Nonlinear stabilization for a class of time delay systems via inverse optimality approach

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Nonlinear stabilization for a class of time delay systems via inverse optimality approach Patricio Ordaz a, Omar-Jacobo Santos-Sánchez a,n, Liliam Rodríguez-Guerrero b, Alberto González-Facundo c a

Research Center on Technology of Information and Systems (CITIS) of the Autonomous University of Hidalgo State (UAEH), Mexico Automatic Control Department of the Center for Research and Advanced Studies of the National Polytechnic Institute (CINVESTAV-IPN), Mexico c Electric and Electronic Engineering Department, Minatitlan Institute of Technology, Mexico b

art ic l e i nf o

a b s t r a c t

Article history: Received 20 January 2016 Received in revised form 29 October 2016 Accepted 27 November 2016

This paper is devoted to obtain a stabilizing optimal nonlinear controller based on the well known Control Lyapunov-Krasovskii Functional (CLKF) approach, aimed to solve the inverse optimality problem for a class of nonlinear time delay systems. To determine sufficient conditions for the Bellman's equation solution of the system under consideration, the CLKF and the inverse optimality approach are considered in this paper. In comparison with previous results, this scheme allows us to obtain less conservative controllers, implying energy saving (in terms of average power consumption for a specific thermoelectrical process). Sufficient delay-independent criteria in terms of CLKF is obtained such that the closed-loop nonlinear time-delay system is guaranteed to be local Asymptotically Stable. To illustrate the effectiveness of the theoretical results, a comparative study with an industrial PID controller tuned by the Ziegler-Nichols methodology (Z-N) and a Robust-PID tuned by using the D-partition method is presented by online experimental tests for an atmospheric drying process with time delay in its dynamics. & 2016 ISA. Published by Elsevier Ltd. All rights reserved.

Keywords: Control Lyapunov-Krasovskii functional Inverse optimality Time delay control system Atmospheric tomatoes drying process

1. Introduction Nowadays the control problem for systems with time delay in the time domain analysis can be studied by using two different features; the Lyapunov-Krasovskii and the Lyapunov-Razumikhin approaches [1–3]. The Lyapunov-Krasovskii functionals plays an important role in stability analysis of time delay systems and design of nonlinear or robust delay-dependent controllers [4–8]. For the case of the stability analysis for discrete-time two-dimensional nonlinear systems one can cite to [9] and for the continuous case the stability analysis of a class of continuous-time multidimensional nonlinear systems in the Roesser form can be see in [10], both cases the use of Lyapunov stability theory is fundamental. Also, the inverse optimality approach has enticed a great interest because the study of this tool gives nonlinear optimal controllers [11,12]. In the last decade, for the time delay systems case, the solution of the inverse optimality problem has been presented by using the so called control Lyapunov-Razumikhin functional or control Lyapunov-Krasovskii functional [13–15]. Certainly, if there exists a CLKF, then there is a continuous control law, such that the trajectories of the system converge n

Corresponding author. E-mail address: [email protected] (O.-J. Santos-Sánchez).

asymptotically to the desired operation point at least in the local point of view [1,2,16]. So, the Control-Lyapunov Functions concept combined with Lyapunov-Krasovskii functionals plays an important role to synthesize the so-called inverse optimality for time delay systems [13,15]. For scalar nonlinear affine time delay systems the problem is studied by [14], where an explicit formula of an optimal nonlinear controller is given, however it is assume that the CLKF is known. In [17] the robust guaranteed cost control for singular Markovian jump linear systems with time-varying delays are addressed. Based on the Linear Matrix Inequalities (LMI) Approach, the existence condition of the guaranteed cost state feedback controller was proposed [17], simulations results were presented. In [18] the H∞ performance and control problems for the linear systems with interval state or input delays and disturbances were investigated, simulation results were presented. The works of [19,20] remark that the most common controller for industry applications is the PID (more than 90% ), and the 20% is used with default factory gains. Moreover, the PID software packages for industrial applications use a PID Loop-Optimizer based on linear models with a time delay in the input system (see [20,21]). In many instances the PID control is tuned to satisfies features on the time domain like overshoot, a criteria of steady state error, among others leaving aside the power consumption of the system [19,22]. However, better performance (respect to signal

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2

analysis) does not entail a saving in energy consumption. On the other hand, some drying process has a heat chamber and a recycling hot air tube, although reduce the cost in the energy consumption this recycling hot air loop induces a time delay in its dynamics [15]. In [23] a nonlinear predictive control was presented to determine the optimal drying profile for a drying process. However, due the obtained model is very complicated, the optimization problem (energy consumption and drying time) was solved by using a genetic algorithm, simulation results were presented. In this paper, in contrast with [15], where complete type CLKF arises, an inverse optimal control by means reduced type CLKF for tomatoes drying process is presented. Although [15] presents an attractive and neat analysis, the control is applicable only to particular estimated model. In practice, the study of controllers designed by complete-type functionals approach provides nonlinear controllers with less robustness than reduced-type functionals [15,24]. Therefore, to avoid the above problems and for sake to conclude with local Uniform Asymptotical Stability one of the contributions of this work is the use of reduced type CLKF. The essential point of this: in context of energy saving, is in the application of controllers that minimize the energy. To show the performance of the proposed controller, an experimental comparison with PID controller tuned by the Ziegler-Nichols method [22] and Robust-PID tuned by using the D-partition method [25], is presented. The theoretical and experimental results presented in this paper, help to bridge the optimal nonlinear control for time delay systems and applied control gap. In fact, our theoretical results are verified in a dryer prototype, and industrial PID controller is used to do a comparative analysis between a non optimized control strategy (PID controller) and the proposed nonlinear optimal control.

Definition 1. The standard uniform norm is given as [26]:

∥ φ ∥h ≔ sup ∥ φ (θ )∥ θ∈[−h,0]

The underlying stability theory for time delay systems for Lyapunov-Krasovskii functionals is provided by the next theorem. Theorem 1 (Lyapunov-Krasovskii Stability Theorem [16]). Suppose the trivial solution of

ẋ (t ) = f0 (x (t ), x (t − h)), φ (θ ) = 0¯ h , θ ∈ [ − h, 0]

(2)

is an equilibrium, and that v1, v2, ω: are continuous non→ decreasing functions, additionally v1 (s) and v2 (s) are positive for s > 0, and v1 (0) = v2 (0) = 0. If there exists a continuous differentiable functional V (xt ) : n × C →  such that:

+

+

v1 (∥ φ (0)∥) ≤ V (xt ) ≤ v2 (∥ φ ∥h )

(3)

and

V̇ (xt )

(2)

≤ − ω (∥ φ (0)∥)

(4)

then the trivial solution of (1) is stable. If additionally ω (s) > 0, for s > 0, then it is asymptotically stable. From this result, and using the classical concept of ControlLyapunov Function, [11,27,28], the time derivative of functional V (xt ) along of the trajectories (1), can be written as (see for instance [15]):

d V (xt ) dt ψ0 (xt ) =

= ψ0 (xt ) + ψ1⊤ (xt ) u (1)

∂V f , ∂x 0

ψ1⊤ (xt ) =

∂V f ∂x 1

(5)

where ψ1: n × m ×  → m is a real valued vector function and ψ0: n ×  →  is a negative definite function.

1.1. Structure of the paper The outline of the paper is as follows: In Section 2 we present basic statements and useful definitions. The next section the specific time delay system description and the problem formulation are presented. In Section 4, the proposed CLKF and the feedback compensator design based on the inverse optimality approach are presented. In the Section 5, numerical aspects of the experimental example and the highlights of this work are presented. Finally, concluding remarks of this paper are presented in Section 6.

Definition 2. A functional V (xt ) is called as CLKF for the system (1), if there exist functions u, ω (s) > 0, for s > 0 and support functions v1 (s) and v2 (s) such that 3 and

d V (xt ) dt

= ψ0 (xt ) + ψ1⊤ (xt ) u ≤ − ω (∥ xt ∥) (1)

hold. Remark 1. Previous definition implies that

ψ1 (xt ) = 0⟹ψ0 (xt ) ≤ − ω (∥ xt ∥)

(6)

2. Preliminaries

is fulfilled.

In this section we present some important properties and definitions needed to establish the main results of this work. A classical representation of affine nonlinear time-delay system is given by:

So, the control u which satisfies Definition 2 must be synthesize by using the inverse optimality approach, for instance in the delay free case one can cite to [11,12], and for the delayed case see for instance [14,15]. These CLKF definitions and time-delay concepts will be useful for establishing the control design that we are interested to solve.

ẋ (t ) = f0 (xt ) + f1 (xt ) u x (θ ) = φ (θ ), xt ≔x (t + θ ), n

with θ ∈ [ − h, 0]

(1) + ,

where x (t ) ∈ is the state space vector at time t ∈ and 0 < h is the time-delay.  is a set [ − h, 0] of real valued continuous functions over [ − h, 0]. f0 : n ×  → n is a nonlinear function, and f1 : n × m ×  → n × m realizing the actuator-mapping, u ∈ m is the control input, the functions f0 and f1 satisfy the Lipschitz condition (at least locally). φ (θ ) ∈  ⊆ n is the minimal information to build the solution x (t , φ , t0 ). The trivial solution of (1) is given by φ (θ ) = 0¯ h and u = 0¯ , then f0 (φ) = 0¯ .

2.1. Inverse optimality approach for affine nonlinear time delay systems In this section we recall important aspects of the inverse optimality approach for time delay nonlinear systems [14,15,24]. The inverse optimality problem is different from of optimal control problem, in fact, for the inverse optimal control problem, assuming that a functional V (xt ) is a CLKF of the affine nonlinear system (1), without solving the Hamilton Jacobi Bellman equation, it seeks

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to find the control law u* such that the performance index

⎛ min ⎜  (xt , u) = u ⎝

∫0

∞

mapping. Hereafter, the next assumptions on the system (1)–(11) are given

⎡⎣ q (xt ) + r (xt ) u⊤u⎤⎦ dt ⎞⎟ ⎠

(7)

reaches a minimum, where q (xt ) and r (xt ) are the strictly positive definite functionals:

q (xt ) = ⎡⎣ ψ1⊤ (xt ) ψ1 (xt ) ⎤⎦ + r (xt ) = dr (xt ) =

1⎡ ⊤ ⎣ ψ1 (xt ) ψ1 (xt ) ⎤⎦ 4

dr (xt ) ψ1⊤ (xt ) ψ1 (xt )

⎡⎣ ψ0 (xt ) ⎦ + ⎡⎣ ψ ⊤ (xt ) ψ1 (xt ) ⎤⎦2 , 1 ⎤2

⎡⎣ ψ0 (xt ) ⎤⎦2 + ⎡⎣ ψ ⊤ (xt ) ψ1 (xt ) ⎤⎦2 . 1

(8)

Notice that these functionals are explicitly given by the knowledge of the CLKF V (xt ). According with the dynamic programming approach, the Hamilton Jacobi Bellman equation related to system (1) and the performance index (7), can be established as

⎛ dV (x ) t min ⎜⎜ u ⎝ dt

(1)

⎞ + q (xt ) + r (xt ) u⊤u⎟⎟ = 0, ⎠

(9)

clearly the CLKF V (xt ) is the Bellman functional in the last equation. The functional (5) is substituted in (9) and by using the Variational Calculus Fundamental Theorem we get the control law u as follows:

u* = −

1 ψ1 (xt ) . 2 r (xt )

It is not hard to calculate the second derivative of (9) with respect to u and to obtain that 2r (xt ) > 0, so the control law u* represents a minimum for (xt , u). Moreover, by the direct substitution of the control u*, the time derivative of the CLKF V (xt ) and the functions q (xt ) and r (xt ) on the Bellman Equation, it could be verified that the Bellman equation is satisfied, then the control u* is optimal. However, a singular point could appear in the control law when the function ψ1 (xt ) = 0 for some x ≠ 0, this implies that r (x ) = 0. The singularity in the control law must be removed as explained in the following proposition: Proposition 1. [15,24] Suppose that the functional V (xt ) is a CLKF for system (1), then the optimal control law

⎧ 1 ψ1 (xt ) ⎪− , ψ1 (xt ) ≠ 0 u (t ) = ⎨ 2 r (xt ) , ⎪ 0, ψ ( x ) = 0 ⎩ 1 t

(i) The nonlinearities f and g belong to the wider class of the Lipschitz functions (at least locally), such that

∥ f (x (t ))∥2 ≤ α12 ∥ x (t )∥2 ∥ g (x (t − h))∥2 ≤ α22 ∥ x (t − h)∥2

,

+ ψ0 (xt ) +

3

(10)

minimizes the performance index (7) and stabilizes to system (1). Remark 2. Notice that it is important to verify the continuity of the control law u(t) in zero. For our study case, it will be done later.

3. System description and problem formulation Here we propose an Inverse-Optimality Control based on CLKF design for a specific class of time-delay systems governed by (1), where

f0 ≔A 0x (t ) + A1x (t − h) + f (x (t )) + g (x (t − h))

(12)

for some 0 < α1 < ∞, 0 < α2 < ∞. (ii) There exists a positive definite functional V (xt ) ∈  such that the Definition 2 is fulfilled, at least, locally for some ψ1 (xt ) ∈ m and negative definite ψ0 (xt ) ∈  which depends on the constant scalars α1 and α2. (iii) There exist scalars α3, β ∈ + , such that

α3 ∥ xt ∥h ≤ ∥ 2x⊤ (t ) PB (xt )∥h = ∥ ψ1⊤ (xt )∥h ≤ β ∥ xt ∥h ,

(13)

this assumption is satisfied at least in the local sense. In the case of the Inverse Optimality approach for time delay systems, to obtain an admissible control u ∈  ⊆ m the Definition 2 must be satisfied. The set  belongs to a class of piecewise continuous functions which stabilize the system (1)–(11) and minimizes the performance index given by (7), subject to (1)–(11), where the arguments of (7) are given by the Eq. (8). It is not hard to see that the functional q (xt ) is positive definite when ψ1 (xt ) ≠ 0. Now, the denominator dr (xt ) of r (xt ) is positive definite when ψ1 (xt ) ≠ 0: notice that ψ0 (xt ) < 0 (Assumption ii) and assume that dr (xt ) ≤ 0, it follows that

⎡⎣ ψ0 (xt ) ⎤⎦2 + ⎡⎣ ψ ⊤ (xt ) ψ1 (xt ) ⎤⎦2 ≤ − ψ0 (xt ), observe 1 that −ψ0 (xt ) > 0 is fulfilled. Squaring on both sides follows that:

ψ1⊤ (xt ) ψ1 (xt ) +

⎡⎣ ψ ⊤ (xt ) ψ (xt ) ⎤⎦2 + ψ ⊤ (xt ) ψ (xt ) ⎡⎣ ψ (xt ) ⎤⎦2 + ⎡⎣ ψ ⊤ (xt ) ψ (xt ) ⎤⎦2 ≤ 0, 1 1 0 1 1 1 1 which is a contradiction. Then dr (xt ) > 0 when ψ0 (xt ) < 0 and ψ1 (xt ) ≠ 0. Now, the optimization problem can be written as in the Inverse Optimality approach as

 *(xt , u) = min (xt , u) u∈ 

(14)

Notice that, the optimization problem (14) is associated with the derivative of the functional V (xt ), then the Hamilton-Jacobi-Bellman's (HJB) equation under performance index (7) is given by

⎧ ⎪ d min ⎨ V (xt ) u∈  ⎪ ⎩ dt

(11)

⎫ ⎪ + q (xt ) + r (xt )∥ u ∥2 ⎬ =0 ⎪ ⎭

(15)

if the functional V (xt ) satisfies the optimization problem (14), then it is so-called as Bellman's functional. Now, we are ready to formulate the problem that we solved here. Problem formulation:The problem lies in finding a sufficient condition to prove the existence of the reduced type CLKF for the dynamic model (1)–(11). Furthermore, to obtain the functionals ψ0 (xt ) and ψ1 (xt ) such that u ∈  minimizes the control problem (14).

(11)

4. On inverse optimal control problem based on reduced type CLKF

f , g : n → n are nonlinear continuous vector functions. A 0 , A1 ∈ n × n are real matrices realizing the linear part of the dynamics (11). B (xt ) ∈ n × m is the matrix realizing the actuator-

In this work the cost functional of the HJB Eq. (15) is provided by (7), where the functional V (xt ) is proposed as the following format:

f1≔B (xt ) ∈

n × m

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V (xt )≔x⊤ (t ) Px (t ) +

t

∫t − h x⊤ (s) Qx (s) ds

(16)

where P and Q are n × n nonnegative matrices. It is easy to see that:1

λmin (P)∥ x (t )∥2 ≤ V (xt ) ≤ { λmax (P) + λmax (Q) } ‖xt ‖2h and it may be a Lyapunov Krasovskii functional for our study case. Now, the analysis of the functional (16) gives us the main theoretical result of this work. Proposition 2. Given positive scalars α1, α2 and the time delay system (1)–(11), under the assumptions (i) - (iii), if the set (ε1, ε2, P, Q) is the solution of the next matrix inequality:

⎡ ⊤ P, ⎢ PA 0 + A 0 P + Q PA1, ⎢ + ε1α 2 In , 1 ⎢ W0≔⎢ A⊤P, ε2 α22 In − Q, 0 n × n, 1 ⎢ − ε1In × n , 0 n × n, ⎢ P, ⎢⎣ P, 0 n × n, 0 n × n,

⎤ ⎥ ⎥ ⎥ ⎥<0 0n×n ⎥ 0n×n ⎥ − ε2 In × n ⎥⎦

(17)

ψ0 (xt ) = z⊤ (t , t − h) W0z (t , t − h) (18)

and

(19)

Proof of proposition 2:. Notice that (16) is bounded by the lower support function λmin (P)∥ x (t )∥2 and by upper support {λmax (P) + λmax (Q)}‖xt ‖h2. The time derivative of (16) along the trajectories (1)–(11) yields to

V̇ (xt ) = 2x⊤ (t ) Pẋ (t ) + x⊤ (t ) Qx (t ) − x⊤ (t − h) Qx (t − h) = x⊤ (t ) ⎡⎣ PA 0 + A ⊤P + Q⎤⎦ x (t ) + 2x⊤ (t ) PB (xt ) u 0

+

2x⊤ (t ) P ⎡⎣ A1x (t

It is easy proving that the previous inequality is negative definite by using the property (12) and from the S-procedure (see for instance chapter 12, pp. 195-196 of [29]) that z⊤ (t , t − h) Wz (t , by means that the t − h) + ε1α12 ∥ x (t )∥2 + ε2 α22 ∥ x (t − h)∥ < 0 matrix W is negative definite too. Now, if W < 0 then,

0 > F0 = z⊤ (t , t − h) Wz (t , t − h) 0 > F1 = − ∥ x (t )∥2 , 0 > F2 = − ∥ x (t − h)∥2 if there exist positive scalars τ1 ≥ 0 , τ 2 ≥ 0 , where τ1≔ε1α12 and τ 2≔ε2 α22 such that

holds, then the functional proven. □

then, the functional (16) is a CLKF (at least in the local point of view), where

ψ1⊤ (xt )≔2x⊤ (t ) BP

+ ε2 α22 ∥ x (t − h)∥ + 2x⊤ (t ) PB (xt ) u

ψ0 (xt )≔F0 + τ1F1 + τ 2 F2 < 0

P

z (t , t − h)≔[x⊤ (t ), x⊤ (t − h), f ⊤ (x (t )), g⊤ (x (t − h))]

V̇ (xt ) ≤ z⊤ (t , t − h) Wz (t , t − h) + ε1α12 ∥ x (t )∥2

− h) + f (x (t )) + g (x (t − h)) ⎤⎦

V (xt ) is a CLFK, and proposition is

The synthesis of control u ∈  which satisfies the Definition 2, is presented bellow. The correct selection of the parameters ε1 and ε2 in (17) define the existence of the Lyapunov-Krasovskii functional. The selection of these parameters lies directly with the conclusion of local or global stability. Furthermore the set of solution of (17), if it exists, implies the existence of a family of controllers that can be adjusted by fixing some parameters, for instance the parameters α1 and α2 or ε1 and ε2. For the systems under consideration, with strictly negative definite time derivative V̇ (xt ), turns out with Asymptotic Stability at the zero solution of the system (1)–(11). From the previous result, and in a sake of applying the Inverse Optimality approach, immediately the next corollary is fulfilled. Corollary 1. Under the assumption of Proposition 2, the CLKF gives the solution of the Bellman's equation (15). Consequently the optimal control law u* yields to

u* = −

1 ψ (xt ) 2r (xt ) 1

(20)

− x⊤ (t − h) Qx (t − h) using the extended vector z (t , t − h)≔⎡⎣ x⊤ (t ), the previous equation is rewritten as:

⊤ x⊤ (t − h) , f ⊤ (x (t )) , g⊤ (x (t − h)) ⎤⎦ ,

⎡ PA + A ⊤P + Q, PA , P, P ⎤ 0 1 0 ⎥ ⎢ ⊤ ⎢ − Q, 0 n × n, 0 n × n ⎥ A1 P, V̇ (xt ) = z⊤ (t , t − h) ⎢ ⎥ z (t , t − h) P, 0 n × n, 0 n × n, 0 n × n ⎥ ⎢ ⎢⎣ P, 0 n × n, 0 n × n, 0 n × n ⎥⎦ +

2x⊤ (t ) PB (xt ) u

adding and subtracting ε1f ⊤ (x (t )) f (x (t )) and ε2 g⊤ (x (t − h)) g (x (t − h)) for 0 < ε1 < ∞ and 0 < ε2 < ∞, we get

V̇ (xt ) = z⊤ (t , t − h) Wz (t , t − h) + 2x⊤ (t ) PB (xt ) u + ε1f

⊤ (x (t )) f

(x (t )) + ε2

g⊤ (x (t

− h)) g (x (t − h))

with

⎡ PA + A ⊤P + Q, PA , P, P ⎤ 0 1 0 ⎥ ⎢ ⎢ − Q, 0 n × n, 0 n × n ⎥ A1⊤P, W=⎢ ⎥ P, 0 n × n, − ε1In , 0 n × n ⎥ ⎢ ⎢⎣ P, 0 n × n, 0 n × n, − ε1In ⎥⎦ just noticed that from property (12), the following is achieved: 1

The notation λmin (A) and λmax (A) define the minimum and maximum eigenvalue of square matrix A , respectively.

To ensure smoothness of the resulting optimal control law away from the origin (i.e. to avoid the apparent singular solution ψ1⊤ (xt ) ψ1 (xt ) = 0 see for instance (8) of the inverse optimality approach), the optimal control law (20) is continuous at x (t ) = 0 if the CLKF V (xt ) satisfies the small control property [7]. Thus, the norm of the control law (20) is given by 2 ψ02 ( xt* ) + ⎡⎣ ψ1⊤ ( xt* ) ψ1 ( xt* ) ⎤⎦ ,

ϰ(xt* ) = u* = 2

(

ψ1 ( xt* ) ψ1⊤ ( xt* ) ψ1 ( xt* ) + ψ0 ( xt* ) + ϰ(xt* ) ψ1⊤

)

( xt* ) ψ1( xt* )

(21)

ψ1⊤ (xt* )

where ψ0 (xt* ) and ≠ 0 are given by (18) and (19), respectively. Moreover, the term ψ0 ( xt* ) satisfies 2

ψ0 (xt* ) ≤ α¯1 xt* h , α¯1 = 2

{

P

(

for

xt*

h

< δ, 0 < α¯1

}

A 0 + A1 + α1 + α2 ) + ε1α12 + ε2 α22 + Q

Hence from (21), and from Assumption (iii), the functional ψ1⊤ (xt* ) satisfies the following inequality

u* ≤ L

‖xt* ‖3h 4 (β2 + βα¯1) = L xt* h , L = 2 * ‖xt ‖h α32

Since, the term in the numerator has a higher order than the denominator, the convergence rate of the numerator is greater, therefore u* (t ) ≤ L xt* h , and xt* h < δ ( ε )⟹ u* (t ) < ε , with

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ε

δ (ε ) = L . This implies that the functional V (xt* ) satisfies the small control property (see [14,8]), and we conclude that the optimal control law u* is continuous at x* (t ) = 0. Remark 3. The continuity proof of the control law (20) can be development according to the ideas given in [14] for the scalar case, which corresponds to our experimental platform considered here. The small control property can be written as

lim sup φ →0

ψ0 (φ) ≤ 0, ψ1 (φ)

so, without loos of generality and by using the assumption (i), the next statement is fulfilled

⎧ ⎪ 1 γ lim sup ⎨ α3 xt* h→ 0 ⎪ ⎩ xt*

(

⎧ * ⎫ ⎪ xt ⎪ α¯1 ⎬ ⎨ lim sup = ⎪ α3 xt* h→ 0 ⎪ xt* h⎭ ⎩

Fig. 2. Physical process.

2⎫

⎪ α¯ ⎬ = 1 lim sup xt* α ⎪ 3 xt* h→ 0 h⎭ h

h

=0

)

γ ≔ − 2PA 0 + Q + ε12 α12 x2 (t ) − ε1α12 x2 (t − h) + ε1f 2 (x (t )) + ε2 g 2 (x (t − h)) − 2x⊤ (t − h) PA1x (t ) − 2f ⊤ (x (t )) Px (t ) − 2g⊤ (x (t − h)) Px (t ) Thus the optimal control law has the next smooth structure ⊤ ⎧ ⎪ u whenψ (xt ) ψ1 (xt ) ≠ 0 1 u* = ⎨ ⊤ ⎪ ⎩ 0 whenψ1 (xt ) ψ1 (xt ) = 0

Fig. 3. Robust stability zone for a PID.

(22)

Note that the implementation of this algorithm is by computing out-line the matrix W0 ∈ 4n × 4n of the CLKF algorithm for the optimization problem (14). In contrast with [15] where the matrix inequality has more than 6 parameters to be solved, here the parameters are only (ε1, ε2, P , Q ) and it facilitate the use of Matlab toolbox Sedumi.

5. Atmospheric tomato drying control Tomato must be the most commonly vegetable produced in the world. Safe storage over an extended period is one of the main problems in tomatoes production [30,31]. For this reason, several drying methods have been applied to tomatoes, from the simplest ones as solar and sun drying to the most elaborated and expensive, as a microwave, or freeze drying [30,32–34]. Hot air drying must be one of the most common form of food preservation and extends the food shelf-life (this process is depicted in Fig. 1). The major goal in drying agricultural products is the abatement of the moisture content to a level which permit high retention of nutritional and sensory attributes [23]. On the food technology, it is well known that the nutrient retention in the atmospheric dried process for the tomatoes is at the temperature of 60 and 70 °C with an air flow rate of 1.5 m/s [30,31,35]. Since the heat equation can be studied as semi-linear time delay equation (see for instance

[36]), the drying process presented here is a good example to illustrate the control algorithm (20). The hot air drying prototype used in this work for the experimental test (which emulate an industrial dehydrator process) is depicted in Figs. 1–2. Our aim is to obtain the conditions where the optimal controller (22) can be applied to the atmospheric tomatoes drying process depicted in Fig. 2. The process variable to be regulated is the temperature of the air flow, and this is controlled by applying voltage to an electric resistance. The pipe in the chamber recycles the hot air into the drying process induce a state delay. In this process the system dynamics (11) are given as the scalar case, where:

a 0 = − 0.060362075717421, a1 = 0.055313698583632 b = 0.050480383855756 f (x (t )) = cx2 (t ) + ex3 (t ), c = 0.050480383855756,

g (x (t − h)) = dx2 (t − h) + kx3 (t − h) d = − 0.051638345496030

e = − 0.087851668591611,

k = 0.04504677583610,

whose parameters and dynamics are given by identification of polynomial function [15]. The time delay of the drying process is h = 10 seconds. Notice that, form assumption (12) the atmospheric drying process is locally Lipschitz, where

α1≔0.0022, α2≔0.0026 Therefore, the problem is obtain the set of solution (ε1, ε2, P, Q), if it exists, such that Proposition 2 holds. This means that the Formula (22) is a solution of the Bellman's equation (15) under performance index (7). More sinuosity, the control law (22) is optimal if there exists a solution of the LMI problem 17. In order to compute a solution of this LMI, the Matlab toolbox, SeDuMi-YalMiP is used. Using the given physical values of system drying process, the numerical values are computed as:

P = 1.2013 × 103,

Q = 68.7668,

for the positive definite scalars Fig. 1. Diagram process.

ε1 = 2.9796 × 105,

ε2 = 2.9659 × 105

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Fig. 4. Experimental results for r ¼60. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)

Fig. 5. Experimental test for r ¼70. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)

where the matrix spectrum yields to

Table 1 Average power in Kilowatthour of each test.

σ (W) =

Temperature

Inv-Opt

Robust-PID

PID tuned by Z-N

60° 70°

0.40590 0.94253

0.94806 1.2943

1.1473 2.7665

Table 2 Energy efficiency (Inverse optimality Versus the PID's, in percent). Temperature

Inv-Opt Vs Robust-PID

Inv-Opt Vs PID tuned by Z-N

60° 70°

42.81 72.82

35.38 34.07

Temperature

Inv-Opt Vs Robust-PID

Inv-Opt Vs PID tuned by Z-N

60° 70°

198.1131 128.5418

270.9182 666.5049

Table 3 Saving per year (in US-dollars).

thus, the linear matrix inequality (17) has the form

⎡ − 0.0008, 0.0007, 0.0120, 0.0120 ⎤ ⎥ ⎢ 0.0007, − 0.0007, 0, 0 ⎥ × 105 W=⎢ ⎥ ⎢ 0.0120, − 2.9796, 0, 0 ⎥ ⎢ ⎣ 0.0120, − 2.9659⎦ 0, 0,

{ −2.9796 × 105,

}

− 2.9659 × 105, − 134.0604 − 1.1483

The numerical results are implemented via LabVIEW environment with National Instruments USB-6008 data acquisition device. An electrical grid is used as a heat source, and the temperature is regulated by applying a voltage. The control process is given by regulation of the desired temperature which is given by the Heaviside step function defined by the following equation:

⎧ 1 if ts ≥ 0 H (ts )≔⎨ ⎩ 0 if ts < 0 In order to show the performance of the control algorithm proposed in this work, we present, the regulation at the best temperatures for the nutrient retention in the dried atmospheric process for the tomatoes. Two different set points are introduced, the first one is at the temperature of r1 = 60 H (ts ), and the second one at r2 = 70 H (ts ) in Celsius degrees. Moreover, a comparison with Robust-PID and PID tuned by Z-N algorithm is presented [22,25]. Although the model presented here is a single input-single output nonlinear system, the application of the exact feedback linearization is not considered. The reason is that the feedback linearization requires the exact knowledge of the mathematical model. In this work, the mathematical model is not available to be used, then we use an estimated model by polynomial interpolation which means that the considered model is an approximation of the real one. The implementation of the PID's is by using a Honeywell DC1040 industrial PID with maximum precise of ±1 °C of cold junction compensation, ±5% of maximum deviation in the linear output 4–20 mA, automatic compensation of dead zone, and J type

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thermocouple of extended rate in the input. This type of controller is widely used in the industry due to facilities of programming and accuracy. The gains for the Z-N algorithm are k p = 46.2, kd = 92.4 , and k i = 5.77. The Robust-PID is tuned by the D-partition method [25,37], where the stability region for each gain are Kp ∈ [ − 1, 60.48], Ki ∈ [0, 0.7402] and Kd ∈ [0, 38.5116]. Fig. 3 depicts the robust stability zone for the PID controller. For the Robust-PID implementation the PID gains are tuned as k p = 2, kd = 8, and k i = 0.2. The control objective is to reach the system temperature at ri (ts ), for i¼1,2, in Celsius degrees, for time interval ts ∈ [0, 1500) seconds. The initial conditions of the system for the control process are approximately at x0 = 20° Celsius degrees. For the realtime experimental test, we have obtained the following results. Figs. 4–5 present the current state response (temperature in Celsius degrees) and the control signal (in volts) for both references r1 = 60Hs (t ) and r2 = 70Hs (t ), respectively. The system response, the control signal, and the power system is depicted for each setpoint. ” shows the solution of the control problem by The blue line “ applying the algorithm presented here. The Robust-PID control ”. Finally, the red response is represented by the green line “ line “ ” depicts the PID control response where the PID gains are tuned by Ziegler-Nichols method. Evidently, the performance of the optimal control is better than the PID's, in energy consumption terms. Table 1 is introduced to support the previous assertion. Here is remarkable to ensure that the steady state error criteria for the ±2% is at time t¼450, t ¼520 for the optimal control and Robust-PID, respectively. For the first reference r1, the PID tuned by Z-N methodology converges to 57 Celsius degrees. For the second experiment, by considering the ±2% criteria; the trajectory of the closed loop system with the optimal control, converges at t¼ 480, for the Robust-PID at t ¼492, finally the PID tuned by Z-N methodology at t¼550 seconds. In what follows, we shall summarize some general guidelines concerning to energy saving. Table 2 shows the energy saving. In contrast with [24], where the save up respect to the PID controller is around to 16%, the energy saving of the proposed controller here is more than 30% (for almost all experimental results). In many cases is more than 40%. Alternatively, to illustrate this result, an economic savings instance for an industrial application is presented. The average retail price of electricity to ultimate industry customers in November of 2015 for the Average Retail Prices of Electricity is 7.32 Cents per Kilowatthour [38]. Table 3 presents the total energy saving using one atmospheric drying process for industrial proposes (16-hours per day and 6-days per week). To achieve a sustainable level of energy saving relative to control design, the industry of dehydration may need to undergo a restructuring of the control process, whereby energy consumption is reduced and saving is increased. So, from the main contribution of this work, the advantages must be summarized in the next points:

 Unlike to previous results [15,24], the region of convergence





when the system operates successfully, is given by only one model, it indicates a lower conservatism by the using of reduced CLKF. In fact, in contrast with [15], to achieve the stabilization at different operation point the model does not require the system identification at each point. Since a reduced type functional is used here, the Hurwitz stability is not required in the linear part of the model to ensure stability in the Lyapunov-Krasovskii sense (see for instance [1,2]). Here we do not need the construction of a Lyapunov matrix for the stability analysis. Which require an advanced knowledge on

7

Lyapunov Matrix theory.

 The control problem is relaxed to a simple reduced CLKF. Meaning that the methodology proposed in this work take into account the adequate system information to achieve the wanted reference. However, it is probably that the experimental results could be improved by using of a more complex functional, which takes into account time varying delays, uncertain parameters, unmodeled nonlinear dynamics, etc.

6. Concluding remarks In this paper, the development of nonlinear optimal controller based on Control-Lyapunov Functions concept and LyapunovKrasovskii functionals for time delay systems is presented. Moreover, a class of optimal nonlinear time delays feedback control scheme for atmospheric drying process which guarantees a local uniform asymptotic stability has been done. Experimental validation was conducted with Proposition 2, and the effectiveness of the regulation of a dehydration process has been presented. Here we have demonstrated how to obtain the sufficient conditions to extend the inverse optimality approach for a class of time delay nonlinear systems (see expressions (8), (22), (18) and (19)). Even more, we have used the dynamic programming, and CLKF approaches to guarantee the uniform asymptotic convergence (at least in local point of view). The results show a considerable decrease of the energy consumption in the dehydration process, obtained by the proposed nonlinear optimal control. Furthermore, the controller proposed here achieves to reduce energy consumption respect to the controller used in [24]. To compare the control scheme with controllers based on PID's, the energy saving in many cases is more than 40%. In terms of saving using the atmospheric drying process, for industrial applications, is more than $198 US-dollars per year.

Acknowledgement This work was supported by CONACYT-Mexico, project 239371, and CONACYT-Mexico National Posdoctoral fellowship 2014-I.

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