Editing Sliding mode control (section)

==Sliding mode observer==
Sliding mode control can be used in the design of [[state observer]]s. These non-linear high-gain observers have the ability to bring coordinates of the estimator error dynamics to zero in finite time. Additionally, switched-mode observers have attractive measurement noise resilience that is similar to a [[Kalman filter]].<ref name="UtkinGS99">{{Cite book|title=Sliding Mode Control in Electromechanical Systems|last1=Utkin|first1=Vadim|last2=Guldner|first2=Jürgen|last3=Shi|first3=Jingxin|year=1999|publisher=Taylor & Francis, Inc.|location=Philadelphia, PA|isbn=978-0-7484-0116-1}}</ref><ref name="Drakunov83">{{Cite journal|last=Drakunov|first=S.V.|title=An adaptive quasioptimal filter with discontinuous parameters|journal=Automation and Remote Control|year=1983|volume=44|issue=9|pages=1167&ndash;1175}}</ref> For simplicity, the example here uses a traditional sliding mode modification of a [[Luenberger observer]] for an [[LTI system]]. In these sliding mode observers, the order of the observer dynamics are reduced by one when the system enters the sliding mode. In this particular example, the estimator error for a single estimated state is brought to zero in finite time, and after that time the other estimator errors decay exponentially to zero. However, as first described by Drakunov,<ref name="Drakunov92">{{Cite book|last=Drakunov|first=S.V.|chapter=Sliding-mode observers based on equivalent control method |title=&#91;1992&#93; Proceedings of the 31st IEEE Conference on Decision and Control|year=1992|issue=Tucson, Arizona, 16–18 December|pages=[https://archive.org/details/proceedingsofthe0003unse/page/2368 2368–2370]|isbn=978-0-7803-0872-5|doi=10.1109/CDC.1992.371368|s2cid=120072463|url=https://works.bepress.com/cgi/viewcontent.cgi?article=1003&context=sergey_v_drakunov|chapter-url=https://archive.org/details/proceedingsofthe0003unse/page/2368}}</ref> a [[State observer#Switched observers|sliding mode observer for non-linear systems]] can be built that brings the estimation error for all estimated states to zero in a finite (and arbitrarily small) time.

Here, consider the LTI system
:<math>\begin{cases}
\dot{\mathbf{x}} = A \mathbf{x} + B \mathbf{u}\\
y = \begin{bmatrix}1 & 0 & 0 & \cdots & \end{bmatrix} \mathbf{x} = x_1 \end{cases}</math>
where state vector <math>\mathbf{x} \triangleq (x_1, x_2, \dots, x_n) \in \mathbb{R}^n</math>, <math>\mathbf{u} \triangleq (u_1, u_2, \dots, u_r) \in \mathbb{R}^r</math> is a vector of inputs, and output {{mvar|y}} is a scalar equal to the first state of the <math>\mathbf{x}</math> state vector. Let
:<math>A \triangleq \begin{bmatrix} a_{11} & A_{12} \\ A_{21} & A_{22} \end{bmatrix}</math>
where
* <math>a_{11}</math> is a scalar representing the influence of the first state <math>x_1</math> on itself,
* <math>A_{21} \in \mathbb{R}^{(n-1)}</math> is a row vector corresponding to the influence of the first state on the other states,
* <math>A_{22} \in \mathbb{R}^{(n-1) \times (n-1)}</math> is a matrix representing the influence of the other states on themselves, and
* <math>A_{12} \in \mathbb{R}^{1\times(n-1)}</math> is a column vector representing the influence of the other states on the first state.

The goal is to design a high-gain state observer that estimates the state vector <math>\mathbf{x}</math> using only information from the measurement <math>y=x_1</math>. Hence, let the vector <math>\hat{\mathbf{x}} = (\hat{x}_1,\hat{x}_2,\dots,\hat{x}_n) \in \mathbb{R}^n</math> be the estimates of the {{mvar|n}} states. The observer takes the form
:<math>\dot{\hat{\mathbf{x}}} = A \hat{\mathbf{x}} + B \mathbf{u} + L v(\hat{x}_1 - x_1)</math>
where <math>v: \R \to \R</math> is a nonlinear function of the error between estimated state <math>\hat{x}_1</math> and the output <math>y=x_1</math>, and <math>L \in \mathbb{R}^n</math> is an observer gain vector that serves a similar purpose as in the typical linear [[state observer|Luenberger observer]]. Likewise, let
:<math>L = \begin{bmatrix} -1 \\ L_{2} \end{bmatrix}</math>
where <math>L_2 \in \mathbb{R}^{(n-1)}</math> is a column vector. Additionally, let <math>\mathbf{e} = (e_1, e_2, \dots, e_n) \in \mathbb{R}^n</math> be the state estimator error. That is, <math>\mathbf{e} = \hat{\mathbf{x}} - \mathbf{x}</math>. The error dynamics are then
:<math>\begin{align}
\dot{\mathbf{e}}
&= \dot{\hat{\mathbf{x}}} - \dot{\mathbf{x}}\\
&= A \hat{\mathbf{x}} + B \mathbf{u} + L v(\hat{x}_1 - x_1) 
 - A \mathbf{x} - B \mathbf{u}\\
&= A (\hat{\mathbf{x}} - \mathbf{x}) + L v(\hat{x}_1 - x_1)\\
&= A \mathbf{e} + L v(e_1)
\end{align}</math>
where <math>e_1 = \hat{x}_1 - x_1</math> is the estimator error for the first state estimate. The nonlinear control law {{mvar|v}} can be designed to enforce the sliding manifold
:<math>0 = \hat{x}_1 - x_1</math>
so that estimate <math>\hat{x}_1</math> tracks the real state <math>x_1</math> after some finite time (i.e., <math>\hat{x}_1 = x_1</math>). Hence, the sliding mode control switching function
:<math>\sigma(\hat{x}_1,\hat{x}) \triangleq e_1 = \hat{x}_1 - x_1.</math>
To attain the sliding manifold, <math>\dot{\sigma}</math> and <math>\sigma</math> must always have opposite signs (i.e., <math>\sigma \dot{\sigma} < 0</math> for [[almost everywhere|essentially]] all <math>\mathbf{x}</math>). However,
:<math>
\dot{\sigma} = \dot{e}_1
= a_{11} e_1 + A_{12} \mathbf{e}_2 - v( e_1 )
= a_{11} e_1 + A_{12} \mathbf{e}_2 - v( \sigma )
</math>
where <math>\mathbf{e}_2 \triangleq (e_2, e_3, \ldots, e_n) \in \mathbb{R}^{(n-1)}</math> is the collection of the estimator errors for all of the unmeasured states. To ensure that <math>\sigma \dot{\sigma} < 0</math>, let
:<math>v( \sigma ) = M \operatorname{sgn}(\sigma)</math>
where
:<math>M > \max\{ |a_{11} e_1 + A_{12} \mathbf{e}_2| \}.</math>
That is, positive constant {{mvar|M}} must be greater than a scaled version of the maximum possible estimator errors for the system (i.e., the initial errors, which are assumed to be bounded so that {{mvar|M}} can be picked large enough; al). If {{mvar|M}} is sufficiently large, it can be assumed that the system achieves <math>e_1 = 0</math> (i.e., <math>\hat{x}_1 = x_1</math>). Because <math>e_1</math> is constant (i.e., 0) along this manifold, <math>\dot{e}_1 = 0</math> as well. Hence, the discontinuous control <math>v(\sigma)</math> may be replaced with the equivalent continuous control <math>v_{\text{eq}}</math> where
:<math>
0 = \dot{\sigma} = a_{11} \mathord{\overbrace{e_1}^{ {} = 0 }} + A_{12} \mathbf{e}_2 - \mathord{\overbrace{v_{\text{eq}}}^{v(\sigma)}}
= A_{12} \mathbf{e}_2 - v_{\text{eq}}.</math>
So
:<math>
\mathord{\underbrace{v_{\text{eq}}}_{\text{scalar}}} = \mathord{\underbrace{A_{12}}_{1 \times (n-1) \atop \text{ vector}}} \mathord{\underbrace{\mathbf{e}_2}_{(n-1) \times 1 \atop \text{ vector}}}.
</math>
This equivalent control <math>v_{\text{eq}}</math> represents the contribution from the other <math>(n-1)</math> states to the trajectory of the output state <math>x_1</math>. In particular, the row <math>A_{12}</math> acts like an output vector for the error subsystem
:<math>
\mathord{\overbrace{
\begin{bmatrix}
\dot{e}_2\\
\dot{e}_3\\
\vdots\\
\dot{e}_n
\end{bmatrix}
}^{\dot{\mathbf{e}}_2}}
=
A_2
\mathord{\overbrace{
\begin{bmatrix}
e_2\\
e_3\\
\vdots\\
e_n
\end{bmatrix}
}^{\mathbf{e}_2}}
+
L_2 v(e_1)
=
A_2
\mathbf{e}_2
+
L_2 v_{\text{eq}}
=
A_2
\mathbf{e}_2
+
L_2 A_{12} \mathbf{e}_2
= ( A_2 + L_2 A_{12} ) \mathbf{e}_2.
</math>
So, to ensure the estimator error <math>\mathbf{e}_2</math> for the unmeasured states converges to zero, the <math>(n-1)\times 1</math> vector <math>L_2</math> must be chosen so that the <math>(n-1)\times (n-1)</math> matrix <math>( A_2 + L_2 A_{12} )</math> is [[Hurwitz-stable matrix|Hurwitz]] (i.e., the real part of each of its [[eigenvalue]]s must be negative). Hence, provided that it is [[Observability|observable]], this <math>\mathbf{e}_2</math> system can be stabilized in exactly the same way as a typical linear [[state observer]] when <math>A_{12}</math> is viewed as the output matrix (i.e., "{{mvar|C}}"). That is, the <math>v_{\text{eq}}</math> equivalent control provides measurement information about the unmeasured states that can continually move their estimates asymptotically closer to them. Meanwhile, the discontinuous control <math>v = M \operatorname{sgn}( \hat{x}_1 - x )</math> forces the estimate of the measured state to have zero error in finite time. Additionally, white zero-mean symmetric measurement noise (e.g., [[Normal distribution|Gaussian noise]]) only affects the switching frequency of the control {{mvar|v}}, and hence the noise will have little effect on the equivalent sliding mode control <math>v_{\text{eq}}</math>. Hence, the sliding mode observer has [[Kalman filter]]&ndash;like features.<ref name="Drakunov83"/>

The final version of the observer is thus
:<math>\begin{align}
\dot{\hat{\mathbf{x}}}
&= A \hat{\mathbf{x}} + B \mathbf{u} + L M \operatorname{sgn}(\hat{x}_1 - x_1)\\
&= A \hat{\mathbf{x}} + B \mathbf{u} + \begin{bmatrix} -1\\L_2 \end{bmatrix} M \operatorname{sgn}(\hat{x}_1 - x_1)\\
&= A \hat{\mathbf{x}} + B \mathbf{u} + \begin{bmatrix} -M\\L_2 M\end{bmatrix} \operatorname{sgn}(\hat{x}_1 - x_1)\\
&= A \hat{\mathbf{x}} + \begin{bmatrix} B & \begin{bmatrix} -M\\L_2 M\end{bmatrix} \end{bmatrix} \begin{bmatrix} \mathbf{u} \\ \operatorname{sgn}(\hat{x}_1 - x_1) \end{bmatrix}\\
&= A_{\text{obs}} \hat{\mathbf{x}} + B_{\text{obs}} \mathbf{u}_{\text{obs}}
\end{align}</math>
where
* <math>A_{\text{obs}} \triangleq A,</math>
* <math>B_{\text{obs}} \triangleq \begin{bmatrix} B & \begin{bmatrix} -M\\L_2 M\end{bmatrix} \end{bmatrix},</math> and
* <math>u_{\text{obs}} \triangleq \begin{bmatrix} \mathbf{u} \\ \operatorname{sgn}(\hat{x}_1 - x_1) \end{bmatrix}.</math>
That is, by augmenting the control vector <math>\mathbf{u}</math> with the switching function <math>\operatorname{sgn}(\hat{x}_1-x_1)</math>, the sliding mode observer can be implemented as an LTI system. That is, the discontinuous signal <math>\operatorname{sgn}(\hat{x}_1-x_1)</math> is viewed as a control ''input'' to the 2-input LTI system.

For simplicity, this example assumes that the sliding mode observer has access to a measurement of a single state (i.e., output <math>y=x_1</math>). However, a similar procedure can be used to design a sliding mode observer for a vector of weighted combinations of states (i.e., when output <math>\mathbf{y} = C \mathbf{x}</math> uses a generic matrix {{mvar|C}}). In each case, the sliding mode will be the manifold where the estimated output <math>\hat{\mathbf{y}}</math> follows the measured output <math>\mathbf{y}</math> with zero error (i.e., the manifold where <math>\sigma(\mathbf{x}) \triangleq \hat{\mathbf{y}} - \mathbf{y} = \mathbf{0}</math>).