Systems and Control Theory Lecture Notes. Laura Giarré

Transcription

1 Systems and Control Theory Lecture Notes Laura Giarré L. Giarré

2 Lesson 5: State Space Systems State Dimension Infinite-Dimensional systems State-space model (nonlinear) LTI State Space model Linearization L. Giarré- Systems and Control Theory

3 State of a System If a system is causal, in order to compute its output at a given time t 0, we need to know only the input signal over (, t 0 ]. (Similarly for DT systems.) A lot of information! Can we summarize it with something more manageable? (memory!) Definition ( State) The state x(t 1 ) of a causal system at time t 1 is the information needed, together with the input u between times t 1 and t 2, to uniquely predict the output at time t 2,forall t 2 t 1. The state of the system at a given time summarizes the whole history of the past inputs to predict the output at future times. The state of a system is a vector in the Euclidean space R n. L. Giarré- Systems and Control Theory

4 State Property of DT state-space models Given the initial state x(t0) and input u(t) for t 0 t t f (with t 0 and t f arbitrary), we can compute the output y(t) for t 0 t t f and the state x(t) for t 0 t t f Thus, the state at any time t 0 summarizes everything about the past that is relevant to the future. Keeping in mind this fact that the state variables are the memory variables (or, in more physical situations, the energy storage variables) of a system, often guides us quickly to good choices of state variables in any given context. L. Giarré- Systems and Control Theory

5 Dimension of a System The choice of a state for a system is not unique ( there are infinite choices, or realizations). However, there are come choices of state which are preferable to others; in particular, we can look at minimal realizations. Definition ( Dimension of a system ) We define the dimension of a causal system as the minimal number of variables sufficient to describe the system s state (i.e., the dimension of the smallest state vector). We will deal mostly with finite-dimensional systems, i.e., systems which can be described with a finite number of variables. L. Giarré- Systems and Control Theory

6 Infinite-Dimensional Systems Even though we will not address infinite-dimensional systems in detail, some examples are very useful: (CT) Time-delay systems: Consider the very simple time delay S T, defined as a continuous-time system such that its input and outputs are related by y(t) =u(t T ). In order to predict the output at times after t, the knowledge of the input for times in (t T, t] is necessary. PDE-driven systems: Many systems in engineering, arising, e.g., in structural control and flow control applications, can only be described exactly using a continuum of state variables (stress, displacement, pressure, temperature, etc.). These are infinite-dimensional systems. L. Giarré- Systems and Control Theory

7 State-Space model Finite-dimensional linear systems can always be modeled using a set of differential (or difference) equations as follows: Definition (Continuous-time State-Space Models) d x(t) =ẋ(t) =f (x(t), u(t), t) dt y(t) =g(x(t), u(t), t); Definition (Discrete-time State-Space Models) x(k + 1) =f (x(k), u(k), k) y(k) =g(x(k), u(k), k); L. Giarré- Systems and Control Theory

8 State-Space model (linear) If the system satisfies Linearity: Definition (Continuous-time State-Space Models) d x(t) =ẋ(t) =A(t)x(t)+B(t)u(t) dt y(t) =C(t)x(t)+D(t)u(t); Definition (Discrete-time State-Space Models) x(k + 1) =A(k)x(k)+B(k)u(k) y(k) =C(k)x(k)+D(k)u(k); The matrices appearing in the above formulas are in general functions of time, and have the correct dimensions. L. Giarré- Systems and Control Theory

9 LTI State-Space model If the system satisfies Linearity and Time Invariance (LTI systems) an n is the dimension of the state vector: LTI TC systems: LTI TD systems: d x(t) =ẋ(t) =Ax(t)+Bu(t) dt y(t) =Cx(t)+Du(t); x(k + 1) =Ax(k)+Bu(k) y(k) =Cx(k)+Du(k); A R n n, B R n m, C R p n, D R m p, x R n 1, y R p 1, u R m 1. L. Giarré- Systems and Control Theory

10 Finite-Dimensional linear Systems Recall the definition of a linear system. Essentially, a system is linear if the linear combination of two inputs generates an output that is the linear combination of the outputs generated by the two individual inputs. The definition of a state allows us to summarize the past inputs into the state, i.e., u(t), t (similar formulas hold for the DT case.) { x(t 0 ), u(t), t t 0, We can extend the definition of linear systems L. Giarré- Systems and Control Theory

11 Finite-Dimensional linear Systems Definition (Linear system (again)) A system is said a Linear System if, for any u 1, u 2, t 0, x 0,1, x 0,2, and any two real numbers α, β, the following are satisfied: { x(t 0 ) = x 0,1, u(t) = u 1 (t), t t 0, y 1 { x(t 0 ) = x 0,2, u(t) = u 2 (t), t t 0, y 2 { x(t 0 ) = αx 0,1 + βx 0,2, u(t) = αu 1 (t)+βu 2 (t), t t 0, αy 1 + βy 2 Similar formulas hold for the discrete-time case. L. Giarré- Systems and Control Theory

12 Linearization 1 Linear models frequently arise as description of small perturbations away from a nominal solution of the system Suppose x o (t), u o (t) and y o (t) constitute a nominal solution of the system, i.e. a collection of CT signals that jointly satisfy the state space equations. Let the input and the initial condition be perturbed from their nominal values x(0) =x o (0)+δx(0), u(t) =u o (t)+δu(t) Let the state trajectory accordingly be perturbed x(t) =x o (t)+δx(t) Substituting them and performing a Taylor expansion to first-order terms, we find L. Giarré- Systems and Control Theory

13 Linearization 2 δẋ(t) δy(t) [ ] f x [ ] g x o o δx(t)+ δx(t)+ [ ] f u [ ] g u o o δu(t) δu(t) Where the n n matrix [ f x ] o denotes the Jacobian of f (...) with respect to x and the other matrices are defined accordingly. The ij-th entry is the partial derivative of f i with respect to x j. The subscript o indicates that the Jacobian are evaluated along the nominal trajectories. L. Giarré- Systems and Control Theory

14 Linearization 3 The linearized model is evidently linear: δẋ(t) =A l (t)δx(t)+b l (t)δu(t) δy(t) =C l (t)δx(t)+d l (t)δu(t) where [ ] f A l (t) = x [ ] g C l (t) = x o o [ ] f B l (t) = u [ ] g D l (t) = u o o Similar formulas hold for the discrete-time case. L. Giarré- Systems and Control Theory

15 Examples System ẋ(t) =tx 2 (t) ẋ(t) =x 2 (t) ẋ(t) =tx(t) ẋ(t) =(cost)x(t) ẋ(t) =x(t) Type NLTV NLTI LTV LPV LTI L. Giarré- Systems and Control Theory

16 Thanks DIEF- Tel: giarre.wordpress.com