CN110824913B - Non-fragile control method of periodic segmented time-varying system based on matrix polynomial - Google Patents

Non-fragile control method of periodic segmented time-varying system based on matrix polynomial Download PDF

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CN110824913B
CN110824913B CN201910798192.8A CN201910798192A CN110824913B CN 110824913 B CN110824913 B CN 110824913B CN 201910798192 A CN201910798192 A CN 201910798192A CN 110824913 B CN110824913 B CN 110824913B
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李攀硕
刘云
鲁仁全
徐雍
李鸿一
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Guangdong University of Technology
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Abstract

The invention discloses a non-fragile control method of a periodic segmented time-varying system based on a matrix polynomial, which aims to enable a designed controller to tolerate the influence caused by disturbance of an uncertain time-varying controller to enable a periodically switched subsystem to be a time-varying subsystem, and performs H on the continuous time periodic segmented time-varying system based on a Lyapunov function of a Lyapunov matrix polynomial with continuous time variation∞Analyzing the performance; then based on H∞Performance, considering controller additive and multiplicative disturbance with norm-bounded constraint, designs non-fragile H∞The controller enables the closed-loop system to be robust and stable and have good performance under the condition that the controller is disturbed within a certain range; according to the invention, due to the use of the matrix polynomial, higher dimensionality and more free variables are introduced, so that the designed controller is easy to solve, can be obtained by convex optimization directly based on LMI, does not need iteration, and is easy to implement in engineering.

Description

Non-fragile control method of periodic segmented time-varying system based on matrix polynomial
Technical Field
The invention relates to the technical field of periodic control, in particular to a non-fragile control method of a periodic segmented time-varying system based on a matrix polynomial.
Background
The periodic system is a dynamic system with periodic characteristics, and in practical application, the periodic system exists widely in the fields of engineering, nature, economy and the like. For example, vibration attenuation in helicopter rotor-blade systems, magnetic attitude control in spacecraft and satellites, bio-predation, and the like are frequently occurring in the fields of physical dynamics, economics, and finance, electrical power, biological systems, and the like. Due to the ubiquitous nature of time-varying periodic systems, the significance and value of research in analyzing their characteristics and control has been great, and since the last 50 centuries, a great deal of research has been devoted to periodic systems. However, due to its complex time-varying nature, many control problems are still difficult to thoroughly study. Until recently, the periodic segmented system approach was considered to be an effective approach to the periodic system and is widely used, and the periodic segmented system can not only approximate to the periodic time-varying system, but also has many applications in itself, such as periodic segmented voltages and currents applied in mechanical systems with ideal diodes, coulomb friction, switch mode DC-DC converters, and so on. Specifically, a dynamic system of the periodic system can be modeled by a periodic variation coefficient, wherein the periodic system under discrete time can be effectively processed by a Lifting technology, and the modeling of the continuous time periodic time-varying system is difficult to obtain a closed-loop form solution of a Floquet factor, so that the stability condition of the system is difficult to directly obtain and extend to a control problem based on a Lyapunov-Floquet theory. Therefore, a large number of methods such as approximation and conversion are started to study the periodic system, but the studies can only be carried out to a stability analysis part, and a controller cannot be directly designed, so that the problem of the periodic system cannot be solved effectively. This has not allowed the study of periodic systems to be effectively solved until it has been proposed to approximate a periodic system using a periodic piecewise system model. A periodic segmented system can also be considered a special case of a switching system, which consists of subsystems with a defined dwell time and a limited number of fixed switching sequences during a basic period. Each subsystem is approximated by an average model over the corresponding subinterval, or calculated directly based on the values of periodic constant variables. The research of the time-lag-free period segmented system is widely concerned, and has good results in the aspects of stability analysis and controller synthesis.
However, in practical engineering applications, components are aged due to the influence of physical characteristics of industrial equipment and control components or environmental factors, so that parameters of the controller are changed within a certain range, namely gain disturbance of the uncertain controller occurs. Small perturbations in the controller parameters are likely to cause poor system performance or even damage to the system. These phenomena are widespread, for example, the inherent inaccuracy of analog-to-digital conversion, limited word size, actuator degradation, rounding errors in numerical calculations and adjustments of parameters during controller implementation can lead to control inaccuracies. Therefore, when controller disturbance occurs, the performance and stability of the system cannot be guaranteed by the current periodic segmented time-varying controller.
Disclosure of Invention
The invention aims to overcome the defects of the prior art and provide a non-fragile control method of a periodic segmented time-varying system based on a matrix polynomial, which can ensure that a closed-loop system can still keep good system performance when the gain disturbance of a controller with uncertain parameters occurs, and has good robustness and non-fragility and great engineering application value.
The purpose of the invention is realized by the following technical scheme:
a non-fragile control method of a matrix polynomial based periodically segmented time varying system, comprising the steps of:
s1, first consider a continuous time period piecewise time varying system, the state space expression of which is shown in the following formula:
Figure BDA0002181547520000031
z(t)＝Ci(t)x(t)+Di(t)u(t)+Dwi(t)w(t)；
s2, the continuous time period segments the time-varying system matrix Ai(t),Bi(t),Bwi(t),Ci(t),Di(t),Dwi(T) satisfies the periodicity, the fundamental period being TpAnd are all given by time-varying linear interpolation formulas, as shown in the following formulas:
Figure BDA0002181547520000032
wherein A isi,Bi,Bwi,Ci,Di,DwiI is 0, 1.. times.i is 1, 2.. times.s, which is a constant matrix, Ti＝ti-ti-1Is the residence time;
s3, constructing a Lyapunov function with continuous time-varying matrix polynomial according to the Lyapunov stability criterion, wherein the Lyapunov function is in the form of V (t) Vi(t)＝xT(t)Pi(t) x (t), wherein the Lyapunov matrix polynomial Pi(t) is:
Figure BDA0002181547520000033
s4, utilizing Lyapunov stability criterion to obtain sufficient conditions of the stability of the continuous time period segmented time-varying system;
s5, analyzing and deducing the negative qualitative general theorem of the matrix polynomial based on the Lyapunov function with the continuous time-varying matrix polynomial, and carrying out inequality scaling by combining the Coppel inequality to obtain H∞The sufficient condition of the performance is satisfied, and the following conditions are satisfied:
Figure BDA0002181547520000041
wherein the content of the first and second substances,
Figure BDA0002181547520000042
s6, obtaining H in S5∞Designing corresponding non-fragile H on the basis of the performance index∞A controller; two disturbance forms of additive and multiplicative gains of the controller are considered:
norm-constrained additive perturbation form: Δ Ki(t)＝MFi(t)Ni(t),i＝1,2,...,S,
Norm-constrained multiplicative perturbation form:
Figure BDA0002181547520000043
wherein the content of the first and second substances,
Figure BDA0002181547520000044
is time-varying, defined as:
Figure BDA0002181547520000045
Figure BDA0002181547520000046
and N isi,Ni+1,
Figure BDA0002181547520000047
A constant matrix of appropriate dimensions, M a constant matrix of known appropriate dimensions, f (t) satisfying a norm constraint: fi(t)Fi(t)TI is less than or equal to I, and t is more than or equal to 0; wherein the disturbance parameter Ni(t),
Figure BDA0002181547520000048
Is time-varying rather than stationary, which results in a time-varying disturbance matrix Δ Ki(t) becomes more flexible, so that the problem under consideration can be guaranteed to have a rich disturbance dynamic; the new closed loop system under disturbance is then regained:
Figure BDA0002181547520000049
z(t)＝Cci(t)x(t)+Dwi(t)w(t)；
wherein A isci(t)＝Ai(t)+Bi(t)(Ki(t)+ΔKi(t)),Cci(t)＝Ci(t)+Di(t)(Ki(t)+ΔKi(t))；
S7, aiming at the closed-loop system obtained in S6, based on the Lyapunov function with continuous time-varying matrix polynomial, utilizing an LMI method, combining the negative qualitative and positive qualitative general lemma conditions of the matrix polynomial, and then introducing a matrix polynomial variable Qi(t),Wi(t) carrying out variable replacement, using Schur's complementary theorem, scaling by inequality, and finally obtaining non-fragile H under the condition of additive and multiplicative controller gain disturbance∞A controller, the designed control being solved by convex optimization;
s8, considering the time-varying controller gain delta Ki(t) ≡ 0, the nominal H without controller disturbance is designed∞A controller; and separately considering the introduction of continuous and discontinuous polynomial matrix variables Qi(t), and a continuous polynomial matrix variable Wi(t), deriving and obtaining the nominal H under two conditions by using methods such as variable replacement, Schur's complement theorem, inequality scaling and the like∞Controller, controller gain Ki(t)＝Qi(t)Wi -1(t), which can be obtained by convex optimization solution;
and S9, carrying out Matlab numerical simulation verification on the obtained controller:
(1) under the condition of norm bounded additive disturbance, selecting disturbance parameters as follows:
Figure BDA0002181547520000051
(2) under the condition of norm bounded additive disturbance, selecting disturbance parameters as follows:
Figure BDA0002181547520000052
wherein Fi(t) is a value randomly selected within the range of (-1,1), i.e. the condition F is satisfiedi1 2(t)+Fi2 2(t) is less than or equal to 1, random value taking is carried out, and ideal effects are obtained after dozens of verification;
s10, refer to the simulation program of S9Taking into account the disturbance Δ K thereiniAnd (t) when the constant time is zero, carrying out simulation verification to obtain a simulation result.
Compared with the prior art, the invention has the following beneficial effects:
the switching subsystem is a time-varying subsystem, original dynamic characteristics of a plurality of periodic systems are reserved, and meanwhile, a matrix polynomial method is adopted, so that more free variables can be obtained by introducing a high-dimensional matrix, and the solution is facilitated; in addition, in view of the vulnerability of the control of the periodic segmented system at present, the additive and multiplicative gain disturbances with uncertain norm constraints of the controller are considered respectively, so that the closed-loop system can still keep good performance and simultaneously has stable robustness under the condition that the controller is disturbed in a certain range, and the problems of analysis and control of the system with periodic time-varying complex characteristics in the current engineering application can be effectively solved.
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FIG. 1 is a flow chart of the present invention;
FIG. 2 shows an open ring system and a continuous nominal H of the present invention∞A state change comparison chart of a closed loop system under the controller;
FIG. 3 shows a continuous nominal H according to the present invention∞A graph of the variation of the controller gain (over one cycle);
FIG. 4 shows an open loop system and a discontinuous nominal H according to the present invention∞A state change comparison chart of a closed loop system under the controller;
FIG. 5 shows a discontinuous nominal H according to the present invention∞A graph of the variation of the controller gain (over one cycle);
FIG. 6 shows the non-vulnerability under multiplicative perturbation of the invention∞A state change map of the controller;
FIG. 7 shows the non-vulnerability under multiplicative perturbation of the invention∞A graph of the variation of the controller gain (over one cycle);
FIG. 8 is a graph of non-vulnerability H under additive perturbation of the present invention∞A state change map of the controller;
FIG. 9 shows non-vulnerability H under additive perturbation of the present invention∞Graph of the variation of the controller gain (over one cycle).
Detailed Description
The present invention will be described in further detail with reference to examples and drawings, but the present invention is not limited thereto.
The invention provides a non-fragile control method of a periodic segmented time-varying system based on a matrix polynomial, which can still enable the system to have good performance and stable robustness under the condition that a controller of the periodic segmented time-varying system under continuous time is disturbed in a certain range, and meanwhile, the designed controller is easier to solve and is beneficial to implementation in application. Firstly, constructing a Lyapunov function with a continuous time-varying n-dimensional matrix polynomial, and then obtaining a system H by using a general theorem about negative and positive natures of the matrix polynomial and a system stability condition through a Lyapunov stability criterion∞Sufficient condition of performance.
Based on completion H∞And (3) analyzing the performance, namely respectively obtaining a continuous-time periodic segmented time-varying system by using a variable substitution method and related lemmas under the consideration of additive and multiplicative gain disturbance of norm-bounded constraint of the controller and obtaining a non-fragile H under the uncertainty disturbance of the additive and multiplicative controller gains∞The condition of the controller; meanwhile, under the condition of zero disturbance, under the two conditions that the gain of the controller is continuously time-varying and discontinuously time-varying, the nominal H without the controller disturbance is respectively designed∞A controller; the results of the simulation comparisons are as follows: (1) for non-fragile H∞Controller and nominal H∞The controller compares and finds non-fragile H∞Under the condition of disturbance, the system is still robust and stable, the value of the performance index gamma is ideal, and the non-fragile H is proved∞The advantages of the controller; (2) in the case of a controller without disturbance, the continuous and discontinuous nominal H are compared∞Controller, finding nominal H with discontinuous controller gain∞The control effect of the controller has very weak performance advantage which can be almost ignored. In fact, nominal H with continuous controller gain∞The controller is actually easier to implement in engineering applications. ThroughMultiple simulation experiments are carried out, the results are compared, the effectiveness of the obtained results is proved, and therefore the non-fragile H which can enable the closed-loop system to still keep good system performance under the condition that the gain of the controller is uncertain and disturbed is obtained∞A controller for solving problems associated with engineering applications.
Specifically, as shown in fig. 1 to 9, a non-fragile control method for a periodic piecewise time varying system based on a matrix polynomial includes the following steps:
s1, first consider a continuous time period piecewise time varying system, the state space expression of which is shown in the following formula:
Figure BDA0002181547520000081
z(t)＝Ci(t)x(t)+Di(t)u(t)+Dwi(t)w(t)；
s2, the continuous time period segmentation time-varying system matrix Ai(t),Bi(t),Bwi(t),Ci(t),Di(t),Dwi(T) satisfies the periodicity, the fundamental period being TpAnd are all given by time-varying linear interpolation formulas, as shown in the following formulas:
Figure BDA0002181547520000082
wherein A isi,Bi,Bwi,Ci,Di,DwiI is 0, 1.. times.i is 1, 2.. times.s, which is a constant matrix, Ti＝ti-ti-1Is the residence time;
s3, according to Lyapunov stability criterion, constructing a Lyapunov function with continuous time-varying matrix polynomial, wherein the form is V (t) Vi(t)＝xT(t)Pi(t) x (t), wherein the Lyapunov matrix polynomial Pi(t) is:
Figure BDA0002181547520000083
s4, through a related derivation proof, a sufficient condition of the stability of the continuous time period segmented time-varying system can be obtained by utilizing a Lyapunov stability criterion;
s5, analyzing and deducing the negative qualitative general theorem of the matrix polynomial based on the Lyapunov function with the continuous time-varying matrix polynomial, and carrying out inequality scaling by combining the Coppel inequality to obtain H∞The sufficient condition of the performance is satisfied, and the following conditions are satisfied:
Figure BDA0002181547520000091
wherein the content of the first and second substances,
Figure BDA0002181547520000092
s6, obtaining H in S5∞Designing corresponding non-fragile H on the basis of the performance index∞A controller; two disturbance forms of additive and multiplicative gains of the controller are considered:
norm-constrained additive perturbation form: Δ Ki(t)＝MFi(t)Ni(t),i＝1,2,...,S,
Norm-constrained multiplicative perturbation form:
Figure BDA0002181547520000093
wherein, N (t),
Figure BDA0002181547520000094
is time-varying, defined as:
Figure BDA0002181547520000095
Figure BDA0002181547520000096
and N isi,Ni+1,
Figure BDA0002181547520000097
A constant matrix of appropriate dimensions, M a constant matrix of known appropriate dimensions, f (t) satisfying a norm constraint: fi(t)Fi(t)TI is less than or equal to I, and t is more than or equal to 0; wherein the disturbance parameter Ni(t),
Figure BDA0002181547520000098
Is time-varying rather than stationary, which results in a time-varying disturbance matrix Δ Ki(t) becomes more flexible, so that the problem under consideration can be guaranteed to have a rich disturbance dynamic; the new closed loop system under disturbance is then regained:
Figure BDA0002181547520000099
z(t)＝Cci(t)x(t)+Dwi(t)w(t)；
wherein, Aci(t)＝Ai(t)+Bi(t)(Ki(t)+ΔKi(t)),Cci(t)＝Ci(t)+Di(t)(Ki(t)+ΔKi(t))；
S7, aiming at the closed-loop system obtained in S6, based on the Lyapunov function with continuous time-varying matrix polynomial, utilizing an LMI method, combining the negative qualitative and positive qualitative general lemma conditions of the matrix polynomial, and then introducing a matrix polynomial variable Qi(t),Wi(t) carrying out variable substitution, using Schur's complementary theorem and inequality scaling, and finally obtaining non-fragile H under the condition of additive and multiplicative controller gain disturbance∞A controller, the designed control being solved by convex optimization;
s8, considering the time-varying controller gain delta Ki(t) ≡ 0, the nominal H without controller disturbance is designed∞A controller; and separately considering the introduction of continuous and discontinuous polynomial matrix variables Qi(t), and a continuous polynomial matrix variable Wi(t), substitution with variables, SThe nominal H under two conditions is obtained by deduction through methods such as the churn compensation theorem and inequality scaling∞Controller, controller gain Ki(t)＝Qi(t)Wi -1(t), which can be obtained by convex optimization solution;
and S9, carrying out Matlab numerical simulation verification on the obtained controller:
(1) under the condition of norm bounded additive disturbance, selecting disturbance parameters as follows:
Figure BDA0002181547520000101
(2) under the condition of norm bounded additive disturbance, selecting disturbance parameters as follows:
Figure BDA0002181547520000102
wherein Fi(t) is a value randomly selected within the range of (-1,1), i.e. the condition F is satisfiedi1 2(t)+Fi2 2(t) is less than or equal to 1, random value taking is carried out, and ideal effects are obtained after dozens of verification;
here, two examples of random values are given:
(1) under additive disturbance:
Figure BDA0002181547520000103
(2) multiplicative disturbance:
Figure BDA0002181547520000104
simulation results are obtained respectively as shown in fig. 6 to 9.
S10, referring to the simulation program of S9, consider the disturbance Δ K thereiniAnd (t) when the constant value is zero, performing simulation verification to obtain a simulation result, as shown in FIGS. 2-5.
From a comparison of simulation results, it can be seen that H is not fragile∞The controller has good robust characteristic and non-fragility, has strong tolerance capability to the gain disturbance of the controller in practical application, and has obvious engineering value.
The invention provides a non-fragile control method of a periodic segmented time-varying system based on a matrix polynomial, which is characterized in that in order that a designed controller can tolerate the influence caused by disturbance of an uncertain time-varying controller, a periodically switched subsystem is a time-varying subsystem, and an H-shaped continuous-time periodic segmented time-varying system is subjected to an H-shaped continuous-time periodic segmented time-varying system based on a Lyapunov function of a Lyapunov matrix polynomial with continuous time variation∞Analyzing the performance; then based on H∞Performance, considering controller additive and multiplicative disturbance with norm-bounded constraint, designs non-fragile H∞The controller enables the closed-loop system to be robust and stable and have good performance under the condition that the controller is disturbed within a certain range; according to the invention, due to the use of the matrix polynomial, higher dimensionality and more free variables are introduced, so that the designed controller is easy to solve, can be obtained by convex optimization directly based on LMI, does not need iteration, and is easy to implement in engineering.
The control method provided by the invention is repeatedly and strictly proved and inferred, the effectiveness of the control method is verified through multiple simulation experiments, and the control method can be used in engineering application with the periodic time-varying complex characteristic.
When the method is actually applied in engineering, the accuracy of characteristic quantization data needs to be ensured as much as possible in the process of quantizing the dynamic characteristics of an actual system so as to reduce errors; the invention considers that the controller can keep good system performance and can be robust and stable under the condition of certain change of parameters within a certain range due to aging or other reasons. If the controller is damaged by human factors or other inefficacy force factors, the controller fails; in practical application, it is necessary to ensure that the quantized data of the system characteristics are as accurate as possible, and the controller is not damaged by human factors or other inequality factors.
The switching subsystem is a time-varying subsystem, original dynamic characteristics of a plurality of periodic systems are reserved, and meanwhile, a matrix polynomial method is adopted, so that more free variables can be obtained by introducing a high-dimensional matrix, and the solution is facilitated; in addition, in view of the vulnerability of the control of the periodic segmented system at present, the additive and multiplicative gain disturbances with uncertain norm constraints of the controller are considered respectively, so that the closed-loop system can still keep good performance and simultaneously has stable robustness under the condition that the controller is disturbed in a certain range, and the problems of analysis and control of the system with periodic time-varying complex characteristics in the current engineering application can be effectively solved.
The present invention is not limited to the above embodiments, and any other changes, modifications, substitutions, combinations, and simplifications which do not depart from the spirit and principle of the present invention should be construed as equivalents and are included in the scope of the present invention.

Claims (1)

1. A non-fragile control method for a matrix polynomial based periodically segmented time varying system, comprising the steps of:
s1, first consider a continuous time period piecewise time varying system, the state space expression of which is shown in the following formula:
Figure FDA0003523023020000011
z(t)＝Ci(t)X(t)+Di(t)u(t)+Dwi(t)W(t)；
s2, the continuous time period segmentation time-varying system matrix Ai(t),Bi(t),Bwi(t),Ci(t),Di(t),Dwi(T) satisfies the periodicity, the fundamental period being TpAnd are all given by time-varying linear interpolation formulas, as shown in the following formulas:
Figure FDA0003523023020000012
Ai，Bi，Bwi，Ci，Di，Dwiis a constant matrix, where l =0, 1max，i=0，1，...，S，Ti＝ti-ti-1Is the residence time;
s3, constructing a Lyapunov function with continuous time-varying matrix polynomial according to the Lyapunov stability criterion, wherein the Lyapunov function is in the form of V (t) Vi(t)＝xT(t)Pi(t) x (t), wherein the Lyapunov matrix polynomial Pi(t) is:
Figure FDA0003523023020000013
s4, utilizing Lyapunov stability criterion to obtain sufficient conditions of the stability of the continuous time period segmented time-varying system;
s5, based on Lyapunov function with continuous time-varying matrix polynomial, using general theory of negative nature of matrix polynomial to analyze and deduce, and combining Coppel inequality to scale inequality to obtain H∞The sufficient condition of the performance is satisfied, and the following conditions are satisfied:
Figure FDA0003523023020000021
wherein the content of the first and second substances,
Figure FDA0003523023020000022
s6, obtaining H in S5∞Designing corresponding non-fragile H on the basis of the performance index∞A controller; two disturbance forms of additive and multiplicative gains of the controller are considered:
norm constrained additive perturbation form: Δ Ki(t)＝MFi(t)Ni(t),i＝1,2,K,S,
Norm-constrained multiplicative perturbation form:
Figure FDA0003523023020000023
wherein, N (t),
Figure FDA0003523023020000024
is time-varying, defined as:
Figure FDA0003523023020000025
Figure FDA0003523023020000026
and N isi,Ni+1,
Figure FDA0003523023020000027
A constant matrix of appropriate dimensions, M a constant matrix of known appropriate dimensions, f (t) satisfying a norm constraint: f (t)TI is less than or equal to I, and t is more than or equal to 0; wherein the disturbance parameter Ni(t),
Figure FDA0003523023020000028
Is time-varying rather than stationary, which causes the profile controller gain perturbation Δ Ki(t) the matrix becomes more flexible, so that the problem under consideration can be guaranteed to have rich disturbance dynamics; the new closed loop system under disturbance is then regained:
Figure FDA0003523023020000031
z(t)＝Cci(t)x(t)+Dwi(t)w(t)；
wherein the content of the first and second substances,
Aci(t)＝Ai(t)+Bi(t)(Ki(t)+ΔKi(t)),Cci(t)＝Ci(t)+Di(t)(Ki(t)+ΔKi(t))；
s7, aiming at the closed-loop system obtained in S6, based on the Lyapunov function with continuous time-varying matrix polynomial, by combining with LMI method, with negative qualitative and positive qualitative general lemma condition of matrix polynomial, then introducing matrix polynomial variable Qi(t),Wi(t) carrying out variable replacement, using Schur's complementary theorem, scaling by inequality, and finally obtaining non-fragile H under the condition of additive and multiplicative controller gain disturbance∞A controller, the designed control being solved by convex optimization;
s8, considering the controller gain disturbance delta KiNominal H in case of (t) ≡ 0, i.e. without controller disturbances∞A controller; and separately considering the introduction of continuous and discontinuous matrix polynomial variables Qdc，i(t), and a continuous matrix polynomial variable Wc，i(t) deriving and obtaining the nominal H under two conditions by using methods of variable substitution, Schur's complement theorem and inequality scaling∞Controller, controller gain Ki(t)＝Qi(t)Wi -1(t), which can be obtained by convex optimization solution;
and S9, carrying out Matlab numerical simulation verification on the obtained controller:
(1) under the condition of norm bounded additive disturbance, selecting disturbance parameters as follows:
M＝1.6,
Figure FDA0003523023020000032
Fi(t)＝[Fi1(t) Fi2]
(2) under the condition of norm bounded additive disturbance, selecting disturbance parameters as follows:
M＝0.04,
Figure FDA0003523023020000033
Fi(t)＝[Fi1(t) Fi2(t)]
wherein Fi(t) is a value randomly selected within the range of (-1,1), i.e. the condition F is satisfiedi1 2(t)+Fi2 2(t) is less than or equal to 1, random value taking is carried out, and ideal effects are obtained after dozens of verification;
s10, referring to the simulation program of S9, consider the controller gain disturbance Δ K thereiniAnd (t) when the constant time is zero, carrying out simulation verification to obtain a simulation result.
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