Adaptive Control Chapter 13: Multimodel adaptive control with switching
Chapter 13: Multimodel adaptive control with switching
Outline 1. Introduction 2. Multimodel Adaptive Control with Switching 3. Stability of the Adaptive System 4. Stability of the Switching System 5. Stability with Minimum Dwell Time 6. Application to the Flexible Transmission System 7. Effects of the Design Parameters I. D. Landau, A. Karimi: “A Course on Adaptive Control”- 5 2
Introduction Consider controller design for a system with very large uncertainty Robust Control : A fixed controller does not necessarily exist that stabilizes the system, or if it exists, it does not give good performances. Adaptive Control : The classical adaptive control gives unacceptable transient adaptation for large and fast parameter variation. Solution: Multimodel adaptive control ◮ Classical multimodel adaptive control ◮ Robust multimodel adaptive control ◮ Multimodel adaptive control with switching ◮ Multimodel adaptive control with switching and tuning I. D. Landau, A. Karimi: “A Course on Adaptive Control”- 5 3
Classical multimodel adaptive control I. D. Landau, A. Karimi: “A Course on Adaptive Control”- 5 4
Robust multimodel adaptive control I. D. Landau, A. Karimi: “A Course on Adaptive Control”- 5 5
Robust multimodel adaptive control ◮ Very similar to the classical multimodel adaptive control. ◮ The controllers are robust with respect to unmodeled dynamics and uses output feedback instead of state-feedback. ◮ The control input is the weighted sum of the outputs of the controllers (no switching). N � u ( t ) = P i ( t ) u i ( t ) i =1 where P i ( t ) is the posterior probability of i -th estimator. ◮ The posterior probabilities are computed as: � β k e − 1 k ( t +1) S − 1 � 2 r ′ r k ( t +1) k P k ( t + 1) = P k ( t ) i =1 β i e − 1 i ( t +1) S − 1 � N 2 r ′ r i ( t +1) P i ( t ) i ◮ The stability of this scheme is not guaranteed. I. D. Landau, A. Karimi: “A Course on Adaptive Control”- 5 6
Multimodel adaptive control with switching I. D. Landau, A. Karimi: “A Course on Adaptive Control”- 5 7
Multimodel adaptive control with switching and tuning ˆ y 0 ε 0 ✲ ✲ ✲ ˆ P - + ✻ y 1 ˆ ε 1 ✲ ✲ ✲ P 1 - + ✻ ˆ y 2 ε 2 ✲ ✲ ✲ u 0 P 2 ✲ - C (ˆ θ ) + ✻ ✇ ˆ u 1 y n ε n ✲ ✲ ✲ ✲ ✲ C 1 P n - + ✻ ✮ u 2 ✲ ✲ y C 2 ✲ ✲ u Plant u n ✲ ✸ C n After a parameter variation (a large estimation error) ◮ First the controller corresponding to the closest model (fixed model) is chosen (switching). ◮ Then the adaptive model is initialized with the parameter of this model and will be adapted (tuning). I. D. Landau, A. Karimi: “A Course on Adaptive Control”- 5 8
Structure of Multimodel Adaptive Control with Switching Plant: LTI-SISO (for analysis) with parametric uncertainty and unmodelled dynamics: � P ( θ ) θ ∈ Θ where P ( θ ) = P 0 ( θ )[1 + W 2 ∆] with � ∆ � ∞ < 1. Other type of uncertainty can also be considered. Multi-Estimator: Kalman filters, fixed models, adaptive models. If Θ is a finite set of n models, these models can be used as estimators (output-error estimator). If Θ is infinite but compact, a finite set of n models with and adaptive model can be used. Multi-Controller: We suppose that for each P ( θ ) there exists C ( θ ) in the multi-controlller set that stabilizes P ( θ ) and satisfies the desired performances (the controllers are robust with respect to unmodelled dynamics). I. D. Landau, A. Karimi: “A Course on Adaptive Control”- 5 9
Structure of Multimodel Adaptive Control with Switching Monitoring Signal: is a function of the estimation error to indicate the best estimator at each time. t J i ( t ) = αε 2 � e − λ ( t − j ) ε 2 i ( t ) + β i ( j ) j =0 with λ > 0 a forgetting factor, α ≥ 0 and β > 0 weightings for instantaneous and past errors. Switching Logic: Based on the monitoring signal, a switching signal σ ( t ) is computed that indicates which control input should be applied to the real plant. To avoid chattering, a minimum dwell-time between two consecutive switchings or a hysteresis is considered. The dwell-time and hysteresis play an important role on the stability of the switching system. I. D. Landau, A. Karimi: “A Course on Adaptive Control”- 5 10
Switching Logic Start Start ✛ ✛ ❄ ❄ σ ( t ) = arg min J i ( t ) σ ( t ) = arg min J i ( t ) i i ✲ ❄ ❄ Wait T d No J σ ≤ (1 + h ) J i Yes Hysteresis Dwell-time ◮ A large value for T d may deteriorate the performance and a small value can lead to instability. ◮ With hysteresis, large errors are rapidly detected and a better controller is chosen. However, the algorithm does not switch to a better controller in the set if the performance improvement is not significant. I. D. Landau, A. Karimi: “A Course on Adaptive Control”- 5 11
Stability of Adaptive Control with Switching Trivial case: ◮ No unmodelled dynamics and no noise, ◮ the set of models is finite, ◮ parameters of one of the estimators matches those of the plant model, ◮ plant is detectable. Main steps toward stability: 1. One of the estimation errors (say ε k ) goes to zero. 2. ε σ ( t ) = y ( t ) − y σ ( t ) goes to zero as well. 3. After a finite time τ switching stops ( σ ( τ ) = k t ≥ τ ). 4. If ε k goes to zero, θ k will be equal to θ and the controller C k stabilizes the plant P ( θ ): ( Certainty equivalence stabilization theorem ) I. D. Landau, A. Karimi: “A Course on Adaptive Control”- 5 12
Stability of Adaptive Control with Switching Assumptions: Presence of unmodelled dynamics and noise. Existence of some“good” estimators in the multi-estimator block. The plant P is detectable. 1. ε k for some k is small. 2. ε σ is small (because of a“good” monitoring signal). 3. All closed-loop signals and states are bounded if: The injected system is stable. y u ✲ Plant Injected System ❄ ✲ y σ ✲ ε σ ✲ + Multi-Controller Multi-Estimator - u σ ✻ ✻ ❄ y ✛ + + I. D. Landau, A. Karimi: “A Course on Adaptive Control”- 5 13
Stability of Switching Systems Each controller stabilizes the corresponding model in the multi-estimator for frozen σ . Question: Is the injected system stable for a time-varying switching signal σ ( t )? Is f σ ( x ) stable? No I. D. Landau, A. Karimi: “A Course on Adaptive Control”- 5 14
Stability of Switching Systems Consider a set of stable systems: x = A 1 x , ˙ x = A 2 x , ˙ . . . x = A n x ˙ then ˙ x = A σ x is stable if : ◮ A 1 to A n have a common Lyapunov matrix ( quadratic stability ). ◮ If the minimum time between two switchings is greater than T d ( minimum dwell time ). ◮ If the number of switching in the interval ( t , T ) does not grow faster than linearly with T ( average dwell time ). N σ ( t , T ) ≤ N 0 + T − t ∀ T ≥ t ≥ 0 T d N 0 = 1 implies that σ cannot switch twice on any interval shorter than T d . I. D. Landau, A. Karimi: “A Course on Adaptive Control”- 5 15
Stability of Switching Systems Common Lyapunov Matrix : The existence of a common Lyapunov matrix for A 1 , . . . , A n guarantees the stability of A σ . This can be verified by a set of Linear Matrix Inequalities (LMI): Continuous-time Discrete-time A T A T 1 P + PA 1 < 0 1 PA 1 − P < 0 A T A T 2 P + PA 2 < 0 2 PA 2 − P < 0 . . . . . . A T A T n P + PA n < 0 n PA n − P < 0 ◮ The stability is guaranteed for arbitrary fast switching. ◮ The stability condition is too conservative. I. D. Landau, A. Karimi: “A Course on Adaptive Control”- 5 16
Stability of Switching Systems by Minimum Dwell Time Theorem : Assume that for some T d > 0 there exists a set of positive definite matrix { P 1 , P 2 , . . . , P n } such that: A T i P i + P i A i < 0 ∀ i = 1 , . . . , N and e A T i T d P j e A i T d − P i < 0 ∀ i � = j = 1 , . . . , N Then any switching signal σ ( t ) ∈ { 1 , 2 , . . . , N } with t k +1 − t k ≥ T d makes the equilibrium solution x = 0 of x ( t ) = A σ ( t ) x ( t ) ˙ x (0) = x 0 globally asymptotically stable. ◮ The first group of LMIs are always feasible because A 1 to A n are stable. ◮ The second LMIs are always feasible if T d is large enough. ◮ T d can be minimized using LMIs. I. D. Landau, A. Karimi: “A Course on Adaptive Control”- 5 17
Stability of Switching Systems by Minimum Dwell Time Proof : Consider the Lyapunov function V ( x ( t )) = x T ( t ) P σ x ( t ). We should show that for any t k +1 = t k + T k with T k ≥ T d > 0 we have V ( x ( t + k )) < V ( x ( t )). Assume that σ ( t ) = i for t ∈ [ t k t k +1 ) and σ ( t k +1 ) = j . We have : x T ( t k +1 ) P j x ( t k +1 ) V ( x ( t + 1)) = x T ( t k ) e A T i T k P j e A i T k x ( t k ) = x T ( t k ) e A T i ( T k − T d ) P i e A i ( T k − T d ) x ( t k ) < x T ( t k ) P i x ( t k ) < < V ( x ( t k )) ◮ This can be proved for discrete-time systems as well. ◮ If A 1 to A n are quadratically stable, i.e. P = P 1 = P 2 = . . . = P n then the LMIs are feasible for any T d > 0. I. D. Landau, A. Karimi: “A Course on Adaptive Control”- 5 18
Recommend
More recommend
