## 1. Introduction

Second order autonomous systems are key systems in the study of non linear systems because their solution trajectories can be represented by curves in the plane (Khalil, 2002), which helps in the development of control strategies through the understanding of their dynamical behaviour. Such autonomous systems are often obtained when considering feedback control strategies, because the closed loop system might be rewritten in terms of the state system and perturbation terms, which are function of the state as well. Thus, analyses of stability properties of second order autonomous systems and their convergence are areas of interest on the control community.

Moreover, several applications consider nonlinear second order systems; there are various examples of this:

In mechanical systems the pendulum, the inverted pendulum, the translational oscillator with rotational actuator (TORA) and the mass-spring systems;

In electrical systems there are examples such as the tunnel diode circuit, some electronic oscillators as the negative-resistance twin-tunnel-diode circuit; and finally

Other type of these systems are mechanical-electrical-electronic combinations, for example a two degree of freedom (DOF) robot arm or a mobile planar robot and among every degree of freedom on a robotic structure can be represented by a second order nonlinear system.

Therefore, due to the wide applications in second order nonlinear systems, several control laws have been proposed, which comprises from simple ones, like linear controllers, to the more complex, like sliding mode, backstepping approach, output-input feedback linearization, among others (Khalil, 2002).

Despite the development of several control strategies for nonlinear second order systems, it is not surprising that for several years and even nowadays the classical PID controllers have been widely used in technical and industrial applications and even on research fields. This is due to the good understanding that engineers have of them. Moreover, the PID controllers have several important functions: provide feedback, has the ability to eliminate steady state offset through integral action, and it can anticipate the future through derivative action.

PID controllers are sufficient for many control problems, particularly when system dynamics are favourable and the performance requirements are moderate. These types of controllers are important elements of distributed control system. Many useful features of PID control are considered trade secrets, (Astrom and Hagglund, 1995). To build complicated automation systems in widely production systems as energy, transportation and manufacturing, PID control is often combined with logic, sequential machines, selectors and simple function blocks. And even advanced techniques as model predictive control is encountered to be organized in hierarchically, where PID control is used in the lower level. Therefore, it can be inferred that PID control is a key ingredient in control engineering.

For the above reasons several authors have developed PID control strategies for nonlinear systems, this is the case of (Ortega, Loria and Kelly, 1995) that designed an asymptotically stable proportional plus integral regulator with position feedback for robots with uncertain payload that results in a PI^{2}D regulator. In the work of (Kelly, 1998), the author proposed a simple PD feedback control plus integral action of a nonlinear function of position errors of robot manipulators, that resulted effective on the control of this class of second nonlinear systems and it is known as PD control with gravity compensation. Also PID modifications for control of robot manipulators are proposed at the work of (Loria, Lefeber and Nijmeijer, 2000), where global asymptotic stability is proven. In process control a kind of PI^{2} compensator was developed in the work of (Belanger and Luyben, 1997) as a low frequency compensator, due to the additional double integral compensation rejects the effects of ramp-like disturbances; and in the work of (Monroy-Loperena, Cervantes, Morales and Alvarez-Ramirez, 1999), a parametrization of the PI^{2} controller in terms of a nominal closed-loop and disturbance estimation constants is obtained, despite both works are on the process control field, their analysis comprises second order plants.

In the present work a class of nonlinear second order system is consider, where the control input can be consider as result of state feedback, that in the case of second order systems is equivalent to a PD controller, meanwhile double integral action is provided when the two state errors are consider, both regulation and tracking cases are considered.

Stability analysis is developed and tuning gain conditions for asymptotic convergence are provided. A comparison study against PID type controller is presented for two examples: a simple pendulum and a 2 DOF robot arm. Simulation results confirm the stability and convergence properties that are predicted by the stability analysis, which is based on Lyapunov theory. Finally, the chapter closes with some conclusions.

## 2. Problem formulation

Two cases are considered in this work, first regulation to a constant reference is boarded, second tracking a time varying reference is studied; in both cases stability and tuning gain conditions are provided.

### 2.1. Regulation

Consider the following type of second order system:

Where

The control objective is to regulate the state

where

The nominal control is designed as a feedback state control plus a type of double integral control and is provided in the following equation

Control (3) provides an extra integral action with the integration of the state

The closed-loop system (4) has a unique equilibrium point in

In the following a stability analysis for the regulation case is determined.

#### 2.1.1.Stability analysis for the regulation case

Consider the following position error vector

Provided that the gains

Theorem 1

Consider the autonomous dynamic second order system given by (5), which represents the closed loop error dynamics obtained from system (1) with the control law (2), and the nominal PI^{2}D controller given by (3). The autonomous dynamic system (5) converge asymptotically to its equilibrium point

Proof:

Consider the position error vector

where

The time derivative of the Lyapunov function (7) is function of the closed loop error dynamics (5), and it is given by

(9) |

Thus, a straightforward simplification of the time derivative of the Lyapunov function is to cancel the crossed error terms

On the other hand, to guarantee that

For

At this point, it is guaranteed that the solutions

Such a condition is clearly over satisfied by the condition

Therefore, if conditions given by (6) at Theorem 1 are satisfied, it implies that

Furthermore, the definition of

Notice that the positive condition on the coefficient of the term

The last two conditions imply that

That is conservatively satisfied by the condition

Therefore, if the conditions given by (6) are satisfied, the Lyapunov function results on a sum of quadratic terms

for positive parameters

Since the definition of the matrix entries (8) allows cancellation of all cross error terms on the time derivative of the Lyapunov function (7), then along the position error solutions, it follows that

To ensure that

which is satisfied by the condition

Thus for

Such that, the above conditions are satisfied by considering those of Theorem 1, equation (6).

Therefore, by satisfying conditions (6) it can be guaranteed that all coefficients of the derivative of the Lyapunov function

Thus, it can be concluded that the closed loop system dynamic (5) is stable and the error vector

Remark 1

The conditions stated at Theorem 1, equations (6) are rather conservative in order to guarantee stability and asymptotic convergence of the closed loop errors. The conditions (6) are only sufficient but not necessary to guarantee the stability of the system.

Remark 2

Because full cancellation of the system dynamics function

#### 2.1.2. Stability analysis for the regulation case with non vanishing perturbation

In case that no full cancellation of

### 2.2. Tracking

In the case of tracking, the problem statement is now to ensure that the sate vector

#### 2.2.1. Stability analysis for the tracking case

Similar to the regulation case, the following position error vector

Remark 3

The second integral action proposed in the nominal controllers, (3) for regulation, and (11) for tracking case, can be interpreted as a composed measured output function, such that this action helps the controller by integrating the velocity errors. When all non linearity is cancelled the integral action converges to zero, yielding asymptotic stability of the complete state of the system. If not all nonlinear dynamics is cancelled, or there is perturbation on the system, which depends on the state, then it is expected that the integral action would act as estimator of such perturbation, and combined with suitable large control gains, it would render ultimate uniformly boundedness of the closed loop states.

## 3. Results

In this section two systems are consider, a simple pendulum with mass concentrated and a 2 DOF planar robot. First the pendulum system results are showed.

### 3.1. Simple pendulum system at regulation

Consider the dynamic model of a simple pendulum, with mass concentrated at the end of the pendulum and frictionless, given by

where

The proposed PI^{2}D is applied and compared against a PID control that also considers full dynamic compensation, i.e. the classical PID is programmed as follows

The comparative results are shown in Figure 1. The control gains were tuned accordingly to conditions given by (6), see Table 1, such that it was considered that:^{2}D controllers is shown in Figure 1; the performance of the double integral action on the PID proposed by the nominal controller (3) shows faster and overdamped convergence to the reference ^{2}D controller without increasing the control action significantly.

For the sake of comparison another simulation is developed considering imperfect model cancellation, in this case due to pendulum parameters uncertainty considered for the definition of the controller (2). The nominal model parameters are those of Table 1, while the control parameters are

The obtained simulation results are shown in Figure 2, where also a change in reference signal is considered from^{2}D controller proposed by (2) and (3) also responds faster that the classical PID with dynamic cancellation, besides the control actions are similar in magnitude and shape as shown in Figure 2.

### 3.2. Simple pendulum system at tracking

A periodic reference given by^{2}D with perfect dynamic compensation, the PI^{2}D controller shows faster convergence to the desired trajectory than the PID control, nonetheless both control actions are similar in magnitude and shape, this shows that a small change on the control action might render better convergence performance, in such a case the double integral action of the PI^{2}D controller plays a key role in improving the closed loop system performance

To close with the pendulum example, uncertainty on the parameters is considered, such that there is no cancellation of the function^{2}D controller shows better performance that the PID case, i.e faster convergence (less than 4 seconds), requiring minimum changes on the control action magnitude and shape, as shown on the below plot of Figure 4, where the control actions are similar to those of Figure 3, which implies that the control gains absorbed the model parameter uncertainties on parameter

### 3.3. A 2 DOF planar robot at regulation

The dynamic model of a 2 DOF serial rigid robot manipulator without friction is considered, and it is represented by

Where

Property 1.- The inertia matrix is a positive symmetric matrix satisfying

Property 2.- The gravity vector

From the generalized 2 DOF dynamic system, eq. (13), each DOF is rewritten as a nonlinear second order system as follows.

With ^{2}D controller of the form given by (2) and (3) is designed and compared against a PID, similar to section 3.1, for both regulation and tracking tasks.

From Figure (5) to Figure (7), the closed loop with dynamic compensation is presented, where the angular position, the regulation error and the control input, are depicted. The PI^{2}D controller shows better behaviour and faster response than the PID. The controller gains for both DOF of the robot are listed at Table 1. The desired reference is

To test the proposed controller robustness against model and parameter uncertainty, it was considered unperfected dynamic compensation, for both links a sign change on the inertia terms corresponding to the function ^{2}D controller behaves faster and with a smaller control effort than the PID control.

### 3.4. A 2 DOF planar robot at tracking

For the tracking case study a simple periodical signal given by

is tested. First perfect cancellation is considered, and then unperfected cancellation of the robot dynamics is taken into account. The control gains are the same as those listed at Table 1. Figures (11) to (13) show the system closed loop performance with perfect dynamic compensation, where the angular position, the regulation error and the control input, respectively, are depicted. The PI^{2}D controller shows a better behaviour and faster response than the PID, both with dynamical compensation.

To test the proposed controller robustness against model and parameter uncertainty, it was considered imperfect dynamic compensation considering as in the regulation case a sign change in^{2}D controller behaves faster and with a smaller control effort than the PID control.

## 4. Conclusions

The proposed controller represents a version of the classical PID controller, where an extra feedback signal and integral term is added. The proposed PI^{2}D controller shows better performance and convergence properties than the PID. The stability analysis yields easy and direct control gain tuning guidelines, which guarantee asymptotic convergence of the closed loop system.

As future work the proposed controller will be implemented at a real robot system, it is expected that the experimental results confirm the simulated ones, besides that it is well know that an integral action renders robustness against signal noise, by filtering it.