Chapter 11 Digital Control Systems Topics to be covered: - Discrete-time systems - z-Transforms - State Equation models of discrete systems (continuous  differential equations) (discrete  difference equations)

Digital Control Systems Simulation Diagrams and Flow Graphs: (How to make an electrical simulation of a system.) For a discrete-time simulation, the main operation is the delay operator. (Recall that for a continuous-time system, we work with integrators. A delay operator is simpler, and, as a result, cheaper.) We can also represent the delay as a z-1 operator. ECE 4923/6923 Chapter 11 Part 2

Digital Control Systems Example: Draw a simulation diagram for the difference equation: m(k) = e(k) – e(k-1) – m(k-1) Components required: 2 delay elements and 1 summer (This is a special purpose computer, which can only solve the given equation.) Alternate representation (in SFG form): Note that the z-1 and the -1 paths can be combined in a “self-loop” ECE 4923/6923 Chapter 11 Part 2

Digital Control Systems Consider a general nth order difference equation of the following form: By using the real translation theorem, we get: This leads to the transfer function: The following two pages show a simulation diagram and a flow graph for this system. ECE 4923/6923 Chapter 11 Part 2

Digital Control Systems For a general nth order difference equation, we can draw the following simulation diagram: ECE 4923/6923 Chapter 11 Part 2

Digital Control Systems As we have seen before, the SFG form is generally much more compact and is easier to draw... ECE 4923/6923 Chapter 11 Part 2

Digital Control Systems State Variables - Consider the following TF and the corresponding simulation diagram and SFG: ECE 4923/6923 Chapter 11 Part 2

Digital Control Systems Alternative representations: ECE 4923/6923 Chapter 11 Part 2

Digital Control Systems State Variables – Let the outputs of each delay be a state, and we can write state variable equations as: and (Phase Variable form of the State Equations) xn xn-1 x1 ECE 4923/6923 Chapter 11 Part 2

Digital Control Systems Note that these state equations can be more compactly written as: (Note that these give a linear, time-invariant discrete system description.) A somewhat more general form is the linear time-varying form of the state equations: ECE 4923/6923 Chapter 11 Part 2

Digital Control Systems Example: Write State Equations for a system with the TF (Technique: generate a SFG, label outputs of delays as states, and then write equations.) (see board for details…) ECE 4923/6923 Chapter 11 Part 2

Digital Control Systems Solution of State Equations: Plug in each value of k: x(1) = A x(0) + B u(0) x(2) = A x(1) + B u(1) = A2 x(0) + AB u(0) + B u(1) x(3) = A3 x(0) + A2B u(0) + AB u(1) + B u(2) . . . x(n) = An x(0) + An-1B u(0) + An-2B u(1) + . . . + AB u(n-2) + B u(n-1) So, ECE 4923/6923 Chapter 11 Part 2

Digital Control Systems Solution of State Equations: By z-Transforms: x(n+1) = A x(n) + B u(n) The term multiplying the initial condition is the z-transform of the state transition matrix and ECE 4923/6923 Chapter 11 Part 2

Digital Control Systems Example: Draw a simulation diagram: And assign states (from right to left): ECE 4923/6923 Chapter 11 Part 2

Digital Control Systems Example, continued With states assigned, we can then write out the specific state equations: In matrix form, they end up as: ECE 4923/6923 Chapter 11 Part 2

Digital Control Systems Example, continued To solve, we can use the sequential technique: (assume that x(0)=0 and u(k)=1, for all k. ECE 4923/6923 Chapter 11 Part 2

Digital Control Systems Example, continued . . .or, by z-transforms: Recall that: So, with x(0)=0, we get and Since we get ECE 4923/6923 Chapter 11 Part 2

Digital Control Systems Transfer Functions (TF’s) and State Variables For the SISO case, we have So, with zero IC’s (since we are looking at TF’s): and So, we end up with ECE 4923/6923 Chapter 11 Part 2

Digital Control Systems Example: Using a previous example setup, we get Since D = 0, the transfer function is: ECE 4923/6923 Chapter 11 Part 2

Digital Control Systems Note: If you don’t really like computing matrix inverses, you can alternatively use the following: This method allows us to compute the TF without the need to determine a matrix inverse. For our previous example, ECE 4923/6923 Chapter 11 Part 2

Digital Control Systems Finally, we can use MATLAB to automate much of the work: Example: num=[0 1 0.95]; den=[1 -1.9 0.93]; printsys(num,den,’z’) [A,B,C,D]=tf2ss(num,den) Results: Then, we can use: dstep(A,B,C,D) (MATLAB chooses # of points) dimpulse(A,B,C,D) dstep(A,B,C,D,IU,50) (we specify 50 points) dimpulse(A,B,C,D,IU,50) (IU is # of input, only one in this case) ECE 4923/6923 Chapter 11 Part 2

Digital Control Systems More MATLAB: Still using: But note that if we specify: which is different from before, we find that: [num,den]=ss2tf(A,B,C,D) gives num=[0 1 0.95] den = [1 -1.9 0.93] which is the SAME TF as before! What gives???? Answer: These are two different forms of the state equations, but are equivalent. Note that there are an infinite number of valid state equations for any given system. ECE 4923/6923 Chapter 11 Part 2