Linear-Quadratic Control System Design

Transcription

1 Linear-Quadratic Control Sytem Deign Robert Stengel Optimal Control and Etimation MAE 546 Princeton Univerity, 218! Control ytem configuration! Proportional-integral! Proportional-integral-filtering! Model following! Root locu analyi Copyright 218 by Robert Stengel. All right reerved. For educational ue only Sytem Equilibrium at Deired Output Recall = FΔx *+GΔu*+LΔw * Δy* = H x Δx *+H u Δu*+H w Δw * B 11 B 12 B 21 B 22 = F G H x H u 1 Equilibrium olution Δx* = B 12 ( B 11 L + B 12 H w )Δw * Δu* = B 22 ( B 21 L + B 22 H w )Δw * B 11 = F 1 where GB 21 + I n B 12 = F 1 GB 22 B 21 = B 22 H x F 1 1 B 22 = H x F 1 G + H u 2

2 Non-Zero Steady-State Regulation with Proportional LQ Regulator Contant command input provide equilibrium tate and control value Control law with command input Δu(t) = Δu*(t) C Δx t = B 22 Δy * C Δx t Δx *( t) B 12 Δy * = ( B 22 + CB 12 )Δy * CΔx t! C F Δy * C B Δx t 3 LQ Regulator with Forward Gain Matrix Δu(t) = Δu*(t) C Δx t = C F Δy * C B Δx t Δx *( t) C F! B 22 + CB 12 C B! C Input = Deired Output 4

3 LQ-PI Command Repone Block Diagram Integrate error in deired (commanded) repone 5 Formulating Proportional-Integral Control a a Linear-Quadratic Problem LTI ytem with command input Δ!x(t) = FΔx(t) + GΔu(t) = H x Δx *+H u Δu* Deired teady-tate repone to command Δx* = B 12 Δu* = B 22 Perturbation from deired repone Δ!x(t) = Δx(t) Δx * Δ!u(t) = Δu(t) Δu* Δ!y t = Δy( t) 6

4 LQ Proportional-Integral (PI) Control with Command Input Δ ξ t t = Δ y(t)dt = Δ!" x(t) = FΔ!x(t) + GΔ!u(t) Δ! " ξ( t) = Hx Δ!x(t) + H u Δ!u(t) t Integral tate [ H x Δ x(t) + H u Δ u(t) ]dt Augmented dynamic ytem, referenced to deired teady tate Δ x(t) Δ ξ( t) = F H x Δ χ(t) = Fχ Δ χ(t) + G χ Δ u(t) Δ x(t) Δξ ( t) Δ χ(t) + G H u Δ x(t) Δξ ( t) Δ u(t) 7 PI Augmented Cot Function 1 min J = min Δ!u Δ!u 2 = min Δ!u 1 2 Δ!x T (t)q x Δ!x(t) + Δ ξ! T ( t)q ξ Δ ξ! t Δ!χ T (t) Q x Q ξ + Δ!u T (t)rδ!u(t) Δ!χ(t) + Δ!u T (t)rδ!u(t) dt dt ubject to Δ!" χ(t) = Fχ Δ"χ(t) + G χ Δ"u(t) Δ χ(t) Δ x(t) Δξ ( t) 8

5 LQ Proportional-Integral (PI) Control with Command Input The cot function i minimized by 1 Δ!u(t) = R χ M T χ + G T χ P χss Δ!χ ( t) = C χ Δ!χ ( t) The control ignal include the error between the commanded and actual repone [ ] Δu(t) Δu* = C χ Δχ(t) Δχ t = C P [ Δx(t) Δx *] C I Δy t dt t = C P [ Δx(t) Δx *] C I H x Δx + H u Δu dt 9 LQ Proportional-Integral (PI) Control with Command Input The cot function i minimized by a control law of the form Δu(t) = ( B 22 + C P B 12 ) C P Δx(t) + C I Δy( t) dt t = C F C P Δx(t) + C I Δy t dt t 1

6 Integrating Action Set Equilibrium Command Error to Zero The cloed-loop ytem Δ x(t) Δ ξ( t) = ( F GC P ) GC I ( H x H u C P ) H u C I Δ x(t) Δξ ( t) i table [ ] t [ ] t = Δy( t) Δ!x(t) = Δx(t) Δx * Δ!u(t) = Δu(t) Δu* Δ!y t Therefore t Δx(t) Δx * t Δu(t) Δu* Δy t t t 11 Equilibrium Error Due to Contant Diturbance i Zero Equilibrium repone to contant diturbance i contant Δ x *(t) Δξ ( t) = ( F GC P ) GC I ( H x H u C P ) H u C I 1 L Δw * Output aymptotically approache it deired value Δx(t) Δx * t Δu(t) Δu*+Δu w* Δy t t t 12

7 Example: Open-Loop Repone of a 2 nd -Order Sytem, with and w/o Contant Diturbance ζ =.2, ω n = 6.28 rad/ Δ x = FΔx + G, = 1 Δ x = FΔx + G + GΔw*, = 1, Δw* = 2 13 Example: Open-Loop and LQ Control of 2 nd -Order Sytem Step input, with and without LQ Control, No Diturbance Start Goal 1 Q = 1 R = 1 Δ x = FΔx + G, = 1 Δ x = ( F GC B )Δx + GC F, = 1 14

8 Example: LQ Control, with and without Diturbance Step Input, with and without Diturbance Start Goal Δ x = ( F GC B )Δx + GC F, = 1 Δ x = ( F GC B )Δx + GC F + GΔw, = 1, Δw = 2 15 Example: Open-Loop, LQ, and LQ Proportional-Integral Control of 2 nd -Order Sytem 16

9 Example: Open-Loop, LQ, and LQ Proportional-Integral Control of 2 nd -Order Sytem Δ x = FΔx + G + GΔw * Δ x = ( F GC B )Δx GC I Δξ + GC F Δ!x = ( F GC B )Δx GC I Δξ +GC F + GΔw * Δ!x = ( F GC B )Δx +GC F + GΔw * Start Goal Q = R = 1 17 Proportional-Integral-Filter (PIF) Controller! Augment dynamic model to include! Integration of command-repone error! Low-pa filtering of actuator input Δ!" x(t) Δ!"u(t) Δ! " ξ( t) = F G H x H u Δ!x(t) Δ!u(t) Δ ξ! ( t) + I v Δ!v(t) Δ!v(t) = Δv(t) = Δv(t) " Δ!#u(t) Δ!" χ(t) = Fχ Δ!χ(t) + I v Δv(t) Δ!χ(t) " Δ!x(t) Δ!u t Δ! ξ t 18

10 min Δv 1 J = min Δv 2 PIF Augmented Cot Function 1 = min Δv 2 Δ!x T (t)q x Δ!x(t) + Δ ξ! T ( t)q ξ Δ ξ! t Q x Δ!χ T (t) R u Q ξ ubject to Δ!" χ(t) = Fχ Δ"χ(t) + I v Δv(t) + Δ!u T (t)r u Δ!u(t) + Δv T (t)r v Δv(t) Δ!χ(t) + Δv T (t)r v Δv(t) dt Optimal Control Law 1 Δv(t) = R χ M T χ + G T χ P χss Δ!χ ( t) = C χ Δ!χ ( t) dt 19 Integral Command Compenation Optimal PIF Control Law Low-Pa Filter Δv(t) = Δ!u(t) = C F (t) C P Δx(t) C I Δ ξ " ( t) C C Δu(t) Δu(t) = t C F (τ ) C P Δx(τ ) C I Δ ξ! τ C C Δu(τ ) dt Pure integration (high low-frequency gain) Low-pa filtering for mooth actuator command Lead (derivative) compenation Zero teady-tate error Satifie Bode gain criterion for atifactory repone 2

11 Digital-Adaptive LQ-PIF Flight Tet in NASA CH-47 Helicopter 21 LQ Model-Following Control 22

12 Implicit Model-Following LQ Regulator Actual and Ideal Model Δ!x(t) = FΔx(t) + GΔu(t) Δ!x M (t) = F M Δx M (t) min Δu J = min Δu # min Δu Δu(t) = Δu C (t) C IMF Δx t = C F (t) C IMF Δx t T { Δ!"x(t) Δ!"x M (t) QM Δ!"x(t) Δ!"x M (t) } dt Δ!x T (t) Δ!u T (t) LQ control hift cloed-loop root toward deired value Q M T Cloed-loop Control M R IMF Δ!x(t) Δ!u(t) Δ!x(t) = FΔx(t) + G Δu C (t) C IMF Δx t = [ F GC IMF ]Δx( t) + GΔu C (t) dt 23 Explicit Model Following Model Simulation! Model of the ideal ytem i explicitly included in the control law! Could have lower dimenion than actual ytem! Here, we aume dimenion are the ame Δ x(t) Δ xm (t) = F F M Δ x(t) Δ x M (t) + G G M Δ u(t) Δ u M (t) Δ χ(t) = Fχ Δ χ(t) + G χ Δ u(t) Control law force actual ytem to mimic the ideal ytem 24

13 Explicit Model Following Output vector = error between actual and ideal tate vector Δ y t Δ x(t) Δ x M (t) = I I Δ x(t) Δ x M (t) Output vector cot function J == 1 2 Δ y T (t)qδ y(t) + Δ u T (t)rδ u(t) dt J == 1 2 Δ x T (t) Δ x M T (t) Q Q Q Q Δ x(t) Δ x M (t) + Δ ut (t)rδ u(t) dt 25 Algebraic Riccati Equation = F T F M T P 11 P 12 P 12 P 22 P 11 P 12 P 12 P 22 F F M Q Q Q Q + P 11 P 12 P 12 P 22 GR 1 G T P 11 P 12 P 12 P 22 Three equation Firt i the LQ Riccati equation for the actual ytem; it olve for P 11, determine feedback gain, and tabilize phyical ytem = F T P 11 P 11 F Q + P 11 GR 1 G T P 11 Second olve for P 12 = ( F T P 11 P 11 GR 1 G T )P 12 F M + Q Third olve for P 22 = F M T P 22 P 22 F M Q + P 12 T GR 1 G T P 12 26

14 Explicit Model Following C = R 1 G T P 11 P 12 Feedback gain i independent of the forward gain Therefore, it determine the tability and bandwidth of the actual ytem P 12 T = R 1 G T P 11 P 12 = C B C M P 22 Forward gain, C F and C M, act a a pre-filter that hape the command input to have ideal ytem input property 27 Cloed-Loop Root Location for Implicit and Explicit Model Following Implicit Model- Following Root Location Explicit Model- Following Root Location Implicit model-following ytem ha n root n LQ cloed-loop root approach root of ideal ytem Relatively mall feedback gain Explicit model-following ytem ha (n + 1) to 2n root n LQ cloed-loop root forced to large, fat value 1 to n ideal ytem root pecified a input to the LQ compenator Relatively large feedback gain However, neither approach guarantee perfect model following 28

15 Goal for Perfect Model Following (Erzberger, 1968) Same condition applie for implicit or explicit model following Plant and ideal model may have different dimenion Δ!x(t) = FΔx(t) + GΔu(t), dim(δx) = ( n 1), dim(δu) = m 1 Δy = HΔx, dim(δy) = l 1 Δ!x M (t) = F M Δx M (t), dim(δx M ) = ( l 1), l n Objective: Output approximate model dynamic Δ!y = F M Δy 29 Condition for Perfect Model Following (Erzberger, 1968) Deired output dynamic HG( HG) L ( F M H HF)Δx = ( F M H HF)Δx HG( HG) L I l HΔ!x = F M HΔx Equate plant dynamic to output dynamic HGΔu = ( F M H HF)Δx Solve for control uing left peudoinvere Δu = ( HG) L ( F M H HF)Δx Model matching i perfect if or ( F MH HF)Δx = 3

16 Comment on LTI Model Following In principle, if perfect-model-following condition i atified, control gain matrix can be defined by algebraic olution Δu = ( HG) L ( F M H HF)Δx In practice, algebraic olution may not be atifactory Condition may not be atified; approximate olution required Stability may not be aured Weighted peudoinvere may be deired to elect favored control Integral compenation may be required to reduce teady-tate error IMF or PMF LQ tructure may be preferred 31 Root Locu Analyi 32

17 Root Locu Analyi of Control Effect on Sytem Dynamic Graphical depiction of control effect on location of eigenvalue of F (or root of the characteritic polynomial) Evan rule for root locu contruction Locu: the et of all point whoe location i determined by tated condition (Webter Dictionary) 33 Root Loci for Angle and Rate Feedback Variation of root a a calar gain, c i, goe from to Example: DC motor control % Root Locu of DC Motor Angle Control F = [ 1; ]; G = [;1]; Hx1 = [1 ]; % Angle Output Hx2 = [ 1]; % Angular Rate Output Sy1 Sy2 = (F,G,Hx1,); = (F,G,Hx2,); rlocu(sy1), grid figure rlocu(sy2), grid 34

18 Angle Control Gain, c 1, Variation Root Loci for Angle and Rate Feedback Rate Control Gain, c 2, Variation 35 Effect of Parameter Variation on Root Location Walter R. Evan Example: Characteritic equation of aircraft longitudinal motion Δ Lon () = 4 + a a a 1 + a = λ 1 λ 3 * * ( λ 2 )( λ 3 )( λ 4 ) = ( λ 1 ) λ 1 λ ( + 2ζ SP ω nsp + ω nsp ) = = ζ P ω np + ω np What effect would variation in a i have on the location (or locu) of root? Let root locu gain = k = c i = a i (jut a notation change) Option 1: Vary k and calculate root for each new value Option 2: Apply Evan Rule of Root Locu Contruction 36

19 Effect of a Variation on Longitudinal Root Location Example: k = a [ ] d() + kn() ( λ 2 )( λ 3 )( λ 4 ) = Δ Lon () = 4 + a a a 1 + k = λ 1 where d() = 4 + a a a 1 n() = 1 d ( λ ' 2 )( λ ' 3 )( λ ' 4 ) = λ ' 1 : Polynomial in : Polynomial in n 37 Effect of a 1 Variation on Longitudinal Root Location Example: k = a 1 Δ Lon () = 4 + a a k + a d() + kn() ( λ 2 )( λ 3 )( λ 4 ) = = λ 1 where d() = 4 + a a a n() = ( λ ' 2 )( λ ' 3 )( λ ' 4 ) = λ ' 1 38

20 Three Equivalent Expreion for the Polynomial d() + k n() = 1+ k n() d() = k n() d() = 1 = (1)e jπ (rad ) = (1)e j18(deg) 39 Example: Effect of a Variation Original 4 th -order polynomial Δ Lon () = = Example: k = a Δ() = 4 + a a a 1 + a + k + k = 4 + a a a 1 = 3 + a a 2 + a = k = 1 + k 4

21 Example: Effect of a 1 Variation Example: k = a 1 Δ() = 4 + a a a 1 + a = 4 + a a k + a + k = 4 + a a a = k k = The Root Locu Criterion All point on the locu of root mut atify the equation k[n()/d()] = 1 Phae angle( 1) = ±18 deg Number of root (or pole) of the denominator = n Number of zero of the numerator = q k = a : k n() d() = k a a a 1 = 1 k = a 1 : k n() d() = k = a a a Number of root = 4 Number of zero = (n q) = 4 Number of root = 4 Number of zero = 1 (n q) = 3 Spirule Manual graphical contruction of the root locu Invented by Walter Evan 42

22 Origin of Root (for k = ) Origin of the root are the Pole of d() Δ() = d() + kn() d() k Poitive a Variation Poitive a 1 Variation 43 Detination of Root (for k -> ± ) q root go to the zero of n() d() + kn() k = d() k + n() n() k No zero when k = a One zero at origin when k = a 1 44

23 Detination of Root (for k -> ± ) n q (n q) root go to infinite radiu from the origin d() + kn() n() = d() n() + k k ± ± R ± n q = Re j18 or Re j 36 k + k = Re j18 ( n q) k + or Re j 36 n q k 4 root to infinite radiu 3 root to infinite radiu 45 (n q) Root Approach Aymptote a k > ± Aymptote angle for poitive k θ(rad) = π + 2mπ n q, m =,1,...,(n q) 1 Aymptote angle for negative k θ(rad) = 2mπ, m =,1,...,(n q) 1 n q 46

24 Origin of Aymptote = Center of Gravity "c.g." = n i=1 q σ λi σ z j n q j =1 47 Root Locu on Real Axi Locu on real axi k > : Any egment with odd number of pole and zero to the right k < : Any egment with even number of pole and zero to the right 48

25 Next Time: Modal Propertie of LQ Regulator 49 Supplemental Material 5

26 Truncation and Reidualization 51 Reduction of Dynamic Model Order Separation of high-order model into looely coupled or decoupled lower order approximation [Rigid body] + [Flexible mode] Chemical/biological proce with fat and low reaction Economic ytem with local and global component Social network with large and mall cluter Δ!x fat Δ!x low fat = F fat F low low F fat F low = F f mall mall F Δx fat Δx low Δx f Δx + G fat G low low G fat G low fat + G f mall mall G Δu fat Δu low Δu f Δu 52

27 Truncation of a Dynamic Model Dynamic model order reduction when Two mode are only lightly coupled Time cale of motion are far apart Forcing term are largely independent Δ!x f f Δ!x = F f F Δx f f F f F Δx + G f G Δu f G f G Δu F f = mall mall Δx f F Δx + G f mall Δu f mall G Δu F f Δx f F Δx + G f Δu f G Δu Approximation: Mode can be analyzed and control ytem can be deigned eparately Δ x f = F f Δx f + G f Δu f Δ x = F Δx + G Δu 53 Reidualization of a Dynamic Model Dynamic model order reduction when Two mode are coupled Time cale of motion are eparated Fat mode i table Δ!x f Δ!x f = F f F F f F F f = mall Δx f Δx mall Δx f F Δx Approximation: Motion can be analyzed eparately uing different clock Fat mode reache teady tate intantaneouly on lowmode time cale Slow mode produce lowly changing bia diturbance on fat-mode time cale f + G f G G f G Δu f Δu + G f mall Δu f mall G Δu 54

28 Reidualized Fat Mode Fat mode dynamic Δ!x f = F f Δx f + G f Δu f + ( F f Δx + G f Δu ) Bia If fat mode i not table, it could be tabilized by inner loop control Δ!x f = F f Δx f + G f ( Δu c C f Δx f ) Bia + F f Δx + G f Δu = ( F f G f C f )Δx f + G f Δu fc Fat Mode Inner Loop Control Law Bia + F f Δx + G f Δu 55 Fat Mode in Quai-Steady State Aume that fat mode reache teady tate on a time cale that i hort compared to the low mode F f Δx f + F f Δx + G f Δu f + G f Δu Δ!x = F f Δx f + F Δx + G Δu + G f Δu f Algebraic olution for fat variable F f Δx f + F f Δx + G f Δu f + G f Δu F f Δx f = F f Δx G f Δu f G f Δu Δx f = F 1 f ( F f Δx + G f Δu f + G f Δu ) 56

29 Δ!x = F f Reidualized Slow Mode Subtitute quai-teady fat variable in differential equation for low variable F f 1 = F F f F 1 f f F ( F f Δx + G f Δu f + G f Δu ) + F Δx + G Δu + G Δu f f Δx + G F f F 1 f f G Δu + G f F f F 1 f G f Reidualized equation for low variable Δ x = F NEW Δx + G NEW Δu f Δu Control law can be deigned for reduced-order low model, auming inner loop ha been tabilized eparately Slow Mode Outer Loop Control Law Δu f 57 Firt Example: k = a +.21 k = 1 58

30 Second Example: k = a 1 k = Spirule Intruction 6