In this section, we introduce the adaptive computed torque control method, and derive an adaptation law to estimate the unknown parameters. The control of nonlinear systems with unknown parameters is traditionally approached as an adaptive control problem. Adaptive control is one of the ideas conceived in the 1950's which has firmly remained in the mainstream of research activity with hundreds of papers and several books published every year. One reason for the rapid growth and continuing popularity of adaptive control is its clearly defined goal: to control plants with unknown parameters. Adaptive control has been most successful for plant models in which the unknown parameters appear linearly. But in many mechanical systems, the unknown parameters appear in a nonlinear manner. For such systems we define parameter functions, P , such that the system have a linear relationship with respect to these parameter functions. Fortunately, such a linear parameterization can be achieved in most situations of practical interest (Kristic et al., 1995). We only consider such systems throughout this work.

In the linear parameterization process, we partition the system into a model-based portion and a servo portion. The result is that the system's parameters appear only in the modelbased portion, and the servo portion is independent of these parameters. This partitioning involves the determination of parameter functions P , such that the error dynamics is linear in the parameter functions. When this is possible, one can write

where W is a n x k matrix, called the regression matrix, and P is a k x 1 vector, representing the parameter function estimation errors and is defined by P = P - P . Once the parameterization process is done successfully, one can employ the following adaptation law to estimate the parameter functions.

Proposition 2.2 For a system with either constant or slowly varying unknown parameters, the adaptation law

estimates the parameter functions, such that the error dynamics of Eq. (8) becomes stable. Definitions of r and Y are given in the following proof.

The error dynamics is given by Eq. (8). Substituting for T from the linear parameterization law, Eq. (9), we have q + Kvq + Kpq = H-1 W(q, q, q)P, (11)

The aim of the adaptation law is to estimate the parameter functions P , so as to make the right hand side of the above equation approach zero, i.e. by making P approach zero. One can write Eq. (11) in state space form by defining the state vector X as

X = [X1,X2,...,Xn]T, X, 4 [qi q ] T , and the output vector Y as

Y A q + o q, o =diag[^2J..], where O is the filtering matrix, and Y represents the vector of filtered errors. The values of (/>j must be chosen such that the transfer function

* + 0j is strictly positive real (SPR)1. Therefore

Having written the error dynamics in state space form, we employ a Lyapunov-based approach to derive the adaptation law. Consider the following Lyapunov candidate, v =xtpx+PT r-1P, (12)

where p is a positive definite matrix, and r =diag[Y1,Y2v,Yr] with yi >0. Taking the time derivative of (12) yields

Substitution of the state space equations of error dynamics into (13) results

This equation can further be simplified, by adopting the following lemma. Lemma 2.1 (Kalman-Yakubovich-Popov) Consider a controllable linear time-invariant system x = Ax + bu

1A transfer function h (p) is positive real if

Re[h(p)] > 0 for all Re[p] > 0 It is strictly positive real if h (p -s) is positive real for some s > 0 .

The transfer function h (p) = c[pI - A]-1b is SPR if, and only if, there exist positive definite matrices p and Q such that

According to the above lemma, one can write (AtP + pA) = -q in Eq. (14). The adaptation law is found by setting the first term on the right side of (14) equal to zero

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