CN108333929A - Based on the mixing H for recalling approaching Shu Fangfa2/H∞Controller synthesis method - Google Patents
Based on the mixing H for recalling approaching Shu Fangfa2/H∞Controller synthesis method Download PDFInfo
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Abstract
The present invention relates to based on the mixing H for recalling approaching Shu Fangfa2/H∞Controller synthesis method, the integrated approach, which can utilize, recalls approaching Shu Fangfa solutions mixing H2/H∞Controlling model, the approaching Shu Fangfa are exactly mixing H2/H∞Controlling model is converted into Unconstrained optimization and is solved.The present invention solves mixing H using approaching Shu Fangfa is recalled2/H∞Model has good numerical result, the systems stabilisation performance that the controller that optimal value and optimal solution are all had good convergence, and designed can be quickly.
Description
Technical field
The invention belongs to intelligent Manufacturing Technology fields, are related to a kind of based on the mixing H for recalling approaching Shu Fangfa2/H∞Control
Device integrated approach.
Background technology
In intelligent manufacturing system, integrated and self organization ability of the research object towards entire manufacturing environment uses
It is necessary and effective that the control strategy that modern control theory provides, which carrys out aid decision making person,.In Optimal Control Theory two most often
Index is H2And H∞Norm, correspondingly Control System Design also have H2And H∞Method.H2Performance refer to so that system for
Bounded composes the gain size under interference effect, H2Control problem is the control system based on internal stability, makes closed loop transfer function,
Its control effect place one's entire reliance upon description controlled device H2Norm minimum, to reach best system performance.And H∞Property
Can refer to stable case of the system under bounded energy interference, H∞Control problem mainly considers the robust stability of system, does not examine
Other indexs of worry system.The advantage of comprehensive two kinds of design methods respectively, produces mixing H2/H∞Control method, and obtain fast
Speed development.Research mixing H2/H∞The common method of control problem is to solve for Riccati the and Lyapunov equations of three couplings,
It is difficult to realize in engineering.With the rise of linear matrix inequality and nonlinear localized modes, problem can be converted into double
Optimization problem under linear inequality constraint solves, and because Lyapunov variables can be generated with the increase of system scale
The growth of secondary rate, this can cause existing computational methods to be quickly invalidated.In recent years, some scholars have studied target in model
The property that function and constraint function contain solves the problem using non-smooth blade.
Invention content
For overcome the deficiencies in the prior art, it is proposed that the mixing H based on approaching Shu Fangfa2/H∞Control method, the side
Method is applied in intelligent manufacturing system, in systems, it is often desired to which closed-loop system is stable, while also wanting system that can expire
The requirement of other aspect of performance of foot.Therefore H is introduced2And H∞Norm thereby produces mixing H2/H∞Planning.H of the present invention∞
Constraint is a compound maximum eigenvalue problem, other than intelligence control system, is all had in each field such as physics, statistics
Very important application.
Mix H2/H∞Plan model is：
WhereinIndicate H2The transmission function in closed-loop characteristic channel,Indicate H∞The transmission function in robust channel.
Notice that f (K) is a continuously differentiable function, and
The vectorization of decision variable K is denoted as x below.Constrained optimization problem with maximum eigenvalue constraint or with half
Above-mentioned optimization problem can be converted to by concludeing a contract or treaty the optimization problem of beam.
The technical scheme is that：Based on the mixing H for recalling approaching Shu Fangfa2/H∞Controller synthesis method, the control
Device integrated approach processed can utilize an approaching Shu Fangfa to solve the mixing H stated by following formula2/H∞Controller model：
Wherein, H∞Indicate that a kind of optimum control makes object function obtain the measurement of extreme value, H under H infinity norm2It indicates
A kind of optimum control is in H2So that object function obtains the measurement of extreme value, H under norm2And H∞Show the intelligent manufacturing system external world
The size of influence under disturbance and model uncertain condition to controller output,Indicate H2Closed-loop characteristic channel
Transmission function,Indicate H∞The transmission function in robust channel, f (x) are a continuously differentiable functions,
Wherein,It is state vector,It is control variable,It is to measure output, nx、ny、nuFor just
Integer, ω∞And ω2Respectively H∞And H2The relevant external disturbance input vector of performance indicator, z∞And z2Respectively H∞And H2Performance
The relevant controlled vector of index, ω2→z2It is H2Performance channel, ω∞→z∞It is H∞Performance channel, (T∞(x, j ω)HIndicate (T∞
The conjugate transposition of (x, j ω), ω are variable, λ1Indicate the maximum eigenvalue of Hermitian matrixes；
The approaching Shu Fangfa is：The mixing H2/H∞Controller model is converted into the Unconstrained optimization of following formula statement
It is solved,
Min F (y, x)
Wherein,
υ ＞ 0 are preset parameters,That is,It is taken compared with 0
The maximum；The feasible stable point of minimization F (y, x) problem is then solved using approaching Shu FangfaI.e.Meet
And It indicates in H∞Under a norm,The stable point of referred to as former problem,It indicatesIt is right
The subdifferential of first variable；Specifically solution procedure is：
Definition
Wherein, α, A are F (y, x respectivelyk) in xkThe functional value and subdifferential at place, the element (α, A) in G is F (y, xk) letter
The beam information pair of numerical value and subdifferential, GlIt is the subset of G；
It takes and determines parameter m1＞ m2＞ 0,0 ＜ μ of minimum step threshold valuemin＜+∞.
Step 0. (initialization outer loop) selection initial point x1∈Rn, RnReal number space, outer loop counter are tieed up for n
It is set to k=1；
Step 1. (outer loop) ifThen stop；Otherwise enter interior loop；
Step 2. (initialization interior loop) internal layer counter is set to l=1, chooses initial approaching parameter μ1＞ 0；Choose beam
The subset G of set Gl, take about object function f (y) and constraint function g (y) in xkBeam information at point is to (α0(xk), A0(xk))
WithIt enables
Step 3. (sounds out point to generate) the following subproblem of solution
Obtain local minimum point yl+1, calculate multiplier ρl+1, collection element pair can be obtained in this wayMeet
Wherein (αi(xk), Ai(xk))∈Gl.Estimated slippage is calculated again
Step 4. (acceptance test) if
Go to step 5；Otherwise remember yl+1It is zero step, takes μl+1≥μl, go to step 6；
Step 5. (backtracking test) if
Then remember yl+1It is walked to decline；Enable xk+1=yl+1, k adds 1, goes to step 1. and otherwise remember yl+1It is walked for backtracking, enables μl+1=μl
+2μmin, Gl+1=Gl, l adds 1, goes to step 3；
Step 6. (increase tangent plane sum aggregate information to) selection makes the linear approximation of the right end branch of improvement function reach most
Big available beam information is to (αl+1(xk), Al+1(xk)), enable (αl+1(xk), Al+1(xk))∈Gl+1, then enable collection information pairL adds 1, goes to step 3.
Beneficial effects of the present invention
Mixing H is solved using the method for the invention2/H∞The advantage of the approaching Shu Fangfa of backtracking of control problem：
1) by mixing H2/H∞The research of the object function and constraint function of control problem, makes mixing H2/H∞Control is asked
Topic is preferably solved.
2) it is analyzed using to the local stability point for improving function, the solution constrained optimization for being utilized improvement function is asked
The Theory of Stability of topic.
3) linear approximation for improving function by construction provides a kind of solution mixing H2/H∞The method of Controlling model.
Description of the drawings
Fig. 1 is using the method for the invention in vehicle hanging H2/H∞Control problem schematic diagram；
The optimal H that Fig. 2 is acquired using the method for the invention2/H∞The step response diagram of controller.
Specific implementation mode
Shown in referring to Fig. 1 and Fig. 2, the state space system of following form is considered：
Wherein,It is state vector,It is control variable,It is to measure output.ω∞And ω2Respectively
For H∞And H2The relevant external disturbance input vector of performance indicator, z∞And z2Respectively H∞And H2Performance indicator is relevant by steering
Amount, A is systematic observation matrix, B∞And B2Respectively ω∞And ω2The gain matrix of input, B input matrixes in order to control, C∞、D∞With
D∞uRespectively and H∞The relevant state variable of performance indicator, the weight matrix of disturbance output and control input.C2And D2uRespectively with
H2The weight matrix of the relevant state variable of performance indicator and control input.ω2→z2It is H2Performance channel, ω∞→z∞It is H∞Performance
Channel.
Mix H2/H∞The target of control is to find an output feedback controller u=Ky so that closed-loop system meets following
Specification：
(1) internal stability.K asymptotically stabilities system P in the closed.
(2) fixed H∞Performance.H∞Performance levelNo more than γ∞, i.e.,
(3) optimal H2Performance.H is minimized in the stability controller K of satisfaction (1) and (2)2PerformanceMinimize
Wherein j is imaginary unit, in (2)Indicate H∞The transmission function in robust channel, σmaxIt indicates
The maximum singular value of Hermitian matrixes, in (3)Indicate H2The transmission function in closed-loop characteristic channel.Trace
The trace function of representing matrix,It indicatesConjugate transposition.Notice that f (K) is a company here
Continuous differentiable function,
Wherein λ1Indicate the maximum eigenvalue of Hermitian matrixes,It indicates's
Conjugate transposition.Then the method will solve an Optimized model with composite characteristic value constraint.
As known from the above, the controller synthesis method can utilize an approaching Shu Fangfa to solve the mixing stated by following formula
H2/H∞Controller model：
Wherein, H∞Indicate that a kind of optimum control makes object function obtain the measurement of extreme value, H under H infinity norm2It indicates
A kind of optimum control is in H2So that object function obtains the measurement of extreme value, H under norm2And H∞Show the intelligent manufacturing system external world
The size of influence under disturbance and model uncertain condition to controller output,Indicate H2Closed-loop characteristic channel
Transmission function,Indicate H∞The transmission function in robust channel, f (x) are a continuously differentiable functions,
Wherein,It is state vector,It is control variable,It is to measure output, nx、ny、nuFor just
Integer, ω∞And ω2Respectively H∞And H2The relevant external disturbance input vector of performance indicator, z∞And z2Respectively H∞And H2Performance
The relevant controlled vector of index, ω2→z2It is H2Performance channel, ω∞→z∞It is H∞Performance channel, (T∞(x, j ω)HIndicate (T∞
The conjugate transposition of (x, j ω), ω are variable, λ1Indicate the maximum eigenvalue of Hermitian matrixes；
The approaching Shu Fangfa is：The mixing H2/H∞Controller model is converted into the Unconstrained optimization of following formula statement
It is solved,
Min F (y, x)
Wherein,
υ ＞ 0 are preset parameters,For indicator function, ifWhen more than 0, value 1, otherwise
It is 0；The feasible stable point of minimization F (y, x) problem is then solved using approaching Shu FangfaI.e.MeetAnd It indicates in H∞Under a norm,The stable point of referred to as former problem,It indicatesTo
The subdifferential of one variable；Specifically solution procedure is：
Definition
Wherein, α, A are F (y, x respectivelyk) in xkThe functional value and subdifferential at place, the element (α, A) in G is F (y, xk) letter
The beam information pair of numerical value and subdifferential, GlIt is the subset of G；
It takes and determines parameter m1＞ m2＞ 0,0 ＜ μ of minimum step threshold valuemin＜+∞.
Step 0. (initialization outer loop) selection initial point x1∈Rn, RnReal number space, outer loop counter are tieed up for n
It is set to k=1；
Step 1. (outer loop) ifThen stop；Otherwise enter interior loop；
Step 2. (initialization interior loop) internal layer counter is set to l=1, chooses initial approaching parameter μ1＞ 0；Choose beam
The subset G of set Gl, take about object function f (y) and constraint function g (y) in xkBeam information at point is to (α0(xk), A0(xk))
WithIt enables
Step 3. (sounds out point to generate) the following subproblem of solution
Obtain local minimum point yl+1, calculate multiplier ρl+1, collection element pair can be obtained in this wayMeet
Wherein (αi(xk), Ai(xk))∈Gl.Estimated slippage is calculated again
Step 4. (acceptance test) if
Go to step 5；Otherwise remember yl+1It is zero step, takes μl+1≥μl, go to step 6；
Step 5. (backtracking test) if
Then remember yl+1It is walked to decline；Enable xk+1=yl+1, k adds 1, goes to step 1. and otherwise remember yl+1It is walked for backtracking, enables μl+1=μl
+2μmin, Gl+1=Gl, l adds 1, goes to step 3；
Step 6. (increase tangent plane sum aggregate information to) selection makes the linear approximation of the right end branch of improvement function reach most
Big available beam information is to (αl+1(xk), Al+1(xk)), enable (αl+1(xk), Al+1(xk))∈Gl+1, then enable collection information pairL adds 1, goes to step 3.
The method of the invention is applied to a kind of mixing H2/H∞In Controlling model-vehicle hanging model.The number of vehicle hanging
The main behavioral characteristics of model display, such as the weight of support vehicle are learned, carries out keeping stablizing when different operation, provide quite
Comfort level, minimize road disturbance strength caused by influence etc..This example research is " a quarter " shown in figure one
Auto model, this be early stage propose can analyze the simplest model of associated dynamic performance.The foundation of dynamical equation is to pass through
Simple dynamic balance.State variable x1, x2Respectively from the unsprung mass M of equilibrium stateusDisplacement and from the spring of equilibrium state
Mass MspDisplacement, x3, x4Respectively x1, x2Derivative.The state-space representation of system is
We take H in this example2And H∞Channel is same channel ω → z.Wherein γ∞It is taken as 5.225,
Kus=1.559 × 105, Mus=28.58, Msp=288.9.
If static controller is u=Ky, we take the biography for being updated to and obtaining closed-loop system in state space system in this example
Delivery function introduces state space data
A (K)=A+B2KC2, B2(K)=B∞(K)=B1
C2(K)=C∞(K)=C1+D12KC2, D2(K)=D∞(K)=0
Then the transmission function of closed loop channel ω → z is
T2(K, s)=C2(K)(sI-A(K))-1B2(K),
T∞(K, s)=C∞(K)(sI-A(K))-1B∞(K)
Wherein s is frequency domain variable, and s=j ω are enabled in calculating.Then object function, that is, H2Norm square is
F (K)=Tr (B2(K)TX(K)B2(K))=Tr (C2(K)Y(K)C2(K)T)
Wherein Tr is writing a Chinese character in simplified form for Trace, the mark of representing matrix.The transposition of subscript T representing matrixes.
X (K) and Y (K) is respectively following two Lyapunov non trivial solutions：
A(K)TX(K)+X(K)A(K)+C2(K)TC2(K)=0,
A(K)Y(K)+Y(K)A(K)T+B2(K)B2(K)T=0.
Constraint function g (K) i.e. H∞Norm square is
The subdifferential for acquiring object function and constraint function again is respectively
Wherein co indicates the convex closure of set.Then improving function is
F(K+, K) and=max { f (K+)-f(K)-μ[g(K)-5.2252]+, (g (K+)-5.2252)-[g(K)-5.2252]+}
Here K+Indicate next iteration point based on K.The feasible of the problem is obtained by the iteration of approaching Shu Fangfa
Local stability point is K=[40,403 2408], corresponding H2Norm is | | T2(K)||2=34.480, H∞Norm is | | T∞(K)|
|∞=5.2212.
Fig. 1 gives the schematic diagram of vehicle hanging model, and Fig. 2 gives the step response of the static controller acquired, by
The controller acquired known to Fig. 2 can make to reach stable in the closed-loop system short time.In conclusion the present invention is approaching using recalling
Shu Fangfa solves mixing H2/H∞Controlling model has good numerical result, and the controller acquired can make closed-loop system steady
Fixed and system performance reaches minimum.The method of the invention can be used in different application scenarios.
Claims (2)
1. based on the mixing H for recalling approaching Shu Fangfa2/H∞Controller synthesis method, it is characterized in that：The controller synthesis method
The mixing H stated by following formula can be solved using an approaching Shu Fangfa2/H∞Controller model：
Wherein, H∞Indicate that a kind of optimum control makes object function obtain the measurement of extreme value, H under H infinity norm2Indicate a kind of
Optimum control is in H2So that object function obtains the measurement of extreme value, H under norm2And H∞Show intelligent manufacturing system external disturbance
With under model uncertain condition to controller output influence size,Indicate H2The biography in closed-loop characteristic channel
Delivery function,Indicate H∞The transmission function in robust channel, f (x) are a continuously differentiable functions,
Wherein,It is state vector,It is control variable,It is to measure output, nx、ny、nuFor positive integer,
ω∞And ω2Respectively H∞And H2The relevant external disturbance input vector of performance indicator, z∞And z2Respectively H∞And H2Performance indicator
Relevant controlled vector, ω2→z2It is H2Performance channel, ω∞→z∞It is H∞Performance channel, (T∞(x, j ω)HIndicate (T∞(x, j
Conjugate transposition ω), ω are variable, λ1Indicate the maximum eigenvalue of Hermitian matrixes；
The approaching Shu Fangfa is：The mixing H2/H∞The Unconstrained optimization that controller model is converted into following formula statement is asked
Solution,
Min F (y, x)
Wherein,
υ ＞ 0 are preset parameters,That is,Maximum is taken compared with 0
Person；The feasible stable point of minimization F (y, x) problem is then solved using approaching Shu FangfaI.e.MeetAndIt indicates in H∞Under a norm,The stable point of referred to as former problem,It indicatesIt is right
The subdifferential of first variable.
2. according to claim 1 based on the mixing H for recalling approaching Shu Fangfa2/H∞Controller synthesis method, it is characterized in that：
The specific solution procedure of the approaching Shu Fangfa is：
Step 0. defines
Wherein, α, A are F (y, x respectivelyk) in xkThe functional value and subdifferential at place, the element (α, A) in G is F (y, xk) functional value
With the beam information pair of subdifferential, GlIt is the subset of G；It takes and determines parameter m1＞ m2＞ 0,0 ＜ μ of minimum step threshold valuemin＜+∞, selection
Initial point x1∈Rn, RnReal number space is tieed up for n, outer loop counter is set to k=1；
Step 1, ifThen stop；Otherwise enter interior loop；
Step 2, internal layer counter is set to l=1, chooses initial approaching parameter μ1＞ 0；Choose the subset G that constriction closes Gl, take about
Object function f (y) and constraint function g (y) are in xkBeam information at point is to (α0(xk), A0(xk)) and
It enables
Step 3, following subproblem is solved
Obtain local minimum point yl+1, calculate multiplier ρl+1, collection element pair can be obtained in this wayMeet
Wherein (αi(xk), Ai(xk))∈Gl；Estimated slippage is calculated again
Step 4, judge following formula, if
Go to step 5；Otherwise remember yl+1It is zero step, takes μl+1≥μl, go to step 6；
Step 5, judge following formula, if
Then remember yl+1It is walked to decline；Enable xk+1=yl+1, k adds 1, goes to step 1. and otherwise remember yl+1It is walked for backtracking, enables μl+1=μl+2
μmin, Gl+1=Gl, l adds 1, goes to step 3；
Step 6, selection makes the linear approximation of the right end branch of improvement function reach maximum available beam information to (αl+1(xk), Al+1
(xk)), enable (αl+1(xk), Al+1(xk))∈Gl+1, then enable collection information pairL adds 1, goes to step
Rapid 3.
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| CN117706932A (en) * | 2023-12-18 | 2024-03-15 | 兰州理工大学 | H ∞ Design method of mu comprehensive mixed dispersion controller |
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| CN117706932A (en) * | 2023-12-18 | 2024-03-15 | 兰州理工大学 | H ∞ Design method of mu comprehensive mixed dispersion controller |
| CN117706932B (en) * | 2023-12-18 | 2024-05-28 | 兰州理工大学 | H∞Design method of mu comprehensive mixed dispersion controller |
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