aa r X i v : . [ c s . S Y ] O c t SPARSITY METHODS FOR NETWORKED CONTROL .
MASAAKI NAGAHARA
Abstract.
In this presentation, we introduce sparsity methods for networkedcontrol systems and show the effectiveness of sparse control. In networkedcontrol, efficient data transmission is important since transmission delay anderror can critically deteriorate the stability and performance. We will showthat this problem is solved by sparse control designed by recent sparse opti-mization methods. Introduction
Sparsity methods, called compressed sensing or sparse representation , have beenrecently introduced in signal processing [4]. The methods are also applied to com-munications, see a survey paper [7].A sparse vector is a vector that has very few non-zero entries compared with thevector size. Compressed sensing takes advantage of sparsity property of signals insome domain (e.g. the Fourier domain), for efficiently reconstructing such signalsfrom very few measurements by finding a solution of underdetermined linear equa-tions. To solve such underdetermined linear equations, one can adopt an L -normminimization [3] to achieve the sparsity, or a matching pursuit to find a sparsesolution in a greedy way [8].Sparsity methods are very recently applied to solving problems in control sys-tems such as predictive networked control [9, 11], hands-off control [10], actuatorscheduling [1], and security in cyber-physical systems [5], to name a few. In thispresentation, we introduce sparsity methods for networked control systems anddiscuss the effectiveness of the sparse control.2. Problem Formulation
We here consider linear and time-invariant (LTI) plant models of the form(1) d x ( t ) dt = A x ( t ) + B u ( t ) , t ∈ [0 , T ] , where x ( t ) = [ x ( t ) , . . . , x n ( t )] ⊤ ∈ R n is the state, u ( t ) = [ u ( t ) , . . . , u m ( t )] ⊤ ∈ R m is the control input, and T ∈ (0 , ∞ ) is the length of the control horizon.The control { u ( t ) : t ∈ [0 , T ] } is chosen to drive the state x ( t ) from a giveninitial state x (0) = x to the origin in time T , that is, x ( T ) = . Also, the control u ( t ) is constrained in magnitude by(2) k u ( t ) k ∞ ≤ , ∀ t ∈ [0 , T ] . We call a control { u ( t ) : t ∈ [0 , T ] } admissible if it satisfies (2) and the resultantstate x ( t ) from (1) satisfies boundary conditions x (0) = x and x ( T ) = . We Key words and phrases. networked control, sparsity, optimal control, hands-off control.SUBMITTED TO IEICE SMARTCOM2014 denote by U the set of all admissible controls. Among all admissible controls inthe set U , we consider a control that maximizes the time interval over which thecontrol u ( t ) is exactly zero. Such a control is called a maximum hands-off control ,the problem of which is described as follows: Problem 1 (Maximum Hands-Off Control) . Find an admissible control { u ( t ) : t ∈ [0 , T ] } ∈ U that minimizes (3) J ( u ) , m X i =1 λ i k u i k L = m X i =1 λ i Z T φ ( u i ( t )) dt, where λ > , . . . , λ m > are given weights and φ ( u ) , ( , if u = 0 , , if u = 0 . This problem is quite hard to solve since the cost function is highly nonlinear andnon-convex. To overcome the non-convexity, we introduce the following L -optimalcontrol problem: Problem 2 ( L -Optimal Control) . Find an admissible control { u ( t ) : t ∈ [0 , T ] } ∈U that minimizes (4) J ( u ) , m X i =1 λ i k u i k L = m X i =1 λ i Z T | u i ( t ) | dt. This problem is a classical L optimal control problem, also known as fuel-optimal control problem, and can be easily solved [2, Chap. 8]. Moreover, thefollowing theorem shows the equivalence between the two control problems [10]: Theorem 1.
Assume that ( A, B ) is controllable . Assume also that Problem 2has at least one solution . Then the set of the solutions of Problem 1 (maximumhands-off control) is equivalent to the set of the solutions of Problem 2 ( L -optimalcontrol). Networked Control
Let us assume that the conditions in Theorem 1 hold. Then, the maximumhands-off control takes only 3 values, {− , , } , and the value changes discon-tinuously. This property, called “bang-bang control,” benefits networked controlsystems since the control value can be represented in only 2 bits. Moreover, thenumber of switching times is bounded as shown in the following proposition: Proposition 1.
Assume that ( A, B ) is controllable and A is nonsingular. Let ω bethe largest imaginary part of the eigenvalues of A . Then, the maximum hands-offcontrol is a piecewise constant signal, with values − , , and , with no switchesfrom +1 to − or − to +1 , and with nm (1 + T ω/π ) discontinuities at most. For the definition of controllability, see [2, Sect. 4-15]. (
A, B ) is controllable iffrank[
B, AB, . . . , BA n − ] = n . A linear system that is controllable and with nonsingular A is called a metanormal system[6]. PARSITY METHODS FOR NETWORKED CONTROL 3
Proof:
Theorem 1 combined with Theorem 3.2 of [6] gives the results. (cid:3)
Let us consider a networked system where we should send the control signal u ( t )on time interval [ kT, ( k + 1) T ) at every sampling time kT , k = 0 , , , . . . . FromProposition 1, we use 1 bit for representing the change of the control values, and b bits for representing each switching time. From Theorem 1, we need in total1 + 2 nmb (cid:18) T ωπ (cid:19) [bit]to represent the maximum hands-off control on time interval [0 , T ], or1 T + 2 nmb (cid:18) T + ωπ (cid:19) [bps] , which is much smaller than representing a general signal on [0 , T ]. This is anadvantage of the maximum hands-off control for networked control systems.4. Conclusion
In this presentation, we have introduced maximum hands-off control (the sparsestcontrol) and shown that this control is equivalent to L -optimal control under someassumptions on the optimal control problem. The maximum hands-off control hasa “bang-bang” property, which is very advantageous to networked control systemsin view of compressed data representation. References [1] R. P. Aguilera, R. Delgado, D. Dolz, and J. C. Ag¨uero, “Quadratic MPC with ℓ -inputconstraint,” IFAC 19th World Congress , pp. 10888–10893, Aug. 2014.[2] M. Athans and P. L. Falb,
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IEEE Trans. Automatic Control , vol. 59, no. 6, pp. 1454–1467, Jun. 2014.[6] O. H´ajek, “ L -optimization in linear systems with bounded controls,” Journal of Optimiza-tion Theory and Applications , vol. 29, no. 3, pp. 409–436, Nov. 1979.[7] K. Hayashi, M. Nagahara, and T. Tanaka, “A user’s guide to compressed sensing for commu-nications systems,”
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IEEETrans. Signal Processing , vol. 41, pp. 3397–3415, Nov. 1993.[9] M. Nagahara and D. E. Quevedo, “Sparse representations for packetized predictive networkedcontrol, ”
IFAC 18th World Congress , pp. 84–89, Aug. 2011.[10] M. Nagahara, D. E. Quevedo, and D. Neˇsi´c, Maximum hands-off control and L optimality, Proc. of 52nd IEEE CDC , 2013.[11] M. Nagahara, D. E. Quevedo, and J. Østergaard, “Sparse packetized predictive control fornetworked control over erasure channels,”
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