Outline. Model Predictive Control Short Course Introduction. The model predictive control framework. The power of abstraction

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1 Model Predictive Control Short Corse Introdction James B. Rawlings Michael J. Risbeck Nishith R. Patel Department of Chemical and Biological Engineering Copright c 7 b James B. Rawlings Milwakee, WI Agst 8 9, 7 Otline Overview and Indstrial Impact of MPC Dnamic Modeling Continos linear differential eqations Continos transfer fnctions Discrete linear difference eqations Software for model conversions 3 Predictive Control Compared to Classical PID Control Difficlt dnamics Constraints Mltivariable interactions 4 Smmar JCI 7 MPC short corse / 7 JCI 7 MPC short corse / 7 The power of abstraction The model predictive control framework process dx = f (x, ) dt = g(x, ) actators sensors sensors Reconcile the past MH Estimate Measrement Forecast the ftre Forecast MPC control actators t JCI 7 MPC short corse 3 / 7 JCI 7 MPC short corse 4 / 7

2 Predictive control State estimation Reconcile the past Forecast the ftre Reconcile the past Forecast the ftre MH Estimate Measrement Forecast MPC control MH Estimate Measrement Forecast MPC control sensors sensors actators actators t t T min sp g(x, ) Q + sp R dt (t) min g(x, ) R + ẋ f (x, x,w(t) ) Q dt T ẋ = f (x, ) x() = x (given) = g(x, ) JCI 7 MPC short corse 5 / 7 ẋ = f (x, ) + w (process noise) = g(x, ) + v (measrement noise) JCI 7 MPC short corse 6 / 7 Indstrial impact of the research Large indstrial sccess stor! Planning and Schedling Stead State Optimization Validation Controller Plant Model Update Reconciliation Two laer strctre Stead-state laer RTO optimizes stead-state model Optimal setpoints passed to dnamic laer Dnamic laer Controller tracks the setpoints Linear MPC (replaces mltiloop PID) Linear MPC and ethlene manfactring Nmber of MPC applications in ethlene: 8 to Credits 5 to 8 M$/r (7) Achieved primaril b increased on-spec prodct, decreased energ se Eastman Chemical experience with MPC First MPC implemented in 996 Crrentl 55-6 MPC applications of varing complexit 3-5 M$/ear increased profit de to increased throghpt (8) Praxair experience with MPC Praxair crrentl has more than 5 MPC installations 6 M$/ear increased profit (8) JCI 7 MPC short corse 7 / 7 JCI 7 MPC short corse 8 / 7

3 Impact for 3 ethlene plants (Starks and Arrieta, 7) Broader indstrial impact (Qin and Badgwell, 3) Hdrocarbons AC&O 7 We re Doing it For the Mone $6,, $5,, $4,, $3,, $,, $,, $ Q 3Q Q 3Q Q 3Q Q 3 3Q 3 Q 4 3Q 4 Q 5 3Q 5 Q6 3Q6 Cmlative $6,, $5,, $4,, $3,, $,, $,, $ Qarterl Advanced Control & Optimization JCI 7 MPC short corse 9 / 7 Area Aspen Honewell Adersa PCL MDC Total Technolog Hi-Spec Refining Petrochemicals Chemicals 3 44 Plp and Paper Air & Gas Utilit Mining/Metallrg Food Processing Polmer Frnaces Aerospace/Defense Atomotive Unclassified Total First App. DMC:985 PCT:984 IDCOM:973 PCL: SMOC: IDCOM-M:987 RMPCT:99 HIECON: OPC:987 Largest App 63x83 5x85-3x - JCI 7 MPC short corse / 7 Linear differential eqations SISO and sstem order The linear continos model is a set of n first-order linear differential eqations in which dx = Ax + B dt = Cx + D () x state n-vector maniplated inpt m-vector measred otpt p-vector dx = Ax + B dt = Cx + D For single inpt, single otpt (SISO) sstems, m = and p =, bt n, known as the order of the sstem, ma be greater than one (even for SISO sstems). The matrix D is often zero. It represents a direct connection from the inpt to the otpt. If the transfer fnction discssed below has nmerator and denominator with eqal orders, then D. JCI 7 MPC short corse / 7 JCI 7 MPC short corse / 7

4 Transfer fnction model Converting between ODEs and transfer fnctions In the Laplace transform domain, we consider the Laplace transform of the inpt (t), denoted (s), and the otpt (t), denoted (s). The linear model is then the transfer fnction G(s) relating the inpt to the otpt (s) (s) = G(s)(s) G(s) (s) Example (First-order sstem) We are probabl alread familiar with the first-order sstem and its transfer fnction G(s) = k τs + with gain k and constant τ. Find the corresponding A, B, C, D in the domain differential eqations. JCI 7 MPC short corse 3 / 7 JCI 7 MPC short corse 4 / 7 First-order sstem First-order sstem Soltion First take the relation Rearrange to (s) = G(s)(s) = k τs + (s) (τs + )(s) = k(s) Invert back to the domain sing L(d/dt) = s(s) () with deviation variables so x() = and () = Cx() =. τ d dt + (t) = k(t) Soltion Identif x(t) = (t) and have the state space form dx dt = τ x + k τ dx = Ax + B dt = x = Cx + D We see that the differential eqation is first order, n =, and that A = B = k C = D = τ τ JCI 7 MPC short corse 5 / 7 JCI 7 MPC short corse 6 / 7

5 Higher-order sstem Second-order sstem Next we work an example where the order of the differential eqation is higher than one. Example (Second-order sstem) Consider the general second-order sstem corresponding to the transfer fnction k G(s) = τ s + ζτs + with gain k, free constant τ, and damping coefficient ζ. Find the corresponding A, B, C, D in the domain differential eqations. Soltion We proceed as in the previos example and note that L ( d ) dt = s (s) () s(). If the sstem is initiall at stead state, in deviation variables we have that () = () =, giving L ( d ) dt = s (s), and L ( d ) dt = s(s). Sbstitting these into (s) = G(s)(s) and mltipling both sides b the denominator of G(s) gives τ d d + ζτ dt dt + = k JCI 7 MPC short corse 7 / 7 JCI 7 MPC short corse 8 / 7 Second-order sstem Second-order sstem Soltion Note that to convert this to a set of first-order differential eqations, we reqire two states. Let x = d/dt and x =. Then the second-order differential eqation above can be expressed as τ dx dt = ζτx x + k and we add the second differential eqation that defines the relationship between x and x dx dt = x Soltion We can rearrange these two differential eqations into the state space form of () and write them as a single vector differential eqation d dt [ x x ] = [ ζ τ τ = [ ] [ ] x x and we have identified the coefficients [ ζ ] [ k ] A = τ τ B = τ ] [ x x ] + [ k ] τ C = [ ] D = JCI 7 MPC short corse 9 / 7 JCI 7 MPC short corse / 7

6 Discrete linear difference eqations Formlas are exact! For discrete sstems, we have a sample, denoted, and we are interested in the state, inpt and otpt of the sstem onl at the sample s, t = k. The linear difference eqation that represents the behavior at the sample s is given b x(k + ) = A d x(k) + B d (k) (k) = C d x(k) + D d (k) () The following formlas let s convert from the continos differential eqation model to the discrete model difference eqation model Note that if (t) is a constant between samples (known as a zero-order hold), then these formlas are exact and there is no approximation error sch as we wold have if we sed an Eler method to approximate the derivative in (). Note also that if A is singlar (one or more of its eigenvales are zero), A does not exist and the formla for B d is not defined. Yo can obtain B d from the following relationship that does not reqire A ( [ ] ) [ ] A B Ad B exp = d I A d = e A B d = A (e A I n )B C d = C D d = D (3) JCI 7 MPC short corse / 7 JCI 7 MPC short corse / 7 Convert the first-order sstem to discrete Example 3 (First-order discrete sstem) Convert the first-order continos sstem with constant τ and gain k to the discrete sstem with sample. Soltion We have alread fond the CT state space model is:a = /τ, B = k/τ, C =, D =. Using the CT to DT conversion formlas (3) gives A d = e /τ B d = k( e /τ ) C d = D d = JCI 7 MPC short corse 3 / 7 Wh discrete models? Note that the discrete model is extremel convenient for compter simlation. An example SISO simlation for a second-order sstem wold be: nts = ; tfin = nts*delta; tot = linspace (, tfin, nts)'; x = zeros(, nts); = zeros(, nts); = sin(w*tot); xp = x; for k = : nts x(:,k) = xp; % take measrement (k) = Cd*x(:,k); % pdate process xp = Ad*xp + Bd*(k); end JCI 7 MPC short corse 4 / 7

7 Software for model conversions The conversions between model forms can be carried ot convenientl in Matlab or Octave. Usefl fnctions: ss, cd, and tf. ss = ss(a,b,c,d). Enter a general sstem sing a continos state space model. D is sall zero. dss = cd(ss, delta). Convert from a continos sstem to a discrete sstem sing sample delta. 3 Ad = dss.a, Bd = dss.b, Cd = dss.c, Dd = dss.d commands extract the state space A d, B d, C d, D d state space matrices of the sstem dss. Note that the state space model is in continos, (), or discrete, (), depending on whether the sstem dss is in continos or discrete. 4 ss = tf(nm, den). Convert from a transfer fnction model to a continos sstem. The vectors nm and den are the coefficients of the s polnomial in the nmerator and denominator of the transfer fnction. JCI 7 MPC short corse 5 / 7 More fine print Use the help fnction at the Matlab or Octave prompt to find ot more options for the calling and retrn argments for these fnctions. Note that o mst se expm and not exp in Matlab or Octave when exponentiating a matrix in the formla for A d and B d. JCI 7 MPC short corse 6 / 7 PID feedback control Consider the process g and feedback controller g c with otpt distrbance d and setpoint sp d Predictive control (t) sp g c g x(t) PID controller takes control action based on crrent tracking error and integral (and derivative) of tracking error, (t) = k c ( e(t) + τ I t ) e(t )dt d + τ d dt e(t), e = sp The control engineer adjsts directl tning parameters k c, τ i, τ d. Model. k Past k Present k + Ftre k + x(k + ) = Ax(k) + B(k) (k) = Cx(k) t vale of control objective JCI 7 MPC short corse 7 / 7 JCI 7 MPC short corse 8 / 7

8 Predictive control Predictive control Stage cost. Control Objective. Constraints. Optimization. V (x, ) = N j= l(x(j), (j) D(k) d, k =,,... N Hx(k) h, k =,,... N sbject to the model and constraints. min V (x, ) l(x, ) = sp Cx(j) Q + (j) (j ) S The matrices Q (p p) and S (m m) are the primar tning parameters in this approach, and the are sall chosen to be diagonal matrices. Let e = sp Cx. Then each term above is a sm of sqares [ ] e e... e p Q Q... Qp e e... = e p Q e + Q e + + Q p e p JCI 7 MPC short corse 9 / 7 JCI 7 MPC short corse 3 / 7 Predictive control MPC feedback control If o wish to have tight control in certain otpts in the -vector, o choose the corresponding diagonal elements of Q large. If o wish to se onl small maniplated variable changes in some actator, o choose the corresponding diagonal elements of S large. The loop responds qickl (tight control) when Q s elements are large relative to S s and responds slowl (loose control) with the reverse tning. Q S S Q Q i Q j S i S j Aggressive control Catios control Otpt (sensor) i is more important than otpt j Inpt (actator) i is more expensive than inpt j The control law is the otcome of the optimization. We do not (sall) have a simple closed form like in PID. The tning parameters are Q, S, i.e., the weights on inpts and otpts Prediction horizon N is not a tning parameter. We choose it large enogh to see the effects of the inpts on the otpts, bt small enogh to have a tractable comptation. In favorable cases we can make N =. We can also se some of the constraint parameters as additional tning parameters (rate of change constraints) JCI 7 MPC short corse 3 / 7 JCI 7 MPC short corse 3 / 7

9 Difficlt dnamics Time delas Nonminimm phase sstems ( delas, right half-plane zeros), and high order sstems limit the gain that can be sed in simple feedback controllers withot casing instabilit or excessive control action. Predictive control does not sffer from this disadvantage becase the model-based forecasting enables the controller to accont for the inherent limitations in the process response. Consider the following first order sstem with dela sp g(s) = g c k τs + e θs, k =, τ =, θ = 5 g d The large dela compared to constant severel limits the achievable closed-loop performance of this sstem. Consider a nit step change in setpoint, sp, at t =. What wold we expect an ideal controller to do in this sitation? The phsicall intitive answer is to take immediatel the control action reqired to drive the sstem withot dela to the desired setpoint and then ignore the large tracking error while waiting for the sstem s dela to pass; then the otpt shold rise qickl to the setpoint. The predictive controller has this inherent dela compensation becase the forecast enables it to see that it shold wait for the dela to pass before taking frther control action. JCI 7 MPC short corse 33 / 7 JCI 7 MPC short corse 34 / 7 Predictive control of sstem with dela The predictive controller behaves precisel as we expect an ideal controller to behave. It takes large control action earl to achieve the setpoint and then does nothing frther ntil the dela passes Figre : MPC applied to sstem with large dela, g(s) = /(s + )e 5s, Q =, R =, S = ; nit step setpoint change at t =. PID control Withot a model, the PID controller is blind to the dela and mst take control action based on crrent tracking error and integral (and derivative) of tracking error, (t) = k c ( e(t) + τ I t ) e(t )dt d + τ d dt e(t), e = sp Even if we choose k c and τ I carefll (we ll set τ d = ), there is onl so mch that we can expect. JCI 7 MPC short corse 35 / 7 JCI 7 MPC short corse 36 / 7

10 PID control of sstem with dela PID control of sstem with dela The response of a reasonabl well tned PID controller Figre : PID applied to sstem with large dela, g(s) = /(s + )e 5s, k c =.3, τ I =.5; nit step setpoint change at t =. Notice that the control action does not have the same strctre as the MPC controller. Control action starts small and increases as the integral of the tracking error increases. After the dela passes, the sstem response is still slggish and does not reach setpoint ntil t = 5. JCI 7 MPC short corse 37 / 7 JCI 7 MPC short corse 38 / 7 Retning the PID controller Tne the PID controller more aggressivel, i.e., increase the gain, in order to speed p the closed-loop response. Consider dobling the gain, k c (and dobling τ I ) Figre 3: PID applied to sstem with large dela, g(s) = /(s + )e 5s, k c =.6, τ I = 5.; nit step setpoint change at t =. Constraints Constraints are present in ever phsical process: valves satrate fll open and fll closed, safet considerations stiplate constraints on process temperatres and pressres, and the like. Even if the process is designed so that the constraints are not active at the designed stead state, market conditions and process operations change over so that plants ma be operated against constraints in some normal ftre operating condition. Model predictive control is particlarl sefl for controlling constrained sstems, becase the constraints can be added explicitl to the on-line optimization problem. JCI 7 MPC short corse 39 / 7 JCI 7 MPC short corse 4 / 7

11 SISO example with inpt constraints First consider a simple SISO sstem, first order withot dela, g(s) = k τs +, k =, τ = A PID controller has no troble controlling this simple sstem Figre 4: PID applied to nconstrained first order sstem, g(s) = /(s + ), k c = /k, τ I = τ. Setpoint tracking. JCI 7 MPC short corse 4 / 7 PID distrbance rejection Let s consider the response to a distrbance: plse distrbance of magnitde at the process otpt that lasts ntil t = Figre 5: PID applied to nconstrained first order sstem, g(s) = /(s + ), k c = /k, τ I = τ. Distrbance rejection. JCI 7 MPC short corse 4 / 7 PID distrbance rejection constraints Given the same plse distrbance, now the controller satrates the valve at fll closed and there is stead-state tracking error ntil t = and the distrbance goes to zero Figre 6: PID applied to constrained first order sstem, g(s) = /(s + ), k c = /k, τ I = τ,. Distrbance rejection; integrator windp after t =. PID constraints and integrator windp This phenomenon is known as integrator (or reset) windp. Can o explain wh the integrator is the clprit in this windp phenomenon? We also know that we can easil prevent the windp problem in this SISO case. Can o explain how to modif this PID controller to avoid windp? Bt the anti-windp strategies are not clear in complex mltivariable problems when mltiple active constraints are possible. JCI 7 MPC short corse 43 / 7 JCI 7 MPC short corse 44 / 7

12 Simple PID anti-reset windp PID control with anti-reset windp for i = : nts %% take measrement (i) = x(i); %% compte PI control err = sp(i) - (i); tr = Kc*(err + /tai*interr(i)); %% satration limits on the valve -lim <= <= lim if (tr >= lim) (i) = lim; elseif (tr <= -lim ) (i) = -lim; else (i) = tr; end %% pdate process and integral (sm) of tracking error if (i == nts) break end x(i+) = A*x(i) + B*(i); %% stop integrating if valve is satrated (anti-reset windp) if ( (i) == tr ) interr(i+) = interr(i) + err; else interr(i+) = interr(i); end end Figre 7: PID applied to constrained first order sstem with anti-reset windp, g(s) = /(s + ), k c = /k, τ I = τ,. Distrbance rejection; anti-reset windp after t =. JCI 7 MPC short corse 45 / 7 JCI 7 MPC short corse 46 / 7 MPC control of constrained sstem MPC control of constrained sstem B contrast, consider the MPC controller behavior Figre 8: MPC applied to constrained first-order sstem, g(s) = /(s + ), Q =,R =,S =,N =,. Distrbance rejection; no integrator windp after t =. JCI 7 MPC short corse 47 / 7 Notice that the MPC controller qickl moves the otpt back to setpoint at t = after the distrbance goes to zero. There is no windp problem in an MPC controller. One does not need to design anti-windp strategies. The inflence of constraints is forecast correctl and handled atomaticall b an MPC controller. That is one of the trl attractive featres of MPC for indstrial practice. JCI 7 MPC short corse 48 / 7

13 Mltivariable interactions Relative gain arra (RGA) A significant advantage of MPC comes to light if we examine strongl interacting mltiinpt mltiotpt (MIMO) sstems, which we know can case problems for mltiloop SISO PID controllers. Consider a two-inpt, two-otpt process with the following transfer fnction (Ognnaike and Ra, 994, pp ). ] G(s) = [ s+ s+ s+ 4 s+ The RGA for this process is given b [ ].8. Λ =..8 The RGA indicates that pairing, is best considering the stead-state interactions. The next figre shows how two SISO PID controllers manage a nit setpoint change in the first otpt. JCI 7 MPC short corse 49 / 7 JCI 7 MPC short corse 5 / 7 Mltiloop PID RGA pairing Mltiloop PID dnamic considerations A classic case of two controllers fighting with each other Figre 9: PID control with stead-state pairing (, ), k c =, τ I =, k c =, τ I = 3; setpoint change in. G(s) = [ s+ s+ s+ 4 s+ This pairing is not attractive from the dnamic perspective. The mch slower inpt is driving (τ = ) when a ten s faster inpt,, cold have been sed (τ = ). Similarl, the slow inpt,, drives (τ = ) when the first inpt is ten s faster (τ = ). Dnamic considerations wold favor switching to the pairing,. ] JCI 7 MPC short corse 5 / 7 JCI 7 MPC short corse 5 / 7

14 Mltiloop PID dnamic pairing Mltiloop PID pairing smmar Figre : PID control with dnamic pairing (, ), k c = 5, τ I =, k c = 5, τ I = 3; setpoint change in. For this case, dnamics carr the da and the pairing based on speed of response exhibits better setpoint tracking than the pairing based on stead-state interactions. These points are all clear for this example, bt the will not be so clear if we start to move the two constants closer to each other and the dnamic separation of scales is not so large. JCI 7 MPC short corse 53 / 7 JCI 7 MPC short corse 54 / 7 MPC of mltivariable sstem Let s see what happens if we refse to choose pairings and allow a mltivariable MPC controller to se its model to sort things ot atomaticall Figre : Mlti-variable MPC control, Q = I, S =. Setpoint change. JCI 7 MPC short corse 55 / 7 MPC of mltivariable sstem Notice that the performance is essentiall deadbeat (i.e. immediate) in moving otpt to its new setpoint while not distrbing otpt. The MPC controller ses its model to forecast exactl what inpt trajectories in both inpts are reqired to track exactl the setpoint in both otpts. Interactions are not an isse becase the model forecast acconts for them. (Model error is discssed later.) Performance is as good or better than the best performance we can achieve sing PID with the best tning constants and the best pairing. No significant tning effort for the MPC controller is reqired to achieve this good nominal performance. We sed Q = I, S = for this deadbeat response. As we expect, we will have to detne this controller (increase S to se less inpt) if we want it to accommodate sensor noise, and mismatch between the plant and the model. JCI 7 MPC short corse 56 / 7

15 Effect of tning parameters: move sppression with S Effect of tning parameters: move sppression with S The effect of increasing the S penalt from to I is shown next The S penalt affects the optimization tradeoff between the otpt tracking error and the maniplation of the inpt. Notice the otpt response has slowed down and is no longer deadbeat, and that the maniplated inpt is moving less aggressivel. Figre : Mlti-variable infinite horizon MPC control, Q = I, S = I. Setpoint change. JCI 7 MPC short corse 57 / 7 JCI 7 MPC short corse 58 / 7 Constraints in mltivariable sstems MPC offers frther flexibilit to shape this tradeoff between otpt tracking error and maniplated inpt. Let s now assme that the inpts are scaled so that valve satration occrs at = ±. The second inpt violates this constraint in the previos simlation. A PID controller wold clip this signal and satrate the valve. The MPC controller can be told abot these constraints and the can be treated natrall b the on-line optimization. The reslts are shown in the next figre. Constraints in mltivariable sstems The otpt tracking error is not seriosl affected while the MPC controller dedces what changes to make in the inpt trajector to accont for the valve satration Figre 3: Mlti-variable infinite horizon MPC control with inpt constraints, Q = I, S = I, min =, max =. Setpoint change. JCI 7 MPC short corse 59 / 7 JCI 7 MPC short corse 6 / 7

16 Rate of change constraints Rate of change constraints Another practicall motivated method for shaping the tradeoff between tracking and control action is to constrain rather than penalize the rate of change of the inpt. Practitioners or operators ma be comfortable with some rates at which the valves ma be moved between each controller exection, bt ma not know how to translate that speed into an appropriate S penalt withot extensive simlation. In these sitations constraining rather than (or in addition to) penalizing the rate of change of the inpt is simpler to implement. Assme the initial portion of the inpt maniplation is nacceptable to the process operators, and the wold be more comfortable if changed not more than. in each sample,.. Adding this constraint to the MPC problem prodces the reslts in the next figre JCI 7 MPC short corse 6 / 7 JCI 7 MPC short corse 6 / 7 Rate of change constraints Rate of change constraints The inpt now slows down significantl in the earl portion of the setpoint change Figre 4: Mlti-variable infinite horizon MPC control with rate of change constraints, Q = I, S = I, max =.. Setpoint change. The rate of change constraint becomes binding in both inpts. The slower inpt maniplation in trn affects the speed of response in the otpt. This kind of simlation allows one to see directl what penalt mst be paid in the speed of otpt tracking if one wishes to impose this rate constraint on the inpt. JCI 7 MPC short corse 63 / 7 JCI 7 MPC short corse 64 / 7

17 Smmar of MPC compared to PID Smmar of MPC compared to PID In this part of the corse we have seen that MPC offers significant advantages compared to PID control in the following sitations Difficlt dnamics: delas, right half plane zeros, high order sstems. Tightl constrained sstems: valve satration, rate of change constraints, avoidance of integrator windp. 3 Strongl interacting mltivariable sstems. No need for pairing, and tre mltivariable control. Of corse MPC does not solve all control problems. The biggest disadvantage of MPC, or an model-based control sstem, is the significant effort reqired to obtain a process model and tne the controller to handle a variet of plant conditions withot hman intervention and constant re-tning and maintenance. Given the sccessfl implementation of MPC in the process indstries, however, we can conclde that this modeling effort can pa off in improved profitabilit, prodct qalit and safet. The decision abot when to implement MPC verss simple, well-tned PID controllers reqires an assessment of the achievable process improvements verss the and expense of the modeling effort. JCI 7 MPC short corse 65 / 7 JCI 7 MPC short corse 66 / 7 MPC can be sed with nonlinear models Finall, nothing restricts the ideas of MPC to linear models. Practitioners are now prsing the implementation of MPC based on nonlinear, fndamental chemical process models. dx/dt = f (x, ) = g(x, ) The on-line optimization becomes significantl more complex, bt toda s comptational capabilities are alread p to the task of implementation for reasonabl sized process nits. The bottlenecks at this point are, again, obtaining nonlinear process models from first principles and operating data, and algorithms for reliabl solving the nonlinear MPC optimization. JCI 7 MPC short corse 67 / 7 Lab Exercises Solve the examples in this section and reprodce the figres. Exercise. (Rawlings and Mane, 9) Example 4 Define a state vector and realize the following models as state space models b hand. One shold do a few b hand to nderstand what the Octave or Matlab calls are doing. Answer the following qestions. What is the connection between the poles of G and the state space description? For what kinds of G(s) does one obtain a nonzero D matrix? What is the order and gain of these sstems? Is there a connection between order and the nmbers of inpts and otpts? a G(s) = s + b G(s) = c G(s) = s + 3s + (s + )(3s + ) d (k + ) = (k) + (k) e (k + ) = a (k) + a (k ) + b (k) + b (k ) JCI 7 MPC short corse 68 / 7

18 Lab Exercises Frther reading I Exercise. Example 5 Find minimal realizations of the state space models o fond b hand in Exercise 4. Use Octave or Matlab for compting minimal realizations. Were an of or hand realizations nonminimal? B. A. Ognnaike and W. H. Ra. Process Dnamics, Modeling, and Control. Oxford Universit Press, New York, 994. S. J. Qin and T. A. Badgwell. A srve of indstrial model predictive control technolog. Control Eng. Pract., (7): , 3. J. B. Rawlings and D. Q. Mane. Model Predictive Control: Theor and Design. Nob Hill Pblishing, Madison, WI, pages, ISBN D. M. Starks and E. Arrieta. Maintaining AC&O applications, sstaining the gain. In Proceedings of National AIChE Spring Meeting, Hoston, Texas, April 7. JCI 7 MPC short corse 69 / 7 JCI 7 MPC short corse 7 / 7