Control Systems and Adaptive Process. Design, and control methods and strategies 1

Controllability and observability Controllability Consider the system of n states and p inputs with constant matrices and. The states equation, or the pair (A, B), is said to be controllable if for any initial state and any final state, there is an input that transfers the state x from x 0 to x 1 in finite time. Otherwise, the equation (1.1), or the pair (A, B), is said non- controllable. 2

Controllability and observability Controllability You can determine if the system is controllable by examining the algebraic condition: Matrix A has dimension n x n and B n x 1. For systems with multiple input matrix B is n × m, where m is the number of inputs. For a system of single-input single-output, controllability matrix P c is described in terms of A and B as: which is an n x n matrix, therefore, if the determinant of P c is not zero, the system is controllable. 3

Controllability and observability Controllability Example: consider the system from which we have that The determinant of P c = 1 ≠ 0, so that the system is controllable. 4

Controllability and observability Controllability test The following statements are equivalent: 1. The pair, is controllable 2. The controllability matrix is of rank n (full row rank). 3. Matrix n x n is nonsingular for all t > 0. 5

Controllability and observability Minimum energy control Control spending minimum energy to bring the system from state x 0 to state x 1 at time t 1, in the sense that, for other control ũ(t) to make the same transfer, is always true that: It is observed that the minimum control power is greater when the distance between x 0 and x 1 is greater, and the transfer time t 1 is lower x 0 y x 1. 6

Controllability and observability Controllability PBH tests 7 The Popov-Belevitch Hautaus (PBH) tests have interesting geometric interpretations used to analyze the controllability in the form of Jordan. There are two types of test, of eigenvectors and of rank. 1. Eigenvectors test: The pair (A,B) is not controllable if and only if there is a left eigenvector of A such that 2. Rank PBH test: Pair (A,B) is controllable if and only if

Controllability and observability Controllability PBH tests Controllability and similarity transformation: invariance theorem regarding controllability coordinate changes.. Controllability is an invariant property with respect to equivalence transformations (coordinate changes). 8

Controllability and observability Observability 9 All poles of a closed-loop system can be placed arbitrarily in the complex plane if and only if the system is observable and controllable. Observability refers to the possibility of estimating a state variable. According to R. Dorf, a system is completely observable if and only if there exists a finite time T such that the initial state x(0) can be determined from the observation of history y(t) given the control u(t).

Controllability and observability Observability Considering the system of one input and one output where C is a row vector 1 x n and x is a column vector n x 1. This system is fully observable when the determinant of the observability matrix P o is nonzero, where which is a matrix of n x n. 10

Controllability and observability Observability Example: Consider the system Therefore and Is thus obtained the determinant of P o = 1 and the system is fully observable. Note that the determination of the observability matrix does not use matrices B and D. 11

Controllability and observability Observability The concept of observability is dual to controllability. Tries to find out the possibility of estimating the system state from the knowledge of the output. Consider the steady linear system This state equation (1.2) is observable if for any unknown initial state x(0), there is a finite time t 1 such that the knowledge of the input u and the output y on the interval [0,t 1 ] is sufficient to determine uniquely the initial state x(0). Otherwise the system is not observable. 12

Controllability and observability State variables For a given system, they exist plenty of possible sets of state variables. However, all possible sets must consist of the same number of state variables and the defined variables must be fully independent. Understanding as independent variable that whose value cannot be expressed in terms of the other variables; which implies that the initial values of each of the chosen state variables may be assigned freely. 13

Controllability and observability State variables E xample, in a system such as shown in figure 3.1 may be taken as state variables the speed ẏ(t) of the mass M and the force ky(t) in the spring; the strength in the spring and the displacement y(t) of the mass may not be taken, since the former is equal to the second multiplied by the constant K. Another valid alternative would be to take as state variables of the system the speed ẏ(t) and the displacement y(t) of the mass. 14

Controllability and observability State variables 15 General methods for the selection of the state variables of a system: -Method of physical variables: the selection of the state variables is performed based on the energy storage elements existing in the system. -Method of phase variables. -Jordan canonical form.

Controllability and observability State variables Linear systems with variable parameters: in a system whose dynamic behavior is characterized by This equation can be represented by the following state and output equations Calculating coefficients B i (t) by means of 16

Controllability and observability State variables Obtaining the transfer function from the state equations: – The transfer matrix or function of a linear time-invariant system can be obtained from the state equations of the system by applying the Laplace transform. 17

Bibliography R. Dorf, R. Bishop: Modern control systems. Class notes ETSII. UNED Interesting links do/node4.html#SECTION