step3.c File Reference

Sample controlling a 3rd order system across a unit step in the setpoint. More...

Include dependency graph for step3.c:

Functions
static void	advance (const double h, const double a[restrict static 3], const double b[restrict static 1], const double u[restrict static 1], double y[restrict static 3])
	Advance the temporal state of a model given by transfer function \( \frac{y(s)}{u(s)} = \frac{b_0}{s^3 + a_2 s^2 + a_1 s + a_0} \).
static void	print_usage (const char *arg0, FILE *out)
	Print usage on the given stream.
int	main (int argc, char *argv[])
	Control the process with transfer function \( \frac{y(s)}{u(s)} = \frac{b_0}{s^3 + a_2 s^2 + a_1 s + a_0} \) across a unit step change in setpoint value.

Variables
static const double	default_a [3] = {1, 3, 3}
	Default process parameters.
static const double	default_b [1] = {1}
	Default process parameters.
static const double	default_f = 0.01
	Default filter time scale.
static const double	default_kd = 1
	Default derivative gain.
static const double	default_ki = 1
	Default integration gain.
static const double	default_kp = 1
	Default proportional gain.
static const double	default_r = 1
	Default reference value.
static const double	default_t = 1
	Default time step size.
static const double	default_T = 25
	Default final time.

Detailed Description

Sample controlling a 3rd order system across a unit step in the setpoint.

This sample can be used to test controller behavior against known good results. For example, those presented in Figure 10.2 within Chapter 10 of Astrom and Murray.

Function Documentation

advance()

static void advance ( const double  h, const double  a[restrict static 3], const double  b[restrict static 1], const double  u[restrict static 1], double  y[restrict static 3]  )

static

Advance the temporal state of a model given by transfer function \( \frac{y(s)}{u(s)} = \frac{b_0}{s^3 + a_2 s^2 + a_1 s + a_0} \).

Given the process transfer function

\begin{align} \frac{y(s)}{u(s)} = \frac{b_0}{s^3 + a_2 s^2 + a_1 s + a_0} \end{align}

a matching state space model consisting of 1st order differential equations can be derived with the form

\begin{align} \frac{\mathrm{d}}{\mathrm{d}t} \begin{bmatrix} y_0(t) \\ y_1(t) \\ y_2(t) \end{bmatrix} &= \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ -a_0 & -a_1 & -a_2 \end{bmatrix} \begin{bmatrix} y_0(t) \\ y_1(t) \\ y_2(t) \end{bmatrix} + \begin{bmatrix} 0 \\ 0 \\ b_0 \end{bmatrix} u(t) \end{align}

for constants \(a_0\), \(a_1\), \(a_2\), and \(b_0\) and time-varying input data \(u(t)\). Using a semi-implicit Euler integration scheme,

\begin{align} \vec{y}\left(t_{i+1}\right) &= \vec{y}\left(t_i\right) + h \vec{f}\left(\vec{y}\left(t_{i+1}\right), u\left(t_i\right)\right) , \end{align}

yields a constant-coefficient linear problem for advancing by time \(h\):

\begin{align} \begin{bmatrix} 1 & -h & 0 \\ 0 & 1 & -h \\ ha_0 & ha_1 & 1+ha_2 \end{bmatrix} \begin{bmatrix} y_0(t_{i+1})\\y_1(t_{i+1})\\y_2(t_{i+1}) \end{bmatrix} &= \begin{bmatrix} y_0(t_i) \\ y_1(t_i) \\ y_2(t_i) \end{bmatrix} + \begin{bmatrix} 0 \\ 0 \\ b_0 \end{bmatrix} u(t_i) \end{align}

Left multiplying by the matrix cofactor and dividing by the determinant gives a form amenable to computation,

\begin{align} \begin{bmatrix} y_0(t_{i+1})\\y_1(t_{i+1})\\y_2(t_{i+1}) \end{bmatrix} &= \frac{ \begin{bmatrix} h (a_2+a_1 h)+1 & h (a_2 h+1) & h^2 \\ -a_0 h^2 & a_2 h+1 & h \\ -a_0 h & -h (a_1+a_0 h) & 1 \end{bmatrix} \left( \begin{bmatrix} y_0(t_i) \\ y_1(t_i) \\ y_2(t_i) \end{bmatrix} + \begin{bmatrix} 0 \\ 0 \\ b_0 \end{bmatrix} u(t_i) \right) }{h (h (a_0 h+a_1)+a_2)+1} . \end{align}

This routine advances \(\vec{y}(t)\) to \(\vec{y}(t + h)\) using the above result.

Parameters

[in]	h	Time step \(h\) to be taken.
[in]	a	Coefficients \(a_0\), \(a_1\), and \(a_2\).
[in]	b	Coefficient \(b_0\).
[in]	u	Input \(u(t)\).
[in,out]	y	On input, state \(y_0(t)\), \(y_1(t)\), and \(y_2(t)\). On output, state \(y_0(t+h)\), \(y_1(t+h)\), and \(y_2(t+h)\).

{
    const double rhs[3] = {
        y[0],
        y[1],
        y[2] + b[0]*u[0]
    };
    const double cof[3][3] = {
        { h*(a[2] + a[1]*h) + 1 , h*(a[2]*h + 1)     , h*h },
        { -a[0]*h*h             , a[2]*h + 1         , h   },
        { -a[0]*h               , -h*(a[1] + a[0]*h) , 1   }
    };
    const double det = h*(h*(a[0]*h + a[1]) + a[2]) + 1;
 
    for (int i = 0; i < 3; ++i) {
        y[i] = 0;
        for (int j = 0; j < 3; ++j) {
            y[i] += cof[i][j]*rhs[j];
        }
        y[i] /= det;
    }
}

print_usage()

static void print_usage ( const char *  arg0, FILE *  out  )

static

Print usage on the given stream.

{
    fprintf(out, "Usage: %s [OPTION...]\n", arg0);
    fprintf(out, "Control 3rd-order system across a setpoint step change.\n");
    fprintf(out, "Output is tab-delimited t, u, y[0], y[1], y[2].\n");
    fputc('\n', out);
    fprintf(out, "Process transfer function "
                    "y(s)/u(s) = b0 / (s^3 + a2 s^2 + a1 s + a0):\n");
    fprintf(out, "  -0 a0\t\tSet coefficient a0 (default %g)\n", default_a[0]);
    fprintf(out, "  -1 a1\t\tSet coefficient a1 (default %g)\n", default_a[1]);
    fprintf(out, "  -2 a2\t\tSet coefficient a2 (default %g)\n", default_a[2]);
    fprintf(out, "  -b b0\t\tSet coefficient b0 (default %g)\n", default_b[0]);
    fputc('\n', out);
    fprintf(out, "Term-by-term, parallel-form PID settings\n");
    fprintf(out, "  -p kp\t\tProportional gain  (default %g)\n", default_kp);
    fprintf(out, "  -i ki\t\tIntegral gain      (default %g)\n", default_ki);
    fprintf(out, "  -d kd\t\tDerivative gain    (default %g)\n", default_kd);
    fprintf(out, "  -f Tf\t\tFilter time scale  (default %g)\n", default_f);
    fprintf(out, "  -r sp\t\tReference value    (default %g)\n", default_r);
    fputc('\n', out);
    fprintf(out, "Miscellaneous:\n");
    fprintf(out, "  -r r \t\tAdjust setpoint    (default %g)\n", default_r);
    fprintf(out, "  -t dt\t\tSet time step size (default %g)\n", default_t);
    fprintf(out, "  -T Tf\t\tSet final time     (default %g)\n", default_T);
    fprintf(out, "  -h\t\tDisplay this help and exit\n");
}

main()

int main	(	int	argc,
		char *	argv[]
	)

Control the process with transfer function \( \frac{y(s)}{u(s)} = \frac{b_0}{s^3 + a_2 s^2 + a_1 s + a_0} \) across a unit step change in setpoint value.

That is, just prior to time zero process state \(y(t)\), reference value \(r(t)\), actuator signal \(u(t)\), and all of their derivatives are zero. At time zero, step change \(r(t) = 1\) is introduced. The transfer function, in conjunction with the controller dynamics, determines the controlled system response.

{
    // Establish mutable settings
    double a[3] = {default_a[0], default_a[1], default_a[2]};
    double b[1] = {default_b[0]};
    double f    = default_f;
    double kd   = default_kd;
    double ki   = default_ki;
    double kp   = default_kp;
    double r    = default_r;
    double t    = default_t;
    double T    = default_T;
 
    // Process incoming arguments
    static const char optstring[] = "0:1:2:b:d:f:i:p:r:t:T:h";
    for (int opt; -1 != (opt = getopt(argc, argv, optstring));) {
        switch (opt) {
        case '0': a[0] = atof(optarg);          break;
        case '1': a[1] = atof(optarg);          break;
        case '2': a[2] = atof(optarg);          break;
        case 'b': b[0] = atof(optarg);          break;
        case 'd': kd   = atof(optarg);          break;
        case 'f': f    = atof(optarg);          break;
        case 'i': ki   = atof(optarg);          break;
        case 'p': kp   = atof(optarg);          break;
        case 'r': r    = atof(optarg);          break;
        case 't': t    = atof(optarg);          break;
        case 'T': T    = atof(optarg);          break;
        case 'h': print_usage(argv[0], stdout); return EXIT_SUCCESS;
        default:  print_usage(argv[0], stderr); return EXIT_FAILURE;
        }
    }
 
    // Avoid infinite loops by sanitizing inputs
    if (t <= 0) {
        fprintf(stderr, "Step size t must be strictly positive\n");
        return EXIT_FAILURE;
    }
    if (T <= 0) {
        fprintf(stderr, "Final time T must be strictly positive\n");
        return EXIT_FAILURE;
    }
 
    // Initialize state
    double u[1] = {0};        // Actuator signal
    double v[1] = {0};        // Control signal
    double y[3] = {0, 0, 0};  // Model state
 
    // Initialize controller setting PID parameters from kp, ki, and kd
    struct helm_state h;
    helm_reset(&h);
    h.kp = kp;         // Unified gain
    h.Td = kd / h.kp;  // Convert to derivative time scale
    h.Tf = f;          // Astrom and Murray p.308 suggests (h.Td / 2--20)
    h.Ti = h.kp / ki;  // Convert to integral time scale
 
    // Simulate controlled model, outputting status after each step
    helm_approach(&h);
    for (size_t i = 0; i*t < T+t;) {
        advance((++i*t > T ? T - (i-1)*t : t), a, b, u, y);      // Advance
        printf("%-22.16g\t%-22.16g\t%-22.16g\t%-22.16g\t%-22.16g\n",
               i*t > T ? T : i*t, u[0], y[0], y[1], y[2]);       // Output
        v[0] += helm_steady(&h, t, r, u[0], v[0], y[0]);         // Control
        u[0]  = v[0];                                            // Ideal
    }
 
    return EXIT_SUCCESS;
}

Variable Documentation

default_a

const double default_a[3] = {1, 3, 3}

static

Default process parameters.

{1, 3, 3}; 

default_b

const double default_b[1] = {1}

static

Default process parameters.

{1};