An anti-windup control of manufacturing lines: performance analysis

Transcription

1 An anti-windup control of manufacturing lines: performance analysis A. Pogromsky Systems Engineering Group Eindhoven University of Technology The Netherlands In cooperation with R. A. van den Berg, W. van den Bremer, J. E. Rooda January 11, Bremen, Workshop on Mathematical modeling of transport and production logistics Systems Engineering Group 1/32

2 Outline Introduction A continuous-time model of manufacturing machines Control of manufacturing machines Problem statement PI control Integrator windup Integrator anti-windup Performance analysis of Lur e systems Control of manufacturing lines Future research Systems Engineering Group 2/32

3 Introduction Introduction Customer Supply chain Manufacturer Systems Engineering Group 3/32

4 Introduction This presentation focuses on manufacturing systems. Related problems: Supply chains Parallel calculations Multiprocessor embedded systems Biological networks Systems Engineering Group 4/32

5 ẋ = f (x, w) x(t) R n, w(t) R m. Following ideas of Demidovich (1967) The system is uniformly convergent for a class of inputs W if for every w( ) W there is a solution x(t, t 0, x 0 ) such that x(t) is bounded on (, + ) x(t) is globally uniformly asymptotically stable Att: x(t) is defined on the whole time axis Systems Engineering Group 5/32

6 Properties of uniformly convergent systems x(t) is unique (due to uniformity) if w(t) is periodic, so is x(t) a cascade of uniformly convergent systems is uniformly convergent (due to boundedness assumption); even though each system is quadratically convergent the cascade is not necessarily quadratically convergent Systems Engineering Group 6/32

7 A continuous-time model of manufacturing machines Customer Supply chain Manufacturer Systems Engineering Group 7/32

8 Let v(t) be the production speed and y(t) is the cumulative output ẏ = v Constraints: v(t) 0 v(t) v max in a line: y i 1 y i, otherwise v i = 0 (the buffer in front of ith machine should be nonempty) Systems Engineering Group 8/32

9 Control of manufacturing machines Problem statement The output of the system should follow demand y d (t). tracking lim y(t) y d(t) = 0 t approximate tracking for some accuracy > 0 y d (t) is the current demand lim sup y(t) y d (t) t y d (t) = u d t + y d0 + r(t) where 0 u d v max and r(t) is the fluctuation on the market Systems Engineering Group 9/32

10 PI control t v(t) = k p (y(t) y d (t)) k i (y(s) y d (s)) ds If a control contains an integral, there is no static error, i.e. if r(t) 0 the tracking goal is achieved asymptotically; if r(t) 0 the tracking 0 properties can be analyzed by linear control theory (Bode plots). Saturation = integrator windup Systems Engineering Group 10/32

11 Introduction multi temp 2007/11/15 :16 page 1 #1 Problem: more than one steady state solutions can co-exist u Time Systems Engineering Group 11/32

12 AWsystem temp 2007/11/15 : page 1 #1 Introduction Integrator anti-windup r k P + + yc u k I / s w 1 s y k A If the saturation is active, an extra signal prevents from the windup. What is an anti-windup? Different problem statements co-exist (finite L 2 gain, finite incremental L 2 gain, etc.) Our focus: the system should be uniformly convergent with (r, w) as an input. Systems Engineering Group 12/32

13 Introduction AWsystem temp 2007/11/15 : page 1 #1 Important observation: the original model with constraints on v can be represented in the form r k P + + yc u k I / s w 1 s y k A with w = v max /2 u d and the saturation nonlinearity u = v max 2 sat(y c), sat(y c ) := sign(y c ) min{1, y c } The control problem: y(t) approximately follows r(t) (market fluctuation). Systems Engineering Group /32

14 If k I, k p, k a > 0, r(t) is uniformly continuous, w(t) < 1 and v max k a k p > 2 the system is uniformly convergent. The proof is based on the Lypunov function V = (x 1 x 2 ) P(x 1 x 2 ) with positive definite P = P > 0, and x 1 (t), x 2 (t) being two different solutions (in an appropriate coordinate system) The derivative of V satisfies V α(t)v, t0 +T t 0 α(s)ds > 0 α is integrally separated from zero uniformly in t 0 and uniformly with respect to the initial conditions from any given compact set. Systems Engineering Group 14/32

15 Performance analysis of Lur e systems If a system is uniformly convergent it has a unique bounded (on the whole time axis) GUAS solution x(t). It allows to pose a problem of performance analysis transient performance: how fast any x(t) converges to x(t) steady state performance: properties of x(t) We focus on the steady state performance with harmonic r(t) = b sin(ωt) computer simulation (accurate, numerically inefficient) describing functions, Galerkin approximation (numerically efficient, approximate) Systems Engineering Group 15/32

16 Lur e system: Incremental sector condition ẋ = Ax Bφ(y) + Fu y = Cx + Du 0 φ(y 1) φ(y 2 ) y 1 y 2 µ An approximate system ξ ζ = Aξ BKζ + Fu = Cξ + Du Systems Engineering Group 16/32

17 The gain K is to be chosen to minimize that is J := 1 T ( T K = 0 T 0 [φ( ζ (t)) K ζ (t)] 2 dt, ) 1 T ζ 2 (t)dt 0 If u = b sin ωt, ζ (t) = a sin(ωt + ψ) φ( ζ (t)) ζ (t)dt. If φ is odd, K(a) = 2 π φ(a sin θ) sin θdθ. πa 0 Systems Engineering Group 17/32

18 Approximation: ξ ζ = Aξ BK(a)ζ + Fb sin ωt = Cξ + Du Harmonic balance equation (HBE): 1 + K(a)G(iω) 2 a 2 = C(iωI n A) 1 F + D 2 b 2, where G(iω) = C(iωI n A) 1 B. Question: given b, ω, is the amplitude a determined in a unique way? Answer: check the FDI for the frequency of excitation: µre G(iω) > 1. Idea of the proof: the left hand side of HBE should be a monotonically increasing function of a. Systems Engineering Group 18/32

19 FDI should be satisfied for one frequency. Systems Engineering Group 19/32

20 Accuracy of harmonic linearization Problem: estimate the difference between z(t) = H x(t) and η(t) = H ξ (t) ρ 1 := ρ 2 := sup C(ikωI n A + µ k=3,5,... 2 BC) 1 B sup H(ikωI n A + µ k=3,5,... 2 BC) 1 B (A, B) is controllable, (A, C) is observable. HBE has a unique positive real solution a(b, ω) ρ 1 µ < 2 φ is an odd function Systems Engineering Group 20/32

21 Then x(t) is the only 2π/ω periodic solution the following estimate is true ω 2π 2π/ω 0 [ z(t) η(t)] 2 dt γ 2 v 2 (a(b, ω)), where v 2 (a) = 1 2π and 2π 0 [ 2 π π 0 φ(a sin θ) sin θdθ sin ϑ φ(a sin ϑ)] 2 dϑ γ = 2ρ 2 2 µρ 1. Idea of the proof: contraction mapping argument Systems Engineering Group 21/32

22 A continuous-time... Illustrative example antiwindup2 temp 2007/2/27 14:24 page 1 #1 reference K P K I / s s output K A K i = 20, K p = 10, K a = 0, ω = 1 K(a) = ( 2 π sin 1 ( 1 a 1, a 1 ) 1 1a, a > 1 2 ) + 1 a Systems Engineering Group 22/32

23 Linear approximation k A =0.5 Bound k A =0.5 Experimental k A =0.5 Linear approximation k A =0 Bound k A =0 Experimental k A = ω [rad/s] Accuracy of harmonic linearization. Blue AW, green no AW. Systems Engineering Group 23/32

24 Introduction multi temp 2007/11/15 :16 page 1 #1 If the conditions of theorem are not satisfied the describing function method can be misleading due to possible subharmonic solutioins u Time Systems Engineering Group 24/32

25 Control of manufacturing lines The last machine (N) in the line should follow y d (t). The jth machine should follow y d (t) + γ j (y d (t) y j+1 (t)) With such a coupling neglecting positivity constraints imposed on the buffers one gets a cascade system. Recall that a cascade of uniformly convergent systems is uniformly convergent, hence the analysis of the manufacturing line can be performed mutatis mutandis. Systems Engineering Group 25/32

26 Implementation issue The controller produces a continuous command v j (t) (the speed of production, jth machine). This signal should be converted into on-off form v PWM j (t) (similarly to pulse-width modulation) so that T 0 v j (t)dt T 0 v PWM j (t)dt A minimal time for an on -phase is t 0 j - the time required for the jth machine to process a lot. Systems Engineering Group 26/32

27 Introduction Results of computer simulation (4 machines in a line) I Systems Engineering Group 27/32

28 Results of computer simulation (4 machines in a line) Systems Engineering Group 28/32

29 Results of computer simulation (4 machines in a line) Bufffer 1,2 Bufffer 2, r(t) Bufffer 3, Time Systems Engineering Group 29/32

30 Future research. From manufacturing lines towards manufacturing networks Assembling, parallel machines: To study separately topology of the network and individual machine dynanics Passivity-based approach, similarly to 1. A. Pogromsky, G. Santoboni and H. Nijmeijer, Partial synchronization: from symmetry towards stability, Physica D, A. Pogromsky, A partial synchronization theorem, submitted Systems Engineering Group 30/32

31 Constraints on communication between the machines: Discretization, batching, finite capacity of the information channels, drops Shannon-like theorems for control of networks (See A. Matveev, A. Savkin, Estimation and Control over Communication Networks, Birkhäuser Boston, 2008) Reentrant systems: To extend results of J.A.W.M. van Eekelen, A.A.J. Lefeber, J.E. Rooda Systems Engineering Group 31/32

32 A simple continuous-time model of a manufacturing machine An anti-windup control of systems with saturation Performance analysis of systems with saturation in frequency domain Extension to manufacturing lines Computer simulation for a more detailed model Systems Engineering Group 32/32