Fragility of controllers Robust and non-fragile control systems - PowerPoint PPT Presentation

5/16/2018 2 1 Fragility of controllers Robust and non-fragile control systems Robust and non-fragile control systems 5/7/2018 5/7/2018 Any controller should be able to tolerate some uncertainty in its coefficients (stability and performance) Inherent imprecision in analog-digital and digital-analog conversion  Finite resolution measuring instruments  Round off errors in numerical computations  Finite word length   Inaccuracies in controller implementation Example (fragility of 𝜈 Based controller) Example (fragility of 𝜈 Based controller): 3 4 Robust and non-fragile control systems Robust and non-fragile control systems A robust controller for an electromagnetic suspension system is designed 5/7/2018 5/7/2018 by using the 𝜈 synthesis technique[1] : Transfer function coefficients of the controller ρ ∶ 𝑚 � Parametric stability margin around the nominal point The normalized ratio of change in controller coefficients required to destabilize the closed loop is: 1

5/16/2018 Non-fragile controller design based on Example results 5 6 centroid of admissible regions Robust and non-fragile control systems Robust and non-fragile control systems 5/7/2018 5/7/2018  In [1] optimum and robust controllers, designed by using the H2, H1, l1, and 𝜈 formulations, can produce extremely fragile controllers Being situated away from the boundaries of the stability region in the controller  Badly chosen optimization criteria => controller parameters that are parameter space, controllers designed based on the centroid method are m mathematically ill-posed both robust and non-fragile.  The stability regions in the parameter space of higher order systems have “ instability holes ” and the optimization algorithm can stuff the controller Non-fragility criterion=minimum distance to the boundary of stability region parameter into tight spots close to these holes Non-convex stability region: Finding the center of the largest convex region  Good gain and phase margins are not necessarily reliable indicators of robustness. 𝟐 However, poor gain and/or phase margins are accurate indicators of Optimization problem : 𝒏𝒃𝒚  𝐧𝐛𝐲 (𝒔 ℂ ) = 𝒍 �(𝒜) � 𝑯 fragileness!  Controller sensitivity which may be important in other non-optimal design Example[2]: IPDT process or a FOPDT techniques as well. Non-fragile PID controller in the viewpoint of the center of mass ~ Ziegler–Nichols Non-fragile controller design based on Non-fragile controller design based on 7 8 pole sensitivity minimization pole sensitivity minimization Robust and non-fragile control systems Robust and non-fragile control systems 5/7/2018 5/7/2018 A new measure: ► Controller fragility will depend upon the particular realization of the controller. ► Handles the fragility problem by minimization of the eigenvalue sensitivity to The measure is a weighted sum of a 2-norm of the sensitivity of the individual controller parameter perturbations. closed loop system pole/eigenvalues to perturbations in the controller ► The eigenvalues pairs closest to the imaginary axis can be weighted more parameters. heavily ► Numerical method for obtaining the solution ► Parameter uncertainty is small so that first-order perturbation equations can be obtained 𝑦 � : 𝑗 = 1, … , 𝑜 � the controllers parameters 2

5/16/2018 9 10 Normal matrix Pseudospectra Robust and non-fragile control systems Robust and non-fragile control systems 5/7/2018 5/7/2018 ↔ 𝐵𝐵 ∗ = 𝐵 ∗ A  A is normal  A is diagonalizable by a unitary matrix  There exists one complete and orthogonal set of eigenvectors of A  Eigenvalues describe the behavior of dynamical systems  While this is only true for normal matrix  For nonnormal matrices eigenvalue analysis proves to be misleading Pseudospectra 11 Pseudospectra(examples) 12 Robust and non-fragile control systems Robust and non-fragile control systems 5/7/2018 5/7/2018 𝜁 − pseudospectrum of A is the set of 𝑨 ∈ 𝐷 1 ST def. (𝑨 − 𝐵) �� = 𝜁 �� 2 nd def. 𝑨 ∈ 𝜏 𝐵 + 𝐹 𝑔𝑝𝑠 𝐹 ∈ 𝐷 �×� , 𝐹 < 𝜁 3 rd def. (𝑨 − 𝐵)𝜉 𝑔𝑝𝑠 𝐹 ∈ 𝐷 � , < 𝜁 𝜉 = 1 3

5/16/2018 State space representation of structured Fragility criterion 13 14 perturbation Robust and non-fragile control systems Robust and non-fragile control systems 5/7/2018 5/7/2018 [5] Where 𝐻(𝑡 � ) �� = � �(�(� � )) is equal to the norm of the smallest perturbation that � 𝑡 � is closed loop eigenvalue of the system Robust stability and performance 15 16 The stability threshold Robust and non-fragile control systems Robust and non-fragile control systems 5/7/2018 5/7/2018 4

5/16/2018 17 Simulation results 18 Robust and nonfragile control system Robust and non-fragile control systems Robust and non-fragile control systems 5/7/2018 5/7/2018 𝑈 � + 𝜍 1 𝜍 ∶ Weighting function 𝐾 = (importance of controller fragility) 𝑌 � 𝑈 � + 𝜍 � Minimize 𝐾 = � � � Conclusions: 19 18 Simulation results Robust and non-fragile control systems Robust and non-fragile control systems 5/7/2018 5/7/2018  Since modern control theory results in higher-order controllers for complicated large scale MIMO systems, controller designs based on fragility and nonnormal matrices would maintain stability for perturbations.  Adding the stability threshold to objective function is proved to be effective through several examples with regards to pseudospectra plot 𝑇𝑢𝑠𝑣𝑑𝑢𝑣𝑠𝑓𝑒 𝑞𝑡𝑓𝑣𝑒𝑝𝑡𝑞𝑓𝑑𝑢𝑠𝑏 𝑝𝑔 𝐵 − 𝑑𝑚 𝑇𝑢𝑠𝑣𝑑𝑢𝑣𝑠𝑓𝑒 𝑞𝑡𝑓𝑣𝑒𝑝𝑡𝑞𝑓𝑑𝑢𝑠𝑏 𝑝𝑔 𝐵 − 𝑑𝑚 (𝐼𝑗𝑜𝑔 𝑡𝑧𝑜 𝑛𝑝𝑒𝑗𝑔𝑗𝑓𝑒 𝑑𝑝𝑜𝑢𝑠𝑝𝑚𝑚𝑓𝑠) (𝐼𝑗𝑜𝑔 𝑡𝑧𝑜 𝑑𝑝𝑜𝑢𝑠𝑝𝑚𝑚𝑓𝑠) 5

5/16/2018 References Challenges: 19 [1] L. H. Keel, P. Bhattacharyya, “Robust, fragile or optimal?”, American Control Conference, 1997. Robust and non-fragile control systems 5/7/2018 Proceedings of the 1997. Vol. 2. IEEE, 1997. [2] Bahavarnia, M, Tavazoei, MS. ”A new view to Ziegler–Nichols step response tuning method: analytic non- fragility justification.” J Process Contr 2013; 23: 23–33  Formulating fragility criterion as an convex problem [3] Hamid Zargaran, “Application of Pseudo spectra in synthesis of robust and fault-tolerant control systems”,  Finding the optimum value of G(jw) �� M.Sc. Thesis  Perturbation to other state matrices [4] L. N. Trefethen and M. Embree, Spectra and pseudospectra: the behavior of nonnormal matrices and operators, Princeton University Press, 2005 [5] D. Hinrichsen and K. Bernd Kelb, Spectral value sets: a graphical tool for robustness analysis, Systems Control Letters, vol. 21, no. 2, pp. 127-136, 1993. Thanks! Any questions? 6