Model (In-)Validation from a H and µ perspective

Transcription

1 Model (In-)Validation from a H and µ perspectie Wolfgang Reinelt Department of Electrical Engineering Linköping Uniersity, S Linköping, Seden WWW: olle@isy.li.se Febrary 1999 REGLERTEKNIK AUTOMATIC CONTROL LINKÖPING Report no.: LiTH-ISY-R-2186 Technical reports from the Atomatic Control grop in Linköping are aailable by anonymos ftp at the address ftp.control.isy.li.se. This report is contained in the portable docment format file 2186.pdf.

2 Model (In-)Validation from a H and µ perspectie Wolfgang Reinelt Smmary for ISY, Abstract We gie a short oerie on methods of Model (In-)Validation, that fit to the robst control frameork. The idea is that the mismatch beteen a measred datm and an expected datm is explained by a distrbance signal and an error model, representing nmodelled dynamics. The key qestion is if there exists a pair (, ), sfficiently small, that can prodce the measred datm. In particlar, e ie the different approaches by Smith et.al. and Poolla et.al., their nmerical soltion and gien examples. 1 Problem Setp The general setp for robst control is depicted in figre 1(a): a generalized plant P ith the inpts control signal and distrbance and an error, representing nmodelled dynamics. The plant P is gien (modelling, identification) and e hae a measred datm ( meas,y meas ). The qestion is: Does the datm fit to the model? The main idea is that the (possible) mismatch beteeen a measred datm and the expected (otpt-)datm is explained by a distrbance signal and an error model. The relation beteen inpts and otpt is gien by ULFT: ( z y ) ( ) P11 P = 12 P 13 P 21 P 22 P 23 y = ( P 21 (I P 11 ) 1 [P 12,P 13 ]+[P 22,P 23 ] ) ( =: P ) =: F U (P, ) (1) Figre 1(b) shos a simplification of the general case: a eighted additie error. As a special case, it disables the feedback (P 11 = 0) and fixes the location of the distrbance (P 12 =0). Theseto simplifications make the optimization problem (presented in the next section) conex. We state the Problem, treated in the folloing sections: ( Problem-Definition: Sppose the setp in figre 1(b). The (scaled) model P (plant and eights) is gien, also gien a measred datm ( meas,y meas ). Do there exist 1and 1sothateqn.(2)holds? ) (2) Scaling is possible by exploiting F U (γp, ) = γf U (P, γ ). Diision of Atomatic Control, Dept of Electrical Engineering, Linköping Uniersity, S Linköping, Seden, olle@isy.li.se, olle/ 1

3 z z P P z P y P 11 P 12 P 13 P 21 P 22 P 23 y P nom (a) General (U)LFT case: generalized plant P and error. (b) Simplified strctre: eighted additie error and nominal plant P nom 2 Three Different Frameorks Figre 1: Model alidation frameork. We hae a certain plant inpt meas ith a measred otpt y meas, a nominal otpt y nom (ithot errors) and a modelled otpt y mod (inclding the error model). The last to signals are calclated by the (error-)model ith inpt meas. All signals are of finite length N. Ideally, the residm r = y meas y nom shold be zero 1. To check the qality of the error model (, ), e compare the measred otpt ith the modelled otpt, i.e. e compare the residm ith y mod y nom. Considering the aboe simplified plant strctre, e get the folloing r = y meas y nom? = ymod y nom = P + P nom meas + P P z }{{ meas P } nom meas = = P + P (3) Notable that there is no need fot P nom to be linear. The lhs of eqn.(3) is knon by measrement of y meas resp. calclation of y nom = P nom meas.intherhs, aries by 1and is gien by = z ( ) z = P z meas (5) The rhs of eqn.(5) is knon, eqn.( ) contains the dynamics-error, hich ill be remoed in the next step. As eqn.( ) mst hold for all 1, the qestion is, if there exists a relation beteen inpt z and otpt? This qestion is ansered by the so-called Extension Theorem: Eqn.( ) holds for a 1, iff the inpt signal z is larger than the otpt signal. Using this, eqn.( ) degenerates to smaller than z (4) 1 that s hy the comptational complexity of the problem depends in operator theoretical sense on the dimension of the kernel of the mapping (, ) r, compare ith eqn.(3). 2

4 and the to degree freedom optimization is redced to a conex optimization problem: minimize ith respect to (3,4,5). The reslt of the optimization is the minimm-norm distrbance, responsible for the gien datm ( meas,y meas ). Finally, e get sfficient inalidation theorems of the folloing kind (dropping technical statements): Inalidation Theorem: The model is inalid, if > 1. The extension theorems for the three frameorks are responsible for the comptaional complexity, becase they increase the size of eqn.(4) in different amonts. 2.1 Discrete Freqency Domain (DFD) Initial ork: Smith [8], oerie [1, sec 3], example [6]. The DFD approach transforms the time domain data, gien in (3,4,5) into freqency domain data by DFT [1, eqns.(5-7)] for all N freqencies. The extension theorem for replacing the ncertainty replaces eqn.(4) eqialently by V n V n Z n Z n, n (DFD 4) The conditions for the optimization are eqns.(3,5) transformed into the freqency domain and eqn.(dfd 4). Exact formlation [1, Lemma 2 + Theorem 3]. Properties/Comments Qadratic objectie + linear constraints no local minima. Een fll sized LFT problems can be soled, applying µ techniqes [8]. The problem remains conex as long as the SSV can be calclated by its pper bond (depends on the nmber of blocks). A comptational example exists [6]: to liqids of different temperatre are mixed in a tank (MIMO, 2 2), to different models are alidated. Application of DFT: signals, hae to be zero for negatie times. No problem for (depends on ), bt for, therefore restriction to static P,see[6,sec2.3]. Comptational complexity N tractable. Another type of problem is posed in [8]: minimize the size of and. The problem is similar to the comptation of µ, bt not soled. To aoid problems ith DFT, e jmp back into the time domain: 2.2 Discrete Time Domain (DTD) Initial ork: Poolla [7], oerie [1, sec 4]. The DTD approach transforms the plse response coefficients of P,P,P z, gien in (3+5) into their associated loer block Toepliz matrices (this a N N matrix) and the signals into appropriate ones (N ector); see [1, eqns.(10+11)]. The extension theorem for replacing the ncertainty replaces eqn.(4) eqialently by V V Z Z (DTD 4) 3

5 here V and Z are the associated loer block Toepliz matrices of the signals (i.e. eqn.(dtd 4) is a matrix ineqality). Exact formlation [1, Theorem 4+5]. Properties/Comments Problem conex as long as Z Z constant ( P 11 = P 12 =0). Generalcase? No restrictions on and for negatie times as in the DFD (appearing from DFT): nonzero can be handled by residm, nonzero be initial conditions of P. Also LTV pertrbations possible in the frameork. This theory orks also for mltidimensional signals. Comptational complexity N 5, not feasible for reasonable data-length, een in LMI formlation [2, sec 4]. Therefore no examples gien 2.3 Sampled Data Domain (SDD) Initial ork: Smith and Dllerd [3] (technical details [5]), eqialent reslts independently deried by Poolla [4], oerie [1, sec 5], example [2]. DTD regards the plant as a prely discrete-time system, ith a bilt-in sampling time T.SDD is based on DTD, bt interprets the plant as a sampled continos system. The separation of plant and sample/hold nit enables s to sbsample, i.e. e se the DTD-machinery for a sbset of or data to get a feasible problem. The theoretical reslts are deried sing the lifting operation. After this transformation, size and appearance of the extension theorem for replacing the ncertainty are similar to the DTD case (inclding transformation to loer block Toepliz matrices): Exact formlation [1, Theorem 7+8]. Properties/Comments ˆV ˆV Ẑ Ẑ (SDD 4) The inalidation theorem gets neccessary and sfficient for T 0 (hich is only of theoretical interest). Same comments as in DTD, bt sbsampling possible. Start ith sbsampling time T sb T and decrease ntil the model is inalid or T sb = T. Example: heating system, SISO [2]. Comptational time ithin the example: model inalid for data-length of N = 64, this iteration-step needed Mflops (2h40mins CPU-time on Ultra1). The final step as the 6th. [2,table1]. 4

6 3 Qestions 1. DTD and SDD in case of fll LFT: conexity is lost. Other soltions? 2. DFD solable for fll LFT problems (µ): implementation? examples? 3. Exploiting sparse strctre in SDD to get a faster implementation [2]? 4. MIMO problems in SDD 5. iff inalidation theorems? 6. Sppose model is inalid becase of min 2 =1.36, ho to adjst the model knoing this ale 1.36? Scaling and bonds in general? 7. All approaches compte the minimm size of for all 1. What abot the qestion: ( k,y k ) gien, mimimm size of and [8, Problem 4.1]? References [1] Smith, R., G.E. Dllerd, S. Rangan and K. Poolla: Model Validation for Dynamically Uncertain Systems. Mathematical Modelling of Systems, Vol.3, No.1 (Janary 1997), pp [2] Dllerd, G.E. and R. Smith: Sampled Data Model Validation: an Algorithm and Experimental Application. Int. J. of Robst and Nonlinear Control, Vol.6, No.9/10 (September/October 1996), pp [3] Smith, R. and G.E. Dllerd: Continos-Time Control Model Validation Using Finite Experimental Data. IEEE Trans. Atomatic Control, Vol.41, No.8 (Agst 1996), pp [4] Rangan, S. and K. Poolla: Time domain alidation for sample-data ncertainty models. IEEE Trans. Atomatic Control, Vol.41, No.7 (Jly 1996), pp [5] Dllerd, G.E. and R. Smith: A Continos-Time Extension Condition. IEEE Trans. Atomatic Control, Vol.41, No.5 (May 1996), pp [6] Smith, R.: Model Validation for Robst Control: an Experimental Process Control Application. Atomatica, Vol.31, No.11 (Noember 1995), pp [7] Poolla, K., P. Khargonekar, A. Tikk, J. Krase and K. Nagpal: A time domain approach to model alidation. IEEE Trans. Atomatic Control, Vol.39, No.5 (May 1994), pp [8] Smith, R. and J.C. Doyle: Model Validation: A Connection Beteen Robst Control and Identification. IEEE Trans. Atomatic Control, Vol.37, No.7 (Jly 1992), pp [9] Kost, R.L., M.K. La and S.P. Boyd: Set-Membership Identification of Systems ith Parametric and Nonparametric Uncertainty. IEEE Trans. Atomatic Control, Vol.37, No.7 (Jly 1992), pp (references in reersed chronological order) 5