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In conjunction with linear general integral control, this paper proposes a fire-new control design technique, named Equal ratio gain technique, and then develops two kinds of control design methods, that is, Decomposition and Synthetic methods, for a class of uncertain nonlinear system. By Routh’s stability criterion, we demonstrate that a canonical system matrix can be designed to be always Hurwitz as any row controller gains, or controller and its integrator gains increase with the same ratio. By solving Lyapunov equation, we demonstrate that 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. By Equal ratio gain technique and Lyapunov method, theorems to ensure semi-globally asymptotic stability are established in terms of some bounded information. Moreover, the striking robustness of linear general integral control and PID control is clearly illustrated by Equal ratio gain technique. Theoretical analysis, design example and simulation results showed that Equal ratio gain technique is a powerful tool to solve the control design problem of uncertain nonlinear system.

The complexity of nonlinear system challenges us to come up with systematic design methods to meet control objectives and specifications. Faced with such challenge, it is clear that we can not expect a particular method to apply to all nonlinear systems [

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, presented by [

Based on Equal ratio gain technique, this paper develops two kinds of systematic methods to design linear general integral control for a class of uncertain nonlinear system, that is, one is Decomposition method and another is Synthetic method. The main contributions are as follows: 1) a canonical system matrix can be designed to be always Hurwitz as any row controller gains, or controller and its integrator gains increase with the same ratio; 2) 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; 3) theorems to ensure semi-globally asymptotic stability are established in terms of some bounded information. Moreover, the striking robustness of linear general integral control and PID control is clearly illustrated by Equal ratio gain technique. All these mean that Equal ratio gain technique is a powerful tool to solve the control design problem of uncertain nonlinear system, and then makes the engineers more easily design a stable controller. Consequently, Equal ratio gain technique has not only the important theoretical significance but also the broad application prospects.

Throughout this paper, we use the notation

The remainder of the paper is organized as follows: Section 2 demonstrates Equal ratio gain technique. Section 3 addresses the control design. Example and simulation are provided in Section 4. Conclusions are given in Section 5.

Consider the following

where

are viewed as the controller and integrator gains, respectively, and then the system matrix

For developing Equal ration gain technique, firstly, we must ensure that the system matrix

For

Step 1: the polynomial of the system matrix

By Routh’s stability criterion, the gains

Step 2: based on the gains

Step 3: by

The demonstration above is only a basic idea to ensure that the system matrix

Case 1: for

By Routh’s stability criterion, if

holds, and then the polynomial (2) is Hurwitz.

Sub-class 1:

By the inequality (4), 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 (6) is Hurwitz.

Sub-class 1:

By the inequality (8), 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

witz stability condition is more and more complex as the order of the system matrix

Theorem 1: There exist

Discussion 1: From the system matrix

Discussion 2: From the polynomial (1), it is not hard to see that that Hurwitz stability of the system matrix

Discussion 3: From the statements above, Hurwitz stability condition is more and more complex as the order of the system matrix

Proposition 1: A canonical system matrix can be designed to be always Hurwitz as any row controller gains, or controller and its integrator gains increase with the same ratio.

By Hurwitz stability condition given by Subsection 2.1, the system matrix

where

The inversion of the system matrix

where the elements

It is well known that the solution

where

and then we have,

Case 1:

It is obvious that

where

Case 2:

It is obvious that

where

From the statements above, it is easy to see that for

the solution of the matrix

by the inversion matrix

that

Theorem 2: If there exist the gains

1)

2)

where

Discussion 4: Theorem 1 and 2 are all obtained by multiplying the controller gains

gains

Discussion 5: From the statements above, the solution of the matrix

Proposition 2: 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.

For testifying the justification of Theorems 1 and 2, and Propositions 1 and 2, we consider a 6-order two variable system matrix

The polynomial of the system matrix

where

The inversion of the system matrix

By the equation

Thus, taking

and then by Routh’s stability criterion, the array of the coefficients of polynomial is,

where

Now, with the help of computer, we have: 1) if

22.43 | 2.72e−1 | 2.34e−2 | |

−2.24 | 6.53e−3 | 3.99e−4 | |

−0.75 | 7.50e−2 | 7.50e−3 | |

0.94 | 7.66e−2 | 7.40e−3 | |

10.86 | 1.34e−1 | 1.16e−2 | |

−3.95 | −4.43e−2 | −4.15e−3 |

−9.42 | −1.59e−1 | −1.49e−2 | |

5.20 | 2.46e−1 | 2.36e−2 | |

−0.84 | −6.66e−2 | −6.40e−3 | |

0.25 | 2.50e−2 | 2.50e−3 | |

−3.95 | −4.43e−2 | −4.15e−3 | |

5.19 | 2.25e−1 | 2.15e−2 |

22.43 | 2.74e−1 | 2.39e−2 | |

−2.24 | 9.96e−2 | 9.55e−3 | |

−0.75 | 7.50e−2 | 7.50e−3 | |

0.94 | 8.98e−2 | 8.57e−3 | |

10.86 | 1.37e−1 | 1.20e−2 | |

−3.95 | 2.85e−2 | 3.01e−3 |

−9.42 | 4.26e−3 | 1.33e−3 | |

5.20 | 3.15e−1 | 3.10e−2 | |

−0.84 | −7.98e−2 | −7.57e−3 | |

0.25 | 2.50e−2 | 2.5e−3 | |

−3.95 | 2.85e−2 | 3.01e−3 | |

5.19 | 2.40e−1 | 2.32e−2 |

From the example above, it is obvious that: 1) for all the six cases, there all exists

Consider the following controllable nonlinear system,

where

tions, and the functions

Assumption 1: There are two unique control inputs

so that

Assumption 2: No loss of generality, suppose that the functions

for all

For the system (10), we develop two kinds of methods to design linear general integral controllers, respectively, that is, Decomposition method and Synthetic method.

The control laws

where

Thus, substituting (21) and (22) into (10), obtain two augmented systems,

By Assumption 1 and choosing

In the same way, we have,

Thus, we ensure that there are two unique solutions

Substituting (25) into (23) and (26) into (24), and then the whole closed-loop system can be rewritten as,

where

and

Moreover, it is worthy to note that the functions

The matrices

can be obtained. Where

with any given positive define symmetric matrices

Thus, using

where

Now, using the inequalities (15)-(20), obtain,

where

Substituting (31) and (32) into (30), and using

where

By Theorems 1 and 2, and Propositions 1 and 2, obtain,

Therefore, there exist

Using the fact that Lyapunov function

negative define function if

Theorem 3: Under Assumptions 1 and 2, if there exist the gains

In the same way, for the case of

Theorem 4: Under Assumptions 1 and 2, if there exist the gains

The control laws

where

In the same way as Subsection 3.1, the closed-loop system can be rewritten as,

where

and

Moreover, by the same way as Subsection 3.1, the functions

The matrix

can be obtained. Where

Thus, using

where

and

Now, using the inequalities (15)-(20), obtain,

where

Substituting (38) and (39) into (37), and using

where

By Theorems 1 and 2, and Propositions 1 and 2, obtain,

Thus, there exist

holds for all

Using the fact that Lyapunov function

Theorem 5: Under Assumptions 1 and 2, if there exist the gains

for all for all

In the same way, for the case of

Theorem 6: Under Assumptions 1 and 2, if there exist the gains

Discussion 6: From Decomposition and Synthetic methods above, it is obvious that: 1) although they are developed with two variable systems, it is not hard to extend them to the multiple variable systems; 2) as the subsystems increase, Decomposition method is simpler and more practical than Synthetic method since we can design the controllers for every subsystems, respectively, and then combine them such that the whole closed-loop system is asymptotically stable; 3) for designing a high performance controller, Synthetic method is more excellent than Decomposition method since we can use all the state variables to design the controller and integrator.

Discussion 7: From the procedure of stability analysis above, it is obvious that so long as the bounded conditions (13)-(20) are satisfied, the asymptotically stable control can be achieved. This shows that the striking feature of linear general integral control, that is, its robustness with respect to

Discussion 8: Form all the statements of Sections 3 and 4, it is not hard to see that although Equal ratio gain technique is demonstrated by a class of special system and linear general integral control, its application is not limited in them and can be extend to solve the other relevant problem since Routh’s stability criterion, Lyapunov equation and Lyapunov method are all universal. For examples: 1) if the system is not given in the form (10), one can find a transformation matrix that takes the given system to this form if the system is controllable; 2) by combining Equal ratio gain technique with Feedback linearization technique, we can achieve the design of nonlinear integral controller; 3) as the integrator gains are equal to zero, the control is reduced to proportional control, and the similar conclusions can still be obtained.

Discussion 9: Although the design procedure above looks quite complicated, there need not abstruse theory since Routh’s stability criterion, Lyapunov equation and Lyapunov method are all simple enough to be presented in the text book and practical enough to have been used in the real-word problem. Therefore, Equal ratio gain technique has not only the important theoretical significance but also the broad application prospects.

Consider the pendulum system [

where

and then it can be verified that

Thus, the closed-loop system can be written as,

where

The normal parameters are

Now, if the gains are taken as

holds for all

By solving the Lyapunov equation

Thus, the asymptotical stability of the closed-loop system can be ensured for all

near system, and then makes the engineers more easily design a stable controller. Consequently, Equal ratio gain technique has not only the important theoretical significance but also the broad application prospects.

In conjunction with linear general integral control, this paper proposes a fire-new control design technique, named Equal ratio gain technique, and then develops two kinds of control design methods, that is, Decomposition and Synthetic methods, for a class of uncertain nonlinear system. The main conclusions are as follows: 1) a canonical system matrix can be designed to be always Hurwitz as any row controller gains, or controller and its integrator gains increase with the same ratio; 2) 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; 3) theorems to ensure semi-globally asymptotic stability are established in terms of some bounded information. Moreover, the striking robustness of linear general integral control and PID control is clearly illustrated by Equal ratio gain technique. All these mean that Equal ratio gain technique is a powerful tool to solve the control design problem of uncertain nonlinear system, and then makes the engineers more easily design a stable controller. Consequently, Equal ratio gain technique has not only the important theoretical significance but also the broad application prospects.

These conclusions above are further confirmed by the design example and simulation results.