Equal Ratio Gain Technique and Its Application in Linear General Integral Control

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.


Introduction
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 B. S. Liu apply to all nonlinear systems [1].Therefore, although there were Linearization techniques, Gain scheduling technique, Singular perturbation technique, feedback linearization technique, sliding mode technique and so on, nonlinear design tools, this paper still develops a new control design technique, named Equal ratio gain technique, in conjunction with linear general integral control since integral control plays an irreplaceable role in the control domain.
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 [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 Lyapunov method.For illustrating the practicability and validity of Equal ratio gain technique and the good robustness of linear general integral control, this paper addresses general integral control design again.
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 x is defined as , and that of matrix A is defined as the corresponding induced norm ( ) 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.

Equal Ratio Gain Technique
Consider the following 1 1 n n + × + system matrix A ,  , α ε and β ε are all positive constants.If 1 i α ε α − and 1 j β ε β − are viewed as the controller and integrator gains, respectively, and then the system matrix A is an 1 n + -order single input single output linear system matrix with linear general integral control.In consideration of the controllable canonical form of linear system, the system matrix A can be called as the controllable canonical form with linear general integral control.
For developing Equal ration gain technique, firstly, we must ensure that the system matrix A is Hurwitz for all 0 α α ε ε * < < and 0 β β ε ε * < < , secondly we need to solve the Lyapunov equation T PA A P Q + = − with any given positive define symmetric matrix Q to obtain the solution of the matrix P .

Hurwitz Stability
For 0 α α ε ε * < < and 0 β β ε ε * < < , Hurwitz stability of the system matrix A can be achieved by Routh's sta- bility criterion, as follows: Step 1: the polynomial of the system matrix A with 1 By Routh's stability criterion, the gains i α and j β can be chosen such that the polynomial (1) is Hurwitz.
Obviously, if i α and j β are all large to zero, and then the necessary condition, that is, the coefficients of the polynomial (1) are all positive, is naturally satisfied.
Step 2: based on the gains i α , j β and Hurwitz stability condition to be obtained by The demonstration above is only a basic idea to ensure that the system matrix A is Hurwitz.For clearly il- lustrating the 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 (1) is, ( ) By Routh's stability criterion, if 3 α , 2 α , 1 α , 2 β , and 1 β are all positive constants, and the following in- equality, ( ) holds, and then the polynomial (2) is Hurwitz.Sub-class 1: 3 α , 2 α and 1 α are multiplied by 1 α ε − , and then substituting them into (3), obtain, ( ) By the inequality (4), obtain, ( ) β and 1 β are multiplied by 1 α ε − , and then substituting them into (3), obtain, ( ) For this sub-class, there are two kinds of cases: and then by the inequality (5), obtain, holds, and then the polynomial ( 6) is Hurwitz.
α and 1 α are multiplied by 1 α ε − , and then substituting them into (7), obtain, ( )( ) ( ) By the inequality (8), obtain, ( )( ) ( ) β and 1 β are multiplied by 1 α ε − , and then substituting them into (7), 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 matrix A , there all exist α ε * such that the system matrix A is Hurwitz for all 0 α α ε ε * < < .Moreover, Hur- witz stability condition is more and more complex as the order of the system matrix A increases.Therefore, for the high order system matrix A , the same result can be still obtained with the help of computer.Thus, we can conclude that the 1 n + -order system matrix A can be designed to be Hurwitz for all 0 α α ε ε * < < and 0 β β ε ε * < < .As a result, the following theorem can be established.Theorem 1: There exist Discussion 1: From the system matrix A , the polynomial (2) and Hurwitz stability condition (3), it is not hard to see that: 1) when 2 0 β = , the control law is reduced to PID control; 2) for 2 0 β > , if 1 α is less than zero, and then Hurwitz stability condition can still be satisfied.However, for PID control, 1 0 α > is the neces- sary condition.This means that linear general integral control is more robust than PID control.Discussion 2: From the polynomial (1), it is not hard to see that that Hurwitz stability of the system matrix A can still be achieved for , , , , 0 , , 0 Therefore, the stability condition of Theorem 1 can be relaxed, but this is useless for the control design.
Discussion 3: From the statements above, Hurwitz stability condition is more and more complex as the order of the system matrix A increases.So, although Theorem 1 is demonstrated by the single variable system ma- trix A , it is easy to extend Theorem 1 to the multiple variable case since there is not any difficulty to ensure that as any row controller gains, or controller and its integrator gains increase with the same ratio, the system matrix A is always Hurwitz in theory, that is, 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 concluded.
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.

Solution of Lyapunov Equation
By Hurwitz stability condition given by Subsection 2.1, the system matrix A can be ensured to be Hurwitz for all 0 α α ε ε * < < and 0 β β ε ε * < < .Thus, by linear system theory, if the system matrix A is Hurwitz, and then for any given positive define symmetric matrix Q there exists a unique positive define symmetric matrix P that satisfies Lyapunov equation T PA A P Q + = − .Thus, the solution of Lyapunov equation can be obtained by skew symmetric matrix approach [10], that is, The inversion of the system matrix A with 1 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, tak- ing Q I = and 2 n = of the system matrix A .Thus, taking 1 ( ) ) It is obvious that 13 s εα and 23 s εα all tend to the constants as 0 α ε → , and then we have, It is obvious that 13 s εαβ and 23 s εαβ all tend to the constants as 0 α ε → , and then we have, P αβ can all be formulated as the linear form on α ε and all tend to zero as 0 α ε → .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 matrix 1 A − (9), n P α and n P αβ can all be formulated as the linear form on α ε for the 1 n + -order system matrix A , and with the help of computer, it can be verified that the solutions of n P εα and n P εαβ still tend to the constants as 0 α ε → .Therefore, for the n+1-order system matrix A , we can conclude that 0 , , 0 < < , and then we have, 1) 0 2) 0  ε − .This is the reason why our method is called as Equal ratio gain technique.Discussion 5: From the statements above, the solution of the matrix S is more and more complex as the or- der of the system matrix A increases.So, although Theorem 2 is demonstrated by taking Q I = and the single variable system matrix A , it is very easy to extend Theorem 2 to any given positive define symmetric ma- trix 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 sys- tem matrix but also the multiple system matrix.Thus, there is the following proposition.
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.

Example
For testifying the justification of Theorems 1 and 2, and Propositions 1 and 2, we consider a 6-order two varia-B.S. Liu ble system matrix A as follows, The polynomial of the system matrix A is, , and .
x y x x y x x y y y x y x y x x y x x y y x y y x y x y x y x y x y x y x y x x y x x y y x y y x y 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 0 1 8 2 0 0 3 1 , and then by Routh's stability criterion, the array of the coefficients of polynomial is,       From the example above, it is obvious that: 1) for all the six cases, there all exists ε * such that the system matrix A is always Hurwitz for all 0 ε ε * < ≤ ; 2) as shown in Tables 1-4, the absolute values of 5i p and 6i p ( )  are all decrease as ε reduces.This not only verifies the results proposed by Subsection 2.1 and 2.2 but also shows that for the high order and multiple variable system matrix A , it is convenient and prac- tical with the help of computer.

Control Design
Consider the following controllable nonlinear system, (  ) ( ) 0 0 0, 0, 0, 0, is the desired equilibrium point, and 0 x u and 0 y u are the steady-state controls that are needed to maintain equilibrium at 0 x y = = .Assumption 2: No loss of generality, suppose that the functions ( )  For system (10), we develop two kinds of methods to design linear general integral controllers, respectively, that is, Decomposition method and Synthetic method.
The matrices x A for all 0 with any given positive define symmetric matrices x Q and y Q , respectively.
Thus, using ( ) ( ) ( ) as Lyapunov function candidate, and then its time derivative along the trajectories of the closed-loop system (27) is, . where By Theorems 1 and 2, and Propositions 1 and 2, obtain, 0 Therefore, there exist Using the fact that Lyapunov function ( ) V η η is a positive define function and its time derivative is a negative define function if 0 Λ > holds, we conclude that the closed-loop system (27) is stable.In fact, = is an exponentially stable equilibrium point of the closed-loop system (27) for all 0 . Moreover, if all assumptions hold globally, and then it is globally exponentially stable.
In the same way, for the case of = is an exponentially stable equilibrium point of the closed-loop system (27) for all 0 . Moreover, if all assumptions hold globally, and then it is globally exponentially stable.

Synthetic Method
The control laws x u and y u are taken as, ( ) ( ) ( ) where In the same way as Subsection 3.1, the closed-loop system can be rewritten as, ( ) where ( ) , , F x y w is an 2 1 n m + + × matrix, all its elements are equal to zero except for , , 0, 0, , , 0, 0, 0, 0, 0, 0,   .Consequently, 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 (41) holds, we conclude that the closed-loop system (35) is stable.In fact, .By invoking LaSalle's invariance principle, it is easy to know that the closed-loop system (35) is exponentially stable.As a result, the following theorem can be established.
Theorem 5: Under Assumptions 1 and 2, if there exist the gains x i α , y i α , x j β and y j β such that the ma- trix A is Hurwitz for all 0 for all for all 0 . Moreover, if all assumptions hold globally, then it is globally exponentially stable.
In the same way, for the case of .Moreover, if all assumptions hold globally, then it is globally exponentially stable.
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 ( ) a powerful tool to solve the control design problem of uncertain nonlinear system, and then makes the engineers more easily design a stable controller.Moreover, for the 2-order system, linear general integral control can be reduced to PID control.Thus, Equal ratio gain technique can clearly explain the reason why PID control has good robustness, too.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 pre- is the steady-state control that is needed to maintain equilibrium at the origin and the control law is, ( ) ( ) Thus, the closed-loop system can be written as, ( ) , 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 magnitude 80 acting on the system input between 15 s and 16 s.
Figure 1 showed the simulation results under normal (solid line) and perturbed (dashed line) cases.The following observations can be made: the system responses under the normal and perturbed cases are almost identical before the additive impulse-like disturbance appears.From the simulation results and design procedure, it is obvious that by Equal ratio gain technique, we can tune a linear general integral controller with good robustness.This demonstrates that not only linear general integral control can effectively deal with the uncertain nonlinearity but also Equal ratio gain technique is a powerful tool to solve the control design problem of uncertain nonli- 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.

Conclusions
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; 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.
smallest and largest eigenvalues, respectively, of a symmetric positive define bounded matrix ( ) A x , for any n x R ∈ .The norm of vector

2 :
a result, the following theorem can be established.TheoremIf there exist the gains 1 1

Discussion 4 :
Theorem 1 and 2 are all obtained by multiplying the controller gains i α and the integrator gains j β by the same ratios 1 α ε − and 1 β ε − , respectively, even under the typical cases 1 β ε − can be equal to 1 or 1 α

αand 6 P
of the system matrix A are multiplied by 1 ε − , and then it is still Hurwitz for all are shown in

1 :
. We want to design the control laws x u and y u such that ( ) ( ), 0 x t y t → as t → ∞ .Assumption There are two unique control inputs

3 :
LaSalle's invariance principle, it is obvious that the closed-loop system (27) is exponentially stable.As a result, we have the following theorem.TheoremUnder Assumptions 1 and 2, if there exist the gains x i the following theorem.Theorem 4: Under Assumptions 1 and 2, if there exist the gains x i . The matrix A can be designed to be Hurwitz for all 0 .Where P is the solution of Lyapunov equation T PA A P Q + = − with any given positive de- fine symmetric matrix Q .Thus, using ( ) V η as Lyapunov function candidate, and then its time derivative along the trajectories of the closed-loop systems (35) is, ( stable equilibrium point of the closed-loop system (35)

Theorem 6 :
Under Assumptions 1 and 2, if there exist the gains x i stable equilibrium point of the closed-loop system (35) for all 0 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 asymptotical stability of the closed-loop system can be ensured for all 0

Table 2 .
Numerical Solutions of 6

Table 3 .
Numerical Solutions of 5

Table 4 .
Numerical Solutions of 6 P for all x 2