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Simultaneous stabilization of polynomial nonlinear systems via density functions
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Simultaneous stabilization of polynomial nonlinear systems via density functions Peyman Kohan-sedgh a,∗, Alireza Khayatian a, Navid Behmanesh-Fard b a School
b Department
of Electrical & Computer Engineering, Shiraz University, Shiraz, Iran of Electrical Engineering, Faculty of Darab Branch, Technical and Vocational University, Fars, Iran
Received 12 August 2018; received in revised form 4 July 2019; accepted 12 November 2019 Available online xxx
Abstract This paper is concerned with simultaneous stabilization of a class of polynomial nonlinear systems with almost stability theory. The theory uses dual Lyapunov or density function criterion which has remarkable advantages over Lyapunov-based simultaneous stabilization. These advantages rely in convexity property of density functions. This property results in controller synthesis problems which are affine with respect to unknown polynomials variables. The results of almost stability are also extended to simultaneous asymptotic stability at the expense of adding some extra affine polynomial inequalities. Numerical method for verification of positivity of multivariate polynomials based on sum of square decomposition is used. Numerical examples and a fault tolerant design scheme are used to show the effectiveness of the proposed methods. © 2019 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
1. Introduction Simultaneous stabilization is the problem of determining a single controller, which simultaneously stabilizes a finite collection of plants. In practice, due to plant uncertainty, plant variation, failure modes or plants with various modes of operation, the simultaneous stabilization problem is frequently encountered. The problem is a subtopic of robust control
∗
Corresponding author. E-mail addresses: [email protected] (P. Kohan-sedgh), [email protected] (A. Khayatian), [email protected] (N. Behmanesh-Fard). https://doi.org/10.1016/j.jfranklin.2019.11.033 0016-0032/© 2019 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
Please cite this article as: P. Kohan-sedgh, A. Khayatian and N. Behmanesh-Fard, Simultaneous stabilization of polynomial nonlinear systems via density functions, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j. jfranklin.2019.11.033
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and a closely related problem is robust stabilization of systems with polytopic or parametric uncertainties. Since the work of [1], simultaneous stabilization has received considerable attention. In [2], complexity of simultaneous stabilization has been discussed and it has been proved that the simultaneous stabilization of three linear systems is rationally undecidable. As a result, the burden of finding linear simultaneous stabilizer is mostly on sufficient conditions and finding methods to reduce conservatism is of high importance. Good literature review on simultaneous stabilization and increasing the feasibility region for linear systems can be found on [3,4] and references there in. It is well-known that the dynamic nonlinearities are inherent in almost all the physical systems. Unfortunately due to complexity of nonlinear systems, linear simultaneous approaches cannot be directly generalized for nonlinear systems and nonlinear control synthesis shall be used in this regard. Several nonlinear control approaches has been developed during past decades [5] and many of which has been generalized or modified to be used in new applications recently [6,7]. However, only a few works has addressed the problem of simultaneous stabilization directly because of difficulties involved in synthesis of nonlinear stabilizers. Ho-Mock-Qai and Dayawansa [8] proved that for any countable family of stabilizable nonlinear systems, a continuous state feedback law, which simultaneously stabilizes the family (non-asymptotically), always exists. Additionally, a sufficient condition for the existence of simultaneously asymptotically stabilizing controllers for a collection of nonlinear systems was provided. Wu [9] presents a method for designing simultaneous stabilizer of a collection single-input non-linear systems based on Control Lyapunov Function (CLF). However, both above mentioned methods are difficult to verify as they depend firstly on finding an asymptotically stabilizing state feedback controller or a CLF for each system, then determining whether some conditions are satisfied. Moreover, the obtained simultaneously asymptotically stabilizing controllers are not easy to implement. Xu et al.[10] addresses the problem of finding simultaneous controller for polynomial nonlinear systems based on state-dependant polynomial Lyapunov functions. The advantages of this method is capability of using convex optimization to verify conditions. However, in order to make the problem convex, the Lyapunov function shall be restricted to states whose dynamics is not directly affected by control inputs. This condition is very restrictive and commonly results in infeasible problem due to increased sufficiency. There are some other control problem closely related to simultaneous stabilization problem. Robust stabilization of uncertain system is one of the kind [11]. Tong [12] has a good survey on adaptive fuzzy control for uncertain nonlinear systems. Liu and Pang [13] focus on the problems of feedback passification and global stabilisation for a class of uncertain switched non-linear systems with parameter uncertainties and structural uncertainties. However, the mentioned approaches result in state dependant switching law. In simultaneous stabilization problem, we are searching for an offline controller and no specific assumption is presumed on nonlinear systems. In recent years, with the rapid development of numerical optimization methods, nonlinear control design based on convex optimization has attracted more attentions. Examples of convex optimizations are LMI-based TS fuzzy models [14] or Lyapunov convex optimization [10]. Another convex optimization approach is dual Lyapunov or density function approach which has used in this paper too. This new convergence and stability theory has been first presented in [15], however, a considerable attention has been paid to methodologies of density functions in recent years [16,17]. This is mainly due to better convexity properties. In Please cite this article as: P. Kohan-sedgh, A. Khayatian and N. Behmanesh-Fard, Simultaneous stabilization of polynomial nonlinear systems via density functions, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j. jfranklin.2019.11.033
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particular, convexity of synthesis problem is a remarkable feature of density functions. This feature allows synthesizing nonlinear controllers by sum-of-squares (SOS) and as a result, the synthesis problem reduces to a semidefinite program with standard LMIs [18]. The density approach guarantees almost stability which means the set of all points that does not converge to origin has zero measure in Lebesgue sense but cannot guarantee the asymptotic stability of the closed loop system. The relation between almost global stability and asymptotic stability is another interesting issue. This problem was first addressed by Rantzer [15] itself. It is shown that it is possible to derive a density function from a Lyapunov function. The result was extended in [19,20], ensure the existence of a density function for almost stability with local asymptotic stability. Masubuchi [21] focus of local results of almost stability theorem, where positive invariance (and stability) is proved for a given subset of the state space. Such local results are useful for nonlinear systems, since control problems are frequently formulated only around a specified point of the state space for nonlinear systems. In this paper, the simultaneous nonlinear feedback stabilizer problem for a set of polynomial systems is considered. Unlike robust stabilization counterparts, there is no assumption on interrelatedness of the systems. Based on almost stability theory and density function convergence criterion, a new synthesis method for finding almost globally simultaneous stabilizer is proposed. The conditions are affine functions of unknown polynomials and hence SOS decomposition can be accommodated to find simultaneous controller. Also a complementary condition is added to insure global simultaneous asymptotic stability. The condition can be used to check asymptotic stability after designing the simultaneous stabilizer or the condition can be added to the synthesis problem to ensure asymptotic stability from the very beginning. Another contribution of this paper is a convex methodology to find local simultaneous stabilizers based on almost stability theory. This method is of great importance when the objective is to stabilize a class of polynomial system around a specified point. Although the theories may seem a little complicated, but the designer can easily set up an SOS decomposition to find the controller and the controller can be easily implemented in real applications. The paper is organized as follows. Section 2 presents problem description and some preliminary definitions and lemmas in almost stability theory. Section 3 contains contribution on almost globally simultaneous stability. Complementary conditions on asymptotic stability will also be introduced in this section. Section 4 is dedicated for local simultaneous almost stability of polynomial systems. Some numerical examples are given in Section 5 to show the effectiveness of the proposed methods. The paper is concluded in Section 6. Notation: the following notation will be used throughout this paper: ∂V ∂V , V : Rn → R ∇V = ... ∂ x1 ∂ xn ∂ f1 ∂ fn ∇. f = + ... + , f : Rn → Rn ∂ x1 ∂ xn An open ball in Rn whose centre is the origin and whose radius is r is denoted by B(r). The property P (x)is true for almost all x ∈ S ⊂ Rn if the Lebesgue measure of the set {x ∈ S|P (x) is f alse}is zero. 2. Preliminaries In this section, Definitions of almost global stability and almost local stability are stated. Related theorems are also reviewed. Please cite this article as: P. Kohan-sedgh, A. Khayatian and N. Behmanesh-Fard, Simultaneous stabilization of polynomial nonlinear systems via density functions, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j. jfranklin.2019.11.033
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Consider a nonlinear system x˙ = f (x),
(1)
where x ∈ Rn and f ∈ C 1 (Rn , Rn ). Assume f (0) = 0and denote φ(t; x0 )the trajectory at the time tthat start at x0 . Definition 1. Consider the system (1). The origin x = 0is said to be almost globally stable (a.g.s.) if almost all trajectories converge to it, i.e. if the set {x0 ∈ Rn | lim φ(t; x0 ) = 0}, t→∞
(2)
has zero Lebesgue measure. Definition 2. Consider the system (1) and suppose that Sis a domain (connected open set) including the origin. The origin x = 0is said to be almost everywhere stable (a.e.s.) over S if φ(t; x0 )uniquely exist for all t ∈ [0, ∞ )and converge to 0 for almost all x0 ∈ S. Clearly almost globally stable and almost everywhere stable over Rn are the same. The following theorem from [15], states conditions for almost global stability of a nonlinear system. Lemma 1. Given the system (1), suppose there exist a non-negative ρ ∈ C 1 (Rn \{0}, R) such that ρ(x) f (x)/|x|is integrable on {x ∈ Rn | |x| ≥ 1}and ∇.( f ρ) (x ) > 0 f or almost all x, (3) then the origin is a.g.s. The non-negative ρ that satisfies both integrability condition and Eq. (3) is said to be density function for system (1). The following theorem, defines conditions under which a pre-defined level set is positively invariant and the system is a.e.s. over that. Lemma 2 [22]. Consider the system (1) and assume the system is analytic near origin. Define open set S(c)as S (c ) := {x ∈ R\{0} : ρ(x ) > c} ∪ {0},
(4)
and S0 (c)as the connected component of S(c)that contains the origin. Then S0 (c) is bounded and the system is a.e.s. over S0 (c) if the following conditions hold: a. b. c. d.
ρ(x) > 0 ∀x ∈ N0 \{0} [∇.( f ρ)](x) > 0 a.e. x ∈ N0 \{0} [∇. f ](x) ≤ 0 ∀x ∈ B(r) There exists a compact set L which includes the origin as an interior point and for which the function ρis bounded on L\N for every open set N such that 0 ∈ N ⊂ L. N0 is some domain including the origin and ρ ∈ C 1 (Rn \{0}, R).
3. Global simultaneous stability In this section, simultaneous stabilization of a collection of nonlinear systems is studied. Consider the following systems x˙ = fi (x ) + gi (x )u, i = 1, ..., r,
(5)
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where fi ∈ C 1 (Rn , Rn ), gi ∈ C 1 (Rn , Rm )and fi (0) = 0. Simultaneous stabilization problem is to find a single state feedback controller u = K (x)such thatx = 0is stable for all r systems. In order to be able to use SOS techniques and simplification, in the followings, fi and gi to polynomial functions will be restricted. However, all methods and theorems in this paper can be easily extended for rational functions with a little modification as it is shown in [23]. Theorem 1. Consider the system x˙ = fi (x) + gi (x)u, i = 1, ..., r where fi and gi are polynomial functions. Suppose that V (x) is a fixed polynomial such that V (x) ≥ 0, V (x) = 0 ⇔ x = 0 and deg (V (x ) ) ≥ deg (R (x ) ) + deg ( fi (x ) ) − 2 deg (V (x ) ) ≥ deg M j (x ) + deg (gi (x ) ) − 2 i = 1, ..., r
,
(6)
j = 1, ..., m Then the r systems are simultaneously a.g.s. withu = K (x) = M(x )/R(x ) if the polynomials R(x) and M(x) exist such that the following conditions hold R ( x ) > 0 ∀x ∈ R n ,
(7)
m ∂ M j (x ) ∂R (x ) ˜ ˜ ˜ fi (x ) + V fi (x )R (x ) + g˜ i j (x ) + Vgi j (x )M j (x ) > 0, ∂x ∂x j=1
(8)
a.e. x ∈ R , i = 1, ..., r, n
M (0 ) = 0, R (0 ) = 1 ,
(9)
Where f˜i (x ) := fi (x )V (x ), g˜ i j (x ) := gi j (x )V (x ), V˜ fi (x ) := ∇. fi (x )V (x ) − ∂V∂x(x ) fi (x ), V˜gi j (x ) := ∇.gi j (x )V (x ) − ∂V∂x(x ) gi j (x ),
(10)
gi j is the jth element of gi , i = 1, ..., rand j = 1, ..., m. Proof. considering state feedback controller, the closed-loop systems are x˙ = fcli (x ), fcli (x ) = fi (x ) + gi (x )K (x ) ,
(11)
i = 1, ..., r M(0) = 0and R(0) = 1 results in K (0) = 0, so the origin is still equilibrium of the closedloop system. Please cite this article as: P. Kohan-sedgh, A. Khayatian and N. Behmanesh-Fard, Simultaneous stabilization of polynomial nonlinear systems via density functions, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j. jfranklin.2019.11.033
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Defining ρ(x ) := R (x )/V (x ) , V (x ) ≥ 0
(12)
then (7) guarantee ρ(x) > 0 ∀x ∈ Rn \{0} . V (x) = 0 ⇔ x = 0guarantee bounded ρexcluding origin. Substituting ρ(x) = R(x )/V (x ) and K (x ) = M(x )/R(x )in [∇.( fcli ρ)](x ) > 0results in Eq. (8) and guarantee Eq. (3) in Lemma 1. It only remains to show integrability condition of the lemma. Note that ρ(x ) fcli (x ) R (x ) fi (x ) gi (x )M (x ) = + , |x | V (x )|x | V (x )|x |
(13)
Since all functions are polynomials and denominators are not zero in specified interval, integrability condition can be translated to some conditions on polynomials’ degree. It can be easily investigated that satisfying (6), will guarantee the integrability condition and the r systems are simultaneously a.g.s. Remark 1. The condition R(0) = 1 can be replaced with R(0) = 0 and does not have any impact on the result of the Theorem 1. R(0) = 1 is selected to simplify the equations in the subsequent theorems and can be relaxed to R(0) = 0 with little change in conditions. Remark 2. In Theorem 1 and other theorems in this paper, V (x)is a fixed scalar polynomial that should be selected by the designer. V (x) is inversely proportional to density function (see Eq. (12)). CLF is also inversely proportional to density function [15]. Hence, a good practice for selecting V (x) is V (x) = V¯ α (x)where V¯ (x)is selected to be the CLF of the linearized system. Then αshould be selected to satisfy (6). [24] offers a systematic method for finding CLF of linear systems. If the CLF cannot be found for the linearized system, then V¯ (x)can be selected to be any positive scalar polynomial satisfying V¯ (x) = 0 ⇔ x = 0and α to satisfy (6). Remark 3. Inequality Eqs. (7) and (8) are affine with respect to M(x)and R(x). Hence one can formulate an SOS problem to find these variable polynomials satisfying Theorem 1 conditions. The strength of Theorem 1 in comparison to other SOS Lyapunov based nonlinear simultaneous problems, lies in the ability to find a convex synthesis problem with less conservatism. In Lyapunov based methods, the set of Lyapunov function and simultaneous stabilizer are not convex and may not even be connected [25]. In order to be able to deal numerically with these problems, sufficient conditions shall be used. As an example, Theorem 1 of [10] restrict Lyapunov functions to only depends on states which are not directly affected by control inputs. Consequently the results are widely conservatism and the feasibility region of the problem is very limited. However, in density function based methods, the set of (u, uρ)is always convex. The only conservatism source in Theorem 1 is selecting common density function for all systems. It is said that the density approach has better convexity properties but the result, i.e. almost stability, is somehow weaker than asymptotic stability. It is of great importance to have a simple verification method for asymptotic stability. In the following the result of [20] will be used not only to find a verification method but also as a synthesis method to guarantee both almost global stability and local asymptotic stability for a set of r systems. Please cite this article as: P. Kohan-sedgh, A. Khayatian and N. Behmanesh-Fard, Simultaneous stabilization of polynomial nonlinear systems via density functions, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j. jfranklin.2019.11.033
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Lemma 3 [20]. Given the system (1), suppose the system admits a density function ρ such that [∇.( f ρ)](x) > 0 and ρ˙ > 0, f or all x ∈ D ⊂ Rn ,
(14)
Then the origin is a.g.s. and locally asymptotically stable. Here D ⊂ R denotes a neighbourhood of the origin except the origin itself. n
The following theorem gives condition for to verify if the designed simultaneous controller in Theorem 1 provides simultaneous asymptotic stability or not. Theorem 2. consider the system x˙ = fi (x) + gi (x)u, i = 1, ..., r where fi and gi are polynomial functions. Suppose there exists polynomials R(x)and M(x) such that the conditions of Theorem 1 hold. Then the set of r systems are simultaneously asymptotically stable if the following inequality holds. ∂ f i (x ) ∂M (x ) Tr + gi ( x ) < 0, i = 1, ..., r (15) ∂x ∂x x=0 Proof. Since conditions of Theorem 1 hold, then Eq. (15) implies ∂ f c li ( x ) Tr < 0, i = 1, ..., r ∂x x=0
(16)
Note that fcli (x) = Ai x + O(x 2 ) and fi and gi are analytic near origin. So Eq. (16) implies ∇. fcli (0) < 0. Since fcli is also continuous in the neighbourhood of the origin, there exist a neighbourhood Dof the origin except the origin itself such that ∇. fcli (x) < 0, f or all x ∈ D. Then ρ˙ = ∇ρ. fcli = ∇. fcli ρ − ρ∇. fcli > 0, (17) f or all x ∈ D Considering Eq. (17), conditions of Lemma 3 are satisfied and then fcli (x)is simultaneously asymptotically stable. This completes the proof. Note that Eq. (15) is also affine with respect to polynomials M(x). This means that the condition (15) can also be added to the synthesis problem to form an SOS problem both guarantee the almost global stability and asymptotic stability. The following theorem concludes this section. Theorem 3. consider the system x˙ = fi (x) + gi (x)u, i = 1, ..., r where fi and gi are polynomial functions. Suppose that V (x) is a fixed polynomial such that V (x) ≥ 0, V (x) = 0 ⇔ x = 0 and satisfies (6). Then the r systems are simultaneously a.g.s. and asymptotically stable withK (x) = M(x )/R(x ) if the polynomials R(x)and M(x)exist such that the conditions (7),(8),(9) and (15) are satisfied. 4. Local simultaneous stability Almost everywhere global stability was discussed in previous sections. In this section, the focus will be on local results where positive invariance and stability are proved for a given subset of the state space. Such local results are useful for nonlinear system, since the control problems are frequently formulated around a specified point of the state space. These local results can guarantee that a given set is positively invariant. Please cite this article as: P. Kohan-sedgh, A. Khayatian and N. Behmanesh-Fard, Simultaneous stabilization of polynomial nonlinear systems via density functions, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j. jfranklin.2019.11.033
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Theorem 4. Consider the system x˙ = fi (x) + gi (x)u, i = 1, ..., r where fi and gi are polynomial functions. Suppose that V (x) is a fixed polynomial such that V (x) ≥ 0and V (x) = 0 ⇔ x = 0.Then the r systems are simultaneously a.e.s. over bounded level set,S0 (c), withu = K (x) = M(x )/R(x ) if the polynomials R(x)and M(x)exist such that the following conditions hold R(x) > 0 ∀x ∈ N0 ,
(18)
m ∂ Mi ( x ) ∂R (x ) ˜ ˜ ˜ fi (x ) + V fi (x )R (x ) + g˜ i j (x ) + Vgi j (x )M j (x ) > 0, ∂x ∂x j=1
(19)
a.e. x ∈ N0 , i = 1, ..., r ∂ f i (x ) ∂M (x ) Tr + gi ( x ) < 0, ∂x ∂x x=0 i = 1, ..., r
(20)
M (0 ) = 0, R (0 ) = 1,
(21)
Where f˜i (x ) := fi (x )V (x ), g˜ i j (x ) := gi j (x )V (x ), ∂V (x ) V˜ fi (x ) := ∇. fi (x )V (x ) − fi (x ), ∂x ∂V (x ) V˜gi j (x ) := ∇.gi j (x )V (x ) − gi j (x ), ∂x gi j is the jth element of gi , i = 1, ..., rand j = 1, ..., m.
(22)
Proof. considering state feedback controller, the closed-loop systems are x˙ = fcli (x ), fcli (x ) = fi (x ) + gi (x )k (x ), i = 1, ..., r
.
(23)
M(0) = 0and R(0) = 0results in K (0) = 0, so the origin is still equilibrium of the closedloop system. Using Lemma 2 and Definition 2, there should be a positive density function ρ ∈ C 1 (Rn \{0}, R). Defining ρ(x) := R(x )/V (x ) and V (x ) ≥ 0, Eq. (7) guarantee condition a of Lemma 2. V (x) = 0 ⇔ x = 0guarantee bounded ρexcluding origin and satisfies condition d of the lemma. Substituting ρ(x) = R(x )/V (x ) and K (x ) = M(x )/R(x )in [∇.( fcli ρ)](x ) > 0results in Eq. (8) and guarantee condition b for i = 1, ..., r. It only remains to prove condition c. Since fcli (x) = Ai x + O(x 2 ), and [∇. fcli ](x) = T r Ai + O(x), then ∂ f i (x ) ∂M (x ) T r Ai = T r + gi ( x ) < 0. (24) ∂x ∂x x=0 Considering fi and gi are analytic near origin, condition c of Lemma 2 is guaranteed for i = 1, ..., r. This completes the proof. Remark: Theorem 4 is affine with respect to polynomials R(x)and M(x). In order to formulate SOS problem, it should be assumed that the set N0 , some domain including the origin, is Please cite this article as: P. Kohan-sedgh, A. Khayatian and N. Behmanesh-Fard, Simultaneous stabilization of polynomial nonlinear systems via density functions, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j. jfranklin.2019.11.033
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given by N0 = {x ∈ Rn : ϕi (x) < 0, i = 1, ..., p} and polynomial ϕi (x)is chosen so that 0 ∈ N0 . Then one can formulate an SOS problem to find parameterized polynomials R(x)and M(x). 5. Some examples In this section, several examples will be given to illustrate the effectiveness of the proposed theorem. The implementations are done in Matlab 7.10.0.499 (R2010a) running on a PC Desktop Intel® Core i3 and 4 GB RAM. SOSTOOLS (ver.3.0) [26] with LMI solver SDPT3 is also used. In order to reduce the number of nonzero terms in resulting polynomials, L1 norm optimization has been performed on polynomial coefficients in all problems [27,28]. Example 1. consider the following systems x˙1 = −8x1 x2 2 − x1 2 x2 + 2x2 3 P1 : x˙2 = x2 u x˙1 = −6x1 x2 2 + x1 2 x2 P2 : x˙2 = −2x1 3 + (x1 + x2 )u
(25)
For the systemP1 , every (x1 , x2 )with x2 = 0is necessarily an equilibrium point even for the closed loop system, so the origin cannot be made asymptotically stable. Considering this point, Lyapunov-based methods will fail to find simultaneous stabilizer, while it is possible to design almost stable simultaneous controller. In order to find a polynomial controller,R(x)is selected to be a constant scalar and M(x) to be a general second order polynomial. In order to satisfy the inequalities (6) of Theorem 1, a general non-negative function V (x) = (x1 2 + x2 2 )3 is selected. Solving the SOS problem of Theorem 1 to find the parameterized polynomialM(x), results M(x) = u(x) = 0.1305x1 2 − 3.1993x2 2
(26)
The phase portrait of closed-loop systems are presented in Fig. 1. This example is chosen to show the strength of this method. The system P1 and P2 cannot be simultaneously stabilized by ordinary methods because no Lyapunov function can be found for these systems and robust control methods are also not applicable because there is no assumption on interrelatedness of the systems. Clearly, the conditions of Theorem 2 have not been met and simultaneous asymptotic stability has not been achieved, however the closed-loop system of P2 is asymptotically stable. Example 2. consider the following systems x˙1 = x2 − x1 3 + x1 2 P1 : x˙2 = u x˙1 = x2 − 3x1 3 + x2 2 P2 : x˙2 = x1 2 − x1 + u
(27)
For the systemP1 , the CLF ρ(x) = 3x1 2 + 2x2 2 + 2x1 x2 can be easily found. The CLF is the first candidate for V (x). In order to satisfy the inequalities (6) and SOS problem of Theorem 1, V (x) = (3x1 2 + 2x2 2 + 2x1 x2 )6 has been chosen with trial and error methods. As the previous example, R(x)is selected to be a constant scalar. Please cite this article as: P. Kohan-sedgh, A. Khayatian and N. Behmanesh-Fard, Simultaneous stabilization of polynomial nonlinear systems via density functions, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j. jfranklin.2019.11.033
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Fig. 1. Phase portrait of closed-loop system in Example 1, (a) P1 (b) P2 .
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In this example, the approach of Theorem 3 is followed to reach asymptotic simultaneous stabilizer. The resulting polynomial controller is u (x ) = M (x ) = −0.6370x2 3 − 0.0456x2 2 − 0.8591x2 − 1.0182x1
(28)
The phase portrait of closed-loop systems are presented in Fig. 2. As it is shown, the origin is asymptotically stable for both systems. It can also be checked that the conditions (15) are well satisfied. Example 3. Fault Tolerant Control As the complexity of the systems are increased, they are more susceptible to failures. A fault-tolerant design enables a system to continue its intended operation, possibly at a reduced level, rather than failing completely, when some part of the system fails. Simultaneous stabilization is a low cost approach in passive fault tolerant control. This example, mainly discuss two kind of faults for the actuator, outage and loss of effectiveness. Consider the following system x˙1 = x2 − x1 3 + x1 2 + u2 PN : (29) x˙2 = u1 + u2 Where x = [ x1 x2 ] ∈ R2 and u = [ u1 u2 ] ∈ R2 . The system is unstable at the origin and it is supposed that the actuator is subject to failure. The aim is to asymptotically stabilize the system under normal operation, denoted by PN as well as asymptotically stabilizing faulty operation conditions, denoted by PF . 5.1. Outage Here, it is supposed that the control input u1 produces zero forces and moment, i.e. x˙1 = x2 − x1 3 + x1 2 + u2 PF : (30) x˙2 = u2 Choosing V (x) = (2x1 2 + x1 x2 + x2 2 )4 and R(x)a constant scalar, it can be found that all condition of Theorem 3 are met by the following controller and consequently both systems are simultaneously asymptotically stabilized. u1 (x ) = M1 (x ) = −0.0884x2 3 − 0.5700x2 − 1.3374x1 u2 (x ) = M2 (x ) = −0.0388x2 − 0.3182x1
(31)
The phase portrait of closed-loop systems in normal and faulty mode are presented in Fig. 3. 5.2. Loss of effectiveness In this case, the actuator gain in a deflection is smaller than the commanded position, i.e. x˙1 = x2 − x1 3 + x1 2 + β1 u2 PF : (32) x˙2 = β2 u1 + β1 u2 Please cite this article as: P. Kohan-sedgh, A. Khayatian and N. Behmanesh-Fard, Simultaneous stabilization of polynomial nonlinear systems via density functions, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j. jfranklin.2019.11.033
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Fig. 2. Phase portrait of closed-loop system in Example 2, (a) P1 (b) P2 .
Where β1 and β2 are deflected values. It is assume that the deflection reduce the controller commands to half of its value; then by selecting β1 and β2 to be either 1 or 0.5, a collection of 4 systems are required to be simultaneously stabilized. Choosing V (x) = (2x1 2 + x1 x2 + x2 2 )4 and R(x)a constant scalar, the configuration of Theorem 3 result in the following simultaneous Please cite this article as: P. Kohan-sedgh, A. Khayatian and N. Behmanesh-Fard, Simultaneous stabilization of polynomial nonlinear systems via density functions, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j. jfranklin.2019.11.033
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Fig. 3. Phase portrait of closed-loop system in Example 3, (a) Normal Mode (b) Faulty Mode.
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Fig. 4. Phase portrait of closed-loop system in Example 3, (a) β1 = 1, β2 = 1 (b) β1 = 1, β2 = 0.5 (c) β1 = 0.5, β2 = 1 (d) β1 = 0.5, β2 = 0.5.
asymptotic stabilizer, u1 (x ) = M1 (x ) = −0.1765x2 3 u2 (x ) = M2 (x ) = −0.6942x1 − 1.0558x2
(33)
The phase portrait of closed-loop systems in normal and faulty mode are presented in Fig. 4. Please cite this article as: P. Kohan-sedgh, A. Khayatian and N. Behmanesh-Fard, Simultaneous stabilization of polynomial nonlinear systems via density functions, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j. jfranklin.2019.11.033
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Fig. 4. Continued
6. Concluding remarks In this paper, the problem of designing simultaneous feedback controllers for polynomial nonlinear systems has been studied. Based on almost stability theory, both local and global stabilization has been studied. The proposed conditions on simultaneous stability are affine with respect to unknown polynomials and hence it is possible to formulate SOS synthesis Please cite this article as: P. Kohan-sedgh, A. Khayatian and N. Behmanesh-Fard, Simultaneous stabilization of polynomial nonlinear systems via density functions, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j. jfranklin.2019.11.033
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problem. In order to check asymptotic stability, some extra conditions on system and variable polynomials has been introduced. These conditions can be used after controller synthesis to check the asymptotic stability or can be used in SOS synthesis problem to guarantee simultaneous asymptotic stabilizer asymptotic stability or can be used in SOS synthesis problem to guarantee simultaneous asymptotic stabilizer from the beginning. Numerical examples and a fault tolerant design has been used to show the effectiveness of the global stabilization method. Regarding local simultaneous stabilization, there are still works to be done to find a good application example. This may be addressed in future works.
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[24] X.S. Cai, Z.Z. Han, Universal construction of control Lyapunov functions for linear systems, Latin Am. Appl. Res. 36 (1) (2006) 15–22. [25] C. Prieur, L. Praly, Uniting local and global controllers, in: Proceedings of the 38th IEEE Conference on Decision and Control (Cat. No. 99CH36304), 2, IEEE, 1999. [26] S. Prajna, A. Papachristodoulou, P.A. Parrilo, Introducing SOSTOOLS: a general purpose sum of squares programming solver, in: Proceedings of the 41st IEEE Conference on Decision and Control, 2002, 1, IEEE, 2002. [27] E.J. Candes, M.B. Wakin, S.P. Boyd, Enhancing sparsity by reweighted 1 minimization, J. Fourier Anal. Appl. 14 (5–6) (2008) 877–905. [28] C.F. Tong, F. Wang, Control synthesis for polynomial nonlinear systems, in: Proceedings of the IEEE International Conference on Automation and Logistics, IEEE, 2009.
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Simultaneous stabilization of polynomial nonlinear systems via density functions Peyman Kohan-sedgh a,∗, Alireza Khayatian a, Navid Behmanesh-Fard b a School
b Department
of Electrical & Computer Engineering, Shiraz University, Shiraz, Iran of Electrical Engineering, Faculty of Darab Branch, Technical and Vocational University, Fars, Iran
Received 12 August 2018; received in revised form 4 July 2019; accepted 12 November 2019 Available online xxx
Abstract This paper is concerned with simultaneous stabilization of a class of polynomial nonlinear systems with almost stability theory. The theory uses dual Lyapunov or density function criterion which has remarkable advantages over Lyapunov-based simultaneous stabilization. These advantages rely in convexity property of density functions. This property results in controller synthesis problems which are affine with respect to unknown polynomials variables. The results of almost stability are also extended to simultaneous asymptotic stability at the expense of adding some extra affine polynomial inequalities. Numerical method for verification of positivity of multivariate polynomials based on sum of square decomposition is used. Numerical examples and a fault tolerant design scheme are used to show the effectiveness of the proposed methods. © 2019 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
1. Introduction Simultaneous stabilization is the problem of determining a single controller, which simultaneously stabilizes a finite collection of plants. In practice, due to plant uncertainty, plant variation, failure modes or plants with various modes of operation, the simultaneous stabilization problem is frequently encountered. The problem is a subtopic of robust control
∗
Corresponding author. E-mail addresses: [email protected] (P. Kohan-sedgh), [email protected] (A. Khayatian), [email protected] (N. Behmanesh-Fard). https://doi.org/10.1016/j.jfranklin.2019.11.033 0016-0032/© 2019 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
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and a closely related problem is robust stabilization of systems with polytopic or parametric uncertainties. Since the work of [1], simultaneous stabilization has received considerable attention. In [2], complexity of simultaneous stabilization has been discussed and it has been proved that the simultaneous stabilization of three linear systems is rationally undecidable. As a result, the burden of finding linear simultaneous stabilizer is mostly on sufficient conditions and finding methods to reduce conservatism is of high importance. Good literature review on simultaneous stabilization and increasing the feasibility region for linear systems can be found on [3,4] and references there in. It is well-known that the dynamic nonlinearities are inherent in almost all the physical systems. Unfortunately due to complexity of nonlinear systems, linear simultaneous approaches cannot be directly generalized for nonlinear systems and nonlinear control synthesis shall be used in this regard. Several nonlinear control approaches has been developed during past decades [5] and many of which has been generalized or modified to be used in new applications recently [6,7]. However, only a few works has addressed the problem of simultaneous stabilization directly because of difficulties involved in synthesis of nonlinear stabilizers. Ho-Mock-Qai and Dayawansa [8] proved that for any countable family of stabilizable nonlinear systems, a continuous state feedback law, which simultaneously stabilizes the family (non-asymptotically), always exists. Additionally, a sufficient condition for the existence of simultaneously asymptotically stabilizing controllers for a collection of nonlinear systems was provided. Wu [9] presents a method for designing simultaneous stabilizer of a collection single-input non-linear systems based on Control Lyapunov Function (CLF). However, both above mentioned methods are difficult to verify as they depend firstly on finding an asymptotically stabilizing state feedback controller or a CLF for each system, then determining whether some conditions are satisfied. Moreover, the obtained simultaneously asymptotically stabilizing controllers are not easy to implement. Xu et al.[10] addresses the problem of finding simultaneous controller for polynomial nonlinear systems based on state-dependant polynomial Lyapunov functions. The advantages of this method is capability of using convex optimization to verify conditions. However, in order to make the problem convex, the Lyapunov function shall be restricted to states whose dynamics is not directly affected by control inputs. This condition is very restrictive and commonly results in infeasible problem due to increased sufficiency. There are some other control problem closely related to simultaneous stabilization problem. Robust stabilization of uncertain system is one of the kind [11]. Tong [12] has a good survey on adaptive fuzzy control for uncertain nonlinear systems. Liu and Pang [13] focus on the problems of feedback passification and global stabilisation for a class of uncertain switched non-linear systems with parameter uncertainties and structural uncertainties. However, the mentioned approaches result in state dependant switching law. In simultaneous stabilization problem, we are searching for an offline controller and no specific assumption is presumed on nonlinear systems. In recent years, with the rapid development of numerical optimization methods, nonlinear control design based on convex optimization has attracted more attentions. Examples of convex optimizations are LMI-based TS fuzzy models [14] or Lyapunov convex optimization [10]. Another convex optimization approach is dual Lyapunov or density function approach which has used in this paper too. This new convergence and stability theory has been first presented in [15], however, a considerable attention has been paid to methodologies of density functions in recent years [16,17]. This is mainly due to better convexity properties. In Please cite this article as: P. Kohan-sedgh, A. Khayatian and N. Behmanesh-Fard, Simultaneous stabilization of polynomial nonlinear systems via density functions, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j. jfranklin.2019.11.033
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particular, convexity of synthesis problem is a remarkable feature of density functions. This feature allows synthesizing nonlinear controllers by sum-of-squares (SOS) and as a result, the synthesis problem reduces to a semidefinite program with standard LMIs [18]. The density approach guarantees almost stability which means the set of all points that does not converge to origin has zero measure in Lebesgue sense but cannot guarantee the asymptotic stability of the closed loop system. The relation between almost global stability and asymptotic stability is another interesting issue. This problem was first addressed by Rantzer [15] itself. It is shown that it is possible to derive a density function from a Lyapunov function. The result was extended in [19,20], ensure the existence of a density function for almost stability with local asymptotic stability. Masubuchi [21] focus of local results of almost stability theorem, where positive invariance (and stability) is proved for a given subset of the state space. Such local results are useful for nonlinear systems, since control problems are frequently formulated only around a specified point of the state space for nonlinear systems. In this paper, the simultaneous nonlinear feedback stabilizer problem for a set of polynomial systems is considered. Unlike robust stabilization counterparts, there is no assumption on interrelatedness of the systems. Based on almost stability theory and density function convergence criterion, a new synthesis method for finding almost globally simultaneous stabilizer is proposed. The conditions are affine functions of unknown polynomials and hence SOS decomposition can be accommodated to find simultaneous controller. Also a complementary condition is added to insure global simultaneous asymptotic stability. The condition can be used to check asymptotic stability after designing the simultaneous stabilizer or the condition can be added to the synthesis problem to ensure asymptotic stability from the very beginning. Another contribution of this paper is a convex methodology to find local simultaneous stabilizers based on almost stability theory. This method is of great importance when the objective is to stabilize a class of polynomial system around a specified point. Although the theories may seem a little complicated, but the designer can easily set up an SOS decomposition to find the controller and the controller can be easily implemented in real applications. The paper is organized as follows. Section 2 presents problem description and some preliminary definitions and lemmas in almost stability theory. Section 3 contains contribution on almost globally simultaneous stability. Complementary conditions on asymptotic stability will also be introduced in this section. Section 4 is dedicated for local simultaneous almost stability of polynomial systems. Some numerical examples are given in Section 5 to show the effectiveness of the proposed methods. The paper is concluded in Section 6. Notation: the following notation will be used throughout this paper: ∂V ∂V , V : Rn → R ∇V = ... ∂ x1 ∂ xn ∂ f1 ∂ fn ∇. f = + ... + , f : Rn → Rn ∂ x1 ∂ xn An open ball in Rn whose centre is the origin and whose radius is r is denoted by B(r). The property P (x)is true for almost all x ∈ S ⊂ Rn if the Lebesgue measure of the set {x ∈ S|P (x) is f alse}is zero. 2. Preliminaries In this section, Definitions of almost global stability and almost local stability are stated. Related theorems are also reviewed. Please cite this article as: P. Kohan-sedgh, A. Khayatian and N. Behmanesh-Fard, Simultaneous stabilization of polynomial nonlinear systems via density functions, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j. jfranklin.2019.11.033
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Consider a nonlinear system x˙ = f (x),
(1)
where x ∈ Rn and f ∈ C 1 (Rn , Rn ). Assume f (0) = 0and denote φ(t; x0 )the trajectory at the time tthat start at x0 . Definition 1. Consider the system (1). The origin x = 0is said to be almost globally stable (a.g.s.) if almost all trajectories converge to it, i.e. if the set {x0 ∈ Rn | lim φ(t; x0 ) = 0}, t→∞
(2)
has zero Lebesgue measure. Definition 2. Consider the system (1) and suppose that Sis a domain (connected open set) including the origin. The origin x = 0is said to be almost everywhere stable (a.e.s.) over S if φ(t; x0 )uniquely exist for all t ∈ [0, ∞ )and converge to 0 for almost all x0 ∈ S. Clearly almost globally stable and almost everywhere stable over Rn are the same. The following theorem from [15], states conditions for almost global stability of a nonlinear system. Lemma 1. Given the system (1), suppose there exist a non-negative ρ ∈ C 1 (Rn \{0}, R) such that ρ(x) f (x)/|x|is integrable on {x ∈ Rn | |x| ≥ 1}and ∇.( f ρ) (x ) > 0 f or almost all x, (3) then the origin is a.g.s. The non-negative ρ that satisfies both integrability condition and Eq. (3) is said to be density function for system (1). The following theorem, defines conditions under which a pre-defined level set is positively invariant and the system is a.e.s. over that. Lemma 2 [22]. Consider the system (1) and assume the system is analytic near origin. Define open set S(c)as S (c ) := {x ∈ R\{0} : ρ(x ) > c} ∪ {0},
(4)
and S0 (c)as the connected component of S(c)that contains the origin. Then S0 (c) is bounded and the system is a.e.s. over S0 (c) if the following conditions hold: a. b. c. d.
ρ(x) > 0 ∀x ∈ N0 \{0} [∇.( f ρ)](x) > 0 a.e. x ∈ N0 \{0} [∇. f ](x) ≤ 0 ∀x ∈ B(r) There exists a compact set L which includes the origin as an interior point and for which the function ρis bounded on L\N for every open set N such that 0 ∈ N ⊂ L. N0 is some domain including the origin and ρ ∈ C 1 (Rn \{0}, R).
3. Global simultaneous stability In this section, simultaneous stabilization of a collection of nonlinear systems is studied. Consider the following systems x˙ = fi (x ) + gi (x )u, i = 1, ..., r,
(5)
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where fi ∈ C 1 (Rn , Rn ), gi ∈ C 1 (Rn , Rm )and fi (0) = 0. Simultaneous stabilization problem is to find a single state feedback controller u = K (x)such thatx = 0is stable for all r systems. In order to be able to use SOS techniques and simplification, in the followings, fi and gi to polynomial functions will be restricted. However, all methods and theorems in this paper can be easily extended for rational functions with a little modification as it is shown in [23]. Theorem 1. Consider the system x˙ = fi (x) + gi (x)u, i = 1, ..., r where fi and gi are polynomial functions. Suppose that V (x) is a fixed polynomial such that V (x) ≥ 0, V (x) = 0 ⇔ x = 0 and deg (V (x ) ) ≥ deg (R (x ) ) + deg ( fi (x ) ) − 2 deg (V (x ) ) ≥ deg M j (x ) + deg (gi (x ) ) − 2 i = 1, ..., r
,
(6)
j = 1, ..., m Then the r systems are simultaneously a.g.s. withu = K (x) = M(x )/R(x ) if the polynomials R(x) and M(x) exist such that the following conditions hold R ( x ) > 0 ∀x ∈ R n ,
(7)
m ∂ M j (x ) ∂R (x ) ˜ ˜ ˜ fi (x ) + V fi (x )R (x ) + g˜ i j (x ) + Vgi j (x )M j (x ) > 0, ∂x ∂x j=1
(8)
a.e. x ∈ R , i = 1, ..., r, n
M (0 ) = 0, R (0 ) = 1 ,
(9)
Where f˜i (x ) := fi (x )V (x ), g˜ i j (x ) := gi j (x )V (x ), V˜ fi (x ) := ∇. fi (x )V (x ) − ∂V∂x(x ) fi (x ), V˜gi j (x ) := ∇.gi j (x )V (x ) − ∂V∂x(x ) gi j (x ),
(10)
gi j is the jth element of gi , i = 1, ..., rand j = 1, ..., m. Proof. considering state feedback controller, the closed-loop systems are x˙ = fcli (x ), fcli (x ) = fi (x ) + gi (x )K (x ) ,
(11)
i = 1, ..., r M(0) = 0and R(0) = 1 results in K (0) = 0, so the origin is still equilibrium of the closedloop system. Please cite this article as: P. Kohan-sedgh, A. Khayatian and N. Behmanesh-Fard, Simultaneous stabilization of polynomial nonlinear systems via density functions, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j. jfranklin.2019.11.033
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Defining ρ(x ) := R (x )/V (x ) , V (x ) ≥ 0
(12)
then (7) guarantee ρ(x) > 0 ∀x ∈ Rn \{0} . V (x) = 0 ⇔ x = 0guarantee bounded ρexcluding origin. Substituting ρ(x) = R(x )/V (x ) and K (x ) = M(x )/R(x )in [∇.( fcli ρ)](x ) > 0results in Eq. (8) and guarantee Eq. (3) in Lemma 1. It only remains to show integrability condition of the lemma. Note that ρ(x ) fcli (x ) R (x ) fi (x ) gi (x )M (x ) = + , |x | V (x )|x | V (x )|x |
(13)
Since all functions are polynomials and denominators are not zero in specified interval, integrability condition can be translated to some conditions on polynomials’ degree. It can be easily investigated that satisfying (6), will guarantee the integrability condition and the r systems are simultaneously a.g.s. Remark 1. The condition R(0) = 1 can be replaced with R(0) = 0 and does not have any impact on the result of the Theorem 1. R(0) = 1 is selected to simplify the equations in the subsequent theorems and can be relaxed to R(0) = 0 with little change in conditions. Remark 2. In Theorem 1 and other theorems in this paper, V (x)is a fixed scalar polynomial that should be selected by the designer. V (x) is inversely proportional to density function (see Eq. (12)). CLF is also inversely proportional to density function [15]. Hence, a good practice for selecting V (x) is V (x) = V¯ α (x)where V¯ (x)is selected to be the CLF of the linearized system. Then αshould be selected to satisfy (6). [24] offers a systematic method for finding CLF of linear systems. If the CLF cannot be found for the linearized system, then V¯ (x)can be selected to be any positive scalar polynomial satisfying V¯ (x) = 0 ⇔ x = 0and α to satisfy (6). Remark 3. Inequality Eqs. (7) and (8) are affine with respect to M(x)and R(x). Hence one can formulate an SOS problem to find these variable polynomials satisfying Theorem 1 conditions. The strength of Theorem 1 in comparison to other SOS Lyapunov based nonlinear simultaneous problems, lies in the ability to find a convex synthesis problem with less conservatism. In Lyapunov based methods, the set of Lyapunov function and simultaneous stabilizer are not convex and may not even be connected [25]. In order to be able to deal numerically with these problems, sufficient conditions shall be used. As an example, Theorem 1 of [10] restrict Lyapunov functions to only depends on states which are not directly affected by control inputs. Consequently the results are widely conservatism and the feasibility region of the problem is very limited. However, in density function based methods, the set of (u, uρ)is always convex. The only conservatism source in Theorem 1 is selecting common density function for all systems. It is said that the density approach has better convexity properties but the result, i.e. almost stability, is somehow weaker than asymptotic stability. It is of great importance to have a simple verification method for asymptotic stability. In the following the result of [20] will be used not only to find a verification method but also as a synthesis method to guarantee both almost global stability and local asymptotic stability for a set of r systems. Please cite this article as: P. Kohan-sedgh, A. Khayatian and N. Behmanesh-Fard, Simultaneous stabilization of polynomial nonlinear systems via density functions, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j. jfranklin.2019.11.033
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Lemma 3 [20]. Given the system (1), suppose the system admits a density function ρ such that [∇.( f ρ)](x) > 0 and ρ˙ > 0, f or all x ∈ D ⊂ Rn ,
(14)
Then the origin is a.g.s. and locally asymptotically stable. Here D ⊂ R denotes a neighbourhood of the origin except the origin itself. n
The following theorem gives condition for to verify if the designed simultaneous controller in Theorem 1 provides simultaneous asymptotic stability or not. Theorem 2. consider the system x˙ = fi (x) + gi (x)u, i = 1, ..., r where fi and gi are polynomial functions. Suppose there exists polynomials R(x)and M(x) such that the conditions of Theorem 1 hold. Then the set of r systems are simultaneously asymptotically stable if the following inequality holds. ∂ f i (x ) ∂M (x ) Tr + gi ( x ) < 0, i = 1, ..., r (15) ∂x ∂x x=0 Proof. Since conditions of Theorem 1 hold, then Eq. (15) implies ∂ f c li ( x ) Tr < 0, i = 1, ..., r ∂x x=0
(16)
Note that fcli (x) = Ai x + O(x 2 ) and fi and gi are analytic near origin. So Eq. (16) implies ∇. fcli (0) < 0. Since fcli is also continuous in the neighbourhood of the origin, there exist a neighbourhood Dof the origin except the origin itself such that ∇. fcli (x) < 0, f or all x ∈ D. Then ρ˙ = ∇ρ. fcli = ∇. fcli ρ − ρ∇. fcli > 0, (17) f or all x ∈ D Considering Eq. (17), conditions of Lemma 3 are satisfied and then fcli (x)is simultaneously asymptotically stable. This completes the proof. Note that Eq. (15) is also affine with respect to polynomials M(x). This means that the condition (15) can also be added to the synthesis problem to form an SOS problem both guarantee the almost global stability and asymptotic stability. The following theorem concludes this section. Theorem 3. consider the system x˙ = fi (x) + gi (x)u, i = 1, ..., r where fi and gi are polynomial functions. Suppose that V (x) is a fixed polynomial such that V (x) ≥ 0, V (x) = 0 ⇔ x = 0 and satisfies (6). Then the r systems are simultaneously a.g.s. and asymptotically stable withK (x) = M(x )/R(x ) if the polynomials R(x)and M(x)exist such that the conditions (7),(8),(9) and (15) are satisfied. 4. Local simultaneous stability Almost everywhere global stability was discussed in previous sections. In this section, the focus will be on local results where positive invariance and stability are proved for a given subset of the state space. Such local results are useful for nonlinear system, since the control problems are frequently formulated around a specified point of the state space. These local results can guarantee that a given set is positively invariant. Please cite this article as: P. Kohan-sedgh, A. Khayatian and N. Behmanesh-Fard, Simultaneous stabilization of polynomial nonlinear systems via density functions, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j. jfranklin.2019.11.033
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Theorem 4. Consider the system x˙ = fi (x) + gi (x)u, i = 1, ..., r where fi and gi are polynomial functions. Suppose that V (x) is a fixed polynomial such that V (x) ≥ 0and V (x) = 0 ⇔ x = 0.Then the r systems are simultaneously a.e.s. over bounded level set,S0 (c), withu = K (x) = M(x )/R(x ) if the polynomials R(x)and M(x)exist such that the following conditions hold R(x) > 0 ∀x ∈ N0 ,
(18)
m ∂ Mi ( x ) ∂R (x ) ˜ ˜ ˜ fi (x ) + V fi (x )R (x ) + g˜ i j (x ) + Vgi j (x )M j (x ) > 0, ∂x ∂x j=1
(19)
a.e. x ∈ N0 , i = 1, ..., r ∂ f i (x ) ∂M (x ) Tr + gi ( x ) < 0, ∂x ∂x x=0 i = 1, ..., r
(20)
M (0 ) = 0, R (0 ) = 1,
(21)
Where f˜i (x ) := fi (x )V (x ), g˜ i j (x ) := gi j (x )V (x ), ∂V (x ) V˜ fi (x ) := ∇. fi (x )V (x ) − fi (x ), ∂x ∂V (x ) V˜gi j (x ) := ∇.gi j (x )V (x ) − gi j (x ), ∂x gi j is the jth element of gi , i = 1, ..., rand j = 1, ..., m.
(22)
Proof. considering state feedback controller, the closed-loop systems are x˙ = fcli (x ), fcli (x ) = fi (x ) + gi (x )k (x ), i = 1, ..., r
.
(23)
M(0) = 0and R(0) = 0results in K (0) = 0, so the origin is still equilibrium of the closedloop system. Using Lemma 2 and Definition 2, there should be a positive density function ρ ∈ C 1 (Rn \{0}, R). Defining ρ(x) := R(x )/V (x ) and V (x ) ≥ 0, Eq. (7) guarantee condition a of Lemma 2. V (x) = 0 ⇔ x = 0guarantee bounded ρexcluding origin and satisfies condition d of the lemma. Substituting ρ(x) = R(x )/V (x ) and K (x ) = M(x )/R(x )in [∇.( fcli ρ)](x ) > 0results in Eq. (8) and guarantee condition b for i = 1, ..., r. It only remains to prove condition c. Since fcli (x) = Ai x + O(x 2 ), and [∇. fcli ](x) = T r Ai + O(x), then ∂ f i (x ) ∂M (x ) T r Ai = T r + gi ( x ) < 0. (24) ∂x ∂x x=0 Considering fi and gi are analytic near origin, condition c of Lemma 2 is guaranteed for i = 1, ..., r. This completes the proof. Remark: Theorem 4 is affine with respect to polynomials R(x)and M(x). In order to formulate SOS problem, it should be assumed that the set N0 , some domain including the origin, is Please cite this article as: P. Kohan-sedgh, A. Khayatian and N. Behmanesh-Fard, Simultaneous stabilization of polynomial nonlinear systems via density functions, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j. jfranklin.2019.11.033
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given by N0 = {x ∈ Rn : ϕi (x) < 0, i = 1, ..., p} and polynomial ϕi (x)is chosen so that 0 ∈ N0 . Then one can formulate an SOS problem to find parameterized polynomials R(x)and M(x). 5. Some examples In this section, several examples will be given to illustrate the effectiveness of the proposed theorem. The implementations are done in Matlab 7.10.0.499 (R2010a) running on a PC Desktop Intel® Core i3 and 4 GB RAM. SOSTOOLS (ver.3.0) [26] with LMI solver SDPT3 is also used. In order to reduce the number of nonzero terms in resulting polynomials, L1 norm optimization has been performed on polynomial coefficients in all problems [27,28]. Example 1. consider the following systems x˙1 = −8x1 x2 2 − x1 2 x2 + 2x2 3 P1 : x˙2 = x2 u x˙1 = −6x1 x2 2 + x1 2 x2 P2 : x˙2 = −2x1 3 + (x1 + x2 )u
(25)
For the systemP1 , every (x1 , x2 )with x2 = 0is necessarily an equilibrium point even for the closed loop system, so the origin cannot be made asymptotically stable. Considering this point, Lyapunov-based methods will fail to find simultaneous stabilizer, while it is possible to design almost stable simultaneous controller. In order to find a polynomial controller,R(x)is selected to be a constant scalar and M(x) to be a general second order polynomial. In order to satisfy the inequalities (6) of Theorem 1, a general non-negative function V (x) = (x1 2 + x2 2 )3 is selected. Solving the SOS problem of Theorem 1 to find the parameterized polynomialM(x), results M(x) = u(x) = 0.1305x1 2 − 3.1993x2 2
(26)
The phase portrait of closed-loop systems are presented in Fig. 1. This example is chosen to show the strength of this method. The system P1 and P2 cannot be simultaneously stabilized by ordinary methods because no Lyapunov function can be found for these systems and robust control methods are also not applicable because there is no assumption on interrelatedness of the systems. Clearly, the conditions of Theorem 2 have not been met and simultaneous asymptotic stability has not been achieved, however the closed-loop system of P2 is asymptotically stable. Example 2. consider the following systems x˙1 = x2 − x1 3 + x1 2 P1 : x˙2 = u x˙1 = x2 − 3x1 3 + x2 2 P2 : x˙2 = x1 2 − x1 + u
(27)
For the systemP1 , the CLF ρ(x) = 3x1 2 + 2x2 2 + 2x1 x2 can be easily found. The CLF is the first candidate for V (x). In order to satisfy the inequalities (6) and SOS problem of Theorem 1, V (x) = (3x1 2 + 2x2 2 + 2x1 x2 )6 has been chosen with trial and error methods. As the previous example, R(x)is selected to be a constant scalar. Please cite this article as: P. Kohan-sedgh, A. Khayatian and N. Behmanesh-Fard, Simultaneous stabilization of polynomial nonlinear systems via density functions, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j. jfranklin.2019.11.033
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Fig. 1. Phase portrait of closed-loop system in Example 1, (a) P1 (b) P2 .
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In this example, the approach of Theorem 3 is followed to reach asymptotic simultaneous stabilizer. The resulting polynomial controller is u (x ) = M (x ) = −0.6370x2 3 − 0.0456x2 2 − 0.8591x2 − 1.0182x1
(28)
The phase portrait of closed-loop systems are presented in Fig. 2. As it is shown, the origin is asymptotically stable for both systems. It can also be checked that the conditions (15) are well satisfied. Example 3. Fault Tolerant Control As the complexity of the systems are increased, they are more susceptible to failures. A fault-tolerant design enables a system to continue its intended operation, possibly at a reduced level, rather than failing completely, when some part of the system fails. Simultaneous stabilization is a low cost approach in passive fault tolerant control. This example, mainly discuss two kind of faults for the actuator, outage and loss of effectiveness. Consider the following system x˙1 = x2 − x1 3 + x1 2 + u2 PN : (29) x˙2 = u1 + u2 Where x = [ x1 x2 ] ∈ R2 and u = [ u1 u2 ] ∈ R2 . The system is unstable at the origin and it is supposed that the actuator is subject to failure. The aim is to asymptotically stabilize the system under normal operation, denoted by PN as well as asymptotically stabilizing faulty operation conditions, denoted by PF . 5.1. Outage Here, it is supposed that the control input u1 produces zero forces and moment, i.e. x˙1 = x2 − x1 3 + x1 2 + u2 PF : (30) x˙2 = u2 Choosing V (x) = (2x1 2 + x1 x2 + x2 2 )4 and R(x)a constant scalar, it can be found that all condition of Theorem 3 are met by the following controller and consequently both systems are simultaneously asymptotically stabilized. u1 (x ) = M1 (x ) = −0.0884x2 3 − 0.5700x2 − 1.3374x1 u2 (x ) = M2 (x ) = −0.0388x2 − 0.3182x1
(31)
The phase portrait of closed-loop systems in normal and faulty mode are presented in Fig. 3. 5.2. Loss of effectiveness In this case, the actuator gain in a deflection is smaller than the commanded position, i.e. x˙1 = x2 − x1 3 + x1 2 + β1 u2 PF : (32) x˙2 = β2 u1 + β1 u2 Please cite this article as: P. Kohan-sedgh, A. Khayatian and N. Behmanesh-Fard, Simultaneous stabilization of polynomial nonlinear systems via density functions, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j. jfranklin.2019.11.033
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Fig. 2. Phase portrait of closed-loop system in Example 2, (a) P1 (b) P2 .
Where β1 and β2 are deflected values. It is assume that the deflection reduce the controller commands to half of its value; then by selecting β1 and β2 to be either 1 or 0.5, a collection of 4 systems are required to be simultaneously stabilized. Choosing V (x) = (2x1 2 + x1 x2 + x2 2 )4 and R(x)a constant scalar, the configuration of Theorem 3 result in the following simultaneous Please cite this article as: P. Kohan-sedgh, A. Khayatian and N. Behmanesh-Fard, Simultaneous stabilization of polynomial nonlinear systems via density functions, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j. jfranklin.2019.11.033
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Fig. 3. Phase portrait of closed-loop system in Example 3, (a) Normal Mode (b) Faulty Mode.
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Fig. 4. Phase portrait of closed-loop system in Example 3, (a) β1 = 1, β2 = 1 (b) β1 = 1, β2 = 0.5 (c) β1 = 0.5, β2 = 1 (d) β1 = 0.5, β2 = 0.5.
asymptotic stabilizer, u1 (x ) = M1 (x ) = −0.1765x2 3 u2 (x ) = M2 (x ) = −0.6942x1 − 1.0558x2
(33)
The phase portrait of closed-loop systems in normal and faulty mode are presented in Fig. 4. Please cite this article as: P. Kohan-sedgh, A. Khayatian and N. Behmanesh-Fard, Simultaneous stabilization of polynomial nonlinear systems via density functions, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j. jfranklin.2019.11.033
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Fig. 4. Continued
6. Concluding remarks In this paper, the problem of designing simultaneous feedback controllers for polynomial nonlinear systems has been studied. Based on almost stability theory, both local and global stabilization has been studied. The proposed conditions on simultaneous stability are affine with respect to unknown polynomials and hence it is possible to formulate SOS synthesis Please cite this article as: P. Kohan-sedgh, A. Khayatian and N. Behmanesh-Fard, Simultaneous stabilization of polynomial nonlinear systems via density functions, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j. jfranklin.2019.11.033
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problem. In order to check asymptotic stability, some extra conditions on system and variable polynomials has been introduced. These conditions can be used after controller synthesis to check the asymptotic stability or can be used in SOS synthesis problem to guarantee simultaneous asymptotic stabilizer asymptotic stability or can be used in SOS synthesis problem to guarantee simultaneous asymptotic stabilizer from the beginning. Numerical examples and a fault tolerant design has been used to show the effectiveness of the global stabilization method. Regarding local simultaneous stabilization, there are still works to be done to find a good application example. This may be addressed in future works.
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Please cite this article as: P. Kohan-sedgh, A. Khayatian and N. Behmanesh-Fard, Simultaneous stabilization of polynomial nonlinear systems via density functions, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j. jfranklin.2019.11.033