ESTIMATION OF THE OUTPUT DEVIATION NORM FOR UNCERTAIN, DISCRETE-TIME NONLINEAR SYSTEMS IN A STATE DEPENDENT FORM

Int. J. Al. Math. Comut. Sci. 2007 Vol. 17 No. 4 505 513 DOI: 10.2478/v10006-007-0042-z ESTIMATION OF THE OUTPUT DEVIATION NORM FOR UNCERTAIN DISCRETE-TIME NONLINEAR SYSTEMS IN A STATE DEPENDENT FORM PRZEMYSŁAW ORŁOWSKI Institute of Control Engineering Szczecin University of Technology ul. Siorsiego 37 70 313 Szczecin Poland e-mail: orzel@s.l Numerical evaluation of the otimal nonlinear robust control requires estimating the imact of arameter uncertainties on the system outut. The main goal of the aer is to roose a method for estimating the norm of an outut trajectory deviation from the nominal trajectory for nonlinear uncertain discrete-time systems. The measure of the deviation allows us to evaluate the robustness of any designed controller. The first art of the aer concerns uncertainty modelling for nonlinear systems given in the state sace deendent form. The method for numerical estimation of the maximal norm of the outut trajectory deviation with alications to robust control synthesis is roosed based on the introduced three-term additive uncertainty model. Theoretical deliberations are comlemented with a numerical water-tan system examle. Keywords: uncertain systems uncertain estimates discrete-time systems nonlinear systems. 1. Introduction Analysis and control synthesis for nonlinear uncertain systems or systems with limited information constitutes a wide area of science and engineering. In recent years a lot of research results Dai et al. 2002) have been ublished on robust control design. The literature can be classified into two categories: the eigenstructure assignment and Riccati-based methods such as H 2 H and μ syntheses Zhou et al. 1996). Other aers focus on simlifications of the nonlinear system e.g. using a describing function analysis Imram et al. 2001) and linearization. Among these multivariable control methods the H technique has a broad base because of its robustness to uncertainties and reliable design algorithms. A very imortant roblem is the selection of aroriate weighting matrices reflecting system stability and erformance Postlethwaite et al. 1990; Yang et al. 1997). When the weighting matrices are regarded as variables the H robust design roblem can be formulated as a multiobjective otimization roblem which needs to simultaneously satisfy design secifications in both the time domain and the frequency domain. This otimization roblem is usually very comlicated with many constraints Tang et al. 1996; Whidborne et al. 1994). The imlementation of the nonlinear otimal control requires solving the Hamilton-Jacobi-Bellman equation Lewis 1986). The imlementation of nonlinear H control requires solving the Hamilton-Jacobi-Isaacs equation Van der Schaft 1992; Basar 1995). The most successful alications of robust control techniques such as μ analysis and synthesis have occurred in roblem domains flexible structures flight control distillation) where there may be substantial uncertainty in the available models the degree of freedom and the dimensions of the inut the outut and the state may be high-dimensional but the basic structure of the system is understood and the uncertainty can be quantified. Nonlinearities are bounded and treated as erturbations on a nominal model or handled by gain scheduling linear oint designs. The concet of state sace artitioning called the iecewise affine linear) PWA PWL) decomosition with resect to control synthesis is extensively studied e.g. in the model redictive control Bacic et al. 2003; Grancharova et al. 2005). The main objective of the aer is to develo a new method for estimating the maximal norm of the timedomain outut trajectory deviation of the uncertain nonlinear discrete-time system with resect to the nonlinear

506 P. Orłowsi system with nominal arameters. It extends the revious aer Orlowsi 2003). The aer concerns the following asects: Transformation of the uncertain nonlinear discretetime system in a general form into a linear timevarying uncertain system in a state deendent form. Modelling an additive uncertainty for nonlinear systems a three-term additive erturbation model for the state deendent form is roosed in Section 3. The method for bound estimation for system matrices based on the concet of PWL decomosition is stated in Section 4. Furthermore a nonlinearfeedbaccontrolis alied to the model with an otimal cost functional. Estimates of the maximal outut trajectory deviation norm are given as two theorems with roofs defined in terms of evolutionary oerators taen from linear time-varying systems theory. Theoretical deliberations are comlemented by a numerical examle of the water tan system. The aroach used in the aer is based on a nown aroximation of the nonlinear system by a linear time varying system. Such an aroach is a very effective method for the synthesis of otimal nonlinear control systems. The nonlinear model redictive control Kouvaritais et al. 1999) is a very efficient iterative method which emloys the otimal control trajectory calculated in the revious time instant. The system is linearized around the trajectory and can be treated as a linear one. The otimal control can be comuted in an iterative way by udating the time-varying aroximation of the nonlinear model calculating a new control and checing whether the convergence condition is satisfied Ordys et al. 1993; Duta et al. 2004; Lee et al. 2002). 2. Model of the Nonlinear System Consider a system described by the following general nonlinear model: x +1 = f x u ) y = g x ). 1) Assume that the system can be transformed into the following nonlinear state sace deendent model: x +1 = Ax )x + Bu )u y = Cx )x. 2) The above descrition is analogous to the classical linear state sace model. Matrix coefficients {a ij } {b ij } and {c ij } can be arbitrary functions of the state i.e. a ij = f ij x)c ij = h ij x) and the inut b ij = g ij u). The model given by 2) covers a class of nonlinear systems for which inut and state functions can be indeendently defined. The inut-deendent matrix Bu ) can be used to reresent inut nonlinearities often modelled by the describing function. State-deendent matrices Cx ) and Ax ) cover resectively outut nonlinearities and internal system nonlinearities. The model can successfully describe well nown nonlinear systems such as a ball and a beam a water tan system etc. The model cannot accurately reresent a mixed inut-state function neither imlicit nor exlicit. For examle such a deendence occurs in the inverted endulum model e.g. F cos φ) wheref is inut force and φ is the endulum angle. 3. Model of Uncertainty Consider the following uncertain nonlinear model of the system: x +1 = A x )x + B u )u y = C x )x. 3) The uncertain system roduces the uncertain state x and the uncertain outut y. Generally they are different from the nominal state x and the nominal outut y and thus y y x x at least for some. Of course they are in general different for the uncertain and nominal systems Ax ) Ax ) A x ) A x ) where x and x are arbitrary nominal and uncertain states resectively. Since the nonlinear system 1) is timeinvariant the system matrices are time-indeendent and they deend on the state/inut only. Therefore the index can be removed from the state and the outut. The stateand inut-deendent matrices can be exanded in a multivariable Taylor series e.g. for a matrix A it is A x )=A x x ) )+ A x x ) 1! + A x ) x x ) 2 +... 4) 2! When the state trajectory error is x ) x ) 1 the series is convergent and it is ossible to rewrite it in the following form: A x )=A x )+ Ar x x ) 5) where Ar satisfies the conditions Ar = A x ) 1! + A x ) x x ) +... 2! 6) A x) =Ax)+ A x). 7) Finally all system matrices have the following additive form: A x ) = Ax )+ A x )+ Ar x ) x x ) 8)

Estimation of the outut deviation norm for uncertain discrete-time nonlinear systems... 507 B u ) = Bu )+ B u )+ Br u ) u u ) 9) C x ) = Cx )+ C x )+ Cr x ) x x ). 10) The model of uncertainty for any erturbed system matrices A B or C consists of three comonents: the one corresonding to the nominal matrix in the nominal state or the inut) e.g. Ax ) an additive erturbation in the nominal state or the inut) e.g. A x ) which does not deend on the deviation from the nominal state or the inut) a differential erturbation in the nominal state or the inut) e.g. Ar x ) which reresents an uncertainty increase in connection with the state or inut) deviation. Generally one does not need to now the additive erturbation matrices A B C and the differential erturbation matrices Ar Br Cr but only has to find their estimates δ A δ B δ C δ Ar δ Br δ Cr. In such a case the following conditions are held for the matrix A: A x ) δ A < 11) Ar x ) δ Ar < 12) where A x ) LRn R n ) Ar x ) LRn R n ) and for the matrices B and C we have B u ) δ B < 13) Br u ) δ Br < 14) C x ) δ C < 15) Cr x ) δ Cr < 16) where B u ) LRm R n ) Br u ) LRm R n ) C x ) LRn R ) Cr x ) LRn R ) = 0 1...N 1. Perturbation norms can be estimated for the nominal state and inut trajectories 11) 16). It is also ossible to estimate erturbation norms for all reachable states and inuts. In such a case the index must be removed from 11) 16). 4. Estimation of the Perturbation Norm The rocedure of state sace artitioning is nown as iecewise affine linear) and it is borrowed from e.g. Bacic et al. 2003; Grancharova et al. 2005). The sace of allowed states can be artitioned into a set of small PWL clusters. The median state of each cluster can be interreted as a woring oint for the linearization of the cluster. The dynamics of the system i.e. the matrices Ax) Bu) and Cx) can be identified locally in the neighbourhood of each woring oint under the assumtion that the system does not change the woring oint. The matrices are nonlinear but time indeendent and therefore the discrete-time index can be omitted. Additive erturbations in the nominal state may be exressed as follows: δ a ij =max a ij x j ) a n ij x j ) ) 17) a x) = { δij a x j) } 18) δij b =max b ij u j ) b n ij u j ) ) 19) b u) = { δiju b j ) } 20) δij c =max c ij x j ) c n ij x j ) ) 21) c x) = { δ c ijx j ) } 22) where the index n denotes the nominal value of the i j) coefficient. Equations 17) 19) and 21) alied to the matrices 18) 20) and 22) allow us to estimate the additive erturbations δ A δ B and δ C given by Eqns. 11) 13) and 15): δ A x) = a x) max A x) ) δij A δ A =max x δa x). 23) Similar relations hold for δ B and δ C. The estimates for differential erturbations δ Ar δ Br and δ Cr can be calculated from differences between nominal matrices and estimates of additive erturbations for different woring oints. The following relations may be used to estimate the norms of differential erturbations: Ax i δ Ar = max ) Ax j ) + δ A x i ) δ A x j ) iji j x i x j 24) Bu i δ Br = max ) Bu j ) + δ B u i ) δ B u j ) iji j u i u j 25) Cx i δ Cr = max ) Cx j ) + δ C x i ) δ C x j ) iji j x i x j 26) 5. Control System Control systems most often have either a linear feedbac controller reresented by a state feedbac or a nonlinear controller for examle fuzzy-logic. A. Mathematical Descrition of Control The most general descrition which can cover many different controllers may be written in a nonlinear state feedbac oerator or matrix form Fx ) as shown in Fig. 1.

508 P. Orłowsi It should be underlined that it is not assumed that all states have to be nown in each time samle. If some states are difficult to determine they can be either estimated by an aroriate state observer or excluded from the feedbac e.g. by simly multilying them by zero in the feedbac Fx). v + u x + Ax)Bx) Cx) Fx) Fig. 1. Closed-loo control system with a state feedbac. On the other hand the state feedbac can be easily converted to an outut feedbac by the inclusion of the outut state deendent matrix Cx) into the state deendent feedbac i.e. F x) =F y Cx))asshowninFig2. The roosed methodology is alicable to the state feedbac oerator but nevertheless it can easily describe the outut feedbac. To model the outut feedbac the outut state deendent matrix must be included in the state feedbac and the diagram from Fig. 2 is equivalent to a classical outut feedbac. Of course for a ractical imlementation of the outut feedbac there must be nown only the outut deendent term F y y). The feedbac oera- v + + u Fx) Ax)Bx) Fyy) Cx) x Cx) Fig. 2. Closed-loo control system with an outut feedbac. tor F describes either a linear feedbac with an invariant vector F or a nonlinear controller with a state deendent feedbac Fy). B. Closed-Loo Model The inut signal can be written as follows: u = v + Fx )x 27) where F LR R m ) and =0 1...N 1. After substituting 27) into 2) the system equations tae the following state deendent form: x +1 =Ax )+Bu )Fx )) x + Bu )v y = Cx )x =0 1...N 1 28) where u is given by 27). y y The system is asymtotically controllable to 0 if the air A B) is stabilizable and the system is invertible Albertini et al. 1994). C. Control Law Let us assume that the cost functional is the worst case norm of the outut trajectory deviation from the given reference trajectory. Generally it can be written as J =max y ) y r ). 29) By alying the following triangle inequality: max y ) y r ) =max y ) y )+y ) y r ) max y ) y ) + y ) y r ) 30) the above functional can be rewritten in the form J =max y ) y ) + y ) y r ) J. 31) The nominal outut deviation norm y ) y r ) can be easily obtained by numerical simulations while the outut uncertainty norm max y ) y ) can only be estimated. The otimization roblem can be formulated as follows: For a given system a fixed reference signal y r a set of ossible inuts v V and a given form of the feedbac function F x a 1...a M )findvaluesa 1...a M which minimize the cost functional J. Due to the conservatism of the estimates for ractical evaluation a weighting factor α is introduced in the following three cost functions: J 2 =max y ) y ) 2 +α y ) y r ) 2 32) J =max y ) y ) +α y ) y r ) 33) J 1 =max y N) y N) 1 +α y N) y r N) 1. 34) 6. Describing a Nonlinear Feedbac System Using Linear Oerators Every linear time-varying system can be described by linear invariant recurrent oerator equations Orłowsi 2001; 2004; 2006). The nonlinear system 2) can be described in a similar manner only in the case of a fixed trajectory inut and initial state vectors: x =NF x 0 ))+L F Bv))) y = CF x. 35)

Estimation of the outut deviation norm for uncertain discrete-time nonlinear systems... 509 For simlicity we introduce oerators L F LR n ) N R n ) N ) and N F LR n R n ) N )definedby 2 L F h)) = N F x 0 )) = i=0 1 j=0 1 j=i+1 A F j) hi)+h 1) 36) A F j)x 0 37) where A F j) =A x j )+Bu j ) F x j )) hi) LR n ) =2 3...N. Alternatively oerators can be rewritten in an equivalent matrix notation. In such a case 35) taes the form ŷ = ĈˆL F ˆBv + Ĉ ˆN F x 0. 38) The oerator ˆL F is given by the following nn nn matrix which simlifies calculating the oerator norm: ˆL F = 0 I A F 1). A F N 2)... A F 1) 0 0 0 0 0 0.. I 0... I 0 0 A F N 2) I 0. 39) The oerators ˆB and Ĉ are resectively nn mn and N nn matrices and have the following diagonal forms: B0) 0 0 ˆB =. 0.. 0 0 0 BN 1) C0) 0 0 Ĉ =. 0.. 0 0 0 CN 1). 40) For an LTV system the oerator ĈˆL F ˆB is a comact and Hilbert-Schmidt oerator from l 2 into l 2 and it mas bounded signals v) V = l 2 [0N] into signals y Y. For SISO systems the oerator ĈˆL F ˆB is an N N matrix. 6.1. Oerator Descrition of the Perturbed System. Theorem 1. The erturbed nonlinear system 2) with the feedbac control 27) a fixed inut and an initial state is always equal to the following equations: x y = L F A x )x ) ) +L F Ar x ) x x )) ) +L F B u )v + Fx )x ) ) +L F Br u ) Fx )x Fx )x )) ) +x 41) = CF x + Cx )x + Crx ) x x ). 42) Proof. The above equations can be roved using mathematical induction. For =2 the state equations 41) and 42) with the feedbac control 27) are x 2 [ A = A F ) F 1 0 + A0 + B0 F 0 x0 + B F 0 + ) ] B0 v0 + B F 1 + B1 + Br1 F 1 x 1 F 1 1) ) x v 1 + A1 + Ar1 x 1 x 1) +B1 F 1 + Br1 F 1 x ) ) 1 F 1 x 1 F 1 x 1 Substituting 41) and 27) in 28) for +1yields x +1 = B F v + A F ) + A + B F ) N F x 0 )+ L F B F v ) ) +L F A x ) ) +L F Ar x x )) ) ) +L F B v + Fx ) ) ) +L F Br Fx Fx ) ) x +1 = N F ) x 0 +1)+ L F B F v ) +1) +L F A x ) +1) +L F Ar x x )) +1) ) +L F B v + Fx ) +1) +L F Br Fx Fx )) +1). Detailed conversions are simle but laborious. A setch of the roof is resented above. 6.2. Outut Perturbation Estimates. The result of Theorem 1 is very useful to find the estimate of

510 P. Orłowsi y ) y ). The hysical interretation of the estimate deends on whether the norm is 2-norm -norm simlified as follows: Assuming that 47) 50) hold the above equation can be or 1-norm. x x L F x δ Axz + δ Arz x x ) Theorem 2. For any additive erturbations A B C and differential erturbations Ar Br Cr with the conditions 11) 16) and δaxz + δ Arz x x ) < L F 1 43) δ Axz + δ Arz x + δ ab ) < L F 1 44) the difference norms x ) x ) Rn ) N y ) y ) R are ) N and x x L F δaoz + δ Axz x ) 1 L F δ Axz + δ Arz x + δ ab ) 45) y y δ C x [ CL F + L F ] δ C +δ Cr )][ δ aoz +δ Axz x + 1 L F δ Axz +δ Arz x +δ ab ) 46) where δ Axz = δ A + δ B F 47) δ Arz = δ Ar + δ Br F 2 48) δ aoz = δ B v 49) δ ab = δ Br v F. 50) Proof. The linear sace with the defined norm satisfies all axioms of a metric sace and thus the triangle inequality follows x x L F B u ) v + Fx )x ) ) + L F Ar x ) x x ) x ) ) + L F A x )x ) ) + L F Br u ) Fx )x Fx )x ) v + Fx )x )) ). Then x x L F δa x + L F δar x x x + L F δ B v + F x ) + L F δbr F x x v + Fx )x. 51) + L F δ ab x x + δ aoz ). 52) The uncertain state norm can be estimated by rearranging 41) and alying the triangle inequality again: x x + L F [ δ Axz x + δ Arz x x x + δ ab x x + δaoz ] x [ 1 δ Axz + δ Arz x x )] x + L F [ δ ab x x + δaoz ] x x + L F [ δ ab x x ] + δaoz 1 δ Axz + δ Arz x x. 53) ) Substituting 53) into 52) yields x x L F δ Axz x x L F δaxz x x L F δarz x x x L F δ Arz x x 2 L F x δ Axz + L F δ ab x x + δaoz ). When the difference x x 2 is small enough it can be neglected and the inequality taes the form x x [ 1 L F δ Axz + δ Arz x + δ ab ) ] L F δaxz x + δ aoz ). 54) It is equivalent to 45). The outut difference trajectory can be we written as y y = Cx )x + Cr x ) x x ) x + C x )x Cx )x = Cx )+ Cr x )x ) x x ) + C x )x y y C + δcr x ) x x +δ C x. 55) Equation 46) can be roven by rearranging Eqns. 42) and 35) and then by substituting the uncertain state norm from 45) which comletes the roof.

Estimation of the outut deviation norm for uncertain discrete-time nonlinear systems... 7. Numerical Examle A Water Tan System 511 oerator F can be estimated from 58) in the following way: Consider the system described by the following onedimensional nonlinear model: x +1 = x at x + bt A A u y = x. 56) The equation describes the level of water x as a function bu x Fig. 3. Water tan system. of time due to the differences between the flow rates into and out of the tan bt u and at x resectively. A is a cross-sectional area of the tan b [b b + ] is an uncertain constant related to the flow rate into the tan a [a a + ] is an uncertain constant related to the flow rate out of the tan u {0 1} is a logical variable which signifies the oen 1) or closed 0) inut valve and T is a samling ariod. A simle scheme of the system is shown in Fig. 3. The nonlinear state sace model can be written using state and inut deendent matrices and Eqn. 3): Ax )= { 1 at x A x for x 0 1 for x =0 Bu )= bt A Cx )=1. 57) The water level can be controlled using a simle bistable controller with hysteresis. The outut of the controller oens or closes the inut valve. The valve oens when the level is lower than L n L and remains oen until the water reaches the level L n +L otherwise the valve is closed. Here L n is a nominal setoint value usually equal to the reference outut y r andl is the width of the one-side hysteresis. The controller has the following mathematical descrition: u i = Fx i 1 if x i <L n L = or x i <L n +L and u i 1 =1 0 otherwise. a x 58) A larger value of L reduces the controller sensitivity and the switching frequency of the valve. The norm of the Fx F =max x 0 x =max 1 x 0 x 1 min x 59). The following substitution can be used to decomose the system into the form 8) 10): A x )=1 a nt A x A x )= a a n) T A x Ar x )= a n T 2Ax x a n = a + + a 2 B u )= b nt A B u )= b b n) T A Br u )=0 60) b n = b + + b 61) 2 C x )=1 C x )=0 Cr x )=0. 62) The arameters assumed for comutations are A = 50 a =4a + =6b =15b + =20T =0.1 s y r = xr = L n =4.5 L =0.5. The comuted estimates and norms are collected in Table 1. Transient resonses simulated for nominal and uncertain systems for N = 1000 x 0 = 0 L n = 4.5 and three different values of L are shown in Fig. 4. The comuted values of the nominal outut deviation norm y ) y r ) and the outut uncertainty norm y ) y ) are annotated on the lot. The functional 32) with α =1isminimized for the most frequent switching L 0. The quality of the estimates is determined by the following estimation error coefficient: ε = y ) y ) estimated y ) y ) simulated 1. 63) The lot of the estimation error as a function of the simulation horizon N and the initial condition x 0 is shown in Fig. 5. The minimal value of ε is 0.37 x 0 =3.5 N=9)and the median value is aroximately equal to 2. For longer time horizons the condition 44) is not satisfied and 45) cannot be used for the estimation of the outut deviation.

512 P. Orłowsi Table 1. Comuted estimates and norms for the water-tan system. ) 0.5 ) 0.5 ) 1 δ A 0.002 min x ) δ Axz 0.002 min x ) +0.01 min x ) ) 1.5 ) 1.5 δ Ar 0.005 min x ) δ Arz 0.005 min x ) ) 1 F min x ) L F 0.636N C =1 x N max x δ B =0.01δ Br =0 v =0 δ C =0δ Cr =0 δ aoz =0δ ab =0 Amlitude 6 5 4 3 2 1 Nominal L=0.1 x -x r =15 Perturbed L=0.1 x -x =36 Nominal L=0.5 x -x r =20 Perturbed L=0.5 x -x =37 Nominal L=1 x -x r =26 Perutrbed L=1 x -x =39 0 0 10 20 30 40 50 60 70 80 90 100 Time s) Fig. 4. Transient resonses and comuted norms for the water tan system. Fig. 5. Estimation error vs. simulation horizon and initial condition for the water tan system. 8. Conclusions The main aim of this aer was to roose a new method for estimating the norm of the outut deviation for uncertain nonlinear discrete-time systems. The method can be an interesting alternative to two existing numerical methods for estimating max y ) y ). The first is to estimate the norm on the basis of simulations of the uncertain system for a secified inut and the initial state on the assumtion of extreme ositive and negative values of erturbation matrices A B C Ar Br Cr. The maximal deviation norm of all simulations is an estimate of the norm max y y. The number of simulations n s grows exonentially with the number of nonzero coefficients of the additive erturbation e.g. n s =2 nzcoeff. Extreme values of arameters do not guarantee a maximal deviation of the outut. Nevertheless the results are often close to the global maximum. The second method taes advantage of numerical otimization methods most of them imlemented in Matlab. Such an algorithm requires considerable comutational ower. Moreover the convergence to the worst-case solution i.e. the maximal norm is not guaranteed. The number of variables is equal to the sum of all nonzero coefficients of additive erturbations. The roosed oerator-based method guarantees that the estimated outut difference norm is not lower than the worst ossible real case norm and requires low comutational ower. The main disadvantage of the method is the conservatism in the estimates. An iterative two-stage rocess can be used to find the otimal control solution. The first stage is to find the aroriate structure of the controller such that the maximal trajectory deviation from the nominal trajectory for the uncertain system y ) y ) is minimal. The second is to find a control which minimizes the deviation of the nominal system from the reference trajectory y ) y r ). The rocedure must be reeated until the assumed accuracy is aroached. References Albertini F. Sontag E.D. 1994): Further results on controllability roerties of discrete-time nonlinear systems. Dynamics and Control Vol. 4 No. 3. 235 253. Basar T. and Bernhard P. 1995): H [ ]-Otimal Control and Related Minimax Design Problems: A Dynamic Game Aroach 2nd Ed. Boston: Birhäuser. Bacic M. Cannon M. Kouvaritais B. 2003): Constrained NMPC via state-sace artitioning for inut affine nonlinear systems. International Journal of Control. Vol. 76 No. 15. 1516 1526.

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