Sliding Mode Control - Sliding Mode Observer

Sliding Mode Observer

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. 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, 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

\begin{align} \dot{\mathbf{x}} &= A \mathbf{x} + B \mathbf{u}\\y &= \begin{bmatrix}1 & 0 & 0 & \cdots & \end{bmatrix} \mathbf{x} = x_1 \end{align}

where state vector, is a vector of inputs, and output is a scalar equal to the first state of the state vector. Let

where

  • is a scalar representing the influence of the first state on itself,
  • is a column vector representing the influence of the other states on the first state,
  • is a matrix representing the influence of the other states on themselves, and
  • is a row vector corresponding to the influence of the first state on the other states.

The goal is to design a high-gain state observer that estimates the state vector using only information from the measurement . Hence, let the vector be the estimates of the states. The observer takes the form

where is a nonlinear function of the error between estimated state and the output, and is an observer gain vector that serves a similar purpose as in the typical linear Luenberger observer. Likewise, let

where is a column vector. Additionally, let be the state estimator error. That is, . The error dynamics are then

\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}

where is the estimator error for the first state estimate. The nonlinear control law can be designed to enforce the sliding manifold

so that estimate tracks the real state after some finite time (i.e., ). Hence, the sliding mode control switching function

To attain the sliding manifold, and must always have opposite signs (i.e., for essentially all ). However,

\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 )

where is the collection of the estimator errors for all of the unmeasured states. To ensure that, let

where

That is, positive constant must be greater that 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 can be picked large enough; al). If is sufficiently large, it can be assumed that the system achieves (i.e., ). Because is constant (i.e., 0) along this manifold, as well. Hence, the discontinuous control may be replaced with the equivalent continuous control where

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}}.

So

\mathord{\overbrace{v_{\text{eq}}}^{\text{scalar}}} = \mathord{\overbrace{A_{12}}^{1 \times (n-1) \text{ vector}}} \mathord{\overbrace{\mathbf{e}_2}^{(n-1) \times 1 \text{ vector}}}.

This equivalent control represents the contribution from the other states to the trajectory of the output state . In particular, the row acts like an output vector for the error subsystem

\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.

So, to ensure the estimator error for the unmeasured states converges to zero, the vector must be chosen so that the matrix is Hurwitz (i.e., the real part of each of its eigenvalues must be negative). Hence, provided that it is observable, this system can be stabilized in exactly the same way as a typical linear state observer when is viewed as the output matrix (i.e., ""). That is, the equivalent control provides measurement information about the unmeasured states that can continually move their estimates asymptotically closer to them. Meanwhile, the discontinuous control 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, and hence the noise will have little effect on the equivalent sliding mode control . Hence, the sliding mode observer has Kalman filter–like features.

The final version of the observer is thus

\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}

where

  • ,
  • , and
  • .

That is, by augmenting the control vector with the switching function, the sliding mode observer can be implemented as an LTI system. That is, the discontinuous signal 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 ). 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 uses a generic matrix ). In each case, the sliding mode will be the manifold where the estimated output follows the measured output with zero error (i.e., the manifold where ).

Read more about this topic:  Sliding Mode Control

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