Keywords

Introduction

The central problem in adaptive control pertains to regulation and tracking of systems in the presence of parametric uncertainties. The classical adaptive control problem solved in 1980 assumed that the underlying transfer function had unknown parameters, but no other uncertainties. No disturbances, delays, time variations in parameters, or unmodeled dynamics were assumed to be present. Under these ideal conditions, it was shown that an adaptive controller can be designed so that the closed-loop system has bounded signals and that asymptotic regulation and tracking were possible.

With asymptotic regulation and tracking achieved under such ideal conditions, the goal of robust adaptive control was to ensure globally bounded signals in the closed-loop adaptive system when the plant was subjected to a variety of nonparametric perturbations such as external disturbances, time-varying parameters, unmodeled dynamics, and time delays. With adaptation in the control parameters in the ideal case accommodating parametric uncertainties, the approaches developed in robust adaptive control focused on developing solutions in the perturbed case to accommodate nonparametric uncertainties and improving on the classical adaptive controller which either underperformed or even exhibited instability with the introduction of nonparametric perturbations.

We briefly present the adaptive control solutions for the ideal case before proceeding with robust adaptive control.

Classical Adaptive Control

Adaptive Control of Plants with State Feedback

One of the very first problems where stable adaptive control was solved was for the case when states are accessible (Narendra and Kudva 1972), with the plant given by (The argument t is suppressed for the sake of convenience, except for emphasis.)

$$\displaystyle\begin{array}{rcl} \dot{x}_{p} = A_{p}x_{p} + b\lambda u& &{}\end{array}$$
(1)

where \(A_{p} \in \mathbb{R}^{n\times n}\) and the scalar λ are unknown parameters with b and the sign of λ known and (A p , b) controllable. An adaptive controller that ensures global boundedness and asymptotic regulation and tracking for such plants is of the form

$$\displaystyle\begin{array}{rcl} u =\theta _{ x}^{T}(t)x_{ p} +\theta _{r}(t)r,& &{}\end{array}$$
(2)

and the adaptive laws for adjusting the unknown parameters are given by

$$\displaystyle\begin{array}{rcl} \dot{\theta } = -\mathrm{sign}(\lambda )\Gamma \omega b_{m}^{T}Pe,& &{}\end{array}$$
(3)

where \(\omega = \left [\begin{array}{*{10}c} x_{p}^{T},&r \end{array} \right ]^{T}\) and \(\theta = \left [\begin{array}{*{10}c} \theta _{x}^{T},&\theta _{r} \end{array} \right ]^{T}\), x m is the state of a reference model

$$\displaystyle\begin{array}{rcl} \dot{x}_{m} = A_{m}x_{m} + br& &{}\end{array}$$
(4)

with A m Hurwitz, and P being the solution of the Lyapunov equation \(A_{m}^{T}P + PA_{m} = -Q\), Q > 0. The reference model in (4) is to be chosen so that certain matching conditions are satisfied, which are of the form

$$\displaystyle\begin{array}{rcl} & A_{p} + b\lambda \theta _{x}^{\star T} = A_{m},\qquad \lambda \theta _{r}^{{\ast}} = 1&{}\end{array}$$
(5)

for some \(\theta ^{{\ast}} = [\theta _{x}^{{\ast}T},\theta _{r}^{{\ast}}]^{T}\). In such a case, the controller in (2) and (3) guarantees stability and ensures that x(t) tracks x m (t). The underlying Lyapunov function is quadratic in e and the parameter error \(\tilde{\theta }=\theta -\theta ^{{\ast}}\), given by

$$\displaystyle\begin{array}{rcl} V = \frac{1} {2}\left (e^{T}Pe +\lambda \tilde{\theta } ^{T}\Gamma ^{-1}\tilde{\theta }\right )& &{}\end{array}$$
(6)

with a negative semi-definite time derivative \(\dot{V }\) given by

$$\displaystyle\begin{array}{rcl} \dot{V } \leq -e^{T}Qe.& &{}\end{array}$$
(7)

Adaptive Control of Plants with Output Feedback

Consider the single-input single-output (SISO) system of equations

$$\displaystyle{ y(t) = W(s)u(t) }$$
(8)

where u ∈ ℜ is the input, y ∈ ℜ the measurable output, and s the differential operator. The transfer function of the plant is parameterized as

$$\displaystyle{ W(s) \triangleq k_{p}\frac{Z(s)} {P(s)} }$$
(9)

where k p is a scalar and Z(s) and P(s) are monic polynomials with deg(Z(s)) < deg(P(s)). The following assumptions will be made throughout:

Assumption 1

W(s) is minimum phase.

Assumption 2

The sign of k p is known.

Assumption 3

The relative degree n∗ and the order of W(s) are known.

The goal is to design a control input u so that the output y in (8) tracks the output y m of the reference system

$$\displaystyle{ y_{m}(t) = W_{m}(s)r(t) \triangleq k_{m}\frac{Z_{m}(s)} {P_{m}(s)}r(t) }$$
(10)

where k m is a scalar and Z m (s) and P m (s) are monic polynomials with W m (s) relative degree n∗.

The structure of the adaptive controller is now presented:

$$\displaystyle\begin{array}{rcl} \dot{\omega }_{1}(t)& = \Lambda \omega _{1} + b_{\lambda }u(t)&{}\end{array}$$
(11)
$$\displaystyle\begin{array}{rcl} \dot{\omega }_{2}(t)& = \Lambda \omega _{2} + b_{\lambda }y(t)&{}\end{array}$$
(12)
$$\displaystyle\begin{array}{rcl} \omega (t)& \triangleq [r(t),\ \omega _{1}^{T}(t),\ y(t),\ \omega _{2}^{T}(t)]^{T}&{}\end{array}$$
(13)
$$\displaystyle\begin{array}{rcl} \theta (t)& \triangleq [k(t),\ \theta _{1}^{T}(t),\ \theta _{0}(t),\ \theta _{2}^{T}(t)]^{T}&{}\end{array}$$
(14)
$$\displaystyle\begin{array}{rcl} u& =\theta ^{T}(t)\omega &{}\end{array}$$
(15)

where \(\Lambda \in \mathfrak{R}^{(n-1)\times (n-1)}\) is Hurwitzx, \(b_{\lambda } \in \mathfrak{R}^{n-1}\), \(\omega _{1},\omega _{2} \in \mathfrak{R}^{n-1}\), and \(\theta \in \mathfrak{R}^{2n}\) is an adaptive gain vector with \(k(t) \in \mathfrak{R}\), θ1(t) ∈ ℜn−1, θ2(t) ∈ ℜn−1, and θ0(t) ∈ ℜ.

The update law for the adaptive parameter differs depending on whether the relative degree of W m (s) is unity or greater than one and can be described as follows:

$$\displaystyle{ \dot{\theta }(t) = -\text{sign}(k_{p})\Gamma e_{y}\omega \qquad n^{{\ast}} = 1 }$$
(16)

and

$$\displaystyle{ \dot{\theta }(t) = -\text{sign}(k_{p})\Gamma \frac{e_{a}\zeta } {1 +\zeta ^{T}\zeta }\qquad n^{{\ast}}\geq 2 }$$
(17)

where \(e_{y} = y - y_{m}\), e a is an augmented error, and ζ is a modified regressor, both of which are determined by the following equations:

$$\displaystyle\begin{array}{rcl} \zeta & = W(s)\omega,\;\;\;\;\omega = [r,\omega _{1}^{T},y,\omega _{2}^{T}]^{T},&{}\end{array}$$
(18)
$$\displaystyle\begin{array}{rcl} e_{2}& =\theta ^{T}\zeta - W(s)[\theta ^{T}\omega ]&{}\end{array}$$
(19)
$$\displaystyle\begin{array}{rcl} e_{a}& = e_{y} + k_{1}(t)e_{2}&{}\end{array}$$
(20)
$$\displaystyle\begin{array}{rcl} \dot{k}_{1}& = -\frac{e_{a}e_{2}} {1+\zeta ^{T}\zeta }&{}\end{array}$$
(21)

The results of Narendra and Annaswamy (2005) guarantee that the above adaptive controller in Eqs. (11)–(21) will guarantee that e y (t) tends to zero as \(t \rightarrow \infty\) with all signals remaining bounded in both the n∗ = 1 and n∗ ≥ 2 cases.

Need for Robust Adaptive Control

When a disturbance η is present, the plant dynamics often is of the form

$$\displaystyle\begin{array}{rcl} \dot{x}_{p} = A_{p}x_{p} + b\lambda (u +\eta (t))& &{}\end{array}$$
(22)

while the reference model and the controller remain the same as in (4) and (2), respectively. This in turn necessitates new tools for the analysis and synthesis of adaptive systems. The main reason for this is the fact that the standard Lyapunov function candidate given by

$$\displaystyle\begin{array}{rcl} V = \frac{1} {2}e^{T}Pe + \frac{1} {2}\lambda \tilde{\theta }^{T}\Gamma ^{-1}\tilde{\theta }& &{}\end{array}$$
(23)

together with the parameter adjustment as in (3) yields a time derivative

$$\displaystyle\begin{array}{rcl} \dot{V } \leq -\frac{1} {2}e^{T}Qe + k_{ 1}\Vert e\Vert d_{0}\qquad k_{1}> 0,& &{}\end{array}$$
(24)

where d0 is an upper bound on the perturbation η. The second term on the right-hand side of (24) causes \(\dot{V }\) to be sign indefinite. This is because V is a function of both e and \(\tilde{\theta }\), and therefore, the second term can be large compared to the first with the second argument of V, \(\tilde{\theta }\), which can be arbitrary, causing \(\dot{V }\) to be sign indefinite. The same property is what caused \(\dot{V }\) to be semi-definite in the ideal case. Hence, in this perturbed case, no guarantees of boundedness can be provided. In fact, it can be shown that if η(t) is chosen in a particular manner, the closed-loop signals can actually be shown to become unbounded, either in the presence of bounded disturbances (Narendra and Annaswamy 2005) or with unmodeled dynamics (Rohrs et al. 1985). This in turn led to the area of robust adaptive control.

Various approaches that have been developed under the rubric of robust adaptive control can be grouped into two categories. The first of these is related to modifications in the adaptive laws so as to ensure boundedness. These changes consist of modifications in the adaptive law (3) as

$$\displaystyle\begin{array}{rcl} \dot{\theta } = -\Gamma \omega (t)b_{m}^{T}Pe -\sigma g(\theta,e)& &{}\end{array}$$
(25)

The problem then reduces to finding a suitable g(θ, e). This is discussed in detail in the next section. The second approach used in adaptive control pertains to the use of a persistently exciting reference signal r. The latter ensures parameter convergence of the adaptive system and therefore exponential stability. This in turn ensures robustness of the overall system. These details are addressed in section “Robust Adaptive Control with Persistently Exciting Reference Input.”

Robust Adaptive Control with Modifications in the Adaptive Law

Robustness to Bounded Disturbances

When a bounded input disturbance η is present, the plant dynamics is changed as

$$\displaystyle\begin{array}{rcl} \dot{x}_{p} = A_{p}x_{p} + b\lambda (u +\eta (t)),& &{}\end{array}$$
(26)

while the reference model and the controller remain the same as in (4) and (2), respectively. As mentioned above, a modification to the adaptive law as in (25) is needed. Over the years, different choices have been suggested for the nonlinear function g(θ, e). For example, these are chosen as

$$\displaystyle\begin{array}{rcl} g(\theta,e) = \left \{\begin{array}{ll} \theta &\mbox{ Ioannou and Sun (2013)} \\ \vert \vert e\vert \vert \theta &\mbox{ Narendra and Annaswamy (2005)} \\ \theta \left (1 - \frac{\vert \vert \theta \vert \vert } {\theta _{\mathrm{max}}} \right )^{2} & \mbox{ Kreisselmeier and Narendra (1982)} \end{array} \right.& &{}\end{array}$$
(27)

where θmax is a known bound on the parameter θ. (One can choose to set σ to zero if \(\Vert \theta \Vert \leq \theta _{\mathrm{max}}\), as is done in Ioannou and Sun (2013), Tsakalis and Ioannou (1987) and many other references in the literature.) An alternate approach that is different from (25) is to not have an additive term \(g(\cdot,\dot{)}\) but rather set \(\dot{\theta }= 0\) whenever the error e is small in some sense. Such a dead zone approach was suggested, for example, in Egardt (1979) and Peterson and Narendra (1982). It can be shown that each one of these choices leads to boundedness, which is described below. Without loss of generality, we assume that λ > 0.

With the same Lyapunov function candidate as in (23), its time derivative now becomes

$$\displaystyle\begin{array}{rcl} \dot{V }& \leq & -\frac{1} {2}e^{T}Qe + k_{ 1}\Vert e\Vert \Vert \eta \Vert \\ & & -\frac{1} {2}\Vert \tilde{\theta }\Vert ^{T}g(\theta,e),\qquad k_{ 1}> 0{}\end{array}$$
(28)

The property of g(. , . ), together with the fact that η is bounded, ensures that \(\dot{V } <0\) outside a compact set \(\Omega\) in the \((e,\tilde{\theta })\) space. This ensures global boundedness of both e and \(\tilde{\theta }\). Boundedness of x p follows.

In all of the above methods, the idea behind adding the term g(e, θ) is this: the parameter θ can drift away from the correct direction due to the term \(k_{1}\Vert e\Vert \Vert \eta \Vert\), and the construction of g(e, θ) is such that it counteracts this drift and keeps the parameter in check, by adding a negative quadratic term in \(\tilde{\theta }\). The boundedness of both e and θ is simultaneously assured in the above since V has a time derivative \(\dot{V }\) that is nonpositive outside a compact set in the \((e,\tilde{\theta })\) space. It should be noted however that this was possible to a large extent because η was bounded, and as a result, the sign-indefinite term remained linear in \(\Vert e\Vert\).

An alternative procedure, originally proposed in Pomet and Praly (1992) and revised and refined in Khalil (2001) and Lavretsky (2010), proceeds in a slightly different manner. Here, the boundedness of θ is first established, independent of the error equation. It should be noted that a similar approach is adopted in the context of output feedback in plants with higher relative degree by using normalization and an augmented error approach (Narendra and Annaswamy 2005). In Khalil (2001) and Lavretsky (2010), no normalization is used but a projection algorithm. This is described below.

The projection algorithm for adjusting the parameter θ is given by

$$\displaystyle\begin{array}{rcl} \begin{array}{ll} \dot{\theta } = \text{Proj}(\theta,y),\qquad \end{array} & &{}\end{array}$$
(29)

where

$$\displaystyle\begin{array}{rcl} \text{Proj}(\theta,y) = \left \{\begin{array}{l} y -\frac{\nabla f(\theta )(\nabla f(\theta ))^{T}} {\parallel \nabla f(\theta )\parallel ^{2}} yf(\theta ) \\ \qquad \qquad \ \mbox{ if}\quad [f(\theta )> 0\bigwedge y^{T}\nabla f(\theta )> 0] \\ y\qquad \qquad \mbox{ otherwise} \end{array} \right.& &{}\end{array}$$
(30)
$$\displaystyle\begin{array}{rcl} y& =& -e^{T}Pb\omega {}\end{array}$$
(31)
$$\displaystyle\begin{array}{rcl} f(\theta )& =& \frac{\parallel \theta \parallel ^{2} -\theta _{\text{max}}^{{\prime}2}} {\varepsilon ^{2} + 2\varepsilon \theta _{\text{max}}^{{\prime}}}{}\end{array}$$
(32)

where θmax′ and \(\varepsilon\) are arbitrary positive constants, and \(\Omega _{0}\) and \(\Omega _{1}\) are defined as

$$\displaystyle\begin{array}{rcl} \begin{array}{ll} \Omega _{0} & =\big\{\theta \in \mathbb{R}^{n}\vert f(\theta ) \leq 0\big\} \\ \Omega _{1} & =\big\{\theta \in \mathbb{R}^{n}\vert f(\theta ) \leq 1\big\}.\end{array} & &{}\end{array}$$
(33)

From the above relations, one can show that

$$\displaystyle{\theta (0) \in \Omega _{0}\Rightarrow\theta (t) \in \Omega _{1}.}$$

In addition,

$$\displaystyle\begin{array}{rcl} \theta _{\text{max}}^{{\prime}}& =& \max _{\theta \in \Omega _{0}}\left (\Vert \theta \Vert \right ),\quad \theta _{\text{max}}\; =\;\max _{\theta \in \Omega _{1}}\left (\Vert \theta \Vert \right ){}\end{array}$$
(34)

where \(\theta _{\text{max}} =\theta _{ \text{max}}^{{\prime}}+\varepsilon\) (Matsutani et al. 2011).

Robustness to Unmodeled Dynamics

One of the major observations in the early eighties was the stark difference between the system signals in the ideal adaptive system and the perturbed adaptive system when the perturbation was due to a commonly present unmodeled dynamics such as those of an actuator used for control implementation. Among other references, the publication in Rohrs et al. (1985) pointed out the fact that when an adaptive controller prescribed for a first-order plant is evaluated with unmodeled dynamics present, instability occurs readily and for a wide range of command signals. A number of solutions have been suggested to alleviate this instability and form the subject matter of this section.

We consider the plant in (26) with an additional unmodeled dynamics so that

$$\displaystyle\begin{array}{rcl} \begin{array}{ll} \dot{x}_{p} =&A_{p}x_{p} + b\lambda v \\ \dot{x}_{\eta } = &A_{\eta }x_{\eta } + b_{\eta }u,\,\,\,\,v =\tilde{ c}_{\eta }^{T}x_{\eta }.\end{array} & &{}\end{array}$$
(35)

where A η is a Hurwitz matrix. If \(\eta = v - u\), then the plant dynamics can be rewritten as

$$\displaystyle{ \dot{x}_{p} = A_{p}x_{p} + b\lambda (u+\eta ) }$$
(36)

Unlike the bounded disturbance case, no upper bound d0 can be assumed to exist as η is a state-dependent disturbance. It is this that causes a huge difference between deriving boundedness in section “Robustness to Bounded Disturbances” and here in section “Robustness to Unmodeled Dynamics.” Significant effort has been extended in the adaptive control community in this regard. These results fall into two categories (i) that assure global boundedness for a narrow class of unmodeled dynamics and (ii) that assure semi-global boundedness for a slightly larger class of unmodeled dynamics. More recently, some results have been obtained that are able to establish global boundedness with minimal restrictions on the unmodeled dynamics. In what follows, we give examples of each of the above two cases as well as the recent results.

Global Boundedness in the Presence of a Small Class of Unmodeled Dynamics

For the plant in (26), under assumptions in (5), the plant can be rewritten as

$$\displaystyle{ \dot{x} = A_{m}x + b\lambda (u +\theta _{ x}^{{\ast}T}x+\eta ) }$$
(37)

where λ and θ x ∗ are unknown, A m and b are known, and \(\eta = v - u\) whose state-space representation can be shown to be of the form

$$\displaystyle\begin{array}{rcl} \dot{x}_{\eta } = A_{\eta }x_{\eta } + b_{\eta }u,\,\,\,\,\eta = c_{\eta }^{T}x_{\eta }& &{}\end{array}$$
(38)

for some vector c η .

For a class of unmodeled dynamics \(\{c_{\eta },A_{\eta },b_{\eta }\}\), if the control input in (2) and the projection algorithm in (29) with y and f(θ) chosen as in (31) and (32) are used, one can guarantee boundedness. In particular, if the inequality

$$\displaystyle{ k\theta _{x,\mathrm{max}}\lambda _{\mathrm{max}}\left (\frac{b_{0}} {\sigma _{A_{\eta }}} \right ) <1 }$$
(39)

is satisfied, where b0 is an upper bound on | | b η | | and σ A denotes the singular value of the matrix A, then boundedness follows. That is, robustness of adaptive controllers can be ensured if the unmodeled dynamics is fast and their zeros are restricted in some sense.

A specific example of such an unmodeled dynamics is given by

$$\displaystyle\begin{array}{rcl} c_{\eta }^{T}(sI - A_{\eta })^{-1}b_{\eta } = \frac{-2\mu s} {1 +\mu s}.& &{}\end{array}$$
(40)

Global Boundedness for a Large Class of Unmodeled Dynamics: A First-Order Example

A different approach can be taken for the problem of unmodeled dynamics which allows a global result, for a class of adaptive systems (Hussain et al. 2013). The main idea here is to use the projection algorithm and use properties of adaptive systems in conjunction with linear time-varying systems. This is presented in this section using a first-order plant.

We consider the control of

$$\displaystyle{ \dot{x}_{p}(t) = a_{p}x_{p}(t) + k_{p}v(t) }$$
(41)

where a p is unknown and k p is known. It is assumed that \(\vert a_{p}\vert \leq \bar{ a}\), where \(\bar{a}\) is a known positive constant. The unmodeled dynamics is given by (38) with

$$\displaystyle{ G_{\eta }(s) \triangleq c_{\eta }^{T}(sI_{ n\mathsf{x}n} - A_{\eta })^{-1}b_{\eta }. }$$
(42)

The goal is to design the control input such that x p (t) follows x m (t) which is specified by the reference model

$$\displaystyle{ \dot{x}_{m}(t) = a_{m}x_{m}(t) + k_{m}r(t) }$$
(43)

where a m < 0 and r(t) is the reference input. The adaptive controller we propose is given by

$$\displaystyle{ u(t) =\theta (t)x_{p}(t) + \frac{k_{m}} {k_{p}} r(t) }$$
(44)

where the parameter θ(t) is updated using a projection algorithm given by

$$\displaystyle\begin{array}{rcl} \dot{\theta }(t)& =& \gamma \mathop{\mathrm{Proj}}\nolimits (\theta (t),-x_{p}(t)(x_{p}(t) - x_{m}(t))), \\ & & \gamma> 0 {}\end{array}$$
(45)

and

$$\displaystyle\begin{array}{rcl} \mathop{\mathrm{Proj}}\nolimits (& \theta,y)\,=\left \{\begin{array}{@{}l@{\quad }l@{}} \frac{\theta _{\max }^{2} -\theta ^{2}} {\theta _{\max }^{2} -\theta _{\max }^{{\prime}2}}y\quad &[\theta \in \Omega _{A},\;y\theta> 0] \\ y \quad &\text{otherwise} \end{array} \right.&{}\end{array}$$
(46)
$$\displaystyle\begin{array}{rcl} \Omega _{0}& =& \{\theta \in \mathbb{R}^{1}\;\vert \; -\theta _{\max }^{{\prime}}\leq \theta \leq \theta _{\max }^{{\prime}}\} \\ \Omega _{1}& =& \{\theta \in \mathbb{R}^{1}\;\vert \; -\theta _{\max }\leq \theta \leq \theta _{\max }\} \\ \Omega _{A}& =& \Omega _{1}\setminus \Omega _{0} {}\end{array}$$
(47)

with positive constants \(\theta _{\max }^{{\prime}}\) and θmax given by

$$\displaystyle{ \theta _{\max }^{{\prime}}> \frac{\bar{a} + \vert a_{m}\vert } {k_{p}},\qquad \theta _{\max } =\theta _{ \max }^{{\prime}} +\varepsilon _{ 0}, }$$
(48)

where \(\varepsilon _{0}\) is an arbitrary constant. It can be shown that if θmax is chosen as in (48), then the closed adaptive system specified by Eqs. (41)–(48) always has guaranteed bounded solutions for a class of unmodeled dynamics G η (s). There is an optimal value of \(\varepsilon _{0}\), however, for which a largest class of G η (s) can be found.

It should be noted that in the Rohrs example in Rohrs et al. (1985), the plant is first order, with \(a_{p} = -1\), and

$$\displaystyle{ G_{\eta } = \frac{w_{n}^{2}} {s^{2} + 2\zeta \omega _{n}s +\omega _{ n}^{2}}, }$$
(49)

for \(\zeta = 1,\;\;\omega _{n} = 15\). It is easy to show that for these values of ζ and ω n , if θmax = 17, then Eq. (48) is satisfied and that the closed-loop system is robust to G η .

In general, for a first-order plant as in (41), it can be shown that the adaptive system is robust for G η for all (ζ, ω n ) that satisfy the following inequalities for all \(\vert a_{p}\vert \leq \overline{a}\):

$$\displaystyle\begin{array}{rcl} \begin{array}{ll} - a_{p}\zeta ^{2} + f(a_{p},\omega _{n})\zeta -\frac{k_{p}\theta _{\max }} {4} &> 0\\ \omega _{ n} &>\omega _{n_{\min }}\end{array} & &{}\end{array}$$
(50)

where

$$\displaystyle\begin{array}{rcl} f(a_{p},\omega _{n})& =& \frac{a_{p}^{2} +\omega _{ n}^{2}} {2\omega _{n}} \\ w_{n_{\min }}& =& \max \left (\frac{\bar{a}} {2\zeta },2\zeta \bar{a},\sqrt{\bar{a}k_{p } \theta _{\max }}\left \{1 + \sqrt{1 - \frac{\bar{a}} {k_{p}\theta _{\max }}}\right \}\right ){}\end{array}$$
(51)

When a time delay τ is present in the plant to be controlled, the plant under consideration can be represented as in (37) where

$$\displaystyle{\eta (t) = u(t-\tau ) - u(t)}$$

Similar results of global boundedness can be derived in this case as well (Matsutani 2013; Matsutani et al. 2012, 2013).

Robust Adaptive Control with Persistently Exciting Reference Input

We return to the plant in (26) with the control input as in (2) and the adaptive law as in (3).When η (t) is bounded with a finite upper bound d0, it can be shown that no modifications are necessary in the adaptive law to ensure boundedness if the reference input is persistently exciting. It can be shown that if the reference input r(t) is such that the vector ω∗ defined as ω∗ = [x m T, r]T is persistently exciting with

$$\displaystyle{\left \vert \frac{1} {T}\int _{t}^{t+T}\omega ^{{\ast}T}(\tau )wd\tau \right \vert \geq \,kd_{ 0}\quad \forall \;t \geq t_{0},\forall w \in \mathbb{R}^{n}}$$

where k, T are finite constants and w is a unit vector, then the adaptive system is well behaved, i.e., has globally bounded solutions (Narendra and Annaswamy 2005).

An alternative approach for achieving robustness has been addressed in Anderson et al. (1986) that addresses local stability in the presence of persistently exciting signals. The starting point for this investigation is (35) but when all states are not accessible. Assuming that an output y = c p Tx p is measurable and a controller as in (11)–(15) and a reference model as in (10) are used, the underlying error equation can be written as

$$\displaystyle{ e_{1} = \overline{W}_{m}(s)\left (\tilde{\theta }^{\,\top }\omega + \overline{\nu }\right ) }$$
(52)

where \(\overline{W}_{m}(s)\) is asymptotically stable, \(\tilde{\theta }\) is the parameter error vector, and \(\overline{\nu }\) is the effect of the unmodeled dynamics η at the output. Suppose the standard adaptive law is used, and as a first step the perturbation \(\overline{\nu }\) is ignored, the underlying error equation and the adaptive law are given by

$$\displaystyle\begin{array}{rcl} e_{1}& =& \overline{W}_{m}(s)\tilde{\theta }^{\,\top }\omega {}\end{array}$$
(53)
$$\displaystyle\begin{array}{rcl} \dot{\tilde{\theta }}& =& -\mu e_{1}\omega,\quad \mu> 0.{}\end{array}$$
(54)

If the origin in the \((e_{1},\tilde{\theta })\) space of (53) and (54) is exponentially stable, all solutions of (52) are bounded for sufficiently small initial conditions and \(\overline{\nu }(t)\). Therefore, the question that is of interest is the set of conditions of persistent excitation that will assure such an exponential stability. This is addressed in Anderson et al. (1986). The underlying tool is the Method of Averaging (Arnold 1982; Hale 1969; Krylov and Bogoliuboff 1943) used in the study of nonlinear oscillations and addresses the stability property of the differential equation

$$\displaystyle{ \dot{x} =\mu f(x,t,\mu ),\qquad x(0) = x_{0} }$$
(55)

where μ is a small parameter. By a process of averaging, the nonautonomous system in (55) is approximated by an autonomous differential equation in x av , an averaged value of x. This autonomous system, which is easier to analyze, can be used to derive stability properties of (55).

In order to use the method of averaging for robust adaptive control, we write Eqs. (53) and (54) as

$$\displaystyle{ \left [\begin{array}{*{10}c} \dot{e}\\ \dot{\tilde{\theta }} \end{array} \right ] = \left [\begin{array}{*{10}c} A &b\omega ^{\top } \\ -\mu \omega h^{\top }& 0 \end{array} \right ]\left [\begin{array}{*{10}c} e\\ \tilde{\theta } \end{array} \right ] }$$
(56)

Theorem 1

Let ω(t) be bounded, almost periodic, and persistently exciting. Then there exists a c∗> 0 such that for all μ ∈ (0,c∗], the origin of (56) is exponentially stable if

$$\displaystyle\begin{array}{rcl} & & \mathfrak{R}\left [\lambda _{i}\left (\int _{0}^{T}\omega (t)\bar{W}_{ m}(s)\omega ^{\top }(t)dt\right )\right ]> 0, \\ & & \quad \forall i = 1,\ldots,n {}\end{array}$$
(57)

and is unstable if

$$\displaystyle\begin{array}{rcl} & & \mathfrak{R}\left [\lambda _{j}\left (\int _{0}^{T}\omega (t)\bar{W}_{ m}(s)\omega ^{\top }(t)dt\right )\right ] <0, \\ & & \quad \text{for some }j = 1,\ldots,n {}\end{array}$$
(58)

In Kokotovic et al. (1985), it is further shown that ω(t) can be expressed at \(\omega (t) =\sum _{ k=-\infty }^{\infty }\Omega (i\nu _{k})\exp (i\nu _{k}t)\) and the inequality in (57) can be satisfied if the condition

$$\displaystyle{ \sum _{k=-\infty }^{\infty }\mathfrak{R}\left [\bar{W}_{ m}(i\nu _{k})\right ]\mathfrak{R}\left [\Omega (i\nu _{k})\bar{\Omega }^{\top }(i\nu _{ k})\right ]> 0 }$$
(59)

is satisfied, where \(\bar{\Omega }(i\nu _{k})\) is the complex conjugate of \(\Omega (i\nu _{k})\). Given a general transfer function \(\bar{W}_{m}(s)\), there exists a large class of functions ω that satisfies (59), even when \(\bar{W}_{m}(s)\) is not SPR.

ω in Theorem 1 is not an independent variable but rather an internal variable of the nonlinear system in (56). Hence, it cannot be shown to be bounded or persistently exciting. If ω∗ represents the signal corresponding to ω in the reference model, it can be made to satisfy (57) by the proper choice of the reference input. Expressing \(\omega =\omega _{{\ast}} +\omega _{e}\), ω will also be bounded, persistently exciting, and satisfy (57) if ω e is small. This can be achieved by choosing the initial conditions e(t0) and \(\tilde{\theta }(t_{0})\) in (56) to be sufficiently small. The conditions of Theorem 1 are then verified, and for a sufficiently small μ, exponential stability of the origin of (56) follows.

Theorem 1 provides conditions for exponential stability and instability when the solutions of the adaptive system are sufficiently close to the tuned solutions. These are very valuable in understanding the stability and instability mechanisms peculiar to adaptive control in the presence of different types of perturbations. Many of these results have been summarized and presented in a unified fashion in Anderson et al. (1986).

Cross-References