THIS paper deals with robust control in the setup associated

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1 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 50, NO 10, OCTOBER Control-Oriented Model Validation and Errors Quantification in the `1 Setup V F Sokolov Abstract A priori information required for robust synthesis includes a nominal model and a model of uncertainty The latter is typically in the form of additive exogenous disturbance and plant perturbations with assumed bounds If these bounds are unknown or too conservative, they have to be estimated from measurement data In this paper, the problem of errors quantification is considered in the framework of the 1 optimal robust control theory associated with the signal space The optimal errors quantification is to find errors bounds that are not falsified by measurement data and provide the minimum value of a given control criterion For model with unstructured uncertainty entering the system in a linear fractional manner, the optimal errors quantification is reduced to quadratic fractional programming For system under coprime factor perturbations, the optimal errors quantification is reduced to linear fractional programming Index Terms Error analysis, model validation, optimal control, robust control, uncertainty I INTRODUCTION THIS paper deals with robust control in the setup associated with the signal space and linear time-invariant nominal models with norm bounded perturbations and bounded exogenous disturbance For general linear fractional transformation (LFT) model structure, basic results on stability and performance robustness in the setup were presented in [7], [8] and optimal robust synthesis was considered in [4] A priori information for optimal robust synthesis includes a nominal model and upper bounds for perturbations and exogenous disturbance Incomplete a priori information gives rise to the following problems in increasing complexity order: model validation, ie, a test of a given nominal model and assumed upper bounds for perturbations and exogenous disturbance against measurement data; errors quantification, ie, data-based estimation of the bounds of perturbations and exogenous disturbance for a given nominal model; identification, ie, both errors quantification and databased estimation of nominal model The model validation problem was studied since the early 1990s both in the setup (see, eg, [12] and [14] for the time-domain approach) and in the setup [12], [13] While the model validation is hard in the setup (see, eg, [3] and [15] for recent progress), it is easier for study in the setup For Manuscript received May 31, 2004; revised March 14, 2005 and May 30, 2005 Recommended by Guest Editor L Ljung This work was supported by the Russian Fund of Basic Research under Grant The author is with the Department of Mathematics, Komi Scientific Center, Syktyvkar , Russia ( sokolov@syktsuru) Digital Object Identifier /TAC general LFT models with structured uncertainty, the model validation in the setup was reduced in [13] to checking feasibility of linear inequalities produced by data The model validation test is a necessary component of the much more complicated problem of errors quantification The latter is unavoidable when the a priori upper bounds for perturbations and exogenous disturbance are too conservative or unknown The same problem arises also when the assumed upper bounds were falsified by the model validation test The solution of the errors quantification problem depends obviously on the choice of the estimation criterion Typical statements of the problem are to minimize the bound for exogenous disturbance under certain fixed upper bounds for perturbations as in [14] or to minimize the maximum of all upper bounds as in [15] Although similar estimation criteria are motivated by original problems of robust synthesis, they are irrespective of control criteria in specific optimal problems Ideally, the estimation criterion for both errors quantification and identification must coincide with the control criterion in a specific problem of optimal robust synthesis As noted in [10], so highly control-oriented approach to estimation problems leads usually to extremely difficult optimization problems Similar control-oriented approach was used in [9] for state estimation and optimal control of linear systems under exogenous disturbances on finite time intervals The purpose of this paper is to study the problem of optimal errors quantification, which is to find errors bounds that are not falsified by measurement data and provide the minimum value of a given control criterion Our prime interest here is with the mathematical complexity of the problem We consider first the LFT model with unstructured uncertainty (one uncertainty block) The uncertainty and exogenous disturbance are assumed to be scaled diagonally and the scaling factors play the role of upper bounds for uncertainty and disturbance (the importance of scaling is discussed in [1]) We show that the model validation test is reduced to the feasibility of data-based quadratic (with respect to the scaling factors) inequalities, which become linear in a particular case when the uncertainty input is a scaled measured signal Then we show that the optimal errors quantification associated with the optimal robust regulation problem is a nonconvex quadratic fractional programming under linear and quadratic constraints This result means that the approximate optimal robust synthesis proposed in [5] for LFT models with unstructured uncertainty becomes much more complicated under incomplete a priori information about upper bounds for uncertainty and disturbance Fortunately, there is an important class of models under coprime factor perturbations, which admit of computationally tractable solution of the optimal errors quantification problem /$ IEEE

2 1502 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 50, NO 10, OCTOBER 2005 The model validation test for such models is obviously reduced to linear programming and our main contribution is that the optimal errors quantification is reduced to linear fractional programming in both regulation and tracking original problems of optimal robust synthesis Since linear fractional programming is reducible to linear programming (see, eg, [2]), the optimal errors quantification problem is easily solvable for models under coprime factor perturbations Notation: for the vector is the space of real vector sequences with elements is the space of real bounded vector sequences with the norm Fig 1 Control system with uncertainty and exogenous disturbance II PROBLEM STATEMENT Consider a model of discrete-time control system defined by the equations is the space of real summable sequences with the norm A map is said to be -stable (stable, for short), if it is causal, maps into, and ( is the gain of for nonlinear ) The terms of system and map are understood as equivalent Any linear time-invariant causal system can be defined by the convolution (1) (3) (4) (5) and depicted in Fig 1 The nominal plant, the controller, and the weighting filters are assumed to be linear time-invariant causal systems The signal is the controlled output, is the measured output, is the control input, represents the unknown bounded exogenous disturbance where the same notation is used for the matrix of impulse responses The system is -stable iff for all The induced -norm of the stable LTI system is defined by (1) and equals The matrix valued function of complex variable is called the transfer function of the system and The optimization problem subject to (2) is called a linear fractional program A similar optimization problem with quadratic numerator, denominator, and constraints is said to be a quadratic fractional program Any optimization problem that is reducible to the solution of a finite number of linear (quadratic) fractional programs is also called linear (quadratic) fractional program for short While the linear fractional program can be solved efficiently either by transforming to linear programming [2] or by interior-point polynomial-time algorithms [11], the quadratic fractional program is generally a very difficult nonconvex optimization problem and is the structured uncertainty (perturbation) is strictly causal The set includes nonlinear and time-varying perturbations The results of this paper remain true if the perturbation is restricted to be linear and time-varying (LTV) Definition: Given a nominal plant, weighting filters, and data, the model is said to be not invalidated by data, if there exist an exogenous disturbance, and a perturbation, such that the (3) and (5) are satisfied on the time interval The model validation problem is to test whether the model is not invalidated by data Remark 1: Without loss of generality, the controlled output is assumed to be measured; otherwise the testing of the second equation in (3) is impossible A solution to the model validation problem has been obtained in [13] For the purpose of further exposition, we present this solution in Section III with necessary details and minor modifications Considering the problem of errors quantification, we assume that, are nonnegative diagonal real matrices to be estimated In other words, the scaling matrices represent the unknown part of the weighting filters, while their know part is absorbed into the nominal system The importance of scaling

3 SOKOLOV: CONTROL-ORIENTED MODEL VALIDATION AND ERRORS QUANTIFICATION 1503 is discussed in [1] Let be a priori set of admissible models, and denote the subset of models that are not invalidated by data Let be a given nominal plant and there exist scaling matrices such that The collection of such scaling matrices is generally not unique and this poses a problem what collection of scaling matrices is preferable A standard idea for making a choice is in terms of some or other minimization of the norms of disturbances and perturbations and typical problem statements are as follows Let, where denotes the identity matrix of order In this case the scaling factors and represent the norms (strictly speaking, the upper bounds for the norms) of exogenous disturbance and perturbation, respectively The optimization approach in [13] is to minimize under a fixed constraint for and vice versa Another optimization approach in [15] is in the minimization of (note that a solution to this problem can be useless in practice in view of the violation of robust stability condition) The aforementioned typical optimization approaches are irrespective of specific problems of optimal robust synthesis In order to state a control-oriented errors quantification problem, introduce the control criterion where is the controlled output of system (3) (5) Control criterion (6) is associated with a regulation problem An example of tracking problem will be considered at the end of Section V Let the nominal plant be stabilizable and be a fixed controller stabilizing The problem of optimal errors quantification for the nominal plant and controller is stated as Problem (7) is generally a hard nonconvex programming and will be considered for two commonly used models The first one is the LFT model (3) (5) with unstructured uncertainty and the second one is a system under coprime factor perturbations We show that the optimal errors quantification amounts to a quadratic fractional programming for the LFT model and to a linear fractional programming for the model under coprime factor perturbations (6) (7) Fig 2 Generic LFT model structure for model validation III MODEL VALIDATION FOR LFT MODEL A generic LFT model structure for model validation is shown in Fig 2 where the signal is included into the measured signal in view of Remark 1 Let be the measured data and Let subsystems in Fig 2 be described on the time interval by their truncated pulse response matrices Similarly to (8), introduce vector notation for unknown signals, respectively Then the system in Fig 2 can be described on the time interval by the matrix equations A Exogenous Disturbance Constraint Rewrite the (10) in the form (8) (9) (10) (11) and consider it as a linear equation in the unknown vector The general solution to this equation is of the form (12) where is a particular solution and spans the input null space of Since the constraint for the exogenous disturbance on the time interval

4 1504 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 50, NO 10, OCTOBER 2005 can be rewritten in the form of linear in inequalities (13) B Structured Uncertainty Constraint A reparametrization of the structured uncertainty constraint is based on the following result Break the vectors and to pieces according the dimensions of the diagonal blocks the structured uncertainty where Lemma 1: Let and There exists a strictly causal LTV map such that if and only if Lemma 1 is a simple multidimensional extension of Lemma 4 in [8] Denote by the matrix of the projector mapping the vector to the vector Denote by to the vector the matrix of the projector mapping the vector Using these notations, the (9), the representation (12) and Lemma 1, one can rewrite the structured uncertainty constraint on the time interval in the form of the following inequalities (piecewise) linear in : of system comprised of the plant and the controller and assumed to be stable The elimination of the controller from the model validation problem is useful for purposes of robust synthesis based on errors quantification Inequalities (14) are nonconvex in, but checking its feasibility can be reduced, by assuming a sign of each linear function in the right-hand sides in (14), to checking feasibility of a family of linear systems To evaluate the computational complexity of such reduction, we formulate the following preliminary lemma Lemma 2: Let and, be linear functions The system of inequalities (15) has a solution iff at least one of linear systems of the order has a solution Proof: For each, the inequality in (15) is equivalent to the statement (16) where denotes the th coordinate of For each, the test of the inequality in (16) is reduced to the test of a pair of linear systems of the order Consequently, the test of the inequality in (15) for a fixed is reduced to the test of linear inequalities of the order It follows now that the test of (15) is reduced to the test of linear systems of the orders Now, we are able to evaluate the complexity of the feasibility test for the system of inequalities (14) Lemma 3: The system of inequalities (14) has a solution iff at least one of linear systems of the order has a solution Proof: Inequalities (14) are of the form (15) where each is associated with a pair from (14) so that, and Then the order of linear systems associated with inequalities (14) is (14) and the number of tested linear systems is C Model Validation Test The obtained reparametrizations of the exogenous disturbance constraint and the structured uncertainty constraint provide the following model validation test Theorem 1: The model in Fig 2 is not invalidated by data if and only if the system of inequalities (13) and (14) has a solution Theorem 1 is similar to the results in Subsection 34, [13] The only difference is that the model in Theorem 1 does not include the controller and may be unstable in contrast to [13] where the model is associated with the closed loop nominal For the LFT model with unstructured uncertainty, the number of tested linear systems is and remains still huge for large A sizeable reduction of the complexity of the model validation test can be achieved by the considering of perturbations with a fixed bounded memory (see [17] and [19] for details on the bounded memory perturbations) In this case the dimensions of linear vector functions in the right-hand side in (14) are for and the number of tested linear systems is less than

5 SOKOLOV: CONTROL-ORIENTED MODEL VALIDATION AND ERRORS QUANTIFICATION 1505 IV OPTIMAL ERRORS QUANTIFICATION FOR LFT MODEL WITH UNSTRUCTURED UNCERTAINTY In this section, we show that the optimal errors quantification for the LFT model with unstructured uncertainty is reduced to a quadratic fractional programming Consider (3) (5) with unstructured uncertainty and the scaling matrices (17) Equations (9) and (10) for the system with the scaling matrices (17) can be presented in the form (18) (19) where ( is the matrix with blocks repeated on the diagonal) We show first that the exogenous disturbance constraint (13) is linear in Let (12) denote the general solution to the system of equalities (11) as before Since the general solution to (19) can be parameterized as follows: and are (piecewise) linear in Now, we present an explicit formula for computing the control criterion (6) Let denote the closed-loop system composed of the plant and the controller We will use the same notation for the matrix of impulse responses of this system The entries of the matrix belong to the space because the controller was assumed to be stabilizing for the nominal plant Breaking the matrix to blocks of appropriate dimensions, rewrite the system (3) (5) in the form Introduce the notation for the th row of matrix Theorem 5 in [7] implies that the system under consideration is robustly stable (that is, is finite) under zero initial conditions if and only if and [19, Th 6] implies that (25) (20) Therefore, and the exogenous disturbance constraint takes the form and is linear in Coordinate representation of this constraint is (21) (26) Let denote the element in -th row and -th column of the matrix For the system with the scaling matrices (17), the robust stability condition (25) and (26) take the form (27) Now consider the constraints for the unstructured uncertainty The parametrization (20) and the (18) come to inequalities (22) that are (piecewise) quadratic in because the matrix contains on the diagonal Remark 2: Consider an important special case where the input of the perturbation is comprised of the measured signals and so that (23) In this case, the unstructured uncertainty constraints take the form (24) (28) Formula (28), condition (27), and Theorem 1 give the following result Theorem 2: Consider the control system (3) (5) with unstructured uncertainty and with the scaling matrices (17) The problem of optimal errors quantification (7) for the nominal plant and the controller is a quadratic fractional program with the cost function (28) under the linear constraints (21), (27), and the quadratic constraints (22) The quadratic constraints (22) are replaced by the linear constraints (24) for the uncertainty (23) Remark 3: The zero initial conditions may be a restrictive assumption in practice The robust stability condition (25) and formula (26) for steady-state robust performance remain true for systems with nonzero initial conditions, if the perturbations are additionally assumed to be of fading or finite memory (see [6] and [19] for details) However, such perturbations do not admit any solution of the model validation problem Consideration of the bounded memory perturbations is an appropriate alternative

6 1506 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 50, NO 10, OCTOBER 2005 for systems with nonzero initial conditions (see [17] and [19] for details) Remark 4: Similar to Theorem 2 result remains true when the scaling matrices and exchange roles so that (29) One can derive similarly that the constraints (21) remain the same and the constraints (25) takes the form is the measured output System (32) and (33) is of the form (3) and (5) where Consider again diagonal scaling matrices and with (34) (35) The unstructured uncertainty constraint (22) becomes and the constraint (24) becomes (30) (36) A solution to the model validation problem for the system (32) and (33) takes a simple form Lemma 4: System (32) and (33) is not invalidated by data if and only if the (piecewise) linear in inequalities (37) (31) The inequalities in (30) and (31) can be rewritten as quadratic and linear in, respectively The control criterion (6) for the scaling matrices (29) takes the form hold for Proof: The statement of Lemma 4 follows obviously from combining the inequalities (13) and (14) associated with into the inequalities (37) (the use of finite parametric nominal ARMA model instead of impulse responses for the description of the nominal model does not need detailed comments) In order to present an explicit formula for the control criterion (6), introduce the transfer matrices from the total disturbance to the output and the control Thus, Theorem 2 can be reformulated for the system with the scaling matrices (29) as well V MODEL UNDER COPRIME FACTOR PERTURBATIONS In this section, we show that the optimal errors quantification can be reduced to a linear fractional programming for model under coprime factor perturbations Consider the system where the total disturbance is of the form (32) (33) and are left coprime polynomial matrices in the backward shift operator The orders of the matrices and are and, respectively, and In view of the representation where is the transfer matrix of controller (4) In order to get sufficiently simple functional dependence of the control criterion (6) with respect to the sacaling factors we have to limit ourselves by one parametric scaling of the exogenous disturbance (38) It follows from [18, Th 3] that the system (32), (33), and (4) is robustly stable under zero initial conditions if and only if and (39) (40) Note that the denominator in (40) is (piecewise) linear in the scaling factors and the uncertainties and are called coprime factor perturbations of the plant (32) Let, that is, the controlled output Then, Lemma 4, (40), and (39) provide the following result Theorem 3: The problem of optimal errors quantification (7) for the model (32) and (33) with the scaling matrices (36) and

7 SOKOLOV: CONTROL-ORIENTED MODEL VALIDATION AND ERRORS QUANTIFICATION 1507 (38) is a linear fractional programming with the cost function (40) under the linear constraints (39) and (37) with Remark 5: The uncertainty (35) is a structured uncertainty entering into the plant (32) in special additive manner in (33) In control literature, the model under coprime factor perturbations is usually considered under unstructured uncertainty (41) In this case, the model validation inequalities (37) are replaced by the inequalities (42) Theorem 3 remains true with the following modifications The robust stability condition takes the form and the formula for the control criterion becomes as follows: (43) (44) For the LFT model with uncertainty blocks, the control criterion equals the maximum of a family of polynomial fractional functions of the induced norms of system transfer functions The degrees of numerators and denominators are generally and, respectively [17] Therefore, the linear fractional representation of the control criterion is not the case even for the simplest, with unstructured uncertainty, LFT model That is why the representations (40) and (44) for the model under coprime factor perturbations is a remarkable fact Any other scaling matrices, eg, when the matrices and exchange roles in (34) or when the matrix is not of the form (38), result in the loss of linear fractional representation and in considerable complication of the optimal errors quantification In particular, linear fractional represention is lost in the presence of measurement noise Remark 6: Consider a special but very important case of the scaling matrices of the form (45) so that only three scaling factors, and are to be estimated This special case is nevertheless more general than the usually studied case of two scaling parameters associated with the unstructured uncertainty and The twoparametric case is of the main interest in numerous problems of model validation, control-oriented identification, and robust adaptive control The problem of optimal errors quantification (7) for the model (32) and (33) with the scaling matrices (45) becomes a linear fractional program of the form (2) where linear constraints are of the form At the same time, the model validation test for the model with unstructured uncertainty (41) takes the form, and is reduced to the testing of linear systems The considerable complication of the model validation test for the accepted model with unstructured uncertainty is a telling argument in favor of the model with structured uncertainty The exponential growth of the complexity disappears in the aforementioned traditional case of two estimated scaling factors and However, the model associated with two parameters is more conservative from the control design point of view Our final result is that the optimal errors quantification in the context of tracking problem for the model (32) and (33) reduces to a linear fractional program as well Let be a given reference signal and be a fixed tracking controller Introduce notation and consider the control criterion where denotes the subset of finite memory perturbations (see [6] for details) Introduce the transfer matrices from the signal to the output and the control Theorem 6 in [19] implies that (46) for the scaling matrices (36) and (38) Since is linear fractional in the scaling factors, the statement of Theorem 3 remains true with the replacement of the cost function (40) by the cost function (46) Stricrly speaking, inequalities (37) must be replaced by those for bounded memory perturbations in view of Remark 3 VI CONCLUSION The gap between robust synthesis and identification is an area of active research for the last two decades In this paper, we addressed this gap from the robust synthesis point of view and treated the control criterion as the estimation criterion No other assumptions on uncertainties than those accepted in robust synthesis have been used The problem of optimal errors quantification in the setup was considered for two popular

8 1508 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 50, NO 10, OCTOBER 2005 model structures For the LFT model with unstructured uncertainty (only one uncertainty block), it was reduced to a nonconvex quadratic fractional program with a huge number of constraints For the model under coprime factor perturbations, the optimal errors quantification was reduced to a linear fractional programming, which is known to be a quasiconvex optimization problem reducible to a linear programming The conclusion is that the LFT model structure, which is commonly used in robust analysis and synthesis problems, is not promising for optimal robust synthesis in the setup under incomplete a priori information, while the model under coprime factor perturbations keeps a hope Iterative alternate minimization over the scaling factors and the controller is discussed in [16] as a heuristic algorithm for data-based robust synthesis under unknown upper bounds for perturbations and exogenous disturbance ACKNOWLEDGMENT The author would like to thank the anonymous reviewers for helpful remarks, in particular for information about [13] REFERENCES [1] S Boyd and C Barratt, Linear Controller Design Limits for Performance Englewood Cliffs, NJ: Prentice-Hall, 1991 [2] S Boyd and L Vandenberghe, Convex Optimization Cambridge, UK: Cambridge Univ Press, 2003 [3] G Dullerud and R Smith, A nonlinear functional approach to LFT model validation, Syst Control Lett, vol 47, pp 1 11, 2002 [4] M H Khammash, M V Salapaka, and T Vanvoorhis, Robust synthesis in ` : A globally optimal solution, IEEE Trans Autom Control, vol 46, no 11, pp , Nov 2001 [5] M H Khammash, Synthesis of globally optimal controllers for robust performance with unstructured uncertainty, IEEE Trans Autom Control, vol 41, no 2, pp , Feb 1996 [6] M H Khammash, Robust steady-state tracking, IEEE Trans Autom Control, vol 40, no 11, pp , Nov 1995 [7] M Khammash and J B Pearson, Analysis and design for robust performance with structured uncertainty, Syst Control Lett, vol 20, pp , 1993 [8], Performance robustness of discrete-time systems with structured uncertainty, IEEE Trans Autom Control, vol 36, no 4, pp , Apr 1991 [9] A B Kurzhanskii, Control and Observation Under Uncertainties (in Russian) Moscow, Russia: Nauka, 1977 [10] P M Mäkilä, J R Partington, and T K Gustafsson, Worst-case control-relevant identification, Automatica, vol 31, no 12, pp , 1995 [11] Y E Nesterov and A S Nemirovskii, An interior-point method for generalized linear-fractional programming, Math Program, vol 69, pp [12] K Poolla, P Khargonekar, A Tikku, J Krause, and K Nagpal, A timedomain approach to model validation, IEEE Trans Autom Control, vol 39, no 5, pp , May 1994 [13] R S Smith, Model Validation and Parameter Identification for Systems in H and `, in Proc Amer Control Conf, 1992, pp [14] R S Smith and J C Doyle, Model validation: A connection between robust control and identification, IEEE Trans Autom Control, vol 37, no 7, pp , Jul 1992 [15] R Smith, G Dullerud, and S Miller, Model validation for nonlinear feedback systems, in Proc 39th IEEE Conf Decision and Control, Sydney, NSW, Australia, 2000, pp [16] V F Sokolov, Iterative design of robust controllers under unknown bounded perturbation norms, Autom Remote Control, vol 66, no 4, pp , 2005 [17], ` robust performance of discrete-time systems with structured uncertainty, Syst Control Lett, vol 42, no 5, pp , 2001 [18], Design of ` suboptimal robust controllers for MIMO plants under coprime factor perturbations, Dokl Math, vol 64, no 3, pp , 2001 [19], Asymptotic robust performance of the discrete tracking system in the ` metric, Autom Remote Control, vol 60, no 1, pp 82 91, 1999 Victor Sokolov was born in Syktyvkar, Russia, in 1952 He received the Diploma in mathematics and the Candidate of Sciences degree (PhD) from Leningrad State University, Leningrad, Russia, in 1974 and 1979, respectively, and the Doctor of Sciences Degree (DrSci) in physics and mathematics from the Institute of Control Sciences of the Russian Academy of Sciences (RAS), Moscow, Russia, in 1998 From 1977 to 2003, he was teaching at Syktyvkar State University Since 2004, he has held the position of Leading Researcher at the Department of Mathematics, Komi Science Center, RAS, and Professor of Mathematics at Syktyvkar State University His research interests include adaptive, robust, and optimal control