Model Reference Adaptive Control of Systems with Gain Scheduled Reference Models

Firstly, a new state feedback model reference adaptive control approach is developed for uncertain systems with gain scheduled reference models in a multi-input multi-output (MIMO) setting. Specifically, adaptive state feedback for output tracking control problem of MIMO nonlinear systems is studied and gain scheduled reference model system is used for generating desired state trajectories. Using convex optimization tools, a common Lyapunov matrix is computed for multiple linearizations near equilibrium and non-equilibrium points of the nonlinear closed loop gain scheduled reference system. This approach guarantees stability of the closed-loop gain scheduled system. Adaptive state feedback control scheme is then developed, and its stability is proven. The resulting closed-loop system is shown to have bounded solutions with bounded tracking error, with the proposed stable gain scheduled reference model. Secondly, the developed control approach is improved for systems with constraints on the control inputs. The resulting closed-loop system is shown to have bounded solutions with bounded tracking error. Sufficient conditions for ultimate boundedness of the closed-loop system are derived. A semi-global stability result is proved with respect to the level of saturation for open-loop unstable plants while the stability result is shown to be global for open-loop stable plants. Thirdly, a decentralized adaptive state feedback control architecture is developed and its stability is proved. Specifically, the resulting closed-loop system is shown to have bounded solutions with bounded tracking error for all the subsystems with the proposed stable gain scheduled reference model. Simulation results are presented for each control architecture. 1 Mathematical Preliminaries 1.1 Projection Operator The definitions and lemmas presented here are mainly adopted from [16, 15, 17]. Definition 1. Consider a convex compact set with a smooth boundary Ωc = {θ ∈ R|f(θ) ≤ c}, 0 ≤ c ≤ 1, (1) where f : R → R is a smooth convex function defined as f(θ) = θθ − θ max θθ max , (2) where θmax is the norm bound imposed on the parameter vector θ, and θ denotes the convergence tolerance of our choice. Let the true value of the parameter θ, denoted by θ∗, belong to Ω0, i.e. θ ∗ ∈ Ω0, the projection ∗Postdoctoral fellow at the School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA 30332, [email protected]. †Assistant Professor at the Mechanical and Aerospace Engineering Department, Missouri University of Science and Technology, Rolla, Mo 65409, [email protected]. 1 ar X iv :1 40 3. 37 38 v1 [ m at h. O C ] 1 5 M ar 2 01 4 operator for two vectors θ, y ∈ R is defined as Proj(θ, y) = { y − Of ||Of || 〈 Of ||Of || , y〉f(θ), if f(θ) > 0 ∧ Of y > 0, y, otherwise, (3) where Of(θ) = ( ∂f(θ) ∂θ1 , ..., ∂f(θ) ∂θn ) ∈ R is the gradient vector of f evaluated at θ and it is computed as Of(θ) = 2θ θθ max , (4) Figure 1 illustrates the projection operator. Figure 1: Illustration of the projection operator [33]. Lemma 1. One important property of the projection operator follows. Given θ∗ ∈ Ω0, (θ − θ∗)T(Proj(θ, y)− y) ≤ 0. (5) Proof. Note that (θ − θ∗)T(Proj(θ, y) − y) = (θ∗ − θ)(y − Proj(θ, y)). For f(θ) > 0 and Ofy > 0, the left-hand side of inequality (5) is (θ∗ − θ) ( y − ( y − Of(θ)(Of(θ)) T ||Of(θ)||2 )) , (6) Since θ∗ ∈ Ω0 and due to the convexity of f(θ), we have (θ∗ − θ)Of(θ) ≤ 0. Hence (θ∗ − θ)Of(θ)(Of(θ))y ||Of(θ)||2 ≤ 0, (7) otherwise Proj(θ, y) = y. Definition 2. The general form of the projection operator is the n × m matrix extension of the vector definition (1). Proj(Θ, Y ) = [Proj(θ1, y1), ...,Proj(θm, ym)], (8) where Θ = [θ1...θm] ∈ Rn×m, Y = [y1...ym] ∈ Rn×m, and F = [f1(θ1)...fm(θm)] ∈ Rm×1, then using definition (1) we have Proj(θj , yj) = { yj − Ofj ||Ofj || 〈 Of j ||Ofj || , yj〉fj(θj), if fj(θj) > 0 ∧ Of T j yj > 0, yj , otherwise, (9) for j = 1 to m. Lemma 2. Let F = [f1(θ1)...fm(θm)] T ∈ Rm×1 be a convex vector function and Θ = [θ1...θm],Θ = [θ∗ 1 ...θ ∗ m], Y = [y1...ym], where Θ,Θ ∗, Y ∈ Rn×m then, trace { (Θ−Θ∗)T(Proj(Θ, Y )− Y ) } ≤ 0. (10)