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Uni Maths > Nonlinear Systems And Control > Flashcards

Nonlinear Systems And Control Flashcards

1
Q

Define Nonlinear system

A

A dynamical system whose behaviour changes over time, often in response to external simulation /forcing

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2
Q

Assumption for generic Nonlinear state space system

A

Solution (x, y) exists and is unique

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3
Q

Comment on the stability of a linear state space system

A

Stability entirely depends on eigenvalues of matrix A

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4
Q

Key features of an Autonomous Nonlinear system

A

No (input or) time dependency
Equilibrium points are real and solutions of f(x)=0
Equilibium point can be shifted to the origin

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5
Q

How to study the stability of an Autonomous Nonlinear system

A

Basically study dx/dt = f(x)

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6
Q

Define BIBO

A

Bounded-Input-Bounded-Output

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7
Q

Do linear or Nonlinear systems have BIBO stability

A

Linear

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8
Q

Isolated equilibrium

A

There exists an epsilon >0 s.t. the open ball bounded by epsilon contains no other equilibrium

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9
Q

Assumptions for making an LTI system controllable

A

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

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10
Q

What does LTI system stand for?

A

Linear Time Invariant

system

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11
Q

Another term for phase plane

A

State plane

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12
Q

At an equilibrium point in the phase plane, what is the gradient/slope?

A

Indeterminate

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13
Q

Trace of matrix A=[ a , b ; c , d ]

A

Tr(A) = a - d

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14
Q

Define SISO

A

Single Input Single Output

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15
Q

What are the poles of the system

A

Eigenvalues of matrix A
OR
roots of the denominator polynomial of the transfer function

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16
Q

Advantages of the State Space method

A
  • 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
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17
Q

Give an example of a matrix triple that has an identical Transfer Function to (A,B,C)

A

( TAT^(-1), TB, CT^(-1) )

T is a square invertible matrix

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18
Q

An uncontrollable state

A

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

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19
Q

Difference between definition for continuous and uniformly continuous

A
  • 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
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20
Q

Banach Space

A

A complete normed vector space

A cauchy sequence of vectors always converge to a distinct limit value.

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21
Q

Slope property from the Lipschitz Condition

A

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.

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22
Q

What is implied if the zero solution is found to not be Lyapunov Stable

A

It is unstable

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23
Q

The case V˙(x) = 0

A

Conservation of energy: the vector field f (x) is orthogonal everywhere to the normal ∂V/∂x to the level surface V(x) = c.

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24
Q

The case V˙(x) < 0, ∀x ∈ D0

A

Dissipation of energy: the vector field f (x) and the normal ∂V/∂x to the level surface V(x) = c makes an obtuse angle.

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25
Q

Symmetric Matrix

A

M=transpose(M)

M is a square matrix

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26
Q

M is a symmetric, positive (semi)definite Matrix …

A

The eigenvalues of M are all greater than (or equal to) zero

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27
Q

A matrix M is positive definite/semidefinite if…

A

Definite: x^(T)Mx > 0 for all x =/= 0.
Semidefinite: x^(T)Mx ≥ 0 for all x

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28
Q

Lyapunov’s Direct Method

A
  • 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.
29
Q

Lyapunov Function

A

V(x) continous
V(0)=0
V(x)>0 for all x in D, x=/=0
V˙(x)≤0 for all x in D ???

30
Q

Positive definite function V(x)

locally

A

V(0) = 0

V(x) > 0 for all x =/= 0 in D

31
Q

Positive semidefinite function V(x)

A

V(0) = 0

V(x) ≥ 0 for all x =/= 0 in D

32
Q

Negative definite function V(x)

A

V(0) = 0

V(x) < 0 for all x =/= 0 in D

33
Q

Negative semidefinite function V(x)

A

V(0) = 0

V(x) ≤ 0 for all x =/= 0 in D

34
Q

Condition for positive (semi)definite function to be defined globally. Also applies to negative.

A

Domain D = R

i.e. the real space

35
Q

Condition for Lyapunov/Asymptotic/Exponential stability to be defined globally

A

Let x be in R^n, not just in D which is a subset of R^n

GLOBALLY ASYMPTOTICALLY STABLE
V(x) →∞ as ||x|| → ∞

36
Q

[A B]^T

A

[A B]^T = B^T A^T

37
Q

Lyapunov Equation

A

A^(T)P + PA + Q = 0
A, P, Q all same size square matrix
P, Q symmetric

38
Q

For the Lyapunov equation, what denotes the ‘generalised energy’

A

If V˙(x) = −x^(T)Qx

ANS: x^(T)Px

39
Q

What term is given to x^(T)Qx with relevance to the Lyapunov Equation, if If V˙(x) = −x^(T)Qx

A

Associated generalised dissipation

40
Q

Hurwitz Matrix

A

Real part of the eigenvalues of the matrix are all negative

41
Q

What is a positive limit set of x(t)

A

The set of all positive limit points of x(t)

42
Q

Difference between an invariant and positively invariant set

A

Positively invariant set: the condition holds for t ≥ 0

Invariant set: the condition holds for all t in R

43
Q

Fundamental Property of Limit set

A

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 → ∞.

44
Q

La Salle’s Theorem/Invariance Principle

A

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 → ∞.

45
Q

Describe an Asymptotically stable equilibrium in terms of positive limits.

A

Asymptotically stable equilibrium is the positive limit set of every solution starting sufficiently near the equilibrium point.

46
Q

Describe a Stable limit cycle in terms of positive limits.

A

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 → ∞.

47
Q

Give the properties of the function V in La Salle’s Theorem

A

Continuously differentiable function
V:D→R
V˙ (x(t)) ≤ 0 in Ω
NEED NOT BE P.D.F

48
Q

In relation to La Salle’s and Barbashin and Krasovskii Theorem, what extra property can enable the origin’s stability to be defined Globally

A

V(-) is radially UNbounded
V→ ∞ as |x|→ ∞
V p.d.f
(If already shown to be L.A.S)

49
Q

Is f(x(t)) Autonomous

A

Yes, f(x(t),t) is NOT

50
Q

Condition for M to contain a periodic orbit

A

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…?

51
Q

True or False, for the P.B. criterion, questions typically state V(x)

A

TRUE

52
Q

What is the Poincare Bendixson Criterion used for

A

The existence of a Periodic Orbit

53
Q

How to rule out the existence of a Limit Cycle

A

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

54
Q

Poincare Bendixson Criterion

A

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)

55
Q

Define an A.S. limit cycle

A

If all trajectories that start near a closed trajectory spiral
toward the closed trajectory as t → ∞, both from the inside and the outside.

56
Q

Another term for an A.S. limit cycle

A

Orbital stability

Since the closed trajectory is itself a periodic orbit rather than an equilibrium point.

57
Q

In the backstepping control method, what is the condition for the function replacing the virtual input?

A

Smooth

58
Q

In the backstepping control method, what is the condition for V(M)

A

smooth
p.d.f
Lyapunov function
dV/dt is n.d.f

59
Q

For the backstepping method, what conditions can lead to the origin being determined as Asymptotically Stable Globally

A

All assumptions hold globally

V(M) and V_c(M,ξ) are radially unbounded

60
Q

Diffeomorphism

A

A continuously differentiable map with a continuously differentiable inverse

61
Q

Minimum-phase system

A

The zero-dynamics of the system are Asymptotically stable

62
Q

Critically minimum-phase system

A

The zero-dynamics of the system are Lyapunov stable

63
Q

Another term for feedback linearizable system

A

input-state linearizable system

64
Q

Define feedback equivalent (in reference to systems)

A

Systems exhibit the same input-state behaviour

65
Q

Zero Dynamics of a nonlinear system

A

The dynamics of a system subject to the constraint that the output (y) for t≥0 is identically 0

66
Q

Controllable matrix pair (A,B)

A

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

67
Q

regulator problem

A

r=0 in control law

u = r -kx = -kx

68
Q

In control design, what do you treat the ‘u’ term as?

A

Free variable

Design freedom

Uni Maths (10 decks)

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