Adaptive Dynamic Surface Control: Stability Analysis Based on LMI

In this paper, quadratic stability of adaptive dynamic surface control for a class of nonlinear systems in strict-feedback form is analyzed in the framework of linear matrix inequality. While the existence of controller gains and filter time constants for semi-global stability was theoretically proved in the literature, it is not sufficient to describe how a set-point value and parameter update laws affect stability and parameter convergence. Thus, it is necessary to provide a systematic analysis method to guarantee both stability and parameter convergence. By deriving the augmented closed-loop error dynamics in linear differential inclusion form, a sufficient condition of the controller gains for stability and parameter convergence is derived in the form of linear matrix inequality. Finally, the quadratic Lyapunov function for its quadratic stability is computed numerically via convex optimization.


Introduction
The dynamic surface control (DSC) is a dynamic extension of multiple sliding surface (MSS) control to overcome the drawback of "explosion of complexity" in backstepping as well as MSS control [1]. The use of a series of dynamic filters enables the controller to be designed sequentially and simple. Furthermore, the existence of controller gains for semi-global stability was theoretically proved in [1]. Recently, an analysis and design method in the framework of convex optimization has been introduced to allow us to find a quadratic Lyapunov function numerically for a class of nonlinear systems called strict-feedback form [2].
This control approach was extended to a class of nonlinear systems where the uncertainty is linearly parameterized, e.g., 1 af in (1) where a is an unknown constant and f 1 is a known nonlinear function. The adaptive backstepping method has been developed [3] and extended to a class of time-delay nonlinear systems [4,5]. As introduced above, the adaptive DSC to solve the problem of "explosion of complexity" has been developed for a class of nonlinear systems and time-delay systems [6,7]. Furthermore, DSC has been combined with adaptive neural network control scheme in the literature [8,9]. However, the useful tools such as linear matrix inequalities (LMIs) are hard to apply to nonlinear system with linearly parameterized uncertainties. There is little work in the literature for LMIs to be used for stability analysis of adaptive nonlinear control problems.
The following example illustrates the design procedure of adaptive dynamic surface control in [6]: , which is called a set-point control problem.
First, define the first error surface as 1 . After taking its derivative along the trajectories of (1) where K is a controller gain and â is the estimate of the unknown parameter a following the update law as propose in [6]: where ρ is a positive constant.
Then, define the second sliding surface as 2 2 2d where x 2d equals 2d x passed through a first-order filter, i.e., where τ is a filter time constant. Similarly, the derivative of S 2 along the trajectories of (1) is , and the control input is derive as x  is calculated from (4) such that It is interesting to remark that the calculation of 2d x  becomes simpler due to the inclusion of the first-order filter while it results in "explosion of complexity" in backstepping.
A next question is how to design a set of controller gains to guarantee stability, e.g., K, τ and ρ in the example. It was proven in [6] that there exist a set of controller gains (K and τ ) to guarantee the stability for stabilization and set-point control problems. However, the performance of the adaptive DSC depends on ρ critically [10]. If a small magnitude of ρ is chosen, the adaption of a in (3) will be slow and the transient error will be large. On the other hand, too large magnitude of ρ will lead to oscillatory estimation of the parameter, thus resulting in the oscillatory error.
Suppose a = 1 in (1), K = 2.5 in (2) and (5), and τ = 0.05 in (4). When ρ is assigned as 1 and 10 respectively, the time responses of x 1 and â are shown is Fig. 1. As explained above, the larger magnitude of ρ results in faster convergence of estimation error of â and tracking error. However, when ρ = 70, the oscillatory estimation of the parameter is shown in Fig. 1. Thus, the tracking error does not converge to zero. Furthermore, if x 1d is changed to a different constant, although it will be discussed later in Section 4, the different time response (e.g., oscillatory estimation) of â may be shown for the same set of K, τ, and ρ. and x 1d guarantee stability and convergence of the parameter estimation error for the given set of a controller gain (K) and a time constant (τ). The main contribution of this paper is to derive the augmented closed-loop error dynamics including parameter estimation errors and filter errors in linear differential inclusion form, and to derive the sufficient condition for stability and parameter convergence. Fur-thermore, the sufficient condition allows us to check stability of the closed-loop system and convergence of estimated parameters by solving the LMI numerically.
Through this paper, we will use the following notation: is an identity matrix in the sense that all diagonal elements are one whatever the dimension of the matrix is. If n n x ∈ ℜ is a vector, diag(x) is a diagonal matrix with the vector x forming the diagonal and diag(x,i) (or diag(x,-i)) is a square matrix of size (n+i) with the vector x forming the i th super-diagonal (or sub-diagonal) stands for a positive definite (or semidefinite) matrix, Tr(X) is the sum of all diagonal entries in X.

Problem Statement
Consider the class of nonlinear system as follows: where a i is an unknown parameter but bounded by a known positive constant is a known nonlinear function in strict-feedback form in the sense that the f i depend only on 1 is bounded on D i [11]. Therefore, there exists a constant 0 i γ > such that 1 ...
for all x on D i .

Design Procedure
An outline of the standard design procedure for the adaptive DSC described in [6,12] is as follows: Define the i th error surface as where x 1d is the constant value. After taking the time derivative of S i along the trajectories of (6), 1 : The surface error S i will converge to zero if , however there is no direct control over the surface dynamics. If x i+1 is considered as the forcing term for the error surface dynamics, then the sliding condition outside some boundary layer is satisfied if 8) and the update law for the parameter estimate is as follows: ˆi where K i is a controller gain and ρ i is a positive gain.
The next step is to force 1 x + passed through a first-order filter, After continuing this procedure for 1 . After taking its derivative, the control input is is calculated from (10) and the update law of ˆn a  is following , n nd nd n n n n n

Augmented Error Dynamics
The augmented closed loop error dynamics is derived for analysis of stability and parameter convergence. After subtracting and adding 1 i ( 1) i d x + , and using u in (11), the closed-loop dynamics in (6) can be written as (8) and the definition of error surfaces, the above equations can be described as a function of errors as follows: is the filter error and is the parameter estimation error multiplied by f i . In addition, we need to consider the augmented error dynamics due to inclusion of a set of the first order low pass filters and the update law for the estimate. After taking a derivative of 1 i ξ + for 1 1 i n ≤ ≤ − , the filter error dynamics is 1 ( 1) where the last equality comes from (10). By taking a derivative of (8), we can write for 2,..., 1 i n = − . Since the derivative of h i is written as is rewritten as the filter error dynamics in (13) is where i = 1,…, n-1, j = 1,…, n, and k = 2,…, n-1. Furthermore, the above error dynamics can be written in matrix form as follows: 11 ( 1) where the vectors are defined as Since the first block matrix in (20) is invertible such that after multiplying the inverse matrix to both sides in (20), the augmented closed-loop error dynamics are written as where the error state  I  A  I  I  A  I  A  T K  T  T   A  I  I  A  A A A is rewritten as follows: Next, we need to determine the upper bound of ω in (22).
Using the assumption in (7), the upper bound of i ω for 1,..., 1 i n = − is for j = 1,…,n-1. Using (12), i x  is written in a function of z as follows:

Quadratic Stability
Since A cl in (24) is not time invariant due to A 21 is (21), both A 21 and A 31 can be decomposed into a steady-state term and a time varying term under the assumption that 0 z → as t → ∞ for the given set of controller gains. That is,    as t→∞. Moreover, if f i satisfies the so called "persistent excitation" [6], i.e., there exist strictly positive constants a i and T such that for any t > 0, 0 i a →  as t→∞. Otherwise, it is not guaranteed for the estimated parameter to converge to the correct value although x 1 →x 1d as t→∞.

Illustrative Example
Consider (1) Thus, the matrix A n is Hurwitz for all ρ i and there exist a solution for LMI (29). The corresponding time responses of x 1 and â are shown in Fig. 3. It is remarked that a smaller gain of τ allows us to enlarge the range of x 1d for n A to be Hurwitz. However, it is well known that 1/τ in the first-order filter is a cut-off frequency and the noise thus may not be attenuated if τ is too small.

Conclusions
The analysis method for stability and parameter convergence of adaptive dynamic surface control was proposed by deriving the augmented closed-loop error dynamics in linear differential inclusion form. The sufficient condition for stability is derived for the given controller gains in the form of linear matrix inequality. It allows us to analyze both quadratic stability and parameter convergence by computing a quadratic Lyapunov function numerically via convex optimization.