We can determine the system metrics and then we can compare those metrics to our specification. If our system meets the specifications we are finished with the design process.
However if the system does not meet the specifications as is typically the case , then suitable controllers and compensators need to be designed and added to the system. Once the controllers and compensators have been designed, the job isn't finished: we need to analyze the new composite system to ensure that the controllers work properly.
Also, we need to ensure that the systems are stable: unstable systems can be dangerous. For proposals, early stage designs, and quick turn around analyses a frequency domain model is often superior to a time domain model. The answer is a steady-state response. Oftentimes the controller is shooting for 0 so the steady-state response is also the residual error that will be the analysis output or metric for report. Frequency domain modeling is a matter of determining the impulse response of a system to a random process.
Note some texts will state that this is only valid for random processes which are stationary. Other texts suggest stationary and ergodic while still others state weakly stationary processes. Some texts do not distinguish between strictly stationary and weakly stationary. From practice, the rule of thumb is if the PSD of the input process is the same from hour to hour and day to day then the input PSD can be used and the above equation is valid.
See a full explanation with example at ControlTheoryPro. Modeling in Control Systems is oftentimes a matter of judgement. This judgement is developed by creating models and learning from other people's models. Here are links to a few of them. Once the system has been properly designed we can prototype our system and test it. Assuming our analysis was correct and our design is good, the prototype should work as expected.
Now we can move on to manufacture and distribute our completed systems.
The modern method of controls uses systems of special state-space equations to model and manipulate systems. The state variable model is broad enough to be useful in describing a wide range of systems, including systems that cannot be adequately described using the Laplace Transform. These chapters will require the reader to have a solid background in linear algebra, and multi-variable calculus. The "Classical" method of controls what we have been studying so far has been based mostly in the transform domain. When we want to control the system in general we use the Laplace transform Z-Transform for digital systems to represent the system, and when we want to examine the frequency characteristics of a system, we use the Fourier Transform.
The question arises, why do we do this?
The Laplace transform is transforming the fact that we are dealing with second-order differential equations. The Laplace transform moves a system out of the time-domain into the complex frequency domain, to study and manipulate our systems as algebraic polynomials instead of linear ODEs. Given the complexity of differential equations, why would we ever want to work in the time domain?
It turns out that to decompose our higher-order differential equations into multiple first-order equations, one can find a new method for easily manipulating the system without having to use integral transforms. The solution to this problem is state variables. By taking our multiple first-order differential equations, and analyzing them in vector form, we can not only do the same things we were doing in the time domain using simple matrix algebra, but now we can easily account for systems with multiple inputs and multiple outputs, without adding much unnecessary complexity.
This demonstrates why the "modern" state-space approach to controls has become popular. Wikipedia has related information at State space controls.
In a state space system, the internal state of the system is explicitly accounted for by an equation known as the state equation. The system output is given in terms of a combination of the current system state, and the current system input, through the output equation. These two equations form a system of equations known collectively as state-space equations. The state-space is the vector space that consists of all the possible internal states of the system.
This text mostly considers linear state space systems, where the state and output equations satisfy the superposition principle and the state space is linear. However, the state-space approach is equally valid for nonlinear systems although some specific methods are not applicable to nonlinear systems. Central to the state-space notation is the idea of a state. A state of a system is the current value of internal elements of the system, that change separately but not completely unrelated to the output of the system.
In essence, the state of a system is an explicit account of the values of the internal system components. Here are some examples:. Consider an electric circuit with both an input and an output terminal. This circuit may contain any number of inductors and capacitors.
Multibody Dynamics Computational Methods and Applications. Showing: 1 - 10 of 27 Showing: 1 - 27 of Optimal control of partial differential equations PDEs is a well-established discipline in mathematics with many interfaces to science and engineering. Item Added:. As a result, F q is determined. The key methodologies combine empirical measures and information-theoretic approaches to derive identification algorithms, provide convergence and convergence speed, establish efficiency of estimation, and explore input design, threshold selection and adaptation, and complexity analysis. Blast Mitigation Experimental and Numerical Studies.
The state variables may represent the magnetic and electric fields of the inductors and capacitors, respectively. Consider a spring-mass-dashpot system. The state variables may represent the compression of the spring, or the acceleration at the dashpot. Consider a chemical reaction where certain reagents are poured into a mixing container, and the output is the amount of the chemical product produced over time.
The state variables may represent the amounts of un-reacted chemicals in the container, or other properties such as the quantity of thermal energy in the container that can serve to facilitate the reaction.
We denote the input variables with u , the output variables with y , and the state variables with x. In essence, we have the following relationship:. Where f x, u is our system. Also, the state variables can change with respect to the current state and the system input:. Where x' is the rate of change of the state variables. We will define f u, x and g u, x in the next chapter. In the Laplace domain, if we want to account for systems with multiple inputs and multiple outputs, we are going to need to rely on the principle of superposition to create a system of simultaneous Laplace equations for each output and each input.
Systems & Control: Foundations & Applications developed methodologies that utilize quantized information in system identification and explores their potential. Buy System Identification with Quantized Observations (Systems & Control: Foundations & Applications) by Le Yi Wang, G. George Yin, Ji-Feng Zhang .
For such systems, the classical approach not only doesn't simplify the situation, but because the systems of equations need to be transformed into the frequency domain first, manipulated, and then transformed back into the time domain, they can actually be more difficult to work with. However, the Laplace domain technique can be combined with the State-Space techniques discussed in the next few chapters to bring out the best features of both techniques.
In a state-space system representation, we have a system of two equations: an equation for determining the state of the system, and another equation for determining the output of the system. We will use the variable y t as the output of the system, x t as the state of the system, and u t as the input of the system. We use the notation x' t note the prime for the first derivative of the state vector of the system, as dependent on the current state of the system and the current input.
Symbolically, we say that there are transforms g and h , that display this relationship:. The first equation shows that the system state change is dependent on the previous system state, the initial state of the system, the time, and the system inputs. The second equation shows that the system output is dependent on the current system state, the system input, and the current time.
If the system state change x' t and the system output y t are linear combinations of the system state and input vectors, then we can say the systems are linear systems, and we can rewrite them in matrix form:. The State Equation shows the relationship between the system's current state and its input, and the future state of the system. The Output Equation shows the relationship between the system state and its input, and the output. These equations show that in a given system, the current output is dependent on the current input and the current state.
The future state is also dependent on the current state and the current input. It is important to note at this point that the state space equations of a particular system are not unique, and there are an infinite number of ways to represent these equations by manipulating the A , B , C and D matrices using row operations. There are a number of "standard forms" for these matrices, however, that make certain computations easier.