Output Feedback Stabilization for a Class of Uncertain High-Order Nonlinear Systems

We investigate output feedback stabilization for a class of high-order nonlinear systems whose output function and nonlinear terms are unknown. First, a smooth state feedback control law is designed by adding a power integrator technique. Next, we design a high-order observer to estimate the unmeasurable state, and allocate gains of the observer one by one in an iterative way. Finally, a dynamic output compensator is achieved such that the closed-loop system converges to the equilibrium point quick. Two examples are provided to demonstrate the effectiveness of the proposed method.


I. INTRODUCTION
It is challenge to investigate output feedback stabilization for nonlinear systems since it involves in observer designs. In order to avoid finite time escape, it is necessary to impose some restrictive conditions on the nonlinear terms when investigating output feedback control for nonlinear uncertain systems in [1]. Global output feedback stabilization for Lipschitz nonlinear systems is presented in [2].
When the nonlinear terms are not precisely known, a feedback domination method is presented, and it is shown that global exponential stabilization can be achieved under the linear growth condition with a priori knowledge of the growth rate in [3]. A homogeneous domination approach is established to handle higher-order nonlinearities, which the linear growth condition is relaxed in [4]. More recent works about global output feedback stabilization for nonlinear systems with uncertain growth and higher-order growth conditions can be found in [5]- [7].
Global output feedback stabilization for a class of homogeneous systems is presented in [8], [9]. For a class of high-order switched nonlinear systems, output-feedback control is also appeared in [10], [11]. For nonlinear systems with unknown output functions, global output feedback stabilization is explored in [12]- [14]. Adaptive output feedback tracking control for uncertain switched nonlinear systems with time delays is investigated in [15], [16]. For nonlinear The associate editor coordinating the review of this manuscript and approving it for publication was Bing Li . stochastic switching systems, two control schemes are presented in [17], [18]. By adding a power integrator technique, global stabilization of high-order lower-triangular systems is shown in [19]. Robust regulation is designed for a chain of power integrators perturbed in [20]. When the output function of the high-order nonlinear systems [8]- [11] is unknown, how to design global output feedback controller becomes much more challenge.
This paper investigates output feedback stabilization for a class of high-order nonlinear systems with unknown output function and nonlinear terms. First, a state feedback law is designed by employing backstepping method and adding a power integrator technique. Then, by an iterative manner, a new observer design is presented, which gains of the observer can be appropriately enlarged to make the error arbitrarily small. Finally, a dynamic output compensator is achieved such that the closed-loop system is globally asymptotically stable quick.
The main contributions of this paper are as follows: (1) a unified method is achieved to deal with high-order nonlinear systems with unknown output function and nonlinear terms; (2) compared with [8]- [11], we deal with a class of uncertain nonlinear systems with weaker constraints on output and nonlinear terms. A new dynamic high-order observer design method is proposed. Gains of the observer are assigned one-by-one by an iterative manner, and can be appropriately enlarged to make the error arbitrarily small; (3) the advantage of dynamic output compensator is that the closed-loop system converges to the equilibrium point quick.

II. SYSTEM DESCRIPTION AND SOME LEMMAS
Consider the nonlinear systeṁ where x = (x 1 , · · · , x n ) T ∈ R n , u ∈ R and y ∈ R are the system state, input and output, respectively, and p ≥ 1 is an odd integer, and h(·) : R → R with h(0) = 0 is unknown continuously differentiable, and the nonlinear terms The nonlinear function h is assumed satisfying Assumption 1 as follows: Assumption 1: There exist two known positive constants θ and θ such that Remark 1: Owing to use sensors in practice, the relationship between the sensor output and state x 1 of the system is always nonlinear, uncertain and time-varying. As shown in [14], the sensor output y is an uncertain nonlinear function of the real displacement x 1 of the working region. However, the derivative of the nonlinear function h(x 1 ) actually is bounded, which implies that (5) is a natural assumption. The simplest function satisfying (5) is a linear output h(x 1 ) = θx 1 with an unknown constant θ if the upper-bound and lower-bounder of θ are known. In addition, some nonlinear output functions are bounded, such as h(x 1 ) = 2x 1 +sin(x 1 ), satisfies Assumption 1 as well.

III. OUTPUT FEEDBACK STABILIZATION DESIGN AND STABILITY ANALYSIS
Under Assumptions 1-2, we will design a output feedback controller for the uncertain system (1)-(4). Theorem 1: If Assumptions 1-2 hold, then the dynamic output compensatorẋ = η(x, y),x ∈ R n u = u(x, y) (12) globally stabilizes system (1)-(4). Proof: The proof consists of three parts. Part 1: Consider state feedback control law design if the state is measurable.
Step 1: Consider the Lyapunov candidate function Under Assumption 1, we obtain the proof is given in the appendix. Under Assumption 2, it can be deduced that system (1)-(4) satisfies the growth conditions where λ i = 2iC ≥ 0, i = 1, · · · , n. With (14) and (15), the time derivative of V 1 along the trajectories of (1)-(4) for n = 1 iṡ Design a smooth state feedback control law where p is a non-negative constant, we geṫ Step 2: Consider the Lyapunov candidate function Let Under the coordinates transform (20), system (1)-(4) for n = 2 is transferred tȯ Using Lemma 4, it is easy to show With the help of Lemma 1, the time derivative of V 2 (·) along the trajectories of (21)-(22) iṡ Clearly, a smooth state feedback controller is chosen as follows Substituting (26) into (25), we havė Step k + 1: Suppose at step k, there exists a global change of coordinates with constants β 1 > 0, · · · , β k−1 > 0, transferring (1)-(4) for n = k into a system of the forṁ And the Lyapunov candidate function is and a smooth state feedback control law is We claim that (36) also holds at step k + 1. To prove the claim, we denote This, together with (31)-(33), yields the augmented systeṁ where Using Lemma 4, by (14), (15), (28)-(30), we can deduce Using (15), (35), (39) and (42), we have

Now, construct the Lyapunov candidate function as
it holdsV Using an almost identical argument as proceeded in step 2, the following inequality can be deduced from (44), (46), and using Lemma 1 Clearly, choose a smooth state feedback control law This completes the inductive step. The inductive argument shows that (36) is true for k = 2, · · · , n − 1.

IV. EXAMPLES
Example 1: Consider a second-order system given bẏ where 2 < d 1 ≤ 3, 0 < d 2 < 0.5 are unknown constants. The linearization of system (86)-(88) is given by which is uncontrollable and unobservable. By Theorem 1, the dynamic output compensator is designed as followṡ with a suitable choice of the parameters β 1 , β 2 , L 1 and L 2 .
In the simulation, d 1 = 2 + rand(1) and d 2 = 0.5rand(1), using Matlab software, we have β 1 = 1, β 2 = 2.2, L 1 = 36, and L 2 = 0.5. Example 2: Consider the electromechanical system in [12] as follows J is the rotor inertia, G is the gravity coefficient, K τ is the coefficient which characterises the electromechanical conversion of armature current to torque. K B is the backemf coefficient, B 0 is the coefficient of viscous friction at the joint, m is the link mass, L 0 is the link length, M 0 is the load mass, R 0 is the radius of the load, q(t) is the angular motor position, I (t) is the motor armature current and L is the armature inductance, R is the armature resistance, and V ε is the input control voltage.
As can be seen from Fig. 2, the proposed output dynamic compensator makes the closed-loop system converge to zero quick, and the time required is less than 1 second. However, as shown in Fig. 3, the closed-loop system does not converge to zero until 4 seconds under the control law design in [12].

V. CONCLUSION
We investigate a class of high-order nonlinear systems whose output function and nonlinear terms are unknown. First, a smooth state feedback control law is designed by adding a power integrator technique. Next, we design a high-order observer to estimate the unmeasurable state by iteratively allocating gains of the observer. Finally, a dynamic output compensator is achieved such that the closed-loop system is globally asymptotically stable. Two examples are provided to demonstrate the effectiveness of the proposed method.

APPENDIX PROOF OF THE INEQUALITY (14)
Since |x 1 | ≥ 0, from Assumption 1, we consider the definite integral of ∂h(x) ∂x from x = 0 to x = x 1 ≥ 0, then In the case when x 1 < 0, similar to A.1, we can obtain the following relation based on (5) with the initial condition h(0) = 0, it can be deduced that So we immediately obtain the inequality (A.3), which completes the proof.