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Deadbeat Response Is L2 0Ptimal Deadbeat Response is l2 0ptimal Vladimír Kučera Fellow IEEE, Fellow IFAC Director, Masaryk Institute of Advanced Studies Professor, Faculty of Electrical Engineering Czech Technical University in Prague Czech Republic http://www.muvs.cvut.cz [email protected] Abstract—A typical linear control strategy in discrete-time origin in k steps by applying an input sequence u0, u1, ..., uk – 1. systems, deadbeat control produces transients that vanish in When R = IRn , the system (A, B) of (1) is said to be finite time. On the other hand, the linear-quadratic control n stabilizes the system and minimizes the l2 norm of its transient reachable. response. Quite surprisingly, it is shown that deadbeat systems Define the integers are l2 optimal, at least for reachable systems. The proof makes use of polynomial matrix fractions and structure theorem for linear time-invariant multivariable qk = dimension Rk − dimension Rk −1 systems, the notions introduced by W.A. Wolovich in the early seventies. and for k = 1, 2, ..., m let The result demonstrates the flexibility offered by the linear- quadratic regulator design and is an exercise in inverse optimality. The linear-quadratic regulator gain is unique, ri = cardinality {qk : qk ≥ i}. whereas the deadbeat feedback gains are not. Only one deadbeat gain is linear-quadratic optimal. An alternative construction of The integers r1 ≥ r2 ≥ ... ≥ rm are the reachability indices of such a gain, based on solving an algebraic Riccati equation, is system (1). thus available. Keywords - linear systems; finite impulse response; l2 norm We further define the controllability subspaces for (1) by minimization C0 = 0, I. DEADBEAT REGULATOR n k Ck = {x ∈ IR : F x ∈ Rk }, k = 1, 2, ... We consider a linear system (A, B) described by the state equation Thus Ck is the set of all states of (1) that can be steered to the origin in k steps by an appropriate control sequence u0, u1, ..., xk +1 = Axk + Buk , k = 0,1, ... (1) n uk – 1. When Cn = IR , the system (A, B) of (1) is said to be controllable. It follows from the definitions that reachability m n where uk ∈ IR and xk ∈ IR . The objective of deadbeat implies controllability and the converse is true whenever A is regulation is to determine a linear state feedback of the form nonsingular. The existence and construction of deadbeat control laws is u = −Lx (2) k k described below. For each k = 1, 2, ... let S1, S2, ..., Sk be a sequence of m × q1, m × q2 , ..., m × qk matrices such that that drives each initial state x0 to the origin in a least number of steps. k −1 range [BS1 ABS2 ... A BS k ] = range Rk . We define the reachability subspaces of system (1) by Therefore S1, S2, ..., Sk serve to select a basis for Rk. R0 = 0, k −1 Theorem 1. [1] There exists a deadbeat control law (2) if Rk = range[B AB ... A B], k = 1, 2, .... and only if the system (A, B) of (1) is controllable. Let Hence Rk is the set of states of (1) that can be reached from the L0 = 0, (3) The original version of the paper was presented at the 3rd IEEE k Lk = Lk −1 + Lk′ (A − BLk −1) , k = 1, 2, ... International Symposium on Communications, Control and Signal Processing, held in St Julians, Malta 2008. where Lk′ satisfies B, C, D) defined by the state equation (1) and the output equation (5) is left invertible if its transfer function −1 k −1 C(zIn − A) B + D has full column normal rank. We further Lk′ [BS1 ABS 2 ... A BS k ] = [0 ... 0 S k ]. define the system matrix as the polynomial matrix Then L = Ln is a deadbeat regulator gain. < ⎡zIn − A − B⎤ The theorem identifies all deadbeat control laws via the S(z) = ⎢ ⎥ recursive procedure (3). Actually the procedure can be ⎣⎢ C D ⎦⎥ terminated in q steps, where q = min {k : Ck +1 = Ck }. The resulting closed-loop system matrix is nilpotent with index q, and say that a complex number ζ is an invariant zero of the system (A, B, C, D) if the rank of S(ζ) is strictly less than the q normal rank of S(z). (A − BL) = 0. (4) Theorem 2. [2] Suppose that the system (A, B) of (1) is If A is nonsingular, the recursive procedure (3) can be shortcut stabilizable. Suppose that the system (A, B, C, D) defined by (1) and (5) is left invertible and also has no invariant zeros on by setting Lq – 1 = 0. Then, the Jordan structure of A – BL the unit circle | z | = 1. Then, there exists a unique LQ regulator comprises m nilpotent blocks [3] of sizes r1, r2, ..., rm and the gain given by index of nilpotency equals q = r1. In fact, this is the least size of Jordan blocks that can be achieved [4] in a reachable system (1) by applying state feedback (2). L = (DT D + BT XB) −1 (BT XA + DT C), (6) II. LINEAR QUADRATIC REGULATOR where X is the largest symmetric nonnegative definite solution We consider a linear system described by the state equation of the algebraic Riccati equation (1), X = AT XA + C TC xk +1 = Axk + Buk , k = 0,1, ... (7) T T T T T −1 T T − (B XA + D C) (D D + B XB) (B XA + D C). < where u ∈ IRm and x ∈ IRn. The objective of LQ regulation k k The assumption of left invertibility for (A, B, C, D) is is to find a linear state feedback of the form (2), required in order for the matrix DTD + BTXB to be nonsingular while the remaining assumptions are required for the existence uk = −Lxk of the requisite solution X of the algebraic Riccati equation. that stabilizes the closed-loop system III. THE DEADBEAT REGULATOR AS AN LQ REGULATOR The aim of this section is to show that deadbeat control xk +1 = (A − BL)xk laws in reachable systems are LQ optimal. This will be done by constructing an output (5) of the system (1) so that the and, for every initial state x0, minimizes the l2 norm resulting LQ regulator gain is a deadbeat gain. Let the system (A, B) of (1) be reachable with reachability 2 ∞ T indices r , r , ..., r . Then there exists a similarity y = ∑k =0 yk yk 1 2 m transformation T that brings the matrices A and B to the p standard reachability form [3], of a specified output yk ∈ IR of the form A′ = TAT −1, B′ = TB (8) yk = Cxk + Duk . (5) where A′ is a top-companion matrix with nonzero entries in The existence and construction of an LQ control law is rows ri, i = 1, 2, ..., m and B′ has nonzero entries only in rows described below. We say that the system (A, B) of (1) is ri and columns j ≥ i, i = 1, 2, ..., m. stabilizable if the system matrices can be transformed to the following form using an appropriate basis: Theorem 3. Suppose that the system (A, B) of (1) is reachable, with reachability indices r1 ≥ r2 ≥ ... ≥ rm and with the matrix B having rank m. Let T be a similarity ⎡A11 A12 ⎤ ⎡B1 ⎤ transformation that brings A and B to the standard reachability A = ⎢ ⎥ , B = ⎢ ⎥ ⎣ 0 A22 ⎦ ⎣ 0 ⎦ form. Then, the feedback gain L that is LQ optimal with respect to C = T and D = 0 in (5) is a deadbeat gain. where the subsystem defined by the pair of matrices (A11, B1) is Proof. Consider the transfer function of system (1) in reachable and A22 is a stable matrix. We say that the system (A, polynomial matrix fraction form −1 −1 and column-degree ordered with column degrees r ≥ r ≥ ... ≥ (zIn − A) B = Q(z)P (z) (9) 1 2 rm, by the equation where P and Q are right coprime polynomial matrices in z of T −1 −1 −1 T respective size m × m and n × m, with P column reduced and F (z )F(z) = [CQ(z ) +DP(z )] [CQ(z) +DP(z)] (12) column-degree ordered with column degrees r1 ≥ r2 ≥ ... ≥ rm. –1 These integers are the reachability indices of (1). in such a way that its inverse F is analytic in the domain | z | ≥ 1. This matrix is referred to as the spectral factor and it is The system (1) being reachable, the matrices zIn – A and B determined uniquely by (12) up to multiplication on the left by are left coprime. It follows from (9) that the denominator a constant orthogonal matrix. matrices zIn – A and P(z) have the same determinant (in fact, the same invariant polynomials). The pair (A, B) being reachable, the matrices A and B can be brought to the reachability standard form (8) using the For any feedback (2) applied to system (1), one obtains similarity transformation matrix T. The corresponding right coprime polynomial fraction matrices are related by ⎡ In 0 ⎤ [zI − A − B] = [zI − (A − BL) − B] n ⎢ ⎥ n P′(z) = P(z), Q′(z) = TQ(z) ⎣− L Im ⎦ and and by the Structure Theorem of Wolovich [5], Q′ has the block-diagonal form ⎡In 0 ⎤ ⎡Q(z)⎤ ⎡ Q(z) ⎤ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥. ⎣ L Im ⎦ ⎣P(z)⎦ ⎣P(z) + LQ(z)⎦ ⎡ 1 ⎤ ⎡ 1 ⎤ ⎡ 1 ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ Then (9) implies that ⎢ z ⎥ ⎢ z ⎥ ⎢ z ⎥ Q′(z) = block-diag [ ⎢ ⎥ , ⎢ ⎥ , ..., ⎢ ⎥ ] . ⎢ M ⎥ ⎢ M ⎥ ⎢ M ⎥ −1 −1 [zIn − (A − BL)] B = Q(z)[P(z) + LQ(z)] . (10) ⎢ r −1⎥ ⎢ r −1⎥ ⎢ r −1⎥ ⎣z 1 ⎦ ⎣z 2 ⎦ ⎣z m ⎦ Thus the closed-loop system transfer function matrices zIn – (A The spectral factorization (12) reads – BL) and B are left coprime while P(z) + LQ(z) and Q(z) are right coprime.
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