Sliding mode control

Sliding mode control can be used in the design of state observers. 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.^[11][12] 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,^[13] a 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


where state vector ${\displaystyle \mathbf {x} \triangleq (x_{1},x_{2},\dots ,x_{n})\in \mathbb {R} ^{n}}$, ${\displaystyle \mathbf {u} \triangleq (u_{1},u_{2},\dots ,u_{r})\in \mathbb {R} ^{r}}$ is a vector of inputs, and output y is a scalar equal to the first state of the ${\displaystyle \mathbf {x} }$ state vector. Let


where


The goal is to design a high-gain state observer that estimates the state vector ${\displaystyle \mathbf {x} }$ using only information from the measurement ${\displaystyle y=x_{1}}$. Hence, let the vector ${\displaystyle {\hat {\mathbf {x} }}=({\hat {x}}_{1},{\hat {x}}_{2},\dots ,{\hat {x}}_{n})\in \mathbb {R} ^{n}}$ be the estimates of the n states. The observer takes the form


where ${\displaystyle v:\mathbb {R} \to \mathbb {R} }$ is a nonlinear function of the error between estimated state ${\displaystyle {\hat {x}}_{1}}$ and the output ${\displaystyle y=x_{1}}$, and ${\displaystyle L\in \mathbb {R} ^{n}}$ is an observer gain vector that serves a similar purpose as in the typical linear Luenberger observer. Likewise, let


where ${\displaystyle L_{2}\in \mathbb {R} ^{(n-1)}}$ is a column vector. Additionally, let ${\displaystyle \mathbf {e} =(e_{1},e_{2},\dots ,e_{n})\in \mathbb {R} ^{n}}$ be the state estimator error. That is, ${\displaystyle \mathbf {e} ={\hat {\mathbf {x} }}-\mathbf {x} }$. The error dynamics are then


where ${\displaystyle e_{1}={\hat {x}}_{1}-x_{1}}$ is the estimator error for the first state estimate. The nonlinear control law v can be designed to enforce the sliding manifold


so that estimate ${\displaystyle {\hat {x}}_{1}}$ tracks the real state ${\displaystyle x_{1}}$ after some finite time (i.e., ${\displaystyle {\hat {x}}_{1}=x_{1}}$). Hence, the sliding mode control switching function


To attain the sliding manifold, ${\displaystyle {\dot {\sigma }}}$ and ${\displaystyle \sigma }$ must always have opposite signs (i.e., ${\displaystyle \sigma {\dot {\sigma }}<0}$ for essentially all ${\displaystyle \mathbf {x} }$). However,


where ${\displaystyle \mathbf {e} _{2}\triangleq (e_{2},e_{3},\ldots ,e_{n})\in \mathbb {R} ^{(n-1)}}$ is the collection of the estimator errors for all of the unmeasured states. To ensure that ${\displaystyle \sigma {\dot {\sigma }}<0}$, let


where


That is, positive constant 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 M can be picked large enough; al). If M is sufficiently large, it can be assumed that the system achieves ${\displaystyle e_{1}=0}$ (i.e., ${\displaystyle {\hat {x}}_{1}=x_{1}}$). Because ${\displaystyle e_{1}}$ is constant (i.e., 0) along this manifold, ${\displaystyle {\dot {e}}_{1}=0}$ as well. Hence, the discontinuous control ${\displaystyle v(\sigma )}$ may be replaced with the equivalent continuous control ${\displaystyle v_{\text{eq}}}$ where


So


This equivalent control ${\displaystyle v_{\text{eq}}}$ represents the contribution from the other ${\displaystyle (n-1)}$ states to the trajectory of the output state ${\displaystyle x_{1}}$. In particular, the row ${\displaystyle A_{12}}$ acts like an output vector for the error subsystem


So, to ensure the estimator error ${\displaystyle \mathbf {e} _{2}}$ for the unmeasured states converges to zero, the ${\displaystyle (n-1)\times 1}$ vector ${\displaystyle L_{2}}$ must be chosen so that the ${\displaystyle (n-1)\times (n-1)}$ matrix ${\displaystyle (A_{2}+L_{2}A_{12})}$ is Hurwitz (i.e., the real part of each of its eigenvalues must be negative). Hence, provided that it is observable, this ${\displaystyle \mathbf {e} _{2}}$ system can be stabilized in exactly the same way as a typical linear state observer when ${\displaystyle A_{12}}$ is viewed as the output matrix (i.e., "C"). That is, the ${\displaystyle v_{\text{eq}}}$ equivalent control provides measurement information about the unmeasured states that can continually move their estimates asymptotically closer to them. Meanwhile, the discontinuous control ${\displaystyle v=M\operatorname {sgn} ({\hat {x}}_{1}-x)}$ forces the estimate of the measured state to have zero error in finite time. Additionally, white zero-mean symmetric measurement noise (e.g., Gaussian noise) only affects the switching frequency of the control v, and hence the noise will have little effect on the equivalent sliding mode control ${\displaystyle v_{\text{eq}}}$. Hence, the sliding mode observer has Kalman filter–like features.^[12]


The final version of the observer is thus


where


That is, by augmenting the control vector ${\displaystyle \mathbf {u} }$ with the switching function ${\displaystyle \operatorname {sgn} ({\hat {x}}_{1}-x_{1})}$, the sliding mode observer can be implemented as an LTI system. That is, the discontinuous signal ${\displaystyle \operatorname {sgn} ({\hat {x}}_{1}-x_{1})}$ 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 ${\displaystyle y=x_{1}}$). 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 ${\displaystyle \mathbf {y} =C\mathbf {x} }$ uses a generic matrix C). In each case, the sliding mode will be the manifold where the estimated output ${\displaystyle {\hat {\mathbf {y} }}}$ follows the measured output ${\displaystyle \mathbf {y} }$ with zero error (i.e., the manifold where ${\displaystyle \sigma (\mathbf {x} )\triangleq {\hat {\mathbf {y} }}-\mathbf {y} =\mathbf {0} }$).