Advanced Control Techniques

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0 16 ∀c).

(No matter how we choose x(t ), that property holds Exercise: Prove that exponential

0

stability =⇒ asymptotical stability.

Let’s show that the stability of a general equilibrium can be studied with the original for

another system. (for a different but equivalent system). ẋ = f (x, t), x is an equilibrium, ie

e

∀t. −

f (x , t) = 0, STATE: [x̃ = x x ], and let’s write the dynamic of x̃. If it is a trajectory, it

e e

must satisfy some dynamic:

˙ −

x̃(t) = ẋ (

x

˙ = 0) = ẋ(t) = f (x(t), t) = f (x̃(t) + x , t)

e e

˙

The dynamic is x̃ = f (x̃ + x , t). Let’s see if the new point is an equilibrium. x̃ = 0 is an

e ˜ ˜

˙

equilibrium of the new system since f (x , t) = 0. The new system is: x̃ = f (x̃, t), where f is a

e

new function such defined: ˜

f (x̃, t) := f (x̃ + x , t)

e

˜

Note that f, f are different functions in their variables. It is important to point out that:

˜ n

6 ∈

f (z, t) = f (z, t), z ! We’ve shown that if we have an equilibrium, we can study the equili-

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brium of the origin for the new system. (Daremo i risultati studiando la stabilità dell’origine ed

altre proprietà, ex.). Observation: TIME-INVARIANT case: ẋ = f (x), x is an equilibrium,

e

˜ ˜

˙

f (x ) = 0; x̃ = f (x̃ + x ) = f (x̃). We get then: f (x̃) = f (x̃ + x ). So, by recap, if I start from

e e e

a time invariant system, the EQUIVALENT SYSTEM is also T. INV!

˜

˙

x̃ = f (x̃)

If I start from a T.INV with an equilibrium, with this transformation we’re playing always with

T.INV systems.

2.1.3 Trajectories Stability

Definition 19. STABILITY OF A TRAJECTORY (INTERNAL STABILITY OF

A TRAJECTORY) A trajectory (x̄(·), u̇(·)) of ẋ = f (x, u, t) (remember that trajectory means

˙ ∀ ∃δ(, |

that x̄(t) = f (x̄(t), ū(t), t) is satisfied) is internally stable if: > 0 t )

0

∀x(t kx(t − kx(t) − ∀t ≥

) with ) x̄(t )k < δ(, t ) =⇒ x̄(t)k < , t

0 0 0 0 0

SAME INPUT, but different CINIT’s. If the trajectory stays in that tube, then it is stable.

ex: Rewrite the previous definition for the equilibrium as a special trajectory.

ẋ(t) = f (x(t), u(t), t), FIXED u. This become: ẋ(t) = f (x(t), ū(t), t) is an UNFORCED

system. If we put an equilibrium in the definition of the stability of a trajectory, then we obtain

the definition of the stability of the equilibrium. Stability of trajectories is a more general

version. Somehow equilibria ARE SPECIAL TRAJECTORIES. The equilibrium is a special

constant trajectory. For the equilibria we’ve seen that we could always study the stability of the

origin for a new equivalent system. Do we can apply the same trick as before for the trajectory?

−

ẋ(t) = f (x, u, t), (x̄(·), ū(·)) is a trajectory. Be: x̃(t) := x(t) x̄(t) (SAME INPUT)! We

get: ˙ ˙

− −

x̃(t) = ẋ(t) x̄(t) = f (x(t), ū(t), t) f (x̄(t), ū(t), t) = (. . . )

The trajectory changes but the INPUT is the same! ˜ ˜

−

(. . . ) = f (x̃(t) + x̄(t), ū(t), t) f (x̄(t), ū(t), t) = f (x̃(t), ū(t), t) = f (x̃(t), t)

So now the question is: is x̃ = 0 an equilibrium?

17

• Yes, of course, it is still valid. It satisfies the condition of being an equilibrium. (. . . ) =

−

f ((x̃(t) = 0) + x̄(t), ū(t), t) f (x̄(t), ū(t), t) = 0;

• What happens if we start from a T.INV system?

ẋ(t) = f (x(t), u(t)) which is TIME-INVARIANT. It is true that we can study the equi-

librium of the origin, but the new system is a TV system! Let’s prove it by playing the

same game as before:

(x̄(·), ū(·)) TRAJECTORY. ˜

˙ − −

x̃(t) = f (x̃(t) + x̄(t), ū(t)) f (x̄(t), ū(t)) = f (x̃, t) := f (x̃(t) + x̄(t), ū(t)) f (x̄(t), ū(t))

˜

˙

So we can also rewrite the system into an equivalent one: x̄(t) = f (x̃(t)), x̃ = 0 is again

an EQUILIBRIUM for the new system, but this system is clearly a TIME VARYING

system since structurally it depends on time, being itself defined as the difference of two

time-(even if not explicitly)-dependant functions! We don’t know whether this resulting

difference depends explicitly on the time, but in general it is true! Even if we started from

a T.INV system, where T.INV is a subclass of TV systems. Even for linear systems, LTI

=⇒ TV is a much more challenging prove.

PUNTI CRUCIALI:

– Possiamo, dato un generico equilibrio/generica traiettoria, studiare la stabilità del-

6

l’origine per il sistema equivalente = sistema iniziale;

– Partendo da un T.INV, tuttavia, applico la trasformazione e studio la stabilità

dell’origine del nuovo sistema, che sarà però un TV!

EXAMPLE: ẋ(t) = A(t)x(t) + B(t)u(t) possibly TV but LINEAR (LTV). Suppose that

for this system we have that (x̄(·), ū(·)) is a TRAJECTORY (even constant, where in this case

∀t). −

(x̄(t), ū(t)) = (x , u ) We defined: x̃(t) = x(t) x̄(t). Let’s write its dynamic:

e e ˙ ˙

−

x̃(t) = ẋ(t) x̄(t) = (. . . )

Si noti anche che (x, x̄) sono traiettorie differenti, giacché pur condividendo il medesimo

ingresso fissato, cambiano le CONDIZIONI INIZIALI!

− − −

(. . . ) = A(t)x(t) + B(t)ū(t) A(t)x̄(t) B(t)ū(t) = A(t)(x(t) x̄(t)) = A(t)x̃(t)

For a LINEAR SYSTEM, if we want to study the stability of a trajectory, we can study the

stability of the origin. In other words, all trajectories share the same stability property. So,

even if in general this is a very grave error, for linear systems, even with a negligible abuse of

notation, it is correct to talk about the STABILITY OF THE SYSTEM. Studiare la stabilità di

qualsiasi traiettoria è equivalente a studiare quindi quella dell’origine per sistemi lineari (anche

TEMPO VARIANTI).

Definition 20. INSTABILITY

∃ ∀δ ∃(t ∃ ≥ kx(t − k kx( − k ≥

> 0 ) t̄ t , with ) x < δ =⇒ t̄) x

0 0 0 e e

18

exercise

Given: 

ẋ = x

1 2



f (x(t)) := g K u

− −

ẋ = sin x x +

2 1 2

 l m ml

g u

⇐=

x = 0 ẋ = 0, [− sin x + = 0].

e

2 1 1e

l ml

(COPPIA CHE BILANCI LA GRAVITA’). Quindi (x , 0) è il punto di equilibrio cercato.

1e

x

1e

[u = mg sin x ]. Quindi la coppia di equilibrio è: ( , u ) nel caso FORCED. Ci si riconduca

e 1e e

0

ad un sistema equivalente che abbia come punto di equilibrio l’origine.

2.1.4 Lyapunov’s Theorem

(STABILITY OF EQUILIBRIA FOR TIME-INVARIANT SYSTEM). Results on the sta-

bility of equilibria in time-invariant systems. Stability of the origin. We need few definitions:

Some preliminar’s mathematical tools

Definition 21. POSITIVE DEFINITE FUNCTION

n 7→

A function V : is positive definite (p.d. as a shorthand, sometimes pd) on some domain

R R

n

⊆ ∈ ∧ ∀x ∈ \ {0}.

D if V (0 D) = 0 V (x) > 0 D

R

V can be a anything.

Definition 22. NEG. DEF

−V ⇐⇒ ∈ ∧ ∀x ∈ \ {0}.

V is neg. def. if is pd V (0 D) = 0 V (x) < 0 D

Definition 23. POSITIVE SEMIDEFINITE FUNCTION

n 7→ ≥

A function V : is positive semidefinite (p.s.d. as a shorthand, sometimes psd) if V (x)

R R

∀x ∈ ∧

0 D V (0) = 0.

Definition 24. NEGATIVE SEMIDEFINITE FUNCTION

n 7→ is negative semidefinite (n.s.d. as a shorthand, sometimes nsd) if

A function V : R R

−V ≤ ∀x ∈ ∧

(x) 0 D V (0) = 0.

Be careful! The difference between pd, psd is that for pd we have that outside the origin

the function is strictly positive, while in the second case it could be zero. We’ll refer with

this notation [V (x) > 0, V (x) < 0] = [V (x) pd, V (x) nd] as a shorthand. The same for the

≥ ≤

semidefinite cases: [V (x) 0, V (x) 0] = [V (x) psd, V (x) nsd].

Let’s give another definition (mathematical).

Definition 25. LIE DERIVATIVE OF A SCALAR FUNCTION wrt a vector FIELD

n n n

7→ 7→

Let V : be a differentiable scalar function and f : a vector field (campo

R R R R

vettoriale). The Lie Derivative of V wrt f is defined as:

∂V (x) n

∈

L V (x) := f (x), x R

f ∂x

It’s taking a scalar function (map), we take the derivative (basically the gradient) of the

function V (basically, the transpose of its gradient) - times this vector field f (x); we’ll see

why it’s very useful. This object here is defined without dynamical systems. It is a function

n 7→ ∈

(scalar) [L V (x) : take a vector X =⇒ associate a number Fornisce in

R R]; R.

f 19

output uno scalare. Let’s suppose we have an UNFORCED T. INVARIANT system ẋ = f (x).

n 7→

(We’ll have some trajectory defined by varying CINIT). Let’s take a function V : and

R R

◦

let’s consider the composite function V (x(t)) = V x. V (x(t)) as my V function. Basically

n

7→ ◦ 7→ ·(t)

x : [0, +∞) , V x : [0, +∞] So new V (x(t)) = is a function of the time. There’s

R R.

a little abuse of notation.

7→ ◦ 7→

t V (x(t)). So it’s a new function, V x : [0, +∞] The Lyapunov’s theorem is a

R.

corner point of ACT. Most of the results will rely on this idea behind the Lyapunov’s theory.

What happens to the trajectory of our system in terms of some energy function. Energy that

decreases as soon as I get from that point. Try to find some energy and try to show that this

n 7→

function decreases. V is defined independently from this system. [V : And gives

R R]!

it a particular number, typically the energy. CORNER STONES. Powerful Idea. The system

trajectory decreases the energy when they evolve. The way to show how this decreases is:

Let’s compute V (x(t)), and let’s compute wrt time the derivative of this SCALAR function

dV (x(t)) , by using the CHAIN RULE:

dt dV (x) ∂V (x) dx(t) ∂V (x)

◦ 7→ | |

V x : [0, +∞] = ( ẋ(t) = )=[ ẋ(t)]

R. x=x(t) x=x(t)

dt ∂x dt ∂x

If x(t) is just a common function, then I cannot say anymore, but if we assume that x :

n

7→

[0, +∞] is a trajectory of ẋ = f (x), then we can say something more! We can say that,

R ∂V (x) ∂V (x)

|

ẋ = f (x(t)): f (x(t)) = [ f (x)]| = L V (x)| = L V (x(t)) = V̇ (x(t));

f f

x=x(t) x=x(t) x=x(t)

∂x ∂x

n 7→ 7→ ∈ ◦ 7→ 7→

REMARK on the function: V : x V (x) V x