Algebraic IDA-PBC for Polynomial Systems with Input Saturation : An SOS-based Approach

The necessity to deal with partial differential equations (PDEs) and the dissipation condition are the main adversities in the application of Interconnection and Damping Assignment Passivity-Based Control (IDA-PBC). Recently, an algebraic solution of IDA-PBC has been explored for a class of affine polynomial systems by using sum of squares (SOS) and semidefinite programming (SDP). In this work, we extend the previous method by incorporating actuator saturation (AS) and two minimization objectives in the SDP. Our results are validated on two polynomial systems.

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 (SIDA-PBC).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.
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.

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 ẋ = f (x) + g(x) u (1) and the target port-Hamiltonian system where X h ⊂ R n x is the state space manifold, U ⊂ R m is the input space, g has full rank, the skew symmetric portion is fulfilled for some F d , H d and full rank left annihilator 2 g ⊥ then the control law transforms (1) into the stable system (2) with Lyapunov function 3) is an algebraic equation. 2The full rank left annihilator g ⊥ is given by g ⊥ (x)g(x) = 0 and rank (g ⊥ ) = n x − m.Consequently, (3) exists iff n x > m.

IDA-PBC FOR POLYNOMIAL SYSTEMS
In this section we summarize Proposition 4-5 from (Cieza and Reger, 2018) for β = 1.The variable β of the aforementioned work can be considered as a scaling factor, which does not alter our main results.

Algebraic IDA-PBC
Let Γ(x), g(x), g ⊥ (x), and g ⊥ (x) f (x) be polynomial functions in and the desired closed loop port-Hamiltonian system where Without loss of generality let argmin H d (x) = 0. Besides, state and input spaces remain as in (1), z : x are full rank polynomial matrices 3 , and F holds the portions Proposition 1 Closing the loop of system (4) with control renders the equilibrium point x = 0 of the closed loop system (5) locally stable for any initial state x(0) ] is a full rank square matrix.5 3 Invertibility of Γ enables (4) to take the usual form ẋ = f (x) + ḡ(x)u and C3 There is a constant matrix P, and polynomial matrices 0 Furthermore, the origin of ( 5) is asymptotically stable if Proof 1 It can be found in (Cieza and Reger, 2018) with a modification of (8) using the Schur complement.

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 x 0 ∈ X P ⊂ X h and assume Λ 1 , g ⊥ , z, N are selected according to C1 and C2.Then there exists a function F 2 (x) that meets C3 if there exists P = P 0 and S 2 (x) 0 such that Lastly, if (10) or (11) are satisfied, then a solution for F 2 with 0 L(x) Proof 2 See (Cieza and Reger, 2018, Prop. 5).
Application of the algorithm starts with adequate selection of Λ 1 , z, g ⊥ and N according to (4), ( 5), C1 and C2.Later, we choose h and solve (for convenience) ( 7), ( 9), ( 10) or ( 11) searching for P (and F 2 in case of Prop. 1) under ( 8) which defines an upper and lower bound on P, namely x 0 ∈ X P ⊂ X h for some given x 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 P, Λ 1 , g ⊥ , N, h and Γ have to satisfy, and it guarantees the existence of a not necessarily polynomial function F 2 .
Hence, if we constrain F 2 to be polynomial, then Proposition 2 is experimentally still a good, but not an unconditionally reliable reference.

MAIN RESULT
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.

Actuator Saturation (AS)
Proposition 3 Let all conditions of Prop. 1 for local (asymptotic) stability be satisfied and assume: C5 There exist polynomial matrices Λ 2 (x) ∈ R m×n z and 0 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 2 ) and P 1 2 P 1 2 = P. Now taking the spectral norm on (15) for x ∈ X P ⊂ X h , i.e. h ≥ 0, and the definition of where last equalities are obtained with (13) and (6).
After solving the conditions of Proposition 1 and 3, we may calculate a control input u b ∈ U b , for any x ∈ X P .Proposition 3 can also be extended to work with Prop. 2 by replacing ( 12) in ( 14).This yields an LMI which is not necessarily polynomial.Therefore, we restrict L to be polynomial and multiply ( 14) on the right with the square non-singular matrix 6 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 u = u b ∈ U b given by (6).Consequently, there is a new (asymptotically) stabilizing control action provided that there exist matrices F 21 (x) ∈ R m×n z and Si 1 ...i m (x) 0, s.t. for all i k ∈ {0, 1} with k = 1 . . .m, where Θ = diag(θ 1 , . . ., θ m ) and e i the ith unity vector.In addition, asymptotic stability is achieved if (9) are satisfied and (17) is strict.
6 Note that using C2, the square matrix [N ] and as a consequence 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 F 2 = F 20 = F 21 then ( 16) is reduced to ( 6) and ( 17) becomes (7) (or ( 10)).Proposition 4 can be easily extended to work with Prop. 2 (instead of 1).In this case, ( 17) is reduced to 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 u sat-i = sat(u x , u, u) and u x = u b + (g g)u δ , where u and u are maximum and minimum values of u b in U b .Figure 1 illustrates the situation for m = 2, g g = I 2 , u x , u b , u sat-i and sets U b , U s .
Figura 1. Relations of u constrained and saturated.
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 θ 1 = θ 2 = • • • = θ m and a new saturation function given by which is also shown in Figure 1.Selection of u sat-n reduce 2 m−1 inequalities and polynomial matrices S in (17).

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 X P ) minimize trace(Y ) Proof 5 The volume of X P is proportional to det(P), see (Boyd et al., 1994, pp. 48-49).In addition, from KL-divergence between two multivariate normal distributions, we obtain the relation trace for any real matrix A 0. Therefore, maximizing the volume of X P with P 0 is equivalent to maximize log(det(P)).Using (21) we enlarge the minimum bound of log(det(P)) by minimization of trace(P −1 ) which is equivalent to Opt. 1 with Schur complement in (20).
This minimization is also used empirically in (Ichihara, 2013).Optimization 1 maximizes the volume of X P by maximizing the minimum bound of P given by Y −1 .Note that searching for the biggest X P does not demand the explicit selection of x 0 (right hand side of ( 8)).
Optimization 2 (Volume minimization of U b ) minimize trace(S u ) subject to S u = constant.
Proof 6 Along the same lines of Optimization 1, except that we consider the upper bound of (21).
Without loss of generality, define , for some function F2 ∈ R m×n z .Then, ( 14) becomes for all x ∈ X h .This shows that minimization of S u (upper bound of u) is equivalent to minimize F2 − Λ 2 and an upper bound of P. As a consequence, it is required to have at least one minimum bound on P (right hand side of (8) or Opt. 1).

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 10 −3 I n z in P 0, 10 −3 I n x −m in (10), and 10 −3 I n in ( 7) and ( 17).The algorithm is processed in Matlab by use of SOSTOOLS and SDPT3, see (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.

Nonlinear Second Order System
We shall test Proposition 1-3 for synthesizing an asymptotically stabilizing constrained controller in the system x is unimodular and C1-C2 are satisfied.Then we select S h = diag(9, 9), S 1 (x) ∈ R 2×2 with polynomials of degree 2 as elements and test Proposition 1 with Optimization 1 (maximizing X P ), obtaining Next, for illustration we select (a minimum bound on P) x 0 = [0, 2] ∈ X P (previously found) and solve (for a new P and F 2 ) the conditions of Prop. 1 and 3 with Opt. 2 (minimization of S u ) for S 3 (x) ∈ R a polynomial of degree 6, resulting in S u = 100.134.
Finally, we evaluate Prop. 1 and 3 with Opt. 1 selecting, for instance, The results can be seen in Figure 2, which shows sets X P ⊂ X h , X P ⊂ U b , and the phase portrait in x 1 -x 2 plane of the closed-loop for 10 extreme initial positions x 0 represented by symbol " * ".Here all trajectories converge to the origin as expected.In addition, Figure 3 illustrates 5 seconds of respective control actions (calculated with ( 6)), which are all constrained in U b .As mentioned in Section 4, we can also use Prop.2-3.Table 1 shows a comparison between both Propositions for x 0 = [0, 2] ∈ X P , S u = 11 2 .We conclude that Prop. 2 yields better optimization results.
For avoiding excessively large u x , we constrain each of the constant elements of F 21 represented by f i j with f i j < 10.The results are illustrated in Figures 4 and 5. Figure 4 shows the states (x 1 scaled for clarity) in closed-loop under initial condition [0, −0.65, 0] = x 0 ∈ X P ⊂ X h .It is clearly seen that all states will converge to the origin.Figure 5 illustrates the first second of u 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.

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: S1 Select Λ 1 , Λ 2 , z, g ⊥ and h.S2 Define S u and calculate u b with Propositions 1 (or 2), 3 and Opt. 1 to maximize the volume of X P .The minimum S u can also be calculated with Opt. 2. 7 The minimum S u can be found similarly as in Example 5.1.S3 Compute u δ with Prop. 4 and P, F 2 found in S2.
S4 Implement the saturation functions u sat-i or u sat-n .
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.

Figura 3 .
Figura 3. Response of control signal u b (u b stays in U b ).