2E1252 Control Theory and Practice Lectre 11: Actator satration and anti wind-p Learning aims After this lectre, yo shold nderstand how satration can case controller states to wind p know how to modify a linear observer to accont for satration be able to interpret the observer modification in a block diagram be able to analyze closed-loop stability sing the small-gain theorem know how to tne the anti-windp gain for a PID controller Mikael Johansson School of Electrical Engineering KTH, Stockholm, Sweden So far Actation is typically limited we have designed linear controllers for linear systems In practically all control systems, the actation is limited (ncertainty models allow s to accont for some nonlinearities) The new block represents a satration 8 >< min if apple min = sat( )= max if max >: otherwise 1
The impact of limited actation Step response of linear system Consider the simple servo model: A nit step response for the linear closed-loop (assming no satration) A controller is which has poles in -13.2444± 13.2255i, and 0.0204 Note: controller is nstable. Step response with satrated control What is wrong? Step responses with actator satration for small and large reference change When actator is satrated, the system operates in open loop (changes in do not affect or controller states while satrated) max Constant inpt make plant and controller states to grow large ( wind p ) particlarly critical when plant or controller is nstable or integrating 2
Understanding what is wrong An improvement Consider observer-based controllers (e.g. LQ, H-infinity, H-2, etc.) d dt ˆx(t) =Aˆx(t)+B(t)+K(y(t) (t) = Lˆx(t) Controller transfer fnction: ŷ(t)) U l (s) = F y (s)y (s) = L(sI A + BL + KC) 1 KY(s) Basic idea of observer: simlate system and correct when ŷ(t) 6= y(t) simlation part does not reflect reality when inpt is satrated! Make sre that observer reflects actal system dynamics d ˆx(t) =Aˆx(t)+B(t)+K(y(t) dt ŷ(t)) (t) = sat( (t)) (t) = Lˆx(t) Modification known as observer-based anti-windp avoids that controller states wind p often enogh to get reasonable performance Step responses with modified observer Analysis: no wind-p in satration Step responses with actator satration for small and large reference change When in satration, (t) =im is constant and controller dynamics is d ˆx(t) =Aˆx(t)+B(t)+K(y(t) ŷ(t)) = dt =(A KC)ˆx(t)+Bim + Ky(t) Stability properties given by A-KC, which is typically stable Taking Laplace transforms, we find U l (s) = L(sI A + KC) 1 KY (s) K(sI A + KC) 1 B im s Smaller overshoot, no longer nstable for large reference changes. 3
Interpretation: feedback from - Interpretation in block diagram To relate modified controller to original, re-write observer as d ˆx(t) =Aˆx(t)+B(t)+K(y(t) ŷ(t)) = dt =(A KC)ˆx(t)+B( (t)+(t) (t)) + K(y(t) ŷ(t)) = =(A BL KC)ˆx(t)+Ky(t)+B((t) (t)) Taking Laplace transforms, we find - W U l (s) = L(sI A + BL + KC) 1 KY (s) L(sI A + BL + KC) 1 B(U(s) U l (s)) = = F y (s)y (s)+w (s)(u(s) U l (s)) The linear control law pls compensation from (t) (t) Basis for many heristic techniqes for anti-windp (more later ) What abot stability? A small-gain analysis We have shown that in absence of satration, the closed-loop system is stable when inpt remains in satration, the controller is stable bt no stability garantees when control moves in and ot of satration! - W M lobal stability can sometimes be ensred sing small-gain theorem To find the linear system M, note that = W ( ) F y ) (I + W ) =(W F y ) := M Since gain of satration nonlinearity is one (cf. Lectre 1), so if kmk 1 < 1 closed-loop stability is ensred 4
Design gidelines Application to PID control 1. Design observer-based controller sing techniqe of choice 2. Modify observer to reflect the presence of satration nonlinearity 3. Attempt to establish stability sing small gain theorem, simlate 4. If nsatisfactory, re-design controller with higher penalty on control Common choice: Rle-of thmb: (tracking anti-reset windp) (T i integral time, T d derivative time of PID) DC Servo nder PID control Servo: PID+anti-windp 5
Smmary The problem with limited actation: New phenomena, not predicted by linear control theory controller, plant states wind p (grow large), feedback loop open Observer-based anti-windp make estimator reflect actal process dynamics ensres that controller states are stable in satration interpretation as feedback from satration error global stability analysis via small-gain theorem Anti-windp for PID control same strctre, heristic compensator, rles-of-thmbs 6