CN107490962A - A kind of servo-drive system method for optimally controlling of data-driven - Google Patents
A kind of servo-drive system method for optimally controlling of data-driven Download PDFInfo
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Abstract
The present invention specifically includes state error feedback control and the adaptive congestion control algorithm based on Policy iteration, wherein state error feedback control can calculate feedback oscillator K in real time using sampled data using a kind of servo-drive system method for optimally controlling of data-driven；Controlled quentity controlled variable and state error amount obtained by the direct use state error feedback control of adaptive congestion control algorithm device are iterated calculating and gradually approached to optimum control amount；This method requires no knowledge about specific system model, it is only necessary to which state error amount can be surveyed.
Description
Technical field
The present invention relates to direct current generator servo-control system, belong to technical field of electromechanical control, and in particular to a kind of data
The servo-drive system method for optimally controlling of driving.
Background technology
In servo system control, due to dynamic friction, load change and the presence of the uncertain factor such as external disturbance,
System model can change.If in addition, unknown parameter in control system be present, to the initial estimate possibility of these parameters
Differ larger with its actual value.Therefore, if can not accurately obtain system model in control process, it is necessary to which research is a kind of not
Control algolithm dependent on system model.
The control algolithm of data-driven directly designs controller using sampled data alternative system model, and wherein data are driven
Dynamic adaptive dynamic programming algorithm is a kind of effective method for optimally controlling.The adaptive thoery of dynamic programming of data-driven is melted
The methods of having closed DP, RL and approximation to function, essence are exactly using online or off-line data, are estimated using approximation to function structure
System performance index function, then obtain optimal control strategy according to the principle of optimization.
The content of the invention
In view of this, Accurate Model problem is unable to for servo-drive system, the present invention uses a kind of servo system of data-driven
System method for optimally controlling, this method require no knowledge about specific system model, it is only necessary to which state error amount can be surveyed.
The present invention mentality of designing be：
If direct current generator servo error equation is：
Wherein, u is controlled quentity controlled variable, and e is state error；
It is as follows to define cost function：
Wherein, r (e, u)=Q (e)+uTRu, Q are positive definite, and R is positive definite symmetric matrices；
Final state of a control error e converges to 0 under conditions of cost function is optimal, using obtained controlled quentity controlled variable u to watching
Dress system is controlled.
Realize that technical scheme is as follows：
A kind of servo-drive system method for optimally controlling of data-driven, include the state error feedback control sum of data-driven
According to adaptive congestion control algorithm two parts of driving：
Part I：The state error feedback control of data-driven
The 101st, k=1, initialization kth -1, system feedback gain K (k-1), the K (k) at k moment are set；
102 on time interval [kT, (k+1) T], will order using the system feedback gain K (k-1) and K (k) in watching
Dress system；
103rd, the state error of sampling servosystem, and calculate state error relation square according to formula (10), (12) to (15)
Battle array A (k), geAnd g (k)u(k)；
A (k)=A0(k)A1(k) (10)
A1(k)=e ((k+1/2) T) (e (kT))+ (12)
A0(k)=e ((k+1) T) (e ((k+1/2) T))+ (13)
Then can obtain：
gu(k)=(A0(k)-A1(k))(K(k)-K(k-1))+ (14)
ge(k)=A0(k)-gu(k)·K(k) (15)
Wherein, e (kT), e ((k+1/2) T) represent kth, (k+1/2) moment system mode error, symbol+Represent matrix
Pseudoinverse；
104th, K (k+1) and K (k+2) are calculated according to formula (11)；
Wherein, λ represents setting step-length, the constant between 0-1；Symbol | | | | the Frobenius norms of representing matrix.
105th, judge whether k is less than given threshold, i.e. whether the kth moment has had arrived at the deadline T1 of setting, if
It is to enter Part II, otherwise makes k=k+1, return to step 102；
In the first portion, state error feedback control amount constantly adjusts while application, the input u of acquisition0And shape
State error information e can be used for the adaptive congestion control algorithm device of next part.
Part II：The adaptive congestion control algorithm of data-driven
J=0 is set, defines φki、uj,Expression formula it is as follows
φki=φi(e(tk))-φi(e(tk-1))
Wherein j is iterations；φi(e), (i=1 ... Nw), ψi(e, u), (i=1 ... Nc), σi(e), (i=
1,...Nl) it is basic function；For weights, u0For state error feedback control amount,For uj(e) estimation
Value,For function uj(e) approximate error between actual value and estimate.
201st, based on the system mode error obtained in Part I, φ is calculatedkiWithThen obtain
202nd, based on obtaining in step 201Weights are calculated by formula (22)
Wherein M is positive integer.
Weights are calculated by formula (23)
Wherein l is weightsAny probable value；
203rd, judgeWhether set up, if so, into step 204, otherwise make j=j+1, return to step
201, ε be given threshold value；
204、For the approximation of optimum control amount, servo-drive system is controlled using it in real time afterwards.
Beneficial effect
This method includes state error feedback control and the adaptive congestion control algorithm based on Policy iteration, wherein state error
Feedback control can calculate feedback oscillator K in real time using sampled data；The direct use state error of adaptive congestion control algorithm device is anti-
The controlled quentity controlled variable and state error amount of feedback control gained are iterated calculating and gradually approached to optimum control amount；This method need not be known
The specific system model in road, it is only necessary to which state error amount can be surveyed.
Brief description of the drawings
Fig. 1 is the servo-drive system method for optimally controlling flow chart of data-driven.
Fig. 2 is initial feedback Gain tuning figure.
Fig. 3 is that MATLAB emulates cost function comparison diagram.
Fig. 4 is MATLAB simulation status track comparison diagram.
Fig. 5 is that the MATLAB simulation status track being applied to two controlled quentity controlled variables in real time after the completion of calculating obtained by system contrasts
Figure.
Embodiment
The present invention is described in detail with instantiation below in conjunction with the accompanying drawings.
The mathematical modeling of certain direct current generator servo-drive system can be write:
Wherein, x (t)=[x1(t),x2(t)]TIt is motor output Angle Position, rotor velocity respectively, the quantity of state is
It is measurable；U is controlled quentity controlled variable；F (x, u) is unknown, meets F (0,0)=0.
Present invention design method for optimally controlling causes motor servo system to reach expectation state xd=[x1d,0]T, that is, track
Step signal, and now ud=0.State error is defined as：E=x-xdAnd e0=x0-xd.Error system equation is：
State of a control error convergence of the invention final is to 0.
For optimum control, it is necessary to which to define cost function as follows：
Wherein r (e, u)=Q (e)+uTRu, Q are positive definite, and R is positive definite symmetric matrices.Make for the cost function as follows
Assuming that：
Assuming that 1：Cost function (3) value is relevant with controlled quentity controlled variable u, for each e0There is unique minimum.
Assuming that 2：Cost function (3) meets for the control input u of any zonal cooling
The servo-drive system optimal control system structure of data-driven is as shown in figure 1, by state error feedback controller and certainly
Optimal controller two parts composition is adapted to, is specifically addressed separately below.
1. state error feedback controller.
For unknown servo-drive system, system information is all lain in state error and input information, is adopted by system information
Sample can establish data below relational expression：
E ((k+1) T)=g (e (kT), u (kT)) (4)
Wherein g (e, u) is unknown, k=0, and 1,2..., T is the sampling period, and formula (4) can be reduced to by omitting T：
E (k+1)=g (e (k), u (k)) (5)
The equation includes state error and input signal, discloses the internal feature of system (2), and it is suitable can be used directly to search
Feedback control amount.In view of system (2) and system (4) represent same system, g (0,0)=0 can be obtained.
In order that state error reaches 0, in arbitrfary point e (k) and Lagrangian mean value theorem is applied at 0 point to formula (5), can be with
Obtain：
E (k+1)=ge(k)e(k)+gu(k)u(k) (6)
Wherein0 ＜ θ ＜ 1.On the other hand, make such as
Lower hypothesis：
Assuming that 3：For any k ∈ N, on each time interval [kT, (k+1) T], matrix guAnd g (k)e(k) keep not
Become.
To ensure that the hypothesis is set up, it should choose sufficiently small sampling period T.
Time interval [kT, (k+1) T] is divided into two parts, at Part I [kT, (k+1/2) T], using feedback oscillator K
(k-1), it is in Part II application feedback oscillator K (k), i.e. design point error feedback control amount：
According to 3 and formula (6) is assumed, it is as follows that state error relational expression can be obtained：
E ((k+1/2) T)=[ge(k)vgu(k) K (k-1)] e (kT)=A1(k)e(kT) (8)
E ((k+1) T)=[ge(k)+gu(k) K (k)] e ((k+1/2) T)=A0(k)e((k+1/2)T) (9)
Therefore can obtain：
E (k+1)=A0(k)A1(k) e (k)=A (k) e (k) (10)
It will be apparent that by adjust A (k) can state error e converge to 0, can by feedback gain K (k)
To adjust A (k), it is as follows to define K (k) regulation rate：
Wherein λ is step-length, 0 ＜ λ ＜ 1, it should choose smaller value and ensure e convergences, but can not be too small to ensure search efficiency；
Symbol | | | | represent Frobenius norms；The pseudoinverse of symbol+represent matrix.When | | A (k) | | when >=1, feedback oscillator K begins
Eventually according to formula (11) constantly regulate, wherein required real-time matrix guAnd g (k)e(k) can ask for the following method.
State error relational matrix A1And A (k)0(k) can be tried to achieve with the state error measured by formula (8) and (9)：
A1(k)=e ((k+1/2) T) (e (kT))+ (12)
A0(k)=e ((k+1) T) (e ((k+1/2) T))+ (13)
Then can obtain：
gu(k)=(A0(k)-A1(k))(K(k)-K(k-1))+ (14)
ge(k)=A0(k)-gu(k)·K(k) (15)
When | | A (k) | | when >=1, feedback oscillator K (k) arrives (15) continuous real-time update according to formula (11)；When | | A (k) | | ＜
When 1, K keeps constant.In this process, state error feedback control amount constantly adjusts while application, the input u of acquisition0
It can be used for the adaptive congestion control algorithm device of next part with state error data e.
2. adaptive congestion control algorithm device.
Using caused data during upper one, the partial adaptivity optimal controller uses adaptive Dynamic Programming strategy
The method of iteration solves servo-drive system optimal control problem.
For above-mentioned state error system, basic Policy iteration method is as follows：
(1) V is solved from following formulaj(e(t))：
R (e, uj(e))+(Ve j(e))T(f (e, uj(e)))=0, Vj(0)=0 (16)
Wherein j is iterations.
(2) controlled quentity controlled variable is calculated by following formula：
Selection time sequence tk, k=0,1 ..., 0 ＜ t of satisfaction1＜ t2＜ ... ＜ tM＜ ∞, wherein M are sufficiently large whole
Number.Defined (3) from cost function：
It can be converted into：
V unknown in above formula is approached with following function structure belowj(e),And uj(e), it is expressed as：
WhereinFor weights；φi(e), (i=
1,...Nw), ψi(e, u), (i=1 ... Nc), σi(e), (i=1 ... Nl) it is basic function；
Respectively three function V are approached with function structurej(e),And uj(e) approximate error.Therefore formula (18) can be write
Make：
WhereinIt is approximate error；
For the sake of convenient, it is defined as follows：
φki=φi(e(tk))-φi(e(tk-1))
Then (19) can be abbreviated as：
Without loss of generality, make the following assumptions：
Assuming that 4：For all uj, existWith γ ＞ 0 so that for allFollowing inequality into
It is vertical：
Therefore weight matrix can be solved by following formula according to formula (20)：
According to formula (17), can solve to obtain weights
Using existing sampled data, constantly iterated to calculate by formula (22) and (23), as t, Nw、NcAnd NlAll level off to nothing
When poor, the respective optimal value of weight convergence, i.e., each unknown function converges to its optimal value.
In summary, the servo-drive system method for optimally controlling of data-driven is as follows：
Part I：The state error feedback control of data-driven
The 101st, k=1, initialization kth -1, system feedback gain K (k-1), the K (k) at k moment are set；
102 on time interval [kT, (k+1) T], will order using the system feedback gain K (k-1) and K (k) in watching
Dress system；
103rd, the state error of sampling servosystem, and calculate state error relation square according to formula (10), (12) to (15)
Battle array A (k), geAnd g (k)u(k)；
A (k)=A0(k)A1(k) (10)
A1(k)=e ((k+1/2) T) (e (kT))+ (12)
A0(k)=e ((k+1) T) (e ((k+1/2) T))+ (13)
Then can obtain：
gu(k)=(A0(k)-A1(k))(K(k)-K(k-1))+ (14)
ge(k)=A0(k)-gu(k)·K(k) (15)
Wherein, e (kT), e ((k+1/2) T) represent kth, (k+1/2) moment system mode error, symbol+Represent matrix
Pseudoinverse；
104th, K (k+1) and K (k+2) are calculated according to formula (11)；
Wherein, λ represents setting step-length, the constant between 0-1
105th, judge whether k is less than given threshold, i.e. whether the kth moment has had arrived at the deadline T1 of setting, if
It is to enter Part II, otherwise makes k=k+1, return to step 102.
Performed Part I obtain a series of system mode error es (kT), e ((k+1/2) T) and system feedback control it is defeated
Enter u0Adaptive congestion control algorithm device for next part.
Part II：The adaptive congestion control algorithm of data-driven
J=0 is set, defines φki、uj,Expression formula it is as follows
φki=φi(e(tk))-φi(e(tk-1))
Wherein j is iterations；φi(e), (i=1 ... Nw), ψi(e, u), (i=1 ... Nc), σi(e), (i=
1,...Nl) it is basic function；For weights；For uj(e) estimate,For function uj(e) actual value
Approximate error between estimate.
201st, based on the system mode error obtained in Part I, φ is calculatedkiWithThen obtain
202nd, based on obtaining in step 201Weights are calculated by formula (22)
Wherein M is positive integer.
Weights are calculated by formula (23)
Wherein l is weightsAny probable value.
203rd, judgeWhether set up, if so, into step 204, otherwise make j=j+1, return to step
201；When returning every time, by what is be calculated in this circulation 202Bring into next circulation step 201 and calculate, ε is given
Threshold value.
204、For the approximation of optimum control amount, servo-drive system is controlled using it in real time afterwards.
The flow chart of whole process is as shown in Figure 2.
The above method is emulated using matlab.Consider the motor servo system with following second-order model：
The model parameter of system is arranged to：θ1=-2, θ2=-5, θ3=1, θ4=0and Ks=900.
Cost function is defined as：
System initial value is arranged to：x0=[- 1 0]T, K (0)=[- 1 0], K (1)=[0-1].
Part I adoption status error feedback controller, the stage are arranged to 1.5s.What Fig. 3 was represented is anti-during being somebody's turn to do
Feedforward gain K adjustment figure line.
Part II uses adaptive congestion control algorithm device, the data directly obtained using Part I, uses polynomial basis
Function, after 17 iteration, obtain final near-optimization controlled quentity controlled variable.Fig. 4 represents initial controlled quentity controlled variable (Part I institute
) and near-optimization controlled quentity controlled variable corresponding to cost function comparison diagram, hence it is evident that it can be seen that the cost letter of near-optimization controlled quentity controlled variable
Number is smaller.Fig. 5 is that two controlled quentity controlled variables are applied into the state trajectory comparison diagram obtained by system in real time after the completion of calculating, hence it is evident that can be with
Find out using near-optimization controlled quentity controlled variable system convergence speed faster.
In summary, presently preferred embodiments of the present invention is these are only, is not intended to limit the scope of the present invention.
Within the spirit and principles of the invention, any modification, equivalent substitution and improvements made etc., it should be included in the present invention's
Within protection domain.
Claims (1)
1. the servo-drive system method for optimally controlling of a kind of data-driven, it is characterised in that the state error including data-driven is anti-
Feedback control and adaptive congestion control algorithm two parts of data-driven：
Part I：The state error feedback control of data-driven
The 101st, k=1, initialization kth -1, system feedback gain K (k-1), the K (k) at k moment are set；
102nd, on time interval [kT, (k+1) T], sequentially using the system feedback gain K (k-1) and K (k) in servo-drive system；
103rd, the state error of sampling servosystem, and calculate state error relational matrix A according to formula (10), (12) to (15)
(k), geAnd g (k)u(k)；
A (k)=A0(k)A1(k) (10)
A1(k)=e ((k+1/2) T) (e (kT))+ (12)
A0(k)=e ((k+1) T) (e ((k+1/2) T))+ (13)
Then can obtain：
gu(k)=(A0(k)-A1(k))(K(k)-K(k-1))+ (14)
ge(k)=A0(k)-gu(k)·K(k) (15)
Wherein, e (kT), e ((k+1/2) T) represent kth, (k+1/2) moment system mode error, symbol+Represent the puppet of matrix
It is inverse；
104th, K (k+1) and K (k+2) are calculated according to formula (11)；
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105th, judge whether k is less than given threshold, if into Part II, otherwise make k=k+1, return to step 102；
Part II：The adaptive congestion control algorithm of data-driven
J=0 is set, defines φki、uj,Expression formula it is as follows
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202nd, based on obtaining in step 201Weights are calculated by formula (22)
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<mi>T</mi>
</msup>
<msubsup>
<mi>&Theta;</mi>
<mi>k</mi>
<mi>j</mi>
</msubsup>
<mo>)</mo>
</mrow>
<mrow>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msup>
<munderover>
<mo>&Sigma;</mo>
<mrow>
<mi>k</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mi>M</mi>
</munderover>
<mrow>
<mo>(</mo>
<msup>
<mrow>
<mo>(</mo>
<msubsup>
<mi>&Theta;</mi>
<mi>k</mi>
<mi>j</mi>
</msubsup>
<mo>)</mo>
</mrow>
<mi>T</mi>
</msup>
<msubsup>
<mo>&Integral;</mo>
<msub>
<mi>t</mi>
<mrow>
<mi>k</mi>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msub>
<msub>
<mi>t</mi>
<mi>k</mi>
</msub>
</msubsup>
<mi>r</mi>
<mo>(</mo>
<mrow>
<mi>e</mi>
<mo>,</mo>
<msup>
<mover>
<mi>u</mi>
<mo>^</mo>
</mover>
<mi>j</mi>
</msup>
<mrow>
<mo>(</mo>
<mi>e</mi>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
<mi>d</mi>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>22</mn>
<mo>)</mo>
</mrow>
</mrow>
Wherein M is positive integer, r (e, u)=Q (e)+uTRu, Q are positive definite, and R is positive definite symmetric matrices；
Weights are calculated by formula (23)
<mrow>
<msup>
<mover>
<mi>l</mi>
<mo>^</mo>
</mover>
<mrow>
<mi>j</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msup>
<mo>=</mo>
<mi>arg</mi>
<munder>
<mrow>
<mi>m</mi>
<mi>i</mi>
<mi>n</mi>
</mrow>
<mi>l</mi>
</munder>
<mo>{</mo>
<msup>
<mover>
<mi>c</mi>
<mo>^</mo>
</mover>
<mi>j</mi>
</msup>
<mi>&psi;</mi>
<mrow>
<mo>(</mo>
<mi>e</mi>
<mo>,</mo>
<mi>l</mi>
<mi>&sigma;</mi>
<mo>(</mo>
<mi>e</mi>
<mo>)</mo>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mi>r</mi>
<mrow>
<mo>(</mo>
<mi>e</mi>
<mo>,</mo>
<mi>l</mi>
<mi>&sigma;</mi>
<mo>(</mo>
<mi>e</mi>
<mo>)</mo>
<mo>)</mo>
</mrow>
<mo>}</mo>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>23</mn>
<mo>)</mo>
</mrow>
</mrow>
Wherein l is weightsAny probable value；
203rd, judgeWhether set up, if so, into step 204, otherwise make j=j+1, return to step 201, ε
For given threshold value；
204、For the approximation of optimum control amount, servo-drive system is controlled using it in real time afterwards.
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