State Space System Modelling

• Dynamical Systems
• Some Intuition About State Space
• Inputs
• Outputs
• State and Memory
• Stability
• Linearisation

Dynamical Systems

Many systems can be modelled using sets of Ordinary Differential Equations (ODEs). For example, in my previous article on the Euler-Lagrange equation, we found that the equations of motion for particles in a system follow the path of least action which turns out to be an ODE. Systems that can be modelled in this way are known as dynamical systems and are ubiquitous across many fields.

Some Intuition About State Space

Consider the ODE

\[x = \frac{dx}{dt}\]

If I know the position of $x$ at some time, then I can calculate the derivative and using the derivative I can update the position over some small amount of time. This allows me to get the derivative at the new position and so on until I have sketched out a path.

Sketching out a simple ODE

Unlike frequency/Laplace methods, state-space equations can be developed for non-linear systems and used to form the basis of non-linear control. Despite tihs, for this post I will focus on linear state-space equations since they are easier to begin with.

A general linear ODE is of the form,

\[\frac{d^nx}{dt^n} = a_0 + a_1 x + ... + a_{n - 1} \frac{d^{n-1}x}{dt^{n-1}}\]

Therefore, if we know the value of all the derivatives up to the $n$th at a specific time we can calculate the $n$th derivative. Using this we can update the $n - 1$th and since we also know the $n - 1$th derivative (since we used to to get the $n$th) we can get the $n - 2$th derivative e.t.c. which means we can update all of the derivatives. Therefore, these derivatives down from the $n - 1$th are known as the state of the system since they contain all of the instantaneous information required to update it over time. If we know what the derivative of each of the states are in terms of the current state, then we know how the system will evolve over time.

For a set of $m$ generalised coordinates we can define the system states using each of their $n - 1$ derivatives in a state vector, $\mathbf{x}$.

\[\mathbf{x} = \begin{pmatrix} q_1 \\ \frac{dq_1}{dt} \\ \vdots \\ \frac{d^{n - 1}q_m}{dt^{n-1}} \end{pmatrix}\]

And then if we the states derivative in terms of the state we can write,

\[\mathbf{\dot{x}} = \begin{pmatrix} \frac{dq_1}{dt} \\ \frac{d^2q_1}{dt^2} \\ \vdots \\ \frac{d^n q_m}{dt^n} \end{pmatrix} = \mathbf{Ax}\]

For some matrix $\mathbf{A} \in \mathbb{R}^{k \times k}$ where $k$ is the total number of states. The same can be said for discrete systems if we know the state at the next time step as a linear function of the current time step. These are known as difference equations as opposed to differential equations.

Inputs

Often, we would like to manipulate or control our system using some external inputs that affect some of the states. For example, we might be able to push the accelerator in a car to apply some force causing it to accelerate. The chosen force will clearly affect how the car moves and will instantaneously affect the car’s velocity. For an effective autonomous car we might like to control the car’s position via this force. The affect of the input, $\mathbf{u}$ on the state derivative can be added in the matrix state space equation.

\[\mathbf{\dot{x}} = \mathbf{Ax + Bu}\]

For $k$ states and $i$ inputs $\mathbf{B} \in \mathbb{R}^{k \times i}$.

Outputs

For the system we may have some desired outputs $\mathbf{Y}$ that are a function of the state and perhaps the current inputs. Therefore, we have

\[\mathbf{Y} = \mathbf{Cx + Du}\]

For $p$ outputs and $k$ states we have $\mathbf{C} \in \mathbb{R}^{p \times k}$ and for $i$ inputs $\mathbf{D} \in \mathbb{R}^{p \times i}$.

State and Memory

Our overall system can now be described using two sets of matrix equations.

\[\mathbf{\dot{x}} = \mathbf{Ax + Bu} \\ \mathbf{Y} = \mathbf{Cx + Du}\]

We can think of the system as having some components with memory that inform the current state and some components that are memoryless and cause an instantaneous output. The second equation demonstrates this well where $\mathbf{x}$ acts as the memory of the system which may cause changes in the output over time based on the memoryless functions parameterised by $\mathbf{A}$ and $\mathbf{B}$.

Systems modelled as a combination of memory elements and memoryless components taken from The Essentials of Linear State-Space Systems by Aplevich

Stability

Assuming that the inputs are bounded and $\mathbf{B}$ has finite entries, the stability of the system will depend on if the state converges to a particular state over time for any initial state. This can be determined by solving the set of ODEs which we know results in a linear combination of exponential functions multiplied by eigenvectors $v_k$ which are themselves a linear combination of the originally defined states. Since there were $k$ states, there will need to be $k$ independent eigenvectors for the general solution.

\[\mathbf{x}(t) = C_1 e^{\lambda_1 t} v_1 + ... + C_k e^{\lambda_k t} v_k\]

Where $\lambda_k$ are the eigenvalues of $\mathbf{A}$ and . These eigenvectors correspond to motions that are independent of each other in the system and are known as the modes of system. Clearly from this equation the system will converge if and only if all of the real parts of the eigenvalues $\lambda_k$ are negative. Incidentally, these eigenvalues are also the poles of the system in the Laplace domain which we know must be negative for stability.

Linearisation

The above formation of linear state-space equations and understanding their stability was predicated on them being linear systems. However, real systems are always non-linear to some extent and some moreso than others. Despite there being techniques to handle non-linear systems, they are not nearly as easy and mature as for linear systems. Fortunately, in many cases, the state of the nonlinear may only vary over a small region and therefore can be linearised around a point. A common example of a nonlinear system is a frictionless pendulum modelled as a point mass with mass $m$ attached to a massless rod of length $L$ that can rotate about a fixed point. It’s equation of motion can be derived as

\[m \ddot{\theta} = -\frac{mg}{L} \sin\theta\]

which is clearly a nonlinear.

A pendulum taken from Britannica

If we are modelling the pendulum for only small angles around some $\theta_d$ then we can use the first order Taylor Series expansion to linearise the system. The Taylor Series expansion around a point $x = a$ is

\[f(x) = \sum_{n = 0}^\infty \frac{f^{(n)}(a)}{n!} (x - a)^n\]

where $f^{(n)}(a)$ is the $n$th derivative of $f$ with respect to $x$ evaluated at $a$. Therefore, for a first order (linear) approximation

\[f(x) \approx f(a) + \frac{df(a)}{dx} x\]

Applying this to the pendulum at $\theta_d$,

\[\ddot{\theta} = -\frac{g}{L} \cos\theta_d \; (\theta - \theta_d)\]

And if linearising around $\theta_d = 0$ then we get

\[\ddot{\theta} = -\frac{g}{L} \theta\]

which is the small angle approximation that many may be familiar with.

In this way, many nonlinear systems can be approximated by linear systems and the same analysis tools may be used for them. For highly nonlinear systems there are other methods which I may discuss in a later post.