Abstract
In this paper, we propose an extension of the binary model reference adaptive control (BMRAC) for uncertain multivariable linear plants with non-uniform arbitrary relative degree. The BMRAC is a robust adaptive strategy which has good transient properties and robustness of sliding mode control with the important advantage of having a continuous control signal free of chattering. The relative degree obstacle is circumvented using a multivariable version of a hybrid estimation scheme, named global robust exact differentiator (GRED). The hybrid estimator switches between robust exact differentiators (RED) based on higher-order sliding modes and lead filters in a way that the exact derivatives are globally obtained in finite time. To improve the robustness and the transient performance of the GRED, we propose a modification of the switching scheme replacing the conventional RED with a non-homogeneous one. Global exact output tracking is obtained with robustness and guaranteed transient performance without requiring stringent symmetry assumptions on the plant high-frequency-gain matrix.
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Notes
In direct adaptive control, the controller parameters are directly updated from an adaptive law.
The tracking of more general reference models could be obtained by simply preshaping the reference signal r through a precompensator at the input of the above model.
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Appendix A Proofs of Theorems
Appendix A Proofs of Theorems
1.1 A.1 Proof of Lemma 1
Applying the change of variables \(\upsilon _i = \zeta _i - f^{(i)}(t)\), \(i=0,1,\ldots ,n\) to the system (30), it follows that
Considering that \(x = |x|{\text {sgn}}\left( x\right) \) and \({\text {sgn}}\left( \upsilon _i - \dot{\upsilon }_{i-1}\right) = {\text {sgn}}\left( \upsilon _{i-1} - \dot{\upsilon }_{i-2}\right) \) for \(i=2,\ldots ,n\), and thus \({\text {sgn}}\left( \upsilon _i - \dot{\upsilon }_{i-1}\right) = {\text {sgn}}\left( \upsilon _0\right) \), for \(i=1,2,\ldots ,n\), the system can be rewritten as
It follows by mathematical induction that the non-recursive form of the system is given by
where
Note that \(\psi _i\) is constant, and \(\phi _i(|\upsilon _0|)\) obeys the following conditions
for \(i=0,1,\ldots ,n\) and some positive constants \(\kappa _1^{[i]}\) and \(\kappa _2^{[i]}\). The first inequality follows from the fact that \(\phi _i\) is bounded, for \(|\upsilon _0| \le 1\), if \(\phi _{i-1}\) is also bounded. Since \(\phi _0\) is bounded under these conditions, then the first inequality follows by mathematical induction. The second inequality can be demonstrated considering that
and the fact that, for \(|\upsilon _0| > 1\), \(\frac{\phi _i}{|\upsilon _0|}\) is bounded if \(\frac{\phi _{i-1}}{|\upsilon _0|}\) is also bounded. Since \(\frac{\phi _0}{|\upsilon _0|}\) is bounded under these conditions, then the second inequality follows by mathematical induction.
The equations \({\dot{\upsilon }}_i = -\left[ \phi _i(|\upsilon _0|) + \psi _i |\upsilon _0|\right] {\text {sgn}}\left( \upsilon _0\right) + \upsilon _{i+1}\), \(i=0,1,\ldots ,n-1\) can be rewritten as
where
The equation \({\dot{\upsilon }}_n = -\left[ \phi _n(|\upsilon _0|) + \psi _n |\upsilon _0|\right] {\text {sgn}}\left( \upsilon _0\right) - f^{(n+1)}(t)\) can be rewritten as
where
Note that, once \(\left| f^{(n+1)}(t) \right| \le K_{n+1} \,\, \forall t\), then \(\left| a_i(\upsilon _0) \right| \le K_{a_i}\) and \(\left| b_i(\upsilon _0) \right| \le K_{b_i}\), for \(i=0,1,\ldots ,n\) and some positive constants \(K_{a_i}\) and \(K_{b_i}\). Defining the complete state vector as \(\varUpsilon = \left[ \begin{array}{cccc} \upsilon _0&\upsilon _1&\ldots&\upsilon _n \end{array}\right] ^T\) , the system (43) can be rewritten as
where
and it follows that \(\Vert b(\varUpsilon ) \Vert \le c_1\) and \(\Vert A(\varUpsilon ) \Vert \le c_2\) for some positive constants \(c_1\) and \(c_2\).
Consider the function
It can be verified that
Consider the comparison equation \(\dot{V_c}(\varUpsilon ) = 2c_2 V_c(\varUpsilon ) + 2c_1 \sqrt{V_c(\varUpsilon )}\). If \(V_c(0) = V(0)\), then \(V(t) \le V_c(t), \forall t \ge 0\). Introducing the new variable \(\chi ^2 = V_c\), it follows that
Considering \(\chi \ne 0\), then \(\frac{\mathrm{d}\chi }{\mathrm{d}t} = c_2 \chi + c_1 \). In this case, it can be shown that
and
Then the function V(t) cannot escape in finite time for any finite constant \(K_{n+1}\), and thus \(\upsilon \in {\mathcal {L}}_{\infty e}\). Furthermore, since the signals \(f^{(i)}(t)\), \(i=0,1,\ldots ,n,\) are bounded, then one can conclude that the state \(\zeta \) cannot escape in finite time. \(\square \)
1.2 A.2 Proof of Theorem 2
The estimate given by the MIMO lead filter and the MIMO RED could be related to \(\xi _y\) in (19) as follows:
where \(\varepsilon _{l}\) and \(\varepsilon _{r}\) are estimation errors. From (44), equation (38) can be rewritten as
From (39), the estimation error \(\varepsilon _g(t)\) can be rewritten as:
where by design \(\beta _{\alpha }({\tilde{\nu }}_{rl}(t))\) is uniformly bounded by
Substituting (45),(46) into (40)–(42), it can be seen that GRED adaptive law is equivalent to lead adaptive law (26)–(28) with an output disturbance \(\left| \! \left| \beta _{\alpha }({\tilde{\nu }}_{rl}(t)) \right| \! \right| \le \varepsilon _M\).
Therefore, Theorem 1 holds if all signals of the GRED-BMRAC system belong to \(L_{\infty e}\). In order to demonstrate that the condition is true, we only have to show that all signals in the MIMO RED system are \(L_{\infty e}\). This property can be proved by contradiction. Suppose that the maximal interval of finiteness of the signals in the MIMO RED is \([0,T_M)\). During this interval, all conditions of Theorem 1 hold, and thus, all signals of the remaining subsystems of the GRED-BMRAC are bounded by a constant, and in particular, \(\left| y_j^{(i)}(t)\right| , i=0,\dots ,\rho _j, \ j = 1,\dots ,M \), from Corollary 2. This leads to a contradiction with Lemma 1, whereby the signals in the MIMO RED could not diverge unboundedly as \(t \rightarrow T_M\). As a consequence of the continuation theorem for differential equations (in Filippov’s theory), \(T_M\) must be \(\infty \), which means that all signals are defined \(\forall t \ge 0\). Thus, Theorem 1 is valid for the GRED-MRAC system and the closed-loop error system with state z is GEpS with respect to a residual set.
Now, we will analyze the ultimate convergence of the GRED-BMRAC. According to Corollary 1, for sufficiently small \(\tau \) and sufficiently large \(\gamma \) the error state z is steered to an invariant compact set \(D_R := \{ z : \left| z(t)\right| < R \}\) in some finite time \(T_1 \ge 0\). Consider the following Lyapunov candidate
whose time derivative is
following the previous steps
Within \(D_R\), \(x_e\) can be upper bounded by \(\left| \! \left| x_e \right| \! \right| \le R\) such that
Thus, it is possible to show that
Since \(\left| \! \left| \varepsilon _l \right| \! \right| \le \left| \! \left| x_\varepsilon \right| \! \right| \), it is straightforward to show that for some finite \(T_2 \ge T_1\), \(\left| \! \left| \varepsilon _l \right| \! \right| \le {{\bar{\varepsilon }}}_l\), where \({{\bar{\varepsilon }}}_l = \tau K_{l}\).
Since the MIMO RED is time-invariant, its initial conditions can be considered to be at \(t=T_1\). From Lemma 1, the initial conditions are finite. If the parameters \(\lambda _i^j\) are adjusted properly, then from Levant (2009) the estimation error \(\varepsilon _{r}(t)\) converges to zero in some finite time \(T_3 > T_1\).
Since \(K_R\) is chosen such that \(\varepsilon _M > {\bar{\varepsilon }}_{l} + \varDelta \) and from (39), it follows that after some finite time \({{\bar{T}}} = \max \{ T_2, T_3\}\) the estimation of \(\sigma \) becomes exact and being made exclusively by the MIMO RED (\(\alpha ({\tilde{\nu }}_{rl}) = 0\)), which implies that \(\varepsilon _g(t) = 0, \forall t \ge {\bar{T}}\).
In this case, the overall error system can be described by
since this system has uniform relative degree one we can apply the result obtained in Yanque et al. (2012). Thus, it is possible to conclude that \(e(t) \rightarrow 0 \). \(\square \)
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Battistel, A., Vidal, P.V.N.M., Nunes, E.V.L. et al. Multivariable Binary MRAC with Guaranteed Transient Performance Using Non-homogeneous Robust Exact Differentiators. J Control Autom Electr Syst 32, 378–389 (2021). https://doi.org/10.1007/s40313-020-00680-y
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DOI: https://doi.org/10.1007/s40313-020-00680-y