Pendulum parameters and control gains.
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 PI2D 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 PI2 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 PI2 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.
Consider the following type of second order system:
Where is the state, is the control input, such that fully actuated systems are considered, is a non linear function that maps the input to the system dynamics, and it is assumed that such function is known and invertible along all solutions of the system,is a nonlinear function that is continuously differentiable, and locally Lipschitz. It is assumed that the state is measurable and thatis known.
The control objective is to regulate the stateto a constant value. The proposed dynamic control considers full cancellation of the system dynamics, and it is given by
whererepresents a nominal feedback control that would be designed to ensure the regulation of (1) to.
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 constant gains are,and and must be positive. The integral action provides an augmented state, therefore system (1) in closed loop with control (2) and (3) is re-written as
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, with,, , such that the closed loop error dynamics (4), which corresponds to an autonomous system, might be rewritten as
Provided that the gains,andare different from zero and positive, it is immediate to obtain that the equilibrium of system (5) corresponds to. On the following stability conditions and tuning guidelines for the control gains, and will be presented.
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 PI2D controller given by (3). The autonomous dynamic system (5) converge asymptotically to its equilibrium point, if the positive control gains, and satisfy the following conditions
Consider the position error vectorand the Lyapunov function
where is a symmetric positive definite matrix, with all entries real and positive for all; in order to simplify the Lyapunov function computation the following conditions are introduced
Thus, a straightforward simplification of the time derivative of the Lyapunov function is to cancel the crossed error terms, which results in given by quadratic error terms. First, in order to cancel the crossed term conditions on can be obtained, and then to cancel the crossed term the matrix entry is defined appropriately, finally by defining the crossed error term is eliminated. So far the conditions on matrix are summarized as follows
On the other hand, to guarantee that, and of the matrix are positive, it is necessary to satisfy the following conditions. For the matrix entry to be positive, it is enough to have that and which are satisfied by conditions (6) of Theorem 1. For it follows that, that is satisfy by the conditions and as stated at (8). Finally, for it follows that, which implies conditions on and, to find out such conditions, the solutions of equation are computed as follows
For to be positive, it is required that, that is satisfied by conditions (6). Then for to be purely real, it is required that the argument of the squared root being positive, i.e., which when it is equal to zero implies the solutions. Thus, by taking the positive part of the solution and considering that it is guaranteed that is real, and condition implies that is real. Notice that all these conditions are satisfied by those stated at Theorem 1, equations (6).
At this point, it is guaranteed that the solutions are real and positive, those to ensure that it is considered that must satisfied
Such a condition is clearly over satisfied by the condition given at Theorem 1, equations (6).
Therefore, if conditions given by (6) at Theorem 1 are satisfied, it implies that, and of the matrix are positive.
Furthermore, the definition of, and stated by (9), yield a time derivative Lyapunov function given by quadratic error terms. Nonetheless it is necessary to check positive definitiveness of the Lyapunov function (7), which after conditions (8) are considered is given by
Notice that the positive condition on the coefficient of the term, i.e. has already been considered for positive definitiveness of. Therefore, the conditions on given by (6) imply that all coefficients of the quadratic terms of are positive. Now to guarantee note that the cross error terms and can be rewritten as part of a quadratic form, for that it is required
The last two conditions imply that and which are satisfied by conditions (6). And the first condition implies to solve the equation for. Similar to the way in which conditions for positive value of the entries of matrix, it follows that
That is conservatively satisfied by the condition given at Theorem 1, equation (6). On the other hand for to be real, it is necessary that, and for to be real it is required that; all these conditions are clearly satisfied by those stated at Theorem 1, equations (6).
Therefore, if the conditions given by (6) are satisfied, the Lyapunov function results on a sum of quadratic terms
for positive parameters; thus concluding that for, and for.
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, it is required that, which implies that
which is satisfied by the condition given at Theorem 1, equations (6). Nonetheless to guaranteed that is real, it follows that, that implies when considering equal to zero, that the solutions are
Thus for to be real it is required that and finally the condition on results on
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 are positive, such that for, and for.
Thus, it can be concluded that the closed loop system dynamic (5) is stable and the error vector converges globally asymptotically to its equilibrium.
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.
Because full cancellation of the system dynamics functionin (1) is assumed by the control law (2), in order to obtain the closed loop error dynamics (5), then the auxiliary polynomial can be considered to obtain a Hurwitz polynomial, and to characterize some properties of the closed loop system.
2.1.2. Stability analysis for the regulation case with non vanishing perturbation
In case that no full cancellation of in (1) can be guaranteed, either because of uncertainties on, , or in the system parameters, convergence of the system to the equilibrium point is not guaranteed. Nonetheless, the Lipschitz condition on, and assuming that is bounded in terms of, i.e for positive, then locally uniformly ultimate boundedness might be proved for large enough control gains, and, see (Khalil, 2002).
In the case of tracking, the problem statement is now to ensure that the sate vector follows a time varying reference; this trajectory is at least twice differentiable, smooth and bounded. For this purpose the control proposed in (2) is considered, but with the nominal controller given by
2.2.1. Stability analysis for the tracking case
Similar to the regulation case, the following position error vectoris defined, with, , , such that the closed loop error dynamics of system (1), with the controller (2) and (11) results in the same dynamic systems given by (5), such that Theorem 1 applies for the tracking case.
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.
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
wherewith, and, with the notation for the mass,for the spring effects,the length of the pendulum, andthe gravity acceleration. The values of the model parameters are presented at Table 1, and the initial condition of the pendulum is.
The proposed PI2D 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:, thus for the selected value, it was obtained that, and after selection of, it was finally obtained that. For the tuned gains listed at Table 1, it follows that the eigenvalues of the closed loop system (5) are the roots of the characteristic polynomial, such that,, and. Therefore, the closed loop system behaves as an overdamped system as shown in Figure 1. The behaviour of the closed-loop system for the PID and PI2D 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 than the PID controller, in which performance it is observed overshoot. Notice however that both input controls are similar in magnitude and shape; this implies better performance of the PI2D 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 control gains and initial conditions are the same as for the case of perfect cancellation.
The obtained simulation results are shown in Figure 2, where also a change in reference signal is considered from[rad] inseconds toin seconds. In the case of non complete dynamic cancellation due to uncertain parameters, it can be seen that the PI2D 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[rad] is considered. The simulation results are shown in Figure 3; the control gains are the same as listed at Table 1. In Figure 3 is depicted both behaviour of the PID and PI2D with perfect dynamic compensation, the PI2D 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 PI2D 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, i.e. the parameters of the controllergiven by (2) are set as,, and; and the controller gains are the same as listed at Table 1. In Figure 4 the comparison results are showed, despite there is no model cancellation, the PI2D 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 as well as the non model cancellation. Notice that the control actions present a sort of chattering that is due to the effort to compensate the no model cancellation
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 are respectively, the joint position, velocity and acceleration vectors in generalized coordinates, is the inertia matrix, is the Coriolis and centrifugal matrix, is the gravity vector and is the input torque vector. The system (13) presents the following properties ( Spong and Vidyasagar, 1989).
Property 1.- The inertia matrix is a positive symmetric matrix satisfying, for all, and some positive constants, where is the 2-dimensional identity matrix.
Property 2.- The gravity vector is bounded for all. That is, there exist positive constants such that for all.
From the generalized 2 DOF dynamic system, eq. (13), each DOF is rewritten as a nonlinear second order system as follows.
With and obtained from rewritten system (13), solving for the acceleration vector and considering the inverse of the inertia matrix. As for the pendulum case a PI2D 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 PI2D 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 is considered and no gravitational compensation was made, meaning that at the controller. The control gains remained the same as for all previous cases. Figures (8) to (10) show the simulation results. Although the inexact compensation, the proposed PI2D 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 PI2D 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, and no compensation on. The control gains remained the same as for all previous cases. Figures (14) to (16) show the simulation results. Although the inexact compensation, the proposed PI2D controller behaves faster and with a smaller control effort than the PID control.
The proposed controller represents a version of the classical PID controller, where an extra feedback signal and integral term is added. The proposed PI2D 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.
The second author acknowledges support from CONACyT via project 133527.