THOUGHTS ON SOME OPTIMAL CONTROL PROBLEMS In memory of U. HELMKE & R. KALMAN

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1 Sde Boker, MARCH 18-25, 2017 p. 1/45 THOUGHTS ON SOME OPTIMAL CONTROL PROBLEMS In memory of U. HELMKE & R. KALMAN Paul A. Fuhrmann Ben-Gurion University of the Negev

2 Things you see from here, you don t see from there. Sde Boker, MARCH 18-25, 2017 p. 2/45

3 WHY THINK ABOUT OPTIMAL CONTROL NOW? Sde Boker, MARCH 18-25, 2017 p. 3/45 Better late than never. Finding the right resolution in which to approach system theory is our goal. If the resolution is too low, the theory may not be practically applicable. On the other hand, if the resolution is too high, we may see the trees but miss the forest. Abstract vs. concrete. Gain better understanding of the subject by clarifying the relations between different aspects of control theory, e.g., robust stabilization, model reduction, optimal regulator and estimation problems. This has the potential of leading to a grand unification of control theory.

4 Sde Boker, MARCH 18-25, 2017 p. 4/45 CONT. Open the possibility of extending methods to other settings as: classes of infinite dimensional systems, special classes of systems (positive real, bounded real. Extend optimal control theory to deal with complexity, that is, to networks of systems, using local optimality results for the nodes as well as the interconnection data.

5 Sde Boker, MARCH 18-25, 2017 p. 5/45 BASIC IDEAS FOR REACHING A TARGET Use coprime factorizations, functional models. Shift/translation realizations. Construct the reachability map R : U X. Derive representation of KerR. Reduce the reachability map to R : U/KerR X. Assuming reachability, this is a module isomorphism. Invert R, using doubly coprime factorizations. For a discrete time system, this leads to time optimal controllers.

6 Sde Boker, MARCH 18-25, 2017 p. 6/45 WHAT HAPPENS WHEN ALL SPACES HAVE HILBERT SPACE STRUCTURE? DUE TO THE AVAILABILITY OF ORTHOGONAL DECOMPOSITION, OPTIMIZATION PROBLEMS ARE GREATLY SIMPLIFIED.

7 FROM ALGEBRA TO ANALYSIS Sde Boker, MARCH 18-25, 2017 p. 7/45 Although the technicalities of polynomial model based system theory for discrete time linear systems over an arbitrary field are vastly different from the Hardy space based theory for some classes of continuous-time systems there are strong algebraic similarities. These, with the help of heavy analytic tools, can be used to extend the algebraic approach to a wide variety of optimal control and estimation problems for several classes of, not necessarily rational, analytic functions. Some of the ideas and results presented owe much to a cooperation with Raimund Ober in the early 1990s and a long term one with Uwe Helmke.

8 Sde Boker, MARCH 18-25, 2017 p. 8/45 SOME REFERENCES A. Beurling, On two problems concerning linear transformations in Hilbert space, Acta Math., 81, (1949, P.D. Lax, Translation invariant subspaces Acta Math., 101, (1959, L. Carleson, Interpolation by bounded analytic functions and the corona problem, Ann. of Math., 76, (1962, D. Sarason, Generalized interpolation in H, Trans. Amer. Math. Soc. 127, P.A. Fuhrmann, On the corona problem and its application to spectral problems in Hilbert space, Trans. Amer. Math. Soc., 132, (1968,

9 Sde Boker, MARCH 18-25, 2017 p. 9/45 MORE REFERENCES P.A. Fuhrmann and R. Ober, A functional approach to LQG balancing, model reduction and robust control, Int. J. Contr. (1993a, 57, P.A. Fuhrmann and R. Ober, On coprime factorizations, T. Ando volume, Birkhauser Verlag, P.A. Fuhrmann, A duality theory for robust control and model reduction, Lin. Alg. Appl., (1994, vols ,

10 Sde Boker, MARCH 18-25, 2017 p. 10/45 SOME ANALOGIES Algebra Analysis L 2 (, ;C m F((z 1 m L 2 (0, ;C m L 2 (,0;C m = F[z] m z 1 F[[z 1 ]] m = H 2 H2 +, whereh2 ± = FL2 (0, F[z] H± F[z] m = X D DF[z] m H+ 2 = H(M r M r H+ 2 F[z]-module structure Submodules D(zF[z] m S(zH+ 2 D(z nonsingular S(z inner H± functional calculus Shift/Translation invariant subspaces

11 Sde Boker, MARCH 18-25, 2017 p. 11/45 Algebra Polynomial Models H G : F[z] m z 1 F[[z 1 ]] p Matrix fractions Shift realization F[z]-homomorphisms Polynomial Bezout equation F[z]-unimodular embedding Inversion Analysis Shifts/Translations H G : H+ 2 H 2 DSS Factorizations Translation realization Intertwining maps, CLT Carleson Corona Thm H -unimodular embedding Inversion

12 REACHABILITY MAP AND OPEN LOOP CONTROL Sde Boker, MARCH 18-25, 2017 p. 12/45 (A,B F n n F n m a reachable pair. Our aim is to compute all open loop control sequences that steer the system from the origin to the stateξ F n. Define the reachability mapr (A,B : F[z] m F n : R s (A,B i=0 u iz i := s i=0 Ai Bu i, u(z = s i=0 u iz i F[ R (A,B u := π zi A Bu, u(z F[z] m KerR (A,B = DF[z] m R := R (A,B X D F[z] m /KerR (A,B u (z = R 1 ξ u(z = u (z+d(zg(z Set of steering controllers is an affine space.

13 Sde Boker, MARCH 18-25, 2017 p. 13/45 DCF & INVERSION OFR (A,B F n n F n m a reachable pair and let (zi A 1 B = N(zD(z 1 be coprime factorizations, with D(z F[z] m m,n(z F[z] n m. ( ( ( Θ(z Ξ(z D(z Ξ(z I 0 B zi A ( D(z Ξ(z N(z Θ(z N(z Θ(z ( Θ(z Ξ(z B zi A = = 0 I ( I 0 0 I Ξ(z(zI A = D(zΞ(z u(z = R 1 ξ = π D Ξξ.

14 Sde Boker, MARCH 18-25, 2017 p. 14/45 THE SHIFT REALIZATION G(z = V(zT(z 1 U(z+W(z = D+C(zI A 1 B A = S T Bξ = π T Uξ, Cf = (VT 1 f 1 D = G(. Realization is reachable T(z and U(z left coprime. Realization is observable T(z and V(z right coprime. Hautus and Kalman criteria.

15 INVERSION FOR HIGH OR- DER SYSTEMS Sde Boker, MARCH 18-25, 2017 p. 15/45 (T(z,U(z F[z] q q F[z] q m a reachable pair, i.e., left coprime, and let T(z 1 U(z = U(zT(z 1 be coprime factorizations, with T(z F[z] m m,u(z F[z] q m. ( Y(z X(z U(z T(z ( T(z X(z U(z Y(z X(zT(z = T(zX(z u(z = R 1 ξ = π T Xξ. = ( I 0 0 I

16 Sde Boker, MARCH 18-25, 2017 p. 16/45 PROGRAM OUTLINE Impulse response/transfer functions as H -homomorphisms. The reduced reachability and observability maps. Hankel operators. Strictly noncyclic (Close to rational functions, Douglas-Shapiro-Shields factorizations. Hankels with large kernels, small image. Beurling-Lax representations in terms of inner functions. Model operators, intertwining maps and the commutant lifting theorem. Inversion. Intertwining maps and reduced Hankel operators.

17 Sde Boker, MARCH 18-25, 2017 p. 17/45 OUTLINE (Contd. Shift/translation realizations. Kernel and image of Hankel operators and their Beurling-Lax representations. Inner functions and their state space representations. Homogeneous Riccati (Lyapunov equation. Normalized coprime factorizations and their unimodular embeddings. Characteristic functions. Optimal control.

18 Sde Boker, MARCH 18-25, 2017 p. 18/45 OUTLINE (Contd. Optimal control and model reduction of some networks of linear systems, from local to global, by interpolation. Weak controllability, rational (AAK approximation and suboptimal control.

19 INNER FUNCTIONS STATE SPACE FORMULAS Sde Boker, MARCH 18-25, 2017 p. 19/45 M + H + & M + = M 1 +. (A,B is reachable and A is stable. Homogeneous Riccati equation. XA+A X = XBB X ( A B M = B X I ( A+BB M 1 X B = B X I

20 STABLE SYSTEMS Any strictly proper, rational functiong(s has a minimal realization of the form G(s = C(sI A 1 B (A, B reachable, (C, A observable. Assuming G + (s H +, then A is stable. Hankel operator G + (s H + H G+ : H 2 H2 + H G+ f = P + G + f Abstract realization: X = H 2 KerH G+ = {KerH G+ } and defining the reachability mapr I : H 2 {KerH G + } and observability map O I : {KerH G+ } H 2 + by Sde Boker, MARCH 18-25, 2017 p. 20/45

21 Sde Boker, MARCH 18-25, 2017 p. 21/45 ABSTRACT REALIZATION X = H 2 KerH G+ = {KerH G+ } R I : H 2 {KerH G+ } R I u = P {KerH G+ } u, u(s H 2 O I : {KerH G+ } H 2 + O I h = H G+ h, h(s {KerH G+ } H G+ = O I R I

22 NORMALIZED COPRIME Sde Boker, MARCH 18-25, 2017 p. 22/45 (DSS FACTORIZATIONS J S = ( I J S NRCF, G H, G = N S MS 1 ( ( ( M S NS I 0 MS = MS 0 0 M S = I N S J S NLCF, G H, G = M 1 S N S ( ( ( I 0 M S MS N S 0 0 N = M S M S = I S

23 Sde Boker, MARCH 18-25, 2017 p. 23/45 DSS FACTORIZATION G + (s = C(sI A 1 B,Astable G + (s = H(sD(s 1 = D(s 1 H(s = (H(sE(s 1 (E(sD(s 1 = M r (snr(s = N l (s M l (s (DSS ( A B M l (s = &M l (s 1 = M l (s C 0 I ( ( A BC0 B A C0 C 0 I B I KerH G+ = H (Ml = {M l H2 } C 0 = B X + XA+A X = XBB X ImH G+ = H + (M r

24 DOUBLY COPRIME Sde Boker, MARCH 18-25, 2017 p. 24/45 FACTORIZATIONS ( Vl (s U l (s N l (s M l (s ( A X 1 + C B CZ + X + I 0 B X + 0 I ( Mr (s U r (s N r (s V r (s, ( = A Z + C Z + X + B C I 0 B Z I ( I 0 0 I A X +XA = XBB X AZ +ZA = ZC CZ. X +,Z + > 0

25 INTERNAL STABILIZATION Sde Boker, MARCH 18-25, 2017 p. 25/45 e y 1 1 G y 2 e 2 K The feedback configuration(g,k is called internally stable if I G K I 1 = (I GK 1 (I K(I GK 1 GK 1 G (I KG 1 H +.

26 INTERNAL STABILIZATION Sde Boker, MARCH 18-25, 2017 p. 26/45 G(s = M 1 l N l = N r Mr 1 K(s = V 1 l U l = U r Vr 1 The following statements are equivalent: (V l (sm r (s+u l (sn r (s 1 H + (M l (sv r (s+n l (su r (s 1 H + V l (sm r (s+u l (sn r (s = I, M l (sv r (s+n l (su r (s = I.

27 DCF and YOULA-KUCERA Sde Boker, MARCH 18-25, 2017 p. 27/45 PARAMETRIZATION ( Vl (s U l (s N l (s M l (s ( Mr (s U r (s N r (s V r (s = ( I 0 0 I K = UV 1 = V 1 U K = (U 0 M r Q(V 0 +N r Q 1 = (V 0 QN l 1 (U 0 +QM l

28 THES-CONTROLLER ANDS-CHARACTERISTIC Sde Boker, MARCH 18-25, 2017 p. 28/45 G + (s = N l (sm l(s = M r (sn r(s There exists a unique stabilizing controller K = V 1 l U l = U r Vr 1 for which R S, the S-characteristic ofg +, defined by, R S (s := M l (su l (s = U r(sm r (s is in H + and strictly proper. KerH RS = MrH 2 ImH RS = H + (M l

29 HANKEL OPERATORS Sde Boker, MARCH 18-25, 2017 p. 29/45 AND INTERTWINING MAPS H (M l M l H + (M l H G+ W W 1 H RS H + (M r M r H (M r wherew : H + (M l H + (M r and W 1 : H + (M r H + (M l are given by Wg = P + Nl g, g(s H +(M l W 1 f = P + Urf, f(s H + (M r.

30 S-CHARACTERISTIC Sde Boker, MARCH 18-25, 2017 p. 30/45 STATE SPACE REALIZATION R S = ξ S (G + = ( A Z+ C B X + 0 Ξ 1 A +AΞ 1 = Z + C CZ + Θ 1 A+A Θ 1 = X + BB X +. ξ S (R S = ( Ξ + = Z+ 1, Θ + = X+ 1 ( A Θ + X + B A B = CZ + Ξ + 0 C 0 = G + (s.

31 DCF Sde Boker, MARCH 18-25, 2017 p. 31/45 STATE SPACE REALIZATION ( Mr (s U r (s = = N r (s V r (s (A +X + BB X + Z + C X + Z + X + B CX+ 1 I 0 B Z+ 1 X+ 1 0 I (A +C CZ + C X + B CZ + I 0 B 0 I,

32 OPTIMAL STEERING TO TARGET Sde Boker, MARCH 18-25, 2017 p. 32/45 ẋ = Ax+Bu, A STABLE 0 = lim t x(t x(0 = ξ C n. R (A,B : L2 (,0;C m C n R (A,B u = 0 e Aτ Bu(τdτ, R, (A,B : Cn L 2 (,0;C m R, (A,B ξ = B e A τ ξ, τ 0, KerR (A,B = {u(t L2 (,0;C m 0 e Aτ Bu(τdτ = 0

33 REACHABILITY GRAMIAN OPTIMAL CONTROLLER Sde Boker, MARCH 18-25, 2017 p. 33/45 W = R (A,B R, (A,B = 0 e Aτ BB e A τ dτ, AW +WA = BB. u (t = B e A t W 1 ξ = B W 1 x (t x (t = We A t W 1 ξ. OPTIMAL CONTROLLER IS A STATE FEEDBACK CONTROLLER

34 NO STABILITY ASSUMPTION Sde Boker, MARCH 18-25, 2017 p. 34/45 G(s = ( A B C 0 minimal realization.

35 NORMALIZED (LQ Sde Boker, MARCH 18-25, 2017 p. 35/45 COPRIME FACTORIZATION J L = ( I 0 0 I J L NRCF, G = N L ML 1 ( ( ( M L NL I 0 ML = I 0 I N L J L NLCF, G = M 1 ( ( ( I 0 M L ML N L 0 I N L L N L = I

36 NORMALIZED COPRIME FACTORIZATION Sde Boker, MARCH 18-25, 2017 p. 36/45 G = N L ML 1 = M 1 L N L ( ( ( V U ML U I 0 = N L M L N L V 0 I ( A BB X B ML = B N X I. L C 0 A X +XA+C C XBB X = 0 ( ( A ZC C B ZC NL M L = C 0 I AZ +ZA +BB ZC CZ = 0

37 Sde Boker, MARCH 18-25, 2017 p. 37/45 THE LQG CONTROLLER R L = M L U L +N L V L H ( UL V L = ( ML N L R + ( N L M L R L = U LM L +V L N L H ( V L U L = R ( N L M L + ( M L N L

38 Sde Boker, MARCH 18-25, 2017 p. 38/45 L-CONTROLLER STATE SPACE EQs ( UL V L = A BB X ZC B X 0 C I ( V L U L = ( A ZC C B ZC B X I 0 K = U L V 1 L = ( A BB X ZC C ZC B X 0

39 Sde Boker, MARCH 18-25, 2017 p. 39/45 THE L-CHARACTERISTIC R L = Φ KS K = S I Φ I KerH R L = {S K H 2 +} ImH R L = {S I H2 }

40 TWO MATRIX COMPLETIONS Sde Boker, MARCH 18-25, 2017 p. 40/45 G = N L ML 1 = M 1 L N ( ( L J1 N L = J 2 M S I ( ( L K1 K 2 = SK M N ( ( V L U L ML U, L UNIMODULAR N L M L N L V ( L ( K1 K Ω K = 2 ML J,Ω N L M I = 1 INNER L N L J 2

41 THE KEY DIAGRAM {S K H 2 +} H R {S I H2 } Z K H (M N H ( N {Ω K H+} 2 {Ω I H2 } Y K Y I h 1 g 1 g 2 H ( N M Z K f = P {ΩK H 2 + } h 2 M ( N M U L V L = P {S I H 2 } (M N = P {SK H 2 + } ( N M f h 1 h 2 g 1 g 2 Y I Sde Boker, MARCH 18-25, 2017 p. 41/45

42 HANKEL TO INTERTWINING MAP Sde Boker, MARCH 18-25, 2017 p. 42/45 {S K H 2 +} H R {SI H2 } {S I H+} 2 Z K Y I H (M N H N M {Ω K H+} 2 {Ω I H2 } {Ω I H+} 2 H Ω I N M ( N M S I

43 {S K H 2 +} {Ω K H 2 +} ( UL V L P {SI H+} 2 Φ I I {S I H 2 +} {Ω K H+} 2 {Ω I H+} 2 P {SI H 2 +} ( K1 K 2 P {ΩI H 2 +} ( I 0 P {ΩI H 2 +} ( 0 I P {ΩI H 2 +} ( N M Sde Boker, MARCH 18-25, 2017 p. 43/45

44 OPTIMIZATION = + H + + PROBLEMS I 1 (1 σ U L V L µ 1 = min Q H + Φ I +S I Q = min Q H + R +Q = min Q H + Φ K +QS K + M N {S K H 2 +} Q U L + M J 1 V L N J 2 R +Q I Q 1 Q 2 µ 1 (1+µ {S I H 2 +} {Ω K H+} 2 {Ω I H+} 2 1 (1+µ µ 1 (1+µ = min Q i H + = min Qi H+ M N µ 1 1+µ 2 = σ 1 (1 σ = min N M + N M 1 Q ij H K 1 K 2 = min Qij H + N ( N M + Q 11 K 1 K2 + + Q 1 Q 1 Q 2 Q 2 S I Q 11 Q 12 Q 21 Q 22 Q 12 Sde Boker, MARCH 18-25, 2017 p. 44/45

45 OPTIMIZATION PROBLEMS II 1 µ n = min Q H = (1 σn ( H + N M ( H + J1 J2 {S K H+} 2 +S K ( ( + U S +S K Q = min Q H + Q 1 Q 2 Q 1 Q 2 R S +Q = min Q H + (1+µ 2 2 n 1 = ( min Qi H + min Qi H + ( (1+µ 2 n 1 2 (1+µ 2 n µ n {S I H 2 +} {Ω K H+} 2 {Ω I H+} 2 1+µ 2 n µ n = 1 σ n (1 σ n 2 µ n 1 2 Φ I R S S K U S +Q I S I +S I ( ( + Q 1 Q 2 R = Q 1 Q 2 = min Qij H + U L ( I U S + M J 1 Q 11 Q 12 V L N J 2 Q 21 Q 22 Sde Boker, MARCH 18-25, 2017 p. 45/45