Docsity
Docsity

Prepare for your exams
Prepare for your exams

Study with the several resources on Docsity


Earn points to download
Earn points to download

Earn points by helping other students or get them with a premium plan


Guidelines and tips
Guidelines and tips

Robust Control: Nominal Controller Design and Min-Max Control, Study notes of Mechanical Engineering

A lecture note from a mechanical engineering & mechanics professor at drexel university, covering the topics of linear quadratic regulator (lqr) problem, min-max control, and solving the riccati equation. It also discusses the stability of lqr and the lyapunov equation. Mathematical equations and explanations for each topic.

Typology: Study notes

Pre 2010

Uploaded on 08/18/2009

koofers-user-l6i
koofers-user-l6i 🇺🇸

4

(1)

10 documents

1 / 19

Toggle sidebar

Related documents


Partial preview of the text

Download Robust Control: Nominal Controller Design and Min-Max Control and more Study notes Mechanical Engineering in PDF only on Docsity! 4/14/2003 Robust Control 5 Nominal Controller Design Continued Harry G. Kwatny Department of Mechanical Engineering & Mechanics Drexel University Outline • The LQR Problem • A Generalization to LQR • Min-Max Control • Solving the Riccati Equation • Classical LQG • Stability Margins & LTR Stability of LQR ~ 2 ( ) ( ) ( ) ( )( ) 1 1 1 1 1 Now consider, , where ( ) ( ) T T T T T T TT T x Ax B BKXBK XB u u Kx x A BK x K R B X XA A X XBR B X Q X A BK A BK X Q XBR B X B R B X XR X X B− − − − − = + = ⇒ = + = − ⇓ + − − + − − + − = − + + + = − − Lyapunov Equation Generalization of LQR ( ) ( ) ( ) ( ) 0 1 1 1 1 1 1 1( ) lim ( ) ( ) 2 ( ) ( ) ( ) ( ) 0, 0 Notice that (complete the square) 2 Define a new control , T T T T T T TT T T T T T T J x x Qx x Su t u Ru d T R Q SR S x Qx x Su u Ru u R Sx R u R Sx x Q SR S x u u R S x x Ax Bu x A BR S x τ τ τ τ τ τ→∞ − − − − − −  = + +  > − ≥ + + = + + + − = + = + ⇒ = − + ∫ ( ) ( ) ( ) ( ) ( ) 1 1 1 1 1 0 1 1( ) lim ( ) ( ) ( ) ( ) T T T T T T TT T T T Bu J x x Q SR S x u Ru d u R B P S x P A BR S A BR S P PBR B P Q SR S T τ τ τ τ τ− − − − − − →∞  = − +  ⇒ = − + − + − − + − ∫ RE & Associated Hamiltonian Matrix ( ) ( ) 1 1 11 12 21 22 0 if is an eigenvalue of , so is , controllable, , observable where there are no eigenvalues on the imaginary axis such T T T T T A BR B A X XA XBR B X Q H Q A H A B A C Q C C T T T T T λ λ − −  −+ − + = ⇒ =  − −  • − • = ⇒   ∃ =     ( ) 1 21 1 1 1 0 that 0 is composed of real Jordan blocks, with Re 0 The positive definite solution is X T T T HT λ − − −Λ  =  Λ  = Λ Λ > Min-Max Control ( ) ( ) ( ) ( ) 2 0 Theorem , ( , ) ( ) ( ) ( ) ( ) ( ) ( ) , , 0 objective: min max ( , ) Suppose has bounded energy, , , , stabilizable, , detectable, then the optimal control s : i ( T T T wu x Ax Bu Ew y Cx J u w y t y t u t u t w t w t dt J u w w t A B A E A C u ρ γ ρ γ ∞ = + + =  = + − > ∫ 2 2 1) ( ), 1the worst case disturbance is ( ) ( ) is the unique, symmetric, nonnegative solution of the Riccati equation: 1 1 T T T T T T t Kx t K B S w t E Sx t S A S SA S BB EE S C C ρ γ ρ γ = = − =   + − − = −    Min-Max Hamiltonian Matrix 2 2 min min min 1 1 1 1 stabilizability/detectability such that there are no pure eigenvalues of if produces LQR solution is the f T T T T T T T T A EE BB A S SA S BB EE S C C H C C A H γ ρ ρ γ γ γ γ γ γ γ  −   + − − = − ⇒ =     − −   ⇒ • ∃ > • →∞ • = min ull state feedback solution all are valid min-max controllers H γ γ ∞ • ≤ < ∞ Min-Max, Continued ( ) 2 2 1 1 Re 0 Re 0 1 1 The closed loop system matrix is , since is destabilizing, there is some stability margin T T T T H A EE S BB S A BB S EE S λ λ γ ρ ρ γ   • ≠ ⇔ + − <    • − LQG Robustness: State Feedback x0x = ( ) 1sI A B−−K− ( ) ( ) ( ) ( ) ( ) ( )( ) 1 1 11 , For diagonal, 1 1: , : (in each channel) 2 3 G s K sI A B K B sI A S s I K sI A B R S j GM PM σ ω ω π − − −− = − − = − Φ Φ = −  = − −  ≤ ∀ ⇒ ∞ Loop break point LQG Robustness: State Feedback 2 ( ){ } ( ){ } 11 min 1 1 1 When is determined via the Ricatti equation, it is known min 1, 0 Similarly, min 1, 0 R V v R v V K S I K j I A B v I C j I A L v σ ω ω ω ω −− = − =    = − − ≥ >     − − ≥ >  LQG Robustness: Output Feedback ( )cG s ( )pG s y0y = ( ) ( ) ( ) ( ) 1 1 No guaranteed margins! p c G s C sI A B G s K s A BK LC L − − = − = − − + +  