Exact Tracking Using Backstepping Control ... - Semantic Scholar

.. . (1) .. .

ferentiator possesses the important features of finite-time exact convergence to the real value of the estimated derivatives and the possibility to provide some degree of smoothness to the estimated signal. The last can be done by the appropriate selection of the order of the differentiator. In this subsection, , are used to denote the scalar differentiator variables. Let be the function to be differentiated, under the assump, the th order differtion that a constant exists such that entiator can be expressed in the following form:

.. . .. . (4) for suitable positive constant coefficients to be chosen recursively large in the given order. A possible selection of the differentiator pa, , , , rameters is , . The following equalities are true after a finite time transient process in the absence of noise (see [17]): (5)

, , , is the control signal, are known nonlinear are known nonsingular masmooth vector fields, , are unknown bounded perturbation trices and terms due to parameter variations and external disturbances with at bounded derivatives with respect to system (1). In parleast of system (1), the th ticular, for any possible trajectory is bounded, i.e., time derivative of where

(2)

The notation is used to represent the signal correspondent to the application of a differentiator of order to the signal . IV. DISTURBANCE IDENTIFICATION The following dynamics is designed for each coordinate (6) Define the state estimation errors as dynamics to the state estimation error is given by

(7)

(3) is a known positive constant, denotes the set of all where derivatives are continuous, differentiable functions whose first denotes the th time and when evaluated along the possible trajectories of derivative of system (1). . The whole state vector and the conThe control output is trol signal are assumed to be known. The control problem is to design a controller such that the output tracks a smooth desired reference with bounded derivatives in spite of the presence of the perturbations . In this paper, the equations are understood in the Filippov sense [16] in order to provide for possibility to use discontinuous signals in the disturbance identification algorithms. Note that Filippov solutions coincide with the usual solutions, when the right-hand sides are continuous. It is assumed also that all considered inputs allow the ex. istence and extension of solutions to the whole semi-axis III. ARBITRARY ORDER EXACT DIFFERENTIATOR The arbitrary-order robust exact differentiator [17] plays an important role to compensate the disturbances to the system (1). This dif-

. The associated

where i.e.,

and

are

vectors

with and

p-components,

Given that the disturbance satisfies (2) and (3), it is possible to obtain the following properties of the state estimation error: (8) (9) In view of (8) and (9), it is possible to estimate the value of , using the high-order sliding mode differentiator described in Section III. In order to allow that the estimated disturbances can be directly applied in the backstepping controller preserving the differentiability of the signals, the order of the differentiator should be carefully selected. Let the gains of the differentiator (4) be chosen such that and, let the initial conditions of the differentiator be equal to zero. The disturbances can be estimated as (10)

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 58, NO. 8, AUGUST 2013

V. BACKSTEPPING CONTROL WITH DISTURBANCE COMPENSATION In this section, the estimated disturbances obtained in Section IV are used in a backstepping design procedure to guarantee exact compensation of the real disturbances and global tracking of the command signal . and , the following virtual Step 1. Defining control is proposed: (11) where is a Hurwitz matrix which provides the rate of convergence of to . The time derivatives of each component of the command can be obtained by application of (4) as signal Step i. The tracking errors are defined as , then the remaining virtual controls are given by

(12) where are Hurwitz matrices. The time derivatives of each compoare obtained by application of (4) as nent of the virtual control . Step n. In this step, the real control is calculated as

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, are chosen sufficiently large Suppose that according to [17]. Then, as a consequence of the Lemma 8 from [17], any solution of (14) satisfies the equalities after a finite-time transient process. It is important to remark that the transient of the disturbance identification process can affect the behavior of the closed loop system, i.e., the control signal (13) depends on the estimated disturbance. However, the transient of the control loop cannot affect the identification process, i.e., the dynamics of the estimation algorithm (7) does not depend on the control. In order to guarantee that the disturbance estimation error will not affect the control loop, it is possible to ensure that all the trajectories of the differential inclusion (14) begins on the sliding-point set . The zero initial conditions of the differentiator (4) and the knowledge in (7) begins from zero of the state, ensure that all the estimations , begins from and that the error variables . Given that the the point-set sliding motion starts from the sliding set, the equality (10) is established since the first moment. Notice that, due that the disturbance is exactly estimated since the first moment, then the estimation does not possess a transient process. This completes the first part of the proof. In the second part of the proof, the convergence of to is studied. , , . Define • Consider the first error coordinate . The Lyapunov candidate is given by (15) Its first derivative is

(13) is a Hurwitz matrix. The time derivatives of each component where are obtained by the application of (4), in of the virtual control . particular The following theorem synthesizes the results of the section. Theorem 1: Let the perturbed system (1) be affected by the disturwhich satisfy the conditions (2) and (3). Using the estibances mated disturbances (10), the controller (13) guarantees the exponential convergence of the control output to the command vector . Proof: First, the convergence of the estimated disturbances to their real values should be proven. Let consider that an instance of the differentiator (4) is applied to each component of the th auxiliary coordinate (7). Define the error , , for all variables , and . Given that the disturbances satisfy the conditions (2) and (3), the evolution of each one of the above defined error coordinates satisfy the following differential inclusions:

(16) In view of (11), and assuming that becomes tive of

Due to being a Hurwitz matrix, the above given inequality is satisfied, then the error converges exponentially to zero, hence converges exponentially to . . A Lya• Consider now the second error coordinate is given by punov function candidate for the pair (17) Its first derivative is computed as

(18) Considering now that obtain

.. .

, the first time deriva-

and substituting (12) into (18), we

(19)

(14)

is Hurwitz, the error system converges exDue that converges exponentially to and ponentially to zero, hence converges exponentially to .

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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 58, NO. 8, AUGUST 2013

Fig. 1. Tracking of signal

Fig. 2. Control signal .

by control output .

• The same procedure can be applied recursively to the remaining . The Lyapunov function candidate for variables until the total error system is

Taking its first derivative and substituting the proposed value of given in (13), we obtain

Fig. 3. Disturbances

It is clear that the error vector exponentially conconverges to , verges to zero. The above implies that converges to , and so on, till converges to and as a consequence converges exponentially to . to zero implies the The exponential convergence of , in spite control output exponentially tracks the command signal of the disturbances. Q.E.D.

and their estimates

,

, 2, 3.

where , , and . Disturbances estimates , and are obtained by application of (10) with and the following design parameters:

VI. EXAMPLE Consider the third-order pertur