CALIFORNIA INSTITUTE OF TECHNOLOGY

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1 CALIFORNIA INSIUE OF ECHNOLOGY Control and Dynaical Systes Course Project CDS 270 Instructor: Eugene Lavretsky, Sring 2007 Project Outline: his roject consists of two flight control designs for itch and rollyaw dynaics of an aircraft / issile, with atched uncertainties, using (LQR PI + Adative) control architectures. Individual rojects are to be selected no later than the 5 th week of classes. Project reorts are required and consist of both written and oral ortions. he written ortion of the reort should be a descrition of the roject using the terinology and notation of the class. It is due the week before the last week of finals. he oral ortion of the reort will be a inute resentation to the class, given during the last week of the ter. 1. Aircraft / Missile Pitch Dynaics Consider aircraft / issile itch dynaics described by the 2 nd order single-inut-singleoutut (SISO) syste: Zα Zδ α 1 V α V = + Λ δ e + f ( ) q q M M u α q M δ A B (1.1) where α is the angle of attack (AOA), q is the itch rate, δ e is the elevator deflection Zα, Zδ, Mα, Mq, Mδ reresent the vehicle (control inut), V is the true airseed, and ( ) stability and control derivatives. It is assued that: a) syste atrices (, ) constant, and b) the syste state vector ( α q) = is known / available on-line. A B are Also in (1.1), an unknown constant Λ > 0 denotes the syste loss-of-control effectiveness araeter, and ( ) f is an unknown nonlinear function, which reresents the syste atched uncertainty.

2 he syste controlled outut is α, that is: ( 1 0) y = = α C (1.2) 2. Aircraft / Missile Roll-Yaw Dynaics Consider aircraft / issile roll-yaw dynaics described by the 3 rd order ulti-inutulti-outut (MIMO) syste with atched uncertainties. Y Y Y β Y Y r δail δ rud 1 β V V V β V V λ1 0 δail = Lβ L Lr + Lδ L f ( ) ail δ rud + 0 λ2 δ rud r N N β N r r Nδ N ail δrud Λ u A B (2.1) In (2.1), β is the angle of sidesli (AOS), and r are the body roll and yaw rates, u = ( δ ) ail δrud is the syste control inut that consists of the aileron δ ail and rudder δ rud, and (, ) A B are constant atrices corised of the vehicle stability and control = β r is known / available on-line. he syste uncertainties are reresented by the unknown constant diagonal atri Λ with λ i > 0, and by the unknown nonlinear vector function derivatives. It is also assued that the syste state vector ( ) 2 ( ) f R. he syste controlled outut is: β y = = C (2.2) 3. Control Proble Forulation and Control Architecture Let y () t R denote a bounded coand for the syste controlled outut y cd R to follow. his task is to be accolished using the syste control inut u R in the for of a full state feedback. Define outut tracking error e y and its integral e yi :

3 e = e = y y (3.1) yi y cd Let n and denote diensions of the syste state and control u, resectively. Consider the etended oen-loo dynaics: e 0 yi C eyi 0 I = + Λ ( u+ f ( ) 0 ) + y n 0 A B n A B Bref cd (3.2) or, equivalently ( ( )) = A+ BΛ u+ f + B y (3.3) ref cd In ters of (3.3), the controlled outut y fro (1.2) can be written as: eyi y = ( 0 C ) = C C (3.4) he control roble of interest is bounded tracking in the resence of the syste f. Secifically, one needs to design the control inut u, so that uncertainties Λ and ( ) the syste controlled outut y tracks a bounded tie-varying coand y cd, with bounded tracking errors, while the rest of the signals in the corresonding closed-loo dynaics reain bounded. Control architecture should consist of a baseline LQR PI controller, augented by a direct odel reference adative control (MRAC). MRAC regressor vector ust contain a set of aroriately chosen Radial Basis Functions (RBF-s). Paraeter adatation laws ust contain robustness odifications such as a) Deadzone, a) Projection Oerator, and c) e od. he resulting itch control law should be designed to recover tracking erforance of the baseline itch controller, in the resence of control failures λ and atched uncertainties f ( ). 4. Baseline Controller: LQR PI Design

4 Setting dynaics. Λ= I, f ( ) = 0 in (3.3), results in the linear tie invariant (LI) baseline = A+ Bu+ B y y = C ref cd (4.1) Assuing constant coand y cd, while using Linear Quadratic Regulator (LQR) ethod, and Proortional + Integral (PI) feedback connections, a baseline control law can be designed. his controller reresents the so-called LQR PI servoechanis. Its design is outlined below. Using the LQR ethod, one can calculate an otial stabilizing controller for: where z = Az+ Bv (4.2) e yi z = =, v= u (4.3) It is well-known that the corresonding LQR solution is given in feedback for: 1 yi u = v= R B Pz = ( KI K P) K e (4.4) In (4.4), P is the unique syetric ositive sei-definite solution of the Riccati equation. 1 A P PA Q PBR B P + + = 0 (4.5) he solution P eists if the weights Q and R are chosen as: with Q ii > 0. ( 11 ) ( ) Pitch: Q= diag Q 0 0, R= 1 Roll / Yaw : Q= diag Q Q 0 0 0, R= I (4.6) Integrating (4.4), yields the baseline LQR PI controller: ( y y) cd ubl = K = KI ey I KP = KI KP (4.7) s

5 where the otial gain atri ( ) K = K K (4.8) I P is artitioned into the integral gain control block-diagra is shown in Figure 3.1. K I and the roortional gain K P. he corresonding () ycd t K I u Plant + s + C y K Figure 3.1. Baseline / Servoechanis LQR PI Control Block-Diagra Baseline control design iterations are erfored by increasing Q 11 (itch ais) and ( 11, 2 2 ) Q Q (roll-yaw aes), until adequate tracking erforance, is achieved. 5. Adative Augentation Design In the resence of the syste uncertainties Λ and f ( ), tracking erforance of the baseline controller (4.7) will often deteriorate. In order to restore it, an MRAC augentation will be added. his rocess consists of a) reference odel definition and b) design of adative laws. It is suggested that the reference odel is defined to reresent baseline closed-loo syste dynaics, which is obtained by substituting the baseline controller (4.7) into the LI syste (4.1). he resulting reference odel dynaics is of the for: = A + B y (5.1) ref ref cd where A = A BK (5.2) ref is Hurwitz by design. It is easy to see that the DC gain fro the coand y cd to the controlled outut y is unity. he adative augentation coonent u ad is chosen in the for:

6 ( ) ˆ ˆ u = K Θ Φ (5.3) ad where 1 ( 1 1) N ( ) ( ) N ( ) Φ = Φ Φ R + (5.4) N { } i= 1 is the regressor vector with N RBF-s Φ i( ) the state of the original syste. Fro (3.2) it follows that he adative araeters. Note that these functions deend on n R, where n n = +. Kˆ R n ( N 1) ( 1 N N+ 1) R Θ= ˆ Θˆ Θˆ Θˆ + (5.5) are udated according to the following robust adative laws: Kˆ Γ Proj Kˆ, e γ e Kˆ, e > e = e - od 0, e ein dead-zone Γ Proj Θˆ, Φ e γ e Θ ˆ, e > e ( ) Θ Θ ˆ Θ= e - od 0, e ein dead-zone in in (5.6) with Dead-zone odification, sooth Projection Oerator, and e odification. Note that in (5.6), = (5.7) e e Pref B reresents the training error, and P ref is the unique syetric ositive definite solution of the Lyaunov algebraic equation A P + P A = Q (5.8) ref ref ref ref ref

7 with a suitably chosen syetric ositive definite weight atri Finally, total control is calculated as: Q ref. u = K ˆ ˆ K +Θ Φ( ) otal Control Baseline Adative Augentation (5.9) 6. Project asks 1) Find (in oen literature) stability and control derivatives data for the itch and the roll-yaw dynaics, (1.1) and (2.1). 2) Using the LQR PI servoechanis design fro Section 4, construct two baseline LQR PI itch and roll-yaw controllers. Note that these are linear controllers. Analyze their stability argins and robustness roerties in frequency doain, (using Bode and / or Nyquist diagras). Evaluate / siulate tracking erforance of the baseline design, without the syste uncertainties. 3) Introduce atched uncertainties, Λ and f ( ), into the syste dynaics, such that the baseline tracking erforance deteriorates, (significantly). 4) Following the design stes fro Section 5, construct two MRAC augentations for the LQR PI baseline itch and the roll-yaw controllers, resectively. 5) Deonstrate closed-loo tracking erforance using various coands and a set of atched uncertainties. 6) Insert gains and tie-delays into the syste control ath and into the syste controlled outut. Calculate (nuerically) gain and tie-delay argins. 7) Write foral reort due the week before the last week of finals. 8) Present your work (15-25 inutes) due the last week of finals. References: [1] Stevens, B.L., Lewis F.L., Aircraft Control and Siulation, John Wiley &Sons, Inc., [2] Brubaugh, R.W., Aircraft Model for the AIAA Controls Design Challenge, Journal of Guidance, Control, and Dynaics, Vol. 17., No. 4, July August, 1994.

8 [3] Lavretsky, E., Hovakiyan, N., Stable Adatation in the Presence of Actuator Constraints with Flight Control Alications, Journal of Guidance, Control, and Dynaics, Vol. 30, No. 2, March Aril 2007.