Techniques - Advanced Control System Design for Aerospace Vehicles - Lecture Slides, Slides of Aeronautical Engineering

Download Techniques - Advanced Control System Design for Aerospace Vehicles - Lecture Slides and more Slides Aeronautical Engineering in PDF only on Docsity! Lecture – 30 Applications Linear Control Design Techniques in Aircraft Control – II Dr. Radhakant Padhi Asst. Professor Dept. of Aerospace Engineering Indian Institute of Science - Bangalore ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore 2 Topics Brief Review of Modern Control Design for Linear Systems Automatic Flight Control Systems: Modern (Time Domain) Designs ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore 5 Question: Can we conclude about nature of the solution, without solving the system model? Answer: YES! Definition: Eigenvalues of A : “Poles” of the system! The nature of the solution is governed only by the locations of its poles 0, (0)X AX X X= = Stability of Linear System ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore 6 Controllability 1If the rank of is , then the system is controllable. n BC B AB A B n −⎡ ⎤⎣ ⎦ Example: u x x x x ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ +⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − − =⎥ ⎦ ⎤ ⎢ ⎣ ⎡ 1 2 20 01 2 1 2 1 ( ) 2 1 0 2 2 2 1 0 2 1 1 2 2 The system is controllable. B B C rank C ⎡ − ⎤ −⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ = =⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥− −⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎣ ⎦ = ∴ Result: ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore 7 Observability ( ) 1If the rank of is , then the system is observable. nT T T T T BO C A C A C n −⎡ ⎤ ⎢ ⎥⎣ ⎦ Example: [ ]1 1 1 2 2 2 1 0 2 1 0 0 2 1 x x x u y x x x −⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤ = + =⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥−⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ( ) 1 1 0 1 1 1 0 0 2 0 0 0 1 2 The system is NOT observable. B B O rank O ⎡ − ⎤ −⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ = =⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥−⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎣ ⎦ = ≠ ∴ Result: ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore 10 Philosophy of Pole Placement Control Design The gain matrix is designed in such a way that K ( ) ( )( ) ( )1 2 1where , , are the desired pole locations. n n sI A BK s s sμ μ μ μ μ − − = − − − ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore 11 Pole Placement Design Steps: Method 1 (low order systems, n ≤ 3) Check controllability Define Substitute this gain in the desired characteristic polynomial equation Solve for by equating the like powers on both sides ( ) ( )1 nsI A BK s sμ μ− + = − − [ ]1 2 3K k k k= 1 2 3, ,k k k ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore 12 Pole Placement Design: Summary of Method 2 (Bass-Gura Approach) Check the controllability condition Form the characteristic polynomial for A find ai’s Find the Transformation matrix Write the desired characteristic polynomial and determine the αi’s The required state feedback gain matrix is 1 2 1 2 1| | n n n n nsI A s a s a s a s a − − −− = + + + + + ( ) ( ) 1 21 1 2n n nn ns s s s sμ μ α α α− −− − = + + + + ( ) ( ) ( ) 11 1 1 1[ ]n n n nK a a a Tα α α −− −= − − − T MW= ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore 15 LQR Design: Necessary Conditions of Optimality Terminal penalty: Hamiltonian: State Equation: Costate Equation: Optimal Control Eq.: Boundary Condition: ( ) ( )1 2 T T TH X Q X U RU AX BUλ= + + + ( ) ( )12 T f f f fX X S Xϕ = X AX BU= + ( ) ( )/ TH X QX Aλ λ= − ∂ ∂ = − + ( ) 1/ 0 TH U U R B λ−∂ ∂ = ⇒ = − ( )/f f f fX S Xλ ϕ= ∂ ∂ = ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore 16 LQR Design: Riccati Equation Riccati equation Boundary condition 1 0T TP PA A P PBR B P Q−+ + − + = ( ) ( )is freef f f f fP t X S X X= ( )f fP t S= ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore 17 LQR Design: Solution Procedure Use the boundary condition and integrate the Riccati Equation backwards from to Store the solution history for the Riccati matrix Compute the optimal control online ( )f fP t S= ft 0t ( )1 TU R B P X K X−= − = − Automatic Flight Control Systems: Time Domain Designs Dr. Radhakant Padhi Asst. Professor Dept. of Aerospace Engineering Indian Institute of Science - Bangalore ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore 21 Applications of Automatic Flight Control Systems Stability Augmentation Systems • Stability enhancement • Handling quality enhancement Cruise Control Systems • Attitude control (to maintain pitch, roll and heading) • Altitude hold (to maintain a desired altitude) • Speed control (to maintain constant speed or Mach no.) Landing Aids • Alignment control (to align wrt. runway centre line) • Glideslope control • Flare control Stability Augmentation Systems Dr. Radhakant Padhi Asst. Professor Dept. of Aerospace Engineering Indian Institute of Science - Bangalore ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore 25 SAS Design for Stability Augmentation Reference: R. C. Nelson, Flight Stability and Automatic Control, McGraw-Hill, 1989. Problem: Determine the feedback gain K that produces the desired stability characteristics. SAS design: • Longitudinal stability augmentation design • Lateral stability augmentation design Note: Handling quality improvement can be done using the same philosophy. ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore 26 Longitudinal SAS Ref.: R. C. Nelson, Flight Stability and Automatic Control, McGraw-Hill, 1989. 0 0 0 0 0 0 1 0 0 u w u w e u w q X BA u X X g u X w Z Z u w Z q M M M q M δ δ δ δ θ θ Δ − Δ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥Δ Δ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥= + Δ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥Δ Δ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥Δ Δ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ The eigenvalues of stability matrix A are the short period and long period roots, which may be unacceptable to the pilot. If unacceptable, then let us design [ ]1 2 3 4 Pilot input P P e e eK X k k k k Xδ δ δΔ = − + Δ = − + Δ ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore 27 Longitudinal SAS Ref.: R. C. Nelson, Flight Stability and Automatic Control, McGraw-Hill, 1989. 1 2 3 4 1 2 0 3 4 1 2 3 4 0 0 1 0 u w u w CL u w q X X k X X k X k g X k Z Z k Z Z k u Z k Z k A M M k M M k M M k M k δ δ δ δ δ δ δ δ δ δ δ δ − − − − −⎡ ⎤ ⎢ ⎥− − − −⎢ ⎥= ⎢ ⎥− − − − ⎢ ⎥ ⎣ ⎦ Augmented CL system matrix Design the gain matrix design to place the eigenvalues at the desired locations following the “Pole placement philosophy” ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore 30 Example: Longitudinal SAS Ref.: R. C. Nelson, Flight Stability and Automatic Control, McGraw-Hill, 1989. Problem: An airplane have poor short-period flying qualities in a particular flight regime. To improve the flying qualities, a stability augmentation system using state feedback is employed is to be employed. Determine the feedback gain so that the airplane’s short-period characteristics are Assume that the original short period dynamics is given by 2.1 2.14sp iλ = − ± 0.334 1 0.027 2.52 0.387 2.6 eq q α α δ Δ − Δ −⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ = + Δ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥Δ − − Δ −⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore 31 Example: Longitudinal SAS Ref.: R. C. Nelson, Flight Stability and Automatic Control, McGraw-Hill, 1989. ( ) 1 2 1 2 2 1 2 1 2 2 Characteristic equation: 0.334 0.027 1 0.027 0 2.52 2.6 0.387 2.6 0.721 0.027 2.6 2.65 2.61 0.8 0 Desired characteristic equation: 4.2 9 0 k k k k k k k k λ λ λ λ λ λ + − − − = − + − + − − + − − = + + = ( ) 1 2 1 2 0.334 0.027 1 0.027 2.52 2.6 0.387 2.6 CLA A BK k k k k = − − + +⎡ ⎤ = ⎢ ⎥− + − +⎣ ⎦ Closed loop matrix ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore 32 ( ) 1 2 1 2 1 2 Pilot input Compare like powers of : 0.721 0.027 2.6 4.2 2.65 2.61 0.8 9 Solving for the gains yields: 2.03 1.318 The state feedback control is given by: 2.03 1.318 + Pe e k k k k k k q λ δ α δ − − = − − = = − = − Δ = Δ + Δ Δ Example: Longitudinal SAS Ref.: R. C. Nelson, Flight Stability and Automatic Control, McGraw-Hill, 1989. ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore 35 ( ) Automatic Pilot input 1 1 2 2 1 Lateral body rate dynamics: Control: Closed loop system matrix: p r a r a p r a r r U A P P p a r r a CL L L L Lp p N N N Nr r U U U CK X U L k c L c L L k c L A A BCK δ δ δ δ δ δ δ δ δ Δ Δ⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤ = +⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥Δ Δ⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎣ ⎦⎣ ⎦ = + = − + − + − = − = ( ) ( ) ( ) 2 1 1 2 2 1 2 r p a r r a r c L N k c N c N N k c N c N δ δ δ δ δ +⎡ ⎤ ⎢ ⎥− + − +⎣ ⎦ Example: Lateral SAS Ref.: R. C. Nelson, Flight Stability and Automatic Control, McGraw-Hill, 1989. ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore 36 ( ) ( ) ( ) ( )( ) ( ) 2 1 2 1 2 1 2 1 2 2 1 2 1 2 1 2 Characteristic equation: 0 0 where , Desired characteristic equation: 0 By equating the like p CL c c p r r c r c p c p c r p p r c a r c a r I A k L k N L N k L N N L k N L L N N L N L L c L c L N c N c Nδ δ δ δ λ λ λ λ λ λ λ λ λ λ λ λ λ − = + + − − + − + − + − = = + = + − − = − + + = 1 2 owers of the feedback gains and can be obtained.k k λ Example: Lateral SAS Ref.: R. C. Nelson, Flight Stability and Automatic Control, McGraw-Hill, 1989. Cruise Control Systems Dr. Radhakant Padhi Asst. Professor Dept. of Aerospace Engineering Indian Institute of Science - Bangalore ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore 40 Guided missiles need roll orientation to be fixed for proper functioning of guidance unit. The objective here is to design a roll autopilot through feedback control. ( ) 0 1 0 0 where 1 1, a p a p a ax x L Lpp L LL LPI I δ δ φφ δ δ ⎡ ⎤⎡ ⎤ ⎡ ⎤⎡ ⎤ = +⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦ ⎛ ⎞∂ ∂= = ⎜ ⎟∂ ∂⎝ ⎠ System dynamics : Example: Roll stabilization system Ref.: R. C. Nelson, Flight Stability and Automatic Control, McGraw-Hill, 1989. ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore 41 2 2 2 max max max0 max max max The quadratic performance index which needs to be minimized is 1 2 the maximum desired roll angle, the maximum roll rate the maxi a a a pJ dt p p δφ φ δ φ δ ∞ ⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎢ ⎥= + +⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦ = = = ∫ 2 max 2 max 2 max mum aileron deflection Comparing the PI with standard form gives Q and R as 1 0 0 1 01 , , , 010 p aa Q R A B L L p δ φ δ ⎡ ⎤ ⎢ ⎥ ⎡ ⎤ ⎡ ⎤⎢ ⎥= = = =⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎣ ⎦⎣ ⎦ ⎢ ⎥ ⎣ ⎦ Example: Roll stabilization system Ref.: R. C. Nelson, Flight Stability and Automatic Control, McGraw-Hill, 1989. ADVANCED CONTROL SYSTEM DESIGN Dr. Radhakant Padhi, AE Dept., IISc-Bangalore 42 1 11 12 12 22 2 2 2 12 max2 max 2 2 11 12 12 22 max 12 22 Algebraic Ricatti Equation: 0 where Substituting matrices , , and into the Ricatti equation 1 0 0 12 2 T T a a p a a p PA A P PBR B P Q p p P p p A B Q R p L p p L p p L p p L p δ δ δ φ δ −+ − + = ⎡ ⎤ = ⎢ ⎥ ⎣ ⎦ − = + − = + + 2 2 222 max2 max 0a ap Lδ δ− = Example: Roll stabilization system Ref.: R. C. Nelson, Flight Stability and Automatic Control, McGraw-Hill, 1989.