Lecture 11 - Introduction to Sliding Mode Control | AOE 5344, Study notes of Aerospace Engineering

Download Lecture 11 - Introduction to Sliding Mode Control | AOE 5344 and more Study notes Aerospace Engineering in PDF only on Docsity! Lecture 11: Introduction to Sliding Mode Control One goal of feedback control is to compensate for uncertainties in the model of the system being controlled. Two types of uncertainty that arise are structured uncertainty (i.e., uncertainty in the parameter values of a model with known structure) and unstructured uncertainty (i.e., uncertainty in the model itself due, for example, to model order reduction). The purpose of robust control is to make explicit consideration of such uncertainties in the control design process in order to achieve some desired control objective. One of the most common techniques for nonlinear robust control design is called sliding mode control. This Lyapunov-based control design technique is similar to backstepping. The presentation here will follow Khalil, Section 14.1. A very different, but equally accessible presentation is given in Chapter 7 of [1]. Consider a system ẋ = f0(x) + δ1(x) + G(x) (u + δ2(x,u)) (1) where x(t) ∈ Rn is the state and u(t) ∈ Rm is the control input. Assume that f0 and G are known, while δ1 and δ2 are terms intended to account for errors in the model structure and parameters. The terms f0 and G define the “nominal system” while δ1 and δ2 represent estimates of the modeling error. We assume that there is a diffeomorphism1 T (x) = ( η ξ ) , such that ∂T ∂x G(x) = ( 0 Ga(x) ) where Ga is a nonsingular, m × m matrix. The two vector components of the transformed coordinates satisfy η(t) ∈ Rn−m and ξ(t) ∈ Rm. Under this change of coordinates, the original system (1) transforms into ( η̇ ξ̇ ) = ( ∂T ∂x ẋ ) ∣ ∣ ∣ x=T −1(η,ξ) = ( ∂T ∂x (f0(x) + δ1(x) + G(x) (u + δ2(x,u))) ) ∣ ∣ ∣ x=T −1(η,ξ) = ( f(η, ξ) + δη(η, ξ) fa(η, ξ) + Ga(η, ξ) (u + δξ(η, ξ,u)) ) (2) (Note that any uncertainty due to δ1 in the ξ̇ equation can be absorbed into δξ because Ga is nonsingular.) Suppose that f , fa and δη vanish when (η, ξ) = (0,0). Treating ξ as an input to the η̇ subsystem, assume that there is a choice ξ = φ(η) with φ(0) = 0 which asymptotically stabilizes the origin of η̇ = f(η,φ(η)) + δη(η,φ(η)). (3) Define z = ξ − φ(η). Recall that, in the backstepping approach, we constructed a Lyapunov function candidate for the origin of the entire system. Differentiating this function, we were able to choose u so that the Lyapunov rate was negative definite. Thus, the control u drove η and z (equivalently, η and ξ) to zero asymptotically. In sliding mode control, we instead choose u to drive z to zero in finite time and to maintain z = 0. Once z = 0, the η dynamics evolve according to (3). The set {(η, ξ) | z(η, ξ) = 0} 1A diffeomorphism is a smoothly differentiable map with a smoothly differentiable inverse. 1 defines a surface within the state space, called the “sliding surface” or the “sliding manifold.” The system’s behavior, restricted to the sliding surface, is called the “sliding mode.” To define a control law to drive z to zero in finite time, we note that ż = fa(η, ξ) + Ga(η, ξ) (u + δξ(η, ξ,u)) − ∂φ ∂η (f(η, ξ) + δη(η, ξ)) . (4) Define u = ueq + G −1 a (η, ξ)v (5) where ueq = G −1 a (η, ξ) ( −fa(η, ξ) + ∂φ ∂η f(η, ξ) ) and where v is an additional control term to be determined. This so-called “equivalent control” is chosen to cancel nominal terms in (4). We do not attempt to cancel the unknown, or poorly known, terms δη and δξ. Substituting the control law (5) into the dynamic equation (4) gives ż = v + ∆(η, ξ,v) where the new “perturbation term” is ∆(η, ξ,v) = Gaδξ − ∂φ ∂η δη. We make the following assumption about the perturbation term: ‖∆(η, ξ,v)‖∞ ≤ ρ(η, ξ) + k‖v‖∞ where ρ(η, ξ) is a known, continuous, non-negative function and where k ∈ [0, 1) is a known constant. This is not a very restrictive assumption, however, it is a quite important one. It describes the accuracy with which we know the dynamic model. Define V = 1 2 ‖z‖2 = 1 2 z21 ︸︷︷︸ V1 + 1 2 z22 ︸︷︷︸ V2 + · · · + 1 2 z2m ︸︷︷︸ Vm as a Lyapunov function candidate for the ż subsystem. Note that V is not a true Lyapunov function because the ż subsystem depends not only on z but also on η. Still, we can use V to find a control law which drives z to zero in finite time irrespective of the η dynamics. The expectation is that η will converge to zero asymptotically once z is zero. Treating each term Vi = 1 2z 2 i of V separately, we have V̇i = d dt ( 1 2 z2i ) = ziżi = zi (vi + ∆i(η, ξ,v)) ≤ zivi + |zi| (ρ(η, ξ) + k‖v‖∞) for each i ∈ {1, . . . ,m}. Define a non-negative function β(η, ξ) which satisfies β(η, ξ) ≥ ρ(η, ξ) + b 2