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Controllability Gramian
In control theory, we may need to find out whether or not a system such as \begin \dot(t) &= \boldsymbol(t) + \boldsymbol(t) \\ \boldsymbol(t) &= \boldsymbol(t) +\boldsymbol(t) \end is controllable, where \boldsymbol, \boldsymbol, \boldsymbol and \boldsymbol are, respectively, n\times n, n\times p, q\times n and q\times p matrices for a system with p inputs, n state variables and q outputs. One of the many ways one can achieve such goal is by the use of the Controllability Gramian. Controllability in LTI Systems Linear Time Invariant (LTI) Systems are those systems in which the parameters \boldsymbol, \boldsymbol, \boldsymbol and \boldsymbol are invariant with respect to time. One can observe if the LTI system is or is not controllable simply by looking at the pair (\boldsymbol,\boldsymbol). Then, we can say that the following statements are equivalent: # The pair (\boldsymbol,\boldsymbol) is controllable. # The n\times n matrix \boldsymbol(t)=\int_^e^\boldsymbole^d\tau=\int_ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Control Theory
Control theory is a field of control engineering and applied mathematics that deals with the control system, control of dynamical systems in engineered processes and machines. The objective is to develop a model or algorithm governing the application of system inputs to drive the system to a desired state, while minimizing any ''delay'', ''overshoot'', or ''steady-state error'' and ensuring a level of control Stability theory, stability; often with the aim to achieve a degree of Optimal control, optimality. To do this, a controller with the requisite corrective behavior is required. This controller monitors the controlled process variable (PV), and compares it with the reference or Setpoint (control system), set point (SP). The difference between actual and desired value of the process variable, called the ''error'' signal, or SP-PV error, is applied as feedback to generate a control action to bring the controlled process variable to the same value as the set point. Other aspects ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Controllability
Controllability is an important property of a control system and plays a crucial role in many regulation problems, such as the stabilization of unstable systems using feedback, tracking problems, obtaining optimal control strategies, or, simply prescribing an input that has a desired effect on the state. Controllability and observability are dual notions. Controllability pertains to regulating the state by a choice of a suitable input, while observability pertains to being able to know the state by observing the output (assuming that the input is also being observed). Broadly speaking, the concept of controllability relates to the ability to steer a system around in its configuration space using only certain admissible manipulations. The exact definition varies depending on the framework or the type of models dealt with. The following are examples of variants of notions of controllability that have been introduced in the systems and control literature: * State controllability ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gramian
In linear algebra, the Gram matrix (or Gramian matrix, Gramian) of a set of vectors v_1,\dots, v_n in an inner product space is the Hermitian matrix of inner products, whose entries are given by the inner product G_ = \left\langle v_i, v_j \right\rangle., p.441, Theorem 7.2.10 If the vectors v_1,\dots, v_n are the columns of matrix X then the Gram matrix is X^\dagger X in the general case that the vector coordinates are complex numbers, which simplifies to X^\top X for the case that the vector coordinates are real numbers. An important application is to compute linear independence: a set of vectors are linearly independent if and only if the Gram determinant (the determinant of the Gram matrix) is non-zero. It is named after Jørgen Pedersen Gram. Examples For finite-dimensional real vectors in \mathbb^n with the usual Euclidean dot product, the Gram matrix is G = V^\top V, where V is a matrix whose columns are the vectors v_k and V^\top is its transpose whose rows are the vectors ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lyapunov Equation
The Lyapunov equation, named after the Russian mathematician Aleksandr Lyapunov, is a matrix equation used in the stability analysis of linear dynamical systems. In particular, the discrete-time Lyapunov equation (also known as Stein equation) for X is :A X A^ - X + Q = 0 where Q is a Hermitian matrix and A^H is the conjugate transpose of A, while the continuous-time Lyapunov equation is :AX + XA^H + Q = 0. Application to stability In the following theorems A, P, Q \in \mathbb^, and P and Q are symmetric. The notation P>0 means that the matrix P is positive definite. Theorem (continuous time version). Given any Q>0, there exists a unique P>0 satisfying A^T P + P A + Q = 0 if and only if the linear system \dot=A x is globally asymptotically stable. The quadratic function V(x)=x^T P x is a Lyapunov function that can be used to verify stability. Theorem (discrete time version). Given any Q>0, there exists a unique P>0 satisfying A^T P A -P + Q = 0 if and only if the linear ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Observability Gramian
In control theory, we may need to find out whether or not a system such as \begin \dot(t)\boldsymbol(t)+\boldsymbol(t)\\ \boldsymbol(t)=\boldsymbol(t)+\boldsymbol(t) \end is observable, where \boldsymbol, \boldsymbol, \boldsymbol and \boldsymbol are, respectively, n\times n, n\times p,q\times n and q\times p matrices. One of the many ways one can achieve such goal is by the use of the Observability Gramian. Observability in LTI Systems Linear Time Invariant (LTI) Systems are those systems in which the parameters \boldsymbol, \boldsymbol, \boldsymbol and \boldsymbol are invariant with respect to time. One can determine if the LTI system is or is not observable simply by looking at the pair (\boldsymbol,\boldsymbol). Then, we can say that the following statements are equivalent: 1. The pair (\boldsymbol,\boldsymbol) is observable. 2. The n\times n matrix \boldsymbol(t)=\int_^e^\boldsymbol^\boldsymbole^d\tau is nonsingular for any t>0. 3. The nq\times n observability matrix ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gramian Matrix
In linear algebra, the Gram matrix (or Gramian matrix, Gramian) of a set of vectors v_1,\dots, v_n in an inner product space is the Hermitian matrix of inner products, whose entries are given by the inner product G_ = \left\langle v_i, v_j \right\rangle., p.441, Theorem 7.2.10 If the vectors v_1,\dots, v_n are the columns of matrix X then the Gram matrix is X^\dagger X in the general case that the vector coordinates are complex numbers, which simplifies to X^\top X for the case that the vector coordinates are real numbers. An important application is to compute linear independence: a set of vectors are linearly independent if and only if the Gram determinant (the determinant of the Gram matrix) is non-zero. It is named after Jørgen Pedersen Gram. Examples For finite-dimensional real vectors in \mathbb^n with the usual Euclidean dot product, the Gram matrix is G = V^\top V, where V is a matrix whose columns are the vectors v_k and V^\top is its transpose whose rows are the vect ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Minimum Energy Control
In control theory, the minimum energy control is the control u(t) that will bring a linear time invariant system to a desired state with a minimum expenditure of energy. Let the linear time invariant (LTI) system be : \dot(t) = A \mathbf(t) + B \mathbf(t) : \mathbf(t) = C \mathbf(t) + D \mathbf(t) with initial state x(t_0)=x_0 . One seeks an input u(t) so that the system will be in the state x_1 at time t_1, and for any other input \bar(t), which also drives the system from x_0 to x_1 at time t_1, the energy expenditure would be larger, i.e., : \int_^ \bar^*(t) \bar(t) dt \ \geq \ \int_^ u^*(t) u(t) dt. To choose this input, first compute the controllability Gramian : W_c(t)=\int_^t e^BB^*e^ d\tau. Assuming W_c is nonsingular (if and only if the system is controllable), the minimum energy control is then : u(t) = -B^*e^W_c^(t_1)^x_0-x_1 Substitution into the solution :x(t)=e^x_0+\int_^e^{A(t-\tau)}Bu(\tau)d\tau verifies the achievement of state x_1 at t_1. See als ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hankel Singular Value
In control theory, Hankel singular values, named after Hermann Hankel, provide a measure of energy for each state in a system. They are the basis for balanced model reduction, in which high energy states are retained while low energy states are discarded. The reduced model retains the important features of the original model. Hankel singular values are calculated as the square roots, , of the eigenvalues, , for the product of the controllability Gramian, ''WC'', and the observability Gramian, ''WO''. Properties * The square of the Hilbert-Schmidt norm of the Hankel operator associated with a linear system is the sum of squares of the Hankel singular values of this system. Moreover, the area enclosed by the oriented Nyquist diagram of an BIBO stable and strictly proper linear system is equal π times the square of the Hilbert-Schmidt norm of the Hankel operator associated with this system. *Hankel singular values also provide the optimal range of analog filters. See also *H ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
