State-space estimators use a model and available sensor measurements to generate an estimate for the state.[^1] When using an estimator, you always need to test it before placing it in either a simulator or an actual control system.[^1] A state-space estimator is defined by the equations for the estimated state and estimated state rate-of-change[^1]
$$\dot{\hat{x}}=A\hat{x}+Bu$$
The measurement prediction is then derived from the second state-space equation[^1]
$$\hat{y}=C\hat{x}$$
The estimation error can be defined by this equation[^1]
$$\tilde{x}=x-\hat{x}$$
We then add a correction term to the estimated model that is the difference between the actual and measured value[^1]
$$\dot{\hat{x}}=A\hat{x}+Bu+L(y-\hat{y})$$
Where L is the estimator gain matrix.[^1] If we plug these rates of change back into the estimation error equation, we get[^1]
$$\dot{\tilde{x}}=(A-LC)\tilde{x}$$
We can set the estimator gain matrix to ensure that the eigenvalues of \((A-LC)\) to the desired location.[^1] This ensures stability as well as performance metrics such as bandwidth, overshoot, and rise time.[^1] You can solve for the Estimator gain for a desired eigenvalue if the system is observable, You can place eigenvalues anywhere you choose.[^1] If the system is non-observable, then you cannot place the eigenvalues wherever you want.[^1] The combined system with controller and estimator may perform differently if the estimator or controller gains are swapped (K and L).[^1] You should keep the estimator faster than the controller to prevent a case where the controller can’t reach a steady state until the estimator has settled, but saturation, actuator limits, and sensor noise may cause one version of the swapped gains to perform better.[^1] The state-space estimators are typically evaluated and implemented in the discrete-time domain.[^1] The computational cost of an estimator increases with the model complexity.[^2]
[[State-Space Regulators]] – have a similar equation structure
Matlab Pole Placement – used to determine the L matrix
Eigenvalues of State-Space System – estimator eigenvalues should be 5-10 times faster than the controller
[[Reduced-Order Estimator]]
Separation Principle
State-Space Control System with Estimator
Testing State-Space Estimators
Discrete State-Space Estimator
Disadvantages of Dynamic Inversion Controllers – require estimation of non-measured states
OGLI Simulation Program – used a mid-course estimator
MFA Control Law – uses state estimators
Battery Management System – the correlation between the open-circuit voltage and the SoC is important for state estimation
Hysteresis-Modulation Technique – does not need an estimator
Iterative Reconstruction – uses a similar estimator to improve CT images
- bevlyMECH4420Lectureb[^1]
- tongOnlineOptimizationBattery2015[^2]