非线性系统第四讲输入输出稳定性

G(s) or f (t , x)
Y (s) y (t )
Topic 4 Input-Output stability
4.1. L-stability 4.2. L-stability of state model 4.3. L2 gain 4.4. The small gain theorem
Topic 4 Input-Output stability
Lemma 5.2 Suppose the assumptions of Theorem 5.5 are satisfied on a domain D Rn that contains the origin, f (x) is continuously differentiable, no solution of x f ( x) can stay identically in other than the origin. the system is smallsignal finite-gain L2 stable.
Nonlinear Systems Theory
Hu Jiangping
School of Automation Engineering University of Electronic Science and Technology of China
E-mail: hjp_lzu@163.com
U ( s) u (t )
Topic 4 Input-Output stability
Topic 4 Input-Output stability
x2 0 f ( x) , G 3 1 ax1 kx2 u
Topic 4 Input-Output stability
If there exists a W(x) such that
We can take
Topic 4 Input-Output stability
Topic 4 Input-Output stability
Lemma 5.1 Suppose the assumptions of Theorem 5.5 are satisfied on a domain D Rn that contains the origin, f (x) is continuously differentiable, and x = 0 is an asymptotically stable equilibrium f ( x). Then, there is kl > 0 such that for each x0 point of x with || x0 || k1 the system
4.4. The small gain theorem
Topic 4 Input-Output stability
4.4 Small gain theorem
Topic 4 Input-Output stability
Topic 4 Input-Output stability
Topic 4 Input-Output stability
Topic 4 Input-Output stability
Topic 4 Input-Output stability
Topic 4 Input-Output stability
Topic 4 Input-Output stability
Topic 4 Input-Output stability
Topic 4 Input-Output stability
Topic 4 Input-Output stability
Topic 4 Input-Output stability
Remark: if the system is causal, we can drop
Topic 4 Input-Output stability
4.1. L-stability
4.2. L-stability of state model 4.3. L2 gain
4.4. The small gain theorem
Topic 4 Input-Output stability
4.3 L2 gain
u (t )
Measure: energy of signal
Topic 4 Input-Output stability
LaSalle's invariance principle
Asymptotically stable
Topic 4 Input-Output stability
Example: f(x)
The condition is not hold globally, we need other lemmas to show the small signal finite gain stability!
x R , u R , f (0) 0, h(0) 0.
n m
Topic 4 Input-Output stability
Remark: in order to obtain V(x), we need solve the following Hamilton-Jacobi inequality
Topic 4 Input-Output stability
Method 1: By using Lemma 5.1 Linearization
Method 2: By using Lemma 5.2
Topic 4 Input-Output stability
4.1. L-stability
4.2. L-stability of state model 4.3. L2 gain
Topic 4 Input-Output stability
4.3 L2 gain
Toຫໍສະໝຸດ Baiduic 4 Input-Output stability
Example: check the finite gain L2 stability of the following system
f ( x) G( x)u, x(0) x0 x y h( x )
is small-signal finite-gain L2 stable.
Topic 4 Input-Output stability
Proof: “x = 0 is an asymptotically stable equilibrium”
Topic 4 Input-Output stability
such that
Topic 4 Input-Output stability
Will the control meet the design objective?
Topic 4 Input-Output stability
Topic 4 Input-Output stability
y (t )
u
T
(t )u (t )dt
Finite gain L2 stability
Topic 4 Input-Output stability
4.3 L2 gain
According to Parseval’s Relation
Fourier transform of the output signal:

1 1 ( ) k k
Remark: Theorem 5.1 cannot be applied since the origin of the system is not exponentially stable
Topic 4 Input-Output stability
Example:
Topic 4 Input-Output stability
Topic 4 Input-Output stability
Topic 4 Input-Output stability
4.1. L-stability
4.2. L-stability of state model 4.3. L2 gain
Example: Let H1 be a linear time-invariant system with a Hurwitz square transfer function matrix
Topic 4 Input-Output stability
Example:
Assume we can design
4.4. The small gain theorem
Topic 4 Input-Output stability
4.2. L-stability of state model
Topic 4 Input-Output stability
Topic 4 Input-Output stability