C fusion era, more and more systems adopt various operating frequencies, which are called multi-rate sampling data systems, and are called multi-rate systems for short. With the development of the current generation of industrial production, there is also a more common type of multi-sampling rate system, i.e., a non-uniform sampling data system in which input signal refresh and/or output signal sampling exhibit unequal time intervals. For example, in the process industries of petroleum, electric power, metallurgy, chemical industry, food, light textile and the like, due to the high price of equipment or the lack of on-line detection, poor reliability, difficult maintenance and the like, some key variables and parameters (such as product components, concentration, Kappa value, melt index and the like) reflecting the quality level of a product can only be obtained through manual sampling and laboratory test analysis, so that the sampling period is long and the sampling time interval is irregular. In a network control system, in order to overcome the negative influence of redundant data on the system performance, a sensor generally adopts a time-driven working mode, the sampling time interval is uniformly changed, and an actuator generally adopts an event-driven working mode, the sampling time interval is random and uncertain. Meanwhile, due to the limitations of transmission distance, network carrying capacity and communication bandwidth, the data cannot avoid the phenomena of time delay, packet loss, disorder and the like in the transmission process, so that the actual sampling frequency is non-uniform.
The non-uniform periodic sampling data system generally has the following characteristics: the measured data is obtained at different sampling moments and different sampling frequencies; controlling the input signal and the output signal to present the characteristics of non-uniformity, non-synchronization, uncertain refreshing time and sampling interval in time sequence; due to certain resource limitation (caused by the bandwidth limitation of a communication network, the measurement limitation of an intelligent instrument, the manual detection time limitation, the calculation capacity limitation and the like), the input signal is delayed; therefore, how to overcome the time variability and uncertainty of the refresh time interval of the input signal in the non-uniform sampling system causes great difficulty in designing the system controller.
Disclosure of Invention
The invention aims to overcome the defects and provides a non-uniform sampling system output feedback control method and a controller based on a switching principle.
In a non-uniform sampling system, the input signal is non-uniformly refreshed and the output signal is uniformly sampled within one frame period, where the input signal refresh interval tends to be time-varying and indeterminate. Therefore, the invention provides a switching system method for processing the problem of uncertainty of refreshing time, which adopts a strategy that the frequency of reading data by a non-uniform retainer is higher than the sampling frequency of a sensor, refreshing time intervals are taken as time delay variables, and system dynamic changes under the condition of different time delay sizes are depicted by different subsystem models, so that an exponential time-varying term is decomposed into a plurality of constant terms, the change law of time delay is converted into a switching law between different subsystem models, and finally, a non-uniform sampling system is converted into a discrete time switching system containing a plurality of limited subsystems. On the basis, the output feedback controller can ensure that the non-uniform sampling system index is stable by using a switching principle.
The technical scheme of the invention is as follows: a non-uniform sampling system output feedback control method based on a switching principle comprises the following steps:
step 1, setting a dynamic process model of a non-uniform sampling system, ScIs a controlled object which is at kT + tikThe moment of reception of a computer-generated discrete input signal u (kT + t)ik) 1,2, p, via a zero-order keeper HΓGenerating a continuous signal u (t) as the controlled object ScThe output y (T) is sampled uniformly with a period T, i.e. within [ kT, kT + T), y (T) ═ y (kT);
characteristics of the zero-order keeper: refresh time interval of tauik(τik:＝tik-t(i-1)k)，tik:＝τ1k+τ2k+...+τik(let t)0k＝0,tpk＝T)，T:＝τ1k+τ2k+...+τpk＝tpkIs the frame period; in a non-uniform sampling system, where the refresh time interval is time-varying and uncertain, the input signal u (t) can be expressed as:
step 2, processing the non-uniform sampling process in the step 1:
1) in a practical non-uniform sampling systemThe input signal is sometimes ductile, i.e. the current input signal and the past history signal are correlated, i.e. u (kT + t)ik)＝u(kT-dT),i＝1,2,d＝0,1,2,...；
2) Taking d as 0,1, i.e.
Wherein tau isk＝t1k；τkIs the refresh time, also represents the time delay (delay for short);
3) the frame period T may be a minimum period T0Of composition, i.e. T0T/N; refresh period inherent to the zeroth order keeper, the refresh period being T0；
4) In the frame period, the input signal is refreshed non-uniformly twice, namely p is 2, and the refreshing time interval is time-varying and uncertain;
the controlled object is described with a lower model,
wherein x (t) ∈ RnIs a state vector, RnA real number vector representing n dimensions, the same as below; u (t) ∈ RmIs a control input; a. thec、BcAnd CcState matrix, input matrix and output matrix with proper dimension;
step 3, controlling the signal u (k) to pass through tauk＝t1After a time delay of-kT, the zero-order keeper reaches the end of a zero-order keeper, and the zero-order keeper is at kT +3T0Reading u (k) at a time, updating the input u (k-1) to u (k), and for any k > 0, recording the time of u (k) and u (k-1) acting on the controlled object as n0(k)T0And n1(k)T0Then there is
n1(k) The value of (a) reflects the size of the time delay;
step 4, discretizing the formula (3) by a sampling period T, and considering the time delay condition to obtain the time delay method
Wherein,
the two input matrices of the discrete model (4) depend on n1(k) And n0(k) When n is the size of1(k) And n0(k) When taking different values, the system (4) will take different forms, and n1(k) And n0(k) All values are in a limited set, therefore, the formula (4) can be regarded as a switching system containing limited subsystems, and R is introduced2Mapping of → R: [ n ] of1(k) n0(k)]→ σ (k), where σ (k) is a non-negative integer; can be expressed as:
[n1(k) n0(k)]＝[0 N]→0
[n1(k) n0(k)]＝[1 N-1]→1
……
[n1(k) n0(k)]＝[N 0]→N
the system (4) can be rewritten to a switched system model,
wherein,σ(k)∈F0is a switching signal, equation (5) is a switching system with N +1 subsystems, with SjJ < th > subsystem represented by (5), j ∈ F0Then σ (k) ═ j indicates that the system (4) resides in the subsystem SjIn the above-mentioned manner,
step 5, consider the controller as an output feedback controller, i.e.
Wherein C isfIn the case of a state feedback matrix, the following closed-loop system model can be obtained by substituting equation (6) into equation (5),
wherein,σ(k)∈F0is a switching signal, at a time delay of taukWhen changed, the system (7) meets the average residence time switching rate, namely the average residence time taIs greater than(mu, lambda is a given constant), that is to say the residence time of the system at each subsystem is longer thanIn case of switching the system (7) from one subsystem to another subsystem, switching between system submodels is realized, and then the control design of the non-uniform sampling system can be completed by means of a switching system control principle.
The invention also provides a non-uniform sampling system output feedback controller based on the switching principle, which is designed by the method, and the output feedback controller comprises the following components:
wherein,σ(k)∈F0is a switching signal.
The invention has the beneficial effects that:
1. the refresh time interval of the non-uniform sampling system is regarded as a time delay variable, the dynamic change of the system under the condition of different time delay sizes is described by different subsystem models, an exponential time-varying term is decomposed into a plurality of constant terms, the change law of the time delay is converted into the switching law between different submodels, the non-uniform sampling system is finally described as a discrete time switching system with a limited subsystem, and the problem of difficult design of a controller caused by uncertain refresh time intervals is solved.
2. Under the condition of meeting the average residence time switching law, the non-uniform sampling system has stable index and good simulation example effect.
3. The overall refreshing condition of the non-uniform sampling system is considered, the non-uniform sampling system can be regarded as the switching of a plurality of subsystems, and the stability of the whole switching system under the rapid control is ensured under the Lyapunov theorem.
Drawings
FIG. 1 is a process diagram of a non-uniform multi-rate sampling system;
FIG. 2 is a diagram of zero order keeper characteristics;
FIG. 3 is a timing diagram of control signals;
FIG. 4 is a schematic diagram of a non-uniform sampling system switching process;
FIG. 5 is a non-uniform refresh interval in an example simulation;
FIG. 6 shows a system x in a simulation example1(k +1) output;
FIG. 7 shows a system x in a simulation example2And (k +1) output.
Detailed Description
The invention will be further illustrated with reference to the following specific examples. It should be understood that these examples are for illustrative purposes only and are not intended to limit the scope of the present invention. Further, it should be understood that various modifications and changes may be made by those skilled in the art after reading the teachings of the present invention, and such equivalents may fall within the scope of the invention as defined in the claims appended hereto.
Example 1
FIG. 1 shows a non-uniform sampling system dynamic process, where ScIs a controlled object which is at kT + tikThe moment of reception of a computer-generated discrete input signal u (kT + t)ik) 1,2, p, via a zero-order keeper HΓGenerating a continuous signal u (t) as the controlled object ScInput and output ofy(T) are uniformly sampled with a period T, i.e., [ kT, kT + T),y(t)＝y(kT)。
FIG. 2 shows the behavior of a zero order keeper with a refresh interval of τik(τik:＝tik-t(i-1)k)，tik:＝τ1k+τ2k+...+τik(let t)0k＝0,tpk＝T)，T:＝τ1k+τ2k+...+τpk＝tpkIs a frameShelf period. In the study, in the non-uniform sampling system, the refresh time interval is time-varying and uncertain, and the input signal u (t) can be expressed as:
according to the actual situation, the non-uniform sampling process is simplified:
1) in practical non-uniform sampling systems, the input signal is typically somewhat ductile, i.e., the current input signal and the past historical signal are correlated, i.e., u (kT + t)ik)＝u(kT-dT),i＝1,2,d＝0,1,2,...。
2) Taking d as 0,1, i.e.
Wherein tau isk＝t1k。τkIs the refresh time and also represents the time lag (delay for short).
3) The frame period T may be a minimum period T0Of composition, i.e. T0T/N. Refresh period inherent to the zeroth order keeper, the refresh period being T0。
4) Within a frame period, the input signal is non-uniformly refreshed twice, i.e., p is 2, and the refresh time interval is time-varying and uncertain.
The four points are the conditions frequently encountered in an actual non-uniform sampling system, such as network delay in a network control system, measurement delay in soft measurement of an industrial process, measurement delay in key variables and parameters of the product quality level in industrial production, and the like.
The controlled object in figure 1 is described in a model,
wherein x (t) ∈ RnIs a state vector; u (t) ∈ RmIs a control input; a. thecAnd BcTwo matrices of appropriate dimensions. From the above assumptions, it is known that 0 ≦ τkT is less than or equal to T, and the control quantity u (kT) and u (kT-T) are abbreviated as u (k) and u (k-1) at most twice in any frame period (the variables are abbreviated as the same in the following description), namely u (k) and u (k-1) act on the controlled object, and the time sequence is shown in FIG. 3.
In FIG. 3, the current control signal is u (k) at the time of the passage of τk＝t1After a time delay of-kT, the zero-order keeper reaches the end of a zero-order keeper, and the zero-order keeper is at kT +3T0U (k) is read at time and the input u (k-1) is updated to u (k). For any k > 0, the time of u (k) and u (k-1) acting on the controlled object are respectively recorded as n0(k)T0And n1(k)T0Then there is
n1(k) The value of (a) reflects the size of the delay.
Discretizing the formula (3) by a sampling period T and considering the time delay condition to obtain the product
Wherein,
the two input matrices of the discrete model (4) depend on n1(k) And n0(k) The size of (2). When n is1(k) And n0(k) When taking different values, the system (4) will take different forms. And n is1(k) And n0(k) All values are in a limited set, therefore, the formula (4) can be regarded as a switching system containing limited subsystems, and R is introduced2Mapping of → R: [ n ] of1(k) n0(k)]→ σ (k), where σ (k) is a non-negative integer. Can be expressed as
[n1(k) n0(k)]＝[0 N]→0
[n1(k) n0(k)]＝[1 N-1]→1
……
[n1(k) n0(k)]＝[N 0]→N
The system (4) can be rewritten to a switched system model,
wherein,σ(k)∈F0is a switching signal. Equation (5) is a switching system having N +1 subsystems. With SjThe first to (5)jSubsystem, j ∈ F0Then σ (k) ═ j indicates that the system (4) resides in the subsystem SjThe above. The handover procedure can be represented by fig. 4.
Consider the controller as an output feedback controller, i.e.
By substituting formula (6) into formula (5), the following closed-loop system model can be obtained,
in the foregoing description, when the time delay τkWhen the system (7) is changed, the system is switched from one subsystem to another subsystem, and the switching between the system submodels is realized, so that the control design of the non-uniform sampling system can be completed by means of the control principle of the switching system.
Example 2 non-uniform sampling System stability analysis
Leading: (Properties of Schur complement) given a symmetric matrix A > 0, a symmetric matrix C > 0, and a matrix B, then A + BTCB < 0 is equivalent to
Or
Theorem: considering a closed-loop non-uniform sampling system (7), if a positive scalar λ (0 < λ ≦ 1), μ ≧ 1 and a symmetric matrix of appropriate dimensionAndso that the inequality and equality below hold,
the system (7) is exponentially stable and has an exponential decay rate
And (3) proving that: suppose k1，k2，……，kiRepresents the switching point of sigma (k) in the interval [0, k), and satisfies 0 < k1＜k2＜...＜ki< k. In the system (7), the subsystem model is
The Lyapunov function was chosen as follows,
wherein P isj＞0，Qj> 0 and Sj＞0，j∈F0And is a undetermined symmetrical positive definite matrix. Consider Vj(k) Edge subsystem ScjThe dynamic trajectory changes. It can be obtained from the formula (13),
the above formula can be arranged into
Wherein,
when phi isjIf < 0, it can be obtained from the formula (14)
Vj(k+1)-λ2Vj(k)≤0 (15)
Due to phijIs < 0, i.e.
This is true.
Order toThen formula (16) can
Is written as
From theory 1, equation (17) can be transformed into
From theory 1, the above formula can be transformed into
Upper type left and right rideCan be equivalent to
Order toThe above formula is equivalent to formula (8).
Further, the compound represented by the formula (15) can be used
Vj(k+1)≤λ2Vj(k) (19)
The formula (19) means Vj(k) J 0, 1.. times.n, decays exponentially along the trajectory of the respective subsystem, i.e. it is a linear function of the trajectory of the subsystem
On the other hand, because
From the formulae (20) and (21)
Since the state of the system (7) does not jump at the switching point, the equation (9) can be used
Taking jitter bound N0When the average residence time is 0, the following equation (22) and (23) can be recursively obtained
Obtainable from formula (24)
β1||x(k)||2≤Vσ(k)(k)≤ρ(λ,ta)2kVσ(0)(0)≤ρ(λ,ta)2kβ2||x0||2(25)
Wherein,further obtainable from formula (25)
Wherein, | | x0And | is an initial value. Inequalities (11) and λ < 1 ensure ρ (λ, t)a) Is less than 1. Thus, the closed-loop non-uniform sampling system is exponentially stable and has an exponential decay rate ρ (λ, t)a). After the syndrome is confirmed.
Note: in practical applications, it is difficult to determine t if the time delay cannot be determinedaAnd thus it is difficult to apply the condition (11). Since ta is not less than 1, ifThe condition (11) is always true regardless of the delay variation and uncertainty. Thus, when the actual delay is unknown, it is usefulInstead of condition (11). At this time, the exponential decay rate ρ ═ λ μ of the closed-loop non-uniform sampling system (7)0.5。
Example 3 simulation example
Consider the following controlled object:
taking the sampling period as T-10 ms, dividing the sampling interval into ten equal parts, i.e. taking N as 10, and taking the period T of reading data by the zeroth-order keeper01 ms. The controller is taken as
The model of the closed-loop non-uniform sampling system can be described by equation (7). Assuming that the maximum delay upper bound tau is 4T0Therefore, the input signal delay is only possible to five values, i.e., τ (k) ∈ {0,1,2,3,4} ms, when the system (7) contains at most five subsystems. Using these five subsystems as SciAnd i is 0,1,2,3, 4. SciIs composed of n1(k) And n0(k) Determine, and [ n1(k)n0(k)]Is [ 010 ] respectively]，[1 9]，[2 8]，[3 7]，[4 6]. Defining a mapping [ n ]1(k) n0(k)]→σ(k)：
[0 10]→0，[1 9]→1，[2 8]→2，[3 7]→3，[4 6]→4
When the time delay tau (k) is 0, the corresponding closed-loop non-uniform sampling system model can be represented by Sc0Description is given; when the time delay tau (k) is epsilon (0T)0]When the internal change is carried out, the corresponding closed-loop non-uniform sampling system model can be represented by Sc1Description is given; and the other subsystems and so on. The minimum feasible linear matrix inequalities (8) and (9) can be obtained by a one-dimensional search algorithm by taking the step length of 0.03 as the step lengthμAnd λ is μ ═ 1.05 and λ ═ 0.96, respectively. The feasible solution problem solver feasp of the Matlab LMI toolbox is adopted, and the feasible solution is obtained by applying theorem 1 as follows:
since λ < μ-0.50.9759, satisfies the condition of note 1, i.e.This is true. Therefore, the corresponding closed-loop non-uniform sampling system can be stabilized by the output feedback controller, and the exponential decay rate of the system satisfies rho ═ lambda mu0.5＝0.9738。
In the simulation, the system initial value is set as x (0) [ 0.10.1]TThe simulation results are shown in fig. 5-7. Where fig. 5 shows a non-uniform refresh time interval or delay variation, it is clear that the uncertain refresh time interval is less than the upper delay bound of 4 ms. X is given in FIGS. 6 and 71(k+1)、x2The (k +1) case, it is clear that the system converges to zero, being asymptotically stable. The simulation results of fig. 5-7 show that the designed output feedback controller can exponentially stabilize the non-uniform sampling system under the condition of meeting the switching condition.
Claims (2)
1. A non-uniform sampling system output feedback control method based on a switching principle is characterized by comprising the following steps:
step 1, setting a dynamic process model of a non-uniform sampling system, ScIs a controlled object which is at kT + tikThe moment of reception of a computer-generated discrete input signal u (kT + t)ik) 1,2, p, via a zero-order keeper HΓGenerating a continuous signal u (t) as the controlled object ScThe output y (T) is sampled uniformly with a period T, i.e. within [ kT, kT + T), y (T) ═ y (kT);
characteristics of the zero-order keeper: refresh time interval of tauik(τik:＝tik-t(i-1)k)，tik:＝τ1k+τ2k+...+τik(let t)0k＝0,tpk＝T)，T:＝τ1k+τ2k+...+τpk＝tpkIs the frame period; in a non-uniform sampling system, where the refresh time interval is time-varying and uncertain, the input signal u (t) can be expressed as:
step 2, processing the non-uniform sampling process in the step 1:
1) in practical non-uniform sampling systems, the input signal is sometimes ductile, i.e., the current input signal and the past history signal are correlated, i.e., u (kT + t)ik)＝u(kT-dT),i＝1,2,d＝0,1,2,...；
2) Taking d as 0,1, i.e.
Wherein tau isk＝t1k，τkIs the refresh time, also represents the time delay (delay for short);
3) the frame period T may be a minimum period T0Of composition, i.e. T0T/N; refresh period inherent to the zeroth order keeper, the refresh period being T0；
4) In the frame period, the input signal is refreshed non-uniformly twice, namely p is 2, and the refreshing time interval is time-varying and uncertain;
the controlled object is described with a lower model,
wherein x (t) ∈ RnIs a state vector, RnA real number vector representing n dimensions, the same as below; u (t))∈RmIs a control input; a. thec、BcAnd CcState matrix, input matrix and output matrix with proper dimension;
step 3, controlling the signal u (k) to pass through tauk＝t1After a time delay of-kT, the zero-order keeper reaches the end of a zero-order keeper, and the zero-order keeper is at kT +3T0Reading u (k) at a time, updating the input u (k-1) to u (k), and for any k > 0, recording the time of u (k) and u (k-1) acting on the controlled object as n0(k)T0And n1(k)T0Then there is
n1(k) The value of (a) reflects the size of the time delay;
step 4, discretizing the formula (3) by a sampling period T, and considering the time delay condition to obtain the time delay method
Wherein,
the two input matrices of the discrete model (4) depend on n1(k) And n0(k) When n is the size of1(k) And n0(k) When taking different values, the system (4) will take different forms, and n1(k) And n0(k) All values are in a limited set, therefore, the formula (4) can be regarded as a switching system containing limited subsystems, and R is introduced2Mapping of → R: [ n ] of1(k)n0(k)]→ σ (k), where σ (k) is a non-negative integer; can be expressed as
[n1(k) n0(k)]＝[0 N]→0
[n1(k) n0(k)]＝[1 N-1]→1
……
[n1(k) n0(k)]＝[N 0]→N
The system (4) can be rewritten to a switched system model,
wherein,σ(k)∈F0is a switching signal, equation (5) is a switching system with N +1 subsystems, with SjJ < th > subsystem represented by (5), j ∈ F0Then σ (k) ═ j indicates that the system (4) resides in the subsystem SjThe above step (1);
step 5, consider the controller as an output feedback controller, i.e.
Wherein C isfIn the case of a state feedback matrix, the following closed-loop system model can be obtained by substituting equation (6) into equation (5),
wherein,σ(k)∈F0is a switching signal, at a time delay of taukWhen changed, the system (7) meets the average residence time switching rate, namely the average residence time taIs greater than(mu, lambda is a given constant), that is to say the residence time of the system at each subsystem is longer thanIn case the system (7) is switched from one subsystem to another subsystem, switching between system submodels is achieved, in which case it is possible to do soThe control design of the non-uniform sampling system is completed by means of a switching system control principle.
2. A non-uniform sampling system output feedback controller based on switching principle is characterized in that: designed using the method of claim 1, the output feedback controller is:
wherein,σ(k)∈F0is a switching signal.
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