Ten years of progress in Identification for Control. Outline

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1 Ten years of progress in Identification for Control Design and Optimization of Restricted Complexity Controllers Grenoble Workshop, January, 2003 Michel Gevers CESAME - UCL, Louvain-la-Neuve, Belgium In collaboration with B.D.O. Anderson, R.R. Bitmead, X. Bombois, B. Codrons, F. De Bruyne, G.C. Goodwin, S. Gunnarsson, R. Hildebrand, H. Hjalmarsson, C. Kulcsar, J. Leblond, A. Lecchini, O. Lequin, L. Ljung, B. Ninness, A.G. Partanen, M.A. Poubelle, G. Scorletti, P.M.J. Van den Hof, and Z. Zang. 1 Outline Outline Why identification for control? A control-relevant nominal model iterative design Iterative Feedback Tuning Robustness analysis of data-based uncertainty sets Control-oriented uncertainty sets Conclusions 2

2 Why identification for control Why identification for control? If the estimated model is exact it is optimal for all applications In general, there is model uncertainty due to Use of limited complexity model structure bias error Finite number of noisy data variance error The quality of the model (and its uncertainty) depend on the design of the identification : experimental conditions, identification criterion, validation criterion, etc Design User s choices!!! 3 Why Identification for control The state of the art around 1990 System identification and robust control had been developed as two separate theories Robust control was based on model uncertainty descriptions that were not data-based Prediction Error Identification had evolved from a search for the true system to the estimation of an approximate model bias error, variance error G 0 (e jω ) G(e jω, ˆθ N ) = G 0 (e jω ) G(e jω, θ ) }{{} bias error + G(e jω, θ ) G(e jω, ˆθ N ) }{{} variance error However, when it came to the design of a controller on the basis of data collected on the system, the dominant idea was : first identify the best model, then design the controller on the basis of that best model. Certainty equivalence principle. Best was not best for control design. 4

3 Why Identification for control The major open problems of the time The model uncertainty descriptions delivered by Prediction Error Identification were not usable by robust control theory. Robust control theory was not taken into account in identification design. The role of the experimental conditions was not well understood. There was very little understanding of the interplay between control design and identification design in a closed loop system. 5 Why Identification for control The relevant feedback loops r t û C t ˆv + t id Ĝ ŷ t - r t + C u G v t t id 0 y t - Identified system Experimental setup r t ū C(Ĝ) t ˆv + t Ĝ ȳ t - r t u C(Ĝ) t v + t G 0 y t - Design loop Achieved loop A game with four players: G 0, Ĝ, C id, C(Ĝ). 6

4 Why Identification for control Identification as a design problem If identification is viewed as approximation, and if one understands the connection between the identification criterion, the quality of the model, and the intended application, then identification can be viewed as a goal-oriented design problem (L. Ljung) Wahlberg and Ljung (1986), Design variables for bias distribution in transfer function estimation Gevers and Ljung (1986), Optimal experiment design with respect to the intended model application This viewpoint was instrumental in the development, from around 1990, of identification for control. An agenda for research was proposed in: Gevers (1991), Connecting identification and robust control: a new challenge, Plenary at 9th IFAC Symp. on System Identification, Budapest. 7 Why Identification for control The chronology of progress A control-relevant nominal model, or understanding bias error manipulation This led to iterative model/controller design ( ) Which in turn led to iterative (model-free) controller design (1994) Robust stability analysis of control-oriented design schemes ( ) Robust performance design of control-oriented model uncertainty sets ( now) 8

5 A control-relevant nominal model : iterative control design A key observation r t + ū C(Ĝ) t Ĝ ȳ t - r t + u C(Ĝ) t G 0 y t - Design loop Achieved loop Ĝ is good for control design if the achieved and designed loop are close. G 0 C 1 + G 0 C ĜC 1 + ĜC = (G 0 Ĝ)CS 0 Ŝ where S 0 = 1 1+G 0 C, Ŝ = 1 1+ĜC. Closed loop PE identification naturally minimizes V (θ) = π π G 0 Ĝ(θ) 2 C id S 0 2 D 2 Φ r dω Problem: C id C(Ĝ) iterative design: C k = C(Ĝ k ) and D k = Ŝ k. 9 A control-relevant nominal model : iterative design Iterative identification and control design schemes Several iterative schemes were developed in the early nineties; they differed by the choice of identification criterion and control performance criterion. Pioneers: Delft, Cesame, Australian National University Adaptive control is replaced by batch processing much easier to analyze This work led to intense new work on closed loop identification Transfer of theory to industrial applications was remarkably fast!! We provided a methodology for something people in industry were waiting for: how to use the data we collect on a controlled process to improve control performance? The result: a restricted complexity control-oriented nominal model. The control objective shapes the bias error distribution. 10

6 Iterative Feedback Tuning How about direct (model-free) controller updates? In 1994, the convergence study of the model-based iterative schemes led to a new scheme, now called Iterative Feedback Tuning, or IFT. Idea : minimize a control performance criterion directly w.r.t. controller parameters using closed loop data only Earlier attempts had failed because such minimization requires a model Key new idea (Hjalmarsson) : the gradient of the control criterion can be computed from signals obtained on the actual closed loop system, by feeding back the output of the closed loop system; hence the name IFT. 11 Iterative Feedback Tuning A brief presentation of IFT Control error : ỹ(ρ) y(ρ) y d = ( ) C(ρ)G0 r y d C(ρ)G 0 Performance criterion : J(ρ) = 1 [ N 2N E (ỹ t (ρ)) 2 + λ Optimal controller : t=1 ρ = arg min ρ J(ρ) C(ρ)G 0 v ] N (u t (ρ)) 2 t=1 12

7 Find ρ that solves 0 = J ρ (ρ) = 1 N E The IFT algorithm Iterative Feedback Tuning [ N N ] ỹ t (ρ) ỹ t (ρ) + λ u t (ρ) u t (ρ) t=1 Controller parameter updates : t=1 ρ i+1 = ρ i γ i R 1 i J ρ (ρ i) J ρ (ρ i) is computed by a judicious filtering of signals obtained from a special experiment This experiment requires feeding back the output at the reference input 13 The impact of IFT Iterative Feedback Tuning IFT has been adopted immediately by the Belgian multinational Solvay Applied to numerous process control and mechanical applications It has given rise to a large number of extensions and variants: Applications to MIMO and to nonlinear systems Application to multirate sampled-data systems Masked IFT: tuning PID controllers with minimum settling time Tuning controllers with constraints using interior point methods IFT with a relaxed reference model for the tuning of nonminimum phase systems Use of randomized algorithms for faster convergence Optimal filtering for accuracy and convergence speed For a recent survey, see H. Hjalmarsson, Iterative Feedback Tuning - An Overview, Int. Journal of Adaptive Control & Signal Processing, Vol. 16, No 5, pp ,

8 Robustness analysis of data-based uncertainty sets So far for nominal design, but what about uncertainty? Understanding the interplay between bias error distribution and control objective led to control-oriented nominal models Can we similarly hope to build control-oriented uncertainty sets from data? We need to understand the interplay between model uncertainty and robust control objectives. 15 Robustness analysis of data-based uncertainty sets Assumptions The robust control paradigm One wants to design a controller C for an unknown system G 0 The system is known to belong to some model uncertainty set D The control design procedure often uses a nominal model Ĝ of G 0, typically the center of D Objectives The controller C designed from Ĝ must stabilize all models in D (and therefore also the true G 0 ) robust stability The performance achieved by C on all models in D must be close to the nominal performance of the [Ĝ, C] loop. robust performance Important observation Achievement of these two objectives hinges as much on the choice of C as on the choice of D!! 16

9 Robustness analysis of data-based uncertainty sets The key role of the uncertainty set The last observation raises several key questions about the construction of uncertainty sets. Can we estimate uncertainty sets D from data? Are they compatible with existing robust analysis and design tools? Are some uncertainty structures better than others? How can we define a control-oriented uncertainty set? Can we tune the estimation of the uncertainty set for robust control design? The last two questions began to be addressed around Robustness analysis of data-based uncertainty sets Estimation of uncertainty sets from data New methods have been developed for the estimation of uncertainty sets from data, based on PE identification methods Combination of PE and worst-case identification We now have robustness tools to relate these uncertainty sets to corresponding sets of stabilizing controllers We have defined quality measures of these uncertainty sets that are related to the size of the corresponding set of stabilizing controllers 18

10 Robustness analysis of data-based uncertainty sets Examples of data-based uncertainty sets Example 1: PE uncertainty set D P E = {G(z, θ) (θ ˆθ) T R(θ ˆθ) < γ} with Ĝ = G(z, ˆθ) Example 2: Youla-parametrization uncertainty set D Y = {G G = N x + D c D x N c, γ} where N x, D x, N c, D c, are stable, rational, proper: Ĝ = N x D x is a nominal model C = N c D c is any stabilizing controller of Ĝ Nice feature of D Y : all models in D Y are stabilized by C. Drawback: the set is a function of the particular controller C chosen for the parametrization. 19 Robustness analysis of data-based uncertainty sets The generic PE model uncertainty set By PE identification methods, one can construct a generic PE model uncertainty set : { D = G(z, δ) G(z, δ) = e + Z } Nδ 1 + Z D δ, δ U = {δ (δ ˆδ) T R(δ ˆδ) < 1} where ˆδ is the parameter estimate resulting from the validation step. R is a symmetric positive definite matrix R k k, proportional to the inverse of the covariance matrix of ˆδ. Z N (z) and Z D (z) are row vectors of size k of known transfer functions. e(z) is either zero or a known transfer function. The true G 0 belongs to D with probability α. We can fix α. 20

11 Robustness analysis of data-based uncertainty sets A robustness theory for PE uncertainty sets We have developed robust analysis tools for these PE uncertainty sets Necessary and sufficient condition to check whether a given controller C stabilizes all models in a PE uncertainty set D Necessary and sufficient condition to check whether a given controller C achieves a given level of performance with all models in D We have defined a quality measure for these PE uncertainty sets that is related to the size of a corresponding set C of stabilizing controllers design tool for the construction of a control-oriented uncertainty set? 21 Robustness analysis of data-based uncertainty sets N & S conditions for controller validation Theorem : (Automatica, 2001) Consider a generic PE uncertainty set D and a controller C(z) = X(z)/Y (z) that stabilizes the nominal model G(z, ˆδ). Then all models in D are stabilized by C(z) if and only if where max ω µ(m D(e jω )) 1, M D (z) = (Z D + X(Z N ez D ) 1 )T Y +ex 1 + (Z D + X(Z N ez D ) Y +ex )ˆδ, T is a square root of the matrix R defining D : R = T T T. φ = T (δ ˆδ), whereby δ U φ 2 < 1. µ(m D (e jω )) is the stability radius of the loop [M D (z) φ]. 22

12 Robustness analysis of data-based uncertainty sets Worst case performance over a PE uncertainty set We consider a generic PE uncertainty set D and a controller C(z) which is validated for stability over D. At a frequency ω we define the worst case performance of all closed loop systems [G(z, δ) C(z)] with G(z, δ) D as J W C (D, C, W l, W r, ω) = max σ [ ] max Wl H(G(e jω, δ), C(e jω ))W r. G(z,δ) D where T (G, C) = GC 1+GC C 1+GC G 1+GC 1 1+GC W l (z) and W r (z) are diagonal weights Note that J W C is a frequency function : it defines a template. 23 Robustness analysis of data-based uncertainty sets Controller validation for robust performance Theorem : (Automatica, 2001) Consider a PE uncertainty region D and a robustly stabilizing controller C. Then at frequency ω: J W C (D, C, W l, W r, ω) = γ opt (ω), where γ opt (ω) is the optimal value of γ for the following standard convex optimization problem involving LMI constraints: minimize γ over γ, τ subject to τ 0 and [ ] [ ] Re(a11 ) Re(a 12 ) R Rˆδ Re(a 12 ) Re(a τ 22) ( Rˆδ) T ˆδ T Rˆδ 1 where the a ij are functions of D and C. < 0 24

13 Control-oriented uncertainty sets What is a control-oriented uncertainty set? No clear consensus yet! Two possible lines of thought: 1. D (1) better than D (2) if the set of controllers C (1) that stabilize D (1) contains the set of controllers C (2) that stabilize D (2). 2. Consider a model uncertainty set D that contains the true system G 0, with Ĝ at its center. Let C be a controller that stabilizes Ĝ, with nominal stability margin bĝ,c and nominal performance J(Ĝ, C). Then we could say that D is tuned for robust control if C stabilizes all models in D the worst case stability margin b D,C is not much smaller than than bĝ,c the worst case performance J W C (D, C) is not much smaller than the nominal performance J(Ĝ, C). 25 Control-oriented uncertainty sets A control-oriented quality measure for an uncertainty set Definition: the Vinnicombe ν-gap (for scalar systems) δ ν (G 1, G 2 ) = max ω G 1 (jω) G 2 (jω) 1 + G1 (jω) G 2 (jω) 2 = max ω κ(g 1, G 2 ) provided some winding number condition is satisfied. Consider a PE model set D and a model Ĝ. Worst case chordal distance at frequency ω between Ĝ and D: κ W C (Ĝ(e jω ), D) = sup G D D κ(ĝ(e jω ), G D (e jω )) Worst case ν-gap between Ĝ and D: δ W C (Ĝ, D) = sup G D D δ ν (Ĝ, G D ) = sup ω κ W C (Ĝ(e jω ), D)) 26

14 Control-oriented uncertainty sets Why is this measure control-oriented? Connection with stabilizing controllers : Let C stabilize Ĝ with stability margin bĝ,c. Then C stabilizes all models in D if and only if δ W C (D) < bĝc. The smaller δ W C (D), the larger is the set C of stabilizing controllers. 27 Control-oriented uncertainty sets How to design control-oriented uncertainty sets? Major topic of present activity on System Identification for control. Two questions have been addressed so far: Comparing uncertainty structures. Given that there are a number of uncertainty structures that can be estimated from data, is one better tuned for robust control than another? (Van den Hof ) Optimal experiment design for PE uncertainty structures. The measure δ W C (D) defines the size of the set of stabilizing controllers. Can we design the identification experiment for the estimation of D in such a way as to minimize δ W C (D)? (Hildebrand & Gevers, SIAM J. Control, 2003) 28

15 Conclusions Conclusions Enormous progress in ten years time Much better understanding of closed loop identification iterative controller retuning the interplay between identification and robust control Workable methods and algorithms have been produced; they have already found their way in industrial applications: EDF, Solvay, ASML (wafer stepper), CSR (sugar mill), Philips (CD player), etc Much work remains to be done on optimal experiment design We are still far from a fully automated procedure. Our main objective for the future: direct controller tuning from data. 29