Domain of Attraction and Guaranteed Cost Control for Nonlinear Quadratic Systems. Part 1: Analysis Francesco Amato, Roberto Ambrosino, Marco Ariola and Alessio Merola
Abstract This is the first paper of a two-parts work devoted to the stability analysis and guaranteed cost control of the class of nonlinear quadratic systems. For a given polytopic region in the state space, a sufficient condition is proposed to check whether an assigned region belongs to the domain of attraction of the zero equilibrium point. It is also shown that this problem is intimately related to the convergence and the computation of quadratic cost functions. Thanks to the main result of the paper, both issues are casted in terms of feasibility problems involving linear matrix inequalities. A meaningful application example, involving the development of optimal strategies for integrated pest management, is illustrated at the end of the paper.
I. INTRODUCTION Nonlinear quadratic systems provide an appropriate tool for modeling phenomena in a wide range of applications, either in engineering (electric power systems, chemical reactors, and robots), or in other areas such as biology, ecology and economics. Over the years, several papers have focused on the estimate of the domain of attraction (DA) of the zero equilibrium point of nonlinear (quadratic, cubic and, more in general, polynomial) autonomous systems. In [?] estimates of the region of attraction of rational control systems with saturating actuators are computed by means of invariant domains associated to rational Lyapunov functions. In [18] a Lyapunov-based procedure is proposed to compute an ellipsoidal estimate of the DA of a second order nonlinear system containing either linear and quadratic or linear and F. Amato and A. Merola are with the School of Computer and Biomedical Engineering, Universit`a degli Studi Magna Græcia di Catanzaro, Campus Universitario di Germaneto, 88100 Catanzaro, Italy
[email protected] R. Ambrosino and M.Ariola are with the Dipartimento per le Tecnologie, Universit`a degli Studi di Napoli Parthenope, Centro Direzionale di Napoli Isola C4, 80143 Napoli, Italy {ambrosino,ariola}@uniparthenope.it
cubic terms of the state. For general n-th order systems, some gridding techniques have been employed in [11], [13], [22]; however, since these methods are computationally heavy, this kind of approach is practically applicable only to low order systems. An alternative way to obtain an estimate of the DA, based on topological considerations, is provided in [10], [17]. More recently, the problem of determining the DA of polynomial systems has been solved by recasting it as a Linear Matrix Inequalities (LMIs) feasibility problem [7], [26], [27]. In the above cited papers, the problem is tackled by means of a two-steps procedure: i) choice of a quadratic Lyapunov function, which proves local asymptotic stability of the equilibrium; ii) computation of the estimate of the DA associated to that particular Lyapunov function. However, the choice of the optimal Lyapunov function is not a trivial task and may severely affect the conservativeness of the estimate. In particular, in [22] it is shown that the computation of the quadratic Lyapunov function which maximizes the volume of the estimate of the DA requires solving a double non-convex optimization problem. A sub-optimal procedure is proposed in [8], [7], where, however, the problem of computing a ‘good’ quadratic Lyapunov function as initial candidate, which is solved by using gradient search algorithms, remains hardly treatable from the numerical point of view, especially in the presence of non-odd polynomial systems (which is the case of quadratic systems). Less conservative results for DA estimation of polynomial systems was developed following the Sum of Squares (SOS) approach. Concerning the earlier results of this approach to the problem, in [9] the DA estimation is recast in terms of SOS conditions, which amount to solve a Linear Matrix Inequality (LMI) feasibility problem. Using the same approach, the work [6] shows how a DA estimate for polynomial systems can be obtained through a family of polynomial Lyapunov functions, whereas both polynomial and composite Lyapunov functions are used in [25] to enlarge the inner estimate of the DA of polynomial systems. More recently, the problem of estimating the DA for rational control systems with saturating actuators has been tackled in the [12] through a differentialalgebraic representation of the rational system dynamics. Some other results on the guaranteed cost control of a class of nonlinear systems are provided in [14], where a guaranteed cost control law is adopted to improuve the rolling capabilities of a bank-to-turn vehicle. It is noteworthy that, for the considered class of nonlinear quadratic systems, it is not necessary to resort to the SOS (or alternative) representations to obtain some LMI conditions. Indeed,
quadratic systems can be handled in their original form, from which results that the problems dealt with here can be tackled directly in our LMI approach. Since the exact determination of the whole DA of the zero equilibrium point of a given quadratic system is a difficult or even impossible task (except for very simple cases), in [3] an approach is proposed to solve the more practical problem of determining whether an assigned polytope containing the origin of the state space belongs to the DA of the equilibrium (see also [21] for an application). A fundamental point is that the approach of [3] optimizes over the whole set of quadratic Lyapunov functions, thus reducing the conservativeness of the previous literature. The proposed technique requires the solution of a Generalized Eigenvalue Problem (GEVP) [5], for which efficient numerical optimization routines exist [15]. This paper is the first of a two-parts work (see also [2]), in which the problem of estimating the DA of a nonlinear quadratic system is extended to the discrete-time context and is linked to the problem of determining a guaranteed bound for a quadratic cost function, given a set of admissible initial conditions. More precisely, here we state a sufficient condition guaranteeing that an assigned polytopic set belongs to the DA of a given discrete-time quadratic system; then another condition is provided to compute a bound for a quadratic cost function over the same polytope. The two problems are strictly related each other, since a finite bound exists if the polytope is included into the DA. This paper is, therefore, more focused on the DA issues, while the second part paper (see [2]) investigates the corresponding design problems, and is more devoted to the optimal control aspects. The motivation for this work stems from the fact that discrete-time systems are becoming more and more important as digital control is replacing classical analog feedback; on the other hand optimal and guaranteed cost control has established in the last decades as probably the most applied control methodology (together with PID regulator theory) in the engineering field (e. g. see applications to aerospace [16], [19], robotics [28], plasma shape control [4], etc.). The advantages of the technique rely on the fact that it is applicable, with no adjoint conceptual difficulty, to multivariable as well as to time-varying systems; moreover it allows optimization over finite-time intervals. The paper is organized as follows. In Section II the problem we deal with is precisely stated and some preliminary results are given. In Section III a sufficient condition which guarantees
that a polytopic set belongs to the DA of a given quadratic system is provided; then a formula to compute a guaranteed bound for a quadratic cost function over the same polytope is given. In Section IV a real engineering example, namely the application of the proposed theory to the field of integrated pest management, is illustrated; finally in Section V some concluding remarks are given. II. PROBLEM STATEMENT AND PRELIMINARY RESULTS We start this section by stating a result which holds for general nonlinear discrete-time systems in the form x(k + 1) = f (x(k)) ,
x(0) = x0 ,
(1)
where x(k) ∈ Rn is the system state. Assume, as usual, that x = 0 is an equilibrium point of system (1). For this system we are interested in guaranteeing that a given quadratic cost function over an infinite horizon in the form J(x0 ) :=
∞
∑ xT (k)Qx(k) ,
(2)
k=0
where Q is a symmetric positive definite matrix, is bounded for an admissible set D, 0 ∈ D, of initial conditions. The set D in the following is referred as Initial Set. It is straightforward to notice that (2) is well posed if D is contained into the Domain of Attraction (DA) of the equilibrium point. The above considerations show that there is an intimate relationship between the estimate of the DA and the computation of a bound for the index (2). The following lemma puts together the two concepts. Lemma 1: Given the Initial Set D, assume there exists a closed set E ⊂ R n , E ⊇ D, such that i) E is an invariant set for system (1); ii) there exists a quadratic Lyapunov function v(x) such that a) v(x) is positive definite on E; b) the rate of change Δv(x) of v(x) computed along the solutions of system (1) is negative definite over E.
Then the equilibrium point x = 0 of system (1) is asymptotically stable and the set E (hence D) is an estimate of the DA of the equilibrium. Moreover if the rate of change Δv(x) satisfies for all x ∈ E iii) xT Qx + Δv(x) < 0 .
(3)
then, for all x0 ∈ E (hence x0 ∈ D), J(x0 ) < v(x0 ). Proof: The first part of the statement follows from classical Lyapunov theory (see for example [20]). Now, given an initial state x0 ∈ E, since E is an invariant set for the nonlinear system we can conclude that, for all k ∈ N, x(k) ∈ E. Therefore, from (3), it follows that for k ∈ N xT (k)Qx(k) < −Δv(x(k)) .
(4)
Moreover, since E is contained into the DA, we are guaranteed that the trajectory starting at x0 converges to zero. Therefore, summing up (4) from 0 to infinity we have ∞
∑x
T
(k)Qx(k)
0, conditions (8b), (8c), (10). Concerning the above Problems, the variable γ is chosen in the interval (0 , 1) with a fine gridding with fixed step Δγ 1. Then a procedure which iteratively solves an LMI optimization problem over the optimization variables P (and κ in Problem 3) is implemented for γ that goes from 0 to 1 with a step Δγ . While for Problems 1 and 2 the procedure stops at the first value of γ verifying the feasibility problems, for Problem 3 the LMI minimization problem is implemented for all γ in (0 , 1) in order to find the minimum of the cost function. The LMI optimization problem is solved using the LMI Matlab optimization toolbox [15]. Remark 4: Theorem 1 is referred to the DA of the zero equilibrium point and to polytopic regions surrounding the origin. It is important to recognize that, when a nonzero equilibrium point of a nonlinear quadratic system is dealt with, the problem can be always reduced to the study of the corresponding properties of the zero state of a suitable fictitious quadratic system. Indeed assume that xe = 0 is an equilibrium point for system (6), then ˜ e )xe = xe . Axe + F(x
(22)
Now let z(k) = x(k) − xe ; by using (22), it is readily seen that z(k + 1) =x(k + 1) − xe
(23)
˜ ˜ e ) · (z(k) + xe) − xe =A · (z(k) + xe ) + F(z(k)) + F(x
(24)
˜ + F(z(k))z(k) ˜ =Az ,
(25)
˜ e ) , which is still a quadratic system in the form (6). with A˜ = A + 2F(x In the following section a real world engineering example, illustrating the applicability of the proposed methodology, is provided.
IV. A DISCRETE TIME QUADRATIC SYSTEM FOR INTEGRATED PEST MANAGEMENT In the field of the agro–ecosystem management, the crops are routinely sprayed with synthetic chemical pesticides in order to achieve the pest control. However, the chemical control approach has also hazardous effects on non-target organisms, including the environment and the food chains. In recent years, Integrated Pest Management (IPM) has been introduced as a more effective and eco–compatible strategy to manage the agricultural pests, with the aim of both preserving the ecosystem biodiversity and limiting the food safety risk. IPM strategies are aimed at keeping the lowest population density of pests which causes economic damages to the crops, with limited recourse to chemical pesticides. To this end, IPM programs integrate chemical control with less hazardous strategies, such as biological control. Within the framework of the biological control, the pest management is accomplished by releasing the natural enemies of the pests, such as predators, parasitoids or pathogens (see, e.g., [23], [29]). In particular, some infective pest individuals, bred in laboratory and then released into the environment, can generate an harmful epidemics over the pest population. IPM programs are planned, over a discrete–time scale, to achieve the optimal economic pest control which both minimizes the pest damage and maximizes the crop profit. In this respect, discrete–time models of interacting populations can be usefully applied for both analyzing and synthesizing optimal control strategies in IPM. For example, Lotka–Volterra (LV) systems with infective disease in the prey can enable to characterize the predatory effect on the pest (prey) along with the spreading of the disease among the pest population. The discrete–time model considered here is based on the continuous LV quadratic system with diseased prey, originally presented in [30], ⎧ ⎪ ⎪ ⎪ ⎨ x1 (k + 1) = x1 (k) + x1 (k)(a − cx3 (k) − λ x2 (k)) − p1x1 (k) + δ1 x2 (k + 1) = x2 (k) + x2 (k)(λ x1(k) − α x3 (k)) − p2x2 (k) − δ2 ⎪ ⎪ ⎪ ⎩ x3 (k + 1) = x3 (k) + x3 (k)(−b + dx1 (k)) − p3 x3 (k) − δ3
,
(26)
where x1 (k) and x2 (k) represent the density of the pest population susceptible to the disease, and of infective individuals, respectively; x 3 (k) denotes the density of predators. All the parameters are taken positive.
The assumptions underlying such a LV system are as follows. i) The habitat for the preys is assumed to be unlimited; therefore, the prey can reproduce exponentially in absence of predators; ii) The infected preys are not able to reproduce; iii) The disease spreads only among the preys, with an incidence rate described by a simple mass action law. The growth of the preys is described by the term ax 1 . The spread of the disease among the preys is described by the term λ x1 x2 . The terms cx1 x3 and α x2 x3 represent the predation effects upon the populations of susceptible and of infective pests, respectively. The predators survive only on the preys and, in absence of preys, the exponential decay of the density of the predators is described by the term bx3 . In the third equation, the term dx1 x3 represents the growth of the predators. The poisoning effects of the pesticide release are also taken into account through the parameters p1 , p2 and p3 . It is reasonable to assume that the agro–ecosystem (26) is subject to some disturbances from the external environment; indeed, the individuals cannot be prevented from entering or leaving the ecosystem. Therefore, the constant exogenous input of new preys is described by δ 1 , whereas
δ2 and δ3 describe the constant loss of infective individuals and of predators, respectively. Assuming the values of the model parameters a = 0.01, b = 0.4, λ = 1, c = 0.4, α = 0.2, d = 0.1, p1 = p2 = 0.3, p3 = 0.1, δ1 = 8.58, δ2 = 1.4 and δ3 = 7.5, the equilibrium point T xe = 2 2 5 is asymptotically stable. In view of the IPM constraints on the preservation of the ecosystem biodiversity, the asymptotic stability is essential to guarantee that all the species coexist at constant population levels. Remark 5: The agro–ecosystem (26) contains both quadratic and constant terms δ 1 , . . . , δ3 . With the values of the model parameters and the chosen equilibrium point, as shown in Remark 4 it is possible to apply the change of variable z = x − x e in order to rewrite system (26) as a quadratic system in form (5).
According to Remark 4, system (26) can be recast in form (5), with ⎛ ⎞ 0 0 ⎜ 1 + a − p1 ⎟ ⎜ ⎟ + 2F(x =⎜ ˜ e) , A A =A ⎟ 0 0 1 − p2 ⎝ ⎠ 0 0 1 − b − p3 ⎛ ⎛ ⎛ ⎞ ⎞ ⎞ λ c λ d 0 ⎟ ⎜ 0 − 2 −2⎟ ⎜0 2 ⎜0 0 2 ⎟ ⎜ λ ⎜λ ⎜ ⎟ ⎟ ⎟ F1 = ⎜− 2 0 0 ⎟ , F2 = ⎜ 2 0 − α2 ⎟ , F3 = ⎜ 0 0 0 ⎟ . ⎝ ⎝ ⎝ ⎠ ⎠ ⎠ d 0 0 0 − α2 0 0 0 − 2c 2
(27)
A fundamental requirement for the planning of an IPM program is that of designing some appropriate control actions (chemical control and/or biological control), taking into account some economic thresholds based on pest population level. Therefore, using the results given in Theorem 1, in the next section we will identify the control parameters of the quadratic model (26) which are critical to restrain the density of the species close to x e . Such control parameters are determined, through a sensitivity analysis, as a result of the optimization of a suitable objective function subject to polytopic constraints (7) related to the economic thresholds. A. GUARANTEED COST OF AGRICULTURAL PEST CONTROL Within the IPM framework, the pest control is achieved taking into account some constraints on the Economic Injury Level (EIL), which is originally defined in [24] as the lowest population density of pests which causes economic damages. The application of the main result of Section III is expected to provide a methodology of analysis of optimal control strategies in IPM. Indeed, the polytope P, whose bounds can be decided on the basis of considerations on EIL, defines the admissible state perturbations which can be tolerated without losing the steady–state x e . The associated worst–case performance cost enables to quantify the robustness of the steady–state against state perturbations, since the cost J can provide a measure of the variations of the densities from their steady–state values. Let us consider system (26) and a quadratic cost function (2), with Q = I3×3 . For the sensitivity analysis which will be performed in the following, the relative impact of the single state variables on the cost index is not of primary interest; therefore, it is sufficient to assign some uniform weights for all the state variables.
The assigned polytope P, expressed in terms of variations of the state variables with respect to the nominal condition, x e , is P := [−0.35, 0.45] × [−0.4, 0.4] × [−0.35, 0.4] .
(28)
An estimate of the worst–case performance index for system (26) can be obtained, according to Problem 2, as the solution of a convex optimization problem subject to LMI conditions in the variables P and κ , and performing a binary search over the parameter γ ; the obtained solution reads
γ = 0.4660 , κ = 0.0568 , ⎛ ⎞ 2.0827 0.0582 0.2030 ⎜ ⎟ ⎜ ⎟ P = ⎜0.0582 1.3614 −0.0654⎟ . ⎝ ⎠ 0.2030 −0.0654 1.7959
(29)
From Theorem 1, the worst–case performance cost is 1/κ = 17.6036. Therefore, it is possible to conclude that: i) The polytope P is included into the DA of x e ; ii) For all the initial conditions belonging to P the performance cost is bounded by 17.6036. The inclusion of P into the DA of xe has been confirmed simulating the state response of system (26) for different initial conditions into P, as shown in Figure 1. It is also remarkable that the state trajectories are contained into the ellipsoid bounded by the level curve of the Lyapunov function xT Px, with P given in (29). In that follows, it is shown how the methodology of Section III can provide a measure of the performance sensitivity of an optimal controlled agro–ecosystem, subject to the polytopic state–space constraints (7), with respect to some perturbations on the model parameters. In order to evaluate the effect of the perturbations of some key parameters on the system performance cost J(x0 ), for a given x0 in P, we adopt the relative sensitivity measure
π 0j ∂ J(x0 ) s= 0 J (x0 ) ∂ π j
j = 1, . . ., n p ,
(30)
which defines the sensitivity as the fractional change in the performance cost resulting from the 0
fractional change in the parameter value π j ; π 0j and J (x0 ) denote the nominal values of the
model parameter and of the worst-case performance cost J(x0 ), respectively. The total number of model parameters used for sensitivity analysis is represented by n p . For example, a positive (negative) value of the relative sensitivity measure indicates that a positive variation of a model parameter yields an increase (decrease) of the performance cost. The relative sensitivity measures for the performance cost of system (26) are obtained for a +1% change of the model parameters λ , α and c. The results of the sensitivity analysis of Tab. I show that a positive variation of the parameter λ is associated to a relative increase of the cost index which, in turn, indicates a degradation of the system performance. The +1% change on the parameters α and c produces a relative decrease of the performance index, yielding some beneficial effects on the minimization of performance cost. Indeed, the state trajectories, starting from P, may exhibit reduced variations before reaching the steady–state operating point. Finally, it is noteworthy that the parameters λ and c have a marked influence on the performance sensitivity; thus suggesting the necessity of an accurate control of those factors involved both in the spread of the disease and the predatory effects on the pest population. V. CONCLUSIONS In this paper we have investigated the stability properties of discrete-time nonlinear quadratic systems. In particular, a sufficient condition, guaranteeing that an assigned polytope in the state space belongs to the domain of attraction of the zero equilibrium point, has been provided. It has been also shown that this problem is related to the convergence and estimation of quadratic cost functions defined on the infinite horizon. The conditions stated in the main theorem have then been casted into the linear matrix inequalities framework, and therefore can be solved via widely available software. To demonstrate the effectiveness of the proposed methodology, an applicative example, concerning the integrated pest approach for agro-ecosystem management, has been presented and discussed. The example shows how the proposed methodology can help the designer in the selection of the control actions necessary to thwart the spread of pests under IPM. On the basis of a quadratic model of the agro–ecosystem, some optimal control parameters have been determined from the solution of a suitable optimization problem subject to polytopic constraints.
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List of Figure Captions Figure 1: The polytope P, the ellipsoid E and some state trajectories, starting from (i)
x0 ∈ P ,
i = 1, . . . , 4, which asymptotically converge to the equilibrium point x e
