| Tracking problem is of great significance in the research of control theory and practical application.Definition tracking error[1]ε(t)= r(t)-y(t)(0.0.12)Based on this expression,there is no overshoot if the symbol for ε(t)does not change withint ∈[0,+∞).r(t)is the reference signal given by the system.y(t)is the output of the system.For many control systems,no overshoot is required.Like some of the more special missions:air refueling,formation performance,precise entry,etc.There is no overshoot in the attitude angle of the aircraft.The other is high precision machine tool,and its feed control needs no overshoot.For the engraving machine,the probe position control must also be without overshoot.So research without overshoot control is very useful for our industry,our defense,and our technology.Therefore,it is necessary to study the condition of system realization without overshoot.This paper firstly introduces research background and present situation of non-overshoot control problems.Then the problem of no overshoot control for linear systems is studied.When the Laplace transformation method is used to solve the differential equations of linear systems,the mathematical model-transfer function of the control system in the complex domain can be obtained.Transfer function can not only characterize the dynamic performance of the system,but also can be used to study the influence of the system structure or parameter change on the system performance.But in the most cases,the input signal of the control system is random.In order to analyze and design easily,we need to assume some basic input function forms.For the linear control system,we usually use the unit step function as a typical input(The step change function with amplitude of 1 appears at the time of t = 0)is feasible in many cases,such as room temperature control system and water level control system,as well as the sudden working state.A control system that changes or is suddenly subjected to constant input.For third-order linear systems(?)[2]the transfer function of SISO(one input end and one output end)of the three order linear system is given.Y(s)/R(s)=K cs3 + bs2 + as + 1/ps3+qs2+rs+1(0.0.13)The Y(s)is the Laplacian transform of the output quantity y(t),The R(s)is the Laplacian transform of the input quantity r(t).The unit step function is used as the input signal.The system tracking error isε(t)= 1-y(t)(0.0.14)There is no overshoot if and only if you have no overshoot for all t ∈[0,+∞),ε(t)≥ 0.The following results are obtained by analyzing the zero and poles of the transfer function:(1)When the transfer function of the system has real poles,a necessary and sufficient condition for the system without overshoot is obtained.(2)When the transfer function of the system has a complex pole,a sufficient condition and two necessary conditions for the system to have no overshoot are obtained.For linear systems with multiple input and output(multiple input terminals and multiple outputs),[3]the state space method is used to consider LTI systems with the following characteristics:(?)where:t ∈ R,x(t)∈ Rn is a state,u(t)∈ Rm is the control input of the system is the control input of the system.y(t)∈ Rp is the output of the system,A,B,C,D is a constant matrix corresponding to the dimension.ε(t)= r-y(t)is a tracking error.If the symbol for ε(t)does not change,the system has no overshoot.By designing the control law,a sufficient condition is obtained for the system without overshoot.For nonlinear systems,there are less studies on the control without overshoot.[4]consider strict feedback nonlinear systems (?)i = xi+1+φi((?)i),i=1,2,...,n-1,(?)n = u + φn((?)n),(0.0.16)y = x1 Where:xi,yand u denote system status,output and input,respectively;(?)i =[x1,...,xi]T;Nonlinera term φi is n-1sub differentiable.The control law u is con-structed by using the improved backstepping method and coordinate transformation is carried out.zi=xi-αi-1(xi-1,t)-r(t)(i-1)(t)(0.0.17)α0 = 0(0.0.18)α1(x1,t)=-c1z1-φ1(0.0.19)(?)u = αn(0.0.21)Among them,c1,...,cn are positive design parameters.coordinate change x → z is smooth and reversible.In the process of coordinate transformation,the parameter ci is introduced,and in[4]it is required that x1(0)<r(0).Tracking error is ε(t)= r(t)-y(t).The system has no overshoot if and only if it is for all t ∈[0,+∞),ε(t)≥ 0 in the z coordinates,the closed loop system is (?)i=-cizi+zi+1,i=1,...,n-1(0.0.22)(?)n=-cnzn At this time,the tracking error ε(t)= r(t)-y(t)=-z1(t),is equivalent to the condition that there is no overshoot in the system t ∈[0,+∞),z1(t)≤0 discussed on different cases,the system realize nonovershooting in different cases with different choice of c(i),the specific as follows:1.When the nonlinear term and initial conditions of the system are not zero,and the reference trajectory is an arbitrary(n-1 differentiable)time function,the choice of c(i.)is only related to the initial conditions and derivative initial value of the reference trajectory;2.When the nonlinear term is zero at zero point and initial condition is zero,c(i.)only depends on the prior value of the reference trajectory derivative.In addition,[1]has changed the control law u based on[4],and obtained the necessary and sufficient conditions of the parameters c(i)for the system and ini-tial value with nonovershooting.Moreover,the condition covers a wider range of parameters.Through studying and studying existing knowledge,the simulation and simu-lation are carried out by practical examples. |
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