Human Friendly Control : an application to Drive by Wire

Size: px

Start display at page:

Download "Human Friendly Control : an application to Drive by Wire"

Lesley Gregory
5 years ago
Views:

1 Human Friendly Control : an application to Drive by Wire By Emmanuel Witrant Master Thesis, Laboratoire d Automatique de Grenoble with Carlos Canudas-de-Wit 2 Juillet 2002 Typeset by FoilTEX

2 Control Objectives safety of the driver and performance passivity and transparency, linear approximation of the plant, optimal multi-objective synthesis using Linear Matrix Inequalities (LMI). 1

3 Outline 1. System Presentation and objectives 2. Control problem expression 3. Control Synthesis 4. Simulations and testing ground 2

4 1. System Presentation and objectives M fm f h v h K fe ve M fs Fig. 1: Schematic principle of the system 3

5 The Plant f ext h + f h + + fm P fs + + fe + f ext e K H Gm Gs E v h ve Fig. 2: Typical Structure of a manipulator 4

6 Control objectives The model of Hu-Salcudean-Loewen, 95 : v h + Z h f h Z th Z d n f.fe + fe Ze + ve = v h /nv Z te Fig. 3: 2 ports representation of an ideal manipulator idéal Use K and n f, n v to shape the desired impedance of the system satisfying the control objectives. 5

7 Coupled Transparency (impedance) Defines a desired mapping between force and speed. v d h = y d (f h + n f f e ) (1) ( 1 ve d = y d f h + n ) f f e n v n v (2) minimize ṽ h. = v d h v h and ṽ e. = v d e v e with adequate filters to set the frequency domain of minimization. 6

8 Passivity Bilateral coupled passivity : < v h, F h >. = 0 v T h F h dt > β (3) < v e, F e >. = That is, Z th and Z te have to be passive. 0 v T e F e dt > β (4) 7

9 2. Control problem expression Given a desired admittance Y d (s), the goal is to find a control K such that : i. Closed loop transparency, Y t (K) = Y d. ii. Closed loop passivity of Z th (K) = Z t (K) + n f n v Z e and Z te (K) = n v n Z t (K) + n v f n Z h, where Z t (K) is the actual impedance of the f system. but it may not have a solution and Z h, Z e are unknown, relax i. and extend ii. : iii. min K Y t (K) Y d, with iv. Y t (K) : F v ESPR. 8

10 3. Control Synthesis State-space representation of the plant P and the control K : Σ P : ẋ = Ax + B w w + Bu z = C z x + D zw w + D z u y = Cx + D w w Σ K : { ζ = AK ζ + B K y u = C K ζ + D K y (5) Criterion specification : ( ) T z = T zw w = vw w Tṽw et [ A Bj C j D j ] = [ A + BD K C BC K B j + BD K F j B K C A K B K F j C j + E j D K C E j C K D j + E j D K F j ] with B j = B w R j, C j = L j C z, D j = L j D zw, E j = L j D z, F j = D w R j. Express the system into two transfer functions ; one for each objective. 9

11 LMI expression :(S.Boyd et al. 94) iii. : equivalent to Tṽw < γ, Bounded Real lemma) : AT P + PA PB 2 C T 2 B T 2 P γi D T 2 C 2 D 2 γi > 0, P > 0 (6) iv. : Positive Real Lemma : ( A T P + PA PB 1 C1 T B1 T P C 1 D1 T D 1 ) > 0, P > 0 (7) 10

12 Singularity problem D 1 singular ( A T P + PA PB 1 C T 1 B T 1 P C 1 D T 1 D 1 ) > 0, P > 0 implies to combine a LMI with a LME. To solve this problem (Khalil, 96) : Introduce a fictitious sector bound non-linearity simulating some uncertainties, use the circle criterion to design K such that the closed-loop fictitious system is SPR, conclude that the original system is SPR. 11

13 ψ w u P z y K Fig. 4: Fictitious system Where ψ is a sector bound non-linearity defined by : [ψ(t)z Q min z] T [ψ(t)z Q max z] 0, t 0, z Γ IR p (8) 12

14 The resulting fictitious system is : Ā Bw B C z Dzw Dz C Dw D = A B w Q min C z B w B (Q max Q min )C z I 0 C D w Q min C z D w 0 (9) The identity matrix in D allows for the use of the positive real lemma. 13

15 Optimal multi-objective synthesis using LMI (Scherer-Gahinet- Chilali, 97) Two LMIs to solve simultaneously, with P > 0, a linearizing change of variables to include the closed-loop expression, A, B et C real and fictitious have to be the same Q min = 0, a unique control K can be found, its order is the same as the system, two parameters still need to be fixed : γ et Q max. Analysis tools passivity : Nyquist plot (SPR), transparency : H norm and Bode plot, 14

16 4. Simulations and testing ground G m (s) was given by : G m (s) = 1 a m s + b m = 1 0, 0222s + 0, 0042 (10) The desired admittance was chosen as : y d = 1 0, 1s + 1 (11) Some good values for the remaining parameters : γ = 0.7, qh max = and qe max =

17 The Control obtained has the form : K = 1 D [ ] K11 K 12 K 13 K 14 K 21 K 22 K 23 K 24 (12) with D and K ii some 4 th order polynomials. 16

18 Passivité vue par le conducteur (fh vh) Imaginary Axis Real Axis Fig. 5: Passivity of the closed-loop system 17

19 0 (vh vhd)/vhd erreur (%) fréquence (rad/sec) Fig. 6: Minimized criterion ṽ 18

20 5 n v = Amplitude temps (s) Fig. 7: v h and v e with a speed factor n v = 1, 5 (input force = sinusoïde + noise) 19

21 Fig. 8: Testing ground 20

22 Conclusions Different approaches were explored for the transparency, the control has been obtained with some tuning parameters, allowing for the freedom of the user, the simulations results are satisfying and the procedure is validated, experimental results were also obtained. 21

Robust Multivariable Control

Robust Multivariable Control Lecture 2 Anders Helmersson anders.helmersson@liu.se ISY/Reglerteknik Linköpings universitet Today s topics Today s topics Norms Today s topics Norms Representation of dynamic systems Today s topics Norms