CN108303876A - The Robust Tracking Control of spring-mass damper - Google Patents
The Robust Tracking Control of spring-mass damper Download PDFInfo
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- 238000010276 construction Methods 0.000 claims abstract description 5
- 239000011159 matrix material Substances 0.000 claims description 30
- 238000011156 evaluation Methods 0.000 claims description 18
- 238000013016 damping Methods 0.000 claims description 5
- 230000004913 activation Effects 0.000 claims description 3
- 230000017105 transposition Effects 0.000 claims description 2
- 230000006870 function Effects 0.000 description 44
- 238000013528 artificial neural network Methods 0.000 description 21
- 238000005457 optimization Methods 0.000 description 7
- 238000013461 design Methods 0.000 description 6
- 230000003190 augmentative effect Effects 0.000 description 5
- 210000002569 neuron Anatomy 0.000 description 5
- 238000005516 engineering process Methods 0.000 description 3
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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
- 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
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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
- G05B13/0265—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric the criterion being a learning criterion
- G05B13/027—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric the criterion being a learning criterion using neural networks only
Abstract
The present invention relates to Control of Nonlinear Systems technical fields, and in particular to a kind of Robust Tracking Control of spring-mass damper system, it is intended to solve the control problem in the case of nonlinear spring-mass damper system is disturbed.The method of the present invention carries out tracing control according to the robust tracking controller of spring-mass damper system；The construction method of tracking control unit includes：Obtain the parameter of spring-mass damper system；It determines the state variable of spring-mass damper system tracing control, establishes and be controlled model；Establish the dynamical equation of augmentation vector；Build utility function and cost function；Using adaptive dynamic programming method, the robust tracking controller of spring-mass damper system is built.The present invention can carry out adaptive robust control to extraneous unknown disturbances, improve spring-mass damper system track following performance.
Description
Technical field
The present invention relates to Control of Nonlinear Systems technical fields, and in particular to a kind of robust of spring-mass damper system
Tracking and controlling method.
Background technology
In control theory and engineering, when controlled device has uncertainty or is disturbed, robustness is assessment control
The major criterion of device performance processed.Since non-linear controlled device would generally be influenced by unknown disturbances, in order to improve nonlinear system
The performance of system control, some robust control methods have been suggested.For example, fuzzy control method, sliding-mode control, based on see
Survey the composite control method etc. of device.
Adaptive Dynamic Programming is a kind of near-optimization control method, including intensified learning and self-adaptive estimation mechanism, one
As realized by neural network, be traditional optimization intelligent control realize.In the case of being disturbed for nonlinear system
Control problem, the optimal solution for obtaining General Nonlinear Systems is difficult to realize, and adaptive dynamic programming method is a kind of effective
The method for designing near-optimization controller.
Invention content
In order to solve the above problem in the prior art, in order to solve nonlinear spring-mass damper system by
Control problem in the case of disturbance, an aspect of of the present present invention propose a kind of Robust tracking control of spring-mass damper system
Method, be applied to spring damper system, which is characterized in that according to the robust tracking controller of spring-mass damper system into
Line trace controls；
The robust tracking controller of the spring-mass damper system, construction method include：
Step 1, obtain the object block mass M of spring-mass damper system, the stiffness coefficient K of spring, system damping C,
Object block position P (t), object block speed V (t)；
Step 2, it determines the state variable x (t) of spring-mass damper system tracing control, establishes and be controlled model；
Step 3, it according to desired pursuit path r (t), calculates the tracking error z (t) of spring-mass damper system and increases
Wide vector ξ (t), establishes the dynamical equation of augmentation vector ξ (t)；
Step 4, the dynamical equation structure utility function U (ξ, t) and cost of the augmentation vector ξ (t) established based on step 3
Function J (ξ, t)；
Step 5, it is based on the utility function U (ξ, t) and the cost function J (ξ, t), using adaptive Dynamic Programming side
Method builds the robust tracking controller of spring-mass damper system
Further, the state variable x (t) is
X (t)=[x1(t),x2(t)]T=[P (t), V (t)]T
Wherein x1(t) it is the first component, x2(t) it is second component, P (t) is object block position, and V (t) is object block speed.
Further, the controlled model constructed in step 2, formula are expressed as
Wherein, u (t) is additional control force.
Further, " dynamical equation of augmentation vector ξ (t) " described in step 3 is：
Wherein, F (ξ (t)) is sytem matrix, and G (ξ (t)) is control matrix, and △ F (ξ (t)) are interference matrix；
Andλf(ξ) is the upper bound of △ F (ξ (t))；
△ f (x (t)) are the external disturbance that spring-mass damper system is subject to, | | △ f (x (t)) | |≤λf(x)。
Further, utility function U described in step 4 (ξ, t) is
U (ξ, u)=ξT(t)Qξ(t)+uT(t)Ru(t)
Wherein,
It is the non-negative diagonal matrix of 2n × 2n dimensions,It is the positive definite matrix of n × n dimensions, R is m × m
The positive definite matrix of dimension.
Further, cost function J described in step 4 (ξ, t) is
Wherein,
For the gradient of cost function J (ξ, t).
Further, the robust tracking controller of the spring-mass damper system built in step 5For
Wherein, GT(ξ) is the transposition of matrix G (ξ (t)),Indicate activation primitive σcThe gradient of (ξ),To comment
Weight matrix of the valence neural network hidden layer to output layer；The evaluation neural network is for obtaining optimal cost function J*(ξ(t))
Approximation
The invention has the advantages that：
(1) present invention devises spring-mass damper system to it is expected rail by using adaptive dynamic programming method
The robust tracking controller of mark carries out spring-mass damper system under conditions of there is interference and uncertainty adaptive
Robust control makes tracking error go to zero, to realize Robust tracking control of the spring-mass damper system to desired trajectory.
(2) present invention is commented by designing the utility function of spring-mass damper tracking error system, cost function, structure
Valence neural network simultaneously adjusts evaluation neural network weight according to cost function, has to the interference that spring-mass damper system is subject to
There is adaptive learning ability, is a kind of Robust Tracking Control with learning ability.
Description of the drawings
Fig. 1 is spring-mass damper system structural schematic diagram；
Fig. 2 is spring-mass damper system Robust tracking control algorithm structure schematic diagram；
Fig. 3 is Robust tracking control system evaluation network weight vectorThe convergence graph of preceding 5 components；
Fig. 4 is the curve graph of Robust tracking control system augmentation vector；
Fig. 5 is the control input of Robust tracking control systemCurve graph.
Specific implementation mode
The preferred embodiment of the present invention described with reference to the accompanying drawings.It will be apparent to a skilled person that this
A little embodiments are used only for explaining the technical principle of the present invention, it is not intended that limit the scope of the invention.
Present invention discusses the spring-mass damper systems comprising spring, quality module composition, are containing outer plus interference
In the case of uncertainty, according to the desired trajectory of frame of reference, robust tracking control is designed using adaptive dynamic programming method
Device processed makes spring-mass damper system can be good at tracking desired trajectory.
A kind of Robust Tracking Control of spring-mass damper system proposed by the present invention is applied to spring-damper
System carries out tracing control according to the robust tracking controller of spring-mass damper system；
The robust tracking controller of the spring-mass damper system, construction method include：
Step 1, obtain the object block mass M of spring-mass damper system, the stiffness coefficient K of spring, system damping C,
Object block position P (t), object block speed V (t)；
Step 2, it determines the state variable x (t) of spring-mass damper system tracing control, establishes and be controlled model；
Step 3, it according to desired pursuit path r (t), calculates the tracking error z (t) of spring-mass damper system and increases
Wide vector ξ (t), establishes the dynamical equation of augmentation vector ξ (t)；
Step 4, the dynamical equation structure utility function U (ξ, t) and cost of the augmentation vector ξ (t) established based on step 3
Function J (ξ, t)；
Step 5, it is based on the utility function U (ξ, t) and the cost function J (ξ, t), using adaptive Dynamic Programming side
Method builds the robust tracking controller of spring-mass damper system
Control method proposed by the present invention is to add interference and uncertainty containing outer, according to the phase of frame of reference
It hopes track, the robust tracking controller of spring-mass damper system is designed using adaptive dynamic programming method, makes spring matter
Amount damper system can be good at track desired trajectory, be meet Control of Nonlinear Systems technology application demand and development become
Gesture.
Below by the present invention robust tracking controller of spring mass damper system construction method carry out by
Step is described in detail.
Step 1, the object block mass M of spring-mass damper system, the stiffness coefficient of spring are obtained by way of measurement
K, damping C, object block position P (t), the object block speed V (t) of system.It is spring-mass damper system structural representation as shown in Figure 1
Figure.
Step 2, it determines the state variable x (t) of spring-mass damper system tracing control, establishes and be controlled model.
The state variable of spring-mass damper system tracing control be object block position P (t), object block speed V (t), therefore
The state variable x (t) of the tracked system constituted is as shown in formula (1)：
X (t)=[x1(t),x2(t)]T=[P (t), V (t)]T (1)
According to Newton interpolation algorithm, the spring-mass damper system of foundation is controlled model such as formula (2), formula (3) institute
Show：
Wherein F (t) is additional control force, is defined as u (t).The state variable x (t) of application definition, spring-mass damper
The controlled model of system can be expressed as formula (4), formula (5)：
Step 3, it according to desired pursuit path r (t), calculates the tracking error z (t) of spring-mass damper system and increases
Wide vector ξ (t), establishes the dynamical equation of augmentation vector ξ (t).It is as follows：
Step 31, to spring-mass damper system, desired pursuit path r (t) meets the differential equation shown in formula (6)：
Wherein, Indicate the real vector of two dimension.It is the derivative of desired trajectory r (t), is continuous letter
Number and
Step 32, the tracking error z (t) of spring-mass damper system is defined as shown in formula (7)：
Z (t)=x (t)-r (t) (7)
Step 33, convolution (4), (5), (6) and (7), in the case that there are certain external disturbance and indeterminate, with
Track error z (t) meets dynamical equation shown in formula (8)：
Wherein△f(x(t))
Indicate the external disturbance that spring-mass damper system is subject to, △ f (x (t)) boundeds and satisfaction | | △ f (x (t)) | |≤λf(x)。
Due to x (t)=z (t)+r (t), augmentation vector ξ (t)=[z is definedT(t),rT(t)]T, it is based on (6) and (8), can obtains
Augmentation vector ξ (t) meets the differential equation shown in formula (9)：
Wherein, F (ξ (t)) is sytem matrix, and G (ξ (t)) is control matrix, and △ F (ξ (t)) are interference matrix, respectively such as formula
(10), shown in formula (11), formula (12).
And Uncertainty △ F (ξ (t)) have the upper bound in the differential equation of augmentation vector ξ (t), are defined as λf(ξ), obtains
Formula (13),
Step 4, the dynamical equation structure utility function U (ξ, t) and cost of the augmentation vector ξ (t) established based on step 3
Function J (ξ, t).
Shown in the nominal system of augmentation vector ξ (t) dynamical equations such as formula (14)
To nominal system (14) design utility function U (ξ, u) as shown in formula (15)：
U (ξ, u)=ξT(t)Qξ(t)+uT(t)Ru(t) (15)
In formulaIt is the non-negative diagonal matrix of 2n × 2n dimensions,It is the positive definite matrix of n × n dimensions, R
It is the positive definite matrix of dimension of m m.
Cost function, expression (16) are further constituted with the utility function of formula (15)：
Wherein, J (ξ (t)) indicates the cost function of augmented system t moment,Indicate the ladder of cost function J (ξ (t))
Degree, Γ (ξ (t)) are taken as
Step 5, it is based on the utility function U (ξ, t) and the cost function J (ξ, t), using adaptive Dynamic Programming side
Method builds the robust tracking controller of spring-mass damper systemMainly include the following steps：
Step 51, according to formula (16), optimal cost function representation is formula (17)：
WhereinIt indicates, by choosing controlled quentity controlled variable u (t), to make
It is minimized, that is, optimal cost function J*(ξ(t)).Meanwhile J (ξ (t)) being made to take J*The u (t) of (ξ (t)) is referred to as optimal
Controlled quentity controlled variable is denoted as u*(t)。
Step 52, optimal control law u*It (t) can be by solving partial differential equationIt acquires, solves optimal control
System rule u*(t) it is expressed as formula (18)：
WhereinIndicate optimal cost function J*The gradient of (ξ (t)).
Step 53, due to optimal cost function J*(ξ (t)) and its gradientIt is difficult to obtain, therefore use adaptive
Dynamic programming method obtains J by evaluating neural network*The approximation of (ξ (t))And obtain optimal control law u*(t)
Approximation
The evaluation neural network of design include n input layer, l hidden neuron, 1 output layer neuron,
Evaluation neural network learning rate is αc, αc>0, excitation function σc(ξ), Indicate the real vector of l dimensions, input layer
Weights to hidden layer are taken as 1, and the ideal weight matrix of hidden layer to output layer is denoted as wc,In-service evaluation neural network
Reconstruct cost function J*(ξ (t)) is expressed as formula (19)
In formula, εc(ξ) is the reconstructed error of neural network.According to formula (19), optimal cost function J*The gradient table of (ξ (t))
It is shown as formula (20)：
Due to ideal weight wcUnknown, the weight matrix for defining hidden layer to the output layer of estimation isIt usesCome
Approximated cost function, as shown in formula (21)：
Wherein, It indicates with evaluation neural network weightApproximate calculation obtains optimal cost letter
Number.According to formula (21), near-optimization cost functionGradient be expressed as formula (22)：
Based on formula (18), (19) and (20) use ideal weight matrix wc, optimal control law u*(t) it is represented by formula
(23)：
Based on formula (18), (21) and (22) use estimation weight matrixNear-optimization control lawIt can indicate
For formula (24)：
Step 54, evaluation neural network obtains J*The approximation of (ξ (t))Definition evaluation neural network error ec
(t) the object function E trained with evaluation neural network weightc(t).By minimizing object function Ec(t), more New Appraisement nerve
The weight vector of networkIt specifically includes：
Step 541, the object function E of definition evaluation neural network weight trainingc(t).According to the theory of optimal control, can obtain
Formula (25)：
It enables
Then
Introduce ideal weight wcIt can obtain formula (26)：
Further obtain formula (27)：
Wherein：
A(ξ)≥0,B(ξ)≥0,εcHDefine ideal weight matrix wcThe reconstructed error of lower evaluation neural network.Using estimating
The weight matrix of meterCarry out approximate expressionIt is denoted asAs shown in formula (28)：
Definition evaluation neural network error be
Due toThenDefinition evaluation nerve net
The weights error vector of networkWithTo indicate ec(t), formula (29) can be obtained：
The object function of the Weight Training of definition evaluation neural network
Step 542, by minimizing object function Ec(t), weightsUpdate rule such as formula (30) shown in：
Wherein αc>0 is the learning rate for evaluating neural network, αs>0 is addition Item adjustment factor. Indicate JsThe gradient of (ξ).Fig. 2 is the structure chart of spring-mass damper system Robust tracking control.
By the near-optimization controller of nominal system (as shown in formula (14)), design without additional interference and under not knowingApplied to comprising additional interference and uncertain augmented system (as shown in formula (9)), realizes additional interference and do not know
Lower spring-mass damper Robust tracking control.
In order to make those skilled in the art more fully understand the present invention with reference to specific embodiments to the spring matter of the present invention
The Robust Tracking Control of amount damper system is described in detail.
According to step 1, the parameter of spring-mass damper system, the object block mass M of spring-mass damper system are measured
=1kg, the stiffness coefficient K=3N/m of spring, the damping C=0.5Ns/m of system, object block initial position P (0)=0.5m, object block
Initial velocity V (0)=0.5m/s.
According to step 2, the state variable of spring-mass damper system tracing control is object block position P (t), object block speed
The tracing control state variable of V (t), composition are x (t)=[x1(t),x2(t)]T=[P (t), V (t)]T, x1(t) it is first point
Amount, x2(t) it is second component.The controlled model of the state variable x (t) defined in this way, spring-mass damper system can
To be expressed as formula (31), formula (32)：
According to step 3, the tracking error z (t) and augmentation vector ξ (t) of spring-mass damper system are obtained.Spring-mass
The desired pursuit path r (t) of damper system meets formula (33)：
The tracking error for defining spring-mass damper system is z (t)=x (t)-r (t).Consider certain indeterminate,
And introduce augmentation vector ξ (t)=[zT(t),rT(t)]T, the dynamical equation for thus establishing augmentation vector ξ (t) is expressed as formula (34)：
△ F (ξ) are the extraneous unknown disturbances that augmented system is subject to.Set the original state of spring-mass damper system as
X (0)=[- 0.5,1.5]T, the original state of spring-mass damper system expectation tracking is r (0)=[0.5,0.5]T, therefore,
Original state ξ (the 0)=[- 1,1,0.5,0.5] of augmented systemT。
First state component ξ of augmented system1(t) initial value is ξ1(0)=- 1, second state component ξ2(t) initial
Value is ξ2(0)=1 ξ, is adjusted1(t) and ξ2(t) to 0, then realize spring-mass damper system to desired locations and speed tracing
Target.
In step 4, it is based on the dynamical equation (34) of augmentation vector ξ (t), design utility function U (ξ, t) and cost function J
(ξ,t).Further comprise following steps：
The nominal system of augmentation vector ξ (t) dynamical equations (34) is expressed as formula (35)
Design utility function is formula (36)：
U (ξ, u)=ξT(t)Qξ(t)+uT(t)u(t) (36)
Wherein Q is 4 × 4 unit matrix, and R=1, it is formula (37) further to constitute cost function：
According to step 5, the spring-mass damper system robust tracking based on adaptive dynamic programming method is designed
Controller.Given evaluation neural network input layer neuron number is n=4, hidden layer neuron number l=10, defeated
It is 1 to go out layer neuron number, learning rate αc=1.2 and addition Item adjustment factor αs=0.01, activation primitive is selected as The weights of input layer to hidden layer are
1, hidden layer is to the weight matrix between output layerBy weights
Training, Fig. 3 illustrate evaluation neural network hidden layer to output layer weight vectorFirst five component
Convergence process, weight vectorFinally converge to [0.5481,0.4121,0,0,0.4476,0,0,0.2030,0.1506,
0.1005]T.According to convergentValue and formula (24), obtain near-optimization control lawApply it to spring matter
In the Robust tracking control for measuring damper system, Fig. 4 is 30s time span inner spring mass damper system augmentation vectors ξ1
(t) and ξ2(t), corresponding tracking error is respectively z1(t)、z2(t), it can be seen that augmentation vector ξ1(t) and ξ2(t) from initial
State is ξ1(0)=- 1, ξ2(0)=1 0 is converged to, shows that spring-mass damper system position and speed arrives separately at expectation
Value.Fig. 5 is that original state is ξ1(0)=- 1, ξ2(0)=1 when, spring-mass damper system tracking control signalSong
Line chart.
Those skilled in the art should be able to recognize that, side described in conjunction with the examples disclosed in the embodiments of the present disclosure
Method step, can be realized with electronic hardware, computer software, or a combination of the two, in order to clearly demonstrate electronic hardware and
The interchangeability of software generally describes each exemplary composition and step according to function in the above description.These
Function is executed with electronic hardware or software mode actually, depends on the specific application and design constraint of technical solution.
Those skilled in the art can use different methods to achieve the described function each specific application, but this reality
Now it should not be considered as beyond the scope of the present invention.
So far, it has been combined preferred embodiment shown in the drawings and describes technical scheme of the present invention, still, this field
Technical staff is it is easily understood that protection scope of the present invention is expressly not limited to these specific implementation modes.Without departing from this
Under the premise of the principle of invention, those skilled in the art can make the relevant technologies feature equivalent change or replacement, these
Technical solution after change or replacement is fallen within protection scope of the present invention.
Claims (7)
1. a kind of Robust Tracking Control of spring-mass damper system, is applied to spring damper system, feature exists
According to the robust tracking controller of spring-mass damper system progress tracing control；
The robust tracking controller of the spring-mass damper system, construction method include：
Step 1, obtain the object block mass M of spring-mass damper system, the stiffness coefficient K of spring, system damping C, object block
Position P (t), object block speed V (t)；
Step 2, it determines the state variable x (t) of spring-mass damper system tracing control, establishes and be controlled model；
Step 3, according to desired pursuit path r (t), calculate spring-mass damper system tracking error z (t) and augmentation to
ξ (t) is measured, the dynamical equation of augmentation vector ξ (t) is established；
Step 4, the dynamical equation structure utility function U (ξ, t) and cost function of the augmentation vector ξ (t) established based on step 3
J(ξ,t)；
Step 5, it is based on the utility function U (ξ, t) and the cost function J (ξ, t), using adaptive dynamic programming method,
Build the robust tracking controller of spring-mass damper system
2. the Robust Tracking Control of spring-mass damper system according to claim 1, which is characterized in that step
The state variable x (t) of the 2 spring-mass damper system tracing controls is
X (t)=[x1(t),x2(t)]T=[P (t), V (t)]T
Wherein x1(t) it is the first component, x2(t) it is second component, P (t) is object block position, and V (t) is object block speed.
3. the Robust Tracking Control of spring-mass damper system according to claim 2, which is characterized in that step
Constructed controlled model, formula are expressed as in 2
Wherein, u (t) is additional control force.
4. the Robust Tracking Control of spring-mass damper system according to claim 3, which is characterized in that step
" dynamical equation of augmentation vector ξ (t) " described in 3 is：
Wherein, F (ξ (t)) is sytem matrix, and G (ξ (t)) is control matrix, and △ F (ξ (t)) are interference matrix；
Andλf(ξ) is the upper bound of △ F (ξ (t))；
△ f (x (t)) are the external disturbance that spring-mass damper system is subject to, | | △ f (x (t)) | |≤λf(x)。
5. the Robust Tracking Control of spring-mass damper system according to claim 4, which is characterized in that step
Utility function U described in 4 (ξ, t) is
U (ξ, u)=ξT(t)Qξ(t)+uT(t)Ru(t)
Wherein,
It is the non-negative diagonal matrix of 2n × 2n dimensions,It is the positive definite matrix of n × n dimensions, R is dimension of m m
Positive definite matrix.
6. the Robust Tracking Control of spring-mass damper system according to claim 5, which is characterized in that step
Cost function J described in 4 (ξ, t) is
Wherein,
For the gradient of cost function J (ξ (t)).
7. the Robust Tracking Control of spring-mass damper system according to claim 6, which is characterized in that step
The robust tracking controller of the spring-mass damper system built in 5For
Wherein, GT(ξ) is the transposition of matrix G (ξ (t)),Indicate activation primitive σcThe gradient of (ξ),For evaluation god
Weight matrix through network hidden layer to output layer.
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| CN109062032A (en) * | 2018-10-19 | 2018-12-21 | 江苏省（扬州）数控机床研究院 | A kind of robot PID impedance control method based on Approximate dynamic inversion |
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