Nonlinear Systems And Control Flashcards
Define Nonlinear system
A dynamical system whose behaviour changes over time, often in response to external simulation /forcing
Assumption for generic Nonlinear state space system
Solution (x, y) exists and is unique
Comment on the stability of a linear state space system
Stability entirely depends on eigenvalues of matrix A
Key features of an Autonomous Nonlinear system
No (input or) time dependency
Equilibrium points are real and solutions of f(x)=0
Equilibium point can be shifted to the origin
How to study the stability of an Autonomous Nonlinear system
Basically study dx/dt = f(x)
Define BIBO
Bounded-Input-Bounded-Output
Do linear or Nonlinear systems have BIBO stability
Linear
Isolated equilibrium
There exists an epsilon >0 s.t. the open ball bounded by epsilon contains no other equilibrium
Assumptions for making an LTI system controllable
1) System is completely state controllable
2) All state variables are measured and available for feedback
3) Control input is unconstrained
4) u has the form u=-kx
What does LTI system stand for?
Linear Time Invariant
system
Another term for phase plane
State plane
At an equilibrium point in the phase plane, what is the gradient/slope?
Indeterminate
Trace of matrix A=[ a , b ; c , d ]
Tr(A) = a - d
Define SISO
Single Input Single Output
What are the poles of the system
Eigenvalues of matrix A
OR
roots of the denominator polynomial of the transfer function
Advantages of the State Space method
- Initial conditions are implicit in the state space approach whereas the transfer function approach assumes a zero initial conditions;
- The representation can be extended to nonlinear cases
- The solution of coupled first order differential equations is easier to determine numerically than the equivalent higher order differential equations;
- Matrix-vector form is simple and can be solved efficiently in programming
Give an example of a matrix triple that has an identical Transfer Function to (A,B,C)
( TAT^(-1), TB, CT^(-1) )
T is a square invertible matrix
An uncontrollable state
Not in the range space of M_c
i.e. not a linear combination of its columns
Likewise, a controllable state would be in the range space of M_c
Difference between definition for continuous and uniformly continuous
- Continuous at x in D, uniformly continuous on D
- Cont delta(eps,x), uniform delta(eps)
- Cont for all y in D, uniform for all x,y in D
Banach Space
A complete normed vector space
A cauchy sequence of vectors always converge to a distinct limit value.
Slope property from the Lipschitz Condition
For f : R → R the Lipschitz condition
|f (x) − f (y)| / |x − y| ≤ L,
implies that any line segment joining two arbitrary points on the graph of f cannot have a slope greater than ±L.
Hence, any function having an infinite slope at a point x0 ∈ D is not Lipschitz continuous at x0.
What is implied if the zero solution is found to not be Lyapunov Stable
It is unstable
The case V˙(x) = 0
Conservation of energy: the vector field f (x) is orthogonal everywhere to the normal ∂V/∂x to the level surface V(x) = c.
The case V˙(x) < 0, ∀x ∈ D0
Dissipation of energy: the vector field f (x) and the normal ∂V/∂x to the level surface V(x) = c makes an obtuse angle.
Symmetric Matrix
M=transpose(M)
M is a square matrix
M is a symmetric, positive (semi)definite Matrix …
The eigenvalues of M are all greater than (or equal to) zero
A matrix M is positive definite/semidefinite if…
Definite: x^(T)Mx > 0 for all x =/= 0.
Semidefinite: x^(T)Mx ≥ 0 for all x
Lyapunov’s Direct Method
- If a continuously differentiable
- positive-definite function
- of the states of a given dynamical system can be constructed for which
- its time rate of change due to perturbations in a neighborhood of the system’s equilibrium is always negative or zero,
- then the system’s equilibrium point is Lyapunov stable.
- If the time rate of change is strictly negative, then the equilibrium point is asymptotically stable.
Lyapunov Function
V(x) continous
V(0)=0
V(x)>0 for all x in D, x=/=0
V˙(x)≤0 for all x in D ???
Positive definite function V(x)
locally
V(0) = 0
V(x) > 0 for all x =/= 0 in D
Positive semidefinite function V(x)
V(0) = 0
V(x) ≥ 0 for all x =/= 0 in D
Negative definite function V(x)
V(0) = 0
V(x) < 0 for all x =/= 0 in D
Negative semidefinite function V(x)
V(0) = 0
V(x) ≤ 0 for all x =/= 0 in D
Condition for positive (semi)definite function to be defined globally. Also applies to negative.
Domain D = R
i.e. the real space
Condition for Lyapunov/Asymptotic/Exponential stability to be defined globally
Let x be in R^n, not just in D which is a subset of R^n
GLOBALLY ASYMPTOTICALLY STABLE
V(x) →∞ as ||x|| → ∞
[A B]^T
[A B]^T = B^T A^T
Lyapunov Equation
A^(T)P + PA + Q = 0
A, P, Q all same size square matrix
P, Q symmetric
For the Lyapunov equation, what denotes the ‘generalised energy’
If V˙(x) = −x^(T)Qx
ANS: x^(T)Px
What term is given to x^(T)Qx with relevance to the Lyapunov Equation, if If V˙(x) = −x^(T)Qx
Associated generalised dissipation
Hurwitz Matrix
Real part of the eigenvalues of the matrix are all negative
What is a positive limit set of x(t)
The set of all positive limit points of x(t)
Difference between an invariant and positively invariant set
Positively invariant set: the condition holds for t ≥ 0
Invariant set: the condition holds for all t in R
Fundamental Property of Limit set
If a solution x(t) of a nonlinear system [x˙=f(x(t)) etc] is bounded and belongs to D for t ≥ 0, then its positive limit set L+ is a nonempty, compact, invariant set. Moreover, x(t) approaches L+ as t → ∞.
La Salle’s Theorem/Invariance Principle
Let Ω ⊂ D be a compact set that is positively invariant with respect to a nonlinear system [x˙=f(x(t)) etc].
Let V : D → R be a continuously differentiable function such that V˙ (x) ≤ 0 in Ω.
Let E be the set of all points in Ω where V˙ (x) = 0.
Let M be the largest invariant set in E.
Then every solution starting in Ω approaches M as t → ∞.
Describe an Asymptotically stable equilibrium in terms of positive limits.
Asymptotically stable equilibrium is the positive limit set of every solution starting sufficiently near the equilibrium point.
Describe a Stable limit cycle in terms of positive limits.
Stable limit cycle is the positive limit set of every solution starting sufficiently near the limit cycle, the solution approaches the limit cycle as t → ∞.
Give the properties of the function V in La Salle’s Theorem
Continuously differentiable function
V:D→R
V˙ (x(t)) ≤ 0 in Ω
NEED NOT BE P.D.F
In relation to La Salle’s and Barbashin and Krasovskii Theorem, what extra property can enable the origin’s stability to be defined Globally
V(-) is radially UNbounded
V→ ∞ as |x|→ ∞
V p.d.f
(If already shown to be L.A.S)
Is f(x(t)) Autonomous
Yes, f(x(t),t) is NOT
Condition for M to contain a periodic orbit
If every trajectory starting in M stays in M for all time, then M contains a periodic orbit (limit cycle) of x’= f(x).
If it does not contain an equilibrium…?
True or False, for the P.B. criterion, questions typically state V(x)
TRUE
What is the Poincare Bendixson Criterion used for
The existence of a Periodic Orbit
How to rule out the existence of a Limit Cycle
If a simply connected region o=D of the plane, the expansion of (inner product of f(x) with grad)
df1/dx1 + df2/dx2 is not 0, and doesn’t change sign, then x’= f(x) has no periodic orbit in D
Poincare Bendixson Criterion
x’= f(x)
Let M be a closed bounded subset of the plane s.t.
Every trajectory starting in M always stays in M for all future time
M contains no equilibrium points OR
M contains only one equilibrium point s.t its Jacobian matrix at this point has eigenvalues with positive real part.
Then M contains a periodic orbit of x’= f(x)
Define an A.S. limit cycle
If all trajectories that start near a closed trajectory spiral
toward the closed trajectory as t → ∞, both from the inside and the outside.
Another term for an A.S. limit cycle
Orbital stability
Since the closed trajectory is itself a periodic orbit rather than an equilibrium point.
In the backstepping control method, what is the condition for the function replacing the virtual input?
Smooth
In the backstepping control method, what is the condition for V(M)
smooth
p.d.f
Lyapunov function
dV/dt is n.d.f
For the backstepping method, what conditions can lead to the origin being determined as Asymptotically Stable Globally
All assumptions hold globally
V(M) and V_c(M,ξ) are radially unbounded
Diffeomorphism
A continuously differentiable map with a continuously differentiable inverse
Minimum-phase system
The zero-dynamics of the system are Asymptotically stable
Critically minimum-phase system
The zero-dynamics of the system are Lyapunov stable
Another term for feedback linearizable system
input-state linearizable system
Define feedback equivalent (in reference to systems)
Systems exhibit the same input-state behaviour
Zero Dynamics of a nonlinear system
The dynamics of a system subject to the constraint that the output (y) for t≥0 is identically 0
Controllable matrix pair (A,B)
A matrix pair (A, B) is said to be controllable iff the eigenvalues of (A − BK) can be assigned arbitrarily by choice of K.
det( M_c ) =/= 0
range( M_c ) = n
regulator problem
r=0 in control law
u = r -kx = -kx
In control design, what do you treat the ‘u’ term as?
Free variable
Design freedom
