Explicit simplified MPC with an adjustment parameter adapted by a fuzzy system

Abstract

In this work, a novel methodology is presented to reduce the computational complexity of applying explicit solution of Model Predictive Control (MPC). The methodology is based on applying the functional principal component analysis, providing a mathematically elegant approach to reduce the complexity of rule-based systems, like piecewise affine systems, allowing the reduction of the number of consequents and combining and merging the antecedents. Thus, the application of MPC is allowed in systems with low computational requirements, such as programmable logic controllers, embedded systems, etc. The proposed design has been validated using an industrial distiller model.

Keywords

Piece wise affine functional principal component analysis model predictive control fuzzy control

1. Introduction

In traditional constrained model predictive control (MPC) [1], the control problem is solved in an on-line optimisation. MPC uses a dynamic model of the process in the optimisation problem to predict the future evolution of the output over a finite horizon to define the optimal control action at each time step by minimising an objective function while satisfying the system constraints. The predicted output vector of a linear system can be expressed as $y = G u + f$ (1) where u is the sequence of future control actions, G the dynamic matrix, f is the free response of the system, that is, the part of the response that does not depend on the future control actions. The formulation of such a constrained optimal control problem can be expressed as: $\begin{matrix} min_{u} J (N_{1}, N_{2}, N_{u}) = & \sum_{j = N_{1}}^{N_{2}} Q [\hat{y} (k + j | k) - w (k + j)]^{2} \\ + \sum_{j = 1}^{N_{u}} R [Δ u (k + j - 1)]^{2} \end{matrix}$ $\begin{matrix} s . t . \\ \hat{y} (k + j | k) = f (Δ u, y_{k + j - 1}, y_{k + j - 2}, . . .) \\ u (k + j) = u_{k + j - 1} + Δ u_{k + j} \\ u_{min} \leq u_{k + j} \leq u_{max} \end{matrix}$ (2) where N₁ and N₂ are the minimum and maximum costing horizons while N_u is the control horizon. $\hat{y} (k + j | k)$ is the prediction of the system output and w (k + j) is the future reference trajectory. $Q, R \in ℝ$ are weighting parameters for the system outputreference tracking and the control effort. Theobjective is to minimize J (N₁, N₂, N_u) in order to compute a future sequence of control actions u (k) , u (k + 1) , . . . that drives the future plant output y (k + j) close towards w (k + j). From the obtained control sequence, only the first control action is applied to the system. In the next control sampling time the complete process is computed and again only the first control action is applied. In this way the MPC is being adjusted at every control sampling time.

One of the characteristics of MPC controllers that makes them indisputably distinguishable from other control techniques is the ability to modify, online, the behavior of the MPC. This is done by using the weighting parameter, especially in practice, the R parameter. As mentioned above, this parameter is the weighting factor that penalizes the control increments in the computation of the objective function. The higher the value of R, the lower the control increments and the lower R, the greater the control increments that the MPC can use when tracking the set point. Hence, a user can modify the behavior of the MPC, making it more reactive or less reactive through the modification of a single parameter and without the need, in general, to compute a new controller, offline, as it happens in another type of controllers. To solve this problem, there are many available and reliable quadratic programming (QP) algorithms, e.g. active set, feasible direction, pivoting methods, etc., [1]. They all use an iterative algorithm, which means that due to the computational burden they are not suitable for every hardware platform. As such, in the past, the main drawback of MPC was that it was restricted to relatively slow response process applications with sampling times in the order of minutes.

In the last decade there has been an effort to accelerate the computation of the MPC. In [2], the implementation aspects are presented and the restriction of the horizon on limited-resource hardware such as fixed-point arithmetics is addressed. Fast model predictive control can be done on-line and off-line. On-line fast MPC is based on the formulation of MPC as a QP problem and an appropriate variable reordering to apply the interior point optimisation algorithms, the active set strategies or the first-order fast gradient methods. Faster on-line optimisation techniques are described in [3 –8]. Off-line MPC or explicit MPC is based on solving the optimisation problem that is parameterised by state, i.e. parametric optimisation, to obtain a pre-computed control law as function of state. Several explicit MPC algorithms have been proposed [9 –13]. In [9], an explicit form of the MPC controller has been proposed by solving a multi-parametric quadratic program (mpQP) transformed from the constrained MPC QP problem, hence on-line computation is dramatically reduced, obtaining a piecewise affine (PWA) controller. Such a function is then composed of numerous distinct affine feedback laws defined over a set of polytopic regions. For further results on explicit MPC, the reader is referred to [14] and the references therein.

The use of explicit MPC can lead to an explosion of regions when it comes to high-order models (e.g. more than 10 states). In [11], the goal-oriented reduction methodology has been proposed to obtain reduced-order models that are efficient for solving the MPC problem to obtain an explicit solution for large-scale systems with fast dynamics. An explicit MPC based on dynamic and multi-parametric programming techniques to decompose the MPC mpQP optimisation problem into a set of smaller stage optimisation subproblems and the optimal control variables for each stage as a function of the systems states has been developed in [12]. Also in [13], an explicit MPC is applied in buildings. Other examples can be found in [15, 16], where a constrained MPC problem is developed using a low range programmable logic controller (PLC). Due to the combinatorial nature of the problem, the explicit MPC synthesis is usually only tractable for low state dimensions. Computing the optimal input on-line then reduces to a mere function evaluation, which can be performed very quickly even with limited computational resources.

It is posible to solve the real-time MPC optimisation by designing a MPC control scheme based on PWA systems. However, if the PWA needs a large number of regions to obtain a good solution, the programming effort and memory capacity of the hardware platform can cause the non-applicability of this methodology. This will end up in preventing the use of predictive control for fast and/or low cost systems. There have been studies in this field, with some methods that can reduce the number of regions of PWA (with some loss of optimality). Another approach for complexity reduction of PWA systems is based on constructing a sub-optimal replacement function of substantially lower complexity [17 –21]. In [22] regions are merged if they share the same expression for the control law. If the PWA function is convex (or if there exists a convex function, defined over the same regions), then the method presented in [23] can be used to reduce the required memory storage. If the original function is non-convex, butcontinuous, its lattice representation [24] can be built, again decreasing the memory consumption. In [25], the performance-lossless replacement was constructed by only considering the regions of the original function, where the control action is not saturated. This can considerably reduce the complexity as the number of unsaturated regions is usually significantly smaller compared to the total number of underlying polytopes over which the original function is defined. In [26] an exploration strategy for subdividing the parameter space, which avoids unnecessary partitioning, has been derived. In [17] a technique by relaxing the Karush-Kuhn-Tucker (KKT) conditions for optimality has been proposed. A rotation of the state space, obtaining a suboptimal control action has been presented in [27]. In [28], a method to reduce complexity has been proposed by taking into account the formulation of the optimisation problem:

$\begin{matrix} J (U, x) & = & \frac{1}{2} U^{T} HU + x^{T} FU + \frac{1}{2} x^{T} Yx \\ s . t . & GU \leq W + Ex \end{matrix}$ (3) where x = (x_t, x_t+1, . . . , x_t+N) is the state for a finite horizon N, U = (u_t, u_t+1, . . . , u_t+N-1) is the input sequence. With proper definitions of the matrices Y,H,F,G,W, and E (see [9]). A singular value decomposition (SVD) analysis is applied in order to replace the terms Ex and F^Tx with approximated linear terms defined on a subspace of the state space.

These techniques, while useful, may not produce a drastic reduction in the number of rules, if an approach to online MPC performance is desired. In [36], a powerful technique, based on Functional Principal Component Analysys (FPCA) has been used to reduce complexity of Fuzzy Systems, decreasing the number of rules in a systematic way.

The main contribution of this work is the application of the reduction technique presented in [36] for PWA systems, in order to reduce complexity of explicit MPC and the use of a fuzzy system to modulate several of them, including an adjustment parameter, which will allow an online adjustment of the behaviour of the MPC to make it more or less reactive and obtain more or less rapid system responses depending on the needs of the process.

Model Predictive Control has been used in conjunction with fuzzy models under the denomination Fuzzy-MPC (FMPC), since the 90s [43 –50], mainly aimed at nonlinear-process control problems. Fuzzy systems have also been used to adjust the MPC parameters setting [51]. In this work a fuzzy system is used to modulate the action of several explicit MPCs. This article is organised as follows: An introduction to PCA and FPCA is given in section 2. Section 3. presents the application of FPCA to PWA systems. The formulation of constrained MPC as PWA controller is shown in section 4. In section 5, an application is used to illustrate the proposed method. Section 6 presents a control scheme, based on the proposed design, permitting an adjustable parameter. Conclusions are given in section 7.

2. Functional principal component analysis

From the last four decades, there has been an important development of functional data analysis (FDA). The main stream of FDA is the FPCA [35, 37]. FPCA is the functional extension of the PCA, therefore before checking FPCA, we will revisit multivariate PCA, that is used to reduce the dimensionality of multivariate data.

2.1. Principal component analysis

PCA is well known in the field of multivariate statistical analysis. Using PCA, the dimensionality of the space variables is reduced. The idea behind PCA is to find the subspace where the data have a high covariance. In Figure 1 there is more variability of the data in two dimensions, therefore, three dimensions are not necessary to represent that data.

Fig. 1

Principal components: C₁ and C₂.

Supposing r variables and N real samples, a real data set is represented by:

$\begin{matrix} x_{11} {\vec{e}}_{1} & + x_{12} {\vec{e}}_{2} & + \dots & + x_{1 r} {\vec{e}}_{r} \\ x_{21} {\vec{e}}_{1} & + x_{22} {\vec{e}}_{2} & + \dots & + x_{2 r} {\vec{e}}_{r} \\ ⋮ & \dots \\ x_{N 1} {\vec{e}}_{1} & + x_{N 2} {\vec{e}}_{2} & + \dots & + x_{Nr} {\vec{e}}_{r}, \end{matrix}$ (4) presenting the data in matrix format:

$X = (\begin{matrix} x_{11} & x_{21} & \dots & x_{N 1} \\ x_{12} & x_{22} & \dots & x_{N 2} \\ ⋮ & ⋮ & \dots & ⋮ \\ x_{1 r} & x_{2 r} & \dots & x_{Nr} \end{matrix}),$ (5) and ${\vec{e}}_{i}$ are the vectors generators of subspace V. The aim is to find a new basis vector to define a new subspace containing the maximum information of the actual data:

$\begin{matrix} w_{11} {\vec{e}}_{1} + w_{12} {\vec{e}}_{2} \dots + w_{1 r} {\vec{e}}_{r} \\ w_{21} {\vec{e}}_{1} + w_{22} {\vec{e}}_{2} \dots + w_{2 r} {\vec{e}}_{r} \\ ⋮ \\ w_{m 1} {\vec{e}}_{1} + w_{m 2} {\vec{e}}_{2} \dots + w_{mr} {\vec{e}}_{r} \end{matrix}} = W^{T} \cdot e,$ (6) where m < r. The matrix W will transform the matrix data X into Y = W^TX.

If μ_Y = E (Y) is the expected value of Y, it is demonstrated that: $μ_{Y} = E (W^{T} X) = W^{T} E (X)$ (7) and the covariance matrix of Y is equal to: $C_{Y} = E {(Y - μ_{Y}) (Y - μ_{Y})^{T}} = W^{T} C_{X} W .$ (8)

To obtain the subspace with maximum variability in the data, the covariance matrix of Y (C_Y) has to be calculated, imposing the orthonormality constraint on it: $W^{T} W = I$ (9) maximising using the Lagrange multipliers technique: $W^{T} C_{X} W - λ (W^{T} W - I) .$ (10) Differentiating (10) and equating to zero: $(C_{X} - λ I) W = 0 .$ (11) The problem is now reduced to calculating the eigenvectors of C_X. Those associated with the most significant eigenvalues, whose components will form the subspace where the data have the greatest variability.

In order to choose the number of principal components, a criterion for choosing the eigenvalues may be the use of an index of variability, defined as follows:

$\frac{\sum_{i = 1}^{h} λ_{i}}{\sum_{i = 1}^{N} λ_{i}} \geq v,$ (12) where h ≤ N and v is the degree of desired information, or variability index (v = 1 means that all the eigenvalues are used). Once the value v is set, the number of eigenvalues h is calculated.

2.2. Functional principal component analysis

The PCA works in a vector space. Working in a space of functions, the analysis will be the FPCA.

Let f₁ (x) , f₁ (x) , . . . , f_n (x) be functions in separable Hilbert space endowed with inner product:

〈 f_{i} | f_{j} 〉 = \int_{0}^{X} f_{i} (x) f_{j} (x) dx \forall f_{i},_{j} \in L^{2} [0, X] .

(13)

If each function f_i (x) may be decomposed in: $f_{i} (x) = \sum_{l = 1}^{L} c_{il} θ_{l} (x) = {c_{i}}^{T} Θ (x),$ (14) where θ_l (x) ∈ L² [0, X] and $c_{il} \in ℝ$ . The mean and covariance functions of f_i, will be: $\bar{f} (x) = E (f (x)) = {\bar{c}}^{T} Θ (x),$ (15) $Cov [f (x), f (s)] = Θ (x)^{T} cov (C) Θ (s),$ (16) where C = {c_il, i = 1, . . . , n, l = 1, . . . , L}. defining the covariance operator as: $\begin{matrix} C (f (x)) = \int_{0}^{X} Cov [f (x), f (s)] f (s) ds, \\ \forall f \in L^{2} [0, X], \forall x, s \in [0, X], \end{matrix}$ (17) where the kernel Cov [f (x) , f (s)] is the covariance function. The covariance operator is positive, selfadjoint and compact [38], thus, using Mercer’s Theorem [39], we may write: $Cov [f (x), f (s)] = \sum_{i = 1}^{\infty} λ_{i} ξ_{i} (x) ξ_{i} (s), \forall x, s \in [0, X],$ (18) where λ₁ > λ₂ > . . . >0 is an enumeration of the eigenvalues of C, and the corresponding orthonormal eigenfunctions are ξ₁, ξ₂, . . .. Thus, they form a complete orthonormal set of solutions of the Fredholm equation: $\int_{0}^{X} Cov [f (x), f (s)] ξ_{i} (s) ds = λ_{i} ξ_{i} (x) .$ (19)

3. FPCA applied to PWA systems

In general, a PWA system can be formulated as:

$\begin{matrix} IF & x \in Θ_{1} & THEN u = f_{1} (x) \\ ELSIF & x \in Θ_{2} & THEN u = f_{2} (x) \\ ⋮ \\ ELSIF & x \in Θ_{N} & THEN u = f_{N} (x) \\ ENDIF \end{matrix}$ (20)

Each region Θ_i is defined by a polytope. Let δ_i be functions defined as: $δ_{i} (x) = {\begin{matrix} 1 & if & x \in Θ_{i} \\ 0 & else . \end{matrix}$ (21)

If N is the number of regions, the PWA system can be expressed as: $u = \sum_{i = 1}^{N} δ_{i} (x) f_{i} (x)$ (22) with $\sum_{i = 1}^{N} δ_{i} = 1$ .

Considering each f_i (x) as a linear combination of the inputs, $u = \sum_{i = 1}^{N} δ_{i} (x) (ρ_{0 i} + ρ_{1 i} x_{1} + . . . + ρ_{ni} x_{n}),$ (23) where n is the number of inputs. Thus, the expression (23) takes a form as: $u (x) = {\tilde{g}}_{0} (x) + {\tilde{g}}_{1} (x) x_{1} + . . . + {\tilde{g}}_{n} (x) x_{n},$ (24) where: ${\tilde{g}}_{i} (x) = \sum_{j = 1}^{N} δ_{j} (x) \cdot ρ_{ji} .$ (25) And for all the regions, $\tilde{g}$ is: $\tilde{g} (x) = [\begin{matrix} {\tilde{g}}_{0} (x) \\ {\tilde{g}}_{1} (x) \\ ⋮ \\ {\tilde{g}}_{n} (x) \end{matrix}] = [\begin{matrix} ρ_{10} & ρ_{20} & \dots & ρ_{N 0} \\ ρ_{11} & ρ_{21} & \dots & ρ_{N 1} \\ ⋮ \\ ρ_{1 n} & ρ_{2 n} & \dots & ρ_{Nn} \end{matrix}] \cdot [\begin{matrix} δ_{0} (x) \\ δ_{1} (x) \\ ⋮ \\ δ_{N} (x) \end{matrix}]$ $\tilde{g} (x) = R \cdot Δ (x) .$ (26) According with (16), the covariance functions of $\tilde{g} (x)$ are: $Cov [\tilde{g} (x), \tilde{g} (s)] = Δ (x)^{T} cov (R) Δ (s) .$ (27) According to (19) and (6), supposing that the eigen functions are: $Γ (x) = [\begin{matrix} γ_{1} (x) \\ γ_{2} (x) \\ ⋮ \\ γ_{h} (x) \end{matrix}] = Δ (x)^{T} \cdot b .$ (28) Thus, taking into account (27): $\begin{matrix} \int_{0}^{X} Cov [\tilde{g} (x), \tilde{g} (s)] \cdot γ (s) ds & = \\ \int_{0}^{X} Δ (x)^{T} cov (R) Δ (s) \cdot Δ (s)^{T} \cdot b ds & = & Δ (x)^{T} cov (R) \cdot W \cdot b \\ cov (R) \cdot W \cdot b & = & λ \cdot b, \end{matrix}$ (29) where: $W = \int_{0}^{X} Δ (s) \cdot Δ (s)^{T} ds$ (30) being γ (x) orthogonal, then $〈 γ_{i} (x), γ_{j} (x) 〉 = b_{i}^{T} \cdot W \cdot b_{j} = 0$ .

Matrix W is symmetric by definition, thus, defining $p = W^{\frac{1}{2}} b$ , $W^{\frac{1}{2}} \cdot cov (R) \cdot W^{\frac{1}{2}} \cdot p = λ \cdot p .$ (31)

A symmetric eigenvalue problem now remains to be solved. Since only one region will be active in each state vector, equation (28) will give only one of the values within b parameter, $γ_{i} (x) \in {b_{i 1}, b_{i, 2}, . . ., b_{iN}} .$ (32)

It may be that there are repeated b_ij values, in such case, the regions associated to those values can be merged, hence reducing the number of regions, i.e. if b_ij = b_ik with j ≠ k, then γ_i (x) = b_i1δ₁ (x) + . . . + b_ij (δ_i (x) + δ_j (x)) + . . . + b_iNδ_N (x), being: $δ_{i} (x) + δ_{j} (x) = {\begin{matrix} 1 & if & x \in Θ_{i} ⋃ Θ_{j} \\ 0 & else . \end{matrix}$ (33)

4. Constrained MPC as a PWA controller

Given a linear system

$\begin{matrix} x (k + 1) = Ax (k) + Bu (k) \\ y (k) = Cx (k) + Du (k) . \end{matrix}$ (34) For the current x, the constrained MPC solves the following optimisation problem such that the optimal control increment Δu is found at each sampling instant. Using the state-state description, the problem posed in (2) becomes: $min_{u} {J (u, r, y (k))}$ (35) $\begin{matrix} s . t . u_{min} \leq u (k + j | k) \leq u_{max}, j = 0, \dots, N_{u} - 1, \\ x (k + j + 1 | k) = Ax (k + j | k) + Bu (k + j), j \geq 0, \\ y (k + j | k) = Cx (k + j | k) + Du (k + j), j \geq 0, \\ u (k + j) = u (k + j - 1) + Δ u (k + j), j \geq 0, \end{matrix}$ where the cost function to be minimised is given by:

$\begin{matrix} J (u, r, y (k)) \\ = \sum_{j = 0}^{N_{y} - 1} [y (k + j | k) - r (k)]^{T} Q [y (k + j | k) - r (k)] \\ + \sum_{j = 0}^{N_{u} - 1} Δ u (k + j)^{T} R Δ u (k + j) \end{matrix}$ (36) and u ≜ [Δu (k) ^T, …, Δu (k + N_u - 1) ^T] ^T, x (k + j|k) is the predicted state at time step k, N_y and N_u are the prediction and control horizons, and Q ≥ 0, R > 0.

The above MPC optimisation problem (35) can be described in the standard QP form, [1], and as shown in [9] such an MPC QP problem can be transformed into:

$\begin{matrix} min_{u} {J (u, χ (k)) = \frac{1}{2} u^{T} H u + χ (k)^{T} F^{T} u} \\ s . t . G u \leq W + S χ (k), \end{matrix}$ (37) where χ (k) is the vector of parameters defined as: $χ (k) = [x (k), u (k - 1), r (k)]^{T} .$ The MPC mpQP problem (4) is solved explicitly, off-line, for all the feasible values of χ of interest, resulting in the solution u^* (χ), which is a continuous piecewise-affine function defined over a polyhedral partition in the χ-space represented as:

$Δ u (k) = f (χ (k)) = {\begin{matrix} K_{1} χ (k) + k_{1}, & if χ (k) \in X_{1} \\ K_{2} χ (k) + k_{2}, & if χ (k) \in X_{2} \\ ⋮ \\ K_{N_{rej}} χ (k) + k_{N_{rej}}, & if χ (k) \in X_{N_{rej}} \end{matrix}$ (38) with a polyhedral partition $P = {X_{1}, \dots, X_{N_{rej}}}$ , where the polyhedral sets are represented by linear inequalities (hyperplanes), $X_{i} = {χ (k) | L_{i} χ (k) \leq l_{i}}, i = 1, \dots, N_{rej} .$ (39) Here, K_i and k_i are the control gain and offset for each region respectively, and N_rej is the number of regions. Consequently, the on-line control algorithm is reduced to a look-up table: the region associated with the current state χ is first determined, and then the optimal control law valid for that region is applied.

5. Example: distillation column

To illustrate the performance of a reduced PWA system, a high purity distillation column is used as an example. The distillation process typically works around an operating point, being identifiable by a linear model. The example is be carried out using the model shown in [40], where the linear model in an operational point is defined as:

$\begin{matrix} [\begin{matrix} {\dot{x}}_{1} \\ {\dot{x}}_{2} \end{matrix}] = & [\begin{matrix} - 0.0133 & 0 \\ 0 & - 0.0133 \end{matrix}] [\begin{matrix} x_{1} \\ x_{2} \end{matrix}] + \\ [\begin{matrix} 0.0117 & 0.0115 \\ 0.0144 & 0.0146 \end{matrix}] [\begin{matrix} u_{1} \\ u_{2} \end{matrix}], \\ [\begin{matrix} y_{1} \\ y_{2} \end{matrix}] & = [\begin{matrix} x_{1} \\ x_{2} \end{matrix}], \end{matrix}$ (40) where y₁ and y₂ are the top and bottom product compositions, respectively, and the inputs, u₁ and u₂, are the reflux flow rate and the boil-up, respectively, as shown in Fig. 2, [41].

Fig. 2

Distillation column.

Considering the reference tracking problem, i.e., the problem of driving the output (product composition) y to track a given reference signal $r \in R^{p}$ by adjusting the control inputs (reflux and boiler flow rates) u under the control input and control increment constraints. The tuning parameters used for deriving the explicit MPC controller are as follows: N_y = 20, N_u = 3, Q = R = I, with a sampling time of 10 min. The control input constraints are given as -2 ≤ u₁ ≤ 2 and -1 ≤ u₂ ≤ 2, respectively.

According to [9, 10], the number of regions obtained for the control law is 140. Applying the developed FPCA above to this application, with the same tuning parameters, only five consequents are obtained for the first manipulated variable and six for the second, $\begin{matrix} {\tilde{g}}_{1} (x) = [\begin{matrix} 0.0177 & - 0.1947 & 0.1562 & 0.0036 & 0.0096 \\ - 0.0207 & 0.2275 & - 0.1822 & 0.0075 & - 0.0103 \\ - 0.1298 & 0.8182 & 0.4014 & - 0.0068 & - 0.0012 \\ - 0.0003 & 0.0035 & - 0.0027 & - 0.0007 & - 0.0004 \\ - 0.0302 & 0.3351 & - 0.2730 & - 0.0245 & 0.0031 \\ 0.0336 & - 0.3704 & 0.3005 & - 0.0195 & - 0.0085 \\ 1.6627 & 0.2191 & 0.0795 & - 0.0067 & - 0.0013 \end{matrix}] \\ \cdot Δ (x) \end{matrix}$

$\begin{matrix} {\tilde{g}}_{2} (x) & = & [\begin{matrix} - 0.0271 & 0.1330 & - 0.1683 & - 0.0030 & 0.0074 & 0.0045 \\ - 0.0322 & 0.1575 & - 0.1992 & 0.0083 & 0.0092 & 0.0027 \\ - 0.0006 & 0.0027 & - 0.0033 & - 0.0029 & - 0.0003 & 0.0032 \\ - 0.4149 & 0.8620 & 0.2987 & 0.0018 & 0.0062 & 0.0035 \\ 0.0455 & - 0.2331 & 0.2968 & 0.0203 & 0.0060 & 0.0038 \\ 0.0504 & - 0.2562 & 0.3258 & - 0.0136 & 0.0088 & 0.0033 \\ 1.3315 & 0.4020 & 0.1321 & 0.0018 & 0.0062 & 0.0035 \end{matrix}] \\ \cdot Δ (x) . \end{matrix}$ Associating regions with the same parameter as it has been seen in (33), the arithmetic operations of antecedents are reduced from 140 to 68. In Fig. 3, it can be observed that the proposed transformation has a similar performance as the the original completed controller.

Fig. 3

PWA Controller vs Simplified PWA controller performance for distillation column.

6. Simplified MPC with constraints

An important drawback of the explicit solution of the MPC is the lack of on-line tuning parameters to the designed controller. It is a weakness in terms of implementation. One possible solution is to design different PWA controllers for different settings parameters and linearly interpolating the control action of the different controllers, e.g. by the inference of a fuzzy controller having the control actions of the various PWA designed for different parameter values as inputs. The problem with this is an increase in complexity, due to the increasing number of rules. Figure 5 shows the scheme of thisstructure.

A Takagi-Sugeno fuzzy system [42] can be chosen to interpolate the different control actions for the distillation column. Only the input R is taken for fuzzyfication. If great accuracy is desired, there should be many PWA controllers programmed into the system and more values of R. As an example, three different PWA for three values of R are designed, giving the number of regions shown in Table 1 and the performance shown in Fig. 4.

Fig. 4

PWA controllers depending on R.

$\begin{matrix} IF R is 1 THEN u = PWA 1 \\ IF R is 5 THEN u = PWA 2 \\ IF R is 10 THEN u = PWA 3 \end{matrix}$ (41)

Selecting a priori, three triangular membership functions, for each value of the adjustment parameter (mf1, mf2 and mf3 according to three precalculated values of R: 1, 5 and 10). The system can be designed with three rules, described by (14): Once the FPCA is applied, a huge reduction in the number of consequents can be observed (see Table 1), and following (33), an association of regions can be done, reducing the antecedents between 45 and 51%.

Table 1

Controllers structure using FPCA

Parameter	Variable	Rules number	Reduced consequents number/reduction(%)	Reduced antecedents number/reduction(%)
R = 1	u1	140	5 (96%)	68 (51%)
R = 1	u2	140	6 (95%)	71 (49%)
R = 5	u1	162	7 (95.6%)	89 (45%)
R = 5	u2	162	5 (97%)	87 (46%)
R = 10	u1	161	5 (97%)	88 (45%)
R = 10	u2	161	6 (96%)	87 (46%)

Following the structure of the system proposed above (see Fig. 5), the parameter R (move suppression) can be adjusted in order to give more or less aggressiveness to the system. Figure 7 shows the performance of the controller when R takes different values between those which are given in Table 1. It is observed that the R value can be changed directly on-line, without the need to have a set of predefined controllers.

Fig. 5

Explicit MPC with FIS structure.

Fig. 6

Control surface depending on each PWA and R.

Fig. 7

FMPC for distillation column depending on R.

7. Conclusions and future works

A novel technique has been applied for PWA systems to reduce the number of consequents and to associated regions in the antecedents of the rules. The technique is based on the application of FPCA to the structure of the PWA. A tuning parameter can be included in the explicit MPC by using a PWA controller per each adjusted parameter, although, the complexity of the system increases. The application of this technique, together with the use of a fuzzy system to combine the different PWA controllers, has succeeded in designing a MPC scheme with constraints and an on-line adjustment parameter, drastically reducing the number of regions of consequents, improving programming time (down 96% the number of consequents to add into the code). The technique applied to the PWA also allows a new merge of regions (not necessarily adjacent), opening the door to future research on reducing their complexity. Another future work aims at the systematic design of the fuzzy system.

Footnotes

Acknowledgements

The authors would like to acknowledge the VI Plan of Research and Transfer of the University of Seville (VI PPIT-US) for funding this work and also the Ministry of Economy and Competitiveness of Spain for the financial support under grant DPI2016-78338-R.

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