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For nonlinear control systems, one would often like to know the domain of attraction (DoA) of an equilibrium point. Often, this domain is difficult to both find and represent computationally. The usual mathematical tool used for analyzing of the region of attraction is Lyapunov’s method. This gives us a sufficient condition for local stability, although it is often difficult to find a Lyapunov function that can be used as a certificate for the whole DoA. Several prior approaches have used quadratic functions, for example [

With recent developments in algebra and sum-of-squares techniques, it is now possible to solve for a Lyapunov function with a more general polynomial form [

A phase-locked loop (PLL) system is a nonlinear system with limited domain of attraction. Due to its importance in communication systems, analyzing and designing a PLL system has attracted many attentions in this field [

In this paper, we utilize the current SOS techniques to analyze the domain-of-attraction of a PLL system. A local Lyapunov function can then be found as the certificate of the DoA using the proposed approach. The Lyapunov function will be further used to improve the stability region and performance of the PLL system. Using this approach, it is possible to design a PLL with a predefined form of controller that has larger domain-of-attraction than the linear design approach. Moreover, since we are designing the system in the nonlinear region, system dynamics outside the linear region can be further refined. Examples of second order PLL systems are used later in this paper to show the effectiveness of this design approach. In the provided examples, we demonstrated that the system designed by the proposed method has 20% larger domain-of-attraction and less overshoot with faster convergent rate than the linear design approach.

This paper is organized as follows. Section 2 contains some preliminary knowledge about SOS techniques. The advection algorithms as well as the SOS methods for finding a local Lyapunov function are stated in Section 3. A brief discussion of the configuration of a PLL can be found in Section 4 and the proposed method for analyzing and designing a PLL controller are presented in Section 5. Then the paper is summarized in Section 6.

The following are some definitions that will be used frequently in this paper. ℝ[^{n}_{1}, ..., _{s}_{+} is used to represent the set of nonnegative real numbers. _{r}

Suppose ^{n}^{1}. Define the 0^{n}^{n}

One feature of the proposed advection algorithm is that the advection problem can be converted into a semidefinite program. The following is a standard form of a semidefinite program.
^{n×n} is symmetric. ^{T}^{n}

The condition of one semi-algebraic set containing another semi-algebraic set is one of the key constraints used in this paper. The following lemma shows that this kind of relationship can be converted to constraints on the coefficients of the polynomials. The proof can be found in [

_{0}, _{1} ∈ Σ _{0}, _{1}, _{0} _{1}

The representation shown in Lemma 1 is one of the simplest cases of Schmüdgen’s Theorem [_{1}, . . ., _{m}^{n}_{i}_{1}, . . ., _{m}^{m}_{v}_{i}

The following result is similar. Given _{0}, _{1} ∈ Σ and ɛ > 0 such that

Usually

Finding a local Lyapunov function is coupled with finding the DoA. Without a clear knowledge of the actual shape of the DoA, it is hard to find a Lyapunov function that can be used to represent the entire DoA [

In this paper, we will consider the following autonomous system
^{n}^{n}^{n}_{+} such that the autonomous system (2) has a unique solution for any

We define the _{t}^{n}^{n}

For any _{t}_{t}^{n}^{n}^{n}

Given _{t}^{n}^{n}_{t}_{t}_{t}

_{1}, _{2} ^{n}_{2} = _{t}g_{1} _{2}) = _{t}_{1})).

Since we are performing advection, we must use an approximation to the flow map _{h}_{h}p_{h}^{n}^{n}^{n}^{n}

Based on the required accuracy of the advection, we could also choose to use higher order Taylor approximation. However, depending on the system dynamics, this usually will lead to the requirement of using higher degree polynomials in the sum-of-squares constraints. The relationship between the accuracy and the degree of polynomials will be further investigated in future work.

The set advection concept is used to estimate the DoA of a system. We use the following definition of the DoA in this paper.

^{n}^{n} is analytic with the flow map, φ, and the origin is asymptotically stable. Define the^{n} such that for any x_{t}_{t→∞} _{t}

The following properties can be easily derived. The detailed proofs can be found in [

_{1} ⊂ _{1}, _{1} _{1} _{2}, _{1} ⊂ _{2} ⊂ _{2} _{2} = _{−h}_{1}.

_{0} _{0} ⊂ _{ε}_{0}.

_{0}, _{1}, _{2}, . . .

_{k}

_{k}_{k+1}

_{n}

For the case of estimating the DoA, we introduce the concept of star-shaped sets. The star-shaped sets have many important properties and can be easily implemented as a semidefinite program. We now start with the first property. The detailed information about the star-shaped set can be found in [

^{n}

Note that a star shaped set

^{n}^{1}

The following lemma shows the connection between strictly star-shaped functions and star-shaped sets.

^{n}

_{+} → ℝ^{n}

The trajectory of

Now we need to show that if ^{1} on [

For the purposes of this paper, we would like to construct a convex set of functions whose sub-level sets are connected. Although the convex set of all convex functions on ℝ^{n}

Here we will state the result of the backward advection algorithm. Given a strictly star-shaped polynomial _{i−1} such that _{i−1}) ⊂ _{i−1}) is bounded and positively invariant, we compute a polynomial _{i}_{h}g_{i}_{i−1}) as follows.

Pick _{i}_{1}, _{2}, _{3}, _{4} ∈ Σ.

Here we introduced an important parameter, _{i}_{i}_{h}_{i}_{i−1}), and _{h−α}_{i}_{i−1}). Hence the parameter _{i}_{i→∞} _{i}

By using the proposed level-set method, one can successfully propagate the system states backward in time. However, a stopping criterion is still needed to terminate the iterations. To detect the convergence of the advected sets, the closeness of two semi-algebraic sets is analyzed. The following result shows that the closeness of two semi-algebraic sets can be estimated by using scaled sets. The detailed proof can be found in [

_{1} _{2} _{1}) ⊂ _{2}), _{2}) _{1}, _{2} ∈ ℝ^{n}_{1} = _{2} ^{n}_{1}),

To determine when the algorithm should terminate, one formulates an optimization problem using Lemma 1 to determine the smallest _{2}), where _{2}(_{1}), Curve 2 is _{2}), and Curve 3 is ∂

We find a local Lyapunov function in order to construct an initial star-shaped positively invariant set. The following result is standard.

^{n}^{n}^{n}^{1} _{0}(_{0}) _{0}) ⊂

One simple approach to finding an initial sub-level set is to find a quadratic Lyapunov function for the linear model of the system, and use a small sub-level set of this quadratic polynomial as the initial set.

An alternative method which often gives a much larger initial set is as follows. Choose a polynomial _{0}, _{1} ∈ Σ such that
_{0} =

After we have found an estimate of the DoA, we can then use Proposition 1 to find a Lyapunov function that can be used to describe the behavior of the system within the DoA. To adequately describe the system behavior, this approach requires that we have a good estimate of the DoA. The following examples show that the level-set algorithm can precisely estimate the DoA.

_{0} = 2^{2} + 2^{2} − 1.

_{0} = 4^{2} + 4^{2} − 1 _{0}, _{2}, _{4}, . . .

_{0})

_{0}(_{i}_{i}_{0}(

Of interest is the behavior of the phase error

_{2}, from _{0}(_{i}

To use the nonlinear design approach, start with a reference design. The reference design used in this paper is the linear model of a PLL system. A Proportional-Integrator (PI) controller is chosen to be the filter, _{1} = _{2} = _{n}_{n}

In this section, the second order PLL controller will be used to demonstrate the nonlinear design approach. The same approach can also be applied to the third order controller design.

Now the advection algorithm can be applied to the PLL nonlinear system. To reduce the required number of iterations, a local Lyapunov function is used as the initial set. After a few iterations of the algorithm, it gives us the estimated domain-of-attraction of the system. The result is shown in

After getting a good estimated DoA, a local Lyapunov function that describes the system behavior can be easily computed using Proposition 1. This local Lyapunov function can also be used to describe the trajectories of the system in the DoA.

The SOS techniques can be applied to the design of a controller. For the PLL system, it is desired to design a system which has a larger DoA or faster converging speed. Here, the objective is to find possible system parameters such that the same Lyapunov function is still valid and the system converges faster. This can be done by solving a semidefinite program.

Suppose _{+} specify the domain of the constraints and _{1}, _{2}, _{3}, _{4} are SOS polynomials.

Since

Besides dynamic performance constraints, noise bandwidth constraints will be applied as well. The noise bandwidth for this PLL system with PI controller has the following form [_{n}

The system phase portrait as well as the estimated stable region are shown in _{n}

A Simulink model is used to compare the nonlinear designed PLL controller with the linear design. In this Simulink model, the sinusoidal input is collapsed by measurement noise and clock noise with zero mean and variances 0.1 and 0.0001, respectively.

Both systems are tested on a NORDNAV R25 software GPS receiver. This receiver collects, down-converts and samples the GPS data by the front end, so that the collected GPS data can be post-processed repeatedly using different tracking-loop filter orders. The results are shown in

It can be seen that the original system has some overshoot and converges around 300 ms. The nonlinear design has much less overshoot and converges about five times faster than the original design.

In this paper, we presented a method of designing a PI controller of a PLL system. This design approach is based on the polynomial nonlinear model of the PLL system. This approach starts with the linear design of the controller and then estimates the DoA of the linear designed system to get the suitable local Lyapunov function for the system. The Lyapunov function is then used as the performance constraints to further refine the performance of the system outside the linear region. The DoA of the initial design can also be extended to get a better pull-in region of the PLL system. This approach gives us a way to design a fixed form controller for a nonlinear system.

A preliminary version of this paper was presented in 16th IEEE Mediterranean Conference on Control and Automation (MED’08) [

Based on “Nonlinear Phase-Locked Loop Design using Semidefinite Programming”, by Ta-Chung Wang, Tsung-Yu Chiou and Sanjay Lall, which was presented at the 16th IEEE Mediterranean Conference on Control and Automation. © 2008 IEEE.

The advection operator _{t}

Example of stopping conditions. Curve 1 is the inner set and curve 2 is the outer set. Curve 3 is the shrunk set of the outer set.

Successive iterates of the level-set algorithm.

Van der Pol oscillator. The left figure shows the sequence of iterations. The right figure shows the final iterate as Curve 4 along with some other results. Curve 1 is ∂ _{0}), when _{0} is obtained through the semidefinite programming based procedure in Section 3.7. For comparison, Curve 2 is the result of [

Sub-level sets of _{0} and _{20}. The dashed-dot curve is the result from [

Basic Configuration of a PLL.

Model of the Phase-Locked Loop.

Result of the original PLL system.

Local Lyapunov level sets of the original PLL system.

Comparison of the domain-of-attraction.

Phase error of Simulink simulation.

Real GPS experimental results.