Design of an amplitude-bounded output feedback adaptive neural ...

Neural Comput & Applic (2014) 25:1313–1328 DOI 10.1007/s00521-014-1612-2

ORIGINAL ARTICLE

Design of an amplitude-bounded output feedback adaptive neural control with guaranteed componentwise performance Mario Luca Fravolini • Giampiero Campa

Received: 2 September 2012 / Accepted: 28 April 2014 / Published online: 27 May 2014  Springer-Verlag London 2014

Abstract This paper proposes a methodology for the design of a mixed output feedback linear and adaptive neural controller that guarantees componentwise boundedness of the tracking error within an a priori specified compact polyhedron for an uncertain nonlinear system. The approach is based on the design of a robust invariant ellipsoidal set where the adaptive neural network (NN) control is modeled as an amplitude-bounded signal. A linear error observer is employed to recover the unmeasured states, and a linear gain controller is used to enforce the containment of the ellipsoidal set within the performance polyhedron. The analysis and design of the observer and linear controller is set up as an LMI problem. The linear observer/controller scheme is then augmented with a general adaptive NN element having the purpose of approximating and compensating for the unknown nonlinearities thus providing performance improvement. The only requirement for the adaptive control signals is that their amplitudes must be confined within pre-specified limits. For this purpose, a novel mechanism called adaptive control redistribution is introduced to manage the adaptive NN control confinement during the online operation. A numerical example is used to illustrate the design methodology.

M. L. Fravolini (&) Department of Engineering, University of Perugia, Via G. Duranti No 93, 06125 Perugia, Italy e-mail: [email protected] G. Campa The Mathworks, 400 Continental Blvd, Suite, El Segundo, CA 600-028, USA e-mail: [email protected]

Keywords Guaranteed componentwise boundedness  Adaptive neuro control  Uncertainty  Linear matrix inequalities  Set invariance

1 Introduction Based on the property of being universal approximators, neural networks (NNs) are often employed as approximating functions having the purpose of compensating for unknown uncertainties in feedback control systems. Approximation-based robust adaptive control of nonlinear systems is today a well-established technique, and relevant findings can be found, e.g., in [1–5]. Despite the recent advancements, many key issues remain still to be addressed; specifically, a very important open issue, associated with neuro adaptive control in presence of unknown nonlinearities, concerns tracking error performance. In the last years, a significant research interest has been focused on the application of well-established design techniques, available for uncertain MIMO linear systems, with the purpose of guaranteeing classical design specifications also for nonlinear neuro adaptive systems. In this context, important results are now available in terms of L1 [6], L2 [7] and H? [8, 9] performance. The above-mentioned nonlinear adaptive control designs, typically, guarantee the convergence of the tracking error to a residual set [10, 11] whose size depends on some design parameters and on some bounded but unknown terms. However, no systematic and easily applicable procedures exist to accurately compute the required bounding set, thus making particularly involved the a priori selection of the aforementioned design parameters in order to force the evolution of the system trajectories within a desired compact domain where the NN approximating

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properties are valid and transient and steady state requirements are guaranteed [12]. In addition, whenever formulas for the bounds are computed, these often represent extremely conservative estimations of the bounding set. Crucially, the fact that the bounds are typically overestimated as function of the norm of the state vector also prevents an analysis of the residual set along specific directions of the state space. In case the uncertainty satisfies specific regularity conditions, or uppers bounds for the NN reconstruction error are known, arbitrary reduction of the tracking error or even asymptotic tracking can be achieved by a large-valued linear control gain matrix K or by robustifying or sliding mode terms [13–15]. Unfortunately, the well-known drawback of these techniques is that tracking performance improvements is achieved by a high gain control action. In practical applications where unmodeled high-frequency dynamics, actuator saturations, measurement noise and delays may occur, high gain control is often considered unsafe, since it might lead to chattering, undesirable transient response, low stability margins and even instability. Furthermore, transitory performances are addressed indirectly so that it is still not immediate how to guarantee desired componentwise requirements. A concrete answer to the above problems was given in [16] where it was proposed a robust adaptive control with guaranteed error bounds for a class of MIMO feedback linearizable nonlinear systems. Very recently, a novel nonlinear adaptive scheme called prescribed performance control was introduced in [12] and [17]. This approach is able to guarantee a priori defined transitory and steady state performances via the redefinition of a specific nonlinear output error transformation. Another performance-oriented approach was also proposed in [18] for a limited class of adaptive nonlinear systems. Within this performance-oriented context, considering a class of nonlinear systems with amplitude-bounded uncertainties, the authors proposed in [19] a novel methodology for the design of a state feedback linear plus neuro adaptive model reference control that guarantees the evolution of the tracking error within an a priori specified compact domain. In this approach, it is assumed that the adaptive control law, by construction, generates amplitude-bounded signals having at least the same maximum magnitude of the uncertainties to be compensated. Under this assumption, the analysis of the original nonlinear system is transformed in the analysis of a linear uncertain systems where both the NN reconstruction error and the adaptive neural control are considered as bounded disturbances. This allowed the analysis of the uncertain system with the theory of the robust invariant ellipsoidal sets [20] generated by quadratic Lyapunov functions [21]. In

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particular in [19], it was shown that componentwise tracking error requirements can be translated into a set of LMI constraints and that an optimal linear state feedback gain controller can be synthesized solving a linear convex optimization problem. The final result is that the linear control ensures the confinement of the error trajectories within the desired performance bounds, while the amplitude-bounded NN control adaptively approximates the unknown uncertainties so that the performance can improve over time during the closed-loop operation. This approach provides a constructive alternative solution to the long-standing problem of guaranteeing performance in nonlinear adaptive control. In [22], the approach was also applied to a performance improvement problem for a complex MIMO avionic system. Other examples of application of LMI-based techniques to tackle different stability problems for NNs in presence of modeling uncertainties can be found in [23–25]. The main contribution of the present study is to demonstrate that the above approach can be extended to the output feedback case. For this purpose, a robust linear error observer, as the one proposed in [26], is employed to recover the unmeasured states while a linear gain controller, based on the observed states, is used to guarantee the desired componentwise requirements. More in details, in the first part of the paper, it is shown that the design of both the linear observer and controller, as well as the tracking error requirements, can be formulated as an overall LMI optimization problem that can be efficiently solved using convex optimization tools [27]. The second part of the paper is instead devoted to the design of a novel amplitude-bounded adaptive NN control. In fact, the scheme assumes that the NN control components are confined within the limits defined in the first phase of the design so that an output ‘‘saturation-like mechanism’’ must be used to enforce such limits. For this reason, as additional contribution, a novel adaptation mechanism was introduced to manage the NN adaptation in presence of output saturation. The approach is based on the novel concept of NN adaptive control redistribution that is a mechanism that, in case a component of the NN control violates a pre-specified limit, decreases the NN weights accordingly so that the saturation limits are never exceeded. This approach generates, by construction, a saturated control, thus eliminating the need of the adaptive control saturations that have been employed in the original formulation [19]. The main advantages of the proposed approach are that componentwise requirements can be easily defined during the design phase and that the design of the NN adaptation mechanism is largely independent from the synthesis of the linear controller and observer.

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A numerical example based on a nonlinear mechanical system in presence of unknown friction forces is used to illustrate the design methodology and to highlight the advantages of the adaptive scheme over a pure linear control design.

2 Problem statement and modeling assumptions Consider the class of uncertain MIMO systems described by the following model: x_ ¼ Ax þ BKðu þ DðxÞÞ y ¼ Cx

ð1Þ

where x 2 Rn91, u 2 Rm91 and y 2 Rp91 are state, input and output vectors, respectively, and A, B and C are the known nominal system matrices. The term D(x) 2 Rm91 is a vector of unknown functions and K is an unknown diagonal, positive matrix representing uncertainty on the control efficiency [28]. The desired response for the system (1) is given by the following reference model: x_m ¼ Am xm þ Br ym ¼ Cxm

ð2Þ

where xm 2 Rn91, ym 2 Rp91 and r 2 Rm91 is the reference command. The reference model is assumed asymptotically stable implying that the matrix Am is Hurwitz. The purpose of this study is the design of an output feedback control strategy such that the components of x track xm with desired bounds while all the signals in the closed-loop system remain bounded. The tracking error is defined as: e = x - xm and the associated error dynamics obtained from (1) and (2) is:   e_ ¼ Am e þ BK u þ D  K1 r þ DA x ey ¼ Ce

ð3Þ

where A = Am ? DA, and DA is a known difference matrix. Assumption-1 The amplitudes of the reference r are bounded: jri ðtÞj  rMi

i ¼ 1; . . .; m

i ¼ 1; . . .; m

i ¼ 1; . . .; m

ð6Þ

where ki ¼ 1  koi and ki ¼ 1 þ koi . The koi represent known upper bounds quantifying the control efficiency. Note that in order for kii to be positive, condition |koi| \ 1 must be verified. This last assumption imposes a not overly restrictive constraint (in the range [0–2]) to the multiplicative control efficiency coefficients kii defined in (6). Further, in case a larger uncertainty would be considered, this could be easily incorporated by defining a proper prescaling matrix Bscale for the original input matrix B so that Bs = Bscale B and by using Bs in lieu of B in (1). Assumption-4 The couple (A, B) is controllable and (A, C) is observable. This guarantees the existence of a control matrix K and of an observer matrix Ko such as the matrices (A-BK) and (A-KoC) are Hurwitz. Remark-1: The class of systems in form (1) is the same considered for example in [28, 29] and [30] and models the dynamics of an uncertain closed-loop MIMO system controlled by a baseline linear controller designed such that, at nominal conditions (D = 0, K = I, u = r), the resulting closed-loop (linear) system guarantees desired performance. Remark-2: The assumption of known bounds for the model uncertainty in (5) is realistic for many systems of interest. For example, a similar assumption was also made in the adaptive neuro control proposed in [13] where the uncertainty was assumed to be bounded by values that could be reasonably estimated; the approach was applied to the control of a nonlinear small-sized electrical drive system with significant friction. In fact, friction forces are an important example of bounded amplitude state-dependent nonlinearities as discussed in [31]. Similarly, in the neuro control proposed in [32], known upper bounds for the modeling uncertainty are assumed, and in the simulation example, these bounds are explicitly given for an uncertain mechanical system. Finally, an interesting application to an aircraft system in presence of unknown actuator failures can be found in [30]. 2.1 The control strategy

ð4Þ

where the rMi are known values. Assumption-2 The components of the uncertainty D(x) are completely unknown but otherwise bounded: jDi ðxÞj  di

0\ki  kii  ki

ð5Þ

where di are known bounds. ~ Assumption-3 The matrix K is defined as: K ¼ I þ K, ~ where K ¼ Ko  Kn and Kn is a diagonal uncertain matrix such that KTn  Kn B I and Ko is a diagonal matrix whose elements kii are such that:

In this study, the control consists of three contributions: uðtÞ ¼ uL ðe^ðtÞÞ þ ua ðxðtÞÞ þ ub ðtÞrðtÞ

ð7Þ

A linear gain controller provides the uL contribution: uL ðtÞ ¼ K  e^ðtÞ

ð8Þ

where K is a gain matrix and e^ðtÞ is the estimation of the state vector provided by an error observer that will be introduced shortly. The controls ua(t) and ub(t) are adaptive contributions that are designed to cancel the uncertainties in (3). Specifically, ua is designed to approximate and

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compensate for D(x), while ubr(t) is designed to compensate for K-1r(t). Substituting (7) and (8) in the error dynamics (3) results in: ~ e_ ¼ Ae þ BðI þ KÞðK e^ þ ua þ ub r þ D  K1 rÞ þ DA xm ~ e^ þ BKðD þ ua  K1 r þ ub rÞ þ DA xm ¼ Ae  BK e^  BKK

ð9Þ ¼ Considering (9), define the overall uncertain terms D   1  K D þ ua  K r þ ub r and Bc Dc ðtÞ ¼ DA xm ðtÞ where Bc is a binary diagonal matrix whose entries are one if the corresponding component of the vector DAxm(t) is different from zero, zero otherwise. The resulting error dynamics becomes: c ~ e^ þ BD  þ Bc D e_ ¼ Ae  BK e^  BKK

ð10Þ

In the following analysis, it will be convenient to express the error dynamics (10) as function of the observer estimation error e~ ¼ e  e^. This substitution results in: ~ þ BKK ~ e~ þ BD  þ Bc D c e_ ¼ ðA  BKÞe þ BK e~  BKKe

juai ðtÞj  di

i ¼ 1; . . .; m

Substituting (14) in (13) gives the explicit bounds: ai ðtÞj  ki ð2di Þ ¼ dai i ¼ 1; . . .; m jD ð15Þ Considering the definition (12), for the components of  Db we have that: bi ðtÞj  j  1 þ kii ubi ðtÞjrMi jD ð16Þ Now, since the ubi are designed to adaptively estimate the bounded positive terms 1/kii defined in (6), it makes sense to assume that the ubi are constrained, by construction, to satisfy the following bounds: 1 k1 i  ubi ðtÞ  ki

i ¼ 1; . . .; m

ð17Þ

Taking into account (17), it is possible to derive the following explicit upper bounds for the components of b j: jD    ki  2koi rMi   ¼ dbi i ¼ 1; . . .; m jDbi ðtÞj  1 þ rMi ¼ ki 1  koi ð18Þ

ey ¼ Ce ð11Þ  is Remark-3: Note that the overall uncertainty vector D not a disturbance vector in the classical sense because it includes not only uncertainty terms but also the contribution of the adaptive signals ua and ub. This is convenient for design purposes, as will be clarified in Sects. 3 and 4. Considering the adaptive control contributions as potential source of disturbance may be regarded as conservative, but it is nevertheless realistic in a worst-case scenario in light of the bounded but sometimes unpredictable response of the adaptive neural control during transients. 2.2 Upper bounds computation for the uncertainties and control signals This section deals with the computation of upper bounds for the uncertain terms in (11) and defines the bounds for  it is convenient to adaptive control signals. Considering D ¼D a þ D b where: split this term as follows: D a ¼ KðD þ ua Þ D ð12Þ b ¼ ðI þ Kub Þr D a are: D ai ¼ kii ðDi þ uai Þ The components of vector D and, considering constraints (5), the following bounds hold: ai ðtÞj  ki ðdi þ juai ðtÞjÞ i ¼ 1; . . .; m jD ð13Þ Now, since the adaptive contributions uai are designed to compensate for the norm-bounded uncertainties in (5), it makes sense to assume that the uai are constrained, by design, to satisfy the following inequalities:

123

ð14Þ

Finally, based on bounds (15) and (18), the upper  are derived as follows: bounds for the components of D   i ðtÞ  dai þ dbi ¼ dDi i ¼ 1; . . .; m D ð19Þ c ðxm ðtÞÞ, several Considering the uncertain vector D methods are available to compute an upper bound vector dc based on the maximal amplitude of the xm vector. In this study, we employed the technique proposed in [33] that relies on the properties of stable linear perturbed system. The resulting componentwise bounds are such that: ci ðtÞj  dci i ¼ 1; . . .; n jD ð20Þ Remark-4: It is underlined that the assumption of the adaptive control amplitude constraints in (14) and (17) originates from the fact that the purpose of the uai and ubi controls is to compensate for bounded uncertainties having known componentwise upper bounds defined in (5) and (6), respectively. Therefore, at this stage, it is assumed that the adaptation algorithm, that will be selected, must be able to generate, by construction, amplitude-bounded controls uai and ubi such that the requirements (14) and (17) are fulfilled. Constraints (15) and (18) descend from these design assumptions. This important issue will be clarified in Sect. 5 where it will be introduced a novel amplitude-bounded adaptive neuro control. 2.3 The tracking error observer The unmeasured error signals of the closed-loop error dynamics are estimated thought a state observer using the approach proposed in [26] for a similar application, that is:

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e^_ ¼ ðA  BKÞ^ e þ DA xm þ Ko ðey  e^y Þ

1317

ð21Þ

where ey = Ce and e^y ¼ C^ e are the output error and the output estimated error, respectively, Ko is the observer gain matrix that will be computed jointly to K, via the LMI design. The observation error dynamics (e~_ ¼ e_  e^_) is easily obtained from (11) and (21) that is: ~ e^ þ BD   Ko Cðe  e^Þ e~_ ¼ A~ e  BKK ~ þ BKK  ~ e~ þ BD e  BKKe ¼ ðA  Ko CÞ~

ð22Þ

3 Bounding set definition and main result The main purpose of the control design is to guarantee the evolution of the tracking error (during the transients and at steady state) within a priori specified compact domain. The method proposed here is based on the theory of robust positive invariant sets generated by quadratic Lyapunov functions [21] where the uncertain terms as well as the bounded adaptive controls are considered as bounded disturbances. Considering the tracking error trajectories these depend on the joint dynamics of the tracking and observation error that are here grouped together: ( ~ þ BKK ~ e~ þ BD  þ Bc D c e_ ¼ ðA  BKÞe þ BK e~  BKKe ~ þ BKK  ~ e~ þ BD e~_ ¼ ðA  Ko CÞ~ e  BKKe ð23Þ The performance analysis of the uncertain system (23) will be based on the following definition of robust positive invariance and of ellipsoidal sets. Definition 1 A set X is said to be robustly positively invariant ( [34] ) for system (23) if, for every initial state ðeð0Þ; e~ð0ÞÞ 2 X, the trajectory of the system ðeðtÞ; e~ðtÞÞ remains in X, for t C 0, for any admissible disturbance  c ðtÞ and KðtÞ. ~ DðtÞ, D Definition 2 Consider a standard quadratic Lyapunov function V ¼ eT P1 e þ e~T P2 e~, with P1 = PT1 [ 0 and P2 = PT2 [ 0 (symmetric positive definite). Associate to V the compact ellipsoidal level set X defined as follows:   X ¼ ðe; e~Þ 2 R2n j eT P1 e þ e~T P2 e~  1; P1 [ 0; P2 [ 0 ð24Þ 3.1 Main result It is now introduced the main result that provides sufficient conditions that guarantee the evolution of the tracking error trajectories within a desired robustly positively invariant ellipsoidal set X defined in (24). The results that follow rely on the following technical lemma: Lemma-1 Given a symmetric matrix G and any matrices M and N of appropriate dimensions, then [34, 35]:

G þ MNn N þ N T NTn M T  0;

for all Nn s:t: NTn Nn  I

if and only if there exist a scalar e [ 0 such that: G þ eMM T þ e1 N T N\0 Theorem 1 Invariance under persistent disturbances Consider the system (23) with bounded disturbance  c ðtÞ and KðtÞ ~ DðtÞ, D and control law (7). If there exist scalars a [ 0, e [ 0, c [ 0, symmetric positive define -1 matrices P1 [ 0 (Q1 = P-1 1 ) and P2 [ 0 (Q2 = P2 ) and matrices X and Y such that: a þ 2

 1 T 2dD dD þ dTc dc \0 e

AQ1 þ Q1 AT  BX  X T BT þ aQ1 þ

6 6 eð2Bo BTo þ BBT þ Bc BTc Þ 6 6 ðBXÞT 6 4 X

ð25Þ 3 BX cQ1 0 X

0

X

T

0

0 0:5eI 0

7 7 7 7\0 7 5

T

X 0 0:5eI

ð26Þ 2

T

T

T

AP2 A þ A P2  YC  C Y þ aP2 4 ðP2 BK00 ÞT I

P2 BK e1 I 0

00

I 0 1

3 5\0

c Q1

ð27Þ then, X is robustly positively invariant for system (23). The control and observer matrices are given by: K = XQ-1 1 and Ko = P-1 Y, respectively. 2 Proof: It is sufficient to show that the time derivative of V is B0 along the boundary (qX) of X for any admissible  D c . In fact, since, by the definition (24), ~ D; uncertainty K; V is constant along qX, if results V_  0 along qX for any  D c then, assuming ðeð0Þ; e~ð0ÞÞ 2 X, the ~ D; admissible K; trajectories cannot escape X for t C 0, implying that X is robustly positively invariant. It can be shown that the requirements of Theorem 1 are satisfied if the following inequalities hold:   V_ ¼ eT P1 A þ AT P1  P1 ðBKÞ  ðBKÞT P1 e ~ e~þ þ 2eT P1 BK e~ þ 2eT P1 BKK c þ  þ 2eT P1 Bc D ~ þ 2eT P1 BD  2eT P1 BKKe   þ e~T P2 A þ AT P2  P2 ðKo CÞ  ðKo CÞT P2 e~þ 0 ~ þ 2~ ~ e~ þ 2~ eT P2 BKK e T P2 B D  2eT P2 BKKe

ð28Þ

for any e; e~ such that: eT P1 e þ e~T P2 e~  1

ð29Þ

In fact, (28) expresses the condition V_  0, while condition (29) defines the boundary qX and the external region outside X. Condition (28) with constraint (29) can be transformed in an equivalent unconstrained MI using the S-procedure [21].

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Applying the S-procedure results that (28) and (29) are satisfied if and only if there exist an a [ 0 such that:   eT P1 A þ AT P1  P1 ðBKÞ  ðBKÞT P1 e þ 2eT P1 BK e~  ~ e~ þ 2eT P1 BKKe ~ þ 2eT P1 BD þ 2eT P1 BKK  c þ e~T P2 A þ AT P2  P2 ðKo CÞ þ 2eT P1 Bc D  ~ þ 2~ ~ e~ ðKo CÞT P2 e~ þ 2eT P2 BKKe eT P2 BKK  þ aðeT P1 e þ e~T P2 e~  1Þ  0 þ 2~ e P2 B D ð30Þ It is now convenient to express the uncertain terms in n ; D cn ~n ; D (30) as function of normalized uncertainties K that are defined as follows: ~T K ~ ðK n n  IÞ n  IÞ T D ðD

c ¼ D cn dc D

 T D ðD cn cn  IÞ

n

M11 6 T 4 ðP1 BKÞ 0

P1 BK M22 0

3 0 7 0 5\0 M33

M11 ¼ P1 A þ AT P  P1 ðBKÞ  ðBKÞT P1 þ aP1 þ þ eð2P1 Bo BTo P1 þ P1 BBT P1 þ P1 Bc BTc P1 Þ þ 2e1 K T K M22 ¼ m22a þ m22b m22a ¼ P2 A þ AT P2  YC  CT Y T þ aP2 þ

T

~¼K ~o K ~n K ¼D n dD D

2

ð31Þ

~o . Substituting relations (31) define also the matrix Bo ¼ BK in (30) results in:   eT P1 A þ AT P1  P1 ðBKÞ  ðBKÞT P1 e þ 2eT P1 BK e~ n dD ~n K e~ þ 2eT P1 Bo K ~n Ke þ 2eT P1 BD þ 2eT P1 Bo K  cn dc þ þ~ þ 2eT P1 Bc D eT P2 A þ AT P2  P2 ðKo CÞ  ~n Ke þ 2~ ~n K e~ e T P2 Bo K ðKo CÞT P2 e~ þ 2eT P2 Bo K n dD þ þaðeT P1 e þ e~T P2 e~  1Þ  0 þ 2~ e T P2 B D

þ eð2P2 Bo BTo P2 þ P2 BBT P2 Þ m22b ¼ 2e1 K T K M33 ¼ a þ e1 ð2dTD dD þ dTc dc Þ ð35Þ where Y = P2Ko. Since matrix (35) is block diagonal, this can be further decomposed into two simpler MIs:

M11 P1 BK \0 ð36Þ T ðP1 BKÞ M22

It can be immediately observed that (37) coincides with the first condition (25) of Theorem 1. Define now the two matrices:

M11 P1 BK M1 ¼ ðP1 BKÞT m22b  cP1

ð32Þ M2 ¼

Defining now the grouped matrices: M ¼ ½eT P1 Bo ; eT P1 Bo ; eT P1 B; eT P1 Bc ; ~ eT P2 Bo ; e~T P2 Bo ; e~T P2 B N ¼ ½ðK e~ÞT ; ðKeÞT ; dTD ; dTc ; ðKeÞT ; ðK e~ÞT ; dTD T ~n ; K ~ n ; Dn ; Dcn ; K ~n ; K ~ n Dn  Nn ¼ diag½K

ð37Þ

M33 \0

0 0

ð38Þ

where c is a positive scalar. Rewriting (36) as function of M1 and M2 gives the equivalent condition: M1 þ M2 \0

ð33Þ

0 m22a þ cP1



ð39Þ

þ 2~ eT P2 Bo BTo P2 e~ þ e~T P2 BBT P2 e~ þ aðeT P1 e þ e~T P2 e~  1Þ  0

Condition (39) is trivially satisfied in case M1 \ 0 and M2 \ 0; therefore, in the following, we provide conditions ensuring M1 \ 0 and M2 \ 0. Applying a congruence transformation to M1 with the matrix diag[Q1, Q1], condition M1 \ 0 is equivalent to: 2 3 AQ1 þ Q1 AT  BX  X T BT þ aQ1 6 7 6 eð2Bo BTo þ BBT þ Bc BTc Þ 7 BX 6 7\0 6 7 1 T T 4 þ e Q1 K KQ1 5 þe1 X T X  cQ1 ðBXÞT

ð34Þ

ð40Þ

It is not difficult to show that the quadratic inequality (34) is equivalent to the following MI condition:

where X = KQ1. Finally, applying the Schur complement to (40) results the following MI:

and applying Lemma-1 to (32) with the M, N, Nn defined in (33), results:   eT P1 A þ AT P1  P1 ðBKÞ  ðBKÞT P1 eþ   þ e~T P2 A þ AT P2  P2 ðKo CÞ  ðKo CÞT P2 e~  1 T T e K K e~ þ 2eT K T Ke þ 2dTD dD þ dTc dc þ 2eT P1 BK e~ þ 2~ e þ e½2eT P1 Bo BTo P1 e þ eT P1 BBT P1 e þ eT P1 Bc BTc P1 e

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1319

2

AQ1 þ Q1 AT  BX  X T BT þ aQ1 6 6 eð2Bo BTo þ BBT þ Bc BTc Þ 6 6 ðBXÞT 6 4X 0

3 BX

XT

0

cQ1 0 X

0 0:5eI 0

XT 0 0:5eI

7 7 7 7\0 7 5

ð41Þ that coincides with the second condition (26) of Theorem 1. Consider now the condition M2 \ 0, this is equivalent to: P2A ? ATP2 - YC - CTYT ? aP2 ? e(2P2BoBTo P2 ? 00 P2BBTP2) ? cP1 \ 0. Defining the matrix K = (2K2o ? 0.5 I) , the above inequality becomes: P2 A þ AT P2  YC  CT Y T þ aP2 þ eðP2 BK00 K00T BT P2 Þ þ cP1 \0 ð42Þ 2

Applying the Schur complement [21] to (42) results in:

AP2 A þ AT P2  YC  C T Y T þ aP2 4 ðP2 BK00 ÞT I

P2 BK00 e1 I 0

I 0 c1 Q1

3

5\0

ð43Þ that coincides with the last (third) condition (27) of Theorem 1. Finally, control and observation matrices can be recovered immediately from the definition of X and Y as follows: K = XQ-1 and Ko = P-1 1 2 Y. This completes the proof of Theorem 1 h 3.2 Tracking error requirements Theorem 1 provides sufficient conditions for X to be robustly positive invariant, but it does not provide conditions to verify whether X satisfies tracking error requirements. The tracking error requirements are here expressed by the following componentwise constraints: jei ðtÞj  dei i ¼ 1; . . .; n

ð44Þ

where dei [ 0 are free design parameters. Defining the vectors gTi = [0,…,1,…,0], where the 1 occupies the i-th position of gi, the constraints (44) can be also expressed as: jgTi ej2  d2ei

i ¼ 1; . . .; n

ð45Þ

Conditions (45) imply that the error trajectory evolves in the compact polyhedral set: n o  2 Pe ¼ e 2 Rn  jgTi ej  d2ei ; i ¼ 1; . . .; n ð46Þ In the synthesis phase, as it will be clarified in Sect. 4, it will be of interest to verify if Pe could be further reduced without violating the other design constraints. For this purpose, it is also defined the l-scaled tracking error polyhedron:

n Pel ¼ e 2 Rn j

jgTi ej2  l2 d2ei ; i ¼ 1; . . .; n

o

ð47Þ

where l [ 0 is a scaling coefficient to be computed; define also Xel as the projection of the set X in the e subspace. In order to fulfill constraints (44), the scaled Xel set (Xe is the projection of the set X in the subspace of the tracking error variables e) must be fully contained in the scaled polyhedron Pel. In [36], it was shown that the ellipsoidal Xel is contained in Pel (Xel , Pel) if the following MIs hold:

l2 d2ei gTi Q1 [ 0 i ¼ 1; . . .; n ð48Þ Q1 gi Q1 The MIs (48) will be used to express the tracking error requirements. Remark-5: The confinement of e within Xel for t C 0 (assuming that ðeð0Þ; e~ð0ÞÞ 2 X) implies also the boundedness of x because x = e ? xm and, by assumption, the reference model is a BIBO asymptotically stable system forced by a bounded reference signal r. Several methods [33] are available to estimate a bounding compact set Dxm containing the trajectories of xm. In addition, assuming constraints (44) satisfied, it is also immediate the computation of the domain Dx which bounds the evolution of x. Remark-6: The previous remark also implies that the uncertain bounds in (5) need to be valid only for x [ Dx. This aspect is practically important for the setup of the NN that will be used for the approximation of D(x) (see Sect. 5) that is to distribute properly the NN neurons over a compact domain DNN such that: Dx ( DNN  ¼ 0; ~ ¼ 0; D Remark-7: In nominal condition that is: K  Dc ¼ 0, inequality (28) simplifies to:   V_ ¼ eT P1 A þ AT P1  P1 ðBKÞ  ðBKÞT P1 e þ 2eT P1 BK e~   þ e~T P2 A þ AT P2  P2 ðKo CÞ  ðKo CÞT P2 e~ 0 It is immediate to verify that the above Lyapunov equation implies the asymptotic stability of the matrix: A ¼ ½ðA  BKÞ BK; ðA  Ko CÞ 0nn . Since A is triangular, this also implies the asymptotic stability of the matrices along its diagonal that is of (A-BK) and of (AKoC) thus ensuring the stability of the nominal error and observation dynamics. It is underlined that the stability of (A-BK) implies also the stabilization of the nominal system dynamics in (1), this implies that the proposed design method can be also applied to open-loop (stabilizable) unstable systems. 4 Requirements verification and controller/observer design Based on the scaled tracking error requirements given in (47), it is now important to verify the existence of solutions

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(Linear controller ? observer) that guarantee these specifications. Formally, we state that the closed-loop system (23) verifies the tracking error requirements if the following two conditions are satisfied: (Problem-1) • •

The ellipsoid X in (24) is robustly positively invariant for the error dynamics (23) according to Theorem 1. The requirements |ei(t)| B l  dei are fulfilled for 0 \ l\1.

The operative procedure for computing the optimal controller/observer leading to the smallest value for l is illustrated in the following section. 4.1 Design guidelines Considering the possible solutions to Problem-1, we are interested in selecting the one that produces the minimal values for the l coefficient in the MIs (48). This design guideline can be formalized as an optimization problem having as cost function J = l2 and as optimization variables the free parameters involved in Theorem 1. Therefore, the following optimization problem is considered: minimize

a [ 0;e [ 0;c [ 0;Q1 [ 0;P2 [ 0;l [ 0;X; Y

subject to MIs

J ¼ l2 ð49Þ

ð25Þð26Þð27Þ; and ð48Þ

It must be noticed that problem (49) is not convex due to the presence of the product between some optimization variables in the MIs. However, it can be also observed that for fixed values of a, e and c, the problem reduces to: minimize

Q1 [ 0;P2 [ 0;l [ 0;X; Y

subject to LMIs

l2 ð50Þ

ð25Þð26Þð27Þ; and ð48Þ

that is linear in the optimization variables and therefore can be solved using standard LMI solvers [27]. As suggested in [37] for a similar parameter-dependent LMI problem, a possible approach is to test the feasibility problem of the LMIs in (50) via a grid search over the parameters space of a, e and c; later, a numerical optimization algorithm (as the fminsearch function of the MATLAB Optimization Toolbox) can be used to refine the optimal grid solution obtaining a solution that provides the smallest value (l*) for l. The optimal controller/observer matrices: K = XQ-1 1 and Ko = P-1 2 Y descend immediately from the solution of (50) computed for l = l*. Remark-8: In the case the optimal solution of problem (50) provides a l* [ 1, this means that a feasible solution exists, but this does not satisfy the requirements (44). Infeasibility of (50), for any value of a, e and c, implies that at least one constraint in (50) cannot be satisfied. In this case, the proposed approach is not able to provide a

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solution to Problem-1 for the current set of uncertainties, performance and design parameters.

5 Adaptive controllers design The positive invariance condition of Theorem 1 and the design/verification procedure of Sect. 4 have been derived under the key assumption of bounded amplitudes for the adaptive contributions ua and ub (see Remark-4). The second part of the paper is thus dedicated to the design of the adaptive contributions ua and ub such that amplitudes constraints (14) and (17) are fulfilled. Considering the control ua, it is designed to adaptively compensate for the uncertainty D(x) relying on the well-known universal approximation property of NNs [38]. This property means that, given a continuous function Di(x):DNN , Rn ? R, where DNN is a compact set, and given an arbitrary small number ei, there is a theoretical optimal NN weight vector W*i 2 Rh and a certain number of neurons h such that the NN h approximation W*T i ri(x) of Di(x) (where ri(x) 2 R is the vector of basis functions) satisfies the condition:   maxx2DNN WiT ri ðxÞ  Di ðxÞ  ei . Note that because the vector x evolves in the set Dx (see Remark-5), then, it can be assumed that DNN : Dy, where Dy : {y = Cx|x 2 Dx}. Therefore, Dy also characterizes the set DNN where the NN neurons are distributed. 5.1 The ua neural network The components of the ua control are defined as follows: ^i ðxÞ ¼ W ^ iT ðtÞri ðxÞ uai ðtÞ ¼ D

i ¼ 1; . . .; m

ð51Þ

^ i ðtÞ provides the adaptive estimation of the optiwhere W mal vector W*i . Since this study deals with an output feedback design, we exploited the approach developed in [39] for recovering the system states from the vector of delayed outputs that is defined as follows: z(t) = [y(t)Ty(t - d)T, …, y(t - ndd)T]T where d represents a time delay and nd the length of the delay line. The components of ua are thus redefined as function of z(t): ^ iT ðtÞri ðzðtÞÞ uai ðtÞ ¼ W

i ¼ 1; . . .; m

ð52Þ

5.2 Weights adaptation and output confinement algorithm through control redistribution for the ua neural network An important feature of the proposed approach is that, as long the controls ua (and ub) fulfill the amplitude constraints (14) and (17), the uncertain closed-loop system has a guaranteed tracking error performance.

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This entails that we are requested to design an adaptation law that guarantees, by construction, the fulfillment of the control amplitude constraints. Classical projection-based adaptation algorithms, although guarantee the boundedness of the norm of the ^ i jj, do not guarantee the fulfillment NN weight vector jjW of the amplitude constrains (14). A simple saturation at the NNi out (for brevity, NNi will be used to indicate the i-th component of the NN) could easily solve the problem; however, this option is not advisable since, when the NNi output is saturated, the NNi adaptation algorithm could potentially continue to increase (windup) the magnitude of adaptation parameters and consequently ‘‘unlearn’’ some of the information stored in the NNi thus vanishing the learning purpose of the NNs. This consideration highlights the fact that the adaptation law will have to deal, directly, with the confinement of the control signals. For this purpose, in this effort, a novel NN adaptation algorithm is proposed, with the purpose of efficiently managing, at the same time, the confinement of the NN weights and the amplitudes of the NN outputs. Before introducing the adaptation algorithm, we define binary variables that indicate the saturation status of the ^ i ðtÞ and of the NNi norm of the NNi weight vectors W output: ^ i norm saturation variables: W 8 ^ i ðtÞjj ¼ MWi < 1 if jjW ^ iT ri ðzÞ cWi ðtÞ ¼ and W ei ðtÞ [ 0 i ¼ 1; . . .; m : 0 otherwise •

ð53Þ In (53), the constant MWi defines the radius of the   2 ^ i  MWi ^i j W ^ iT W where the weight domain DWi W ^ i is allowed to evolve. vector W •

uai output saturation variables:   T  ^ i ri ðzÞ  di 1 if W cai ðtÞ ¼ i ¼ 1; . . .; m 0 otherwise

ð54Þ

During the operation, for each NNi, four configurations (cases) are possible: (1) cWi = 0; cai = 0, (2) cWi = 1; cai = 0, (3) cWi = 0; cai = 1, (4) cWi = 1; cai = 1. The following novel adaptation algorithm is proposed: 8 ( > ei ðtÞ ðcase 1Þ > _^ ðtÞ ¼ ci  ri ðzÞ   > : ^ ^ i ðt Þ ðcase 3 and 4Þ Wi ðtþ Þ ¼ RED W ð55Þ

where ci is the NNi learning rate and the ei ðtÞ is the NNi teaching signal employed for the online adaptive learning ^ i ðtÞ. The following analysis clarifies of the parameters in W the operation of the adaptation law (55): ^ i ðtÞjj is inside Case 1: cWi = 0 indicates that either jjW ^ i ðtÞjj ¼ MWi ); the domain DWi or it is on the boundary (jjW T ^ i ri ðzÞ ei ðtÞ  0 which however, in the latter case results W ^ means that jjWi ðtÞjj does not grow. Therefore, in any of the ^ i ðtÞjj is contained within DWi. Condition two cases jjW cai = 0 indicates that uai is not saturated. Case 2 (cWi = 1; cai = 0): cWi = 1 indicates that ^ iT ri ðzÞ ^ i ðtÞjj ¼ Mwi , and W ei ðtÞ [ 0. In this case, the jjW ^ i ðtÞjj ¼ Mwi . The learning is simply halted, maintaining jjW uai is not saturated. Case 3 and 4: Whatever cWi, results that cai = 1 there^ iT ðtÞri ðzðtÞÞj  di , thus fore the NNi output is such that jW requiring ‘‘de-saturation’’. To avoid amplitude violation for uai, the proposed strategy is to compute, online, the magnitude Dui(t) of the (potential) NNi output violation and, in case a violation is detected at a time instant t-, then the Dui(t-) is redistributed (i.e., subtracted) between the nodes before updating the actual NNi output at time t?. In detail, the amplitude of the violation at t- is defined as Dui(t-) where: Dui ðtÞ ¼ uai ðtÞ  signðuai ðtÞÞ  di , and the control redistribution operation is defined as follows:   Duai ðt Þ ^ ij ðtþ Þ ¼ RED W ^ i ðtþ Þ ¼ W ^ i ðt Þ  P W ð56Þ  j rij ðt Þ It is immediate to verify that following the redistribution ^ iT ðtþ Þri ðtþ Þj ¼ uai ðtþ Þ ¼ di , therefore it will result jW implying that |uai(t)| B di Vt. In practice, the control redistribution is implemented using a weight reset mechanism that, whenever a violation is detected at time instant t- (that is cai(t-) = 1), recal^ ij ðtþ Þ according to (56). Whenever also culates the states W cWi(t ) = 1 the redistribution will also contribute to bring ^ i ðtÞjj back within DWi. jjW 5.3 The NNs online teaching signal An important feature of the proposed algorithm (55) is that it ^ i and of uai guarantees, by construction, the boundedness of W regardless of the particular choice of the NNi teaching signal eðtÞ thus allowing the experimentation of different NN adaptive learning strategies. In this study, we focused on the comparison of 4 classical NN learning strategies rather than considering more complex strategies as, for instance, those proposed in [40, 41]. Since the 4 strategies are well known in the literature, we give only the basic definitions.

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(a)

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State feedback (SF) Only for analysis purpose, it was assumed that the feedback error signal is available for feedback. The NNs teaching vector in (55) is: eðtÞ ¼ eT ðtÞP1 B ð57Þ In (57), the matrix P1 is typically derived from a Lyapunov function design [19]. In this paper, it was employed the P1 produced by the optimization (50).

(b)

Observer error (OE) Approach similar to (57), but it is now assumed that the error signals are provided by the error observer (21). The resulting NNs teaching signal vector in (55) is [26]: eðtÞ ¼ e^T ðtÞP1 B

(c)

(d)

ð58Þ

Feedback error learning (FEL) This approach is based on the well-known feedback error learning strategy proposed by Kawato [42] where: eðtÞ ¼ uL ðtÞ ð59Þ Strictly positive real (SPR) Other well-known output feedback approaches are based on filtered tracking error methods [2] that derive the e signal as the result of a linear filtering of the measured error ey. In this category falls the so-called strictly positive real (SPR) approach [43]. This approach requires the design of a linear matrix filter PSPR(s) such that the filtered transfer matrix PSPR(s) We(s) between the NN outputs and the filtered output e is SPR (We(s) = Ccl(sI - Acl)-1 Bcl represents the closedloop transfer matrix between the system inputs and the output error ey). In this case, the feedback error signal is: ð60Þ eðsÞ ¼ PSPR ðsÞey ðsÞ

5.4 The ub adaptive control A similar design approach was adopted for the control ub ^1 of K-1 withwhich produces the adaptive estimation K out exceeding the limits imposed by (16). This implies that ^¼K ^1 have to the elements h^i of the diagonal matrix H 1 ^ satisfy the constraints k1 i  hi  ki . The components of ub are therefore defined as follows: ubi ¼ h^i

i ¼ 1; . . .; m

ð61Þ

In this case, the weights redistribution mechanism is clearly not necessary since the estimation of the ubi is provided by the scalar parameter h^i . The following adaptation and confinement algorithm is proposed:

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8_ > < h^i ðtÞ ¼ 0 _ h^i ðtÞ ¼ 0 > :_ h^i ðtÞ ¼  chi  ei ðtÞ i ¼ 1; . . .; m

ð h^i ¼ 1=ki ð h^i ¼ 1=ki

and ð ei ðtÞÞ\0Þ and ð ei ðtÞÞ [ 0Þ

ðotherwiseÞ ð62Þ

where chi is a learning rate and the ei ðtÞ are defined in the previous section. Remark-9: Since the amplitudes of adaptive contributions ua and ub are bounded, by design, this entails that tracking performances are guaranteed for any t C 0 provided that eð0Þ; e~ð0Þ 2 X. This property is important from a practical point of view since entails that pretraining is not necessary for the parameters of adaptive controllers.

6 Design example A simulation study was carried out to illustrate the design steps and the potentiality of the approach. The uncertain two-masses-spring-damper mechanical system shown in Fig. 1 was considered. States x1 and x2 represent the position and velocity of mass m1 and states x3 and x4 the position and velocity of mass m2. The dynamics equations of this MIMO uncertain system are [44]: 3 2 2 3 0 1 0 0 2 3 x_1 6 k  k2 c1  c2 k2 c2 7 x1 76 7 6 7 6 1 6 x_2 7 6 m1 6x 7 m1 m1 m1 7 76 2 7 6 7¼6 76 x 7 6 x_ 7 6 0 0 0 1 74 3 5 4 35 6 4 k2 c2 k2 c2 5 x4 x_4 m2 m2 m2 m2 2 3 0 0 ! 6 1 7 "u # 6 m1 0 7 1 7 þ K6 þD 6 0 0 7 u2 4 5 0 m1 2 y ¼ ½x1 ; x3 T 2 3 ð0:15 þ 0:10eð20x2 Þ Þsignðx2 Þ 5 DðxÞ ¼ 4 ð20x4 Þ ð0:10 þ 0:05e Þsignðx4 Þ

" K¼

0:9

0

0

0:9

#

ð63Þ where c1 = c2 = 3 and k1 = k2 = 6 are nominal linear damping and spring coefficients, respectively, and m1 = m2 = 1 are the masses of the two bodies. The unknown uncertainty D(x) models a typical (nonlinear and discontinuous) friction function [45]. The inputs u = [u1, u2]T are the two control forces applied to the masses.

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Fig. 1 The (nominal) two-masses-spring-damper system

6.1 Uncertainty definition and upper bound computation A control efficiency K of 90 % was assumed for the two input channels. The reference model was derived from (63) by setting u = r, K = I, D = 0 and assuming for the reference model damping coefficients c1 = 4 and c2 = 3.1. The reference commands r1 and r2 for the two masses are random amplitude filtered square waves with periods of 30 s and amplitude constraints: jri j  rMi ¼ 1 i ¼ 1; 2. Conservative upper bounds for the uncertainty D(x) in (63) are fixed at: d = [0.3, 0.2]T, while the maximum uncertainty for the control effectiveness was fixed at ±10 % implying that: 0:90 ¼ ki  ki  ki ¼ 1:10 and 0:9091  hi  1:1111. Based on the above settings, upper bounds for the system uncertainties were computed applying (15) and (17) that results in da = [0.6, 0.4]T and db = [0.105, 0.105]T, respectively. The upper bounds dc were computed following the approach proposed in [33] that result in dc ¼ ½0; 0:1827; 0; 0:0372T . 6.2 Componentwise tracking error requirements Baseline tracking error requirements were placed on the positions of the two masses and quantified by |e1(t)| B 1.4 and |e3(t)| B 1.4 resulting in the tracking error polyhedron: Pe13 = {e1, e3|| e1| B 1.4; |e3| B 1.4}. 6.3 LMI-based controller and observer design The design of the linear controller and observer was performed following the guidelines of Sect. 3. Considering the fact that the LMI optimization (50) takes into account also the effects of the adaptive controls ua and ub, this design will be hereafter referred as linear adaptive (LA). The LMI problem (50) was solved by searching for feasible solutions in the domain [0–5] 9 [0–5] 9 [0–5] in the a, e, c space (with a grid resolution 0.02) exploiting the MATLAB Robust Control Toolbox functions to solve the

1323

LMI optimization. The best grid solution, refined through the fminsearch () function of MATLAB, was achieved for [a, e, c]* = [2.75 1.21 0.55] and produced a minimal value for the performance scaling coefficient equal to l* = 0.911. This optimal solution resulted in the following matrixes. The Matrix P1 = Q-1 1 that defines the ellipsoidal invariant set X*el is: P1 = [25.90, 0.73, -15.72, -0.41; 0.73, 1.92, -0.41, -1.05; -15.72, -0.41, 10.18, 0.31; -0.41, -1.05, 0.31, 0.86]; the optimal control and observer matrices are: K = Q-1 1 X = [0.3051, 0.0792, -0.1823, -0.0259; -0.1698, -0.0229, 0.1168 0.0524] and Ko = P-1 2 Y = 1.0e ? 004*[0.0029, 0.0029; 8.1722, 8.2287; 0.0035, 0.0037; 8.2269, 8.2872]. Figure 2a shows the projection of the optimal robust invariant subset X*el in the e1–e3 plane, the nominal (l = 1) performance polyhedron Pe as long as the optimal l-scaled polyhedron P*el. The X*el set represents the smallest invariant set such that the performance requirements are guaranteed. The trajectory shown in Fig. 2a is the tracking error obtained in a 2000s simulation under LA control in the case the SPR approach is used for the NN adaptation. 6.4 Set-up of the adaptive controllers The control laws ua and ub were designed following the procedure of Sect. 5. A radial basis functions (RBF) NN was used to approximate D(x). Since D1(x) depends on x2, the neurons were equally spaced over the domain |x2|B0.3 (this range was derived following a simulative analysis). A total of 81 Gaussian neurons were used for the NN1 with ‘‘width’’ r1 = 0.03. Since the velocity x2 is not measured, this was derived applying the tapped delay approach from the position as follows: x2s(t) = (x1(t) - x1(t - d))/d where d = 0.01 s. The signal z2(t) = x2s(t) is the input to the NN1. An identical structure was used for the NN2 approximating D2(x) whose input is z4(t) = x4s(t). The maximum norm for the NNs weight vectors was fixed at MW1 = MW2 = 10, while, as previously mentioned, the two NNs outputs saturation limits in (14) were fixed at d1 = 0.3 and d2 = 0.2. The NNs learning rates were fixed at cW1 = cW2 = 10. Considering ub, the output saturation limits for h^ are h1min ¼ h2min ¼ 0:90 and h1max ¼ h2max ¼ 1:11, while the learning rates were fixed at ch1 = ch2 = 0.1. 6.5 LMI design for the pure linear case It is interesting to compare the performance of the LA design with those of a simpler ‘‘pure linear’’ (PL) controller K0 obtained by solving problem (50) in case the adaptive control is not used, namely when |ua| = 0 and |ub| = 1.

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Fig. 2 a Linear adaptive approach. The robust invariant ellipsoidal X*el, the baseline rectangular polyhedron Pe (solid, l = 1) and the l-scaled Pel* (l* = 0.911) polyhedron (dotted line). It is also shown the trajectory for a 2000s simulation under LA control. b Pure linear approach. The robust invariant ellipsoidal X*el, the l-scaled Pel* (l* = 0.453) polyhedron (dotted line). It is also shown the trajectory for a 2000s simulation under PL control

It is important to point out that the absence of the ‘‘adaptive disturbance contributions’’ is relevant, at design stage, because it halves the maximum value of the uncertainty da in (15) (da = [0.3, 0.2]T) and reduces db in (17) (db = [0.1, 0.1]T). As a consequence, the optimal PL controller uLo ðtÞ ¼ Ko e^ðtÞ (alone) may produce a smaller polyhedron P*el. In fact, repeating the optimization (50) using the new values for da and db, and maintaining the same bounds for dc, resulted in l* = 0.453 that, de-facto, halves the size of the performance polyhedron achieved in the LA design. At first look, this result may rise the question about the utility of the Linear ? Adaptive scheme. A detailed answer to this important question will be given in the following sections where the actual performances of the LA and the PL controller are compared. Figure 2b shows the robust invariant subset X*el and the lscaled polyhedron P*el in case of the PL design and the tracking error trajectory for a 2000s simulation. 6.6 Performance comparison Simulations were performed to evaluate the proposed method using the 4 NN teaching signals eðtÞ introduced in Sect. 5.3. In detail, the optimal P1, K and Ko matrices computed in Sect. 6.1 were employed for the setup of the SF, OE and FEL approaches. Considering the SPR approach, the closed-loop error transfer function matrix (We(s) = Ccl(sI - Acl)-1Bcl) between the 2 control inputs and the 2 output errors ey) is given by: Acl11 = Am, Acl12 = -BK, Acl21 = KoC, Acl22 = A-BK-KoC, Bcl =

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Table 1 Tracking and control performance for the four NN teaching strategies evaluated in a 2000s simulation Mean (|E3|)

Mean (|u1|)

Mean (|u2|)

NN teaching strategy

Mean (|E1|)

(PL)

100 %

100 %

100 %

100 %

(SF)

26.2

29.3

102.3

102.0

(OE)

51.9

68.2

115.6

94.5

(SPR)

15.4

16.0

97.8

96.6

(FEL)

28.2

30.8

93.5

99.9

(SF) (no adapt)

28.6

32.0

106.9

104.3

SPR (no adapt)

33.90

37.71

108.96

103.28

[B;B] Ccl = [C 0; 0 0]). A simple proportional ? derivative output error filtering approach was employed to impose the SPR property. The selected matrix filter was: PSPR(s) = [(s ? 1) 0; 0 (s ? 1)]. The performance of the four methods was then compared with that of the PL approach. The mean absolute value of the tracking error and of the overall control effort, evaluated for a 2000s simulation, was used to measure the performance. The PL approach was considered as the baseline (performance = 100 %) and the performance of the other methods was expressed as percentage of the baseline. Table 1 reports the qualitative results. Analyzing the table, it is evident that all the experimented adaptive learning methodologies provide significantly better performance than the PL approach. The best performance was achieved by the SPR approach that was able to reduce the mean tracking

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Fig. 3 Components of the adaptive control ua1 and ua2 and of ub1 ¼ h^1 and ub2 ¼ h^2 and amplitude boundary values in the first 300s of simulation (SPR case). It shows the output saturations bits ca. Note that the components of ub vector never saturates, and therefore, the components of the cb binary signals are low all the time

^ 1 ðtÞjj Fig. 4 Evolution of jjW ^ 2 ðtÞjj and the and of jjW corresponding NN weights. The weights’ initial conditions were set to 0. In the figures are also shown the output saturations bits ca

error to 15 % for e1 and to 16 % for e2. For all the approaches, it is worth noting that the improvement was not achieved with a large increment of the overall control effort; in fact, the control activity in the two channels is comparable to that requested by the PL approach. To evaluate the capacity of the adaptive NN controller to effectively learn the uncertainties, additional experiments were carried out by ‘‘freezing’’ the NNs’ weights to the values at the end of the 2000s simulation and repeating

an equal length simulation for validation purpose. The results are reported in the last two rows of Table 1 for the SF and SPR approach. In the SF case, it was observed a modest performance degradation (vs the adaptive case) testifying that the NNs have correctly learned the uncertainty functions in the previous teaching phase. In the SPR case, the performance degrades but sill remains comparable to that of SF in the adaptive case. As final remark, we conclude that although the PL approach, working alone can

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guarantee a smaller worst-case tracking error performance polyhedron; in practice, any one of the four NN adaptive schemes was able to provide a significant actual performance improvement (even in case the adaptation was disabled) without increasing the control activity. The results of this comparative analysis are meaningful and provide a clear answer to the question raised in Sect. 6.2. about the utility of using the LA scheme instead of the simpler PL approach.

Fig. 5 Comparison of tracking error performance for e1 and e3 in the time interval 0–300s under PL and adaptive SPR control

Fig. 6 a Uncertainly D1(x2(t)) (thick line) and its approximation D^1 ð^ x2 ðtÞÞ (thin line) in the interval 1800–2000s. (b) Uncertainly D2(x4(t)) (thick line) and its approximation D^2 ð^ x4 ðtÞÞ (thin line). c Coefficients h1 and h2 and their estimations h^1 and h^2

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6.7 Additional results for the SPR approach This section illustrates in more detail the performance of the novel weights adaptation and output confinement algorithm in case of SPR approach. Figure 3 shows the components of adaptive signals ua and ub along with the corresponding evolution of the output saturation bits. It can be observed that the proposed parameter adaptation algorithm is able to manage successfully the output confinement maintaining the adaptive control signals within the desired ranges. In case an output saturation bit is high, the weights redistribution mechanics is applied. Figure 4 shows the evolution of the norms of the NNs weight vectors and the associated 81 weights for the two components of ua along with the output saturations bits ca in the first 1000s. It can be observed the effects of the weights redistribution does not alter significantly the information stored in the NNs, but introduces only a small local reduction in the magnitude of weights. Figure 5 compares the tracking error performance for e1 and e3 for the PL and the adaptive SPR approach. As the time increases, an evident improvement is observed in the adaptive tracking performance thanks to the online learning capacity of the NNs. As for the NNs’ dynamic approximation performance, Fig. 6a shows a comparison between the uncertainty ^1 ð^ D1(x2(t)) and its approximation D x2 ðtÞÞ provided by the NN1 in the last 200 s of the simulation; similarly Fig. 6b ^2 ð^ x4 ðtÞÞ provided shows D2(x4(t)) and its approximation D

Neural Comput & Applic (2014) 25:1313–1328

by NN2. Figure 6c shows the estimation of the coefficients k11 and k22 associated to the ub control. Analyzing the Fig. 6, it is evident that the proposed learning and confinement algorithm is effective also in providing a satisfactory dynamic approximation of the system uncertainties. It is expected that the approximation accuracy could be also improved by extending the adaptation period.

7 Conclusions This paper introduced a novel approach for designing a mixed linear and neural adaptive output feedback control for uncertain systems with guaranteed transitory componentwise performance. We find the design framework based on robust invariant set theory to be particularly effective because in this context the design of the observer, controller and componentwise tracking error requirements can be formulated as an overall LMI problem. It follows an analysis and design that is rigorous and clear. An important benefit of the scheme is that the robust invariant set containing the error trajectories can be explicitly defined in the design phase, allowing componentwise specifications. In this sense, the proposed technique overcomes some limitations of many existing design methods in adaptive neural control. An additional benefit is that no specific NN structure or parameter adaptation algorithm is required, as long as the NN outputs are confined within the predefined limits. For enforcing this confinement, the novel mechanism of adaptive control redistribution is introduced and different NN teaching strategies based on state and output feedback have been evaluated. A detailed simulation study was carried out to show the design steps and to highlight the benefits of the methodology compared to a pure linear controller design. The comparative analysis revealed clearly that although the linear controller guarantees smaller error bounds, the joint linear and adaptive scheme, thanks to adaptive NN learning property, provides, in practice, a significant better performance that improves over time. Particularly relevant is the fact that the performance improvement is achieved without increasing the overall control activity. All these facts highlighted a fundamental tradeoff that is: the price to pay to have potentially better performances (thanks to the action of the online learning NN element) is a widening of the worst-case performance bounds with respect to a pure linear control design.

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