«State Nulliﬁcation by Memoryless Output Feedback∗ Zvi Artstein† and Gera Weiss‡ Abstract. We examine linear single-input single-output ...»
Springer-Verlag London Ltd. © 2004
Math. Control Signals Systems (2005) 17: 38–56
State Nulliﬁcation by Memoryless Output Feedback∗
Zvi Artstein† and Gera Weiss‡
Abstract. We examine linear single-input single-output ﬁnite-dimensional sys-
tems. It is shown that a continuous time controllable and observable system can
be nulliﬁed utilizing periodic sampling of the output with time-varying linear feed-
back. Almost any sampling rate can be used. The result relies on a characterization of linear output feedback nulliﬁcation of discrete time observable and controllable systems. An algorithm for the nulliﬁcation and an estimate on the time in which the algorithm is concluded are provided.
Key words. Output feedback, Static feedback, Nulliﬁcation, Hold function, Sampled data.
1. Introduction and the Main Results Nullifying the state, namely, driving the state to the origin in ﬁnite time, can be viewed as a sharp form of stabilization (desirable at times due to precision require- ments); it is also of interest for its own sake as a structural problem. In this paper we examine the issue of nullifying a linear system, thus, nulliﬁcation in a given time interval amounts to placing the poles of the associated transformation at the origin. We are interested in utilizing linear output feedback which is memoryless, namely, time-varying static feedback. While efﬁcient algorithms for pole placement invoking dynamic feedback are available, the issue of stabilization with memory- less feedback has been recognized as a subtle problem and was recorded as such in Brockett . The problem has drawn attention in the literature. Some papers, those most relevant to the subject matter of the present contribution, are listed as references below and discussed in the body of the text. In particular, the nulliﬁ- cation result concerning discrete-time systems displayed in the present paper ﬁlls a gap in Aeyels and Willems  where a possibility of pole assignment is offered, provided that the poles are not zero. Theorem D below provides a characterization of the possibility to nullify a controllable and observable discrete-time linear system ∗ Date received: September 26, 2003. Date revised: April 15, 2004. Published online: 7 October 2004 Research supported by grants from the Israel Science Foundation and from the Information Society Technologies Programme of the European Commission.
† Department of Mathematics, The Weizmann Institute of Science, Rehovot, 76100, Israel.
firstname.lastname@example.org. (Incumbent of the Hettie H. Heineman Professorial Chair in Mathematics).
‡ Department of Computer Science and Applied Mathematics, The Weizmann Institute of Science, Rehovot, 76100, Israel. email@example.com.
State Nulliﬁcation by Memoryless Output Feedback with a static linear output feedback. The proof we display is intricate, but it results in a simple algorithm for both checking the condition and nullifying the system; we also provide an estimate for the number of steps in which the algorithm is concluded.
The result in the discrete-time framework is then used to analyze the possibility to nullify a continuous-time system. Our approach pertains to the analysis of sampled-data hold functions, also referred to as deadbeat controls; see the analysis of Kabamba  where a general result concerning generalized sampled-data hold functions is presented. Theorem S below veriﬁes that for all but a discrete set of sample periods the sufﬁcient conditions established in the analysis of the discrete-time case are satisﬁed.
In both the discrete-time case and the continuous-time framework we address single-input single-output systems (even in this case, as mentioned, the proof is quite intricate).
The main results are as follows.
The discrete-time systems we examine are ﬁnite-dimensional linear systems whose dynamics is generated by the equations of the form xj +1 = Axj + buj yj +1 = cxj +1 (1.1) where A is an n × n matrix, b an n-dimensional column vector and c an n-dimensional row vector. In particular, the control u is scalar and the system is single-input single-output. When feedback from the output is invoked, the control is of the form uj = u(cxj ). In particular, when the feedback from the output is linear and memoryless, the dynamics has the form
xj +1 = Axj + kj bcxj. (1.2)
The main problem we examine is the possibility of nullifying the system while utilizing a memoryless linear feedback from the output. Our technique yields some new information about the intimately related property of stabilizing the system while utilizing a linear feedback from the output. We refer to these problems as output nulliﬁcation and output stabilization (they should not be confused with nullifying and stabilizing the output). The nulliﬁability is formalized as follows.
Deﬁnition 1.1. The system (1.1) is memoryless linear output feedback nulliﬁable if a ﬁnite sequence k0, k1,..., kj0 exists such that for any initial condition x0 given at time t = 0 the sequence x1, x2,..., resulting from the dynamics (1.2) satisﬁes xj0 = 0; equivalently, (A + kj0 bc) · · · (A + k1 bc)(A + k0 bc) = 0.
The deﬁnition of linear output feedback stabilization is analogous; for instance, one may require the existence of k0, k1,..., kj0 such that all the eigenvalues of the matrix (A + kj0 bc) · · · (A + k1 bc)(A + k0 bc) have absolute value less than 1.
In order to formulate our main result we recall two notions. First recall the controller canonical form of the system (see e.g. Sontag [11, Deﬁnition 4.1.5]), namely, if the system (1.1) is controllable then after a similarity change of variables the data have the form Z. Artstein and G. Weiss
See Sontag [11, Section 3.4].
State Nulliﬁcation by Memoryless Output Feedback Deﬁnition 1.2. We say that the continuous system (1.4) is h-sample linear output feedback nulliﬁable if the corresponding discrete time system (1.5) is linear output feedback nulliﬁable.
The main result concerning sampled data systems refers to a discrete set of sampling periods. By a discrete set we mean a set whose intersection with any bounded interval is ﬁnite. The result is as follows.
Theorem S. Suppose that the system (1.4) is controllable and observable. The system is h-sample linear output feedback nulliﬁable except for a discrete set of sampling periods h.
The rest of the paper is organized as follows. The proof of the necessity in Theorem D is given in Sect. 2. The sufﬁciency in Theorem D is established in Sect. 3.
The proof has an algorithmic nature and utilizes arguments which yield information also when the sufﬁcient conditions do not hold. The algorithm is fairly simple and is displayed in Sect. 4 along with an estimate on the number of steps it takes to conclude the algorithm. In Sect. 5 we provide some corollaries of Theorem D along with other comments, examples and counterexamples within the framework of discrete time systems. In Sect. 6 we verify Theorem S and provide some comments and examples.
2. The Necessity in Theorem D
The necessity of the condition cAdj(A)b = 0 (equivalently, γ1 = 0 in the controller canonical form) follows from two simple observations.
Det(A + kbc) = Det(A) + kcAdj(A)b (equivalently, Det(A + kbc) = (α1 + kγ1 )(−1)n+1 when A, b, c are given in the controller canonical form (1.3)).
Proof. The claims are straightforward for the canonical form and hold in the general case since all the expressions are invariant under similarity transformations.
(The result for A nonsingular is recorded in Aeyels and Willems  where it is used in the same manner we use it.) Lemma 2.2. If in the controller canonical form γ1 = 0 then α1 = 0 is a necessary condition for observability.
Proof. Suppose that both γ1 and α1 are equal to 0. Consider the vector x0 = (1, 0,..., 0)T (the superscript T signiﬁes transposition). Then Ax = 0. Hence Aj x = 0 for any j 0. If in addition γ1 = 0 then cx = 0 which together with the obvious equalities cAj x = 0 for all j 0 contradict observability.
Completion of the proof of the necessity in Theorem D. As was pointed out after the statement of Theorem D it is enough to establish the result for a system in the canonical form. Now notice that Lemma 2.1 implies that when γ1 = 0 the equality |Det((A + kj bc) · · · (A + k1 bc)(A + k0 bc))| = |α1 |j +1 holds for any sequence k0,..., kj of inputs. Also, under observability Lemma 2.2 implies that whenever γ1 = 0 then α1 = 0. Therefore, if observability holds and γ1 = 0 then no sequence of inputs can generate the zero matrix and the nulliﬁcation property does not hold.
Z. Artstein and G. Weiss Remark 2.3. The condition cAdj(A)b = 0 which characterizes nulliﬁcation is also related to stabilization. In fact, it is a necessary condition for linear output feedback stabilization when |Det(A)| ≥ 1 (which amounts to |α1 | ≥ 1 when the system is in the canonical form). Indeed, when cAdj(A)b = 0 and |Det(A)| ≥ 1 then by Lemma 2.1 the determinant of any matrix generated by a sequence of inputs has absolute value greater than or equal to 1; this prohibits linear output feedback stabilization.
3. The Sufﬁciency in Theorem D
First we simplify the arguments of the main construction and offer a more speciﬁc property which, in turn, guarantees linear output feedback nulliﬁcation.
Proposition 3.1. Suppose that the system (1.1) has the property that for every x0 ∈ R n the equality (A + kj0 bc) · · · (A + k1 bc)(A + k0 bc)x0 = 0 (3.1) holds for an appropriate choice (which may depend on x0 ) of numbers k0,..., kj0.
Then the system is linear output feedback nulliﬁable.
Proof. We choose n linearly independent vectors v1,..., vn. For v1 we utilize (3.1) with x0 = v1 to determine a matrix, say M1, of the form appearing in (3.1) such that M1 v1 = 0. Inductively, for vj +1 we use (3.1) with x0 = Mj · · · M1 vj +1 to determine a matrix, say Mj +1 of the form appearing in (3.1) such that Mj +1 Mj · · · M1 vj +1 = 0.
Clearly, Mn · · · M1 x0 = 0 for every x0 and Mn · · · M1 has the desired form of a feedback from the output. This completes the proof.
With the preceding result the proof of the sufﬁciency would be complete if we prove the following result.
Proposition 3.2. Suppose that the system (1.1) is controllable and observable and that cAdj(A)b = 0. Then for every ﬁxed x0 ∈ R n the equality (A + kj0 bc) · · · (A + k1 bc)(A + k0 bc)x0 = 0 (3.2) holds for a certain sequence (which may depend on x0 ) of numbers k0,..., kj0.
The rest of the section is devoted to the proof of the preceding proposition. The proof utilizes a method of introducing free variables whose possible realizations play a major role. In spite of the apparently indirect method, an outcome of the derivation is a simple algorithm whose characteristics and an estimate of its conclusion time are displayed in the next section.
We ﬁnd it very convenient to prove the result for a system in the controller canonical form. As was pointed out after the statement of Theorem D it is enough to verify the result for a system in this form (see Remark 4.2 on the general case). We start with an algorithm and draw some consequences on its outcome with no reference to the conditions of the proposition. Then we derive some properties of the outcome under the observability assumption. Only then do we utilize the condition γ1 = 0 to verify the nulliﬁcation.
State Nulliﬁcation by Memoryless Output Feedback Let x = (ξ1,..., ξn )T be in R n (the superscript T signiﬁes transposition). A useful observation is that when A is applied to x the result is the vector Ax = (ξ2,..., ξn, ax)T where a = (α1,..., αn ) is the bottom row of A; namely, the ﬁrst n − 1 coordinates of Ax are formed by a shift of the last n − 1 coordinates of x and the last coordinate of Ax is equal to ax.
Construction 3.3. Starting with x0 = (ξ0,1,..., ξ0,n )T we generate the sequence x1, x2,..., as follows.
(i) If cx0 = 0 we deﬁne x1 = Ax0.
(ii) If cx0 = 0 we deﬁne x1 = (ξ0,2,..., ξ0,n, δ1 )T where δ1 is a variable whose value will be determined later; namely, x1 is formed by shifting the last n − 1 coordinates of x0 and adding the free variable δ1 as the n-th coordinate.
Inductively, suppose that xj = (ξj,1,..., ξj,n )T has been constructed.
(i) If cxj = 0 for any choice of numerical values of the free variables δi for i ≤ j, we deﬁne xj +1 = Axj. We denote then the last coordinate of xj +1 by σj +1, namely σj +1 = axj. (The coordinates of xj +1, including σj +1, may still be functions of the free variables δi which have been introduced earlier.) (ii) If cxj = 0 for some numerical realization of δi for i ≤ j we introduce a new free variable δj +1 and deﬁne xj +1 = (ξj,2,..., ξj,n, δj +1 )T ; namely, xj +1 is formed by shifting the last n − 1 coordinates of xj and adding the variable δj +1 as the n-th coordinate.
The free variables will be used in the conclusion of the proof to determine (via the equations δi+1 = (a + ki c)xi ) the desired feedback ki. Meanwhile, the previous construction determines a sequence x0, x1,..., such that the coordinates of xj are either free variables δi or terms σi which are functions of the free variables with index less than j and of the coordinates of x0. Notice that the index i of δi or of σi indicates the ﬁrst time at which the term appears in the construction. In particular, if a free variable δi or a term σi occupies the l-th coordinate of xj then i = j − n + l.
Observation 3.4. A coordinate σj is an afﬁne (namely, linear plus a constant shift) function of the free variables δi for i j.
Proof. Follows easily from the construction.
Next recall that a free variable appears as the n-th coordinate of xj +1 if for at least one realization of the free variables δi for i ≤ j the expression cxj is not zero.
Let f (j ) be the number of the free variables δi appearing in the process for i ≤ j (here f stands for free). We consider the possible realizations of the sequence δi for i ≤ j to be elements in R f (j ), the f (j )-dimensional linear space.