DIGITAL STABILIZATION OF LINEAR CONTINUOUS-TIME PERIODIC PROCESSES WITH PURE DELAY B.P. Lampe E.N. Rosenwasser University of Rostock, Department of Computer Science and Electrical Engineering, D-18051 Rostock, Germany, Email: bernhard.lampe@uni-rostock.de State Marine Technological University, Institute of Automation, Lotsmanskaya str. 3, 190008 Saint Petersburg, Russia, Email: k10@smtu.ru Abstract: : The paper considers the stabilization problem for linear continuoustime T -periodic processes with pure delay by a T -periodic digital controller. Necessary and sufficient conditions for the stabilizability are formulated, and an algorithm for constructing the set of stabilizing controllers is provided. Its application is demonstrated by a simple example. Keywords: Periodic processes, Sampled-data control, Time delay, Stabilizing controllers, Parametrization 1. INTRODUCTION The stabilization problem for continuous LTI processes without pure delay by sampled-data controllers for a non-pathological sampling period T has been considered in (Francis and Georgiou 1988). A polynomial solution for continuous LTI processes and arbitrary sampling period T is given in (Lampe and Rosenwasser 2003). The digital stabilization problem for continuous LTI processes with pure delay is considered in (Lampe and Rosenwasser 2006). The case of continuous T -periodic processes without delay is studied in (Lampe and Rosenwasser 2001). The present paper extends the results from (Lampe and Rosenwasser 2001, Lampe and Rosenwasser 2006) to systems of with pure delay. The main results are formulated in a series of theorems which are given without proofs due to limited space. These proofs succeed by using the ideas in (Lampe and Rosenwasser 2001, Lampe and Rosenwasser 2006). 2. SYSTEM DESCRIPTION AND PROBLEM FORMULATION 1. Consider the control problem for the linear T - periodic process described by the state equation dx dt = A(t)x(t) + B(t)u(t τ) (1) and the output equation y(t) = C(t)x(t). (2) In (1) and (2) u(t), x(t), y(t) are the input, state and output vectors of the dimensions m 1, p 1, n 1, respectively, and A(t) = A(t + T ), B(t) = B(t + T ), C(t) = C(t + T ) are continuous periodic matrices of dimensions p p, p m, n p, respectively. Moreover, in (1) the symbol τ denotes a non-negative constant, determining the pure time-delay of the control signal. Assume the representation τ = dt + θ, (3)
where d is a non-negative integer and 0 θ < T. 2. Suppose that the control signal u(t) is defined by the relations u(t) = µ(t )ψ k, < t < (k + 1)T, (4) where µ(t) is a piecewise continuous function giving the shape of the control impulses, and T is the sampling period which coincides with the period of the process. Moreover, in (4) ψ k, (k = 0, ±1,...) is the control sequence consisting of number vectors. From (3) and (4), it follows u(t τ) = µ(t + T θ)ψ k d 1, < t < + θ, µ(t θ)ψ k d, + θ < t < (k + 1)T. (5) Therefore, state equation (1) can be written in the form As known from (Åström and Wittenmark 1997), only causal B-programs can be realized in real time. Therefore, we will always assume that (10) takes place. 4. Altogether, equations (2) and (6)-(8) build a system of differential-difference equations which is referred to as system S τ. Definition 1. As a solution of the system S τ, we consider the entirety of the continuous vector functions x(t), y(t) and the sequence ψ k, which for all t > 0, k > 0 satisfy equations (2), (6)-(8). Hereinafter, the symbol formulation. indicates the end of a Definition 2. A system S τ is called stable, if for any solution of equations (2), (6)-(8) and for all t > 0, k > 0 the estimates dx dt = A(t)x(t) + B(t)µ(t + T θ)ψ k d 1, < t < + θ, (6) dx dt = A(t)x(t) + B(t)µ(t θ)ψ k d, + θ < t < (k + 1)T. x(t) < c x e ρt, y(t) < c y e ρt, ψ k < c ψ e kρt (11) Below we always propose that the state vector x(t) is a continuous solution of equation (6). In that case, also y(t) is continuous. 3. Assume that the elements of the control sequence ψ k are connected with the output vector y(t) by the relation α 0 ψ k +... α q ψ k q = β 0 y k +... + β q y k q, (7) which is called equation of the backward control program (Åström and Wittenmark 1997), (Rosenwasser and Lampe 2006). Herein, α i, β i (i=0,...,q) are constant matrices of size m m and m n, respectively. In (7), we used y k = y( ), (8) which makes sense, because we assumed that the vector y(t) is continuous at the sampling instants. Introduce the polynomial matrices α(ζ) = α 0 + α 1 ζ +... + α q ζ q, β(ζ) = β 0 + β 1 ζ +... + β q ζ q. (9) At that the pair of matrices α(ζ), β(ζ) is called backward control program, or shortly B-program. The B-program is sometimes also named control algorithm or simply controller. The B-program α(ζ), β(ζ) is called causal, if det α(0) = det α 0 0. (10) are valid, where denotes any norm for number vectors or its associated norm for number matrices. Furthermore, c x, c y, c ψ and ρ are positive constants, where ρ does not depend on the concretely selected solution. Definition 3. A system S τ is called stabilizable, if there exists a causal B-program α(ζ), β(ζ), for which the system S τ becomes stable. In this case, α(ζ), β(ζ) is called a stabilizing B-program. 5. The backward stabilization (B-stabilization) of the system S τ proves to be a fundamental design problem. It can be formulated as follows. B(ackward) stabilization problem For given matrices A(t), B(t), C(t), period T, pure delay τ and form function µ(t), solve the following problems: a) Find the conditions under which for the system S τ, there exists at least one stabilizing B-program. b) Under fulfilled existence conditions, construct the set of all stabilizing B- programs. The present paper provides the general solution to the problems a) and b). So it extends the results from (Lampe and Rosenwasser 2001, Lampe and Rosenwasser 2006) to systems of type S τ.
3. DISCRETE BACKWARD MODEL OF SYSTEM S τ 1. Let H(t) be the p p matrix, satisfying the conditions dh(t) dt = A(t)H(t), H(0) = I p, (12) where I p is the p p identity matrix. Moreover, denote G(t) = H 1 (t). (13) Let M be the monodromy matrix defined by M = H(T ). (14) As known (Yakubovich and Starzhinskii 1975) and therefore, H(t + T ) = H(t)M, (15) G(t + T ) = M 1 G(t). (16) 2. Integrating equations (6) by using (5), we obtain t x(t) = H(t)H 1 ( )x k + + H(t)G(ν) B(ν)µ(t + T θ) dν ψ k d 1, x(t) = H(t)H 1 ( )x k + + t + θ, H(t)G(ν) B(ν)µ(t + T θ) dν ψ k d 1, t where the notation (17) H(t)G(ν)B(ν)µ(t θ) dν ψ k d, + θ < t (k + 1)T, x k = x( ) (18) was used. For t = (k +1)T, we find from (17) with the help of (15) x k+1 = Mx k + M k+1 G(ν) B(ν)µ(ν + T θ) dν ψ k d 1 (19) + M k+1 (k+1)t G(ν) B(ν)µ(t θ) dν ψ k d. A direct calculation, with the help of (16), leads to (k+1)t G(ν)B(ν)µ(ν + T θ)dν = M k M T G(λ + θ)b(λ + θ)µ(λ) dλ, G(ν)B(ν)µ(ν + T θ) dν = M k 0 G(λ + θ)b(λ + θ)µ(λ) dλ. (20) Insert (20) into (19) and rename k by k 1, to obtain x k = Mx k 1 + M 2 Γ 2 ψ k d 2 + MΓ 1 ψ k d 1, (21) where Γ 1 and Γ 2 are the constant matrices Γ 1 = Γ 2 = 0 T G(λ + θ)b(λ + θ)µ(λ) dλ, G(λ + θ)b(λ + θ)µ(λ) dλ. (22) Relation (21) is called the discrete backward model of the state equations. 3. From (2) for t =, we find the equation y k = Γ 0 x k, Γ 0 = C(0), (23) which is called discrete output model. 4. Subjoining control program (7) and (21)-(23), we obtain a system of difference equations x k = Mx k 1 + M 2 Γ 2 ψ k d 2 + MΓ 1 ψ k d 1, y k = Γ 0 x k (24) α 0 ψ k +... + α q ψ k q = β 0 y k +... + β q y k q, which is referred to as discrete backward (B) model of the system S τ, and it is denoted by S b. For det α 0 0, the system S b is normal in the sense of (Rosenwasser and Lampe 2006), therefore, it possesses for k > 0 a unique solution for arbitrary initial conditions ψ 0,..., ψ χ, y 0,..., y q, χ = max{q, d + 1}. (25)
Definition 4. A discrete B-model is called stable, if for arbitrary initial conditions (25) the corresponding solutions of equation (24) satisfy the estimates x k < d x e kηt, y k < d y e kηt, ψ k < d ψ e kηt, (26) where d x, d y, d ψ, η are positive constants, hereby η does not depend on the initial conditions. As in (Rosenwasser and Lampe 2006), it follows that the system S b is stable, if and only if the polynomial matrix Q(ζ, α, β) = I p ζm O pn ζ d+1 M(ζMΓ 2 + Γ 1 ) Γ 0 I n O nm, O mp β(ζ) α(ζ) (27) where O ik means the i k zero matrix, does not possess eigenvalues inside the unit disc or on its border. In what follows, such matrices of the argument ζ are called stable. Definition 5. The system S b is called stabilizable, if there exists a causal program α ρ (ζ), β ρ (ζ) such that the polynomial (ζ, α ρ, β ρ ) = det Q(ζ, α ρ, β ρ ) (28) becomes stable. Besides, we say that the program α ρ (ζ), β ρ (ζ) is B-stabilizing for the system S b. Analogously to (Rosenwasser and Lampe 2006), it can be shown that all B-stabilizing programs α ρ (ζ), β ρ (ζ) are causal. 4. SOLUTION OF B-STABILIZATION PROBLEM 1. The claims below provide the general solution of the B-stabilization problem formulated above. Theorem 1. The set of all stabilizing B-programs for the system S τ coincides with the set of all B- stabilizing controllers for the system S b. 2. Denote L(ζ) = ζ d+1 M(ζMΓ 2 + Γ 1 ). (29) Theorem 2. Let p(ζ) be the greatest common left devisor of the matrices I p ζm and L(ζ), and q(ζ) be the greatest common right devisor of the matrices I p ζm and Γ 0. Then for the stabilizability of the system S τ, it is necessary and sufficient that the matrices p(ζ) and q(ζ) are stable. 3. Introduce the constant matrix Γ 3 = T 0 G(λ + θ)b(λ + θ)µ(λ) dλ. (30) Another form of the stabilizability condition for system S τ provides the next claim. Theorem 3. For the stabilizability of system S τ, it is necessary and sufficient that the pair M, Γ 3 is stabilizable and the pair M, Γ 0 is detectable. 4. The next statement yields the construction of the set of all stabilizing controllers. Theorem 4. Let the system S τ be stabilizable. Then the set of all stabilizing B-programs can be constructed by the following algorithm (1) Build the rational matrix W (ζ) = ζ d+1 Γ 0 (I p ζm) 1 M(ζMΓ 2 + Γ 1 ).(31) (2) For the matrix W (ζ), find a MFD (Kailath 1980) W (ζ) = ζ d+1 a 1 (ζ)b(ζ), (32) where a(ζ), b(ζ) are polynomial matrices, such that for all ζ rank [ a(ζ) ζ d+1 b(ζ) ] = n. (33) (3) Find polynomial matrices α 0 (ζ), β 0 (ζ) satisfying [ ] a(ζ) ζ d+1 b(ζ) det = 1. (34) β 0 (ζ) α 0 (ζ) (4) The set of all causal stabilizing controllers are determined by the relations α(ζ) = D(ζ)α 0 (ζ) ζ d+1 N(ζ)b(ζ) β(ζ) = D(ζ)β 0 (ζ) N(ζ)a(ζ), (35) where D(ζ), N(ζ) are polynomial matrices, hereby N(ζ) is arbitrary but D(ζ) is stable. 5. B-STABILIZING TRANSFER MATRICES 1. The B-program α(ζ), β(ζ) is called nonsingular, if det α(ζ) 0. All stabilizing B- programs are non-singular and possess transfer matrices of the form W d (ζ) = α 1 (ζ)β(ζ). (36)
Definition 6. A rational m n matrix W d (ζ) is called B-stabilizing if it permits a representation of the form (36), where the pair α(ζ), β(ζ) defines a stabilizing B-program. The set of all B-stabilizing transfer matrices is determined by the following theorem. Theorem 5. Let the matrices α 0 (ζ), β 0 (ζ) satisfy relation (34). Then the set of all B-stabilizing transfer matrices can be represented in the form W d (ζ) = (37) [α 0 (ζ) ζ d+1 Φ(ζ)b(ζ)] 1 [β 0 (ζ) Φ(ζ)a(ζ)], where Φ(ζ) is any rational m n transfer matrix, free of poles inside the unit disc or on its border. 6. EXAMPLE 1. Consider the system S τ with the continuoustime process of first order dx(t) dt = sin t cos t 2 x(t) + u(t τ), (38) y(t) = x(t). In the case at hand, we have A(t) = Moreover, we suppose sin t cos t 2, B(t) = 1, (39) C(t) = 1, T = 2π. µ(t) = 1, 0 < t < T. (40) In the given case, we obtain and H(t) = 1, G(t) = 2 cos t (41) 2 cos t M = H(2π) = 1. (42) 2. Using (30) and (41), (42), we find Γ 3 = 4π 0. Due to Γ 0 = 1, we conclude with Theorem 3, that the system under investigation is stabilizable for all τ. 3. For constructing the set of stabilizing controllers, we apply Theorem 4. Besides, for simplifying the calculations, we assume d = 0, 0 < θ < T. Using (22), (39)-(42), we find Γ 1 = 4π 2θ sin θ, Γ 2 = 2θ + sin θ. (43) Hereby, from (31), we obtain W (ζ) = ζ(γ 1 + ζγ 2 ) 1 ζ. (44) Applying (43), it is easy to show that the fraction on the right side of (44) is irreducible. Therefore, we can choose a(ζ) = 1 ζ, b(ζ) = (Γ 1 + ζγ 2 ). (45) Now, relation (34) leads to the Diophantine equation (1 ζ)α 0 (ζ) ζ(γ 1 + ζγ 2 )β 0 (ζ) = 1. (46) A particular solution of equation (46) may be expressed in the form α 0 = 1 + ζ Γ 2 4π, β 0(ζ) = 1 4π. (47) According to (35), the set of all B-stabilizing programs can be represented as ( α(ζ) = D(ζ) 1 + ζ Γ ) 2 ζn(ζ)(γ 1 + ζγ 2 ), 4π (48) β(ζ) = 1 D(ζ) N(ζ)(1 ζ), 4π where N(ζ) is an arbitrary polynomial and D(ζ) is any stable polynomial. The set of B-stabilizing transfer functions, according to (37) has the form W d (ζ) = [ 1 + ζ Γ ] 1 2 4π ζφ(ζ)(γ 1 + ζγ 2 ) [ 1 ] Φ(ζ)(1 ζ). 4π 7. CONCLUSIONS (49) The paper states the stabilization problem for linear continuous-time T -periodic processes with pure delay by a T -periodic digital controller. Pathologic sampling is not excluded and the shape of the hold element is free. Necessary and sufficient conditions for the stabilizability of such systems are formulated using polynomial description. On basis of polynomial methods, an algorithm for constructing the parameterized set of all stabilizing digital controllers is provided. A simple example demonstrates the application of the results. ACKNOWLEDGEMENT The authors are grateful to the German Science Foundation (Deutsche Forschungsgemeinschaft) for financial support.
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