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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.

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 perturbation technique and so on, were presented by [

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

defined as

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.

Consider the following controllable nonlinear system,

where

Assumption 1: There is a unique pair

so that

Assumption 2: Suppose that the functions

for all

Definition 1:

and

hold for all

Definition 2:

and

hold for all

The generic nonlinear integral controller is given as,

where

Thus, substituting (7) into (1), obtain the augmented system,

By Assumption 1 and choosing

Therefore, we ensure that there is a unique solution

Now, by Definition 1, 2 and

where

Thus, substituting (9)-(11) into (8), obtain,

where

And

Moreover, it is worthy to note that the function

For analyzing the stability of closed-loop system (12), we must ensure that the matrix

Hurwitz stability of the matrix

Step 1: the polynomial of the matrix

By Routh’s stability criterion,

Step 2: based on the gains

Step 3: by

It is well known that Hurwitz stability condition is more and more complex as the order of the matrix

Case 1: for

By Routh’s stability criterion, if

holds, and then the polynomial (14) is Hurwitz for all

Sub-class 1:

By the inequality (16), obtain,

Sub-class 2:

For this sub-class, there are two kinds of cases:

1) if

2) if

Case 2: for

By Routh’s stability criterion, if

holds, and then the polynomial (18) is Hurwitz for all

Sub-class 1:

By the inequality (20), obtain,

Sub-class 2:

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

Theorem 1: There exist

Discussion 1: From the statements above, it is easy to see that: 1) the system matrix

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.

The matrix

where

Thus, using

where

Now, using the inequalities (3), (5) and (6), obtain,

where

Substituting (23) into (22), and using

By proposition proposed by [

1)

2)

where

Although there is innumerable

holds for all

Using the fact that Lyapunov function

Theorem 2: Under Assumptions 1 and 2, if the system matrix

and then the equilibrium points

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

For making up the shortage indicated by Discussion 2, a practical nonlinear integral controller is given as,

where

the performance of integral control action.

output.

Assumptions 3: By the definition of the controller (26), it is convenient to suppose that the following inequalities,

hold for all

By the same way as Section 3, we have,

where

and the functions

By the design method proposed by Subsection 3.1, the system matrix

where

Now, using the inequalities (4), (5), (6), (27) and (28), obtain,

where

Substituting (31) into (30), obtain,

By proposition proposed by [

holds for all

the inequality (33) holds for all

Using the fact that Lyapunov function

is stable. In fact,

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

and then the equilibrium point

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

Firstly, by the definitions of

and

Secondly, by the inequality (33), the impermissible minimum of

Finally, by the limitation conditions,

and the iterative method to solve Lyapunov equation,

Discussion 3: 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)

uncertain nonlinear action of

or

Consider the pendulum system [

where

and then it can be verified that

By the practical nonlinear integral controller (26), the control law can be given as,

Thus, it is easy to obtain

where

and

The normal parameters are

and 5, respectively, corresponding to double the mass. Thus, we have

Now, with

holds for all

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.