CN106154839A - Nonlinear system robust adaptive tracking control method based on unknown object track - Google Patents

Abstract

The invention discloses a kind of nonlinear system robust adaptive tracking control method based on unknown object track, including step: step one, the mathematical model of the foundation MIMO nonlinear systems containing actuator failures；Step 2, foundation estimate the model of unknown object ideal trajectory, and the target trajectory of estimation approaches preferable target trajectory to utilize this model to draw；Step 3, design robust adaptive fault-tolerant controller.The present invention can be derived that the unknown object track of estimation so that it is approaches preferable target trajectory；The controller of design is by carrying out matrix decomposition to " virtual control gain " dexterously so that in other control methods, the condition restriction to gain matrix is substantially reduced；Finally, in the case of system exists actuator failures, parameter uncertainty, external disturbance simultaneously and follows the trail of target trajectory the unknown, still can obtain the steady-state behaviour of asymptotic tracking, make closed-loop control system that ambiguous model and unknown disturbances are had robust adaptive effect.

Description

Nonlinear system robust adaptive tracking control method based on unknown object track

Technical field

The present invention relates to Control of Nonlinear Systems field, particularly to self adaptation during a kind of track the unknown following the trail of target Control method.

Background technology

The Tracing Control design of absolutely most dynamical system generally all can assume preferable tracing path be known or Can obtain easily.But in real life, the ideal trajectory referred to not necessarily all is readily available, the most preferably Tracing path cannot be used in Tracing Control design.Additionally, this phenomenon is widely present in various concrete instance, Such as: in Missile Launching Process, in order to avoid the interception of enemy missile, we can deliberately change its preferable track by guided missile, So original preferably tracing path cannot be used in STT missile design；On-the-spot at industrial robot, robot is because of special Different reason can follow the trail of the track of " obscuring ", and now preferably tracing path can not be used in robot control design case.Therefore, How to obtain the ideal trajectory of the unknown, controller designed concerning important, be also simultaneously have the most challenging.

Control method currently for the nonlinear system following the trail of unknown ideal goal track is few in number, the side only deposited Method mainly obtains the relation of unknown track and ideal trajectory, then removes to design controller by this layer of relation, but this is not The method solving how to obtain the ideal trajectory of the unknown.

In real system, actuator failures the most inevitably occurs in particular, when system tracks the unknown ideal goal While track, executor unexpectedly breaks down, and this is the highest for the requirement of controller, cannot with current existing technology Reach good tracking performance, need to redesign a simple in construction, functional, calculate simple controller.

Summary of the invention

In view of this, it is an object of the invention to provide a kind of nonlinear system robust adaptive based on unknown object track Tracking and controlling method, it is followed the trail of the situation of unknown object track for nonlinear system, uses based on expanding Kalman filtering Estimate the preferable target trajectory of Model approximation, thus obtain the target trajectory estimated, recycling matrix decomposition technology and extraction Core function method carries out robust adaptive faults-tolerant control, it is achieved the preferable target trajectory of system output tracking.

Present invention nonlinear system based on unknown object track robust adaptive tracking control method, including following step Rapid:

Step one, the mathematical model of the foundation MIMO nonlinear systems containing actuator failures；

The described MIMO nonlinear systems containing actuator failures has a following state space form:

$y^{\left(n\right)}=F\left(x\right)+G\left(\stackrel{-}{x}\right)u_a+d(x,t),$

X=[x in formula₁ ^T..., x_q ^T]^T∈RⁿIt is the whole state vector of system, wherein, I=1,2 ..., q, and n₁+n₂+…+n_q=n；It is the output of system；F(x)∈R^qIt is Unknown functional vector；It is continuously differentiable unknown function, wherein ；D (x, t) ∈ R^qUncertainty for system is non-linear, u_a∈R^qFor controlling Input vector；

Consider that actuator failures, actual control input u_aThe relation controlling input u with ideal is:

u_a=ρ (t) u+ ε (t),

ρ=diag{ ρ in formula_i}∈R^q×qIt is diagonal matrix, ρ_iFor executor's efficiency factor, and meet 0 ＜h _i≤ρ_i≤ 1,h _iFor ρ_iMinima；ε (t) represents part the most out of control in controlling behavior and is assumed to be bounded；

Step 2, foundation estimate the model of unknown object ideal trajectory, and utilize this model to draw the target trajectory of estimation Approach preferable target trajectory；

For unknown target trajectory, use and go to estimate based on the mathematical model expanding Kalman filtering, and make it approach Preferably target trajectory；

y_d(t)=y_EKF(t)+y^* _guess(t)

In formula, y_d(t)∈R^qEstimated value for unknown object ideal trajectory；y_EKF(t)∈R^qFor with expanding Kalman filtering The ideal goal track of technological prediction；y^* _guess(t)∈R^qThe rough estimate of the ideal trajectory for drawing based on certain known conditions Value, without available known conditions, this value can be 0；y_d ^(j)T () is the j order derivative of ideal trajectoryEstimation Value, y^j _EKF(t) andIt is respectively with expanding Kalman Filter Technology and the corresponding ideal j rank of known conditions prediction Derivative track；The optimal estimation value drawn by expanding Kalman filtering corresponds to the y in model_EKF(t) and y^j _EKF(t)；

Step 3, design robust adaptive fault-tolerant controller；

1) the unknown object track estimated is utilized to obtain tracking error E with system output_m, by obtaining after sliding formwork wave filter New state variable s_m；

2) control gain G and executor's efficiency factor ρ regards as virtual control gain as entirety, virtual control gain is entered Row matrix is decomposed, and obtains known matrix D (x), U (x) and unknown matrix S (x)；Wherein matrix S (x) leads to as indeterminate Cross core function generator, simultaneity factor ambiguous model and external disturbance indeterminate also by core function generator at Reason, obtains the virtual parameter a of the unknown and computable core function

3) core functionSquare by any direct proportion c₁With state variable s obtained before after amplification_mTake advantage of A part for the virtual parameter that long-pending composition is unknown, then deduct the virtual parameter of estimationC₂Times, c₂For any normal number, obtain Virtual parameterDerivative value, be finally integrated computing and obtain the estimated value of virtual parameter a of the unknown

4) state variable s obtained is utilized_mWith core functionThe estimated value of the long-pending virtual parameter a being multiplied by the unknown again -c₁Times, obtain last controller u；

5) control instruction calculated is sent to the executor of nonlinear system by controller u, it is achieved system output tracking Preferably target trajectory.

Beneficial effects of the present invention:

Present invention nonlinear system based on unknown object track robust adaptive tracking control method, it is possible to draw estimation Unknown object track so that it is approach preferable target trajectory.The controller of design is by dexterously to " virtual control gain " Carry out matrix decomposition so that in other control methods, the condition restriction to gain matrix is substantially reduced.By extracting core function Method process nonlinear uncertainty, simplify controller design procedure.Finally, system exist simultaneously actuator failures, Still the stability of asymptotic tracking can be obtained in the case of parameter uncertainty, external disturbance and tracking target trajectory the unknown Can, make closed-loop control system that ambiguous model and unknown disturbances are had robust adaptive effect.

Accompanying drawing explanation

Fig. 1 is the robust adaptive faults-tolerant control principle that the nonlinear system containing actuator failures follows the trail of unknown object track Schematic diagram；

Fig. 2 is based on the unknown object ideal trajectory estimating system schematic diagram expanding Kalman filtering；

Fig. 3 is the design principle figure of robust adaptive fault-tolerant controller；

Fig. 4 is carried out device efficiency factor change curve；

Fig. 5 is estimation target trajectory in x-axis, and ideal trajectory observes track correlation curve figure；

Fig. 6 is estimation target trajectory in y-axis, and ideal trajectory observes track correlation curve figure

Fig. 7 is to estimate target trajectory, ideal trajectory, the two-dimentional correlation curve figure of observation track；

Fig. 8 is estimation target velocity, ideal velocity, observation speed change curve comparison diagram in x-axis；

Fig. 9 is estimation target velocity, ideal velocity, observation speed change curve comparison diagram in y-axis；

Figure 10 is that expectation tracing positional changes over curve chart；

Figure 11 is that expectation tracing positional changes over two dimensional plot；

Figure 12 is expectation track position error curve chart；

Figure 13 is that the control input under controller action changes over curve chart；

Figure 14 is that the systematic parameter under controller action is estimated to change over curve chart.

Detailed description of the invention

The invention will be further described with embodiment below in conjunction with the accompanying drawings.

The present embodiment nonlinear system based on unknown object track robust adaptive tracking control method, including following step Rapid:

Comprise the following steps:

Step one, the mathematical model of the foundation MIMO nonlinear systems containing actuator failures；

The described MIMO nonlinear systems containing actuator failures has a following state space form:

$y^{\left(n\right)}=F\left(x\right)+G\left(\stackrel{\‾}{x}\right)u_a+d(x,t),$

X=[x in formula₁ ^T..., x_q ^T]^T∈RⁿIt is the whole state vector of system, wherein, I=1,2 ..., q, and n₁+n₂+…+n_q=n；It is the output of system；F(x)∈R^qIt is Unknown functional vector；It is continuously differentiable unknown function, wherein ；D (x, t) ∈ R^qUncertainty for system is non-linear, u_a∈R^qFor controlling Input vector；

Consider that actuator failures, actual control input u_aThe relation controlling input u with ideal is:

u_a=ρ (t) u+ ε (t),

ρ=diag{ ρ in formula_i}∈R^q×qIt is diagonal matrix, ρ_iFor executor's efficiency factor, and meet 0 ＜h _i≤ρ_i≤ 1,h _iFor ρ_iMinima；ε (t) represents part the most out of control in controlling behavior and is assumed to be bounded；

Step 2, foundation estimate the model of unknown object ideal trajectory, and utilize this model to draw the target trajectory of estimation Approach preferable target trajectory；

For unknown target trajectory, use and go to estimate based on the mathematical model expanding Kalman filtering, and make it approach Preferably target trajectory；

y_d(t)=y_EKF(t)+y^* _guess(t)

In formula, y_d(t)∈R^qEstimated value for unknown object ideal trajectory；y_EKF(t)∈R^qFor with expanding Kalman filtering The ideal goal track of technological prediction；y^* _guess(t)∈R^qThe rough estimate of the ideal trajectory for drawing based on certain known conditions Value, without available known conditions, this value can be 0；y_d ^(j)T () is the j order derivative of ideal trajectoryEstimation Value, y^j _EKF(t) andIt is respectively with expanding Kalman Filter Technology and the corresponding ideal j rank of known conditions prediction Derivative track；The optimal estimation value drawn by expanding Kalman filtering corresponds to the y in model_EKFAnd y (T)^j _EKF(t)；

Step 3, design robust adaptive fault-tolerant controller；

1) the unknown object track estimated is utilized to obtain tracking error E with system output_m, by obtaining after sliding formwork wave filter New state variable s_m；

2) control gain G and executor's efficiency factor ρ regards as virtual control gain as entirety, virtual control gain is entered Row matrix is decomposed, and obtains known matrix D (x), U (x) and unknown matrix S (x)；Wherein matrix S (x) leads to as indeterminate Cross core function generator, simultaneity factor ambiguous model and external disturbance indeterminate also by core function generator at Reason, obtains the virtual parameter a of the unknown and computable core function

3) core functionSquare by any direct proportion c₁With state variable s obtained before after amplification_mTake advantage of A part for the virtual parameter that long-pending composition is unknown, then deduct the virtual parameter of estimationC₂Times, c₂For any normal number, obtain Virtual parameterDerivative value, be finally integrated computing and obtain the estimated value of virtual parameter a of the unknown

4) state variable s obtained is utilized_mWith core functionThe estimated value of the long-pending virtual parameter a being multiplied by the unknown again -c₁Times, obtain last controller u；

5) control instruction calculated is sent to the executor of nonlinear system by controller u, it is achieved system output tracking Preferably target trajectory.

Nonlinear system robust adaptive tracking control method based on unknown object track to the present embodiment is carried out below Simulating, verifying, for the controller performance designed by examination, provides following simulation example.

Consider following Nonlinear Second Order System:

$\begin{array}{c}\left(7+\cos y\right)\stackrel{\·\·}{x}+\left(4+2\cos y\right)\stackrel{\·\·}{y}-2\stackrel{\·}{y}\sin y\left(2\stackrel{\·}{x}+\stackrel{\·}{y}\right)+2\sin x+d_1(\·)=u_{a1}\\ \left(4+2\cos y\right)\stackrel{\·\·}{x}+y^2\stackrel{\·\·}{y}+2\stackrel{\·}{x}\sin x+\sin \left(x+y\right)+d_2(\·)=u_{a2}\end{array}$

In above formula, u_a1=ρ₁(t)u₁+ε₁(t) and u_a2=ρ₂(t)u₂+ε₂T () is carried out the executor after device breaks down Output, d₁() and d₂() is the indeterminates such as external disturbance.Consider that actuator failures efficiency factor ρ (t) is it is known that set For

${\mathrm{\&rho;}}_1\left(t\right)=\left\{\begin{array}{cc}1& t\&le;50\\ 0.8+0.1\cos \left(\mathrm{\&pi;}\left(t-60\right)/60\right)& 50<t\&le;120\\ 0.7& t>120\end{array}\right.$

${\mathrm{\&rho;}}_2\left(t\right)=\left\{\begin{array}{cc}1& t\&le;40\\ 0.75+0.15\cos \left(\mathrm{\&pi;}\left(t-50\right)/60\right)& 40<t\&le;110\\ 0.6& t>110\end{array}\right.$

As shown in Figure 4.Interference d₁=0.1sin t, d₂=0.1cos0.2 π t.

Assume preferable reference locus X_d=(x_d, y_d)^TThe unknown, draws estimation by the method proposed in the present embodiment now Reference locus.According to expanding Kalman filtering algorithm, it is known that the state equation of reference target track and observational equation be:

$\left\{\begin{array}{c}{Z}_n=f\left(Z_{n-1}\right)+W_{n-1}\\ T_n=g\left(Z_n\right)+V_n\end{array}\right.$

In above formula,For quantity of state,

T_n=[t_1n, 0, t_2n, 0, t_3n, 0, t_4n, 0]^TFor observed quantity, w_n=[w_1n, 0, w_2n, 0, w_3n, 0, w_4n, 0]^TFor state Noise, V_n=[v_1n, 0, v_2n, 0, v_3n, 0, v_4n, 0]^TFor observation noise,

And

Wherein x_n/y_nIt is the position at moment n of x-axis or y-axis, x_fn/y_fnIt is displacement frequency,It is x-axis or y-axis The speed at moment n,It it is corresponding frequency.State-noise covariance is respectively as follows: Q_1n=0.1, Q_2n=0.02, Q_3n=0.11, Q_4n=0.01；Observation noise covariance is respectively as follows: R_1n=0.03, R_2n=0.007, R_3n=0.028, R_4n= 0.008."ball-park" estimate is set to 0, then uses expansion Kalman filtering algorithm, initial value is set to Z₀=[0,0.3,1,0.3, 0.3,0.3,0,0.3]^TAnd P₀=diag{0.01}^T∈R⁸.Finally give simulation result as shown in Fig. 5～Fig. 9.Can from figure To find out, the method proposed in the present embodiment shows good performance on estimation unknown object track so that the mesh of estimation Mark track X_dApproach preferable target trajectory X=[sin (0.3t), cos (0.3t)]^T。

Utilize the estimation target trajectory X obtained_d, then use the control method in the present embodiment, choose controller parameter: c =10, c₁=2.5, c₂=0.02；Initial parameter value is taken as [x, y]=[1,1.5].Obtain the aircraft pursuit course under controller action As shown in Figure 10, as shown in figure 11, aircraft pursuit course error is as shown in figure 12 for two-dimensional tracking curve.It can be seen that propose Controller all show good tracking performance in transient state or stable state.The control input of controller is time dependent Curve is as shown in figure 13.Systematic parameter estimation procedure is as shown in figure 14.

Finally illustrating, above example is only in order to illustrate technical scheme and unrestricted, although with reference to relatively The present invention has been described in detail by good embodiment, it will be understood by those within the art that, can be to the skill of the present invention Art scheme is modified or equivalent, and without deviating from objective and the scope of technical solution of the present invention, it all should be contained at this In the middle of the right of invention.

Claims (1)

1. nonlinear system robust adaptive tracking control method based on unknown object track, it is characterised in that: include following Step:

Step one, the mathematical model of the foundation MIMO nonlinear systems containing actuator failures；

The described MIMO nonlinear systems containing actuator failures has a following state space form:

$y^{\left(n\right)}=F\left(x\right)+G\left(\stackrel{\&OverBar;}{x}\right)u_a+d(x,t),$

X=[x in formula₁ ^T..., x_q ^T]^T∈RⁿIt is the whole state vector of system, whereini =1,2 ..., q, and n₁+n₂+…+n_q=n；It is the output of system；F(x)∈R^qIt is Unknown functional vector；It is continuously differentiable unknown function, wherein D (x, t) ∈ R^qUncertainty for system is non-linear, u_a∈R^qFor controlling Input vector；

Consider that actuator failures, actual control input u_aThe relation controlling input u with ideal is:

u_a=ρ (t) u+ ε (t),

ρ=diag{ ρ in formula_i}∈R^q×qIt is diagonal matrix, ρ_iFor executor's efficiency factor, and meet 0 ＜h _i≤ρ_i≤ 1,h _iFor ρ_i Minima；ε (t) represents part the most out of control in controlling behavior and is assumed to be bounded；

Step 2, foundation estimate the model of unknown object ideal trajectory, and the target trajectory of estimation approaches to utilize this model to draw Preferably target trajectory；

For unknown target trajectory, use and go to estimate based on the mathematical model expanding Kalman filtering, and make it approach ideal Target trajectory；

y_d(t)=y_EKF(t)+y^* _guess(t)

${y_d}^{\left(i\right)}\left(t\right)={y^j}_{EKF}\left(t\right)+{y^{*\left(j\right)}}_{guess}\left(t\right)$

In formula, y_d(t)∈R^qEstimated value for unknown object ideal trajectory；y_EKF(t)∈R^qFor with expanding Kalman Filter Technology The ideal goal track of prediction；y^* _guess(t)∈R^qThe rough estimate value of the ideal trajectory for drawing based on certain known conditions, Without available known conditions, this value can be 0；y_d ^(j)T () is the j order derivative of ideal trajectoryEstimated value, y^j _EKF(t) andIt is respectively the corresponding ideal j rank with expanding Kalman Filter Technology and known conditions prediction to lead Number track；The optimal estimation value drawn by expanding Kalman filtering corresponds to the y in model_EKF(t) and y^j _EKF(t)；

Step 3, design robust adaptive fault-tolerant controller；

1) the unknown object track estimated is utilized to obtain tracking error E with system output_m, new by obtaining after sliding formwork wave filter State variable s_m；

2) control gain G and executor's efficiency factor ρ regards as virtual control gain as entirety, virtual control gain is carried out square Battle array is decomposed, and obtains known matrix D (x), U (x) and unknown matrix S (x)；Wherein matrix S (x) passes through core as indeterminate The heart function generator of more vairable, simultaneity factor ambiguous model and external disturbance indeterminate process also by core function generator, To unknown virtual parameter a and computable core function

3) core functionSquare by any direct proportion c₁With state variable s obtained before after amplification_mProduct composition A part for unknown virtual parameter, then deduct the virtual parameter of estimationC₂Times, c₂For any normal number, obtain virtual ginseng NumberDerivative value, be finally integrated computing and obtain the estimated value of virtual parameter a of the unknown

4) state variable s obtained is utilized_mWith core function-the c of estimated value of the long-pending virtual parameter a being multiplied by the unknown again₁ Times, obtain last controller u；

5) control instruction calculated is sent to the executor of nonlinear system by controller u, it is achieved system output tracking is preferable Target trajectory.