Received 13 May 2015; accepted 11 July 2015; published 14 July 2015

ABSTRACT

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.

Keywords:

General Integral Control, PID control, GI-PD Control, Robust Control, Output Regulation

1. Introduction

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 [1] . Over the last half-century, a great deal of academic and industrial effort has focused on improving PID control, but the trouble, which often suffers a serious loss of performance due to integrator windup, was not resolved in principle before general integral control [2] appeared in 2009.

After that various general integral controls along with the design techniques were presented. For example, general concave integral control [3] , general convex integral control [4] , constructive general bounded integral control [5] and the generalization of the integrator and integral control action [6] were all developed by resorting to an ordinary control along with a known Lyapunov function; general integral control designs based on linear system theory, sliding mode technique, feedback linearization technique, singular perturbation technique, equal ratio gain technique and power ratio gain technique were presented by [7] - [12] , respectively. Although general integral control has developed rapidly in theory, its practical applications have not been reported. Therefore, in consideration of its good control performance, it is appropriate at this time to compare the simplest general integral control (GI-PD) with PID control in order to promote its applications in practice.

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 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, 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.

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 actions, the uncertain nonlinear function is continuous in on the control domain.

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.

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

(7)

where, and are the controller gains; and are the integrator gains.

It is worth to note that although the control law (7) is GI-PD control, it is reduced to PID control as and. Thus, under GI-PD and PID control, the closed-loop system can be written as the same form, that is,

(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.

Now, defining, and substituting (9) into (8), obtain,

(10)

where

and is a matrix, all its elements is equal to zero except for

.

Moreover, it is worthy to note that the function is integrated into, and.

By linear system theory, if the matrix is Hurwitz, and then for any given positive define symmetric matrix, there is a unique positive define symmetric matrix that satisfies Lyapunov equation. Therefore, there exists a quadratic Lyapunov function,

(11)

Thus, using as Lyapunov function candidate, and then its time derivative along the trajectories of the closed-loop systems (10) is,

(12)

where.

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

(13)

where is a positive constant.

Substituting (13) into (12), obtain,

(14)

It is obvious that if

(15)

holds, 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 (15) holds, we conclude that the closed-loop system (10) is stable. In fact, means and. By invoking LaSalle’s invariance principle, it is easy to know that the closed-loop system (10) is exponentially stable.

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 is Hurwitz and another is that the inequality (15) holds. Thus, for comparing GI-PD control with PID control, the differences of two key conditions above must be demonstrated. Moreover, the analysis of PID and GI-PD control action and performance is unnecessary, too. All these are addressed in the following Sections, respectively.

3. Hurwitz Stability

The polynomials of the system matrix under PID control and GI-PD control are,

(16)

(17)

By Routh’s stability criterion and the polynomials (16) and (17), Hurwitz stability conditions of the system matrix under PID control and GI-PD control can be obtained as follows:

Under PID control, if, and are all positive constants, and the inequality,

(18)

holds, and then the system matrix is Hurwitz.

Under GI-PD control, if, and are all positive constants, and the inequality,

(19)

holds, then the system matrix is Hurwitz.

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 manifests itself in two aspects: one is that the gain produces a special term such that the gain is enhanced, and then for achieving Hurwitz stability, it is not necessary to increase the value of, even can be taken as a negative constant; another is that the gain educes another special term such that it makes the inequality (19) holds more easily, and then for achieving Hurwitz stability, it is unnecessary to increase and/or, or decrease.

2) As, if the system matrix with PID control is Hurwitz, and then the one with GI-PD control and must be Hurwitz.

3) The gain is indispensable. For ensuring Hurwitz stability, seems to be unfavorable, but is absolutely favorable.

4) There are two additional gains and in GI-PD control law. Therefore, more information can be exploited to stabilize the system matrix than PID control.

All these means that the system matrix under GI-PD control can be stabilized more easily than PID control.

4. Robustness against Uncertain Nonlinear Actions

For comparing PID control and GI-PD control robustness against uncertain nonlinear actions, we need to solve the Lyapunov equation with any given positive define symmetric matrix to obtain the solution of the matrix.

Under PID control, is,

Under GI-PD control, is,

For the sake of simplicity, we just consider the case of and. Thus, by comparing with, we have,

as

and then by and as, we obtain,

as

where

It is easy to see that there exists such that holds for all, and then by the in-

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, , and along with moderately choosing and, GI-PD control can be designed to have the stronger robustness against the uncertain nonlinear actions than PID control.

Discussion 2: Although the demonstration above aims at a special case, it is not hard to conclude that by synthesizing all the gains, , , and, GI-PD control can be designed to have the stronger robustness with respect to the uncertain nonlinear actions than PID control since more information can be used to decrease the value of.

5. Analysis of Control Action

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. Obviously, the integral control action continues to increase unless the error passes through zero, and then for making the integral control action tends to a constant, the error is usually needed to pass through zero repeatedly. Just the stubborn increase of integral control action results in the integrator windup.

Under GI-PD control, the integrator is. Thus, as, the integral control action does not increased and remains a constant; if the integral control action is large, increases, and the integral control action instantly decreases, and vice versa. This shows that the effect of is an attempt to anticipate and prevent an excess integral control action, and then integrator windup can be removed completely. Moreover, as, the integral control action is equivalent to the accumulation of PD control action. This means that the harmonization of the integral control action and PD control action can be achieved. All these means that GI-PD control has the better control performance than PID control.

6. Example and Simulation

Consider the pendulum system [13] 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.

GI-PD control law is,

It is worth to note that as, the control law above is PID control law. Thus, 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,

(20)

Now, taking, , , , , and, and using Routh’s stability criterion, we have,

(21)

(22)

and then the system matrix under GI-PD control and PID control is Hurwitz. Thus, solving Lyapunov equation, we obtain and, and then by the inequality (15) and the bound condition (20), we have,

(23)

(24)

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 of magnitude 60 acting on the system input between 30s and 31s.

Figure 1 and Figure 2 showed the simulation results under normal and perturbed cases. From the simulation results and design procedure, the following observations can be made: 1) by Hurwitz stability conditions (21) and (22), stability margin of the system matrix under GI-PD control is larger than the one of PID control; 2) by stability conditions (23) and (24), GI-PD control has the stronger robustness with respect to the uncertain nonlinear action than PID control; 3) by Figure 1 and Figure 2, under GI-PD control, no matter normal case or perturbed case, the optimum responses can all be achieved in the whole control domain. However, under PID control, the overshoot is proportional to the initial error and the settling time is long. Due to the above experimental results, it could be concluded that GI-PD control has more broad application prospects than PID control.

7. Conclusion

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.

Cite this paper

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

References