# Stability Analysis of Adaptive Model Predictive Control
Using the Circle and Tsypkin Criteria

Juan A. Paredes and Dennis S. Bernstein  Juan A. Paredes and Dennis S.
Bernstein are with the Department of Aerospace Engineering, University of
Michigan, Ann Arbor, MI, USA. {jparedes<EMAIL_ADDRESS>
###### Abstract

Absolute stability is a technique for analyzing the stability of Lur’e
systems, which arise in diverse applications, such as oscillators with
nonlinear damping or nonlinear stiffness. A special class of Lur’e systems
consists of self-excited systems (SES), in which bounded oscillations arise
from constant inputs. In many cases, SES can be stabilized by linear
controllers, which motivates the present work, where the goal is to evaluate
the effectiveness of adaptive model predictive control for Lur’e systems. In
particular, the present paper considers predictive cost adaptive control
(PCAC), which is equivalent to a linear, time-variant (LTV) controller. A
closed-loop Lur’e system comprised of the positive feedback interconnection of
the Lur’e system and the PCAC-based controller can thus be derived at each
step. In this work, the circle and Tsypkin criteria are used to evaluate the
absolute stability of the closed-loop Lur’e system, where the adaptive
controller is viewed as instantaneously linear time-invariant. When the
controller converges, the absolute stability criteria guarantee global
asymptotic stability of the asymptotic closed-loop dynamics.

##  I Introduction

Absolute stability is a technique for analyzing the stability of Lur’e systems
[1, 2]. A Lur’e system consists of linear dynamics connected in a feedback
loop with a memoryless nonlinear function. Lur’e systems arise in diverse
applications, such as oscillators with nonlinear damping or nonlinear
stiffness. More generally, viewing the nonlinear dynamics $\dot{x}=f(x)$ as
the feedback interconnection of a vector of integrators and the multivariable
memoryless nonlinearity $f,$ all nonlinear systems can be viewed as Lur’e
systems. The stability of a Lur’e system is the subject of absolute stability
theory, which has been developed in continuous time [3, 4, 5, 6, 7, 8, 9, 10,
11, 12, 13, 14, 15, 16, 17, 18, 19] as well as in discrete time [20, 21, 22,
23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41,
42].

A special class of Lur’e systems consists of self-excited systems (SES), which
have the property that bounded oscillations arise from constant inputs [43,
44]. Self-excited phenomena include flutter due to fluid-structure interaction
[45, 46] as well as thermoacoustic oscillation in combustors [47, 48]. A
widely studied example of a self-excited system is the Rijke tube, in which
the nonlinear heat release interacts with the linear acoustic dynamics to
produce pressure oscillations [49, 50, 51].

An interesting aspect of the Rijke tube is the fact that linear controllers
based on linear system identification and linearized analytical models are
often effective for suppressing self-excited oscillations [52, 53, 54, 55, 56,
57, 58]. This observation motivates the present work, where the goal is to
evaluate the effectiveness of adaptive model predictive control for SES
modeled by Lur’e systems. In particular, the present paper considers
predictive cost adaptive control (PCAC), which performs online closed-loop
linear system identification identification; the identified model is then used
as the basis for receding-horizon optimization [59].

In PCAC, system identification is performed during closed-loop operation by
recursive least squares (RLS) [60, 61]. For receding-horizon optimization,
quadratic programming (QP) is used in [59]. Since state and control
constraints are not crucial for SES, the present paper uses the backward
propagating Riccati equation (BPRE) [62, 63] in place of QP. Furthermore, the
resulting BPRE-based controller is equivalent to an output-feedback, model-
based dynamic compensator with time-dependent gain. The closed-loop Lur’e
system comprised of the positive feedback interconnection of the Lur’e system
and the PCAC-based controller can thus be derived at each step. This
observation facilitates the application of absolute stability theory for
analyzing the stability of the closed-loop system. In this work, the discrete-
time circle and Tsypkin criteria [20, 21, 64, 23, 24, 25, 27] are used to
evaluate the absolute stability of the closed-loop Lur’e system, where the
adaptive controller is viewed as instantaneously linear time-invariant. Since
the closed-loop Lur’e system has time-varying linear dynamics, application of
these absolute stability criteria to the instantaneous closed-loop system is
heuristic. Nevertheless, this analysis technique provides insight into the
extent to which the adaptive controller stabilizes the Lur’e system. When the
controller converges, the absolute stability criteria guarantee global
asymptotic stability of the asymptotic closed-loop dynamics.

The contents of the paper are as follows. Section II provides a statement of
the control problem, which involves discrete-time, Lur’e systems. Section III
describes the predictive control law considered in this paper for suppression.
Section IV presents the closed-loop predictive controller and the closed-loop
Lur’e system to apply the circle criterion to determine absolute stability.
Section V presents a numerical example in which the predictive controller
stabilizes the oscillations of a Lur’e system and the stability of the closed-
loop Lur’e system presented in Section IV is evaluated. Finally, Section VI
presents conclusions.

Notation: ${\bf q}\in{\mathbb{C}}$ denotes the forward-shift operator.
$x_{(i)}$ denotes the $i$th component of $x\in{\mathbb{R}}^{n}.$ ${\rm
spr}(G)$ denotes the maximum of the absolute values of the poles of the
discrete-time transfer function $G.$ $\lambda_{\rm min}(H)$ denotes the
minimum eigenvalue of Hermitian matrix $H\in{\mathbb{R}}^{n\times n}.$ The
symmetric matrix $P\in{\mathbb{R}}^{n\times n}$ is positive semidefinite
(resp., positive definite) if all of its eigenvalues are nonnegative (resp.,
positive). $\operatorname{vec}X\in{\mathbb{R}}^{nm}$ denotes the vector formed
by stacking the columns of $X\in{\mathbb{R}}^{n\times m}$, and $\otimes$
denotes the Kronecker product. $I_{n}$ is the $n\times n$ identity matrix, and
$0_{n\times m}$ is the $n\times m$ zeros matrix. For all $r\in{\mathbb{R}},$
${\rm e}r\stackrel{{\scriptstyle\triangle}}{{=}}10^{r}.$

##  II Statement of the Control Problem

We consider the control architecture shown in Figure 1.

Let $G({\bf q})\stackrel{{\scriptstyle\triangle}}{{=}}C({\bf q}I_{n}-A)^{-1}B$
be a discrete-time, linear, time-invariant (LTI) system with $n$th-order
minimal realization $(A,B,C)$ and state $x_{k}\in{\mathbb{R}}^{n}$ at step
$k,$ let $u_{k}\in{\mathbb{R}}^{m}$ be the control, let
$v_{k}\in{\mathbb{R}}^{m}$ be the perturbation, $y_{k}\in{\mathbb{R}}^{p}$ be
the output of $G,$ and let $\gamma\colon{\mathbb{R}}^{p}\to{\mathbb{R}}^{m}.$
Then, for all $k\geq 0,$ the discrete-time Lur’e (DTL) system consisting of
the linear system $G$ and feedback nonlinearity $\gamma$ shown in Figure 1 has
the closed-loop dynamics

$\displaystyle x_{k+1}$ $\displaystyle=Ax_{k}+B(\gamma(y_{k})+u_{k}+v_{k}),$
(1) $\displaystyle y_{k}$ $\displaystyle=Cx_{k}.$ (2)

At each step $k,$ the linear, time-variant (LTV) controller $G_{{\rm
c},k}({\bf q})\stackrel{{\scriptstyle\triangle}}{{=}}C_{{\rm c},k}({\bf
q}I_{n_{\rm c}}-A_{{\rm c},k})^{-1}B_{{\rm c},k}$ with $n_{\rm c}$th-order
minimal realization $(A_{{\rm c},k},B_{{\rm c},k},C_{{\rm c},k})$ and state
$x_{{\rm c},k}\in{\mathbb{R}}^{n_{\rm c}}$ is updated by $y_{k}$ and generates
the control $u_{k}$ such that the dynamics of the controller are given by

$\displaystyle x_{{\rm c},k+1}$ $\displaystyle=\ A_{{\rm c},k}x_{{\rm
c},k}+B_{{\rm c},k}y_{k},$ (3) $\displaystyle u_{k}$ $\displaystyle=\ C_{{\rm
c},k}x_{{\rm c},k}.$ (4)

The objective of the controller is to provide a control signal that minimizes
the output of the DTL system, that is, a control $u_{k}$ such that
$\lim_{k\to\infty}y_{k}=0.$ In this work, PCAC determines $A_{{\rm
c},k},B_{{\rm c},k},C_{{\rm c},k},$ and $x_{{\rm c},k}$ at each step $k.$

At each step $k,$ let the linear dynamics of the closed-loop Lur’e system
shown in Figure 1 be given by $\tilde{G}_{k}({\bf
q})\stackrel{{\scriptstyle\triangle}}{{=}}G({\bf q})\ [I_{m}-G_{{\rm
c},k}({\bf q})G({\bf q})]^{-1}=\tilde{C}({\bf q}I_{n+n_{\rm
c}}-\tilde{A}_{k})^{-1}\tilde{B},$ which is a LTV system arising from the
positive feedback interconnection of $G$ and $G_{{\rm c},k}$ with $(n+n_{\rm
c})$th-order minimal realization $(\tilde{A}_{k},\tilde{B},\tilde{C}),$ state
$\tilde{x}_{k}\stackrel{{\scriptstyle\triangle}}{{=}}\begin{bmatrix}x_{k}^{\mathsf{T}}&x_{{\rm
c},k}^{\mathsf{T}}\end{bmatrix}^{\mathsf{T}},$ and

$\displaystyle\tilde{A}_{k}\stackrel{{\scriptstyle\triangle}}{{=}}\begin{bmatrix}A&BC_{{\rm
c},k}\\\ B_{{\rm c},k}C&A_{{\rm c},k}\end{bmatrix},$
$\displaystyle\tilde{B}\stackrel{{\scriptstyle\triangle}}{{=}}\begin{bmatrix}B^{\mathsf{T}}&0_{m\times
n_{\rm
c}}\end{bmatrix}^{\mathsf{T}},\quad\tilde{C}\stackrel{{\scriptstyle\triangle}}{{=}}\begin{bmatrix}C&0_{p\times
n_{\rm c}}\end{bmatrix}.$

The dynamics of the closed-loop Lur’e system in Figure 1 are thus given by

$\displaystyle\tilde{x}_{k+1}$ $\displaystyle=\
\tilde{A}_{k}\tilde{x}_{k}+\tilde{B}(\gamma(y_{k})+v_{k}),$ (5) $\displaystyle
y_{k}$ $\displaystyle=\ \tilde{C}\tilde{x}_{k}.$ (6)

$+$$G$$G_{{\rm
c},k}$$\gamma$$v_{k}$$y_{k}$$u_{k}$$+$$\tilde{G}_{k}$$\gamma$$v_{k}$$y_{k}$$\bm{\equiv}$

Figure 1: Closed-loop LTV control of the discrete-time Lur’e (DTL) system. The
LTV controller $G_{{\rm c},k}$ is applied to the DTL system consisting of the
linear system $G$ and feedback nonlinearity $\gamma.$ The linear dynamics of
the closed-loop Lur’e system are given by
$\tilde{G}_{k}\stackrel{{\scriptstyle\triangle}}{{=}}G(I_{m}-G_{{\rm
c},k}G)^{-1},$ which is the LTV system arising from the positive feedback
interconnection of $G$ and $G_{{\rm c},k}.$

##  III Predictive Cost Adaptive Control

PCAC is briefly reviewed in this section. Subsection III-A describes the
technique used for online identification, namely, RLS with variable-rate
forgetting based on the F-test [61]. Subsection III-B presents the block
observable canonical form (BOCF), which is used to represent the input-output
dynamics model as a state space model whose state is given explicitly in terms
of inputs, outputs, and model-coefficient estimates. Subsection III-C reviews
the backward-propagating Riccai equation (BPRE) technique for receding-horizon
optimization. Using BOCF, the full-state feedback controller obtained using
BPRE is implementable as an output-feedback dynamic compensator.

###  III-A Online Identification Using Recursive Least Squares with Variable-
Rate Forgetting Based on the F-Test

Let $\hat{n}\geq 0$ and, for all $k\geq 0,$ let $F_{{\rm
m},1,k},\ldots,F_{{\rm m},\hat{n},k}\in{\mathbb{R}}^{p\times p}$ and $G_{{\rm
m},1,k},\ldots,G_{{\rm m},\hat{n},k}\in{\mathbb{R}}^{p\times m}$ be the
coefficient matrices to be estimated using RLS. Furthermore, let
$\hat{y}_{k}\in{\mathbb{R}}^{p}$ be an estimate of $y_{k}$ defined by

$\hat{y}_{k}\stackrel{{\scriptstyle\triangle}}{{=}}-\sum_{i=1}^{\hat{n}}F_{{\rm
m},i,k}y_{k-i}+\sum_{i=1}^{\hat{n}}{G}_{{\rm m},i,k}u_{k-i},$ (7)

where

$\displaystyle y_{-\hat{n}}=\cdots=y_{-1}=0,$ (8) $\displaystyle
u_{-\hat{n}}=\cdots=u_{-1}=0.$ (9)

Using the identity ${\rm vec}(XY)=(Y^{\mathsf{T}}\otimes I){\rm vec}X,$ it
follows from (7) that, for all $k\geq 0,$

$\hat{y}_{k}=\phi_{k}\theta_{k},$ (10)

where

$\displaystyle\theta_{k}\stackrel{{\scriptstyle\triangle}}{{=}}$
$\displaystyle\ \begin{bmatrix}\theta_{F_{\rm
m},k}^{\mathsf{T}}&\theta_{G_{\rm
m},k}^{\mathsf{T}}\end{bmatrix}^{\mathsf{T}}\in{\mathbb{R}}^{\hat{n}p(m+p)},$
(11) $\displaystyle\theta_{F_{\rm
m},k}\stackrel{{\scriptstyle\triangle}}{{=}}$ $\displaystyle\ {\rm
vec}\begin{bmatrix}F_{{\rm m},1,k}&\cdots&F_{{\rm
m},\hat{n},k}\end{bmatrix}\in{\mathbb{R}}^{\hat{n}p^{2}},$ (12)
$\displaystyle\theta_{G_{\rm m},k}\stackrel{{\scriptstyle\triangle}}{{=}}$
$\displaystyle\ {\rm vec}\begin{bmatrix}G_{{\rm m},1,k}&\cdots&G_{{\rm
m},\hat{n},k}\end{bmatrix}\in{\mathbb{R}}^{\hat{n}pm},$ (13)
$\displaystyle\phi_{k}\stackrel{{\scriptstyle\triangle}}{{=}}$
$\displaystyle\begin{bmatrix}-y_{k-1}^{\mathsf{T}}&\cdots&-y_{k-\hat{n}}^{\mathsf{T}}&u_{k-1}^{\mathsf{T}}&\cdots&u_{k-\hat{n}}^{\mathsf{T}}\end{bmatrix}\otimes
I_{p}$ $\displaystyle\in{\mathbb{R}}^{p\times\hat{n}p(m+p)}.$ (14)

To determine the update equations for $\theta_{k}$, for all $k\geq 0$, define
$e_{k}\colon{\mathbb{R}}^{\hat{n}p(m+p)}\to{\mathbb{R}}^{p}$ by

$e_{k}(\bar{\theta})\stackrel{{\scriptstyle\triangle}}{{=}}y_{k}-\phi_{k}\bar{\theta},$
(15)

where $\bar{\theta}\in{\mathbb{R}}^{\hat{n}p(m+p)}.$ Using (10), the
identification error at step $k$ is defined by

$e_{k}(\theta_{k})=y_{k}-\hat{y}_{k}.$ (16)

For all $k\geq 0$, the RLS cumulative cost
$J_{k}\colon{\mathbb{R}}^{\hat{n}p(m+p)}\to[0,\infty)$ is defined by [60]

$J_{k}(\bar{\theta})\stackrel{{\scriptstyle\triangle}}{{=}}\sum_{i=0}^{k}\frac{\rho_{i}}{\rho_{k}}e_{i}^{\mathsf{T}}(\bar{\theta})e_{i}(\bar{\theta})+\frac{1}{\rho_{k}}(\bar{\theta}-\theta_{0})^{\mathsf{T}}\Psi_{0}^{-1}(\bar{\theta}-\theta_{0}),$
(17)

where $\Psi_{0}\in{\mathbb{R}}^{\hat{n}p(m+p)\times\hat{n}p(m+p)}$ is positive
definite, $\theta_{0}\in{\mathbb{R}}^{\hat{n}p(m+p)}$ is the initial estimate
of the coefficient vector, and, for all $i\geq 0,$

$\rho_{i}\stackrel{{\scriptstyle\triangle}}{{=}}\prod_{j=0}^{i}\lambda_{j}^{-1}.$
(18)

For all $j\geq 0$, the parameter $\lambda_{j}\in(0,1]$ is the forgetting
factor defined by
$\lambda_{j}\stackrel{{\scriptstyle\triangle}}{{=}}\beta_{j}^{-1}$, where

$\beta_{j}\stackrel{{\scriptstyle\triangle}}{{=}}\begin{cases}1,&j<\tau_{\rm
d},\\\ 1+\eta\bar{\beta}_{j},&j\geq\tau_{\rm d},\end{cases}$ (19)
$\bar{\beta}_{j}\stackrel{{\scriptstyle\triangle}}{{=}}g(e_{j-\tau_{\rm
d}}(\theta_{j-\tau_{\rm
d}}),\ldots,e_{j}(\theta_{j}))\textbf{1}\big{(}g(e_{j-\tau_{\rm
d}}(\theta_{j-\tau_{\rm d}}),\ldots,e_{j}(\theta_{j}))\big{)},$ (20)

and $\tau_{\rm d}>p$, $\eta>0$, $\textbf{1}\colon{\mathbb{R}}\to\\{0,1\\}$ is
the unit step function, and $g$ is the function of past RLS identification
errors given by (10) in [61] for $p=1$ and (13) in [61] for $p>1.$ Note that
$g$ includes forgetting terms based on the inverse cumulative distribution
function of the F-distribution and depends on $\tau_{\rm d},$ $\tau_{\rm
n}\in[p,\tau_{\rm d}),$ and significance level $\alpha\in(0,1].$

Finally, for all $k\geq 0$, the unique global minimizer of $J_{k}$ is given by
[60]

$\theta_{k+1}=\theta_{k}+\Psi_{k+1}\phi_{k}^{\mathsf{T}}(y_{k}-\phi_{k}\theta_{k}),$
(21)

where

$\displaystyle\Psi_{k+1}$
$\displaystyle\stackrel{{\scriptstyle\triangle}}{{=}}\beta_{k}\Psi_{k}-\beta_{k}\Psi_{k}\phi_{k}^{\mathsf{T}}(\tfrac{1}{\beta_{k}}I_{p}+\phi_{k}\Psi_{k}\phi_{k}^{\mathsf{T}})^{-1}\phi_{k}\Psi_{k},$
(22)

and $\Psi_{0}$ is the performance-regularization weighting in (17). Additional
details concerning RLS with forgetting based on the F-distribution are given
in [61].

###  III-B Input-Output Model and the Block Observable Canonical Form

Considering the estimate $\hat{y}_{k}$ of $y_{k}$ given by (7), it follows
that, for all $k\geq 0,$

$y_{k}\approx-\sum_{i=1}^{\hat{n}}F_{{\rm
m},i,k}y_{k-i}+\sum_{i=1}^{\hat{n}}{G}_{{\rm m},i,k}u_{k-i}.$ (23)

Viewing (23) as an equality, it follows that, for all $k\geq 0,$ the block
observable canonical form (BOCF) state-space realization of (23) is given by
[65]

$\displaystyle x_{{\rm m},k+1}$ $\displaystyle=A_{{\rm m},k}x_{{\rm
m},k}+B_{{\rm m},k}u_{k},$ (24) $\displaystyle y_{k}$ $\displaystyle=C_{\rm
m}x_{{\rm m},k},$ (25)

where

$\displaystyle A_{{\rm
m},k}\stackrel{{\scriptstyle\triangle}}{{=}}\begin{bmatrix}-F_{{\rm
m},1,k+1}&I_{p}&\cdots&\cdots&0_{p\times p}\\\ -F_{{\rm m},2,k+1}&0_{p\times
p}&\ddots&&\vdots\\\ \vdots&{\vdots}&\ddots&\ddots&0_{p\times p}\\\
\vdots&\vdots&&\ddots&I_{p}\\\ -F_{{\rm m},\hat{n},k+1}&0_{p\times
p}&\cdots&\cdots&0_{p\times
p}\end{bmatrix}\in{\mathbb{R}}^{\hat{n}p\times\hat{n}p},$ (26) $\displaystyle
B_{{\rm m},k}\stackrel{{\scriptstyle\triangle}}{{=}}\begin{bmatrix}G_{{\rm
m},1,k+1}\\\ G_{{\rm m},2,k+1}\\\ \vdots\\\ G_{{\rm
m},\hat{n},k+1}\end{bmatrix}\in{\mathbb{R}}^{\hat{n}p\times m},$ (27)
$\displaystyle C_{\rm
m}\stackrel{{\scriptstyle\triangle}}{{=}}\begin{bmatrix}I_{p}&0_{p\times
p}&\cdots&0_{p\times p}\end{bmatrix}\in{\mathbb{R}}^{p\times\hat{n}p},$ (28)

and

$x_{{\rm m},k}\stackrel{{\scriptstyle\triangle}}{{=}}\begin{bmatrix}x_{{\rm
m},k(1)}\\\ \vdots\\\ x_{{\rm
m},k(\hat{n})}\end{bmatrix}\in{\mathbb{R}}^{\hat{n}p},$ (29)

where $x_{{\rm m},k(1)}\stackrel{{\scriptstyle\triangle}}{{=}}y_{k},$ and, for
all $j=2,\ldots,\hat{n},$

$\displaystyle x_{{\rm m},k(j)}\stackrel{{\scriptstyle\triangle}}{{=}}$
$\displaystyle-\sum_{i=1}^{\hat{n}-j+1}F_{{\rm m},i+j-1,k+1}y_{k-i}$
$\displaystyle+\sum_{i=1}^{\hat{n}-j+1}G_{{\rm m},i+j-1,k+1}u_{k-i}.$ (30)

Note that multiplying both sides of (24) by $C_{\rm m}$ and using (25)–(30)
implies that, for all $k\geq 0,$

$\displaystyle y_{k+1}$ $\displaystyle=C_{\rm m}x_{{\rm m},k+1}$
$\displaystyle=C_{\rm m}(A_{{\rm m},k}x_{{\rm m},k}+B_{{\rm m},k}u_{k})$
$\displaystyle=-F_{{\rm m},1,k+1}x_{{\rm m},k(1)}+x_{{\rm m},k(2)}+G_{{\rm
m},1,k+1}u_{k}$ $\displaystyle=-F_{{\rm
m},1,k+1}y_{k}-\sum_{i=1}^{\hat{n}-1}F_{{\rm m},i+1,k+1}y_{k-i}$
$\displaystyle+\sum_{i=1}^{\hat{n}-1}G_{{\rm m},i+1,k+1}u_{k-i}+G_{{\rm
m},1,k+1}u_{k}$ $\displaystyle=-\sum_{i=1}^{\hat{n}}F_{{\rm
m},i,k+1}y_{k+1-i}+\sum_{i=1}^{\hat{n}}G_{{\rm m},i,k+1}u_{k+1-i},$ (31)

which is approximately equivalent to (23) with $k$ in (23) replaced by $k+1$.

###  III-C Receding-Horizon Control with Backward-Propagating Riccati Equation
(BPRE)

In this section, we use receding-horizon optimization to determine the
requested control $u_{k+1}$ and thus the implemented control $u_{k+1}$, as
discussed in Section II. Let $\ell\geq 1$ be the horizon, and, for all $k\geq
0$ and all $j=1,\ldots,\ell,$ consider the state-space prediction model

$x_{{\rm m},k|j+1}=A_{{\rm m},k}x_{{\rm m},k|j}+B_{{\rm m},k}u_{k|j},$ (32)

where $A_{{\rm m},k}$ and $B_{{\rm m},k}$ are given by (26) and (27),
respectively, $x_{{\rm m},k|j}\in{\mathbb{R}}^{\hat{n}p}$ is the $j$-step
predicted state, $u_{k|j}\in{\mathbb{R}}^{m}$ is the $j$-step predicted
control, and the initial conditions are

$x_{{\rm m},k|1}\stackrel{{\scriptstyle\triangle}}{{=}}x_{{\rm m},k+1},\quad
u_{k|1}\stackrel{{\scriptstyle\triangle}}{{=}}u_{k+1}.$ (33)

Note that, at each step $k\geq 0$, after obtaining the measurement $y_{k}$,
$x_{{\rm m},k+1}$ is computed using (24), where $u_{k}$ is the implemented
control at step $k.$ Furthermore, $u_{k+1}$, which is determined below, is the
requested control at step $k+1$. For all $k\geq 0$, define the performance
index

$\displaystyle{\mathcal{J}}_{k}($ $\displaystyle u_{k|1},\ldots,u_{k|\ell})$
$\displaystyle\stackrel{{\scriptstyle\triangle}}{{=}}$ $\displaystyle\
\tfrac{1}{2}\sum_{j=1}^{\ell}(x_{{\rm m},k|j}^{\mathsf{T}}R_{1,k|j}x_{{\rm
m},k|j}+u_{k|j}^{\mathsf{T}}R_{2,k|j}u_{k|j})$
$\displaystyle+\tfrac{1}{2}x_{{\rm
m},k|\ell+1}^{\mathsf{T}}P_{k|\ell+1}x_{{\rm m},k|\ell+1},$ (34)

where the terminal weighting
$P_{k|\ell+1}\in{\mathbb{R}}^{\hat{n}p\times\hat{n}p}$ is positive
semidefinite and, for all $j=1,\ldots,\ell,$
$R_{1,k|j}\in{\mathbb{R}}^{\hat{n}p\times\hat{n}p}$ is the positive
semidefinite state weighting and $R_{2,k|j}\in{\mathbb{R}}^{m\times m}$ is the
positive definite control weighting. The first term in (34) can be written as

$x_{{\rm m},k|j}^{\mathsf{T}}R_{1,k|j}x_{{\rm
m},k|j}=z_{k}^{\mathsf{T}}z_{k},$ (35)

where $z_{k}\in{\mathbb{R}}^{p}$ is defined by

$z_{k}\stackrel{{\scriptstyle\triangle}}{{=}}E_{1,k|j}x_{{\rm m},k|j},$ (36)

and $E_{1,k|j}\in{\mathbb{R}}^{p\times\hat{n}p}$ is defined such that

$R_{1,k|j}=E_{1,k|j}^{\mathsf{T}}E_{1,k|j}.$ (37)

With this notation, $z_{k}$ is the performance variable.

For all $k\geq 0$ and $j=\ell,\ell-1,\ldots,2,$ let $P_{k|j}$ be given by

$\displaystyle P_{k|j}$ $\displaystyle=\ A_{{\rm
m},k}^{\mathsf{T}}P_{k|j+1}\left(A_{{\rm m},k}-B_{{\rm
m},k}\Gamma_{k|j}\right)+R_{1,k|j},$ (38)
$\displaystyle\Gamma_{k|j}\stackrel{{\scriptstyle\triangle}}{{=}}$
$\displaystyle\ (R_{2,k|j}+B_{{\rm m},k}^{\mathsf{T}}P_{k|j+1}B_{{\rm
m},k})^{-1}B_{{\rm m},k}^{\mathsf{T}}P_{k|j+1}A_{{\rm m},k}.$

Then, for all $k\geq 0$ and all $j=1,\ldots,\ell,$ the requested optimal
control is given by

$u_{k|j}=K_{k|j}x_{{\rm m},k|j},$ (39)
$K_{k|j}\stackrel{{\scriptstyle\triangle}}{{=}}-(R_{2,k|j}+B_{{\rm
m},k}^{\mathsf{T}}P_{k|j+1}B_{{\rm m},k})^{-1}B_{{\rm
m},k}^{\mathsf{T}}P_{k|j+1}A_{{\rm m},k}.$

For all $k\geq 0,$ define
$K_{k+1}\stackrel{{\scriptstyle\triangle}}{{=}}K_{k|1}$. Then, for all $k\geq
0,$ (33) and (39) imply that

$\displaystyle u_{k+1}$ $\displaystyle=K_{k+1}x_{{\rm m},k+1},$ (40)

which, combined with (III-C), implies that

$u_{k+1}=-(R_{2,k|1}+B_{{\rm m},k}^{\mathsf{T}}P_{k|2}B_{{\rm
m},k})^{-1}B_{{\rm m},k}^{\mathsf{T}}P_{k|2}A_{{\rm m},k}x_{{\rm m},k+1}.$
(41)

Note that the initial control $u_{0}\in{\mathbb{R}}^{m}$ is not computed and
must be specified. In this work, $u_{0}=0.$ In addition, note that, for all
$k\geq 0,$ the requested control $u_{k+1}$ is computed during the interval
$[kT_{\rm s},(k+1)T_{\rm s})$ using the measurement $y_{k}$ and the
implemented control $u_{k}.$ Furthermore, note that, for all $k\geq 0$ and all
$j=2,\ldots,\ell,$ $K_{k|j}$, $x_{{\rm m},k|j},$ and $u_{k|j}$ need not be
computed, in accordance with receding-horizon control. Finally, for all
examples in this paper, we choose $R_{1,k|j}$, $R_{2,k|j}$, $E_{1,k|j}$ and
$P_{k|\ell+1}$ to be independent of $k$ and $j,$ and we thus write $R_{1}$,
$R_{2}$, $E_{1}$, and $P_{\ell+1},$ respectively.

##  IV Closed-loop Lur’e system for absolute stability analysis

To derive the dynamics of the closed-loop Lur’e system, note that (24) and
(25) imply that

$x_{{\rm m},k+1}=(A_{{\rm m},k}-F_{k}C_{\rm m})x_{{\rm m},k}+B_{{\rm
m},k}u_{k}+F_{k}y_{k},$ (42)

where

$F_{k}\stackrel{{\scriptstyle\triangle}}{{=}}\begin{bmatrix}-F_{{\rm
m},1,k+1}^{\mathsf{T}}&\cdots&-F_{{\rm
m},\hat{n},k+1}^{\mathsf{T}}\end{bmatrix}^{\mathsf{T}}\in{\mathbb{R}}^{\hat{n}p\times
p}.$

Then, it follows from (40) and (42)

$x_{{\rm m},k+1}=(A_{{\rm m},k}-F_{k}C_{\rm m}+B_{{\rm m},k}K_{k})x_{{\rm
m},k}+F_{k}y_{k}.$ (43)

Hence, replacing $n_{\rm c},$ $x_{{\rm c},k},$ (3) and (4) with $\hat{n}p,$
$x_{{\rm m},k},$ (40) and (43), respectively, yields the closed-loop dynamics
of the PCAC-based controller

$\displaystyle x_{{\rm c},k+1}=x_{{\rm m},k+1}$ $\displaystyle=\ A_{{\rm
c},k}x_{{\rm m},k}+B_{{\rm c},k}y_{k},$ (44) $\displaystyle u_{k}$
$\displaystyle=\ C_{{\rm c},k}x_{{\rm m},k},$ (45)

where

$A_{{\rm c},k}\stackrel{{\scriptstyle\triangle}}{{=}}A_{{\rm m},k}-F_{k}C_{\rm
m}+B_{{\rm m},k}K_{k},\ B_{{\rm
c},k}\stackrel{{\scriptstyle\triangle}}{{=}}F_{k},\ C_{{\rm
c},k}\stackrel{{\scriptstyle\triangle}}{{=}}K_{k}.$

Next, we analyze the stability of the closed-loop Lur’e system shown in Figure
1 with the controller (44), (45). Since the feedback controller is LTV,
absolute stability tests such as the circle and Tsypkin criteria, which assume
that the linear dynamics are LTI, are not applicable. However, as a heuristic
criterion, we apply these criteria instantaneously, which provides insight
into the ability of PCAC to stabilize the Lur’e system. When the controller
converges, the absolute stability criteria guarantee global asymptotic
stability of the asymptotic closed-loop dynamics. The absolute stability
criteria used in this work are summarized in the Appendix; in particular,
Theorem .2 is the discrete-time circle criterion, and Theorem .4 is the
Tsypkin criterion. Furthermore, sector-bounded (SB) and diagonal, increasing,
and sector-bounded (DISB) nonlinearities are defined by Definitions .1 and .3,
respectively.

To apply the circle criterion, let $\gamma$ be SB with sector bound
[$M_{1},M_{2}$], and, for all $k\geq 0,$ define

$\displaystyle H_{k}\stackrel{{\scriptstyle\triangle}}{{=}}$ $\displaystyle\
[I_{m}-M_{2}\tilde{G}_{k}][I_{m}-M_{1}\tilde{G}_{k}]^{-1},$
$\displaystyle\alpha_{{\rm CC},k}\stackrel{{\scriptstyle\triangle}}{{=}}$
$\displaystyle\ {\rm spr}(H_{k}),$ $\displaystyle\beta_{{\rm
CC},k}\stackrel{{\scriptstyle\triangle}}{{=}}$
$\displaystyle\min_{\psi\in[0,\pi]}\left(\lambda_{\rm
min}\left[H_{k}(e^{\jmath\psi})+H_{k}^{\mathsf{T}}(e^{-\jmath\psi})\right]\right).$

At step $k,$ if $\alpha_{{\rm CC},k}<1,$ then (CC1) of Theorem .2 is
instantaneously satisfied by the closed-loop Lur’e system, and, if
$\beta_{{\rm CC},k}>0,$ then (CC2) of Theorem .2 is instantaneously satisfied
by the closed-loop Lur’e system. Hence, Theorem .2 implies that, if
$\alpha_{{\rm CC},k}<1$ and $\beta_{{\rm CC},k}>0,$ then the closed-loop Lur’e
system shown in Figure 1 with the PCAC controller satisfies (CC1) and (CC2) at
step $k.$

To apply the Tsypkin criterion, let $m=p,$ let $\gamma$ be DISB with sector
bound [$0,M$], let $N\stackrel{{\scriptstyle\triangle}}{{=}}{\rm
diag}(N_{1},N_{2},\ldots,N_{m})$ be positive definite, and, for all $k\geq 0,$
define

$\displaystyle L_{N,k}({\bf q})\stackrel{{\scriptstyle\triangle}}{{=}}$
$\displaystyle\ M^{-1}-[I_{m}+(1-{\bf q}^{-1})N]\ \tilde{G}_{k}({\bf q}),$
$\displaystyle\zeta_{1,k}\stackrel{{\scriptstyle\triangle}}{{=}}$
$\displaystyle\ {\rm det}(\tilde{C}\tilde{A}_{k}^{-1}\tilde{B}),$
$\displaystyle\zeta_{2,N,k}\stackrel{{\scriptstyle\triangle}}{{=}}$
$\displaystyle\ {\rm rank}({\rm
obsv}((\tilde{A}_{k},\tilde{C}+N\tilde{C}-N\tilde{C}\tilde{A}_{k}^{-1}))),$
$\displaystyle\zeta_{3,N,k}\stackrel{{\scriptstyle\triangle}}{{=}}$
$\displaystyle\lim_{{\bf z}\to\infty}[L_{N}({\bf z})+L_{N}^{\mathsf{T}}({\bf
z})],$ $\displaystyle\alpha_{{\rm
TC},N,k}\stackrel{{\scriptstyle\triangle}}{{=}}$ $\displaystyle\ {\rm
spr}(L_{N,k}),$ $\displaystyle\beta_{{\rm
TC},N,k}\stackrel{{\scriptstyle\triangle}}{{=}}$
$\displaystyle\min_{\psi\in[0,\pi]}\left(\lambda_{\rm
min}\left[L_{N,k}(e^{\jmath\psi})+L_{N,k}^{\mathsf{T}}(e^{-\jmath\psi})\right]\right).$

At step $k,$ if $\zeta_{1,k}\neq 0,$ $\zeta_{2,N,k}=n+\hat{n}p,$ and
$\zeta_{3,N,k}>0,$ then (TC1) of Theorem .4 is instantaneously satisfied by
the closed-loop Lur’e system, if $\alpha_{{\rm TC},N,k}<1,$ then (TC2) of
Theorem .4 is instantaneously satisfied by the closed-loop Lur’e system, and,
if $\beta_{{\rm TC},N,k}>0,$ then (TC3) of Theorem .4 is instantaneously
satisfied by the closed-loop Lur’e system. Hence, Theorem .4 implies that, if
$\zeta_{1,k}\neq 0,$ $\zeta_{2,N,k}=n+\hat{n}p,$ $\zeta_{3,N,k}>0,$
$\alpha_{{\rm TC},N,k}<1$ and $\beta_{{\rm TC},N,k}>0,$ then the closed-loop
Lur’e system shown in Figure 1 with the PCAC controller instantaneously
satisfies (TC1), (TC2), and (TC3) at step $k.$

These conditions are evaluated in Section V to determine whether PCAC yields a
globally asymptotically stable (GAS) closed-loop Lur’e system.

##  V Numerical Example

In this section, PCAC is applied to a DTL system, and the resulting closed-
loop system is analyzed by using absolute stability criteria. In particular,
the circle criterion given by Theorem .2 and the Tsypkin criterion given by
Theorem .4 are used to evaluate the stability of the closed-loop system at
each step. Since the closed-loop Lur’e system has time-varying linear
dynamics, application of these absolute stability criteria is heuristic.
Nevertheless, this analysis technique provides insight into the ability of the
adaptive controller to stabilize the Lur’e system. When the controller
converges, the absolute stability criteria guarantee global asymptotic
stability of the asymptotic closed-loop dynamics.

Example V.1 features a bounded, monotonic nonlinearity that lies in the first
and third quadrants. This system exhibits self-oscillation. Hence, the
simulations begin in open-loop operation to allow the self-excited
oscillations to fully develop before PCAC is applied. In this example, the RLS
hyperparameters are

$\displaystyle\hat{n}=10,\quad\theta_{0}=10^{-10}\,1_{2\hat{n}\times
1},\quad\Psi_{0}=10^{-4}I_{2\hat{n}},$ $\displaystyle\tau_{\rm
n}=40,\quad\tau_{\rm d}=200,\quad\eta=0.1,\quad\alpha=0.001,$

and the BPRE hyperparameters are

$\displaystyle\ell=20,\quad
P_{\ell+1}=\operatorname{diag}(1,0_{1\times\hat{n}-1}),$ $\displaystyle
R_{1}=\operatorname{diag}(1,0_{1\times\hat{n}-1}),\quad R_{2}=10^{-4}.$

The exogenous input $v_{k}$ is used to enhance persistency and thus enhance
the ability of the closed-loop Lur’e system to satisfy the absolute stability
criteria. Three cases are considered, namely, for all $k\geq 0,$ $v_{k}=0,$
$v_{k}=v_{{\rm imp},k},$ and $v_{k}=v_{{\rm rand},k},$ where

$\displaystyle v_{{\rm imp},k}\stackrel{{\scriptstyle\triangle}}{{=}}$
$\displaystyle\begin{cases}1&k\in\\{1000,1400,1800\\}\\\
-1&k\in\\{1200,1600,2000\\}\\\ 0&\mbox{otherwise},\end{cases}$ (46)
$\displaystyle v_{{\rm rand},k}\stackrel{{\scriptstyle\triangle}}{{=}}$
$\displaystyle\begin{cases}\sigma_{k}&k\in[1000,1500]\\\
0&\mbox{otherwise},\end{cases}$ (47)

and $\sigma_{k}$ is a Gaussian random variable with mean 0 and standard
deviation 1.

###### Example V.1

Let $G({\bf q})=({\bf q}-1)/({\bf q}^{2}-{\bf q}+0.5)$ with minimal
realization

$A=\begin{bmatrix}1&-0.5\\\ 1&0\end{bmatrix},\ B=\begin{bmatrix}1\\\
0\end{bmatrix},\ C=\begin{bmatrix}1&-1\end{bmatrix},$ (48)

$x_{0}=1000B,$ and $\gamma(y)=\tanh(y).$ The open-loop response of the DTL
system and the nonlinearity $\gamma$ are shown in Figure 2. Note, $\gamma$ is
SB and DISB with sector bound $[0,1].$

In the case where $v_{k}=0,$ the results for $k\in[0,1000]$ are shown in
Figures 3, 4, and 5. Note that, although the output $y_{k}$ of the closed-loop
DTL system converges to zero, the circle and Tsypkin criteria are not
satisfied at $k=1000.$ Figure 6 shows, nevertheless, that the closed-loop
Lur’e system with the converged LTI linear dynamics $\tilde{G}_{1000}$ has a
large region of attraction.

Next, for $v_{k}=v_{{\rm imp},k},$ the DTL system output is shown in Figures
7, 8, and 9. Note that the output $y_{k}$ of the closed-loop DTL system
converges to zero, and the circle and Tsypkin criteria are satisfied at
$k=3000.$

Finally, for $v_{k}=v_{{\rm rand},k},$ the DTL system output is shown in
Figures 10, 11, and 12. Note that the output $y_{k}$ of the closed-loop DTL
system converges to zero, and the circle and Tsypkin criteria are satisfied at
$k=2700.$ $\huge\diamond$

Figure 2: Example V.1: Open-loop response of the DTL system in Example V.1 for
$k\in[100,150]$. $\gamma$ is SB and DISB with sector bound $[0,1],$ as
indicated by the red, dashed line segments and the green-shaded region.

Figure 3: Example V.1: DTL system output $y$ and the adaptive controller
coefficients $\theta_{k}$ for $x_{0}=1000B$ and $v_{k}=0$. The simulation
transitions from open-loop operation to closed-loop operation at the step
indicated by the vertical, dashed red line.

Figure 4: Example V.1: Evaluation of (CC1) and (CC2) for $x_{0}=1000B$ and
$v_{k}=0.$ Values in red correspond to steps at which $1-\alpha_{{\rm CC},k}$
and $\beta_{{\rm CC},k}$ are nonpositive. Values in blue correspond to steps
at which $1-\alpha_{{\rm CC},k}$ and $\beta_{{\rm CC},k}$ are positive. The
closed-loop system satisfies (CC1) and (CC2) when the values in both plots are
green. For this case, the circle criterion is not satisfied at most steps.

Figure 5: Example V.1: Evaluation of the Tsypkin criterion for $x_{0}=1000B,$
and $v_{k}=0$ with $N=0.08.$ Values not in black correspond to steps at which
$\zeta_{1,k}\neq 0,$ $\zeta_{2,N,k}=n+\hat{n}p,$ and $\zeta_{3,N,k}>0.$ Values
in red correspond to steps at which $1-\alpha_{{\rm TC},N,k}$ and $\beta_{{\rm
TC},N,k}$ are nonpositive. Values in blue correspond to steps at which
$1-\alpha_{{\rm TC},N,k}$ and $\beta_{{\rm TC},N,k}$ are positive. The closed-
loop system satisfies (TC1), (TC2) and (TC3) when the values in both plots are
green. For this case, the Tsypkin criterion is not satisfied at most steps.

Figure 6: Example V.1: (a) shows the output $y$ of the open-loop DTL system
with linear dynamics $G$. (b) shows the output $y$ of the closed-loop DTL
system with linear dynamics $\tilde{G}_{1000}$. Both plots show the response
for all $x_{0}\in\\{-{\rm e}6,-9{\rm e}5,\ldots,9{\rm e}5,{\rm
e}6\\}\times\\{-{\rm e}6,-9{\rm e}5,\ldots,9{\rm e}5,{\rm e}6\\}.$

Figure 7: Example V.1: DTL system output $y$ and the adaptive controller
coefficients $\theta_{k}$ for $x_{0}=1000B$ and $v_{k}=v_{{\rm imp},k}$. The
simulation transitions from open-loop operation to closed-loop operation at
the step indicated by the vertical, dashed red line.

Figure 8: Example V.1: Evaluation of (CC1) and (CC2) for $x_{0}=1000B$ and
$v_{k}=v_{{\rm imp},k}$. Values in red correspond to steps at which
$1-\alpha_{{\rm CC},k}$ and $\beta_{{\rm CC},k}$ are nonpositive. Values in
blue correspond to steps at which $1-\alpha_{{\rm CC},k}$ and $\beta_{{\rm
CC},k}$ are positive. The closed-loop system satisfies (CC1) and (CC2) when
the values in both plots are green. For this case, the circle criterion is
satisfied for all steps $k\geq 2000.$

Figure 9: Example V.1: Evaluation of the Tsypkin criterion for $x_{0}=1000B$
and $v_{k}=v_{{\rm imp},k}$ with $N=0.08.$ Values not in black correspond to
steps at which $\zeta_{1,k}\neq 0,$ $\zeta_{2,N,k}=n+\hat{n}p,$ and
$\zeta_{3,N,k}>0.$ Values in red correspond to steps at which $1-\alpha_{{\rm
TC},N,k}$ and $\beta_{{\rm TC},N,k}$ are nonpositive. Values in blue
correspond to steps at which $1-\alpha_{{\rm TC},N,k}$ and $\beta_{{\rm
TC},N,k}$ are positive. The closed-loop system satisfies (TC1), (TC2) and
(TC3) when the values in both plots are green. For this case, the Tsypkin
criterion is satisfied for all $k\geq 2000$.

Figure 10: Example V.1: DTL system output $y$ and the adaptive controller
coefficients $\theta_{k}$ in the case where $x_{0}=1000B,$ and $v_{k}=v_{{\rm
rand},k}$. The simulation transitions from open-loop operation to closed-loop
operation at the step indicated by the vertical, dashed red line.

Figure 11: Example V.1: Evaluation of (CC1) and (CC2) in the case where
$x_{0}=1000B,$ and $v_{k}=v_{{\rm rand},k}$. Values in red correspond to steps
at which $1-\alpha_{{\rm CC},k}$ and $\beta_{{\rm CC},k}$ are nonpositive.
Values in blue correspond to steps at which $1-\alpha_{{\rm CC},k}$ and
$\beta_{{\rm CC},k}$ are positive. The closed-loop system satisfies (CC1) and
(CC2) when the values in both plots are green. For this case, the circle
criterion is satisfied for all $k\geq 1700.$

Figure 12: Example V.1: Evaluation of the Tsypkin criterion in the case where
$x_{0}=1000B,$ and $v_{k}=v_{{\rm rand},k}$ with $N=0.08.$ Values not in black
correspond to steps at which $\zeta_{1,k}\neq 0,$ $\zeta_{2,N,k}=n+\hat{n}p,$
and $\zeta_{3,N,k}>0.$ Values in red correspond to steps at which
$1-\alpha_{{\rm TC},N,k}$ and $\beta_{{\rm TC},N,k}$ are nonpositive. Values
in blue correspond to steps at which $1-\alpha_{{\rm TC},N,k}$ and
$\beta_{{\rm TC},N,k}$ are positive. The closed-loop system satisfies (TC1),
(TC2) and (TC3) when the values in both plots are green. For this case, the
Tsypkin criterion is satisfied for all $k\geq 1700$.

##  VI Conclusions

This paper analyzed the closed-loop stability of predictive cost adaptive
control for output-feedback of a discrete-time, Lur’e system using absolute
stability criteria. Two absolute stability criteria for discrete-time systems
were considered, namely, the circle criterion and the Tsypkin test. Since the
linear dynamics of the closed-loop Lur’e system are time-varying, these tests
were applied instantaneously as a means for determining whether or not the
adaptive controller is converging to a GAS system. A numerical example showed
that, under additional excitation, the circle and Tsypkin criteria are
satisfied, suggesting that PCAC globally asymptotically stabilizes the DTL
system.

In this application of PCAC, linear-system identification is applied to a
nonlinear system that exhibits self-oscillations; this behavior is not shared
by the linearized system. Future research will thus focus on understanding the
reasons for this effectiveness despite the modeling mismatch.

## Acknowledgments

This research was supported by ONR under grant N00014-18-1-2211 and AFOSR
under grant FA9550-20-1-0028. The authors thank Wassim Haddad for helpful
discussions on absolute stability.

Let $G({\bf q})\stackrel{{\scriptstyle\triangle}}{{=}}C({\bf q}I_{n}-A)^{-1}B$
be a strictly proper, discrete-time transfer function with $n$th-order minimal
realization $(A,B,C)$ and state $x_{k}\in{\mathbb{R}}^{n}$, let
$v_{k}\in{\mathbb{R}}^{m}$ be an exogenous input, let
$y_{k}\in{\mathbb{R}}^{p}$ be the output of $G,$ and let
$\gamma\colon{\mathbb{R}}^{p}\to{\mathbb{R}}^{m}.$ Then, for all $k\geq 0,$
the discrete-time Lur’e (DTL) system composed of the positive feedback
interconnection of $G$ and $\gamma$ shown in Figure .1 has the closed-loop
dynamics

$\displaystyle x_{k+1}$ $\displaystyle=Ax_{k}+B(\gamma(y_{k})+v_{k}),$ (.1)
$\displaystyle y_{k}$ $\displaystyle=Cx_{k}.$ (.2)

$+$$G$$\gamma$$v_{k}$$y_{k}$

1.Figure:  Discrete-time Lur’e (DTL) system composed of the positive feedback
interconnection of $G$ and $\gamma.$

###### Definition .1

Let $\gamma\colon{\mathbb{R}}^{p}\to{\mathbb{R}}^{m}$, and let
$M_{1}\in{\mathbb{R}}^{m\times p}$ and $M_{2}\in{\mathbb{R}}^{m\times p}$ be
such that $M_{2}-M_{1}$ is positive definite. Then, $\gamma$ is sector-bounded
(SB) with sector bound [$M_{1},M_{2}$] if, for all $y\in{\mathbb{R}}^{p},$
$[\gamma(y)-M_{1}y]^{\mathsf{T}}[\gamma(y)-M_{2}y]\leq 0.$

The following result is the circle criterion [25, Thm 5.1].

###### Theorem .2

Assume that $G$ is strictly proper, let $M_{1}\in{\mathbb{R}}^{m\times p}$ and
$M_{2}\in{\mathbb{R}}^{m\times p}$ be such that $M_{2}-M_{1}$ is positive
definite, assume that $\gamma$ is SB with sector bound [$M_{1},M_{2}$], define
$H({\bf q})\stackrel{{\scriptstyle\triangle}}{{=}}[I_{m}-M_{2}G({\bf
q})][I_{m}-M_{1}G({\bf q})]^{-1},$ and assume that the following conditions
are satisfied:

  * (CC1) $\alpha_{\rm CC}\stackrel{{\scriptstyle\triangle}}{{=}}{\rm spr}(H)<1.$

  * (CC2) $\beta_{\rm CC}\stackrel{{\scriptstyle\triangle}}{{=}}\min_{\psi\in[0,\pi]}\left(\lambda_{\rm min}\left[H(e^{\jmath\psi})+H^{\mathsf{T}}(e^{-\jmath\psi})\right]\right)>0.$

Then, the zero solution of (.1), (.2) is GAS.

The following definition considers the case where $m=p.$

###### Definition .3

Let $\gamma\colon{\mathbb{R}}^{m}\to{\mathbb{R}}^{m}$ be continuous, and let
$M\in{\mathbb{R}}^{m\times m}$ be positive definite. Then, $\gamma$ is
diagonal, increasing, and sector-bounded (DISB) with sector bound $[0,M]$ if
the following conditions are satisfied:

  1. $i)$

For all $y\in{\mathbb{R}}^{m},$
$\gamma(y)=[\gamma_{(1)}(y_{(1)})\cdots\gamma_{(m)}(y_{(m)})]^{\mathsf{T}}.$

  2. $ii)$

For all $y\in{\mathbb{R}}^{m},$ $\gamma^{\mathsf{T}}(y)[\gamma(y)-My]\leq 0.$

  3. $iii)$

For all $i\in\\{1,\ldots,m\\}$ and distinct
$\sigma,\hat{\sigma}\in{\mathbb{R}},$ it follows that
$(\gamma_{(i)}(\sigma)-\gamma_{(i)}(\hat{\sigma}))/(\sigma-\hat{\sigma})>0.$

Next is the Tsypkin criterion [27, Thm 3.1].

###### Theorem .4

Assume $m=p,$ $G$ is strictly proper, $\gamma$ is DISB with sector bound
$[0,M],$ let $N\stackrel{{\scriptstyle\triangle}}{{=}}{\rm
diag}(N_{1},\ldots,N_{m})$ be positive definite, define $L_{N}({\bf
q})\stackrel{{\scriptstyle\triangle}}{{=}}M^{-1}-[I_{m}+(1-{\bf
q}^{-1})N]G({\bf q}),$ and assume the following conditions hold:

  * (TC1) ${\rm det}(CA^{-1}B)\neq 0,$ $(A,C+NC-NCA^{-1})$ is observable, and $\lim_{{\bf z}\to\infty}[L_{N}({\bf z})+L_{N}({\bf z})^{\mathsf{T}}]>0.$

  * (TC2) $\alpha_{{\rm TC},N}\stackrel{{\scriptstyle\triangle}}{{=}}{\rm spr}(L_{N})<1.$

  * (TC3) $\beta_{{\rm TC},N}\stackrel{{\scriptstyle\triangle}}{{=}}\min_{\psi\in[0,\pi]}(\lambda_{\rm min}[L_{N}(e^{\jmath\psi})+L_{N}^{\mathsf{T}}(e^{-\jmath\psi})])>0.$

Then, the zero solution of (.1), (.2) is GAS.

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