Nonlinear General Integral Control Design via Equal Ratio Gain Technique ()
1. 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 perturbation 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
and
to indicate the smallest and largest eigenva- lues, respectively, of a symmetric positive define bounded matrix
, for any
. The norm of vector 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 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.
2. Problem Formulation
Consider the following controllable nonlinear system,
(1)
where
is the state;
is the control input;
is a vector of unknown constant parameters and disturbances. The function
is the uncertain nonlinear action, and the uncertain nonlinear function
is continuous in
on the control domain
. We want to design a control law
such that
as
.
Assumption 1: There is a unique pair
that satisfies the equation,
(2)
so that
is the desired equilibrium point and
is the steady-state control that is needed to maintain equilibrium at
.
Assumption 2: Suppose that the functions
and
satisfy the following inequalities,
(3)
(4)
(5)
(6)
for all
and
. where
,
,
,
and
are all positive constants.
Definition 1:
with
,
and
denotes the set of all continuous functions,
such that
![]()
and
![]()
hold for all
and
. Where
is a point on the line segment connecting
to
.
Definition 2:
with
, and
denotes the set of all integrable functions,
such that
![]()
and
![]()
hold for all
. Where
is a point on the line segment connecting
to the origin.
3. Generic Nonlinear Integral Control
The generic nonlinear integral controller is given as,
(7)
where
and
belong to the function sets
and
, respectively,
,
and
are all positive constants.
Thus, substituting (7) into (1), obtain the augmented system,
(8)
By Assumption 1 and choosing
to be large enough, and then setting
and
of the system (8), obtain,
(9)
Therefore, we ensure that there is a unique solution
, and then
is a unique equilibrium point of the closed-loop system (8) in the domain of interest. At the equilibrium point,
, irrespective of the value of
.
Now, by Definition 1, 2 and
,
and
can be written as,
(10)
(11)
where
and
.
Thus, substituting (9)-(11) into (8), obtain,
(12)
where ![]()
![]()
And
is an
matrix, all its elements are equal to zero except for
![]()
Moreover, it is worthy to note that the function
is integrated into
via a change of variable. This has not influence on the results if the inequality (4) holds and it can be taken as
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
is Hurwitz for all
,
,
and
. This can be achieved by Routh’s stability criterion.
3.1. Hurwitz Stability
Hurwitz stability of the matrix
can be achieved by Routh’s stability criterion, which is motivated by the idea [10] , as follows:
Step 1: the polynomial of the matrix
with
is,
(13)
By Routh’s stability criterion,
and
can be chosen such that the polynomial (13) is Hurwitz for all
and
. Obviously, if
and
are all large to zero, and then the necessary condition, that is, the coefficients of the polynomial (13) are all positive, is naturally satisfied.
Step 2: based on the gains
,
and Hurwitz stability condition to be obtained by Step 1, the maximums of
and
, that is,
and
, can be obtained, respectively. Since
and
interact, there exist innumerable
and
. Thus, two kinds of typical cases are interesting, that is, one is that
is evaluated with
; another is that let
, and then
can be obtained together.
Step 3: by
and
obtained by Step 2, check Hurwitz stability of the matrix
for all
and
. If it does not hold, redesign
and
and repeat the previous steps until the matrix
is Hurwitz for all
and
.
It is well known that Hurwitz stability condition is more and more complex as the order of the matrix
increases. Thus, for clearly illustrating the design method above, we consider two kinds of cases, that is,
and
, respectively, as follows:
Case 1: for
, the polynomial (13) is,
(14)
By Routh’s stability criterion, if
,
,
,
, and
are all positive numbers, and the following inequality,
(15)
holds, and then the polynomial (14) is Hurwitz for all
and
.
Sub-class 1:
,
and
are multiplied by
, and then substituting them into (15), obtain,
(16)
By the inequality (16), obtain,
![]()
Sub-class 2:
,
,
,
and
are multiplied by
, and then substituting them into (15), obtain,
(17)
For this sub-class, there are two kinds of cases:
1) if
, and then by the inequality (17), obtain,
![]()
2) if
, and then by the inequality (17), obtain,
![]()
Case 2: for
, the polynomial (13) is,
(18)
By Routh’s stability criterion, if
,
,
,
,
,
and
are all positive numbers, and the following inequality,
(19)
holds, and then the polynomial (18) is Hurwitz for all
and
.
Sub-class 1:
,
,
and
are multiplied by
, and then substituting them into (19), obtain,
(20)
By the inequality (20), obtain,
![]()
Sub-class 2:
,
,
,
,
,
and
are multiplied by
, 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
,
and
or
of the matrix
, there all exist
such that the matrix
is Hurwitz for all
,
and
. Therefore, for the high order matrix
, the same result can be still obtained with the help of computer. Thus, we can conclude that the n+1-order matrix
can be designed to be Hurwitz for all
,
,
and
.
Theorem 1: There exist
and
such that the system matrix
for
is Hurwitz, and then it is still Hurwitz for all
and
.
Discussion 1: From the statements above, it is easy to see that: 1) the system matrix
is an interval matrix; 2) in consideration of the controllable canonical form of linear system, the system matrix
can be called as the controllable canonical interval system matrix; 3) although Theorem 1 is demonstrated by the single variable system matrix
, 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.
3.2. Closed-Loop Stability Analysis
The matrix
can be designed to be Hurwitz for all
,
,
and
. Thus, by linear system theory, if the matrix
is Hurwitz, and then for any given positive define symmetric matrix
, there exists positive define symmetric matrix
that satisfies Lyapunov equation
. Therefore, the solution of Lyapunov equation [11] is,
(21)
where
,
and![]()
Thus, using
as Lyapunov function candidate, and then its time derivative along the trajectories of the closed-loop systems (12) is,
(22)
where
.
Now, using the inequalities (3), (5) and (6), obtain,
(23)
where
is a positive constant.
Substituting (23) into (22), and using
, obtain,
(24)
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) ![]()
as ![]()
2)
as ![]()
where
.
Although there is innumerable
, the maximum
exists and
as
. Thus, there exist
with
, or
such that the following inequality,
(25)
holds for all
with
, or
. Therefore, we have
.
Using the fact that Lyapunov function
is a positive define function and its time derivative is a negative define function if the inequality (25) holds, we conclude that the closed-loop system (12) is stable. In fact,
means
and
. 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
is Hurwitz for all
,
,
and![]()
and then the equilibrium points
and
of the closed-loop system (12) is an exponentially stable for all
with
, or![]()
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
or
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.
4. Practical Nonlinear Integral Control
For making up the shortage indicated by Discussion 2, a practical nonlinear integral controller is given as,
(26)
where
,
,
and
are the slopes of the line segment connecting
to the origin
, and they are utilized to harmonize the actions of
in the controller and integrator, respectively. ![]()
is used to attenuate the uncertain nonlinear action of
. ![]()
is applied to improve
the performance of integral control action. ![]()
is used to reorganize the integrator
output.
,
and
are all positive constants, and
.
Assumptions 3: By the definition of the controller (26), it is convenient to suppose that the following inequalities,
(27)
(28)
hold for all
,
and
, where
and
are all positive constants.
By the same way as Section 3, we have,
(29)
where
,
![]()
is an
matrix, all its elements are equal to zero except for
![]()
and the functions
and
are integrated into
and
, respectively.
By the design method proposed by Subsection 3.1, the system matrix
can be designed to be Hurwitz for all
,
,
,
and
. Thus, by linear system theory, there exists positive define symmetric matrix
that satisfies Lyapunov equation
for any given positive define symmetric matrix
. Therefore, we can utilize
as Lyapunov function candidate, and then its time derivative along the trajectories of the closed-loop system (29) is,
(30)
where
.
Now, using the inequalities (4), (5), (6), (27) and (28), obtain,
(31)
where
is a positive constant.
Substituting (31) into (30), obtain,
(32)
By proposition proposed by [10] (details see Subsection 3.2), for any moment
, there exists
with
, or
such that the inequality,
(33)
holds for all
with
, or
. Consequently, if
the inequality (33) holds for all
, and then we conclude that
holds uniformly in
.
Using the fact that Lyapunov function
is a positive define function and its time derivative is a negative define function if the inequality (33) holds for all
, we conclude that the closed-loop system (29)
is stable. In fact,
means
and
. By invoking LaSalle’s invariance principle, it is easy
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
is Hurwitz for all
,
,
,
and![]()
and then the equilibrium point
and
of the closed-loop system (29) is uniformly exponentially stable for all
with
, or![]()
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
with
, or
. To achieve this objective, the procedure can be summarized as follows:
Firstly, by the definitions of
and
, the instantaneous values
and
can be given as,
![]()
and
![]()
Secondly, by the inequality (33), the impermissible minimum of
is,
![]()
Finally, by the limitation conditions,
with
, or![]()
and the iterative method to solve Lyapunov equation,
![]()
with
, or
can be obtained.
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)
is not greater than
since
can be used to attenuate the
uncertain nonlinear action of
and
can be designed moderately; 3) ![]()
or
is less conservative since they are all evaluated by the instantaneous values
and
. All these not only solve the problem indicated by Discussion 2 but also mean that equal ratio gain technique is a powerful and practical tool to solve the control design problem of dynamics with the nonlinear and uncertain actions.
5. Example and Simulation
Consider the pendulum system [1] described by,
![]()
where
,
is the angle subtended by the rod and the vertical axis, and
is the torque applied to the pendulum. View
as the control input and suppose we want to regulate
to
. Now, taking
,
, the pendulum system can be written as,
![]()
and then it can be verified that
is the steady-state control that is needed to maintain equilibrium at the origin.
By the practical nonlinear integral controller (26), the control law can be given as,
![]()
Thus, it is easy to obtain
,
,
,
and
, and then the closed-loop system can be written as,
![]()
where
,
![]()
and
![]()
The normal parameters are
and
, and in the perturbed case,
and
are reduced to 1
and 5, respectively, corresponding to double the mass. Thus, we have
.
Now, with
,
,
,
,
,
and
, the following inequality,
![]()
holds for all
, and then the matrix
is Hurwitz for all
. Consequently, taking
as the initial value, the simulation is implemented under the normal and perturbed cases, respectively. Moreover, in the perturbed case, we consider an additive impulse-like disturbance
of magnitude 60 acting on the system input between 15 s and 16 s.
Figure 1 and Figure 2 showed the simulation results under normal (solid line) and perturbed (dashed line) cases. The following observations can be made: 1) The instantaneous value
holds for all
,
,
, ![]()
,
,
,
and
. 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.
![]()
Figure 1. The values of
under normal (solid line) and perturbed case (dashed line).
![]()
Figure 2. System output under normal (solid line) and perturbed case (dashed line).
6. 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.