Dynamic Systems Simulation PART I

This course really goes like a train.

Lec 1&2: Mathemetical models

Went through the course plan quickly, surely a lot of topics, a lot of labs, and 3 tests.

Mathematical modelling definitions

Mathematical modelling is

• is the act of building a model.

• a process in which real-life situations and relations in these

  situations are expressed by using mathematics. (Haines and

  Crouch, 2007)

• the art of translating problems from an application area into

  tractable mathematical formulations whose theoretical and numerical analysis provides insight, answers, and guidance useful for the originating application. (A. Neumaier, 2004)

• a cyclical process in which real-life problems are translated into mathematical language, solved within a symbolic system, and the solutions tested back within the real-life system. (Verschaffel, Greer, and De Corte, 2002)

Mathematical model

• a representation in mathematical terms of the behavior of real devices and objects. (What Is Mathematical Modeling? by Clive L. Dym)
• a description of a system using mathematical concepts and language to facilitate proper explanation of a system or to study the effects of different components and to make predictions on patterns of behaviour. (Abramowitz and Stegun, 1968)

The real systems are too complicated. Model shoud be simple as possible, but not simpler. And which can be used to predict the behaviour of the real system.

Simulation

Simulating is the act of using a model for a simulation.

Major steps

1. Formulate problem and model.
2. Construct system model.
3. Conduct simulation experiments.
4. Interpret results.
5. Document study.

6. Implement conclusions.

   X. Validate model.

Types of methematical models

Models can be divided to three types by used information about real objects or processes, according to adversary’s knowledge

• White-box • Black-box • Grey-box

Black box

System which are modelled entirely based on experimental data (input-output measurements) are called black-box models. The user can observe the response (output) of the model for a certain stimulus (input) but has no information about the internal mechanism (principles).

Gray-box

Most of the time, it’s better to benefit from the advantages of both methods. In this case we’ll end up with gray-box models.

Simplification examples2:

• Non-linear parameters or processes are often approximated by

  linear ones.

• Areas of physical detail are often simplified by averaging the localised properties.

• Contact between one part of a model and another is sometimes considered perfect.

Model Classification

• Linear vs Nonlinear
• Static vs Dynamic
• Time-invariant vs Time-variant
• Discrete vs continuous
• Explicit vs Implicit
• Deterministic vs Probabilistic (stochastic). • etc.

Linear & Nonlinear

A linear mathematical model is governed by linear differential equations. A linear model is a model for which the superposition principle can be applied (Additivity and Homogeneity properties).

$$y(\alpha u_1+\beta u_2)=\alpha y(u_1)+\beta y(u_2)$$

As for nonlinear, ones for which the principle of superposition does not apply.

Static & Dynamic

• Static models are not affected with time, only reply on present input.

• Dynamic models are affected with time

Time-invariant & Time-variant

• Time-invariant — the behaviour does not depends on time. If y(t) is a output for u(t), then y(t −τ) is a output for u(t −τ)

• Time-variant — output response depends on moment of observation as well as moment of input signal application.

Discrete & Continuous

• Discrete system is affected by the state variable changes at a discrete point of time.

• Continuous system is affected by the state variable, which changes continuously as a function with time.

Classification by sampling

See page 24, very direct.

---

Lec 3&4: Linear models

Linear time-invariant system (LTI)

As the name suggest, any linear time-invariant (LTI) system has two defining properties: linearity and time invariance.

Linearity

If input u1(t) produces response y1(t), input u2(t) produces response y2(t), …,and uk(t) produces response yk(t), then input $\sum_k c_ku_k(t)$ produces response $\sum_k c_ky_k(t),\ c_1,c_2,\dots,c_k$ Are real scalars.

Time invariance

If input u(t) produces response y(t), then input u(t − τ) produces response y (t − τ ), where τ is a time delay.

N-order differential equations

Linear dynamical models of continuous-time system could be given as n-order differential equations (DE) :

$$y^{(n)}(t)+a_{n-1}y^{n-1}(t)+\dots+a_2\ddot{y}(t)+a_1\dot{y}(t)+a_0y(t)=\\ =b_mu^{(m)}(t)+b_{m-1}u^{(m-1)}(t)+\dots+b_2\ddot{u}(t)+b_1\dot{u}(t)+b_0u(t)$$

Causality condition for DE

The LTI system is called causal, if $n\ge m$

Differential operator and Characteristic equation

Some operator form, and characteristic equation, whoose roots determine a behavior of the system.

That can be demonstrate by a phase portrait.

Behaviour of the second-order LTI system

A phase portrait is a representative set of LTI system‘s solutions, plotted as parametric curves (with t as the parameter) on the Cartesian plane tracing the path of each particular solution ((x,y) = (y(t),y ̇(t)), −∞ < t < ∞.

The Cartesian plane where the phase portrait resides is called the phase plane.

The parametric curves traced by the solutions for specific initial conditions are called phase trajectories.

Second order LTI system

Mainly the classification for different root conditions. Just like old courses or what learn from Dr.Can’s Ch.

1. Roots are all real and negative - Stable

2. Roots are all real and positive - Unstable

3. Both real and is positive and negative respectively - Saddle

4. Both complex and $Re(\lambda_1,\lambda_2)<0$ - Stable focus

5. Both complex and $Re(\lambda_1,\lambda_2)>0$ - Unstable focus

6. Both complex and $Re(\lambda_1,\lambda_2)=0$ - Center

Followed by Laplace transformation and transfer function, which should be very familiar with.

Control System

Feedback control (closed-loop) enables to achieve new parameters of the system.

The modal control is based on this idea. The input signal is set as

$$u(t)=-\bold{K}x(t)$$

where K is k × n matrix of feed-back gain coefficients.

Thus the augmented system is described in State-Space representation as

$$\dot{x}(t)=Ax(t)-BKx(t)\\ =(A-BK)x(t)$$

Here F is a n × n matrix that describe augmented closed-loop system.

Lec 4&5: Nonlinear systems

Definition 1.

The equilibrium point x = 0 of $\dot{x} = f(x)$ is

• stable if for each $\epsilon>0$ There is $\delta>0$ (dependent on $\epsilon$ ) such that

  $$||x(0)||<\delta \Rightarrow ||x(t)||<\epsilon, \forall t \geq 0$$

• Unstable otherwise

Definition 2. Asymptotic stability

Let the origin be an asymptotically stable equilibrium point of the system $x ̇ = f(x)$, where f is a locally Lipschitz function defined over a domain $D⊂\mathbb{R}^n (0∈D)$

Definition 3. Exponential Stability

The equilibrium point x = 0 of x ̇ = f(x) is said to be exponentially stable if

$$||x(t)||\leq k||x(0)||e^{-\lambda t},\ \forall t\geq 0,k\geq 1,\ \lambda>0, \forall||x(0)||<c$$

Some linearization and bifurcation, check slides for more detail.

Lec 6&7: Numerical methods for ordinary differential equations (ODEs)

Intro

Differential equations are an essential tool in a wide range of applications. Many phenomena can be modelled by a relationship between a function and its derivatives.

$$f'(t)=lim_{\triangle t\rightarrow0}\frac{f(t+\triangle t)-f(t)}{\triangle t}$$

Differential equations can be divided into several types namely: • Ordinary Differential Equations
• Partial Differential Equations
• Linear Differential Equations

• Non-linear Differential equations
• Homogeneous Differential Equations
• Non-homogeneous Differential Equations

We will consider ordinary linear homogeneous and non-homogeneous differential equations.

Why we use numerical solutions of differential equations?

• Numerical solution of differential equations used, if impossible to find an analytic solution of the math system.
• Also, computing machines usually work with numerical methods, so programming numerical methods are simpler than programming analytical methods.

The use of high iteration frequencies produces a more realistic simulation than the lower frequency models. But high iteration frequencies produces a harder for computing machines

Euler’s method

The simplest of numerical methods is Euler’s method which is based directly on the geometric interpretation in observation.

$$Y_{n+1} =Y_n +h·f(t_n,Y_n)$$

Error analysis

Consider polynomial of second degree of the Taylor series:

$$f(x)\approx f(a)+\frac{f'(a)}{1!}(x-a)+\frac{f''(a)}{2!}(x-a)^2.$$ $$Y(t_{n+1})=Y(t_n)+hY'(t_n)+\frac{1}{2}h^2Y''(\xi_n),\ \xi\in[t_n,t_{n+1}]\\ Y'(t_{n+1})-y_{n+1}=Y(t_n)-y_n+h[f(t_n,Y(t_n))-f(t_n,y_n)]+\frac{1}{2}h^2Y''(\xi_n)$$

Where $\frac{1}{2}h^2Y’’(\xi_n)$ - the truncation error

$Y(t_n)-y_n+h[f(t_n,Y(t_n))-f(t_n,y_n)]$ - the propagated error

Using the mean value theorem obtain

$$f(t_n,Y(t_n))-f(t_n,y_n)=\frac{\partial f(t_n,\varsigma)}{\partial y}$$

For the global error:

$$e_k\equiv Y(t_k)-y_k,\ k\leq 0, \xi\in[t_n;t_{n+1}]$$

General error

$$e_{n+1}=[1+h\frac{\partial f(t_n,\varsigma_n)}{\partial y}]e_n+\frac{1}{2}h^2Y''(\xi_n)$$

Euler method modifications

If $Y’(t)=f(t,Y(t))$, then

$$Y(t)=Y_0+\int_{t_0}^{t}f(s,Y(s))ds=Y_0,\lim_{||\triangle t||\rightarrow0}\sum_{t=1}^n f(t,Y(t))\triangle t_i.$$

Trapezoidal Riemann sum

$$y_{n+1} =y_n +\frac{h}{2}(f(t_n,y_n)+f(t_{n+1},y_{n+1}))$$

and cuz $y_{n+1}$​ is unknown

$$y_{n+1} =y_n +\frac{h}{2}(f(t_n,y_n)+f(t_{n+1},y_n+hf(t_n,y_n)))$$

Midpont Riemann sum

$$y_{n+1}=y_n+hf(t_n+\frac{h}{2},y_{n+0.5})$$

Remember to replace $y_{n+0.5}$ with $\hat{y}=y_n+\frac{h}{2}f(t_n,y_n)$

Taylor methods

Use the quadratic Taylor approximation

$$Y(t_{n+1})≈Y(t_n)+hY'(t_n)+ \frac{h}{2}Y''(t_n)$$

The truncation error: $T_{n+1}(Y)=\frac{1}{6}h^3Y’’’(\xi_n),\ \xi_n\in[t_n;t_{n+1}]$

See general in page 22.

Error analysis

It’s an asymptotic error formula.

To avoid the need for the higher-order derivatives (in the Taylor method), the Runge - Kutta methods evaluate f(t,y) at more points, while attempting to retain the accuracy of the Taylor approximation.

Let’s take a look at Runge - Kutta method of order 2:

$$y_{n+1}=y_n+hF(t_n,y_n,h), y_0=Y_0$$

See 24 for more detail.

Runge - Kutta methods

The explicit methods

$$y_{n+1}=y_n+\sum_{i=1}^s b_i k_i$$

where

$$k_1=hf(t_n,y_n)\\ k_i=hf(t_n+c_i h,y_n+\sum_{j=1}^{i-1}a_{ij}k_j)$$

Note the construction of matrix of coefficients on page 29.

Numerical methods in Matlab

In a Matlab, the choice of solver depends on many factors:

1. System dynamics (stiff, nonstiff, linear, nonlinear)
2. Solution stability
3. Computation speed
4. Solver robustness

Model states:

• Continuous solvers
  Use numerical integration to compute continuous states of a model at the current time step based on the states at previous time steps and the state derivatives.
• Discrete solvers
  Primarily used for solving purely discrete models.

Computation step size:

• Fixed-step solvers
Solve the model using the same step size from the beginning to the end of the simulation

• Variable-step solvers vary the step size during the simulation

Explicit form of ODE

$$F(x,y,y',\dots,y^{n-1})=y^{n}$$

Implicit form of ODE (usually used in stiff solvers)

$$F(x,y,y',\dots,y^{n-1},y^{n})=0$$

Check the provided diagram for reference. at Page 33.

Just like manual.

Stiff equation

A stiff equation is a differential equation for which certain numerical methods for solving the equation are numerically unstable.

The main idea is that the equation includes some terms that can lead to rapid variation in the solution

Conclusion

1. About methods

   1.1 Euler’s method plotting tangent lines with fixed step h, $Y_{n+1} = Y_n + hf (t_n , Y_n )$. The local error may become smaller, opposite global error only increase

   1.2 Taylor methods use high order derivatives

   1.3 Runge–Kutta methods. The idea of Runge–Kutta methods is to use

   combinations of compositions of the right-side function of the equation to approximate the derivative terms to a required order

2. The use of high iteration frequencies produces a more realistic simulation than the lower frequency models. But high frequencies of iterations requires more computing power.