Non-identifier based adaptive control in mechatronics Lecture notes

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1 Non-identifier based adaptive control in mechatronics Lecture notes SS 2012 July 18, 2012 These notes discuss the content of the lecture of the summer term 2012 in brief form. A more detailed and comprehensive treatment of the material is given in the script Non-identifier based adaptive control in mechatronics Part I: speed control All references to Definitions, Lemmas or Theorems, etc. refer to this script. I am indebted to Florian Larcher who nicely did the typesetting and many of the drawings and provided an excellent first draft of these lecture notes. 1

2 Non-identifier based adaptive control in mechatronics Lecture notes 0 SS2012 Lehrstuhl für Elektrische Antriebssysteme und Leistungselektronik, TUM 2012/04/20 Contents 1 Non-identifier based adaptive control in mechatronics:an introduction 2012/04/ Learning outcomes Motivation: Reference tracking under load CNC-turning machine Standard solutions in industry Motivation for non-identifier based adaptive control Direct and indirect adaptive control Modeling of standard mechatronic systems High-gain adaptive stabilization 2012/04/ Motivation: Structural properties of 1MS speed control? High-gain adaptive stabilization Proof of our first result 2012/05/ Adaptive λ-tracking control and funnel control:an introduction 2012/06/ High-gain adaptive stabilization Motivation for wider system class & more robust controllers Adaptive λ-tracking control Funnel control Funnel speed control of unsaturated 1MS2012/06/ Funnel speed control of saturated 1MS2012/06/29, part Internal model principle2012/06/29 & 2012/07/ Motivation 2012/06/29, part Internal model principle Funnel control with internal model Page 2/41

3 Non-identifier based adaptive control in mechatronics Lecture notes 1 SS2012 Lehrstuhl für Elektrische Antriebssysteme und Leistungselektronik, TUM 2012/04/20 1 Non-identifier based adaptive control in mechatronics: An introduction 2012/04/ Learning outcomes After this course you should be able to evaluate and create models of standard mechatronic systems including dynamic friction understand, analyze and evaluate nonlinear, functional differential equations e.g. existence and uniqueness of a solution understand and apply mathematical tools e.g. Lyapunov s 2. direct method, proof by contradiction/induction,... to prove theoretical results create and design robust non-identifier based adaptive controllers and internal models for speed and position control of mechatronic systems 1.2 Motivation: Reference tracking under load CNC-turning machine Ω y x problem statement reference trajectory Ω ref,x ref,y ref : R 0 R 3 given high-precision position and speed control, e.g. for prescribed λ > 0 challenges t t 0 : et = y ref t yt λ. nonlinear effects e.g. actuator saturation, friction friction and disturbances unknown and varying system parameters only roughly known Page 3/41

4 Non-identifier based adaptive control in mechatronics Lecture notes 1 SS2012 Lehrstuhl für Elektrische Antriebssysteme und Leistungselektronik, TUM 2012/04/ Standard solutions in industry feedforward control disturbances y ref position controller speed controller u actuator mechanical system y ẏ disturbance observer sensors implementation physical system heuristic or empirical controller design trial-and-error time consuming modell-based controller design system identification complex/time consuming general challenge desired tracking accuracy not explicitly included in design t t 0 : et = y ref t yt λ? 1.4 Motivation for non-identifier based adaptive control Motivation 1: desirable to have tools at hand which directly allow to account for desired tracking accuracy, e.g. adaptive λ-tracking control funnel control e0 e = y ref y ψ0 e0 ψt et ψ λ e λ Time t [s] λ t Time t [s] λ-strip funnel ψ0 Motivation 2: to avoid stick-slip major problem due to friction, the simplest approach is high gain in the feedback stiff closed-loop system. Motivation 3: mechatronic systems at least of low order are structurally known: the direction of influence of control input on system output i.e. the sign of the high-frequency gain of the system Page 4/41

5 Non-identifier based adaptive control in mechatronics Lecture notes 1 SS2012 Lehrstuhl für Elektrische Antriebssysteme und Leistungselektronik, TUM 2012/04/20 the time derivative e.g. d 2 dt 2 yt of the system output which is directly affected by the control input i.e. the relative degree of the system. the internal dynamics of the system are input-to-state stable i.e. stability of the zero-dynamics or, for linear systems, the minimum-phase property. Non-identifier based high-gain adaptive control allows to think in system classes robustness: often only the signs of the system parameters must be known prescribe desired accuracy a priori e.g. according to customer specifications reduce friction effects 1.5 Direct and indirect adaptive control reference controller with fixed structure control action system of known structure output measured values identification/ estimation of syst. parameters estimated parameters adaption of controller parameters control objectives controller parameters Figure 1: Indirect adaptive control: The adaptive controller estimates/identifies the system. Controller parameters are adjusted according to the system estimate. Page 5/41

6 Non-identifier based adaptive control in mechatronics Lecture notes 1 SS2012 Lehrstuhl für Elektrische Antriebssysteme und Leistungselektronik, TUM 2012/04/20 reference controller with fixed structure control action system output measured values adaption of controller parameters control objectives controller parameters Figure 2: Direct adaptive control: the adaptive controller does not estimate/identify the system. Controller parameters are adjusted according to output measurement, control objective, control action and reference. Non-identifier based high-gain adaptive control is a direct adaptive control strategy! Typical non-identifier based adaptive controllers are given by: 1. Classical high-gain adaptive control ut = kt y ref t yt }{{} et kt = y ref t yt 2, k0 = k 0 R 2. Adaptive λ-tracking control where bounded noise see Fig. 3 ut = kt y ref t yt }{{} et kt = max{ et λ,0}, k0 = k 0 R, λ > 0 n m W 1, R 0 ;R t 0 c > 0: n m t c should be considered appropriately: choose λ large enough: λ c 3. Funnel control with positive, Lipschitz continuous funnel boundary ψ ut = kt y ref t yt }{{} et where kt = Page 6/41 1 ψt et and ψ : R 0 [λ,

7 Non-identifier based adaptive control in mechatronics Lecture notes 1 SS2012 Lehrstuhl für Elektrische Antriebssysteme und Leistungselektronik, TUM 2012/04/20 yt,n m t c y m = y +n m t c Figure 3: Output measurement y m of output y = 0 deteriorated by exemplary measurement errors and/or noise n m. First motivating examples of the application of non-identifier based adaptive controllers are discussed in Tutorial 1 see Tutorial notes. 1.6 Modeling of standard mechatronic systems Components of mechatronic systems host computer monitoring, control objectives power source e.g. electricity grid, battery actuator current controlled electrical drive physical system mechanics micro-processor with controller implementation power electronics e.g. converter, inverter, etc. electrical machine e.g. AC or DC motor linkage e.g. clutch, gear, shaft work machine e.g. milling head, applicator roll real-time system sensors e.g. current sen., resolver, encoder instrumentation measurement of e.g. current, motor and/or load speed & position For more details on the modeling of the components of standard mechatronic systems see script p Page 7/41

8 Non-identifier based adaptive control in mechatronics Lecture notes 1 SS2012 Lehrstuhl für Elektrische Antriebssysteme und Leistungselektronik, TUM 2012/04/ Stiff servo-system 1MS 1/g r m L ν 2 ω g r +F 2 ω g r 1MS ν 1 ω +F 1 ω u actuator ω = φ k A 1/Θ 1/g r m M ω φ φ/g r satûa u A ω m φ m ṅ m sensors n m The mathematical model of the saturated 1MS is given by d dt xt = Axt+bsat û A ut+ua t +B L yt = c xt, x0 = x 0 R 2 F 1 ωt m L t+f 2 ω g r t, 1MS where system matrix A viscous friction terms are included, input vector b, disturbance input matrix B L, output vector c, disturbances u A, m L, friction operators F 1, F 2 and system parameters are as follow [ ] ν 1 +ν 2 /gr 2 0 kaθ [ A = Θ 1, b =, B L = ] 1 Θ g rθ, c R 2, Θ > 0, 0 0 g r R\{0}, ν 1,ν 2 > 0, û A,k A > 0, u A, m L L R 0 ;R and i {1,2}: F i : CR 0 ;R L loc R 0;R with M Fi := sup{ F i ωt t 0, ω CR 0,R} <. 1MS-Data The state variable xt = ωt, φt represents angular velocity speed and angle position at time t 0 [s] in [rad/s] and [rad], respectively. The symbols have the following physical meaning [dimension]: actuator u A [Nm] and load disturbance m L [Nm] actuator gain k A > 0 [1] inertia Θ > 0 [kgm] gear ratio g r R\{0} [1] mostly known viscous friction coefficients ν 1, ν 2 0 [ Nms rad measurement errors or noise n m [rad] and ṅ m [rad/s] ] Page 8/41

9 Non-identifier based adaptive control in mechatronics Lecture notes 1 SS2012 Lehrstuhl für Elektrische Antriebssysteme und Leistungselektronik, TUM 2012/04/20 ν 1 ω +F 1 ω ν 1 ω Figure 4: Nonlinear dynamic but bounded friction behavior F 1 ω e.g. due to dynamic LuGre friction model with linear, but unbounded viscous friction ν 1 ω: friction is a continuous function of speed ω! In contrast to many text books, friction is modeled as nonlinear, dynamic operator, which here implies that friction is not continuous see Fig. 4. For simplicity, in the following, we consider the nonlinear friction operator F 1 ω as nonlinear, locally Lipschitz continuous function, e.g. ω atanω. More details on nonlinear, dynamic friction modeling can be found in [1, Section 1.4.5]. Page 9/41

10 Non-identifier based adaptive control in mechatronics Lecture notes 2 SS2012 Lehrstuhl für Elektrische Antriebssysteme und Leistungselektronik, TUM 2012/04/27 2 High-gain adaptive stabilization 2012/04/ Motivation: Structural properties of 1MS speed control? For speed control, the stiff servo system 1MS simplifies to without noise ṅ m and friction operator F 1,F 2 ωt = ν 1+ν 2 /gr 2 ωt+ k A satûa ut+ua Θ Θ t f 1ωt m Lt+f 2 ωt gr Θ g rθ, 1MS ω yt = c 1 ωt, ω0 = ω R where Θ > 0, g r R\{0}, signg r known, ν 1,ν 2 0, û A,k A > 0, u A, m L L R 0 ;R, c 1 {1,1/g r }and f 1,f 2 L R 0 ;R and locally Lipschitz continuous. Structural properties of 1MS ω with 1MS ω -Data: relative degree see Definition 2.1 ẏt = c 1 ωt =...+c 1 k A θ sat û A ut+u A t+... = r = 1. } 1MS ω -Data high frequency gain see Definition 2.4 γ 0 := c 1 k A θ where signγ 0 1MSω -Data = signg r is known. minimum-phase? see Definition 2.7 n = 1 = r = 1 = No internal dynamics, hence minimum-phase. 2.2 High-gain adaptive stabilization Considered system class relative-degree-one case Definition 2.1 System class S lin 1. A system of form ẋt=axt+but, yt=c xt n N, x0 = x 0 R n, A,b,c R n n R n R n. } 2.1 is of Class S lin 1 if, and only if, the following system properties sp hold: S lin 1 -sp 1 relative degree is one and sign of the high-frequency gain is known, i.e. γ 0 := c b 0 and signγ 0 is known; S lin 1 -sp 2 it is minimum-phase, i.e. s C 0 : det S lin 1 -sp 3 the regulated output y is available for feedback. Page 10/41 [ ] sin A b c 0, and 0

11 Non-identifier based adaptive control in mechatronics Lecture notes 2 SS2012 Lehrstuhl für Elektrische Antriebssysteme und Leistungselektronik, TUM 2012/04/27 Goal: design single adaptive controller to stabilize any system of S lin 1! Motivation: Root locus controller system u = ky u F 1 s = s+4s+5 s 1 2 s+1 y Structural properties of F 1 s: relative degree pole excess: r = 1 positive high-frequency gain lim s sf 1 s = 1 minimum-phase numerator is Hurwitz imaginary axis real axis Figure 5: Root-locus of F 1 s and u = ky poles, zeros = k > 0 such that closed-loop is stable for all k > k, however k clearly depends on system data and must be known a priori to design a stabilizing controller. Questions can this minimum gain k found online by some suitable gain adaption law? how to achieve kt > k for some t 0? does the adapted gain k remain bounded? Page 11/41

12 Non-identifier based adaptive control in mechatronics Lecture notes 2 SS2012 Lehrstuhl für Elektrische Antriebssysteme und Leistungselektronik, TUM 2012/04/27 do all states here: x : R 0 R 3 remain bounded? is equilibrium at origin asymptotically reached? Our first result Theorem 2.2 High-gain adaptive control for a simple system. Consider the following system given by } ẏt = a 1 yt+a 2 zt+γ 0 ut, y0, z0 = y 0, z 0 R 2, 2.2 żt = a 3 yt a 4 zt a 1,a 2,a 3 R, a 4,γ 0 R >0. The high-gain adaptive controller ut = ktyt where kt = yt 2, k0 = k 0 HG 1 with design parameter k 0 > 0 applied to 2.2 yields a closed-loop initial-value problem with the following properties: i there exists a unique, maximal solution y,z,k : [0,T = R 2 R >0, T 0, ]; ii the solution is global, i.e. T = ; iii all signals are bounded, i.e. y,z L R 0 ;R and k L R 0 ;R >0 ; iv lim t kt = 0 and limt yt, zt = 0 2. Before we can show our first result, we need to understand the concept of solutions of ordinary differential equations ODEs Existence and uniqueness of solutions Consider the initial-value problem IVP given by ordinary differential equation where ẋt = ft,xt, xt 0 = x 0 IVP f: I D R n, n N, is called the right-hand side of IVP with open time interval I R and open, non-empty state domain D R n t 0 I is the initial time and x 0 D is the initial state or condition. Questions: what is a solution of the initial-value problem IVP? do solutions always exist? Page 12/41

13 Non-identifier based adaptive control in mechatronics Lecture notes 2 SS2012 Lehrstuhl für Elektrische Antriebssysteme und Leistungselektronik, TUM 2012/04/27 ẋ = 1 ẋ = 1 Figure 6: An initial value problem without solution. x is there a unique solution? does solution exist on complete interval I R 0? Answers: First answer: A continuously differentiable function x: [t 0,T R n, T = Tt 0,x 0 t 0, ] is called solution of the initial-value problem IVP if it satisfies IVP for all t [t 0,T I and xt 0 = x 0. Second answer: there exist differential equations which do not have a solution in the sense as stated above, see motivating examples below Third answer: there does not necessarily exist a unique solution see motivating examples below Fourth answer: there does not necessarily exist a global solution see motivating examples below Discussion of motivating examples: Example 2.3. ẋt = { 1, xt 0 1, xt < 0, x0 = 0 = no solution, see Fig. 6 recall also the practical simulation exercise. Example 2.4. Try to find a solution: = no unique solution. Example 2.5. ẋt = 2 xt, x0 = 0 = x 1 t = 0 t 0 = α < 0 < β α t 2, t,α] x 2 t = 0, t α,β t β 2, t [β, ẋt = xt 2, x0 = 1 Page 13/41

14 Non-identifier based adaptive control in mechatronics Lecture notes 2 SS2012 Lehrstuhl für Elektrische Antriebssysteme und Leistungselektronik, TUM 2012/04/27 A solution is given by xt = 1. We verify this claim: 1 t initial value: x0 = 1 = 1 = OK 1 d Solution of IVP: dt xt = d 1 dt 1 t 1 = +1 for all t [0,1 = OK 1 t 2 maximal time of existence: T = T1 = 1 = solution not a global! Example 2.6. ẋt = 1+ 1 xt+ xt3 t xt = x 0 t 0 t x 2 0t 2 0 x 2 0t 2 0 1exp2t t 0 t 2, xt 0 = x 0 R, t 0 > 0 Possible singularity in the right-hand side at: x x 2 0t 2 0 = x 2 0t exp2t t 0 = ln 0 t 2 0 = 2t t x 2 0t For x 2 0t 2 0 > 1, the solution explodes and T = Tt 0,x 0 = t 0 +ln x2 0 t2 0 < finite escape x 2 0 t2 0 1 time. A solution of the initial-value problem IVP exists Peano existence theorem and it is unique Picard-Lindelöf theorem, see Theorem D.32 if the right-hand side of IVP satisfies two presuppositions see Definition D.31. Page 14/41

15 Non-identifier based adaptive control in mechatronics Lecture notes 3 SS2012 Lehrstuhl für Elektrische Antriebssysteme und Leistungselektronik, TUM 2012/05/04 3 Proof of our first result 2012/05/04 Proof of Theorem 2.2. The closed-loop system is given by yt a 1 γ 0 ktyt+a 2 zt d zt = a dt 3 yt a 4 zt, kt yt 2 }{{}}{{} ẋ ft,xt=fxt Note that f does not explicitly dependent on t. Step 1: We show that Assertion i holds true. Define y0 y 0 z0 = z 0 R 3 k0 k 0 }{{}}{{} x0 x I = R 0 and D = }{{} R }{{} R R }{{} 0 y z k 0 The right-hand side of the closed loop system 3.3 is i continuous on I D ii and, for any compact set here a ball, see Fig. 7 C D, the following hold: z x x k y Figure 7: Compact ball in R 3. x = y,z,k C M C > 0: for any y,z,k,ỹ, z, k C, we have hint: y +ỹ y + ỹ y2 +z 2 +k }{{} 2 M C x ky kỹ = k ky + ky kỹ y k k + k y ỹ M C k k +M C y ỹ y 2 ỹ 2 = y ỹy +ỹ y ỹ y + ỹ 2M C y ỹ Page 15/41

16 Non-identifier based adaptive control in mechatronics Lecture notes 3 SS2012 Lehrstuhl für Elektrische Antriebssysteme und Leistungselektronik, TUM 2012/05/04 Hence, fy,z,k fỹ, z, k a 1 y ỹ + a 2 z z + γ 0 ky kỹ + + a 3 y ỹ + a 4 z z + y 2 ỹ 2 a 1 + γ 0 M C + a 3 +2M C y ỹ + }{{} c 0 + a 2 + a 4 z z + γ }{{} 0 M C k k }{{} c 1 c 2 y ỹ L C z z = OK k k where L C = c 0 +c 1 +c 2 = f,, is locally Lipschitz continuous. Therefore, in view of Theorem D.32, there exists a unique solution x = y,z,k : [0,T = R 3 with maximal T 0, ]. This completes step 1 of the proof and shows assertion i of Theorem 2.2. Step 2: We show some technical inequalities: Clearly, m > 0, a,b R: a ±2ab = 2 mb + a2 m m +mb2 a2 m +mb2 BF and, for any a 4 > 0, introduce the following Lyapunov candidate: y V : R R = R 0 : = Vy,z = z 1 2 y a 4 z 2 y,z R 2 : 0 1 min{1, 1 2 a 4 } y 2 Vy,z z 1 max{1, 1 2 a 4 } y 2 z From Step 1 we know, that there exists a solution x = y,z,k on [0,T, hence we Page 16/41

17 Non-identifier based adaptive control in mechatronics Lecture notes 3 SS2012 Lehrstuhl für Elektrische Antriebssysteme und Leistungselektronik, TUM 2012/05/04 kt k Figure 8: Assumption: k is unbounded on [0,T. t t may differentiate Vy,z along this solution, i.e. t [0,T : Vyt,zt = ytẏt+ 1 a 4 ztżt = yta 1 γ 0 ktyt+a 2 zt+ 1 a 4 zta 3 yt a 4 zt γ 0 kt a 1 yt 2 + a 2 + a 3 γ 0 kt a a 4 } {{ } a a 2 + a 3 a 4 γ 0 kt a 1 yt m We choose m = 1 arbitrary in BF, then Vyt,zt γ 0 kt a Step 3: We show that k is bounded on [0,T: yt zt zt }{{} 2 b 2 yt zt2 a 2 + a 3 a 4 a 2 + a 3 a m 2 zt2 zt 2. 2 } {{ } high gain: term will eventually get smaller than zero Proof by contradiction: Assume that k is unbounded on [0,T. Since yt zt2 t [0,T: k 0 non-decreasing therefore t 0 t [t,t: kt k := 1 γ 0 a a 2 + a a 4 2. Page 17/41

18 Non-identifier based adaptive control in mechatronics Lecture notes 3 SS2012 Lehrstuhl für Elektrische Antriebssysteme und Leistungselektronik, TUM 2012/05/04 Hence, t [t,t: 1 Vyt,zt }{{} 2 yt2 1 2 zt2 }{{} =: d dt ξt 1 yt 2 2 zt 1 2µ 2 Vyt,zt }{{} =:ξt where µ 2 := 1 max{1, 1 a 4 }. From the Bellman-Gronwall Lemma in its differential form, see Lemma E.46, we conclude that t [t,t: Vyt,zt Vyt,zt exp 2µ 2 t t 3.4 and therefore t [0,T: kt = kt + t kτdτ = kt + t yτ 2 dτ t kt + t µ 1 Vyτ,zτdτ 3.4 which implies that k is bounded on [0,T. t 1 kt + 1 µ 1 Vyt,zt t exp 2µ t 2 τ t dτ <, Step 4: We show that Assertion ii holds true, i.e. T =. Since k is non-decreasing and bounded on [0,T, the limit lim t= T kt = k < exists. Hence, we can find an upper bound for the right-hand side of 3.3 as follows t [0,T : d dt yt σ yt zt where σ > 0 zt Invoking the Bellman-Gronwall Lemma in its differential form, see Lemma E.46 again yields yt y0 zt e σt = T =. z0 Step 5: We show that Assertion iii and iv hold true. From Steps 3 and 4, it follows that Moreover, since 0 yτ 2 }{{} kτ k L R 0 ;R >0. dτ k k 0 < = y L 2 R 0 ;R but y L R 0 ;R. We look at the internal dynamics żt = a 3 yt a 4 zt with solution derived by variation-of- Page 18/41

19 Non-identifier based adaptive control in mechatronics Lecture notes 3 SS2012 Lehrstuhl für Elektrische Antriebssysteme und Leistungselektronik, TUM 2012/05/04 constants t 0: zt = z 0 e a 4t + t e a 4t τ 0 }{{} ht yτ }{{} L 2 R 0 ;R dτ } {{ } L 2 R 0 ;R = z L 2 R 0 ;R. Combining the results above yields d yt = fyt,zt,kt dt zt }{{} L 2 R 0,R 2 = d dt y L 2 R z 0,R 2. We want to show that y, z are bounded. Since y, z L 2 R 0 ;R and ẏ, ż L 2 R 0 ;R, we can apply Lemma E.44: which completes the proof. = lim yt = lim zt = 0 t t lim kt = lim yt 2 = 0 t t = y,z L R 0 ;R, Page 19/41

20 Non-identifier based adaptive control in mechatronics Lecture notes 4 SS2012 Lehrstuhl für Elektrische Antriebssysteme und Leistungselektronik, TUM 2012/06/01 4 Adaptive λ-tracking control and funnel control: An introduction 2012/06/ High-gain adaptive stabilization One can show the following theorem: Theorem High-gain adaptive control for systems of class S1 lin see Theorem Consider a system of class S1 lin given by } ẋt = Axt+but, n N, x0 = x 0 R n, yt = c xt A,b,c R n n R n R n SYS. lin 1 The high-gain adaptive controller ut = signc bktyt where kt = q1 yt q 2, k0 = k 0 HG 1 with design parameters q 1 > 0, q 2 1 and k 0 > 0 applied to SYS lin 1 yields a closed-loop initial-value problem with the following properties: i there exists a unique, maximal solution x,k : [0,T R n R >0, T 0, ]; ii the solution is global, i.e. T = ; iii all signals are bounded, i.e. x L R 0 ;R n and k L R 0 ;R >0 ; iv lim t kt = 0 and limt xt = 0 n. Proof. Proof was not explicitly presented see Proof of Theorem 2.21 on p It is quite similar to the proof of our first result. Drawback of this result: Already the simple nonlinear stiff-servo system 1MS ω with 1MS ω -Data is not element of class S1 lin why?. Hence, a wider system class and/or more robust controllers are needed Motivation for wider system class & more robust controllers Motivation 1: System class S lin 1 is restrictive e.g. 1MS ω / S lin 1 linear no external disturbances possibly piecewise continuous! no nonlinear perturbations Motivation 2: high-gain adaptive stabilizing controller not robust gain drift due to noisy measurements or external disturbances Motivation 3: reference tracking not considered only stabilization, or for certain reference classes, asymptotic tracking with internal models Page 20/41

21 Non-identifier based adaptive control in mechatronics Lecture notes 4 SS2012 Lehrstuhl für Elektrische Antriebssysteme und Leistungselektronik, TUM 2012/06/01 Gain drift Time t [s] a due to noise n m W 2, R 0 ;R u d = 0: k, y Time t [s] b due to disturbance u d L R 0 ;R n m = 0: k and y. Figure 9: Simulation results for closed-loop system 4.5, 4.6 only shown for the first 20 [s]. Simulations are performed in Matlab/Simulink for the following system ẋt = 10xt+ ut+u d t x0 = 1, u, d L R 0 ;R, yt = xt+n m t n m W 2, R 0 ;R } 4.5 and controller ut = ktyt where kt = yt 2, k0 = Adaptive λ-tracking control e0 e lim dist et, [0,λ] t λ λ Time t [s] λ-strip The adaptive λ-tracking controller assures tracking with prescribed asymptotic accuracy, i.e. for all λ > 0, the tracking error et = y ref t yt asymptotically converges into the λ-strip, i.e. lim dist et, [0,λ] = 0 ; t state variable is bounded, i.e. x L R 0 ;R n ; Page 21/41

22 Non-identifier based adaptive control in mechatronics Lecture notes 4 SS2012 Lehrstuhl für Elektrische Antriebssysteme und Leistungselektronik, TUM 2012/06/01 control action is bounded, i.e. u L R 0 ;R Application: Adaptive λ-tracking speed control of unsaturated 1MS ω Theorem 4.1 Adaptive λ-tracking speed control of unsat. 1MS ω. Consider the mechatronic system 1MS ω with 1MS ω -Data and û A = unsaturated actuator. Then, for arbitrary initial value ω 0 R and reference signal y ref W 1, R 0 ;R, the adaptive λ-tracking controller } ut = signg r ktet, where et = y ref t yt LT 1 kt = d λ et 2, k0 = k 0 with design parameters k 0 > 0 and λ > 0 applied to 1MS ω yields a closed-loop initial-value problem with the properties: i there exists a unique, maximal solution ω,k : [0,T R R >0, T 0, ]; ii the solution is global, i.e. T = ; iii all signals are bounded, i.e. ω L R 0 ;R and k L R 0 ;R >0 ; iv the λ-strip is asymptotically reached, i.e. lim t dist et,[0, λ] = 0. Proof. The proof was not explicitly presented. It is a special case of the proof of Theorem 2.27 see p Funnel control ψ0 ψt e0 et ψ = F ψ e λ t Time t [s] funnel F ψ ψ0 The funnel controller assures tracking with prescribed transient accuracy, i.e. λ > 0 ψ W 1, R 0 ;[λ, e0 < ψ0 t 0: et < ψt; state variable is bounded, i.e. x L R 0 ;R n ; control action is bounded, i.e. u L R 0 ;R. Page 22/41

23 Non-identifier based adaptive control in mechatronics Lecture notes 4 SS2012 Lehrstuhl für Elektrische Antriebssysteme und Leistungselektronik, TUM 2012/06/ Motivation for funnel control Motivation 1: Adaptive λ-tracking and high-gain adaptive control exhibit a non-decreasing gain and so, as time tends to infinity, it is likely that e.g. noise sensitivity permanently exceeds an acceptable level if gain adaption is not stopped. Motivation 2: Adaptive λ-tracking control assures tracking with prescribed asymptotic accuracy, however statements on the transient accuracy are not possible; e.g. albeit bounded large overshoots might occur and the λ-strip is not reached in finite time in general Motivation 3: Input saturations are not yet considered e.g. actuator saturation of electrical drives. Unpredictable overshoots and infinite transient time Time t [s] a shortrun: noisy output y +n m, reference y ref, y ref ±λ Time t [s] b longrun: noisy output y +n m, reference y ref, y ref ±λ. Figure 10: Simulation results for closed-loop system 4.5, 4.7 with y ref = 10, λ = k0 = q 1 = 1, q 2 = 2, y0 = 0, noise n m W 1, R 0 ;[ 0.1, 0.1] and disturbance u d = 0. Simulations are performed in Matlab/Simulink for system 4.5 and adaptive λ-tracking controller ut = ktyt where kt = dλ et 2, k0 = 1 = λ. 4.7 where error et = y ref t yt and y ref = Admissible funnels The funnel boundary ψ must be chosen from the following set B 1 := { ψ: R 0 R >0 c > 0 : ψ W 1, R 0,[c, }, 4.8 which allows to introduce the performance funnel with funnel boundary F ψ t = ψt for all t 0. F ψ := { t,e R 0 R e < ψt } 4.9 Page 23/41

24 Non-identifier based adaptive control in mechatronics Lecture notes 4 SS2012 Lehrstuhl für Elektrische Antriebssysteme und Leistungselektronik, TUM 2012/06/01 Examples 4.2. Let T L, T E > 0 [s] and Λ λ > 0. ψ L : R 0 [λ, Λ], t ψ L t := max { Λ t/t L, λ } ψ E : R 0 λ, Λ], 10 8 t ψ E t := Λ λexp t/t E +λ. ψ E ψ L Time t [s] Application: Funnel speed control of unsaturated 1MS ω Theorem 4.3 Funnel speed control of unsaturated 1MS ω. Consider the mechatronic system 1MS ω with 1MS ω -Data and û A = unsaturated actuator. Then, for any funnel boundary ψ B 1, gain scaling function ς B 1, reference signal y ref W 1, R 0 ;R and initial value ω0 = ω 0 R satisfying the funnel controller y ref 0 c 1 ω0 < ψ0, FC 1 -init ut = signg r ktet where et = y ref t yt and kt = ςt ψt et FC 1 applied to 1MS ω yields a closed-loop initial-value problem with the properties i there exists a unique solution ω : [0, T R with maximal T 0, ]; ii the solution ω does not have finite escape time, i.e. T = ; iii the tracking error is uniformly bounded away from the funnel boundary, i.e. ε > 0 t 0 : ψt et ε; iv gain and control action are uniformly bounded, i.e. k,u L R 0 ;R. Page 24/41

25 Non-identifier based adaptive control in mechatronics Lecture notes 5 SS2012 Lehrstuhl für Elektrische Antriebssysteme und Leistungselektronik, TUM 2012/06/22 5 Funnel speed control of unsaturated 1MS 2012/06/22 In this lecture, we discussed an outline which is relevant for the exam of the proof of Theorem 4.3. In the following the full proof is presented. Since we allow for essentially bounded disturbances u A and m L, we must apply an extended existence theory in the sense of Carathéodory see Section D.2 in script: Definition D.35 and Theorem D.36. Proof of Theorem 4.3. Step 1: Some preliminaries. First, note that signg r = sign c 1 k A Θ and, for et = y ref t yt yt = y ref t et = c 1 ωt, a 1 := ν 1 +ν 2 /g 2 r Θ 0 and g: R R, rewrite 1MS ω, FC 1 in error ODE, i.e., ėt = ẏ ref t ẏt 1MSω = ẏ ref t c 1 ωt = ẏ ref t+a 1 c 1 ωt c 1k A Θ }{{} =:γ 0 signg r ktet+u A t = ẏ ref t+a 1 y ref t et γ 0 ktet γ 0 u A t 1 +c 1 g y ref t et + c 1 c 1 g r Θ m Lt. ω gω := f f 1ω 2 Θ + g r Θ, ω g r +c 1 gωt+c 1 m L t g r Θ In more compact form: et = ft, et := a 1 + γ 0 kt et+a 1 y ref t+ẏ ref t γ 0 u A t 1 +c 1 g c 1 y ref t et + c 1 g rθ m Lt CL Step 2: We show Assertion i. In view of the Carathéodory Existence Theorem D.36, we need open state domain D. Trick: introduce augmented system with time state τ, i.e. τt = 1, τ0 = τ 0 = 0. Then, for 1 f : I D R 2 ςt, t,τ,e a 1 γ 0 e+a ψ τ e 1 y ref t+ẏ ref t RHS 1 +c 1 g c 1 y ref t e + c 1 m g rθ Lt γ 0 u A t where I := R 0 and open state domain D := {τ, e R R e < ψ τ }. Hence, for augmented state ˆx := τ, e, we may write unsaturated 1MS ω, FC 1 in standard Page 25/41

26 Non-identifier based adaptive control in mechatronics Lecture notes 5 SS2012 Lehrstuhl für Elektrische Antriebssysteme und Leistungselektronik, TUM 2012/06/22 form, i.e. d dt ˆxt = ft, ˆxt, ˆx0 = τ 0 y ref 0 c 1 ω 0. Note that all exogenous signals are bounded, i.e. u A, ς, ẏ ref, y ref, m L L R 0 ;R and the perturbation is bounded, i.e. g L R 0 ;R. So, for any compact set C D, the following holds M C > 0 τ, e C : τ, e M C m C > 0 τ, e C : min{ψ τ e } m C. To apply Theorem D.36, we have to check whether f, satisfies the Carathéodory conditions car 1,..., car 4 see Definition D.35 on p. 119: car 1 ft, is continuous on D for almost all t I. OK car 2 f,τ,e is measurable e.g. all exogenous signals are piecewise continuous for each fixed τ,e D. OK car 3 f,τ,e is locally integrable on I for each fixed τ,e D, since ft,τ,e 1+a 1 M C + γ 0 ς m C M C +a 1 y ref + ẏ ref + c 1 Θ g + m L + γ 0 u A =: l C <. OK car 4 First note that, since ψ W 1, R 0 ;R is globally Lipschitz, i.e. there exists L ψ > 0 such that ψt 1 ψt 2 L ψ t 1 t 2 for all t 1, t 2 0, the following holds for all τ, e, τ,ẽ C: e ψ τ e ẽ ψ τ ẽ e 1 ψ τ e 1 ψ τ ẽ + 1 ψ τ ẽ e ẽ 5.10 ψ τ ẽ ψ τ +e M C ψ τ e ψ τ ẽ + 1 e ẽ 5.11 m C M C ψ τ ψ τ + e ẽ + 1 e ẽ 5.12 m 2 C m C MC + 1 e ẽ + M C L m 2 C m C m 2 ψ τ τ. SC 1 C Moreover, since f 1 and f 2 are locally Lipschitz due to 1MS ω -Data, there exists L g > 0 such that the following holds for all t, e, t,ẽ C 1 1 g 1 y ref t e g y ref t ẽ L g y ref t e 1 y ref t+ ẽ c 1 c 1 c 1 c 1 c 1 L g c 1 e ẽ SC 2 c 1 Page 26/41

27 Non-identifier based adaptive control in mechatronics Lecture notes 5 SS2012 Lehrstuhl für Elektrische Antriebssysteme und Leistungselektronik, TUM 2012/06/22 Hence, for each compact C I D exists l C > 0, such that for all t,τ,e, t, τ,ẽ }{{}}{{} C: x x ft,τ,e ft, τ,ẽ a 1 e ẽ + γ 0 ςt e ψ τ e ẽ ψ τ ẽ + c g y ref t e g y ref t ẽ c 1 c 1 SC 1,SC 2 τ τ l C. OK e ẽ So all four Carathéodory conditions are satisfied and Theorem D.36 gives a unique solution τ,e : [0,T R 2 with maximal T 0, ]. Moreover, f, is locally essentially bounded, i.e., for any compact A I D, we have t,τ,e A : ft,τ,e RHS 1+ a 1 + ς γ 0 m A M A + a 1 y ref + ẏ ref + c 1 g + c 1 g r Θ m L + γ 0 u A <. Now, if T <, then Theorem D.36 also gives for any compact C D existence of τ t t [0,T such that ˆx t = e t C the solution leaves any compact set. Step 3: We show a key inequality. Note that, in view of Step 2, we have t [0,T : et < ψt ψ and so for a.a. t [0,T : ėt CL, a 1 ψ γ 0 ktet+ a 1 y ref + ẏ ref + c 1 Θ g + c 1 g r Θ m L + γ 0 u A. Hence, for M := a 1 ψ + y ref + ẏ ref + γ 0 u A + c 1 g + 1 g r m L the following holds for a.a. t [0,T : M γ 0 ktet ėt M γ 0 ktet. KI Step 4: For M as in, ς := infςt and λ := inf ψt, t 0 t 0 Page 27/41

28 Non-identifier based adaptive control in mechatronics Lecture notes 5 SS2012 Lehrstuhl für Elektrische Antriebssysteme und Leistungselektronik, TUM 2012/06/22 ε ψ e t 0 t 1 T t Figure 11: Sketch of the construction of the contradiction in Step 4. we show that there exists a positive λ ε min 2, ψ0 e0, γ 0 ςλ 2 M + ψ such that t [0,T : ψt et ε. Seeking a contradiction, assume there exists see Fig. 11 t 1 := min{t [0,T ψt et < ε}. Then, by continuity of ψ e on [0,T, there exists t 0 := max{t [0,t 1 ψt et = ε}. and, for ε > 0 as in, we have t [t 0, t 1 ] : et ψt ε λ λ 2 = λ 2. E Hence, signe is constant on [t 0,t 1 ] [0,T. We only consider the case e > 0 on [t 0,t 1 ], the other case follows analogously. Now, clearly, and integration yields for a.a. t [t 0,t 1 ] : ėt M γ 0 t [t 0,t 1 ] : et et 0 =,E M γ 0 ςλ 2ε This, with the properties of ψ B 1 locally Lipschitz, i.e. t ς ψt et et ψ ėτdτ ψ t t 0. t 0 t t 0 : ψt ψt 0 ψ t t 0, Page 28/41

29 Non-identifier based adaptive control in mechatronics Lecture notes 5 SS2012 Lehrstuhl für Elektrische Antriebssysteme und Leistungselektronik, TUM 2012/06/22 gives t [t 0,t 1 ] : et et 0 ψ t t 0 ψt ψt 0, with which we arrive at the contradiction ε = ψt 0 et 0 ψt et }{{} < ε. by initial assumption ψt et ε for all t [0,T, which completes Step 4. Step 5: We show that Assertions ii, iii and iv hold true. First, we show Assertion ii. For ε > 0 as in, define the compact set C := {t,e [0,T] R e ψt ε} D, where D = {t,e R 0 R e < ψt} as in Step 2. Note that, if T <, C is a compact subset of D. Moreover, t t [0,T] : et C, therefore the whole graph of the solution evolves in C. But this, in view of Theorem D.36, contradicts maximality of T. Hence T =. Now, Assertion iii, i.e. t 0 : ψt et ε, follows from Step 4, which also implies, for any ς L R 0 ;R, that k = L R 0 ;R. This with implies ς ψ e u = k e L R 0 ;R which shows Assertion iv and completes the proof. Page 29/41

30 Non-identifier based adaptive control in mechatronics Lecture notes 6 SS2012 Lehrstuhl für Elektrische Antriebssysteme und Leistungselektronik, TUM 2012/06/29 6 Funnel speed control of saturated 1MS 2012/06/29, part 1 Now, we will apply funnel control for speed control of 1MS ω with actuator saturation. Theorem 6.1 Funnel speed control of saturated 1MS ω. Consider the mechatronic system 1MS ω with 1MS ω -Data. If actuator saturation satisfies û A Θ c 1 k A ψ +M +δ, δ > 0 arbitrary FC 1 -feas where M := ν 1 + ν 2 g r 2 ψ + y ref + ẏref + c 1 f 1 + f 2 + m L, Θ Θ g r then, for any funnel boundary ψ B 1, gain scaling function ς B 1, reference signal y ref W 1, R 0 ;R and initial value ω0 = ω 0 R satisfying y ref 0 c 1 ω0 < ψ0, the funnel controller ut = signg r ktet where et = y ref t yt and kt = ςt ψt et FC 1 applied to 1MS ω yields a closed-loop initial-value problem with the following properties: i there exists a unique solution ω : [0, T R with maximal T 0, ]; ii the solution ω does not have finite escape time, i.e. T = ; iii the tracking error is uniformly bounded away from the funnel boundary, i.e. ε > 0 t 0 : ψt et ε; iv gain and control action are uniformly bounded, i.e. k,u L R 0 ;R. Clearly, to evaluate the feasibility condition FC 1 -feas rough system knowledge is necessary. Moreover, since the feasibility condition is derived by worst case analysis, the upper bound on the actuator saturation û A may be a very conservative and so unrealistic bound. Proof of Theorem 6.1. Step 1: Some preliminaries. We rewrite 1MS ω, FC 1 in error ODE: ėt = a 1 y ref t+et+ẏ ref t c 1k A sat Θ û A signg r ktet+u A t + c 1 1 Θ f 1 c 1 y ref t et + 1 f g rθ 2 1 c 1 g r y ref t et + m Lt g rθ CL Step 2: We show Assertion ii.... skipped; similar as in unsaturated case see Proof of Theorem 4.3. Page 30/41

31 Non-identifier based adaptive control in mechatronics Lecture notes 6 SS2012 Lehrstuhl für Elektrische Antriebssysteme und Leistungselektronik, TUM 2012/06/29 Step 3: We show a key inequality. From Step 2, we have and so t [0,T : et < ψt ψ {}}{ t [0,T : ėt a 1 y ref + ψ + ẏ ref + c [ 1 f 1 Θ + 1 ] g r f 2 + m L c 1k A signg r ktet+u A t }{{} Θ =:γ 0 Hence, t [0,T: M γ 0 satûa ktet u A ėt M γ 0 satûa ktet u A Step 4: For M as above and =:M KI ς := infςt and λ := inf ψt, t 0 t 0 we show that there exists a positive { } λ ε min 2, ψ0 e0, ςλ 2û A + u A such that t [0,T : ψt et ε. Seeking a contradiction see Fig. 11 again, assume Then, by continuity of ψ e on [0,T, and t 1 := min{t [0,T ψt et < ε}. t 0 := max{t [0,t 1 ψt et = ε} t [t 0,t 1 ] : et ψt ε λ λ 2 = λ 2. Hence, signe is constant on [t 0,t 1 ] [0,T. We only consider e > 0 on [t 0,t 1 ]. Clearly, t [t 0,t 1 ] : ėt KI ς M γ 0 satûa ε λ 2 u A FC 1 -feas M γ 0 û A ψ δ. Page 31/41

32 Non-identifier based adaptive control in mechatronics Lecture notes 6 SS2012 Lehrstuhl für Elektrische Antriebssysteme und Leistungselektronik, TUM 2012/06/29 Now, identical arguments as in the proof of Theorem 4.3 show the claim. Step 5: We show that Assertions ii, iii and iv hold true. Again identical arguments as in the proof of Theorem 4.3 show the claim. Page 32/41

33 Non-identifier based adaptive control in mechatronics Lecture notes 7 SS2012 Lehrstuhl für Elektrische Antriebssysteme und Leistungselektronik, TUM 2012/06/29 & 2012/07/13 7 Internal model principle 2012/06/29 & 2012/07/ Motivation 2012/06/29, part 2 With funnel control, we can achieve tracking with prescribed asymptotic accuracy, but the tracking error does not necessarily tend to zero: We are not allowed to choose a funnel boundary which tends to zero recall admissible boundaries in set B 1 on p. 65, hence we cannot guarantee lim t et = 0. Following the internal model principle as stated by Wonham in the 80s: Every good regulator must incorporate a model of the outside world being capable to reduplicate the dynamic structure of the exogenous signals which the regulator is required to process [3, p. 210], we are able to significantly improve the tracking performance. What is the standard solution in industry to achieve steady state accuracy for constant references and/or disturbances? It is a PI controller, which is actually the most simple internal model capable to reduplicate constant signals: transfer function: F PI s = k P + k I s = k P s+k I s, k I, k P > 0 relative degree: n = 1, m = 1 r PI = n m = 0 high frequency gain: γ 0,PI = lim s s r F PI s = lim s kp + k I s = kp minimum-phase, since for all s C 0 : sk P +k I 0 s = k I k P < 0 Now, additionally, consider a linear system given in the frequency domain: Ns transfer function: F S s = γ 0, N, D monic; N Hurwitz Ds relative degree: r = degn degd = 1 high frequency gain: γ 0 known minimum-phase, since N is Hurwitz. Analyzing the serial interconnection of PI controller F PI s and F S s yields an augmented system see Fig. 12: augmented system e FC 1 v u F PI s F S s y internal model system Figure 12: Serial interconnection of PI controller F PI s and system F S s: the augmented system with new control input vt controlled by e.g. funnel controller FC 1. Page 33/41

34 Non-identifier based adaptive control in mechatronics Lecture notes 7 SS2012 Lehrstuhl für Elektrische Antriebssysteme und Leistungselektronik, TUM 2012/06/29 & 2012/07/13 transfer function: F PI sfs = k P s+k I s relative degree: r = 1 γ 0 Ns Ds = k pγ 0 s+ k I kp s Ns Ds high frequency gain: γ 0 = k P γ 0 minimum-phase, sinces+ k I k p andns are Hurwitz= the augmented system is minimumphase. Concluding, the augmented system is again element of system class S lin 1 see Definition 2.19 and hence any high-gain adaptive controller e.g. funnel controller FC 1 for relative-degree-one systems is still applicable for the serial interconnection. 7.2 Internal model principle For non-identifier based adaptive control: augmented system Y ref y ref e high-gain adapt. v internal u controller model system y n m What is an internal model? Which exogenous signals can it reduplicate admissible function class Y ref? How is an internal model designed realized? Does augmented system has same properties as system relative degree one, known sign of high-frequency gain, minimum-phase? Can we achieve asymptotic tracking, i.e. lim t et = 0 for linear systems? Application example: PI-funnel control for speed control of 1MS ω What is an internal model? dynamical linear system purpose to reduplicate a certain class of reference and/or disturbance signals In the following, we only consider the case to reduplicate reference signals y ref : R 0 R the case to reduplicate certain disturbance signals can be treated analogously. An internal model is represented by an linear ordinary differential equation of p order, p N: y p ref t+â p 1 y p 1 ref +...+â 1 ẏ ref t+â 0 y ref t = 0, y p 1 ref 0,...,y ref 0 R p with characteristic polynomial: D IM s = s p +â p 1 s p â 1 s+â 0 R[s], i.e. â i R for all i {1,...,p} Page 34/41

35 Non-identifier based adaptive control in mechatronics Lecture notes 7 SS2012 Lehrstuhl für Elektrische Antriebssysteme und Leistungselektronik, TUM 2012/06/29 & 2012/07/13 From standard linear control theory and experience, we know that a proportional controller can only exhibit a non-zero control action as soon as the control error is non-zero to attenuate e.g. constant disturbances. To circumvent this drawback, typically, a PI controller is introduced which allows to generate a non-zero control action even if the control error is zero! In a more general setup, the same task has an internal model introduced in for a wider exogenous signal class Which exogenous signals can a linear internal model reduplicate? all signals, which are the solutions of the linear ODE, where y ref C R 0 ;R since we are interested in asymptotic tracking, lim t et = 0, we do not consider references which tend to zero: in the following, let lim t y ref t 0. the linear system is assumed to be not asymptotically stable, more precisely, we only consider D IM with roots in the left complex half-plane and on the imaginary axis, i.e., Y ref := y ref C d R 0 ;R D D IM y ref t = 0, IM R[s], monic with. dt {s 0 C D IM s 0 = 0} C 0 }{{} corresponds to How is an internal model designed realized? Goal: Find state space realization of the internal model. Procedure: given y ref Y ref e.g. ramps, sinusoidal, constants or linear combinations thereof, find the Laplace transform: y ref : t R F ref s =... D IM s D IMs design of internal model in frequency domain: F IM s = us vs = N IMs D IM s where N IM s such that r IM = 0 γ0 IM > 0 minimum phase choice of N IM s as follows: p := degd IM = degn IM N IM s = c p s p + +c 1 s+c 0, c p 0 r IM = 0 lim s s 0 F IM s = γ IM 0 > 0 c p = γ IM 0 > 0 N IM s Hurwitz F IM s is minim-phase design of a minimal state space realization of F IM s: ˆxt = ˆxt+ˆbvt, ˆx0 = ˆx 0 R p IM ut = ĉ ˆx+γ 0 vt IM SS Page 35/41

36 Non-identifier based adaptive control in mechatronics Lecture notes 7 SS2012 Lehrstuhl für Elektrische Antriebssysteme und Leistungselektronik, TUM 2012/06/29 & 2012/07/13 with new augmented control input v. To choose matrix Â, vectors ˆb, ĉ and feedthrough γ IM 0 adequately, rewrite F IM s = N IMs D IM s = ˆN IM s D IM s }{{} r=1 +γ IM 0 where N IM s = ˆN IM +γ IM 0 D IM s and ˆN IM s = ĉ p 1 s p 1 + +ĉ 1 s+ĉ 0, ĉ i > 0 for all i {1,...,p 1} and then choose  = R p p, ˆb = R p, ĉ = â 0 â 1 â p 2 â p 1 according to the controllable canonical form Does the augmented system has same properties as system? ĉ 0. ĉ p 1 R p The serial interconnection of IM SS and linear system ẋ = Ax+bu, y = c x x0 = x 0 R n SYS is given by d dt x = ˆx y = c, 0 }{{ p } =:c s =:A s =:b s [ {}} ] {{}}{ xs { }}{ A bĉ x bγ IM + 0  ˆx 0ˆb x s vt IM+SYS with properties see Lemma 2.38: if system SYS has relative degree 1 r n, then so the serial interconnection IM+SYS. the high-frequency gain of IM+SYS has same sign as the high-frequency gain of SYS if SYS is minimum phase, then so IM+SYS IM+SYS S lin 1 Page 36/41

37 Non-identifier based adaptive control in mechatronics Lecture notes 7 SS2012 Lehrstuhl für Elektrische Antriebssysteme und Leistungselektronik, TUM 2012/06/29 & 2012/07/ Can we achieve asymptotic tracking, i.e. lim t et = 0? Can we achieve asymptotic tracking, i.e. lim t et = 0 for some y ref Y ref, if we apply the high gain adaptive tracking controller vt = signγ 0 ktet, where et = y ref t yt HGAC kt = q 1 et q 2, k0 = k 1, q 1 > 0, q 2 1 to linear system IM+SYS? Yes, see Theorem Sketch of proof of Theorem 2.39: Consider ẇ = A s w which has a global solution w on R 0 and w C. In view of Lemma in [2], there exists w ref 0 R n+p, such that ẇt = A s wt, w0 = w ref 0 R n+p y ref t = c s wt. Introducing the tracking error as follows et = y ref t yt #,IM+SYS = c s wt x s t }{{} =:x et # allows to rewrite the tracking problem of HGAC, IM+SYS in x e, i.e. ẋ e t = ẇt ẋ s t = et = c s x e t A sx et {}}{ A s wt A s x s t b s vt, x e 0 = w ref 0 x 0 s R n+p output is now the error. In view of Lemma 2.38, system is element of class S lin 1 r = 1, known high-frequency gain, minimum phase, hence Theorem 2.21 yields lim x et = 0 lim et = lim c s x e t = 0. t t t By introducing the virtual system # which corresponds to the serial interconnection IM+SYS for v = 0, we were able to express the tracking problem HGAC, IM+SYS as stabilization problem, i.e. that the output here: e asymptotically tends to zero. 7.3 Funnel control with internal model use of linear internal models admissible see Lemma 2.40, if relative degree r IM = 0 positive high-frequency gain γ IM 0 > 0 controllable and observable = minimum-phase Page 37/41

38 Non-identifier based adaptive control in mechatronics Lecture notes 7 SS2012 Lehrstuhl für Elektrische Antriebssysteme und Leistungselektronik, TUM 2012/06/29 & 2012/07/13 standard internal model in industry: } ẋ I t = vt, x I 0 = 0 ut = k P vt+k I x I t us = k P s+k I s vs, k P,k I > 0 PI lim t et = 0 not guaranteed in general since ψt λ > 0 for all t 0 and large class of disturbances and references, but tracking performance improves with internal model simulation and measurement results W 1, y ref e funnel v internal model, u controller e.g. PI FC 1 augmented system of class S 1 system of class S 1 y n m PI-Funnel speed control of unsaturated 1MS Theorem 7.1 PI-Funnel speed control of unsaturated 1MS ω. Consider the mechatronic system 1MS ω with 1MS ω -Data und û A =. For any funnel boundary ψ B 1, gain scaling function ς B 1, reference signal y ref W 1, R 0 ;R and initial value ω0 = ω R satisfying y ref 0 c 1 ω0 < ψ0, the PI-funnel controller ẋ I t=ktet, x I 0 = 0 ut=signg r ktet+k I x I t where et = y ref t yt and kt = ςt ψt et PI-FC 1 applied to 1MS ω with known signg r yields a closed-loop initial-value problem with the following properties: i there exists a unique solution ω,x I : [0,T R R with maximal T 0, ]; ii the solution ω,x I does not have finite escape time, i.e. T = ; iii the tracking error is uniformly bounded away from the funnel boundary, i.e. ε > 0 t 0 : ψt et ε; iv gain, control action and integral state are uniformly bounded, i.e.k,u,x I L R 0 ;R. v if steady state is reached, i.e. lim t ωt,ẋ I t = 0, and lim t ẏ ref t = 0, then asymptotic tracking is achieved, i.e. lim t et = 0 and lim t ėt = 0. Proof of Theorem 7.1. Introduce g: R 0 R R, t,ω gt,ω := f m 1ω L t+f 2 Θ g r Θ a 1 := ν 1 + ν 2 g 2 r Θ Page 38/41 ω g r + k A Θ u At,

39 Non-identifier based adaptive control in mechatronics Lecture notes 7 SS2012 Lehrstuhl für Elektrische Antriebssysteme und Leistungselektronik, TUM 2012/06/29 & 2012/07/13 and ẋ I t = vt, x I 0 = 0 ut = vt+k I x I t, k I > 0 } PI In view of 1MS ω -Data, note that g, L R 0 R;R. Now, consider the serial interconnection of PI and 1MS ω unsaturated: û A = : ωt = a 1 ωt+ k A Θ k Ix I t+ k A Θ vt+gt,ωt, ω0 = ω 0 ẋ I t = vt, x I 0 = 0 yt = c 1, 0 ωt x I t or, in a more compact form, d dt =:xt { }} { ωt x I t = =:A yt = c 1, 0 }{{} c =:b [{}} ] { {}}{ k a1 AΘ k I ωt kaθ x I t 1 ωt. x I t vt+ =:B L {}}{ 1 gt,ωt, 0 ω0 = x I 0 ω0 0 Checking properties of system class S 1 : γ 0 := c b = c 1 k AΘ 0 r = 1 signγ 0 = signc b = signg r where signg r is known minimum-phase property? minimum phase. det s a 1 k A Θ k I k AΘ 0 s 1 c = c det[ k AΘ k I k AΘ s 1 = c 1 k A Θ s+k I 0 s C 0 g, L R 0 R;R 0 disturbances & perturbation is bounded Hence, application of funnel controller vt = ktet where kt = is admissible see Theorem Moreover, note that which shows Assertion v. ςt ψt et ẋ I t {}}{ lim ωt,ẋ It = 0 lim kt et = 0 t t }{{} >0 Page 39/41 ]

40 Non-identifier based adaptive control in mechatronics Lecture notes 7 SS2012 Lehrstuhl für Elektrische Antriebssysteme und Leistungselektronik, TUM 2012/06/29 & 2012/07/ Implementation: PI-Funnel speed control of 1MS ω implementation in xpc target laboratory setup with sensor W 1, y ref = ω ref e speed controller u ω ω m = ω +ṅ m ṅ m speed controller #1: Standard PI controller ut = k P et+k I t eτdτ speed controller #2: PI-Funnel controller PI-FC 1 ut = ktet+k I t kτeτdτ where kt = ςt ψt et ω + ṅm [rad/s] ω ref ω ref ± ψ time t[s] a Set-point tracking mm [Nm] k, kp [Nms/rad] e [rad/s] ω + ṅm [rad/s] ω ref ±ψ m L time t[s] b Reference tracking Figure 13: Implementation: Measurement results for standard PI controller PI-Funnel controller PI-FC 1 and Page 40/41

41 Non-identifier based adaptive control in mechatronics Lecture notes 7 SS2012 Lehrstuhl für Elektrische Antriebssysteme und Leistungselektronik, TUM 2012/06/29 & 2012/07/13 References [1] C. M. Hackl. Contributions to high-gain adaptive control in mechatronics. PhD thesis, Lehrstuhl für Elektrische Antriebssyteme & Leistungselektronik, Technische Universität München, [2] A. Ilchmann. Non-Identifier-Based High-Gain Adaptive Control, volume 189 of Lecture notes in control and information sciences. Springer-Verlag, London, [3] W. M. Wonham. Linear Multivariable Control: A Geometric Approach. Number 10 in Applications of Mathematics. Springer-Verlag, Berlin, 3rd edition, Page 41/41