Design of Adaptive PCA Controllers for SISO Systems

Transcription

1 Preprints of the 8th IFAC World Congress Milano (Italy) August 28 - September 2, 2 Design of Adaptive PCA Controllers for SISO Systems L. Brito Palma*, F. Vieira Coito*, P. Sousa Gil*, R. Neves-Silva* *Universidade Nova de Lisboa Faculdade de Ciencias e Tecnologia Dep. Eng. Electrot. ; Uninova; CISUC Portugal (Tel: ; s: LBP,FJVC,PSG,RNS@fct.unl.pt). Abstract: In this paper an approach to design adaptive controllers, for SISO systems, based on principal components analysis (PCA models) is presented. Closed-loop control can be formulated and implemented within the reduced space defined by a PCA model. This PCA controller, results in an integral controller, which can be used as an inferential controller for nonlinear systems. The main contribution of the paper is the adaptive mechanism based on online recursive PCA, using eigenvector perturbation, for process control applications. Some experimental results, obtained with a nonlinear system (the three tank benchmark) are presented, showing the controller performance. Keywords: adaptive control, principal components analysis, nonlinear systems, performance analysis.. INTRODUCTION Modern automation and distributed control systems make possible the collection of large quantities of process data. These data has to be processed to extract important and relevant information, which is commonly saved in the databases. Knowledge should be created from the significant information. Multivariate techniques can be used for data analysis, monitoring and control, while multivariate statistical methods can be used to develop nonparametric models for process monitoring (including fault detection and diagnosis), (Piovoso & Kosanovich, 994; Piovoso, 996; Jackson, 23; Brito Palma, 27), as well as for feedback control in the scores space (Piovoso & Kosanovich, 994; Piovoso, 996; Brito Palma, et al., 2). In processes where redundancy or correlation between variables exists, it is advantageous to reduce the number of variables, preserving an important quantity of original information. Dimensionality reduction techniques can greatly simplify and improve process monitoring procedures, since they project the data into a lower-dimensional space that accurately characterizes the state of the process (Chiang, et al., 2; Jolliffe, 22; Jackson, 23; Brito Palma, 27). The proposed PCA control formulation, in a reduced control space, is analogous to modal control (also called eigenvalueassignment control, (Feng, 999; Magni, 22). In (Akamatsu, et al., 2), PCA control and PI control using CVA (canonical variate analysis) modeling is tested, and compared, on a chemical reactor. A statistical controller based on PCA was applied to the Tennessee Eastman process in (Chen & McAvoy, 996). In order to evaluate the controller s performance, PCA control charts can be applied (Chen & Wang, 29). To implement real-time algorithms based on adaptive (recursive) PCA, an adaptive eigendecomposition of data covariance matrix should be implemented (Champagne, 994). Most of the publications related to recursive PCA were focused on monitoring and fault diagnosis (Li, et al., 2; Jeng, 2). Few publications exist related to adaptive process control based on recursive PCA; one of these references is (Yao, 29). Here, recursive PCA is used for adaptive control purposes (Astrom & Wittenmark, 995; Levine, 996). In this paper two versions of PCA controllers for SISO systems are presented: a controller based on a fixed PCA model (Brito Palma, et al., 2), and an adaptive controller based on an online recursive PCA model, using eigenvector matrix perturbation. The perturbation analysis is used to find an approximate solution to a problem that cannot be solved exactly (Champagne, 994). For real-time applications, fast, adaptive and online algorithms are required to deal with nonlinear systems. 2. PCA MODELS PCA is a multivariate statistical technique. Others statistical techniques include Principal Components Regression (PCR), Partial Least Squares (PLS or Projection to Latent Structures), etc, (Chiang, et al., 2; Jolliffe, 22; Jackson, 23; Brito Palma, 27). PCA models can be used for data compression, process monitoring, fault detection and diagnosis and also for process control as described in the next section. PCA involves several steps (Chiang, et al., 2; Jolliffe, 22; Jackson, 23; Brito Palma, et al., 25). First, the original data Xo should be scaled, i.e., mean centered, and often normalized by the standard deviation. From the scaled data X R n m, the covariance matrix is computed by the relationship (), where n is the number of observations (samples) and m is the number of process variables. Copyright by the International Federation of Automatic Control (IFAC) 5483

2 Preprints of the 8th IFAC World Congress Milano (Italy) August 28 - September 2, 2 S = n- XT X PCA captures the variability of the data X. It determines loading vectors (orthogonal vectors) ordered by the amount of variance explained in the loading vector directions. The loading vectors are computed by solving the stationary points of an optimization problem (Chiang, et al., 2; Jolliffe, 22; Jackson, 23), i.e., solving a singular value decomposition (SVD) of the sample covariance matrix S S = n- XT X = V Λ V T (2) The diagnonal matrix Λ = Σ T Σ, Λ R m m, contains the non-negative real singular values of decreasing magnitude obeying the relation () λ λ 2 λ m (3) When the goal is to minimize the effect of random noise that corrupt the PCA representation, and to optimally capture the variations of data, then only the loading vectors corresponding to the a largest singular values must be retained in the PCA model; a is the number of principal components that captures the most important information, that can be computed according a certain explained variance (usually greater than 8 % or 9%) or other approaches (Chiang, et al., 2; Jolliffe, 22; Jackson, 23). PCA projects the observation space into two subspaces: the scores subspace and the residual subspace. Selecting the columns of the loading matrix P R m a to correspond to the loading vectors V R m m associated with the a largest singular values, the projections of the observation data X R n m into the lower-dimensional space are contained in the scores matrix T R n a T = X P (4) The projection of T back into the m-dimensional observation space is given by The residual matrix E is given by X = T P T (5) E = X - X (6) The residual matrix E captures the variations in the observation space spanned by the loading vectors associated with the "m - a" smallest singular values. The subspaces spanned by X and E are usually denominated scores space and residual space, respectively. 3. PROCESS CONTROL BASED ON FIXED PCA MODELS First, the controller design is presented based on a fixed PCA model, inspired on the previous work developed by Piovoso et al. (994, 996). Next, a gain is incorporated on the controller equation, and some techniques for controller tuning are described (Brito Palma, et al., 2). 3. PCA Controller Design Many industrial plants (chemical processes, etc) have a large number of exogenous variables and a few manipulated ones. The exogenous variables indicate the process state, and the manipulated variables control directly the measured quantities and indirectly the unmeasured quantities. Some of these unmeasured quantities are related to the final product quality, so they should be carefully monitored and controlled. The main idea is to develop a PCA model to represent the desired process region in the score space, and then design a controller in the score space that maintains operation within this region. The control moves in the score space are then mapped to the real variable space and implemented on the process. The process is kept within the desired region provided that the PCA model has correctly established the relationship between exogenous variables and manipulated variables. This proposed control formulation is analogous to modal control also called eigenvalue-assignment control, (Magni, 22). Next, the controller design approach proposed by Piovoso et. al. (994; 996) is presented. Let X be composed of two types of variables (exogenous and manipulated) X = [X ex X mp ] (7) The development of a PCA model yields X = [X ex X mp ] = T P T + E (8) The process output signal can be decomposed in where f v is the output residual. x d = T q T + f v (9) The equivalent controller set point in the scores space is obtained from (), where Ψ is the pseudo-inverse. t sp = x d,sp (q T ) Ψ () The score vector, t, can be computed from the projection of x onto the matrix of eigenvectors P, obeying t = x P () 5484

3 Preprints of the 8th IFAC World Congress Milano (Italy) August 28 - September 2, 2 In the scores space the error between the desired set point and the online scores associated with x at discrete time k is given by t = t sp - t (2) The error in the scores space can be reconstructed as an error in the X space by x = t P T (3) The relationship between the exogenous and manipulated variables in the score space must be defined. Consider the partition of the matrix of eigenvectors P as P T =[P ex P mp ] (4) The relationship between exogenous and manipulated variables is given by (5), where Λ p is the matrix of coefficients that defines this relationship in the scores space. P mp = P ex Λ p (5) So, the matrix Λ p can be computed using the pseudoinverse Ψ Λ p = P ex Ψ Pmp (6) The architecture of the PCA controller used in this work is depicted in Fig. (Brito Palma, et al., 2). The block P a represents the plant, C is the PCA controller, and Q is given by x t sp d, sp Q + Q = (q T ) Ψ (7) t x + C mp P a - t Fig.. Architecture of the PCA controller. 3.2 Implementation The PCA model is a steady-state model that results in an integral PCA controller. The implementation presented will be detailed assuming, without loss of generality, a linear SISO process, modelled by an autoregressive ARX inputoutput model. First, an ARX model structure should be selected by means of a system identification algorithm such as least-squares, principal components regression, etc (Brito Palma, 27). In this work, a relay controller was used to capture the nominal data around a certain operating point. Let s assume a low order ARX model, ARX(na=2, nb=, nd=). The matrix X R n m, mean centered, will have lines of vectors Q x d (regressor for each discrete time k) given by the expression x(k) = η(k) = [y(k) y(k-) y(k-2) u(k-)]. In this work, the values used for training are n = 2, m = 4; n is the number of data samples and m is the number of columns in the regressor η(k). The sampling time used was Ts = s. The number of exogenous variables is r = 3, the number of manipulated variables is p = m - r =, and the number of principal components is a = 2. The covariance matrix is computed from (), and the SVD decomposition (2) gives the eigenvalues and the eigenvectors. The loading matrix P R m a corresponds to the loading vectors V R m m associated with the a largest singular values. The scores matrix T R n a is computed based on (4). The process output is given by x d (k) = y(k) and q is given by q = (T T x d ) T (8) The decomposition of transposed P matrix permits to obtain the others matrices (4). The matrix Λ p is computed using (6). The previous computations were obtained offline. In online operation the next computations should be performed for each discrete time k. First, assign the reference signal to x d,sp (k). Next, compute the set point in the scores space using (). Compute the scores in the reduced 2D space from t(k) = x d (k) (q T ) Ψ (9) The control error in the scores space is given by (2). The control error in the X space is given by (3). Finally, the increment in the manipulated variable can be computed from x m (k) = t(k) P mp (2) In the incremental form, assuming u(k) = x mp (k), the control action is given by x mp (k) = x mp (k-) + K c x m (k) (2) The controller gain, K c, was introduced to enable tuning the closed-loop dynamics (Brito Palma, et al., 2); this gain can be found from pole placement techniques (Astrom & Hagglund, 988; Astrom & Wittenmark, 997). Another alternative tuning method proposed by Brito Palma, et al. (2 ) is the use of a relay controller to generate the data matrix X. Next, assume a gain around the unit, for input / output data normalized in the range [; ]. Many closed-loop simulations, in linear and nonlinear systems, show that this approach gives good performance for open-loop stable systems, as will be presented in the experimental results section. 5485

4 Preprints of the 8th IFAC World Congress Milano (Italy) August 28 - September 2, 2 4. PROCESS CONTROL BASED ON ADAPTIVE PCA MODELS Here, an adaptive version of the PCA controller presented in section 3, is described. The main idea is to compute online, recursively, the eigenvector matrix P(k) and the eigenvalue matrix Λ(k). The eigenvector matrix is computed accordingly (22), were I is an identity matrix, P v is a perturbation matrix and N p (k) is a diagonal normalization matrix (Peddaneni, et al., 24; Yao, et al., 29). P(k) = P(k-) (I + P v ) N p (k) (22) The recursive PCA algorithm implemented, detailed in Appendix A, was based on previous works (Peddaneni, et al., 24; Yao, et al., 29); a stabilization mechanism of the algorithm was implemented accordingly the ideas proposed by Jeng (2). The recursive PCA algorithm is initialized with the information obtained from the linear PCA model. Assuming that the eigenvector (loadings) matrix is computed online, P(k), for each time instant k, the controller described in section 3 can be implemented, now, in an adaptive way. The general architecture of the proposed adaptive PCA controller is depicted in Fig. 2, based on the architecture (Fig. ) and on the recursive PCA algorithm (Appendix A). The proposed PCA controllers do not need a process model in the design stage; this is their main advantage. At the present, the main drawback is to guarantee stability of the closed-loop system. Different experiments had shown a good performance of the adaptive PCA controller for different models and real plants (nonlinear water tank system, linear and nonlinear DC motor, etc). The tuning parameters allows the designer to adjust the behaviour of the closed loop system. In the next section, experimental results obtained with the three tank benchmark are presented. 5. EXPERIMENTAL RESULTS The results presented here were obtained with a nonlinear system (the three-tank benchmark, Fig. 3). 5. The Three-Tank Benchmark The three-tank benchmark was developed within the European Cosy project (Fig. 3) The detailed process description can be found in (Heiming & Lunze, 999). t(k) Q(k) P mp (k) K c (k) e(k) + u(k) r(k) - + z - y(k) x(k) δ Recursive PCA Algorithm z - λ Matrix decomposition P(k) Fig. 2. Architecture of the adaptive PCA controller. The performance of the adaptive PCA controller depends mainly on: a) the memory depth memory λ; λ presents a trade-off between tracking capability and estimation variance (a fixed value equals to /3 was used, for a memory of 3 samples = 3 seconds); b) the stabilization parameter δ of the recursive PCA algorithm (a value equals to.5 was selected); c) the controller gain K c (k) should be a small positive value around one, for data normalized in the range [; ]; a value of.5 was used in the experiments. Fig. 3. The three-tank benchmark. 5.2 Experimental Results The simulation results were obtained using the Matlab environment. It was considered that the valve V3 is opened and the set point for tank T is.33 (in normalized values, range [; ]). In Figure 4 can be observed, from top to bottom, the reference signal (set point, r) and the process output (y, level in tank T), the control error (e), the control action (u), the Alfa parameter α(k) and the controller gain K c, and finally the estimated closed-loop poles for an ARX(2,,) model. The switching between controllers (fixed PCA and adaptive PCA) occurs at time instant 2 s. At time t(k) = 42 s a set-point change occurs from.33 to.5. A dither signal has been added to the set-point signal in order to permit estimate online the ARX models. 5486

5 Preprints of the 8th IFAC World Congress Milano (Italy) August 28 - September 2, 2 r,y e u Af,Kc p Time [s] Fig. 4. Experimental results for nominal operation with setpoint change. In order to compare the performance of the PCA controllers with other controllers some experiments were done. In Table are presented the results for the performance index (MSE) of the control error, for different set-points (SP). The controllers are: a) the linear PCA controller (section 3); b) the adaptive PCA controller (section 4); c) the classical PI controller (incremental version with gains K p = 6.7, T i = 2. s) autotuned with the relay feedback approach (Astrom & Hagglund, 988); d) the adaptive optimal polynomial LQG controller with design parameter r q =. (Levine, 996). The PCA controllers present reasonable performances compared to other controllers. In the future an optimization algorithm to adjust the gain K c (k) should be incorporated in the PCA controllers, in order to optimize their performance. Table. Comparison of controller s performance (MSE: mean squared error). MSE( -3 ) SP:.33 SP:.33.5 SP:.33.7 PCA Adp_PCA PI LQG In order to analyse the robustness of the adaptive PCA controller in terms of faults the next experiment (Fig. 5) show the behaviour when a leak in tank T occurs at time instant 36 s. As expected a saturation of the control action occurs and also the degradation of the persistent excitation conditions. r,y e u Af,Kc p Time [s] Fig. 5. Experimental results for faulty situation (leak in tank T). 6. CONCLUSIONS In this paper, a PCA controller design approach for SISO systems was presented. Two types of PCA controllers were described: a controller based on a fixed PCA model and an adaptive controller based on recursive PCA using eigenvector perturbation matrix. Algorithms for controller implementation, with an adjustable gain, were described. Some tuning rules were proposed for both controllers. The PCA controllers reveal good performances, to deal with linear and nonlinear systems, compared to other controllers (PI and LQG). The incorporated controller gain allows adjusting the closed-loop dynamics. Some pointers for future work are: a) test both PCA controllers with different benchmark systems with special features (deadtime, pure time-delays, different kinds of nonlinearities, etc); b) analyze the adaptive PCA controller performance for different gains, for different forgetting factors (memory depth parameters), and for different number of principal components; c) try to implement neural adaptive PCA controllers for nonlinear systems and time-varying systems; d) incorporate an optimization algorithm in order to adjust the controller gain and guarantee stability; e) establish theoretically the stability conditions. 5487

6 Preprints of the 8th IFAC World Congress Milano (Italy) August 28 - September 2, 2 REFERENCES Akamatsu, K., S. Lakshminarayanan, H. Manako, H. Takada, T. Satou, S. Shah (2), Data-based control of an industrial tubular reactor, in Control Eng. Practice, 8, pp Astrom, K., T. Hagglund (988), Automatic Tuning of PID Controllers, Instrument Society of America. Astrom, K., B. Wittenmark (995), Adaptive Control, Addison-Wesley. Astrom, K., B. Wittenmark (997), Computer Control Systems, Prentice-Hall. Brito Palma, L., F. Vieira Coito, R. Neves-Silva (25), Diagnosis of Parametric Faults Based on Identification and Statistical Methods, in 44th IEEE Conference on Decision and Control, and European Control Conference, Joint CDC ECC, Dec. 2-5, Seville, Spain, 25. Brito Palma, L. (27), Fault Detection, Diagnosis and Fault Tolerance Approaches in Dynamic Systems based on Black-Box Models, Phd Thesis, Universidade Nova de Lisboa (UNL-FCT-DEE), Portugal. Brito Palma, L., F. Vieira Coito, P. Sousa Gil, R. Neves-Silva (2), Process Control based on PCA Models, 5th IEEE Int. Conf. on Emerging Technologies and Factory Automation, Setp. 3-6, Univ. of the Basque Country, Bilbao - Spain. Champagne, B. (994), Adaptive Eigendecomposition of Data Covariance Matrices based on First-Order Perturbations, IEEE Transactions on Signal Processing, vol. 42, no.. Chen, G., T. J. McAvoy (996), Process Control Utilizing Data Based Multivariate Statistical Models, in Canadian Journal of Chemical Eng., Vol. 74, pp Chen, J., W. Wang (29), Performance Assessment of Multivariable Control Systems Using PCA Control Charts, in 4th IEEE Conf. on Industrial Electronics and Applications (ICIEA), China. Chiang, L., E. Russell, R. Braatz (2), Fault Detection and Diagnosis in Industrial Systems, Springer-Verlag. Feng, G., R. Lozano (999), Adaptive Control Systems, Newnes. Heiming, B., J. Lunze (999), Definition of the Three-Tank Benchmark Problem for Controller Reconfiguration, in Proc. of the European Control Conference ECC 99, Karlsruhe, Germany. Jackson, J. (23), A User s Guide to Principal Components, Wiley. Jeng, J. (2), Adaptive Process Monitoring using Efficient Recursive PCA and Moving Window PCA Algorithms, in Journal of the Taiwan Institute of Chemical Engineers, No. 4, pp Jolliffe, I. (22), Principal Component Analysis, Springer. Levine, W. (996), The Control Handbook, CRC Press & IEEE Press. Li, W., H. Yue, S. Valle-Cervantes, S. Qin (2), Recursive PCA for Adaptive Process Monitoring, in Journal of Process Control, No., pp Magni, J. (22), Robust Modal Control with a Toolbox for Use with Matlab, Kluwer. Peddaneni, H., D. Erdogmus, Y. Rao, A. Hedge, J. Principe (24), Recursive Principal Components Analysis using Eigenvector Matrix Perturbation, Proc. Of IEEE Workshop on Machine Learning for Signal Processing, São Luís Maranhão, Brazil. Piovoso, M., K. Kosanovich (994), Applications of Multivariate Statistical Methods to Process Monitoring and Controller Design, in Int. Journal of Control, Vol. 59, No. 3, pp Piovoso, M. (996), The Use of Multivariate Statistics in Process Control, in The Control Handbook (edited by W. Levine), pp , CRC Press & IEEE Press. Yao, J., X. Liu, X. Zhu (29), Reduced Dimension Control Based on Online Recursive Principal Component Analysis, in Proc. of American Control Conference, St. Louis, USA. Appendix A. THE RECURSIVE PCA ALGORITHM WITH A STABILIZATION MECHANISM (Peddaneni, et al., 24; Yao, et al., 29; Jeng, 2). Initialize the orthonormal eigenvector matrix P = I and the diagonal eigenvalue matrix Λ = diag(r N ) where R N is the estimated covariance matrix using N input samples. 2. At each time instant k do the following tasks: a. Get input sample vector (ARX(2,,) regressor) x(k) = η(k) = [y(k) y(k-) y(k-2) u(k-)]; b. Set memory depth parameter λ(k) [; ]; for a fixed value assume λ = / N s, where N s is the number of samples; c. Compute α(k) = P(k-) T x(k); d. Compute eigenvalue perturbation matrix P Λ and eigenvector perturbation matrix P v accordingly the following equations: α 2 i = (i,i) th element of P Λ = (i,i) th element of P v α i α j max(δ λ,λ j + α 2 j -λ i - α 2 i ) = (i,j)th element of P v i > j α i α j min(-δ λ,λ j + α 2 j -λ i - α 2 i ) = (i,j)th element of P v i < j The variables α n are obtained from α(k). The variables λ n are obtained from P Λ. The variable δ should be a small positive number to guarantee stability. e. Update eigenvector and eigenvalue matrices: P u (k) = P(k-) (I + P v ) Λ u (k) = ( - λ(k)) Λ(k-) + λ(k) P Λ f. Normalize the norms of eigenvector estimates by P(k) = P u (k) N p (k), where N p (k) is a diagonal normalization matrix containing the inverses of the norms of each column of P u (k); g. Correct eigenvalue estimates by Λ(k) = Λ u (k) N Λ (k) where N Λ (k) is a diagonal matrix containing the squared norms of the columns of Λ u (k). 5488