State Space, Part 1: Introduction to State-Space Equations

From the series: State Space

In this video, I’m going to introduce state-space equations, the model representation of choice for modern control. However, before we begin, a quick warning: We’re not going to go into depth on the mathematics of state space. After this video you’re probably not going to be able to write out the equations for any arbitrary system. But what I think you’ll get is a good intuitive understanding of these equations, which can be a foundation that you can build on as you learn more. At the very least, it might clear up a few lingering questions you might have. So with that in mind, let’s get to it. I’m Brian, and welcome to a MATLAB Tech Talk.

I think an interesting thing about dynamic systems is that they can be represented by ordinary differential equations. This is because they have this property that how the system is changing at any given time is a function of its current state.

If we zip through the derivation for a spring-mass system real quick, you can see we end up with a differential equation. Here, the variable p is position, and the second derivative with respect to time is acceleration. The way the system is changing—acceleration—is a function of the current state, position. If this dynamic system was initialized with some energy it would continue to move on its own because of this relationship between the derivative of the state and the state itself. Think about initializing this system by stretching the spring and letting go. Acceleration changes velocity, which changes position, which then changes acceleration, and it’s going to oscillate back and forth without you having to add any external inputs. Since there is no damping in this system, the energy is conserved and this acceleration-velocity-position cycle will continue forever.

For any arbitrary dynamic system, if we look at how the energy changes by analyzing the relationship between its states and their derivatives, we can make claims about things like the stability of the system. Is the energy being dissipated over time? Then the system is stable, and the faster the energy is dissipated, the more stable it is. If the energy is growing unbounded over time, even for just a part of it, then the system as a whole is unstable.

Stability is an inherent property of a system simply from the way that the states and their derivatives are connected to each other. The way the system moves, however, can also be influenced by external energy being added or removed over time. So the derivatives of a dynamic system are a function of both the current state as well as any external inputs.

And state-space representation is simply a repackaging of the high-order differential equations into a set of first-order differential equations that focuses on this relationship. This repackaging can make the system easier to analyze because we can look at the underlying behavior of the interconnected system as well as how the system is affected by external inputs, even multiple external inputs, which is really powerful. 

There are also a lot of control techniques that are built on state-space models like Kalman filtering, LQR control, robust control, and model predictive control, to name a few. There are also numerical solution benefits when simulating your system as a state-space model since you build the vector of derivatives first, and then integrate the whole vector to get the states. So, all in all, they’re pretty awesome and worth learning.

Let’s take a moment to look at the state-space equations in more detail and try to make a little sense of them. For this, we’ll focus on the continuous form for a linear, time-invariant system. There are two equations. The first one is the state equation, which we’ve already developed a little intuition around, and the second equation is the output equation, which I’ll get to in just a bit.

State-space representation is built around the state vector, x. This is a vector of all of the state variables. How the state vector changes, or the derivative of the state vector, is a linear combination of the current state plus a linear combination of the external inputs.

In this way, we’re not trying to look at the entire dynamics of the system end to end like we would with a high-order differential equation or a transfer function, but rather focus on the dynamics of each state variable and how those variables relate to each other. 

How does the first state change as a function of all of the states and all of the inputs? How does the second state change as a function of all of the states and the inputs? And so on. And since we’re only ever looking at the first derivative of each state, we end up with a set of first-order differential equations.

What we’re left with is a system of linear equations that we can package into matrix form.  And if linear algebra is good for anything, it’s definitely solving a system of linear equations.

It is this repackaging into matrix form that gives us access to a lot of mathematical tools that we didn’t have before.

The matrix form of the state equation has two matrices, A and B. The A matrix describes how the internal states are all connected to each other, the underlying dynamics of the system; and the B matrix describes how the inputs enter into the system—which states are they affecting?

Now let’s look at the output equation. Y is a vector of the outputs of the system, or the part that you’re interested in knowing. The important thing to realize is that the outputs are not necessarily the state variables. The C matrix describes how the states are combined to get the outputs, and the D matrix is used to allow the inputs to bypass the system altogether and feedforward to the output.

And real simple example of the D matrix in action would be the state-space representation of a simple multiplier, a gain block. The output equals the input times the gain, so in this way the A, B, and C matrices are all 0 since there are no derivatives and no states and the D matrix is the value of the gain.

The importance of the output equation will become obvious when we start feeding back the outputs to develop a closed-loop controller for our system. But for now, it’s enough to understand the structure of the equation and in the next video on feedback control, we’ll go into more depth on the relationship between the output and state equations.

As you might have noticed, the state variables are the key to the whole thing. They show up in three places and are the part of this representation that I think is the hardest to conceptualize. Therefore, understanding what they are is really important. The state variables are the minimum set of variables that fully describe the system. Fully described means that we have enough information about the changing part of the system, or the variable part, that we can predict the system’s future behavior. 

This will make more sense if we go back to the spring-mass system and suss out what the state variables are by starting with no information about the system. If I draw the spring and mass like this, and asked you to predict where the system will be in one second, what information would you need?

Well, you’ll want the system constants like the spring constant and mass, and we’d want to know what the external force is that’s acting on the mass, so the inputs, but this only tells you part of the story because you don’t know what you might call the initial conditions for the variable parts of the system. If the mass is currently moving and how hard is the spring currently pulling on the mass. If we knew those two starting conditions, along with the system constants and inputs, then we’d be able to predict where the mass would be after one second.

So let’s consider the first question: Is the mass currently moving? We would know this from the velocity of the mass and velocity is variable since the mass would be oscillating back and forth, assuming there was some energy in this system.  We need velocity in order to predict the future position so we can say that it is a state variable; it’s necessary.

Now let’s turn to the initial spring force. Force is also variable, but let’s think about how we’d derive it. It’s the spring constant times the distance that the spring is stretched. So if we have that distance, or the position of the mass relative to the unstretched length of spring, then we’d be able to get spring force. So the minimum number of variables we need to fully describe this system is two: position and velocity. And since there are two, this is a second-order system.

So a natural question now might be: Why not more states? Why not three?

To answer that, let’s consider acceleration. It’s a variable since it’s changing along with position and velocity. However, we don’t need to know acceleration to fully describe the system; acceleration is a result of force acting on a mass, and force in this case is a byproduct of the position of the mass. Therefore, we get acceleration through position and adding acceleration to your list of state variables is extra information.

Something that might seem strange at first is that if we add a damper into this system, we’re not increasing the order of this system, because we don’t need any additional state variables to account for this additional element. The force from the damper is determined by the velocity of the mass, which, again, we already have in our state variable list. So both of these systems have the same number of state variables: two.

A way to verify you have the right the number of states needed to describe a system is to count the energy storage devices that your system has. I’m not going to go into too much detail here because I think Dr. Rick Hill has done a great job of explaining it in his YouTube video that I’ve linked in the description below. But the general idea is this: A dynamic system stores energy in various ways. In our spring mass system, there are two energy storage elements: the spring, which stores potential energy, and the mass, which stores kinetic energy.

The dynamic system moves and changes on its own simply because the energy that exists within the system transfers back and forth between storage elements and eventually dissipates through heat, which is ultimately lost to the environment. At any given time, if we know how much energy is stored in each element, then we know it’s state and have the information needed to fully understand the state of the system. The states don’t represent the energy. They’re just needed to determine the energy.

For the spring-mass system, we know the energy in the mass from the velocity of the mass, and we know the energy in the spring from the position, or from how far the spring is stretched. And if we know how those energies are connected within the system—or, another way of putting it, how each state absorbs and loses energy based on the current state of the system—then we fully understand the system itself.

And this is essentially what state-space representation is doing for us. Looking at the state variables—how the energy is stored—and combining them according to how energy is transferred between states, and adding in any additional energy from the external sources.

With a few special caveats that Professor Hill describes, that is why the number of energy storage elements in your system equals the number of states. So our spring-mass-damper is a second-order system still, since a damper doesn’t store energy; it just dissipates it.

So we know the order has to be two, but why not two different states? Why position and velocity? Well, they don’t have to be. Here’s where things can get a little confusing. The state variables, position and velocity, can be thought of as coordinate system within the state space. Go over five position units and up three velocity units. This point in the state space is the current state of the system, and the state vector describes that location.

But we can describe this location in the state space with any two linearly independent variables. Some arbitrary state 1 and state 2. We’ll end up with the same system state, same total energy, just with different coordinates.

And we do this redefining of state variables all the time with state-space representation. This is what we’re doing, for example, when we diagonalize the A matrix to get it into so-called modal form. These two models represent the exact same system, we’ve just chosen a different set of coordinates, a different basis, for our state vector. So state vector x is made up of position and velocity, and state vector z is made up of two modal states and these particular states change as a function of just themselves. That is, state one does not rely on the current value of state 2, and vice versa. And if we were able to write out the state-space representation directly in modal form, then maybe we would.  There’s nothing really special about position and velocity for state variables other than they are quantities that make intuitive sense to us.

However, since the states are no longer defined as position and velocity, the C matrix needed to change as well if we wanted to keep the outputs of the system as position and velocity. In this way, we can see the benefit of the output equation. We’re not forced to output the states exactly as we define them, but rather have the ability to linearly combine them into variables that we’re interested in.

Okay, that’s where I’m going to leave this video for now. I hope this has helped you think about the state-space equations in a slightly different way. Rather than thinking of them as a set of matrices that repackage the model into arbitrary first-order differential equations, we can think about them as writing out the dynamics for each state variable separately. And the dynamics are tied to how energy is stored and transferred within the system. By doing this, we get the benefit of being able to understand how these internal states change and relate to each other, how the energy is moving around, and we get to use the powerful tools in linear algebra to analyze and solve the equations.

MATLAB and Simulink are great environments to work with state-space systems. You can convert to state space from a transfer function, manipulate the matrices, simulate them, and design and analyze control systems using a number of built-in tools. I’m going to be using MATLAB to demonstrate control techniques with state-space equations for the rest of this series. If you’re not familiar with using State Space in MATLAB, or if you are looking for a quick refresher, I’ve linked a few videos in the description that are worth checking out. In the next video in this series, I’ll go through the design of a simple feedback control system using state-space representation.

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