Sistemas com saturação no controle - VII. VII. Discussões e extensões

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1 1 Sistemas com saturação no controle VII. Discussões e extensões Sophie Tarbouriech LAAS - CNRS, Toulouse, France Colaboradores principais : J.-M. Gomes da Silva Jr (UFRGS), G. Garcia (LAAS-CNRS), I. Queinnec (LAAS-CNRS)

2 Outline 2 Few comments and discussions relative to analysis and synthesis of saturated systems 1. General question 2. Complexity analysis 3. Main remarks 4. Some applications 5. Some extensions 6. Further extensions

3 1. General question 3 Question Is it the effective saturation useful to enlarge the ellipsoidal estimates of the domain of attraction? [Iwasaki, SCL 2002] NO The larger ellipsoidal stability set is achieved for a linear control law (unsaturated) [Gomes da Silva Jr. et al, IEEE-TAC 2003] YES The larger ellipsoidal stability set is achieved for a linear control law (u = Kx), but in this case K is such that the eigenvalues of (A + BK) have real part ɛ 0 poor performance. If time-domain performances constraints are taken into account, the best ellipsoidal domain is obtained with saturating control laws. Trade-offs: performance size of the stability region.

4 2. Complexity analysis 4 Saturation regions approaches: BMI (if the set E(c) is unknown) LMI otherwise 3 m constraints Polytopic LDI approaches: BMI (approach 1) LMI (approach 2) 2 m + m constraints Sector Nonlinearities approaches: BMI (approach 1) LMI (approach 2) 1+m constraints

5 2. Complexity analysis 5 Relative to the complexity analysis of the different approaches, the computation of the number of lines and number of variables (elements of the variables) of the LMIs considered is crucial. Indeed, the computation becomes more complex as these numbers increase. In the contexts of the polytopic LDI approaches (1 and 2) and saturation region approaches, one can verify that the complexity analysis exponentially increases. In the context of the sector nonlinearities approach, the complexity only linearly increases. The approach 2 in the context of sector nonlinearities approaches is very interesting and allows to obtain less complex numerical procedures. Generally, the sector nonlinearities approaches allow to obtain larger domains of stability, but there is no mathematical proof they outclass the other ones.

6 Saturation regions approaches: Useful only for stability analysis. Conditions difficult to test. Conservatism of the method. Polytopic LDI approaches: 3. Main remarks 6 Synthesis conditions similar to analysis condition with the classical change of variables K = YW 1. Approach 1 allows: to deal with global asymptotic stability; to consider LPV problems in the discrete-time case. Approach 2 main advantage is that it provides LMIs conditions. Sector Nonlinearities approaches: Synthesis conditions similar to analysis condition with the classical change of variables K = YW 1. The computation of global stabilizing gains is direct. Lure type domains of stability can be considered, but only approximation can be plotted.

7 4. Some applications 7 Aeronautics (GARTEUR projects): avoidance of PIO Polytopic LDI approaches (past project) Sector nonlinearities approaches (current project) Spatial (Launcher Ariane 5+) Sector nonlinearities approaches Telecommunication networks Sector nonlinearities approaches

8 5. Some extensions 8 Discrete-time systems Control laws via output feedback Static Dynamic It is possible to deal with performance taken into account through LMI regions For example by using Finsler Lemma to do multi-objective Additive disturbance and L 2 problems. Systems subject to uncertainties Time-delay systems Other limitations can be considered such as limitations on the state: Linear limitations Saturation

9 6. Further extensions 9 An interesting problem is to consider systems presenting nested saturations. Such functions appear in: The modelling of nonlinear actuators actuators presenting both amplitude and dynamics saturations [Tarbouriech et al], [Bateman & Lin] The modelling of systems presenting both actuator and sensor amplitude saturation [Tarbouriech et al] The context of some nonlinear control laws [Sontag], [Sussman], [Castelan et al] The forwarding techniques for cascade systems [Teel], [Kokotovic]

10 6. Further extensions 10 To study such systems we use an extension of Lemma 4 of chapter III. Lemma 1 A generic nonlinearity ψ(v) =sat(v) v satisfies the sector condition: ψ(v) T [ψ(v)+w] 0, v, w S(v 0 ) (1) where T is a positive diagonal matrix and the set S(u 0 ) isapolyhedralsetdefinedas follows: S(v 0 )={v R m,w R m ; v 0(i) v (i) w (i) v 0(i), i =1,...,m}

11 6. Further extensions 11 Example. Consider the system: ẋ = A 2 x + B 2 sat u2 (A 1 x + B 1 sat u1 (A 0 x)) This system reads: ẋ =[A 2 + B 2 (A 1 + B 1 A 0 )]x + B 2 ψ 2 + B 2 B 1 ψ 1 with ψ 1 = sat u1 (A 0 x) A 0 x ψ 2 = sat u2 (A 1 x + B 1 sat u1 (A 0 x)) (A 1 x + B 1 sat u1 (A 0 x)) = sat u2 ((A 1 + B 1 A 0 )x + B 1 ψ 1 ) ((A 1 + B 1 A 0 )x + B 1 ψ 1 ) Relation (1) applies for ψ 1 with: Relation (1) applies for ψ 2 with: T = T 1 ; v = A 0 x ; w = G 0 x ; v 0 = u 1 T = T 2 ; v =(A 1 + B 1 A 0 )x + B 1 ψ 1 ; w = G 1 x + G 2 ψ 1 ; v 0 = u 2

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