Nonlinear General Integral Control Design via Equal Ratio Gain Technique

This paper proposes two kinds of nonlinear general integral controllers, that is, one is generic and another is practical, for a class of uncertain nonlinear system. By extending equal ratio gain technique to a canonical interval system matrix and using Lyapunov method, theorems to ensure regionally as well as semi-globally asymptotic stability are established in terms of some bounded information. Moreover, for the practical nonlinear integral controller, a real time method to evaluate the equal ratio coefficient is proposed such that its value can be chosen moderately. Theoretical analysis and simulation results demonstrated that not only nonlinear general integral control can effectively deal with the uncertain nonlinear system but also equal ratio gain technique is a powerful and practical tool to solve the 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 appears.However, nonlinear general integral control design is not trivial matter because it depends on not only the uncertain nonlinear actions and disturbances but also the nonlinear control actions.Therefore, it is of important significance to develop the design method for nonlinear general integral control.
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 perturba-tion technique and so on, 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 for a class of uncertain nonlinear system.
All these general integral controllers above constitute only a minute portion of general integral control, and therefore lack generalization.Moreover, in consideration of the complexity of nonlinear system, it is clear that we can not expect that a particular integral controller has the high control performance for all nonlinear system.Thus, the generalization of general integral controller naturally appears since for all nonlinear system, we can not enumerate all the categories of integral controllers with high control performance.It is not hard to know that this is a very valuable and challenging problem, and equal ratio gain technique can be used to deal with this trouble since it is a powerful and practical tool to solve the nonlinear control design problem.
Motivated by the cognition above, this paper proposes a generic nonlinear integral controller and a practical nonlinear integral controller for a class of uncertain nonlinear system.The main contributions are that: 1) By defining two function sets, the generalization of general integral controller is achieved; 2) 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; 3) Theorems to ensure regionally as well as semi-globally asymptotic stability is established in terms of some bounded information.Moreover, for the practical nonlinear integral controller, a real time method to evaluate the equal ratio coefficient is proposed such that its value can be chosen moderately.Theoretical analysis and simulation results demonstrated that not only nonlinear general integral control can effectively deal with the uncertain nonlinear system but also equal ratio gain technique is a powerful and practical tool to solve the control design problem of dynamics with the nonlinear and uncertain actions.
Throughout this paper, we use the notation λ to indicate the smallest and largest eigenvalues, respectively, of a symmetric positive define bounded matrix ( ) , 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 definition.Sections 3 and 4 present the generic and practical nonlinear integral controllers along with their design method, respectively.Example 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 a control law 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 = .
Assumption 2: Suppose that the functions ( ) , f x w and ( ) , g x w satisfy the following inequalities, ( ) ( ) , 0, x y x y x y z z x y z z u x y u x y u x y u y x y y x y hold for all ( ) denotes the set of all integrable functions, ( ) Where z is a point on the line segment connecting x to the origin.

Generic Nonlinear Integral Control
The generic nonlinear integral controller is given as, ( ) ( ) ( ) where ( ) Thus, substituting (7) into (1), obtain the augmented system, ( ) ( ) ( ) ( ) ( ) By Assumption 1 and choosing 1 α ε − to be large enough, and then setting 0 x =  and 0 x = of the system (8), obtain, ( ) ( ) ( ) Therefore, we ensure that there is a unique solution 0 σ , and then ( ) 0 0,σ is a unique equilibrium point of the closed-loop system (8) in the domain of interest.At the equilibrium point, 0 x = , irrespective of the value of w .Now, by Definition 1, 2 and ( ) µ σ can be written as, ( ) ( ) ( ) where ( ) = .Thus, substituting ( 9)-( 11) into (8), obtain, ( ) where Moreover, it is worthy to note that the function ( ) , g x w is integrated into α ε via a change of variable.This has not influence on the results if the inequality (4) holds and it can be taken as m g in the design.Therefore, it is omitted in all the following demonstrations.
For analyzing the stability of closed-loop system (12), we must ensure that the matrix A is Hurwitz for all This can be achieved by Routh's stability criterion.

Hurwitz Stability
Hurwitz stability of the matrix A can be achieved by Routh's stability criterion, which is motivated by the idea [10], as follows: Step 1: the polynomial of the matrix A with 1 By Routh's stability criterion, i α and j β can be chosen such that the polynomial (13) is Hurwitz for all α and j β are all large to zero, and then the nec- essary condition, that is, the coefficients of the polynomial (13) are all positive, is naturally satisfied.
Step 2: based on the gains i α , j β and Hurwitz stability condition to be obtained by Step 1, the maximums of α ε and β ε , that is, α ε * and β ε * , can be obtained, respectively.Since α ε and β ε interact, there exist innumerable α ε * and β ε * .Thus, two kinds of typical cases are interesting, that is, one is that α ε * is evaluated with 1 It is well known that Hurwitz stability condition is more and more complex as the order of the matrix A in- creases.Thus, for clearly illustrating the design method above, we consider two kinds of cases, that is, 2 n = and 3 n = , respectively, as follows: Case 1: for 2 n = , the polynomial (13) is, ( ) holds, and then the polynomial ( 14) is Hurwitz for all 0 α and 1 α are multiplied by 1 α ε − , and then substituting them into (15), obtain, ( ) By the inequality (16), obtain, ( ) β and 1 β are multiplied by 1 α ε − , and then substituting them into (15), obtain, ( ) For this sub-class, there are two kinds of cases: , and then by the inequality (17), obtain,  ) ( ) β and 1 β are all positive numbers, and the fol- lowing inequality, ( )( ) ( ) holds, and then the polynomial (18) is Hurwitz for all 0 α , 2 α and 1 α are multiplied by 1 α ε − , and then substituting them into (19), obtain, ( )( ) ( ) By the inequality (20), obtain, β and 1 β are multiplied by 1 α ε − , and then substituting them into (19), obtain,

(
)( ) ( ) For this sub-class, although the situation is complex, a moderate solution can still be obtained, that is, ( ) From the demonstration above, it is obvious that for 2 n = , 3 n = and 1 Therefore, for the high order matrix A , the same result can be still obtained with the help of computer.Thus, we can conclude that the n+1-order matrix A can be designed to be Hurwitz for all < < .Discussion 1: From the statements above, it is easy to see that: 1) the system matrix A is an interval matrix; 2) in consideration of the controllable canonical form of linear system, the system matrix A can be called as the controllable canonical interval system matrix; 3) although Theorem 1 is demonstrated by the single variable system matrix A , it is easy to extend Theorem 1 to the multiple variable case since Routh's stability criterion applies to not only the single variable system matrix but also the multiple variable one.Thus, the following proposition can be established.
Proposition 1: 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.

Closed-Loop Stability Analysis
The matrix A can be designed to be Hurwitz for all 0 Thus, by linear system theory, if the matrix A is Hurwitz, and then for any given positive define symmetric matrix Q , there exists positive define symmetric matrix P that satisfies Lyapunov equation Now, using the inequalities (3), ( 5) and ( 6), obtain, where x f κ is a positive constant.Substituting (23) into (22), and using x η ≤ , obtain, By proposition proposed by [10], that is, 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, we have, 1) 0 Although there is innumerable n P , the maximum n M P exists and 0 .Therefore, we have ( ) Using the fact that Lyapunov function ( ) V η is a positive define function and its time derivative is a nega- tive define function if the inequality (25) holds, we conclude that the closed-loop system (12) is stable.In fact, By invoking LaSalle's invariance principle, it is easy to know that the closed-loop system (12) is exponentially stable.As a result, the following theorem can be established.
Theorem 2: Under Assumptions 1 and 2, if the system matrix A is Hurwitz for all and then the equilibrium points 0 x = and 0 σ σ = of the closed-loop system (12) is an exponentially stable for all Moreover, if all assumptions hold globally, then it is globally exponentially stable.Remark 1: From the statements of Subsections 3.1 and 3.2, it is to see that: by extending equal ratio gain technique to a canonical interval system matrix and using Lyapunov method, the asymptotic stability of the uncertain nonlinear system with generic nonlinear integral control can be ensured in terms of some bounded information.This shows that not only nonlinear general integral control can effectively deal with the uncertain nonlinear system but also equal ratio gain technique is a powerful tool to solve the control design problem of dynamics with the nonlinear and uncertain actions.
Discussion 2: From the statements above, it is obvious that: although the generalization of nonlinear general integral control is achieved by defining two function sets, there are two unavoidable drawbacks, that is, one is that the controller ( 7) is too generic such that it is shortage of pertinence; another is that it is difficulty to obtain the less conservative * * α = such that it is shortage of practicability.All these mean that Theorem 2 has only theoretical significance and not practical significance.Therefore, a practical nonlinear integral controller along with a new design method is proposed to solve these troubles in the next Section.

Practical Nonlinear Integral Control
For making up the shortage indicated by Discussion 2, a practical nonlinear integral controller is given as, where ( ) . Assumptions 3: By the definition of the controller (26), it is convenient to suppose that the following inequalities, ( ) ( ) hold for all where is an 1 1 n + × matrix, all its elements are equal to zero except for By the design method proposed by Subsection 3.1, the system matrix A can be designed to be Hurwitz for all ( )  .Thus, by linear system theory, there exists positive define symmetric matrix P that satisfies Lyapunov equation where Now, using the inequalities (4), ( 5), ( 6), ( 27) and ( 28), obtain, where f η κ is a positive constant.
Substituting (31) into (30), By proposition proposed by [10] (details see Subsection 3.2), for any moment t , there exists ( ) holds for all ( ) ( ) to know that the closed-loop system (29) is uniformly exponentially stable.As a result, we have the following theorem.
Theorem 3: Under Assumptions 1, 2 and 3, if the system matrix A is Hurwitz for all ( ) and then the equilibrium point 0 x = and 0 σ σ = of the closed-loop system (29) is uniformly exponentially stable for all ( ) ( ) Moreover, if all assumptions hold globally, and then it is globally uniformly exponentially stable.Now, the design task is to provide a method to evaluate the instantaneous value ( ) To achieve this objective, the procedure can be summarized as follows: Firstly, by the definitions of ( ) Secondly, by the inequality (33), the impermissible minimum of ( ) Finally, by the limitation conditions, ( ) and the iterative method to solve Lyapunov equation, ( ) ( ) ( ) ( ) ( ) 6.68 β ≤ < and 2 1 3 β < ≤ , and then the closed-loop system can be written as, The normal parameters are  .This shows that the closed-loop system is uniformly asymptotic stable and the equal ratio coefficient can be used to improve the conservatism.2) The system responses are almost identical before the additive impulse-like disturbance appears.This means that by equal ratio gain technique, we can tune a nonlinear general integral controller with good robustness and high control performance.All these demonstrate that not only nonlinear general integral control can effectively deal with the uncertain nonlinear system but also equal ratio gain technique is a powerful and practical tool to solve the control design problem of dynamics with the nonlinear and uncertain actions.

Conclusions
This paper proposes a generic nonlinear integral controller and a practical nonlinear integral controller for a class of uncertain nonlinear system.The main contributions are that: 1) By defining two function sets, the generalization of general integral controller is achieved; 2) 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; 3) Theorems to ensure regionally as well as semi-globally asymptotic stability are established in terms of some bounded information.Moreover, for the practical nonlinear integral controller, a real time method to evaluate the equal ratio coefficient is proposed such that its value can be chosen moderately.
Theoretical analysis and simulation results demonstrated that not only nonlinear general integral control can effectively deal with the uncertain nonlinear system but also equal ratio gain technique is a powerful and practical tool to solve the control design problem of dynamics with the nonlinear and uncertain actions.
a vector of unknown constant parameters and disturbances.The function ( ) , f x w is the uncertain nonlinear action, and the uncertain nonlinear function ( ) , g x w is continuous in ( ) , x w on the control domain

,
u x σ and ( ) v x belong to the function sets u F and v F , respectively, by α ε * and β ε * obtained by Step 2, check Hurwitz stability of the matrix A for all 0 it does not hold, redesign i α and j β and repeat the previous steps until the matrix A is Hurwitz for all 0 and then by the inequality (17), obtain, α ε * = ∞ Case 2: for 3 n = , the polynomial (13) is,

T
PA A P Q + = − .Therefore, the solution of Lyapunov equation[11] is, candidate, and then its time derivative along the trajectories of the closed-loop systems (12) is, ( are utilized to harmonize the actions of i x in the controller and integrator, respectively.( ) reorganize the integrator output.σ α , α ε and β ε are all positive constants, and constants.By the same way as Section 3, we have, and ( ) µ σ are integrated into α ε and i β , respectively.

T
any given positive define symmetric matrix Q .Therefore, we can utilize ( ) T V P η η η = as Lyapunov function candidate, and then its time derivative along the trajectories of the closed-loop system (29) is, ( , and then we conclude that ( ) 0V η ≤  holds uniformly in t .Using the fact that Lyapunov function ( )V η is a positive define function and its time derivative is a nega- tive define function if the inequality (33) holds for all [ ) 0, t ∈ ∞ , we conclude that the closed-loop system (29) is stable.In fact, ( ) 0 invoking LaSalle's invariance principle, it is easy and in the perturbed case, b and c are reduced to 1 and 5, respectively, corresponding to double the mass.Thus, we have and then the matrix A is Hurwitz for all 0 value, the simulation is implemented under the normal and perturbed cases, respectively.Moreover, in the perturbed case, we consider an additive impulse-like disturbance ( ) d t of magni- tude 60 acting on the system input between 15 s and 16 s.

Figure 1 and
Figure2showed the simulation results under normal (solid line) and perturbed (dashed line) cases.The following observations can be made: 1) The instantaneous value

Figure 2 .
Figure 2. System output under normal (solid line) and perturbed case (dashed line).
From the statements above, it is easy to see that: 1) all the component of the nonlinear integral controller (26) have the clear actions; 2) f