1. INTRODUCTION

Over the last decade, Interconnection and Damping Assignment Passivity-Based Control (IDA-PBC) has experienced increasing practice due its wide applicability (^{Petrovi´c et al., 2001}; ^{Batlle et al., 2004}, ^{2007}; ^{Ortega and García-Canseco, 2004}; ^{Li et al., 2010}; ^{Astolfi and Ortega, 2001}; ^{Fujimoto et al., 2001}; ^{Renton et al., 2012}; ^{Li et al., 2013}; ^{Astolfi et al., 2002a}; ^{Xue and Zhiyong, 2017}). The standard IDA-PBC method requires a two step procedure: energy shaping and damping injection. The first depends on the solution of partial differential equations (PDEs) and the second together with zero state detectability (ZSD) of the closed-loop guarantees asymptotic stability.

In order to simplify the PDE problem ^{Viola et al. (2007)} have introduced a change of coordinates and a modification of the target dynamics. With the objective to completely avoid PDEs, the following leading methods have been proposed: constructive procedures (^{Donaire et al., 2016a}; ^{Borja et al., 2016}; ^{Romero et al., 2017}), implicit port-Hamiltonian representation (^{Macchelli, 2014}; ^{Castaños and Gromov, 2016}) and an algebraic approach (^{Fujimoto and Sugie, 2001}; ^{Batlle et al., 2007}; ^{Nunna et al., 2015}). In addition, it has been shown in (^{Batlle et al., 2007}; ^{Donaire et al., 2016b}) that a two step IDA-PBC may be restrictive in some cases, thus introducing a single step procedure (SIDAPBC). Furthermore, dissipation in the under-actuated degrees of freedom, see (^{Gómez-Estern and Van der Schaft, 2004}), may also turn out an obstacle for the implementation of IDA-PBC on some systems, e.g. on the cart-pole system (^{Delgado and Kotyczka, 2014}).

It is well-known that actuator saturation (AS) can cause performance losses or even lead to closed-loop instability. In this context, ^{Åström et al. (2008)}; ^{Escobar et al. (1999)} have studied PBC with AS on two specific systems. ^{Sun et al. (2009)}; ^{Wei and Yuzhen (2010)} analyze stability for saturation in the damping injection term. A variable structure approach to energy shaping for a class of Port-Hamiltonians system is developed in (^{Macchelli, 2002}; ^{Macchelli et al., 2003}). Besides, ^{Sprangers et al. (2015)} studied a reinforcement learning method for energy shaping which shows robust properties under AS.

In polynomial systems the sum of squares (SOS) approach with semidefinite programming (SDP) allows to synthesize Lyapunov functions (^{Parrilo, 2000}), optimal (^{Ichihara, 2009}), robust (^{Zhu et al., 2018}; ^{Jennawasin et al., 2010}), fuzzy (^{Wibowo et al., 2014}; ^{Yu and Wang, 2013}) and AS-focused controllers (^{Jennawasin et al., 2012}; ^{Valmorbida et al., 2013}; ^{Ichihara, 2013}), among others. This AS controllers, calculated with SOS, use the polytope representation introduced in (^{Hu and Lin, 2001}) for linear system with multiple input saturation.

For a class of polynomial affine systems, lately, ^{Cieza and Reger (2018)} have presented an algebraic method the conditions of which are met by means of SOS and SDP solutions. The method solves the typical problems of IDA-PBC at the expense of an adequate parametrization and selection of the Hamiltonian. To the best knowledge of the authors there is no definitive solution to the AS controller design problem with IDA-PBC. The underlying work shall now also incorporate AS and two minimization objectives in the SDP solver, extending the algorithm of ^{Cieza and Reger (2018)}.

The work is organized as follows. We summarize the concepts of IDA-PBC for nonlinear affine systems in Section 2. Section 3 recapitulates the algebraic method of ^{Cieza and Reger (2018)}. In Section 4 we solve the AS problem in the algebraic approach using additionally two minimization objectives. We discuss the application of SOS methodology and verify our results in Section 5, applying the approach on two polynomial systems. Finally, we draw our conclusions in Section 6.

2. IDA-PBC FOR AFFINE NONLINEAR SYSTEMS

Let us recall the IDA-PBC approach for nonlinear affine systems introduced by ^{Ortega and García-Canseco (2004)}. Consider the system

and the target port-Hamiltonian system

where

is fulfilled for some

transforms (1) into the stable system (2) with Lyapunov function Hd. Asymptotic stability of ^{Astolfi et al., 2002b}; ^{Sepulchre et al., 1997}).

3. IDA-PBC FOR POLYNOMIAL SYSTEMS

In this section we summarize Proposition 4–5 from (^{Cieza and Reger, 2018}) for

3.1 Algebraic IDA-PBC

Let

and the desired closed loop port-Hamiltonian system

where

Without loss of generality let

**Proposition 1** Closing the loop of system (4) with control

renders the equilibrium point

C1 There exist polynomial functions

C2 There is a polynomial function _{n} represents the identity matrix of size n.)

C3 There is a constant matrix P, and polynomial matrices _{2}(x) such that

Furthermore, the origin of (5) is asymptotically stable if

for

**Proof 1** It can be found in (^{Cieza and Reger, 2018}) with a modification of (8) using the Schur complement.

3.2 Existence of u_{b}

The next proposition provides a sufficient condition for the existence of F_{2}, i.e. the existence of the (asymptotically) stabilizing control law (6).

**Proposition 2** Consider

with

Lastly, if (10) or (11) are satisfied, then a solution for F_{2} with

**Proof 2** See (^{Cieza and Reger, 2018}, Prop. 5).

Application of the algorithm starts with adequate selection of _{2} in case of Prop. 1) under (8) which defines an upper and lower bound on P, namely _{0}.

In comparison, Proposition 2 requires the solution of smaller LMIs to calculate the parametrized function F_{2}, see (12), or to guarantee its existence, whereas Prop. 1 defines F_{2} as general function s.t. (7) (and (9)) is satisfied, which grants more flexibility at the expense of computational cost. In order to use SOS with SDP we force F_{2} (in Prop. 1) to be a polynomial function of some selected degree.

Proposition 2 can also be used as a fast indicator such that Proposition 1 will work. Note that Proposition 2 contains the minimal conditions that _{2}. Hence, if we constrain F_{2} to be polynomial, then Proposition 2 is experimentally still a good, but not an unconditionally reliable reference.

4. MAIN RESULTL

In view of Proposition 1 and 2, we extend the results of ^{Cieza and Reger (2018)} to consider actuator saturation (AS). In addition, we define two possible minimizations (optimization objectives) for the SDP.

4.1 Actuator Saturation (AS)

**Proposition 3** Let all conditions of Prop. 1 for local (asymptotic) stability be satisfied and assume:

C5 There exist polynomial matrices

Then the stabilizing control law (6) is restricted to

**Proof 3** Multiplying (14) on both sides by adequate matrices and using the Schur complement yields

where last equalities are obtained with (13) and (6).

After solving the conditions of Proposition 1 and 3, we may calculate a control input

and on the left by its transpose. This results in conditions that can be solved by means of SOS + SDP

Following the works of ^{Hu and Lin (2001)}; ^{Valmorbida et al. (2013)}; ^{Ichihara (2013)}, among others, we may use the polytope or polytopic saturation model within the algebraic IDA-PBC, as phrased in the following proposition.

Proposition 4 Let the conditions of Propositions 1 and 3 be satisfied for some system of the form (4) resulting in some matrices P, F_{2} and a locally (asymptotically) stabilizing constrained controller

provided that there exist matrices

**Proof 4** Define

^{Cieza and Reger, 2018}). Then (7) results in

and rewritten as a sum of positive semidefinite polynomial functions (convex set) this yields

Proposition 4 implies that if there is a solution to the conditions of Propositions 1 and 3 with (17), then there also exists an (asymptotically) stabilizing control law (16). In addition, if

In the same way as in (^{Hu and Lin, 2001}; ^{Valmorbida et al., 2013}; ^{Ichihara, 2013}) for multiple input systems, we can adopt the independent input saturation given by

Here, we also observed that in order to have AS, independent input saturation (u_{sat-i}) is not the only solution. Therefore, to simplify (17), we select

which is also shown in Figure 1. Selection of

4.2 Optimization Objectives in SDP

Proposition 1–3 only guarantee a solution for P and F_{2}(x) without any performance or optimization goal in the SDP. In addition, we may set the following simple objectives:

**Optimization 1 (Volume maximization** of XP)

**Proof 5** The volume of XP is proportional to ^{Boyd et al., 1994, pp. 48-49}). In addition, from KL-divergence between two multivariate normal distributions, we obtain the relation

for any real matrix

This minimization is also used empirically in (Ichihara, 2013). Optimization 1 maximizes the volume of XP by maximizing the minimum bound of P given by Y^{-1}. Note that searching for the biggest XP does not demand the explicit selection of x_{0} (right hand side of (8)).

**Optimization 2 (Volume minimization** of Ub)

**Proof 6** Along the same lines of Optimization 1, except that we consider the upper bound of (21).

Without loss of generality, define

for all _{u} (upper bound of u) is equivalent to minimize

5. SIMULATIONS

It is well-known that the SOS property is a sufficient condition for checking the non-negativity of a polynomial function (^{Parrilo, 2000}). For this reason, we may search for positive semidefinite matrices that are matrix SOS polynomials in Propositions 1–4. To guarantee strict inequalities in the SDP solver, we add ^{Papachristodoulou et al., 2016}). For details on the transformation from SOS to SDP see (^{Parrilo, 2000}).)

In the following examples we search for asymptotically stabilizing controllers wrt. two systems using the results of Proposition 1–4. Values presented in this paper have been rounded to three decimals for better visibility.

5.1 Nonlinear Second Order System

We shall test Proposition 1–3 for synthesizing an asymptotically stabilizing constrained controller in the system

First, we pick

Next, for illustration we select (a minimum bound on P) _{2}) the conditions of Prop. 1 and 3 with Opt. 2 (minimization of S_{u}) for _{u} = 100.134.

Finally, we evaluate Prop. 1 and 3 with Opt. 1 selecting, for instance, _{b}. As mentioned in Section 4, we can also use Prop. 2–3. Table 1 shows a comparison between both Propositions for

5.2 Third Order Multiple Input System with AS

Now, we consider a third order system given by

with AS using

and

_{1} scaled for clarity) in closed-loop under initial condition _{sat-n}. Note that u_{2} is saturated, obviously, without compromising stability. Furthermore, using Prop. 2 in this system gives worse optimization results, which shows that the selection of the best Proposition (1 or 2) is system dependent.

6. CONCLUSION

In this paper we provide an algebraic solution for IDA-PBC that is able to resolve the problem of actuator saturation. To this end, we restrict the design to a class of polynomial systems that yield conditions which are solvable with SOS and SDP. The presented algorithm requires the following steps:

Additionally, we enjoy features as: no need to solve a PDE, dissipation in design, and one step IDA-PBC. Simulations of two polynomial example systems validate our approach.