Integral control [1] plays an important role in control system design because it ensures asymptotic tracking and disturbance rejection. In the presence of the parametric uncertainties and unknown constant disturbances, integral control can still preserve the stability of the closedloop system and create an equilibrium point at which the tracking error is zero. The main task of the integral controller is to stabilize this point, which is challenging because it depends on uncertain parameters and unknown disturbances.

The design of integral control for general linear systems was done in the 1970’s in the work of Davision, Francis, and others [2,3]. In the early 1990’s, Isidori and Byrnes [4] extended integral control to nonlinear systems. Their results, however, were local. Regional and semiglobal results for integral control appeared later in the work of Khalil [1,5]. These papers dealt with minimumphase input-output linearizable systems and designed output feedback control using high-gain observers and the tool of saturating the control outside a compact region of interest. All these design methods above for integral control are achieved by using a conventional integrator, where is the controlled output and is a constant reference. In 2009, we originated General Integral Control in [6], which presented a unified framework for General Integral Control, some general integrator, and the necessary conditions and basic principles for designing a general integrator. Based on linear system theory, we presented a systematic design method for General Integral Control [7] with a linear integrator in 2012. The results, however, were local. In 2012, regional and semi-global results were proposed in [8], which presented a nonlinear integrator shaped by sliding mode manifold. And then, General Integral Control design was achieved by sliding mode technique and linear system theory.

In this paper, based on feedback linearization technique, we present a systematic design method for General Integral Control. The main contributions are as follows: 1) A new integral control strategy along with a class of firenew integrator is proposed; 2) By using linear system theory and Lyapunov method along with LaSalle’s invariance principle, the conditions on the control gains to ensure regionally as well as semi-globally asymptotic stability are provided.

Throughout this paper, we use the notation and to indicate the smallest and largest eigenvalues, respectively, of a symmetric positive define bounded matrix, for any. The norm of vector is defined as, and that of matrix

is defined as the corresponding induced norm

.

The remainder of the paper is organized as follows. Section 2 describes the system under consideration, assumptions, and General Integral Control law proposed here. Section 3 addresses the systematic design method of General Integral Control. Example and simulation are provided in Sections 4. Conclusions are presented in Section 5.

Consider the feedback linearizable system,

where is the state, is the control input, is a vector of unknown constant parameters and disturbances. The function and are continuous in on a domain,. The inequality, holds for all and.

Assumption 1: Suppose that there is a unique pair that satisfies the equation,

so that is the steady-state control that is needed to maintain equilibrium at the origin.

Assumption 2: Suppose that there is a diffeomorphism

such that contains the origin and satisfies the partial differential equations,

where is controllable, and are all known nonlinear functions, is nonsingular for all

, and are the uncertain terms of the system (1), which arises from several practical reasons such as model simplification, parameter uncertainty, computational errors.

The change of variables transforms the system (1) into the form,

For stabilizing the system (4), we need to include “integral action” in the control law. Therefore, General Integral Controller is proposed as follows,

where and are all positive define matrices.

Thus, substituting (5) into (4) to obtain the augmented system,

By setting, and in (6), we obtain,

By Assumption 1 and choosing nonsingular to counteract the constant uncertainties, we ensure that there is a unique solution, , and then is a unique equilibrium point of the closed-loop system (6) in a domain of interest. At the equilibrium point, , irrespective of the value of.

Remark 1: From the control law (5), it is not hard to see that the integrator to be shaped by diffeomorphism is a fire new integrator. And then, it resulted in a class of fire-new general integral controller and design method.

Now, the design task is to provide the conditions on the gain matrices and such that is an asymptotically stable equilibrium point of the closedloop system (6) in the control domain of interest, which is not a trivial task because the closed-loop system (6) depends on the unknown vector, the uncertain terms

and. In the next section, we will propose a systematic design method to this dilemma.

For analyzing the stability of the closed-loop system (6), we substitute (7) into (6) and obtain,

where

For stabilizing the system (8), we need to know some bound information on. This results in the following assumptions.

Assumption 3: By Assumption 2 and the function

and are continuous in on a domain, and, it is reasonable to suppose that the following inequalities hold,

Thus, by the definition of and (9)~(13)we obtain,

where, , , and are all positive constants.

Setting, the equation (8) can be rewritten as,

where

and

Based on linear system theory  [5], we can choose and such that is Hurwitz, and then for any given positive define symmetric matrix there exists a unique positive define symmetric matrix that satisfied Lyapunov equation (16). Consequently, there exists a quadratic Lyapunov function,.

We use as a Lyapunov function candidate. Obviously, is positive define.

Therefore, our task is to show that its time derivative along the trajectories of the closed-loop system (8) is negative define, which is given by,

Now, by definition of, we have and, and then inequality (14) can be rewritten as,

where.

Substituting (18) into (17) and taking, we obtain,

Using the fact that Lyapunov function

is a positive define function and its time derivative is a negative define function if the inequality (19) holds, we conclude that the closed-loop system (8) is stable. In fact,

means and. By invoking the LaSalle’s invariance principle [5], it is easy to know that the closed-loop system (8) is asymptotically stable in the control domain of interest. This established the following theorem.  

Theorem 1: Let be a unique equilibrium point for the closed-loop system (8). If Assumptions 1 ~ 3 hold and there exist the gain matrices and such that the inequality (19) holds, and then the closedloop system (8) is asymptotically stable in the domain

. Moreover, if all assumptions hold globally, and then it is globally asymptotically stable.

Remark 2: From the analysis procedure above, it is obvious that the distinct feature of this design method is:

1) Just the integrator is taken as, , we can easily use linear system theory to analyze the stability of the closed-loop system; 2) Just the integral control action is introduced, we can use the linear growth bound to estimate the impact of the uncertain term, on the system stability. All of them provide an ingenious solution for designing a stable general integral controller.

Discussion 1: Compared with General Integral Control design proposed by [7,8], it is easy to see that: 1) The design method proposed here can cancel the central nonlinear action via feedback linearization; 2) When the bound of the system uncertainty is fairly estimated we can design a stable general integral controller with the lesser conservativeness. All those mean that general integral control design method proposed here can more effectively deal with nonlinearity and uncertainty of dynamics, and then makes the engineers more easily design a stable controller.

Consider the pendulum system [5] 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

. Taking, , we can write the pendulum system as,

It can be easily seen that the system (20) is feedback linearizable with. Thus, general integral controller can be taken as,

where and are the nominal values of and.

Substituting (21) into (20), we obtain,

where

.

For the closed-loop system (22), taking as the nominal values of and, and using the design method proposed here, general integral controller can be taken as,

Thus, regulation will be achieved for all.

In simulation, the normal parameters are and. In the perturbed case, and are reduced

to 0.5 and 5, respectively, corresponding to doubling of the mass. Moreover, we consider an additive impulselike disturbance of magnitude 30 acting on the system input between 11 s and 12 s.

Figure 1 showed the simulation results under normal (solid line) and perturbed (dashed line) cases. The following observations can be made: under normal and perturbed cases, the optimum response in the whole domain of interest can all be achieved by a set of the same control gains, even under the case that the payload is changed abruptly. This demonstrates that general integral control proposed here has strong robustness, fast convergence and good flexibility, and then can more effectively deal with unknown exogenous disturbances, nonlinearity and uncertainties of dynamics and makes the engineers more easily design a high performance controller.

Based on the feedback linearization technique, we present a systematic design method for General Integral Control. The main contributions are as follows: 1) A new integral control strategy along with a class of fire-new integrator is proposed; 2) By using the linear system theory and Lyapunov method along with LaSalle’s invariance principle, the conditions on the control gains to ensure regionally as well as semi-globally asymptotic stability are provided.

In this paper, only one design method for General Integral Control was presented. It is clear that we cannot expect one particular procedure to apply to all system. Therefore, new design techniques for General Integral Control are needed to solve wider theoretical and practical problems.