A Novel Constraint Narrowing Technique for MIMO Unstable System

Frequency response data collection can be a boon for modeling of MIMO uncertain plant. System stability can be assessed either by transfer function or by state-space method. Both will arrive at matrix transformation and further decision approach. Both can be considered for diagonalization of matrix. It is a proven fact that when the matrix is diagonalized the elements of the principle diagonal are the Eigen values and these Eigen values are closed loop poles from which stability can be assessed. The feature of such a diagonal matrix is that its principle diagonal elements contain gains of all the feedback paths. Singular value decomposition is used here for diagonalization. Singular value decomposition technique has been demonstrated by many authors but, application of PCA with Euclidian norm has not been paid attention so far. The systems numerical array is fed to a digital processing tool such as Mat lab and SVD-PCA (Singular Value Decomposition- Principal Component Analysis) is applied to determine the reduction of disturbance or noise and to provide minimum sensitivity and error correction. There are Hull, Box and KB consistency narrowing techniques used previously and the idea is extended further and an SVD-PCA-Norm technique which is now referred as LA criteria has been demonstrated here.


INTRODUCTION
Good performance of control system is the result of combination of feed-forward and feedback control systems. Stability is the constraint applicable to feedback control due to the uncertainty in tracking and measured noise filtering, whereas sensor availability and modeling errors limit the performance of feed-forward system.
Generally, a 2-DOF is selected for demonstration in which the output of the plant and reference are available to the control system. The number of degree of freedom is defined as the number of closed loop transfer function that can be designed independently. In 2-DOF closed loop systems, there are transfer functions from disturbance to output and reference to output can be designed independently.
Many control requirements are assessed in frequency domain. The ability of the control system to reject the disturbances whose frequency components are concentrated on a certain band determines its performance. It is a proven fact that the effective control band is the one whose worst-case sensitivity is below 6 dB which indicates a minimum attenuation of 50% of output disturbance. In PCA actually very few components are selected which is as good as rejecting frequency components in a particular band and thus amounts to 50% of disturbance rejection. For disturbance rejection a comparison of the worst case open loop response and the closed loop response will determine how effective control design has been. In other words, there are finite set of constraints which specify which value combination from given variable domains are admitted and the value combination satisfying all constraints, that means rounding off errors and this has been done by PCA-Euclidian Norm.

II. DESIGN CONSTRINATS AND SATISFACTION
Referring to above inferences a conclusion is reached whereby High gain, low sensitivity and larger bandwidth provides stability. In this paper a LA criterion is demonstrated which suggest application of SVD-PCA Norm technique to reduce noise and disturbance and uncontrolled variables elimination by pairing.

III.
INFERENCES AND VALIDATIONS This section brings out the proven inferences and its validation with respect to LA criteria.

A.
Sensitivity must be minimized to get good disturbance rejection The plant, the controller and the pre-filter are the components of a control system which are governed by noted If N(s) = 0, then desired output is achieved as Y(s) = F(s)R(s); the output follows the reference input. That implies s+2.5 =0; a single root by which the system is said to be stable. For N(s) = F(s) R(s) = 0 then A pole at s= +0.5 implies that output response to a disturbance is unstable; this is because the characteristic equation does not include pole-zero cancellation.
If ≈ 0, then output response to disturbance is good and ( ) ( ) ≈ 1 implies that set point tracking occurs. The constraint on sensitivity and complementary sensitivity is that S(s) + T(s) = 1. High loop gain at low frequency and low gain at high frequency are some of the inferences related to sensitivity and complementary sensitivity which requires that:  For tracking of reference signal and good rejection of disturbance it is required that S(s) ≈0; T(s) ≈ 1, which can be met by having loop gain say |L(s)| ≥ 1  To prevent propagation of measurement noise to the error and output signals it is required that T(s) ≈0 => S(s) ≈ 1which is met by having loop gain say |L(s)| ≤ 1  In general, |L(s)| ≥ 1 s required at low frequency and |L(s)| ≤ 1 is required at high frequency.

B.
Maximum Gain corresponds to the Eigen Vector associated with maximum Eigen Value. In the above equation (1) to make N(s) = 0 the system gain must be maximum and maximum gain corresponds to the eigenvector associated to the maximum Eigen value. Let us consider a 2x2 matrix to assess this concept: The characteristics equation is given by | A-λI | = 0, arranging the equation we get, Putting λ= 6 and λ = 1in the above equation we get And hence the Eigen vectors are (4, 1). The maximum Eigen vector corresponding to the maximum Eigen value is 4 and hence the maximum gain is 4. The maximum gain implies minimum sensitivity which is required for stability of a system.

C.
Rejection of frequency components in a particular band amounts to minimum 50% of attenuation of Noise and Disturbance. To assess, consider the following matrix As can be seen the SVD-PCA gives out compressed data array amounting to minimum 50% reduction and hence a comprehensive reduction of noise disturbance. It is a proven fact that the effective control band is the one whose worstcase sensitivity is below 6 db which indicates a minimum attenuation of 50% of output disturbance. In PCA actually very few components are selected which is as good as rejecting frequency components in a particular band and thus amounts to 50% of disturbance rejection. SVD matrix provides three matrices U, the row matrix, Σ, the diagonal matrix V, the column matrix maximum, minimum gain and its ratio the condition number N. The first column of matrix V from controller output transfer matrix is the combination of manipulation with highest effect on the control objective and the first column of matrix U from disturbance output matrix points out better measure of controlled variables. Minimum condition number N (close to 1) must be achieved to have gain and sensitivity stability. The condition number with K is 197.8571, the condition number with V is 0.9999 and it has been referred to as V can be fed back to attain stability and its CN number is near unity, which can contribute to stability and the condition number with PCA is 1. The SVD-PCA are helping to bring down the highest condition number from 197.8571 to near unity. An SVD gives an idea of system matrix acting upon an input at which is achieved by norm.  Another inference is that the larger the condition number of the diagonal matrix the more unstable is the system. In the above example Σ is the diagonal matrix and its condition number N≈ 198which is quite large and the system oscillates and hence it should be kept as low as possible to attain stability. D. One Transfer Function is enough to assess the stability The stability can be drawn at by Nyquist plot and in that if Eigen value locus does not encircle the point (-1, 0) the MIMO system is closed loop stable. The state space representation of a system in standard format is: X = Ax + Bu Y= Cx + Du The equations above gives rise to four matrices namely; A -State Matrix; B -Control Matrix; C -Output Matrix and D -Transmission Matrix. For closed loop stability only one transfer matrix must be checked instead of four. Consider an example   Zero steady state error is obtained for a constant reference signal or disturbance signal by having low frequency slope of loop gain |L(s)| at -20db/dec and for a linearly increasing reference signal or disturbance signal a low frequency slope of -40 db/dec is required. Adequate phase margin must be provided which can be achieved by having the slope of the magnitude curve at the gain crossover frequency at -20db/dec.
Bandwidth indicates the frequency range for which satisfactory set point tracking occurs and it should be large enough for speedy response. The performance specifications of closed loop control system are robust stability for which high gain and low sensitivity must be maintained and robust tracking for which bandwidth must be high.

V.
CONCLUSION The paper demonstrates how LA criteria propose to solve closed loop stability at constrained intervals. The constraints may represent various bounds of control system stability. Uncontrolled systems behavior can be fed in the shape of numerical array to the processing tool such as Matlab. The SVD-PCA-NORM (LA criteria) is applied to the array until satisfactory compressibility is observed and the ultimate result is reduction in measurement noise and disturbance. The challenge of this criterion now lies in extending the idea to unknown source of disturbance and measurement noise, tracking and sensitivity minimization along with gain maximization when calculations goes unpredictable. Our first paper laid the foundation for this research work, whereas this paper illustrates the conceptual ground work and the upcoming paper will demonstrates this LA criterion for a real time dynamic system.