CN103792848A - Longitudinal flight model cluster man-machine closed-loop composite root locus multi-stage PID robust controller design method - Google Patents

Abstract

The invention provides a longitudinal flight model cluster man-machine closed-loop composite root locus multi-stage PID robust controller design method. According to the method, a model cluster composed of amplitude-frequency properties and phase-frequency properties within a whole envelope is directly determined and obtained through a sweep-frequency flight test under the condition that different heights and Mach numbers are given; according to military standard requirements for amplitude-frequency margins and phase margins within a flight envelope, closed-loop pole distribution limiting indicators under a corresponding root locus description are given, and the stage number and parameter values of a multi-stage PID robust controller are determined by additionally arranging a multi-stage PID controller according to the closed-loop pole distribution limiting indicators and a model identification method in system identification within the whole envelope of a flying machine; a low-altitude flight controller is designed by starting from the concept of closed-loop pole distribution limiting under the root locus description, wherein the low-altitude flight controller accords with the whole flight envelop and is free of pilot induced oscillation, low in overshoot and stable.

Description

The multistage PID robust Controller Design of the compound root locus of Longitudinal Flight model cluster man-machine loop method

Technical field

The present invention relates to a kind of controller of aircraft method for designing, particularly the multistage PID robust Controller Design of the compound root locus of Longitudinal Flight model cluster man-machine loop method, belongs to the category such as observation and control technology and flight mechanics.

Background technology

The control of aircraft landing process plays an important role to flight safety; Because flying speed in aircraft landing process changes greatly, even also can face strong nonlinearity problem according to longitudinal model; On the other hand, there is the phenomenons such as saturated, dead band in the control vane of aircraft; Consider from flight safety, when hedgehopping (as take off/land), controller must guarantee that system has certain stability margin, non-overshoot and stationarity, like this, just make hedgehopping controller design very complicated, can not directly apply mechanically existing control theory and carry out the design of aircraft control.

In the design of modern practical flight controller, a small part adopts state-space method to design, and great majority still adopt the classical frequency domain method take PID as representative and carry out controller design against Nyquist Array Method as the modern frequency method of representative.Modern control theory is take state-space method as feature, take analytical Calculation as Main Means, to realize performance index as optimum modern control theory, then have and developed method for optimally controlling, model reference control method, self-adaptation control method, dynamic inversion control method, feedback linearization method, directly nonlinear optimization control, variable-gain control method, neural network control method, fuzzy control method, a series of controller design methods such as robust control method and several different methods combination control, the scientific paper of delivering is ten hundreds of, for example Ghasemi A in 2011 has designed reentry vehicle (the Ghasemi A of Adaptive Fuzzy Sliding Mode Control, Moradi M, Menhaj M B.Adaptive Fuzzy Sliding Mode Control Design for a Low-Lift Reentry Vehicle[J] .Journal of Aerospace Engineering, 2011, 25 (2): 210-216), Babaei A R in 2013 is that non-minimum phase and Nonlinear Flight device have designed fuzzy sliding mode tracking control robot pilot (Babaei A R, Mortazavi M, Moradi M H.Fuzzy sliding mode autopilot design for nonminimum phase and nonlinear UAV[J] .Journal of Intelligent and Fuzzy Systems, 2013, 24 (3): 499-509), a lot of research only rests on the Utopian simulation study stage, and there are three problems in this design: (1), owing to cannot carrying out the extreme low-altitude handling and stability experiment of aircraft, is difficult to obtain the mathematical model of accurate controlled device, (2) stability margin stipulating for army's mark etc. is evaluated the important performance indexes of flight control system, and state-space method far can be expressed with obvious form unlike classical frequency method, (3) too complicated, the constraint of not considering working control device and state of flight of controller architecture, the controller of design physically can not be realized.

The scholar Rosenbrock of Britain systematically, study in a creative way in the design that how frequency domain method is generalized to multi-variable system and gone, utilize matrix diagonal dominance concept, Multivariable is converted into the design problem of the single-variable system of the classical approach that can know with people, in succession there is Mayne sequence return difference method later, MacFarlane System with Characteristic Locus Method, the methods such as Owens dyadic expansion, common feature is many input more than one outputs, the design of serious associated multi-variable system between loop, turn to the design problem of a series of single-variable systems, and then can select a certain classical approach (frequency response method of Nyquist and Bode, the root-locus technique of Evans etc.) design of completion system, above-mentioned these methods retain and have inherited the advantage of classic graphic-arts technique, do not require accurate especially mathematical model, easily meet the restriction in engineering.Particularly, in the time that employing has the conversational computer-aided design system of people's one machine of graphic display terminal to realize, can give full play to deviser's experience and wisdom, design and both meet quality requirements, be again controller physically attainable, simple in structure; (tall and big far away, sieve becomes, Shen Hui, Hu Dewen, Flexible Satellite Attitude Decoupling Controller Design Using Multiple Variable Frequency Domain Method, aerospace journal, 2007, Vol.28 (2), pp442-447 multivariate frequency method have been carried out improving research both at home and abroad; Xiong Ke, Xia Zhixun, Guo Zhenyun, the hypersonic cruise vehicle multivariable frequency domain approach of banked turn Decoupling design, plays arrow and guidance journal, 2011, Vol.31 (3), pp25-28) still, when this method for designing can taking into account system uncertain problem, conservative property is excessive, under aircraft control vane limited case, can not obtain rational design result.

In the development of high performance airplane, evaluate the quality of an airplane flight quality, not only depend on aircraft itself and driver-operated dynamics, also depend between driver and aircraft and highly as one man coordinate, and the rationality that between driver and Advanced Aircraft flight control system, function is distributed.Until after 1980, the American army mark of evaluating flying qualifies of aircraft still exists major defect, does not consider that driver is in the effect of handling in loop, thus thus the evaluation of gained and Aviatrix take a flight test after the result of gained still have certain gap.In recent years, developing the closed loop criterion (Neil 2 Smith's criterions) that a kind of driver of having participates in system, but how to realize so far still without efficient algorithm.Neil-Smith criterion proposed in 1970, and it is a closed loop following in elevation criterion.The method that it considers a problem is: in the time that driver drives an airplane and aircraft form a closed-loop system, driver handles like a cork and just can reach specific airmark, flight quality is good.In order to obtain the evaluation opinion to aircraft consistent with driver, in theoretical analysis, driver must be included.Conventionally, the mathematical model of driving behavior is nonlinear, may be discrete, but in the time that research has the manipulating objects of stability, useful approximate model is still linear.Shown by a large amount of flight practices and simulation study, the task that driver's behavior will be completed by his psychological characteristic, physiological property, surrounding environment, control system, manipulating objects decides, although driver has feature separately, but completing in single aerial mission, most of drivers' action can be described by completely specified mathematical model, it is the mean state of the lot of experiments of driving behavior, very approaching with actual conditions, the therefore present man-machine loop's characteristic pilot model that mostly adopts following form:

$Y_p\left(s\right)=K_p\frac{T_{D}s+1}{T_{I}s+1}{e}^{-\mathrm{\τs}}$

Estimate man-machine loop's characteristic with system open-loop transfer function or frequency characteristic.

Wherein: K _pfor the static gain of driver's link, inherent delay characteristic, the T that τ is driver _dfor driver's lead compensation time constant, T _ifor driver's lag compensation time constant; After this model adds, that in flight controller, will consider that the slower driver of reaction brings brings out oscillation problem, in thru-flight envelope curve, the slower driver of reaction is allowed to controller design does not also have systematic method, just has part Study to single state of flight.

In sum, current control method can't change at dummy vehicle, design non-driver bring out that vibration, overshoot are little, low-latitude flying controller stably according to the stability margin index in full flight envelope.

Summary of the invention

Can not in the situation that full flight envelope inner model changes greatly, design at aircraft and meet the technological deficiency that the non-driver of the stability margin index in full flight envelope brings out vibration, little, the steady low-latitude flying controller of overshoot in order to overcome existing method, the invention provides the multistage PID robust Controller Design of the compound root locus of a kind of Longitudinal Flight model cluster man-machine loop method, the method directly determines by frequency sweep flight test the model cluster that the amplitude-frequency that obtains in full envelope curve and phase-frequency characteristic form under given differing heights, Mach number condition; According to the amplitude-frequency nargin in flight envelope and the mark requirement of phase margin army, provide the Distribution of Closed Loop Poles restriction index under corresponding root locus description, by adding the identification Method in multistage PID controller the restriction index of the Distribution of Closed Loop Poles in the full envelope curve of aircraft and System Discrimination to determine multistage PID robust controller sum of series parameter value; Describing from root locus concept that Distribution of Closed Loop Poles limits designs the non-driver that meets full flight envelope and brings out that vibration, overshoot are little, low-latitude flying controller stably.

The technical solution adopted for the present invention to solve the technical problems: the multistage PID robust Controller Design of the compound root locus of a kind of Longitudinal Flight model cluster man-machine loop method, is characterized in comprising the following steps:

1, under given differing heights, Mach number by frequency sweep flight test directly by allowing amplitude-frequency and phase-frequency characteristic in the full envelope curve of flight to form elevating rudder in the full envelope curve of aircraft and the model cluster of flying height, between the aircraft elevating rudder of correspondence and flying height, open-loop transfer function bunch is described as:

$G_0\left(S\right)=\frac{e^{-\mathrm{\σ}(h,M)s}K(h,M)A(h,M,s)}{B(h,M,s)}[1+\mathrm{\Δ}\left(s\right)]$

Wherein

A(h,M,s)=s ^m+a _m-1(h,M)s ^m-1+a _m-2(h,M)s ^m-2+…+a ₁(h,M)s+a ₀(h,M)、

B (h, M, s)=s ⁿ+ b _n-1(h, M) s ^n-1+ b _n-2(h, M) s ^n-2+ ... + b ₁(h, M) s+b ₀(h, M) is polynomial expression, and s is the variable after laplace transform conventional in transport function, h, and M is respectively flying height and Mach number, and σ (h, M) is the time delay of pitch channel, and K (h, M) is with h, the gain that M changes, a _l(h, M), l=0,1,2 ..., m-1 be in polynomial expression A (h, M, s) with h, M change coefficient bunch, b _i(h, M), i=0,1,2 ..., n-1 be in polynomial expression B (h, M, s) with h, M change coefficient bunch, the indeterminate that △ k (s) is model;

Pilot model while considering man-machine loop's characteristic:

$Y_p\left(s\right)=K_p\frac{T_{D}s+1}{T_{I}s+1}{e}^{-\mathrm{\τs}}$

Estimate man-machine loop's characteristic with system open-loop transfer function or frequency characteristic;

Wherein: K _pfor the static gain of driver's link, inherent delay characteristic, the T that τ is driver _dfor driver's lead compensation time constant, T _ifor driver's lag compensation time constant;

Like this, the open loop models of man-machine system just becomes:

$G\left(s\right)=K_p\frac{T_{D}s+1}{T_{I}s+1}{e}^{-\mathrm{\τs}}\frac{e^{-\mathrm{\σ}(h,M)s}K(h,M)A(h,M,s)}{B(h,M,s)}[1+\mathrm{\Δk}\left(s\right)]$

2, the transport function of the multistage PID controller of candidate is:

$G_c\left(s\right)=\underset{i=1}{\overset{N}{\mathrm{\Π}}}[k_p\left(i\right){+k}_I\left(i\right)/s+k_D\left(i\right)\·s]$

In formula, k _cfor constant gain to be determined, N is integer, represents the progression of multistage PID controller to be determined, k _p(i), k _i(i), k _d(i) i=1,2 ..., N is constant to be determined;

Add after multistage PID controller, the open-loop transfer function of whole system is:

$G\left(s\right)G_c\left(s\right)=K_p\frac{T_{D}s+1}{T_{I}s+1}{e}^{-\mathrm{\τs}}\frac{e^{-\mathrm{\σ}(h,M)s}K(h,M)A(h,M,s)}{B(h,M,s)}[1+\mathrm{\Δk}\left(s\right)]\underset{i=1}{\overset{N}{\mathrm{\Π}}}{[k}_p\left(i\right)+k_I\left(i\right)/s+k_D\left(i\right)\·s]$

Corresponding root locus equation is:

$\begin{array}{c}{K}_p{e}^{-\mathrm{\τs}}{e}^{-\mathrm{\σ}(h,M)s}K(h,M)A(h,M,s)\·(T_{D}s+1)[1+\mathrm{\Δk}\left(s\right)]\underset{i=1}{\overset{N}{\mathrm{\Π}}}[k_p\left(i\right)\·s+k_I\left(i\right)+k_D\left(i\right)\·s^2];\\ =-s\·(T_{I}s+1)\·B(h,M,s)\end{array}$

3, establish s=σ+j ω, wherein: the real part that σ is s, the imaginary part that ω is s, j is the imaginary part of symbol; The stability margin index of system is set as:

wherein, be non-zero real, ξ gives fixed number; Set up according to flight test or wind tunnel test the lagging phase angle that model indeterminate causes

radian, amplitude

the stability margin index of system is adjusted into:

with

wherein, △ _mand △ _abe whole real number;

Like this, the stability margin index of system can be converted into: according to

$\begin{array}{c}\{{K_p{e}^{-\mathrm{\τs}}{e}^{-\mathrm{\σ}(h,M)s}K(h,M)A(h,M,s)\·(T_{D}s+1)[1+\mathrm{\Δk}\left(s\right)]\underset{i=1}{\overset{N}{\mathrm{\Π}}}[k_p\left(i\right)\·s+k_I\left(i\right)+k_D\left(i\right)\·s^2]\}}_{s=\mathrm{\σ}+\mathrm{j\ω}}\\ =-{\{s\·(T_{I}s+1)\·B(h,M,s)\}}_{s=\mathrm{\σ}+\mathrm{j\ω}}\end{array}$

Or

$\left\{\begin{array}{c}\mathrm{Re}\left\{{\{K_p{e}^{-\mathrm{\τs}}{e}^{-\mathrm{\σ}(h,M)s}k(h,M)A(h,M,s)\·(T_{D}s+1)[1+\mathrm{\Δk}\left(s\right)\left]\underset{i=1}{\overset{N}{\mathrm{\Π}}}\right[k_p\left(i\right)\·s+k_I\left(i\right)+k_D\left(i\right)\·s^2]{\}}_{}}_{s=\mathrm{\σ}+\mathrm{j\ω}}\right\}\\ =-\mathrm{Re}\left\{{\{s\·(T_{I}s+1)\·B(h,M,s)\}}_{s=\mathrm{\σ}+\mathrm{j\ω}}\right\}\\ \mathrm{Im}\left\{{\{K_p{e}^{-\mathrm{\τs}}{e}^{-\mathrm{\σ}(h,M)s}K(h,M)A(h,M,s)\·(T_{D}s+1)[1+\mathrm{\Δk}\left(s\right)\left]\underset{i=1}{\overset{N}{\mathrm{\Π}}}\right[k_p\left(i\right)\·s+K_I\left(i\right)+K_D\left(i\right)\·s^2\left]\right\}}_{s=\mathrm{\σ}+\mathrm{j\ω}}\right\}\\ =\mathrm{Im}\left\{{\{s\·(T_{I}s+1\·B(h,M,s)1\}}_{s=\mathrm{\σ}+\mathrm{j\ω}}\right\}\end{array}\right.$

The root locus obtaining must meet

according under this index and maximum likelihood criterion or the common constraint of other criterion, can determine according to the maximum likelihood method in system model Structure Identification or discrimination method progression N, the constant k of multistage PID controller _p(i), k _i(i), k _d(i) i=1,2 ..., N.

The invention has the beneficial effects as follows: the concept of the Distribution of Closed Loop Poles restriction from root locus is described, by adding multistage PID controller, in full flight envelope, according to meeting that given Distribution of Closed Loop Poles restriction requires and identification Method is determined the parameter of multistage PID robust controller, design the non-driver that meets full flight envelope and bring out that vibration, overshoot are little, low-latitude flying controller stably.

Below in conjunction with embodiment, the present invention is elaborated.

Embodiment

1, under given differing heights, Mach number, use Linear chirp

(f ₀for initial frequency, f ₁for cutoff frequency, r=(f ₁-f ₀)/T, T is the frequency sweep time) or logarithm swept-frequency signal f (t)=A (t) sin{2 π f ₀/ r[ex _p(rt)-1] } (f ₀for initial frequency, f ₁for cutoff frequency, r=ln (f ₁/ f ₀)/T, T is the frequency sweep time) aircraft is encouraged, amplitude-frequency and phase-frequency characteristic in the full envelope curve that can directly obtain allowing to fly, the elevating rudder in the full envelope curve of formation aircraft and the model cluster of flying height, between corresponding aircraft elevating rudder and flying height, open-loop transfer function bunch is described as:

$G_0\left(S\right)=\frac{e^{-\mathrm{\σ}(h,M)s}K(h,M)A(h,M,s)}{B(h,M,s)}[1+\mathrm{\Δ}\left(s\right)]$

Wherein

A(h,M,s)=s ^m+a _m-1(h,M)s ^m-1+a _m-2(h,M)s ^m-2+…+a ₁(h,M)s+a ₀(h,M)、

B (h, M, s)=s ⁿ+ b _n-1(h, M) s ^n-1+ b _n-2(h, M) s ^n-2+ ... + b ₁(h, M) s+b ₀(h, M) is polynomial expression, and s is the variable after laplace transform conventional in transport function, h, and M is respectively flying height and Mach number, and σ (h, M) is the time delay of pitch channel, and K (h, M) is with h, the gain that M changes, a _l(h, M), l=0,1,2 ..., m-1 be in polynomial expression A (h, M, s) with h, M change coefficient bunch, b _i(h, M), i=0,1,2 ..., n-1 be in polynomial expression B (h, M, s) with h, M change coefficient bunch, the indeterminate that △ k (s) is model;

Pilot model while considering man-machine loop's characteristic:

$Y_p\left(s\right)=K_p\frac{T_{D}s+1}{T_{I}s+1}{e}^{-\mathrm{\τs}}$

Estimate man-machine loop's characteristic with system open-loop transfer function or frequency characteristic;

Wherein: K _pfor the static gain of driver's link, inherent delay characteristic, the T that τ is driver _dfor driver's lead compensation time constant, T _ifor driver's lag compensation time constant;

Like this, man-machine system open loop models just becomes:

$G\left(s\right)=K_p\frac{T_{D}s+1}{T_{I}s+1}{e}^{-\mathrm{\τs}}\frac{e^{-\mathrm{\σ}(h,M)s}K(h,M)A(h,M,s)}{B(h,M,s)}[1+\mathrm{\Δk}\left(s\right)]$

2, the transport function of the multistage PID controller of candidate is:

$G_c\left(s\right)=\underset{i=1}{\overset{N}{\mathrm{\Π}}}[k_p\left(i\right){+k}_I\left(i\right)/s+k_D\left(i\right)\·s]$

In formula, k _cfor constant gain to be determined, N is integer, represents the progression of multistage PID controller to be determined, k _p(i), k _i(i), k _d(i) i=1,2 ..., N is constant to be determined;

Add after multistage PID controller, the open-loop transfer function of whole system is:

$G\left(s\right)G_c\left(s\right)=K_p\frac{T_{D}s+1}{T_{I}s+1}{e}^{-\mathrm{\τs}}\frac{e^{-\mathrm{\σ}(h,M)s}K(h,M)A(h,M,s)}{B(h,M,s)}[1+\mathrm{\Δk}\left(s\right)]\underset{i=1}{\overset{N}{\mathrm{\Π}}}{[k}_p\left(i\right)+k_I\left(i\right)/s+k_D\left(i\right)\·s]$

Corresponding root locus equation is:

$\begin{array}{c}{K}_p{e}^{-\mathrm{\τs}}{e}^{-\mathrm{\σ}(h,M)s}K(h,M)A(h,M,s)\·(T_{D}s+1)[1+\mathrm{\Δk}\left(s\right)]\underset{i=1}{\overset{N}{\mathrm{\Π}}}[k_p\left(i\right)\·s+k_I\left(i\right)+k_D\left(i\right)\·s^2];\\ =-s\·(T_{I}s+1)\·B(h,M,s)\end{array}$

3, establish s=σ+j ω, wherein: the real part that σ is s, the imaginary part that ω is s, j is the imaginary part of symbol; The stability margin index of system is set as:

wherein, be non-zero real, ξ gives fixed number; Set up according to flight test or wind tunnel test the lagging phase angle that model indeterminate causes

radian, amplitude

the stability margin index of system is adjusted into:

with

wherein, △ _mand △ _abe whole real number;

Like this, the stability margin index of system can be converted into: according to

$\begin{array}{c}\{{K_p{e}^{-\mathrm{\τs}}{e}^{-\mathrm{\σ}(h,M)s}K(h,M)A(h,M,s)\·(T_{D}s+1)[1+\mathrm{\Δk}\left(s\right)]\underset{i=1}{\overset{N}{\mathrm{\Π}}}[k_p\left(i\right)\·s+k_I\left(i\right)+k_D\left(i\right)\·s^2]\}}_{s=\mathrm{\σ}+\mathrm{j\ω}}\\ =-{\{s\·(T_{I}s+1)\·B(h,M,s)\}}_{s=\mathrm{\σ}+\mathrm{j\ω}}\end{array}$

Or

$\left\{\begin{array}{c}\mathrm{Re}\left\{{\{K_p{e}^{-\mathrm{\τs}}{e}^{-\mathrm{\σ}(h,M)s}k(h,M)A(h,M,s)\·(T_{D}s+1)[1+\mathrm{\Δk}\left(s\right)\left]\underset{i=1}{\overset{N}{\mathrm{\Π}}}\right[k_p\left(i\right)\·s+k_I\left(i\right)+k_D\left(i\right)\·s^2]{\}}_{}}_{s=\mathrm{\σ}+\mathrm{j\ω}}\right\}\\ =-\mathrm{Re}\left\{{\{s\·(T_{I}s+1)\·B(h,M,s)\}}_{s=\mathrm{\σ}+\mathrm{j\ω}}\right\}\\ \mathrm{Im}\left\{{\{K_p{e}^{-\mathrm{\τs}}{e}^{-\mathrm{\σ}(h,M)s}K(h,M)A(h,M,s)\·(T_{D}s+1)[1+\mathrm{\Δk}\left(s\right)\left]\underset{i=1}{\overset{N}{\mathrm{\Π}}}\right[k_p\left(i\right)\·s+K_I\left(i\right)+K_D\left(i\right)\·s^2\left]\right\}}_{s=\mathrm{\σ}+\mathrm{j\ω}}\right\}\\ =\mathrm{Im}\left\{{\{s\·(T_{I}s+1\·B(h,M,s)1\}}_{s=\mathrm{\σ}+\mathrm{j\ω}}\right\}\end{array}\right.$

The root locus obtaining must meet

with

according under this index and maximum likelihood criterion or the common constraint of other criterion, can determine according to the maximum likelihood method in system model Structure Identification or discrimination method progression N, the constant k of multistage PID controller _p(i), k _i(i), k _d(i) i=1,2 ..., N.

Claims (1)

1. the multistage PID robust Controller Design of the compound root locus of a Longitudinal Flight model cluster man-machine loop method, is characterized in comprising the following steps:

1) under given differing heights, Mach number by frequency sweep flight test directly by allowing amplitude-frequency and phase-frequency characteristic in the full envelope curve of flight to form elevating rudder in the full envelope curve of aircraft and the model cluster of flying height, between the aircraft elevating rudder of correspondence and flying height, open-loop transfer function bunch is described as:

$G_0\left(S\right)=\frac{e^{-\mathrm{\σ}(h,M)s}K(h,M)A(h,M,s)}{B(h,M,s)}[1+\mathrm{\Δ}\left(s\right)]$

Wherein

A(h,M,s)=s ^m+a _m-1(h,M)s ^m-1+a _m-2(h,M)s ^m-2+…+a ₁(h,M)s+a ₀(h,M)、

B (h, M, s)=s ⁿ+ b _n-1(h, M) s ^n-1+ b _n-2(h, M) s ^n-2+ ... + b ₁(h, M) s+b ₀(h, M) is polynomial expression, and s is the variable after laplace transform conventional in transport function, h, and M is respectively flying height and Mach number, and σ (h, M) is the time delay of pitch channel, and K (h, M) is with h, the gain that M changes, a _l(h, M), l=0,1,2 ..., m-1 be in polynomial expression A (h, M, s) with h, M change coefficient bunch, b _i(h, M), i=0,1,2 ..., n-1 be in polynomial expression B (h, M, s) with h, M change coefficient bunch, the indeterminate that △ k (s) is model;

Pilot model while considering man-machine loop's characteristic:

$Y_p\left(s\right)=K_p\frac{T_{D}s+1}{T_{I}s+1}{e}^{-\mathrm{\τs}}$

Estimate man-machine loop's characteristic with system open-loop transfer function or frequency characteristic;

Wherein: K _pfor the static gain of driver's link, inherent delay characteristic, the T that τ is driver _dfor driver's lead compensation time constant, T _ifor driver's lag compensation time constant;

Like this, the open loop models of man-machine system just becomes:

$G\left(s\right)=K_p\frac{T_{D}s+1}{T_{I}s+1}{e}^{-\mathrm{\τs}}\frac{e^{-\mathrm{\σ}(h,M)s}K(h,M)A(h,M,s)}{B(h,M,s)}[1+\mathrm{\Δk}\left(s\right)]$

2) transport function of the multistage PID controller of candidate is:

$G_c\left(s\right)=\underset{i=1}{\overset{N}{\mathrm{\Π}}}[k_p\left(i\right){+k}_I\left(i\right)/s+k_D\left(i\right)\·s]$

In formula, k _cfor constant gain to be determined, N is integer, represents the progression of multistage PID controller to be determined, k _p(i), k _i(i), k _d(i) i=1,2 ..., N is constant to be determined;

Add after multistage PID controller, the open-loop transfer function of whole system is:

$G\left(s\right)G_c\left(s\right)=K_p\frac{T_{D}s+1}{T_{I}s+1}{e}^{-\mathrm{\τs}}\frac{e^{-\mathrm{\σ}(h,M)s}K(h,M)A(h,M,s)}{B(h,M,s)}[1+\mathrm{\Δk}\left(s\right)]\underset{i=1}{\overset{N}{\mathrm{\Π}}}{[k}_p\left(i\right)+k_I\left(i\right)/s+k_D\left(i\right)\·s]$

Corresponding root locus equation is:

$\begin{array}{c}{K}_p{e}^{-\mathrm{\τs}}{e}^{-\mathrm{\σ}(h,M)s}K(h,M)A(h,M,s)\·(T_{D}s+1)[1+\mathrm{\Δk}\left(s\right)]\underset{i=1}{\overset{N}{\mathrm{\Π}}}[k_p\left(i\right)\·s+k_I\left(i\right)+k_D\left(i\right)\·s^2];\\ =-s\·(T_{I}s+1)\·B(h,M,s)\end{array}$

3) establish s=σ+j ω, wherein: the real part that σ is s, the imaginary part that ω is s, j is the imaginary part of symbol; The stability margin index of system is set as:

wherein, be non-zero real, ξ gives fixed number; Set up according to flight test or wind tunnel test the lagging phase angle that model indeterminate causes radian, amplitude

the stability margin index of system is adjusted into:

with

wherein, △ _mand △ _abe whole real number;

Like this, the stability margin index of system can be converted into: according to

$\begin{array}{c}\{{K_p{e}^{-\mathrm{\τs}}{e}^{-\mathrm{\σ}(h,M)s}K(h,M)A(h,M,s)\·(T_{D}s+1)[1+\mathrm{\Δk}\left(s\right)]\underset{i=1}{\overset{N}{\mathrm{\Π}}}[k_p\left(i\right)\·s+k_I\left(i\right)+k_D\left(i\right)\·s^2]\}}_{s=\mathrm{\σ}+\mathrm{j\ω}}\\ =-{\{s\·(T_{I}s+1)\·B(h,M,s)\}}_{s=\mathrm{\σ}+\mathrm{j\ω}}\end{array}$

Or

$\left\{\begin{array}{c}\mathrm{Re}\left\{{\{K_p{e}^{-\mathrm{\τs}}{e}^{-\mathrm{\σ}(h,M)s}k(h,M)A(h,M,s)\·(T_{D}s+1)[1+\mathrm{\Δk}\left(s\right)\left]\underset{i=1}{\overset{N}{\mathrm{\Π}}}\right[k_p\left(i\right)\·s+k_I\left(i\right)+k_D\left(i\right)\·s^2]{\}}_{}}_{s=\mathrm{\σ}+\mathrm{j\ω}}\right\}\\ =-\mathrm{Re}\left\{{\{s\·(T_{I}s+1)\·B(h,M,s)\}}_{s=\mathrm{\σ}+\mathrm{j\ω}}\right\}\\ \mathrm{Im}\left\{{\{K_p{e}^{-\mathrm{\τs}}{e}^{-\mathrm{\σ}(h,M)s}K(h,M)A(h,M,s)\·(T_{D}s+1)[1+\mathrm{\Δk}\left(s\right)\left]\underset{i=1}{\overset{N}{\mathrm{\Π}}}\right[k_p\left(i\right)\·s+K_I\left(i\right)+K_D\left(i\right)\·s^2\left]\right\}}_{s=\mathrm{\σ}+\mathrm{j\ω}}\right\}\\ =\mathrm{Im}\left\{{\{s\·(T_{I}s+1\·B(h,M,s)1\}}_{s=\mathrm{\σ}+\mathrm{j\ω}}\right\}\end{array}\right.$ The root locus obtaining must meet

according under this index and maximum likelihood criterion or the common constraint of other criterion, can determine according to the maximum likelihood method in system model Structure Identification or discrimination method progression N, the constant k of multistage PID controller _p(i), k _i(i), k _d(i) i=1,2 ..., N.