Semigroup formalism and internal model control for a heat exchanger

01/10/2017
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Semigroup formalism and internal model control for a heat exchanger

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1 Semigroup formalism and internal model control for a heat exchanger Touré Y., Josserand L. LVR UPRES EA 2078, University of Orléans, 63 avenue de Lattre de Tassigny, 18020 Bourges Cedex, France. Abstract— A particular control design for a distributed parameter system is discussed in this lecture. It concerns a direct method design for the control of a heat exchange laboratory experimental process. The aim is to show that semigroup approach is suitable for control synthesis in state-space representation using the perturbation theory of linear operators and semigroups. This allows to use an efficient control structure like Internal Model Control for the boundary control of a real process. Key Words: Heat exchanger, Internal Model Control, Semigroup, Distributed Parameter System, Boundary control. I. INTRODUCTION It is now known that direct methods are suitable for control design of infinite dimensional systems. This means that the partial differen- tial equations are directly used for the system analysis and control design. An approximation in finite dimension area is done only at the end for implementation purpose. In state space representation of infinite dimensional systems, the semigroup approach is well suited for the system analysis and enables one to use well-known concepts like open loop or closed loop design, stability, regulation, etc. However, some care must be taken regarding the same concepts given in finite dimensional area. The aim in this paper is to generalize the finite dimensional controller called Internal Model Control (IMC), with a closed loop like struc- ture, for a boundary control design for a trajectory tracking objective. This control design is addressed to a multivariable double pipe heat exchanger experimental process. In section II, the mathematical model of the heat exchanger is given as a set of partial differential equations and the state space representation with linear operators in a Hilbert space is given to state the boundary Youssoufi.Toure@bourges.univ-orleans.fr Laurence.Josserand@bourges.univ-orleans.fr control problem. So, since the underlying aim is also to show some similarities with finite dimensional systems, many recalls are done both in finite dimension and in infinite dimension areas. In section III, the open loop system is characterized with infinite dimensional concepts. The fourth section is devoted to the Internal Model Control design. It is used for the boundary control design and called Internal Model Boundary Control (IMBC). An extended linear operator is then introduced to show that the closed system can be viewed as a bounded perturbation (of the operators and the semigroup) of the open loop system. In the fifth section, we give some fundamental results from perturbation theory, as established by Kato since the seventies, for semigroup and system operator pertubation in the closed loop IMBC. The closed loop system stability is achieved using the spectral perturbation theory. As a consequence, a design condition is given for the controller parameters and an asymtotic tracking (regulation) is given. According to these results, an experimental tuning framework is given and the application to the heat exchanger boundary control is depicted in the last section with real plant results. II. MATHEMATICAL MODEL OF THE HEAT EXCHANGER PROCESS A. The process y1 y2         
  0 cold fluid heat fluid exchanger 1 cold fluid heat fluid exchanger 2 Fig. 1 DOUBLE PIPE HEAT EXCHANGER 2 The experimental heat exchanger process is a cascade of two coupled heat exchangers (see Fig. 1). The first is a parallel flow exchanger, the second is a counterflow exchanger. Before entering the second exchanger, the fluid of the internal tube, which is common to both exchangers and which has to be heated, is in contact with the environment. The deviations of the temperature fields around a stationary profile are used to describe the process:  in the inner tube,  and  ! in the outer tubes. The environmental temperature is assumed constant. Accordingly, " , its deviation from the stationary value, is zero. The velocities #$# , and # in the three tubes may all be different, while the other physical parameters are supposed to be equal for all fluids, and constant. There are the heat capacity % p, the density & , and the thermal conductivity ' . The mathematical model is obtained through energy balances. Considering sufficiently small variations, the linear mathematical model is : (  ( *),+ # $ (  (  - ' &% p ( "  (  " -  ./ 0 21 /  / + 3465  ) 78:9 ;: (   ( ,),+ #  (   (  - ' &% p ( "   (  " - 1   +   <4653" ) 78=; +?>   " ) 7 4A@ ; +A> ; - >2B (  (  ) #  (  (  - ' &% p ( "  (  " - 1   +    465  ) ; - > C9 ;: with 1 /ED 7 on 5GF!H0 / , IJ ),K L9M . (Here the arguments  and  are omitted for the sake of readability.) The initial conditions are N7 ) 7O  7P ) 7NQ  7P ) 7 The inflow temperatures of the outer tubes are the two boundary control inputs R  and R2" : 7 ) 7C (  (  S S S S "T ) 7   7N ) R  UV ( P (  S S S S TVWYX ) 7 Z9!;[ ) R " UV (  (  S S S S T\2X ) 7 The control objective concerns the inner tube temperature between the exchangers, ]U ) N;3 , and its outflow temperature ] " U ) NZ9!;[ . These two temperatures should follow prescribed trajecto- ries, such as to achieve a new stationary regime, for instance. B. State space representation and semigroup formalism 1) State space representation: A state space form ^ _ U )a` d _ U on 5b for Ccd7 of the heat exchanger model can be introduced with _ U ) ef f f g  Ph    iLj j j k  ` d ) ef f f g ` d 1  1  1  ` d"" 7 1  7 ` d iLj j j k with ` d ) + # $ ( (  -ml ( " (  " + 1  + 1 " + 1  ` d"" ) + #  ( (  -ml ( " (  " + 1  ` d ) #  ( (  -nl ( " (  " + 1  where l ) 'Vo U&% p and 5 ) 5bGpq5 " pm5[ . Similarly, the control is described by rts _ U )au s REU with rts _ U ) K 9!vxwy e g{zY| W2}N~ \}€    z | "TYW2}N~ "T\}E€    i ka‚  u s RtU ) RtU ) e g R  U R " U i k valid on ƒ ) ( 5 , the boundary of 5 . The output is defined by ]2U ) e g ]  U ] " U i k )…„ _ U ) K 9v wy e g z†| TYWVXW2}N~ TYWVX!\}E€  z | "TYW2}N~ "TN\}‡€  i k ‚  Here z | ˆ‰ WV}Š~ ˆ‰ \}€ ) ‹Œ  ŒŽ K for 4@  $ + v‘’ $ - v B 7 for 6“4@  $ + v‘’ $ - v B ’v”cd7 where v is a small positive constant. It is worth noting that the input R and the output ] are lumped quantities, with values in • " , while the state evolves in an (infinite dimensional) function space – . As most often, – is taken to be the space of square integrable functions, which is a Hilbert space. Here – ) ;:"5  2pm;:"53"hpm;:"5   ) –  pm–"=pn–  and for —t˜ ) Z—  —‡"!’—  L’— 4™– the inner product is š —t›E:—‡œžGŸ )…  y Ÿ —t›—‡œ ‚ 5 / and the norm is ¡ — ¡ " ž ) š —=—‡ž ) š —  :—  ž  - š —‡"¢=—‡"hž  - š —  =—  ž  3 Finally, to complete the definition of ` d, its domain of definition is specified as £  ` d )x¤ —¥4¦–¨§©— / ’—Eª / are absolutely continuous —Eª ª / 4¦; " 5 / L«—=h7P ) 7N’—Eª Z9 ;: ) 7N —Eª" ; +?>  ) 7’—Eª ; - >  ) 7†¬ 2) Some recalls on semigroup formalism : finite dimensional case to infinite dimensional case.: Consider the following finite dimensional linear time invariant system: ‹Œ  ŒŽ ^ _ U )a` _ U‘­Y:c®7 _ 7P ) _ $ where _ U=4¦•¢¯ , ` 4Š•¢¯†°¯ . The solution is given by _ U )a±²2³ _ $ The exponential matrix then defines a one parameter group if the parameter :4 +µ´  - ´  , and only a semigroup for C4@ 7 - ´  . From linear systems theory, it is known that this semigroup has the following properties: i) ± ²t¶·³ \2¸¹ )a± ²2³ ± ² ¸ for all L’º©cd7 ii) ± ²t¶ $ ¹ )¼» ¯ iii) ½ ¶·¾¿À ¹ ½ ³ )a`b± ²2³ )a± ²2³ ` iv) ÁÃÂ·Ä ³ÆÅ $Ç  ³  ± ²2³ +?» ¯ È )…` ÈÉ­2ȏ4¦•¢¯ All these properties can be easily checked since the opertor ± ²2³ can be defined simply by the Taylor series: ± ²2³ )ËÊ . ¯ 0 $ K Ì‡Í  ¯ ` ¯ which converges absolutely for any number (complex number, in large) t. Now, if the operator ` works in a more general Hilbert space than • , say – , some care must be taken. For a bounded operator ` , (i.e. the smallest number Î such that §·§ ` _ §·§ žÐÏ Îѧ·§ _ §·§ ž for all _ 4 £  ` :Òd– is bounded), all previous formulas form i) to iv) can be used. For an unbounded operator, we consider the more convenient one for most applications, the closed operator ` (i.e. for any sequences _ ¯ 4 £  `  such that _ ¯¥Ó _ and ` _ ¯AÓ ] , then _ belongs to £  `  and ` _ ) ] ); in other words, if one supposes that ` form – to Ô , two Hilbert spaces, the graph of ` , Տ `  )×Ö  _  ` _ L _ Ò £  ` Ø is a closed linear manifold of –ÚÙ*Ô . Then, for the semigroup properties i) to iii), replace ± ²2³ by Û ² U and add the fact that £  `  is dense in – . The last property iv) means that ` is the infinitesimal generator of the semigroup Û ² U : £  `  )×Ö _ 46–n§†Á·Â·Ä ³ÆÅ $Ç K  UÛ ² U +¥» ¯  _ exists and is egal to ` _ Ø The finite dimensional linear control system ‹Œ  ŒŽ ^ _ U )a` _ U - u RtUL…­†Ccd7N _ NÜU=Ò®• ¯ _ 7P ) _ $¢4 £  `  has the classical solution _ U )ݱ ²2³ _ $ -   ³$ ± ²t¶·³ WV¸¹ u RtZº ‚ º . The generalisation to an infinite dimensional linear control system is immediate: ‹Œ  ŒŽ ^ _ U )a` _ U -mÞ UL…­YCcd7 _ U:Òq– _ 7P ) _ $ 4 £  `  (1) where ` is a closed linear operator, has the following classical solution for continuous fonction Þ : _ U ) Û ² U _ $ - w ³ $ Û ² U + º Þ Zº ‚ º (2) C. Boundary control semigroup formulation. Summarizing the above state space model of the heat exchanger process, one has: ‹ŒŒŒŒŒ  ŒŒŒŒŒŽ ^ _ U )a` ½ _ U on 5b3Ccd7 r s _ U )au s RtU on ƒ ) ( 5b3Ccd7 _ 7P ) _ $ in £  `  (3) the output is given by: ]2U )…„ _ UL×:ß®7 This boundary control system may be formulated in the classical state space control system representation like in Eq. (1). Following an approach initiated by Fattorini in 1968 [9] (see also [2]), a change of variables and operators allows a change of representation: ‹  Ž ^ —:U )a` —:U +à ^ R‡ULQ—:UC4 £  ` d L:c®7 —:7 ) _ 7 +Aà Rt7 (4) where á ` is the “extension operator” of ` d, which means ` —:U ) ` d —:U for all —®4 £  `  and £  `  )ÑÖ —®4 £  ` d©§ r b — ) 7Ø , ` is assumed closed and densely defined in – . á à is the bounded “distribution operator” describing the action of the boundary control on the state: à 4?âbäã:–å , the set of bounded operators from ã in – (U=• " for the heat exchanger exemple) such that à R…4 £  ` dL r b  à R2 )æu dRE­VRa4çã , 4 and à is choosen such that it leaves the operator ` ½ unchanged (i.e., im  à :ÒdèéÜê ` ½  ). á The change of variables is: _ U ) —:U - à RtU According to Eq.(2), the abstract boundary control Eq.(4) has the following classical solution : —:U ) Û ² U—:7P + w ³ $ Û ² U + º à ^ RtZº ‚ º where ^ R is continuous and where the operator ` is assumed to be an infinitesimal generator of the semigroup Û ² . Now, it is necessary to characterize the open loop system in order to check its well-posedness and its stability. III. CHARACTERIZATION OF THE OPEN LOOP SYSTEM A. Open loop semigroup According to the abstract boundary control system, the open loop system is ‹Œ  ŒŽ ^ —:U )a` —:UëCcd7 —:7 ) — $ in £  `  (5) with the solution —:U ) Û ² U— $ 4 £  `  where ` is assumed an infinitesimal generator of the „ $ -semigroup Û ² . Proposition 3.1: the corresponding semigroup of the open loop system of the heat exchanger process given in (3), (4) and (5) is a contraction semigroup. Proof : A simple criterium for a densely defined closed operator to be a generator of contraction semigroup is the dissipativity property which is recalled as follows : Proposition 3.2: [24], [4] Let ` be a densely defined closed operator in a Hilbert space – , if á®ì ` _  _ c Ï 7 for all _ 4 £  `  á®ì `©í ]V]îc Ï 7 for all ]
4 £  `©í  then ` generates a contraction „ $ -semigroup on – , ¡ Û ² U—t$ ¡ ž,Ï ¡ — $ ¡ . According to open loop equation (4), the operator ` is `a)a`  -ðï , and the adjoint operator is ` í )a` í  -qï í where: `  ) ef f f g + #$¢ñ ñ ˆ -nl ñ  ñ ˆ  7 7 7 + # 3ñ ñ ˆ -nl ñ  ñ ˆ  7 7 7 #©ñ ñ ˆ -nl ñ  ñ ˆ  i j j j k ` í  ) ef f f g #!$ ñ ñ ˆ -l ñ  ñ ˆ  7 7 7 # 3ñ ñ ˆ -ml ñ  ñ ˆ  7 7 7 + #! ñ ñ ˆ -nl ñ  ñ ˆ  iLj j j k (6) ï ) ï í ) ef f f g + 1  + 1 " + 1  1  1  1  + 1  7 1  7 + 1  iLj j j k since ï is a symmetric negative definite matrix. Then ì ` —:UL—:U:c3ž ) ì `  —:UL—:U:c3ž - ì ï —:UL’—:U:c3ž and, ï being negative definite ì ï —:ULò—:U:c3ž Ï + &ó ¡ —:U ¡ " ž  where &ó is the smallest absolute value of an eigenvalue of ï . Using the definition of `  and integrating by parts ì `  —:UL’—:U:c Ï + l ¡ ( —  (  ¡ " ž  + l ¡ ( —‡" (  ¡ " ž  + l ¡ ( —  (  ¡ " ž  and ì `  —:UL—:U:c Ï + l ¡õô — ¡ " ž (7) It follows that ` is dissipative (analogous computations show the same to hold true for ` í (6) [36], [31]), and it generates a semigroup Û ² U such that ¡ Û ² U— $ ¡ ž Ï ¡ — $ ¡ ž8ö­2— $ 4 £  `  B. Exponential stability Corollary 3.3: The open loop system is an exponentially stable system. Proof : The exponential stability can be seen following a Lyapunov like approach : using the Lyapunov functional vîU )  " ¡ —:U ¡ " ž , one has ^ vîU ) š ` ’—:ULµ—:U ž - š ï —:UL3—:U ž The previous dissipativity calculations (7) and the Cauchy-Schwartz inequality are used for —¥4 £  `  , — ˜ ) Z—=¢— " —t : ¡ —  U ¡ " ž  Ï Z9!;= " 9 ¡ ( —= (  ¡ " ž   ¡ —‡"U ¡ " ž  Ï ; +å>  " 9 ¡ ( —‡" (  ¡ " ž   ¡ —tPU ¡ " ž  Ï ; +A>  " 9 ¡ ( —  (  ¡ " ž  5 which can be restated as ¡ — ¡ " ž Ï ¶ "T¹  " ¡õô —:U ¡ " ž8÷ Now š `  —:UL’—:U Ï + 9 l Z9 ;: " ¡ —:U ¡ " ž and ^ v
U Ï +  "ø ¶ "T¹  - &ób ¡ — ¡ " ž , such that, with l “ ) 7 , it follows ^ vîU ) K 9 ‚ ¡ —:U ¡ " ž ‚  ì + & ó ¡ —:U ¡ " ž hence Á·ùîú ¡ —:U ¡ " žbû ì + 9& ó  - const. or ¡ —:U ¡ ž ì %õ$ ± WVüLý ³ with some constant % $ . Using the formal solution —:U ) Û ² U—:7P of the open loop system (5), one has : ¡ Û ² U— $ ¡ ž ì % $ ± WYüÜý ³  ¡ Û ² U— $ ¡ ž ì ¡ — $ ¡ ž ± WVüLý ³  ¡ Û ² U ¡ ž ì ± WYü ý ³ which means that it is an exponentially stable contraction semigroup. The control objective is now to achieve a stable trajectory tracking. The internal model control like structure is used to achieve this goal. IV. INTERNAL MODEL BOUNDARY CONTROL (IMBC) A. Structure of the closed loop system and control system Internal model control (IMC) is a well-known synthesis method for linear time invariant systems. It is here generalized to infinite dimensional systems with a little modification from the original structure. Formally, this is not difficult because input and output have a finite number of components. The method described in the following is of the same spirit as work of Pohjolainen, who generalized control design methods known from finite dimension to the infinite dimensional context. y(t) - - Model Process þ ÿ À  ÿ À ÿ À v ÿ À  Lÿ À ÿ À  ÿ À  Control synthesis Fig. 2 INTERNAL MODEL CONTROL STRUCTURE. The IMC developed consists in two parts: á a classical closed loop system on the model of the plant Zΐ¸h á the IMC structure with reference model ZÎ
 . In this structure, one considers: á one time invariant finite dimensional linear system, called Î f, ‹  Ž ^ _ f U )a` f _ f U - u f ± U ] f U )…„ f _ f U (with _ f 7 ) 7 ), which is used to filter the output error ±µ) ] + ] s (this signal represents modelling error and output disturbance and it is supposed to be non persistent and bounded); á another time invariant finite dimensional linear system, called Î r, ‹  Ž ^ _ r U )a` r _ r U - u r#VU  U )…„ r _ r U (with _ r 7 ) 7 ), used as a pre-filter generating the reference signal  ; á and, finally, a stabilizer based on feedback of the error  )  + ] + ] f. The feedback is chosen to be of integral type: RtU )  i w ³ $ V ‚ N (8) with U )  U + ]U + ] f U:4Š• . Its design parameters are  , a real positive constant, and  i, a matrix mapping from •  into • . The control problem considered consists in making the output ] s of Î s track a prescribed (bounded) trajectory  U which is generated by Î r. Moreover, the closed loop is required to be stable. B. State space model of IMBC 1) Extended control system: Considering all state space represen- tations in the above IMBC, one has: ef f f g ^ —:U ^ _
!U ^ _ U i j j j k ) ef f f g ` 7 7 7 `
7 7 7 `  i j j j k ef f f g —:U _
U _ U i j j j k + ef f f g à 7 7 i j j j k ^ RtU - ef f f g 7 7 u
7 7 u  i j j j k e g #VU ± U i k with ef f f g —:7 _
7 _  7P i j j j k ) ef f f g _ 7P +¥à Rt7P 7 7 i j j j k 6 or ‹ŒŒŒ  ŒŒŒŽ ^_ U )  ` _ +  à ^ RtU -   
    e g #VU ± U i k  _ 7 ) _ $ with _ ) ef f f g —:U _
U _  U iLj j j k  – ) – pm•  pm•  «  à ) ef f f g à 7 7 i j j j k 4¦;ò@ •    – B «  
) ef f f g 7 u
7 iLj j j k «    ) ef f f g 7 7 u  iLj j j k 4¦;ò@ •  pq•    – B « All subsystems in the IMBC structure are taken as exponentially stable linear systems. Introducing the extended open loop operator  `a) ef f f g ` 7 7 7 `
7 7 7 `  iLj j j k defined on £   ` a4  – . It follows that  ` is a generator of an exponentially stable semigroup (i.e. there exists positive constants Î and  such that §·§ Û  ² Uõ§·§ Ï Î ± W"! ³ ­ŠCßd7 ) 2) Closed loop system: Using the integral type control law (8) and introducing # U ) w ³ $ V ‚  # 46•   one has ^ # U )  U + ]  U + ]2U ) „
_
!U +„  _  U +n„ _ U Using the preceding definitions one obtains ^ # U )  „ _ U +n„¢à REU where  „…) @ +µ„ „
+6„  B 4Š;3@  –•  B . Finally, using an extended state space vector: _"$ U ) @ _ U # U B 4*– $ )  – pm•   one obtains the closed loop state space representation: ‹  Ž ^ _ a U ) % _ a U - u Þ UL _ a 7P ) _ a0 where uÝ) e g   7 i k 4*âb@ •  °  – a B  Þ U ) e g #VU ± U i k  % )a` e -  ` ¶  ¹ e -  " ` ¶ "¹ e (9) and ` e ) e g  ` 7  „ 7 i k   „…) & +µ„ „
+µ„ (' 4*âb@  –•  B ` ¶  ¹ e ) e g +  à  i  „ 7 7 +µ„¢à  i i k  ` ¶ "¹ e ) e g 7  à  i „¢à  i 7 7 i k Operators ` ¶  ¹ e and ` ¶ "¹ e are bounded. This follows from the defi- nition of the operators they are composed of. It is assumed that the reference input #VU is a stable signal or a constant step function and the error signal (disturbances or gap between the model and the plant) is a non persistant signal, so one has: ­"Écd7N*)  cd7 such that §·§ Þ U + Þ Zºõ§·§ ì Ñ­ŠL’º¢cq  (10) Now, control synthesis consists to find control law parameters  and  / such that the above closed loop system is stable and regulation objectives are fulfilled. This can be achieved using the perturbation theory for linear operators and semigroups. V. STUDY OF THE CLOSED LOOP IMBC SYSTEM This part deals with the stability and the regulation studies of the closed loop system. Firstly, the closed loop system can be viewed as a perturbation of the open loop system from perturbation theory point of view. Secondly, a design condition is given for the controller parameters (  and  / ), as a direct consequence of some fundamental results from operators and spectral perturbation theory. A. Some results from analytic perturbation theory The closed loop operator % (9) can be viewed as a family of %    operators while  lies in a domain à $ . Recall its expression : %    )a` ¾ -  ` ¶  ¹ ¾ -  " ` ¶ "¹ ¾ (11) with ` ¾ ) % 7 and £  %    ) £  ` ¾  (recall that ` ¶  ¹ ¾ and ` ¶ "¹ ¾ are bounded). Definition 5.1: [18] A family `    of closed operators defined on a Banach space – for  in a domain à $ 4,+ - is said to be holomorphic of type ( ` ) if i) £  `    ) £ is independant of  , ii) `    _ is holomorphic for  4 à $ for every _ 4 £ . 7 In this case, `    _ has a Taylor expansion at each  4 à $ . For example, if  ) 7 belongs to à $ , one can write this developpement around zero: `    _ )…` _ -   ` ¶  ¹ _ -  " ` ¶ "¹ _ - ÷·÷Ã÷  _ 4 £ (12) which is convergent serie in a disk ¡  ¡ ì  independant of _ ; `¼) ` 7P and ` ¶ ¯ ¹ are linear operators in – with the same domain. The following theorem gives a criterion for this above property. Theorem 5.2: [18] Let ` be a closable operator in a Hilbert space – with £  `  ) £ . Let ` ¶ ¯ ¹  Ì ) K L9 ÷·÷ be operators in – with domains containing £ . Let there be positive constants .V0/!% such that : ¡ `ɶ ¯ ¹ _ ¡ Ï % ¯ W  . ¡ _ ¡ - / ¡ ` _ ¡ L _ 4 £  Ì ),K ’9 ÷·÷Ã÷ then the series (12) defines an operator `    with domain £ for ¡  ¡ ì  1 . The second part of recalls concerns the spectrum perturbations results [18]. Suppose that the spectrum of a closed operator ` , say 2‘ `  contains a bounded part 3 ª separated from the rest 3 ª ª in such a way that a rectifiable, simple closed curve ƒ can be drawn so as to enclose an open set containing 3 ª in its interior and 3 ª ª in its exterior. Definition 5.3: [18], [4] Let 2‘ `  such that it can be represented in the manner describe above. Then, ` is said to satisfy the spectrum decomposition assumption. Remark 5.4: The bounded part of 2‘ `  will constist, in general, of a finite number of eigenvalues of ` . For the holomorphic family introduced here, we have the following result. Theorem 5.5: [18] If a family `    of closed operators in – , depending on  holomorphically, has a spectrum consisting of two parts, then the subspaces of – corresponding to the separated parts also depend on  holomorphically. More precisely, this theorem means that if `    is holomorphic near  ) 7 and 2‘ ` 7 is separated into two parts separated by a closed curve ƒ in the manner of definition (5.3), then, for sufficiently small ¡  ¡ , 2= `    is likewise separated by ƒ . Before considering the closed loop operator in (11), notice that %    can also be viewed as a “bounded” perturbation of the extended operator ` ¾ . So some results on the change of the spectrum under relatively bounded perturbation are needed. Theorem 5.6: [18]Let ` be a closed operator in – and let u an operator in – which is ` -bounded (i.e. there exists positive constants .V0/ such that ¡ u _ ¡ ž Ï . ¡ _ ¡ ž - / ¡ ` _ ¡ ž for _ 4 £  ` CÒ £  u  ). If ºõR54 68759 . ¡ •È `  ¡ - / ¡ ` •8È `  ¡  ì K where ƒ is the curve which separates the spectrum of ` according to definition (5.5), then the spectrum of the operator : )…` - u is seprarated by ƒ in the same way. Theorem 5.7: [18] Let Û be a closed operator in – and let ` be an operator in – which is Û -bounded, so that £  ` <; £ UÛb . If there is a point È of &2UÛb such that : . ¡ •ÈNÛb ¡ - / ¡ Ûµ•ÈäÛ¢ ¡ ì K then : ) Û - ` is closed and È4*&2=:G , with : ¡ •È:G ¡ Ï ¡ •ÈNÛb ¡ @ K[+ . ¡ •ÈÛb ¡ + / ¡ Ûµ•ÈNÛb ¡ B W  Recall that the operator •È `  is the resolvent operator of ` : •È `  )  `®+ È »  W  B. Characterization of the closed loop operator By construction, the spectrum of the closed loop operator %    is a perturbation of the spectrum of ` ¾ . And the spectrum 2‘ ` ¾  of ` ¾ ) % 7 is, also by construction: 2= ` ¾  ) 2=  ` > Ö 7PØ Now, since  ` is a generator of an exponentially stable semigroup, 2‘  ` [4?+ - W [12], [29] and ºõR54 Ö • ±  ' L« ' 4@2=  `  ) 2V$ ì 7Ø . As a consequence, the eigenvalue 0 can be separated form the remainder 2‘  `  of the spectrum by a vertical straight line: 0 σ Im Re σ l ( A B ) Fig. 3 SPECTRUM OF THE OPERATOR C ¾ . More precisely and according to the definition (5.5), a curve ƒ can be drawn as follows to separate the spectrum of % 7P : 8 0 Im Re σ Γ r l σ( D C ) Fig. 4 SPECTRUM DECOMPOSITION OF C ¾ . Then, by construction : ƒ¥Òm&2 % 7P®­  c®7 This means that ­2ȏ4¦ƒ‘畏È % 7 exists and is bounded. The problem is now to study the conservation of this spectrum decomposition property for the closed loop operator. It leads to the characterization of the parameter  of the control law such that : ƒAÒm&2 %   ¼­  cd7 Proposition 5.8: There exists a value   $FE of the tuning param- eter  such that, if a closed oriented curve ƒ surrounds an open set containing the origin and encloses 2‘ ` ¾  in its exterior, then the same separation holds for the closed loop system 2‘ %  for all  such that: 7 Ï  ì   $FE ÷ Proof : G Set : %    )…` ¾ - u    with u    )  ` ¶  ¹ ¾ -  " ` ¶ "¹ ¾ . G The theorem 5.2 allows to specify the ` ¾ -boundness of the operator u    such that : ¡ u    _ ¡ Ï   K3+ %   W  . ¡ _ ¡ - / ¡ ` ¾ _ ¡  (13) with S S S S S S the intrinsic-bound : .   K3+ %   W  ) . ª the ` ¾ -bound : /   K[+ %   W  ) / ª G If È*4¥ƒ is chosen such that the condition (13) of the theorem 5.6 holds, then ƒ always separates 2 Ö %    )…` ¾ - u   Ø and the maximal value of  in the following, can be found : ºõR54 68759 . ª ¡ •È ` ¾  ¡ - / ª ¡ ` ¾ •È ` ¾  ¡  ì K with the expressions of . ª and / ª , one has :  ì   $HE )JI I Ì 68759 . ¡ •8È ` ¾  ¡ - / ¡ ` ¾ •È ` ¾  ¡ - %h W  G More precisely, we have : ¡ u    _ ¡ ) ¡  `ɶ  ¹ ¾ _ -  " `ɶ "¹ ¾ _ ¡ Ï §  § ¡ `ɶ  ¹ ¾ _ ¡ -  " ¡ ` ¶ "¹ ¾ _ ¡ let . )KI . _ Ö ¡ ` ¶  ¹ ¾ ¡ « ¡ ` ¶ "¹ ¾ ¡ Ø , then : ¡ u    _ ¡ Ï §  §· K3+ §  §  W  . ¡ _ ¡ So with the relation (13), we have : / ) 7 , % )ÝK and finally :  ì   $FE )LI I Ì 68759 . ¡ •ÈN ` ¾  ¡ - K  W  with . )KI . _ Ö ¡ ` ¶  ¹ ¾ ¡ « ¡ ` ¶ "¹ ¾ ¡ Ø . The second step is to verify that ƒ?Òq&2 %    for all  cd7 . For that, one just has to apply the theorem 5.7 under the above calculations which verify the first condition of the theorem. C. Closed loop system stability The stability is a direct consequence of the following result due to Pohjolainen [28]. Theorem 5.9: [28] Assume that:  . ÌNM @ „¢àB2) 4E if  / 4¦;ò•  =•   is chosen such that: O @ 2= +µ„¢à  /  B ì 7 then the closed loop system is stable for all 7 Ï  ì   $FE (   $FE in (??)). For the proof see [28], [13], [14], [16], [15], [17]. D. Result of regulation As a consequence of the stability property of the closed loop system, the regulation or the tracking problem, can be achieved under the IMBC structure. Proposition 5.10: Let the closed loop system % in (11), be a generator of an exponentially stable semigroup, then the IMBC system has the following asymptotic behavior: P I I ³ÆÅ Ê @ ]P¸!U +  U BY) 7 9 Proof Recall that the closed loop system has the following solution: _ $ U ) ÛNQµU _ $ 7 - w ³ $ ÛNQòU + º u Þ Zº ‚ º since Û Q is a stable semigroup: ¡ Û Q U _"$ 7P ¡ žSR Ó ³ÆÅ Ê 7 the second term can be written as w ³ $ Û Q U + º u Þ Zº ‚ º ) w ³ $ Û Q U + º u  Þ Zº + Þ U ‚ º - w ³ $ ÛNQµU + º u Þ U ‚ º ) w ³ $ Û Q U + º u  Þ Zº  + Þ U ‚ º + % W  u Þ U - % W  ÛNQµU u Þ U )×w ³ $ ÛNQòU + º u  Þ Zº  + Þ U ‚ º + % W  u Þ U Using (10), one obtains: P I I ³ÆÅ Ê _"$ U ),+ P I I ³ÆÅ Ê % W  u Þ U or P I I ³ÆÅ Ê @ % _"$ U - u Þ U BY) P I I ³UÅ Ê ^ _"$ U ) 7 which means that P I I ³ÆÅ Ê ^ # U ) 7 or P I I ³ÆÅ Ê U ) P I I ³ÆÅ Ê @  U + ]  U + ]2U BV) 7 And finally, using the IMBC structure, one has P I I ³UÅ Ê @  U + ]¸!U BV) 7 ÷ VI. CONTROLLER TUNING AND EXPERIMENTAL RESULTS According to the method sketched in the previous section, the only operator not yet explicitly defined is à . For simulation purposes its definition ( r b  à R2 )…u dRE im  à :ÒqèPéÜê ` d ) can be used [6], [16]. For an open-loop stable system, as studied here, also an experimental determination is possible. Controller tuning can then be performed in the following way: 1) The “gain matrix” of the system with I ) 9 inputs and 4 ) 9 outputs is experimentally determined. To this end, in I ) 9 experiments a step change is successively applied to each of the inputs, while keeping the other input constant, and while mesuring the asymptotic values of the 4 ) 9 outputs. As (small) variations are considered in the model, this means applying R )  R  )   R  ~  7P in the first, and R )  R2" ) 7N  R2"L~ "h in the second experiment. Here  R  ~  and  R2"L~ " are constant values. Using —:U ) Û A U—:7 + w ³ $ Û A U + º à ^ R‡Zº ‚ º _ U +Aà RtU ) Û A UÜ _ 7 +à Rt7 + w ³ $ Û A U + º à ^ RtZº ‚ º it follows, for RtU )  R / , I )ÝK L9 , that ^ RtU D 7 , and Rt7P ) 7 together with ]2U ) „ _ U imply ]2U ³ÆÅ \ Ê ) „¢à  R / . Therefore, indeed, one experimentally identifies the operator „¢à 46âb•  •   with rank 4 ) 9 (cf. (5.9)). 2) The controller parameter  i 4 âb•  •   is chosen as to get stability 2=@ +µ„¢à  i B ì 7 (cf. (5.9)), starting tuning with  i )UT„¢à W   T c®7 . 3) The scalar parameter  is increased until the closed loop system approaches instability, cf proposition (5.8). Fig. 5 Experimental results on the heat exchanger Applying this method to the heat exchanger under consideration led to the experimental results reported in Figure 5 (obtained on 10 a process situated at LAGEP, Lyon, France). In the beginning the system is at steady state (]  ) 9WV(X C and ]" ) M!75X C). Then a reference signal  U ) v is applied, which corresponds to successive up- and downwards step responses of a second order linear system (VWX C step through the pre-filter Î r in Figure 2), as depicted in Figure 5. The same reference  U is used for both measured outputs (simultaneously), which are the temperatures ] s U and ] s"U . This is done despite the fact that these output temperatures are (physically) coupled and despite the obvious delay between them. As can be seen on that figure, nonetheless both outputs follow the reference in a satisfactory fashion. VII. SUMMARY In this lecture, we have given a real example of the generalization of a control design known from finite dimensional system, namely the IMC, to infinite dimensional system with boundary control. The mathematical model used in this work can be easily viewed as a general form for many physical processes in local behavior (continuous distillation columns, absorption columns, tubular endothermic reactors, etc). 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