## Dynamic Estimation Introduction

## Main.DynamicEstimation History

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plt.show() (:sourceend:) (:divend:)

The approach can also be extended to multiple data sets and when the experimental values are at different times.

(:toggle hide solution3b button show="Show GEKKO Solution":) (:div id=solution3b:)

(:source lang=python:) from gekko import GEKKO import numpy as np import pandas as pd import matplotlib.pyplot as plt

- data set 1

t_data1 = [0.0, 0.1, 0.2, 0.4, 0.8, 1.00] x_data1 = [2.0, 1.6, 1.2, 0.7, 0.3, 0.15]

- data set 2

t_data2 = [0.0, 0.15, 0.25, 0.45, 0.85, 0.95] x_data2 = [3.6, 2.25, 1.75, 1.00, 0.35, 0.20]

- combine with dataframe join

data1 = pd.DataFrame({'Time':t_data1,'x1':x_data1}) data2 = pd.DataFrame({'Time':t_data2,'x2':x_data2}) data1.set_index('Time', inplace=True) data2.set_index('Time', inplace=True) data = data1.join(data2,how='outer') print(data.head())

- indicate which points are measured

z1 = (data['x1']==data['x1']).astype(int) # 0 if NaN z2 = (data['x2']==data['x2']).astype(int) # 1 if number

- replace NaN with any number (0)

data.fillna(0,inplace=True)

m = GEKKO(remote=False)

- measurements

xm = m.Array(m.Param,2) xm[0].value = data['x1'].values xm[1].value = data['x2'].values

- index for objective (0=not measured, 1=measured)

zm = m.Array(m.Param,2) zm[0].value=z1 zm[1].value=z2

m.time = data.index x = m.Array(m.Var,2) # fit to measurement x[0].value=x_data1[0]; x[1].value=x_data2[0]

k = m.FV(); k.STATUS = 1 # adjustable parameter for i in range(2):

m.free_initial(x[i]) # calculate initial condition m.Equation(x[i].dt()== -k * x[i]) # differential equations m.Minimize(zm[i]*(x[i]-xm[i])**2) # objectives

m.options.IMODE = 5 # dynamic estimation m.options.NODES = 2 # collocation nodes m.solve(disp=True) # solve k = k.value[0] print('k = '+str(k))

- plot solution

plt.plot(m.time,x[0].value,'b.--',label='Predicted 1') plt.plot(m.time,x[1].value,'r.--',label='Predicted 2') plt.plot(t_data1,x_data1,'bx',label='Measured 1') plt.plot(t_data2,x_data2,'rx',label='Measured 2') plt.legend(); plt.xlabel('Time'); plt.ylabel('Value') plt.xlabel('Time');

x_data = [2.0,1.6,1.2,0.7,0.3,0.15,0.1, 0.05,0.03,0.02,0.015,0.01]

x_data = [2.0,1.6,1.2,0.7,0.3,0.15,0.1,0.05,0.03,0.02,0.015,0.01]

Use an initial condition of `x=2` that matches the data.

Use an initial condition of `x=2` that matches the data. Create new states `y=dx/dt` and `z=dy/dt` for the higher order derivative terms.

$$\frac{dx}{dt} = y$$ $$\frac{dy}{dt} = z$$ $$\frac{dz}{dt} = az+by+cx+d$$

plt.legend() plt.xlabel('Time'), plt.ylabel('Value') plt.show() (:sourceend:) (:divend:)

#### Dynamic Parameter Estimation Example 4

Estimate the parameter `a,b,c,d` in the differential equation:

$$\frac{d^3x}{dt^3} = a\frac{d^2x}{dt^2}+b\frac{dx}{dt}+c x+d$$

by minimizing the error between the predicted and measured `x` values. The `x` values are measured at the following time intervals.

(:source lang=python:) t_data = [0,0.1,0.2,0.4,0.8,1,1.5,2,2.5,3,3.5,4] x_data = [2.0,1.6,1.2,0.7,0.3,0.15,0.1, 0.05,0.03,0.02,0.015,0.01] (:sourceend:)

Use an initial condition of `x=2` that matches the data.

(:toggle hide solution4 button show="Show GEKKO Solution":) (:div id=solution4:)

(:source lang=python:) from gekko import GEKKO

t_data = [0,0.1,0.2,0.4,0.8,1,1.5,2,2.5,3,3.5,4] x_data = [2.0,1.6,1.2,0.7,0.3,0.15,0.1, 0.05,0.03,0.02,0.015,0.01]

m = GEKKO() m.time = t_data

- states

x = m.CV(value=x_data); x.FSTATUS = 1 # fit to measurement y,z = m.Array(m.Var,2,value=0)

- adjustable parameters

a,b,c,d = m.Array(m.FV,4) a.STATUS=1; b.STATUS=1; c.STATUS=1; d.STATUS=1

- differential equation
- Original: x
*' = a*x*+ b x' + c x + d - Transform: y = x'
- z = y'
- z' = a*z + b*y + c*x + d

m.Equations([y==x.dt(),z==y.dt()]) m.Equation(z.dt()==a*z+b*y+c*x+d) # differential equation

m.options.IMODE = 5 # dynamic estimation m.options.NODES = 3 # collocation nodes m.solve(disp=False) # display solver output print(a.value[0],b.value[0],c.value[0],d.value[0])

import matplotlib.pyplot as plt # plot solution plt.plot(m.time,x.value,'bo',label='Predicted') plt.plot(m.time,x_data,'rx',label='Measured')

$$\frac{x}{dt} = -k\,x$$

$$\frac{dx}{dt} = -k\,x$$

Use an initial condition of `x=2` that matches the data. Verify the solution of `x` with the analytic expression `x(t)=2 exp(-k\,t)`.

Use an initial condition of `x=2` that matches the data. Verify the solution of `x` with the analytic expression `x(t)=2 exp(-k t)`.

by matching the predicted `x` to the measured `x` data:

by minimizing the error between the predicted and measured `x` values. The `x` values are measured at the following time intervals.

time = [0, 0.1, 0.2, 0.4, 0.8, 1] x_data = [2.0081, 1.5512, 1.1903, 0.7160, 0.2562, 0.1495]

t_data = [0, 0.1, 0.2, 0.4, 0.8, 1] x_data = [2.0, 1.6, 1.2, 0.7, 0.3, 0.15]

Verify the solution of `x` with the analytic expression `x(t)=exp(-k\,t)`.

Use an initial condition of `x=2` that matches the data. Verify the solution of `x` with the analytic expression `x(t)=2 exp(-k\,t)`.

(:toggle hide solution3 button show="Show GEKKO Solution":) (:div id=solution3:)

(:source lang=python:) from gekko import GEKKO

t_data = [0, 0.1, 0.2, 0.4, 0.8, 1] x_data = [2.0, 1.6, 1.2, 0.7, 0.3, 0.15]

m = GEKKO(remote=False) m.time = t_data x = m.CV(value=x_data); x.FSTATUS = 1 # fit to measurement k = m.FV(); k.STATUS = 1 # adjustable parameter m.Equation(x.dt()== -k * x) # differential equation

m.options.IMODE = 5 # dynamic estimation m.options.NODES = 5 # collocation nodes m.solve(disp=False) # display solver output k = k.value[0]

import numpy as np import matplotlib.pyplot as plt # plot solution plt.plot(m.time,x.value,'bo', label='Predicted (k=str(np.round(k,2)))') plt.plot(m.time,x_data,'rx',label='Measured')

- plot exact solution

t = np.linspace(0,1); xe = 2*np.exp(-k*t) plt.plot(t,xe,'k:',label='Exact Solution') plt.legend() plt.xlabel('Time'), plt.ylabel('Value') plt.show() (:sourceend:) (:divend:)

#### Dynamic Parameter Estimation Example 3

Estimate the parameter `k` in the exponential decay equation:

$$\frac{x}{dt} = -k\,x$$

by matching the predicted `x` to the measured `x` data:

(:source lang=python:) time = [0, 0.1, 0.2, 0.4, 0.8, 1] x_data = [2.0081, 1.5512, 1.1903, 0.7160, 0.2562, 0.1495] (:sourceend:)

Verify the solution of `x` with the analytic expression `x(t)=exp(-k\,t)`.

#### Dynamic Parameter Estimation Example

#### Dynamic Parameter Estimation Example 1

(:html:) <iframe width="560" height="315" src="https://www.youtube.com/embed/eXco8_3MmjI" frameborder="0" allowfullscreen></iframe> (:htmlend:)

#### Dynamic Parameter Estimation Example 2

<iframe width="560" height="315" src="https://www.youtube.com/embed/y0ERNz5Kms8?rel=0" frameborder="0" allowfullscreen></iframe>

<iframe width="560" height="315" src="https://www.youtube.com/embed/umxAfu44kWo?rel=0" frameborder="0" allowfullscreen></iframe>

#### Numerical Solution Tutorial for Dynamic Parameter Estimation

#### Dynamic Parameter Estimation Example

#### Numerical Estimation Introduction

#### Numerical Solution Tutorial for Dynamic Parameter Estimation

#### Numerical Estimation Introduction

A method to solve dynamic estimation is by numerically integrating the dynamic model at discrete time intervals, much like measuring a physical system at particular time points. The numerical solution is compared to measured values and the difference is minimized by adjusting parameters in the model. Excel, MATLAB, Python, and Simulink are used in the following example to both solve the differential equations that describe the velocity of a vehicle as well as minimize an objective function.

(:html:) <iframe width="560" height="315" src="https://www.youtube.com/embed/y0ERNz5Kms8?rel=0" frameborder="0" allowfullscreen></iframe> (:htmlend:)

(:html:) <iframe width="560" height="315" src="https://www.youtube.com/embed/UTxYp0VTe-E?rel=0" frameborder="0" allowfullscreen></iframe> (:htmlend:)

(:title Estimation Introduction:)

(:title Dynamic Estimation Introduction:)

(:title Estimation Introduction:) (:keywords dynamic data, validation, estimation, simulation, modeling language, differential, algebraic, tutorial:) (:description Dynamic estimation for use in real-time or off-line dynamic simulators and controllers:)

Dynamic estimation is a method to align data and model predictions for time-varying systems. Dynamic models and data rarely align perfectly because of several factors including limiting assumptions that were used to build the model, incorrect model parameters, data that is corrupted by measurement noise, instrument calibration problems, measurement delay, and many other factors. All of these factors may cause mismatch between predicted and measured values.

The focus of this section is to develop methods with dynamic optimization to realign model predictions and measured values with the goal of estimating states and parameters. Another focus of this section is to understand model structure that can lead to poorly observable parameters and determine confidence regions for parameter estimates. The uncertainty analysis serves to not only predict unmeasured quantities but also to relate a confidence in those predictions.

Dynamic estimation algorithms optimize model predictions over a prior time horizon of measurements. These state and parameter values may then be used to update the model for improved forward prediction in time to anticipate future dynamic events. The updated model allows dynamic optimization or control actions with increased confidence.