CN107203139B - Stabilization control method of multi-input multi-output nonlinear differential algebra subsystem - Google Patents

Stabilization control method of multi-input multi-output nonlinear differential algebra subsystem Download PDF

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CN107203139B
CN107203139B CN201710549039.2A CN201710549039A CN107203139B CN 107203139 B CN107203139 B CN 107203139B CN 201710549039 A CN201710549039 A CN 201710549039A CN 107203139 B CN107203139 B CN 107203139B
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臧强
石瑶瑶
胡凯
潘慧敏
司云腾
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Nanjing Efei Scientific Instrument Co ltd
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Abstract

A stabilization control method of a multi-input multi-output nonlinear differential algebraic subsystem comprises the following steps: establishing a model of a multi-input multi-output nonlinear differential algebra subsystem; equivalently converting the model into a multi-input multi-output nonlinear ordinary differential system through differential homomorphism and state feedback, wherein the multi-input multi-output nonlinear ordinary differential system comprises each group of ordinary differential subsystems; and obtaining subsystem controllers corresponding to the ordinary differential subsystems through a backstepping method, and forming a system stabilizing controller by each group of subsystem controllers and state feedback. The invention has wider range of the controlled object, simpler and more practical backstepping method, and good effect when the model and the method are applied to the power system, thus leading the system to be more stable.

Description

Stabilization control method of multi-input multi-output nonlinear differential algebra subsystem
Technical Field
The invention belongs to the field of engineering control, and particularly relates to a stabilization control method of a multi-input multi-output nonlinear differential algebra subsystem based on a backstepping method.
Background
The stabilizing controller is converted by adding controlled input into the subsystem controller, and can convert the nonlinear differential algebra subsystem into a nonlinear ordinary differential system by substituting the nonlinear ordinary differential algebra subsystem, which plays a crucial role in the development of the power system, however, many existing achievements relate to a single-input single-output nonlinear differential algebra subsystem. In practical application, the controlled object is often described by a multi-input multi-output nonlinear differential algebra subsystem. The multi-input multi-output nonlinear differential algebraic subsystem has the following advantages: 1. the single-input single-output model has no universality; 2. the design of the multi-input multi-output stabilizing controller is expanded from the design of the controller of each subsystem to the design of the stabilizing controller of the whole large system, so that the large system is gradually stabilized; 3. the effect of zero dynamics is not considered. Through the search of the design of the stabilizing controller of the multi-input multi-output nonlinear differential algebra subsystem, no papers and patents related to the aspect are found.
Disclosure of Invention
Aiming at the defects in the prior art, the invention provides a stabilization control method of a multi-input multi-output nonlinear differential algebra subsystem.
In order to achieve the purpose, the invention adopts the following technical scheme:
establishing a model of a multi-input multi-output nonlinear differential algebra subsystem;
equivalently converting the model into a multi-input multi-output nonlinear ordinary differential system through differential homomorphism and state feedback, wherein the multi-input multi-output nonlinear ordinary differential system comprises each group of ordinary differential subsystems;
and obtaining subsystem controllers corresponding to the ordinary differential subsystems through a backstepping method, and forming a system stabilizing controller by each group of subsystem controllers and state feedback.
In order to optimize the technical scheme, the specific measures adopted further comprise:
establishing a model of a multi-input multi-output nonlinear differential algebra subsystem:
Figure BDA0001343017370000011
wherein, x ∈ Rn，z∈RlX and z are differential variables and algebraic variables respectively; u. ofi∈R，yi∈ R, i ═ 1, …, m, m is a positive integer tending to infinity, i is a variable subscript, uiAnd yiRespectively a control input and a control output;
Figure BDA0001343017370000021
for relating input variables, o is the dimension of the variable, f ∈ Rn，si∈Rn，g∈Rl，hi∈ R are smooth maps and n, l are variable dimensions.
The model is equivalently converted into a multi-input multi-output nonlinear ordinary differential system:
the vector relative order (gamma) of the multi-input multi-output nonlinear differential algebra subsystem is proposed1，γ2，…，γm)，γ1+γ2+…+γmN is the system dimension, γ1，…，γmRelative rank of each group of subsystems;
obtaining a differential homoembryo:
Figure BDA0001343017370000022
wherein the content of the first and second substances,
Figure BDA0001343017370000023
the expression sign of each group of variables of differential isoembryo, chi and g are algebraic equations, L represents lie derivative operation, gammaiFor the consistent relative rank of the ith subsystem,
Figure BDA0001343017370000024
Inrepresenting an n-order identity matrix;
obtaining state feedback:
Figure BDA0001343017370000025
wherein u is1，…，umFor each group of control inputs, r1…rmFor each set of sub-system controllers,
Figure BDA0001343017370000026
for non-linear terms in each set of system equations, i.e.
Figure BDA0001343017370000031
Therefore, the multi-input multi-output nonlinear differential algebraic subsystem is equivalently converted into a multi-input multi-output nonlinear ordinary differential system.
The step of obtaining the system stabilization controller comprises:
firstly, obtaining a subsystem controller corresponding to each ordinary differential subsystem:
for the qth sub-system
Figure BDA0001343017370000032
q is an integer between 0 and m:
defining error variables
Figure BDA0001343017370000033
Figure BDA0001343017370000034
Is gammaq-1 virtual controller, selecting the Lyapunov function as
Figure BDA0001343017370000035
Obtaining a subsystem controller r of the qth subsystemqI.e. by
Figure BDA0001343017370000036
Wherein the content of the first and second substances,
Figure BDA0001343017370000037
is a design parameter;
thereby obtaining a controller of the multi-input multi-output nonlinear differential algebra subsystem:
Figure BDA0001343017370000038
Figure BDA0001343017370000039
for a design parameter greater than 0 a and,
Figure BDA00013430173700000310
is an error variable;
and combining each group of subsystem controllers with state feedback to form a system stabilizing controller:
Figure BDA0001343017370000041
wherein u is1，…，umFor each group control input, h1，h2，…，hmIs output for each group.
The invention has the beneficial effects that: based on the prior single-input single-output nonlinear differential algebra subsystem stabilizing controller, the stabilizing controller of the multi-input multi-output nonlinear differential algebra subsystem is provided, so that the range of a controlled object is wider, the backstepping method is simpler and more practical, the model and the method have good effects when being applied to an electric power system, and the system tends to be more stable.
Drawings
Fig. 1 is a diagram of a design process of a calm controller of the present invention.
FIG. 2 is a subsystem state diagram of the present invention.
Detailed Description
The present invention will now be described in further detail with reference to the accompanying drawings.
As shown in fig. 1, the mimo nonlinear differential algebraic subsystem is equivalently converted into the mimo nonlinear ordinary differential system through differential homomorphism and state feedback, and then a stabilizing controller of a closed-loop large system is designed by using a backstepping method, so that the large system is gradually stabilized. The main structural module comprises: the system comprises a multi-input multi-output nonlinear differential algebra subsystem, a multi-input multi-output nonlinear ordinary differential system, a system stabilizing controller, a nonlinear differential algebra subsystem module, a nonlinear ordinary differential subsystem module and a subsystem controller module. When the multi-input multi-output nonlinear differential algebraic subsystem module has vector relative orders, the multi-input multi-output nonlinear differential algebraic subsystem module can be converted into the multi-input multi-output nonlinear ordinary differential system module through one differential homomorphism and state feedback. Each group of nonlinear ordinary differential subsystem modules can be designed into a subsystem controller module of each group of subsystem modules through a backstepping method. The large-system stabilizing controller module comprises all groups of subsystem controller modules, all the groups of subsystem controller modules and state feedback form the large-system stabilizing controller, and the multiple-input multiple-output nonlinear differential algebraic subsystem module is gradually stabilized through stabilizing control for derivation of a large-system Lyapunov function.
As can be seen in connection with fig. 2, the subsystem is associated with the rest of the large system by interface variables, the subsystem controller is composed of interface variable feedback, internal variable feedback and output feedback, the interface variables are generated by the effect of the rest of the large system on the subsystem and the effect of the subsystem on the rest of the large system.
The design method of the multi-input multi-output nonlinear differential algebraic subsystem stabilizing controller specifically comprises the following working steps.
First, model establishment
Modeling of multiple-input multiple-output non-linear differential algebraic subsystems, i.e.
Figure BDA0001343017370000051
Wherein x ∈ Rn，z∈RlAre respectively differential variable, algebraic variable, ui∈R，yi∈ R, i is 1, …, m (m is a positive integer tending to infinity, i is a variable subscript) is a control input and a control output respectively,
Figure BDA0001343017370000053
(o is the variable dimension) is the associated input variable reflecting the influence of the rest of the larger system on the subsystem, f ∈ Rn，si∈Rn，g∈Rl，hi∈ R (n, l are variable dimensions) are smooth mappings, while the multi-input multi-output nonlinear differential algebraic subsystem is required to be exponential 1, i.e. the algebraic equation g (-) is always full rank with respect to the Jacobian matrix for the algebraic variable z, and secondly, the input variables are correlated
Figure BDA0001343017370000054
And its sufficient order derivative are both locally bounded measurable signals.
In the specific embodiment, a dual-input dual-output nonlinear differential algebraic subsystem model of the synchronous generator is established, namely
Figure BDA0001343017370000052
Wherein the differential variable x ═ ω, E'q，PH)TThe brackets respectively indicate the power angle of the synchronous generator, the rotating speed of a generator rotor, q-axis transient potential and the output power of a high-pressure cylinder of the steam turbine; choosing algebraic variable z ═ (P)t，θU，Id，Iq)TIn the brackets are respectively the generator active power, the generator bus voltage phase angle and the generator endD-axis and q-axis components of the current; the associated input variable is selected as
Figure BDA0001343017370000055
The brackets are respectively the stator current of the generator and the reactive power of the generator; in the normal operating range of the generator, It，QtAnd
Figure BDA0001343017370000056
are all locally bounded testable; control input u ═ E (E)f，Uc)TThe brackets respectively indicate excitation electromotive force and valve opening; the control output is selected as h (·) ═ (V)tAnd brackets are the generator end voltage and the generator power angle respectively; h is mechanical moment of inertia, D is damping coefficient, omega0Is synchronous angular velocity, T'd0Is the transient time constant, x, of the field windingd，xq，x′d，x′qD-axis and q-axis synchronous reactance and transient reactance, rαIs armature resistance, CH，CML，Pm0The power distribution coefficients of the high, medium and low pressure cylinders, the total output power of the steam turbine and TH∑The equivalent time constants of the high-pressure cylinder of the steam turbine comprise the time constant of the high-pressure cylinder and the time constant of a hydraulic engine of the high-pressure cylinder.
Two, proposing vector relative order
The following symbols are defined, wherein F represents an operation, InRepresenting an identity matrix of order n, i.e.
Figure BDA0001343017370000061
If it is paired with
Figure BDA0001343017370000062
And i, j-1, …, m, k-0, …, γi-2, having the following equation (i, j are subscripts, respectively, k is 0 to γ)iA positive integer between-2, γiFor the uniform relative order of the ith subsystem, L represents the lie derivative operation):
1、
Figure BDA0001343017370000063
wherein
Figure BDA0001343017370000064
2. Matrix array
Figure BDA0001343017370000065
Is non-singular;
then the integer vector (gamma)1，γ2，…，γm) Is the vector relative order of the multi-input multi-output nonlinear differential algebra subsystem.
In a specific embodiment, the vector relative order of the synchronous generator dual-input dual-output nonlinear differential algebraic subsystem model is (1, 3).
Third, model transformation
When the multi-input multi-output nonlinear differential algebra subsystem has a vector relative order (gamma)1，…，γm) And gamma is1+γ2+…+γmN (n is the system dimension, γ)1，…，γmRelative order of each set of subsystems), then there is one differential isoembryo:
Figure BDA0001343017370000071
and there is one state feedback:
Figure BDA0001343017370000072
so that the large system can be equivalently converted into a multi-input multi-output nonlinear ordinary differential system, wherein,
Figure BDA0001343017370000074
representing symbols of each group of variables of the differential isoembryo, wherein x and g are algebraic equations; u. of1，…，umFor each group of control inputs, r1…rmFor each set of sub-system controllers,
Figure BDA0001343017370000075
for non-linear terms in each set of system equations, i.e.
Figure BDA0001343017370000073
In a specific embodiment, a synchronous generator dual-input dual-output nonlinear differential algebraic subsystem model selects a differential homomorphism:
Figure BDA0001343017370000081
and a state feedback:
Figure BDA0001343017370000082
the dual-input dual-output nonlinear differential algebraic subsystem of the synchronous generator is converted into a dual-input dual-output nonlinear ordinary differential system, namely
Figure BDA0001343017370000083
Design of stabilizing controller
Designing a subsystem controller of each subsystem, and for a qth subsystem (q is an integer between 0 and m):
Figure BDA0001343017370000084
step 4-1:
defining an error variable eq，1＝ξq，1，eq，2＝ξq，2-αq，1Wherein αq，1Is the first virtual controller to be designed. To eq，1The derivation is carried out to obtain the result,
Figure BDA0001343017370000085
selecting a first Lyapunov function
Figure BDA0001343017370000086
Then
Figure BDA0001343017370000087
And is
Figure BDA0001343017370000088
Wherein c isq，1> 0 is a design parameter. … …
Step 4-k (k is 2 to γ)q-an integer between 1):
definition of
Figure BDA0001343017370000091
Selecting Lyapunov functions
Figure BDA0001343017370000092
Then α is obtained by calculationq，kAnd
Figure BDA0001343017370000093
αq，kfor the k-th virtual controller, the virtual controller,
Figure BDA0001343017370000094
is the kth derivative of the Lyapunov function, cq，i> 0, i ═ 1, …, k is the design parameter, integer i is the parameter subscript, eq，k，eq，k+1As a function of error, i.e.
Figure BDA0001343017370000095
……
Step 4-. gamma.q：
Defining error variables
Figure BDA0001343017370000096
Figure BDA0001343017370000097
Is gammaq-1 virtual controller, selecting the Lyapunov function as
Figure BDA0001343017370000098
Then the subsystem controller r of the qth subsystem is availableqI.e. by
Figure BDA0001343017370000099
Wherein the content of the first and second substances,
Figure BDA00013430173700000910
in order to design the parameters of the device,
Figure BDA00013430173700000911
for error variables, a controller of a multi-input multi-output nonlinear differential algebraic subsystem is then available, i.e.
Figure BDA00013430173700000912
Wherein
Figure BDA00013430173700000913
For a design parameter greater than 0 a and,
Figure BDA00013430173700000914
is an error variable.
Combining each set of subsystem controllers with state feedback constitutes a large system calm controller, i.e. a large system calm controller
Figure BDA00013430173700000915
Wherein u is1，…，umFor each group control input, h1，h2，…，hmFor each group of output, the multiple-input multiple-output nonlinear differential algebraic subsystem is known to be asymptotically stable by differentiating the large system Lyapunov function.
In the large-scale system stabilizing controller, the power supply,
Figure BDA0001343017370000101
by a matrix
Figure BDA0001343017370000102
The inverse of the matrix may be obtained,
Figure BDA0001343017370000103
from step 4-. gammaqIt is possible to obtain,
Figure BDA0001343017370000104
and
Figure BDA0001343017370000105
derived from the state feedback in step 3.
In a specific embodiment, a controller of a dual-input dual-output nonlinear differential algebra subsystem of a synchronous generator is as follows:
Figure BDA0001343017370000106
substituting the controller into state feedback, wherein c1，1，c2，3For a parameter to be designed greater than 0, e1，1，e2，2，e2，3For error variable, the stabilizing controller of the dual-input dual-output nonlinear differential algebraic subsystem of the synchronous generator is as follows:
Figure BDA0001343017370000107
by derivation of a large-system Lyapunov function, the dual-input dual-output nonlinear differential algebraic subsystem of the synchronous generator is asymptotically stable.
The above is only a preferred embodiment of the present invention, and the protection scope of the present invention is not limited to the above-mentioned embodiments, and all technical solutions belonging to the idea of the present invention belong to the protection scope of the present invention. It should be noted that modifications and embellishments within the scope of the invention may be made by those skilled in the art without departing from the principle of the invention.

Claims (1)

1. A stabilization control method of a multi-input multi-output nonlinear differential algebraic subsystem comprises the following steps:
1) establishing a dual-input dual-output nonlinear differential algebraic subsystem model of the synchronous generator, i.e.
Figure FDA0002551754010000011
yi＝hi,i＝1,2
Wherein the differential variable x ═ ω, E'q,PH)TThe brackets respectively indicate the power angle of the synchronous generator, the rotating speed of a generator rotor, q-axis transient potential and the output power of a high-pressure cylinder of the steam turbine; choosing algebraic variable z ═ (P)t,θU,Id,Iq)TThe brackets are respectively the active power of the generator, the voltage phase angle of the generator bus and the d-axis and q-axis components of the generator terminal current; the associated input variable is selected as
Figure FDA0002551754010000012
The brackets are respectively the stator current of the generator and the reactive power of the generator; in the normal operating range of the generator, It，QtAnd
Figure FDA0002551754010000013
are all locally bounded testable; control input u ═ E (E)f,Uc)TThe brackets respectively indicate excitation electromotive force and valve opening; the control output is selected as h (·) ═ (V)tAnd brackets are the generator end voltage and the generator power angle respectively; h is mechanical moment of inertia, D is damping coefficient, omega0Is synchronous angular velocity, T'd0Is the transient time constant, x, of the field windingd,xq,x′d,x′qD-axis and q-axis synchronous reactance and transient reactance, rαIs armature resistance, CH,CML,Pm0The power distribution coefficients of the high pressure cylinder, the medium pressure cylinder and the low pressure cylinder and the total output work of the steam turbine are respectivelyRate, THΣThe equivalent time constants of the high-pressure cylinder of the steam turbine comprise the time constant of the high-pressure cylinder and the time constant of a hydraulic engine of the high-pressure cylinder;
2) the model is equivalently converted into a dual-input dual-output nonlinear ordinary differential system through differential homomorphism and state feedback, the dual-input dual-output nonlinear ordinary differential system comprises various groups of ordinary differential subsystems, and the method specifically comprises the following steps:
selecting a differential homomorphism for a synchronous generator double-input double-output nonlinear differential algebraic subsystem model:
Figure FDA0002551754010000021
and a state feedback:
Figure FDA0002551754010000022
wherein, ξ1,1,…,ξ2,3The expression sign of each group of variables of differential isoembryo, x and g are algebraic equations, L represents lie derivative operation, u is1,u2For each group of control inputs, r1,r2For each set of subsystem controllers, β1,1,β2,3Is a non-linear term in each set of system formula;
the dual-input dual-output nonlinear differential algebraic subsystem of the synchronous generator is converted into a dual-input dual-output nonlinear ordinary differential system, namely
Figure FDA0002551754010000023
3) The subsystem controller corresponding to the ordinary differential subsystem is obtained through a backstepping method, and each group of subsystem controllers and state feedback are combined to obtain a system stabilizing controller, which specifically comprises the following steps:
firstly, obtaining a subsystem controller corresponding to each ordinary differential subsystem:
for the qth sub-system
Figure FDA0002551754010000024
q is an integer between 0 and m:
defining error variables
Figure FDA0002551754010000025
Figure FDA0002551754010000026
Is gammaq-1 virtual controller, selecting the Lyapunov function as
Figure FDA0002551754010000027
Obtaining a subsystem controller r of the qth subsystemqI.e. by
Figure FDA0002551754010000028
Wherein the content of the first and second substances,
Figure FDA0002551754010000029
is a design parameter;
thereby obtaining a controller of the multi-input multi-output nonlinear differential algebra subsystem:
Figure FDA0002551754010000031
wherein the content of the first and second substances,
Figure FDA0002551754010000032
for a design parameter greater than 0 a and,
Figure FDA0002551754010000033
is an error variable; (gamma. rays)1,γ2,…,γm) Vector relative order of the multi-input multi-output nonlinear differential algebra subsystem;
and combining each group of subsystem controllers with state feedback to form a system stabilizing controller:
Figure FDA0002551754010000034
wherein u is1,…,umFor each group control input, h1,h2,…,hmOutputting for each group;
the controller of the double-input double-output nonlinear differential algebraic subsystem of the synchronous generator obtained from the above steps is as follows:
Figure FDA0002551754010000035
substituting the controller into state feedback, the stabilizing controller of the double-input double-output nonlinear differential algebraic subsystem of the synchronous generator is as follows:
Figure FDA0002551754010000036
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