JOURNAL OF INDUSTRIAL AND MANAGEMENT OPTIMIZATION Volume 9, Number 3, July 2013
doi:10.3934/jimo.2013.9.579 pp. 579–593
ON THE ROBUST CONTROL DESIGN FOR A CLASS OF NONLINEARLY AFFINE CONTROL SYSTEMS: THE ATTRACTIVE ELLIPSOID APPROACH
Vadim Azhmyakov, Alex Poznyak and Omar Gonzalez Department of Control and Automation, CINVESTAV, Av Instituto Politecnico Nacional 2508 Mexico D.F., Mexico
(Communicated by Huanshui Zhang) Abstract. This paper is devoted to a problem of robust control design for a class of continuous-time dynamic systems with bounded uncertainties. We study a family of nonlinearly affine control systems and develop a computational extension of the conventional invariant ellipsoid techniques. The obtained method can be considered as a powerful numerical approach that makes it possible to design a concrete stabilizing control strategies for the resulting closed-loop systems. The design procedure for this feedback-type control is based on the classic Lyapunov-type stability analysis of invariant sets for the given dynamic system. We study the necessary theoretic basis and propose a computational algorithm that guarantee some minimality properties of the stability/attractivity regions for dynamic systems under consideration. The complete solution procedure contains an auxiliary LMI-constrained optimization problem. The effectiveness of the proposed robust control design is illustrated by a numerical example.
1. Introduction. The Lyapunov-based optimization techniques provide a useful theoretic tool not only for the classic stability test but also in connection with the several types of robust control design procedures (see [1, 8, 10, 12, 19, 20, 21, 22, 26, 28, 35, 36, 41]). Various feedback control problems for the input-output systems with bounded disturbances have been recognized as challenging tasks in the modern control engineering (see e.g., [9, 13, 15, 16, 17, 20, 23, 25, 26, 27, 29, 33, 34, 35, 37, 38, 39]). For example, the newly elaborated theoretic and numeric extensions of the classical technique of optimal rejections of bounded disturbances was examined in [7, 31, 32, 40] and [14]. Given the view of a control system with a class of restricted, additive uncertainties as nonlinearly affine model, it is quite intuitive to exploit Lyapunov-type tools in a constructive design procedure of stabilizing controls. In this direction, one of the main tools for generating a practically stable trajectory is the invariant ellipsoid method. We refer to [5, 10, 24, 25, 27, 31, 32, 33, 35] for some basic theoretical details and also for further references. In our paper, we investigate a particular family of nonlinearly affine control systems with the abovementioned type of additive uncertainties. The nonlinear differential equations in our contribution are equipped with so called quasi-Lipschitz right-hand sides. We 2010 Mathematics Subject Classification. Primary: 93C10, 37N40; Secondary: 93C41. Key words and phrases. Affine dynamic systems, the invariant ellipsoid method, LMI-based optimization.
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are mainly interested in creating of effective control algorithms and to establish some effective numerical extensions of the propose robust control design schemes. Note that the robust control algorithms we propose are based on the simple linear feedback methodology. Recently, design of general robust control processes governed by various types of dynamical models have attracted a lot of attention, thus both theoretical results and applications were developed, (see e.g., [7, 10, 13, 14, 15, 16, 17, 19, 20, 23, 29, 34, 37, 38, 39]). The control design strategy to be discussed in this work is constitutes an extension of the classic invariant ellipsoid approach. Recall that a set in the state space is said to be positively invariant for a given dynamical system if any trajectory initiated in this set remains inside the set at all future time instants [25, 28]. An abstract existence question of an invariant set for a general dynamical system constitutes a very sophisticated mathematical question. Under some technical assumptions related to the structure of the closed-loop system, it is possible to characterize an attractive set constructively. In the modern control engineering this set is practically chosen in the form of an ellipsoid in the state space of the given system (see [10]). Note that in this paper we propose a numerically tractable schemes that make it possible to construct a minimal size Lyapunov stable invariant ellipsoid for the closed-loop control system. This is a significant advantage in comparison with the conventional and some heavy-implementable (Lyapunovbased) stabilization approaches discussed in [13, 23, 29, 38, 39]. Concretely, we deal here with a particular class of nonlinearly affine control systems that contain additive uncertainties. The a priory unknown nonlinear righthand sides of the state equations in our contribution are modeled by the quasiLipschitz functions (strictly defined later). This general modelling framework and the specific quasi-Lipschitz requirement make it possible to interpret the methodology proposed in this paper as a robust control approach not only to one selected dynamic system but also to a wide class of systems specified by the differential equations with the above-mentioned right-hand sides characterization. To put it another way, we propose a control algorithm that stabilize (in a practical sense) a family of dynamic systems under consideration. The resulting stable ellipsoidal invariant set we obtain possesses some optimal properties and is used constructively in the main design procedure of a feedback control. It is not surprisingly, that a robust synthesis problem for a quasi-linear system and the minimal size properties of the obtained invariant set can be usually expressed in a form of an equivalent auxiliary LMI-constrained optimization problem. We refer to [11, 10, 27, 31, 32, 33, 35]) for some similar ideas and technics. A minimization approach to the class of nonlinearly affine input-output systems is used in our contribution in combination with the classical full-order linear dynamic output controllers. Roughly speaking, we choose the necessary controller parameters (gain matrices) that minimize the size of an attractive ellipsoid related to the resulting closed-loop control system. The remainder of our paper is organized as follows. Section 2 contains the problem formulation and some preliminary concepts and facts. Section 3 discusses an effective robust feedback control design procedure based on the the minimal attractive ellipsoid method. Section 4 is devoted to the corresponding computational technique that extends the obtained theoretic schemes. We propose a reduction of the controller design problem to an auxiliary LMI-constrained optimization procedure and propose some constructive solution procedures. Finally, we develop
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an implementable numerical algorithm and consider an illustrative computational example. Section 5 summarizes our paper. 2. Problem formulation. Consider the following nonlinearly affine control system ˜ x(t) ˙ = f (x(t)) + Bu(t) + ξ(t), t > 0, x(0) = x0 ,
(1)
y(t) = h(x(t)) + η˜(t), where x(t) ∈ Rn is the state vector, u(t) ∈ Rm is the control input and y(t) ∈ Rk is the measured output. Here f : Rn → Rn , h : Rn → Rn are some nonlinear function and B ∈ Rn×m is a given matrix. The components ξ˜ and η˜ in (1) are interpreted as system uncertainties and are assumed to be uniformly bounded ˜ ||ξ(t)|| η (t)||Kη ≤ 1 ∀t ∈ R+ . Kξ ≤ 1, ||˜ By ||·||Kξ and ||·||Kη we denote the weighted (by some positive-defined matrices Kξ and Kη ) Euclidean norms. We also consider some additional technical assumptions: • The right hand sides in (1) possess the so-called quasi-Lipschitz property (see e.g., [5, 27, 35]) ˜ − Ax||2 ≤ f0 + f1 ||x||2 , ||ξ(t, x)||2 = ||f (x) + ξ(t) ||η(t, x)||2 = ||h(x) + η˜ − Cx||2 ≤ h0 + h1 kxk2 ,
(2)
where t ∈ R+ , x ∈ Rn and ˜ − Ax, ξ(t, x) := f (x) + ξ(t) η(t, x) := h(x) + η˜(t) − Cx. Here A ∈ Rn×n , C ∈ Rk×n are some appropriate (in the sense of (2)) matrices. • The pair (A, B) is controllable and the pair (A, C) is observable (both the controllability/observability statements are related to a linearization of system (1), see [23, 34]); • The matrix B T B is invertible. Note that we deal with a general class of so-called quasi-Lipschitz uncertain systems (see the Fig. 1) since it includes the nonlinearities of the both types, namely, the bounded discontinuous (as, for example, in the sliding mode control) containing some sign-terms and Lipschitz-type as well. This class is very general and includes practically all nonlinear models considered in engineering applications. In fact, the estimate (2) is a non-quadratic characteristic, but can be defined as a quasi-affine (the nonlinearity) may increase not quicker then an affine one. The initial system (1) can now be rewritten in the following equivalent form x(t) ˙ = Ax(t) + Bu(t) + ξ(t, x(t)), ∀t > 0, x(0) = x0 ,
(3)
y(t) = Cx(t) + η(t, x(t)). A control system of the type (1) is usually associated with a set of feasible control functions u(·). An admissible control strategy for the resulting system (3) here is chosen from the class of so called full-order linear dynamic output controllers (see
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VADIM AZHMYAKOV, ALEX POZNYAK AND OMAR GONZALEZ
15
10
5
f(x) 0
−5
−10
−15 −4
−3
−2
−1
0
1
2
3
4
Figure 1. The quasi-Lipshitz functions [34] for details). The structure of that type of controllers is determined by the conditions u(t) = Cr xr (t) + Dr y(t), where xr (t) ∈ Rn is a “controller state”, Cr ∈ Rm×n and Dr ∈ Rm×k are “gain” matrices. Following the methodology of dynamic output feedback controllers, we now determine the controller state introduced above as a solution of the following auxiliary initial value problem x˙ r (r) = Ar xr (t) + Br y(t),
(4)
xr (0) = xr0 ,
with a suitable initial condition xr0 . Moreover, we have the additional parameters (matrices) Ar ∈ Rn×n and Br ∈ Rn×k associated with this controller design. We refer to [12, 13, 20, 26, 34] for some related basic concepts and technical details. The robust-type systems design by a feedback control law under consideration is now equivalent to an adequate selection procedure of the gain (the system/controller) matrix Ar B r Θ := . Cr Dr We call Θ a system/controller matrix. Let use also introduce the following additional notation: xc := (x, xr )T , x0 := (x0 , xr0 )T , z := (η, ξ)T and Ac :=
A + BDr C Br C
BCr Ar
Bc :=
BDr Br
I 0
.
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Evidently, xc , x0 ∈ R2n , z ∈ Rk+n . Using the above notation, we have Ac ∈ R2n×2n and moreover, Bc ∈ R2n×(k+n) . In this paper, we are interested in a constructive design procedure for an admissible stabilizing feedback control strategy. We understand the stabilizing property in a practical sense (specified below) for the closed-loop variant of the given control system (1) x˙ c (t) = Ac xc (t) + Bc z(t, x), xc (0) = x0 .
(5)
The above-mentioned “practical stability” terminology need to be characterized exactly in every concrete situation. We use a theoretic extension of the conventional invariant ellipsoid concept for this purpose (see [5, 27, 33, 35]). Let us consider the state space of system (5) and introduce an ellipsoid with the center at the origin E := {xc ∈ R2n xTc P xc ≤ 1}, where P is a symmetric positive defined 2n × 2n-matrix. Our aim is to generate a feedback control strategy (5) such that E to be a quasi-invariant set of the closedloop dynamic system (5). The practical stability is understood here as a property of the obtained ellipsoid E to have a “minimal size”. This property can be formalized by the following minimization problem min tr P −1 subject to P ∈ Γ1 (¯ x0 , xc (·)),
(6)
Θ ∈ Γ2 where xc (·) is a solution of the closed-loop system (5), Γ1 (¯ x0 , xc (·)) is a set of symmetric and positive defined 2n-matrices that guarantee the following attractive property of E limt↑∞ xc (t) ∈ E. Evidently, Γ1 (¯ x0 , xc (·)) is characterized by the dynamics of the control process under consideration. In this sense we use the above notation Γ1 (¯ x0 , xc (·)) (the dependence of x ¯0 and xc (·)). Moreover, Γ2 is a subset of the space of (n + m) × (n + k)dimensional (block) matrices. This set describe the admissible system/controller matrices Θ. Clearly, problem (6) is equivalent to a constrained maximization of N tr P over the same set of constraints Γ1 Γ2 . This last formulation guarantees the minimal “size” of the generated attractive ellipsoid E. We assume that the above abstract minimization problem (6) has an optimal ˆ Note that this solution is characterized by the above pair of masolution {Pˆ , Θ}. trices, namely, by an optimal matrix of the minimal-size ellipsoid E, and also by an optimal system/controller matrix Θ that determines the dynamic of the system. Let us note that (6) is a specific nonlinear optimization problem. The existence of an optimal solutionN to (6) constitutes a sophisticated general question (see e.g., [30, 36]). The set Γ1 Γ2 in (6) is a set of restrictions that defines the class of admissible matrices P and Θ such that E has the property to be attractive for the corresponding closed-loop system (5). Our next step N is motivated by a possible constructive characterization of the restrictions Γ1 Γ2 . In the next section we use the Lyapunov-type methodology for this purpose. We will next rewrite (6) in an equivalent constructive form and apply a specific relaxation procedure for the concrete numerical treatment of the resulting minimization problem.
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3. The attractive ellipsoid techniques. Robust control design procedures based on the conventional invariant ellipsoid method usually incorporate a Lyapunov analysis of the corresponding sets in the state space. We refer to [1, 5, 28, 33, 34, 35, 40, 41] for some classical stability theorems associated with the classic dynamic systems. For the purpose of the extended ellipsoid techniques (the attractive ellipsoids) studied in this contribution we firstly formulate an auxiliary technical result. Theorem 3.1. Let a function Υ : R+ → R+ satisfies the following differential inequality ˙ Υ(t) ≤ −αΥ(t) + β, (7) where α > 0 and β ≥ 0. Then limt↑∞ Υ(t) ≤ β/α. This theorem can be proved by a direct calculation applied to a specially constructed “cutting function” and using some standard properties of the continuous functions, namely, the Weierstrass Theorem. We refer to [34] for the corresponding proof. As next let us recall the standard Λ-matrix inequality (see also [34] for mathematical details). Theorem 3.2. For any matrices X, Y ∈ Rn×m and any symmetric positive definite matrix Λ ∈ Rn×n the following inequalities hold X T Y + Y T X ≤ X T ΛX + Y T Λ−1 Y and (X + Y )T (X + Y ) ≤ X T (I + Λ)X + Y T (I + Λ−1 )Y. Note that in an one-dimensional case the inequality from Theorem 3.2 constitutes a trivial fact. We use the above two technical results in the proof of our main theorem. Let us now consider the following quadratic “energetic” Lyapunov-like function V : Rn → R+ , V (xc (t)) = xc (t)T P xc (t). This function is considered on the trajectories xc (·) of the closed-loop system (5). Note that V (·) is similar to a classic Lyapunov function for linear systems. In contrast to the linear systems theory, the function V (·) can only guaranty the attracting properties of a concrete set in the state space. We are now ready to reformulate the abstract minimization problem (6) in a constructive form. Let α > 0 and β > 0 some constants from the above Theorem 3.1. Using this result and the introduced energetic function V (·), we get the equivalent representation of (6) minimize tr P −1 /β subject to P > 0, P T = P, ( xTc (t)P xc (t) ≤ 1 if xc (0) ∈ E dV ˙ c (·))(t) ≤ −αV (xc (t)) + β if xc (0) ∈ R2n \ E dt (xc (·), x
(8)
Θ ∈ Γ2 , where xc (·) is a solution of the closed-loop system (5). The first significative restriction in (8) determines an attractive ellipsoid E explicitly (the initial condition belongs to E). One can easy to see that the second optional condition from (8) also implies the attractive property in consequence of Theorem 3.1. In that case the initial condition is outside of E. The optional constraint in (8) characterizes constructively the set Γ1 (¯ x0 , xc (·)) in the abstract formulation (6). An optimal solution
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ˆ and introduced in Section 2. The ellipsoid E associated of (8) is denoted by {Pˆ , Θ} with Pˆ is called a minimal attractive ellipsoid. Let us now introduce the additional notation Kξ 0 Λ := . 0 Kη We are now able to formulate our main theoretic result. Theorem 3.3. Consider (5) and assume that all conditions from Section 2 are ˆ Then the fulfilled. Assume that the optimization problem (6) has a solution {Pˆ , Θ}. ellipsoid E determined by the following matrix α P := Pˆ , β where β = h0 + f0 and α is computed from the auxiliary optimization problem minimize tr P −1 /β subject to P > 0, P T = P, S11 S12 W = ≤ 0, with T S12 S22 α α S11 := P (Ac + I) + (Ac + I)T P + (h1 + f1 )I, 2 2 S12 = S21 := P Bc , S22 := −Λ,
(9)
Θ ∈ Γ2 , is a minimal attractive ellipsoid for (5). Proof. Our aim is to show that the composed function Υ(t) := V (xc (t)) satisfies inequality (8) from Theorem 3.1. We have V˙ (xc (t)) = 2xTc (t)P x˙ c (t).
(10)
Using (5) and (10), we obtain 2xTc (t)P x˙ c (t) = 2xTc (t)P [Ac xc (t) + Bc z(t, xc (t))] = xTc (t)[P Ac + ATc P ]xc (t) + 2xTc (t)P Bc z(t, x(t)).
(11)
The application of Theorem 3.2 to the right-hand side of (11) involves the following inequality 2xTc (t)P Bc z(t, x(t)) = 2((P Bc )T xc (t), z(t, x(t))) ≤ xTc (t)[P Bc Λ(P Bc )T ]xc (t) + z(t, x(t))T Λ−1 z(t, x(t)). Hence V˙ (xc (t)) ≤ xTc (t) × [P Ac + ATc P + P (Bc ΛBcT )P + αP ]xc (t)− | {z } W0
αxTc (t)P xc (t)
T
+ z(t, x(t)) Λ
Using the representation for Λ T
z(t, x(t)) Λ
−1
−1
−1
z(t, x(t)).
, we derive the next estimation
z(t, x(t)) = ||η(t, x(t))||2Kη + ||ξ(t, x(t))||2Kξ ≤
(h0 + f0 ) + (h1 + f1 )||xc (t)||2 . The last one implies the estimation V˙ (xc (t)) ≤ −αV (xc (t)) + xT (t)W xc (t) + β c
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VADIM AZHMYAKOV, ALEX POZNYAK AND OMAR GONZALEZ
for the concrete constant β determined above. Here we use the notation α α W := W0 + (h1 + f1 )I = P (Ac + I) + (Ac + I)T P + P Bc ΛBcT P + (h1 + f1 )I. 2 2 The natural requirement W ≤ 0 implies V˙ (xc (t)) ≤ −ˆ αV (xc (t)) + β. From the auxiliary Theorem 3.1 it follows that limt↑∞ Υ(t) = limt↑∞ V (xc (t)) ≤ β/ˆ α
(12)
From the obtained estimation (12) we deduce the resulting relation α ˆ limt↑∞ xTc (t)Pˆ xc (t) ≤ 1. β As a simple consequence of the last one we finally obtain α ˆ T x (t)Pˆ xc (t) ≤ 1 + β c limt↑∞ V (xc (t)) =
for all > 0 and for t > T (), where T () > 0 is a big enough number. Note that the last inequality is true for every initial point x ¯0 from R2n . As we have ˆ seen the element P of the solution pair to (9) guarantees the corresponding miniˆ ˆ mality property of the attractive ellipsoid E determined by P = α β P . The proof is completed The proved theoretical result characterizes a specific attractive ellipsoid E , namely, the minimal-size ellipsoid for system (5). This attractive ellipsoidal region is obtained as a solution of an auxiliary minimization problem. In fact, we specify the initial optimization problem (6) by a constructive way. Otherwise, the main question, namely, the question of the corresponding robust control strategy that guarantees the robustness property ( we also call it the practical stability property) of the closed-loop system (5) in the sense of this attractive set is still an open question. Roughly speaking, we now need to give a concrete description of the constraints set Γ2 (see (8) and (9)) and also elaborate an effective computational rule for the optimal selection the system/controller matrix Θ ∈ Γ2 . Theorem 3.3 constitutes a main theoretic result of the robust control methodology we propose. In contrast to the invariant ellipsoid-based stabilization techniques proposed in [10, 13, 19, 24, 25], our technique involves a specific minimization problem that guarantee some minimal size properties of the constructed stable invariant set. Moreover, the optimization requirement given by (6) makes it possible to apply the well-developed (theoretically and numerically) mathematical programming algorithms in the robust control design procedure we consider. On the other hand our approach is a formal extension of the methods from [31, 32]. This extension is related to a wide class of right-hand sides (so called quasi-Lipschitz right-hand sides) of the basic differential equations in (1). The same is also true with respect to the more general formal controller schemes we use, namely, the dynamic controllers determined by (4). Let us introduce some additional notations 0 B 0 0 F1 := F2 := . I 0 I 0 We now are interested in a relaxed representation of the inequality constraints from the obtained Theorem 3.3. The necessity of an adequate relaxation to the given
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minimization problem (9) is motivated by the bilinear (nonlinear) nature of the matrix inequality W < 0 from (9). Recall that various relaxation techniques are widely used in the conventional optimization theory (see e. g., [30] for motivation). These techniques can usually simplify the computational techniques for a sophisticated optimization problem in the original form. Our next result realize the above-mentioned relaxation idea for the constrained optimization problem (9). In that case we replace one the bilinear matrix inequality by a suitable LMI (the linear matrix inequality). Theorem 3.4. Under assumptions of Theorem 3.3 the bilinear-type matrix inequality constraint from (9) can be relaxed to the following LMI Z11 Z12 ˜ W := ≤ 0, (13) T Z12 Z22 where α α Z11 := P (A0 + I) + Y G + (A0 + I)T P + GT Y T + (h1 + f1 )I, 2 2 A 0 0 I 0 I A0 := G := B0 := . 0 0 C 0 0 0 and Z12 := P B0 + Y F2 , Z22 := −Λ, relaxed optimization problem
Y := P F1 Θ. Assume that the following
minimize tr P −1 /β subject to P > 0, P T = P, ˜ 0 calculated from (9). Then the ellipsoid determined by the matrix α P = P˜ , β where β > 0 is determined in Theorem 3.3, is a minimal attractive ellipsoid associated with the given system (5). Moreover, the corresponding optimal system/controller matrix Θ can be calculated as follows Θ = (F1T F1 )−1 F1T P˜ −1 Y˜ . Proof. By the definition 0 T F1 F1 = BT
I 0
0 I
B 0
=
I 0
(15)
0 BT B
> 0.
Consider the original bilinear inequality constraint from Theorem 3.3 χ P Bc W = ≤0 BcT P −Λ where
(16)
α α I) + (Ac + I)T P + (h1 + f1 )I. 2 2 The unknown parameter T heta can be calculated from (16). Using the additional notation introduced above, the system matrices Ac and Bc can now be rewritten in the equivalent form χ := P (Ac +
Ac = A0 + F1 ΘG Bc = B0 + F1 ΘF2 .
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VADIM AZHMYAKOV, ALEX POZNYAK AND OMAR GONZALEZ
Replacing Ac and Bc in (16), we finally obtain the LMI-inequality constraints χ P B 0 + Y F2 ˜ W = ≤ 0, B0T P + F2T Y T −λ where α α I) + (A0 + I)T P + Y G + GT Y T + (h1 + f1 )I. 2 2 To omit a possible non-invertibility of the above introduced matrix F1 , we consider the symmetrized matrix F1T F1 . Recall that under condition formulated for matrix B (the invertibility of the product B T B, see Section 2), the matrix F1T F1 is also invertible. Thus we finally obtain the constructive relation (15). The proof is finished. χ := P (A0 +
ˆ satisfies the inNote that the associated optimal system/controller matrix Θ equality constraints in the original problem (9). The problem (14) is a formal “extension” of the original optimization problem (9) with the bilinear-type matrix inequalities. The numerical solution of (9) can be processed by applications of the usual optimization techniques implemented, for example, in the standard MATLAB procedures. The main conceptual question is: in which a sense the relaxed LMI constraints (13) approximates the original bilinear matrix inequality from (9). Evidently, the above question indicates the relation between the original problem (9) and the relaxed version (14). The answer to this important question is given in our next result. Theorem 3.5. Every solution {Y˜ , P˜ } to the auxiliary relaxed optimization problem (14) is also an optimal solution for (9). The proof of this theorem is a consequence of the following observation: for a given α > 0 the relation for Z11 in (13) is a simple linear factorization of the bilinear element S11 of the matrix W in (9). From the existence of an optimal solution {Y˜ , P˜ } to (14) and the definition of Y it also follows the relation (15) for the optimal Θ. Using this system/controller matrix, we can compute Ac , Bc and finally the matrix W . Now it is easy to see that P˜ and α satisfy the inequality W < 0. That means: the pair {Y˜ , P˜ } corresponds to the optimal solution of the original (non-relaxed) problem (9). In fact, the “relaxation” procedure (14) does not represent a real extension of the original problem (9). Therefore, the proposed relaxation approach constitutes an under-relaxation of the original optimization problem. This observation makes it possible to consider the reformulated problem (14) as a constructive numerical basis for the computational treatment of (9). The obtained LMI (13) characterizes as usual a polygonal region (in a suitable space) that is included into the original region specified by the original bilinear-type constraints in (9). The resulting relaxed optimization problem (14) can be described as a LMIconstrained optimization problem with a non-linear objective functional. This auxiliary optimization problem provides a basis for an effective numerical solution procedure to the initial problem (6) from Section 2. We can conclude that the abstract sets of constraints Γ1 (¯ x0 , xc (·)) and Γ2 the theoretic setting (6) are now completely specified. In this work, we follow the robust or practical stability methodology (see e.g., [5, 27, 31, 32, 33, 35]) and compute P such that the size of the corresponding attractive ellipsoid E will be minimal. This minimizing problem was considered
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under some natural restrictions for the dynamical variables, namely, for the system/controller matrix Θ. The resulting feedback control strategy posses some natural robustness properties related to the above-mentioned ellipsoidal region in the state space. Moreover, using the generated minimal attractive ellipsoid E one can also construct an invariant ellipsoid E related to the given closed-loop system. This invariant ellipsoidal set can be defined as E := E + , where > 0 is small enough. In fact, the proposed approach is a theoretic and computational generalization of the classic invariant ellipsoid technique for the class of nonlinearly affine systems with uncertainties. The emergency of this generalized invariant ellipsoid method can provide a new perspective on some important practical control problems. 4. Computational aspects. This section is devoted to some numerical results for the auxiliary optimization problem (14). From the computational point of view one needs to describe an implementable numerical scheme for this optimization problem. The solution procedure used in the following example is based on the standard MATLAB toolbox for LMI-constrained optimization. Consider the model given by the strongly non-linear differential equations x˙ 1 = k1 Signx2 + ξe1 (t) (17)
x˙ 2 = k2 Signx1 + ξe2 (t) + u y = x1 + x2 + ηe(t) where
k1 = k2 = −1, ξei (t) = ξi sin(100t) and ηe(t) = ξη cos(100t). We also choose ξi = ξη = 0.1. The given system (17) can be represented in the form x(t) ˙ = Ax(t) + Bu(t) + ξ(t, x(t)) y(t) = Cx(t) + η(t, x(t)) with the correspondingly defined system matrices 0 1 0 A= ; B= ; C= 1 1 0 1
1
We now investigate two two cases, namely, the given system with the initial conditions inside E (Case (1)) and the alternative variant (Case (2)). For the Case (1) we put x0 = (−0.05, 0.01)T . Case 2 is characterized by the following vector of initial conditions x0 = (−4, 5)T . The right hand sides of the original affine system (2) are determined here by the following constants f0 = 0.002 h0 = 0.038 and f1 = 0.01 h1 = 0.01. The calculated system/controller matrix is determined by the matrices −0.5056 1.53e−6 1.2e−5 Ar = Br = −1.12e−2 −0.5055 9.56e−6 −15.21e−2 Cr = Dr = −4.13 −15.22e−2
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VADIM AZHMYAKOV, ALEX POZNYAK AND OMAR GONZALEZ
Let us note that matrix Ar in this example is a stable matrix. The matrices Ac and Bc can be computed correspondingly (see Section 2) 0 1 0 0 −3.13 −4.13 −0.15 −0.15 Ac = 0.12e−4 0.12e−4 −4.09 −1.66e−6 0.095e−4 0.095e−4 −0.011 −0.505
1 0 0 1 0 0 0 0
0 −4.13 Bc = 0.12e−4 0.095e−4
Using the numerical optimization algorithm based on the standard MATLAB routines for LMI-constrained minimization problems, we obtain the computational results for the above cases, namely, for Case (1) and Case (2). These results are presented on Fig. 2 - Fig. 3.
0.04
0.02
x2
0
−0.02
−0.04
−0.06
−0.08 −0.08
−0.06
−0.04
−0.02
0 x1
0.02
0.04
0.06
0.08
Figure 2. The Case (1): the minimal attractive ellipsoid
0.04
x2
0.02
0
−0.02
−0.04
−0.06 −0.1
−0.08
−0.06
−0.04
−0.02
0 x
0.02
0.04
0.06
1
Figure 3. The Case (2): the minimal attractive ellipsoid
0.08
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The optimal (minimal) attractive ellipsoid in both cases is characterized by the following matrix 6.8886 6.2217 2.74e−7 −2.86e−6 6.2217 8.9937 −7.9e−6 −1.62e−5 P = 2.74e−7 −7.9e−6 1.1151 −3.2e−6 −2.86e−6 −1.62e−5 −3.2e−6 1.1151 The presented example of a nonlinear system illustrates numerical effectiveness and potential practical applicability of the proposed robust control design methodology for the class of dynamic models of the type (1). Note that in contrast to some heavy-implementable robust control algorithms from [10, 13, 21, 37, 38, 39, 40], the approach proposed in this contribution is based on the well-developed computational techniques of the nonlinear programming. In comparison to [31, 32] we crucially generalize the class of dynamic systems under consideration. Let us mention that the above example studies a differential equation with a formally discontinuous right-hand side. The possibility to incorporate the discontinuous systems into the developed methodology is a formal consequence of the general quasi-Lipschitz properties of the right-hand sides we consider. Finally, let us note that the theoretic and computational techniques we propose can also be interpreted as a robust stabilization algorithm associated with a family of systems of the type (1) with undefined but quasi-Lipschitz function f (·) that satisfies all the formal conditions from Section 2. This possibility to stabilize a class of dynamic models with uncertainties also constitutes an evident advantage of the elaborated approach. 5. Concluding remarks. The paper proposes a new computational technique for the robust control design associated with a class of nonlinearly affine dynamic systems with uncertainties. This analytic method can also provide a conceptual basis for the corresponding implementable control algorithms. The design procedure is based on an extension of the conventional invariant ellipsoid method and incorporates an auxiliary nonlinear LMI-constrained optimization problem. As a result of this optimization procedure in combination with the adequate numerical methods we obtain a stable attractive ellipsoid with some minimal properties (a minimal “size” ellipsoid) that can be interpreted as a region of maximal robustness with respect to the above-mentioned types of additive uncertainties. The analytic technique and the computational scheme proposed in our paper make it possible to generate the practically stabilizing control law in the form of a simple linear feedback. The effectiveness of the control design method is demonstrated by a illustrative numerical example with strong (discontinuous) nonlinearities. The proposed minimization approach used in this contribution is considered under some natural LMI-type restrictions for the unknown system/controller matrix. The proposed approach constitutes a conceptual implementable extension of the classic invariant ellipsoid approach to the class of nonlinearly affine systems in the presence of bounded uncertainties. The emergency of this generalized invariant ellipsoid method can provide a new computational perspective on some important real-world problems of control engineering. Finally note that the presented version of the attractive ellipsoid method can also be applied to some alternative classes of control systems. It seems to be possible to use the elaborated methodology as an additional tool in robust control design of hybrid systems (consult e. g., [2, 4, 6]) and for the practical stabilization of
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