MRAS参考模型自适应控制

 Key Problem: determine the adjustment mechanism so that a desired performance is obtained

 T wo Methods:

 Design controller to drive plant response to mimic ideal response (error = y plant-y model => 0)

Designer chooses: reference model, controller structure,

 Form cost function:

 Can chose different cost functions

 EX:

From cost function and MIT rule, control law can be formed

 EX: Adaptation of feedforward gain

Adjustment Mechanism

y model

u y plant u c Π

Πθ

Reference Model

Plant

-

+

 For system where is unknown

 Goal: Make it look like

using plant (note, plant model is scalar multiplied by plant)

Controller:

 Choose cost function:

 Write equation for error: Calculate sensitivity derivative: Apply MIT rule:

 Gives block diagram:

u c

considered tuning parameter

 NOTE: MIT rule does not guarantee error convergence or stability

usually kept small

 Based upon the concept of energy and the relation of stored energy with system stability.

Example: Consider a mass spring damper system .

The dynamics of this system is expressed as A corresponding state model is

 At any instant, the total energy V in the system consists of ◦ kinetic energy of the moving mass and

◦ potential energy stored in the spring

 Thus

Thus

This means that total energy is positive unless the system is at rest at the equilibrium point x

The rate of change of energy is given by

 Lyapunov’s direct method makes use of a Lyapunov function

of the state may be thought as a generalized energy.

 When a system is described mathematically, it may not be clear what “energy” means.

Consider a zero-input system described by the state equation

where x is an n

 Assume that the system has only one equilibrium point

 The system of (1) is

Consider a spherical region of radius “r ” about an

equilibrium state x || x-x e || defined as follows: Let And let x −x e =

 An equilibrium state x e of the system of (1) is said to be stable in the sense of Lyapunov if, corresponding to each

that trajectories starting in

(ε) as t increases indefinitely.

 The real number

also depends on t

 Choose the region

 An equilibrium state x e of the system of (1) is said to be asymptotically stable if it is stable in the sense of Liapunov and if every solution starting within converges, without leaving

indefinitely.

 Note that asymptotic stability is a local concept. A knowledge of largest region of asymptotic stability is usually necessary.

attraction. It is that part of the state space in which asymptotically stable trajectories originate. trajectory originating in the domain of attraction is asymptotically stable.