自适应控制(研究生经典教材)

2024年1月29日发(作者：)

自适应控制

Adaptive control

1. 关于控制

2. 关于自适应控制

3. 模型参考自适应控制

4. 自校正控制

5. 自适应替代方案

6. 预测控制

参考文献

主要章节内容说明：

第一部分：

第一章 自适应律的设计

§1.参数最优化方法

§2.基于Lyapunov稳定性理论的方法

§3.超稳定性理论在自适应控制中的应用

第二章 误差模型

§ra误差模型

§2.增广矩阵

§3.线性误差模型

第三章 MRAC的设计和实现

第四章 小结

第二部分：

第一章 模型辨识及控制器设计

§1.系统模型：CARMA模型

§2.参数估计：LS法

§3.控制器的设计方法：利用传递函数模型

§4.自校正

第二章 最小方差自校正控制

§1.最小方差自校正调节器

§2.广义最小方差自校正控制

第三章 极点配置自校正控制

§1.间接自校正

§2.直接自校正

1. About control engineering education

1) control curriculum basic concept

(1) dynamic system

 The processes and plants that are controlled have responses that evolve

in time with memory of past responses

 The most common mathematical tool used to describe dynamic system is

the ordinary differential equation (ODE).

 First approximate the equation as linear and time-invariant. Then

extensions can be made from this foundation that are nonlinear 、time-varying、sampled-data、distributed parameter and so on.

 Method of building model (or equation )

a) Idea of writing equations of motion based on the physics and

chemistry of the situation.

b) That of system identification based on experimental data.

 Part of understanding the dynamical system requires understanding the

performance limitations and expectation of the system.

2. stability

With stability, the system can at least be used

 Classical control design method, are based on a stability test.

Root locus 根轨迹

Bode‟s frequency response 波特图

Nyquist stability criterion 奈奎斯特判据

 Optimal control, especially linear-quadratic Gaussian (LQG) control (线性二次型高斯问题) was always haunted by the fact that method did not

include a guarantee of margin of stability.

The theory and techniques of robust (鲁棒)design have been developed

as alternative to LQG

 In the realm of nonlinear control, including adaptive control, it is

common practice to base the design on Lyapunov function in order to be

able to guarantee stability of final result.

3. feedback

Many open-loop devices such as programmable logic controllers (PLC) are in use,

their design and use are not part of control engineering.

 The introduction of feedback brings costs as well as benefits. Among the

costs are need for both actuators and sensors, especially sensors.

 Actuator defines the control authority and set the limits of speed in

dynamic response.

 Sensor via their inevitable noise, limit the ultimate(最终) accuracy of

control within these limits, feedback affords the benefit of improved

dynamic response and stability margins, improved disturbance

rejection(拒绝) ，and improved robustness to parameter variability.

 The trade off between costs and benefits of feedback is at the center of

control design.

4. Dynamic compensation

 In beginning there was PID compensation, today remaining a widely used

element of control, especially in the process control.

 Other compensation approaches : lead-and-log networks (超前-滞后)

observer-based compensators include : pole placement, LQG designs.

 Of increasing interest are designs capable of including trade-off among

stability, dynamic response and parameter robustness.

Include: Q parameterization, adaptive schemes.

Such as self-tuning regulators, neural-network-based-controllers.

二、historical perspectives (透视)

 Most of early control manifestations appear as simple on-off (bang-bang)

controllers with empirical (实验；经验性的) setting much dependent upon

experience.

 The following advances such as Routhis and Hurwitz stability analysis

(1877).

Lyapunov‟s state model and nonlinear stability criteria(判据) (1890) .

Sperry‟s early work on gyroscope and autopilots (1910), and Sikorsky‟s

work on ship steering (1923)

Take differential equation, Heaviside operators and Laplace transform as

their tools.

 电机工程(electrical engineering)

The largely changed in the late 1920s and 1930s with Black‟s development

of the feedback electronic amplifier, Bush‟s differential analyzer, Nyquist‟s

stability criterion and Bode‟s frequency response methods.

The electrical engineering problems faced usually had vary complex albeit

mostly linear model and had arbitrary (独立的；随机的) and wide-ringing

dynamics.

 过程控制(process control in chemical engineering)

Most of the progress controlled were complex and highly nonlinear, but

usually had relatively docile (易于处理的) dynamics.

One major outcome of this type of work was Ziegler-Nichols‟ PID

thres-term controller. This control approach is still in use today, worldwide

with relatively minor modifications and upgrades (including sampled data

PID controllers with feed forward control, anti-integrator-windup

controllers :抗积分饱和，and fuzzy logic implementations).

 机械工程(mechanical engineering)

The application of controls in mechanical engineering dealt mostly in the

beginning with mechanism controls, such as servomechanisms, governors

and robots.

Some typical control application areas now include manufacturing process

controls, vehicle dynamic and safety control, biomedical devices and genetic

process research.

Some early methodological outcomes were the olden burger-Kahenbuger

describing function method of equivalent linearization, and minimum-time,

bang-bang control.

 航空工程（aeronautical engineering ）

The problems were generally a hybrid (混合) of well-modeled mechanics

plus marginally understood fluid dynamics. The models were often weakly

nonlinear, and the dynamics were sometimes unstable.

Major contributions to framework of controls as discipline were Evan‟s root

locus (1948) and gain-scheduling.

 Additional major contributions to growth of the discipline of control over the

last 30-40 years have tended to be independent of traditional disciplines.

Examples include:

Pontryagin‟s maximum principle (1956) 庞特里金

Bellman‟s dynamic programming (1957)贝尔曼

Kalman‟s optimal estimation (1960)

And the recent advances in robust control.

三、 Abstract thoughts on curriculum

 The possibilities for topic to teach are sufficiently great. If one tries to

present proofs of all theoretical results. One is in danger of giving the

students many mathematical details with little physical intuition or

appreciation for the purposes for which the system is designed.

 Control is based on two distinct streams of thought. One stream is physical

and discipline-based. Because one must always be controlling some thing.

The other stream is mathematics-based, because the basis concepts of

stability and feedback are fundamentally abstract concepts best expressed

mathematically. This duality(两重性) has raised, over the years, regular

complaints about the „gap‟ between theory and practice.

 The control curriculum typically begins with one or two courses designed to

present an overview of control based on linear, constant, ODE models,

s-plane and Nyquist‟s stability ideas, SISO feedback and PID, lead-lay and

pole-placement compensation.

These introductory courses can then be followed by courses in linear system

theory, digital of control, optimal control, advanced theory of feedback, and

system identification.

四、 Main control courses

 Introduction to control

Lumped system theory

Nonlinear control

Optimal control

Adaptive control

Robot control

Digital control

Modeling and simulation

Advanced theory

Stochastic processes

Large scale multivariable system

Manufacturing system

Fuzzy logic Neural Networks

外文期刊：《Automatic》IFAC 国际自动控制联合会

Computer and control abstracts

IEEE translations on Automatic control

Automation

 Specialized experimental courses

 Intelligent control

Application of Artificial Intelligence

Simulation and optimization of lager scale systems robust control

 System identification

 Microcomputer-based control system

Discrete-event systems

Parallel and Distributed computation

Numerical optimization methods

Numerical system theory

 Top key works from 1963-1995 in IIAC

Adaptive control 305

Optimal control 277

Identification 255

Parameter estimation 244

Stability 217

Linear system 184

Non-linear systems 168

Robust control 158

Discrete-time systems 143

Multivariable systems 140

Robustness 140

Multivariable systems control systems 110

Optimization 110

Computer control 104

Large-scale systems 103

Kalman filter 102

Modeling 107

为什么自适应 《Astrom》chapter 1

 反馈可以消除扰动。 P5，例1.1

但是：过程本身

 有些时变过程不可控 如：G(s)kp，kp为正时，系统可控；kp为s负时，则不可控；

 非线性问题，P10例1.4，不同操作水平下，效果不同；

 时变过程，P11例1.5，不同流量下滞后和时间常数也不同；

2. Adaptive control

-----introduction

(1) What is an adaptive control

The definition of an adaptive system continues to be multifaceted

and cannot be compressed into a simple statement without loss of

vital content.

Here are four representative samples:

 Definition 1 (during 1960s)

An adaptive system is a system which is provided with means of

continuously monitoring its own performance in relation to a given

figure of merit (最优值) or optimal condition and a means of

modifying its own parameters by a closed-loop action so as to

approach this optimum.

Or: An adaptive control system is defined as a feedback control

system intelligent enough to adjust its characteristics in a changing

environment so as to operate in an optimum manner according to

some specified criterion( 指标) .

 Definition 2

Adaptive control processes is defined as belonging to the last of a

series of three stages in evolution (演化) of control processes.

When the dynamical systems are described by ordinary vector

differential equations of the form:

x(t)f(x(t),u(t),,t)y(t)h(x(t),,t) (1.1)

x(t)[x1(t),x2(t),,xn(t)]TRnstateparameterimput

[1,2,,r]TRrWhere

u(t)[u1(t),u2(t),,up(t)]TRpy(t)[y1(t),y2(t),,ym(t)]TRmoutputDeterministic control process 确定性控制

This corresponds to the case where it is a control vector and the

designer has complete knowledge of function f(·) as well as the

parameter vector

.

Stochastic control process 随机控制

Where some of the inputs are random processes or when some of

the parameters are unknown with know distribution.

Adaptive control process 自适应控制

The third stage is when even less information about the process is

available. In such situations, the controller has to learn to improve if

performance through the observation of the outputs of the process.

As the process unfolds (发展，显露)，additional information

becomes available and improved decisions become possible.

(„Bellman·R and Kalaba·R” on adaptive control process” ;IRE

transaction on automatic control ;1-9nov, 1959)

 Definition 3

According to Zadeh (“on the definition of adaptivity “. In

proceeding of the IEEE 51: 569-570. March 1963) the difficulty in

defining the notion (概念) of adaptivity is due to a lack of a dear

differentiation between the external manifestations (现象，表现) of

adaptive behavior and the internal mechanism by which it is

achieved. Zadeh concentrated on the former and expressed

adaptation in mathematical terms:

The system A is adaptive with respect to(关于)

{Sr}and

w if it

performs acceptably (可实现，可接受) well with every source in

the family

{Sr},

rT, that is

Prw. More compactly A is

adaptive with respect to T and

w if it maps T to

w where:

The performance of system A be denoted by P and let

w denote

the set of acceptable performances. Let

{Sr} denote a family of

time functions indexed by

r, to which the system A is subjected

the resultant (结果的，综合的) performance is denoted by

Pr.

 Definition 4

In 1963 Trural defined an adaptive system as one that is designed

from an adaptive viewpoint.

Adaptation, like beauty, is only in the eye of the designer.

For example: feedback

Cascade control 串级控制

(2) How to understand adaptive control

1) 主要特点：

a) 通过辨识（包括对系统结构、参数、性能指标等的辨识）而获得自适应能力。

b) 两种时间尺度的过程并存（即系统状态的变化—快过程；和参数的变化和调整——慢过程）。

2) 参数自适应与结构自适应系统

(Parameter Adaptive and structurally Adaptive )

If a dynamical system is described by Eq(1.1 ), uncertainty may

arise due to a lack of knowledge of some of the system or noise

parameters that are elements of a parameter vector or some of the

function fi(·). In both cases, other elements of the parameter vector

or the function fi(·), may be varied appropriately to achieve the

desired control. The former are said to be parameter adaptive, while

the latter are said to be structurally adaptive.

While parameter adaptive system can also result in(引起) structural

changes, as for example:

When certain parameters assume zero values, we shall ignore

such special cases and be concerned entirely with parametric

adaptation.

(3) 辨识与控制（identification and control）

The understanding or identification of a plant (or process) and

controlling are tow distinct activities.

Identification of the process characteristic alone dose not result in

satisfactory control while attempts to control the unknown plant

without identification may result on poor response.

Hence, the controller A in an automatic control system with

incomplete information regarding the mant B must simultaneously

solve two problems that are closely related but different in character.

Feldbaum (“optimal control system” New York: Academic press,

1965) referred to this as dual control. First, on the basis of the

information collected, the controller must determine the

characteristics and state of the plant B. Second on [4] P48 dual

control: 在有些随机系统中，加到系统上的控制有两个效应——当前控制即影响受控过程的状态，也影响未来状态的不确定性，这就是所谓的控制的双重效应。从理论上讲，自然希望控制器的控制作用，不仅能把受控过程的状态纳入预定轨迹，而且还能提高未知参数的估计精度，这样的控制叫双重控制。

The basis of this acquired knowledge, if has to determine what

actions are necessary for successful control.

However, not every estimation scheme followed by a suitable control

action will result in optimal or even stable behavior of the overall

system. Hence the identification and control procedures have to be

blended carefully to achieve the desired objective. The adaptive

control scheme described in the chapters following can be considered

special case where successful dual control has been realized.

There are two different approaches for solution of the adaptive

control problem earlier.

 Indirect control (or explicit identification)

The plant parameters are estimated on-line and the control

parameters are adjusted based on these estimates.

 Direct control (or implicit identification)

No effort is made to identify the parameters but the control

parameters are directly adjusted to improve a performance index.

Although the basis for parameter adjustment using the two methods

is guite dear, global stability of the result out nonlinear system

cannot be demonstrated. This fact led to a search for alternate

methods for designing stable adaptive controllers. However, the

importance of these early methods lies in the fact that many of the

ideas presented in the following chapters can be traced back to

them.

两种最常用的方法：

(4) (Two well known methods currently in vogue(流行))

 As shown in Fig 2.1

uprocessOr plantyp+-ymThe aim of control may be quantitatively stated as the determination

of the input U to keep the error

ecypym between process

output

yp and a desired output

ym within prescribed values.

If

ym is a constant, the problem is one of regulation around this

value. When

yma function of time, the problem is is referred to as

tracking.

1) when the characteristics of the process P are completely known,

the former involves the determination of a controller to stabilize

the feedback loop around the set point. In latter case, a suitable

controller structure may be employed and control parameters

determined so as to minimize a performance index based on the

errorec. As we know, powerful analytical techniques based on

the optimization of quadratic performance indices are available

when the differential equations describing the behavior of the

process are linear and are known a priori.

2) When the characteristics of the process are unknown, both

regulation and tracking can be viewed as adaptive control

problems.

Our interest will be in determining suitable controllers for these

two cases, when it is known a priori that the process is linear but

contains unknown parameter.

Model reference adaptive system (MRAS) and self-tuning

regulator (STR) are two classes of systems that achieve this

objective.

 （区分）辨识模型与参考模型

(Identification model and reference model)

它们分别使用在辨识控制场合。如图：

plantuIdentify

model(1)

The identification problem

yp+-y^prRefer

modelym-+yp

uplant(2) The control problem

a) 假设plant 为LTI，辨识问题要解决的是，找到一个Model （主要是参数未知）。使在同样input （U）条件下，yptends to

ypasymptotically.此时得到的模型称identification model.

b) 在control problem 中，首先是选一个Reference model (包括结构和参数structure and parameter ) 该model 可以是线性的，也可以是非线性的，只要它的outputs能准确反映所要的outputs剩下的事情就是如何找一个plant的input(u) 使ypym.

 MRAC中direct and indirect control

Output error

ecis directly used to adjust controllerc().

Mym-+uPyp

recC(ɵ)ec

Identification error

eiis used to adjust the identification parameter

turn is used to adjust

c().

p,which in

Mreiym-+Identfy

M(p)+ei-ecpC(ɵ)uPypIndirect adaptive control

The choices of the reference model, identifier and controller structure for

different classes of problems, and the generation of a stable adaptive law in each

case, are our following subjects.

 STR

STRS consist of a parameter estimator, a linear controller, and a block which

determines the control parameters from the estimated parameters.

The principal approach of the STR is indirect control where the parameters of

the plant are estimated prior to determining the control parameters.

Explicit STRS consist of an explicit estimation of the process to be controlled

followed by a turning of the regulator parameters, implicit STRs are based on

implicit estimation of the process and a direct update of the regulator

parameters.

设计Identify

model

r+-uC(θ)Py

 Compare MRAS with STR

The MRAS concerned with the optimal control of deterministic

servomechanisms, in contrast to this, self-tuning regulators evolved during the

study of stochastic regulator problems.

四、自适应控制与领域的关系

自适应控制不是一个成熟的领域，采用的方法带有一种特定的性质。工具也取自不同领域。缺乏系统性，它与许多领域有联系：非线性系统、稳定性理论、随机控制、参数估计、最优化、线性系统、计算机控制、采样系统、控制设计等等。

自适应系统本质是非线性的，所以它强烈依赖于非线性系统理论，稳定性理论是自适应控制的一个关键部分。因为自适应系统中的时标是分离的，故与奇异摄动理论和平均理论有关。考察自适应系统时，一种方法是把它看成参数估计和控制的结合，故与随机控制和参数估计有联系，当然自适应系统实际都用计算机，所以采样系统的知识也是必须的。

模型参考自适应控制问题的提法：

根据获得的有关受控对象及参考模型的信息（状态、输出、误差、输入等）设计一个自适应控制律，按照该控制律自动地调整控制器的可调参数（参数自适应）或形成辅助输入信号（信号综合自适应），使可调系统的动态特性尽量接近理想的参数模型的动态特性。

3

模型参考自适应控制

第一章 自适应律的设计(Adaptive law)

这种系统性能要求不是用一个指标系数来表达，而是直接通过一个参数模型的输出来表达。

设计这类自适应控制系统的核心问题是如何综合自适应调正律，即自适应机构所应遵循的算法。

关于自适应律的设计主要有三种方法：

（1） 局部参数最优化方法（简称MIT法）；

（2） 基于Lyapunov 稳定性理论的方法；

（3） 基于popov超稳定理论的方法；

§1.局部参数最优化方法

常用的参数最优化方法有：梯度法gradient method

牛顿核实逊法、变尺度法、共轭梯度法等

实现具体方案有：可调增益方案：单参数

反馈补偿器

前馈及反馈补偿器（此二者适用于多个参数同时可调）

下面以MIT 首先提出的单参数增益可调系统为例阐明此法基本思想。（奥斯本

Osburn ． et ．al 1961）

进行局部参数最优化设计时，一般有如下假设：

1) 可调系统参数变化不大，在一定领域内缓慢变化；

2) 自适应速度较低，当然比可调系统参数变化相比要快，但比可调系统阶跃响应相比是很慢的。

设受控对象(plant or process)传递函数为：

yp(s)R(s)M(s)KmN(s)D(s)(32)

式中Km根据希望的动态响应来确定。

设可调系统(controller)仅为一个可调的前馈增益Kc。

取performance index （性能指标）为

tJe12()d0(33)

式中e1ymyp为输出广义误差(generalized error)

MIT 方法的中心思想是：设计调节Kc的自适应律，使(3-3)式J最小。

假设Kc由两部分组成，一是固定部分，作为可调增益的初值，用Kc0表示，另一是可变增益部分，用Kc表示。

首先求出J对Kc的梯度

teJ2e11d0KcKc(34)

梯度法是在以性能指标J所构成的超曲面上对Kc进行，沿负梯度方向，按搜索步长B0依次搜索，即

则Kc调整值应为KcB0JKc(35)

式中，B0为步长，最终适当选定的正常数。

一步调整后Kc值为

KcKc(0)B0J (3-6)

Kc进一步求J，对(3-6)式求导（对Kc对时间的导数）

KcedJdtKcB0()B0(2e11d)dtKcdt0KceB02e11Kc (3-7)

如图，已知由R(s)到e1(s)的transfer function

rKcKmN(s) D(s)KpN(s)D(s)图3-1

yme1yp

e1ymyp

KcB02e1ypKc

可称为敏感度函数，因为系统中一般存在高频干扰，故构成系统时应避免微分元件。

e1(s)ym(s)yp(s)KmN(s)N(s)R(s)KcKpR(s)D(s)D(s)(38)e(s)N(s)1(KmKcKp)R(s)D(s)即：D(s)e1(s)(KmKcKp)两边对Kc求导，得

N(s) (3-9)

D(s)D(s)e1KpN(s)R(s) (3-10)

Kc

(3-10)(3-2)得Kpe1ym(s) (3-11)

KcKmKpe1(t)或ym(s) (3-12)

KcKm代入(3-7)

KcB02e1(t)KpKmym(t) (3-13)

令B2B0KpKm，则KcB2e()t()ymt (3-14)

1这即为可调参数Kc的自适应律。MIT自适应控制结构图，如下自适应机构包括一个乘法器和积分器。

Mr-+e1XpKcuPypɵ图3-2

其数学模型可归结为：

B

输出误差（误差方程）D(s)e1(s)(KmKcKp)N(s)R(s)

模型输出（参考模型）D(s)ym(s)KmN(s)R(s) (3-15)

自适应律KcBe1(t)ym(t)

问题：自适应律中B2B0KpKm，Kp未知，如何得B？

因为假设Kp变化很慢，可以认为在自适应过程中Kp不变。

综合这种自适应律所需信号都是容易获得的，

因为这里利用的是输出误差e1，而不是状态偏差e，这是MIT方案的主要优点。它的缺点也是明显的：

（1） 没有以稳定性理论作依据，故不能保证系统是稳定的。当自适应增益过大，或者当参考输入太大时，都可能导致系统不稳定，所以得到的自适应律后，还要对整个系统进行稳定性检验，这是一种兼顾稳定性和优化性能指标的设计方法。只能有效地应用于低阶系统。

（2） 参数最优化，要不断寻优，，需一定的搜索时间，也即需有较长的自适应调整时间。

例题，习题

《现代控制理论及其应用》周凤岐编 电子科大 P386

P(s)Kp设11Tps11Tms，式中TpTmT

M(s)Km根据MIT自适应方案，设计自适应控制系统，并检验其稳定性。

解：自适应控制系统结构图如图3-2，其数学模型，

输出误差：Te1e1(KmKcKp)r (3-16)

模型输出：TymymKmr (3-17)

自适应律

KcB1emy (3-18)

检验稳定性：

设r(t)r0，由式(3-17)，得

ym(t)Kmr0(1etT) (3-19)

对式(3-16)两边求导：Te1e1KcKpr0 (3-20)

将(3-18)代入

Te1e1BKpr0yme10 (3-21)

ymTe1e1 将(3-19)代入上式(3-21)

BKpr0e12p0tT得Te1e1BKrKm(1e)e10

Te1的系数BKpr02Km(1et)0 ，系统是稳定的

由于参考模型是稳定的，即(3-19)中，当t时，ymKmr0

练习 [9] P387

设对象为二阶系统，其传递函数P(s)Kpa2sa1s12，a1,a2为常值，Kp受环境影响而改变，选取参考模型传递函数为m(s)Km，式按MIT法设计自适2a2sa1s1应系统，并验证稳定性。

可得出结论：

1) MRAC稳定性不仅与该系统结构有关，而且同外加扰动信号的类型和大小有关。

2) 并且Kp较大时，也会使系统不稳定。

3) 与步长B0也有关系，B0太长稳定性变差，太短搜索时间太长。

§2基于Lyapunov稳定性理论的方法

2.1 Lyapunov stability

一、definitions（定义） (1892年)

Let a system be described by the (nonlinear) differential equation.

xf(x,t),x(t0)x0 (2-1)

其中，x为系统的状态，xRn; 若

f(x,t)A(t)x(t),A(t)Rnn (2-2)

称系统为线性的；否则为非线性的。若f()不依赖于时间t，称系统为是不变系统。Solution (方程的解)可表示为x(t;x0,t0),(tt0). 如果状态空间中，存在某一状态xe，满足f(xe,t)0对所有t0；则xe就是系统的一个平衡点(equilibrium

state). 将状态空间的原点取着系统的平衡点，（因为任何孤立的平衡点，均可通过坐标的变换，将其移到坐标原点）。这样连续系统的平衡状态表达式一般写为：f(0,t)0,(t0)

The following definitions pertain to(关于) some of the basic notions in Lyapunov

stability of such an equilibrium state .

Definition 1: (stable)

The equilibrium state

x0 of Eg (2-1) is said to be stable if for every

0and

t00, there exists a

(,t0)0such that (使得)

x0(,t0),implies that

x(t)(对所有)

tt0。

这就是李亚普诺夫意义下的稳定。

此处x(t)又可写成x(t;t0,x0)是方程（2-1）从(t0,x0)开始的解。

Fig 1 为稳定性的平面M可表示：

S(ɵ)ɵS(δ)x00δX(t)

Figure 1

Definition 2 一致稳定

The equilibrium state

x0of Eg (2-1) is said to be uniformly stable if in Definition

1 ,is independent of initial time to:

23x(1xt) is not. For example,xx is uniformly stable and

xxt

Definition 3 渐进稳定

The equilibrium state

x0 of Eq (2-1) is said to be asymptotically stable if it is

x(t)0

stable (definition l) and

limxFig 2

S(ɵ)ɵS(δ)x00δX(t)Figure 2

经典控制理论中的稳定性定义与渐进稳定意义相同，渐进稳定性等价于工程意义上稳定性。

Definition 4 一致渐进稳定

The equilibrium stable

x0 of Eq (2-1) is uniformly asymptotically stable (u.a.s) if

it is uniformly stable and for some

10and every

20,there exists

T(1,2)0,such that

fx01,then

x(t)2,for all

tt0T.

Fig 3

x0X(t)0ɵ2ɵ1Figure 3

不稳定：若平衡状态xe既不是渐进稳定的，也不是稳定的，即无论规定多么小，由S()以外，称此平衡状态xe为不稳定的。

Definition 5 大范围（全局）渐进稳定，

While stability implies that solution(解) lies near the equilibrium state, asymptotic

stability implies that the solution tends to the equilibrium as

tand uniform

asymptotic stability implies that the convergence of the solution is independent of the

initial time.

If

lim()in definition above, the stability is said to hold in the large or the

equilibrium state is globally stable.

If definition 4 holds for every

10,x0in Eg (2-1) is uniformly

asymptotically stable in large (u.a.s.1).

x0 是全局渐进稳定，即对所有x0Rn，存在limx(t)0。

t

Definition 6 指数（收敛）稳定

以上的稳定定义中，对收敛速率没有定量，下面用指数收敛来定义稳定：

The equilibrium state

x0 of Eg(2-1) is exponentially stable if there exist constants

m0 and

a0 such that

x(t)me(tt0)x0,

tt0for all to and initial state

x0 in a certain neighborhood B of the origin. (即：x0B)

Exponential stability of a system always implies uniform asymptotic stability the

converse is not true in general. (For examplexx3) In the case of linear system,

however, uniform asymptotic stability is equivalent to exponential stability. Also, all

stability properties hold in the large for linear systems.

If the system is autonomous （自治的），all stability properties are uniform.

If

BRn in Definition 6. Then equilibrium state is globally exponentially stable.

注意几点：

（1） 上诉采用诸如x形式的范数，也可用其它形式的范数（norm）根据范数的等价性（即各种范数下考虑向量序列的收敛问题时，表现出明显的一致性），运动是否稳定与范数选择无关。

（2） 运动的稳定性与“系统的”稳定性。

运动的稳定性研究系统（2-1）任何一个解x(t;x0,t0)附近的行为，它可以化为平衡状态xe的稳定性，若某一平衡状态的充分小的邻域内不存在别的平衡状态，称该平衡状态为孤立的平衡状态，对于孤立的平衡状态，总可以通过适当的坐标变换，将xe变换到状态空间原点，即xe0。

对于形如（2-2）的线性定常系统xAx，若A为非奇异矩阵，则它有唯一的孤立平衡状态xe0，所以关于平衡状态的稳定性和“系统的”稳定性完全一致。如果某系统有不止一个孤立的平衡状态，则可能它的运动关于其中一个平衡状态是稳定的，而关于另一个则是不稳定的，这种情况下就无所谓“系统的”稳定性。

（3） 工程上通常认为渐近稳定性质比稳定性质更重要。



二、李亚普诺夫稳定性分析

Lyapunov 把分析常微分方程（组）稳定性的全部方法归纳为两类：

第一类方法：先求出常微分方程（组）的解，而后分析其解运动的稳定性，称为间接法；

第二类方法：不必求解常微分方程，而提供出解运动稳定性的信息，称为直接法。

＊Lyapunov’s indirect method （第一方法）

第一定理（Theorem 1）： 若系统线性化后，系统特征方程的所有的根均为负实数或实数部分为负的复数，则原系统的运动不但是稳定的而且是渐近稳定的。线性化过程中被忽略的高于一阶的项也不会使运动变成不稳定。

第二定理（Theorem 2）若线性化后系统特征方程的诸根中，只要有一个为正实数或实部为正的复数，则系统的运动就是不稳定的，被忽略的高于一阶的项也不会使运动变为稳定。

若线性化后特征方程的诸根中，有一些是实部为零的，其余均具有负实部，则原系统运动的稳定与否与被忽略的高阶项有关，这种情况下不可能按照线性化后的方程来判断稳定性，必须分析原始的非线性模型。

实际系统一般是非线性的，Lyapunov 第一方法使线性化研究方法有坚实可信的理论基础。

故有：线性系统稳定的充分必要条件：微分方程的特征方程的全部根都是负实数或实部为负的复数，或微分方程组的系统矩阵A（Yacobian 雅克比）的所有特征值均具有负的实部。

工程实践中，可不必解出特征方程就能判别是否有右半平面的根，这类方法，称之为稳定性判别。如：Routh 判据、Hurwitz判据等。

＊Lyapunov’s Direct Method (第二方法)

1、标量函数（scalar function）的定号性

正定性（positive-definite）：标量函数V(x)在域S中对于x0有V(x)0或，称V(x)在域S内正定。

负定性（negative-definite）：V(x)对于x0有V(x)0(V(x)0)及V(0)0；

若V(x)负半定，则V(x)为正半定

不定性：V(x)对于x0可正可负。

2、李亚普诺夫函数（能量函数）

（1）概念的引出：一个振动系统的能量若随时间而衰减，该系统最终会达到平衡状态，如何描述能量的变化，Lyapunov 虚构一个能量函数V(x,t)，表示能量与状态x、时间t有关，它是一个标量函数，由于能量总大于零，故为正定函数。

V(x,t)的选取不是唯一的，经验与技巧很重要。

关于V(x,t)的正定性的提法：标量函数V(x,t)在域S中对于tt0及x0有V(x,t)0及V(0,t)0，称V(x,t)在域S内正定。

（2）很多情况下，线性系统下李亚普诺夫函数可取为二次型

二次型及其定号性：

n个变量x1,x2,,xn的二次齐次多项式

V(x1,x2,,xn)a11x12a12x1x2a1nx1xna21x1x22a22x2a2nx2xnan1x1xnan2x2xn

2annxn称为二次型。

其中，aik(i,k1,2,,n)是二次型的系数，用矩阵形式表示如下：

a11aV(x)[x1,x2,,xn]21an1a12a1nx1xa22a2n2XTpX

an2annxn设aikaki，则p为对称矩阵（权矩阵）。p的秩则为二次型的秩。二次型V(x)或对称矩阵p的定号性由塞尔维斯特准则判定（Sylvester）.

a) 若p的各主子行列式均大于零，即

a11a110,a11a12aa120,,12a22a1na12an1a22an2a2nann0，则p为正定，即V(x)为正定；

b) 若p得各顺序主子行列式正负相间，则p为负定；

c) 若主子行列式会有等于零，则为正半定，或负半定；

d) 不符以上情况者，为不定。

3、Lyapunov 稳定性基本定理2.1

设不受外部作用的系统运动方程和平衡状态如下：

f(x,t)x （1）

xe0如果在平衡状态附近可找到单值标量函数V(x,t)，对状态向量x的每个分量均存在一阶偏导数（first partial derivatives with respect toxi），而xi(i1,2,,n)V(x,t)及其对时间导数满足下列条件：

<1>

V(x,t)0，即V(x)正定；

<2>

V(x,t)[vx1vx21xvx2]0，即V(x,t)负定。

xnnx则equilibrium state

xe0是asymptotically stable. 并称V(x,t) system (1) 的一个Lyapunov function. (卡尔曼) 增加了<3>，即：

<3>

limV(x,t)，则xe0是大范围渐进稳定。

x

Lyapunov 基本定理中，要求V(x,t)为负定，在许多场合下常常是构造函数V(x,t)的困难所在。

进一步下面有宽松一些的定理：

定理2-2

如果V(x,t)和V(x,t)满足

<1>

V(x,t)0 为正定

<2>

V(x,t)0 为负半定

则xe0是稳定的。反之，

<1>

V(x,t)0，<2>

V(x,t)0，则xe0不稳定。

选取不同的李氏函数（Lyapunov函数），可能使问题分析得出不同的结果。

练习：设系统方程为：



01x1x2 或

xx

11x2xx12试确定系统平衡状态的稳定性

解：原点(0,0)为给定系统的唯一平衡状态点。

(1) 选取标准型二次函数

22 (正定)

V(x,t)x1x22V(x,t)2xx12x2x22x1x22x1x22x21则

22x2当x10,x20(即xe0)

V(x,t)0，

但x10,x20时，(即非xe0点)

V(x,t)0

故V(x,t)为负半定

可确定系统在平衡状态xe0处是稳定的。但是否渐进稳定还需进一步确定。也即除xe0处是否存在x10,x20，使得V(x,t)0。

采用反推法：

令V(x,t)2x20，必要求x2在tt0时恒等于0，也即x20，以状态方程22x2x1x2来看，若存在平衡状态x20，现x20，则x1必为0。表明V(x,t)只可能在原点(x10,x20)处恒等于0.

所以系统在xe0平衡状态处是渐近稳定的。

又x时，V(x,t)

故在xe0处是大范围内渐近稳定。

(2) 若选取另一正定李氏函数

31x1222V(x,t)[x1,x2](xx)2xx12120

12x2则V(x,t)2(x1x2)(x1x2)4x1x12x2x2

222(x1x2)0上式显然说明V(x,t)负定；又x时，V(x,t)，所以系统在原点xe0处的

平衡状态是大范围渐近稳定。

定理2.2可判定稳定，如要判定是否渐近稳定，还需进一步研究确定。

通过上面练习在定理2.2后增加附加条件，可判定渐近稳定。

定理2.3

对于系统(1)，若

<1>

V(x,t)0 为正定

<2>V(x,t)0 为负半定

<3>若V(x,t)在方程(1)的非零解状态运动轨迹线上不恒为0，则平衡状态xe0是渐近稳定的；反之，系统可以保持在一个稳定的等幅振荡状态上，即是稳定的。

注意：上述定理只给出了系统稳定的充分条件，并非必要条件。应用李亚普诺夫稳定定理，关键是构造函数V(x,t)。

具体方法：

a) 先试构造V(x,t)是正定的，然后考察V(x,t)；

b) 先给出V(x,t)是负定的，然后确定V(x,t)正定；

c) 先使V(x,t)正定，然后从系统稳定性要求出发，推导出对系统的限制。



2.2 基于Lyapunov 理论设计自适应律

1、利用Lyapunov Direct Method 来综合§1节讨论过的自适应律。

为表述上的简便：

设Model:

km

1sT Plant:

kp1sT (更简化一步，kp1)

重新画出结构图

r(t)km1sTyme1kckp1sTyp1s图3-3

已知：e1ymyp

B

e1ymyp，即P30，(1-15)输出误差方程

e1ymyp11(kmrym)(kpkcryp)TT则 (2-2-1)

1r(ymyp)(kmkc)TTer1xTTkym(s)mR(s)1sTymTymkmrym

1(kmrym)T令

xkmkc

则

xkc，(根据§1节M.I.T自适应律，xkcBe1ym)

现在利用Lyapunov 再设计自适应律。



选择如下Lyapunov 函数：

2V(x,t)e1x2,(0) (2-2-2)

这是一个正定函数。

V2e1e12xx (2-2-3)

将(2-2-1)代入，判断输出误差方程的稳定性：

1rV2e1(e1x)2xxTT (2-2-4)

22re12e1x2xxTT2r2欲使V负定，或负半定，取e1x2xx,Ve12

TT1e1r (2-2-5)

即xT1B 由于xkc，令T得自适应律：kcBe1r (2-2-6)

注意控制结构图与M.I.T法的区别。

显然这种设计方法，在(e1,x)平面上，点e10,x0是稳定的。但是否是渐近稳定？

当然不是：因为可能存在e10,而x0的情况，使V0。例如，当r(t)0，虽e10，但x0为常数。

因此为了在t时，e1和

2、进一步将上述Lyapunov 综合法推广至高阶系统。比如，用1b1sb2s代替1sT组成二阶系统。这时，选择Lyapunov 函数如下：

x均趋向于零。r(t)在某种意义上必须是充分激励的( Persistency Excitation)[Morgan and Narendra 1978]

V(x,t)b12b122eex0 正定

112b2b22b12b11e (2-2-7)

eee12xx211b2b2V(x,t)又e1ymyp

ymyp

ey1em

y (2-2-8)

p

m由ykmybm (2-2-9)

rm1yb2b2b2ypb1kcprp (2-2-10) 同理yyb2b2b2(2.2-9)、(2.2-10)代入(2.2-8)

e1br1myp)

(kmkc)(ymyp)1(yb2b2b2br11

xe11eb2b2b2 (2.2-11)

代入(2.2-7)

2b1b1r12b1ee1)2xx

Ve(xee11211b2b2b2b2b22b1222b112re1x2xx

2eb2b2令2b11x2xx0

re2b2b1r

e21b22即x2b1e120 负半定

得V2b2xkmkc



kxcb1re1

自适应律kc2b2令Bb1

b22Bre1 (2.2-12)

则kc1，而不是e1。这将带来不希望的噪声，一般不采用这在这种情况下，使用的是e1的微分e种方案。

那么在组成自适应反馈回路时，怎样才能做到只用e1调整kc，而不需要e1的微分。

正实传递函数在解决这个问题时，有着重要的作用。