Performance Improvements in Extremum Seeking Control M. Guay September 30, 2016, LCCC Process Control Worshop, Lund
1 Background 2 Perturbation based ESC Basic perturbation based ESC Proportional-integral ESC 3 Recursive least-squares approach RLS Proportional integral ESC 4 Discrete-time systems 5 Distributed network optimization 6 Concluding Remarks and Perspective
Introduction Extremum seeking is a real-time optimization technique. Parameter Plant Control Estimation Real-Time Optimization Figure : Basic RTO loop. RTO is a supervisory system designed to monitor and improve process performance. It uses process data to move the process to operating points that are optimal wrt a meaningful user-defined metric 3 / 59
Introduction In most applications, RTO exploits process models and optimization techniques to compute optimal steady-state operating conditions ◮ Control objectives vs. Optimization objectives Success of RTO relies on ◮ the accuracy of the (steady-state) model ◮ robustness of the RTO approach ◮ flexibility of the control system In the absence of accurate process descriptions (model-based) RTO yields erratic results Successful RTO requires integrated solutions. 4 / 59
Introduction Extremum Seeking Control (ESC) is a model free technique that relies on minimal assumptions concerning: ◮ the process model ◮ the objective function ◮ the constraints ESC only requires the measurement of the objective function and the constraints Considerable appeal in practice ◮ Achieves RTO objectives without the need for complex model-based formulations. 5 / 59
Introduction Extremum-seeking control (ESC) has been the subject of considerable research effort over the last decade. Mechanism dates back to the 1920s [Leblanc, 1922] ◮ Objective is to drive a system to the optimum of a measured variable of interest [Tan et al., 2010] Revived interest in the field was primarily sparked by Krstic and co-workers [Krstic and Wang, 2000] ◮ Provided an elegant proof of the convergence of a standard perturbation based ESC for a general class of nonlinear systems 6 / 59
Introduction Basic ESC objectives: Given an (unknown) nonlinear dynamical system and (unknown) measured cost function: x ˙ = f ( x, u ) (1) = h ( x ) (2) y The objective is to steer the system to the equilibrium x ∗ and u ∗ that achieves the minimum value of y (= h ( x ∗ )) . 7 / 59
Problem Definition The objective is to steer the system to the equilibrium x ∗ and u ∗ that achieves the minimum value of y (= h ( x ∗ )) . ◮ The equilibrium (or steady-state) map is the n dimensional vector π ( u ) which is such that: f ( π ( u ) , u ) = 0 . ◮ The equilibrium cost function is given by: y = h ( π ( u )) = ℓ ( u ) (3) ◮ The problem is to find the minimizer u ∗ of y = ℓ ( u ∗ ) . 8 / 59
Problem Definition x 2 π ( u ) h ( π ( u )) x 1 y ` ( u ) u u ∗ 9 / 59
Basic ESC Loop x = f ( x, u ) ˙ y = h ( x ) ω l s − k s + ω l s + ω h s 10 / 59
Basic ESC Loop Closed-loop dynamics are: x ˙ = f ( x, � u ( t ) + a sin( ωt )) ˙ u � = − ωkξ − ωω l ξ + ωω l ˙ ξ = a ( h ( x ) − η ) sin( ωt ) η ˙ = − ωω h η + ωω h h ( x ) . Tuning parameters are: ◮ k the adaptation gain ◮ a the dither signal amplitude ◮ ω the dither signal frequency ◮ ω l and ω h the low-pass and high-pass filter parameters 11 / 59
Basic ESC loop The stability analysis [Krstic and Wang, 2000] relies on two components: 1 an averaging analysis of the persistently perturbed ESC loop 2 a time-scale separation of ESC closed-loop dynamics between the system dynamics and the quasi steady-state extremum-seeking task. This is a very powerful and very general result. Analysis confirms properties: small a , small ω , small k . Convergence is slow with limited robustness. 12 / 59
Proportional Integral ESC Limitations associated with the two time-scale approach to ESC remains problematic. Two (or more) time-scale assumption is required to ensure that optimization operates at a quasi steady-state time-scale Convergence is very slow. Limits applicability in practice. Improvement in transient performance are possible: Standard ESC is an integral controller → Performance limitation Add proportional action. 13 / 59
Proportional Integral ESC x = f ( x, u ) ˙ y = h ( x ) − 1 τ I s ω h s s + ω h − k 1 a sin( ωt ) a sin( ωt ) 14 / 59
Proportional Integral ESC Proposed PI-ESC algorithm: x = f ( x ) + g ( x ) u ˙ v = − ω h v + y ˙ u = − 1 ˙ ( − ω 2 � h v + ω h y ) sin( ωt ) τ I u = − k a ( − ω 2 h v + ω h y ) sin( ωt ) + � u + a sin( ωt ) . Tuning parameters: ◮ k and τ I are the proportional and integral gain ◮ a and ω are the dither amplitude and frequency ◮ ω h ( >> ω ) is the high-pass filter parameter. 15 / 59
Proportional Integral ESC Theorem 1 Consider the nonlinear closed-loop PIESC system with cost function y = h ( x ) . Let Assumptions 1, 2, 3 and 4 hold. Then 1 there exists a τ ∗ I such that for all τ I > τ ∗ I the trajectories of the nonlinear system converge to an O (1 /ω ) neighbourhood of the unknown optimum equilibrium, x ∗ = π ( u ∗ ) , 2 there exists ω ∗ > 0 such that, for any ω > ω ∗ , the unknown optimum is a practically stable equilibrium of the PIESC system with a region of attraction whose size grows with the ratio a k , 3 � x − x ∗ � enters an O ( 1 ω ) + O ( k ωa ) + O ( a ω ) neighbourhood of the u − u ∗ � enters an O ( 1 ωaτ I ) + O ( a 1 origin and � � ω ) + O ( τ I ω ) of the origin. 16 / 59
Proportional Integral ESC Proof of theorem demonstrates that: ◮ the proportional action minimizes the impact of the time scale separation ◮ the integral action acts as a standard perturbation based ESC ◮ Combined action guarantees stabilization of the unknown equilibrium ◮ With fast convergence Impact of dither signal is inversely proportional to the frequency Size of ROA is proportional to a k . PIESC acts as a dynamic output feedback nonlinear controller. 17 / 59
Example 1 We consider the following dynamical system taken from Guay and Zhang [2003]: x 1 = x 2 ˙ 1 + x 2 + u x 2 = − x 2 + x 2 ˙ 1 The cost function to be minimized is given by: y = − 1 − x 1 + x 2 1 . the optimum cost is y ∗ = − 1 . 25 and occurs at u ∗ = − 0 . 5 , x ∗ 1 = 0 . 5 , x ∗ 2 = 0 . 25 The tuning parameters are chosen as: k = 10 , τ I = 0 . 1 , a = 10 , ω = 100 with ω h = 1000 . Outperforms the model-based approach of Guay and Zhang [2003] 18 / 59
Example 1 −1 1 −1.05 0.5 −1.1 y ˆ u 0 −1.15 −0.5 −1.2 −1.25 −1 0 5 10 0 5 10 t t 0.8 30 20 0.6 x 1 , x 2 10 0.4 u 0 0.2 −10 0 −20 0 5 10 0 5 10 t t 19 / 59
RLS Proportional Integral ESC Parameterize ˙ y as: y = θ 0 + θ 1 u = φ T θ ˙ (4) where φ = [1 , u T ] T and θ = [ θ 0 , θ T 1 ] T . θ 0 and θ 1 are unknown time-varying parameters. Proposed PI-ESC given by: u = − k � θ 1 + � u + d ( t ) u = − k ˙ � � θ 1 τ I where ◮ � θ 1 is the estimation of θ 1 . ◮ k is the proportional gain ◮ τ I is the integral time constant. 20 / 59
Parameter Estimation The proposed time-varying parameter estimation scheme consists of an output prediction mechanism. θ + Ke + c T ˙ φ T � ˙ � y � = θ (5) − Kc T + φ T c T ˙ = (6) ˙ η � = − K � η. (7) where � θ are parameter estimates y and � θ = θ − � e = y − � θ K is a positive constant to be assigned c ∈ R p is the solution of the differential equation: 21 / 59
Parameter Estimation The parameter estimation law is given by: Σ − 1 = − Σ − 1 cc T Σ − 1 + k T Σ − 1 − δ Σ − 2 ˙ (8) with initial condition Σ − 1 ( t 0 ) = 1 α I , and the parameter update law: η ) − δ ˙ θ ( t 0 ) = θ 0 ∈ Θ 0 , � � � θ = Proj (Σ − 1 ( c ( e − � θ ) , Θ 0 ) , (9) 2 where δ is a positive constant. Proj { φ, � θ } denotes a Lipschitz projection operator Krstic et al. [1995] such that θ } T � θ ≤ − φ T � − Proj { φ, � θ, (10) θ ( t 0 ) ∈ Θ 0 = � ⇒ � θ ∈ Θ , ∀ t ≥ t 0 (11) where Θ � B ( � θ, z θ ) , 22 / 59
Parameter Estimation Assumption 4: There exists constants α 1 > 0 and T > 0 such that � t + T c ( τ ) c ( τ ) T dτ ≥ α 1 I (12) t � ∀ t > 0 . Theorem 1 Let Assumptions 1 to 4 hold. Consider the extremum-seeking controller and the parameter estimation algorithm. Then there exists tuning parameters k , k T , K and τ ∗ I such that for all τ I > τ ∗ I . the system converges exponentially to an O ( D/τ I ) neighbourhood of the minimizer x ∗ of the measured cost function y . 23 / 59
Example 2 Consider the following system x 1 = x 2 ˙ x 2 = − x 1 − x 2 + u ˙ with the following cost function: y = 4 + ( x 1 − 1 . 5) 2 + x 2 2 . Tuning parameters: k T = 20 , K = 20 I , k = 0 . 25 and τ I = 0 . 15 . d ( t ) = 0 . 1 sin(10 t ) . θ (0) = [0 , − 1] T , The initial conditions are � x 1 (0) = x 2 (0) = u (0) = 0 . 24 / 59
Example 2 3 x 1 x 2 2.5 2 1.5 1 x 1 x 2 0.5 0 −0.5 −1 −1.5 −2 0 10 20 30 40 50 60 70 80 90 100 t Figure : State trajectories as a function of time. 25 / 59
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