Indirect Adaptive Robust Control of Uncertain Systems with Unknown Asymmetric Input Deadband Using a Smooth Inverse

In this paper, we present an indirect adaptive robust controller (IARC) for output tracking of a class of uncertain nonlinear systems with unknown input asymmetric deadband in presence of uncertain nonlinearities and parametric uncertainties. Most of the parameter adaptation algorithms, such as, gradienttype and least squares-type require that the unknown parameters of a system appear in affine with known regressor functions globally. However, deadband nonlinearity can not be represented in those global linear parametric form. Therefore, the existing parameter estimation algorithms for deadband focus on some approximate linear parametric model. Hence, even in absence of any other uncertain nonlinearities and disturbances, these algorithms can never achieve asymptotic tracking. Departing from those approximate deadband estimation, we design an indirect parameter estimation algorithm with online condition monitoring. This parameter estimation algorithm in conjunction with a well-designed robust controller and a deadband inverse function can be used to obtain asymptotic tracking without restoring to discontinuous control law. With this strong result in our repertoire, we proceed to design a smooth deadband inverse (SDI) function to avoid certain problems during implementation, e.g, control input chattering and significant appearance of highfrequency dynamics. The effect of such an approximation on the L2-norm of output tracking error is analytically determined. We also show that while operating away from the deadband, the pr0posed controller even with an SDI can achieve asymptotic ∗Address all correspondence to this author. tracking. In presence of disturbances and other uncertain nonlinearities, the proposed IARC controller attains guaranteed transient performance and final tracking accuracy. INTRODUCTION Deadband is a static ‘memoryless’ nonsmooth nonlinearities present in many practical systems, e.g., hydraulic servo value, DC servo motors, mechanical connections and piezoelectric translators. This nonlinearity characterizes the insensitivity of output to the small input values. If the parameters of the deadband are known, it is straight forward to construct a perfect inverse function so that the effect of deadband can be compensated by a well-designed controller. When the parameters of the deadband are unknown, this problem has to be tackled from an adaptive and robust control framework. An adaptive deadband inverse was developed in [1] using unrealistic assumptions, such as, set certainty equivalence. With removal of such assumptions in [2, 3] lead to bounded output tracking, but it could not achieve asymptotic tracking. In [4], even though asymptotic adaptive cancelation of unknown deadband was possible, a much stronger condition of both deadband input and output measurement was made. In most of the applications, only input to the deadband is measured. In [5–7], heuristic approaches such as fuzzy logic and neural networks were used to construct basis functions for deadband nonlinearity compensation. These controllers could achieve only bounded output tracking and therefore, there was no further theoretical improvement in the obtained results. In [8, 9], the deadband was modeled as 1 Copyright c © 2009 by ASME a combination of linear input with either an unknown constant gain for symmetric deadband [8] or a time-varying gain for unknown gain for non-symmetric ones [9] and a bounded input disturbance. With this formulation, the traditional robust control techniques were applied to obtain bounded output tracking error. However, this approach treated deadband as a bounded disturbance without considering the deadband characteristics; thus, leaving an opportunity for possible performance improvement by explicitly considering the deadband effect. Though deadband compensation has been studied extensively as reviewed above, the available results in the literature suffer from undesirable characteristics on two accounts. Firstly, none of them can achieve asymptotic tracking without measurement of deadband output. The main reason for this stems from the fact that it is not possible to obtain a global parameterized model affine with unknown parameters. The usual parameter adaptation algorithms, such as, gradient-type and least squares type, depend on the parametric affine structure to derive all of its nice asymptotic properties. However, it was noticed in this work that even though a global linear parameterization of deadband is not possible, deadband can still be linearly parameterized for most of the operating range away from the deadband. Using indirect adaptive designs and online-condition monitoring, if we estimates the parameters only when the linear parametrization is valid, then accurate parameter estimates can be obtained and perfect adaptive inversion is possible. For such an estimator to work harmoniously with a controller, the workings of controller and estimator module have to be independent of each other. Therefore, we will use an indirect adaptive robust control (IARC) approach [10] in this paper to achieve complete controller-estimator separation. Secondly, the use discontinuous deadband inverse during adaptation was not desirable in practice, as it may lead to control input chattering. In [2], a soft inverse was proposed, but its effect on controller performance was not analyzed explicitly. In [11, 12], a smooth deadband inverse function was proposed for reducing the control chattering. Even though it was identified that the nonsmoothness of deadband is a local problem, the proposed smooth inverses ignores this fact and smooths out the deadband inverse globally; thus, by-passing the problem of deadband completely. As a result of this global smoothening, asymptotic tracking can not be obtained. Furthermore, it was not clear how the smoothening of control affect the controller performance. The second drawback of the existing adaptive designs will be overcome by using a new novel SDI function. The proposed SDI approximates the nonsmooth deadband inverse (NDI) locally and mimics its behavior at large (i.e., when the working range is beyond the deadband). During the actual controller design, the impact of approximating the deadband inverse on output tracking performance is explicitly considered and the controller is designed to minimize its effect. PROBLEM FORMULATION The deadband function DB(v) can be mathematically represented as follow [13]: u(t) = DB(v(t)) =  mr(v(t)−br) v(t)≥ br 0 bl < v(t) < br ml(v(t)−bl) v(t)≤ bl , (1) where br ≥ 0, bl ≤ 0 and mr > 0 and ml > 0 are the gains across the deadband in positive and negative directions respectively. This paper considers the same class of nonlinear system preceded by an asymmetric deadband nonlinearity as in [9, 12]. System Model The system can be represented in the following canonical form ẋi = xi+1, i = {1,2, · · · ,(n−1)} ẋn = DB(v)+ p ∑ i=1 θiYi(x̄n, t)+∆(x̄n, t) (2) where x1 is the system output, Yi, i = 1, · · · , p are some known continuous nonlinear functions, x̄i = [x1, ...,xi] is the vector of the first i states, θ = [θ1, ...,θp] represents the vector of other unknown parameters, ∆(x̄n, t) is the uncertain nonlinearity, v(t) is the actual control input to be designed, and u = DB(v) represents the output of the deadband. For notationalsimplicity, let θb ∈ Rp+4 be the vector of all unknown constant parameters, i.e., θb = [θ1, · · · ,θp,mr,mrbr,ml ,mlbl ]T . The following nomenclature is used throughout this paper: •̂ is used to denote the estimate of •, •̃ is used to denote the parameter estimation error of •, e.g., θ̃ = θ̂−θ. Assumption 1. The unknown parameter vector θ is within a known bounded convex set Ωθ. Without loss of generality, it is assumed that ∀θ∈Ωθ, θimin ≤ θi ≤ θimax, i = 1, . . . , p+4 where θimin, θimax are some known constants. Furthermore, the signs of mr, br, ml and bl are assumed to be known. Assumption 2. The uncertain nonlinearity ∆(x̄n, t) can be bounded by |∆(x̄n, t)| ≤ δ(x̄n, t), where δ(x̄n) is a known positive function. Assumption 3. The output of the deadband u = DB(v) is not available for measurement. The control objective is to synthesize a control law for the input u(t) and a parameter estimation law for the unknown parameter vector θ and unknown deadband parameters mr, ml , br and bl so that: (i) all the signals of the resulting closed system are bounded, (ii) the output x1 tracks the desired output trajectory x1d(t) with a guaranteed transient performance and final tracking accuracy. 2 Copyright c © 2009 by ASME Construction Of Smooth Deadband Inverse The usual practice of dealing with deadband function is to compensate the deadband effect employing a deadband inverse function as follows [2]: v(t) = DI(u(t)) =  u(t)+mrbr mr u(t)≥ 0 u(t)+mlbl ml u(t)≤ 0 (3) Noticing that u(t) > 0⇒ v(t) > 0 and u(t) < 0⇒ v(t) < 0, we can parameterize inverse deadband function as follows v(t) = DI(u(t)) = Hr(u) [ u+mrbr mr ] +Hl(u) [ u+mlbl ml ] (4) where Hr(u) = { 0 if u≤ 0 1 if u > 0 , Hl(u) = { 0 if u≥ 0 1 if u < 0 . The proposed nonsmooth deadband inverse (NDI) function DI(u) can achieve total deadband nonlinearity compensation, when the deadband parameters were completely known. Though the above ideal deadband inverse look natural and theoretically beautiful, it suffers from following practical issues due to its discontinuity of at u = 0. 1. Implementing NDI in practice is extremely sensitive to measurement noise. For input deadband type of problem, u = 0 may be steady-state value for many regulation type of problem such as the point-to-point positioning in precise motion control. As such, it is one of the critical operating points, where problems due to measurement noise are most severe due to almost zero signal-to-noise ratio. 2. The equivalent gain of NDI for very small signals around origin can be thought of as infinite, which makes it impossible to analyze the effect of even small amount of implementation imperfection such as neglected high frequency dynamics. In [12, 14], following smooth deadband inverse (SDI) function was proposed as follows: v = SDI(u) = φr(u) [ u+mrbr mr ] +φl(u) [ u+mlbl ml ] (5) where φr(u) and φl(u) are smooth indicator functions defined as φr(u) = e u/k eu/e0 +e−u/k , φl(u) = e −u/k eu/e0 +e−u/k , where k > 0 is chosen by designer. It was not clear how to make an appropriate choice for k from a quantitative output performance point of view. The effect of this approximation was not considered explicitly on the performance of the system output. Moreover, we can see that in the idealizing case, when we know the deadband parameters completely, we still can not achieve perfect inversion. The main reason is that the smooth inverse function SDI(•)approximates the actual inverse globally, i.e., for all values of u ∈ R. These drawbacks point to the fact that the approximation used in (5) does not represent the solution to the control problem of aforementioned nonsmooth nonlinearity; rather, the approximation is used globally to bypass the nonsmooth nonlinearity problem altogether. In this paper, we propose the following deadband inverse function SDI(u)1: v =SDI(u) = Nr(u) [ u+mrbr mr ] +Nl(u) [ u+mlbl ml ] , (6) where Nr(u) is a nth order smooth curve as defined in (7), n being an odd integer. Nr(u) =  0 if u <−ε n−1 2 ∑ i=0 ( n i ) (ε−u)i(ε+u)n−i 2nεn if |u| ≤ ε