Definición de exposiciones deterioradas y determinación de las correcciones de valor

Σ2 :

˙x = f (x, u) y = h(x)

(3.13)

is feedback equivalent to a passive system with positive definite storage function V (x) are derived. These conditions are developed to exploit the following interesting stabilizing property of passive systems. Assume that Σ2 is passive and zero-state ob-

servable. This means that if the output h(x) = 0 is zero, then the state is identically zero. With this property the following theorem states that the system is globally stabilized purely by output feedback.

Definition 3.3.1. [60][Theorem 14.4] If Σ2 is

(i) passive with a radially unbounded positive definite storage function and (ii) zero-state observable

then the origin x = 0 can be globally stabilized by u = −φ(y), where φ is any locally Lipschitz function such that φ(0) = 0 and yTφ(y) > 0 for all y 6= 0.

The control in Definition 3.3.1 has been formulated to ensure the passivity condi- tion in Definition 3.2.1 holds globally. Then the zero-state observable property helps conclude that the origin is the largest invariant set and hence the global equilibrium

of the closed-loop system. In order to use this powerful result for control design, con- ditions under which systems can be made passive need to be studied. The first result toward this end, studies the relative degree of a passive system. Relative degree of a system is number of times the output must be differentiated for the input to appear explicitly. The following definition expresses this condition using Lie derivatives. Definition 3.3.2. The system Σ2 is said to have a relative degree (r1, r2, . . . , rm) at

a point (x0, u0) if:

(i) _∂u∂ Lk_fhi(x) = 0 for all 1 ≤ i ≤ m, x in the neighbourhood of x0 and all u in

the neighbourhood of u0 and all k < ri,

(ii) _∂u∂ [Lri

f hi(x)]

(x0,u0)6= 0.

Note the relative degree of a nonlinear system is a local concept defined about the point (x0, u0) and also depends on the domain of control. This dependence is

a result of the non-affinity of the system. Next a lemma is derived that will help determine the relative degree of Σ2.

Lemma 3.3. Origin belongs to the set Ω1 given in Definition 3.2.2.

Proof. Consider the open-loop system Σ2. The necessary condition for passivity with

positive definite storage function is

Lf0V (x) ≤ 0.

This indicates that the system is stable in the Lyapunov sense. Further, by Laselle’s theorem [64] it is known that the state of this open-loop system will enter the set {x ∈ Rn _{: L}

f0V (x) = 0}. This is exactly the set Ω1 in Definition 3.2.2. This result

Further, the set Ω1 contains the invariant sets of the system. Since origin is the

fixed-point of the system Σ1, it is concluded that it belongs to the set Ω1. This

completes the proof.

The next theorem analyzes the relative degree of the passive system Σ2.

Theorem 3.4. Suppose Σ2 is passive with a C2 storage function V which is positive

definite. If g0(0) and _∂x∂h(0) have full rank, then Σ2 has relative degree (1, 1, . . . , 1)

at (x = 0, u = 0).

Proof. The relative degree of Σ2 is one if

∂ ˙y ∂u (0, 0) is non-singular, or ∂ ˙y ∂u(0, 0) = ( ∂h ∂xg0(x) + ∂ ∂u " ∂h ∂x " _m X i=1 uiRi(x, u) # u #) (0, 0) = ∂h ∂xg0(0) (3.14) = Lg0h(0)

are m × m and non-singular. The above relations are obtained by using the smooth property of the vector fields. Hence conditions for which (3.14) holds true need to be determined. This is carried out in the following two steps.

Firstly, since Σ2 is passive, it satisfies the necessary conditions given in Defini-

tion 3.2.2. But property (ii) in Definition 3.2.2 is defined only for set Ω1. Hence the

first step in the proof is to show that origin belongs to this set. This has been shown in Lemma 3.3. Thus, from property (ii) of Definition 3.2.2

∂ ∂x  gT₀(x)∂V ∂x  g0(x) = ∂h ∂xg0(x) (3.15)

is satisfied at x = 0. Differentiating and using the fact ∂V

g₀T(0)∂

2_V

∂x2(0)g0(0) =

∂h

∂xg0(0). (3.16)

The rest of the proof proceeds similar to Proposition 2.44 given in [63]. The Hessian

∂2_V

∂x2(0) is symmetric positive definite by properties of the storage function and can

be factored as RT_{R with some matrix R. Then,}

gT₀(0)RTR(0)g0(0) =

∂h

∂xg0(0). (3.17)

Since ∂h_∂x(0) = gT

0(0)RTR(0) is assumed to be full rank, Rg0(0) has full rank. Hence

it is concluded that ∂h_∂xg0(0) is m × m and full rank. This completes the proof.

Remark 3.3.1. For an affine system, the conditions of Definition 3.2.2 are satisfied for all control inputs. Since the relative degree for an affine system does not depend on input, Theorem 3.4 consequently reduces to Proposition 2.44 [63].

The next result examines the nature of the zero dynamics of Σ2.

Theorem 3.5. Suppose Σ2 is passive with a C2 storage function V which is positive

definite. If g0(0) and ∂h_∂x(0) have full rank, then zero dynamics of Σ2 locally exist

about (x = 0, u = 0) and is weakly minimum phase.

Proof. From Theorem 3.4, Σ2has a well-defined relative degree and local zero dynam-

ics exist. Let the set Ω2 = {x ∈ Rn : h(x) = 0} define the points on the zero-output

manifold. By definition of Σ2 this set contains the origin. By Lemma 3.3 origin is

also contained in the set Ω1. Thus, in order to study the local nature of the zero

dynamics about the origin, only those state trajectories that fall in the intersection set Ω2T Ω1 need to be considered. On these set of points properties (i) through (ii)

˙ V = Lf (x,u)V = Lf0V + Lg0V u + u T_L R(x,u)V u (3.18) = uTLR(x,u)V u.

By Definition 3.2.1, for passive systems ˙V ≤ yT_{u. Furthermore, this condition}

becomes ˙V ≤ 0 on the set Ω2T Ω1. This inference along with condition (3.18) implies

that the origin is Lyapunov stable and hence zero dynamics is weakly minimum phase. This completes the proof.

Theorems 3.4 and 3.5 together give the necessary conditions for feedback equiv- alence to a passive system. This result is summarized by the following theorem. Theorem 3.6. Suppose g0(0) and ∂h_∂x(0) have full rank. The necessary conditions

for transforming Σ2 into a passive system with C2 positive definite storage function

V using static state-feedback compensation are: (i) Σ2 has relative degree {1, 1, . . . , 1} and

(ii) is weakly minimum phase

Proof. From Theorem 3.4 and Theorem 3.5 it is known that the resulting system will have relative degree (1, 1, . . .) with weakly minimum phase zero dynamics. Further, it is well understood that relative degree and zero dynamics are invariant under static feedback [66][Lemma 2.4]. Hence the conditions in the proof follow.

Theorem 3.6 extends the powerful feedback equivalence approach to general non- linear systems. It provides necessary conditions for a system to be made passive by feedback under mild restrictions. The equivalent theorem for affine systems de- rived in [58] shows that Theorem 3.6 is also sufficient for feedback passivity. But

the topological and nonlinear nature of non-affine systems hinders this result to be sufficient.