Control of Complex Nonlinear Dynamic Rational Systems

Transcription

1 Control of Comlex Nonlinear Dynamic Rational Systems Quanmin Zhu,, Li Liu *, Weicun Zhang, and Shaoyuan Li 3 School of Automation and Electrical Engineering, University of Science and Technology Beiing, Beiing 00083, China Deartment of Engineering Design and Mathematics, University of the West of England, Frenchay Camus, Coldharbour Lane, Bristol, BS6 QY, UK 3 Deartment of Automation, Shanghai Jiao Tong University, 800 Dongchuan Rd., Minhang, Shanghai, 0040, China *Corresonding author: liuli@ustb.edu.cn Abstract: Nonlinear rational systems/models, also known as total nonlinear dynamic systems/models, in exression of a ratio of two olynomials, has roots in describing general engineering lants and chemical reaction rocesses. The maor challenge issue in control of such system is the control inut embedded in its denominator olynomials. With extensive searching, it could not find any systematic aroach in designing this class of control systems directly from its model structure. This study exands U-model based aroach to establish a latform for the first layer of feedback control and the second layer of adative control of the nonlinear rational systems, which, in rincile, searates control system design (without involving a lant model) and controller outut determination (with solving inversion of the lant U-model). This rocedure makes it ossible to achieve closed loo control of nonlinear systems with linear erformance (transient resonse and steady state accuracy). For the conditions using the aroach, this study resents the associated stability and convergence analyses. Simulation studies are erformed to show off the characteristics of the develoed rocedure in numerical tests, and to give the general guidelines for alications. Keywords: Rational systems/models, total nonlinear systems/models, U-model based control(u-control), control comlex dynamics, adative control, arameter estimation/otimisation. Introduction This section ustifies the reasons for designing controllers for rational models by introducing model exression and reresentations, achieved results in model identification, and a critical review of controller designing aroaches... Nonlinear dynamic rational systems Definition []: Assign a trilet X, f, h, X is an irreducible real affine variety, f, h are maing functions. m r A system, with inut U and outut Y, is defined as olynomial/rational, while the functions r f f U and h: X both on X are, maings from inut sace to state sace and from state sace to outut sace resectively, olynomial/rational. That is for olynomial systems hi A for all i,, rwhere A is the algebra of all olynomials on the variety X and for rational systems hi Q for all i,, rwhere Q is the algebra of all rational functions on the variety X. For a single-inut and single-outut (SISO) nonlinear dynamic rational system, it can be generally modelled with a ratio of two olynomials [, ]. N ( k) N ( Yk, U k, Ek ) y( k) e( k) e( k) D ( k) D ( Y, U, E ) num n den d k k k ( k) n ek ( ) ( k) d (.) where yk ( ), uk ( ), and ek ( ) denote measured outut, inut and model error/noise/uncertainties at time instant k(,, ) resectively. N ( k) and D ( k ) are real valued and smooth numerator and denominator olynomials resectively. Y n k y( k ),..., y( k n), n U k u( k ),..., u( k n), and n Ek e( k ),..., e( k n) denote the delayed oututs inuts, and model noises resectively. n ( k) and d ( k) for regression terms, n and d are the coefficients, and num and den for numbers of total regression terms of the olynomials resectively. The maor roerties of the rational model (.) are summarised below: It is also defined as a total nonlinear model [] as it covers many different linear and nonlinear models as its subsets (such as NARMAX (Nonlinear AutoRegressive Moving Average with exogenous inut) models [3] and intelligent models for neuro-fuzzy systems [4]. Rational systems have been observed in general engineering, chemical rocesses, hysics, biological reactions, and

2 econometrics, for examle, rational models are a class of mechanistic models in describing catalytic reactions in chemical kinetics [5, 6], metabolic, signal and genetic networks in systems biology [], and movement of satellite in earth orbit []. There have also been reorts of rational modelling alications [7-9]. This is more concise in structure than a olynomial; the examle below uses a Taylor series exansion to aroximate a simle rational model below. sin( uk ( )) y k u k y k y k y ( k3) 4 ( ) sin( ( ))[ ( 3) ( 3) ] (.) The other characteristic of the rational models is the ower in quick change of the model outut while inut has small variations. Consider a simle system outut below yk ( ) uk ( ) (.3) Clearly the model outut will be dramatically increased, as the inut uk ( ) aroaches. This comes from function of the denominator. Introducing a denominator olynomial, makes model concise in describing comlexity and add more functions in describing nonlinearities. On the other side, contrast to olynomial systems, this makes identification and control system design noticeably different and more difficult with the inherent nonlinear arameters and control inuts []. Therefore, it requests comrehensive studies of this class of systems in theoretical and alication asects. This study takes the ioneer ste towards to the control of the rational systems.. Model identification Model identification has been relatively mutual to some extent. So far, the identification asect has gone through data-driven model structure detection, arameters estimation and model validation from noise contaminated inut and outut data. The maor work on rational model identification is summarised in the following categories. Linear Least Squares (LLS) algorithms for arameter estimation: Extended LLS estimator [0]; Recursive LLS estimator []; Orthogonal LLS structure detector and estimator []; Fast orthogonal algorithm [3]; Imlicit least squares algorithm [4]. Nonlinear least square algorithms: Prediction error estimator [5]; Globally consistent nonlinear least squares estimator [6]. Other algorithms include the following categories. Back Proagation (BP) algorithm [7]; Enhanced Linear Kalman Filter (EnLKF) [8]. Model validation: Higher order correlation tests [9]. Omni-directional cross-correlation tests [0]. A summary of the reresentative ublications till 05 can be found in a survey of rational model identification []..3. Controller design As surveyed above, rational models have been increasingly used to reresent nonlinear dynamic lants. Consequently, the control system design should have been considered on agenda in the follow u studies. However, u to now, there is no reference found for designing such controllers directly referred to the model analytical knowledge. The aramount difficulty is that art of the controller outut is embedded in the denominator olynomial D ( k ). For examle: yk ( ) y( k ) y( k ) u( k ) 0. u ( k ) y ( k ) 0. u ( k ). With extensive investigations through maor academic ublication searching engines, it can be concluded that this study is the first trial with analytical aroaches to design a controller for rational systems. Regarding controller design aroaches ossibly referred to the rational systems, these could be the reduction of rational model structure comlexity, which are neural network models, linear aroximation models, linearization, and iterative learning control, and U-model enhanced control. A brief critical review of the aroaches is resented below. Neural controller []: this is robably the first ublication relating to control of rational models. However, the design aroach has merely used rational models as extreme nonlinear examles, it has not designed controllers by taking the model structure into consideration (even known in advance), excet taking the models as the reresentatives of comlex nonlinear dynamic systems. Piecewise linearization [, 3] around oerating oints has been widely studied to simlify controller designed rocedures when lants are subect to mild nonlinear dynamics. It should be mentioned that a grou of iecewise linear models can be admitted as a linear model, with varying order and arameters in different oerating intervals. The romising roerty is using linear control design strategies directly. However, it could induce inaccuracy and dynamic uncertainty because of ignoring some inherent nonlinearities from their original nonlinear reresentations. Further this method may also increase comutational burden/comlexity while over barrowing iecewise linear intervals to match severe nonlinearities. Point-wise linearisation has been claimed by neural network based control and/or adative control, which uses linear models to aroximate redominant dynamics around an oerating oint or every inut outut dynamic gain at each time instance, and then emloys a neural network to determine the error induced by the linearisation [4, 5]. Once again, it uses linear control systems design to construct nonlinear control systems. However, this involves on-line neural network learning or online model iden arameter estimation, and therefore the constructed nonlinear control system is oerated under adative rinciles (the controller arameters are udated with the neural network outut), even for deterministic nonlinear lants. The other related issue is the selection of neural network toology, which has no systematic rocedure available to find the best fitted neural network reresentative.

3 Feedback linearization is a well-develoed subect [6]. A general SISO nonlinear system is described as x f ( x) g( x) u y h( x) (.4) where x is the state vector, u and y are inut and outut resectively. f (), g() and h() are real valued and smooth maing functions. With this model structure a series of analogies with some fundamental features of linear control systems have been established, which rovides a very useful concet in design of nonlinear control systems using linear design methodologies. Obviously the model has u in an exlicit osition. The studied nonlinear rational model has no such exlicit exression for inut u to be designed, and this immediately reveals that the methodologies rooted in the aroach, although useful references, are not directly alicable in designing control of nonlinear rational systems. The other inut-outut linearization techniques [7] have had similar requirements for an exlicit u exression and secial skills for state variable transformation. Iterative learning/data-driven control/model-free control is another ossible control system design methodology in avoiding model structure comlexity. The aroaches do not require clear lant model structure, but still need lants with some mild conditions in control [8, 9]. Again, if a rational model is available, it is wasteful without using the model information in the control system design. It is believed, articularly for man-made engineering systems/roducts (built u by rules/models), that any reetitive rocess and motion has model exist in oeration even though the model is yet to identify. U-model based control has claimed to radically relieve the deendence of lant model oriented design foundation. The use of the lant model is effectively reduced as a reference for converting to U-model and accordingly to work out the control outut [30]. U-model based control assumes the feasibility of using linear system design rocedures to design the control of nonlinear dynamic lants with assigned resonse erformances. The U-model control latform is illustrated in Fig.. The U-model systematically converts smooth (olynomial and extended including transcendental functions) models, derived from rinciles or identified from measurements, into a tye of u-based model to equivalently describe lant inut/outut relationshi, so that it establishes a general latform to facilitate control system design and dynamic inversion. It should be mentioned that there is nothing lost with the derived U-models from the original nonlinear models. The difference between the two tyes of model exressions is that those original nonlinear models could be obtained from rinciles, such as Newton s law, or identified directly from measured data, the U-models are derived from the original models in control design oriented exressions. Regarding the U-control (U-model based control) research status, Zhu and Guo [3] have brought forward a fundamental framework in terms of ole lacement control for nonlinear systems. More recently, U- control has been exanded to General Predictive Control [3] and Sliding Mode Control [30]. To accommodate U-control of state sace models, a Backsteing algorithm is being exanded to extract the controller outut within multi loo U-models. With the nature of searating control system design (secifying closed loo erformance) and controller outut calculation (by resolving lant dynamic inversion through U-model), it can be forecast that the other classical control issues could be similarly formulated within a general and concise framework. Linear control system design Fig..U-model based control systems design.4. Organisation of the study The remaining study is organised in five maor sections. Section is used to define a generic framework of control oriented U-model for reresenting smooth nonlinear dynamic lants. It is then exanded with including rational model and transcendental functions as its subsets to lay a basis for alying linear control system design techniques. Section 3 rooses a general ole lacement controller for nonlinear rational systems within the U-model framework. Section 4 shows design of adative UPPC for the control of stochastic nonlinear rational systems. Section 5 tests a number of tyical rational systems with the develoed rocedures and show the exemlary rocedures for otential users.. U-model --- a generic framework of control oriented nonlinear lant models.. U-model foundation --- olynomial [30] Consider a general olynomial descrition of. y( k) N ( Y U ) L i( k) i i0 U-model k k where yk ( ) and uk ( ) Polynomial and state sace lant models (.) denote the lant outut and inut at time instant k(,, ) resectively. N () is a real valued and smooth olynomial function, n Y k y( k ),..., y( k n) and n Uk u( k ),..., u( k n) denote the delayed oututs and inuts resectively. i ( k) denotes the 3 model structure variables, e.g. u( k ) y ( k ), u( k ) u ( k 3), y( k ) y( k 3), and i denote the coefficients. To convert the above olynomial into U-model, which is a olynomial with argument of control inut uk ( ) (also called controller outut while talking about control system design), it gives [30] 3

4 M y( k) ( k) u ( k ) (.) 0 where degree M is of controller outut uk ( ), M ( k) ( k) ( k) is the time varying 0 M arameter vector, a function of absorbing ast inuts Uk, oututs Yt, and arameters n in the original olynomial. An examle illustrating the conversion to U-model from an ordinary olynomial is shown here. Consider a olynomial, y( k) 0. y( k ) y( k 3) 0.5 u( k ) u( k 3) 0.9 y( k ) u ( k ) (.3) Rearrange olynomial (.3) with y k k k u k t u k (.4) ( ) 0 ( ) ( ) ( ) ( ) ( ) where 0 ( k ) 0. y ( k ) y ( k 3), ( k ) 0.5 u ( k 3), and ( k ) 0.9 y ( k ). Clearly, the time varying ( k) is absorbing the ast inuts/oututs and arameters of the original olynomial, associated with u ( k ). L M Proerty : Assign : a U-maing to convert classical olynomial exression of (.) to its U- exression of (.) and the inverse be, that is f (, ) f ( u, ) (.5) i i Thus it has good maing roerties [30]... U-mode --- rational With reference to (.), its deterministic arametric rational exression is given below. N yk ( ) D (*) (*) num den ( k) n d n ( k) d (.6) olynomial. Similarly, ( k) is a function of ast inuts i Uk and oututs Yk, and arameters d in the denominator olynomial. M and L are the degrees of the model inut uk ( ) in numerator and denominator resectively. Here is a simle examle to show the conversion of Insection of (.8), it gives y( k) y( k ) uk ( ) 0 (.9) y( k) ( k) u( t ) ( k) (.0) where ( k) 0( k) y( k ). In the following sections of the controller design, it is required making dynamic inversion of (.8) in way of rootsolving. There are many standard root-solving algorithms for such olynomial equations [30]. Remark : Comared with olynomial U-realisation, it can be noted that rational model U-realisation is an imlicit exression of yk ( ) due to the multilicative item y( k) D ( k )..3. U-model --- extended To describe more general nonlinear terms including those transcendental functions, define the extended U-model below y( k) f ( u( k )) f ( u( k )) (.) b a where fb( u( k )) and fa ( u( k )) are smooth functions. In general, these can be exressed as f ( u( k )) f ( u( k )) b f ( u( k )) f ( u( k )) a b a (.) Here is a simle examle to show its U-model reresentation, consider Its U-model realisation can be determined by removing the denominator to the left hand side of (.6), it gives y( k) D (*) N (*) (.7) Convert (.7) into U-model form to yield y( k )sin( u( k )) yk ( ) cos ( uk ( )) For its U-model of (.), it gives f u k k k u k b ( ( )) 0( ) ( )cos ( ( )) f ( u( k )) ( k)sin( u( k ) a (.3) (.4) L M i i (.8) i0 0 y( k) ( k) u ( k ) ( k) u ( k ) where ( k) is a function of ast inuts Uk and oututs Yk, and arameters n in the numerator where 0( k) ( k) ( k) y( k ) 3. Pole lacement controller A show case of the design rocedure 4

5 The control obective is, for a desired traectory vk ( ), determine a control inut ut () to drive the underlying system outut yk ( ) to follow the desired traectory vk ( ) with an accetable erformance (such as transient resonse and steady-state error), while all the inuts and oututs of the control system are bounded within the ermitted ranges. 3.. U-control system design In general, there are three stes in the U-control system design routine: Form a roer linear feedback control system structure, as shown in Fig.. The controller, in the dashed line block, is consist of two functions, invariant controller G and dynamic inverter G. The lant model is G. c Design invariant controller Gc by linear control system aroach. By letting G, therefore, G, and secifying the desired closed loo transfer function G, it G gives Gc and the invariant controller outut vt () is G the desired outut while the lant model is a unit constant. Determine dynamic inverter G to work out the controller outut uk ( ). Assuming the lant model is bounded-inut/bounded-outut (BIBO) stable and the inverse of G exist, exressing the lant model G in forms of U-model, letting y( k) v( k) in the U-model, it gives model (.) in exression of v( k) f ( u( k )) f ( u( k )). To determine control inut ut ( ) is to find the inverse by resolving the equation of v( k) f ( u( k )) f ( u( k )) 0. r - Gc v Fig..U-model control system It should be noted that the arrow line from the lant to the dashed line block reresents the U-model udate from the lant model at each time instance. Proosition : Generality, U-model based control allows once-off design for all linear and nonlinear olynomial models. This is due to the controller G c design being indeendent of model G. Proosition : Simlicity, U-model based control requires no reeated comutation if a lant model is changed. Again, this is due to the controller G c design being once-off and indeendent of model G, and changes to lant model G only changing the U-model to resolve different roots. In comarison, almost all classical and modern control aroaches are lant model based designs, that is, the controller design is a function of both system erformance and lant, accordingly if the lant model is changed, and controller must be redesigned. b b G ( U model) a a u G y Proosition 3: Feasibility for controller design of rational systems, this is can be roved directly from roosition and U-realisation of the rational model in (.8). In formality, U-adative control is very similar to deterministic U-model control. The difference is that the lant model is required to be estimated or udated online in adative control. For simlicity, but not losing generality, in formulation of U-model --- olynomial, once the invariant controller is designed, the real controller outut can be determined by letting M v( k) ( k) u ( k ) (3.) 0 Then resolving one of the roots from M v( k) ( k) u ( k ) 0 (3.) Stability and robust analysis of U-model control systems There are two tyical situations: Ideal case: deterministic systems without modelling error and disturbance. Non-ideal case: deterministic systems with modelling errors and/or disturbance. Theorem : Bounded-inut, bounded-outut (BIBO) stability of deterministic U-model control systems Regarding the U-model control system shown in Fig., it is BIBO stable and tracks the bounded reference signal r roerly while the following conditions are satisfied: (i) Invariant controller Gc is closed-loo stable, that is, all oles of the closed loo are located with the unit circle. (ii) Plant model G is a bounded-inut/boundedoutut (BIBO). (iii) The inverse of the lant model G exits. Proof: With reference to Fig., it has G G from the conditions (ii) and (iii). Accordingly, the closed Gc loo transfer function is given in terms of, which is Gc stable from (i) and thus the tracking erformance is given by rgc. Gc Remark : This establishes a framework for designing control for both linear and nonlinear dynamic lants. It is feasible, simle, general, and with no reetition of controller design on changes to the lant model, excet the comutation of the inversion of the changed lant U- model olynomial. In the other words, this is a new methodology for minimising the comlexity induced by the lant model in control system design, which is articularly imortant for nonlinear lants. U-model, as a universal dynamic inverter, is the key to achieve the gaols. Theorem : BIBO Stability of uncertain U-model control systems 5

6 G Regarding the U-model control system structured in Fig., modelling error and/or disturbance U () t can be treated as an external disturbance as shown in Fig. 3. It is BIBO stable and tracking the reference signal with a bounded error while the following conditions are satisfied: (i) Invariant controller Gc is closed loo stable. (ii) Plant model is a bounded-inut/boundedoutut (BIBO). (iii) The inverse of the lant model G exits. (iv) The uer bound of modelling error and/or disturbance U () t is satisfied with the conditions of small gain robust stability [4]. Proof: In Fig. 3, G G this gives rgc U y. ( Gc ) ( Gc ) Then the stability of Fig. 3 is the same as in Fig. while the uer bound U () t is satisfied with the small gain robust stability criterion. Remark 3: It should be noted that the tracking error is U determined by ; therefore, a roerly designed G c ( G c ) will have a degree of robustness against uncertainties/disturbance. r - Fig. 3. Uncertain U-model control system 3. Design of ole lacement controller A classical aroach [33] has been selected to formulate the U-model enhanced ole lacement controller (UPPC) [30, 3]. Here a further refined version of UPPC is resented. Within U-model framework, closed loo control system erformance is indeendently secified without involving the lant model. Therefore, the classical version involving lant model can be simlified as below. and G c v G ( U model) Rv( k) Tr( k) Sy( k) (3.) n n R z r z rn m m 0 m l l 0 l T t z t z t S s z s z s (3.) with rk ( ) for reference, vk ( ) for invariant controller G c outut, and yk ( ) for lant outut. The olynomials R, S, and T, with backward shift oerator z and roer orders (n, m, and l), are used to secify closed loo control system erformance. To guarantee the control system realistically imlementable, secify u G U + y O( S) O( R) l n O( T) O( R) m n (3.3) where the oerator O(*) Order (*) denotes the order of the concerned linear olynomial. With reference to (3.), two control roles can be assigned with negative feedback R for stabilising closed S loo system with requested dynamics and feedforward T R for reducing steady state errors. The structured control system is shown in Fig. 4. r(k) u(k-) v(k) y ( k) Linear or NL Rv(k)=Tw(k)-Sy(k) (v(k),u(k-)) lant Fig. 4. Structured UPPC control system For designing an invariant controller, let v( t) y( t) in (3.), thus it gives the closed loo transfer function T T y( k) r( k) r( k) R S A c (3.4) Accordingly, the required design task is to assign the closed loo denominator olynomial A and the numerator olynomial T. It should be noted that after c Ac secified (by customers and/or designers), a routine for resolving Diohantine is needed to workout the arameters of olynomials R and S from the following relationshi R S (3.5) To achieve zero steady state, Tcan be designed with A c T A c () (3.6) The detailed design rocedure and examles can be refereed to [3]. Remark 4: Comared with classical ole lacement control design rocedures [33], the UPPC is more concise and indeendent of the lant model, which results in the UPPC being generalised to any lant model structure and once off designed. For each different lant model, this task is merely the resolving of the U-model to obtain one of the roots as the oerational controller outut. The relevant comarison details can be referred in [30]. 4. U-model based ole lacement control with adative arameter estimation U-model based adative control schematic diagram is shown in Fig. 5. Again, this U-model adative control is different from those classical adative/self-tuning control aroaches in terms of control structure. The feedback controller arameters are not tuned and thereafter fixed: the only adatation is to udate U-model arameters to 6

7 accommodate the lant model arameter variation and/or external disturbance, which is consistent with roositions, and 4. r - Fig. 5. Adative U-model control system In general, an adative control system can be considered as a two layer system, that is, Layer : conventional feedback control; Layer : adatation loo. In this study, the UPPC resented in section 3 is selected to form conventional feedback control. Thus this section mainly develos this adatation loo formulation. In recursive formulation, there are two ways to estimate the U-model arameters in the adatation loo. Indirect arameter estimation: Estimate the original rational model arameters ( ( k), ( k) ) first and then convert into U-model arameters ( k). The challenging issue is that classical recursive least squares estimation algorithms give biased estimates and recursive rational model estimators need noise variance information in advance [, 8]. Direct arameter estimation: Estimate the U-model arameters ( k) directly. The challenging issue is that the arameters ( k), while converted from a rational model, are time varying at every samling time. It has been roved [34] that for time-varying stochastic models, the arameter estimation errors (PEE) with the well-known forgetting factor least-squares (FFLS) algorithm are bounded and the FFLS is caable of reducing the squared measurement error (the difference between measured outut and model redicted outut) even the time-varying arameter estimate are not converged to their real values. In this study a FFLS estimator [35] is selected with the following formulations T ( k) y( k) ( k) ( k ) U Gc v G ( U model) P( k ) ( k) Kk ( ) T ( k) P( k ) ( k) ( k) ( k ) K( k) ( k) n U T P( k) [ K( k) ( k)] P( k ) (4.) where vector ˆ( ) ˆ ˆ ˆ 0( ) ( ) ( ) T M k k k M k is the estimate of ( k), ( k) is the error, that is, the difference between the measured outut and the model redicted outut, K( t) ( M ) is the weighting factor vector indicating the effect of () t to change the arameter d u G U- modela rest y M vector, ( ) ( ) ( ) T M k u k u k is the inut vector at time k-, is forgetting factor (a number less than, e.g or 0.95, reresents a trade-off between fast tracking and noisy estimate), the smaller value of, the quicker the information in revious data will be ( M) ( M) forgotten, and Pk ( ) is the covariance matrix. In resenting the stability of the roosed adative U-control, exand the virtual equivalent system (VES) concet and methodology [36] for the analysis, which is an alternative insight and udgement of the stability/convergence for adative control systems. Following the similar arguments as shown before, we assume G G, and the invariant controller Gc is well defined to stabilise conventional feedback control systems and track the bounded reference signal in terms of mean squares. Then for a slow time varying arameter model (because it is converted from its original time invariant arameter model referred to (.) and (.)), the U-model arameter estimation errors U () t are bounded with FFLS or the other recursive algorithms [34, 37]. In this case, using Fig. 3 again, knowing U () t includes U-model arameter estimation errors as. Hence, in terms of VES, the adative control system can be treated as a summation of two subsystems of y y y rg c U Gc Gc ( ) ( ) (4.) As U () t is bounded, the adative control system is stable and the tracking control error will converge to a bounded comact set around zero, whose size deends on the ultimate bounds of estimation error U. Remark 5: U-model rovides a latform for simlifying control system design and VES rovides a latform for simlifying the analysis of stability and convergence of general adative control systems. 5. Simulation studies Four case studies have been conducted to initially validate the new design rocedure. It should be made clear that there is no other comarison result can be rovided as this is the first study in controlling of such nonlinear rational systems. As described before, the design is slit into two stages, design invariant control G c (thus vk ( ) ) by ole lacement) and determination of the controller outut uk ( ) by resolving lant U-model equation. To design the ole lacement controller, assign the characteristic equation Ac z z (5.) Factorisation of (5.) gives the closed loo oles as i, this gives a decayed oscillatory 7

8 resonse ( n 0.7 ), which is a commonly used dynamic resonse index. For steady state error erformance, making its error zero gives T A c () (5.) From the causality condition, secify the structures of R and S with R z rz r S s z s 0 (5.3) Form a Diohantine equation with olynomials A R S [30] to yield c r s r s (5.4) To determine the control inut u(k-), form a U- model equation from (5.9) as where 3 0 ( k) ( k) u( k ) ( k) u ( k ) 3 ( k) u ( k ) 0 (5.0) ( k) v( k)( y ( k )) ( k) 0.5 y( k ) 0 ( k) v( t) ( k) 3 (5.) In this simulation, the oeration time length was configured with 400 samling oints, and reference was a sequence of multi-amlitude stes. The achieved outut resonse and controller outut are shown in Fig. 6(a) and Fig. 6(b) resectively. To make olynomial R stable and having the requested resonse, assign r 0.06, r , which gives two oles ( z0.05) ( z 0.0). Then the coefficients of olynomial S are resolved in the Diohantine equation of (5.4) as follows. s.605 s (5.5) 0 Consequently controller (3.) can be recursively imlemented to calculate the virtual controller outut vt () v( k ) 0.06 v( k) v( k ) 0.76 r( k ).605 y( k) y( k ) (5.6) (a) Plant outut resonse 5.. Case --- feasibility test of U-control of nonlinear rational systems Consider a rational system modelled by yk ( ) y( k ) u( k ) u ( k ) y ( k ) u ( k ) (5.7) where y(k) is the model outut, u(k) is the inut of the model or controller outut. This is used to test deterministic feedback control. The model structure has been tyically investigated in system identification. Accordingly, its U- realisation can be exressed as y k y k u k y k u k u k 3 ( ) ( ) ( ) 0.5 ( ) ( ) ( ) To obtain the dynamic inverter G (5.8) outut, that is, the controller outut ut (), let y( k) v( k), then it gives rise to v k y k u k y k u k u k 3 ( ) ( ) ( ) 0.5 ( ) ( ) ( ) (5.9) (b) Control inut Fig. 6. Plant outut and control inut 5.. Case --- test of external disturbance Consider a stochastic rational system modelled by y( k ) u( k ) u ( k ) y ( k ) u ( k ) y( k) e( k) (5.) where y(k) is the model outut, u(k) is the inut of the model or controller outut resectively, and ek ( ) is Gaussian noise reresenting unknown an disturbance acting on the controlled lant outut. This case study was used to test adative feedback control. The feedback control loo has been designed as in 8

9 Case, that is, all configurations for feedback control were ket as those used in Case. For the adatation loo, the disturbance was configured with e( k) ~ N (0, 0.0), the 6 initial covariance matrix with P( k) 0 I4, the forgetting factor with 0.95 to deal with fast time varying arameter estimation, the initial arameter vector was randomly assigned with ˆ (0) ˆ ˆ ˆ ˆ 0(0) (0) 3 (0) 4 (0) , and the inut vector was secified with 3 ( k) u( k ) u ( k ) u ( k ) T. The achieved outut resonse and controller outut are shown in Fig. 7(a) and Fig. 7(b) resectively. T T k 50 ak ( ) 0.5 otherwise (5.4) The adation loo, secified as in Case, was used to follow the lant model internal structure variation. The achieved outut resonse and controller outut are shown in Fig. 8(a) and Fig. 8(b) resectively. Insecting the simulation results, the outut of the systems are seen to track the reference signals after a short transient hase. U-model arameter estimation is shown in Fig. 9. It should be noted that this estimated arameter vector is to achieve smaller squared error between measured outut and model redicted outut. Therefore the estimates are not converged to those real time varying arameters in the U-model. In the future studies to deal with time varying arameter estimation will be conducted in terms of reducing both squared measurement errors and squared dynamic errors [40]. (a) Plant outut (a) Plant outut (b) Control inut Fig. 7. Plant outut and control inut 5.3. Case test of internal arameter variation The same model structure as Case is used, but the arameter associated with y(k-) u(k-) is time varying reresenting internal arameter disturbances, such as worn arts in mechanical and electrical systems. (b) Control inut Fig. 8. Plant outut and control inut a k y k u k u k yk ( ) 3 ( ) ( ) ( ) ( ) y ( k ) u ( k ) (5.3) In simulation, all the setus were the same as those used in Case. The arameter variation was configured as 9

10 Fig. 9. U-model arameter estimates (a) Plant outut 5.4. Case feasibility test of U-control of extended nonlinear rational systems This study is used to test the U-control of extended rational systems with transcendental inut and delayed outut. 0.5 y( k ) sin( u( k )) u( k ) yk ( ) ex( y ( k )) (5.5) where yk ( ) is the model outut, uk ( ) is the inut of the model or controller outut. Accordingly, the extended U- model can be exressed as y k y k y k u k u k ( ) ex( ( )) 0.5 ( ) sin( ( )) ( ) (5.6) With the same controller designed in (5.6) above, assign the outut yk ( ) of (5.6) with the desired outut vk ( ) of (5.6) gives v k y k y k u k u k ( ) ex( ( )) 0.5 ( ) sin( ( )) ( ) Therefore the control inut uk ( ) can be solved by v k y k y k u k u k (5.7) ( ) ex( ( )) 0.5 ( ) sin( ( )) ( ) 0 (5.7) The achieved outut resonse and controller outut are shown in Fig. 0(a) and Fig. 0(b) resectively. Once again the comutational exeriment confirms the feasibility of U- control. (b) Control inut Fig. 0. Plant outut and control inut 6. Conclusions A fundamental question is raised in this study and those for the other U-model enhanced controls: after two generations of lant model (olynomial and state sace) centered control system design research/alications, what is the next generation of develoment? Should the research for new model structures continue, or should control systems be designed without such lant model requirements (ossibly imlying searation of control system design and controller outut determination)? One of the feasible choices in the future rogression could be the U-control design methodology, which radically reduces the comlexity of lant model oriented design methods. The roosed U-control method rovides a latform with ) a universal control oriented structure to reresent existing models; ) searating closed control system design from lant model structure (no matter linear or nonlinear, olynomial or state sace); 3) all welldeveloed linear control system design methods can be exanded in arallel to nonlinear lant models, 4) a sulementary to all existing control design methods. Accordingly, this study is a show case using the U-model framework to design the control of the nonlinear rational 0

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