Nonlinear Disturbance Observers for EL Systems

Mohammadi, A., Marquez, H. J., & Tavakoli, M. (2017). Nonlinear disturbance observers: Design and applications to Euler-Lagrange systems. IEEE Control Systems Magazine, 37(4), 50-72.

DOBs possess several promising features in control applications [10]. The most important feature is the addon or patch feature. Disturbance feedforward compensation can be employed as an add-on to a previously designed feedback control.

In general, DOB design has been carried out using linear system techniques. To overcome the limitations of linear DOBs (LDOBs) in the presence of highly nonlinear and coupled dynamics, researchers have started investigating nonlinear DOBs (NDOBs) for systems with nonlinear dynamics during the past decade. One possible way to categorize NDOBs is to compare their underlying dynamical equations to state observers. Based on this criterion, NDOBs can be categorized into two general classes, namely, basic and sliding-mode based. Basic NDOBs are extensively used in mechatronics and robotics applications and generalize the LDOB. Sliding-mode-like NDOBs are appealing in applications where the disturbance-estimation error needs to converge in finite time. The presence of discontinuities in the sliding-mode-based NDOBs, which may give rise to the chattering phenomenon, makes their analysis harder.

By exploiting the special structure of dynamical equations of robotic systems, NDOB design takes a special form. In this article, the focus of applications is on robotic systems possessing EL dynamics.

Nonlinear Disturbance Observer

Nonlinear Disturbance Observer Structure and Assumptions on Disturbances

Consider the following nonlinear-control affine system where is the state vector, is the control input vector, is the lumped disturbance vector, and is the output vector. The lumped disturbance vector d is assumed to lump the effect of unknown disturbances. The functions f, g1, and g2 in (1) are assumed to be smooth. Additionally, the functions and represent the nominal model of the system.

The problem to solve is to design an observer, called the DOB, to estimate the lumped disturbance vector d from the state and input vectors. The output of the DOB can then be used for feedforward compensation of the disturbances.

An NDOB structure, that can be employed to estimate the lumped disturbance vector is where is the internal state vector of the DOB, is the estimated disturbance vector, and is called the auxiliary vector of the NDOB. The matrix is called the NDOB gain matrix. This section shows that there should exist a certain relation between the auxiliary vector and the NDOB gain .

The disturbance-tracking error is defined to be Before presenting the disturbance-tracking error dynamics of the NDOB, a brief sketch of the derivation of this NDOB is provided.

Taking the derivative of and assuming yields . It is desired for the estimation error to asymptotically converge to zero. Therefore, it is desirable to have where and is some positive scalar. We also have . Assuming that has full column rank for all , the gain matrix where is a left inverse of , that is, Hence, we have Accordingly, Before proceeding further, note that if the derivative of the state is assumed to be available, then design is completed at this stage. Defining the variable and inserting into the above equation gives Therefore, if we have Taking the derivative of , the tracking error dynamics can be computed as With the fact that , it can be seen that the disturbance-tracking error dynamics are governed by It is assumed that the observer gain has been designed to satisfy uniformly with respect to for some positive constant . In general, the design of the NDOB gain matrix is not trivial. The structure of must be known to design the gain matrix . Therefore, determining this gain depends on the application under study.

In the case of EL systems and due to the special structure of the underlying dynamical equations, such a gain matrix can be found. Under the assumption that has full row rank for all x, which is also the case for EL systems under study in this article, an expression for is given in To analyze how the NDOB achieves disturbance tracking, the Lyapunov candidate function is considered. Taking the derivative of yields Some prior knowledge regarding the lumped disturbance dynamics is required to analyze further the performance of an NDOB in terms of stability and disturbance tracking. In this article, it is assumed that there exists a positive real constant (which need not be known precisely) such that Note that the constant ~ does not need to be known precisely; the NDOB stability analysis only requires knowing some upper bounds on .

Finally, it is assumed that the disturbance d satisfies the matching condition; namely, there exists a smooth function such that In other words, the lumped disturbance and the control input act on the system through the same channel. Finally, it is assumed that holds uniformly with respect to for some positive constant .

To investigate the add-on feature of NDOBs, the following DOB-based control input is considered: The nominal control input is designed for the system without disturbances, namely, the nominal control system Under the matching condition and using the DOB-based control input, yields

Brief Review of Nonlinear Systems

Uniform ultimate boundedness is a useful concept in nonlinear systems literature that can be effectively used for control design because it provides a practical notion of stability under certain assumptions.

Definition: A solution of the nonlinear system with initial condition is said to be uniformly ultimately bounded with respect to a set if there exists a constant , dependent on and , such that for all .

In a closed-loop system, it is desired that the set and the convergence time depend on control parameters such that, with proper design, and can be made arbitrarily small. Moreover, if the domain of attraction can be arbitrarily enlarged by tuning control parameters, it is said that the closed-loop system is semiglobally practically stable.

Theorem S1 gives a sufficient condition for having the uniform ultimate boundedness property in terms of the existence of a Lyapunov function.

Theorem S1 (Uniform Ultimate Boundedness)

Let be the open ball centered at the origin with radius and be a continuously differentiable function such that

for all and all . Take such that and . Then, for every initial state , the trajectories of the system converge with an exponential rate proportional to to a ball centered at the origin with a radius proportional to . Moreover, these trajectories enter the ball in a finite time dependent on and , and remain within the ball after .

Theorem S2 guarantees the existence of suitable Lyapunov functions when the system is exponentially stable.

Theorem S2 (Converse 逆 Lyapunov Theorem)

Let be an equilibrium point for the nonlinear system , where is continuously differentiable, is the open ball centered at the origin with radius , and the Jacobian matrix is bounded on , uniformly in . Let , , and be positive constants with . Let be the open ball centered at the origin with radius . If the trajectories of the system satisfy for all and all , then then, there exists a function that satisfies the inequalities for some positive constants , and .

One of the fundamental concepts in analyzing the stability of nonlinear systems is that of passivity. Passive systems, such as linear circuits that only contain positive resistors, possess desirable stability properties. Consider the nonlinear-control affine system The system is said to be passive if there exists a nonnegative differentiable function, called the storage function, , with , and nonnegative continuous function , with , such that, for all admissible inputs and initial conditions , the inequality holds. A weaker notion of passivity that is used in this article is that of quasi-passivity (or emipassivity). System is said to be semipassive in , if there exists a nonnegative function ; is open, connected, and invariant such that for all admissible inputs, for all initial conditions in , and for all time instants for which the solutions exist; and the inequality holds, where the function is nonnegative outside the ball for some positive constant . If , the system is called semipassive.

Semi /Quasi-passivity of NDOBs

Taking the derivative of , we have

The first thing to notice is that it immediately follows from the uniform ultimate boundedness theorem that for all , the disturbance-tracking error converges with an exponential rate, proportional to , to a ball centered at the origin with a radius proportional to . Since the ultimate bound and convergence rate of the NDOB can be changed by tuning the design parameter , it is said to be practically stable. Indeed, increasing the parameter results in better disturbance tracking response; that is, both convergence rate and accuracy of the DOB are improved when is increased. It is remarked that when , the disturbance-tracking error ed will converge to zero asymptotically with an exponential rate.

To analyze the semi/quasi-passivity property of the nonlinear system in the presence of disturbances, the candidate Lyapunov function is considered, where is the candidate Lyapunov function for the nominal control system , for which the inequality ​ holds. Taking the derivative of yields To proceed further, Young’s inequality is used as It can be seen that Therefore, when the augmented nonlinear system with NDOB is a semi/quasi-passive system with input , output , and According to the inequalities, the damping injection coefficient can be chosen to be as small as desired, namely, by choosing the constant to be arbitrarily small, provided that the coefficient , which is the convergence rate of disturbance-tracking error, is chosen to be sufficiently large. However, there exists a tradeoff between the NDOB gain and noise amplification.

It is assumed that the control input makes the origin of the nominal closed-loop system globally exponentially stable and .

When the control input makes the origin of the nominal closed-loop system , where There exists a function that satisfies To analyze the convergence properties of NDOB error dynamics together with the exponentially stabilizing nominal control input , the candidate Lyapunov function It can be seen that Taking the derivative of yields Using Young’s inequality, it can be seen that The disturbance-tracking error and the state converge with an exponential rate, proportional to , to ab all centered at the origin with a radius proportional to .

NDOB Design for Euler –Lagrange Systems

Brief Review of Euler–Lagrange Systems

The model of a fully actuated EL system with an -dimensional configuration space is where is the vector of generalized coordinates, is vector of the generalized velocities, and is the vector of control inputs. The Lagrangian function is a smooth function and assumed to have the form where is the kinetic energy function and is the potential function.

In the case of mechanical systems, s the sum of mechanical kinetic energies. is the generalized inertia matrix and is assumed to be symmetric and positive definite.

We have and is the vector of Coriolis and centrifugal forces. Also, are the forces generated by potential fields such as the gravitational field.

Following the EL systems literature, it is assumed that the EL model under study has the following properties:

P1): The inertia matrix satisfies where and are some positive constants and it holds uniformly with respect to .

P2): The matrix is skew-symmetric; namely, for all .

P3): The matrix is bounded in and linear in ; namely, for all and some positive constant .

P4): Let be an arbitrary vector. There exists a linear parameterization for EL models of the form where is a regressor matrix of known functions and is a vector containing EL system parameters.

The model of a fully actuated EL system with DOFs and states is given by The Lagrangian function is a smooth function and assumed to have the form Then, we have Note that the EL dynamics can be written in the nonlinear-control affine form with where is the dimension of the state space.

NDOB Structure for EL systems

The lumped disturbance vector is assumed to satisfy for all , for some positive constant . This class of disturbances includes, but is not limited to, harmonics and external disturbances.

In EL systems, the NDOB dynamic equations take the form Defining the disturbance-tracking error to be , its governing dynamics can be computed as To remove dependence of the disturbance-tracking error dynamics on acceleration vector , should hold. On the other hand, should hold between the NDOB gain and auxiliary vector.

Since is a function of , it necessarily follows that Therefore, the gain matrix can have the form where X is a constant symmetric and positive definite matrix to be determined. For simplicity of exposition,the matrix is considered to be equal to , where and is the positive constant in Property P1 of EL systems. Using the above gain gives Following from Property P1 of EL systems and the NDOB gain matrix structure, with is

Augmented dynamics of EL systems with NDOBs

To suppress the adverse effects of disturbances, the following NDOB-based control input is applied to the EL system It follows that the augmented EL dynamics with NDOB and control input are

Semi/Quasi-Passivity Property of Augmented EL Systems with NDOBs

In this section, it is assumed that the nominal control input has the form of conventional passivity-based controllers designed for EL systems; namely, it can be written as To analyze the behavior of the augmented EL system and NDOB, the storage function is considered. It can be shown that The design parameters and should be chosen such that cd becomes positive.

Since this NDOB possesses the add-on feature, there is no need to modify any previously designed passivity-based controllers for EL systems integrated with this NDOB.