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Chapter State and Output eedbac This hapter describ es ho feedbac can used shap the lo cal eha vior of system. Both state and output feedbac are discussed. The concepts of reac habilit and observ abilit are in tro duced and it is sho wn ho states can estimated from measuremen ts of the input and the output. 5.1 In tro duction The idea of using feedbac to shap the dynamic eha vior as discussed in broad terms in Section 1.4. In this hapter will discuss this in detail for linear systems. In particular it will sho wn that under certain conditions it is ossible to assign the system eigen alues to arbitrary alues feedbac k, allo wing us to \design" the dynamics of the system. The state of dynamical system is collection of ariables that ermits prediction of the future dev elopmen of system. In this hapter will explore the idea of con trolling system through feedbac of the state. will assume that the system to con trolled is describ ed linear state mo del and has single input (for simplicit y). The feedbac con trol will dev elop ed step step using one single idea: the ositioning of closed lo op eigen alues in desired lo cations. It turns out that the con troller has ery in teresting structure that applies to man design metho ds. This hapter ma therefore view ed as protot yp of man analytical design metho ds. If the state of system is not ailable for direct measuremen t, it is often ossible to determine the state reasoning ab out the state through our kno wledge of the dynamics and more limited measuremen ts. This is done building an \observ er" that uses measuremen ts of the inputs and outputs of linear system, along with mo del of the system dynamics, to 109

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110 CHAPTER 5. ST TE AND OUTPUT FEEDBA CK estimate the state. The details of the analysis and designs in this hapter are carried out for systems with one input and one output, but it turns out that the structure of the con troller and the forms of the equations are exactly the same for systems with man inputs and man outputs. There are also man other design tec hniques that giv con trollers with the same structure. harac- teristic feature of con troller with state feedbac and an observ er is that the complexit of the con troller is giv en the complexit of the system to con trolled. Th us the con troller actually con tains mo del of the sys- tem. This is an example of the internal mo del principle whic sa ys that con troller should ha an in ternal mo del of the con trolled system. 5.2 Reac habilit egin disregarding the output measuremen ts and fo cus on the ev olu- tion of the state whic is giv en dx dt Ax u; (5.1) where is an matrix and an matrix. fundamen tal question is if it is ossible to ˇnd con trol signals so that an oin in the state space can reac hed. First observ that ossible equilibria for constan con trols are giv en Ax bu This means that ossible equilibria lies in one (or ossibly higher) dimen- sional subspace. If the matrix is in ertible this subspace is spanned Ev en if ossible equilibria lie in one dimensional subspace it ma still ossible to reac all oin ts in the state space transien tly explore this will ˇrst giv heuristic argumen based on formal calculations with impulse functions. When the initial state is zero the resp onse of the state to unit step in the input is giv en d (5.2) The deriv ativ of unit step function is the impulse function ), whic ma regarded as function whic is zero ev erywhere except at the origin

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5.2. REA CHABILITY 111 and with the prop ert that dt The resp onse of the system to impulse function is th us the deriv ativ of (5.2) dx dt At Similarly ˇnd that the resp onse to the deriv ativ of impulse function is dt Ae At The input 1) th us giv es the state At Ae At At At Hence, righ after the initial time 0, denoted 0+, ha (0+) AB The righ hand is linear com bination of the columns of the matrix AB (5.3) reac an arbitrary oin in the state space th us require that there are linear indep enden columns of the matrix The matrix is called the achability matrix An input consisting of sum of impulse functions and their deriv ativ es is ery violen signal. see that an arbitrary oin can reac hed with smo other signals can also argue as follo w. Assuming that the initial condition is zero, the state of linear system is giv en d A d It follo ws from the theory of matrix functions that A A

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112 CHAPTER 5. ST TE AND OUTPUT FEEDBA CK and ˇnd that d AB d d Again observ that the righ hand side is linear com bination of the columns of the reac habilit matrix giv en (5.3) illustrate examples. Example 5.1 (R achability of the Inverte Pendulum). Consider the in erted endulum example in tro duced in Example 3.5. The nonlinear equations of motion are giv en in equation (3.5) dx dt sin cos Linearizing this system ab out 0, the linearized mo del ecomes dx dt x: (5.4) The dynamics matrix and the con trol matrix are The reac habilit matrix is (5.5) This matrix has full rank and can conclude that the system is reac hable. This implies that can mo the system from an initial state to an ˇnal state and, in particular, that can alw ys ˇnd an input to bring the system from an initial state to the equilibrium. Example 5.2 (System in achable Canonic al orm). Next will consider system in achable anonic al form dz dt Az

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5.2. REA CHABILITY 113 sho that is full rank, sho that the in erse of the reac habilit matrix exists and is giv en (5.6) sho this consider the pro duct where AB The ectors satisfy the relation and iterating this relation ˇnd that whic sho ws that the matrix (5.6) is indeed the in erse of Systems That Are Not Reac hable It is useful of ha an in tuitiv understanding of the mec hanisms that mak system unreac hable. An example of suc system is giv en in Figure 5.1. The system consists of iden tical systems with the same input. The in tuition can also demonstrated analytically demonstrate this simple example.

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114 CHAPTER 5. ST TE AND OUTPUT FEEDBA CK Figure 5.1: non-reac hable system. Example 5.3 (Non-r achable System). Assume that the systems in Figure 5.1 are of ˇrst order. The complete system is then describ ed dx dt dx dt The reac habilit matrix is This matrix is singular and the system is not reac hable. One implication of this is that if and start with the same alue, it is nev er ossible to ˇnd an input whic causes them to ha di˛eren alues. Similarly if they start with di˛eren alues, no input will able to driv them oth to zero. Co ordinate Changes It is in teresting to in estigate ho the reac habilit matrix transforms when the co ordinates are hanged. Consider the system in (5.1). Assume that the co ordinates are hanged to As sho wn in the last hapter, that the dynamics matrix and the con trol matrix for the transformed system are AT The reac habilit matrix for the transformed system then ecomes

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5.3. ST TE FEEDBA CK 115 ha AT AB AT AT AT The reac habilit matrix for the transformed system is th us AB (5.7) This form ula is useful for ˇnding the transformation matrix that con erts system in to reac hable canonical form (using from Example 5.2). 5.3 State eedbac Consider system describ ed the linear di˛eren tial equation dx dt Ax (5.8) The output is the ariable that are in terested in con trolling. egin with it is assumed that all comp onen ts of the state ector are measured. Since the state at time con tains all information necessary to predict the future eha vior of the system, the most general time in arian con trol la is function of the state, i.e. )) If the feedbac is restricted to linear, it can written as (5.9) where is the reference alue. The negativ sign is simply con en tion to indicate that negativ feedbac is the normal situation. The closed lo op system obtained when the feedbac (5.8) is applied to the system (5.9) is giv en dx dt (5.10)

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116 CHAPTER 5. ST TE AND OUTPUT FEEDBA CK It will attempted to determine the feedbac gain so that the closed lo op system has the haracteristic olynomial (5.11) This con trol problem is called the eigen alue assignmen problem or the ole placemen problem (w will deˇne \p oles" more formally in later hapter). Examples will start considering few examples that giv insigh in to the nature of the problem. Example 5.4 (The Double Inte gr ator). The double in tegrator is describ ed dx dt In tro ducing the feedbac the closed lo op system ecomes dx dt (5.12) The closed lo op system has the haracteristic olynomial det Assume it is desired to ha feedbac that giv es closed lo op system with the haracteristic olynomial Comparing this with the haracteristic olynomial of the closed lo op system ˇnd ˇnd that the feedbac gains should hosen as ha unit steady state gain the parameter ust equal to The con trol la can th us written as

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5.3. ST TE FEEDBA CK 117 In the next example will encoun ter some diculties. Example 5.5 (A Unr achable System). Consider the system dx dt with the con trol la The closed lo op system is dx dt This system has the haracteristic olynomial det This olynomial has zeros at and One closed lo op eigen alue is th us alw ys equal to and it is not ossible to obtain an arbitrary haracteristic olynomial. This example sho ws that the eigen alue placemen problem cannot al- ys solv ed. An analysis of the equation describing the system sho ws that the state is not reac hable. It is th us clear that some conditions on the system are required. The reac hable canonical form has the prop ert that the parameters of the system are the co ecien ts of the haracteristic equation. It is therefore natural to consider systems on this form when solving the eigen alue place- men problem. In the next example in estigate the case when the system is in reac hable canonical form. Example 5.6 (System in achable Canonic al orm). Consider system in reac hable canonical form, i.e, dz dt Az (5.13)

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118 CHAPTER 5. ST TE AND OUTPUT FEEDBA CK The op en lo op system has the haracteristic olynomial det Expanding the determinan the last ro ˇnd that the follo wing re- cursiv equation for the determinan t: sD It follo ws from this equation that useful prop ert of the system describ ed (5.13) is th us that the co ef- ˇcien ts of the haracteristic olynomial app ear in the ˇrst ro w. Since the all elemen ts of the -matrix except the ˇrst ro are zero it follo ws that the state feedbac only hanges the ˇrst ro of the -matrix. It is th us straigh forw ard to see ho the closed lo op eigen alues are hanged the feedbac k. In tro duce the con trol la (5.14) The closed lo op system then ecomes dz dt (5.15) The feedbac th us hanges the elemen ts of the ˇrst ro of the matrix, whic corresp onds to the parameters of the haracteristic equation. The closed lo op system th us has the haracteristic olynomial

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5.3. ST TE FEEDBA CK 119 Requiring this olynomial to equal to the desired closed lo op olynomial (5.11) ˇnd that the con troller gains should hosen as This feedbac simply replace the parameters in the system (5.15) The feedbac gain for system in reac hable canonical form is th us (5.16) ha unit steady state gain the parameter should hosen as (5.17) Notice that it is essen tial to kno the precise alues of parameters and in order to obtain the correct steady state gain. The steady state gain is th us obtained precise calibration. This is ery di˛eren from obtaining the correct steady state alue in tegral action, whic shall see in later hapters. th us ˇnd that it is easy to solv the eigen alue placemen problem when the system has the structure giv en (5.13) The General Case solv the problem in the general case, simply hange co ordinates so that the system is in reac hable canonical form. Consider the system (5.8) Change the co ordinates linear transformation so that the transformed system is in reac hable canonical form (5.13) or suc system the feedbac is giv en (5.14) where the co ecien ts are giv en (5.16) ransforming bac to the original co ordinates giv es the feedbac It no remains to ˇnd the transformation. do this observ that the reac habilit matrices ha the prop ert AB

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120 CHAPTER 5. ST TE AND OUTPUT FEEDBA CK The transformation matrix is th us giv en (5.18) and the feedbac gain can written as (5.19) Notice that the matrix is giv en (5.6) The feedforw ard gain is giv en equation (5.17). The results obtained can summarized as follo ws. The or em 5.1 (Pole-plac ement by State db ack). Consider the system given by quation (5.8) dx dt Ax with one input and one output. If the system is achable ther exits fe db ack that gives close lo op system with the char acteristic olynomial The fe db ack gain is given by wher ar the ecients of the char acteristic olynomial of the matrix and the matric es and ar given by AB emark 5.1 (A mathematic al interpr etation). Notice that the eigen alue placemen problem can form ulated abstractly as the follo wing algebraic problem. Giv en an matrix and an matrix ˇnd matrix suc that the matrix has prescrib ed eigen alues.

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5.4. OBSER ABILITY 121 Computing the eedbac Gain ha th us obtained solution to the problem and the feedbac has een describ ed closed form solution. or simple problems it is easy to solv the problem the follo wing simple pro cedure: In tro duce the elemen ts of as unkno wn ariables. Compute the haracteristic olynomial det( sI Equate co ecien ts of equal ers of to the co ecien ts of the desired haracteristic olynomial This giv es system of linear equations to determine The equations can alw ys solv ed if the system is observ able. Example 5.4 is ypical illustrations. or systems of higher order it is more con enien to use equation (5.19) this can also used for umeric computations. Ho ev er, for large systems this is not sound umerically ecause it in olv es computation of the harac- teristic olynomial of matrix and computations of high ers of matrices. Both op erations lead to loss of umerical accuracy or this reason there are other metho ds that are etter umerically In MA TLAB the state feedbac can computed the pro cedures acker or place 5.4 Observ abilit In Section 5.3 it as sho wn that it as ossible to ˇnd feedbac that giv es desired closed lo op eigen alues pro vided that the system is reac hable and that all states ere measured. It is highly unrealistic to assume that all states are measured. In this section will in estigate ho the state can estimated using the mathematical mo del and few measuremen ts. It will sho wn that the computation of the states can done dynamical systems. Suc systems will called observers Consider system describ ed dx dt Ax (5.20) where is the state, the input, and the measured output. The problem of determining the state of the system from its inputs and outputs will

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122 CHAPTER 5. ST TE AND OUTPUT FEEDBA CK considered. It will assumed that there is only one measured signal, i.e. that the signal is scalar and that is (ro w) ector. Observ abilit When discussing reac habilit neglected the output and fo cused on the state. will no discuss related problem where will neglect the input and instead fo cus on the output. Consider the system dx dt Ax (5.21) will no in estigate if it is ossible to determine the state from observ a- tions of the output. This is clearly problem of signiˇcan practical in terest, ecause it will tell if the sensors are sucien t. The output itself giv es the pro jection of the state on ectors that are ro ws of the matrix The problem can clearly solv ed if the matrix is in ertible. If the matrix is not in ertible can tak deriv ativ es of the output to obtain dy dt dx dt Ax: rom the deriv ativ of the output th us get the pro jections of the state on ectors whic are ro ws of the matrix Pro ceeding in this get dy dt dt dt (5.22) th us ˇnd that the state can determined if the matrix (5.23) has indep enden ro ws. Notice that ecause of the Ca yley-Hamilton equa- tion it is not orth while to con tin ue and tak deriv ativ es of order higher

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5.4. OBSER ABILITY 123 y Figure 5.2: non-observ able system. than 1. The matrix is called the observ abilit matrix. system is called observ able if the observ abilit matrix has full rank. illustrate with an example. Example 5.7 (Observability of the Inverte Pendulum). The linearized mo del of in erted endulum around the uprigh osition is describ ed (5.4) The matrices and are The observ abilit matrix is whic has full rank. It is th us ossible to compute the state from mea- suremen of the angle. The calculation can easily extended to systems with inputs. The state is then giv en linear com bination of inputs and outputs and their higher deriv ativ es. Di˛eren tiation can giv ery large errors when there is measure- men noise and the metho is therefore not ery practical particularly when deriv ativ es of high order app ear. metho that orks with inputs will giv en the next section. Non-Observ able System It is useful to ha an understanding of the mec hanisms that mak system unobserv able. Suc system is sho wn in Figure 5.2. Next will consider

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124 CHAPTER 5. ST TE AND OUTPUT FEEDBA CK the system in observable anonic al form i.e. dz dt straigh forw ard but tedious calculation sho ws that the in erse of the observ abilit matrix has simple form. It is giv en This matrix is alw ys in ertible. The system is comp osed of iden ti- cal systems whose outputs are added. It seems in tuitiv ely clear that it is not ossible to deduce the states from the output. This can also seen formally Co ordinate Changes It is in teresting to in estigate ho the observ abilit matrix transforms when the co ordinates are hanged. Consider the system in equation (5.21). As- sume that the co ordinates are hanged to It follo ws from linear algebra that the dynamics matrix and the output matrix are giv en AT The observ abilit matrix for the transformed system then ecomes

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5.5. OBSER VERS 125 ha AT AT AT AT AT and ˇnd that the observ abilit matrix for the transformed system has the prop ert (5.24) This form ula is ery useful for ˇnding the transformation matrix 5.5 Observ ers or system go erned equation (5.20) can attempt to determine the state simply sim ulating the equations with the correct input. An estimate of the state is then giv en dt (5.25) ˇnd the prop erties of this estimate, in tro duce the estimation error It follo ws from (5.20) and (5.25) that dt If matrix has all its eigen alues in the left half plane, the error will th us go to zero. Equation (5.25) is th us dynamical system whose output con erges to the state of the system (5.20) The observ er giv en (5.25) uses only the pro cess input the measured signal do es not app ear in the equation. It ust also required that the system is stable. will therefore attempt to mo dify the observ er so that

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126 CHAPTER 5. ST TE AND OUTPUT FEEDBA CK the output is used and that it will ork for unstable systems. Consider the follo wing observ er dt (5.26) This can considered as generalization of (5.25) eedbac from the measured output is pro vided adding the term ). Notice that is the output that is predicted the observ er. in estigate the observ er (5.26) form the error It follo ws from (5.20) and (5.26) that dt LC If the matrix can hosen in suc that the matrix LC has eigen alues with negativ real parts, the error will go to zero. The con- ergence rate is determined an appropriate selection of the eigen alues. The problem of determining the matrix suc that LC has pre- scrib ed eigen alues is ery similar to the eigen alue placemen problem that as solv ed ab e. In fact, if observ that the eigen alues of the matrix and its transp ose are the same, ˇnd that could determine suc that has giv en eigen alues. First notice that the problem can solv ed if the matrix 1) is in ertible. Notice that this matrix is the transp ose of the observ abilit matrix for the system (5.20) of the system. Assume it is desired that the haracteristic olynomial of the matrix LC is It follo ws from Remark 5.1 of Theorem 5.1 that the solution is giv en

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5.5. OBSER VERS 127 where is the observ abilit matrix and is the observ abilit matrix of the system of the system dz dt whic is the observ able canonical form of the system (5.20) ransp osing the form ula for obtain The result is summarized the follo wing theorem. The or em 5.2 (Observer design by eigenvalue plac ement). Consider the sys- tem given by dx dt Ax wher output is sc alar. Assume that the system is observable. The dynamic al system dt with chosen as (5.27) wher the matric es and ar given by

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128 CHAPTER 5. ST TE AND OUTPUT FEEDBA CK PSfrag replacemen ts Figure 5.3: Blo diagram of the observ er. Notice that the observ er con tains cop of the pro cess. Then the observer err or is governe by di˛er ential quation having the char acteristic olynomial emark 5.2 The dynamical system (5.26) is called an observ er for (the states of the) system (5.20) ecause it will generate an appro ximation of the states of the system from its inputs and outputs. emark 5.3 The theorem can deriv ed transforming the system to observ able canonical form and solving the problem for system in this form. emark 5.4 Notice that ha giv en observ ers, one based on pure dif- feren tiation (5.22) and another describ ed the di˛eren tial equation (5.26) There are also other forms of observ ers. In terpretation of the Observ er The observ er is dynamical system whose inputs are pro cess input and pro cess output The rate of hange of the estimate is comp osed of terms. One term is the rate of hange computed from the mo del with substituted for The other term is prop ortional to the di˛erence et een measured output and its estimate The estimator gain is matrix that tells ho the error is eigh ted and distributed among the states. The observ er th us com bines measuremen ts with dynamical mo del of the system. blo diagram of the observ er is sho wn in Figure 5.3

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5.5. OBSER VERS 129 Dualit Notice the similarit et een the problems of ˇnding state feedbac and ˇnding the observ er. The ey is that oth of these problems are equiv alen to the same algebraic problem. In eigen alue placemen it is attempted to ˇnd so that has giv en eigen alues. or the observ er design it is instead attempted to ˇnd so that LC has giv en eigen alues. The follo wing equiv alence can established et een the problems The similarit et een design of state feedbac and observ ers also means that the same computer co de can used for oth problems. Computing the Observ er Gain The observ er gain can computed in sev eral di˛eren ys. or simple problems it is con enien to in tro duce the elemen ts of as unkno wn param- eters, determine the haracteristic olynomial of the observ er det LC and iden tify it with the desired haracteristic olynomial. Another alterna- tiv is to use the fact that the observ er gain can obtained insp ection if the system is in observ able canonical form. In the general case the ob- serv er gain is then obtained transformation to the canonical form. There are also reliable umerical algorithms. They are iden tical to the algorithms for computing the state feedbac k. The pro cedures are illustrated few examples. Example 5.8 (The Double Inte gr ator). The double in tegrator is describ ed dx dt The observ abilit matrix is

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130 CHAPTER 5. ST TE AND OUTPUT FEEDBA CK i.e. the iden tit matrix. The system is th us observ able and the problem can solv ed. ha LC It has the haracteristic olynomial det LC det Assume that it is desired to ha an observ er with the haracteristic oly- nomial The observ er gains should hosen as The observ er is then dt 5.6 Output eedbac In this section will consider the same system as in the previous sections, i.e. the th order system describ ed dx dt Ax (5.28) where only the output is measured. As efore it will assumed that and are scalars. It is also assumed that the system is reac hable and observ able. In Section 5.3 had found feedbac for the case that all states could measured and in Section 5.4 ha presen ted dev elop ed an observ er that can generate estimates of the state based on inputs and outputs. In this section will com bine the ideas of these sections to ˇnd an feedbac whic giv es desired closed lo op eigen alues for systems where only outputs are ailable for feedbac k.

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5.6. OUTPUT FEEDBA CK 131 If all states are not measurable, it seems reasonable to try the feedbac (5.29) where is the output of an observ er of the state (5.26), i.e. dt (5.30) Since the system (5.28) and the observ er (5.30) oth are of order the closed lo op system is th us of order The states of the system are and The ev olution of the states is describ ed equations (5.28) (5.29)(5.30) analyze the closed lo op system, the state ariable is replace (5.31) Subtraction of (5.28) from (5.28) giv es dt Ax LC LC In tro ducing from (5.29) in to this equation and using (5.31) to eliminate giv es dx dt Ax Ax Ax The closed lo op system is th us go erned dt LC (5.32) Since the matrix on the righ t-hand side is blo diagonal, ˇnd that the haracteristic olynomial of the closed lo op system is det sI det sI LC This olynomial is pro duct of terms, where the ˇrst is the harac- teristic olynomial of the closed lo op system obtained with state feedbac and the other is the haracteristic olynomial of the observ er error. The feedbac (5.29) that as motiv ated heuristically th us pro vides ery neat solution to the eigen alue placemen problem. The result is summarized as follo ws.

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132 CHAPTER 5. ST TE AND OUTPUT FEEDBA CK The or em 5.3 (Pole plac ement by output fe db ack). Consider the system dx dt Ax The ontr ol ler describ by dt gives close lo op system with the char acteristic olynomial det sI det sI LC This olynomial an assigne arbitr ary ots if the system is observable and achable. emark 5.5 Notice that the haracteristic olynomial is of order and that it can naturally separated in to factors, one det sI asso ciated with the state feedbac and the other det sI LC with the observ er. emark 5.6 The con troller has strong in tuitiv app eal. It can though of as comp osed of parts, one state feedbac and one observ er. The feedbac gain can computed as if all state ariables can measured. The In ternal Mo del Principle blo diagram of the con troller is sho wn in Figure 5.4. Notice that the con troller con tains dynamical mo del of the plan t. This is called the in ternal mo del principle. Notice that the dynamics of the con troller is due to the observ er. The con troller can view ed as dynamical system with input and output dt LC Ly (5.33) The con troller has the transfer function sI LC (5.34)

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5.7. INTEGRAL CTION 133 PSfrag replacemen ts Pro cess Observ er Figure 5.4: Blo diagram of con troller whic com bines state feedbac with an observ er. 5.7 In tegral Action The con troller based on state feedbac ac hiev es the correct steady state re- sp onse to reference signals careful calibration of the gain and it lac ks the nice prop ert of in tegral con trol. It is then natural to ask wh the the eautiful theory of state feedbac and observ ers do es not automatically giv con trollers with in tegral action. This is consequence of the assump- tions made when deriving the analytical design metho whic will no in estigate. When using an analytical design metho d, ostulate criteria and sp ec- iˇcations, and the con troller is then consequence of the assumptions. In this case the problem is the mo del (5.8). This mo del assumes implicitly that the system is erfectly calibrated in the sense that the output is zero when the input is zero. In practice it is ery dicult to obtain suc mo del. Consider, for example, hemical pro cess con trol problem where the output is temp erature and the con trol ariable is large rust alv e. The mo del (5.8) then implies that kno exactly ho to osition the alv to get sp eciˇed outlet temp erature|indeed, highly unrealistic assumption.

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134 CHAPTER 5. ST TE AND OUTPUT FEEDBA CK Ha ving understo the dicult it is not to hard to hange the mo del. By mo difying the mo del to dx dt Ax x; (5.35) where is an unkno wn constan t, can can capture the idea that the mo del is no longer erfectly calibrated. This mo del is called mo del with an input disturbance. Another ossibilit is to use the mo del dx dt Ax where is an unkno wn constan t. This is mo del with an output disturbance. It will no sho wn that straigh tforw ard design of an output feedbac for this system do es indeed giv in tegral action. Both disturbance mo dels will pro duce con trollers with in tegral action. will start in estigating the case of an input disturbance. This is little more con enien for us ecause it ˇts the con trol goal of ˇnding con troller that driv es the state to zero. The mo del with an input disturbance can con enien tly brough in to the framew ork of state feedbac k. do this, ˇrst observ that is an unkno wn constan whic can describ ed dv dt bring the system in to the standard format simply in tro duce the dis- turbance as an extra state ariable. The state of the system is th us This is also called state augmen tation. Using the augmen ted state the mo del (5.35) can written as dt (5.36) Notice that the disturbance state is not reac hable. If the disturbance can measured, the state feedbac is then (5.37)

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5.8. GENERAL CONTR OLLER STR UCTURE 135 The disturbance state is not reac hable. The the e˛ect of the disturbance on the system can, ho ev er, eliminated ho osing 1. If the distur- bance is kno wn the con trol la ab can in terpreted as com bination of feedbac from the system state and feedforw ard from measured distur- bance. It is not realistic to assume that the disturbance can measured and will instead replace the states estimates. The feedbac la then ecomes This means that feedbac is based on estimates of the state and the distur- bance. There are man other ys to in tro duce in tegral action. 5.8 General Con troller Structure So far reference signals ha een in tro duced simply adding it to the state feedbac k. more sophisticated of doing this is sho wn the blo dia- gram in Figure 5.5, where the con troller consists of three parts: an observ er that computes estimates of the states based on mo del and measured pro- cess inputs and outputs, state feedbac and tra jectory generator that generates the desired eha vior of all states and feedforw ard signal The signal is suc that it generates the desired eha vior of the states when applied to the system, under ideal conditions of no disturbances and no mo deling errors. The con troller is said to ha two de gr es of fr dom ecause the resp onse to command signals and disturbances are decoupled. Disturbance resp onses are go erned the observ er and the state feedbac and the resp onse to command signal is go erned the feedforw ard. get some insigh in to the eha vior of the system let us discuss what happ ens when the command signal is hanged. ˇx the ideas let us assume that the system is in equilibrium with the observ er state equal to the pro cess state When the command signal is hanged feedforw ard signal is generated. This signal has the prop ert that the pro cess output giv es the desired output when the feedforw ard signal is applied to the system. The pro cess state hanges in resp onse to the feedforw ard signal. The ob- serv er trac ks the state erfectly ecause the initial state as correct. The estimated state will equal to the desired state and the feedbac signal is zero. If there are some disturbances or some mo deling errors the feedbac signal will di˛eren from zero and attempt to correct the situation. The con troller giv en in Figure 5.5 is ery general structure. There are man ys to generate the feedforw ard signal and there are also man dif-

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136 CHAPTER 5. ST TE AND OUTPUT FEEDBA CK PSfrag replacemen ts ra jectory eedforw ard Generator Pro cess State eedbac Observ er fb Figure 5.5: Blo diagram of con troller based on structure with degrees of freedom. The con troller consists of command signal generator, state feedbac and an observ er. feren ys to compute the feedbac gains and the gain of the observ er. The system in Figure 5.5 is an example of the internal mo del principle whic sa ys that con troller should con tain mo del of the system to con trolled and the disturbances action on the system. Computer Implemen tation The con trollers obtained so far ha een describ ed ordinary di˛eren tial equations. They can implemen ted directly using analog computers. Since most con trollers are implemen ted using digital computers will brie—y discuss ho this can done. The computer ypically op erates erio dically signals from the sensors are sampled and con erted to digital form the A/D con erter, the con trol signal is computed, con erted to analog form for the actuators, as sho wn in Figure 1.3 on page 5. illustrate the main principles consider the con troller describ ed equations (5.29) and (5.30), i.e. dt The ˇrst equation whic only consists of additions and ultiplications can implemen ted directly in computer. The second equation has to appro ximated. simple is to replace the deriv ativ di˛erence +1 ))

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5.9. EXER CISES 137 where are the sampling instan ts and +1 is the sampling erio d. Rewriting the equation get +1 )) (5.38) The calculation of the state only requires addition and ultiplication and can easily done computer. pseudo co de for the program that runs in the digital computer is "Control algorithm main loop r=adin(ch1) "read setpoint from ch1 y=adin(ch2) "read process variable from ch2 u=C*x+Kr*r "compute control variable daout(ch1) "set analog output ch1 x=x+h*(A*x+B*u+L*(y-C*x)) "update state estimate The program runs erio dically Notice that the um er of computations et een reading the analog input and setting th analog output has een minimized. The state is up dated after the analog output has een set. The program has one states The hoice of sampling erio requires some care. or linear systems the di˛erence appro ximation can oided ob- serving that the con trol signal is constan er the sampling erio d. An exact theory for this can dev elop ed. Doing this get con trol la that is iden tical to (5.38) but with sligh tly di˛eren co ecien ts. There are sev eral practical issues that also ust dealt with. or ex- ample it is necessary to ˇlter signal efore it is sampled so that the ˇltered signal has little frequency con ten ab where is the sampling fre- quency If con trollers with in tegral action are used it is necessary to pro vide protection so that the in tegral do es not ecome to large when the actuator saturates. Care ust also tak en so that parameter hanges do not cause disturbances. Some of these issues are discussed in Chapter 10. 5.9 Exercises 1. Consider system on reac hable canonical form. Sho that the in erse of the reac habilit matrix is giv en (5.39)

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138 CHAPTER 5. ST TE AND OUTPUT FEEDBA CK

Both state and output feedbac are discussed The concepts of reac habilit and observ abilit are in tro duced and it is sho wn ho states can estimated from measuremen ts of the input and the output 51 In tro duction The idea of using feedbac to shap t ID: 23943

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Chapter State and Output eedbac This hapter describ es ho feedbac can used shap the lo cal eha vior of system. Both state and output feedbac are discussed. The concepts of reac habilit and observ abilit are in tro duced and it is sho wn ho states can estimated from measuremen ts of the input and the output. 5.1 In tro duction The idea of using feedbac to shap the dynamic eha vior as discussed in broad terms in Section 1.4. In this hapter will discuss this in detail for linear systems. In particular it will sho wn that under certain conditions it is ossible to assign the system eigen alues to arbitrary alues feedbac k, allo wing us to \design" the dynamics of the system. The state of dynamical system is collection of ariables that ermits prediction of the future dev elopmen of system. In this hapter will explore the idea of con trolling system through feedbac of the state. will assume that the system to con trolled is describ ed linear state mo del and has single input (for simplicit y). The feedbac con trol will dev elop ed step step using one single idea: the ositioning of closed lo op eigen alues in desired lo cations. It turns out that the con troller has ery in teresting structure that applies to man design metho ds. This hapter ma therefore view ed as protot yp of man analytical design metho ds. If the state of system is not ailable for direct measuremen t, it is often ossible to determine the state reasoning ab out the state through our kno wledge of the dynamics and more limited measuremen ts. This is done building an \observ er" that uses measuremen ts of the inputs and outputs of linear system, along with mo del of the system dynamics, to 109

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110 CHAPTER 5. ST TE AND OUTPUT FEEDBA CK estimate the state. The details of the analysis and designs in this hapter are carried out for systems with one input and one output, but it turns out that the structure of the con troller and the forms of the equations are exactly the same for systems with man inputs and man outputs. There are also man other design tec hniques that giv con trollers with the same structure. harac- teristic feature of con troller with state feedbac and an observ er is that the complexit of the con troller is giv en the complexit of the system to con trolled. Th us the con troller actually con tains mo del of the sys- tem. This is an example of the internal mo del principle whic sa ys that con troller should ha an in ternal mo del of the con trolled system. 5.2 Reac habilit egin disregarding the output measuremen ts and fo cus on the ev olu- tion of the state whic is giv en dx dt Ax u; (5.1) where is an matrix and an matrix. fundamen tal question is if it is ossible to ˇnd con trol signals so that an oin in the state space can reac hed. First observ that ossible equilibria for constan con trols are giv en Ax bu This means that ossible equilibria lies in one (or ossibly higher) dimen- sional subspace. If the matrix is in ertible this subspace is spanned Ev en if ossible equilibria lie in one dimensional subspace it ma still ossible to reac all oin ts in the state space transien tly explore this will ˇrst giv heuristic argumen based on formal calculations with impulse functions. When the initial state is zero the resp onse of the state to unit step in the input is giv en d (5.2) The deriv ativ of unit step function is the impulse function ), whic ma regarded as function whic is zero ev erywhere except at the origin

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5.2. REA CHABILITY 111 and with the prop ert that dt The resp onse of the system to impulse function is th us the deriv ativ of (5.2) dx dt At Similarly ˇnd that the resp onse to the deriv ativ of impulse function is dt Ae At The input 1) th us giv es the state At Ae At At At Hence, righ after the initial time 0, denoted 0+, ha (0+) AB The righ hand is linear com bination of the columns of the matrix AB (5.3) reac an arbitrary oin in the state space th us require that there are linear indep enden columns of the matrix The matrix is called the achability matrix An input consisting of sum of impulse functions and their deriv ativ es is ery violen signal. see that an arbitrary oin can reac hed with smo other signals can also argue as follo w. Assuming that the initial condition is zero, the state of linear system is giv en d A d It follo ws from the theory of matrix functions that A A

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112 CHAPTER 5. ST TE AND OUTPUT FEEDBA CK and ˇnd that d AB d d Again observ that the righ hand side is linear com bination of the columns of the reac habilit matrix giv en (5.3) illustrate examples. Example 5.1 (R achability of the Inverte Pendulum). Consider the in erted endulum example in tro duced in Example 3.5. The nonlinear equations of motion are giv en in equation (3.5) dx dt sin cos Linearizing this system ab out 0, the linearized mo del ecomes dx dt x: (5.4) The dynamics matrix and the con trol matrix are The reac habilit matrix is (5.5) This matrix has full rank and can conclude that the system is reac hable. This implies that can mo the system from an initial state to an ˇnal state and, in particular, that can alw ys ˇnd an input to bring the system from an initial state to the equilibrium. Example 5.2 (System in achable Canonic al orm). Next will consider system in achable anonic al form dz dt Az

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5.2. REA CHABILITY 113 sho that is full rank, sho that the in erse of the reac habilit matrix exists and is giv en (5.6) sho this consider the pro duct where AB The ectors satisfy the relation and iterating this relation ˇnd that whic sho ws that the matrix (5.6) is indeed the in erse of Systems That Are Not Reac hable It is useful of ha an in tuitiv understanding of the mec hanisms that mak system unreac hable. An example of suc system is giv en in Figure 5.1. The system consists of iden tical systems with the same input. The in tuition can also demonstrated analytically demonstrate this simple example.

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114 CHAPTER 5. ST TE AND OUTPUT FEEDBA CK Figure 5.1: non-reac hable system. Example 5.3 (Non-r achable System). Assume that the systems in Figure 5.1 are of ˇrst order. The complete system is then describ ed dx dt dx dt The reac habilit matrix is This matrix is singular and the system is not reac hable. One implication of this is that if and start with the same alue, it is nev er ossible to ˇnd an input whic causes them to ha di˛eren alues. Similarly if they start with di˛eren alues, no input will able to driv them oth to zero. Co ordinate Changes It is in teresting to in estigate ho the reac habilit matrix transforms when the co ordinates are hanged. Consider the system in (5.1). Assume that the co ordinates are hanged to As sho wn in the last hapter, that the dynamics matrix and the con trol matrix for the transformed system are AT The reac habilit matrix for the transformed system then ecomes

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5.3. ST TE FEEDBA CK 115 ha AT AB AT AT AT The reac habilit matrix for the transformed system is th us AB (5.7) This form ula is useful for ˇnding the transformation matrix that con erts system in to reac hable canonical form (using from Example 5.2). 5.3 State eedbac Consider system describ ed the linear di˛eren tial equation dx dt Ax (5.8) The output is the ariable that are in terested in con trolling. egin with it is assumed that all comp onen ts of the state ector are measured. Since the state at time con tains all information necessary to predict the future eha vior of the system, the most general time in arian con trol la is function of the state, i.e. )) If the feedbac is restricted to linear, it can written as (5.9) where is the reference alue. The negativ sign is simply con en tion to indicate that negativ feedbac is the normal situation. The closed lo op system obtained when the feedbac (5.8) is applied to the system (5.9) is giv en dx dt (5.10)

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116 CHAPTER 5. ST TE AND OUTPUT FEEDBA CK It will attempted to determine the feedbac gain so that the closed lo op system has the haracteristic olynomial (5.11) This con trol problem is called the eigen alue assignmen problem or the ole placemen problem (w will deˇne \p oles" more formally in later hapter). Examples will start considering few examples that giv insigh in to the nature of the problem. Example 5.4 (The Double Inte gr ator). The double in tegrator is describ ed dx dt In tro ducing the feedbac the closed lo op system ecomes dx dt (5.12) The closed lo op system has the haracteristic olynomial det Assume it is desired to ha feedbac that giv es closed lo op system with the haracteristic olynomial Comparing this with the haracteristic olynomial of the closed lo op system ˇnd ˇnd that the feedbac gains should hosen as ha unit steady state gain the parameter ust equal to The con trol la can th us written as

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5.3. ST TE FEEDBA CK 117 In the next example will encoun ter some diculties. Example 5.5 (A Unr achable System). Consider the system dx dt with the con trol la The closed lo op system is dx dt This system has the haracteristic olynomial det This olynomial has zeros at and One closed lo op eigen alue is th us alw ys equal to and it is not ossible to obtain an arbitrary haracteristic olynomial. This example sho ws that the eigen alue placemen problem cannot al- ys solv ed. An analysis of the equation describing the system sho ws that the state is not reac hable. It is th us clear that some conditions on the system are required. The reac hable canonical form has the prop ert that the parameters of the system are the co ecien ts of the haracteristic equation. It is therefore natural to consider systems on this form when solving the eigen alue place- men problem. In the next example in estigate the case when the system is in reac hable canonical form. Example 5.6 (System in achable Canonic al orm). Consider system in reac hable canonical form, i.e, dz dt Az (5.13)

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118 CHAPTER 5. ST TE AND OUTPUT FEEDBA CK The op en lo op system has the haracteristic olynomial det Expanding the determinan the last ro ˇnd that the follo wing re- cursiv equation for the determinan t: sD It follo ws from this equation that useful prop ert of the system describ ed (5.13) is th us that the co ef- ˇcien ts of the haracteristic olynomial app ear in the ˇrst ro w. Since the all elemen ts of the -matrix except the ˇrst ro are zero it follo ws that the state feedbac only hanges the ˇrst ro of the -matrix. It is th us straigh forw ard to see ho the closed lo op eigen alues are hanged the feedbac k. In tro duce the con trol la (5.14) The closed lo op system then ecomes dz dt (5.15) The feedbac th us hanges the elemen ts of the ˇrst ro of the matrix, whic corresp onds to the parameters of the haracteristic equation. The closed lo op system th us has the haracteristic olynomial

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5.3. ST TE FEEDBA CK 119 Requiring this olynomial to equal to the desired closed lo op olynomial (5.11) ˇnd that the con troller gains should hosen as This feedbac simply replace the parameters in the system (5.15) The feedbac gain for system in reac hable canonical form is th us (5.16) ha unit steady state gain the parameter should hosen as (5.17) Notice that it is essen tial to kno the precise alues of parameters and in order to obtain the correct steady state gain. The steady state gain is th us obtained precise calibration. This is ery di˛eren from obtaining the correct steady state alue in tegral action, whic shall see in later hapters. th us ˇnd that it is easy to solv the eigen alue placemen problem when the system has the structure giv en (5.13) The General Case solv the problem in the general case, simply hange co ordinates so that the system is in reac hable canonical form. Consider the system (5.8) Change the co ordinates linear transformation so that the transformed system is in reac hable canonical form (5.13) or suc system the feedbac is giv en (5.14) where the co ecien ts are giv en (5.16) ransforming bac to the original co ordinates giv es the feedbac It no remains to ˇnd the transformation. do this observ that the reac habilit matrices ha the prop ert AB

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120 CHAPTER 5. ST TE AND OUTPUT FEEDBA CK The transformation matrix is th us giv en (5.18) and the feedbac gain can written as (5.19) Notice that the matrix is giv en (5.6) The feedforw ard gain is giv en equation (5.17). The results obtained can summarized as follo ws. The or em 5.1 (Pole-plac ement by State db ack). Consider the system given by quation (5.8) dx dt Ax with one input and one output. If the system is achable ther exits fe db ack that gives close lo op system with the char acteristic olynomial The fe db ack gain is given by wher ar the ecients of the char acteristic olynomial of the matrix and the matric es and ar given by AB emark 5.1 (A mathematic al interpr etation). Notice that the eigen alue placemen problem can form ulated abstractly as the follo wing algebraic problem. Giv en an matrix and an matrix ˇnd matrix suc that the matrix has prescrib ed eigen alues.

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5.4. OBSER ABILITY 121 Computing the eedbac Gain ha th us obtained solution to the problem and the feedbac has een describ ed closed form solution. or simple problems it is easy to solv the problem the follo wing simple pro cedure: In tro duce the elemen ts of as unkno wn ariables. Compute the haracteristic olynomial det( sI Equate co ecien ts of equal ers of to the co ecien ts of the desired haracteristic olynomial This giv es system of linear equations to determine The equations can alw ys solv ed if the system is observ able. Example 5.4 is ypical illustrations. or systems of higher order it is more con enien to use equation (5.19) this can also used for umeric computations. Ho ev er, for large systems this is not sound umerically ecause it in olv es computation of the harac- teristic olynomial of matrix and computations of high ers of matrices. Both op erations lead to loss of umerical accuracy or this reason there are other metho ds that are etter umerically In MA TLAB the state feedbac can computed the pro cedures acker or place 5.4 Observ abilit In Section 5.3 it as sho wn that it as ossible to ˇnd feedbac that giv es desired closed lo op eigen alues pro vided that the system is reac hable and that all states ere measured. It is highly unrealistic to assume that all states are measured. In this section will in estigate ho the state can estimated using the mathematical mo del and few measuremen ts. It will sho wn that the computation of the states can done dynamical systems. Suc systems will called observers Consider system describ ed dx dt Ax (5.20) where is the state, the input, and the measured output. The problem of determining the state of the system from its inputs and outputs will

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122 CHAPTER 5. ST TE AND OUTPUT FEEDBA CK considered. It will assumed that there is only one measured signal, i.e. that the signal is scalar and that is (ro w) ector. Observ abilit When discussing reac habilit neglected the output and fo cused on the state. will no discuss related problem where will neglect the input and instead fo cus on the output. Consider the system dx dt Ax (5.21) will no in estigate if it is ossible to determine the state from observ a- tions of the output. This is clearly problem of signiˇcan practical in terest, ecause it will tell if the sensors are sucien t. The output itself giv es the pro jection of the state on ectors that are ro ws of the matrix The problem can clearly solv ed if the matrix is in ertible. If the matrix is not in ertible can tak deriv ativ es of the output to obtain dy dt dx dt Ax: rom the deriv ativ of the output th us get the pro jections of the state on ectors whic are ro ws of the matrix Pro ceeding in this get dy dt dt dt (5.22) th us ˇnd that the state can determined if the matrix (5.23) has indep enden ro ws. Notice that ecause of the Ca yley-Hamilton equa- tion it is not orth while to con tin ue and tak deriv ativ es of order higher

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5.4. OBSER ABILITY 123 y Figure 5.2: non-observ able system. than 1. The matrix is called the observ abilit matrix. system is called observ able if the observ abilit matrix has full rank. illustrate with an example. Example 5.7 (Observability of the Inverte Pendulum). The linearized mo del of in erted endulum around the uprigh osition is describ ed (5.4) The matrices and are The observ abilit matrix is whic has full rank. It is th us ossible to compute the state from mea- suremen of the angle. The calculation can easily extended to systems with inputs. The state is then giv en linear com bination of inputs and outputs and their higher deriv ativ es. Di˛eren tiation can giv ery large errors when there is measure- men noise and the metho is therefore not ery practical particularly when deriv ativ es of high order app ear. metho that orks with inputs will giv en the next section. Non-Observ able System It is useful to ha an understanding of the mec hanisms that mak system unobserv able. Suc system is sho wn in Figure 5.2. Next will consider

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124 CHAPTER 5. ST TE AND OUTPUT FEEDBA CK the system in observable anonic al form i.e. dz dt straigh forw ard but tedious calculation sho ws that the in erse of the observ abilit matrix has simple form. It is giv en This matrix is alw ys in ertible. The system is comp osed of iden ti- cal systems whose outputs are added. It seems in tuitiv ely clear that it is not ossible to deduce the states from the output. This can also seen formally Co ordinate Changes It is in teresting to in estigate ho the observ abilit matrix transforms when the co ordinates are hanged. Consider the system in equation (5.21). As- sume that the co ordinates are hanged to It follo ws from linear algebra that the dynamics matrix and the output matrix are giv en AT The observ abilit matrix for the transformed system then ecomes

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5.5. OBSER VERS 125 ha AT AT AT AT AT and ˇnd that the observ abilit matrix for the transformed system has the prop ert (5.24) This form ula is ery useful for ˇnding the transformation matrix 5.5 Observ ers or system go erned equation (5.20) can attempt to determine the state simply sim ulating the equations with the correct input. An estimate of the state is then giv en dt (5.25) ˇnd the prop erties of this estimate, in tro duce the estimation error It follo ws from (5.20) and (5.25) that dt If matrix has all its eigen alues in the left half plane, the error will th us go to zero. Equation (5.25) is th us dynamical system whose output con erges to the state of the system (5.20) The observ er giv en (5.25) uses only the pro cess input the measured signal do es not app ear in the equation. It ust also required that the system is stable. will therefore attempt to mo dify the observ er so that

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126 CHAPTER 5. ST TE AND OUTPUT FEEDBA CK the output is used and that it will ork for unstable systems. Consider the follo wing observ er dt (5.26) This can considered as generalization of (5.25) eedbac from the measured output is pro vided adding the term ). Notice that is the output that is predicted the observ er. in estigate the observ er (5.26) form the error It follo ws from (5.20) and (5.26) that dt LC If the matrix can hosen in suc that the matrix LC has eigen alues with negativ real parts, the error will go to zero. The con- ergence rate is determined an appropriate selection of the eigen alues. The problem of determining the matrix suc that LC has pre- scrib ed eigen alues is ery similar to the eigen alue placemen problem that as solv ed ab e. In fact, if observ that the eigen alues of the matrix and its transp ose are the same, ˇnd that could determine suc that has giv en eigen alues. First notice that the problem can solv ed if the matrix 1) is in ertible. Notice that this matrix is the transp ose of the observ abilit matrix for the system (5.20) of the system. Assume it is desired that the haracteristic olynomial of the matrix LC is It follo ws from Remark 5.1 of Theorem 5.1 that the solution is giv en

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5.5. OBSER VERS 127 where is the observ abilit matrix and is the observ abilit matrix of the system of the system dz dt whic is the observ able canonical form of the system (5.20) ransp osing the form ula for obtain The result is summarized the follo wing theorem. The or em 5.2 (Observer design by eigenvalue plac ement). Consider the sys- tem given by dx dt Ax wher output is sc alar. Assume that the system is observable. The dynamic al system dt with chosen as (5.27) wher the matric es and ar given by

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128 CHAPTER 5. ST TE AND OUTPUT FEEDBA CK PSfrag replacemen ts Figure 5.3: Blo diagram of the observ er. Notice that the observ er con tains cop of the pro cess. Then the observer err or is governe by di˛er ential quation having the char acteristic olynomial emark 5.2 The dynamical system (5.26) is called an observ er for (the states of the) system (5.20) ecause it will generate an appro ximation of the states of the system from its inputs and outputs. emark 5.3 The theorem can deriv ed transforming the system to observ able canonical form and solving the problem for system in this form. emark 5.4 Notice that ha giv en observ ers, one based on pure dif- feren tiation (5.22) and another describ ed the di˛eren tial equation (5.26) There are also other forms of observ ers. In terpretation of the Observ er The observ er is dynamical system whose inputs are pro cess input and pro cess output The rate of hange of the estimate is comp osed of terms. One term is the rate of hange computed from the mo del with substituted for The other term is prop ortional to the di˛erence et een measured output and its estimate The estimator gain is matrix that tells ho the error is eigh ted and distributed among the states. The observ er th us com bines measuremen ts with dynamical mo del of the system. blo diagram of the observ er is sho wn in Figure 5.3

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5.5. OBSER VERS 129 Dualit Notice the similarit et een the problems of ˇnding state feedbac and ˇnding the observ er. The ey is that oth of these problems are equiv alen to the same algebraic problem. In eigen alue placemen it is attempted to ˇnd so that has giv en eigen alues. or the observ er design it is instead attempted to ˇnd so that LC has giv en eigen alues. The follo wing equiv alence can established et een the problems The similarit et een design of state feedbac and observ ers also means that the same computer co de can used for oth problems. Computing the Observ er Gain The observ er gain can computed in sev eral di˛eren ys. or simple problems it is con enien to in tro duce the elemen ts of as unkno wn param- eters, determine the haracteristic olynomial of the observ er det LC and iden tify it with the desired haracteristic olynomial. Another alterna- tiv is to use the fact that the observ er gain can obtained insp ection if the system is in observ able canonical form. In the general case the ob- serv er gain is then obtained transformation to the canonical form. There are also reliable umerical algorithms. They are iden tical to the algorithms for computing the state feedbac k. The pro cedures are illustrated few examples. Example 5.8 (The Double Inte gr ator). The double in tegrator is describ ed dx dt The observ abilit matrix is

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130 CHAPTER 5. ST TE AND OUTPUT FEEDBA CK i.e. the iden tit matrix. The system is th us observ able and the problem can solv ed. ha LC It has the haracteristic olynomial det LC det Assume that it is desired to ha an observ er with the haracteristic oly- nomial The observ er gains should hosen as The observ er is then dt 5.6 Output eedbac In this section will consider the same system as in the previous sections, i.e. the th order system describ ed dx dt Ax (5.28) where only the output is measured. As efore it will assumed that and are scalars. It is also assumed that the system is reac hable and observ able. In Section 5.3 had found feedbac for the case that all states could measured and in Section 5.4 ha presen ted dev elop ed an observ er that can generate estimates of the state based on inputs and outputs. In this section will com bine the ideas of these sections to ˇnd an feedbac whic giv es desired closed lo op eigen alues for systems where only outputs are ailable for feedbac k.

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5.6. OUTPUT FEEDBA CK 131 If all states are not measurable, it seems reasonable to try the feedbac (5.29) where is the output of an observ er of the state (5.26), i.e. dt (5.30) Since the system (5.28) and the observ er (5.30) oth are of order the closed lo op system is th us of order The states of the system are and The ev olution of the states is describ ed equations (5.28) (5.29)(5.30) analyze the closed lo op system, the state ariable is replace (5.31) Subtraction of (5.28) from (5.28) giv es dt Ax LC LC In tro ducing from (5.29) in to this equation and using (5.31) to eliminate giv es dx dt Ax Ax Ax The closed lo op system is th us go erned dt LC (5.32) Since the matrix on the righ t-hand side is blo diagonal, ˇnd that the haracteristic olynomial of the closed lo op system is det sI det sI LC This olynomial is pro duct of terms, where the ˇrst is the harac- teristic olynomial of the closed lo op system obtained with state feedbac and the other is the haracteristic olynomial of the observ er error. The feedbac (5.29) that as motiv ated heuristically th us pro vides ery neat solution to the eigen alue placemen problem. The result is summarized as follo ws.

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132 CHAPTER 5. ST TE AND OUTPUT FEEDBA CK The or em 5.3 (Pole plac ement by output fe db ack). Consider the system dx dt Ax The ontr ol ler describ by dt gives close lo op system with the char acteristic olynomial det sI det sI LC This olynomial an assigne arbitr ary ots if the system is observable and achable. emark 5.5 Notice that the haracteristic olynomial is of order and that it can naturally separated in to factors, one det sI asso ciated with the state feedbac and the other det sI LC with the observ er. emark 5.6 The con troller has strong in tuitiv app eal. It can though of as comp osed of parts, one state feedbac and one observ er. The feedbac gain can computed as if all state ariables can measured. The In ternal Mo del Principle blo diagram of the con troller is sho wn in Figure 5.4. Notice that the con troller con tains dynamical mo del of the plan t. This is called the in ternal mo del principle. Notice that the dynamics of the con troller is due to the observ er. The con troller can view ed as dynamical system with input and output dt LC Ly (5.33) The con troller has the transfer function sI LC (5.34)

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5.7. INTEGRAL CTION 133 PSfrag replacemen ts Pro cess Observ er Figure 5.4: Blo diagram of con troller whic com bines state feedbac with an observ er. 5.7 In tegral Action The con troller based on state feedbac ac hiev es the correct steady state re- sp onse to reference signals careful calibration of the gain and it lac ks the nice prop ert of in tegral con trol. It is then natural to ask wh the the eautiful theory of state feedbac and observ ers do es not automatically giv con trollers with in tegral action. This is consequence of the assump- tions made when deriving the analytical design metho whic will no in estigate. When using an analytical design metho d, ostulate criteria and sp ec- iˇcations, and the con troller is then consequence of the assumptions. In this case the problem is the mo del (5.8). This mo del assumes implicitly that the system is erfectly calibrated in the sense that the output is zero when the input is zero. In practice it is ery dicult to obtain suc mo del. Consider, for example, hemical pro cess con trol problem where the output is temp erature and the con trol ariable is large rust alv e. The mo del (5.8) then implies that kno exactly ho to osition the alv to get sp eciˇed outlet temp erature|indeed, highly unrealistic assumption.

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134 CHAPTER 5. ST TE AND OUTPUT FEEDBA CK Ha ving understo the dicult it is not to hard to hange the mo del. By mo difying the mo del to dx dt Ax x; (5.35) where is an unkno wn constan t, can can capture the idea that the mo del is no longer erfectly calibrated. This mo del is called mo del with an input disturbance. Another ossibilit is to use the mo del dx dt Ax where is an unkno wn constan t. This is mo del with an output disturbance. It will no sho wn that straigh tforw ard design of an output feedbac for this system do es indeed giv in tegral action. Both disturbance mo dels will pro duce con trollers with in tegral action. will start in estigating the case of an input disturbance. This is little more con enien for us ecause it ˇts the con trol goal of ˇnding con troller that driv es the state to zero. The mo del with an input disturbance can con enien tly brough in to the framew ork of state feedbac k. do this, ˇrst observ that is an unkno wn constan whic can describ ed dv dt bring the system in to the standard format simply in tro duce the dis- turbance as an extra state ariable. The state of the system is th us This is also called state augmen tation. Using the augmen ted state the mo del (5.35) can written as dt (5.36) Notice that the disturbance state is not reac hable. If the disturbance can measured, the state feedbac is then (5.37)

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5.8. GENERAL CONTR OLLER STR UCTURE 135 The disturbance state is not reac hable. The the e˛ect of the disturbance on the system can, ho ev er, eliminated ho osing 1. If the distur- bance is kno wn the con trol la ab can in terpreted as com bination of feedbac from the system state and feedforw ard from measured distur- bance. It is not realistic to assume that the disturbance can measured and will instead replace the states estimates. The feedbac la then ecomes This means that feedbac is based on estimates of the state and the distur- bance. There are man other ys to in tro duce in tegral action. 5.8 General Con troller Structure So far reference signals ha een in tro duced simply adding it to the state feedbac k. more sophisticated of doing this is sho wn the blo dia- gram in Figure 5.5, where the con troller consists of three parts: an observ er that computes estimates of the states based on mo del and measured pro- cess inputs and outputs, state feedbac and tra jectory generator that generates the desired eha vior of all states and feedforw ard signal The signal is suc that it generates the desired eha vior of the states when applied to the system, under ideal conditions of no disturbances and no mo deling errors. The con troller is said to ha two de gr es of fr dom ecause the resp onse to command signals and disturbances are decoupled. Disturbance resp onses are go erned the observ er and the state feedbac and the resp onse to command signal is go erned the feedforw ard. get some insigh in to the eha vior of the system let us discuss what happ ens when the command signal is hanged. ˇx the ideas let us assume that the system is in equilibrium with the observ er state equal to the pro cess state When the command signal is hanged feedforw ard signal is generated. This signal has the prop ert that the pro cess output giv es the desired output when the feedforw ard signal is applied to the system. The pro cess state hanges in resp onse to the feedforw ard signal. The ob- serv er trac ks the state erfectly ecause the initial state as correct. The estimated state will equal to the desired state and the feedbac signal is zero. If there are some disturbances or some mo deling errors the feedbac signal will di˛eren from zero and attempt to correct the situation. The con troller giv en in Figure 5.5 is ery general structure. There are man ys to generate the feedforw ard signal and there are also man dif-

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136 CHAPTER 5. ST TE AND OUTPUT FEEDBA CK PSfrag replacemen ts ra jectory eedforw ard Generator Pro cess State eedbac Observ er fb Figure 5.5: Blo diagram of con troller based on structure with degrees of freedom. The con troller consists of command signal generator, state feedbac and an observ er. feren ys to compute the feedbac gains and the gain of the observ er. The system in Figure 5.5 is an example of the internal mo del principle whic sa ys that con troller should con tain mo del of the system to con trolled and the disturbances action on the system. Computer Implemen tation The con trollers obtained so far ha een describ ed ordinary di˛eren tial equations. They can implemen ted directly using analog computers. Since most con trollers are implemen ted using digital computers will brie—y discuss ho this can done. The computer ypically op erates erio dically signals from the sensors are sampled and con erted to digital form the A/D con erter, the con trol signal is computed, con erted to analog form for the actuators, as sho wn in Figure 1.3 on page 5. illustrate the main principles consider the con troller describ ed equations (5.29) and (5.30), i.e. dt The ˇrst equation whic only consists of additions and ultiplications can implemen ted directly in computer. The second equation has to appro ximated. simple is to replace the deriv ativ di˛erence +1 ))

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5.9. EXER CISES 137 where are the sampling instan ts and +1 is the sampling erio d. Rewriting the equation get +1 )) (5.38) The calculation of the state only requires addition and ultiplication and can easily done computer. pseudo co de for the program that runs in the digital computer is "Control algorithm main loop r=adin(ch1) "read setpoint from ch1 y=adin(ch2) "read process variable from ch2 u=C*x+Kr*r "compute control variable daout(ch1) "set analog output ch1 x=x+h*(A*x+B*u+L*(y-C*x)) "update state estimate The program runs erio dically Notice that the um er of computations et een reading the analog input and setting th analog output has een minimized. The state is up dated after the analog output has een set. The program has one states The hoice of sampling erio requires some care. or linear systems the di˛erence appro ximation can oided ob- serving that the con trol signal is constan er the sampling erio d. An exact theory for this can dev elop ed. Doing this get con trol la that is iden tical to (5.38) but with sligh tly di˛eren co ecien ts. There are sev eral practical issues that also ust dealt with. or ex- ample it is necessary to ˇlter signal efore it is sampled so that the ˇltered signal has little frequency con ten ab where is the sampling fre- quency If con trollers with in tegral action are used it is necessary to pro vide protection so that the in tegral do es not ecome to large when the actuator saturates. Care ust also tak en so that parameter hanges do not cause disturbances. Some of these issues are discussed in Chapter 10. 5.9 Exercises 1. Consider system on reac hable canonical form. Sho that the in erse of the reac habilit matrix is giv en (5.39)

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138 CHAPTER 5. ST TE AND OUTPUT FEEDBA CK

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