Design Global Extremum Seeking Control with ... - Semantic Scholar
51st IEEE Conference on Decision and Control December 10-13, 2012. Maui, Hawaii, USA
Global Extremum Seeking Control with Sliding Modes for Output-Feedback Global Tracking of Nonlinear Systems* Chun Yin1,2 , Brandon Stark2 , Shou-ming Zhong1 and YangQuan Chen2 Abstract— In this paper, a novel design of an ExtremumSeeking Controller (ESC) based on Sliding Mode Control (SMC) for a class of Single-Input-Single-Output (SISO) uncertain or unknown nonlinear systems is proposed. The corrective term of the proposed observer is used to estimate the uncertain system. Global exact tracking using only output-feedback is proven with a tanh periodic switching function. The tanh periodic switching function is shown to provide a smaller vicinity of the desired extremum point compared to the traditional sgn periodic switching function. Numerical simulation results are presented to demonstrate the advantages and effectiveness of the proposed control strategy.
The rest of the paper is organized as follows. In Section II, the problem is formulated and the necessary notation is established. The design of the sliding mode ESC is presented in Section III. In Section IV, an example system is presented and the results of the numerical simulations are shown. Finally, concluding remarks are presented in Section V. II. P ROBLEM F ORMULATION Consider the nonlinear system described by
I. INTRODUCTION In many applications, the optimum plant efficiency occurs at a singular extremum point of a nonlinear system. The control objective is to maintain the extremum point in realtime without a priori knowledge of the system. Over the last several decades, many control strategies have been proposed, including perturbation and averaging ([1], [2], [3]) and more recently, sliding mode control ([4], [5], [6], [7], [8], [9], [10], [11], [12], [13]). In [10], the authors developed a ¨ uner’s pesliding mode ESC based on the Drakunov-Ozg¨ riodic switching function. In their proposed controller, at least one of the sliding surfaces would be a stable sliding surface, independent of control direction. In [13], the authors enhanced the sliding mode ESC approach by utilizing an ¨ uner periodic output-feedback version of the Drakunov-Ozg¨ switching function. The authors proved that global exact tracking was achievable for uncertain plants with unknown control direction. In this paper, the controller design found in [13] is further enhanced and is proven to reduce the oscillations around the optimal point. The oscillations of order O(γ0 ) around the optimal point y ∗ can be reduced by replacing the sgn periodic switching function with the tanh periodic switching function and the inclusion of a first norm observer. Previous works have shown that the tanh operator can be used to improve the performance, but a rigorous proof is presented in this paper. Numerical simulations are used to demonstrate the advantages and effectiveness of the proposed method. *This work was supported by National Basic Research Program of China (2010CB732501) and the National Natural Science Foundation of China (NSFC-60873102). 1 School of Mathematics Science, University of Electronic Science and Technology of China, Chengdu 611731, P. R. China ([email protected]) 2 Mechatronics, Embedded Systems and Automation (MESA) Lab, School of Engineering, University of California, Merced, 5200 North Lake Road, Merced, CA 95343, USA ([email protected], [email protected])
978-1-4673-2064-1/12/$31.00 ©2012 IEEE
x˙ = f (x, t) + g(x, t)u,
(1)
y = h(x),
(2)
where x = [x1 , x2 , · · · , xn ] ∈ Rn is the state vector, u ∈ R is the control input, y ∈ R is the measured output, f , g and h are unknown but continuous functions. In the state vector x, not all xi are assumed to be available for measurement. Without loss of generality, the following assumptions are made: Assumption 1: The nonlinear uncertain functions f , g and h are locally Lipschitz continuous in x, piecewise continuous and uniformly in t and sufficiently smooth. Assumption 1 is a necessary condition to prove the existence and uniqueness of a solution. For each solution, there exists a time interval [0, tM ) that the solution is valid, where tM may be finite or infinite. In order to solve the problem of the unavailable states, an observer is proposed. Proposition 1: To estimate the system (1), the following first order norm observer is proposed ˆ˙ = −˜ ˆ + ρ0 (x, t), x ωx
(3)
where ρ0 (x, t) is the input, ω ˜ is a positive constant and ρ0 (x, t) is a non-negative function continuous in x, piecewise ˆ is the output such that continuous and upperbounded in t. x ˆ (0) and x(0), for each initial state x ||x(t)|| ≤ ||ˆ x(t)|| + c0 ,
∀t ∈ [0, tM ),
(4)
where c0 := ω0 (||ˆ x(0)|| + ||x(0)||)e−¯ωt and ω0 , ω ¯ are positive constants. It is assumed that the system (1) is ISS [14] and thus can utilize such a norm observer and the plant is minimum-phase. Assumption 2: The system (1) admits a known norm observer (3). Assumption 3: There exists a positive constant g such that ¯ 0 < g ≤ ||g(x, t)||, ∀t ∈ [0, tM ). (5) ¯ From this assumption, the uncertain nonlinear function g(x, t) is bounded away from zero. The system (1) has
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relative degree one due to g(x, t) is not 0 for all x and t. Remark 1: In this paper, the goal is to solve the minimum seeking problem. Thus the global real-time optimization control problem, i.e., minimization of (2) under (1) can be stated as follows: (I) Optimal operating point: Assume that there exists a unique point x∗ such that y ∗ = h(x∗ ) is the minimum of the function y = h(x). (II) Uncertainty assumption: Suppose that x∗ , h(·) and its gradient are unknown to the control designer. (III) Global extremum seeking control based on sliding modes: The control problem consists in determining the control input u to steer the system to achieve to the minimum point y ∗ = h(x∗ ) and remain on such point as close as possible. The following assumption is used to derive the uncertainty bounds for control design. Assumption 4: All parameters of the uncertain system belong to a compact set Ω. The output of the system (1) has a unique extremum (minimum) point in the compact set Ω by using the notation h0 (x) = dh(x)/dx and h00 (x) = d2 h(x)/dx2 . The following assumption is necessary for the output function. Assumption 5: There exists a unique point x∗ ∈ R such that h0 (x∗ ) = 0 and h00 (x∗ ) > 0. For any given δ > 0, let Dδ = {x : ||x − x∗ || < δ/2} is called δ-vicinity of x∗ , there exists a positive constant ε such that |h0 (x)| > ε, ∀x ∈ / Dδ , in which δ can be made arbitrary small by allowing a smaller ε. According (1) and (2), the first time derivative of the output is derived as follows, y˙ = h0 f + kp u,
(6)
where kp is the coefficient of u which appears in the first time derivative of the output y and it is obtained as kp = h0 g. From Assumption 5, the following equality can be derived as (7) |kp | ≥ kp , ¯ where the lower bound kp ≤ gε is a positive constant. In order to derive a ¯norm ¯ bound for the term h0 f , the following assumption is given. ¯ η1 , and Assumption 6: There exist known functions h, ψ1 (x, t) such that ||f (x, t)|| ≤ η1 (||x||) + ψ1 (x, t) and ¯ |h0 | ≤ h(||x||), where η1 ∈ K∞ and is locally Lipschitz, and ψ1 (x, t) is a known non-negative function continuous in x, piecewise continuous and upperbounded in t. This assumption is not restrictive since h0 is assumed to be smooth and f is locally Lipschitz continuous in x. Furthermore, the dominating function η1 and ψ1 force strict growth condition only with respect to the time-dependence. Thus, polynomial nonlinearities in x can be dealt with.
tends asymptotically to zero (exact tracking), starting from any initial conditions and maintaining uniform closed loop signal boundedness and where yr is the desired trajectory. This control law is designed based on output-feedback version of tanh periodic switching function method. To ensure that the output y of the system is steered to reach the minimum point y ∗ , the output-feedback ESC with periodic switching function is proposed as π τ (t) , (9) u = φ(t)tanh sin γ0 where φ(t) is the designed modulation function and is continuous in t, where Z t τ (t) = γ1 e(t) + γ2 sgn(e(ς))dς, (10) 0
and γi > 0, (i = 0, 1, 2) are appropriate constants. For analysis convenience, the desired trajectory yr is generated by y˙ r (t) = −kr , yr (0) = yr0 ,
(11)
with kr is a positive constant and yr0 is a design constant. The model output can saturate at a rough known norm lower bound of y ∗ without effecting the control performance in order to avoid an unbounded reference signal yr (t). The modulation function φ(t) is derived such that y(t) tracks yr (t) as long as possible. This dictates that y(t) is forced to reach the vicinity of the minimum y ∗ = h(x∗ ) and remains close to y ∗ . Thus, the function φ(t) is designed so that a sliding mode τ˙ (t) = 0 occurs in finite time on one of the manifolds τ (t) = lγ0 , with some integer l. According to (10), the following equation can be obtained e(t) ˙ = −(γ2 /γ1 )sgn(e(t)).
(12)
Thus, e tends to converge to zero. The output y will track yr , while y remains outside of the small vicinity of y ∗ = h(x∗ ) where the high frequency gain (HFG) is not zero. In contrast, once y reaches the vicinity of the minimum value y ∗ = h(x∗ ), the controllability is lost since the HFG will tend to zero. Thus, y will not continue to track yr . However, the output y reach the vicinity of the optimum point y ∗ as expected. It can be shown that the convergence rate of x to the δ-vicinity Dδ is a function of γ1 , γ2 . Although Dδ is not positively invariant, the designed control law u will guarantee that x will remain close to x∗ . It does not imply that x remains in Dδ . But it will be shown that y remains close to the minimum point y ∗ in the proof of Theorem 1. A. Modulation function design
The goal of the control law is to design a control law u, via output feedback and without the knowledge of the system control direction, so that
In this subsection, the modulation function φ(t) will be designed. From Assumption 6, it can be inferred that ||f || ≤ η1 (||ˆ x|| + c0 ) + ψ1 (x, t). Utilizing that ψ(a + b) < ψ(2a)+ψ(2b), ∀a, b ≥ 0 and ∀ψ ∈ K∞ , η1 (||ˆ x||+c0 ) can be rewritten as η1 (||ˆ x||+c0 ) < η1 (2||ˆ x||)+η1 (2c0 ), (η1 ∈ K∞ ). Thus, the following inequality can be obtained
e(t) = y(t) − yr (t),
||f || ≤ η1 (2||ˆ x||) + η1 (2c0 ) + ψ1 (x, t).
III. D ESIGN EXTREMUM - SEEKING CONTROLLER WITH SLIDING MODE
(8) 7114
(13)
From the definition of c0 in the Proposition 1, c0 is uniformly bounded. Therefore, since η1 is assumed locally Lipschitz in Assumption 6, there exists a positive constant l1 such that η1 (2c0 ) ≤ 2l1 c0 = 2l1 ω0 (||ˆ x(0)|| + ||x(0)||)e−¯ωt . Hence, (13) can be rewritten as ||f || ≤ η1 (2||ˆ x||) + ψ1 (x, t) + 2l1 c0 .
(14)
Furthermore, the first term h0 f of the right-hand side in (6) satisfies the following condition 0
0
0 2
Fig. 1.
W1 is solid line when ks < 0.
Fig. 2.
W2 is solid line when ks > 0.
2
|h f | ≤ |h |(η1 (2||ˆ x||) + ψ1 (x, t)) + |h | + (2l1 c0 ) . (15) The time derivative of τ (t) is obtained from (8) as τ˙ (t) = γ1 kp u + ξs ,
(16)
0
where ξs := γ1 h f − γ1 kr + γ2 sgn(e). Let k = γ1 kp , such that τ˙ (t) = k(u + ω), in which ω = ξs /k. It is clear that the modulation function φ is designed such that φ ≥ |ω| + α, in which α is an arbitrary positive constant. According to (16), the norm bounds of the parameter ω is defined as |ω| ≤ ω ˆ + ς/|k|, (17) ¯ f˜ + γ1 h ¯ 2 + γ1 kr + γ2 )/k and ς = where ω ˆ ≤ (γ1 h −¯ (2l1 ω0 (||ˆ x(0)|| + ||x(0)||)e ωt )2 , in which f˜ =¯η1 (2||ˆ x||) + ψ1 (x, t), k = γ1 kp . One possible modulation function im¯ is proposed ¯ plementation such that φ| tanh(sin(πs/γ0 ))| ≥ |ω| + α holds in the following theorem, finite-time escape is avoided and realization of the s-sliding modes is guaranteed. Theorem 1: Consider the system (1)-(2), (8) with the control law (9) and that outside the δ-vicinity, φ(t) in (9) satisfies the following equality ¯ f˜ + γ1 h ¯ 2 + γ1 kr + γ2 ) φ tanh2 [sin[(π/γ0 )τ ] := [(γ1 h +||yt ||e−α1 t ]/k + α, (18) ¯ where α1 is an arbitrary positive constant and yt is an arbitrary vector. While x ∈ / Dδ , one has: (i) no finite-time escape occurs in the system signals (tM → +∞) and (ii) a sliding mode on sliding manifold s(t) = lγ0 is achieved in finite time for some integer l. Proof. Consider the following non-negative functions (Figs. (1-2)) Z τ π ς dς, W2 (τ ) = γm − W1 , W1 (τ ) = tanh sin γ 0 0 (19) where γm denotes the maximum value of W1 (τ ). Where W1 (τ (t)) and τ (t) are both differentiable, the time derivatives of W1 and W2 are given by π π 2 ˙ W1 = k φ tanh sin τ + ω tanh sin τ , γ0 γ0 ˙ 2 = −W ˙ 1, W (20) ˙ 1 = (∂W1 /∂τ )τ˙ was utilized. It where the relationship W can be seen that both inequalities are the same, ˙ 1 ≤ k{φ tanh2 [sin[(π/γ0 )τ ]} + |k||ω|, W ˙ 2 ≤ k{φ tanh2 [sin[(π/γ0 )τ ]} + |k||ω|, W (21)
since | tanh(sin[(π/γ0 )τ )| ≤ 1. According to (17), |k||ω| ≤ |k|ˆ ω + ς. They can be further broken down into, ˙ 1 ≤ −|k|{φ tanh2 [sin[(π/γ0 )τ ] − ω W ˆ } + ς, if sgn(k) < 0, 2 ˙ 2 ≤ −|k|{φ tanh [sin[(π/γ0 )τ ] − ω W ˆ } + ς, if sgn(k) > 0, (22) when inclu Consequently, by definition of k and k, the ¯ relationship −|k| ≤ −k can be obtained from (9). According ¯ to (18), it is easy to conclude that ˙ 1 ≤ −||yt ||e−α1 t − kα + ς, if sgn(k) < 0, (23) W ¯ ˙ 2 ≤ −||yt ||e−α1 t − kα + ς, if sgn(k) > 0, (24) W ¯ holds almost everywhere, with α ≥ 0, α1 > 0. The exponential term with rate α1 performs as a forgetting factor which allows a less conservative modulation function design. It is fundamental to avoid finite-time escape in the system signals. Property (i): Assume by contradiction that |s(t)| escapes in some finite time t1 ∈ [0, tM ). From (14), it can be seen that e(t) and y(t) also can escapes at t = t1 . Hence, there exists t2 ∈ [0, t1 ) such that ||yt || ≥ eα1 t [α2 − kα + ς], ¯ from where is an arbitrary non-negative constant. Moreover, inequalities (23) and (24), one has ˙ 1 ≤ −α2 or W ˙ 2 ≤ −α2 , ∀t ∈ [t2 , t1 ), W
(25)
independently of sgn(k). Since τ (t) is absolute continuous and escape in t = t1 , there exists te ∈ [t2 , t1 ) and an integer lτ such that τ (te ) = lτ γ0 . Therefore, W1 (te ) = 0 (if ls is an even number) or W2 (te ) = 0 (if ls is an odd number). It is clear that W1 (t) = 0, ∀t ∈ [te , t1 ) or W2 (t) = 0, ∀t ∈ [te , t1 ) from (25) in this interval (Figs. 1,2). With some abuse of notation, Wi (τ (t)) was replaced by Wi (t), for i = 1, 2. Consequently, τ (t) = lτ r0 is uniformly bounded ∀t ∈ [te , t1 ), i.e., a contradiction. Thus, the signals
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τ, e and y cannot escape in finite time. Moreover, from Assumption 3 and Assumption 5, one can further conclude ˆ and all closed loop signals cannot escape in finite that x, x time (tM → +∞). Property (ii): According to Property (i) and ς decreases exponentially, it is clear that there exists a finite time t¯ ≥ 0 ˙ i ≤ −αt¯, ∀t ≥ t¯ and 0 < αt¯ < kα, for such that W ¯ [15], i = 1 or i = 2. Using the Comparison Lemma ¯ ¯ ¯ Wi (t) ≤ −αt¯(t − t) + Wi (t), ∀t ≥ t. Consequently, there exists a finite time t∗ ≥ t¯ such that Wi (t) = 0, ∀t ≥ t∗ . Moreover, the corresponding points τ = lγ0 for which W1 (τ ) = 0(W2 (τ ) = 0) occur only for even (or odd) value of l (Fig. 1,2). In the neighborhood of the points τ = lγ0 , sgn(tanh(sin(πs/γ0 ))) = sgn(tanh(τ − lγ0 )) for l is an even number or sgn(tanh(sin(πτ /γ0 ))) = sgn(− tanh(τ − lγ0 )) for l is an odd number. Thus, there exists two positive functions p1 (t), p2 (t) (i.e. p1 (t), p2 (t) > 0, ∀t) such that tanh(sin((π/γ0 )τ )) = (tanh(τ − lγ0 ))p1 and tanh(sin((π/γ0 )τ )) = −(tanh(τ − lγ0 ))p2 . Now, the following inequality can be derived, either for l is an even number (sgn(k) < 0) or l is an odd number (sgn(k) > 0). Selecting a lyapunov candidate function, V = 0.5(τ − lγ0 )2 , and computing its derivative with respect to time, π τ + ω)) V˙ = (τ −lγ0 )(k(φ tanh sin γ0 (s−lγ0 )(k(φ tanh(τ −lγ0 )p1 +ω)), if sgn(k) < 0, = −(s−lγ0 )(k(φ tanh(τ −lγ0 )p2 +ω)), if sgn(k) > 0, ≤ 0. (26) Hence, a sliding mode occurs in finite time on one of the manifolds τ = lγ0 , independently of sgn(k). Remark 2: In the Theorem 2, the exponential term with rate acts like a forgetting factor which allows a less conservative modulation function design. Note that the functional norm term is fundamental to avoid finite-time escape of the system signals. B. Main result for extremum-seeking control based on sliding mode In this subsection, the output-feedback ESC with tanh periodic switching function (9) and (10) with the appropriate constants and a modulation function φ that satisfies (18) is shown to guarantee the state reaches to the δ-vicinity of the unknown minimizer x∗ defined in Assumption 2. Theorem 2: Consider the system (1), with output or cost function (2), control law (9) and (10), a modulation function φ that satisfies (18) and reference trajectory (11). If the Assumptions (1-5) hold, then there exists two properties: (i) δ-vicinity Dδ in Assumption 2 is globally attractive and can be reached in finite time and (ii) for ε sufficiently small, the oscillations around the minimum value y ∗ of y can be made of order O(γ0 ) with γ0 from (9). Moreover, all signals in the closed-loop system remain uniformly bounded except for τ (t) which is only an argument of a sgn function in (10). Proof: The derivative of the cost function y = h(x) does not vanish (h0 (x) 6= 0, ∀x ∈ Dδ ), while x is outside the δ-vicinity. Therefore, a lower norm bound kp for kp can be ¯
derived from the lower bound given in Assumption 5 which is valid globally. A lower norm bound k for k can be also obtained (k = γ1 kp , k = γ1 kp ). When x ¯stays outside the δ¯ escape ¯ occurs for the system signals vicinity, no finite-time and a sliding mode on τ = lγ0 can occur in finite time for some integer l from Theorem 1. Now, the proof of the properties (i) and (ii) of Theorem 2 will be given as follows: (i) Attractiveness of Dδ : It is assumed that x remains outside the δ-vicinity for all t ∈ [0, tM ) (i.e. x ∈ / Dδ , ∀t ∈ [0, tM )). Since the sliding mode on τ = lγ0 can occur in finite time, there exists a finite time tf such that τ˙ = 0. Therefore, from (12) that e˙ = −(γ2 /γ1 )sgn(e), ∀t ≥ tf . The error e = y − yr tends to zero. However, it is apparent that the cost function y = h(x) has a minimum value y ∗ and the reference signal yr (t) strictly decreases with time. So one has that yr < y ∗ ≤ y and sgn(e) = 1 when t is large enough. Then, it is assured that y decreases with the negative rate (y˙ = −kr −(γ2 /γ1 )sgn(e) < 0), that is, the cost function y = h(x) must to approach the minimum value y ∗ . Hence, x goes to δ-vicinity, which is a contradiction. Thus, the δ-vicinity is attained in some finite time. Consequently, x remains or oscillates around Dδ and similarly y with respect to some small vicinity of y ∗ , ∀t large enough. These oscillations appear due to the loss of control strength as k → 0 (kp → 0) and the recurrent changes in the HFG sign at the extremum point (x∗ , y ∗ ) where k = 0 (k = γ0 h0 f ). During these oscillations, τ can go from one sliding manifold τ = lγ0 to another manifold. In such cases, y could start oscillations around y ∗ with decreasing minimum amplitude. These oscillations can be made ultimately of order O(γ0 ) with γ0 from (9). (ii) Oscillations of order O(γ0 ) around y ∗ : According to the Assumption 5, δ can be made arbitrary small so that |y − y ∗ | = O(γ0 ) when x ∈ Dδ . Thus, if x remains in Dδ , ∀t, the corresponding neighborhood of y ∗ can be made of order O(γ0 ) with an appropriate ε. It is shown that, if x oscillates around Dδ , |y − y ∗ | = O(γ0 ) also holds since x is outside after leaving the set Dδ . There exists some finite time t˜ > 0 such that sgn(e) = 1, ∀t ≥ t˜, since the cost function y = h(x) has a minimum value y ∗ and yr (t) strictly decreases with time. Suppose that x reaches the frontier of δ-vicinity Dδ (from inside) at some time t˜1 > t˜ and τ (t) is not in a sliding mode when t = t˜1 . Note that Dδ is invariant when in a sliding mode. From (10), it is clear that Z t τ (t) = γ1 y(t) − γ1 yr (t) + γ2 sgn(e(τ ))dτ + C, (27) t˜
R t˜
where C = 0 sgn(e(τ ))dτ is a constant. Define τ˜(t) := τ (t) − τ (t˜1 ) and y˜(t) := y(t) − y(t˜1 ), τ˜(t) = γ1 y˜(t) + (γ1 ks + γ2 )(t − t˜1 ),
(28)
where ks = 0 if yr is saturated and ks = kr , otherwise. From the above equation, it is seen that
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|˜ τ (t)| ≤ γ1 |˜ y (t)| + (γ1 ks + γ2 )(t − t˜1 ), ∀t ≥ t˜1 ,
(29)
Define t˜2 as the first time when τ (t) reaches the next sliding manifold τ (t) = τ (t˜2 ) (independent of whether x is inside or outside Dδ ) and t˜3 as the first time when x reaches the frontier of Dδ again (from outside). It is seen that t˜2 ≥ t˜1 > t˜, t˜3 ≥ t˜1 > t˜. Then, there are two cases: (i) x reaches the frontier of Dδ with τ (t) in sliding motion (t˜3 > t˜2 ) and (ii) x reaches the frontier of Dδ with τ (t) not in sliding mode (t˜3 ≤ t˜2 ). For case (i), let t ∈ [t˜1 , t˜3 ] and consider t ∈ [t˜1 , t˜2 ) such that τ (t) is not in a sliding motion during this time interval. Hence, it is clear that there exists some integer l such that lγ0 < τ (t) < (l + 1)γ0 . Otherwise, sliding mode occurs at τ (t) = lγ0 or τ (t) = (l + 1)γ0 according to Theorem 1. Consequently, τ˜(t) = τ (t) − τ (t˜1 ) is of order O(γ0 ), ∀t ∈ [t˜1 , t˜2 ). Moreover, it is obvious that 0 < | tanh(sin(πτ (t)/γ0 ))| ≤ 1, ∀t ∈ [t˜1 , t˜2 ). Allowing t˜ to be large enough so that the exponential term ς in has decreased to an arbitrarily small value, the conditions φ ≥ |ω|| tanh(sin(πτ (t)/γ0 ))| + α are satisfied. Since lγ0 < τ (t) < (l + 1)γ0 for all t ∈ [t˜1 , t˜2 ), they guarantee an appropriate positive constant α so that |τ˙ (t)| ≥ k|u + ω| ≥ k(φ| tanh(sin(πτ (t)/γ0 ))| − |ω|) ≥ α ˜ in which α ˜¯ = kα is a ¯positive constant. Thus, (t − t˜ ) ≤ |˜ τ |/˜ α, ∀t ∈ [t˜1 , t˜¯2 ) and 1 thus (t − t˜1 ) is of order O(γ0 ), ∀t ∈ [t˜1 , t˜2 ). According to (24), y(t) − y(t˜1 ) is of order O(γ0 ), ∀t ∈ [t˜1 , t˜2 ). Moreover, by continuity, y(t) − y(t˜1 ) is also of order O(γ0 ), ∀t ∈ [t˜1 , t˜2 ]. Next, consider t ∈ [t˜2 , t˜3 ] where τ (t) is in sliding motion during this interval. Therefore, τ˙ (t) = 0. From (12), one has that y˙ = −kr − (γ2 /γ1 )sgn(e) < 0 and y(t) is strictly decreasing, ∀t ∈ [t˜2 , t˜3 ]. Hence, y(t) − y(t˜1 ) is also of order O(γ0 ), ∀t ∈ [t˜2 , t˜3 ], since y(t) approaches the minimum value y ∗ during that latter interval. Since this is also valid for the interval, it is proven that the oscillation outside Dδ is of order O(γ0 ) in case (i). For case (ii), let t ∈ [t˜1 , t˜2 ], where τ (t) is not in a sliding motion during this time interval. Therefore, following the first part of the proof of case (i), it is directly seen that y(t) − y(t˜1 ) is of order O(γ0 ), ∀t ∈ [t˜1 , t˜2 ]. From (20), lim |τ (t)| = +∞.
t→+∞
However, considering τ (t) is only a modified time scale for the argument of the sin function in the controller (9), this does not affect the uniformly norm boundedness (UB) of other signals. The boundedness of y stated above implies that the signal x is UB, according to the continuity of the uncertain output function y = h(x). Likewise, the other closed-loop signals are UB from Proposition 1. Corollary 1 : Consider the output function y = h(x) that has a unique unknown minimum value y ∗ = h(x∗ ), such that x∗ is in the interior of some closed interval [a, b]. If the assumptions in Theorem 2 hold, then only semi-global practical convergence of the closed-loop system is obtained in the sense that the domain of attraction can be arbitrarily increased by reducing ε.
Fig. 3. (a) Time responses of the output plant y (green line) and the output model yr (blue line) via our controller. The output plant tends to the minimum value y ∗ = −2; (b) Time responses of τ showing the oscillations of x around x∗ = 3 which minimizes y considering γ0 = 0.05.
IV. N UMERICAL SIMULATIONS The following example is used to show the applicability of the proposed ESC controller based on sliding mode law. For simplicity, the proposed control scheme is utilized to derive a robust one-dimensional nonderivative optimizer based on periodic switching functional and sliding modes. Example: Consider a function y = h(x) = −12x/(9+x2 ) to be minimized in the interval of interest [a, b] = [0, 10]. It has a minimum y ∗ = h(x∗ ) = −2 at x = x∗ = 3. Since h0 (x) tends to zero as x → ∞, Assumption 5 is not globally satisfied. Hence, only local convergence to the optimal value might be expected. Yet, the convergence was verified in all simulations starting with x = 0, which is realistic situation in an ABS braking scenario. In following section, the design of the proposed nonderivative optimizer is processed. Consider the following auxiliary order nonlinear system x˙ = u,
y = h(x) = −12x/(9 + x2 ),
(30)
where y˙ = kp u, kp = h0 (x) can be regarded as the HFGs, x is the integrator output and the initial condition is (x0 , y0 ) = (1, −1.3). In this example, x is not used for feedback even though it is available; only y is utilized. The tracking signal yr is selected as in (11) and, in the example, yr was saturated at -150. The control law (9) and (10) is used with constant modulation function φ(t) = (γ1 + γ2 + kr )/ε + α. Let the parameters γ0 = 0.05, γ1 = 0.5, γ2 = 0.1, kr = 0.5 and α = 0.1 in the simulations. The lower bounded ε = 0.1γ0 in and initialize γ0 with not so small value. Then, γ0 is decreased until the variance of y is ultimately small, that is, |y − y ∗ | → O(γ0 ). In Fig. 3, y does not track yr till y reaches the vicinity of the minimizer x∗ = 3. The exact tracking is not obtained but y is locked at some γ0 -neighborhood of y ∗ = −2 and yr will decrease until the saturation value -150, and time responses of τ showing the oscillations of x around x∗ = 3 which minimizes y considering γ0 = 0.05. The comparison of the output y under the proposed control law and the output y1 under the normal sliding mode based extremum seeking control with sgn periodic switching function in [13]
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point can be guaranteed. It is possible to have better tracking performance by utilizing the proposed control law. Numerical simulation results were presented to show the effectiveness of the proposed method. R EFERENCES
Fig. 4. Times responses of the output y under our control and the output y1 under the control with sgn periodic switching function.
Fig. 5. (a) Times responses of the output y under the proposed control and the output y1 under the control with sgn periodic switching function, when γ0 = 0.15; (b) Time responses of τ showing the oscillations of x around x∗ = 3 which minimizes y considering γ0 = 0.15.
[1] M. Krsti´ c and H.-H. Wang. Stability analysis of extremum seeking feedback for general nonlinear systems. Automatica, 36(2):595-601, 2000. [2] C. Zhang and R. Ord´ on ˜ez. Robust and adaptive design of numerical optimization-based extremum seeking control. Automatica, 45,634646 (2009). [3] M. Guay and T. Zhang. Adaptive extremum seeking control of nonlinear dynamic systems with parametric uncertainties. Automatica, 39(7):1283-1293, 2003. [4] S. K. Korovin and V. I. Utkin. Using sliding modes in static optimization and nonlinear programming. Automatica, 10:525-532, 1974. [5] A. B. Will, S. Hui and S. H. Zak. Sliding mode wheel slip controller for an antilock braking system. International Journal of Vehicle Design, 19(4):523-539, 1998. [6] T. R. Oliveira, A. J. Peixoto, E. V. L. Nunes, and L. Hsu. Control of uncertain nonlinear systems with arbitrary relative degree and unknown control direction using sliding modes. International Journal of Adaptive Control and Signal Processing, 21:692-707, 2007. [7] A. Levant. Higher-order sliding modes, differentiation and outputfeedback control. International Journal of Control, 76(9):924-941, 2003. [8] T. R. Oliveira, A. J. Peixoto, E. V. L. Nunes and L. Hsu. Control of uncertain nonlinear systems with arbitrary relative degree and unknown control direction using sliding modes. International Journal of Adaptive Control and Signal Processing, 21:692-707, 2007. [9] C. Yin, S. Zhong and W. Chen. Design of sliding mode controller for a class of fractional-order chaotic systems. Commun. Nonlinear Sci Numer. Simulat., 17:356-366, 2010. ¨ Ozg¨ ¨ uner. Optimization of Nonlinear System [10] S. Drakunov and U. Output via Sliding Mode Approach. In Proceedings of the IEEE International Workshop on Variable Structure and Lyapunov Control of Uncertain Dynamical Systems, pp. 61-62, 1992. ¨ Ozg¨ ¨ uner, P. Dix, and B. Ashrafi. ABS control using [11] S. Drakunov, U. optimum search via sliding modes. IEEE Transactions on Control Systems Technology, 3(1):79-85, 1995. ¨ Ozg¨ ¨ uner. Stability and performance improvement of [12] Y. Pan and U. extremum seeking control with sliding mode. International Journal of Control, 76:968-85 2003. [13] T. R. Oliveira, L. Hsub and A. J. Peixoto. Output-feedback global tracking for unknown control direction plants with application to extremum-seeking control. Automatica, 47:2029-2038, 2011. [14] M. Krichman, E. D. Sontag and Y. Wang. Input-output-to-state stability. SIAM Journal on Control and Optimization, 39(6):1874-1928, 2001. [15] H. K. Khalil. Nonlinear Systems (3rd ed.). Prentice Hall, 2002.
is depicted in Fig. 4. As shown in Fig. 4, the comparison of the output y via the proposed control and y1 via control in [13] when γ0 = 0.15. The oscillations around can be reduced as desired only by reducing the distance γ0 between the manifolds. In Fig. 5, the time evolution of τ (t) and also the changes between the τ -manifolds every time x crosses x∗ = 3, when γ0 = 0.15, can be seen. From the simulation results, it shows that the obtained theoretic results can have better tracking performance. V. CONCLUSIONS A new output-feedback ESC based on sliding mode law in which tanh periodic switching function is used and norm state observation was utilized to observe the uncertain nonlinear system was proposed. Using the proposed method, finite-time convergence of the output cost function of the system to a small neighborhood of the extremum (minimum) 7118
Global Extremum Seeking Control with Sliding Modes for Output-Feedback Global Tracking of Nonlinear Systems* Chun Yin1,2 , Brandon Stark2 , Shou-ming Zhong1 and YangQuan Chen2 Abstract— In this paper, a novel design of an ExtremumSeeking Controller (ESC) based on Sliding Mode Control (SMC) for a class of Single-Input-Single-Output (SISO) uncertain or unknown nonlinear systems is proposed. The corrective term of the proposed observer is used to estimate the uncertain system. Global exact tracking using only output-feedback is proven with a tanh periodic switching function. The tanh periodic switching function is shown to provide a smaller vicinity of the desired extremum point compared to the traditional sgn periodic switching function. Numerical simulation results are presented to demonstrate the advantages and effectiveness of the proposed control strategy.
The rest of the paper is organized as follows. In Section II, the problem is formulated and the necessary notation is established. The design of the sliding mode ESC is presented in Section III. In Section IV, an example system is presented and the results of the numerical simulations are shown. Finally, concluding remarks are presented in Section V. II. P ROBLEM F ORMULATION Consider the nonlinear system described by
I. INTRODUCTION In many applications, the optimum plant efficiency occurs at a singular extremum point of a nonlinear system. The control objective is to maintain the extremum point in realtime without a priori knowledge of the system. Over the last several decades, many control strategies have been proposed, including perturbation and averaging ([1], [2], [3]) and more recently, sliding mode control ([4], [5], [6], [7], [8], [9], [10], [11], [12], [13]). In [10], the authors developed a ¨ uner’s pesliding mode ESC based on the Drakunov-Ozg¨ riodic switching function. In their proposed controller, at least one of the sliding surfaces would be a stable sliding surface, independent of control direction. In [13], the authors enhanced the sliding mode ESC approach by utilizing an ¨ uner periodic output-feedback version of the Drakunov-Ozg¨ switching function. The authors proved that global exact tracking was achievable for uncertain plants with unknown control direction. In this paper, the controller design found in [13] is further enhanced and is proven to reduce the oscillations around the optimal point. The oscillations of order O(γ0 ) around the optimal point y ∗ can be reduced by replacing the sgn periodic switching function with the tanh periodic switching function and the inclusion of a first norm observer. Previous works have shown that the tanh operator can be used to improve the performance, but a rigorous proof is presented in this paper. Numerical simulations are used to demonstrate the advantages and effectiveness of the proposed method. *This work was supported by National Basic Research Program of China (2010CB732501) and the National Natural Science Foundation of China (NSFC-60873102). 1 School of Mathematics Science, University of Electronic Science and Technology of China, Chengdu 611731, P. R. China ([email protected]) 2 Mechatronics, Embedded Systems and Automation (MESA) Lab, School of Engineering, University of California, Merced, 5200 North Lake Road, Merced, CA 95343, USA ([email protected], [email protected])
978-1-4673-2064-1/12/$31.00 ©2012 IEEE
x˙ = f (x, t) + g(x, t)u,
(1)
y = h(x),
(2)
where x = [x1 , x2 , · · · , xn ] ∈ Rn is the state vector, u ∈ R is the control input, y ∈ R is the measured output, f , g and h are unknown but continuous functions. In the state vector x, not all xi are assumed to be available for measurement. Without loss of generality, the following assumptions are made: Assumption 1: The nonlinear uncertain functions f , g and h are locally Lipschitz continuous in x, piecewise continuous and uniformly in t and sufficiently smooth. Assumption 1 is a necessary condition to prove the existence and uniqueness of a solution. For each solution, there exists a time interval [0, tM ) that the solution is valid, where tM may be finite or infinite. In order to solve the problem of the unavailable states, an observer is proposed. Proposition 1: To estimate the system (1), the following first order norm observer is proposed ˆ˙ = −˜ ˆ + ρ0 (x, t), x ωx
(3)
where ρ0 (x, t) is the input, ω ˜ is a positive constant and ρ0 (x, t) is a non-negative function continuous in x, piecewise ˆ is the output such that continuous and upperbounded in t. x ˆ (0) and x(0), for each initial state x ||x(t)|| ≤ ||ˆ x(t)|| + c0 ,
∀t ∈ [0, tM ),
(4)
where c0 := ω0 (||ˆ x(0)|| + ||x(0)||)e−¯ωt and ω0 , ω ¯ are positive constants. It is assumed that the system (1) is ISS [14] and thus can utilize such a norm observer and the plant is minimum-phase. Assumption 2: The system (1) admits a known norm observer (3). Assumption 3: There exists a positive constant g such that ¯ 0 < g ≤ ||g(x, t)||, ∀t ∈ [0, tM ). (5) ¯ From this assumption, the uncertain nonlinear function g(x, t) is bounded away from zero. The system (1) has
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relative degree one due to g(x, t) is not 0 for all x and t. Remark 1: In this paper, the goal is to solve the minimum seeking problem. Thus the global real-time optimization control problem, i.e., minimization of (2) under (1) can be stated as follows: (I) Optimal operating point: Assume that there exists a unique point x∗ such that y ∗ = h(x∗ ) is the minimum of the function y = h(x). (II) Uncertainty assumption: Suppose that x∗ , h(·) and its gradient are unknown to the control designer. (III) Global extremum seeking control based on sliding modes: The control problem consists in determining the control input u to steer the system to achieve to the minimum point y ∗ = h(x∗ ) and remain on such point as close as possible. The following assumption is used to derive the uncertainty bounds for control design. Assumption 4: All parameters of the uncertain system belong to a compact set Ω. The output of the system (1) has a unique extremum (minimum) point in the compact set Ω by using the notation h0 (x) = dh(x)/dx and h00 (x) = d2 h(x)/dx2 . The following assumption is necessary for the output function. Assumption 5: There exists a unique point x∗ ∈ R such that h0 (x∗ ) = 0 and h00 (x∗ ) > 0. For any given δ > 0, let Dδ = {x : ||x − x∗ || < δ/2} is called δ-vicinity of x∗ , there exists a positive constant ε such that |h0 (x)| > ε, ∀x ∈ / Dδ , in which δ can be made arbitrary small by allowing a smaller ε. According (1) and (2), the first time derivative of the output is derived as follows, y˙ = h0 f + kp u,
(6)
where kp is the coefficient of u which appears in the first time derivative of the output y and it is obtained as kp = h0 g. From Assumption 5, the following equality can be derived as (7) |kp | ≥ kp , ¯ where the lower bound kp ≤ gε is a positive constant. In order to derive a ¯norm ¯ bound for the term h0 f , the following assumption is given. ¯ η1 , and Assumption 6: There exist known functions h, ψ1 (x, t) such that ||f (x, t)|| ≤ η1 (||x||) + ψ1 (x, t) and ¯ |h0 | ≤ h(||x||), where η1 ∈ K∞ and is locally Lipschitz, and ψ1 (x, t) is a known non-negative function continuous in x, piecewise continuous and upperbounded in t. This assumption is not restrictive since h0 is assumed to be smooth and f is locally Lipschitz continuous in x. Furthermore, the dominating function η1 and ψ1 force strict growth condition only with respect to the time-dependence. Thus, polynomial nonlinearities in x can be dealt with.
tends asymptotically to zero (exact tracking), starting from any initial conditions and maintaining uniform closed loop signal boundedness and where yr is the desired trajectory. This control law is designed based on output-feedback version of tanh periodic switching function method. To ensure that the output y of the system is steered to reach the minimum point y ∗ , the output-feedback ESC with periodic switching function is proposed as π τ (t) , (9) u = φ(t)tanh sin γ0 where φ(t) is the designed modulation function and is continuous in t, where Z t τ (t) = γ1 e(t) + γ2 sgn(e(ς))dς, (10) 0
and γi > 0, (i = 0, 1, 2) are appropriate constants. For analysis convenience, the desired trajectory yr is generated by y˙ r (t) = −kr , yr (0) = yr0 ,
(11)
with kr is a positive constant and yr0 is a design constant. The model output can saturate at a rough known norm lower bound of y ∗ without effecting the control performance in order to avoid an unbounded reference signal yr (t). The modulation function φ(t) is derived such that y(t) tracks yr (t) as long as possible. This dictates that y(t) is forced to reach the vicinity of the minimum y ∗ = h(x∗ ) and remains close to y ∗ . Thus, the function φ(t) is designed so that a sliding mode τ˙ (t) = 0 occurs in finite time on one of the manifolds τ (t) = lγ0 , with some integer l. According to (10), the following equation can be obtained e(t) ˙ = −(γ2 /γ1 )sgn(e(t)).
(12)
Thus, e tends to converge to zero. The output y will track yr , while y remains outside of the small vicinity of y ∗ = h(x∗ ) where the high frequency gain (HFG) is not zero. In contrast, once y reaches the vicinity of the minimum value y ∗ = h(x∗ ), the controllability is lost since the HFG will tend to zero. Thus, y will not continue to track yr . However, the output y reach the vicinity of the optimum point y ∗ as expected. It can be shown that the convergence rate of x to the δ-vicinity Dδ is a function of γ1 , γ2 . Although Dδ is not positively invariant, the designed control law u will guarantee that x will remain close to x∗ . It does not imply that x remains in Dδ . But it will be shown that y remains close to the minimum point y ∗ in the proof of Theorem 1. A. Modulation function design
The goal of the control law is to design a control law u, via output feedback and without the knowledge of the system control direction, so that
In this subsection, the modulation function φ(t) will be designed. From Assumption 6, it can be inferred that ||f || ≤ η1 (||ˆ x|| + c0 ) + ψ1 (x, t). Utilizing that ψ(a + b) < ψ(2a)+ψ(2b), ∀a, b ≥ 0 and ∀ψ ∈ K∞ , η1 (||ˆ x||+c0 ) can be rewritten as η1 (||ˆ x||+c0 ) < η1 (2||ˆ x||)+η1 (2c0 ), (η1 ∈ K∞ ). Thus, the following inequality can be obtained
e(t) = y(t) − yr (t),
||f || ≤ η1 (2||ˆ x||) + η1 (2c0 ) + ψ1 (x, t).
III. D ESIGN EXTREMUM - SEEKING CONTROLLER WITH SLIDING MODE
(8) 7114
(13)
From the definition of c0 in the Proposition 1, c0 is uniformly bounded. Therefore, since η1 is assumed locally Lipschitz in Assumption 6, there exists a positive constant l1 such that η1 (2c0 ) ≤ 2l1 c0 = 2l1 ω0 (||ˆ x(0)|| + ||x(0)||)e−¯ωt . Hence, (13) can be rewritten as ||f || ≤ η1 (2||ˆ x||) + ψ1 (x, t) + 2l1 c0 .
(14)
Furthermore, the first term h0 f of the right-hand side in (6) satisfies the following condition 0
0
0 2
Fig. 1.
W1 is solid line when ks < 0.
Fig. 2.
W2 is solid line when ks > 0.
2
|h f | ≤ |h |(η1 (2||ˆ x||) + ψ1 (x, t)) + |h | + (2l1 c0 ) . (15) The time derivative of τ (t) is obtained from (8) as τ˙ (t) = γ1 kp u + ξs ,
(16)
0
where ξs := γ1 h f − γ1 kr + γ2 sgn(e). Let k = γ1 kp , such that τ˙ (t) = k(u + ω), in which ω = ξs /k. It is clear that the modulation function φ is designed such that φ ≥ |ω| + α, in which α is an arbitrary positive constant. According to (16), the norm bounds of the parameter ω is defined as |ω| ≤ ω ˆ + ς/|k|, (17) ¯ f˜ + γ1 h ¯ 2 + γ1 kr + γ2 )/k and ς = where ω ˆ ≤ (γ1 h −¯ (2l1 ω0 (||ˆ x(0)|| + ||x(0)||)e ωt )2 , in which f˜ =¯η1 (2||ˆ x||) + ψ1 (x, t), k = γ1 kp . One possible modulation function im¯ is proposed ¯ plementation such that φ| tanh(sin(πs/γ0 ))| ≥ |ω| + α holds in the following theorem, finite-time escape is avoided and realization of the s-sliding modes is guaranteed. Theorem 1: Consider the system (1)-(2), (8) with the control law (9) and that outside the δ-vicinity, φ(t) in (9) satisfies the following equality ¯ f˜ + γ1 h ¯ 2 + γ1 kr + γ2 ) φ tanh2 [sin[(π/γ0 )τ ] := [(γ1 h +||yt ||e−α1 t ]/k + α, (18) ¯ where α1 is an arbitrary positive constant and yt is an arbitrary vector. While x ∈ / Dδ , one has: (i) no finite-time escape occurs in the system signals (tM → +∞) and (ii) a sliding mode on sliding manifold s(t) = lγ0 is achieved in finite time for some integer l. Proof. Consider the following non-negative functions (Figs. (1-2)) Z τ π ς dς, W2 (τ ) = γm − W1 , W1 (τ ) = tanh sin γ 0 0 (19) where γm denotes the maximum value of W1 (τ ). Where W1 (τ (t)) and τ (t) are both differentiable, the time derivatives of W1 and W2 are given by π π 2 ˙ W1 = k φ tanh sin τ + ω tanh sin τ , γ0 γ0 ˙ 2 = −W ˙ 1, W (20) ˙ 1 = (∂W1 /∂τ )τ˙ was utilized. It where the relationship W can be seen that both inequalities are the same, ˙ 1 ≤ k{φ tanh2 [sin[(π/γ0 )τ ]} + |k||ω|, W ˙ 2 ≤ k{φ tanh2 [sin[(π/γ0 )τ ]} + |k||ω|, W (21)
since | tanh(sin[(π/γ0 )τ )| ≤ 1. According to (17), |k||ω| ≤ |k|ˆ ω + ς. They can be further broken down into, ˙ 1 ≤ −|k|{φ tanh2 [sin[(π/γ0 )τ ] − ω W ˆ } + ς, if sgn(k) < 0, 2 ˙ 2 ≤ −|k|{φ tanh [sin[(π/γ0 )τ ] − ω W ˆ } + ς, if sgn(k) > 0, (22) when inclu Consequently, by definition of k and k, the ¯ relationship −|k| ≤ −k can be obtained from (9). According ¯ to (18), it is easy to conclude that ˙ 1 ≤ −||yt ||e−α1 t − kα + ς, if sgn(k) < 0, (23) W ¯ ˙ 2 ≤ −||yt ||e−α1 t − kα + ς, if sgn(k) > 0, (24) W ¯ holds almost everywhere, with α ≥ 0, α1 > 0. The exponential term with rate α1 performs as a forgetting factor which allows a less conservative modulation function design. It is fundamental to avoid finite-time escape in the system signals. Property (i): Assume by contradiction that |s(t)| escapes in some finite time t1 ∈ [0, tM ). From (14), it can be seen that e(t) and y(t) also can escapes at t = t1 . Hence, there exists t2 ∈ [0, t1 ) such that ||yt || ≥ eα1 t [α2 − kα + ς], ¯ from where is an arbitrary non-negative constant. Moreover, inequalities (23) and (24), one has ˙ 1 ≤ −α2 or W ˙ 2 ≤ −α2 , ∀t ∈ [t2 , t1 ), W
(25)
independently of sgn(k). Since τ (t) is absolute continuous and escape in t = t1 , there exists te ∈ [t2 , t1 ) and an integer lτ such that τ (te ) = lτ γ0 . Therefore, W1 (te ) = 0 (if ls is an even number) or W2 (te ) = 0 (if ls is an odd number). It is clear that W1 (t) = 0, ∀t ∈ [te , t1 ) or W2 (t) = 0, ∀t ∈ [te , t1 ) from (25) in this interval (Figs. 1,2). With some abuse of notation, Wi (τ (t)) was replaced by Wi (t), for i = 1, 2. Consequently, τ (t) = lτ r0 is uniformly bounded ∀t ∈ [te , t1 ), i.e., a contradiction. Thus, the signals
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τ, e and y cannot escape in finite time. Moreover, from Assumption 3 and Assumption 5, one can further conclude ˆ and all closed loop signals cannot escape in finite that x, x time (tM → +∞). Property (ii): According to Property (i) and ς decreases exponentially, it is clear that there exists a finite time t¯ ≥ 0 ˙ i ≤ −αt¯, ∀t ≥ t¯ and 0 < αt¯ < kα, for such that W ¯ [15], i = 1 or i = 2. Using the Comparison Lemma ¯ ¯ ¯ Wi (t) ≤ −αt¯(t − t) + Wi (t), ∀t ≥ t. Consequently, there exists a finite time t∗ ≥ t¯ such that Wi (t) = 0, ∀t ≥ t∗ . Moreover, the corresponding points τ = lγ0 for which W1 (τ ) = 0(W2 (τ ) = 0) occur only for even (or odd) value of l (Fig. 1,2). In the neighborhood of the points τ = lγ0 , sgn(tanh(sin(πs/γ0 ))) = sgn(tanh(τ − lγ0 )) for l is an even number or sgn(tanh(sin(πτ /γ0 ))) = sgn(− tanh(τ − lγ0 )) for l is an odd number. Thus, there exists two positive functions p1 (t), p2 (t) (i.e. p1 (t), p2 (t) > 0, ∀t) such that tanh(sin((π/γ0 )τ )) = (tanh(τ − lγ0 ))p1 and tanh(sin((π/γ0 )τ )) = −(tanh(τ − lγ0 ))p2 . Now, the following inequality can be derived, either for l is an even number (sgn(k) < 0) or l is an odd number (sgn(k) > 0). Selecting a lyapunov candidate function, V = 0.5(τ − lγ0 )2 , and computing its derivative with respect to time, π τ + ω)) V˙ = (τ −lγ0 )(k(φ tanh sin γ0 (s−lγ0 )(k(φ tanh(τ −lγ0 )p1 +ω)), if sgn(k) < 0, = −(s−lγ0 )(k(φ tanh(τ −lγ0 )p2 +ω)), if sgn(k) > 0, ≤ 0. (26) Hence, a sliding mode occurs in finite time on one of the manifolds τ = lγ0 , independently of sgn(k). Remark 2: In the Theorem 2, the exponential term with rate acts like a forgetting factor which allows a less conservative modulation function design. Note that the functional norm term is fundamental to avoid finite-time escape of the system signals. B. Main result for extremum-seeking control based on sliding mode In this subsection, the output-feedback ESC with tanh periodic switching function (9) and (10) with the appropriate constants and a modulation function φ that satisfies (18) is shown to guarantee the state reaches to the δ-vicinity of the unknown minimizer x∗ defined in Assumption 2. Theorem 2: Consider the system (1), with output or cost function (2), control law (9) and (10), a modulation function φ that satisfies (18) and reference trajectory (11). If the Assumptions (1-5) hold, then there exists two properties: (i) δ-vicinity Dδ in Assumption 2 is globally attractive and can be reached in finite time and (ii) for ε sufficiently small, the oscillations around the minimum value y ∗ of y can be made of order O(γ0 ) with γ0 from (9). Moreover, all signals in the closed-loop system remain uniformly bounded except for τ (t) which is only an argument of a sgn function in (10). Proof: The derivative of the cost function y = h(x) does not vanish (h0 (x) 6= 0, ∀x ∈ Dδ ), while x is outside the δ-vicinity. Therefore, a lower norm bound kp for kp can be ¯
derived from the lower bound given in Assumption 5 which is valid globally. A lower norm bound k for k can be also obtained (k = γ1 kp , k = γ1 kp ). When x ¯stays outside the δ¯ escape ¯ occurs for the system signals vicinity, no finite-time and a sliding mode on τ = lγ0 can occur in finite time for some integer l from Theorem 1. Now, the proof of the properties (i) and (ii) of Theorem 2 will be given as follows: (i) Attractiveness of Dδ : It is assumed that x remains outside the δ-vicinity for all t ∈ [0, tM ) (i.e. x ∈ / Dδ , ∀t ∈ [0, tM )). Since the sliding mode on τ = lγ0 can occur in finite time, there exists a finite time tf such that τ˙ = 0. Therefore, from (12) that e˙ = −(γ2 /γ1 )sgn(e), ∀t ≥ tf . The error e = y − yr tends to zero. However, it is apparent that the cost function y = h(x) has a minimum value y ∗ and the reference signal yr (t) strictly decreases with time. So one has that yr < y ∗ ≤ y and sgn(e) = 1 when t is large enough. Then, it is assured that y decreases with the negative rate (y˙ = −kr −(γ2 /γ1 )sgn(e) < 0), that is, the cost function y = h(x) must to approach the minimum value y ∗ . Hence, x goes to δ-vicinity, which is a contradiction. Thus, the δ-vicinity is attained in some finite time. Consequently, x remains or oscillates around Dδ and similarly y with respect to some small vicinity of y ∗ , ∀t large enough. These oscillations appear due to the loss of control strength as k → 0 (kp → 0) and the recurrent changes in the HFG sign at the extremum point (x∗ , y ∗ ) where k = 0 (k = γ0 h0 f ). During these oscillations, τ can go from one sliding manifold τ = lγ0 to another manifold. In such cases, y could start oscillations around y ∗ with decreasing minimum amplitude. These oscillations can be made ultimately of order O(γ0 ) with γ0 from (9). (ii) Oscillations of order O(γ0 ) around y ∗ : According to the Assumption 5, δ can be made arbitrary small so that |y − y ∗ | = O(γ0 ) when x ∈ Dδ . Thus, if x remains in Dδ , ∀t, the corresponding neighborhood of y ∗ can be made of order O(γ0 ) with an appropriate ε. It is shown that, if x oscillates around Dδ , |y − y ∗ | = O(γ0 ) also holds since x is outside after leaving the set Dδ . There exists some finite time t˜ > 0 such that sgn(e) = 1, ∀t ≥ t˜, since the cost function y = h(x) has a minimum value y ∗ and yr (t) strictly decreases with time. Suppose that x reaches the frontier of δ-vicinity Dδ (from inside) at some time t˜1 > t˜ and τ (t) is not in a sliding mode when t = t˜1 . Note that Dδ is invariant when in a sliding mode. From (10), it is clear that Z t τ (t) = γ1 y(t) − γ1 yr (t) + γ2 sgn(e(τ ))dτ + C, (27) t˜
R t˜
where C = 0 sgn(e(τ ))dτ is a constant. Define τ˜(t) := τ (t) − τ (t˜1 ) and y˜(t) := y(t) − y(t˜1 ), τ˜(t) = γ1 y˜(t) + (γ1 ks + γ2 )(t − t˜1 ),
(28)
where ks = 0 if yr is saturated and ks = kr , otherwise. From the above equation, it is seen that
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|˜ τ (t)| ≤ γ1 |˜ y (t)| + (γ1 ks + γ2 )(t − t˜1 ), ∀t ≥ t˜1 ,
(29)
Define t˜2 as the first time when τ (t) reaches the next sliding manifold τ (t) = τ (t˜2 ) (independent of whether x is inside or outside Dδ ) and t˜3 as the first time when x reaches the frontier of Dδ again (from outside). It is seen that t˜2 ≥ t˜1 > t˜, t˜3 ≥ t˜1 > t˜. Then, there are two cases: (i) x reaches the frontier of Dδ with τ (t) in sliding motion (t˜3 > t˜2 ) and (ii) x reaches the frontier of Dδ with τ (t) not in sliding mode (t˜3 ≤ t˜2 ). For case (i), let t ∈ [t˜1 , t˜3 ] and consider t ∈ [t˜1 , t˜2 ) such that τ (t) is not in a sliding motion during this time interval. Hence, it is clear that there exists some integer l such that lγ0 < τ (t) < (l + 1)γ0 . Otherwise, sliding mode occurs at τ (t) = lγ0 or τ (t) = (l + 1)γ0 according to Theorem 1. Consequently, τ˜(t) = τ (t) − τ (t˜1 ) is of order O(γ0 ), ∀t ∈ [t˜1 , t˜2 ). Moreover, it is obvious that 0 < | tanh(sin(πτ (t)/γ0 ))| ≤ 1, ∀t ∈ [t˜1 , t˜2 ). Allowing t˜ to be large enough so that the exponential term ς in has decreased to an arbitrarily small value, the conditions φ ≥ |ω|| tanh(sin(πτ (t)/γ0 ))| + α are satisfied. Since lγ0 < τ (t) < (l + 1)γ0 for all t ∈ [t˜1 , t˜2 ), they guarantee an appropriate positive constant α so that |τ˙ (t)| ≥ k|u + ω| ≥ k(φ| tanh(sin(πτ (t)/γ0 ))| − |ω|) ≥ α ˜ in which α ˜¯ = kα is a ¯positive constant. Thus, (t − t˜ ) ≤ |˜ τ |/˜ α, ∀t ∈ [t˜1 , t˜¯2 ) and 1 thus (t − t˜1 ) is of order O(γ0 ), ∀t ∈ [t˜1 , t˜2 ). According to (24), y(t) − y(t˜1 ) is of order O(γ0 ), ∀t ∈ [t˜1 , t˜2 ). Moreover, by continuity, y(t) − y(t˜1 ) is also of order O(γ0 ), ∀t ∈ [t˜1 , t˜2 ]. Next, consider t ∈ [t˜2 , t˜3 ] where τ (t) is in sliding motion during this interval. Therefore, τ˙ (t) = 0. From (12), one has that y˙ = −kr − (γ2 /γ1 )sgn(e) < 0 and y(t) is strictly decreasing, ∀t ∈ [t˜2 , t˜3 ]. Hence, y(t) − y(t˜1 ) is also of order O(γ0 ), ∀t ∈ [t˜2 , t˜3 ], since y(t) approaches the minimum value y ∗ during that latter interval. Since this is also valid for the interval, it is proven that the oscillation outside Dδ is of order O(γ0 ) in case (i). For case (ii), let t ∈ [t˜1 , t˜2 ], where τ (t) is not in a sliding motion during this time interval. Therefore, following the first part of the proof of case (i), it is directly seen that y(t) − y(t˜1 ) is of order O(γ0 ), ∀t ∈ [t˜1 , t˜2 ]. From (20), lim |τ (t)| = +∞.
t→+∞
However, considering τ (t) is only a modified time scale for the argument of the sin function in the controller (9), this does not affect the uniformly norm boundedness (UB) of other signals. The boundedness of y stated above implies that the signal x is UB, according to the continuity of the uncertain output function y = h(x). Likewise, the other closed-loop signals are UB from Proposition 1. Corollary 1 : Consider the output function y = h(x) that has a unique unknown minimum value y ∗ = h(x∗ ), such that x∗ is in the interior of some closed interval [a, b]. If the assumptions in Theorem 2 hold, then only semi-global practical convergence of the closed-loop system is obtained in the sense that the domain of attraction can be arbitrarily increased by reducing ε.
Fig. 3. (a) Time responses of the output plant y (green line) and the output model yr (blue line) via our controller. The output plant tends to the minimum value y ∗ = −2; (b) Time responses of τ showing the oscillations of x around x∗ = 3 which minimizes y considering γ0 = 0.05.
IV. N UMERICAL SIMULATIONS The following example is used to show the applicability of the proposed ESC controller based on sliding mode law. For simplicity, the proposed control scheme is utilized to derive a robust one-dimensional nonderivative optimizer based on periodic switching functional and sliding modes. Example: Consider a function y = h(x) = −12x/(9+x2 ) to be minimized in the interval of interest [a, b] = [0, 10]. It has a minimum y ∗ = h(x∗ ) = −2 at x = x∗ = 3. Since h0 (x) tends to zero as x → ∞, Assumption 5 is not globally satisfied. Hence, only local convergence to the optimal value might be expected. Yet, the convergence was verified in all simulations starting with x = 0, which is realistic situation in an ABS braking scenario. In following section, the design of the proposed nonderivative optimizer is processed. Consider the following auxiliary order nonlinear system x˙ = u,
y = h(x) = −12x/(9 + x2 ),
(30)
where y˙ = kp u, kp = h0 (x) can be regarded as the HFGs, x is the integrator output and the initial condition is (x0 , y0 ) = (1, −1.3). In this example, x is not used for feedback even though it is available; only y is utilized. The tracking signal yr is selected as in (11) and, in the example, yr was saturated at -150. The control law (9) and (10) is used with constant modulation function φ(t) = (γ1 + γ2 + kr )/ε + α. Let the parameters γ0 = 0.05, γ1 = 0.5, γ2 = 0.1, kr = 0.5 and α = 0.1 in the simulations. The lower bounded ε = 0.1γ0 in and initialize γ0 with not so small value. Then, γ0 is decreased until the variance of y is ultimately small, that is, |y − y ∗ | → O(γ0 ). In Fig. 3, y does not track yr till y reaches the vicinity of the minimizer x∗ = 3. The exact tracking is not obtained but y is locked at some γ0 -neighborhood of y ∗ = −2 and yr will decrease until the saturation value -150, and time responses of τ showing the oscillations of x around x∗ = 3 which minimizes y considering γ0 = 0.05. The comparison of the output y under the proposed control law and the output y1 under the normal sliding mode based extremum seeking control with sgn periodic switching function in [13]
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point can be guaranteed. It is possible to have better tracking performance by utilizing the proposed control law. Numerical simulation results were presented to show the effectiveness of the proposed method. R EFERENCES
Fig. 4. Times responses of the output y under our control and the output y1 under the control with sgn periodic switching function.
Fig. 5. (a) Times responses of the output y under the proposed control and the output y1 under the control with sgn periodic switching function, when γ0 = 0.15; (b) Time responses of τ showing the oscillations of x around x∗ = 3 which minimizes y considering γ0 = 0.15.
[1] M. Krsti´ c and H.-H. Wang. Stability analysis of extremum seeking feedback for general nonlinear systems. Automatica, 36(2):595-601, 2000. [2] C. Zhang and R. Ord´ on ˜ez. Robust and adaptive design of numerical optimization-based extremum seeking control. Automatica, 45,634646 (2009). [3] M. Guay and T. Zhang. Adaptive extremum seeking control of nonlinear dynamic systems with parametric uncertainties. Automatica, 39(7):1283-1293, 2003. [4] S. K. Korovin and V. I. Utkin. Using sliding modes in static optimization and nonlinear programming. Automatica, 10:525-532, 1974. [5] A. B. Will, S. Hui and S. H. Zak. Sliding mode wheel slip controller for an antilock braking system. International Journal of Vehicle Design, 19(4):523-539, 1998. [6] T. R. Oliveira, A. J. Peixoto, E. V. L. Nunes, and L. Hsu. Control of uncertain nonlinear systems with arbitrary relative degree and unknown control direction using sliding modes. International Journal of Adaptive Control and Signal Processing, 21:692-707, 2007. [7] A. Levant. Higher-order sliding modes, differentiation and outputfeedback control. International Journal of Control, 76(9):924-941, 2003. [8] T. R. Oliveira, A. J. Peixoto, E. V. L. Nunes and L. Hsu. Control of uncertain nonlinear systems with arbitrary relative degree and unknown control direction using sliding modes. International Journal of Adaptive Control and Signal Processing, 21:692-707, 2007. [9] C. Yin, S. Zhong and W. Chen. Design of sliding mode controller for a class of fractional-order chaotic systems. Commun. Nonlinear Sci Numer. Simulat., 17:356-366, 2010. ¨ Ozg¨ ¨ uner. Optimization of Nonlinear System [10] S. Drakunov and U. Output via Sliding Mode Approach. In Proceedings of the IEEE International Workshop on Variable Structure and Lyapunov Control of Uncertain Dynamical Systems, pp. 61-62, 1992. ¨ Ozg¨ ¨ uner, P. Dix, and B. Ashrafi. ABS control using [11] S. Drakunov, U. optimum search via sliding modes. IEEE Transactions on Control Systems Technology, 3(1):79-85, 1995. ¨ Ozg¨ ¨ uner. Stability and performance improvement of [12] Y. Pan and U. extremum seeking control with sliding mode. International Journal of Control, 76:968-85 2003. [13] T. R. Oliveira, L. Hsub and A. J. Peixoto. Output-feedback global tracking for unknown control direction plants with application to extremum-seeking control. Automatica, 47:2029-2038, 2011. [14] M. Krichman, E. D. Sontag and Y. Wang. Input-output-to-state stability. SIAM Journal on Control and Optimization, 39(6):1874-1928, 2001. [15] H. K. Khalil. Nonlinear Systems (3rd ed.). Prentice Hall, 2002.
is depicted in Fig. 4. As shown in Fig. 4, the comparison of the output y via the proposed control and y1 via control in [13] when γ0 = 0.15. The oscillations around can be reduced as desired only by reducing the distance γ0 between the manifolds. In Fig. 5, the time evolution of τ (t) and also the changes between the τ -manifolds every time x crosses x∗ = 3, when γ0 = 0.15, can be seen. From the simulation results, it shows that the obtained theoretic results can have better tracking performance. V. CONCLUSIONS A new output-feedback ESC based on sliding mode law in which tanh periodic switching function is used and norm state observation was utilized to observe the uncertain nonlinear system was proposed. Using the proposed method, finite-time convergence of the output cost function of the system to a small neighborhood of the extremum (minimum) 7118
