CN1877105B

CN1877105B - Automobile longitudinal acceleration tracking control method

Info

Publication number: CN1877105B
Application number: CN200610089496XA
Authority: CN
Inventors: 李克强; 高锋; 王建强; 罗禹贡; 连小珉
Original assignee: Tsinghua University
Current assignee: Tsinghua University
Priority date: 2006-06-30
Filing date: 2006-06-30
Publication date: 2012-02-08
Anticipated expiration: 2026-06-30
Also published as: CN1877105A

Abstract

A vehicle longitudinal acceleration tracking control method belongs to the technical field of vehicle longitudinal dynamics control. It is characterized in that it is an estimator designed using the elements in the set of multiplicative uncertain models of the controlled object composed of the reverse and forward dynamics models of the car, and the current value of the model error between each model and the actual object Estimate the system gain of the input signal, and use the estimated system gain as a switching index function, select the corresponding controller online from the controller set designed according to the elements in the multiplicative uncertainty model set, and The throttle opening of the engine is controlled to realize the tracking control of the longitudinal acceleration of the vehicle. When the parameters of the vehicle and the environment are relatively uncertain, the method of the invention can effectively control the longitudinal acceleration of the vehicle, and has better stability and tracking performance.

Description

A vehicle longitudinal acceleration tracking control method

技术领域： Technical field:

一种汽车纵向加速度跟踪控制方法属于汽车纵向动力学控制技术领域。A vehicle longitudinal acceleration tracking control method belongs to the technical field of vehicle longitudinal dynamics control.

背景技术： Background technique:

汽车纵向运动控制系统通常被设计成分层结构：上层控制器根据相对车距和车速输出期望加速度，设计时主要考虑驾驶员特性、队列稳定性和交通流等问题；下层加速度跟踪控制器通过对执行机构的控制使汽车实际加速度跟踪期望值，设计时主要考虑车辆动力学问题。汽车纵向加速度跟踪控制是汽车纵向运动控制的关键技术之一。文献1(Mikael Persson，etc.带起停功能的自适应巡航控制系统控制器设计，Proceedings of the 1999 IEEEInternational Conference on Control Application，1999)将车辆纵向动力学特性近似为线性系统，采用PI方法设计加速度跟踪控制器。由于车辆纵向动力学特性具有非线性，近似为线性系统的方法很难在所有工作点都取得较好的效果。文献2(R.Mayr，基于反馈线性化方法的汽车职能巡航控制系统设计，Proceedings of the American Control Conference，1994)首先采用精确线性化方法将非线性模型转化为线性模型，然后基于线性化模型设计了汽车纵向加速度控制器。文献3(J.K.Hedrick，面向自动驾驶车辆的非线性控制器设计，UKACC International Conference on Control，1998)在建立非线性车辆纵向动力学模型基础上，采用滑模控制方法设计了汽车纵向加速度跟踪滑模控制器。文献4(Kyongsu Yi，etc.基于电控真空助力器的车间距和车速控制系统设计，JSAE Review，22，2001)、文献5(YangBin，etc.带起停功能的巡航控制系统加速度跟踪控制器设计，Proceedings of the 2004IEEE International Conference on Networking，Sensing & Control，2004)和文献6(侯德藻等，基于模型匹配方法的汽车主动避撞下位控制系统，汽车工程，25(4)，2003)都首先通过逆动力学模型来消除传动系统的非线性，将由逆向和正向车辆纵向动力学模型组成的新系统近似为线性系统并对线性系统特性进行辨识基础上，文献4采用PI方法，文献5采用滑模方法，文献6采用前馈为模型匹配控制器反馈为H_∞控制器的二自由度结构分别设计了相应的加速度跟踪控制器。Vehicle longitudinal motion control systems are usually designed in a hierarchical structure: the upper layer controller outputs the desired acceleration according to the relative vehicle distance and vehicle speed, and the design mainly considers driver characteristics, queue stability, and traffic flow; the lower layer acceleration tracking controller implements The control of the mechanism makes the actual acceleration of the car track the expected value, and the vehicle dynamics problem is mainly considered in the design. Vehicle longitudinal acceleration tracking control is one of the key technologies of vehicle longitudinal motion control. Document 1 (Mikael Persson, etc. Adaptive cruise control system controller design with start-stop function, Proceedings of the 1999 IEEEInternational Conference on Control Application, 1999) approximates the longitudinal dynamics of the vehicle as a linear system, and uses the PI method to design the acceleration Track controller. Due to the nonlinearity of the longitudinal dynamics of the vehicle, it is difficult to obtain good results at all operating points by the method of approximating the linear system. Literature 2 (R.Mayr, Design of Automotive Functional Cruise Control System Based on Feedback Linearization Method, Proceedings of the American Control Conference, 1994) first uses the exact linearization method to convert the nonlinear model into a linear model, and then designs based on the linearized model A car longitudinal acceleration controller. Document 3 (JK Hedrick, Nonlinear Controller Design for Autonomous Vehicles, UKACC International Conference on Control, 1998) based on the establishment of a nonlinear vehicle longitudinal dynamics model, a sliding mode control method for vehicle longitudinal acceleration tracking was designed using the sliding mode control method device. Document 4 (Kyongsu Yi, etc. Design of inter-vehicle distance and vehicle speed control system based on electronically controlled vacuum booster, JSAE Review, 22, 2001), Document 5 (YangBin, etc. Acceleration tracking controller for cruise control system with start-stop function Design, Proceedings of the 2004IEEE International Conference on Networking, Sensing & Control, 2004) and Document 6 (Hou Dezao et al., Vehicle Active Collision Avoidance Lower Control System Based on Model Matching Method, Automotive Engineering, 25(4), 2003) are first passed The inverse dynamics model is used to eliminate the nonlinearity of the transmission system, and the new system composed of the inverse and forward vehicle longitudinal dynamics models is approximated as a linear system and on the basis of identifying the characteristics of the linear system, the PI method is used in Document 4, and the sliding mode is used in Document 5 Method, literature 6 adopts the feed-forward as the model matching controller and the feedback as the H _∞ controller's two-degree-of-freedom structure to design the corresponding acceleration tracking controllers.

上述控制方法需要比较准确的车辆模型，即使H_∞等鲁棒控制方法也不能处理很大的模型不确定性。但在实际行驶条件下，由于建模过程中不可避免的未建模动态特性和参数变化造成的模型误差会使车辆纵向动力学特性在很大一个范围内发生变化，使得通过一个固定参数的控制器很难在保证闭环稳定性的同时获得较好的加速度跟踪性能。The above control methods require a relatively accurate vehicle model, and even robust control methods such as H _∞ cannot handle large model uncertainties. However, under actual driving conditions, the model error caused by the inevitable unmodeled dynamic characteristics and parameter changes in the modeling process will cause the longitudinal dynamic characteristics of the vehicle to change in a large range, making the control through a fixed parameter It is difficult for the controller to obtain better acceleration tracking performance while ensuring closed-loop stability.

发明内容： Invention content:

本发明的目的在于，提出一种汽车纵向加速度跟踪控制方法。该方法根据一定的切换规则在线地从控制器集合中选择合适的控制器，通过对节气门开度的控制实现汽车纵向加速度的跟踪控制，使得汽车纵向动力学特性在很大范围内发生变化时，能够同时保证闭环系统稳定性和较好的跟踪性能。The object of the present invention is to propose a method for tracking and controlling the longitudinal acceleration of the vehicle. According to a certain switching rule, the method selects the appropriate controller online from the controller set, and realizes the tracking control of the longitudinal acceleration of the vehicle through the control of the throttle opening, so that when the longitudinal dynamic characteristics of the vehicle change in a wide range , which can ensure the stability of the closed-loop system and better tracking performance at the same time.

本发明的特征在于，它是利用由汽车的逆向和正向动力学模型组成的被控对象的乘性不确定模型集合中的元素设计的估计器，对每个模型与实际对象之间的模型误差对当前输入信号的系统增益进行估计，并以估计得到的该系统增益作为切换指标函数，在线地从根据所述乘性不确定模型集合中的元素而设计的控制器集合中选择相应的控制器，对发动机节气门开度进行控制，实现汽车纵向加速度的跟踪控制；The present invention is characterized in that it is an estimator designed using the elements in the set of multiplicative uncertain models of the controlled object consisting of the reverse and forward dynamics models of the car, and the model error between each model and the actual object Estimate the system gain of the current input signal, and use the estimated system gain as the switching index function, and select the corresponding controller online from the controller set designed according to the elements in the multiplicative uncertainty model set , to control the throttle opening of the engine to realize the tracking control of the longitudinal acceleration of the vehicle;

该方法含有在整车控制器中执行的以下步骤：The method contains the following steps performed in the vehicle controller:

1)初始化：1) Initialization:

车辆参数和汽车行驶环境的参数；Vehicle parameters and parameters of the vehicle driving environment;

δ：指数衰减系数，其范围为：0.1～1；δ: Exponential attenuation coefficient, its range is: 0.1～1;

u：加速度控制量，其初始值为0；u: Acceleration control amount, its initial value is 0;

与被控对象乘性不确定模型集合中的元素对应的估计器集合为：The set of estimators corresponding to the elements in the multiplicative uncertainty model set of the plant is:

${\overset{\cdot \cdot}{x x}}_{E E. 11} = = {A A}_{E E. 11} {x x}_{E E. 11} + + {B B}_{E E. 11} u u,, {\overset{\cdot &Center Dot;}{x x}}_{E E. 22} = = {A A}_{E E. 22} {x x}_{E E. 22} + + {B B}_{E E. 22} u u,,$

e_i＝a_i-a＝C_E1ix_E1+D_E1iu-a，z_i＝C_E2ix_E2+D_E2iu，i＝1…N，e _i =a _i -a=C _E1i x _E1 +D _E1i ua, z _i =C _E2i x _E2 +D _E2i u, i=1...N,

式(13)中，x_E1和x_E2为与被控对象乘性不确定模型集合中元素对应的估计器状态，其初始值为0；N是控制器个数，A_E1，B_E1，A_E2，B_E2，C_E1i，D_E1i，C_E2i，D_E2i是估计器的状态矩阵；In formula (13), x _E1 and x _E2 are the estimator states corresponding to the elements in the multiplicative uncertain model set of the plant, and their initial value is 0; N is the number of controllers, A _E1 , B _E1 , A _E2 , B _E2 , C _E1i , D _E1i , C _E2i , D _E2i are the state matrix of the estimator;

与被控对象乘性不确定模型集合中的元素对应的控制器集合C为：The controller set C corresponding to the elements in the multiplicative uncertain model set of the controlled object is:

$C C = = {{{K K}_{i i} : : \begin{matrix} {\overset{\cdot &Center Dot;}{x x}}_{C C} = = {A A}_{Ci Ci} {x x}_{C C} + + {B B}_{Ci Ci} (({a a}_{des des} - - a a)) \\ u u = = {C C}_{Ci Ci} {x x}_{C C} + + {D D.}_{Ci Ci} (({a a}_{des des} - - a a)) \end{matrix},, i i = = 11 \cdot &Center Dot; \cdot &Center Dot; \cdot &Center Dot; N N}}$

式(19)中，K_i为控制器集合C的第i个控制器，x_c是控制器集合的状态，其初始值为0；N是控制器个数，A_Ci，B_Ci，C_Ci，D_Ci是状态矩阵；控制器集合C中元素满足：In formula (19), K _i is the i-th controller of the controller set C, x _c is the state of the controller set, and its initial value is 0; N is the number of controllers, A _Ci , B _Ci , C _Ci , D _Ci is the state matrix; the elements in the controller set C satisfy:

${| | | | Σ Σ (({A A}_{i i},, {B B}_{i i},, {C C}_{11 i i},, {D D.}_{11 i i})) | | | |}_{\infty \infty}^{δ δ} < < η η < < \frac{11}{γ γ},, {| | | | {W W}_{per per} ((s the s)) Σ Σ (({A A}_{i i},, {B B}_{i i} {C C}_{22 i i},, {D D.}_{22 i i})) | | | |}_{\infty \infty}^{δ δ} < < β β,, i i = = 11 \cdot &Center Dot; \cdot \cdot \cdot &Center Dot; N N,,$

${A A}_{σ σ} = = [\begin{matrix} {A A}_{E E. 11} - - {Φ Φ}_{σ σ} {B B}_{E E. 11} {D D.}_{Cσ Cσ} {C C}_{E E. 11 σ σ} & 00 & {Φ Φ}_{σ σ} {B B}_{E E. 11} {C C}_{Cσ Cσ} \\ - - {Φ Φ}_{σ σ} {B B}_{E E. 22} {D D.}_{Cσ Cσ} {C C}_{E E. 11 σ σ} & {A A}_{E E. 22} & {Φ Φ}_{σ σ} {B B}_{E E. 22} {C C}_{Cσ Cσ} \\ - - {Φ Φ}_{σ σ} {B B}_{Cσ Cσ} {C C}_{E E. 11 σ σ} & 00 & {A A}_{Cσ Cσ} - - {Φ Φ}_{σ σ} {B B}_{Cσ Cσ} {D D.}_{E E. 11 σ σ} {C C}_{Cσ Cσ} \end{matrix}],, {B B}_{σ σ} = = {Φ Φ}_{σ σ} [\begin{matrix} {B B}_{E E. 11} {D D.}_{Cσ Cσ} \\ {B B}_{E E. 22} {D D.}_{Cσ Cσ} \\ {B B}_{Cσ Cσ} \end{matrix}],,$

C_1σ＝[-Φ_σD_E2σD_CσC_E1σ C_E2σ Φ_σD_E2σC_Cσ]，D_1σ＝Φ_σD_E2σD_Cσ，C _1σ = [-Φ _σ D _E2σ D _Cσ C _E1σ C _E2σ Φ _σ D _E2σ C _Cσ ], D _1σ = Φ _σ D _E2σ D _Cσ ,

D_2σ＝Φ_σ，C_2σ＝-Φ_σ[C_E1σ 0 D_E1σC_Cσ]；D _2σ = Φ _σ , C _2σ = -Φ _σ [C _E1σ 0 D _E1σ C _Cσ ];

其中W_per(s)为性能指标加权函数，符号∑(A，B，C，D)表示由A、B、C、D做系数矩阵的状态空间模型，满足：Among them, W _per (s) is the weighting function of the performance index, and the symbol ∑(A, B, C, D) represents the state space model with A, B, C, D as the coefficient matrix, which satisfies:

$\frac{{| | | | {W W}_{per per} ((s the s)) q q | | | |}_{22}^{δ δ}}{{| | | | {a a}_{des des} | | | |}_{22}^{δ δ}} < < \frac{β β}{11 - - ηγ ηγ};;$

2)采集汽车加速度a，期望汽车加速度a_des；2) Acquisition of vehicle acceleration a, expected vehicle acceleration a _des ;

3)根据被控对象的乘性不确定模型集合设计的估计器的系数矩阵计算估计误差e_i和不确定部分输入z_i：3) According to the coefficient matrix of the estimator designed by the set of multiplicative uncertain models of the plant, calculate the estimation error e _i and the input z _i of the uncertain part:

${\overset{\cdot &Center Dot;}{x x}}_{E E. 11} = = {A A}_{E E. 11} {x x}_{E E. 11} + + {B B}_{E E. 11} u u,, {\overset{\cdot &Center Dot;}{x x}}_{E E. 22} = = {A A}_{E E. 22} {x x}_{E E. 22} + + {B B}_{E E. 22} u u,,$

e_i＝C_E1ix_E1+D_E1iu-a，z_i＝C_E2ix_E2+D_E2iu，i＝1…N， (1)e _i =C _E1i x _E1 +D _E1i ua, z _i =C _E2i x _E2 +D _E2i u, i=1...N, (1)

其中N为被控对象乘性不确定模型集合中元素的个数；Where N is the number of elements in the multiplicative uncertainty model set of the controlled object;

4)采用下式计算乘性不确定模型集合中的N个模型的切换指标：4) Use the following formula to calculate the switching index of N models in the multiplicative uncertainty model set:

${J J}_{i i} ((t t)) = = \frac{{| | | | {e e}_{i i} | | | |}_{22}^{δ δ}}{{| | | | {z z}_{i i} | | | |}_{22}^{δ δ}},, i i = = 11 \cdot &Center Dot; \cdot &Center Dot; \cdot &Center Dot; N N - - - - - - ((22))$

5)上述N个切换指标与控制器集合 $C = {K_{i} : \begin{matrix} {\overset{\cdot}{x}}_{C} = A_{Ci} x_{C} + B_{Ci} (a_{des} - a) \\ u = C_{Ci} x_{C} + D_{Ci} (a_{des} - a) \end{matrix}, i = 1 \cdot \cdot \cdot N}$ 中的N个元素按照i的顺序一一对应，选择上述N个切换指标中最小的切换指标J_k(t)对应的控制器计算控制量u：5) The above N switching indicators and controller set $C = {K_{i} : \begin{matrix} {\overset{&Center Dot;}{x}}_{C} = A_{Ci} x_{C} + B_{Ci} (a_{des} - a) \\ u = C_{Ci} x_{C} + {D.}_{Ci} (a_{des} - a) \end{matrix}, i = 1 &Center Dot; &Center Dot; &Center Dot; N}$ The N elements in are corresponding to each other in the order of i, and the controller corresponding to the smallest switching index J _k (t) among the above N switching indexes is selected to calculate the control quantity u:

$C C = = {{{K K}_{σ σ} : : \begin{matrix} {\overset{\cdot \cdot}{x x}}_{C C} = = {A A}_{Ck C} {x x}_{C C} + + {B B}_{Ck C} (({a a}_{des des} - - a a)) \\ u u = = {C C}_{Ck C} {x x}_{C C} + + {D D.}_{Ck C} (({a a}_{des des} - - a a)) \end{matrix}}}$

6)按照逆汽车纵向动力学模型计算节气门开度θ：6) Calculate the throttle opening θ according to the inverse vehicle longitudinal dynamics model:

${T T}_{edes edes} = = \frac{r r}{{R R}_{d d} {R R}_{} g g 00 {η η}_{00}} (({M m}_{00} u u + + {C C}_{D D.} {Av Av}^{22} + + {M m}_{00} {gf gf}_{00}))$

θ＝MAP^-1(T_edes，ω_e) (3)θ=MAP ^-1 (T _edes , ω _e ) (3)

其中包含下述车辆参数和汽车行驶环境的参数：It contains the following vehicle parameters and parameters of the vehicle driving environment:

v为车速，T_edes为期望的发动机扭矩，ω_e为发动机转速，θ为节气门开度，r为车轮半径，R_d为主减速器速比，R_g0为变速器速比的公称值，η₀为传动系统机械效率的公称值，M₀为整车质量的公称值，C_D为空气阻力系数，g为重力常数，f₀为滚动阻力系数的公称值，A为迎风面积；MAP^-1(■)为发动机逆扭矩特性图；v is the vehicle speed, T _edes is the expected engine torque, ω _e is the engine speed, θ is the throttle opening, r is the wheel radius, R _d is the speed ratio of the main reducer, R _g0 is the nominal value of the transmission speed ratio, η ₀ is the nominal value of the mechanical efficiency of the transmission system, M ₀ is the nominal value of the vehicle mass, C _D is the air resistance coefficient, g is the gravity constant, f ₀ is the nominal value of the rolling resistance coefficient, A is the windward area; MAP ^-1 (■) is the reverse torque characteristic diagram of the engine;

7)根据上述计算得到的节气门开度θ对汽车的发动机进行控制，并将控制量u值返回第2)步继续进行计算和控制。7) Control the engine of the car according to the throttle opening θ obtained by the above calculation, and return the value of the control variable u to step 2) to continue calculation and control.

所述被控对象的乘性不确定模型集合的构造步骤为：The construction steps of the multiplicative uncertain model set of the controlled object are:

第一步：将车辆和环境参数的可能变化范围离散成有限个参数点，设为M个；在每一个参数点利用过程辨识的方法利用输入输出数据辨识得到被控对象的模型，这样共可得到M个传递函数，H_i(s)，i＝1...M；The first step: discretize the possible variation range of the vehicle and environmental parameters into a limited number of parameter points, set as M; at each parameter point, use the method of process identification and use the input and output data to identify the model of the controlled object, so that a total of Obtain M transfer functions, H _i (s), i=1...M;

第二步：采用M个传递函数在频域上覆盖一定的区域，根据拟采用的公称模型的个数，设为N，将该区域平均分成N块，表示为Ω_i，i＝1...N，选择每块区域中心的传递函数作为公称模型，得到N个公称模型G_i(s)，i＝1...N；The second step: use M transfer functions to cover a certain area in the frequency domain, set N according to the number of nominal models to be used, and divide the area into N blocks on average, expressed as Ω _i , i=1.. .N, select the transfer function of the center of each area as the nominal model, and obtain N nominal models G _i (s), i=1...N;

第三步：针对第一步中得到的每个区域，利用下式计算该区域中所有传递函数与该区域对应的公称模型的模型误差：Step 3: For each area obtained in the first step, use the following formula to calculate the model error of all transfer functions in this area and the nominal model corresponding to this area:

${ΔH ΔH}_{i i} ((s the s)) = = \frac{{H h}_{i i} ((s the s)) - - {G G}_{i i} ((s the s))}{{G G}_{i i} ((s the s))},, {&ForAll; &ForAll; H h}_{i i} ((s the s)) &Element; &Element; {Ω Ω}_{j j},, - - - - - - ((33))$

根据计算得到的模型误差，选取模型误差加权函数W_j(s)在所有频率上满足According to the calculated model error, select the model error weighting function W _j (s) to satisfy at all frequencies

$| | {W W}_{j j} ((s the s - - 0.5 0.5 δ δ)) | | > > \frac{11}{γ γ} | | {ΔH ΔH}_{i i} ((s the s - - 0.5 0.5 δ δ)) | |,, &ForAll; &ForAll; {H h}_{i i} ((s the s)) &Element; &Element; {Ω Ω}_{j j},, - - - - - - ((44))$

其中δ为指数衰减系数，是一个常数；符号|□|表示传递函数的幅值，这样便得到描述区域Ω_j的乘性不确定性模型：where δ is the exponential decay coefficient, which is a constant; the symbol |□| represents the magnitude of the transfer function, so that the multiplicative uncertainty model describing the region Ω _j is obtained:

$P P = = {{{P P}_{i i} ((s the s)) = = [[11 + + {Δ Δ}_{i i} {W W}_{i i} ((s the s))]] {G G}_{i i} ((s the s)),, {| | | | {Δ Δ}_{i i} | | | |}_{\infty \infty}^{δ δ} < < γ γ,, i i = = 11,, \cdot &Center Dot; \cdot &Center Dot; \cdot &Center Dot;,, N N}} - - - - - - ((55))$

其中γ是被控对象模型误差的上限；W_i(s)是具有相同特征多项式的模型误差加权函数；G_i(s)是具有相同特征多项式的公称模型；Δ_i为被控对象的模型不确定部分；其大小由(4)保证；N是模型集合中模型的个数；符号‖□‖_∞ ^δ表示系统的L_2δ增益。Where γ is the upper limit of the model error of the controlled object; W _i (s) is the model error weighting function with the same characteristic polynomial; G _i (s) is the nominal model with the same characteristic polynomial; Δ _i is the model of the controlled object Determine the part; its size is guaranteed by (4); N is the number of models in the model set; the symbol ‖□ _‖∞ ^δ represents the L _2δ gain of the system.

试验证明：在汽车和环境参数具有较大不确定时，本发明中的方法可以对汽车纵向加速度进行有效的控制，具有较好的稳定性和跟踪性能。The test proves that the method of the invention can effectively control the longitudinal acceleration of the vehicle when the parameters of the vehicle and the environment are relatively uncertain, and has good stability and tracking performance.

附图说明： Description of drawings:

图1，汽车纵向加速度跟踪控制方法的整体结构图；Fig. 1, the overall structural diagram of automobile longitudinal acceleration tracking control method;

图2，被控对象结构图；Figure 2, the structure diagram of the controlled object;

图3，切换逻辑流程图；Figure 3, switching logic flow chart;

图4，汽车纵向加速度跟踪控制方法流程图；Fig. 4, the flow chart of automobile longitudinal acceleration tracking control method;

图5，国内捷达AT型轿车发动机逆扭矩特性图；Figure 5, the reverse torque characteristic diagram of the domestic Jetta AT car engine;

图6，被控对象频域响应特性覆盖范围，其中(a)是幅频特性图，(b)是相频特性图；Figure 6, the frequency-domain response characteristics coverage of the controlled object, where (a) is the amplitude-frequency characteristic diagram, and (b) is the phase-frequency characteristic diagram;

图7，汽车纵向加速度跟踪控制方法仿真结果，其中(a)汽车纵向加速度响应曲线，(b)切换信号响应曲线，(c)汽车速度响应曲线，(d)节气门开度响应曲线；Figure 7, the simulation results of the vehicle longitudinal acceleration tracking control method, wherein (a) the vehicle longitudinal acceleration response curve, (b) the switching signal response curve, (c) the vehicle speed response curve, (d) the throttle opening response curve;

图8，汽车纵向加速度跟踪控制方法实验曲线，其中(a)汽车纵向加速度响应曲线，(b)切换信号响应，(c)汽车速度响应曲线，(d)节气门开度响应。Figure 8, the experimental curve of the vehicle longitudinal acceleration tracking control method, in which (a) the vehicle longitudinal acceleration response curve, (b) the switching signal response, (c) the vehicle speed response curve, (d) the throttle opening response.

具体实施方式： Detailed ways:

如图1所示，是本发明的汽车纵向加速度跟踪控制方法的整体结构。a_des为期望的汽车加速度；a为实际的汽车加速度；q＝a_des-a为实际汽车加速度跟踪期望加速度的跟踪误差；J_i(t)，i＝1...N为模型集合中各元素与被控对象之间的模型误差对当前信号的系统增益，N是被控对象乘性不确定模型集合中元素的个数；σ为切换信号，编号与σ一致的控制器将被连接到控制回路中；u为根据控制器计算出的控制量。整个控制系统包括被控对象、估计器、切换逻辑和控制器集合四部分。估计器根据控制量u和汽车加速度实时地估计每个控制器对应的切换指标，切换逻辑根据估计器输出的切换指标选择切换指标最小的控制器对应的编号作为输出，根据切换逻辑输出的控制器编号从控制器集合中选择对应的控制器计算控制量u，根据u由汽车逆纵向动力学模型即可得到油门开度，从而对发动机进行控制。As shown in Figure 1, it is the overall structure of the vehicle longitudinal acceleration tracking control method of the present invention. a _des is the expected acceleration of the vehicle; a is the actual acceleration of the vehicle; q=a _des -a is the tracking error of the actual vehicle acceleration tracking the expected acceleration; J _i (t), i=1...N is each The model error between the element and the controlled object is the system gain of the current signal, N is the number of elements in the multiplicative uncertain model set of the controlled object; σ is the switching signal, and the controller whose number is consistent with σ will be connected to In the control loop; u is the control quantity calculated by the controller. The whole control system includes four parts: controlled object, estimator, switching logic and controller set. The estimator estimates the switching index corresponding to each controller in real time according to the control variable u and the vehicle acceleration, and the switching logic selects the number corresponding to the controller with the smallest switching index as output according to the switching index output by the estimator, and outputs the controller according to the switching logic The number selects the corresponding controller from the controller set to calculate the control variable u, and according to u, the throttle opening can be obtained from the vehicle inverse longitudinal dynamics model, so as to control the engine.

下面将分别介绍各部分的原理和结构。The principle and structure of each part will be introduced respectively below.

(A)被控对象及其乘性不确定性模型集合描述(A) Description of the controlled object and its multiplicative uncertainty model set

由于汽车中的发动机等具有比较严重的非线性特性，参考文献4、文献5和文献6，本发明中也采用一个逆汽车纵向动力学模型来消除非线性特性。得到如图2所示的由逆汽车纵向动力学特性和汽车组成的被控对象。Since the engine in the automobile has relatively serious nonlinear characteristics, references 4, 5 and 6, an inverse vehicle longitudinal dynamics model is also used in the present invention to eliminate the nonlinear characteristics. The controlled object composed of the inverse vehicle longitudinal dynamics and the vehicle is obtained as shown in Figure 2.

图2中，v为车速，T_edes为期望的发动机扭矩，ω_e为发动机转速，θ为节气门开度。对于给定的加速度u，根据汽车阻力驱动力平衡方程(余志生，汽车理论，北京：机械工业出版社，2000年第3版)可以计算得到期望的发动机输出扭矩：In Figure 2, v is the vehicle speed, T _edes is the desired engine torque, ω _e is the engine speed, and θ is the throttle opening. For a given acceleration u, the expected engine output torque can be calculated according to the automobile resistance driving force balance equation (Yu Zhisheng, Automobile Theory, Beijing: Mechanical Industry Press, 3rd edition in 2000):

${T T}_{edes edes} = = \frac{r r}{{R R}_{d d} {R R}_{g g 00} {η η}_{00}} (({M m}_{00} u u + + {C C}_{D D.} {Av Av}^{22} + + {M m}_{00} g g {f f}_{00})),, - - - - - - ((11))$

其中r为车轮半径，R_d为主减速器速比，R_g0为变速器速比的公称值，η₀为传动系统机械效率的公称值，M₀为整车质量的公称值，C_D为空气阻力系数，g为重力常数，f₀为滚动阻力系数的公称值，A为迎风面积。由于实际行驶过程中，档位、传动系统机械效率、整车质量、滚动阻力是随环境变化的，而且很难对这些参数值进行实时的检测，所以在控制过程中，只能根据其公称值利用(1)对期望的发动机扭矩进行计算。根据期望发动机扭矩和发动机转速，利用发动机逆扭矩特性图可以得到发动机的节气门开度：Where r is the radius of the wheel, R _d is the speed ratio of the main reducer, R _g0 is the nominal value of the transmission speed ratio, _η0 is the nominal value of the mechanical efficiency of the transmission system, _M0 is the nominal value of the vehicle mass, _CD is the air Drag coefficient, g is the gravity constant, f ₀ is the nominal value of the rolling resistance coefficient, and A is the windward area. In the actual driving process, the gear position, the mechanical efficiency of the transmission system, the vehicle quality, and the rolling resistance change with the environment, and it is difficult to detect these parameter values in real time, so in the control process, only according to their nominal values Use (1) to calculate the desired engine torque. According to the desired engine torque and engine speed, the throttle opening of the engine can be obtained by using the engine reverse torque characteristic map:

θ＝MAP^-1(T_edes，ω_e)， (2)θ = MAP ^-1 (T _edes , ω _e ), (2)

其中MAP^-1(■)为发动机逆扭矩特性图，该图表示了发动机节气门开度同发动机转速、期望发动机扭矩之间的关系，该图可以从发动机供应商获取。(1)和(2)即为汽车逆纵向动力学模型。Among them, MAP ^-1 (■) is the engine reverse torque characteristic map, which shows the relationship between the engine throttle opening, engine speed and expected engine torque, and the map can be obtained from the engine supplier. (1) and (2) are vehicle inverse longitudinal dynamics models.

实际行驶条件下，由于车辆和环境参数的变化及建模过程中存在的未建模动态特性会使被控对象的动态特性在很大范围内发生变化。仅采用一个模型对被控对象进行描述会造成很大的建模误差，所以本发明中采用多个乘性不确定模型组成模型集合来描述被控对象。乘性不确定模型集合的建立方法有多种，下面介绍文献7(X.Rong Li，etc.面向多模型控制的模型集合设计一般方法，IEEE Transactions on Automatic Control，2005，50(9))和文献8(侯德藻，汽车主动避撞系统研究[博士学位论文]，2004，清华大学汽车工程系)中介绍的方法：Under actual driving conditions, the dynamic characteristics of the controlled object will change in a wide range due to the changes of vehicle and environmental parameters and the unmodeled dynamic characteristics in the modeling process. Using only one model to describe the controlled object will cause a large modeling error, so in the present invention, multiple multiplicative uncertain models are used to form a model set to describe the controlled object. There are many methods for establishing multiplicative uncertain model sets. The following introduces literature 7 (X.Rong Li, etc. General method for model set design for multi-model control, IEEE Transactions on Automatic Control, 2005, 50(9)) and The method introduced in Document 8 (Hou Dezao, Research on Automobile Active Collision Avoidance System [PhD Dissertation], 2004, Department of Automotive Engineering, Tsinghua University):

第一步：将车辆和环境参数(如汽车的质量、发动机的时间常数、档位的速比、坡度、风速等)的可能变化范围离散成有限个参数点，设为M个。在每一个参数点利用过程辨识的方法(如方崇智，等，过程辨识，北京：清华大学出版社，1988)利用输入输出数据辨识得到被控对象的模型，这样共可得到M个传递函数，H_i(s)，i＝1...M。Step 1: Discretize the possible variation range of vehicle and environmental parameters (such as vehicle mass, engine time constant, gear ratio, slope, wind speed, etc.) into a limited number of parameter points, set as M. At each parameter point, the method of process identification (such as Fang Chongzhi, et al., Process Identification, Beijing: Tsinghua University Press, 1988) is used to identify the model of the controlled object using input and output data, so that a total of M transfer functions can be obtained. H _i (s), i=1...M.

第二步：M个传递函数在频域上会覆盖一定的区域，根据拟采用的公称模型的个数，设为N，将该区域平均分成N块，表示为Ω_i，i＝1...N，选择每块区域中心的传递函数作为公称模型，从而得到N个公称模型G_i(s)，i＝1...N。The second step: M transfer functions will cover a certain area in the frequency domain. According to the number of nominal models to be used, set N, and divide the area into N blocks on average, expressed as Ω _i , i=1.. .N, select the transfer function of the center of each area as the nominal model, so as to obtain N nominal models G _i (s), i=1...N.

第三步：针对第二步中得到的每个区域，以Ω_j为例，利用下式计算该区域中所有传递函数与该区域对应的公称模型的模型误差：Step 3: For each area obtained in the second step, taking Ω _j as an example, use the following formula to calculate the model error of all transfer functions in this area and the nominal model corresponding to this area:

${ΔH ΔH}_{i i} ((s the s)) = = \frac{{H h}_{i i} ((s the s)) - - {G G}_{j j} ((s the s))}{{G G}_{j j} ((s the s))},, {&ForAll; &ForAll; H h}_{i i} ((s the s)) &Element; &Element; {Ω Ω}_{j j},, - - - - - - ((33))$

其中δ为指数衰减系数，是一个常数，其物理意义在后面阐述；γ为常数，反应了乘性不确定性的大小，通常取为1；符号|□|表示传递函数的幅值。这样便得到描述区域Ω_j的乘性不确定性模型：Among them, δ is an exponential decay coefficient, which is a constant, and its physical meaning will be explained later; γ is a constant, which reflects the magnitude of multiplicative uncertainty, and is usually taken as 1; the symbol |□| indicates the magnitude of the transfer function. In this way, the multiplicative uncertainty model describing the region Ω _j is obtained:

${P P}_{j j} ((s the s)) = = [[11 + + {Δ Δ}_{j j} {W W}_{j j} ((s the s))]] {G G}_{j j} ((s the s)),, {| | | | {Δ Δ}_{j j} | | | |}_{\infty \infty}^{δ δ} < < γ γ,, - - - - - - ((55))$

其中Δ_j为模型不确定部分，其大小由(4)保证；符号‖□‖_∞ ^δ表示系统的L_2δ增益，当采用L_2δ范数来衡量信号大小时，它反应了系统对输入信号的放大倍数，L_2δ范数‖□‖₂ ^δ定义为：Among them, Δ _j is the uncertain part of the model, and its size is guaranteed by (4); the symbol ‖□ _‖∞ ^δ represents the L _2δ gain of the system. When the L _2δ norm is used to measure the signal size, it reflects the system’s response to the input signal Magnification, L _2δ norm ‖□‖ ₂ ^δ is defined as:

${| | | | u u | | | |}_{22}^{δ δ} = = \sqrt{{&Integral; &Integral;}_{00}^{t t} {e e}^{- - δ δ ((t t - - τ τ))} {u u}^{22} ((τ τ)) dτ dτ},, - - - - - - ((66))$

其中u为时域信号，δ意义与(4)式中的一致，从(6)可以看出δ反应了过去时刻的数据按指数衰减的速率。直接用采用(6)式计算L_2δ范数比较复杂，根据定义可知信号的L_2δ范数是信号u的平方与e^-δt的卷积的平方根，则可以采用下式计算信号的L_2δ范数：Where u is the time-domain signal, and the meaning of δ is consistent with that in (4). It can be seen from (6) that δ reflects the exponential decay rate of the past data. It is more complicated to directly calculate the L _2δ norm by using formula (6). According to the definition, the L _2δ norm of the signal is the square root of the convolution of the square of the signal u and e ^-δt , and the following formula can be used to calculate the L _2δ norm of the signal number:

${| | | | u u | | | |}_{22}^{δ δ} = = \sqrt{\frac{11}{s the s + + δ δ} {u u}^{22}} . . - - - - - - ((77))$

由于对象特性覆盖的区域被划分成N块，针对每块通过上面运算都可得到对应的乘性不确定模型，从而得到描述被控对象特性的乘性不确定性模型集合：Since the area covered by the characteristics of the object is divided into N blocks, the corresponding multiplicative uncertainty model can be obtained through the above operations for each block, and thus a set of multiplicative uncertainty models describing the characteristics of the controlled object can be obtained:

$P P = = {{{P P}_{i i} ((s the s)) = = [[11 + + {Δ Δ}_{i i} {W W}_{i i} ((s the s))]] {G G}_{i i} ((s the s)),, {| | | | {Δ Δ}_{i i} | | | |}_{\infty \infty}^{δ δ} < < γ γ,, i i = = 11,, \cdot &Center Dot; \cdot &Center Dot; \cdot &Center Dot;,, N N}} . . - - - - - - ((88))$

上面在进行N个公称模型的传递函数和N个模型误差加权函数选择时，要使得：N个公称模型G_i(s)具有相同的特征多项式，N个模型误差加权函数W_i(s)具有相同的特征多项式。When selecting the transfer functions of N nominal models and N model error weighting functions above, it is necessary to make: N nominal models G _i (s) have the same characteristic polynomial, and N model error weighting functions W _i (s) have the same characteristic polynomial.

(B)估计器(B) Estimator

估计器的作用是根据被控对象的输入输出信息，对模型集合中各元素与对象的模型不确定部分的大小进行估计。假设被控对象由如下乘性不确定模型描述：The role of the estimator is to estimate the size of the model uncertain part of each element and object in the model set according to the input and output information of the controlled object. Suppose the controlled object is described by the following multiplicative uncertainty model:

a＝[1+ΔW(s)]G(s)u。 (9)a=[1+ΔW(s)]G(s)u. (9)

若G(s)稳定，则可以采用下式对被控对象的输出进行估计：If G(s) is stable, the output of the controlled object can be estimated by the following formula:

$\overset{^^}{a a} = = G G ((s the s)) u u . . - - - - - - ((1010))$

(9)减(10)可以得到被控对象输出估计误差：(9) minus (10) can get the output estimation error of the controlled object:

$e e = = \overset{^^}{a a} - - a a = = - - ΔW ΔW ((s the s)) G G ((s the s)) u u . . - - - - - - ((1111))$

由(10)可以看出模型不确定部分的输出为估计误差e，输入为It can be seen from (10) that the output of the uncertain part of the model is the estimated error e, and the input is

z＝W(s)G(s)u。 (12)z=W(s)G(s)u. (12)

利用(11)和(12)可以对模型不确定部分对当前输入信号的L_2δ增益进行估计。Using (11) and (12), the uncertain part of the model can be used to estimate the L _2δ gain of the current input signal.

基于上述考虑，参照文献7(吴麟，自动控制原理(下册)，北京：清华大学出版社，1992年第1版)中的可控规范型的转化方法得到(A)部分设计的乘性不确定模型集合中各元素对应的可控规范型形式，并将其作为估计器：Based on the above considerations, refer to the transformation method of the controllable normative type in Document 7 (Wu Lin, Principles of Automatic Control (Volume 2), Beijing: Tsinghua University Press, 1st Edition, 1992) to obtain the multiplicative nature of the design in part (A) Determine the controllable canonical form corresponding to each element of the model ensemble as an estimator:

${\overset{\cdot &Center Dot;}{x x}}_{E E. 11} = = {A A}_{E E. 11} {x x}_{E E. 11} + + {B B}_{E E. 11} u u,, {\overset{\cdot \cdot}{x x}}_{E E. 22} = = {A A}_{E E. 22} {x x}_{E E. 22} + + {B B}_{E E. 22} u u,,$

e_i＝a_i-a＝C_E1ix_E1+D_E1iu-a，z_i＝C_E2ix_E2+D_E2iu，i＝1…N， (13)e _i =a _i -a=C _E1i x _E1 +D _E1i ua, z _i =C _E2i x _E2 +D _E2i u, i=1...N, (13)

其中a_i为基于模型P_i(s)设计的对象输出估计器对被控对象输出的估计值，e_i为对应输出估计误差，z_i为基于模型P_i(s)设计的公称模型与对象之间模型不确定性部分输出的估计值。根据文献7，可知(13)中的状态方程的系数矩阵要满足：where a _i is the estimated value of the output of the controlled object by the object output estimator designed based on the model P _i (s), e _i is the corresponding output estimation error, z _i is the nominal model and object designed based on the model P _i (s) Estimates of the output from the uncertainty part of the model. According to literature 7, it can be seen that the coefficient matrix of the state equation in (13) should satisfy:

C_E1i(sI-A_E1)^-1B_E1+D_E1i＝G_i(s)，C _E1i (sI-A _E1 ) ^-1 B _E1 +D _E1i = G _i (s),

C_E2i(sI-A_E2)^-1B_E2+D_E2i＝W_i(s)G_i(s)，i＝1…N。 (14)C _E2i (sI−A _E2 ) ⁻¹ B _E2 +D _E2i =W _i (s)G _i (s), i=1...N. (14)

由(13)和前面的分析，可以采用下式对模型集合中各元素与对象之间的模型不确定部分的L_2δ增益，即为切换指标，进行计算：From (13) and the previous analysis, the L _2δ gain of the uncertain part of the model between each element in the model set and the object, that is, the switching index, can be calculated by using the following formula:

${J J}_{i i} ((t t)) = = \frac{{| | | | {e e}_{i i} | | | |}_{22}^{δ δ}}{{| | | | {z z}_{i i} | | | |}_{22}^{δ δ}},, i i = = 11 \cdot \cdot \cdot &Center Dot; \cdot \cdot N N . . - - - - - - ((1515))$

J_i(t)反应了模型P_i(s)与被控对象之间模型不确定部分对当前输入信号的L_2δ增益。J _i (t) reflects the L _2δ gain of the model uncertain part between the model P _i (s) and the controlled object to the current input signal.

(C)切换逻辑(C) switching logic

切换逻辑的作用根据估计器输出的切换指标J_i(t)，i＝1...N从控制器集合中选择控制器对被控对象进行控制，其输出即为选择的控制器的编号。与被控对象之间模型不确定部分最小的模型应该与对象最接近，那么基于该模型设计的控制器应该能够获得较好的控制效果。基于上述考虑，采用图3所示的切换逻辑，图中，J_σ(t)为当前控制器对应的切换指标，若当前控制器对应的切换指标为最小时，继续采用当前控制器控制；若当前控制器对应的切换指标不为最小，则选择最小的切换指标对应的控制器进行控制。k表示时刻t切换指标最小的模型编号。The role of switching logic According to the switching index J _i (t) output by the estimator, i=1...N selects a controller from the controller set to control the controlled object, and the output is the number of the selected controller. The model with the smallest model uncertainty between the controlled object and the controlled object should be the closest to the object, so the controller designed based on this model should be able to obtain better control effect. Based on the above considerations, the switching logic shown in Figure 3 is adopted. In the figure, J _σ(t) is the switching index corresponding to the current controller. If the switching index corresponding to the current controller is the smallest, the current controller will continue to be used for control; if The switching index corresponding to the current controller is not the smallest, and the controller corresponding to the smallest switching index is selected for control. k represents the model number with the smallest switching index at time t.

(D)控制器集合(D) Collection of controllers

根据估计误差的定义有According to the definition of estimation error, there is

a＝a_σ-e_σ。 (16)a=a _σ -e _σ . (16)

将(16)代入(13)可以得到Substitute (16) into (13) to get

a＝C_E1σx_E1+D_E1σu-e_σ，z_σ＝C_E2σx_E2+D_E2σu。 (17)a=C _E1σ x _E1 +D _E1σ ue _σ , z _σ =C _E2σ x _E2 +D _E2σ u. (17)

在(17)描述的系统中，将e_σ作为模型不确定部分引起的扰动，z_σ作为模型不确定部分的输入，设计的切换规则可以保证下式成立。In the system described in (17), e _σ is used as the disturbance caused by the uncertain part of the model, z _σ is used as the input of the uncertain part of the model, and the designed switching rules can ensure that the following formula holds.

$\frac{| | | | {e e}_{σ σ} | | | |}{| | | | {z z}_{σ σ} | | | |} < < γ γ . . - - - - - - ((1818))$

设控制器集合为：Let the collection of controllers be:

$C C = = {{{K K}_{i i} : : \begin{matrix} {\overset{\cdot \cdot}{x x}}_{C C} = = {A A}_{Ci Ci} {x x}_{C C} + + {B B}_{Ci Ci} (({a a}_{des des} - - a a)) \\ u u = = {C C}_{Ci Ci} {x x}_{C C} + + {D D.}_{Ci Ci} (({a a}_{des des} - - a a)) \end{matrix},, i i = = 11 \cdot &Center Dot; \cdot &Center Dot; \cdot &Center Dot; N N}} . . - - - - - - ((1919))$

则由当前选择的控制器K_σ与系统(17)组成的闭环系统为：Then the closed-loop system composed of the currently selected controller K _σ and the system (17) is:

$\overset{\cdot}{x} = A_{σ} x + B_{σ} (a_{des} + e_{σ}),$ z_σ＝C_1σx+D_1σ(a_des+e_σ)，q＝C_2σx+D_2σ(a_des+e_σ)， (20) $\overset{&Center Dot;}{x} = A_{σ} x + B_{σ} (a_{des} + e_{σ}),$ z _σ =C _1σ x+D _1σ (a _des +e _σ ), q=C _2σ x+D _2σ (a _des +e _σ ), (20)

其中 $x = [\begin{matrix} x_{E 1} \\ x_{E 2} \\ x_{C} \end{matrix}],$ $A_{σ} = [\begin{matrix} A_{E 1} - Φ_{σ} B_{E 1} D_{Cσ} C_{E 1 σ} & 0 & Φ_{σ} B_{E 1} C_{Cσ} \\ - Φ_{σ} B_{E 2} D_{Cσ} C_{E 1 σ} & A_{E 2} & Φ_{σ} B_{E 2} C_{Cσ} \\ - Φ_{σ} B_{Cσ} C_{E 1 σ} & 0 & A_{Cσ} - Φ_{σ} B_{Cσ} D_{E 1 σ} C_{Cσ} \end{matrix}],$ $B_{σ} = Φ_{σ} [\begin{matrix} B_{E 1} D_{Cσ} \\ B_{E 2} D_{Cσ} \\ B_{Cσ} \end{matrix}],$ in $x = [\begin{matrix} x_{E. 1} \\ x_{E. 2} \\ x_{C} \end{matrix}],$ $A_{σ} = [\begin{matrix} A_{E. 1} - Φ_{σ} B_{E. 1} {D.}_{Cσ} C_{E. 1 σ} & 0 & Φ_{σ} B_{E. 1} C_{Cσ} \\ - Φ_{σ} B_{E. 2} {D.}_{Cσ} C_{E. 1 σ} & A_{E. 2} & Φ_{σ} B_{E. 2} C_{Cσ} \\ - Φ_{σ} B_{Cσ} C_{E. 1 σ} & 0 & A_{Cσ} - Φ_{σ} B_{Cσ} {D.}_{E. 1 σ} C_{Cσ} \end{matrix}],$ $B_{σ} = Φ_{σ} [\begin{matrix} B_{E. 1} {D.}_{Cσ} \\ B_{E. 2} {D.}_{Cσ} \\ B_{Cσ} \end{matrix}],$

C_1σ＝[-Φ_σD_E2σD_CσC_E1σ C_E2σ Φ_σD_E2σC_Cσ]，D_1σ＝Φ_σD_E2σD_Cσ，D_2σ＝Φ_σ，C_2σ＝-Φ_σ[C_E1σ 0 D_E1σC_Cσ]。C _1σ = [-Φ _σ D _E2σ D _Cσ C _E1σ C _E2σ Φ _σ D _E2σ C _Cσ ], D _1σ = Φ _σ D _E2σ D _Cσ , D _2σ = Φ _σ , C _2σ = -Φ _σ [C _E1σ 0 D _E1σ C _Cσ ].

若控制器集合C中元素满足If the elements in the controller set C satisfy

${| | | | Σ Σ (({A A}_{i i},, {B B}_{i i},, {C C}_{11 i i},, {D D.}_{11 i i})) | | | |}_{\infty \infty}^{δ δ} < < η η < < \frac{11}{γ γ},, {| | | | {W W}_{per per} ((s the s)) Σ Σ (({A A}_{i i},, {B B}_{i i} {C C}_{22 i i},, {D D.}_{22 i i})) | | | |}_{\infty \infty}^{δ δ} < < β β,, i i = = 11 \cdot &Center Dot; \cdot &Center Dot; \cdot &Center Dot; N N,, - - - - - - ((21 twenty one))$

其中W_per(s)为性能指标加权函数，符号∑(A，B，C，D)表示由A、B、C、D做系数矩阵的状态空间模型，那么：Where W _per (s) is the performance index weighting function, the symbol ∑ (A, B, C, D) represents the state space model with A, B, C, D as the coefficient matrix, then:

$\frac{{| | | | {W W}_{per per} ((s the s)) q q | | | |}_{22}^{δ δ}}{{| | | | {a a}_{des des} | | | |}_{22}^{δ δ}} < < \frac{β β}{11 - - ηγ ηγ},, - - - - - - ((22 twenty two))$

即本发明中的加速度控制系统的加速度跟踪误差q对期望加速度a_des引起的扰动具有一定抑制能力，从而保证一定的跟踪性能。此外，若信号a_des有界，根据(22)可知加速度跟踪误差q也有界，从而汽车的加速度信号a也有界(冯纯伯，等，鲁棒控制系统设计，南京：东南大学，1995年第1版)。That is, the acceleration tracking error q of the acceleration control system in the present invention has a certain ability to suppress the disturbance caused by the desired acceleration a _des , thereby ensuring a certain tracking performance. In addition, if the signal a _des is bounded, according to (22), it can be known that the acceleration tracking error q is also bounded, so the acceleration signal a of the car is also bounded (Feng Chunbo, et al., Robust Control System Design, Nanjing: Southeast University, 1995, 1st edition ).

根据切换逻辑输出的切换信号，选择编号与σ一致的控制器集合中元素的系数矩阵对控制量进行计算。According to the switching signal output by the switching logic, select the coefficient matrix of the elements in the controller set whose number is consistent with σ to calculate the control quantity.

本方法的流程图如图4所示，需要说明的是，本方法中，只用到了根据被控对象的模型集合而设计的控制器集合与估计器，而不需要在每次流程的运行中都建立被控对象模型和设计控制器与估计器，只要根据设计结果将控制器与估计器的系数矩阵进行初始化即可。方法流程如下：The flow chart of this method is shown in Figure 4. It should be noted that in this method, only the controller set and estimator designed according to the model set of the controlled object are used, and it is not necessary to Both establish the controlled object model and design the controller and estimator, as long as the coefficient matrix of the controller and estimator are initialized according to the design results. The method flow is as follows:

2)初始化：2) Initialization:

与被控对象乘性不确定模型集合中的元素对应的控制器状态矩阵A_Ci，B_Ci，C_Ci，D_Ci；Controller state matrices A _Ci , B _Ci , C _Ci , D _Ci corresponding to the elements in the multiplicative uncertainty model set of the controlled object;

与被控对象乘性不确定模型集合中的元素对应的估计器状态矩阵A_E1，B_E1，A_E2，B_E2，C_E1i，D_E1i，C_E2i，D_E2i；Estimator state matrices A _E1 , B _E1 , A E2 , B _E2 , C _E1i , D _E1i , C _E2i _, D _E2i corresponding to the elements in the multiplicative uncertainty model set of the plant;

x_E1和x_E2为与被控对象乘性不确定模型集合中元素对应的估计器状态，其初始值为0；x _E1 and x _E2 are the estimator states corresponding to the elements in the multiplicative uncertain model set of the plant, and their initial value is 0;

x_C为与被控对象乘性不确定模型集合中的元素对应的控制器状态，其初始值为0；x _C is the controller state corresponding to the elements in the multiplicative uncertain model set of the controlled object, and its initial value is 0;

e_i＝C_E1ix_E1+D_E1iu-a，z_i＝C_E2ix_E2+D_E2iu，i＝1…N，e _i =C _E1i x _E1 +D _E1i ua, z _i =C _E2i x _E2 +D _E2i u, i=1...N,

其中N为被控对象乘性不确定模型集合中元素的个数；Where N is the number of elements in the multiplicative uncertain model set of the controlled object;

4)采用式 $J_{i} (t) = \frac{{| | e_{i} | |}_{2}^{δ}}{{| | z_{i} | |}_{2}^{δ}},$ i＝1…N计算乘性不确定模型集合中的N个模型的切换指标：4) adoption $J_{i} (t) = \frac{{| | e_{i} | |}_{2}^{δ}}{{| | z_{i} | |}_{2}^{δ}},$ i=1...N Calculate the switching index of N models in the set of multiplicative uncertain models:

实际计算时可采用(15)式所示的方法计算信号的L_2δ范数。In actual calculation, the method shown in (15) can be used to calculate the L _2δ norm of the signal.

5)上述N个切换指标与控制器集合 $C = {K_{i} : \begin{matrix} {\overset{\cdot}{x}}_{C} = A_{Ci} x_{C} + B_{Ci} (a_{des} - a) \\ u = C_{Ci} x_{C} + D_{Ci} (a_{des} - a) \end{matrix}, i = 1 \cdot \cdot \cdot N}$ 中的N个元素按照i的顺序一一对应，选择上述N个切换指标中最小的切换指标J_k(t)对应的控制器计算控制量u：5) The above N switching indicators and controller set $C = {K_{i} : \begin{matrix} {\overset{\cdot}{x}}_{C} = A_{Ci} x_{C} + B_{Ci} (a_{des} - a) \\ u = C_{Ci} x_{C} + {D.}_{Ci} (a_{des} - a) \end{matrix}, i = 1 &Center Dot; &Center Dot; &Center Dot; N}$ The N elements in are corresponding to each other in the order of i, and the controller corresponding to the smallest switching index J _k (t) among the above N switching indexes is selected to calculate the control quantity u:

$C C = = {{{K K}_{σ σ} : : \begin{matrix} {\overset{\cdot &Center Dot;}{x x}}_{C C} = = {A A}_{Ck C} {x x}_{C C} + + {B B}_{Ck C} (({a a}_{des des} - - a a)) \\ u u = = {C C}_{Ck C} {x x}_{C C} + + {D D.}_{Ck C} (({a a}_{des des} - - a a)) \end{matrix}}}$

${T T}_{edes edes} = = \frac{r r}{{R R}_{d d} {R R}_{g g 00} {η η}_{00}} (({M m}_{00} u u + + {C C}_{D D.} {Av Av}^{22} + + {M m}_{00} g g {f f}_{00}))$

θ＝MAP^-1(T_edes，ω_e)θ=MAP ^-1 (T _edes , ω _e )

v为车速，T_edes为期望的发动机扭矩，ω_e为发动机转速，θ为节气门开度， r为车轮半径，R_d为主减速器速比，R_g0为变速器速比的公称值，η₀为传动系统机械效率的公称值，M₀为整车质量的公称值，C_D为空气阻力系数，g为重力常数，f₀为滚动阻力系数的公称值，A为迎风面积；MAP^-1(■)为发动机逆扭矩特性图；v is the vehicle speed, T _edes is the expected engine torque, ω _e is the engine speed, θ is the throttle opening, r is the wheel radius, R _d is the speed ratio of the main reducer, R _g0 is the nominal value of the transmission speed ratio, η ₀ is the nominal value of the mechanical efficiency of the transmission system, M ₀ is the nominal value of the vehicle mass, C _D is the air resistance coefficient, g is the gravity constant, f ₀ is the nominal value of the rolling resistance coefficient, A is the windward area; MAP ^-1 (■) is the reverse torque characteristic diagram of the engine;

7)根据计算得到的节气门开度θ对汽车的发动机进行控制，并将控制量u值返回第2)步继续进行计算和控制。7) Control the engine of the car according to the calculated throttle opening θ, and return the value of the control variable u to step 2) to continue calculation and control.

具体实施例： Specific examples:

针对国内捷达AT型轿车，其相关参数的变化范围及公称值如表1所示。For the domestic Jetta AT car, the variation range and nominal value of relevant parameters are shown in Table 1.

表1，汽车和环境参数的变化范围及对应的公称值Table 1, Variation range of vehicle and environmental parameters and corresponding nominal values

发动机的逆力矩特性如图5所示。The reverse torque characteristics of the engine are shown in Fig. 5.

根据(A)部分的方法得到被控对象的频域特性覆盖范围如图6所示。将被控对象频域响应特性覆盖范围分为四部分，得到如下的乘性不确定模型集合：According to the method in part (A), the frequency domain characteristic coverage of the controlled object is shown in Figure 6. Divide the frequency-domain response characteristic coverage of the controlled object into four parts, and get the following multiplicative uncertainty model set:

$P P = = {{{P P}_{i i} ((s the s)) = = [[11 + + {Δ Δ}_{i i} {W W}_{i i} ((s the s))]] {G G}_{i i} ((s the s)),, {| | | | {Δ Δ}_{i i} | | | |}_{\infty \infty}^{δ δ} < < 11,, i i = = 11,, \cdot &Center Dot; \cdot &Center Dot; \cdot &Center Dot;,, 44}},, - - - - - - ((23 twenty three))$

其中模型误差加权函数 $W (s) = \frac{2.1 s + 2.478}{s + 5.1},$ 指数衰减系数δ＝0.4，公称模型如表2所示。where the model error weighting function $W (the s) = \frac{2.1 the s + 2.478}{the s + 5.1},$ The exponential decay coefficient δ=0.4, and the nominal model is shown in Table 2.

表2，公称模型传递函数Table 2, Nominal Model Transfer Functions

根据模型集合，得到估计器的系数矩阵为：According to the model set, the coefficient matrix of the estimator is obtained as:

A_E1＝-3.33， $A_{E 2} = [\begin{matrix} 0 & - 17 \\ 1 & - 8.43 \end{matrix}],$ $B_{E 1} = 1, B_{E 2} = [\begin{matrix} 1 \\ 0 \end{matrix}],$ C_E11＝6.2367，C_E12＝3.314，C_E13＝2.3013，C_E14＝1.703，C_E21＝[13.097-94.997]，C_E22＝[6.959-50.479]，C_E23＝[4.833-35.054]，C_E24＝[3.576-25.94]。A _E1 = -3.33, $A_{E. 2} = [\begin{matrix} 0 & - 17 \\ 1 & - 8.43 \end{matrix}],$ $B_{E. 1} = 1, B_{E. 2} = [\begin{matrix} 1 \\ 0 \end{matrix}],$ C _E11 =6.2367, C _E12 =3.314, C _E13 =2.3013, C _E14 =1.703, C _E21 =[13.097-94.997], C _E22 =[6.959-50.479], C _E23 =[4.833-35.054], C _E24 = [3.576-25.94].

(24) (twenty four)

根据模型集合P利用Matlab的LMI工具箱计算得到的控制器集合C的元素为：According to the model set P, the elements of the controller set C calculated by using the LMI toolbox of Matlab are:

${K K}_{11} ((s the s)) = = \frac{137.14 137.14 ((s the s + + 4.9 4.9)) ((s the s + + 3.133 3.133))}{s the s ((s the s + + 41.85 41.85)) ((s the s + + 45.7 45.7))},,$ ${K K}_{22} ((s the s)) = = \frac{233.41 233.41 ((s the s + + 4.9 4.9)) ((s the s + + 3.133 3.133))}{s the s ((s the s + + 80.06 80.06)) ((s the s + + 21.42 21.42))},,$

${K K}_{33} = = \frac{572.97 572.97 ((s the s + + 4.9 4.9)) ((s the s + + 3.133 3.133))}{s the s ((s the s + + 29.63 29.63)) ((s the s + + 99.3 99.3))},,$ ${K K}_{44} ((s the s)) = = \frac{283.35 283.35 ((s the s + + 4.9 4.9)) ((s the s + + 3.133 3.133))}{s the s ((s the s + + 54.15 54.15)) ((s the s + + 19.89 19.89))},, - - - - - - ((2525))$

控制器集合设计过程中取性能指标加权函数 $W_{per} (s) = \frac{0.1 s + 1.1}{s},$ 常数η＝β＝0.7。汽车纵向加速度跟踪控制方法的仿真结果如图7所示。从图7(a)可以看出整个过程中，除25s和75s左右由于汽车换档造成的冲击外，实际的汽车加速度可以较好地跟踪期望加速度。其中虚线表示实际的汽车加速度a，实线表示期望的汽车加速度a_des。从图7(b)可以看出，随着汽车纵向特性的变化，不同的控制器会被切换到反馈回路中，说明本发明中的切换指标函数可以对控制器的潜在性能进行评价。(c)和(d)分别为汽车速度和节气门开度响应曲线。Taking performance index weighting function in the process of controller set design $W_{per} (the s) = \frac{0.1 the s + 1.1}{the s},$ The constant η=β=0.7. The simulation results of the vehicle longitudinal acceleration tracking control method are shown in Figure 7. It can be seen from Figure 7(a) that during the whole process, except for the shock caused by the car shifting around 25s and 75s, the actual car acceleration can track the expected acceleration well. The dashed line represents the actual vehicle acceleration a, and the solid line represents the desired vehicle acceleration a _des . It can be seen from Fig. 7(b) that as the longitudinal characteristics of the vehicle change, different controllers will be switched into the feedback loop, indicating that the switching index function in the present invention can evaluate the potential performance of the controller. (c) and (d) are the vehicle speed and throttle opening response curves, respectively.

汽车纵向加速度跟踪控制方法的实车实验结果图8所示。结果表明随着汽车纵向动力学特性的变化，相应的控制器会被切换到反馈回路中。整个控制过程中，实际加速度可以很好地跟踪期望加速度。The actual vehicle test results of the vehicle longitudinal acceleration tracking control method are shown in Fig. 8. The results show that as the vehicle's longitudinal dynamics change, the corresponding controller is switched into a feedback loop. Throughout the control process, the actual acceleration can track the desired acceleration well.

附录1Appendix 1

由控制器满足的条件(式21)得到控制系统性能指标(式22)的说明：The description of the control system performance index (formula 22) is obtained from the conditions satisfied by the controller (formula 21):

不失一般性，假设当前对象可以用P_j(s)描述，而连接在反馈回路中的控制器为K_k(s)。根据对象模型(5)和估计器方程(13)得到：Without loss of generality, assume that the current object can be described by P _j (s), and the controller connected in the feedback loop is K _k (s). According to the object model (5) and the estimator equation (13):

e_j＝-Δ_jW_j(s)G_j(s)u，z_j＝W_j(s)G_j(s)u。 (26)e _j =-Δ _j W _j (s)G _j (s)u, z _j =W _j (s)G _j (s)u. (26)

由于 ${| | Δ_{j} | |}_{\infty}^{δ} < γ,$ 可以得到：because ${| | Δ_{j} | |}_{\infty}^{δ} < γ,$ can get:

${J J}_{j j} ((t t)) = = {| | | | {e e}_{j j} | | | |}_{22}^{δ δ} / / {| | | | {z z}_{j j} | | | |}_{22}^{δ δ} < < γ γ . . - - - - - - ((2727))$

切换逻辑S选择切换指标最小的模型对应的控制器，而根据前面假设，当前控制器为K_k(s)，则有：The switching logic S selects the controller corresponding to the model with the smallest switching index, and according to the previous assumption, the current controller is K _k (s), then:

${J J}_{k k} ((t t)) = = {| | | | {e e}_{k k} | | | |}_{22}^{δ δ} / / {| | | | {z z}_{k k} | | | |}_{22}^{δ δ} \leq \leq {J J}_{j j} ((t t)) < < γ γ . . - - - - - - ((2828))$

根据闭环系统方程(20)，可以得到：According to the closed-loop system equation (20), it can be obtained:

z_k＝∑(A_k，B_k，C_1k，D_1k)(a_des+e_k)。 (29)z _k =Σ(A _k , B _k , C _1k , D _1k )(a _des +e _k ). (29)

由(28)和控制器集合满足的条件(21)有The condition (21) satisfied by (28) and the controller set has

$| | | | {e e}_{k k} | | | | < < γ γ {| | | | {z z}_{k k} | | | |}_{22}^{δ δ} \leq \leq γη γη | | | | {a a}_{des des} + + {e e}_{k k} | | | | . . - - - - - - ((3030))$

由e_k＝a_des+e_k-a_des，再利用三角不等式可以得到：From e _k ＝a _des +e _k -a _des , and then use the triangle inequality to get:

$| | | | {a a}_{des des} + + {e e}_{k k} | | | | < < \frac{11}{11 - - γη γη} {| | | | {a a}_{des des} | | | |}_{22}^{δ δ} . . - - - - - - ((3131))$

另一方面，由闭环系统方程(20)可以得到：On the other hand, from the closed-loop system equation (20), we can get:

W_per(s)q＝W_per(s)∑(A_k，B_k，C_2k，D_2k)(a_des+e_k)。 (32)W _per (s)q=W _per (s)∑(A _k , B _k , C _2k , D _2k )(a _des +e _k ). (32)

由控制器集合满足的条件(21)有The condition (21) satisfied by the set of controllers has

${| | | | {W W}_{per per} ((s the s)) q q | | | |}_{22}^{δ δ} \leq \leq β β {| | | | {a a}_{des des} + + {e e}_{k k} | | | |}_{22}^{δ δ} . . - - - - - - ((3333))$

将(31)代入(33)可以得到不等式：Substituting (31) into (33) yields the inequality:

$\frac{{| | | | {W W}_{per per} ((s the s)) q q | | | |}_{22}^{δ δ}}{{| | | | {a a}_{des des} | | | |}_{22}^{δ δ}} \leq \leq \frac{β β}{11 - - γη γη} . . - - - - - - ((3434))$

Claims

1. automobile longitudinal acceleration tracking control method; It is characterized in that; It is the estimator that utilizes the property the taken advantage of ambiguous model of the controlled device of being made up of the reverse of automobile and the positive power model element in gathering to design; Model error between each model and the practical object is estimated the system gain of current input signal, and with this system gain of estimating to obtain as switching target function, from the controller set that designs according to the element the said property the taken advantage of ambiguous model set, select corresponding controller online; The engine air throttle aperture is controlled, realized the tracking Control of automobile longitudinal acceleration;

This method contains the following steps of in entire car controller, carrying out:

1) initialization:

The parameter of vehicle parameter and automobile running environment;

δ: the exponential decay coefficient, its scope is: 0.1～1;

U: Acceleration Control amount, its initial value are 0;

With the corresponding estimator set of element in the set of the controlled device property taken advantage of ambiguous model be:

{\overset{\cdot}{x}}_{E 1} = A_{E 1} x_{E 1} + B_{E 1} u,

{\overset{\cdot}{x}}_{E 2} = A_{E 2} x_{E 2} + B_{E 2} u,

e _i＝a _i-a＝C _E1ix _E1+D _E1iu-a，z _i＝C _E2ix _E2+D _E2iu，i＝1…N，

In the formula (13), x _E1And x _E2For with the controlled device property taken advantage of ambiguous model set in the corresponding estimator state of element, its initial value is 0; N is the controller number, A _E1, B _E1, A _E2, B _E2, C _E1i, D _Eli, C _E2i, D _E2iIt is the state matrix of estimator;

With the corresponding controller set C of element in the set of the controlled device property taken advantage of ambiguous model be:

C = \{\begin{matrix} K_{i} : \end{matrix} \begin{matrix} {\overset{\cdot}{x}}_{C} = A_{Ci} x_{C} + B_{Ci} (a_{des} - a) \\ u = C_{Ci} x_{C} + D_{Ci} (a_{des} - a) \end{matrix}, i = 1 \cdot \cdot \cdot N\}

In the formula (19), K _iBe i the controller of controller set C, x _cBe the state of controller set, its initial value is 0; N is the controller number, A _Ci, B _Ci, C _Ci, D _CiIt is state matrix; Element satisfies among the controller set C:

{| | Σ (A_{i}, B_{i}, C_{1 i}, D_{1 i}) | |}_{\infty}^{δ} < η < \frac{1}{γ},

{| | W_{per} (s) Σ (A_{i}, B_{i}, C_{2 i}, D_{2 i}) | |}_{\infty}^{δ} < β,

i＝1…N，

A_{σ} = [\begin{matrix} A_{E 1} - Φ_{σ} B_{E 1} D_{Cσ} C_{E 1 σ} & 0 & Φ_{σ} B_{E 1} C_{Cσ} \\ - Φ_{σ} B_{E 2} D_{Cσ} C_{E 1 σ} & A_{E 2} & Φ_{σ} B_{E 2} C_{Cσ} \\ - Φ_{σ} B_{Cσ} C_{E 1 σ} & 0 & A_{Cσ} - Φ_{σ} B_{Cσ} D_{E 1 σ} C_{Cσ} \end{matrix}],

B_{σ} = Φ_{σ} [\begin{matrix} B_{E 1} D_{Cσ} \\ B_{E 2} D_{Cσ} \\ B_{Cσ} \end{matrix}],

C _1σ＝[-Φ _σD _E2σD _CσC _E1σ?C _E2σ?Φ _σD _E2σC _Cσ]，D _1σ＝Φ _σD _E2σD _Cσ，

D _2σ＝Φ _σ，C _2σ＝-Φ _σ[C _E1σ?0?D _E1σC _Cσ]；

W wherein _Per(s) be the performance index weighting function, the symbol ∑ (A, B, C, D) expression is satisfied by the state-space model that A, B, C, D do matrix of the coefficients:

\frac{{| | W_{per} (s) q | |}_{2}^{δ}}{{| | a_{des} | |}_{2}^{δ}} < \frac{β}{1 - ηγ};

2) gather pickup a, expectation pickup a _Des

3) matrix of the coefficients according to the estimator of the property the taken advantage of ambiguous model of controlled device set design calculates evaluated error e _iReally part is not imported z _i

{\overset{\cdot}{x}}_{E 1} = A_{E 1} x_{E 1} + B_{E 1} u,

{\overset{\cdot}{x}}_{E 2} = A_{E 2} x_{E 2} + B_{E 2} u,

e _i＝C _E1ix _E1+D _E1iu-a，z _i＝C _E2ix _E2+D _E2iu，i＝1…N， (1)

Wherein N is the number of element in the set of the controlled device property taken advantage of ambiguous model;

4) the switching index of N model in the set of the employing computes property taken advantage of ambiguous model:

J_{i} (t) = \frac{{| | e_{i} | |}_{2}^{δ}}{{| | z_{i} | |}_{2}^{δ}}, i = 1 \cdot \cdot \cdot N - - - (2)

5) above-mentioned N switches index and controller set

C = \{\begin{matrix} K_{i} : \end{matrix} \begin{matrix} {\overset{\cdot}{x}}_{C} = A_{Ci} x_{C} + B_{Ci} (a_{Des} - a) \\ u = C_{Ci} x_{C} + D_{Ci} (a_{Des} - a) \end{matrix}, i = 1 \cdot \cdot \cdot N\}

In N element

Order according to i is corresponding one by one, selects above-mentioned N to switch switching index J minimum in the index _k(t) the controller compute control amount u of correspondence:

C = \{\begin{matrix} K_{σ} \end{matrix} : \begin{matrix} {\overset{\cdot}{x}}_{C} = A_{Ck} x_{C} + B_{Ck} (a_{des} - a) \\ u = C_{Ck} x_{C} + D_{Ck} (a_{des} - a) \end{matrix}\}

6) calculate throttle according to contrary automobile longitudinal dynamic model:

T_{edes} = \frac{r}{R_{d} R_{g 0} η_{0}} (M_{0} u + C_{D} {Av}^{2} + M_{0} {gf}_{0})

θ＝MAP ^-1(T _edes，ω _e) (3)

The parameter that wherein comprises following vehicle parameter and automobile running environment:

V is the speed of a motor vehicle, T _EdesBe the Engine torque of expectation, ω _eBe engine speed, θ is a throttle opening, and r is a radius of wheel, R _dBe main reducing gear speed ratio, R _G0Be the nominal value of transmission gear ratio, η ₀Be the nominal value of transmission system mechanical efficiency, M ₀Be the nominal value of complete vehicle quality, C _DBe coefficient of air resistance, g is a gravity constant, f ₀Be the nominal value of coefficient of rolling resistance, A is a wind-exposuring area; MAP ^-1(■) be the contrary torque characteristics figure of motor;

7) throttle that obtains according to aforementioned calculation is controlled the motor of automobile, and controlled quentity controlled variable u value is returned the 2nd) go on foot and proceed to calculate and control.

2. automobile longitudinal acceleration tracking control method as claimed in claim 1 is characterized in that, the constitution step of the property the taken advantage of ambiguous model set of said controlled device is:

The first step: the possible excursion of vehicle and enviromental parameter is separated into limited parameter point, is made as M; Utilize the method for process identification to utilize the inputoutput data identification to obtain the model of controlled device at each parameter point, can obtain M transfer function so altogether, H _i(s), i=1...M;

Second step: adopt M transfer function on frequency domain, to cover certain zone, the number according to intending the nominal model that adopts is made as N, and this zone leveling is divided into the N piece, is expressed as Ω _i, i=1...N, the transfer function of selecting every regional center obtains N nominal model G as the nominal model _i(s), i=1...N;

The 3rd step: to each zone that obtains in the first step, utilize computes should the zone in the model error of all transfer functions nominal model corresponding with this zone:

Δ H_{i} (s) = \frac{H_{i} (s) - G_{j} (s)}{G_{j} (s)}, &ForAll; H_{i} (s) &Element; Ω_{j}, - - - (3)

According to the model error that calculates, choose model error weighting function W _i(s) on all frequencies, satisfy

| W_{j} (s - 0.5 δ) | > \frac{1}{γ} | {ΔH}_{i} (s - 0.5 δ) |, &ForAll; H_{i} (s) &Element; Ω_{j}, - - - (4)

Wherein δ is the exponential decay coefficient, is a constant; Symbol | | the amplitude of expression transfer function so just obtains describing regional Ω _jThe property taken advantage of uncertainty models:

P = {P_{i} (s) = [1 + Δ_{i} W_{i} (s)] G_{i} (s), {| | Δ_{i} | |}_{\infty}^{δ} < γ, i = 1, \cdot \cdot \cdot, N} - - - (5)

Wherein γ is the upper limit of plant model error; W _i(s) be to have the polynomial model error weighting function of same characteristic features; G _i(s) be to have the polynomial nominal model of same characteristic features; Δ _iBe the uncertain part of the model of controlled device; Its size is guaranteed by (4); N is the number of model in the model set; Symbol || || _∞ ^δThe L of expression system _{2 δ}Gain.