1. Determine the control law. This is to set up the locations of poles for the closed loop system so that transient characteristics are met (i.e. rise time, overshoot, etc.)

2. Design an estimator (if all states are not available). The estimator will estimate states that are not measured (which is quite rare in real world applications).

3. Combine the control law and estimator. This is basically taking place of the controller D that we've been working with. all inputs going to the Plant from here are based on estimated states rather than actual state measurements.

4. Introduce the reference input to the system, ensuring the plant tracks the input properly. This is done by controlling the zeros of the closed loop transfer function, since all the poles are set by steps 1-3.

Overall, this seems to simplify the craziness of trying to understand state feedback for me. I don't pretend to grasp this yet, but this provides a good structure for me to start.

This approach ensures stability, improves response time, and minimizes steady-state errors by continuously adjusting the input. The feedback loop measures or estimates the state and feeds it back to the controller to counter disturbances. Designing an optimal K and ensuring full system controllability is essential, as performance is limited without controllability.

Source: Section 7.5

Source: Dalle-engine and GPT

State Feedback Design For a state-space system to be stable (i.e., to ensure all states converge to desired values over time), it must be controllable. If any state is uncontrollable, we can't use feedback to stabilize or influence it, which might prevent achieving overall stability or desired performance.

We can check for controllability using the controllability matrix that is given as

C= [B AB A^2B… A^n−1B]

The system is said to be controllable only if the rank of C is the same as the number of states n.

Observability is the idea of designing a feedback-like system to compute the states from the available measurements; in general, observability is the capacity to infer details about every mode of the system by observing only the outputs that are perceived. Unobservability results when some mode or subsystem is disconnected physically from the output and therefore no longer appears in the measurements.

The mathematical test for determining observability can be done by computing the observability matrix.

O= [​C; CA; CA^2;… ;C^An−1​]​

The system is said to be observable only if the rank of O is the same as the number of states n.

In conclusion, controllability ensures we can design a controller to manipulate all states as needed.

Observability allows us to design an observer/estimator to reconstruct all states, making feedback control feasible even if not all states are directly measurable.

Source: Franklin, Gene, F. et al. Feedback Control of Dynamic Systems ( section 7.5 and 7.7), GPT, Controllability of LTI Systems, MEEM 5715.

I found it fascinating that the Kalman Filter optimally estimates states even in the presence of noise, leveraging both system models and measurements. This blend of prediction and correction helps minimize errors, and it's widely applied in fields ranging from robotics to aerospace.

A question I have is: How does the performance of a state estimator vary depending on system complexity or noise characteristics, and are there particular strategies to improve estimator accuracy under challenging conditions like a highly non linear system?

Example:

Industrial Automation: In process control (like in chemical plants or steel manufacturing), state estimators can infer concentrations, temperatures, or flow rates in various parts of a system when direct measurement isn't feasible. This enables better control and optimization of industrial processes.

Reference: Ogata, Katsuhiko. "Modern control engineering." (2020).

and GPT

1.) Good pole design needs to eliminate undesirable aspects of open loop response creating a good system response

2.) Good pole design needs to limit controller effort

With this in mind, there are two methods for selecting pole locations that will adequately control your system.

Dominant Second Order Poles Method

This method uses dominant poles to create the desired system response, and then places other poles far away from the dominant poles which will not impact system response but can impact controller effort. The goal in this method is to select 2 dominant poles that will meet system requirements.

Symmetric Root Locus Method

This method is designed to meet the balance between controller effort and system response. This is done by minimizing the equation for the performance index and using the SRL equation to find the optimal stable poles.

One disadvantage to this is that each state needs a measurement. Without this, you can't have state feedback control. However, there are times where measuring everything is not an option due to other factors. There are a lot of estimation techniques out there to try to satisfy this criteria without direct measurements, but to me it seems a little dangerous to rely on those too heavily. I think an example of this would be our cart in lab. We have the ability to only measure two states of the system, the position of the cart and the position of the rod. In order to use state feedback control for this system, we would need the estimations of the velocity of the cart and the angular velocity of the rod since we have 4 states.

The textbook discussed two main estimator designs full-order and partial order estimators. These estimators both work in similar ways but change the order of the estimator for the system. I am curious to see how these estimator schemes work with estimating things like stiction.

However the aforementioned method is not a robust method to track the reference input signal, because our plant may have some dynamic elements which we have missed during modelling. Moreover the parameter values (like A,B,C and D matrices of state space model) may change over time due to system aging or external disturbances.

Hence we need a more robust way to achieve reference signal tracking, this can be achieved by adding a integrator block in the feedforward loop of the closed loop signal and multiplying it by a gain Ka, as shown in the figure above.

By adding this integrator block it can be proved that the new system completely rejects step disturbance inputs, and perfectly tracks step reference inputs (provided that the system has no pole on the imaginary line and there is no pole and zero cancellation).

Hence integral action on state feedback control provides us a robust method to track step reference inputs.

Written By: - Kunal Deshmukh

(Reference: Dr. Shangyan Zou's Lecture Notes for MEEM5715: Linear Systems)

The observability of the system is defined as the ability of the engineer defining the controller to find the states of the system through measurable data. In other words the system can be defined through it's outputs if the observability is high. The way this is found is if an observability matrix(O=[C CA CA^2...CA^(n-1)]) has all independent columns, much like the controllability matrix. A system is uncontrollable if it does not fulfill these requirements, indicating an unmeasurable part of the system outputs.

To me, observability was much easier to visualize than controllability. The book provides an example of a case where a physical system measures an output that is a derivative of a select state variable. If this variable itself isn't measured directly as well as the derivative, a constant of integration will be present and can potentially limit your system ability.

Section 7.4.2 of the text provides an example of an electric circuit with current as the input and voltage as the output. The circuit's controllability and observability are affected by a number of elements, including inductance (L), capacitance (C), and resistance (R). By examining these characteristics, we may ascertain whether each system mode is influenced by the input and discernible from the output, which is essential for developing strong and dependable feedback systems.

When direct measurement of states is challenging, such as in nuclear power plants when some variables are unavailable due to severe conditions, an estimator is used. In these cases, an estimator that continuously adjusts itself based on measurement feedback gradually reduces the difference (or error) between the estimated and true states over time. This error reduction, achieved by using a proportionate gain in the estimator to efficiently match the model's predictions with actual system behavior, enables reliable control performance even in complex scenarios.

In section 7.7 in the estimator design it mentions the fact that there some disadvantages. One such is the cost of sensors. Imagine that you have a system that to get a full state description you need to have the definition of 6 states. This means that you need at least 6 different sensors. This by itself will be expensive not to mention that you will also need space on your system to put them on. If you are working with something relativity small you start running into space constraints with the existence of the sensor itself. In another case the environment may prohibit the use of a sensor. In my internship experience this has come up in the form of harsh environment. The situation is where there was an encoder on a machine that was under brine spray and would destroy the encoder costing more money and reduce efficiency.

That said, state feedback has some great pluses as well. Due to the nature of the state feedback its use results in a faster transient response. The biggest plus is state feedback's ability to be used in non-linear systems which greatly increases its capability.

These topics and others are covered across section 7.5-7.8

Create an estimator if not all states are accessible. In real-world applications, it is uncommon for the estimator to estimate states that are not measured.

Integrate the estimator and control law. In essence, this replaces the controller D that we have been using. Instead of using actual state measurements, all inputs that are sent to the plant from this location are based on estimated states.

Make sure the plant tracks the reference input correctly before adding it to the system. Since steps 1-3 set all of the poles, this is accomplished by managing the closed loop transfer function's zeros.

In general, this appears to make the absurdity of attempting to comprehend state feedback easier for me to understand. This gives me a solid framework to begin with, albeit I don't pretend to understand it yet.

References: State observer - Wikipedia, Luenberger Observer - (Nonlinear Control Systems) - Vocab, Definition, Explanations | Fiveable

According to section 7.11, in state feedback control, adding an estimator can reduce stability robustness, affecting phase margin (PM) and gain margin (GM). LTR counteracts this by modifying the estimator to regain some Linear Quadratic Regulator (LQR) robustness, particularly effective in minimum-phase systems. This process places some estimator poles near the plant’s zeros and moves others into the left-half plane (LHP), shaping the loop gain to approximate the LQR’s stability properties.

LTR Mechanism and Trade-offs

LTR allows feedback controllers to reach desired sensitivity functions at key points in the feedback system. However, this often increases sensitivity to sensor noise. One approach is making estimator poles fast to achieve a loop gain close to that of LQR, essentially inverting the plant transfer function. For nonminimum-phase systems, LTR can still improve robustness, though full recovery is limited by right-half plane (RHP) zeros.

LTR Example From Text : Satellite Attitude Control

In a satellite system example, an LQR controller is designed with feedback gain KKK, then refined using LTR for varying values of q (e.g., 1, 10, 100). As q increases, the loop gain better resembles the ideal LQR loop gain but leads to higher actuator activity under sensor noise, suggesting a trade-off between robustness and actuator wear.

One thing I found interesting is that "The newly designed control system may have worse sensor noise sensitivity properties. Intuitively, one can think of making (some of) the estimator poles arbitrarily fast so the loop gain is approximately that of LQR." (7.11). It makes me consider that sometimes there is 'no free lunch' and you always have to make trade offs as a controller designer.

Question:

LTR is supposed to be especially effective in minimum-phase systems, but I want to know why nonminimum-phase systems limit full recovery?

Sources: Section 7.11 of the textbook.

For the poles when using the symetric root locus method, the equation 1 + ρ(N(-s)N(s)/D(-s)N(s)) = 0. By solving for s, the optimal stable pole can be found. This requires the value ρ to be selected. The value of ρ is selected based on the desired trade off between a fast response and control effort. In the same way as the dominant second order method, control gains can then be found with the acker function in Matlab.

A question I have is if state estimators can be used in conjunction with the actual states for the system, as a "backup"? This would be for if a sensor fails, or if the control system experiences a large latency.

From the reading, the main disadvantage of state feedback is that all states of the system are used in the closed loop transfer function either coming from an estimator/observer or hardware sensors. Hardware sensors can be either unfeasible or cumbersome to install while state estimators have the potential to introduce error and instability to the system if implemented improperly. Another disadvantage of state feedback is its complexity. Compared to PID controllers, state feedback is significantly more complex, requiring values or estimates for each state, and involves significant and complex matrix math. This complexity appears to be a tradeoff for the level of control that state feedback can apply. The book reading talks about many of the different aspects of a response that can be accounted for, many of which are not available if PID was to be chosen.

Another cool advantage is how controllers can take an unstable system and make it stable by manipulating the feedback signals. The textbook explains how this method is especially powerful for complex systems with multiple inputs and outputs, which makes it super valuable in real-world applications like robotics or aircraft control systems. By knowing the poles of the plant, you can design a controller that negates any unstable poles and makes sure the system does what you want it to.

The most obvious disadvantage of feedback control systems is that they are often much more difficult to implement than a feedforward system. Feedback systems (that are fully controllable) require at least 1 sensor for every output that the system has, but likely will use much more than that. It will also need a way to receive input signals from the operator of the system. Also, if sensors fail or are inaccurate, you can easily lose control of the system.

Another disadvantage is that controller parameters need to be tested and set very carefully because otherwise, it can make the system unstable. If the feedforward system is inaccurate, it will be off by however much it needs to be adjusted. It might result in a failed part or a steady-state error (depending on what the plant is). However, if a feedback system is inaccurate, it will attempt to self-correct and if that self-correction is too much, it can cause the system to self-destruct.

I still have a few questions:

1. For many applications, it seems like there are just too many problems with feedback control. If I wanted to punch a hole in a specific spot in a piece of sheet metal, I think I would want to use dead-reckoning instead of feedback control. How do engineers decide to use state feedback control as opposed to meticulously crafting a precise machine?

2. What are some less common industries that use feedback control? The technology seems very broadly useful and there seems to be money in automation for almost any industry. Do industries like farming, fabrication, and mining use state feedback control and how/how much?