Motivation. From SS to TF (review) Realization: From TF to SS. MECH468 Modern Control Engineering MECH550P Foundations in Control Engineering
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1 MECH468 Modern Control Engineering MECH550P Foundations in Control Engineering Realization Dr. Ryozo Nagamune Department of Mechanical Engineering University of British Columbia 2008/09 MECH468/550P 1 Motivation Now, we have learned two ways to describe LTI systems, i.e., TF and SS models. TF: SS: To use analysis & design techniques for SS models, one may want to transform a TF model to an equivalent SS model. How? More generally, what is the relationship between two models? 2008/09 MECH468/550P 2 From SS to TF (review) CT LTI SS model Realization: From TF to SS Given a rational proper transfer matrix find matrices (A,B,C,D) s.t. Laplace transform with x(0)=0 Memorize this! Rationality: : (polynomial)/(polynomial) Rational Non-rational Properness: deg(num)<= )<=deg(den) Proper Non-proper 2008/09 MECH468/550P /09 MECH468/550P 4 1
2 Remarks For one SS model, there is a unique TF. For one TF model, there are infinitely many SS! First step for realization Always extract D-matrix D first! (A,B,C,D) unique G(s) Ex constant Strictly proper infinitely many Coordinate transformation T Order difference After realization, check the correctness by recovering the original transfer matrix. constant After extracting D, find (A,B,C) s.t. Strictly proper 2008/09 MECH468/550P /09 MECH468/550P 6 Controllable canonical form SISO example Companion matrix The following form (and its transpose) of a matrix is called companion matrix (form): or equivalently Important property of a companion matrix 2008/09 MECH468/550P /09 MECH468/550P 8 2
3 Ex.1 SISO example Controllable canonical form MIMO case Least common denominator Ex /09 MECH468/550P /09 MECH468/550P 10 Transfer matrix MIMO example Remark Controllable canonical realization is always controllable (but not always observable). Why? Note that the size of A-matrix A is four. 2008/09 MECH468/550P /09 MECH468/550P 12 3
4 Observable canonical form SISO example Ex.1 SISO example Ex.2 or equivalently 2008/09 MECH468/550P /09 MECH468/550P 14 Observable canonical form MIMO case Transfer matrix MIMO example Least common denominator Note that the size of A-matrix A is two, not four! Q: : What is the smallest size of A? (Minimal realization: next lecture) 2008/09 MECH468/550P /09 MECH468/550P 16 4
5 Remark Note the duality between controllable and observable canonical form. Observable canonical realization is always observable (but not always controllable). Why? Review So far, we considered the realizations of the following transfer matrix: Controllable canonical form with order 4 Observable canonical form with order /09 MECH468/550P /09 MECH468/550P 18 Topics from now on Realization of the smallest order How to characterize such realization? How to obtain such realization? Some terminologies: Minimal realization of G(s): Realization (A,B,C,D) of G that has the smallest dimension of A-matrix. A McMillan degree of G(s): The dimension of A of the minimal realization. (This indicates the complexity of a system G(s).) Why minimal realization? Easy to Analyze (understand) the system Design a controller Implement a controller Computationally less demanding in both design and implementation Higher reliability Few parts to go wrong in the hardware Few bugs to fix in the software 2008/09 MECH468/550P /09 MECH468/550P 20 5
6 Two important facts on minimal realization Fact 1: A realization (A,B,C,D) is minimal if and only if (A,B) is controllable and (A,C) is observable. Nonminimal realization examples Ex Controllable canonical form Is it observable? No! Fact 2: All minimal realizations of G(s) ) are related by coordinate transformations. Observable canonical form Is it controllable? No! (Sketches of proofs are given in the appendix.) 2008/09 MECH468/550P /09 MECH468/550P 22 How to obtain minimal realization SISO (Single-Input Input-Single-Output) case Remove common factors from numerator and denominator of G(s). Then, realize G in a controllable (or observable) canonical form. How to obtain minimal realization SIMO case (G(s( G(s) ) is a column vector) Use the controllable canonical form. C.C.F. MISO case (G(s( G(s) ) is a row vector) Use the observable canonical form. Observable! Minimal! McMillan degree /09 MECH468/550P /09 MECH468/550P 24 6
7 How to obtain minimal realization MIMO (Multiple-Input Input-Multiple-Output) case 1. Realize G(s) ) in some non-minimal canonical form. 2. Use the Kalman decomposition to remove uncontrollable/unobservable parts. ( minreal.m( minreal.m ) Kalman decomposition (review) Every SS model can be transformed by z=tx for some appropriate T into a canonical form: Remark: Unfortunately, it is generally hard to compute the Kalman decomposition by hand. Remark: There is another famous algorithm, called Ho s s algorithm,, to compute a minimal realization. (Not covered in this course.) 2008/09 MECH468/550P /09 MECH468/550P 26 Kalman decomposition (review) Conceptual figure (Not block diagram!) System TF Example Order /09 MECH468/550P /09 MECH468/550P 28 7
8 Minimal realization Example (cont d) Controllable! Example (cont d) After realization, check the correctness by recovering the original transfer matrix. (1,1) (2,1) Observable! (1,2) Order 3 Minimal! (2,2) 2008/09 MECH468/550P /09 MECH468/550P 30 Summary Realization (In Matlab, ss.m) Controllable canonical form Observability canonical form Minimal realization Realization for DT systems is exactly the same as that for CT systems. Exercises Obtain controllable canonical realizations for the following transfer matrices. By hand-calculation, check the correctness of your answers. 2008/09 MECH468/550P /09 MECH468/550P 32 8
9 Exercises Obtain realizations in observable canonical forms for the following transfer matrices. Exercises By both hand-calculation and Matlab,, obtain minimal realization for: Using the obtained realizations above, find realizations of the following TFs. By hand-calculation, check the correctness by recovering the original transfer matrix. 2008/09 MECH468/550P /09 MECH468/550P 34 TF Appendix: Derivation of controllable canonical form Appendix: Derivation (cont d) Introduce state variables Introduce a variable x(s) Then, We have By inverse Laplace transform, done! 2008/09 MECH468/550P /09 MECH468/550P 36 9
10 TF Appendix: Derivation of observable canonical form Appendix: Derivation (cont d) Draw block-diagram Rewrite I/O relation as By introducing state variables as in the figure, done! 2008/09 MECH468/550P /09 MECH468/550P 38 Appendix: Proof of Fact 1 only if -part : If (A,B) is not controllable or (A,C) is not observable, due to Kalman decomposition theorem, there is a realization with less state dimension. if -part : Suppose (A,B,C) controllable & observable. Appendix: Proof of Fact 2 Given a rational proper transfer matrix, assume two minimal realizations: Contradiction! Define T as Then, due to One can verify the following by using (by ) 2008/09 MECH468/550P /09 MECH468/550P 40 10
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