CN110989358A - Control method of under-actuated system - Google Patents

Control method of under-actuated system Download PDF

Info

Publication number
CN110989358A
CN110989358A CN201911326840.6A CN201911326840A CN110989358A CN 110989358 A CN110989358 A CN 110989358A CN 201911326840 A CN201911326840 A CN 201911326840A CN 110989358 A CN110989358 A CN 110989358A
Authority
CN
China
Prior art keywords
model
output
input
full
under
Prior art date
Legal status (The legal status is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the status listed.)
Granted
Application number
CN201911326840.6A
Other languages
Chinese (zh)
Other versions
CN110989358B (en
Inventor
王蕊
高磊
孙辉
Current Assignee (The listed assignees may be inaccurate. Google has not performed a legal analysis and makes no representation or warranty as to the accuracy of the list.)
Civil Aviation University of China
Original Assignee
Civil Aviation University of China
Priority date (The priority date is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the date listed.)
Filing date
Publication date
Application filed by Civil Aviation University of China filed Critical Civil Aviation University of China
Priority to CN201911326840.6A priority Critical patent/CN110989358B/en
Publication of CN110989358A publication Critical patent/CN110989358A/en
Application granted granted Critical
Publication of CN110989358B publication Critical patent/CN110989358B/en
Active legal-status Critical Current
Anticipated expiration legal-status Critical

Links

Images

Classifications

    • G—PHYSICS
    • G05—CONTROLLING; REGULATING
    • G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
    • G05B13/00—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion
    • G05B13/02—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric
    • G05B13/04—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric involving the use of models or simulators
    • G05B13/042—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric involving the use of models or simulators in which a parameter or coefficient is automatically adjusted to optimise the performance

Landscapes

  • Engineering & Computer Science (AREA)
  • Health & Medical Sciences (AREA)
  • Artificial Intelligence (AREA)
  • Computer Vision & Pattern Recognition (AREA)
  • Evolutionary Computation (AREA)
  • Medical Informatics (AREA)
  • Software Systems (AREA)
  • Physics & Mathematics (AREA)
  • General Physics & Mathematics (AREA)
  • Automation & Control Theory (AREA)
  • Feedback Control In General (AREA)

Abstract

The invention relates to a control method of an under-actuated system, which comprises the following steps: constructing a preset under-actuated system, wherein a dynamic model between an input vector and an output vector exists in the preset under-actuated system, and converting the dynamic model into a state space expression based on model mapping; decoupling the state space expression into at least one full-drive subsystem through coordinate transformation; determining time-varying parameters of the full-drive subsystem, and establishing a U model; determining a desired output by the U model through a method of linear pole arrangement; and determining the control input quantity of the full-drive subsystem through backstepping iteration to form a U-model-based decoupling control closed-loop system.

Description

Control method of under-actuated system
Technical Field
The invention relates to the technical field of control, in particular to a control method of an under-actuated system.
Background
In the field of control, under-actuated systems are ubiquitous. The underdrive of the system refers to the fact that the dimension of the control input vector is smaller than the dimension of the system degree of freedom. Typically represented by aerospace vehicles, bat systems (inverted pendulum, TORA (translational), with a robotic actuator), robots, and most surface and underwater vessels [1-6 ]. The system saves partial drivers, reduces the complexity and energy consumption of the system, effectively reduces the structural weight and reduces the cost. However, the internal dynamics of the under-actuated system is complex, the coupling degree is high, the nonlinearity is strong, and great challenges are brought to control. Therefore, the research of the control method of the underactuated system has important value.
Disclosure of Invention
The invention provides a control method of an under-actuated system, which is used for solving the problems of complex internal dynamic characteristics, high coupling degree, strong nonlinearity and difficult control of the under-actuated system. A method for controlling an under-actuated system, comprising:
constructing a preset under-actuated system, wherein a dynamic model between an input vector and an output vector exists in the preset under-actuated system, and converting the dynamic model into a state space expression based on model mapping;
decoupling the state space expression into at least one full-drive subsystem through coordinate transformation;
determining time-varying parameters of the full-drive subsystem, and establishing a U model taking a pseudo linear expression as a representation form;
determining a desired output by the U model through a method of linear pole arrangement;
and determining the control input quantity of the full-drive subsystem through backstepping iteration to form a U-model-based decoupling control closed-loop system.
Further: constructing the under-actuated system, namely constructing a dynamic model of the under-actuated system on any m-dimensional input vector and n-dimensional output vector, wherein the method comprises the following steps:
Figure BDA0002328601920000021
wherein i ═ 1,2,3, … … n, j ═ 1,2,3, … … m; y isi＝[y1,y2,...yn]T∈RnIs the n-dimensional degree of freedom vector of the system,
Figure BDA0002328601920000022
is yiThe s-order partial derivative over time t;
Figure BDA0002328601920000023
is the m-dimensional control input vector of the system,fi(*),gijall are smooth nonlinear functions.
Further: the state space expression obtained by the dynamic model based on model mapping is as follows:
Figure BDA0002328601920000024
further: the state space expression is obtained through coordinate transformation, and the step of obtaining the full-drive subsystem is as follows:
decoupling the state space expression in a coordinate transformation mode to obtain a new coordinate system:
with the m-th entry g in the n-th degree of freedomnmumEliminating the mth input item g in the remaining n-1 degrees of freedom, respectively, as a toolimum(i＝1,2,...,n-1)
Similarly, the m-1 st entry g in the n-1 st degree of freedomn-1,m-1um-1To target, the m-1 st entry g in the remaining n-1 degrees of freedom is eliminatedim-1um-1(i＝1,2,...,n-2,n)
Repeating the calculating step m times until each degree of freedom only corresponds to one input item;
Figure BDA0002328601920000031
wherein, in the new coordinate system
Figure BDA0002328601920000032
Is given as an input quantity ujThe associated state quantity.
Further: and transforming the full-drive subsystem into a nonlinear function mode, acquiring the time-varying parameters of the full-drive subsystem, and expressing the time-varying parameters by a U model:
the U model of expressions in pseudolinear form: if the full drive subsystem is converted to a non-linear system expressed in polynomial form, such as:
yu(k)＝fu(yu(k-1),...,yu(k-ny),uu(k-1),...,uu(k-nu))；
wherein u isu(k) e.R is the input variable of the controlled object, nuIs uu(k) Power of yu(k) e.R is the output variable of the controlled object (likewise the output variable of the controller), nyIs yu(k) Power of fu(k) Is a reversible mapping in a discrete time domain, where k ∈ N+；
Combining the input and output quantity related terms into a time-varying parameter lambdap(k-1), deriving the pseudolinearity expression of the U model:
Figure BDA0002328601920000041
wherein r ∈ N+Is an input variable u of the controlled objectuPower of (k-1), λp(k-1) is a time-varying parameter term consisting of (u)u(k-2),...,uu(k-nu),yu(k-1),...,yu(k-ny) ) is expressed by the product of (a);
u model expressed in a state space approach:
if the full drive subsystem is converted to an n-order single-input single-output nonlinear discrete-time state space model,
x(k)＝f(x(k-1),u(k-1))
y(k)＝h(x(k))
wherein, x (k) e R, u (k) e R, y (k) e R are respectively the state variable of the controlled object, the system input quantity and the system output quantity, f (, h) is a smooth nonlinear function. According to the pseudo-linear expression of the U model, a U model space state expression can be written as follows:
Figure BDA0002328601920000042
wherein r isiN is xp+1The power of (k-1) and u (k-1).λip(k-1) is a time-varying parameter (p 0.. r.) incorporating a past-state termi)。
Further: a method of determining a desired output from the U model in the linear pole configuration:
determining, by a pole control configurator, that the U model is: au (k) ═ bw (k) — cy (k)
Wherein U (k) is the expected output value, w (k) is the reference value of the output, A, B, C are the values with
A polynomial of a feed forward operator l, the polynomial being:
Figure BDA0002328601920000051
wherein α, gamma are the orders of the A, B, C polynomials, respectively;
determining an analytical relationship between the output y (k) and the reference input w (k):
Figure BDA0002328601920000052
let AcCharacteristic polynomials for closed-loop transfer functions, in AcIn the case of a + C, the performance index of the control system is determined.
Further: the step of determining the control input quantity of the full-drive subsystem through backstepping iteration comprises the following steps: with the expected output determined, in Newton's overlap
The formula is obtained by substitution calculation:
Figure BDA0002328601920000053
wherein u (k-1) is the real output quantity, xn(k-1)...x1And (k-1) establishing a U-model-based decoupling control closed-loop system for each state quantity under the common use of pole output and root finding to obtain a closed-loop system from output to reference output.
Further: the U model decoupling control closed loop system comprises:
outputting, by the under-actuated system, at least one of the fully-actuated subsystems;
then the U model is established by the full-drive subsystem;
outputting the U model by the full-drive subsystem, sequentially passing through the pole control configurator and the root-finding controller, returning to the full-drive subsystem, and outputting, wherein,
the pole allocation controller brings an output reference value into the pole allocation controller and outputs an expected output value;
an expected output value is brought in front of the root finding controller; and outputs a true
Outputting the value; the real output value is brought into the full-drive subsystem again to obtain a value
And (6) outputting.
Additional features and advantages of the invention will be set forth in the description which follows, and in part will be obvious from the description, or may be learned by practice of the invention. The objectives and other advantages of the invention will be realized and attained by the structure particularly pointed out in the written description and drawings.
The technical solution of the present invention is further described in detail by the accompanying drawings and embodiments.
Drawings
The accompanying drawings, which are included to provide a further understanding of the invention and are incorporated in and constitute a part of this specification, illustrate embodiments of the invention and together with the description serve to explain the principles of the invention and not to limit the invention.
In the drawings:
fig. 1 is a flowchart of a method for controlling an under-actuated system according to an embodiment of the present invention;
FIG. 2 is a decoupling controller algorithm based on a U model according to an embodiment of the present invention;
FIG. 3 is a schematic diagram of a U-model-based decoupling control closed-loop system in an embodiment of the present invention;
FIG. 4 is a block diagram of a primary inverted pendulum according to an embodiment of the inverted pendulum system of the present invention;
figure 5 illustrates parameters x of the inverted pendulum system in an embodiment of the inverted pendulum system of the present invention,
Figure BDA0002328601920000061
θ,
Figure BDA0002328601920000062
a variation graph;
FIG. 6 is a diagram illustrating the variation of the output u of the inverted pendulum system controller according to an embodiment of the present invention;
FIG. 7 is a VTOL system in an embodiment of the present invention;
FIG. 8 is a diagram illustrating changes in VTOL system parameters based on the U model method in an embodiment of the present invention;
FIG. 9 is a diagram illustrating the output U of the VTOL system controller based on the U model method according to an embodiment of the present invention1,u2A variation graph;
fig. 10 is a comparison diagram of the VTOL system based on the U model and the VTOL system of the conventional sliding mode method according to the embodiment of the present invention.
Detailed Description
The preferred embodiments of the present invention will be described in conjunction with the accompanying drawings, and it will be understood that they are described herein for the purpose of illustration and explanation and not limitation.
FIG. 1 shows a flow chart of a control method of the present invention: a method of controlling an under-actuated system, comprising:
constructing a preset under-actuated system, wherein a dynamic model between an input vector and an output vector exists in the preset under-actuated system, and converting the dynamic model into a state space expression based on model mapping;
decoupling the state space expression into at least one full-drive subsystem through coordinate transformation;
determining time-varying parameters of the full-drive subsystem, and establishing a U model taking a pseudo linear expression as a representation form;
determining a desired output by the U model through a method of linear pole arrangement;
and determining the control input quantity of the full-drive subsystem through backstepping iteration to form a U-model-based decoupling control closed-loop system.
The decoupling control closed-loop system based on the U model is formed through model mapping, decoupling, U transformation, U model and pole configuration controller, and root-finding backstepping iteration, and the under-actuated system is controlled. The design algorithm of the under-actuated system controller is divided into four steps, as shown in fig. 2, which are decoupling, U transformation, design of the controller by a linear state space method and solving of the output quantity of the controller. Firstly, decoupling the input and output relationship of the system, and decomposing the input and output relationship into subsystems with input variables corresponding to the degrees of freedom. And then carrying out U conversion on the model, converting the model into a mathematical expression form of the U model, and selecting a proper method to carry out linear design on the pseudo-linear U model according to the design thought of the U model. And finally, solving a state space equation by a backstepping iteration method to obtain a solution output by the controller.
The invention has the beneficial effects that: the method can perform stabilization control or track tracking control on the under-actuated system, and can ensure that the closed-loop performance index of the under-actuated system meets the expected effect and the system is stable by prescribing the expected pole in the pole configuration method. Based on the invariant controller theory of the U model, the controller algorithm designed in the patent can be applied to objects with different complexity degrees, can effectively realize a stabilized control target and a track tracking control target, and avoids repeated design aiming at different controlled objects.
As an embodiment of the present invention: constructing the under-actuated system, namely constructing a dynamic model of the under-actuated system on any m-dimensional input vector and n-dimensional output vector, wherein the method comprises the following steps:
Figure BDA0002328601920000081
wherein i ═ 1,2,3, … … n, j ═ 1,2,3, … … m; y isi＝[y1,y2,...yn]T∈RnIs the n-dimensional degree of freedom vector of the system,
Figure BDA0002328601920000082
is yiThe s-order partial derivative over time t; u. ofj＝[u1,u2,...um]T∈RmIs the m-dimensional control input vector of the system, fi(*),gijAll are smooth nonlinear functions.
The principle of the embodiment is based on the establishment of a dynamic model of an under-actuated system, and a smooth nonlinear function is established by setting an output vector and an input vector.
The beneficial effect of this embodiment lies in: a linear function capable of carrying out space mapping can be established, the dynamic model can be mapped on a state space, coordinate transformation is carried out through an expression of the state space, and the U transformation difficulty is reduced.
As an embodiment of the present invention: the state space expression obtained by the dynamic model based on model mapping is as follows:
Figure BDA0002328601920000091
the spatial expression is obtained by mapping the dynamic model with a state space.
The principle of the embodiment is as follows: and based on the mapping mode of the state space, mapping the dynamic model on the state space to obtain a state space expression.
The method has the advantages that the state space expression capable of carrying out coordinate transformation can be obtained, and decoupling can be facilitated.
As an embodiment of the present invention: the method for obtaining the full-drive subsystem through coordinate transformation by the state space expression comprises the following steps:
decoupling the state space expression into at least one full-drive subsystem through coordinate transformation, and realizing the following steps:
with the m-th entry g in the n-th degree of freedomnmumEliminating the mth input item g in the remaining n-1 degrees of freedom, respectively, as a toolimum(i＝1,2,...,n-1)
Similarly, the m-1 st entry g in the n-1 st degree of freedomn-1,m-1um-1To target, the m-1 st entry g in the remaining n-1 degrees of freedom is eliminatedim-1um-1(i＝1,2,...,n-2,n)
Repeating the calculating step m times until each degree of freedom only corresponds to one input item;
Figure BDA0002328601920000101
wherein, in the new coordinate system
Figure BDA0002328601920000102
Is given as an input quantity ujThe associated state quantity.
As another embodiment of the present embodiment: selecting a second-order two-input three-output under-actuated system with an output dimension s of 2, a degree of freedom n of 3 and a control input dimension m of 2, wherein the system state space can be expressed as
Figure BDA0002328601920000103
The decoupled coordinate changes are as follows:
Figure BDA0002328601920000104
similarly, it can be seen that, through coordinate transformation, the original system is decoupled into a full-drive subsystem with the input vector dimension being the same as the system degree of freedom (both 1)
Figure BDA0002328601920000111
The principle of the invention is as follows: and (3) eliminating m-1 input items in the rest n-degrees of freedom by taking m input items in n degrees of freedom as tools, and then repeating the calculation steps until one degree of freedom corresponds to one input item. And then decoupling is carried out through coordinate transformation to obtain a new coordinate system which can be expressed through a formula.
The invention has the beneficial effects that: the calculation steps of eliminating m-1 input items in the rest n-free pairs are carried out by repeatedly using m input items in n degrees of freedom as tools to obtain a coordinate system which is decoupled by a coordinate transformation mode and corresponds to one degree of freedom and one input item, the input and the output in the coordinate system can be in one-to-one correspondence, and the under-actuated system can be decoupled into a plurality of full-actuated subsystems. The controller convenient to design can control any full-drive subsystem, and then control the under-actuated system.
As an embodiment of the present invention: obtaining a time-varying parameter by transforming the full-drive subsystem into a nonlinear function through the time-varying parameter of the full-drive subsystem, and expressing the time-varying parameter by a U model:
the U model of expressions in pseudolinear form: if the full drive subsystem is converted to a non-linear system expressed in polynomial form, such as:
yu(k)＝fu(yu(k-1),...,yu(k-ny),uu(k-1),...,uu(k-nu))；
wherein u isu(k) e.R is the input variable of the controlled object, nuIs uu(k) Power of yu(k) e.R is the output variable of the controlled object (likewise the output variable of the controller), nyIs yu(k) Power of fu(k) Is a reversible mapping in a discrete time domain, where k ∈ N+；
Combining the input and output quantity related terms into a time-varying parameter lambdap(k-1), deriving the pseudolinearity expression of the U model:
Figure BDA0002328601920000121
wherein r ∈ N+Is an input variable u of the controlled objectuPower of (k-1), λp(k-1) is a time-varying parameter term consisting of (u)u(k-2),...,uu(k-nu),yu(k-1),...,yu(k-ny) ) is expressed by the product of (a);
u model of system expressed by state space method:
if the full driver is converted to an n-order single-input single-output nonlinear discrete-time state space model,
x(k)＝f(x(k-1),u(k-1))
y(k)＝h(x(k))
wherein, x (k) e R, u (k) e R, y (k) e R are respectively the state variable of the controlled object, the system input quantity and the system output quantity, f (, h) is a smooth nonlinear function. According to the pseudo-linear expression of the U model, a U model space state expression can be written as follows:
Figure BDA0002328601920000122
wherein r isiN is xp+1Power of (k-1) and u (k-1), lambdaip(k-1) is a time-varying parameter (p 0.. r.) incorporating a past-state termi)。
The principle of the embodiment is as follows: in this embodiment, a U model taking a pseudo linear expression as a representation form is established by a method of separating controller design from object inversion and by time-varying parameters of a full-drive subsystem, and is also a nonlinear system U model in a polynomial form. And obtaining a state space building U model through a U model expression according to the pseudo linear expression, wherein the U model is also a single-input single-output nonlinear discrete time state space model and is expressed by a space state expression.
The beneficial effect of this embodiment lies in: in the design of a controller, a U model is established, and under the condition that the attributes such as the dynamic and transient performances, the input-output relation and the like of the original system are not changed, the transformation mode provides a case for a large class of nonlinear systems to use a linear control means; meanwhile, the U model is displayed as a nonlinear function in a polynomial form and a single-input single-output form, so that the U model is conveniently displayed.
As an embodiment of the present invention: a method of determining a desired output using the U model with a linear pole configuration:
determining, by a pole control configurator, that the U model is: au (k) ═ bw (k) — cy (k)
Where u (k) is the desired output value, w (k) is the reference value of the output, a, B, and C are polynomials with the feed forward operator l, said polynomials being:
Figure BDA0002328601920000131
wherein α, gamma are the orders of the A, B, C polynomials, respectively;
determining an analytical relationship between the output y (k) and the reference input w (k):
Figure BDA0002328601920000132
let AcCharacteristic polynomials for closed-loop transfer functions, in AcIn the case of a + C, the performance index of the control system is determined.
The principle of the embodiment is as follows: under the requirement of certain performance indexes, firstly, a closed-loop characteristic equation A is subjectedcDesigning in advance, solving A, B and C by using Diophanthinequality, and finally determining the design method of the pole allocation controller by U (k).
Under the condition that the desired output U (k) for good system performance requirements has been determined, the output of the controller can be found by the following formula, and an analytical formula of the output y (k) and the reference input w (k) can be obtained by defining the U model.
The beneficial effect of this embodiment lies in: by the pole placement method, a desired output is provided for the controller to be set up.
As an embodiment of the present invention: the step of determining the control input quantity of the full-drive subsystem through backstepping iteration comprises the following steps: from the desired output that has been determined, the formula is calculated by newton's iteration:
Figure BDA0002328601920000141
wherein u (k-1) is the real output quantity, xn(k-1)...x1(k-1) represents each state quantity.
The principle of the embodiment is as follows: in the case where the pseudo input U (k) is determined for the U-state space, the true input amount U (k-1) and the respective state amounts x can be determined by solving each state equation in turn through inverse iteration by making y (k) equal to U (k) and kn(k-1)...x1(k-1)。
And under the combined action of the pole configuration controller and the root finding controller, the controller can be regarded as a generalized controller of the decoupled system, and the output u (k-1) of the controller is the control input quantity of the decoupled full-drive subsystem.
The beneficial effect of this embodiment lies in: a base as shown in figure 3 is established
And the U model is used for controlling a closed loop system in a decoupling mode.
As an embodiment of the present invention: decoupling control closed loop system of U model
The method comprises the following steps:
outputting, by the under-actuated system, at least one of the fully-actuated subsystems;
then the U model is established by the full-drive subsystem;
the U model is output by the full-drive subsystem, passes through the pole control configurator and the root finding controller in sequence, returns to the full-drive subsystem and is output, wherein,
the pole allocation controller brings an output reference value into the pole allocation controller and outputs an expected output value;
an expected output value is brought in front of the root finding controller; and outputting a true output value; and the real output value is then brought into the full-drive subsystem to obtain an output.
The principle of the embodiment is as follows: and the output U (k-1) of the controller is the control input quantity of the fully-driven subsystem after decoupling, and finally, a decoupling control closed-loop system based on a U model is obtained.
The beneficial effect of this embodiment lies in: and generating a closed-loop control system, mapping the under-actuated system into at least one full-drive subsystem, and obtaining output corresponding to the full-drive subsystem through a U model after U transformation through the closed loop.
As an embodiment of the present invention: the method of the invention was verified using an inverted pendulum system.
In summary, the under-actuated system of the present invention can be decoupled into a fully-actuated subsystem, and for pseudo linear objects after U transformation, the control stability depends on the distribution of closed-loop poles of the system, and the dynamic performance of the system is also closely related to the positions of the closed-loop poles. A pole configuration feedback control law is selected, and the closed-loop pole is located at an expected position (the z plane is in a unit circle) by prescribing an expected system characteristic equation in advance, so that the closed-loop performance of the configured system can not only guarantee index requirements, but also guarantee the closed-loop stability of the system. In addition, when the generalized controller output is solved, the solution of the Newton iterative equation can be always obtained through the limiting conditions, the controller can ensure the stability under the condition of the solution, and similarly, the closed-loop feedback control system can also realize the stable performance requirement.
Verification of the examples: as shown in FIG. 4, the primary inverted pendulum is a classic under-actuated system, which is composed of a trolley and a pendulum rod arranged on the trolley.
Set the mass of the trolley as M1Mass of pendulum is M2The position of the trolley is x, the included angle between the pendulum and the vertical direction is theta, and the pendulum length is L1，L1＝2L2，I＝M2L2 2The moment of inertia of the pendulum about its center of gravity is/3 and u is the control input (here a force in the horizontal direction to the vehicle). The linear one-pole inverted pendulum mechanical model is
Figure BDA0002328601920000161
A control target: by inputting a control force u in the horizontal direction to the trolley, the trolley is driven from one to the otherThe initial position is set to move to the zero position, and the swing rod is kept vertically not to fall down, that is, the order of theta → 0,
Figure BDA0002328601920000162
x→0，
Figure BDA0002328601920000163
the controller design of the system is divided into four steps according to a controller design algorithm based on a U model.
Step 1: decoupled coordinate transformation
First, the system is rewritten as:
Figure BDA0002328601920000171
the corresponding coordinate transformation can be obtained as:
Figure BDA0002328601920000172
therefore, the original system is decoupled into a new system gamma with single input and single output according to the formula tillage step, and the input quantity u is only connected with the state quantity ξ2Correlation
Figure BDA0002328601920000173
Wherein the content of the first and second substances,
Figure BDA0002328601920000174
Figure BDA0002328601920000175
Figure BDA0002328601920000176
so that the control target of the original system is changed to order z1→0，z2→0，ξ1→0，ξ2→0。
Step 2: perform U conversion
Observing the above formula, the decoupled system can be divided into two subsystems, namely a subsystem gamma for determining the position of the trolley1And a subsystem gamma for determining the pendulum angle2。
For subsystem Γ1Its U is changed into
Figure BDA0002328601920000181
Wherein psi11＝1，ψ21＝T1x11Where z is2And x11Related, x11Is a subsystem F1Is input.
For subsystem Γ2Its U is changed into
Figure BDA0002328601920000182
Wherein psi31＝T2x11，ψ41＝T3。
And step 3: controller design based on U model
According to the formula pole control configurator, the natural frequency is selected to be 1rad/s and the damping ratio is selected to design a characteristic equation, namely Ac＝l2-1.3205l+0.4966，B＝Ac(1) 0.1761. In particular, let a be the convergence of the numerical values in order to ensure that U (k) is solved1＝-0.9,a2At 0.009, the corresponding C can also be determined from the diamantineequalification, and finally u (k) can be determined.
And 4, step 4: controller output solving
For subsystem Γ1Let U equal to z1And calculating pseudo input x by backstepping iteration11(ii) a For subsystem Γ2Let U equal to ξ1Similarly, the real input amount u of the inverted pendulum system is obtained in a reverse step.
Simulation experiments were performed on the control system in the MATLAB software environment, and various parameter settings of the inverted pendulum system are listed in table 1, and the results are shown in fig. 5 and fig. 6.
TABLE 1 initial value settings for parameters and variables of inverted pendulum system
Figure BDA0002328601920000191
As can be seen from fig. 5, under the influence of the decoupling controller, the four states x of the system,
Figure BDA0002328601920000192
θ,
Figure BDA0002328601920000193
after a certain time of adjustment, the output u of the controller is stable and converged to 0, as shown in fig. 6, and is also adjusted to 0 for a short time, so that the system is stable and meets the expected performance requirement.
As another example of validation: verifying the under-actuated system based on the U model through a VTOL system:
in order to verify the universality of the universal decoupling control algorithm of the under-actuated system based on the U model, an under-actuated VTOL aircraft system with three degrees of freedom and two system input quantities is selected, and the U model-based pole configuration controller design method which is the same as that of an inverted pendulum is used for carrying out experiments in an MATLAB simulation environment.
As shown in the VTOL system of figure 6, the dynamic equation can be expressed by combining Newton's law of mechanics
Figure BDA0002328601920000194
Assuming that the VTOL moves in a vertical plane (x, y), u1And u2For control input, the thrust torque and the rolling torque of the bottom of the aircraft are represented respectively; g is the acceleration of gravity; theta is a rolling angle (included angle of the wing relative to the horizontal line); ε is a description u1And u2Coefficient of coupling relationship therebetween, and ≠ 0.
To apply the decoupling controller design algorithm based on the U model provided by the invention, firstly, a dynamic model of the decoupling controller is converted into a state space form, wherein the designed state variables are as follows:
Figure BDA0002328601920000201
the system in the above equation can be represented as:
Figure BDA0002328601920000202
the control target is to control the rate u by design1And u2And the trajectory tracking of x and y is realized, and the theta is ensured to be calm within a certain range.
Similarly, with the decoupling control algorithm based on the U model proposed in chapter ii, the controller design of the VTOL aircraft system is divided into the following four steps:
step 1: decoupled coordinate transformation
Because the system is not consistent with a general algorithm form, the system is subjected to certain transformation and is divided into the following two subsystems for decoupling firstly
Figure BDA0002328601920000203
Figure BDA0002328601920000204
For the subsystem shown in the above formula, the order
Figure BDA0002328601920000211
Then the new variable is column written according to the decoupling algorithm
Figure BDA0002328601920000212
To z2The first derivative is obtained
Figure BDA0002328601920000213
Likewise, for subsystem, order
Figure BDA0002328601920000214
According to a decoupling algorithm, new variables can be written in columns
Figure BDA0002328601920000215
To s2The first derivative is obtained
Figure BDA0002328601920000216
Order to
Figure BDA0002328601920000217
The original VTOL system can be decoupled into a new system Θ having three single-input single-output subsystems
Figure BDA0002328601920000221
The control target is changed to pass the design u1，u2So that the new system is in a new coordinate system (z)1,s1) To perform tracking tasks and guarantee ξ1The calming of (1).
Step 2: u conversion
The decoupled new system is divided into three subsystems theta1，Θ2，Θ3. Wherein the subsystem Θ1,Θ2As a track control system, the subsystem Θ3Is a flight attitude control system.
For subsystem Θ1The state variable z1,z2Pseudo control input amount v1Therefore, according to the U transformation rule, the subsystem theta1In the form of U transformation of
Figure BDA0002328601920000222
Wherein psi11＝1，ψ21＝1。
For subsystem Θ2State variable s1,s2Pseudo input control quantity v2The U model form can be expressed according to the U transformation rule
Figure BDA0002328601920000223
Wherein psi31＝1，ψ40＝-g，ψ41＝1。
For subsystem Θ3The state variable is ξ1,ξ2The real input quantity of the system is u2For the same reason, the U transformation form is
Figure BDA0002328601920000224
Wherein psi51＝1，ψ61＝1。
And step 3: controller design based on U model
The same controller as in example 1, according to the above formula, only the pole is adjusted here, and the characteristic equation is selected as Ac＝l2-0.01l+0.01，B＝Ac(1) 1. Similarly, to ensure the convergence of the numerical values of U (k) solution, let a1＝-0.9，a2At 0.009, the corresponding C can also be determined from the diophaninequality, and finally u (k) is determined.
And 4, step 4: controller output solving
For subsystem Θ1Let U equal to z1And v is obtained by backstepping iteration1(ii) a Similarly, subsystem Θ2Let U be s1And v can be obtained in a reverse step2. Can be obtained by
Figure BDA0002328601920000231
Into the above-mentioned pair s2The first derivative can be calculated to obtain a formula, and the expected value of the rolling angle can be obtained
Figure BDA0002328601920000232
Substituting into pole allocation design formula, determining U (k) according to the same pole allocation method, and substituting into subsystem theta3Let U equal to ξ1Calculating u in reverse steps2。
Thus, the output u of the controller1And u2The solution is complete.
The initial values of the VTOL system are set in Table 2, g is 9.8m/s2, and epsilon is 0.01. The x variable and the y variable are respectively made to track a sine function with the amplitude value of 1 and the phase of 0rad, and the rolling angle theta of the sine function is guaranteed to be kept stable within a certain range.
TABLE 2 initial value settings of variables of VTOL system
Figure BDA0002328601920000233
Under the simulation environment of MATLAB, based on the set conditions and system parameters, two groups of simulation experiments are performed under the condition that the same initial conditions and the same parameters are the same and the same control target is achieved, namely a U-model-based decoupling control algorithm experiment and a traditional sliding mode control algorithm experiment are performed respectively, and the results are compared and analyzed.
The simulation results of the VTOL system obtained under the decoupling control algorithm based on the U model in fig. 8 and fig. 9 are shown. As can be seen from FIG. 8, the two states x and y follow the sinusoidal signal well, and the following error is approximately 0. The range of the roll angle θ is stabilized within 0.12, and the stabilization effect expected in the experiment is achieved.
Fig. 10 shows a control design of a VTOL system in a conventional sliding mode method, in which an SMC method overshoots a U model in tracking an x variable, and in tracking a y variable, the U model method overshoots a large value at an initial time, but can still better track a sinusoidal curve by recursion over time, and in stabilizing control over θ, the SMC method can also ensure the stabilization of θ, but the variation range of the SMC method is inferior to the U model method.
Through comparison of the two verification implementation control methods, it can be clearly seen that compared with the traditional sliding mode control algorithm, the decoupling control algorithm constructed in the method has the advantages that the performance meets the design requirement, the effect is stronger than that of an SMC method on certain indexes, and the stability of the system can be well ensured.
It will be apparent to those skilled in the art that various changes and modifications may be made in the present invention without departing from the spirit and scope of the invention. Thus, if such modifications and variations of the present invention fall within the scope of the claims of the present invention and their equivalents, the present invention is also intended to include such modifications and variations.

Claims (8)

1. A method for controlling an under-actuated system, comprising:
constructing a preset under-actuated system, wherein a dynamic model between an input vector and an output vector exists in the preset under-actuated system, and converting the dynamic model into a state space expression based on model mapping;
decoupling the state space expression into at least one full-drive subsystem through coordinate transformation;
transforming the full-drive subsystem into a nonlinear function form, acquiring time-varying parameters of the full-drive subsystem, and generating a U model;
determining expected output by the U model through a linear pole configuration method;
and performing backstepping iteration root solving on the full-drive subsystem, determining the control input quantity and the state quantity of the full-drive subsystem, and forming a U-model-based decoupling control closed-loop system according to the expected output.
2. The control method of the under-actuated system according to claim 1, characterized in that: the method comprises the following steps of constructing a preset under-actuated system, wherein a dynamic model between an input vector and an output vector exists, and setting an arbitrary m-dimensional input vector and an arbitrary n-dimensional output vector to obtain the following expression:
Figure FDA0002328601910000011
wherein i ═ 1,2,3, … … n, j ═ 1,2,3, … … m; y isi＝[y1,y2,...yn]T∈RnIs the n-dimensional degree of freedom vector of the system,
Figure FDA0002328601910000012
is yiThe s-order partial derivative over time t; u. ofj＝[u1,u2,...um]T∈RmIs the m-dimensional control input vector of the system, fi(*),gij(all) are smooth non-linear functions, wherein,
fi(xi) is a non-linear function related to the vector of degrees of freedom;
gij(. x) is a non-linear function related to the input vector.
3. The method of claim 1, wherein the model-based mapping transforms the dynamical model into a state space expression:
Figure FDA0002328601910000021
wherein the content of the first and second substances,
Figure FDA0002328601910000022
mapping state variables on a state space for the dynamical model;
and mapping the space expression to a state space by the dynamic model to obtain the state space expression.
4. The method for controlling an under-actuated system according to claim 1, wherein the state space expression is decoupled into at least one fully-actuated subsystem through coordinate transformation by the steps of:
with the m-th entry g in the n-th degree of freedomnmumEliminating the mth input item g in the remaining n-1 degrees of freedom, respectively, as a toolimum(i＝1,2,...,n-1)；
Similarly, the m-1 st entry g in the n-1 st degree of freedomn-1,m-1um-1To target, the m-1 st entry g in the remaining n-1 degrees of freedom is eliminatedim-1um-1(i＝1,2,...,n-2,n)；
Repeating the calculating step m times until each degree of freedom only corresponds to one input item;
decoupling the state space expression in a coordinate transformation mode to obtain a new coordinate system:
Figure FDA0002328601910000031
wherein, in the new coordinate system
Figure FDA0002328601910000032
Figure FDA0002328601910000033
Is given as an input quantity ujThe associated state quantity.
5. The method for controlling an under-actuated system according to claim 1, wherein the method for transforming the full-drive subsystem into a form of a nonlinear function, and obtaining the time-varying parameters of the full-drive subsystem, which are expressed by a U-model, comprises the following steps:
the U model of expressions in pseudolinear form: if the full drive subsystem is converted to a non-linear system expressed in polynomial form, such as:
yu(k)＝fu(yu(k-1),...,yu(k-ny),uu(k-1),...,uu(k-nu))；
wherein u isu(k) e.R is the input variable of the controlled object, nuIs uu(k) Power of yu(k) e.R is the output variable of the controlled object (likewise the output variable of the controller), nyIs yu(k) Power of fu(k) Is a reversible mapping in a discrete time domain, where k ∈ N+；
Combining the input and output quantity related terms into a time-varying parameter lambdap(k-1), deriving the pseudolinearity expression of the U model:
Figure FDA0002328601910000034
wherein r ∈ N+Is an input variable u of the controlled objectuPower of (k-1), λp(k-1) is a time-varying parameter term consisting of (u)u(k-2),...,uu(k-nu),yu(k-1),...,yu(k-ny) ) is expressed by the product of (a);
u model expressed in a state space approach:
converting into an n-order single-input single-output nonlinear discrete time state space model,
x(k)＝f(x(k-1),u(k-1))
y(k)＝h(x(k))
wherein, x (k) e R, U (k) e R, y (k) e R are respectively the state variable of the controlled object, the system input quantity and the system output quantity, f (·), h (·) is a smooth nonlinear function, and a U model space state expression is written according to the pseudo linear expression of the U model:
Figure FDA0002328601910000041
Figure FDA0002328601910000042
Figure FDA0002328601910000043
Figure FDA0002328601910000044
y(k)＝h(x(k))
wherein r isiN is xp+1Power of (k-1) and u (k-1), lambdaip(k-1) is a time-varying parameter (p 0.. r.) incorporating a past-state termi)。
6. The method for controlling an under-actuated system according to claim 1, wherein the method for determining the desired output by the U model in the linear pole configuration comprises:
applied to the U model by a pole control configurator, represented as: au (k) ═ bw (k) — cy (k)
And setting U (k) as an expected output value, and w (k) as an output reference value to obtain a u model expression: au (k) ═ bw (k) — cy (k)
A, B and C are polynomials with a feedforward operator l, and the polynomials are as follows:
Figure FDA0002328601910000051
wherein α, gamma are the orders of the A, B, C polynomials, respectively;
determining an analytical relationship between the output y (k) and the reference input w (k):
Figure FDA0002328601910000052
let AcCharacteristic polynomials for closed-loop transfer functions, in AcIn the case of a + C, the desired output of the control system is determined.
7. The method for controlling an under-actuated system according to claim 1, wherein the fully-driven subsystem determines the control input quantity and the state quantity of the fully-driven subsystem through backstepping iteration, and forms a U-model-based decoupled control closed-loop system according to the expected output, and comprises the following steps:
with the desired output determined, an iterative root calculation is performed in the root controller to obtain:
Figure FDA0002328601910000053
wherein u (k-1) is the real output quantity, xn(k-1)...x1(k-1) represents each state quantity;
and under the common use of the pole control configurator and the root finding controller, the real input quantity and the state quantity are introduced, a decoupling control closed-loop system based on a U model is established, and a closed-loop system from output to reference output is obtained.
8. The method for controlling an under-actuated system according to claim 7, wherein the decoupling control closed-loop system of the U model comprises: constructing a preset under-actuated system;
outputting, by the under-actuated system, at least one of the fully-actuated subsystems;
the U model is established by the full-drive subsystem;
the full-drive subsystem outputs the U model, sequentially passes through the pole control configurator and the root finding controller, returns to the full-drive subsystem and outputs the U model, wherein,
the pole allocation controller is used for carrying in a reference value of the output in front and outputting an expected output value;
the root finding controller brings one expected output value into front; and outputting a true output value; and the real output value is then brought into the full-drive subsystem to obtain an output.
CN201911326840.6A 2019-12-20 2019-12-20 Control method of under-actuated system Active CN110989358B (en)

Priority Applications (1)

Application Number Priority Date Filing Date Title
CN201911326840.6A CN110989358B (en) 2019-12-20 2019-12-20 Control method of under-actuated system

Applications Claiming Priority (1)

Application Number Priority Date Filing Date Title
CN201911326840.6A CN110989358B (en) 2019-12-20 2019-12-20 Control method of under-actuated system

Publications (2)

Publication Number Publication Date
CN110989358A true CN110989358A (en) 2020-04-10
CN110989358B CN110989358B (en) 2022-03-29

Family

ID=70074324

Family Applications (1)

Application Number Title Priority Date Filing Date
CN201911326840.6A Active CN110989358B (en) 2019-12-20 2019-12-20 Control method of under-actuated system

Country Status (1)

Country Link
CN (1) CN110989358B (en)

Cited By (2)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN111913391A (en) * 2020-08-12 2020-11-10 深圳职业技术学院 Method for stabilizing self-adaptive control discrete time non-minimum phase system
CN114114917A (en) * 2021-11-24 2022-03-01 哈尔滨工业大学 Backstepping control method based on pole allocation

Citations (2)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN108972560A (en) * 2018-08-23 2018-12-11 北京邮电大学 A kind of activation lacking mechanical arm Hierarchical sliding mode control method based on fuzzy optimization
CN109116732A (en) * 2018-08-02 2019-01-01 哈尔滨工程大学 A kind of drive lacking unmanned boat sliding formwork stabilized control method stable based on Hurwitz

Patent Citations (2)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN109116732A (en) * 2018-08-02 2019-01-01 哈尔滨工程大学 A kind of drive lacking unmanned boat sliding formwork stabilized control method stable based on Hurwitz
CN108972560A (en) * 2018-08-23 2018-12-11 北京邮电大学 A kind of activation lacking mechanical arm Hierarchical sliding mode control method based on fuzzy optimization

Non-Patent Citations (1)

* Cited by examiner, † Cited by third party
Title
NUR AFANDE ALI HUSSAIN等: "Underactuated nonlinear adaptive control approach using U-model for multivariable underwater glider control parameters", 《2016 IEEE INTERNATIONAL CONFERENCE ON UNDERWATER SYSTEM TECHNOLOGY: THEORY AND APPLICATIONS(USYS)》 *

Cited By (3)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN111913391A (en) * 2020-08-12 2020-11-10 深圳职业技术学院 Method for stabilizing self-adaptive control discrete time non-minimum phase system
CN111913391B (en) * 2020-08-12 2022-05-24 深圳职业技术学院 Method for stabilizing self-adaptive control discrete time non-minimum phase system
CN114114917A (en) * 2021-11-24 2022-03-01 哈尔滨工业大学 Backstepping control method based on pole allocation

Also Published As

Publication number Publication date
CN110989358B (en) 2022-03-29

Similar Documents

Publication Publication Date Title
Lee et al. Constrained autonomous precision landing via dual quaternions and model predictive control
Korayem et al. Finite-time state-dependent Riccati equation for time-varying nonaffine systems: Rigid and flexible joint manipulator control
Rudra et al. Nonlinear state feedback controller design for underactuated mechanical system: A modified block backstepping approach
Figueroa et al. Reinforcement learning for balancing a flying inverted pendulum
Lee Geometric controls for a tethered quadrotor UAV
CN112987567B (en) Fixed time self-adaptive neural network sliding mode control method of nonlinear system
Liu et al. Prescribed performance tracking control of a free-flying flexible-joint space robot with disturbances under input saturation
ul Amin et al. An adaptive trajectory tracking control of four rotor hover vehicle using extended normalized radial basis function network
Bu et al. Guaranteeing preselected tracking quality for air-breathing hypersonic non-affine models with an unknown control direction via concise neural control
CN106874617B (en) Efficient helicopter maneuvering flight quality grade evaluation method
CN110989358B (en) Control method of under-actuated system
Man et al. Nonlinear control design for a class of underactuated systems
Teng et al. An error-state model predictive control on connected matrix Lie groups for legged robot control
Selfridge et al. Multivariable output feedback MRAC for a quadrotor UAV
Zou et al. Spacecraft attitude control using two control torques
De Stefano et al. An energy-based approach for the multi-rate control of a manipulator on an actuated base
Hovakimyan et al. Adaptive dynamic inversion via time-scale separation
Heydari et al. Finite-horizon input-constrained nonlinear optimal control using single network adaptive critics
Zhang et al. Extreme learning machine assisted adaptive control of a quadrotor helicopter
Anderson et al. Predictive guidance and control for a tail-sitting unmanned aerial vehicle
Li et al. Almost disturbance decoupling control of MIMO nonlinear system subject to feedback linearization and a feedforward neural network: application to half-car active suspension system
Karthika et al. Nonlinear optimal control of a two wheeled self balancing robot
CN111176117B (en) Fuzzy self-adaptive elastic control method of unmanned helicopter
Necsulescu et al. Neural network based feedback linearization control of an unmanned aerial vehicle
Mahmoud et al. Linear and nonlinear stabilizing control of quadrotor UAV

Legal Events

Date Code Title Description
PB01 Publication
PB01 Publication
SE01 Entry into force of request for substantive examination
SE01 Entry into force of request for substantive examination
GR01 Patent grant
GR01 Patent grant