On the Structure of Quantized-Control Quantized-Observation Systems

Copyright © IFAC Theory and Application of Digital Control New Delhi, India 1982

ON THE STRUCTURE OF QUANTIZED-CONTROL QUANTIZED-OBSERVATION SYSTEMS J. Miyamichi Faculty of Engineering, Utsunomiya University, Utsunomiya-shi 321, Japan

Abstract. Systems whose control inputs and observation outputs can take only discrete-level values are called quantized systems. In this paper some problems concerning reachability or observability of quantized systems are investigated for linear, finite-dimensional and discrete-time systems. A system is said to be almost reachable if reachable states are dense in the state space. The state space of a quantized control system can be decomposed into three subspaces which are discretely reachable subspace, almost reachable subspace and almost reachable in finite time subspace. The expansion behavior of reachable states of quantized control systems is illustrated by an example. Concerning observability, i t is shown that there exists no universal input for quantized control system and some conditions under which any two distinct states are distinguishable from quantized outputs are obtained. Also almost reachability for the quantized bounded control system and for the system defined over Q is studied. Keywords. Linear systems; quantized inputs; quantized bounded inputs; output quantization; almost reachability; observability; canonical structure.

INTRODUCTION

are basic questions when we are dealing with digital systems.

Nowadays, in computing control schemes for real world continuous system, we shall use a digital computer and so we are forced to treat them as a discrete-time system. We can make measurements of input values of the system at discrete times and also can only generate distinct control signals at discrete times.

Di~crete-time systems whose control inputs and observation outputs can take only given discrete-level values are called quantizedcontrol quantized-observation systems or simply quantized systems.

In digital systems, not only the time set but

also input. and output values of the system are specified in some discrete manner, since they are represented using finite length words. Discrete-level control and discrete-level observation are common in the area of digital controls. Many investigations have been made so far on discrete-time continuous-valued systems (usual discrete-time dynamical systems) and it is well known that basic properties such as reachability, observability and realization are common both to continuous-time and discrete-time systems. On the other hand, we have poor knowledge about quantized level systems (discrete-time discrete-valued systems) especially about the system behavior under the condition that their control input and observation output are quanti zed into discrete level values . "To what extent can we control the system using quanti zed control inputs ?" and "To what extent can we estimate the jnternal state using quantized observation outputs ?" 65

Anzai (1972,1974) introduced the concept of quantized control system and investigated the relationship between controllability properties and system structure for some restricted class of quanti zed control systems. Miyarnichi (1976,1977a,1977b) then extended his work to more general multi-input rnultioutput linear systems, leading to a canonical structure of quantized control system which clearly shows how well a system can be controlled by quanti zed control inputs. He also made some investigations(1981) on observability of quantized system and its relationship to system structure for linear finite-dimensional systems. It was found that both reachability and observability properties of a quanti zed system are closely related to a Q-linear dependence of the system coefficients. The purpose of this paper is to clarify the situation we shall encounter when we only be able to utilize digitized informations as control inputs and observation outputs . We shall represent some investigations on reachability and observability of these quantized systems.

66

J. Miyamichi

Input u(k)

Output y(k) Physical System S x(k+l)~ Ax(k)+ Bu(k)

We shall use the Euclidian norm as distance measure on the state space. Terms such as closure, isolated point, accumulation point, dense, ... are used with respect to this norm.

y(k)- Cx(k) continuous-level system

Integer Vector

quantlzation

Fig. 1.

we have a relation ax+by+ ... +cz = 0. Otherwise they are R-linearly (Q-linearly) independent. Note that two numbers 1 and 12 (as one dimensional vectors) are Q-linearly independent but R-linearly dependent.

A Z-module in Rn is a set of vectors which is closed under addition and multiplication by integer. Next three lemmas are fundamental in the following discussion. Integer Vector

Quantized system.

LEMMA 1. Let a,b, ... ,c be vectors in Rn which are Q-linearly independent but R-linearly dependent, then the Z-module generated by a,b, ... ,c has at least one accumulation point. LEMMA 2. Infinitely generated Z-module in Rn has at least one accumulation point.

We shall be concerned in this paper with linear, finite-dimensional and discrete-time quanti zed-control quanti zed observation systems --- whose control inputs are restricted to integer vectors and whose observation outputs are quanti zed into integer vectors --- of the type x(k+l) Ax(k) + Bu(k) (1) y(k) q[ Cx(k) ] where A, B, C are matrices whose elements are real numbers and whose size is n x n, n x m and p x n, respectively, and x(k) E Rn: state vector u(k) E Zm: integer vector(input) y(k) E ZP: integer vector(quantized output) q[ ] coordinatewise quantization. A quanti zed system is illustrated in Fig. 1. Even if a physical system S can be controlled fairly well by continuous-level inputs, it can be expected neither it is controllable by quanti zed control inputs nor it is observable when some quantization processes are introduced to the output observation. In the following it will be assumed that the continuous level system S = (A,B,C) :the physical system or real world system to be controlled is both controllable and observable in the usual sense (i.e. by continuous-level inputs and quantization free outputs).

LEMMA 3. (Bourbaki,1963) Let M be a closed Z-module in n-dimensional Euclidian space, M is a union of equally spaced parallel linear manifolds. In other words, there is a basis { a } i= 1,2, ... ,n such that M can be i represented as follows: M={

x

I

d

k

x = L tia. + L n.aj , tiER, nJ.EZ } i=l 1 j=d+l J

Let C[ M ] be t~ subspace spanned by a ,a 2 , l ... ,ad then C[ M ] is the maximum subspace of ~ contained in M. LEMMA 4. (Kronecker's approximation theorem, Bourbaki,1963) t t Let el=(l,O, ... ,0) , e 2=(0,1,0, ... ,0) , ... , en=(O, ... ,O,l)t be the natural basis of Rn and let x=(x ,x , ... ,x )t be a vector in Rn. n l 2 Let M be a Z-module generated by e ,e 2 , ... ,e n l and x, M is dense in Rn i f and only if xl ,x 2 , ... ,xn and 1 (the number one) are Q-linearly independent. QUANTIZED CONTROL SYSTEM

We shall use Z ( Q,R ) to denote the set of all integers (of all rational numbers and of all real numbers, respectively). An algebraic integer is a solution of any finite order algebraic equation with integral coefficients whose leading coefficient is 1.

A state x is said to be "reachable" if there is an (integer valued) input sequence '.vhich carries the system from zero to x. Let ~ be a set of all states that are reachable by input sequences of length k and let Moo be their union, a set of all reachable states. Both ~ and Moo are Z-modules and the latter is in general infinitely generated. The set of all reachable states is countable and so no quanti zed control system is reachable in the usual sense.

Vectors x,y, ... ,z in Rn (n-dimensional Euclidian space) are said to be R-linearly (Q-linearly) dependent if for some real (rational) numbers a,b, ... ,c ,not all zero,

The best situation for reachability of quantized control system is that within any neighborhood of x, there is at least one reachable state, in which case the state x

PRELIMINARIES

Qua nti zed- Contro l Qu a nt ized- Ob serva ti on Sy st ems

67

is said to be "almost reachable" . If there is an integer N such that within any neigh- . borhood of x there is at least one state belonging to ~ , x is said to be "almost

•

reachable in finite time".

•

If we denote the closure of M by M we can get the following: DEFINITION 5. A state x is "almost reachable" if x EM. A state x is "almost reachable in finite t k " if x E ~ for some N. It is easy to show that both Moo and

~

are closed Z- module in Rn and , by lemma 3, almost reachable states are distributed within the linear manifolds which are parallel and equally spaced to each other . It also can be seen (Miyamichi , 1976 , 1977b) that both c[ Moo ] and C[ ~ ] are A- invariant subspaces and the latter is a subspace of the former . It was shown (Miyamichi , 1977b , 198l ) that every quantized control system admits a canoni cal decomposition into three subsystems which are discretely reachable system, almost reachab l e but not almost reachable in finite time system and almost reachable in finite time system. The fo llowings are one dimensional exampl es of these three types of quantized control systems . (a) Discret ely reachab l e : x(k+l) = 2 x(k) + u(k) The set of all reachabl e states Moo coincides to Z which is discrete set in R. (b) Almost reachable but not almost r eachable in finite time : 1

x(k+l) = 2 x(k) + u(k) The set of all reachabl e states M is dense in R. Longer control time is ne~essary if we need higher control accuracy. (c) Almost reachable in finite time: x(k+l) = /2 x(k) + u(k) Moo M2 and we can control· this system in two unit of time with any degree of accuracy . We shall show the expansion behavior of the set of reachable states ~ in the next: EXAMPLE 6 . Consider the following three dimensional quantized control system in canonical form.

Xlk+ll<[ ~

!

: the subspace in whi ch r eachable stat es are isolated points. :A- invariant subspace i n which every state is a lmost reachabl e .

~

MS

Fig . 2.

Expasi on of the set of reachabl e states . A-invariant subspace in which every stat e i s almost reachabl e in finite time .

The expansion behavior of illu~ trated in Fig . 2. For i=1, 2, 3, Mi = Mi are isolated points .

~

(k=1, ... , 6) i s

and reachable states The set M4 is dense in

the lines parallel to x - axis and C[ M4 ] = X3 3 is A- invariant subspace . The set Moo is dense in the pl anes parallel to x - x pl ane 2 3 which is C[ Moo ] and also A- invariant . THEOREM 7. (CANONICAL STRUCTURE OF QUANTIZED CONTROL SYSTEM) By suitably choosing a basis , every quanti zed control system can be represented as fo l lows . Al

,I I I

, 0 , 0

Bl

x(k+l )= A : A2 : 0 2l

x(k) + B2

____ 1 ____ 1_---

* where Al , Bl A ,A ,B 2l 2 2

I I I I

* ! A3

B3

1 1 I

Xl

----1----1- - --

u(k)

X 2

X3

matrices with integer el ements ; matrices with r ati onal el ement s . A2 has no algebraic integer s as i ts characteristic values; matri ces with r eal elements .

68

J. Miyamichi

FURTHER DISCUSSIONS

control inputs for every integer N.

In this section we shall discuss some proper-

Almost reachability of a system over Q

ties of quantized control systems which are useful to understand the behavior of quantized systems more deeply. First we shall consider the case in which there imposed one more restriction on our control inputs, i.e. a control problem with bounded and quantized inputs. Next we shall show that when we are dealing with a system defined over the field of rational numbers almost reachability of the system depends on a system matrix A only and not on a input matrix B. Finally we shall refer to a problem of system realization.

Suppose that an object we want to control is represented as a dynamical system over Q, i.e. all matrix elements of A and B are rational numbers. This can happen quite often in the area of digital control where every quantity is represented as finite length word. In this case, as is shown in the following discussion, almost reachability of the system fully depends on the state transition matrix A (especially on its algebraic nature) and not on the input matrix B.

Control with bounded input It is well known that a finite dimensional linear dynamical system is controllable by bounded (but continuous level) inputs if all the characteristic values of A have magnitudes less than unity. But as is shown in the following example this condition is not suffi.cient for a quantized system to be almost reachable by bounded inputs even if it is almost reachable by quantized (but not bounded) inputs. When continuous level inputs are available we can apply ,if necessary, very small control input to the system and in this fashion we can effect little change in the state space. On the other hand, if the control input values are restricted to discrete levels, we cannot make an input value arbitrary small. In quantized control system, small quantities are produced in a very tricky way; they are the differences of two large quantities. For example, by sui tabl~ choosing two integers M and N, we can make 12 N + M arbitrary small. Under the condition of bounded inputs we may fail to control the system since above tricky mechanism cannot work any This is true as is shown in the more. following simple :

EXAMPLE 8. Let us consider the following one dimensional quantized control system. x(k+l) = ax(k) + u(k) Suppose that the admissible control input values are { +1, 0, -l}. Reachable states are dense in the interval [ -1 f +1 ] containing the origin only if 1/3 < a 1 < 1 . This comes from the fact that when a is in that range we have an inequality for some k k 1;1 < lal 2+ ial 3+ + la: Too small values of a do not confirm almost reachability. In the particular case of a=O , the reachable states are only +1, 0, -1 and the system cannot be almost reachable in any sense. This type of difficulties cannot arise when control inputs are not bounded, since if a state x is reachable by quantized inputs a state Nx is also reachable by quanti zed

Every coefficient (matrix element) is Qlinearly dependent and the situations defined by Lerma 1 or Lerma 4 cannot occur. The subspace X3 in the canonical structure of quantized control system disappear and no quantized control system can be "almost reachable in finite time". The set of all reachable states M is dense in the state space only when it is infinitely generated Z-module. It is easily shown that M is finitely generated if and only if all t~e characteristic values of A are algebraic integers. (This is true without assuming that A is a matrix over Q.) We can summarize above consideration in the form of next: THEOREM. 9. Let A and B in Eq.l be matrices over Q. The system is "almost reachable" if and only if A has no algebraic integers as its characteristic values. The key point of theorem 8 is in the fact that if an algebraic integer a is a characteristic value of some rational matrix A then every conjugate of a is also characterristic value of A. COROLLARY. If A is nonsingular and if all its characteristic values have magnitudes less than unity, the system is almost reachable. Realization of discretely reachable system We can show as a consequence of above structure theorem that if we are given an external description of a system S (input-output relations) which has a remarkable nature that for every integer valued inputs it always emits integer valued outputs, we can realize it as a system over Z. Notice that this problem has been solved for more general systems (i.e. systems over Noetherian integral domain, Y.Rouchaleau, B. Wyrnan and R.E. Kalman,1972) .

A Hankel matrix that represents input-output behavior of the system contains only integer elements. Since it is realizable, by using Ho's algorithm for example, we can get a realization of S over rational field. From our structure theorem of quantized control system, the state space consists of two

Quantized-Control Quantized-Observation Systems

subspaces --- discretely reachable subspace Suppose and almost reachable subspace. that two reachable states x and x* are close to each other, then the corresponding output sequences have almost same values which is possible only when they are exactly same since they are integer valued and it follows that x = x*. This shows that every reachable state is isolated point and the system has no almost reachable subspace. STATE OBSERVATION USING QUANTIZED OUTPur INFORMATION For any input sequence { u(i) }, i= O,l, ... ,k and for any initial states x and x* we have: k-l x(k) = Akx + E Ak-l-~u(i) i=O k-l x*(k) = Akx*+ E Ak-l-~u(i). i=O

(2)

The second term is the same in both cases, and the outputs are equal if the first terms are equal. Two states are not distinguishable if and only if they yield the same response to all zero sequences. This is true only if quantization free outputs are available. When output quantization is introduced two different states may yield the same response. Also it is not necessary true that zero input sequences alone suffices to test observability. We introduce the following: DEFINITION 10. Two states x and x* are said to be "zero-equivalent" if q [ Cx J= q [ Cx* J, i.e. if two current observation outputs are same. We can distinguish two initial states x and x* if and only if x(k) and x*(k) are not zeroequivalent to each other for some k. First we shall consider observability of a quantized output system by zero input response ( the response to all zero input sequence ). For a state x with very small norm to be distinguishable from zero state x must grow up to a vector x* such that q[ Cx* J I 0, since the zero input response of the zero state is all zero sequence. This leads to the following: THEOREM 11. Every pair of distinct states of quantized output system is distinguishable by quantized zero-input response if and only if all the characteristic values of A have the magnitudes greater than unity. Next we shall consider the role of the system inputs. The cardinality consideration of the equivalence classes defined by an input sequence leads to the following: LEMMA 12. For every finite input sequence { u(k) }, k= 0,1, ... there exists a pair of states x and x* such that the corresponding

69

quantized outputs y (k) (initial state = x) and y*(k) (initial state = x*) are same. Lemma 12 says that no quantized output system A universal input is has universal input. an input sequence u with the property that, if two states x and x* can be distinguishable by some input, they can be distinguished by u. It has been shown(Sussman,1979) that universal inputs always exist for a large class of smooth systems. System inputs are useful in the case that we have partial informations about the internal states of the system beforhand. If we know that the system is either in a state x or in a state x* we can determine the internal state by constructing an input sequence This which distinguishes these two states. is possible only if we can control the second term in Eq. 2 in such a manner that x(k) and x*(k) belong to the different zero-equivalence classes. It is easily shown that we can distinguish every two initial states by quantized output information if continuous level inputs are available. Difficulties arise when control inputs are restricted to integer vectors. DEFINITION 13. Two states x and x* are "distinguishable by quantized inputs" if there is an integer valued input sequence which causes different outputs for x and x*. A quanti zed system is "observable by quantized outputs" if every two states x and x* (x I x*) are distinguishable by quantized inputs. In the following we shall investigate observability for each of three types of quanti zed control system.

Discretely Reachable System Let A and B are matrices containing only integer elements. Every reachable state is a point represented by an integer vector. The second term of Eq. 2 can be any integer vector but cannot be noninteger vector. Observability of the system fully depends on the structure of the output matrix C as shown in the following: THEOREM 14. Let S be a quantized system defined by Eq.l with A and B matrices over Z and C amatrix over R. S is observable by quantized outputs if and only if the Zmodule generated by e ,e , ... ,e (natural l 2 p bases) and c ,c , ... ,c (column vectors of n l 2 C ) is dense in RP. We can use Lemma 4 above and its extension (Mlyam1chi,1976,1981) to check if a finitely generated Z-module is dense or not. Immediate consequence of the Theorem is the following: COROLLARY. Let A and B are matrices over Z. If C is over Q the system cannot be observable by quantized outputs.

J. Miyarnichi

70

Almost Reachable System Let A and B are matrices over the field of rational numbers. The almost reachable in finite time subspace is zero and 1\ = has a lattice structure for every k. Every reachable state is Q-linearly dependent to a set of n vectors of Mn (n is the dimension of the system) which span the state space.

f\

The above discussion done for discretely reachable systems is still true in this case. The difference is as follows . The sequence of the set of reachable states ~ ,M2 , .. . are strictly increasing and the mesh of 1\ is finer than that of Mi if k is larger than i . The distance between nearest reachable states decreases to zero as k increases to infinity. We can control the second term of Eq . 2 with any accuracy if sufficiently long control time is available. Does this imply that the system is observable ? Unfortunatel y this is not the case as is shown in the following: EXAMPLE. 15. Let us consider the following one-input one- output one-dimensional system which is almost reachable by integer inputs : 1 x(k+l) = 2 x(k) + u(k)

y(k)

= q[ x(k) ] :quantization by rounding.

It is easil y shown that two initial states x = 8/10 and x*= 9/10 cannot be distinguished to each other by integer inputs. More generally, if the output matrix C has no irrational elements the system cannot be observable by quantized outputs. THEOREM 16 . If A,B and C are matrices over Q there exists a pair of initial states which cannot be distinguishable by quantized outputs. In spite of the advantage that the mesh of reachable states becomes finer if we spend l onger control time, the observability criterion of the system is identical to that for the discretely reachable systems (Theorem 14 ) . Notice that, though we may fail to construct an input sequence that distinguishes two initial states x and x*, there are no need to distinguish these two states. If the system is almost reachable two indistinguishable states always approach to each other. LEMMA. 17. Let S be an almost reachable system. If two states x and x* are not distinguishable by quantized outputs then x(k) - x*(k) tends to zero vector as k tends to infinity. Almost Reachable in Finite Time System

When S is almost reachable in finite time we can control the second term of Eq. 2 with any precision by some control inputs of length less than 2n (n is the dimension of the system). This leads to the foll owing: THEOREM. 18 . Every almost reachable in finite time system is observable by quantized outputs. CONCLUSION The problems of control and state observation using quantized control inputs and quantized observation outputs are studied for linear, finite-dimensional and discrete- time systems whose admissible control inputs are integer valued vectors and its output informati ons can be obtained only in quanti zed form. Every quantized control system admits a canonical decomposition into three subsystems which are discretely reachable system, almost reachable but not almost reachable in finite time system and almost reachable in finite time system. With the aid of this canonical structure theorem,basic questions on quanti zed systems ___ "to what extent can we control the system using quanti zed control inputs ?" and "to what extent can we estimate the internal state using quantized observation outputs ?" --- are solved. REFERENCES

Anzai, Y. (1972). Concept of controllability of quantized control systems . Keio Engineering Report, ..Q, 141- 159, Keio University , Tokyo. Anzai, Y. (1974). A note on reachability of discrete- time quanti zed control system . IEEE Trans. Autom . Contrl, 19, 575- 577. Bourbaki, N. (1963) . Elementsde Mathematigue . Topologie Generale , Hermann , Paris. Chap. 7. Miyamichi, J., and T. Fukao (1976) . On reachability of quantized control systems. J. Soc. Instrum . & Control Eng., 12. 145-150 . (in Japanese) Miyamichi , J. (1977a) . On control lability of quantized control systems. J. Soc. Instrum. & Control Eng.,~, 330- 335 . (in Japanese) Miyamichi, J . (1977b) . Canonical form of quantized control system. J . Soc . Instrum. & Control Eng.,]J, 439- 444. (in Japanese) Miyamichi , J . (1981) . State observation from quantized input-output information . Proc. IFAC VIIIth Triennial World Congress Kyoto, Japan . Rouchaleau , Y., B. Wyman, and R.E . KalrrRn (1972). Algebraic structure of linear dynamical systems Ill: Realization theory over a commutative ring . Proc . Nat. Acad . Sci. U.S .A., ~ 3404- 3406 . Sussman , H.J . (1979) Single-input observability of continuous- time systems. Math. Syst. Theory., 11., 371- 393. --