A note on the causality of singular linear discrete time systems
aa r X i v : . [ m a t h . R A ] J un A note on the causality of singular linear discrete timesystems
Christos Tsegkis School of Engineering, The University of Edinburgh, UK
Abstract:
In this article we study the causality of non-homogeneous linear singular dis-crete time systems whose coefficients are square constant matrices. By assuming that theinput vector changes only at equally space sampling instants we provide properties forcausality between state and inputs and causality between output and inputs.
Keywords : causality, singular, system.
In this article we shall be concerned with the non-homogeneous singular discrete timesystem of the form
F Y k +1 = GY k + BV k X k = CY k (1)with known initial conditions Y k (2)where F, G ∈ M ( n × n ; F ), Y k ∈ M ( n × F ) (i.e., the algebra of square matrices withelements in the field F ), X k ∈ M ( m × F ), B ∈ M ( n × l ; F ) and C ∈ M ( m × n ; F ). Forthe sake of simplicity, we set M n = M ( n × n ; F ) and M nm = M ( n × m ; F ). We assumethat the system (1) is singular, i.e. the matrix F is singular and that the input vector V k changes only at equally space sampling instants. Many authors have studied discrete timesystems, see and their applications, see [1-9, 12, 13, 16-18, 20-32]. In this article we studythe causality of these systems. The results of this paper can be applied also in systemsof fractional nabla difference equations, see [10, 11]. In addition they are very useful forapplications in many mathematical models using systems of difference equations existingin the literature, see [14, 15, 29-32]. Definition 1.1.
Given
F, G ∈ M nm and an indeterminate s ∈ F , the matrix pencil sF − G is called regular when m = n and det( sF − G ) = 0. In any other case, the pencilwill be called singular.In this article, we consider the case that the pencil is regular . The class of the pencil sF − G is characterized by a uniquely defined element, known as a complex Weierstrasscanonical form, sF w − Q w , see [19, 24], specified by the complete set of invariants of sF − G .This is the set of elementary divisors (e.d.). In the case of a regular matrix pencil, wehave e.d. of the following type: e.d. of the type ( s − a j ) p j , are called finite elementary divisors (f.e.d.), where a j isa finite eigenavalue of algebraic multiplicity p j • e.d. of the type ˆ s q are called infinite elementary divisors (i.e.d.), where q thealgebraic multiplicity of the infinite eigenvaluesWe assume that P νi =1 p j = p and p + q = n .Let B , B , . . . , B n be elements of M n . The direct sum of them denoted by B ⊕ B ⊕ . . . ⊕ B n is the blockdiag (cid:2) B B . . . B n (cid:3) . From the regularity of sF − G , thereexist nonsingular matrices P, Q ∈ M n such that P F Q = F w = I p ⊕ H q and P GQ = G w = J p ⊕ I q Where sF w − Q w is the complex Weierstrass form of the regular pencil sF-G and is definedby sF w − Q w := sI p − J p ⊕ sH q − I q , where the first normal Jordan type element is uniquelydefined by the set of the finite eigenvalues.( s − a ) p , . . . , ( s − a ν ) p ν of sF − G . The second block has the form sI p − J p := sI p − J p ( a ) ⊕ . . . ⊕ sI p ν − J p ν ( a ν )And also the q blocks of the third uniquely defined block sH q − I q correspond to theinfinite eigenvalues ˆ s q , . . . , ˆ s q σ , σ X j =1 q j = q of sF − G and has the form sH q − I q := sH q − I q ⊕ . . . ⊕ sH q σ − I q σ Thus, H q is a nilpotent element of M n with index ˜ q = max { q j : j = 1 , , . . . , σ } , where H ˜ qq = 0 q,q , and J p j ( a j ) , H q j are defined as J p j ( a j ) = a j . . . a j . . . . . . a j
10 0 . . . a j ∈ M p j , H q j = . . . . . . . . . . . . ∈ M q j . For algorithms about the computations of the jordan matrices see [19, 24, 28]. The solution of a singular linear discretetime system
In this subsection we obtain formulas for the solutions of LMDEs with regular matrix pen-cil and we give necessary and sufficient conditions for existence and uniqueness of solutions.
Theorem 2.1.
Consider the system (1), (2) and let the p linear independent (gener-alized) eigenvectors of the finite eigenvalues of the pencil sF-G be the columns of a matrix Q p . Then the solution is unique if and only if Y k ∈ colspanQ p + QD k (3)Moreover the analytic solution is given by Y k = Q p J k − k p Z pk + QD k (4)where D k = " P k − i =0 J k − i − p B p V i − P q ∗ − i =0 H iq B q V k + i and PB= (cid:20) B p B q (cid:21) , with B p ∈ M pn , B q ∈ M qn . Proof.
Consider the transformation Y k = QZ k Substituting the previous expression into (1) we obtain
F QZ k +1 = GQZ k + BV k . Whereby, multiplying by P, we arrive at F w Z k +1 = G w Z k + P BV k . Moreover, we can write Z k as Z k = (cid:20) Z pk Z qK (cid:21) . Taking into account the above expressions,we arrive easily at two subsystems of (1). The subsystem Z pk +1 = J p Z pk + B p V k (5)and the subsystem H q Z qk +1 = Z qk + B q V k (6)The subsystem (5) has the unique solution Z pk = J k − k p Z pk + k − X i =0 J k − i − p B p V i , k ≥ k , (7)see [1, 4, 11, 12]). By applying the Zeta transform we get the solution of subsystem (6) Z qk = − q ∗ − X i =0 H iq B q V k + i (8) et Q = (cid:2) Q p Q q (cid:3) , where Q p ∈ M np , Q q ∈ M nq the matrices with columns the p, qgeneralized eigenvectors of the finite and infinite eigenvalues respectively. Then we obtain Y k = QZ k = [ Q p Q q ] " J k − k p Z pk + P k − i =0 J k − i − p B p V i − P q ∗ − i =0 H iq B q V k + i Y k = Q p J k − k p Z pk + Q p k − X i =0 J k − i − p B p V i − Q q − q ∗ − X i =0 H iq B q V k + i .Y k = Q p J k − k p Z pk + QD k The solution that exists if and only if Y k = Q p Z pk + QD k or Y k ∈ colspanQ p + QD k Generally for systems of type (1) we define the notion of causality, which is properly de-fined bellow
Definition 3.1.
The non-homogeneous singular continuous system (1) is called casual, ifits state Y k , for any k > k is determined completely by initial state Y k and former inputs V k , V k +1 , ..., V k . Otherwise it is called noncausal.Discrete time normal systems are characterized by the property of causality. Next wewill study the causality in a singular system of the form (1). Causality between state and inputs
Proposition 3.1.
In system (1) causality between state and inputs exists if and only if H q B q = 0 q,l Proof.
From (8) it is clear that the state Z k and obviously Y k for any k ≥ k is tobe determined by former inputs if and only if H iq B q = 0 q,l for every i = 1 , , ..., q ∗ − H q B q = 0 q,l . Causality between output and inputs
Proposition 3.2.
In system (1) causality between output and input exists if and only if CQ q H iq B q = 0 m,l (9) or every i = 1 , , ..., q ∗ − Proof.
The solution of the state equation of the system (1) is given by Theorem 2.1. Y k = Q p J k − k p Z pk + Q p k − X i =0 J k − i − p B p V i − Q q q ∗ − X i =0 H iq B q V k + i Setting the expression of Y k in the state output relation X k = CY k we take X k = CQ p J k − k p Z pk + CQ p k − X i =0 J k − i − p B p V i − CQ q q ∗ − X i =0 H iq B q V k + i (10)From the above expression it is clear that non causality is due to the existence of the term P q ∗ − i =0 CQ q H iq B q V k + i . So the causal relationship between X k and V k exists if and only if CQ q H iq B q = 0 m,l for every i = 1 , , ..., q ∗ − C (cid:2) Q q H q B q ... Q q H q ∗ − q B q (cid:3) = 0 m,q ∗ nl (11)So the following Proposition is obvious. Proposition 3.3.
The system (1) is causal if and only if every column of the matrix (cid:2) Q q H q B q ... Q q H q ∗ − q B q (cid:3) lies in the right nullspace of the matrix C . Remark 3.1.
If the system pencil sF − G has no infinite eigenvalues then the ma-trix Q q = 0 n,q . So the relation (11) is satisfied and we have causality between inputs andoutputs of the system. Conclusions
Having shown that the solution of the discrete time system of the form (1) exists if theinitial conditions (2) belong to the set (3) and is given by the formula (4), we prove thatin system (1) causality between state and inputs and causality between output and inputsexists under necessary and sufficient conditions.
Acknowledgments
I would like to express my sincere gratitude to Professor G.I. Kalogeropoulos and Dr. I.K.Dassios for their helpful and fruitful discussions that improved this article.
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