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Adaptive Dynamic Output Feedback Neural Network Control of Uncertain MIMO Nonlinear Systems with Prescribed Performance Artemis K. Kostarigka and George A. Rovithakis, Senior Member, IEEE
Abstract— An adaptive dynamic output feedback neural network controller for a class of multi-input/multi-output affine in the control uncertain nonlinear systems is designed, capable of guaranteeing prescribed performance bounds on the system’s output as well as boundedness of all other closed loop signals. It is proved that simply guaranteeing a boundedness property for the states of a specifically defined augmented closed loop system is necessary and sufficient to solve the problem under consideration. The proposed dynamic controller is of switching type. However, its continuity is guaranteed, thus alleviating any issues related to the existence and uniqueness of solutions. Simulations on a planar two-link articulated manipulator illustrate the approach. Index Terms— Neural network adaptive control, output feedback, prescribed performance.
I. I NTRODUCTION
G
UARANTEEING performance quality on a system’s output response is an important and pretentious task in control engineering. Owing to the serious technical issues raised whenever performance specifications are incorporated in closed-loop design, control solutions are hard to meet, even when the system structure is considered known. Traditionally, quality of performance is approached through minimizing certain functionals [1]–[3]. Unfortunately, however, no obvious and a priori specified connection exists between the aforementioned performance functionals and trajectory oriented metrics such as maximum overshoot, convergence rate, and steady-state error. In an attempt to establish such a direct link, the prescribed performance control (PPC) paradigm was recently proposed [4]–[6], to guarantee, for various classes of uncertain nonlinear systems, convergence of the output tracking error to a predefined arbitrarily small residual set, with convergence rate no less than a prespecified value, while exhibiting maximum overshoot less than some sufficiently small preassigned constant. In the same spirit, application of PPC to adaptive force/position control of robotic manipulators was provided in [7]. The above-mentioned PPC results were derived under a full state measurement requirement. In real-world applications though, such an assumption is rarely satisfied. The necessity of Manuscript received February 1, 2011; revised October 6, 2011; accepted October 7, 2011. Date of publication December 15, 2011; date of current version January 5, 2012. The authors are with the Department of Electrical and Computer Engineering, Aristotle University of Thessaloniki, Thessaloniki 54124, Greece (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TNNLS.2011.2178448
developing PPC designs for general classes of uncertain nonlinear systems incorporating output feedback only led to [8], in which an approximate passivation approach was realized, imposing, however, restrictive structural constraints/properties. Even without prescribed performance, designing output feedback controllers for uncertain nonlinear systems is a highly sophisticated problem. To overcome the full state measurement obstacle, constructive control solutions rely mostly on observers for state estimation. Unfortunately, the use of observers introduces further restrictions on the relative degree of the output [9], as well as on the form of the allowed plant uncertainties. Representative works in this direction include single-input/single-output feedback linearizable systems [10], [11], special classes of strict feedback systems [12], non-affine in the control nonlinear systems in normal form [13], [14] and in pure feedback form [15], as well as systems in the output feedback form [16], [17] and stochastic systems [18]. Needless to mention that constructing observers for more general nonlinear and uncertain system structures is a challenging task and their incorporation in closed-loop system analysis is still considered a major open control problem. In this paper, an adaptive dynamic output feedback controller is proposed, which, without resorting to an observer construction, is capable of achieving the prescribed performance for the output and uniform boundedness of all other signals in the closed loop, for a class of multi-input/multi-output (MIMO) affine in the control uncertain nonlinear systems. To overcome the amount of uncertainties present, linearly parameterized neural networks are used as approximation models. It is proved that guaranteeing a boundedness property for the states of a specifically defined augmented closed loop system is necessary and sufficient to solve the problem under consideration. Even though the developed controller is of switching type, its continuity is guaranteed, thus alleviating any issues related to the existence and uniqueness of solutions. With respect to [8], the structural constraints/properties required are significantly relaxed owing to the control solution proposed. This paper is organized as follows. Necessary definitions and preliminary results are summarized in Section II. The prescribed performance output feedback control (PPOFC) problem is stated in Section III. Section IV is focused on the design of a dynamic output feedback neural network controller capable of guaranteeing the prescribed performance of the output tracking errors, as well as the boundedness of all other
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closed loop signals. Simulation results on a planar two-link articulated manipulator are provided in Section V. Finally, we conclude in Section VI.
ei(0) ≥ 0
ρi∞ ei(t) 0
II. D EFINITIONS AND P RELIMINARIES A function V : Rn → R+ is said to be positive definite if V (x) is positive for all nonzero x and is zero at x = 0. A function V : Rn → R+ is said to be proper or radially unbounded if V (x) tends to +∞ as |x| → +∞. A function f (·) is said to be of class C k if the derivatives f , f , . . . , f (k) 0 exist and are continuous. In particular, the class C consists of all continuous functions. A continuous function γ : [0, α) → R+ is said to belong to class K if it is strictly increasing and γ (0) = 0. It is said to belong to class K∞ if α = ∞ and γ (r ) → ∞ as r → ∞. If 1 , 2 are two subsets of Rn and 2 ⊂ 1 , 1 /2 denotes the set of all x ∈ 1 satisfying x∈ / 2 . Consider the system x˙ = f 0 (x, u), x ∈ Rn , u ∈ Rq y = h 0 (x), y ∈ Rm
(1)
whose solution x(t; t0 , x 0 ) is defined on [t0 , ∞) for each initial state x 0 , input u, and t0 ∈ R+ . Definition 1 [19]: The solutions of (1) are uniformly ultimately bounded (u.u.b) (with bound B) if there exists a B > 0 and if corresponding to any a > 0 and t0 ∈ R+ , there exists a T = T (a) > 0 (independent of t0 ) such that |x 0 | < a implies |x(t; t0 , x 0 )| < B for all t ≥ t0 + T. Theorem 1 [19]: Assume that (1) possesses unique solutions for all x 0 ∈ Rn and t0 ∈ R+ . If there exists a function V (t, x) defined on |x| ≥ R (where R may be large) and t ∈ [t0 , ∞) with continuous first-order partial derivatives with respect to x, t and if there exist functions ϕ1 , ϕ2 ∈ K∞ and ϕ3 ∈ K defined on [t0 , ∞), such that: 1) ϕ1 (|x|) ≤ V (t, x) ≤ ϕ2 (|x|); 2) V˙ (t, x) ≤ −ϕ3 (|x|) for all |x| ≥ R and t ∈ [t0 , ∞); then the solutions of (1) are u.u.b. A. Linear in the Weights Neural Networks In this paper, we shall be using linear-in-the-weights neural networks of the form z = W Z (v 0 ), where v 0 ∈ Rn2 and z ∈ Rn1 are the neural net input and output, respectively, W is an L−dimensional vector of synaptic weights, and Z (v 0 ) is an L × n 1 matrix of regressor terms. The regressor terms may contain high-order connections of sigmoid functions [20], radial basis functions (RBFs) with fixed centers and widths [21], and shifted sigmoids [22], thus forming high-order neural networks (HONNs), RBFs, and shifted sigmoidal neural networks, respectively. Another class of linear-in-the-weights neural nets is the cerebellar model articulation controller network, which mainly uses B-splines in Z (v 0 ) [23]. An important and well-known property shared among the aforementioned neural approximating structures is the following (see also the references above). Density property: For each continuous function F: Rn2 → n R 1 , and for every ε0 ≥ 0, there exist an integer L and an
0 0 −ρi∞
ei(t)
ρi(t) −Miρi(t) Miρi(t) −ρi(t)
ei(0) ≤ 0 0 time Fig. 1.
Graphical illustration of the prescribed performance definition (2).
optimal synaptic weight vector W ∗ ∈ R L such that sup F (v 0 ) − W ∗ Z (v 0 ) ≤ ε0 v 0 ∈
where ⊂ Rn2 is a given compact set. In other words, if the number of the regressor terms L is sufficiently large, then there exist weight values W ∗ such that W ∗ Z (v 0 ) can approximate F (v 0 ) to any degree of accuracy, in a given compact set. This property allows us to focus on linear-in-the-weights neural networks without loss of generality in terms of approximation error. This, in turn, will make it easier to prove basic systems properties such as stability and robustness. B. Prescribed Performance Preliminaries For completeness and compactness of presentation, this subsection summarizes preliminary knowledge on prescribed performance originally stated in [4]. In that respect, consider a generic tracking error e (t) = [e1 (t) . . . em (t)] ∈ Rm . Prescribed performance is achieved if each element ei (t) , i = 1, . . . , m evolves strictly within a predefined region that is bounded by a decaying function of time. The mathematical expression of prescribed performance is given, ∀t ≥ 0, by the following inequalities: −Mi ρi (t) < ei (t) < ρi (t), ei (0) ≥ 0 i = 1, . . . , m −ρi (t) < ei (t) < Mi ρi (t), ei (0) ≤ 0 (2) where 0 ≤ Mi ≤ 1, i = 1, . . . , m and ρi (t), i = 1, . . . , m are bounded, smooth, strictly positive, and decreasing functions satisfying limt →∞ ρi (t) = ρi∞ > 0, i = 1, . . . , m, called performance functions [4]. As (2) implies, only one set of the performance bounds is employed, and specifically the one associated with the sign of ei (0). The aforementioned statements are clearly illustrated in Fig. 1, for an exponential performance function ρi (t) = (ρi0 − ρi∞ ) exp (−li t) + ρi∞ , i = 1, . . . , m, with ρi0 , ρi∞ , li , i = 1, . . . , m strictly positive constants. The constant ρi0 = ρi (0), i = 1, . . . , m is selected such that (2) is satisfied at t = 0 (i.e., ρi (0) > ei (0)
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in case ei (0) ≥ 0 or ρi (0) > −ei (0) in case ei (0) ≤ 0). The constant ρi∞ = limt →∞ ρi (t) , i = 1, . . . , m represents the maximum allowable size of ei (t) at the steady state and can be set arbitrarily small to a value reflecting the resolution of the measurement device, thus achieving practical convergence of ei (t) to zero. Furthermore, the decreasing rate of ρi (t) , i = 1, . . . , m, which is related to the constant li , i = 1, . . . , m in this case, introduces a lower bound on the required speed of convergence of ei (t). Moreover, the maximum overshoot is prescribed less than Mi ρi (0) , i = 1, . . . , m, which may even become zero by setting Mi = 0, i = 1, . . . , m. Thus, the appropriate selection of the performance function ρi (t) , i = 1, . . . , m, as well as of the constant Mi , i = 1, . . . , m, imposes performance characteristics for the tracking error ei (t) , i = 1, . . . , m. To introduce prescribed performance, an error transformation is incorporated modulating the tracking error element ei (t) , i = 1, . . . , m with respect to the required performance bounds imposed by ρi (t), Mi , i = 1, . . . , m. More specifically, we define εi (t) = Ti
ei (t) , ρi (t)
i = 1, . . . , m
(3)
where εi (t) , i = 1, . . . , m is the transformed error and Ti (·) , i = 1, . . . , m is a smooth strictly increasing function defining an onto mapping Ti : (−Mi , 1) → (−∞, ∞), Ti : (−1, Mi ) → (−∞, ∞),
ei (0) ≥ 0 ei (0) ≤ 0
i = 1, . . . , m.
(4) For example, in case ei (0) ≥ 0, i = 1, . . . , m, a candidate transformation function could be Ti (ei (t)/ρi (t)) = ln((Mi + ei (t)/ρi (t))/(1 − ei (t)/ρi (t))), i = 1, . . . , m. Otherwise, in case ei (0) ≤ 0, we could select Ti (ei (t)/ρi (t)) = ln((1 + ei (t)/ρi (t))/(Mi − ei (t)/ρi (t))), i = 1, . . . , m. A graphical illustration of the transformation function for both cases is provided in Fig. 2. Moreover, as (4) implies and the aforementioned example clarifies, the choice of the mapping Ti (·) , i = 1, . . . , m, depends only on the sign of ei (0) , i = 1, . . . , m. Notice also that, since ρi (0) , i = 1, . . . , m is selected such that (2) is satisfied at t = 0, the initial condition of the transformed error εi (0) , i = 1, . . . , m is finite owing to (4). Furthermore, the case ei (0) = 0 clearly satisfies (2) at t = 0 with the choice Mi = 0, i = 1, . . . , m, since otherwise (i.e., Mi = 0) εi (0) , i = 1, . . . , m becomes infinite. Thus, in such a case the transformation function can be selected to satisfy either Ti : (−Mi , 1) → (−∞, ∞) , i = 1, . . . , m, or Ti : (−1, Mi ) → (−∞, ∞) , i = 1, . . . , m. Remark 1: Notice that the magnitude of εi (t) , i = 1, . . . , m does not affect the evolution of ei (t) , i = 1, . . . , m, which is solely prescribed by (2) and thus by the selection of the performance function ρi (t) , i = 1, . . . , m, as well as of the constant Mi , i = 1, . . . , m. In other words, if εi (t), i = 1, . . . , m is guaranteed bounded for all t ≥ 0, then prescribed performance for ei (t), i = 1, . . . , m is achieved ∀t ≥ 0.
ei(0) ≥ 0
εi
εi
0
0
−Mi Fig. 2.
ei(0) ≤ 0
0
ei/ρi
1
−1
0 Mi
ei/ρi
Graphical illustration of a transformation function.
III. P ROBLEM S TATEMENT In this paper, we shall consider uncertain MIMO systems of the form x˙ = f (x) + g(x)u () y = h(x) where x ∈ Rn is the state, y ∈ Rm is the measured output, u(t) ∈ Rq is the control input, h(x) ∈ Rm is an unknown continuous, and C 1 function, while f (x) ∈ Rn , g(x) ∈ Rn×q are locally Lipschitz unknown functions. The state x is assumed to be unavailable for measurement. Our goal is to design a controller utilizing only measurable signals, capable of regulating the output tracking error e(t) = y(t) − yr (t) to a neighborhood of zero with prescribed performance while guaranteeing the uniform boundedness of all other signals in the closed loop. For the desired output trajectory yr (t), we assume that it is bounded with bounded first-order time derivatives. More formally, our problem is defined as follows. Definition 2: The PPOFC problem for () is solvable if there exists a bounded and possibly parameterized dynamic output feedback continuous control law to render the states of () u.u.b with respect to some compact sets and the output tracking error elements ei (t), i = 1, . . . , m to satisfy λli ρi (t) ≤ ei (t) ≤ λui ρi (t), i = 1, . . . , m
∀t ≥ 0
(5)
for some constants λli , λui obeying −Mi < λli < λui < 1, in case ei (0) ≥ 0 −1 < λli < λui < Mi , in case ei (0) ≤ 0
(6) (7)
for all i = 1, . . . , m. To continue, we differentiate (3) with respect to time to obtain ρ˙i ε˙ i = ri y˙i − y˙ri − ei , i = 1, . . . , m ρi
KOSTARIGKA AND ROVITHAKIS: NEURAL NETWORK CONTROL OF UNCERTAIN MIMO NONLINEAR SYSTEMS
or in matrix form ∂h ε˙ = r x˙ − ν = r L f h(x) + L g h(x)u − ν ∂x
(8)
where ε = [ ε1 . . . εm ] , r = diag([ r1 . . . rm ]), ν = [ ν1 . . . νm ] , with ri = (1/ρi )∂ Ti /∂(ei /ρi ) and νi = y˙ri + ei ρ˙i /ρi , i = 1, . . . , m and consider the augmented system x˙ = f (x) + g(x)u . (a ) ε˙ = r L f h(x) + L g h(x)u − ν To pursue our problem, we shall consider continuous dynamic output feedback controllers of the form u = u(ξ, ε) ξ˙ = φ(ξ, ε),
ξ ∈R . n
(9) (10)
Incorporating the dynamic controller (9), (10) to (a ), we obtain the following augmented closed-loop system: ⎫ x˙ = f (x) + g(x)u(ξ, ε) ⎬ ξ˙ = φ(ξ, (ac ) ε) ⎭ ε˙ = r L f h(x) + L g h(x)u(ξ, ε) − ν which is defined over the set U = {x ∈ Rn , ξ ∈ Rn , ε ∈ Rm : h(x) − yr (t) ∈ Ee,t ∪ ∂Ee,t }, where Ee,t is the set defined for each t ≥ 0 via (2) and ∂Ee,t represents its boundary. Remark 2: The augmented closed loop system (ac ) is said to be initially well defined if (x(0), ξ(0), ε(0)) ∈ U . To guarantee such a property, we have to ensure that (2) is satisfied at t = 0. In this direction, notice that, for any given initial condition ξ(0) and any (unknown) initial state x(0), the corresponding initial output measurement y(0) leads to a measurable initial output tracking error value e(0) = y(0) − yr (0). However, given e(0) = [e1 (0) . . . em (0)] , we can always select ρi (0) and Mi , i = 1, . . . , m to satisfy (2) at t = 0, which, owing to (4), further leads to ε(0) = [ε1 (0) . . . εm (0)] ∈ L∞ , thus establishing an initially well defined (ac ). The following theorem introduces necessary and sufficient conditions for the solution of the PPOFC problem. Theorem 1: Consider the original uncertain system () along with the dynamic output feedback controller (9), (10) and the corresponding initially well-defined augmented closedloop system (ac ). The PPOFC problem for () admits a solution if and only if all states of (ac ) are u.u.b with respect to some arbitrary sets. Proof: (⇒) If the PPOFC problem for () admits a solution, then there exists an admissible continuous dynamic output feedback control input u of the form (9), (10) rendering the state (x, ξ ) of closed loop () u.u.b with respect to some arbitrary compact sets Ux , Uξ , respectively, and (5) holds true ∀t ≥ 0, for some unknown constants λli , λui , i = 1, . . . , m obeying (6) and (7). Employing (3)–(7) elementwise, we arrive at −∞ < Ti (λli ) ≤ εi (t) ≤ Ti (λui ) < ∞, i = 1, . . . , m ∀t ≥ 0. ]
(11)
Hence ε(t) = [ε1 . . . εm is u.u.b with respect to the set Uε ≡ {ε ∈ Rm : Ti (λli ) ≤ εi (t) ≤ Ti (λui ), i = 1, . . . , m}, thereby proving the u.u.b of all states of (ac ).
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(⇐) As noticed earlier in Remark 2, we can always select ρi (0), λli , λui , i = 1, . . . , m such that (ac ) is initially well defined (i.e., (x(0), ξ(0), ε(0)) ∈ U ). Furthermore, there exists an admissible continuous dynamic output feedback control law to guarantee that x(t), ξ(t), ε(t) are u.u.b. with respect to the compact sets Ux , Uξ , Uε , respectively. From the u.u.b of ε(t) and owing to (4), there exist some unknown constants εi , εi , i = 1, . . . , m such that εi ≤ εi (t) ≤ εi , i = 1, . . . , m for all t ≥ 0. Utilizing the inverse transformation Ti−1 (·), i = 1, . . . , m which is well defined as ε ∈ L∞ , we arrive at (5) with λli = Ti−1 (εi ), λui = Ti−1 (εi ), i = 1, . . . , m. Therefore, as the states x, ξ of the closed loop system () are u.u.b. and (5) holds, through a bounded and continuous dynamic output feedback controller, the PPOFC problem for () admits a solution.
IV. M AIN R ESULTS Let us assume that the PPOFC problem admits a solution for the original uncertain system (). Owing to Theorem 1, this means that there exists a continuous dynamic output feedback controller u 0 (ξ, ε), ξ˙ = φ0 (ξ, ε), such that the states x, ξ, ε of the augmented closed loop system (ac ) are u.u.b with respect to some arbitrary sets Ux , Uξ , Uε , respectively. Therefore, there exists a Lyapunov function V (x, ξ, ε) to satisfy Theorem II. To proceed, we consider (with no loss of generality) the decomposition V (x, ξ, ε) = Va (x) + Vb (ξ, ε) and obtain for the Lyapunov function derivative the relation V˙ (x, ξ, ε) = L f Va (x) + L g Va (x)u 0 (ξ, ε) ∂ Vb (ξ, ε) ∂ Vb (ξ, ε) + φ0 (ξ, ε) + r ∂ξ ∂ε × L f h(x) + L g h(x)u 0 (ξ, ε) − ν ≤ −ηx (|x|) − ηξ (|ξ |) − ηε (|ε|) ∀ (x, ξ, ε) ∈ U /UX
(12)
where UX ≡ Ux × Uξ × Uε and ηx (·), ηξ (·), ηε (·) are some K∞ −functions. Notice that UX ⊂ U . This is true because of the fact that, whenever (x, ξ, ε) ∈ UX , we have ε ∈ L∞ . Therefore invoking (4) we conclude that h(x) − yr (t) ∈ Ee,t ∀(x, ξ, ε) ∈ UX , which means that UX is a subset of U . Moreover, differentiating V (x, ξ, ε) along the solutions of (ac ) yields ∂ Vb (ξ, ε) ˙ ∂ Vb (ξ, ε) ∂ Va (x) x˙ + ε˙ V˙ (x, ξ, ε) = ξ+ ∂x ∂ξ ∂ε ∂ Vb (ξ, ε) = L f Va (x) + L g Va (x)u + φ(ξ, ε) ∂ξ ∂ Vb (ξ, ε) r L f h(x) + L g h(x)u − ν . + ∂ε Adding and subtracting
∂ Vb (ξ, ε) ∂ Vb (ξ, ε) r L g h(x) u 0 + φ0 (ξ, ε) L g Va (x) + ∂ε ∂ξ
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we obtain
A. Dynamic Controller Design
V˙ (x, ξ, ε) = L f Va (x) + L g Va (x)u 0 + +
∂ Vb (ξ, ε)
∂ Vb (ξ, ε)
∂ξ
φ0 (ξ, ε)
∂ε
G(x, ξ, ε) = L g Va (x) +
∂ Vb (ξ, ε) ∂ε
r L g h(x)
∂ Vb (ξ, ε) φ0 (ξ, ε) ∂ξ ∂ Vb (ξ, ε) + r L f h(x) + L g h(x)u 0 − ν . ∂ε
I(x, ξ, ε, r, ν) = L f Va (x)+ L g Va (x)u 0 +
However, owing to (12) I(x, ξ, ε, r, ν) ≤ −
u = Wˆ u Z u (ξ, ε) ξ˙ = φˆ o (ξ, ε, Wˆ o ) + φσ
r L f h(x) + L g h(x)u 0 − ν
∂ Vb (ξ, ε) r L g h(x) (u − u 0 ) + L g Va (x) + ∂ε ∂ Vb (ξ, ε) + (φ(ξ, ε) − φ0 (ξ, ε)) ∂ξ = I(x, ξ, ε, r, ν) + G(x, ξ, ε)(u − u 0 ) ∂ Vb (ξ, ε) + (φ(ξ, ε) − φ0 (ξ, ε)) ∂ξ where
The control input is designed as follows:
ηi (| i |) ∀ (x, ξ, ε) ∈ UX .
i=x,ξ,ε
Therefore V˙ (x, ξ, ε) becomes ηi (| i |) + G(x, ξ, ε)(u − u 0 ) V˙ (x, ξ, ε) ≤ − i=x,ξ,ε
∂ Vb (ξ, ε) + (φ(ξ, ε) − φ0 (ξ, ε)) , ∀ (x, ξ, ε) ∈ U /UX . ∂ξ (13) Because they are unknown, the terms u 0 (ξ, ε), ∂ Vb (ξ, ε)/∂ξ , and φ0 (ξ, ε) shall be approximated by suitable neural network structures. In this direction, by applying the neural network density property we conclude the existence of constant but unknown weight vectors Wu∗ ∈ R L u ×q , Wν∗ ∈ R L ν , Wo∗ ∈ R L o ×n , continuous regressor terms Z u (ξ, ε) ∈ R L u , Z ν (ξ, ε) ∈ R L ν ×n , Z o (ξ, ε) ∈ R L o and ωu (ξ, ε), ων (ξ, ε), ωo (ξ, ε), denoting the approximation errors such that u 0 (ξ, ε) = Wu∗ Z u (ξ, ε) + ωu (ξ, ε) ∈ Rq ∂ Vb (ξ, ε) (ξ, ε) = Wν∗ Z ν (ξ, ε) + ων (ξ, ε) ∈ R1×n ∂ξ φ0 (ξ, ε) = Wo∗ Z o (ξ, ε) + ωo (ξ, ε) ∈ Rn (14) ∀ (ξ, ε) ∈ ac ≡ ξ × ε ⊂ Rn × Rm where ξ , ε are arbitrary compact sets. Such a substitution is possible because of the continuity and the boundedness of u 0 (ξ, ε), ∂ Vb (ξ, ε)/∂ξ and φ0 (ξ, ε) ∀ (ξ, ε) ∈ ac . Moreover, on the generic compact set ac , the approximation errors can be suitably bounded as |ωu (ξ, ε)| ≤ ω¯ u , |ων (ξ, ε)| ≤ ω¯ ν , |ωo (ξ, ε)| ≤ ω¯ o , where ω¯ u , ω¯ ν , and ω¯ o are some unknown bounds.
∈ Rq
(15)
∈R
(16)
n
with φσ switching according to a switching strategy σ (t), taking values in {I, II} with a(t) + γ (|ε|, |ξ |) + δ u (17) φI = −b ξ, ε, Wˆ ν 2 ˆ |b(ξ, ε, Wν )| a(t) + γ (|ε|, |ξ |) + δ u (18) φII = −b ξ, ε, Wˆ ν δb2 and φˆo (ξ, ε, Wˆ o ) = Wˆ o Z o (ξ, ε) ∈ Rn b(ξ, ε, Wˆ ν ) = Wˆ ν Z ν (ξ, ε) ∈ R1×n .
(19) (20)
In (17)–(20), γ (|ε|, |ξ |) = γξ (|ξ |) + γε (|ε|), where γξ (·), γε (·) are K−functions, δu , δb > 0 are design constants, while Wˆ u , Wˆ o , Wˆ ν are estimates of Wu∗ , Wo∗ , Wν∗ , defined in (14), which are generated through the update laws W˙ˆ u = −Wˆ u + Z u (ξ, ε) · K u ˙ Wˆ o = −Wˆ o − Z o (ξ, ε) · b(ξ, ε, Wˆ ν ) W˙ˆ ν = −Wˆ ν + Z ν (ξ, ε) · φσ (t)
(21) (22) (23)
where in (21) K u ∈ R1×q is a constant matrix. Furthermore, a(t) is generated by a(t) ˙ = −a(t) − ϕa ( j )
(24)
where ϕa ( j ) is a strictly increasing sequence satisfying lim j →∞ ϕa ( j ) → ∞ with the index j = 0, 1, 2, . . . , denoting the number of times the following equality is satisfied: H = a(t) + γ (|ε|, |ξ |) +
δu
c j +1 a(t) + γ (|ε|, |ξ |) + δu −|b(ξ, ε, Wˆ ν )|2 = 0 (25) δb2
while σ (t) = II with c > 1. Moreover, ϕa ( j ) = 0 when σ (t) = I. Before we proceed and analyze the proposed control law, let us first present the switching strategy σ (t). Switching strategy: We define the sets R1 = (ξ, ε, Wˆ ν ) : |b(ξ, ε, Wˆ ν )| ≤ δb R2 = (ξ, ε, Wˆ ν ) : |b(ξ, ε, Wˆ ν )| > δb . We initiate at (ξ(0), ε(0), Wˆ ν (0)) ∈ R2 and allocate σ (t) = I whenever (ξ, ε, Wˆ ν ) ∈ R2 . The switching variable σ (t) remains in that state until (ξ, ε, Wˆ ν ) ∈ R1 , at which time σ (t) = II. Consequently, σ (t) remains unaltered until (ξ, ε, Wˆ ν ) ∈ R2 . Summarizing, the switching strategy is defined as follows: I, if (ξ, ε, Wˆ ν ) ∈ R2 (26) σ (t) = II, if (ξ, ε, Wˆ ν ) ∈ R1 . To continue, let us first define the set X ≡ x × ac ⊂ U ⊂ Rn × Rn × Rm , where ac ⊂ Rn × Rm is the compact
KOSTARIGKA AND ROVITHAKIS: NEURAL NETWORK CONTROL OF UNCERTAIN MIMO NONLINEAR SYSTEMS
set where the neural network approximation capabilities hold and x ⊂ Rn is a compact set. It is reasonable to define X ⊆ U , as outside of U , (ac ) does not exist. Lemma 1: The controller (15)–(26) guarantees ∀(x, ξ, ε) ∈ X : 1) H ≤ 0 whenever φ = φII , and 2) the sequence ϕa ( j ) is bounded. Proof: 1) Following the switching strategy, at t = 0 we start and operate with φ = φI until |b(ξ, ε, Wˆ ν )| = δb , at which time we switch from φI to φII . Then (25) becomes H = −δu + (δu /c j +1) < 0 since c j +1 is a strictly increasing sequence of j, with c > 1. Hence, it is guaranteed that H < 0 when we first operate with φII . To continue, let T j denote the corresponding time instants (25) is satisfied, [i.e., T1 is the time instant (25) is satisfied for the first time, T2 for the second time, etc.]. Notice that H remains negative ∀t ∈ [T j −1 , T j ) and for all j, until the time instant T j , when H = 0. Hence, at that time, (25) yields |b(ξ, ε, Wˆ ν )|2 (a(t) + γ (|ε|, |ξ |) + δu ) + a(t) + γ (|ε|, |ξ |) δb2 δu (27) = − j +1 . c However, immediately after, at t = T j+ , the value of c j +1 increases to [c j +1]n (i.e., [c j +1]n > c j +1 ) as j increases by 1. Thus, H (T j+ ) = −(δu /c j +1 ) + (δu /[c j +1)]n < 0. Therefore H ≤ 0 whenever φ = φII . 2) We shall prove the second part of this lemma by contradiction. Let us assume that j can be infinite. Assuming that lim j →∞ a(t) = +∞, then since lim j →∞ ϕa ( j ) = +∞, we ˙ = −∞, which contradicts obtain from (24) that lim j →∞ a(t) the supposition. Hence, a(t) either tends to −∞ or it remains bounded. To two cases. proceed, we distinguish Case 1 lim j →∞ a(t) = −∞ : We have proved in the first part of this lemma that H < 0 whenever |b(ξ, ε, Wˆ ν )| = δb . Hence, owing to the continuity of H with respect to |b(ξ, ε, Wˆ ν )|2 , we conclude that there exists a sufficiently small positive constant such that H < 0, ∀ |b(ξ, ε, Wˆ ν )|2 ≥ δb2 −, thus formulating a no-event zone. Alternatively, H = 0 (while φ = φII ) may occur only when −
1−
|b(ξ, ε, Wˆ ν )|2 > 2 > 0. 2 δb δb
(28)
Hence, from (25), lim j →∞ H = −∞, which is a contradiction since H = 0 whenever criterion (25) is satisfied. Case 2 (a(t) ∈ L∞ ): Since lim j →∞ ϕa ( j ) = +∞, (24) yields lim j →∞ a(t) ˙ = −∞. Notice further that |a(t)| + γ (|ε|, |ξ |) + δu . (29) δb and γ (|ε|, |ξ |) ∈ K, (29) yields φII ∈ L∞ .
|φII (t)| ≤ Since a(t) ∈ L∞ To continue
˙ ε, Wˆ ν )=W˙ˆ ν Z ν (ξ, ε)+ Wˆ ν ∂ Z ν (ξ, ε) ξ˙ + Wˆ ν ∂ Z ν (ξ, ε) ε˙ b(ξ, ∂ξ ∂ε ∂ Z ν (ξ, ε) =W˙ˆ ν Z ν (ξ, ε) + Wˆ ν ∂ξ ∂ Z ν (ξ, ε) ε˙ . (30) × Wˆ o Z o (ξ, ε)+φII + Wˆ ν ∂ε
143
Notice that (21) and (23) are in the bounded-input boundedoutput (BIBO) form with Z u (ξ, ε), Z ν (ξ, ε) bounded by construction and therefore Wˆ u , Wˆ ν ∈ L∞ , which in turn proves that u(ξ, ε), b(ξ, ε, Wˆ ν ) ∈ L∞ . Moreover, since Z o (ξ, ε) is bounded by construction, we conclude that (22) is also in the BIBO form leading to Wˆ o ∈ L∞ . Furthermore, (23) yields W˙ˆ ν ∈ L∞ . From (8), we have (31) ε˙ = r L f h(x) + L g h(x)u(ξ, ε) − ν . By assumption L f h(x), L g h(x) ∈ L∞ , ∀x ∈ x , where x is an arbitrary compact set. Further, ρi , ∂ Ti /∂(ei /ρi ) (∂ Z ν (ξ, ε)/∂ξ ), (∂ Z ν (ξ, ε)/∂ε) ∈ L∞ by construction. Hence, r, ν ∈ L∞ ∀(ξ, ε) ∈ ac . Therefore, from (30) we ˙ ε, Wˆ ν ) ∈ L∞ , ∀(ξ, ε) ∈ ac . Differentiating (25) have b(ξ, with respect to time, we obtain 2 |b| H˙ = 1 − 2 (a(t) ˙ + γ˙ (|ε|, |ξ |)) δb b b˙ (32) −2 2 a(t) + γ (|ε|, |ξ |) + δu . δb To continue, since H = 0 (while φ = φII ) may occur only outside the no event zone where (28) holds, it can be easily concluded from (32) that lim j →∞ H˙ = −∞, which contradicts the fact that H < 0 and tends to zero whenever (25) occurs, while φ = φII . Remark 3: Lemma 1 proves that H ≤ 0 whenever φσ = φII and that j is piecewise continuous with respect to time and finite. Hence, the solutions of (16), (23), and (24) exist and are continuous whenever σ (t) = II. The same triviality holds when σ (t) = I as well. The above stated arguments, together with the fact that φI = φII whenever |b(ξ, ε, Wˆ ν )| = δb , lead to the continuity of φσ . As a direct consequence, the dynamic output feedback controller (15)–(26) is continuous, guaranteeing [24] the existence and uniqueness of the solutions of (ac ) and consequently of (). Lemma 2: The continuous dynamic controller (15)–(26) guarantees, ∀ (ξ, ε) ∈ ac that Wˆ u , Wˆ ν , Wˆ o , a(t), and u(t) ∈ L∞ . Proof: From Lemma 1, ϕa is bounded. Therefore a(t) ∈ L∞ as well, since it is generated by the BIBO differential equation (24). Moreover Z u (ξ, ε) is bounded by construction. Hence, since (21) is in a BIBO Form, we obtain Wˆ u ∈ L∞ , which in turn proves that u(ξ, ε) ∈ L∞ . Furthermore γ (|ε|, |ξ |) is bounded by construction. To proceed, notice that whenever σ (t) = I, we have |b(ξ, ε, Wˆ ν )| ≥ δb and |φI (t)| ≤
|a(t)| + γ (|ε|, |ξ |) + δu . |b(ξ, ε, Wˆ ν )|
(33)
From the aforementioned analysis, we straightforwardly conclude that |φI (t)| ∈ L∞ . However, whenever σ (t) = II, we have |b(ξ, ε, Wˆ ν )| < δb and |φII (t)| ≤
|a(t)| + γ (|ε|, |ξ |) + δu . δb
(34)
Hence, |φII (t)| ∈ L∞ . Summarizing, |φσ (t)| ∈ L∞ , ∀t ≥ 0. Moreover, since Z ν (ξ, ε) is bounded by construction, we conclude that (23) is in BIBO form and thus that Wˆ ν ∈ L∞ ,
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IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 23, NO. 1, JANUARY 2012
which in turn proves that b(ξ, ε, Wˆ ν ) ∈ L∞ . Finally, since Z o (ξ, ε) is bounded by construction, we conclude that (22) is also in BIBO form, leading to Wˆ o ∈ L∞ .
L˙ becomes L˙ ≤ −
ηi (| i |) + G(x, ξ, ε) · W˜ u Z u (ξ, ε)
i=x,ξ,ε
+ |G(x, ξ, ε)|2 + tr W˜ u W˙˜ u + W˜ o W˙˜ o + W˜ ν W˙˜ ν
B. Output Feedback Stabilization Analysis Let us consider the augmented Lyapunov-like function L(x, ξ, ε, W˜ u , W˜ ν , W˜ o ) = V (x, ξ, ε) 1 1 + tr |W˜ u |2 + |W˜ o |2 + |W˜ ν |2 (35) 2 2 where V (x, ξ, ε) is the Lyapunov function defined earlier and W˜ u , W˜ ν , W˜ o are parameter errors defined as W˜ u = Wˆ u − Wu∗ , W˜ ν = Wˆ ν − Wν∗ , W˜ o = Wˆ o − Wo∗ . The properties of the proposed controller are summarized in the next theorem. Theorem 3: Consider the uncertain system () for which the PPOFC problem admits a solution, the augmented closed loop system (ac ), and the dynamic output feedback control law (15)–(26). If there exist a positive constant ζ¯u and appropriate positive definite functions η¯ x (·), η¯ ξ (·), η¯ ε (·), such that 2 ∂ Va (x) ∂ Vb (ξ, ε) − g(x)+ r L g h(x) ηi (| i |) + ζ¯u ∂x ∂ε i=x,ξ,ε ≤− η¯ i (| i |) (36) i=x,ξ,ε
with V (x, ξ, ε) = Va (x) + Vb (ξ, ε) being an unknown control Lyapunov function for (ac ) that satisfies (12), then the controller (15)–(26) solves the PPOFC problem for (), ∀(x, ξ, ε) ∈ X ⊆ U . Proof: According to Theorem 1, we need to establish u.u.b of (x, ξ, ε) with respect to some arbitrary sets. In that respect, we distinguish two cases. Case 1 (σ (t) = I): Differentiating (35) along the solutions of (ac ) and using (13) and (14), we obtain (A), shown at the bottom of the page. Employing the fact that −G(x, ξ, ε) · ωu (ξ, ε) ≤ |G(x, ξ, ε)| · |ωu (ξ, ε)| ≤ |G(x, ξ, ε)|2 +
1 |ωu (ξ, ε)|2 4
−W˜ ν + Wˆ ν Z ν (ξ, ε)·φI (t)Z ν (ξ, ε) · φI (t) +Wˆ ν Z ν (ξ, ε) · W˜ o Z o (ξ, ε) + WI where 1 |ωu (ξ, ε)|2 − Wˆ ν Z ν (ξ, ε) · ωo (ξ, ε) + ων (ξ, ε) · φI (t) 4 − W˜ ν Z ν (ξ, ε) − ων (ξ, ε) W˜ o Z o (ξ, ε) − ωo (ξ, ε) ≤
1 2 ω¯ + |W¯ ν | ζν ω¯ o + ω¯ ν φ¯ I + W¯ ν + |Wν∗ | ζν + ω¯ ν 4 u × W¯ o + |Wo∗ | ζo + ω¯ o = WI . (37)
In (37), ζi , i = u, ν, o are the constant unknown upper bounds of the sigmoid terms Z i (ξ, ε), i = u, ν, o, respectively, which are bounded by construction, and W¯ i , i = u, ν, o are the unknown upper bounds of the weights Wˆ i , i = u, ν, o, bounded because of Lemma 2. Moreover, φ¯I is an upper bound of the switching term φI , also proved to be bounded in Lemma 2. Finally, owing to the neural network density property, whenever (ξ, ε) ∈ ac , the approximation errors are upper bounded by unknown but small bounds as follows: |ωu (ξ, ε)| ≤ ω¯ u , |ων (ξ, ε)| ≤ ω¯ ν , |ωo (ξ, ε)| ≤ ω¯ o . Using (20)–(23) we obtain (B), shown at the top of the following page. After using the identities W˜ Wˆ = (1/2)|W˜ |2 + (1/2)|Wˆ |2 − (1/2)|W ∗ |2 , tr{W˜ Wˆ } = (1/2) |W˜ |2 + (1/2)|Wˆ |2 − (1/2)|W ∗ |2 , and the trace property tr{yx } = x y, x, y ∈ Rn , we get ηi (| i |) + G(x, ξ, ε) · W˜ u Z u (ξ, ε) + |G(x, ξ, ε)|2 L˙ ≤ − i=x,ξ,ε
−
1 ˜ 2 |Wi | + |Wˆ i |2 − |Wi∗ |2 2
i=u,ν,o +K u · W˜ u Z u (ξ, ε)
+ b(ξ, ε, Wˆ ν ) · φI (t) + WI .
⎫ (φ(ξ, ε) − φ0 (ξ, ε)) + tr W˜ u W˙˜ u + W˜ o W˙˜ o + W˜ ν W˙˜ ν ⎪ ⎪ ⎪ ⎪ i=x,ξ,ε ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ˙ ˙ ∗ ⎪ ˆ ˜ ˜ ˜ ˜ ≤− ηi (| i |) + G(x, ξ, ε) Wu Z u (ξ, ε) − Wu Z u (ξ, ε) − ωu (ξ, ε) + tr Wu Wu + Wo Wo ⎪ ⎪ ⎪ i=x,ξ,ε ⎪ ⎪ ⎪ ⎪ ˙ ∗ ∗ ⎪ +W˜ ν W˜ ν + Wν Z ν (ξ, ε) + ων (ξ, ε) Wˆ o Z o (ξ, ε) + φI (t) − Wo Z o (ξ, ε) − ωo (ξ, ε) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ˙ ˙ ˙ ηi (| i |) + G(x, ξ, ε) W˜ u Z u (ξ, ε) − ωu (ξ, ε) + tr W˜ u W˜ u + W˜ o W˜ o + W˜ ν W˜ ν ≤− . i=x,ξ,ε ⎪ ⎪ ⎪ ⎪ ⎪ + (Wˆ ν − W˜ ν ) Z ν (ξ, ε) + ων (ξ, ε) W˜ o Z o (ξ, ε) + φI (t) − ωo (ξ, ε) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ˙ ˙ ˙ ⎪ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ⎪ ≤− ηi (| i |) + G(x, ξ, ε) Wu Z u (ξ, ε) − ωu (ξ, ε) + tr Wu Wu + Wo Wo + Wν Wν ⎪ ⎪ ⎪ i=x,ξ,ε ⎪ ⎪ ⎪ ⎪ +Wˆ ν Z ν (ξ, ε) · W˜ o Z o (ξ, ε) + Wˆ ν Z ν (ξ, ε) · φI (t) − Wˆ ν Z ν (ξ, ε) · ωo (ξ, ε) − W˜ ν Z ν (ξ, ε) · φI (t) ⎪ ⎪ ⎪ ⎪ ⎭ ˜ ˜ +ων (ξ, ε) · φI (t) − Wν Z ν (ξ, ε) − ων (ξ, ε) Wo Z o (ξ, ε) − ωo (ξ, ε)
L˙ ≤ −
ηi (| i |) + G(x, ξ, ε)(u − u 0 ) +
∂ Vb (ξ,ε) ∂ξ
(A)
KOSTARIGKA AND ROVITHAKIS: NEURAL NETWORK CONTROL OF UNCERTAIN MIMO NONLINEAR SYSTEMS
L˙ ≤ −
i=x,ξ,ε
ηi (| i |) + G(x, ξ, ε) · W˜ u Z u (ξ, ε) + |G(x, ξ, ε)|2 + tr W˜ u W˙˜ u + W˜ o W˙˜ o + W˜ ν W˙˜ ν
+ b(ξ, ε, Wˆ ν ) · W˜ o Z o (ξ, ε) + b(ξ, ε, Wˆ ν ) · φI (t) − W˜ ν Z ν (ξ, ε) · φI (t) + WI ≤− ηi (| i |) + G(x, ξ, ε) · W˜ u Z u (ξ, ε) + |G(x, ξ, ε)|2 + tr W˜ u −Wˆ u + Z u (ξ, ε) · K u i=x,ξ,ε +tr W˜ o −Wˆ o − Z o (ξ, ε) · b(ξ, ε, Wˆ ν ) + W˜ ν −Wˆ ν + Z ν (ξ, ε) · φI (t) + b(ξ, ε, Wˆ ν ) · W˜ o Z o (ξ, ε) + b(ξ, ε, Wˆ ν ) · φI (t) − W˜ ν Z ν (ξ, ε) · φI (t) + WI
145
⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2 ˜ ˜ ˜ ˜ ˆ ˆ ˆ ⎪ ≤− ηi (| i |) + G(x, ξ, ε) · Wu Z u (ξ, ε) + |G(x, ξ, ε)| − tr Wu Wu − tr Wo Wo − Wν Wν ⎪ ⎪ ⎪ ⎪ i=x,ξ,ε ⎪ ⎪ ⎪ ⎪ ⎪ + tr W˜ u Z u (ξ, ε) · K u − tr W˜ o Z o (ξ, ε) · b(ξ, ε, Wˆ ν ) + W˜ ν Z ν (ξ, ε) · φI (t) ⎪ ⎪ ⎪ ⎪ ⎭ + b(ξ, ε, Wˆ ν ) · W˜ o Z o (ξ, ε) + b(ξ, ε, Wˆ ν ) · φI (t) − W˜ ν Z ν (ξ, ε) · φI (t) + WI Moreover, using the inequalities G(x, ξ, ε)· W˜ u Z u (ξ, ε) ≤ |G(x, ξ, ε)| · |W˜ u | · |Z u (ξ, ε)| 2 |W˜ u | ≤ |G(x, ξ, ε)| ·|Z u (ξ, ε)| + 4 K u · W˜ u Z u (ξ, ε) ≤ |K u | · |W˜ u | · |Z u (ξ, ε)| 2
2
2 |W˜ u | ≤ |K u | · |Z u (ξ, ε)| + 4 2
we have L˙ ≤ −
2
ηi (| i |) + |G(x, ξ, ε)|2 |Z u (ξ, ε)|2 + 1
i=x,ξ,ε
1 ˜ 2 |Wi | + |Wˆ i |2 − |Wi∗ |2 − 2 i=ν,o 1 ˜ 2 − |Wu | + |Wˆ u |2 − |Wu∗ |2 + |K u |2 |Z u (ξ, ε)|2 2 2 |W˜ u | + + b(ξ, ε, Wˆ ν ) · φI (t) + WI . 2 Dropping the negative terms −(1/2) i=ν,o (|W˜ i |2 +|Wˆ i |2 ), −(1/2)|Wˆ u |2 , we obtain L˙ ≤ − ηi (| i |) + |G(x, ξ, ε)|2 |Z u (ξ, ε)|2 + 1 i=x,ξ,ε
+ b(ξ, ε, Wˆ ν ) · φI (t) + μ¯ I where
1 ∗ 2 |Wi | WI + |K u |2 |Z u (ξ, ε)|2 + 2 i=u,ν,o 1 ∗ 2 |Wi | = μ¯ I . ≤ WI + |K u |2 ζu2 + 2
i=x,ξ,ε
a(t) + γ (|ε|, |ξ |) + δu + μ¯ I −|b(ξ, ε, Wˆ ν )|2 |b(ξ, ε, Wˆ ν )|2 ≤− ηi (| i |) + |G(x, ξ, ε)|2 |Z u (ξ, ε)|2 + 1 i=x,ξ,ε
−a(t) − γ (|ε|, |ξ |) − δu + μ¯ I .
(B)
Finally, dropping the negative term −δu and exploiting Lemma 1 to guarantee the boundedness of a(t), we arrive at L˙ ≤ − ηi (| i |) − γ (|ε|, |ξ |) i=x,ξ,ε
+ |G(x, ξ, ε)|2 |Z u (ξ, ε)|2 + 1 + μI
(38)
where μI = μ¯ I + sup {a(t)} . Case 2 (σ (t) = II): Using the weight update laws (21)–(23) with σ = II, following similar steps as in Case 1 and after adding and subtracting a(t), we arrive at ηi (| i |) + |G(x, ξ, ε)|2 |Z u (ξ, ε)|2 + 1 L˙ ≤ − i=x,ξ,ε
+b(ξ, ε, Wˆ ν ) · φII (t) + a(t) + μII 2 2 where μ II ∗ 2= WII + |K u | ζu + sup {a(t)} + (1/2) i=u,ν,o |Wi | , with
1 |ωu (ξ, ε)|2 − Wˆ ν Z ν (ξ, ε) · ωo (ξ, ε) + ων (ξ, ε) · φI (t) 4 − W˜ ν Z ν (ξ, ε) − ων (ξ, ε) W˜ o Z o (ξ, ε) − ωo (ξ, ε) ≤
1 2 ω¯ u + |Wˆ ν | ζν ω¯ o + ω¯ ν φ¯ II + W¯ ν + |Wν∗ | ζν + ω¯ ν 4 × W¯ o + |Wo∗ | ζo + ω¯ o = WII . (39)
Substituting φII from (18), we obtain L˙ ≤ − ηi (| i |) + |G(x, ξ, ε)|2 |Z u (ξ, ε)|2 + 1 i=x,ξ,ε
−|b(ξ, ε, Wˆ ν )|2
i=u,ν,o
Substituting φI from (17), we get L˙ ≤ − ηi (| i |) + |G(x, ξ, ε)|2 |Z u (ξ, ε)|2 + 1
.
a(t) + γ (|ε|, |ξ |) + δu + a(t) + μII . δb2
However, owing to (27) −|b(ξ, ε, Wˆ ν )|2 leading to L˙ ≤ −
a(t) + γ (|ε|, |ξ |) + δu + a(t) ≤ −γ (|ε|, |ξ |) δb2
ηi (| i |) − γ (|ε|, |ξ |)
i=x,ξ,ε
+|G(x, ξ, ε)|2 |Z u (ξ, ε)|2 + 1 + μII .
(40)
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2
q˙1(t) (rad/s)
q1(t)/qr1(t)(rad)
2
1.5
1
0
2
4 6 time (sec)
8
0
2
4 6 time (sec)
8
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4 6 time (sec)
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4 6 time (sec)
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2
4 6 time (sec)
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1 0
2
q˙2(t) (rad/s)
q2(t)/qr2(t)(rad)
0
−1
10
2.5
1.5
1
−1 −2
0
2
4 6 time (sec)
8
−3
10
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2
2.5
1 ξ2(t)
ξ1(t)
1
2
0 1.5 2
4 6 time (sec)
8
3
1
2.5
0
2
1
−1 −2
1.5
Fig. 3.
−1
10
ξ4(t)
ξ3(t)
0
0
2
4 6 time (sec)
8
10
−3
System response (solid line: state trajectories; dashed line: reference trajectories).
Using (38) and (40), we conclude that in both cases ηi (| i |) − γ (|ε|, |ξ |) + ζ¯u |G(x, ξ, ε)|2 + μ (41) L˙ ≤ − i=x,ξ,ε
where μ = max (μI , μII ) and |Z u (ξ, ε)|2 + 1 ≤ ζu2 + 1 = ζ¯u . Employing (36) and recalling the definition of γ (|ε|, |ξ |) = γξ (|ξ |) + γε (|ε|), L˙ finally becomes L˙ ≤ − η¯ i (| i |) − γ (|ε|, |ξ |) + μ i=x,ξ,ε
≤ −η¯ x (|x|) − γξ (|ξ |) − γε (|ε|) + μ. −1 Hence, L˙ ≤ 0 whenever |x(t)| > η¯ −1 x (μ) , |ξ(t)| > γξ (μ), −1 or |ε(t)| > γε (μ) , from which we conclude the u.u.b of
x, ξ, ε with respect to the sets Ex = x ∈ Rn : |x(t)| ≤ η¯ −1 x (μ) Eξ = ξ ∈ Rn : |ξ(t)| ≤ γξ−1 (μ) Eε = ε ∈ R : |ε(t)| ≤ γε−1 (μ)
(42) (43) (44)
respectively. The results presented are valid ∀(x, ξ, ε) ∈ X ⊆ U , where X ≡ x × ac with ac ⊂ Rn × Rm the compact set in which the approximation capabilities of the linear-in-the-weights neural networks hold and x ⊂ Rn is a compact set. It is therefore mandatory to prove that, under certain achievable conditions, the aforementioned
KOSTARIGKA AND ROVITHAKIS: NEURAL NETWORK CONTROL OF UNCERTAIN MIMO NONLINEAR SYSTEMS
0
7 × 10−3
−0.4
0
6
−0.6
−5
5
−0.8
−10 6
−1 0
1
2
3
6.5
4 5 6 time (sec)
7
7
7.5
8
8
9
10
ˆ ˆ ˆ |Wu|,|Wo|,|Wν|
e1(t)(rad)
−0.2
1.5
4 3 2 1
10
e2(t)(rad)
147
× 10−3
0
1
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0 6
6.5
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7.5
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5 6 time (sec)
7
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Fig. 5. Evolution of the Euclidean neural network weight norms (solid line: | Wˆ u |, dashed line: |Wˆ o |, dotted line: |Wˆ ν |).
0 0
1
2
3
4 5 6 time (sec)
7
10
Fig. 4. Output tracking errors along with their corresponding performance bounds. The subplots represent details at the steady state (solid lines: tracking errors; dashed line: prescribed output error bounds).
¯ = property holds ∀t ≥ 0. To proceed, we define {x, ξ, ε, W˜ u , W˜ ν , W˜ o : L(x, ξ, ε, W˜ u , W˜ ν , W˜ o ) ≤ cω }, where the constant cω satisfies cω ≥ cω + μ > cω with cω being the largest positive constant for which cω (V ) ⊆ X , where cω (V ) is a compact set defined as cω (V ) ≡ {(x, ξ, ε) ∈ Rn × Rn × Rm : V (x, ξ, ε) ≤ cω } . Given that EW is the u.u.b region of the neural network weights (W˜ u , W˜ ν , W˜ o ), proved to be bounded in Lemma 2, it can ¯ since in such a be easily verified that cω (V ) × EW ⊂ , case L(x, ξ, ε, W˜ u , W˜ ν , W˜ o ) ≤ cω , ∀(x, ξ, ε, W˜ u , W˜ ν , W˜ o ) ∈ cω (V )×EW . From the proof of Lemma 2 it can be seen that, whenever (ξ, ε) ∈ ac , for any selection of δu , ϕa , γξ (·), γε (·) we can always choose δb to upper bound |φσ (t)| and thus to keep μ constant. To continue, notice that the control functions γξ (·), γε (·) can be chosen to render the sets Eξ , Eε arbitrarily small and therefore to ensure that Eξ ×Eε ⊂ ac . Furthermore, the K∞ −function η¯ x (·), introduced through (12) and (36), should lead to Ex ⊂ x . Hence, X ≡ Ex × Eξ × Eε ⊆ U . As ¯ thus a consequence X ⊂ cω (V ) ⊆ X and X × EW ⊂ , ¯ Therefore, for establishing ≡ (cω (V ) ∪ X ) × EW ⊂ . ¯ it follows any (x(0), ξ(0), ε(0), W˜ u (0), W˜ ν (0), W˜ o (0)) ∈ , that L is bounded from above by cω since L˙ ≤ 0 in the set . As a result (x, ξ, ε, W ˜ u , W˜ ν , W˜ o ) ∈ , ¯ ¯ ∀t ≥ 0. Hence, / (x, ξ, ε) cannot escape X and consequently U , at any time. Remark 4: Inequality (36) is a sufficient stabilizability condition for (ac ). It practically states that the continuous dynamic output feedback controller u 0 (ξ, ε), ξ˙ = φ0 (ξ, ε), which guarantees that the states of (ac ) are u.u.b. with respect to some sets, satisfies (12), with the positive definite function i=x,ξ,ε ηi (|i |) being strictly greater than ζ¯u |(∂ Va (x)/∂ x)g(x) + (∂ Vb (ξ, ε)/∂ε)r L g h(x)|2 . In other words, (36) provides a qualitative description of the relation that connects the class of original systems () for which the PPOFC problem admits a solution and the controller that
generates the solutions. As the terms engaged in (36) are unknown, its a priori verification is difficult. Fortunately, this statement is relaxed by noticing that (36) is required only for analysis and not for controller design. V. S IMULATION R ESULTS Consider a planar two-link articulated manipulator whose position can be described by a 2-D vector q = [ q1 q2 ] of joint angles and whose actuator inputs consist of a 2-D vector u = [ u 1 u 2 ] of torques applied at the manipulator joints. The dynamics of this simple manipulator is strongly nonlinear and can be written in the general form M(q)q¨ + C(q, q) ˙ q˙ + G(q) = u where
M(q) =
M11 M12 M12 M22
(45)
M11 = Iz1 + Iz2 + m 1r12 + m 2 (l12 + r22 ) + 2m 2l1r2 c2 M12 = Iz2 + m 2r22 + m 2l1r2 c2 M22 = Iz2 + m 2r22 is the 2 × 2 manipulator inertia matrix
−m 2l1r2 s2 q˙2 + k1 −m 2l1r2 s2 (q˙1 + q˙2 ) C(q, q) ˙ = m 2l1r2 s2 q˙1 k2 is a 2 × 2 matrix of centripetal and Coriolis torques, and
m 1 gr1 c1 + m 2 g(l1 c1 + r2 c12 ) G(q) = m 2 gr2 c12 is the 2-D vector of gravitational torques. In the above, we have used the following notation: c1 = cos(q1 ), c12 = cos(q1 + q2 −
π 2 ),
c2 = cos (q2 − π2 ) s2 = sin(q2 − π2 ).
For simulation purposes, we have employed Link masses (kg): Link lengths (m): Inertia (kg m 2 ): Friction (N m):
m 1 = 3.2 l1 = 0.5 Iz1 = 0.96 k1 = 1
m2 = 2 l2 = 0.4 Iz2 = 0.81 k2 = 1.
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5
30
0.05 0 −0.05 −0.1 −0.15 −0.2 −0.25
0
10 0
−5
−10
H
u1(t) (nm)
20
−20 −30
−10 0
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4 5 6 time (sec)
7
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0.45 0.5 0.55 0.6
10 −15
5 −20
u2(t) (nm)
0 −5
−15
Fig. 6.
1
2
3
4
5 6 time (sec)
7
8
9
10
Fig. 7. Evolution of H. The subplot presents details of the events H = 0 appearing while operating with φσ (t) = φII (t).
−10
−20
0
0
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4 5 6 time (sec)
7
8
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Demanded control effort.
Furthermore, g = 10 m/s2 is gravity acceleration, Which is taken as constant, and r1 = (l1 /2), r2 = (l2 /2). The considered task is for the system output to track the following reference signal:
90 30 π + π cos(t) qr1 (t) 180 . = 180 90 30 qr2 (t) 180 π − 180 π cos(t) The initial state conditions are 80 q1 (0) q2 (0) q˙1 (0) q˙2 (0) = 180 π
in Fig. 4. Variation of the Euclidean norms of the weight vectors Wˆ u , Wˆ o , Wˆ ν are shown in Fig. 5. Obviously, all output trajectories converge to their desired references with prescribed performance, and all closed-loop states are kept uniformly bounded, thus solving the PPOFC problem for the considered robotic system. It is worth mentioning that the problem is solved without requiring excessive control effort as Fig. 6 clearly demonstrates. Details on the operation of the proposed switching term φσ are provided in Fig. 7, through the evolution of the H function. Notice that the number of the events H = 0, while σ (t) = II, is finite (as theory predicts). VI. C ONCLUSION
130 180 π
00
.
Performance specifications: For the output errors e1 = q1 − qr1 , e2 = q2 − qr2 we desire to achieve convergence rate no more than e−2t , steady-state error no more than (π/360), and no overshoot. The aforementioned specifications are prescribed via the following performance functions: 1 60 1 π− π e−2t + π, i = 1, 2 (46) ρi (t) = 180 360 360 where ρ0i = (60/180)π, i = 1, 2 was chosen in order for the problem to be well defined, in other words to guarantee that (5) is satisfied at t = 0. Moreover, since no overshoot is required, we choose Mi = 0, i = 1, 2. To implement the control (15)–(26), we have used HONNs, whose structure was selected according to a trial-and-error procedure. The weight vectors Wˆ u (t) ∈ R20×2 , Wˆ o (t) ∈ R10×4 , and Wˆ ν (t) ∈ R5×4 were randomly initialized in [0, 10]. The initial value of a(t) was taken a(0) = 1, while ϕa ( j ) = j. Furthermore, we have selected c0 = 1, c = 1.1, K u = [1 1] , while γξ i (·) = γεi (·) = 10(1 − e−10(·)), i = 1, 2. Finally, concerning the switching procedure, the following constants have been used: δb = 3, δu = 1. The response of the proposed adaptive output feedback control scheme is demonstrated in Figs. 3–7. In Fig. 3, the robot joint angles, the respective velocities, and the ξ −states are plotted, while the evolution of the output tracking errors along with their corresponding performance bounds is shown
An adaptive dynamic output feedback controller for a fairly general class of MIMO uncertain nonlinear systems (those that are affine in the control) has been presented, which was capable of guaranteeing convergence of the output tracking error to a predefined arbitrarily small residual set, with convergence rate no less than a prespecified value, while exhibiting maximum overshoot less than some sufficiently small preassigned constant. To overcome the amount of uncertainties present, linearly parameterized neural networks were used as approximation models. It was proved that guaranteeing a boundedness property for the states of a specifically defined augmented closed-loop system was the necessary and sufficient condition for solving the considered problem. Even though the designed dynamic output feedback controller was of switching type, its continuity was guaranteed, thus alleviating any problems related to the existence and uniqueness of solutions. Simulations on a planar two-link articulated manipulator illustrated the approach. R EFERENCES [1] J. T. Spooner, M. Maggiore, R. Ordonez, and K. M. Passino, Stable Adaptive Control and Estimation for Nonlinear Systems-Neural and Fuzzy Approximation Techniques. New York: Wiley, 2002. [2] Y.-C. Chang, “An adaptive H∞ tracking control for a class of nonlinear multiple-input multiple-output (MIMO) systems,” IEEE Trans. Automat. Control, vol. 46, no. 9, pp. 1432–1437, Sep. 2001. [3] G. Bianchini, R. Genesio, A. Parenti, and A. Tesi, “Global H∞ controllers for a class of nonlinear systems,” IEEE Trans. Automat. Control, vol. 49, no. 2, pp. 244–249, Feb. 2004.
KOSTARIGKA AND ROVITHAKIS: NEURAL NETWORK CONTROL OF UNCERTAIN MIMO NONLINEAR SYSTEMS
[4] C. Bechlioulis and G. A. Rovithakis, “Robust adaptive control of feedback linearizable MIMO nonlinear systems with prescribed performance,” IEEE Trans. Automat. Control, vol. 53, no. 9, pp. 2090–2099, Oct. 2008. [5] C. P. Bechlioulis and G. A. Rovithakis, “Adaptive control with guaranteed transient and steady state tracking error bounds for strict feedback systems,” Automatica, vol. 45, no. 2, pp. 532–538, 2009. [6] C. Bechlioulis and G. A. Rovithakis, “Prescribed performance adaptive control for multi-input multi-output affine in the control nonlinear systems,” IEEE Trans. Automat. Control, vol. 55, no. 5, pp. 1220–1226, May 2010. [7] C. P. Bechlioulis, Z. Doulgeri, and G. A. Rovithakis, “Neuro-adaptive force/position control with prescribed performance and guaranteed contact maintenance,” IEEE Trans. Neural Netw., vol. 21, no. 12, pp. 1857– 1868, Dec. 2010. [8] A. K. Kostarigka and G. A. Rovithakis, “Prescribed performance output feedback control: An approximate passivation approach,” in Proc. 18th Mediter. Conf. Control Automat., Marrakech, Morocco, 2010, pp. 11–16. [9] N. Hovakimyan, F. Nardi, A. J. Calise, and H. Lee, “Adaptive output feedback control of a class of non-linear systems using neural networks,” Int. J. Control, vol. 74, no. 12, pp. 1161–1169, Aug. 2001. [10] S. Seshagiri and H. Khalil, “Output feedback control of nonlinear systems using RBF neural networks,” IEEE Trans. Neural Netw., vol. 11, no. 1, pp. 69–79, Jan. 2000. [11] N. Kim and A. J. Calise, “Several extensions in methods for adaptive output feedback control,” IEEE Trans. Neural Netw., vol. 18, no. 2, pp. 482–494, Mar. 2007. [12] Y.-J. Liu, S.-C. Tong, D. Wang, T.-S. Li, and C. L. P. Chen, “Adaptive neural output feedback controller design with reduced-order observer for a class of uncertain nonlinear SISO systems,” IEEE Trans. Neural Netw., vol. 22, no. 8, pp. 1328–1334, Aug. 2011. [13] S. S. Ge and J. Zhang, “Neural-network control of nonaffine nonlinear system with zero dynamics by state and output feedback,” IEEE Trans. Neural Netw., vol. 14, no. 4, pp. 900–918, Jul. 2003. [14] N. Hovakimyan, F. Nardi, A. Calise, and N. Kim, “Adaptive output feedback control of uncertain nonlinear systems using single-hiddenlayer neural networks,” IEEE Trans. Neural Netw., vol. 13, no. 6, pp. 1420–1431, Nov. 2002. [15] Y.-J. Liu, C. L. P. Chen, G.-X. Wen, and S. Tong, “Adaptive neural output feedback tracking control for a class of uncertain discrete-time nonlinear systems,” IEEE Trans. Neural, vol. 22, no. 7, pp. 1162–1167, Jul. 2011. [16] J. Y. Choi and J. A. Farrell, “Adaptive observer backstepping control using neural networks,” IEEE Trans. Neural Netw., vol. 12, no. 5, pp. 1103–1112, Sep. 2001. [17] B. Ren, S. S. Ge, K. P. Tee, and T. H. Lee, “Adaptive neural control for output feedback nonlinear systems using a Barrier Lyapunov function,” IEEE Trans. Neural Netw., vol. 21, no. 8, pp. 1339–1345, Aug. 2010. [18] S. Battilotti and A. De Santis, “Robust output feedback control of nonlinear stochastic systems using neural networks,” IEEE Trans. Neural Netw., vol. 14, no. 1, pp. 103–116, Jan. 2003. [19] P. A. Ioannou and J. Sun, Robust Adaptive Control. NJ: Prentice Hall, 1996.
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[20] G. A. Rovithakis and M. A. Christodoulou, Adaptive Control with Recurrent High-Order Neural Networks. New York: Springer-Verlag, 2000. [21] M. M. Gupta and D. H. Rao, Neuro-Control Systems: Theory and Applications. New York: Springer-Verlag, 1994. [22] G. Cybenko, “Approximations by superpositions of a sigmoidal function,” Math. Control Signals Syst., vol. 2, no. 4, pp. 303–314, 1989. [23] S. H. Lane, D. A. Handelman, and J. J. Gelfand, “Theory and developement of higher-order CMAC neural networks,” IEEE Control Syst. Mag., vol. 12, no. 2, pp. 23–30, Apr. 1992. [24] M. M. Polycarpou and P. A. Ioannou, “On the existence and uniqueness of solutions in adaptive control systems,” IEEE Trans Automat. Control, vol. 38, no. 3, pp. 474–479, Mar. 1993.
Artemis K. Kostarigka was born in Larissa, Greece, in 1980. She received the B.Sc. degree in physics and the M.Sc. degree in control systems theory from the Aristotle University of Thessaloniki, Thessaloniki, Greece, in 2003 and 2005, respectively. She is currently pursuing the Ph.D. degree with the Department of Electrical and Computer Engineering, Division of Electronics and Computer Engineering, Aristotle University of Thessaloniki. Her current research interests include nonlinear robust adaptive controls and control of unknown systems using neural networks.
George A. Rovithakis (S’89–M’98–SM’02) received the Diploma degree in electrical engineering from the Aristotle University of Thessaloniki, Thessaloniki, Greece, in 1990, and the M.S. and Ph.D. degrees in electronic and computer engineering from the Technical University of Crete, Crete, Greece, in 1994 and 1995, respectively. He was a Visiting Assistant Professor with the Department of Electronic and Computer Engineering, Technical University of Crete, from 1995 to 2002. He joined the Aristotle University of Thessaloniki, where he is currently an Associate Professor in the Department of Electrical and Computer Engineering. He has authored or co-authored two books and several book chapters and has published over 110 papers in scientific journals and conference proceedings. His current research interests include nonlinear robust adaptive controls and neural networks for identification and control of uncertain systems. Dr. Rovithakis was a former Associate Editor of the IEEE T RANSACTIONS ON N EURAL N ETWORKS . He is a member of the IEEE Control Systems Society Conference Editorial Board and a member of the Technical Chamber of Greece.
IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 23, NO. 1, JANUARY 2012
Adaptive Dynamic Output Feedback Neural Network Control of Uncertain MIMO Nonlinear Systems with Prescribed Performance Artemis K. Kostarigka and George A. Rovithakis, Senior Member, IEEE
Abstract— An adaptive dynamic output feedback neural network controller for a class of multi-input/multi-output affine in the control uncertain nonlinear systems is designed, capable of guaranteeing prescribed performance bounds on the system’s output as well as boundedness of all other closed loop signals. It is proved that simply guaranteeing a boundedness property for the states of a specifically defined augmented closed loop system is necessary and sufficient to solve the problem under consideration. The proposed dynamic controller is of switching type. However, its continuity is guaranteed, thus alleviating any issues related to the existence and uniqueness of solutions. Simulations on a planar two-link articulated manipulator illustrate the approach. Index Terms— Neural network adaptive control, output feedback, prescribed performance.
I. I NTRODUCTION
G
UARANTEEING performance quality on a system’s output response is an important and pretentious task in control engineering. Owing to the serious technical issues raised whenever performance specifications are incorporated in closed-loop design, control solutions are hard to meet, even when the system structure is considered known. Traditionally, quality of performance is approached through minimizing certain functionals [1]–[3]. Unfortunately, however, no obvious and a priori specified connection exists between the aforementioned performance functionals and trajectory oriented metrics such as maximum overshoot, convergence rate, and steady-state error. In an attempt to establish such a direct link, the prescribed performance control (PPC) paradigm was recently proposed [4]–[6], to guarantee, for various classes of uncertain nonlinear systems, convergence of the output tracking error to a predefined arbitrarily small residual set, with convergence rate no less than a prespecified value, while exhibiting maximum overshoot less than some sufficiently small preassigned constant. In the same spirit, application of PPC to adaptive force/position control of robotic manipulators was provided in [7]. The above-mentioned PPC results were derived under a full state measurement requirement. In real-world applications though, such an assumption is rarely satisfied. The necessity of Manuscript received February 1, 2011; revised October 6, 2011; accepted October 7, 2011. Date of publication December 15, 2011; date of current version January 5, 2012. The authors are with the Department of Electrical and Computer Engineering, Aristotle University of Thessaloniki, Thessaloniki 54124, Greece (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TNNLS.2011.2178448
developing PPC designs for general classes of uncertain nonlinear systems incorporating output feedback only led to [8], in which an approximate passivation approach was realized, imposing, however, restrictive structural constraints/properties. Even without prescribed performance, designing output feedback controllers for uncertain nonlinear systems is a highly sophisticated problem. To overcome the full state measurement obstacle, constructive control solutions rely mostly on observers for state estimation. Unfortunately, the use of observers introduces further restrictions on the relative degree of the output [9], as well as on the form of the allowed plant uncertainties. Representative works in this direction include single-input/single-output feedback linearizable systems [10], [11], special classes of strict feedback systems [12], non-affine in the control nonlinear systems in normal form [13], [14] and in pure feedback form [15], as well as systems in the output feedback form [16], [17] and stochastic systems [18]. Needless to mention that constructing observers for more general nonlinear and uncertain system structures is a challenging task and their incorporation in closed-loop system analysis is still considered a major open control problem. In this paper, an adaptive dynamic output feedback controller is proposed, which, without resorting to an observer construction, is capable of achieving the prescribed performance for the output and uniform boundedness of all other signals in the closed loop, for a class of multi-input/multi-output (MIMO) affine in the control uncertain nonlinear systems. To overcome the amount of uncertainties present, linearly parameterized neural networks are used as approximation models. It is proved that guaranteeing a boundedness property for the states of a specifically defined augmented closed loop system is necessary and sufficient to solve the problem under consideration. Even though the developed controller is of switching type, its continuity is guaranteed, thus alleviating any issues related to the existence and uniqueness of solutions. With respect to [8], the structural constraints/properties required are significantly relaxed owing to the control solution proposed. This paper is organized as follows. Necessary definitions and preliminary results are summarized in Section II. The prescribed performance output feedback control (PPOFC) problem is stated in Section III. Section IV is focused on the design of a dynamic output feedback neural network controller capable of guaranteeing the prescribed performance of the output tracking errors, as well as the boundedness of all other
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closed loop signals. Simulation results on a planar two-link articulated manipulator are provided in Section V. Finally, we conclude in Section VI.
ei(0) ≥ 0
ρi∞ ei(t) 0
II. D EFINITIONS AND P RELIMINARIES A function V : Rn → R+ is said to be positive definite if V (x) is positive for all nonzero x and is zero at x = 0. A function V : Rn → R+ is said to be proper or radially unbounded if V (x) tends to +∞ as |x| → +∞. A function f (·) is said to be of class C k if the derivatives f , f , . . . , f (k) 0 exist and are continuous. In particular, the class C consists of all continuous functions. A continuous function γ : [0, α) → R+ is said to belong to class K if it is strictly increasing and γ (0) = 0. It is said to belong to class K∞ if α = ∞ and γ (r ) → ∞ as r → ∞. If 1 , 2 are two subsets of Rn and 2 ⊂ 1 , 1 /2 denotes the set of all x ∈ 1 satisfying x∈ / 2 . Consider the system x˙ = f 0 (x, u), x ∈ Rn , u ∈ Rq y = h 0 (x), y ∈ Rm
(1)
whose solution x(t; t0 , x 0 ) is defined on [t0 , ∞) for each initial state x 0 , input u, and t0 ∈ R+ . Definition 1 [19]: The solutions of (1) are uniformly ultimately bounded (u.u.b) (with bound B) if there exists a B > 0 and if corresponding to any a > 0 and t0 ∈ R+ , there exists a T = T (a) > 0 (independent of t0 ) such that |x 0 | < a implies |x(t; t0 , x 0 )| < B for all t ≥ t0 + T. Theorem 1 [19]: Assume that (1) possesses unique solutions for all x 0 ∈ Rn and t0 ∈ R+ . If there exists a function V (t, x) defined on |x| ≥ R (where R may be large) and t ∈ [t0 , ∞) with continuous first-order partial derivatives with respect to x, t and if there exist functions ϕ1 , ϕ2 ∈ K∞ and ϕ3 ∈ K defined on [t0 , ∞), such that: 1) ϕ1 (|x|) ≤ V (t, x) ≤ ϕ2 (|x|); 2) V˙ (t, x) ≤ −ϕ3 (|x|) for all |x| ≥ R and t ∈ [t0 , ∞); then the solutions of (1) are u.u.b. A. Linear in the Weights Neural Networks In this paper, we shall be using linear-in-the-weights neural networks of the form z = W Z (v 0 ), where v 0 ∈ Rn2 and z ∈ Rn1 are the neural net input and output, respectively, W is an L−dimensional vector of synaptic weights, and Z (v 0 ) is an L × n 1 matrix of regressor terms. The regressor terms may contain high-order connections of sigmoid functions [20], radial basis functions (RBFs) with fixed centers and widths [21], and shifted sigmoids [22], thus forming high-order neural networks (HONNs), RBFs, and shifted sigmoidal neural networks, respectively. Another class of linear-in-the-weights neural nets is the cerebellar model articulation controller network, which mainly uses B-splines in Z (v 0 ) [23]. An important and well-known property shared among the aforementioned neural approximating structures is the following (see also the references above). Density property: For each continuous function F: Rn2 → n R 1 , and for every ε0 ≥ 0, there exist an integer L and an
0 0 −ρi∞
ei(t)
ρi(t) −Miρi(t) Miρi(t) −ρi(t)
ei(0) ≤ 0 0 time Fig. 1.
Graphical illustration of the prescribed performance definition (2).
optimal synaptic weight vector W ∗ ∈ R L such that sup F (v 0 ) − W ∗ Z (v 0 ) ≤ ε0 v 0 ∈
where ⊂ Rn2 is a given compact set. In other words, if the number of the regressor terms L is sufficiently large, then there exist weight values W ∗ such that W ∗ Z (v 0 ) can approximate F (v 0 ) to any degree of accuracy, in a given compact set. This property allows us to focus on linear-in-the-weights neural networks without loss of generality in terms of approximation error. This, in turn, will make it easier to prove basic systems properties such as stability and robustness. B. Prescribed Performance Preliminaries For completeness and compactness of presentation, this subsection summarizes preliminary knowledge on prescribed performance originally stated in [4]. In that respect, consider a generic tracking error e (t) = [e1 (t) . . . em (t)] ∈ Rm . Prescribed performance is achieved if each element ei (t) , i = 1, . . . , m evolves strictly within a predefined region that is bounded by a decaying function of time. The mathematical expression of prescribed performance is given, ∀t ≥ 0, by the following inequalities: −Mi ρi (t) < ei (t) < ρi (t), ei (0) ≥ 0 i = 1, . . . , m −ρi (t) < ei (t) < Mi ρi (t), ei (0) ≤ 0 (2) where 0 ≤ Mi ≤ 1, i = 1, . . . , m and ρi (t), i = 1, . . . , m are bounded, smooth, strictly positive, and decreasing functions satisfying limt →∞ ρi (t) = ρi∞ > 0, i = 1, . . . , m, called performance functions [4]. As (2) implies, only one set of the performance bounds is employed, and specifically the one associated with the sign of ei (0). The aforementioned statements are clearly illustrated in Fig. 1, for an exponential performance function ρi (t) = (ρi0 − ρi∞ ) exp (−li t) + ρi∞ , i = 1, . . . , m, with ρi0 , ρi∞ , li , i = 1, . . . , m strictly positive constants. The constant ρi0 = ρi (0), i = 1, . . . , m is selected such that (2) is satisfied at t = 0 (i.e., ρi (0) > ei (0)
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in case ei (0) ≥ 0 or ρi (0) > −ei (0) in case ei (0) ≤ 0). The constant ρi∞ = limt →∞ ρi (t) , i = 1, . . . , m represents the maximum allowable size of ei (t) at the steady state and can be set arbitrarily small to a value reflecting the resolution of the measurement device, thus achieving practical convergence of ei (t) to zero. Furthermore, the decreasing rate of ρi (t) , i = 1, . . . , m, which is related to the constant li , i = 1, . . . , m in this case, introduces a lower bound on the required speed of convergence of ei (t). Moreover, the maximum overshoot is prescribed less than Mi ρi (0) , i = 1, . . . , m, which may even become zero by setting Mi = 0, i = 1, . . . , m. Thus, the appropriate selection of the performance function ρi (t) , i = 1, . . . , m, as well as of the constant Mi , i = 1, . . . , m, imposes performance characteristics for the tracking error ei (t) , i = 1, . . . , m. To introduce prescribed performance, an error transformation is incorporated modulating the tracking error element ei (t) , i = 1, . . . , m with respect to the required performance bounds imposed by ρi (t), Mi , i = 1, . . . , m. More specifically, we define εi (t) = Ti
ei (t) , ρi (t)
i = 1, . . . , m
(3)
where εi (t) , i = 1, . . . , m is the transformed error and Ti (·) , i = 1, . . . , m is a smooth strictly increasing function defining an onto mapping Ti : (−Mi , 1) → (−∞, ∞), Ti : (−1, Mi ) → (−∞, ∞),
ei (0) ≥ 0 ei (0) ≤ 0
i = 1, . . . , m.
(4) For example, in case ei (0) ≥ 0, i = 1, . . . , m, a candidate transformation function could be Ti (ei (t)/ρi (t)) = ln((Mi + ei (t)/ρi (t))/(1 − ei (t)/ρi (t))), i = 1, . . . , m. Otherwise, in case ei (0) ≤ 0, we could select Ti (ei (t)/ρi (t)) = ln((1 + ei (t)/ρi (t))/(Mi − ei (t)/ρi (t))), i = 1, . . . , m. A graphical illustration of the transformation function for both cases is provided in Fig. 2. Moreover, as (4) implies and the aforementioned example clarifies, the choice of the mapping Ti (·) , i = 1, . . . , m, depends only on the sign of ei (0) , i = 1, . . . , m. Notice also that, since ρi (0) , i = 1, . . . , m is selected such that (2) is satisfied at t = 0, the initial condition of the transformed error εi (0) , i = 1, . . . , m is finite owing to (4). Furthermore, the case ei (0) = 0 clearly satisfies (2) at t = 0 with the choice Mi = 0, i = 1, . . . , m, since otherwise (i.e., Mi = 0) εi (0) , i = 1, . . . , m becomes infinite. Thus, in such a case the transformation function can be selected to satisfy either Ti : (−Mi , 1) → (−∞, ∞) , i = 1, . . . , m, or Ti : (−1, Mi ) → (−∞, ∞) , i = 1, . . . , m. Remark 1: Notice that the magnitude of εi (t) , i = 1, . . . , m does not affect the evolution of ei (t) , i = 1, . . . , m, which is solely prescribed by (2) and thus by the selection of the performance function ρi (t) , i = 1, . . . , m, as well as of the constant Mi , i = 1, . . . , m. In other words, if εi (t), i = 1, . . . , m is guaranteed bounded for all t ≥ 0, then prescribed performance for ei (t), i = 1, . . . , m is achieved ∀t ≥ 0.
ei(0) ≥ 0
εi
εi
0
0
−Mi Fig. 2.
ei(0) ≤ 0
0
ei/ρi
1
−1
0 Mi
ei/ρi
Graphical illustration of a transformation function.
III. P ROBLEM S TATEMENT In this paper, we shall consider uncertain MIMO systems of the form x˙ = f (x) + g(x)u () y = h(x) where x ∈ Rn is the state, y ∈ Rm is the measured output, u(t) ∈ Rq is the control input, h(x) ∈ Rm is an unknown continuous, and C 1 function, while f (x) ∈ Rn , g(x) ∈ Rn×q are locally Lipschitz unknown functions. The state x is assumed to be unavailable for measurement. Our goal is to design a controller utilizing only measurable signals, capable of regulating the output tracking error e(t) = y(t) − yr (t) to a neighborhood of zero with prescribed performance while guaranteeing the uniform boundedness of all other signals in the closed loop. For the desired output trajectory yr (t), we assume that it is bounded with bounded first-order time derivatives. More formally, our problem is defined as follows. Definition 2: The PPOFC problem for () is solvable if there exists a bounded and possibly parameterized dynamic output feedback continuous control law to render the states of () u.u.b with respect to some compact sets and the output tracking error elements ei (t), i = 1, . . . , m to satisfy λli ρi (t) ≤ ei (t) ≤ λui ρi (t), i = 1, . . . , m
∀t ≥ 0
(5)
for some constants λli , λui obeying −Mi < λli < λui < 1, in case ei (0) ≥ 0 −1 < λli < λui < Mi , in case ei (0) ≤ 0
(6) (7)
for all i = 1, . . . , m. To continue, we differentiate (3) with respect to time to obtain ρ˙i ε˙ i = ri y˙i − y˙ri − ei , i = 1, . . . , m ρi
KOSTARIGKA AND ROVITHAKIS: NEURAL NETWORK CONTROL OF UNCERTAIN MIMO NONLINEAR SYSTEMS
or in matrix form ∂h ε˙ = r x˙ − ν = r L f h(x) + L g h(x)u − ν ∂x
(8)
where ε = [ ε1 . . . εm ] , r = diag([ r1 . . . rm ]), ν = [ ν1 . . . νm ] , with ri = (1/ρi )∂ Ti /∂(ei /ρi ) and νi = y˙ri + ei ρ˙i /ρi , i = 1, . . . , m and consider the augmented system x˙ = f (x) + g(x)u . (a ) ε˙ = r L f h(x) + L g h(x)u − ν To pursue our problem, we shall consider continuous dynamic output feedback controllers of the form u = u(ξ, ε) ξ˙ = φ(ξ, ε),
ξ ∈R . n
(9) (10)
Incorporating the dynamic controller (9), (10) to (a ), we obtain the following augmented closed-loop system: ⎫ x˙ = f (x) + g(x)u(ξ, ε) ⎬ ξ˙ = φ(ξ, (ac ) ε) ⎭ ε˙ = r L f h(x) + L g h(x)u(ξ, ε) − ν which is defined over the set U = {x ∈ Rn , ξ ∈ Rn , ε ∈ Rm : h(x) − yr (t) ∈ Ee,t ∪ ∂Ee,t }, where Ee,t is the set defined for each t ≥ 0 via (2) and ∂Ee,t represents its boundary. Remark 2: The augmented closed loop system (ac ) is said to be initially well defined if (x(0), ξ(0), ε(0)) ∈ U . To guarantee such a property, we have to ensure that (2) is satisfied at t = 0. In this direction, notice that, for any given initial condition ξ(0) and any (unknown) initial state x(0), the corresponding initial output measurement y(0) leads to a measurable initial output tracking error value e(0) = y(0) − yr (0). However, given e(0) = [e1 (0) . . . em (0)] , we can always select ρi (0) and Mi , i = 1, . . . , m to satisfy (2) at t = 0, which, owing to (4), further leads to ε(0) = [ε1 (0) . . . εm (0)] ∈ L∞ , thus establishing an initially well defined (ac ). The following theorem introduces necessary and sufficient conditions for the solution of the PPOFC problem. Theorem 1: Consider the original uncertain system () along with the dynamic output feedback controller (9), (10) and the corresponding initially well-defined augmented closedloop system (ac ). The PPOFC problem for () admits a solution if and only if all states of (ac ) are u.u.b with respect to some arbitrary sets. Proof: (⇒) If the PPOFC problem for () admits a solution, then there exists an admissible continuous dynamic output feedback control input u of the form (9), (10) rendering the state (x, ξ ) of closed loop () u.u.b with respect to some arbitrary compact sets Ux , Uξ , respectively, and (5) holds true ∀t ≥ 0, for some unknown constants λli , λui , i = 1, . . . , m obeying (6) and (7). Employing (3)–(7) elementwise, we arrive at −∞ < Ti (λli ) ≤ εi (t) ≤ Ti (λui ) < ∞, i = 1, . . . , m ∀t ≥ 0. ]
(11)
Hence ε(t) = [ε1 . . . εm is u.u.b with respect to the set Uε ≡ {ε ∈ Rm : Ti (λli ) ≤ εi (t) ≤ Ti (λui ), i = 1, . . . , m}, thereby proving the u.u.b of all states of (ac ).
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(⇐) As noticed earlier in Remark 2, we can always select ρi (0), λli , λui , i = 1, . . . , m such that (ac ) is initially well defined (i.e., (x(0), ξ(0), ε(0)) ∈ U ). Furthermore, there exists an admissible continuous dynamic output feedback control law to guarantee that x(t), ξ(t), ε(t) are u.u.b. with respect to the compact sets Ux , Uξ , Uε , respectively. From the u.u.b of ε(t) and owing to (4), there exist some unknown constants εi , εi , i = 1, . . . , m such that εi ≤ εi (t) ≤ εi , i = 1, . . . , m for all t ≥ 0. Utilizing the inverse transformation Ti−1 (·), i = 1, . . . , m which is well defined as ε ∈ L∞ , we arrive at (5) with λli = Ti−1 (εi ), λui = Ti−1 (εi ), i = 1, . . . , m. Therefore, as the states x, ξ of the closed loop system () are u.u.b. and (5) holds, through a bounded and continuous dynamic output feedback controller, the PPOFC problem for () admits a solution.
IV. M AIN R ESULTS Let us assume that the PPOFC problem admits a solution for the original uncertain system (). Owing to Theorem 1, this means that there exists a continuous dynamic output feedback controller u 0 (ξ, ε), ξ˙ = φ0 (ξ, ε), such that the states x, ξ, ε of the augmented closed loop system (ac ) are u.u.b with respect to some arbitrary sets Ux , Uξ , Uε , respectively. Therefore, there exists a Lyapunov function V (x, ξ, ε) to satisfy Theorem II. To proceed, we consider (with no loss of generality) the decomposition V (x, ξ, ε) = Va (x) + Vb (ξ, ε) and obtain for the Lyapunov function derivative the relation V˙ (x, ξ, ε) = L f Va (x) + L g Va (x)u 0 (ξ, ε) ∂ Vb (ξ, ε) ∂ Vb (ξ, ε) + φ0 (ξ, ε) + r ∂ξ ∂ε × L f h(x) + L g h(x)u 0 (ξ, ε) − ν ≤ −ηx (|x|) − ηξ (|ξ |) − ηε (|ε|) ∀ (x, ξ, ε) ∈ U /UX
(12)
where UX ≡ Ux × Uξ × Uε and ηx (·), ηξ (·), ηε (·) are some K∞ −functions. Notice that UX ⊂ U . This is true because of the fact that, whenever (x, ξ, ε) ∈ UX , we have ε ∈ L∞ . Therefore invoking (4) we conclude that h(x) − yr (t) ∈ Ee,t ∀(x, ξ, ε) ∈ UX , which means that UX is a subset of U . Moreover, differentiating V (x, ξ, ε) along the solutions of (ac ) yields ∂ Vb (ξ, ε) ˙ ∂ Vb (ξ, ε) ∂ Va (x) x˙ + ε˙ V˙ (x, ξ, ε) = ξ+ ∂x ∂ξ ∂ε ∂ Vb (ξ, ε) = L f Va (x) + L g Va (x)u + φ(ξ, ε) ∂ξ ∂ Vb (ξ, ε) r L f h(x) + L g h(x)u − ν . + ∂ε Adding and subtracting
∂ Vb (ξ, ε) ∂ Vb (ξ, ε) r L g h(x) u 0 + φ0 (ξ, ε) L g Va (x) + ∂ε ∂ξ
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we obtain
A. Dynamic Controller Design
V˙ (x, ξ, ε) = L f Va (x) + L g Va (x)u 0 + +
∂ Vb (ξ, ε)
∂ Vb (ξ, ε)
∂ξ
φ0 (ξ, ε)
∂ε
G(x, ξ, ε) = L g Va (x) +
∂ Vb (ξ, ε) ∂ε
r L g h(x)
∂ Vb (ξ, ε) φ0 (ξ, ε) ∂ξ ∂ Vb (ξ, ε) + r L f h(x) + L g h(x)u 0 − ν . ∂ε
I(x, ξ, ε, r, ν) = L f Va (x)+ L g Va (x)u 0 +
However, owing to (12) I(x, ξ, ε, r, ν) ≤ −
u = Wˆ u Z u (ξ, ε) ξ˙ = φˆ o (ξ, ε, Wˆ o ) + φσ
r L f h(x) + L g h(x)u 0 − ν
∂ Vb (ξ, ε) r L g h(x) (u − u 0 ) + L g Va (x) + ∂ε ∂ Vb (ξ, ε) + (φ(ξ, ε) − φ0 (ξ, ε)) ∂ξ = I(x, ξ, ε, r, ν) + G(x, ξ, ε)(u − u 0 ) ∂ Vb (ξ, ε) + (φ(ξ, ε) − φ0 (ξ, ε)) ∂ξ where
The control input is designed as follows:
ηi (| i |) ∀ (x, ξ, ε) ∈ UX .
i=x,ξ,ε
Therefore V˙ (x, ξ, ε) becomes ηi (| i |) + G(x, ξ, ε)(u − u 0 ) V˙ (x, ξ, ε) ≤ − i=x,ξ,ε
∂ Vb (ξ, ε) + (φ(ξ, ε) − φ0 (ξ, ε)) , ∀ (x, ξ, ε) ∈ U /UX . ∂ξ (13) Because they are unknown, the terms u 0 (ξ, ε), ∂ Vb (ξ, ε)/∂ξ , and φ0 (ξ, ε) shall be approximated by suitable neural network structures. In this direction, by applying the neural network density property we conclude the existence of constant but unknown weight vectors Wu∗ ∈ R L u ×q , Wν∗ ∈ R L ν , Wo∗ ∈ R L o ×n , continuous regressor terms Z u (ξ, ε) ∈ R L u , Z ν (ξ, ε) ∈ R L ν ×n , Z o (ξ, ε) ∈ R L o and ωu (ξ, ε), ων (ξ, ε), ωo (ξ, ε), denoting the approximation errors such that u 0 (ξ, ε) = Wu∗ Z u (ξ, ε) + ωu (ξ, ε) ∈ Rq ∂ Vb (ξ, ε) (ξ, ε) = Wν∗ Z ν (ξ, ε) + ων (ξ, ε) ∈ R1×n ∂ξ φ0 (ξ, ε) = Wo∗ Z o (ξ, ε) + ωo (ξ, ε) ∈ Rn (14) ∀ (ξ, ε) ∈ ac ≡ ξ × ε ⊂ Rn × Rm where ξ , ε are arbitrary compact sets. Such a substitution is possible because of the continuity and the boundedness of u 0 (ξ, ε), ∂ Vb (ξ, ε)/∂ξ and φ0 (ξ, ε) ∀ (ξ, ε) ∈ ac . Moreover, on the generic compact set ac , the approximation errors can be suitably bounded as |ωu (ξ, ε)| ≤ ω¯ u , |ων (ξ, ε)| ≤ ω¯ ν , |ωo (ξ, ε)| ≤ ω¯ o , where ω¯ u , ω¯ ν , and ω¯ o are some unknown bounds.
∈ Rq
(15)
∈R
(16)
n
with φσ switching according to a switching strategy σ (t), taking values in {I, II} with a(t) + γ (|ε|, |ξ |) + δ u (17) φI = −b ξ, ε, Wˆ ν 2 ˆ |b(ξ, ε, Wν )| a(t) + γ (|ε|, |ξ |) + δ u (18) φII = −b ξ, ε, Wˆ ν δb2 and φˆo (ξ, ε, Wˆ o ) = Wˆ o Z o (ξ, ε) ∈ Rn b(ξ, ε, Wˆ ν ) = Wˆ ν Z ν (ξ, ε) ∈ R1×n .
(19) (20)
In (17)–(20), γ (|ε|, |ξ |) = γξ (|ξ |) + γε (|ε|), where γξ (·), γε (·) are K−functions, δu , δb > 0 are design constants, while Wˆ u , Wˆ o , Wˆ ν are estimates of Wu∗ , Wo∗ , Wν∗ , defined in (14), which are generated through the update laws W˙ˆ u = −Wˆ u + Z u (ξ, ε) · K u ˙ Wˆ o = −Wˆ o − Z o (ξ, ε) · b(ξ, ε, Wˆ ν ) W˙ˆ ν = −Wˆ ν + Z ν (ξ, ε) · φσ (t)
(21) (22) (23)
where in (21) K u ∈ R1×q is a constant matrix. Furthermore, a(t) is generated by a(t) ˙ = −a(t) − ϕa ( j )
(24)
where ϕa ( j ) is a strictly increasing sequence satisfying lim j →∞ ϕa ( j ) → ∞ with the index j = 0, 1, 2, . . . , denoting the number of times the following equality is satisfied: H = a(t) + γ (|ε|, |ξ |) +
δu
c j +1 a(t) + γ (|ε|, |ξ |) + δu −|b(ξ, ε, Wˆ ν )|2 = 0 (25) δb2
while σ (t) = II with c > 1. Moreover, ϕa ( j ) = 0 when σ (t) = I. Before we proceed and analyze the proposed control law, let us first present the switching strategy σ (t). Switching strategy: We define the sets R1 = (ξ, ε, Wˆ ν ) : |b(ξ, ε, Wˆ ν )| ≤ δb R2 = (ξ, ε, Wˆ ν ) : |b(ξ, ε, Wˆ ν )| > δb . We initiate at (ξ(0), ε(0), Wˆ ν (0)) ∈ R2 and allocate σ (t) = I whenever (ξ, ε, Wˆ ν ) ∈ R2 . The switching variable σ (t) remains in that state until (ξ, ε, Wˆ ν ) ∈ R1 , at which time σ (t) = II. Consequently, σ (t) remains unaltered until (ξ, ε, Wˆ ν ) ∈ R2 . Summarizing, the switching strategy is defined as follows: I, if (ξ, ε, Wˆ ν ) ∈ R2 (26) σ (t) = II, if (ξ, ε, Wˆ ν ) ∈ R1 . To continue, let us first define the set X ≡ x × ac ⊂ U ⊂ Rn × Rn × Rm , where ac ⊂ Rn × Rm is the compact
KOSTARIGKA AND ROVITHAKIS: NEURAL NETWORK CONTROL OF UNCERTAIN MIMO NONLINEAR SYSTEMS
set where the neural network approximation capabilities hold and x ⊂ Rn is a compact set. It is reasonable to define X ⊆ U , as outside of U , (ac ) does not exist. Lemma 1: The controller (15)–(26) guarantees ∀(x, ξ, ε) ∈ X : 1) H ≤ 0 whenever φ = φII , and 2) the sequence ϕa ( j ) is bounded. Proof: 1) Following the switching strategy, at t = 0 we start and operate with φ = φI until |b(ξ, ε, Wˆ ν )| = δb , at which time we switch from φI to φII . Then (25) becomes H = −δu + (δu /c j +1) < 0 since c j +1 is a strictly increasing sequence of j, with c > 1. Hence, it is guaranteed that H < 0 when we first operate with φII . To continue, let T j denote the corresponding time instants (25) is satisfied, [i.e., T1 is the time instant (25) is satisfied for the first time, T2 for the second time, etc.]. Notice that H remains negative ∀t ∈ [T j −1 , T j ) and for all j, until the time instant T j , when H = 0. Hence, at that time, (25) yields |b(ξ, ε, Wˆ ν )|2 (a(t) + γ (|ε|, |ξ |) + δu ) + a(t) + γ (|ε|, |ξ |) δb2 δu (27) = − j +1 . c However, immediately after, at t = T j+ , the value of c j +1 increases to [c j +1]n (i.e., [c j +1]n > c j +1 ) as j increases by 1. Thus, H (T j+ ) = −(δu /c j +1 ) + (δu /[c j +1)]n < 0. Therefore H ≤ 0 whenever φ = φII . 2) We shall prove the second part of this lemma by contradiction. Let us assume that j can be infinite. Assuming that lim j →∞ a(t) = +∞, then since lim j →∞ ϕa ( j ) = +∞, we ˙ = −∞, which contradicts obtain from (24) that lim j →∞ a(t) the supposition. Hence, a(t) either tends to −∞ or it remains bounded. To two cases. proceed, we distinguish Case 1 lim j →∞ a(t) = −∞ : We have proved in the first part of this lemma that H < 0 whenever |b(ξ, ε, Wˆ ν )| = δb . Hence, owing to the continuity of H with respect to |b(ξ, ε, Wˆ ν )|2 , we conclude that there exists a sufficiently small positive constant such that H < 0, ∀ |b(ξ, ε, Wˆ ν )|2 ≥ δb2 −, thus formulating a no-event zone. Alternatively, H = 0 (while φ = φII ) may occur only when −
1−
|b(ξ, ε, Wˆ ν )|2 > 2 > 0. 2 δb δb
(28)
Hence, from (25), lim j →∞ H = −∞, which is a contradiction since H = 0 whenever criterion (25) is satisfied. Case 2 (a(t) ∈ L∞ ): Since lim j →∞ ϕa ( j ) = +∞, (24) yields lim j →∞ a(t) ˙ = −∞. Notice further that |a(t)| + γ (|ε|, |ξ |) + δu . (29) δb and γ (|ε|, |ξ |) ∈ K, (29) yields φII ∈ L∞ .
|φII (t)| ≤ Since a(t) ∈ L∞ To continue
˙ ε, Wˆ ν )=W˙ˆ ν Z ν (ξ, ε)+ Wˆ ν ∂ Z ν (ξ, ε) ξ˙ + Wˆ ν ∂ Z ν (ξ, ε) ε˙ b(ξ, ∂ξ ∂ε ∂ Z ν (ξ, ε) =W˙ˆ ν Z ν (ξ, ε) + Wˆ ν ∂ξ ∂ Z ν (ξ, ε) ε˙ . (30) × Wˆ o Z o (ξ, ε)+φII + Wˆ ν ∂ε
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Notice that (21) and (23) are in the bounded-input boundedoutput (BIBO) form with Z u (ξ, ε), Z ν (ξ, ε) bounded by construction and therefore Wˆ u , Wˆ ν ∈ L∞ , which in turn proves that u(ξ, ε), b(ξ, ε, Wˆ ν ) ∈ L∞ . Moreover, since Z o (ξ, ε) is bounded by construction, we conclude that (22) is also in the BIBO form leading to Wˆ o ∈ L∞ . Furthermore, (23) yields W˙ˆ ν ∈ L∞ . From (8), we have (31) ε˙ = r L f h(x) + L g h(x)u(ξ, ε) − ν . By assumption L f h(x), L g h(x) ∈ L∞ , ∀x ∈ x , where x is an arbitrary compact set. Further, ρi , ∂ Ti /∂(ei /ρi ) (∂ Z ν (ξ, ε)/∂ξ ), (∂ Z ν (ξ, ε)/∂ε) ∈ L∞ by construction. Hence, r, ν ∈ L∞ ∀(ξ, ε) ∈ ac . Therefore, from (30) we ˙ ε, Wˆ ν ) ∈ L∞ , ∀(ξ, ε) ∈ ac . Differentiating (25) have b(ξ, with respect to time, we obtain 2 |b| H˙ = 1 − 2 (a(t) ˙ + γ˙ (|ε|, |ξ |)) δb b b˙ (32) −2 2 a(t) + γ (|ε|, |ξ |) + δu . δb To continue, since H = 0 (while φ = φII ) may occur only outside the no event zone where (28) holds, it can be easily concluded from (32) that lim j →∞ H˙ = −∞, which contradicts the fact that H < 0 and tends to zero whenever (25) occurs, while φ = φII . Remark 3: Lemma 1 proves that H ≤ 0 whenever φσ = φII and that j is piecewise continuous with respect to time and finite. Hence, the solutions of (16), (23), and (24) exist and are continuous whenever σ (t) = II. The same triviality holds when σ (t) = I as well. The above stated arguments, together with the fact that φI = φII whenever |b(ξ, ε, Wˆ ν )| = δb , lead to the continuity of φσ . As a direct consequence, the dynamic output feedback controller (15)–(26) is continuous, guaranteeing [24] the existence and uniqueness of the solutions of (ac ) and consequently of (). Lemma 2: The continuous dynamic controller (15)–(26) guarantees, ∀ (ξ, ε) ∈ ac that Wˆ u , Wˆ ν , Wˆ o , a(t), and u(t) ∈ L∞ . Proof: From Lemma 1, ϕa is bounded. Therefore a(t) ∈ L∞ as well, since it is generated by the BIBO differential equation (24). Moreover Z u (ξ, ε) is bounded by construction. Hence, since (21) is in a BIBO Form, we obtain Wˆ u ∈ L∞ , which in turn proves that u(ξ, ε) ∈ L∞ . Furthermore γ (|ε|, |ξ |) is bounded by construction. To proceed, notice that whenever σ (t) = I, we have |b(ξ, ε, Wˆ ν )| ≥ δb and |φI (t)| ≤
|a(t)| + γ (|ε|, |ξ |) + δu . |b(ξ, ε, Wˆ ν )|
(33)
From the aforementioned analysis, we straightforwardly conclude that |φI (t)| ∈ L∞ . However, whenever σ (t) = II, we have |b(ξ, ε, Wˆ ν )| < δb and |φII (t)| ≤
|a(t)| + γ (|ε|, |ξ |) + δu . δb
(34)
Hence, |φII (t)| ∈ L∞ . Summarizing, |φσ (t)| ∈ L∞ , ∀t ≥ 0. Moreover, since Z ν (ξ, ε) is bounded by construction, we conclude that (23) is in BIBO form and thus that Wˆ ν ∈ L∞ ,
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which in turn proves that b(ξ, ε, Wˆ ν ) ∈ L∞ . Finally, since Z o (ξ, ε) is bounded by construction, we conclude that (22) is also in BIBO form, leading to Wˆ o ∈ L∞ .
L˙ becomes L˙ ≤ −
ηi (| i |) + G(x, ξ, ε) · W˜ u Z u (ξ, ε)
i=x,ξ,ε
+ |G(x, ξ, ε)|2 + tr W˜ u W˙˜ u + W˜ o W˙˜ o + W˜ ν W˙˜ ν
B. Output Feedback Stabilization Analysis Let us consider the augmented Lyapunov-like function L(x, ξ, ε, W˜ u , W˜ ν , W˜ o ) = V (x, ξ, ε) 1 1 + tr |W˜ u |2 + |W˜ o |2 + |W˜ ν |2 (35) 2 2 where V (x, ξ, ε) is the Lyapunov function defined earlier and W˜ u , W˜ ν , W˜ o are parameter errors defined as W˜ u = Wˆ u − Wu∗ , W˜ ν = Wˆ ν − Wν∗ , W˜ o = Wˆ o − Wo∗ . The properties of the proposed controller are summarized in the next theorem. Theorem 3: Consider the uncertain system () for which the PPOFC problem admits a solution, the augmented closed loop system (ac ), and the dynamic output feedback control law (15)–(26). If there exist a positive constant ζ¯u and appropriate positive definite functions η¯ x (·), η¯ ξ (·), η¯ ε (·), such that 2 ∂ Va (x) ∂ Vb (ξ, ε) − g(x)+ r L g h(x) ηi (| i |) + ζ¯u ∂x ∂ε i=x,ξ,ε ≤− η¯ i (| i |) (36) i=x,ξ,ε
with V (x, ξ, ε) = Va (x) + Vb (ξ, ε) being an unknown control Lyapunov function for (ac ) that satisfies (12), then the controller (15)–(26) solves the PPOFC problem for (), ∀(x, ξ, ε) ∈ X ⊆ U . Proof: According to Theorem 1, we need to establish u.u.b of (x, ξ, ε) with respect to some arbitrary sets. In that respect, we distinguish two cases. Case 1 (σ (t) = I): Differentiating (35) along the solutions of (ac ) and using (13) and (14), we obtain (A), shown at the bottom of the page. Employing the fact that −G(x, ξ, ε) · ωu (ξ, ε) ≤ |G(x, ξ, ε)| · |ωu (ξ, ε)| ≤ |G(x, ξ, ε)|2 +
1 |ωu (ξ, ε)|2 4
−W˜ ν + Wˆ ν Z ν (ξ, ε)·φI (t)Z ν (ξ, ε) · φI (t) +Wˆ ν Z ν (ξ, ε) · W˜ o Z o (ξ, ε) + WI where 1 |ωu (ξ, ε)|2 − Wˆ ν Z ν (ξ, ε) · ωo (ξ, ε) + ων (ξ, ε) · φI (t) 4 − W˜ ν Z ν (ξ, ε) − ων (ξ, ε) W˜ o Z o (ξ, ε) − ωo (ξ, ε) ≤
1 2 ω¯ + |W¯ ν | ζν ω¯ o + ω¯ ν φ¯ I + W¯ ν + |Wν∗ | ζν + ω¯ ν 4 u × W¯ o + |Wo∗ | ζo + ω¯ o = WI . (37)
In (37), ζi , i = u, ν, o are the constant unknown upper bounds of the sigmoid terms Z i (ξ, ε), i = u, ν, o, respectively, which are bounded by construction, and W¯ i , i = u, ν, o are the unknown upper bounds of the weights Wˆ i , i = u, ν, o, bounded because of Lemma 2. Moreover, φ¯I is an upper bound of the switching term φI , also proved to be bounded in Lemma 2. Finally, owing to the neural network density property, whenever (ξ, ε) ∈ ac , the approximation errors are upper bounded by unknown but small bounds as follows: |ωu (ξ, ε)| ≤ ω¯ u , |ων (ξ, ε)| ≤ ω¯ ν , |ωo (ξ, ε)| ≤ ω¯ o . Using (20)–(23) we obtain (B), shown at the top of the following page. After using the identities W˜ Wˆ = (1/2)|W˜ |2 + (1/2)|Wˆ |2 − (1/2)|W ∗ |2 , tr{W˜ Wˆ } = (1/2) |W˜ |2 + (1/2)|Wˆ |2 − (1/2)|W ∗ |2 , and the trace property tr{yx } = x y, x, y ∈ Rn , we get ηi (| i |) + G(x, ξ, ε) · W˜ u Z u (ξ, ε) + |G(x, ξ, ε)|2 L˙ ≤ − i=x,ξ,ε
−
1 ˜ 2 |Wi | + |Wˆ i |2 − |Wi∗ |2 2
i=u,ν,o +K u · W˜ u Z u (ξ, ε)
+ b(ξ, ε, Wˆ ν ) · φI (t) + WI .
⎫ (φ(ξ, ε) − φ0 (ξ, ε)) + tr W˜ u W˙˜ u + W˜ o W˙˜ o + W˜ ν W˙˜ ν ⎪ ⎪ ⎪ ⎪ i=x,ξ,ε ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ˙ ˙ ∗ ⎪ ˆ ˜ ˜ ˜ ˜ ≤− ηi (| i |) + G(x, ξ, ε) Wu Z u (ξ, ε) − Wu Z u (ξ, ε) − ωu (ξ, ε) + tr Wu Wu + Wo Wo ⎪ ⎪ ⎪ i=x,ξ,ε ⎪ ⎪ ⎪ ⎪ ˙ ∗ ∗ ⎪ +W˜ ν W˜ ν + Wν Z ν (ξ, ε) + ων (ξ, ε) Wˆ o Z o (ξ, ε) + φI (t) − Wo Z o (ξ, ε) − ωo (ξ, ε) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ˙ ˙ ˙ ηi (| i |) + G(x, ξ, ε) W˜ u Z u (ξ, ε) − ωu (ξ, ε) + tr W˜ u W˜ u + W˜ o W˜ o + W˜ ν W˜ ν ≤− . i=x,ξ,ε ⎪ ⎪ ⎪ ⎪ ⎪ + (Wˆ ν − W˜ ν ) Z ν (ξ, ε) + ων (ξ, ε) W˜ o Z o (ξ, ε) + φI (t) − ωo (ξ, ε) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ˙ ˙ ˙ ⎪ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ⎪ ≤− ηi (| i |) + G(x, ξ, ε) Wu Z u (ξ, ε) − ωu (ξ, ε) + tr Wu Wu + Wo Wo + Wν Wν ⎪ ⎪ ⎪ i=x,ξ,ε ⎪ ⎪ ⎪ ⎪ +Wˆ ν Z ν (ξ, ε) · W˜ o Z o (ξ, ε) + Wˆ ν Z ν (ξ, ε) · φI (t) − Wˆ ν Z ν (ξ, ε) · ωo (ξ, ε) − W˜ ν Z ν (ξ, ε) · φI (t) ⎪ ⎪ ⎪ ⎪ ⎭ ˜ ˜ +ων (ξ, ε) · φI (t) − Wν Z ν (ξ, ε) − ων (ξ, ε) Wo Z o (ξ, ε) − ωo (ξ, ε)
L˙ ≤ −
ηi (| i |) + G(x, ξ, ε)(u − u 0 ) +
∂ Vb (ξ,ε) ∂ξ
(A)
KOSTARIGKA AND ROVITHAKIS: NEURAL NETWORK CONTROL OF UNCERTAIN MIMO NONLINEAR SYSTEMS
L˙ ≤ −
i=x,ξ,ε
ηi (| i |) + G(x, ξ, ε) · W˜ u Z u (ξ, ε) + |G(x, ξ, ε)|2 + tr W˜ u W˙˜ u + W˜ o W˙˜ o + W˜ ν W˙˜ ν
+ b(ξ, ε, Wˆ ν ) · W˜ o Z o (ξ, ε) + b(ξ, ε, Wˆ ν ) · φI (t) − W˜ ν Z ν (ξ, ε) · φI (t) + WI ≤− ηi (| i |) + G(x, ξ, ε) · W˜ u Z u (ξ, ε) + |G(x, ξ, ε)|2 + tr W˜ u −Wˆ u + Z u (ξ, ε) · K u i=x,ξ,ε +tr W˜ o −Wˆ o − Z o (ξ, ε) · b(ξ, ε, Wˆ ν ) + W˜ ν −Wˆ ν + Z ν (ξ, ε) · φI (t) + b(ξ, ε, Wˆ ν ) · W˜ o Z o (ξ, ε) + b(ξ, ε, Wˆ ν ) · φI (t) − W˜ ν Z ν (ξ, ε) · φI (t) + WI
145
⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2 ˜ ˜ ˜ ˜ ˆ ˆ ˆ ⎪ ≤− ηi (| i |) + G(x, ξ, ε) · Wu Z u (ξ, ε) + |G(x, ξ, ε)| − tr Wu Wu − tr Wo Wo − Wν Wν ⎪ ⎪ ⎪ ⎪ i=x,ξ,ε ⎪ ⎪ ⎪ ⎪ ⎪ + tr W˜ u Z u (ξ, ε) · K u − tr W˜ o Z o (ξ, ε) · b(ξ, ε, Wˆ ν ) + W˜ ν Z ν (ξ, ε) · φI (t) ⎪ ⎪ ⎪ ⎪ ⎭ + b(ξ, ε, Wˆ ν ) · W˜ o Z o (ξ, ε) + b(ξ, ε, Wˆ ν ) · φI (t) − W˜ ν Z ν (ξ, ε) · φI (t) + WI Moreover, using the inequalities G(x, ξ, ε)· W˜ u Z u (ξ, ε) ≤ |G(x, ξ, ε)| · |W˜ u | · |Z u (ξ, ε)| 2 |W˜ u | ≤ |G(x, ξ, ε)| ·|Z u (ξ, ε)| + 4 K u · W˜ u Z u (ξ, ε) ≤ |K u | · |W˜ u | · |Z u (ξ, ε)| 2
2
2 |W˜ u | ≤ |K u | · |Z u (ξ, ε)| + 4 2
we have L˙ ≤ −
2
ηi (| i |) + |G(x, ξ, ε)|2 |Z u (ξ, ε)|2 + 1
i=x,ξ,ε
1 ˜ 2 |Wi | + |Wˆ i |2 − |Wi∗ |2 − 2 i=ν,o 1 ˜ 2 − |Wu | + |Wˆ u |2 − |Wu∗ |2 + |K u |2 |Z u (ξ, ε)|2 2 2 |W˜ u | + + b(ξ, ε, Wˆ ν ) · φI (t) + WI . 2 Dropping the negative terms −(1/2) i=ν,o (|W˜ i |2 +|Wˆ i |2 ), −(1/2)|Wˆ u |2 , we obtain L˙ ≤ − ηi (| i |) + |G(x, ξ, ε)|2 |Z u (ξ, ε)|2 + 1 i=x,ξ,ε
+ b(ξ, ε, Wˆ ν ) · φI (t) + μ¯ I where
1 ∗ 2 |Wi | WI + |K u |2 |Z u (ξ, ε)|2 + 2 i=u,ν,o 1 ∗ 2 |Wi | = μ¯ I . ≤ WI + |K u |2 ζu2 + 2
i=x,ξ,ε
a(t) + γ (|ε|, |ξ |) + δu + μ¯ I −|b(ξ, ε, Wˆ ν )|2 |b(ξ, ε, Wˆ ν )|2 ≤− ηi (| i |) + |G(x, ξ, ε)|2 |Z u (ξ, ε)|2 + 1 i=x,ξ,ε
−a(t) − γ (|ε|, |ξ |) − δu + μ¯ I .
(B)
Finally, dropping the negative term −δu and exploiting Lemma 1 to guarantee the boundedness of a(t), we arrive at L˙ ≤ − ηi (| i |) − γ (|ε|, |ξ |) i=x,ξ,ε
+ |G(x, ξ, ε)|2 |Z u (ξ, ε)|2 + 1 + μI
(38)
where μI = μ¯ I + sup {a(t)} . Case 2 (σ (t) = II): Using the weight update laws (21)–(23) with σ = II, following similar steps as in Case 1 and after adding and subtracting a(t), we arrive at ηi (| i |) + |G(x, ξ, ε)|2 |Z u (ξ, ε)|2 + 1 L˙ ≤ − i=x,ξ,ε
+b(ξ, ε, Wˆ ν ) · φII (t) + a(t) + μII 2 2 where μ II ∗ 2= WII + |K u | ζu + sup {a(t)} + (1/2) i=u,ν,o |Wi | , with
1 |ωu (ξ, ε)|2 − Wˆ ν Z ν (ξ, ε) · ωo (ξ, ε) + ων (ξ, ε) · φI (t) 4 − W˜ ν Z ν (ξ, ε) − ων (ξ, ε) W˜ o Z o (ξ, ε) − ωo (ξ, ε) ≤
1 2 ω¯ u + |Wˆ ν | ζν ω¯ o + ω¯ ν φ¯ II + W¯ ν + |Wν∗ | ζν + ω¯ ν 4 × W¯ o + |Wo∗ | ζo + ω¯ o = WII . (39)
Substituting φII from (18), we obtain L˙ ≤ − ηi (| i |) + |G(x, ξ, ε)|2 |Z u (ξ, ε)|2 + 1 i=x,ξ,ε
−|b(ξ, ε, Wˆ ν )|2
i=u,ν,o
Substituting φI from (17), we get L˙ ≤ − ηi (| i |) + |G(x, ξ, ε)|2 |Z u (ξ, ε)|2 + 1
.
a(t) + γ (|ε|, |ξ |) + δu + a(t) + μII . δb2
However, owing to (27) −|b(ξ, ε, Wˆ ν )|2 leading to L˙ ≤ −
a(t) + γ (|ε|, |ξ |) + δu + a(t) ≤ −γ (|ε|, |ξ |) δb2
ηi (| i |) − γ (|ε|, |ξ |)
i=x,ξ,ε
+|G(x, ξ, ε)|2 |Z u (ξ, ε)|2 + 1 + μII .
(40)
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2
q˙1(t) (rad/s)
q1(t)/qr1(t)(rad)
2
1.5
1
0
2
4 6 time (sec)
8
0
2
4 6 time (sec)
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0
2
4 6 time (sec)
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2
4 6 time (sec)
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0
2
4 6 time (sec)
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10
1 0
2
q˙2(t) (rad/s)
q2(t)/qr2(t)(rad)
0
−1
10
2.5
1.5
1
−1 −2
0
2
4 6 time (sec)
8
−3
10
3
2
2.5
1 ξ2(t)
ξ1(t)
1
2
0 1.5 2
4 6 time (sec)
8
3
1
2.5
0
2
1
−1 −2
1.5
Fig. 3.
−1
10
ξ4(t)
ξ3(t)
0
0
2
4 6 time (sec)
8
10
−3
System response (solid line: state trajectories; dashed line: reference trajectories).
Using (38) and (40), we conclude that in both cases ηi (| i |) − γ (|ε|, |ξ |) + ζ¯u |G(x, ξ, ε)|2 + μ (41) L˙ ≤ − i=x,ξ,ε
where μ = max (μI , μII ) and |Z u (ξ, ε)|2 + 1 ≤ ζu2 + 1 = ζ¯u . Employing (36) and recalling the definition of γ (|ε|, |ξ |) = γξ (|ξ |) + γε (|ε|), L˙ finally becomes L˙ ≤ − η¯ i (| i |) − γ (|ε|, |ξ |) + μ i=x,ξ,ε
≤ −η¯ x (|x|) − γξ (|ξ |) − γε (|ε|) + μ. −1 Hence, L˙ ≤ 0 whenever |x(t)| > η¯ −1 x (μ) , |ξ(t)| > γξ (μ), −1 or |ε(t)| > γε (μ) , from which we conclude the u.u.b of
x, ξ, ε with respect to the sets Ex = x ∈ Rn : |x(t)| ≤ η¯ −1 x (μ) Eξ = ξ ∈ Rn : |ξ(t)| ≤ γξ−1 (μ) Eε = ε ∈ R : |ε(t)| ≤ γε−1 (μ)
(42) (43) (44)
respectively. The results presented are valid ∀(x, ξ, ε) ∈ X ⊆ U , where X ≡ x × ac with ac ⊂ Rn × Rm the compact set in which the approximation capabilities of the linear-in-the-weights neural networks hold and x ⊂ Rn is a compact set. It is therefore mandatory to prove that, under certain achievable conditions, the aforementioned
KOSTARIGKA AND ROVITHAKIS: NEURAL NETWORK CONTROL OF UNCERTAIN MIMO NONLINEAR SYSTEMS
0
7 × 10−3
−0.4
0
6
−0.6
−5
5
−0.8
−10 6
−1 0
1
2
3
6.5
4 5 6 time (sec)
7
7
7.5
8
8
9
10
ˆ ˆ ˆ |Wu|,|Wo|,|Wν|
e1(t)(rad)
−0.2
1.5
4 3 2 1
10
e2(t)(rad)
147
× 10−3
0
1
0
1
2
3
4
5
0.5
0 6
6.5
7
7.5
8
8
9
5 6 time (sec)
7
8
9
10
Fig. 5. Evolution of the Euclidean neural network weight norms (solid line: | Wˆ u |, dashed line: |Wˆ o |, dotted line: |Wˆ ν |).
0 0
1
2
3
4 5 6 time (sec)
7
10
Fig. 4. Output tracking errors along with their corresponding performance bounds. The subplots represent details at the steady state (solid lines: tracking errors; dashed line: prescribed output error bounds).
¯ = property holds ∀t ≥ 0. To proceed, we define {x, ξ, ε, W˜ u , W˜ ν , W˜ o : L(x, ξ, ε, W˜ u , W˜ ν , W˜ o ) ≤ cω }, where the constant cω satisfies cω ≥ cω + μ > cω with cω being the largest positive constant for which cω (V ) ⊆ X , where cω (V ) is a compact set defined as cω (V ) ≡ {(x, ξ, ε) ∈ Rn × Rn × Rm : V (x, ξ, ε) ≤ cω } . Given that EW is the u.u.b region of the neural network weights (W˜ u , W˜ ν , W˜ o ), proved to be bounded in Lemma 2, it can ¯ since in such a be easily verified that cω (V ) × EW ⊂ , case L(x, ξ, ε, W˜ u , W˜ ν , W˜ o ) ≤ cω , ∀(x, ξ, ε, W˜ u , W˜ ν , W˜ o ) ∈ cω (V )×EW . From the proof of Lemma 2 it can be seen that, whenever (ξ, ε) ∈ ac , for any selection of δu , ϕa , γξ (·), γε (·) we can always choose δb to upper bound |φσ (t)| and thus to keep μ constant. To continue, notice that the control functions γξ (·), γε (·) can be chosen to render the sets Eξ , Eε arbitrarily small and therefore to ensure that Eξ ×Eε ⊂ ac . Furthermore, the K∞ −function η¯ x (·), introduced through (12) and (36), should lead to Ex ⊂ x . Hence, X ≡ Ex × Eξ × Eε ⊆ U . As ¯ thus a consequence X ⊂ cω (V ) ⊆ X and X × EW ⊂ , ¯ Therefore, for establishing ≡ (cω (V ) ∪ X ) × EW ⊂ . ¯ it follows any (x(0), ξ(0), ε(0), W˜ u (0), W˜ ν (0), W˜ o (0)) ∈ , that L is bounded from above by cω since L˙ ≤ 0 in the set . As a result (x, ξ, ε, W ˜ u , W˜ ν , W˜ o ) ∈ , ¯ ¯ ∀t ≥ 0. Hence, / (x, ξ, ε) cannot escape X and consequently U , at any time. Remark 4: Inequality (36) is a sufficient stabilizability condition for (ac ). It practically states that the continuous dynamic output feedback controller u 0 (ξ, ε), ξ˙ = φ0 (ξ, ε), which guarantees that the states of (ac ) are u.u.b. with respect to some sets, satisfies (12), with the positive definite function i=x,ξ,ε ηi (|i |) being strictly greater than ζ¯u |(∂ Va (x)/∂ x)g(x) + (∂ Vb (ξ, ε)/∂ε)r L g h(x)|2 . In other words, (36) provides a qualitative description of the relation that connects the class of original systems () for which the PPOFC problem admits a solution and the controller that
generates the solutions. As the terms engaged in (36) are unknown, its a priori verification is difficult. Fortunately, this statement is relaxed by noticing that (36) is required only for analysis and not for controller design. V. S IMULATION R ESULTS Consider a planar two-link articulated manipulator whose position can be described by a 2-D vector q = [ q1 q2 ] of joint angles and whose actuator inputs consist of a 2-D vector u = [ u 1 u 2 ] of torques applied at the manipulator joints. The dynamics of this simple manipulator is strongly nonlinear and can be written in the general form M(q)q¨ + C(q, q) ˙ q˙ + G(q) = u where
M(q) =
M11 M12 M12 M22
(45)
M11 = Iz1 + Iz2 + m 1r12 + m 2 (l12 + r22 ) + 2m 2l1r2 c2 M12 = Iz2 + m 2r22 + m 2l1r2 c2 M22 = Iz2 + m 2r22 is the 2 × 2 manipulator inertia matrix
−m 2l1r2 s2 q˙2 + k1 −m 2l1r2 s2 (q˙1 + q˙2 ) C(q, q) ˙ = m 2l1r2 s2 q˙1 k2 is a 2 × 2 matrix of centripetal and Coriolis torques, and
m 1 gr1 c1 + m 2 g(l1 c1 + r2 c12 ) G(q) = m 2 gr2 c12 is the 2-D vector of gravitational torques. In the above, we have used the following notation: c1 = cos(q1 ), c12 = cos(q1 + q2 −
π 2 ),
c2 = cos (q2 − π2 ) s2 = sin(q2 − π2 ).
For simulation purposes, we have employed Link masses (kg): Link lengths (m): Inertia (kg m 2 ): Friction (N m):
m 1 = 3.2 l1 = 0.5 Iz1 = 0.96 k1 = 1
m2 = 2 l2 = 0.4 Iz2 = 0.81 k2 = 1.
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5
30
0.05 0 −0.05 −0.1 −0.15 −0.2 −0.25
0
10 0
−5
−10
H
u1(t) (nm)
20
−20 −30
−10 0
1
2
3
4 5 6 time (sec)
7
8
9
0.45 0.5 0.55 0.6
10 −15
5 −20
u2(t) (nm)
0 −5
−15
Fig. 6.
1
2
3
4
5 6 time (sec)
7
8
9
10
Fig. 7. Evolution of H. The subplot presents details of the events H = 0 appearing while operating with φσ (t) = φII (t).
−10
−20
0
0
1
2
3
4 5 6 time (sec)
7
8
9
10
Demanded control effort.
Furthermore, g = 10 m/s2 is gravity acceleration, Which is taken as constant, and r1 = (l1 /2), r2 = (l2 /2). The considered task is for the system output to track the following reference signal:
90 30 π + π cos(t) qr1 (t) 180 . = 180 90 30 qr2 (t) 180 π − 180 π cos(t) The initial state conditions are 80 q1 (0) q2 (0) q˙1 (0) q˙2 (0) = 180 π
in Fig. 4. Variation of the Euclidean norms of the weight vectors Wˆ u , Wˆ o , Wˆ ν are shown in Fig. 5. Obviously, all output trajectories converge to their desired references with prescribed performance, and all closed-loop states are kept uniformly bounded, thus solving the PPOFC problem for the considered robotic system. It is worth mentioning that the problem is solved without requiring excessive control effort as Fig. 6 clearly demonstrates. Details on the operation of the proposed switching term φσ are provided in Fig. 7, through the evolution of the H function. Notice that the number of the events H = 0, while σ (t) = II, is finite (as theory predicts). VI. C ONCLUSION
130 180 π
00
.
Performance specifications: For the output errors e1 = q1 − qr1 , e2 = q2 − qr2 we desire to achieve convergence rate no more than e−2t , steady-state error no more than (π/360), and no overshoot. The aforementioned specifications are prescribed via the following performance functions: 1 60 1 π− π e−2t + π, i = 1, 2 (46) ρi (t) = 180 360 360 where ρ0i = (60/180)π, i = 1, 2 was chosen in order for the problem to be well defined, in other words to guarantee that (5) is satisfied at t = 0. Moreover, since no overshoot is required, we choose Mi = 0, i = 1, 2. To implement the control (15)–(26), we have used HONNs, whose structure was selected according to a trial-and-error procedure. The weight vectors Wˆ u (t) ∈ R20×2 , Wˆ o (t) ∈ R10×4 , and Wˆ ν (t) ∈ R5×4 were randomly initialized in [0, 10]. The initial value of a(t) was taken a(0) = 1, while ϕa ( j ) = j. Furthermore, we have selected c0 = 1, c = 1.1, K u = [1 1] , while γξ i (·) = γεi (·) = 10(1 − e−10(·)), i = 1, 2. Finally, concerning the switching procedure, the following constants have been used: δb = 3, δu = 1. The response of the proposed adaptive output feedback control scheme is demonstrated in Figs. 3–7. In Fig. 3, the robot joint angles, the respective velocities, and the ξ −states are plotted, while the evolution of the output tracking errors along with their corresponding performance bounds is shown
An adaptive dynamic output feedback controller for a fairly general class of MIMO uncertain nonlinear systems (those that are affine in the control) has been presented, which was capable of guaranteeing convergence of the output tracking error to a predefined arbitrarily small residual set, with convergence rate no less than a prespecified value, while exhibiting maximum overshoot less than some sufficiently small preassigned constant. To overcome the amount of uncertainties present, linearly parameterized neural networks were used as approximation models. It was proved that guaranteeing a boundedness property for the states of a specifically defined augmented closed-loop system was the necessary and sufficient condition for solving the considered problem. Even though the designed dynamic output feedback controller was of switching type, its continuity was guaranteed, thus alleviating any problems related to the existence and uniqueness of solutions. Simulations on a planar two-link articulated manipulator illustrated the approach. R EFERENCES [1] J. T. Spooner, M. Maggiore, R. Ordonez, and K. M. Passino, Stable Adaptive Control and Estimation for Nonlinear Systems-Neural and Fuzzy Approximation Techniques. New York: Wiley, 2002. [2] Y.-C. Chang, “An adaptive H∞ tracking control for a class of nonlinear multiple-input multiple-output (MIMO) systems,” IEEE Trans. Automat. Control, vol. 46, no. 9, pp. 1432–1437, Sep. 2001. [3] G. Bianchini, R. Genesio, A. Parenti, and A. Tesi, “Global H∞ controllers for a class of nonlinear systems,” IEEE Trans. Automat. Control, vol. 49, no. 2, pp. 244–249, Feb. 2004.
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Artemis K. Kostarigka was born in Larissa, Greece, in 1980. She received the B.Sc. degree in physics and the M.Sc. degree in control systems theory from the Aristotle University of Thessaloniki, Thessaloniki, Greece, in 2003 and 2005, respectively. She is currently pursuing the Ph.D. degree with the Department of Electrical and Computer Engineering, Division of Electronics and Computer Engineering, Aristotle University of Thessaloniki. Her current research interests include nonlinear robust adaptive controls and control of unknown systems using neural networks.
George A. Rovithakis (S’89–M’98–SM’02) received the Diploma degree in electrical engineering from the Aristotle University of Thessaloniki, Thessaloniki, Greece, in 1990, and the M.S. and Ph.D. degrees in electronic and computer engineering from the Technical University of Crete, Crete, Greece, in 1994 and 1995, respectively. He was a Visiting Assistant Professor with the Department of Electronic and Computer Engineering, Technical University of Crete, from 1995 to 2002. He joined the Aristotle University of Thessaloniki, where he is currently an Associate Professor in the Department of Electrical and Computer Engineering. He has authored or co-authored two books and several book chapters and has published over 110 papers in scientific journals and conference proceedings. His current research interests include nonlinear robust adaptive controls and neural networks for identification and control of uncertain systems. Dr. Rovithakis was a former Associate Editor of the IEEE T RANSACTIONS ON N EURAL N ETWORKS . He is a member of the IEEE Control Systems Society Conference Editorial Board and a member of the Technical Chamber of Greece.
