CONTROL OF SYSTEMS WITH ASYMMETRIC BOUNDS USING LINEAR PROGRAMMING : APPLICATION TO A HYDROGEN REFORMER

This paper studies controller design for feedback systems in the presence of asymmetrically bounded signals, using a case study. An asymmetric objective functional is used to consider the asymmetrically bounded signals, which makes possible to derive a linear programming problem. Solving this LP makes possible to design controllers that minimize certain outputs, fulfilling at the same time hard constraints on certain signals. The method is presented by application to a hydrogen reformer, a system in petrochemical plants that produces hydrogen from hydrocarbons: a mixed sensitivity problem is stated and solved, with an additional constraint given by the asymmetric limitations on the magnitude and rate of the control signal, and the asymmetricity in the disturbances.


Introduction
In many control engineering problems, there are constrained variables (in magnitude, rate, acceleration, etc.), whose limits are not symmetric with respect to the nominal working point.This paper studies a novel extension of the 1 control theory, to directly handle the asymmetrical aspect of bounded signals.
The topic of designing control systems that maintain stability (and performance) in the presence of these constraints is a topic of continuing interest (see, e.g., Eun et al. [7], Mesquine et al. [9], and references therein), because real plants always present constraints on signals (e.g., there is always saturation of the control signal due to the limited range of the actuators).Most of the approaches proposed in the literature are based on constraint avoidance: preventing the saturation, the closed-loop system stays in a region of linear behavior.
From the available approaches, we select the 1 norm optimization concept (Dahleh and Diaz-Bobillo [3]), because the synthesis problem is solved using linear programming, and applications to real process have already been published (Malaterre and Kammash [8]; Tadeo and Grimble [14]; Tadeo et al. [16]), albeit for the symmetric case.In Vidyasagar [17] this problem was first formulated.Some basic results on this theory were presented in Dahleh and Pearson [4,5].These results showed that the 1 -optimal control problem could be stated as linear programming problem.Since then, many other results and extensions have been presented (Blanchini and Sznaier [2], Sokolov [13]), but no extension to asymmetric signals has been published.
Then, this paper studies a novel extension of the 1 control theory, to directly handle the asymmetrical aspect of bounded signals.The authors introduced this extension from a purely theoretical point of view in a previous paper (Naib et al. [11]), following the methods proposed for positive invariance (Dambrine et al. [6]; Benzaouia and Burgat [1]).This proposed extension to asymmetric signals is necessary because when the considered signal u is constrained to evolve in an asymmetrical domain (i.e., signal u is such that −u min ≤ u(k) ≤ u max , where u min ≥ 0; u max ≥ 0; u min = u max ), in this case the peakto-peak norm of the signal u (by definition, gives a quite conservative information on the signal amplitude, so a new functional must be defined that takes separately into account the positive and negative deviations of the signals.This paper presents how the functional defined in (Naib et al. [11]) can be extended to a multiblock synthesis problem for a real engineering problem: the control of a hydrogen reformer plant in petrochemical industry, which prompted us the development of this technique.
The paper is organized as follows.Section 2 gives a description of the hydrogen reformer plant.Section 3 presents the reformulation of the 1 optimal control theory taking into account the asymmetrical aspect of the considered signals.Section 4 is devoted to the application of this theoretical issue to the hydrogen reformer.Some conclusions end this paper.

Description of the hydrogen reformer control problem
The development of this extension of 1 was prompted by a process control problem in an oil refinery, which had asymmetric constraints on the control signal and asymmetric disturbances.This process is the so-called hydrogen reformer: the objective of this system, in petrochemical plants, is to produce hydrogen from desulphurized hydrocarbons by catalysis (Tadeo and Grimble [14]).A schematic diagram of the system is shown in Figure 2.1.To generate the hydrogen, the hydrocarbons are mixed with superheated steam before entering the reformer tubes, where a nickel catalyst, heated at high temperature (about 750 C), produces the hydrogen.The high temperature that is reached to speed up the reaction is produced by burning fuel in the reformer.The fuel flow must be controlled to maintain the desired catalyst temperature, despite the presence of asymmetric disturbances and the fact that nominal flow is not near the middle of the allowed flow range.
The control system should maintain the desired catalyst temperature by modifying the amount of fuel that is fed to the reformer.The catalyst temperature and the fuel flow are measured for the implementation of the cascaded control structure shown in Figure 2.2.
In the real process, there are several disturbances, such as varying feed flow, fuel gas quality, steam temperature, and so forth.The most important disturbance is from the steam temperature variation, which modifies the catalyst temperature directly.The control system must attenuate this (asymmetric) disturbance as much as possible by acting on the fuel flow set point (which is also asymmetric).
From a control engineering point of view, the control design problem is to design an advanced controller for this system that could improve the disturbance rejection properties, taking into account the nonsymmetric constraints on the amplitude and rate of the controlled variable (fuel flow).
First the existing control system was investigated and the slave flow control loop was considered adequate, so only the design of a master controller for the outer loop will be presented.
Summing up, reducing the existing slave loop, a schematic of the controller structure used for design is shown in Figure 2.3, which will be the topic of the rest of the paper.The main objective is to obtain tight control of the output, taking into account that the disturbance is asymmetric and the control signal is limited in amplitude nonsymmetrically.

1 analysis using asymmetric signals
This section presents the objective functional proposed by the authors (Naib et al. [11]), which replaces the peak-to-peak-norm normally used in the 1 analysis when asymmetric signals are used.For simplicity reasons, the method is presented for the singleinput/single-output case, although it is very simple to extend it to the multivariable case, as it is discussed later on.

Motivation.
First, let us take a look in the 1 norm of a transfer function G representing a SISO discrete linear time-invariant system governed by an asymmetrical control signal w (z = Gw).It is well known that and {φ i , i = 0,...,∞} denotes the impulse response of the transfer function One drawback of using this definition of the 1 norm is that the ∞ norm of a signal only retains the worst-case deviation of the signal, without taking into account if the worst-case negative deviation is different from the worst-case positive deviation.This increases the conservativeness of the controllers designed based on minimizing the 1 norm.

Asymmetric objective functional.
To solve this drawback, the following objective functional (denoted by ϕ(w)) was proposed by the authors (Naib et al. [11]), to give information on the signal amplitude, retaining information on the asymmetricity, and being valid for multivariable systems: where w + = max{0,max k (w(k))} and w − = max{0,−min k (w(k))}.
M. Naib et al. 5 By extension for a vector of signals z ∈ R n , we denote Hence, ϕ(z) ≤ Z, where Z = [ zmax zmin ] is equivalent to −z min ≤ z ≤ z max , the inequalities considered componentwise.In the proposed context, this objective functional is used to handle the asymmetrical aspect of any set of bounded signals, as it is now presented.

Asymmetric analysis.
The following result, which is an extension to MIMO systems of the theorem proposed in Naib et al. [11], gives necessary and sufficient conditions for a vector of signals to avoid the constraints, that is, ϕ(z) ≤ Z for any bounded vector of inputs ϕ(w) ≤ W such that z = Mw.
Theorem 3.1.The output vector z ∈ R n of an LTI system is constrained in the nonsymmetrical domain defined by ϕ(z) ≤ Z for any input vector w ∈ R m such that ϕ(w) ≤ W if and only if where and {φ l ∈ R n×m , l = 0,...,∞} denote the elements of the impulse response of the matrix transfer function M of the system (i.e., z = Mw).
Remark 3.2.If the constraints were symmetric (w min = w max and z min = z max ), then the second set of rows in the definition of Ψ is redundant, and the standard 1 condition problem is obtained: ∞ l=0 (φ + l + φ − l )w max ≤ z max .

Controller design for asymmetric signals.
The following controller design technique is now discussed: calculate a stabilizing controller K that minimizes a certain generalized output vector z 1 when the generalized input signal w is bounded in magnitude and hard limitations are imposed on an output vector z 2 (in the hydrogen reformer w corresponds to a vector composed by the reference and the disturbance signals, z 2 is the control signal, and z 1 corresponds to the measured output deviations).This problem can be expressed mathematically as follows: We denote by M(K) the matrix transfer function from the generalized input vector w to the generalized output vector z 1 .This M(K) is usually composed of several characteristic (matrix) transfer functions, such as the sensitivity S = (1 + KG) −1 , the control 6 Control with asymmetric bounds of a hydrogen reformer sensitivity M = KS, the complementary sensitivity T = I − S, and so forth, and maybe weighted by transfer functions W i given by the design specifications, as is usual in robust control.Then, using the result in the previous section, (3.6) can be expressed as calculating a stabilizing controller K that solves the following minimization problem: (3.7) The good news is that (3.7) can be transformed to a LP problem, following a method parallel to that already proposed for solving the following 1 optimization: where M 1 and M 2 are the matrix transfer functions from w to z 1 and z 2 , respectively.Unfortunately the description of the resulting LP problem in the general case is too cumbersome: as it corresponds to small modifications on the 1 optimization technique only a short sketch is now presented for a one-block SISO problem, where only the main modifications are discussed.The detailed example in next section should clarify any remaining issue.
The original 1 optimization problem (3.8) can be solved by separating the positive and negative terms of the corresponding matrix of impulse responses (Dahleh and Pearson [5]): The following LP problem is then obtained: The first set of constraints in (3.10) are the so-called norm constraints; the second set correspond to the signal constraints, with additional interpolation, feasibility, and rank constraints that are equivalent for both problems, so they are omitted for clarity.
In the presence of asymmetric signal, the proposed functional can be used, being only necessary to replace the norm constraint in (3.10) given by the 1 norm with the constraint given the objective functional presented in the previous section, obtaining the following LP: where ε = [ε + , ε − ] t , and ε + ,ε − ∈ R m are column vectors with dimensions equal to the number of generalized outputs in z 1 : Following the usual method in robust control, the control objectives can be achieved by solving (3.6), where z 2 ≡ u, z 1 ≡ y, and w ≡ [W1r W2d ] t .This can be expressed as the following mixed sensitivity problem.
Search among all the internally stabilizing controllers for the one which minimize the following cost function: where S = 1/(1 + KG) represents the sensitivity, K is the controller, T = KG/(1 + KG) is the complementary sensitivity, and W 1 and W 2 are weighting transfer functions.
Since the control signal is given by u = KS(r − d), then the constraints are respected if the following inequality is fulfilled: where D = [0.1,1.9] t represents the constraint vector on the signal −d.
To simplify the transformation into an LP, in this study the optimization problem is expressed directly on the coefficients of the impulse response of the considered transfer functions (Dahleh and Pearson [5], Tadeo and Grimble [14]): if we denote the impulse responses of W 1 S, W 2 T, and KS, respectively, to be φ 1 , φ 2 , and φ 3 , then, using the previous formalism, our problem can be rewritten as To solve this problem, it is necessary to add some feasibility constraint equalities to make the link between φ 1 ,φ 2 , and φ 3 : These equalities can be rewritten as where W 1 = nw 1 /dw 1 ,W 2 = nw 2 /dw 2 , and G = ng/dg, with nw 1 , dw 1 , nw 2 , dw 2 , ng, and dg coprime polynomials.Further, additional constraints (or namely interpolation constraints) must be added to ensure the internal stability of the system, as in standard 1 optimization.These constraints mean that any unstable pole (resp., zero) of the plant cannot be cancelled by any zero (resp., pole) of the controller.If we denote by {q k , k = 1,...,n} the unstable zeros of the plant and by {p l , l = 1,...,m} its unstable poles, then z=pl should tend to infinity. (4.9) In this problem, taking into account the feasibility constraints, the interpolation constraints are redundant.
The obtained linear programming problem has infinite number of variables and constraints.In order to obtain a finite-dimensional problem, this semi-infinite linear programming problem is truncated as is frequently done in 1 optimization.The dimension of the new linear programming problem, denoted by μ, can be calculated from the dual problem.Finally, we have a linear programming problem in 3μ + 4 variables which can be summarized as follows: minc t x s.t.Ax ≤ b, A eq x = b eq , (4.10) where x is formed by the variables Matrices A, A eq and vectors b and b eq are given according to the linear programming problem defined in (4.6).

Controller calculation.
The linear model of the plant that will be used to design the controller is (4.11) The model was obtained in transfer-function form from the input (u: fuel-flow setpoint) to the output (y: outlet temperature) and includes the flow controller dynamics.In this case, a simplified model was previously obtained by modeling and identification using data from the real plant (Shakoor,[12]).
Note that this plant has a stable nonminimum phase model with three zeros outside the unit circle, at z = 15.45 and z = 0.79 ± 1.57 j.The disturbance model W d was also identified using the available data from the real plant and it is given by (4.12) The weights W 1 and W 2 were selected based on a tradeoff between the specifications obtained from the system analysis and the control effort.These weights are, respectively, the discrete-time equivalents of the following continuous-time transfer functions: (0.5s + 0.005)/(s + 10 −6 ) and (50s + 0.5)/(0.1s+ 1) for a sampling time T s = 30.
This linear problem can be solved with the "linprog" optimization function of Matlab, for increasing values of μ until no improvement is obtained.The obtained impulse responses for μ = 20 are shown in Figure 4.1.
The characteristic transfer functions S, T, and M can be computed from any φ i .The frequency responses are shown in Figure 4.2.
The controller can be calculated from the relation that defines any of the characteristic transfer functions.For example, using the relation which defines the sensitivity (S = (1 + KG) −1 ), one can compute the corresponding regulator from K = G −1 (S −1 − 1), is depicted on the original controller.It is possible to see that over the 4th order the two controllers give a similar response, so a 4th-order controller is selected.

Conclusion
In this paper, a new way to handle asymmetrical signals has been presented.Using a new asymmetric objective functional, instead of the usual peak-to-peak norm, necessary and sufficient conditions are given to check if some performance specifications are fulfilled.
In the next step, the analysis and the synthesis problems are also formulated with respect to this new objective functional.As in the 1 -optimal control, this approach leads to a linear programming problem which can be solved by available optimization tools.Compared with the linear problem obtained using standard 1 -optimal control, no additional variables are added to the primal problem, but the number of constraints is doubled, to cope with the asymmetricity.The technique has been demonstrated on a system from petrochemical industry: a hydrogen reformer.

Figure 4 . 3 .
Figure 4.3.Frequency response of the optimal controller K and the reduced-order controller K r .