The geometric approach and Kalman regulator

Transcription

1 Kalman filtering: past and future Bologna, September 4, 2002 The geometric approach and Kalman regulator Giovanni MARRO DEIS, University of Bologna, Italy

2 The purpose of this talk is to show that not only modern system and control theory was initiated in the early sixties by a remarkable set of scientific contributions by R.E. Kalman, but has also been driven during its development by the analysis and increasing insight into several of his very basic and general results. Thus the settlement of system and control theory in the last forty years followed a path that was strongly traced and influenced by R.E. Kalman. I will try to point out the main landmarks of this path. Of course, this will be done from a very personal, subjective standpoint: since my research field has been almost solely the geometric approach, I will try to explain the impact of Kalman s results on the geometric approach setting and, conversely, how Kalman s results (mainly the Kalman regulator) may be interpreted from the geometric approach standpoint. Relating Kalman regulator and filter to the geometric approach is a relatively recent trend in the scientific literature (H2 optimal control) which, in my opinion, is also very suitable for educational aims. I will try to present my feelings in a conversational way, with a limited use of mathematical manipulations and formulas. Since I am rather sensitive to practical applications, the available algorithms in Matlab, standard or home made, will be quoted throughout the talk. 1

3 Early references (controllability, observability, filtering and the LQR problem) Kalman Contribution to the theory of optimal control, Boletin de la Societed Mathematica Mexicana, Kalman A new appraoch to linear filtering and prediction problems, Transactions of the ASME, Journal of Basic Engineering, march Kalman On the general theory of control systems, Proceedings of the first IFAC Congress, vol. 1, Butterworth, London, Kalman Mathematical description of linear dynamical systems, SIAM Journal Control, vol. 1, no. 2, Kalman, Ho and Narendra Controllability of linear dynamical systems, Contribution to Differential Equations, vol. 1, no. 2, The seminal papers (25 in the last forty years) are shown in red. 2

4 Early references (geometric approach - before Wonham and Morse) Basile, Laschi and Marro Invarianza controllata e non interazione nello spazio degli stati, L Elettrotecnica, vol.56., n.1, Basile and Marro Controlled and conditioned invariant subspaces in linear system theory, Journal of Optimization Theory and Applications, vol. 3, n. 5, Basile and Marro On the observability of linear time-invariant systems with unknown inputs, Journal of Optimization Theory and Applications, vol. 3, n. 6, Basile and Marro L invarianza rispetto ai disturbi studiata nello spazio degli stati, Rendiconti della LXX Riunione Annuale AEI, paper , Laschi and Marro Alcune considerazioni sull osservabilità dei sistemi dinamici con ingressi inaccessibili, Rendiconti della LXX Riunione Annuale AEI, paper ,

5 Early references (geometric approach - Wonham and Morse s first paper) Wonham and Morse Decoupling and pole assignment in linear multivariable systems: a geometric approach, SIAM Journal on Control, vol. 8, n. 1, Books (geometric approach) Wonham Linear Multivariable Control A Geometric Approach, Springer Verlag, Basile and Marro Controlled and Conditioned Invariants in Linear System Theory, Prentice Hall, Trentelman, Stoorvogel and Hautus Control Theory for Linear Systems, Springer Verlag,

6 The topics herein considered Kalman-derived framework controllability observability Kalman regulator (LQR, H 2 ) Kalman dual filter Kalman filter previewed signal optimal decoupling delayed filter (smoother) H 2 -optimal model following H 2 -optimal model observer Geometric framework controlled invariants conditioned invariants disturbance decoupling problem measured signal decoupling unknown-input state observer previewed signal decoupling del. unknown-input state observer exact model following exact model observer primal problem dual problem 5

7 1 - Controllability and Observability Controllability and observability are basic properties of the dynamic systems. They are usually referred to state solvability of control and/or observation (filtering) problems. They were introduced by Kalman around 1960, and involved some other basic concepts, also concomitantly studied by Kalman, like duality; relation between continuous and discrete-time systems (sampling theorem). Controllability and observability carried some important further achievements, like theorems on pole assignment and controlled and conditioned invariants, the basic tools of the geometric approach. In fact, they are the first step towards a geometric picture of the dynamic systems evolution in time, where the concepts of subspace and invariant subspace give a significant insight, and algebra of subspaces provides a solid algorithmic substance. Let us briefly recall what a trajectory in the state space means. 6

8 An example of a dynamic system: an electric motor v i amplifier K v a motor v c ω, ϑ v a (t) = R a i a (t)+l di a a dt (t)+k 1 ω(t) c m (t) = Bω(t)+J dω dt (t)+c r(t) ω(t) = dϑ dt (t) with c m (t)=k 2 i a (t). h u Σ y c r ẋ(t) = Ax(t)+Bu(t)+Hh(t) y(t) = Cx(t)+Du(t)+Gh(t) where x := [i a ωϑ] T, u := v i, h := c r, y := ϑ and B = A = R a/l a k 1 /L a 0 k 2 /J B/J K/L a 0 0, H =, 0 1/J 0, C = [ ], D =0, G =0 7

9 State-space models: continuous-time systems ẋ(t) = Ax(t)+Bu(t) y(t) = Cx(t) (x X = R n, u R p, y R q ) with feedthrough term Du(t) ẋ(t) = Ax(t)+Bu(t) y(t) = Cx(t)+Du(t) X C =kerc x(t) ẋ(t) L(t) Fig A state-space trajectory. A dynamic system without the feedthrough term is said to be purely dynamic. The linear variety L(t)=Ax(t)+Bu(t) represents the locus of all the state velocities due to the control action u(t). 8

10 State space models: discrete-time systems without feedthrough term x(k+1) = A d x(k)+b d u(k) y(k) = C d x(k) with feedthrough term x(k+1) = A d x(k)+b d u(k) y(k) = C d x(k)+d d u(t) From continuous to discrete-time The control action is applied stepwise and the output of the system is accordingly sampled with the same sampling time T. Referring to an equivalent discrete-time system provides a significant insight into the behavior of continuous-time systems. 0 t, k T 9

11 Controllability and observability Refer to the continuous-time dynamic system ẋ(t) = Ax(t)+Bu(t) y(t) = Cx(t) [+Du(t)] R X Let B := imb. The reachability subspace of (A, B), i.e., the set of all the states that can be reached from the origin in any finite time by means of control actions, is the minimal A-invariant containing B, orr =minj (A, B). If R = X, the pair (A, B) issaidtobecompletely controllable. Let C :=kerc. The unobservability subspace of (A, C), i.e., the set of all the initial states that cannot be recognized from the output function, is the maximal A-invariant contained in C, or Q =maxj (A, C). If Q = {0}, (A, C) is said to be completely observable. The above definitions also apply in the discrete-time case. 10

12 Algorihms and dualities Consider the sequence Z 1 = B Z i = B + A Z i 1 i =2, 3,... Consider the sequence Z 1 = C Z i = C A 1 Z i 1 i =2, 3,... minj (A, B) is obtained when the sequence stops, i.e., when (Z ρ+1 = Z ρ ). This value of ρ is called the controllability index. maxj (A, C) is obtained when the sequence stops, i.e., when (Z ρ+1 = Z ρ ). This value of ρ is called the observability index. The following dualities hold: maxj (A, C) = ( minj ( A T, C )) minj (A, B) = ( maxj ( A T, B )) where the symbol denotes the orthogonal complement. 11

13 Some system properties related to controllability and observability The sampling theorem. (Kalman, 1960) Let (A, B) be controllable. The corresponding zero-order hold pair (A d,b d ) is controllable if the spectrum of A does not contain eigenvalues whose imaginary part is a multiple of π/t, wheret is the sampling time. The deadbeat control u 0 ρ k In the discrete-time case the minimumtime control from the origin to a given final state is performed through a deadbeat type of control action and the minimum time is equal at most to the controllability index ρ. In the continuous-time case this problem does not admit any solution (the minimum time should be zero and the control should be a distribution). 12

14 Some consequences of controllability and observability: pole assignment State feedback Output injection v u y + + Σ u Σ y F x G ẋ(t) = (A + BF) x(t)+bv(t) y(t) = Cx(t) ẋ(t) = (A + GC) x(t)+bu(t) y(t) = Cx(t) Thepoleassigmenttheorem.The eigenvalues of A + BF are arbitrarily assignable by a suitable choice of F if and only if the system is completely controllable and those of A + GC are arbitrarily assignable by a suitable choice of G if and only if the system is completely observable. 13

15 Other types of systems (signal processors) u delay y u FIR y Fig The delay and the FIR system. - Continuous-time: y(t) = y(t) =u(t t 0 ) tf 0 W (τ) u(t τ) dτ where W (τ), τ [0,t f ], is a q p real matrix of time functions, referred to as the gain of the FIR system, while [0,t f ] is called the window of the FIR system. - Discrete-time: y(k) = y(k) =u(k k 0 ) k f l=0 W (l) u(k l) where W (k), k [0,k f ], is a q p real matrix of time functions, referred to as the gain of the FIR system, while [0,k f ] is called the window of the FIR system. 14

16 Duality System Σ : (A, B, C, D); FIR system Σ : W (τ); Dual system Σ T : (A T,C T,B T,D T ). Dual FIR system Σ T : W T (τ). u Σ 1 Σ 2 Σ 3 y ū Σ T 3 Σ T 1 Σ T ȳ ẋ 1 (t) ẋ 2 (t) ẋ 3 (t) y(t) = A B 1 0 A 2 0 B 2 B 3,1 C 1 B 3,2 C 2 A C 3 0 x 1 (t) x 2 (t) x 3 (t) u(t) x 1 (t) x 2 (t) x 3 (t) ȳ(t) = A T 1 0 C T 1 BT 3,1 0 0 A T 2 C T 2 BT 3, A T 3 C T 3 B T 1 B T x 1 (t) x 2 (t) x 3 (t) ū(t) The overall dual system is obtained by reversing the order of serially connected systems and interchanging branching points with summing junctions and vice versa. 15

17 A possible use of FIR systems Controlling a stable system Σ to a given final state x 0 δ(t) Σ c u Σ x + - e Observing an unknown initial state of a stable system Σ x 0 δ(t) Σ y Σ o x + - e e x 0 e x 0 τ t τ t The above problems are dual to each other: Σ c and Σ o can be profitably realized with FIRs. The window of the FIR system on the left is easily computed by solving a finite-time Kalman regulator problem for the reverse-time system of Σ. 16

18 Computational support with Matlab A subspace Y is represented by an orthonormal basis matrix Y (such that Y =imy ). The operation on subspaces related to controllability and observability are Z = sums(x,y) Sum of subspaces. Z = ints(x,y) Intersection of subspaces. Y = ortco(x) Orthogonal complementation of a subspace. Y = invt(a,x) Inverse transform of a subspace. Q = ker(a) Kernel of a matrix. Q = mininv(a,x) Min A-invariant containing imx. Q = maxinv(a,x) Max A-invariant contained in imx. [P,Q] = stabi(a,x) Matrices for the internal and external stability of the A-invariant imx. P = place(a,b,p) Pole assignment by state feedback (Matlab). The geometric approach software in Matlab was initiated with a diskette enclosed with the 1992 Basile and Marro book. It is now freely downloadable from 17

19 2 - Controlled and Conditioned Invariants Controlled and conditioned invariants are the tools that extend the concept of invariance in linear systems. They make the definition of some further system properties, besides controllability and observability, and the solution of some basic problems, very straightforward. Roughly speaking, the geometric approach studies conditions under which the Kalman-derived problems (regulator and filter) admit zero-cost solutions. But this interpretation was not so clear at the beginning. It became clear when the minimum H 2 norm interpretation of the Kalman regulator and filter was introduced. The geometric approach has been settled through more than 30 years of contributions by several researchers. It gives insight into many interesting properties of systems, but has not yet been collected into a simple, easy to read, and complete treatise. Controlled and conditioned invariants are often referred to as (A, B)-invariants and (C, A)-invariants in the literature, since Wonham gave them these new names in 1974 (five years after their introduction). 18

20 Definitions and algorithms Given a linear map A : X X and a subspace B X, a subspace V X is an (A, B)-controlled invariant if A V V+ B The set of all the (A, B)-controlled invariants contained in a given subspace C is closed with respect to the sum, hence has a maximum, which will be referred to as V =maxv (A, B, C). It is computed with the sequence V 1 = C V i = C A 1 (V i 1 + B) i =2, 3,... maxv (A, B, C) is obtained when the sequence stops (V i+1 = V i ). Given a linear map A : X X and a subspace C X, a subspace S X is an (A, C)-conditioned invariant if A (S C) S The set of all the (A, C)-conditioned invariants containing a given subspace B is closed with respect to the intersection, hence has a minimum, which will be referred to as S =mins (A, C, B). It is computed with the sequence S 1 = B S i = B + A (S i 1 C) i =2, 3,... mins (A, C, B) is obtained when the sequence stops (S i+1 = S i ). 19

21 Invariance by state feedback and output injection Let B be a basis matrix of B: amatrix F exists such that (A + BF) V V if and only if V is an (A, B)-controlled invariant. Hence a controlled invariant can be made a simple invariant by state feedback. V is said to be internally stabilizable if it can also be internally stabilized as an (A + BF)-invariant by the matrix F. Let C be a matrix such that C =kerc: amatrixg exists such that (A + GC) S S if and only if S is an (A, C)-conditioned invariant. Hence a conditioned invariant can be made a simple invariant by output injection. S is said to be externally stabilizable if it can also be externally stabilized as an (A +GC)-invariant by the matrix G. Internal stabilizability of controlled invariants and external stabilizability of conditioned invariants can easily be checked and stabilizing feedback matrices F and G can easily be obtained by using algorithms based on suitable changes of bases in the state space. 20

22 The meaning of controlled and conditioned invariants x(0) V X ẋ(t) = Ax(t)+Bu(t) y(t) = Cx(t) x(k+1) = A d x(k)+b d u(k) y(k) = C d x(k) Let V =maxv (A, B, C), V d =maxv (A d, B d, C d ), S d =mins (A d, C d, B d ). The meaning of V or Vd : it is the maximal subspace of the state space where it is possible to follow state trajectories invisible at the output. In fact, a state trajectory can be maintained on a subspace V X if and only if it is an (A, B) or an (A d, B d )-controlled invariant. The meaning of Sd : it is the maximal subspace of the state space reachable from the origin in at most ρ steps with trajectories having all the states but the last one belonging to kerc d, hence invisible at the output. The integer ρ is the number of iterations required for the algorithm of Sd to converge. 21

23 Properties of dynamic systems: invariant zeros Consider the following figure unstable zero V Recall: V =maxv (A, B, C) S =mins (A, C, B) R V stable zero Let R V denote the maximum reachable subspace on V (which is poleassignable with state feedback). It has been shown (Morse, 1973) that R V = V S Let F be such that (A + BF) V V (A + BF) R V R V. The invariant zeros of system (A, B, C) or (A d,b d,c d ) are the internal unassignable eigenvalues of the (A + BF)-invariant V. A dynamic system is said to be minimum phase if all its invariant zeros are stable. 22

24 Properties of dynamic systems: left and right invertibility, relative degree A system is said to be left-invertible or, simply, invertible if, starting from the zero state, for any admissible output function y(t), t [0,t 1 ] t 1 > 0 or y(k), k [0,k 1 ], k 1 n there exists a unique corresponding input function u(t), t [0,t 1 ) or u(k), k [0,k 1 1]. The left invertibility condition is V S = {0} A system is said to be right-invertible or functionally controllable if there exists an integer ρ 1 such that, given any output function y(t), t [0,t 1 ], t 1 > 0 with ρ-th derivative piecewise continuous and such that y(0) = 0,..., y (ρ) (0) = 0, or y(k), k [0,k 1 ], k 1 ρ such that y(k)=0, k [0,ρ 1], there exists at least one corresponding input function u(t), t [0,t 1 ) or u(k), k [0,k 1 1]. The minimum value of ρ satisfying the above statement is called the relative degree of the system. The right invertibility condition is V + S = X and the relative degree is the least integer such that V + S ρ = X in the conditioned invariant algorithm. 23

25 Extension to systems with feedthrough Extension to non purely dynamic systems of the above definitions and properties can be obtained through a simple contrivance (state extension). v u Σ e integrators or delays u Σ Σ y y z Σ e integrators or delays Refer to the second figure: system Σ e is modeled by ż(t)=y(t) and the overall system by with ˆx(t) = y(t) = [ ] x ˆx := u [ ] B ˆB := D Â ˆx(t)+ ˆB v(t) Ĉ ˆx(t) Â := [ A 0 C 0 ] Ĉ := [ 0 I q ] The addition of integrators at inputs or outputs does not affect the system right and left invertibility, while the relative degree of (Â, ˆB,Ĉ) must be simply reduced by 1 to be referred to(a, B, C, D). The controlled invariants obtained by the above state extension are called output nulling subspaces (Anderson, 1976). 24

26 u Σ y An output nulling subspace is simply a controlled invariant V such that there exists a state feedback F satisfying F x (A + BF) V V V ker (C + DF) so that the output is identically zero for any initial state on V. The seven properties of dynamic systems Stability (internal and external) Controllability Observability Left invertibility Right invertibility Relative degree Minimality of phase 25

27 Computational support with Matlab Q = mainco(a,b,x) Maximal (A, imb)-controlled invariant contained in im X. Q = miinco(a,c,x) Minimal (A, imc)-conditioned invariant containing im X. [P,Q] = stabv(a,b,x) Matrices for the internal and external stabilizability of the (A,im B)-controlled invariant im X. [Q,F] = vstar(a,b,c,[d]) V, maximal output nulling controlled invariant of (A,B,C,[D]) and stabilizing state feedback matrix F. [Q,G] = sstar(a,b,c,[d]) S, minimal conditioned invariant dual of V and stabilizing output injection matrix G. R = rvstar(a,b,c,[d]) Reachable set on V. z = gazero(a,b,c,[d] Invariant zeros of (A,B,C,[D]). F = effesta(a,b,x) Stabilizing state feedback for the (A,B)-controlled invariant X. 26

28 3 - The disturbance decoupling problem The exact disturbance decoupling problem by state feedback is the basic problem of the geometric approach, and it was studied by Basile and Marro as one of the earliest applications of the new concepts in It was first approached without the stability requirement. Disturbance decoupling with the stability requirement was satisfactorily settled around by Basile, Marro and Schumacher in 1982, by using self-bounded controlled invariants. When the disturbance is measurable we have a milder solvability condition and we can apply a feedforward unit. This is the exact counterpart of the Kalman dual filter, and admits a dual, the unknown-input exact observer of the state or of a linear function of the state, which is the counterpart of the Kalman filter. 27

29 The structural condition d u F Σ Let us consider the system ẋ(t) = Ax(t)+Bu(t)+Hh(t) e(t) = Ex(t) where u denotes the manipulable input, d the disturbance input. Let B := imb, H := imh, E := kere. The disturbance decoupling problem is: determine, if possible, a state feedback matrix F such that disturbance h has no influence on output e. x e The system with state feedback is described by ẋ(t) = (A + BF) x(t)+hh(t) e(t) = Ex(t) It behaves as requested if and only if its reachable set by d, i.e., the minimum (A + BF)-invariant containing H, is contained in E. Let V(B,E) := max V(A, B, E). Since any (A + BF)-invariant is an (A, B)- controlled invariant, the inaccessible disturbance decoupling problem has a solution if and only if H V (B,E) 28

30 The conditions with stability This is a necessary and sufficient structural condition and does not ensure internal stability. If stability is requested, we have the disturbance decoupling problem with stability. Stability is easily handled by using selfbounded controlled invariants. Assume that (A, B) is stabilizable (i.e., that R =minj (A, B) is externally stable) and let V m := V (B,E) S (E,B+H) This is the minimum self-bounded (A, B)-controlled invariant containing H. It is the reachable set on V(B,E) corresponding to a control action through both inputs u and h. The following theorem yields the solution. If V m is not internally stabilizable no other (A, B)-controlled invariant internally stabilizable and containing H exists (Basile-Marro-Schumacher, ). Hence we have obtained the following result: the disturbance decoupling problem with stability admits a solution if and only if H V (B,E) V m is internally stabilizable If the above conditions are satisfied, a solution is provided by a state feedback matrix such that (A + BF) V m V m and σ(a + BF) is stable. 29

31 If the state is not accessible, disturbance decoupling may be achieved through a dynamic unit similar to a state observer. This is called the disturbance decoupling problem with dynamic measurement feedback and was also solved at the beginning of the eighties (Willems and Commault, 1982). Extension to systems with feedthough terms The disturbance decoupling problem for systems with feedthrough terms, like can easily be handled by state extension. ẋ(t) = Ax(t)+Bu(t)+Hh(t) e(t) = Ex(t)+Du(t)+Gh(t) 30

32 4 - The Kalman Regulator (LQR, H 2 ) The study of the Kalman linear-quadratic regulator (LQR) is the central topic of most courses and treatises on advanced control systems. See, for instance, the books by Kwakernaak and Sivan (1973), Anderson and Moore (1989), Syrmos and Lewis (1995). More recently, in the nineties, a certain attention was given to the H 2 optimal control, which is substantially a rehash of the LQR with some standard and well settled problems of the geometric approach (for instance, disturbance decoupling with output feedback). Feedthrough is not present in general, so that the standard Riccati-based solutions are not implementable and the existence of optimal solution is not ensured. Books on this subject are by Stoorvogel (1992), Saberi, Sannuti and Chen (1995). The computational method they use to solve the H 2 -optimal problem are linear matrix inequalities (LMI), supported by a special coordinate basis that points out the geometric features of the systems dealt with. An alternative route, which will be briefly presented in the following, is to treat the singular and cheap problems, where feedthrough is not present, by directly referring to the Hamiltonian system, which can be considered as a generic dynamic system, with all the previously described features. 31

33 The standard LQR problem Consider the continuous-time system ẋ(t) =Ax(t)+Bu(t), x(0) = x 0 with the performance index ( J = x(t) T Qx(t)+u(t) T Ru(t)+ 0 2 x(t) T Nu(t) ) dt [ ] Q N with N T 0, R>0. R Consider the discrete-time system x(k +1)=A d x(k)+b d u(k), x(0) = x 0 with the performance index ( J d = x(k) T Q d x(k)+u(k) T R d u(k)+ k=0 2 x(k) T N d u(k) ) [ ] Qd N with d Nd T 0, R R d > 0. d The LQR problem: Find a control u(t), t [0, ) oru(k), k [0, ) such that J or J d is minimal. If R>0orR d > 0 the control problem is said to be regular,ifr 0orR d 0itis said to be singular,ifr =0 or R d = 0 it is said to be cheap. 32

34 The solution through the Hamiltonian system The continuous-time Hamiltonian system: [ ] [ ][ ẋ(t) A 0 x(t) = ṗ(t) 2 Q A T p(t) 0 = [ 2 N T B ] [ T x(t) p(t) The discrete time Hamiltonian system: [ ] [ ][ ] [ x(k+1) A 0 x(k) = p(k+1) 2 A T Q A T + p(k) ] ] + [ B 2 N ] + [ 2 R ] u(t) B 2 A T N ] 0 = [ 2 B T A T Q +2N T B T A T ] [ x(k) p(k) ] u(k) u(t) + [ 2 R 2 B T A T N ] u(k) NOTE: In both cases the Hamiltonian system is a quadruple (Â, ˆB, Ĉ, ˆD) whose output must be maintained at zero for the given initial state x 0. Hence the LQR problem is easily interpreted as a standard disturbance localization problem of the geometric approach. The results of the geometric approach also give a significant insight into its solvability in the singular and cheap cases. 33

35 The solution is achieved as follows: 1 - Regular case. Derive, from the algebraic condition, u(t) = (2R) 1 ( 2 N T x(t)+b T p(t) ) or u(k) = ( 2 R 2 B T A T N ) 1 (( 2 B T A T Q +2N T) x(k) B T A T p(k) ) and substitute in the differential equations of state and costate. An overall autonomous differential system is obtained whose eigenvalues are stable-unstable by pairs. Let W (2n n) be a basis matrix of the stable invariant subspace, so that [ ][ ] [ ][ ] x(t) W1 x(k) W1,d α(t) or α(k) p(t) W 2 p(k) W 2,d hence p(t)=sx(t) or p(k)=s d x(k), with S := W 2 W1 1 or S d := W 2,d W 1 1,d, and, by substitution in the previous equations, u(t)=kx(t) oru(k)=k d x(k) withk or K d suitably defined. Through a Lyapunov-Riccati equation, the optimal cost is also derived as x T 0 Sx 0 or x T 0 S d x 0. NOTE: There are simple contrivances to deal with A 1 and A T in the discrete-time case when A is singular and/or the system is not completely controllable (see, for instance, Marro, Prattichizzo, Zattoni, 2002). 34

36 x 0 δ(t) u K Σ Hence the regular LQR problem can be solved by state feedback, as shown in the figure. h y u K 0 Σ x x y Reformulation in H 2 terms By using a standard matrix decomposition: [ ] Q N M T M = N T R and taking [ C D ] = M, the problem on hand can be restated as a minimum H 2 norm problem from input h to output y for the system ẋ(t) = Ax(t)+Bu(t)+Hh(t) y(t) = Cx(t)+Du(t) A similar procedure can be applied in the discrete-time case. ( ( ( ( 1/2 )) 1/2 G 2 = tr g(t) g T (t) dt)) and G d 2 = tr g d (k) gd T (k) dt are the expressions of the H 2 norms in terms of the impulse responses g(t) org d (k). k=0 35

37 1 - Singular and cheap cases. These can be handled by applying the geometric approach to the Hamiltonian system (Â, ˆB, Ĉ, ˆD). The LQR Problem admits a solution if and only if there exists an internally stable output nulling (Â, ˆB)-controlled invariant of the overall Hamiltonian system whose projection on the state space of the original system contains the initial state x(0). This projection is defined as { [ ] } x P (ˆV) = x : ˆV p It can be proven that the internal unassignable eigenvalues of ˆV, the maximal output nulling controlled invariant of (Â, ˆB, Ĉ, ˆD) are stable-unstable by pairs. Hence a solution of the LQR Problem is obtained as follows: 1. compute ˆV ; 2. compute a matrix ˆK such that (Â + ˆB ˆK)ˆV ˆV and the assignable eigenvalues (those internal to RˆV ) are stable; 3. compute ˆV s, the maximum internally stabilizable (Â + ˆB ˆK)-invariant contained in ˆV ; 4. if x(0) P (ˆV s ) the problem admits a solution K, that is easily computable as a function of ˆV s and ˆK; if not, the problem has no solution. 36

38 The above procedure also provides a state feedback matrix K corresponding to the minimum H 2 norm from h to y. This immediately follows from the expression of the H 2 norm in terms of the impulse response. In fact, the impulse response corresponds to the set of initial states defined by the column vectors of matrix H. Thus, the minimum H 2 norm disturbance decoupling problem from h to y has a solution if and only if H P (ˆV s ) with H =imh On the other hand, it can be proven that in the discrete-time case this problem is always solvable, since P (ˆV s ) has dimension n. dead-beat 0 ρ regular trajectory A typical control sequence in the discrete-time case is shown in the figure: as the sampling time approaches zero, the dead-beat segment approaches a distribution, which is not obtainable with state feedback. For this reason solvability of the H 2 optimal decoupling problem is more restricted in the continuous-time case. 37

39 The usefulness of the geometric tools in the LQR problem Hence the geometric approach tools can profitably be applied to the Hamiltanian system for solving the LQR problem, particularly in the singular and cheap cases. For instance, the dead-beat subarc in the previous figure is also avoided in the discrete-time case if ˆV s is restricted, like in the continuous-time case, by discarding the zero eigenvalues of ˆV, whose number equals the dimension of Ŝ. Some features of the Hamiltonian system can be expressed in geometric terms, like: if the original system is left-invertible, the Hamiltonian system is both left and right-invertible. A significant information on the features of the LQR problem in the singular and cheap cases is provided by analyzing the number of invariant zeros of the Hamiltonian system, stable-unstable by pairs: the stable zeros are the modes of the solution. In conclusion, the basic system properties that can be expressed in geometric terms, like left and right invertibility, relative degree and minimality of phase yield useful information also in the LQR problem. 38

40 Dealing with non left-invertible systems The computational support with Matlab for the infinite-time LQR problem consists of two very efficient routines (Laub, 1974), available in the standard Control System Toolbox: [S,L,K] = care(a.b,q,r,n) Solution of the continuous-time LQR problem. [S,L,K] = dare(a.b,q,r,n) Solution of the discrete-time LQR problem. The routine care only works in the regular case, while dare also works in the cheap and singular cases (hence providing the dead-beat subarc). Both routines do not work if the system is not left-invertible. This is due to the non-uniqueness of the solution: in fact, the eigenvalues on R V are arbitrarily assignable, since they do not affect the performance index. v G K u F h Σ x y In the figure it is shown how the geometric approach can integrate these standard LQR routines to provide handling of the degrees of freedom in computing the state feedback matrix K when the system is not left-invertible. 39

41 Equivalence between the LQR-H 2 problem and disturbance decoupling Consider the LQR-H 2 problem X ẋ(t) =Ax(t)+Bu(t)+Hh(t) C 1 C with the cost y(t) =Cx(t)+Du(t) J = 0 y T (t) y(t) dt The following basic result (Stoorvogel, 1992) states a very direct correspondence between the LQR-H 2 optimal control problem and the geometric approach: there exist matrices C 1, D 1 such that, if the output equation of the given system is replaced by y(t) =C 1 x(t)+d 1 u(t) the optimal control problem becomes zero-cost, i.e., it is transformed into a standard disturbance decoupling problem solvable with geometric tools. This result is rather intuitive: the optimal state feedback provides motion on a controlled invariant that can be made output nulling by changing the output matrices (see the figure, referring to the cheap case). 40

42 Further problems which could be treated in this context Kalman-derived framework controllability observability Kalman regulator (LQR, H 2 ) Kalman dual filter Kalman filter previewed signal optimal decoupling delayed filter (smoother) H 2 -optimal model following H 2 -optimal model observer Geometric framework controlled invariants conditioned invariants disturbance decoupling problem measured signal decoupling unknown-input state observer previewed signal decoupling del. unknown-input state observer exact model following exact model observer primal problem dual problem 41

43 References Morse Structural invariants of linear multivariable systems, SIAM J. Control, vol 1, no 3, Anderson Output nulling invariants and controllability subspaces, Proceedings of the 6th IFAC Congress, paper 43.6, Basile and Marro Self-bounded controlled invariant subspaces: a straightforward approach to constrained controllability, J. of Optimization Theory and Applic., vol 38, no 1, Schumacher On a conjecture of Basile and Marro J. of Optimization Theory and Applications, vol 41, no 2, Willems and Commault Disturbance decoupling with measurement feedback with stability or pole placement, SIAM J. of Control and Optimization, vol 19, no 4,

44 Kwakernaak and Sivan Linear optimal control systems, John Wiley & Sons, Anderson and Moore Optimal Control: linear quadratic methods Prentice Hall International, Lewis and Syrmos Optimal control, John Wiley & Sons, Stoorvogel The singular H 2 control problem, Automatica, vol. 28, no 3, The H control problem: a state space approach, Prentice Hall International, Saberi, Sannuti and Chen H 2 optimal control, Prentice Hall International, Marro, Prattichizzo and Zattoni A geometric insight into the discrete-time cheap and singular LQR problems, IEEE Trans. Automatic Control, vol 47, no 1, Arnold and Laub Generalized eigenproblem algorithms and software for algebraic Riccati equation, Proceedings IEEE, vol 72,