CN106681144A - Method for designing reduced-order observer of inverted pendulum - Google Patents

Method for designing reduced-order observer of inverted pendulum Download PDF

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CN106681144A
CN106681144A CN201611245424.XA CN201611245424A CN106681144A CN 106681144 A CN106681144 A CN 106681144A CN 201611245424 A CN201611245424 A CN 201611245424A CN 106681144 A CN106681144 A CN 106681144A
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inverted pendulum
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state
observer
control system
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李跃
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Deqing Shangwei Electromechanical Technology Co Ltd
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Deqing Shangwei Electromechanical Technology Co Ltd
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    • 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

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Abstract

The invention relates to the technical field of inverted pendulum control, in particular to a method for designing a reduced-order observer of an inverted pendulum. The method includes the steps that a movement differential equation set of an inverted pendulum control system is obtained and subjected to linear processing to obtain a state space equation of a controlled object; the reduced-order observer is constructed in the inverted pendulum control system and used for observing a state quantity which cannot be directly measured in the inverted pendulum; a feedback controller based on the reduced-order observer is designed, so that the inverted pendulum control system is stable. The method is capable of observing unmeasured state quantities in the inverted pendulum control system, so the observer is more targeted, the computation amount and design complexity are further reduced, and the concept of an observer design method is widened.

Description

A kind of reduced order observer method for designing of inverted pendulum
Technical field
The present invention relates to inverted pendulum control technical field, and in particular to a kind of reduced order observer method for designing of inverted pendulum.
Background technology
At present, inverted pendulum is an absolute instability, order high, multivariable, the nonlinear system of close coupling, from inverted pendulum Proposition is got up, and be applied to various control methods on inverted pendulum by people, and achieves good control effect.In engineer applied In, the research of inverted pendulum has real value, such as robot ambulation problem, the pose adjustment problem of rocket launching process, sea very much , all there is very big similitude the problems such as stability contorting of upper drilling platform with inverted pendulum, so the research of inverted pendulum control method is right The industries such as military project, space flight and robot field have very big directive function.
In to the design of reversible pendulum system controller, observer is highly important.In the control to inverted pendulum, lead to Frequently with PID control, STATE FEEDBACK CONTROL, Based Intelligent Control and other advanced control methods.But in Control System of Inverted Pendulum In some quantity of states be immesurable, the current most number controller of engineering field all uses feedback of status, and in actual feelings Some states are but unknown in condition, are now accomplished by observer reconfiguration system.Though such method is obtained in solving control system The problem that can not survey quantity of state is taken, but because its amount of calculation is extremely complex and no specific aim, therefore this kind of method is in inverted pendulum Existing defects in the control of system.
The content of the invention
For the deficiency of above-mentioned technology, the present invention devises a kind of reduced order observer method for designing of inverted pendulum.First obtain The differential equations of the Control System of Inverted Pendulum, and after being carried out linearization process, obtain the state of controlled device Space equation；A reduced order observer is constructed in the Control System of Inverted Pendulum, being used for observing in inverted pendulum can not be straight Connect the quantity of state that measurement is obtained；Finally designing a feedback controller based on reduced order observer makes Control System of Inverted Pendulum town It is fixed.The method is not only realized to that can not survey the observation of quantity of state in Control System of Inverted Pendulum, and observer is more directed to Property, reduce further the complexity of amount of calculation and design.
A kind of reduced order observer method for designing of inverted pendulum, specifically includes：
1) differential equations of the Control System of Inverted Pendulum, are obtained, and after being carried out linearization process, is obtained The state space equation of controlled device；
2) reduced order observer, is constructed in the Control System of Inverted Pendulum, being used for observing can not in inverted pendulum The quantity of state that direct measurement is obtained；
3) feedback controller based on reduced order observer, is constructed in the Control System of Inverted Pendulum；
Further, step 1) in, the method for obtaining the state space equation of controlled device is comprised the following steps：
(1.1) after it have ignored air drag and various frictional force, single inverted pendulum system can be abstracted into small The system of car and homogeneous bar composition, its differential equations are as follows：
Wherein, M is dolly quality, and m is swing rod quality, and b is dolly coefficient of friction, and l is swing rod center of rotation to barycenter Length, I is swing rod inertia, and F is the power being added on dolly, and x is small truck position, and θ is the angle of swing rod and vertical downward direction, N With the component both horizontally and vertically that P is dolly and swing rod interaction force；
Calculated and linearization process by by formula (1), can be obtained：
Wherein, φ be swing rod and vertically upward between angle；
(1.2) by formula (2), the state space equation that can obtain controlled device is as follows：
Wherein, x is displacement and speed, the angle of swing rod and the angle speed of the displacement of the quantity of state including dolly of the inverted pendulum Degree, u is the control gain of controlled device, and y is exported for the system of controlled device, and A is the dynamic matrix of controlled device, and B is controlled The input matrix of object, C is the output matrix of controlled device.
Further, step 2) in, a reduced order observer is constructed in the Control System of Inverted Pendulum, specifically include：
(2.1) according to formula (3), the quantity of state x of inverted pendulum is resolved into two parts：
Wherein, xaIt is the measured state amount in inverted pendulum, xbQuantity of state can not be surveyed in for inverted pendulum.
Make y=xa (5)
By the transition spatial decomposition of inverted pendulum into following form：
Wherein AaaAnd AabIt is the dynamic matrix of measured state amount, AbaAnd AbbFor the dynamic matrix of quantity of state, B can not be surveyedaFor The input matrix of measured state amount, BbFor the input matrix of quantity of state can not be surveyed.
(2.2) by formula (4), (5), (6) are set up with xbIt is the state-space model of quantity of state：
Choosing observation limit λ again, and obtain estimation by formula (7) can not survey quantity of state xbObserver：
Wherein, L is observer gain.
Further, step 3) in, a feedback based on reduced order observer is constructed in the Control System of Inverted Pendulum Controller；Specifically include：
(3.1) define：xb- Ly=w (9)
Can selecting system closed-loop pole z, and by formula (8), (9) are to the feedback controller based on reduced order observer：
Wherein,It is the quantity of state of reduced order observer, KaIt is the feedback oscillator of inverted pendulum measured state amount, KbIt is inverted pendulum The feedback oscillator of quantity of state can not be surveyed,
(3.2) the control gain u described by formula (10) makes Control System of Inverted Pendulum calm, and by reduced order observer The quantity of state that can not survey in Control System of Inverted Pendulum realizes observation.
The present invention devises a kind of reduced order observer method for designing of inverted pendulum.The method is not only realized to inverted pendulum control The observation of quantity of state can not be surveyed in system processed, and makes observer more targeted, reduce further amount of calculation with design Complexity, has widened the thinking of Design of Observer method.
Brief description of the drawings
Fig. 1 is the real-time experiment design sketch of reversible pendulum system control of the present invention.
Specific embodiment
To become apparent from the object, technical solutions and advantages of the present invention, below in conjunction with the accompanying drawings with actual experiment data pair Technical scheme is further described.
The present invention devises a kind of reduced order observer method for designing of inverted pendulum.First obtain the Control System of Inverted Pendulum Differential equations, and after being carried out linearization process, obtain the state space equation of controlled device；In the inverted pendulum A reduced order observer is constructed in control system, is used for observing in inverted pendulum the quantity of state for not directly measuring and obtaining；Most Designing a feedback controller based on reduced order observer afterwards makes Control System of Inverted Pendulum calm.The method is not only realized to falling Can not survey the observation of quantity of state in vertical oscillation control system, and make observer more targeted, reduce further amount of calculation and The complexity of design.
A kind of reduced order observer method for designing of inverted pendulum, specifically includes：
1) differential equations of the Control System of Inverted Pendulum, are obtained, and after being carried out linearization process, is obtained The state space equation of controlled device；
2) reduced order observer, is constructed in the Control System of Inverted Pendulum, being used for observing can not in inverted pendulum The quantity of state that direct measurement is obtained；
3) feedback controller based on reduced order observer, is constructed in the Control System of Inverted Pendulum；
Further, step 1) in, the method for obtaining the state space equation of controlled device is comprised the following steps：
(1.1) after it have ignored air drag and various frictional force, single inverted pendulum system can be abstracted into small The system of car and homogeneous bar composition, its differential equations are as follows：
Wherein, dolly quality is M=1.096kg, and swing rod quality is m=0.109kg, and dolly coefficient of friction is b=0.1N/ M/sec, the length of swing rod center of rotation to barycenter is l=0.25m, and swing rod inertia is I=0.00223kg*m*m, and gravity accelerates It is g=9.8m/s to spend2, F is the power being added on dolly, and x is small truck position, and θ is the angle of swing rod and vertical downward direction, N and P It is dolly and the component both horizontally and vertically of swing rod interaction force；
Calculated and linearization process by by formula (1), can be obtained：
Wherein, φ be swing rod and vertically upward between angle；
(1.2) by formula (2), the state space equation that can obtain controlled device is as follows：
Wherein, x is displacement and speed, the angle of swing rod and the angle speed of the displacement of the quantity of state including dolly of the inverted pendulum Degree, u is the control gain of controlled device, and y is exported for the system of controlled device, and the dynamic matrix of controlled device isThe input matrix of controlled device isThe output matrix of controlled device is
Further, step 2) in, a reduced order observer is constructed in the Control System of Inverted Pendulum, specifically include：
(2.1) according to formula (3), the quantity of state x of inverted pendulum is resolved into two parts：
Wherein, xaIt is the measured state amount in inverted pendulum, xbQuantity of state can not be surveyed in for inverted pendulum.
Make y=xa (5)
By the transition spatial decomposition of inverted pendulum into following form：
The dynamic matrix of wherein measured state amount isWithThe dynamic of quantity of state can not be surveyed Matrix isWithThe input matrix of measured state amount isQuantity of state can not be surveyed Input matrix is
(2.2) by formula (4), (5), (6) are set up with xbIt is the state-space model of quantity of state：
Observation limit λ is chosen again1=-10, λ2=-10, and estimation is obtained by formula (7) quantity of state x can not be surveyedbObserver：
Wherein, observer gain is
Further, step 3) in, a feedback based on reduced order observer is constructed in the Control System of Inverted Pendulum Controller；Specifically include：
(3.1) define：xb- Ly=w (9)
Can selecting system closed-loop pole z1,2=-3 ± 3j, z3=-5, z4=-8, and by formula (8), (9) to based on depression of order The feedback controller of observer is：
Wherein,It is the quantity of state of reduced order observer, the feedback oscillator of inverted pendulum measured state amount is Ka=[- 24.4898 63.2966], the feedback oscillator that inverted pendulum can not survey quantity of state is Kb=[- 16.1224 11.7075],
(3.2) as shown in figure 1, making Control System of Inverted Pendulum calm by the control gain u that formula (10) is described, and pass through Reduced order observer to Control System of Inverted Pendulum in the quantity of state that can not survey realize observation.
The present invention devises a kind of reduced order observer method for designing of inverted pendulum.The method is not only realized to inverted pendulum control The observation of quantity of state can not be surveyed in system processed, and makes observer more targeted, reduce further amount of calculation with design Complexity, has widened the thinking of Design of Observer method.
Described in detail above in association with accompanying drawing and set forth embodiments of the present invention, but be not limited to aforesaid way. In the ken that those skilled in the art possesses, as long as based on design of the invention, various changes can also be made Change and improve.

Claims (1)

1. a kind of reduced order observer method for designing of inverted pendulum, comprises the following steps：
1) differential equations of the Control System of Inverted Pendulum, are obtained, and after being carried out linearization process, is controlled The state space equation of object；Specifically include：
(1.1) after it have ignored air drag and various frictional force, can by single inverted pendulum system be abstracted into dolly and The system of homogeneous bar composition, its differential equations are as follows：
M x ·· = F - b x · - N N = m d 2 dt 2 ( x + l s i n θ ) P - m g = m d 2 dt 2 ( l c o s θ ) - P l sin θ - N l cos θ = I θ ·· - - - ( 1 )
Wherein, M is dolly quality, and m is swing rod quality, and b is dolly coefficient of friction, and l is length of the swing rod center of rotation to barycenter, I is swing rod inertia, and F is the power being added on dolly, and x is small truck position, and θ is the angle of swing rod and vertical downward direction, and N and P are The component both horizontally and vertically of dolly and swing rod interaction force；
Calculated and linearization process by by formula (1), can be obtained：
( I + ml 2 ) φ ·· - m g l φ = m l x ·· ( M + m ) x ·· + b x · - m l φ ·· = u - - - ( 2 )
Wherein, φ be swing rod and vertically upward between angle；
(1.2) by formula (2), the state space equation that can obtain controlled device is as follows：
x · = A x + B u y = C x - - - ( 3 )
Wherein, x is displacement and speed, the angle and angular speed of swing rod, u of the displacement of the quantity of state including dolly of the inverted pendulum It is the control gain of controlled device, y is exported for the system of controlled device, and A is the dynamic matrix of controlled device, and B is controlled device Input matrix, C for controlled device output matrix；
2) reduced order observer, is constructed in the Control System of Inverted Pendulum, is used for observing in inverted pendulum not directly The quantity of state that measurement is obtained；Specifically include：
(2.1) according to formula (3), the quantity of state x of inverted pendulum is resolved into two parts：
x = x a x b - - - ( 4 )
Wherein, xaIt is the measured state amount in inverted pendulum, xbQuantity of state can not be surveyed in for inverted pendulum；
Make y=xa (5)
By the transition spatial decomposition of inverted pendulum into following form：
x · a x · b = A a a A a b A b a A b b x a x b + B a B b u - - - ( 6 )
Wherein AaaAnd AabIt is the dynamic matrix of measured state amount, AbaAnd AbbFor the dynamic matrix of quantity of state, B can not be surveyedaFor that can survey The input matrix of quantity of state, BbFor the input matrix of quantity of state can not be surveyed；
(2.2) by formula (4), (5), (6) are set up with xbIt is the state-space model of quantity of state：
x · b = A b b x b + ( A b a x a + B b u ) x · a - A a a x a - B a u = A a b x b - - - ( 7 )
Choosing observation limit λ again, and obtain estimation by formula (7) can not survey quantity of state xbObserver：
x ~ · b - L y · = ( A b b - LA a b ) ( x ~ b - L y ) + [ ( A b b - LA a b ) L + A b a - LA a a ] y + ( B b - LB a ) u - - - ( 8 )
Wherein, L is observer gain；
3) feedback controller based on reduced order observer, is constructed in the Control System of Inverted Pendulum；Specifically include：
(3.1) define：xb- Ly=w (9)
Can selecting system closed-loop pole z, and by formula (8), (9) are to the feedback controller based on reduced order observer：
w ~ · = ( A ^ - F ^ K b ) w ~ + [ B ^ - F ^ ( K a + K b L ) ] y u = - K b w ~ - ( K a + K b L ) y - - - ( 10 )
Wherein,It is the quantity of state of reduced order observer, KaIt is the feedback oscillator of inverted pendulum measured state amount, KbFor inverted pendulum can not be surveyed The feedback oscillator of quantity of state,
(3.2) the control gain u described by formula (10) makes Control System of Inverted Pendulum calm, and by reduced order observer to falling The quantity of state that can not survey in vertical oscillation control system realizes observation.
CN201611245424.XA 2016-12-29 2016-12-29 Method for designing reduced-order observer of inverted pendulum Pending CN106681144A (en)

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Cited By (2)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN112083653A (en) * 2020-09-09 2020-12-15 中国人民解放军火箭军工程大学 Active disturbance rejection control design method of inverted pendulum system based on differential flatness
CN113050493A (en) * 2021-03-19 2021-06-29 大连理工大学 Output feedback control method for inverted pendulum system of trolley in networked environment

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Cited By (3)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN112083653A (en) * 2020-09-09 2020-12-15 中国人民解放军火箭军工程大学 Active disturbance rejection control design method of inverted pendulum system based on differential flatness
CN112083653B (en) * 2020-09-09 2022-11-11 中国人民解放军火箭军工程大学 Active disturbance rejection control design method of inverted pendulum system based on differential flatness
CN113050493A (en) * 2021-03-19 2021-06-29 大连理工大学 Output feedback control method for inverted pendulum system of trolley in networked environment

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Application publication date: 20170517