State Estimation chibum@seoultech.ac.kr
Outline Dominant pole design Symmetric root locus
State estimation We are able to place the CLPs arbitrarily by feeding back all the states: u = Kx. But these may not all be available for measurement. Hence we seek to estimate them by observing the plan input and output, and use the estimated states in the control law: Using our model of the plant, we could make open-loop estimates:
State estimation However, our model may not be precise we don't know the plant's initial conditions the plant may be subject to unmodelled disturbances hence the estimated state is likely to diverge from the actual state We therefore introduce feedback to try to get the estimated output ŷ to track the measured output y
State estimation is a vector of feedback gains Thus, we form our estimates from equations to be implemented in estimator
Estimator dynamics Given the plant input u and measured output y, the state estimate evolves according to: Our plant model is Subtracting, and using the output equation, gives: Denote the error in the state estimate by Then the error dynamics are: That is, given the assumptions of a perfect plant model, and no disturbances, this suggests that the estimation error can be made to go to zero in a stable and rapid manner by a suitable choice of L
Estimator dynamics The estimated states should then continue to track the plant states with zero error In reality there will be estimation errors: plant disturbances model imperfections These errors can be kept small if the eigenvalues of the error system are well damped and relatively fast compared with the closed-loop plant dynamics Characteristic polynomial for error dynamics: Desired characteristic polynomial: Provided system is observable, we can solve for L so that a L (s) = α e (s), using the same techniques as for K
Benefit of observer canonical form The solution for the estimator gain matrix L is trivial if the equations are in observer canonical form The system matrix for the error equation is then: This is still in observer c.f., and has the char. poly Compare with desired char. poly: Hence, by inspection: L 1 = β 1 a 1, L 2 = β 2 a 2, L 3 = β 3 a 3
Ackermann's formula Again, Ackermann's formula can be used to effectively implement the process of: transformation of an observable system to observer canonical form design of estimator gains by inspection transformation back to original states c.f. MATLAB: Specify desired estimator poles, dep L = acker(a, C, dep) L = place(a, C, dep)
Relationships between canonical forms: Duality Observer canonical form Controller canonical form Observability canonical form Controllability canonical form
Regulator with estimated states Plant: Control law: Estimator:
State equations for closed-loop system Plant: Estimator: Control law: Combining these equations: Thus the closed-loop state equations are:
State equations for closed-loop system These equations may also be written in terms of the estimation error We had: Subtracting: Substituting in the first equation yields: With this alternative set of state variables the closed-loop system equations are:
State equations for closed-loop system The closed-loop characteristic polynomial is thus: which is block triangular. Hence: Thus, the poles of the CL system consist of the poles obtained by full state feedback through K, together with the estimator poles determined by the selection of L That is, the control law and the estimator can be designed independently of each other Separation Principle
Ex: regulation of water level in two-tank system The system can be described For
Ex: regulation of water level in two-tank system Specs: Required closed-loop time constants are 1 = 0.2 s, 2 = 0.05 s; i.e., desired CL char. poly is Control law: Demonstrate Ackermann's formula:
Ex: regulation of water level in two-tank system Estimator: Try estim. poles = 2 x controller poles; i.e., desired CL char. poly is α = (s +10)(s + 40) = s 2 + 50s + 400 Demonstrate Ackermann's formula:
Ex: regulation of water level in two-tank system Complete system Equivalent feedback compensator: Transfer function for compensator:
Example 7.25
Reduced-order estimators In the examples so far, we have estimated all the states In the situation where we can measure some of the states directly, we do not need to reconstruct them in an estimator note, however, that if the measurements are noisy, a full-state estimator will smooth these data, as well as reconstruct the unmeasured states If the measurements are reliable, then a reduced order estimator will be more accurate, because some of its outputs will be direct measurements A reduced-order estimator will be of lower order, thus requiring less computational power
Reduced-order estimators Plant model: possibly multiple measurements Partition states into measured x a and estimated x b That is:
Reduced-order estimators Hence, observer: Now, is in principle known, but derivative term is a problem: difficult to realize So, introduce a change in variable:
Reduced-order estimators
Error dynamics Estimation error That is, Hence, design L r as before, by selecting poles for A bb LrA ab Design: Lr = place(a_bb, A_ab, dep)
Example 7.26
Selection of estimator poles CL char. poly: The estimator poles (roots of a L (s) = 0) are usually chosen to be between 2 to 6 times faster than the controller poles (roots of a K (s) = 0) Clearly there is no direct effect of choosing faster estimator poles on control effort. However, consider the effect of sensor noise. Assume that the sensed output is error dynamics then become:. The estimation Thus, the higher estimator gains required to produce faster estimator dynamics will further amplify the sensor noise, thereby corrupting the state estimates
Symmetric root locus for SISO system estimator Estimator dynamics with 'process noise' w and 'sensor noise' v : The process noise w can represent unknown disturbances and errors in the plant model parameters Recall the estimator equations: If w is large: our plant model is uncertain, and hence a poor predictor we would put greater emphasis on the sensor data to correct the model predictions larger L, faster estimator poles If v is large: our measurements are uncertain, and hence provide poor corrections we would put greater emphasis on the model predictions smaller L, slower estimator poles
Symmetric root locus for SISO system estimator In optimal estimation theory the process noise is modeled as white noise with variance the sensor noise is modeled as white noise with variance The estimator pole locations which will minimize the variance of the state estimation error can be found from the solution of the symmetric root locus (SRL) equation: where is a measure of the plant model uncertainty relative to the measurement uncertainty, and is the plant transfer function from the process noise input to the measured output If the process noise w and control input u are additive (Bw = B), the same SRL can be used for controller and estimator pole selection