Session outline. Introduction to Feedback Control. The Idea of Feedback. Automatic control. Basic setting. The feedback principle

Transcription

1 Session otline Intodction to Feedback Contol Kal-Eik Åzen, Anton Cevin Feedback and feedfowad PID Contol State-space models Tansfe fnction models Contol design sing pole placement State feedback and obseves 2 The Idea of Feedback Atomatic contol Feedback: Compae the actal eslt with the desied eslt. Take actions based on the diffeence. A seemingl simple idea that is temendosl powefl. Use of feedback has often been evoltiona. Feedback is also called closed loop contol. The opposite is feedfowad o open loop contol: make a plan and execte it. Feedback and feedfowad ae ke ideas in the discipline of contol. Use of models and feedback Activities: Modeling Analsis and simlation Contol design Implementation Inpt Distbance Otpt Pocess Contolle Refeence 3 4 Basic setting The feedback pinciple Model Contol Pocess Distbance A ve powefl idea, that often leads to evoltiona changes in the wa sstems ae designed. The pima paadigm in atomatic contol. Ref. signal e Feedback Σ Pocess Contolle Mst handle two tasks: Follow efeence signals, Compensate fo distbances How to do seveal things with the contol signal 5 Base coective action on an eo that has occed Closed loop 6

2 Popeties of feedback + Redces inflence of distbances + Redces effect of pocess vaiations + Does not eqie exact models Feeds senso noise into the sstem Ma lead to instabilit, e.g.: if the contolle has too high gain if the feedback loop contains too lage time delas Ref. signal The feedfowad pinciple Feedfowad contolle Measable Distbance Pocess Take coective action befoe an eo has occed Mease the distbance and compensate fo it Use the fact that the efeence signal is known and adjst the contol signal to the efeence signal Open loop 7 8 Popeties of feedfowad Example: Cise contol sing feedfowad + Redces effect of distbances that cannot be edced b feedback + Allows faste set-point changes, withot intodcing contol eos Reqies good models Reqies stable sstems Open loop Poblems? Desied speed Table Ca Meased speed 9 Example: Cise contol sing feedback Exempel: Cise contol sing feedback and feedfowad Desied speed Eo Σ Contolle Thottle Ca Meased speed Table Desied speed Σ Contolle Σ Ca Meased speed Closed loop Simple contolle: Eo > : incease thottle Eo < : decease thottle Both poactive and eactive 2

3 Example: Segwa The sevo poblem Focs on efeence vale changes: Distbance Model Contol Pocess Tpical design citeia: Rise time, T Oveshoot, M. M 2p e Wh is this pocess moe difficlt to contol? Unstable dnamics 3 Settling time, T s Stead-state eo, e... T T s 4 t The eglato poblem Session otline Focs on pocess distbances: Model Contol Tpical design citeia: Otpt vaiance Contol signal vaiance Pobabilit densit.5 Test limit Pocess Distbance Set point fo eglato with low vaiance Set point fo eglato with high vaiance Feedback and feedfowad PID Contol State-space models Tansfe fnction models Contol design sing pole placement State feedback and obseves Pocess otpt 5 6 Example: Oven On/off contol = 2 C Σ e Contolle Oven (t) = { min, e(t) < max, e(t) > actal tempeate desied tempeate heating element powe ( ) Oscillations 8

4 Popotional contol Popotional contol P-contolle: (t) =Ke(t) (K gain) Inceased gain K: Stationa eo Smalle stationa eo Lage oscillations 2 Popotional Integal contol ( ) PI-contolle: (t) =K e(t)+ t e(s)ds T i (T i integal time) Popotional Integal contol Smalle integal time T i (lage integal action): No stationa eo Lage oscillations Popotional Integal Deivative contol PID-contolle: ( ) (t) =K e(t)+ t T i d de(t) (T d deivative time) PID: Pesent, past, and fte e D I P t P-pat: needed fo fast esponse I-pat: needed to emove stationa eo D-pat: ma be needed to stabilize the pocess The deivative pat edces oscillations 23 24

5 Session otline Feedback and feedfowad PID Contol State-space models Tansfe fnction models Contol design sing pole placement State feedback and obseves Dnamical sstems Sstem Static sstem: (t) = f ( (t) ) (The otpt at time t onl depends on the inpt at time t.) Dnamical sstem: (t) = f ( x(), [, t] ) 25 (The otpt at time t depends on the initial state x() and the inpt fom time to t.) 26 Linea sstems We will mainl deal with linea, time-invaiant (LTI) sstems Fo linea sstems, the pinciple of speposition holds: Sstem 2 2 Sstem Nonlinea sstems Almost all eal sstems ae nonlinea limited inpt and otpt signals nonlinea pocess geomet fiction, tblence,... Can be lineaized aond an opeating point If thee is feedback, a simple linea model is often enogh Bt, alwas emembe the limitations of the model! α + β 2 α + β 2 Sstem State space fom Standad sstem foms A nmbe of fist-ode diffeential eqations Descibes what happens inside the sstem and how inpts and otpt ae connected to this Nmeicall speio The heitage of mechanics Tansfe fnction fom The tansfom of a highe-ode linea diffeential eqation Descibes the elationship between the inpt and the otpt The sstem is a black box Compact notation, convenient fo hand calclations The heitage of electical engineeing 29 Nonlinea state-space model: State Space Models x dx = f (x,...,x n, ). dx n = f n (x,...,x n, ) = (x,...,x n, ) Sstem Linea state-space model: dx = a x a n x n + b. dx n = a n x a nn x n + b n = c x c n x n + d 3

6 State Space Models Intodce vectos and matices fo compact notation: x x =. x n Example: Pendlm n sstem ode Nonlinea state-space model: dx = f (x, ) = (x, ) Linea state-space model: dx = Ax + B = Cx + D 3 Nonlinea state-space model (x = angle, x 2 = angla velocit): ẋ = x 2 ẋ 2 = ω 2 sin x + kcos x = x whee ω = l is the natal feqenc of the pendlm. 32 Lineaization A nonlinea sstem can be lineaized aond an eqilibim point, whee it holds dx = f (x, )= Make fist-ode Talo appoximations of f and aond (x, ): f (x, ) f (x, ) + f }{{} x = (x, ) (x, ) }{{} x = + (x, ) (x, ) (x x )+ f (x, )( ) (x x )+ (x, )( ) Lineaization Intodce new vaiables Δx = x x, Δ = och Δ = The sstem can now be witten as dδx = dx f = f (x, ) x Δ = (x, ) x In matix fom: (x, ) (x, ) dδx = AΔx + BΔ Δ = CΔx + DΔ Δx + f (x, )Δ Δx + (x, )Δ Example Pendlm Lineaize ẋ = x 2 = f (x, x 2, ) ẋ 2 = ω 2 sin x + kcos x = f 2 (x, x 2, ) = x = (x, x 2, ) aond the ppe (nstable) eqilibim x = π, x 2 =, =. The lineaized sstem is given b dδx = AΔx + BΔ Δ = CΔx + DΔ whee Δx = x x, Δ =, Δ = and A = f f x x 2 f 2 f 2 x x 2 = ω 2 B = C = f f 2 x (x, ) (x, ) = = k cos x x 2 = (x, ) ω 2 cos x k sin x (x, ) (x, ) = k 35 D = = 36

7 Solving the sstem eqation The soltion to the sstem eqation {ẋ = Ax + B Stabilit concepts Stable = Cx + D is given b x(t) =e At x()+ (t) =Ce At x()+ t t e A(t τ ) B(τ )dτ Ce A(t τ ) B(τ )dτ + D(t) Unstable Asmptoticall stable Stabilit definitions Assme ẋ = Ax, x() =x The sstem is stable if x(t) is limited fo all x. The sstem is asmptoticall stable if x(t) fo all x. The sstem is nstable if x(t) is nlimited fo some x. Stabilit citeia {ẋ = Ax x() =x x(t) =x e At The behavio of the soltion depends on the eigenvales of A All eigenvales have negative eal pat: As. stab. Some eigenvale has positive eal pat: Unstable No eigenvales with positive eal pat and no mltiple eigenvales on the imagina axis: Stable 39 4 Session otline Feedback and feedfowad PID Contol State-space models Tansfe fnction models Contol design sing pole placement State feedback and obseves Tansfe fnction fom Std the sstem in the (complex) feqenc domain: U(s) Y(s) G(s) U(s) Laplace tansfom of (t) Y(s) Laplace tansfom of (t) G(s) tansfe fnction Y(s) =G(s)U(s) 4 (if the initial state is assmed to be zeo) 42

8 Some opeatos/signals and thei Laplace tansfoms Definition: L f = F(s) = ( ) df Deivative: L = sf(s) ( ) Integal: L f = s F(s) Diac implse: Lδ = Step fnction: Lθ = s e st f (t) Ramp fnction: L(tθ) = s 2 Exponential fnction: L(e at θ)= s a 43 Fom tansfe fnction to state space fom {ẋ = Ax + B x() = = Cx + D { sx(s) =AX(s)+BU(s) Y(s) =CX(s)+DU(s) Y(s) = [ C(sI A) B + D ] U(s) G(s) =C(sI A) B + D = p(s) q(s) q(s) = det(si A) is called chaacteistic polnomial 44 Often, Poles and zeos The oots of p(s) ae called zeos The oots of q(s) ae called poles Note that G(s) = p(s) q(s) Poles of G(s) Eigenvales of A Calclating sstem esponses. Find the tansfe fnction G(s) of the sstem 2. Find the Laplace tansfom U(s) of the inpt (t) 3. Y(s) =G(s)U(s) 4. Use invese Laplace tansfom to find (t) Calclating sstem esponses Example: Compte the step esponse of G(s) = s+ Inpt: U(s) =L {θ(t)} = s Otpt: Y(s) =G(s)U(s) = s(s+) Otpt in the time domain: { } (t) =L = e t s(s + ) G(s) = s + a Step esponse of fist-ode sstems G(s) = s + a = T + st Time constant: T = a Static gain: G() =/a step esponse (t) = a ( e at ) 47 48

9 Step esponse of second-ode sstems Block diagams Real poles: G(s) = (s + a)(s + b) step esponse: (t) = ae bt be at ab ab(b a) U G G 2 Y Y = G 2 G U Complex poles: ω 2 G(s) = s 2 + 2ζωs + ω 2 U G Σ Y Y = ( G + G 2 ) U step esponse: (t) = ζ 2 sin(ω ζ 2 t + φ) G 2 φ = accosζ ω = ndamped feqenc (ω > ) ζ = elative damping ( < ζ < ) 49 5 U Σ G G 2 Y. Feqenc esponse Y = G 2 G ( U Y) Y( + G 2 G )=G 2 G U Y = G 2G + G 2 G U Given a stable sstem G(s), the inpt (t) = sinω t will, afte a tansient, give the otpt ( ) (t) = G(iω ) sin ω t + ag G(iω ) The stead-state otpt is also sinsoidal 5 52 Bode diagam Daw G(iω ) as a fnction of ω (in log-log scale) Amplitde/magnitde/gain diagam ag G(iω ) as a fnction of ω (in log-lin scale) Phase/angle diagam d(t) Example: low-pass filte + (t) =(t) G(s) = s + G(iω )= iω + G(iω ) = ω 2 + ag G(iω )= actan ω 53 54

10 Example: low-pass filte Bode Diagam Nqist Diagam Daw G(iω ) in a pola diagam when ω goes fom to Im G(iω) Magnitde (abs) Ultimate point 2 ϕ a Re G(iω) Phase (deg) 45 ω Feqenc (ad/sec) Im Re G(iω ) Example of Nqist Diagam G(s) = s + G(iω )= iω + = iω ω 2 + Session otline Feedback and feedfowad PID Contol State-space models Tansfe fnction models Contol design sing pole placement State feedback and obseves Small ω : G(iω ) Lage ω : G(iω ) ω i 2 ω Closed-loop contol Analsis of the standad feedback loop efeence Contolle contol Pocess distbances measement Σ e C(s) P(s) Pima goals of the contolle: Follow the efeence Reject distbances C(s): contolle P(s): pocess Closed-loop tansfe fnction (fom to ): Y = PC + PC R 59 Contol design: Choose C to get the desied behavio! 6

11 Example cise contol Assme that the elationship between the thottle and the speed is given b d = P(s) = 5 s +.2 Fist t to eglate the speed with a P-contolle: whee e(t) =(t) (t) (t) =Ke(t) The closed-loop tansfe fnction is given b 5 PC + PC = K s K = 5K s K s+.2 The gain K affects the pole of the closed-loop sstem the static gain of the closed-loop sstem 6 62 Simlation of the contol sstem with diffeent vales of K: Speed Thottle K= K= K=.3 K=.3 K=. Now t a PI-contolle: t ) (t) =K (e(t)+ Ti e(τ )dτ ( U(s) =K + ) st }{{ i } C(s) E(s) K= Time Stationa eo The closed-loop tansfe fnction is given b 5 PC + PC = K( ) + s+.2 st i 5K ( ) s + T + 5 K( ) = i + s+.2 st i s 2 +(5K +.2)s + 5K T i The poles of the closed-loop sstem depend on K and T i The static gain of the closed-loop sstem is alwas Simlation of the contol sstem with = 2, K =.3 and diffeent vales of T i : Speed Thottle Ti=.5 Ti=2 Ti=.5 Ti= Ti=2 Ti= Time 65 No stationa eo 66

12 Whee to place the poles? Stabilit nde Feedback Pole placement accoding to the chaacteistic polnomial q(s) = s 2 + 2ζω s + ω 2 : Im Σ G ϕ ω Re The closed loop sstem is asmptoticall stable if and onl if all the zeos of + G (s) Lage ω faste sstem esponse Smalle ϕ bette damping (elative damping ζ = cosϕ). (Common choice: ζ = cos 45 =.7) 67 lies in the left half plane. 68 The Nqist Citeion Example If G (s) is stable then the closed loop sstem [ + G (s)] is stable if and onl if the the Nqist cveg(iω ) does not encicle. G (s) = G P (s)g R (s), i.e. modif G P sch that the Nqist cve does not encicle 69 G (iω )= = = Σ K s(s+)(s+2) K iω ( + iω )(2 + iω ) Ki( iω )(2 iω ) ω ( + ω 2 )(4 + ω 2 ) = Ki(2 ω 2 3iω ) ω ( + ω 2 )(4 + ω 2 ) 3K ( + ω 2 )(4 + ω 2 ) + i K(ω 2 2) ω ( + ω 2 )(4 + ω 2 ) 7 Stabilit fo the closed loop sstem Amplitde and phase magins G (iω ) Amplitde magina m ag G(iω )=8, G(iω ) = A m Phase magin φ m G(iω c ) =, ag G(iω c )=φ m 8 Stable if K < 6. G (i 2)= 3K 3 6 = K 6 7 (Rles of Thmb: A m [2, 6], φ m [3,6 ]) 72

13 Magins in the Bode diagam Session otline Feedback and feedfowad PID Contol State-space models Tansfe fnction models Contol design sing pole placement State feedback and obseves Session otline Feedback and feedfowad PID Contol State-space models Tansfe fnction models Contol design sing pole placement State feedback and obseves State feedback Pocess: dx = Ax + B = Cx Assme that the fll state vecto x is measable. Contol law: = Lx + l l Σ P L x Closed-loop sstem: dx =(A BL)x + Bl = Cx The closed loop poles ae given b det(si A + BL)= Tning: L is chosen to give the desied poles l is chosen to give the static gain fom to Example - Inveted pendlm State vaiables x =, x 2 = ẏ dx = x + = x 77 78

14 Detemine a state feedback law (assme = ) = Lx = l l 2 x x 2 Simlation fom x() =[.75 ] T :.5 = x sch that the closed-loop chaacteistic polnomial becomes s 2 +.4s +..5 x 2 Closed-loop poles: s det(si A + BL)= + l s + l = s2 + l 2 s + l 2 A compaison with the desied polnomial gives l = 2 l 2 = Obseve It is most often not possible to mease the fll state vecto x. The state can then be estimated sing an obseve: Obseve: d ˆx = A ˆx + B + K( ŷ ) ŷ = C ˆx P Obseve ˆx Dnamics of the estimation eo x = x ˆx: d x = Ax + B A ˆx B KC(x ˆx) =(A KC) x Obseve poles: Tning: det(si A + KC)= 8 K is chosen to give the desied poles fast poles fast convegence ˆx x bt sensitive to noise 82 slow poles slow convegence bt obst The complete contolle (obseve + state feedback) is given b d ˆx = A ˆx + B + K( C ˆx) = L ˆx The tansfe fnction of the contolle is given b C(s) = L(sI A + BL+ KC) K State feedback fom estimated states: x 2 = x