Output Feedback Nonlinear General Integral Control

This paper proposes an output feedback nonlinear general integral controller for a class of uncertain nonlinear system. By solving Lyapunov equation, we demonstrate a new proposition on Equal ratio gain technique. By using Equal ratio gain technique, Singular perturbation technique and Lyapunov method, theorem to ensure regionally as well as semi-globally exponential stability is established in terms of some bounded information. Moreover, a real time method to evaluate the ratio coefficients of controller and observer are proposed such that their values can be chosen moderately. Theoretical analysis and simulation results show that not only output feedback nonlinear general integral control has the striking robustness but also the organic combination of Equal ratio gain technique and Singular perturbation technique constitutes a powerful tool to solve the output feedback control design problem of dynamics with the nonlinear and uncertain actions.


Introduction
Integral control [1] plays an important role in practice because it ensures asymptotic tracking and disturbance rejection when exogenous signals are constants or planting parametric uncertainties appear.However, output feedback nonlinear general integral control design is not a trivial matter because it depends on not only the uncertain nonlinear actions, disturbances and nonlinear control actions but also the uncertain estimation error dynamics.Therefore, it is of important significance to develop the design method for output feedback nonlinear general integral control since some states cannot be measured in practice.
For general integral control design, there were various design methods, such as general integral control design based on linear system theory, sliding mode technique, feedback linearization technique and singular perturbation technique and so on, which were presented by [2]- [5], respectively.In addition, general concave integral control [6], general convex integral control [7], constructive general bounded integral control [8] and the generalization of the integrator and integral control action [9] were all developed by using Lyapunov method and resorting to a known stable control law.Equal ratio gain technique firstly was proposed by [10] and was used to address the linear general integral control design.After that Equal ratio gain technique was extended to the canonical interval system matrix [11] and was used to deal with nonlinear general integral control design.All these design methods and general integral controls above are all based on the state feedback.Presently, output feedback general integral control along with its design method has not been developed.
Motivated by the cognition above, this paper proposes an output feedback nonlinear general integral controller for a class of uncertain nonlinear system.The main contributions are that: 1) as any row integrator and its controller gains of a canonical interval system matrix tend to infinity with the same ratio, if it is always Hurwitz, and then the same row solutions of Lyapunov equation all tend to zero; 2) theorem to ensure regionally as well as semi-globally exponential stability is established in terms of some bounded information; 3) a real time method to evaluate the ratio coefficients of controller and observer are proposed such that their values can be chosen moderately.Moreover, theoretical analysis and simulation results show that not only output feedback nonlinear general integral control has the striking robustness but also the organic combination of Equal ratio gain technique and Singular perturbation technique constitutes a powerful tool to solve the output feedback control design problem of dynamics with the nonlinear and uncertain actions.
Throughout this paper, we use the notation , and that of matrix A is defined as the corresponding induced norm ( ) The remainder of the paper is organized as follows: Section 2 describes the system under consideration, assumption and output feedback nonlinear general integral control.Section 3 demonstrates a new proposition on Equal ratio gain technique.Section 4 addresses the design method.Examples and simulation are provided in Section 5. Conclusions are presented in Section 6.

Problem Formulation
Consider the following controllable nonlinear system, ( ) ( ) where ⊂ × .We want to design an output feedback control law u such that ( ) Assumption 1: There is a unique pair ( ) 0 0,u that satisfies the equation, so that 0 x = is the desired equilibrium point and 0 u is the steady-state control that is needed to maintain equilibrium at 0 x = , irrespective of the value of w .
Assumption 2: Suppose that the functions ( ) , f x w and ( ) , g x w satisfy the following inequalities, ( ) ( ) , 0, for all γ are all positive constants.For the purpose of this paper, it is convenient to introduce the following definition.Definition 1: ∈ denotes the set of all continuous differential increasing function [12], ( ) , such that ( ) where  stands for the absolute value.Figure 1 depicts the example curves for the functions belonging to the function set F Φ .For instance, for all x R ∈ , the functions, ( ) sinh x , ax and so on, all belong to function set F Φ .
The output feedback nonlinear general integral controller [11] and observer [12] are given as, ) where ˆn x R ∈ is the estimated state; ˆl w R ∈ is the prescient constant parameters and disturbances; µ , ε , σ α and j h ( ) ( ) are the slopes of the line segment connecting ˆi x to the origin ( ) ( ) Φ  belongs to the function set F Φ .Assumptions 3: By the definition of controller (7), it is convenient to suppose that the following inequalities, ( ) ( ) ( ) ( ) , 0, , ( ) ( ) hold for all ( )  , the controller (7) can be written as,
, and substituting ( 14) and ( 15) into ( 12) and ( 13), respectively, the whole closed-loop system can be rewritten as, , 0, f x w g x w x f w g x w g x w g w f w g w and ( ) By Assumptions 2 and 3, the uncertain terms 1 δ , 2 δ , 3 δ , 1 ∆ and 2 ∆ satisfy the linear growth bound, ( )

Propositions on Equal Ratio Gain Technique
Equal ratio gain technique is firstly proposed by [10] and is extended to the canonical interval system matrix in [11].For analyzing the stability of the closed-loop system (16), it is necessary to review two important propositions on Equal ratio gain technique as follows.
Proposition 1 [10]: as any row controller gains, or controller and its integrator gains of a canonical system matrix tend to infinity with the same ratio, if it is always Hurwitz, and then the same row solutions of Lyapunov equation all tend to zero.
Proposition 2 [11]: a canonical interval system matrix can be designed to be Hurwitz as any row controller gains, or controller and its integrator gains increase with the same ratio.
Based on two Propositions above, it is not enough to analyze the stability of the closed-loop system (16).So, a new proposition on Equal ratio gain technique is demonstrated in the next two subsections.

New Proposition
Consider the following controllable canonical interval system matrix A, and µ is a positive constant.
By Proposition 2, the interval system matrix A can be designed to be Hurwitz for all 0 and 0 µ µ * < < .Thus, for any given positive define symmetric matrix Q there exists a unique positive define symmetric matrix P that satisfies Lyapunov equation , and the solution of Lyapunov equation can be obtained by skew symmetric matrix approach [13], that is,

The inversion of the matrix
where the elements * are omitted since it is useless to achieve our object.The interesting reader can evaluate them by It is well known that the solution P of Lyapunov equation is more and more complex as the order of the system matrix A increases.Therefore, for clearly showing the results, we consider a simple case, that is, taking where ) ( ) ) and then we have,  ( ) ( ) It is obvious that 12 s µ , 23 s µ and 13 s µ all tend to the constants as 0 µ → , and then we have, B. S. Liu .From the statements above, it is easy to see that for 2 n = of the system matrix A, 3 P can be formulated as the linear form on µ and tends to zero as 0 µ → .Moreover, the solution of the matrix S is more and more complex as the order of the system matrix A increases.Thus, by the inversion of system matrix 1 A − (22), 1 n P + can be formulated as the linear form on µ for the 1 n + -order system matrix A, and with the help of computer, it can be verified the solution of 1 n P µ + still tends to the constant as 0 µ → .Therefore, for the 1 n + -order system matrix A, we can conclude that 1 0 n P + → as 0 µ → .As a result, the following theorem can be established.
Theorem 1: If the interval system matrix A is Hurwitz for all 0 < < , and then we have, Discussion 1: From the statements above, the solution of the matrix S is more and more complex as the order of the system matrix A increases.So, although Theorem 1 is demonstrated by taking Q I = and the single variable system matrix A, it is very easy to extend Theorem 1 to any given positive define symmetric matrix Q and the multiple variable system matrix A with the help of computer since there is not any difficulty to obtain the solution of the matrix S in theory, that is, Lyapunov equation applies to not only the single system matrix but also the multiple system matrix.Thus, there is the following proposition.
Proposition 3: as any row integrator and its controller gains of a canonical interval system matrix tend to infinity with the same ratio, if it is always Hurwitz, and then the same row solutions of Lyapunov equation all tend to zero.

Example
For testifying the justification of Theorem 1 and Proposition 3, we consider a 6-order two variable system matrix A as follows, A The inversion of the system matrix A is,   , it is very easy to obtain the fifteen linear equations with fifteen elements of the matrix S. So, it is omitted.
Thus, taking 0 0 0 0 1 0 0 0 0 0 1 8 2 0 0 3 1 2 7 0 0 1 5 . Now, by Routh's stability criterion and with the help of computer, we have: 1) if are all decrease as µ reduces; 2) although the result above is obtained by a constant sys- tem matrix, it is easy to be extended to the interval system matrix.This not only verifies the justification of Theorem 1 and Proposition 3 but also shows that for the high order and multiple variable system matrix, it is convenient and practical with the help of computer.

Stability Analysis
The asymptotic stability of the closed-loop system (16) can be achieved by Equal ratio gain technique and Singular perturbation technique as follows: By Proposition 2 [11], the interval system matrix z A can be designed to be Hurwitz for all and by choosing j h ( ) , the matrix A η can be designed to be Hurwitz, too.Thus, by linear system theory, two quadratic Lyapunov functions, with any given positive define symmetric matrices z Q and Q η , respectively.
Using ( ) ( ) ( ) ( ) as Lyapunov function candidate, and then its time derivative along the trajectories of the closed-loop system (16) is, F z e and ( ) , where ( ) , 2 ) The right-hand side of the inequality ( 27) is a quadratic form, which is negative define when, ( 1 This is equivalent to, ( By the dependence of d ε on d , it is obvious that the maximum of d ε occurs at ( ) ] and is given by, Although zn P µ and 1 zn P µ + are dependent on x , they are fixed for any given moment t and all tend to the constants as 0 µ → , and then there exists  By invoking LaSalle's invariance principle, it is easy to know that the closed-loop system (16) is uniformly exponentially stable.As a result, we have the following theorem.
Theorem 2: Under Assumptions 1, 2 and 3, if the matrix A η is Hurwitz and the interval system matrix z A is Hurwitz for all 0 µ µ * < < , 0    Discussion 2: From the procedure of stability analysis above, it is obvious that: although ( ) can be chosen arbitrarily small.Thus, so long as the bounded conditions (17) -( 21) are satisfied, the asymptotically stable control can be achieved.This shows that the striking feature of output feedback nonlinear general integral control, that is, its robustness with respect to the nonlinearities, uncertainties and disturbances from the real system, control input and estimated error dynamics, is clearly demonstrated by Equal ratio gain technique and Singular perturbation technique.This means that the organic combination of Equal ratio gain technique and Singular perturbation technique constitutes a powerful tool to solve the output feedback control design problem of dynamics with the nonlinear and uncertain actions. is the steady-state control that is needed to maintain equilibrium at the origin.
The nonlinear general integral controller and the integral observer can be given as, Thus, it is easy to obtain  .This shows that the closed-loop system is uniformly asymptotic stable.2) the optimum responses are almost identical before the additive impulse-like disturbance appears.This means that by Equal ratio gain technique and Singular perturbation technique, we can tune an output feedback nonlinear general integral controller with good robustness and high control performance.All these demonstrate that output feedback nonlinear general integral control has the striking robustness, that is, so long as the bounded conditions are satisfied, the asymptotically stable control can be achieved, but also the organic combination of Equal ratio gain technique and Singular perturbation technique constitutes a powerful and practical tool to solve the output feedback control design problem of dynamics with the nonlinear and uncertain actions.

Conclusions
This paper proposes an output feedback nonlinear general integral controller for a class of uncertain nonlinear system.The main contributions are that: 1) as any row integrator and its controller gains of a canonical interval system matrix tend to infinity with the same ratio, if it is always Hurwitz, and then the same row solutions of Lyapunov equation all tend to zero; 2) theorem to ensure regionally as well as semi-globally exponential stability is established in terms of some bounded information; 3) a real time method to evaluate the ratio coefficients of controller and observer are proposed such that their values can be chosen moderately.
Theoretical analysis and simulation results show that not only output feedback nonlinear general integral control has the striking robustness but also the organic combination of Equal ratio gain technique and Singular perturbation technique constitutes a powerful tool to solve the output feedback control design problem of dynamics with the nonlinear and uncertain actions.
smallest and largest eigenvalues, respectively, of a symmetric positive define bounded matrix ( ) a vector of unknown constant parameters and disturbances.The uncertain nonlinear functions ( )

Figure 1 .
Figure 1.Example curves for the functions belonging to the function set F Φ .

∆
are all positive constants.


holds uniformly in t .Using the fact that Lyapunov function ( ) , V z η is a positive define function and its time derivative is a nega- tive define function if 0 Λ > holds for all [ ) 0, t ∈ ∞ , we conclude that the closed-loop system (16) is stable.In fact, for given ε .
the iterative method to solve the inequality (30),

Figure 2 .
Figure 2. The values of 100ε under normal (solid line) and perturbed case (dashed line).

Figure 3 .
Figure 3. System output under normal (solid line) and perturbed case (dashed line).

Table 2 .
From the example above, it is obvious that: 1) as shown in

Table 1 .
Numerical Solutions of 3

Table 2 .
Numerical Solutions of 4