Adaptive Control 28h Contents . Read the book in advancell . Exercises 14h Int roduct ion L1 1. Indirect STR Self-tuning Reg ulators STR L3, L4 Syst ems L5, L8 3. Auto-tunIng Nonlinear St a bility T heory L6, L7 Pro ject 30 h Analysis of Adaptive S L9..L11 Combine wit h real-time systems Auto matic Tuning L12 Present at io n Gain Scheduling L12 Products L13 A Perspective L14 e s ntio Adapt to adiust behavior to new circum- nces What is Adaptive Control? Any alt eration in struct ure or function of Introduct ion ism to make it better fitted to survive and multiply in its enviro nment . Linear feed back . Effects of p rocess variat io ns Change in respo nse of sensory organs to chang ed envi ro nment al conditions · Adaptive Schemes A slow usually uncons cio us mo difi cation of The Adaptive Co nt rol Pro blem individual and so cial act ivity in adjust ment to Learn to acquire knowledge or skill by st udy, Problem Ada pt at ion and feed back? C K. J. Ast ro m and B. Wittenmark
Adaptive Control Lectures 28 h Read the book in advance!! Exercises 14h Labs 1. Indirect STR 2. Direct STR 3. Auto-tuning Project 30 h { Combine with real-time systems { Presentation Computer simulation { Start tomorrow!!! Examination Feedback Use the web! Contents Introduction L1 Parameter Estimation L1. L2 Self-tuning Regulators STR L3, L4 Model Reference Adaptive Systems L5, L8 Nonlinear Stability Theory L6, L7 Analysis of Adaptive Systems L9 .. L11 Automatic Tuning L12 Gain Scheduling L12 Implementation L13 Products L13 A Perspective L14 What is Adaptive Control? Introduction Linear Feedback E ects of Process Variations Adaptive Schemes The Adaptive Control Problem Conclusions Semantics Adapt to adjust behavior to new circumstances. Any alteration in structure or function of an organism to make it better tted to survive and multiply in its environment. Change in response of sensory organs to changed environmental conditions. A slow usually unconscious modi cation of individual and social activity in adjustment to cultural surroundings. Learn to acquire knowledge or skill by study, instruction or experience. Problem: Adaptation and feedback? c K. J. Åström and B. Wittenmark 1
Adaptive Control Brief History Adapt at ion and feed back Early flig ht co nt rol systems 1955 Truxal 1959: Desig ned from an adaptive Dy namic program ming Bellman 1957 Dual cont rol Feld baum 1960 . EEE CSS Committee 1973 System ident ificat io n 1965- Self-organizing co nt rol process SOC Learning co nt rol Tsypkin 1971 Parameter adaptive soc Perfor ma nce adaptive soc Algorit hms MRAS STR 1970 Learning co nt rol system Stability analysis 1980 Prag mat ically: A special class of nonlinear Robust ness 1985 cont rol systems Ind ust rial products 1982 Linear Feed back-2DOF Feedforward Feedback Process An Adaptive Control system G adjustment Controller Setpoint Two-degree-of-freedo m struct ure(FB+FF) Controller Plant The sensit iv ity funct io n and co mplement ary sensitivity function 1+G, Gb 1+L Yol(s) PgB 1+ GpGrb 1+L Reg ular feed back loop L= GPGfb is the loop transfer function. Para meter adjust ment loop s+T=l and dT 1 dG C K. J. Ast ro m and B. Wittenmark
Adaptive Control Adaptation and feedback Truxal 1959:Designed from an adaptive view point IEEE CSS Committee 1973: { Self-organizing control process SOC { Parameter adaptive SOC { Performance adaptive SOC { Learning control system Pragmatically: A special class of nonlinear control systems Brief History Early ight control systems 1955 { Dynamic programming Bellman 1957 Dual control Feldbaum 1960 System identi cation 1965 { Learning control Tsypkin 1971 Algorithms MRAS STR 1970 { Stability analysis 1980 { Robustness 1985 { Industrial products 1982 { An Adaptive Control System Parameter adjustment Controller Plant Controller parameters Control signal Output Setpoint Notice two loops Regular feedback loop Parameter adjustment loop Linear Feedback - 2DOF Feedforward Process u y Feedback −1 uc ym Gfb Gff Σ Gp Two-degree-of-freedom structure (FB+FF) The sensitivity function and complementary sensitivity function S = 1 1 + GpGf b = 1 1 + L = Ycl (s) Yol (s) T = GpGf b 1 + GpGf b = L 1 + L L = GpGf b is the loop transfer function, S + T = 1 and dT T = 1 1 + GpGf b dGp Gp = S c K. J. Åström and B. Wittenmark 2
Judging process variations from Judging Process Variations from Open Loop Data Open Loop Data Open loo p chang es drast ically but little change ose (s+1)(s+a Judging process variations from Judging Process Variations from Open Loop Data Open loop dat a Open loo p res po nse changes little but drastic change in closed loop Go(s 400(1-s7) (s+1)(s+20(1+Ts) Frequency frad/s Frequency lrad C K J. Ast ro m and B.Wittenmark
Judging Process Variations from Open Loop Data Open loop changes drastically but little change in closed loop. G0(s) = 1 (s + 1)(s + a) 0 100 200 300 0 100 200 300 0 5 10 0.0 0.5 1.0 Time Time Judging Process Variations from Open Loop Data 10−3 10−2 10−1 100 101 10−2 100 102 104 Magnitude 10−3 10−2 10−1 100 101 −200 −100 0 Frequency [rad/s] Phase [deg] 10−3 10−2 10−1 100 101 10−1 100 101 Magnitude 10−3 10−2 10−1 100 101 −200 −100 0 Frequency [rad/s] Phase [deg] Judging Process Variations from Open Loop Data Open loop response changes little but drastic change in closed loop. G0(s) = 400(1 sT ) (s + 1)(s + 20)(1 + T s) 012345 0.0 0.5 1.0 0 0.2 0.4 0.6 0.8 1 0 1 Time Time Judging Process Variations from Open Loop Data 10−1 100 101 102 103 10−2 100 102 Magnitude 10−1 100 101 102 103 −400 −200 0 Frequency [rad/s] Phase [deg] 10−1 100 101 102 103 10−2 100 102 Magnitude 10−1 100 101 102 103 −400 −200 0 Frequency [rad/s] Phase [deg] c K. J. Åström and B. Wittenmark 3
Nonlinear Actuators Flow and speed variations PI controller Process (∑)-x Valve characteristics dc(t) q(t)(cin(t-T)-c(t)) where T= Va/q(t)and T=vm/q(t) Time Flight Control Changing Distur bances te noise a+2 coas +a a21品2aa]c+0 (a)t out put error 00 ut p (c It put er C K. J. Ast ro m and B. Wittenmark
Nonlinear Actuators y - 1 u v Σ PI controller Valve Process f(⋅) G0 (s) K 1 + 1 Tis −1 uc Valve characteristics v = f (u) = u4 u 0 0 10 20 30 40 0.2 0.3 0 10 20 30 40 1.0 1.1 0 10 20 30 40 5.0 5.2 Time Time Time uc y uc y y uc Flow and Speed Variations cin Vd Vm c Vm dc(t) dt = q(t) (cin(t ) c(t)) where = Vd=q(t) and T = Vm=q(t) 0 5 10 15 20 0.0 0.5 1.0 0 5 10 15 20 0.0 0.5 1.0 1.5 Time Time Flight Control α V θ q = ˙ θ Nz δ e dx dt = 0 @ a11 a12 a13 a21 a22 a23 0 0 a 1 A x + 0 @ b1 0 a 1 A u 0 0.4 0.8 1.2 1.6 2.0 2.4 80 60 40 20 0 1 2 3 4 Mach number Altitude (x1000 ft) Changing Disturbances b0s2 + b1s + b2 s2 + ω e 2 1 s + 1 y White noise ωs s2 + 2ζωs + ω2 Σ 0 200 400 600 −1 1 0 200 400 600 −1 1 0 200 400 600 −1 1 Time Time Time (a) Output error (b) Output error (c) Output error c K. J. Åström and B. Wittenmark 4
ain sche Gain schedule Adaptive schemes e gain Scheduling Command Model Reference adaptive co nt rol mras Controller Process Self-t uning Regulator STR Cert ainty Equivalet . dual Co nt rol es ● Product ion rate Mach num ber and dynamic pressure Self-Tuning Regulator STR Model Reference Adaptive control MRAS Model Estimation Controller parameters mechanism Controller Controller Plant Certainty equivalence Parameter est imat io Linear feed back fro m e = y-ym is not gradient met hods adequate for para meter adjust ment Least squares Cont rol d Pole pla LQG C K J. Ast ro m and B.Wittenmark
Adaptive Schemes Gain Scheduling Model Reference Adaptive Control MRAS Self-tuning Regulator STR Certainty Equivalence Dual Control Gain Scheduling Process schedule Gain Output Control signal Controller parameters Operating condition Command signal Controller Example of scheduling variables Production rate Machine speed Mach number and dynamic pressure Model Reference Adaptive Control MRAS Adjustment mechanism u Model Controller parameters Plant y Controller ym uc Linear feedback from e = y ym is not adequate for parameter adjustment! The MIT rule d dt = e @e @ Self-Tuning Regulator STR Process parameters Controller design Estimation Controller Process Controller parameters Reference Input Output Specification Self-tuning regulator Certainty Equivalence Parameter estimation Gradient methods Least squares Control design methods PID Pole placement LQG c K. J. Åström and B. Wittenmark 5
he Ada ptive Cont rol Probler Dual Control P Certainty equiva Dual co ntre e Cont roller struct ure Nonlin e rt ai nty equivalet St ate model Cont rol sho uld be directing as well as put Out put Model investig ating Intent io nal pert ur bat io n to obtain bett dj Concept ually very int eresting Specificat io ns Unfort unately very co mplicated Situation dependent? Optimality Applications inclusions Adaptive co nt rol deals wit h Process dynamics Variat ions in pro cess dynamics in disturbanc varying Constant aptive syst ems are no nline Use a controller with troller with Reasonably well understood theoretically varying parameters constant parameters Emerging as an ind ust rially viable techno ogy variations · Applicat io ns Auto matic tuni ng Use an adaptive Use gain scheduling ng dapt ation(FB and FF) · Automatic Tuning hallenging pro bl · Gain Scheduling Cont Adapt at io Adaptation rates C K. J. Ast ro m and B. Wittenmark
Dual Control Nonlinear u y control law Process Calculation of hyperstate Hyperstate u c No certainty equivalence Control should be directing as well as investigating! Intentional perturbation to obtain better information Conceptually very interesting Unfortunately very complicated The Adaptive Control Problem Principles { Certainty Equivalence { Caution { Dual Control Controller structure { Linear Nonlinear { State Model { Input Output Model Control Design Method Parameter Adjustment Method Speci cations { Situation dependent? { Optimality Applications Process dynamics Varying Constant Use a controller with varying parameters Use a controller with constant parameters Unpredictable variations Predictable variations Use an adaptive controller Use gain scheduling Automatic Tuning Gain Scheduling Continuous Adaptation Feedback and Feedforward Conclusions Adaptive control deals with { Variations in process dynamics { Variations in disturbances Adaptive systems are nonlinear Reasonably well understood theoretically Emerging as an industrially viable technology Applications { Automatic tuning { Gain scheduling { Continuous adaptation (FB and FF) Many challenging problems remain { Dual control { Nonlinear systems { Adaptation rates c K. J. Åström and B. Wittenmark 6