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In conjunction with a second order uncertain nonlinear system, this paper makes some comparisons between PID control and general-integral-proportional-derivative (GI-PD) control; that is, by Routh’s stability criterion, we demonstrate that the system matrix under GI-PD control can be stabilized more easily; by linear system theory and Lyapunov method, we demonstrate that GI-PD control can deal with the uncertain nonlinearity more effectively; by analyzing and comparing the integral control action, we demonstrate that GI-PD control has the better control performance. Design example and simulation results verify the justification of our conclusions again. All these mean that GI-PD control has the stronger robustness and higher control performance than PID control. Consequently, GI-PD control has broader application prospects than PID control.

Proportional-integral-derivative (PID) control is certainly the most widely used control strategy today. It is estimated that over 90% of control loops employ PID control [

After that various general integral controls along with the design techniques were presented. For example, general concave integral control [

Motivated by the cognition above, in conjunction with a second order uncertain nonlinear system, this paper makes some comparisons between PID control and GI-PD control. The main contributions are: under GI-PD control, it is demonstrated that: 1) the system matrix can be stabilized more easily; 2) it is more effective to deal with the uncertain nonlinear actions; 3) the trouble caused by integrator windup is resolved in principle, and then it has the better control performance; 4) the harmonization of the integral control action and PD control action can be achieved. Moreover, design example and simulation results verify the justification of our conclusions again. All these mean that GI-PD control has the stronger robustness and higher control performance than PID control. Consequently, GI-PD control has broader application prospects than PID control.

Throughout this paper, we use the notation

The remainder of the paper is organized as follows: Section 2 describes the system under consideration, assumption, and stability analysis of the closed-loop system. Section 3 compares Hurwitz stability of the system matrix. Section 4 demonstrates the robustness against the uncertain nonlinearity. Section 5 analyzes the control action. Example and simulation are provided in Section 6. Conclusions are presented in Section 7.

Consider the following controllable nonlinear system,

where

Assumption 1: There is a unique pair

so that

Assumption 2: Suppose that the functions

for all

For comparing PID and GI-PD control, the control law is taken as,

where

It is worth to note that although the control law (7) is GI-PD control, it is reduced to PID control as

By assumption 1 and choosing

Therefore, we ensure that there is a unique solution

Now, defining

where

and

Moreover, it is worthy to note that the function

By linear system theory, if the matrix

Thus, using

where

Now, using the inequalities (3), (5), (6) and definition of

where

Substituting (13) into (12), obtain,

It is obvious that if

holds, we have

Using the fact that Lyapunov function

Discussion 1: From the demonstration above, it is obvious that: for ensuring that the closed-loop system is exponentially stable, two key conditions are indispensable, that is, one is that the system matrix

The polynomials of the system matrix

By Routh’s stability criterion and the polynomials (16) and (17), Hurwitz stability conditions of the system matrix

Under PID control, if

holds, and then the system matrix

Under GI-PD control, if

holds, then the system matrix

Compared with Hurwitz stability conditions of PID control, the one of GI-PD control has the following features:

1) The striking feature is that the role of gain

2) As

3) The gain

4) There are two additional gains

All these means that the system matrix

For comparing PID control and GI-PD control robustness against uncertain nonlinear actions, we need to solve the Lyapunov equation

Under PID control,

Under GI-PD control,

For the sake of simplicity, we just consider the case of

and then by

where

It is easy to see that there exists

equality (15), we can conclude that GI-PD control is more effective to deal with the uncertain nonlinear actions than PID control. This means that under the case of the same gains

Discussion 2: Although the demonstration above aims at a special case, it is not hard to conclude that by synthesizing all the gains

No matter PID control or GI-PD control, Proportional and Derivative control actions are all identical, that is:

Proportional control action is proportional to the error. If the error is small, its corrective effect is small, and vice versa.

Derivative control action is proportional to the rate at which the error is changing. Its corrective effect attempts to anticipate a large error and prevent this future error.

Compared with PID control, the main difference of GI-PD control is the integrator, that is, the error derivative is introduced into the integrator. This lead to an important change of the integral control action, that is,

Under PID control, the integrator is

Under GI-PD control, the integrator is

Consider the pendulum system [

where

and then it can be verified that

GI-PD control law is,

It is worth to note that as

where

and

The normal parameters are

Now, taking

and then the system matrix

Thus, Under PID and GI-PD control, the asymptotical stability of the whole closed-loop system can all be ensured. Consequently, the simulations are implemented under the normal and perturbed cases, respectively. Moreover, in the perturbed case, we consider an additive impulse-like disturbance

In conjunction with a second order uncertain nonlinear system, this paper makes some comparisons between PID control and GI-PD control. The main contributions are: under GI-PD control, it is demonstrated that: 1) the system matrix can be stabilized more easily; 2) it is more effective to deal with the uncertain nonlinear actions; 3) the trouble caused by integrator windup is resolved in principle, and then it has the better control performance; 4) the harmonization of the integral control action and PD control action can be achieved. Moreover, design example and simulation results verify the justification of our conclusions again. All these means that GI-PD control has the stronger robustness and higher control performance than PID control. Consequently, GI-PD control has broader application prospects than PID control.

BaishunLiu,BinHe,XiangqianLuo, (2015) On the Comparisons of PID and GI-PD Control. Engineering,07,387-394. doi: 10.4236/eng.2015.77035