Explicit model predictive control of gas-liquid. separation plant via orthogonal search tree. partitioning Trondheim, Norway

Transcription

1 Explicit model predictive control of gas-liquid separation plant via orthogonal search tree partitioning Alexandra Grancharova,,, Tor A. Johansen, 3, Juš Kocijan Department of Engineering Cybernetics, Norwegian University of Science and Technology 749 Trondheim, Norway Department of Systems and Control, Jozef Stefan Institute, Jamova 39 Ljubljana, Slovenia Nova Gorica Polytechnic, Vipavska 3, 5 Nova Gorica, Slovenia,, 4 Present address: Institute of Control and System Research, Bulgarian Academy of Sciences, Acad. G. Bonchev str., Bl., P.O.Box 79, Sofia 3, Bulgaria Corresponding author: alexandra@icsr.bas.bg 3 Tor.Arne.Johansen@itk.ntnu.no 4 jus.kocijan@ijs.si of 3

2 Abstract Solutions to constrained linear model predictive control (MPC) problems can be pre-computed off-line in an explicit form as a piecewise linear (PWL) state feedback defined on a polyhedral partition of the state space. This admits implementation at high sampling frequencies in real-time systems with high reliability and low software complexity. Recently, algorithms that determine an approximate explicit PWL state feedback solution by imposing an orthogonal search tree structure on the partition, have been developed, and it has been shown that they may offer computational advantages. This paper considers the application of an approximate approach to the design of an explicit model predictive controller for a two-input two-output laboratory gas-liquid separation plant, including experimental evaluation. The approximate explicit MPC controller achieves performance close to that of the conventional MPC, but requires only a fraction of the real-time computational machinery, thus leading to fast and reliable computations. Keywords: Constrained linear model predictive control; Multi-parametric quadratic programming; Gasliquid separator of 3

3 . Introduction. Model predictive control (MPC) has become the accepted methodology to solve complex control problems related to process industries (Garcia et al., 987; Biegler & Rawlings, 99; Mayne et al., ). It allows the design of multi-input multi-output (MIMO) control systems that minimize a quadratic performance index in the presence of input and output constraints imposed on the system. Recently, several methods for explicit solution of MPC problems have been developed. The main motivation behind explicit MPC is that an explicit state feedback law avoids the need for real-time optimization, and is therefore potentially useful for applications where MPC has not traditionally been used. In addition to embedded system applications, this technology is also suitable for certain small-scale process control applications. In particular, low-level control of fairly simple unit processes is typically implemented using conventional PI/PID control rather than MPC. Some reasons are the need for high reliability, fast processing and low software complexity, as these are typically executed in a highly reliable computer environment. On the other hand, explicit MPC is ideally suited for such problems as it offers the benefits of constrained MIMO control with low complexity and high reliability for such smallscale problems. In (Bemporad et al., ; Pistikopoulos et al., ) it was recognized that the constrained linear MPC problem can be posed as a multi-parametric quadratic program (mp-qp), when the state is viewed as a parameter to the problem. It was shown that the solution (the control input) has an explicit representation as a piecewise linear (PWL) state feedback on a polyhedral partition of the state space, see also (Bemporad et al., ; Johansen et al., ; Seron et al., ; Tøndel et al., 3), and they develop an mp-qp algorithm to compute this function. Recently, in (Johansen & Grancharova, ; Johansen & Grancharova, 3; Grancharova & Johansen, ), algorithms that determine an approximate explicit PWL state feedback solution by imposing an orthogonal search tree structure on the partition, have been developed. It has been shown that they may lead to more efficient real-time computations. The present paper considers the application of these algorithms to the design of an explicit model predictive controller for a laboratory gas-liquid separation plant, including experimental evaluation. A preliminary version of this paper is given in (Grancharova et al., 3). 3 3 of 3

4 . Approximate approach to explicit MPC... Explicit MPC and exact mp-qp. Formulating a linear MPC problem as an mp-qp is briefly described below, see (Bemporad et al., ; Pistikopoulos et al., ) for further details. Consider the linear system: where n x ( t), n m B, C u ( t) p n m, and x( t + ) Ax( t) + Bu( t) y( t) Cx( t) y( t) p are the state, input and output variable. Also, () n n A, and (A, B) is a controllable pair. It is assumed that a full measurement of the state x(t) is available at the current time t. Then, for the current x (t), MPC solves the optimization problem: subject to: with the cost function given by: V y u x x y t t ( x( t)) min J ( U, x( t)) () U { u,..., u } min min t+ k t x( t) t+ k+ t y u t+ k t t+ k Ax Cx t+ k t t y u t+ k t max max t + N, k, k,..., N, k,,..., N + Bu t+ k, k T [ ] k T T x t+ tqxt+ k t + u t+ k Rut+ k + x t+ N t t N t N J ( U, x( t)) Px + k and symmetric R >, Q, P >. The column vector T T T s U ( u t,..., ut+ N ), s mn (3) (4), is the optimization vector and the final cost matrix P may be taken as the solution of the discrete-time algebraic Riccati equation (Ogata, 995): With the assumption that no constraints are active for T T T T T P A PA A PB( B PB + R) ( A PB) + Q (5) k N criterion, and the MPC is stabilizing (Chmielewski & Manousiouthakis, 996). It is shown in (Bemporad et al., ) that by substituting: this corresponds to an infinite horizon LQ k k j x t+ k t A x( t) + A But+ k j (6) j in the optimization problem defined by (), (3) and (4), this can be rewritten in the form: 4 4 of 3

5 V ( x( t)) x T T ( t) Yx( t) + min U HU + x U GU W + Ex( t) subject to T ( t) FU (7) Here, H H T >, and H, F, Y, G, W, E are easily obtained from Q, R, and () (6) (since only the optimizer U is needed, the term involving Y is usually removed from (7)). Further, by defining z T U + H F x( t), problem (Bemporad et al., ): s z, the optimization problem (7) is transformed into the following equivalent T T where V z ( x) V ( x) x ( Y FH F )x and S T V z ( x) min z Hz z (8) subject to Gz W + Sx F T E + GH. The vector x is the current state, which can be treated as a vector of parameters. The number of inequalities is denoted q and the number of free variables is s mn. Then s s H, q s G, q W, S q n. It has been shown in (Bemporad et al., ; Pistikopoulos et al., ) that the optimization problem (8) is a multiparametric quadratic program (mp-qp) and its solution can be found in an explicit form z z (x) as a PWL function of x defined over a polyhedral partition of the parameter space. Algorithms for iteratively constructing a polyhedral partition of the state space and computing the PWL solution are given in (Bemporad et al., ; Pistikopoulos et al., ; Bemporad et al., ; Johansen et al., ; Seron et al., ; Tøndel et al., 3)... Approximate mp-qp algorithm. In (Johansen & Grancharova, ; Johansen & Grancharova, 3), an algorithm that determines an approximate explicit PWL state feedback solution of possible lower complexity is developed. The idea is to require that the state space partition is represented as a binary search tree (quad-tree (Bentley, 975), cf. Fig. ), i.e. to consist of orthogonal hypercubes organized in a hierarchical data-structure. This allows extremely fast real-time search. When searching the tree, only n scalar comparisons are required at each level. In this paper, an improved version of the approximate mp-qp algorithm (Johansen & Grancharova, ; Johansen & Grancharova, 3) is used which is based on a k - d tree partition of the 5 5 of 3

6 state space (Fig. ) as a more flexible and powerful alternative to the generalized quad-tree (Fig. ). With the k - d tree (Bentley, 975), a hyper-rectangle is split into two equal parts and thus only one scalar comparison is required at each level when searching the tree. Also, the k - d tree allows the incorporation of heuristic rules that split the hyper-rectangle at the axis along which the change of error is maximal (before splitting). The approximate mp-qp algorithm is based on the cost function approximation error defined as: ( x) Vˆ ( x) V ( x) (9) where ˆ T T Vz ( x) zˆ ( x) Hzˆ ( x) and Vz ( x) z ( x) Hz ( x) are respectively the sub-optimal and the optimal costs, and zˆ ( x) and z (x) are the sub-optimal and the optimal PWL solutions. The main idea of the algorithm is to compute the solution of the problem (8) at the n vertices of a considered hyperrectangle X by solving up to z z n QPs. Based on these solutions, a feasible local linear approximation zˆ ( x) to the PWL optimal solution z (x), valid in the whole hyper-rectangle X, is computed by using the following result (Bemporad & Filippi, ): Lemma. Consider the bounded polyhedron f n X X f with vertices { v v,..., v M} feasible set: X { x( t) Usatisfying(3) }). If K and g solve the QP: subject to: K, g M T ( z ( v i ) K v i g ) H( z ( v i ) K v i g ) min G i ( K v + g ) S v + W, i {,,..., M}, (here X f is the () i i () then the least squares approximation z ˆ ( x) K x + g is feasible for the mp-qp (8) for all x X. If the maximal approximation error in the hyper-rectangle X is smaller than some prescribed tolerance >, no further refinement of X is needed. Otherwise, X is partitioned into two hyper-rectangles and the procedure described above is repeated for each of these. The upper bound the approximation error is determined by using the method proposed in (Johansen & Grancharova, 3). of In order to reduce the complexity of the partition, the heuristic rule described in (Grancharova & Johansen, ) is applied when splitting the hyper-rectangle X. The rule attempts to split the hyper- 6 6 of 3

7 rectangle at the axis along which the change of the approximation error is maximal (before splitting), because it is reasonable to hope this is how the largest reduction of the error can be made. It has been shown in (Grancharova & Johansen, ) that the use of such heuristics reduces the complexity of the partition significantly. The complexity is further reduced by implementing control input trajectory parameterization with less degrees of freedom in order to reduce the dimension of the optimization problem (Tøndel & Johansen, ). The most common approach is to pre-determine the time-instants at which the control input ui is allowed to change (input blocking). The following approximate mp-qp algorithm is taken from (Johansen & Grancharova, 3): Algorithm (approximate mp-qp) Step. Initialize the partition to the whole hyper-rectangle, i.e. P { X} unexplored. Step. Select any unexplored hyper-rectangle Step 3. Compute the solution to the QP (8) for x fixed to each of the. Mark the hyper-rectangle X as X P. If no such hyper-rectangle exists, go to step 8. X. If all QPs have a feasible solution, go to step 5. Otherwise, go to step 4. n vertices of the hyper-rectangle Step 4. Compute the size of X using some metric. If it is smaller than some given tolerance, mark X infeasible and explored. Go to step. Otherwise, go to step 7. Step 5. Compute an affine state feedback ẑ using Lemma, as an approximation to be used in X. If no feasible solution was found, go to step 7. Step 6. Compute the error bound in X. If, mark X as explored and feasible and go to step. Step 7. Split the hyper-rectangle X into two hyper-rectangles X and X by applying the heuristic splitting rule from (Grancharova & Johansen, ). Mark them unexplored, remove X from P, add X and X to P, and go to step. Step 8. If necessary, split the hyper-rectangles containing the origin such that ( x) z is optimal everywhere in these hyper-rectangles. Terminate. 7 7 of 3

8 This algorithm will terminate with a PWL function that is an approximation to the PWL exact solution and is defined on an inner approximation a union of hyper-rectangles. X f of the set X f X. The set X f is represented as 3. Model of the gas-liquid separation plant. We consider a sub-process within a semi-industrial installation which is used for reduction of NO x in effluent gases and technological waste water treatment by means of neutralisation with CO contained in flue gases (VranLiM et al., 995). The role of the separation unit is to capture flue gases under low pressure from effluent channels by means of water flow and to carry them over under high enough pressure to the downstream (neutralisation) stage. The separation unit is shown in Fig. 3 (VranLiM et al., 995). The flue gases coming from the effluent channels are pooled by the water flow into the water circulation pipe through the injector I. The water flow is generated by the pump P (water ring). The speed of the pump is kept constant. The pump feeds the mixture of water and gas into the separator R where gas is separated from water. Hence the accumulated gas in R forms a sort of gas cushion with increased internal pressure. Owing to this pressure, flue gas is blown out from R into the next neutralisation unit. On the other side the cushion forces water to circulate back to the reservoir R. The quantity of water in the circuit is constant. If for some reason additional water is needed, the water supply path through the valve V 5 is utilised. The complete process model can be described by the following expressions (VranLiM et al., 995) (the meaning of the variables is given in the Appendix): v V Valve V : K K R [ l/(s bar)] () v if r > v if r < (3) r otherwise v V Valve V : K K R [ l/(s bar) ] (4) v r max if r > r max v if r < (5) r otherwise r.865 (6) max 8 8 of 3

9 The speed limit of valves positions: v max if r > v max v v min if r < v min (7) r otherwise v.66[s ] (8) max v.33[s ] (9) min Air flow through valve V : K p [ l/s] () Water flow through valve V : K p + K ( h h ) [ /s] () W R l Water flow to the separator R : w.644 [ l /s] () Air flow to the separator R : + p p [ /s] (3) The change of water level in R is: The change of water level in R is: The change of air pressure inside R is: The air volume inside R is: dp dt dh dt dh air dt air air 6 l ( w ) K F [m/s] (4) S ( w ) K F [m/s] (5) S [ p ( air ) K F + ( p + p )( w ) K F ][bar/s] (6) V 3 ( h h ) S (.5 ) [m ] V S R (7) h A linearized model (VranLiM et al., 995) can be obtained from the existing non-linear model: $ % p! $ % p! $ % v " " + " #% h A c B c #% h #% v where % p and % h denote the deviations of separator gas pressure p and liquid level h from the steady-state values ( % p p ps, % h h hs ), and % v and % v are respectively the deviations of the two valves positions v and v ( % v v vs, % v v vs ). The elements of matrices Ac and Bc! (8) are computed as follows (VranLiM et al., 995): 9 9 of 3

10 a K V F s $ " p " # + (! ) Ks & K s air ( p + ps ) (9) ) & ps ' pv s where K, s K s a a a b K V F s K S F K S F $ " "# ( p + p ) $ K " "# p s V s $ K s K " "# p s w V s!! K s p K w V s ( R ) F s s ln V Vs! (3) (3) (3) K p K p (33) K b + (34) ( p p ) K p ( R ) F s s V s ln V Vs b (35) b K K p (36) ( R ) F s V s ln V S and Vs denote the steady-state values of K, K and V, respectively. The steady-state pressure on the valve V is marked as p pv s and is equal to: ( h h ) [bar] p + K (37) V s w R By using the above equations, a linear continuous-time model is obtained that corresponds to the following steady state: p s.5 [bar], hs.4 [m], vs.45, vs.746 From the continuous-time model, a linear discrete-time model corresponding to sampling interval T s s is obtained by applying the exact discretization using zero-order hold (the control inputs are assumed piecewise constant over the sampling period) (Ogata, 995). The state and control matrices of the discrete-time model are: (38) $ ! $ A ", B # "! (39) # -.3 of 3

11 The state variables are x % p [bar] and x % h [m], and the control variables are u % v and u % v. The following input and rate constraints are imposed on the valve positions v and v : v, v. 865 (4).33 v.66,.33 v. 66 (4) which by taking into account the steady state values (38) are represented as the following constraints on the control inputs u and u : -.45 u ( t + k).5848, u ( t + k).63, k,,..., N - (4).33T u ( t + k) u ( t + k ).66T, i,, k,,..., N - s i i In order to avoid the steady state offset of the MPC controller, two more states are added to the model (39), which take into account the integral error: x s (43) t + ) x ( t) + T x ( ), x t + ) x ( t) + T x ( ) (44) 3( 3 s t 4 ( 4 s t Thus, the linear discrete-time model of the gas-liquid separation unit becomes: $ ! $ ! " " " A, " -.3 B (45) " T s " " " # Ts # 4. Real-time performance of explicit MPC controller for the gas-liquid separation plant. plant. 4.. Design of approximate explicit MPC controller for the gas-liquid separation The approximate mp-qp approach described in section is applied to design an explicit MPC controller for the gas-liquid separation plant. The controller minimizes the cost function (4) subject to the system equation (45) and the input constraints (4). In (4), P is chosen as the solution of the discrete-time algebraic Riccati equation and the cost matrices are: Q diag{.5,,.5,.}, R diag{, } (46) The horizon is N 5 and the time instants at which the input variables can change are: of 3

12 N u [ ] (47) N u [ ] (48) which makes totally 36 optimization variables. With this MPC problem, numerous tests showed that the rate constraints (43) almost never become active and there are no practical benefits of including them in the problem formulation. Also, in order to limit the complexity of the explicit MPC by keeping the dimension of the state space low, the rate constraints were only ensured via rate limiters rather than directly included in the optimization problem. The state space to be partitioned is 4-dimensional and is defined by X [.5,.5] [.,.] [ 3,3] [, 6]. The size of the regions on each of the state variables is restricted to be larger than R x., R x. 4, R x 3. 6 and R x The approximation tolerance > is chosen according to the approach in (Johansen & Grancharova, 3) and it depends T.5 on X such that + x Ax T, where x arg min x Ax. The MPC controller has 693 regions in xx its state space partition and 4 levels of search. With one scalar comparison required at each level of the k-d tree, 4 arithmetic operations are required in the worst case to determine which region the state belongs to. Totally, 4 arithmetic operations are needed in real-time to compute the two control inputs (4 comparisons, 8 multiplications and 8 additions). For comparison, the average computation time with the conventional MPC approach is. sec for.8 GHz Pentium 4 computer, which indicates that millions of arithmetic operations are required in real-time to solve the QP problem (8). 4.. Environment for the real-time experiments. The real-time experiments were pursued in the environment schematically shown in Fig. 4. This environment encompasses supervisory control on two levels: upper level with Factory Link SCADA system and lower procedural and basic control levels implemented in two PLCs. This is one of the possible configurations of control, which can be found in industry. User-friendly experimentation with the process plant is enabled through interface with Matlab/Simulink environment. This interface enables PLC access with Matlab/Simulink using DDE protocol via Serial Communication Link RS3 or TCP/IPv4 over Ethernet IEEE8.3. of 3

13 Control algorithms for experimentation can be prepared in Matlab code or as Simulink blocks and extended with functions/blocks, which access PLC. This interface also enables user-friendly data acquisition for Matlab users. In our case the control schemes were put together as Simulink blocks and tested at the plant operating points as described in the previous section Real-time performance of the explicit MPC controller. In Fig. 5 to 8, the real-time performance of the approximate explicit MPC controller, in closed-loop with the plant is shown. The initial condition is p ().3 [bar], h ().5[m] and the set point is p p.5 [bar], h h.4[m]. The experimental results (the solid line) are compared with the s s exact MPC trajectory (the dotted line) computed by solving the optimization problem at each time instant, based on the process model and with the simulated approximate trajectory (the dashed line). The latter two curves (the dotted curve and the dashed curve) are difficult to distinguish since they are almost matching. It can be seen from the figures that the explicit MPC controller brings the plant to the desired set-point despite of the error in the steady state process model and the transient performance is close to that of the optimal trajectory. The set point changes are handled by using the new set-point values and h when determining the values of the state variables h are the measured variables. p p x and h h x p, where p and It can be seen from Fig. 6 that there is a slight chattering of the signal for the second valve. This can be explained as follows. The signal we depict has come out of analog digital converter which has bit A/D converter resolution that means approximately. % of quantisation noise on the range to which was used in our case. Afterwards this signal has gone through the controller with gain of about which amplified the quantisation noise to about % (as it can be seen in the figure). This is not a problem for the valve and actually even helps beating the small dead zone contained in valve and makes it react faster when the change of control signal occurs. v In Fig. 9 and Fig., the computed approximate PWL feedback laws for valve positions v and are shown, together with the exact control laws computed by solving the QP problem (8) on a grid. The corresponding sub-optimal and optimal cost functions are given in Fig.. It can be noticed from Fig. that for h.5[m ] the cost function makes a jump. This artifact is due to the implementation of 3 3 of 3

14 the anti-windup strategy. The cost function is computed by taking into account also the integral error (cf. equation (44)), which is reset by applying the anti-windup strategy in (Åström & Wittenmark, 997). Table shows statistics about the performance decrease J as well as the errors u and u in the two control inputs with the approximate explicit MPC controller, based on simulations for 36 combinations of initial and target states. The errors J, u and u are computed as the relative deviations (measured in %) of the sub-optimal cost function and control input values (respectively J ˆ( x ) and uˆ i ( x), i, ) from the optimal ones (respectively ( x J ) and u i ( x), i, ): Jˆ( x) J ( x) J ( x) % (49) J ( x) In (5), i i uˆ i ( x) ui ( x) u i %, i, (5) u u max, i min, i u min, and u max,, i, are the lower and upper bounds in the control input constraints (4). It can be seen from Table and from Fig. 5 to that the performance decrease with the approximate explicit MPC is not prohibitive, and the experimental results for initial condition p ().3 [bar], s s h ().5[m] and set point p p.5 [bar], h h.4[m] correspond to a typical amount of sub-optimality. Another approximate explicit MPC controller for the gas-liquid separation plant has been designed by applying the method based on the control input approximation error (Grancharova & Johansen, ). The experiments showed that this controller has performance very similar to that of the approximate explicit MPC controller which minimizes the cost function approximation error (described above). 5. Conclusions. An approximate explicit MPC approach has been experimentally tested on a two-input two-output laboratory gas-liquid separation plant. The approach achieves performance close to that of conventional MPC, but requires only a fraction of the real-time computational machinery, leading to fast and reliable computations. The main limitation of the approach is that the complexity of state space partition tends to 4 4 of 3

15 increase rapidly with the state space dimension, which makes the approach unsuitable for larger problems. A convex nonlinear extension of the approximate explicit MPC approach has recently been reported by Johansen (4). Appendix List of the constants and variables related to the gas-liquid separation unit (VranLiM et al., 995): Constant Value Unit Description S.3 m Cross-section area of R S.3 m Cross-section area of R p.33 bar Normal atmospheric pressure h.5 m Height of the separator R R h m Height of the reservoir R R K 75. l/(s bar) Flow coefficient of valve V R V The open-close flow ratio of valve V K.74 l/(s bar) Flow coefficient of valve V R V The open-close flow ratio of valve V K.98 bar/m The proportional factor between the water level in [m] and w F pressure in [bar] K e -3 m 3 /l Proportional factor between the flow in m 3 /s and l/s.644 l/s Water flow to the R w 6.46 l/s Air flow to R (at the atmospheric pressure) air -.65 l/(sbar) The proportional factor between air flow and pressure air inside R r max The maximum value of the variable v.66 s - Maximum speed of valves opening max min v (the position of V ) v -.33 s - Maximum speed of valves closing 5 5 of 3

16 Variable Unit Description l/s Air flow to the separator R air l/s Air flow from the separator R l/s Water flow from the separator R p bar Excess of gas pressure above the normal atmospheric pressure inside R p bar Absolute air pressure inside R : p p + p ) p bar s Steady-state value of pressure pv bar s Steady-state pressure on the valve V h m Water level inside R h m s Steady-state value of water level h m Water level inside R V m 3 Gas volume inside R V m 3 Steady-state value of gas volume V s m kg Mass of the gas inside R K /(s bar ) l Flow coefficient of valve V ( p used for linearization K s l/(s bar ) Steady-state value of the flow coefficient K K l/(s bar ) Flow coefficient of valve V K s l/(s bar ) Steady-state value of the flow coefficient K v - Actual position of the valve V h used for linearization % v - Deviations from the steady-state % v v vs r - Valve command signal (desired position of v - Actual position of valve V % v - Deviations from the steady-state % v v vs r - Valve command signal (desired position of V ) V ) 6 6 of 3

17 Acknowledgements We would like to thank to Boštjan WeLko for his assistance when performing the experiments on the gasliquid separation plant and to Dr. Damir VranLiM for his useful advices. This work was sponsored by the European Commission through the Research Training Network MAC ( Multi-Agent Control: Probabilistic reasoning, optimal coordination, stability analysis and controller design for intelligent hybrid systems, HPRN-CT-999-7). References Åström, K., & Wittenmark, B. (997). Computer-controlled systems. Theory and design. Prentice-Hall, Inc., Upper Saddle River, New Jersey. Bemporad, A., Morari, M., Dua, V., & Pistikopoulos, E. N. (). The explicit solution of model predictive control via multiparametric quadratic programming. Proc. of American Control Conference, Chicago, Bemporad, A., Morari, M., Dua, V., & Pistikopoulos, E. N. (). The explicit linear quadratic regulator for constrained systems. Automatica, 38, 3-. Bemporad, A., & Filippi, C. (). Suboptimal explicit MPC via approximate quadratic programming. Proc. of IEEE Conf. Decision and Control, Orlando, Bentley, J. L. (975). Multidimensional binary search trees used for associative searching. Communications of the ACM, 8, Biegler, L. T., & Rawlings, J. B. (99). Optimization approaches to nonlinear model predictive control. Proc. of Chemical Process Control IV, Chmielewski, D., & Manousiouthakis, V. (996). On constrained infinite-time linear quadratic optimal control. Systems and Control Letters, 9, 9. Garcia, C. E., Prett, D. M., & Morari, M. (987). Model predictive control: Theory and practice, A survey. Automatica, 5, Grancharova, A., & Johansen, T. A. (). Approximate explicit model predictive control incorporating heuristics. Proc. of IEEE International Symposium on Computer Aided Control System Design, Glasgow, Scotland, U.K., of 3

18 Grancharova, A., Johansen, T. A., & Kocijan, J. (3). Explicit model predictive control of gas-liquid separation plant. Proc. of European Control Conference, Cambridge, U.K. Johansen, T. A., Petersen, I., & Slupphaug, O. (). Explicit sub-optimal linear quadratic regulation with state and input constraints. Automatica, 38, 99-. Johansen, T. A., & Grancharova, A. (). Approximate explicit model predictive control implemented via orthogonal search tree partitioning. Proc. of 5-th IFAC World Congress, Barcelona, Spain, session T-We-M7. Johansen, T. A., & Grancharova, A. (3). Approximate explicit constrained linear model predictive control via orthogonal search tree. IEEE Trans. Automatic Control, 48, Johansen, T. A. (4). Approximate explicit receding horizon control of constrained nonlinear systems. Automatica, 4, Mayne, D. Q., Rawlings, J. B., Rao, C. V., & Scokaert, P. O. M. (). Constrained model predictive control: Stability and optimality. Automatica, 36, Ogata, K. (995). Discrete-time control systems. Prentice-Hall, Inc., Englewood Cliffs, New Jersey. Pistikopoulos, E. N., Dua, V., Bozinis, N. A., Bemporad, A. & Morari, M. (). On-line optimization via off-line parametric optimization tools. Computers & Chemical Engineering, 4, Seron, M., De Dona, J. A., & Goodwin, G. C. (). Global analytical model predictive control with input constraints. Proc. of IEEE Conf. Decision and Control, Sydney, TuA5. Tøndel, P., Johansen, T. A., & Bemporad, A. (3). An algorithm for multi-parametric quadratic programming and explicit MPC solutions. Automatica, 39, Tøndel, P., & Johansen, T. A. (). Complexity reduction in explicit linear model predictive control. Proc. of 5-th IFAC World Congress, Barcelona, Spain, session T-We-M7. VranLiM, D., JuriLiM, D., & PetrovLiL, J. (995). Measurements and mathematical modelling of a semiindustrial liquid-gas separator for the purpose of fault diagnosis. Report DP-76, University of Ljubljana, J. Stefan Institute, Ljubljana. 8 8 of 3

19 List of Tables Table. The average and the maximal approximation errors in the cost function and in the two control inputs for the approximate explicit MPC controller. 9 9 of 3

20 List of Figures Fig.. Quad-tree partition. Fig.. k d tree partition. Fig. 3. Process scheme of the separation unit. Fig. 4. Scheme showing environment for control and experimentation. Fig. 5. Trajectory of v (position of valve ). Fig. 6. Trajectory of v (position of valve ). Fig. 7. Trajectory of p (pressure in the separator). Fig. 8. Trajectory of h (liquid level in the separator). Fig. 9. Piecewise linear approximate feedback control law (top) and the exact feedback control law (bottom) for valve position v. Fig.. Piecewise linear approximate feedback control law (top) and the exact feedback control law (bottom) for valve position v. Fig.. Sub-optimal cost function (top) and optimal cost function (bottom). of 3

21 Table. Error in cost function J [%] Error in u u [%] Error in u u [%] Average Maximal of 3

22 Fig.. of 3

23 Fig of 3

24 Fig of 3

25 Personal computer FactoryLink SCADA for basic control SCADA for proc. control Personal computer Matlab-Simulink DDE MelDDE server R S3 Ethernet Fig. 4. PLC Mitsubishi QAS-S Procedural control Batch process control Continuous process control PLC Mitsubishi ASH-S Basic control Equipment + Supervision and control Process OS IS Mitsubishi NetB 5 5 of 3

26 .7 v time [s] Fig. 5. experiments simulated exact MPC simulated approximate explicit MPC 6 6 of 3

27 .88 v experiments simulated exact MPC time [s] Fig. 6. simulated approximate explicit MPC 7 7 of 3

28 .8 p [bar] experiments simulated exact MPC. 5 5 time [s] Fig. 7. simulated approximate explicit MPC 8 8 of 3

29 .5 h [m] experiments simulated exact MPC time [s] Fig. 8. simulated approximate explicit MPC 9 9 of 3

30 v (p,h ) p. v (p,h ) p.4. Fig h h of 3

31 v (p,h ) p.8. v (p,h ) p Fig h h of 3

32 J(p,h ) p.8. J (p,h ) 5 5. p Fig h h of 3