On the Practical Output Feedback Stabilization for Nonlinear Uncertain Systems

In this paper, we treat the problem of output feedback stabilization of nonlinear uncertain systems. We propose an output feedback controller that guarantees global uniform practical stability of the closed loop system.


Introduction
The problem of stabilization for uncertain systems has been widely investigated for many years [1][2][3][4][5][6][7][8][9][10][11].In these studies, the origin was not supposed to be an equilibrium point of the uncertain system.So we can no longer expect to design a controller that guarantee the stability of the origin as an equilibrium point.In [1], a class of state feedback controls is proposed in order to guarantee uniform ultimate boundedness of every system response within an arbitrarily small neighborhood of the zero state.[5], [9] and [6] presented controllers that guarantee exponential stability of a ball containing the origin of the state space and the radius of this ball can be made arbitrary small.In order to study uncertain dynamical systems, the authors in [12] introduced the notion of input to state practical stability.In [13], the concept of input to state practical stability is extended to stochastic case and an output feedback controller is proposed for a class of stochastic nonlinear systems with uncertain nonlinear functions.
Most of the recent nonlinear controllers are designed for an uncertain system that has a nominal linear part and the controller is designed based on the knowledge of the upper bound, possibly time varying and state dependent, of the uncertainties vector norm.Another class of uncertain systems which has also received considerable attention, namely systems with nominal part which is affine in the control.Such a class of systems is important because it may represent many physical systems.In [2,8,10] authors investigated the state feedback stabilization problem for these systems.In this paper, we will synthesis an output feedback controller for this class of systems.It should be noted that output feedback stabilization problem for uncertain system with linear nominal part has been discussed in ( [3, 4, 7, 11]).Under the assumption that the uncertain part is bounded by a known function that depends only on the output, they construct an output feedback controller that guarantees global exponential stability of the closed loop system.Here, we will suppose that the unknown part is bounded by a function that depends on the input and the output.We will design an output feedback controller that guarantees global uniform practical stability of the closed loop system.In Section 2, we recall the definition of global uniform practical stability and we give a sufficient condition to assure it.In Section 3, we state the main result.Throughout this paper .denotes the Euclidean norm of R n .

Practical stability
Consider a system described by with t ∈ R + is the time and x ∈ R n is the state.As a first step, we need to recall what is meant by global uniform practical stability of (1).For r ≥ 0, denote B r = {x ∈ R n / x ≤ r}.
Definition 1.The system (1) is said globally uniformly practically stable if there exists r ≥ 0 such that: (i) for all ε > r, there exists δ = δ(ε) > 0 such that, for all t 0 ≥ 0, (ii) for all ε > r and c > 0, there exists T (ε, c) > 0 such that, for all t 0 ≥ 0, The origin may not be an equilibrium point of the system (1).But, since the B r ball is an attractor and 0 ∈ B r , it follows that at least zero cannot be a globally unstable equilibrium provided that zero is an equilibrium but it could be a locally unstable equilibrium, even when the ball is an attractor for r > 0. If system (1) satisfies the requirements of Definition 1 with r = 0, then it is globally uniformly asymptotically stable.
The following theorem gives sufficient conditions to assure global practical stability.Its proof can be deduced from [1].It uses the following comparison function definitions.
) and a small positive real number ̺ such that the following inequalities hold for all t ∈ R + and x ∈ R n .
Remark 1.Consider a control system having x as state and u as input: where f is a smooth function.This system is said to be input to state practical stable (see [12]) if there exist a function β of class KL (ie β : [0, +∞[×[0, +∞[ → [0, +∞[, such that for each fixed t, the function β(., t) is of class K and for each fixed s, the function β(s, .) is non increasing and tends to zero at infty), a function γ of class K and a nonnegative constant d such that for any initial condition x(0) and each measurable essentially bounded control u the associated solution x(t) exists and satisfies It is clear that if system (Σ) is input to state practical stable then it is globally uniformly practically stable in the sense of Definition 1 when u = 0.

Output feedback stabilization
Throughout this paper, we deal with uncertain dynamical systems described by where t ∈ R + is the time, x(t) ∈ R n is the state, u(t) ∈ R m is the control input and y ∈ R p is the output.f : R + × R n → R n , g : R + × R n → R n×m and h : R + × R n → R p are known functions satisfying f (t, 0) = 0 and h(t, 0) = 0 for all t ∈ R + .f and g are supposed to be locally Lipschitz in x and continuous in t.The function ξ : R + ×R n ×R m → R m represents uncertainties in the plant.The nominal system corresponding to system (2) is given by Our aim is to design an output feedback controller such that system (2) is globally practically stable.We consider the following assumptions pertaining to system (2).
(A1) There exist nonnegative real scalar functions ρ 1 (., .),ρ 2 (., .), with In order to design an output feedback variable structure control, the author in [14] consider uncertain systems with linear nominal part and such that the unknown function satisfies assumption (A1).
We will consider the problem of choosing an output feedback u(t, y) such that, for all uncertainties satisfying the assumption (A1), system (2) is globally uniformly practically stable.We will assume that the nominal system (3) is globally asymptotically stabilizable.Indeed, we suppose that the assumptions below are fulfilled.
We have the following result.
Theorem 2. Consider an uncertain system described by (2) satisfying assumptions (A1), (A2) and (A3) and subject to the control given by (7).Then the resulting closed-loop system is globally uniformly practically stable.
Proof.First, note that controller (7) satisfies u(t, y) ≤ u 0 (t, y) + u 1 (t, y) + u 2 (t, y) We will use the function V as a Lyapunov function candidate for the closed-loop system.Its derivative along the trajectories of ( 2) is given by Taking into account assumption (A1) we have By assumption (A3) and ( 10) we have On the one hand, using ( 8) and the fact that we get ϕ T (t, y)u 1 (t, y) + ρ 1 (t, y) 1 + θ 1 (t, y) u 0 (t, y) ϕ(t, y) ≤ ε 1 .
On the other hand, using (9) and the fact that we get So, by assumption (A2), ( 11) and ( 12) we obtain the following upper bound on we deduce by Theorem 1 that the closed loop system is globally practically stable.

Remark 2. Since
That is the controller (7) insures that solutions of the closed loop system converge towards an arbitrarily small neighborhood of the origin.Remark 3. It should be remarked that when the nominal system is linear, i.e. f (t, x) = Ax, g(t, x) = B and h(t, x) = Cx, the result of Theorem 2 will recover that of [7].
In the rest of this section, we give a class nominal systems satisfying assumptions (A2) and (A3).We consider systems (2) with nominal autonomous part described by where x ∈ R n , u, y ∈ R m , f and the m columns of g are smooth, f (0) = 0 and h(0) = 0. We will suppose that the uncertainties satisfy assumption (A1).For the nominal system (13) we will make the following assumption.
Here, we consider the output feedback stabilization problem for uncertain system (2) with nominal autonomous part satisfying assumptions (A1) and (A4).We propose the following controller where where V is the Lyapunov function given by assumption (A4) and φ is any nonlinearity satisfying condition (14).
We have the following corollary of Theorem 2.
Corollary 1.Consider an uncertain system described by (2) satisfying assumptions (A1) and (A4) and subject to the control given by (15).Then the resulting closed-loop system is globally practically stable.

Example
As an example of systems satisfying assumptions (A1)-(A3), let us consider the following planar system This system is of the form (2) with We note that assumption (A1) is fulfilled with it is shown in [16] that the nominal system in closed loop with the controller u 0 (t, y) = −y(t) = −x 2 (t) is globally uniformly exponentially stable.It is readily seen that assumption (A2) is satisfied with Thus assumption (A3) is also fulfilled with ϕ(t, y) = 2 1 + e −2t y.

Conclusion
In this paper, the problem of output feedback stabilization for nonlinear uncertain systems with nominal part that is affine in the control is investigated.A controller that assures global uniform practical stability of the closed-loop system is proposed, that is, the solutions of the closed-loop system converge towards an arbitrary small neighborhood of the origin.A special case of systems is also considered, namely systems with a strictly passive nominal part.