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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.

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Pre 2010

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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 − − = − = − − + +