An MPC Approach to Transient Control of Liquid-Propellant Rocket Engines

An MPC Approac h to T ransien t Con trol of Liquid-Prop ellan t Ro c k et Engines Sergio P´ erez-Ro ca ∗ , ∗∗∗ Julien Marzat ∗ ´ Emilien Fla yac ∗ H´ el` ene Piet-Lahanier ∗ Nicolas Langlois ∗∗ F ran¸ cois F arago ∗∗∗ Marco Galeotta ∗∗∗ Serge Le Gonidec ∗∗∗∗ ∗ DTIS, ONERA, Universit ´ e Paris-Saclay, Chemin de la Hunier e, 91120 Palaise au, F r anc e (e-mails: { ser gio.p er ez r o c a, julien.marzat, emilien.flayac, helene.piet-lahanier } @oner a.fr) ∗∗ Normandie Universit´ e, UNIR OUEN, ESIGELEC, IRSEEM, R ouen, F r anc e (e-mail: nic olas.langlois@esigele c.fr) ∗∗∗ CNES - Dir e ction des L anc eurs, 52 Rue Jac ques Hil lair et, 75612 Paris, F r anc e (e-mails: { fr anc ois.far ago, mar c o.gale otta } @cnes.fr) ∗∗∗∗ A rianeGr oup SAS, F or ˆ et de V ernon, 27208 V ernon, F r anc e, (e-mail: ser ge.le-gonide c@ariane.gr oup) Abstract: The curren t context of launc hers reusability requires the impro v ement of con trol algorithms for their liquid-prop ellan t ro c ket engines. Their transien t phases are generally still p erformed in op en lo op. In this pap er, it is aimed at enhancing the control performance and robustness during the fully contin uous phase of the start-up transient of a generic gas-generator cycle. The main con trol goals concern end-state tracking in terms of combustion-c ham b er pressure and c hambers mixture ratios, as w ell as the v erification of a set of hard op erational constrain ts. A con troller based on a nonlinear preprocessor and on linear MPC (Model-Predictive Control) has b een syn thesised, making use of nonlinear state-space models of the engine. The former generates the full-state reference to be trac k ed while the latter achiev es the aforemen tioned goals with sufficient accuracy and v erifying constrain ts for the required pressure lev els. Robustness considerations are included in the MPC algorithm via an epigraph formulation of the minimax robust optimisation problem, where a finite set of p erturbation scenarios is considered. Keywor ds: Liquid-prop ellan t ro c ket engines, mo del predictiv e and optimisation-based control, trac king, control of constrained systems, robustness, transients. 1. INTR ODUCTION In the current con text of launc her vehicles design, reusabil- it y is considered as a ma jor factor. F rom the automatic con trol p ersp ectiv e, the p oten tial need for reusable liquid- prop ellan t rock et engines (LPRE) implies stronger robust- ness requirements than controlling exp endable ones due to their multi-restart and thrust-modulation capabilities. The classical m ultiv ariable con trol of main-stage LPRE had attained a reduced throttling env elop e (70%-120%) in test b enc hes. At real flights, only the nominal op er- ating point is generally targeted. In the future Europ ean Pr ometheus engine, it is aimed at throttling down to 30% (Baio cco P . and Bonnal C. (2016)). Thus, an enlarged v a- lidit y domain for reusability has to b e conceiv ed. At least, it b ecomes crucial to maintain tracking and robustness at those lo w throttle levels, where physical phenomena are more difficult to an ticipate. The main control problem in these m ultiv ariable systems primarily consists in tracking set-p oin ts in com bustion- c hamber pressure and mixture ratio, whose references stem from launcher needs. Con trol-v alv es op ening angles are adjusted in order to adapt engine’s op erating point while resp ecting some constraints. The most common control approac hes identified in the literature rely on linearised mo dels ab out op erating p oints for synthesising steady- state controllers, most of them based on PID techniques (suc h as Nemeth E. et al. (1991)). Generally , initial MIMO (Multi Input Multi Output) syste ms are considered de- coupled in to dominant SISO (Single Input Single Output) subsystems. Off-line optimisation strategies ha ve also been carried out (Dai X. and Ray A. (1996)). More complex approac hes present in the literature, incorp orating some nonlinear (Lorenzo C.F. et al. (2001)), h ybrid (Musgra ve J.L. et al. (1996)) or robust (Saudemont R. and Le Go- nidec S. (2000)) techniques, enhance certain asp ects of p erformance and robustness. And in the ev en t that a comp onen t fails, some reconfiguration-con trol strategies ha ve b een prop osed (Musgrav e J.L. et al. (1996)). T o the b est of our kno wledge, there are no publications whic h consider not only the steady state but also the demanding transien t phases at the same level of p erfor- mance and robustness, as reviewed in P´ erez-Ro ca S. et al. (2019). Pre-defined sequences of engine op eration (start- up and sh utdown), are traditionally managed in op en loop with narrow correction margins. They consist in an initial succession of discrete even ts including v alves op enings and cham b ers ignitions. Once these commands hav e all b een executed, the second part of the transient, which is completely contin uous, takes place un til the steady state is achiev ed. The main reasons for p erforming op en-loop (OL) control in the first (discrete-even t) phase, explained in Nemeth E. et al. (1991), are controllabilit y and observ- abilit y issues at v ery low mass flows. T ransient control through v alves starts to be plausible once all even ts ha ve finished. This observ ation has been considered in this pa- p er, where only the second part of the start-up transient, fully con tinuous, is controlled. The ob jectiv e of this work is to con trol the start-up transien t of a pump-fed LPRE. In this case, a gas- generator-cycle engine is targeted. Concretely , it is aimed at achieving combustion-pressure and mixture-ratio end- state tracking while complying with hard op erational con- strain ts, mainly concerning mixture ratios, turb opumps rotational speeds and v alves actuators angular velocities. The control strategy presented in this pap er is based on Mo del Predictiv e Con trol (MPC), whic h is accompanied b y a prepro cessor for full-state reference generation. This metho d was selected as the most appropriate for this kind of complex systems with hard op erational constraints, as in tro duced in the next sections. Indeed, it is more and more used in industry and can be extended for instance with robustness (Mayne D.Q. et al. (2000)) or hybrid considerations, whic h will b e in teresting for future work on this topic. This pap er is organised as follows. Section 2 serves as a recapitulation of mo delling considerations published in a previous pap er. The state-space system used in the follo wing sections is presented there. Section 3 describ es the control strategy carried out, whic h mainly consists in the use of MPC tec hniques. Section 4 depicts results and includes their analysis. Finally Section 5 serves as a conclusion. 2. MODELLING The mo delling strategy used in this pap er was introduced in P´ erez-Roca S. et al. (2018). Several t yp es of mo dels are emplo yed in the control loop in this paper. Concerning the plan t on which the con trol is exerted, a simulator of the real plan t was constructed in first place. This sim ulator, whose structure is built component-wise, already considers the basic thermo-fluid-dynamics and mechanics of LPRE elemen ts: mass, energy and momen tum conserv ation equa- tions. The engine considered in this paper, represen tativ e of the V ulc ain 1 , presen ts a gas-generator (GG) cycle. In Fig. 1, its main comp onen ts are depicted and the main acron yms are summarised. It consists in a LO X/LH 2 (liquid oxygen as o xidiser, liquid hydrogen as fuel) en- gine. The hot-gas flo w necessary to drive turbines comes from a GG, a small combustion cham ber that receives a small p ortion of the main propellant flo w. The actuators considered in this pap er are five con tinuously-con trollable v alv es (VCH, VCO, VGH, VGO and VGC). Apart from those, there are tw o discrete actuators: one binary igniter ( i C C ) and one binary starter ( i GG ). How ev er, they are considered as active in this pap er in order to treat the con tinuous part ot transients (up from 1 . 5 s after start command), where only con tinuous con trol tak es place. The GG starter injects hot gas in to that cavit y during less than 1 . 5 s so as to start driving turbines. V alves angles ( α ), which hav e a nonlinear but direct relation to sections Fig. 1. V ulc ain 1 simplified flow plan ( A ), control the flows to the main combustion c hamber (V CH and VCO), to the GG (VGH, V GO), and to the o xidiser turbine (V GC). The latter consists in the main con tribution in determining mixture ratio ( M R ), which is defined as the quotient betw een o xidiser and fuel mass flo w rates M R = ˙ m ox / ˙ m f u . This ratio, a ma jor p erformance indicator in LPRE, is established at three levels: at an engine’s global lev el ( M R P I ), taking pumped prop ellants in to account; in the combustion c hamber ( M R C C ) and in the GG ( M R GG ). The simulator was then translated into a nonlinear state- space model b y joining components equations symboli- cally . A t this stage, having already carried out certain sim- plifications with respect to the initial sim ulator, the mo del is referred to as c omplex NLSS (nonlinear state-space) or f c ( x , u ). Ho wev er, this mo del w as to o cumbersome for con- trol design. Hence, it w as further reduced until achieving the here-called simplifie d NLSS such that ˙ x = f s ( x , u ), more tractable for its manipulation and deriv ation of con- trol la ws. The cost of these simplifications, such as the consideration of constant thermodynamic prop erties, is the increase of modelling error. A difference in comparison with P´ erez-Ro ca S. et al. (2018) is the consideration of ca v- it y temperatures as correlated functions of the respective M R , thereby shortening the state vector. Besides, in this pap er, all equations, states and con trol ha v e been rendered non-dimensional with respect to the nominal equilibrium v alues. In terms of notation, the presence of a tilde ( ˜ ) on top of a quantit y means that it is dimensional and its absence means the con trary . The num b er of states is n = 12 and m = 5 is the num b er of control inputs. Here, the state v ector x , of b oth NLSS, comprises the tw o turb opumps sp eeds ω H and ω O , the four pressures in the system ( p C C of combustion c hamber, p GG of the GG, p LT H for hydrogen-turbine inlet cavit y and p V GC for o xygen-turbine inlet ca vity) and six mass flo ws, including the ones streaming through control v alv es ( ˙ m V C H , ˙ m V C O , ˙ m V GH , ˙ m V GO and ˙ m V GC ) and the one streaming through the hydrogen-turbine inlet pip e ˙ m LT H . x = [ ω H ω O p C C p GG p LT H p V GC ˙ m LT H ˙ m V C H ˙ m V C O ˙ m V GH ˙ m V GO ˙ m V GC ] T . (1) The states with greater tracking relev ance are incorp o- rated in to a reduced state vector x z : x z = [ p C C ˙ m V C H ˙ m V C O ˙ m V GH ˙ m V GO ] T . (2) The control input u con tains the sections of the fiv e con trol v alv es: u = [ A V C H A V C O A V GH A V GO A V GC ] T . (3) Mo delling error is sp ecially present in mass flows, which can presen t a mismatch of 10 to 25% at each step of simplification (simulator, f c , f s and linearised mo dels). Errors in the other states are generally b elo w 10% at eac h step. The state is assumed to b e fully measurable in the real engine. This is a realistic assumption for ω and p . Ho wev er, measuring some mass flo ws would be problematic in terms of engine design. Mass flows are generally not measured in LPRE, but estimated through pressure, temp erature and v olumetric flo w measurements. This estimation pro cess is deemed p erfect in this pap er. 3. CONTR OLLER DESIGN The goal of the con troller is to drive the state to w ards a desired reference x r at the end of the start-up transient, with a sp ecial fo cus on having a small tracking error in x z . At the same time, a set of hard constrain ts on x and u has to b e met throughout the transien t. This second goal is somewhat more imp ortan t than the former in order to a void excessive temp eratures, p or ω during engine’s op er- ation. The duration of the start-up transient until reac hing the reference is required to range b et ween 2 and 4 s , whic h allo ws the system to cop e with p ossible perturbations or uncertain ties while complying with constrain ts. A concrete reference tra jectory is not imp osed. The designed controller comprises tw o main parts. The main comp onen t is the MPC block, which receives a full- state reference and drives the system to it while satisfying constrain ts. The other part of the controller consists in a prepro cessing blo ck, whic h serves to generate the full reference vector x r fed to the MPC. The whole control diagram is depicted in Fig.2. The remaining elements in that diagram are the following. T o the right there is the sim ulation of the ro c k et engine (complex sim ulator), run at 10 − 5 s for capturing fast dynamics and for b eing robust to n umerical stiffness. The inputs of that simulator and of the state-space mo del used for control are v alv e sections u . Ho w ever, the actuators mo del (in ternal v alve actuators) requires an input in terms of α . That is the reason why there is a conv ersion blo c k, c haracterised by static and monotone nonlinear functions. The MPC controller pro- vides v alve sections that are then translated into angles. The cause for considering v alv e actuators as a separate en tity is the fact that they represent an internal serv o- lo op, in which the angular p osition of the v alve is tuned b y means of a hydraulic or electrical actuator, modelled as a second-order system. 3.1 Pr epr o c essing The prepro cessing blo ck serv es as an off-line reference generator for the MPC controller. This is required b ecause the set of reference commands derived from launcher needs is not sufficient to provide a complete-state target equilibrium p oin t to the engine controller. Hence, a w ay of restoring a full state v ector from those data is necessary . In addition, without this x r , the p osterior MPC controller w ould not attain the trac king goal with high precision. This is mainly due to the fact that f s is linearised ab out x r , as explained in Section 3.2. As said b efore, x presen ts t welv e states, and there are four reference inputs: p C C,r , M R P I ,r , M R C C,r and M R GG,r . Moreo ver, the last three, in contrast to p C C , do not directly corresp ond to states in the mo del. They establish relations b et w een ˜ ˙ m i . In a first place, thanks to the selected pressure and M R C C,r , the c hoked-flo w static equation ˙ m C C = p C C A th C ∗ (4) can provide ˙ m V C H and ˙ m V C O , where A th is the throat area and C ∗ is the characteristic velocity , dep endent on M R itself. Then, the rest of states at equilibrium are computed by solving the following ov erdetermined system of nonlinear equations:                ˙ x = f c ( x r , u r ) = 0 \ ( ˙ p C C = 0) ˜ ˙ m V C O + ˜ ˙ m V GO ˜ ˙ m V C H + ˜ ˙ m V GH = M R P I ,r ˜ ˙ m V GO ˜ ˙ m V GH = M R GG,r ˜ ˙ m V GH + ˜ ˙ m V GO = ˜ ˙ m LT H + ˜ ˙ m V GC . (5) The first equations force the ODE to be at equilibrium, the second and the third ones determine the M R and the last one enforces the equilibrium of P ˜ ˙ m i in the GG. The ODE for ˙ p C C is remov ed since it is completely dep endent on the reference inputs, not providing additional information. This resolution is p erformed numerically via nonlinear least squares due to the una v ailability of an analytic solution of the system, of either f c or f s . The complex mo del has b een chosen to increase accuracy . 3.2 MPC algorithm MPC predicts the future system b ehaviour along a hori- zon, and optimises con trol inputs according to a cost function generally related to a reference tra jectory or to an end state. In this pap er, the dynamic mo del used in the state-feedback MPC controller is considered as a linearisation of f s ab out the previously computed ( x r , u r ) and as a zero-order hold discretisation at ∆ t = 10 ms (due to computational constrain ts): ∆x k +1 = A d ( x r , u r ) ∆x k + B d ( x r , u r ) ∆u k (6) Th us, in linear terms, the goal of the con troller is to find the set of ∆u = u − u r that driv es the state to ∆x = x − x r = 0 . The matrix A d is stable for all the physically feasible x r , which is a particularity of GG-cycle LPRE. In this MPC section, in order to lighten notation, all x and u refer to v ariations with resp ect to the equilibrium p oin t. The approach carried out is partially based on the quasi-infinite horizon (QIH) approac h by Chen H. and Allgo ew er F. (1998), b ecause it presents pro ofs for guaran teed stability and end-state reachabilit y of MPC b y incorp orating the notion of a terminal region. The MPC driv es the system to that region, where a fictitious lo cal controller K p erforms the precise tracking at the end of the state prediction horizon, N p + 1. Ho wev er, in MPC only the first computed control, u M P C ≡ u 1 , is transmitted to the plant. Hence, the real role of the fictitious feedback u N p +1 = K x N p +1 is to compute the P Fig. 2. Con trol-lo op diagram matrix of a Lyapuno v function V ( x ) = x T P x , b y solving the follo wing Lyapuno v equation: ( A K + κI ) T P + P ( A K + κI ) = − Q K − K T R K K. (7) In (7), the comp ound of the linear system with a simple LQR feedback controller is considered, A K = A c + B c K (where A c is the con tinuous coun terpart of A d ), κ ∈ R + (satisfying κ < − λ max ( A K )) and Q K and R K are p ositiv e definite symmetric matrices Q K ∈ R n × n , R K ∈ R m × m . The computed P ∈ R n × n serv es to add an additional terminal-region term in the MPC cost. In addition, an inte- gral action is also included to enforce a more precise trac k- ing on x z . Those in tegral decision v ariables are denoted b y z and pres en t a corresp onding weigh t matrix S ∈ R n z × n z in the cost, whose diagonal is [1 , 0 . 1 , 0 . 1 , 0 . 1 , 0 . 1]. Th us, the MPC cost J is defined as: (8) J ( x , u , z ) =   N p X k =1 x T k Q x k + N u X k =1 u T k R u k + N p X k =1 z T k S z k   ∆ t + x T N p +1 P x N p +1 , whic h consists in the traditional quadratic cost on states and controls plus the integral and terminal costs, with a prediction horizon N p = 10 steps (0 . 1 s ) and a con trol horizon N u = 5. Implicitly , the last control u N u is used for k ≥ N u . F urther extensions of these horizons did not im- pro ve the solutions in terms of tracking or constraints sat- isfaction. Q and R are p ositiv e-definite symmetric w eight- ing matrices Q ∈ R n × n , R ∈ R m × m , whose diagonals ha ve b een computed off-line via Kriging-based black-box optimisation as in Marzat J. et al. (2010). The criterion for that weigh t selection concerns the minimisation of static error and o vershoot in simulations. F urthermore, the first steps to w ards a robust consideration of the problem hav e b een implemen ted. The minimisation of the previous J under constraints is not robust. Indeed, robustness to parameters and initial conditions v ariations, p erturbations and modelling error is v ery imp ortan t in this application. Robust MPC approaches generally mak e use of the minimax optimisation, which minimises the w orst- case scenario. A generic expression of this problem is the follo wing, in which w represents disturbance (Ma yne D.Q. et al. (2000)): min u max w J ( x , u ) s.t. x ∈ X ∀ w ∈ W n u ∈ U ∀ w ∈ W n (9) Ho wev er, solving (9) for all p ossible p erturbations is to o computationally costly for this application. Hence, it has b een opted for choosing a finite set of disturbance scenarios (in a similar manner to Calafiore G.C. and F agiano L. (2013)) and for solving an equiv alen t formulation based on Lo efberg J. (2003). Concretely , it consists in minimising γ ∈ R + via an epigraph formulation. In this pap er, that γ constrains the J of the original problem ev aluated at sev eral p erturb ed states propagations x i : x i = [ x i, 1 , ..., x i,k , ..., x i,N p +1 ] T , i ∈ I x i,k +1 = A d x i,k + B d u k + w i,k , k ∈ [0 , N p + 1] , (10) where w i,k are certain selected p erturbation v ectors be- longing to W = { w i,k , i ∈ I , k ∈ [0 , N p + 1] } . I is a finite set, which serv es to index the considered p erturbation cases. Indeed, the epigraph form ulation allo ws to en tirely shift the robustness considerations in to the list of con- strain ts. Therefore, only a smo oth conv ex nonlinear pro- gramme (NLP) is required, whic h is more computationally tractable than (9). The minimisation problem prop osed here, in which decision v ariables are extended to consider all x i , is describ ed b elo w: min x i , u , z i ,γ γ (11) s.t. J ( x i , u , z i ) ≤ γ ∀ i ∈ I x i ∈ X, u ∈ U ∀ i ∈ I A ineq [ x i u ] T ≤ b ineq ∀ i ∈ I A eq [ x i u ] T = b i,eq ∀ i ∈ I x T i,N p +1 P x i,N p +1 ≤ α P ∀ i ∈ I z i,k +1 = z i,k + ∆ tK I x z ,i,k ∀ i ∈ I , k ∈ [0 , N p ] . X and U are the allo wable sets for states and control (com- pact subsets of R n ( N p +1) and R mN u resp ectiv ely). The set U for the first con trol u M P C is specially constrained to comply with actuators capacit y (Luo Y. et al. (2004)): u M P C ∈ [max( U , u 0 − ˙ u max ∆ t ) , min( U , u 0 + ˙ u max ∆ t )] , where u 0 is the previous-step con trol (w arm start is per- formed) and ˙ u max is the maximum sectional velocity of v alv es. Regarding the rest of constraints, (11) con tains equalit y constrain ts (defined b y A eq and b i,eq ) for lin- ear dynamics (10) and also linear inequality constraints (defined b y A ineq and b ineq ), for complying with M R and actuators sectional-velocity b ounds at all x i . In re- lation to the terminal region, a constan t α P refers to the neigh b ourho od in which the Ly apunov term of J is constrained in a nonlinear w ay (further details on the QIH metho d in Chen H. and Allgo ew er F. (1998)). The differen t w i,k represen t the v arious wa ys in which the system can ev olve after an unknown perturbation or uncertaint y in the state, and hence it is proposed to estimate them b y analysing the mo des of the system (eigenv ectors of A c ). The total num ber of p erturbation cases I = { 1 , 2 , 3 } corresp onds to a subset of the eigenv ectors. In this man- ner, the structural information of A c is used to define unfa vourable disturbance scenarios, similarly to Y eda v alli R.K. (1985). The mo dulus of the vectors w i,k is kept equal to 0 . 1. It is important to emphasise the fact that the resulting u obtained in (11) has b een confronted to all these p erturbation scenarios and that all propagated p erturbed states m ust comply with all constrain ts, thereby impro ving the robustness of the controller. This approac h with equality constraints within an uncertain problem is only v alid b ecause of the finite choice of w i,k . The last line in constraints corresponds to the integrator dynamics (San tos L.O. et al. (2001)), where K I is a gain matrix computed off-line in the same manner as Q and R . 4. ANAL YSIS OF RESUL TS The interior-point optimisation softw are IPOPT (W aec hter A. and Biegler L.T. (2006)) has b een used to solv e this smo oth conv ex NLP within the MA TLAB environmen t. Sim ulations of the previously presented control lo op are run from 1 . 5 s until 3 s after the start command, that is to sa y , during the time windo w in whic h con tinuous control is p ossible in engine start-up transient. Mixture ratios nat- urally start from v alues very far from the allow able area, due to the low initial mass flows that hinder the definition of quotients. Indeed, c hambers are not physically ignited during the first instants (even if igniters are active); hence, M R are not relev ant there. Fig. 3 depicts the results of p C C trac king for three op erating points: p C C,r = 1 (nominal), p C C,r = 0 . 7 (minimum for this engine) and p C C,r = 1 . 2 (maxim um). At all three points, the reference mixture ratios remain the same M R C C,r = 6, M R GG,r = 1 and M R P I ,r = 5 . 25. M R tracking for the nominal case is depicted in Fig. 4. T racking is achiev ed with sufficien t accuracy in p C C for 1.5 2 2.5 3 Time [s] 0 0.5 1 1.5 Pressure [-] Combustion-chamber pressure tracking p CC,nom p CC,min p CC,max p CC,ref,nom p CC,ref,min p CC,ref,max Fig. 3. T racking results in p C C for p C C,r = 1 (nominal), p C C,r = 0 . 7 (minim um) and p C C,r = 1 . 2 (maxim um) all cases (under 0 . 7%) and with little error in M R (under 0 . 3% in nominal, under 1 . 7% in off-nominal) while resp ect- ing constraints up from the time when it is considered feasible and acceptable to resp ect them in practice (1 . 9 s ). The o vershoot and oscillations present b efore achieving the final trac king are generated by the exogenous influence of the GG-starter input mass flo w, which is not taken into accoun t in the linearised mo del. Ov ersho ot is more pro- nounced in the minimum case since the relativ e influence of the starter is more elev ated. The con troller is able to ac hieve that trac king performance after random initial conditions coming from the sequen- tial transient, whereas constraints-v erification time oscil- lates some hundredths of seconds. Computational times in MA TLAB are of the order of ten times longer than real time, which do es not rule out a future real-mac hine implemen tation. 4.1 Comp arison with op en lo op and other line ar c ontr ol lers T able 1 summarises the comparison b et ween this closed- lo op (CL) prop osal and OL sim ulations in terms of some p erformance indicators. The nominal OL is engine’s orig- T able 1. Performance-indicators comparison b et w een this CL prop osal and OL at the three selected op erating p oin ts Op erating p oin t Nominal Minim um Maximum Indicator OL CL OL CL OL CL Settling time (99%) [ s ] 2.8 2.51 2.67 2.55 2.69 2.53 Overshoot (% in p C C ) 6.31 5.04 15.1 11.46 3.34 4.04 Constraints verification [ s ] 1.81 1.8 1.83 1.76 1.77 1.81 p C C static error (%) 0.25 0.26 2.8 0.26 0.34 0.67 M R C C static error (%) 0.17 0.01 2.58 1.38 3.18 1.37 M R GG static error (%) 1.39 0.05 1.31 0.69 1.23 0.59 M R P I static error (%) 1.43 0.3 2.84 0.85 3.41 1.64 inal command, whic h is precisely tuned for the standard case. The minim um and maximum OL commands hav e b een computed by means of the prepro cessor explained in Section 3.1. The impro vemen t with resp ect to OL is not dramatic, it is even w orse in some indicators. Nonetheless, the real gain of this CL MPC control appears for op erating p oin ts differen t from the nominal, where multiv ariable trac king was difficult to achiev e with high p erformance while resp ecting constraints during the transient. Moreo ver, other linear con trol methods hav e b een tested on the same plant, such as simple PID and LQR con- trollers. T racking results of these con trollers are go od in 1.5 2 2.5 3 Time [s] 0 2 4 6 8 10 Mixture ratio [-] Mixture-ratios tracking and constraints MR CC,max MR GG,max MR min MR CC,nom MR GG,nom MR PI,nom MR CC,ref MR GG,ref MR PI,ref Fig. 4. T racking results in M R for p C C,r = 1 (nominal) 1.5 2 2.5 3 Time [s] 0 0.5 1 1.5 Rotational speed [-] Shaft rotational speeds H ,max O,max H,MPC O,MPC H,PID O,PID H,LQR O,LQR Fig. 5. Rotational sp eeds ω H and ω O for p C C,r = 1 . 2 with MPC, PID and LQR con trollers some of the reference v ariables (under 0 . 0001% in p C C ), but not for all of them simultaneously . Moreov er, there are no guarantees of complying with all the constraints in this problem. Hence, when aiming at tracking off- nominal points, constrain ts are indeed highly violated. F or instance, while throttling up until p C C = 1 . 2, the system con trolled by PID or LQR has the tendency to surpass rotational sp eeds b ounds, as depicted in Fig. 5, whereas MPC resp ects them. 5. CONCLUSION The control of the transien t phases of liquid-prop ellan t ro c k et engines has traditionally b een p erformed in open lo op due to its highly nonlinear b ehaviour. This work has sough t to impro ve the control of the fully con tinuous part of the start-up of a gas-generator-cycle LPRE, whose v alves can b e adjusted for controlling pressure in the main c hamber and mass-flo w mixture ratios. An MPC con troller has b een synthesised on that phase for tracking com bustion-cham b er pressure and mixture ratios while resp ecting a set of hard op erational constrain ts. This con troller is accompanied b y a prepro cessor that serves to pro vide a full-state reference built from launcher needs, b y making use of a nonlinear state-space mo del of the engine. The linear MPC con troller with integral action is able to track that end-state reference with sufficient accuracy and constrain ts are respected when necessary . Robustness, vital in this application with p ossible p erturbations and in ternal-parameter v ariations, is tak en into account for a given set of p erturbation scenarios. The costly nested minimax optimisation of typical robust MPC approaches has b een rewritten as the minimisation of a scalar cost. In future work, other wa ys of p osing this robustness consideration globally will b e inv estigated. The tracking of a predefined tra jectory will also b e studied. A more extensiv e v alidation study with resp ect to p erturbation cases will b e carried out. REFERENCES Baio cco P . and Bonnal C. (2016). 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