Observer Design for an Inertia Wheel Pendulum with Static Frictionfootnoteinfo

[footnoteinfo]This work has been supported by the COMET-K2 Center of the Linz Center of Mechatronics (LCM) funded by the Austrian federal government and the federal state of Upper Austria.

First]L. Ecker
First]K. Schlacher
First]M. Schöberl

[First]Institute of Automatic Control and Control Systems Technology, Johannes Kepler University Linz,  Altenberger Str. 69, A-4040 Linz, Austria (e-mail: {lukas.ecker, kurt.schlacher, markus.schoeberl}@jku.at).

A state observer design for the inertia wheel pendulum considering static friction of the actuated inertia disc is presented.
The frictional force is modeled by the Stribeck effect, with two separate differential equations describing the sticking and non-sticking system, respectively.
The transition between the two scenarios is in general determined by a static friction condition.
Three proposed observers are designed for both the sticking and non-sticking model.
Depending on the probability of the respective model given a current measurement, the more likely is selected for state estimation.
The performance is demonstrated on a laboratory demonstrator, where the observers including the proposed probabilistic model selection are compared.

Nonlinear Observer Design, Static Friction, State Observation, Inertia Wheel Pendulum, Bayesian Statistics

§ INTRODUCTION

State estimation is a common method for determining non-measurable and unknown system states.
Just as the state estimation problem is generally more difficult in the nonlinear scenario than in the linear scenario, the same is true for the distinction between infinite- and finite-dimensional systems, see [Meurer et al., 2005].
Moreover, in particular systems governed by a family of differential equations with algebraic switching conditions, i. e., switched or hybrid systems, see, e. g., [Liberzon, 2003], require special attention.
Switching conditions, as encountered when modeling mechanical systems with static friction, are generally state-dependent.
Thus, depending on the current values of the state, the system behaves according to a different differential equation.
This is usually not a problem when evaluating the initial value problem, since the state variables required for the switching condition are known from the simulation.
However, for the state estimation problem, the state-dependent condition is critical because, except for certain measured quantities, only the estimated quantities are known to the observer. Depending on which differential equation is used, different estimates result, which in turn have different implications for the choice of the actual model for the observer.
The present paper focuses on the observer design for the inertia wheel pendulum, a nonlinear mechanical benchmark example in control theory as found in, e. g., [Spong et al., 1999], with switching sticking and non-sticking subsystem.
The stiction of the wheel, which typically occurs during pendulum tests in real experimental setups, is modeled as an alternation between two models, with each described by a separate differential equation.
An extended Kalman filter (EKF), a nonlinear observer with linear error dynamics ($\text{NO}_\text{L}$), and a nonlinear observer with passive error dynamics ($\text{NO}_\text{P}$), are designed and evaluated on the laboratory demonstrator.
The widely used EKF, see, e. g., [Simon, 2006], is based on a sampled-data system representation and stochastic methods, while the $\text{NO}_\text{L}$ and $\text{NO}_\text{P}$ exploit system theoretic approaches such as linearization by output injection, see, e. g., [Krener and Isidori, 1983], and passivity properties,  see, e. g., [Schaft, 2017].
Therefore, the observer with passive error dynamics exploits a port-Hamiltonian representation of the inertia wheel pendulum, see, e. g.,[Ortega et al., 2001], which allows to inject a dissipative behavior of the error system between state and estimate.
The port-Hamiltonian framework has proven useful for both finite and infinite dimensions in system analysis and in the design of controllers and observers, see, e. g., [Macchelli et al., 2005], [Schöberl and Siuka, 2014], [Malzer et al., 2020], and [Ecker et al., 2021].

The paper considers a finite-dimensional switched system.
The main difficulties that arise in the transition between the sticking and non-sticking model for the observer are addressed by probabilistic model selection. Instead of the sole evaluation of the switching condition with estimates from either the sticking or the non-sticking model, the marginal likelihood of the two models given the obtained measurement is used. The Bayesian factor, a ratio of the likelihoods of the competing models, see, e. g., [Bernardo and Smith, 2008], determines which model is used for the prediction of the state.
The paper is structured as follows. In Section 2, the model of the inertia wheel pendulum including static friction of the wheel is briefly derived. Section 3, is dedicated to the design of the observers as well as the presentation of the probabilistic model selection. Finally, in Section 4, the performance of the proposed observers are demonstrated on a laboratory model.

§ MATHEMATICAL MODELING

This section is dedicated to the mathematical modeling of the inertia wheel pendulum. Since the mechanical system is a common example in control theory, see, e. g.,
[Spong et al., 1999], a detailed derivation of the equation of motion is omitted and more attention is devoted to the modeling of the static friction.

§.§ Inertia Wheel Pendulum

The model of the inertia wheel pendulum under consideration, see, Fig. <ref>, consists of a rigid, ball-bearing pendulum with actuated wheel at one end and counterweight at the opposite end.
The absolute angle of the pendulum is denoted by $\varphi_1$ and the relative angle of the wheel by $\varphi_2$.
The wheel is driven by a subordinate torque-controlled DC electric motor.
The electrical dynamics, which are much faster compared to the dynamics of the mechanical system, are neglected.
Therefore, the motor torque $M$ is considered as the input of the model of the inertia wheel pendulum.
The governing equations of motion with regard to the generalized coordinates $q = [q^\alpha] = [\varphi_1,\varphi_2]^T$ and velocities $\dot{q} = [\dot{q}^\alpha] =  [\dot{\varphi_1},\dot{\varphi_2}]^T$ are derived by means of the Euler-Lagrange equations

\begin{equation}
\label{equ:lagrange_equation}
\frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial L}{\partial \dot{q}^\alpha} - \frac{\partial L}{\partial q^\alpha} = Q_\alpha, \quad \alpha = 1,2\,,
\end{equation}

where the Lagrange function $L = T - V$ is the difference between the kinetic energy $T = \frac{1}{2} \theta_1 \dot{\varphi}_1^2 + \frac{1}{2} \theta_2 \left(\dot{\varphi}_1 + \dot{\varphi}_2\right)^2$ and the potential energy $V = a \cos(\varphi_1)$.
The moments of inertia of the subsystem pendulum are summarized by the parameter $\theta_{1}$, whereas the moment of inertia of the wheel with respect to the center of gravity is described by $\theta_2$.
Moreover, the factor $a$ covers the components of the potential energy  of the overall system.
The viscous damping between the pendulum and the base as well as the pendulum and the inertia wheel are taken into account.
The damping terms are assumed to be proportional to the respective angular velocity, i. e., $d_\alpha \dot{\varphi}_\alpha $, which together with the input moment $M$ and friction moment $M_S$ form the terms of the generalized forces $Q = [Q_\alpha] = [ -d_1\dot{\varphi}_1, -d_2 \dot{\varphi}_2 + M {- M_S}]^T$.
Evaluating the Euler-Lagrange equations (<ref>), the equations of motion correspond to the second order differential equation system

\begin{equation}\small
\begin{bmatrix}
\theta_1 + \theta_2 & \theta_2\\
\theta_2 & \theta_2
\end{bmatrix}
\begin{bmatrix}
\ddot{\varphi}_{1}\\
\ddot{\varphi}_{2}
\end{bmatrix}
\begin{bmatrix}
a \sin(\varphi_1)\\
\end{bmatrix}
\begin{bmatrix}
-d_1 \dot{\varphi_1} \\ -d_2 \dot{\varphi_2} + M {- M_S}
\end{bmatrix}.
\label{equ:iwp_eom}
\end{equation}

Furthermore, the equations can be rewritten as a system of first order differential equations with state $x = [\varphi_{1} ,\varphi_{2} , \omega_1 , \omega_2 ]^T$, where $\omega_\alpha = \dot{\varphi}_\alpha$. Since the resulting system does not depend on the angle $\varphi_2$, the  equation $\dot{\varphi}_2 = \omega_2$ is omitted and only the three dimensional system

\begin{equation}\hspace*{-0.2cm}
\label{equ:non_adherent_system}
\begin{bmatrix}
\dot{\varphi}_{1}\\
\dot{\omega}_{1}\\
\dot{\omega}_{2}
\end{bmatrix}
\begin{bmatrix}
\omega_{1}\\
\frac{1}{\theta_{1}} \left( a\sin \left( \varphi_{1} \right) -d_{1}\omega_{1} +d_{2}\omega_{2}-M {+ M_S}\right)\\
-\frac{a}{\theta_{1}}\sin(\varphi_{1}) + \frac{d_1}{\theta_{1}}  \omega_{1} +\frac{1}{\theta_{c}} (M {- M_S} - d_2 \omega_{2})
\end{bmatrix}
\end{equation}

with state $x = [\varphi_{1}, \omega_1, \omega_2 ]^T$ and abbreviation $\theta_c = \frac{\theta_{1} \theta_{2}}{\theta_{1}+\theta_{2}}$ is considered.

Schematic of the inertia wheel pendulum.

§.§ Static friction

The present model is not yet satisfactory for the state observation of the inertia wheel pendulum.
The occurring discrepancy between the mathematical model and real measured data, which become apparent during the validation (see Section <ref>), indicate that crucial effects are not taken into account by the model (<ref>).
Depending on the trajectory of the system, i. e., usually the drop down or swing up of the pendulum, different disturbances are crucial. In the swing-up experiment, the static friction of the pendulum dominates, while in the drop-down experiment, the static friction of the wheel dominates. However, the present paper focuses on the observer design with the pendulum angle as measured quantity. Since a direct measurement of the wheel angle is not considered, the static friction of the wheel is more critical for the observer design and therefore the main emphasis of this work.
The modeling of the static friction of the inertia wheel is realized by a structural switch between two differential equations.
The friction force is described as a function of the friction velocity by the Stribeck curve

\begin{align*}
M_S(\omega_2) = r_C \,\mathrm{sgn}(\omega_2) + (r_S-r_C) e^{-\left(\frac{\omega_2}{\omega_{2,0}}\right)^2}\mathrm{sgn}(\omega_2)
\end{align*}

where $r_C$ corresponds to the Coulomb friction and $r_S$ to the Stiction parameter. The negative slope friction for a small transition of velocity from zero is referred to as the Stribeck effect, see [Ellis, 2003]. Note that viscous damping has already been taken into account in the derivation of the mathematical model. Static friction, i. e., $\omega_2 = 0$ and $\dot{\omega}_2 = 0$, must meet the condition

\begin{align}\label{equ:stiction_condition}{\left|\frac{\theta_2}{\theta_1 +\theta2}\left(-a \sin(\varphi_{1})+d_1\omega_1\right)+M\right| < r_S.}
%\frac{-a}{\theta_{1}}\sin(\varphi_{1}) + \frac{d_1}{\theta_{1}}  \omega_{1} +\frac{\theta_{1} + \theta_2}{\theta_1 \theta_2} M  < r_S.
\end{align}

The model of the inertia wheel pendulum in stiction with $\omega_{2} = 0$ and $M_S = 0$ is given by

\begin{equation}\label{equ:adherent_system}%\hspace*{-0.32cm}
\begin{bmatrix}
\dot{\varphi}_{1}\\
\dot{\omega}_{1}\\
\dot{\omega}_{2}
\end{bmatrix}
\begin{bmatrix}
\omega_{1}\\
{\frac{1}{\theta_{1}} \left( a\sin \left( \varphi_{1} \right) -d_{1}\omega_{1} \right)}\\
\end{bmatrix},
\end{equation}

where the associated Lagrange function is equivalent to

\begin{equation}%\hspace*{-0.32cm}
\label{equ:stiction_lagrangian}
L = \frac{1}{2}\theta_{1}\dot{\varphi}_1^2 - a\cos(\varphi_{1}).
\end{equation}

§.§ Port-Hamiltonian Representation

The subsequent nonlinear observer design with passive error dynamics will exploit a port-Hamiltonian representation of the mechanical system. By means of the regular Legendre mapping the Hamiltonian $H = \sum_{\alpha = 1}^{n}p_\alpha \dot{q}^\alpha - L$ of the non-sticking system (<ref>) corresponds to

\begin{equation}
H = \frac{1}{2}\frac{1}{\theta_{1}}\left(p_1-p_2\right)^2 + \frac{1}{2}\frac{1}{\theta_{2}}p_2^2 + a \cos(\varphi_{1})
\end{equation}

with the generalized momenta $p_\alpha = \frac{\partial L}{\partial \dot{q}^\alpha}$ introduced as

\begin{align}
\begin{bmatrix}
\end{bmatrix}
\begin{bmatrix}
\theta_1 + \theta_2 & \theta_2 \\
\theta_2 & \theta_2
\end{bmatrix}
\begin{bmatrix}
\dot{\varphi}_{1}\\
\dot{\varphi}_{2}
\end{bmatrix}.
\end{align}

The Hamiltonian formalism $\dot{p}_\alpha = \frac{\partial L }{\partial q^\alpha} + Q_\alpha$ allows to represent the pendulum as port-Hamiltonian system

\begin{equation}  \label{equ:ph_non_sticking}
\dot{x} = \left(J - R\right) \left(\frac{\partial H}{\partial x}\right)^T + G u
\end{equation}

with state $x = [{\varphi_{1}},{p}_1, {p}_2]^T$, input $u = M - M_S$, partial derivatives ${\partial_x H} = {\partial/ \partial_x H}= [-a \sin(\varphi_1),
-(p_1-p_2)/{\theta_1} + p_2/\theta_{2}
]$ and matrices

\begin{equation*}
{J=-J^T} =
\begin{bmatrix}
0 & 1 & 0 \\
-1 & 0 & 0 \\
0 & 0 & 0
\end{bmatrix},\;
{R=R^T} =
\begin{bmatrix}
0 & 0 & 0 \\
0 & d_1 & 0 \\
0 & 0 & d_2
\end{bmatrix}
G =
\begin{bmatrix} 0 \\ 0\\ 1	\end{bmatrix}.
\end{equation*}

Proceeding similarly, the port-Hamiltonian representation of the inertia wheel pendulum in stiction with Lagrangian (<ref>), generalized momentum $p_1 = \frac{\partial L}{\partial \dot{\varphi}_1} =  \theta_{1} \dot{\varphi}^1$ and Hamiltonian $H = \frac{1}{2}\frac{1}{\theta_{1}}\dot{p}_{1}^2 + a\cos(\varphi_{1})$ is given by

\begin{equation} \label{equ:ph_sticking}
\underbrace{\begin{bmatrix}
\dot{\varphi}_1\\
\dot{p}_{1}
\end{bmatrix}}_{\dot{x}}
\underbrace{	\begin{bmatrix}
0 & 1 \\
-1 & 0
\end{bmatrix}}_{J=-J^T}}
\underbrace{\begin{bmatrix}
d_1 & 0 \\
0 & 0
\end{bmatrix}}_{R=R^T}
\biggl)\cdot
\underbrace{	\biggl(
\begin{matrix}
-a \sin(\varphi_1)\\
\frac{1}{\theta_1}p_1

\end{matrix}
\biggl)}_{\left(\frac{\partial H}{\partial x}\right)^T}.
\end{equation}

§ OBSERVER DESIGN

The presented observer design focuses on the estimation of the state $x = [\varphi_{1},\omega_{1},\omega_{2}]^T$ with the pendulum angle $\varphi_{1}$ as measured output. For this purpose, three nonlinear state estimators for the inertia wheel pendulum are introduced.

§.§ Extended Kalman Filter

The Kalman filter is an optimal state observer for linear systems with Gaussian noise. Under the assumption that both the probability density function of the initial state and  the noise are Gaussian distributed, the first two stochastic moments, i. e., the expectation and the variance, are sufficient for a stochastic representation.

The extended Kalman filter is a nonlinear version of the Kalman filter based on the linearization of the dynamics about the current estimate. Since the EKF is de facto standard in the theory of nonlinear state estimation, a detailed derivation, as found in [Simon, 2006], is omitted.
The time-continuous models (<ref>) and (<ref>) are discretized using a fourth-order Runge-Kutta schema with sampling time $T_a = 5\,\mathrm{ms}$, i. e., considered are the equations

\begin{align}
x_{k+1} &= f_k(x_k,u_k,w_k) \\
y_k &= h_k(x_k,v_k)
\end{align}

with time index $k$, system state
$x_k = [\varphi_{1},\omega_{1},\omega_{2}]^T$, output $y_k  = \varphi_{1}$, input $u_k = M $ as well as process noise $w_k $ and measurement noise $v_k$.
The estimation of the state is denoted with $\hat{x}_k$.
Depending on the number of measurements included, it is referred to as an a posteriori or a priori estimate. The a priori state $\hat{x}_k^-$ considers all measurements up to the previous time $k-1$, i. e., $Y_{k-1} = \{y_0,\ldots,y_{k-1}\}$, where as the a posteriori state $\hat{x}_k^+$ includes all measurements up to actual time $k$, i. e., $Y_{k} = \{y_0,\ldots,y_{k}\}$. The Taylor series expansions of the sampled-data system (<ref>) along the estimates corresponds to the linearised system

\begin{align}
x_{k+1} &= F_k x_k + L_k w_k + \tilde{u}_k \\
y_{k} &= H_k x_k + M_k v_k
\end{align}

with Jacobian matrices $ F_k = \left. \frac{\partial f_k}{\partial x}  \right \vert_{\hat{x}^+_k}$,  $L_k = \left. \frac{\partial f_k}{\partial w}  \right \vert_{\hat{x}^+_k}$ $
H_k = \left. \frac{\partial h_k}{\partial x}  \right \vert_{\hat{x}^-_k}$, $
M_k = \left. \frac{\partial h_k}{\partial v}  \right \vert_{\hat{x}^-_k}$
and input $\tilde{u}_k = f_k(\hat{x}_k^+, u_k, 0) - F_k \hat{x}^+_k$.
Assuming mean-free Gaussian process and measurement noise, i. e., $ w_k \sim (0,Q_k)$ and $v_k \sim (0,R_k)$, the prediction step is defined as
$P^-_{k+1} = F_k P_k^+ F_k^T + L_k Q_k L_k^T$ and $\hat{x}^-_{k+1} = f_k(\hat{x}_k,u_k,0)$

while the measurement update can be expressed by

\begin{align}
K_k &= P^-_k H_k^T(H_k P^-_k H_k^T + M_k R_k M_k^T)^{-1}\\
\hat{x}^+_{k} &=  \hat{x}^-_k + K_k \left(y_k - h_k(\hat{x}^-_k,0)\right)\\
P^+_k &= P^-_k - K_k H_k P^-_k.
\end{align}

§.§ Nonlinear Observer with linear error dynamics

The nonlinear observer with linear error dynamics is based on the idea of a coordinate change that splits the system into a linear and a nonlinear term, where the nonlinear part may only depend on input and output. This linearization by output injection, see [Krener and Isidori, 1983] and [Krener and Respondek, 1985], has the advantage that if the observer is chosen as a copy of the plant in adapted coordinates, the nonlinear terms cancel. The nonlinear time-continuous models (<ref>) and (<ref>) can be represented by a regular coordinate transformation $z = \Phi(x)$ and output $y$ as
\dot{z} = A z + \alpha(u,y),
where $\alpha(u,y)$ corresponds to the nonlinear term that depends only on the input $u$ and output $y$. In the case of the inertia wheel pendulum, the nonlinearities occur exclusively in the measurable state $y = \varphi_{1}$, which is why a desired splitting is already possible with $z = \Phi(x) = x$. Therefore, the decomposition of the inertia wheel pendulum in the non-sticking scenario with original coordinates is given by

\begin{equation*}
\underbrace{
\begin{bmatrix}
\dot{\varphi}_1\\
\dot{\omega}_1 \\
\dot{\omega}_2
\end{bmatrix}}_{\dot{x}}
\underbrace{\renewcommand{\arraystretch}{1.2}
\begin{bmatrix}
0& 1 & 0\\
0&-\frac {d_{1} }{\theta_1}& \frac {d_{2} }{\theta_1} \\
0&\frac{d_1}{\theta_1}&-\frac{d_2}{\theta_c} \\
\end{bmatrix}}_{A}
\underbrace{
\begin{bmatrix}
\varphi_1\\
\omega_1 \\
\omega_2 \\
\end{bmatrix}}_{x}
\underbrace{\renewcommand{\arraystretch}{1.2}
\begin{bmatrix}
0 \\
\frac{a}{\theta_{1}}\sin(\varphi_1) - \frac{M-M_S}{\theta_{1}} \\
-\frac{a}{\theta_{1}}\sin(\varphi_1) + \frac{M-M_S}{\theta_{c}}
\end{bmatrix}}_{\alpha(u,y)},
\end{equation*}

where the decomposition of the sticking model is given by

\begin{equation*}
\underbrace{
\begin{bmatrix}
\dot{\varphi}_1\\
\dot{\omega}_1
\end{bmatrix}}_{\dot{x}}
\underbrace{\renewcommand{\arraystretch}{1.2}
\begin{bmatrix}
0& 1 \\
0&-\frac {d_{1} }{\theta_1}  \\
\end{bmatrix}}_{A}
\underbrace{
\begin{bmatrix}
\varphi_1\\
\omega_1 \\
\end{bmatrix}}_{x}
\underbrace{\renewcommand{\arraystretch}{1.2}
\begin{bmatrix}
0 \\
\frac{a}{\theta_{1}}\sin(\varphi_1)  \\
\end{bmatrix}}_{\alpha(u,y)}.
\end{equation*}

The nonlinear observer with state $\hat{x}$ is composed of a copy of the plant and linear correction term

\begin{equation}\label{equ:nlowled}
\dot{\hat{x}} = A\hat{x} + \alpha(u,y) + K(y-\hat{y})
\end{equation}

where $\hat{y}$ corresponds to the estimated linear output $\hat{y} = C \hat{x}$  and $K$ represents the correction coefficients of the observer.
The dynamics of the linear error $e = x - \hat{x}$ are thus given by

\begin{equation}\label{equ:continious_linear_error_dynamics}
\dot{e} = (A-KC)e.
\end{equation}

Both the linear subsystem of the sticking and non-sticking models are observable, allowing observer design methods from linear control theory such as pole placement or optimal designs based on the solution of the algebraic Riccati equation, see, e. g., [Simon, 2006].
The proposed nonlinear observer with linear error dynamics (<ref>) is implemented quasi-continuous.
The previously described design approach can also be carried out for the sampled-data system $x_{k+1} = A_dx_k + G_d \alpha(u_k,y_k)$  with $A_d = e^{A T_a}$ and $G_d = \int_{0}^{T_a}e^{A \tau}\mathrm{d}{\tau}$, where $\alpha(u_k,y_k)$ is assumed to be constant in the time interval $t_k$ and $t_{k+1}$.
Note that since the state and hence the output are quantities of a continuous process, the term $\alpha(u,y)$ is generally not constant over the sampling period.
However, given sufficient sampling time, satisfactory observer performance can be achieved even with this non-exact approximation to the sampled-data system.
Nevertheless, the nonlinear discrete observer given by

\begin{equation}\label{equ:nlowled_sampled_data}
\hat{x}_{k+1} = A_d \hat{x}_k + G_d \alpha(u_k,y_k)+ K_d(y_k-\hat{y}_k)
\end{equation}

with correction term $K_d$ also leads to an error dynamics

\begin{equation}\label{equ:discrete_linear_error_dynamics}
e_{k+1} = (A_d-K_d C)e_k
\end{equation}

with linear behavior. Both the linear subsystem of the sticking and of the non-sticking model are observable.
Finally, the nonlinear observer is designed by an appropriate choice of $K$ or $K_d$, with the objective of stabilizing the linear error dynamics (<ref>) and (<ref>), respectively.

§.§ Nonlinear Observer with passive error dynamics

The underlying idea is to design a nonlinear observer that induces a passive behavior of the error of state and estimation. Exploiting the port-Hamiltonian representation of the system, the passive error dynamics is derived with the observer as a copy of the plant. In addition, a compensation term allows the imposition of an appropriate desired Hamiltonian function $H_d$ of the error system, and a measurable collocated output can be used for the injection of damping behavior. In this way, the asymptotic stability of the error system for the model in stiction and the model not in stiction should be ensured. Considering the port-Hamiltonian representation  (<ref>) and (<ref>), the nonlinear observer with state $\hat{x}$ and system output $y = h(x)$ is introduced as

\begin{align}\label{equ:ph_observer}
\dot{\hat{x}} &= \left(J-R\right)\left(\partial_{\hat{x}} H(\hat{x}) + \Phi(\hat{x},y)\right)^T + G u - G_o u_o,
\end{align}

where $\Phi(\hat{x},y)$ corresponds to the compensation term
and $G_o u_o$  to the observer input used for the damping injection. Since the compensation term $\Phi(\hat{x},y)$ and the input $G_o u_o$ are quantities of the observer, they should only depend on the estimation $\hat{x}$ and the measurement $y$. As mentioned, the compensation term will be used to induce the desired Hamiltonian function $H_d$. For this purpose the compensation term is determined as
\begin{equation*}
\Phi = \left(\partial_x H(x)-\partial_{\hat{x}} H(\hat{x})- \partial_e H_d(e)\right)^T,
\end{equation*}
where $ e = x - \hat{x}$ corresponds to the considered linear error between state and estimate.
As a result of the proposed observer (<ref>) and the compensation term, the error dynamic is given by

\begin{align*}
\dot{{e}} &= \left(J-R\right)\underbrace{\left(\partial_x H(x)-\partial_{\hat{x}} H(\hat{x})- \Phi(\hat{x},y)\right)^T}_{\left(\partial_e H_d(e)\right)^T}  + G_o u_o,
\end{align*}

where the evolution of the Hamiltonian corresponds to

\begin{align*}
\dot{H_d}(e) = -\left(\partial_e H_d(e)\right) R \left(\partial_e H_d(e)\right)^T + \left(\partial_e H_d(e)\right) G_o u_o.
\end{align*}
For the implementation of the observer, an appropriate positive definite Hamilton function  $H_d>0$ must ensure that the compensation term depends only on the estimate $\hat{x}$ and the output $y$, i. e., $\Phi(\hat{x},y)$ holds.
This functional dependency can be verified by means of exterior algebra, where $\mathrm{d}$ refers to the exterior derivative with respect to the coordinates $x$ and $\wedge$ to the exterior product.
Therefore, considering in general $m$ output equations, the entries of the compensation term $\Phi^i$ defined by $H_d$ with $1\leq i \leq n$ must satisfy the condition
\mathrm{d}\Phi^i \wedge \mathrm{d} h^1 \wedge \ldots \wedge \mathrm{d}h^m = 0.
In order to verify the asymptotic stability of the error system, not only the positive definiteness of $H_d$ but also the negative definiteness of $\dot{H}_d$ must be ensured.
Since for mechanical systems in general only $R \ge 0$ holds, see (<ref>) and (<ref>), and thus $\dot{H}_d \le 0$, additional dissipative behavior has to be included to force a steady decrease of the observation error.
Therefore, a desired matrix $R_d = R_d^T = R + \bar{R} > 0$
must match the condition

\begin{multline*}
\hspace*{-0.4cm} -\left(\partial_e H_d(e)\right) R \left(\partial_e H_d(e)\right)^T + \underbrace{\left(\partial_e H_d(e)\right) G_o u_o}_{F(\hat{x},u,y)} \\ {=} -\left(\partial_e H_d(e)\right) R_d \left(\partial_e H_d(e)\right)^T,
\end{multline*}

which allows the observer input with

\begin{align*}
G_o u_o = F(\hat{x},u,y) = -\bar{R}\left(\partial_e H_d(e)\right)^T
\end{align*}

asymptotic stabilize the error dynamics. Note that $G_0 u_0$ as part of the observer must also depend only on $\hat{x}$, $u$ and $y$.
Considering the inertia wheel pendulum with measured output $y = h(x) = \varphi_{1}$, the positive definite function
H_d(e) = \frac{\alpha}{2}\tilde{\varphi}_{1}^2 + \frac{1}{2 \theta_1}\left(\tilde{p}_1-\tilde{p}_2\right)^2 + \frac{1}{2 \theta_2} \tilde{p}_2^2
with $\alpha > 0$ serves as proper desired Hamiltonian, since $\mathrm{d}\Phi^i \wedge \mathrm{d} h = 0$ holds. Likewise, the matching condition is fulfilled with

\begin{align*}
G_o u_o = -\bar{R}\left(\partial_e H_d(e)\right)^T = -\begin{bmatrix}
\beta & 0 & 0\\
0 & 0 & 0\\
0 & 0 & 0
\end{bmatrix}
\left(
\begin{matrix}
\alpha \tilde{\varphi}_1\\
. \\
\end{matrix}
\right),
\end{align*}

where also simultaneously $R_d = R + \bar{R} > 0$ ensures the desired behavior of steady decreasing estimation error. The desired Hamiltonian for the model in stiction is set to
$H_d(e) = \frac{\alpha}{2}\tilde{\varphi}_{1}^2 + \frac{1}{2 \theta_1} \tilde{p}_1^2$
and the observer input to

\begin{align*}
G_o u_o = -\bar{R}\left(\partial_e H_d(e)\right)^T = -\begin{bmatrix}
\beta &  0\\
0 & 0 \\
\end{bmatrix}
\left(
\begin{matrix}
\alpha \tilde{\varphi}_1\\
. \\
\end{matrix}
\right).
\end{align*}

The nonlinear observer with passive error dynamics (<ref>) is implemented quasi-continuous.

§.§ Model selection

This section presents an approach for the implementation of a switching law between two models in the scope of a observer design with no complete knowledge of the state. Since the state dependence of the derived switching condition in the observer must be evaluated with estimates, a correct selection of the currently correct model for the observer is not always possible. Therefore, a probabilistic model selection based on the Bayesian factor is proposed. The basic idea is that with the current measurement obtained, the probability of both models is determined and the more likely one is used for prediction. The output is interpreted as a random variable $Y$ because a perturbation $v_k$ with probability density function $p_V(v_k)$ enters the output equation with $y_k = h(x_k) + v_k$. The non-sticking model (<ref>) is referred to as $M_{1}$, and the sticking model (<ref>) is referred to as $M_2$, where $f_{M_1,k}$ and $f_{M_2,k}$ denote their corresponding sampled-data systems. In principle, this methodology is not limited to two models. Nevertheless, considered are the two models $M_1$ and $M_2$ as well as the a posterior estimate of the previous time step $\hat{x}_{k-1}$ as well as the current measurement $y_k$. The marginal likelihoods of the two models given the obtained measurement in ratio results in the so-called Bayes factor

\begin{equation}
K = \frac{\mathrm{Pr}(M_1 | Y = y_k)}{\mathrm{Pr}(M_2 | Y = y_k)}.
\end{equation}

Therefore, when $K>1$, it is more likely that the measurement $y_k$ stems from the model $M_1$, and vice versa, when $K<1$, the model $M_2$ is to be preferred. With equal probable models, i. e., $\mathrm{Pr}(M_1) = \mathrm{Pr}(M_2)$ and Bayes' theorem, see, e. g., [Papoulis and Pillai, 2002],

\begin{equation}
\mathrm{Pr}(M | Y = y_k) = \frac{p_Y(y_k|M)\mathrm{Pr}(M)}{p_Y(y_k)},
\end{equation}

where $p_Y(y_k) = p_Y(y_k|M_1)\mathrm{Pr}(M_1) + p_Y(y_k|M_2)\mathrm{Pr}(M_2)$ denotes the probability density function of the output, the ratio of the Bayes factor can be simplified to

\begin{equation}
K = \frac{p_Y(y_k|M_1)}{p_Y(y_k|M_2)}.
\end{equation}

For the computational implementation the conditional probability density function can be expressed in terms of the probability density of the perturbation as

\begin{equation}
p_Y(y_k|M_d) = p_V(v_k) \circ \left(y_k - h(\hat{x}_{M_d,k})\right)
\end{equation}

with $\hat{x}_{M_d,k} = f_{M_d,k}(\hat{x}_{k-1})$.

§ EXPERIMENTAL RESULTS

This section is dedicated to the validation of the proposed models and the presentation of the observer performance by the drop-down experiment of the inertia pendulum.

§.§ Validation

The validation should highlight the discrepancy between the model without static friction and the real measured data in order to justify the additional effort for modeling static friction. Furthermore, the validation allowed to identify the parameters of the Stribeck curve as well as the viscous damping coefficients. The parameters of the static friction model as well as the experimental determined parameters of the inertia wheel pendulum are listed in Table <ref>. Note that the stiction of the pendulum as well as the counter-torque due to the supply lines to the drive wheel are not taken into account. However, in the course of validation, the initialization error of the incremental encoder was  compensated by calibration.

Parameters of the inertia wheel pendulum and the static friction model.

3lModel parameters        3lStatic friction model

2lParameter      Value           2lParameter      Value

$a$     [□□]    0.15535         $r_C$   [□□]    0.0024

$\theta_{1}$     [□]     0.05045        $r_S$    [□□]   0.0026

$\theta_{2}$     [□]     0.00113        $w_{2,0}$   []   0.0501

$d_1$     [□]     0.00885

$d_2$     [□]     0.00015

The reference measurement data, i. e., the angles of pendulum and wheel, of the real laboratory model were recorded by two incremental encoders. The associated angular velocities were determined by gate time measurement.
The following Figure <ref> shows the validation of both the model without static friction and the model with static friction and structure switch based on (<ref>). While the results of measurement and simulation of both models hardly differ for $\varphi_{1}$ and $\omega_{1}$, significant discrepancies between measurement and the model without static friction occur for $\omega_{2}$.
In contrast, the model with static friction reproduces the real behavior of the pendulum very well. Therefore, it is also used for the observer design, since the estimation performance depends to a high degree on the model accuracy.

Validation of the inertia wheel pendulum: Comparison of the  mathematical models considering static friction (<ref>) and non static friction (<ref>) with measurements from the laboratory demonstrator (<ref>).

§.§ Observer performance

The three proposed observers with probabilistic model switching are presented in the context of the drop-down experiment, i. e., free swinging down of the inertia pendulum to the lower stable rest position.
The observer parameters were not optimized, since a direct comparison was not intended, rather the principle suitability should be demonstrated.
Apart from the introductory overview, see Fig. <ref>, the observer behavior with respect to the angular velocity $\omega_{2}$ is investigated in detail, since the pendulum angle is used as a measurand and the static friction particularly affects the wheel.
The transient behavior of the estimates results from the intentionally miss initialized observers.
The experiment starts in the state $x_0 = [0,0,0]^T$, while the observers are initialized with $\hat{x}_0 = [\SI{-\pi/10 }{\radian}, \SI{1}{\radian \per \second},\SI{1}{\radian \per \second}]^T$. The covariances of the Gaussian distributed quantities for the EKF and the $\text{NO}_\text{L}$, see Section <ref>, are set to $P = \text{diag}(0.00165, 0.01, 0.1)$, $Q = \text{diag}(0, 0.01, 0.1)$ and $R = 0.001$.

Overview of the three observers: Comparison of the EKF (<ref>), $\text{NO}_\text{L}$ (<ref>), and $\text{NO}_\text{P}$ (<ref>) estimates $\varphi_{1}$ and $\omega_{1}$ with measurements from the laboratory demonstrator (<ref>).

The nonlinear observer with linear error dynamics has been implemented as sampled-data system (<ref>).
Designed as a time-invariant linear-quadratic estimator, the correction term of the error dynamics is obtained as Kalman gain.
Assuming that the measurement is not disturbed, i. e., $R = 0$, the solution of the algebraic Riccati equation yields observer parameter that lead to a dead-beat behavior. The parameter of nonlinear observer with passive error dynamics are set to $\alpha = 10$ and $\beta = 5$.
The observer estimates in Fig. <ref> are nearly identical, except for the transient responses.
The estimates of the wheel angular velocity $\hat{\omega}_2$ are presented in Fig. <ref>.
The results of the observers are divided into three separate graphs to illustrate the change in adhesion of the wheel.
In every estimation step, the observers first determine the probability that the system is in stiction and not in stiction given the latest measurement.
The most likely model is subsequently applied for the estimation of the state.
In Fig. <ref>, the current model used for estimation is indicated by the background color, with a red area (<ref>) marking the sticking model and a green area (<ref>) marking the no-sticking model.
All three observers provide satisfactory results, especially with respect to their combination with the probabilistic model selection.

Estimation of the wheel angular velocity: Comparison of the proposed observer with probabilistic model selection (<ref>) and measurements from the laboratory demonstrator (<ref>).

§ CONCLUSION

In this contribution, three observers for the nonlinear model of inertia wheel pendulum with switching stiction of the wheel have been presented. Beside an extended Kalman Filter, a nonlinear observer with linear as well as a nonlinear observer with passive error dynamics are proposed. As the considered system is described by two differential equations, representing a sticking and a non-sticking model, the switching in the observer is implemented by means of a probabilistic model selection based on Bayes factor.
The performance of the observers is verified on a laboratory demonstrator.

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# On-CMOS High-Throughput Multi-Modal Amperometric DNA Analysis with
Distributed Thermal Regulation

Hamed M. Jafari, Xilin Liu, and Roman Genov

###### Abstract

Accurate temperature regulation is critical for amperometric DNA analysis to
achieve high fidelity, reliability, and throughput. In this work, a 9$\times$6
cell array of mixed-signal CMOS distributed temperature regulators for on-CMOS
multi-modal amperometric DNA analysis is presented. Three DNA analysis methods
are supported, including constant potential amperometry (CPA), cyclic
voltammetry (CV), and impedance spectroscopy (IS). In-cell heating and
temperature sensing elements are implemented in standard CMOS technology
without post-processing. Using proportional-integral-derivative (PID) control,
the local temperature can be regulated to within ±0.5∘C of any desired value
between 20∘C and 90∘C. The two computationally intensive operations in the PID
algorithm, multiplication, and subtraction, are performed by an in-cell dual-
slope multiplying ADC in the mixed-signal domain, resulting in a small area
and low power consumption. Over 95% of the circuit blocks are synergistically
shared among the four operating modes, including CPA, CV, IS, and the proposed
temperature regulation mode. A 3mm$\times$3mm CMOS prototype fabricated in a
0.13$\mu$m CMOS technology has been fully experimentally characterized. Each
channel occupies an area of 0.06mm2 and consumes 42$\mu$W from a 1.2V supply.
The proposed distributed temperature regulation design and the mixed-signal
PID implementation can be applied to a wide range of sensory and other
applications.

###### Index Terms:

Temperature regulation, mixed-signal IC, PID control, on-CMOS DNA analysis,
circuit sharing

##  I Introduction

Figure 1: (a) Conceptual block diagram of one of the temperature-regulated
amperometric DNA analysis channels. (b) Illustration of the basic operations
of the three amperometric DNA analysis modes supported in this work: constant
potential amperometry (CPA), cyclic voltammetry (CV), and impedance
spectroscopy (IS).

Conventional optical DNA analysis methods typically require sample flushing,
thus cannot be implemented in real-time [3]. Amperometric electrochemical DNA
analysis supports real-time operation [4]. Recently, several CMOS system-on-
chip (SoC) designs for amperometric electrochemical DNA analysis have been
developed [5, 6, 7, 12, 13, 8, 9, 11]. On-CMOS DNA analysis permits
advantageous high-density, low-noise, low-cost monolithic integration of
various modules for sensing, signal processing, communication, and power
management [10]. Fig. 1 illustrates the operating principles of amperometric
DNA analysis. First, the surface of the working electrode (WE) is
functionalized with probe DNA [5]. Then, a voltage is applied between the
reference electrode (RE) and the WE. The voltage waveform is set to be a
constant voltage for constant potential amperometry (CPA), a bidirectional
ramp for cyclic voltammetry (CV), and a sinusoid for impedance spectroscopy
(IS). The hybridization of the probe DNA with the target DNA alters the
surface properties of the WE, such as the impedance or the surface charges.
This variation will result in a waveform change of the redox current that
indicates the kinetics and thermodynamics of the chemical reactions [14].

Temperature is a key constraint in all DNA analysis methods. Studies have
shown that DNA hybridization is highly dependent on the temperature [17, 16,
15]. For a temperature increase of 10∘C, the redox current will drop
approximately by 10% [17]. Additionally, time-varying temperature control in
the range of 20∘C to 90∘C is required for DNA multiplication through
polymerase chain reaction (PCR). Therefore, a temperature regulation system is
of great benefit for DNA sensing applications. In practice, off-array CMOS
processing modules often cause temperature variation across the DNA analysis
array, and do not scale well to large sensing arrays. Thus, distributed in-
channel temperature sensing and regulation are necessary. In addition, a
channel-level thermal regulation scheme permits the generation of spatial
temperature gradients for advanced non-uniform DNA analysis [18, 19].

Figure 2: Four modes of operation of the proposed multi-modal DNA analysis
SoC. (a) CPA configuration [29], (b) CV configuration [28], (c) IS
configuration [30], and (d) temperature regulation configuration (the focus of
this work). The blocks with a thick outline and highlighted in light green are
synergistically shared in all modes, with over 95% of all area reused in each
operating mode.

Several temperature regulation designs for chemical sensing applications have
been reported [20, 21, 6, 22, 8, 9, 11]. These designs typically use resistive
heating and sensing elements. Paper [20] presents a temperature sensing and
regulation microsystem for biological applications that achieves a single
fixed temperature of 37∘C on the backside of the chip, with an off-chip
digital proportional-integral-derivative (PID) controller. Paper [21] presents
a 3$\times$3 micro-hotplate array for a protein-sensing application. In this
design, the temperature of the chip is regulated up to 45∘C with an off-chip
digital PID controller. Paper [6] presents a 16-channel, frequency-shift CMOS
magnetic biosensor array for a protein-sensing application. This design
consists of an array of in-channel heaters, with a single temperature sensor
and an off-chip analog PID controller. Paper [22] presents a digital CMOS
micro-hotplate array architecture for a gas-sensing application. The array
consists of 3$\times$3 gas-sensing sites. Each site consists of a heater, a
temperature sensor, and a delta-sigma modulated PID controller. The micro-
hotplate achieves an operating temperature of up to 350∘C. A 40-channel DNA
analysis SoC based on ISFET sensors for rapid point-of-care DNA detection is
reported in [7]. This design includes in-channel heaters, temperature sensing
elements, and a dedicated ADC. A fully digital on-chip PID controller is
utilized for temperature regulation. In summary, most existing closed-loop
temperature regulation microsystems use off-chip controllers, with few
exceptions [7, 9]. Conventional on-chip controller designs often require a
large silicon area along with high power consumption, which is prohibitive for
most on-chip electrochemical sensing applications.

In this work, we present a compact, distributed temperature regulator design
implemented in 0.13$\mu$m standard CMOS technology. Fig. 1 depicts the block
diagram of the proposed temperature-regulated electrochemical cell. It
consists of a temperature sensor, a heater, a closed-loop regulator, a working
electrode, a reference electrode, and a potentiostat. We chose the PID
algorithm for the closed-loop regulation because of its high efficacy in
temperature control systems [23, 24]. PID controllers have been implemented
on-chip in analog [25, 26] and digital domains [27]. Here we introduced a
novel mixed-signal design that yields a compact integration with low power
dissipation. This design permits distributed temperature regulation in large-
scale parallel sensing applications.

The presented temperature controller is a part of a highly-integrated multi-
functional SoC that performs on-chip amperometric electrochemical DNA
analysis. The overall SoC consists of 54 sensing channels with 600 gold
microelectrodes. It performs label-free DNA analysis and pH sensing for
prostate cancer detection. During the design, we synergistically share the
circuits among different operation modes to achieve high integration and low
resource overhead. We have previously reported on the cyclic voltammetry DNA
sensing modality in [28]. We briefly discussed how the DNA sensing and the
temperature control circuits are integrated in [29]. The impedance modality
was introduced by us in [30]. In this paper, we presented a unified review of
how all of these sensing modalities are implemented on the same IC with
minimal overhead, and focus on presenting the details of the distributed
mixed-signal temperature regulator, including the system architecture,
algorithm design, circuit implementation, and experimental results. To the
best of our knowledge, this work presents the first distributed temperature
regulation design with on-chip PID control for DNA analysis. This design
permits an accurate thermal control for compensating the temperature variation
across the DNA analysis array while potentially enabling advanced temperature
gradient generation. We developed a new way of implementing PID control using
mixed-signal circuits (e.g., multiplying ADC), which avoids the
computationally intensive multiplications of the PID algorithm in the digital
domain, rendering compact silicon area and low power consumption. The proposed
design is required for the three sensory modalities: CPA, CA, and IS, and
reuses over 95% of their circuit implementation. It can be used in a wide
range of applications beyond DNA analysis, including lab-on-chip designs, in-
vitro systems, and implantable medical devices (e.g., retinal prosthesis
array).

The rest of the paper is organized as follows. Section II overviews the multi-
modal DNA analysis SoC. Section III provides background on temperature
regulation principles. Section IV presents the VLSI architecture of the
temperature controller. Section V details the circuit implementation. Section
VI shows the experimental results obtained from the fabricated 0.13$\mu$m CMOS
prototype. Finally, Section VII concludes the paper.

##  II Multi-modal DNA Analysis SoC with Synergistic Circuit Sharing

Fig. 2 shows an overview of the presented multi-modal DNA analysis SoC, which
can be configured to perform CPA (Fig. 2(a) [29]), CV (Fig. 2(b) [28]), IS
(Fig. 2(c) [30]), and temperature regulation (Fig. 2(d)).

The SoC includes a 9$\times$6 array of low-power low-noise sensory circuit
cells, a programmable waveform generator, on-chip SRAM memories, a shared PID
controller, a digital PWM, an all-digital ultra-wideband (UWB) transmitter,
and on-chip biasing and clock generation circuits. Each sensory cell includes
a current-to-digital conversion channel, an array of DNA sensors, a pH sensor,
several BJTs to generate the CTAT and PTAT currents, and a heater for
temperature regulation. Each cell also integrates a bias voltage generator,
clock generation circuitry, and SRAM memory that can be programmed
individually.

Each current-to-digital conversion channel consists of either a transimpedance
amplifier (TIA) or a current conveyor (CC) and a dual-slope multiplying ADC
(MADC). The dual-slope MADC multiplies the sensor’s response with a set of
digital coefficients, and outputs the corresponding digital word. The digital
output of each channel is serialized on the chip and then wirelessly
transmitted using the all-digital UWB transmitter [28].

The SoC reuses a number of blocks (denoted by bold outlines and light green
color in Fig. 2) in different configurations. The circuit sharing design
yields a significant saving in the integration area. Specifically, the
circuits that occupy 95% of the in-cell area are shared with other modes of
operation. For completeness, all key functionalities of the SoC are summarized
in this section. For the detailed implementation of the CPA, CV and IS, please
refer to our previous publications in [29], [28] and [30], respectively.

Figure 3: (a) A pair of BJTs for generating $\Delta$VBE and VBE, (b)
temperature dependence of the base-emitter voltage, (c) generation of PTAT and
CTAT current, (d) conceptual plot of the proposed temperature dependent
current generation, and (e) utilization of a current-to-digital channel for
temperature measurements.

###  II-A Constant Potential Amperometry (CPA) Mode

Fig. 2(a) shows the functional blocks for pH sensing using CPA analysis.
Sensing pH is required as a part of DNA analysis. The in-channel pH sensor is
implemented by a floating gate PFET, with the gate connected to the top metal
layer for forming a floating gate electrode. The CMOS passivation layer is
used as the pH-sensitive membrane, which gives a linear pH response with a
sensitivity of about 56mV/pH, depending on the stoichiometry. The pH sensors
are directly interfaced to the current recording channels, with gate voltage
set by the on-chip reference electrode. In this configuration, both Vgs and
Vds of the pH sensors are fixed. Changes in the pH level will affect the PMOS
threshold voltage, resulting in the changes in the drain current, which is
digitized by the recording channel. The chip temperature needs to be held at a
constant value within ±0.5∘C.

###  II-B Cyclic Voltammetry (CV) Mode

Fig. 2(b) shows the functional blocks for CV analysis. In this mode, each
channel is multiplexed among a bank of in-cell DNA sensors. The sensors are
interrogated by the on-chip programmable waveform generator that is shared
among all cells. The CC front-end is utilized in this configuration. The
digital data representing the stimulation waveform properties are stored in
the on-chip SRAM. The current-to-digital channel quantizes the input current
and outputs the corresponding digital word. No computation is required in this
case and the MADC digital multiplication coefficients are set to one. The chip
temperature is held at a constant value within ±0.5∘C.

###  II-C Impedance Spectroscopy (IS) Mode

Fig. 2(c) shows the functional blocks for IS analysis. In this mode, a
frequency response analysis (FRA) algorithm is used to extract the real and
imaginary components of the biosensor impedance [30]. The computationally
intensive operations required by the FRA algorithm are performed by the in-
channel MADC in the mixed-signal domain. The waveform generator produces a
variable frequency sinusoidal interrogation waveform for driving the reference
electrode, and front-end TIA acquires the DNA sensor’s response. The MADC
multiplies the sensor’s response with a set of digital FRA algorithm
coefficients (stored in the multiplication coefficient SRAM). Next, the MADC
accumulates the results over one period of the interrogation signal
(integration) for extracting the impedance. In this mode, the cell temperature
is also held constant within ±0.5∘C.

###  II-D Temperature Regulation Mode

It is clear that each of the three sensory modes requires accurate temperature
control. Thus, the temperature regulation function is the main focus of this
paper. Fig. 2(d) shows the functional blocks. In this mode, the SoC utilizes
the current-to-digital channels to measure temperature. The on-chip PID
controller is used to regulate each cell temperature. On-chip SRAMs are
utilized to store PID coefficients and multiplication coefficients. The
channel measures temperature by taking the ratio of a CTAT to PTAT currents as
described in Section II. The measured temperature is fed to the on-chip PID
controller. The PID controller regulates the 2D chip temperature profile by
modulating the in-cell heaters. The multiplication and subtraction operations
required by the PID algorithm are performed by the in-channel MADC in the
mixed-signal domain, yielding a small silicon area and low power consumption.

##  III Temperature Control Principles

###  III-A Temperature regulation

The block diagram of the temperature regulation loop is shown in Fig. 4. It
consists of a heater, a temperature sensor, a PID controller, and an actuator.
The actuator uses digital pulse-width modulation (PWM) for controlling the
heater.

Figure 4: Block diagram of the PID control of the temperature regulation loop.

The continuous-time representation of a PID controller is given by

$\displaystyle\centering
u(t)=K_{p}e(t)+K_{i}\displaystyle\int^{t}_{0}e(t)dt+K_{d}\frac{d}{dt}e(t)\@add@centering$
(1)

where Kp is the proportional gain, Ki is the integral gain, Kd is the
derivative gain, e(t) is the error value, and u(t) is the actuation value. A
discrete-time representation of (1) in Z-domain is given by

$\displaystyle\centering
D(z)=\frac{u(z)}{e(z)}=K_{p}+K_{i}\frac{T_{s}}{2}\frac{z+1}{z-1}+\frac{K_{d}}{T_{s}}\frac{z-1}{z}\@add@centering$
(2)

where Ts is the sampling period. Eq. (2) can be implemented numerically as
follows

$\displaystyle\centering\vspace{0.005in}u(k)=u(k-1)+c_{0}e(k)+c_{1}e(k-1)+c_{2}e(k-2)\@add@centering$
(3)

where k is the discrete time index, and c0, c1, and c2 are the PID control
coefficients.

During the operation, the PID controller calculates the error value e(k) and
attempts to minimize it by adjusting u(k), which sets the PWM duty cycle of
the actuator.

Figure 5: Top-level VLSI architecture of temperature regulation loop (shaded
boxes are in-cell).

###  III-B Temperature sensing

Temperature can be measured by taking the ratio of two signals that are
linearly dependent on it. In CMOS technologies, these signals can be derived
from the base-emitter voltage of substrate pnp transistors [31]. These
transistors use the p-substrate as the collector, the p+-diffusion as the
emitter, and an N-well as the base. Fig. 3(a) shows the generation of two
linearly temperature dependent signals. One is generated from the base-emitter
voltage (VBE) of a single pnp transistor, the other is generated from the
difference between the base-emitter voltages ($\Delta$VBE) of two pnp
transistors biased at different collector current levels.

The temperature dependence of the VBE can be derived from the exponential
relation between the collector current IC and the base-emitter voltage as
follows.

$\displaystyle\centering V_{BE}(T)\@add@centering$ $\displaystyle=$
$\displaystyle V_{g0}(1-\frac{T}{T_{r}})+\frac{T}{T_{r}}V_{BE}T_{r}$ (4)
$\displaystyle-$ $\displaystyle
n\frac{KT}{q}ln\frac{T}{T_{r}}+\frac{KT}{q}ln\frac{I_{C}T}{I_{C}T_{r}}$

where Vg0 is the extrapolated bandgap voltage at 0 Kelvin, n is a process-
dependent constant, K is Boltzmann constant, q is the electron charge, and Tr
is an arbitrary reference temperature [32]. As illustrated in Fig. 3(b), VBE
is complementary to absolute temperature (CTAT) with a typical slope of - 2
mV/K. The nonlinearity is represented by the last two terms of (4), which is
negligible in the temperature range of 20${}^{\circ}C$ to 90${}^{\circ}C$
required by DNA analysis.

The slope of the base-emitter voltage depends on the absolute value of the
collector current. This dependence can be used to generate a voltage that is
proportional to absolute temperature (PTAT). The difference between the base-
emitter voltages of two pnp transistors biased at two different collector
currents can be expressed as follows

$\displaystyle\centering\Delta V_{BE}(T)\@add@centering$ $\displaystyle=$
$\displaystyle V_{BE2}(T)-V_{BE1}(T)$ (5) $\displaystyle=$
$\displaystyle\frac{KT}{q}ln(\frac{I_{C2}}{I_{C1}})$

where IC1 and IC2 are the collector currents of the two pnp transistors.

Conventional temperature sensors measure the ratio of the PTAT signal to a
temperature-independent reference signal. The temperature-independent
reference signal can be generated by adding VBE to a scaled version of
$\Delta$VBE [32, 33, 34, 31, 35]. The scaling factor $\alpha$ is used to match
the temperature dependence (with an opposite sign) of VBE and $\Delta$VBE.
However, the generation of the temperature-independent reference signal adds
complexity to the circuit implementation, which is not suitable for a channel-
level implementation. Although distributing a global temperature-independent
signal is feasible, it places challenges in the signal routing if a voltage
signal is used, and requires accurate matching or time multiplexing if a
current signal is used. In this work, we eliminate these issues by directly
taking the ratio of PTAT and CTAT signals, which have an approximately linear
relationship with temperature within the range of 20∘C to 90∘C. The PTAT and
CTAT voltages are converted to currents using resistors R1 and R2. The
$\alpha\Delta$VBE/R2 is used as the reference the dual-slope MADC and the
VBE/R1 is used as the input signal, as illustrated in Fig. 3 (e). As a result,
the dual-slope MADC inherently digitizes the ratio of $\alpha\Delta$VBE/R2 to
VBE/R1 [28]. The digitized output number represents the temperature. The
factor $\alpha$, the resistance R1 and R2 are set such that both the input and
the reference currents utilize the full dynamic range of the channel over the
operating temperature range.

##  IV VLSI Architecture

The top-level VLSI architecture of the temperature regulation loop is shown in
Fig. 5. The regulation loop consists of CTAT and PTAT BJTs, PTAT and CTAT
current sources, a current-to-digital channel, a 7-bit SRAM bank with an
adder, a 12-bit digital PWM, and an in-cell heater. The in-cell BJTs are
interfaced sequentially to the current sources that generate IPTAT and ICTAT.
ICTAT is used as the input to the channel, and IPTAT is used as the reference
current. The channel determines the ratio of ICTAT to IPTAT. Both currents are
scaled such that their magnitude fits within the channel dynamic range over
the operating temperature range. The ratio of ICTAT to IPTAT results in a
linear transfer characteristic (versus temperature) [31]. The channel dual-
slope MADC performs three tasks. First, it computes the ratio of ICTAT to
IPTAT. Next, it multiplies this ratio by the PID coefficients, (c0, c1, c2),
and finally it subtracts the results from the scaled version of the desired
temperature, (c0t(k), c1t(k), c2t(k)). The result of these three operations is
the computation of cne(k), as shown in Fig. 5. Next, these values are stored
in the on-chip 7-bit SRAM bank. An on-chip adder adds the appropriate values
according to Eq.(3) and computes u(k), which sets the duty cycle of the signal
generated by the digital PWM. The duty cycle of this signal sets the amount of
time that the in-cell heater is on, thus regulating the temperature. The in-
cell heater was implemented by a PMOS transistor with a width and length of
100$\mu$m and 0.13$\mu$m, respectively, and a 15$\Omega$ load.

The timing diagram of the PID regulation loop is shown in Fig. 6. The channel
dual-slope MADC performs three conversion cycles during one cycle of the PID
controller, thus eliminating the need for three parallel ADCs. As shown in
Fig. 6, in the first conversion cycle of the PID controller, the ADC computes
c0e(0), c1e(0), and c2e(0). Next, u(0) is calculated from these values, and
the duty cycle of the digital PWM is set accordingly. Consecutive values of
c0e(n), c1e(n), c${}_{2}e(n)$, and u(n) are generated in the same manner.

Figure 6: Timing diagram of the temperature regulation loop.

##  V Circuit Implementation

###  V-A Current-to-Digital Conversion Channel

The top-level VLSI architecture of the current-to-digital conversion channel
is shown in Fig. 7.

Figure 7: Top-level VLSI architecture of one temperature sensing channel and
the timing diagram of the dual-slope MADC.

Each channel consists of a chopper-stabilized current conveyor [36], a dual-
slope MADC [37], control logic, on-chip memories, and a counter. The dual-
slope MADC performs both multiplication and subtraction, as required by the
PID algorithm.

Figure 8: Circuit diagrams of (a) the CTAT sensor and current source, (b) the
PTAT sensor and current source, and (c) the chopper-stabilized folded-cascode
OTA in the CTAT (OTA1) and PTAT (OTA2) current sources.

The timing diagram of the MADC for a typical conversion cycle is shown in Fig.
7. The dual-slope MADC digitizes the input signal in two phases: charging
phase T1 and discharging phase T2. The ratio of the duration of the charging
phase to the discharging phase represents the input signal level. As shown in
Fig. 7, to implement multiplication of the input current by a digital PID
coefficient cn, the duration of the charging phase is scaled with a constant
coefficient cn$<$1 [37]. In this case, by counting the time cnT2, a digital
representation of cnICTAT can be obtained.

As shown in Fig. 7, the counter is first reset. At the same time, the PID
multiplication coefficient, cn, is loaded into the in-channel latch (S3=1).
There is a large channel-to-channel variation between the BJT current output.
To compensate for this variation, the channel counter is pre-loaded with a
digital calibration value (stored in the in-channel calibration SRAM) at the
rising edge of the RESET signal. Next, the in-channel counter counts up from
this value to time cnT1, and the input current ICTAT is integrated onto the
capacitor CINT. After time cnT1, the voltage on the capacitor is held constant
for a fixed time interval TRD. During this time, the digital representation of
the desired temperature that is scaled by the PID coefficients cnt(k) is
loaded into the counter. During time cnT2, the integrating capacitor is
discharged with the PTAT current source. During time cnT2, the counter counts
down, and the final value of the counter is available at the falling edge of
S2 signal. By counting down from cnt(k) in the second phase of the conversion
cycle, the MADC performs subtraction of cny(k) from cnt(k) (as shown in Fig.
5) thus computing the error signal cne(k).

The front-end bidirectional current conveyor is implemented by a PMOS and an
NMOS transistor connected in the feedback of an OTA [38]. The negative
feedback ensures a known potential at the working electrode is set by the
voltage at the non-inverting input of the OTA. It also enables the current
conveyor to source and sink an input current without the need for a DC offset
current. The OTA is implemented as a folded-cascode amplifier with a PMOS
input pair. Chopper-stabilization is utilized to reduce the OTA flicker noise
and offset and dynamic element matching is utilized to reduce the output
current mirrors mismatch [38].

The ADC comparator is implemented with three stages of pre-amplifiers with a
total gain of 60dB, and a high-speed latch as the last stage. The first stage
of the comparator is implemented as a cross-coupled diode-connected gain stage
to provide a moderate gain with high bandwidth. Chopper-stabilization
suppresses the input offset and ensures a 9-bit accuracy. The second and third
stages are identical to the first one but with no chopping. The high-speed
latch is implemented with an NMOS input pair gain stage and an NMOS-PMOS
cross-coupled load to provide high accuracy, low offset and a high bandwidth
[30].

###  V-B CTAT and PTAT Current Sources

A simplified circuit diagram of the CTAT current source is shown in Fig. 8(a).
To generate a digitally programmable base-emitter voltage VBEtrim
(compensating for chip-to-chip variations), a PTAT current (generated by a
separate on-chip biasing circuit) is passed through a 6-bit programmable
resistor connected in series with the diode-connected pnp transistor Q1. The
current V${}_{BEtrim}/$R1 is generated by a voltage-to-current converter, as
shown in Fig. 8(a). A large resistor (R1$>$100M$\Omega$) is required to ensure
that the CTAT current is kept relatively low (20nA) and stays within the
dynamic range of the ADC. This would result in a large integration area. To
reduce the size of the resistor, a current mirror with a 10:1 ratio is
utilized. A 10:1 dynamic element matching is used to reduce the effect of the
current mirror mismatch. In this work R1 is set to 1.5M$\Omega$.

A simplified circuit diagram of the PTAT current source is shown in Fig. 8(b).
The $\Delta$VBE is generated by determining the difference between the VBE of
the two substrate pnp transistors that are biased at 3:1 collector current
ratio. The voltage-to-current converter consisting of OTA2 and the NMOS
feedback transistor generates $\Delta$VBE across R2 (split into R2$/$3 and
2R2$/$3). In this work R2 is set to 100k$\Omega$. This results in the desired
output current $\Delta$V${}_{BE}/$R2. The resistor R${}_{2}/$3 is added in
series with the base of the Q3 transistor in order to ensure that the base
current of the Q3 does not affect the output current.

The main causes of inaccuracy in the current sources are the input offset and
the flicker noise of OTAs and the mismatch of the bias current mirror.
Internal OTA chopping is utilized to reduce the effect of flicker noise and
input offset voltage [39]. OTA1 and OTA2 adopt the same circuit topology, as
shown in Fig. 8(c). The transistor sizes are given in Table I. The chopper
switches are placed at the input of the OTA and below the cascoded NMOS
transistors in the current mirror. This placement reduces the flicker noise
and DC offsets caused by the input pair transistors and the NMOS current
mirror transistors. Dynamic element matching at 100Hz, by the means of the
chopper switches in series between the current sources and the resistors, is
utilized to improve the matching.

TABLE I: OTA Transistor Sizing Transistor |  W/L $(\mu m)$
---|---
$M_{1,2}$ | $9\times 4/1$
$M_{3,4}$ | $4\times 1.5/4$
$M_{5,6}$ | $5\times 1.5/1$
$M_{7,8}$ | $10\times 1.5/1$
$M_{9,10}$ | $4\times 1.5/4$
$M_{11}$ | $11\times 1.5/2$

The simulated input-referred noise of OTA1 in cases where the chopper is
disabled and enabled is shown in Fig. 9.

Figure 9: Simulated input-referred noise spectrum of the OTA with chopper
enabled and disabled from 0.01Hz to 1kHz.

The integrated input-referred noise from 0.01Hz to 1kHz is 0.11pA when the
chopper is disabled and 0.06pA when the chopper is enabled. The contribution
of each transistor to the total input-referred noise is shown in Fig. 10. When
the chopper is disabled, the main contributions to the input-referred flicker
noise are from the OTA current mirror transistors M3 and M4, and from the
input pair transistors M1 and M2. When the chopper is enabled, the current
mirror transistors M9 and M10 are the main contributors to the input-referred
flicker noise.

(a)

(b)

Figure 10: Simulated OTA noise. (a) Thermal noise contribution, and (b)
flicker noise contribution.

###  V-C Digital Pulse Width Modulator

In the temperature regulation loop, a digital PWM sets the duty cycle of the
pulse for controlling the in-channel heater. The achievable discrete set of
the duty cycle settings of the pulse depends on the digital PWM resolution. In
this work, a 12-bit digital PWM architecture is selected. The design is based
on a hybrid delay-line/counter architecture in [40]. The block diagram of the
PWM is shown in Fig. 11. In this architecture, a 7-bit counter and 32-stage
ring oscillator are used to achieve the 12-bit resolution. At the beginning of
a switching cycle, the output set-reset (SR) flip-flop is set, and the PWM
output pulse goes high. The pulse that propagates through the ring at the
oscillation frequency serves as the clock for the counter. At the time when
the counter output matches the top most significant bits of the digital input
(nc), and a pulse reaches the tap selected by the least significant bits (nd),
the output flip-flop is reset, and the output pulse goes low [40].

Figure 11: Block diagram of the digital pulse width modulator.

##  VI Experimental Results

###  VI-A Chip Micrograph

The micrograph of the fabricated multi-modal amperometric DNA analysis SoC is
shown in Fig. 12.

Figure 12: Micrograph of the wireless temperature-regulated DNA analysis SoC
fabricated in 0.13$\mu$m CMOS technology.

The 54 cells are arranged in a 9$\times$6 array on a 3mm$\times$3mm CMOS die.
Each cell consists of a current conveyor, a dual-slope MADC, an in-cell bias
and clock generator, a pH sensor, a heater, and temperature sensing BJTs. The
PID controller, CTAT and PTAT current sources, and the digital PWM are located
in the top section of the die. The electrodes are postprocessed with 2D and 3D
Au top layers and several metal underlayers [28].

###  VI-B Temperature Regulation Testing Results

The output digital code of the ADC versus temperature, from 20∘C to 90∘C, is
shown in Fig. 13.

Figure 13: Experimentally measured digital output of the ADC v.s. temperature.

The temperature sensor archives a linear transfer function over the operating
temperature range. Fig. 14 shows the measured temperature error of seven dies
from one wafer, which operated at a supply voltage of 1.2V. After a one-point
calibration for compensating for the BJT variation across channels, the
temperature error across the seven dies was less than ±0.5∘C within the
temperature range of 20∘C to 90∘C.

Figure 14: Temperature regulation error experimentally measured on seven dies
from one wafer.

The digital PWM is characterized by sweeping the digital input code from 0 to
4095 and clocking the digital PWM at 2.5MHz. The experimental results of Fig.
15 show the measured duty ratio of the output pulses as a function of the
12-bits digital input. The minimum (4$\%$) and the maximum (96$\%$) duty
ratios are set by design. The digital PWM achieves a linear transfer function
over the programmable digital input code with a 0.82 percent maximum error.
The digital PWM achieves a minimum duty cycle resolution of 0.1$\mu$S.

Figure 15: Experimentally measured transfer function of the pulse-width-
modulator.

An external heater was utilized to set the chip temperature at 50∘C and the
temperature at each channel was recorded to study the effect of channel-to-
channel mismatch. Experimentally measured temperature from 54 channels in one
chip is shown in Fig. 16. The mean digital output temperature and the
corresponding standard deviation are 49.83∘C and 0.20∘C, respectively.

Figure 16: Experimentally measured temperature from 54 channels in one chip.

An example of the temperature regulation cycle in liquid (5mL 1M potassium
phosphate buffer solution), with steps at 35∘C, 45∘C, 55∘C, and 65∘C, is shown
in Fig. 17. The solid line is the chip temperature and the dashed line is the
desired temperature. It takes roughly 10 seconds to achieve an increase of 5∘C
in the chip. The measured absolute value of the relative error of the PID
regulation loop over the operating temperature range is shown in Fig. 18. The
absolute value of the error stays below 0.75∘C over the operating temperature
range.

Figure 17: Transient responses of temperature regulation with target
temperatures at 35∘C, 45∘C, 55∘C, and 65∘C. Figure 18: Measured absolute value
of the relative error of the PID regulation loop.

Table II provides a summary of the experimentally measured characteristics of
the system for temperature regulation.

TABLE II: Experimentally Measured Characteristics Technology |  0.13$\mu$m CMOS
---|---
Supply Voltage | 1.2V
Area |  3mm$\times$3mm
Array Dimensions |  9$\times$6 channels
Channel Size |  300$\mu$m$\times$200$\mu$m
Sensitivity | 8.6pA
Power Consumption (System) |
SRAM |  1.3$\mu$W
PID Controller |  21$\mu$W
Temperature Core |  12$\mu$W
Heater |  270mW @ 90∘C
Power Consumption (Channel) |
Current conveyor |  8$\mu$W
Comparator |  19$\mu$W
Biasing |  4$\mu$W
Digital |  11$\mu$W
Total (Channel) |  42$\mu$W

###  VI-C Multi-modal Amperometric DNA Analysis Results

The dynamic performance of the cell was measured with a 15Hz sinusoidal input
current at a full-scale current of 400nA. The ADC was clocked at 10MHz. A SNR
of 56.9dB was measured, leading to an effective number of bits (ENOB) of 9.3.
The distortion was limited by a -58.7dB HD2 due to the single-ended
architecture of the ADC.

Figure 19: Experimental characterization of pH sensing in the CPA mode. The
3$\sigma$ error bars from 20 measurements are shown.

Fig. 19 shows the experimental characterization of pH sensing in the CPA mode.
The 3$\sigma$ error bars from 20 measurements are shown in the figure. The
measured current leads to a linear response of 1.8nA/pH within ±2 relative to
a pH of 7.

Figure 20: Experimental measurements of 5µM prostate cancer DNA using CV from
0 to -700mV at a scan rate of 100mV/s. The 3$\sigma$ error bars show 20
measurements from 3 chips. Figure 21: Experimental measurements of 5µM
prostate cancer DNA using IS from 0.1Hz to 10kHz. The 3$\sigma$ error bars
show 20 measurements from 3 chips.

Fig. 20 shows the experimental measurements of prostate cancer DNA in the CV
mode. On-chip gold electrodes were used for the CV scans, which were performed
in a 20µM potassium ferrocyanide solution. Oxidation and reduction peaks are
clearly visible in the CV scans using 5µM single stranded DNA modified
electrodes. This is because of the negatively charged film created by
thiolated DNA probes when they repell the negatively charged electrochemical
reporter. Adding a 5µM non-complementary DNA target does not change the CV
signal, which indicates that non-specific adsorption is negligible. On the
other hand, adding a 5µM complementary target to the chip leads to the
creation of double stranded DNA on the biosensing electrode, which indicates
additional negative charge and elimination of ferrocyanide’s redox peak. The
error bars (from 3 chips) in the figure show a detection noise margin of
around 4nA.

Fig. 21 shows the experimental measurements of prostate cancer DNA using the
IS mode. The IS frequency was from 0.1Hz to 10kHz. The recordings of 20µM
potassium ferrocyanide in the cases of 5µM single stranded DNA, addition of
5µM non-complementary DNA, and addition of 5µM complementary DNA target are
shown. The 3$\sigma$ error bars show 20 measurements from 3 chips. The
impedance spectrum of DNA probe and non-complementary target are fairly
similar as expected. The addition of the complementary DNA probe results in a
change in the impedance spectrum.

Table III provides a comparative analysis of the presented design and existing
temperature regulated biochemical sensory microsystems. The presented work
supports advantageous multi-modal DNA analysis with the most flexible
temperature regulation scheme. Our design also achieves a compact channel
area, thanks to the synergistic circuit sharing design.

TABLE III: Comparative Analysis of Temperature-Regulated Biosensory Microsystems |  |  ISSCC 2010 [6] |  | JSSC
---
2017 [13]
JSSC 2019 [8] |  | TBioCAS
---
2020 [9]
| TBioCAS
---
2020 [11]
This Work
Modes | CPA |  No | No |  No | No | Yes | Yes
CV |  No | No |  Yes | No | Yes | Yes
IS |  No | No |  Yes | No | Yes | Yes
Temp. Reg. |  Yes | Yes |  Yes | Yes | Yes | Yes
Process | CMOS Process |  0.13µm | 0.25µm |  0.25µm | 0.18µm | 0.6µm | 0.13µm
Power |  165mW | 118mW |  256mW | 2.7mW | 1.242mW | 0.35mW
Supply Voltage |  1.2V | 2.5V |  2.5V | 3.3V | 5V | 1.2V
Chip Area |  7.5mm2 | 63mm2 |  10.24mm2 | 15mm2 | 361mm2 | 9mm2
Channel | Electrode Count |  16 | 1,024 |  1,024 | 1,225 | 16,064 | 600
Electrode Type |  2D | 2D |  2D | 2D | 2D | 3D
Dynamic Range |  55dB | 116dB |  93dB | N/A | N/A | 128dB
Conversion Rate |  N/A | 50Hz |  50Hz | 17.06Hz | 100kHz | 10kHz
Sensitivity |  0.3Hz | 20fA |  280fA | N/A | 42.8pA | 8.6pA
Signal Generator |  Yes | No |  Yes | Yes | Yes | Yes
Signal Frequency |  1GHz | N/A |  50Hz | N/A | N/A | 10kHz
Wireless |  No | No |  No | No | No | UWB
Temp. Regul. | Channel |  1 | 13 |  13 | 49 | 2 | 54
Control Method |  PID | N/A |  N/A | On/off | N/A | PID
Implementation |  Off-chip | Off-chip |  Off-chip | On-chip | Off-chip | On-chip
Signal Domain |  Analog | N/A |  N/A | Digital | N/A | Mixed-signal
Accuracy |  1∘C | 0.3∘C |  N/A | 1∘C | 0.5∘C | ±0.5∘C
|  |  |  |  |  |  |

##  VII Conclusions

A 9$\times$6 cell array (54-channel) mixed-signal CMOS temperature-regulated
distributed DNA-sensing SoC was developed. The SoC reuses circuitry to perform
CPA, CA, IS, and temperature regulation. The on-chip, in-channel heating and
temperature sensing elements were implemented in standard CMOS without any
post-processing. Local temperature can be regulated to within ±0.5°C of any
desired point between 20∘C and 90∘C using PID control. The overall design
achieved a very low power consumption at 42$\mu$W per channel from a 1.2V
supply. Each channel occupies a silicon area of only 0.06mm2. This design can
be used in a wide range of applications where temperature regulation is
desirable. The proposed circuit sharing methodology can also be beneficial for
future highly integrated SoC designs.

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