"state" in state space analysis?

Question

What is meant by "state"in state space analysis? Is it something related to initial conditions such as in RLC circuit in attached snap,we have two states
1)Current through inductor
2) Voltage across capacitor

Answer 1

At a given time this circuit has several solutions.

Of course, once all voltages and currents are stabilised, there is only one solution ( current through L1 and R1 = 0.01A, C2 voltage = 1 volt).

But otherwise, when you apply power to this circuit, the values of the currents and voltages at a given time depends on how the power was applied : When ? Suddenly ? Gradually ? etc...

Fortunately one can describe the behavior of this circuit without mentioning all possible histories. It suffices to describe the state of the circuit with just enough details to predict its evolution.

It the case of your circuit 2 numerical values are necessary and sufficient to predict the future, for example the current through inductor and the voltage across the capacitor. A lot of other variables can be used.

Answer 2

I'd prefer to say we have one state, with two components I_L and V_C, but basically, yes.

Each component of the state is something that stores energy.

It's related to the initial conditions in that it's the state at t=0.

Answer 3

Explaining state-space from an introduction is long. I'm no expert in this topic and I don't even consider complete the following answer, but here I go (I suggest you watch Eugene's video I shared in the comments.)

Brief introduction to the concept of state-space

A system or device has in general many variables. Using laws and theorems, we mathematically model the system. Usually you'll end up with differential equations. Remember in linear algebra where you could solve \$n\$ simultaneous linear algebraic equations with \$n\$ unknowns? Somewhat similarly, we could solve linear constant-coefficient ordinary differential equations. If we have many unknown functions, we'll need just as many linearly-indepedent ODEs to find all of the unknown functions.

When first obtaining the ODEs that describe the model of the system under study, you must have sufficient equations for the number of unknowns. However, usually a system can have more variables than the minimum needed to find a solution. As an example, consider your circuit, where there's a current for the inductor (or voltage source), the capacitor, and the resistor; also, there's a voltage for the inductor, and the capacitor (or resistor). Counting these variables we have 5 unknowns. We'd need 5 equations! But what if we chose less variables? I mean, yes, the circuit still has 5 variables, but we could transform the circuit into the Laplace domain and then apply loop analysis to the two loops. We'd end up now with only 2 equations and 2 unknowns.

My point in the previous paragraph is that while a system usually has too many variables, to actually find a solution you may need less.

What we do in state space analysis is somewhat similar. While the system has a lot of variables, we don't need to solve for all of them simultaneously. As others have said, in the case of circuits, the variables we choose are the inductor currents and the capacitor voltages. These are called the state variables, and a vector containing all of them is called the state vector. The n-dimensional space whose variables are the state variables is called the state-space.

Why do we select capacitor voltages and inductor currents as state variables?

For an nth order differential equation that completely describes a system, we need to choose n state variables. But do you remember that in an inductor \$v(t) = L \, \text{d}i(t)/\text{d}t\$ and in a capacitor \$i(t) = C \, \text{d}v(t)/\text{d}t\$? Thus, when we apply KVL or KCL, we'll get one term containing a derivative for each inductor or each capacitor. In other words, the number of capacitors plus the number of inductors determine the order of the differential equation we need to solve for! In fact, first- and second-order circuits are called like that precisely because they have only one or two, respectively, storage elements (that we can't combine to simplify.) Maybe now you see why it makes sense we choose the capacitor voltages and inductor currents as state variables.

How can we be sure that only with the capacitor voltages and inductor currents, we can solve the circuit?

Unfortunately I don't know how to explain this intuitively, but I think you can get the point. Remember when you solved many first-order circuits? What did you do first? You first solved for the inductor current or the capacitor voltage, and after that you could find the rest of the variables.

In the case of general (not in parallel nor in series) second-order circuits, you first had to pose the equations by applying Kirchhoff's laws along with the elements equations. After that you had to manipulate the differential equations until you got only one ODE with only one unknown. Now you would solve the equation.

But in state-space applied to a second-order circuit, you don't have to manipulate the ODEs until you get only one. The reason is because in SS we use matrix equations. In this example, you'd get matrices and vectors with two rows (which means two equations), since second-order circuits have two storage elements.

What's the meaning of state in state-space?

I'm not 100% sure of the following point of view.

To answer the above question, first let's think about derivatives and definite integrals for a moment. Derivatives represent rate of change. If a car has a high velocity (derivative of vector position), that for the sake of simplicity we consider it constant, at any instant, we could know where the car would be after that. For example, if the car's speed is 4 m/s at some instant \$t_1\$, then we know that one second latter at \$t_2 = t_1 + 1 \text{ s}\$ the car will have moved 4 meters in the same direction it has been moving. So, we can somehow predict the future, at least in the short term (one second in this case.)

Definite integrals represent the sum of signed areas. Usually, we have definite integrals in which we're integrating with respect to time, but also the upper limit is time. These definite integrals represent the sum of signed areas up to this instant in time. For example, in an inductor the current through it is proportional to the signed area of the current from \$-\infty\$ up to the present \$t\$. This means that the current in an inductor depends on the past values of its voltage, and so it is also called a device with memory. So, in a sense, with definite integrals where the lower limit is \$-\infty\$ and the upper limit is \$t\$, we can somehow know about the past of the given quantity.

To summarize, derivatives allows us to predict the future, and integrals allows us to know the past. (In fact, if you ever take a class of control theory, you could think about P [proportional] controllers as "meters that give present values", D [derivative] controllers as "prophets that predict the future", and I [integral] controllers as "people who life in the past.")

Why did I explain all of this? Realize that derivatives allows us to predict the future. So, if we solve an equation containing derivatives, we can know the value of the quantities in the future. In other words, we can know the value for \$ t > 0\$. But in an electric circuit, which variables are derivatives? Inductor currents and capacitor voltages! Thus we choose them as state variables.

In state-space, state refers to the past, present and future of the system. How do we know the past, if derivatives only predict the future? Well, this is where initial conditions comes in. Remember that not only you need an ODE to solve a system or circuit, but also you need to know the initial conditions. You see, you don't actually need to know all the past of the variables you're solving for (inductor currents and capacitor voltages in our case), instead it's sufficient to know their value at the time we define as \$t=0\$. These values are called the initial conditions which I'm sure you're familiar with.

Note: Keep in mind throughout my explanation I've limited myself to LTI systems or circuits, but state space can be applied to non-linear time-variant MIMO systems with initial conditions. In this case you'll also need numerical methods.