A universal tool for solving different control problems Ulle - PowerPoint PPT Presentation

A universal tool for solving different control problems ¨ Ulle Kotta kotta@cc.ioc.ee Institute of Cybernetics at TUT A universal tool for solving different control problems – p. 1/28

� � � Contents Three control problems Algebraic formalism Solutions and examples A universal tool for solving different control problems – p. 2/28

� � � Three control problems Analysis Controllability (Accessibility) Synthesis Linearization by static state feedback Modelling Realization in classical state-space form A universal tool for solving different control problems – p. 3/28

✒ ✔ ✡ ✎ � ✕ � ☛ ✡ ✏ ✝ ✒ ✓ ✑ ✓ ✑ ✝ ✌ ✞ ✍ ✌✓ ✌ � ✁ ✂ ✄ ☎ ✆ ✠ ✡ ☛ ☞ ✆ ✝ ✞ ✡ Accessibility System ✝✟✞ is said to be accessible if it does not have any nonzero autonomous element Function is said to be an autonomous element if there exists an integer and a nonzero meromorphic function such that A universal tool for solving different control problems – p. 4/28

✆ ✝ � ✞ ✡ ✌ ✆ ✝ ✡ ✎ ✝ ☛ ✆ ☎ ✡ ✝ ✝ ✞ ✠ ✡ ☛ ✝ ✝ ✞ ☞ ☛ ✆ ✆ ✁ ✂ ✄ ✆ ✝ ✞ ✠ ✡ ☛ ☞ ✡ ✡ ✡ ✌ ✍ ✝ ✞ ✡ ✎ ✍ ✝ ✞ ✡ Linearization ✝✟✞ ✝✟✞ ✝✟✞ A universal tool for solving different control problems – p. 5/28

✂ ✌ � ✠ ✡ ✎ ✆ ✝ ✞ ✠ ✡ ☛ ☞ ✆ ☎ ✡ ✍ ✆ ✝ ✞ ✡ ✎ � ✝ ✞ ✡ ☛ ✄ ☞ ✆ ✝ ✞ ✁ ✍ ✎ � ✝ ✁ ✡ ☛ ☞ � ✝ ✞ ✡ ✌ ✓ ✓ ✓ ✌ ✝ ✌ ✞ ✁ ✂ ✠ ✡ ✌ ✍ ✝ ✞ ✡ ✌ ✓ ✓ ✓ ✡ State-space realization ✝✟✞ ✝✟✞ ✝✟✞ NB!!! A universal tool for solving different control problems – p. 6/28

✆ ✄ ✟ ✌ ✞ ✌ ✍ ✍ ✌✍ ✆ ✂ ✎ ✞ ✌ ✠ ✏ ✎ ✝ ✟ ✄ ✝ ✏ ☛ ✍ ✆ ✠ ✂ ✓ ✌ ✘ ✌ ✍ ✌✍ ✆✑ ✌✍ ✡ ✝ ✆ ✍ ✍ � ✒ ✂ ✌ ✜ ✝ ✞ ✝ ✆ ✌ ✡ ✞ ✝ ✝ � ✡ ☛ ✡ ✠ ✞ ✝ ✝ � ✒ ✡ � ✍ ✠ ✍ ✌✍ ✆ ✘ ✚ ✝ ✆ ✌ � ✁ ✆ ✝ ✟ ✁ ✞ ✝ ✆ ✑ ✌ A universal tool for solving different control problems – p. 7/28 an extended state-space system ✂☎✄ ✠✛✚ Extended system ✠✛✚ Associate to ✂☎✄ ✠✙✘ ✂☎✄ ✡☞☛ ✂✗✖ ✂☎✄ ✂☎✄ ✓✕✔

✌ ✝ ✞ ✡ � ✍ ✝ ✞ ✡ ✡ ☛ ✂ ✝ ✆ ✆ ✝ ✞ ✡ ✌ ✍ ✝ ✞ ✡ ✡ ✝ ✝ ✍ ✡ ✒ � ✡ � ✆ ✝ ✕ ✡ ✌ ✍ ✡ ✂ ✌ ✞ ✕ ✁ � ✠ ✒ ☎ ✞ ✝ ✒ ✌ The forward-shift operator Let denote the field of meromorphic functions in a finite number of variables ✝✟✞ The forward-shift operator is defined by is one-to-one, if the system is submersive. A universal tool for solving different control problems – p. 8/28

☛ ✡ ☛ ✡ ✕ ✠ ✡ ✡ ✒ ✄ ✂ ✁ � ✁ ✠ ✍ ✞ ✌ ✆ ✝ ✝ ✡ ✍ ✌ ✆ ✝ ✝ ✕ ✂ ✁ � ✝ Submersivity The system is said to be submersive , if ✄✆☎ ✝ ✟✞ A universal tool for solving different control problems – p. 9/28

✝ ☛ ✆ ✆ ✡ ✝ ✆ ☛ � ✁ ✂ ✆ ☎ ✕ ✠ ✕ ✂ ✠ ✕ ✕ ✝ ✆ ✝ ✄ ✕ ✆ � ✁ ☛ ✞ ✆ � ✂ ☛ ✂ ✁ ✂ ✆ � ☎ ☛ ✆ ✁ ✆ ✂ � ✁ ✁ Submersivity - example ✆ ✄✂ ✄✆☎ ✆ ✄✂ The system is not submersive A universal tool for solving different control problems – p. 10/28

� ✝ ✝ ✡ ✆ ✝ ✝ ✡ ✝ ✝ � � ✆ ✘ ✁ ✏ ✟ ✝ ✝ � ✏ Submersivity - example 1. Consider the following function ✁✄✂ ✁✄☎ ✞✠✟ 2. is not defined A universal tool for solving different control problems – p. 11/28

✒ ✝ ✌ ✒ ✡ Difference field If is one-to-one then pair is a difference field A universal tool for solving different control problems – p. 12/28

✂ � ☛ ✍ � ✆ ☎ ☛ ✆ ✂ � ✆ � ✁ ✆ ✁ ☎ ☛ ☎ ✆ ✆ ☛ ✂ ✆ ✁ ✆ ✆ � ✁ ✍ ✁ ✁ ✆ � ✂ ☛ ✆ ☎ ✍ ✁ Inversive closure It is not guaranteed that each element in has a pre-image. Example ✆ ✄✂ But for and there are no pre-images in . A universal tool for solving different control problems – p. 13/28

✁ ✄ ✆ ☛ � ☎ ✂ ✁ ✝ ☞ ✌ ✁ ✍ ☛ � ☎ ✂ � ✝ � ☛ ✡ ✁ ☎ ✂ ✁ ✝ ✄ ☞ ✁ ✍ ✆ ☛ � ☎ ✂ ✁ ✝ ✄ ✁ ✄ ☛ ☛ ✎ ✍ ✍ ✌✍ ✂ ✎ ✌ ✏ ✝ ✄ ✄ ☛ ✆ ✁ ☎ ✁ ✆ ✁ ✌ ✄ ✍ ☎ ✝ ✡ ✍ ✌ ✆ ✝ ✝ ✄ ☎ ✂ ✁ � ☎ ✂ ✆ ✞ ✆✝ � Construction of the inversive closure The idea is to construct a backward shift operator in order to enlarge domain of state-equation to the negative time instances . ✡ ✠✟ A universal tool for solving different control problems – p. 14/28

✝ ✁ ✁ ✝ ✂ ✁ ✁ ✘ ✞ ✝ ✂ ✜ ☎ ✌ ✞ ✂ ✁ ✝ ✘ ✜ ✁ ☎ ✞ � ✘ ✠ ✜ ✍ ✂ ✠ ✝ ✎ ✄ ✌ ✆ ✆ ✞ ✁ ✌ ✝ ☞ ✆ ✘ ✁ ✁ ✌ ☎ ✁ ✁ ✂ ☎ ✁ ✏ ✄ ✂ ✁ ✠ ✟ ✁ ✁ ✞ ✝ ✆ ☎ ✂ ☎ ✁ ✎ ✆ ✁ ✁ ✂ � ✝ ✠ ✝ ✄ ✆ ✏ ✟ ✆ ✠ ✌ ✁ ✂ ☛ ✝ ✁ ✁ ✡ ✠ ✝ ✞ ✝ ✆ ☎ ✄ ✂ ✁ ✟ ✟ ✁ ✞ ✎ Construction of the inversive closure Define such that Example - continuation ✁✄☎ ✂☎✄ A universal tool for solving different control problems – p. 15/28

� ✄ ✄ � ✄ ☎ ✄ ☛ ✑ ✄ ✒ ✌ ☎ ✄ ☛ ✝ ✒ ✑ ✄ ✡ ☎ ✑ ☎ � � ☎ ☛ ✄ ✝ ✁ ✂ ☎ ☛ � ✑ ✁ ✑ ✂ ✁ ✂ ✒ ☎ ✄ ✌ Vector space over Forward shift operator induces in a forward shift operator A universal tool for solving different control problems – p. 16/28

✆ ✁ ✁ ✆ ☛ � ☎ ✂ ✁ ✝ ✄ ✂ ✁ ☎ ☎ � ✂ ✄ � ✁ ✂ ✡ ☛ � � ✂ ✁ ☛ � ✁ Relative degree The relative degree of a one form If such an integer does not exist set A universal tool for solving different control problems – p. 17/28

� ✂ � � ✕ ☛ ✁ ✁ � ☛ ✁ ✂ ☛ ☛ ✁ � � is said to be exact if is said to be closed if Exact one-form is always closed. Closed form is not always exact globally but always exact locally. The subspace of differential one-forms is said to be completely integrable if it has a basis which consists of closed differential one forms only. A universal tool for solving different control problems – p. 18/28

✌✍ ✍ ✍ ✌✍ ☎ � ✝ ✌ � ✌ ☞ ✌ ✍ ✍ � ✏ ✝ ✍ � ☛ ✁ ✟ ✝ ✆ ☎ ✄ ✂ ✝ ✝ � ✄ ✂ ✁ � ✌✍ ✟ ✏ ✝ ☎ ☎ ✝ ☎ ✌ ✍ ✍ ✌✍ ✝ � ✆ ✝ ✄ ✁✂ ✝ � ✟ ✟ ✞ ✝ ☎ ✝ ☎ ✡ ✍ ✍ ✍ ✡ ☎ ✟ ✡ ✠ ☎ ✟ ✍ ✌ � � Frobenius theorem ✝ ✆☎ Let is the subspace of . is completely integrable iff If Frobenius theorem is satisfied then such that A universal tool for solving different control problems – p. 19/28

✠ ✂ � ✁ ☎ ☛ � ✞ ✁ ✂ ✞ ✞ ✞ ✁ ✌ ✄ ✂ ✁ ✄ � ✁ ☛ ✁ ✄ � ✂ ☛ � ✁ ✞ ✕ ✁ ☛ ✄ ✝ ✁ ✂ ☎ � ☛ ✆ ✝ ✡ ✁ ✁ ✁ � ✁ ☛ � ✁ ✂ ✁ ✁ ✁ ✂ ✞ The sequence of subspaces contains only one-forms of relative degree equal or greater than . A universal tool for solving different control problems – p. 20/28

✌✓ � ✌ ✓ ✓ � ✠ ☛ � ✌ ✁ ✁ ✠ ✠ � � ✁ ✕ � ☛ ☎ ✁ ✞ ✁ Solutions The system is accessible iff . The system is linearizable by static state feedback iff are completely integrable. The system admits classical state-space realization iff are completely integrable. A universal tool for solving different control problems – p. 21/28

✌ � ✞ ✂ ✁ ✆ ☛ ✂ ✆ ☎ ✁ ✂ ✁ ✝ ✄ ☛ ☎ ✂ ✆ ✆ ☛ ✆ ✁ ✡ ✠ ☎ ✂ ✆ ✝ ☛ � ✁ ✆ ☎ ✂ ✁ ✝ ✄ ☛ ☎ ✂ ☛ ✁ ✁ ✠ ✂ ✆ ☛ ✂ � ✆ ✂ ✡ ☎ ✆ ✂ ✆ ✝ ✁ ✆ ☛ ✁ � ✝ ✂ ✆ ✍ ☛ ☎ ✂ ✁ ✝ ✄ ☛ ✂ ☎ ☎ ✆ ☛ ☎ � ✆ ☎ ✡ ✠ ✂ Accessibility - example The system is not accessible. A universal tool for solving different control problems – p. 22/28