CN108663937B - Non-minimum phase linear system regulation control method - Google Patents

Non-minimum phase linear system regulation control method Download PDF

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CN108663937B
CN108663937B CN201810435850.2A CN201810435850A CN108663937B CN 108663937 B CN108663937 B CN 108663937B CN 201810435850 A CN201810435850 A CN 201810435850A CN 108663937 B CN108663937 B CN 108663937B
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CN108663937A (en
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宋永端
黄秀财
何鎏
赖俊峰
谭觅
高辉
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Chongqing University
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Abstract

The invention relates to a non-minimum phase linear system regulation control method, which comprises the following steps: s1: establishing a non-minimum phase linear system model; s2: designing a closed-loop auxiliary system of the non-minimum-phase linear system in the step S1; s3: and setting a controller to stably control the non-minimum phase linear system. The invention provides an output feedback strategy aiming at an unknown non-minimum phase linear system based on an integral control and high gain control method, has certain practicability and is effectively applied to a vehicle-mounted inverted pendulum system.

Description

Non-minimum phase linear system regulation control method
Technical Field
The invention relates to a control method, in particular to a non-minimum phase linear system regulation control method.
Background
Although control methods for non-minimum phase linear systems are endless, the study of output regulation control for unknown non-minimum phase linear systems is not thorough. This is because, 1) there are uncertain system parameters in the unknown non-minimum phase system, and most of the related control methods are based on the known system model, and there is no ability for the unknown system; 2) most of the existing research methods aiming at output regulation control are based on state feedback, but are ineffective for systems with unknown states; and 3) although there is a correlation method that can implement output feedback control on an unknown non-minimum phase linear system, the system response under output feedback is worse than that under state feedback.
Disclosure of Invention
Aiming at the defects of the prior art, the invention provides a control method based on integral control and high-gain control and combined with an extended high-gain observer, comprehensively solves the problems and successfully applies the method to a vehicle-mounted inverted pendulum system.
In order to achieve the purpose, the invention adopts the following technical scheme: a non-minimum phase linear system regulation control method comprises
The method comprises the following steps:
s1: establishing a non-minimum phase linear system model;
s2: designing a closed-loop auxiliary system of the non-minimum-phase linear system in the step S1;
s3: and setting a controller to stably control the non-minimum phase linear system.
As an improvement, the process of establishing the non-minimum phase linear system model in step S1 is as follows:
s1 a: the linear system is set as follows:
Figure GDA0001690334290000011
wherein the content of the first and second substances,
Figure GDA0001690334290000012
in order to be a state of the linear system,
Figure GDA0001690334290000013
in order to control the input of the electronic device,
Figure GDA0001690334290000014
in order to adjust the error, the error is adjusted,
Figure GDA0001690334290000015
for the unknown constant and the interference vector,
Figure GDA0001690334290000016
in order to be compact, the device is provided with a plurality of small-sized and compact parts,
Figure GDA0001690334290000017
is a real number set; s1 b: to achieve output regulation tracking of a non-minimum phase linear system model,
let limt→∞e(t)＝0 (1.2)；
S1 c: the linear system (1.1) satisfies the following conditions:
the first condition is as follows: (A, B) is stabilizable and (A, C) is detectable;
and a second condition: matrix array
Figure GDA0001690334290000021
A full rank;
and (3) carrying out a third condition: the relative order of the system is rho, and rho is in the form of [1, n ]]And incorporating CAρ-1B＝1；
S1 d: by the setting of step S1b, there is a unique steady state solution (x)s,us) So that
Figure GDA0001690334290000022
Definition of
Figure GDA0001690334290000023
The linear system (1.1) can be re-expressed as:
Figure GDA0001690334290000024
wherein χ (ω) usIs constant and represents the ideal input for maintaining the system stable, and for all
Figure GDA0001690334290000025
|χ(ω)|≤χ0；
S1 e: after coordinate transformation, the system (1.4) can be transformed into the following expression:
Figure GDA0001690334290000026
wherein the content of the first and second substances,
Figure GDA0001690334290000027
showing the state of the endomembrane system,
Figure GDA0001690334290000028
is the outer membrane state of the system;
f, H and G all represent a matrix, which is determined by a system structure, the characteristic root of the matrix F is a zero point of { A, B, C }, and the matrix F does not satisfy Hurwitz; ρ represents the relative order of a non-minimum phase linear system and satisfies ρ e [1, n ].
aiI 1., ρ is a constant coefficient, which is determined by the system structure;
the relation (1.5) is a non-minimum phase linear system model.
As an improvement, the procedure of designing the closed-loop auxiliary system in step S2 is as follows:
s2 a: in the non-minimum phase linear system model (1.5), the
Figure GDA0001690334290000029
Viewed as an output, xiρControl input u considered as auxiliaryaThe auxiliary systems of the system (1.5) can then be selected as:
Figure GDA00016903342900000210
wherein, yaRepresenting the output of the auxiliary system;
the auxiliary system (1.6) can be stabilized by output feedback control, and the controller can be designed for the auxiliary system (1.6)
Figure GDA0001690334290000031
Wherein
Figure GDA0001690334290000032
In order to assist in the state of the system,
Figure GDA0001690334290000033
and
Figure GDA0001690334290000034
to design a matrix;
in combination with equation (1.6) and equation (1.7), the closed-loop auxiliary system is:
Figure GDA0001690334290000035
by definition of X ═ eta, xi1,...,ξρ-1,z]TThe system (1.8) may be further expressed as:
Figure GDA0001690334290000036
wherein the content of the first and second substances,
Figure GDA0001690334290000037
B0＝[0,0,…,0,0,M0]T (1.11)
C0＝[0,1,…,0,0,0]T (1.12)。
as a modification, the controller designed in step S3 includes a state feedback controller and an output feedback controller.
As a modification, the state feedback controller in step S3 is as follows:
Figure GDA0001690334290000038
wherein, sigma represents integral variables kappa and mu as design parameters;
Figure GDA0001690334290000041
matrix A0,B0And C0Is a constant matrix and
Figure GDA0001690334290000042
as a modification, the output feedback controller design process in step S3 is as follows:
1) ξ for non-minimum phase linear system through extended high gain observer2To xiρAnd a virtual output y of the auxiliary controlleraAnd estimating, wherein the high-gain observer is as follows:
Figure GDA0001690334290000043
wherein>0 is a small constant, γ1To gammaρSelected as normal numbers and made polynomial
qρ+1+γ1qρ+…+γρq+γρ+1＝0 (1.25)
All the characteristic roots of (a) have a negative real part;
2) based on the high-gain observer, the output feedback controller is designed as follows:
Figure GDA0001690334290000044
wherein KyAnd KμRespectively determining the virtual output signals as normal numbers
Figure GDA0001690334290000045
And a saturation threshold for the control input u.
The invention has at least the following beneficial effects:
the invention provides an output feedback strategy aiming at an unknown non-minimum phase linear system based on an integral control and high gain control method, which can ensure that:
a) under the conditions that unknown parameters exist in the system and the system is interfered by an external constant value, the stability of the system is kept;
b) the system state, unknown external interference and an unknown model are estimated by adopting an extended high-gain observer, so that the output feedback control of an unknown non-minimum phase linear system can be realized;
c) controlling to design the integral parameter, the high-gain parameter and the observer parameter to be large enough (or small enough), and recovering the system performance under the output feedback to the level when the state feedback is carried out;
d) the method has certain practicability and is effectively applied to a vehicle-mounted inverted pendulum system.
Drawings
Fig. 1 shows the recovery of the performance of the auxiliary system in example 1: the solid line represents the secondary system response; the dashed or dotted lines represent the high gain system response (without integrator).
Fig. 2 is the high gain system response in example 1: the solid line represents the high gain system response (without integrator, k ═ 0); the dashed, dotted or dotted line represents the high gain system response (with the integrator).
Fig. 3 shows the performance recovery of the state feedback system in embodiment 1: the dotted or dashed line represents output feedback control; the solid line represents the state feedback control.
FIG. 4 shows the system response and control inputs under state feedback in example 1: the dashed line represents the method of the invention; the solid line represents the prior art method.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention will be described in further detail with reference to the accompanying drawings in conjunction with the following detailed description. It should be understood that the description is intended to be exemplary only, and is not intended to limit the scope of the present invention. Moreover, in the following description, descriptions of well-known structures and techniques are omitted so as to not unnecessarily obscure the concepts of the present invention.
A non-minimum phase linear system regulation control method comprises the following steps:
s1: establishing a non-minimum phase linear system model;
s2: designing a closed-loop auxiliary system of the non-minimum-phase linear system in the step S1; our control objective is to design an output feedback controller such that all signals in a closed loop system are bounded and lim is madet→∞e (t) is 0. To achieve this control objective, we introduce an auxiliary system of linear systems (1.5).
S3: and setting a controller to stably control the non-minimum phase linear system.
Specifically, the process of establishing the non-minimum phase linear system model in step S1 is as follows:
s1 a: the linear system is set as follows:
Figure GDA0001690334290000051
wherein the content of the first and second substances,
Figure GDA0001690334290000052
in order to be a state of the linear system,
Figure GDA0001690334290000053
in order to control the input of the electronic device,
Figure GDA0001690334290000054
in order to adjust the error, the error is adjusted,
Figure GDA0001690334290000055
for the unknown constant and the interference vector,
Figure GDA0001690334290000056
in order to be compact, the device is provided with a plurality of small-sized and compact parts,
Figure GDA0001690334290000057
is a real number set; s1 b: to achieve output regulation tracking of a non-minimum phase linear system model,
let limt→∞e(t)＝0 (1.2)；
S1 c: the linear system (1.1) satisfies the following conditions:
the first condition is as follows: (A, B) is stabilizable and (A, C) is detectable;
and a second condition: matrix array
Figure GDA0001690334290000061
A full rank;
and (3) carrying out a third condition: the relative order of the system is rho, and rho is in the form of [1, n ]]And incorporating CAρ-1If B is 1, without loss of generality, if CAρ-1B is known;
s1 d: by the setting of step S1b, there is a unique steady state solution (x)s,us) So that
Figure GDA0001690334290000062
Definition of
Figure GDA0001690334290000063
The linear system (1.1) can be re-expressed as:
Figure GDA0001690334290000064
wherein χ (ω) usIs constant and represents the ideal input for maintaining the system stable, and for all
Figure GDA0001690334290000065
|χ(ω)|≤χ0；
S1 e: after coordinate transformation, the system (1.4) can be transformed into the following expression:
Figure GDA0001690334290000066
wherein the content of the first and second substances,
Figure GDA0001690334290000067
showing the state of the endomembrane system,
Figure GDA0001690334290000068
is the outer membrane state of the system; f, H and G all represent a matrix, which is determined by a system structure, the characteristic root of the matrix F is a zero point of { A, B, C }, and the matrix F does not satisfy Hurwitz; rho represents the relative order of a non-minimum phase linear system and satisfies rho e [1, n]；
aiI 1., ρ is a constant coefficient, which is determined by the system structure;
the relation (1.5) is a non-minimum phase linear system model.
Wherein the content of the first and second substances,
Figure GDA0001690334290000069
for the new system state, the characteristic root of matrix F is the zero of { A, B, C }. Meanwhile, condition two indicates that all the characteristic roots of the matrix F are not at the origin. Here we consider the case where the matrix F does not satisfy Hurwitz, i.e. the system (1.5) is a non-minimum phase system.
Specifically, the process of designing the closed-loop auxiliary system in step S2 is as follows:
s2 a: in the non-minimum phase linear system model (1.5), the
Figure GDA0001690334290000071
Viewed as an output, xiρControl input u considered as auxiliaryaThe auxiliary systems of the system (1.5) can then be selected as:
Figure GDA0001690334290000072
wherein, yaRepresenting the output of the auxiliary system; the auxiliary system (1.6) can be stabilized by output feedback control, and the controller can be designed for the auxiliary system (1.6)
Figure GDA0001690334290000073
Wherein
Figure GDA0001690334290000074
In order to assist in the state of the system,
Figure GDA0001690334290000075
and
Figure GDA0001690334290000076
to design a matrix;
we call equation (1.7) the secondary controller of the system (1.6). In combination with equation (1.6) and equation (1.7), the closed-loop auxiliary system is:
Figure GDA0001690334290000077
by definition of X ═ eta, xi1,...,ξρ-1,z]TThe system (1.8) may be further expressed as:
Figure GDA0001690334290000078
wherein the content of the first and second substances,
Figure GDA0001690334290000079
B0＝[0,0,…,0,0,M0]T (1.11)；
C0＝[0,1,…,0,0,0]T (1.12)。
preferably, the controller designed in step S3 includes a state feedback controller and an output feedback controller.
Specifically, the state feedback controller in step S3 is as follows:
by designing a high-gain feedback controller and using the singular perturbation theory, we will show that the performance of the whole system (1.5) under high-gain control can be very close to that of the auxiliary system (1.6). The high-gain feedback controller is designed as follows:
Figure GDA0001690334290000081
where μ >0 is a small constant. When χ (ω) ≠ 0, the controller (1.19) is unable to zero the steady-state error of the system (1.5). To converge e (t) to zero, we add a slow integration element in the controller as follows:
Figure GDA0001690334290000082
wherein κ>0 is a small constant which is a constant number,
Figure GDA0001690334290000083
is defined by formula (1.13).
Definition of
Figure GDA0001690334290000084
Let us assume that
Figure GDA0001690334290000085
It is known to provide for subsequent decisions on the direction of the integral control. Due to the matrix A0,B0And C0Is a constant matrix and
Figure GDA0001690334290000086
Figure GDA0001690334290000087
must be strictly positive or strictly negative.
Figure GDA0001690334290000088
Wherein, sigma represents integral variables kappa and mu as design parameters;
Figure GDA0001690334290000089
matrix A0,B0And C0Is a constant matrix and
Figure GDA00016903342900000810
by defining the following variables:
s＝ξρ-N0z,υ＝χ+σ (1.22)；
the dynamic equation of the closed loop system under the action of the controller (3.21) is
Figure GDA00016903342900000811
Wherein "major" represents some constant value matrix independent of μ and κ. When μ and κ are sufficiently small, the closed-loop system (1.23) has a triple time scale. According to the singular perturbation correlation theory, matrix
Figure GDA0001690334290000098
Has a characteristic root close to-1/mu + aρ(1-N0M0)，
Figure GDA0001690334290000091
And matrix A0The characteristic root of (2). Thus, when μ and κ are sufficiently small, A0Is a Hurwitz matrix, thereby ensuring that the closed loop system (1.23) has a unique exponential stable equilibrium point. Integration element in feedback controller (1.21)
Figure GDA0001690334290000092
It can be ensured that the adjustment error at the equilibrium point is 0, limt→∞e(t)＝0。
Specifically, the output feedback controller design process in step S3 is as follows:
1) extended high gain observer pairXi in non-minimum phase linear system2To xiρAnd a virtual output y of the auxiliary controlleraAnd estimating, wherein the high-gain observer is as follows:
Figure GDA0001690334290000093
wherein>0 is a small constant, γ1To gammaρSelected as normal numbers and made polynomial
qρ+1+γ1qρ+…+γρq+γρ+1＝0 (1.25)；
All the characteristic roots of (a) have a negative real part;
2) based on the high-gain observer, the output feedback controller is designed as follows:
Figure GDA0001690334290000094
wherein KyAnd KμRespectively determining the virtual output signals as normal numbers
Figure GDA0001690334290000095
And a saturation threshold for the control input u. To pair
Figure GDA0001690334290000096
And u are saturated to avoid the peaking phenomenon caused during the transient response phase using a high gain observer. KyAnd KμCan be based on the state feedback controller (1.21), yaAnd u is determined in the attraction domain. Since the control input u has a saturation characteristic, it is good for
Figure GDA0001690334290000097
Has global stability, so by selecting small enough, we can ensure that the system performance under the output feedback can be restored to the level of the state feedback.
Example 1: the method is applied to the vehicle-mounted inverted pendulum model
Consider the following vehicle-mounted inverted pendulum model:
Figure GDA0001690334290000101
where q is the position of the vehicle, θ represents the angle of the simple pendulum to the vertical, u represents the control input, and χ (ω) ω is the constant disturbance applied to the actuator. The system parameters can be as shown in Table 1.1, where the mass m of the simple pendulumpUncertain but remaining within a certain range, while we assume the mass m of the vehiclecAre known.
TABLE 1.1 vehicle-mounted inverted pendulum model parameter comparison table
Parameter(s) Physical significance Value of
mc Mass of vehicle 0.82kg
mp Mass of simple pendulum [0.19，0.23]kg
h Length of simple pendulum 0.61m
g Acceleration of gravity 9.81m/s2
Linear approximation is performed on formula (1.27) at θ ═ 0, and definition is performed
Figure GDA0001690334290000102
The system (1.27) can be converted into a linear model as follows
Figure GDA0001690334290000103
The zero dynamics of the system are
Figure GDA0001690334290000104
Note that the system (1.30) is unstable at the origin, so the original system (1.29) is a non-minimum phase system. Our control objective is to use a unique measurable signal xi in the case of a system (1.29) subject to external interference1A feedback controller is designed such that the regulation error e (t) converges exponentially to 0.
Therefore, we first need to find the auxiliary system of the system (1.29) and design the corresponding auxiliary controller, and then assume the system state ξ1And xi2And- (m)pgη1+χ(ω))/mcIn the known case, a state feedback controller is designed, and finally xi is estimated by using an extended high gain observer2And- (m)pgη1+χ(ω))/mcAnd designing an output feedback controller.
Problem of assistance
The auxiliary system corresponding to the system (1.29) is
Figure GDA0001690334290000111
Wherein
Figure GDA0001690334290000112
Is not controllable (in Matlab, the available code ctrb
Figure GDA0001690334290000113
Detection) of the signal,
Figure GDA0001690334290000114
is not observable (in Matlab, the available code obsv
Figure GDA0001690334290000115
Detection). To design an observer-based controller for the system (1.31), we define ua＝kξ1+νaThen the auxiliary system can be re-represented as
Figure GDA0001690334290000116
With k 2, we have a system (1.32) that is both controllable and observable. In this case, the observer-based controller can be designed as
Figure GDA0001690334290000117
Wherein the matrix
Figure GDA0001690334290000121
And
Figure GDA0001690334290000122
are respectively designed as
Figure GDA0001690334290000123
Thereby ensuring
Figure GDA0001690334290000124
And
Figure GDA0001690334290000125
all satisfy the Hurwitz condition. By definition
ζ＝[η1 η2 ξ1 zT]T (1.35)；
The closed-loop auxiliary system is given by
Figure GDA0001690334290000126
Wherein the matrix
Figure GDA0001690334290000127
Satisfying the Hurwitz condition, steady state gain from χ (ω) to e (t)
Figure GDA0001690334290000128
According to formula (1.13) with respect to
Figure GDA0001690334290000129
By definition of (1), we can further obtain
Figure GDA00016903342900001210
Controller design and simulation results
A) State feedback
The state feedback controller for the system (1.29) is designed to
Figure GDA00016903342900001211
Wherein
Figure GDA00016903342900001212
And
Figure GDA00016903342900001213
both μ and κ are positive small constants.
We will simulate the system with Simulink in Matlab. The initial values of all the variables of the selected system are as follows: eta1(0)＝η2(0)＝ξ2(0)＝z1(0)＝z2(0)＝z3(0)＝σ(0)＝0，ξ1(0) -20, the control parameters are set to: ω is 0.1, μ ∈ {0.1,0.01,0.001}, and κ ∈ (10)-4,5×10-5,10-5). The results of the simulation are shown in fig. 1 and 2.
Firstly, the following high-gain state feedback controller is adopted without adding an integral link in the controller
Figure GDA00016903342900001214
And (5) simulating the system. The simulation results are shown in fig. 1, from which it can be observed that: when mu is smaller, the system performance under the action of the controller is closer to the performance of the auxiliary system, but the regulation error e (t) converges to a non-zero constant value because the system does not add an integration element. Subsequently, setting μ to 0.001, we add an integration element to the controller and verify the performance of the controller (1.37) by adjusting the value of the parameter κ. The simulation results are shown in fig. 2: a) when κ is 0, i.e. no integrating loop is added, the controller (1.37) cannot achieve exponential adjustment because ξ1Converge to a non-zero constant (this point is also analyzable in fig. 1); b) when the integral link is added, xi1The true exponent converges to zero; c) as the value of κ decreases, the integral control can restore the performance of the system during the rise Phase (Rising Phase) when acted upon by the high gain controller, but at the expense of extending the Settling Time of the system.
B) Output feedback
When the state of the system is unknown, we can estimate it by using a high gain observer, i.e. in the controller, the state ξ2And unknown item yaWill be respectively estimated by them
Figure GDA0001690334290000131
And
Figure GDA0001690334290000132
instead. The high-gain observer is designed as follows:
Figure GDA0001690334290000133
wherein the parameters>0 needs to be designed small enough, parameter γ1To gamma3Is selected such that3+γ1λ2+γ2λ+γ3Meets the Hur witz condition. Based on the observer (1.39), the corresponding output feedback controller is designed as
Figure GDA0001690334290000134
By adjusting the parameters, we will restore the performance of the system when the state feedback controller is active. The initial value of the observer is selected as:
Figure GDA0001690334290000135
it is worth emphasizing that
Figure GDA0001690334290000136
Should be different from ξ1(0). The controller parameters are selected as: gamma ray1＝6,γ2＝11,γ3＝6,κ＝0.0001,∈{0.001，0.0005,0.0001}，Kμ＝33000，Ky5000. Wherein, KμAnd KyAccording to u and y, respectivelyaThe amplitude value in the state feedback control is selected, and the values of other controller parameters are not changed, namely the same as in the state feedback control. The simulation results are shown in fig. 3, from which it can be seen that: when reduced, the system performance under state feedback is gradually restored. Practically, when 10-4In this case, the system performance under the state feedback control is substantially indistinguishable from the system performance under the output feedback control.
The above description is only an embodiment of the present invention, but the scope of the present invention is not limited thereto, and any person skilled in the art can easily conceive of changes or substitutions within the technical scope of the present invention, and all such changes or substitutions are included in the scope of the present invention. Therefore, the protection scope of the present invention shall be subject to the protection scope of the claims.

Claims (5)

1. A non-minimum phase linear system regulation control method is characterized in that: the method comprises the following steps:
s1: establishing a non-minimum phase linear system model, wherein the process of establishing the non-minimum phase linear system model is as follows:
s1 a: the linear system is set as follows:
Figure FDA0002680117520000011
wherein the content of the first and second substances,
Figure FDA0002680117520000012
in order to be a state of the linear system,
Figure FDA0002680117520000013
in order to control the input of the electronic device,
Figure FDA0002680117520000014
in order to adjust the error, the error is adjusted,
Figure FDA0002680117520000015
for the unknown constant and the interference vector,
Figure FDA0002680117520000016
in order to be compact, the device is provided with a plurality of small-sized and compact parts,
Figure FDA0002680117520000017
is a real number set; s1 b: to achieve output regulation tracking of a non-minimum phase linear system model,
let limt→∞e(t)＝0 (1.2)；
S1 c: the linear system (1.1) satisfies the following conditions:
the first condition is as follows: (A, B) is stabilizable and (A, C) is detectable;
and a second condition: matrix array
Figure FDA0002680117520000018
A full rank;
and (3) carrying out a third condition: the relative order of the system is rho, and rho is in the form of [1, n ]]And incorporating CAρ-1B＝1；
S1 d: by the setting of step S1b, there is a unique steady state solution (x)s,us) So that
Figure FDA0002680117520000019
Definition of
Figure FDA00026801175200000110
The linear system (1.1) can be re-expressed as:
Figure FDA00026801175200000111
wherein χ (ω) usIs constant and represents the ideal input for maintaining the system stable, and for all
Figure FDA00026801175200000112
|χ(ω)|≤χ0；
S1 e: after coordinate transformation, the system (1.4) can be transformed into the following expression:
Figure FDA00026801175200000113
wherein the content of the first and second substances,
Figure FDA00026801175200000114
showing the state of the endomembrane system,
Figure FDA00026801175200000115
is the outer membrane state of the system;
f, H and G all represent a matrix, which is determined by a system structure, the characteristic root of the matrix F is a zero point of { A, B, C }, and the matrix F does not satisfy Hurwitz; rho represents the relative order of a non-minimum phase linear system and satisfies rho epsilon [1, n ];
aii is 1, and K is a constant coefficient and is determined by the system structure;
the relation (1.5) is a non-minimum phase linear system model;
s2: designing a closed-loop auxiliary system of the non-minimum-phase linear system in the step S1;
s3: and setting a controller to stably control the non-minimum phase linear system.
2. The non-minimum phase linear system regulation control method of claim 1 wherein: the process of designing the closed-loop auxiliary system in step S2 is as follows:
s2 a: in the non-minimum phase linear system model (1.5), the
Figure FDA0002680117520000021
Viewed as an output, xiρControl input u considered as auxiliaryaThe auxiliary systems of the system (1.5) can then be selected as:
Figure FDA0002680117520000022
wherein, yaRepresenting the output of the auxiliary system;
the auxiliary system (1.6) can be stabilized by output feedback control, and the controller can be designed for the auxiliary system (1.6)
Figure FDA0002680117520000023
Wherein
Figure FDA0002680117520000024
In order to assist in the state of the system,
Figure FDA0002680117520000025
and
Figure FDA0002680117520000026
to design a matrix;
in combination with equation (1.6) and equation (1.7), the closed-loop auxiliary system is:
Figure FDA0002680117520000027
by definition of X ═ eta, xi1,K,ξρ-1,z]TThe system (1.8) may be further expressed as:
Figure FDA0002680117520000028
wherein the content of the first and second substances,
Figure FDA0002680117520000031
B0＝[0,0,...,0,0,M0]T (1.11)
C0＝[0,1,...,0,0,0]T (1.12)。
3. the non-minimum phase linear system regulation control method of claim 1 wherein: the controller designed in step S3 includes a state feedback controller and an output feedback controller.
4. A non-minimum phase linear system regulation control method as claimed in claim 3 wherein: the state feedback controller in step S3 is as follows:
Figure FDA0002680117520000032
wherein, sigma represents integral variables kappa and mu as design parameters;
Figure FDA0002680117520000033
matrix A0,B0And C0Is a constant matrix and
Figure FDA0002680117520000034
5. a non-minimum phase linear system regulation control method as claimed in claim 3 wherein: the output feedback controller design process in step S3 is as follows:
1) ξ for non-minimum phase linear system through extended high gain observer2To xiρAnd a virtual output y of the auxiliary controlleraAnd estimating, wherein the high-gain observer is as follows:
Figure FDA0002680117520000035
where >0 is a small constant, gamma1To gammaρSelected as normal numbers and made polynomial
qρ+1+γ1qρ+L+γρq+γρ+1＝0 (1.25)
All the characteristic roots of (a) have a negative real part;
2) based on the high-gain observer, the output feedback controller is designed as follows:
Figure FDA0002680117520000041
wherein KyAnd KμRespectively determining the virtual output signals as normal numbers
Figure FDA0002680117520000042
And a saturation threshold for the control input u.
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