Adaptive control of uncertain systems with gain scheduled reference models

Abstract

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.

Adaptive control of uncertain systems with gain scheduled reference models and constrained control inputs

“…The gains in adaptive laws Γ ∈ (n+2m)×(n+2m) , Γ ∆ ∈ m×m are positive definite matrices Γ = Γ T > 0 and Γ ∆ = Γ T ∆ > 0. [19] for detailed proof. Remark 9: The proof of Theorem 2 showed the boundedness of e v (t), however it can not guarantee the boundedness of the tracking error e(t).…”

“…and error e(t) is in the order of ||e(t)|| = O[sup τ ≤t ||∆v(τ )||].Proof: See[19] for detailed proof.Remark 10: Theorem 3 implies that if the initial conditions of the state and the parameter error lie within certain bounds, then the adaptive system will have bounded solutions. The local nature of the result for unstable systems is because of the saturation limits on the control input.…”