Lecture 6

Stability, Asymptotic Stability, Instability
Lyapunov Theorem on Asymptotic Stability • Theorem 1: Consider a system described by
! = f ( x, t ), f (0, t ) = 0 " t x
Adaptive Control Applications
Example: Roll Dynamics
(Model Reference Adaptive Control)
• Uncertain Roll dynamics:
&= Lp p + Lδδail p
– p is roll rate, – δail is aileron position – Lp , Lδail are unknown damping, aileron effectiveness
B - = b0 , B + = B / b0
B+ = 1, B- = B
Conditions
• Two causality conditions:
deg Ac ³ 2 deg A - 1 deg Am - deg Bm ³ deg A - deg B = d0
• Compatibility conditions:
Ax + By = C
Diophantine Equation
• How to guarantee a solutions in polynomial case?
– A and B must be relatively prime!!
• Solving the Diophantine Equation
Lecture 6
Self Tuning Regulators
Dr. ZHENG YI Email: yizheng@ Dec. 16, 2015
MIT Rule to Lyapunov transition • Several Problems were encountered in the usage of the MIT rule. • Also, it was not possible in general to prove closed loop stability, or convergence of the output error to zero. • A new way of redesigning adaptive systems using Lyapunov theory. • This was based on Lyapunov stability theorems, so that stable and provably convergent model reference schemes were obtained. • The update laws are similar to that of the MIT Rule, with the sensitivity functions replaced by other functions. • The theme was to generate parameter adjustment rule which guarantee stability.
BT BR y= uc + v AR + BS AR + BS
• Closed-loop characteristic polynomial:
AR + BS = Ac
Diophantine Equation
• A Diophantine equation is an equation in which only integer solutions are allowed. • The number of unknowns is larger than the number of equations - indefinite equations. • Problems
T R
• The process is described by the SISO system:
Pole Placement Design
Ay = B ( u + v )
• A general linear controller is described by:
Ru = Tuc - Sy
• Then we get the closed-loop system :
deg Am = deg A deg Bm = deg B deg Ao = deg A - deg B + - 1
' Bm = B - Bm
Example 3.1
• Transfer function:
1 G(s) = s( s + 1)
• With h=0.5, pulse transfer operator is :
ˆ K δ
ep → 0
parameter adaptation loop
• Adaptive control provides Lyapunov stability • Design is based on Lyapunov Theorem (2nd method) • Yields closed-loop asymptotic tracking with all remaining signals bounded in the presence of system uncertainties
Lyapunov Stability of Linear Systems
• Consider the system described by
! = Ax x
where x is a n´1 state vector and A is an (n´n) constant nonsingular matrix. • A necessary and sufficient condition for the equilibrium state x=0 to be asymptotically stable in the large is that, given any positive definite Hermitian ( or any positive definite real symmetric) matrix Q, there exists a positive definite Hermitian (or a positive definite real symmetric) matrix P such that
AT P + PA = -Q
• The scalar function is a Lyapunov function for this x T Px system.
Summary of Lyapunov rule for MRAC
Adaptation of Feed forward gain
(
)
• Flying Qualities Model:
p = Lm p + Lm δ (t )
m p m
.
m , L – (Lm are desired damping, control effectiveness p δ – ) δ (t ) is a reference input, (pilot stick, guidance command) e p (t ) = ( p (t ) − pm (t )) → 0 – roll rate tracking error:
δ
• Adaptive Roll Control: ⎧Kˆ = −γ p ( p − p ) ⎪ p p m , ( γ p , γδ ⎨ γ δ δ (t )( p − p m ) ⎪ δ =− ⎩K
ˆ
ail
ˆ p + Kˆ δ δail = K δ p
ail
>0
parameter adaptation laws
4
Motivating Example: Roll Dynamics
(Block-Diagram)
desired flying qualities model
δ (t )
unknown plant
Lm δ s + Lmp
Lδ ail s + Lp ˆp K
pm p
roll tracking error
Ac = Ao Am B +
AR¢ + B - S = Ao Am
' T = Ao Bm
R = R¢B+
Minimum-degree Pole Placement (MDPP)
• Step1: Factor B as B = B+ B-, where is B+ monic.
• Step2: Find the solution R ' and S with deg S < deg A from.
• If there exists a scalar function , having continuous V ( x, t ) first partial derivatives and satisfying the conditions • V ( x, t ) is positive definite & • V ( x, t ) is negative definite then the equilibrium state at the origin is uniformly asymptotically stable. • If in addition, V(x,t)®¥ as ||x|| ®¥, then therigin is uniformly asymptotically stable in the large.