On the presence of equilibrium points in PI control systems with send-on-delta sampling

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1 211 5th IEEE Conference on Decson and Control and Euroean Control Conference (CDC-ECC) Orlando, FL, USA, December 12-15, 211 On the resence of equlbrum onts n PI control systems wth send-on-delta samlng Manuel Besch, Antono Vsol, Sebastán Dormdo and José Sánchez Abstract Ths aer deals wth the resence of equlbrum onts and lmt-cycles n a PI control system wth send-on-delta samlng. In artcular, suffcent condtons on the controller arameters for the exstence of equlbrum onts are gven. These condtons can be usefully exloted for the tunng of the event-based PI controller arameters, thus makng the overall desgn easer. Smulaton results are rovded as llustratve examles. I. INTRODUCTION It s well known that n some rocesses a small steadystate control error of the rocess outut around the set-ont does not consttute a hard desgn constrant but, however, the reducton of the nformaton exchanged between the agents that take art n the control loo (sensors, controllers, actuators) s one of the tghtest requrements. Indeed, the reducton of the nformaton flow s a relevant ssue esecally when there are constrants on the communcaton rate, for nstance when data are exchanged n a dstrbuted control system by wred or wreless networks [9], [7], [1]. In these stuatons, cuttng down the traffc load s a key ont because the more traffc, the hgher ossblty of lost data and stochastc tme delays. Ths revents the occurrence of large latences and delay jtter and the CPU utlzaton s also reduced. In artcular, a framework where the reducton of the exchanged traffc s an essental ssue s n wreless networks and secally usng battery-owered or lmted comutatonal ower devces [21], [1]. Therefore, the hgher nformaton flow reducton, the hgher decrease of comutng oeratons and transmssons, and thus the longest lfetme of batteres. In ths context, one of the most convenent strateges s the use of event-based samlng and control aroaches. Indeed, durng last years event-based samlng and control technques have been addressed by a large number of researchers (see, for examle, [3], [19], [6]) also n the context of Proortonal-Integral-Dervatve (PID) controllers [2], [11], [5], [13]. Between the dfferent event-based samlng strateges, one of the most common s the so called send-ondelta samlng (also known as deadband samlng) where feedback control actons are comuted when the rocess outut s outsde a certan detecton band located around the set-ont value [16], [12]. Once the rocess s nsde the detecton band, new control actons are not roduced untl the rocess leaves the regon as a consequence of dsturbances or of a change of the set-ont value. The controller emloyed s usually a PID controller wth varable samlng erod [15], [17], [18]. Actually, (tme-based) PID controllers are the most emloyed controllers n ndustry owng to ther advantageous costbeneft rato. In fact, they are caable to rovde a satsfactory erformance for many rocesses and the settngs of ther arameters are relatvely easy also because of the large number of tunng rules that are avalable [8]. However, n event-based control the events occur asynchronously and therefore the tunng of the PID controller arameters s n general more challengng, as the tmng of the events nfluences the system erformance and lmt cycles may arse [14] (note that the resence of lmt cycles s a tycal roblem n general event-based control systems [4]). Further, n addton to the PID gans, the threshold values emloyed n the control algorthm (see Secton 2) have also to be tuned. Indeed, the tunng of a PID controller wth deadband samlng has not been exlctly addressed n the lterature untl now, at least to the authors knowledge. In ths aer, suffcent condtons on the controller arameters for the exstence of equlbrum onts are gven. On the other hand, startng from them, condtons for the exstence of a lmt cycle can be easly derved. The cases of eventbased P, I and a PI controllers are consdered (the dervatve acton s not consdered because ts mlementaton s very crtcal wth a varable, and ossbly long, samlng erod). It s beleved that these condtons can be usefully emloyed for the tunng of the overall controller. The aer s organzed as follows. In Secton 2 the roblem s formulated. Suffcent condtons for the resence of equlbrum onts (or, from another ont of vew, the condtons for the resence of a lmt cycle) are gven, wth llustratve examles, n Secton 3, 4 and 5 for the P, I, and PI case resectvely. Conclusons are exlaned n Secton 6. II. PROBLEM FORMULATION We consder the control scheme shown n Fgure 1 where the rocess s descrbed by the followng general transfer functon G(s) = bmsm + +b 1s+b a ns n + +a 1s+a e Ls. (1) where L s the dead tme, b and m < n. We assume that all the oles belong to the oen left half lane wth the exceton of a ossble ole at the orgn, namely, we can have a self-regulatng or a non self-regulatng (ntegral) rocess. The control acton u(t) s generated by an event-based PI controller where the roortonal and ntegral actons are enabled once the rocess outut s outsde a redefned band around the set-ont value r. In artcular, a send-ondelta samlng strategy s aled. The roortonal acton /11/$ IEEE 7843

2 u P (t) = K e(t) (where K s the roortonal gan) s recalculated every tme that the aboslute error s greater than and the absolute value of the dfference between the current error e(t) = r(t) y(t) and the error n the last crossng e(t P l ) s greater than, that s: e(t) e(t P l ). (2) The ntegral acton u I (t) = K IE(t) (where K s the ntegral gan) s calculated every tme that IE(t) IE(t I l ) (3) where the ntegral of the error IE(t) s defned as { 1 f e > ε IE(t) = e(t)f(e(t))dt, f(e) = f e ε (4) where ε s the desred maxmum error (deadband). It s worth stressng that, from a ractcal ont of vew, the consdered event-based control aroach s mlemented by samlng the rocess varable and by evaluatng the event-based condtons as fast as ossble. Ths s called the comound aroach or the fast samlng aroach where the asynchronous events are resynchronzed by usng a fast erodc samlng [7]. Thus, by denotng as T the samlng erod of the sensor, the followng overall control algorthm can be outlned. Sensor unt 1) f r y > then set e (t) = r y; else set e (t) = ; 2) f e (t) e (t P l ) > then generate a P event by sendng e (t) to the control unt and set e (t P l ) = e (t); 3) f r y > ε then set e (t) = r y; else set e (t) = ; 4) set IE = IE +Te (t); 5) f IE(t) IE(t I l ) > then generate an I event by sendng IE(t) to the control unt and set IE(t I l ) = IE(t). Control unt 1) f a P event s receved, then set u P (t) = K e (t); 2) f a I event s receved, then set u I (t) = K IE(t); 3) set u(t) = u P (t)+u I (t). Note that f K = then a roortonal controller results whle f K = then an ntegral controller results. It aears that, wth resect to a standard tme-based PI controller, the roosed algorthm has more arameters to tune. Indeed, n addton to the roortonal and ntegral gans the threshold values,, and ε have to be sutably selected. The am of the followng sectons s to defne condtons on the controller arameters whch allow the system to have at least an equlbrum ont for every value of the setont r and of the constant dsturbance amltude D. In fact, equlbrum onts whch deend on a lmted set of set-ont values r and dsturbance amltudes D have not ractcal relevance, because n general ther values are not known a r Sensor Unt Fg. 1. D Event Control u(t) y(t) G(s) Unt Scheme of the event-based control system. e>(j 1) e>j e>(j+1) e>(j+2) Pj 1 Pj Pj+1 e<(j 2) e<(j 1) e<j e<(j+1) Fg. 2. P event unt automaton. ror. Note that no consderatons are done wth resect to the regon of the attracton of the equlbrum onts. Thus, the resence of an equlbrum ont does not mly that t wll be attaned by the control system. III. EVENT-BASED PROPORTIONAL CONTROLLER In a roortonal event-based control strategy, the behavor of the sensor unt can be descrbed as an automaton. In fact, t s ossble to defne a state P j, where e(t) [(j 1),(j+ 1) ]; when the error reaches the threshold (j +1) the automaton jums to the state P j+1 and the sensor unt sends to the controller unt the value (j + 1) ; as soon as the error s less than (j 1) the automaton jums to the state P j 1 and the sensor sends the value (j 1). Fgure 2 shows the automaton reresentaton. In a generc state P j the control system s an oen-loo system wth a constant control varable u P j, therefore t s ossble to defne a steady-state oututy ss,j = G() ( u P j +D). Ths outut s an equlbrum ont f e ss,j [(j 1),(j +1) ]. In fact, as Fgure 3 shows, f the latter condton s not satsfed, the automaton would jum n another state. It s mortant to notce that e ss,j [(j 1),(j+1) ] s a necessary condton only of the exstence of the equlbrum ont, because t does not gve nformaton about the regon of attracton of the equlbrum ont. In fact, the system can reach the equlbrum ont wth a fnte number of transtons, or can admt a lmt cycle whch nvolves two or more states, as a consequence of a erodc sequence of events that are generated because of the characterstcs of the transent resonse of the system. Ths asect s exemlfed n Fgure 4, where the equlbrum ont exsts but the system does not attan t. The necessary condton can be used to fnd suffcent condtons of the values of the controller arameters for whch there s at least one ossble equlbrum ont, ndfferently by the values ofd. By takng nto account that the control acton, wth an event-based roortonal controller, can assume the followng values u P ss,j = K e (t) = K j, j Z (5) the followng roostons can be stated. 7844

3 (j+1) j State transton d no equlbrum onts KK (1+KK ) j f s an equlbrum ont j +1 s an equlbrum ont f Fg. 5. Relatonsh between equlbrums and d (j 1) (j+1) j (j 1) Fg. 3. Case wthout an equlbrum ont. State transtons Fg. 4. Case wth an equlbrum ont whch s not attaned. Sold thck lne: evoluton of the error n the state P j. Dashed thck lne: evoluton of the error f the frst event would not occur. Sold thn lne: evoluton of the error after the frst event. Prooston 1: If the rocess (1) has a ole at the orgn, there are values of D for whch there does not exst an equlbrum ont. Proof. In order for the rocess outut to be constant the steady-state rocess nut should be null. As t s u P ss,j + D = f D s not exactly a multle of K there are no equlbrum onts. Prooston 2: If the rocess (1) s asymtotcally stable, then a suffcent condton for the resence of an equlbrum ont for all the values of r and D s K 1/K where K = G(). Proof. If the rocess s self-regulatng, then the steady-state oututs y ss,j are y ss,j = K(jK +D), j Z. Thus, y ss,j can be an equlbrum ont f: that s, (j 1) e ss,j = r y ss,j (j +1), r jkk KD (j 1), r jkk KD (j +1). These condtons can be rewrtten as (1+KK ) + (1+KK ), (1+KK ). j r KD j r KD (1+KK ) The term r KD 1+KK corresonds to the steady-state error that would be obtaned by usng a roortonal controller wth erodc samlng. The quantty r KD can be exressed as r KD r KD = j f (1+KK ) +d wth j f = (1+KK ) Z and d = r KD j f (1+KK ) [,(1+KK ) ], where d s, roughly seakng, the unquantzable art of r KD. In ths way, the condtons (7) can be rewrtten as: j j f + j j f + (6) (7) d+ (1+KK ) (8) d (1+KK ). (9) It s mortant to notce that, wth the frst condton, t s ossble to exclude all states P j wth j > j f + 1. In fact, consderng a state P jf +h, wth h 2 the condton (8) becomes: and therefore j f +h j f + d+ (1+KK ) h(1+kk ) < (1+KK ) d, whch s absurd because d [,(1+KK ) ]. In the same way the state P j wth j < j f can be excluded by alyng the second condton. In fact, consderng the state P jf h, wth h 1 the condton (9) becomes: or equally: j f h j f + d (1+KK ) d h(1+kk ) +, whch s absurd because d [,(1+KK ) ]. From the revous consderatons, there are only two ossble canddates to be equlbrum states, namely P jf and P jf +1. Consderng the frst one, the necessary condtons (8)-(9) can be wrtten, resectvely, as: j f j f + whch s always true, and j f j f + d+ (1+KK ), (1) d (1+KK ), (11) whch s true f d. Consderng the state P jf +1, the necessary condtons are: j f +1 j f + whch s verfed f d KK, and j f +1 j f + d+ (1+KK ) (12) d (1+KK ) (13) whch s always true. As already stressed, we need to fnd values of K and whch allow the controlled system to have at least one equlbrum ont ndfferently to the value of d. By lookng the revous equatons, t s easy to note that: f d, then P jf s a equlbrum state; f d KK, then P jf +1 s an equlbrum state; f < d < KK, then there s the absence of equlbrum onts. To avod the occurrence of the thrd condton, t s necessary to chosekk <, thereforekk 1. The condtons are summarzed n Fgure 5. Remark 1. In ths secton we consder that the send-on-delta samled error assumes only values multle of (see for nstance (5)). Ths assumton seems contradctory wth the defnton (2), however t s clear that alyng (2) f the error 7845

4 IE>( 1) IE> IE>(+1) IE>(+2) IE<( 2) I j 1 I j I j+1 Fg. 6. IE< I event unt automaton. IE<(+1) s a contnuous sgnal, then the send-on-delta samled error assumes values whch are the sum of a multle of and an ntal constant term whch can be neglected by consderng t as a art of the dsturbance D. Smlar consderatons are aled n Sectons IV-V. IV. EVENT-BASED INTEGRAL CONTROLLER By followng a smlar reasonng aled to the P case, the sensor unt of an event-based ntegral controller has a behavor whch can be descrbed as an automaton. In ths automaton the state I s assumed when IE = [ 1,+1] wth > ; the automaton jums to the uer state f IE ( + 1) and to the downer state f IE ( 1). When a transton occurs, the sensor sends the new value to the controller unt. Fgure 6 shows the automaton reresentaton. Also n ths case, when the system remans n a state, the control system s an oen-loo system wth a constant nut u I, therefore t s ossble to defne a steadystate outut y ss, = G() ( u I +D). Ths s an equlbrum ont f e ss, [ ε,ε]. Ths necessary condton s easy to exlan by notng that f t s false then IE changes ts value contnuously. As for the P controller, t can be used to fnd the values of the controller arameters for whch there s at least one equlbrum ont. By takng nto account that the control acton of an event-based ntegral controller can assume the followng values: u I ss, = K IE = K, Z the followng roostons can be stated. Prooston 3: If the rocess (1) as a ole at the orgn, there are values of D for whch there does not exsts an equlbrum ont. Proof. If the rocess (1) s non self-regulatng, then the steady-state control varable should be null, that s u I ss, + D =. Thus, f D s not exactly a multle of K there are no equlbrum onts. Prooston 4: If the rocess (1) s asymtotcally stable, then a suffcent condton for the resence of an equlbrum ont for all the values of r and D s K (2ε)/(K ) where K = G(). Proof. If the rocess s self-regulatng, the steady-state oututs y ss, are: y ss, = K(K +D) wth Z Thus y ss, s an equlbrum ont f or equvalently ε e ss,j = r y ss, ε KK +r KD ε KK +r KD ε These condtons can be rewrtten as r KD KK + ε KK r KD KK ε KK Redefnng the roduct KK as αε, the term r KD can be exressed as r KD = f αε+d wth f = r KD αε Z and d = r KD f αε [,αε]. In ths way, the condtons can be rewrtten as: or: d αε + 1 α + f d αε 1 α + f d (α( f ) 1)ε (14) d (α( f )+1)ε (15) Condton (14) s certanly true f ( f ) 1/α, and t s certanly false f ( f ) (1+1/α). Conversely, condton (15) s certanly true f ( f ) (1 1/α), and t s certanly false f ( f ) 1/α. Also n ths controller, t s mortant to fnd values of the arameters whch allow the controlled system to have at least a ossble equlbrum ont ndeendently from the value of d. The revous condtons are verfed together f 1/α (1 1/α) or, equvalently, f α 2, that s K (2ε)/(K ). If ths condton s not verfed an equlbrum ont can exst for artcular values of d (that s, of D). It s worth notng that the number of equlbrum onts ncreases asαdecreases. Some llustratve cases are outlned. Choosng α = 2, f d < ε, then I f s an equlbrum state; f d > ε, then I f +1 s an equlbrum state; f d = ε, then both I f and I f +1 are equlbrum states. Choosng α = 1, I f and I f +1 are equlbrum states for every values of d. Choosng α = 1/2, I f, I f +1, I f +2 and I f 1 are equlbrum states for every values of d. V. EVENT-BASED PI CONTROLLER If the chosen control strategy s an event-based PI controller, t s ossble to defne an automaton where a generc state S,j s the combnaton between a state P j on the roortonal art and a state I on the ntegral art. Fgure 7 shows the automaton reresentaton. In a generc state S,j the control system s an oen-loo system wth a constant nutu,j, therefore s ossble to defne a steady-state outut y ss,,j = G()(u,j +D). Ths outut corresonds to an equlbrum ont f e ss,,j [(j 1),(j + 1) ] and e ss,,j [ ε,ε]. Note that the value of should be greater than ε, because when the error s less than the maxmum desrable error, namely e [ ε,ε], the new value of the control varable must not be comuted. By takng nto account that for the event-based PI control strategy the control acton can assume the followng values u ss,,j = K +jk,,j Z we can state the followng roostons. 7846

5 IE>(+2) IE>(+2) e>j e>(j+1) e>(j+2) TABLE I SUFFICIENT CONDITIONS FOR EVENT BASED PI α > 2 Equlbrums onts exst only for some values of D, otherwse there s a lmt cycle. 1 < α 2 There are equlbrum onts wth j =. < α 1 There are equlbrum onts wth j =,1, 1. e<(j 1) IE< s +1,j s +1,j+1 e<j IE>(+1) IE< e>j e>(j+1) e>(j+2) IE>(+1) e<(j+1) where f1 = K(K D)+r αε Z and d = K(K D) + d f1 αε [,αε]. In ths way, the condtons (16) can be rewrtten as ( f1 )αε+d mn(ε,2 ) = ε (17) ( f1 )αε+d mn(ε,) = (18) Condtons (17) and (18) can be exressed as e<(j 1) s,j Fg. 7. IE> e<j s,j+1 IE> PI event unt automaton. e<(j+1) Prooston 5: If the rocess (1) as a ole at the orgn, there are values of D for whch there does not exsts an equlbrum ont. Proof. If the rocess s non self-regulatng, the steady-state rocess nut should be null, that s, u ss,,j +D =. Thus, f D s not exactly a multle of a combnaton of K and K there are not equlbrum onts and the system certanly has a lmt cycle. Prooston 6: If the rocess (1) s asymtotcally stable, then a suffcent condton for the resence of an equlbrum ont for all the values of r and D s K (2ε)/(K ). Proof. If the rocess s asymtotcally stable, the steady-state oututs y ss,,j are y ss,,j = K(K +jk +D) wth,j Z These oututs are equlbrum onts f KK jkk +r KD ε KK jkk +r KD ε (16) KK jkk +r KD (j 1) KK jkk +r KD (j +1) Hence, by consderng > ε, there are only three values of j whch satsfy all the condtons: 1, and 1. When j =, the condtons on the steady-state error are equal to the ntegral case, therefore, recallng that KK = αε, there s at least an equlbrum ont f α 2. When j = 1 the steady-state errors are e ss,, 1 = KK +K(K P D)+r, Z The term K(K D)+r can be exressed as K(K P D)+r = f1 αε+d d ( f1 )αε ε (19) d ( f1 )αε (2) Condton (19) s certanly true f ( f2 ) 1/α and s certanly false f ( f2 ) (1+1/α), whle condton (2) s certanly true f ( f2 ) 1 and s certanly false f ( f2 ). Prevous equatons are certanly both verfed f 1 α > 1, or equally α < 1. When j = 1 the steady-state oututs are y ss,, 1 = KK +K( K P D)+r, Z The term r K(K +D) can be exressed as: r K(K P +D) = f2 αε+d wth f2 = r K(K +D) αε Z and d = r K(K + D) f2 αε [,αε]. In ths way, the four condtons (16) can be rewrtten as: ( f2 )αε+d ε (21) ( f2 )αε+d ε (22) ( f2 )αε+d (23) ( f2 )αε+d 2 (24) Condtons (21) and (24) are always verfed, whle condtons (22)-(23) can be exressed as: d ( f1 )αε (25) d ε+( f1 )αε (26) Condton (25) s certanly true f ( f ) and certanly false f ( f2 ) (1 1/α) whle condton (26) s certanly true f ( f2 ) (1 1/α) and certanly false f ( f2 ) < 1/α. Both condtons are therefore verfed f (1 1/α) >,.e., α < 1. Results are summarzed n Table 1. Remark 2. Note that n an event-based PI controller, the steady-state suffcent condtons for the exstence of at least an equlbrum ont concern only the roduct KK = αε and there are no condtons on the roortonal arameters K and (rovded > ε). Note also that the condtons on the ntegral art are the same as the event-based ntegral controller case. 7847

6 rocess outut automata states tme tme Fg. 8. Smulaton wth α = 2.1. Dashed lne: evoluton of the automaton P j. Sold lne: evoluton of the automaton I rocess outut automata states tme tme Fg. 9. Smulaton wth α = 2.. Dashed lne: evoluton of the automaton P j. Sold lne: evoluton of the automaton I VI. ILLUSTRATIVE EXAMPLE In ths secton, the unt ste resonse of a second-orderlus-dead-tme rocess (wth K = 1) controlled by an eventbased P controller s analyzed. In artcular, two cases are resented, the frst where α = 2.1 (therefore the suffcent condton s not verfed), and the second where α = 2. (therefore the condton s verfed). Both the cases have the same K = 1. The other arameters are set as ε =.1, =.15, K = 3 and D = The consdered rocess s G(s) = 1 s 2 +3s+1 e.4s (27) The rocess outut and the evoluton of the automaton S,j are resented for the two cases n Fgures 8 and 9. Remark 3. It s worth stressng that from the above analyss t can be straghtforwardly deduced that, from a ractcal ont of vew, t s very lkely that a lmt cycle occurs f α > 2 (ths does not haen just for secfc values of D, see Table 1). VII. CONCLUSIONS In ths work condtons on the exstence of equlbrum onts and lmt cycles are nvestgated. In artcular, condtons on the arameters of P, I and PI controllers based on send-on delta samlng are resented. These condtons allow the controlled system to have at least one equlbrum ont ndfferently from the value of the constant load dsturbance. Another mortant result that has been resented s the resence of lmt cycle n rocesses wth a ole at the orgn of the comlex lane. Ths fact s caused by the quantzed nature of the controller, whch can not exactly comensate the constant load dsturbance. Ths s relevant because ntegral (non self-regulatng) rocess are frequently encountered n the rocess ndustry and ther control has been wdely nvestgated [2]. REFERENCES [1] G. 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