Sufficient Conditions for Controllability and Observability of Serial and Parallel Concatenated Linear Systems

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1 Sufficient Conditions for Controllability and Observability of Serial and Parallel Concatenated Linear Systems M. I. García-Planas J. L. Domínguez-García and L. E. Um Abstract This paper deals with the sufficient conditions for controllability and observability characters of finitedimensional linear continuous-time-invariant systems of serial and parallel concatenated systems. The obtained conditions depend on the controllability and observability of the systems and in some cases the functional output-controllability of the first one. Keywords Linear systems serial composite systemsparallel composite systems controllability observability functional-output controllability. I. INTRODUCTION It is well known that many physical problems as for example electrical networks multibody systems chemical engineering convolutional codes among others see [4] [5] [8] for example the most popular and the most frequently used mathematical model for its description is the state space representation in the form } ẋt Axt + But yt Cxt + Dut where A M n R B M n m R C M p n R and D M p m R These linear systems can be described with an input-output relation called transfer function obtained by applying Laplace transformation at x 0 to equation } sx AX + BU Y CX + DU obtaining the following relation Hs CsI n A B + D M.I. García-Planas is with the Department of Matemàtica Aplicada I Universitat Politècnica de Catalunya 0808 Barcelona Spain e- mail: maria.isabel.garcia@upc.edu J.L. Domínguez-García is Catalonia Institute for Energy Research and L.R. Um is Université Mohammed V. ISSN: In engineering problems a system is sometimes built by interconnecting some other systems this kind of systems are called concatenated or composite systems. Let ẋ i A i x i + B i u i y i C i x i + D i u i for i be two systems that can be connected in different ways. The most common are the following: a Serialized one after the other so that the input information u y t. Consequently A 0 cx B ẋ + B C A x y x D C C + D x D u. B D u 3 Another model of serial concatenation is the systematic serial concatenation. In this case we consider the same kind of systems than the case of serial concatenation systems. A 0 x ẋ + B C A x C 0 x y + D C C x B u B D u. D D D 4 b The second concatenated model that we will study is the parallel concatenation. So that the input information is u t u t ut and the output is yt y t + y t. Consequently ẋ A 0 x B + u 0 A x B y x C C + D x + D u. 5 Obviously both in the case of serial concatenation as parallel the sizes of the systems must be adequate to make sense the concatenation. The transfer functions for concatenated systems are

2 i serial concatenated case: Hs H s H s ii systematic serial concatenated case: H Hs s H s H s iii parallel concatenated case: Hs H s + H s. Controllability and observability are of important and fundamental properties of control systems. The controllability concept of a dynamical standard system is largely studied by several authors and under many different points of view see [] [4] [7] [5] [6] for example. Nevertheless controllability for the output vector of a system has been less treated see [5] [8] [3] for example. The functional output-controllability generally means that the system can steer output of dynamical system along the arbitrary given curve over any interval of time independently of its state vector. A similar but least essentially restrictive condition is the pointwise output-controllability [8]. It is well known the importance of observable systems because it is possible to use observers for to track and reconstruct states of a model [3]. Roughly speaking observability means the possibility of identifying the internal state of a system from measurements of the outputs. In this paper controllability and observability for serial and parallel composite systems are analyzed and a sufficient conditions are presented. II. PRELIMINARIES In this paper it is considered the state space system introduced in equation } ẋt Axt + But yt Cxt + Dut where x is the state vector y is the output vector u is the input or control vector A M n R is the state matrix B M n m R is the input matrix C M p n R is the output matrix and D M p m R. For simplicity we will write the systems by a triple of matrices A B C D. ISSN: A. Controllability The most frequently used fundamental definition of controllability for linear control systems with constant coefficients is the following. Definition.: Dynamical system is said to be controllable if for every initial condition x0 and every vector x R n there exist a finite time t and control ut R m t [0 t ] such that xt x. This definition requires only that any initial state x0 can be steered to any final state x at time t. However the trajectory of the dynamical system between 0 and t is not specified. Furthermore there is no constraints posed on the control vector ut and the state vector xt. In order to formulate easily computable algebraic controllability criteria let us introduce the so-called controllability matrix C which is known as controllability matrix and defined as follows. C B AB A B... A n B. 6 Theorem.: Dynamical system is controllable if and only if C. Or equivalently test de Hautus [4] if and only if si n A B for all s C 7 Observe that s C the algebraic closure of R. B. Observability The observability character deals with the question of if it is possible to determine the state from the input and output does not knowing the initial state. Definition.: Dynamical system is said to be observable on [0 t ] if for all inputs ut and outputs yt t [0 t ] the state x0 x 0 can be uniquely determined. There is a duality between controllability and observability. Definition.3: Given the system A B C D we call adjoint system to the system A t C t B t D t Theorem. Duality: i The system A B C D is controllable if and only if A t C t B t D t is observable ii The system A B C D is observable if and only if A t C t B t D t is controllable As a consequence we can formulate an easily computable algebraic observability criteria introducing the so-called observability matrix O which

3 is known as observability matrix and defined as follows. C O 8. n Theorem.3: Dynamical system is observable if and only if O. or equivalently test de Hautus [4] if and only if si A for all C 9 C C. Functional output-controllability The functional output-controllability generally means that the system can steer output of dynamical system independently of its state vector. Concretely Definition.4: A system is functional outputcontrollable if and only if its output can be steered along the arbitrary given curve over any interval of time. It means that if it is given any output y d t t 0 there exists t and a control u t t 0 such that for any t t yt y d t. Proposition. [4]: A system is functional output-controllable if and only CsI n A B + D p in the field of rational functions. A necessary and sufficient condition for functional output-controllability is Proposition. [4] [6]: si A B + p C D Test for functional output-controllability: The functional output-controllability can be computed by means of the of a constant matrix in the following manner Theorem.4 [8]: The system A B C D is functional output-controllable if and only if oc f A B C D + p. C D CB D B CB D.... n n B... B CB D The null terms are not written in the matrix. Remark.: We call oc i A B C D or simply oc i if confusion is not possible the following matrix oc i C D CB D B CB.... i i B... B CB D i. i If the system A B C D is functional outputcontrollable then the matrices oc i have full row for all 0 i n. ii If the matrix oc n has full row it is not necessary that the matrix oc n has full row. Example.: Let A B C D a system with 0 0 A B C and D 0. C D CB D C D CB D B CB D III. CONTROLLABILITY OF COMPOSITE SYSTEMS A. Controllability of serial composite systems We will try to characterize the controllability of serial concatenated systems in terms of the original systems. Let A B C D the serial concatenated system of the systems A i B i C i D i i defined in 3. The controllability character for serial concatenated systems can be described in terms of the both systems using the Hautus test in the following manner. Theorem 3.: A serial concatenated system is controllable if and only if 0 B B C si n A B D + n s C. Corollary 3.: A necessary condition for controllability of serial concatenated system is that the pair A B be controllable. ISSN:

4 Corollary 3.: A necessary condition for controllability of serial concatenated system is that the pair A B where B B C B D be controllable. Corollary 3.3: A necessary condition for controllability of serial concatenated system is that the pair A B be controllable. Proof: si n A B C B D si n A B B I min si n A So C D B I n C. D si n A B C B D si n A B n. Example 3.: Let A B C D be a serial concatenated system of A B C D and A B C D where and A 0 B 3 C 4 D A 0 B 0 0 C 0 D the concatenated serial system A B C D of both is: A 0 A B C A 0 0 B B 3 B D 0 0 C D C C 4 0 D D D In this case is it easy to observe that all systems are controllable this can be easily observed from the ISSN: Hautus representation of our serial controllability matrix si n A B s 3 s C si n A B s s C s 0 0 B B C si n A B D s s 3 s C 0 s 0 0 Nevertheless the corollary only gives us a necessary condition but not sufficient as we can see in the following example. Example 3.: Let A B C D a serial concatenated system of A B C D and A B C D where A B C D and A 0 B C D. Both systems are controllable because of s s C s s C but the serial concatenated system A B C D where 0 A B 0 C D. is not controllable because of si A B { for all s 0 for s 0. s 0 s Remark 3.: Obviously if the matrix B B D has full row then the concatenated serial system A B C D is controllable. A sufficient condition is obtained in the case where SpecA SpecA. Spectrum of A i SpecA i is the set of eigenvalues of matrix A i for i. Proposition 3.: Let A B C D and A B C D be the systems with SpecA SpecA. If the pairs A B

5 and A B are controllable and the system A B C D is functional output controllable then the serial concatenated system is controllable. Proof: For all s / SpecA SpecA we have that B C si n A + n so B B C si n A B D + n. Theorem 3.: A systematic serial controllability concatenated system of A i B i C i D i i is controllable if and only if. siδ A 0 B B C si δ A B D + n s C We observe that this result coincides with the case of serial concatenation. Then all results about controllability of serial concatenation are valid for systematic serial concatenation and vice-versa. If s 0 SpecA taking into account that SpecA SpecA we have that s 0 I n A then n A B B C s 0 I n A B D n A B B C B D s 0 I n A s 0 I n A B + n. But taking into account that A B is controllable we have s 0 I n A B. Then n A B B C s 0 I n A B D +n. Finally if s 0 SpecA we have that s 0 I n A then n A B B C s 0 I n A B D n + s 0 I A B In 0 C s 0 I A B + D a for all s 0 SpecA and knowing that A B is controllable and A B C D is functional output-controllable we have that a + n. Remark 3.: Example 3 shows that if C s 0 I n A B + D has not full for all s 0 SpecA the result 3. is not true. For systematic serial concatenated systems the controllability character can be described in terms of the both systems using the Hautus test in the following manner. ISSN: B. Controllability of parallel composite systems As in the serial concatenated case we will try to characterize the observability of serial concatenated systems in terms of the original systems. Theorem 3.3: The parallel concatenated system A B C D is controllable if and only if the following matrix B ni n A B + n s C. Proposition 3.: A necessary condition for controllability of parallel concatenated system is that the pairs A B and A B are controllable Unfortunately this condition is necessary but not sufficient. As we can see in this particular case: Example 3.3: Let A i B i C i D i i be two systems with 0 A B 0 0 C D A B 0 0 C D 0. 0 The pair A B is controllable because B A B 0 and A B is also controllable because 0 B A B However the parallel concatenated model

6 0 0 0 A B C D is not controllable because B AB A B A 3 B < 4 A sufficient condition is obtained in the case where SpecA SpecA. Proposition 3.3: Let A B C D and A B C D be two systems such that SpecA SpecA. If the pairs A B and A B are controllable then the parallel concatenated system is controllable. Proof: For all s / SpecA SpecA we have that sin A + n so B si n A B + n. If s 0 SpecA taking into account that SpecA SpecA we have that s 0 I n A then n A B s 0 I n A B n A B B s 0 I n A s 0 I n A B + n. But taking into account that A B is controllable we have s 0 I n A B. Then n A B s 0 I n A B + n. Analogously if s 0 SpecA we have that s 0 I n A then n A B s 0 I n A B + n for all s 0 SpecA. Remark 3.3: Example 3.3 shows that if SpecA SpecA the result 3.3 is not true. ISSN: IV. OBSERVABILITY OF COMPOSITE SYSTEMS A. Observability of serial composite systems The serial observability concatenated character is obtained from the Hautus test: Theorem 4.: si n A 0 B C si n A + n s C D C C Corollary 4.: A necessary condition for observability of concatenated system is that the pair A C be observable. Corollary 4.: A necessary condition for observability of concatenated system is that the pair A C with C B C be observable. D C Corollary 4.3: A necessary condition for observability of concatenated system is that the pair A C be observable. Proof: si n A B C D C I n B D si n A C C min I n B si n A D C C Then si n A B C n C. D C In this specific case a sufficient condition is obtained after observation of the spectrum of both matrices A and A. Proposition 4.: Let A B C D and A B C D be two systems with SpecA SpecA. If the pairs A C and A C are observable with B having full column then the serial concatenated system is observable. Proof: Let us suppose that we have: s / SpecA SpecA ; then 0 B C si n A + n s C

7 which means that si n A 0 B C si n A + n s C D C C Let us consider s 0 SpecA ; knowing that SpecA SpecA then s 0 I n A. Then s 0I n A 0 B C s 0 I n A D C C n + n A C + n since A C is observable. Let us consider s 0 SpecA ; knowing that SpecA SpecA then s 0 I n A. s 0I n A 0 B C s 0 I n A D C C Then n + s 0I n A B C D C If we suppose that B has full column we can see that: s 0I n A B C D C I n B s 0I n A D s 0 I n A C 0 D C C C since A C is observable Which means that: si n A 0 B C si n A + n D C C s C when SpecA SpecA. In the systematic serial concatenated system A B C D obtained from the systems A B C D and A B C D the observability concatenated character is obtained from the Hautus test: ISSN: Theorem 4.: The systematic serial concatenated system A B C D is observable if and only if the following relation holds. si n A 0 B C si n A C 0 + n s C D C C After this theorem it is obvious the following sufficient condition for observability of systematic serial concatenated system. Corollary 4.4: A sufficient condition for observability of systematic serial concatenated system is that the serial concatenated system is. Proof: n + n si n A 0 B C zi n A D C C si n A 0 B C si n A C 0 n + n. D C C We have the following result. Theorem 4.3: A necessary and sufficient condition for observability of systematic serial concatenated system is that both systems are observable. Proof: si n A 0 B C si n A C 0 D C C si n A 0 C 0 B C si n A. D C C Example 4.: Let us consider the systems A B C D A B C D two systems to be concatenated in a serial systematic form and A B C D and A B C D as follows: A B C 0 D 3

8 and A 3 0 B C D 0 4 Then both systems are observable; C C A C A C 0 4 C A C A The concatenated system A B C D is observable because: A B C D C and Then the systematic serial concatenated matrix is observable. The example has been computed using Matlab. B. Observability of parallel composite systems In parallel concatenated model A B C D obtained from the systems A B C D and A B C D the observability matrix is: C. n +n C C C A C A C A. C A C A n +n C A n +n 0 So we have the following theorem. Theorem 4.4: The parallel concatenated system B C D is observable if and only if the matrix 0 has full. Using the Hautus test we have Theorem 4.5: The parallel concatenated system A B C D is observable if and only if the following matrix si n A 0 0 si n A + n s C C C which means that: Proposition 4.: A necessary condition for observability of parallel concatenated system is that the pairs A C and A C are observable. Nevertheless this condition is not sufficient as we can see in the following example. Example 4.: Let A B C D be the parallel concatenated system of the following realizations. The first system is A B C D with A B 0 0 C 0 and 0 0 D and the second one is the system A B C D with 0 A 0 B C 0 D 0. Both systems are observable: C C A C A 0 ISSN:

9 C C A C A 3 However the parallel concatenated model is not observable: < Proposition 4.3: Let A B C D and A B C D be two systems where SpecA SpecA. If the pairs A C and A C are observable then the parallel concatenated system is observable. Proof: Analogous to 3.3. [0] M.I. García-Planas El M. Souidi L. E. Um Convolutional systems under linear systems point of view. Analysis of outputcontrollability. Wseas Transcations on Mathematics 4 pp [] M.I. García-Planas El M. Souidi L. E. Um Convolutional Codes under Control Theory Point of View. Analysis of Output- Observability. Recent Advances in Circuits Communications and Signal Processing. Proceedings of the th International Conference on Signal Processing Robotics and Automation ISPRA 3 03 pp [] M.I. García-Planas S. Tarragona Analysis of functional outputcontrollability of time-invariant singular linear systems. Proceedings of Cedya 03. [3] A. Germani S. Monaco Functional output-controlability for linear systems on hilbert Systems & Control Letters 5 pp [4] M. Hautus Controllability and observability condition for linear autonomous systems Proceedings of Nedderlandse Akademie voor Wetenschappen Series A 7 pp [5] P. Kundur Power System Stability and Control McGraw-Hill New York 994. [6] Yan Wu Controllability and Observability of Matrix Differential Algebraic Equations International Journal of Circuits Systems and Signal Processing. 5 3 pp V. CONCLUSION In this paper a sufficient condition for controllability and observability characters of serial and parallel concatenated finite-dimensional linear continuous-time-invariant systems are obtained. REFERENCES [] P. Campillo A. Devesa V. Herranz C. Perea Modelization of Turbo Encoder from Linear System Point of View. Proceedings of the 0th International Conference on Computational and Mathematical Methods in Science and Engineering CMMSE 00. [] F. Cardetti M. GordinaA note on local controllability on lie groups Systems & Control Letters 57 pp [3] P. Chaupa J. Novák P. Januška State Space MPC Using State Observers International Journal of Circuits Systems and Signal Processing 8 pp [4] C. Chen Introduction to Linear System Theory Holt Rinehart and Winston Inc. New York 970. [5] J. Domínguez-García M. García-Planas Output controllability analysis of fixed speed wind turbine in: 5th Physcon 0. [6] P. Ferreira On degenerate systems International Journal of Control 4 4 pp [7] M.I. García-Planas Holomorphic generic families of singular systems under feedback and derivative feedback International Journal of Mathematical Models and Methods in Applied Sciences pp [8] M.I. García-Planas J. Domínguez-García Alternative tests for functional and pointwise output-controllability of linear timeinvariant systems. Systems & Control letters. 6 5 pp [9] M.I. García-Planas J. Domínguez-García B. Mediano-Valiente Input observability analysis of fixed speed wind turbine. Mathematical Modelling and Simulation in Applied Sciences. Proceedings of the 3rd International Conference on Energy Environment Devices Systems Communications Computers IN- EEE. Rovaniemi Finland April 8-0 pp Maria Isabel Garcia-Planas joined the Department of Applied Mathematics at the Universitat Politecnica de Catalunya Barcelona Spain in 98. Her work had been centred on Linear Algebra Systems and Control Theory. She has authored over a hundred papers and serves on the referee on several Journals and International Conferences. José Luis Domínguez-García works in the Catalonia Institute for Energy Research Sant Adrià de Besòs Barcelona Spain. His reserach had been centred on coordinated control of Hybrid AC- MultiTerminal DC grids for offshore wind power transmission and Control Theory. Laurence E. Um studies in the Département d Informatique at the Unoversit e Mohammed V Rabat Morocco. Her Phd reserach had been centred on Codes Theory Systems and Control Theory. ISSN: