A conjecture on sustained oscillations for a closed-loop heat equation

A conjecture on sustained oscillations for a closed-loop heat equation C.I. Byrnes, D.S. Gilliam Abstract The conjecture in this paper represents an initial step aimed toward understanding and shaping the dynamics of sustained oscillations in nonlinear feedback systems for distributed parameter systems. Periodic phenomena are pervasive in nature and in engineered systems. Electronic devices producing stable periodic signals underly both the electrification of our world and wireless communications. After reviewing the analysis of classical voltage controlled oscillators, we describe a recent extension of this design [1] to a fairly general class of lumped models for the loop filter, which is the design element in a phase-lockedloop This naturally leads to the analysis and design of nonlinear feedback controllers which we believe creates a self-sustained oscillation in a particularly simple distributed parameter feedback system. We provide some mathematical and physical basis, as well as a simulation, for our conjecture. In particular, we expect this conjecture to be proved using Leray-Schauder Fixed Point theory. We also believe there are intriguing applications for similar constructions for hyperbolic systems, such as the design of a sustained oscillation whose phase could be adapted on-line for noise cancellation. This research was supported in part by a Grant from the AFOSR. Electrical and Systems Engineering, Washington University St. Louis, MO 63130, Mathematics and Statistics, Texas Tech University, Lubbock, TX 79409,

1 Introduction A phase-locked loop (PLL) is a basic electronic component used in the transmission of stable periodic signals representing bits in a communication channel. For example, every cell-phone transmits signals at two distinct, stable radio frequencies, one for 0 s and the other for 1 s. A PLL consists of three components: a phase detector (PD), a voltage controlled oscillator (VCO) and a (low-pass) loop filter (LF), each of which can be described in terms of a mathematical model. There is a sustained, or self-excited, oscillation in a PLL whose phase is tuned to asymptotically match the phase of a received signal which is to be tracked. The limit cycle in a PLL is produced by the cascade of the the LF precompensator with the VCO (an integrator) filtered by a nonlinearity determined by the type of PD. More explicitly, the output of the VCO is compared by the PD with the periodic signal to be tracked, producing a nonlinear function h(φ) of the phase error φ and then passed through the LF precompensator which determines the input to the VCO. If the PD employed in the circuit is a multiplier, then h(φ) = sin(φ) and the LF remains as the controller to be designed. In typical commercial hardware, the LF is chosen to be first order; i.e., with a current-voltage pair represented as the forcing term and solution of a first order differential equation. The nonlinear dynamics of this class of PLL s has been derived and analyzed in [3]. For the case of a PD multiplier and a lag filter, the closed loop error dynamics take the form of a damped nonlinear pendulum [6] and is therefore stable. After reviewing the analysis of classical voltage controlled oscillators in Section 2, in Section 3 we describe a recent extension of this design [1] to a fairly general class of lumped linear models for the loop filter. These models include the mathematical models for many RLC circuits. In [1], the existence of periodic orbit, for certain small parameters is also established for loop filters modeled by lumped nonlinear minimum phase systems having relative degree 1. In particular, we believe that the existence of sustained oscillations for loop filters in a nonlinear connections with an integrator (e.g., VCO s) is a general property of systems with asymptotically stable zero dynamics and nonzero instantaneous gain. Our conjecture represents the first expression of this belief for distributed parameter systems.

2 The voltage controlled oscillator A phase-locked loop (PLL) is a basic electronic component used in the transmission of stable periodic signals representing bits in a communication channel. For example, every cell-phone transmits signals at two distinct, stable radio frequencies, one for 0 s and the other for 1 s. A PLL consists of three components: a phase detector (PD), a voltage controlled oscillator (VCO) and a (low-pass) loop filter (LF), each of which can be described in terms of a mathematical model. There is a sustained, or self-excited, oscillation in a PLL whose phase is tuned to asymptotically match the phase of a received signal which is to be tracked. The limit cycle in a PLL is produced by the cascade of the the LF precompensator with the VCO (an integrator) filtered by a nonlinearity determined by the type of PD. More explicitly, the output of the VCO is compared by the PD with the periodic signal to be tracked, producing a nonlinear function h(φ) of the phase error φ and then passed through the LF precompensator which determines the input to the VCO. If the PD employed in the circuit is a multiplier, then h(φ) = sin(φ) and the LF remains as the controller to be designed. In typical commercial hardware, the LF is chosen to be first order. The nonlinear dynamics of this class of PLL s has been derived and analyzed in [3]. For the case of a PD multiplier and a lag filter, the closed loop error dynamics take the form of a damped nonlinear pendulum [6] and is therefore stable. Since the design and performance of a PLL relies on the fact that the coupling of the VCO with the LF precompensator has a sustained oscillation even when the input signal is zero, we want to analyze this simpler situation in more detail. When the loop filter is designed as a lag filter, the LF dynamics are ẏ = λy + u, (1) and the VCO dynamics and coupling are given [5] by θ = y (2) u = α + a sinθ, (3) where α > a, and λ > 0. In particular, the vector field X defined by (1)-(3) leaves the submanifold

M R S 1 0 < α a λy α + a positively invariant and satisfies X, dθ > 0 on M Since M A, there exists a limit cycle in M by the Poincaré-Bendixson Theorem. Voltage Controlled Oscillator 3 The voltage controlled oscillator with an n-dimensional minimum phase loop filter We consider systems of the form dx dt = Ax + bu, y = cx (4) dθ dt = y (5) where x R n, u, y R and A, b, c are real matrices of the appropriate sizes. We shall assume that the system has relative degree 1, i.e. that cb 0, and is minimum phase, i.e., that the Laplace transform g(s) = c(si A) 1 b satisfies g(s) = 0 = Re(s) < 0. We construct a feedback law u as a superposition u(x, θ) = u 1 (x) + u 2 (θ), (6)

where the first feedback law is u 1 (x) = (cb) 1 ( cax y + α). (7) Implementing u 1 yields a closed loop system (4), (7) for which ẏ = y + α, so that lim t + y(t) = α for all initial values y(0). Since the system is minimum phase there exists an x α R n for (4), (7), with lim t + x(t) = x α for all initial conditions x(0). In particular, cx α = α. Setting e α = x x α we can express the closed loop-system (4), (7) as ė α = Ãe α (8) where à = A + (cb) 1 b( ca c) is independent of α. From the theory of (A, b)-invariant subspaces [7] it follows that à has spectrum σ(ã) = { 1} g 1 (0). In particular, e α = 0 is an asymptotically stable equilibrium for (8) and therefore there exists P = P T > 0 such that V (e α ) = e T αpe α is a Lyapunov function for (8), independent of α, satisfying V (e α ) = e α 2. Finally, implementing the feedback law (6) where u 2 (θ) = a sin(θ) (9) yields a closed loop vector field X (α,a) on M = R n S 1. Computing V along trajectories of (4)-(7), (9) we obtain V (e α ) e α 2 + ac e α sinθ e α 2 + ac e α, (10) where c > 0 is independent of α. In particular, there exists κ a 0 such that the solid ellipsoid M (a,α) = V 1 (, κ a ), is positively invariant. Since M (a,α) has principal axes of fixed length and origin x α, by choosing α 0 we can ensure that y > 0 on M (a,α). As in the one-dimensional case, consider the submanifold M = M (a,α) S 1 R n S 1. Since y > 0 on M the solid ellipsoid M (a,α) is a crooss-section for the closed-loop dynamics and therefore we may define a Poincaré P : M (a,α) M (a,α). By Brower s Fixed Point Theorem, P has a fixed point and therefore the closed-loop system (4)-(7), (9) has a periodic orbit.

4 The voltage controlled oscillator with a controlled heat equation loop filter We now consider the one dimensional heat equation. In this example the system z t (x, t) = z xx (x, t), z(x, 0) = ϕ(x) (11) z(0, t) = 0, (12) Bz(t) = z x (1, t) = u(t), boundary input (13) y(t) = Cz(t) = z(1, t), boundary output (14) θ t (t) = y(t), θ(0) = θ 0. (15) For 0 < a < α, we introduce the control u given by u(t) = y + α + a cos(θ). (16) Using a lumped approximation (for small values of the Biot constant), Newton s law of cooling simplifies to an ODE and the boundary condition (13) simplifies to ż(1, t) = hu(t) for h > 0. Normalizing h = 1 and implementing the control law (16), the dynamics of the boundary system (13)-(16) reduces to the VCO (1)-(3). In this case, there exists a smooth initial condition φ so that y(0) = φ(1) generates a periodic forcing term in the boundary condition (13) and hence to a periodic solution the system (11)-(16). This is, of course, a heuristic argument depending on an approximation which becomes better as the heat conductivity of the material gets larger or the length of the bar gets smaller. Nonetheless, for a numerical example where we have set a = 1 and α = 2 the plot of y(t) depicted in Figure 1 suggests that the system output is indeed a periodic function of t. to Returning to the DPS model, it can be shown that the system is equivalent z t = Az + Bf(θ), z(0) = ϕ y(t) = Cz(t) = z(1, t), θ t (t) = y(t), θ(0) = θ 0, where f(θ) = α + a cos(θ), B = δ x=1 and A is given by A = d 2 /dx 2 with D(A) = {ϕ H 2 (0, 1) : ϕ(0) = 0, ϕ (1) + ϕ(1) = 0}. A is an unbounded self-adjoint

operator and generates an exponentially stable semigroup in Z = L 2 (0, 1) and Z 1 = H 1 (0, 1). Furthermore the output operator C of point evaluation is a bounded operator from Z 1 to R. Plot of y(t) with a = 1 and α = 2 Periodicity is also suggested by the following plot of z(x, t). Plot of solution surface with a = 1 and α = 2 We claim that the closed convex set C Z 1 defined by the inequality α a < y < α+a is positively invariant. Indeed, if α+a < y then Bz(t) = z x (1, t) < 0 and,

by Fourier s law heat flows out of the bar until y α + a. A similar argument for y < α a justifies our claim. In particular, the hybrid DPS/ODE system defines a semi-flow on the toroidal cylinder Z 1 S 1, with C S 1 being a positively invariant subset. We note that for trajectories in C S 1, θ( ) is a monotonically increasing function of t. Choosing the initial condition θ(0) = 0, a comparison argument shows that for such trajectories, θ(t) (α a)t. In particular, a trajectory with an initial condition in S = C {θ = 0} returns to S at a least time, so that we can define a Poincaré (first-return) map P : S S. From the Sobolev Embedding Theorem, we expect that P is a compact, continuous map. However, in order to complete the proof that P has a fixed point, and hence that there exists a periodic orbit, one needs (as in Section 3) to construct a Lyapunov function with bounded sublevel sets on Z 1 or at least on C. Again, by Fourier s law we expect that this should be defined in terms of heat as an energy function. If a convex Lyapunov function V can be found, then one should be able to apply Schauder s Fixed Point Theorem. More generally, if the sublevel sets of V are contractible - as they always are in finite dimensions nonlinear cases, then we expect one should be able to apply Leray Index Theory [4] to show the existence of a periodic orbit. Acknowledgment: This research was supported in part by a Grant from the AFOSR.

Bibliography [1] Byrnes, C. I. and Brockett, R. W., Vector Fields, Angular One-forms and Periodic Orbits, submitted to Annals of Math. [2] Byrnes, C. I., Gilliam, D.S. and He, J., Root Locus Methods for Boundary Feedback Control of a Class of Distributed Parameter Systems SIAM J. Control and Opt., 32 (1994) 1364-1427. [3] Endo, T. and Chua, L. O., Chaos from phase-locked loops, IEEE Trans. on Circuits and Systems 35 (1988) 987-1003. [4] Granas, A., The Leray-Schauder index and the fixed point theory for arbitrary ANRs, Bull. de la Soc. Math. France 100 (1972) 209-228. [5] Hoppensteadt, F.C., Analysis and Simulation of Chaotic Systems, 2nd Ed. Springer 2000. [6] Khalil, H., Nonlinear Systems, Prentice Hall, Upper Saddle River, NJ, 1996. [7] Wonham, W. M., Linear Multivariable Control: a Geometric Approach, Springer-Verlag, New York, 1979.