WIENER MODELS OF DIRECTION-DEPENDENT DYNAMIC SYSTEMS. Singleton Park, Swansea, SA2 8PP, UK. University of Warwick, Coventry, CV4 7AL, UK

Transcription

1 Copyight IFAC 5th Tiennial Wold Congess, Bacelona, Spain WIEER MOELS OF IRECTIO-EPEET YAMIC SYSTEMS H. A. Bake, A. H. Tan and K. R. Godfey epatment of Electical and Electonic Engineeing, Univesity of Wales, Swansea, Singleton Pak, Swansea, SA 8PP, UK ivision of Electical and Electonic Engineeing, School of Engineeing, Univesity of Wawick, Coventy, CV4 7AL, UK Abstact: iection-dependent dynamic systems ae defined, and Wiene models fo them ae descibed. Fo fist-ode systems with pseudo-andom binay inputs, optimising the model paametes by coss-coelation function matching methods based on analysis gives excellent esults. Fo fist-ode systems with invese-epeat pseudo-andom binay inputs, optimisation by discete Fouie tansfom matching and output matching methods also give excellent esults. These optimisation methods may be extended to diectiondependent dynamic systems that cannot be analysed, as illustated by a fist-ode system with a pseudo-andom tenay input, and a second-ode system with an invese-epeat pseudo-andom binay input. Copyight IFAC Keywods: iscete Fouie tansfoms; Fist-ode systems; Identification; Modelling; onlinea models; onlinea systems; Optimisation; Pseudo-andom sequences.. ITROUCTIO In diection-dependent dynamic systems, the system dynamics when the system output is inceasing ae diffeent fom those when the output is deceasing. Such systems ae not eadily amenable to analysis, and theoetical solutions ae known only fo cases in which the system dynamics in each diection ae fistode linea and the system inputs ae binay. The paucity of analytical esults fo these systems contasts shaply with thei abundance in industy, whee examples include steam-aising plants (Godfey and Biggs, 97), gas tubines, chemical pocesses and nuclea eactos (Godfey and Mooe, 974), distillation columns (Tune et al., 996), and automotive suspensions and tyes (Tan and Godfey, ). In the absence of methods fo dealing with these systems analytically, inteest tuns natually to modelling them in foms known to be suitable fo analysis. Fo nonlinea systems of this kind, models based on the Voltea functional seies (Voltea, 9) have taditionally been deemed suitable. This seies is known to convege fo block-oiented models, whee the blocks epesent subsystems that eithe have linea dynamics o ae static nonlineaities (Billiant, 958), and models of this kind ae theefoe used hee. Fom among the many diffeent kinds available, the Wiene model in which dynamic linea subsystems pecede static powe law nonlineaities is chosen because it is paticulaly suitable fo use with binay inputs (Bake et al., a), and these ae the only inputs fo which analytical esults ae available fo the outputs and coss-coelation functions of fistode diection-dependent systems (Tan and Godfey, ). The appoach adopted hee is to commence with the most geneal fom of a diection-dependent system capable of being analysed, which is a fist-ode system in which the values of both the gain and the time constant ae diection-dependent. A Wiene model is then developed fo the system, using analytical esults obtained when both the system and the model have a pseudo-andom binay input. The most effective modelling methods ae then applied to examples of diection-dependent systems fo which no theoetical methods ae available. These ae a fist-ode system with a pseudo-andom tenay input, and a second-ode system with an inveseepeat pseudo-andom binay input.. IRECTIO-EPEET SYSTEMS Recent wok by Tan and Godfey () has led to new and impotant esults fo diection-dependent systems. In paticula, analytical expessions have been obtained fo the outputs and coss-coelation functions of fist-ode systems with diectiondependent gains and time constants when the inputs

2 ae eithe maximum-length pseudo-andom binay signals o invese-epeat pseudo-andom binay signals. Hee the tansfe functions ae taken to be K U when the diection of the system output y(t) + st U is positive, and K + st when it is negative. A system fo which K U = 4, K =, T U = T, T = T and T = is used fo illustation in this pape. The system input u(t) is taken to be eithe a maximum-length pseudo-andom binay signal with peiod T geneated using a constant clock-pulse inteval of T fom a maximum-length binay sequence s(i) with peiod and chaacteistic polynomial f(), o the invese-epeat maximum length pseudo-andom binay signal with peiod T deived fom it by inveting altenate values. In both cases, the zeo elements of s(i) ae conveted to and the unity elements of s(i) ae conveted to +. A pseudoandom binay signal fo which = 7 and f ( ) = is used fo illustation in this pape. Of paticula impotance is the discete peiodic cosscoelation function φ (it ) between the system uy input u(t) and the steady-state system output y(t), given by uy ( it ) = y( T ) u(( i) T ) φ () = The coss-coelation function is the sum of an infinite numbe of components, fo which the analytical expessions involve fou paametes and an impulse esponse function: a = (exp( T / T ) + exp( T / T )) / b = (exp( T / T ) exp( T / T )) / U U c = exp( T / T ))( exp( T / T )) ( U A = ( K U + K ) / Ac i wc ( it ) = a fo i and = fo i < a The significant components of the system cosscoelation function ae as follows. Constant component φ uy = K () This component is simply a bias. Linea component + φ uy( it ) = wc ( it ) () This component, which is the most significant, is the + impulse esponse w C ( it ) scaled by. The Ac equivalent tansfe function has gain KC = ( a) and time constant T = T / ln(a). C Quadatic component + uy ( it ) = b a wc (( i i ) T ) = i si i = si si φ (4) whee is defined though and is i detemined by dividing the polynomial by the chaacteistic polynomial f () until the singletem emainde is obtained. This component is the sum of scaled and shifted eplicas of the impulse esponse, with the moe significant eplicas obtained when =,,, 4,.... Fo the chaacteistic polynomial used hee, the coesponding shifts ae i = 89, 5,,,.... Cubic component φ s q + q+ uy ( it ) = b a wc (( i iq, ) T ) q= = i iq, i q, = si si q s i q q q+ dividing the polynomial by the chaacteistic polynomial f () until the single-tem i q, (5) whee is defined though and is detemined by emainde is obtained. This component is also the sum of scaled and shifted eplicas of the linea impulse esponse. Fo the chaacteistic polynomial used hee, the moe significant eplicas ae at shifts i, = 59, i, = 85, i, =, i, = 7, i, = 8, and i, = 95. If the system input is the invese-epeat maximumlength pseudo-andom binay signal, then the oddode components in the new coss-coelation function ae identical to those above and the evenode components ae zeo, except fo a vey small zeo-ode tem that is an oscillatoy bias (Tan and Godfey, ).. WIEER MOEL The Wiene model of the system is shown in Fig.. As the highest-ode component of any significance in the system coss-coelation function is cubic, the model paths ae esticted to those of ode to. The discete peiodic coss-coelation function φ uv (it ) between the model input u(t) and the steadystate model output v(t) is the sum of fou components.

3 K Cubic component q + ( a) q+ φ uv( it) = 6 a w (( i i, ) T) q a q= = u(t) K +st K +st ( ) sgn(b) v(t) + + K whee i q, a + a + ( a) K w ( it) w ( it) (9) a is the same as in Section. This component is the sum of scaled and shifted eplicas of the impulse esponse w ( it ), togethe with a scaled eplica of the impulse esponse w ( it ) and a bias. K +st ( ) Fig.. Wiene model of fist-ode diectiondependent system The analytical expession fo the component coesponding to the -th ode path involves one paamete and two impulse esponse functions (Bake and Obidegwu, 97): a = exp( T / T ) i w ( it ) = K ( a ) a fo i and = fo i < i w ( it ) = K ( a ) a fo i and = fo i < The components of the model coss-coelation function ae as follows. Constant component φ K uv = (6) This component is simply a bias. Linea component + K φ uv ( it ) = w ( it ) (7) This component is the impulse esponse w ( it ) scaled by +, with a bias. Quadatic component φ + a uv ( it ) = [sgn( b)] a w (( i i ) T ) + a = K + [sgn( b)] (8) whee i is the same as in Section. This component is the sum of scaled and shifted eplicas of the impulse esponse w ( it ), with a bias. If the system input is the invese-epeat maximumlength pseudo-andom binay signal, then the oddode components in the new coss-coelation function ae identical to those above with the biases emoved and the even-ode components ae zeo, except fo a vey small zeo-ode tem that is an oscillatoy bias (Bake and Obidegwu, 97). 4. MOEL OPTIMISATIO WITH PSEUO- RAOM BIARY IPUTS The Wiene model that best epesents the diectiondependent system is obtained when its paametes ae set to thei optimal values. As the system and model coss-coelation functions have simila components, the coss-coelation component (CCC) matching method may be used (Bake et al., b; Bake et al., ). The constant components ae simply chosen to equate the biases. The linea components ae the most significant, and these ae matched when K = and K C T = T C. The quadatic components ae the next most significant, and the most consistent methods fo matching them involves minimising the sum of eithe the squaes o the moduli of the diffeences between them. These both give simila esults, so that only the fome will be used hee. Fo this, the MATLAB Optimization Toolbox (Coleman et al., 999) can be used to obtain the optimal values of K and a that give K, a = = Acb a a + K ( a ) a + The cubic components, which ae the least significant, can be matched in the same way, by obtaining the optimal values of and a that give K, a q q= K = = 6 ( ) Acb q+ + q+ + a K a a a

4 Fig.. Coss-coelation functions of fist-ode diection-dependent system (o) and Wiene model (+) with pseudo-andom binay input This method epesents the limit of the analytical appoach, but the esults obtained ae excellent. Fig. shows the coss-coelation functions of the system. The coespondence between the linea components is stiking, as is that between the quadatic components in which the moe significant eplicas at shifts of 89T, 5T, T, T, etc., ae easily seen. The cubic components ae less obvious. The optimal model paametes obtained by this method ae given in Table. Table Optimal paametes fo Wiene model of fistode diection-dependent system with pseudoandom binay input Model Paametes Method K K T K T K T CCC CCF OUT In contast to the completely analytical appoach above, two patly analytical methods may also be used to obtain the optimal model paametes. Both methods use the esults obtained by Tan and Godfey () to compute the system output values at the clock-pulse epochs. Fo a given set of model paametes, each of the model path output values at the same epochs is also computed and summed to obtain the model output values. In the coss-coelation function (CCF) matching method the coss-coelation functions of the system and the model ae computed and matched using the MATLAB Optimization Toolbox to obtain the optimal values of,, T,, T, and T that give K, K, T, K, T, K, T K K K K mimimum [ φ ( it ) φ ( it )] uy As seen fom Table, the optimal values obtained by this method ae simila to those obtained by the component matching method. uv Fig.. Outputs of fist-ode diection-dependent system (o) and Wiene model (+) with pseudoandom binay input In the output (OUT) matching method, the system and model output values ae matched by obtaining the optimal values of K, K, T, K, T, K and T that give mimimum K, K, T, K, T, K, T i = [ y ( it ) v( it )] Fig. shows the vey close match between the system and model outputs obtained by this method. As seen fom Table, the optimal values obtained ae simila to those obtained by the coss-coelation function methods. 5. MOEL OPTIMISATIO WITH IVERSE- REPEAT PSEUO-RAOM BIARY IPUTS If the input is now taken to be the invese-epeat maximum-length pseudo-andom binay signal with peiod T, then matching the odd-ode components of the system and model coss-coelation functions by the methods in Section 4 gives optimal values fo K, T, K and T simila to those in Table. Howeve, the even-ode components ae zeo except fo the vey small zeo-ode tem, so no meaningful infomation is available fo the optimal values of K, K, and T. Fotunately, the use of an invese-epeat input confes popeties on the output that can be exploited to give two impoved matching methods. The system and model outputs may be sepaated into odd and even components by subtacting o adding membes sepaated by a half-peiod. In the output (OUT) matching method, the odd components of the system and model outputs ae matched to obtain the optimal values of,, and T that give K, T, K, T k = K T K [ y ( i) y( i + ) v( i) + v( i + )] and the even components ae matched to obtain the optimal values of, and T that give K, K, T K K [ y ( i) + y( i + ) v( i) v( i + )]

5 4.5 As seen fom Tables and, the optimal values obtained by this method ae simila to those obtained using the CCC, CCF and OUT methods with a pseudo-andom binay input MULTI-LEVEL IPUTS A HIGHER-ORER SYSTEMS Fig. 4. Outputs of fist-ode diection-dependent system (o) and Wiene model (+) with inveseepeat pseudo-andom binay input Fig. 4 shows the vey close match between the system and model outputs that is obtained by this method. As seen fom Tables and, the optimal paamete values obtained ae simila to those obtained with a pseudo-andom binay input. Table Optimal paametes fo Wiene model of fistode diection-dependent system with invese-epeat pseudo-andom binay input Model paametes method K K T K T K T CCF OUT FT Odd and even hamonics in the system output ae attibutable to the odd-ode and even-ode nonlineaities of the system espectively. In the discete Fouie tansfom (FT) matching method, the odd-ode and even-ode nonlineaities ae matched independently though the odd and even membes of the discete Fouie tansfom of a peiod of the output (Bake and Godfey, 999). The system output tansfom Y (k) is given by Y ( k) = y( it )exp( π ki / ) () and the model output tansfom V (k) is given by V ( k) = v( it ) exp( πki / ) () the odd membes of the system and model tansfoms yields the optimal values of,, K and T K, T, K, T that give k = Y (k + ) V (k + ) K T and matching the even membes yields the optimal values of, and T that give K, K, T K K k = Y (k) V (k) o analytical solutions ae known fo diectiondependent systems that eithe have inputs with moe than two levels o ae of ode geate than one, so fo those systems coss-coelation component matching cannot be used and coss-coelation function matching becomes consideably less attactive. Output matching and output tansfom matching with invese-epeat pseudo-andom inputs ae theefoe pefeed and the methods descibed in Section 5 can be applied in both cases. As an example of a system with a multi-level input, the same fist-ode system as befoe is consideed, but with an invese-epeat maximum-length pseudoandom tenay input with levels, and +, and peiod = 4. Fig. 5 shows that a close match between the system and model outputs can be obtained by the output matching method. Table shows that the optimal paamete values obtained by the output matching and output tansfom matching methods ae emakably consistent, and that the cubic component in the system and the model is vey small. Table. Optimal paametes fo Wiene model of fist-ode diection-dependent system with inveseepeat pseudo-andom tenay input Model paametes method K K T K T K T OUT FT Fig. 5. Outputs of fist-ode diection-dependent system (o) and Wiene model (+) with inveseepeat pseudo-andom tenay input

6 Fig. 6. Outputs of second-ode diection-dependent system (equal gains) (o) and Wiene model (+) with invese-epeat pseudo-andom binay input As an example of a highe-ode system, the tansfe function of the system dynamics is taken to be ( s with T U = T when the diection of the + T U ) system output is positive, and ( + s T ) with T = T when it is negative. In the Wiene model, the tansfe function in the -th ode path is taken to K be. The choice of citical damping fo the + st ) ( linea dynamics in the Wiene model paths eflects the natue of the system dynamics, and educes the numbe of paametes to be optimised, but is not necessaily optimal. Fig. 6 shows that a close match between the system and model outputs can be obtained by the output matching method. Table 4 shows that the optimal paamete values obtained by the output matching and output tansfom matching methods ae easonably consistent. Table 4. Optimal paametes fo Wiene model of second-ode diection-dependent system with invese-epeat pseudo-andom binay input Model Paametes Method K K T K T K T OUT FT COCLUSIOS As diection-dependent dynamic systems can be analysed in only simple cases, it is useful to model them in foms that ae amenable to analysis, and Wiene models with paths of ode to have been shown to be suitable fo this pupose. Methods fo obtaining optimal values of the model paametes have been developed fo those cases that can be analysed, which ae fist-ode diection-dependent systems with binay inputs. The methods involve matching appopiate functions o chaacteistics of the systems and models with eithe pseudo-andom inputs o invese-epeat pseudo-andom inputs. Coss-coelation function matching methods give excellent esults with pseudo-andom inputs, but cannot give complete esults with invese-epeat pseudo-andom inputs. Output matching and discete Fouie tansfom matching methods give excellent esults with invese-epeat pseudo-andom inputs, and ae to be pefeed in this application. These methods have also been used with good esults in two cases that cannot be analysed, which ae a fist-ode system with a pseudo-andom tenay input and a secondode system with an invese-epeat binay input. REFERECES Bake, H.A. and K.R. Godfey (999). System identification with multi-level peiodic petubation signals, Contol Engineeing Pactice, 7, Bake, H.A. and S.. Obidegwu (97). Effects of nonlineaities on the measuement of weighting functions by cosscoelation using pseudoandom signals, IEE Poc.,, 9-. Bake, H.A., K.R. Godfey and A.J. Tucke (a). onlinea system identification with multi-level petubation signals, Poc. th IFAC Symposium on System Identification SYSI, Santa Babaa, Califonia, June. Bake, H.A., K.R. Godfey and A.H. Tan (b). Identification of systems with diection-dependent dynamics, Poc. 9 th IEEE Confeence on ecision and Contol, Sydney, Austalia, ecembe, Bake, H.A., K.R. Godfey and A.H. Tan (). Optimised Wiene models fo diection-dependent dynamic systems, Poc. Ameican Contol Confeence, Washington,.C., USA, June, Billiant, M.B. (958). Theoy of the analysis of nonlinea systems, MIT, Reseach Laboatoy of Electonics, Repot 45. Coleman, T., M.A. Banch and A. Gace (999). Optimization toolbox fo use with MATLAB, The MathWoks Inc., atick, Massachusetts. Godfey, K.R. and P.A.. Biggs (97). Identification of pocesses with diectiondependent dynamic esponses, IEE Poc., 9, Godfey, K.R. and.j. Mooe (974). Identification of pocesses having diection-dependent esponses, with gas-tubine engine applications, Automatica,, Tan, A.H. and K.R. Godfey (). Identification of pocesses with diection-dependent dynamics, IEE Poc. Contol Theoy Appl., 48, Tune, P., G. Montague and J. Mois (996). onlinea and diection-dependent dynamic pocess modelling using neual netwoks, IEE Poc. Contol Theoy Appl., 4,