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Онлайн-доступ к журналу: http: / / mathizv.isu.ru
Серия «Математика»
2019. Т. 30. С. 31-44
УДК 517.977
MSG 93D15, 93С05, 49J15
DOI https://doi.org/10.26516/1997-7670.2019.30.31
Stabilization of Coupled Linear Systems Via Bounded Distributed Feedbacks
N. M. Dmitruk
Belarusian State University, Minsk, Republic of Belarus
Abstract. This article deals with a stabilization problem for a team of linear interconnected systems via bounded feedbacks. Effective approaches to stabilization of constrained systems from model predictive control theory are developed for the decentralized case when each system of the group is controlled by its local controller. We propose formulations of local optimal control problems and an algorithm based on them that constructs a distributed feedback guaranteeing asymptotic stability of the group.
Keywords: stabilization, feedback, distributed control.
In recent, years, control problems for teams of interacting dynamical systems has received a significant attention from the research community [6; 7], which is motivated by a large amount of practical applications — these are control problems for teams of mobile robots, unmanned aerial vehicles, energy systems, transport systems, etc. In such applications classical control theory methods may not be applicable, since they assume centralized control of the whole team, often representing a large-scale system. Besides, they do not account for networked or communication restrictions (e.g., delays in the communication between systems) within the team. In these cases distributed control techniques are needed.
One approach to tackle stabilization problems, popular in theoretical research and in practice, is Model Predictive Control [15] (MPC), and Distributed Model Predictive Control (DMPC) [10; 11] for interconnected systems. Within the DMPC framework many approaches have been proposed for systems with coupled dynamics [9; 16] and multi-agent, systems
1. Introduction
[13; 14]. The underlying idea is to break a large-scale control problem into sub-problems (local problems) where only inputs of the local system are optimized. For multi-agent systems a stabilization problem is most studied [11; 13], however, other control objectives, such as consensus and synchronization, are also of great practical interest (see e.g. [14]).
In this paper we consider a stabilization problem for a team of linear time-invariant systems with coupled dynamics subject to delays in communication between the systems. The goal is to achieve asymptotic stability [2] of the team via distributed feedback control. The proposed algorithms develop ideas of centralized model predictive control methods based on linear programming [8; 15], related stabilization methods based on optimal damping problems [1; 5] and distributed feedback control schemes for optimal control problems developed in [3; 4; 12]. The focus is on constructing local optimization problems and analyzing information to be communicated between the systems in order to establish a rather small amount of data that is sufficient for the algorithm implementation.
2. Problem formulation
We consider a team of q linear time-invariant control systems with coupled dynamics of the form
±i = AiXi + Aiixi + BiUi> = Xi0> (2-1)
where Xi = Xi(t) € Rrai denotes the state, Ui = Ui(t) € Rri denotes the control of the i-th system at time t, i € I = {1,2, ...,<?}, Ai = An, Bi, Aij, j € U = I \ {¿}, i € I, are given matrices of respective dimensions. The matrix Ai characterizes system's self-dynamics, Bi is the input of the i-th system, the matrices A^ characterize dynamical coupling between the systems in the team.
As feasible inputs we use sampled-data functions Ui(t), t > 0, with the sampling time h > 0: Ui(t) = Ui(s), t € [s, s + h[, s € T^ = {0, h,...}.
Along with the team of systems (2.1) we consider its representation
x = Ax + Bu, x(0) = xo, (2.2)
where x = (xj,... ,x^)T € Rn, u = (uj,... ,v,g)T € Rr, A = (Aij,i,j € I), B = diag(Bi, i € I) is block diagonal, n = Y^i&in»> r = Y^i&i rt-
Let £>(j) = (0,..., Bj, ... 0)T € RraXi-i. In the following we assume that for each i € I the pair (A, B^) is controllable in the class of sampled-data inputs.
A function u(x), x € Rra, is called a discrete feedback if for each xq the trajectory x(t), t > 0, of the closed-loop system
x = Ax + Bu(x), x(0) = xo, (2.3)
is a recursive solution of the linear equation x = Ax + Bu(x(s)), x(s) = x(s — 0) (x(—0) = xo), on the intervals t € [s,s + h[, s € T^. Obviously, the closed-loop system (2.3) has a unique solution.
Let D denote a region around the origin x = 0, and let L > 0 define a feasible input set in the form IJ = {u £ Rr : ||«||oo < L}.
Definition 1. [1] A discrete feedback u(x), x £ D, is called a bounded stabilizing discrete feed,back for (2.2) if: 1) u(x) £ U, x £ D; u{0) = 0; 2) for every xq £ D the states x{s), s £ Th, of system (2.3) stay in D; 3) the trivial solution x(t) = 0, t > 0, is asymptotically stable in D.
Obviously, the feedback u(x), x £ D, with properties l)-3) is not uniquely defined. In this paper we consider two approaches. In Section 3 we review centralized stabilization based on MPC methods [8; 15] and related works [1; 5]. In Section 4 we propose a new approach to distributed stabilization which combines the methods from [1;5] and ideas from [3;4; 12].
Throughout this paper the following notations are used: x(t\\to, xo, «(•)) denotes the state at time instant t\ of system (2.2) with the initial condition x(to) = Xo and input u(-) = (u(t),t £ [io,ii]); F(t) = eAt, t > 0; Fi(t) £ IPnxni ¿eno^es corresponding block of the matrix F(t), i.e. F(t) =
(F\ (t),.. .,Fq(t))-, D{s) = F(tf -s-h) Jq F(t)Bdt, A(s) = F(tf - s -h) Jo Fi(t)Bidt-, 1 p — (1,..., 1) € Kp. The following norms of the vector y £\Rp are used: ||y||co = max{|yi|,..., \yp\}, \\y\\2Q = yTQy, Q > 0.
3. Centralized stabilization
In the centralized stabilization case the team has a single central controller, which, based on the team's current state, chooses a control input for all systems in the team. In what follows the team's current state is denoted by x*(t) and is assumed to be available for complete and accurate measurements. We stress that it may differ from the states of the mathematical model (2.2) due to inaccuracies of mathematical modeling, presence of disturbances, and other uncertainties.
As discussed in the introduction, a popular approach to solving stabilization problems is MPC. The overall idea of all MPC methods is based on repetitive solution at each current discrete time instant t £ th the so-called predictive optimal control problem subject to a finite time interval [0,t/] (tf = Nh, N £ N), and the initial condition for the predictive model (2.2) coinciding with the current state x*(t). The general formulation of the predictive problem (in the centralized case) has the form
J°(x*(t))= min J(u),
x = Ax + Bu, x(0) = x*(t), (3.1)
x(tf) £ Xf, u(t) £ U, t £ [0,tf],
where J (и) is some cost; Xf с D is the terminal set, {0} € Xf. Problem (3.1) may also include path constraints if those are imposed on the transient trajectories.
Let и°(£|ж*(т)), t € [0, tf], denote the optimal open-loop input of problem (3.1).
The MPC algorithm can be described as follows: at each time instant r € T/j, the controller solves the optimal control problem (3.1) and feeds the first value и°(0|ж*(т)) of its optimal open-loop input to the team on the interval [т,т + h[. As a result, we obtain the so-called discrete feedback realization
u*(t) =и°(х*(т)) :=u°(0\x*(t)), t £[t,t + h[, reTh.
Asymptotic stability of the closed-loop system is achieved by a proper choice of the cost J(u) of the predictive problem (3.1) and a suitable terminal condition at time instant tf. Significant attention in the literature is given to MPC schemes with quadratic costs J(u) of the form
J(u)= j '\\x(t)\\Q + \Ht)\\2Rdt+\\x(tf)\\2P, Q,R,P> 0, Jo
and terminal sets Xf being ellipsoids for wich there exists a local linear feedback щос(х) = Kx € U, x € Xf, such that A + В К is a Hurwitz matrix. This approach allows us to reduce problem (3.1) to a quadratically constrained quadratic program and to solve it efficiently, e.g. using the interior-point methods. Simplest approaches (see, e.g. [15]) use the terminal constraint x(tf) = 0, i.e. Xf = {0}.
For linear systems a linear cost of the form
J(u)= [f \\Qx(t)\\oo + \\Ru(t)\\oo dt + ||Рж(£/)||оо Jo
is popular, and allows to reduce the predictive problem (3.1) to a multi-parametric linear program [8].
In this paper bounded stabilizing feedbacks are constructed according to the approach in [1], where we use
J(u) = max Mt)\\oo, Xf = {0}. t&[o,tf]
Hence the predictive problem is the optimal damping problem. For an arbitrary initial state z € Rra it has the form
V{z) : p(z) = min max ||«(i)||oo) (3.2)
u t&[o,tf]
x = Ax + Bu, x(0) = z, x(tf) = 0.
Under the controllability assumption of Section 2 and for a control horizon tf > nh/r every problem P(z) of family (3.2) has a solution that is denoted by u0(t|z), t € [0,tf].
Let D = {z € Rn : p(z) < L}. In the following we assume that x0 € D. In [1;5] the following result is proved
Proposition 1. A function u0(x) = u0(0|x), x € D, is a bounded stabilizing feedback for system (2.2).
Algorithm 1 (centralized stabilizing control construction):
1) Set t = 0, x*(t) = x0.
2) Find an optimal open-loop input u0(t|x*(T)), t € [0,tf], to the problem P (x* (t )).
3) Apply input u*(t) = u0(x*(t)) := u0(0|x*(t)), t € [t,t + h[ to system (2.2).
4) Set t := t + h, return to Step 2).
The algorithm for centralized stabilization is specified as Algorithm 1. Now we briefly discuss how the problem P(x*(t)) is solved during Step 2) of Algorithm 1. In the class of sampled-data inputs it is equivalent to a linear program
p*(t) = minp, Du = —g(x*(t)), u — pirn < 0, —u — pirn < 0, (3.3)
p,u
where p € R, u = (uT(0),uT(h),...,uT(tf — h))T € RrN, g(z) = F(tf)z; D = (D(0), D(h),... , D(tf — h)) € RnxrN.
The linear program (3.3) has rN+1 variables, n equality constraints and 2rN inequality constraints. The paper [1] proposes to reduce the problem dimension using the following change of variables: £ = 1/p, v = u/p (p > 0 for x* (t) = 0). This results in a linear program
£ * (t ) = max g(x* (t ))£ + Dv = 0, ||v|U < 1, (3.4)
with rN +1 variables, n equality constraints and geometric constraints for the variables v. Now we have p* (t) = 1/£*(t). Since problem (3.2) has to be solved at each t € Th for the current state x*(t) in time less than h, solving problem (3.4) is preferable to solving problem (3.3).
Using the optimal damping problem (3.2) for predictions as compared to the classical MPC approaches allows us to propose a rather simple approach to distributed feedback control, which is described in the next section.
4. Distributed stabilization
In the distributed stabilization case it is assumed that each system (2.1) has its own (local) controller that generates a (distributed) bounded stabilizing feedback only for the associated system. The local controller constructs the inputs as in centralized case, i.e. on the base of solution of the (local) predictive optimal control problem. This problem is denoted by Vi(xi, z), where Xi is the state of the г-th system, z is information on other systems' behavior. In the particular control process it is assumed that by the current time instant r € T^ \ {0} information arriving from systems к € U consists of 1) the state х*к(т — h), and 2) the input и*к(т — h) applied to system к at the previous time т — h.
The above assumption means that communication is delayed by one sampling period h, and the current position of the control process, as it is available to the г-th controller at time instant r, is (ж*(т), z*(r)), where z*(r) = {xt(T-h),ut(T-h),keI}.
The solution of the problem Vi(r) := Vi(x*(т), z*(т)) is denoted by uf(t\r), t € [0,tf], and is referred to as the local optimal open-loop input of system г predicted at time r. For every r we define an overall open-loop input ud{t\r) = к € I), t € [0,tf], as an input composed of all
local optimal open-loop inputs.
Following [3] and taking into account the centralized problem V(x*(t)) formulation, the local predictive problem Vi(r) of the г-th controller at time r € Th \ {0} is formulated in the form
Vi{r) : pi(r) = min max ||«i(i)||oo, (4.1)
щ te[o,tf]
Xi — AiXi / A-ijXj J3iUi,
^J&h J J
xk = Akxk + V . Akjxj + Bkvdk{t + h\r -h), к €
Xi{0) = х*(т), xk(0) = хлк(т\т - h), к € h, x{tf) = 0.
In problem (4.1) the optimization variable is the input щ, and the inputs uk of all other systems к € h are held as fixed parameters equal to their local optimal open-loop inputs uf(t\r — h), t € [h,tf], of problems Vk(T — h), predicted at the previous time r — h and assumed trivial on the intervals [tf,tf + h]: uk(t\r — h) :=0,te [tf,tf + h], к € h-
The initial state of the г-th system in problem (4.1) is its current state x*(t). For the initial states of all other systems к £ U we use the components хк(т\т — h) of the state хл(т\т — h)= x(t\r — h, х*(т — h),u*(T — h)) of system (2.2) with the initial condition x{r — h) = x*(r — h) and the input u(t) = и*(t - h), t £ [t - h,r].
The proposed formulation (4.1) has a drawback: to form the constraints the controller of the г-th system needs to know the local optimal open-loop
inputs uf(t\r — h) on the whole interval [0,t/]. In the following we show that information communicated during the control process can be reduced to the data z*(r) chosen above.
We note that the terminal state x(tf) of system (4.1) can be represented as a sum
x(tf)=x(tf) + Y,keIiydk(r), (4.2)
where
yf(r) = Fk(tf)xdk(T\T - h) + f f Fk(tf- t)Bkudk(t + h\r - h)dt, kel,
Jo
and x(tf) is the terminal state of the following system
Xi = AiXi + ^ ^ ^ A-ijXj + BiUi, Xi(0) = Xi (t), xk = Akxk + V . Akjxj, xk(0) = 0, k € h.
It is easy to establish that for u*(t — h) = ud(0\r — h) the following equality holds
Efce/^(r) = F(hMtf\0,x*(r - h),ud(-\r - h)). (4.3)
Let t = h, and ud(t\0) = u°(t\x0), t € [0,tf}. Obviously,
J2vi(h) = F(h)x(tf\0,x*(0),ud(-\0)) = 0, fee/
since ud(-|0) is the solution of the centralized problem V(xq).
Assume that for some t € th\ {0,h} we have J2keiyi(T) = Then (4.2) takes the form x(tf) = x(tf) — yf{r) that allows us to reformulate the problem Vi(r) in the equivalent form
Vi{r) : pi(r) = min max ||Uj(i)||oo, (4.4)
t€[0,t/]
Xi — AiXi / AijXj B^i,
^J&h J J
xk = Akxk +yi.T
AkjXj,
*—'j&ik
xm = x*{t), xk(0) = 0, x(tf) = yd{r). The following equalities hold x(tf\0,x*(r),ud(-\r)) = Y/keIx(tf\0,x*k(r),udk(-\T)) = Efce/^(r) =°-Hence, from (4.3) we have J2k£iyk(T + h) = 0.
Summarizing, we have established that 1) the overall open-loop input ud{t\r), t € [0,t/] steers system (2.2) from x(0) = x*{t) to the origin in time tf, i.e. it is feasible in the centralized problem V(x*(t)).
2) For all t eth\ {0} the equality T,k€ivk(t) = 0 holds-
3) The local problem may be formulated as (4.4).
Obviously, to form problem (4.4) the i-th controller needs the states x*k{r — h) and applied inputs u*k{r — h) from other systems k € which in turn composes information z*(r).
When solving problem (4.4) numerically, one has to reduce it to a linear program
ii{r) = ma^(r)^ + Avj = 0, IH^ < 1, (4.5)
where & € R, v* = (vf(0),vf(h),..., vj(tf - h))T € RriN, A = (A(0), Di{h),..., Di(tf - h)) e RnxriN,
g%{t)=Fl{tf)x*{t)-yi{t).
Note that (4.5) has ViN +1 variables and n equality constraints, and its dimension does not depend on the number of systems in the team.
The algorithm for distributed stabilization is specified as Algorithm 2. The algorithm constructs the distributed feedback ud(x, z) = (uf(xi, z), i € I) as a function of a position (x,z), and its realization in a particular control process
u*(t)=uf(x*(T),z*(T)):=uf(0\T), te[T,T + h[, TGTh, iel. (4.6)
Algorithm 2 (distributed stabilizing control construction):
1) Set r = 0, x*(t) = xq.
2) Find a solution u°(t\xo), t € [0,tf], to the centralized problem P{xo). For each к € I set uf(t\0) = u°k(t\x0), t € [0,i/].
For each system i € I (in parallel):
3) Apply input u*(t) = uf(x*(T),z*(T)) := ^(0|r), t € [r,T + h[.
4) Communicate х*(т), и*(т) to all systems к £ U.
5) Set r := r + h, and obtain current state measurement ж*(т).
6) Solve problem (4.4) and find uf(t\r), t € [0,i/]. Return to Step 3).
Proposition 2. A distributed feedback ud(x, z) is a bounded stabilizing feedback for the overall system (2.2) with the region of attraction D.
Proof. 1. We need to establish that an input
(ud(t + h\r-h), ^[O/f- Я (4.7)
is a feasible open-loop input of the problem Vi(r). Indeed, since system (2.2) is not affected by disturbances, we have xd(r\t — h) = x*(t). By construction of yf(r) in problem (4.4), the input Ui(t), t € [0,t/[, satisfies the terminal constraint, which means that Vi(r) is feasible.
2. Function (4.7) in the problem Vi(r) has the cost equal to pi(r — h) that is not less than the optimal value pi(r) of this problem. Therefore, for alHe/ we have pi(r) < pi(r — h).
3. The cost of the cenralized problem V(x*(t)) at the input ud(-\r) = {udk{-\r),k € I) equals to pd{r) = max^;PfcCr), that yields pd{r) < pd(r — h), r = 2h, 3h,...; pd(h) < p(x0).
This implies that 1) any trajectory of system (2.2) with xq € D and input (4.6), i € I, stays in D, and 2) u*(t) € U.
4. Following [1], we show that if x*(t) / 0 then pd(r + i/) < pd(r), i.e. at least after N steps of Algorithm 2 we obtain a strict decrease of the cost of the problem V(x*(t)). Assume the opposite: for any I = 1,N the equality pd{r) = pd(r + Ih) holds. Then there exists io € / such that pd{r) = pf (t) = pf (t + Ih), I = 1, N, and the optimal open-loop input in the problem 7\>(t + Ih) is a function Ui0(t) = uf (t + lh\r), t € [0, tf — lh[, uio (t) = 0, t € [tf — Ih, tf[. For the time instant r + tf we obtain the optimality of the trivial input, which implies that pd{r) = pd(r + tf) = pf0(r + tf) = 0, and therefore, x*(t) = x*(t + tf) = 0. Further proof repeats the arguments given in [1]. □
5. Examples
To illustrate and compare the stabilization algorithms based on centralized and distributed feedbacks we consider two examples.
Example 1. Consider the team of two systems:
¿1=1 11 ft ] + I i ft Va+foVi' i5-1)
x-11 0 J 1 V -1 0 x
¿2 = ( 25 J )«2 + ( _0°25 o + ( 0°5
Choose the following parameters: L = 2, tf = 3, h = 0.1. As initial states at time instant r = 0 we choose £i(0) = x\o = (—1,0)T, «2(0) = ^20 = (0.5,0)T. We verify that the initial state xq = (xf0,X20)T of system (5.1) is in the domain D by solving the problem V(xq), and obtaining p(x0) = 1.52197255 < L.
On the base of the predictive problem P(x*(t)), r € T^, Algorithm 1 constructed a realization of the centralized stabilizing feedback. On the
Figure 1. Trajectories and realizations of stabilized and distributed feedbacks in
example 1
base of two predictive problems V\ (r), V2 (т), т € Th, Algorithm 2 constructed a realization of the distributed stabilizing feedback for the example under consideration.
Figure 1 presents fragments (for 0 < t < 20) of the trajectories and feedback realizations. Solid lines correspond to centralized solution and dash lines correspond to distributed solution.
For the centralized stabilization the neighborhood ||ж*(т)|| < 10-5 was reached by the time instant r = 37.5, by that time the total control impulse was 8.3651. For the distributed stabilization the same neighborhood was
2
cs
H 0
l^jUv--
10
•H 0
10 5
S 0
0 5 10 15 0 t 10 m^Vw-0 -10 5 10 15 0 t - 0 -10 1 1 1 -l 5 10 15 0 t ■«? 0 5 10 15 0 t ifffi^/^ -5 5 10 15 t
0 5 10 15 0 t 10 5 S 0 [' 5 10 15 0 t 10 5 S 0 -5 5 10 15 0 t 10 -51 S 0 -5 5 10 15 0 t 10 5 pím^— £ o -5 1 5 10 15 t
0 5 10 15 0 5 10 15 0 5 10 15 0 5 10 15 0 5 10 15
t t t t t
Figure 2. Trajectories and realizations of stabilized and distributed feedbacks in
example 2
rH 0
H
reached faster, by the time instant t = 28.5. This is due to the control inputs of greater amplitude. By the time instant t = 28.5 the total control impulse was already 8.9213.
Example 2. Consider a team consisting of five coupled oscillating systems:
x1 = —2kx1 + kx2 + ui, (5.2)
xi = —2kxi + kxi-1 + kxi+1 + u, i = 2, 3, 4, x5 = —2kx5 + kx4 + u5.
We choose the following parameters' values: k = 10, L = 10, tf = 3, h = 0.1. We assume that at time t = 0 all systems are stationary in different states: xi(0) = —4, x2(0) = 3, xa(0) = 0, x4(0) = 2, x5(0) = —3. The initial state belongs to the region D since p(x0) = 8.6621 < L.
Figure 2 presents fragments (for 0 < t < 15) of the trajectories and feedback realizations. In the centralized control process team (5.2) reached the neighborhood ||x*(t)|| < 10-5 by the time instant t = 42.2, the total input impulse was 125.7846. In the distributed control process same neighborhood was reached in comparable time, by the time instant t = 42.3, however, the total impulse of the distributed inputs was 162.9104.
n. m. dmitruk 6. Conclusion
This paper proposes an algorithm for constructing distributed bounded stabilizing feedbacks for a team of linear time-invariant coupled systems. The algorithm is based on a parallel solution at each time instant of local predictive optimal control problems associated with each system in the team and having lower dimensions compared to the centralized predictive problem. Local problem's solution yields the feedback realization only for the respective system. The proposed algorithm guarantees feasibility of all local problems during the control process, communication of a small amount of data, and asymptotic stability of the overall team.
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Natalia Dmitruk, Candidate of Sciences (Physics and Mathematics), Associate Professor, Belarusian State University, 4, Nezavisimosti av., Minsk, 220030, Republic of Belarus, tel.: +375 (17) 2095074, e-mail: dmitrukn@bsu.by, ORCID iD https://orcid.org/0000-0003-1845-4927
Received 29.10.19
Стабилизация линейных взаимосвязанных систем ограниченными децентрализованными обратными связями
Н. М. Дмитрук
Белорусский государственный университет, Минск, Республика Беларусь
Аннотация. Рассматривается задача стабилизации группы линейных взаимосвязанных систем ограниченными обратными связями. Эффективные подходы к стабилизации при наличии ограничений из теории управления по прогнозирующей модели развиваются на децентрализованный случай, когда каждая система группы управляется своим локальным регулятором. Предлагаются формулировки локальных задач оптимального управления и основанный на них алгоритм, который строит децентрализованную обратную связь, обеспечивающую асимптотическую устойчивость группы.
Ключевые слова: стабилизация, обратная связь, децентрализованное управление.
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Наталия Михайловна Дмитрук, кандидат физико-математических наук, доцент, факультет прикладной математики и информатики, Белорусский государственный университет, Республика Беларусь, 220030, г. Минск, просп. Независимости, 4; тел.: +375 (17) 2095074, e-mail: dmitrukn@bsu.by, ORCID iD https://orcid.org/0000-0003-1845-4927
Поступила в редакцию 29.10.19
